LMFDB - Embedded newform 441.6.a.be.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,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, 6, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$70.7292645375$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 146x^{6} + 5453x^{4} - 40868x^{2} + 3844 $ x^8 - 146x^6 + 5453x^4 - 40868*x^2 + 3844
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.308653$ 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)

$f(q)$	$=$	$q+8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} +16.5098 q^{8} +O(q^{10})$	$q+8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} +16.5098 q^{8} +141.556 q^{10} -114.280 q^{11} -205.240 q^{13} -954.857 q^{16} -757.277 q^{17} -1013.65 q^{19} +592.838 q^{20} -928.635 q^{22} -916.299 q^{23} -2821.54 q^{25} -1667.78 q^{26} -1095.47 q^{29} +8233.32 q^{31} -8287.47 q^{32} -6153.62 q^{34} -10716.4 q^{37} -8236.94 q^{38} +287.604 q^{40} +18758.1 q^{41} -4643.49 q^{43} -3889.13 q^{44} -7445.84 q^{46} -13969.4 q^{47} -22927.8 q^{50} -6984.69 q^{52} -29306.0 q^{53} -1990.77 q^{55} -8901.78 q^{58} -30378.0 q^{59} +18658.2 q^{61} +66903.9 q^{62} -36788.5 q^{64} -3575.32 q^{65} +19933.4 q^{67} -25771.4 q^{68} -57338.5 q^{71} -60194.4 q^{73} -87081.2 q^{74} -34496.4 q^{76} +35715.5 q^{79} -16633.7 q^{80} +152428. q^{82} +86641.2 q^{83} -13191.9 q^{85} -37733.0 q^{86} -1886.73 q^{88} -42941.7 q^{89} -31183.2 q^{92} -113515. q^{94} -17658.0 q^{95} +20619.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{4}+O(q^{10}) $ 8 * q + 20 * q^4	$ 8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100}) $ 8 * q + 20 * q^4 - 828 * q^16 - 2384 * q^22 - 2392 * q^25 - 19136 * q^37 - 41184 * q^43 - 13152 * q^46 - 88872 * q^58 - 210812 * q^64 - 42336 * q^67 - 251072 * q^79 - 567664 * q^85 - 88752 * q^88

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 $

$2$	8.12599	1.43649	0.718243	−	0.695792i	$-0.244947\pi$
$2$	8.12599	1.43649	0.718243	+	0.695792i	$0.244947\pi$
$3$	0	0
$3$	0	0
$4$	34.0317	1.06349
$5$	17.4201	0.311621	0.155811	−	0.987787i	$-0.450201\pi$
$5$	17.4201	0.311621	0.155811	+	0.987787i	$0.450201\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	16.5098	0.0912047
$9$	0	0
$10$	141.556	0.447639
$11$	−114.280	−0.284765	−0.142383	−	0.989812i	$-0.545476\pi$
$11$	−114.280	−0.284765	−0.142383	+	0.989812i	$0.545476\pi$
$12$	0	0
$13$	−205.240	−0.336825	−0.168413	−	0.985717i	$-0.553864\pi$
$13$	−205.240	−0.336825	−0.168413	+	0.985717i	$0.553864\pi$
$14$	0	0
$15$	0	0
$16$	−954.857	−0.932477
$17$	−757.277	−0.635524	−0.317762	−	0.948170i	$-0.602931\pi$
$17$	−757.277	−0.635524	−0.317762	+	0.948170i	$0.602931\pi$
$18$	0	0
$19$	−1013.65	−0.644178	−0.322089	−	0.946709i	$-0.604385\pi$
$19$	−1013.65	−0.644178	−0.322089	+	0.946709i	$0.604385\pi$
$20$	592.838	0.331406
$21$	0	0
$22$	−928.635	−0.409061
$23$	−916.299	−0.361175	−0.180587	−	0.983559i	$-0.557800\pi$
$23$	−916.299	−0.361175	−0.180587	+	0.983559i	$0.557800\pi$
$24$	0	0
$25$	−2821.54	−0.902892
$26$	−1667.78	−0.483845
$27$	0	0
$28$	0	0
$29$	−1095.47	−0.241883	−0.120942	−	0.992660i	$-0.538591\pi$
$29$	−1095.47	−0.241883	−0.120942	+	0.992660i	$0.538591\pi$
$30$	0	0
$31$	8233.32	1.53876	0.769380	−	0.638791i	$-0.220565\pi$
$31$	8233.32	1.53876	0.769380	+	0.638791i	$0.220565\pi$
$32$	−8287.47	−1.43070
$33$	0	0
$34$	−6153.62	−0.912922
$35$	0	0
$36$	0	0
$37$	−10716.4	−1.28690	−0.643448	−	0.765490i	$-0.722497\pi$
$37$	−10716.4	−1.28690	−0.643448	+	0.765490i	$0.722497\pi$
$38$	−8236.94	−0.925352
$39$	0	0
$40$	287.604	0.0284213
$41$	18758.1	1.74273	0.871365	−	0.490636i	$-0.163235\pi$
$41$	18758.1	1.74273	0.871365	+	0.490636i	$0.163235\pi$
$42$	0	0
$43$	−4643.49	−0.382978	−0.191489	−	0.981495i	$-0.561332\pi$
$43$	−4643.49	−0.382978	−0.191489	+	0.981495i	$0.561332\pi$
$44$	−3889.13	−0.302845
$45$	0	0
$46$	−7445.84	−0.518823
$47$	−13969.4	−0.922428	−0.461214	−	0.887289i	$-0.652586\pi$
$47$	−13969.4	−0.922428	−0.461214	+	0.887289i	$0.652586\pi$
$48$	0	0
$49$	0	0
$50$	−22927.8	−1.29699
$51$	0	0
$52$	−6984.69	−0.358211
$53$	−29306.0	−1.43307	−0.716533	−	0.697553i	$-0.754272\pi$
$53$	−29306.0	−1.43307	−0.716533	+	0.697553i	$0.754272\pi$
$54$	0	0
$55$	−1990.77	−0.0887388
$56$	0	0
$57$	0	0
$58$	−8901.78	−0.347462
$59$	−30378.0	−1.13613	−0.568067	−	0.822983i	$-0.692308\pi$
$59$	−30378.0	−1.13613	−0.568067	+	0.822983i	$0.692308\pi$
$60$	0	0
$61$	18658.2	0.642014	0.321007	−	0.947077i	$-0.395979\pi$
$61$	18658.2	0.642014	0.321007	+	0.947077i	$0.395979\pi$
$62$	66903.9	2.21041
$63$	0	0
$64$	−36788.5	−1.12270
$65$	−3575.32	−0.104962
$66$	0	0
$67$	19933.4	0.542493	0.271246	−	0.962510i	$-0.412564\pi$
$67$	19933.4	0.542493	0.271246	+	0.962510i	$0.412564\pi$
$68$	−25771.4	−0.675875
$69$	0	0
$70$	0	0
$71$	−57338.5	−1.34990	−0.674948	−	0.737866i	$-0.735834\pi$
$71$	−57338.5	−1.34990	−0.674948	+	0.737866i	$0.735834\pi$
$72$	0	0
$73$	−60194.4	−1.32205	−0.661027	−	0.750362i	$-0.729879\pi$
$73$	−60194.4	−1.32205	−0.661027	+	0.750362i	$0.729879\pi$
$74$	−87081.2	−1.84861
$75$	0	0
$76$	−34496.4	−0.685078
$77$	0	0
$78$	0	0
$79$	35715.5	0.643857	0.321928	−	0.946764i	$-0.395669\pi$
$79$	35715.5	0.643857	0.321928	+	0.946764i	$0.395669\pi$
$80$	−16633.7	−0.290580
$81$	0	0
$82$	152428.	2.50341
$83$	86641.2	1.38048	0.690239	−	0.723582i	$-0.257506\pi$
$83$	86641.2	1.38048	0.690239	+	0.723582i	$0.257506\pi$
$84$	0	0
$85$	−13191.9	−0.198043
$86$	−37733.0	−0.550142
$87$	0	0
$88$	−1886.73	−0.0259719
$89$	−42941.7	−0.574651	−0.287326	−	0.957833i	$-0.592766\pi$
$89$	−42941.7	−0.574651	−0.287326	+	0.957833i	$0.592766\pi$
$90$	0	0
$91$	0	0
$92$	−31183.2	−0.384107
$93$	0	0
$94$	−113515.	−1.32506
$95$	−17658.0	−0.200739
$96$	0	0
$97$	20619.4	0.222508	0.111254	−	0.993792i	$-0.464513\pi$
$97$	20619.4	0.222508	0.111254	+	0.993792i	$0.464513\pi$
$98$	0	0
$99$	0	0
$100$	−96021.8	−0.960218
$101$	−106588.	−1.03970	−0.519848	−	0.854259i	$-0.674012\pi$
$101$	−106588.	−1.03970	−0.519848	+	0.854259i	$0.674012\pi$
$102$	0	0
$103$	87392.9	0.811677	0.405839	−	0.913945i	$-0.366980\pi$
$103$	87392.9	0.811677	0.405839	+	0.913945i	$0.366980\pi$
$104$	−3388.48	−0.0307201
$105$	0	0
$106$	−238140.	−2.05858
$107$	39306.9	0.331902	0.165951	−	0.986134i	$-0.446931\pi$
$107$	39306.9	0.331902	0.165951	+	0.986134i	$0.446931\pi$
$108$	0	0
$109$	−44341.3	−0.357472	−0.178736	−	0.983897i	$-0.557201\pi$
$109$	−44341.3	−0.357472	−0.178736	+	0.983897i	$0.557201\pi$
$110$	−16177.0	−0.127472
$111$	0	0
$112$	0	0
$113$	64665.1	0.476402	0.238201	−	0.971216i	$-0.423442\pi$
$113$	64665.1	0.476402	0.238201	+	0.971216i	$0.423442\pi$
$114$	0	0
$115$	−15962.1	−0.112550
$116$	−37280.7	−0.257241
$117$	0	0
$118$	−246851.	−1.63204
$119$	0	0
$120$	0	0
$121$	−147991.	−0.918909
$122$	151616.	0.922244
$123$	0	0
$124$	280194.	1.63646
$125$	−103590.	−0.592981
$126$	0	0
$127$	−155448.	−0.855215	−0.427607	−	0.903965i	$-0.640643\pi$
$127$	−155448.	−0.855215	−0.427607	+	0.903965i	$0.640643\pi$
$128$	−33744.0	−0.182042
$129$	0	0
$130$	−29053.0	−0.150776
$131$	189150.	0.963002	0.481501	−	0.876446i	$-0.340092\pi$
$131$	189150.	0.963002	0.481501	+	0.876446i	$0.340092\pi$
$132$	0	0
$133$	0	0
$134$	161978.	0.779283
$135$	0	0
$136$	−12502.5	−0.0579628
$137$	344445.	1.56790	0.783950	−	0.620824i	$-0.213202\pi$
$137$	344445.	1.56790	0.783950	+	0.620824i	$0.213202\pi$
$138$	0	0
$139$	−270646.	−1.18813	−0.594065	−	0.804417i	$-0.702478\pi$
$139$	−270646.	−1.18813	−0.594065	+	0.804417i	$0.702478\pi$
$140$	0	0
$141$	0	0
$142$	−465932.	−1.93911
$143$	23454.8	0.0959161
$144$	0	0
$145$	−19083.2	−0.0753759
$146$	−489139.	−1.89911
$147$	0	0
$148$	−364697.	−1.36860
$149$	131911.	0.486759	0.243379	−	0.969931i	$-0.421744\pi$
$149$	131911.	0.486759	0.243379	+	0.969931i	$0.421744\pi$
$150$	0	0
$151$	−322961.	−1.15268	−0.576338	−	0.817211i	$-0.695519\pi$
$151$	−322961.	−1.15268	−0.576338	+	0.817211i	$0.695519\pi$
$152$	−16735.2	−0.0587521
$153$	0	0
$154$	0	0
$155$	143426.	0.479510
$156$	0	0
$157$	181881.	0.588894	0.294447	−	0.955668i	$-0.404865\pi$
$157$	181881.	0.588894	0.294447	+	0.955668i	$0.404865\pi$
$158$	290224.	0.924891
$159$	0	0
$160$	−144369.	−0.445835
$161$	0	0
$162$	0	0
$163$	−196117.	−0.578157	−0.289078	−	0.957305i	$-0.593349\pi$
$163$	−196117.	−0.578157	−0.289078	+	0.957305i	$0.593349\pi$
$164$	638372.	1.85338
$165$	0	0
$166$	704046.	1.98304
$167$	−293276.	−0.813741	−0.406870	−	0.913486i	$-0.633380\pi$
$167$	−293276.	−0.813741	−0.406870	+	0.913486i	$0.633380\pi$
$168$	0	0
$169$	−329169.	−0.886549
$170$	−107197.	−0.284486
$171$	0	0
$172$	−158026.	−0.407294
$173$	442828.	1.12492	0.562458	−	0.826826i	$-0.309856\pi$
$173$	442828.	1.12492	0.562458	+	0.826826i	$0.309856\pi$
$174$	0	0
$175$	0	0
$176$	109121.	0.265537
$177$	0	0
$178$	−348944.	−0.825479
$179$	661446.	1.54298	0.771492	−	0.636239i	$-0.219511\pi$
$179$	661446.	1.54298	0.771492	+	0.636239i	$0.219511\pi$
$180$	0	0
$181$	139700.	0.316956	0.158478	−	0.987363i	$-0.449341\pi$
$181$	139700.	0.316956	0.158478	+	0.987363i	$0.449341\pi$
$182$	0	0
$183$	0	0
$184$	−15127.9	−0.0329409
$185$	−186681.	−0.401024
$186$	0	0
$187$	86541.2	0.180975
$188$	−475402.	−0.980995
$189$	0	0
$190$	−143489.	−0.288359
$191$	751907.	1.49135	0.745677	−	0.666308i	$-0.232126\pi$
$191$	751907.	1.49135	0.745677	+	0.666308i	$0.232126\pi$
$192$	0	0
$193$	464929.	0.898449	0.449224	−	0.893419i	$-0.351700\pi$
$193$	464929.	0.898449	0.449224	+	0.893419i	$0.351700\pi$
$194$	167553.	0.319630
$195$	0	0
$196$	0	0
$197$	518493.	0.951870	0.475935	−	0.879480i	$-0.342110\pi$
$197$	518493.	0.951870	0.475935	+	0.879480i	$0.342110\pi$
$198$	0	0
$199$	−590287.	−1.05665	−0.528324	−	0.849043i	$-0.677179\pi$
$199$	−590287.	−1.05665	−0.528324	+	0.849043i	$0.677179\pi$
$200$	−46583.1	−0.0823480
$201$	0	0
$202$	−866137.	−1.49351
$203$	0	0
$204$	0	0
$205$	326770.	0.543071
$206$	710154.	1.16596
$207$	0	0
$208$	195975.	0.314082
$209$	115840.	0.183439
$210$	0	0
$211$	207037.	0.320142	0.160071	−	0.987106i	$-0.448828\pi$
$211$	207037.	0.320142	0.160071	+	0.987106i	$0.448828\pi$
$212$	−997333.	−1.52405
$213$	0	0
$214$	319408.	0.476772
$215$	−80890.3	−0.119344
$216$	0	0
$217$	0	0
$218$	−360317.	−0.513503
$219$	0	0
$220$	−67749.2	−0.0943730
$221$	155424.	0.214061
$222$	0	0
$223$	1.23347e6	1.66098	0.830491	−	0.557033i	$-0.188060\pi$
$223$	1.23347e6	1.66098	0.830491	+	0.557033i	$0.188060\pi$
$224$	0	0
$225$	0	0
$226$	525468.	0.684345
$227$	−1.12039e6	−1.44312	−0.721562	−	0.692350i	$-0.756576\pi$
$227$	−1.12039e6	−1.44312	−0.721562	+	0.692350i	$0.756576\pi$
$228$	0	0
$229$	412271.	0.519510	0.259755	−	0.965675i	$-0.416358\pi$
$229$	412271.	0.519510	0.259755	+	0.965675i	$0.416358\pi$
$230$	−129708.	−0.161676
$231$	0	0
$232$	−18086.0	−0.0220609
$233$	836448.	1.00937	0.504684	−	0.863304i	$-0.331609\pi$
$233$	836448.	1.00937	0.504684	+	0.863304i	$0.331609\pi$
$234$	0	0
$235$	−243349.	−0.287448
$236$	−1.03382e6	−1.20827
$237$	0	0
$238$	0	0
$239$	−319694.	−0.362026	−0.181013	−	0.983481i	$-0.557938\pi$
$239$	−319694.	−0.362026	−0.181013	+	0.983481i	$0.557938\pi$
$240$	0	0
$241$	−965937.	−1.07129	−0.535644	−	0.844444i	$-0.679931\pi$
$241$	−965937.	−1.07129	−0.535644	+	0.844444i	$0.679931\pi$
$242$	−1.20258e6	−1.32000
$243$	0	0
$244$	634970.	0.682776
$245$	0	0
$246$	0	0
$247$	208043.	0.216975
$248$	135931.	0.140342
$249$	0	0
$250$	−841768.	−0.851809
$251$	1.60171e6	1.60472	0.802361	−	0.596839i	$-0.203577\pi$
$251$	1.60171e6	1.60472	0.802361	+	0.596839i	$0.203577\pi$
$252$	0	0
$253$	104714.	0.102850
$254$	−1.26317e6	−1.22850
$255$	0	0
$256$	903029.	0.861196
$257$	1.08451e6	1.02424	0.512120	−	0.858914i	$-0.328860\pi$
$257$	1.08451e6	1.02424	0.512120	+	0.858914i	$0.328860\pi$
$258$	0	0
$259$	0	0
$260$	−121674.	−0.111626
$261$	0	0
$262$	1.53703e6	1.38334
$263$	−1.04604e6	−0.932517	−0.466259	−	0.884648i	$-0.654398\pi$
$263$	−1.04604e6	−0.932517	−0.466259	+	0.884648i	$0.654398\pi$
$264$	0	0
$265$	−510514.	−0.446574
$266$	0	0
$267$	0	0
$268$	678368.	0.576937
$269$	−1.41567e6	−1.19284	−0.596419	−	0.802673i	$-0.703410\pi$
$269$	−1.41567e6	−1.19284	−0.596419	+	0.802673i	$0.703410\pi$
$270$	0	0
$271$	−884078.	−0.731252	−0.365626	−	0.930762i	$-0.619145\pi$
$271$	−884078.	−0.731252	−0.365626	+	0.930762i	$0.619145\pi$
$272$	723091.	0.592612
$273$	0	0
$274$	2.79896e6	2.25227
$275$	322444.	0.257112
$276$	0	0
$277$	−1.77062e6	−1.38652	−0.693258	−	0.720690i	$-0.743825\pi$
$277$	−1.77062e6	−1.38652	−0.693258	+	0.720690i	$0.743825\pi$
$278$	−2.19926e6	−1.70673
$279$	0	0
$280$	0	0
$281$	−802195.	−0.606058	−0.303029	−	0.952981i	$-0.597998\pi$
$281$	−802195.	−0.606058	−0.303029	+	0.952981i	$0.597998\pi$
$282$	0	0
$283$	1.81693e6	1.34856	0.674282	−	0.738474i	$-0.264453\pi$
$283$	1.81693e6	1.34856	0.674282	+	0.738474i	$0.264453\pi$
$284$	−1.95133e6	−1.43560
$285$	0	0
$286$	190593.	0.137782
$287$	0	0
$288$	0	0
$289$	−846389.	−0.596109
$290$	−155070.	−0.108276
$291$	0	0
$292$	−2.04852e6	−1.40599
$293$	2.34122e6	1.59321	0.796605	−	0.604500i	$-0.206627\pi$
$293$	2.34122e6	1.59321	0.796605	+	0.604500i	$0.206627\pi$
$294$	0	0
$295$	−529189.	−0.354043
$296$	−176925.	−0.117371
$297$	0	0
$298$	1.07190e6	0.699222
$299$	188062.	0.121653
$300$	0	0
$301$	0	0
$302$	−2.62438e6	−1.65580
$303$	0	0
$304$	967894.	0.600681
$305$	325028.	0.200065
$306$	0	0
$307$	855540.	0.518077	0.259039	−	0.965867i	$-0.416594\pi$
$307$	855540.	0.518077	0.259039	+	0.965867i	$0.416594\pi$
$308$	0	0
$309$	0	0
$310$	1.16548e6	0.688809
$311$	−2.58071e6	−1.51300	−0.756498	−	0.653996i	$-0.773091\pi$
$311$	−2.58071e6	−1.51300	−0.756498	+	0.653996i	$0.773091\pi$
$312$	0	0
$313$	−3.03363e6	−1.75026	−0.875129	−	0.483889i	$-0.839224\pi$
$313$	−3.03363e6	−1.75026	−0.875129	+	0.483889i	$0.839224\pi$
$314$	1.47796e6	0.845938
$315$	0	0
$316$	1.21546e6	0.684736
$317$	751440.	0.419997	0.209999	−	0.977702i	$-0.432654\pi$
$317$	751440.	0.419997	0.209999	+	0.977702i	$0.432654\pi$
$318$	0	0
$319$	125190.	0.0688799
$320$	−640861.	−0.349856
$321$	0	0
$322$	0	0
$323$	767616.	0.409391
$324$	0	0
$325$	579094.	0.304117
$326$	−1.59364e6	−0.830514
$327$	0	0
$328$	309693.	0.158945
$329$	0	0
$330$	0	0
$331$	−726862.	−0.364655	−0.182327	−	0.983238i	$-0.558363\pi$
$331$	−726862.	−0.364655	−0.182327	+	0.983238i	$0.558363\pi$
$332$	2.94855e6	1.46813
$333$	0	0
$334$	−2.38316e6	−1.16893
$335$	347243.	0.169052
$336$	0	0
$337$	650213.	0.311875	0.155938	−	0.987767i	$-0.450160\pi$
$337$	650213.	0.311875	0.155938	+	0.987767i	$0.450160\pi$
$338$	−2.67483e6	−1.27351
$339$	0	0
$340$	−448942.	−0.210617
$341$	−940900.	−0.438185
$342$	0	0
$343$	0	0
$344$	−76663.2	−0.0349294
$345$	0	0
$346$	3.59842e6	1.61593
$347$	60293.3	0.0268810	0.0134405	−	0.999910i	$-0.495722\pi$
$347$	60293.3	0.0268810	0.0134405	+	0.999910i	$0.495722\pi$
$348$	0	0
$349$	3.61218e6	1.58747	0.793735	−	0.608263i	$-0.208134\pi$
$349$	3.61218e6	1.58747	0.793735	+	0.608263i	$0.208134\pi$
$350$	0	0
$351$	0	0
$352$	947088.	0.407412
$353$	−681893.	−0.291259	−0.145629	−	0.989339i	$-0.546521\pi$
$353$	−681893.	−0.291259	−0.145629	+	0.989339i	$0.546521\pi$
$354$	0	0
$355$	−998844.	−0.420656
$356$	−1.46138e6	−0.611137
$357$	0	0
$358$	5.37490e6	2.21648
$359$	3.14620e6	1.28840	0.644200	−	0.764857i	$-0.277190\pi$
$359$	3.14620e6	1.28840	0.644200	+	0.764857i	$0.277190\pi$
$360$	0	0
$361$	−1.44860e6	−0.585035
$362$	1.13520e6	0.455303
$363$	0	0
$364$	0	0
$365$	−1.04860e6	−0.411980
$366$	0	0
$367$	−2.97099e6	−1.15143	−0.575714	−	0.817651i	$-0.695276\pi$
$367$	−2.97099e6	−1.15143	−0.575714	+	0.817651i	$0.695276\pi$
$368$	874934.	0.336787
$369$	0	0
$370$	−1.51697e6	−0.576066
$371$	0	0
$372$	0	0
$373$	3.00412e6	1.11801	0.559005	−	0.829164i	$-0.311183\pi$
$373$	3.00412e6	1.11801	0.559005	+	0.829164i	$0.311183\pi$
$374$	703233.	0.259968
$375$	0	0
$376$	−230632.	−0.0841298
$377$	224835.	0.0814723
$378$	0	0
$379$	−754946.	−0.269972	−0.134986	−	0.990848i	$-0.543099\pi$
$379$	−754946.	−0.269972	−0.134986	+	0.990848i	$0.543099\pi$
$380$	−600932.	−0.213485
$381$	0	0
$382$	6.10999e6	2.14231
$383$	3.66829e6	1.27781	0.638905	−	0.769286i	$-0.279388\pi$
$383$	3.66829e6	1.27781	0.638905	+	0.769286i	$0.279388\pi$
$384$	0	0
$385$	0	0
$386$	3.77801e6	1.29061
$387$	0	0
$388$	701712.	0.236635
$389$	3.00960e6	1.00841	0.504203	−	0.863585i	$-0.331786\pi$
$389$	3.00960e6	1.00841	0.504203	+	0.863585i	$0.331786\pi$
$390$	0	0
$391$	693892.	0.229536
$392$	0	0
$393$	0	0
$394$	4.21327e6	1.36735
$395$	622170.	0.200639
$396$	0	0
$397$	287204.	0.0914564	0.0457282	−	0.998954i	$-0.485439\pi$
$397$	287204.	0.0914564	0.0457282	+	0.998954i	$0.485439\pi$
$398$	−4.79666e6	−1.51786
$399$	0	0
$400$	2.69416e6	0.841927
$401$	−863171.	−0.268063	−0.134031	−	0.990977i	$-0.542792\pi$
$401$	−863171.	−0.268063	−0.134031	+	0.990977i	$0.542792\pi$
$402$	0	0
$403$	−1.68981e6	−0.518293
$404$	−3.62739e6	−1.10571
$405$	0	0
$406$	0	0
$407$	1.22466e6	0.366463
$408$	0	0
$409$	−2.82932e6	−0.836321	−0.418161	−	0.908373i	$-0.637325\pi$
$409$	−2.82932e6	−0.836321	−0.418161	+	0.908373i	$0.637325\pi$
$410$	2.65533e6	0.780114
$411$	0	0
$412$	2.97413e6	0.863212
$413$	0	0
$414$	0	0
$415$	1.50930e6	0.430186
$416$	1.70092e6	0.481894
$417$	0	0
$418$	941314.	0.263508
$419$	−7.08132e6	−1.97051	−0.985256	−	0.171089i	$-0.945272\pi$
$419$	−7.08132e6	−1.97051	−0.985256	+	0.171089i	$0.945272\pi$
$420$	0	0
$421$	−2.95296e6	−0.811994	−0.405997	−	0.913874i	$-0.633076\pi$
$421$	−2.95296e6	−0.811994	−0.405997	+	0.913874i	$0.633076\pi$
$422$	1.68238e6	0.459879
$423$	0	0
$424$	−483836.	−0.130702
$425$	2.13668e6	0.573810
$426$	0	0
$427$	0	0
$428$	1.33768e6	0.352975
$429$	0	0
$430$	−657314.	−0.171436
$431$	4.47340e6	1.15996	0.579981	−	0.814630i	$-0.303060\pi$
$431$	4.47340e6	1.15996	0.579981	+	0.814630i	$0.303060\pi$
$432$	0	0
$433$	5.11279e6	1.31050	0.655252	−	0.755411i	$-0.272563\pi$
$433$	5.11279e6	1.31050	0.655252	+	0.755411i	$0.272563\pi$
$434$	0	0
$435$	0	0
$436$	−1.50901e6	−0.380168
$437$	928810.	0.232661
$438$	0	0
$439$	−590666.	−0.146279	−0.0731393	−	0.997322i	$-0.523302\pi$
$439$	−590666.	−0.146279	−0.0731393	+	0.997322i	$0.523302\pi$
$440$	−32867.2	−0.00809340
$441$	0	0
$442$	1.26297e6	0.307495
$443$	160765.	0.0389209	0.0194605	−	0.999811i	$-0.493805\pi$
$443$	160765.	0.0389209	0.0194605	+	0.999811i	$0.493805\pi$
$444$	0	0
$445$	−748051.	−0.179073
$446$	1.00231e7	2.38598
$447$	0	0
$448$	0	0
$449$	7.98274e6	1.86869	0.934343	−	0.356376i	$-0.115988\pi$
$449$	7.98274e6	1.86869	0.934343	+	0.356376i	$0.115988\pi$
$450$	0	0
$451$	−2.14367e6	−0.496269
$452$	2.20067e6	0.506650
$453$	0	0
$454$	−9.10426e6	−2.07303
$455$	0	0
$456$	0	0
$457$	−1.22463e6	−0.274292	−0.137146	−	0.990551i	$-0.543793\pi$
$457$	−1.22463e6	−0.274292	−0.137146	+	0.990551i	$0.543793\pi$
$458$	3.35011e6	0.746269
$459$	0	0
$460$	−543217.	−0.119696
$461$	−1.99446e6	−0.437091	−0.218546	−	0.975827i	$-0.570131\pi$
$461$	−1.99446e6	−0.437091	−0.218546	+	0.975827i	$0.570131\pi$
$462$	0	0
$463$	−125144.	−0.0271304	−0.0135652	−	0.999908i	$-0.504318\pi$
$463$	−125144.	−0.0271304	−0.0135652	+	0.999908i	$0.504318\pi$
$464$	1.04602e6	0.225550
$465$	0	0
$466$	6.79697e6	1.44994
$467$	−7.06372e6	−1.49879	−0.749396	−	0.662122i	$-0.769656\pi$
$467$	−7.06372e6	−1.49879	−0.749396	+	0.662122i	$0.769656\pi$
$468$	0	0
$469$	0	0
$470$	−1.97745e6	−0.412915
$471$	0	0
$472$	−501535.	−0.103621
$473$	530656.	0.109059
$474$	0	0
$475$	2.86006e6	0.581623
$476$	0	0
$477$	0	0
$478$	−2.59783e6	−0.520045
$479$	4.64810e6	0.925629	0.462814	−	0.886455i	$-0.346840\pi$
$479$	4.64810e6	0.925629	0.462814	+	0.886455i	$0.346840\pi$
$480$	0	0
$481$	2.19943e6	0.433459
$482$	−7.84919e6	−1.53889
$483$	0	0
$484$	−5.03640e6	−0.977252
$485$	359192.	0.0693382
$486$	0	0
$487$	−5.57906e6	−1.06595	−0.532977	−	0.846130i	$-0.678927\pi$
$487$	−5.57906e6	−1.06595	−0.532977	+	0.846130i	$0.678927\pi$
$488$	308043.	0.0585547
$489$	0	0
$490$	0	0
$491$	−5.81454e6	−1.08846	−0.544229	−	0.838937i	$-0.683178\pi$
$491$	−5.81454e6	−1.08846	−0.544229	+	0.838937i	$0.683178\pi$
$492$	0	0
$493$	829574.	0.153723
$494$	1.69055e6	0.311682
$495$	0	0
$496$	−7.86164e6	−1.43486
$497$	0	0
$498$	0	0
$499$	−7.17797e6	−1.29048	−0.645238	−	0.763982i	$-0.723242\pi$
$499$	−7.17797e6	−1.29048	−0.645238	+	0.763982i	$0.723242\pi$
$500$	−3.52533e6	−0.630631
$501$	0	0
$502$	1.30155e7	2.30516
$503$	3.00238e6	0.529109	0.264555	−	0.964371i	$-0.414775\pi$
$503$	3.00238e6	0.529109	0.264555	+	0.964371i	$0.414775\pi$
$504$	0	0
$505$	−1.85679e6	−0.323991
$506$	850907.	0.147743
$507$	0	0
$508$	−5.29016e6	−0.909514
$509$	3.52386e6	0.602870	0.301435	−	0.953487i	$-0.402534\pi$
$509$	3.52386e6	0.602870	0.301435	+	0.953487i	$0.402534\pi$
$510$	0	0
$511$	0	0
$512$	8.41781e6	1.41914
$513$	0	0
$514$	8.81274e6	1.47131
$515$	1.52240e6	0.252936
$516$	0	0
$517$	1.59641e6	0.262675
$518$	0	0
$519$	0	0
$520$	−59027.9	−0.00957302
$521$	3.20451e6	0.517210	0.258605	−	0.965983i	$-0.416737\pi$
$521$	3.20451e6	0.517210	0.258605	+	0.965983i	$0.416737\pi$
$522$	0	0
$523$	−8.14584e6	−1.30221	−0.651106	−	0.758987i	$-0.725695\pi$
$523$	−8.14584e6	−1.30221	−0.651106	+	0.758987i	$0.725695\pi$
$524$	6.43709e6	1.02414
$525$	0	0
$526$	−8.50007e6	−1.33955
$527$	−6.23490e6	−0.977920
$528$	0	0
$529$	−5.59674e6	−0.869553
$530$	−4.14843e6	−0.641497
$531$	0	0
$532$	0	0
$533$	−3.84993e6	−0.586995
$534$	0	0
$535$	684733.	0.103428
$536$	329097.	0.0494779
$537$	0	0
$538$	−1.15037e7	−1.71349
$539$	0	0
$540$	0	0
$541$	−9.33721e6	−1.37159	−0.685794	−	0.727795i	$-0.740545\pi$
$541$	−9.33721e6	−1.37159	−0.685794	+	0.727795i	$0.740545\pi$
$542$	−7.18401e6	−1.05043
$543$	0	0
$544$	6.27591e6	0.909242
$545$	−772432.	−0.111396
$546$	0	0
$547$	−7.27605e6	−1.03975	−0.519873	−	0.854244i	$-0.674021\pi$
$547$	−7.27605e6	−1.03975	−0.519873	+	0.854244i	$0.674021\pi$
$548$	1.17221e7	1.66745
$549$	0	0
$550$	2.62018e6	0.369338
$551$	1.11043e6	0.155816
$552$	0	0
$553$	0	0
$554$	−1.43880e7	−1.99171
$555$	0	0
$556$	−9.21054e6	−1.26357
$557$	−9.10577e6	−1.24359	−0.621797	−	0.783178i	$-0.713597\pi$
$557$	−9.10577e6	−1.24359	−0.621797	+	0.783178i	$0.713597\pi$
$558$	0	0
$559$	953032.	0.128997
$560$	0	0
$561$	0	0
$562$	−6.51863e6	−0.870594
$563$	−1.93485e6	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$563$	−1.93485e6	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$564$	0	0
$565$	1.12648e6	0.148457
$566$	1.47643e7	1.93719
$567$	0	0
$568$	−946648.	−0.123117
$569$	−8.73948e6	−1.13163	−0.565816	−	0.824532i	$-0.691439\pi$
$569$	−8.73948e6	−1.13163	−0.565816	+	0.824532i	$0.691439\pi$
$570$	0	0
$571$	−1.03137e7	−1.32380	−0.661902	−	0.749591i	$-0.730250\pi$
$571$	−1.03137e7	−1.32380	−0.661902	+	0.749591i	$0.730250\pi$
$572$	798207.	0.102006
$573$	0	0
$574$	0	0
$575$	2.58537e6	0.326102
$576$	0	0
$577$	−5.56614e6	−0.696009	−0.348004	−	0.937493i	$-0.613141\pi$
$577$	−5.56614e6	−0.696009	−0.348004	+	0.937493i	$0.613141\pi$
$578$	−6.87775e6	−0.856302
$579$	0	0
$580$	−649436.	−0.0801616
$581$	0	0
$582$	0	0
$583$	3.34907e6	0.408087
$584$	−993799.	−0.120578
$585$	0	0
$586$	1.90247e7	2.28862
$587$	1.57310e6	0.188435	0.0942173	−	0.995552i	$-0.469965\pi$
$587$	1.57310e6	0.188435	0.0942173	+	0.995552i	$0.469965\pi$
$588$	0	0
$589$	−8.34574e6	−0.991235
$590$	−4.30019e6	−0.508578
$591$	0	0
$592$	1.02326e7	1.20000
$593$	−1.62461e7	−1.89720	−0.948599	−	0.316481i	$-0.897499\pi$
$593$	−1.62461e7	−1.89720	−0.948599	+	0.316481i	$0.897499\pi$
$594$	0	0
$595$	0	0
$596$	4.48914e6	0.517664
$597$	0	0
$598$	1.52819e6	0.174753
$599$	−1.69388e7	−1.92893	−0.964465	−	0.264210i	$-0.914889\pi$
$599$	−1.69388e7	−1.92893	−0.964465	+	0.264210i	$0.914889\pi$
$600$	0	0
$601$	−700155.	−0.0790693	−0.0395347	−	0.999218i	$-0.512588\pi$
$601$	−700155.	−0.0790693	−0.0395347	+	0.999218i	$0.512588\pi$
$602$	0	0
$603$	0	0
$604$	−1.09909e7	−1.22586
$605$	−2.57803e6	−0.286351
$606$	0	0
$607$	3.13306e6	0.345141	0.172570	−	0.984997i	$-0.444793\pi$
$607$	3.13306e6	0.345141	0.172570	+	0.984997i	$0.444793\pi$
$608$	8.40063e6	0.921622
$609$	0	0
$610$	2.64118e6	0.287391
$611$	2.86708e6	0.310697
$612$	0	0
$613$	2.65140e6	0.284986	0.142493	−	0.989796i	$-0.454488\pi$
$613$	2.65140e6	0.284986	0.142493	+	0.989796i	$0.454488\pi$
$614$	6.95211e6	0.744210
$615$	0	0
$616$	0	0
$617$	−1.47657e7	−1.56149	−0.780747	−	0.624848i	$-0.785161\pi$
$617$	−1.47657e7	−1.56149	−0.780747	+	0.624848i	$0.785161\pi$
$618$	0	0
$619$	−8.33398e6	−0.874230	−0.437115	−	0.899406i	$-0.644000\pi$
$619$	−8.33398e6	−0.874230	−0.437115	+	0.899406i	$0.644000\pi$
$620$	4.88103e6	0.509955
$621$	0	0
$622$	−2.09708e7	−2.17340
$623$	0	0
$624$	0	0
$625$	7.01276e6	0.718107
$626$	−2.46513e7	−2.51422
$627$	0	0
$628$	6.18971e6	0.626284
$629$	8.11526e6	0.817854
$630$	0	0
$631$	−1.17243e7	−1.17223	−0.586116	−	0.810227i	$-0.699344\pi$
$631$	−1.17243e7	−1.17223	−0.586116	+	0.810227i	$0.699344\pi$
$632$	589657.	0.0587228
$633$	0	0
$634$	6.10620e6	0.603320
$635$	−2.70792e6	−0.266503
$636$	0	0
$637$	0	0
$638$	1.01729e6	0.0989449
$639$	0	0
$640$	−587825.	−0.0567281
$641$	−8.58605e6	−0.825369	−0.412685	−	0.910874i	$-0.635409\pi$
$641$	−8.58605e6	−0.825369	−0.412685	+	0.910874i	$0.635409\pi$
$642$	0	0
$643$	1.87633e7	1.78970	0.894852	−	0.446364i	$-0.147281\pi$
$643$	1.87633e7	1.78970	0.894852	+	0.446364i	$0.147281\pi$
$644$	0	0
$645$	0	0
$646$	6.23764e6	0.588084
$647$	1.59559e7	1.49851	0.749255	−	0.662282i	$-0.230412\pi$
$647$	1.59559e7	1.49851	0.749255	+	0.662282i	$0.230412\pi$
$648$	0	0
$649$	3.47159e6	0.323531
$650$	4.70571e6	0.436860
$651$	0	0
$652$	−6.67419e6	−0.614865
$653$	−1.36524e7	−1.25293	−0.626464	−	0.779451i	$-0.715498\pi$
$653$	−1.36524e7	−1.25293	−0.626464	+	0.779451i	$0.715498\pi$
$654$	0	0
$655$	3.29501e6	0.300092
$656$	−1.79113e7	−1.62506
$657$	0	0
$658$	0	0
$659$	−4.32613e6	−0.388049	−0.194024	−	0.980997i	$-0.562154\pi$
$659$	−4.32613e6	−0.388049	−0.194024	+	0.980997i	$0.562154\pi$
$660$	0	0
$661$	−1.97317e7	−1.75655	−0.878275	−	0.478157i	$-0.841305\pi$
$661$	−1.97317e7	−1.75655	−0.878275	+	0.478157i	$0.841305\pi$
$662$	−5.90647e6	−0.523821
$663$	0	0
$664$	1.43043e6	0.125906
$665$	0	0
$666$	0	0
$667$	1.00378e6	0.0873621
$668$	−9.98070e6	−0.865406
$669$	0	0
$670$	2.82169e6	0.242841
$671$	−2.13225e6	−0.182823
$672$	0	0
$673$	9.88089e6	0.840927	0.420464	−	0.907309i	$-0.361867\pi$
$673$	9.88089e6	0.840927	0.420464	+	0.907309i	$0.361867\pi$
$674$	5.28362e6	0.448004
$675$	0	0
$676$	−1.12022e7	−0.942837
$677$	−2.22205e6	−0.186329	−0.0931647	−	0.995651i	$-0.529698\pi$
$677$	−2.22205e6	−0.186329	−0.0931647	+	0.995651i	$0.529698\pi$
$678$	0	0
$679$	0	0
$680$	−217795.	−0.0180624
$681$	0	0
$682$	−7.64575e6	−0.629447
$683$	−5.54335e6	−0.454695	−0.227347	−	0.973814i	$-0.573005\pi$
$683$	−5.54335e6	−0.454695	−0.227347	+	0.973814i	$0.573005\pi$
$684$	0	0
$685$	6.00028e6	0.488591
$686$	0	0
$687$	0	0
$688$	4.43387e6	0.357118
$689$	6.01477e6	0.482693
$690$	0	0
$691$	1.43508e7	1.14335	0.571677	−	0.820478i	$-0.306293\pi$
$691$	1.43508e7	1.14335	0.571677	+	0.820478i	$0.306293\pi$
$692$	1.50702e7	1.19634
$693$	0	0
$694$	489943.	0.0386142
$695$	−4.71469e6	−0.370246
$696$	0	0
$697$	−1.42051e7	−1.10755
$698$	2.93525e7	2.28038
$699$	0	0
$700$	0	0
$701$	1.50894e7	1.15978	0.579891	−	0.814694i	$-0.303095\pi$
$701$	1.50894e7	1.15978	0.579891	+	0.814694i	$0.303095\pi$
$702$	0	0
$703$	1.08627e7	0.828990
$704$	4.20417e6	0.319705
$705$	0	0
$706$	−5.54105e6	−0.418389
$707$	0	0
$708$	0	0
$709$	1.04016e7	0.777115	0.388557	−	0.921425i	$-0.372974\pi$
$709$	1.04016e7	0.777115	0.388557	+	0.921425i	$0.372974\pi$
$710$	−8.11660e6	−0.604266
$711$	0	0
$712$	−708960.	−0.0524109
$713$	−7.54419e6	−0.555762
$714$	0	0
$715$	408586.	0.0298895
$716$	2.25101e7	1.64095
$717$	0	0
$718$	2.55660e7	1.85077
$719$	−1.11331e7	−0.803145	−0.401572	−	0.915827i	$-0.631536\pi$
$719$	−1.11331e7	−0.803145	−0.401572	+	0.915827i	$0.631536\pi$
$720$	0	0
$721$	0	0
$722$	−1.17713e7	−0.840395
$723$	0	0
$724$	4.75422e6	0.337080
$725$	3.09091e6	0.218394
$726$	0	0
$727$	−8.95327e6	−0.628269	−0.314135	−	0.949378i	$-0.601714\pi$
$727$	−8.95327e6	−0.628269	−0.314135	+	0.949378i	$0.601714\pi$
$728$	0	0
$729$	0	0
$730$	−8.52088e6	−0.591803
$731$	3.51641e6	0.243392
$732$	0	0
$733$	−2.60536e7	−1.79105	−0.895524	−	0.445014i	$-0.853199\pi$
$733$	−2.60536e7	−1.79105	−0.895524	+	0.445014i	$0.853199\pi$
$734$	−2.41423e7	−1.65401
$735$	0	0
$736$	7.59380e6	0.516731
$737$	−2.27798e6	−0.154483
$738$	0	0
$739$	−5.97111e6	−0.402202	−0.201101	−	0.979571i	$-0.564452\pi$
$739$	−5.97111e6	−0.402202	−0.201101	+	0.979571i	$0.564452\pi$
$740$	−6.35307e6	−0.426486
$741$	0	0
$742$	0	0
$743$	−6.80330e6	−0.452114	−0.226057	−	0.974114i	$-0.572583\pi$
$743$	−6.80330e6	−0.452114	−0.226057	+	0.974114i	$0.572583\pi$
$744$	0	0
$745$	2.29790e6	0.151684
$746$	2.44115e7	1.60601
$747$	0	0
$748$	2.94515e6	0.192466
$749$	0	0
$750$	0	0
$751$	−2.12648e7	−1.37582	−0.687911	−	0.725795i	$-0.741472\pi$
$751$	−2.12648e7	−1.37582	−0.687911	+	0.725795i	$0.741472\pi$
$752$	1.33388e7	0.860143
$753$	0	0
$754$	1.82701e6	0.117034
$755$	−5.62602e6	−0.359198
$756$	0	0
$757$	−9.49238e6	−0.602054	−0.301027	−	0.953616i	$-0.597329\pi$
$757$	−9.49238e6	−0.602054	−0.301027	+	0.953616i	$0.597329\pi$
$758$	−6.13469e6	−0.387810
$759$	0	0
$760$	−291530.	−0.0183084
$761$	1.71806e7	1.07542	0.537710	−	0.843130i	$-0.319290\pi$
$761$	1.71806e7	1.07542	0.537710	+	0.843130i	$0.319290\pi$
$762$	0	0
$763$	0	0
$764$	2.55887e7	1.58604
$765$	0	0
$766$	2.98085e7	1.83556
$767$	6.23480e6	0.382678
$768$	0	0
$769$	1.39968e7	0.853517	0.426758	−	0.904366i	$-0.359655\pi$
$769$	1.39968e7	0.853517	0.426758	+	0.904366i	$0.359655\pi$
$770$	0	0
$771$	0	0
$772$	1.58223e7	0.955493
$773$	2.54449e7	1.53162	0.765812	−	0.643065i	$-0.222337\pi$
$773$	2.54449e7	1.53162	0.765812	+	0.643065i	$0.222337\pi$
$774$	0	0
$775$	−2.32306e7	−1.38933
$776$	340422.	0.0202938
$777$	0	0
$778$	2.44560e7	1.44856
$779$	−1.90143e7	−1.12263
$780$	0	0
$781$	6.55261e6	0.384403
$782$	5.63856e6	0.329725
$783$	0	0
$784$	0	0
$785$	3.16839e6	0.183512
$786$	0	0
$787$	9.95632e6	0.573010	0.286505	−	0.958079i	$-0.407507\pi$
$787$	9.95632e6	0.573010	0.286505	+	0.958079i	$0.407507\pi$
$788$	1.76452e7	1.01231
$789$	0	0
$790$	5.05575e6	0.288216
$791$	0	0
$792$	0	0
$793$	−3.82941e6	−0.216246
$794$	2.33382e6	0.131376
$795$	0	0
$796$	−2.00885e7	−1.12374
$797$	−2.19815e7	−1.22578	−0.612888	−	0.790170i	$-0.709992\pi$
$797$	−2.19815e7	−1.22578	−0.612888	+	0.790170i	$0.709992\pi$
$798$	0	0
$799$	1.05787e7	0.586226
$800$	2.33834e7	1.29176
$801$	0	0
$802$	−7.01412e6	−0.385068
$803$	6.87899e6	0.376475
$804$	0	0
$805$	0	0
$806$	−1.37314e7	−0.744521
$807$	0	0
$808$	−1.75976e6	−0.0948253
$809$	1.01820e7	0.546969	0.273485	−	0.961876i	$-0.411824\pi$
$809$	1.01820e7	0.546969	0.273485	+	0.961876i	$0.411824\pi$
$810$	0	0
$811$	−1.06750e7	−0.569923	−0.284961	−	0.958539i	$-0.591981\pi$
$811$	−1.06750e7	−0.569923	−0.284961	+	0.958539i	$0.591981\pi$
$812$	0	0
$813$	0	0
$814$	9.95160e6	0.526419
$815$	−3.41638e6	−0.180166
$816$	0	0
$817$	4.70689e6	0.246706
$818$	−2.29910e7	−1.20136
$819$	0	0
$820$	1.11205e7	0.577552
$821$	2.22260e7	1.15081	0.575404	−	0.817869i	$-0.304845\pi$
$821$	2.22260e7	1.15081	0.575404	+	0.817869i	$0.304845\pi$
$822$	0	0
$823$	−2.42908e7	−1.25009	−0.625046	−	0.780588i	$-0.714920\pi$
$823$	−2.42908e7	−1.25009	−0.625046	+	0.780588i	$0.714920\pi$
$824$	1.44284e6	0.0740288
$825$	0	0
$826$	0	0
$827$	2.66722e7	1.35611	0.678055	−	0.735011i	$-0.262823\pi$
$827$	2.66722e7	1.35611	0.678055	+	0.735011i	$0.262823\pi$
$828$	0	0
$829$	6.78413e6	0.342853	0.171427	−	0.985197i	$-0.445162\pi$
$829$	6.78413e6	0.342853	0.171427	+	0.985197i	$0.445162\pi$
$830$	1.22646e7	0.617956
$831$	0	0
$832$	7.55049e6	0.378152
$833$	0	0
$834$	0	0
$835$	−5.10892e6	−0.253579
$836$	3.94223e6	0.195086
$837$	0	0
$838$	−5.75427e7	−2.83061
$839$	−1.88975e7	−0.926830	−0.463415	−	0.886141i	$-0.653376\pi$
$839$	−1.88975e7	−0.926830	−0.463415	+	0.886141i	$0.653376\pi$
$840$	0	0
$841$	−1.93111e7	−0.941493
$842$	−2.39958e7	−1.16642
$843$	0	0
$844$	7.04584e6	0.340468
$845$	−5.73418e6	−0.276267
$846$	0	0
$847$	0	0
$848$	2.79830e7	1.33630
$849$	0	0
$850$	1.73627e7	0.824270
$851$	9.81941e6	0.464795
$852$	0	0
$853$	1.69205e7	0.796235	0.398118	−	0.917334i	$-0.369664\pi$
$853$	1.69205e7	0.796235	0.398118	+	0.917334i	$0.369664\pi$
$854$	0	0
$855$	0	0
$856$	648950.	0.0302710
$857$	2.87564e7	1.33747	0.668733	−	0.743502i	$-0.266837\pi$
$857$	2.87564e7	1.33747	0.668733	+	0.743502i	$0.266837\pi$
$858$	0	0
$859$	−8.92517e6	−0.412699	−0.206350	−	0.978478i	$-0.566158\pi$
$859$	−8.92517e6	−0.412699	−0.206350	+	0.978478i	$0.566158\pi$
$860$	−2.75284e6	−0.126921
$861$	0	0
$862$	3.63508e7	1.66627
$863$	7.18205e6	0.328262	0.164131	−	0.986439i	$-0.447518\pi$
$863$	7.18205e6	0.328262	0.164131	+	0.986439i	$0.447518\pi$
$864$	0	0
$865$	7.71413e6	0.350547
$866$	4.15465e7	1.88252
$867$	0	0
$868$	0	0
$869$	−4.08155e6	−0.183348
$870$	0	0
$871$	−4.09114e6	−0.182725
$872$	−732066.	−0.0326031
$873$	0	0
$874$	7.54750e6	0.334214
$875$	0	0
$876$	0	0
$877$	5.05341e6	0.221863	0.110932	−	0.993828i	$-0.464617\pi$
$877$	5.05341e6	0.221863	0.110932	+	0.993828i	$0.464617\pi$
$878$	−4.79975e6	−0.210127
$879$	0	0
$880$	1.90090e6	0.0827469
$881$	−3.15002e6	−0.136733	−0.0683665	−	0.997660i	$-0.521779\pi$
$881$	−3.15002e6	−0.136733	−0.0683665	+	0.997660i	$0.521779\pi$
$882$	0	0
$883$	3.33141e7	1.43789	0.718946	−	0.695066i	$-0.244625\pi$
$883$	3.33141e7	1.43789	0.718946	+	0.695066i	$0.244625\pi$
$884$	5.28934e6	0.227652
$885$	0	0
$886$	1.30638e6	0.0559094
$887$	−2.04549e6	−0.0872946	−0.0436473	−	0.999047i	$-0.513898\pi$
$887$	−2.04549e6	−0.0872946	−0.0436473	+	0.999047i	$0.513898\pi$
$888$	0	0
$889$	0	0
$890$	−6.07866e6	−0.257237
$891$	0	0
$892$	4.19770e7	1.76644
$893$	1.41601e7	0.594208
$894$	0	0
$895$	1.15225e7	0.480827
$896$	0	0
$897$	0	0
$898$	6.48677e7	2.68434
$899$	−9.01936e6	−0.372200
$900$	0	0
$901$	2.21927e7	0.910749
$902$	−1.74195e7	−0.712883
$903$	0	0
$904$	1.06761e6	0.0434502
$905$	2.43359e6	0.0987702
$906$	0	0
$907$	2.69774e7	1.08888	0.544442	−	0.838798i	$-0.316741\pi$
$907$	2.69774e7	1.08888	0.544442	+	0.838798i	$0.316741\pi$
$908$	−3.81287e7	−1.53475
$909$	0	0
$910$	0	0
$911$	−5.42821e6	−0.216701	−0.108350	−	0.994113i	$-0.534557\pi$
$911$	−5.42821e6	−0.216701	−0.108350	+	0.994113i	$0.534557\pi$
$912$	0	0
$913$	−9.90132e6	−0.393112
$914$	−9.95129e6	−0.394016
$915$	0	0
$916$	1.40303e7	0.552495
$917$	0	0
$918$	0	0
$919$	9.41696e6	0.367809	0.183904	−	0.982944i	$-0.441126\pi$
$919$	9.41696e6	0.367809	0.183904	+	0.982944i	$0.441126\pi$
$920$	−263531.	−0.0102651
$921$	0	0
$922$	−1.62069e7	−0.627876
$923$	1.17682e7	0.454679
$924$	0	0
$925$	3.02367e7	1.16193
$926$	−1.01692e6	−0.0389724
$927$	0	0
$928$	9.07868e6	0.346061
$929$	−8.43871e6	−0.320802	−0.160401	−	0.987052i	$-0.551279\pi$
$929$	−8.43871e6	−0.320802	−0.160401	+	0.987052i	$0.551279\pi$
$930$	0	0
$931$	0	0
$932$	2.84658e7	1.07345
$933$	0	0
$934$	−5.73997e7	−2.15299
$935$	1.50756e6	0.0563957
$936$	0	0
$937$	−683755.	−0.0254420	−0.0127210	−	0.999919i	$-0.504049\pi$
$937$	−683755.	−0.0254420	−0.0127210	+	0.999919i	$0.504049\pi$
$938$	0	0
$939$	0	0
$940$	−8.28158e6	−0.305699
$941$	−4.97436e7	−1.83132	−0.915658	−	0.401958i	$-0.868330\pi$
$941$	−4.97436e7	−1.83132	−0.915658	+	0.401958i	$0.868330\pi$
$942$	0	0
$943$	−1.71881e7	−0.629430
$944$	2.90066e7	1.05942
$945$	0	0
$946$	4.31211e6	0.156661
$947$	−2.21967e7	−0.804292	−0.402146	−	0.915575i	$-0.631736\pi$
$947$	−2.21967e7	−0.804292	−0.402146	+	0.915575i	$0.631736\pi$
$948$	0	0
$949$	1.23543e7	0.445301
$950$	2.32408e7	0.835493
$951$	0	0
$952$	0	0
$953$	−1.34934e7	−0.481270	−0.240635	−	0.970616i	$-0.577356\pi$
$953$	−1.34934e7	−0.481270	−0.240635	+	0.970616i	$0.577356\pi$
$954$	0	0
$955$	1.30983e7	0.464737
$956$	−1.08797e7	−0.385012
$957$	0	0
$958$	3.77704e7	1.32965
$959$	0	0
$960$	0	0
$961$	3.91585e7	1.36778
$962$	1.78726e7	0.622658
$963$	0	0
$964$	−3.28725e7	−1.13931
$965$	8.09913e6	0.279976
$966$	0	0
$967$	−1.59641e7	−0.549008	−0.274504	−	0.961586i	$-0.588514\pi$
$967$	−1.59641e7	−0.549008	−0.274504	+	0.961586i	$0.588514\pi$
$968$	−2.44331e6	−0.0838088
$969$	0	0
$970$	2.91879e6	0.0996033
$971$	2.03921e7	0.694087	0.347043	−	0.937849i	$-0.387186\pi$
$971$	2.03921e7	0.694087	0.347043	+	0.937849i	$0.387186\pi$
$972$	0	0
$973$	0	0
$974$	−4.53354e7	−1.53123
$975$	0	0
$976$	−1.78159e7	−0.598663
$977$	−2.62595e7	−0.880136	−0.440068	−	0.897964i	$-0.645046\pi$
$977$	−2.62595e7	−0.880136	−0.440068	+	0.897964i	$0.645046\pi$
$978$	0	0
$979$	4.90736e6	0.163641
$980$	0	0
$981$	0	0
$982$	−4.72489e7	−1.56355
$983$	−8.01076e6	−0.264417	−0.132209	−	0.991222i	$-0.542207\pi$
$983$	−8.01076e6	−0.264417	−0.132209	+	0.991222i	$0.542207\pi$
$984$	0	0
$985$	9.03223e6	0.296623
$986$	6.74111e6	0.220820
$987$	0	0
$988$	7.08006e6	0.230751
$989$	4.25483e6	0.138322
$990$	0	0
$991$	−1.60083e7	−0.517798	−0.258899	−	0.965904i	$-0.583360\pi$
$991$	−1.60083e7	−0.517798	−0.258899	+	0.965904i	$0.583360\pi$
$992$	−6.82334e7	−2.20150
$993$	0	0
$994$	0	0
$995$	−1.02829e7	−0.329274
$996$	0	0
$997$	−2.97034e7	−0.946386	−0.473193	−	0.880959i	$-0.656899\pi$
$997$	−2.97034e7	−0.946386	−0.473193	+	0.880959i	$0.656899\pi$
$998$	−5.83281e7	−1.85375
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.be.1.8	yes	8
3.2	odd	2	inner	441.6.a.be.1.1	✓	8
7.6	odd	2	inner	441.6.a.be.1.7	yes	8
21.20	even	2	inner	441.6.a.be.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.6.a.be.1.1	✓	8	3.2	odd	2	inner
441.6.a.be.1.2	yes	8	21.20	even	2	inner
441.6.a.be.1.7	yes	8	7.6	odd	2	inner
441.6.a.be.1.8	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} +16.5098 q^{8} +O(q^{10})\)	\(q+8.12599 q^{2} +34.0317 q^{4} +17.4201 q^{5} +16.5098 q^{8} +141.556 q^{10} -114.280 q^{11} -205.240 q^{13} -954.857 q^{16} -757.277 q^{17} -1013.65 q^{19} +592.838 q^{20} -928.635 q^{22} -916.299 q^{23} -2821.54 q^{25} -1667.78 q^{26} -1095.47 q^{29} +8233.32 q^{31} -8287.47 q^{32} -6153.62 q^{34} -10716.4 q^{37} -8236.94 q^{38} +287.604 q^{40} +18758.1 q^{41} -4643.49 q^{43} -3889.13 q^{44} -7445.84 q^{46} -13969.4 q^{47} -22927.8 q^{50} -6984.69 q^{52} -29306.0 q^{53} -1990.77 q^{55} -8901.78 q^{58} -30378.0 q^{59} +18658.2 q^{61} +66903.9 q^{62} -36788.5 q^{64} -3575.32 q^{65} +19933.4 q^{67} -25771.4 q^{68} -57338.5 q^{71} -60194.4 q^{73} -87081.2 q^{74} -34496.4 q^{76} +35715.5 q^{79} -16633.7 q^{80} +152428. q^{82} +86641.2 q^{83} -13191.9 q^{85} -37733.0 q^{86} -1886.73 q^{88} -42941.7 q^{89} -31183.2 q^{92} -113515. q^{94} -17658.0 q^{95} +20619.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{4}+O(q^{10}) \) 8 * q + 20 * q^4	\( 8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100}) \) 8 * q + 20 * q^4 - 828 * q^16 - 2384 * q^22 - 2392 * q^25 - 19136 * q^37 - 41184 * q^43 - 13152 * q^46 - 88872 * q^58 - 210812 * q^64 - 42336 * q^67 - 251072 * q^79 - 567664 * q^85 - 88752 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more