LMFDB - Embedded newform 4600.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.a (trivial)

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:	yes
Analytic conductor:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.791953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2 $ x^5 - 2x^4 - 7x^3 + 7x^2 + 9x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.336890$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33689 q^{3} -3.16736 q^{7} -1.21273 q^{9} +O(q^{10})$	$q-1.33689 q^{3} -3.16736 q^{7} -1.21273 q^{9} -0.0955192 q^{11} +1.44343 q^{13} -2.29775 q^{17} +7.00371 q^{19} +4.23441 q^{21} -1.00000 q^{23} +5.63195 q^{27} +5.39076 q^{29} -0.584488 q^{31} +0.127699 q^{33} -9.29985 q^{37} -1.92970 q^{39} -2.86534 q^{41} +9.50208 q^{43} +7.09353 q^{47} +3.03218 q^{49} +3.07184 q^{51} -7.73922 q^{53} -9.36319 q^{57} +13.6426 q^{59} +0.234413 q^{61} +3.84114 q^{63} +7.49729 q^{67} +1.33689 q^{69} -5.18426 q^{71} -1.52384 q^{73} +0.302544 q^{77} -3.04068 q^{79} -3.89112 q^{81} -15.9909 q^{83} -7.20685 q^{87} +5.53325 q^{89} -4.57186 q^{91} +0.781396 q^{93} +2.58305 q^{97} +0.115839 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + q^{7} + 4 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 + q^7 + 4 * q^9	$ 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49} - 5 q^{51} - 18 q^{53} - 27 q^{57} + 23 q^{59} - 26 q^{61} - 5 q^{63} - 3 q^{67} + 3 q^{69} - 2 q^{71} - 4 q^{73} - 15 q^{77} + 43 q^{79} - 3 q^{81} - 30 q^{83} - 27 q^{87} + 15 q^{89} - 19 q^{91} + 15 q^{93} - 8 q^{97} + 37 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + q^7 + 4 * q^9 - 4 * q^11 + q^13 - 5 * q^17 + 4 * q^19 - 6 * q^21 - 5 * q^23 - 6 * q^27 - 11 * q^29 + 4 * q^31 - 13 * q^33 - 6 * q^37 + 31 * q^39 - 8 * q^41 - 3 * q^43 - 2 * q^47 - 2 * q^49 - 5 * q^51 - 18 * q^53 - 27 * q^57 + 23 * q^59 - 26 * q^61 - 5 * q^63 - 3 * q^67 + 3 * q^69 - 2 * q^71 - 4 * q^73 - 15 * q^77 + 43 * q^79 - 3 * q^81 - 30 * q^83 - 27 * q^87 + 15 * q^89 - 19 * q^91 + 15 * q^93 - 8 * q^97 + 37 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.33689	−0.771854	−0.385927	−	0.922529i	$-0.626118\pi$
$3$	−1.33689	−0.771854	−0.385927	+	0.922529i	$0.626118\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.16736	−1.19715	−0.598575	−	0.801067i	$-0.704266\pi$
$7$	−3.16736	−1.19715	−0.598575	+	0.801067i	$0.704266\pi$
$8$	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
$13$	1.44343	0.400335	0.200167	−	0.979762i	$-0.435851\pi$
$13$	1.44343	0.400335	0.200167	+	0.979762i	$0.435851\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.29775	−0.557287	−0.278643	−	0.960395i	$-0.589885\pi$
$17$	−2.29775	−0.557287	−0.278643	+	0.960395i	$0.589885\pi$
$18$	0	0
$19$	7.00371	1.60676	0.803381	−	0.595466i	$-0.203032\pi$
$19$	7.00371	1.60676	0.803381	+	0.595466i	$0.203032\pi$
$20$	0	0
$21$	4.23441	0.924025
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.63195	1.08387
$28$	0	0
$29$	5.39076	1.00104	0.500519	−	0.865725i	$-0.333142\pi$
$29$	5.39076	1.00104	0.500519	+	0.865725i	$0.333142\pi$
$30$	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
$33$	0.127699	0.0222295
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.29985	−1.52889	−0.764443	−	0.644691i	$-0.776986\pi$
$37$	−9.29985	−1.52889	−0.764443	+	0.644691i	$0.776986\pi$
$38$	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
$43$	9.50208	1.44905	0.724527	−	0.689246i	$-0.242058\pi$
$43$	9.50208	1.44905	0.724527	+	0.689246i	$0.242058\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.09353	1.03470	0.517349	−	0.855775i	$-0.326919\pi$
$47$	7.09353	1.03470	0.517349	+	0.855775i	$0.326919\pi$
$48$	0	0
$49$	3.03218	0.433168
$50$	0	0
$51$	3.07184	0.430144
$52$	0	0
$53$	−7.73922	−1.06306	−0.531532	−	0.847038i	$-0.678383\pi$
$53$	−7.73922	−1.06306	−0.531532	+	0.847038i	$0.678383\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.36319	−1.24018
$58$	0	0
$59$	13.6426	1.77612	0.888059	−	0.459730i	$-0.152054\pi$
$59$	13.6426	1.77612	0.888059	+	0.459730i	$0.152054\pi$
$60$	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
$63$	3.84114	0.483938
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.49729	0.915940	0.457970	−	0.888968i	$-0.348577\pi$
$67$	7.49729	0.915940	0.457970	+	0.888968i	$0.348577\pi$
$68$	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
$73$	−1.52384	−0.178352	−0.0891760	−	0.996016i	$-0.528423\pi$
$73$	−1.52384	−0.178352	−0.0891760	+	0.996016i	$0.528423\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.302544	0.0344781
$78$	0	0
$79$	−3.04068	−0.342103	−0.171052	−	0.985262i	$-0.554717\pi$
$79$	−3.04068	−0.342103	−0.171052	+	0.985262i	$0.554717\pi$
$80$	0	0
$81$	−3.89112	−0.432347
$82$	0	0
$83$	−15.9909	−1.75523	−0.877613	−	0.479369i	$-0.840865\pi$
$83$	−15.9909	−1.75523	−0.877613	+	0.479369i	$0.840865\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.20685	−0.772655
$88$	0	0
$89$	5.53325	0.586523	0.293261	−	0.956032i	$-0.405259\pi$
$89$	5.53325	0.586523	0.293261	+	0.956032i	$0.405259\pi$
$90$	0	0
$91$	−4.57186	−0.479261
$92$	0	0
$93$	0.781396	0.0810270
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.58305	0.262269	0.131135	−	0.991365i	$-0.458138\pi$
$97$	2.58305	0.262269	0.131135	+	0.991365i	$0.458138\pi$
$98$	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
$103$	−17.6668	−1.74076	−0.870378	−	0.492383i	$-0.836126\pi$
$103$	−17.6668	−1.74076	−0.870378	+	0.492383i	$0.836126\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.83635	−0.564221	−0.282111	−	0.959382i	$-0.591035\pi$
$107$	−5.83635	−0.564221	−0.282111	+	0.959382i	$0.591035\pi$
$108$	0	0
$109$	7.02096	0.672486	0.336243	−	0.941775i	$-0.390844\pi$
$109$	7.02096	0.672486	0.336243	+	0.941775i	$0.390844\pi$
$110$	0	0
$111$	12.4329	1.18008
$112$	0	0
$113$	−17.1003	−1.60866	−0.804328	−	0.594185i	$-0.797475\pi$
$113$	−17.1003	−1.60866	−0.804328	+	0.594185i	$0.797475\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.75048	−0.161832
$118$	0	0
$119$	7.27781	0.667156
$120$	0	0
$121$	−10.9909	−0.999171
$122$	0	0
$123$	3.83065	0.345398
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.8050	−0.958787	−0.479394	−	0.877600i	$-0.659143\pi$
$127$	−10.8050	−0.958787	−0.479394	+	0.877600i	$0.659143\pi$
$128$	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
$133$	−22.1833	−1.92354
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.5322	−1.58331	−0.791655	−	0.610969i	$-0.790780\pi$
$137$	−18.5322	−1.58331	−0.791655	+	0.610969i	$0.790780\pi$
$138$	0	0
$139$	7.42618	0.629880	0.314940	−	0.949112i	$-0.398016\pi$
$139$	7.42618	0.629880	0.314940	+	0.949112i	$0.398016\pi$
$140$	0	0
$141$	−9.48327	−0.798635
$142$	0	0
$143$	−0.137875	−0.0115297
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.05369	−0.334343
$148$	0	0
$149$	−4.53968	−0.371905	−0.185952	−	0.982559i	$-0.559537\pi$
$149$	−4.53968	−0.371905	−0.185952	+	0.982559i	$0.559537\pi$
$150$	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
$153$	2.78654	0.225279
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.1721	0.971439	0.485719	−	0.874115i	$-0.338558\pi$
$157$	12.1721	0.971439	0.485719	+	0.874115i	$0.338558\pi$
$158$	0	0
$159$	10.3465	0.820529
$160$	0	0
$161$	3.16736	0.249623
$162$	0	0
$163$	20.2201	1.58376	0.791879	−	0.610677i	$-0.209103\pi$
$163$	20.2201	1.58376	0.791879	+	0.610677i	$0.209103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.04709	0.622702	0.311351	−	0.950295i	$-0.399218\pi$
$167$	8.04709	0.622702	0.311351	+	0.950295i	$0.399218\pi$
$168$	0	0
$169$	−10.9165	−0.839732
$170$	0	0
$171$	−8.49358	−0.649520
$172$	0	0
$173$	16.8394	1.28028	0.640139	−	0.768259i	$-0.278877\pi$
$173$	16.8394	1.28028	0.640139	+	0.768259i	$0.278877\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.2387	−1.37090
$178$	0	0
$179$	23.7641	1.77621	0.888105	−	0.459641i	$-0.152022\pi$
$179$	23.7641	1.77621	0.888105	+	0.459641i	$0.152022\pi$
$180$	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
$183$	−0.313385	−0.0231661
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.219479	0.0160499
$188$	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
$193$	7.22267	0.519899	0.259949	−	0.965622i	$-0.416294\pi$
$193$	7.22267	0.519899	0.259949	+	0.965622i	$0.416294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.0019	−0.926349	−0.463174	−	0.886267i	$-0.653290\pi$
$197$	−13.0019	−0.926349	−0.463174	+	0.886267i	$0.653290\pi$
$198$	0	0
$199$	−1.84114	−0.130515	−0.0652575	−	0.997868i	$-0.520787\pi$
$199$	−1.84114	−0.130515	−0.0652575	+	0.997868i	$0.520787\pi$
$200$	0	0
$201$	−10.0231	−0.706972
$202$	0	0
$203$	−17.0745	−1.19839
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.21273	0.0842903
$208$	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
$213$	6.93078	0.474889
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.85128	0.125673
$218$	0	0
$219$	2.03721	0.137662
$220$	0	0
$221$	−3.31664	−0.223101
$222$	0	0
$223$	−4.04924	−0.271157	−0.135579	−	0.990767i	$-0.543289\pi$
$223$	−4.04924	−0.271157	−0.135579	+	0.990767i	$0.543289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.3916	−1.15432	−0.577162	−	0.816630i	$-0.695840\pi$
$227$	−17.3916	−1.15432	−0.577162	+	0.816630i	$0.695840\pi$
$228$	0	0
$229$	−3.51761	−0.232450	−0.116225	−	0.993223i	$-0.537079\pi$
$229$	−3.51761	−0.232450	−0.116225	+	0.993223i	$0.537079\pi$
$230$	0	0
$231$	−0.404468	−0.0266120
$232$	0	0
$233$	10.7910	0.706941	0.353471	−	0.935446i	$-0.385001\pi$
$233$	10.7910	0.706941	0.353471	+	0.935446i	$0.385001\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.06506	0.264054
$238$	0	0
$239$	−2.86944	−0.185608	−0.0928042	−	0.995684i	$-0.529583\pi$
$239$	−2.86944	−0.185608	−0.0928042	+	0.995684i	$0.529583\pi$
$240$	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
$243$	−11.6939	−0.750161
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.1093	0.643242
$248$	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
$253$	0.0955192	0.00600524
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.47742	−0.528807	−0.264404	−	0.964412i	$-0.585175\pi$
$257$	−8.47742	−0.528807	−0.264404	+	0.964412i	$0.585175\pi$
$258$	0	0
$259$	29.4560	1.83031
$260$	0	0
$261$	−6.53751	−0.404662
$262$	0	0
$263$	−18.5118	−1.14149	−0.570745	−	0.821128i	$-0.693345\pi$
$263$	−18.5118	−1.14149	−0.570745	+	0.821128i	$0.693345\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.39734	−0.452710
$268$	0	0
$269$	−25.4583	−1.55222	−0.776109	−	0.630599i	$-0.782809\pi$
$269$	−25.4583	−1.55222	−0.776109	+	0.630599i	$0.782809\pi$
$270$	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
$273$	6.11207	0.369919
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−21.4281	−1.28749	−0.643744	−	0.765241i	$-0.722620\pi$
$277$	−21.4281	−1.28749	−0.643744	+	0.765241i	$0.722620\pi$
$278$	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
$283$	13.0369	0.774966	0.387483	−	0.921877i	$-0.373345\pi$
$283$	13.0369	0.774966	0.387483	+	0.921877i	$0.373345\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.07558	0.535715
$288$	0	0
$289$	−11.7203	−0.689431
$290$	0	0
$291$	−3.45325	−0.202433
$292$	0	0
$293$	6.74772	0.394206	0.197103	−	0.980383i	$-0.436847\pi$
$293$	6.74772	0.394206	0.197103	+	0.980383i	$0.436847\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.537959	−0.0312156
$298$	0	0
$299$	−1.44343	−0.0834755
$300$	0	0
$301$	−30.0965	−1.73474
$302$	0	0
$303$	8.89854	0.511208
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.6667	−0.894143	−0.447071	−	0.894498i	$-0.647533\pi$
$307$	−15.6667	−0.894143	−0.447071	+	0.894498i	$0.647533\pi$
$308$	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
$313$	−15.4898	−0.875534	−0.437767	−	0.899088i	$-0.644231\pi$
$313$	−15.4898	−0.875534	−0.437767	+	0.899088i	$0.644231\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.5803	0.594251	0.297125	−	0.954838i	$-0.403972\pi$
$317$	10.5803	0.594251	0.297125	+	0.954838i	$0.403972\pi$
$318$	0	0
$319$	−0.514921	−0.0288300
$320$	0	0
$321$	7.80256	0.435496
$322$	0	0
$323$	−16.0928	−0.895427
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.38625	−0.519061
$328$	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
$333$	11.2782	0.618040
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.1297	−1.80469	−0.902345	−	0.431014i	$-0.858156\pi$
$337$	−33.1297	−1.80469	−0.902345	+	0.431014i	$0.858156\pi$
$338$	0	0
$339$	22.8611	1.24165
$340$	0	0
$341$	0.0558298	0.00302335
$342$	0	0
$343$	12.5675	0.678582
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.7579	−0.845926	−0.422963	−	0.906147i	$-0.639010\pi$
$347$	−15.7579	−0.845926	−0.422963	+	0.906147i	$0.639010\pi$
$348$	0	0
$349$	−14.5904	−0.781004	−0.390502	−	0.920602i	$-0.627699\pi$
$349$	−14.5904	−0.781004	−0.390502	+	0.920602i	$0.627699\pi$
$350$	0	0
$351$	8.12931	0.433910
$352$	0	0
$353$	28.0097	1.49080	0.745402	−	0.666615i	$-0.232258\pi$
$353$	28.0097	1.49080	0.745402	+	0.666615i	$0.232258\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−9.72964	−0.514947
$358$	0	0
$359$	0.214689	0.0113308	0.00566542	−	0.999984i	$-0.498197\pi$
$359$	0.214689	0.0113308	0.00566542	+	0.999984i	$0.498197\pi$
$360$	0	0
$361$	30.0520	1.58168
$362$	0	0
$363$	14.6936	0.771213
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.0197	−1.09722	−0.548610	−	0.836079i	$-0.684843\pi$
$367$	−21.0197	−1.09722	−0.548610	+	0.836079i	$0.684843\pi$
$368$	0	0
$369$	3.47488	0.180895
$370$	0	0
$371$	24.5129	1.27265
$372$	0	0
$373$	10.2558	0.531023	0.265511	−	0.964108i	$-0.414459\pi$
$373$	10.2558	0.531023	0.265511	+	0.964108i	$0.414459\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.78116	0.400750
$378$	0	0
$379$	−20.6274	−1.05956	−0.529780	−	0.848135i	$-0.677726\pi$
$379$	−20.6274	−1.05956	−0.529780	+	0.848135i	$0.677726\pi$
$380$	0	0
$381$	14.4451	0.740044
$382$	0	0
$383$	−18.7419	−0.957667	−0.478833	−	0.877906i	$-0.658940\pi$
$383$	−18.7419	−0.957667	−0.478833	+	0.877906i	$0.658940\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.5234	−0.585769
$388$	0	0
$389$	−3.07719	−0.156020	−0.0780100	−	0.996953i	$-0.524857\pi$
$389$	−3.07719	−0.156020	−0.0780100	+	0.996953i	$0.524857\pi$
$390$	0	0
$391$	2.29775	0.116202
$392$	0	0
$393$	11.2995	0.569986
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.4670	−0.575513	−0.287756	−	0.957704i	$-0.592909\pi$
$397$	−11.4670	−0.575513	−0.287756	+	0.957704i	$0.592909\pi$
$398$	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
$403$	−0.843666	−0.0420260
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.888314	0.0440321
$408$	0	0
$409$	−0.851429	−0.0421005	−0.0210502	−	0.999778i	$-0.506701\pi$
$409$	−0.851429	−0.0421005	−0.0210502	+	0.999778i	$0.506701\pi$
$410$	0	0
$411$	24.7755	1.22208
$412$	0	0
$413$	−43.2111	−2.12628
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.92799	−0.486176
$418$	0	0
$419$	0.680728	0.0332557	0.0166279	−	0.999862i	$-0.494707\pi$
$419$	0.680728	0.0332557	0.0166279	+	0.999862i	$0.494707\pi$
$420$	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
$423$	−8.60251	−0.418268
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.742472	−0.0359307
$428$	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
$433$	13.2632	0.637390	0.318695	−	0.947857i	$-0.396755\pi$
$433$	13.2632	0.637390	0.318695	+	0.947857i	$0.396755\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.00371	−0.335033
$438$	0	0
$439$	18.7740	0.896035	0.448018	−	0.894025i	$-0.352130\pi$
$439$	18.7740	0.896035	0.448018	+	0.894025i	$0.352130\pi$
$440$	0	0
$441$	−3.67720	−0.175105
$442$	0	0
$443$	−5.49270	−0.260966	−0.130483	−	0.991451i	$-0.541653\pi$
$443$	−5.49270	−0.260966	−0.130483	+	0.991451i	$0.541653\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.06905	0.287056
$448$	0	0
$449$	13.1705	0.621553	0.310777	−	0.950483i	$-0.399411\pi$
$449$	13.1705	0.621553	0.310777	+	0.950483i	$0.399411\pi$
$450$	0	0
$451$	0.273695	0.0128878
$452$	0	0
$453$	−0.00902890	−0.000424215 0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.9676	0.980821	0.490410	−	0.871492i	$-0.336847\pi$
$457$	20.9676	0.980821	0.490410	+	0.871492i	$0.336847\pi$
$458$	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
$463$	9.49565	0.441300	0.220650	−	0.975353i	$-0.429182\pi$
$463$	9.49565	0.441300	0.220650	+	0.975353i	$0.429182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.9066	0.643518	0.321759	−	0.946822i	$-0.395726\pi$
$467$	13.9066	0.643518	0.321759	+	0.946822i	$0.395726\pi$
$468$	0	0
$469$	−23.7466	−1.09652
$470$	0	0
$471$	−16.2727	−0.749809
$472$	0	0
$473$	−0.907631	−0.0417329
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.38555	0.429735
$478$	0	0
$479$	3.66220	0.167330	0.0836652	−	0.996494i	$-0.473337\pi$
$479$	3.66220	0.167330	0.0836652	+	0.996494i	$0.473337\pi$
$480$	0	0
$481$	−13.4237	−0.612066
$482$	0	0
$483$	−4.23441	−0.192672
$484$	0	0
$485$	0	0
$486$	0	0
$487$	28.9654	1.31255	0.656274	−	0.754523i	$-0.272132\pi$
$487$	28.9654	1.31255	0.656274	+	0.754523i	$0.272132\pi$
$488$	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
$493$	−12.3866	−0.557866
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.4204	0.736557
$498$	0	0
$499$	−26.6914	−1.19487	−0.597436	−	0.801917i	$-0.703814\pi$
$499$	−26.6914	−1.19487	−0.597436	+	0.801917i	$0.703814\pi$
$500$	0	0
$501$	−10.7581	−0.480635
$502$	0	0
$503$	−18.2182	−0.812309	−0.406154	−	0.913804i	$-0.633130\pi$
$503$	−18.2182	−0.812309	−0.406154	+	0.913804i	$0.633130\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.5942	0.648150
$508$	0	0
$509$	−31.0138	−1.37466	−0.687332	−	0.726344i	$-0.741218\pi$
$509$	−31.0138	−1.37466	−0.687332	+	0.726344i	$0.741218\pi$
$510$	0	0
$511$	4.82655	0.213514
$512$	0	0
$513$	39.4446	1.74152
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.677568	−0.0297994
$518$	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
$523$	10.3111	0.450873	0.225437	−	0.974258i	$-0.427619\pi$
$523$	10.3111	0.450873	0.225437	+	0.974258i	$0.427619\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.34301	0.0585024
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−16.5448	−0.717981
$532$	0	0
$533$	−4.13592	−0.179146
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−31.7699	−1.37097
$538$	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
$543$	4.19139	0.179870
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.47986	0.106031	0.0530156	−	0.998594i	$-0.483117\pi$
$547$	2.47986	0.106031	0.0530156	+	0.998594i	$0.483117\pi$
$548$	0	0
$549$	−0.284279	−0.0121327
$550$	0	0
$551$	37.7553	1.60843
$552$	0	0
$553$	9.63094	0.409549
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.9814	0.973755	0.486877	−	0.873470i	$-0.338136\pi$
$557$	22.9814	0.973755	0.486877	+	0.873470i	$0.338136\pi$
$558$	0	0
$559$	13.7156	0.580107
$560$	0	0
$561$	−0.293420	−0.0123882
$562$	0	0
$563$	−18.4939	−0.779427	−0.389713	−	0.920936i	$-0.627426\pi$
$563$	−18.4939	−0.779427	−0.389713	+	0.920936i	$0.627426\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.3246	0.517584
$568$	0	0
$569$	35.6081	1.49277	0.746385	−	0.665514i	$-0.231788\pi$
$569$	35.6081	1.49277	0.746385	+	0.665514i	$0.231788\pi$
$570$	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
$573$	−8.25645	−0.344918
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−19.0378	−0.792554	−0.396277	−	0.918131i	$-0.629698\pi$
$577$	−19.0378	−0.792554	−0.396277	+	0.918131i	$0.629698\pi$
$578$	0	0
$579$	−9.65591	−0.401286
$580$	0	0
$581$	50.6489	2.10127
$582$	0	0
$583$	0.739244	0.0306163
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.29062	0.300916	0.150458	−	0.988616i	$-0.451925\pi$
$587$	7.29062	0.300916	0.150458	+	0.988616i	$0.451925\pi$
$588$	0	0
$589$	−4.09358	−0.168673
$590$	0	0
$591$	17.3821	0.715006
$592$	0	0
$593$	10.5832	0.434598	0.217299	−	0.976105i	$-0.430275\pi$
$593$	10.5832	0.434598	0.217299	+	0.976105i	$0.430275\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.46140	0.100739
$598$	0	0
$599$	−5.63520	−0.230248	−0.115124	−	0.993351i	$-0.536727\pi$
$599$	−5.63520	−0.230248	−0.115124	+	0.993351i	$0.536727\pi$
$600$	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
$603$	−9.09216	−0.370261
$604$	0	0
$605$	0	0
$606$	0	0
$607$	22.0996	0.896996	0.448498	−	0.893784i	$-0.351959\pi$
$607$	22.0996	0.896996	0.448498	+	0.893784i	$0.351959\pi$
$608$	0	0
$609$	22.8267	0.924984
$610$	0	0
$611$	10.2390	0.414225
$612$	0	0
$613$	5.08532	0.205394	0.102697	−	0.994713i	$-0.467253\pi$
$613$	5.08532	0.205394	0.102697	+	0.994713i	$0.467253\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.63073	0.307202	0.153601	−	0.988133i	$-0.450913\pi$
$617$	7.63073	0.307202	0.153601	+	0.988133i	$0.450913\pi$
$618$	0	0
$619$	38.5591	1.54982	0.774910	−	0.632072i	$-0.217795\pi$
$619$	38.5591	1.54982	0.774910	+	0.632072i	$0.217795\pi$
$620$	0	0
$621$	−5.63195	−0.226002
$622$	0	0
$623$	−17.5258	−0.702156
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.894364	0.0357175
$628$	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
$633$	33.2607	1.32199
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.37673	0.173412
$638$	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
$643$	5.06670	0.199811	0.0999055	−	0.994997i	$-0.468146\pi$
$643$	5.06670	0.199811	0.0999055	+	0.994997i	$0.468146\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.0429	1.45631	0.728153	−	0.685414i	$-0.240379\pi$
$647$	37.0429	1.45631	0.728153	+	0.685414i	$0.240379\pi$
$648$	0	0
$649$	−1.30313	−0.0511524
$650$	0	0
$651$	−2.47496	−0.0970014
$652$	0	0
$653$	−11.9884	−0.469143	−0.234572	−	0.972099i	$-0.575369\pi$
$653$	−11.9884	−0.469143	−0.234572	+	0.972099i	$0.575369\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.84800	0.0720974
$658$	0	0
$659$	−2.82513	−0.110051	−0.0550257	−	0.998485i	$-0.517524\pi$
$659$	−2.82513	−0.110051	−0.0550257	+	0.998485i	$0.517524\pi$
$660$	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
$663$	4.43398	0.172202
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.39076	−0.208731
$668$	0	0
$669$	5.41339	0.209294
$670$	0	0
$671$	−0.0223910	−0.000864393 0
$672$	0	0
$673$	−19.0620	−0.734784	−0.367392	−	0.930066i	$-0.619749\pi$
$673$	−19.0620	−0.734784	−0.367392	+	0.930066i	$0.619749\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.54346	−0.0977533	−0.0488766	−	0.998805i	$-0.515564\pi$
$677$	−2.54346	−0.0977533	−0.0488766	+	0.998805i	$0.515564\pi$
$678$	0	0
$679$	−8.18146	−0.313975
$680$	0	0
$681$	23.2507	0.890969
$682$	0	0
$683$	0.377109	0.0144297	0.00721483	−	0.999974i	$-0.497703\pi$
$683$	0.377109	0.0144297	0.00721483	+	0.999974i	$0.497703\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.70266	0.179418
$688$	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
$693$	−0.366903	−0.0139375
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.58385	0.249381
$698$	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
$703$	−65.1335	−2.45656
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.0824	0.792887
$708$	0	0
$709$	−45.9062	−1.72404	−0.862022	−	0.506870i	$-0.830802\pi$
$709$	−45.9062	−1.72404	−0.862022	+	0.506870i	$0.830802\pi$
$710$	0	0
$711$	3.68751	0.138293
$712$	0	0
$713$	0.584488	0.0218892
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.83612	0.143263
$718$	0	0
$719$	−50.4975	−1.88324	−0.941619	−	0.336682i	$-0.890695\pi$
$719$	−50.4975	−1.88324	−0.941619	+	0.336682i	$0.890695\pi$
$720$	0	0
$721$	55.9570	2.08395
$722$	0	0
$723$	−36.1609	−1.34484
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.0273	0.705685	0.352842	−	0.935683i	$-0.385215\pi$
$727$	19.0273	0.705685	0.352842	+	0.935683i	$0.385215\pi$
$728$	0	0
$729$	27.3067	1.01136
$730$	0	0
$731$	−21.8334	−0.807539
$732$	0	0
$733$	42.1027	1.55510	0.777549	−	0.628822i	$-0.216463\pi$
$733$	42.1027	1.55510	0.777549	+	0.628822i	$0.216463\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.716135	−0.0263792
$738$	0	0
$739$	47.9212	1.76281	0.881405	−	0.472361i	$-0.156598\pi$
$739$	47.9212	1.76281	0.881405	+	0.472361i	$0.156598\pi$
$740$	0	0
$741$	−13.5151	−0.496489
$742$	0	0
$743$	−21.3492	−0.783225	−0.391613	−	0.920130i	$-0.628083\pi$
$743$	−21.3492	−0.783225	−0.391613	+	0.920130i	$0.628083\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	19.3925	0.709536
$748$	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
$753$	15.7478	0.573881
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.74990	−0.318020	−0.159010	−	0.987277i	$-0.550830\pi$
$757$	−8.74990	−0.318020	−0.159010	+	0.987277i	$0.550830\pi$
$758$	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
$763$	−22.2379	−0.805066
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.6921	0.711041
$768$	0	0
$769$	−31.3229	−1.12953	−0.564766	−	0.825251i	$-0.691034\pi$
$769$	−31.3229	−1.12953	−0.564766	+	0.825251i	$0.691034\pi$
$770$	0	0
$771$	11.3334	0.408162
$772$	0	0
$773$	−32.3308	−1.16286	−0.581429	−	0.813597i	$-0.697506\pi$
$773$	−32.3308	−1.16286	−0.581429	+	0.813597i	$0.697506\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−39.3794	−1.41273
$778$	0	0
$779$	−20.0680	−0.719012
$780$	0	0
$781$	0.495196	0.0177195
$782$	0	0
$783$	30.3605	1.08499
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.5712	0.661990	0.330995	−	0.943632i	$-0.392616\pi$
$787$	18.5712	0.661990	0.330995	+	0.943632i	$0.392616\pi$
$788$	0	0
$789$	24.7483	0.881063
$790$	0	0
$791$	54.1627	1.92580
$792$	0	0
$793$	0.338358	0.0120155
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.1985	0.857154	0.428577	−	0.903505i	$-0.359015\pi$
$797$	24.1985	0.857154	0.428577	+	0.903505i	$0.359015\pi$
$798$	0	0
$799$	−16.2992	−0.576624
$800$	0	0
$801$	−6.71031	−0.237097
$802$	0	0
$803$	0.145556	0.00513656
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.0349	1.19809
$808$	0	0
$809$	−30.4574	−1.07082	−0.535412	−	0.844591i	$-0.679844\pi$
$809$	−30.4574	−1.07082	−0.535412	+	0.844591i	$0.679844\pi$
$810$	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
$813$	18.9587	0.664910
$814$	0	0
$815$	0	0
$816$	0	0
$817$	66.5499	2.32829
$818$	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
$823$	42.9810	1.49822	0.749111	−	0.662444i	$-0.230481\pi$
$823$	42.9810	1.49822	0.749111	+	0.662444i	$0.230481\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.6777	−1.79701	−0.898505	−	0.438963i	$-0.855346\pi$
$827$	−51.6777	−1.79701	−0.898505	+	0.438963i	$0.855346\pi$
$828$	0	0
$829$	25.6228	0.889915	0.444958	−	0.895552i	$-0.353219\pi$
$829$	25.6228	0.889915	0.444958	+	0.895552i	$0.353219\pi$
$830$	0	0
$831$	28.6470	0.993753
$832$	0	0
$833$	−6.96720	−0.241399
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.29181	−0.113781
$838$	0	0
$839$	−53.0313	−1.83084	−0.915422	−	0.402495i	$-0.868143\pi$
$839$	−53.0313	−1.83084	−0.915422	+	0.402495i	$0.868143\pi$
$840$	0	0
$841$	0.0602571	0.00207783
$842$	0	0
$843$	11.9021	0.409929
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.8121	1.19616
$848$	0	0
$849$	−17.4290	−0.598160
$850$	0	0
$851$	9.29985	0.318795
$852$	0	0
$853$	−13.8668	−0.474791	−0.237395	−	0.971413i	$-0.576294\pi$
$853$	−13.8668	−0.474791	−0.237395	+	0.971413i	$0.576294\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.3518	−0.558567	−0.279283	−	0.960209i	$-0.590097\pi$
$857$	−16.3518	−0.558567	−0.279283	+	0.960209i	$0.590097\pi$
$858$	0	0
$859$	42.1443	1.43795	0.718973	−	0.695038i	$-0.244612\pi$
$859$	42.1443	1.43795	0.718973	+	0.695038i	$0.244612\pi$
$860$	0	0
$861$	−12.1331	−0.413493
$862$	0	0
$863$	15.2130	0.517856	0.258928	−	0.965897i	$-0.416631\pi$
$863$	15.2130	0.517856	0.258928	+	0.965897i	$0.416631\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.6688	0.532140
$868$	0	0
$869$	0.290443	0.00985262
$870$	0	0
$871$	10.8218	0.366683
$872$	0	0
$873$	−3.13253	−0.106020
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.6934	1.81310	0.906549	−	0.422101i	$-0.138707\pi$
$877$	53.6934	1.81310	0.906549	+	0.422101i	$0.138707\pi$
$878$	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
$883$	19.9787	0.672337	0.336169	−	0.941802i	$-0.390869\pi$
$883$	19.9787	0.672337	0.336169	+	0.941802i	$0.390869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51.9251	−1.74347	−0.871737	−	0.489974i	$-0.837006\pi$
$887$	−51.9251	−1.74347	−0.871737	+	0.489974i	$0.837006\pi$
$888$	0	0
$889$	34.2233	1.14781
$890$	0	0
$891$	0.371676	0.0124516
$892$	0	0
$893$	49.6810	1.66251
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.92970	0.0644309
$898$	0	0
$899$	−3.15083	−0.105086
$900$	0	0
$901$	17.7828	0.592431
$902$	0	0
$903$	40.2358	1.33896
$904$	0	0
$905$	0	0
$906$	0	0
$907$	6.64335	0.220589	0.110294	−	0.993899i	$-0.464821\pi$
$907$	6.64335	0.220589	0.110294	+	0.993899i	$0.464821\pi$
$908$	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
$913$	1.52744	0.0505507
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.7709	0.884052
$918$	0	0
$919$	41.2068	1.35929	0.679644	−	0.733542i	$-0.262134\pi$
$919$	41.2068	1.35929	0.679644	+	0.733542i	$0.262134\pi$
$920$	0	0
$921$	20.9446	0.690148
$922$	0	0
$923$	−7.48310	−0.246309
$924$	0	0
$925$	0	0
$926$	0	0
$927$	21.4249	0.703687
$928$	0	0
$929$	−4.86715	−0.159686	−0.0798431	−	0.996807i	$-0.525442\pi$
$929$	−4.86715	−0.159686	−0.0798431	+	0.996807i	$0.525442\pi$
$930$	0	0
$931$	21.2365	0.695999
$932$	0	0
$933$	−11.0684	−0.362364
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.44469	0.112533	0.0562666	−	0.998416i	$-0.482080\pi$
$937$	3.44469	0.112533	0.0562666	+	0.998416i	$0.482080\pi$
$938$	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
$943$	2.86534	0.0933084
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.17298	0.168099	0.0840496	−	0.996462i	$-0.473215\pi$
$947$	5.17298	0.168099	0.0840496	+	0.996462i	$0.473215\pi$
$948$	0	0
$949$	−2.19955	−0.0714005
$950$	0	0
$951$	−14.1447	−0.458675
$952$	0	0
$953$	−3.41090	−0.110490	−0.0552449	−	0.998473i	$-0.517594\pi$
$953$	−3.41090	−0.110490	−0.0552449	+	0.998473i	$0.517594\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.688392	0.0222526
$958$	0	0
$959$	58.6981	1.89546
$960$	0	0
$961$	−30.6584	−0.988980
$962$	0	0
$963$	7.07789	0.228082
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.5047	−0.562914	−0.281457	−	0.959574i	$-0.590818\pi$
$967$	−17.5047	−0.562914	−0.281457	+	0.959574i	$0.590818\pi$
$968$	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
$973$	−23.5214	−0.754062
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.3634	1.00340	0.501702	−	0.865040i	$-0.332707\pi$
$977$	31.3634	1.00340	0.501702	+	0.865040i	$0.332707\pi$
$978$	0	0
$979$	−0.528531	−0.0168919
$980$	0	0
$981$	−8.51450	−0.271847
$982$	0	0
$983$	36.4262	1.16182	0.580908	−	0.813969i	$-0.302698\pi$
$983$	36.4262	1.16182	0.580908	+	0.813969i	$0.302698\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.0369	0.956086
$988$	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
$993$	1.77587	0.0563556
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.0896	0.762925	0.381463	−	0.924384i	$-0.375420\pi$
$997$	24.0896	0.762925	0.381463	+	0.924384i	$0.375420\pi$
$998$	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.a.bc.1.3	✓	5
4.3	odd	2		9200.2.a.cw.1.3		5
5.2	odd	4		4600.2.e.v.4049.7		10
5.3	odd	4		4600.2.e.v.4049.4		10
5.4	even	2		4600.2.a.bg.1.3	yes	5
20.19	odd	2		9200.2.a.cs.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.3	✓	5	1.1	even	1	trivial
4600.2.a.bg.1.3	yes	5	5.4	even	2
4600.2.e.v.4049.4		10	5.3	odd	4
4600.2.e.v.4049.7		10	5.2	odd	4
9200.2.a.cs.1.3		5	20.19	odd	2
9200.2.a.cw.1.3		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-1.33689 q^{3} -3.16736 q^{7} -1.21273 q^{9} +O(q^{10})\)	\(q-1.33689 q^{3} -3.16736 q^{7} -1.21273 q^{9} -0.0955192 q^{11} +1.44343 q^{13} -2.29775 q^{17} +7.00371 q^{19} +4.23441 q^{21} -1.00000 q^{23} +5.63195 q^{27} +5.39076 q^{29} -0.584488 q^{31} +0.127699 q^{33} -9.29985 q^{37} -1.92970 q^{39} -2.86534 q^{41} +9.50208 q^{43} +7.09353 q^{47} +3.03218 q^{49} +3.07184 q^{51} -7.73922 q^{53} -9.36319 q^{57} +13.6426 q^{59} +0.234413 q^{61} +3.84114 q^{63} +7.49729 q^{67} +1.33689 q^{69} -5.18426 q^{71} -1.52384 q^{73} +0.302544 q^{77} -3.04068 q^{79} -3.89112 q^{81} -15.9909 q^{83} -7.20685 q^{87} +5.53325 q^{89} -4.57186 q^{91} +0.781396 q^{93} +2.58305 q^{97} +0.115839 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 + q^7 + 4 * q^9	\( 5 q - 3 q^{3} + q^{7} + 4 q^{9} - 4 q^{11} + q^{13} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} + 4 q^{31} - 13 q^{33} - 6 q^{37} + 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49} - 5 q^{51} - 18 q^{53} - 27 q^{57} + 23 q^{59} - 26 q^{61} - 5 q^{63} - 3 q^{67} + 3 q^{69} - 2 q^{71} - 4 q^{73} - 15 q^{77} + 43 q^{79} - 3 q^{81} - 30 q^{83} - 27 q^{87} + 15 q^{89} - 19 q^{91} + 15 q^{93} - 8 q^{97} + 37 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + q^7 + 4 * q^9 - 4 * q^11 + q^13 - 5 * q^17 + 4 * q^19 - 6 * q^21 - 5 * q^23 - 6 * q^27 - 11 * q^29 + 4 * q^31 - 13 * q^33 - 6 * q^37 + 31 * q^39 - 8 * q^41 - 3 * q^43 - 2 * q^47 - 2 * q^49 - 5 * q^51 - 18 * q^53 - 27 * q^57 + 23 * q^59 - 26 * q^61 - 5 * q^63 - 3 * q^67 + 3 * q^69 - 2 * q^71 - 4 * q^73 - 15 * q^77 + 43 * q^79 - 3 * q^81 - 30 * q^83 - 27 * q^87 + 15 * q^89 - 19 * q^91 + 15 * q^93 - 8 * q^97 + 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more