LMFDB - Embedded newform 76.8.e.a.49.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(45,76)]

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, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, 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:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.5
Character	$\chi$	$=$	76.49
Dual form			76.8.e.a.45.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.42876 + 5.93879i) q^{3} +(-200.054 + 346.504i) q^{5} -1032.75 q^{7} +(1069.99 + 1853.27i) q^{9} +O(q^{10})$	\(q+(-3.42876 + 5.93879i) q^{3} +(-200.054 + 346.504i) q^{5} -1032.75 q^{7} +(1069.99 + 1853.27i) q^{9} +309.913 q^{11} +(-4892.02 - 8473.22i) q^{13} +(-1371.88 - 2376.16i) q^{15} +(9807.53 - 16987.1i) q^{17} +(19379.0 + 22766.8i) q^{19} +(3541.04 - 6133.25i) q^{21} +(-2635.65 - 4565.08i) q^{23} +(-40980.8 - 70980.9i) q^{25} -29672.3 q^{27} +(-94034.0 - 162872. i) q^{29} -59256.8 q^{31} +(-1062.62 + 1840.51i) q^{33} +(206605. - 357850. i) q^{35} +61832.4 q^{37} +67094.2 q^{39} +(188128. - 325847. i) q^{41} +(276316. - 478594. i) q^{43} -856221. q^{45} +(-390472. - 676317. i) q^{47} +243019. q^{49} +(67255.3 + 116490. i) q^{51} +(-505468. - 875497. i) q^{53} +(-61999.5 + 107386. i) q^{55} +(-201653. + 37025.9i) q^{57} +(-884350. + 1.53174e6i) q^{59} +(795991. + 1.37870e6i) q^{61} +(-1.10502e6 - 1.91396e6i) q^{63} +3.91467e6 q^{65} +(557626. + 965836. i) q^{67} +36148.1 q^{69} +(-1.33615e6 + 2.31428e6i) q^{71} +(2.07183e6 - 3.58851e6i) q^{73} +562054. q^{75} -320062. q^{77} +(-1.55161e6 + 2.68747e6i) q^{79} +(-2.23832e6 + 3.87689e6i) q^{81} -5.31855e6 q^{83} +(3.92407e6 + 6.79669e6i) q^{85} +1.28968e6 q^{87} +(4.01342e6 + 6.95144e6i) q^{89} +(5.05221e6 + 8.75068e6i) q^{91} +(203177. - 351914. i) q^{93} +(-1.17656e7 + 2.16031e6i) q^{95} +(-3.68244e6 + 6.37817e6i) q^{97} +(331603. + 574354. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−3.42876	+	5.93879i	−0.0733184	+	0.126991i	−0.900354	−	0.435159i	$-0.856692\pi$
$3$	−3.42876	+	5.93879i	−0.0733184	+	0.126991i	0.827035	+	0.562150i	$0.190026\pi$
$4$		0			0
$5$	−200.054	+	346.504i	−0.715735	+	1.23969i	0.246940	+	0.969031i	$0.420575\pi$
$5$	−200.054	+	346.504i	−0.715735	+	1.23969i	−0.962675	+	0.270659i	$0.912758\pi$
$6$		0			0
$7$	−1032.75			−1.13802			−0.569010	−	0.822331i	$-0.692673\pi$
$7$	−1032.75			−1.13802			−0.569010	+	0.822331i	$0.692673\pi$
$8$		0			0
$9$	1069.99	+	1853.27i	0.489249	+	0.847404i
$10$		0			0
$11$	309.913			0.0702047			0.0351023	−	0.999384i	$-0.488824\pi$
$11$	309.913			0.0702047			0.0351023	+	0.999384i	$0.488824\pi$
$12$		0			0
$13$	−4892.02	−	8473.22i	−0.617570	−	1.06966i	−0.989928	−	0.141573i	$-0.954784\pi$
$13$	−4892.02	−	8473.22i	−0.617570	−	1.06966i	0.372358	−	0.928089i	$-0.378549\pi$
$14$		0			0
$15$	−1371.88	−	2376.16i	−0.104953	−	0.181784i
$16$		0			0
$17$	9807.53	−	16987.1i	0.484159	−	0.838589i	−0.515675	−	0.856784i	$-0.672459\pi$
$17$	9807.53	−	16987.1i	0.484159	−	0.838589i	0.999834	+	0.0181957i	$0.00579218\pi$
$18$		0			0
$19$	19379.0	+	22766.8i	0.648177	+	0.761490i
$19$	19379.0	+	22766.8i	0.648177	+	0.761490i
$20$		0			0
$21$	3541.04	−	6133.25i	0.0834378	−	0.144518i
$22$		0			0
$23$	−2635.65	−	4565.08i	−0.0451690	−	0.0782351i	0.842557	−	0.538607i	$-0.181049\pi$
$23$	−2635.65	−	4565.08i	−0.0451690	−	0.0782351i	−0.887726	+	0.460372i	$0.847716\pi$
$24$		0			0
$25$	−40980.8	−	70980.9i	−0.524554	−	0.908555i
$26$		0			0
$27$	−29672.3			−0.290120
$28$		0			0
$29$	−94034.0	−	162872.i	−0.715965	−	1.24009i	−0.962586	−	0.270976i	$-0.912654\pi$
$29$	−94034.0	−	162872.i	−0.715965	−	1.24009i	0.246621	−	0.969112i	$-0.420680\pi$
$30$		0			0
$31$	−59256.8			−0.357250			−0.178625	−	0.983917i	$-0.557165\pi$
$31$	−59256.8			−0.357250			−0.178625	+	0.983917i	$0.557165\pi$
$32$		0			0
$33$	−1062.62	+	1840.51i	−0.00514729	+	0.00891537i
$34$		0			0
$35$	206605.	−	357850.i	0.814521	−	1.41079i
$36$		0			0
$37$	61832.4			0.200683			0.100341	−	0.994953i	$-0.468006\pi$
$37$	61832.4			0.200683			0.100341	+	0.994953i	$0.468006\pi$
$38$		0			0
$39$	67094.2			0.181117
$40$		0			0
$41$	188128.	−	325847.i	0.426295	−	0.738364i	−0.570246	−	0.821474i	$-0.693152\pi$
$41$	188128.	−	325847.i	0.426295	−	0.738364i	0.996540	+	0.0831103i	$0.0264854\pi$
$42$		0			0
$43$	276316.	−	478594.i	0.529989	−	0.917968i	−0.469399	−	0.882986i	$-0.655529\pi$
$43$	276316.	−	478594.i	0.529989	−	0.917968i	0.999388	−	0.0349820i	$-0.0111374\pi$
$44$		0			0
$45$	−856221.			−1.40069
$46$		0			0
$47$	−390472.	−	676317.i	−0.548589	−	0.950184i	−0.998372	−	0.0570461i	$-0.981832\pi$
$47$	−390472.	−	676317.i	−0.548589	−	0.950184i	0.449782	−	0.893138i	$-0.351502\pi$
$48$		0			0
$49$	243019.			0.295090
$50$		0			0
$51$	67255.3	+	116490.i	0.0709955	+	0.122968i
$52$		0			0
$53$	−505468.	−	875497.i	−0.466368	−	0.807773i	0.532894	−	0.846182i	$-0.321104\pi$
$53$	−505468.	−	875497.i	−0.466368	−	0.807773i	−0.999262	+	0.0384091i	$0.987771\pi$
$54$		0			0
$55$	−61999.5	+	107386.i	−0.0502480	+	0.0870321i
$56$		0			0
$57$	−201653.	+	37025.9i	−0.144226	+	0.0264816i
$58$		0			0
$59$	−884350.	+	1.53174e6i	−0.560586	+	0.970963i	0.436859	+	0.899530i	$0.356091\pi$
$59$	−884350.	+	1.53174e6i	−0.560586	+	0.970963i	−0.997445	+	0.0714335i	$0.977243\pi$
$60$		0			0
$61$	795991.	+	1.37870e6i	0.449008	+	0.777704i	0.998322	−	0.0579120i	$-0.0184443\pi$
$61$	795991.	+	1.37870e6i	0.449008	+	0.777704i	−0.549314	+	0.835616i	$0.685111\pi$
$62$		0			0
$63$	−1.10502e6	−	1.91396e6i	−0.556775	−	0.964363i
$64$		0			0
$65$	3.91467e6			1.76807
$66$		0			0
$67$	557626.	+	965836.i	0.226507	+	0.392321i	0.956770	−	0.290844i	$-0.0939362\pi$
$67$	557626.	+	965836.i	0.226507	+	0.392321i	−0.730264	+	0.683165i	$0.760603\pi$
$68$		0			0
$69$	36148.1			0.0132469
$70$		0			0
$71$	−1.33615e6	+	2.31428e6i	−0.443048	+	0.767381i	−0.997914	−	0.0645583i	$-0.979436\pi$
$71$	−1.33615e6	+	2.31428e6i	−0.443048	+	0.767381i	0.554866	+	0.831940i	$0.312769\pi$
$72$		0			0
$73$	2.07183e6	−	3.58851e6i	0.623338	−	1.07965i	−0.365522	−	0.930803i	$-0.619109\pi$
$73$	2.07183e6	−	3.58851e6i	0.623338	−	1.07965i	0.988860	−	0.148850i	$-0.0475573\pi$
$74$		0			0
$75$	562054.			0.153838
$76$		0			0
$77$	−320062.			−0.0798943
$78$		0			0
$79$	−1.55161e6	+	2.68747e6i	−0.354070	+	0.613267i	−0.986958	−	0.160976i	$-0.948536\pi$
$79$	−1.55161e6	+	2.68747e6i	−0.354070	+	0.613267i	0.632889	+	0.774243i	$0.281869\pi$
$80$		0			0
$81$	−2.23832e6	+	3.87689e6i	−0.467978	+	0.810561i
$82$		0			0
$83$	−5.31855e6			−1.02099			−0.510493	−	0.859882i	$-0.670537\pi$
$83$	−5.31855e6			−1.02099			−0.510493	+	0.859882i	$0.670537\pi$
$84$		0			0
$85$	3.92407e6	+	6.79669e6i	0.693060	+	1.20042i
$86$		0			0
$87$	1.28968e6			0.209974
$88$		0			0
$89$	4.01342e6	+	6.95144e6i	0.603461	+	1.04522i	0.992293	+	0.123916i	$0.0395454\pi$
$89$	4.01342e6	+	6.95144e6i	0.603461	+	1.04522i	−0.388832	+	0.921309i	$0.627121\pi$
$90$		0			0
$91$	5.05221e6	+	8.75068e6i	0.702807	+	1.21730i
$92$		0			0
$93$	203177.	−	351914.i	0.0261930	−	0.0453676i
$94$		0			0
$95$	−1.17656e7	+	2.16031e6i	−1.40793	+	0.258513i
$96$		0			0
$97$	−3.68244e6	+	6.37817e6i	−0.409670	+	0.709570i	−0.994853	−	0.101332i	$-0.967690\pi$
$97$	−3.68244e6	+	6.37817e6i	−0.409670	+	0.709570i	0.585182	+	0.810902i	$0.301023\pi$
$98$		0			0
$99$	331603.	+	574354.i	0.0343476	+	0.0594917i
$100$		0			0
$101$	−9.41116e6	−	1.63006e7i	−0.908905	−	1.57427i	−0.815588	−	0.578633i	$-0.803586\pi$
$101$	−9.41116e6	−	1.63006e7i	−0.908905	−	1.57427i	−0.0933167	−	0.995636i	$-0.529747\pi$
$102$		0			0
$103$	8.64481e6			0.779517			0.389758	−	0.920917i	$-0.372559\pi$
$103$	8.64481e6			0.779517			0.389758	+	0.920917i	$0.372559\pi$
$104$		0			0
$105$	1.41680e6	+	2.45397e6i	0.119439	+	0.206874i
$106$		0			0
$107$	1.53786e7			1.21359			0.606797	−	0.794857i	$-0.292454\pi$
$107$	1.53786e7			1.21359			0.606797	+	0.794857i	$0.292454\pi$
$108$		0			0
$109$	1.06313e7	−	1.84140e7i	0.786310	−	1.36193i	−0.141903	−	0.989881i	$-0.545322\pi$
$109$	1.06313e7	−	1.84140e7i	0.786310	−	1.36193i	0.928213	−	0.372049i	$-0.121345\pi$
$110$		0			0
$111$	−212009.	+	367210.i	−0.0147137	+	0.0254849i
$112$		0			0
$113$	−2.30223e6			−0.150098			−0.0750490	−	0.997180i	$-0.523911\pi$
$113$	−2.30223e6			−0.150098			−0.0750490	+	0.997180i	$0.523911\pi$
$114$		0			0
$115$	2.10909e6			0.129316
$116$		0			0
$117$	1.04688e7	−	1.81325e7i	0.604291	−	1.04666i
$118$		0			0
$119$	−1.01287e7	+	1.75434e7i	−0.550983	+	0.954331i
$120$		0			0
$121$	−1.93911e7			−0.995071
$122$		0			0
$123$	1.29009e6	+	2.23450e6i	0.0625104	+	0.108271i
$124$		0			0
$125$	1.53507e6			0.0702980
$126$		0			0
$127$	−1.09606e7	−	1.89843e7i	−0.474810	−	0.822395i	0.524774	−	0.851242i	$-0.324150\pi$
$127$	−1.09606e7	−	1.89843e7i	−0.474810	−	0.822395i	−0.999584	+	0.0288466i	$0.990817\pi$
$128$		0			0
$129$	1.89485e6	+	3.28197e6i	0.0777159	+	0.134608i
$130$		0			0
$131$	−1.33047e7	+	2.30444e7i	−0.517077	+	0.895604i	0.482726	+	0.875771i	$0.339647\pi$
$131$	−1.33047e7	+	2.30444e7i	−0.517077	+	0.895604i	−0.999803	+	0.0198325i	$0.993687\pi$
$132$		0			0
$133$	−2.00136e7	−	2.35123e7i	−0.737639	−	0.866591i
$134$		0			0
$135$	5.93607e6	−	1.02816e7i	0.207649	−	0.359659i
$136$		0			0
$137$	−698694.	−	1.21017e6i	−0.0232148	−	0.0402092i	0.854185	−	0.519970i	$-0.174057\pi$
$137$	−698694.	−	1.21017e6i	−0.0232148	−	0.0402092i	−0.877399	+	0.479761i	$0.840723\pi$
$138$		0			0
$139$	−2.62493e7	−	4.54652e7i	−0.829022	−	1.43591i	−0.898806	−	0.438347i	$-0.855564\pi$
$139$	−2.62493e7	−	4.54652e7i	−0.829022	−	1.43591i	0.0697835	−	0.997562i	$-0.477769\pi$
$140$		0			0
$141$	5.35534e6			0.160887
$142$		0			0
$143$	−1.51610e6	−	2.62597e6i	−0.0433563	−	0.0750953i
$144$		0			0
$145$	7.52476e7			2.04977
$146$		0			0
$147$	−833255.	+	1.44324e6i	−0.0216355	+	0.0374738i
$148$		0			0
$149$	3.17294e7	−	5.49570e7i	0.785797	−	1.36104i	−0.142725	−	0.989762i	$-0.545586\pi$
$149$	3.17294e7	−	5.49570e7i	0.785797	−	1.36104i	0.928522	−	0.371278i	$-0.121080\pi$
$150$		0			0
$151$	−8.06708e7			−1.90676			−0.953382	−	0.301765i	$-0.902424\pi$
$151$	−8.06708e7			−1.90676			−0.953382	+	0.301765i	$0.902424\pi$
$152$		0			0
$153$	4.19757e7			0.947498
$154$		0			0
$155$	1.18546e7	−	2.05327e7i	0.255697	−	0.442879i
$156$		0			0
$157$	1.81104e7	−	3.13681e7i	0.373490	−	0.646904i	−0.616609	−	0.787269i	$-0.711494\pi$
$157$	1.81104e7	−	3.13681e7i	0.373490	−	0.646904i	0.990100	+	0.140365i	$0.0448276\pi$
$158$		0			0
$159$	6.93252e6			0.136773
$160$		0			0
$161$	2.72196e6	+	4.71457e6i	0.0514033	+	0.0890331i
$162$		0			0
$163$	−9.29480e7			−1.68106			−0.840530	−	0.541764i	$-0.817756\pi$
$163$	−9.29480e7			−1.68106			−0.840530	+	0.541764i	$0.817756\pi$
$164$		0			0
$165$	−425163.	−	736404.i	−0.00736820	−	0.0127621i
$166$		0			0
$167$	3.19347e7	+	5.53124e7i	0.530584	+	0.918999i	0.999363	+	0.0356836i	$0.0113608\pi$
$167$	3.19347e7	+	5.53124e7i	0.530584	+	0.918999i	−0.468779	+	0.883316i	$0.655306\pi$
$168$		0			0
$169$	−1.64894e7	+	2.85605e7i	−0.262786	+	0.455158i
$170$		0			0
$171$	−2.14578e7	+	6.02747e7i	−0.328169	+	0.921826i
$172$		0			0
$173$	2.20485e7	−	3.81890e7i	0.323755	−	0.560761i	−0.657504	−	0.753451i	$-0.728388\pi$
$173$	2.20485e7	−	3.81890e7i	0.323755	−	0.560761i	0.981260	+	0.192690i	$0.0617212\pi$
$174$		0			0
$175$	4.23227e7	+	7.33051e7i	0.596954	+	1.03395i
$176$		0			0
$177$	−6.06445e6	−	1.05039e7i	−0.0822025	−	0.142379i
$178$		0			0
$179$	−8.53869e7			−1.11277			−0.556386	−	0.830924i	$-0.687812\pi$
$179$	−8.53869e7			−1.11277			−0.556386	+	0.830924i	$0.687812\pi$
$180$		0			0
$181$	5.01761e7	+	8.69076e7i	0.628959	+	1.08939i	0.987761	+	0.155975i	$0.0498519\pi$
$181$	5.01761e7	+	8.69076e7i	0.628959	+	1.08939i	−0.358802	+	0.933414i	$0.616815\pi$
$182$		0			0
$183$	−1.09170e7			−0.131682
$184$		0			0
$185$	−1.23698e7	+	2.14252e7i	−0.143636	+	0.248785i
$186$		0			0
$187$	3.03948e6	−	5.26454e6i	0.0339903	−	0.0588728i
$188$		0			0
$189$	3.06439e7			0.330163
$190$		0			0
$191$	4.76457e7			0.494774			0.247387	−	0.968917i	$-0.420428\pi$
$191$	4.76457e7			0.494774			0.247387	+	0.968917i	$0.420428\pi$
$192$		0			0
$193$	1.96096e7	−	3.39648e7i	0.196344	−	0.340078i	−0.750996	−	0.660306i	$-0.770426\pi$
$193$	1.96096e7	−	3.39648e7i	0.196344	−	0.340078i	0.947340	+	0.320229i	$0.103760\pi$
$194$		0			0
$195$	−1.34225e7	+	2.32484e7i	−0.129632	+	0.224529i
$196$		0			0
$197$	−8.97786e7			−0.836644			−0.418322	−	0.908299i	$-0.637382\pi$
$197$	−8.97786e7			−0.836644			−0.418322	+	0.908299i	$0.637382\pi$
$198$		0			0
$199$	−2.75156e7	−	4.76585e7i	−0.247511	−	0.428701i	0.715324	−	0.698793i	$-0.246279\pi$
$199$	−2.75156e7	−	4.76585e7i	−0.247511	−	0.428701i	−0.962835	+	0.270092i	$0.912946\pi$
$200$		0			0
$201$	−7.64786e6			−0.0664284
$202$		0			0
$203$	9.71132e7	+	1.68205e8i	0.814783	+	1.41125i
$204$		0			0
$205$	7.52715e7	+	1.30374e8i	0.610228	+	1.05695i
$206$		0			0
$207$	5.64023e6	−	9.76916e6i	0.0441978	−	0.0765528i
$208$		0			0
$209$	6.00581e6	+	7.05573e6i	0.0455051	+	0.0534601i
$210$		0			0
$211$	6.72518e7	−	1.16484e8i	0.492850	−	0.853642i	−0.507116	−	0.861878i	$-0.669288\pi$
$211$	6.72518e7	−	1.16484e8i	0.492850	−	0.853642i	0.999966	+	0.00823595i	$0.00262161\pi$
$212$		0			0
$213$	−9.16267e6	−	1.58702e7i	−0.0649671	−	0.112526i
$214$		0			0
$215$	1.10556e8	+	1.91489e8i	0.758664	+	1.31405i
$216$		0			0
$217$	6.11972e7			0.406558
$218$		0			0
$219$	1.42076e7	+	2.46083e7i	0.0914043	+	0.158317i
$220$		0			0
$221$	−1.91914e8			−1.19601
$222$		0			0
$223$	−1.00298e8	+	1.73722e8i	−0.605657	+	1.04903i	0.386291	+	0.922377i	$0.373756\pi$
$223$	−1.00298e8	+	1.73722e8i	−0.605657	+	1.04903i	−0.991947	+	0.126651i	$0.959577\pi$
$224$		0			0
$225$	8.76979e7	−	1.51897e8i	0.513275	−	0.889019i
$226$		0			0
$227$	−3.02654e8			−1.71734			−0.858670	−	0.512528i	$-0.828709\pi$
$227$	−3.02654e8			−1.71734			−0.858670	+	0.512528i	$0.828709\pi$
$228$		0			0
$229$	−2.02274e7			−0.111306			−0.0556528	−	0.998450i	$-0.517724\pi$
$229$	−2.02274e7			−0.111306			−0.0556528	+	0.998450i	$0.517724\pi$
$230$		0			0
$231$	1.09741e6	−	1.90078e6i	0.00585772	−	0.0101459i
$232$		0			0
$233$	−2.72219e7	+	4.71496e7i	−0.140985	+	0.244193i	−0.927868	−	0.372910i	$-0.878360\pi$
$233$	−2.72219e7	+	4.71496e7i	−0.140985	+	0.244193i	0.786883	+	0.617102i	$0.211694\pi$
$234$		0			0
$235$	3.12462e8			1.57058
$236$		0			0
$237$	−1.06402e7	−	1.84294e7i	−0.0519196	−	0.0899274i
$238$		0			0
$239$	−9.18021e7			−0.434970			−0.217485	−	0.976064i	$-0.569785\pi$
$239$	−9.18021e7			−0.434970			−0.217485	+	0.976064i	$0.569785\pi$
$240$		0			0
$241$	−4.72321e7	−	8.18084e7i	−0.217359	−	0.376477i	0.736641	−	0.676284i	$-0.236411\pi$
$241$	−4.72321e7	−	8.18084e7i	−0.217359	−	0.376477i	−0.954000	+	0.299808i	$0.903078\pi$
$242$		0			0
$243$	−4.77960e7	−	8.27852e7i	−0.213683	−	0.370110i
$244$		0			0
$245$	−4.86170e7	+	8.42071e7i	−0.211206	+	0.365820i
$246$		0			0
$247$	9.81056e7	−	2.75578e8i	0.414242	−	1.16360i
$248$		0			0
$249$	1.82360e7	−	3.15857e7i	0.0748570	−	0.129656i
$250$		0			0
$251$	−1.06661e8	−	1.84742e8i	−0.425743	−	0.737409i	0.570746	−	0.821126i	$-0.306654\pi$
$251$	−1.06661e8	−	1.84742e8i	−0.425743	−	0.737409i	−0.996490	+	0.0837176i	$0.973321\pi$
$252$		0			0
$253$	−816824.	−	1.41478e6i	−0.00317108	−	0.00549247i
$254$		0			0
$255$	−5.38188e7			−0.203256
$256$		0			0
$257$	−9.94297e7	−	1.72217e8i	−0.365385	−	0.632865i	0.623453	−	0.781861i	$-0.285729\pi$
$257$	−9.94297e7	−	1.72217e8i	−0.365385	−	0.632865i	−0.988838	+	0.148996i	$0.952396\pi$
$258$		0			0
$259$	−6.38571e7			−0.228381
$260$		0			0
$261$	2.01230e8	−	3.48541e8i	0.700570	−	1.21342i
$262$		0			0
$263$	−2.15071e7	+	3.72514e7i	−0.0729016	+	0.126269i	−0.900172	−	0.435535i	$-0.856559\pi$
$263$	−2.15071e7	+	3.72514e7i	−0.0729016	+	0.126269i	0.827270	+	0.561804i	$0.189893\pi$
$264$		0			0
$265$	4.04484e8			1.33518
$266$		0			0
$267$	−5.50442e7			−0.176979
$268$		0			0
$269$	−1.78542e8	+	3.09244e8i	−0.559253	+	0.968654i	0.438306	+	0.898826i	$0.355579\pi$
$269$	−1.78542e8	+	3.09244e8i	−0.559253	+	0.968654i	−0.997559	+	0.0698285i	$0.977755\pi$
$270$		0			0
$271$	1.93309e8	−	3.34821e8i	0.590010	−	1.02193i	−0.404221	−	0.914662i	$-0.632457\pi$
$271$	1.93309e8	−	3.34821e8i	0.590010	−	1.02193i	0.994230	−	0.107265i	$-0.0342095\pi$
$272$		0			0
$273$	−6.92912e7			−0.206115
$274$		0			0
$275$	−1.27005e7	−	2.19979e7i	−0.0368262	−	0.0637848i
$276$		0			0
$277$	4.54226e8			1.28408			0.642042	−	0.766670i	$-0.278088\pi$
$277$	4.54226e8			1.28408			0.642042	+	0.766670i	$0.278088\pi$
$278$		0			0
$279$	−6.34040e7	−	1.09819e8i	−0.174784	−	0.302735i
$280$		0			0
$281$	−1.53164e8	−	2.65289e8i	−0.411800	−	0.713258i	0.583287	−	0.812266i	$-0.301766\pi$
$281$	−1.53164e8	−	2.65289e8i	−0.411800	−	0.713258i	−0.995087	+	0.0990084i	$0.968433\pi$
$282$		0			0
$283$	2.15788e8	−	3.73756e8i	0.565946	−	0.980247i	−0.431015	−	0.902345i	$-0.641845\pi$
$283$	2.15788e8	−	3.73756e8i	0.565946	−	0.980247i	0.996961	−	0.0779020i	$-0.0248221\pi$
$284$		0			0
$285$	2.75119e7	−	7.72808e7i	0.0703985	−	0.197749i
$286$		0			0
$287$	−1.94288e8	+	3.36517e8i	−0.485132	+	0.840273i
$288$		0			0
$289$	1.27942e7	+	2.21601e7i	0.0311795	+	0.0540045i
$290$		0			0
$291$	−2.52524e7	−	4.37385e7i	−0.0600727	−	0.104049i
$292$		0			0
$293$	−5.86914e8			−1.36313			−0.681565	−	0.731757i	$-0.738700\pi$
$293$	−5.86914e8			−1.36313			−0.681565	+	0.731757i	$0.738700\pi$
$294$		0			0
$295$	−3.53836e8	−	6.12862e8i	−0.802462	−	1.38991i
$296$		0			0
$297$	−9.19585e6			−0.0203678
$298$		0			0
$299$	−2.57873e7	+	4.46649e7i	−0.0557901	+	0.0966313i
$300$		0			0
$301$	−2.85364e8	+	4.94266e8i	−0.603139	+	1.04467i
$302$		0			0
$303$	1.29074e8			0.266558
$304$		0			0
$305$	−6.36965e8			−1.28548
$306$		0			0
$307$	−2.69068e8	+	4.66039e8i	−0.530735	+	0.919260i	0.468622	+	0.883399i	$0.344751\pi$
$307$	−2.69068e8	+	4.66039e8i	−0.530735	+	0.919260i	−0.999357	+	0.0358608i	$0.988583\pi$
$308$		0			0
$309$	−2.96410e7	+	5.13397e7i	−0.0571529	+	0.0989917i
$310$		0			0
$311$	4.78266e8			0.901589			0.450794	−	0.892628i	$-0.351141\pi$
$311$	4.78266e8			0.901589			0.450794	+	0.892628i	$0.351141\pi$
$312$		0			0
$313$	4.75443e7	+	8.23492e7i	0.0876382	+	0.151794i	0.906512	−	0.422179i	$-0.138735\pi$
$313$	4.75443e7	+	8.23492e7i	0.0876382	+	0.151794i	−0.818874	+	0.573973i	$0.805401\pi$
$314$		0			0
$315$	8.84258e8			1.59401
$316$		0			0
$317$	3.70472e7	+	6.41676e7i	0.0653202	+	0.113138i	0.896836	−	0.442363i	$-0.145860\pi$
$317$	3.70472e7	+	6.41676e7i	0.0653202	+	0.113138i	−0.831516	+	0.555501i	$0.812526\pi$
$318$		0			0
$319$	−2.91424e7	−	5.04761e7i	−0.0502641	−	0.0870600i
$320$		0			0
$321$	−5.27295e7	+	9.13302e7i	−0.0889787	+	0.154116i
$322$		0			0
$323$	5.76802e8	−	1.05908e8i	0.952397	−	0.174871i
$324$		0			0
$325$	−4.00958e8	+	6.94479e8i	−0.647898	+	1.12219i
$326$		0			0
$327$	7.29044e7	+	1.26274e8i	0.115302	+	0.199709i
$328$		0			0
$329$	4.03258e8	+	6.98463e8i	0.624305	+	1.08133i
$330$		0			0
$331$	8.16278e8			1.23720			0.618601	−	0.785706i	$-0.287700\pi$
$331$	8.16278e8			1.23720			0.618601	+	0.785706i	$0.287700\pi$
$332$		0			0
$333$	6.61599e7	+	1.14592e8i	0.0981839	+	0.170059i
$334$		0			0
$335$	−4.46221e8			−0.648475
$336$		0			0
$337$	4.17261e8	−	7.22717e8i	0.593886	−	1.02864i	−0.399818	−	0.916595i	$-0.630927\pi$
$337$	4.17261e8	−	7.22717e8i	0.593886	−	1.02864i	0.993703	−	0.112045i	$-0.0357402\pi$
$338$		0			0
$339$	7.89381e6	−	1.36725e7i	0.0110049	−	0.0190611i
$340$		0			0
$341$	−1.83645e7			−0.0250806
$342$		0			0
$343$	5.99533e8			0.802202
$344$		0			0
$345$	−7.23158e6	+	1.25255e7i	−0.00948126	+	0.0164220i
$346$		0			0
$347$	−4.55250e8	+	7.88517e8i	−0.584921	+	1.01311i	0.409965	+	0.912101i	$0.365541\pi$
$347$	−4.55250e8	+	7.88517e8i	−0.584921	+	1.01311i	−0.994885	+	0.101011i	$0.967792\pi$
$348$		0			0
$349$	−1.50247e9			−1.89199			−0.945993	−	0.324187i	$-0.894909\pi$
$349$	−1.50247e9			−1.89199			−0.945993	+	0.324187i	$0.894909\pi$
$350$		0			0
$351$	1.45158e8	+	2.51420e8i	0.179170	+	0.310331i
$352$		0			0
$353$	−5.97618e7			−0.0723123			−0.0361561	−	0.999346i	$-0.511511\pi$
$353$	−5.97618e7			−0.0723123			−0.0361561	+	0.999346i	$0.511511\pi$
$354$		0			0
$355$	−5.34604e8	−	9.25962e8i	−0.634210	−	1.09848i
$356$		0			0
$357$	−6.94576e7	−	1.20304e8i	−0.0807944	−	0.139940i
$358$		0			0
$359$	2.57575e8	−	4.46133e8i	0.293814	−	0.508902i	−0.680894	−	0.732382i	$-0.738409\pi$
$359$	2.57575e8	−	4.46133e8i	0.293814	−	0.508902i	0.974708	+	0.223480i	$0.0717419\pi$
$360$		0			0
$361$	−1.42781e8	+	8.82395e8i	−0.159733	+	0.987160i
$362$		0			0
$363$	6.64875e7	−	1.15160e8i	0.0729570	−	0.126365i
$364$		0			0
$365$	8.28955e8	+	1.43579e9i	0.892290	+	1.54549i
$366$		0			0
$367$	7.77565e7	+	1.34678e8i	0.0821118	+	0.142222i	0.904157	−	0.427201i	$-0.140500\pi$
$367$	7.77565e7	+	1.34678e8i	0.0821118	+	0.142222i	−0.822045	+	0.569422i	$0.807167\pi$
$368$		0			0
$369$	8.05178e8			0.834256
$370$		0			0
$371$	5.22020e8	+	9.04165e8i	0.530736	+	0.919262i
$372$		0			0
$373$	−1.06099e9			−1.05860			−0.529298	−	0.848436i	$-0.677545\pi$
$373$	−1.06099e9			−1.05860			−0.529298	+	0.848436i	$0.677545\pi$
$374$		0			0
$375$	−5.26339e6	+	9.11645e6i	−0.00515414	+	0.00892722i
$376$		0			0
$377$	−9.20032e8	+	1.59354e9i	−0.884318	+	1.53168i
$378$		0			0
$379$	1.33602e8			0.126060			0.0630298	−	0.998012i	$-0.479924\pi$
$379$	1.33602e8			0.126060			0.0630298	+	0.998012i	$0.479924\pi$
$380$		0			0
$381$	1.50325e8			0.139249
$382$		0			0
$383$	−8.70073e8	+	1.50701e9i	−0.791334	+	1.37063i	0.133806	+	0.991007i	$0.457280\pi$
$383$	−8.70073e8	+	1.50701e9i	−0.791334	+	1.37063i	−0.925141	+	0.379624i	$0.876053\pi$
$384$		0			0
$385$	6.40296e7	−	1.10903e8i	0.0571832	−	0.0990442i
$386$		0			0
$387$	1.18262e9			1.03719
$388$		0			0
$389$	−7.04123e8	−	1.21958e9i	−0.606492	−	1.05047i	−0.991814	−	0.127693i	$-0.959243\pi$
$389$	−7.04123e8	−	1.21958e9i	−0.606492	−	1.05047i	0.385322	−	0.922782i	$-0.374090\pi$
$390$		0			0
$391$	−1.03397e8			−0.0874760
$392$		0			0
$393$	−9.12372e7	−	1.58028e8i	−0.0758225	−	0.131328i
$394$		0			0
$395$	−6.20813e8	−	1.07528e9i	−0.506840	−	0.877873i
$396$		0			0
$397$	1.05321e9	−	1.82422e9i	0.844791	−	1.46322i	−0.0410111	−	0.999159i	$-0.513058\pi$
$397$	1.05321e9	−	1.82422e9i	0.844791	−	1.46322i	0.885802	−	0.464063i	$-0.153609\pi$
$398$		0			0
$399$	2.08256e8	−	3.82383e7i	0.164132	−	0.0301365i
$400$		0			0
$401$	−1.95631e8	+	3.38844e8i	−0.151507	+	0.262418i	−0.931782	−	0.363019i	$-0.881746\pi$
$401$	−1.95631e8	+	3.38844e8i	−0.151507	+	0.262418i	0.780275	+	0.625437i	$0.215079\pi$
$402$		0			0
$403$	2.89885e8	+	5.02096e8i	0.220627	+	0.382137i
$404$		0			0
$405$	−8.95571e8	−	1.55118e9i	−0.669896	−	1.16029i
$406$		0			0
$407$	1.91627e7			0.0140889
$408$		0			0
$409$	6.98307e8	+	1.20950e9i	0.504678	+	0.874128i	0.999985	+	0.00541045i	$0.00172221\pi$
$409$	6.98307e8	+	1.20950e9i	0.504678	+	0.874128i	−0.495307	+	0.868718i	$0.664944\pi$
$410$		0			0
$411$	9.58262e6			0.00680829
$412$		0			0
$413$	9.13308e8	−	1.58190e9i	0.637958	−	1.10498i
$414$		0			0
$415$	1.06400e9	−	1.84290e9i	0.730755	−	1.26571i
$416$		0			0
$417$	3.60011e8			0.243130
$418$		0			0
$419$	−1.68975e9			−1.12221			−0.561103	−	0.827746i	$-0.689623\pi$
$419$	−1.68975e9			−1.12221			−0.561103	+	0.827746i	$0.689623\pi$
$420$		0			0
$421$	5.77764e8	−	1.00072e9i	0.377366	−	0.653618i	−0.613312	−	0.789841i	$-0.710163\pi$
$421$	5.77764e8	−	1.00072e9i	0.377366	−	0.653618i	0.990678	+	0.136223i	$0.0434964\pi$
$422$		0			0
$423$	8.35600e8	−	1.44730e9i	0.536793	−	0.929753i
$424$		0			0
$425$	−1.60768e9			−1.01587
$426$		0			0
$427$	−8.22056e8	−	1.42384e9i	−0.510980	−	0.885043i
$428$		0			0
$429$	2.07934e7			0.0127153
$430$		0			0
$431$	−4.49150e7	−	7.77950e7i	−0.0270222	−	0.0468038i	0.852198	−	0.523219i	$-0.175269\pi$
$431$	−4.49150e7	−	7.77950e7i	−0.0270222	−	0.0468038i	−0.879220	+	0.476416i	$0.841936\pi$
$432$		0			0
$433$	1.37930e9	+	2.38903e9i	0.816493	+	1.41421i	0.908251	+	0.418426i	$0.137418\pi$
$433$	1.37930e9	+	2.38903e9i	0.816493	+	1.41421i	−0.0917580	+	0.995781i	$0.529249\pi$
$434$		0			0
$435$	−2.58006e8	+	4.46879e8i	−0.150286	+	0.260302i
$436$		0			0
$437$	5.28560e7	−	1.48472e8i	0.0302977	−	0.0851059i
$438$		0			0
$439$	1.01886e9	−	1.76472e9i	0.574762	−	0.995517i	−0.421305	−	0.906919i	$-0.638428\pi$
$439$	1.01886e9	−	1.76472e9i	0.574762	−	0.995517i	0.996067	−	0.0885983i	$-0.0282388\pi$
$440$		0			0
$441$	2.60027e8	+	4.50381e8i	0.144372	+	0.250060i
$442$		0			0
$443$	−5.65429e8	−	9.79352e8i	−0.309005	−	0.535212i	0.669140	−	0.743136i	$-0.266663\pi$
$443$	−5.65429e8	−	9.79352e8i	−0.309005	−	0.535212i	−0.978145	+	0.207924i	$0.933329\pi$
$444$		0			0
$445$	−3.21160e9			−1.72767
$446$		0			0
$447$	2.17585e8	+	3.76869e8i	0.115227	+	0.199579i
$448$		0			0
$449$	3.23894e8			0.168865			0.0844326	−	0.996429i	$-0.473092\pi$
$449$	3.23894e8			0.168865			0.0844326	+	0.996429i	$0.473092\pi$
$450$		0			0
$451$	5.83034e7	−	1.00984e8i	0.0299279	−	0.0518366i
$452$		0			0
$453$	2.76601e8	−	4.79087e8i	0.139801	−	0.242142i
$454$		0			0
$455$	−4.04286e9			−2.01210
$456$		0			0
$457$	3.60252e9			1.76563			0.882815	−	0.469721i	$-0.155646\pi$
$457$	3.60252e9			1.76563			0.882815	+	0.469721i	$0.155646\pi$
$458$		0			0
$459$	−2.91012e8	+	5.04048e8i	−0.140465	+	0.243292i
$460$		0			0
$461$	−1.12127e9	+	1.94210e9i	−0.533038	+	0.923248i	0.466218	+	0.884670i	$0.345616\pi$
$461$	−1.12127e9	+	1.94210e9i	−0.533038	+	0.923248i	−0.999256	+	0.0385785i	$0.987717\pi$
$462$		0			0
$463$	6.97166e8			0.326440			0.163220	−	0.986590i	$-0.447812\pi$
$463$	6.97166e8			0.326440			0.163220	+	0.986590i	$0.447812\pi$
$464$		0			0
$465$	8.12930e7	+	1.40804e8i	0.0374945	+	0.0649424i
$466$		0			0
$467$	−1.42577e9			−0.647798			−0.323899	−	0.946092i	$-0.604994\pi$
$467$	−1.42577e9			−0.647798			−0.323899	+	0.946092i	$0.604994\pi$
$468$		0			0
$469$	−5.75885e8	−	9.97462e8i	−0.257769	−	0.446469i
$470$		0			0
$471$	1.24193e8	+	2.15108e8i	0.0547674	+	0.0948599i
$472$		0			0
$473$	8.56342e7	−	1.48323e8i	0.0372077	−	0.0644457i
$474$		0			0
$475$	8.21839e8	−	2.30854e9i	0.351851	−	0.988347i
$476$		0			0
$477$	1.08169e9	−	1.87354e9i	0.456340	−	0.790404i
$478$		0			0
$479$	1.32428e9	+	2.29372e9i	0.550562	+	0.953601i	0.998234	+	0.0594031i	$0.0189197\pi$
$479$	1.32428e9	+	2.29372e9i	0.550562	+	0.953601i	−0.447672	+	0.894198i	$0.647747\pi$
$480$		0			0
$481$	−3.02485e8	−	5.23920e8i	−0.123936	−	0.214663i
$482$		0			0
$483$	−3.73318e7			−0.0150752
$484$		0			0
$485$	−1.47337e9	−	2.55196e9i	−0.586431	−	1.01573i
$486$		0			0
$487$	2.84312e9			1.11543			0.557717	−	0.830031i	$-0.311678\pi$
$487$	2.84312e9			1.11543			0.557717	+	0.830031i	$0.311678\pi$
$488$		0			0
$489$	3.18696e8	−	5.51998e8i	0.123253	−	0.213480i
$490$		0			0
$491$	8.76747e8	−	1.51857e9i	0.334264	−	0.578962i	−0.649079	−	0.760721i	$-0.724846\pi$
$491$	8.76747e8	−	1.51857e9i	0.334264	−	0.578962i	0.983343	+	0.181759i	$0.0581790\pi$
$492$		0			0
$493$	−3.68896e9			−1.38657
$494$		0			0
$495$	−2.65355e8			−0.0983351
$496$		0			0
$497$	1.37990e9	−	2.39006e9i	0.504197	−	0.873295i
$498$		0			0
$499$	−1.62927e9	+	2.82198e9i	−0.587005	+	1.01672i	0.407617	+	0.913153i	$0.366360\pi$
$499$	−1.62927e9	+	2.82198e9i	−0.587005	+	1.01672i	−0.994622	+	0.103569i	$0.966974\pi$
$500$		0			0
$501$	−4.37985e8			−0.155606
$502$		0			0
$503$	7.24667e8	+	1.25516e9i	0.253893	+	0.439756i	0.964594	−	0.263738i	$-0.0849556\pi$
$503$	7.24667e8	+	1.25516e9i	0.253893	+	0.439756i	−0.710701	+	0.703494i	$0.751622\pi$
$504$		0			0
$505$	7.53097e9			2.60214
$506$		0			0
$507$	−1.13076e8	−	1.95854e8i	−0.0385340	−	0.0667429i
$508$		0			0
$509$	2.51350e9	+	4.35350e9i	0.844824	+	1.46328i	0.885774	+	0.464117i	$0.153628\pi$
$509$	2.51350e9	+	4.35350e9i	0.844824	+	1.46328i	−0.0409504	+	0.999161i	$0.513039\pi$
$510$		0			0
$511$	−2.13967e9	+	3.70602e9i	−0.709371	+	1.22867i
$512$		0			0
$513$	−5.75020e8	−	6.75543e8i	−0.188049	−	0.220924i
$514$		0			0
$515$	−1.72943e9	+	2.99546e9i	−0.557928	+	0.966359i
$516$		0			0
$517$	−1.21012e8	−	2.09600e8i	−0.0385135	−	0.0667074i
$518$		0			0
$519$	1.51198e8	+	2.61882e8i	0.0474744	+	0.0822281i
$520$		0			0
$521$	−2.39494e9			−0.741930			−0.370965	−	0.928647i	$-0.620973\pi$
$521$	−2.39494e9			−0.741930			−0.370965	+	0.928647i	$0.620973\pi$
$522$		0			0
$523$	3.02362e9	+	5.23707e9i	0.924213	+	1.60078i	0.792822	+	0.609453i	$0.208611\pi$
$523$	3.02362e9	+	5.23707e9i	0.924213	+	1.60078i	0.131391	+	0.991331i	$0.458056\pi$
$524$		0			0
$525$	−5.80458e8			−0.175071
$526$		0			0
$527$	−5.81163e8	+	1.00660e9i	−0.172966	+	0.299586i
$528$		0			0
$529$	1.68852e9	−	2.92460e9i	0.495920	−	0.858958i
$530$		0			0
$531$	−3.78497e9			−1.09706
$532$		0			0
$533$	−3.68130e9			−1.05307
$534$		0			0
$535$	−3.07655e9	+	5.32874e9i	−0.868612	+	1.50448i
$536$		0			0
$537$	2.92771e8	−	5.07095e8i	0.0815866	−	0.141312i
$538$		0			0
$539$	7.53149e7			0.0207167
$540$		0			0
$541$	−1.58330e9	−	2.74236e9i	−0.429906	−	0.744618i	0.566959	−	0.823746i	$-0.308120\pi$
$541$	−1.58330e9	−	2.74236e9i	−0.429906	−	0.744618i	−0.996864	+	0.0791278i	$0.974786\pi$
$542$		0			0
$543$	−6.88168e8			−0.184457
$544$		0			0
$545$	4.25368e9	+	7.36758e9i	1.12558	+	1.94956i
$546$		0			0
$547$	3.08848e9	+	5.34941e9i	0.806844	+	1.39749i	0.915039	+	0.403365i	$0.132159\pi$
$547$	3.08848e9	+	5.34941e9i	0.806844	+	1.39749i	−0.108195	+	0.994130i	$0.534507\pi$
$548$		0			0
$549$	−1.70340e9	+	2.95038e9i	−0.439353	+	0.760981i
$550$		0			0
$551$	1.88578e9	−	5.29714e9i	0.480242	−	1.34900i
$552$		0			0
$553$	1.60242e9	−	2.77547e9i	0.402938	−	0.697910i
$554$		0			0
$555$	−8.48264e7	−	1.46924e8i	−0.0210623	−	0.0364810i
$556$		0			0
$557$	−5.51752e7	−	9.55662e7i	−0.0135285	−	0.0234321i	0.859182	−	0.511670i	$-0.170973\pi$
$557$	−5.51752e7	−	9.55662e7i	−0.0135285	−	0.0234321i	−0.872710	+	0.488238i	$0.837640\pi$
$558$		0			0
$559$	−5.40698e9			−1.30922
$560$		0			0
$561$	2.08433e7	+	3.61017e7i	0.00498422	+	0.00863292i
$562$		0			0
$563$	1.31030e9			0.309451			0.154725	−	0.987958i	$-0.450551\pi$
$563$	1.31030e9			0.309451			0.154725	+	0.987958i	$0.450551\pi$
$564$		0			0
$565$	4.60571e8	−	7.97733e8i	0.107430	−	0.186075i
$566$		0			0
$567$	2.31162e9	−	4.00384e9i	0.532568	−	0.922435i
$568$		0			0
$569$	−8.76132e8			−0.199378			−0.0996889	−	0.995019i	$-0.531785\pi$
$569$	−8.76132e8			−0.199378			−0.0996889	+	0.995019i	$0.531785\pi$
$570$		0			0
$571$	−5.88405e8			−0.132267			−0.0661333	−	0.997811i	$-0.521066\pi$
$571$	−5.88405e8			−0.132267			−0.0661333	+	0.997811i	$0.521066\pi$
$572$		0			0
$573$	−1.63366e8	+	2.82958e8i	−0.0362760	+	0.0628319i
$574$		0			0
$575$	−2.16022e8	+	3.74162e8i	−0.0473872	+	0.0820771i
$576$		0			0
$577$	2.83427e9			0.614222			0.307111	−	0.951674i	$-0.400638\pi$
$577$	2.83427e9			0.614222			0.307111	+	0.951674i	$0.400638\pi$
$578$		0			0
$579$	1.34473e8	+	2.32914e8i	0.0287912	+	0.0498679i
$580$		0			0
$581$	5.49270e9			1.16190
$582$		0			0
$583$	−1.56651e8	−	2.71328e8i	−0.0327412	−	0.0567094i
$584$		0			0
$585$	4.18865e9	+	7.25495e9i	0.865025	+	1.49827i
$586$		0			0
$587$	4.48056e9	−	7.76056e9i	0.914322	−	1.58365i	0.106430	−	0.994320i	$-0.466058\pi$
$587$	4.48056e9	−	7.76056e9i	0.914322	−	1.58365i	0.807891	−	0.589332i	$-0.200609\pi$
$588$		0			0
$589$	−1.14834e9	−	1.34909e9i	−0.231561	−	0.272042i
$590$		0			0
$591$	3.07829e8	−	5.33176e8i	0.0613414	−	0.106246i
$592$		0			0
$593$	2.23594e9	+	3.87277e9i	0.440321	+	0.762658i	0.997713	−	0.0675913i	$-0.0215314\pi$
$593$	2.23594e9	+	3.87277e9i	0.440321	+	0.762658i	−0.557392	+	0.830249i	$0.688198\pi$
$594$		0			0
$595$	−4.05257e9	−	7.01925e9i	−0.788716	−	1.36610i
$596$		0			0
$597$	3.77378e8			0.0725883
$598$		0			0
$599$	2.07531e9	+	3.59454e9i	0.394538	+	0.683359i	0.993042	−	0.117760i	$-0.0375714\pi$
$599$	2.07531e9	+	3.59454e9i	0.394538	+	0.683359i	−0.598504	+	0.801120i	$0.704238\pi$
$600$		0			0
$601$	−4.71806e9			−0.886549			−0.443274	−	0.896386i	$-0.646183\pi$
$601$	−4.71806e9			−0.886549			−0.443274	+	0.896386i	$0.646183\pi$
$602$		0			0
$603$	−1.19330e9	+	2.06686e9i	−0.221636	+	0.383885i
$604$		0			0
$605$	3.87927e9	−	6.71910e9i	0.712208	−	1.23358i
$606$		0			0
$607$	−9.08059e9			−1.64799			−0.823993	−	0.566599i	$-0.808259\pi$
$607$	−9.08059e9			−1.64799			−0.823993	+	0.566599i	$0.808259\pi$
$608$		0			0
$609$	−1.33191e9			−0.238954
$610$		0			0
$611$	−3.82039e9	+	6.61711e9i	−0.677584	+	1.17361i
$612$		0			0
$613$	−1.14764e9	+	1.98778e9i	−0.201231	+	0.348543i	−0.948925	−	0.315500i	$-0.897828\pi$
$613$	−1.14764e9	+	1.98778e9i	−0.201231	+	0.348543i	0.747694	+	0.664043i	$0.231161\pi$
$614$		0			0
$615$	−1.03235e9			−0.178964
$616$		0			0
$617$	−3.24054e9	−	5.61278e9i	−0.555417	−	0.962010i	−0.997871	−	0.0652190i	$-0.979225\pi$
$617$	−3.24054e9	−	5.61278e9i	−0.555417	−	0.962010i	0.442454	−	0.896791i	$-0.354108\pi$
$618$		0			0
$619$	−3.59214e9			−0.608746			−0.304373	−	0.952553i	$-0.598447\pi$
$619$	−3.59214e9			−0.608746			−0.304373	+	0.952553i	$0.598447\pi$
$620$		0			0
$621$	7.82059e7	+	1.35457e8i	0.0131045	+	0.0226976i
$622$		0			0
$623$	−4.14483e9	−	7.17906e9i	−0.686751	−	1.18949i
$624$		0			0
$625$	2.89453e9	−	5.01347e9i	0.474240	−	0.821407i
$626$		0			0
$627$	−6.24950e7	+	1.14748e7i	−0.0101253	+	0.00185913i
$628$		0			0
$629$	6.06423e8	−	1.05036e9i	0.0971625	−	0.168290i
$630$		0			0
$631$	4.87462e9	+	8.44309e9i	0.772393	+	1.33782i	0.936248	+	0.351339i	$0.114274\pi$
$631$	4.87462e9	+	8.44309e9i	0.772393	+	1.33782i	−0.163856	+	0.986484i	$0.552393\pi$
$632$		0			0
$633$	4.61181e8	+	7.98788e8i	0.0722700	+	0.125175i
$634$		0			0
$635$	8.77083e9			1.35935
$636$		0			0
$637$	−1.18885e9	−	2.05916e9i	−0.182239	−	0.315647i
$638$		0			0
$639$	−5.71865e9			−0.867043
$640$		0			0
$641$	5.87557e9	−	1.01768e10i	0.881144	−	1.52619i	0.0310726	−	0.999517i	$-0.490108\pi$
$641$	5.87557e9	−	1.01768e10i	0.881144	−	1.52619i	0.850071	−	0.526668i	$-0.176559\pi$
$642$		0			0
$643$	−7.74384e8	+	1.34127e9i	−0.114873	+	0.198966i	−0.917729	−	0.397207i	$-0.869979\pi$
$643$	−7.74384e8	+	1.34127e9i	−0.114873	+	0.198966i	0.802856	+	0.596173i	$0.203313\pi$
$644$		0			0
$645$	−1.51629e9			−0.222496
$646$		0			0
$647$	1.16031e9			0.168426			0.0842131	−	0.996448i	$-0.473162\pi$
$647$	1.16031e9			0.168426			0.0842131	+	0.996448i	$0.473162\pi$
$648$		0			0
$649$	−2.74072e8	+	4.74707e8i	−0.0393558	+	0.0681662i
$650$		0			0
$651$	−2.09830e8	+	3.63437e8i	−0.0298082	+	0.0516292i
$652$		0			0
$653$	−1.03338e10			−1.45233			−0.726164	−	0.687522i	$-0.758698\pi$
$653$	−1.03338e10			−1.45233			−0.726164	+	0.687522i	$0.758698\pi$
$654$		0			0
$655$	−5.32332e9	−	9.22026e9i	−0.740181	−	1.28203i
$656$		0			0
$657$	8.86732e9			1.21987
$658$		0			0
$659$	−5.82319e9	−	1.00861e10i	−0.792614	−	1.37285i	−0.924343	−	0.381562i	$-0.875386\pi$
$659$	−5.82319e9	−	1.00861e10i	−0.792614	−	1.37285i	0.131729	−	0.991286i	$-0.457947\pi$
$660$		0			0
$661$	5.29073e8	+	9.16381e8i	0.0712542	+	0.123416i	0.899451	−	0.437021i	$-0.143967\pi$
$661$	5.29073e8	+	9.16381e8i	0.0712542	+	0.123416i	−0.828197	+	0.560437i	$0.810633\pi$
$662$		0			0
$663$	6.58028e8	−	1.13974e9i	0.0876894	−	0.151883i
$664$		0			0
$665$	1.21509e10	−	2.23105e9i	1.60226	−	0.294194i
$666$		0			0
$667$	−4.95682e8	+	8.58546e8i	−0.0646789	+	0.112027i
$668$		0			0
$669$	−6.87798e8	−	1.19130e9i	−0.0888115	−	0.153826i
$670$		0			0
$671$	2.46688e8	+	4.27277e8i	0.0315224	+	0.0545985i
$672$		0			0
$673$	−6.15132e9			−0.777886			−0.388943	−	0.921262i	$-0.627160\pi$
$673$	−6.15132e9			−0.777886			−0.388943	+	0.921262i	$0.627160\pi$
$674$		0			0
$675$	1.21600e9	+	2.10617e9i	0.152184	+	0.263590i
$676$		0			0
$677$	−1.01802e10			−1.26095			−0.630474	−	0.776210i	$-0.717140\pi$
$677$	−1.01802e10			−1.26095			−0.630474	+	0.776210i	$0.717140\pi$
$678$		0			0
$679$	3.80302e9	−	6.58702e9i	0.466213	−	0.807505i
$680$		0			0
$681$	1.03773e9	−	1.79740e9i	0.125913	−	0.218087i
$682$		0			0
$683$	3.10947e9			0.373435			0.186717	−	0.982414i	$-0.440215\pi$
$683$	3.10947e9			0.373435			0.186717	+	0.982414i	$0.440215\pi$
$684$		0			0
$685$	5.59107e8			0.0664626
$686$		0			0
$687$	6.93551e7	−	1.20127e8i	0.00816074	−	0.0141348i
$688$		0			0
$689$	−4.94552e9	+	8.56589e9i	−0.576030	+	0.997712i
$690$		0			0
$691$	2.68784e9			0.309907			0.154953	−	0.987922i	$-0.450477\pi$
$691$	2.68784e9			0.309907			0.154953	+	0.987922i	$0.450477\pi$
$692$		0			0
$693$	−3.42462e8	−	5.93161e8i	−0.0390882	−	0.0677028i
$694$		0			0
$695$	2.10051e10			2.37344
$696$		0			0
$697$	−3.69014e9	−	6.39151e9i	−0.412789	−	0.714971i
$698$		0			0
$699$	−1.86674e8	−	3.23330e8i	−0.0206735	−	0.0358076i
$700$		0			0
$701$	5.86197e9	−	1.01532e10i	0.642733	−	1.11325i	−0.342088	−	0.939668i	$-0.611134\pi$
$701$	5.86197e9	−	1.01532e10i	0.642733	−	1.11325i	0.984820	−	0.173577i	$-0.0555327\pi$
$702$		0			0
$703$	1.19825e9	+	1.40773e9i	0.130078	+	0.152818i
$704$		0			0
$705$	−1.07136e9	+	1.85565e9i	−0.115152	+	0.199450i
$706$		0			0
$707$	9.71933e9	+	1.68344e10i	1.03435	+	1.79155i
$708$		0			0
$709$	2.19422e9	+	3.80050e9i	0.231216	+	0.400479i	0.958166	−	0.286212i	$-0.0923961\pi$
$709$	2.19422e9	+	3.80050e9i	0.231216	+	0.400479i	−0.726950	+	0.686690i	$0.759063\pi$
$710$		0			0
$711$	−6.64083e9			−0.692913
$712$		0			0
$713$	1.56180e8	+	2.70512e8i	0.0161366	+	0.0279495i
$714$		0			0
$715$	1.21321e9			0.124127
$716$		0			0
$717$	3.14767e8	−	5.45193e8i	0.0318913	−	0.0552374i
$718$		0			0
$719$	−9.26409e9	+	1.60459e10i	−0.929505	+	1.60995i	−0.145354	+	0.989380i	$0.546432\pi$
$719$	−9.26409e9	+	1.60459e10i	−0.929505	+	1.60995i	−0.784151	+	0.620570i	$0.786901\pi$
$720$		0			0
$721$	−8.92789e9			−0.887106
$722$		0			0
$723$	6.47790e8			0.0637456
$724$		0			0
$725$	−7.70718e9	+	1.33492e10i	−0.751126	+	1.30099i
$726$		0			0
$727$	7.79368e9	−	1.34990e10i	0.752267	−	1.30296i	−0.194455	−	0.980911i	$-0.562294\pi$
$727$	7.79368e9	−	1.34990e10i	0.752267	−	1.30296i	0.946722	−	0.322053i	$-0.104373\pi$
$728$		0			0
$729$	−9.13490e9			−0.873288
$730$		0			0
$731$	−5.41996e9	−	9.38765e9i	−0.513198	−	0.888886i
$732$		0			0
$733$	−4.52039e9			−0.423948			−0.211974	−	0.977275i	$-0.567989\pi$
$733$	−4.52039e9			−0.423948			−0.211974	+	0.977275i	$0.567989\pi$
$734$		0			0
$735$	−3.33392e8	−	5.77452e8i	−0.0309706	−	0.0536427i
$736$		0			0
$737$	1.72816e8	+	2.99326e8i	0.0159018	+	0.0275428i
$738$		0			0
$739$	−8.41317e8	+	1.45720e9i	−0.0766839	+	0.132820i	−0.901817	−	0.432118i	$-0.857767\pi$
$739$	−8.41317e8	+	1.45720e9i	−0.0766839	+	0.132820i	0.825133	+	0.564938i	$0.191100\pi$
$740$		0			0
$741$	1.30022e9	+	1.52752e9i	0.117396	+	0.137919i
$742$		0			0
$743$	−4.92734e9	+	8.53440e9i	−0.440709	+	0.763330i	−0.997742	−	0.0671601i	$-0.978606\pi$
$743$	−4.92734e9	+	8.53440e9i	−0.440709	+	0.763330i	0.557033	+	0.830490i	$0.311940\pi$
$744$		0			0
$745$	1.26952e10	+	2.19887e10i	1.12485	+	1.94829i
$746$		0			0
$747$	−5.69078e9	−	9.85671e9i	−0.499516	−	0.865187i
$748$		0			0
$749$	−1.58822e10			−1.38109
$750$		0			0
$751$	8.43842e9	+	1.46158e10i	0.726979	+	1.25916i	0.958154	+	0.286252i	$0.0924094\pi$
$751$	8.43842e9	+	1.46158e10i	0.726979	+	1.25916i	−0.231176	+	0.972912i	$0.574257\pi$
$752$		0			0
$753$	1.46286e9			0.124859
$754$		0			0
$755$	1.61385e10	−	2.79528e10i	1.36474	−	2.36380i
$756$		0			0
$757$	−7.68849e9	+	1.33168e10i	−0.644177	+	1.11575i	0.340314	+	0.940312i	$0.389467\pi$
$757$	−7.68849e9	+	1.33168e10i	−0.644177	+	1.11575i	−0.984491	+	0.175435i	$0.943867\pi$
$758$		0			0
$759$	1.12028e7			0.000929993
$760$		0			0
$761$	3.94746e9			0.324692			0.162346	−	0.986734i	$-0.448094\pi$
$761$	3.94746e9			0.324692			0.162346	+	0.986734i	$0.448094\pi$
$762$		0			0
$763$	−1.09794e10	+	1.90169e10i	−0.894837	+	1.54990i
$764$		0			0
$765$	−8.39742e9	+	1.45447e10i	−0.678158	+	1.17460i
$766$		0			0
$767$	1.73050e10			1.38480
$768$		0			0
$769$	7.06474e9	+	1.22365e10i	0.560214	+	0.970320i	0.997477	+	0.0709859i	$0.0226145\pi$
$769$	7.06474e9	+	1.22365e10i	0.560214	+	0.970320i	−0.437263	+	0.899334i	$0.644052\pi$
$770$		0			0
$771$	1.36368e9			0.107158
$772$		0			0
$773$	−9.44191e9	−	1.63539e10i	−0.735244	−	1.27348i	−0.954616	−	0.297839i	$-0.903734\pi$
$773$	−9.44191e9	−	1.63539e10i	−0.735244	−	1.27348i	0.219372	−	0.975641i	$-0.429599\pi$
$774$		0			0
$775$	2.42839e9	+	4.20610e9i	0.187397	+	0.324581i
$776$		0			0
$777$	2.18951e8	−	3.79234e8i	0.0167445	−	0.0290024i
$778$		0			0
$779$	1.10642e10	−	2.03152e9i	0.838571	−	0.153972i
$780$		0			0
$781$	−4.14091e8	+	7.17226e8i	−0.0311040	+	0.0538738i
$782$		0			0
$783$	2.79021e9	+	4.83278e9i	0.207716	+	0.359775i
$784$		0			0
$785$	7.24612e9	+	1.25507e10i	0.534641	+	0.926025i
$786$		0			0
$787$	1.09229e10			0.798779			0.399390	−	0.916781i	$-0.369222\pi$
$787$	1.09229e10			0.798779			0.399390	+	0.916781i	$0.369222\pi$
$788$		0			0
$789$	−1.47485e8	−	2.55452e8i	−0.0106900	−	0.0185157i
$790$		0			0
$791$	2.37762e9			0.170815
$792$		0			0
$793$	7.78800e9	−	1.34892e10i	0.554587	−	0.960573i
$794$		0			0
$795$	−1.38688e9	+	2.40215e9i	−0.0978935	+	0.169557i
$796$		0			0
$797$	−1.26385e9			−0.0884280			−0.0442140	−	0.999022i	$-0.514078\pi$
$797$	−1.26385e9			−0.0884280			−0.0442140	+	0.999022i	$0.514078\pi$
$798$		0			0
$799$	−1.53183e10			−1.06242
$800$		0			0
$801$	−8.58861e9	+	1.48759e10i	−0.590485	+	1.02275i
$802$		0			0
$803$	6.42087e8	−	1.11213e9i	0.0437612	−	0.0757967i
$804$		0			0
$805$	−2.17816e9			−0.147165
$806$		0			0
$807$	−1.22436e9	−	2.12065e9i	−0.0820070	−	0.142040i
$808$		0			0
$809$	1.99516e10			1.32482			0.662412	−	0.749140i	$-0.269533\pi$
$809$	1.99516e10			1.32482			0.662412	+	0.749140i	$0.269533\pi$
$810$		0			0
$811$	−4.42506e9	−	7.66442e9i	−0.291303	−	0.504552i	0.682815	−	0.730592i	$-0.260756\pi$
$811$	−4.42506e9	−	7.66442e9i	−0.291303	−	0.504552i	−0.974118	+	0.226039i	$0.927422\pi$
$812$		0			0
$813$	1.32562e9	+	2.29604e9i	0.0865171	+	0.149852i
$814$		0			0
$815$	1.85946e10	−	3.22068e10i	1.20319	−	2.08399i
$816$		0			0
$817$	1.62508e10	−	2.98384e9i	1.04255	−	0.191425i
$818$		0			0
$819$	−1.08116e10	+	1.87262e10i	−0.687695	+	1.19112i
$820$		0			0
$821$	−5.44573e9	−	9.43229e9i	−0.343444	−	0.594862i	0.641626	−	0.767018i	$-0.278260\pi$
$821$	−5.44573e9	−	9.43229e9i	−0.343444	−	0.594862i	−0.985070	+	0.172156i	$0.944927\pi$
$822$		0			0
$823$	2.20282e8	+	3.81540e8i	0.0137746	+	0.0238583i	0.872831	−	0.488023i	$-0.162282\pi$
$823$	2.20282e8	+	3.81540e8i	0.0137746	+	0.0238583i	−0.859056	+	0.511882i	$0.828949\pi$
$824$		0			0
$825$	1.74188e8			0.0108001
$826$		0			0
$827$	−7.60272e9	−	1.31683e10i	−0.467412	−	0.809581i	0.531895	−	0.846810i	$-0.321480\pi$
$827$	−7.60272e9	−	1.31683e10i	−0.467412	−	0.809581i	−0.999307	+	0.0372295i	$0.988147\pi$
$828$		0			0
$829$	−2.34670e10			−1.43060			−0.715298	−	0.698820i	$-0.753709\pi$
$829$	−2.34670e10			−1.43060			−0.715298	+	0.698820i	$0.753709\pi$
$830$		0			0
$831$	−1.55743e9	+	2.69755e9i	−0.0941469	+	0.163067i
$832$		0			0
$833$	2.38342e9	−	4.12820e9i	0.142871	−	0.247459i
$834$		0			0
$835$	−2.55546e10			−1.51903
$836$		0			0
$837$	1.75829e9			0.103646
$838$		0			0
$839$	−7.84074e9	+	1.35806e10i	−0.458343	+	0.793873i	−0.998874	−	0.0474514i	$-0.984890\pi$
$839$	−7.84074e9	+	1.35806e10i	−0.458343	+	0.793873i	0.540531	+	0.841324i	$0.318223\pi$
$840$		0			0
$841$	−9.05985e9	+	1.56921e10i	−0.525213	+	0.909695i
$842$		0			0
$843$	2.10066e9			0.120770
$844$		0			0
$845$	−6.59755e9	−	1.14273e10i	−0.376170	−	0.651545i
$846$		0			0
$847$	2.00261e10			1.13241
$848$		0			0
$849$	1.47977e9	+	2.56304e9i	0.0829884	+	0.143740i
$850$		0			0
$851$	−1.62969e8	−	2.82270e8i	−0.00906465	−	0.0157004i
$852$		0			0
$853$	−1.96010e9	+	3.39499e9i	−0.108133	+	0.187291i	−0.915014	−	0.403423i	$-0.867820\pi$
$853$	−1.96010e9	+	3.39499e9i	−0.108133	+	0.187291i	0.806881	+	0.590714i	$0.201154\pi$
$854$		0			0
$855$	−1.65927e10	−	1.94934e10i	−0.907896	−	1.06661i
$856$		0			0
$857$	9.16873e9	−	1.58807e10i	0.497595	−	0.861861i	−0.502401	−	0.864635i	$-0.667550\pi$
$857$	9.16873e9	−	1.58807e10i	0.497595	−	0.861861i	0.999996	+	0.00277432i	$0.000883094\pi$
$858$		0			0
$859$	6.89177e9	+	1.19369e10i	0.370984	+	0.642562i	0.989717	−	0.143038i	$-0.0456873\pi$
$859$	6.89177e9	+	1.19369e10i	0.370984	+	0.642562i	−0.618734	+	0.785601i	$0.712354\pi$
$860$		0			0
$861$	−1.33234e9	−	2.30767e9i	−0.0711381	−	0.123215i
$862$		0			0
$863$	−6.63513e9			−0.351408			−0.175704	−	0.984443i	$-0.556220\pi$
$863$	−6.63513e9			−0.351408			−0.175704	+	0.984443i	$0.556220\pi$
$864$		0			0
$865$	8.82177e9	+	1.52798e10i	0.463446	+	0.802712i
$866$		0			0
$867$	−1.75473e8			−0.00914413
$868$		0			0
$869$	−4.80866e8	+	8.32884e8i	−0.0248573	+	0.0430542i
$870$		0			0
$871$	5.45583e9	−	9.44977e9i	0.279767	−	0.484571i
$872$		0			0
$873$	−1.57607e10			−0.801723
$874$		0			0
$875$	−1.58534e9			−0.0800006
$876$		0			0
$877$	1.03218e10	−	1.78779e10i	0.516722	−	0.894989i	−0.483089	−	0.875571i	$-0.660485\pi$
$877$	1.03218e10	−	1.78779e10i	0.516722	−	0.894989i	0.999811	−	0.0194178i	$-0.00618128\pi$
$878$		0			0
$879$	2.01239e9	−	3.48556e9i	0.0999425	−	0.173106i
$880$		0			0
$881$	9.87740e9			0.486661			0.243331	−	0.969943i	$-0.421760\pi$
$881$	9.87740e9			0.486661			0.243331	+	0.969943i	$0.421760\pi$
$882$		0			0
$883$	−1.85403e10	−	3.21128e10i	−0.906264	−	1.56969i	−0.819212	−	0.573491i	$-0.805589\pi$
$883$	−1.85403e10	−	3.21128e10i	−0.906264	−	1.56969i	−0.0870520	−	0.996204i	$-0.527745\pi$
$884$		0			0
$885$	4.85287e9			0.235341
$886$		0			0
$887$	1.21684e10	+	2.10763e10i	0.585466	+	1.01406i	0.994817	+	0.101680i	$0.0324218\pi$
$887$	1.21684e10	+	2.10763e10i	0.585466	+	1.01406i	−0.409351	+	0.912377i	$0.634245\pi$
$888$		0			0
$889$	1.13195e10	+	1.96059e10i	0.540343	+	0.935902i
$890$		0			0
$891$	−6.93686e8	+	1.20150e9i	−0.0328542	+	0.0569052i
$892$		0			0
$893$	7.83061e9	−	2.19961e10i	0.367973	−	1.03363i
$894$		0			0
$895$	1.70820e10	−	2.95869e10i	0.796450	−	1.37949i
$896$		0			0
$897$	−1.76837e8	−	3.06291e8i	−0.00818088	−	0.0141697i
$898$		0			0
$899$	5.57216e9	+	9.65126e9i	0.255779	+	0.443022i
$900$		0			0
$901$	−1.98296e10			−0.903185
$902$		0			0
$903$	−1.95689e9	−	3.38944e9i	−0.0884423	−	0.153187i
$904$		0			0
$905$	−4.01518e10			−1.80067
$906$		0			0
$907$	1.43237e10	−	2.48094e10i	0.637426	−	1.10405i	−0.348570	−	0.937283i	$-0.613332\pi$
$907$	1.43237e10	−	2.48094e10i	0.637426	−	1.10405i	0.985996	−	0.166771i	$-0.0533342\pi$
$908$		0			0
$909$	2.01396e10	−	3.48829e10i	0.889361	−	1.54042i
$910$		0			0
$911$	9.21467e9			0.403799			0.201900	−	0.979406i	$-0.435288\pi$
$911$	9.21467e9			0.403799			0.201900	+	0.979406i	$0.435288\pi$
$912$		0			0
$913$	−1.64829e9			−0.0716780
$914$		0			0
$915$	2.18400e9	−	3.78280e9i	0.0942495	−	0.163245i
$916$		0			0
$917$	1.37404e10	−	2.37990e10i	0.588444	−	1.01922i
$918$		0			0
$919$	−4.19815e10			−1.78424			−0.892120	−	0.451798i	$-0.850783\pi$
$919$	−4.19815e10			−1.78424			−0.892120	+	0.451798i	$0.850783\pi$
$920$		0			0
$921$	−1.84514e9	−	3.19587e9i	−0.0778252	−	0.134797i
$922$		0			0
$923$	2.61459e10			1.09445
$924$		0			0
$925$	−2.53394e9	−	4.38892e9i	−0.105269	−	0.182331i
$926$		0			0
$927$	9.24984e9	+	1.60212e10i	0.381378	+	0.660565i
$928$		0			0
$929$	−5.67307e9	+	9.82605e9i	−0.232147	+	0.402091i	−0.958440	−	0.285295i	$-0.907908\pi$
$929$	−5.67307e9	+	9.82605e9i	−0.232147	+	0.402091i	0.726293	+	0.687386i	$0.241242\pi$
$930$		0			0
$931$	4.70947e9	+	5.53277e9i	0.191271	+	0.224708i
$932$		0			0
$933$	−1.63986e9	+	2.84032e9i	−0.0661030	+	0.114494i
$934$		0			0
$935$	1.21612e9	+	2.10639e9i	0.0486561	+	0.0842748i
$936$		0			0
$937$	−1.35639e10	−	2.34934e10i	−0.538638	−	0.932948i	−0.998978	−	0.0452054i	$-0.985606\pi$
$937$	−1.35639e10	−	2.34934e10i	−0.538638	−	0.932948i	0.460340	−	0.887743i	$-0.347728\pi$
$938$		0			0
$939$	−6.52072e8			−0.0257020
$940$		0			0
$941$	−2.68415e9	−	4.64909e9i	−0.105013	−	0.181888i	0.808730	−	0.588179i	$-0.200155\pi$
$941$	−2.68415e9	−	4.64909e9i	−0.105013	−	0.181888i	−0.913744	+	0.406291i	$0.866822\pi$
$942$		0			0
$943$	−1.98336e9			−0.0770212
$944$		0			0
$945$	−6.13045e9	+	1.06182e10i	−0.236309	+	0.409300i
$946$		0			0
$947$	2.31120e10	−	4.00312e10i	0.884327	−	1.53170i	0.0378444	−	0.999284i	$-0.487951\pi$
$947$	2.31120e10	−	4.00312e10i	0.884327	−	1.53170i	0.846483	−	0.532416i	$-0.178716\pi$
$948$		0			0
$949$	−4.05417e10			−1.53982
$950$		0			0
$951$	−5.08103e8			−0.0191567
$952$		0			0
$953$	−2.31060e10	+	4.00208e10i	−0.864769	+	1.49782i	0.00250756	+	0.999997i	$0.499202\pi$
$953$	−2.31060e10	+	4.00208e10i	−0.864769	+	1.49782i	−0.867276	+	0.497827i	$0.834132\pi$
$954$		0			0
$955$	−9.53171e9	+	1.65094e10i	−0.354127	+	0.613366i
$956$		0			0
$957$	3.99689e8			0.0147411
$958$		0			0
$959$	7.21573e8	+	1.24980e9i	0.0264189	+	0.0457589i
$960$		0			0
$961$	−2.40012e10			−0.872372
$962$		0			0
$963$	1.64549e10	+	2.85007e10i	0.593749	+	1.02840i
$964$		0			0
$965$	7.84595e9	+	1.35896e10i	0.281061	+	0.486811i
$966$		0			0
$967$	2.91059e9	−	5.04128e9i	0.103511	−	0.179287i	−0.809618	−	0.586958i	$-0.800326\pi$
$967$	2.91059e9	−	5.04128e9i	0.103511	−	0.179287i	0.913129	+	0.407671i	$0.133659\pi$
$968$		0			0
$969$	−1.34875e9	+	3.78864e9i	−0.0476211	+	0.133767i
$970$		0			0
$971$	−1.77878e10	+	3.08094e10i	−0.623526	+	1.07998i	0.365298	+	0.930891i	$0.380967\pi$
$971$	−1.77878e10	+	3.08094e10i	−0.623526	+	1.07998i	−0.988824	+	0.149088i	$0.952366\pi$
$972$		0			0
$973$	2.71089e10	+	4.69539e10i	0.943444	+	1.63409i
$974$		0			0
$975$	−2.74958e9	−	4.76241e9i	−0.0950057	−	0.164555i
$976$		0			0
$977$	1.86368e10			0.639353			0.319677	−	0.947527i	$-0.396426\pi$
$977$	1.86368e10			0.639353			0.319677	+	0.947527i	$0.396426\pi$
$978$		0			0
$979$	1.24381e9	+	2.15434e9i	0.0423658	+	0.0733797i
$980$		0			0
$981$	4.55015e10			1.53881
$982$		0			0
$983$	−2.54863e10	+	4.41437e10i	−0.855796	+	1.48228i	0.0201083	+	0.999798i	$0.493599\pi$
$983$	−2.54863e10	+	4.41437e10i	−0.855796	+	1.48228i	−0.875904	+	0.482485i	$0.839734\pi$
$984$		0			0
$985$	1.79606e10	−	3.11086e10i	0.598816	−	1.03718i
$986$		0			0
$987$	−5.53070e9			−0.183092
$988$		0			0
$989$	−2.91310e9			−0.0957564
$990$		0			0
$991$	1.57841e9	−	2.73389e9i	0.0515185	−	0.0892326i	−0.839116	−	0.543952i	$-0.816927\pi$
$991$	1.57841e9	−	2.73389e9i	0.0515185	−	0.0892326i	0.890635	+	0.454720i	$0.150261\pi$
$992$		0			0
$993$	−2.79882e9	+	4.84770e9i	−0.0907096	+	0.157114i
$994$		0			0
$995$	2.20185e10			0.708609
$996$		0			0
$997$	3.66012e9	+	6.33951e9i	0.116967	+	0.202592i	0.918564	−	0.395272i	$-0.129350\pi$
$997$	3.66012e9	+	6.33951e9i	0.116967	+	0.202592i	−0.801598	+	0.597864i	$0.796016\pi$
$998$		0			0
$999$	−1.83471e9			−0.0582222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.49.5	yes	22
19.7	even	3	inner	76.8.e.a.45.5	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.5	✓	22	19.7	even	3	inner
76.8.e.a.49.5	yes	22	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.42876 + 5.93879i) q^{3} +(-200.054 + 346.504i) q^{5} -1032.75 q^{7} +(1069.99 + 1853.27i) q^{9} +O(q^{10})\)	\(q+(-3.42876 + 5.93879i) q^{3} +(-200.054 + 346.504i) q^{5} -1032.75 q^{7} +(1069.99 + 1853.27i) q^{9} +309.913 q^{11} +(-4892.02 - 8473.22i) q^{13} +(-1371.88 - 2376.16i) q^{15} +(9807.53 - 16987.1i) q^{17} +(19379.0 + 22766.8i) q^{19} +(3541.04 - 6133.25i) q^{21} +(-2635.65 - 4565.08i) q^{23} +(-40980.8 - 70980.9i) q^{25} -29672.3 q^{27} +(-94034.0 - 162872. i) q^{29} -59256.8 q^{31} +(-1062.62 + 1840.51i) q^{33} +(206605. - 357850. i) q^{35} +61832.4 q^{37} +67094.2 q^{39} +(188128. - 325847. i) q^{41} +(276316. - 478594. i) q^{43} -856221. q^{45} +(-390472. - 676317. i) q^{47} +243019. q^{49} +(67255.3 + 116490. i) q^{51} +(-505468. - 875497. i) q^{53} +(-61999.5 + 107386. i) q^{55} +(-201653. + 37025.9i) q^{57} +(-884350. + 1.53174e6i) q^{59} +(795991. + 1.37870e6i) q^{61} +(-1.10502e6 - 1.91396e6i) q^{63} +3.91467e6 q^{65} +(557626. + 965836. i) q^{67} +36148.1 q^{69} +(-1.33615e6 + 2.31428e6i) q^{71} +(2.07183e6 - 3.58851e6i) q^{73} +562054. q^{75} -320062. q^{77} +(-1.55161e6 + 2.68747e6i) q^{79} +(-2.23832e6 + 3.87689e6i) q^{81} -5.31855e6 q^{83} +(3.92407e6 + 6.79669e6i) q^{85} +1.28968e6 q^{87} +(4.01342e6 + 6.95144e6i) q^{89} +(5.05221e6 + 8.75068e6i) q^{91} +(203177. - 351914. i) q^{93} +(-1.17656e7 + 2.16031e6i) q^{95} +(-3.68244e6 + 6.37817e6i) q^{97} +(331603. + 574354. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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