LMFDB - Embedded newform 400.6.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,6,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,0,-8,0,0,0,8,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.26209$ 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)

$f(q)$	$=$	$q+11.5242 q^{3} -27.0483 q^{7} -110.193 q^{9} -226.008 q^{11} +511.257 q^{13} +387.387 q^{17} +1335.93 q^{19} -311.710 q^{21} -545.369 q^{23} -4070.26 q^{27} -4637.58 q^{29} -2991.56 q^{31} -2604.55 q^{33} -1263.70 q^{37} +5891.82 q^{39} -17197.6 q^{41} +16592.0 q^{43} -13036.0 q^{47} -16075.4 q^{49} +4464.31 q^{51} -28994.7 q^{53} +15395.4 q^{57} +34429.9 q^{59} -24149.1 q^{61} +2980.55 q^{63} -29389.7 q^{67} -6284.93 q^{69} -9064.32 q^{71} -55528.7 q^{73} +6113.13 q^{77} +101587. q^{79} -20129.4 q^{81} -73240.4 q^{83} -53444.3 q^{87} -42498.5 q^{89} -13828.7 q^{91} -34475.2 q^{93} +10565.9 q^{97} +24904.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} + 8 q^{7} + 28 q^{9} + 200 q^{11} - 592 q^{13} + 278 q^{17} + 840 q^{19} - 996 q^{21} + 1952 q^{23} - 2024 q^{27} - 4680 q^{29} + 5008 q^{31} - 10922 q^{33} + 12500 q^{37} + 27432 q^{39} - 5334 q^{41}+ \cdots + 83776 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9 + 200 * q^11 - 592 * q^13 + 278 * q^17 + 840 * q^19 - 996 * q^21 + 1952 * q^23 - 2024 * q^27 - 4680 * q^29 + 5008 * q^31 - 10922 * q^33 + 12500 * q^37 + 27432 * q^39 - 5334 * q^41 - 224 * q^43 - 26072 * q^47 - 31654 * q^49 + 6600 * q^51 - 46812 * q^53 + 25078 * q^57 + 81776 * q^59 - 46932 * q^61 + 7824 * q^63 - 68808 * q^67 - 55044 * q^69 - 7448 * q^71 - 108822 * q^73 + 21044 * q^77 + 108104 * q^79 - 93662 * q^81 - 27224 * q^83 - 52616 * q^87 + 70990 * q^89 - 52496 * q^91 - 190660 * q^93 - 96852 * q^97 + 83776 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	11.5242	0.739276	0.369638	−	0.929176i	$-0.379482\pi$
$3$	11.5242	0.739276	0.369638	+	0.929176i	$0.379482\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−27.0483	−0.208639	−0.104320	−	0.994544i	$-0.533266\pi$
$7$	−27.0483	−0.208639	−0.104320	+	0.994544i	$0.533266\pi$
$8$	0	0
$9$	−110.193	−0.453471
$10$	0	0
$11$	−226.008	−0.563173	−0.281586	−	0.959536i	$-0.590861\pi$
$11$	−226.008	−0.563173	−0.281586	+	0.959536i	$0.590861\pi$
$12$	0	0
$13$	511.257	0.839037	0.419518	−	0.907747i	$-0.362199\pi$
$13$	511.257	0.839037	0.419518	+	0.907747i	$0.362199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	387.387	0.325104	0.162552	−	0.986700i	$-0.448027\pi$
$17$	387.387	0.325104	0.162552	+	0.986700i	$0.448027\pi$
$18$	0	0
$19$	1335.93	0.848982	0.424491	−	0.905432i	$-0.360453\pi$
$19$	1335.93	0.848982	0.424491	+	0.905432i	$0.360453\pi$
$20$	0	0
$21$	−311.710	−0.154242
$22$	0	0
$23$	−545.369	−0.214967	−0.107483	−	0.994207i	$-0.534279\pi$
$23$	−545.369	−0.214967	−0.107483	+	0.994207i	$0.534279\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4070.26	−1.07452
$28$	0	0
$29$	−4637.58	−1.02399	−0.511996	−	0.858988i	$-0.671094\pi$
$29$	−4637.58	−1.02399	−0.511996	+	0.858988i	$0.671094\pi$
$30$	0	0
$31$	−2991.56	−0.559105	−0.279552	−	0.960130i	$-0.590186\pi$
$31$	−2991.56	−0.559105	−0.279552	+	0.960130i	$0.590186\pi$
$32$	0	0
$33$	−2604.55	−0.416340
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1263.70	−0.151754	−0.0758770	−	0.997117i	$-0.524176\pi$
$37$	−1263.70	−0.151754	−0.0758770	+	0.997117i	$0.524176\pi$
$38$	0	0
$39$	5891.82	0.620280
$40$	0	0
$41$	−17197.6	−1.59775	−0.798875	−	0.601497i	$-0.794571\pi$
$41$	−17197.6	−1.59775	−0.798875	+	0.601497i	$0.794571\pi$
$42$	0	0
$43$	16592.0	1.36845	0.684223	−	0.729272i	$-0.260141\pi$
$43$	16592.0	1.36845	0.684223	+	0.729272i	$0.260141\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13036.0	−0.860795	−0.430397	−	0.902639i	$-0.641627\pi$
$47$	−13036.0	−0.860795	−0.430397	+	0.902639i	$0.641627\pi$
$48$	0	0
$49$	−16075.4	−0.956470
$50$	0	0
$51$	4464.31	0.240342
$52$	0	0
$53$	−28994.7	−1.41785	−0.708923	−	0.705286i	$-0.750819\pi$
$53$	−28994.7	−1.41785	−0.708923	+	0.705286i	$0.750819\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15395.4	0.627632
$58$	0	0
$59$	34429.9	1.28768	0.643838	−	0.765162i	$-0.277341\pi$
$59$	34429.9	1.28768	0.643838	+	0.765162i	$0.277341\pi$
$60$	0	0
$61$	−24149.1	−0.830952	−0.415476	−	0.909604i	$-0.636385\pi$
$61$	−24149.1	−0.830952	−0.415476	+	0.909604i	$0.636385\pi$
$62$	0	0
$63$	2980.55	0.0946117
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−29389.7	−0.799849	−0.399925	−	0.916548i	$-0.630964\pi$
$67$	−29389.7	−0.799849	−0.399925	+	0.916548i	$0.630964\pi$
$68$	0	0
$69$	−6284.93	−0.158920
$70$	0	0
$71$	−9064.32	−0.213397	−0.106699	−	0.994291i	$-0.534028\pi$
$71$	−9064.32	−0.213397	−0.106699	+	0.994291i	$0.534028\pi$
$72$	0	0
$73$	−55528.7	−1.21958	−0.609791	−	0.792563i	$-0.708746\pi$
$73$	−55528.7	−1.21958	−0.609791	+	0.792563i	$0.708746\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6113.13	0.117500
$78$	0	0
$79$	101587.	1.83135	0.915673	−	0.401924i	$-0.131658\pi$
$79$	101587.	1.83135	0.915673	+	0.401924i	$0.131658\pi$
$80$	0	0
$81$	−20129.4	−0.340893
$82$	0	0
$83$	−73240.4	−1.16696	−0.583479	−	0.812128i	$-0.698309\pi$
$83$	−73240.4	−1.16696	−0.583479	+	0.812128i	$0.698309\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−53444.3	−0.757012
$88$	0	0
$89$	−42498.5	−0.568719	−0.284360	−	0.958718i	$-0.591781\pi$
$89$	−42498.5	−0.568719	−0.284360	+	0.958718i	$0.591781\pi$
$90$	0	0
$91$	−13828.7	−0.175056
$92$	0	0
$93$	−34475.2	−0.413333
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10565.9	0.114019	0.0570093	−	0.998374i	$-0.481844\pi$
$97$	10565.9	0.114019	0.0570093	+	0.998374i	$0.481844\pi$
$98$	0	0
$99$	24904.6	0.255382
$100$	0	0
$101$	46486.0	0.453439	0.226719	−	0.973960i	$-0.427200\pi$
$101$	46486.0	0.453439	0.226719	+	0.973960i	$0.427200\pi$
$102$	0	0
$103$	−119526.	−1.11012	−0.555059	−	0.831811i	$-0.687304\pi$
$103$	−119526.	−1.11012	−0.555059	+	0.831811i	$0.687304\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	22922.4	0.193553	0.0967765	−	0.995306i	$-0.469147\pi$
$107$	22922.4	0.193553	0.0967765	+	0.995306i	$0.469147\pi$
$108$	0	0
$109$	210121.	1.69396	0.846980	−	0.531624i	$-0.178418\pi$
$109$	210121.	1.69396	0.846980	+	0.531624i	$0.178418\pi$
$110$	0	0
$111$	−14563.1	−0.112188
$112$	0	0
$113$	−203886.	−1.50208	−0.751039	−	0.660258i	$-0.770447\pi$
$113$	−203886.	−1.50208	−0.751039	+	0.660258i	$0.770447\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−56337.2	−0.380479
$118$	0	0
$119$	−10478.2	−0.0678294
$120$	0	0
$121$	−109972.	−0.682837
$122$	0	0
$123$	−198188.	−1.18118
$124$	0	0
$125$	0	0
$126$	0	0
$127$	175022.	0.962905	0.481453	−	0.876472i	$-0.340109\pi$
$127$	175022.	0.962905	0.481453	+	0.876472i	$0.340109\pi$
$128$	0	0
$129$	191209.	1.01166
$130$	0	0
$131$	−149218.	−0.759701	−0.379850	−	0.925048i	$-0.624025\pi$
$131$	−149218.	−0.759701	−0.379850	+	0.925048i	$0.624025\pi$
$132$	0	0
$133$	−36134.6	−0.177131
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−212180.	−0.965836	−0.482918	−	0.875666i	$-0.660423\pi$
$137$	−212180.	−0.965836	−0.482918	+	0.875666i	$0.660423\pi$
$138$	0	0
$139$	−77082.5	−0.338391	−0.169195	−	0.985583i	$-0.554117\pi$
$139$	−77082.5	−0.338391	−0.169195	+	0.985583i	$0.554117\pi$
$140$	0	0
$141$	−150229.	−0.636365
$142$	0	0
$143$	−115548.	−0.472522
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−185256.	−0.707095
$148$	0	0
$149$	470139.	1.73485	0.867423	−	0.497571i	$-0.165775\pi$
$149$	470139.	1.73485	0.867423	+	0.497571i	$0.165775\pi$
$150$	0	0
$151$	−311118.	−1.11041	−0.555205	−	0.831714i	$-0.687360\pi$
$151$	−311118.	−1.11041	−0.555205	+	0.831714i	$0.687360\pi$
$152$	0	0
$153$	−42687.5	−0.147425
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−543649.	−1.76023	−0.880115	−	0.474760i	$-0.842535\pi$
$157$	−543649.	−1.76023	−0.880115	+	0.474760i	$0.842535\pi$
$158$	0	0
$159$	−334140.	−1.04818
$160$	0	0
$161$	14751.3	0.0448504
$162$	0	0
$163$	−298673.	−0.880495	−0.440248	−	0.897876i	$-0.645109\pi$
$163$	−298673.	−0.880495	−0.440248	+	0.897876i	$0.645109\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	65486.6	0.181703	0.0908513	−	0.995864i	$-0.471041\pi$
$167$	65486.6	0.181703	0.0908513	+	0.995864i	$0.471041\pi$
$168$	0	0
$169$	−109909.	−0.296017
$170$	0	0
$171$	−147210.	−0.384989
$172$	0	0
$173$	355092.	0.902040	0.451020	−	0.892514i	$-0.351060\pi$
$173$	355092.	0.902040	0.451020	+	0.892514i	$0.351060\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	396777.	0.951947
$178$	0	0
$179$	39265.7	0.0915970	0.0457985	−	0.998951i	$-0.485417\pi$
$179$	39265.7	0.0915970	0.0457985	+	0.998951i	$0.485417\pi$
$180$	0	0
$181$	99638.2	0.226063	0.113031	−	0.993591i	$-0.463944\pi$
$181$	99638.2	0.226063	0.113031	+	0.993591i	$0.463944\pi$
$182$	0	0
$183$	−278298.	−0.614303
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−87552.4	−0.183090
$188$	0	0
$189$	110094.	0.224186
$190$	0	0
$191$	−380336.	−0.754369	−0.377185	−	0.926138i	$-0.623108\pi$
$191$	−380336.	−0.754369	−0.377185	+	0.926138i	$0.623108\pi$
$192$	0	0
$193$	126401.	0.244262	0.122131	−	0.992514i	$-0.461027\pi$
$193$	126401.	0.244262	0.122131	+	0.992514i	$0.461027\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−326136.	−0.598733	−0.299366	−	0.954138i	$-0.596775\pi$
$197$	−326136.	−0.598733	−0.299366	+	0.954138i	$0.596775\pi$
$198$	0	0
$199$	498559.	0.892451	0.446225	−	0.894921i	$-0.352768\pi$
$199$	498559.	0.892451	0.446225	+	0.894921i	$0.352768\pi$
$200$	0	0
$201$	−338692.	−0.591309
$202$	0	0
$203$	125439.	0.213645
$204$	0	0
$205$	0	0
$206$	0	0
$207$	60096.1	0.0974811
$208$	0	0
$209$	−301930.	−0.478123
$210$	0	0
$211$	310763.	0.480532	0.240266	−	0.970707i	$-0.422765\pi$
$211$	310763.	0.480532	0.240266	+	0.970707i	$0.422765\pi$
$212$	0	0
$213$	−104459.	−0.157760
$214$	0	0
$215$	0	0
$216$	0	0
$217$	80916.7	0.116651
$218$	0	0
$219$	−639923.	−0.901607
$220$	0	0
$221$	198054.	0.272774
$222$	0	0
$223$	627285.	0.844700	0.422350	−	0.906433i	$-0.361205\pi$
$223$	627285.	0.844700	0.422350	+	0.906433i	$0.361205\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−450131.	−0.579794	−0.289897	−	0.957058i	$-0.593621\pi$
$227$	−450131.	−0.579794	−0.289897	+	0.957058i	$0.593621\pi$
$228$	0	0
$229$	−461485.	−0.581526	−0.290763	−	0.956795i	$-0.593909\pi$
$229$	−461485.	−0.581526	−0.290763	+	0.956795i	$0.593909\pi$
$230$	0	0
$231$	70448.8	0.0868648
$232$	0	0
$233$	−675812.	−0.815523	−0.407761	−	0.913089i	$-0.633690\pi$
$233$	−675812.	−0.815523	−0.407761	+	0.913089i	$0.633690\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.17071e6	1.35387
$238$	0	0
$239$	1.00730e6	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$239$	1.00730e6	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$240$	0	0
$241$	−232091.	−0.257404	−0.128702	−	0.991683i	$-0.541081\pi$
$241$	−232091.	−0.257404	−0.128702	+	0.991683i	$0.541081\pi$
$242$	0	0
$243$	757099.	0.822502
$244$	0	0
$245$	0	0
$246$	0	0
$247$	683002.	0.712327
$248$	0	0
$249$	−844035.	−0.862704
$250$	0	0
$251$	1.77096e6	1.77429	0.887143	−	0.461494i	$-0.152686\pi$
$251$	1.77096e6	1.77429	0.887143	+	0.461494i	$0.152686\pi$
$252$	0	0
$253$	123258.	0.121063
$254$	0	0
$255$	0	0
$256$	0	0
$257$	528458.	0.499088	0.249544	−	0.968363i	$-0.419719\pi$
$257$	528458.	0.499088	0.249544	+	0.968363i	$0.419719\pi$
$258$	0	0
$259$	34181.0	0.0316618
$260$	0	0
$261$	511030.	0.464350
$262$	0	0
$263$	1.69907e6	1.51469	0.757344	−	0.653016i	$-0.226497\pi$
$263$	1.69907e6	1.51469	0.757344	+	0.653016i	$0.226497\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−489760.	−0.420441
$268$	0	0
$269$	−648393.	−0.546333	−0.273167	−	0.961967i	$-0.588071\pi$
$269$	−648393.	−0.546333	−0.273167	+	0.961967i	$0.588071\pi$
$270$	0	0
$271$	1.94947e6	1.61248	0.806239	−	0.591590i	$-0.201500\pi$
$271$	1.94947e6	1.61248	0.806239	+	0.591590i	$0.201500\pi$
$272$	0	0
$273$	−159364.	−0.129415
$274$	0	0
$275$	0	0
$276$	0	0
$277$	313195.	0.245254	0.122627	−	0.992453i	$-0.460868\pi$
$277$	313195.	0.245254	0.122627	+	0.992453i	$0.460868\pi$
$278$	0	0
$279$	329650.	0.253538
$280$	0	0
$281$	1.72743e6	1.30507	0.652537	−	0.757757i	$-0.273705\pi$
$281$	1.72743e6	1.30507	0.652537	+	0.757757i	$0.273705\pi$
$282$	0	0
$283$	205142.	0.152261	0.0761306	−	0.997098i	$-0.475743\pi$
$283$	205142.	0.152261	0.0761306	+	0.997098i	$0.475743\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	465167.	0.333353
$288$	0	0
$289$	−1.26979e6	−0.894307
$290$	0	0
$291$	121763.	0.0842912
$292$	0	0
$293$	390309.	0.265607	0.132804	−	0.991142i	$-0.457602\pi$
$293$	390309.	0.265607	0.132804	+	0.991142i	$0.457602\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	919911.	0.605138
$298$	0	0
$299$	−278824.	−0.180365
$300$	0	0
$301$	−448787.	−0.285511
$302$	0	0
$303$	535713.	0.335217
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.10708e6	−0.670401	−0.335201	−	0.942147i	$-0.608804\pi$
$307$	−1.10708e6	−0.670401	−0.335201	+	0.942147i	$0.608804\pi$
$308$	0	0
$309$	−1.37744e6	−0.820684
$310$	0	0
$311$	−2.19106e6	−1.28455	−0.642277	−	0.766472i	$-0.722010\pi$
$311$	−2.19106e6	−1.28455	−0.642277	+	0.766472i	$0.722010\pi$
$312$	0	0
$313$	2.89050e6	1.66768	0.833840	−	0.552006i	$-0.186138\pi$
$313$	2.89050e6	1.66768	0.833840	+	0.552006i	$0.186138\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.46652e6	0.819671	0.409835	−	0.912160i	$-0.365586\pi$
$317$	1.46652e6	0.819671	0.409835	+	0.912160i	$0.365586\pi$
$318$	0	0
$319$	1.04813e6	0.576684
$320$	0	0
$321$	264161.	0.143089
$322$	0	0
$323$	517520.	0.276008
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.42147e6	1.25230
$328$	0	0
$329$	352602.	0.179595
$330$	0	0
$331$	2.23500e6	1.12126	0.560632	−	0.828065i	$-0.310558\pi$
$331$	2.23500e6	1.12126	0.560632	+	0.828065i	$0.310558\pi$
$332$	0	0
$333$	139251.	0.0688160
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.48851e6	−0.713966	−0.356983	−	0.934111i	$-0.616195\pi$
$337$	−1.48851e6	−0.713966	−0.356983	+	0.934111i	$0.616195\pi$
$338$	0	0
$339$	−2.34962e6	−1.11045
$340$	0	0
$341$	676115.	0.314872
$342$	0	0
$343$	889414.	0.408196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.34245e6	−1.04435	−0.522177	−	0.852837i	$-0.674880\pi$
$347$	−2.34245e6	−1.04435	−0.522177	+	0.852837i	$0.674880\pi$
$348$	0	0
$349$	−119901.	−0.0526940	−0.0263470	−	0.999653i	$-0.508387\pi$
$349$	−119901.	−0.0526940	−0.0263470	+	0.999653i	$0.508387\pi$
$350$	0	0
$351$	−2.08095e6	−0.901559
$352$	0	0
$353$	2.49528e6	1.06582	0.532909	−	0.846173i	$-0.321099\pi$
$353$	2.49528e6	1.06582	0.532909	+	0.846173i	$0.321099\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−120752.	−0.0501447
$358$	0	0
$359$	−3.43398e6	−1.40625	−0.703124	−	0.711068i	$-0.748212\pi$
$359$	−3.43398e6	−1.40625	−0.703124	+	0.711068i	$0.748212\pi$
$360$	0	0
$361$	−691400.	−0.279230
$362$	0	0
$363$	−1.26733e6	−0.504805
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.67310e6	1.81109	0.905545	−	0.424249i	$-0.139462\pi$
$367$	4.67310e6	1.81109	0.905545	+	0.424249i	$0.139462\pi$
$368$	0	0
$369$	1.89507e6	0.724533
$370$	0	0
$371$	784259.	0.295818
$372$	0	0
$373$	2.03766e6	0.758333	0.379167	−	0.925328i	$-0.376211\pi$
$373$	2.03766e6	0.758333	0.379167	+	0.925328i	$0.376211\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.37099e6	−0.859166
$378$	0	0
$379$	−1.78978e6	−0.640032	−0.320016	−	0.947412i	$-0.603688\pi$
$379$	−1.78978e6	−0.640032	−0.320016	+	0.947412i	$0.603688\pi$
$380$	0	0
$381$	2.01698e6	0.711853
$382$	0	0
$383$	2.04521e6	0.712429	0.356214	−	0.934404i	$-0.384067\pi$
$383$	2.04521e6	0.712429	0.356214	+	0.934404i	$0.384067\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.82833e6	−0.620551
$388$	0	0
$389$	234191.	0.0784687	0.0392344	−	0.999230i	$-0.487508\pi$
$389$	234191.	0.0784687	0.0392344	+	0.999230i	$0.487508\pi$
$390$	0	0
$391$	−211269.	−0.0698865
$392$	0	0
$393$	−1.71961e6	−0.561629
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.81706e6	−0.897058	−0.448529	−	0.893768i	$-0.648052\pi$
$397$	−2.81706e6	−0.897058	−0.448529	+	0.893768i	$0.648052\pi$
$398$	0	0
$399$	−416421.	−0.130949
$400$	0	0
$401$	735044.	0.228272	0.114136	−	0.993465i	$-0.463590\pi$
$401$	735044.	0.228272	0.114136	+	0.993465i	$0.463590\pi$
$402$	0	0
$403$	−1.52946e6	−0.469109
$404$	0	0
$405$	0	0
$406$	0	0
$407$	285606.	0.0854636
$408$	0	0
$409$	6.08894e6	1.79984	0.899918	−	0.436058i	$-0.143626\pi$
$409$	6.08894e6	1.79984	0.899918	+	0.436058i	$0.143626\pi$
$410$	0	0
$411$	−2.44520e6	−0.714020
$412$	0	0
$413$	−931273.	−0.268659
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−888312.	−0.250164
$418$	0	0
$419$	−5.65964e6	−1.57490	−0.787452	−	0.616376i	$-0.788600\pi$
$419$	−5.65964e6	−1.57490	−0.787452	+	0.616376i	$0.788600\pi$
$420$	0	0
$421$	−6.34902e6	−1.74583	−0.872914	−	0.487873i	$-0.837773\pi$
$421$	−6.34902e6	−1.74583	−0.872914	+	0.487873i	$0.837773\pi$
$422$	0	0
$423$	1.43648e6	0.390345
$424$	0	0
$425$	0	0
$426$	0	0
$427$	653192.	0.173369
$428$	0	0
$429$	−1.33160e6	−0.349325
$430$	0	0
$431$	−3.03516e6	−0.787024	−0.393512	−	0.919320i	$-0.628740\pi$
$431$	−3.03516e6	−0.787024	−0.393512	+	0.919320i	$0.628740\pi$
$432$	0	0
$433$	3.50633e6	0.898739	0.449369	−	0.893346i	$-0.351649\pi$
$433$	3.50633e6	0.898739	0.449369	+	0.893346i	$0.351649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−728573.	−0.182503
$438$	0	0
$439$	6.08849e6	1.50782	0.753908	−	0.656980i	$-0.228166\pi$
$439$	6.08849e6	1.50782	0.753908	+	0.656980i	$0.228166\pi$
$440$	0	0
$441$	1.77140e6	0.433731
$442$	0	0
$443$	−6.03357e6	−1.46071	−0.730357	−	0.683066i	$-0.760646\pi$
$443$	−6.03357e6	−1.46071	−0.730357	+	0.683066i	$0.760646\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.41797e6	1.28253
$448$	0	0
$449$	3.97943e6	0.931548	0.465774	−	0.884904i	$-0.345776\pi$
$449$	3.97943e6	0.931548	0.465774	+	0.884904i	$0.345776\pi$
$450$	0	0
$451$	3.88680e6	0.899809
$452$	0	0
$453$	−3.58538e6	−0.820899
$454$	0	0
$455$	0	0
$456$	0	0
$457$	430533.	0.0964308	0.0482154	−	0.998837i	$-0.484647\pi$
$457$	430533.	0.0964308	0.0482154	+	0.998837i	$0.484647\pi$
$458$	0	0
$459$	−1.57677e6	−0.349330
$460$	0	0
$461$	3.80383e6	0.833622	0.416811	−	0.908993i	$-0.363148\pi$
$461$	3.80383e6	0.833622	0.416811	+	0.908993i	$0.363148\pi$
$462$	0	0
$463$	−7.62853e6	−1.65382	−0.826911	−	0.562333i	$-0.809904\pi$
$463$	−7.62853e6	−1.65382	−0.826911	+	0.562333i	$0.809904\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.05102e6	−1.92046	−0.960230	−	0.279209i	$-0.909928\pi$
$467$	−9.05102e6	−1.92046	−0.960230	+	0.279209i	$0.909928\pi$
$468$	0	0
$469$	794943.	0.166880
$470$	0	0
$471$	−6.26511e6	−1.30130
$472$	0	0
$473$	−3.74992e6	−0.770672
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.19502e6	0.642952
$478$	0	0
$479$	8.51621e6	1.69593	0.847964	−	0.530054i	$-0.177828\pi$
$479$	8.51621e6	1.69593	0.847964	+	0.530054i	$0.177828\pi$
$480$	0	0
$481$	−646076.	−0.127327
$482$	0	0
$483$	169997.	0.0331569
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.65770e6	0.507789	0.253895	−	0.967232i	$-0.418288\pi$
$487$	2.65770e6	0.507789	0.253895	+	0.967232i	$0.418288\pi$
$488$	0	0
$489$	−3.44196e6	−0.650929
$490$	0	0
$491$	5.05116e6	0.945557	0.472779	−	0.881181i	$-0.343251\pi$
$491$	5.05116e6	0.945557	0.472779	+	0.881181i	$0.343251\pi$
$492$	0	0
$493$	−1.79654e6	−0.332904
$494$	0	0
$495$	0	0
$496$	0	0
$497$	245175.	0.0445230
$498$	0	0
$499$	2.15003e6	0.386538	0.193269	−	0.981146i	$-0.438091\pi$
$499$	2.15003e6	0.386538	0.193269	+	0.981146i	$0.438091\pi$
$500$	0	0
$501$	754679.	0.134328
$502$	0	0
$503$	1.79475e6	0.316289	0.158144	−	0.987416i	$-0.449449\pi$
$503$	1.79475e6	0.316289	0.158144	+	0.987416i	$0.449449\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.26661e6	−0.218839
$508$	0	0
$509$	697940.	0.119405	0.0597027	−	0.998216i	$-0.480985\pi$
$509$	697940.	0.119405	0.0597027	+	0.998216i	$0.480985\pi$
$510$	0	0
$511$	1.50196e6	0.254452
$512$	0	0
$513$	−5.43757e6	−0.912245
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.94624e6	0.484776
$518$	0	0
$519$	4.09214e6	0.666857
$520$	0	0
$521$	−1.14928e6	−0.185495	−0.0927475	−	0.995690i	$-0.529565\pi$
$521$	−1.14928e6	−0.185495	−0.0927475	+	0.995690i	$0.529565\pi$
$522$	0	0
$523$	−7.22071e6	−1.15432	−0.577159	−	0.816632i	$-0.695839\pi$
$523$	−7.22071e6	−1.15432	−0.577159	+	0.816632i	$0.695839\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.15889e6	−0.181767
$528$	0	0
$529$	−6.13892e6	−0.953789
$530$	0	0
$531$	−3.79395e6	−0.583923
$532$	0	0
$533$	−8.79241e6	−1.34057
$534$	0	0
$535$	0	0
$536$	0	0
$537$	452505.	0.0677155
$538$	0	0
$539$	3.63316e6	0.538657
$540$	0	0
$541$	−1.24720e7	−1.83207	−0.916035	−	0.401099i	$-0.868628\pi$
$541$	−1.24720e7	−1.83207	−0.916035	+	0.401099i	$0.868628\pi$
$542$	0	0
$543$	1.14825e6	0.167123
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.58966e6	−1.22746	−0.613730	−	0.789516i	$-0.710332\pi$
$547$	−8.58966e6	−1.22746	−0.613730	+	0.789516i	$0.710332\pi$
$548$	0	0
$549$	2.66107e6	0.376812
$550$	0	0
$551$	−6.19546e6	−0.869350
$552$	0	0
$553$	−2.74776e6	−0.382090
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.38207e6	−0.461896	−0.230948	−	0.972966i	$-0.574183\pi$
$557$	−3.38207e6	−0.461896	−0.230948	+	0.972966i	$0.574183\pi$
$558$	0	0
$559$	8.48278e6	1.14818
$560$	0	0
$561$	−1.00897e6	−0.135354
$562$	0	0
$563$	1.50247e6	0.199772	0.0998859	−	0.994999i	$-0.468152\pi$
$563$	1.50247e6	0.199772	0.0998859	+	0.994999i	$0.468152\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	544468.	0.0711237
$568$	0	0
$569$	1.87838e6	0.243222	0.121611	−	0.992578i	$-0.461194\pi$
$569$	1.87838e6	0.243222	0.121611	+	0.992578i	$0.461194\pi$
$570$	0	0
$571$	1.49628e7	1.92053	0.960267	−	0.279082i	$-0.0900301\pi$
$571$	1.49628e7	1.92053	0.960267	+	0.279082i	$0.0900301\pi$
$572$	0	0
$573$	−4.38306e6	−0.557687
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.03660e6	−0.504750	−0.252375	−	0.967630i	$-0.581212\pi$
$577$	−4.03660e6	−0.504750	−0.252375	+	0.967630i	$0.581212\pi$
$578$	0	0
$579$	1.45666e6	0.180577
$580$	0	0
$581$	1.98103e6	0.243473
$582$	0	0
$583$	6.55303e6	0.798492
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.57983e7	−1.89241	−0.946203	−	0.323574i	$-0.895115\pi$
$587$	−1.57983e7	−1.89241	−0.946203	+	0.323574i	$0.895115\pi$
$588$	0	0
$589$	−3.99650e6	−0.474670
$590$	0	0
$591$	−3.75845e6	−0.442629
$592$	0	0
$593$	−1.37947e7	−1.61092	−0.805462	−	0.592647i	$-0.798083\pi$
$593$	−1.37947e7	−1.61092	−0.805462	+	0.592647i	$0.798083\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.74549e6	0.659768
$598$	0	0
$599$	814106.	0.0927073	0.0463537	−	0.998925i	$-0.485240\pi$
$599$	814106.	0.0927073	0.0463537	+	0.998925i	$0.485240\pi$
$600$	0	0
$601$	−1.55613e7	−1.75736	−0.878679	−	0.477413i	$-0.841574\pi$
$601$	−1.55613e7	−1.75736	−0.878679	+	0.477413i	$0.841574\pi$
$602$	0	0
$603$	3.23855e6	0.362708
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.63191e7	1.79773	0.898865	−	0.438226i	$-0.144393\pi$
$607$	1.63191e7	1.79773	0.898865	+	0.438226i	$0.144393\pi$
$608$	0	0
$609$	1.44558e6	0.157942
$610$	0	0
$611$	−6.66475e6	−0.722239
$612$	0	0
$613$	5.38233e6	0.578520	0.289260	−	0.957250i	$-0.406591\pi$
$613$	5.38233e6	0.578520	0.289260	+	0.957250i	$0.406591\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.83565e6	0.617130	0.308565	−	0.951203i	$-0.400151\pi$
$617$	5.83565e6	0.617130	0.308565	+	0.951203i	$0.400151\pi$
$618$	0	0
$619$	3.36289e6	0.352765	0.176383	−	0.984322i	$-0.443560\pi$
$619$	3.36289e6	0.352765	0.176383	+	0.984322i	$0.443560\pi$
$620$	0	0
$621$	2.21980e6	0.230985
$622$	0	0
$623$	1.14951e6	0.118657
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.47949e6	−0.353465
$628$	0	0
$629$	−489541.	−0.0493358
$630$	0	0
$631$	−8.10425e6	−0.810288	−0.405144	−	0.914253i	$-0.632779\pi$
$631$	−8.10425e6	−0.810288	−0.405144	+	0.914253i	$0.632779\pi$
$632$	0	0
$633$	3.58128e6	0.355246
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.21866e6	−0.802513
$638$	0	0
$639$	998828.	0.0967695
$640$	0	0
$641$	7.08523e6	0.681097	0.340548	−	0.940227i	$-0.389387\pi$
$641$	7.08523e6	0.681097	0.340548	+	0.940227i	$0.389387\pi$
$642$	0	0
$643$	−5.72946e6	−0.546495	−0.273247	−	0.961944i	$-0.588098\pi$
$643$	−5.72946e6	−0.546495	−0.273247	+	0.961944i	$0.588098\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.17751e6	−0.204503	−0.102251	−	0.994759i	$-0.532605\pi$
$647$	−2.17751e6	−0.204503	−0.102251	+	0.994759i	$0.532605\pi$
$648$	0	0
$649$	−7.78143e6	−0.725183
$650$	0	0
$651$	932498.	0.0862374
$652$	0	0
$653$	1.42048e7	1.30363	0.651813	−	0.758379i	$-0.274009\pi$
$653$	1.42048e7	1.30363	0.651813	+	0.758379i	$0.274009\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.11890e6	0.553044
$658$	0	0
$659$	−1.54033e7	−1.38166	−0.690829	−	0.723018i	$-0.742754\pi$
$659$	−1.54033e7	−1.38166	−0.690829	+	0.723018i	$0.742754\pi$
$660$	0	0
$661$	−544048.	−0.0484322	−0.0242161	−	0.999707i	$-0.507709\pi$
$661$	−544048.	−0.0484322	−0.0242161	+	0.999707i	$0.507709\pi$
$662$	0	0
$663$	2.28241e6	0.201656
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.52919e6	0.220124
$668$	0	0
$669$	7.22894e6	0.624467
$670$	0	0
$671$	5.45787e6	0.467969
$672$	0	0
$673$	1.37093e7	1.16675	0.583374	−	0.812203i	$-0.301732\pi$
$673$	1.37093e7	1.16675	0.583374	+	0.812203i	$0.301732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.49910e7	−1.25707	−0.628536	−	0.777781i	$-0.716345\pi$
$677$	−1.49910e7	−1.25707	−0.628536	+	0.777781i	$0.716345\pi$
$678$	0	0
$679$	−285789.	−0.0237887
$680$	0	0
$681$	−5.18738e6	−0.428628
$682$	0	0
$683$	−3.47996e6	−0.285445	−0.142723	−	0.989763i	$-0.545586\pi$
$683$	−3.47996e6	−0.285445	−0.142723	+	0.989763i	$0.545586\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.31824e6	−0.429908
$688$	0	0
$689$	−1.48237e7	−1.18962
$690$	0	0
$691$	−5.09436e6	−0.405877	−0.202939	−	0.979191i	$-0.565049\pi$
$691$	−5.09436e6	−0.405877	−0.202939	+	0.979191i	$0.565049\pi$
$692$	0	0
$693$	−673627.	−0.0532827
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.66213e6	−0.519435
$698$	0	0
$699$	−7.78817e6	−0.602896
$700$	0	0
$701$	1.75839e7	1.35151	0.675755	−	0.737126i	$-0.263818\pi$
$701$	1.75839e7	1.35151	0.675755	+	0.737126i	$0.263818\pi$
$702$	0	0
$703$	−1.68821e6	−0.128836
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.25737e6	−0.0946051
$708$	0	0
$709$	1.35383e7	1.01146	0.505730	−	0.862692i	$-0.331223\pi$
$709$	1.35383e7	1.01146	0.505730	+	0.862692i	$0.331223\pi$
$710$	0	0
$711$	−1.11942e7	−0.830462
$712$	0	0
$713$	1.63150e6	0.120189
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.16083e7	0.843275
$718$	0	0
$719$	−1.27911e7	−0.922756	−0.461378	−	0.887204i	$-0.652645\pi$
$719$	−1.27911e7	−0.922756	−0.461378	+	0.887204i	$0.652645\pi$
$720$	0	0
$721$	3.23298e6	0.231614
$722$	0	0
$723$	−2.67466e6	−0.190293
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.11509e7	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$727$	2.11509e7	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$728$	0	0
$729$	1.36164e7	0.948949
$730$	0	0
$731$	6.42753e6	0.444888
$732$	0	0
$733$	1.23015e7	0.845667	0.422833	−	0.906207i	$-0.361036\pi$
$733$	1.23015e7	0.845667	0.422833	+	0.906207i	$0.361036\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.64230e6	0.450453
$738$	0	0
$739$	−1.04425e7	−0.703383	−0.351691	−	0.936116i	$-0.614393\pi$
$739$	−1.04425e7	−0.703383	−0.351691	+	0.936116i	$0.614393\pi$
$740$	0	0
$741$	7.87103e6	0.526606
$742$	0	0
$743$	3.63296e6	0.241429	0.120714	−	0.992687i	$-0.461481\pi$
$743$	3.63296e6	0.241429	0.120714	+	0.992687i	$0.461481\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.07060e6	0.529181
$748$	0	0
$749$	−620012.	−0.0403827
$750$	0	0
$751$	1.77346e7	1.14741	0.573707	−	0.819060i	$-0.305505\pi$
$751$	1.77346e7	1.14741	0.573707	+	0.819060i	$0.305505\pi$
$752$	0	0
$753$	2.04088e7	1.31169
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.89094e6	−0.500483	−0.250241	−	0.968183i	$-0.580510\pi$
$757$	−7.89094e6	−0.500483	−0.250241	+	0.968183i	$0.580510\pi$
$758$	0	0
$759$	1.42044e6	0.0894992
$760$	0	0
$761$	1.98455e7	1.24223	0.621114	−	0.783721i	$-0.286681\pi$
$761$	1.98455e7	1.24223	0.621114	+	0.783721i	$0.286681\pi$
$762$	0	0
$763$	−5.68343e6	−0.353426
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.76026e7	1.08041
$768$	0	0
$769$	3.89976e6	0.237806	0.118903	−	0.992906i	$-0.462062\pi$
$769$	3.89976e6	0.237806	0.118903	+	0.992906i	$0.462062\pi$
$770$	0	0
$771$	6.09004e6	0.368964
$772$	0	0
$773$	−2.33736e6	−0.140695	−0.0703474	−	0.997523i	$-0.522411\pi$
$773$	−2.33736e6	−0.140695	−0.0703474	+	0.997523i	$0.522411\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	393908.	0.0234068
$778$	0	0
$779$	−2.29748e7	−1.35646
$780$	0	0
$781$	2.04860e6	0.120180
$782$	0	0
$783$	1.88762e7	1.10030
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.52903e7	0.879993	0.439997	−	0.897999i	$-0.354980\pi$
$787$	1.52903e7	0.879993	0.439997	+	0.897999i	$0.354980\pi$
$788$	0	0
$789$	1.95804e7	1.11977
$790$	0	0
$791$	5.51479e6	0.313392
$792$	0	0
$793$	−1.23464e7	−0.697199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.25972e7	−0.702468	−0.351234	−	0.936288i	$-0.614238\pi$
$797$	−1.25972e7	−0.702468	−0.351234	+	0.936288i	$0.614238\pi$
$798$	0	0
$799$	−5.04997e6	−0.279848
$800$	0	0
$801$	4.68305e6	0.257898
$802$	0	0
$803$	1.25499e7	0.686835
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.47219e6	−0.403891
$808$	0	0
$809$	−1.13476e7	−0.609584	−0.304792	−	0.952419i	$-0.598587\pi$
$809$	−1.13476e7	−0.609584	−0.304792	+	0.952419i	$0.598587\pi$
$810$	0	0
$811$	1.91000e7	1.01972	0.509860	−	0.860257i	$-0.329697\pi$
$811$	1.91000e7	1.01972	0.509860	+	0.860257i	$0.329697\pi$
$812$	0	0
$813$	2.24661e7	1.19207
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.21657e7	1.16179
$818$	0	0
$819$	1.52383e6	0.0793827
$820$	0	0
$821$	1.25668e6	0.0650678	0.0325339	−	0.999471i	$-0.489642\pi$
$821$	1.25668e6	0.0650678	0.0325339	+	0.999471i	$0.489642\pi$
$822$	0	0
$823$	6.95590e6	0.357976	0.178988	−	0.983851i	$-0.442718\pi$
$823$	6.95590e6	0.357976	0.178988	+	0.983851i	$0.442718\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.04638e6	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$827$	9.04638e6	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$828$	0	0
$829$	−1.54026e7	−0.778407	−0.389203	−	0.921152i	$-0.627250\pi$
$829$	−1.54026e7	−0.778407	−0.389203	+	0.921152i	$0.627250\pi$
$830$	0	0
$831$	3.60932e6	0.181310
$832$	0	0
$833$	−6.22739e6	−0.310952
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.21764e7	0.600767
$838$	0	0
$839$	−1.75941e7	−0.862903	−0.431451	−	0.902136i	$-0.641998\pi$
$839$	−1.75941e7	−0.862903	−0.431451	+	0.902136i	$0.641998\pi$
$840$	0	0
$841$	995979.	0.0485580
$842$	0	0
$843$	1.99072e7	0.964809
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.97455e6	0.142466
$848$	0	0
$849$	2.36410e6	0.112563
$850$	0	0
$851$	689183.	0.0326220
$852$	0	0
$853$	2.06755e7	0.972934	0.486467	−	0.873699i	$-0.338285\pi$
$853$	2.06755e7	0.972934	0.486467	+	0.873699i	$0.338285\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.16205e7	−1.00557	−0.502786	−	0.864411i	$-0.667692\pi$
$857$	−2.16205e7	−1.00557	−0.502786	+	0.864411i	$0.667692\pi$
$858$	0	0
$859$	1.70061e7	0.786362	0.393181	−	0.919461i	$-0.371375\pi$
$859$	1.70061e7	0.786362	0.393181	+	0.919461i	$0.371375\pi$
$860$	0	0
$861$	5.36067e6	0.246440
$862$	0	0
$863$	−2.96043e7	−1.35309	−0.676547	−	0.736399i	$-0.736525\pi$
$863$	−2.96043e7	−1.35309	−0.676547	+	0.736399i	$0.736525\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.46333e7	−0.661140
$868$	0	0
$869$	−2.29594e7	−1.03136
$870$	0	0
$871$	−1.50257e7	−0.671103
$872$	0	0
$873$	−1.16429e6	−0.0517041
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.77691e7	1.21917	0.609583	−	0.792722i	$-0.291337\pi$
$877$	2.77691e7	1.21917	0.609583	+	0.792722i	$0.291337\pi$
$878$	0	0
$879$	4.49799e6	0.196357
$880$	0	0
$881$	1.16824e7	0.507099	0.253550	−	0.967322i	$-0.418402\pi$
$881$	1.16824e7	0.507099	0.253550	+	0.967322i	$0.418402\pi$
$882$	0	0
$883$	2.66919e7	1.15207	0.576034	−	0.817426i	$-0.304600\pi$
$883$	2.66919e7	1.15207	0.576034	+	0.817426i	$0.304600\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.95714e7	−1.68878	−0.844388	−	0.535733i	$-0.820036\pi$
$887$	−3.95714e7	−1.68878	−0.844388	+	0.535733i	$0.820036\pi$
$888$	0	0
$889$	−4.73406e6	−0.200900
$890$	0	0
$891$	4.54940e6	0.191982
$892$	0	0
$893$	−1.74151e7	−0.730799
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.21321e6	−0.133339
$898$	0	0
$899$	1.38736e7	0.572518
$900$	0	0
$901$	−1.12322e7	−0.460948
$902$	0	0
$903$	−5.17189e6	−0.211072
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.79635e7	1.12869	0.564343	−	0.825541i	$-0.309130\pi$
$907$	2.79635e7	1.12869	0.564343	+	0.825541i	$0.309130\pi$
$908$	0	0
$909$	−5.12245e6	−0.205621
$910$	0	0
$911$	1.87296e7	0.747710	0.373855	−	0.927487i	$-0.378036\pi$
$911$	1.87296e7	0.747710	0.373855	+	0.927487i	$0.378036\pi$
$912$	0	0
$913$	1.65529e7	0.657199
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.03610e6	0.158503
$918$	0	0
$919$	7.77613e6	0.303721	0.151861	−	0.988402i	$-0.451474\pi$
$919$	7.77613e6	0.303721	0.151861	+	0.988402i	$0.451474\pi$
$920$	0	0
$921$	−1.27582e7	−0.495612
$922$	0	0
$923$	−4.63420e6	−0.179048
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.31710e7	0.503406
$928$	0	0
$929$	−3.88269e7	−1.47602	−0.738011	−	0.674788i	$-0.764235\pi$
$929$	−3.88269e7	−1.47602	−0.738011	+	0.674788i	$0.764235\pi$
$930$	0	0
$931$	−2.14755e7	−0.812026
$932$	0	0
$933$	−2.52501e7	−0.949640
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.75302e7	0.652288	0.326144	−	0.945320i	$-0.394251\pi$
$937$	1.75302e7	0.652288	0.326144	+	0.945320i	$0.394251\pi$
$938$	0	0
$939$	3.33107e7	1.23288
$940$	0	0
$941$	−4.29638e6	−0.158172	−0.0790859	−	0.996868i	$-0.525200\pi$
$941$	−4.29638e6	−0.158172	−0.0790859	+	0.996868i	$0.525200\pi$
$942$	0	0
$943$	9.37905e6	0.343463
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.43637e7	−1.60751	−0.803754	−	0.594962i	$-0.797167\pi$
$947$	−4.43637e7	−1.60751	−0.803754	+	0.594962i	$0.797167\pi$
$948$	0	0
$949$	−2.83895e7	−1.02327
$950$	0	0
$951$	1.69004e7	0.605963
$952$	0	0
$953$	−5.71990e6	−0.204012	−0.102006	−	0.994784i	$-0.532526\pi$
$953$	−5.71990e6	−0.204012	−0.102006	+	0.994784i	$0.532526\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.20788e7	0.426329
$958$	0	0
$959$	5.73912e6	0.201511
$960$	0	0
$961$	−1.96797e7	−0.687402
$962$	0	0
$963$	−2.52589e6	−0.0877706
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.00548e7	−1.03359	−0.516794	−	0.856110i	$-0.672875\pi$
$967$	−3.00548e7	−1.03359	−0.516794	+	0.856110i	$0.672875\pi$
$968$	0	0
$969$	5.96399e6	0.204046
$970$	0	0
$971$	−1.89662e7	−0.645552	−0.322776	−	0.946475i	$-0.604616\pi$
$971$	−1.89662e7	−0.645552	−0.322776	+	0.946475i	$0.604616\pi$
$972$	0	0
$973$	2.08495e6	0.0706015
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.45541e7	−0.822975	−0.411488	−	0.911415i	$-0.634991\pi$
$977$	−2.45541e7	−0.822975	−0.411488	+	0.911415i	$0.634991\pi$
$978$	0	0
$979$	9.60498e6	0.320287
$980$	0	0
$981$	−2.31540e7	−0.768162
$982$	0	0
$983$	3.35539e7	1.10754	0.553769	−	0.832670i	$-0.313189\pi$
$983$	3.35539e7	1.10754	0.553769	+	0.832670i	$0.313189\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.06345e6	0.132771
$988$	0	0
$989$	−9.04877e6	−0.294170
$990$	0	0
$991$	−2.05230e7	−0.663830	−0.331915	−	0.943309i	$-0.607695\pi$
$991$	−2.05230e7	−0.663830	−0.331915	+	0.943309i	$0.607695\pi$
$992$	0	0
$993$	2.57565e7	0.828924
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.25729e7	−0.400589	−0.200294	−	0.979736i	$-0.564190\pi$
$997$	−1.25729e7	−0.400589	−0.200294	+	0.979736i	$0.564190\pi$
$998$	0	0
$999$	5.14359e6	0.163062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.r.1.2		2
4.3	odd	2		200.6.a.f.1.1	yes	2
5.2	odd	4		400.6.c.o.49.2		4
5.3	odd	4		400.6.c.o.49.3		4
5.4	even	2		400.6.a.u.1.1		2
20.3	even	4		200.6.c.f.49.2		4
20.7	even	4		200.6.c.f.49.3		4
20.19	odd	2		200.6.a.e.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.a.e.1.2	✓	2	20.19	odd	2
200.6.a.f.1.1	yes	2	4.3	odd	2
200.6.c.f.49.2		4	20.3	even	4
200.6.c.f.49.3		4	20.7	even	4
400.6.a.r.1.2		2	1.1	even	1	trivial
400.6.a.u.1.1		2	5.4	even	2
400.6.c.o.49.2		4	5.2	odd	4
400.6.c.o.49.3		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+11.5242 q^{3} -27.0483 q^{7} -110.193 q^{9} -226.008 q^{11} +511.257 q^{13} +387.387 q^{17} +1335.93 q^{19} -311.710 q^{21} -545.369 q^{23} -4070.26 q^{27} -4637.58 q^{29} -2991.56 q^{31} -2604.55 q^{33} -1263.70 q^{37} +5891.82 q^{39} -17197.6 q^{41} +16592.0 q^{43} -13036.0 q^{47} -16075.4 q^{49} +4464.31 q^{51} -28994.7 q^{53} +15395.4 q^{57} +34429.9 q^{59} -24149.1 q^{61} +2980.55 q^{63} -29389.7 q^{67} -6284.93 q^{69} -9064.32 q^{71} -55528.7 q^{73} +6113.13 q^{77} +101587. q^{79} -20129.4 q^{81} -73240.4 q^{83} -53444.3 q^{87} -42498.5 q^{89} -13828.7 q^{91} -34475.2 q^{93} +10565.9 q^{97} +24904.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} + 8 q^{7} + 28 q^{9} + 200 q^{11} - 592 q^{13} + 278 q^{17} + 840 q^{19} - 996 q^{21} + 1952 q^{23} - 2024 q^{27} - 4680 q^{29} + 5008 q^{31} - 10922 q^{33} + 12500 q^{37} + 27432 q^{39} - 5334 q^{41}+ \cdots + 83776 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 + 8 * q^7 + 28 * q^9 + 200 * q^11 - 592 * q^13 + 278 * q^17 + 840 * q^19 - 996 * q^21 + 1952 * q^23 - 2024 * q^27 - 4680 * q^29 + 5008 * q^31 - 10922 * q^33 + 12500 * q^37 + 27432 * q^39 - 5334 * q^41 - 224 * q^43 - 26072 * q^47 - 31654 * q^49 + 6600 * q^51 - 46812 * q^53 + 25078 * q^57 + 81776 * q^59 - 46932 * q^61 + 7824 * q^63 - 68808 * q^67 - 55044 * q^69 - 7448 * q^71 - 108822 * q^73 + 21044 * q^77 + 108104 * q^79 - 93662 * q^81 - 27224 * q^83 - 52616 * q^87 + 70990 * q^89 - 52496 * q^91 - 190660 * q^93 - 96852 * q^97 + 83776 * q^99

Introduction

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