LMFDB - Embedded newform 320.4.d.d.161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,4,Mod(161,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.161"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.36185170176.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 38x^{6} + 517x^{4} + 2976x^{2} + 6084 $ x^8 + 38x^6 + 517x^4 + 2976*x^2 + 6084
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-3.81392i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.4.d.d.161.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.35989i q^{3} -5.00000i q^{5} -10.7522 q^{7} -42.8877 q^{9} -5.01534i q^{11} +50.0632i q^{13} -41.7994 q^{15} -136.304 q^{17} -41.8926i q^{19} +89.8871i q^{21} +121.937 q^{23} -25.0000 q^{25} +132.819i q^{27} +190.193i q^{29} -135.790 q^{31} -41.9277 q^{33} +53.7610i q^{35} -399.595i q^{37} +418.523 q^{39} +46.2759 q^{41} +385.530i q^{43} +214.438i q^{45} +436.584 q^{47} -227.390 q^{49} +1139.49i q^{51} -18.1818i q^{53} -25.0767 q^{55} -350.218 q^{57} +191.746i q^{59} +833.525i q^{61} +461.137 q^{63} +250.316 q^{65} -446.397i q^{67} -1019.38i q^{69} -1047.68 q^{71} -57.5478 q^{73} +208.997i q^{75} +53.9259i q^{77} -697.709 q^{79} -47.6132 q^{81} -1244.50i q^{83} +681.521i q^{85} +1589.99 q^{87} -1205.19 q^{89} -538.289i q^{91} +1135.19i q^{93} -209.463 q^{95} -755.602 q^{97} +215.097i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 64 q^{7} - 104 q^{9} - 96 q^{17} + 416 q^{23} - 200 q^{25} - 64 q^{31} + 944 q^{33} + 2176 q^{39} + 208 q^{41} + 1216 q^{47} - 760 q^{49} + 160 q^{55} - 16 q^{57} + 32 q^{63} + 800 q^{65} + 1344 q^{71}+ \cdots - 896 q^{97}+O(q^{100}) $ 8 * q + 64 * q^7 - 104 * q^9 - 96 * q^17 + 416 * q^23 - 200 * q^25 - 64 * q^31 + 944 * q^33 + 2176 * q^39 + 208 * q^41 + 1216 * q^47 - 760 * q^49 + 160 * q^55 - 16 * q^57 + 32 * q^63 + 800 * q^65 + 1344 * q^71 + 576 * q^73 - 3712 * q^79 - 3352 * q^81 + 5600 * q^87 + 1296 * q^89 - 2080 * q^95 - 896 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 8.35989i	− 1.60886i	−0.594047	−	0.804430i	$-0.702471\pi$
$3$	− 8.35989i	− 1.60886i	0.594047	−	0.804430i	$-0.297529\pi$
$4$	0	0
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	−10.7522	−0.580564	−0.290282	−	0.956941i	$-0.593749\pi$
$7$	−10.7522	−0.580564	−0.290282	+	0.956941i	$0.593749\pi$
$8$	0	0
$9$	−42.8877	−1.58843
$10$	0	0
$11$	− 5.01534i	− 0.137471i	−0.997635	−	0.0687356i	$-0.978104\pi$
$11$	− 5.01534i	− 0.137471i	0.997635	−	0.0687356i	$-0.0218965\pi$
$12$	0	0
$13$	50.0632i	1.06808i	0.845459	+	0.534040i	$0.179327\pi$
$13$	50.0632i	1.06808i	−0.845459	+	0.534040i	$0.820673\pi$
$14$	0	0
$15$	−41.7994	−0.719504
$16$	0	0
$17$	−136.304	−1.94462	−0.972312	−	0.233687i	$-0.924921\pi$
$17$	−136.304	−1.94462	−0.972312	+	0.233687i	$0.924921\pi$
$18$	0	0
$19$	− 41.8926i	− 0.505833i	−0.967488	−	0.252917i	$-0.918610\pi$
$19$	− 41.8926i	− 0.505833i	0.967488	−	0.252917i	$-0.0813898\pi$
$20$	0	0
$21$	89.8871i	0.934046i
$22$	0	0
$23$	121.937	1.10547	0.552733	−	0.833358i	$-0.313585\pi$
$23$	121.937	1.10547	0.552733	+	0.833358i	$0.313585\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	132.819i	0.946707i
$28$	0	0
$29$	190.193i	1.21786i	0.793224	+	0.608930i	$0.208401\pi$
$29$	190.193i	1.21786i	−0.793224	+	0.608930i	$0.791599\pi$
$30$	0	0
$31$	−135.790	−0.786729	−0.393364	−	0.919383i	$-0.628689\pi$
$31$	−135.790	−0.786729	−0.393364	+	0.919383i	$0.628689\pi$
$32$	0	0
$33$	−41.9277	−0.221172
$34$	0	0
$35$	53.7610i	0.259636i
$36$	0	0
$37$	− 399.595i	− 1.77549i	−0.460337	−	0.887744i	$-0.652272\pi$
$37$	− 399.595i	− 1.77549i	0.460337	−	0.887744i	$-0.347728\pi$
$38$	0	0
$39$	418.523	1.71839
$40$	0	0
$41$	46.2759	0.176270	0.0881352	−	0.996109i	$-0.471909\pi$
$41$	46.2759	0.176270	0.0881352	+	0.996109i	$0.471909\pi$
$42$	0	0
$43$	385.530i	1.36728i	0.729822	+	0.683638i	$0.239603\pi$
$43$	385.530i	1.36728i	−0.729822	+	0.683638i	$0.760397\pi$
$44$	0	0
$45$	214.438i	0.710369i
$46$	0	0
$47$	436.584	1.35494	0.677472	−	0.735549i	$-0.263076\pi$
$47$	436.584	1.35494	0.677472	+	0.735549i	$0.263076\pi$
$48$	0	0
$49$	−227.390	−0.662946
$50$	0	0
$51$	1139.49i	3.12863i
$52$	0	0
$53$	− 18.1818i	− 0.0471218i	−0.999722	−	0.0235609i	$-0.992500\pi$
$53$	− 18.1818i	− 0.0471218i	0.999722	−	0.0235609i	$-0.00750037\pi$
$54$	0	0
$55$	−25.0767	−0.0614790
$56$	0	0
$57$	−350.218	−0.813815
$58$	0	0
$59$	191.746i	0.423105i	0.977367	+	0.211553i	$0.0678520\pi$
$59$	191.746i	0.423105i	−0.977367	+	0.211553i	$0.932148\pi$
$60$	0	0
$61$	833.525i	1.74954i	0.484538	+	0.874770i	$0.338988\pi$
$61$	833.525i	1.74954i	−0.484538	+	0.874770i	$0.661012\pi$
$62$	0	0
$63$	461.137	0.922187
$64$	0	0
$65$	250.316	0.477660
$66$	0	0
$67$	− 446.397i	− 0.813970i	−0.913435	−	0.406985i	$-0.866580\pi$
$67$	− 446.397i	− 0.813970i	0.913435	−	0.406985i	$-0.133420\pi$
$68$	0	0
$69$	− 1019.38i	− 1.77854i
$70$	0	0
$71$	−1047.68	−1.75123	−0.875614	−	0.483012i	$-0.839543\pi$
$71$	−1047.68	−1.75123	−0.875614	+	0.483012i	$0.839543\pi$
$72$	0	0
$73$	−57.5478	−0.0922665	−0.0461333	−	0.998935i	$-0.514690\pi$
$73$	−57.5478	−0.0922665	−0.0461333	+	0.998935i	$0.514690\pi$
$74$	0	0
$75$	208.997i	0.321772i
$76$	0	0
$77$	53.9259i	0.0798108i
$78$	0	0
$79$	−697.709	−0.993651	−0.496826	−	0.867850i	$-0.665501\pi$
$79$	−697.709	−0.993651	−0.496826	+	0.867850i	$0.665501\pi$
$80$	0	0
$81$	−47.6132	−0.0653130
$82$	0	0
$83$	− 1244.50i	− 1.64581i	−0.568182	−	0.822903i	$-0.692353\pi$
$83$	− 1244.50i	− 1.64581i	0.568182	−	0.822903i	$-0.307647\pi$
$84$	0	0
$85$	681.521i	0.869662i
$86$	0	0
$87$	1589.99	1.95937
$88$	0	0
$89$	−1205.19	−1.43539	−0.717695	−	0.696358i	$-0.754803\pi$
$89$	−1205.19	−1.43539	−0.717695	+	0.696358i	$0.754803\pi$
$90$	0	0
$91$	− 538.289i	− 0.620088i
$92$	0	0
$93$	1135.19i	1.26574i
$94$	0	0
$95$	−209.463	−0.226215
$96$	0	0
$97$	−755.602	−0.790925	−0.395463	−	0.918482i	$-0.629416\pi$
$97$	−755.602	−0.790925	−0.395463	+	0.918482i	$0.629416\pi$
$98$	0	0
$99$	215.097i	0.218364i
$100$	0	0
$101$	447.030i	0.440407i	0.975454	+	0.220204i	$0.0706722\pi$
$101$	447.030i	0.440407i	−0.975454	+	0.220204i	$0.929328\pi$
$102$	0	0
$103$	567.470	0.542859	0.271429	−	0.962458i	$-0.412504\pi$
$103$	567.470	0.542859	0.271429	+	0.962458i	$0.412504\pi$
$104$	0	0
$105$	449.435	0.417718
$106$	0	0
$107$	− 1150.37i	− 1.03935i	−0.854365	−	0.519674i	$-0.826053\pi$
$107$	− 1150.37i	− 1.03935i	0.854365	−	0.519674i	$-0.173947\pi$
$108$	0	0
$109$	− 607.574i	− 0.533899i	−0.963710	−	0.266950i	$-0.913984\pi$
$109$	− 607.574i	− 0.533899i	0.963710	−	0.266950i	$-0.0860157\pi$
$110$	0	0
$111$	−3340.57	−2.85651
$112$	0	0
$113$	−319.435	−0.265929	−0.132964	−	0.991121i	$-0.542450\pi$
$113$	−319.435	−0.265929	−0.132964	+	0.991121i	$0.542450\pi$
$114$	0	0
$115$	− 609.687i	− 0.494380i
$116$	0	0
$117$	− 2147.09i	− 1.69657i
$118$	0	0
$119$	1465.57	1.12898
$120$	0	0
$121$	1305.85	0.981102
$122$	0	0
$123$	− 386.862i	− 0.283595i
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	−2134.56	−1.49143	−0.745713	−	0.666267i	$-0.767891\pi$
$127$	−2134.56	−1.49143	−0.745713	+	0.666267i	$0.767891\pi$
$128$	0	0
$129$	3222.99	2.19976
$130$	0	0
$131$	906.871i	0.604837i	0.953175	+	0.302419i	$0.0977941\pi$
$131$	906.871i	0.604837i	−0.953175	+	0.302419i	$0.902206\pi$
$132$	0	0
$133$	450.438i	0.293668i
$134$	0	0
$135$	664.097	0.423380
$136$	0	0
$137$	119.624	0.0745998	0.0372999	−	0.999304i	$-0.488124\pi$
$137$	119.624	0.0745998	0.0372999	+	0.999304i	$0.488124\pi$
$138$	0	0
$139$	− 2046.09i	− 1.24854i	−0.781210	−	0.624269i	$-0.785397\pi$
$139$	− 2046.09i	− 1.24854i	0.781210	−	0.624269i	$-0.214603\pi$
$140$	0	0
$141$	− 3649.79i	− 2.17991i
$142$	0	0
$143$	251.084	0.146830
$144$	0	0
$145$	950.965	0.544644
$146$	0	0
$147$	1900.96i	1.06659i
$148$	0	0
$149$	1453.15i	0.798969i	0.916740	+	0.399485i	$0.130811\pi$
$149$	1453.15i	0.798969i	−0.916740	+	0.399485i	$0.869189\pi$
$150$	0	0
$151$	893.396	0.481480	0.240740	−	0.970590i	$-0.422610\pi$
$151$	893.396	0.481480	0.240740	+	0.970590i	$0.422610\pi$
$152$	0	0
$153$	5845.77	3.08891
$154$	0	0
$155$	678.950i	0.351836i
$156$	0	0
$157$	− 2770.38i	− 1.40828i	−0.710060	−	0.704141i	$-0.751332\pi$
$157$	− 2770.38i	− 1.40828i	0.710060	−	0.704141i	$-0.248668\pi$
$158$	0	0
$159$	−151.997	−0.0758125
$160$	0	0
$161$	−1311.10	−0.641794
$162$	0	0
$163$	− 2808.16i	− 1.34940i	−0.738093	−	0.674699i	$-0.764273\pi$
$163$	− 2808.16i	− 1.34940i	0.738093	−	0.674699i	$-0.235727\pi$
$164$	0	0
$165$	209.639i	0.0989111i
$166$	0	0
$167$	−2674.32	−1.23919	−0.619595	−	0.784921i	$-0.712703\pi$
$167$	−2674.32	−1.23919	−0.619595	+	0.784921i	$0.712703\pi$
$168$	0	0
$169$	−309.323	−0.140793
$170$	0	0
$171$	1796.68i	0.803482i
$172$	0	0
$173$	1303.63i	0.572907i	0.958094	+	0.286453i	$0.0924763\pi$
$173$	1303.63i	0.572907i	−0.958094	+	0.286453i	$0.907524\pi$
$174$	0	0
$175$	268.805	0.116113
$176$	0	0
$177$	1602.98	0.680718
$178$	0	0
$179$	478.502i	0.199804i	0.994997	+	0.0999019i	$0.0318529\pi$
$179$	478.502i	0.199804i	−0.994997	+	0.0999019i	$0.968147\pi$
$180$	0	0
$181$	− 1752.48i	− 0.719672i	−0.933016	−	0.359836i	$-0.882833\pi$
$181$	− 1752.48i	− 0.719672i	0.933016	−	0.359836i	$-0.117167\pi$
$182$	0	0
$183$	6968.17	2.81477
$184$	0	0
$185$	−1997.98	−0.794022
$186$	0	0
$187$	683.612i	0.267330i
$188$	0	0
$189$	− 1428.10i	− 0.549624i
$190$	0	0
$191$	−37.8904	−0.0143542	−0.00717711	−	0.999974i	$-0.502285\pi$
$191$	−37.8904	−0.0143542	−0.00717711	+	0.999974i	$0.502285\pi$
$192$	0	0
$193$	−500.910	−0.186820	−0.0934101	−	0.995628i	$-0.529777\pi$
$193$	−500.910	−0.186820	−0.0934101	+	0.995628i	$0.529777\pi$
$194$	0	0
$195$	− 2092.61i	− 0.768488i
$196$	0	0
$197$	2564.22i	0.927375i	0.885999	+	0.463687i	$0.153474\pi$
$197$	2564.22i	0.927375i	−0.885999	+	0.463687i	$0.846526\pi$
$198$	0	0
$199$	−4069.97	−1.44981	−0.724905	−	0.688848i	$-0.758117\pi$
$199$	−4069.97	−1.44981	−0.724905	+	0.688848i	$0.758117\pi$
$200$	0	0
$201$	−3731.82	−1.30957
$202$	0	0
$203$	− 2044.99i	− 0.707046i
$204$	0	0
$205$	− 231.380i	− 0.0788305i
$206$	0	0
$207$	−5229.62	−1.75596
$208$	0	0
$209$	−210.106	−0.0695375
$210$	0	0
$211$	− 4167.30i	− 1.35966i	−0.733368	−	0.679832i	$-0.762053\pi$
$211$	− 4167.30i	− 1.35966i	0.733368	−	0.679832i	$-0.237947\pi$
$212$	0	0
$213$	8758.52i	2.81748i
$214$	0	0
$215$	1927.65	0.611464
$216$	0	0
$217$	1460.04	0.456746
$218$	0	0
$219$	481.093i	0.148444i
$220$	0	0
$221$	− 6823.82i	− 2.07701i
$222$	0	0
$223$	−1141.43	−0.342761	−0.171381	−	0.985205i	$-0.554823\pi$
$223$	−1141.43	−0.342761	−0.171381	+	0.985205i	$0.554823\pi$
$224$	0	0
$225$	1072.19	0.317687
$226$	0	0
$227$	− 410.051i	− 0.119894i	−0.998202	−	0.0599472i	$-0.980907\pi$
$227$	− 410.051i	− 0.119894i	0.998202	−	0.0599472i	$-0.0190932\pi$
$228$	0	0
$229$	1332.92i	0.384636i	0.981333	+	0.192318i	$0.0616005\pi$
$229$	1332.92i	0.384636i	−0.981333	+	0.192318i	$0.938400\pi$
$230$	0	0
$231$	450.815	0.128404
$232$	0	0
$233$	1495.89	0.420597	0.210298	−	0.977637i	$-0.432556\pi$
$233$	1495.89	0.420597	0.210298	+	0.977637i	$0.432556\pi$
$234$	0	0
$235$	− 2182.92i	− 0.605949i
$236$	0	0
$237$	5832.77i	1.59865i
$238$	0	0
$239$	−5059.50	−1.36934	−0.684669	−	0.728854i	$-0.740053\pi$
$239$	−5059.50	−1.36934	−0.684669	+	0.728854i	$0.740053\pi$
$240$	0	0
$241$	−6645.52	−1.77625	−0.888124	−	0.459604i	$-0.847991\pi$
$241$	−6645.52	−1.77625	−0.888124	+	0.459604i	$0.847991\pi$
$242$	0	0
$243$	3984.16i	1.05179i
$244$	0	0
$245$	1136.95i	0.296478i
$246$	0	0
$247$	2097.28	0.540270
$248$	0	0
$249$	−10403.9	−2.64787
$250$	0	0
$251$	1788.72i	0.449812i	0.974380	+	0.224906i	$0.0722075\pi$
$251$	1788.72i	0.449812i	−0.974380	+	0.224906i	$0.927792\pi$
$252$	0	0
$253$	− 611.559i	− 0.151970i
$254$	0	0
$255$	5697.43	1.39917
$256$	0	0
$257$	−5281.41	−1.28189	−0.640944	−	0.767587i	$-0.721457\pi$
$257$	−5281.41	−1.28189	−0.640944	+	0.767587i	$0.721457\pi$
$258$	0	0
$259$	4296.52i	1.03078i
$260$	0	0
$261$	− 8156.94i	− 1.93449i
$262$	0	0
$263$	3527.45	0.827042	0.413521	−	0.910495i	$-0.364299\pi$
$263$	3527.45	0.827042	0.413521	+	0.910495i	$0.364299\pi$
$264$	0	0
$265$	−90.9088	−0.0210735
$266$	0	0
$267$	10075.2i	2.30934i
$268$	0	0
$269$	5325.40i	1.20705i	0.797345	+	0.603523i	$0.206237\pi$
$269$	5325.40i	1.20705i	−0.797345	+	0.603523i	$0.793763\pi$
$270$	0	0
$271$	−1940.33	−0.434932	−0.217466	−	0.976068i	$-0.569779\pi$
$271$	−1940.33	−0.434932	−0.217466	+	0.976068i	$0.569779\pi$
$272$	0	0
$273$	−4500.03	−0.997635
$274$	0	0
$275$	125.384i	0.0274942i
$276$	0	0
$277$	5247.58i	1.13825i	0.822249	+	0.569127i	$0.192719\pi$
$277$	5247.58i	1.13825i	−0.822249	+	0.569127i	$0.807281\pi$
$278$	0	0
$279$	5823.72	1.24967
$280$	0	0
$281$	−137.594	−0.0292106	−0.0146053	−	0.999893i	$-0.504649\pi$
$281$	−137.594	−0.0292106	−0.0146053	+	0.999893i	$0.504649\pi$
$282$	0	0
$283$	− 4758.16i	− 0.999445i	−0.866185	−	0.499723i	$-0.833435\pi$
$283$	− 4758.16i	− 0.999445i	0.866185	−	0.499723i	$-0.166565\pi$
$284$	0	0
$285$	1751.09i	0.363949i
$286$	0	0
$287$	−497.568	−0.102336
$288$	0	0
$289$	13665.8	2.78156
$290$	0	0
$291$	6316.75i	1.27249i
$292$	0	0
$293$	− 276.941i	− 0.0552186i	−0.999619	−	0.0276093i	$-0.991211\pi$
$293$	− 276.941i	− 0.0552186i	0.999619	−	0.0276093i	$-0.00878943\pi$
$294$	0	0
$295$	958.730	0.189218
$296$	0	0
$297$	666.135	0.130145
$298$	0	0
$299$	6104.58i	1.18073i
$300$	0	0
$301$	− 4145.30i	− 0.793790i
$302$	0	0
$303$	3737.12	0.708554
$304$	0	0
$305$	4167.62	0.782418
$306$	0	0
$307$	− 2453.33i	− 0.456087i	−0.973651	−	0.228044i	$-0.926767\pi$
$307$	− 2453.33i	− 0.456087i	0.973651	−	0.228044i	$-0.0732329\pi$
$308$	0	0
$309$	− 4743.98i	− 0.873385i
$310$	0	0
$311$	7555.82	1.37766	0.688829	−	0.724924i	$-0.258125\pi$
$311$	7555.82	1.37766	0.688829	+	0.724924i	$0.258125\pi$
$312$	0	0
$313$	9164.97	1.65506	0.827531	−	0.561419i	$-0.189745\pi$
$313$	9164.97	1.65506	0.827531	+	0.561419i	$0.189745\pi$
$314$	0	0
$315$	− 2305.68i	− 0.412414i
$316$	0	0
$317$	− 3805.97i	− 0.674335i	−0.941445	−	0.337168i	$-0.890531\pi$
$317$	− 3805.97i	− 0.674335i	0.941445	−	0.337168i	$-0.109469\pi$
$318$	0	0
$319$	953.883	0.167421
$320$	0	0
$321$	−9616.94	−1.67217
$322$	0	0
$323$	5710.14i	0.983655i
$324$	0	0
$325$	− 1251.58i	− 0.213616i
$326$	0	0
$327$	−5079.25	−0.858969
$328$	0	0
$329$	−4694.23	−0.786631
$330$	0	0
$331$	970.099i	0.161092i	0.996751	+	0.0805460i	$0.0256664\pi$
$331$	970.099i	0.161092i	−0.996751	+	0.0805460i	$0.974334\pi$
$332$	0	0
$333$	17137.7i	2.82024i
$334$	0	0
$335$	−2231.98	−0.364019
$336$	0	0
$337$	7538.49	1.21854	0.609270	−	0.792963i	$-0.291463\pi$
$337$	7538.49	1.21854	0.609270	+	0.792963i	$0.291463\pi$
$338$	0	0
$339$	2670.44i	0.427842i
$340$	0	0
$341$	681.033i	0.108153i
$342$	0	0
$343$	6132.95	0.965446
$344$	0	0
$345$	−5096.92	−0.795388
$346$	0	0
$347$	10580.1i	1.63680i	0.574647	+	0.818402i	$0.305139\pi$
$347$	10580.1i	1.63680i	−0.574647	+	0.818402i	$0.694861\pi$
$348$	0	0
$349$	− 2830.85i	− 0.434189i	−0.976151	−	0.217095i	$-0.930342\pi$
$349$	− 2830.85i	− 0.434189i	0.976151	−	0.217095i	$-0.0696580\pi$
$350$	0	0
$351$	−6649.36	−1.01116
$352$	0	0
$353$	−5165.74	−0.778880	−0.389440	−	0.921052i	$-0.627331\pi$
$353$	−5165.74	−0.778880	−0.389440	+	0.921052i	$0.627331\pi$
$354$	0	0
$355$	5238.42i	0.783173i
$356$	0	0
$357$	− 12252.0i	− 1.81637i
$358$	0	0
$359$	−378.859	−0.0556976	−0.0278488	−	0.999612i	$-0.508866\pi$
$359$	−378.859	−0.0556976	−0.0278488	+	0.999612i	$0.508866\pi$
$360$	0	0
$361$	5104.01	0.744133
$362$	0	0
$363$	− 10916.7i	− 1.57846i
$364$	0	0
$365$	287.739i	0.0412629i
$366$	0	0
$367$	−3124.61	−0.444423	−0.222211	−	0.974999i	$-0.571328\pi$
$367$	−3124.61	−0.444423	−0.222211	+	0.974999i	$0.571328\pi$
$368$	0	0
$369$	−1984.67	−0.279994
$370$	0	0
$371$	195.494i	0.0273572i
$372$	0	0
$373$	− 2391.67i	− 0.332000i	−0.986126	−	0.166000i	$-0.946915\pi$
$373$	− 2391.67i	− 0.332000i	0.986126	−	0.166000i	$-0.0530852\pi$
$374$	0	0
$375$	1044.99	0.143901
$376$	0	0
$377$	−9521.67	−1.30077
$378$	0	0
$379$	− 10338.4i	− 1.40118i	−0.713565	−	0.700589i	$-0.752921\pi$
$379$	− 10338.4i	− 1.40118i	0.713565	−	0.700589i	$-0.247079\pi$
$380$	0	0
$381$	17844.6i	2.39950i
$382$	0	0
$383$	9766.89	1.30304	0.651521	−	0.758631i	$-0.274131\pi$
$383$	9766.89	1.30304	0.651521	+	0.758631i	$0.274131\pi$
$384$	0	0
$385$	269.630	0.0356925
$386$	0	0
$387$	− 16534.5i	− 2.17183i
$388$	0	0
$389$	2515.15i	0.327823i	0.986475	+	0.163911i	$0.0524111\pi$
$389$	2515.15i	0.327823i	−0.986475	+	0.163911i	$0.947589\pi$
$390$	0	0
$391$	−16620.6	−2.14972
$392$	0	0
$393$	7581.34	0.973099
$394$	0	0
$395$	3488.55i	0.444374i
$396$	0	0
$397$	− 6659.30i	− 0.841866i	−0.907092	−	0.420933i	$-0.861703\pi$
$397$	− 6659.30i	− 0.841866i	0.907092	−	0.420933i	$-0.138297\pi$
$398$	0	0
$399$	3765.61	0.472472
$400$	0	0
$401$	3038.40	0.378380	0.189190	−	0.981941i	$-0.439414\pi$
$401$	3038.40	0.378380	0.189190	+	0.981941i	$0.439414\pi$
$402$	0	0
$403$	− 6798.08i	− 0.840289i
$404$	0	0
$405$	238.066i	0.0292088i
$406$	0	0
$407$	−2004.11	−0.244078
$408$	0	0
$409$	661.959	0.0800288	0.0400144	−	0.999199i	$-0.487260\pi$
$409$	661.959	0.0800288	0.0400144	+	0.999199i	$0.487260\pi$
$410$	0	0
$411$	− 1000.04i	− 0.120021i
$412$	0	0
$413$	− 2061.69i	− 0.245640i
$414$	0	0
$415$	−6222.51	−0.736026
$416$	0	0
$417$	−17105.0	−2.00872
$418$	0	0
$419$	− 6731.33i	− 0.784838i	−0.919787	−	0.392419i	$-0.871638\pi$
$419$	− 6731.33i	− 0.784838i	0.919787	−	0.392419i	$-0.128362\pi$
$420$	0	0
$421$	− 11912.4i	− 1.37904i	−0.724267	−	0.689520i	$-0.757822\pi$
$421$	− 11912.4i	− 1.37904i	0.724267	−	0.689520i	$-0.242178\pi$
$422$	0	0
$423$	−18724.1	−2.15224
$424$	0	0
$425$	3407.60	0.388925
$426$	0	0
$427$	− 8962.22i	− 1.01572i
$428$	0	0
$429$	− 2099.03i	− 0.236229i
$430$	0	0
$431$	2912.87	0.325541	0.162770	−	0.986664i	$-0.447957\pi$
$431$	2912.87	0.325541	0.162770	+	0.986664i	$0.447957\pi$
$432$	0	0
$433$	−4426.43	−0.491272	−0.245636	−	0.969362i	$-0.578997\pi$
$433$	−4426.43	−0.491272	−0.245636	+	0.969362i	$0.578997\pi$
$434$	0	0
$435$	− 7949.96i	− 0.876256i
$436$	0	0
$437$	− 5108.28i	− 0.559181i
$438$	0	0
$439$	−6158.90	−0.669586	−0.334793	−	0.942292i	$-0.608666\pi$
$439$	−6158.90	−0.669586	−0.334793	+	0.942292i	$0.608666\pi$
$440$	0	0
$441$	9752.25	1.05305
$442$	0	0
$443$	− 2261.94i	− 0.242591i	−0.992616	−	0.121296i	$-0.961295\pi$
$443$	− 2261.94i	− 0.242591i	0.992616	−	0.121296i	$-0.0387049\pi$
$444$	0	0
$445$	6025.94i	0.641925i
$446$	0	0
$447$	12148.1	1.28543
$448$	0	0
$449$	13142.3	1.38134	0.690670	−	0.723170i	$-0.257316\pi$
$449$	13142.3	1.38134	0.690670	+	0.723170i	$0.257316\pi$
$450$	0	0
$451$	− 232.090i	− 0.0242321i
$452$	0	0
$453$	− 7468.69i	− 0.774635i
$454$	0	0
$455$	−2691.44	−0.277312
$456$	0	0
$457$	13126.9	1.34365	0.671825	−	0.740710i	$-0.265511\pi$
$457$	13126.9	1.34365	0.671825	+	0.740710i	$0.265511\pi$
$458$	0	0
$459$	− 18103.8i	− 1.84099i
$460$	0	0
$461$	8978.82i	0.907126i	0.891224	+	0.453563i	$0.149847\pi$
$461$	8978.82i	0.907126i	−0.891224	+	0.453563i	$0.850153\pi$
$462$	0	0
$463$	1729.19	0.173569	0.0867844	−	0.996227i	$-0.472341\pi$
$463$	1729.19	0.173569	0.0867844	+	0.996227i	$0.472341\pi$
$464$	0	0
$465$	5675.94	0.566055
$466$	0	0
$467$	− 1151.88i	− 0.114138i	−0.998370	−	0.0570691i	$-0.981824\pi$
$467$	− 1151.88i	− 0.114138i	0.998370	−	0.0570691i	$-0.0181755\pi$
$468$	0	0
$469$	4799.74i	0.472562i
$470$	0	0
$471$	−23160.1	−2.26573
$472$	0	0
$473$	1933.57	0.187961
$474$	0	0
$475$	1047.32i	0.101167i
$476$	0	0
$477$	779.774i	0.0748499i
$478$	0	0
$479$	7174.74	0.684389	0.342194	−	0.939629i	$-0.388830\pi$
$479$	7174.74	0.684389	0.342194	+	0.939629i	$0.388830\pi$
$480$	0	0
$481$	20005.0	1.89636
$482$	0	0
$483$	10960.6i	1.03256i
$484$	0	0
$485$	3778.01i	0.353713i
$486$	0	0
$487$	15644.8	1.45572	0.727858	−	0.685728i	$-0.240516\pi$
$487$	15644.8	1.45572	0.727858	+	0.685728i	$0.240516\pi$
$488$	0	0
$489$	−23475.9	−2.17099
$490$	0	0
$491$	10780.4i	0.990861i	0.868647	+	0.495431i	$0.164990\pi$
$491$	10780.4i	0.990861i	−0.868647	+	0.495431i	$0.835010\pi$
$492$	0	0
$493$	− 25924.1i	− 2.36828i
$494$	0	0
$495$	1075.48	0.0976553
$496$	0	0
$497$	11264.9	1.01670
$498$	0	0
$499$	17443.2i	1.56486i	0.622738	+	0.782430i	$0.286020\pi$
$499$	17443.2i	1.56486i	−0.622738	+	0.782430i	$0.713980\pi$
$500$	0	0
$501$	22357.0i	1.99369i
$502$	0	0
$503$	−227.632	−0.0201782	−0.0100891	−	0.999949i	$-0.503212\pi$
$503$	−227.632	−0.0201782	−0.0100891	+	0.999949i	$0.503212\pi$
$504$	0	0
$505$	2235.15	0.196956
$506$	0	0
$507$	2585.90i	0.226517i
$508$	0	0
$509$	− 11385.4i	− 0.991449i	−0.868480	−	0.495725i	$-0.834903\pi$
$509$	− 11385.4i	− 0.991449i	0.868480	−	0.495725i	$-0.165097\pi$
$510$	0	0
$511$	618.765	0.0535666
$512$	0	0
$513$	5564.15	0.478876
$514$	0	0
$515$	− 2837.35i	− 0.242774i
$516$	0	0
$517$	− 2189.62i	− 0.186266i
$518$	0	0
$519$	10898.2	0.921727
$520$	0	0
$521$	4729.20	0.397678	0.198839	−	0.980032i	$-0.436283\pi$
$521$	4729.20	0.397678	0.198839	+	0.980032i	$0.436283\pi$
$522$	0	0
$523$	7431.63i	0.621343i	0.950517	+	0.310671i	$0.100554\pi$
$523$	7431.63i	0.621343i	−0.950517	+	0.310671i	$0.899446\pi$
$524$	0	0
$525$	− 2247.18i	− 0.186809i
$526$	0	0
$527$	18508.7	1.52989
$528$	0	0
$529$	2701.75	0.222056
$530$	0	0
$531$	− 8223.55i	− 0.672075i
$532$	0	0
$533$	2316.72i	0.188271i
$534$	0	0
$535$	−5751.84	−0.464810
$536$	0	0
$537$	4000.22	0.321457
$538$	0	0
$539$	1140.44i	0.0911359i
$540$	0	0
$541$	− 9217.87i	− 0.732546i	−0.930508	−	0.366273i	$-0.880634\pi$
$541$	− 9217.87i	− 0.732546i	0.930508	−	0.366273i	$-0.119366\pi$
$542$	0	0
$543$	−14650.5	−1.15785
$544$	0	0
$545$	−3037.87	−0.238767
$546$	0	0
$547$	− 15650.6i	− 1.22335i	−0.791109	−	0.611676i	$-0.790496\pi$
$547$	− 15650.6i	− 1.22335i	0.791109	−	0.611676i	$-0.209504\pi$
$548$	0	0
$549$	− 35748.0i	− 2.77903i
$550$	0	0
$551$	7967.69	0.616034
$552$	0	0
$553$	7501.90	0.576878
$554$	0	0
$555$	16702.9i	1.27747i
$556$	0	0
$557$	− 7433.98i	− 0.565508i	−0.959193	−	0.282754i	$-0.908752\pi$
$557$	− 7433.98i	− 0.565508i	0.959193	−	0.282754i	$-0.0912479\pi$
$558$	0	0
$559$	−19300.9	−1.46036
$560$	0	0
$561$	5714.92	0.430096
$562$	0	0
$563$	1421.27i	0.106394i	0.998584	+	0.0531968i	$0.0169411\pi$
$563$	1421.27i	0.106394i	−0.998584	+	0.0531968i	$0.983059\pi$
$564$	0	0
$565$	1597.18i	0.118927i
$566$	0	0
$567$	511.946	0.0379183
$568$	0	0
$569$	15981.1	1.17744	0.588718	−	0.808338i	$-0.299633\pi$
$569$	15981.1	1.17744	0.588718	+	0.808338i	$0.299633\pi$
$570$	0	0
$571$	− 9091.60i	− 0.666325i	−0.942869	−	0.333162i	$-0.891884\pi$
$571$	− 9091.60i	− 0.666325i	0.942869	−	0.333162i	$-0.108116\pi$
$572$	0	0
$573$	316.760i	0.0230939i
$574$	0	0
$575$	−3048.44	−0.221093
$576$	0	0
$577$	−1353.02	−0.0976205	−0.0488102	−	0.998808i	$-0.515543\pi$
$577$	−1353.02	−0.0976205	−0.0488102	+	0.998808i	$0.515543\pi$
$578$	0	0
$579$	4187.55i	0.300568i
$580$	0	0
$581$	13381.1i	0.955495i
$582$	0	0
$583$	−91.1878	−0.00647790
$584$	0	0
$585$	−10735.5	−0.758730
$586$	0	0
$587$	− 3417.61i	− 0.240306i	−0.992755	−	0.120153i	$-0.961661\pi$
$587$	− 3417.61i	− 0.240306i	0.992755	−	0.120153i	$-0.0383385\pi$
$588$	0	0
$589$	5688.60i	0.397953i
$590$	0	0
$591$	21436.6	1.49202
$592$	0	0
$593$	−25023.3	−1.73286	−0.866429	−	0.499300i	$-0.833591\pi$
$593$	−25023.3	−1.73286	−0.866429	+	0.499300i	$0.833591\pi$
$594$	0	0
$595$	− 7327.84i	− 0.504894i
$596$	0	0
$597$	34024.5i	2.33254i
$598$	0	0
$599$	−25035.5	−1.70772	−0.853859	−	0.520505i	$-0.825744\pi$
$599$	−25035.5	−1.70772	−0.853859	+	0.520505i	$0.825744\pi$
$600$	0	0
$601$	−4856.86	−0.329643	−0.164821	−	0.986323i	$-0.552705\pi$
$601$	−4856.86	−0.329643	−0.164821	+	0.986323i	$0.552705\pi$
$602$	0	0
$603$	19144.9i	1.29294i
$604$	0	0
$605$	− 6529.23i	− 0.438762i
$606$	0	0
$607$	−1308.96	−0.0875271	−0.0437636	−	0.999042i	$-0.513935\pi$
$607$	−1308.96	−0.0875271	−0.0437636	+	0.999042i	$0.513935\pi$
$608$	0	0
$609$	−17095.9	−1.13754
$610$	0	0
$611$	21856.8i	1.44719i
$612$	0	0
$613$	− 1202.11i	− 0.0792050i	−0.999216	−	0.0396025i	$-0.987391\pi$
$613$	− 1202.11i	− 0.0792050i	0.999216	−	0.0396025i	$-0.0126092\pi$
$614$	0	0
$615$	−1934.31	−0.126827
$616$	0	0
$617$	−16198.1	−1.05690	−0.528451	−	0.848964i	$-0.677227\pi$
$617$	−16198.1	−1.05690	−0.528451	+	0.848964i	$0.677227\pi$
$618$	0	0
$619$	− 14666.6i	− 0.952341i	−0.879353	−	0.476171i	$-0.842024\pi$
$619$	− 14666.6i	− 0.952341i	0.879353	−	0.476171i	$-0.157976\pi$
$620$	0	0
$621$	16195.7i	1.04655i
$622$	0	0
$623$	12958.4	0.833335
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1756.46i	0.111876i
$628$	0	0
$629$	54466.5i	3.45266i
$630$	0	0
$631$	24235.9	1.52903	0.764513	−	0.644609i	$-0.222980\pi$
$631$	24235.9	1.52903	0.764513	+	0.644609i	$0.222980\pi$
$632$	0	0
$633$	−34838.2	−2.18751
$634$	0	0
$635$	10672.8i	0.666986i
$636$	0	0
$637$	− 11383.9i	− 0.708079i
$638$	0	0
$639$	44932.7	2.78171
$640$	0	0
$641$	−28099.8	−1.73148	−0.865739	−	0.500496i	$-0.833151\pi$
$641$	−28099.8	−1.73148	−0.865739	+	0.500496i	$0.833151\pi$
$642$	0	0
$643$	− 23582.0i	− 1.44632i	−0.690679	−	0.723161i	$-0.742688\pi$
$643$	− 23582.0i	− 1.44632i	0.690679	−	0.723161i	$-0.257312\pi$
$644$	0	0
$645$	− 16115.0i	− 0.983761i
$646$	0	0
$647$	−28370.0	−1.72386	−0.861931	−	0.507026i	$-0.830745\pi$
$647$	−28370.0	−1.72386	−0.861931	+	0.507026i	$0.830745\pi$
$648$	0	0
$649$	961.673	0.0581648
$650$	0	0
$651$	− 12205.8i	− 0.734841i
$652$	0	0
$653$	− 7401.42i	− 0.443553i	−0.975098	−	0.221776i	$-0.928814\pi$
$653$	− 7401.42i	− 0.443553i	0.975098	−	0.221776i	$-0.0711855\pi$
$654$	0	0
$655$	4534.36	0.270491
$656$	0	0
$657$	2468.09	0.146559
$658$	0	0
$659$	24004.4i	1.41893i	0.704739	+	0.709467i	$0.251064\pi$
$659$	24004.4i	1.41893i	−0.704739	+	0.709467i	$0.748936\pi$
$660$	0	0
$661$	23605.5i	1.38903i	0.719479	+	0.694514i	$0.244381\pi$
$661$	23605.5i	1.38903i	−0.719479	+	0.694514i	$0.755619\pi$
$662$	0	0
$663$	−57046.3	−3.34162
$664$	0	0
$665$	2252.19	0.131332
$666$	0	0
$667$	23191.7i	1.34630i
$668$	0	0
$669$	9542.22i	0.551455i
$670$	0	0
$671$	4180.41	0.240511
$672$	0	0
$673$	5322.84	0.304874	0.152437	−	0.988313i	$-0.451288\pi$
$673$	5322.84	0.304874	0.152437	+	0.988313i	$0.451288\pi$
$674$	0	0
$675$	− 3320.48i	− 0.189341i
$676$	0	0
$677$	− 2413.64i	− 0.137022i	−0.997650	−	0.0685109i	$-0.978175\pi$
$677$	− 2413.64i	− 0.137022i	0.997650	−	0.0685109i	$-0.0218248\pi$
$678$	0	0
$679$	8124.38	0.459183
$680$	0	0
$681$	−3427.98	−0.192894
$682$	0	0
$683$	3124.52i	0.175046i	0.996162	+	0.0875230i	$0.0278951\pi$
$683$	3124.52i	0.175046i	−0.996162	+	0.0875230i	$0.972105\pi$
$684$	0	0
$685$	− 598.121i	− 0.0333621i
$686$	0	0
$687$	11143.0	0.618826
$688$	0	0
$689$	910.237	0.0503299
$690$	0	0
$691$	− 24273.9i	− 1.33636i	−0.744000	−	0.668179i	$-0.767074\pi$
$691$	− 24273.9i	− 1.33636i	0.744000	−	0.668179i	$-0.232926\pi$
$692$	0	0
$693$	− 2312.76i	− 0.126774i
$694$	0	0
$695$	−10230.4	−0.558363
$696$	0	0
$697$	−6307.60	−0.342780
$698$	0	0
$699$	− 12505.5i	− 0.676682i
$700$	0	0
$701$	− 25663.7i	− 1.38275i	−0.722498	−	0.691373i	$-0.757006\pi$
$701$	− 25663.7i	− 1.38275i	0.722498	−	0.691373i	$-0.242994\pi$
$702$	0	0
$703$	−16740.1	−0.898101
$704$	0	0
$705$	−18249.0	−0.974888
$706$	0	0
$707$	− 4806.55i	− 0.255685i
$708$	0	0
$709$	8367.54i	0.443230i	0.975134	+	0.221615i	$0.0711327\pi$
$709$	8367.54i	0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$710$	0	0
$711$	29923.2	1.57835
$712$	0	0
$713$	−16557.9	−0.869702
$714$	0	0
$715$	− 1255.42i	− 0.0656644i
$716$	0	0
$717$	42296.8i	2.20307i
$718$	0	0
$719$	27244.1	1.41312	0.706561	−	0.707653i	$-0.250246\pi$
$719$	27244.1	1.41312	0.706561	+	0.707653i	$0.250246\pi$
$720$	0	0
$721$	−6101.55	−0.315164
$722$	0	0
$723$	55555.8i	2.85774i
$724$	0	0
$725$	− 4754.83i	− 0.243572i
$726$	0	0
$727$	−25294.2	−1.29038	−0.645191	−	0.764021i	$-0.723222\pi$
$727$	−25294.2	−1.29038	−0.645191	+	0.764021i	$0.723222\pi$
$728$	0	0
$729$	32021.6	1.62687
$730$	0	0
$731$	− 52549.4i	− 2.65884i
$732$	0	0
$733$	− 22904.8i	− 1.15417i	−0.816683	−	0.577086i	$-0.804190\pi$
$733$	− 22904.8i	− 1.15417i	0.816683	−	0.577086i	$-0.195810\pi$
$734$	0	0
$735$	9504.79	0.476992
$736$	0	0
$737$	−2238.83	−0.111897
$738$	0	0
$739$	18129.2i	0.902427i	0.892416	+	0.451214i	$0.149009\pi$
$739$	18129.2i	0.902427i	−0.892416	+	0.451214i	$0.850991\pi$
$740$	0	0
$741$	− 17533.0i	− 0.869219i
$742$	0	0
$743$	31079.4	1.53458	0.767289	−	0.641302i	$-0.221605\pi$
$743$	31079.4	1.53458	0.767289	+	0.641302i	$0.221605\pi$
$744$	0	0
$745$	7265.74	0.357310
$746$	0	0
$747$	53373.8i	2.61425i
$748$	0	0
$749$	12369.0i	0.603408i
$750$	0	0
$751$	37097.6	1.80254	0.901271	−	0.433255i	$-0.142635\pi$
$751$	37097.6	1.80254	0.901271	+	0.433255i	$0.142635\pi$
$752$	0	0
$753$	14953.5	0.723685
$754$	0	0
$755$	− 4466.98i	− 0.215325i
$756$	0	0
$757$	14835.9i	0.712309i	0.934427	+	0.356155i	$0.115912\pi$
$757$	14835.9i	0.712309i	−0.934427	+	0.356155i	$0.884088\pi$
$758$	0	0
$759$	−5112.56	−0.244498
$760$	0	0
$761$	523.995	0.0249603	0.0124802	−	0.999922i	$-0.496027\pi$
$761$	523.995	0.0249603	0.0124802	+	0.999922i	$0.496027\pi$
$762$	0	0
$763$	6532.75i	0.309962i
$764$	0	0
$765$	− 29228.8i	− 1.38140i
$766$	0	0
$767$	−9599.42	−0.451910
$768$	0	0
$769$	−9483.99	−0.444735	−0.222368	−	0.974963i	$-0.571378\pi$
$769$	−9483.99	−0.444735	−0.222368	+	0.974963i	$0.571378\pi$
$770$	0	0
$771$	44152.0i	2.06238i
$772$	0	0
$773$	− 26421.5i	− 1.22939i	−0.788767	−	0.614693i	$-0.789280\pi$
$773$	− 26421.5i	− 1.22939i	0.788767	−	0.614693i	$-0.210720\pi$
$774$	0	0
$775$	3394.75	0.157346
$776$	0	0
$777$	35918.5	1.65839
$778$	0	0
$779$	− 1938.62i	− 0.0891634i
$780$	0	0
$781$	5254.49i	0.240743i
$782$	0	0
$783$	−25261.3	−1.15296
$784$	0	0
$785$	−13851.9	−0.629803
$786$	0	0
$787$	− 8485.12i	− 0.384322i	−0.981363	−	0.192161i	$-0.938450\pi$
$787$	− 8485.12i	− 0.384322i	0.981363	−	0.192161i	$-0.0615496\pi$
$788$	0	0
$789$	− 29489.1i	− 1.33060i
$790$	0	0
$791$	3434.63	0.154389
$792$	0	0
$793$	−41728.9	−1.86865
$794$	0	0
$795$	759.987i	0.0339044i
$796$	0	0
$797$	27287.5i	1.21276i	0.795174	+	0.606381i	$0.207380\pi$
$797$	27287.5i	1.21276i	−0.795174	+	0.606381i	$0.792620\pi$
$798$	0	0
$799$	−59508.2	−2.63485
$800$	0	0
$801$	51687.7	2.28002
$802$	0	0
$803$	288.622i	0.0126840i
$804$	0	0
$805$	6555.48i	0.287019i
$806$	0	0
$807$	44519.8	1.94197
$808$	0	0
$809$	9999.47	0.434564	0.217282	−	0.976109i	$-0.430281\pi$
$809$	9999.47	0.434564	0.217282	+	0.976109i	$0.430281\pi$
$810$	0	0
$811$	− 9408.95i	− 0.407390i	−0.979034	−	0.203695i	$-0.934705\pi$
$811$	− 9408.95i	− 0.407390i	0.979034	−	0.203695i	$-0.0652950\pi$
$812$	0	0
$813$	16220.9i	0.699745i
$814$	0	0
$815$	−14040.8	−0.603469
$816$	0	0
$817$	16150.9	0.691613
$818$	0	0
$819$	23086.0i	0.984969i
$820$	0	0
$821$	− 21215.6i	− 0.901861i	−0.892559	−	0.450931i	$-0.851092\pi$
$821$	− 21215.6i	− 0.901861i	0.892559	−	0.450931i	$-0.148908\pi$
$822$	0	0
$823$	4645.90	0.196775	0.0983875	−	0.995148i	$-0.468632\pi$
$823$	4645.90	0.196775	0.0983875	+	0.995148i	$0.468632\pi$
$824$	0	0
$825$	1048.19	0.0442344
$826$	0	0
$827$	27578.3i	1.15960i	0.814758	+	0.579802i	$0.196870\pi$
$827$	27578.3i	1.15960i	−0.814758	+	0.579802i	$0.803130\pi$
$828$	0	0
$829$	− 213.593i	− 0.00894859i	−0.999990	−	0.00447430i	$-0.998576\pi$
$829$	− 213.593i	− 0.00894859i	0.999990	−	0.00447430i	$-0.00142422\pi$
$830$	0	0
$831$	43869.2	1.83129
$832$	0	0
$833$	30994.2	1.28918
$834$	0	0
$835$	13371.6i	0.554183i
$836$	0	0
$837$	− 18035.5i	− 0.744802i
$838$	0	0
$839$	−25030.5	−1.02997	−0.514987	−	0.857198i	$-0.672203\pi$
$839$	−25030.5	−1.02997	−0.514987	+	0.857198i	$0.672203\pi$
$840$	0	0
$841$	−11784.4	−0.483184
$842$	0	0
$843$	1150.27i	0.0469959i
$844$	0	0
$845$	1546.61i	0.0629647i
$846$	0	0
$847$	−14040.7	−0.569592
$848$	0	0
$849$	−39777.6	−1.60797
$850$	0	0
$851$	− 48725.6i	− 1.96274i
$852$	0	0
$853$	− 9629.81i	− 0.386540i	−0.981146	−	0.193270i	$-0.938091\pi$
$853$	− 9629.81i	− 0.386540i	0.981146	−	0.193270i	$-0.0619093\pi$
$854$	0	0
$855$	8983.39	0.359328
$856$	0	0
$857$	18302.5	0.729524	0.364762	−	0.931101i	$-0.381150\pi$
$857$	18302.5	0.729524	0.364762	+	0.931101i	$0.381150\pi$
$858$	0	0
$859$	5445.35i	0.216290i	0.994135	+	0.108145i	$0.0344911\pi$
$859$	5445.35i	0.216290i	−0.994135	+	0.108145i	$0.965509\pi$
$860$	0	0
$861$	4159.61i	0.164645i
$862$	0	0
$863$	−32428.2	−1.27911	−0.639554	−	0.768746i	$-0.720881\pi$
$863$	−32428.2	−1.27911	−0.639554	+	0.768746i	$0.720881\pi$
$864$	0	0
$865$	6518.13	0.256212
$866$	0	0
$867$	− 114245.i	− 4.47515i
$868$	0	0
$869$	3499.25i	0.136598i
$870$	0	0
$871$	22348.0	0.869385
$872$	0	0
$873$	32406.0	1.25633
$874$	0	0
$875$	− 1344.02i	− 0.0519272i
$876$	0	0
$877$	42987.0i	1.65515i	0.561354	+	0.827575i	$0.310280\pi$
$877$	42987.0i	1.65515i	−0.561354	+	0.827575i	$0.689720\pi$
$878$	0	0
$879$	−2315.19	−0.0888391
$880$	0	0
$881$	781.459	0.0298843	0.0149421	−	0.999888i	$-0.495244\pi$
$881$	781.459	0.0298843	0.0149421	+	0.999888i	$0.495244\pi$
$882$	0	0
$883$	5419.96i	0.206564i	0.994652	+	0.103282i	$0.0329344\pi$
$883$	5419.96i	0.206564i	−0.994652	+	0.103282i	$0.967066\pi$
$884$	0	0
$885$	− 8014.88i	− 0.304426i
$886$	0	0
$887$	2876.60	0.108892	0.0544458	−	0.998517i	$-0.482661\pi$
$887$	2876.60	0.108892	0.0544458	+	0.998517i	$0.482661\pi$
$888$	0	0
$889$	22951.2	0.865868
$890$	0	0
$891$	238.796i	0.00897865i
$892$	0	0
$893$	− 18289.7i	− 0.685375i
$894$	0	0
$895$	2392.51	0.0893550
$896$	0	0
$897$	51033.6	1.89962
$898$	0	0
$899$	− 25826.3i	− 0.958126i
$900$	0	0
$901$	2478.25i	0.0916342i
$902$	0	0
$903$	−34654.2	−1.27710
$904$	0	0
$905$	−8762.39	−0.321847
$906$	0	0
$907$	− 23255.3i	− 0.851354i	−0.904875	−	0.425677i	$-0.860036\pi$
$907$	− 23255.3i	− 0.851354i	0.904875	−	0.425677i	$-0.139964\pi$
$908$	0	0
$909$	− 19172.1i	− 0.699558i
$910$	0	0
$911$	−22041.2	−0.801600	−0.400800	−	0.916166i	$-0.631268\pi$
$911$	−22041.2	−0.801600	−0.400800	+	0.916166i	$0.631268\pi$
$912$	0	0
$913$	−6241.61	−0.226251
$914$	0	0
$915$	− 34840.9i	− 1.25880i
$916$	0	0
$917$	− 9750.85i	− 0.351147i
$918$	0	0
$919$	−6046.43	−0.217033	−0.108517	−	0.994095i	$-0.534610\pi$
$919$	−6046.43	−0.217033	−0.108517	+	0.994095i	$0.534610\pi$
$920$	0	0
$921$	−20509.5	−0.733781
$922$	0	0
$923$	− 52450.4i	− 1.87045i
$924$	0	0
$925$	9989.88i	0.355098i
$926$	0	0
$927$	−24337.5	−0.862295
$928$	0	0
$929$	−22153.3	−0.782374	−0.391187	−	0.920311i	$-0.627935\pi$
$929$	−22153.3	−0.782374	−0.391187	+	0.920311i	$0.627935\pi$
$930$	0	0
$931$	9525.98i	0.335340i
$932$	0	0
$933$	− 63165.8i	− 2.21646i
$934$	0	0
$935$	3418.06	0.119554
$936$	0	0
$937$	−44131.6	−1.53865	−0.769326	−	0.638856i	$-0.779408\pi$
$937$	−44131.6	−1.53865	−0.769326	+	0.638856i	$0.779408\pi$
$938$	0	0
$939$	− 76618.1i	− 2.66277i
$940$	0	0
$941$	33006.1i	1.14343i	0.820452	+	0.571715i	$0.193722\pi$
$941$	33006.1i	1.14343i	−0.820452	+	0.571715i	$0.806278\pi$
$942$	0	0
$943$	5642.77	0.194861
$944$	0	0
$945$	−7140.50	−0.245799
$946$	0	0
$947$	20759.6i	0.712351i	0.934419	+	0.356176i	$0.115919\pi$
$947$	20759.6i	0.712351i	−0.934419	+	0.356176i	$0.884081\pi$
$948$	0	0
$949$	− 2881.03i	− 0.0985480i
$950$	0	0
$951$	−31817.4	−1.08491
$952$	0	0
$953$	−22426.4	−0.762289	−0.381145	−	0.924515i	$-0.624470\pi$
$953$	−22426.4	−0.762289	−0.381145	+	0.924515i	$0.624470\pi$
$954$	0	0
$955$	189.452i	0.00641940i
$956$	0	0
$957$	− 7974.36i	− 0.269357i
$958$	0	0
$959$	−1286.22	−0.0433100
$960$	0	0
$961$	−11352.1	−0.381058
$962$	0	0
$963$	49336.6i	1.65093i
$964$	0	0
$965$	2504.55i	0.0835485i
$966$	0	0
$967$	30927.5	1.02850	0.514251	−	0.857640i	$-0.328070\pi$
$967$	30927.5	1.02850	0.514251	+	0.857640i	$0.328070\pi$
$968$	0	0
$969$	47736.1	1.58256
$970$	0	0
$971$	23937.9i	0.791149i	0.918434	+	0.395574i	$0.129454\pi$
$971$	23937.9i	0.791149i	−0.918434	+	0.395574i	$0.870546\pi$
$972$	0	0
$973$	21999.9i	0.724856i
$974$	0	0
$975$	−10463.1	−0.343678
$976$	0	0
$977$	−23689.0	−0.775720	−0.387860	−	0.921718i	$-0.626786\pi$
$977$	−23689.0	−0.775720	−0.387860	+	0.921718i	$0.626786\pi$
$978$	0	0
$979$	6044.43i	0.197325i
$980$	0	0
$981$	26057.4i	0.848063i
$982$	0	0
$983$	−22716.1	−0.737063	−0.368531	−	0.929615i	$-0.620139\pi$
$983$	−22716.1	−0.737063	−0.368531	+	0.929615i	$0.620139\pi$
$984$	0	0
$985$	12821.1	0.414735
$986$	0	0
$987$	39243.3i	1.26558i
$988$	0	0
$989$	47010.6i	1.51148i
$990$	0	0
$991$	−55337.3	−1.77381	−0.886905	−	0.461951i	$-0.847149\pi$
$991$	−55337.3	−1.77381	−0.886905	+	0.461951i	$0.847149\pi$
$992$	0	0
$993$	8109.92	0.259175
$994$	0	0
$995$	20349.8i	0.648375i
$996$	0	0
$997$	− 10707.9i	− 0.340143i	−0.985432	−	0.170072i	$-0.945600\pi$
$997$	− 10707.9i	− 0.340143i	0.985432	−	0.170072i	$-0.0543999\pi$
$998$	0	0
$999$	53074.0	1.68087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.d.d.161.1	yes	8
4.3	odd	2		320.4.d.c.161.8	yes	8
8.3	odd	2		320.4.d.c.161.1	✓	8
8.5	even	2	inner	320.4.d.d.161.8	yes	8
16.3	odd	4		1280.4.a.u.1.1		4
16.5	even	4		1280.4.a.v.1.1		4
16.11	odd	4		1280.4.a.z.1.4		4
16.13	even	4		1280.4.a.q.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.4.d.c.161.1	✓	8	8.3	odd	2
320.4.d.c.161.8	yes	8	4.3	odd	2
320.4.d.d.161.1	yes	8	1.1	even	1	trivial
320.4.d.d.161.8	yes	8	8.5	even	2	inner
1280.4.a.q.1.4		4	16.13	even	4
1280.4.a.u.1.1		4	16.3	odd	4
1280.4.a.v.1.1		4	16.5	even	4
1280.4.a.z.1.4		4	16.11	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.35989i q^{3} -5.00000i q^{5} -10.7522 q^{7} -42.8877 q^{9} -5.01534i q^{11} +50.0632i q^{13} -41.7994 q^{15} -136.304 q^{17} -41.8926i q^{19} +89.8871i q^{21} +121.937 q^{23} -25.0000 q^{25} +132.819i q^{27} +190.193i q^{29} -135.790 q^{31} -41.9277 q^{33} +53.7610i q^{35} -399.595i q^{37} +418.523 q^{39} +46.2759 q^{41} +385.530i q^{43} +214.438i q^{45} +436.584 q^{47} -227.390 q^{49} +1139.49i q^{51} -18.1818i q^{53} -25.0767 q^{55} -350.218 q^{57} +191.746i q^{59} +833.525i q^{61} +461.137 q^{63} +250.316 q^{65} -446.397i q^{67} -1019.38i q^{69} -1047.68 q^{71} -57.5478 q^{73} +208.997i q^{75} +53.9259i q^{77} -697.709 q^{79} -47.6132 q^{81} -1244.50i q^{83} +681.521i q^{85} +1589.99 q^{87} -1205.19 q^{89} -538.289i q^{91} +1135.19i q^{93} -209.463 q^{95} -755.602 q^{97} +215.097i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 64 q^{7} - 104 q^{9} - 96 q^{17} + 416 q^{23} - 200 q^{25} - 64 q^{31} + 944 q^{33} + 2176 q^{39} + 208 q^{41} + 1216 q^{47} - 760 q^{49} + 160 q^{55} - 16 q^{57} + 32 q^{63} + 800 q^{65} + 1344 q^{71}+ \cdots - 896 q^{97}+O(q^{100}) \) 8 * q + 64 * q^7 - 104 * q^9 - 96 * q^17 + 416 * q^23 - 200 * q^25 - 64 * q^31 + 944 * q^33 + 2176 * q^39 + 208 * q^41 + 1216 * q^47 - 760 * q^49 + 160 * q^55 - 16 * q^57 + 32 * q^63 + 800 * q^65 + 1344 * q^71 + 576 * q^73 - 3712 * q^79 - 3352 * q^81 + 5600 * q^87 + 1296 * q^89 - 2080 * q^95 - 896 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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