LMFDB - Embedded newform 800.6.d.c.401.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(401,800)]

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, 0]))

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

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.d (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^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.19
Root			$-3.90102 + 0.884346i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.6.d.c.401.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.4343i q^{3} -56.4938 q^{7} -403.904 q^{9} +O(q^{10})$	$q+25.4343i q^{3} -56.4938 q^{7} -403.904 q^{9} +261.019i q^{11} -720.631i q^{13} +1876.44 q^{17} +1992.33i q^{19} -1436.88i q^{21} +2570.29 q^{23} -4092.49i q^{27} +1700.16i q^{29} +7734.68 q^{31} -6638.83 q^{33} -12228.1i q^{37} +18328.8 q^{39} +14979.3 q^{41} +18113.9i q^{43} +2141.03 q^{47} -13615.4 q^{49} +47726.0i q^{51} -1605.71i q^{53} -50673.5 q^{57} -2680.90i q^{59} +44521.9i q^{61} +22818.1 q^{63} -12486.0i q^{67} +65373.7i q^{69} -8189.38 q^{71} +41082.7 q^{73} -14746.0i q^{77} -46325.9 q^{79} +5940.95 q^{81} +61655.4i q^{83} -43242.3 q^{87} +53205.4 q^{89} +40711.2i q^{91} +196726. i q^{93} +39211.8 q^{97} -105427. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 196 q^{7} - 1620 q^{9}+O(q^{10}) $ 20 * q - 196 * q^7 - 1620 * q^9	$ 20 q - 196 q^{7} - 1620 q^{9} - 4676 q^{23} - 7160 q^{31} - 5672 q^{33} + 44904 q^{39} + 11608 q^{41} + 44180 q^{47} + 18756 q^{49} - 5032 q^{57} + 240620 q^{63} + 200312 q^{71} + 105136 q^{73} - 282080 q^{79} + 65172 q^{81} - 332592 q^{87} - 3160 q^{89} - 147376 q^{97}+O(q^{100}) $ 20 * q - 196 * q^7 - 1620 * q^9 - 4676 * q^23 - 7160 * q^31 - 5672 * q^33 + 44904 * q^39 + 11608 * q^41 + 44180 * q^47 + 18756 * q^49 - 5032 * q^57 + 240620 * q^63 + 200312 * q^71 + 105136 * q^73 - 282080 * q^79 + 65172 * q^81 - 332592 * q^87 - 3160 * q^89 - 147376 * q^97

Display 100 coefficients 10 coefficients

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
$3$	25.4343i	1.63161i	0.578326	+	0.815806i	$0.303706\pi$
$3$	25.4343i	1.63161i	−0.578326	+	0.815806i	$0.696294\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−56.4938	−0.435769	−0.217884	−	0.975975i	$-0.569916\pi$
$7$	−56.4938	−0.435769	−0.217884	+	0.975975i	$0.569916\pi$
$8$	0	0
$9$	−403.904	−1.66216
$10$	0	0
$11$	261.019i	0.650414i	0.945643	+	0.325207i	$0.105434\pi$
$11$	261.019i	0.650414i	−0.945643	+	0.325207i	$0.894566\pi$
$12$	0	0
$13$	− 720.631i	− 1.18265i	−0.806435	−	0.591323i	$-0.798606\pi$
$13$	− 720.631i	− 1.18265i	0.806435	−	0.591323i	$-0.201394\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1876.44	1.57476	0.787378	−	0.616471i	$-0.211438\pi$
$17$	1876.44	1.57476	0.787378	+	0.616471i	$0.211438\pi$
$18$	0	0
$19$	1992.33i	1.26613i	0.774100	+	0.633063i	$0.218203\pi$
$19$	1992.33i	1.26613i	−0.774100	+	0.633063i	$0.781797\pi$
$20$	0	0
$21$	− 1436.88i	− 0.711005i
$22$	0	0
$23$	2570.29	1.01313	0.506563	−	0.862203i	$-0.330916\pi$
$23$	2570.29	1.01313	0.506563	+	0.862203i	$0.330916\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4092.49i	− 1.08038i
$28$	0	0
$29$	1700.16i	0.375400i	0.982226	+	0.187700i	$0.0601032\pi$
$29$	1700.16i	0.375400i	−0.982226	+	0.187700i	$0.939897\pi$
$30$	0	0
$31$	7734.68	1.44557	0.722783	−	0.691075i	$-0.242863\pi$
$31$	7734.68	1.44557	0.722783	+	0.691075i	$0.242863\pi$
$32$	0	0
$33$	−6638.83	−1.06122
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 12228.1i	− 1.46844i	−0.678913	−	0.734218i	$-0.737549\pi$
$37$	− 12228.1i	− 1.46844i	0.678913	−	0.734218i	$-0.262451\pi$
$38$	0	0
$39$	18328.8	1.92962
$40$	0	0
$41$	14979.3	1.39166	0.695830	−	0.718206i	$-0.255037\pi$
$41$	14979.3	1.39166	0.695830	+	0.718206i	$0.255037\pi$
$42$	0	0
$43$	18113.9i	1.49397i	0.664842	+	0.746984i	$0.268499\pi$
$43$	18113.9i	1.49397i	−0.664842	+	0.746984i	$0.731501\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2141.03	0.141377	0.0706885	−	0.997498i	$-0.477480\pi$
$47$	2141.03	0.141377	0.0706885	+	0.997498i	$0.477480\pi$
$48$	0	0
$49$	−13615.4	−0.810106
$50$	0	0
$51$	47726.0i	2.56939i
$52$	0	0
$53$	− 1605.71i	− 0.0785192i	−0.999229	−	0.0392596i	$-0.987500\pi$
$53$	− 1605.71i	− 0.0785192i	0.999229	−	0.0392596i	$-0.0124999\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−50673.5	−2.06583
$58$	0	0
$59$	− 2680.90i	− 0.100265i	−0.998743	−	0.0501327i	$-0.984036\pi$
$59$	− 2680.90i	− 0.100265i	0.998743	−	0.0501327i	$-0.0159644\pi$
$60$	0	0
$61$	44521.9i	1.53197i	0.642860	+	0.765984i	$0.277748\pi$
$61$	44521.9i	1.53197i	−0.642860	+	0.765984i	$0.722252\pi$
$62$	0	0
$63$	22818.1	0.724316
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12486.0i	− 0.339809i	−0.985461	−	0.169905i	$-0.945654\pi$
$67$	− 12486.0i	− 0.339809i	0.985461	−	0.169905i	$-0.0543460\pi$
$68$	0	0
$69$	65373.7i	1.65303i
$70$	0	0
$71$	−8189.38	−0.192799	−0.0963996	−	0.995343i	$-0.530733\pi$
$71$	−8189.38	−0.192799	−0.0963996	+	0.995343i	$0.530733\pi$
$72$	0	0
$73$	41082.7	0.902302	0.451151	−	0.892448i	$-0.351014\pi$
$73$	41082.7	0.902302	0.451151	+	0.892448i	$0.351014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 14746.0i	− 0.283430i
$78$	0	0
$79$	−46325.9	−0.835134	−0.417567	−	0.908646i	$-0.637117\pi$
$79$	−46325.9	−0.835134	−0.417567	+	0.908646i	$0.637117\pi$
$80$	0	0
$81$	5940.95	0.100611
$82$	0	0
$83$	61655.4i	0.982372i	0.871055	+	0.491186i	$0.163436\pi$
$83$	61655.4i	0.982372i	−0.871055	+	0.491186i	$0.836564\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−43242.3	−0.612507
$88$	0	0
$89$	53205.4	0.712001	0.356000	−	0.934486i	$-0.384140\pi$
$89$	53205.4	0.712001	0.356000	+	0.934486i	$0.384140\pi$
$90$	0	0
$91$	40711.2i	0.515360i
$92$	0	0
$93$	196726.i	2.35860i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	39211.8	0.423143	0.211571	−	0.977363i	$-0.432142\pi$
$97$	39211.8	0.423143	0.211571	+	0.977363i	$0.432142\pi$
$98$	0	0
$99$	− 105427.i	− 1.08109i
$100$	0	0
$101$	− 41893.0i	− 0.408637i	−0.978904	−	0.204319i	$-0.934502\pi$
$101$	− 41893.0i	− 0.408637i	0.978904	−	0.204319i	$-0.0654979\pi$
$102$	0	0
$103$	−118358.	−1.09927	−0.549635	−	0.835405i	$-0.685233\pi$
$103$	−118358.	−1.09927	−0.549635	+	0.835405i	$0.685233\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 147978.i	− 1.24951i	−0.780822	−	0.624754i	$-0.785199\pi$
$107$	− 147978.i	− 1.24951i	0.780822	−	0.624754i	$-0.214801\pi$
$108$	0	0
$109$	126538.i	1.02013i	0.860135	+	0.510066i	$0.170379\pi$
$109$	126538.i	1.02013i	−0.860135	+	0.510066i	$0.829621\pi$
$110$	0	0
$111$	311014.	2.39592
$112$	0	0
$113$	221898.	1.63478	0.817388	−	0.576088i	$-0.195421\pi$
$113$	221898.	1.63478	0.817388	+	0.576088i	$0.195421\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	291066.i	1.96574i
$118$	0	0
$119$	−106007.	−0.686229
$120$	0	0
$121$	92920.2	0.576961
$122$	0	0
$123$	380989.i	2.27065i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	237825.	1.30842	0.654212	−	0.756312i	$-0.273001\pi$
$127$	237825.	1.30842	0.654212	+	0.756312i	$0.273001\pi$
$128$	0	0
$129$	−460715.	−2.43758
$130$	0	0
$131$	− 151213.i	− 0.769856i	−0.922947	−	0.384928i	$-0.874226\pi$
$131$	− 151213.i	− 0.769856i	0.922947	−	0.384928i	$-0.125774\pi$
$132$	0	0
$133$	− 112554.i	− 0.551738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−163216.	−0.742954	−0.371477	−	0.928442i	$-0.621148\pi$
$137$	−163216.	−0.742954	−0.371477	+	0.928442i	$0.621148\pi$
$138$	0	0
$139$	7490.33i	0.0328824i	0.999865	+	0.0164412i	$0.00523364\pi$
$139$	7490.33i	0.0328824i	−0.999865	+	0.0164412i	$0.994766\pi$
$140$	0	0
$141$	54455.7i	0.230672i
$142$	0	0
$143$	188098.	0.769209
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 346300.i	− 1.32178i
$148$	0	0
$149$	35543.3i	0.131157i	0.997847	+	0.0655786i	$0.0208893\pi$
$149$	35543.3i	0.131157i	−0.997847	+	0.0655786i	$0.979111\pi$
$150$	0	0
$151$	−549802.	−1.96229	−0.981147	−	0.193263i	$-0.938093\pi$
$151$	−549802.	−1.96229	−0.981147	+	0.193263i	$0.938093\pi$
$152$	0	0
$153$	−757903.	−2.61749
$154$	0	0
$155$	0	0
$156$	0	0
$157$	252420.i	0.817287i	0.912694	+	0.408643i	$0.133998\pi$
$157$	252420.i	0.817287i	−0.912694	+	0.408643i	$0.866002\pi$
$158$	0	0
$159$	40840.0	0.128113
$160$	0	0
$161$	−145206.	−0.441488
$162$	0	0
$163$	− 383218.i	− 1.12974i	−0.825182	−	0.564868i	$-0.808927\pi$
$163$	− 383218.i	− 1.12974i	0.825182	−	0.564868i	$-0.191073\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	108418.	0.300821	0.150411	−	0.988624i	$-0.451940\pi$
$167$	108418.	0.300821	0.150411	+	0.988624i	$0.451940\pi$
$168$	0	0
$169$	−148016.	−0.398650
$170$	0	0
$171$	− 804710.i	− 2.10450i
$172$	0	0
$173$	− 305932.i	− 0.777157i	−0.921416	−	0.388579i	$-0.872966\pi$
$173$	− 305932.i	− 0.777157i	0.921416	−	0.388579i	$-0.127034\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	68186.9	0.163594
$178$	0	0
$179$	209868.i	0.489568i	0.969578	+	0.244784i	$0.0787170\pi$
$179$	209868.i	0.489568i	−0.969578	+	0.244784i	$0.921283\pi$
$180$	0	0
$181$	212990.i	0.483239i	0.970371	+	0.241620i	$0.0776786\pi$
$181$	212990.i	0.483239i	−0.970371	+	0.241620i	$0.922321\pi$
$182$	0	0
$183$	−1.13239e6	−2.49958
$184$	0	0
$185$	0	0
$186$	0	0
$187$	489787.i	1.02424i
$188$	0	0
$189$	231201.i	0.470798i
$190$	0	0
$191$	177246.	0.351555	0.175777	−	0.984430i	$-0.443756\pi$
$191$	177246.	0.351555	0.175777	+	0.984430i	$0.443756\pi$
$192$	0	0
$193$	−758117.	−1.46502	−0.732509	−	0.680758i	$-0.761651\pi$
$193$	−758117.	−1.46502	−0.732509	+	0.680758i	$0.761651\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	353509.i	0.648985i	0.945888	+	0.324492i	$0.105193\pi$
$197$	353509.i	0.648985i	−0.945888	+	0.324492i	$0.894807\pi$
$198$	0	0
$199$	−233027.	−0.417132	−0.208566	−	0.978008i	$-0.566880\pi$
$199$	−233027.	−0.417132	−0.208566	+	0.978008i	$0.566880\pi$
$200$	0	0
$201$	317572.	0.554437
$202$	0	0
$203$	− 96048.4i	− 0.163587i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.03815e6	−1.68397
$208$	0	0
$209$	−520035.	−0.823507
$210$	0	0
$211$	401222.i	0.620410i	0.950670	+	0.310205i	$0.100398\pi$
$211$	401222.i	0.620410i	−0.950670	+	0.310205i	$0.899602\pi$
$212$	0	0
$213$	− 208291.i	− 0.314573i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−436961.	−0.629932
$218$	0	0
$219$	1.04491e6i	1.47221i
$220$	0	0
$221$	− 1.35222e6i	− 1.86238i
$222$	0	0
$223$	−475659.	−0.640521	−0.320260	−	0.947330i	$-0.603770\pi$
$223$	−475659.	−0.640521	−0.320260	+	0.947330i	$0.603770\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28559.0i	− 0.0367856i	−0.999831	−	0.0183928i	$-0.994145\pi$
$227$	− 28559.0i	− 0.0367856i	0.999831	−	0.0183928i	$-0.00585494\pi$
$228$	0	0
$229$	969736.i	1.22198i	0.791638	+	0.610991i	$0.209229\pi$
$229$	969736.i	1.22198i	−0.791638	+	0.610991i	$0.790771\pi$
$230$	0	0
$231$	375053.	0.462448
$232$	0	0
$233$	16005.7	0.0193146	0.00965728	−	0.999953i	$-0.496926\pi$
$233$	16005.7	0.0193146	0.00965728	+	0.999953i	$0.496926\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.17827e6i	− 1.36261i
$238$	0	0
$239$	−1.26598e6	−1.43361	−0.716807	−	0.697272i	$-0.754397\pi$
$239$	−1.26598e6	−1.43361	−0.716807	+	0.697272i	$0.754397\pi$
$240$	0	0
$241$	414590.	0.459807	0.229904	−	0.973213i	$-0.426159\pi$
$241$	414590.	0.459807	0.229904	+	0.973213i	$0.426159\pi$
$242$	0	0
$243$	− 843371.i	− 0.916227i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.43573e6	1.49738
$248$	0	0
$249$	−1.56816e6	−1.60285
$250$	0	0
$251$	− 184150.i	− 0.184496i	−0.995736	−	0.0922479i	$-0.970595\pi$
$251$	− 184150.i	− 0.184496i	0.995736	−	0.0922479i	$-0.0294052\pi$
$252$	0	0
$253$	670895.i	0.658951i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	846268.	0.799236	0.399618	−	0.916682i	$-0.369143\pi$
$257$	846268.	0.799236	0.399618	+	0.916682i	$0.369143\pi$
$258$	0	0
$259$	690813.i	0.639899i
$260$	0	0
$261$	− 686701.i	− 0.623974i
$262$	0	0
$263$	−1.53385e6	−1.36740	−0.683698	−	0.729765i	$-0.739629\pi$
$263$	−1.53385e6	−1.36740	−0.683698	+	0.729765i	$0.739629\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.35324e6i	1.16171i
$268$	0	0
$269$	646714.i	0.544918i	0.962167	+	0.272459i	$0.0878370\pi$
$269$	646714.i	0.544918i	−0.962167	+	0.272459i	$0.912163\pi$
$270$	0	0
$271$	1.58318e6	1.30950	0.654752	−	0.755844i	$-0.272773\pi$
$271$	1.58318e6	1.30950	0.654752	+	0.755844i	$0.272773\pi$
$272$	0	0
$273$	−1.03546e6	−0.840867
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.62475e6i	− 1.27229i	−0.771568	−	0.636147i	$-0.780527\pi$
$277$	− 1.62475e6i	− 1.27229i	0.771568	−	0.636147i	$-0.219473\pi$
$278$	0	0
$279$	−3.12407e6	−2.40276
$280$	0	0
$281$	−1.48375e6	−1.12097	−0.560487	−	0.828163i	$-0.689386\pi$
$281$	−1.48375e6	−1.12097	−0.560487	+	0.828163i	$0.689386\pi$
$282$	0	0
$283$	1.18244e6i	0.877634i	0.898577	+	0.438817i	$0.144602\pi$
$283$	1.18244e6i	0.877634i	−0.898577	+	0.438817i	$0.855398\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−846241.	−0.606442
$288$	0	0
$289$	2.10118e6	1.47985
$290$	0	0
$291$	997325.i	0.690405i
$292$	0	0
$293$	887981.i	0.604275i	0.953264	+	0.302137i	$0.0977002\pi$
$293$	887981.i	0.604275i	−0.953264	+	0.302137i	$0.902300\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.06822e6	0.702697
$298$	0	0
$299$	− 1.85223e6i	− 1.19817i
$300$	0	0
$301$	− 1.02332e6i	− 0.651024i
$302$	0	0
$303$	1.06552e6	0.666738
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.47690e6i	0.894346i	0.894447	+	0.447173i	$0.147569\pi$
$307$	1.47690e6i	0.894346i	−0.894447	+	0.447173i	$0.852431\pi$
$308$	0	0
$309$	− 3.01035e6i	− 1.79358i
$310$	0	0
$311$	−364521.	−0.213708	−0.106854	−	0.994275i	$-0.534078\pi$
$311$	−364521.	−0.213708	−0.106854	+	0.994275i	$0.534078\pi$
$312$	0	0
$313$	−324246.	−0.187074	−0.0935371	−	0.995616i	$-0.529817\pi$
$313$	−324246.	−0.187074	−0.0935371	+	0.995616i	$0.529817\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.55670e6i	0.870074i	0.900413	+	0.435037i	$0.143265\pi$
$317$	1.55670e6i	0.870074i	−0.900413	+	0.435037i	$0.856735\pi$
$318$	0	0
$319$	−443773.	−0.244165
$320$	0	0
$321$	3.76373e6	2.03871
$322$	0	0
$323$	3.73849e6i	1.99384i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.21842e6	−1.66446
$328$	0	0
$329$	−120955.	−0.0616077
$330$	0	0
$331$	558769.i	0.280325i	0.990128	+	0.140163i	$0.0447626\pi$
$331$	558769.i	0.280325i	−0.990128	+	0.140163i	$0.955237\pi$
$332$	0	0
$333$	4.93899e6i	2.44077i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.18320e6	−1.04717	−0.523587	−	0.851972i	$-0.675407\pi$
$337$	−2.18320e6	−1.04717	−0.523587	+	0.851972i	$0.675407\pi$
$338$	0	0
$339$	5.64384e6i	2.66732i
$340$	0	0
$341$	2.01890e6i	0.940216i
$342$	0	0
$343$	1.71868e6	0.788787
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.53924e6i	1.13209i	0.824375	+	0.566043i	$0.191527\pi$
$347$	2.53924e6i	1.13209i	−0.824375	+	0.566043i	$0.808473\pi$
$348$	0	0
$349$	2.58452e6i	1.13584i	0.823085	+	0.567918i	$0.192251\pi$
$349$	2.58452e6i	1.13584i	−0.823085	+	0.567918i	$0.807749\pi$
$350$	0	0
$351$	−2.94918e6	−1.27771
$352$	0	0
$353$	−284338.	−0.121450	−0.0607250	−	0.998155i	$-0.519341\pi$
$353$	−284338.	−0.121450	−0.0607250	+	0.998155i	$0.519341\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.69623e6i	− 1.11966i
$358$	0	0
$359$	1.97109e6	0.807179	0.403590	−	0.914940i	$-0.367762\pi$
$359$	1.97109e6	0.807179	0.403590	+	0.914940i	$0.367762\pi$
$360$	0	0
$361$	−1.49328e6	−0.603077
$362$	0	0
$363$	2.36336e6i	0.941377i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.04179e6	0.403754	0.201877	−	0.979411i	$-0.435296\pi$
$367$	1.04179e6	0.403754	0.201877	+	0.979411i	$0.435296\pi$
$368$	0	0
$369$	−6.05022e6	−2.31316
$370$	0	0
$371$	90712.4i	0.0342162i
$372$	0	0
$373$	1.58767e6i	0.590866i	0.955363	+	0.295433i	$0.0954639\pi$
$373$	1.58767e6i	0.590866i	−0.955363	+	0.295433i	$0.904536\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.22519e6	0.443965
$378$	0	0
$379$	− 995922.i	− 0.356145i	−0.984017	−	0.178073i	$-0.943014\pi$
$379$	− 995922.i	− 0.356145i	0.984017	−	0.178073i	$-0.0569862\pi$
$380$	0	0
$381$	6.04892e6i	2.13484i
$382$	0	0
$383$	1.53418e6	0.534415	0.267208	−	0.963639i	$-0.413899\pi$
$383$	1.53418e6	0.534415	0.267208	+	0.963639i	$0.413899\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 7.31629e6i	− 2.48321i
$388$	0	0
$389$	− 4.70941e6i	− 1.57795i	−0.614428	−	0.788973i	$-0.710613\pi$
$389$	− 4.70941e6i	− 1.57795i	0.614428	−	0.788973i	$-0.289387\pi$
$390$	0	0
$391$	4.82301e6	1.59542
$392$	0	0
$393$	3.84599e6	1.25611
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 485420.i	− 0.154576i	−0.997009	−	0.0772879i	$-0.975374\pi$
$397$	− 485420.i	− 0.154576i	0.997009	−	0.0772879i	$-0.0246261\pi$
$398$	0	0
$399$	2.86274e6	0.900223
$400$	0	0
$401$	1.73402e6	0.538508	0.269254	−	0.963069i	$-0.413223\pi$
$401$	1.73402e6	0.538508	0.269254	+	0.963069i	$0.413223\pi$
$402$	0	0
$403$	− 5.57385e6i	− 1.70959i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.19177e6	0.955092
$408$	0	0
$409$	−853587.	−0.252313	−0.126156	−	0.992010i	$-0.540264\pi$
$409$	−853587.	−0.252313	−0.126156	+	0.992010i	$0.540264\pi$
$410$	0	0
$411$	− 4.15129e6i	− 1.21221i
$412$	0	0
$413$	151454.i	0.0436925i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−190511.	−0.0536514
$418$	0	0
$419$	3.86903e6i	1.07663i	0.842743	+	0.538316i	$0.180939\pi$
$419$	3.86903e6i	1.07663i	−0.842743	+	0.538316i	$0.819061\pi$
$420$	0	0
$421$	1.15014e6i	0.316260i	0.987418	+	0.158130i	$0.0505464\pi$
$421$	1.15014e6i	0.316260i	−0.987418	+	0.158130i	$0.949454\pi$
$422$	0	0
$423$	−864773.	−0.234991
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.51522e6i	− 0.667583i
$428$	0	0
$429$	4.78415e6i	1.25505i
$430$	0	0
$431$	3.09078e6	0.801448	0.400724	−	0.916199i	$-0.368759\pi$
$431$	3.09078e6	0.801448	0.400724	+	0.916199i	$0.368759\pi$
$432$	0	0
$433$	−2.47892e6	−0.635394	−0.317697	−	0.948192i	$-0.602910\pi$
$433$	−2.47892e6	−0.635394	−0.317697	+	0.948192i	$0.602910\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.12087e6i	1.28275i
$438$	0	0
$439$	−997159.	−0.246947	−0.123473	−	0.992348i	$-0.539403\pi$
$439$	−997159.	−0.246947	−0.123473	+	0.992348i	$0.539403\pi$
$440$	0	0
$441$	5.49934e6	1.34652
$442$	0	0
$443$	2.10966e6i	0.510744i	0.966843	+	0.255372i	$0.0821980\pi$
$443$	2.10966e6i	0.510744i	−0.966843	+	0.255372i	$0.917802\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−904020.	−0.213998
$448$	0	0
$449$	6.24963e6	1.46298	0.731490	−	0.681852i	$-0.238825\pi$
$449$	6.24963e6	1.46298	0.731490	+	0.681852i	$0.238825\pi$
$450$	0	0
$451$	3.90989e6i	0.905156i
$452$	0	0
$453$	− 1.39838e7i	− 3.20170i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.87669e6	−0.420340	−0.210170	−	0.977665i	$-0.567402\pi$
$457$	−1.87669e6	−0.420340	−0.210170	+	0.977665i	$0.567402\pi$
$458$	0	0
$459$	− 7.67933e6i	− 1.70134i
$460$	0	0
$461$	− 8.54777e6i	− 1.87327i	−0.350307	−	0.936635i	$-0.613923\pi$
$461$	− 8.54777e6i	− 1.87327i	0.350307	−	0.936635i	$-0.386077\pi$
$462$	0	0
$463$	−7.55869e6	−1.63868	−0.819340	−	0.573308i	$-0.805660\pi$
$463$	−7.55869e6	−1.63868	−0.819340	+	0.573308i	$0.805660\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.29127e6i	0.698346i	0.937058	+	0.349173i	$0.113538\pi$
$467$	3.29127e6i	0.698346i	−0.937058	+	0.349173i	$0.886462\pi$
$468$	0	0
$469$	705380.i	0.148078i
$470$	0	0
$471$	−6.42013e6	−1.33349
$472$	0	0
$473$	−4.72807e6	−0.971698
$474$	0	0
$475$	0	0
$476$	0	0
$477$	648551.i	0.130511i
$478$	0	0
$479$	−4.95610e6	−0.986965	−0.493482	−	0.869756i	$-0.664276\pi$
$479$	−4.95610e6	−0.986965	−0.493482	+	0.869756i	$0.664276\pi$
$480$	0	0
$481$	−8.81196e6	−1.73664
$482$	0	0
$483$	− 3.69321e6i	− 0.720338i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.56942e6	−1.44624	−0.723120	−	0.690723i	$-0.757293\pi$
$487$	−7.56942e6	−1.44624	−0.723120	+	0.690723i	$0.757293\pi$
$488$	0	0
$489$	9.74688e6	1.84329
$490$	0	0
$491$	− 1.25015e6i	− 0.234023i	−0.993131	−	0.117012i	$-0.962668\pi$
$491$	− 1.25015e6i	− 0.234023i	0.993131	−	0.117012i	$-0.0373315\pi$
$492$	0	0
$493$	3.19025e6i	0.591163i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	462649.	0.0840158
$498$	0	0
$499$	− 5.59295e6i	− 1.00552i	−0.864427	−	0.502758i	$-0.832319\pi$
$499$	− 5.59295e6i	− 1.00552i	0.864427	−	0.502758i	$-0.167681\pi$
$500$	0	0
$501$	2.75753e6i	0.490824i
$502$	0	0
$503$	9.76813e6	1.72144	0.860719	−	0.509080i	$-0.170014\pi$
$503$	9.76813e6	1.72144	0.860719	+	0.509080i	$0.170014\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.76469e6i	− 0.650443i
$508$	0	0
$509$	9.05091e6i	1.54845i	0.632909	+	0.774226i	$0.281861\pi$
$509$	9.05091e6i	1.54845i	−0.632909	+	0.774226i	$0.718139\pi$
$510$	0	0
$511$	−2.32092e6	−0.393195
$512$	0	0
$513$	8.15359e6	1.36790
$514$	0	0
$515$	0	0
$516$	0	0
$517$	558850.i	0.0919536i
$518$	0	0
$519$	7.78116e6	1.26802
$520$	0	0
$521$	−7.68287e6	−1.24002	−0.620011	−	0.784593i	$-0.712872\pi$
$521$	−7.68287e6	−1.24002	−0.620011	+	0.784593i	$0.712872\pi$
$522$	0	0
$523$	− 8.45353e6i	− 1.35140i	−0.737177	−	0.675700i	$-0.763841\pi$
$523$	− 8.45353e6i	− 1.35140i	0.737177	−	0.675700i	$-0.236159\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.45137e7	2.27641
$528$	0	0
$529$	170070.	0.0264234
$530$	0	0
$531$	1.08283e6i	0.166657i
$532$	0	0
$533$	− 1.07946e7i	− 1.64584i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.33784e6	−0.798785
$538$	0	0
$539$	− 3.55389e6i	− 0.526904i
$540$	0	0
$541$	− 4.67406e6i	− 0.686596i	−0.939227	−	0.343298i	$-0.888456\pi$
$541$	− 4.67406e6i	− 0.686596i	0.939227	−	0.343298i	$-0.111544\pi$
$542$	0	0
$543$	−5.41725e6	−0.788459
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.86478e6i	− 0.266477i	−0.991084	−	0.133238i	$-0.957462\pi$
$547$	− 1.86478e6i	− 0.266477i	0.991084	−	0.133238i	$-0.0425376\pi$
$548$	0	0
$549$	− 1.79826e7i	− 2.54637i
$550$	0	0
$551$	−3.38727e6	−0.475304
$552$	0	0
$553$	2.61713e6	0.363925
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.40472e6i	1.28442i	0.766528	+	0.642211i	$0.221983\pi$
$557$	9.40472e6i	1.28442i	−0.766528	+	0.642211i	$0.778017\pi$
$558$	0	0
$559$	1.30535e7	1.76683
$560$	0	0
$561$	−1.24574e7	−1.67117
$562$	0	0
$563$	− 849619.i	− 0.112967i	−0.998404	−	0.0564837i	$-0.982011\pi$
$563$	− 849619.i	− 0.112967i	0.998404	−	0.0564837i	$-0.0179889\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−335627.	−0.0438429
$568$	0	0
$569$	−5.41948e6	−0.701741	−0.350870	−	0.936424i	$-0.614114\pi$
$569$	−5.41948e6	−0.701741	−0.350870	+	0.936424i	$0.614114\pi$
$570$	0	0
$571$	− 331372.i	− 0.0425329i	−0.999774	−	0.0212664i	$-0.993230\pi$
$571$	− 331372.i	− 0.0425329i	0.999774	−	0.0212664i	$-0.00676983\pi$
$572$	0	0
$573$	4.50813e6i	0.573601i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.51001e7	1.88817	0.944086	−	0.329701i	$-0.106948\pi$
$577$	1.51001e7	1.88817	0.944086	+	0.329701i	$0.106948\pi$
$578$	0	0
$579$	− 1.92822e7i	− 2.39034i
$580$	0	0
$581$	− 3.48315e6i	− 0.428087i
$582$	0	0
$583$	419119.	0.0510700
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.51509e7i	1.81486i	0.420199	+	0.907432i	$0.361960\pi$
$587$	1.51509e7i	1.81486i	−0.420199	+	0.907432i	$0.638040\pi$
$588$	0	0
$589$	1.54100e7i	1.83027i
$590$	0	0
$591$	−8.99125e6	−1.05889
$592$	0	0
$593$	1.46568e7	1.71160	0.855800	−	0.517307i	$-0.173066\pi$
$593$	1.46568e7	1.71160	0.855800	+	0.517307i	$0.173066\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.92688e6i	− 0.680597i
$598$	0	0
$599$	5.14552e6	0.585952	0.292976	−	0.956120i	$-0.405354\pi$
$599$	5.14552e6	0.585952	0.292976	+	0.956120i	$0.405354\pi$
$600$	0	0
$601$	9.03954e6	1.02085	0.510423	−	0.859923i	$-0.329489\pi$
$601$	9.03954e6	1.02085	0.510423	+	0.859923i	$0.329489\pi$
$602$	0	0
$603$	5.04314e6i	0.564817i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.25949e7	1.38747	0.693733	−	0.720232i	$-0.255965\pi$
$607$	1.25949e7	1.38747	0.693733	+	0.720232i	$0.255965\pi$
$608$	0	0
$609$	2.44292e6	0.266911
$610$	0	0
$611$	− 1.54290e6i	− 0.167199i
$612$	0	0
$613$	− 8.66424e6i	− 0.931278i	−0.884975	−	0.465639i	$-0.845825\pi$
$613$	− 8.66424e6i	− 0.931278i	0.884975	−	0.465639i	$-0.154175\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.86089e6	−0.725551	−0.362775	−	0.931877i	$-0.618171\pi$
$617$	−6.86089e6	−0.725551	−0.362775	+	0.931877i	$0.618171\pi$
$618$	0	0
$619$	3.44552e6i	0.361434i	0.983535	+	0.180717i	$0.0578417\pi$
$619$	3.44552e6i	0.361434i	−0.983535	+	0.180717i	$0.942158\pi$
$620$	0	0
$621$	− 1.05189e7i	− 1.09457i
$622$	0	0
$623$	−3.00578e6	−0.310268
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.32267e7i	− 1.34364i
$628$	0	0
$629$	− 2.29454e7i	− 2.31243i
$630$	0	0
$631$	7.79725e6	0.779593	0.389797	−	0.920901i	$-0.372545\pi$
$631$	7.79725e6	0.779593	0.389797	+	0.920901i	$0.372545\pi$
$632$	0	0
$633$	−1.02048e7	−1.01227
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.81171e6i	0.958068i
$638$	0	0
$639$	3.30773e6	0.320463
$640$	0	0
$641$	7.82134e6	0.751859	0.375929	−	0.926648i	$-0.377324\pi$
$641$	7.82134e6	0.751859	0.375929	+	0.926648i	$0.377324\pi$
$642$	0	0
$643$	− 1.35325e7i	− 1.29078i	−0.763854	−	0.645389i	$-0.776695\pi$
$643$	− 1.35325e7i	− 1.29078i	0.763854	−	0.645389i	$-0.223305\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.34237e6	−0.126070	−0.0630352	−	0.998011i	$-0.520078\pi$
$647$	−1.34237e6	−0.126070	−0.0630352	+	0.998011i	$0.520078\pi$
$648$	0	0
$649$	699766.	0.0652140
$650$	0	0
$651$	− 1.11138e7i	− 1.02780i
$652$	0	0
$653$	− 5.36490e6i	− 0.492355i	−0.969225	−	0.246178i	$-0.920825\pi$
$653$	− 5.36490e6i	− 0.492355i	0.969225	−	0.246178i	$-0.0791747\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.65935e7	−1.49977
$658$	0	0
$659$	8.49067e6i	0.761603i	0.924657	+	0.380801i	$0.124352\pi$
$659$	8.49067e6i	0.761603i	−0.924657	+	0.380801i	$0.875648\pi$
$660$	0	0
$661$	9.80254e6i	0.872640i	0.899792	+	0.436320i	$0.143718\pi$
$661$	9.80254e6i	0.872640i	−0.899792	+	0.436320i	$0.856282\pi$
$662$	0	0
$663$	3.43929e7	3.03868
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.36990e6i	0.380327i
$668$	0	0
$669$	− 1.20981e7i	− 1.04508i
$670$	0	0
$671$	−1.16211e7	−0.996413
$672$	0	0
$673$	−7.99241e6	−0.680205	−0.340103	−	0.940388i	$-0.610462\pi$
$673$	−7.99241e6	−0.680205	−0.340103	+	0.940388i	$0.610462\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.50891e6i	− 0.713514i	−0.934197	−	0.356757i	$-0.883882\pi$
$677$	− 8.50891e6i	− 0.713514i	0.934197	−	0.356757i	$-0.116118\pi$
$678$	0	0
$679$	−2.21522e6	−0.184392
$680$	0	0
$681$	726378.	0.0600198
$682$	0	0
$683$	1.39302e7i	1.14263i	0.820732	+	0.571314i	$0.193566\pi$
$683$	1.39302e7i	1.14263i	−0.820732	+	0.571314i	$0.806434\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.46646e7	−1.99380
$688$	0	0
$689$	−1.15712e6	−0.0928604
$690$	0	0
$691$	1.79579e6i	0.143074i	0.997438	+	0.0715371i	$0.0227904\pi$
$691$	1.79579e6i	0.143074i	−0.997438	+	0.0715371i	$0.977210\pi$
$692$	0	0
$693$	5.95595e6i	0.471106i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.81079e7	2.19152
$698$	0	0
$699$	407094.i	0.0315139i
$700$	0	0
$701$	1.76628e7i	1.35758i	0.734334	+	0.678789i	$0.237495\pi$
$701$	1.76628e7i	1.35758i	−0.734334	+	0.678789i	$0.762505\pi$
$702$	0	0
$703$	2.43624e7	1.85923
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.36670e6i	0.178071i
$708$	0	0
$709$	1.67397e7i	1.25064i	0.780369	+	0.625319i	$0.215031\pi$
$709$	1.67397e7i	1.25064i	−0.780369	+	0.625319i	$0.784969\pi$
$710$	0	0
$711$	1.87112e7	1.38812
$712$	0	0
$713$	1.98804e7	1.46454
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.21993e7i	− 2.33910i
$718$	0	0
$719$	1.41699e7	1.02222	0.511111	−	0.859515i	$-0.329234\pi$
$719$	1.41699e7	1.02222	0.511111	+	0.859515i	$0.329234\pi$
$720$	0	0
$721$	6.68649e6	0.479027
$722$	0	0
$723$	1.05448e7i	0.750227i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.78412e6	0.265539	0.132770	−	0.991147i	$-0.457613\pi$
$727$	3.78412e6	0.265539	0.132770	+	0.991147i	$0.457613\pi$
$728$	0	0
$729$	2.28942e7	1.59554
$730$	0	0
$731$	3.39897e7i	2.35263i
$732$	0	0
$733$	2.58722e7i	1.77858i	0.457345	+	0.889290i	$0.348801\pi$
$733$	2.58722e7i	1.77858i	−0.457345	+	0.889290i	$0.651199\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.25907e6	0.221017
$738$	0	0
$739$	5.89215e6i	0.396883i	0.980113	+	0.198441i	$0.0635880\pi$
$739$	5.89215e6i	0.396883i	−0.980113	+	0.198441i	$0.936412\pi$
$740$	0	0
$741$	3.65169e7i	2.44314i
$742$	0	0
$743$	−2.47551e7	−1.64510	−0.822551	−	0.568692i	$-0.807450\pi$
$743$	−2.47551e7	−1.64510	−0.822551	+	0.568692i	$0.807450\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.49029e7i	− 1.63286i
$748$	0	0
$749$	8.35986e6i	0.544496i
$750$	0	0
$751$	−1.98371e6	−0.128345	−0.0641724	−	0.997939i	$-0.520441\pi$
$751$	−1.98371e6	−0.128345	−0.0641724	+	0.997939i	$0.520441\pi$
$752$	0	0
$753$	4.68372e6	0.301026
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.85647e7i	1.17746i	0.808328	+	0.588732i	$0.200373\pi$
$757$	1.85647e7i	1.17746i	−0.808328	+	0.588732i	$0.799627\pi$
$758$	0	0
$759$	−1.70638e7	−1.07515
$760$	0	0
$761$	−5.16348e6	−0.323207	−0.161603	−	0.986856i	$-0.551667\pi$
$761$	−5.16348e6	−0.323207	−0.161603	+	0.986856i	$0.551667\pi$
$762$	0	0
$763$	− 7.14864e6i	− 0.444542i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.93194e6	−0.118578
$768$	0	0
$769$	3.01408e7	1.83797	0.918985	−	0.394293i	$-0.129011\pi$
$769$	3.01408e7	1.83797	0.918985	+	0.394293i	$0.129011\pi$
$770$	0	0
$771$	2.15242e7i	1.30404i
$772$	0	0
$773$	− 1.51718e7i	− 0.913246i	−0.889660	−	0.456623i	$-0.849059\pi$
$773$	− 1.51718e7i	− 0.913246i	0.889660	−	0.456623i	$-0.150941\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.75704e7	−1.04407
$778$	0	0
$779$	2.98438e7i	1.76202i
$780$	0	0
$781$	− 2.13758e6i	− 0.125399i
$782$	0	0
$783$	6.95788e6	0.405576
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.46106e7i	0.840874i	0.907322	+	0.420437i	$0.138123\pi$
$787$	1.46106e7i	0.840874i	−0.907322	+	0.420437i	$0.861877\pi$
$788$	0	0
$789$	− 3.90125e7i	− 2.23106i
$790$	0	0
$791$	−1.25359e7	−0.712384
$792$	0	0
$793$	3.20839e7	1.81177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.81785e6i	0.268663i	0.990936	+	0.134331i	$0.0428887\pi$
$797$	4.81785e6i	0.268663i	−0.990936	+	0.134331i	$0.957111\pi$
$798$	0	0
$799$	4.01753e6	0.222634
$800$	0	0
$801$	−2.14899e7	−1.18346
$802$	0	0
$803$	1.07234e7i	0.586870i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.64487e7	−0.889095
$808$	0	0
$809$	−267715.	−0.0143814	−0.00719072	−	0.999974i	$-0.502289\pi$
$809$	−267715.	−0.0143814	−0.00719072	+	0.999974i	$0.502289\pi$
$810$	0	0
$811$	− 1.10232e7i	− 0.588514i	−0.955726	−	0.294257i	$-0.904928\pi$
$811$	− 1.10232e7i	− 0.588514i	0.955726	−	0.294257i	$-0.0950722\pi$
$812$	0	0
$813$	4.02671e7i	2.13660i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.60889e7	−1.89155
$818$	0	0
$819$	− 1.64434e7i	− 0.856609i
$820$	0	0
$821$	− 7.52183e6i	− 0.389462i	−0.980857	−	0.194731i	$-0.937617\pi$
$821$	− 7.52183e6i	− 0.389462i	0.980857	−	0.194731i	$-0.0623834\pi$
$822$	0	0
$823$	−2.86933e7	−1.47666	−0.738331	−	0.674439i	$-0.764386\pi$
$823$	−2.86933e7	−1.47666	−0.738331	+	0.674439i	$0.764386\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.90830e7i	− 0.970248i	−0.874445	−	0.485124i	$-0.838774\pi$
$827$	− 1.90830e7i	− 0.970248i	0.874445	−	0.485124i	$-0.161226\pi$
$828$	0	0
$829$	2.31916e7i	1.17205i	0.810295	+	0.586023i	$0.199307\pi$
$829$	2.31916e7i	1.17205i	−0.810295	+	0.586023i	$0.800693\pi$
$830$	0	0
$831$	4.13245e7	2.07589
$832$	0	0
$833$	−2.55486e7	−1.27572
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.16541e7i	− 1.56177i
$838$	0	0
$839$	887580.	0.0435314	0.0217657	−	0.999763i	$-0.493071\pi$
$839$	887580.	0.0435314	0.0217657	+	0.999763i	$0.493071\pi$
$840$	0	0
$841$	1.76206e7	0.859075
$842$	0	0
$843$	− 3.77382e7i	− 1.82899i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.24942e6	−0.251422
$848$	0	0
$849$	−3.00746e7	−1.43196
$850$	0	0
$851$	− 3.14299e7i	− 1.48771i
$852$	0	0
$853$	− 1.39449e7i	− 0.656208i	−0.944642	−	0.328104i	$-0.893590\pi$
$853$	− 1.39449e7i	− 0.656208i	0.944642	−	0.328104i	$-0.106410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.58423e7	1.66703	0.833516	−	0.552495i	$-0.186324\pi$
$857$	3.58423e7	1.66703	0.833516	+	0.552495i	$0.186324\pi$
$858$	0	0
$859$	767053.i	0.0354685i	0.999843	+	0.0177342i	$0.00564528\pi$
$859$	767053.i	0.0354685i	−0.999843	+	0.0177342i	$0.994355\pi$
$860$	0	0
$861$	− 2.15236e7i	− 0.989478i
$862$	0	0
$863$	−2.90420e7	−1.32739	−0.663697	−	0.748002i	$-0.731013\pi$
$863$	−2.90420e7	−1.32739	−0.663697	+	0.748002i	$0.731013\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.34421e7i	2.41455i
$868$	0	0
$869$	− 1.20919e7i	− 0.543183i
$870$	0	0
$871$	−8.99778e6	−0.401874
$872$	0	0
$873$	−1.58378e7	−0.703330
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 9.62715e6i	− 0.422667i	−0.977414	−	0.211334i	$-0.932219\pi$
$877$	− 9.62715e6i	− 0.422667i	0.977414	−	0.211334i	$-0.0677807\pi$
$878$	0	0
$879$	−2.25852e7	−0.985942
$880$	0	0
$881$	3.88627e7	1.68691	0.843457	−	0.537196i	$-0.180517\pi$
$881$	3.88627e7	1.68691	0.843457	+	0.537196i	$0.180517\pi$
$882$	0	0
$883$	− 4.29023e6i	− 0.185173i	−0.995705	−	0.0925867i	$-0.970486\pi$
$883$	− 4.29023e6i	− 0.185173i	0.995705	−	0.0925867i	$-0.0295135\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.30883e7	−0.558564	−0.279282	−	0.960209i	$-0.590096\pi$
$887$	−1.30883e7	−0.558564	−0.279282	+	0.960209i	$0.590096\pi$
$888$	0	0
$889$	−1.34356e7	−0.570170
$890$	0	0
$891$	1.55070e6i	0.0654386i
$892$	0	0
$893$	4.26564e6i	0.179001i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.71103e7	1.95495
$898$	0	0
$899$	1.31502e7i	0.542665i
$900$	0	0
$901$	− 3.01301e6i	− 0.123649i
$902$	0	0
$903$	2.60276e7	1.06222
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.36895e7i	0.956175i	0.878312	+	0.478087i	$0.158670\pi$
$907$	2.36895e7i	0.956175i	−0.878312	+	0.478087i	$0.841330\pi$
$908$	0	0
$909$	1.69208e7i	0.679220i
$910$	0	0
$911$	−1.27323e7	−0.508289	−0.254144	−	0.967166i	$-0.581794\pi$
$911$	−1.27323e7	−0.508289	−0.254144	+	0.967166i	$0.581794\pi$
$912$	0	0
$913$	−1.60932e7	−0.638948
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.54258e6i	0.335479i
$918$	0	0
$919$	−3.36064e7	−1.31260	−0.656302	−	0.754499i	$-0.727880\pi$
$919$	−3.36064e7	−1.31260	−0.656302	+	0.754499i	$0.727880\pi$
$920$	0	0
$921$	−3.75640e7	−1.45923
$922$	0	0
$923$	5.90152e6i	0.228013i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.78053e7	1.82716
$928$	0	0
$929$	2.44326e7	0.928816	0.464408	−	0.885621i	$-0.346267\pi$
$929$	2.44326e7	0.928816	0.464408	+	0.885621i	$0.346267\pi$
$930$	0	0
$931$	− 2.71265e7i	− 1.02570i
$932$	0	0
$933$	− 9.27134e6i	− 0.348689i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.46226e7	0.916188	0.458094	−	0.888904i	$-0.348532\pi$
$937$	2.46226e7	0.916188	0.458094	+	0.888904i	$0.348532\pi$
$938$	0	0
$939$	− 8.24698e6i	− 0.305233i
$940$	0	0
$941$	− 3.64578e7i	− 1.34220i	−0.741368	−	0.671098i	$-0.765823\pi$
$941$	− 3.64578e7i	− 1.34220i	0.741368	−	0.671098i	$-0.234177\pi$
$942$	0	0
$943$	3.85013e7	1.40993
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.49302e7i	− 0.540992i	−0.962721	−	0.270496i	$-0.912812\pi$
$947$	− 1.49302e7i	− 0.540992i	0.962721	−	0.270496i	$-0.0871876\pi$
$948$	0	0
$949$	− 2.96055e7i	− 1.06710i
$950$	0	0
$951$	−3.95935e7	−1.41962
$952$	0	0
$953$	−2.82853e7	−1.00885	−0.504426	−	0.863455i	$-0.668296\pi$
$953$	−2.82853e7	−1.00885	−0.504426	+	0.863455i	$0.668296\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1.12871e7i	− 0.398383i
$958$	0	0
$959$	9.22071e6	0.323756
$960$	0	0
$961$	3.11960e7	1.08966
$962$	0	0
$963$	5.97691e7i	2.07688i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.47503e7	1.19507	0.597533	−	0.801844i	$-0.296148\pi$
$967$	3.47503e7	1.19507	0.597533	+	0.801844i	$0.296148\pi$
$968$	0	0
$969$	−9.50860e7	−3.25317
$970$	0	0
$971$	4.65499e7i	1.58442i	0.610247	+	0.792211i	$0.291070\pi$
$971$	4.65499e7i	1.58442i	−0.610247	+	0.792211i	$0.708930\pi$
$972$	0	0
$973$	− 423158.i	− 0.0143291i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.15712e7	−1.39334	−0.696668	−	0.717394i	$-0.745335\pi$
$977$	−4.15712e7	−1.39334	−0.696668	+	0.717394i	$0.745335\pi$
$978$	0	0
$979$	1.38876e7i	0.463096i
$980$	0	0
$981$	− 5.11094e7i	− 1.69562i
$982$	0	0
$983$	−2.55186e7	−0.842313	−0.421157	−	0.906988i	$-0.638376\pi$
$983$	−2.55186e7	−0.842313	−0.421157	+	0.906988i	$0.638376\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 3.07641e6i	− 0.100520i
$988$	0	0
$989$	4.65581e7i	1.51358i
$990$	0	0
$991$	3.10296e7	1.00367	0.501836	−	0.864963i	$-0.332658\pi$
$991$	3.10296e7	1.00367	0.501836	+	0.864963i	$0.332658\pi$
$992$	0	0
$993$	−1.42119e7	−0.457382
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.66036e7i	0.529011i	0.964384	+	0.264505i	$0.0852087\pi$
$997$	1.66036e7i	0.529011i	−0.964384	+	0.264505i	$0.914791\pi$
$998$	0	0
$999$	−5.00435e7	−1.58648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.d.c.401.19		20
4.3	odd	2		200.6.d.b.101.18		20
5.2	odd	4		800.6.f.b.49.19		20
5.3	odd	4		800.6.f.c.49.2		20
5.4	even	2		160.6.d.a.81.2		20
8.3	odd	2		200.6.d.b.101.17		20
8.5	even	2	inner	800.6.d.c.401.2		20
20.3	even	4		200.6.f.b.149.13		20
20.7	even	4		200.6.f.c.149.8		20
20.19	odd	2		40.6.d.a.21.3	✓	20
40.3	even	4		200.6.f.c.149.7		20
40.13	odd	4		800.6.f.b.49.20		20
40.19	odd	2		40.6.d.a.21.4	yes	20
40.27	even	4		200.6.f.b.149.14		20
40.29	even	2		160.6.d.a.81.19		20
40.37	odd	4		800.6.f.c.49.1		20
60.59	even	2		360.6.k.b.181.18		20
120.59	even	2		360.6.k.b.181.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.3	✓	20	20.19	odd	2
40.6.d.a.21.4	yes	20	40.19	odd	2
160.6.d.a.81.2		20	5.4	even	2
160.6.d.a.81.19		20	40.29	even	2
200.6.d.b.101.17		20	8.3	odd	2
200.6.d.b.101.18		20	4.3	odd	2
200.6.f.b.149.13		20	20.3	even	4
200.6.f.b.149.14		20	40.27	even	4
200.6.f.c.149.7		20	40.3	even	4
200.6.f.c.149.8		20	20.7	even	4
360.6.k.b.181.17		20	120.59	even	2
360.6.k.b.181.18		20	60.59	even	2
800.6.d.c.401.2		20	8.5	even	2	inner
800.6.d.c.401.19		20	1.1	even	1	trivial
800.6.f.b.49.19		20	5.2	odd	4
800.6.f.b.49.20		20	40.13	odd	4
800.6.f.c.49.1		20	40.37	odd	4
800.6.f.c.49.2		20	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+25.4343i q^{3} -56.4938 q^{7} -403.904 q^{9} +O(q^{10})\)	\(q+25.4343i q^{3} -56.4938 q^{7} -403.904 q^{9} +261.019i q^{11} -720.631i q^{13} +1876.44 q^{17} +1992.33i q^{19} -1436.88i q^{21} +2570.29 q^{23} -4092.49i q^{27} +1700.16i q^{29} +7734.68 q^{31} -6638.83 q^{33} -12228.1i q^{37} +18328.8 q^{39} +14979.3 q^{41} +18113.9i q^{43} +2141.03 q^{47} -13615.4 q^{49} +47726.0i q^{51} -1605.71i q^{53} -50673.5 q^{57} -2680.90i q^{59} +44521.9i q^{61} +22818.1 q^{63} -12486.0i q^{67} +65373.7i q^{69} -8189.38 q^{71} +41082.7 q^{73} -14746.0i q^{77} -46325.9 q^{79} +5940.95 q^{81} +61655.4i q^{83} -43242.3 q^{87} +53205.4 q^{89} +40711.2i q^{91} +196726. i q^{93} +39211.8 q^{97} -105427. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 20 * q - 196 * q^7 - 1620 * q^9	\( 20 q - 196 q^{7} - 1620 q^{9} - 4676 q^{23} - 7160 q^{31} - 5672 q^{33} + 44904 q^{39} + 11608 q^{41} + 44180 q^{47} + 18756 q^{49} - 5032 q^{57} + 240620 q^{63} + 200312 q^{71} + 105136 q^{73} - 282080 q^{79} + 65172 q^{81} - 332592 q^{87} - 3160 q^{89} - 147376 q^{97}+O(q^{100}) \) 20 * q - 196 * q^7 - 1620 * q^9 - 4676 * q^23 - 7160 * q^31 - 5672 * q^33 + 44904 * q^39 + 11608 * q^41 + 44180 * q^47 + 18756 * q^49 - 5032 * q^57 + 240620 * q^63 + 200312 * q^71 + 105136 * q^73 - 282080 * q^79 + 65172 * q^81 - 332592 * q^87 - 3160 * q^89 - 147376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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