LMFDB - Embedded newform 4640.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4640,2,Mod(1,4640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4640, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4640 = 2^{5} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4640.a (trivial)

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:	yes
Analytic conductor:	$37.0505865379$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} -1.00000 q^{5} +3.85410 q^{7} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} -1.00000 q^{5} +3.85410 q^{7} -0.381966 q^{9} +1.23607 q^{11} +6.09017 q^{13} +1.61803 q^{15} -1.38197 q^{17} +7.23607 q^{19} -6.23607 q^{21} -0.854102 q^{23} +1.00000 q^{25} +5.47214 q^{27} +1.00000 q^{29} +0.618034 q^{31} -2.00000 q^{33} -3.85410 q^{35} +4.76393 q^{37} -9.85410 q^{39} +9.70820 q^{41} -5.38197 q^{43} +0.381966 q^{45} -8.00000 q^{47} +7.85410 q^{49} +2.23607 q^{51} -6.32624 q^{53} -1.23607 q^{55} -11.7082 q^{57} -11.6180 q^{59} +8.85410 q^{61} -1.47214 q^{63} -6.09017 q^{65} +6.47214 q^{67} +1.38197 q^{69} -4.94427 q^{71} +13.0902 q^{73} -1.61803 q^{75} +4.76393 q^{77} -14.0902 q^{79} -7.70820 q^{81} +2.29180 q^{83} +1.38197 q^{85} -1.61803 q^{87} +11.7082 q^{89} +23.4721 q^{91} -1.00000 q^{93} -7.23607 q^{95} -15.7984 q^{97} -0.472136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^5 + q^7 - 3 * q^9	$ 2 q - q^{3} - 2 q^{5} + q^{7} - 3 q^{9} - 2 q^{11} + q^{13} + q^{15} - 5 q^{17} + 10 q^{19} - 8 q^{21} + 5 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - q^{31} - 4 q^{33} - q^{35} + 14 q^{37} - 13 q^{39} + 6 q^{41} - 13 q^{43} + 3 q^{45} - 16 q^{47} + 9 q^{49} + 3 q^{53} + 2 q^{55} - 10 q^{57} - 21 q^{59} + 11 q^{61} + 6 q^{63} - q^{65} + 4 q^{67} + 5 q^{69} + 8 q^{71} + 15 q^{73} - q^{75} + 14 q^{77} - 17 q^{79} - 2 q^{81} + 18 q^{83} + 5 q^{85} - q^{87} + 10 q^{89} + 38 q^{91} - 2 q^{93} - 10 q^{95} - 7 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + q^7 - 3 * q^9 - 2 * q^11 + q^13 + q^15 - 5 * q^17 + 10 * q^19 - 8 * q^21 + 5 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - q^31 - 4 * q^33 - q^35 + 14 * q^37 - 13 * q^39 + 6 * q^41 - 13 * q^43 + 3 * q^45 - 16 * q^47 + 9 * q^49 + 3 * q^53 + 2 * q^55 - 10 * q^57 - 21 * q^59 + 11 * q^61 + 6 * q^63 - q^65 + 4 * q^67 + 5 * q^69 + 8 * q^71 + 15 * q^73 - q^75 + 14 * q^77 - 17 * q^79 - 2 * q^81 + 18 * q^83 + 5 * q^85 - q^87 + 10 * q^89 + 38 * q^91 - 2 * q^93 - 10 * q^95 - 7 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.85410	1.45671	0.728357	−	0.685198i	$-0.240284\pi$
$7$	3.85410	1.45671	0.728357	+	0.685198i	$0.240284\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	6.09017	1.68911	0.844555	−	0.535469i	$-0.179865\pi$
$13$	6.09017	1.68911	0.844555	+	0.535469i	$0.179865\pi$
$14$	0	0
$15$	1.61803	0.417775
$16$	0	0
$17$	−1.38197	−0.335176	−0.167588	−	0.985857i	$-0.553598\pi$
$17$	−1.38197	−0.335176	−0.167588	+	0.985857i	$0.553598\pi$
$18$	0	0
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	−6.23607	−1.36082
$22$	0	0
$23$	−0.854102	−0.178093	−0.0890463	−	0.996027i	$-0.528382\pi$
$23$	−0.854102	−0.178093	−0.0890463	+	0.996027i	$0.528382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.618034	0.111002	0.0555011	−	0.998459i	$-0.482324\pi$
$31$	0.618034	0.111002	0.0555011	+	0.998459i	$0.482324\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−3.85410	−0.651462
$36$	0	0
$37$	4.76393	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$37$	4.76393	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$38$	0	0
$39$	−9.85410	−1.57792
$40$	0	0
$41$	9.70820	1.51617	0.758083	−	0.652158i	$-0.226136\pi$
$41$	9.70820	1.51617	0.758083	+	0.652158i	$0.226136\pi$
$42$	0	0
$43$	−5.38197	−0.820742	−0.410371	−	0.911919i	$-0.634601\pi$
$43$	−5.38197	−0.820742	−0.410371	+	0.911919i	$0.634601\pi$
$44$	0	0
$45$	0.381966	0.0569401
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	2.23607	0.313112
$52$	0	0
$53$	−6.32624	−0.868976	−0.434488	−	0.900678i	$-0.643071\pi$
$53$	−6.32624	−0.868976	−0.434488	+	0.900678i	$0.643071\pi$
$54$	0	0
$55$	−1.23607	−0.166671
$56$	0	0
$57$	−11.7082	−1.55079
$58$	0	0
$59$	−11.6180	−1.51254	−0.756270	−	0.654260i	$-0.772980\pi$
$59$	−11.6180	−1.51254	−0.756270	+	0.654260i	$0.772980\pi$
$60$	0	0
$61$	8.85410	1.13365	0.566826	−	0.823838i	$-0.308171\pi$
$61$	8.85410	1.13365	0.566826	+	0.823838i	$0.308171\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	−6.09017	−0.755393
$66$	0	0
$67$	6.47214	0.790697	0.395349	−	0.918531i	$-0.370624\pi$
$67$	6.47214	0.790697	0.395349	+	0.918531i	$0.370624\pi$
$68$	0	0
$69$	1.38197	0.166369
$70$	0	0
$71$	−4.94427	−0.586777	−0.293389	−	0.955993i	$-0.594783\pi$
$71$	−4.94427	−0.586777	−0.293389	+	0.955993i	$0.594783\pi$
$72$	0	0
$73$	13.0902	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$73$	13.0902	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$74$	0	0
$75$	−1.61803	−0.186834
$76$	0	0
$77$	4.76393	0.542900
$78$	0	0
$79$	−14.0902	−1.58527	−0.792634	−	0.609698i	$-0.791291\pi$
$79$	−14.0902	−1.58527	−0.792634	+	0.609698i	$0.791291\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	2.29180	0.251557	0.125779	−	0.992058i	$-0.459857\pi$
$83$	2.29180	0.251557	0.125779	+	0.992058i	$0.459857\pi$
$84$	0	0
$85$	1.38197	0.149895
$86$	0	0
$87$	−1.61803	−0.173471
$88$	0	0
$89$	11.7082	1.24107	0.620534	−	0.784180i	$-0.286916\pi$
$89$	11.7082	1.24107	0.620534	+	0.784180i	$0.286916\pi$
$90$	0	0
$91$	23.4721	2.46055
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−7.23607	−0.742405
$96$	0	0
$97$	−15.7984	−1.60408	−0.802041	−	0.597269i	$-0.796252\pi$
$97$	−15.7984	−1.60408	−0.802041	+	0.597269i	$0.796252\pi$
$98$	0	0
$99$	−0.472136	−0.0474514
$100$	0	0
$101$	4.09017	0.406987	0.203494	−	0.979076i	$-0.434770\pi$
$101$	4.09017	0.406987	0.203494	+	0.979076i	$0.434770\pi$
$102$	0	0
$103$	−8.94427	−0.881305	−0.440653	−	0.897678i	$-0.645253\pi$
$103$	−8.94427	−0.881305	−0.440653	+	0.897678i	$0.645253\pi$
$104$	0	0
$105$	6.23607	0.608578
$106$	0	0
$107$	13.7082	1.32522	0.662611	−	0.748964i	$-0.269448\pi$
$107$	13.7082	1.32522	0.662611	+	0.748964i	$0.269448\pi$
$108$	0	0
$109$	−1.70820	−0.163616	−0.0818081	−	0.996648i	$-0.526069\pi$
$109$	−1.70820	−0.163616	−0.0818081	+	0.996648i	$0.526069\pi$
$110$	0	0
$111$	−7.70820	−0.731630
$112$	0	0
$113$	−15.7984	−1.48619	−0.743093	−	0.669188i	$-0.766642\pi$
$113$	−15.7984	−1.48619	−0.743093	+	0.669188i	$0.766642\pi$
$114$	0	0
$115$	0.854102	0.0796454
$116$	0	0
$117$	−2.32624	−0.215061
$118$	0	0
$119$	−5.32624	−0.488255
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	−15.7082	−1.41636
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.9443	1.50356	0.751780	−	0.659413i	$-0.229195\pi$
$127$	16.9443	1.50356	0.751780	+	0.659413i	$0.229195\pi$
$128$	0	0
$129$	8.70820	0.766715
$130$	0	0
$131$	−1.05573	−0.0922394	−0.0461197	−	0.998936i	$-0.514686\pi$
$131$	−1.05573	−0.0922394	−0.0461197	+	0.998936i	$0.514686\pi$
$132$	0	0
$133$	27.8885	2.41824
$134$	0	0
$135$	−5.47214	−0.470966
$136$	0	0
$137$	−6.85410	−0.585585	−0.292793	−	0.956176i	$-0.594585\pi$
$137$	−6.85410	−0.585585	−0.292793	+	0.956176i	$0.594585\pi$
$138$	0	0
$139$	3.38197	0.286855	0.143427	−	0.989661i	$-0.454188\pi$
$139$	3.38197	0.286855	0.143427	+	0.989661i	$0.454188\pi$
$140$	0	0
$141$	12.9443	1.09010
$142$	0	0
$143$	7.52786	0.629512
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−12.7082	−1.04815
$148$	0	0
$149$	−6.18034	−0.506313	−0.253157	−	0.967425i	$-0.581469\pi$
$149$	−6.18034	−0.506313	−0.253157	+	0.967425i	$0.581469\pi$
$150$	0	0
$151$	−3.23607	−0.263347	−0.131674	−	0.991293i	$-0.542035\pi$
$151$	−3.23607	−0.263347	−0.131674	+	0.991293i	$0.542035\pi$
$152$	0	0
$153$	0.527864	0.0426753
$154$	0	0
$155$	−0.618034	−0.0496417
$156$	0	0
$157$	22.9443	1.83115	0.915576	−	0.402145i	$-0.131735\pi$
$157$	22.9443	1.83115	0.915576	+	0.402145i	$0.131735\pi$
$158$	0	0
$159$	10.2361	0.811773
$160$	0	0
$161$	−3.29180	−0.259430
$162$	0	0
$163$	−22.4721	−1.76015	−0.880077	−	0.474831i	$-0.842509\pi$
$163$	−22.4721	−1.76015	−0.880077	+	0.474831i	$0.842509\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−6.61803	−0.512119	−0.256059	−	0.966661i	$-0.582424\pi$
$167$	−6.61803	−0.512119	−0.256059	+	0.966661i	$0.582424\pi$
$168$	0	0
$169$	24.0902	1.85309
$170$	0	0
$171$	−2.76393	−0.211363
$172$	0	0
$173$	19.7984	1.50524	0.752621	−	0.658454i	$-0.228789\pi$
$173$	19.7984	1.50524	0.752621	+	0.658454i	$0.228789\pi$
$174$	0	0
$175$	3.85410	0.291343
$176$	0	0
$177$	18.7984	1.41297
$178$	0	0
$179$	10.0902	0.754175	0.377087	−	0.926178i	$-0.376926\pi$
$179$	10.0902	0.754175	0.377087	+	0.926178i	$0.376926\pi$
$180$	0	0
$181$	13.7082	1.01892	0.509461	−	0.860494i	$-0.329845\pi$
$181$	13.7082	1.01892	0.509461	+	0.860494i	$0.329845\pi$
$182$	0	0
$183$	−14.3262	−1.05903
$184$	0	0
$185$	−4.76393	−0.350251
$186$	0	0
$187$	−1.70820	−0.124916
$188$	0	0
$189$	21.0902	1.53408
$190$	0	0
$191$	−14.1459	−1.02356	−0.511781	−	0.859116i	$-0.671014\pi$
$191$	−14.1459	−1.02356	−0.511781	+	0.859116i	$0.671014\pi$
$192$	0	0
$193$	6.85410	0.493369	0.246685	−	0.969096i	$-0.420659\pi$
$193$	6.85410	0.493369	0.246685	+	0.969096i	$0.420659\pi$
$194$	0	0
$195$	9.85410	0.705667
$196$	0	0
$197$	12.1459	0.865359	0.432680	−	0.901548i	$-0.357568\pi$
$197$	12.1459	0.865359	0.432680	+	0.901548i	$0.357568\pi$
$198$	0	0
$199$	−13.2361	−0.938280	−0.469140	−	0.883124i	$-0.655436\pi$
$199$	−13.2361	−0.938280	−0.469140	+	0.883124i	$0.655436\pi$
$200$	0	0
$201$	−10.4721	−0.738648
$202$	0	0
$203$	3.85410	0.270505
$204$	0	0
$205$	−9.70820	−0.678050
$206$	0	0
$207$	0.326238	0.0226751
$208$	0	0
$209$	8.94427	0.618688
$210$	0	0
$211$	10.7639	0.741020	0.370510	−	0.928829i	$-0.379183\pi$
$211$	10.7639	0.741020	0.370510	+	0.928829i	$0.379183\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	5.38197	0.367047
$216$	0	0
$217$	2.38197	0.161698
$218$	0	0
$219$	−21.1803	−1.43123
$220$	0	0
$221$	−8.41641	−0.566149
$222$	0	0
$223$	21.9787	1.47180	0.735902	−	0.677088i	$-0.236759\pi$
$223$	21.9787	1.47180	0.735902	+	0.677088i	$0.236759\pi$
$224$	0	0
$225$	−0.381966	−0.0254644
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−7.09017	−0.468532	−0.234266	−	0.972173i	$-0.575269\pi$
$229$	−7.09017	−0.468532	−0.234266	+	0.972173i	$0.575269\pi$
$230$	0	0
$231$	−7.70820	−0.507163
$232$	0	0
$233$	7.52786	0.493167	0.246583	−	0.969122i	$-0.420692\pi$
$233$	7.52786	0.493167	0.246583	+	0.969122i	$0.420692\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	22.7984	1.48091
$238$	0	0
$239$	20.4721	1.32423	0.662116	−	0.749401i	$-0.269659\pi$
$239$	20.4721	1.32423	0.662116	+	0.749401i	$0.269659\pi$
$240$	0	0
$241$	−25.7984	−1.66182	−0.830910	−	0.556407i	$-0.812179\pi$
$241$	−25.7984	−1.66182	−0.830910	+	0.556407i	$0.812179\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	−7.85410	−0.501780
$246$	0	0
$247$	44.0689	2.80404
$248$	0	0
$249$	−3.70820	−0.234998
$250$	0	0
$251$	4.18034	0.263861	0.131930	−	0.991259i	$-0.457882\pi$
$251$	4.18034	0.263861	0.131930	+	0.991259i	$0.457882\pi$
$252$	0	0
$253$	−1.05573	−0.0663731
$254$	0	0
$255$	−2.23607	−0.140028
$256$	0	0
$257$	30.0689	1.87565	0.937823	−	0.347115i	$-0.112839\pi$
$257$	30.0689	1.87565	0.937823	+	0.347115i	$0.112839\pi$
$258$	0	0
$259$	18.3607	1.14088
$260$	0	0
$261$	−0.381966	−0.0236431
$262$	0	0
$263$	−26.4721	−1.63234	−0.816171	−	0.577811i	$-0.803907\pi$
$263$	−26.4721	−1.63234	−0.816171	+	0.577811i	$0.803907\pi$
$264$	0	0
$265$	6.32624	0.388618
$266$	0	0
$267$	−18.9443	−1.15937
$268$	0	0
$269$	6.43769	0.392513	0.196257	−	0.980553i	$-0.437121\pi$
$269$	6.43769	0.392513	0.196257	+	0.980553i	$0.437121\pi$
$270$	0	0
$271$	11.4164	0.693497	0.346749	−	0.937958i	$-0.387286\pi$
$271$	11.4164	0.693497	0.346749	+	0.937958i	$0.387286\pi$
$272$	0	0
$273$	−37.9787	−2.29858
$274$	0	0
$275$	1.23607	0.0745377
$276$	0	0
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	0	0
$279$	−0.236068	−0.0141330
$280$	0	0
$281$	1.85410	0.110606	0.0553032	−	0.998470i	$-0.482387\pi$
$281$	1.85410	0.110606	0.0553032	+	0.998470i	$0.482387\pi$
$282$	0	0
$283$	−0.180340	−0.0107201	−0.00536005	−	0.999986i	$-0.501706\pi$
$283$	−0.180340	−0.0107201	−0.00536005	+	0.999986i	$0.501706\pi$
$284$	0	0
$285$	11.7082	0.693534
$286$	0	0
$287$	37.4164	2.20862
$288$	0	0
$289$	−15.0902	−0.887657
$290$	0	0
$291$	25.5623	1.49849
$292$	0	0
$293$	−8.18034	−0.477901	−0.238950	−	0.971032i	$-0.576803\pi$
$293$	−8.18034	−0.477901	−0.238950	+	0.971032i	$0.576803\pi$
$294$	0	0
$295$	11.6180	0.676428
$296$	0	0
$297$	6.76393	0.392483
$298$	0	0
$299$	−5.20163	−0.300818
$300$	0	0
$301$	−20.7426	−1.19559
$302$	0	0
$303$	−6.61803	−0.380196
$304$	0	0
$305$	−8.85410	−0.506984
$306$	0	0
$307$	−18.4721	−1.05426	−0.527130	−	0.849785i	$-0.676732\pi$
$307$	−18.4721	−1.05426	−0.527130	+	0.849785i	$0.676732\pi$
$308$	0	0
$309$	14.4721	0.823291
$310$	0	0
$311$	−9.32624	−0.528842	−0.264421	−	0.964407i	$-0.585181\pi$
$311$	−9.32624	−0.528842	−0.264421	+	0.964407i	$0.585181\pi$
$312$	0	0
$313$	31.8885	1.80245	0.901224	−	0.433355i	$-0.142670\pi$
$313$	31.8885	1.80245	0.901224	+	0.433355i	$0.142670\pi$
$314$	0	0
$315$	1.47214	0.0829455
$316$	0	0
$317$	−9.23607	−0.518749	−0.259375	−	0.965777i	$-0.583516\pi$
$317$	−9.23607	−0.518749	−0.259375	+	0.965777i	$0.583516\pi$
$318$	0	0
$319$	1.23607	0.0692065
$320$	0	0
$321$	−22.1803	−1.23799
$322$	0	0
$323$	−10.0000	−0.556415
$324$	0	0
$325$	6.09017	0.337822
$326$	0	0
$327$	2.76393	0.152846
$328$	0	0
$329$	−30.8328	−1.69987
$330$	0	0
$331$	−23.7082	−1.30312	−0.651560	−	0.758597i	$-0.725885\pi$
$331$	−23.7082	−1.30312	−0.651560	+	0.758597i	$0.725885\pi$
$332$	0	0
$333$	−1.81966	−0.0997168
$334$	0	0
$335$	−6.47214	−0.353611
$336$	0	0
$337$	23.8541	1.29942	0.649708	−	0.760184i	$-0.274891\pi$
$337$	23.8541	1.29942	0.649708	+	0.760184i	$0.274891\pi$
$338$	0	0
$339$	25.5623	1.38835
$340$	0	0
$341$	0.763932	0.0413692
$342$	0	0
$343$	3.29180	0.177740
$344$	0	0
$345$	−1.38197	−0.0744025
$346$	0	0
$347$	35.2361	1.89157	0.945786	−	0.324792i	$-0.105294\pi$
$347$	35.2361	1.89157	0.945786	+	0.324792i	$0.105294\pi$
$348$	0	0
$349$	4.94427	0.264661	0.132330	−	0.991206i	$-0.457754\pi$
$349$	4.94427	0.264661	0.132330	+	0.991206i	$0.457754\pi$
$350$	0	0
$351$	33.3262	1.77882
$352$	0	0
$353$	−9.52786	−0.507117	−0.253559	−	0.967320i	$-0.581601\pi$
$353$	−9.52786	−0.507117	−0.253559	+	0.967320i	$0.581601\pi$
$354$	0	0
$355$	4.94427	0.262415
$356$	0	0
$357$	8.61803	0.456115
$358$	0	0
$359$	26.0902	1.37699	0.688493	−	0.725243i	$-0.258272\pi$
$359$	26.0902	1.37699	0.688493	+	0.725243i	$0.258272\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	0	0
$363$	15.3262	0.804419
$364$	0	0
$365$	−13.0902	−0.685171
$366$	0	0
$367$	4.18034	0.218212	0.109106	−	0.994030i	$-0.465201\pi$
$367$	4.18034	0.218212	0.109106	+	0.994030i	$0.465201\pi$
$368$	0	0
$369$	−3.70820	−0.193041
$370$	0	0
$371$	−24.3820	−1.26585
$372$	0	0
$373$	−7.27051	−0.376453	−0.188226	−	0.982126i	$-0.560274\pi$
$373$	−7.27051	−0.376453	−0.188226	+	0.982126i	$0.560274\pi$
$374$	0	0
$375$	1.61803	0.0835549
$376$	0	0
$377$	6.09017	0.313660
$378$	0	0
$379$	−20.0689	−1.03087	−0.515435	−	0.856929i	$-0.672370\pi$
$379$	−20.0689	−1.03087	−0.515435	+	0.856929i	$0.672370\pi$
$380$	0	0
$381$	−27.4164	−1.40459
$382$	0	0
$383$	7.27051	0.371506	0.185753	−	0.982596i	$-0.440528\pi$
$383$	7.27051	0.371506	0.185753	+	0.982596i	$0.440528\pi$
$384$	0	0
$385$	−4.76393	−0.242792
$386$	0	0
$387$	2.05573	0.104499
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	1.18034	0.0596924
$392$	0	0
$393$	1.70820	0.0861675
$394$	0	0
$395$	14.0902	0.708953
$396$	0	0
$397$	11.8541	0.594940	0.297470	−	0.954731i	$-0.403857\pi$
$397$	11.8541	0.594940	0.297470	+	0.954731i	$0.403857\pi$
$398$	0	0
$399$	−45.1246	−2.25906
$400$	0	0
$401$	−7.32624	−0.365855	−0.182927	−	0.983126i	$-0.558557\pi$
$401$	−7.32624	−0.365855	−0.182927	+	0.983126i	$0.558557\pi$
$402$	0	0
$403$	3.76393	0.187495
$404$	0	0
$405$	7.70820	0.383024
$406$	0	0
$407$	5.88854	0.291884
$408$	0	0
$409$	−19.4164	−0.960080	−0.480040	−	0.877247i	$-0.659378\pi$
$409$	−19.4164	−0.960080	−0.480040	+	0.877247i	$0.659378\pi$
$410$	0	0
$411$	11.0902	0.547038
$412$	0	0
$413$	−44.7771	−2.20334
$414$	0	0
$415$	−2.29180	−0.112500
$416$	0	0
$417$	−5.47214	−0.267972
$418$	0	0
$419$	15.0902	0.737203	0.368602	−	0.929587i	$-0.379837\pi$
$419$	15.0902	0.737203	0.368602	+	0.929587i	$0.379837\pi$
$420$	0	0
$421$	16.4721	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$422$	0	0
$423$	3.05573	0.148575
$424$	0	0
$425$	−1.38197	−0.0670352
$426$	0	0
$427$	34.1246	1.65141
$428$	0	0
$429$	−12.1803	−0.588072
$430$	0	0
$431$	10.2918	0.495738	0.247869	−	0.968794i	$-0.420270\pi$
$431$	10.2918	0.495738	0.247869	+	0.968794i	$0.420270\pi$
$432$	0	0
$433$	−4.47214	−0.214917	−0.107459	−	0.994210i	$-0.534271\pi$
$433$	−4.47214	−0.214917	−0.107459	+	0.994210i	$0.534271\pi$
$434$	0	0
$435$	1.61803	0.0775788
$436$	0	0
$437$	−6.18034	−0.295646
$438$	0	0
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−8.27051	−0.392944	−0.196472	−	0.980509i	$-0.562948\pi$
$443$	−8.27051	−0.392944	−0.196472	+	0.980509i	$0.562948\pi$
$444$	0	0
$445$	−11.7082	−0.555022
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	34.0689	1.60781	0.803905	−	0.594758i	$-0.202752\pi$
$449$	34.0689	1.60781	0.803905	+	0.594758i	$0.202752\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	5.23607	0.246012
$454$	0	0
$455$	−23.4721	−1.10039
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	−7.56231	−0.352978
$460$	0	0
$461$	−3.67376	−0.171104	−0.0855521	−	0.996334i	$-0.527265\pi$
$461$	−3.67376	−0.171104	−0.0855521	+	0.996334i	$0.527265\pi$
$462$	0	0
$463$	14.8328	0.689339	0.344670	−	0.938724i	$-0.387991\pi$
$463$	14.8328	0.689339	0.344670	+	0.938724i	$0.387991\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−38.3262	−1.77353	−0.886763	−	0.462224i	$-0.847052\pi$
$467$	−38.3262	−1.77353	−0.886763	+	0.462224i	$0.847052\pi$
$468$	0	0
$469$	24.9443	1.15182
$470$	0	0
$471$	−37.1246	−1.71061
$472$	0	0
$473$	−6.65248	−0.305881
$474$	0	0
$475$	7.23607	0.332014
$476$	0	0
$477$	2.41641	0.110640
$478$	0	0
$479$	16.3262	0.745965	0.372982	−	0.927838i	$-0.378335\pi$
$479$	16.3262	0.745965	0.372982	+	0.927838i	$0.378335\pi$
$480$	0	0
$481$	29.0132	1.32289
$482$	0	0
$483$	5.32624	0.242352
$484$	0	0
$485$	15.7984	0.717367
$486$	0	0
$487$	12.2705	0.556030	0.278015	−	0.960577i	$-0.410324\pi$
$487$	12.2705	0.556030	0.278015	+	0.960577i	$0.410324\pi$
$488$	0	0
$489$	36.3607	1.64429
$490$	0	0
$491$	21.3050	0.961479	0.480740	−	0.876863i	$-0.340368\pi$
$491$	21.3050	0.961479	0.480740	+	0.876863i	$0.340368\pi$
$492$	0	0
$493$	−1.38197	−0.0622406
$494$	0	0
$495$	0.472136	0.0212209
$496$	0	0
$497$	−19.0557	−0.854766
$498$	0	0
$499$	−17.3262	−0.775629	−0.387814	−	0.921737i	$-0.626770\pi$
$499$	−17.3262	−0.775629	−0.387814	+	0.921737i	$0.626770\pi$
$500$	0	0
$501$	10.7082	0.478407
$502$	0	0
$503$	20.0000	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$504$	0	0
$505$	−4.09017	−0.182010
$506$	0	0
$507$	−38.9787	−1.73111
$508$	0	0
$509$	9.70820	0.430309	0.215154	−	0.976580i	$-0.430975\pi$
$509$	9.70820	0.430309	0.215154	+	0.976580i	$0.430975\pi$
$510$	0	0
$511$	50.4508	2.23181
$512$	0	0
$513$	39.5967	1.74824
$514$	0	0
$515$	8.94427	0.394132
$516$	0	0
$517$	−9.88854	−0.434898
$518$	0	0
$519$	−32.0344	−1.40616
$520$	0	0
$521$	−19.6738	−0.861923	−0.430962	−	0.902370i	$-0.641826\pi$
$521$	−19.6738	−0.861923	−0.430962	+	0.902370i	$0.641826\pi$
$522$	0	0
$523$	12.1803	0.532609	0.266305	−	0.963889i	$-0.414197\pi$
$523$	12.1803	0.532609	0.266305	+	0.963889i	$0.414197\pi$
$524$	0	0
$525$	−6.23607	−0.272164
$526$	0	0
$527$	−0.854102	−0.0372053
$528$	0	0
$529$	−22.2705	−0.968283
$530$	0	0
$531$	4.43769	0.192580
$532$	0	0
$533$	59.1246	2.56097
$534$	0	0
$535$	−13.7082	−0.592657
$536$	0	0
$537$	−16.3262	−0.704529
$538$	0	0
$539$	9.70820	0.418162
$540$	0	0
$541$	−24.0344	−1.03332	−0.516661	−	0.856190i	$-0.672825\pi$
$541$	−24.0344	−1.03332	−0.516661	+	0.856190i	$0.672825\pi$
$542$	0	0
$543$	−22.1803	−0.951849
$544$	0	0
$545$	1.70820	0.0731714
$546$	0	0
$547$	−21.1246	−0.903223	−0.451612	−	0.892215i	$-0.649151\pi$
$547$	−21.1246	−0.903223	−0.451612	+	0.892215i	$0.649151\pi$
$548$	0	0
$549$	−3.38197	−0.144339
$550$	0	0
$551$	7.23607	0.308267
$552$	0	0
$553$	−54.3050	−2.30928
$554$	0	0
$555$	7.70820	0.327195
$556$	0	0
$557$	30.8541	1.30733	0.653665	−	0.756784i	$-0.273230\pi$
$557$	30.8541	1.30733	0.653665	+	0.756784i	$0.273230\pi$
$558$	0	0
$559$	−32.7771	−1.38632
$560$	0	0
$561$	2.76393	0.116693
$562$	0	0
$563$	−45.3951	−1.91318	−0.956588	−	0.291443i	$-0.905865\pi$
$563$	−45.3951	−1.91318	−0.956588	+	0.291443i	$0.905865\pi$
$564$	0	0
$565$	15.7984	0.664643
$566$	0	0
$567$	−29.7082	−1.24763
$568$	0	0
$569$	−12.3607	−0.518187	−0.259093	−	0.965852i	$-0.583424\pi$
$569$	−12.3607	−0.518187	−0.259093	+	0.965852i	$0.583424\pi$
$570$	0	0
$571$	−26.8541	−1.12381	−0.561905	−	0.827202i	$-0.689931\pi$
$571$	−26.8541	−1.12381	−0.561905	+	0.827202i	$0.689931\pi$
$572$	0	0
$573$	22.8885	0.956183
$574$	0	0
$575$	−0.854102	−0.0356185
$576$	0	0
$577$	−29.7426	−1.23820	−0.619101	−	0.785311i	$-0.712503\pi$
$577$	−29.7426	−1.23820	−0.619101	+	0.785311i	$0.712503\pi$
$578$	0	0
$579$	−11.0902	−0.460892
$580$	0	0
$581$	8.83282	0.366447
$582$	0	0
$583$	−7.81966	−0.323857
$584$	0	0
$585$	2.32624	0.0961781
$586$	0	0
$587$	44.8328	1.85045	0.925224	−	0.379421i	$-0.123877\pi$
$587$	44.8328	1.85045	0.925224	+	0.379421i	$0.123877\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	−19.6525	−0.808395
$592$	0	0
$593$	24.6525	1.01236	0.506178	−	0.862429i	$-0.331058\pi$
$593$	24.6525	1.01236	0.506178	+	0.862429i	$0.331058\pi$
$594$	0	0
$595$	5.32624	0.218354
$596$	0	0
$597$	21.4164	0.876515
$598$	0	0
$599$	−16.3262	−0.667072	−0.333536	−	0.942737i	$-0.608242\pi$
$599$	−16.3262	−0.667072	−0.333536	+	0.942737i	$0.608242\pi$
$600$	0	0
$601$	−46.5410	−1.89845	−0.949224	−	0.314601i	$-0.898129\pi$
$601$	−46.5410	−1.89845	−0.949224	+	0.314601i	$0.898129\pi$
$602$	0	0
$603$	−2.47214	−0.100673
$604$	0	0
$605$	9.47214	0.385097
$606$	0	0
$607$	6.47214	0.262696	0.131348	−	0.991336i	$-0.458069\pi$
$607$	6.47214	0.262696	0.131348	+	0.991336i	$0.458069\pi$
$608$	0	0
$609$	−6.23607	−0.252698
$610$	0	0
$611$	−48.7214	−1.97106
$612$	0	0
$613$	−9.74265	−0.393502	−0.196751	−	0.980454i	$-0.563039\pi$
$613$	−9.74265	−0.393502	−0.196751	+	0.980454i	$0.563039\pi$
$614$	0	0
$615$	15.7082	0.633416
$616$	0	0
$617$	−7.20163	−0.289927	−0.144963	−	0.989437i	$-0.546306\pi$
$617$	−7.20163	−0.289927	−0.144963	+	0.989437i	$0.546306\pi$
$618$	0	0
$619$	−29.7771	−1.19684	−0.598421	−	0.801182i	$-0.704205\pi$
$619$	−29.7771	−1.19684	−0.598421	+	0.801182i	$0.704205\pi$
$620$	0	0
$621$	−4.67376	−0.187552
$622$	0	0
$623$	45.1246	1.80788
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−14.4721	−0.577961
$628$	0	0
$629$	−6.58359	−0.262505
$630$	0	0
$631$	−6.36068	−0.253215	−0.126607	−	0.991953i	$-0.540409\pi$
$631$	−6.36068	−0.253215	−0.126607	+	0.991953i	$0.540409\pi$
$632$	0	0
$633$	−17.4164	−0.692240
$634$	0	0
$635$	−16.9443	−0.672413
$636$	0	0
$637$	47.8328	1.89521
$638$	0	0
$639$	1.88854	0.0747096
$640$	0	0
$641$	−43.2361	−1.70772	−0.853861	−	0.520501i	$-0.825745\pi$
$641$	−43.2361	−1.70772	−0.853861	+	0.520501i	$0.825745\pi$
$642$	0	0
$643$	−12.2918	−0.484741	−0.242371	−	0.970184i	$-0.577925\pi$
$643$	−12.2918	−0.484741	−0.242371	+	0.970184i	$0.577925\pi$
$644$	0	0
$645$	−8.70820	−0.342885
$646$	0	0
$647$	−25.3050	−0.994840	−0.497420	−	0.867510i	$-0.665719\pi$
$647$	−25.3050	−0.994840	−0.497420	+	0.867510i	$0.665719\pi$
$648$	0	0
$649$	−14.3607	−0.563706
$650$	0	0
$651$	−3.85410	−0.151054
$652$	0	0
$653$	−9.59675	−0.375550	−0.187775	−	0.982212i	$-0.560128\pi$
$653$	−9.59675	−0.375550	−0.187775	+	0.982212i	$0.560128\pi$
$654$	0	0
$655$	1.05573	0.0412507
$656$	0	0
$657$	−5.00000	−0.195069
$658$	0	0
$659$	−38.1803	−1.48729	−0.743647	−	0.668572i	$-0.766906\pi$
$659$	−38.1803	−1.48729	−0.743647	+	0.668572i	$0.766906\pi$
$660$	0	0
$661$	−21.5967	−0.840016	−0.420008	−	0.907520i	$-0.637973\pi$
$661$	−21.5967	−0.840016	−0.420008	+	0.907520i	$0.637973\pi$
$662$	0	0
$663$	13.6180	0.528881
$664$	0	0
$665$	−27.8885	−1.08147
$666$	0	0
$667$	−0.854102	−0.0330710
$668$	0	0
$669$	−35.5623	−1.37492
$670$	0	0
$671$	10.9443	0.422499
$672$	0	0
$673$	23.2361	0.895685	0.447842	−	0.894113i	$-0.352193\pi$
$673$	23.2361	0.895685	0.447842	+	0.894113i	$0.352193\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	21.8885	0.841245	0.420623	−	0.907236i	$-0.361812\pi$
$677$	21.8885	0.841245	0.420623	+	0.907236i	$0.361812\pi$
$678$	0	0
$679$	−60.8885	−2.33669
$680$	0	0
$681$	−3.23607	−0.124006
$682$	0	0
$683$	46.6525	1.78511	0.892554	−	0.450941i	$-0.148912\pi$
$683$	46.6525	1.78511	0.892554	+	0.450941i	$0.148912\pi$
$684$	0	0
$685$	6.85410	0.261882
$686$	0	0
$687$	11.4721	0.437689
$688$	0	0
$689$	−38.5279	−1.46779
$690$	0	0
$691$	−16.0344	−0.609979	−0.304989	−	0.952356i	$-0.598653\pi$
$691$	−16.0344	−0.609979	−0.304989	+	0.952356i	$0.598653\pi$
$692$	0	0
$693$	−1.81966	−0.0691232
$694$	0	0
$695$	−3.38197	−0.128285
$696$	0	0
$697$	−13.4164	−0.508183
$698$	0	0
$699$	−12.1803	−0.460703
$700$	0	0
$701$	27.1246	1.02448	0.512241	−	0.858842i	$-0.328815\pi$
$701$	27.1246	1.02448	0.512241	+	0.858842i	$0.328815\pi$
$702$	0	0
$703$	34.4721	1.30014
$704$	0	0
$705$	−12.9443	−0.487509
$706$	0	0
$707$	15.7639	0.592864
$708$	0	0
$709$	−31.7082	−1.19083	−0.595413	−	0.803420i	$-0.703012\pi$
$709$	−31.7082	−1.19083	−0.595413	+	0.803420i	$0.703012\pi$
$710$	0	0
$711$	5.38197	0.201839
$712$	0	0
$713$	−0.527864	−0.0197687
$714$	0	0
$715$	−7.52786	−0.281526
$716$	0	0
$717$	−33.1246	−1.23706
$718$	0	0
$719$	−10.6525	−0.397270	−0.198635	−	0.980074i	$-0.563651\pi$
$719$	−10.6525	−0.397270	−0.198635	+	0.980074i	$0.563651\pi$
$720$	0	0
$721$	−34.4721	−1.28381
$722$	0	0
$723$	41.7426	1.55243
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	49.2361	1.82606	0.913032	−	0.407887i	$-0.133734\pi$
$727$	49.2361	1.82606	0.913032	+	0.407887i	$0.133734\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	7.43769	0.275093
$732$	0	0
$733$	−35.7771	−1.32146	−0.660728	−	0.750625i	$-0.729753\pi$
$733$	−35.7771	−1.32146	−0.660728	+	0.750625i	$0.729753\pi$
$734$	0	0
$735$	12.7082	0.468749
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	23.4164	0.861386	0.430693	−	0.902498i	$-0.358269\pi$
$739$	23.4164	0.861386	0.430693	+	0.902498i	$0.358269\pi$
$740$	0	0
$741$	−71.3050	−2.61945
$742$	0	0
$743$	19.4164	0.712319	0.356159	−	0.934425i	$-0.384086\pi$
$743$	19.4164	0.712319	0.356159	+	0.934425i	$0.384086\pi$
$744$	0	0
$745$	6.18034	0.226430
$746$	0	0
$747$	−0.875388	−0.0320288
$748$	0	0
$749$	52.8328	1.93047
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−6.76393	−0.246491
$754$	0	0
$755$	3.23607	0.117773
$756$	0	0
$757$	−0.180340	−0.00655456	−0.00327728	−	0.999995i	$-0.501043\pi$
$757$	−0.180340	−0.00655456	−0.00327728	+	0.999995i	$0.501043\pi$
$758$	0	0
$759$	1.70820	0.0620039
$760$	0	0
$761$	4.03444	0.146248	0.0731242	−	0.997323i	$-0.476703\pi$
$761$	4.03444	0.146248	0.0731242	+	0.997323i	$0.476703\pi$
$762$	0	0
$763$	−6.58359	−0.238342
$764$	0	0
$765$	−0.527864	−0.0190850
$766$	0	0
$767$	−70.7558	−2.55484
$768$	0	0
$769$	19.3475	0.697690	0.348845	−	0.937181i	$-0.386574\pi$
$769$	19.3475	0.697690	0.348845	+	0.937181i	$0.386574\pi$
$770$	0	0
$771$	−48.6525	−1.75218
$772$	0	0
$773$	0.832816	0.0299543	0.0149771	−	0.999888i	$-0.495232\pi$
$773$	0.832816	0.0299543	0.0149771	+	0.999888i	$0.495232\pi$
$774$	0	0
$775$	0.618034	0.0222004
$776$	0	0
$777$	−29.7082	−1.06578
$778$	0	0
$779$	70.2492	2.51694
$780$	0	0
$781$	−6.11146	−0.218685
$782$	0	0
$783$	5.47214	0.195558
$784$	0	0
$785$	−22.9443	−0.818916
$786$	0	0
$787$	−13.3475	−0.475788	−0.237894	−	0.971291i	$-0.576457\pi$
$787$	−13.3475	−0.475788	−0.237894	+	0.971291i	$0.576457\pi$
$788$	0	0
$789$	42.8328	1.52489
$790$	0	0
$791$	−60.8885	−2.16495
$792$	0	0
$793$	53.9230	1.91486
$794$	0	0
$795$	−10.2361	−0.363036
$796$	0	0
$797$	−22.8328	−0.808780	−0.404390	−	0.914587i	$-0.632516\pi$
$797$	−22.8328	−0.808780	−0.404390	+	0.914587i	$0.632516\pi$
$798$	0	0
$799$	11.0557	0.391124
$800$	0	0
$801$	−4.47214	−0.158015
$802$	0	0
$803$	16.1803	0.570992
$804$	0	0
$805$	3.29180	0.116021
$806$	0	0
$807$	−10.4164	−0.366675
$808$	0	0
$809$	−15.4164	−0.542012	−0.271006	−	0.962578i	$-0.587356\pi$
$809$	−15.4164	−0.542012	−0.271006	+	0.962578i	$0.587356\pi$
$810$	0	0
$811$	−16.3262	−0.573292	−0.286646	−	0.958037i	$-0.592540\pi$
$811$	−16.3262	−0.573292	−0.286646	+	0.958037i	$0.592540\pi$
$812$	0	0
$813$	−18.4721	−0.647846
$814$	0	0
$815$	22.4721	0.787165
$816$	0	0
$817$	−38.9443	−1.36249
$818$	0	0
$819$	−8.96556	−0.313282
$820$	0	0
$821$	24.0689	0.840010	0.420005	−	0.907522i	$-0.362028\pi$
$821$	24.0689	0.840010	0.420005	+	0.907522i	$0.362028\pi$
$822$	0	0
$823$	25.7771	0.898533	0.449266	−	0.893398i	$-0.351685\pi$
$823$	25.7771	0.898533	0.449266	+	0.893398i	$0.351685\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−13.0902	−0.455190	−0.227595	−	0.973756i	$-0.573086\pi$
$827$	−13.0902	−0.455190	−0.227595	+	0.973756i	$0.573086\pi$
$828$	0	0
$829$	−21.2016	−0.736363	−0.368181	−	0.929754i	$-0.620019\pi$
$829$	−21.2016	−0.736363	−0.368181	+	0.929754i	$0.620019\pi$
$830$	0	0
$831$	−48.5410	−1.68387
$832$	0	0
$833$	−10.8541	−0.376072
$834$	0	0
$835$	6.61803	0.229027
$836$	0	0
$837$	3.38197	0.116898
$838$	0	0
$839$	16.3607	0.564833	0.282417	−	0.959292i	$-0.408864\pi$
$839$	16.3607	0.564833	0.282417	+	0.959292i	$0.408864\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.00000	−0.103325
$844$	0	0
$845$	−24.0902	−0.828727
$846$	0	0
$847$	−36.5066	−1.25438
$848$	0	0
$849$	0.291796	0.0100144
$850$	0	0
$851$	−4.06888	−0.139480
$852$	0	0
$853$	24.1803	0.827919	0.413960	−	0.910295i	$-0.364145\pi$
$853$	24.1803	0.827919	0.413960	+	0.910295i	$0.364145\pi$
$854$	0	0
$855$	2.76393	0.0945245
$856$	0	0
$857$	20.2918	0.693155	0.346577	−	0.938021i	$-0.387344\pi$
$857$	20.2918	0.693155	0.346577	+	0.938021i	$0.387344\pi$
$858$	0	0
$859$	14.6525	0.499936	0.249968	−	0.968254i	$-0.419580\pi$
$859$	14.6525	0.499936	0.249968	+	0.968254i	$0.419580\pi$
$860$	0	0
$861$	−60.5410	−2.06323
$862$	0	0
$863$	−25.0344	−0.852182	−0.426091	−	0.904680i	$-0.640110\pi$
$863$	−25.0344	−0.852182	−0.426091	+	0.904680i	$0.640110\pi$
$864$	0	0
$865$	−19.7984	−0.673165
$866$	0	0
$867$	24.4164	0.829225
$868$	0	0
$869$	−17.4164	−0.590811
$870$	0	0
$871$	39.4164	1.33557
$872$	0	0
$873$	6.03444	0.204235
$874$	0	0
$875$	−3.85410	−0.130292
$876$	0	0
$877$	48.3820	1.63374	0.816871	−	0.576820i	$-0.195707\pi$
$877$	48.3820	1.63374	0.816871	+	0.576820i	$0.195707\pi$
$878$	0	0
$879$	13.2361	0.446441
$880$	0	0
$881$	−12.1803	−0.410366	−0.205183	−	0.978724i	$-0.565779\pi$
$881$	−12.1803	−0.410366	−0.205183	+	0.978724i	$0.565779\pi$
$882$	0	0
$883$	−27.5967	−0.928705	−0.464352	−	0.885651i	$-0.653713\pi$
$883$	−27.5967	−0.928705	−0.464352	+	0.885651i	$0.653713\pi$
$884$	0	0
$885$	−18.7984	−0.631900
$886$	0	0
$887$	4.58359	0.153902	0.0769510	−	0.997035i	$-0.475482\pi$
$887$	4.58359	0.153902	0.0769510	+	0.997035i	$0.475482\pi$
$888$	0	0
$889$	65.3050	2.19026
$890$	0	0
$891$	−9.52786	−0.319195
$892$	0	0
$893$	−57.8885	−1.93717
$894$	0	0
$895$	−10.0902	−0.337277
$896$	0	0
$897$	8.41641	0.281016
$898$	0	0
$899$	0.618034	0.0206126
$900$	0	0
$901$	8.74265	0.291260
$902$	0	0
$903$	33.5623	1.11688
$904$	0	0
$905$	−13.7082	−0.455676
$906$	0	0
$907$	−37.0902	−1.23156	−0.615779	−	0.787919i	$-0.711159\pi$
$907$	−37.0902	−1.23156	−0.615779	+	0.787919i	$0.711159\pi$
$908$	0	0
$909$	−1.56231	−0.0518184
$910$	0	0
$911$	5.72949	0.189826	0.0949132	−	0.995486i	$-0.469743\pi$
$911$	5.72949	0.189826	0.0949132	+	0.995486i	$0.469743\pi$
$912$	0	0
$913$	2.83282	0.0937525
$914$	0	0
$915$	14.3262	0.473611
$916$	0	0
$917$	−4.06888	−0.134366
$918$	0	0
$919$	17.5279	0.578191	0.289095	−	0.957300i	$-0.406646\pi$
$919$	17.5279	0.578191	0.289095	+	0.957300i	$0.406646\pi$
$920$	0	0
$921$	29.8885	0.984861
$922$	0	0
$923$	−30.1115	−0.991131
$924$	0	0
$925$	4.76393	0.156637
$926$	0	0
$927$	3.41641	0.112210
$928$	0	0
$929$	11.4508	0.375690	0.187845	−	0.982199i	$-0.439850\pi$
$929$	11.4508	0.375690	0.187845	+	0.982199i	$0.439850\pi$
$930$	0	0
$931$	56.8328	1.86262
$932$	0	0
$933$	15.0902	0.494030
$934$	0	0
$935$	1.70820	0.0558642
$936$	0	0
$937$	2.87539	0.0939348	0.0469674	−	0.998896i	$-0.485044\pi$
$937$	2.87539	0.0939348	0.0469674	+	0.998896i	$0.485044\pi$
$938$	0	0
$939$	−51.5967	−1.68380
$940$	0	0
$941$	−12.3607	−0.402947	−0.201473	−	0.979494i	$-0.564573\pi$
$941$	−12.3607	−0.402947	−0.201473	+	0.979494i	$0.564573\pi$
$942$	0	0
$943$	−8.29180	−0.270018
$944$	0	0
$945$	−21.0902	−0.686063
$946$	0	0
$947$	43.9098	1.42688	0.713439	−	0.700717i	$-0.247137\pi$
$947$	43.9098	1.42688	0.713439	+	0.700717i	$0.247137\pi$
$948$	0	0
$949$	79.7214	2.58786
$950$	0	0
$951$	14.9443	0.484601
$952$	0	0
$953$	−40.8328	−1.32270	−0.661352	−	0.750075i	$-0.730017\pi$
$953$	−40.8328	−1.32270	−0.661352	+	0.750075i	$0.730017\pi$
$954$	0	0
$955$	14.1459	0.457751
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	−26.4164	−0.853030
$960$	0	0
$961$	−30.6180	−0.987679
$962$	0	0
$963$	−5.23607	−0.168730
$964$	0	0
$965$	−6.85410	−0.220641
$966$	0	0
$967$	25.5967	0.823136	0.411568	−	0.911379i	$-0.364981\pi$
$967$	25.5967	0.823136	0.411568	+	0.911379i	$0.364981\pi$
$968$	0	0
$969$	16.1803	0.519787
$970$	0	0
$971$	−6.06888	−0.194760	−0.0973799	−	0.995247i	$-0.531046\pi$
$971$	−6.06888	−0.194760	−0.0973799	+	0.995247i	$0.531046\pi$
$972$	0	0
$973$	13.0344	0.417865
$974$	0	0
$975$	−9.85410	−0.315584
$976$	0	0
$977$	−20.6525	−0.660731	−0.330366	−	0.943853i	$-0.607172\pi$
$977$	−20.6525	−0.660731	−0.330366	+	0.943853i	$0.607172\pi$
$978$	0	0
$979$	14.4721	0.462531
$980$	0	0
$981$	0.652476	0.0208320
$982$	0	0
$983$	−30.9443	−0.986969	−0.493484	−	0.869755i	$-0.664277\pi$
$983$	−30.9443	−0.986969	−0.493484	+	0.869755i	$0.664277\pi$
$984$	0	0
$985$	−12.1459	−0.387000
$986$	0	0
$987$	49.8885	1.58797
$988$	0	0
$989$	4.59675	0.146168
$990$	0	0
$991$	48.0689	1.52696	0.763479	−	0.645832i	$-0.223490\pi$
$991$	48.0689	1.52696	0.763479	+	0.645832i	$0.223490\pi$
$992$	0	0
$993$	38.3607	1.21734
$994$	0	0
$995$	13.2361	0.419612
$996$	0	0
$997$	21.8197	0.691036	0.345518	−	0.938412i	$-0.387703\pi$
$997$	21.8197	0.691036	0.345518	+	0.938412i	$0.387703\pi$
$998$	0	0
$999$	26.0689	0.824783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4640.2.a.g.1.1	✓	2
4.3	odd	2		4640.2.a.i.1.2	yes	2
8.3	odd	2		9280.2.a.y.1.1		2
8.5	even	2		9280.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.g.1.1	✓	2	1.1	even	1	trivial
4640.2.a.i.1.2	yes	2	4.3	odd	2
9280.2.a.y.1.1		2	8.3	odd	2
9280.2.a.bd.1.2		2	8.5	even	2

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:	\( N \)	\(=\)	\( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} -1.00000 q^{5} +3.85410 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} -1.00000 q^{5} +3.85410 q^{7} -0.381966 q^{9} +1.23607 q^{11} +6.09017 q^{13} +1.61803 q^{15} -1.38197 q^{17} +7.23607 q^{19} -6.23607 q^{21} -0.854102 q^{23} +1.00000 q^{25} +5.47214 q^{27} +1.00000 q^{29} +0.618034 q^{31} -2.00000 q^{33} -3.85410 q^{35} +4.76393 q^{37} -9.85410 q^{39} +9.70820 q^{41} -5.38197 q^{43} +0.381966 q^{45} -8.00000 q^{47} +7.85410 q^{49} +2.23607 q^{51} -6.32624 q^{53} -1.23607 q^{55} -11.7082 q^{57} -11.6180 q^{59} +8.85410 q^{61} -1.47214 q^{63} -6.09017 q^{65} +6.47214 q^{67} +1.38197 q^{69} -4.94427 q^{71} +13.0902 q^{73} -1.61803 q^{75} +4.76393 q^{77} -14.0902 q^{79} -7.70820 q^{81} +2.29180 q^{83} +1.38197 q^{85} -1.61803 q^{87} +11.7082 q^{89} +23.4721 q^{91} -1.00000 q^{93} -7.23607 q^{95} -15.7984 q^{97} -0.472136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^5 + q^7 - 3 * q^9	\( 2 q - q^{3} - 2 q^{5} + q^{7} - 3 q^{9} - 2 q^{11} + q^{13} + q^{15} - 5 q^{17} + 10 q^{19} - 8 q^{21} + 5 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - q^{31} - 4 q^{33} - q^{35} + 14 q^{37} - 13 q^{39} + 6 q^{41} - 13 q^{43} + 3 q^{45} - 16 q^{47} + 9 q^{49} + 3 q^{53} + 2 q^{55} - 10 q^{57} - 21 q^{59} + 11 q^{61} + 6 q^{63} - q^{65} + 4 q^{67} + 5 q^{69} + 8 q^{71} + 15 q^{73} - q^{75} + 14 q^{77} - 17 q^{79} - 2 q^{81} + 18 q^{83} + 5 q^{85} - q^{87} + 10 q^{89} + 38 q^{91} - 2 q^{93} - 10 q^{95} - 7 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + q^7 - 3 * q^9 - 2 * q^11 + q^13 + q^15 - 5 * q^17 + 10 * q^19 - 8 * q^21 + 5 * q^23 + 2 * q^25 + 2 * q^27 + 2 * q^29 - q^31 - 4 * q^33 - q^35 + 14 * q^37 - 13 * q^39 + 6 * q^41 - 13 * q^43 + 3 * q^45 - 16 * q^47 + 9 * q^49 + 3 * q^53 + 2 * q^55 - 10 * q^57 - 21 * q^59 + 11 * q^61 + 6 * q^63 - q^65 + 4 * q^67 + 5 * q^69 + 8 * q^71 + 15 * q^73 - q^75 + 14 * q^77 - 17 * q^79 - 2 * q^81 + 18 * q^83 + 5 * q^85 - q^87 + 10 * q^89 + 38 * q^91 - 2 * q^93 - 10 * q^95 - 7 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more