LMFDB - Embedded newform 800.6.f.b.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{93}\cdot 3^{4}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$0.236693 - 3.99299i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.6.f.b.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.0521 q^{3} +103.624i q^{7} +384.607 q^{9} -740.776i q^{11} +892.067 q^{13} -1138.81i q^{17} -1155.46i q^{19} -2596.00i q^{21} -1602.57i q^{23} -3547.55 q^{27} -2158.47i q^{29} -4955.24 q^{31} +18558.0i q^{33} +4403.89 q^{37} -22348.1 q^{39} +3780.62 q^{41} +13068.3 q^{43} -8000.58i q^{47} +6069.06 q^{49} +28529.6i q^{51} +34313.6 q^{53} +28946.6i q^{57} +22065.0i q^{59} +2822.53i q^{61} +39854.5i q^{63} -54981.5 q^{67} +40147.8i q^{69} -42879.2 q^{71} -20893.2i q^{73} +76762.2 q^{77} +30227.6 q^{79} -4586.03 q^{81} +93949.9 q^{83} +54074.1i q^{87} -1178.06 q^{89} +92439.5i q^{91} +124139. q^{93} -74100.8i q^{97} -284908. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 36 q^{3} + 1620 q^{9} - 11664 q^{27} - 7160 q^{31} - 3608 q^{37} - 44904 q^{39} + 11608 q^{41} + 51772 q^{43} - 18756 q^{49} + 928 q^{53} + 161604 q^{67} + 200312 q^{71} + 26008 q^{77} + 282080 q^{79}+ \cdots + 293472 q^{93}+O(q^{100}) $ 20 * q - 36 * q^3 + 1620 * q^9 - 11664 * q^27 - 7160 * q^31 - 3608 * q^37 - 44904 * q^39 + 11608 * q^41 + 51772 * q^43 - 18756 * q^49 + 928 * q^53 + 161604 * q^67 + 200312 * q^71 + 26008 * q^77 + 282080 * q^79 + 65172 * q^81 + 99092 * q^83 + 3160 * q^89 + 293472 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−25.0521	−1.60709	−0.803546	−	0.595243i	$-0.797056\pi$
$3$	−25.0521	−1.60709	−0.803546	+	0.595243i	$0.797056\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	103.624i	0.799310i	0.916666	+	0.399655i	$0.130870\pi$
$7$	103.624i	0.799310i	−0.916666	+	0.399655i	$0.869130\pi$
$8$	0	0
$9$	384.607	1.58274
$10$	0	0
$11$	− 740.776i	− 1.84589i	−0.384935	−	0.922944i	$-0.625776\pi$
$11$	− 740.776i	− 1.84589i	0.384935	−	0.922944i	$-0.374224\pi$
$12$	0	0
$13$	892.067	1.46399	0.731996	−	0.681309i	$-0.238589\pi$
$13$	892.067	1.46399	0.731996	+	0.681309i	$0.238589\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1138.81i	− 0.955717i	−0.878437	−	0.477859i	$-0.841413\pi$
$17$	− 1138.81i	− 0.955717i	0.878437	−	0.477859i	$-0.158587\pi$
$18$	0	0
$19$	− 1155.46i	− 0.734294i	−0.930163	−	0.367147i	$-0.880335\pi$
$19$	− 1155.46i	− 0.734294i	0.930163	−	0.367147i	$-0.119665\pi$
$20$	0	0
$21$	− 2596.00i	− 1.28457i
$22$	0	0
$23$	− 1602.57i	− 0.631682i	−0.948812	−	0.315841i	$-0.897713\pi$
$23$	− 1602.57i	− 0.631682i	0.948812	−	0.315841i	$-0.102287\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3547.55	−0.936524
$28$	0	0
$29$	− 2158.47i	− 0.476596i	−0.971192	−	0.238298i	$-0.923410\pi$
$29$	− 2158.47i	− 0.476596i	0.971192	−	0.238298i	$-0.0765896\pi$
$30$	0	0
$31$	−4955.24	−0.926106	−0.463053	−	0.886331i	$-0.653246\pi$
$31$	−4955.24	−0.926106	−0.463053	+	0.886331i	$0.653246\pi$
$32$	0	0
$33$	18558.0i	2.96651i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4403.89	0.528850	0.264425	−	0.964406i	$-0.414818\pi$
$37$	4403.89	0.528850	0.264425	+	0.964406i	$0.414818\pi$
$38$	0	0
$39$	−22348.1	−2.35277
$40$	0	0
$41$	3780.62	0.351240	0.175620	−	0.984458i	$-0.443807\pi$
$41$	3780.62	0.351240	0.175620	+	0.984458i	$0.443807\pi$
$42$	0	0
$43$	13068.3	1.07783	0.538913	−	0.842361i	$-0.318835\pi$
$43$	13068.3	1.07783	0.538913	+	0.842361i	$0.318835\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8000.58i	− 0.528296i	−0.964482	−	0.264148i	$-0.914909\pi$
$47$	− 8000.58i	− 0.528296i	0.964482	−	0.264148i	$-0.0850907\pi$
$48$	0	0
$49$	6069.06	0.361103
$50$	0	0
$51$	28529.6i	1.53593i
$52$	0	0
$53$	34313.6	1.67794	0.838971	−	0.544175i	$-0.183157\pi$
$53$	34313.6	1.67794	0.838971	+	0.544175i	$0.183157\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	28946.6i	1.18008i
$58$	0	0
$59$	22065.0i	0.825227i	0.910906	+	0.412613i	$0.135384\pi$
$59$	22065.0i	0.825227i	−0.910906	+	0.412613i	$0.864616\pi$
$60$	0	0
$61$	2822.53i	0.0971212i	0.998820	+	0.0485606i	$0.0154634\pi$
$61$	2822.53i	0.0971212i	−0.998820	+	0.0485606i	$0.984537\pi$
$62$	0	0
$63$	39854.5i	1.26510i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−54981.5	−1.49634	−0.748168	−	0.663509i	$-0.769066\pi$
$67$	−54981.5	−1.49634	−0.748168	+	0.663509i	$0.769066\pi$
$68$	0	0
$69$	40147.8i	1.01517i
$70$	0	0
$71$	−42879.2	−1.00949	−0.504744	−	0.863269i	$-0.668413\pi$
$71$	−42879.2	−1.00949	−0.504744	+	0.863269i	$0.668413\pi$
$72$	0	0
$73$	− 20893.2i	− 0.458879i	−0.973323	−	0.229440i	$-0.926311\pi$
$73$	− 20893.2i	− 0.458879i	0.973323	−	0.229440i	$-0.0736894\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	76762.2	1.47544
$78$	0	0
$79$	30227.6	0.544925	0.272462	−	0.962166i	$-0.412162\pi$
$79$	30227.6	0.544925	0.272462	+	0.962166i	$0.412162\pi$
$80$	0	0
$81$	−4586.03	−0.0776648
$82$	0	0
$83$	93949.9	1.49693	0.748465	−	0.663175i	$-0.230791\pi$
$83$	93949.9	1.49693	0.748465	+	0.663175i	$0.230791\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	54074.1i	0.765934i
$88$	0	0
$89$	−1178.06	−0.0157650	−0.00788248	−	0.999969i	$-0.502509\pi$
$89$	−1178.06	−0.0157650	−0.00788248	+	0.999969i	$0.502509\pi$
$90$	0	0
$91$	92439.5i	1.17018i
$92$	0	0
$93$	124139.	1.48834
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 74100.8i	− 0.799638i	−0.916594	−	0.399819i	$-0.869073\pi$
$97$	− 74100.8i	− 0.799638i	0.916594	−	0.399819i	$-0.130927\pi$
$98$	0	0
$99$	− 284908.i	− 2.92157i
$100$	0	0
$101$	31104.3i	0.303401i	0.988427	+	0.151700i	$0.0484749\pi$
$101$	31104.3i	0.303401i	−0.988427	+	0.151700i	$0.951525\pi$
$102$	0	0
$103$	140298.i	1.30305i	0.758629	+	0.651523i	$0.225870\pi$
$103$	140298.i	1.30305i	−0.758629	+	0.651523i	$0.774130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	34792.6	0.293784	0.146892	−	0.989153i	$-0.453073\pi$
$107$	34792.6	0.293784	0.146892	+	0.989153i	$0.453073\pi$
$108$	0	0
$109$	− 83759.2i	− 0.675252i	−0.941280	−	0.337626i	$-0.890376\pi$
$109$	− 83759.2i	− 0.675252i	0.941280	−	0.337626i	$-0.109624\pi$
$110$	0	0
$111$	−110327.	−0.849910
$112$	0	0
$113$	57262.1i	0.421863i	0.977501	+	0.210931i	$0.0676497\pi$
$113$	57262.1i	0.421863i	−0.977501	+	0.210931i	$0.932350\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	343095.	2.31713
$118$	0	0
$119$	118008.	0.763914
$120$	0	0
$121$	−387698.	−2.40730
$122$	0	0
$123$	−94712.5	−0.564475
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8848.12i	− 0.0486790i	−0.999704	−	0.0243395i	$-0.992252\pi$
$127$	− 8848.12i	− 0.0486790i	0.999704	−	0.0243395i	$-0.00774827\pi$
$128$	0	0
$129$	−327389.	−1.73217
$130$	0	0
$131$	− 110458.i	− 0.562365i	−0.959654	−	0.281183i	$-0.909273\pi$
$131$	− 110458.i	− 0.562365i	0.959654	−	0.281183i	$-0.0907267\pi$
$132$	0	0
$133$	119733.	0.586929
$134$	0	0
$135$	0	0
$136$	0	0
$137$	304365.i	1.38546i	0.721199	+	0.692728i	$0.243591\pi$
$137$	304365.i	1.38546i	−0.721199	+	0.692728i	$0.756409\pi$
$138$	0	0
$139$	− 59379.5i	− 0.260675i	−0.991470	−	0.130338i	$-0.958394\pi$
$139$	− 59379.5i	− 0.260675i	0.991470	−	0.130338i	$-0.0416061\pi$
$140$	0	0
$141$	200431.i	0.849020i
$142$	0	0
$143$	− 660822.i	− 2.70237i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−152043.	−0.580326
$148$	0	0
$149$	233744.i	0.862529i	0.902225	+	0.431265i	$0.141932\pi$
$149$	233744.i	0.862529i	−0.902225	+	0.431265i	$0.858068\pi$
$150$	0	0
$151$	89068.4	0.317893	0.158947	−	0.987287i	$-0.449190\pi$
$151$	89068.4	0.317893	0.158947	+	0.987287i	$0.449190\pi$
$152$	0	0
$153$	− 437994.i	− 1.51266i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−608909.	−1.97153	−0.985764	−	0.168132i	$-0.946227\pi$
$157$	−608909.	−1.97153	−0.985764	+	0.168132i	$0.946227\pi$
$158$	0	0
$159$	−859628.	−2.69661
$160$	0	0
$161$	166065.	0.504910
$162$	0	0
$163$	−172801.	−0.509421	−0.254711	−	0.967017i	$-0.581980\pi$
$163$	−172801.	−0.509421	−0.254711	+	0.967017i	$0.581980\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	441506.i	1.22503i	0.790461	+	0.612513i	$0.209841\pi$
$167$	441506.i	1.22503i	−0.790461	+	0.612513i	$0.790159\pi$
$168$	0	0
$169$	424490.	1.14327
$170$	0	0
$171$	− 444397.i	− 1.16220i
$172$	0	0
$173$	−281414.	−0.714874	−0.357437	−	0.933937i	$-0.616349\pi$
$173$	−281414.	−0.714874	−0.357437	+	0.933937i	$0.616349\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 552774.i	− 1.32622i
$178$	0	0
$179$	− 805768.i	− 1.87965i	−0.341655	−	0.939825i	$-0.610987\pi$
$179$	− 805768.i	− 1.87965i	0.341655	−	0.939825i	$-0.389013\pi$
$180$	0	0
$181$	− 610344.i	− 1.38477i	−0.721528	−	0.692385i	$-0.756560\pi$
$181$	− 610344.i	− 1.38477i	0.721528	−	0.692385i	$-0.243440\pi$
$182$	0	0
$183$	− 70710.3i	− 0.156083i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−843604.	−1.76415
$188$	0	0
$189$	− 367611.i	− 0.748573i
$190$	0	0
$191$	845737.	1.67746	0.838729	−	0.544549i	$-0.183299\pi$
$191$	845737.	1.67746	0.838729	+	0.544549i	$0.183299\pi$
$192$	0	0
$193$	− 184089.i	− 0.355742i	−0.984054	−	0.177871i	$-0.943079\pi$
$193$	− 184089.i	− 0.355742i	0.984054	−	0.177871i	$-0.0569210\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	390522.	0.716935	0.358468	−	0.933542i	$-0.383299\pi$
$197$	390522.	0.716935	0.358468	+	0.933542i	$0.383299\pi$
$198$	0	0
$199$	−584263.	−1.04587	−0.522933	−	0.852374i	$-0.675162\pi$
$199$	−584263.	−1.04587	−0.522933	+	0.852374i	$0.675162\pi$
$200$	0	0
$201$	1.37740e6	2.40475
$202$	0	0
$203$	223669.	0.380948
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 616361.i	− 0.999791i
$208$	0	0
$209$	−855935.	−1.35542
$210$	0	0
$211$	− 428258.i	− 0.662215i	−0.943593	−	0.331107i	$-0.892578\pi$
$211$	− 428258.i	− 0.662215i	0.943593	−	0.331107i	$-0.107422\pi$
$212$	0	0
$213$	1.07421e6	1.62234
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 513482.i	− 0.740246i
$218$	0	0
$219$	523419.i	0.737461i
$220$	0	0
$221$	− 1.01590e6i	− 1.39916i
$222$	0	0
$223$	− 955618.i	− 1.28683i	−0.765517	−	0.643416i	$-0.777516\pi$
$223$	− 955618.i	− 1.28683i	0.765517	−	0.643416i	$-0.222484\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	866233.	1.11576	0.557879	−	0.829922i	$-0.311615\pi$
$227$	866233.	1.11576	0.557879	+	0.829922i	$0.311615\pi$
$228$	0	0
$229$	− 654035.i	− 0.824161i	−0.911147	−	0.412080i	$-0.864802\pi$
$229$	− 654035.i	− 0.824161i	0.911147	−	0.412080i	$-0.135198\pi$
$230$	0	0
$231$	−1.92305e6	−2.37116
$232$	0	0
$233$	− 820513.i	− 0.990138i	−0.868854	−	0.495069i	$-0.835143\pi$
$233$	− 820513.i	− 0.990138i	0.868854	−	0.495069i	$-0.164857\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−757265.	−0.875744
$238$	0	0
$239$	867314.	0.982159	0.491079	−	0.871115i	$-0.336603\pi$
$239$	867314.	0.982159	0.491079	+	0.871115i	$0.336603\pi$
$240$	0	0
$241$	−1.65377e6	−1.83414	−0.917072	−	0.398722i	$-0.869454\pi$
$241$	−1.65377e6	−1.83414	−0.917072	+	0.398722i	$0.869454\pi$
$242$	0	0
$243$	976943.	1.06134
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.03075e6i	− 1.07500i
$248$	0	0
$249$	−2.35364e6	−2.40570
$250$	0	0
$251$	− 860879.i	− 0.862497i	−0.902233	−	0.431249i	$-0.858073\pi$
$251$	− 860879.i	− 0.862497i	0.902233	−	0.431249i	$-0.141927\pi$
$252$	0	0
$253$	−1.18715e6	−1.16601
$254$	0	0
$255$	0	0
$256$	0	0
$257$	765335.i	0.722801i	0.932411	+	0.361401i	$0.117701\pi$
$257$	765335.i	0.722801i	−0.932411	+	0.361401i	$0.882299\pi$
$258$	0	0
$259$	456349.i	0.422715i
$260$	0	0
$261$	− 830162.i	− 0.754330i
$262$	0	0
$263$	− 260746.i	− 0.232449i	−0.993223	−	0.116225i	$-0.962921\pi$
$263$	− 260746.i	− 0.232449i	0.993223	−	0.116225i	$-0.0370792\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	29512.9	0.0253357
$268$	0	0
$269$	− 1.12412e6i	− 0.947175i	−0.880747	−	0.473587i	$-0.842959\pi$
$269$	− 1.12412e6i	− 0.947175i	0.880747	−	0.473587i	$-0.157041\pi$
$270$	0	0
$271$	1.07277e6	0.887330	0.443665	−	0.896193i	$-0.353678\pi$
$271$	1.07277e6	0.887330	0.443665	+	0.896193i	$0.353678\pi$
$272$	0	0
$273$	− 2.31580e6i	− 1.88059i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−615080.	−0.481651	−0.240825	−	0.970568i	$-0.577418\pi$
$277$	−615080.	−0.481651	−0.240825	+	0.970568i	$0.577418\pi$
$278$	0	0
$279$	−1.90582e6	−1.46579
$280$	0	0
$281$	−652198.	−0.492736	−0.246368	−	0.969176i	$-0.579237\pi$
$281$	−652198.	−0.492736	−0.246368	+	0.969176i	$0.579237\pi$
$282$	0	0
$283$	908596.	0.674380	0.337190	−	0.941437i	$-0.390524\pi$
$283$	908596.	0.674380	0.337190	+	0.941437i	$0.390524\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	391763.i	0.280750i
$288$	0	0
$289$	122967.	0.0866050
$290$	0	0
$291$	1.85638e6i	1.28509i
$292$	0	0
$293$	514047.	0.349811	0.174906	−	0.984585i	$-0.444038\pi$
$293$	514047.	0.349811	0.174906	+	0.984585i	$0.444038\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.62794e6i	1.72872i
$298$	0	0
$299$	− 1.42960e6i	− 0.924778i
$300$	0	0
$301$	1.35419e6i	0.861517i
$302$	0	0
$303$	− 779227.i	− 0.487593i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.46693e6	0.888309	0.444155	−	0.895950i	$-0.353504\pi$
$307$	1.46693e6	0.888309	0.444155	+	0.895950i	$0.353504\pi$
$308$	0	0
$309$	− 3.51477e6i	− 2.09411i
$310$	0	0
$311$	−547919.	−0.321229	−0.160615	−	0.987017i	$-0.551348\pi$
$311$	−547919.	−0.321229	−0.160615	+	0.987017i	$0.551348\pi$
$312$	0	0
$313$	1.70857e6i	0.985762i	0.870097	+	0.492881i	$0.164056\pi$
$313$	1.70857e6i	0.985762i	−0.870097	+	0.492881i	$0.835944\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	978608.	0.546966	0.273483	−	0.961877i	$-0.411824\pi$
$317$	978608.	0.546966	0.273483	+	0.961877i	$0.411824\pi$
$318$	0	0
$319$	−1.59894e6	−0.879743
$320$	0	0
$321$	−871627.	−0.472137
$322$	0	0
$323$	−1.31585e6	−0.701777
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.09834e6i	1.08519i
$328$	0	0
$329$	829053.	0.422272
$330$	0	0
$331$	345907.i	0.173536i	0.996229	+	0.0867679i	$0.0276539\pi$
$331$	345907.i	0.173536i	−0.996229	+	0.0867679i	$0.972346\pi$
$332$	0	0
$333$	1.69377e6	0.837034
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 376513.i	− 0.180595i	−0.995915	−	0.0902975i	$-0.971218\pi$
$337$	− 376513.i	− 0.180595i	0.995915	−	0.0902975i	$-0.0287818\pi$
$338$	0	0
$339$	− 1.43454e6i	− 0.677972i
$340$	0	0
$341$	3.67073e6i	1.70949i
$342$	0	0
$343$	2.37051e6i	1.08794i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	164656.	0.0734099	0.0367049	−	0.999326i	$-0.488314\pi$
$347$	164656.	0.0734099	0.0367049	+	0.999326i	$0.488314\pi$
$348$	0	0
$349$	− 35942.6i	− 0.0157959i	−0.999969	−	0.00789797i	$-0.997486\pi$
$349$	− 35942.6i	− 0.0157959i	0.999969	−	0.00789797i	$-0.00251403\pi$
$350$	0	0
$351$	−3.16465e6	−1.37106
$352$	0	0
$353$	1.39430e6i	0.595550i	0.954636	+	0.297775i	$0.0962445\pi$
$353$	1.39430e6i	0.595550i	−0.954636	+	0.297775i	$0.903756\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.95635e6	−1.22768
$358$	0	0
$359$	3.50589e6	1.43569	0.717847	−	0.696201i	$-0.245128\pi$
$359$	3.50589e6	1.43569	0.717847	+	0.696201i	$0.245128\pi$
$360$	0	0
$361$	1.14102e6	0.460812
$362$	0	0
$363$	9.71265e6	3.86875
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.64440e6i	1.41241i	0.708007	+	0.706206i	$0.249595\pi$
$367$	3.64440e6i	1.41241i	−0.708007	+	0.706206i	$0.750405\pi$
$368$	0	0
$369$	1.45405e6	0.555923
$370$	0	0
$371$	3.55572e6i	1.34120i
$372$	0	0
$373$	4.19252e6	1.56028	0.780142	−	0.625603i	$-0.215147\pi$
$373$	4.19252e6	1.56028	0.780142	+	0.625603i	$0.215147\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.92550e6i	− 0.697734i
$378$	0	0
$379$	2.31089e6i	0.826384i	0.910644	+	0.413192i	$0.135586\pi$
$379$	2.31089e6i	0.826384i	−0.910644	+	0.413192i	$0.864414\pi$
$380$	0	0
$381$	221664.i	0.0782316i
$382$	0	0
$383$	− 1.18240e6i	− 0.411876i	−0.978565	−	0.205938i	$-0.933976\pi$
$383$	− 1.18240e6i	− 0.411876i	0.978565	−	0.205938i	$-0.0660245\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.02617e6	1.70592
$388$	0	0
$389$	− 5.09219e6i	− 1.70620i	−0.521745	−	0.853102i	$-0.674719\pi$
$389$	− 5.09219e6i	− 1.70620i	0.521745	−	0.853102i	$-0.325281\pi$
$390$	0	0
$391$	−1.82503e6	−0.603709
$392$	0	0
$393$	2.76720e6i	0.903773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	63006.9	0.0200637	0.0100319	−	0.999950i	$-0.496807\pi$
$397$	63006.9	0.0200637	0.0100319	+	0.999950i	$0.496807\pi$
$398$	0	0
$399$	−2.99957e6	−0.943248
$400$	0	0
$401$	−4.48960e6	−1.39427	−0.697135	−	0.716940i	$-0.745542\pi$
$401$	−4.48960e6	−1.39427	−0.697135	+	0.716940i	$0.745542\pi$
$402$	0	0
$403$	−4.42041e6	−1.35581
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.26230e6i	− 0.976197i
$408$	0	0
$409$	−2.03416e6	−0.601281	−0.300641	−	0.953738i	$-0.597200\pi$
$409$	−2.03416e6	−0.601281	−0.300641	+	0.953738i	$0.597200\pi$
$410$	0	0
$411$	− 7.62497e6i	− 2.22656i
$412$	0	0
$413$	−2.28646e6	−0.659612
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.48758e6i	0.418929i
$418$	0	0
$419$	− 1.53143e6i	− 0.426150i	−0.977036	−	0.213075i	$-0.931652\pi$
$419$	− 1.53143e6i	− 0.426150i	0.977036	−	0.213075i	$-0.0683479\pi$
$420$	0	0
$421$	− 4.86857e6i	− 1.33874i	−0.742929	−	0.669370i	$-0.766564\pi$
$421$	− 4.86857e6i	− 1.33874i	0.742929	−	0.669370i	$-0.233436\pi$
$422$	0	0
$423$	− 3.07708e6i	− 0.836157i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−292482.	−0.0776300
$428$	0	0
$429$	1.65550e7i	4.34295i
$430$	0	0
$431$	−537754.	−0.139441	−0.0697204	−	0.997567i	$-0.522211\pi$
$431$	−537754.	−0.139441	−0.0697204	+	0.997567i	$0.522211\pi$
$432$	0	0
$433$	1.48235e6i	0.379954i	0.981789	+	0.189977i	$0.0608414\pi$
$433$	1.48235e6i	0.379954i	−0.981789	+	0.189977i	$0.939159\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.85171e6	−0.463840
$438$	0	0
$439$	−2.49966e6	−0.619041	−0.309520	−	0.950893i	$-0.600168\pi$
$439$	−2.49966e6	−0.619041	−0.309520	+	0.950893i	$0.600168\pi$
$440$	0	0
$441$	2.33420e6	0.571534
$442$	0	0
$443$	−5.00804e6	−1.21244	−0.606218	−	0.795299i	$-0.707314\pi$
$443$	−5.00804e6	−1.21244	−0.606218	+	0.795299i	$0.707314\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.85576e6i	− 1.38616i
$448$	0	0
$449$	−3.04025e6	−0.711694	−0.355847	−	0.934544i	$-0.615808\pi$
$449$	−3.04025e6	−0.711694	−0.355847	+	0.934544i	$0.615808\pi$
$450$	0	0
$451$	− 2.80059e6i	− 0.648349i
$452$	0	0
$453$	−2.23135e6	−0.510884
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.20014e6i	− 1.61269i	−0.591446	−	0.806345i	$-0.701443\pi$
$457$	− 7.20014e6i	− 1.61269i	0.591446	−	0.806345i	$-0.298557\pi$
$458$	0	0
$459$	4.03999e6i	0.895052i
$460$	0	0
$461$	7.87732e6i	1.72634i	0.504914	+	0.863170i	$0.331524\pi$
$461$	7.87732e6i	1.72634i	−0.504914	+	0.863170i	$0.668476\pi$
$462$	0	0
$463$	6.18812e6i	1.34155i	0.741662	+	0.670774i	$0.234038\pi$
$463$	6.18812e6i	1.34155i	−0.741662	+	0.670774i	$0.765962\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.21143e6	−1.53013	−0.765066	−	0.643952i	$-0.777294\pi$
$467$	−7.21143e6	−1.53013	−0.765066	+	0.643952i	$0.777294\pi$
$468$	0	0
$469$	− 5.69740e6i	− 1.19604i
$470$	0	0
$471$	1.52544e7	3.16843
$472$	0	0
$473$	− 9.68070e6i	− 1.98955i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.31973e7	2.65575
$478$	0	0
$479$	4.05783e6	0.808081	0.404040	−	0.914741i	$-0.367605\pi$
$479$	4.05783e6	0.808081	0.404040	+	0.914741i	$0.367605\pi$
$480$	0	0
$481$	3.92856e6	0.774232
$482$	0	0
$483$	−4.16028e6	−0.811437
$484$	0	0
$485$	0	0
$486$	0	0
$487$	121792.i	0.0232700i	0.999932	+	0.0116350i	$0.00370361\pi$
$487$	121792.i	0.0232700i	−0.999932	+	0.0116350i	$0.996296\pi$
$488$	0	0
$489$	4.32902e6	0.818687
$490$	0	0
$491$	2.80649e6i	0.525363i	0.964883	+	0.262682i	$0.0846069\pi$
$491$	2.80649e6i	0.525363i	−0.964883	+	0.262682i	$0.915393\pi$
$492$	0	0
$493$	−2.45809e6	−0.455491
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.44332e6i	− 0.806894i
$498$	0	0
$499$	− 2.59476e6i	− 0.466494i	−0.972418	−	0.233247i	$-0.925065\pi$
$499$	− 2.59476e6i	− 0.466494i	0.972418	−	0.233247i	$-0.0749351\pi$
$500$	0	0
$501$	− 1.10606e7i	− 1.96873i
$502$	0	0
$503$	− 1.04638e6i	− 0.184404i	−0.995740	−	0.0922021i	$-0.970609\pi$
$503$	− 1.04638e6i	− 0.184404i	0.995740	−	0.0922021i	$-0.0293906\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.06344e7	−1.83735
$508$	0	0
$509$	8.38369e6i	1.43430i	0.696918	+	0.717151i	$0.254554\pi$
$509$	8.38369e6i	1.43430i	−0.696918	+	0.717151i	$0.745446\pi$
$510$	0	0
$511$	2.16504e6	0.366787
$512$	0	0
$513$	4.09904e6i	0.687684i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.92664e6	−0.975174
$518$	0	0
$519$	7.05000e6	1.14887
$520$	0	0
$521$	−4.38397e6	−0.707577	−0.353788	−	0.935325i	$-0.615107\pi$
$521$	−4.38397e6	−0.707577	−0.353788	+	0.935325i	$0.615107\pi$
$522$	0	0
$523$	−915789.	−0.146400	−0.0732000	−	0.997317i	$-0.523321\pi$
$523$	−915789.	−0.146400	−0.0732000	+	0.997317i	$0.523321\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.64309e6i	0.885096i
$528$	0	0
$529$	3.86810e6	0.600978
$530$	0	0
$531$	8.48634e6i	1.30612i
$532$	0	0
$533$	3.37257e6	0.514212
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.01862e7i	3.02077i
$538$	0	0
$539$	− 4.49581e6i	− 0.666556i
$540$	0	0
$541$	− 6.98594e6i	− 1.02620i	−0.858329	−	0.513100i	$-0.828497\pi$
$541$	− 6.98594e6i	− 1.02620i	0.858329	−	0.513100i	$-0.171503\pi$
$542$	0	0
$543$	1.52904e7i	2.22545i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.72947e6	−0.675841	−0.337920	−	0.941175i	$-0.609723\pi$
$547$	−4.72947e6	−0.675841	−0.337920	+	0.941175i	$0.609723\pi$
$548$	0	0
$549$	1.08556e6i	0.153718i
$550$	0	0
$551$	−2.49402e6	−0.349962
$552$	0	0
$553$	3.13231e6i	0.435564i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.26312e6	−0.445651	−0.222825	−	0.974858i	$-0.571528\pi$
$557$	−3.26312e6	−0.445651	−0.222825	+	0.974858i	$0.571528\pi$
$558$	0	0
$559$	1.16578e7	1.57793
$560$	0	0
$561$	2.11340e7	2.83514
$562$	0	0
$563$	−5.81898e6	−0.773706	−0.386853	−	0.922141i	$-0.626438\pi$
$563$	−5.81898e6	−0.773706	−0.386853	+	0.922141i	$0.626438\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 475223.i	− 0.0620782i
$568$	0	0
$569$	9.08770e6	1.17672	0.588360	−	0.808599i	$-0.299774\pi$
$569$	9.08770e6	1.17672	0.588360	+	0.808599i	$0.299774\pi$
$570$	0	0
$571$	258870.i	0.0332270i	0.999862	+	0.0166135i	$0.00528849\pi$
$571$	258870.i	0.0332270i	−0.999862	+	0.0166135i	$0.994712\pi$
$572$	0	0
$573$	−2.11875e7	−2.69583
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.45070e6i	− 0.306444i	−0.988192	−	0.153222i	$-0.951035\pi$
$577$	− 2.45070e6i	− 0.306444i	0.988192	−	0.153222i	$-0.0489650\pi$
$578$	0	0
$579$	4.61182e6i	0.571710i
$580$	0	0
$581$	9.73547e6i	1.19651i
$582$	0	0
$583$	− 2.54187e7i	− 3.09729i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.61982e6	−0.433602	−0.216801	−	0.976216i	$-0.569562\pi$
$587$	−3.61982e6	−0.433602	−0.216801	+	0.976216i	$0.569562\pi$
$588$	0	0
$589$	5.72557e6i	0.680034i
$590$	0	0
$591$	−9.78339e6	−1.15218
$592$	0	0
$593$	− 7.07347e6i	− 0.826030i	−0.910724	−	0.413015i	$-0.864476\pi$
$593$	− 7.07347e6i	− 0.826030i	0.910724	−	0.413015i	$-0.135524\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.46370e7	1.68080
$598$	0	0
$599$	−2.71813e6	−0.309530	−0.154765	−	0.987951i	$-0.549462\pi$
$599$	−2.71813e6	−0.309530	−0.154765	+	0.987951i	$0.549462\pi$
$600$	0	0
$601$	1.67171e6	0.188788	0.0943942	−	0.995535i	$-0.469909\pi$
$601$	1.67171e6	0.188788	0.0943942	+	0.995535i	$0.469909\pi$
$602$	0	0
$603$	−2.11462e7	−2.36832
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.35562e6i	− 0.259498i	−0.991547	−	0.129749i	$-0.958583\pi$
$607$	− 2.35562e6i	− 0.259498i	0.991547	−	0.129749i	$-0.0414171\pi$
$608$	0	0
$609$	−5.60338e6	−0.612219
$610$	0	0
$611$	− 7.13705e6i	− 0.773421i
$612$	0	0
$613$	1.88749e6	0.202877	0.101439	−	0.994842i	$-0.467655\pi$
$613$	1.88749e6	0.202877	0.101439	+	0.994842i	$0.467655\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.45446e7i	− 1.53812i	−0.639178	−	0.769058i	$-0.720725\pi$
$617$	− 1.45446e7i	− 1.53812i	0.639178	−	0.769058i	$-0.279275\pi$
$618$	0	0
$619$	1.08497e7i	1.13813i	0.822292	+	0.569066i	$0.192695\pi$
$619$	1.08497e7i	1.13813i	−0.822292	+	0.569066i	$0.807305\pi$
$620$	0	0
$621$	5.68521e6i	0.591585i
$622$	0	0
$623$	− 122075.i	− 0.0126011i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.14430e7	2.17829
$628$	0	0
$629$	− 5.01520e6i	− 0.505431i
$630$	0	0
$631$	−3.76561e6	−0.376497	−0.188249	−	0.982121i	$-0.560281\pi$
$631$	−3.76561e6	−0.376497	−0.188249	+	0.982121i	$0.560281\pi$
$632$	0	0
$633$	1.07287e7i	1.06424i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.41400e6	0.528652
$638$	0	0
$639$	−1.64917e7	−1.59776
$640$	0	0
$641$	−223711.	−0.0215052	−0.0107526	−	0.999942i	$-0.503423\pi$
$641$	−223711.	−0.0215052	−0.0107526	+	0.999942i	$0.503423\pi$
$642$	0	0
$643$	−2.53011e6	−0.241330	−0.120665	−	0.992693i	$-0.538503\pi$
$643$	−2.53011e6	−0.241330	−0.120665	+	0.992693i	$0.538503\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.44825e6i	− 0.793425i	−0.917943	−	0.396713i	$-0.870151\pi$
$647$	− 8.44825e6i	− 0.793425i	0.917943	−	0.396713i	$-0.129849\pi$
$648$	0	0
$649$	1.63452e7	1.52328
$650$	0	0
$651$	1.28638e7i	1.18964i
$652$	0	0
$653$	−1.72303e7	−1.58128	−0.790641	−	0.612280i	$-0.790253\pi$
$653$	−1.72303e7	−1.58128	−0.790641	+	0.612280i	$0.790253\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 8.03568e6i	− 0.726289i
$658$	0	0
$659$	− 5.43243e6i	− 0.487282i	−0.969865	−	0.243641i	$-0.921658\pi$
$659$	− 5.43243e6i	− 0.487282i	0.969865	−	0.243641i	$-0.0783419\pi$
$660$	0	0
$661$	− 1.12770e7i	− 1.00390i	−0.864897	−	0.501949i	$-0.832616\pi$
$661$	− 1.12770e7i	− 1.00390i	0.864897	−	0.501949i	$-0.167384\pi$
$662$	0	0
$663$	2.54503e7i	2.24858i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.45911e6	−0.301057
$668$	0	0
$669$	2.39402e7i	2.06806i
$670$	0	0
$671$	2.09086e6	0.179275
$672$	0	0
$673$	1.15475e7i	0.982769i	0.870943	+	0.491384i	$0.163509\pi$
$673$	1.15475e7i	0.982769i	−0.870943	+	0.491384i	$0.836491\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.72804e6	0.731889	0.365944	−	0.930637i	$-0.380746\pi$
$677$	8.72804e6	0.731889	0.365944	+	0.930637i	$0.380746\pi$
$678$	0	0
$679$	7.67862e6	0.639159
$680$	0	0
$681$	−2.17009e7	−1.79313
$682$	0	0
$683$	5.37492e6	0.440880	0.220440	−	0.975401i	$-0.429251\pi$
$683$	5.37492e6	0.440880	0.220440	+	0.975401i	$0.429251\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.63849e7i	1.32450i
$688$	0	0
$689$	3.06101e7	2.45650
$690$	0	0
$691$	1.03106e7i	0.821467i	0.911756	+	0.410733i	$0.134727\pi$
$691$	1.03106e7i	0.821467i	−0.911756	+	0.410733i	$0.865273\pi$
$692$	0	0
$693$	2.95233e7	2.33524
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.30541e6i	− 0.335686i
$698$	0	0
$699$	2.05556e7i	1.59124i
$700$	0	0
$701$	7.87993e6i	0.605658i	0.953045	+	0.302829i	$0.0979311\pi$
$701$	7.87993e6i	0.605658i	−0.953045	+	0.302829i	$0.902069\pi$
$702$	0	0
$703$	− 5.08851e6i	− 0.388331i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.22315e6	−0.242511
$708$	0	0
$709$	− 2.06172e7i	− 1.54033i	−0.637846	−	0.770164i	$-0.720174\pi$
$709$	− 2.06172e7i	− 1.54033i	0.637846	−	0.770164i	$-0.279826\pi$
$710$	0	0
$711$	1.16258e7	0.862476
$712$	0	0
$713$	7.94115e6i	0.585005i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.17280e7	−1.57842
$718$	0	0
$719$	1.37162e7	0.989490	0.494745	−	0.869038i	$-0.335262\pi$
$719$	1.37162e7	0.989490	0.494745	+	0.869038i	$0.335262\pi$
$720$	0	0
$721$	−1.45383e7	−1.04154
$722$	0	0
$723$	4.14305e7	2.94764
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2.33681e7i	− 1.63979i	−0.572514	−	0.819895i	$-0.694032\pi$
$727$	− 2.33681e7i	− 1.63979i	0.572514	−	0.819895i	$-0.305968\pi$
$728$	0	0
$729$	−2.33601e7	−1.62800
$730$	0	0
$731$	− 1.48823e7i	− 1.03010i
$732$	0	0
$733$	2.94411e6	0.202392	0.101196	−	0.994866i	$-0.467733\pi$
$733$	2.94411e6	0.202392	0.101196	+	0.994866i	$0.467733\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.07289e7i	2.76207i
$738$	0	0
$739$	− 3.66027e6i	− 0.246548i	−0.992373	−	0.123274i	$-0.960661\pi$
$739$	− 3.66027e6i	− 0.246548i	0.992373	−	0.123274i	$-0.0393395\pi$
$740$	0	0
$741$	2.58223e7i	1.72763i
$742$	0	0
$743$	2.60824e7i	1.73331i	0.498912	+	0.866653i	$0.333733\pi$
$743$	2.60824e7i	1.73331i	−0.498912	+	0.866653i	$0.666267\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.61338e7	2.36926
$748$	0	0
$749$	3.60535e6i	0.234824i
$750$	0	0
$751$	1.03196e7	0.667673	0.333836	−	0.942631i	$-0.391657\pi$
$751$	1.03196e7	0.667673	0.333836	+	0.942631i	$0.391657\pi$
$752$	0	0
$753$	2.15668e7i	1.38611i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.70313e7	−1.71446	−0.857228	−	0.514936i	$-0.827816\pi$
$757$	−2.70313e7	−1.71446	−0.857228	+	0.514936i	$0.827816\pi$
$758$	0	0
$759$	2.97405e7	1.87389
$760$	0	0
$761$	1.29548e7	0.810905	0.405452	−	0.914116i	$-0.367114\pi$
$761$	1.29548e7	0.810905	0.405452	+	0.914116i	$0.367114\pi$
$762$	0	0
$763$	8.67946e6	0.539736
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.96834e7i	1.20813i
$768$	0	0
$769$	−8.35551e6	−0.509515	−0.254757	−	0.967005i	$-0.581996\pi$
$769$	−8.35551e6	−0.509515	−0.254757	+	0.967005i	$0.581996\pi$
$770$	0	0
$771$	− 1.91732e7i	− 1.16161i
$772$	0	0
$773$	3.12215e7	1.87934	0.939668	−	0.342087i	$-0.111134\pi$
$773$	3.12215e7	1.87934	0.939668	+	0.342087i	$0.111134\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.14325e7i	− 0.679342i
$778$	0	0
$779$	− 4.36835e6i	− 0.257913i
$780$	0	0
$781$	3.17639e7i	1.86340i
$782$	0	0
$783$	7.65727e6i	0.446344i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.14664e7	−0.659920	−0.329960	−	0.943995i	$-0.607035\pi$
$787$	−1.14664e7	−0.659920	−0.329960	+	0.943995i	$0.607035\pi$
$788$	0	0
$789$	6.53222e6i	0.373567i
$790$	0	0
$791$	−5.93373e6	−0.337199
$792$	0	0
$793$	2.51789e6i	0.142185i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.28998e7	−1.27699	−0.638493	−	0.769628i	$-0.720442\pi$
$797$	−2.28998e7	−1.27699	−0.638493	+	0.769628i	$0.720442\pi$
$798$	0	0
$799$	−9.11115e6	−0.504901
$800$	0	0
$801$	−453090.	−0.0249519
$802$	0	0
$803$	−1.54772e7	−0.847040
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.81614e7i	1.52220i
$808$	0	0
$809$	−2.82761e7	−1.51896	−0.759482	−	0.650528i	$-0.774548\pi$
$809$	−2.82761e7	−1.51896	−0.759482	+	0.650528i	$0.774548\pi$
$810$	0	0
$811$	2.97657e7i	1.58915i	0.607167	+	0.794574i	$0.292306\pi$
$811$	2.97657e7i	1.58915i	−0.607167	+	0.794574i	$0.707694\pi$
$812$	0	0
$813$	−2.68752e7	−1.42602
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.50999e7i	− 0.791441i
$818$	0	0
$819$	3.55529e7i	1.85210i
$820$	0	0
$821$	− 2.33663e7i	− 1.20985i	−0.796282	−	0.604926i	$-0.793203\pi$
$821$	− 2.33663e7i	− 1.20985i	0.796282	−	0.604926i	$-0.206797\pi$
$822$	0	0
$823$	3.90864e6i	0.201153i	0.994929	+	0.100576i	$0.0320687\pi$
$823$	3.90864e6i	0.201153i	−0.994929	+	0.100576i	$0.967931\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.34407e7	0.683371	0.341686	−	0.939814i	$-0.389002\pi$
$827$	1.34407e7	0.683371	0.341686	+	0.939814i	$0.389002\pi$
$828$	0	0
$829$	− 2.39273e7i	− 1.20923i	−0.796519	−	0.604613i	$-0.793328\pi$
$829$	− 2.39273e7i	− 1.20923i	0.796519	−	0.604613i	$-0.206672\pi$
$830$	0	0
$831$	1.54090e7	0.774057
$832$	0	0
$833$	− 6.91151e6i	− 0.345112i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.75790e7	0.867320
$838$	0	0
$839$	−4.44572e6	−0.218041	−0.109020	−	0.994040i	$-0.534771\pi$
$839$	−4.44572e6	−0.218041	−0.109020	+	0.994040i	$0.534771\pi$
$840$	0	0
$841$	1.58522e7	0.772856
$842$	0	0
$843$	1.63389e7	0.791871
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.01748e7i	− 1.92418i
$848$	0	0
$849$	−2.27622e7	−1.08379
$850$	0	0
$851$	− 7.05756e6i	− 0.334065i
$852$	0	0
$853$	−2.57954e7	−1.21386	−0.606931	−	0.794754i	$-0.707600\pi$
$853$	−2.57954e7	−1.21386	−0.606931	+	0.794754i	$0.707600\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.79012e7i	− 0.832590i	−0.909230	−	0.416295i	$-0.863328\pi$
$857$	− 1.79012e7i	− 0.832590i	0.909230	−	0.416295i	$-0.136672\pi$
$858$	0	0
$859$	− 2.50747e7i	− 1.15945i	−0.814811	−	0.579727i	$-0.803159\pi$
$859$	− 2.50747e7i	− 1.15945i	0.814811	−	0.579727i	$-0.196841\pi$
$860$	0	0
$861$	− 9.81449e6i	− 0.451190i
$862$	0	0
$863$	− 8.98511e6i	− 0.410673i	−0.978691	−	0.205337i	$-0.934171\pi$
$863$	− 8.98511e6i	− 0.410673i	0.978691	−	0.205337i	$-0.0658289\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.08057e6	−0.139182
$868$	0	0
$869$	− 2.23919e7i	− 1.00587i
$870$	0	0
$871$	−4.90471e7	−2.19063
$872$	0	0
$873$	− 2.84997e7i	− 1.26562i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.08381e7	−0.475834	−0.237917	−	0.971286i	$-0.576465\pi$
$877$	−1.08381e7	−0.475834	−0.237917	+	0.971286i	$0.576465\pi$
$878$	0	0
$879$	−1.28780e7	−0.562179
$880$	0	0
$881$	−816876.	−0.0354582	−0.0177291	−	0.999843i	$-0.505644\pi$
$881$	−816876.	−0.0354582	−0.0177291	+	0.999843i	$0.505644\pi$
$882$	0	0
$883$	−2.50842e6	−0.108268	−0.0541339	−	0.998534i	$-0.517240\pi$
$883$	−2.50842e6	−0.108268	−0.0541339	+	0.998534i	$0.517240\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.58294e7i	− 0.675545i	−0.941228	−	0.337773i	$-0.890327\pi$
$887$	− 1.58294e7i	− 0.675545i	0.941228	−	0.337773i	$-0.109673\pi$
$888$	0	0
$889$	916878.	0.0389096
$890$	0	0
$891$	3.39722e6i	0.143360i
$892$	0	0
$893$	−9.24434e6	−0.387924
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.58145e7i	1.48620i
$898$	0	0
$899$	1.06957e7i	0.441379i
$900$	0	0
$901$	− 3.90768e7i	− 1.60364i
$902$	0	0
$903$	− 3.39253e7i	− 1.38454i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.36328e7	0.953887	0.476944	−	0.878934i	$-0.341745\pi$
$907$	2.36328e7	0.953887	0.476944	+	0.878934i	$0.341745\pi$
$908$	0	0
$909$	1.19629e7i	0.480206i
$910$	0	0
$911$	9.17484e6	0.366271	0.183136	−	0.983088i	$-0.441375\pi$
$911$	9.17484e6	0.366271	0.183136	+	0.983088i	$0.441375\pi$
$912$	0	0
$913$	− 6.95959e7i	− 2.76316i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.14461e7	0.449504
$918$	0	0
$919$	−3.58059e7	−1.39851	−0.699255	−	0.714872i	$-0.746485\pi$
$919$	−3.58059e7	−1.39851	−0.699255	+	0.714872i	$0.746485\pi$
$920$	0	0
$921$	−3.67497e7	−1.42759
$922$	0	0
$923$	−3.82511e7	−1.47788
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.39597e7i	2.06239i
$928$	0	0
$929$	2.41031e7	0.916291	0.458146	−	0.888877i	$-0.348514\pi$
$929$	2.41031e7	0.916291	0.458146	+	0.888877i	$0.348514\pi$
$930$	0	0
$931$	− 7.01254e6i	− 0.265156i
$932$	0	0
$933$	1.37265e7	0.516245
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.84615e7i	0.686938i	0.939164	+	0.343469i	$0.111602\pi$
$937$	1.84615e7i	0.686938i	−0.939164	+	0.343469i	$0.888398\pi$
$938$	0	0
$939$	− 4.28032e7i	− 1.58421i
$940$	0	0
$941$	− 3.10319e6i	− 0.114244i	−0.998367	−	0.0571220i	$-0.981808\pi$
$941$	− 3.10319e6i	− 0.114244i	0.998367	−	0.0571220i	$-0.0181924\pi$
$942$	0	0
$943$	− 6.05873e6i	− 0.221872i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.89572e7	−1.77395	−0.886975	−	0.461817i	$-0.847198\pi$
$947$	−4.89572e7	−1.77395	−0.886975	+	0.461817i	$0.847198\pi$
$948$	0	0
$949$	− 1.86382e7i	− 0.671796i
$950$	0	0
$951$	−2.45162e7	−0.879025
$952$	0	0
$953$	3.94753e7i	1.40797i	0.710216	+	0.703984i	$0.248597\pi$
$953$	3.94753e7i	1.40797i	−0.710216	+	0.703984i	$0.751403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.00568e7	1.41383
$958$	0	0
$959$	−3.15395e7	−1.10741
$960$	0	0
$961$	−4.07470e6	−0.142327
$962$	0	0
$963$	1.33815e7	0.464984
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.23447e7i	− 0.768438i	−0.923242	−	0.384219i	$-0.874471\pi$
$967$	− 2.23447e7i	− 0.768438i	0.923242	−	0.384219i	$-0.125529\pi$
$968$	0	0
$969$	3.29647e7	1.12782
$970$	0	0
$971$	− 2.63033e7i	− 0.895288i	−0.894212	−	0.447644i	$-0.852263\pi$
$971$	− 2.63033e7i	− 0.895288i	0.894212	−	0.447644i	$-0.147737\pi$
$972$	0	0
$973$	6.15315e6	0.208360
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.78338e7i	− 1.60324i	−0.597835	−	0.801619i	$-0.703972\pi$
$977$	− 4.78338e7i	− 1.60324i	0.597835	−	0.801619i	$-0.296028\pi$
$978$	0	0
$979$	872679.i	0.0291003i
$980$	0	0
$981$	− 3.22144e7i	− 1.06875i
$982$	0	0
$983$	− 4.88361e7i	− 1.61197i	−0.591935	−	0.805986i	$-0.701636\pi$
$983$	− 4.88361e7i	− 1.61197i	0.591935	−	0.805986i	$-0.298364\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.07695e7	−0.678630
$988$	0	0
$989$	− 2.09429e7i	− 0.680843i
$990$	0	0
$991$	−7.49231e6	−0.242344	−0.121172	−	0.992632i	$-0.538665\pi$
$991$	−7.49231e6	−0.242344	−0.121172	+	0.992632i	$0.538665\pi$
$992$	0	0
$993$	− 8.66569e6i	− 0.278888i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.46209e7	0.465841	0.232920	−	0.972496i	$-0.425172\pi$
$997$	1.46209e7	0.465841	0.232920	+	0.972496i	$0.425172\pi$
$998$	0	0
$999$	−1.56230e7	−0.495280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.b.49.4		20
4.3	odd	2		200.6.f.c.149.5		20
5.2	odd	4		160.6.d.a.81.18		20
5.3	odd	4		800.6.d.c.401.3		20
5.4	even	2		800.6.f.c.49.17		20
8.3	odd	2		200.6.f.b.149.15		20
8.5	even	2		800.6.f.c.49.18		20
20.3	even	4		200.6.d.b.101.6		20
20.7	even	4		40.6.d.a.21.15	✓	20
20.19	odd	2		200.6.f.b.149.16		20
40.3	even	4		200.6.d.b.101.5		20
40.13	odd	4		800.6.d.c.401.18		20
40.19	odd	2		200.6.f.c.149.6		20
40.27	even	4		40.6.d.a.21.16	yes	20
40.29	even	2	inner	800.6.f.b.49.3		20
40.37	odd	4		160.6.d.a.81.3		20
60.47	odd	4		360.6.k.b.181.6		20
120.107	odd	4		360.6.k.b.181.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.15	✓	20	20.7	even	4
40.6.d.a.21.16	yes	20	40.27	even	4
160.6.d.a.81.3		20	40.37	odd	4
160.6.d.a.81.18		20	5.2	odd	4
200.6.d.b.101.5		20	40.3	even	4
200.6.d.b.101.6		20	20.3	even	4
200.6.f.b.149.15		20	8.3	odd	2
200.6.f.b.149.16		20	20.19	odd	2
200.6.f.c.149.5		20	4.3	odd	2
200.6.f.c.149.6		20	40.19	odd	2
360.6.k.b.181.5		20	120.107	odd	4
360.6.k.b.181.6		20	60.47	odd	4
800.6.d.c.401.3		20	5.3	odd	4
800.6.d.c.401.18		20	40.13	odd	4
800.6.f.b.49.3		20	40.29	even	2	inner
800.6.f.b.49.4		20	1.1	even	1	trivial
800.6.f.c.49.17		20	5.4	even	2
800.6.f.c.49.18		20	8.5	even	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-25.0521 q^{3} +103.624i q^{7} +384.607 q^{9} -740.776i q^{11} +892.067 q^{13} -1138.81i q^{17} -1155.46i q^{19} -2596.00i q^{21} -1602.57i q^{23} -3547.55 q^{27} -2158.47i q^{29} -4955.24 q^{31} +18558.0i q^{33} +4403.89 q^{37} -22348.1 q^{39} +3780.62 q^{41} +13068.3 q^{43} -8000.58i q^{47} +6069.06 q^{49} +28529.6i q^{51} +34313.6 q^{53} +28946.6i q^{57} +22065.0i q^{59} +2822.53i q^{61} +39854.5i q^{63} -54981.5 q^{67} +40147.8i q^{69} -42879.2 q^{71} -20893.2i q^{73} +76762.2 q^{77} +30227.6 q^{79} -4586.03 q^{81} +93949.9 q^{83} +54074.1i q^{87} -1178.06 q^{89} +92439.5i q^{91} +124139. q^{93} -74100.8i q^{97} -284908. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 36 q^{3} + 1620 q^{9} - 11664 q^{27} - 7160 q^{31} - 3608 q^{37} - 44904 q^{39} + 11608 q^{41} + 51772 q^{43} - 18756 q^{49} + 928 q^{53} + 161604 q^{67} + 200312 q^{71} + 26008 q^{77} + 282080 q^{79}+ \cdots + 293472 q^{93}+O(q^{100}) \) 20 * q - 36 * q^3 + 1620 * q^9 - 11664 * q^27 - 7160 * q^31 - 3608 * q^37 - 44904 * q^39 + 11608 * q^41 + 51772 * q^43 - 18756 * q^49 + 928 * q^53 + 161604 * q^67 + 200312 * q^71 + 26008 * q^77 + 282080 * q^79 + 65172 * q^81 + 99092 * q^83 + 3160 * q^89 + 293472 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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