LMFDB - Embedded newform 608.4.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,4,Mod(1,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("608.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	608.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.8731612835$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 $ x^5 - 2x^4 - 28x^3 - 8x^2 + 73x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.31928$ of defining polynomial
Character	$\chi$	$=$	608.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.67449 q^{3} -7.12795 q^{5} -0.511313 q^{7} -13.4981 q^{9} +17.5714 q^{11} +52.1247 q^{13} +26.1915 q^{15} +68.1582 q^{17} -19.0000 q^{19} +1.87881 q^{21} +48.8262 q^{23} -74.1924 q^{25} +148.810 q^{27} -87.1627 q^{29} +53.2537 q^{31} -64.5659 q^{33} +3.64461 q^{35} -199.635 q^{37} -191.532 q^{39} -372.432 q^{41} +416.493 q^{43} +96.2140 q^{45} -221.394 q^{47} -342.739 q^{49} -250.446 q^{51} +107.677 q^{53} -125.248 q^{55} +69.8153 q^{57} +155.785 q^{59} -229.061 q^{61} +6.90178 q^{63} -371.542 q^{65} +580.065 q^{67} -179.411 q^{69} -38.2227 q^{71} -803.910 q^{73} +272.619 q^{75} -8.98449 q^{77} -1034.63 q^{79} -182.350 q^{81} -1053.94 q^{83} -485.828 q^{85} +320.278 q^{87} -641.103 q^{89} -26.6521 q^{91} -195.680 q^{93} +135.431 q^{95} +19.9998 q^{97} -237.181 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 + 5 * q^5 - 7 * q^7 - 5 * q^9 + 13 * q^11 - 72 * q^13 - 72 * q^15 - 59 * q^17 - 95 * q^19 - 224 * q^21 - 52 * q^23 - 86 * q^25 + 54 * q^27 - 128 * q^29 + 110 * q^31 - 68 * q^33 - 45 * q^35 - 436 * q^37 - 356 * q^39 - 804 * q^41 - 143 * q^43 - 579 * q^45 - 661 * q^47 - 406 * q^49 + 570 * q^51 - 898 * q^53 - 17 * q^55 - 114 * q^57 + 196 * q^59 - 1079 * q^61 - 143 * q^63 - 1632 * q^65 + 832 * q^67 - 1644 * q^69 - 834 * q^71 - 1375 * q^73 + 1054 * q^75 - 1473 * q^77 - 154 * q^79 - 1859 * q^81 + 2224 * q^83 - 1807 * q^85 - 2004 * q^87 - 542 * q^89 + 1890 * q^91 - 2788 * q^93 - 95 * q^95 - 1528 * q^97 + 601 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.67449	−0.707155	−0.353578	−	0.935405i	$-0.615035\pi$
$3$	−3.67449	−0.707155	−0.353578	+	0.935405i	$0.615035\pi$
$4$	0	0
$5$	−7.12795	−0.637543	−0.318771	−	0.947832i	$-0.603270\pi$
$5$	−7.12795	−0.637543	−0.318771	+	0.947832i	$0.603270\pi$
$6$	0	0
$7$	−0.511313	−0.0276083	−0.0138042	−	0.999905i	$-0.504394\pi$
$7$	−0.511313	−0.0276083	−0.0138042	+	0.999905i	$0.504394\pi$
$8$	0	0
$9$	−13.4981	−0.499931
$10$	0	0
$11$	17.5714	0.481634	0.240817	−	0.970570i	$-0.422585\pi$
$11$	17.5714	0.481634	0.240817	+	0.970570i	$0.422585\pi$
$12$	0	0
$13$	52.1247	1.11206	0.556030	−	0.831162i	$-0.312324\pi$
$13$	52.1247	1.11206	0.556030	+	0.831162i	$0.312324\pi$
$14$	0	0
$15$	26.1915	0.450842
$16$	0	0
$17$	68.1582	0.972399	0.486200	−	0.873848i	$-0.338383\pi$
$17$	68.1582	0.972399	0.486200	+	0.873848i	$0.338383\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	1.87881	0.0195234
$22$	0	0
$23$	48.8262	0.442651	0.221325	−	0.975200i	$-0.428962\pi$
$23$	48.8262	0.442651	0.221325	+	0.975200i	$0.428962\pi$
$24$	0	0
$25$	−74.1924	−0.593539
$26$	0	0
$27$	148.810	1.06068
$28$	0	0
$29$	−87.1627	−0.558128	−0.279064	−	0.960273i	$-0.590024\pi$
$29$	−87.1627	−0.558128	−0.279064	+	0.960273i	$0.590024\pi$
$30$	0	0
$31$	53.2537	0.308537	0.154269	−	0.988029i	$-0.450698\pi$
$31$	53.2537	0.308537	0.154269	+	0.988029i	$0.450698\pi$
$32$	0	0
$33$	−64.5659	−0.340590
$34$	0	0
$35$	3.64461	0.0176015
$36$	0	0
$37$	−199.635	−0.887020	−0.443510	−	0.896269i	$-0.646267\pi$
$37$	−199.635	−0.887020	−0.443510	+	0.896269i	$0.646267\pi$
$38$	0	0
$39$	−191.532	−0.786400
$40$	0	0
$41$	−372.432	−1.41864	−0.709318	−	0.704889i	$-0.750997\pi$
$41$	−372.432	−1.41864	−0.709318	+	0.704889i	$0.750997\pi$
$42$	0	0
$43$	416.493	1.47708	0.738542	−	0.674207i	$-0.235515\pi$
$43$	416.493	1.47708	0.738542	+	0.674207i	$0.235515\pi$
$44$	0	0
$45$	96.2140	0.318728
$46$	0	0
$47$	−221.394	−0.687098	−0.343549	−	0.939135i	$-0.611629\pi$
$47$	−221.394	−0.687098	−0.343549	+	0.939135i	$0.611629\pi$
$48$	0	0
$49$	−342.739	−0.999238
$50$	0	0
$51$	−250.446	−0.687637
$52$	0	0
$53$	107.677	0.279066	0.139533	−	0.990217i	$-0.455440\pi$
$53$	107.677	0.279066	0.139533	+	0.990217i	$0.455440\pi$
$54$	0	0
$55$	−125.248	−0.307063
$56$	0	0
$57$	69.8153	0.162233
$58$	0	0
$59$	155.785	0.343753	0.171877	−	0.985118i	$-0.445017\pi$
$59$	155.785	0.343753	0.171877	+	0.985118i	$0.445017\pi$
$60$	0	0
$61$	−229.061	−0.480791	−0.240395	−	0.970675i	$-0.577277\pi$
$61$	−229.061	−0.480791	−0.240395	+	0.970675i	$0.577277\pi$
$62$	0	0
$63$	6.90178	0.0138023
$64$	0	0
$65$	−371.542	−0.708987
$66$	0	0
$67$	580.065	1.05770	0.528852	−	0.848714i	$-0.322623\pi$
$67$	580.065	1.05770	0.528852	+	0.848714i	$0.322623\pi$
$68$	0	0
$69$	−179.411	−0.313023
$70$	0	0
$71$	−38.2227	−0.0638901	−0.0319451	−	0.999490i	$-0.510170\pi$
$71$	−38.2227	−0.0638901	−0.0319451	+	0.999490i	$0.510170\pi$
$72$	0	0
$73$	−803.910	−1.28891	−0.644456	−	0.764641i	$-0.722916\pi$
$73$	−803.910	−1.28891	−0.644456	+	0.764641i	$0.722916\pi$
$74$	0	0
$75$	272.619	0.419724
$76$	0	0
$77$	−8.98449	−0.0132971
$78$	0	0
$79$	−1034.63	−1.47348	−0.736739	−	0.676177i	$-0.763635\pi$
$79$	−1034.63	−1.47348	−0.736739	+	0.676177i	$0.763635\pi$
$80$	0	0
$81$	−182.350	−0.250137
$82$	0	0
$83$	−1053.94	−1.39379	−0.696897	−	0.717171i	$-0.745436\pi$
$83$	−1053.94	−1.39379	−0.696897	+	0.717171i	$0.745436\pi$
$84$	0	0
$85$	−485.828	−0.619946
$86$	0	0
$87$	320.278	0.394683
$88$	0	0
$89$	−641.103	−0.763560	−0.381780	−	0.924253i	$-0.624689\pi$
$89$	−641.103	−0.763560	−0.381780	+	0.924253i	$0.624689\pi$
$90$	0	0
$91$	−26.6521	−0.0307021
$92$	0	0
$93$	−195.680	−0.218184
$94$	0	0
$95$	135.431	0.146262
$96$	0	0
$97$	19.9998	0.0209348	0.0104674	−	0.999945i	$-0.496668\pi$
$97$	19.9998	0.0209348	0.0104674	+	0.999945i	$0.496668\pi$
$98$	0	0
$99$	−237.181	−0.240784
$100$	0	0
$101$	−473.527	−0.466512	−0.233256	−	0.972415i	$-0.574938\pi$
$101$	−473.527	−0.466512	−0.233256	+	0.972415i	$0.574938\pi$
$102$	0	0
$103$	−289.467	−0.276913	−0.138456	−	0.990369i	$-0.544214\pi$
$103$	−289.467	−0.276913	−0.138456	+	0.990369i	$0.544214\pi$
$104$	0	0
$105$	−13.3921	−0.0124470
$106$	0	0
$107$	−390.982	−0.353249	−0.176625	−	0.984278i	$-0.556518\pi$
$107$	−390.982	−0.353249	−0.176625	+	0.984278i	$0.556518\pi$
$108$	0	0
$109$	−1268.65	−1.11481	−0.557407	−	0.830239i	$-0.688204\pi$
$109$	−1268.65	−1.11481	−0.557407	+	0.830239i	$0.688204\pi$
$110$	0	0
$111$	733.555	0.627261
$112$	0	0
$113$	1100.46	0.916132	0.458066	−	0.888918i	$-0.348542\pi$
$113$	1100.46	0.916132	0.458066	+	0.888918i	$0.348542\pi$
$114$	0	0
$115$	−348.030	−0.282209
$116$	0	0
$117$	−703.587	−0.555954
$118$	0	0
$119$	−34.8502	−0.0268463
$120$	0	0
$121$	−1022.25	−0.768028
$122$	0	0
$123$	1368.50	1.00320
$124$	0	0
$125$	1419.83	1.01595
$126$	0	0
$127$	−2799.28	−1.95587	−0.977935	−	0.208907i	$-0.933009\pi$
$127$	−2799.28	−1.95587	−0.977935	+	0.208907i	$0.933009\pi$
$128$	0	0
$129$	−1530.40	−1.04453
$130$	0	0
$131$	691.265	0.461039	0.230520	−	0.973068i	$-0.425957\pi$
$131$	691.265	0.461039	0.230520	+	0.973068i	$0.425957\pi$
$132$	0	0
$133$	9.71495	0.00633378
$134$	0	0
$135$	−1060.71	−0.676232
$136$	0	0
$137$	1044.00	0.651057	0.325528	−	0.945532i	$-0.394458\pi$
$137$	1044.00	0.651057	0.325528	+	0.945532i	$0.394458\pi$
$138$	0	0
$139$	1663.13	1.01486	0.507429	−	0.861694i	$-0.330596\pi$
$139$	1663.13	1.01486	0.507429	+	0.861694i	$0.330596\pi$
$140$	0	0
$141$	813.509	0.485885
$142$	0	0
$143$	915.904	0.535607
$144$	0	0
$145$	621.291	0.355830
$146$	0	0
$147$	1259.39	0.706616
$148$	0	0
$149$	−887.849	−0.488157	−0.244079	−	0.969755i	$-0.578485\pi$
$149$	−887.849	−0.488157	−0.244079	+	0.969755i	$0.578485\pi$
$150$	0	0
$151$	−55.6469	−0.0299899	−0.0149950	−	0.999888i	$-0.504773\pi$
$151$	−55.6469	−0.0299899	−0.0149950	+	0.999888i	$0.504773\pi$
$152$	0	0
$153$	−920.009	−0.486133
$154$	0	0
$155$	−379.590	−0.196706
$156$	0	0
$157$	−1643.09	−0.835239	−0.417620	−	0.908622i	$-0.637136\pi$
$157$	−1643.09	−0.835239	−0.417620	+	0.908622i	$0.637136\pi$
$158$	0	0
$159$	−395.656	−0.197343
$160$	0	0
$161$	−24.9655	−0.0122208
$162$	0	0
$163$	2241.78	1.07724	0.538619	−	0.842549i	$-0.318946\pi$
$163$	2241.78	1.07724	0.538619	+	0.842549i	$0.318946\pi$
$164$	0	0
$165$	460.222	0.217141
$166$	0	0
$167$	−2543.96	−1.17879	−0.589393	−	0.807847i	$-0.700633\pi$
$167$	−2543.96	−1.17879	−0.589393	+	0.807847i	$0.700633\pi$
$168$	0	0
$169$	519.985	0.236680
$170$	0	0
$171$	256.465	0.114692
$172$	0	0
$173$	−1084.28	−0.476512	−0.238256	−	0.971202i	$-0.576576\pi$
$173$	−1084.28	−0.476512	−0.238256	+	0.971202i	$0.576576\pi$
$174$	0	0
$175$	37.9356	0.0163866
$176$	0	0
$177$	−572.429	−0.243087
$178$	0	0
$179$	2053.26	0.857362	0.428681	−	0.903456i	$-0.358978\pi$
$179$	2053.26	0.857362	0.428681	+	0.903456i	$0.358978\pi$
$180$	0	0
$181$	−2066.62	−0.848676	−0.424338	−	0.905504i	$-0.639493\pi$
$181$	−2066.62	−0.848676	−0.424338	+	0.905504i	$0.639493\pi$
$182$	0	0
$183$	841.681	0.339994
$184$	0	0
$185$	1422.99	0.565513
$186$	0	0
$187$	1197.64	0.468341
$188$	0	0
$189$	−76.0885	−0.0292837
$190$	0	0
$191$	1250.64	0.473787	0.236893	−	0.971536i	$-0.423871\pi$
$191$	1250.64	0.473787	0.236893	+	0.971536i	$0.423871\pi$
$192$	0	0
$193$	2815.01	1.04989	0.524945	−	0.851136i	$-0.324086\pi$
$193$	2815.01	1.04989	0.524945	+	0.851136i	$0.324086\pi$
$194$	0	0
$195$	1365.23	0.501364
$196$	0	0
$197$	−593.857	−0.214774	−0.107387	−	0.994217i	$-0.534248\pi$
$197$	−593.857	−0.214774	−0.107387	+	0.994217i	$0.534248\pi$
$198$	0	0
$199$	−15.4805	−0.00551450	−0.00275725	−	0.999996i	$-0.500878\pi$
$199$	−15.4805	−0.00551450	−0.00275725	+	0.999996i	$0.500878\pi$
$200$	0	0
$201$	−2131.44	−0.747961
$202$	0	0
$203$	44.5674	0.0154090
$204$	0	0
$205$	2654.67	0.904441
$206$	0	0
$207$	−659.063	−0.221295
$208$	0	0
$209$	−333.857	−0.110495
$210$	0	0
$211$	−2228.10	−0.726961	−0.363481	−	0.931602i	$-0.618412\pi$
$211$	−2228.10	−0.726961	−0.363481	+	0.931602i	$0.618412\pi$
$212$	0	0
$213$	140.449	0.0451802
$214$	0	0
$215$	−2968.74	−0.941704
$216$	0	0
$217$	−27.2293	−0.00851819
$218$	0	0
$219$	2953.96	0.911461
$220$	0	0
$221$	3552.73	1.08137
$222$	0	0
$223$	−2068.17	−0.621054	−0.310527	−	0.950565i	$-0.600506\pi$
$223$	−2068.17	−0.621054	−0.310527	+	0.950565i	$0.600506\pi$
$224$	0	0
$225$	1001.46	0.296729
$226$	0	0
$227$	591.946	0.173079	0.0865393	−	0.996248i	$-0.472419\pi$
$227$	591.946	0.173079	0.0865393	+	0.996248i	$0.472419\pi$
$228$	0	0
$229$	−2519.07	−0.726920	−0.363460	−	0.931610i	$-0.618405\pi$
$229$	−2519.07	−0.726920	−0.363460	+	0.931610i	$0.618405\pi$
$230$	0	0
$231$	33.0134	0.00940313
$232$	0	0
$233$	3985.94	1.12072	0.560359	−	0.828250i	$-0.310663\pi$
$233$	3985.94	1.12072	0.560359	+	0.828250i	$0.310663\pi$
$234$	0	0
$235$	1578.08	0.438054
$236$	0	0
$237$	3801.73	1.04198
$238$	0	0
$239$	−2371.73	−0.641901	−0.320951	−	0.947096i	$-0.604002\pi$
$239$	−2371.73	−0.641901	−0.320951	+	0.947096i	$0.604002\pi$
$240$	0	0
$241$	5295.59	1.41543	0.707715	−	0.706498i	$-0.249726\pi$
$241$	5295.59	1.41543	0.707715	+	0.706498i	$0.249726\pi$
$242$	0	0
$243$	−3347.82	−0.883798
$244$	0	0
$245$	2443.02	0.637057
$246$	0	0
$247$	−990.369	−0.255124
$248$	0	0
$249$	3872.69	0.985629
$250$	0	0
$251$	5766.76	1.45018	0.725088	−	0.688656i	$-0.241799\pi$
$251$	5766.76	1.45018	0.725088	+	0.688656i	$0.241799\pi$
$252$	0	0
$253$	857.945	0.213196
$254$	0	0
$255$	1785.17	0.438398
$256$	0	0
$257$	3631.88	0.881518	0.440759	−	0.897625i	$-0.354709\pi$
$257$	3631.88	0.881518	0.440759	+	0.897625i	$0.354709\pi$
$258$	0	0
$259$	102.076	0.0244891
$260$	0	0
$261$	1176.53	0.279025
$262$	0	0
$263$	−5848.72	−1.37128	−0.685641	−	0.727940i	$-0.740478\pi$
$263$	−5848.72	−1.37128	−0.685641	+	0.727940i	$0.740478\pi$
$264$	0	0
$265$	−767.513	−0.177917
$266$	0	0
$267$	2355.73	0.539955
$268$	0	0
$269$	−4283.31	−0.970848	−0.485424	−	0.874279i	$-0.661335\pi$
$269$	−4283.31	−0.970848	−0.485424	+	0.874279i	$0.661335\pi$
$270$	0	0
$271$	−2350.55	−0.526885	−0.263443	−	0.964675i	$-0.584858\pi$
$271$	−2350.55	−0.526885	−0.263443	+	0.964675i	$0.584858\pi$
$272$	0	0
$273$	97.9327	0.0217112
$274$	0	0
$275$	−1303.66	−0.285869
$276$	0	0
$277$	7901.74	1.71397	0.856985	−	0.515342i	$-0.172335\pi$
$277$	7901.74	1.71397	0.856985	+	0.515342i	$0.172335\pi$
$278$	0	0
$279$	−718.826	−0.154247
$280$	0	0
$281$	−1555.29	−0.330180	−0.165090	−	0.986279i	$-0.552791\pi$
$281$	−1555.29	−0.330180	−0.165090	+	0.986279i	$0.552791\pi$
$282$	0	0
$283$	−3037.74	−0.638073	−0.319036	−	0.947742i	$-0.603359\pi$
$283$	−3037.74	−0.638073	−0.319036	+	0.947742i	$0.603359\pi$
$284$	0	0
$285$	−497.639	−0.103430
$286$	0	0
$287$	190.429	0.0391662
$288$	0	0
$289$	−267.461	−0.0544394
$290$	0	0
$291$	−73.4892	−0.0148042
$292$	0	0
$293$	4477.08	0.892676	0.446338	−	0.894864i	$-0.352728\pi$
$293$	4477.08	0.892676	0.446338	+	0.894864i	$0.352728\pi$
$294$	0	0
$295$	−1110.43	−0.219157
$296$	0	0
$297$	2614.80	0.510862
$298$	0	0
$299$	2545.05	0.492254
$300$	0	0
$301$	−212.959	−0.0407798
$302$	0	0
$303$	1739.97	0.329896
$304$	0	0
$305$	1632.73	0.306525
$306$	0	0
$307$	−1580.87	−0.293893	−0.146947	−	0.989144i	$-0.546945\pi$
$307$	−1580.87	−0.293893	−0.146947	+	0.989144i	$0.546945\pi$
$308$	0	0
$309$	1063.64	0.195820
$310$	0	0
$311$	3241.23	0.590975	0.295488	−	0.955347i	$-0.404518\pi$
$311$	3241.23	0.590975	0.295488	+	0.955347i	$0.404518\pi$
$312$	0	0
$313$	−9579.34	−1.72989	−0.864946	−	0.501865i	$-0.832647\pi$
$313$	−9579.34	−1.72989	−0.864946	+	0.501865i	$0.832647\pi$
$314$	0	0
$315$	−49.1955	−0.00879954
$316$	0	0
$317$	5152.39	0.912894	0.456447	−	0.889751i	$-0.349122\pi$
$317$	5152.39	0.912894	0.456447	+	0.889751i	$0.349122\pi$
$318$	0	0
$319$	−1531.57	−0.268813
$320$	0	0
$321$	1436.66	0.249802
$322$	0	0
$323$	−1295.01	−0.223084
$324$	0	0
$325$	−3867.26	−0.660052
$326$	0	0
$327$	4661.64	0.788347
$328$	0	0
$329$	113.202	0.0189696
$330$	0	0
$331$	−4909.62	−0.815278	−0.407639	−	0.913143i	$-0.633648\pi$
$331$	−4909.62	−0.815278	−0.407639	+	0.913143i	$0.633648\pi$
$332$	0	0
$333$	2694.70	0.443449
$334$	0	0
$335$	−4134.67	−0.674332
$336$	0	0
$337$	−5662.83	−0.915353	−0.457677	−	0.889119i	$-0.651318\pi$
$337$	−5662.83	−0.915353	−0.457677	+	0.889119i	$0.651318\pi$
$338$	0	0
$339$	−4043.64	−0.647847
$340$	0	0
$341$	935.743	0.148602
$342$	0	0
$343$	350.627	0.0551956
$344$	0	0
$345$	1278.83	0.199565
$346$	0	0
$347$	−12194.6	−1.88658	−0.943289	−	0.331973i	$-0.892286\pi$
$347$	−12194.6	−1.88658	−0.943289	+	0.331973i	$0.892286\pi$
$348$	0	0
$349$	2887.35	0.442854	0.221427	−	0.975177i	$-0.428929\pi$
$349$	2887.35	0.442854	0.221427	+	0.975177i	$0.428929\pi$
$350$	0	0
$351$	7756.67	1.17955
$352$	0	0
$353$	1828.06	0.275632	0.137816	−	0.990458i	$-0.455992\pi$
$353$	1828.06	0.275632	0.137816	+	0.990458i	$0.455992\pi$
$354$	0	0
$355$	272.449	0.0407327
$356$	0	0
$357$	128.057	0.0189845
$358$	0	0
$359$	−2027.89	−0.298128	−0.149064	−	0.988828i	$-0.547626\pi$
$359$	−2027.89	−0.298128	−0.149064	+	0.988828i	$0.547626\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	3756.23	0.543115
$364$	0	0
$365$	5730.23	0.821737
$366$	0	0
$367$	3450.93	0.490836	0.245418	−	0.969417i	$-0.421075\pi$
$367$	3450.93	0.490836	0.245418	+	0.969417i	$0.421075\pi$
$368$	0	0
$369$	5027.14	0.709220
$370$	0	0
$371$	−55.0565	−0.00770455
$372$	0	0
$373$	−5713.27	−0.793089	−0.396544	−	0.918016i	$-0.629791\pi$
$373$	−5713.27	−0.793089	−0.396544	+	0.918016i	$0.629791\pi$
$374$	0	0
$375$	−5217.16	−0.718434
$376$	0	0
$377$	−4543.33	−0.620672
$378$	0	0
$379$	7828.71	1.06104	0.530519	−	0.847673i	$-0.321997\pi$
$379$	7828.71	1.06104	0.530519	+	0.847673i	$0.321997\pi$
$380$	0	0
$381$	10285.9	1.38310
$382$	0	0
$383$	11760.3	1.56898	0.784492	−	0.620139i	$-0.212924\pi$
$383$	11760.3	1.56898	0.784492	+	0.620139i	$0.212924\pi$
$384$	0	0
$385$	64.0410	0.00847748
$386$	0	0
$387$	−5621.89	−0.738441
$388$	0	0
$389$	−6717.76	−0.875589	−0.437795	−	0.899075i	$-0.644240\pi$
$389$	−6717.76	−0.875589	−0.437795	+	0.899075i	$0.644240\pi$
$390$	0	0
$391$	3327.90	0.430433
$392$	0	0
$393$	−2540.05	−0.326026
$394$	0	0
$395$	7374.78	0.939406
$396$	0	0
$397$	−12360.2	−1.56257	−0.781287	−	0.624172i	$-0.785436\pi$
$397$	−12360.2	−1.56257	−0.781287	+	0.624172i	$0.785436\pi$
$398$	0	0
$399$	−35.6975	−0.00447897
$400$	0	0
$401$	2006.45	0.249869	0.124934	−	0.992165i	$-0.460128\pi$
$401$	2006.45	0.249869	0.124934	+	0.992165i	$0.460128\pi$
$402$	0	0
$403$	2775.83	0.343112
$404$	0	0
$405$	1299.78	0.159473
$406$	0	0
$407$	−3507.86	−0.427219
$408$	0	0
$409$	5154.98	0.623221	0.311611	−	0.950210i	$-0.399132\pi$
$409$	5154.98	0.623221	0.311611	+	0.950210i	$0.399132\pi$
$410$	0	0
$411$	−3836.16	−0.460398
$412$	0	0
$413$	−79.6548	−0.00949045
$414$	0	0
$415$	7512.42	0.888603
$416$	0	0
$417$	−6111.17	−0.717662
$418$	0	0
$419$	5037.60	0.587357	0.293679	−	0.955904i	$-0.405120\pi$
$419$	5037.60	0.587357	0.293679	+	0.955904i	$0.405120\pi$
$420$	0	0
$421$	−4025.51	−0.466013	−0.233006	−	0.972475i	$-0.574856\pi$
$421$	−4025.51	−0.466013	−0.233006	+	0.972475i	$0.574856\pi$
$422$	0	0
$423$	2988.41	0.343502
$424$	0	0
$425$	−5056.82	−0.577157
$426$	0	0
$427$	117.122	0.0132738
$428$	0	0
$429$	−3365.48	−0.378757
$430$	0	0
$431$	−2863.81	−0.320057	−0.160029	−	0.987112i	$-0.551159\pi$
$431$	−2863.81	−0.320057	−0.160029	+	0.987112i	$0.551159\pi$
$432$	0	0
$433$	−6617.66	−0.734468	−0.367234	−	0.930129i	$-0.619695\pi$
$433$	−6617.66	−0.734468	−0.367234	+	0.930129i	$0.619695\pi$
$434$	0	0
$435$	−2282.93	−0.251627
$436$	0	0
$437$	−927.698	−0.101551
$438$	0	0
$439$	−8810.03	−0.957813	−0.478907	−	0.877866i	$-0.658967\pi$
$439$	−8810.03	−0.957813	−0.478907	+	0.877866i	$0.658967\pi$
$440$	0	0
$441$	4626.33	0.499550
$442$	0	0
$443$	202.487	0.0217165	0.0108583	−	0.999941i	$-0.496544\pi$
$443$	202.487	0.0217165	0.0108583	+	0.999941i	$0.496544\pi$
$444$	0	0
$445$	4569.75	0.486802
$446$	0	0
$447$	3262.39	0.345203
$448$	0	0
$449$	4619.26	0.485516	0.242758	−	0.970087i	$-0.421948\pi$
$449$	4619.26	0.485516	0.242758	+	0.970087i	$0.421948\pi$
$450$	0	0
$451$	−6544.15	−0.683264
$452$	0	0
$453$	204.474	0.0212075
$454$	0	0
$455$	189.974	0.0195739
$456$	0	0
$457$	−8196.37	−0.838972	−0.419486	−	0.907762i	$-0.637790\pi$
$457$	−8196.37	−0.838972	−0.419486	+	0.907762i	$0.637790\pi$
$458$	0	0
$459$	10142.6	1.03141
$460$	0	0
$461$	−1269.48	−0.128255	−0.0641277	−	0.997942i	$-0.520426\pi$
$461$	−1269.48	−0.128255	−0.0641277	+	0.997942i	$0.520426\pi$
$462$	0	0
$463$	8502.91	0.853485	0.426743	−	0.904373i	$-0.359661\pi$
$463$	8502.91	0.853485	0.426743	+	0.904373i	$0.359661\pi$
$464$	0	0
$465$	1394.80	0.139101
$466$	0	0
$467$	−8132.55	−0.805845	−0.402923	−	0.915234i	$-0.632006\pi$
$467$	−8132.55	−0.805845	−0.402923	+	0.915234i	$0.632006\pi$
$468$	0	0
$469$	−296.595	−0.0292014
$470$	0	0
$471$	6037.50	0.590644
$472$	0	0
$473$	7318.37	0.711414
$474$	0	0
$475$	1409.66	0.136167
$476$	0	0
$477$	−1453.43	−0.139514
$478$	0	0
$479$	15965.6	1.52293	0.761466	−	0.648205i	$-0.224480\pi$
$479$	15965.6	1.52293	0.761466	+	0.648205i	$0.224480\pi$
$480$	0	0
$481$	−10405.9	−0.986421
$482$	0	0
$483$	91.7353	0.00864203
$484$	0	0
$485$	−142.558	−0.0133468
$486$	0	0
$487$	−11358.3	−1.05687	−0.528435	−	0.848974i	$-0.677221\pi$
$487$	−11358.3	−1.05687	−0.528435	+	0.848974i	$0.677221\pi$
$488$	0	0
$489$	−8237.40	−0.761775
$490$	0	0
$491$	−5256.31	−0.483124	−0.241562	−	0.970385i	$-0.577660\pi$
$491$	−5256.31	−0.483124	−0.241562	+	0.970385i	$0.577660\pi$
$492$	0	0
$493$	−5940.85	−0.542723
$494$	0	0
$495$	1690.62	0.153510
$496$	0	0
$497$	19.5438	0.00176390
$498$	0	0
$499$	−11667.7	−1.04673	−0.523364	−	0.852109i	$-0.675323\pi$
$499$	−11667.7	−1.04673	−0.523364	+	0.852109i	$0.675323\pi$
$500$	0	0
$501$	9347.73	0.833585
$502$	0	0
$503$	−16902.2	−1.49828	−0.749139	−	0.662413i	$-0.769533\pi$
$503$	−16902.2	−1.49828	−0.749139	+	0.662413i	$0.769533\pi$
$504$	0	0
$505$	3375.27	0.297421
$506$	0	0
$507$	−1910.68	−0.167369
$508$	0	0
$509$	2828.32	0.246293	0.123147	−	0.992388i	$-0.460701\pi$
$509$	2828.32	0.246293	0.123147	+	0.992388i	$0.460701\pi$
$510$	0	0
$511$	411.050	0.0355847
$512$	0	0
$513$	−2827.39	−0.243338
$514$	0	0
$515$	2063.30	0.176544
$516$	0	0
$517$	−3890.20	−0.330930
$518$	0	0
$519$	3984.18	0.336968
$520$	0	0
$521$	6552.16	0.550970	0.275485	−	0.961305i	$-0.411162\pi$
$521$	6552.16	0.550970	0.275485	+	0.961305i	$0.411162\pi$
$522$	0	0
$523$	7136.05	0.596631	0.298315	−	0.954467i	$-0.403575\pi$
$523$	7136.05	0.596631	0.298315	+	0.954467i	$0.403575\pi$
$524$	0	0
$525$	−139.394	−0.0115879
$526$	0	0
$527$	3629.68	0.300021
$528$	0	0
$529$	−9783.00	−0.804060
$530$	0	0
$531$	−2102.80	−0.171853
$532$	0	0
$533$	−19412.9	−1.57761
$534$	0	0
$535$	2786.90	0.225212
$536$	0	0
$537$	−7544.68	−0.606288
$538$	0	0
$539$	−6022.40	−0.481267
$540$	0	0
$541$	−22860.0	−1.81669	−0.908343	−	0.418227i	$-0.862652\pi$
$541$	−22860.0	−1.81669	−0.908343	+	0.418227i	$0.862652\pi$
$542$	0	0
$543$	7593.76	0.600146
$544$	0	0
$545$	9042.88	0.710742
$546$	0	0
$547$	−9176.33	−0.717279	−0.358639	−	0.933476i	$-0.616759\pi$
$547$	−9176.33	−0.717279	−0.358639	+	0.933476i	$0.616759\pi$
$548$	0	0
$549$	3091.90	0.240362
$550$	0	0
$551$	1656.09	0.128043
$552$	0	0
$553$	529.019	0.0406803
$554$	0	0
$555$	−5228.74	−0.399906
$556$	0	0
$557$	−23966.7	−1.82316	−0.911581	−	0.411120i	$-0.865138\pi$
$557$	−23966.7	−1.82316	−0.911581	+	0.411120i	$0.865138\pi$
$558$	0	0
$559$	21709.6	1.64261
$560$	0	0
$561$	−4400.70	−0.331190
$562$	0	0
$563$	−1356.83	−0.101569	−0.0507847	−	0.998710i	$-0.516172\pi$
$563$	−1356.83	−0.101569	−0.0507847	+	0.998710i	$0.516172\pi$
$564$	0	0
$565$	−7844.04	−0.584073
$566$	0	0
$567$	93.2381	0.00690588
$568$	0	0
$569$	−16869.9	−1.24292	−0.621461	−	0.783445i	$-0.713461\pi$
$569$	−16869.9	−1.24292	−0.621461	+	0.783445i	$0.713461\pi$
$570$	0	0
$571$	−19377.3	−1.42016	−0.710082	−	0.704119i	$-0.751342\pi$
$571$	−19377.3	−1.42016	−0.710082	+	0.704119i	$0.751342\pi$
$572$	0	0
$573$	−4595.47	−0.335041
$574$	0	0
$575$	−3622.53	−0.262730
$576$	0	0
$577$	−16335.8	−1.17863	−0.589313	−	0.807905i	$-0.700602\pi$
$577$	−16335.8	−1.17863	−0.589313	+	0.807905i	$0.700602\pi$
$578$	0	0
$579$	−10343.7	−0.742435
$580$	0	0
$581$	538.893	0.0384803
$582$	0	0
$583$	1892.03	0.134408
$584$	0	0
$585$	5015.13	0.354445
$586$	0	0
$587$	−9216.05	−0.648019	−0.324009	−	0.946054i	$-0.605031\pi$
$587$	−9216.05	−0.648019	−0.324009	+	0.946054i	$0.605031\pi$
$588$	0	0
$589$	−1011.82	−0.0707833
$590$	0	0
$591$	2182.12	0.151879
$592$	0	0
$593$	−7176.76	−0.496988	−0.248494	−	0.968633i	$-0.579936\pi$
$593$	−7176.76	−0.496988	−0.248494	+	0.968633i	$0.579936\pi$
$594$	0	0
$595$	248.410	0.0171157
$596$	0	0
$597$	56.8830	0.00389961
$598$	0	0
$599$	3445.62	0.235032	0.117516	−	0.993071i	$-0.462507\pi$
$599$	3445.62	0.235032	0.117516	+	0.993071i	$0.462507\pi$
$600$	0	0
$601$	17940.3	1.21764	0.608820	−	0.793308i	$-0.291643\pi$
$601$	17940.3	1.21764	0.608820	+	0.793308i	$0.291643\pi$
$602$	0	0
$603$	−7829.80	−0.528779
$604$	0	0
$605$	7286.51	0.489651
$606$	0	0
$607$	14232.0	0.951664	0.475832	−	0.879536i	$-0.342147\pi$
$607$	14232.0	0.951664	0.475832	+	0.879536i	$0.342147\pi$
$608$	0	0
$609$	−163.763	−0.0108965
$610$	0	0
$611$	−11540.1	−0.764095
$612$	0	0
$613$	19440.3	1.28089	0.640446	−	0.768003i	$-0.278749\pi$
$613$	19440.3	1.28089	0.640446	+	0.768003i	$0.278749\pi$
$614$	0	0
$615$	−9754.56	−0.639580
$616$	0	0
$617$	−359.820	−0.0234778	−0.0117389	−	0.999931i	$-0.503737\pi$
$617$	−359.820	−0.0234778	−0.0117389	+	0.999931i	$0.503737\pi$
$618$	0	0
$619$	18561.5	1.20525	0.602624	−	0.798026i	$-0.294122\pi$
$619$	18561.5	1.20525	0.602624	+	0.798026i	$0.294122\pi$
$620$	0	0
$621$	7265.82	0.469513
$622$	0	0
$623$	327.805	0.0210806
$624$	0	0
$625$	−846.443	−0.0541723
$626$	0	0
$627$	1226.75	0.0781368
$628$	0	0
$629$	−13606.7	−0.862538
$630$	0	0
$631$	−9684.27	−0.610974	−0.305487	−	0.952196i	$-0.598819\pi$
$631$	−9684.27	−0.610974	−0.305487	+	0.952196i	$0.598819\pi$
$632$	0	0
$633$	8187.13	0.514075
$634$	0	0
$635$	19953.1	1.24695
$636$	0	0
$637$	−17865.1	−1.11121
$638$	0	0
$639$	515.935	0.0319407
$640$	0	0
$641$	−4576.97	−0.282027	−0.141014	−	0.990008i	$-0.545036\pi$
$641$	−4576.97	−0.282027	−0.141014	+	0.990008i	$0.545036\pi$
$642$	0	0
$643$	−9722.95	−0.596323	−0.298161	−	0.954515i	$-0.596373\pi$
$643$	−9722.95	−0.596323	−0.298161	+	0.954515i	$0.596373\pi$
$644$	0	0
$645$	10908.6	0.665931
$646$	0	0
$647$	−9150.86	−0.556040	−0.278020	−	0.960575i	$-0.589678\pi$
$647$	−9150.86	−0.556040	−0.278020	+	0.960575i	$0.589678\pi$
$648$	0	0
$649$	2737.36	0.165563
$650$	0	0
$651$	100.054	0.00602369
$652$	0	0
$653$	−9202.67	−0.551498	−0.275749	−	0.961230i	$-0.588926\pi$
$653$	−9202.67	−0.551498	−0.275749	+	0.961230i	$0.588926\pi$
$654$	0	0
$655$	−4927.30	−0.293932
$656$	0	0
$657$	10851.3	0.644367
$658$	0	0
$659$	−5007.55	−0.296004	−0.148002	−	0.988987i	$-0.547284\pi$
$659$	−5007.55	−0.296004	−0.148002	+	0.988987i	$0.547284\pi$
$660$	0	0
$661$	−21922.2	−1.28998	−0.644989	−	0.764192i	$-0.723138\pi$
$661$	−21922.2	−1.28998	−0.644989	+	0.764192i	$0.723138\pi$
$662$	0	0
$663$	−13054.4	−0.764695
$664$	0	0
$665$	−69.2477	−0.00403806
$666$	0	0
$667$	−4255.82	−0.247056
$668$	0	0
$669$	7599.47	0.439182
$670$	0	0
$671$	−4024.92	−0.231565
$672$	0	0
$673$	13736.9	0.786806	0.393403	−	0.919366i	$-0.371298\pi$
$673$	13736.9	0.786806	0.393403	+	0.919366i	$0.371298\pi$
$674$	0	0
$675$	−11040.6	−0.629558
$676$	0	0
$677$	20015.0	1.13625	0.568123	−	0.822944i	$-0.307670\pi$
$677$	20015.0	1.13625	0.568123	+	0.822944i	$0.307670\pi$
$678$	0	0
$679$	−10.2262	−0.000577975 0
$680$	0	0
$681$	−2175.10	−0.122393
$682$	0	0
$683$	23674.0	1.32630	0.663148	−	0.748488i	$-0.269220\pi$
$683$	23674.0	1.32630	0.663148	+	0.748488i	$0.269220\pi$
$684$	0	0
$685$	−7441.56	−0.415077
$686$	0	0
$687$	9256.29	0.514046
$688$	0	0
$689$	5612.61	0.310339
$690$	0	0
$691$	2796.62	0.153963	0.0769815	−	0.997033i	$-0.475472\pi$
$691$	2796.62	0.153963	0.0769815	+	0.997033i	$0.475472\pi$
$692$	0	0
$693$	121.274	0.00664765
$694$	0	0
$695$	−11854.7	−0.647015
$696$	0	0
$697$	−25384.3	−1.37948
$698$	0	0
$699$	−14646.3	−0.792522
$700$	0	0
$701$	−222.627	−0.0119950	−0.00599750	−	0.999982i	$-0.501909\pi$
$701$	−222.627	−0.0119950	−0.00599750	+	0.999982i	$0.501909\pi$
$702$	0	0
$703$	3793.06	0.203496
$704$	0	0
$705$	−5798.65	−0.309773
$706$	0	0
$707$	242.121	0.0128796
$708$	0	0
$709$	−1068.75	−0.0566119	−0.0283059	−	0.999599i	$-0.509011\pi$
$709$	−1068.75	−0.0566119	−0.0283059	+	0.999599i	$0.509011\pi$
$710$	0	0
$711$	13965.6	0.736638
$712$	0	0
$713$	2600.18	0.136574
$714$	0	0
$715$	−6528.52	−0.341472
$716$	0	0
$717$	8714.89	0.453924
$718$	0	0
$719$	27102.1	1.40575	0.702876	−	0.711312i	$-0.251899\pi$
$719$	27102.1	1.40575	0.702876	+	0.711312i	$0.251899\pi$
$720$	0	0
$721$	148.008	0.00764509
$722$	0	0
$723$	−19458.6	−1.00093
$724$	0	0
$725$	6466.81	0.331271
$726$	0	0
$727$	6285.65	0.320663	0.160331	−	0.987063i	$-0.448744\pi$
$727$	6285.65	0.320663	0.160331	+	0.987063i	$0.448744\pi$
$728$	0	0
$729$	17225.0	0.875120
$730$	0	0
$731$	28387.4	1.43632
$732$	0	0
$733$	1594.38	0.0803406	0.0401703	−	0.999193i	$-0.487210\pi$
$733$	1594.38	0.0803406	0.0401703	+	0.999193i	$0.487210\pi$
$734$	0	0
$735$	−8976.85	−0.450498
$736$	0	0
$737$	10192.6	0.509427
$738$	0	0
$739$	−11152.4	−0.555141	−0.277571	−	0.960705i	$-0.589529\pi$
$739$	−11152.4	−0.555141	−0.277571	+	0.960705i	$0.589529\pi$
$740$	0	0
$741$	3639.10	0.180413
$742$	0	0
$743$	35565.1	1.75606	0.878032	−	0.478601i	$-0.158856\pi$
$743$	35565.1	1.75606	0.878032	+	0.478601i	$0.158856\pi$
$744$	0	0
$745$	6328.54	0.311221
$746$	0	0
$747$	14226.2	0.696801
$748$	0	0
$749$	199.915	0.00975263
$750$	0	0
$751$	33922.2	1.64825	0.824126	−	0.566406i	$-0.191667\pi$
$751$	33922.2	1.64825	0.824126	+	0.566406i	$0.191667\pi$
$752$	0	0
$753$	−21189.9	−1.02550
$754$	0	0
$755$	396.648	0.0191199
$756$	0	0
$757$	−18859.2	−0.905479	−0.452740	−	0.891643i	$-0.649553\pi$
$757$	−18859.2	−0.905479	−0.452740	+	0.891643i	$0.649553\pi$
$758$	0	0
$759$	−3152.51	−0.150763
$760$	0	0
$761$	15707.4	0.748217	0.374109	−	0.927385i	$-0.377949\pi$
$761$	15707.4	0.748217	0.374109	+	0.927385i	$0.377949\pi$
$762$	0	0
$763$	648.679	0.0307782
$764$	0	0
$765$	6557.78	0.309931
$766$	0	0
$767$	8120.23	0.382275
$768$	0	0
$769$	1292.73	0.0606202	0.0303101	−	0.999541i	$-0.490351\pi$
$769$	1292.73	0.0606202	0.0303101	+	0.999541i	$0.490351\pi$
$770$	0	0
$771$	−13345.3	−0.623370
$772$	0	0
$773$	38483.0	1.79060	0.895301	−	0.445461i	$-0.146960\pi$
$773$	38483.0	1.79060	0.895301	+	0.445461i	$0.146960\pi$
$774$	0	0
$775$	−3951.02	−0.183129
$776$	0	0
$777$	−375.077	−0.0173176
$778$	0	0
$779$	7076.20	0.325457
$780$	0	0
$781$	−671.626	−0.0307717
$782$	0	0
$783$	−12970.7	−0.591997
$784$	0	0
$785$	11711.8	0.532501
$786$	0	0
$787$	−18911.7	−0.856579	−0.428290	−	0.903641i	$-0.640884\pi$
$787$	−18911.7	−0.856579	−0.428290	+	0.903641i	$0.640884\pi$
$788$	0	0
$789$	21491.0	0.969710
$790$	0	0
$791$	−562.682	−0.0252929
$792$	0	0
$793$	−11939.7	−0.534668
$794$	0	0
$795$	2820.22	0.125815
$796$	0	0
$797$	35376.1	1.57225	0.786127	−	0.618065i	$-0.212083\pi$
$797$	35376.1	1.57225	0.786127	+	0.618065i	$0.212083\pi$
$798$	0	0
$799$	−15089.8	−0.668134
$800$	0	0
$801$	8653.71	0.381727
$802$	0	0
$803$	−14125.8	−0.620784
$804$	0	0
$805$	177.953	0.00779131
$806$	0	0
$807$	15739.0	0.686541
$808$	0	0
$809$	29057.9	1.26282	0.631411	−	0.775448i	$-0.282476\pi$
$809$	29057.9	1.26282	0.631411	+	0.775448i	$0.282476\pi$
$810$	0	0
$811$	25476.2	1.10307	0.551534	−	0.834152i	$-0.314043\pi$
$811$	25476.2	1.10307	0.551534	+	0.834152i	$0.314043\pi$
$812$	0	0
$813$	8637.07	0.372590
$814$	0	0
$815$	−15979.3	−0.686786
$816$	0	0
$817$	−7913.37	−0.338866
$818$	0	0
$819$	359.753	0.0153490
$820$	0	0
$821$	41186.3	1.75081	0.875403	−	0.483394i	$-0.160596\pi$
$821$	41186.3	1.75081	0.875403	+	0.483394i	$0.160596\pi$
$822$	0	0
$823$	−9004.24	−0.381371	−0.190685	−	0.981651i	$-0.561071\pi$
$823$	−9004.24	−0.381371	−0.190685	+	0.981651i	$0.561071\pi$
$824$	0	0
$825$	4790.30	0.202154
$826$	0	0
$827$	11943.7	0.502204	0.251102	−	0.967961i	$-0.419207\pi$
$827$	11943.7	0.502204	0.251102	+	0.967961i	$0.419207\pi$
$828$	0	0
$829$	869.327	0.0364209	0.0182105	−	0.999834i	$-0.494203\pi$
$829$	869.327	0.0364209	0.0182105	+	0.999834i	$0.494203\pi$
$830$	0	0
$831$	−29034.8	−1.21204
$832$	0	0
$833$	−23360.4	−0.971658
$834$	0	0
$835$	18133.2	0.751526
$836$	0	0
$837$	7924.68	0.327261
$838$	0	0
$839$	−1265.71	−0.0520824	−0.0260412	−	0.999661i	$-0.508290\pi$
$839$	−1265.71	−0.0520824	−0.0260412	+	0.999661i	$0.508290\pi$
$840$	0	0
$841$	−16791.7	−0.688493
$842$	0	0
$843$	5714.88	0.233489
$844$	0	0
$845$	−3706.43	−0.150893
$846$	0	0
$847$	522.688	0.0212040
$848$	0	0
$849$	11162.1	0.451217
$850$	0	0
$851$	−9747.40	−0.392640
$852$	0	0
$853$	15823.2	0.635143	0.317572	−	0.948234i	$-0.397133\pi$
$853$	15823.2	0.635143	0.317572	+	0.948234i	$0.397133\pi$
$854$	0	0
$855$	−1828.07	−0.0731211
$856$	0	0
$857$	36570.0	1.45765	0.728825	−	0.684700i	$-0.240067\pi$
$857$	36570.0	1.45765	0.728825	+	0.684700i	$0.240067\pi$
$858$	0	0
$859$	18900.5	0.750731	0.375366	−	0.926877i	$-0.377517\pi$
$859$	18900.5	0.750731	0.375366	+	0.926877i	$0.377517\pi$
$860$	0	0
$861$	−699.730	−0.0276966
$862$	0	0
$863$	30643.8	1.20872	0.604361	−	0.796710i	$-0.293428\pi$
$863$	30643.8	1.20872	0.604361	+	0.796710i	$0.293428\pi$
$864$	0	0
$865$	7728.71	0.303797
$866$	0	0
$867$	982.780	0.0384971
$868$	0	0
$869$	−18179.9	−0.709678
$870$	0	0
$871$	30235.7	1.17623
$872$	0	0
$873$	−269.961	−0.0104660
$874$	0	0
$875$	−725.979	−0.0280487
$876$	0	0
$877$	−33839.9	−1.30295	−0.651477	−	0.758668i	$-0.725850\pi$
$877$	−33839.9	−1.30295	−0.651477	+	0.758668i	$0.725850\pi$
$878$	0	0
$879$	−16451.0	−0.631261
$880$	0	0
$881$	−3158.38	−0.120781	−0.0603907	−	0.998175i	$-0.519235\pi$
$881$	−3158.38	−0.120781	−0.0603907	+	0.998175i	$0.519235\pi$
$882$	0	0
$883$	39820.3	1.51762	0.758810	−	0.651312i	$-0.225781\pi$
$883$	39820.3	1.51762	0.758810	+	0.651312i	$0.225781\pi$
$884$	0	0
$885$	4080.24	0.154978
$886$	0	0
$887$	−50176.9	−1.89941	−0.949705	−	0.313146i	$-0.898617\pi$
$887$	−50176.9	−1.89941	−0.949705	+	0.313146i	$0.898617\pi$
$888$	0	0
$889$	1431.31	0.0539983
$890$	0	0
$891$	−3204.15	−0.120475
$892$	0	0
$893$	4206.48	0.157631
$894$	0	0
$895$	−14635.5	−0.546605
$896$	0	0
$897$	−9351.76	−0.348100
$898$	0	0
$899$	−4641.74	−0.172203
$900$	0	0
$901$	7339.04	0.271364
$902$	0	0
$903$	782.513	0.0288377
$904$	0	0
$905$	14730.7	0.541068
$906$	0	0
$907$	−42737.3	−1.56457	−0.782287	−	0.622918i	$-0.785947\pi$
$907$	−42737.3	−1.56457	−0.782287	+	0.622918i	$0.785947\pi$
$908$	0	0
$909$	6391.74	0.233224
$910$	0	0
$911$	−47034.3	−1.71056	−0.855278	−	0.518170i	$-0.826614\pi$
$911$	−47034.3	−1.71056	−0.855278	+	0.518170i	$0.826614\pi$
$912$	0	0
$913$	−18519.2	−0.671299
$914$	0	0
$915$	−5999.46	−0.216761
$916$	0	0
$917$	−353.453	−0.0127285
$918$	0	0
$919$	35256.7	1.26552	0.632759	−	0.774349i	$-0.281922\pi$
$919$	35256.7	1.26552	0.632759	+	0.774349i	$0.281922\pi$
$920$	0	0
$921$	5808.90	0.207828
$922$	0	0
$923$	−1992.35	−0.0710497
$924$	0	0
$925$	14811.4	0.526481
$926$	0	0
$927$	3907.26	0.138437
$928$	0	0
$929$	3588.82	0.126744	0.0633720	−	0.997990i	$-0.479815\pi$
$929$	3588.82	0.126744	0.0633720	+	0.997990i	$0.479815\pi$
$930$	0	0
$931$	6512.03	0.229241
$932$	0	0
$933$	−11909.9	−0.417911
$934$	0	0
$935$	−8536.68	−0.298587
$936$	0	0
$937$	−13741.6	−0.479101	−0.239551	−	0.970884i	$-0.577000\pi$
$937$	−13741.6	−0.479101	−0.239551	+	0.970884i	$0.577000\pi$
$938$	0	0
$939$	35199.1	1.22330
$940$	0	0
$941$	31893.3	1.10488	0.552440	−	0.833553i	$-0.313697\pi$
$941$	31893.3	1.10488	0.552440	+	0.833553i	$0.313697\pi$
$942$	0	0
$943$	−18184.4	−0.627960
$944$	0	0
$945$	542.355	0.0186696
$946$	0	0
$947$	22540.6	0.773465	0.386733	−	0.922192i	$-0.373604\pi$
$947$	22540.6	0.773465	0.386733	+	0.922192i	$0.373604\pi$
$948$	0	0
$949$	−41903.6	−1.43335
$950$	0	0
$951$	−18932.4	−0.645558
$952$	0	0
$953$	−2204.23	−0.0749233	−0.0374617	−	0.999298i	$-0.511927\pi$
$953$	−2204.23	−0.0749233	−0.0374617	+	0.999298i	$0.511927\pi$
$954$	0	0
$955$	−8914.51	−0.302060
$956$	0	0
$957$	5627.74	0.190093
$958$	0	0
$959$	−533.810	−0.0179746
$960$	0	0
$961$	−26955.0	−0.904805
$962$	0	0
$963$	5277.54	0.176600
$964$	0	0
$965$	−20065.2	−0.669350
$966$	0	0
$967$	−56851.1	−1.89060	−0.945299	−	0.326205i	$-0.894230\pi$
$967$	−56851.1	−1.89060	−0.945299	+	0.326205i	$0.894230\pi$
$968$	0	0
$969$	4758.48	0.157755
$970$	0	0
$971$	48267.9	1.59525	0.797627	−	0.603151i	$-0.206088\pi$
$971$	48267.9	1.59525	0.797627	+	0.603151i	$0.206088\pi$
$972$	0	0
$973$	−850.383	−0.0280185
$974$	0	0
$975$	14210.2	0.466759
$976$	0	0
$977$	32914.4	1.07781	0.538907	−	0.842366i	$-0.318838\pi$
$977$	32914.4	1.07781	0.538907	+	0.842366i	$0.318838\pi$
$978$	0	0
$979$	−11265.1	−0.367757
$980$	0	0
$981$	17124.4	0.557331
$982$	0	0
$983$	−29352.6	−0.952394	−0.476197	−	0.879338i	$-0.657985\pi$
$983$	−29352.6	−0.952394	−0.476197	+	0.879338i	$0.657985\pi$
$984$	0	0
$985$	4232.98	0.136928
$986$	0	0
$987$	−415.958	−0.0134145
$988$	0	0
$989$	20335.8	0.653832
$990$	0	0
$991$	−2460.63	−0.0788744	−0.0394372	−	0.999222i	$-0.512557\pi$
$991$	−2460.63	−0.0788744	−0.0394372	+	0.999222i	$0.512557\pi$
$992$	0	0
$993$	18040.3	0.576528
$994$	0	0
$995$	110.344	0.00351573
$996$	0	0
$997$	971.621	0.0308641	0.0154321	−	0.999881i	$-0.495088\pi$
$997$	971.621	0.0308641	0.0154321	+	0.999881i	$0.495088\pi$
$998$	0	0
$999$	−29707.6	−0.940849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.a.h.1.2	yes	5
4.3	odd	2		608.4.a.e.1.4	✓	5
8.3	odd	2		1216.4.a.bd.1.2		5
8.5	even	2		1216.4.a.y.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.e.1.4	✓	5	4.3	odd	2
608.4.a.h.1.2	yes	5	1.1	even	1	trivial
1216.4.a.y.1.4		5	8.5	even	2
1216.4.a.bd.1.2		5	8.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	608.a (trivial)

\(f(q)\)	\(=\)	\(q-3.67449 q^{3} -7.12795 q^{5} -0.511313 q^{7} -13.4981 q^{9} +17.5714 q^{11} +52.1247 q^{13} +26.1915 q^{15} +68.1582 q^{17} -19.0000 q^{19} +1.87881 q^{21} +48.8262 q^{23} -74.1924 q^{25} +148.810 q^{27} -87.1627 q^{29} +53.2537 q^{31} -64.5659 q^{33} +3.64461 q^{35} -199.635 q^{37} -191.532 q^{39} -372.432 q^{41} +416.493 q^{43} +96.2140 q^{45} -221.394 q^{47} -342.739 q^{49} -250.446 q^{51} +107.677 q^{53} -125.248 q^{55} +69.8153 q^{57} +155.785 q^{59} -229.061 q^{61} +6.90178 q^{63} -371.542 q^{65} +580.065 q^{67} -179.411 q^{69} -38.2227 q^{71} -803.910 q^{73} +272.619 q^{75} -8.98449 q^{77} -1034.63 q^{79} -182.350 q^{81} -1053.94 q^{83} -485.828 q^{85} +320.278 q^{87} -641.103 q^{89} -26.6521 q^{91} -195.680 q^{93} +135.431 q^{95} +19.9998 q^{97} -237.181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 + 5 * q^5 - 7 * q^7 - 5 * q^9 + 13 * q^11 - 72 * q^13 - 72 * q^15 - 59 * q^17 - 95 * q^19 - 224 * q^21 - 52 * q^23 - 86 * q^25 + 54 * q^27 - 128 * q^29 + 110 * q^31 - 68 * q^33 - 45 * q^35 - 436 * q^37 - 356 * q^39 - 804 * q^41 - 143 * q^43 - 579 * q^45 - 661 * q^47 - 406 * q^49 + 570 * q^51 - 898 * q^53 - 17 * q^55 - 114 * q^57 + 196 * q^59 - 1079 * q^61 - 143 * q^63 - 1632 * q^65 + 832 * q^67 - 1644 * q^69 - 834 * q^71 - 1375 * q^73 + 1054 * q^75 - 1473 * q^77 - 154 * q^79 - 1859 * q^81 + 2224 * q^83 - 1807 * q^85 - 2004 * q^87 - 542 * q^89 + 1890 * q^91 - 2788 * q^93 - 95 * q^95 - 1528 * q^97 + 601 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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