LMFDB - Embedded newform 400.6.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.47217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 38x - 24 $ x^3 - x^2 - 38*x - 24
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.649919$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.6278 q^{3} -107.252 q^{7} +269.020 q^{9} +O(q^{10})$	$q+22.6278 q^{3} -107.252 q^{7} +269.020 q^{9} +290.903 q^{11} -519.583 q^{13} -2031.17 q^{17} -1709.34 q^{19} -2426.89 q^{21} +4355.36 q^{23} +588.766 q^{27} +4791.56 q^{29} -7597.89 q^{31} +6582.51 q^{33} -5260.87 q^{37} -11757.0 q^{39} -10109.1 q^{41} -23414.4 q^{43} -3319.92 q^{47} -5303.91 q^{49} -45960.9 q^{51} +28948.8 q^{53} -38678.7 q^{57} -14553.4 q^{59} +24379.4 q^{61} -28853.0 q^{63} -5931.14 q^{67} +98552.5 q^{69} +14407.4 q^{71} -28153.4 q^{73} -31200.1 q^{77} -21655.0 q^{79} -52049.2 q^{81} -37344.7 q^{83} +108423. q^{87} +90586.5 q^{89} +55726.6 q^{91} -171924. q^{93} +73882.4 q^{97} +78258.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) $ 3 * q + q^3 - 70 * q^7 + 154 * q^9	$ 3 q + q^{3} - 70 q^{7} + 154 q^{9} + 19 q^{11} + 196 q^{13} - 1223 q^{17} - 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} - 7442 q^{31} + 6969 q^{33} - 14766 q^{37} - 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} - 43165 q^{51} + 12878 q^{53} - 77791 q^{57} - 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} - 86676 q^{71} - 8305 q^{73} - 106822 q^{77} - 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} - 149224 q^{91} - 196470 q^{93} - 76342 q^{97} + 149130 q^{99}+O(q^{100}) $ 3 * q + q^3 - 70 * q^7 + 154 * q^9 + 19 * q^11 + 196 * q^13 - 1223 * q^17 - 1221 * q^19 + 958 * q^21 - 2490 * q^23 + 4123 * q^27 + 11912 * q^29 - 7442 * q^31 + 6969 * q^33 - 14766 * q^37 - 27396 * q^39 + 3223 * q^41 - 41060 * q^43 + 29188 * q^47 + 66423 * q^49 - 43165 * q^51 + 12878 * q^53 - 77791 * q^57 - 64912 * q^59 + 22478 * q^61 - 112916 * q^63 + 26499 * q^67 + 178642 * q^69 - 86676 * q^71 - 8305 * q^73 - 106822 * q^77 - 21982 * q^79 - 71453 * q^81 - 213353 * q^83 + 14952 * q^87 + 182381 * q^89 - 149224 * q^91 - 196470 * q^93 - 76342 * q^97 + 149130 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	22.6278	1.45158	0.725789	−	0.687918i	$-0.241475\pi$
$3$	22.6278	1.45158	0.725789	+	0.687918i	$0.241475\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−107.252	−0.827298	−0.413649	−	0.910436i	$-0.635746\pi$
$7$	−107.252	−0.827298	−0.413649	+	0.910436i	$0.635746\pi$
$8$	0	0
$9$	269.020	1.10708
$10$	0	0
$11$	290.903	0.724881	0.362440	−	0.932007i	$-0.381944\pi$
$11$	290.903	0.724881	0.362440	+	0.932007i	$0.381944\pi$
$12$	0	0
$13$	−519.583	−0.852701	−0.426350	−	0.904558i	$-0.640201\pi$
$13$	−519.583	−0.852701	−0.426350	+	0.904558i	$0.640201\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2031.17	−1.70460	−0.852301	−	0.523051i	$-0.824794\pi$
$17$	−2031.17	−1.70460	−0.852301	+	0.523051i	$0.824794\pi$
$18$	0	0
$19$	−1709.34	−1.08629	−0.543144	−	0.839640i	$-0.682766\pi$
$19$	−1709.34	−1.08629	−0.543144	+	0.839640i	$0.682766\pi$
$20$	0	0
$21$	−2426.89	−1.20089
$22$	0	0
$23$	4355.36	1.71674	0.858370	−	0.513030i	$-0.171477\pi$
$23$	4355.36	1.71674	0.858370	+	0.513030i	$0.171477\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	588.766	0.155430
$28$	0	0
$29$	4791.56	1.05799	0.528996	−	0.848624i	$-0.322569\pi$
$29$	4791.56	1.05799	0.528996	+	0.848624i	$0.322569\pi$
$30$	0	0
$31$	−7597.89	−1.42000	−0.710000	−	0.704201i	$-0.751305\pi$
$31$	−7597.89	−1.42000	−0.710000	+	0.704201i	$0.751305\pi$
$32$	0	0
$33$	6582.51	1.05222
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5260.87	−0.631761	−0.315881	−	0.948799i	$-0.602300\pi$
$37$	−5260.87	−0.631761	−0.315881	+	0.948799i	$0.602300\pi$
$38$	0	0
$39$	−11757.0	−1.23776
$40$	0	0
$41$	−10109.1	−0.939185	−0.469593	−	0.882883i	$-0.655599\pi$
$41$	−10109.1	−0.939185	−0.469593	+	0.882883i	$0.655599\pi$
$42$	0	0
$43$	−23414.4	−1.93113	−0.965566	−	0.260158i	$-0.916225\pi$
$43$	−23414.4	−1.93113	−0.965566	+	0.260158i	$0.916225\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3319.92	−0.219222	−0.109611	−	0.993975i	$-0.534960\pi$
$47$	−3319.92	−0.219222	−0.109611	+	0.993975i	$0.534960\pi$
$48$	0	0
$49$	−5303.91	−0.315578
$50$	0	0
$51$	−45960.9	−2.47436
$52$	0	0
$53$	28948.8	1.41560	0.707800	−	0.706413i	$-0.249688\pi$
$53$	28948.8	1.41560	0.707800	+	0.706413i	$0.249688\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−38678.7	−1.57683
$58$	0	0
$59$	−14553.4	−0.544295	−0.272147	−	0.962256i	$-0.587734\pi$
$59$	−14553.4	−0.544295	−0.272147	+	0.962256i	$0.587734\pi$
$60$	0	0
$61$	24379.4	0.838876	0.419438	−	0.907784i	$-0.362227\pi$
$61$	24379.4	0.838876	0.419438	+	0.907784i	$0.362227\pi$
$62$	0	0
$63$	−28853.0	−0.915882
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5931.14	−0.161418	−0.0807089	−	0.996738i	$-0.525718\pi$
$67$	−5931.14	−0.161418	−0.0807089	+	0.996738i	$0.525718\pi$
$68$	0	0
$69$	98552.5	2.49198
$70$	0	0
$71$	14407.4	0.339188	0.169594	−	0.985514i	$-0.445754\pi$
$71$	14407.4	0.339188	0.169594	+	0.985514i	$0.445754\pi$
$72$	0	0
$73$	−28153.4	−0.618334	−0.309167	−	0.951008i	$-0.600050\pi$
$73$	−28153.4	−0.618334	−0.309167	+	0.951008i	$0.600050\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−31200.1	−0.599693
$78$	0	0
$79$	−21655.0	−0.390383	−0.195191	−	0.980765i	$-0.562533\pi$
$79$	−21655.0	−0.390383	−0.195191	+	0.980765i	$0.562533\pi$
$80$	0	0
$81$	−52049.2	−0.881458
$82$	0	0
$83$	−37344.7	−0.595022	−0.297511	−	0.954718i	$-0.596157\pi$
$83$	−37344.7	−0.595022	−0.297511	+	0.954718i	$0.596157\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	108423.	1.53576
$88$	0	0
$89$	90586.5	1.21224	0.606120	−	0.795374i	$-0.292725\pi$
$89$	90586.5	1.21224	0.606120	+	0.795374i	$0.292725\pi$
$90$	0	0
$91$	55726.6	0.705438
$92$	0	0
$93$	−171924.	−2.06124
$94$	0	0
$95$	0	0
$96$	0	0
$97$	73882.4	0.797281	0.398641	−	0.917107i	$-0.369482\pi$
$97$	73882.4	0.797281	0.398641	+	0.917107i	$0.369482\pi$
$98$	0	0
$99$	78258.6	0.802498
$100$	0	0
$101$	62790.5	0.612478	0.306239	−	0.951955i	$-0.400929\pi$
$101$	62790.5	0.612478	0.306239	+	0.951955i	$0.400929\pi$
$102$	0	0
$103$	−135312.	−1.25673	−0.628365	−	0.777919i	$-0.716276\pi$
$103$	−135312.	−1.25673	−0.628365	+	0.777919i	$0.716276\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−27359.0	−0.231015	−0.115507	−	0.993307i	$-0.536849\pi$
$107$	−27359.0	−0.231015	−0.115507	+	0.993307i	$0.536849\pi$
$108$	0	0
$109$	119952.	0.967036	0.483518	−	0.875334i	$-0.339359\pi$
$109$	119952.	0.967036	0.483518	+	0.875334i	$0.339359\pi$
$110$	0	0
$111$	−119042.	−0.917050
$112$	0	0
$113$	−230568.	−1.69864	−0.849322	−	0.527875i	$-0.822989\pi$
$113$	−230568.	−1.69864	−0.849322	+	0.527875i	$0.822989\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−139778.	−0.944005
$118$	0	0
$119$	217848.	1.41021
$120$	0	0
$121$	−76426.4	−0.474548
$122$	0	0
$123$	−228746.	−1.36330
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−321572.	−1.76917	−0.884584	−	0.466381i	$-0.845558\pi$
$127$	−321572.	−1.76917	−0.884584	+	0.466381i	$0.845558\pi$
$128$	0	0
$129$	−529818.	−2.80319
$130$	0	0
$131$	103701.	0.527966	0.263983	−	0.964527i	$-0.414964\pi$
$131$	103701.	0.527966	0.263983	+	0.964527i	$0.414964\pi$
$132$	0	0
$133$	183331.	0.898684
$134$	0	0
$135$	0	0
$136$	0	0
$137$	400259.	1.82197	0.910983	−	0.412444i	$-0.135325\pi$
$137$	400259.	1.82197	0.910983	+	0.412444i	$0.135325\pi$
$138$	0	0
$139$	180530.	0.792523	0.396262	−	0.918138i	$-0.370307\pi$
$139$	180530.	0.792523	0.396262	+	0.918138i	$0.370307\pi$
$140$	0	0
$141$	−75122.7	−0.318217
$142$	0	0
$143$	−151148.	−0.618106
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−120016.	−0.458085
$148$	0	0
$149$	308066.	1.13678	0.568392	−	0.822758i	$-0.307566\pi$
$149$	308066.	1.13678	0.568392	+	0.822758i	$0.307566\pi$
$150$	0	0
$151$	−293359.	−1.04703	−0.523513	−	0.852018i	$-0.675379\pi$
$151$	−293359.	−1.04703	−0.523513	+	0.852018i	$0.675379\pi$
$152$	0	0
$153$	−546423.	−1.88713
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−276321.	−0.894673	−0.447336	−	0.894366i	$-0.647627\pi$
$157$	−276321.	−0.894673	−0.447336	+	0.894366i	$0.647627\pi$
$158$	0	0
$159$	655049.	2.05485
$160$	0	0
$161$	−467123.	−1.42026
$162$	0	0
$163$	557207.	1.64266	0.821330	−	0.570454i	$-0.193233\pi$
$163$	557207.	1.64266	0.821330	+	0.570454i	$0.193233\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−217034.	−0.602193	−0.301097	−	0.953594i	$-0.597353\pi$
$167$	−217034.	−0.602193	−0.301097	+	0.953594i	$0.597353\pi$
$168$	0	0
$169$	−101326.	−0.272902
$170$	0	0
$171$	−459846.	−1.20260
$172$	0	0
$173$	152453.	0.387276	0.193638	−	0.981073i	$-0.437971\pi$
$173$	152453.	0.387276	0.193638	+	0.981073i	$0.437971\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−329312.	−0.790086
$178$	0	0
$179$	−183447.	−0.427935	−0.213968	−	0.976841i	$-0.568639\pi$
$179$	−183447.	−0.427935	−0.213968	+	0.976841i	$0.568639\pi$
$180$	0	0
$181$	−808678.	−1.83476	−0.917380	−	0.398013i	$-0.869700\pi$
$181$	−808678.	−1.83476	−0.917380	+	0.398013i	$0.869700\pi$
$182$	0	0
$183$	551653.	1.21769
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−590872.	−1.23563
$188$	0	0
$189$	−63146.6	−0.128587
$190$	0	0
$191$	429516.	0.851913	0.425957	−	0.904744i	$-0.359938\pi$
$191$	429516.	0.851913	0.425957	+	0.904744i	$0.359938\pi$
$192$	0	0
$193$	−610774.	−1.18029	−0.590143	−	0.807299i	$-0.700929\pi$
$193$	−610774.	−1.18029	−0.590143	+	0.807299i	$0.700929\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−260379.	−0.478014	−0.239007	−	0.971018i	$-0.576822\pi$
$197$	−260379.	−0.478014	−0.239007	+	0.971018i	$0.576822\pi$
$198$	0	0
$199$	−763493.	−1.36670	−0.683349	−	0.730092i	$-0.739477\pi$
$199$	−763493.	−1.36670	−0.683349	+	0.730092i	$0.739477\pi$
$200$	0	0
$201$	−134209.	−0.234310
$202$	0	0
$203$	−513907.	−0.875275
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.17168e6	1.90056
$208$	0	0
$209$	−497253.	−0.787429
$210$	0	0
$211$	189676.	0.293297	0.146648	−	0.989189i	$-0.453151\pi$
$211$	189676.	0.293297	0.146648	+	0.989189i	$0.453151\pi$
$212$	0	0
$213$	326009.	0.492357
$214$	0	0
$215$	0	0
$216$	0	0
$217$	814892.	1.17476
$218$	0	0
$219$	−637050.	−0.897560
$220$	0	0
$221$	1.05536e6	1.45352
$222$	0	0
$223$	50555.3	0.0680776	0.0340388	−	0.999421i	$-0.489163\pi$
$223$	50555.3	0.0680776	0.0340388	+	0.999421i	$0.489163\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−641272.	−0.825996	−0.412998	−	0.910732i	$-0.635518\pi$
$227$	−641272.	−0.825996	−0.412998	+	0.910732i	$0.635518\pi$
$228$	0	0
$229$	1.20712e6	1.52111	0.760554	−	0.649275i	$-0.224927\pi$
$229$	1.20712e6	1.52111	0.760554	+	0.649275i	$0.224927\pi$
$230$	0	0
$231$	−705990.	−0.870500
$232$	0	0
$233$	1.06486e6	1.28500	0.642501	−	0.766285i	$-0.277897\pi$
$233$	1.06486e6	1.28500	0.642501	+	0.766285i	$0.277897\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−490006.	−0.566671
$238$	0	0
$239$	358903.	0.406426	0.203213	−	0.979135i	$-0.434862\pi$
$239$	358903.	0.406426	0.203213	+	0.979135i	$0.434862\pi$
$240$	0	0
$241$	1.58522e6	1.75811	0.879056	−	0.476719i	$-0.158174\pi$
$241$	1.58522e6	1.75811	0.879056	+	0.476719i	$0.158174\pi$
$242$	0	0
$243$	−1.32083e6	−1.43493
$244$	0	0
$245$	0	0
$246$	0	0
$247$	888145.	0.926278
$248$	0	0
$249$	−845029.	−0.863721
$250$	0	0
$251$	1.09079e6	1.09284	0.546418	−	0.837513i	$-0.315991\pi$
$251$	1.09079e6	1.09284	0.546418	+	0.837513i	$0.315991\pi$
$252$	0	0
$253$	1.26699e6	1.24443
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−140503.	−0.132694	−0.0663472	−	0.997797i	$-0.521134\pi$
$257$	−140503.	−0.132694	−0.0663472	+	0.997797i	$0.521134\pi$
$258$	0	0
$259$	564241.	0.522655
$260$	0	0
$261$	1.28902e6	1.17128
$262$	0	0
$263$	−1688.29	−0.00150508	−0.000752538	−	1.00000i	$-0.500240\pi$
$263$	−1688.29	−0.00150508	−0.000752538		1.00000i	$0.500240\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.04978e6	1.75966
$268$	0	0
$269$	1.03111e6	0.868810	0.434405	−	0.900718i	$-0.356959\pi$
$269$	1.03111e6	0.868810	0.434405	+	0.900718i	$0.356959\pi$
$270$	0	0
$271$	−152323.	−0.125992	−0.0629959	−	0.998014i	$-0.520066\pi$
$271$	−152323.	−0.125992	−0.0629959	+	0.998014i	$0.520066\pi$
$272$	0	0
$273$	1.26097e6	1.02400
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.45824e6	1.14190	0.570952	−	0.820983i	$-0.306574\pi$
$277$	1.45824e6	1.14190	0.570952	+	0.820983i	$0.306574\pi$
$278$	0	0
$279$	−2.04398e6	−1.57205
$280$	0	0
$281$	−509886.	−0.385218	−0.192609	−	0.981276i	$-0.561695\pi$
$281$	−509886.	−0.385218	−0.192609	+	0.981276i	$0.561695\pi$
$282$	0	0
$283$	139645.	0.103648	0.0518239	−	0.998656i	$-0.483497\pi$
$283$	139645.	0.103648	0.0518239	+	0.998656i	$0.483497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.08422e6	0.776987
$288$	0	0
$289$	2.70578e6	1.90567
$290$	0	0
$291$	1.67180e6	1.15732
$292$	0	0
$293$	−316162.	−0.215150	−0.107575	−	0.994197i	$-0.534309\pi$
$293$	−316162.	−0.215150	−0.107575	+	0.994197i	$0.534309\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	171274.	0.112668
$298$	0	0
$299$	−2.26297e6	−1.46387
$300$	0	0
$301$	2.51125e6	1.59762
$302$	0	0
$303$	1.42081e6	0.889059
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.12429e6	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$307$	−2.12429e6	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$308$	0	0
$309$	−3.06181e6	−1.82424
$310$	0	0
$311$	1.00247e6	0.587720	0.293860	−	0.955848i	$-0.405060\pi$
$311$	1.00247e6	0.587720	0.293860	+	0.955848i	$0.405060\pi$
$312$	0	0
$313$	268434.	0.154874	0.0774368	−	0.996997i	$-0.475326\pi$
$313$	268434.	0.154874	0.0774368	+	0.996997i	$0.475326\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.47341e6	−0.823521	−0.411761	−	0.911292i	$-0.635086\pi$
$317$	−1.47341e6	−0.823521	−0.411761	+	0.911292i	$0.635086\pi$
$318$	0	0
$319$	1.39388e6	0.766918
$320$	0	0
$321$	−619074.	−0.335336
$322$	0	0
$323$	3.47196e6	1.85169
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.71426e6	1.40373
$328$	0	0
$329$	356070.	0.181362
$330$	0	0
$331$	1.40915e6	0.706948	0.353474	−	0.935444i	$-0.385000\pi$
$331$	1.40915e6	0.706948	0.353474	+	0.935444i	$0.385000\pi$
$332$	0	0
$333$	−1.41528e6	−0.699408
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.00961e6	−0.963909	−0.481955	−	0.876196i	$-0.660073\pi$
$337$	−2.00961e6	−0.963909	−0.481955	+	0.876196i	$0.660073\pi$
$338$	0	0
$339$	−5.21725e6	−2.46571
$340$	0	0
$341$	−2.21025e6	−1.02933
$342$	0	0
$343$	2.37145e6	1.08838
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.51970e6	−0.677538	−0.338769	−	0.940870i	$-0.610011\pi$
$347$	−1.51970e6	−0.677538	−0.338769	+	0.940870i	$0.610011\pi$
$348$	0	0
$349$	−3.65792e6	−1.60757	−0.803786	−	0.594919i	$-0.797184\pi$
$349$	−3.65792e6	−1.60757	−0.803786	+	0.594919i	$0.797184\pi$
$350$	0	0
$351$	−305913.	−0.132535
$352$	0	0
$353$	3.13110e6	1.33739	0.668697	−	0.743535i	$-0.266852\pi$
$353$	3.13110e6	1.33739	0.668697	+	0.743535i	$0.266852\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.92942e6	2.04704
$358$	0	0
$359$	4.69349e6	1.92203	0.961013	−	0.276503i	$-0.0891754\pi$
$359$	4.69349e6	1.92203	0.961013	+	0.276503i	$0.0891754\pi$
$360$	0	0
$361$	445748.	0.180020
$362$	0	0
$363$	−1.72936e6	−0.688843
$364$	0	0
$365$	0	0
$366$	0	0
$367$	66533.6	0.0257855	0.0128928	−	0.999917i	$-0.495896\pi$
$367$	66533.6	0.0257855	0.0128928	+	0.999917i	$0.495896\pi$
$368$	0	0
$369$	−2.71954e6	−1.03975
$370$	0	0
$371$	−3.10483e6	−1.17112
$372$	0	0
$373$	−436576.	−0.162475	−0.0812376	−	0.996695i	$-0.525887\pi$
$373$	−436576.	−0.162475	−0.0812376	+	0.996695i	$0.525887\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.48962e6	−0.902150
$378$	0	0
$379$	640711.	0.229120	0.114560	−	0.993416i	$-0.463454\pi$
$379$	640711.	0.229120	0.114560	+	0.993416i	$0.463454\pi$
$380$	0	0
$381$	−7.27648e6	−2.56808
$382$	0	0
$383$	476798.	0.166088	0.0830438	−	0.996546i	$-0.473536\pi$
$383$	476798.	0.166088	0.0830438	+	0.996546i	$0.473536\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.29893e6	−2.13791
$388$	0	0
$389$	−2.43449e6	−0.815707	−0.407854	−	0.913047i	$-0.633723\pi$
$389$	−2.43449e6	−0.815707	−0.407854	+	0.913047i	$0.633723\pi$
$390$	0	0
$391$	−8.84647e6	−2.92636
$392$	0	0
$393$	2.34654e6	0.766384
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.69898e6	1.17789	0.588946	−	0.808172i	$-0.299543\pi$
$397$	3.69898e6	1.17789	0.588946	+	0.808172i	$0.299543\pi$
$398$	0	0
$399$	4.14839e6	1.30451
$400$	0	0
$401$	3.17381e6	0.985643	0.492822	−	0.870130i	$-0.335966\pi$
$401$	3.17381e6	0.985643	0.492822	+	0.870130i	$0.335966\pi$
$402$	0	0
$403$	3.94773e6	1.21084
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.53040e6	−0.457952
$408$	0	0
$409$	1.92926e6	0.570272	0.285136	−	0.958487i	$-0.407961\pi$
$409$	1.92926e6	0.570272	0.285136	+	0.958487i	$0.407961\pi$
$410$	0	0
$411$	9.05701e6	2.64472
$412$	0	0
$413$	1.56089e6	0.450294
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.08500e6	1.15041
$418$	0	0
$419$	5.28590e6	1.47090	0.735451	−	0.677578i	$-0.236970\pi$
$419$	5.28590e6	1.47090	0.735451	+	0.677578i	$0.236970\pi$
$420$	0	0
$421$	4.63926e6	1.27568	0.637842	−	0.770167i	$-0.279827\pi$
$421$	4.63926e6	1.27568	0.637842	+	0.770167i	$0.279827\pi$
$422$	0	0
$423$	−893124.	−0.242695
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.61475e6	−0.694001
$428$	0	0
$429$	−3.42016e6	−0.897229
$430$	0	0
$431$	5.05569e6	1.31095	0.655477	−	0.755215i	$-0.272468\pi$
$431$	5.05569e6	1.31095	0.655477	+	0.755215i	$0.272468\pi$
$432$	0	0
$433$	−3.02308e6	−0.774873	−0.387437	−	0.921896i	$-0.626639\pi$
$433$	−3.02308e6	−0.774873	−0.387437	+	0.921896i	$0.626639\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.44480e6	−1.86487
$438$	0	0
$439$	−493181.	−0.122136	−0.0610682	−	0.998134i	$-0.519451\pi$
$439$	−493181.	−0.122136	−0.0610682	+	0.998134i	$0.519451\pi$
$440$	0	0
$441$	−1.42686e6	−0.349368
$442$	0	0
$443$	−5.86117e6	−1.41898	−0.709488	−	0.704717i	$-0.751074\pi$
$443$	−5.86117e6	−1.41898	−0.709488	+	0.704717i	$0.751074\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.97086e6	1.65013
$448$	0	0
$449$	983425.	0.230211	0.115105	−	0.993353i	$-0.463279\pi$
$449$	983425.	0.230211	0.115105	+	0.993353i	$0.463279\pi$
$450$	0	0
$451$	−2.94076e6	−0.680798
$452$	0	0
$453$	−6.63808e6	−1.51984
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.78555e6	−0.847887	−0.423944	−	0.905689i	$-0.639355\pi$
$457$	−3.78555e6	−0.847887	−0.423944	+	0.905689i	$0.639355\pi$
$458$	0	0
$459$	−1.19588e6	−0.264946
$460$	0	0
$461$	−5.84423e6	−1.28078	−0.640391	−	0.768049i	$-0.721228\pi$
$461$	−5.84423e6	−1.28078	−0.640391	+	0.768049i	$0.721228\pi$
$462$	0	0
$463$	4.65701e6	1.00961	0.504807	−	0.863232i	$-0.331564\pi$
$463$	4.65701e6	1.00961	0.504807	+	0.863232i	$0.331564\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.35873e6	−0.500479	−0.250239	−	0.968184i	$-0.580509\pi$
$467$	−2.35873e6	−0.500479	−0.250239	+	0.968184i	$0.580509\pi$
$468$	0	0
$469$	636130.	0.133541
$470$	0	0
$471$	−6.25254e6	−1.29869
$472$	0	0
$473$	−6.81132e6	−1.39984
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.78779e6	1.56718
$478$	0	0
$479$	−1.87288e6	−0.372968	−0.186484	−	0.982458i	$-0.559709\pi$
$479$	−1.87288e6	−0.372968	−0.186484	+	0.982458i	$0.559709\pi$
$480$	0	0
$481$	2.73346e6	0.538703
$482$	0	0
$483$	−1.05700e7	−2.06161
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.94339e6	0.371311	0.185655	−	0.982615i	$-0.440559\pi$
$487$	1.94339e6	0.371311	0.185655	+	0.982615i	$0.440559\pi$
$488$	0	0
$489$	1.26084e7	2.38445
$490$	0	0
$491$	8.02560e6	1.50236	0.751179	−	0.660098i	$-0.229485\pi$
$491$	8.02560e6	1.50236	0.751179	+	0.660098i	$0.229485\pi$
$492$	0	0
$493$	−9.73246e6	−1.80346
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.54523e6	−0.280609
$498$	0	0
$499$	−3.53659e6	−0.635818	−0.317909	−	0.948121i	$-0.602981\pi$
$499$	−3.53659e6	−0.635818	−0.317909	+	0.948121i	$0.602981\pi$
$500$	0	0
$501$	−4.91101e6	−0.874130
$502$	0	0
$503$	2.81866e6	0.496733	0.248367	−	0.968666i	$-0.420106\pi$
$503$	2.81866e6	0.496733	0.248367	+	0.968666i	$0.420106\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.29280e6	−0.396138
$508$	0	0
$509$	538808.	0.0921806	0.0460903	−	0.998937i	$-0.485324\pi$
$509$	538808.	0.0921806	0.0460903	+	0.998937i	$0.485324\pi$
$510$	0	0
$511$	3.01952e6	0.511547
$512$	0	0
$513$	−1.00640e6	−0.168841
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−965776.	−0.158910
$518$	0	0
$519$	3.44968e6	0.562160
$520$	0	0
$521$	5.25606e6	0.848332	0.424166	−	0.905584i	$-0.360567\pi$
$521$	5.25606e6	0.848332	0.424166	+	0.905584i	$0.360567\pi$
$522$	0	0
$523$	−5.36466e6	−0.857606	−0.428803	−	0.903398i	$-0.641065\pi$
$523$	−5.36466e6	−0.857606	−0.428803	+	0.903398i	$0.641065\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.54326e7	2.42054
$528$	0	0
$529$	1.25328e7	1.94720
$530$	0	0
$531$	−3.91514e6	−0.602576
$532$	0	0
$533$	5.25250e6	0.800844
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.15101e6	−0.621181
$538$	0	0
$539$	−1.54292e6	−0.228756
$540$	0	0
$541$	1.05037e7	1.54294	0.771471	−	0.636264i	$-0.219521\pi$
$541$	1.05037e7	1.54294	0.771471	+	0.636264i	$0.219521\pi$
$542$	0	0
$543$	−1.82986e7	−2.66330
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.64670e6	−1.09271	−0.546356	−	0.837553i	$-0.683985\pi$
$547$	−7.64670e6	−1.09271	−0.546356	+	0.837553i	$0.683985\pi$
$548$	0	0
$549$	6.55853e6	0.928700
$550$	0	0
$551$	−8.19042e6	−1.14928
$552$	0	0
$553$	2.32255e6	0.322963
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.93235e6	−0.946766	−0.473383	−	0.880857i	$-0.656967\pi$
$557$	−6.93235e6	−0.946766	−0.473383	+	0.880857i	$0.656967\pi$
$558$	0	0
$559$	1.21657e7	1.64668
$560$	0	0
$561$	−1.33702e7	−1.79362
$562$	0	0
$563$	−7.26755e6	−0.966311	−0.483156	−	0.875535i	$-0.660509\pi$
$563$	−7.26755e6	−0.966311	−0.483156	+	0.875535i	$0.660509\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.58241e6	0.729229
$568$	0	0
$569$	−1.33658e7	−1.73067	−0.865336	−	0.501193i	$-0.832895\pi$
$569$	−1.33658e7	−1.73067	−0.865336	+	0.501193i	$0.832895\pi$
$570$	0	0
$571$	−9.84651e6	−1.26384	−0.631920	−	0.775034i	$-0.717733\pi$
$571$	−9.84651e6	−1.26384	−0.631920	+	0.775034i	$0.717733\pi$
$572$	0	0
$573$	9.71901e6	1.23662
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9.93536e6	1.24235	0.621175	−	0.783672i	$-0.286656\pi$
$577$	9.93536e6	1.24235	0.621175	+	0.783672i	$0.286656\pi$
$578$	0	0
$579$	−1.38205e7	−1.71328
$580$	0	0
$581$	4.00531e6	0.492261
$582$	0	0
$583$	8.42129e6	1.02614
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.55552e6	−1.02483	−0.512414	−	0.858738i	$-0.671249\pi$
$587$	−8.55552e6	−1.02483	−0.512414	+	0.858738i	$0.671249\pi$
$588$	0	0
$589$	1.29874e7	1.54253
$590$	0	0
$591$	−5.89182e6	−0.693874
$592$	0	0
$593$	3.07038e6	0.358555	0.179277	−	0.983799i	$-0.442624\pi$
$593$	3.07038e6	0.358555	0.179277	+	0.983799i	$0.442624\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.72762e7	−1.98387
$598$	0	0
$599$	−174294.	−0.0198480	−0.00992398	−	0.999951i	$-0.503159\pi$
$599$	−174294.	−0.0198480	−0.00992398	+	0.999951i	$0.503159\pi$
$600$	0	0
$601$	3.73604e6	0.421915	0.210958	−	0.977495i	$-0.432342\pi$
$601$	3.73604e6	0.421915	0.210958	+	0.977495i	$0.432342\pi$
$602$	0	0
$603$	−1.59559e6	−0.178702
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.02370e6	−0.333094	−0.166547	−	0.986033i	$-0.553262\pi$
$607$	−3.02370e6	−0.333094	−0.166547	+	0.986033i	$0.553262\pi$
$608$	0	0
$609$	−1.16286e7	−1.27053
$610$	0	0
$611$	1.72498e6	0.186930
$612$	0	0
$613$	−3.07263e6	−0.330263	−0.165131	−	0.986272i	$-0.552805\pi$
$613$	−3.07263e6	−0.330263	−0.165131	+	0.986272i	$0.552805\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.60221e6	−0.698195	−0.349097	−	0.937086i	$-0.613512\pi$
$617$	−6.60221e6	−0.698195	−0.349097	+	0.937086i	$0.613512\pi$
$618$	0	0
$619$	−2.68055e6	−0.281189	−0.140594	−	0.990067i	$-0.544901\pi$
$619$	−2.68055e6	−0.281189	−0.140594	+	0.990067i	$0.544901\pi$
$620$	0	0
$621$	2.56429e6	0.266832
$622$	0	0
$623$	−9.71562e6	−1.00288
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.12518e7	−1.14301
$628$	0	0
$629$	1.06857e7	1.07690
$630$	0	0
$631$	−1.72975e7	−1.72946	−0.864728	−	0.502240i	$-0.832509\pi$
$631$	−1.72975e7	−1.72946	−0.864728	+	0.502240i	$0.832509\pi$
$632$	0	0
$633$	4.29197e6	0.425743
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.75582e6	0.269093
$638$	0	0
$639$	3.87588e6	0.375507
$640$	0	0
$641$	−1.50990e7	−1.45145	−0.725725	−	0.687984i	$-0.758496\pi$
$641$	−1.50990e7	−1.45145	−0.725725	+	0.687984i	$0.758496\pi$
$642$	0	0
$643$	8.40104e6	0.801319	0.400660	−	0.916227i	$-0.368781\pi$
$643$	8.40104e6	0.801319	0.400660	+	0.916227i	$0.368781\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.69105e7	−1.58816	−0.794080	−	0.607813i	$-0.792047\pi$
$647$	−1.69105e7	−1.58816	−0.794080	+	0.607813i	$0.792047\pi$
$648$	0	0
$649$	−4.23362e6	−0.394549
$650$	0	0
$651$	1.84393e7	1.70526
$652$	0	0
$653$	3.38875e6	0.310997	0.155499	−	0.987836i	$-0.450302\pi$
$653$	3.38875e6	0.310997	0.155499	+	0.987836i	$0.450302\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.57381e6	−0.684543
$658$	0	0
$659$	−1.21397e7	−1.08891	−0.544457	−	0.838789i	$-0.683264\pi$
$659$	−1.21397e7	−1.08891	−0.544457	+	0.838789i	$0.683264\pi$
$660$	0	0
$661$	−1.67377e7	−1.49002	−0.745009	−	0.667055i	$-0.767555\pi$
$661$	−1.67377e7	−1.49002	−0.745009	+	0.667055i	$0.767555\pi$
$662$	0	0
$663$	2.38805e7	2.10989
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.08690e7	1.81630
$668$	0	0
$669$	1.14396e6	0.0988199
$670$	0	0
$671$	7.09204e6	0.608085
$672$	0	0
$673$	−7.36101e6	−0.626469	−0.313235	−	0.949676i	$-0.601413\pi$
$673$	−7.36101e6	−0.626469	−0.313235	+	0.949676i	$0.601413\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.68650e6	0.476841	0.238420	−	0.971162i	$-0.423370\pi$
$677$	5.68650e6	0.476841	0.238420	+	0.971162i	$0.423370\pi$
$678$	0	0
$679$	−7.92407e6	−0.659589
$680$	0	0
$681$	−1.45106e7	−1.19900
$682$	0	0
$683$	−229728.	−0.0188435	−0.00942175	−	0.999956i	$-0.502999\pi$
$683$	−229728.	−0.0188435	−0.00942175	+	0.999956i	$0.502999\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.73144e7	2.20801
$688$	0	0
$689$	−1.50413e7	−1.20708
$690$	0	0
$691$	−1.32222e7	−1.05344	−0.526720	−	0.850039i	$-0.676578\pi$
$691$	−1.32222e7	−1.05344	−0.526720	+	0.850039i	$0.676578\pi$
$692$	0	0
$693$	−8.39343e6	−0.663906
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.05332e7	1.60094
$698$	0	0
$699$	2.40955e7	1.86528
$700$	0	0
$701$	8.59857e6	0.660893	0.330447	−	0.943825i	$-0.392801\pi$
$701$	8.59857e6	0.660893	0.330447	+	0.943825i	$0.392801\pi$
$702$	0	0
$703$	8.99261e6	0.686274
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.73443e6	−0.506702
$708$	0	0
$709$	−1.00967e7	−0.754336	−0.377168	−	0.926145i	$-0.623102\pi$
$709$	−1.00967e7	−0.754336	−0.377168	+	0.926145i	$0.623102\pi$
$710$	0	0
$711$	−5.82562e6	−0.432184
$712$	0	0
$713$	−3.30916e7	−2.43777
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.12120e6	0.589959
$718$	0	0
$719$	−3.34106e6	−0.241025	−0.120513	−	0.992712i	$-0.538454\pi$
$719$	−3.34106e6	−0.241025	−0.120513	+	0.992712i	$0.538454\pi$
$720$	0	0
$721$	1.45125e7	1.03969
$722$	0	0
$723$	3.58701e7	2.55203
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.35160e7	−0.948447	−0.474224	−	0.880404i	$-0.657271\pi$
$727$	−1.35160e7	−0.948447	−0.474224	+	0.880404i	$0.657271\pi$
$728$	0	0
$729$	−1.72396e7	−1.20146
$730$	0	0
$731$	4.75585e7	3.29181
$732$	0	0
$733$	2.86786e7	1.97150	0.985752	−	0.168202i	$-0.0537962\pi$
$733$	2.86786e7	1.97150	0.985752	+	0.168202i	$0.0537962\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.72539e6	−0.117009
$738$	0	0
$739$	−1.81256e7	−1.22090	−0.610452	−	0.792053i	$-0.709012\pi$
$739$	−1.81256e7	−1.22090	−0.610452	+	0.792053i	$0.709012\pi$
$740$	0	0
$741$	2.00968e7	1.34456
$742$	0	0
$743$	−7.08132e6	−0.470590	−0.235295	−	0.971924i	$-0.575606\pi$
$743$	−7.08132e6	−0.470590	−0.235295	+	0.971924i	$0.575606\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.00464e7	−0.658735
$748$	0	0
$749$	2.93431e6	0.191118
$750$	0	0
$751$	4.00260e6	0.258966	0.129483	−	0.991582i	$-0.458668\pi$
$751$	4.00260e6	0.258966	0.129483	+	0.991582i	$0.458668\pi$
$752$	0	0
$753$	2.46821e7	1.58634
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.71922e7	1.09041	0.545207	−	0.838302i	$-0.316451\pi$
$757$	1.71922e7	1.09041	0.545207	+	0.838302i	$0.316451\pi$
$758$	0	0
$759$	2.86692e7	1.80639
$760$	0	0
$761$	1.98543e6	0.124277	0.0621387	−	0.998068i	$-0.480208\pi$
$761$	1.98543e6	0.124277	0.0621387	+	0.998068i	$0.480208\pi$
$762$	0	0
$763$	−1.28652e7	−0.800027
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.56169e6	0.464120
$768$	0	0
$769$	−1.30522e7	−0.795915	−0.397958	−	0.917404i	$-0.630281\pi$
$769$	−1.30522e7	−0.795915	−0.397958	+	0.917404i	$0.630281\pi$
$770$	0	0
$771$	−3.17928e6	−0.192616
$772$	0	0
$773$	1.33637e7	0.804408	0.402204	−	0.915550i	$-0.368244\pi$
$773$	1.33637e7	0.804408	0.402204	+	0.915550i	$0.368244\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.27676e7	0.758674
$778$	0	0
$779$	1.72798e7	1.02023
$780$	0	0
$781$	4.19116e6	0.245871
$782$	0	0
$783$	2.82111e6	0.164443
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.84001e7	−1.63449	−0.817246	−	0.576289i	$-0.804500\pi$
$787$	−2.84001e7	−1.63449	−0.817246	+	0.576289i	$0.804500\pi$
$788$	0	0
$789$	−38202.4	−0.00218473
$790$	0	0
$791$	2.47290e7	1.40529
$792$	0	0
$793$	−1.26671e7	−0.715310
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.56843e7	−1.43226	−0.716130	−	0.697967i	$-0.754088\pi$
$797$	−2.56843e7	−1.43226	−0.716130	+	0.697967i	$0.754088\pi$
$798$	0	0
$799$	6.74332e6	0.373686
$800$	0	0
$801$	2.43695e7	1.34204
$802$	0	0
$803$	−8.18990e6	−0.448219
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.33318e7	1.26114
$808$	0	0
$809$	−1.39936e7	−0.751721	−0.375861	−	0.926676i	$-0.622653\pi$
$809$	−1.39936e7	−0.751721	−0.375861	+	0.926676i	$0.622653\pi$
$810$	0	0
$811$	−6.58958e6	−0.351808	−0.175904	−	0.984407i	$-0.556285\pi$
$811$	−6.58958e6	−0.351808	−0.175904	+	0.984407i	$0.556285\pi$
$812$	0	0
$813$	−3.44674e6	−0.182887
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00232e7	2.09776
$818$	0	0
$819$	1.49915e7	0.780973
$820$	0	0
$821$	8.99703e6	0.465845	0.232922	−	0.972495i	$-0.425171\pi$
$821$	8.99703e6	0.465845	0.232922	+	0.972495i	$0.425171\pi$
$822$	0	0
$823$	5.23574e6	0.269450	0.134725	−	0.990883i	$-0.456985\pi$
$823$	5.23574e6	0.269450	0.134725	+	0.990883i	$0.456985\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−315583.	−0.0160454	−0.00802268	−	0.999968i	$-0.502554\pi$
$827$	−315583.	−0.0160454	−0.00802268	+	0.999968i	$0.502554\pi$
$828$	0	0
$829$	−8.50851e6	−0.429999	−0.214999	−	0.976614i	$-0.568975\pi$
$829$	−8.50851e6	−0.429999	−0.214999	+	0.976614i	$0.568975\pi$
$830$	0	0
$831$	3.29968e7	1.65756
$832$	0	0
$833$	1.07731e7	0.537934
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.47338e6	−0.220710
$838$	0	0
$839$	−2.91278e7	−1.42858	−0.714288	−	0.699852i	$-0.753249\pi$
$839$	−2.91278e7	−1.42858	−0.714288	+	0.699852i	$0.753249\pi$
$840$	0	0
$841$	2.44793e6	0.119346
$842$	0	0
$843$	−1.15376e7	−0.559174
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.19692e6	0.392593
$848$	0	0
$849$	3.15987e6	0.150453
$850$	0	0
$851$	−2.29130e7	−1.08457
$852$	0	0
$853$	−5.70286e6	−0.268361	−0.134181	−	0.990957i	$-0.542840\pi$
$853$	−5.70286e6	−0.268361	−0.134181	+	0.990957i	$0.542840\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.92322e7	1.35959	0.679797	−	0.733400i	$-0.262068\pi$
$857$	2.92322e7	1.35959	0.679797	+	0.733400i	$0.262068\pi$
$858$	0	0
$859$	−1.74800e7	−0.808274	−0.404137	−	0.914698i	$-0.632428\pi$
$859$	−1.74800e7	−0.808274	−0.404137	+	0.914698i	$0.632428\pi$
$860$	0	0
$861$	2.45336e7	1.12786
$862$	0	0
$863$	5.53579e6	0.253018	0.126509	−	0.991965i	$-0.459623\pi$
$863$	5.53579e6	0.253018	0.126509	+	0.991965i	$0.459623\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.12260e7	2.76623
$868$	0	0
$869$	−6.29951e6	−0.282981
$870$	0	0
$871$	3.08172e6	0.137641
$872$	0	0
$873$	1.98758e7	0.882651
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.87087e6	−0.213849	−0.106925	−	0.994267i	$-0.534100\pi$
$877$	−4.87087e6	−0.213849	−0.106925	+	0.994267i	$0.534100\pi$
$878$	0	0
$879$	−7.15407e6	−0.312306
$880$	0	0
$881$	2.25286e7	0.977900	0.488950	−	0.872312i	$-0.337380\pi$
$881$	2.25286e7	0.977900	0.488950	+	0.872312i	$0.337380\pi$
$882$	0	0
$883$	2.27462e7	0.981762	0.490881	−	0.871227i	$-0.336675\pi$
$883$	2.27462e7	0.981762	0.490881	+	0.871227i	$0.336675\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.52256e7	−1.07655	−0.538274	−	0.842770i	$-0.680923\pi$
$887$	−2.52256e7	−1.07655	−0.538274	+	0.842770i	$0.680923\pi$
$888$	0	0
$889$	3.44894e7	1.46363
$890$	0	0
$891$	−1.51413e7	−0.638952
$892$	0	0
$893$	5.67488e6	0.238138
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.12062e7	−2.12491
$898$	0	0
$899$	−3.64058e7	−1.50235
$900$	0	0
$901$	−5.87998e7	−2.41304
$902$	0	0
$903$	5.68242e7	2.31907
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.51254e7	1.01413	0.507066	−	0.861907i	$-0.330730\pi$
$907$	2.51254e7	1.01413	0.507066	+	0.861907i	$0.330730\pi$
$908$	0	0
$909$	1.68919e7	0.678060
$910$	0	0
$911$	−2.81254e6	−0.112280	−0.0561400	−	0.998423i	$-0.517879\pi$
$911$	−2.81254e6	−0.112280	−0.0561400	+	0.998423i	$0.517879\pi$
$912$	0	0
$913$	−1.08637e7	−0.431320
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.11222e7	−0.436785
$918$	0	0
$919$	−3.18718e7	−1.24485	−0.622426	−	0.782679i	$-0.713853\pi$
$919$	−3.18718e7	−1.24485	−0.622426	+	0.782679i	$0.713853\pi$
$920$	0	0
$921$	−4.80680e7	−1.86727
$922$	0	0
$923$	−7.48585e6	−0.289226
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.64015e7	−1.39130
$928$	0	0
$929$	−3.38523e6	−0.128691	−0.0643455	−	0.997928i	$-0.520496\pi$
$929$	−3.38523e6	−0.128691	−0.0643455	+	0.997928i	$0.520496\pi$
$930$	0	0
$931$	9.06619e6	0.342808
$932$	0	0
$933$	2.26838e7	0.853122
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.33776e7	−1.61405	−0.807025	−	0.590517i	$-0.798924\pi$
$937$	−4.33776e7	−1.61405	−0.807025	+	0.590517i	$0.798924\pi$
$938$	0	0
$939$	6.07409e6	0.224811
$940$	0	0
$941$	−3.32026e7	−1.22236	−0.611178	−	0.791493i	$-0.709304\pi$
$941$	−3.32026e7	−1.22236	−0.611178	+	0.791493i	$0.709304\pi$
$942$	0	0
$943$	−4.40286e7	−1.61234
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.31754e7	−0.477407	−0.238704	−	0.971092i	$-0.576722\pi$
$947$	−1.31754e7	−0.477407	−0.238704	+	0.971092i	$0.576722\pi$
$948$	0	0
$949$	1.46280e7	0.527254
$950$	0	0
$951$	−3.33400e7	−1.19540
$952$	0	0
$953$	1.24126e7	0.442721	0.221360	−	0.975192i	$-0.428950\pi$
$953$	1.24126e7	0.442721	0.221360	+	0.975192i	$0.428950\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.15405e7	1.11324
$958$	0	0
$959$	−4.29288e7	−1.50731
$960$	0	0
$961$	2.90987e7	1.01640
$962$	0	0
$963$	−7.36009e6	−0.255751
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.35785e7	1.15477	0.577385	−	0.816472i	$-0.304073\pi$
$967$	3.35785e7	1.15477	0.577385	+	0.816472i	$0.304073\pi$
$968$	0	0
$969$	7.85629e7	2.68787
$970$	0	0
$971$	−5.27945e6	−0.179697	−0.0898485	−	0.995955i	$-0.528638\pi$
$971$	−5.27945e6	−0.179697	−0.0898485	+	0.995955i	$0.528638\pi$
$972$	0	0
$973$	−1.93623e7	−0.655653
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.27469e7	−0.427237	−0.213619	−	0.976917i	$-0.568525\pi$
$977$	−1.27469e7	−0.427237	−0.213619	+	0.976917i	$0.568525\pi$
$978$	0	0
$979$	2.63519e7	0.878729
$980$	0	0
$981$	3.22695e7	1.07058
$982$	0	0
$983$	−2.38142e7	−0.786052	−0.393026	−	0.919527i	$-0.628572\pi$
$983$	−2.38142e7	−0.786052	−0.393026	+	0.919527i	$0.628572\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.05710e6	0.263261
$988$	0	0
$989$	−1.01978e8	−3.31525
$990$	0	0
$991$	−1.23214e7	−0.398545	−0.199273	−	0.979944i	$-0.563858\pi$
$991$	−1.23214e7	−0.398545	−0.199273	+	0.979944i	$0.563858\pi$
$992$	0	0
$993$	3.18861e7	1.02619
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.52318e6	−0.239697	−0.119849	−	0.992792i	$-0.538241\pi$
$997$	−7.52318e6	−0.239697	−0.119849	+	0.992792i	$0.538241\pi$
$998$	0	0
$999$	−3.09742e6	−0.0981943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.y.1.3		3
4.3	odd	2		200.6.a.h.1.1	✓	3
5.2	odd	4		400.6.c.p.49.1		6
5.3	odd	4		400.6.c.p.49.6		6
5.4	even	2		400.6.a.x.1.1		3
20.3	even	4		200.6.c.g.49.1		6
20.7	even	4		200.6.c.g.49.6		6
20.19	odd	2		200.6.a.i.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.a.h.1.1	✓	3	4.3	odd	2
200.6.a.i.1.3	yes	3	20.19	odd	2
200.6.c.g.49.1		6	20.3	even	4
200.6.c.g.49.6		6	20.7	even	4
400.6.a.x.1.1		3	5.4	even	2
400.6.a.y.1.3		3	1.1	even	1	trivial
400.6.c.p.49.1		6	5.2	odd	4
400.6.c.p.49.6		6	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+22.6278 q^{3} -107.252 q^{7} +269.020 q^{9} +O(q^{10})\)	\(q+22.6278 q^{3} -107.252 q^{7} +269.020 q^{9} +290.903 q^{11} -519.583 q^{13} -2031.17 q^{17} -1709.34 q^{19} -2426.89 q^{21} +4355.36 q^{23} +588.766 q^{27} +4791.56 q^{29} -7597.89 q^{31} +6582.51 q^{33} -5260.87 q^{37} -11757.0 q^{39} -10109.1 q^{41} -23414.4 q^{43} -3319.92 q^{47} -5303.91 q^{49} -45960.9 q^{51} +28948.8 q^{53} -38678.7 q^{57} -14553.4 q^{59} +24379.4 q^{61} -28853.0 q^{63} -5931.14 q^{67} +98552.5 q^{69} +14407.4 q^{71} -28153.4 q^{73} -31200.1 q^{77} -21655.0 q^{79} -52049.2 q^{81} -37344.7 q^{83} +108423. q^{87} +90586.5 q^{89} +55726.6 q^{91} -171924. q^{93} +73882.4 q^{97} +78258.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 70 q^{7} + 154 q^{9}+O(q^{10}) \) 3 * q + q^3 - 70 * q^7 + 154 * q^9	\( 3 q + q^{3} - 70 q^{7} + 154 q^{9} + 19 q^{11} + 196 q^{13} - 1223 q^{17} - 1221 q^{19} + 958 q^{21} - 2490 q^{23} + 4123 q^{27} + 11912 q^{29} - 7442 q^{31} + 6969 q^{33} - 14766 q^{37} - 27396 q^{39} + 3223 q^{41} - 41060 q^{43} + 29188 q^{47} + 66423 q^{49} - 43165 q^{51} + 12878 q^{53} - 77791 q^{57} - 64912 q^{59} + 22478 q^{61} - 112916 q^{63} + 26499 q^{67} + 178642 q^{69} - 86676 q^{71} - 8305 q^{73} - 106822 q^{77} - 21982 q^{79} - 71453 q^{81} - 213353 q^{83} + 14952 q^{87} + 182381 q^{89} - 149224 q^{91} - 196470 q^{93} - 76342 q^{97} + 149130 q^{99}+O(q^{100}) \) 3 * q + q^3 - 70 * q^7 + 154 * q^9 + 19 * q^11 + 196 * q^13 - 1223 * q^17 - 1221 * q^19 + 958 * q^21 - 2490 * q^23 + 4123 * q^27 + 11912 * q^29 - 7442 * q^31 + 6969 * q^33 - 14766 * q^37 - 27396 * q^39 + 3223 * q^41 - 41060 * q^43 + 29188 * q^47 + 66423 * q^49 - 43165 * q^51 + 12878 * q^53 - 77791 * q^57 - 64912 * q^59 + 22478 * q^61 - 112916 * q^63 + 26499 * q^67 + 178642 * q^69 - 86676 * q^71 - 8305 * q^73 - 106822 * q^77 - 21982 * q^79 - 71453 * q^81 - 213353 * q^83 + 14952 * q^87 + 182381 * q^89 - 149224 * q^91 - 196470 * q^93 - 76342 * q^97 + 149130 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more