LMFDB - Embedded newform 441.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.57516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 24x + 6 $ x^3 - x^2 - 24*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.30829$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.30829 q^{2} +20.1780 q^{4} +5.56140 q^{5} -64.6443 q^{8} +O(q^{10})$	$q-5.30829 q^{2} +20.1780 q^{4} +5.56140 q^{5} -64.6443 q^{8} -29.5215 q^{10} +13.9174 q^{11} -38.6718 q^{13} +181.727 q^{16} +43.4788 q^{17} +109.028 q^{19} +112.218 q^{20} -73.8775 q^{22} +74.8778 q^{23} -94.0708 q^{25} +205.281 q^{26} +72.3589 q^{29} -64.0431 q^{31} -447.507 q^{32} -230.798 q^{34} +188.727 q^{37} -578.751 q^{38} -359.513 q^{40} -24.7923 q^{41} -243.881 q^{43} +280.825 q^{44} -397.474 q^{46} -620.549 q^{47} +499.356 q^{50} -780.319 q^{52} +287.839 q^{53} +77.4001 q^{55} -384.102 q^{58} +525.051 q^{59} +383.436 q^{61} +339.960 q^{62} +921.681 q^{64} -215.069 q^{65} +198.117 q^{67} +877.314 q^{68} -785.432 q^{71} +331.141 q^{73} -1001.82 q^{74} +2199.96 q^{76} +437.647 q^{79} +1010.66 q^{80} +131.605 q^{82} +241.241 q^{83} +241.803 q^{85} +1294.59 q^{86} -899.680 q^{88} +1585.54 q^{89} +1510.88 q^{92} +3294.05 q^{94} +606.347 q^{95} -79.2754 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 25 q^{4} + 11 q^{5} - 39 q^{8}+O(q^{10}) $ 3 * q - q^2 + 25 * q^4 + 11 * q^5 - 39 * q^8	$ 3 q - q^{2} + 25 q^{4} + 11 q^{5} - 39 q^{8} + 55 q^{10} - 35 q^{11} - 62 q^{13} + 241 q^{16} + 48 q^{17} + 202 q^{19} + 439 q^{20} - 7 q^{22} - 216 q^{23} + 130 q^{25} + 274 q^{26} - 53 q^{29} + 95 q^{31} - 683 q^{32} + 24 q^{34} + 262 q^{37} - 398 q^{38} - 21 q^{40} + 244 q^{41} + 360 q^{43} + 905 q^{44} - 1056 q^{46} - 210 q^{47} + 1378 q^{50} - 324 q^{52} - 393 q^{53} + 1031 q^{55} - 1249 q^{58} + 1143 q^{59} + 70 q^{61} + 1059 q^{62} - 399 q^{64} + 472 q^{65} - 628 q^{67} + 1944 q^{68} - 318 q^{71} - 988 q^{73} - 1002 q^{74} + 2340 q^{76} + 861 q^{79} + 175 q^{80} - 124 q^{82} + 519 q^{83} + 1800 q^{85} + 3208 q^{86} - 891 q^{88} + 1766 q^{89} + 672 q^{92} + 3294 q^{94} + 736 q^{95} - 19 q^{97}+O(q^{100}) $ 3 * q - q^2 + 25 * q^4 + 11 * q^5 - 39 * q^8 + 55 * q^10 - 35 * q^11 - 62 * q^13 + 241 * q^16 + 48 * q^17 + 202 * q^19 + 439 * q^20 - 7 * q^22 - 216 * q^23 + 130 * q^25 + 274 * q^26 - 53 * q^29 + 95 * q^31 - 683 * q^32 + 24 * q^34 + 262 * q^37 - 398 * q^38 - 21 * q^40 + 244 * q^41 + 360 * q^43 + 905 * q^44 - 1056 * q^46 - 210 * q^47 + 1378 * q^50 - 324 * q^52 - 393 * q^53 + 1031 * q^55 - 1249 * q^58 + 1143 * q^59 + 70 * q^61 + 1059 * q^62 - 399 * q^64 + 472 * q^65 - 628 * q^67 + 1944 * q^68 - 318 * q^71 - 988 * q^73 - 1002 * q^74 + 2340 * q^76 + 861 * q^79 + 175 * q^80 - 124 * q^82 + 519 * q^83 + 1800 * q^85 + 3208 * q^86 - 891 * q^88 + 1766 * q^89 + 672 * q^92 + 3294 * q^94 + 736 * q^95 - 19 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.30829	−1.87677	−0.938383	−	0.345598i	$-0.887676\pi$
$2$	−5.30829	−1.87677	−0.938383	+	0.345598i	$0.887676\pi$
$3$	0	0
$3$	0	0
$4$	20.1780	2.52225
$5$	5.56140	0.497427	0.248713	−	0.968577i	$-0.419992\pi$
$5$	5.56140	0.497427	0.248713	+	0.968577i	$0.419992\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.6443	−2.85690
$9$	0	0
$10$	−29.5215	−0.933553
$11$	13.9174	0.381477	0.190738	−	0.981641i	$-0.438912\pi$
$11$	13.9174	0.381477	0.190738	+	0.981641i	$0.438912\pi$
$12$	0	0
$13$	−38.6718	−0.825048	−0.412524	−	0.910947i	$-0.635353\pi$
$13$	−38.6718	−0.825048	−0.412524	+	0.910947i	$0.635353\pi$
$14$	0	0
$15$	0	0
$16$	181.727	2.83949
$17$	43.4788	0.620303	0.310152	−	0.950687i	$-0.399620\pi$
$17$	43.4788	0.620303	0.310152	+	0.950687i	$0.399620\pi$
$18$	0	0
$19$	109.028	1.31646	0.658228	−	0.752818i	$-0.271306\pi$
$19$	109.028	1.31646	0.658228	+	0.752818i	$0.271306\pi$
$20$	112.218	1.25463
$21$	0	0
$22$	−73.8775	−0.715943
$23$	74.8778	0.678831	0.339415	−	0.940637i	$-0.389771\pi$
$23$	74.8778	0.678831	0.339415	+	0.940637i	$0.389771\pi$
$24$	0	0
$25$	−94.0708	−0.752567
$26$	205.281	1.54842
$27$	0	0
$28$	0	0
$29$	72.3589	0.463335	0.231667	−	0.972795i	$-0.425582\pi$
$29$	72.3589	0.463335	0.231667	+	0.972795i	$0.425582\pi$
$30$	0	0
$31$	−64.0431	−0.371048	−0.185524	−	0.982640i	$-0.559398\pi$
$31$	−64.0431	−0.371048	−0.185524	+	0.982640i	$0.559398\pi$
$32$	−447.507	−2.47215
$33$	0	0
$34$	−230.798	−1.16416
$35$	0	0
$36$	0	0
$37$	188.727	0.838556	0.419278	−	0.907858i	$-0.362283\pi$
$37$	188.727	0.838556	0.419278	+	0.907858i	$0.362283\pi$
$38$	−578.751	−2.47068
$39$	0	0
$40$	−359.513	−1.42110
$41$	−24.7923	−0.0944367	−0.0472184	−	0.998885i	$-0.515036\pi$
$41$	−24.7923	−0.0944367	−0.0472184	+	0.998885i	$0.515036\pi$
$42$	0	0
$43$	−243.881	−0.864920	−0.432460	−	0.901653i	$-0.642354\pi$
$43$	−243.881	−0.864920	−0.432460	+	0.901653i	$0.642354\pi$
$44$	280.825	0.962180
$45$	0	0
$46$	−397.474	−1.27401
$47$	−620.549	−1.92588	−0.962940	−	0.269717i	$-0.913070\pi$
$47$	−620.549	−1.92588	−0.962940	+	0.269717i	$0.913070\pi$
$48$	0	0
$49$	0	0
$50$	499.356	1.41239
$51$	0	0
$52$	−780.319	−2.08098
$53$	287.839	0.745995	0.372997	−	0.927832i	$-0.378330\pi$
$53$	287.839	0.745995	0.372997	+	0.927832i	$0.378330\pi$
$54$	0	0
$55$	77.4001	0.189757
$56$	0	0
$57$	0	0
$58$	−384.102	−0.869571
$59$	525.051	1.15857	0.579287	−	0.815124i	$-0.303331\pi$
$59$	525.051	1.15857	0.579287	+	0.815124i	$0.303331\pi$
$60$	0	0
$61$	383.436	0.804818	0.402409	−	0.915460i	$-0.368173\pi$
$61$	383.436	0.804818	0.402409	+	0.915460i	$0.368173\pi$
$62$	339.960	0.696369
$63$	0	0
$64$	921.681	1.80016
$65$	−215.069	−0.410401
$66$	0	0
$67$	198.117	0.361251	0.180625	−	0.983552i	$-0.442188\pi$
$67$	198.117	0.361251	0.180625	+	0.983552i	$0.442188\pi$
$68$	877.314	1.56456
$69$	0	0
$70$	0	0
$71$	−785.432	−1.31287	−0.656434	−	0.754384i	$-0.727936\pi$
$71$	−785.432	−1.31287	−0.656434	+	0.754384i	$0.727936\pi$
$72$	0	0
$73$	331.141	0.530919	0.265459	−	0.964122i	$-0.414476\pi$
$73$	331.141	0.530919	0.265459	+	0.964122i	$0.414476\pi$
$74$	−1001.82	−1.57377
$75$	0	0
$76$	2199.96	3.32043
$77$	0	0
$78$	0	0
$79$	437.647	0.623280	0.311640	−	0.950200i	$-0.399122\pi$
$79$	437.647	0.623280	0.311640	+	0.950200i	$0.399122\pi$
$80$	1010.66	1.41244
$81$	0	0
$82$	131.605	0.177236
$83$	241.241	0.319032	0.159516	−	0.987195i	$-0.449007\pi$
$83$	241.241	0.319032	0.159516	+	0.987195i	$0.449007\pi$
$84$	0	0
$85$	241.803	0.308555
$86$	1294.59	1.62325
$87$	0	0
$88$	−899.680	−1.08984
$89$	1585.54	1.88840	0.944198	−	0.329378i	$-0.106839\pi$
$89$	1585.54	1.88840	0.944198	+	0.329378i	$0.106839\pi$
$90$	0	0
$91$	0	0
$92$	1510.88	1.71218
$93$	0	0
$94$	3294.05	3.61442
$95$	606.347	0.654841
$96$	0	0
$97$	−79.2754	−0.0829814	−0.0414907	−	0.999139i	$-0.513211\pi$
$97$	−79.2754	−0.0829814	−0.0414907	+	0.999139i	$0.513211\pi$
$98$	0	0
$99$	0	0
$100$	−1898.16	−1.89816
$101$	1154.97	1.13786	0.568931	−	0.822385i	$-0.307357\pi$
$101$	1154.97	1.13786	0.568931	+	0.822385i	$0.307357\pi$
$102$	0	0
$103$	1444.86	1.38220	0.691098	−	0.722761i	$-0.257127\pi$
$103$	1444.86	1.38220	0.691098	+	0.722761i	$0.257127\pi$
$104$	2499.91	2.35708
$105$	0	0
$106$	−1527.93	−1.40006
$107$	−990.960	−0.895325	−0.447662	−	0.894203i	$-0.647743\pi$
$107$	−990.960	−0.895325	−0.447662	+	0.894203i	$0.647743\pi$
$108$	0	0
$109$	1953.17	1.71633	0.858164	−	0.513376i	$-0.171605\pi$
$109$	1953.17	1.71633	0.858164	+	0.513376i	$0.171605\pi$
$110$	−410.862	−0.356129
$111$	0	0
$112$	0	0
$113$	−672.882	−0.560172	−0.280086	−	0.959975i	$-0.590363\pi$
$113$	−672.882	−0.560172	−0.280086	+	0.959975i	$0.590363\pi$
$114$	0	0
$115$	416.426	0.337669
$116$	1460.06	1.16865
$117$	0	0
$118$	−2787.13	−2.17437
$119$	0	0
$120$	0	0
$121$	−1137.31	−0.854475
$122$	−2035.39	−1.51045
$123$	0	0
$124$	−1292.26	−0.935874
$125$	−1218.34	−0.871773
$126$	0	0
$127$	175.815	0.122843	0.0614216	−	0.998112i	$-0.480437\pi$
$127$	175.815	0.122843	0.0614216	+	0.998112i	$0.480437\pi$
$128$	−1312.50	−0.906325
$129$	0	0
$130$	1141.65	0.770226
$131$	1125.93	0.750939	0.375470	−	0.926835i	$-0.377481\pi$
$131$	1125.93	0.750939	0.375470	+	0.926835i	$0.377481\pi$
$132$	0	0
$133$	0	0
$134$	−1051.66	−0.677983
$135$	0	0
$136$	−2810.66	−1.77215
$137$	−1868.70	−1.16536	−0.582678	−	0.812703i	$-0.697995\pi$
$137$	−1868.70	−1.16536	−0.582678	+	0.812703i	$0.697995\pi$
$138$	0	0
$139$	2817.19	1.71907	0.859537	−	0.511074i	$-0.170752\pi$
$139$	2817.19	1.71907	0.859537	+	0.511074i	$0.170752\pi$
$140$	0	0
$141$	0	0
$142$	4169.30	2.46394
$143$	−538.210	−0.314737
$144$	0	0
$145$	402.417	0.230475
$146$	−1757.79	−0.996410
$147$	0	0
$148$	3808.14	2.11505
$149$	−1800.33	−0.989855	−0.494928	−	0.868934i	$-0.664805\pi$
$149$	−1800.33	−0.989855	−0.494928	+	0.868934i	$0.664805\pi$
$150$	0	0
$151$	−452.984	−0.244128	−0.122064	−	0.992522i	$-0.538951\pi$
$151$	−452.984	−0.244128	−0.122064	+	0.992522i	$0.538951\pi$
$152$	−7048.03	−3.76099
$153$	0	0
$154$	0	0
$155$	−356.169	−0.184569
$156$	0	0
$157$	1863.66	0.947364	0.473682	−	0.880696i	$-0.342925\pi$
$157$	1863.66	0.947364	0.473682	+	0.880696i	$0.342925\pi$
$158$	−2323.16	−1.16975
$159$	0	0
$160$	−2488.77	−1.22971
$161$	0	0
$162$	0	0
$163$	2321.14	1.11537	0.557686	−	0.830052i	$-0.311689\pi$
$163$	2321.14	1.11537	0.557686	+	0.830052i	$0.311689\pi$
$164$	−500.259	−0.238193
$165$	0	0
$166$	−1280.58	−0.598749
$167$	3211.62	1.48816	0.744079	−	0.668092i	$-0.232889\pi$
$167$	3211.62	1.48816	0.744079	+	0.668092i	$0.232889\pi$
$168$	0	0
$169$	−701.494	−0.319296
$170$	−1283.56	−0.579086
$171$	0	0
$172$	−4921.04	−2.18154
$173$	214.277	0.0941687	0.0470844	−	0.998891i	$-0.485007\pi$
$173$	214.277	0.0941687	0.0470844	+	0.998891i	$0.485007\pi$
$174$	0	0
$175$	0	0
$176$	2529.17	1.08320
$177$	0	0
$178$	−8416.53	−3.54408
$179$	2437.22	1.01769	0.508845	−	0.860858i	$-0.330073\pi$
$179$	2437.22	1.01769	0.508845	+	0.860858i	$0.330073\pi$
$180$	0	0
$181$	248.631	0.102103	0.0510514	−	0.998696i	$-0.483743\pi$
$181$	248.631	0.102103	0.0510514	+	0.998696i	$0.483743\pi$
$182$	0	0
$183$	0	0
$184$	−4840.43	−1.93935
$185$	1049.59	0.417120
$186$	0	0
$187$	605.110	0.236631
$188$	−12521.4	−4.85755
$189$	0	0
$190$	−3218.67	−1.22898
$191$	4313.08	1.63394	0.816972	−	0.576677i	$-0.195651\pi$
$191$	4313.08	1.63394	0.816972	+	0.576677i	$0.195651\pi$
$192$	0	0
$193$	2060.85	0.768618	0.384309	−	0.923205i	$-0.374440\pi$
$193$	2060.85	0.768618	0.384309	+	0.923205i	$0.374440\pi$
$194$	420.817	0.155737
$195$	0	0
$196$	0	0
$197$	1666.09	0.602557	0.301279	−	0.953536i	$-0.402587\pi$
$197$	1666.09	0.602557	0.301279	+	0.953536i	$0.402587\pi$
$198$	0	0
$199$	1087.53	0.387403	0.193702	−	0.981061i	$-0.437951\pi$
$199$	1087.53	0.387403	0.193702	+	0.981061i	$0.437951\pi$
$200$	6081.15	2.15001
$201$	0	0
$202$	−6130.94	−2.13550
$203$	0	0
$204$	0	0
$205$	−137.880	−0.0469754
$206$	−7669.74	−2.59406
$207$	0	0
$208$	−7027.72	−2.34271
$209$	1517.38	0.502198
$210$	0	0
$211$	−4676.47	−1.52579	−0.762895	−	0.646522i	$-0.776223\pi$
$211$	−4676.47	−1.52579	−0.762895	+	0.646522i	$0.776223\pi$
$212$	5808.01	1.88158
$213$	0	0
$214$	5260.31	1.68031
$215$	−1356.32	−0.430234
$216$	0	0
$217$	0	0
$218$	−10368.0	−3.22114
$219$	0	0
$220$	1561.78	0.478614
$221$	−1681.40	−0.511780
$222$	0	0
$223$	−3246.03	−0.974754	−0.487377	−	0.873192i	$-0.662046\pi$
$223$	−3246.03	−0.974754	−0.487377	+	0.873192i	$0.662046\pi$
$224$	0	0
$225$	0	0
$226$	3571.86	1.05131
$227$	5138.16	1.50234	0.751171	−	0.660108i	$-0.229490\pi$
$227$	5138.16	1.50234	0.751171	+	0.660108i	$0.229490\pi$
$228$	0	0
$229$	−614.806	−0.177413	−0.0887064	−	0.996058i	$-0.528273\pi$
$229$	−614.806	−0.177413	−0.0887064	+	0.996058i	$0.528273\pi$
$230$	−2210.51	−0.633725
$231$	0	0
$232$	−4677.59	−1.32370
$233$	−2827.42	−0.794979	−0.397490	−	0.917607i	$-0.630119\pi$
$233$	−2827.42	−0.794979	−0.397490	+	0.917607i	$0.630119\pi$
$234$	0	0
$235$	−3451.12	−0.957984
$236$	10594.5	2.92221
$237$	0	0
$238$	0	0
$239$	3432.45	0.928983	0.464491	−	0.885578i	$-0.346237\pi$
$239$	3432.45	0.928983	0.464491	+	0.885578i	$0.346237\pi$
$240$	0	0
$241$	2636.11	0.704593	0.352296	−	0.935888i	$-0.385401\pi$
$241$	2636.11	0.704593	0.352296	+	0.935888i	$0.385401\pi$
$242$	6037.16	1.60365
$243$	0	0
$244$	7736.96	2.02995
$245$	0	0
$246$	0	0
$247$	−4216.30	−1.08614
$248$	4140.02	1.06005
$249$	0	0
$250$	6467.31	1.63611
$251$	2057.57	0.517422	0.258711	−	0.965955i	$-0.416702\pi$
$251$	2057.57	0.517422	0.258711	+	0.965955i	$0.416702\pi$
$252$	0	0
$253$	1042.10	0.258958
$254$	−933.279	−0.230548
$255$	0	0
$256$	−406.321	−0.0991996
$257$	2150.21	0.521892	0.260946	−	0.965353i	$-0.415966\pi$
$257$	2150.21	0.521892	0.260946	+	0.965353i	$0.415966\pi$
$258$	0	0
$259$	0	0
$260$	−4339.66	−1.03513
$261$	0	0
$262$	−5976.77	−1.40934
$263$	4590.15	1.07620	0.538100	−	0.842881i	$-0.319142\pi$
$263$	4590.15	1.07620	0.538100	+	0.842881i	$0.319142\pi$
$264$	0	0
$265$	1600.79	0.371078
$266$	0	0
$267$	0	0
$268$	3997.59	0.911164
$269$	−379.378	−0.0859891	−0.0429945	−	0.999075i	$-0.513690\pi$
$269$	−379.378	−0.0859891	−0.0429945	+	0.999075i	$0.513690\pi$
$270$	0	0
$271$	5368.84	1.20345	0.601723	−	0.798705i	$-0.294481\pi$
$271$	5368.84	1.20345	0.601723	+	0.798705i	$0.294481\pi$
$272$	7901.28	1.76134
$273$	0	0
$274$	9919.61	2.18710
$275$	−1309.22	−0.287087
$276$	0	0
$277$	−4781.60	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$277$	−4781.60	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$278$	−14954.5	−3.22630
$279$	0	0
$280$	0	0
$281$	2076.57	0.440845	0.220423	−	0.975404i	$-0.429256\pi$
$281$	2076.57	0.440845	0.220423	+	0.975404i	$0.429256\pi$
$282$	0	0
$283$	−2557.62	−0.537224	−0.268612	−	0.963248i	$-0.586565\pi$
$283$	−2557.62	−0.537224	−0.268612	+	0.963248i	$0.586565\pi$
$284$	−15848.4	−3.31138
$285$	0	0
$286$	2856.97	0.590687
$287$	0	0
$288$	0	0
$289$	−3022.60	−0.615224
$290$	−2136.15	−0.432548
$291$	0	0
$292$	6681.75	1.33911
$293$	560.049	0.111667	0.0558335	−	0.998440i	$-0.482218\pi$
$293$	560.049	0.111667	0.0558335	+	0.998440i	$0.482218\pi$
$294$	0	0
$295$	2920.02	0.576305
$296$	−12200.2	−2.39567
$297$	0	0
$298$	9556.66	1.85773
$299$	−2895.66	−0.560068
$300$	0	0
$301$	0	0
$302$	2404.57	0.458171
$303$	0	0
$304$	19813.3	3.73806
$305$	2132.44	0.400338
$306$	0	0
$307$	−3653.02	−0.679117	−0.339558	−	0.940585i	$-0.610278\pi$
$307$	−3653.02	−0.679117	−0.339558	+	0.940585i	$0.610278\pi$
$308$	0	0
$309$	0	0
$310$	1890.65	0.346393
$311$	3492.27	0.636747	0.318374	−	0.947965i	$-0.396863\pi$
$311$	3492.27	0.636747	0.318374	+	0.947965i	$0.396863\pi$
$312$	0	0
$313$	−8712.09	−1.57328	−0.786640	−	0.617412i	$-0.788181\pi$
$313$	−8712.09	−1.57328	−0.786640	+	0.617412i	$0.788181\pi$
$314$	−9892.85	−1.77798
$315$	0	0
$316$	8830.83	1.57207
$317$	−1940.33	−0.343785	−0.171892	−	0.985116i	$-0.554988\pi$
$317$	−1940.33	−0.343785	−0.171892	+	0.985116i	$0.554988\pi$
$318$	0	0
$319$	1007.05	0.176752
$320$	5125.84	0.895447
$321$	0	0
$322$	0	0
$323$	4740.39	0.816602
$324$	0	0
$325$	3637.89	0.620904
$326$	−12321.3	−2.09329
$327$	0	0
$328$	1602.68	0.269797
$329$	0	0
$330$	0	0
$331$	−5731.51	−0.951759	−0.475879	−	0.879510i	$-0.657870\pi$
$331$	−5731.51	−0.951759	−0.475879	+	0.879510i	$0.657870\pi$
$332$	4867.76	0.804678
$333$	0	0
$334$	−17048.2	−2.79292
$335$	1101.81	0.179696
$336$	0	0
$337$	2403.74	0.388547	0.194273	−	0.980947i	$-0.437765\pi$
$337$	2403.74	0.388547	0.194273	+	0.980947i	$0.437765\pi$
$338$	3723.73	0.599244
$339$	0	0
$340$	4879.09	0.778253
$341$	−891.312	−0.141546
$342$	0	0
$343$	0	0
$344$	15765.6	2.47099
$345$	0	0
$346$	−1137.45	−0.176733
$347$	3336.44	0.516166	0.258083	−	0.966123i	$-0.416909\pi$
$347$	3336.44	0.516166	0.258083	+	0.966123i	$0.416909\pi$
$348$	0	0
$349$	2424.54	0.371870	0.185935	−	0.982562i	$-0.440469\pi$
$349$	2424.54	0.371870	0.185935	+	0.982562i	$0.440469\pi$
$350$	0	0
$351$	0	0
$352$	−6228.12	−0.943069
$353$	−12403.1	−1.87012	−0.935059	−	0.354491i	$-0.884654\pi$
$353$	−12403.1	−1.87012	−0.935059	+	0.354491i	$0.884654\pi$
$354$	0	0
$355$	−4368.10	−0.653055
$356$	31993.1	4.76300
$357$	0	0
$358$	−12937.5	−1.90997
$359$	−1353.84	−0.199034	−0.0995168	−	0.995036i	$-0.531730\pi$
$359$	−1353.84	−0.199034	−0.0995168	+	0.995036i	$0.531730\pi$
$360$	0	0
$361$	5028.05	0.733059
$362$	−1319.81	−0.191623
$363$	0	0
$364$	0	0
$365$	1841.61	0.264093
$366$	0	0
$367$	1378.06	0.196006	0.0980031	−	0.995186i	$-0.468754\pi$
$367$	1378.06	0.196006	0.0980031	+	0.995186i	$0.468754\pi$
$368$	13607.3	1.92753
$369$	0	0
$370$	−5571.52	−0.782837
$371$	0	0
$372$	0	0
$373$	5456.92	0.757503	0.378752	−	0.925498i	$-0.376353\pi$
$373$	5456.92	0.757503	0.378752	+	0.925498i	$0.376353\pi$
$374$	−3212.10	−0.444101
$375$	0	0
$376$	40115.0	5.50205
$377$	−2798.25	−0.382273
$378$	0	0
$379$	554.675	0.0751761	0.0375881	−	0.999293i	$-0.488033\pi$
$379$	554.675	0.0751761	0.0375881	+	0.999293i	$0.488033\pi$
$380$	12234.9	1.65167
$381$	0	0
$382$	−22895.1	−3.06653
$383$	5860.66	0.781895	0.390948	−	0.920413i	$-0.372147\pi$
$383$	5860.66	0.781895	0.390948	+	0.920413i	$0.372147\pi$
$384$	0	0
$385$	0	0
$386$	−10939.6	−1.44252
$387$	0	0
$388$	−1599.62	−0.209300
$389$	−7778.86	−1.01389	−0.506946	−	0.861978i	$-0.669226\pi$
$389$	−7778.86	−1.01389	−0.506946	+	0.861978i	$0.669226\pi$
$390$	0	0
$391$	3255.60	0.421081
$392$	0	0
$393$	0	0
$394$	−8844.08	−1.13086
$395$	2433.93	0.310036
$396$	0	0
$397$	−8027.88	−1.01488	−0.507440	−	0.861687i	$-0.669408\pi$
$397$	−8027.88	−1.01488	−0.507440	+	0.861687i	$0.669408\pi$
$398$	−5772.95	−0.727065
$399$	0	0
$400$	−17095.2	−2.13690
$401$	−779.980	−0.0971330	−0.0485665	−	0.998820i	$-0.515465\pi$
$401$	−779.980	−0.0971330	−0.0485665	+	0.998820i	$0.515465\pi$
$402$	0	0
$403$	2476.66	0.306132
$404$	23305.0	2.86997
$405$	0	0
$406$	0	0
$407$	2626.59	0.319890
$408$	0	0
$409$	−14692.5	−1.77628	−0.888139	−	0.459575i	$-0.848002\pi$
$409$	−14692.5	−1.77628	−0.888139	+	0.459575i	$0.848002\pi$
$410$	731.907	0.0881617
$411$	0	0
$412$	29154.4	3.48624
$413$	0	0
$414$	0	0
$415$	1341.64	0.158695
$416$	17305.9	2.03964
$417$	0	0
$418$	−8054.70	−0.942508
$419$	3370.31	0.392960	0.196480	−	0.980508i	$-0.437049\pi$
$419$	3370.31	0.392960	0.196480	+	0.980508i	$0.437049\pi$
$420$	0	0
$421$	15651.0	1.81184	0.905919	−	0.423450i	$-0.139181\pi$
$421$	15651.0	1.81184	0.905919	+	0.423450i	$0.139181\pi$
$422$	24824.1	2.86355
$423$	0	0
$424$	−18607.2	−2.13123
$425$	−4090.08	−0.466819
$426$	0	0
$427$	0	0
$428$	−19995.6	−2.25823
$429$	0	0
$430$	7199.75	0.807449
$431$	−4888.12	−0.546294	−0.273147	−	0.961972i	$-0.588064\pi$
$431$	−4888.12	−0.546294	−0.273147	+	0.961972i	$0.588064\pi$
$432$	0	0
$433$	5255.73	0.583313	0.291656	−	0.956523i	$-0.405794\pi$
$433$	5255.73	0.583313	0.291656	+	0.956523i	$0.405794\pi$
$434$	0	0
$435$	0	0
$436$	39411.0	4.32901
$437$	8163.76	0.893651
$438$	0	0
$439$	824.977	0.0896902	0.0448451	−	0.998994i	$-0.485721\pi$
$439$	824.977	0.0896902	0.0448451	+	0.998994i	$0.485721\pi$
$440$	−5003.48	−0.542117
$441$	0	0
$442$	8925.37	0.960490
$443$	−13027.3	−1.39717	−0.698583	−	0.715529i	$-0.746186\pi$
$443$	−13027.3	−1.39717	−0.698583	+	0.715529i	$0.746186\pi$
$444$	0	0
$445$	8817.84	0.939339
$446$	17230.9	1.82939
$447$	0	0
$448$	0	0
$449$	−16526.1	−1.73700	−0.868500	−	0.495689i	$-0.834916\pi$
$449$	−16526.1	−1.73700	−0.868500	+	0.495689i	$0.834916\pi$
$450$	0	0
$451$	−345.044	−0.0360254
$452$	−13577.4	−1.41289
$453$	0	0
$454$	−27274.9	−2.81954
$455$	0	0
$456$	0	0
$457$	−3710.82	−0.379836	−0.189918	−	0.981800i	$-0.560822\pi$
$457$	−3710.82	−0.379836	−0.189918	+	0.981800i	$0.560822\pi$
$458$	3263.57	0.332962
$459$	0	0
$460$	8402.63	0.851684
$461$	−9714.00	−0.981401	−0.490701	−	0.871328i	$-0.663259\pi$
$461$	−9714.00	−0.981401	−0.490701	+	0.871328i	$0.663259\pi$
$462$	0	0
$463$	−43.2780	−0.00434406	−0.00217203	−	0.999998i	$-0.500691\pi$
$463$	−43.2780	−0.00434406	−0.00217203	+	0.999998i	$0.500691\pi$
$464$	13149.6	1.31563
$465$	0	0
$466$	15008.8	1.49199
$467$	−1533.24	−0.151927	−0.0759633	−	0.997111i	$-0.524203\pi$
$467$	−1533.24	−0.151927	−0.0759633	+	0.997111i	$0.524203\pi$
$468$	0	0
$469$	0	0
$470$	18319.6	1.79791
$471$	0	0
$472$	−33941.6	−3.30993
$473$	−3394.19	−0.329947
$474$	0	0
$475$	−10256.3	−0.990722
$476$	0	0
$477$	0	0
$478$	−18220.5	−1.74348
$479$	−7035.37	−0.671095	−0.335547	−	0.942023i	$-0.608921\pi$
$479$	−7035.37	−0.671095	−0.335547	+	0.942023i	$0.608921\pi$
$480$	0	0
$481$	−7298.42	−0.691849
$482$	−13993.3	−1.32236
$483$	0	0
$484$	−22948.6	−2.15520
$485$	−440.882	−0.0412772
$486$	0	0
$487$	−15371.3	−1.43026	−0.715132	−	0.698989i	$-0.753634\pi$
$487$	−15371.3	−1.43026	−0.715132	+	0.698989i	$0.753634\pi$
$488$	−24786.9	−2.29929
$489$	0	0
$490$	0	0
$491$	2393.35	0.219980	0.109990	−	0.993933i	$-0.464918\pi$
$491$	2393.35	0.219980	0.109990	+	0.993933i	$0.464918\pi$
$492$	0	0
$493$	3146.08	0.287408
$494$	22381.3	2.03843
$495$	0	0
$496$	−11638.4	−1.05359
$497$	0	0
$498$	0	0
$499$	693.520	0.0622169	0.0311084	−	0.999516i	$-0.490096\pi$
$499$	693.520	0.0622169	0.0311084	+	0.999516i	$0.490096\pi$
$500$	−24583.7	−2.19883
$501$	0	0
$502$	−10922.2	−0.971079
$503$	−8646.95	−0.766498	−0.383249	−	0.923645i	$-0.625195\pi$
$503$	−8646.95	−0.766498	−0.383249	+	0.923645i	$0.625195\pi$
$504$	0	0
$505$	6423.27	0.566003
$506$	−5531.79	−0.486004
$507$	0	0
$508$	3547.60	0.309841
$509$	−15500.9	−1.34983	−0.674916	−	0.737895i	$-0.735820\pi$
$509$	−15500.9	−1.34983	−0.674916	+	0.737895i	$0.735820\pi$
$510$	0	0
$511$	0	0
$512$	12656.9	1.09250
$513$	0	0
$514$	−11413.9	−0.979469
$515$	8035.44	0.687541
$516$	0	0
$517$	−8636.41	−0.734678
$518$	0	0
$519$	0	0
$520$	13903.0	1.17248
$521$	864.707	0.0727131	0.0363565	−	0.999339i	$-0.488425\pi$
$521$	864.707	0.0727131	0.0363565	+	0.999339i	$0.488425\pi$
$522$	0	0
$523$	6255.61	0.523019	0.261509	−	0.965201i	$-0.415780\pi$
$523$	6255.61	0.523019	0.261509	+	0.965201i	$0.415780\pi$
$524$	22719.0	1.89406
$525$	0	0
$526$	−24365.9	−2.01978
$527$	−2784.51	−0.230162
$528$	0	0
$529$	−6560.31	−0.539189
$530$	−8497.45	−0.696426
$531$	0	0
$532$	0	0
$533$	958.762	0.0779148
$534$	0	0
$535$	−5511.13	−0.445358
$536$	−12807.1	−1.03206
$537$	0	0
$538$	2013.85	0.161381
$539$	0	0
$540$	0	0
$541$	143.871	0.0114334	0.00571671	−	0.999984i	$-0.498180\pi$
$541$	143.871	0.0114334	0.00571671	+	0.999984i	$0.498180\pi$
$542$	−28499.4	−2.25858
$543$	0	0
$544$	−19457.1	−1.53348
$545$	10862.4	0.853747
$546$	0	0
$547$	5455.65	0.426448	0.213224	−	0.977003i	$-0.431604\pi$
$547$	5455.65	0.426448	0.213224	+	0.977003i	$0.431604\pi$
$548$	−37706.6	−2.93932
$549$	0	0
$550$	6949.72	0.538795
$551$	7889.13	0.609960
$552$	0	0
$553$	0	0
$554$	25382.2	1.94654
$555$	0	0
$556$	56845.3	4.33593
$557$	−24809.9	−1.88730	−0.943652	−	0.330940i	$-0.892634\pi$
$557$	−24809.9	−1.88730	−0.943652	+	0.330940i	$0.892634\pi$
$558$	0	0
$559$	9431.33	0.713600
$560$	0	0
$561$	0	0
$562$	−11023.0	−0.827363
$563$	16369.8	1.22541	0.612705	−	0.790312i	$-0.290081\pi$
$563$	16369.8	1.22541	0.612705	+	0.790312i	$0.290081\pi$
$564$	0	0
$565$	−3742.17	−0.278645
$566$	13576.6	1.00824
$567$	0	0
$568$	50773.7	3.75074
$569$	18450.6	1.35938	0.679691	−	0.733498i	$-0.262114\pi$
$569$	18450.6	1.35938	0.679691	+	0.733498i	$0.262114\pi$
$570$	0	0
$571$	−7108.69	−0.520997	−0.260499	−	0.965474i	$-0.583887\pi$
$571$	−7108.69	−0.520997	−0.260499	+	0.965474i	$0.583887\pi$
$572$	−10860.0	−0.793844
$573$	0	0
$574$	0	0
$575$	−7043.82	−0.510865
$576$	0	0
$577$	−7594.17	−0.547919	−0.273960	−	0.961741i	$-0.588333\pi$
$577$	−7594.17	−0.547919	−0.273960	+	0.961741i	$0.588333\pi$
$578$	16044.8	1.15463
$579$	0	0
$580$	8119.96	0.581315
$581$	0	0
$582$	0	0
$583$	4005.96	0.284580
$584$	−21406.4	−1.51678
$585$	0	0
$586$	−2972.91	−0.209573
$587$	−1763.34	−0.123988	−0.0619939	−	0.998077i	$-0.519746\pi$
$587$	−1763.34	−0.123988	−0.0619939	+	0.998077i	$0.519746\pi$
$588$	0	0
$589$	−6982.47	−0.488468
$590$	−15500.3	−1.08159
$591$	0	0
$592$	34296.9	2.38107
$593$	−12316.1	−0.852889	−0.426445	−	0.904514i	$-0.640234\pi$
$593$	−12316.1	−0.852889	−0.426445	+	0.904514i	$0.640234\pi$
$594$	0	0
$595$	0	0
$596$	−36327.0	−2.49666
$597$	0	0
$598$	15371.0	1.05112
$599$	8903.40	0.607317	0.303659	−	0.952781i	$-0.401792\pi$
$599$	8903.40	0.607317	0.303659	+	0.952781i	$0.401792\pi$
$600$	0	0
$601$	19157.1	1.30022	0.650112	−	0.759838i	$-0.274722\pi$
$601$	19157.1	1.30022	0.650112	+	0.759838i	$0.274722\pi$
$602$	0	0
$603$	0	0
$604$	−9140.31	−0.615751
$605$	−6325.02	−0.425039
$606$	0	0
$607$	7569.93	0.506184	0.253092	−	0.967442i	$-0.418552\pi$
$607$	7569.93	0.506184	0.253092	+	0.967442i	$0.418552\pi$
$608$	−48790.7	−3.25448
$609$	0	0
$610$	−11319.6	−0.751340
$611$	23997.7	1.58894
$612$	0	0
$613$	2907.13	0.191546	0.0957730	−	0.995403i	$-0.469468\pi$
$613$	2907.13	0.191546	0.0957730	+	0.995403i	$0.469468\pi$
$614$	19391.3	1.27454
$615$	0	0
$616$	0	0
$617$	12510.9	0.816320	0.408160	−	0.912910i	$-0.366171\pi$
$617$	12510.9	0.816320	0.408160	+	0.912910i	$0.366171\pi$
$618$	0	0
$619$	−10065.6	−0.653585	−0.326792	−	0.945096i	$-0.605968\pi$
$619$	−10065.6	−0.653585	−0.326792	+	0.945096i	$0.605968\pi$
$620$	−7186.78	−0.465529
$621$	0	0
$622$	−18538.0	−1.19503
$623$	0	0
$624$	0	0
$625$	4983.18	0.318923
$626$	46246.4	2.95268
$627$	0	0
$628$	37604.9	2.38949
$629$	8205.63	0.520159
$630$	0	0
$631$	−25146.6	−1.58648	−0.793242	−	0.608907i	$-0.791608\pi$
$631$	−25146.6	−1.58648	−0.793242	+	0.608907i	$0.791608\pi$
$632$	−28291.4	−1.78065
$633$	0	0
$634$	10299.8	0.645203
$635$	977.779	0.0611055
$636$	0	0
$637$	0	0
$638$	−5345.70	−0.331721
$639$	0	0
$640$	−7299.33	−0.450830
$641$	28958.9	1.78441	0.892206	−	0.451629i	$-0.149157\pi$
$641$	28958.9	1.78441	0.892206	+	0.451629i	$0.149157\pi$
$642$	0	0
$643$	−7341.90	−0.450290	−0.225145	−	0.974325i	$-0.572286\pi$
$643$	−7341.90	−0.450290	−0.225145	+	0.974325i	$0.572286\pi$
$644$	0	0
$645$	0	0
$646$	−25163.4	−1.53257
$647$	6071.98	0.368955	0.184478	−	0.982837i	$-0.440941\pi$
$647$	6071.98	0.368955	0.184478	+	0.982837i	$0.440941\pi$
$648$	0	0
$649$	7307.33	0.441969
$650$	−19311.0	−1.16529
$651$	0	0
$652$	46835.9	2.81324
$653$	−26262.1	−1.57384	−0.786920	−	0.617056i	$-0.788325\pi$
$653$	−26262.1	−1.57384	−0.786920	+	0.617056i	$0.788325\pi$
$654$	0	0
$655$	6261.75	0.373537
$656$	−4505.44	−0.268152
$657$	0	0
$658$	0	0
$659$	−26130.1	−1.54459	−0.772296	−	0.635263i	$-0.780892\pi$
$659$	−26130.1	−1.54459	−0.772296	+	0.635263i	$0.780892\pi$
$660$	0	0
$661$	11925.5	0.701737	0.350868	−	0.936425i	$-0.385886\pi$
$661$	11925.5	0.701737	0.350868	+	0.936425i	$0.385886\pi$
$662$	30424.5	1.78623
$663$	0	0
$664$	−15594.9	−0.911444
$665$	0	0
$666$	0	0
$667$	5418.08	0.314526
$668$	64804.0	3.75350
$669$	0	0
$670$	−5848.71	−0.337247
$671$	5336.42	0.307019
$672$	0	0
$673$	−6359.85	−0.364271	−0.182135	−	0.983273i	$-0.558301\pi$
$673$	−6359.85	−0.364271	−0.182135	+	0.983273i	$0.558301\pi$
$674$	−12759.8	−0.729211
$675$	0	0
$676$	−14154.7	−0.805344
$677$	8561.61	0.486041	0.243020	−	0.970021i	$-0.421862\pi$
$677$	8561.61	0.486041	0.243020	+	0.970021i	$0.421862\pi$
$678$	0	0
$679$	0	0
$680$	−15631.2	−0.881513
$681$	0	0
$682$	4731.34	0.265649
$683$	6705.88	0.375686	0.187843	−	0.982199i	$-0.439850\pi$
$683$	6705.88	0.375686	0.187843	+	0.982199i	$0.439850\pi$
$684$	0	0
$685$	−10392.6	−0.579679
$686$	0	0
$687$	0	0
$688$	−44319.9	−2.45593
$689$	−11131.2	−0.615481
$690$	0	0
$691$	25330.6	1.39453	0.697267	−	0.716811i	$-0.254399\pi$
$691$	25330.6	1.39453	0.697267	+	0.716811i	$0.254399\pi$
$692$	4323.68	0.237517
$693$	0	0
$694$	−17710.8	−0.968723
$695$	15667.5	0.855113
$696$	0	0
$697$	−1077.94	−0.0585794
$698$	−12870.2	−0.697912
$699$	0	0
$700$	0	0
$701$	27184.1	1.46467	0.732333	−	0.680947i	$-0.238432\pi$
$701$	27184.1	1.46467	0.732333	+	0.680947i	$0.238432\pi$
$702$	0	0
$703$	20576.5	1.10392
$704$	12827.4	0.686719
$705$	0	0
$706$	65839.5	3.50977
$707$	0	0
$708$	0	0
$709$	16145.5	0.855228	0.427614	−	0.903961i	$-0.359354\pi$
$709$	16145.5	0.855228	0.427614	+	0.903961i	$0.359354\pi$
$710$	23187.2	1.22563
$711$	0	0
$712$	−102496.	−5.39496
$713$	−4795.41	−0.251879
$714$	0	0
$715$	−2993.20	−0.156558
$716$	49178.2	2.56687
$717$	0	0
$718$	7186.59	0.373539
$719$	17297.5	0.897200	0.448600	−	0.893733i	$-0.351923\pi$
$719$	17297.5	0.897200	0.448600	+	0.893733i	$0.351923\pi$
$720$	0	0
$721$	0	0
$722$	−26690.4	−1.37578
$723$	0	0
$724$	5016.87	0.257528
$725$	−6806.86	−0.348690
$726$	0	0
$727$	−3514.71	−0.179303	−0.0896516	−	0.995973i	$-0.528575\pi$
$727$	−3514.71	−0.179303	−0.0896516	+	0.995973i	$0.528575\pi$
$728$	0	0
$729$	0	0
$730$	−9775.78	−0.495641
$731$	−10603.7	−0.536513
$732$	0	0
$733$	27511.2	1.38629	0.693144	−	0.720799i	$-0.256225\pi$
$733$	27511.2	1.38629	0.693144	+	0.720799i	$0.256225\pi$
$734$	−7315.16	−0.367858
$735$	0	0
$736$	−33508.4	−1.67817
$737$	2757.26	0.137809
$738$	0	0
$739$	16101.6	0.801497	0.400749	−	0.916188i	$-0.368750\pi$
$739$	16101.6	0.801497	0.400749	+	0.916188i	$0.368750\pi$
$740$	21178.6	1.05208
$741$	0	0
$742$	0	0
$743$	−14682.4	−0.724961	−0.362480	−	0.931991i	$-0.618070\pi$
$743$	−14682.4	−0.724961	−0.362480	+	0.931991i	$0.618070\pi$
$744$	0	0
$745$	−10012.3	−0.492380
$746$	−28966.9	−1.42166
$747$	0	0
$748$	12209.9	0.596843
$749$	0	0
$750$	0	0
$751$	−7273.06	−0.353393	−0.176696	−	0.984265i	$-0.556541\pi$
$751$	−7273.06	−0.353393	−0.176696	+	0.984265i	$0.556541\pi$
$752$	−112771.	−5.46851
$753$	0	0
$754$	14853.9	0.717437
$755$	−2519.23	−0.121436
$756$	0	0
$757$	8505.93	0.408393	0.204196	−	0.978930i	$-0.434542\pi$
$757$	8505.93	0.408393	0.204196	+	0.978930i	$0.434542\pi$
$758$	−2944.38	−0.141088
$759$	0	0
$760$	−39196.9	−1.87082
$761$	14217.7	0.677256	0.338628	−	0.940920i	$-0.390037\pi$
$761$	14217.7	0.677256	0.338628	+	0.940920i	$0.390037\pi$
$762$	0	0
$763$	0	0
$764$	87029.3	4.12121
$765$	0	0
$766$	−31110.1	−1.46743
$767$	−20304.7	−0.955879
$768$	0	0
$769$	−16379.1	−0.768068	−0.384034	−	0.923319i	$-0.625466\pi$
$769$	−16379.1	−0.768068	−0.384034	+	0.923319i	$0.625466\pi$
$770$	0	0
$771$	0	0
$772$	41583.8	1.93865
$773$	−39896.7	−1.85638	−0.928192	−	0.372102i	$-0.878637\pi$
$773$	−39896.7	−1.85638	−0.928192	+	0.372102i	$0.878637\pi$
$774$	0	0
$775$	6024.59	0.279238
$776$	5124.70	0.237070
$777$	0	0
$778$	41292.5	1.90284
$779$	−2703.05	−0.124322
$780$	0	0
$781$	−10931.1	−0.500829
$782$	−17281.7	−0.790270
$783$	0	0
$784$	0	0
$785$	10364.5	0.471244
$786$	0	0
$787$	33128.5	1.50051	0.750257	−	0.661146i	$-0.229930\pi$
$787$	33128.5	1.50051	0.750257	+	0.661146i	$0.229930\pi$
$788$	33618.3	1.51980
$789$	0	0
$790$	−12920.0	−0.581865
$791$	0	0
$792$	0	0
$793$	−14828.1	−0.664013
$794$	42614.3	1.90469
$795$	0	0
$796$	21944.2	0.977127
$797$	−17851.5	−0.793390	−0.396695	−	0.917951i	$-0.629843\pi$
$797$	−17851.5	−0.793390	−0.396695	+	0.917951i	$0.629843\pi$
$798$	0	0
$799$	−26980.7	−1.19463
$800$	42097.4	1.86046
$801$	0	0
$802$	4140.36	0.182296
$803$	4608.61	0.202533
$804$	0	0
$805$	0	0
$806$	−13146.8	−0.574538
$807$	0	0
$808$	−74662.5	−3.25076
$809$	5057.03	0.219772	0.109886	−	0.993944i	$-0.464951\pi$
$809$	5057.03	0.219772	0.109886	+	0.993944i	$0.464951\pi$
$810$	0	0
$811$	17535.4	0.759251	0.379626	−	0.925140i	$-0.376053\pi$
$811$	17535.4	0.759251	0.379626	+	0.925140i	$0.376053\pi$
$812$	0	0
$813$	0	0
$814$	−13942.7	−0.600358
$815$	12908.8	0.554815
$816$	0	0
$817$	−26589.8	−1.13863
$818$	77992.1	3.33366
$819$	0	0
$820$	−2782.14	−0.118484
$821$	18700.8	0.794959	0.397480	−	0.917611i	$-0.369885\pi$
$821$	18700.8	0.794959	0.397480	+	0.917611i	$0.369885\pi$
$822$	0	0
$823$	22222.6	0.941230	0.470615	−	0.882339i	$-0.344032\pi$
$823$	22222.6	0.941230	0.470615	+	0.882339i	$0.344032\pi$
$824$	−93402.0	−3.94880
$825$	0	0
$826$	0	0
$827$	−25178.9	−1.05872	−0.529358	−	0.848399i	$-0.677567\pi$
$827$	−25178.9	−1.05872	−0.529358	+	0.848399i	$0.677567\pi$
$828$	0	0
$829$	−12278.9	−0.514432	−0.257216	−	0.966354i	$-0.582805\pi$
$829$	−12278.9	−0.514432	−0.257216	+	0.966354i	$0.582805\pi$
$830$	−7121.81	−0.297834
$831$	0	0
$832$	−35643.1	−1.48522
$833$	0	0
$834$	0	0
$835$	17861.1	0.740249
$836$	30617.7	1.26667
$837$	0	0
$838$	−17890.6	−0.737494
$839$	−25765.0	−1.06020	−0.530098	−	0.847936i	$-0.677845\pi$
$839$	−25765.0	−1.06020	−0.530098	+	0.847936i	$0.677845\pi$
$840$	0	0
$841$	−19153.2	−0.785321
$842$	−83080.2	−3.40040
$843$	0	0
$844$	−94361.8	−3.84842
$845$	−3901.29	−0.158826
$846$	0	0
$847$	0	0
$848$	52308.2	2.11824
$849$	0	0
$850$	21711.4	0.876111
$851$	14131.5	0.569238
$852$	0	0
$853$	−37864.5	−1.51988	−0.759939	−	0.649995i	$-0.774771\pi$
$853$	−37864.5	−1.51988	−0.759939	+	0.649995i	$0.774771\pi$
$854$	0	0
$855$	0	0
$856$	64060.0	2.55786
$857$	−29208.7	−1.16424	−0.582118	−	0.813104i	$-0.697776\pi$
$857$	−29208.7	−1.16424	−0.582118	+	0.813104i	$0.697776\pi$
$858$	0	0
$859$	−34902.9	−1.38635	−0.693173	−	0.720771i	$-0.743788\pi$
$859$	−34902.9	−1.38635	−0.693173	+	0.720771i	$0.743788\pi$
$860$	−27367.8	−1.08516
$861$	0	0
$862$	25947.6	1.02527
$863$	13589.3	0.536021	0.268011	−	0.963416i	$-0.413634\pi$
$863$	13589.3	0.536021	0.268011	+	0.963416i	$0.413634\pi$
$864$	0	0
$865$	1191.68	0.0468420
$866$	−27899.0	−1.09474
$867$	0	0
$868$	0	0
$869$	6090.89	0.237767
$870$	0	0
$871$	−7661.52	−0.298049
$872$	−126261.	−4.90338
$873$	0	0
$874$	−43335.7	−1.67717
$875$	0	0
$876$	0	0
$877$	−2379.75	−0.0916288	−0.0458144	−	0.998950i	$-0.514588\pi$
$877$	−2379.75	−0.0916288	−0.0458144	+	0.998950i	$0.514588\pi$
$878$	−4379.22	−0.168327
$879$	0	0
$880$	14065.7	0.538812
$881$	−24235.5	−0.926803	−0.463401	−	0.886148i	$-0.653371\pi$
$881$	−24235.5	−0.926803	−0.463401	+	0.886148i	$0.653371\pi$
$882$	0	0
$883$	−9844.13	−0.375177	−0.187589	−	0.982248i	$-0.560067\pi$
$883$	−9844.13	−0.375177	−0.187589	+	0.982248i	$0.560067\pi$
$884$	−33927.3	−1.29084
$885$	0	0
$886$	69152.6	2.62215
$887$	28609.9	1.08300	0.541502	−	0.840699i	$-0.317856\pi$
$887$	28609.9	1.08300	0.541502	+	0.840699i	$0.317856\pi$
$888$	0	0
$889$	0	0
$890$	−46807.7	−1.76292
$891$	0	0
$892$	−65498.4	−2.45857
$893$	−67657.0	−2.53534
$894$	0	0
$895$	13554.4	0.506226
$896$	0	0
$897$	0	0
$898$	87725.2	3.25994
$899$	−4634.09	−0.171919
$900$	0	0
$901$	12514.9	0.462743
$902$	1831.59	0.0676113
$903$	0	0
$904$	43498.0	1.60036
$905$	1382.74	0.0507886
$906$	0	0
$907$	44578.3	1.63197	0.815986	−	0.578071i	$-0.196194\pi$
$907$	44578.3	1.63197	0.815986	+	0.578071i	$0.196194\pi$
$908$	103678.	3.78928
$909$	0	0
$910$	0	0
$911$	−45870.6	−1.66823	−0.834116	−	0.551589i	$-0.814022\pi$
$911$	−45870.6	−1.66823	−0.834116	+	0.551589i	$0.814022\pi$
$912$	0	0
$913$	3357.45	0.121703
$914$	19698.1	0.712863
$915$	0	0
$916$	−12405.6	−0.447479
$917$	0	0
$918$	0	0
$919$	31088.3	1.11590	0.557948	−	0.829876i	$-0.311589\pi$
$919$	31088.3	1.11590	0.557948	+	0.829876i	$0.311589\pi$
$920$	−26919.6	−0.964686
$921$	0	0
$922$	51564.8	1.84186
$923$	30374.0	1.08318
$924$	0	0
$925$	−17753.7	−0.631069
$926$	229.732	0.00815278
$927$	0	0
$928$	−32381.1	−1.14543
$929$	−42094.5	−1.48663	−0.743313	−	0.668943i	$-0.766747\pi$
$929$	−42094.5	−1.48663	−0.743313	+	0.668943i	$0.766747\pi$
$930$	0	0
$931$	0	0
$932$	−57051.5	−2.00513
$933$	0	0
$934$	8138.87	0.285130
$935$	3365.26	0.117707
$936$	0	0
$937$	44385.1	1.54749	0.773745	−	0.633497i	$-0.218381\pi$
$937$	44385.1	1.54749	0.773745	+	0.633497i	$0.218381\pi$
$938$	0	0
$939$	0	0
$940$	−69636.6	−2.41627
$941$	40991.6	1.42007	0.710036	−	0.704165i	$-0.248679\pi$
$941$	40991.6	1.42007	0.710036	+	0.704165i	$0.248679\pi$
$942$	0	0
$943$	−1856.39	−0.0641066
$944$	95416.1	3.28976
$945$	0	0
$946$	18017.4	0.619233
$947$	52622.4	1.80570	0.902851	−	0.429955i	$-0.141470\pi$
$947$	52622.4	1.80570	0.902851	+	0.429955i	$0.141470\pi$
$948$	0	0
$949$	−12805.8	−0.438034
$950$	54443.6	1.85935
$951$	0	0
$952$	0	0
$953$	10798.1	0.367035	0.183517	−	0.983016i	$-0.441252\pi$
$953$	10798.1	0.367035	0.183517	+	0.983016i	$0.441252\pi$
$954$	0	0
$955$	23986.8	0.812768
$956$	69260.0	2.34313
$957$	0	0
$958$	37345.8	1.25949
$959$	0	0
$960$	0	0
$961$	−25689.5	−0.862324
$962$	38742.2	1.29844
$963$	0	0
$964$	53191.4	1.77716
$965$	11461.2	0.382331
$966$	0	0
$967$	15648.9	0.520408	0.260204	−	0.965554i	$-0.416210\pi$
$967$	15648.9	0.520408	0.260204	+	0.965554i	$0.416210\pi$
$968$	73520.4	2.44115
$969$	0	0
$970$	2340.33	0.0774675
$971$	47259.8	1.56194	0.780968	−	0.624571i	$-0.214726\pi$
$971$	47259.8	1.56194	0.780968	+	0.624571i	$0.214726\pi$
$972$	0	0
$973$	0	0
$974$	81595.2	2.68427
$975$	0	0
$976$	69680.7	2.28527
$977$	48966.5	1.60346	0.801728	−	0.597689i	$-0.203914\pi$
$977$	48966.5	1.60346	0.801728	+	0.597689i	$0.203914\pi$
$978$	0	0
$979$	22066.6	0.720380
$980$	0	0
$981$	0	0
$982$	−12704.6	−0.412851
$983$	−19111.3	−0.620097	−0.310049	−	0.950721i	$-0.600345\pi$
$983$	−19111.3	−0.620097	−0.310049	+	0.950721i	$0.600345\pi$
$984$	0	0
$985$	9265.77	0.299728
$986$	−16700.3	−0.539397
$987$	0	0
$988$	−85076.4	−2.73951
$989$	−18261.3	−0.587134
$990$	0	0
$991$	−54102.5	−1.73423	−0.867115	−	0.498107i	$-0.834029\pi$
$991$	−54102.5	−1.73423	−0.867115	+	0.498107i	$0.834029\pi$
$992$	28659.7	0.917286
$993$	0	0
$994$	0	0
$995$	6048.21	0.192705
$996$	0	0
$997$	−9192.80	−0.292015	−0.146008	−	0.989283i	$-0.546642\pi$
$997$	−9192.80	−0.292015	−0.146008	+	0.989283i	$0.546642\pi$
$998$	−3681.41	−0.116766
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.t.1.1		3
3.2	odd	2		147.4.a.m.1.3		3
7.2	even	3		441.4.e.w.361.3		6
7.3	odd	6		63.4.e.c.37.3		6
7.4	even	3		441.4.e.w.226.3		6
7.5	odd	6		63.4.e.c.46.3		6
7.6	odd	2		441.4.a.s.1.1		3
12.11	even	2		2352.4.a.cg.1.2		3
21.2	odd	6		147.4.e.n.67.1		6
21.5	even	6		21.4.e.b.4.1	✓	6
21.11	odd	6		147.4.e.n.79.1		6
21.17	even	6		21.4.e.b.16.1	yes	6
21.20	even	2		147.4.a.l.1.3		3
84.47	odd	6		336.4.q.k.193.2		6
84.59	odd	6		336.4.q.k.289.2		6
84.83	odd	2		2352.4.a.ci.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.b.4.1	✓	6	21.5	even	6
21.4.e.b.16.1	yes	6	21.17	even	6
63.4.e.c.37.3		6	7.3	odd	6
63.4.e.c.46.3		6	7.5	odd	6
147.4.a.l.1.3		3	21.20	even	2
147.4.a.m.1.3		3	3.2	odd	2
147.4.e.n.67.1		6	21.2	odd	6
147.4.e.n.79.1		6	21.11	odd	6
336.4.q.k.193.2		6	84.47	odd	6
336.4.q.k.289.2		6	84.59	odd	6
441.4.a.s.1.1		3	7.6	odd	2
441.4.a.t.1.1		3	1.1	even	1	trivial
441.4.e.w.226.3		6	7.4	even	3
441.4.e.w.361.3		6	7.2	even	3
2352.4.a.cg.1.2		3	12.11	even	2
2352.4.a.ci.1.2		3	84.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-5.30829 q^{2} +20.1780 q^{4} +5.56140 q^{5} -64.6443 q^{8} +O(q^{10})\)	\(q-5.30829 q^{2} +20.1780 q^{4} +5.56140 q^{5} -64.6443 q^{8} -29.5215 q^{10} +13.9174 q^{11} -38.6718 q^{13} +181.727 q^{16} +43.4788 q^{17} +109.028 q^{19} +112.218 q^{20} -73.8775 q^{22} +74.8778 q^{23} -94.0708 q^{25} +205.281 q^{26} +72.3589 q^{29} -64.0431 q^{31} -447.507 q^{32} -230.798 q^{34} +188.727 q^{37} -578.751 q^{38} -359.513 q^{40} -24.7923 q^{41} -243.881 q^{43} +280.825 q^{44} -397.474 q^{46} -620.549 q^{47} +499.356 q^{50} -780.319 q^{52} +287.839 q^{53} +77.4001 q^{55} -384.102 q^{58} +525.051 q^{59} +383.436 q^{61} +339.960 q^{62} +921.681 q^{64} -215.069 q^{65} +198.117 q^{67} +877.314 q^{68} -785.432 q^{71} +331.141 q^{73} -1001.82 q^{74} +2199.96 q^{76} +437.647 q^{79} +1010.66 q^{80} +131.605 q^{82} +241.241 q^{83} +241.803 q^{85} +1294.59 q^{86} -899.680 q^{88} +1585.54 q^{89} +1510.88 q^{92} +3294.05 q^{94} +606.347 q^{95} -79.2754 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 25 q^{4} + 11 q^{5} - 39 q^{8}+O(q^{10}) \) 3 * q - q^2 + 25 * q^4 + 11 * q^5 - 39 * q^8	\( 3 q - q^{2} + 25 q^{4} + 11 q^{5} - 39 q^{8} + 55 q^{10} - 35 q^{11} - 62 q^{13} + 241 q^{16} + 48 q^{17} + 202 q^{19} + 439 q^{20} - 7 q^{22} - 216 q^{23} + 130 q^{25} + 274 q^{26} - 53 q^{29} + 95 q^{31} - 683 q^{32} + 24 q^{34} + 262 q^{37} - 398 q^{38} - 21 q^{40} + 244 q^{41} + 360 q^{43} + 905 q^{44} - 1056 q^{46} - 210 q^{47} + 1378 q^{50} - 324 q^{52} - 393 q^{53} + 1031 q^{55} - 1249 q^{58} + 1143 q^{59} + 70 q^{61} + 1059 q^{62} - 399 q^{64} + 472 q^{65} - 628 q^{67} + 1944 q^{68} - 318 q^{71} - 988 q^{73} - 1002 q^{74} + 2340 q^{76} + 861 q^{79} + 175 q^{80} - 124 q^{82} + 519 q^{83} + 1800 q^{85} + 3208 q^{86} - 891 q^{88} + 1766 q^{89} + 672 q^{92} + 3294 q^{94} + 736 q^{95} - 19 q^{97}+O(q^{100}) \) 3 * q - q^2 + 25 * q^4 + 11 * q^5 - 39 * q^8 + 55 * q^10 - 35 * q^11 - 62 * q^13 + 241 * q^16 + 48 * q^17 + 202 * q^19 + 439 * q^20 - 7 * q^22 - 216 * q^23 + 130 * q^25 + 274 * q^26 - 53 * q^29 + 95 * q^31 - 683 * q^32 + 24 * q^34 + 262 * q^37 - 398 * q^38 - 21 * q^40 + 244 * q^41 + 360 * q^43 + 905 * q^44 - 1056 * q^46 - 210 * q^47 + 1378 * q^50 - 324 * q^52 - 393 * q^53 + 1031 * q^55 - 1249 * q^58 + 1143 * q^59 + 70 * q^61 + 1059 * q^62 - 399 * q^64 + 472 * q^65 - 628 * q^67 + 1944 * q^68 - 318 * q^71 - 988 * q^73 - 1002 * q^74 + 2340 * q^76 + 861 * q^79 + 175 * q^80 - 124 * q^82 + 519 * q^83 + 1800 * q^85 + 3208 * q^86 - 891 * q^88 + 1766 * q^89 + 672 * q^92 + 3294 * q^94 + 736 * q^95 - 19 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more