LMFDB - Embedded newform 1216.4.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$0$
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:	no (minimal twist has level 608)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.31928$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-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	$-0.396730\pi$
$5$	7.12795	0.637543	0.318771	+	0.947832i	$0.396730\pi$
$6$	0	0
$7$	0.511313	0.0276083	0.0138042	−	0.999905i	$-0.495606\pi$
$7$	0.511313	0.0276083	0.0138042	+	0.999905i	$0.495606\pi$
$8$	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	$-0.687676\pi$
$13$	−52.1247	−1.11206	−0.556030	+	0.831162i	$0.687676\pi$
$14$	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	$-0.571038\pi$
$23$	−48.8262	−0.442651	−0.221325	+	0.975200i	$0.571038\pi$
$24$	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	$-0.409976\pi$
$29$	87.1627	0.558128	0.279064	+	0.960273i	$0.409976\pi$
$30$	0	0
$31$	−53.2537	−0.308537	−0.154269	−	0.988029i	$-0.549302\pi$
$31$	−53.2537	−0.308537	−0.154269	+	0.988029i	$0.549302\pi$
$32$	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	$-0.353733\pi$
$37$	199.635	0.887020	0.443510	+	0.896269i	$0.353733\pi$
$38$	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	$-0.388371\pi$
$47$	221.394	0.687098	0.343549	+	0.939135i	$0.388371\pi$
$48$	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	$-0.544560\pi$
$53$	−107.677	−0.279066	−0.139533	+	0.990217i	$0.544560\pi$
$54$	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	$-0.422723\pi$
$61$	229.061	0.480791	0.240395	+	0.970675i	$0.422723\pi$
$62$	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	$-0.489830\pi$
$71$	38.2227	0.0638901	0.0319451	+	0.999490i	$0.489830\pi$
$72$	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	$-0.236365\pi$
$79$	1034.63	1.47348	0.736739	+	0.676177i	$0.236365\pi$
$80$	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	$-0.425062\pi$
$101$	473.527	0.466512	0.233256	+	0.972415i	$0.425062\pi$
$102$	0	0
$103$	289.467	0.276913	0.138456	−	0.990369i	$-0.455786\pi$
$103$	289.467	0.276913	0.138456	+	0.990369i	$0.455786\pi$
$104$	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	$-0.311796\pi$
$109$	1268.65	1.11481	0.557407	+	0.830239i	$0.311796\pi$
$110$	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	$-0.0669907\pi$
$127$	2799.28	1.95587	0.977935	+	0.208907i	$0.0669907\pi$
$128$	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	$-0.421515\pi$
$149$	887.849	0.488157	0.244079	+	0.969755i	$0.421515\pi$
$150$	0	0
$151$	55.6469	0.0299899	0.0149950	−	0.999888i	$-0.495227\pi$
$151$	55.6469	0.0299899	0.0149950	+	0.999888i	$0.495227\pi$
$152$	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	$-0.362864\pi$
$157$	1643.09	0.835239	0.417620	+	0.908622i	$0.362864\pi$
$158$	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	$-0.299367\pi$
$167$	2543.96	1.17879	0.589393	+	0.807847i	$0.299367\pi$
$168$	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	$-0.423424\pi$
$173$	1084.28	0.476512	0.238256	+	0.971202i	$0.423424\pi$
$174$	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	$-0.360507\pi$
$181$	2066.62	0.848676	0.424338	+	0.905504i	$0.360507\pi$
$182$	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	$-0.576129\pi$
$191$	−1250.64	−0.473787	−0.236893	+	0.971536i	$0.576129\pi$
$192$	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	$-0.465752\pi$
$197$	593.857	0.214774	0.107387	+	0.994217i	$0.465752\pi$
$198$	0	0
$199$	15.4805	0.00551450	0.00275725	−	0.999996i	$-0.499122\pi$
$199$	15.4805	0.00551450	0.00275725	+	0.999996i	$0.499122\pi$
$200$	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	$-0.399494\pi$
$223$	2068.17	0.621054	0.310527	+	0.950565i	$0.399494\pi$
$224$	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	$-0.381595\pi$
$229$	2519.07	0.726920	0.363460	+	0.931610i	$0.381595\pi$
$230$	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	$-0.395998\pi$
$239$	2371.73	0.641901	0.320951	+	0.947096i	$0.395998\pi$
$240$	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	$-0.259522\pi$
$263$	5848.72	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$264$	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	$-0.338665\pi$
$269$	4283.31	0.970848	0.485424	+	0.874279i	$0.338665\pi$
$270$	0	0
$271$	2350.55	0.526885	0.263443	−	0.964675i	$-0.415142\pi$
$271$	2350.55	0.526885	0.263443	+	0.964675i	$0.415142\pi$
$272$	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	$-0.827665\pi$
$277$	−7901.74	−1.71397	−0.856985	+	0.515342i	$0.827665\pi$
$278$	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	$-0.647272\pi$
$293$	−4477.08	−0.892676	−0.446338	+	0.894864i	$0.647272\pi$
$294$	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	$-0.595482\pi$
$311$	−3241.23	−0.590975	−0.295488	+	0.955347i	$0.595482\pi$
$312$	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	$-0.650878\pi$
$317$	−5152.39	−0.912894	−0.456447	+	0.889751i	$0.650878\pi$
$318$	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	$-0.571071\pi$
$349$	−2887.35	−0.442854	−0.221427	+	0.975177i	$0.571071\pi$
$350$	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	$-0.452374\pi$
$359$	2027.89	0.298128	0.149064	+	0.988828i	$0.452374\pi$
$360$	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	$-0.578925\pi$
$367$	−3450.93	−0.490836	−0.245418	+	0.969417i	$0.578925\pi$
$368$	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	$-0.370209\pi$
$373$	5713.27	0.793089	0.396544	+	0.918016i	$0.370209\pi$
$374$	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	$-0.787076\pi$
$383$	−11760.3	−1.56898	−0.784492	+	0.620139i	$0.787076\pi$
$384$	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	$-0.355760\pi$
$389$	6717.76	0.875589	0.437795	+	0.899075i	$0.355760\pi$
$390$	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	$-0.214564\pi$
$397$	12360.2	1.56257	0.781287	+	0.624172i	$0.214564\pi$
$398$	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	$-0.425144\pi$
$421$	4025.51	0.466013	0.233006	+	0.972475i	$0.425144\pi$
$422$	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	$-0.448841\pi$
$431$	2863.81	0.320057	0.160029	+	0.987112i	$0.448841\pi$
$432$	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	$-0.341033\pi$
$439$	8810.03	0.957813	0.478907	+	0.877866i	$0.341033\pi$
$440$	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	$-0.479574\pi$
$461$	1269.48	0.128255	0.0641277	+	0.997942i	$0.479574\pi$
$462$	0	0
$463$	−8502.91	−0.853485	−0.426743	−	0.904373i	$-0.640339\pi$
$463$	−8502.91	−0.853485	−0.426743	+	0.904373i	$0.640339\pi$
$464$	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	$-0.775520\pi$
$479$	−15965.6	−1.52293	−0.761466	+	0.648205i	$0.775520\pi$
$480$	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	$-0.322779\pi$
$487$	11358.3	1.05687	0.528435	+	0.848974i	$0.322779\pi$
$488$	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	$-0.230467\pi$
$503$	16902.2	1.49828	0.749139	+	0.662413i	$0.230467\pi$
$504$	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	$-0.539299\pi$
$509$	−2828.32	−0.246293	−0.123147	+	0.992388i	$0.539299\pi$
$510$	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	$-0.137348\pi$
$541$	22860.0	1.81669	0.908343	+	0.418227i	$0.137348\pi$
$542$	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	$-0.134862\pi$
$557$	23966.7	1.82316	0.911581	+	0.411120i	$0.134862\pi$
$558$	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	$-0.537493\pi$
$599$	−3445.62	−0.235032	−0.117516	+	0.993071i	$0.537493\pi$
$600$	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	$-0.657853\pi$
$607$	−14232.0	−0.951664	−0.475832	+	0.879536i	$0.657853\pi$
$608$	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	$-0.721251\pi$
$613$	−19440.3	−1.28089	−0.640446	+	0.768003i	$0.721251\pi$
$614$	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	$-0.401181\pi$
$631$	9684.27	0.610974	0.305487	+	0.952196i	$0.401181\pi$
$632$	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	$-0.410322\pi$
$647$	9150.86	0.556040	0.278020	+	0.960575i	$0.410322\pi$
$648$	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	$-0.411074\pi$
$653$	9202.67	0.551498	0.275749	+	0.961230i	$0.411074\pi$
$654$	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	$-0.276862\pi$
$661$	21922.2	1.28998	0.644989	+	0.764192i	$0.276862\pi$
$662$	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	$-0.692330\pi$
$677$	−20015.0	−1.13625	−0.568123	+	0.822944i	$0.692330\pi$
$678$	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	$-0.498091\pi$
$701$	222.627	0.0119950	0.00599750	+	0.999982i	$0.498091\pi$
$702$	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	$-0.490989\pi$
$709$	1068.75	0.0566119	0.0283059	+	0.999599i	$0.490989\pi$
$710$	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	$-0.748101\pi$
$719$	−27102.1	−1.40575	−0.702876	+	0.711312i	$0.748101\pi$
$720$	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	$-0.551256\pi$
$727$	−6285.65	−0.320663	−0.160331	+	0.987063i	$0.551256\pi$
$728$	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	$-0.512790\pi$
$733$	−1594.38	−0.0803406	−0.0401703	+	0.999193i	$0.512790\pi$
$734$	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	$-0.841144\pi$
$743$	−35565.1	−1.75606	−0.878032	+	0.478601i	$0.841144\pi$
$744$	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	$-0.808333\pi$
$751$	−33922.2	−1.64825	−0.824126	+	0.566406i	$0.808333\pi$
$752$	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	$-0.350447\pi$
$757$	18859.2	0.905479	0.452740	+	0.891643i	$0.350447\pi$
$758$	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	$-0.853040\pi$
$773$	−38483.0	−1.79060	−0.895301	+	0.445461i	$0.853040\pi$
$774$	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	$-0.787917\pi$
$797$	−35376.1	−1.57225	−0.786127	+	0.618065i	$0.787917\pi$
$798$	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	$-0.839404\pi$
$821$	−41186.3	−1.75081	−0.875403	+	0.483394i	$0.839404\pi$
$822$	0	0
$823$	9004.24	0.381371	0.190685	−	0.981651i	$-0.438929\pi$
$823$	9004.24	0.381371	0.190685	+	0.981651i	$0.438929\pi$
$824$	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	$-0.505797\pi$
$829$	−869.327	−0.0364209	−0.0182105	+	0.999834i	$0.505797\pi$
$830$	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	$-0.491710\pi$
$839$	1265.71	0.0520824	0.0260412	+	0.999661i	$0.491710\pi$
$840$	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	$-0.602867\pi$
$853$	−15823.2	−0.635143	−0.317572	+	0.948234i	$0.602867\pi$
$854$	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	$-0.706572\pi$
$863$	−30643.8	−1.20872	−0.604361	+	0.796710i	$0.706572\pi$
$864$	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	$-0.274150\pi$
$877$	33839.9	1.30295	0.651477	+	0.758668i	$0.274150\pi$
$878$	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	$-0.101383\pi$
$887$	50176.9	1.89941	0.949705	+	0.313146i	$0.101383\pi$
$888$	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	$-0.173386\pi$
$911$	47034.3	1.71056	0.855278	+	0.518170i	$0.173386\pi$
$912$	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	$-0.718078\pi$
$919$	−35256.7	−1.26552	−0.632759	+	0.774349i	$0.718078\pi$
$920$	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	$-0.686303\pi$
$941$	−31893.3	−1.10488	−0.552440	+	0.833553i	$0.686303\pi$
$942$	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	$-0.105770\pi$
$967$	56851.1	1.89060	0.945299	+	0.326205i	$0.105770\pi$
$968$	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	$-0.342015\pi$
$983$	29352.6	0.952394	0.476197	+	0.879338i	$0.342015\pi$
$984$	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	$-0.487443\pi$
$991$	2460.63	0.0788744	0.0394372	+	0.999222i	$0.487443\pi$
$992$	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	$-0.504912\pi$
$997$	−971.621	−0.0308641	−0.0154321	+	0.999881i	$0.504912\pi$
$998$	0	0
$999$	29707.6	0.940849

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.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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