LMFDB - Embedded newform 4600.2.e.v.4049.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 $ x^10 + 18x^8 + 103x^6 + 239x^4 + 197x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.7
Root			$-3.11721i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.v.4049.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33689i q^{3} -3.16736i q^{7} +1.21273 q^{9} +O(q^{10})$	$q+1.33689i q^{3} -3.16736i q^{7} +1.21273 q^{9} -0.0955192 q^{11} -1.44343i q^{13} -2.29775i q^{17} -7.00371 q^{19} +4.23441 q^{21} +1.00000i q^{23} +5.63195i q^{27} -5.39076 q^{29} -0.584488 q^{31} -0.127699i q^{33} -9.29985i q^{37} +1.92970 q^{39} -2.86534 q^{41} -9.50208i q^{43} +7.09353i q^{47} -3.03218 q^{49} +3.07184 q^{51} +7.73922i q^{53} -9.36319i q^{57} -13.6426 q^{59} +0.234413 q^{61} -3.84114i q^{63} +7.49729i q^{67} -1.33689 q^{69} -5.18426 q^{71} +1.52384i q^{73} +0.302544i q^{77} +3.04068 q^{79} -3.89112 q^{81} +15.9909i q^{83} -7.20685i q^{87} -5.53325 q^{89} -4.57186 q^{91} -0.781396i q^{93} +2.58305i q^{97} -0.115839 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{9}+O(q^{10}) $ 10 * q - 8 * q^9	$ 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) $ 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.33689i	0.771854i	0.922529	+	0.385927i	$0.126118\pi$
$3$	1.33689i	0.771854i	−0.922529	+	0.385927i	$0.873882\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.16736i	− 1.19715i	−0.801067	−	0.598575i	$-0.795734\pi$
$7$	− 3.16736i	− 1.19715i	0.801067	−	0.598575i	$-0.204266\pi$
$8$	0	0
$9$	1.21273	0.404242
$10$	0	0
$11$	−0.0955192	−0.0288001	−0.0144001	−	0.999896i	$-0.504584\pi$
$11$	−0.0955192	−0.0288001	−0.0144001	+	0.999896i	$0.504584\pi$
$12$	0	0
$13$	− 1.44343i	− 0.400335i	−0.979762	−	0.200167i	$-0.935851\pi$
$13$	− 1.44343i	− 0.400335i	0.979762	−	0.200167i	$-0.0641486\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.29775i	− 0.557287i	−0.960395	−	0.278643i	$-0.910115\pi$
$17$	− 2.29775i	− 0.557287i	0.960395	−	0.278643i	$-0.0898848\pi$
$18$	0	0
$19$	−7.00371	−1.60676	−0.803381	−	0.595466i	$-0.796968\pi$
$19$	−7.00371	−1.60676	−0.803381	+	0.595466i	$0.796968\pi$
$20$	0	0
$21$	4.23441	0.924025
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.63195i	1.08387i
$28$	0	0
$29$	−5.39076	−1.00104	−0.500519	−	0.865725i	$-0.666858\pi$
$29$	−5.39076	−1.00104	−0.500519	+	0.865725i	$0.666858\pi$
$30$	0	0
$31$	−0.584488	−0.104977	−0.0524886	−	0.998622i	$-0.516715\pi$
$31$	−0.584488	−0.104977	−0.0524886	+	0.998622i	$0.516715\pi$
$32$	0	0
$33$	− 0.127699i	− 0.0222295i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.29985i	− 1.52889i	−0.644691	−	0.764443i	$-0.723014\pi$
$37$	− 9.29985i	− 1.52889i	0.644691	−	0.764443i	$-0.276986\pi$
$38$	0	0
$39$	1.92970	0.309000
$40$	0	0
$41$	−2.86534	−0.447492	−0.223746	−	0.974648i	$-0.571829\pi$
$41$	−2.86534	−0.447492	−0.223746	+	0.974648i	$0.571829\pi$
$42$	0	0
$43$	− 9.50208i	− 1.44905i	−0.689246	−	0.724527i	$-0.742058\pi$
$43$	− 9.50208i	− 1.44905i	0.689246	−	0.724527i	$-0.257942\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.09353i	1.03470i	0.855775	+	0.517349i	$0.173081\pi$
$47$	7.09353i	1.03470i	−0.855775	+	0.517349i	$0.826919\pi$
$48$	0	0
$49$	−3.03218	−0.433168
$50$	0	0
$51$	3.07184	0.430144
$52$	0	0
$53$	7.73922i	1.06306i	0.847038	+	0.531532i	$0.178383\pi$
$53$	7.73922i	1.06306i	−0.847038	+	0.531532i	$0.821617\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 9.36319i	− 1.24018i
$58$	0	0
$59$	−13.6426	−1.77612	−0.888059	−	0.459730i	$-0.847946\pi$
$59$	−13.6426	−1.77612	−0.888059	+	0.459730i	$0.847946\pi$
$60$	0	0
$61$	0.234413	0.0300135	0.0150068	−	0.999887i	$-0.495223\pi$
$61$	0.234413	0.0300135	0.0150068	+	0.999887i	$0.495223\pi$
$62$	0	0
$63$	− 3.84114i	− 0.483938i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.49729i	0.915940i	0.888968	+	0.457970i	$0.151423\pi$
$67$	7.49729i	0.915940i	−0.888968	+	0.457970i	$0.848577\pi$
$68$	0	0
$69$	−1.33689	−0.160943
$70$	0	0
$71$	−5.18426	−0.615258	−0.307629	−	0.951506i	$-0.599536\pi$
$71$	−5.18426	−0.615258	−0.307629	+	0.951506i	$0.599536\pi$
$72$	0	0
$73$	1.52384i	0.178352i	0.996016	+	0.0891760i	$0.0284234\pi$
$73$	1.52384i	0.178352i	−0.996016	+	0.0891760i	$0.971577\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.302544i	0.0344781i
$78$	0	0
$79$	3.04068	0.342103	0.171052	−	0.985262i	$-0.445283\pi$
$79$	3.04068	0.342103	0.171052	+	0.985262i	$0.445283\pi$
$80$	0	0
$81$	−3.89112	−0.432347
$82$	0	0
$83$	15.9909i	1.75523i	0.479369	+	0.877613i	$0.340865\pi$
$83$	15.9909i	1.75523i	−0.479369	+	0.877613i	$0.659135\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.20685i	− 0.772655i
$88$	0	0
$89$	−5.53325	−0.586523	−0.293261	−	0.956032i	$-0.594741\pi$
$89$	−5.53325	−0.586523	−0.293261	+	0.956032i	$0.594741\pi$
$90$	0	0
$91$	−4.57186	−0.479261
$92$	0	0
$93$	− 0.781396i	− 0.0810270i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.58305i	0.262269i	0.991365	+	0.131135i	$0.0418620\pi$
$97$	2.58305i	0.262269i	−0.991365	+	0.131135i	$0.958138\pi$
$98$	0	0
$99$	−0.115839	−0.0116422
$100$	0	0
$101$	−6.65615	−0.662312	−0.331156	−	0.943576i	$-0.607439\pi$
$101$	−6.65615	−0.662312	−0.331156	+	0.943576i	$0.607439\pi$
$102$	0	0
$103$	17.6668i	1.74076i	0.492383	+	0.870378i	$0.336126\pi$
$103$	17.6668i	1.74076i	−0.492383	+	0.870378i	$0.663874\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.83635i	− 0.564221i	−0.959382	−	0.282111i	$-0.908965\pi$
$107$	− 5.83635i	− 0.564221i	0.959382	−	0.282111i	$-0.0910345\pi$
$108$	0	0
$109$	−7.02096	−0.672486	−0.336243	−	0.941775i	$-0.609156\pi$
$109$	−7.02096	−0.672486	−0.336243	+	0.941775i	$0.609156\pi$
$110$	0	0
$111$	12.4329	1.18008
$112$	0	0
$113$	17.1003i	1.60866i	0.594185	+	0.804328i	$0.297475\pi$
$113$	17.1003i	1.60866i	−0.594185	+	0.804328i	$0.702525\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.75048i	− 0.161832i
$118$	0	0
$119$	−7.27781	−0.667156
$120$	0	0
$121$	−10.9909	−0.999171
$122$	0	0
$123$	− 3.83065i	− 0.345398i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.8050i	− 0.958787i	−0.877600	−	0.479394i	$-0.840857\pi$
$127$	− 10.8050i	− 0.958787i	0.877600	−	0.479394i	$-0.159143\pi$
$128$	0	0
$129$	12.7032	1.11846
$130$	0	0
$131$	−8.45211	−0.738464	−0.369232	−	0.929337i	$-0.620379\pi$
$131$	−8.45211	−0.738464	−0.369232	+	0.929337i	$0.620379\pi$
$132$	0	0
$133$	22.1833i	1.92354i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.5322i	− 1.58331i	−0.610969	−	0.791655i	$-0.709220\pi$
$137$	− 18.5322i	− 1.58331i	0.610969	−	0.791655i	$-0.290780\pi$
$138$	0	0
$139$	−7.42618	−0.629880	−0.314940	−	0.949112i	$-0.601984\pi$
$139$	−7.42618	−0.629880	−0.314940	+	0.949112i	$0.601984\pi$
$140$	0	0
$141$	−9.48327	−0.798635
$142$	0	0
$143$	0.137875i	0.0115297i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.05369i	− 0.334343i
$148$	0	0
$149$	4.53968	0.371905	0.185952	−	0.982559i	$-0.440463\pi$
$149$	4.53968	0.371905	0.185952	+	0.982559i	$0.440463\pi$
$150$	0	0
$151$	0.00675366	0.000549605 0	0.000274803	−	1.00000i	$-0.499913\pi$
$151$	0.00675366	0.000549605 0	0.000274803		1.00000i	$0.499913\pi$
$152$	0	0
$153$	− 2.78654i	− 0.225279i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.1721i	0.971439i	0.874115	+	0.485719i	$0.161442\pi$
$157$	12.1721i	0.971439i	−0.874115	+	0.485719i	$0.838558\pi$
$158$	0	0
$159$	−10.3465	−0.820529
$160$	0	0
$161$	3.16736	0.249623
$162$	0	0
$163$	− 20.2201i	− 1.58376i	−0.610677	−	0.791879i	$-0.709103\pi$
$163$	− 20.2201i	− 1.58376i	0.610677	−	0.791879i	$-0.290897\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.04709i	0.622702i	0.950295	+	0.311351i	$0.100782\pi$
$167$	8.04709i	0.622702i	−0.950295	+	0.311351i	$0.899218\pi$
$168$	0	0
$169$	10.9165	0.839732
$170$	0	0
$171$	−8.49358	−0.649520
$172$	0	0
$173$	− 16.8394i	− 1.28028i	−0.768259	−	0.640139i	$-0.778877\pi$
$173$	− 16.8394i	− 1.28028i	0.768259	−	0.640139i	$-0.221123\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 18.2387i	− 1.37090i
$178$	0	0
$179$	−23.7641	−1.77621	−0.888105	−	0.459641i	$-0.847978\pi$
$179$	−23.7641	−1.77621	−0.888105	+	0.459641i	$0.847978\pi$
$180$	0	0
$181$	−3.13518	−0.233036	−0.116518	−	0.993189i	$-0.537173\pi$
$181$	−3.13518	−0.233036	−0.116518	+	0.993189i	$0.537173\pi$
$182$	0	0
$183$	0.313385i	0.0231661i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.219479i	0.0160499i
$188$	0	0
$189$	17.8384	1.29755
$190$	0	0
$191$	6.17586	0.446870	0.223435	−	0.974719i	$-0.428273\pi$
$191$	6.17586	0.446870	0.223435	+	0.974719i	$0.428273\pi$
$192$	0	0
$193$	− 7.22267i	− 0.519899i	−0.965622	−	0.259949i	$-0.916294\pi$
$193$	− 7.22267i	− 0.519899i	0.965622	−	0.259949i	$-0.0837059\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.0019i	− 0.926349i	−0.886267	−	0.463174i	$-0.846710\pi$
$197$	− 13.0019i	− 0.926349i	0.886267	−	0.463174i	$-0.153290\pi$
$198$	0	0
$199$	1.84114	0.130515	0.0652575	−	0.997868i	$-0.479213\pi$
$199$	1.84114	0.130515	0.0652575	+	0.997868i	$0.479213\pi$
$200$	0	0
$201$	−10.0231	−0.706972
$202$	0	0
$203$	17.0745i	1.19839i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.21273i	0.0842903i
$208$	0	0
$209$	0.668989	0.0462749
$210$	0	0
$211$	−24.8791	−1.71275	−0.856375	−	0.516354i	$-0.827289\pi$
$211$	−24.8791	−1.71275	−0.856375	+	0.516354i	$0.827289\pi$
$212$	0	0
$213$	− 6.93078i	− 0.474889i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.85128i	0.125673i
$218$	0	0
$219$	−2.03721	−0.137662
$220$	0	0
$221$	−3.31664	−0.223101
$222$	0	0
$223$	4.04924i	0.271157i	0.990767	+	0.135579i	$0.0432894\pi$
$223$	4.04924i	0.271157i	−0.990767	+	0.135579i	$0.956711\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.3916i	− 1.15432i	−0.816630	−	0.577162i	$-0.804160\pi$
$227$	− 17.3916i	− 1.15432i	0.816630	−	0.577162i	$-0.195840\pi$
$228$	0	0
$229$	3.51761	0.232450	0.116225	−	0.993223i	$-0.462921\pi$
$229$	3.51761	0.232450	0.116225	+	0.993223i	$0.462921\pi$
$230$	0	0
$231$	−0.404468	−0.0266120
$232$	0	0
$233$	− 10.7910i	− 0.706941i	−0.935446	−	0.353471i	$-0.885001\pi$
$233$	− 10.7910i	− 0.706941i	0.935446	−	0.353471i	$-0.114999\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.06506i	0.264054i
$238$	0	0
$239$	2.86944	0.185608	0.0928042	−	0.995684i	$-0.470417\pi$
$239$	2.86944	0.185608	0.0928042	+	0.995684i	$0.470417\pi$
$240$	0	0
$241$	27.0486	1.74235	0.871176	−	0.490972i	$-0.163358\pi$
$241$	27.0486	1.74235	0.871176	+	0.490972i	$0.163358\pi$
$242$	0	0
$243$	11.6939i	0.750161i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.1093i	0.643242i
$248$	0	0
$249$	−21.3780	−1.35478
$250$	0	0
$251$	−11.7794	−0.743510	−0.371755	−	0.928331i	$-0.621244\pi$
$251$	−11.7794	−0.743510	−0.371755	+	0.928331i	$0.621244\pi$
$252$	0	0
$253$	− 0.0955192i	− 0.00600524i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.47742i	− 0.528807i	−0.964412	−	0.264404i	$-0.914825\pi$
$257$	− 8.47742i	− 0.528807i	0.964412	−	0.264404i	$-0.0851751\pi$
$258$	0	0
$259$	−29.4560	−1.83031
$260$	0	0
$261$	−6.53751	−0.404662
$262$	0	0
$263$	18.5118i	1.14149i	0.821128	+	0.570745i	$0.193345\pi$
$263$	18.5118i	1.14149i	−0.821128	+	0.570745i	$0.806655\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 7.39734i	− 0.452710i
$268$	0	0
$269$	25.4583	1.55222	0.776109	−	0.630599i	$-0.217191\pi$
$269$	25.4583	1.55222	0.776109	+	0.630599i	$0.217191\pi$
$270$	0	0
$271$	−14.1812	−0.861446	−0.430723	−	0.902484i	$-0.641741\pi$
$271$	−14.1812	−0.861446	−0.430723	+	0.902484i	$0.641741\pi$
$272$	0	0
$273$	− 6.11207i	− 0.369919i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 21.4281i	− 1.28749i	−0.765241	−	0.643744i	$-0.777380\pi$
$277$	− 21.4281i	− 1.28749i	0.765241	−	0.643744i	$-0.222620\pi$
$278$	0	0
$279$	−0.708823	−0.0424361
$280$	0	0
$281$	−8.90281	−0.531097	−0.265548	−	0.964098i	$-0.585553\pi$
$281$	−8.90281	−0.531097	−0.265548	+	0.964098i	$0.585553\pi$
$282$	0	0
$283$	− 13.0369i	− 0.774966i	−0.921877	−	0.387483i	$-0.873345\pi$
$283$	− 13.0369i	− 0.774966i	0.921877	−	0.387483i	$-0.126655\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.07558i	0.535715i
$288$	0	0
$289$	11.7203	0.689431
$290$	0	0
$291$	−3.45325	−0.202433
$292$	0	0
$293$	− 6.74772i	− 0.394206i	−0.980383	−	0.197103i	$-0.936847\pi$
$293$	− 6.74772i	− 0.394206i	0.980383	−	0.197103i	$-0.0631533\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 0.537959i	− 0.0312156i
$298$	0	0
$299$	1.44343	0.0834755
$300$	0	0
$301$	−30.0965	−1.73474
$302$	0	0
$303$	− 8.89854i	− 0.511208i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.6667i	− 0.894143i	−0.894498	−	0.447071i	$-0.852467\pi$
$307$	− 15.6667i	− 0.894143i	0.894498	−	0.447071i	$-0.147533\pi$
$308$	0	0
$309$	−23.6185	−1.34361
$310$	0	0
$311$	8.27922	0.469472	0.234736	−	0.972059i	$-0.424577\pi$
$311$	8.27922	0.469472	0.234736	+	0.972059i	$0.424577\pi$
$312$	0	0
$313$	15.4898i	0.875534i	0.899088	+	0.437767i	$0.144231\pi$
$313$	15.4898i	0.875534i	−0.899088	+	0.437767i	$0.855769\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.5803i	0.594251i	0.954838	+	0.297125i	$0.0960279\pi$
$317$	10.5803i	0.594251i	−0.954838	+	0.297125i	$0.903972\pi$
$318$	0	0
$319$	0.514921	0.0288300
$320$	0	0
$321$	7.80256	0.435496
$322$	0	0
$323$	16.0928i	0.895427i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.38625i	− 0.519061i
$328$	0	0
$329$	22.4678	1.23869
$330$	0	0
$331$	−1.32836	−0.0730133	−0.0365067	−	0.999333i	$-0.511623\pi$
$331$	−1.32836	−0.0730133	−0.0365067	+	0.999333i	$0.511623\pi$
$332$	0	0
$333$	− 11.2782i	− 0.618040i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 33.1297i	− 1.80469i	−0.431014	−	0.902345i	$-0.641844\pi$
$337$	− 33.1297i	− 1.80469i	0.431014	−	0.902345i	$-0.358156\pi$
$338$	0	0
$339$	−22.8611	−1.24165
$340$	0	0
$341$	0.0558298	0.00302335
$342$	0	0
$343$	− 12.5675i	− 0.678582i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.7579i	− 0.845926i	−0.906147	−	0.422963i	$-0.860990\pi$
$347$	− 15.7579i	− 0.845926i	0.906147	−	0.422963i	$-0.139010\pi$
$348$	0	0
$349$	14.5904	0.781004	0.390502	−	0.920602i	$-0.372301\pi$
$349$	14.5904	0.781004	0.390502	+	0.920602i	$0.372301\pi$
$350$	0	0
$351$	8.12931	0.433910
$352$	0	0
$353$	− 28.0097i	− 1.49080i	−0.666615	−	0.745402i	$-0.732258\pi$
$353$	− 28.0097i	− 1.49080i	0.666615	−	0.745402i	$-0.267742\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 9.72964i	− 0.514947i
$358$	0	0
$359$	−0.214689	−0.0113308	−0.00566542	−	0.999984i	$-0.501803\pi$
$359$	−0.214689	−0.0113308	−0.00566542	+	0.999984i	$0.501803\pi$
$360$	0	0
$361$	30.0520	1.58168
$362$	0	0
$363$	− 14.6936i	− 0.771213i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 21.0197i	− 1.09722i	−0.836079	−	0.548610i	$-0.815157\pi$
$367$	− 21.0197i	− 1.09722i	0.836079	−	0.548610i	$-0.184843\pi$
$368$	0	0
$369$	−3.47488	−0.180895
$370$	0	0
$371$	24.5129	1.27265
$372$	0	0
$373$	− 10.2558i	− 0.531023i	−0.964108	−	0.265511i	$-0.914459\pi$
$373$	− 10.2558i	− 0.531023i	0.964108	−	0.265511i	$-0.0855408\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.78116i	0.400750i
$378$	0	0
$379$	20.6274	1.05956	0.529780	−	0.848135i	$-0.322274\pi$
$379$	20.6274	1.05956	0.529780	+	0.848135i	$0.322274\pi$
$380$	0	0
$381$	14.4451	0.740044
$382$	0	0
$383$	18.7419i	0.957667i	0.877906	+	0.478833i	$0.158940\pi$
$383$	18.7419i	0.957667i	−0.877906	+	0.478833i	$0.841060\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 11.5234i	− 0.585769i
$388$	0	0
$389$	3.07719	0.156020	0.0780100	−	0.996953i	$-0.475143\pi$
$389$	3.07719	0.156020	0.0780100	+	0.996953i	$0.475143\pi$
$390$	0	0
$391$	2.29775	0.116202
$392$	0	0
$393$	− 11.2995i	− 0.569986i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 11.4670i	− 0.575513i	−0.957704	−	0.287756i	$-0.907091\pi$
$397$	− 11.4670i	− 0.575513i	0.957704	−	0.287756i	$-0.0929093\pi$
$398$	0	0
$399$	−29.6566	−1.48469
$400$	0	0
$401$	14.1227	0.705252	0.352626	−	0.935764i	$-0.385289\pi$
$401$	14.1227	0.705252	0.352626	+	0.935764i	$0.385289\pi$
$402$	0	0
$403$	0.843666i	0.0420260i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.888314i	0.0440321i
$408$	0	0
$409$	0.851429	0.0421005	0.0210502	−	0.999778i	$-0.493299\pi$
$409$	0.851429	0.0421005	0.0210502	+	0.999778i	$0.493299\pi$
$410$	0	0
$411$	24.7755	1.22208
$412$	0	0
$413$	43.2111i	2.12628i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.92799i	− 0.486176i
$418$	0	0
$419$	−0.680728	−0.0332557	−0.0166279	−	0.999862i	$-0.505293\pi$
$419$	−0.680728	−0.0332557	−0.0166279	+	0.999862i	$0.505293\pi$
$420$	0	0
$421$	−35.9671	−1.75293	−0.876466	−	0.481465i	$-0.840105\pi$
$421$	−35.9671	−1.75293	−0.876466	+	0.481465i	$0.840105\pi$
$422$	0	0
$423$	8.60251i	0.418268i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.742472i	− 0.0359307i
$428$	0	0
$429$	−0.184324	−0.00889923
$430$	0	0
$431$	15.2881	0.736402	0.368201	−	0.929746i	$-0.379974\pi$
$431$	15.2881	0.736402	0.368201	+	0.929746i	$0.379974\pi$
$432$	0	0
$433$	− 13.2632i	− 0.637390i	−0.947857	−	0.318695i	$-0.896755\pi$
$433$	− 13.2632i	− 0.637390i	0.947857	−	0.318695i	$-0.103245\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.00371i	− 0.335033i
$438$	0	0
$439$	−18.7740	−0.896035	−0.448018	−	0.894025i	$-0.647870\pi$
$439$	−18.7740	−0.896035	−0.448018	+	0.894025i	$0.647870\pi$
$440$	0	0
$441$	−3.67720	−0.175105
$442$	0	0
$443$	5.49270i	0.260966i	0.991451	+	0.130483i	$0.0416528\pi$
$443$	5.49270i	0.260966i	−0.991451	+	0.130483i	$0.958347\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.06905i	0.287056i
$448$	0	0
$449$	−13.1705	−0.621553	−0.310777	−	0.950483i	$-0.600589\pi$
$449$	−13.1705	−0.621553	−0.310777	+	0.950483i	$0.600589\pi$
$450$	0	0
$451$	0.273695	0.0128878
$452$	0	0
$453$	0.00902890i	0 0.000424215i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.9676i	0.980821i	0.871492	+	0.490410i	$0.163153\pi$
$457$	20.9676i	0.980821i	−0.871492	+	0.490410i	$0.836847\pi$
$458$	0	0
$459$	12.9408	0.604026
$460$	0	0
$461$	6.27335	0.292179	0.146089	−	0.989271i	$-0.453331\pi$
$461$	6.27335	0.292179	0.146089	+	0.989271i	$0.453331\pi$
$462$	0	0
$463$	− 9.49565i	− 0.441300i	−0.975353	−	0.220650i	$-0.929182\pi$
$463$	− 9.49565i	− 0.441300i	0.975353	−	0.220650i	$-0.0708179\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.9066i	0.643518i	0.946822	+	0.321759i	$0.104274\pi$
$467$	13.9066i	0.643518i	−0.946822	+	0.321759i	$0.895726\pi$
$468$	0	0
$469$	23.7466	1.09652
$470$	0	0
$471$	−16.2727	−0.749809
$472$	0	0
$473$	0.907631i	0.0417329i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.38555i	0.429735i
$478$	0	0
$479$	−3.66220	−0.167330	−0.0836652	−	0.996494i	$-0.526663\pi$
$479$	−3.66220	−0.167330	−0.0836652	+	0.996494i	$0.526663\pi$
$480$	0	0
$481$	−13.4237	−0.612066
$482$	0	0
$483$	4.23441i	0.192672i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	28.9654i	1.31255i	0.754523	+	0.656274i	$0.227868\pi$
$487$	28.9654i	1.31255i	−0.754523	+	0.656274i	$0.772132\pi$
$488$	0	0
$489$	27.0320	1.22243
$490$	0	0
$491$	−37.8991	−1.71036	−0.855182	−	0.518328i	$-0.826554\pi$
$491$	−37.8991	−1.71036	−0.855182	+	0.518328i	$0.826554\pi$
$492$	0	0
$493$	12.3866i	0.557866i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.4204i	0.736557i
$498$	0	0
$499$	26.6914	1.19487	0.597436	−	0.801917i	$-0.296186\pi$
$499$	26.6914	1.19487	0.597436	+	0.801917i	$0.296186\pi$
$500$	0	0
$501$	−10.7581	−0.480635
$502$	0	0
$503$	18.2182i	0.812309i	0.913804	+	0.406154i	$0.133130\pi$
$503$	18.2182i	0.812309i	−0.913804	+	0.406154i	$0.866870\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.5942i	0.648150i
$508$	0	0
$509$	31.0138	1.37466	0.687332	−	0.726344i	$-0.258782\pi$
$509$	31.0138	1.37466	0.687332	+	0.726344i	$0.258782\pi$
$510$	0	0
$511$	4.82655	0.213514
$512$	0	0
$513$	− 39.4446i	− 1.74152i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 0.677568i	− 0.0297994i
$518$	0	0
$519$	22.5125	0.988188
$520$	0	0
$521$	−39.9824	−1.75166	−0.875831	−	0.482618i	$-0.839686\pi$
$521$	−39.9824	−1.75166	−0.875831	+	0.482618i	$0.839686\pi$
$522$	0	0
$523$	− 10.3111i	− 0.450873i	−0.974258	−	0.225437i	$-0.927619\pi$
$523$	− 10.3111i	− 0.450873i	0.974258	−	0.225437i	$-0.0723808\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.34301i	0.0585024i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−16.5448	−0.717981
$532$	0	0
$533$	4.13592i	0.179146i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 31.7699i	− 1.37097i
$538$	0	0
$539$	0.289631	0.0124753
$540$	0	0
$541$	−42.5472	−1.82925	−0.914623	−	0.404307i	$-0.867513\pi$
$541$	−42.5472	−1.82925	−0.914623	+	0.404307i	$0.867513\pi$
$542$	0	0
$543$	− 4.19139i	− 0.179870i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.47986i	0.106031i	0.998594	+	0.0530156i	$0.0168833\pi$
$547$	2.47986i	0.106031i	−0.998594	+	0.0530156i	$0.983117\pi$
$548$	0	0
$549$	0.284279	0.0121327
$550$	0	0
$551$	37.7553	1.60843
$552$	0	0
$553$	− 9.63094i	− 0.409549i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.9814i	0.973755i	0.873470	+	0.486877i	$0.161864\pi$
$557$	22.9814i	0.973755i	−0.873470	+	0.486877i	$0.838136\pi$
$558$	0	0
$559$	−13.7156	−0.580107
$560$	0	0
$561$	−0.293420	−0.0123882
$562$	0	0
$563$	18.4939i	0.779427i	0.920936	+	0.389713i	$0.127426\pi$
$563$	18.4939i	0.779427i	−0.920936	+	0.389713i	$0.872574\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.3246i	0.517584i
$568$	0	0
$569$	−35.6081	−1.49277	−0.746385	−	0.665514i	$-0.768212\pi$
$569$	−35.6081	−1.49277	−0.746385	+	0.665514i	$0.768212\pi$
$570$	0	0
$571$	20.5827	0.861359	0.430680	−	0.902505i	$-0.358274\pi$
$571$	20.5827	0.861359	0.430680	+	0.902505i	$0.358274\pi$
$572$	0	0
$573$	8.25645i	0.344918i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19.0378i	− 0.792554i	−0.918131	−	0.396277i	$-0.870302\pi$
$577$	− 19.0378i	− 0.792554i	0.918131	−	0.396277i	$-0.129698\pi$
$578$	0	0
$579$	9.65591	0.401286
$580$	0	0
$581$	50.6489	2.10127
$582$	0	0
$583$	− 0.739244i	− 0.0306163i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.29062i	0.300916i	0.988616	+	0.150458i	$0.0480748\pi$
$587$	7.29062i	0.300916i	−0.988616	+	0.150458i	$0.951925\pi$
$588$	0	0
$589$	4.09358	0.168673
$590$	0	0
$591$	17.3821	0.715006
$592$	0	0
$593$	− 10.5832i	− 0.434598i	−0.976105	−	0.217299i	$-0.930275\pi$
$593$	− 10.5832i	− 0.434598i	0.976105	−	0.217299i	$-0.0697247\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.46140i	0.100739i
$598$	0	0
$599$	5.63520	0.230248	0.115124	−	0.993351i	$-0.463273\pi$
$599$	5.63520	0.230248	0.115124	+	0.993351i	$0.463273\pi$
$600$	0	0
$601$	−17.7093	−0.722377	−0.361188	−	0.932493i	$-0.617629\pi$
$601$	−17.7093	−0.722377	−0.361188	+	0.932493i	$0.617629\pi$
$602$	0	0
$603$	9.09216i	0.370261i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.0996i	0.896996i	0.893784	+	0.448498i	$0.148041\pi$
$607$	22.0996i	0.896996i	−0.893784	+	0.448498i	$0.851959\pi$
$608$	0	0
$609$	−22.8267	−0.924984
$610$	0	0
$611$	10.2390	0.414225
$612$	0	0
$613$	− 5.08532i	− 0.205394i	−0.994713	−	0.102697i	$-0.967253\pi$
$613$	− 5.08532i	− 0.205394i	0.994713	−	0.102697i	$-0.0327472\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.63073i	0.307202i	0.988133	+	0.153601i	$0.0490870\pi$
$617$	7.63073i	0.307202i	−0.988133	+	0.153601i	$0.950913\pi$
$618$	0	0
$619$	−38.5591	−1.54982	−0.774910	−	0.632072i	$-0.782205\pi$
$619$	−38.5591	−1.54982	−0.774910	+	0.632072i	$0.782205\pi$
$620$	0	0
$621$	−5.63195	−0.226002
$622$	0	0
$623$	17.5258i	0.702156i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.894364i	0.0357175i
$628$	0	0
$629$	−21.3688	−0.852028
$630$	0	0
$631$	−1.86751	−0.0743444	−0.0371722	−	0.999309i	$-0.511835\pi$
$631$	−1.86751	−0.0743444	−0.0371722	+	0.999309i	$0.511835\pi$
$632$	0	0
$633$	− 33.2607i	− 1.32199i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.37673i	0.173412i
$638$	0	0
$639$	−6.28708	−0.248713
$640$	0	0
$641$	−1.91236	−0.0755338	−0.0377669	−	0.999287i	$-0.512024\pi$
$641$	−1.91236	−0.0755338	−0.0377669	+	0.999287i	$0.512024\pi$
$642$	0	0
$643$	− 5.06670i	− 0.199811i	−0.994997	−	0.0999055i	$-0.968146\pi$
$643$	− 5.06670i	− 0.199811i	0.994997	−	0.0999055i	$-0.0318541\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.0429i	1.45631i	0.685414	+	0.728153i	$0.259621\pi$
$647$	37.0429i	1.45631i	−0.685414	+	0.728153i	$0.740379\pi$
$648$	0	0
$649$	1.30313	0.0511524
$650$	0	0
$651$	−2.47496	−0.0970014
$652$	0	0
$653$	11.9884i	0.469143i	0.972099	+	0.234572i	$0.0753687\pi$
$653$	11.9884i	0.469143i	−0.972099	+	0.234572i	$0.924631\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.84800i	0.0720974i
$658$	0	0
$659$	2.82513	0.110051	0.0550257	−	0.998485i	$-0.482476\pi$
$659$	2.82513	0.110051	0.0550257	+	0.998485i	$0.482476\pi$
$660$	0	0
$661$	−26.5004	−1.03075	−0.515374	−	0.856965i	$-0.672347\pi$
$661$	−26.5004	−1.03075	−0.515374	+	0.856965i	$0.672347\pi$
$662$	0	0
$663$	− 4.43398i	− 0.172202i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.39076i	− 0.208731i
$668$	0	0
$669$	−5.41339	−0.209294
$670$	0	0
$671$	−0.0223910	−0.000864393 0
$672$	0	0
$673$	19.0620i	0.734784i	0.930066	+	0.367392i	$0.119749\pi$
$673$	19.0620i	0.734784i	−0.930066	+	0.367392i	$0.880251\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.54346i	− 0.0977533i	−0.998805	−	0.0488766i	$-0.984436\pi$
$677$	− 2.54346i	− 0.0977533i	0.998805	−	0.0488766i	$-0.0155641\pi$
$678$	0	0
$679$	8.18146	0.313975
$680$	0	0
$681$	23.2507	0.890969
$682$	0	0
$683$	− 0.377109i	− 0.0144297i	−0.999974	−	0.00721483i	$-0.997703\pi$
$683$	− 0.377109i	− 0.0144297i	0.999974	−	0.00721483i	$-0.00229657\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.70266i	0.179418i
$688$	0	0
$689$	11.1710	0.425581
$690$	0	0
$691$	7.24743	0.275705	0.137853	−	0.990453i	$-0.455980\pi$
$691$	7.24743	0.275705	0.137853	+	0.990453i	$0.455980\pi$
$692$	0	0
$693$	0.366903i	0.0139375i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.58385i	0.249381i
$698$	0	0
$699$	14.4264	0.545655
$700$	0	0
$701$	−15.7381	−0.594420	−0.297210	−	0.954812i	$-0.596056\pi$
$701$	−15.7381	−0.594420	−0.297210	+	0.954812i	$0.596056\pi$
$702$	0	0
$703$	65.1335i	2.45656i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.0824i	0.792887i
$708$	0	0
$709$	45.9062	1.72404	0.862022	−	0.506870i	$-0.169198\pi$
$709$	45.9062	1.72404	0.862022	+	0.506870i	$0.169198\pi$
$710$	0	0
$711$	3.68751	0.138293
$712$	0	0
$713$	− 0.584488i	− 0.0218892i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.83612i	0.143263i
$718$	0	0
$719$	50.4975	1.88324	0.941619	−	0.336682i	$-0.109305\pi$
$719$	50.4975	1.88324	0.941619	+	0.336682i	$0.109305\pi$
$720$	0	0
$721$	55.9570	2.08395
$722$	0	0
$723$	36.1609i	1.34484i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.0273i	0.705685i	0.935683	+	0.352842i	$0.114785\pi$
$727$	19.0273i	0.705685i	−0.935683	+	0.352842i	$0.885215\pi$
$728$	0	0
$729$	−27.3067	−1.01136
$730$	0	0
$731$	−21.8334	−0.807539
$732$	0	0
$733$	− 42.1027i	− 1.55510i	−0.628822	−	0.777549i	$-0.716463\pi$
$733$	− 42.1027i	− 1.55510i	0.628822	−	0.777549i	$-0.283537\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 0.716135i	− 0.0263792i
$738$	0	0
$739$	−47.9212	−1.76281	−0.881405	−	0.472361i	$-0.843402\pi$
$739$	−47.9212	−1.76281	−0.881405	+	0.472361i	$0.843402\pi$
$740$	0	0
$741$	−13.5151	−0.496489
$742$	0	0
$743$	21.3492i	0.783225i	0.920130	+	0.391613i	$0.128083\pi$
$743$	21.3492i	0.783225i	−0.920130	+	0.391613i	$0.871917\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	19.3925i	0.709536i
$748$	0	0
$749$	−18.4858	−0.675458
$750$	0	0
$751$	−27.8218	−1.01523	−0.507617	−	0.861583i	$-0.669473\pi$
$751$	−27.8218	−1.01523	−0.507617	+	0.861583i	$0.669473\pi$
$752$	0	0
$753$	− 15.7478i	− 0.573881i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 8.74990i	− 0.318020i	−0.987277	−	0.159010i	$-0.949170\pi$
$757$	− 8.74990i	− 0.318020i	0.987277	−	0.159010i	$-0.0508303\pi$
$758$	0	0
$759$	0.127699	0.00463517
$760$	0	0
$761$	21.9923	0.797221	0.398610	−	0.917120i	$-0.369493\pi$
$761$	21.9923	0.797221	0.398610	+	0.917120i	$0.369493\pi$
$762$	0	0
$763$	22.2379i	0.805066i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.6921i	0.711041i
$768$	0	0
$769$	31.3229	1.12953	0.564766	−	0.825251i	$-0.308966\pi$
$769$	31.3229	1.12953	0.564766	+	0.825251i	$0.308966\pi$
$770$	0	0
$771$	11.3334	0.408162
$772$	0	0
$773$	32.3308i	1.16286i	0.813597	+	0.581429i	$0.197506\pi$
$773$	32.3308i	1.16286i	−0.813597	+	0.581429i	$0.802494\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 39.3794i	− 1.41273i
$778$	0	0
$779$	20.0680	0.719012
$780$	0	0
$781$	0.495196	0.0177195
$782$	0	0
$783$	− 30.3605i	− 1.08499i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.5712i	0.661990i	0.943632	+	0.330995i	$0.107384\pi$
$787$	18.5712i	0.661990i	−0.943632	+	0.330995i	$0.892616\pi$
$788$	0	0
$789$	−24.7483	−0.881063
$790$	0	0
$791$	54.1627	1.92580
$792$	0	0
$793$	− 0.338358i	− 0.0120155i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.1985i	0.857154i	0.903505	+	0.428577i	$0.140985\pi$
$797$	24.1985i	0.857154i	−0.903505	+	0.428577i	$0.859015\pi$
$798$	0	0
$799$	16.2992	0.576624
$800$	0	0
$801$	−6.71031	−0.237097
$802$	0	0
$803$	− 0.145556i	− 0.00513656i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.0349i	1.19809i
$808$	0	0
$809$	30.4574	1.07082	0.535412	−	0.844591i	$-0.320156\pi$
$809$	30.4574	1.07082	0.535412	+	0.844591i	$0.320156\pi$
$810$	0	0
$811$	4.05626	0.142435	0.0712174	−	0.997461i	$-0.477312\pi$
$811$	4.05626	0.142435	0.0712174	+	0.997461i	$0.477312\pi$
$812$	0	0
$813$	− 18.9587i	− 0.664910i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	66.5499i	2.32829i
$818$	0	0
$819$	−5.54441	−0.193737
$820$	0	0
$821$	19.0615	0.665252	0.332626	−	0.943059i	$-0.392065\pi$
$821$	19.0615	0.665252	0.332626	+	0.943059i	$0.392065\pi$
$822$	0	0
$823$	− 42.9810i	− 1.49822i	−0.662444	−	0.749111i	$-0.730481\pi$
$823$	− 42.9810i	− 1.49822i	0.662444	−	0.749111i	$-0.269519\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 51.6777i	− 1.79701i	−0.438963	−	0.898505i	$-0.644654\pi$
$827$	− 51.6777i	− 1.79701i	0.438963	−	0.898505i	$-0.355346\pi$
$828$	0	0
$829$	−25.6228	−0.889915	−0.444958	−	0.895552i	$-0.646781\pi$
$829$	−25.6228	−0.889915	−0.444958	+	0.895552i	$0.646781\pi$
$830$	0	0
$831$	28.6470	0.993753
$832$	0	0
$833$	6.96720i	0.241399i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.29181i	− 0.113781i
$838$	0	0
$839$	53.0313	1.83084	0.915422	−	0.402495i	$-0.131857\pi$
$839$	53.0313	1.83084	0.915422	+	0.402495i	$0.131857\pi$
$840$	0	0
$841$	0.0602571	0.00207783
$842$	0	0
$843$	− 11.9021i	− 0.409929i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.8121i	1.19616i
$848$	0	0
$849$	17.4290	0.598160
$850$	0	0
$851$	9.29985	0.318795
$852$	0	0
$853$	13.8668i	0.474791i	0.971413	+	0.237395i	$0.0762937\pi$
$853$	13.8668i	0.474791i	−0.971413	+	0.237395i	$0.923706\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 16.3518i	− 0.558567i	−0.960209	−	0.279283i	$-0.909903\pi$
$857$	− 16.3518i	− 0.558567i	0.960209	−	0.279283i	$-0.0900969\pi$
$858$	0	0
$859$	−42.1443	−1.43795	−0.718973	−	0.695038i	$-0.755388\pi$
$859$	−42.1443	−1.43795	−0.718973	+	0.695038i	$0.755388\pi$
$860$	0	0
$861$	−12.1331	−0.413493
$862$	0	0
$863$	− 15.2130i	− 0.517856i	−0.965897	−	0.258928i	$-0.916631\pi$
$863$	− 15.2130i	− 0.517856i	0.965897	−	0.258928i	$-0.0833692\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.6688i	0.532140i
$868$	0	0
$869$	−0.290443	−0.00985262
$870$	0	0
$871$	10.8218	0.366683
$872$	0	0
$873$	3.13253i	0.106020i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.6934i	1.81310i	0.422101	+	0.906549i	$0.361293\pi$
$877$	53.6934i	1.81310i	−0.422101	+	0.906549i	$0.638707\pi$
$878$	0	0
$879$	9.02096	0.304269
$880$	0	0
$881$	46.5787	1.56928	0.784638	−	0.619954i	$-0.212849\pi$
$881$	46.5787	1.56928	0.784638	+	0.619954i	$0.212849\pi$
$882$	0	0
$883$	− 19.9787i	− 0.672337i	−0.941802	−	0.336169i	$-0.890869\pi$
$883$	− 19.9787i	− 0.672337i	0.941802	−	0.336169i	$-0.109131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 51.9251i	− 1.74347i	−0.489974	−	0.871737i	$-0.662994\pi$
$887$	− 51.9251i	− 1.74347i	0.489974	−	0.871737i	$-0.337006\pi$
$888$	0	0
$889$	−34.2233	−1.14781
$890$	0	0
$891$	0.371676	0.0124516
$892$	0	0
$893$	− 49.6810i	− 1.66251i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.92970i	0.0644309i
$898$	0	0
$899$	3.15083	0.105086
$900$	0	0
$901$	17.7828	0.592431
$902$	0	0
$903$	− 40.2358i	− 1.33896i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.64335i	0.220589i	0.993899	+	0.110294i	$0.0351794\pi$
$907$	6.64335i	0.220589i	−0.993899	+	0.110294i	$0.964821\pi$
$908$	0	0
$909$	−8.07209	−0.267734
$910$	0	0
$911$	15.2064	0.503811	0.251905	−	0.967752i	$-0.418943\pi$
$911$	15.2064	0.503811	0.251905	+	0.967752i	$0.418943\pi$
$912$	0	0
$913$	− 1.52744i	− 0.0505507i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.7709i	0.884052i
$918$	0	0
$919$	−41.2068	−1.35929	−0.679644	−	0.733542i	$-0.737866\pi$
$919$	−41.2068	−1.35929	−0.679644	+	0.733542i	$0.737866\pi$
$920$	0	0
$921$	20.9446	0.690148
$922$	0	0
$923$	7.48310i	0.246309i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	21.4249i	0.703687i
$928$	0	0
$929$	4.86715	0.159686	0.0798431	−	0.996807i	$-0.474558\pi$
$929$	4.86715	0.159686	0.0798431	+	0.996807i	$0.474558\pi$
$930$	0	0
$931$	21.2365	0.695999
$932$	0	0
$933$	11.0684i	0.362364i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.44469i	0.112533i	0.998416	+	0.0562666i	$0.0179197\pi$
$937$	3.44469i	0.112533i	−0.998416	+	0.0562666i	$0.982080\pi$
$938$	0	0
$939$	−20.7081	−0.675784
$940$	0	0
$941$	−34.8591	−1.13637	−0.568187	−	0.822900i	$-0.692355\pi$
$941$	−34.8591	−1.13637	−0.568187	+	0.822900i	$0.692355\pi$
$942$	0	0
$943$	− 2.86534i	− 0.0933084i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.17298i	0.168099i	0.996462	+	0.0840496i	$0.0267854\pi$
$947$	5.17298i	0.168099i	−0.996462	+	0.0840496i	$0.973215\pi$
$948$	0	0
$949$	2.19955	0.0714005
$950$	0	0
$951$	−14.1447	−0.458675
$952$	0	0
$953$	3.41090i	0.110490i	0.998473	+	0.0552449i	$0.0175940\pi$
$953$	3.41090i	0.110490i	−0.998473	+	0.0552449i	$0.982406\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.688392i	0.0222526i
$958$	0	0
$959$	−58.6981	−1.89546
$960$	0	0
$961$	−30.6584	−0.988980
$962$	0	0
$963$	− 7.07789i	− 0.228082i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 17.5047i	− 0.562914i	−0.959574	−	0.281457i	$-0.909182\pi$
$967$	− 17.5047i	− 0.562914i	0.959574	−	0.281457i	$-0.0908176\pi$
$968$	0	0
$969$	−21.5143	−0.691139
$970$	0	0
$971$	35.9954	1.15515	0.577574	−	0.816338i	$-0.303999\pi$
$971$	35.9954	1.15515	0.577574	+	0.816338i	$0.303999\pi$
$972$	0	0
$973$	23.5214i	0.754062i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.3634i	1.00340i	0.865040	+	0.501702i	$0.167293\pi$
$977$	31.3634i	1.00340i	−0.865040	+	0.501702i	$0.832707\pi$
$978$	0	0
$979$	0.528531	0.0168919
$980$	0	0
$981$	−8.51450	−0.271847
$982$	0	0
$983$	− 36.4262i	− 1.16182i	−0.813969	−	0.580908i	$-0.802698\pi$
$983$	− 36.4262i	− 1.16182i	0.813969	−	0.580908i	$-0.197302\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.0369i	0.956086i
$988$	0	0
$989$	9.50208	0.302149
$990$	0	0
$991$	6.22316	0.197685	0.0988426	−	0.995103i	$-0.468486\pi$
$991$	6.22316	0.197685	0.0988426	+	0.995103i	$0.468486\pi$
$992$	0	0
$993$	− 1.77587i	− 0.0563556i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.0896i	0.762925i	0.924384	+	0.381463i	$0.124580\pi$
$997$	24.0896i	0.762925i	−0.924384	+	0.381463i	$0.875420\pi$
$998$	0	0
$999$	52.3763	1.65711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.v.4049.7		10
5.2	odd	4		4600.2.a.bg.1.3	yes	5
5.3	odd	4		4600.2.a.bc.1.3	✓	5
5.4	even	2	inner	4600.2.e.v.4049.4		10
20.3	even	4		9200.2.a.cw.1.3		5
20.7	even	4		9200.2.a.cs.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.3	✓	5	5.3	odd	4
4600.2.a.bg.1.3	yes	5	5.2	odd	4
4600.2.e.v.4049.4		10	5.4	even	2	inner
4600.2.e.v.4049.7		10	1.1	even	1	trivial
9200.2.a.cs.1.3		5	20.7	even	4
9200.2.a.cw.1.3		5	20.3	even	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.33689i q^{3} -3.16736i q^{7} +1.21273 q^{9} +O(q^{10})\)	\(q+1.33689i q^{3} -3.16736i q^{7} +1.21273 q^{9} -0.0955192 q^{11} -1.44343i q^{13} -2.29775i q^{17} -7.00371 q^{19} +4.23441 q^{21} +1.00000i q^{23} +5.63195i q^{27} -5.39076 q^{29} -0.584488 q^{31} -0.127699i q^{33} -9.29985i q^{37} +1.92970 q^{39} -2.86534 q^{41} -9.50208i q^{43} +7.09353i q^{47} -3.03218 q^{49} +3.07184 q^{51} +7.73922i q^{53} -9.36319i q^{57} -13.6426 q^{59} +0.234413 q^{61} -3.84114i q^{63} +7.49729i q^{67} -1.33689 q^{69} -5.18426 q^{71} +1.52384i q^{73} +0.302544i q^{77} +3.04068 q^{79} -3.89112 q^{81} +15.9909i q^{83} -7.20685i q^{87} -5.53325 q^{89} -4.57186 q^{91} -0.781396i q^{93} +2.58305i q^{97} -0.115839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{9}+O(q^{10}) \) 10 * q - 8 * q^9	\( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) \) 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more