LMFDB - Embedded newform 76.7.h.a.69.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,7,Mod(65,76)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("76.65");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	76.h (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.4841103551$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 $ x^20 + 9460x^18 + 36670708x^16 + 75655761912x^14 + 90488614064544x^12 + 64290781142478432x^10 + 26741969359067432592x^8 + 6092313787108272225600x^6 + 632563499034568748983104x^4 + 15152233769669432690883072*x^2 + 103372598905826219735270400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{8}\cdot 19^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			69.1
Root			$50.9288i$ of defining polynomial
Character	$\chi$	$=$	76.69
Dual form			76.7.h.a.65.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-45.6057 + 26.3304i) q^{3} +(-106.246 - 184.024i) q^{5} -29.3098 q^{7} +(1022.08 - 1770.30i) q^{9} +O(q^{10})$	\(q+(-45.6057 + 26.3304i) q^{3} +(-106.246 - 184.024i) q^{5} -29.3098 q^{7} +(1022.08 - 1770.30i) q^{9} -1233.12 q^{11} +(-1236.43 - 713.855i) q^{13} +(9690.85 + 5595.02i) q^{15} +(-1071.44 - 1855.78i) q^{17} +(-4826.72 + 4873.26i) q^{19} +(1336.69 - 771.740i) q^{21} +(-4325.77 + 7492.46i) q^{23} +(-14764.0 + 25572.0i) q^{25} +69258.0i q^{27} +(15974.6 + 9222.95i) q^{29} -42379.6i q^{31} +(56237.2 - 32468.6i) q^{33} +(3114.05 + 5393.69i) q^{35} -4244.26i q^{37} +75184.5 q^{39} +(67848.9 - 39172.6i) q^{41} +(73392.5 + 127120. i) q^{43} -434370. q^{45} +(73415.4 - 127159. i) q^{47} -116790. q^{49} +(97727.1 + 56422.8i) q^{51} +(50241.7 + 29007.1i) q^{53} +(131014. + 226923. i) q^{55} +(91810.5 - 349338. i) q^{57} +(-7982.10 + 4608.47i) q^{59} +(-122097. + 211479. i) q^{61} +(-29957.1 + 51887.2i) q^{63} +303378. i q^{65} +(-393125. - 226971. i) q^{67} -455598. i q^{69} +(39272.6 - 22674.0i) q^{71} +(328579. + 569116. i) q^{73} -1.55497e6i q^{75} +36142.4 q^{77} +(312579. - 180467. i) q^{79} +(-1.07849e6 - 1.86801e6i) q^{81} +229403. q^{83} +(-227672. + 394339. i) q^{85} -971378. q^{87} +(90395.3 + 52189.8i) q^{89} +(36239.6 + 20922.9i) q^{91} +(1.11587e6 + 1.93275e6i) q^{93} +(1.40962e6 + 370465. i) q^{95} +(426941. - 246494. i) q^{97} +(-1.26035e6 + 2.18299e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) $ 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9	$ 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) $ 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9 - 3644 * q^11 - 7140 * q^13 + 9168 * q^15 + 1132 * q^17 + 2110 * q^19 - 8748 * q^21 + 832 * q^23 - 27698 * q^25 - 10920 * q^29 - 30306 * q^33 + 4172 * q^35 + 81144 * q^39 + 109206 * q^41 + 110740 * q^43 - 785440 * q^45 + 107080 * q^47 + 136092 * q^49 + 199872 * q^51 + 254796 * q^53 + 354840 * q^55 + 212268 * q^57 - 610638 * q^59 + 47864 * q^61 - 254476 * q^63 - 839562 * q^67 + 366660 * q^71 + 854482 * q^73 + 763088 * q^77 + 1718592 * q^79 - 1054142 * q^81 + 439612 * q^83 - 400236 * q^85 - 1604736 * q^87 + 478032 * q^89 + 599856 * q^91 + 829380 * q^93 - 1055660 * q^95 - 191286 * q^97 - 2336728 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−45.6057	+	26.3304i	−1.68910	+	0.975202i	−0.733891	+	0.679268i	$0.762298\pi$
$3$	−45.6057	+	26.3304i	−1.68910	+	0.975202i	−0.955208	+	0.295934i	$0.904369\pi$
$4$		0			0
$5$	−106.246	−	184.024i	−0.849969	−	1.47219i	−0.881235	−	0.472678i	$-0.843287\pi$
$5$	−106.246	−	184.024i	−0.849969	−	1.47219i	0.0312659	−	0.999511i	$-0.490046\pi$
$6$		0			0
$7$	−29.3098			−0.0854513			−0.0427256	−	0.999087i	$-0.513604\pi$
$7$	−29.3098			−0.0854513			−0.0427256	+	0.999087i	$0.513604\pi$
$8$		0			0
$9$	1022.08	−	1770.30i	1.40204	−	2.42840i
$10$		0			0
$11$	−1233.12			−0.926460			−0.463230	−	0.886238i	$-0.653310\pi$
$11$	−1233.12			−0.926460			−0.463230	+	0.886238i	$0.653310\pi$
$12$		0			0
$13$	−1236.43	−	713.855i	−0.562783	−	0.324923i	0.191479	−	0.981497i	$-0.438672\pi$
$13$	−1236.43	−	713.855i	−0.562783	−	0.324923i	−0.754262	+	0.656574i	$0.772005\pi$
$14$		0			0
$15$	9690.85	+	5595.02i	2.87136	+	1.65778i
$16$		0			0
$17$	−1071.44	−	1855.78i	−0.218082	−	0.377729i	0.736140	−	0.676830i	$-0.236647\pi$
$17$	−1071.44	−	1855.78i	−0.218082	−	0.377729i	−0.954222	+	0.299101i	$0.903313\pi$
$18$		0			0
$19$	−4826.72	+	4873.26i	−0.703706	+	0.710492i
$19$	−4826.72	+	4873.26i	−0.703706	+	0.710492i
$20$		0			0
$21$	1336.69	−	771.740i	0.144336	−	0.0833322i
$22$		0			0
$23$	−4325.77	+	7492.46i	−0.355533	+	0.615802i	−0.987209	−	0.159431i	$-0.949034\pi$
$23$	−4325.77	+	7492.46i	−0.355533	+	0.615802i	0.631676	+	0.775233i	$0.282367\pi$
$24$		0			0
$25$	−14764.0	+	25572.0i	−0.944895	+	1.63661i
$26$		0			0
$27$			69258.0i			3.51867i
$28$		0			0
$29$	15974.6	+	9222.95i	0.654993	+	0.378160i	0.790366	−	0.612634i	$-0.209890\pi$
$29$	15974.6	+	9222.95i	0.654993	+	0.378160i	−0.135374	+	0.990795i	$0.543223\pi$
$30$		0			0
$31$		−	42379.6i		−	1.42256i	−0.702906	−	0.711282i	$-0.748115\pi$
$31$		−	42379.6i		−	1.42256i	0.702906	−	0.711282i	$-0.251885\pi$
$32$		0			0
$33$	56237.2	−	32468.6i	1.56488	−	0.903485i
$34$		0			0
$35$	3114.05	+	5393.69i	0.0726309	+	0.125800i
$36$		0			0
$37$		−	4244.26i		−	0.0837909i	−0.999122	−	0.0418954i	$-0.986660\pi$
$37$		−	4244.26i		−	0.0837909i	0.999122	−	0.0418954i	$-0.0133396\pi$
$38$		0			0
$39$	75184.5			1.26746
$40$		0			0
$41$	67848.9	−	39172.6i	0.984444	−	0.568369i	0.0808351	−	0.996727i	$-0.474241\pi$
$41$	67848.9	−	39172.6i	0.984444	−	0.568369i	0.903609	+	0.428359i	$0.140908\pi$
$42$		0			0
$43$	73392.5	+	127120.i	0.923095	+	1.59885i	0.794597	+	0.607138i	$0.207682\pi$
$43$	73392.5	+	127120.i	0.923095	+	1.59885i	0.128498	+	0.991710i	$0.458984\pi$
$44$		0			0
$45$	−434370.			−4.76675
$46$		0			0
$47$	73415.4	−	127159.i	0.707121	−	1.22477i	−0.258800	−	0.965931i	$-0.583327\pi$
$47$	73415.4	−	127159.i	0.707121	−	1.22477i	0.965921	−	0.258838i	$-0.0833396\pi$
$48$		0			0
$49$	−116790.			−0.992698
$50$		0			0
$51$	97727.1	+	56422.8i	0.736724	+	0.425348i
$52$		0			0
$53$	50241.7	+	29007.1i	0.337471	+	0.194839i	0.659153	−	0.752009i	$-0.270915\pi$
$53$	50241.7	+	29007.1i	0.337471	+	0.194839i	−0.321682	+	0.946848i	$0.604248\pi$
$54$		0			0
$55$	131014.	+	226923.i	0.787462	+	1.36392i
$56$		0			0
$57$	91810.5	−	349338.i	0.495756	−	1.88635i
$58$		0			0
$59$	−7982.10	+	4608.47i	−0.0388652	+	0.0224389i	−0.519307	−	0.854588i	$-0.673810\pi$
$59$	−7982.10	+	4608.47i	−0.0388652	+	0.0224389i	0.480441	+	0.877027i	$0.340476\pi$
$60$		0			0
$61$	−122097.	+	211479.i	−0.537919	+	0.931703i	0.461097	+	0.887350i	$0.347456\pi$
$61$	−122097.	+	211479.i	−0.537919	+	0.931703i	−0.999016	+	0.0443530i	$0.985877\pi$
$62$		0			0
$63$	−29957.1	+	51887.2i	−0.119806	+	0.207510i
$64$		0			0
$65$			303378.i			1.10470i
$66$		0			0
$67$	−393125.	−	226971.i	−1.30709	−	0.754650i	−0.325482	−	0.945548i	$-0.605526\pi$
$67$	−393125.	−	226971.i	−1.30709	−	0.754650i	−0.981610	+	0.190899i	$0.938860\pi$
$68$		0			0
$69$		−	455598.i		−	1.38687i
$70$		0			0
$71$	39272.6	−	22674.0i	0.109727	−	0.0633510i	−0.444132	−	0.895961i	$-0.646488\pi$
$71$	39272.6	−	22674.0i	0.109727	−	0.0633510i	0.553859	+	0.832610i	$0.313155\pi$
$72$		0			0
$73$	328579.	+	569116.i	0.844640	+	1.46296i	0.885933	+	0.463813i	$0.153519\pi$
$73$	328579.	+	569116.i	0.844640	+	1.46296i	−0.0412933	+	0.999147i	$0.513148\pi$
$74$		0			0
$75$		−	1.55497e6i		−	3.68585i
$76$		0			0
$77$	36142.4			0.0791672
$78$		0			0
$79$	312579.	−	180467.i	0.633983	−	0.366031i	−0.148310	−	0.988941i	$-0.547383\pi$
$79$	312579.	−	180467.i	0.633983	−	0.366031i	0.782293	+	0.622910i	$0.214050\pi$
$80$		0			0
$81$	−1.07849e6	−	1.86801e6i	−2.02938	−	3.51499i
$82$		0			0
$83$	229403.			0.401203			0.200601	−	0.979673i	$-0.435710\pi$
$83$	229403.			0.401203			0.200601	+	0.979673i	$0.435710\pi$
$84$		0			0
$85$	−227672.	+	394339.i	−0.370726	+	0.642116i
$86$		0			0
$87$	−971378.			−1.47513
$88$		0			0
$89$	90395.3	+	52189.8i	0.128226	+	0.0740313i	0.562741	−	0.826633i	$-0.309747\pi$
$89$	90395.3	+	52189.8i	0.128226	+	0.0740313i	−0.434515	+	0.900665i	$0.643080\pi$
$90$		0			0
$91$	36239.6	+	20922.9i	0.0480905	+	0.0277651i
$92$		0			0
$93$	1.11587e6	+	1.93275e6i	1.38729	+	2.40285i
$94$		0			0
$95$	1.40962e6	+	370465.i	1.64411	+	0.432092i
$96$		0			0
$97$	426941.	−	246494.i	0.467792	−	0.270080i	−0.247523	−	0.968882i	$-0.579617\pi$
$97$	426941.	−	246494.i	0.467792	−	0.270080i	0.715315	+	0.698802i	$0.246283\pi$
$98$		0			0
$99$	−1.26035e6	+	2.18299e6i	−1.29893	+	2.24981i
$100$		0			0
$101$	347717.	−	602263.i	0.337491	−	0.584551i	−0.646469	−	0.762940i	$-0.723755\pi$
$101$	347717.	−	602263.i	0.337491	−	0.584551i	0.983960	+	0.178389i	$0.0570886\pi$
$102$		0			0
$103$		−	401838.i		−	0.367739i	−0.982951	−	0.183870i	$-0.941138\pi$
$103$		−	401838.i		−	0.367739i	0.982951	−	0.183870i	$-0.0588624\pi$
$104$		0			0
$105$	−284037.	−	163989.i	−0.245362	−	0.141660i
$106$		0			0
$107$			1.44086e6i			1.17617i	0.808798	+	0.588087i	$0.200119\pi$
$107$			1.44086e6i			1.17617i	−0.808798	+	0.588087i	$0.799881\pi$
$108$		0			0
$109$	740278.	−	427400.i	0.571630	−	0.330031i	−0.186170	−	0.982518i	$-0.559607\pi$
$109$	740278.	−	427400.i	0.571630	−	0.330031i	0.757800	+	0.652487i	$0.226274\pi$
$110$		0			0
$111$	111753.	+	193562.i	0.0817130	+	0.141531i
$112$		0			0
$113$			1.94585e6i			1.34857i	0.738470	+	0.674286i	$0.235548\pi$
$113$			1.94585e6i			1.34857i	−0.738470	+	0.674286i	$0.764452\pi$
$114$		0			0
$115$	1.83839e6			1.20877
$116$		0			0
$117$	−2.52748e6	+	1.45924e6i	−1.57808	+	0.911108i
$118$		0			0
$119$	31403.6	+	54392.5i	0.0186354	+	0.0322774i
$120$		0			0
$121$	−250981.			−0.141672
$122$		0			0
$123$	−2.06286e6	+	3.57298e6i	−1.10855	+	1.92006i
$124$		0			0
$125$	2.95427e6			1.51259
$126$		0			0
$127$	597645.	+	345050.i	0.291764	+	0.168450i	0.638737	−	0.769425i	$-0.279457\pi$
$127$	597645.	+	345050.i	0.291764	+	0.168450i	−0.346973	+	0.937875i	$0.612790\pi$
$128$		0			0
$129$	−6.69423e6	−	3.86492e6i	−3.11840	−	1.80041i
$130$		0			0
$131$	−1.79203e6	−	3.10389e6i	−0.797135	−	1.38068i	−0.921474	−	0.388439i	$-0.873014\pi$
$131$	−1.79203e6	−	3.10389e6i	−0.797135	−	1.38068i	0.124339	−	0.992240i	$-0.460319\pi$
$132$		0			0
$133$	141470.	−	142834.i	0.0601325	−	0.0607124i
$134$		0			0
$135$	1.27451e7	−	7.35840e6i	5.18015	−	2.99076i
$136$		0			0
$137$	−721921.	+	1.25040e6i	−0.280755	+	0.486282i	−0.971571	−	0.236749i	$-0.923918\pi$
$137$	−721921.	+	1.25040e6i	−0.280755	+	0.486282i	0.690816	+	0.723031i	$0.257252\pi$
$138$		0			0
$139$	128379.	−	222360.i	0.0478025	−	0.0827964i	−0.841134	−	0.540827i	$-0.818112\pi$
$139$	128379.	−	222360.i	0.0478025	−	0.0827964i	0.888937	+	0.458030i	$0.151445\pi$
$140$		0			0
$141$			7.73224e6i			2.75834i
$142$		0			0
$143$	1.52467e6	+	880268.i	0.521396	+	0.301028i
$144$		0			0
$145$		−	3.91961e6i		−	1.28570i
$146$		0			0
$147$	5.32628e6	−	3.07513e6i	1.67677	−	0.968081i
$148$		0			0
$149$	107357.	+	185948.i	0.0324542	+	0.0562124i	0.881796	−	0.471630i	$-0.156334\pi$
$149$	107357.	+	185948.i	0.0324542	+	0.0562124i	−0.849342	+	0.527843i	$0.823001\pi$
$150$		0			0
$151$			4.15531e6i			1.20690i	0.797400	+	0.603452i	$0.206208\pi$
$151$			4.15531e6i			1.20690i	−0.797400	+	0.603452i	$0.793792\pi$
$152$		0			0
$153$	−4.38039e6			−1.22304
$154$		0			0
$155$	−7.79885e6	+	4.50267e6i	−2.09428	+	1.20914i
$156$		0			0
$157$	−1.14394e6	−	1.98136e6i	−0.295599	−	0.511993i	0.679525	−	0.733652i	$-0.262186\pi$
$157$	−1.14394e6	−	1.98136e6i	−0.295599	−	0.511993i	−0.975124	+	0.221660i	$0.928853\pi$
$158$		0			0
$159$	−3.05508e6			−0.760030
$160$		0			0
$161$	126787.	−	219602.i	0.0303808	−	0.0526210i
$162$		0			0
$163$	−2.32928e6			−0.537848			−0.268924	−	0.963161i	$-0.586668\pi$
$163$	−2.32928e6			−0.537848			−0.268924	+	0.963161i	$0.586668\pi$
$164$		0			0
$165$	−1.19500e7	−	6.89932e6i	−2.66020	−	1.53587i
$166$		0			0
$167$	4.83313e6	+	2.79041e6i	1.03772	+	0.599126i	0.919186	−	0.393824i	$-0.128848\pi$
$167$	4.83313e6	+	2.79041e6i	1.03772	+	0.599126i	0.118531	+	0.992950i	$0.462181\pi$
$168$		0			0
$169$	−1.39423e6	−	2.41487e6i	−0.288850	−	0.500303i
$170$		0			0
$171$	3.69384e6	+	1.35256e7i	0.738736	+	2.70501i
$172$		0			0
$173$	8.86113e6	−	5.11598e6i	1.71140	−	0.988076i	0.778715	−	0.627378i	$-0.215872\pi$
$173$	8.86113e6	−	5.11598e6i	1.71140	−	0.988076i	0.932683	−	0.360697i	$-0.117461\pi$
$174$		0			0
$175$	432729.	−	749509.i	0.0807424	−	0.139850i
$176$		0			0
$177$	242686.	−	420345.i	0.0437648	−	0.0758029i
$178$		0			0
$179$		−	4.63247e6i		−	0.807706i	−0.914824	−	0.403853i	$-0.867671\pi$
$179$		−	4.63247e6i		−	0.807706i	0.914824	−	0.403853i	$-0.132329\pi$
$180$		0			0
$181$	−5.73559e6	−	3.31145e6i	−0.967258	−	0.558447i	−0.0688591	−	0.997626i	$-0.521936\pi$
$181$	−5.73559e6	−	3.31145e6i	−0.967258	−	0.558447i	−0.898399	+	0.439179i	$0.855269\pi$
$182$		0			0
$183$		−	1.28595e7i		−	2.09832i
$184$		0			0
$185$	−781044.	+	450936.i	−0.123356	+	0.0712197i
$186$		0			0
$187$	1.32121e6	+	2.28840e6i	0.202044	+	0.349951i
$188$		0			0
$189$		−	2.02994e6i		−	0.300675i
$190$		0			0
$191$	5.76811e6			0.827815			0.413908	−	0.910319i	$-0.364164\pi$
$191$	5.76811e6			0.827815			0.413908	+	0.910319i	$0.364164\pi$
$192$		0			0
$193$	4.02567e6	−	2.32422e6i	0.559972	−	0.323300i	−0.193162	−	0.981167i	$-0.561874\pi$
$193$	4.02567e6	−	2.32422e6i	0.559972	−	0.323300i	0.753134	+	0.657867i	$0.228541\pi$
$194$		0			0
$195$	−7.98807e6	−	1.38357e7i	−1.07730	−	1.86594i
$196$		0			0
$197$	−3.33501e6			−0.436213			−0.218107	−	0.975925i	$-0.569988\pi$
$197$	−3.33501e6			−0.436213			−0.218107	+	0.975925i	$0.569988\pi$
$198$		0			0
$199$	−6.68036e6	+	1.15707e7i	−0.847697	+	1.46825i	0.0355621	+	0.999367i	$0.488678\pi$
$199$	−6.68036e6	+	1.15707e7i	−0.847697	+	1.46825i	−0.883259	+	0.468886i	$0.844655\pi$
$200$		0			0
$201$	2.39050e7			2.94374
$202$		0			0
$203$	−468213.	−	270323.i	−0.0559700	−	0.0323143i
$204$		0			0
$205$	−1.44174e7	−	8.32387e6i	−1.67349	−	0.966192i
$206$		0			0
$207$	8.84262e6	+	1.53159e7i	0.996942	+	1.72675i
$208$		0			0
$209$	5.95191e6	−	6.00931e6i	0.651955	−	0.658242i
$210$		0			0
$211$	−6.58278e6	+	3.80057e6i	−0.700748	+	0.404577i	−0.807626	−	0.589695i	$-0.799248\pi$
$211$	−6.58278e6	+	3.80057e6i	−0.700748	+	0.404577i	0.106878	+	0.994272i	$0.465915\pi$
$212$		0			0
$213$	−1.19404e6	+	2.06813e6i	−0.123560	+	0.214012i
$214$		0			0
$215$	1.55953e7	−	2.70119e7i	1.56920	−	2.71794i
$216$		0			0
$217$			1.24214e6i			0.121560i
$218$		0			0
$219$	−2.99702e7	−	1.73033e7i	−2.85336	−	1.64739i
$220$		0			0
$221$			3.05940e6i			0.283439i
$222$		0			0
$223$	−1.19332e7	+	6.88964e6i	−1.07607	+	0.621272i	−0.929835	−	0.367977i	$-0.880050\pi$
$223$	−1.19332e7	+	6.88964e6i	−1.07607	+	0.621272i	−0.146240	+	0.989249i	$0.546717\pi$
$224$		0			0
$225$	3.01801e7	+	5.22734e7i	2.64955	+	4.58916i
$226$		0			0
$227$			1.49319e6i			0.127655i	0.997961	+	0.0638275i	$0.0203307\pi$
$227$			1.49319e6i			0.127655i	−0.997961	+	0.0638275i	$0.979669\pi$
$228$		0			0
$229$	−2.22506e6			−0.185283			−0.0926415	−	0.995700i	$-0.529531\pi$
$229$	−2.22506e6			−0.185283			−0.0926415	+	0.995700i	$0.529531\pi$
$230$		0			0
$231$	−1.64830e6	+	951646.i	−0.133721	+	0.0772040i
$232$		0			0
$233$	7.78735e6	+	1.34881e7i	0.615633	+	1.06631i	0.990273	+	0.139137i	$0.0444329\pi$
$233$	7.78735e6	+	1.34881e7i	0.615633	+	1.06631i	−0.374640	+	0.927170i	$0.622234\pi$
$234$		0			0
$235$	−3.12004e7			−2.40412
$236$		0			0
$237$	−9.50357e6	+	1.64607e7i	−0.713907	+	1.23652i
$238$		0			0
$239$	−1.04770e7			−0.767438			−0.383719	−	0.923450i	$-0.625357\pi$
$239$	−1.04770e7			−0.767438			−0.383719	+	0.923450i	$0.625357\pi$
$240$		0			0
$241$	−2.36496e6	−	1.36541e6i	−0.168956	−	0.0975467i	0.413138	−	0.910669i	$-0.364433\pi$
$241$	−2.36496e6	−	1.36541e6i	−0.168956	−	0.0975467i	−0.582093	+	0.813122i	$0.697766\pi$
$242$		0			0
$243$	5.46461e7	+	3.15500e7i	3.80838	+	2.19877i
$244$		0			0
$245$	1.24085e7	+	2.14921e7i	0.843763	+	1.46144i
$246$		0			0
$247$	9.44672e6	−	2.57989e6i	0.626888	−	0.171202i
$248$		0			0
$249$	−1.04621e7	+	6.04027e6i	−0.677671	+	0.391254i
$250$		0			0
$251$	−8.42322e6	+	1.45894e7i	−0.532668	+	0.922609i	0.466604	+	0.884466i	$0.345477\pi$
$251$	−8.42322e6	+	1.45894e7i	−0.532668	+	0.922609i	−0.999272	+	0.0381422i	$0.987856\pi$
$252$		0			0
$253$	5.33419e6	−	9.23909e6i	0.329387	−	0.570516i
$254$		0			0
$255$		−	2.39788e7i		−	1.44613i
$256$		0			0
$257$	−1.52289e7	−	8.79242e6i	−0.897159	−	0.517975i	−0.0208820	−	0.999782i	$-0.506647\pi$
$257$	−1.52289e7	−	8.79242e6i	−0.897159	−	0.517975i	−0.876278	+	0.481807i	$0.839981\pi$
$258$		0			0
$259$			124398.i			0.00716004i
$260$		0			0
$261$	3.26548e7	−	1.88533e7i	1.83665	−	1.06039i
$262$		0			0
$263$	5.12678e6	+	8.87985e6i	0.281824	+	0.488133i	0.971834	−	0.235667i	$-0.0757273\pi$
$263$	5.12678e6	+	8.87985e6i	0.281824	+	0.488133i	−0.690010	+	0.723800i	$0.742394\pi$
$264$		0			0
$265$		−	1.23276e7i		−	0.662429i
$266$		0			0
$267$	−5.49672e6			−0.288782
$268$		0			0
$269$	2.04053e7	−	1.17810e7i	1.04830	−	0.605237i	0.126128	−	0.992014i	$-0.459745\pi$
$269$	2.04053e7	−	1.17810e7i	1.04830	−	0.605237i	0.922173	+	0.386777i	$0.126412\pi$
$270$		0			0
$271$	8.07647e6	+	1.39889e7i	0.405802	+	0.702869i	0.994414	−	0.105546i	$-0.0336591\pi$
$271$	8.07647e6	+	1.39889e7i	0.405802	+	0.702869i	−0.588613	+	0.808415i	$0.700326\pi$
$272$		0			0
$273$	−2.20364e6			−0.108306
$274$		0			0
$275$	1.82057e7	−	3.15333e7i	0.875407	−	1.51625i
$276$		0			0
$277$	1.54131e7			0.725187			0.362593	−	0.931947i	$-0.381891\pi$
$277$	1.54131e7			0.725187			0.362593	+	0.931947i	$0.381891\pi$
$278$		0			0
$279$	−7.50248e7	−	4.33156e7i	−3.45455	−	1.99449i
$280$		0			0
$281$	3.25426e6	+	1.87885e6i	0.146667	+	0.0846784i	0.571538	−	0.820576i	$-0.306347\pi$
$281$	3.25426e6	+	1.87885e6i	0.146667	+	0.0846784i	−0.424871	+	0.905254i	$0.639680\pi$
$282$		0			0
$283$	−4.10431e6	−	7.10888e6i	−0.181084	−	0.313648i	0.761166	−	0.648558i	$-0.224627\pi$
$283$	−4.10431e6	−	7.10888e6i	−0.181084	−	0.313648i	−0.942250	+	0.334910i	$0.891294\pi$
$284$		0			0
$285$	−7.40410e7	+	2.02205e7i	−3.19844	+	0.873489i
$286$		0			0
$287$	−1.98864e6	+	1.14814e6i	−0.0841220	+	0.0485678i
$288$		0			0
$289$	9.77283e6	−	1.69270e7i	0.404881	−	0.701274i
$290$		0			0
$291$	−1.29806e7	+	2.24831e7i	−0.526764	+	0.912383i
$292$		0			0
$293$			1.93946e7i			0.771040i	0.922699	+	0.385520i	$0.125978\pi$
$293$			1.93946e7i			0.771040i	−0.922699	+	0.385520i	$0.874022\pi$
$294$		0			0
$295$	1.69614e6	+	979264.i	0.0660685	+	0.0381447i
$296$		0			0
$297$		−	8.54033e7i		−	3.25991i
$298$		0			0
$299$	1.06971e7	−	6.17595e6i	0.400176	−	0.231042i
$300$		0			0
$301$	−2.15112e6	−	3.72585e6i	−0.0788796	−	0.136624i
$302$		0			0
$303$			3.66222e7i			1.31649i
$304$		0			0
$305$	5.18895e7			1.82886
$306$		0			0
$307$	1.80540e7	−	1.04235e7i	0.623961	−	0.360244i	−0.154448	−	0.988001i	$-0.549360\pi$
$307$	1.80540e7	−	1.04235e7i	0.623961	−	0.360244i	0.778410	+	0.627757i	$0.216027\pi$
$308$		0			0
$309$	1.05806e7	+	1.83261e7i	0.358620	+	0.621148i
$310$		0			0
$311$	−7.30209e6			−0.242754			−0.121377	−	0.992606i	$-0.538731\pi$
$311$	−7.30209e6			−0.242754			−0.121377	+	0.992606i	$0.538731\pi$
$312$		0			0
$313$	−1.01402e7	+	1.75634e7i	−0.330685	+	0.572763i	−0.982646	−	0.185489i	$-0.940613\pi$
$313$	−1.01402e7	+	1.75634e7i	−0.330685	+	0.572763i	0.651962	+	0.758252i	$0.273946\pi$
$314$		0			0
$315$	1.27313e7			0.407325
$316$		0			0
$317$	2.20020e7	+	1.27029e7i	0.690693	+	0.398772i	0.803872	−	0.594803i	$-0.202770\pi$
$317$	2.20020e7	+	1.27029e7i	0.690693	+	0.398772i	−0.113179	+	0.993575i	$0.536103\pi$
$318$		0			0
$319$	−1.96986e7	−	1.13730e7i	−0.606825	−	0.350350i
$320$		0			0
$321$	−3.79386e7	−	6.57115e7i	−1.14701	−	1.98667i
$322$		0			0
$323$	1.42152e7	+	3.73594e6i	0.421839	+	0.110865i
$324$		0			0
$325$	3.65094e7	−	2.10787e7i	1.06354	−	0.614036i
$326$		0			0
$327$	−2.25072e7	+	3.89837e7i	−0.643694	+	1.11491i
$328$		0			0
$329$	−2.15179e6	+	3.72701e6i	−0.0604244	+	0.104658i
$330$		0			0
$331$		−	3.85917e6i		−	0.106417i	−0.998583	−	0.0532083i	$-0.983055\pi$
$331$		−	3.85917e6i		−	0.106417i	0.998583	−	0.0532083i	$-0.0169447\pi$
$332$		0			0
$333$	−7.51363e6	−	4.33799e6i	−0.203478	−	0.117478i
$334$		0			0
$335$			9.64590e7i			2.56572i
$336$		0			0
$337$	−3.35572e7	+	1.93742e7i	−0.876790	+	0.506215i	−0.869599	−	0.493759i	$-0.835622\pi$
$337$	−3.35572e7	+	1.93742e7i	−0.876790	+	0.506215i	−0.00719130	+	0.999974i	$0.502289\pi$
$338$		0			0
$339$	−5.12351e7	−	8.87418e7i	−1.31513	−	2.27787i
$340$		0			0
$341$			5.22591e7i			1.31795i
$342$		0			0
$343$	6.87135e6			0.170279
$344$		0			0
$345$	−8.38409e7	+	4.84055e7i	−2.04173	+	1.17879i
$346$		0			0
$347$	1.61942e6	+	2.80491e6i	0.0387588	+	0.0671321i	0.884754	−	0.466058i	$-0.154326\pi$
$347$	1.61942e6	+	2.80491e6i	0.0387588	+	0.0671321i	−0.845995	+	0.533190i	$0.820993\pi$
$348$		0			0
$349$	−1.99100e7			−0.468377			−0.234189	−	0.972191i	$-0.575243\pi$
$349$	−1.99100e7			−0.468377			−0.234189	+	0.972191i	$0.575243\pi$
$350$		0			0
$351$	4.94402e7	−	8.56330e7i	1.14330	−	1.98025i
$352$		0			0
$353$	6.13846e7			1.39552			0.697759	−	0.716333i	$-0.254181\pi$
$353$	6.13846e7			1.39552			0.697759	+	0.716333i	$0.254181\pi$
$354$		0			0
$355$	−8.34512e6	−	4.81806e6i	−0.186529	−	0.107693i
$356$		0			0
$357$	−2.86436e6	−	1.65374e6i	−0.0629540	−	0.0363465i
$358$		0			0
$359$	1.46393e7	+	2.53561e7i	0.316401	+	0.548023i	0.979734	−	0.200301i	$-0.0641920\pi$
$359$	1.46393e7	+	2.53561e7i	0.316401	+	0.548023i	−0.663333	+	0.748324i	$0.730859\pi$
$360$		0			0
$361$	−451488.	−	4.70437e7i	−0.00959676	−	0.999954i
$362$		0			0
$363$	1.14461e7	−	6.60843e6i	0.239298	−	0.138159i
$364$		0			0
$365$	6.98206e7	−	1.20933e8i	1.43584	−	2.48694i
$366$		0			0
$367$	1.14095e7	−	1.97619e7i	0.230818	−	0.399788i	−0.727231	−	0.686393i	$-0.759193\pi$
$367$	1.14095e7	−	1.97619e7i	0.230818	−	0.399788i	0.958049	+	0.286604i	$0.0925265\pi$
$368$		0			0
$369$		−	1.60151e8i		−	3.18750i
$370$		0			0
$371$	−1.47257e6	−	850191.i	−0.0288374	−	0.0166493i
$372$		0			0
$373$			7.45444e7i			1.43644i	0.695814	+	0.718222i	$0.255044\pi$
$373$			7.45444e7i			1.43644i	−0.695814	+	0.718222i	$0.744956\pi$
$374$		0			0
$375$	−1.34732e8	+	7.77873e7i	−2.55491	+	1.47508i
$376$		0			0
$377$	−1.31677e7	−	2.28071e7i	−0.245746	−	0.425644i
$378$		0			0
$379$			7.65294e7i			1.40576i	0.711309	+	0.702879i	$0.248102\pi$
$379$			7.65294e7i			1.40576i	−0.711309	+	0.702879i	$0.751898\pi$
$380$		0			0
$381$	−3.63413e7			−0.657091
$382$		0			0
$383$	−5.94308e6	+	3.43124e6i	−0.105783	+	0.0610738i	−0.551958	−	0.833872i	$-0.686119\pi$
$383$	−5.94308e6	+	3.43124e6i	−0.105783	+	0.0610738i	0.446175	+	0.894946i	$0.352786\pi$
$384$		0			0
$385$	−3.83999e6	−	6.65106e6i	−0.0672896	−	0.116549i
$386$		0			0
$387$	3.00054e8			5.17685
$388$		0			0
$389$	−4.08770e7	+	7.08011e7i	−0.694434	+	1.20279i	0.275938	+	0.961176i	$0.411012\pi$
$389$	−4.08770e7	+	7.08011e7i	−0.694434	+	1.20279i	−0.970371	+	0.241619i	$0.922322\pi$
$390$		0			0
$391$	1.85392e7			0.310141
$392$		0			0
$393$	1.63454e8	+	9.43700e7i	2.69288	+	1.55474i
$394$		0			0
$395$	−6.64205e7	−	3.83479e7i	−1.07773	−	0.622229i
$396$		0			0
$397$	2.20743e7	+	3.82339e7i	0.352790	+	0.611050i	0.986737	−	0.162326i	$-0.0518997\pi$
$397$	2.20743e7	+	3.82339e7i	0.352790	+	0.611050i	−0.633947	+	0.773376i	$0.718566\pi$
$398$		0			0
$399$	−2.69095e6	+	1.02390e7i	−0.0423630	+	0.161191i
$400$		0			0
$401$	3.66171e7	−	2.11409e7i	0.567873	−	0.327862i	−0.188426	−	0.982087i	$-0.560339\pi$
$401$	3.66171e7	−	2.11409e7i	0.567873	−	0.327862i	0.756299	+	0.654226i	$0.227005\pi$
$402$		0			0
$403$	−3.02529e7	+	5.23996e7i	−0.462224	+	0.800595i
$404$		0			0
$405$	−2.29172e8	+	3.96937e8i	−3.44982	+	5.97526i
$406$		0			0
$407$			5.23367e6i			0.0776289i
$408$		0			0
$409$	3.25975e7	+	1.88202e7i	0.476447	+	0.275077i	0.718935	−	0.695078i	$-0.244630\pi$
$409$	3.25975e7	+	1.88202e7i	0.476447	+	0.275077i	−0.242487	+	0.970155i	$0.577963\pi$
$410$		0			0
$411$		−	7.60340e7i		−	1.09517i
$412$		0			0
$413$	233954.	−	135073.i	0.00332108	−	0.00191743i
$414$		0			0
$415$	−2.43731e7	−	4.22155e7i	−0.341010	−	0.590647i
$416$		0			0
$417$			1.35211e7i			0.186468i
$418$		0			0
$419$	4.32766e7			0.588316			0.294158	−	0.955757i	$-0.404961\pi$
$419$	4.32766e7			0.588316			0.294158	+	0.955757i	$0.404961\pi$
$420$		0			0
$421$	6.03662e7	−	3.48525e7i	0.808999	−	0.467076i	−0.0376095	−	0.999293i	$-0.511974\pi$
$421$	6.03662e7	−	3.48525e7i	0.808999	−	0.467076i	0.846608	+	0.532217i	$0.178641\pi$
$422$		0			0
$423$	−1.50074e8	−	2.59935e8i	−1.98282	−	3.43434i
$424$		0			0
$425$	6.32746e7			0.824257
$426$		0			0
$427$	3.57865e6	−	6.19840e6i	0.0459658	−	0.0796152i
$428$		0			0
$429$	−9.27114e7			−1.17425
$430$		0			0
$431$	1.03241e8	+	5.96065e7i	1.28950	+	0.744495i	0.978566	−	0.205935i	$-0.0660237\pi$
$431$	1.03241e8	+	5.96065e7i	1.28950	+	0.744495i	0.310937	+	0.950430i	$0.399357\pi$
$432$		0			0
$433$	4.66058e7	+	2.69079e7i	0.574086	+	0.331449i	0.758780	−	0.651348i	$-0.225796\pi$
$433$	4.66058e7	+	2.69079e7i	0.574086	+	0.331449i	−0.184694	+	0.982796i	$0.559129\pi$
$434$		0			0
$435$	1.03205e8	+	1.78757e8i	1.25382	+	2.17167i
$436$		0			0
$437$	−1.56334e7	−	5.72446e7i	−0.187331	−	0.685947i
$438$		0			0
$439$	−1.09139e8	+	6.30114e7i	−1.28999	+	0.744775i	−0.978652	−	0.205524i	$-0.934110\pi$
$439$	−1.09139e8	+	6.30114e7i	−1.28999	+	0.744775i	−0.311337	+	0.950300i	$0.600777\pi$
$440$		0			0
$441$	−1.19369e8	+	2.06754e8i	−1.39180	+	2.41067i
$442$		0			0
$443$	1.16110e7	−	2.01109e7i	0.133555	−	0.231323i	−0.791490	−	0.611182i	$-0.790694\pi$
$443$	1.16110e7	−	2.01109e7i	0.133555	−	0.231323i	0.925044	+	0.379859i	$0.124028\pi$
$444$		0			0
$445$		−	2.21798e7i		−	0.251697i
$446$		0			0
$447$	−9.79217e6	−	5.65351e6i	−0.109637	−	0.0632988i
$448$		0			0
$449$		−	7.66539e6i		−	0.0846828i	−0.999103	−	0.0423414i	$-0.986518\pi$
$449$		−	7.66539e6i		−	0.0846828i	0.999103	−	0.0423414i	$-0.0134817\pi$
$450$		0			0
$451$	−8.36657e7	+	4.83044e7i	−0.912048	+	0.526571i
$452$		0			0
$453$	−1.09411e8	−	1.89506e8i	−1.17697	−	2.03858i
$454$		0			0
$455$		−	8.89193e6i		−	0.0943978i
$456$		0			0
$457$	5.20935e7			0.545802			0.272901	−	0.962042i	$-0.412017\pi$
$457$	5.20935e7			0.545802			0.272901	+	0.962042i	$0.412017\pi$
$458$		0			0
$459$	1.28528e8	−	7.42055e7i	1.32910	−	0.767358i
$460$		0			0
$461$	6.56153e7	+	1.13649e8i	0.669734	+	1.16001i	0.977978	+	0.208707i	$0.0669253\pi$
$461$	6.56153e7	+	1.13649e8i	0.669734	+	1.16001i	−0.308244	+	0.951307i	$0.599741\pi$
$462$		0			0
$463$	−1.40977e8			−1.42039			−0.710194	−	0.704006i	$-0.751393\pi$
$463$	−1.40977e8			−1.42039			−0.710194	+	0.704006i	$0.751393\pi$
$464$		0			0
$465$	2.37115e8	−	4.10695e8i	2.35830	−	4.08470i
$466$		0			0
$467$	−9.34464e7			−0.917512			−0.458756	−	0.888562i	$-0.651705\pi$
$467$	−9.34464e7			−0.917512			−0.458756	+	0.888562i	$0.651705\pi$
$468$		0			0
$469$	1.15224e7	+	6.65246e6i	0.111693	+	0.0644857i
$470$		0			0
$471$	1.04340e8	+	6.02408e7i	0.998592	+	0.576538i
$472$		0			0
$473$	−9.05016e7	−	1.56753e8i	−0.855211	−	1.48127i
$474$		0			0
$475$	−5.33573e7	−	1.95377e8i	−0.497867	−	1.82303i
$476$		0			0
$477$	1.02703e8	−	5.92954e7i	0.946295	−	0.546344i
$478$		0			0
$479$	2.55023e7	−	4.41712e7i	0.232045	−	0.401914i	−0.726365	−	0.687309i	$-0.758792\pi$
$479$	2.55023e7	−	4.41712e7i	0.232045	−	0.401914i	0.958410	+	0.285396i	$0.0921250\pi$
$480$		0			0
$481$	−3.02979e6	+	5.24775e6i	−0.0272256	+	0.0471561i
$482$		0			0
$483$			1.33535e7i			0.118509i
$484$		0			0
$485$	−9.07217e7	−	5.23782e7i	−0.795217	−	0.459119i
$486$		0			0
$487$			7.27504e7i			0.629867i	0.949114	+	0.314933i	$0.101982\pi$
$487$			7.27504e7i			0.629867i	−0.949114	+	0.314933i	$0.898018\pi$
$488$		0			0
$489$	1.06229e8	−	6.13311e7i	0.908479	−	0.524511i
$490$		0			0
$491$	−5.34853e7	−	9.26392e7i	−0.451845	−	0.782619i	0.546656	−	0.837358i	$-0.315901\pi$
$491$	−5.34853e7	−	9.26392e7i	−0.451845	−	0.782619i	−0.998501	+	0.0547388i	$0.982567\pi$
$492$		0			0
$493$		−	3.95272e7i		−	0.329880i
$494$		0			0
$495$	5.35630e8			4.41620
$496$		0			0
$497$	−1.15107e6	+	664571.i	−0.00937633	+	0.00541343i
$498$		0			0
$499$	2.08693e6	+	3.61466e6i	0.0167960	+	0.0290915i	0.874301	−	0.485384i	$-0.161320\pi$
$499$	2.08693e6	+	3.61466e6i	0.0167960	+	0.0290915i	−0.857505	+	0.514475i	$0.827987\pi$
$500$		0			0
$501$	−2.93891e8			−2.33708
$502$		0			0
$503$	−3.27062e7	+	5.66487e7i	−0.256996	+	0.445129i	−0.965436	−	0.260642i	$-0.916066\pi$
$503$	−3.27062e7	+	5.66487e7i	−0.256996	+	0.445129i	0.708440	+	0.705771i	$0.249399\pi$
$504$		0			0
$505$	−1.47774e8			−1.14743
$506$		0			0
$507$	1.27169e8	+	7.34211e7i	0.975794	+	0.563375i
$508$		0			0
$509$	−2.01366e8	−	1.16259e8i	−1.52698	−	0.881603i	−0.999486	−	0.0320454i	$-0.989798\pi$
$509$	−2.01366e8	−	1.16259e8i	−1.52698	−	0.881603i	−0.527495	−	0.849558i	$-0.676869\pi$
$510$		0			0
$511$	−9.63059e6	−	1.66807e7i	−0.0721756	−	0.125012i
$512$		0			0
$513$	−3.37513e8	−	3.34289e8i	−2.49999	−	2.47611i
$514$		0			0
$515$	−7.39478e7	+	4.26938e7i	−0.541382	+	0.312567i
$516$		0			0
$517$	−9.05299e7	+	1.56802e8i	−0.655119	+	1.13470i
$518$		0			0
$519$	−2.69412e8	+	4.66635e8i	−1.92715	+	3.33792i
$520$		0			0
$521$			1.30208e8i			0.920714i	0.887734	+	0.460357i	$0.152279\pi$
$521$			1.30208e8i			0.920714i	−0.887734	+	0.460357i	$0.847721\pi$
$522$		0			0
$523$	−6.22735e7	−	3.59536e7i	−0.435309	−	0.251326i	0.266297	−	0.963891i	$-0.414200\pi$
$523$	−6.22735e7	−	3.59536e7i	−0.435309	−	0.251326i	−0.701606	+	0.712565i	$0.747533\pi$
$524$		0			0
$525$			4.55758e7i			0.314961i
$526$		0			0
$527$	−7.86473e7	+	4.54071e7i	−0.537344	+	0.310235i
$528$		0			0
$529$	3.65933e7	+	6.33815e7i	0.247192	+	0.428149i
$530$		0			0
$531$			1.88410e7i			0.125840i
$532$		0			0
$533$	−1.11854e8			−0.738704
$534$		0			0
$535$	2.65153e8	−	1.53086e8i	1.73155	−	0.999711i
$536$		0			0
$537$	1.21975e8	+	2.11267e8i	0.787677	+	1.36430i
$538$		0			0
$539$	1.44016e8			0.919695
$540$		0			0
$541$	1.33532e8	−	2.31285e8i	0.843324	−	1.46068i	−0.0437454	−	0.999043i	$-0.513929\pi$
$541$	1.33532e8	−	2.31285e8i	0.843324	−	1.46068i	0.887069	−	0.461637i	$-0.152738\pi$
$542$		0			0
$543$	3.48767e8			2.17839
$544$		0			0
$545$	−1.57303e8	−	9.08191e7i	−0.971736	−	0.561032i
$546$		0			0
$547$	7.19217e7	+	4.15240e7i	0.439438	+	0.253710i	0.703359	−	0.710834i	$-0.251682\pi$
$547$	7.19217e7	+	4.15240e7i	0.439438	+	0.253710i	−0.263921	+	0.964544i	$0.585016\pi$
$548$		0			0
$549$	2.49588e8	+	4.32299e8i	1.50836	+	2.61256i
$550$		0			0
$551$	−1.22051e8	+	3.33319e7i	−0.729602	+	0.199253i
$552$		0			0
$553$	−9.16161e6	+	5.28946e6i	−0.0541747	+	0.0312778i
$554$		0			0
$555$	2.37467e7	−	4.11305e7i	0.138907	−	0.240594i
$556$		0			0
$557$	−6.51247e7	+	1.12799e8i	−0.376860	+	0.652740i	−0.990604	−	0.136765i	$-0.956329\pi$
$557$	−6.51247e7	+	1.12799e8i	−0.376860	+	0.652740i	0.613744	+	0.789505i	$0.289663\pi$
$558$		0			0
$559$		−	2.09567e8i		−	1.19974i
$560$		0			0
$561$	−1.20509e8	−	6.95760e7i	−0.682545	−	0.394067i
$562$		0			0
$563$		−	2.55431e8i		−	1.43136i	−0.698429	−	0.715679i	$-0.746117\pi$
$563$		−	2.55431e8i		−	1.43136i	0.698429	−	0.715679i	$-0.253883\pi$
$564$		0			0
$565$	3.58083e8	−	2.06739e8i	1.98535	−	1.14624i
$566$		0			0
$567$	3.16104e7	+	5.47509e7i	0.173413	+	0.300360i
$568$		0			0
$569$			1.34820e8i			0.731841i	0.930646	+	0.365920i	$0.119246\pi$
$569$			1.34820e8i			0.731841i	−0.930646	+	0.365920i	$0.880754\pi$
$570$		0			0
$571$	1.08447e8			0.582517			0.291259	−	0.956644i	$-0.405926\pi$
$571$	1.08447e8			0.582517			0.291259	+	0.956644i	$0.405926\pi$
$572$		0			0
$573$	−2.63059e8	+	1.51877e8i	−1.39826	+	0.807287i
$574$		0			0
$575$	−1.27731e8	−	2.21237e8i	−0.671883	−	1.16374i
$576$		0			0
$577$	−1.15661e8			−0.602086			−0.301043	−	0.953611i	$-0.597335\pi$
$577$	−1.15661e8			−0.602086			−0.301043	+	0.953611i	$0.597335\pi$
$578$		0			0
$579$	−1.22396e8	+	2.11996e8i	−0.630566	+	1.09217i
$580$		0			0
$581$	−6.72374e6			−0.0342833
$582$		0			0
$583$	−6.19540e7	−	3.57692e7i	−0.312654	−	0.180511i
$584$		0			0
$585$	5.37070e8	+	3.10078e8i	2.68265	+	1.54883i
$586$		0			0
$587$	5.91039e7	+	1.02371e8i	0.292215	+	0.506131i	0.974333	−	0.225111i	$-0.0722745\pi$
$587$	5.91039e7	+	1.02371e8i	0.292215	+	0.506131i	−0.682118	+	0.731242i	$0.738941\pi$
$588$		0			0
$589$	2.06527e8	+	2.04554e8i	1.01072	+	1.00107i
$590$		0			0
$591$	1.52096e8	−	8.78124e7i	0.736807	−	0.425396i
$592$		0			0
$593$	−1.09887e8	+	1.90329e8i	−0.526963	+	0.912727i	0.472543	+	0.881308i	$0.343336\pi$
$593$	−1.09887e8	+	1.90329e8i	−0.526963	+	0.912727i	−0.999506	+	0.0314196i	$0.989997\pi$
$594$		0			0
$595$	6.67301e6	−	1.15580e7i	0.0316790	−	0.0548696i
$596$		0			0
$597$		−	7.03587e8i		−	3.30670i
$598$		0			0
$599$	−1.52648e8	−	8.81316e7i	−0.710251	−	0.410064i	0.100903	−	0.994896i	$-0.467827\pi$
$599$	−1.52648e8	−	8.81316e7i	−0.710251	−	0.410064i	−0.811154	+	0.584833i	$0.801160\pi$
$600$		0			0
$601$			8.90617e7i			0.410268i	0.978734	+	0.205134i	$0.0657630\pi$
$601$			8.90617e7i			0.410268i	−0.978734	+	0.205134i	$0.934237\pi$
$602$		0			0
$603$	−8.03614e8	+	4.63967e8i	−3.66518	+	2.11609i
$604$		0			0
$605$	2.66657e7	+	4.61864e7i	0.120417	+	0.208568i
$606$		0			0
$607$		−	3.31481e8i		−	1.48215i	−0.671422	−	0.741075i	$-0.734316\pi$
$607$		−	3.31481e8i		−	1.48215i	0.671422	−	0.741075i	$-0.265684\pi$
$608$		0			0
$609$	2.84709e7			0.126052
$610$		0			0
$611$	−1.81547e8	+	1.04816e8i	−0.795911	+	0.459519i
$612$		0			0
$613$	1.44103e7	+	2.49594e7i	0.0625593	+	0.108356i	0.895609	−	0.444843i	$-0.146740\pi$
$613$	1.44103e7	+	2.49594e7i	0.0625593	+	0.108356i	−0.833049	+	0.553199i	$0.813407\pi$
$614$		0			0
$615$	8.76684e8			3.76893
$616$		0			0
$617$	8.81885e7	−	1.52747e8i	0.375454	−	0.650305i	−0.614941	−	0.788573i	$-0.710820\pi$
$617$	8.81885e7	−	1.52747e8i	0.375454	−	0.650305i	0.990395	+	0.138268i	$0.0441536\pi$
$618$		0			0
$619$	1.79968e8			0.758794			0.379397	−	0.925234i	$-0.376131\pi$
$619$	1.79968e8			0.758794			0.379397	+	0.925234i	$0.376131\pi$
$620$		0			0
$621$	−5.18913e8	−	2.99595e8i	−2.16680	−	1.25101i
$622$		0			0
$623$	−2.64947e6	−	1.52967e6i	−0.0109571	−	0.00632607i
$624$		0			0
$625$	−8.31927e7	−	1.44094e8i	−0.340757	−	0.590209i
$626$		0			0
$627$	−1.13213e8	+	4.30775e8i	−0.459298	+	1.74762i
$628$		0			0
$629$	−7.87642e6	+	4.54745e6i	−0.0316502	+	0.0182733i
$630$		0			0
$631$	5.06395e6	−	8.77102e6i	0.0201559	−	0.0349110i	−0.855772	−	0.517354i	$-0.826917\pi$
$631$	5.06395e6	−	8.77102e6i	0.0201559	−	0.0349110i	0.875927	+	0.482443i	$0.160250\pi$
$632$		0			0
$633$	2.00141e8	−	3.46655e8i	0.789088	−	1.36674i
$634$		0			0
$635$		−	1.46641e8i		−	0.572709i
$636$		0			0
$637$	1.44403e8	+	8.33711e7i	0.558673	+	0.322550i
$638$		0			0
$639$		−	9.26992e7i		−	0.355282i
$640$		0			0
$641$	−2.70193e8	+	1.55996e8i	−1.02589	+	0.592296i	−0.915804	−	0.401626i	$-0.868445\pi$
$641$	−2.70193e8	+	1.55996e8i	−1.02589	+	0.592296i	−0.110083	+	0.993922i	$0.535112\pi$
$642$		0			0
$643$	1.67942e8	+	2.90884e8i	0.631723	+	1.09418i	0.987199	+	0.159491i	$0.0509854\pi$
$643$	1.67942e8	+	2.90884e8i	0.631723	+	1.09418i	−0.355476	+	0.934685i	$0.615681\pi$
$644$		0			0
$645$			1.64253e9i			6.12116i
$646$		0			0
$647$	−2.06589e8			−0.762770			−0.381385	−	0.924416i	$-0.624553\pi$
$647$	−2.06589e8			−0.762770			−0.381385	+	0.924416i	$0.624553\pi$
$648$		0			0
$649$	9.84288e6	−	5.68279e6i	0.0360071	−	0.0207887i
$650$		0			0
$651$	−3.27060e7	−	5.66485e7i	−0.118545	−	0.205327i
$652$		0			0
$653$	−8.31685e7			−0.298689			−0.149344	−	0.988785i	$-0.547716\pi$
$653$	−8.31685e7			−0.298689			−0.149344	+	0.988785i	$0.547716\pi$
$654$		0			0
$655$	−3.80793e8	+	6.59553e8i	−1.35508	+	2.34707i
$656$		0			0
$657$	1.34334e9			4.73687
$658$		0			0
$659$	1.81887e8	+	1.05013e8i	0.635544	+	0.366932i	0.782896	−	0.622152i	$-0.213742\pi$
$659$	1.81887e8	+	1.05013e8i	0.635544	+	0.366932i	−0.147352	+	0.989084i	$0.547075\pi$
$660$		0			0
$661$	3.63055e6	+	2.09610e6i	0.0125710	+	0.00725785i	0.506272	−	0.862374i	$-0.331023\pi$
$661$	3.63055e6	+	2.09610e6i	0.0125710	+	0.00725785i	−0.493701	+	0.869632i	$0.664356\pi$
$662$		0			0
$663$	−8.05554e7	−	1.39526e8i	−0.276410	−	0.478757i
$664$		0			0
$665$	−4.13155e7	−	1.08582e7i	−0.140491	−	0.0369228i
$666$		0			0
$667$	−1.38205e8	+	7.97928e7i	−0.465743	+	0.268897i
$668$		0			0
$669$	3.62814e8	−	6.28413e8i	1.21173	−	2.09878i
$670$		0			0
$671$	1.50560e8	−	2.60778e8i	0.498360	−	0.863185i
$672$		0			0
$673$			1.80245e8i			0.591315i	0.955294	+	0.295658i	$0.0955388\pi$
$673$			1.80245e8i			0.591315i	−0.955294	+	0.295658i	$0.904461\pi$
$674$		0			0
$675$	−1.77106e9	−	1.02252e9i	−5.75868	−	3.32477i
$676$		0			0
$677$		−	1.70990e7i		−	0.0551067i	−0.999620	−	0.0275534i	$-0.991228\pi$
$677$		−	1.70990e7i		−	0.0551067i	0.999620	−	0.0275534i	$-0.00877161\pi$
$678$		0			0
$679$	−1.25135e7	+	7.22470e6i	−0.0399734	+	0.0230787i
$680$		0			0
$681$	−3.93164e7	−	6.80980e7i	−0.124489	−	0.215622i
$682$		0			0
$683$			2.82504e8i			0.886670i	0.896356	+	0.443335i	$0.146205\pi$
$683$			2.82504e8i			0.886670i	−0.896356	+	0.443335i	$0.853795\pi$
$684$		0			0
$685$	3.06805e8			0.954533
$686$		0			0
$687$	1.01475e8	−	5.85869e7i	0.312961	−	0.180688i
$688$		0			0
$689$	−4.14137e7	−	7.17307e7i	−0.126615	−	0.219304i
$690$		0			0
$691$	5.21610e7			0.158093			0.0790464	−	0.996871i	$-0.474812\pi$
$691$	5.21610e7			0.158093			0.0790464	+	0.996871i	$0.474812\pi$
$692$		0			0
$693$	3.69406e7	−	6.39830e7i	0.110995	−	0.192249i
$694$		0			0
$695$	−5.45592e7			−0.162523
$696$		0			0
$697$	−1.45391e8	−	8.39418e7i	−0.429379	−	0.247902i
$698$		0			0
$699$	−7.10294e8	−	4.10089e8i	−2.07973	−	1.20073i
$700$		0			0
$701$	−2.29400e8	−	3.97333e8i	−0.665948	−	1.15346i	−0.979027	−	0.203729i	$-0.934694\pi$
$701$	−2.29400e8	−	3.97333e8i	−0.665948	−	1.15346i	0.313080	−	0.949727i	$-0.398639\pi$
$702$		0			0
$703$	2.06834e7	+	2.04858e7i	0.0595327	+	0.0589641i
$704$		0			0
$705$	1.42292e9	−	8.21521e8i	4.06080	−	2.34451i
$706$		0			0
$707$	−1.01915e7	+	1.76522e7i	−0.0288390	+	0.0499506i
$708$		0			0
$709$	−1.51093e8	+	2.61702e8i	−0.423943	+	0.734290i	−0.996321	−	0.0856995i	$-0.972688\pi$
$709$	−1.51093e8	+	2.61702e8i	−0.423943	+	0.734290i	0.572378	+	0.819990i	$0.306021\pi$
$710$		0			0
$711$		−	7.37812e8i		−	2.05275i
$712$		0			0
$713$	3.17528e8	+	1.83325e8i	0.876018	+	0.505769i
$714$		0			0
$715$		−	3.74100e8i		−	1.02346i
$716$		0			0
$717$	4.77811e8	−	2.75864e8i	1.29628	−	0.748407i
$718$		0			0
$719$	1.32874e8	+	2.30144e8i	0.357480	+	0.619174i	0.987539	−	0.157373i	$-0.0503026\pi$
$719$	1.32874e8	+	2.30144e8i	0.357480	+	0.619174i	−0.630059	+	0.776547i	$0.716969\pi$
$720$		0			0
$721$			1.17778e7i			0.0314238i
$722$		0			0
$723$	1.43808e8			0.380511
$724$		0			0
$725$	−4.71698e8	+	2.72335e8i	−1.23780	+	0.714643i
$726$		0			0
$727$	−1.30595e8	−	2.26198e8i	−0.339879	−	0.588688i	0.644530	−	0.764579i	$-0.277053\pi$
$727$	−1.30595e8	−	2.26198e8i	−0.339879	−	0.588688i	−0.984410	+	0.175890i	$0.943720\pi$
$728$		0			0
$729$	−1.75045e9			−4.51822
$730$		0			0
$731$	1.57271e8	−	2.72401e8i	0.402621	−	0.697359i
$732$		0			0
$733$	−4.89100e8			−1.24190			−0.620948	−	0.783851i	$-0.713252\pi$
$733$	−4.89100e8			−1.24190			−0.620948	+	0.783851i	$0.713252\pi$
$734$		0			0
$735$	−1.13179e9	−	6.53442e8i	−2.85040	−	1.64568i
$736$		0			0
$737$	4.84769e8	+	2.79882e8i	1.21097	+	0.699153i
$738$		0			0
$739$	2.97068e8	+	5.14537e8i	0.736076	+	1.27492i	0.954250	+	0.299011i	$0.0966566\pi$
$739$	2.97068e8	+	5.14537e8i	0.736076	+	1.27492i	−0.218174	+	0.975910i	$0.570010\pi$
$740$		0			0
$741$	−3.62894e8	+	3.66394e8i	−0.891920	+	0.900521i
$742$		0			0
$743$	3.59351e8	−	2.07471e8i	0.876098	−	0.505815i	0.00672784	−	0.999977i	$-0.497858\pi$
$743$	3.59351e8	−	2.07471e8i	0.876098	−	0.505815i	0.869370	+	0.494162i	$0.164525\pi$
$744$		0			0
$745$	2.28125e7	−	3.95124e7i	0.0551702	−	0.0955575i
$746$		0			0
$747$	2.34469e8	−	4.06112e8i	0.562501	−	0.974281i
$748$		0			0
$749$		−	4.22314e7i		−	0.100505i
$750$		0			0
$751$	3.08591e8	+	1.78165e8i	0.728557	+	0.420633i	0.817894	−	0.575369i	$-0.195141\pi$
$751$	3.08591e8	+	1.78165e8i	0.728557	+	0.420633i	−0.0893368	+	0.996001i	$0.528475\pi$
$752$		0			0
$753$		−	8.87148e8i		−	2.07784i
$754$		0			0
$755$	7.64675e8	−	4.41486e8i	1.77679	−	1.02583i
$756$		0			0
$757$	4.00583e8	+	6.93830e8i	0.923431	+	1.59943i	0.794065	+	0.607833i	$0.207961\pi$
$757$	4.00583e8	+	6.93830e8i	0.923431	+	1.59943i	0.129367	+	0.991597i	$0.458706\pi$
$758$		0			0
$759$			5.61806e8i			1.28488i
$760$		0			0
$761$	−3.90770e8			−0.886681			−0.443340	−	0.896353i	$-0.646207\pi$
$761$	−3.90770e8			−0.886681			−0.443340	+	0.896353i	$0.646207\pi$
$762$		0			0
$763$	−2.16974e7	+	1.25270e7i	−0.0488465	+	0.0282016i
$764$		0			0
$765$	4.65400e8	+	8.06096e8i	1.03954	+	1.80054i
$766$		0			0
$767$	1.31591e7			0.0291636
$768$		0			0
$769$	3.44452e8	−	5.96608e8i	0.757443	−	1.31193i	−0.186708	−	0.982415i	$-0.559782\pi$
$769$	3.44452e8	−	5.96608e8i	0.757443	−	1.31193i	0.944151	−	0.329514i	$-0.106885\pi$
$770$		0			0
$771$	9.26033e8			2.02052
$772$		0			0
$773$	−3.65594e8	−	2.11076e8i	−0.791517	−	0.456983i	0.0489793	−	0.998800i	$-0.484403\pi$
$773$	−3.65594e8	−	2.11076e8i	−0.791517	−	0.456983i	−0.840496	+	0.541817i	$0.817736\pi$
$774$		0			0
$775$	1.08373e9	+	6.25692e8i	2.32818	+	1.34417i
$776$		0			0
$777$	−3.27546e6	−	5.67327e6i	−0.00698248	−	0.0120940i
$778$		0			0
$779$	−1.36589e8	+	5.19720e8i	−0.288937	+	1.09940i
$780$		0			0
$781$	−4.84277e7	+	2.79598e7i	−0.101658	+	0.0586922i
$782$		0			0
$783$	−6.38763e8	+	1.10637e9i	−1.33062	+	2.30470i
$784$		0			0
$785$	−2.43078e8	+	4.21023e8i	−0.502500	+	0.870356i
$786$		0			0
$787$			5.74087e8i			1.17775i	0.808224	+	0.588875i	$0.200429\pi$
$787$			5.74087e8i			1.17775i	−0.808224	+	0.588875i	$0.799571\pi$
$788$		0			0
$789$	−4.67621e8	−	2.69981e8i	−0.952056	−	0.549670i
$790$		0			0
$791$		−	5.70324e7i		−	0.115237i
$792$		0			0
$793$	3.01931e8	−	1.74320e8i	0.605463	−	0.349564i
$794$		0			0
$795$	3.24590e8	+	5.62207e8i	0.646002	+	1.11891i
$796$		0			0
$797$		−	4.40620e8i		−	0.870340i	−0.900348	−	0.435170i	$-0.856688\pi$
$797$		−	4.40620e8i		−	0.870340i	0.900348	−	0.435170i	$-0.143312\pi$
$798$		0			0
$799$	−3.14640e8			−0.616841
$800$		0			0
$801$	1.84783e8	−	1.06685e8i	0.359555	−	0.207589i
$802$		0			0
$803$	−4.05177e8	−	7.01787e8i	−0.782525	−	1.35537i
$804$		0			0
$805$	−5.38827e7			−0.103291
$806$		0			0
$807$	−6.20398e8	+	1.07456e9i	−1.18046	+	2.04461i
$808$		0			0
$809$	−5.39215e8			−1.01839			−0.509197	−	0.860650i	$-0.670058\pi$
$809$	−5.39215e8			−1.01839			−0.509197	+	0.860650i	$0.670058\pi$
$810$		0			0
$811$	3.24724e8	+	1.87480e8i	0.608768	+	0.351472i	0.772483	−	0.635035i	$-0.219014\pi$
$811$	3.24724e8	+	1.87480e8i	0.608768	+	0.351472i	−0.163715	+	0.986508i	$0.552348\pi$
$812$		0			0
$813$	−7.36666e8	−	4.25314e8i	−1.37088	−	0.791477i
$814$		0			0
$815$	2.47477e8	+	4.28644e8i	0.457154	+	0.791815i
$816$		0			0
$817$	−9.73732e8	−	2.55909e8i	−1.78555	−	0.469267i
$818$		0			0
$819$	7.40799e7	−	4.27701e7i	0.134849	−	0.0778553i
$820$		0			0
$821$	−3.43904e8	+	5.95659e8i	−0.621452	+	1.07639i	0.367764	+	0.929919i	$0.380123\pi$
$821$	−3.43904e8	+	5.95659e8i	−0.621452	+	1.07639i	−0.989216	+	0.146467i	$0.953210\pi$
$822$		0			0
$823$	−9.89198e7	+	1.71334e8i	−0.177453	+	0.307358i	−0.941007	−	0.338386i	$-0.890119\pi$
$823$	−9.89198e7	+	1.71334e8i	−0.177453	+	0.307358i	0.763554	+	0.645744i	$0.223452\pi$
$824$		0			0
$825$			1.91746e9i			3.41479i
$826$		0			0
$827$	1.81354e8	+	1.04705e8i	0.320635	+	0.185118i	0.651675	−	0.758498i	$-0.274066\pi$
$827$	1.81354e8	+	1.04705e8i	0.320635	+	0.185118i	−0.331041	+	0.943616i	$0.607400\pi$
$828$		0			0
$829$		−	1.00441e9i		−	1.76298i	−0.472203	−	0.881490i	$-0.656541\pi$
$829$		−	1.00441e9i		−	1.76298i	0.472203	−	0.881490i	$-0.343459\pi$
$830$		0			0
$831$	−7.02924e8	+	4.05833e8i	−1.22491	+	0.707203i
$832$		0			0
$833$	1.25133e8	+	2.16737e8i	0.216489	+	0.374971i
$834$		0			0
$835$		−	1.18588e9i		−	2.03696i
$836$		0			0
$837$	2.93513e9			5.00554
$838$		0			0
$839$	1.37154e8	−	7.91861e7i	0.232233	−	0.134080i	−0.379369	−	0.925246i	$-0.623859\pi$
$839$	1.37154e8	−	7.91861e7i	0.232233	−	0.134080i	0.611602	+	0.791166i	$0.290526\pi$
$840$		0			0
$841$	−1.27286e8	−	2.20466e8i	−0.213990	−	0.370641i
$842$		0			0
$843$	−1.97884e8			−0.330314
$844$		0			0
$845$	−2.96262e8	+	5.13141e8i	−0.491028	+	0.850485i
$846$		0			0
$847$	7.35619e6			0.0121061
$848$		0			0
$849$	3.74360e8	+	2.16137e8i	0.611739	+	0.353188i
$850$		0			0
$851$	3.17999e7	+	1.83597e7i	0.0515986	+	0.0297904i
$852$		0			0
$853$	4.06930e8	+	7.04824e8i	0.655652	+	1.13562i	0.981730	+	0.190279i	$0.0609393\pi$
$853$	4.06930e8	+	7.04824e8i	0.655652	+	1.13562i	−0.326078	+	0.945343i	$0.605727\pi$
$854$		0			0
$855$	2.09658e9	−	2.11680e9i	3.35439	−	3.38674i
$856$		0			0
$857$	5.03084e8	−	2.90455e8i	0.799278	−	0.461463i	−0.0439409	−	0.999034i	$-0.513991\pi$
$857$	5.03084e8	−	2.90455e8i	0.799278	−	0.461463i	0.843218	+	0.537571i	$0.180658\pi$
$858$		0			0
$859$	−3.95261e8	+	6.84613e8i	−0.623598	+	1.08010i	0.365212	+	0.930924i	$0.380997\pi$
$859$	−3.95261e8	+	6.84613e8i	−0.623598	+	1.08010i	−0.988810	+	0.149179i	$0.952337\pi$
$860$		0			0
$861$	6.04620e7	−	1.04723e8i	0.0947269	−	0.164072i
$862$		0			0
$863$		−	1.17877e9i		−	1.83398i	−0.398907	−	0.916992i	$-0.630610\pi$
$863$		−	1.17877e9i		−	1.83398i	0.398907	−	0.916992i	$-0.369390\pi$
$864$		0			0
$865$	−1.88292e9	−	1.08711e9i	−2.90927	−	1.67967i
$866$		0			0
$867$			1.02929e9i			1.57936i
$868$		0			0
$869$	−3.85446e8	+	2.22538e8i	−0.587360	+	0.339113i
$870$		0			0
$871$	3.24048e8	+	5.61268e8i	0.490406	+	0.849408i
$872$		0			0
$873$		−	1.00775e9i		−	1.51465i
$874$		0			0
$875$	−8.65890e7			−0.129252
$876$		0			0
$877$	1.15470e9	−	6.66664e8i	1.71186	−	0.988344i	0.779814	−	0.626011i	$-0.215314\pi$
$877$	1.15470e9	−	6.66664e8i	1.71186	−	0.988344i	0.932049	−	0.362333i	$-0.118020\pi$
$878$		0			0
$879$	−5.10667e8	−	8.84502e8i	−0.751920	−	1.30236i
$880$		0			0
$881$	2.51326e8			0.367545			0.183772	−	0.982969i	$-0.441169\pi$
$881$	2.51326e8			0.367545			0.183772	+	0.982969i	$0.441169\pi$
$882$		0			0
$883$	5.12698e8	−	8.88020e8i	0.744697	−	1.28985i	−0.205639	−	0.978628i	$-0.565927\pi$
$883$	5.12698e8	−	8.88020e8i	0.744697	−	1.28985i	0.950336	−	0.311226i	$-0.100739\pi$
$884$		0			0
$885$	−1.03138e8			−0.148795
$886$		0			0
$887$	−3.98793e8	−	2.30243e8i	−0.571448	−	0.329926i	0.186280	−	0.982497i	$-0.440357\pi$
$887$	−3.98793e8	−	2.30243e8i	−0.571448	−	0.329926i	−0.757727	+	0.652571i	$0.773690\pi$
$888$		0			0
$889$	−1.75168e7	−	1.01133e7i	−0.0249316	−	0.0143943i
$890$		0			0
$891$	1.32991e9	+	2.30347e9i	1.88014	+	3.25649i
$892$		0			0
$893$	2.65325e8	+	9.71534e8i	0.372583	+	1.36428i
$894$		0			0
$895$	−8.52484e8	+	4.92182e8i	−1.18910	+	0.686525i
$896$		0			0
$897$	−3.25231e8	+	5.63317e8i	−0.450625	+	0.780505i
$898$		0			0
$899$	3.90865e8	−	6.76998e8i	0.537957	−	0.931770i
$900$		0			0
$901$		−	1.24317e8i		−	0.169964i
$902$		0			0
$903$	1.96206e8	+	1.13280e8i	0.266471	+	0.153847i
$904$		0			0
$905$			1.40731e9i			1.89865i
$906$		0			0
$907$	6.09116e8	−	3.51673e8i	0.816353	−	0.471322i	−0.0328041	−	0.999462i	$-0.510444\pi$
$907$	6.09116e8	−	3.51673e8i	0.816353	−	0.471322i	0.849157	+	0.528140i	$0.177110\pi$
$908$		0			0
$909$	−7.10792e8	−	1.23113e9i	−0.946348	−	1.63912i
$910$		0			0
$911$			5.23305e8i			0.692149i	0.938207	+	0.346074i	$0.112486\pi$
$911$			5.23305e8i			0.692149i	−0.938207	+	0.346074i	$0.887514\pi$
$912$		0			0
$913$	−2.82881e8			−0.371698
$914$		0			0
$915$	−2.36645e9	+	1.36627e9i	−3.08912	+	1.78350i
$916$		0			0
$917$	5.25241e7	+	9.09744e7i	0.0681162	+	0.117981i
$918$		0			0
$919$	−8.61834e8			−1.11039			−0.555197	−	0.831719i	$-0.687357\pi$
$919$	−8.61834e8			−1.11039			−0.555197	+	0.831719i	$0.687357\pi$
$920$		0			0
$921$	−5.48909e8	+	9.50738e8i	−0.702621	+	1.21698i
$922$		0			0
$923$	−6.47439e7			−0.0823368
$924$		0			0
$925$	1.08534e8	+	6.26622e7i	0.137133	+	0.0791736i
$926$		0			0
$927$	−7.11376e8	−	4.10713e8i	−0.893017	−	0.515584i
$928$		0			0
$929$	−4.41263e8	−	7.64290e8i	−0.550364	−	0.953259i	−0.998248	−	0.0591674i	$-0.981155\pi$
$929$	−4.41263e8	−	7.64290e8i	−0.550364	−	0.953259i	0.447884	−	0.894092i	$-0.352178\pi$
$930$		0			0
$931$	5.63712e8	−	5.69148e8i	0.698567	−	0.705304i
$932$		0			0
$933$	3.33017e8	−	1.92267e8i	0.410035	−	0.236734i
$934$		0			0
$935$	2.80746e8	−	4.86267e8i	0.343462	−	0.594894i
$936$		0			0
$937$	−1.03224e8	+	1.78789e8i	−0.125476	+	0.217331i	−0.921919	−	0.387383i	$-0.873379\pi$
$937$	−1.03224e8	+	1.78789e8i	−0.125476	+	0.217331i	0.796443	+	0.604714i	$0.206713\pi$
$938$		0			0
$939$		−	1.06799e9i		−	1.28994i
$940$		0			0
$941$	9.01333e8	+	5.20385e8i	1.08172	+	0.624533i	0.931361	−	0.364098i	$-0.118623\pi$
$941$	9.01333e8	+	5.20385e8i	1.08172	+	0.624533i	0.150363	+	0.988631i	$0.451956\pi$
$942$		0			0
$943$			6.77806e8i			0.808296i
$944$		0			0
$945$	−3.73557e8	+	2.15673e8i	−0.442651	+	0.255564i
$946$		0			0
$947$	−6.09878e8	−	1.05634e9i	−0.718113	−	1.24381i	−0.961747	−	0.273941i	$-0.911673\pi$
$947$	−6.09878e8	−	1.05634e9i	−0.718113	−	1.24381i	0.243633	−	0.969867i	$-0.421661\pi$
$948$		0			0
$949$		−	9.38233e8i		−	1.09777i
$950$		0			0
$951$	−1.33789e9			−1.55553
$952$		0			0
$953$	8.84221e8	−	5.10505e8i	1.02160	−	0.589822i	0.107035	−	0.994255i	$-0.465864\pi$
$953$	8.84221e8	−	5.10505e8i	1.02160	−	0.589822i	0.914568	+	0.404433i	$0.132531\pi$
$954$		0			0
$955$	−6.12839e8	−	1.06147e9i	−0.703617	−	1.21870i
$956$		0			0
$957$	1.19782e9			1.36665
$958$		0			0
$959$	2.11593e7	−	3.66491e7i	0.0239909	−	0.0415534i
$960$		0			0
$961$	−9.08529e8			−1.02369
$962$		0			0
$963$	2.55076e9	+	1.47268e9i	2.85622	+	1.64904i
$964$		0			0
$965$	−8.55424e8	−	4.93879e8i	−0.951918	−	0.549590i
$966$		0			0
$967$	−6.83063e7	−	1.18310e8i	−0.0755408	−	0.130840i	0.825781	−	0.563992i	$-0.190735\pi$
$967$	−6.83063e7	−	1.18310e8i	−0.0755408	−	0.130840i	−0.901321	+	0.433151i	$0.857402\pi$
$968$		0			0
$969$	−7.46664e8	+	2.03913e8i	−0.820642	+	0.224116i
$970$		0			0
$971$	6.81790e8	−	3.93632e8i	0.744720	−	0.429964i	−0.0790631	−	0.996870i	$-0.525193\pi$
$971$	6.81790e8	−	3.93632e8i	0.744720	−	0.429964i	0.823783	+	0.566905i	$0.191860\pi$
$972$		0			0
$973$	−3.76277e6	+	6.51731e6i	−0.00408479	+	0.00707506i
$974$		0			0
$975$	−1.11002e9	+	1.92262e9i	−1.19762	+	2.07433i
$976$		0			0
$977$			1.58606e8i			0.170073i	0.996378	+	0.0850366i	$0.0271007\pi$
$977$			1.58606e8i			0.170073i	−0.996378	+	0.0850366i	$0.972899\pi$
$978$		0			0
$979$	−1.11468e8	−	6.43561e7i	−0.118796	−	0.0685870i
$980$		0			0
$981$		−	1.74735e9i		−	1.85086i
$982$		0			0
$983$	4.69551e8	−	2.71095e8i	0.494336	−	0.285405i	−0.232035	−	0.972707i	$-0.574539\pi$
$983$	4.69551e8	−	2.71095e8i	0.494336	−	0.285405i	0.726371	+	0.687302i	$0.241205\pi$
$984$		0			0
$985$	3.54332e8	+	6.13721e8i	0.370768	+	0.642189i
$986$		0			0
$987$		−	2.26630e8i		−	0.235704i
$988$		0			0
$989$	−1.26992e9			−1.31276
$990$		0			0
$991$	3.24842e8	−	1.87547e8i	0.333773	−	0.192704i	−0.323742	−	0.946145i	$-0.604941\pi$
$991$	3.24842e8	−	1.87547e8i	0.333773	−	0.192704i	0.657515	+	0.753442i	$0.271608\pi$
$992$		0			0
$993$	1.01614e8	+	1.76000e8i	0.103778	+	0.179748i
$994$		0			0
$995$	2.83905e9			2.88206
$996$		0			0
$997$	9.39954e7	−	1.62805e8i	0.0948464	−	0.164279i	−0.814698	−	0.579885i	$-0.803097\pi$
$997$	9.39954e7	−	1.62805e8i	0.0948464	−	0.164279i	0.909544	+	0.415607i	$0.136431\pi$
$998$		0			0
$999$	2.93949e8			0.294833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.h.a.69.1	yes	20
3.2	odd	2		684.7.y.c.145.10		20
19.8	odd	6	inner	76.7.h.a.65.1	✓	20
57.8	even	6		684.7.y.c.217.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.h.a.65.1	✓	20	19.8	odd	6	inner
76.7.h.a.69.1	yes	20	1.1	even	1	trivial
684.7.y.c.145.10		20	3.2	odd	2
684.7.y.c.217.10		20	57.8	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-45.6057 + 26.3304i) q^{3} +(-106.246 - 184.024i) q^{5} -29.3098 q^{7} +(1022.08 - 1770.30i) q^{9} +O(q^{10})\)	\(q+(-45.6057 + 26.3304i) q^{3} +(-106.246 - 184.024i) q^{5} -29.3098 q^{7} +(1022.08 - 1770.30i) q^{9} -1233.12 q^{11} +(-1236.43 - 713.855i) q^{13} +(9690.85 + 5595.02i) q^{15} +(-1071.44 - 1855.78i) q^{17} +(-4826.72 + 4873.26i) q^{19} +(1336.69 - 771.740i) q^{21} +(-4325.77 + 7492.46i) q^{23} +(-14764.0 + 25572.0i) q^{25} +69258.0i q^{27} +(15974.6 + 9222.95i) q^{29} -42379.6i q^{31} +(56237.2 - 32468.6i) q^{33} +(3114.05 + 5393.69i) q^{35} -4244.26i q^{37} +75184.5 q^{39} +(67848.9 - 39172.6i) q^{41} +(73392.5 + 127120. i) q^{43} -434370. q^{45} +(73415.4 - 127159. i) q^{47} -116790. q^{49} +(97727.1 + 56422.8i) q^{51} +(50241.7 + 29007.1i) q^{53} +(131014. + 226923. i) q^{55} +(91810.5 - 349338. i) q^{57} +(-7982.10 + 4608.47i) q^{59} +(-122097. + 211479. i) q^{61} +(-29957.1 + 51887.2i) q^{63} +303378. i q^{65} +(-393125. - 226971. i) q^{67} -455598. i q^{69} +(39272.6 - 22674.0i) q^{71} +(328579. + 569116. i) q^{73} -1.55497e6i q^{75} +36142.4 q^{77} +(312579. - 180467. i) q^{79} +(-1.07849e6 - 1.86801e6i) q^{81} +229403. q^{83} +(-227672. + 394339. i) q^{85} -971378. q^{87} +(90395.3 + 52189.8i) q^{89} +(36239.6 + 20922.9i) q^{91} +(1.11587e6 + 1.93275e6i) q^{93} +(1.40962e6 + 370465. i) q^{95} +(426941. - 246494. i) q^{97} +(-1.26035e6 + 2.18299e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) \) 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9	\( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) \) 20 * q - 30 * q^3 - 56 * q^5 + 464 * q^7 + 2200 * q^9 - 3644 * q^11 - 7140 * q^13 + 9168 * q^15 + 1132 * q^17 + 2110 * q^19 - 8748 * q^21 + 832 * q^23 - 27698 * q^25 - 10920 * q^29 - 30306 * q^33 + 4172 * q^35 + 81144 * q^39 + 109206 * q^41 + 110740 * q^43 - 785440 * q^45 + 107080 * q^47 + 136092 * q^49 + 199872 * q^51 + 254796 * q^53 + 354840 * q^55 + 212268 * q^57 - 610638 * q^59 + 47864 * q^61 - 254476 * q^63 - 839562 * q^67 + 366660 * q^71 + 854482 * q^73 + 763088 * q^77 + 1718592 * q^79 - 1054142 * q^81 + 439612 * q^83 - 400236 * q^85 - 1604736 * q^87 + 478032 * q^89 + 599856 * q^91 + 829380 * q^93 - 1055660 * q^95 - 191286 * q^97 - 2336728 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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