LMFDB - Embedded newform 169.6.b.b.168.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,6,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not 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:	$27.1048655484$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 201x^{4} + 10512x^{2} + 65536 $ x^6 + 201x^4 + 10512x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.3
Root			$-2.68079i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.6.b.b.168.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.68079i q^{2} +13.5120 q^{3} +10.0902 q^{4} -15.4272i q^{5} -63.2466i q^{6} +12.5359i q^{7} -197.015i q^{8} -60.4272 q^{9} +O(q^{10})$	\(q-4.68079i q^{2} +13.5120 q^{3} +10.0902 q^{4} -15.4272i q^{5} -63.2466i q^{6} +12.5359i q^{7} -197.015i q^{8} -60.4272 q^{9} -72.2115 q^{10} -271.483i q^{11} +136.338 q^{12} +58.6777 q^{14} -208.451i q^{15} -599.303 q^{16} -1710.35 q^{17} +282.847i q^{18} -2235.01i q^{19} -155.663i q^{20} +169.384i q^{21} -1270.76 q^{22} +966.314 q^{23} -2662.06i q^{24} +2887.00 q^{25} -4099.89 q^{27} +126.489i q^{28} +4916.69 q^{29} -975.718 q^{30} -2318.90i q^{31} -3499.28i q^{32} -3668.26i q^{33} +8005.78i q^{34} +193.393 q^{35} -609.721 q^{36} -13463.8i q^{37} -10461.6 q^{38} -3039.39 q^{40} -1646.31i q^{41} +792.850 q^{42} +9529.76 q^{43} -2739.31i q^{44} +932.222i q^{45} -4523.12i q^{46} +22144.0i q^{47} -8097.75 q^{48} +16649.9 q^{49} -13513.5i q^{50} -23110.1 q^{51} -30677.6 q^{53} +19190.8i q^{54} -4188.22 q^{55} +2469.76 q^{56} -30199.3i q^{57} -23014.0i q^{58} +37271.7i q^{59} -2103.31i q^{60} -30506.8 q^{61} -10854.3 q^{62} -757.506i q^{63} -35557.1 q^{64} -17170.4 q^{66} -44159.3i q^{67} -17257.7 q^{68} +13056.8 q^{69} -905.233i q^{70} +16879.6i q^{71} +11905.1i q^{72} +13170.7i q^{73} -63021.2 q^{74} +39009.0 q^{75} -22551.6i q^{76} +3403.27 q^{77} +19848.9 q^{79} +9245.56i q^{80} -40713.7 q^{81} -7706.02 q^{82} +22216.4i q^{83} +1709.11i q^{84} +26385.8i q^{85} -44606.8i q^{86} +66434.0 q^{87} -53486.3 q^{88} -3877.28i q^{89} +4363.54 q^{90} +9750.28 q^{92} -31332.8i q^{93} +103652. q^{94} -34479.9 q^{95} -47282.1i q^{96} -15571.2i q^{97} -77934.5i q^{98} +16405.0i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9}+O(q^{10}) $ 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9	$ 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9} + 2582 q^{10} + 2182 q^{12} - 1586 q^{14} + 5570 q^{16} - 1816 q^{17} - 6820 q^{22} - 7248 q^{23} - 4046 q^{25} - 8552 q^{27} - 17516 q^{29} - 3434 q^{30} + 8696 q^{35} + 9356 q^{36} - 67572 q^{38} - 87018 q^{40} + 27206 q^{42} - 4064 q^{43} - 152446 q^{48} + 89142 q^{49} - 26936 q^{51} - 25140 q^{53} + 70624 q^{55} + 98574 q^{56} - 25508 q^{61} + 80680 q^{62} - 234786 q^{64} - 224708 q^{66} - 98350 q^{68} + 40224 q^{69} - 263038 q^{74} - 47096 q^{75} + 42104 q^{77} - 123744 q^{79} - 216442 q^{81} - 284520 q^{82} + 278144 q^{87} - 316020 q^{88} - 202516 q^{90} + 601728 q^{92} + 145686 q^{94} + 393120 q^{95}+O(q^{100}) $ 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9 + 2582 * q^10 + 2182 * q^12 - 1586 * q^14 + 5570 * q^16 - 1816 * q^17 - 6820 * q^22 - 7248 * q^23 - 4046 * q^25 - 8552 * q^27 - 17516 * q^29 - 3434 * q^30 + 8696 * q^35 + 9356 * q^36 - 67572 * q^38 - 87018 * q^40 + 27206 * q^42 - 4064 * q^43 - 152446 * q^48 + 89142 * q^49 - 26936 * q^51 - 25140 * q^53 + 70624 * q^55 + 98574 * q^56 - 25508 * q^61 + 80680 * q^62 - 234786 * q^64 - 224708 * q^66 - 98350 * q^68 + 40224 * q^69 - 263038 * q^74 - 47096 * q^75 + 42104 * q^77 - 123744 * q^79 - 216442 * q^81 - 284520 * q^82 + 278144 * q^87 - 316020 * q^88 - 202516 * q^90 + 601728 * q^92 + 145686 * q^94 + 393120 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 4.68079i	− 0.827455i	−0.910401	−	0.413728i	$-0.864227\pi$
$2$	− 4.68079i	− 0.827455i	0.910401	−	0.413728i	$-0.135773\pi$
$3$	13.5120	0.866792	0.433396	−	0.901204i	$-0.357315\pi$
$3$	13.5120	0.866792	0.433396	+	0.901204i	$0.357315\pi$
$4$	10.0902	0.315318
$5$	− 15.4272i	− 0.275970i	−0.990434	−	0.137985i	$-0.955937\pi$
$5$	− 15.4272i	− 0.275970i	0.990434	−	0.137985i	$-0.0440626\pi$
$6$	− 63.2466i	− 0.717231i
$7$	12.5359i	0.0966961i	0.998831	+	0.0483480i	$0.0153957\pi$
$7$	12.5359i	0.0966961i	−0.998831	+	0.0483480i	$0.984604\pi$
$8$	− 197.015i	− 1.08837i
$9$	−60.4272	−0.248672
$10$	−72.2115	−0.228353
$11$	− 271.483i	− 0.676489i	−0.941058	−	0.338245i	$-0.890167\pi$
$11$	− 271.483i	− 0.676489i	0.941058	−	0.338245i	$-0.109833\pi$
$12$	136.338	0.273315
$13$	0	0
$13$	0	0
$14$	58.6777	0.0800116
$15$	− 208.451i	− 0.239209i
$16$	−599.303	−0.585256
$17$	−1710.35	−1.43536	−0.717681	−	0.696372i	$-0.754797\pi$
$17$	−1710.35	−1.43536	−0.717681	+	0.696372i	$0.754797\pi$
$18$	282.847i	0.205765i
$19$	− 2235.01i	− 1.42035i	−0.704025	−	0.710175i	$-0.748616\pi$
$19$	− 2235.01i	− 1.42035i	0.704025	−	0.710175i	$-0.251384\pi$
$20$	− 155.663i	− 0.0870184i
$21$	169.384i	0.0838154i
$22$	−1270.76	−0.559764
$23$	966.314	0.380889	0.190445	−	0.981698i	$-0.439007\pi$
$23$	966.314	0.380889	0.190445	+	0.981698i	$0.439007\pi$
$24$	− 2662.06i	− 0.943388i
$25$	2887.00	0.923841
$26$	0	0
$27$	−4099.89	−1.08234
$28$	126.489i	0.0304900i
$29$	4916.69	1.08562	0.542810	−	0.839856i	$-0.317361\pi$
$29$	4916.69	1.08562	0.542810	+	0.839856i	$0.317361\pi$
$30$	−975.718	−0.197934
$31$	− 2318.90i	− 0.433388i	−0.976240	−	0.216694i	$-0.930473\pi$
$31$	− 2318.90i	− 0.433388i	0.976240	−	0.216694i	$-0.0695275\pi$
$32$	− 3499.28i	− 0.604093i
$33$	− 3668.26i	− 0.586375i
$34$	8005.78i	1.18770i
$35$	193.393	0.0266852
$36$	−609.721	−0.0784107
$37$	− 13463.8i	− 1.61683i	−0.588616	−	0.808413i	$-0.700327\pi$
$37$	− 13463.8i	− 1.61683i	0.588616	−	0.808413i	$-0.299673\pi$
$38$	−10461.6	−1.17528
$39$	0	0
$40$	−3039.39	−0.300357
$41$	− 1646.31i	− 0.152950i	−0.997071	−	0.0764752i	$-0.975633\pi$
$41$	− 1646.31i	− 0.152950i	0.997071	−	0.0764752i	$-0.0243666\pi$
$42$	792.850	0.0693534
$43$	9529.76	0.785979	0.392990	−	0.919543i	$-0.371441\pi$
$43$	9529.76	0.785979	0.392990	+	0.919543i	$0.371441\pi$
$44$	− 2739.31i	− 0.213309i
$45$	932.222i	0.0686259i
$46$	− 4523.12i	− 0.315169i
$47$	22144.0i	1.46222i	0.682262	+	0.731108i	$0.260997\pi$
$47$	22144.0i	1.46222i	−0.682262	+	0.731108i	$0.739003\pi$
$48$	−8097.75	−0.507296
$49$	16649.9	0.990650
$50$	− 13513.5i	− 0.764437i
$51$	−23110.1	−1.24416
$52$	0	0
$53$	−30677.6	−1.50014	−0.750069	−	0.661360i	$-0.769980\pi$
$53$	−30677.6	−1.50014	−0.750069	+	0.661360i	$0.769980\pi$
$54$	19190.8i	0.895587i
$55$	−4188.22	−0.186691
$56$	2469.76	0.105241
$57$	− 30199.3i	− 1.23115i
$58$	− 23014.0i	− 0.898301i
$59$	37271.7i	1.39396i	0.717092	+	0.696978i	$0.245473\pi$
$59$	37271.7i	1.39396i	−0.717092	+	0.696978i	$0.754527\pi$
$60$	− 2103.31i	− 0.0754268i
$61$	−30506.8	−1.04972	−0.524858	−	0.851190i	$-0.675882\pi$
$61$	−30506.8	−1.04972	−0.524858	+	0.851190i	$0.675882\pi$
$62$	−10854.3	−0.358610
$63$	− 757.506i	− 0.0240456i
$64$	−35557.1	−1.08512
$65$	0	0
$66$	−17170.4	−0.485199
$67$	− 44159.3i	− 1.20181i	−0.799321	−	0.600904i	$-0.794807\pi$
$67$	− 44159.3i	− 1.20181i	0.799321	−	0.600904i	$-0.205193\pi$
$68$	−17257.7	−0.452596
$69$	13056.8	0.330152
$70$	− 905.233i	− 0.0220808i
$71$	16879.6i	0.397391i	0.980061	+	0.198695i	$0.0636704\pi$
$71$	16879.6i	0.397391i	−0.980061	+	0.198695i	$0.936330\pi$
$72$	11905.1i	0.270646i
$73$	13170.7i	0.289268i	0.989485	+	0.144634i	$0.0462005\pi$
$73$	13170.7i	0.289268i	−0.989485	+	0.144634i	$0.953800\pi$
$74$	−63021.2	−1.33785
$75$	39009.0	0.800778
$76$	− 22551.6i	− 0.447862i
$77$	3403.27	0.0654138
$78$	0	0
$79$	19848.9	0.357823	0.178912	−	0.983865i	$-0.442742\pi$
$79$	19848.9	0.357823	0.178912	+	0.983865i	$0.442742\pi$
$80$	9245.56i	0.161513i
$81$	−40713.7	−0.689491
$82$	−7706.02	−0.126560
$83$	22216.4i	0.353980i	0.984213	+	0.176990i	$0.0566360\pi$
$83$	22216.4i	0.353980i	−0.984213	+	0.176990i	$0.943364\pi$
$84$	1709.11i	0.0264285i
$85$	26385.8i	0.396117i
$86$	− 44606.8i	− 0.650362i
$87$	66434.0	0.941006
$88$	−53486.3	−0.736268
$89$	− 3877.28i	− 0.0518863i	−0.999663	−	0.0259431i	$-0.991741\pi$
$89$	− 3877.28i	− 0.0518863i	0.999663	−	0.0259431i	$-0.00825888\pi$
$90$	4363.54	0.0567849
$91$	0	0
$92$	9750.28	0.120101
$93$	− 31332.8i	− 0.375658i
$94$	103652.	1.20992
$95$	−34479.9	−0.391974
$96$	− 47282.1i	− 0.523623i
$97$	− 15571.2i	− 0.168033i	−0.996464	−	0.0840164i	$-0.973225\pi$
$97$	− 15571.2i	− 0.168033i	0.996464	−	0.0840164i	$-0.0267748\pi$
$98$	− 77934.5i	− 0.819718i
$99$	16405.0i	0.168224i
$100$	29130.4	0.291304
$101$	73698.5	0.718878	0.359439	−	0.933169i	$-0.382968\pi$
$101$	73698.5	0.718878	0.359439	+	0.933169i	$0.382968\pi$
$102$	108174.i	1.02949i
$103$	158913.	1.47593	0.737964	−	0.674840i	$-0.235787\pi$
$103$	158913.	1.47593	0.737964	+	0.674840i	$0.235787\pi$
$104$	0	0
$105$	2613.12	0.0231305
$106$	143595.i	1.24130i
$107$	21338.9	0.180182	0.0900912	−	0.995934i	$-0.471284\pi$
$107$	21338.9	0.180182	0.0900912	+	0.995934i	$0.471284\pi$
$108$	−41368.7	−0.341281
$109$	118147.i	0.952484i	0.879314	+	0.476242i	$0.158001\pi$
$109$	118147.i	0.952484i	−0.879314	+	0.476242i	$0.841999\pi$
$110$	19604.2i	0.154478i
$111$	− 181922.i	− 1.40145i
$112$	− 7512.77i	− 0.0565920i
$113$	230515.	1.69825	0.849126	−	0.528190i	$-0.177129\pi$
$113$	230515.	1.69825	0.849126	+	0.528190i	$0.177129\pi$
$114$	−141357.	−1.01872
$115$	− 14907.5i	− 0.105114i
$116$	49610.3	0.342315
$117$	0	0
$118$	174461.	1.15344
$119$	− 21440.6i	− 0.138794i
$120$	−41068.2	−0.260347
$121$	87348.0	0.542363
$122$	142796.i	0.868593i
$123$	− 22244.8i	− 0.132576i
$124$	− 23398.1i	− 0.136655i
$125$	− 92748.3i	− 0.530922i
$126$	−3545.73	−0.0198966
$127$	−121835.	−0.670292	−0.335146	−	0.942166i	$-0.608786\pi$
$127$	−121835.	−0.670292	−0.335146	+	0.942166i	$0.608786\pi$
$128$	54458.4i	0.293792i
$129$	128766.	0.681280
$130$	0	0
$131$	158065.	0.804745	0.402372	−	0.915476i	$-0.368186\pi$
$131$	158065.	0.804745	0.402372	+	0.915476i	$0.368186\pi$
$132$	− 37013.4i	− 0.184895i
$133$	28017.7	0.137342
$134$	−206701.	−0.994443
$135$	63249.8i	0.298693i
$136$	336965.i	1.56220i
$137$	230949.i	1.05127i	0.850710	+	0.525636i	$0.176173\pi$
$137$	230949.i	1.05127i	−0.850710	+	0.525636i	$0.823827\pi$
$138$	− 61116.1i	− 0.273186i
$139$	243865.	1.07057	0.535283	−	0.844673i	$-0.320205\pi$
$139$	243865.	1.07057	0.535283	+	0.844673i	$0.320205\pi$
$140$	1951.37	0.00841433
$141$	299209.i	1.26744i
$142$	79010.1	0.328823
$143$	0	0
$144$	36214.2	0.145537
$145$	− 75850.7i	− 0.299598i
$146$	61649.1	0.239356
$147$	224972.	0.858687
$148$	− 135852.i	− 0.509814i
$149$	− 229447.i	− 0.846676i	−0.905972	−	0.423338i	$-0.860858\pi$
$149$	− 229447.i	− 0.846676i	0.905972	−	0.423338i	$-0.139142\pi$
$150$	− 182593.i	− 0.662607i
$151$	− 473597.i	− 1.69031i	−0.534519	−	0.845156i	$-0.679507\pi$
$151$	− 473597.i	− 1.69031i	0.534519	−	0.845156i	$-0.320493\pi$
$152$	−440331.	−1.54586
$153$	103351.	0.356934
$154$	− 15930.0i	− 0.0541270i
$155$	−35774.1	−0.119602
$156$	0	0
$157$	−58729.3	−0.190154	−0.0950770	−	0.995470i	$-0.530310\pi$
$157$	−58729.3	−0.190154	−0.0950770	+	0.995470i	$0.530310\pi$
$158$	− 92908.6i	− 0.296083i
$159$	−414514.	−1.30031
$160$	−53984.1	−0.166712
$161$	12113.6i	0.0368305i
$162$	190573.i	0.570523i
$163$	430921.i	1.27037i	0.772362	+	0.635183i	$0.219075\pi$
$163$	430921.i	1.27037i	−0.772362	+	0.635183i	$0.780925\pi$
$164$	− 16611.5i	− 0.0482281i
$165$	−56591.0	−0.161822
$166$	103990.	0.292903
$167$	436606.i	1.21143i	0.795681	+	0.605715i	$0.207113\pi$
$167$	436606.i	1.21143i	−0.795681	+	0.605715i	$0.792887\pi$
$168$	33371.2	0.0912218
$169$	0	0
$170$	123507.	0.327769
$171$	135055.i	0.353201i
$172$	96157.0	0.247833
$173$	−80120.5	−0.203530	−0.101765	−	0.994808i	$-0.532449\pi$
$173$	−80120.5	−0.203530	−0.101765	+	0.994808i	$0.532449\pi$
$174$	− 310964.i	− 0.778640i
$175$	36191.0i	0.0893317i
$176$	162700.i	0.395920i
$177$	503613.i	1.20827i
$178$	−18148.7	−0.0429335
$179$	264728.	0.617542	0.308771	−	0.951136i	$-0.400082\pi$
$179$	264728.	0.617542	0.308771	+	0.951136i	$0.400082\pi$
$180$	9406.29i	0.0216390i
$181$	−752932.	−1.70828	−0.854140	−	0.520043i	$-0.825916\pi$
$181$	−752932.	−1.70828	−0.854140	+	0.520043i	$0.825916\pi$
$182$	0	0
$183$	−412206.	−0.909885
$184$	− 190379.i	− 0.414547i
$185$	−207709.	−0.446195
$186$	−146662.	−0.310840
$187$	464330.i	0.971007i
$188$	223437.i	0.461063i
$189$	− 51395.7i	− 0.104658i
$190$	161393.i	0.324341i
$191$	−12931.7	−0.0256490	−0.0128245	−	0.999918i	$-0.504082\pi$
$191$	−12931.7	−0.0256490	−0.0128245	+	0.999918i	$0.504082\pi$
$192$	−480446.	−0.940570
$193$	− 949823.i	− 1.83548i	−0.397184	−	0.917739i	$-0.630012\pi$
$193$	− 949823.i	− 1.83548i	0.397184	−	0.917739i	$-0.369988\pi$
$194$	−72885.8	−0.139040
$195$	0	0
$196$	168000.	0.312370
$197$	330461.i	0.606674i	0.952883	+	0.303337i	$0.0981008\pi$
$197$	330461.i	0.606674i	−0.952883	+	0.303337i	$0.901899\pi$
$198$	76788.2	0.139197
$199$	1.04928e6	1.87828	0.939140	−	0.343535i	$-0.111624\pi$
$199$	1.04928e6	1.87828	0.939140	+	0.343535i	$0.111624\pi$
$200$	− 568784.i	− 1.00548i
$201$	− 596679.i	− 1.04172i
$202$	− 344967.i	− 0.594839i
$203$	61634.9i	0.104975i
$204$	−233185.	−0.392307
$205$	−25397.9	−0.0422098
$206$	− 743837.i	− 1.22126i
$207$	−58391.6	−0.0947163
$208$	0	0
$209$	−606767.	−0.960851
$210$	− 12231.5i	− 0.0191395i
$211$	846030.	1.30822	0.654108	−	0.756401i	$-0.273044\pi$
$211$	846030.	1.30822	0.654108	+	0.756401i	$0.273044\pi$
$212$	−309542.	−0.473021
$213$	228077.i	0.344455i
$214$	− 99882.9i	− 0.149093i
$215$	− 147018.i	− 0.216907i
$216$	807742.i	1.17798i
$217$	29069.4	0.0419070
$218$	553023.	0.788138
$219$	177961.i	0.250735i
$220$	−42259.9	−0.0588670
$221$	0	0
$222$	−851540.	−1.15964
$223$	− 225529.i	− 0.303697i	−0.988404	−	0.151849i	$-0.951477\pi$
$223$	− 225529.i	− 0.303697i	0.988404	−	0.151849i	$-0.0485226\pi$
$224$	43866.5	0.0584134
$225$	−174453.	−0.229733
$226$	− 1.07899e6i	− 1.40523i
$227$	− 962314.i	− 1.23952i	−0.784793	−	0.619758i	$-0.787231\pi$
$227$	− 962314.i	− 1.23952i	0.784793	−	0.619758i	$-0.212769\pi$
$228$	− 304717.i	− 0.388203i
$229$	761024.i	0.958981i	0.877547	+	0.479490i	$0.159178\pi$
$229$	761024.i	0.958981i	−0.877547	+	0.479490i	$0.840822\pi$
$230$	−69779.0	−0.0869771
$231$	45984.8	0.0567002
$232$	− 968663.i	− 1.18155i
$233$	−71340.8	−0.0860890	−0.0430445	−	0.999073i	$-0.513706\pi$
$233$	−71340.8	−0.0860890	−0.0430445	+	0.999073i	$0.513706\pi$
$234$	0	0
$235$	341620.	0.403528
$236$	376078.i	0.439540i
$237$	268197.	0.310159
$238$	−100359.	−0.114846
$239$	764041.i	0.865211i	0.901583	+	0.432605i	$0.142406\pi$
$239$	764041.i	0.865211i	−0.901583	+	0.432605i	$0.857594\pi$
$240$	124925.i	0.139998i
$241$	− 329309.i	− 0.365226i	−0.983185	−	0.182613i	$-0.941545\pi$
$241$	− 329309.i	− 0.365226i	0.983185	−	0.182613i	$-0.0584555\pi$
$242$	− 408858.i	− 0.448781i
$243$	446152.	0.484693
$244$	−307819.	−0.330994
$245$	− 256861.i	− 0.273390i
$246$	−104123.	−0.109701
$247$	0	0
$248$	−456859.	−0.471686
$249$	300187.i	0.306827i
$250$	−434136.	−0.439314
$251$	−1.49318e6	−1.49598	−0.747992	−	0.663708i	$-0.768982\pi$
$251$	−1.49318e6	−1.49598	−0.747992	+	0.663708i	$0.768982\pi$
$252$	− 7643.38i	− 0.00758200i
$253$	− 262338.i	− 0.257667i
$254$	570286.i	0.554637i
$255$	356524.i	0.343351i
$256$	−882919.	−0.842017
$257$	1.22254e6	1.15460	0.577299	−	0.816533i	$-0.304107\pi$
$257$	1.22254e6	1.15460	0.577299	+	0.816533i	$0.304107\pi$
$258$	− 602726.i	− 0.563729i
$259$	168780.	0.156341
$260$	0	0
$261$	−297102.	−0.269963
$262$	− 739871.i	− 0.665890i
$263$	−747717.	−0.666573	−0.333287	−	0.942826i	$-0.608158\pi$
$263$	−747717.	−0.666573	−0.333287	+	0.942826i	$0.608158\pi$
$264$	−722704.	−0.638191
$265$	473269.i	0.413993i
$266$	− 131145.i	− 0.113644i
$267$	− 52389.6i	− 0.0449746i
$268$	− 445576.i	− 0.378952i
$269$	−1.65153e6	−1.39157	−0.695786	−	0.718249i	$-0.744944\pi$
$269$	−1.65153e6	−1.39157	−0.695786	+	0.718249i	$0.744944\pi$
$270$	296059.	0.247155
$271$	1.00669e6i	0.832667i	0.909212	+	0.416333i	$0.136685\pi$
$271$	1.00669e6i	0.832667i	−0.909212	+	0.416333i	$0.863315\pi$
$272$	1.02501e6	0.840055
$273$	0	0
$274$	1.08103e6	0.869880
$275$	− 783772.i	− 0.624968i
$276$	131745.	0.104103
$277$	2.15076e6	1.68419	0.842097	−	0.539327i	$-0.181321\pi$
$277$	2.15076e6	1.68419	0.842097	+	0.539327i	$0.181321\pi$
$278$	− 1.14148e6i	− 0.885844i
$279$	140124.i	0.107771i
$280$	− 38101.4i	− 0.0290433i
$281$	− 974647.i	− 0.736345i	−0.929758	−	0.368172i	$-0.879984\pi$
$281$	− 974647.i	− 0.736345i	0.929758	−	0.368172i	$-0.120016\pi$
$282$	1.40053e6	1.04875
$283$	−1.09326e6	−0.811439	−0.405719	−	0.913998i	$-0.632979\pi$
$283$	−1.09326e6	−0.811439	−0.405719	+	0.913998i	$0.632979\pi$
$284$	170319.i	0.125304i
$285$	−465891.	−0.339760
$286$	0	0
$287$	20637.8	0.0147897
$288$	211452.i	0.150221i
$289$	1.50543e6	1.06027
$290$	−355041.	−0.247904
$291$	− 210398.i	− 0.145649i
$292$	132894.i	0.0912114i
$293$	2.18586e6i	1.48749i	0.668465	+	0.743744i	$0.266952\pi$
$293$	2.18586e6i	1.48749i	−0.668465	+	0.743744i	$0.733048\pi$
$294$	− 1.05305e6i	− 0.710525i
$295$	574998.	0.384690
$296$	−2.65258e6	−1.75970
$297$	1.11305e6i	0.732190i
$298$	−1.07400e6	−0.700586
$299$	0	0
$300$	393608.	0.252500
$301$	119464.i	0.0760011i
$302$	−2.21681e6	−1.39866
$303$	995810.	0.623117
$304$	1.33945e6i	0.831269i
$305$	470634.i	0.289690i
$306$	− 483767.i	− 0.295347i
$307$	− 751336.i	− 0.454976i	−0.973781	−	0.227488i	$-0.926949\pi$
$307$	− 751336.i	− 0.454976i	0.973781	−	0.227488i	$-0.0730512\pi$
$308$	34339.6	0.0206262
$309$	2.14722e6	1.27932
$310$	167451.i	0.0989655i
$311$	1.26077e6	0.739154	0.369577	−	0.929200i	$-0.379503\pi$
$311$	1.26077e6	0.739154	0.369577	+	0.929200i	$0.379503\pi$
$312$	0	0
$313$	−514267.	−0.296707	−0.148354	−	0.988934i	$-0.547397\pi$
$313$	−514267.	−0.296707	−0.148354	+	0.988934i	$0.547397\pi$
$314$	274900.i	0.157344i
$315$	−11686.2	−0.00663585
$316$	200279.	0.112828
$317$	586709.i	0.327925i	0.986467	+	0.163963i	$0.0524276\pi$
$317$	586709.i	0.327925i	−0.986467	+	0.163963i	$0.947572\pi$
$318$	1.94025e6i	1.07595i
$319$	− 1.33480e6i	− 0.734410i
$320$	548546.i	0.299460i
$321$	288330.	0.156181
$322$	56701.1	0.0304756
$323$	3.82264e6i	2.03872i
$324$	−410809.	−0.217409
$325$	0	0
$326$	2.01705e6	1.05117
$327$	1.59640e6i	0.825606i
$328$	−324348.	−0.166466
$329$	−277594.	−0.141391
$330$	264891.i	0.133900i
$331$	3.72419e6i	1.86837i	0.356795	+	0.934183i	$0.383869\pi$
$331$	3.72419e6i	1.86837i	−0.356795	+	0.934183i	$0.616131\pi$
$332$	224168.i	0.111616i
$333$	813579.i	0.402059i
$334$	2.04366e6	1.00240
$335$	−681254.	−0.331663
$336$	− 101512.i	− 0.0490535i
$337$	−398907.	−0.191336	−0.0956680	−	0.995413i	$-0.530499\pi$
$337$	−398907.	−0.191336	−0.0956680	+	0.995413i	$0.530499\pi$
$338$	0	0
$339$	3.11470e6	1.47203
$340$	266238.i	0.124903i
$341$	−629541.	−0.293183
$342$	632166.	0.292258
$343$	419410.i	0.192488i
$344$	− 1.87751e6i	− 0.855433i
$345$	− 201430.i	− 0.0911120i
$346$	375028.i	0.168412i
$347$	989902.	0.441335	0.220668	−	0.975349i	$-0.429176\pi$
$347$	989902.	0.441335	0.220668	+	0.975349i	$0.429176\pi$
$348$	670331.	0.296716
$349$	− 135162.i	− 0.0594008i	−0.999559	−	0.0297004i	$-0.990545\pi$
$349$	− 135162.i	− 0.0594008i	0.999559	−	0.0297004i	$-0.00945533\pi$
$350$	169403.	0.0739180
$351$	0	0
$352$	−949995.	−0.408663
$353$	− 1.51107e6i	− 0.645427i	−0.946497	−	0.322714i	$-0.895405\pi$
$353$	− 1.51107e6i	− 0.645427i	0.946497	−	0.322714i	$-0.104595\pi$
$354$	2.35731e6	0.999789
$355$	260406.	0.109668
$356$	− 39122.5i	− 0.0163607i
$357$	− 289705.i	− 0.120305i
$358$	− 1.23914e6i	− 0.510989i
$359$	− 2.73777e6i	− 1.12114i	−0.828106	−	0.560571i	$-0.810582\pi$
$359$	− 2.73777e6i	− 1.12114i	0.828106	−	0.560571i	$-0.189418\pi$
$360$	183662.	0.0746901
$361$	−2.51916e6	−1.01739
$362$	3.52432e6i	1.41352i
$363$	1.18024e6	0.470116
$364$	0	0
$365$	203186.	0.0798293
$366$	1.92945e6i	0.752889i
$367$	167036.	0.0647359	0.0323679	−	0.999476i	$-0.489695\pi$
$367$	167036.	0.0647359	0.0323679	+	0.999476i	$0.489695\pi$
$368$	−579114.	−0.222918
$369$	99481.6i	0.0380344i
$370$	972241.i	0.369207i
$371$	− 384569.i	− 0.145057i
$372$	− 316154.i	− 0.118452i
$373$	3.00861e6	1.11968	0.559839	−	0.828601i	$-0.310863\pi$
$373$	3.00861e6	1.11968	0.559839	+	0.828601i	$0.310863\pi$
$374$	2.17343e6	0.803465
$375$	− 1.25321e6i	− 0.460199i
$376$	4.36271e6	1.59143
$377$	0	0
$378$	−240572.	−0.0865997
$379$	− 1.34991e6i	− 0.482733i	−0.970434	−	0.241366i	$-0.922404\pi$
$379$	− 1.34991e6i	− 0.482733i	0.970434	−	0.241366i	$-0.0775956\pi$
$380$	−347909.	−0.123596
$381$	−1.64623e6	−0.581004
$382$	60530.4i	0.0212234i
$383$	− 794657.i	− 0.276811i	−0.990376	−	0.138405i	$-0.955802\pi$
$383$	− 794657.i	− 0.276811i	0.990376	−	0.138405i	$-0.0441977\pi$
$384$	735839.i	0.254656i
$385$	− 52502.9i	− 0.0180523i
$386$	−4.44592e6	−1.51878
$387$	−575857.	−0.195451
$388$	− 157117.i	− 0.0529838i
$389$	2.17772e6	0.729671	0.364836	−	0.931072i	$-0.381125\pi$
$389$	2.17772e6	0.729671	0.364836	+	0.931072i	$0.381125\pi$
$390$	0	0
$391$	−1.65273e6	−0.546714
$392$	− 3.28028e6i	− 1.07819i
$393$	2.13577e6	0.697546
$394$	1.54682e6	0.501995
$395$	− 306213.i	− 0.0987486i
$396$	165529.i	0.0530440i
$397$	− 156276.i	− 0.0497639i	−0.999690	−	0.0248820i	$-0.992079\pi$
$397$	− 156276.i	− 0.0497639i	0.999690	−	0.0248820i	$-0.00792099\pi$
$398$	− 4.91148e6i	− 1.55419i
$399$	378574.	0.119047
$400$	−1.73019e6	−0.540684
$401$	− 5.89967e6i	− 1.83217i	−0.400981	−	0.916087i	$-0.631331\pi$
$401$	− 5.89967e6i	− 1.83217i	0.400981	−	0.916087i	$-0.368669\pi$
$402$	−2.79293e6	−0.861975
$403$	0	0
$404$	743631.	0.226675
$405$	628099.i	0.190279i
$406$	288500.	0.0868622
$407$	−3.65519e6	−1.09376
$408$	4.55305e6i	1.35410i
$409$	− 3.42439e6i	− 1.01222i	−0.862469	−	0.506110i	$-0.831083\pi$
$409$	− 3.42439e6i	− 1.01222i	0.862469	−	0.506110i	$-0.168917\pi$
$410$	118882.i	0.0349267i
$411$	3.12057e6i	0.911234i
$412$	1.60346e6	0.465387
$413$	−467233.	−0.134790
$414$	273319.i	0.0783735i
$415$	342737.	0.0976879
$416$	0	0
$417$	3.29510e6	0.927957
$418$	2.84015e6i	0.795061i
$419$	−6.68499e6	−1.86022	−0.930112	−	0.367275i	$-0.880291\pi$
$419$	−6.68499e6	−1.86022	−0.930112	+	0.367275i	$0.880291\pi$
$420$	26366.8	0.00729348
$421$	− 3.30508e6i	− 0.908816i	−0.890794	−	0.454408i	$-0.849851\pi$
$421$	− 3.30508e6i	− 0.908816i	0.890794	−	0.454408i	$-0.150149\pi$
$422$	− 3.96009e6i	− 1.08249i
$423$	− 1.33810e6i	− 0.363612i
$424$	6.04395e6i	1.63270i
$425$	−4.93777e6	−1.32605
$426$	1.06758e6	0.285021
$427$	− 382428.i	− 0.101503i
$428$	215313.	0.0568147
$429$	0	0
$430$	−688159.	−0.179481
$431$	− 980594.i	− 0.254270i	−0.991885	−	0.127135i	$-0.959422\pi$
$431$	− 980594.i	− 0.254270i	0.991885	−	0.127135i	$-0.0405782\pi$
$432$	2.45708e6	0.633446
$433$	−5.32853e6	−1.36580	−0.682900	−	0.730511i	$-0.739282\pi$
$433$	−5.32853e6	−1.36580	−0.682900	+	0.730511i	$0.739282\pi$
$434$	− 136068.i	− 0.0346761i
$435$	− 1.02489e6i	− 0.259689i
$436$	1.19213e6i	0.300336i
$437$	− 2.15972e6i	− 0.540996i
$438$	833000.	0.207472
$439$	2.09243e6	0.518191	0.259095	−	0.965852i	$-0.416576\pi$
$439$	2.09243e6	0.518191	0.259095	+	0.965852i	$0.416576\pi$
$440$	825144.i	0.203188i
$441$	−1.00610e6	−0.246346
$442$	0	0
$443$	−6.53140e6	−1.58124	−0.790619	−	0.612309i	$-0.790241\pi$
$443$	−6.53140e6	−1.58124	−0.790619	+	0.612309i	$0.790241\pi$
$444$	− 1.83563e6i	− 0.441903i
$445$	−59815.6	−0.0143191
$446$	−1.05566e6	−0.251296
$447$	− 3.10028e6i	− 0.733892i
$448$	− 445738.i	− 0.104926i
$449$	2.01357e6i	0.471359i	0.971831	+	0.235679i	$0.0757315\pi$
$449$	2.01357e6i	0.471359i	−0.971831	+	0.235679i	$0.924268\pi$
$450$	816580.i	0.190094i
$451$	−446944.	−0.103469
$452$	2.32593e6	0.535490
$453$	− 6.39923e6i	− 1.46515i
$454$	−4.50439e6	−1.02564
$455$	0	0
$456$	−5.94973e6	−1.33994
$457$	4.81967e6i	1.07951i	0.841822	+	0.539755i	$0.181483\pi$
$457$	4.81967e6i	1.07951i	−0.841822	+	0.539755i	$0.818517\pi$
$458$	3.56220e6	0.793513
$459$	7.01224e6	1.55355
$460$	− 150420.i	− 0.0331444i
$461$	3.19042e6i	0.699190i	0.936901	+	0.349595i	$0.113681\pi$
$461$	3.19042e6i	0.699190i	−0.936901	+	0.349595i	$0.886319\pi$
$462$	− 215245.i	− 0.0469168i
$463$	2.58922e6i	0.561327i	0.959806	+	0.280664i	$0.0905546\pi$
$463$	2.58922e6i	0.561327i	−0.959806	+	0.280664i	$0.909445\pi$
$464$	−2.94658e6	−0.635366
$465$	−483378.	−0.103670
$466$	333931.i	0.0712348i
$467$	−3.29476e6	−0.699087	−0.349543	−	0.936920i	$-0.613663\pi$
$467$	−3.29476e6	−0.699087	−0.349543	+	0.936920i	$0.613663\pi$
$468$	0	0
$469$	553575.	0.116210
$470$	− 1.59905e6i	− 0.333901i
$471$	−793547.	−0.164824
$472$	7.34310e6	1.51714
$473$	− 2.58717e6i	− 0.531706i
$474$	− 1.25538e6i	− 0.256642i
$475$	− 6.45247e6i	− 1.31218i
$476$	− 216340.i	− 0.0437642i
$477$	1.85376e6	0.373042
$478$	3.57632e6	0.715923
$479$	− 2.23321e6i	− 0.444725i	−0.974964	−	0.222362i	$-0.928623\pi$
$479$	− 2.23321e6i	− 0.444725i	0.974964	−	0.222362i	$-0.0713768\pi$
$480$	−729431.	−0.144504
$481$	0	0
$482$	−1.54143e6	−0.302208
$483$	163678.i	0.0319244i
$484$	881357.	0.171017
$485$	−240221.	−0.0463720
$486$	− 2.08834e6i	− 0.401062i
$487$	− 871932.i	− 0.166594i	−0.996525	−	0.0832972i	$-0.973455\pi$
$487$	− 871932.i	− 0.166594i	0.996525	−	0.0832972i	$-0.0265451\pi$
$488$	6.01031e6i	1.14248i
$489$	5.82259e6i	1.10114i
$490$	−1.20231e6	−0.226218
$491$	−4.42227e6	−0.827830	−0.413915	−	0.910316i	$-0.635839\pi$
$491$	−4.42227e6	−0.827830	−0.413915	+	0.910316i	$0.635839\pi$
$492$	− 224454.i	− 0.0418037i
$493$	−8.40924e6	−1.55826
$494$	0	0
$495$	253082.	0.0464247
$496$	1.38972e6i	0.253643i
$497$	−211601.	−0.0384261
$498$	1.40511e6	0.253886
$499$	− 6.42088e6i	− 1.15436i	−0.816615	−	0.577182i	$-0.804152\pi$
$499$	− 6.42088e6i	− 1.15436i	0.816615	−	0.577182i	$-0.195848\pi$
$500$	− 935847.i	− 0.167409i
$501$	5.89940e6i	1.05006i
$502$	6.98925e6i	1.23786i
$503$	−4.18617e6	−0.737729	−0.368865	−	0.929483i	$-0.620253\pi$
$503$	−4.18617e6	−0.737729	−0.368865	+	0.929483i	$0.620253\pi$
$504$	−149240.	−0.0261704
$505$	− 1.13696e6i	− 0.198389i
$506$	−1.22795e6	−0.213208
$507$	0	0
$508$	−1.22934e6	−0.211355
$509$	− 6.78875e6i	− 1.16144i	−0.814105	−	0.580718i	$-0.802772\pi$
$509$	− 6.78875e6i	− 1.16144i	0.814105	−	0.580718i	$-0.197228\pi$
$510$	1.66882e6	0.284108
$511$	−165105.	−0.0279711
$512$	5.87543e6i	0.990523i
$513$	9.16330e6i	1.53730i
$514$	− 5.72246e6i	− 0.955378i
$515$	− 2.45158e6i	− 0.407312i
$516$	1.29927e6	0.214820
$517$	6.01172e6	0.989173
$518$	− 790025.i	− 0.129365i
$519$	−1.08258e6	−0.176418
$520$	0	0
$521$	4.90522e6	0.791707	0.395853	−	0.918314i	$-0.370449\pi$
$521$	4.90522e6	0.791707	0.395853	+	0.918314i	$0.370449\pi$
$522$	1.39067e6i	0.223382i
$523$	765448.	0.122366	0.0611831	−	0.998127i	$-0.480513\pi$
$523$	765448.	0.122366	0.0611831	+	0.998127i	$0.480513\pi$
$524$	1.59491e6	0.253751
$525$	489011.i	0.0774320i
$526$	3.49991e6i	0.551559i
$527$	3.96612e6i	0.622070i
$528$	2.19840e6i	0.343180i
$529$	−5.50258e6	−0.854923
$530$	2.21527e6	0.342561
$531$	− 2.25222e6i	− 0.346637i
$532$	282704.	0.0433065
$533$	0	0
$534$	−245225.	−0.0372145
$535$	− 329199.i	− 0.0497249i
$536$	−8.70007e6	−1.30801
$537$	3.57699e6	0.535281
$538$	7.73047e6i	1.15146i
$539$	− 4.52015e6i	− 0.670164i
$540$	638202.i	0.0941833i
$541$	− 1.15421e6i	− 0.169548i	−0.996400	−	0.0847741i	$-0.972983\pi$
$541$	− 1.15421e6i	− 0.169548i	0.996400	−	0.0847741i	$-0.0270169\pi$
$542$	4.71209e6	0.688994
$543$	−1.01736e7	−1.48072
$544$	5.98498e6i	0.867093i
$545$	1.82268e6	0.262857
$546$	0	0
$547$	−7.71012e6	−1.10178	−0.550888	−	0.834580i	$-0.685711\pi$
$547$	−7.71012e6	−1.10178	−0.550888	+	0.834580i	$0.685711\pi$
$548$	2.33032e6i	0.331485i
$549$	1.84344e6	0.261034
$550$	−3.66867e6	−0.517133
$551$	− 1.09888e7i	− 1.54196i
$552$	− 2.57239e6i	− 0.359326i
$553$	248823.i	0.0346001i
$554$	− 1.00672e7i	− 1.39359i
$555$	−2.80655e6	−0.386759
$556$	2.46065e6	0.337569
$557$	2.04843e6i	0.279759i	0.990169	+	0.139879i	$0.0446715\pi$
$557$	2.04843e6i	0.279759i	−0.990169	+	0.139879i	$0.955328\pi$
$558$	655894.	0.0891760
$559$	0	0
$560$	−115901.	−0.0156177
$561$	6.27400e6i	0.841661i
$562$	−4.56212e6	−0.609292
$563$	−1.21731e7	−1.61856	−0.809282	−	0.587421i	$-0.800143\pi$
$563$	−1.21731e7	−1.61856	−0.809282	+	0.587421i	$0.800143\pi$
$564$	3.01907e6i	0.399646i
$565$	− 3.55619e6i	− 0.468667i
$566$	5.11730e6i	0.671429i
$567$	− 510381.i	− 0.0666710i
$568$	3.32555e6	0.432507
$569$	3.14525e6	0.407262	0.203631	−	0.979048i	$-0.434726\pi$
$569$	3.14525e6	0.407262	0.203631	+	0.979048i	$0.434726\pi$
$570$	2.18074e6i	0.281136i
$571$	8.24961e6	1.05887	0.529436	−	0.848350i	$-0.322404\pi$
$571$	8.24961e6	1.05887	0.529436	+	0.848350i	$0.322404\pi$
$572$	0	0
$573$	−174732.	−0.0222324
$574$	− 96601.5i	− 0.0122378i
$575$	2.78975e6	0.351881
$576$	2.14862e6	0.269838
$577$	4.39359e6i	0.549389i	0.961531	+	0.274695i	$0.0885768\pi$
$577$	4.39359e6i	0.549389i	−0.961531	+	0.274695i	$0.911423\pi$
$578$	− 7.04659e6i	− 0.877323i
$579$	− 1.28340e7i	− 1.59098i
$580$	− 765347.i	− 0.0944688i
$581$	−278502.	−0.0342285
$582$	−984829.	−0.120518
$583$	8.32843e6i	1.01483i
$584$	2.59482e6	0.314830
$585$	0	0
$586$	1.02316e7	1.23083
$587$	− 9.36277e6i	− 1.12153i	−0.827977	−	0.560763i	$-0.810508\pi$
$587$	− 9.36277e6i	− 1.12153i	0.827977	−	0.560763i	$-0.189492\pi$
$588$	2.27001e6	0.270760
$589$	−5.18276e6	−0.615563
$590$	− 2.69145e6i	− 0.318314i
$591$	4.46518e6i	0.525860i
$592$	8.06889e6i	0.946258i
$593$	− 1.54425e7i	− 1.80336i	−0.432408	−	0.901678i	$-0.642336\pi$
$593$	− 1.54425e7i	− 1.80336i	0.432408	−	0.901678i	$-0.357664\pi$
$594$	5.20996e6	0.605854
$595$	−330769.	−0.0383030
$596$	− 2.31516e6i	− 0.266972i
$597$	1.41779e7	1.62808
$598$	0	0
$599$	1.26786e7	1.44379	0.721897	−	0.692001i	$-0.243271\pi$
$599$	1.26786e7	1.44379	0.721897	+	0.692001i	$0.243271\pi$
$600$	− 7.68538e6i	− 0.871540i
$601$	6.71074e6	0.757851	0.378926	−	0.925427i	$-0.376294\pi$
$601$	6.71074e6	0.757851	0.378926	+	0.925427i	$0.376294\pi$
$602$	559185.	0.0628875
$603$	2.66842e6i	0.298856i
$604$	− 4.77868e6i	− 0.532986i
$605$	− 1.34754e6i	− 0.149676i
$606$	− 4.66118e6i	− 0.515602i
$607$	3.00733e6	0.331291	0.165646	−	0.986185i	$-0.447029\pi$
$607$	3.00733e6	0.331291	0.165646	+	0.986185i	$0.447029\pi$
$608$	−7.82093e6	−0.858024
$609$	832807.i	0.0909916i
$610$	2.20294e6	0.239706
$611$	0	0
$612$	1.04283e6	0.112548
$613$	1.11037e7i	1.19349i	0.802432	+	0.596743i	$0.203539\pi$
$613$	1.11037e7i	1.19349i	−0.802432	+	0.596743i	$0.796461\pi$
$614$	−3.51685e6	−0.376472
$615$	−343175.	−0.0365871
$616$	− 670496.i	− 0.0711942i
$617$	9.36760e6i	0.990639i	0.868711	+	0.495319i	$0.164949\pi$
$617$	9.36760e6i	0.990639i	−0.868711	+	0.495319i	$0.835051\pi$
$618$	− 1.00507e7i	− 1.05858i
$619$	6.92281e6i	0.726200i	0.931750	+	0.363100i	$0.118282\pi$
$619$	6.92281e6i	0.726200i	−0.931750	+	0.363100i	$0.881718\pi$
$620$	−360967.	−0.0377128
$621$	−3.96178e6	−0.412251
$622$	− 5.90140e6i	− 0.611617i
$623$	48605.0	0.00501720
$624$	0	0
$625$	7.59103e6	0.777322
$626$	2.40718e6i	0.245512i
$627$	−8.19860e6	−0.832858
$628$	−592589.	−0.0599590
$629$	2.30278e7i	2.32073i
$630$	54700.7i	0.00549087i
$631$	− 6.70011e6i	− 0.669898i	−0.942236	−	0.334949i	$-0.891281\pi$
$631$	− 6.70011e6i	− 0.669898i	0.942236	−	0.334949i	$-0.108719\pi$
$632$	− 3.91054e6i	− 0.389443i
$633$	1.14315e7	1.13395
$634$	2.74626e6	0.271343
$635$	1.87958e6i	0.184981i
$636$	−4.18252e6	−0.410010
$637$	0	0
$638$	−6.24790e6	−0.607691
$639$	− 1.01999e6i	− 0.0988197i
$640$	840140.	0.0810777
$641$	−7.49696e6	−0.720677	−0.360338	−	0.932822i	$-0.617339\pi$
$641$	−7.49696e6	−0.720677	−0.360338	+	0.932822i	$0.617339\pi$
$642$	− 1.34961e6i	− 0.129232i
$643$	1.32735e7i	1.26607i	0.774122	+	0.633036i	$0.218192\pi$
$643$	1.32735e7i	1.26607i	−0.774122	+	0.633036i	$0.781808\pi$
$644$	122228.i	0.0116133i
$645$	− 1.98649e6i	− 0.188013i
$646$	1.78930e7	1.68695
$647$	−4.28044e6	−0.402001	−0.201001	−	0.979591i	$-0.564419\pi$
$647$	−4.28044e6	−0.402001	−0.201001	+	0.979591i	$0.564419\pi$
$648$	8.02124e6i	0.750419i
$649$	1.01186e7	0.942996
$650$	0	0
$651$	392784.	0.0363246
$652$	4.34807e6i	0.400569i
$653$	1.36414e7	1.25192	0.625958	−	0.779856i	$-0.284708\pi$
$653$	1.36414e7	1.25192	0.625958	+	0.779856i	$0.284708\pi$
$654$	7.47243e6	0.683152
$655$	− 2.43850e6i	− 0.222085i
$656$	986635.i	0.0895153i
$657$	− 795866.i	− 0.0719327i
$658$	1.29936e6i	0.116994i
$659$	−1.12829e7	−1.01206	−0.506029	−	0.862516i	$-0.668887\pi$
$659$	−1.12829e7	−1.01206	−0.506029	+	0.862516i	$0.668887\pi$
$660$	−571014.	−0.0510254
$661$	8.93032e6i	0.794993i	0.917604	+	0.397497i	$0.130121\pi$
$661$	8.93032e6i	0.794993i	−0.917604	+	0.397497i	$0.869879\pi$
$662$	1.74322e7	1.54599
$663$	0	0
$664$	4.37698e6	0.385260
$665$	− 432235.i	− 0.0379023i
$666$	3.80820e6	0.332685
$667$	4.75106e6	0.413501
$668$	4.40543e6i	0.381986i
$669$	− 3.04734e6i	− 0.263242i
$670$	3.18881e6i	0.274436i
$671$	8.28207e6i	0.710121i
$672$	592722.	0.0506323
$673$	575561.	0.0489840	0.0244920	−	0.999700i	$-0.492203\pi$
$673$	575561.	0.0489840	0.0244920	+	0.999700i	$0.492203\pi$
$674$	1.86720e6i	0.158322i
$675$	−1.18364e7	−0.999908
$676$	0	0
$677$	1.25361e7	1.05121	0.525607	−	0.850727i	$-0.323838\pi$
$677$	1.25361e7	1.05121	0.525607	+	0.850727i	$0.323838\pi$
$678$	− 1.45793e7i	− 1.21804i
$679$	195199.	0.0162481
$680$	5.19842e6	0.431121
$681$	− 1.30027e7i	− 1.07440i
$682$	2.94675e6i	0.242595i
$683$	723570.i	0.0593511i	0.999560	+	0.0296756i	$0.00944741\pi$
$683$	723570.i	0.0593511i	−0.999560	+	0.0296756i	$0.990553\pi$
$684$	1.36273e6i	0.111371i
$685$	3.56290e6	0.290120
$686$	1.96317e6	0.159275
$687$	1.02829e7i	0.831237i
$688$	−5.71121e6	−0.459999
$689$	0	0
$690$	−942850.	−0.0753911
$691$	8.17633e6i	0.651424i	0.945469	+	0.325712i	$0.105604\pi$
$691$	8.17633e6i	0.651424i	−0.945469	+	0.325712i	$0.894396\pi$
$692$	−808431.	−0.0641767
$693$	−205650.	−0.0162666
$694$	− 4.63353e6i	− 0.365185i
$695$	− 3.76216e6i	− 0.295444i
$696$	− 1.30885e7i	− 1.02416i
$697$	2.81575e6i	0.219539i
$698$	−632668.	−0.0491515
$699$	−963953.	−0.0746213
$700$	365174.i	0.0281679i
$701$	−1.21420e6	−0.0933246	−0.0466623	−	0.998911i	$-0.514858\pi$
$701$	−1.21420e6	−0.0933246	−0.0466623	+	0.998911i	$0.514858\pi$
$702$	0	0
$703$	−3.00917e7	−2.29646
$704$	9.65314e6i	0.734069i
$705$	4.61595e6	0.349775
$706$	−7.07300e6	−0.534062
$707$	923873.i	0.0695126i
$708$	5.08155e6i	0.380990i
$709$	1.29331e7i	0.966244i	0.875553	+	0.483122i	$0.160497\pi$
$709$	1.29331e7i	0.966244i	−0.875553	+	0.483122i	$0.839503\pi$
$710$	− 1.21890e6i	− 0.0907453i
$711$	−1.19941e6	−0.0889805
$712$	−763884.	−0.0564713
$713$	− 2.24078e6i	− 0.165073i
$714$	−1.35605e6	−0.0995474
$715$	0	0
$716$	2.67115e6	0.194722
$717$	1.03237e7i	0.749958i
$718$	−1.28149e7	−0.927695
$719$	1.91637e7	1.38248	0.691238	−	0.722627i	$-0.257066\pi$
$719$	1.91637e7	1.38248	0.691238	+	0.722627i	$0.257066\pi$
$720$	− 558683.i	− 0.0401637i
$721$	1.99210e6i	0.142716i
$722$	1.17917e7i	0.841847i
$723$	− 4.44961e6i	− 0.316575i
$724$	−7.59721e6	−0.538652
$725$	1.41945e7	1.00294
$726$	− 5.52447e6i	− 0.388999i
$727$	−1.68355e7	−1.18138	−0.590690	−	0.806898i	$-0.701145\pi$
$727$	−1.68355e7	−1.18138	−0.590690	+	0.806898i	$0.701145\pi$
$728$	0	0
$729$	1.59218e7	1.10962
$730$	− 951073.i	− 0.0660551i
$731$	−1.62992e7	−1.12817
$732$	−4.15923e6	−0.286903
$733$	1.19840e7i	0.823840i	0.911220	+	0.411920i	$0.135142\pi$
$733$	1.19840e7i	0.823840i	−0.911220	+	0.411920i	$0.864858\pi$
$734$	− 781861.i	− 0.0535660i
$735$	− 3.47069e6i	− 0.236972i
$736$	− 3.38141e6i	− 0.230093i
$737$	−1.19885e7	−0.813011
$738$	465653.	0.0314718
$739$	− 3.40243e6i	− 0.229181i	−0.993413	−	0.114590i	$-0.963444\pi$
$739$	− 3.40243e6i	− 0.229181i	0.993413	−	0.114590i	$-0.0365556\pi$
$740$	−2.09582e6	−0.140694
$741$	0	0
$742$	−1.80009e6	−0.120028
$743$	− 2.11492e7i	− 1.40547i	−0.711451	−	0.702736i	$-0.751961\pi$
$743$	− 2.11492e7i	− 1.40547i	0.711451	−	0.702736i	$-0.248039\pi$
$744$	−6.17305e6	−0.408853
$745$	−3.53973e6	−0.233657
$746$	− 1.40827e7i	− 0.926484i
$747$	− 1.34248e6i	− 0.0880248i
$748$	4.68517e6i	0.306176i
$749$	267501.i	0.0174229i
$750$	−5.86602e6	−0.380794
$751$	−7.78493e6	−0.503680	−0.251840	−	0.967769i	$-0.581036\pi$
$751$	−7.78493e6	−0.503680	−0.251840	+	0.967769i	$0.581036\pi$
$752$	− 1.32710e7i	− 0.855771i
$753$	−2.01757e7	−1.29671
$754$	0	0
$755$	−7.30628e6	−0.466476
$756$	− 518591.i	− 0.0330005i
$757$	1.27698e7	0.809927	0.404964	−	0.914333i	$-0.367284\pi$
$757$	1.27698e7	0.809927	0.404964	+	0.914333i	$0.367284\pi$
$758$	−6.31865e6	−0.399440
$759$	− 3.54469e6i	− 0.223344i
$760$	6.79307e6i	0.426611i
$761$	2.33534e7i	1.46180i	0.682482	+	0.730902i	$0.260900\pi$
$761$	2.33534e7i	1.46180i	−0.682482	+	0.730902i	$0.739100\pi$
$762$	7.70568e6i	0.480755i
$763$	−1.48108e6	−0.0921015
$764$	−130483.	−0.00808760
$765$	− 1.59442e6i	− 0.0985031i
$766$	−3.71963e6	−0.229048
$767$	0	0
$768$	−1.19300e7	−0.729854
$769$	− 1.37985e7i	− 0.841424i	−0.907194	−	0.420712i	$-0.861780\pi$
$769$	− 1.37985e7i	− 0.841424i	0.907194	−	0.420712i	$-0.138220\pi$
$770$	−245755.	−0.0149374
$771$	1.65189e7	1.00080
$772$	− 9.58388e6i	− 0.578760i
$773$	− 3.68424e6i	− 0.221768i	−0.993833	−	0.110884i	$-0.964632\pi$
$773$	− 3.68424e6i	− 0.221768i	0.993833	−	0.110884i	$-0.0353682\pi$
$774$	2.69547e6i	0.161727i
$775$	− 6.69466e6i	− 0.400382i
$776$	−3.06777e6	−0.182881
$777$	2.28055e6	0.135515
$778$	− 1.01934e7i	− 0.603770i
$779$	−3.67951e6	−0.217243
$780$	0	0
$781$	4.58254e6	0.268830
$782$	7.73609e6i	0.452381i
$783$	−2.01579e7	−1.17501
$784$	−9.97830e6	−0.579784
$785$	906028.i	0.0524768i
$786$	− 9.99710e6i	− 0.577188i
$787$	− 8.19567e6i	− 0.471680i	−0.971792	−	0.235840i	$-0.924216\pi$
$787$	− 8.19567e6i	− 0.471680i	0.971792	−	0.235840i	$-0.0757841\pi$
$788$	3.33442e6i	0.191295i
$789$	−1.01031e7	−0.577780
$790$	−1.43332e6	−0.0817100
$791$	2.88970e6i	0.164214i
$792$	3.23203e6	0.183089
$793$	0	0
$794$	−731493.	−0.0411774
$795$	6.39478e6i	0.358846i
$796$	1.05875e7	0.592256
$797$	−4.58294e6	−0.255563	−0.127782	−	0.991802i	$-0.540786\pi$
$797$	−4.58294e6	−0.255563	−0.127782	+	0.991802i	$0.540786\pi$
$798$	− 1.77203e6i	− 0.0985061i
$799$	− 3.78739e7i	− 2.09881i
$800$	− 1.01024e7i	− 0.558086i
$801$	234293.i	0.0129026i
$802$	−2.76151e7	−1.51604
$803$	3.57561e6	0.195687
$804$	− 6.02059e6i	− 0.328473i
$805$	186878.	0.0101641
$806$	0	0
$807$	−2.23154e7	−1.20620
$808$	− 1.45197e7i	− 0.782402i
$809$	188019.	0.0101002	0.00505012	−	0.999987i	$-0.498392\pi$
$809$	188019.	0.0101002	0.00505012	+	0.999987i	$0.498392\pi$
$810$	2.94000e6	0.157447
$811$	1.22901e6i	0.0656153i	0.999462	+	0.0328076i	$0.0104449\pi$
$811$	1.22901e6i	0.0656153i	−0.999462	+	0.0328076i	$0.989555\pi$
$812$	621907.i	0.0331005i
$813$	1.36023e7i	0.721749i
$814$	1.71092e7i	0.905041i
$815$	6.64791e6	0.350583
$816$	1.38499e7	0.728153
$817$	− 2.12991e7i	− 1.11636i
$818$	−1.60289e7	−0.837566
$819$	0	0
$820$	−256269.	−0.0133095
$821$	2.66298e7i	1.37883i	0.724367	+	0.689414i	$0.242132\pi$
$821$	2.66298e7i	1.37883i	−0.724367	+	0.689414i	$0.757868\pi$
$822$	1.46068e7	0.754005
$823$	5.69729e6	0.293203	0.146602	−	0.989196i	$-0.453166\pi$
$823$	5.69729e6	0.293203	0.146602	+	0.989196i	$0.453166\pi$
$824$	− 3.13082e7i	− 1.60635i
$825$	− 1.05903e7i	− 0.541717i
$826$	2.18702e6i	0.111533i
$827$	1.66404e7i	0.846059i	0.906116	+	0.423029i	$0.139033\pi$
$827$	1.66404e7i	0.846059i	−0.906116	+	0.423029i	$0.860967\pi$
$828$	−589182.	−0.0298658
$829$	1.54371e7	0.780150	0.390075	−	0.920783i	$-0.372449\pi$
$829$	1.54371e7	0.780150	0.390075	+	0.920783i	$0.372449\pi$
$830$	− 1.60428e6i	− 0.0808323i
$831$	2.90609e7	1.45985
$832$	0	0
$833$	−2.84770e7	−1.42194
$834$	− 1.54237e7i	− 0.767843i
$835$	6.73561e6	0.334319
$836$	−6.12238e6	−0.302974
$837$	9.50723e6i	0.469073i
$838$	3.12910e7i	1.53925i
$839$	− 3.24533e7i	− 1.59167i	−0.605512	−	0.795836i	$-0.707032\pi$
$839$	− 3.24533e7i	− 1.59167i	0.605512	−	0.795836i	$-0.292968\pi$
$840$	− 514824.i	− 0.0251745i
$841$	3.66266e6	0.178569
$842$	−1.54704e7	−0.752005
$843$	− 1.31694e7i	− 0.638258i
$844$	8.53659e6	0.412504
$845$	0	0
$846$	−6.26337e6	−0.300872
$847$	1.09498e6i	0.0524443i
$848$	1.83851e7	0.877965
$849$	−1.47720e7	−0.703349
$850$	2.31127e7i	1.09724i
$851$	− 1.30103e7i	− 0.615832i
$852$	2.30134e6i	0.108613i
$853$	− 6.03451e6i	− 0.283968i	−0.989869	−	0.141984i	$-0.954652\pi$
$853$	− 6.03451e6i	− 0.283968i	0.989869	−	0.141984i	$-0.0453482\pi$
$854$	−1.79007e6	−0.0839895
$855$	2.08352e6	0.0974728
$856$	− 4.20409e6i	− 0.196104i
$857$	−5.81976e6	−0.270678	−0.135339	−	0.990799i	$-0.543212\pi$
$857$	−5.81976e6	−0.270678	−0.135339	+	0.990799i	$0.543212\pi$
$858$	0	0
$859$	−3.59792e6	−0.166367	−0.0831837	−	0.996534i	$-0.526509\pi$
$859$	−3.59792e6	−0.166367	−0.0831837	+	0.996534i	$0.526509\pi$
$860$	− 1.48343e6i	− 0.0683946i
$861$	278858.	0.0128196
$862$	−4.58996e6	−0.210397
$863$	− 1.60627e6i	− 0.0734163i	−0.999326	−	0.0367082i	$-0.988313\pi$
$863$	− 1.60627e6i	− 0.0734163i	0.999326	−	0.0367082i	$-0.0116872\pi$
$864$	1.43467e7i	0.653833i
$865$	1.23604e6i	0.0561682i
$866$	2.49417e7i	1.13014i
$867$	2.03413e7	0.919031
$868$	293315.	0.0132140
$869$	− 5.38864e6i	− 0.242064i
$870$	−4.79730e6	−0.214881
$871$	0	0
$872$	2.32769e7	1.03665
$873$	940926.i	0.0417850i
$874$	−1.01092e7	−0.447650
$875$	1.16268e6	0.0513381
$876$	1.79566e6i	0.0790613i
$877$	− 3.41995e7i	− 1.50148i	−0.660597	−	0.750741i	$-0.729697\pi$
$877$	− 3.41995e7i	− 1.50148i	0.660597	−	0.750741i	$-0.270303\pi$
$878$	− 9.79423e6i	− 0.428780i
$879$	2.95352e7i	1.28934i
$880$	2.51001e6	0.109262
$881$	1.24162e7	0.538952	0.269476	−	0.963007i	$-0.413149\pi$
$881$	1.24162e7	0.538952	0.269476	+	0.963007i	$0.413149\pi$
$882$	4.70936e6i	0.203841i
$883$	−6.43332e6	−0.277673	−0.138836	−	0.990315i	$-0.544336\pi$
$883$	−6.43332e6	−0.277673	−0.138836	+	0.990315i	$0.544336\pi$
$884$	0	0
$885$	7.76934e6	0.333446
$886$	3.05721e7i	1.30840i
$887$	2.29845e6	0.0980901	0.0490450	−	0.998797i	$-0.484382\pi$
$887$	2.29845e6	0.0980901	0.0490450	+	0.998797i	$0.484382\pi$
$888$	−3.58415e7	−1.52529
$889$	− 1.52731e6i	− 0.0648146i
$890$	279984.i	0.0118484i
$891$	1.10531e7i	0.466433i
$892$	− 2.27563e6i	− 0.0957613i
$893$	4.94921e7	2.07686
$894$	−1.45118e7	−0.607263
$895$	− 4.08400e6i	− 0.170423i
$896$	−682682.	−0.0284085
$897$	0	0
$898$	9.42512e6	0.390028
$899$	− 1.14013e7i	− 0.470495i
$900$	−1.76027e6	−0.0724389
$901$	5.24692e7	2.15324
$902$	2.09205e6i	0.0856162i
$903$	1.61419e6i	0.0658771i
$904$	− 4.54149e7i	− 1.84832i
$905$	1.16156e7i	0.471434i
$906$	−2.99534e7	−1.21235
$907$	−2.14917e7	−0.867466	−0.433733	−	0.901041i	$-0.642804\pi$
$907$	−2.14917e7	−0.867466	−0.433733	+	0.901041i	$0.642804\pi$
$908$	− 9.70992e6i	− 0.390842i
$909$	−4.45339e6	−0.178764
$910$	0	0
$911$	−5.71982e6	−0.228342	−0.114171	−	0.993461i	$-0.536421\pi$
$911$	−5.71982e6	−0.228342	−0.114171	+	0.993461i	$0.536421\pi$
$912$	1.80985e7i	0.720537i
$913$	6.03138e6	0.239464
$914$	2.25599e7	0.893246
$915$	6.35918e6i	0.251101i
$916$	7.67887e6i	0.302384i
$917$	1.98148e6i	0.0778157i
$918$	− 3.28228e7i	− 1.28549i
$919$	960581.	0.0375185	0.0187592	−	0.999824i	$-0.494028\pi$
$919$	960581.	0.0375185	0.0187592	+	0.999824i	$0.494028\pi$
$920$	−2.93701e6	−0.114403
$921$	− 1.01520e7i	− 0.394369i
$922$	1.49337e7	0.578548
$923$	0	0
$924$	463995.	0.0178786
$925$	− 3.88700e7i	− 1.49369i
$926$	1.21196e7	0.464473
$927$	−9.60264e6	−0.367021
$928$	− 1.72049e7i	− 0.655815i
$929$	3.06213e7i	1.16408i	0.813159	+	0.582042i	$0.197746\pi$
$929$	3.06213e7i	1.16408i	−0.813159	+	0.582042i	$0.802254\pi$
$930$	2.26259e6i	0.0857825i
$931$	− 3.72126e7i	− 1.40707i
$932$	−719841.	−0.0271454
$933$	1.70355e7	0.640693
$934$	1.54221e7i	0.578463i
$935$	7.16331e6	0.267969
$936$	0	0
$937$	4.23178e7	1.57462	0.787308	−	0.616560i	$-0.211474\pi$
$937$	4.23178e7	1.57462	0.787308	+	0.616560i	$0.211474\pi$
$938$	− 2.59117e6i	− 0.0961587i
$939$	−6.94875e6	−0.257183
$940$	3.44701e6	0.127240
$941$	4.23161e7i	1.55787i	0.627104	+	0.778935i	$0.284240\pi$
$941$	4.23161e7i	1.55787i	−0.627104	+	0.778935i	$0.715760\pi$
$942$	3.71443e6i	0.136384i
$943$	− 1.59085e6i	− 0.0582572i
$944$	− 2.23370e7i	− 0.815822i
$945$	−792891.	−0.0288824
$946$	−1.21100e7	−0.439963
$947$	4.30217e7i	1.55888i	0.626478	+	0.779439i	$0.284496\pi$
$947$	4.30217e7i	1.55888i	−0.626478	+	0.779439i	$0.715504\pi$
$948$	2.70616e6	0.0977986
$949$	0	0
$950$	−3.02027e7	−1.08577
$951$	7.92759e6i	0.284243i
$952$	−4.22414e6	−0.151059
$953$	−1.86099e7	−0.663761	−0.331881	−	0.943321i	$-0.607683\pi$
$953$	−1.86099e7	−0.663761	−0.331881	+	0.943321i	$0.607683\pi$
$954$	− 8.67706e6i	− 0.308675i
$955$	199499.i	0.00707836i
$956$	7.70931e6i	0.272817i
$957$	− 1.80357e7i	− 0.636580i
$958$	−1.04532e7	−0.367990
$959$	−2.89515e6	−0.101654
$960$	7.41193e6i	0.259569i
$961$	2.32519e7	0.812174
$962$	0	0
$963$	−1.28945e6	−0.0448062
$964$	− 3.32279e6i	− 0.115162i
$965$	−1.46531e7	−0.506537
$966$	766143.	0.0264160
$967$	− 4.43067e7i	− 1.52371i	−0.647745	−	0.761857i	$-0.724288\pi$
$967$	− 4.43067e7i	− 1.52371i	0.647745	−	0.761857i	$-0.275712\pi$
$968$	− 1.72089e7i	− 0.590289i
$969$	5.16513e7i	1.76714i
$970$	1.12442e6i	0.0383707i
$971$	2.29703e7	0.781842	0.390921	−	0.920424i	$-0.372157\pi$
$971$	2.29703e7	0.781842	0.390921	+	0.920424i	$0.372157\pi$
$972$	4.50175e6	0.152833
$973$	3.05706e6i	0.103519i
$974$	−4.08133e6	−0.137849
$975$	0	0
$976$	1.82828e7	0.614353
$977$	3.96457e7i	1.32880i	0.747378	+	0.664400i	$0.231313\pi$
$977$	3.96457e7i	1.32880i	−0.747378	+	0.664400i	$0.768687\pi$
$978$	2.72543e7	0.911146
$979$	−1.05262e6	−0.0351005
$980$	− 2.59177e6i	− 0.0862047i
$981$	− 7.13932e6i	− 0.236856i
$982$	2.06997e7i	0.684992i
$983$	− 2.31666e6i	− 0.0764677i	−0.999269	−	0.0382339i	$-0.987827\pi$
$983$	− 2.31666e6i	− 0.0764677i	0.999269	−	0.0382339i	$-0.0121732\pi$
$984$	−4.38257e6	−0.144292
$985$	5.09809e6	0.167424
$986$	3.93619e7i	1.28939i
$987$	−3.75084e6	−0.122556
$988$	0	0
$989$	9.20874e6	0.299371
$990$	− 1.18463e6i	− 0.0384143i
$991$	−5.55999e7	−1.79841	−0.899207	−	0.437523i	$-0.855856\pi$
$991$	−5.55999e7	−1.79841	−0.899207	+	0.437523i	$0.855856\pi$
$992$	−8.11448e6	−0.261807
$993$	5.03211e7i	1.61948i
$994$	990459.i	0.0317959i
$995$	− 1.61875e7i	− 0.518349i
$996$	3.02894e6i	0.0967481i
$997$	7.03932e6	0.224281	0.112141	−	0.993692i	$-0.464229\pi$
$997$	7.03932e6	0.224281	0.112141	+	0.993692i	$0.464229\pi$
$998$	−3.00548e7	−0.955185
$999$	5.52001e7i	1.74995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.6.b.b.168.3		6
13.5	odd	4		169.6.a.b.1.2		3
13.8	odd	4		13.6.a.b.1.2	✓	3
13.12	even	2	inner	169.6.b.b.168.4		6
39.8	even	4		117.6.a.d.1.2		3
52.47	even	4		208.6.a.j.1.1		3
65.8	even	4		325.6.b.c.274.3		6
65.34	odd	4		325.6.a.c.1.2		3
65.47	even	4		325.6.b.c.274.4		6
91.34	even	4		637.6.a.b.1.2		3
104.21	odd	4		832.6.a.s.1.1		3
104.99	even	4		832.6.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.2	✓	3	13.8	odd	4
117.6.a.d.1.2		3	39.8	even	4
169.6.a.b.1.2		3	13.5	odd	4
169.6.b.b.168.3		6	1.1	even	1	trivial
169.6.b.b.168.4		6	13.12	even	2	inner
208.6.a.j.1.1		3	52.47	even	4
325.6.a.c.1.2		3	65.34	odd	4
325.6.b.c.274.3		6	65.8	even	4
325.6.b.c.274.4		6	65.47	even	4
637.6.a.b.1.2		3	91.34	even	4
832.6.a.s.1.1		3	104.21	odd	4
832.6.a.t.1.3		3	104.99	even	4

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.68079i q^{2} +13.5120 q^{3} +10.0902 q^{4} -15.4272i q^{5} -63.2466i q^{6} +12.5359i q^{7} -197.015i q^{8} -60.4272 q^{9} +O(q^{10})\)	\(q-4.68079i q^{2} +13.5120 q^{3} +10.0902 q^{4} -15.4272i q^{5} -63.2466i q^{6} +12.5359i q^{7} -197.015i q^{8} -60.4272 q^{9} -72.2115 q^{10} -271.483i q^{11} +136.338 q^{12} +58.6777 q^{14} -208.451i q^{15} -599.303 q^{16} -1710.35 q^{17} +282.847i q^{18} -2235.01i q^{19} -155.663i q^{20} +169.384i q^{21} -1270.76 q^{22} +966.314 q^{23} -2662.06i q^{24} +2887.00 q^{25} -4099.89 q^{27} +126.489i q^{28} +4916.69 q^{29} -975.718 q^{30} -2318.90i q^{31} -3499.28i q^{32} -3668.26i q^{33} +8005.78i q^{34} +193.393 q^{35} -609.721 q^{36} -13463.8i q^{37} -10461.6 q^{38} -3039.39 q^{40} -1646.31i q^{41} +792.850 q^{42} +9529.76 q^{43} -2739.31i q^{44} +932.222i q^{45} -4523.12i q^{46} +22144.0i q^{47} -8097.75 q^{48} +16649.9 q^{49} -13513.5i q^{50} -23110.1 q^{51} -30677.6 q^{53} +19190.8i q^{54} -4188.22 q^{55} +2469.76 q^{56} -30199.3i q^{57} -23014.0i q^{58} +37271.7i q^{59} -2103.31i q^{60} -30506.8 q^{61} -10854.3 q^{62} -757.506i q^{63} -35557.1 q^{64} -17170.4 q^{66} -44159.3i q^{67} -17257.7 q^{68} +13056.8 q^{69} -905.233i q^{70} +16879.6i q^{71} +11905.1i q^{72} +13170.7i q^{73} -63021.2 q^{74} +39009.0 q^{75} -22551.6i q^{76} +3403.27 q^{77} +19848.9 q^{79} +9245.56i q^{80} -40713.7 q^{81} -7706.02 q^{82} +22216.4i q^{83} +1709.11i q^{84} +26385.8i q^{85} -44606.8i q^{86} +66434.0 q^{87} -53486.3 q^{88} -3877.28i q^{89} +4363.54 q^{90} +9750.28 q^{92} -31332.8i q^{93} +103652. q^{94} -34479.9 q^{95} -47282.1i q^{96} -15571.2i q^{97} -77934.5i q^{98} +16405.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9}+O(q^{10}) \) 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9	\( 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9} + 2582 q^{10} + 2182 q^{12} - 1586 q^{14} + 5570 q^{16} - 1816 q^{17} - 6820 q^{22} - 7248 q^{23} - 4046 q^{25} - 8552 q^{27} - 17516 q^{29} - 3434 q^{30} + 8696 q^{35} + 9356 q^{36} - 67572 q^{38} - 87018 q^{40} + 27206 q^{42} - 4064 q^{43} - 152446 q^{48} + 89142 q^{49} - 26936 q^{51} - 25140 q^{53} + 70624 q^{55} + 98574 q^{56} - 25508 q^{61} + 80680 q^{62} - 234786 q^{64} - 224708 q^{66} - 98350 q^{68} + 40224 q^{69} - 263038 q^{74} - 47096 q^{75} + 42104 q^{77} - 123744 q^{79} - 216442 q^{81} - 284520 q^{82} + 278144 q^{87} - 316020 q^{88} - 202516 q^{90} + 601728 q^{92} + 145686 q^{94} + 393120 q^{95}+O(q^{100}) \) 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9 + 2582 * q^10 + 2182 * q^12 - 1586 * q^14 + 5570 * q^16 - 1816 * q^17 - 6820 * q^22 - 7248 * q^23 - 4046 * q^25 - 8552 * q^27 - 17516 * q^29 - 3434 * q^30 + 8696 * q^35 + 9356 * q^36 - 67572 * q^38 - 87018 * q^40 + 27206 * q^42 - 4064 * q^43 - 152446 * q^48 + 89142 * q^49 - 26936 * q^51 - 25140 * q^53 + 70624 * q^55 + 98574 * q^56 - 25508 * q^61 + 80680 * q^62 - 234786 * q^64 - 224708 * q^66 - 98350 * q^68 + 40224 * q^69 - 263038 * q^74 - 47096 * q^75 + 42104 * q^77 - 123744 * q^79 - 216442 * q^81 - 284520 * q^82 + 278144 * q^87 - 316020 * q^88 - 202516 * q^90 + 601728 * q^92 + 145686 * q^94 + 393120 * q^95

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