LMFDB - Embedded newform 200.6.c.f.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(49,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("200.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{241})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 121x^{2} + 3600 $ x^4 + 121*x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$8.26209i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.6.c.f.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.5242i q^{3} -27.0483i q^{7} +110.193 q^{9} +O(q^{10})$	$q-11.5242i q^{3} -27.0483i q^{7} +110.193 q^{9} +226.008 q^{11} +511.257i q^{13} -387.387i q^{17} +1335.93 q^{19} -311.710 q^{21} +545.369i q^{23} -4070.26i q^{27} +4637.58 q^{29} +2991.56 q^{31} -2604.55i q^{33} +1263.70i q^{37} +5891.82 q^{39} -17197.6 q^{41} -16592.0i q^{43} -13036.0i q^{47} +16075.4 q^{49} -4464.31 q^{51} -28994.7i q^{53} -15395.4i q^{57} +34429.9 q^{59} -24149.1 q^{61} -2980.55i q^{63} -29389.7i q^{67} +6284.93 q^{69} +9064.32 q^{71} -55528.7i q^{73} -6113.13i q^{77} +101587. q^{79} -20129.4 q^{81} +73240.4i q^{83} -53444.3i q^{87} +42498.5 q^{89} +13828.7 q^{91} -34475.2i q^{93} -10565.9i q^{97} +24904.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 56 q^{9}+O(q^{10}) $ 4 * q - 56 * q^9	$ 4 q - 56 q^{9} - 400 q^{11} + 1680 q^{19} - 1992 q^{21} + 9360 q^{29} - 10016 q^{31} + 54864 q^{39} - 10668 q^{41} + 63308 q^{49} - 13200 q^{51} + 163552 q^{59} - 93864 q^{61} + 110088 q^{69} + 14896 q^{71} + 216208 q^{79} - 187324 q^{81} - 141980 q^{89} + 104992 q^{91} + 167552 q^{99}+O(q^{100}) $ 4 * q - 56 * q^9 - 400 * q^11 + 1680 * q^19 - 1992 * q^21 + 9360 * q^29 - 10016 * q^31 + 54864 * q^39 - 10668 * q^41 + 63308 * q^49 - 13200 * q^51 + 163552 * q^59 - 93864 * q^61 + 110088 * q^69 + 14896 * q^71 + 216208 * q^79 - 187324 * q^81 - 141980 * q^89 + 104992 * q^91 + 167552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 11.5242i	− 0.739276i	−0.929176	−	0.369638i	$-0.879482\pi$
$3$	− 11.5242i	− 0.739276i	0.929176	−	0.369638i	$-0.120518\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 27.0483i	− 0.208639i	−0.994544	−	0.104320i	$-0.966734\pi$
$7$	− 27.0483i	− 0.208639i	0.994544	−	0.104320i	$-0.0332665\pi$
$8$	0	0
$9$	110.193	0.453471
$10$	0	0
$11$	226.008	0.563173	0.281586	−	0.959536i	$-0.409139\pi$
$11$	226.008	0.563173	0.281586	+	0.959536i	$0.409139\pi$
$12$	0	0
$13$	511.257i	0.839037i	0.907747	+	0.419518i	$0.137801\pi$
$13$	511.257i	0.839037i	−0.907747	+	0.419518i	$0.862199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 387.387i	− 0.325104i	−0.986700	−	0.162552i	$-0.948027\pi$
$17$	− 387.387i	− 0.325104i	0.986700	−	0.162552i	$-0.0519725\pi$
$18$	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
$23$	545.369i	0.214967i	0.994207	+	0.107483i	$0.0342792\pi$
$23$	545.369i	0.214967i	−0.994207	+	0.107483i	$0.965721\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4070.26i	− 1.07452i
$28$	0	0
$29$	4637.58	1.02399	0.511996	−	0.858988i	$-0.328906\pi$
$29$	4637.58	1.02399	0.511996	+	0.858988i	$0.328906\pi$
$30$	0	0
$31$	2991.56	0.559105	0.279552	−	0.960130i	$-0.409814\pi$
$31$	2991.56	0.559105	0.279552	+	0.960130i	$0.409814\pi$
$32$	0	0
$33$	− 2604.55i	− 0.416340i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1263.70i	0.151754i	0.997117	+	0.0758770i	$0.0241756\pi$
$37$	1263.70i	0.151754i	−0.997117	+	0.0758770i	$0.975824\pi$
$38$	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
$43$	− 16592.0i	− 1.36845i	−0.729272	−	0.684223i	$-0.760141\pi$
$43$	− 16592.0i	− 1.36845i	0.729272	−	0.684223i	$-0.239859\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13036.0i	− 0.860795i	−0.902639	−	0.430397i	$-0.858373\pi$
$47$	− 13036.0i	− 0.860795i	0.902639	−	0.430397i	$-0.141627\pi$
$48$	0	0
$49$	16075.4	0.956470
$50$	0	0
$51$	−4464.31	−0.240342
$52$	0	0
$53$	− 28994.7i	− 1.41785i	−0.705286	−	0.708923i	$-0.749181\pi$
$53$	− 28994.7i	− 1.41785i	0.705286	−	0.708923i	$-0.250819\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 15395.4i	− 0.627632i
$58$	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
$63$	− 2980.55i	− 0.0946117i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 29389.7i	− 0.799849i	−0.916548	−	0.399925i	$-0.869036\pi$
$67$	− 29389.7i	− 0.799849i	0.916548	−	0.399925i	$-0.130964\pi$
$68$	0	0
$69$	6284.93	0.158920
$70$	0	0
$71$	9064.32	0.213397	0.106699	−	0.994291i	$-0.465972\pi$
$71$	9064.32	0.213397	0.106699	+	0.994291i	$0.465972\pi$
$72$	0	0
$73$	− 55528.7i	− 1.21958i	−0.792563	−	0.609791i	$-0.791254\pi$
$73$	− 55528.7i	− 1.21958i	0.792563	−	0.609791i	$-0.208746\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6113.13i	− 0.117500i
$78$	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
$83$	73240.4i	1.16696i	0.812128	+	0.583479i	$0.198309\pi$
$83$	73240.4i	1.16696i	−0.812128	+	0.583479i	$0.801691\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 53444.3i	− 0.757012i
$88$	0	0
$89$	42498.5	0.568719	0.284360	−	0.958718i	$-0.408219\pi$
$89$	42498.5	0.568719	0.284360	+	0.958718i	$0.408219\pi$
$90$	0	0
$91$	13828.7	0.175056
$92$	0	0
$93$	− 34475.2i	− 0.413333i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10565.9i	− 0.114019i	−0.998374	−	0.0570093i	$-0.981844\pi$
$97$	− 10565.9i	− 0.114019i	0.998374	−	0.0570093i	$-0.0181565\pi$
$98$	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.i	1.11012i	0.831811	+	0.555059i	$0.187304\pi$
$103$	119526.i	1.11012i	−0.831811	+	0.555059i	$0.812696\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	22922.4i	0.193553i	0.995306	+	0.0967765i	$0.0308532\pi$
$107$	22922.4i	0.193553i	−0.995306	+	0.0967765i	$0.969147\pi$
$108$	0	0
$109$	−210121.	−1.69396	−0.846980	−	0.531624i	$-0.821582\pi$
$109$	−210121.	−1.69396	−0.846980	+	0.531624i	$0.821582\pi$
$110$	0	0
$111$	14563.1	0.112188
$112$	0	0
$113$	− 203886.i	− 1.50208i	−0.660258	−	0.751039i	$-0.729553\pi$
$113$	− 203886.i	− 1.50208i	0.660258	−	0.751039i	$-0.270447\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	56337.2i	0.380479i
$118$	0	0
$119$	−10478.2	−0.0678294
$120$	0	0
$121$	−109972.	−0.682837
$122$	0	0
$123$	198188.i	1.18118i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	175022.i	0.962905i	0.876472	+	0.481453i	$0.159891\pi$
$127$	175022.i	0.962905i	−0.876472	+	0.481453i	$0.840109\pi$
$128$	0	0
$129$	−191209.	−1.01166
$130$	0	0
$131$	149218.	0.759701	0.379850	−	0.925048i	$-0.375975\pi$
$131$	149218.	0.759701	0.379850	+	0.925048i	$0.375975\pi$
$132$	0	0
$133$	− 36134.6i	− 0.177131i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	212180.i	0.965836i	0.875666	+	0.482918i	$0.160423\pi$
$137$	212180.i	0.965836i	−0.875666	+	0.482918i	$0.839577\pi$
$138$	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.i	0.472522i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 185256.i	− 0.707095i
$148$	0	0
$149$	−470139.	−1.73485	−0.867423	−	0.497571i	$-0.834225\pi$
$149$	−470139.	−1.73485	−0.867423	+	0.497571i	$0.834225\pi$
$150$	0	0
$151$	311118.	1.11041	0.555205	−	0.831714i	$-0.312640\pi$
$151$	311118.	1.11041	0.555205	+	0.831714i	$0.312640\pi$
$152$	0	0
$153$	− 42687.5i	− 0.147425i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	543649.i	1.76023i	0.474760	+	0.880115i	$0.342535\pi$
$157$	543649.i	1.76023i	−0.474760	+	0.880115i	$0.657465\pi$
$158$	0	0
$159$	−334140.	−1.04818
$160$	0	0
$161$	14751.3	0.0448504
$162$	0	0
$163$	298673.i	0.880495i	0.897876	+	0.440248i	$0.145109\pi$
$163$	298673.i	0.880495i	−0.897876	+	0.440248i	$0.854891\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	65486.6i	0.181703i	0.995864	+	0.0908513i	$0.0289588\pi$
$167$	65486.6i	0.181703i	−0.995864	+	0.0908513i	$0.971041\pi$
$168$	0	0
$169$	109909.	0.296017
$170$	0	0
$171$	147210.	0.384989
$172$	0	0
$173$	355092.i	0.902040i	0.892514	+	0.451020i	$0.148940\pi$
$173$	355092.i	0.902040i	−0.892514	+	0.451020i	$0.851060\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 396777.i	− 0.951947i
$178$	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.i	0.614303i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 87552.4i	− 0.183090i
$188$	0	0
$189$	−110094.	−0.224186
$190$	0	0
$191$	380336.	0.754369	0.377185	−	0.926138i	$-0.376892\pi$
$191$	380336.	0.754369	0.377185	+	0.926138i	$0.376892\pi$
$192$	0	0
$193$	126401.i	0.244262i	0.992514	+	0.122131i	$0.0389728\pi$
$193$	126401.i	0.244262i	−0.992514	+	0.122131i	$0.961027\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	326136.i	0.598733i	0.954138	+	0.299366i	$0.0967753\pi$
$197$	326136.i	0.598733i	−0.954138	+	0.299366i	$0.903225\pi$
$198$	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.i	− 0.213645i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	60096.1i	0.0974811i
$208$	0	0
$209$	301930.	0.478123
$210$	0	0
$211$	−310763.	−0.480532	−0.240266	−	0.970707i	$-0.577235\pi$
$211$	−310763.	−0.480532	−0.240266	+	0.970707i	$0.577235\pi$
$212$	0	0
$213$	− 104459.i	− 0.157760i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 80916.7i	− 0.116651i
$218$	0	0
$219$	−639923.	−0.901607
$220$	0	0
$221$	198054.	0.272774
$222$	0	0
$223$	− 627285.i	− 0.844700i	−0.906433	−	0.422350i	$-0.861205\pi$
$223$	− 627285.i	− 0.844700i	0.906433	−	0.422350i	$-0.138795\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 450131.i	− 0.579794i	−0.957058	−	0.289897i	$-0.906379\pi$
$227$	− 450131.i	− 0.579794i	0.957058	−	0.289897i	$-0.0936211\pi$
$228$	0	0
$229$	461485.	0.581526	0.290763	−	0.956795i	$-0.406091\pi$
$229$	461485.	0.581526	0.290763	+	0.956795i	$0.406091\pi$
$230$	0	0
$231$	−70448.8	−0.0868648
$232$	0	0
$233$	− 675812.i	− 0.815523i	−0.913089	−	0.407761i	$-0.866310\pi$
$233$	− 675812.i	− 0.815523i	0.913089	−	0.407761i	$-0.133690\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.17071e6i	− 1.35387i
$238$	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.i	− 0.822502i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	683002.i	0.712327i
$248$	0	0
$249$	844035.	0.862704
$250$	0	0
$251$	−1.77096e6	−1.77429	−0.887143	−	0.461494i	$-0.847314\pi$
$251$	−1.77096e6	−1.77429	−0.887143	+	0.461494i	$0.847314\pi$
$252$	0	0
$253$	123258.i	0.121063i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 528458.i	− 0.499088i	−0.968363	−	0.249544i	$-0.919719\pi$
$257$	− 528458.i	− 0.499088i	0.968363	−	0.249544i	$-0.0802808\pi$
$258$	0	0
$259$	34181.0	0.0316618
$260$	0	0
$261$	511030.	0.464350
$262$	0	0
$263$	− 1.69907e6i	− 1.51469i	−0.653016	−	0.757344i	$-0.726497\pi$
$263$	− 1.69907e6i	− 1.51469i	0.653016	−	0.757344i	$-0.273503\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 489760.i	− 0.420441i
$268$	0	0
$269$	648393.	0.546333	0.273167	−	0.961967i	$-0.411929\pi$
$269$	648393.	0.546333	0.273167	+	0.961967i	$0.411929\pi$
$270$	0	0
$271$	−1.94947e6	−1.61248	−0.806239	−	0.591590i	$-0.798500\pi$
$271$	−1.94947e6	−1.61248	−0.806239	+	0.591590i	$0.798500\pi$
$272$	0	0
$273$	− 159364.i	− 0.129415i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 313195.i	− 0.245254i	−0.992453	−	0.122627i	$-0.960868\pi$
$277$	− 313195.i	− 0.245254i	0.992453	−	0.122627i	$-0.0391319\pi$
$278$	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.i	− 0.152261i	−0.997098	−	0.0761306i	$-0.975743\pi$
$283$	− 205142.i	− 0.152261i	0.997098	−	0.0761306i	$-0.0242566\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	465167.i	0.333353i
$288$	0	0
$289$	1.26979e6	0.894307
$290$	0	0
$291$	−121763.	−0.0842912
$292$	0	0
$293$	390309.i	0.265607i	0.991142	+	0.132804i	$0.0423979\pi$
$293$	390309.i	0.265607i	−0.991142	+	0.132804i	$0.957602\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 919911.i	− 0.605138i
$298$	0	0
$299$	−278824.	−0.180365
$300$	0	0
$301$	−448787.	−0.285511
$302$	0	0
$303$	− 535713.i	− 0.335217i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.10708e6i	− 0.670401i	−0.942147	−	0.335201i	$-0.891196\pi$
$307$	− 1.10708e6i	− 0.670401i	0.942147	−	0.335201i	$-0.108804\pi$
$308$	0	0
$309$	1.37744e6	0.820684
$310$	0	0
$311$	2.19106e6	1.28455	0.642277	−	0.766472i	$-0.277990\pi$
$311$	2.19106e6	1.28455	0.642277	+	0.766472i	$0.277990\pi$
$312$	0	0
$313$	2.89050e6i	1.66768i	0.552006	+	0.833840i	$0.313862\pi$
$313$	2.89050e6i	1.66768i	−0.552006	+	0.833840i	$0.686138\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.46652e6i	− 0.819671i	−0.912160	−	0.409835i	$-0.865586\pi$
$317$	− 1.46652e6i	− 0.819671i	0.912160	−	0.409835i	$-0.134414\pi$
$318$	0	0
$319$	1.04813e6	0.576684
$320$	0	0
$321$	264161.	0.143089
$322$	0	0
$323$	− 517520.i	− 0.276008i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.42147e6i	1.25230i
$328$	0	0
$329$	−352602.	−0.179595
$330$	0	0
$331$	−2.23500e6	−1.12126	−0.560632	−	0.828065i	$-0.689442\pi$
$331$	−2.23500e6	−1.12126	−0.560632	+	0.828065i	$0.689442\pi$
$332$	0	0
$333$	139251.i	0.0688160i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.48851e6i	0.713966i	0.934111	+	0.356983i	$0.116195\pi$
$337$	1.48851e6i	0.713966i	−0.934111	+	0.356983i	$0.883805\pi$
$338$	0	0
$339$	−2.34962e6	−1.11045
$340$	0	0
$341$	676115.	0.314872
$342$	0	0
$343$	− 889414.i	− 0.408196i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.34245e6i	− 1.04435i	−0.852837	−	0.522177i	$-0.825120\pi$
$347$	− 2.34245e6i	− 1.04435i	0.852837	−	0.522177i	$-0.174880\pi$
$348$	0	0
$349$	119901.	0.0526940	0.0263470	−	0.999653i	$-0.491613\pi$
$349$	119901.	0.0526940	0.0263470	+	0.999653i	$0.491613\pi$
$350$	0	0
$351$	2.08095e6	0.901559
$352$	0	0
$353$	2.49528e6i	1.06582i	0.846173	+	0.532909i	$0.178901\pi$
$353$	2.49528e6i	1.06582i	−0.846173	+	0.532909i	$0.821099\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	120752.i	0.0501447i
$358$	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
$363$	1.26733e6i	0.504805i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.67310e6i	1.81109i	0.424249	+	0.905545i	$0.360538\pi$
$367$	4.67310e6i	1.81109i	−0.424249	+	0.905545i	$0.639462\pi$
$368$	0	0
$369$	−1.89507e6	−0.724533
$370$	0	0
$371$	−784259.	−0.295818
$372$	0	0
$373$	2.03766e6i	0.758333i	0.925328	+	0.379167i	$0.123789\pi$
$373$	2.03766e6i	0.758333i	−0.925328	+	0.379167i	$0.876211\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.37099e6i	0.859166i
$378$	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
$383$	− 2.04521e6i	− 0.712429i	−0.934404	−	0.356214i	$-0.884067\pi$
$383$	− 2.04521e6i	− 0.712429i	0.934404	−	0.356214i	$-0.115933\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.82833e6i	− 0.620551i
$388$	0	0
$389$	−234191.	−0.0784687	−0.0392344	−	0.999230i	$-0.512492\pi$
$389$	−234191.	−0.0784687	−0.0392344	+	0.999230i	$0.512492\pi$
$390$	0	0
$391$	211269.	0.0698865
$392$	0	0
$393$	− 1.71961e6i	− 0.561629i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.81706e6i	0.897058i	0.893768	+	0.448529i	$0.148052\pi$
$397$	2.81706e6i	0.897058i	−0.893768	+	0.448529i	$0.851948\pi$
$398$	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
$403$	1.52946e6i	0.469109i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	285606.i	0.0854636i
$408$	0	0
$409$	−6.08894e6	−1.79984	−0.899918	−	0.436058i	$-0.856374\pi$
$409$	−6.08894e6	−1.79984	−0.899918	+	0.436058i	$0.856374\pi$
$410$	0	0
$411$	2.44520e6	0.714020
$412$	0	0
$413$	− 931273.i	− 0.268659i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	888312.i	0.250164i
$418$	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
$423$	− 1.43648e6i	− 0.390345i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	653192.i	0.173369i
$428$	0	0
$429$	1.33160e6	0.349325
$430$	0	0
$431$	3.03516e6	0.787024	0.393512	−	0.919320i	$-0.371260\pi$
$431$	3.03516e6	0.787024	0.393512	+	0.919320i	$0.371260\pi$
$432$	0	0
$433$	3.50633e6i	0.898739i	0.893346	+	0.449369i	$0.148351\pi$
$433$	3.50633e6i	0.898739i	−0.893346	+	0.449369i	$0.851649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	728573.i	0.182503i
$438$	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
$443$	6.03357e6i	1.46071i	0.683066	+	0.730357i	$0.260646\pi$
$443$	6.03357e6i	1.46071i	−0.683066	+	0.730357i	$0.739354\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.41797e6i	1.28253i
$448$	0	0
$449$	−3.97943e6	−0.931548	−0.465774	−	0.884904i	$-0.654224\pi$
$449$	−3.97943e6	−0.931548	−0.465774	+	0.884904i	$0.654224\pi$
$450$	0	0
$451$	−3.88680e6	−0.899809
$452$	0	0
$453$	− 3.58538e6i	− 0.820899i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 430533.i	− 0.0964308i	−0.998837	−	0.0482154i	$-0.984647\pi$
$457$	− 430533.i	− 0.0964308i	0.998837	−	0.0482154i	$-0.0153534\pi$
$458$	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
$463$	7.62853e6i	1.65382i	0.562333	+	0.826911i	$0.309904\pi$
$463$	7.62853e6i	1.65382i	−0.562333	+	0.826911i	$0.690096\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.05102e6i	− 1.92046i	−0.279209	−	0.960230i	$-0.590072\pi$
$467$	− 9.05102e6i	− 1.92046i	0.279209	−	0.960230i	$-0.409928\pi$
$468$	0	0
$469$	−794943.	−0.166880
$470$	0	0
$471$	6.26511e6	1.30130
$472$	0	0
$473$	− 3.74992e6i	− 0.770672i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.19502e6i	− 0.642952i
$478$	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.i	− 0.0331569i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.65770e6i	0.507789i	0.967232	+	0.253895i	$0.0817117\pi$
$487$	2.65770e6i	0.507789i	−0.967232	+	0.253895i	$0.918288\pi$
$488$	0	0
$489$	3.44196e6	0.650929
$490$	0	0
$491$	−5.05116e6	−0.945557	−0.472779	−	0.881181i	$-0.656749\pi$
$491$	−5.05116e6	−0.945557	−0.472779	+	0.881181i	$0.656749\pi$
$492$	0	0
$493$	− 1.79654e6i	− 0.332904i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 245175.i	− 0.0445230i
$498$	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
$503$	− 1.79475e6i	− 0.316289i	−0.987416	−	0.158144i	$-0.949449\pi$
$503$	− 1.79475e6i	− 0.316289i	0.987416	−	0.158144i	$-0.0505511\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.26661e6i	− 0.218839i
$508$	0	0
$509$	−697940.	−0.119405	−0.0597027	−	0.998216i	$-0.519015\pi$
$509$	−697940.	−0.119405	−0.0597027	+	0.998216i	$0.519015\pi$
$510$	0	0
$511$	−1.50196e6	−0.254452
$512$	0	0
$513$	− 5.43757e6i	− 0.912245i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.94624e6i	− 0.484776i
$518$	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
$523$	7.22071e6i	1.15432i	0.816632	+	0.577159i	$0.195839\pi$
$523$	7.22071e6i	1.15432i	−0.816632	+	0.577159i	$0.804161\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.15889e6i	− 0.181767i
$528$	0	0
$529$	6.13892e6	0.953789
$530$	0	0
$531$	3.79395e6	0.583923
$532$	0	0
$533$	− 8.79241e6i	− 1.34057i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 452505.i	− 0.0677155i
$538$	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
$543$	− 1.14825e6i	− 0.167123i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.58966e6i	− 1.22746i	−0.789516	−	0.613730i	$-0.789668\pi$
$547$	− 8.58966e6i	− 1.22746i	0.789516	−	0.613730i	$-0.210332\pi$
$548$	0	0
$549$	−2.66107e6	−0.376812
$550$	0	0
$551$	6.19546e6	0.869350
$552$	0	0
$553$	− 2.74776e6i	− 0.382090i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.38207e6i	0.461896i	0.972966	+	0.230948i	$0.0741827\pi$
$557$	3.38207e6i	0.461896i	−0.972966	+	0.230948i	$0.925817\pi$
$558$	0	0
$559$	8.48278e6	1.14818
$560$	0	0
$561$	−1.00897e6	−0.135354
$562$	0	0
$563$	− 1.50247e6i	− 0.199772i	−0.994999	−	0.0998859i	$-0.968152\pi$
$563$	− 1.50247e6i	− 0.199772i	0.994999	−	0.0998859i	$-0.0318478\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	544468.i	0.0711237i
$568$	0	0
$569$	−1.87838e6	−0.243222	−0.121611	−	0.992578i	$-0.538806\pi$
$569$	−1.87838e6	−0.243222	−0.121611	+	0.992578i	$0.538806\pi$
$570$	0	0
$571$	−1.49628e7	−1.92053	−0.960267	−	0.279082i	$-0.909970\pi$
$571$	−1.49628e7	−1.92053	−0.960267	+	0.279082i	$0.909970\pi$
$572$	0	0
$573$	− 4.38306e6i	− 0.557687i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.03660e6i	0.504750i	0.967630	+	0.252375i	$0.0812117\pi$
$577$	4.03660e6i	0.504750i	−0.967630	+	0.252375i	$0.918788\pi$
$578$	0	0
$579$	1.45666e6	0.180577
$580$	0	0
$581$	1.98103e6	0.243473
$582$	0	0
$583$	− 6.55303e6i	− 0.798492i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.57983e7i	− 1.89241i	−0.323574	−	0.946203i	$-0.604885\pi$
$587$	− 1.57983e7i	− 1.89241i	0.323574	−	0.946203i	$-0.395115\pi$
$588$	0	0
$589$	3.99650e6	0.474670
$590$	0	0
$591$	3.75845e6	0.442629
$592$	0	0
$593$	− 1.37947e7i	− 1.61092i	−0.592647	−	0.805462i	$-0.701917\pi$
$593$	− 1.37947e7i	− 1.61092i	0.592647	−	0.805462i	$-0.298083\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.74549e6i	− 0.659768i
$598$	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
$603$	− 3.23855e6i	− 0.362708i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.63191e7i	1.79773i	0.438226	+	0.898865i	$0.355607\pi$
$607$	1.63191e7i	1.79773i	−0.438226	+	0.898865i	$0.644393\pi$
$608$	0	0
$609$	−1.44558e6	−0.157942
$610$	0	0
$611$	6.66475e6	0.722239
$612$	0	0
$613$	5.38233e6i	0.578520i	0.957250	+	0.289260i	$0.0934093\pi$
$613$	5.38233e6i	0.578520i	−0.957250	+	0.289260i	$0.906591\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.83565e6i	− 0.617130i	−0.951203	−	0.308565i	$-0.900151\pi$
$617$	− 5.83565e6i	− 0.617130i	0.951203	−	0.308565i	$-0.0998487\pi$
$618$	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
$623$	− 1.14951e6i	− 0.118657i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.47949e6i	− 0.353465i
$628$	0	0
$629$	489541.	0.0493358
$630$	0	0
$631$	8.10425e6	0.810288	0.405144	−	0.914253i	$-0.367221\pi$
$631$	8.10425e6	0.810288	0.405144	+	0.914253i	$0.367221\pi$
$632$	0	0
$633$	3.58128e6i	0.355246i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.21866e6i	0.802513i
$638$	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
$643$	5.72946e6i	0.546495i	0.961944	+	0.273247i	$0.0880978\pi$
$643$	5.72946e6i	0.546495i	−0.961944	+	0.273247i	$0.911902\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.17751e6i	− 0.204503i	−0.994759	−	0.102251i	$-0.967395\pi$
$647$	− 2.17751e6i	− 0.204503i	0.994759	−	0.102251i	$-0.0326046\pi$
$648$	0	0
$649$	7.78143e6	0.725183
$650$	0	0
$651$	−932498.	−0.0862374
$652$	0	0
$653$	1.42048e7i	1.30363i	0.758379	+	0.651813i	$0.225991\pi$
$653$	1.42048e7i	1.30363i	−0.758379	+	0.651813i	$0.774009\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.11890e6i	− 0.553044i
$658$	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
$663$	− 2.28241e6i	− 0.201656i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.52919e6i	0.220124i
$668$	0	0
$669$	−7.22894e6	−0.624467
$670$	0	0
$671$	−5.45787e6	−0.467969
$672$	0	0
$673$	1.37093e7i	1.16675i	0.812203	+	0.583374i	$0.198268\pi$
$673$	1.37093e7i	1.16675i	−0.812203	+	0.583374i	$0.801732\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.49910e7i	1.25707i	0.777781	+	0.628536i	$0.216345\pi$
$677$	1.49910e7i	1.25707i	−0.777781	+	0.628536i	$0.783655\pi$
$678$	0	0
$679$	−285789.	−0.0237887
$680$	0	0
$681$	−5.18738e6	−0.428628
$682$	0	0
$683$	3.47996e6i	0.285445i	0.989763	+	0.142723i	$0.0455857\pi$
$683$	3.47996e6i	0.285445i	−0.989763	+	0.142723i	$0.954414\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.31824e6i	− 0.429908i
$688$	0	0
$689$	1.48237e7	1.18962
$690$	0	0
$691$	5.09436e6	0.405877	0.202939	−	0.979191i	$-0.434951\pi$
$691$	5.09436e6	0.405877	0.202939	+	0.979191i	$0.434951\pi$
$692$	0	0
$693$	− 673627.i	− 0.0532827i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.66213e6i	0.519435i
$698$	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
$703$	1.68821e6i	0.128836i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.25737e6i	− 0.0946051i
$708$	0	0
$709$	−1.35383e7	−1.01146	−0.505730	−	0.862692i	$-0.668777\pi$
$709$	−1.35383e7	−1.01146	−0.505730	+	0.862692i	$0.668777\pi$
$710$	0	0
$711$	1.11942e7	0.830462
$712$	0	0
$713$	1.63150e6i	0.120189i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.16083e7i	− 0.843275i
$718$	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
$723$	2.67466e6i	0.190293i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.11509e7i	1.48420i	0.670289	+	0.742100i	$0.266170\pi$
$727$	2.11509e7i	1.48420i	−0.670289	+	0.742100i	$0.733830\pi$
$728$	0	0
$729$	−1.36164e7	−0.948949
$730$	0	0
$731$	−6.42753e6	−0.444888
$732$	0	0
$733$	1.23015e7i	0.845667i	0.906207	+	0.422833i	$0.138964\pi$
$733$	1.23015e7i	0.845667i	−0.906207	+	0.422833i	$0.861036\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6.64230e6i	− 0.450453i
$738$	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
$743$	− 3.63296e6i	− 0.241429i	−0.992687	−	0.120714i	$-0.961481\pi$
$743$	− 3.63296e6i	− 0.241429i	0.992687	−	0.120714i	$-0.0385185\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.07060e6i	0.529181i
$748$	0	0
$749$	620012.	0.0403827
$750$	0	0
$751$	−1.77346e7	−1.14741	−0.573707	−	0.819060i	$-0.694495\pi$
$751$	−1.77346e7	−1.14741	−0.573707	+	0.819060i	$0.694495\pi$
$752$	0	0
$753$	2.04088e7i	1.31169i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.89094e6i	0.500483i	0.968183	+	0.250241i	$0.0805100\pi$
$757$	7.89094e6i	0.500483i	−0.968183	+	0.250241i	$0.919490\pi$
$758$	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
$763$	5.68343e6i	0.353426i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.76026e7i	1.08041i
$768$	0	0
$769$	−3.89976e6	−0.237806	−0.118903	−	0.992906i	$-0.537938\pi$
$769$	−3.89976e6	−0.237806	−0.118903	+	0.992906i	$0.537938\pi$
$770$	0	0
$771$	−6.09004e6	−0.368964
$772$	0	0
$773$	− 2.33736e6i	− 0.140695i	−0.997523	−	0.0703474i	$-0.977589\pi$
$773$	− 2.33736e6i	− 0.140695i	0.997523	−	0.0703474i	$-0.0224108\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 393908.i	− 0.0234068i
$778$	0	0
$779$	−2.29748e7	−1.35646
$780$	0	0
$781$	2.04860e6	0.120180
$782$	0	0
$783$	− 1.88762e7i	− 1.10030i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.52903e7i	0.879993i	0.897999	+	0.439997i	$0.145020\pi$
$787$	1.52903e7i	0.879993i	−0.897999	+	0.439997i	$0.854980\pi$
$788$	0	0
$789$	−1.95804e7	−1.11977
$790$	0	0
$791$	−5.51479e6	−0.313392
$792$	0	0
$793$	− 1.23464e7i	− 0.697199i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.25972e7i	0.702468i	0.936288	+	0.351234i	$0.114238\pi$
$797$	1.25972e7i	0.702468i	−0.936288	+	0.351234i	$0.885762\pi$
$798$	0	0
$799$	−5.04997e6	−0.279848
$800$	0	0
$801$	4.68305e6	0.257898
$802$	0	0
$803$	− 1.25499e7i	− 0.686835i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.47219e6i	− 0.403891i
$808$	0	0
$809$	1.13476e7	0.609584	0.304792	−	0.952419i	$-0.401413\pi$
$809$	1.13476e7	0.609584	0.304792	+	0.952419i	$0.401413\pi$
$810$	0	0
$811$	−1.91000e7	−1.01972	−0.509860	−	0.860257i	$-0.670303\pi$
$811$	−1.91000e7	−1.01972	−0.509860	+	0.860257i	$0.670303\pi$
$812$	0	0
$813$	2.24661e7i	1.19207i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.21657e7i	− 1.16179i
$818$	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
$823$	− 6.95590e6i	− 0.357976i	−0.983851	−	0.178988i	$-0.942718\pi$
$823$	− 6.95590e6i	− 0.357976i	0.983851	−	0.178988i	$-0.0572823\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.04638e6i	0.459950i	0.973197	+	0.229975i	$0.0738645\pi$
$827$	9.04638e6i	0.459950i	−0.973197	+	0.229975i	$0.926136\pi$
$828$	0	0
$829$	1.54026e7	0.778407	0.389203	−	0.921152i	$-0.372750\pi$
$829$	1.54026e7	0.778407	0.389203	+	0.921152i	$0.372750\pi$
$830$	0	0
$831$	−3.60932e6	−0.181310
$832$	0	0
$833$	− 6.22739e6i	− 0.310952i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.21764e7i	− 0.600767i
$838$	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
$843$	− 1.99072e7i	− 0.964809i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.97455e6i	0.142466i
$848$	0	0
$849$	−2.36410e6	−0.112563
$850$	0	0
$851$	−689183.	−0.0326220
$852$	0	0
$853$	2.06755e7i	0.972934i	0.873699	+	0.486467i	$0.161715\pi$
$853$	2.06755e7i	0.972934i	−0.873699	+	0.486467i	$0.838285\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.16205e7i	1.00557i	0.864411	+	0.502786i	$0.167692\pi$
$857$	2.16205e7i	1.00557i	−0.864411	+	0.502786i	$0.832308\pi$
$858$	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
$863$	2.96043e7i	1.35309i	0.736399	+	0.676547i	$0.236525\pi$
$863$	2.96043e7i	1.35309i	−0.736399	+	0.676547i	$0.763475\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.46333e7i	− 0.661140i
$868$	0	0
$869$	2.29594e7	1.03136
$870$	0	0
$871$	1.50257e7	0.671103
$872$	0	0
$873$	− 1.16429e6i	− 0.0517041i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.77691e7i	− 1.21917i	−0.792722	−	0.609583i	$-0.791337\pi$
$877$	− 2.77691e7i	− 1.21917i	0.792722	−	0.609583i	$-0.208663\pi$
$878$	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
$883$	− 2.66919e7i	− 1.15207i	−0.817426	−	0.576034i	$-0.804600\pi$
$883$	− 2.66919e7i	− 1.15207i	0.817426	−	0.576034i	$-0.195400\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.95714e7i	− 1.68878i	−0.535733	−	0.844388i	$-0.679964\pi$
$887$	− 3.95714e7i	− 1.68878i	0.535733	−	0.844388i	$-0.320036\pi$
$888$	0	0
$889$	4.73406e6	0.200900
$890$	0	0
$891$	−4.54940e6	−0.191982
$892$	0	0
$893$	− 1.74151e7i	− 0.730799i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.21321e6i	0.133339i
$898$	0	0
$899$	1.38736e7	0.572518
$900$	0	0
$901$	−1.12322e7	−0.460948
$902$	0	0
$903$	5.17189e6i	0.211072i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.79635e7i	1.12869i	0.825541	+	0.564343i	$0.190870\pi$
$907$	2.79635e7i	1.12869i	−0.825541	+	0.564343i	$0.809130\pi$
$908$	0	0
$909$	5.12245e6	0.205621
$910$	0	0
$911$	−1.87296e7	−0.747710	−0.373855	−	0.927487i	$-0.621964\pi$
$911$	−1.87296e7	−0.747710	−0.373855	+	0.927487i	$0.621964\pi$
$912$	0	0
$913$	1.65529e7i	0.657199i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.03610e6i	− 0.158503i
$918$	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
$923$	4.63420e6i	0.179048i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.31710e7i	0.503406i
$928$	0	0
$929$	3.88269e7	1.47602	0.738011	−	0.674788i	$-0.235765\pi$
$929$	3.88269e7	1.47602	0.738011	+	0.674788i	$0.235765\pi$
$930$	0	0
$931$	2.14755e7	0.812026
$932$	0	0
$933$	− 2.52501e7i	− 0.949640i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.75302e7i	− 0.652288i	−0.945320	−	0.326144i	$-0.894251\pi$
$937$	− 1.75302e7i	− 0.652288i	0.945320	−	0.326144i	$-0.105749\pi$
$938$	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
$943$	− 9.37905e6i	− 0.343463i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.43637e7i	− 1.60751i	−0.594962	−	0.803754i	$-0.702833\pi$
$947$	− 4.43637e7i	− 1.60751i	0.594962	−	0.803754i	$-0.297167\pi$
$948$	0	0
$949$	2.83895e7	1.02327
$950$	0	0
$951$	−1.69004e7	−0.605963
$952$	0	0
$953$	− 5.71990e6i	− 0.204012i	−0.994784	−	0.102006i	$-0.967474\pi$
$953$	− 5.71990e6i	− 0.204012i	0.994784	−	0.102006i	$-0.0325261\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.20788e7i	− 0.426329i
$958$	0	0
$959$	5.73912e6	0.201511
$960$	0	0
$961$	−1.96797e7	−0.687402
$962$	0	0
$963$	2.52589e6i	0.0877706i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.00548e7i	− 1.03359i	−0.856110	−	0.516794i	$-0.827125\pi$
$967$	− 3.00548e7i	− 1.03359i	0.856110	−	0.516794i	$-0.172875\pi$
$968$	0	0
$969$	−5.96399e6	−0.204046
$970$	0	0
$971$	1.89662e7	0.645552	0.322776	−	0.946475i	$-0.395384\pi$
$971$	1.89662e7	0.645552	0.322776	+	0.946475i	$0.395384\pi$
$972$	0	0
$973$	2.08495e6i	0.0706015i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.45541e7i	0.822975i	0.911415	+	0.411488i	$0.134991\pi$
$977$	2.45541e7i	0.822975i	−0.911415	+	0.411488i	$0.865009\pi$
$978$	0	0
$979$	9.60498e6	0.320287
$980$	0	0
$981$	−2.31540e7	−0.768162
$982$	0	0
$983$	− 3.35539e7i	− 1.10754i	−0.832670	−	0.553769i	$-0.813189\pi$
$983$	− 3.35539e7i	− 1.10754i	0.832670	−	0.553769i	$-0.186811\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.06345e6i	0.132771i
$988$	0	0
$989$	9.04877e6	0.294170
$990$	0	0
$991$	2.05230e7	0.663830	0.331915	−	0.943309i	$-0.392305\pi$
$991$	2.05230e7	0.663830	0.331915	+	0.943309i	$0.392305\pi$
$992$	0	0
$993$	2.57565e7i	0.828924i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.25729e7i	0.400589i	0.979736	+	0.200294i	$0.0641898\pi$
$997$	1.25729e7i	0.400589i	−0.979736	+	0.200294i	$0.935810\pi$
$998$	0	0
$999$	5.14359e6	0.163062

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-11.5242i q^{3} -27.0483i q^{7} +110.193 q^{9} +O(q^{10})\)	\(q-11.5242i q^{3} -27.0483i q^{7} +110.193 q^{9} +226.008 q^{11} +511.257i q^{13} -387.387i q^{17} +1335.93 q^{19} -311.710 q^{21} +545.369i q^{23} -4070.26i q^{27} +4637.58 q^{29} +2991.56 q^{31} -2604.55i q^{33} +1263.70i q^{37} +5891.82 q^{39} -17197.6 q^{41} -16592.0i q^{43} -13036.0i q^{47} +16075.4 q^{49} -4464.31 q^{51} -28994.7i q^{53} -15395.4i q^{57} +34429.9 q^{59} -24149.1 q^{61} -2980.55i q^{63} -29389.7i q^{67} +6284.93 q^{69} +9064.32 q^{71} -55528.7i q^{73} -6113.13i q^{77} +101587. q^{79} -20129.4 q^{81} +73240.4i q^{83} -53444.3i q^{87} +42498.5 q^{89} +13828.7 q^{91} -34475.2i q^{93} -10565.9i q^{97} +24904.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 56 q^{9}+O(q^{10}) \) 4 * q - 56 * q^9	\( 4 q - 56 q^{9} - 400 q^{11} + 1680 q^{19} - 1992 q^{21} + 9360 q^{29} - 10016 q^{31} + 54864 q^{39} - 10668 q^{41} + 63308 q^{49} - 13200 q^{51} + 163552 q^{59} - 93864 q^{61} + 110088 q^{69} + 14896 q^{71} + 216208 q^{79} - 187324 q^{81} - 141980 q^{89} + 104992 q^{91} + 167552 q^{99}+O(q^{100}) \) 4 * q - 56 * q^9 - 400 * q^11 + 1680 * q^19 - 1992 * q^21 + 9360 * q^29 - 10016 * q^31 + 54864 * q^39 - 10668 * q^41 + 63308 * q^49 - 13200 * q^51 + 163552 * q^59 - 93864 * q^61 + 110088 * q^69 + 14896 * q^71 + 216208 * q^79 - 187324 * q^81 - 141980 * q^89 + 104992 * q^91 + 167552 * q^99

Introduction

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