LMFDB - Embedded newform 4012.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 $ x^12 - 4x^11 - 13x^10 + 60x^9 + 48x^8 - 289x^7 - 89x^6 + 602x^5 + 161x^4 - 555x^3 - 197x^2 + 181*x + 82
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.43162$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43162 q^{3} -0.659543 q^{5} +4.52406 q^{7} +2.91279 q^{9} +O(q^{10})$	$q-2.43162 q^{3} -0.659543 q^{5} +4.52406 q^{7} +2.91279 q^{9} +0.926475 q^{11} +3.23177 q^{13} +1.60376 q^{15} -1.00000 q^{17} -5.97818 q^{19} -11.0008 q^{21} +1.60970 q^{23} -4.56500 q^{25} +0.212065 q^{27} -9.97474 q^{29} -9.56778 q^{31} -2.25284 q^{33} -2.98381 q^{35} +7.27511 q^{37} -7.85845 q^{39} -5.10111 q^{41} +3.90210 q^{43} -1.92111 q^{45} -10.6913 q^{47} +13.4671 q^{49} +2.43162 q^{51} +6.72672 q^{53} -0.611049 q^{55} +14.5367 q^{57} +1.00000 q^{59} +11.7849 q^{61} +13.1776 q^{63} -2.13149 q^{65} +8.56754 q^{67} -3.91418 q^{69} +3.27076 q^{71} -10.1628 q^{73} +11.1004 q^{75} +4.19143 q^{77} +1.91728 q^{79} -9.25403 q^{81} -7.51691 q^{83} +0.659543 q^{85} +24.2548 q^{87} -10.3308 q^{89} +14.6207 q^{91} +23.2652 q^{93} +3.94287 q^{95} -8.67239 q^{97} +2.69862 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * 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
$3$	−2.43162	−1.40390	−0.701949	−	0.712227i	$-0.747687\pi$
$3$	−2.43162	−1.40390	−0.701949	+	0.712227i	$0.747687\pi$
$4$	0	0
$5$	−0.659543	−0.294956	−0.147478	−	0.989065i	$-0.547116\pi$
$5$	−0.659543	−0.294956	−0.147478	+	0.989065i	$0.547116\pi$
$6$	0	0
$7$	4.52406	1.70993	0.854967	−	0.518682i	$-0.173577\pi$
$7$	4.52406	1.70993	0.854967	+	0.518682i	$0.173577\pi$
$8$	0	0
$9$	2.91279	0.970930
$10$	0	0
$11$	0.926475	0.279343	0.139671	−	0.990198i	$-0.455395\pi$
$11$	0.926475	0.279343	0.139671	+	0.990198i	$0.455395\pi$
$12$	0	0
$13$	3.23177	0.896333	0.448166	−	0.893950i	$-0.352077\pi$
$13$	3.23177	0.896333	0.448166	+	0.893950i	$0.352077\pi$
$14$	0	0
$15$	1.60376	0.414089
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.97818	−1.37149	−0.685745	−	0.727842i	$-0.740523\pi$
$19$	−5.97818	−1.37149	−0.685745	+	0.727842i	$0.740523\pi$
$20$	0	0
$21$	−11.0008	−2.40057
$22$	0	0
$23$	1.60970	0.335646	0.167823	−	0.985817i	$-0.446326\pi$
$23$	1.60970	0.335646	0.167823	+	0.985817i	$0.446326\pi$
$24$	0	0
$25$	−4.56500	−0.913001
$26$	0	0
$27$	0.212065	0.0408120
$28$	0	0
$29$	−9.97474	−1.85226	−0.926131	−	0.377202i	$-0.876886\pi$
$29$	−9.97474	−1.85226	−0.926131	+	0.377202i	$0.876886\pi$
$30$	0	0
$31$	−9.56778	−1.71842	−0.859212	−	0.511620i	$-0.829046\pi$
$31$	−9.56778	−1.71842	−0.859212	+	0.511620i	$0.829046\pi$
$32$	0	0
$33$	−2.25284	−0.392169
$34$	0	0
$35$	−2.98381	−0.504356
$36$	0	0
$37$	7.27511	1.19602	0.598010	−	0.801488i	$-0.295958\pi$
$37$	7.27511	1.19602	0.598010	+	0.801488i	$0.295958\pi$
$38$	0	0
$39$	−7.85845	−1.25836
$40$	0	0
$41$	−5.10111	−0.796660	−0.398330	−	0.917242i	$-0.630410\pi$
$41$	−5.10111	−0.796660	−0.398330	+	0.917242i	$0.630410\pi$
$42$	0	0
$43$	3.90210	0.595065	0.297533	−	0.954712i	$-0.403836\pi$
$43$	3.90210	0.595065	0.297533	+	0.954712i	$0.403836\pi$
$44$	0	0
$45$	−1.92111	−0.286382
$46$	0	0
$47$	−10.6913	−1.55948	−0.779742	−	0.626101i	$-0.784650\pi$
$47$	−10.6913	−1.55948	−0.779742	+	0.626101i	$0.784650\pi$
$48$	0	0
$49$	13.4671	1.92388
$50$	0	0
$51$	2.43162	0.340495
$52$	0	0
$53$	6.72672	0.923987	0.461993	−	0.886883i	$-0.347134\pi$
$53$	6.72672	0.923987	0.461993	+	0.886883i	$0.347134\pi$
$54$	0	0
$55$	−0.611049	−0.0823939
$56$	0	0
$57$	14.5367	1.92543
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	11.7849	1.50890	0.754451	−	0.656357i	$-0.227903\pi$
$61$	11.7849	1.50890	0.754451	+	0.656357i	$0.227903\pi$
$62$	0	0
$63$	13.1776	1.66023
$64$	0	0
$65$	−2.13149	−0.264379
$66$	0	0
$67$	8.56754	1.04669	0.523346	−	0.852121i	$-0.324684\pi$
$67$	8.56754	1.04669	0.523346	+	0.852121i	$0.324684\pi$
$68$	0	0
$69$	−3.91418	−0.471212
$70$	0	0
$71$	3.27076	0.388168	0.194084	−	0.980985i	$-0.437827\pi$
$71$	3.27076	0.388168	0.194084	+	0.980985i	$0.437827\pi$
$72$	0	0
$73$	−10.1628	−1.18946	−0.594731	−	0.803925i	$-0.702741\pi$
$73$	−10.1628	−1.18946	−0.594731	+	0.803925i	$0.702741\pi$
$74$	0	0
$75$	11.1004	1.28176
$76$	0	0
$77$	4.19143	0.477658
$78$	0	0
$79$	1.91728	0.215711	0.107855	−	0.994167i	$-0.465602\pi$
$79$	1.91728	0.215711	0.107855	+	0.994167i	$0.465602\pi$
$80$	0	0
$81$	−9.25403	−1.02823
$82$	0	0
$83$	−7.51691	−0.825089	−0.412544	−	0.910938i	$-0.635360\pi$
$83$	−7.51691	−0.825089	−0.412544	+	0.910938i	$0.635360\pi$
$84$	0	0
$85$	0.659543	0.0715374
$86$	0	0
$87$	24.2548	2.60039
$88$	0	0
$89$	−10.3308	−1.09507	−0.547534	−	0.836784i	$-0.684433\pi$
$89$	−10.3308	−1.09507	−0.547534	+	0.836784i	$0.684433\pi$
$90$	0	0
$91$	14.6207	1.53267
$92$	0	0
$93$	23.2652	2.41249
$94$	0	0
$95$	3.94287	0.404530
$96$	0	0
$97$	−8.67239	−0.880548	−0.440274	−	0.897864i	$-0.645119\pi$
$97$	−8.67239	−0.880548	−0.440274	+	0.897864i	$0.645119\pi$
$98$	0	0
$99$	2.69862	0.271222
$100$	0	0
$101$	−0.575188	−0.0572333	−0.0286167	−	0.999590i	$-0.509110\pi$
$101$	−0.575188	−0.0572333	−0.0286167	+	0.999590i	$0.509110\pi$
$102$	0	0
$103$	0.613304	0.0604307	0.0302153	−	0.999543i	$-0.490381\pi$
$103$	0.613304	0.0604307	0.0302153	+	0.999543i	$0.490381\pi$
$104$	0	0
$105$	7.25550	0.708065
$106$	0	0
$107$	−1.21744	−0.117694	−0.0588471	−	0.998267i	$-0.518742\pi$
$107$	−1.21744	−0.117694	−0.0588471	+	0.998267i	$0.518742\pi$
$108$	0	0
$109$	−16.7472	−1.60409	−0.802044	−	0.597265i	$-0.796254\pi$
$109$	−16.7472	−1.60409	−0.802044	+	0.597265i	$0.796254\pi$
$110$	0	0
$111$	−17.6903	−1.67909
$112$	0	0
$113$	−1.50728	−0.141793	−0.0708964	−	0.997484i	$-0.522586\pi$
$113$	−1.50728	−0.141793	−0.0708964	+	0.997484i	$0.522586\pi$
$114$	0	0
$115$	−1.06167	−0.0990008
$116$	0	0
$117$	9.41347	0.870276
$118$	0	0
$119$	−4.52406	−0.414720
$120$	0	0
$121$	−10.1416	−0.921968
$122$	0	0
$123$	12.4040	1.11843
$124$	0	0
$125$	6.30853	0.564252
$126$	0	0
$127$	−6.53260	−0.579674	−0.289837	−	0.957076i	$-0.593601\pi$
$127$	−6.53260	−0.579674	−0.289837	+	0.957076i	$0.593601\pi$
$128$	0	0
$129$	−9.48844	−0.835411
$130$	0	0
$131$	−15.7524	−1.37629	−0.688145	−	0.725573i	$-0.741575\pi$
$131$	−15.7524	−1.37629	−0.688145	+	0.725573i	$0.741575\pi$
$132$	0	0
$133$	−27.0457	−2.34516
$134$	0	0
$135$	−0.139866	−0.0120378
$136$	0	0
$137$	16.4626	1.40650	0.703249	−	0.710943i	$-0.251732\pi$
$137$	16.4626	1.40650	0.703249	+	0.710943i	$0.251732\pi$
$138$	0	0
$139$	−12.9745	−1.10048	−0.550241	−	0.835006i	$-0.685464\pi$
$139$	−12.9745	−1.10048	−0.550241	+	0.835006i	$0.685464\pi$
$140$	0	0
$141$	25.9972	2.18936
$142$	0	0
$143$	2.99416	0.250384
$144$	0	0
$145$	6.57876	0.546337
$146$	0	0
$147$	−32.7470	−2.70093
$148$	0	0
$149$	12.1896	0.998613	0.499307	−	0.866425i	$-0.333588\pi$
$149$	12.1896	0.998613	0.499307	+	0.866425i	$0.333588\pi$
$150$	0	0
$151$	−10.1800	−0.828440	−0.414220	−	0.910177i	$-0.635946\pi$
$151$	−10.1800	−0.828440	−0.414220	+	0.910177i	$0.635946\pi$
$152$	0	0
$153$	−2.91279	−0.235485
$154$	0	0
$155$	6.31036	0.506860
$156$	0	0
$157$	24.4088	1.94803	0.974017	−	0.226475i	$-0.0727202\pi$
$157$	24.4088	1.94803	0.974017	+	0.226475i	$0.0727202\pi$
$158$	0	0
$159$	−16.3569	−1.29718
$160$	0	0
$161$	7.28238	0.573932
$162$	0	0
$163$	12.7751	1.00062	0.500310	−	0.865846i	$-0.333219\pi$
$163$	12.7751	1.00062	0.500310	+	0.865846i	$0.333219\pi$
$164$	0	0
$165$	1.48584	0.115673
$166$	0	0
$167$	−13.8864	−1.07456	−0.537282	−	0.843403i	$-0.680549\pi$
$167$	−13.8864	−1.07456	−0.537282	+	0.843403i	$0.680549\pi$
$168$	0	0
$169$	−2.55564	−0.196588
$170$	0	0
$171$	−17.4132	−1.33162
$172$	0	0
$173$	21.9279	1.66714	0.833572	−	0.552410i	$-0.186292\pi$
$173$	21.9279	1.66714	0.833572	+	0.552410i	$0.186292\pi$
$174$	0	0
$175$	−20.6524	−1.56117
$176$	0	0
$177$	−2.43162	−0.182772
$178$	0	0
$179$	−18.0677	−1.35044	−0.675220	−	0.737616i	$-0.735951\pi$
$179$	−18.0677	−1.35044	−0.675220	+	0.737616i	$0.735951\pi$
$180$	0	0
$181$	1.16416	0.0865311	0.0432656	−	0.999064i	$-0.486224\pi$
$181$	1.16416	0.0865311	0.0432656	+	0.999064i	$0.486224\pi$
$182$	0	0
$183$	−28.6564	−2.11834
$184$	0	0
$185$	−4.79824	−0.352774
$186$	0	0
$187$	−0.926475	−0.0677505
$188$	0	0
$189$	0.959397	0.0697858
$190$	0	0
$191$	−3.22483	−0.233340	−0.116670	−	0.993171i	$-0.537222\pi$
$191$	−3.22483	−0.233340	−0.116670	+	0.993171i	$0.537222\pi$
$192$	0	0
$193$	8.61564	0.620167	0.310084	−	0.950709i	$-0.399643\pi$
$193$	8.61564	0.620167	0.310084	+	0.950709i	$0.399643\pi$
$194$	0	0
$195$	5.18298	0.371161
$196$	0	0
$197$	−20.4722	−1.45859	−0.729294	−	0.684201i	$-0.760151\pi$
$197$	−20.4722	−1.45859	−0.729294	+	0.684201i	$0.760151\pi$
$198$	0	0
$199$	−8.87410	−0.629068	−0.314534	−	0.949246i	$-0.601848\pi$
$199$	−8.87410	−0.629068	−0.314534	+	0.949246i	$0.601848\pi$
$200$	0	0
$201$	−20.8330	−1.46945
$202$	0	0
$203$	−45.1263	−3.16725
$204$	0	0
$205$	3.36440	0.234980
$206$	0	0
$207$	4.68871	0.325888
$208$	0	0
$209$	−5.53864	−0.383115
$210$	0	0
$211$	13.3596	0.919714	0.459857	−	0.887993i	$-0.347901\pi$
$211$	13.3596	0.919714	0.459857	+	0.887993i	$0.347901\pi$
$212$	0	0
$213$	−7.95326	−0.544948
$214$	0	0
$215$	−2.57360	−0.175518
$216$	0	0
$217$	−43.2852	−2.93839
$218$	0	0
$219$	24.7120	1.66988
$220$	0	0
$221$	−3.23177	−0.217393
$222$	0	0
$223$	3.53538	0.236747	0.118373	−	0.992969i	$-0.462232\pi$
$223$	3.53538	0.236747	0.118373	+	0.992969i	$0.462232\pi$
$224$	0	0
$225$	−13.2969	−0.886459
$226$	0	0
$227$	−1.71094	−0.113559	−0.0567795	−	0.998387i	$-0.518083\pi$
$227$	−1.71094	−0.113559	−0.0567795	+	0.998387i	$0.518083\pi$
$228$	0	0
$229$	−17.1742	−1.13491	−0.567453	−	0.823406i	$-0.692071\pi$
$229$	−17.1742	−1.13491	−0.567453	+	0.823406i	$0.692071\pi$
$230$	0	0
$231$	−10.1920	−0.670583
$232$	0	0
$233$	2.81952	0.184713	0.0923563	−	0.995726i	$-0.470560\pi$
$233$	2.81952	0.184713	0.0923563	+	0.995726i	$0.470560\pi$
$234$	0	0
$235$	7.05136	0.459980
$236$	0	0
$237$	−4.66209	−0.302836
$238$	0	0
$239$	−30.2810	−1.95872	−0.979359	−	0.202128i	$-0.935214\pi$
$239$	−30.2810	−1.95872	−0.979359	+	0.202128i	$0.935214\pi$
$240$	0	0
$241$	28.8307	1.85715	0.928573	−	0.371150i	$-0.121037\pi$
$241$	28.8307	1.85715	0.928573	+	0.371150i	$0.121037\pi$
$242$	0	0
$243$	21.8661	1.40271
$244$	0	0
$245$	−8.88215	−0.567460
$246$	0	0
$247$	−19.3201	−1.22931
$248$	0	0
$249$	18.2783	1.15834
$250$	0	0
$251$	−10.4833	−0.661703	−0.330851	−	0.943683i	$-0.607336\pi$
$251$	−10.4833	−0.661703	−0.330851	+	0.943683i	$0.607336\pi$
$252$	0	0
$253$	1.49135	0.0937601
$254$	0	0
$255$	−1.60376	−0.100431
$256$	0	0
$257$	17.3095	1.07974	0.539868	−	0.841750i	$-0.318474\pi$
$257$	17.3095	1.07974	0.539868	+	0.841750i	$0.318474\pi$
$258$	0	0
$259$	32.9130	2.04512
$260$	0	0
$261$	−29.0543	−1.79842
$262$	0	0
$263$	−17.9275	−1.10546	−0.552729	−	0.833361i	$-0.686413\pi$
$263$	−17.9275	−1.10546	−0.552729	+	0.833361i	$0.686413\pi$
$264$	0	0
$265$	−4.43656	−0.272536
$266$	0	0
$267$	25.1207	1.53736
$268$	0	0
$269$	7.25146	0.442129	0.221065	−	0.975259i	$-0.429047\pi$
$269$	7.25146	0.442129	0.221065	+	0.975259i	$0.429047\pi$
$270$	0	0
$271$	−3.27173	−0.198743	−0.0993716	−	0.995050i	$-0.531683\pi$
$271$	−3.27173	−0.198743	−0.0993716	+	0.995050i	$0.531683\pi$
$272$	0	0
$273$	−35.5521	−2.15171
$274$	0	0
$275$	−4.22936	−0.255040
$276$	0	0
$277$	32.3349	1.94282	0.971408	−	0.237415i	$-0.0763000\pi$
$277$	32.3349	1.94282	0.971408	+	0.237415i	$0.0763000\pi$
$278$	0	0
$279$	−27.8689	−1.66847
$280$	0	0
$281$	21.3290	1.27238	0.636190	−	0.771533i	$-0.280510\pi$
$281$	21.3290	1.27238	0.636190	+	0.771533i	$0.280510\pi$
$282$	0	0
$283$	4.94883	0.294177	0.147089	−	0.989123i	$-0.453010\pi$
$283$	4.94883	0.294177	0.147089	+	0.989123i	$0.453010\pi$
$284$	0	0
$285$	−9.58756	−0.567918
$286$	0	0
$287$	−23.0777	−1.36224
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	21.0880	1.23620
$292$	0	0
$293$	−14.0323	−0.819774	−0.409887	−	0.912136i	$-0.634432\pi$
$293$	−14.0323	−0.819774	−0.409887	+	0.912136i	$0.634432\pi$
$294$	0	0
$295$	−0.659543	−0.0384001
$296$	0	0
$297$	0.196473	0.0114005
$298$	0	0
$299$	5.20218	0.300850
$300$	0	0
$301$	17.6534	1.01752
$302$	0	0
$303$	1.39864	0.0803497
$304$	0	0
$305$	−7.77264	−0.445060
$306$	0	0
$307$	−8.50528	−0.485422	−0.242711	−	0.970099i	$-0.578037\pi$
$307$	−8.50528	−0.485422	−0.242711	+	0.970099i	$0.578037\pi$
$308$	0	0
$309$	−1.49132	−0.0848385
$310$	0	0
$311$	−30.3572	−1.72140	−0.860699	−	0.509115i	$-0.829973\pi$
$311$	−30.3572	−1.72140	−0.860699	+	0.509115i	$0.829973\pi$
$312$	0	0
$313$	1.10775	0.0626137	0.0313068	−	0.999510i	$-0.490033\pi$
$313$	1.10775	0.0626137	0.0313068	+	0.999510i	$0.490033\pi$
$314$	0	0
$315$	−8.69121	−0.489694
$316$	0	0
$317$	−15.8068	−0.887798	−0.443899	−	0.896077i	$-0.646405\pi$
$317$	−15.8068	−0.887798	−0.443899	+	0.896077i	$0.646405\pi$
$318$	0	0
$319$	−9.24134	−0.517416
$320$	0	0
$321$	2.96035	0.165231
$322$	0	0
$323$	5.97818	0.332635
$324$	0	0
$325$	−14.7531	−0.818352
$326$	0	0
$327$	40.7228	2.25198
$328$	0	0
$329$	−48.3680	−2.66662
$330$	0	0
$331$	−13.8429	−0.760875	−0.380437	−	0.924807i	$-0.624227\pi$
$331$	−13.8429	−0.760875	−0.380437	+	0.924807i	$0.624227\pi$
$332$	0	0
$333$	21.1909	1.16125
$334$	0	0
$335$	−5.65065	−0.308728
$336$	0	0
$337$	−14.7012	−0.800826	−0.400413	−	0.916335i	$-0.631133\pi$
$337$	−14.7012	−0.800826	−0.400413	+	0.916335i	$0.631133\pi$
$338$	0	0
$339$	3.66513	0.199063
$340$	0	0
$341$	−8.86430	−0.480029
$342$	0	0
$343$	29.2577	1.57977
$344$	0	0
$345$	2.58157	0.138987
$346$	0	0
$347$	−24.7768	−1.33009	−0.665044	−	0.746804i	$-0.731587\pi$
$347$	−24.7768	−1.33009	−0.665044	+	0.746804i	$0.731587\pi$
$348$	0	0
$349$	−33.3293	−1.78408	−0.892038	−	0.451960i	$-0.850725\pi$
$349$	−33.3293	−1.78408	−0.892038	+	0.451960i	$0.850725\pi$
$350$	0	0
$351$	0.685347	0.0365811
$352$	0	0
$353$	−2.86087	−0.152269	−0.0761343	−	0.997098i	$-0.524258\pi$
$353$	−2.86087	−0.152269	−0.0761343	+	0.997098i	$0.524258\pi$
$354$	0	0
$355$	−2.15721	−0.114493
$356$	0	0
$357$	11.0008	0.582225
$358$	0	0
$359$	−16.4655	−0.869014	−0.434507	−	0.900668i	$-0.643077\pi$
$359$	−16.4655	−0.869014	−0.434507	+	0.900668i	$0.643077\pi$
$360$	0	0
$361$	16.7387	0.880983
$362$	0	0
$363$	24.6607	1.29435
$364$	0	0
$365$	6.70277	0.350839
$366$	0	0
$367$	4.06526	0.212205	0.106102	−	0.994355i	$-0.466163\pi$
$367$	4.06526	0.212205	0.106102	+	0.994355i	$0.466163\pi$
$368$	0	0
$369$	−14.8585	−0.773500
$370$	0	0
$371$	30.4321	1.57996
$372$	0	0
$373$	−1.59492	−0.0825818	−0.0412909	−	0.999147i	$-0.513147\pi$
$373$	−1.59492	−0.0825818	−0.0412909	+	0.999147i	$0.513147\pi$
$374$	0	0
$375$	−15.3400	−0.792152
$376$	0	0
$377$	−32.2361	−1.66024
$378$	0	0
$379$	−6.89588	−0.354218	−0.177109	−	0.984191i	$-0.556674\pi$
$379$	−6.89588	−0.354218	−0.177109	+	0.984191i	$0.556674\pi$
$380$	0	0
$381$	15.8848	0.813804
$382$	0	0
$383$	−25.7469	−1.31560	−0.657802	−	0.753191i	$-0.728514\pi$
$383$	−25.7469	−1.31560	−0.657802	+	0.753191i	$0.728514\pi$
$384$	0	0
$385$	−2.76443	−0.140888
$386$	0	0
$387$	11.3660	0.577766
$388$	0	0
$389$	20.2675	1.02760	0.513801	−	0.857910i	$-0.328237\pi$
$389$	20.2675	1.02760	0.513801	+	0.857910i	$0.328237\pi$
$390$	0	0
$391$	−1.60970	−0.0814060
$392$	0	0
$393$	38.3038	1.93217
$394$	0	0
$395$	−1.26453	−0.0636252
$396$	0	0
$397$	−1.26935	−0.0637069	−0.0318534	−	0.999493i	$-0.510141\pi$
$397$	−1.26935	−0.0637069	−0.0318534	+	0.999493i	$0.510141\pi$
$398$	0	0
$399$	65.7649	3.29236
$400$	0	0
$401$	4.91694	0.245540	0.122770	−	0.992435i	$-0.460822\pi$
$401$	4.91694	0.245540	0.122770	+	0.992435i	$0.460822\pi$
$402$	0	0
$403$	−30.9209	−1.54028
$404$	0	0
$405$	6.10343	0.303282
$406$	0	0
$407$	6.74020	0.334099
$408$	0	0
$409$	12.0913	0.597877	0.298939	−	0.954272i	$-0.403367\pi$
$409$	12.0913	0.597877	0.298939	+	0.954272i	$0.403367\pi$
$410$	0	0
$411$	−40.0309	−1.97458
$412$	0	0
$413$	4.52406	0.222615
$414$	0	0
$415$	4.95772	0.243365
$416$	0	0
$417$	31.5491	1.54497
$418$	0	0
$419$	19.4653	0.950944	0.475472	−	0.879731i	$-0.342277\pi$
$419$	19.4653	0.950944	0.475472	+	0.879731i	$0.342277\pi$
$420$	0	0
$421$	−14.5271	−0.708007	−0.354003	−	0.935244i	$-0.615180\pi$
$421$	−14.5271	−0.708007	−0.354003	+	0.935244i	$0.615180\pi$
$422$	0	0
$423$	−31.1415	−1.51415
$424$	0	0
$425$	4.56500	0.221435
$426$	0	0
$427$	53.3156	2.58012
$428$	0	0
$429$	−7.28066	−0.351513
$430$	0	0
$431$	−18.3916	−0.885894	−0.442947	−	0.896548i	$-0.646067\pi$
$431$	−18.3916	−0.885894	−0.442947	+	0.896548i	$0.646067\pi$
$432$	0	0
$433$	−22.5998	−1.08608	−0.543038	−	0.839708i	$-0.682726\pi$
$433$	−22.5998	−1.08608	−0.543038	+	0.839708i	$0.682726\pi$
$434$	0	0
$435$	−15.9971	−0.767001
$436$	0	0
$437$	−9.62308	−0.460334
$438$	0	0
$439$	−1.53199	−0.0731179	−0.0365590	−	0.999331i	$-0.511640\pi$
$439$	−1.53199	−0.0731179	−0.0365590	+	0.999331i	$0.511640\pi$
$440$	0	0
$441$	39.2269	1.86795
$442$	0	0
$443$	−38.5765	−1.83282	−0.916412	−	0.400236i	$-0.868928\pi$
$443$	−38.5765	−1.83282	−0.916412	+	0.400236i	$0.868928\pi$
$444$	0	0
$445$	6.81363	0.322997
$446$	0	0
$447$	−29.6406	−1.40195
$448$	0	0
$449$	2.33464	0.110178	0.0550892	−	0.998481i	$-0.482456\pi$
$449$	2.33464	0.110178	0.0550892	+	0.998481i	$0.482456\pi$
$450$	0	0
$451$	−4.72605	−0.222541
$452$	0	0
$453$	24.7540	1.16305
$454$	0	0
$455$	−9.64300	−0.452071
$456$	0	0
$457$	31.7737	1.48631	0.743155	−	0.669119i	$-0.233329\pi$
$457$	31.7737	1.48631	0.743155	+	0.669119i	$0.233329\pi$
$458$	0	0
$459$	−0.212065	−0.00989836
$460$	0	0
$461$	−29.1108	−1.35582	−0.677912	−	0.735143i	$-0.737115\pi$
$461$	−29.1108	−1.35582	−0.677912	+	0.735143i	$0.737115\pi$
$462$	0	0
$463$	−26.7217	−1.24186	−0.620932	−	0.783865i	$-0.713246\pi$
$463$	−26.7217	−1.24186	−0.620932	+	0.783865i	$0.713246\pi$
$464$	0	0
$465$	−15.3444	−0.711580
$466$	0	0
$467$	6.05007	0.279964	0.139982	−	0.990154i	$-0.455296\pi$
$467$	6.05007	0.279964	0.139982	+	0.990154i	$0.455296\pi$
$468$	0	0
$469$	38.7601	1.78977
$470$	0	0
$471$	−59.3530	−2.73484
$472$	0	0
$473$	3.61520	0.166227
$474$	0	0
$475$	27.2904	1.25217
$476$	0	0
$477$	19.5935	0.897126
$478$	0	0
$479$	−10.7376	−0.490611	−0.245306	−	0.969446i	$-0.578888\pi$
$479$	−10.7376	−0.490611	−0.245306	+	0.969446i	$0.578888\pi$
$480$	0	0
$481$	23.5115	1.07203
$482$	0	0
$483$	−17.7080	−0.805742
$484$	0	0
$485$	5.71981	0.259723
$486$	0	0
$487$	8.22178	0.372564	0.186282	−	0.982496i	$-0.440356\pi$
$487$	8.22178	0.372564	0.186282	+	0.982496i	$0.440356\pi$
$488$	0	0
$489$	−31.0641	−1.40477
$490$	0	0
$491$	19.3071	0.871318	0.435659	−	0.900112i	$-0.356515\pi$
$491$	19.3071	0.871318	0.435659	+	0.900112i	$0.356515\pi$
$492$	0	0
$493$	9.97474	0.449240
$494$	0	0
$495$	−1.77986	−0.0799987
$496$	0	0
$497$	14.7971	0.663742
$498$	0	0
$499$	13.5287	0.605626	0.302813	−	0.953050i	$-0.402074\pi$
$499$	13.5287	0.605626	0.302813	+	0.953050i	$0.402074\pi$
$500$	0	0
$501$	33.7665	1.50858
$502$	0	0
$503$	19.0595	0.849823	0.424912	−	0.905235i	$-0.360305\pi$
$503$	19.0595	0.849823	0.424912	+	0.905235i	$0.360305\pi$
$504$	0	0
$505$	0.379361	0.0168813
$506$	0	0
$507$	6.21435	0.275989
$508$	0	0
$509$	4.86980	0.215850	0.107925	−	0.994159i	$-0.465579\pi$
$509$	4.86980	0.215850	0.107925	+	0.994159i	$0.465579\pi$
$510$	0	0
$511$	−45.9770	−2.03390
$512$	0	0
$513$	−1.26777	−0.0559732
$514$	0	0
$515$	−0.404500	−0.0178244
$516$	0	0
$517$	−9.90520	−0.435630
$518$	0	0
$519$	−53.3203	−2.34050
$520$	0	0
$521$	31.7482	1.39091	0.695457	−	0.718568i	$-0.255202\pi$
$521$	31.7482	1.39091	0.695457	+	0.718568i	$0.255202\pi$
$522$	0	0
$523$	28.4375	1.24348	0.621742	−	0.783222i	$-0.286425\pi$
$523$	28.4375	1.24348	0.621742	+	0.783222i	$0.286425\pi$
$524$	0	0
$525$	50.2187	2.19173
$526$	0	0
$527$	9.56778	0.416779
$528$	0	0
$529$	−20.4089	−0.887342
$530$	0	0
$531$	2.91279	0.126404
$532$	0	0
$533$	−16.4856	−0.714072
$534$	0	0
$535$	0.802952	0.0347146
$536$	0	0
$537$	43.9337	1.89588
$538$	0	0
$539$	12.4770	0.537421
$540$	0	0
$541$	−28.9579	−1.24500	−0.622498	−	0.782621i	$-0.713882\pi$
$541$	−28.9579	−1.24500	−0.622498	+	0.782621i	$0.713882\pi$
$542$	0	0
$543$	−2.83079	−0.121481
$544$	0	0
$545$	11.0455	0.473136
$546$	0	0
$547$	18.8154	0.804489	0.402245	−	0.915532i	$-0.368230\pi$
$547$	18.8154	0.804489	0.402245	+	0.915532i	$0.368230\pi$
$548$	0	0
$549$	34.3269	1.46504
$550$	0	0
$551$	59.6308	2.54036
$552$	0	0
$553$	8.67388	0.368851
$554$	0	0
$555$	11.6675	0.495259
$556$	0	0
$557$	−21.6557	−0.917580	−0.458790	−	0.888545i	$-0.651717\pi$
$557$	−21.6557	−0.917580	−0.458790	+	0.888545i	$0.651717\pi$
$558$	0	0
$559$	12.6107	0.533376
$560$	0	0
$561$	2.25284	0.0951148
$562$	0	0
$563$	−9.14610	−0.385462	−0.192731	−	0.981252i	$-0.561734\pi$
$563$	−9.14610	−0.385462	−0.192731	+	0.981252i	$0.561734\pi$
$564$	0	0
$565$	0.994114	0.0418227
$566$	0	0
$567$	−41.8658	−1.75820
$568$	0	0
$569$	−9.79514	−0.410634	−0.205317	−	0.978696i	$-0.565823\pi$
$569$	−9.79514	−0.410634	−0.205317	+	0.978696i	$0.565823\pi$
$570$	0	0
$571$	41.9362	1.75497	0.877487	−	0.479600i	$-0.159218\pi$
$571$	41.9362	1.75497	0.877487	+	0.479600i	$0.159218\pi$
$572$	0	0
$573$	7.84156	0.327586
$574$	0	0
$575$	−7.34828	−0.306445
$576$	0	0
$577$	−4.46746	−0.185983	−0.0929913	−	0.995667i	$-0.529643\pi$
$577$	−4.46746	−0.185983	−0.0929913	+	0.995667i	$0.529643\pi$
$578$	0	0
$579$	−20.9500	−0.870651
$580$	0	0
$581$	−34.0070	−1.41085
$582$	0	0
$583$	6.23214	0.258109
$584$	0	0
$585$	−6.20859	−0.256693
$586$	0	0
$587$	−36.6889	−1.51431	−0.757157	−	0.653233i	$-0.773412\pi$
$587$	−36.6889	−1.51431	−0.757157	+	0.653233i	$0.773412\pi$
$588$	0	0
$589$	57.1979	2.35680
$590$	0	0
$591$	49.7808	2.04771
$592$	0	0
$593$	10.9356	0.449070	0.224535	−	0.974466i	$-0.427914\pi$
$593$	10.9356	0.449070	0.224535	+	0.974466i	$0.427914\pi$
$594$	0	0
$595$	2.98381	0.122324
$596$	0	0
$597$	21.5785	0.883148
$598$	0	0
$599$	−11.4308	−0.467051	−0.233526	−	0.972351i	$-0.575026\pi$
$599$	−11.4308	−0.467051	−0.233526	+	0.972351i	$0.575026\pi$
$600$	0	0
$601$	9.92517	0.404856	0.202428	−	0.979297i	$-0.435117\pi$
$601$	9.92517	0.404856	0.202428	+	0.979297i	$0.435117\pi$
$602$	0	0
$603$	24.9554	1.01626
$604$	0	0
$605$	6.68885	0.271940
$606$	0	0
$607$	31.4164	1.27515	0.637575	−	0.770388i	$-0.279938\pi$
$607$	31.4164	1.27515	0.637575	+	0.770388i	$0.279938\pi$
$608$	0	0
$609$	109.730	4.44649
$610$	0	0
$611$	−34.5518	−1.39782
$612$	0	0
$613$	−31.5185	−1.27302	−0.636509	−	0.771269i	$-0.719622\pi$
$613$	−31.5185	−1.27302	−0.636509	+	0.771269i	$0.719622\pi$
$614$	0	0
$615$	−8.18095	−0.329888
$616$	0	0
$617$	−16.5466	−0.666139	−0.333070	−	0.942902i	$-0.608084\pi$
$617$	−16.5466	−0.666139	−0.333070	+	0.942902i	$0.608084\pi$
$618$	0	0
$619$	−44.9036	−1.80483	−0.902414	−	0.430870i	$-0.858207\pi$
$619$	−44.9036	−1.80483	−0.902414	+	0.430870i	$0.858207\pi$
$620$	0	0
$621$	0.341361	0.0136984
$622$	0	0
$623$	−46.7374	−1.87249
$624$	0	0
$625$	18.6643	0.746571
$626$	0	0
$627$	13.4679	0.537855
$628$	0	0
$629$	−7.27511	−0.290078
$630$	0	0
$631$	42.7887	1.70339	0.851696	−	0.524036i	$-0.175574\pi$
$631$	42.7887	1.70339	0.851696	+	0.524036i	$0.175574\pi$
$632$	0	0
$633$	−32.4855	−1.29118
$634$	0	0
$635$	4.30853	0.170979
$636$	0	0
$637$	43.5227	1.72443
$638$	0	0
$639$	9.52704	0.376884
$640$	0	0
$641$	21.2110	0.837785	0.418893	−	0.908036i	$-0.362418\pi$
$641$	21.2110	0.837785	0.418893	+	0.908036i	$0.362418\pi$
$642$	0	0
$643$	−34.5875	−1.36400	−0.682000	−	0.731353i	$-0.738889\pi$
$643$	−34.5875	−1.36400	−0.682000	+	0.731353i	$0.738889\pi$
$644$	0	0
$645$	6.25803	0.246410
$646$	0	0
$647$	31.0218	1.21959	0.609797	−	0.792558i	$-0.291251\pi$
$647$	31.0218	1.21959	0.609797	+	0.792558i	$0.291251\pi$
$648$	0	0
$649$	0.926475	0.0363673
$650$	0	0
$651$	105.253	4.12520
$652$	0	0
$653$	18.2139	0.712766	0.356383	−	0.934340i	$-0.384010\pi$
$653$	18.2139	0.712766	0.356383	+	0.934340i	$0.384010\pi$
$654$	0	0
$655$	10.3894	0.405946
$656$	0	0
$657$	−29.6020	−1.15488
$658$	0	0
$659$	49.8036	1.94007	0.970036	−	0.242963i	$-0.0781194\pi$
$659$	49.8036	1.94007	0.970036	+	0.242963i	$0.0781194\pi$
$660$	0	0
$661$	29.2053	1.13596	0.567978	−	0.823044i	$-0.307726\pi$
$661$	29.2053	1.13596	0.567978	+	0.823044i	$0.307726\pi$
$662$	0	0
$663$	7.85845	0.305197
$664$	0	0
$665$	17.8378	0.691719
$666$	0	0
$667$	−16.0563	−0.621704
$668$	0	0
$669$	−8.59672	−0.332368
$670$	0	0
$671$	10.9184	0.421501
$672$	0	0
$673$	−23.2178	−0.894980	−0.447490	−	0.894289i	$-0.647682\pi$
$673$	−23.2178	−0.894980	−0.447490	+	0.894289i	$0.647682\pi$
$674$	0	0
$675$	−0.968079	−0.0372614
$676$	0	0
$677$	19.0796	0.733289	0.366644	−	0.930361i	$-0.380507\pi$
$677$	19.0796	0.733289	0.366644	+	0.930361i	$0.380507\pi$
$678$	0	0
$679$	−39.2344	−1.50568
$680$	0	0
$681$	4.16036	0.159425
$682$	0	0
$683$	18.2093	0.696758	0.348379	−	0.937354i	$-0.386732\pi$
$683$	18.2093	0.696758	0.348379	+	0.937354i	$0.386732\pi$
$684$	0	0
$685$	−10.8578	−0.414856
$686$	0	0
$687$	41.7613	1.59329
$688$	0	0
$689$	21.7392	0.828199
$690$	0	0
$691$	−27.4291	−1.04345	−0.521727	−	0.853113i	$-0.674712\pi$
$691$	−27.4291	−1.04345	−0.521727	+	0.853113i	$0.674712\pi$
$692$	0	0
$693$	12.2087	0.463772
$694$	0	0
$695$	8.55723	0.324594
$696$	0	0
$697$	5.10111	0.193218
$698$	0	0
$699$	−6.85600	−0.259318
$700$	0	0
$701$	2.48782	0.0939635	0.0469818	−	0.998896i	$-0.485040\pi$
$701$	2.48782	0.0939635	0.0469818	+	0.998896i	$0.485040\pi$
$702$	0	0
$703$	−43.4919	−1.64033
$704$	0	0
$705$	−17.1462	−0.645765
$706$	0	0
$707$	−2.60219	−0.0978652
$708$	0	0
$709$	15.3157	0.575195	0.287597	−	0.957751i	$-0.407144\pi$
$709$	15.3157	0.575195	0.287597	+	0.957751i	$0.407144\pi$
$710$	0	0
$711$	5.58462	0.209440
$712$	0	0
$713$	−15.4012	−0.576781
$714$	0	0
$715$	−1.97477	−0.0738523
$716$	0	0
$717$	73.6321	2.74984
$718$	0	0
$719$	−8.68177	−0.323775	−0.161888	−	0.986809i	$-0.551758\pi$
$719$	−8.68177	−0.323775	−0.161888	+	0.986809i	$0.551758\pi$
$720$	0	0
$721$	2.77463	0.103332
$722$	0	0
$723$	−70.1053	−2.60724
$724$	0	0
$725$	45.5347	1.69112
$726$	0	0
$727$	−16.6679	−0.618177	−0.309089	−	0.951033i	$-0.600024\pi$
$727$	−16.6679	−0.618177	−0.309089	+	0.951033i	$0.600024\pi$
$728$	0	0
$729$	−25.4080	−0.941039
$730$	0	0
$731$	−3.90210	−0.144325
$732$	0	0
$733$	−21.7078	−0.801795	−0.400898	−	0.916123i	$-0.631302\pi$
$733$	−21.7078	−0.801795	−0.400898	+	0.916123i	$0.631302\pi$
$734$	0	0
$735$	21.5980	0.796656
$736$	0	0
$737$	7.93760	0.292385
$738$	0	0
$739$	−28.9300	−1.06421	−0.532104	−	0.846679i	$-0.678599\pi$
$739$	−28.9300	−1.06421	−0.532104	+	0.846679i	$0.678599\pi$
$740$	0	0
$741$	46.9793	1.72583
$742$	0	0
$743$	37.1684	1.36358	0.681789	−	0.731549i	$-0.261202\pi$
$743$	37.1684	1.36358	0.681789	+	0.731549i	$0.261202\pi$
$744$	0	0
$745$	−8.03958	−0.294547
$746$	0	0
$747$	−21.8952	−0.801103
$748$	0	0
$749$	−5.50776	−0.201249
$750$	0	0
$751$	18.3083	0.668081	0.334040	−	0.942559i	$-0.391588\pi$
$751$	18.3083	0.668081	0.334040	+	0.942559i	$0.391588\pi$
$752$	0	0
$753$	25.4915	0.928963
$754$	0	0
$755$	6.71417	0.244354
$756$	0	0
$757$	−43.3774	−1.57658	−0.788289	−	0.615305i	$-0.789033\pi$
$757$	−43.3774	−1.57658	−0.788289	+	0.615305i	$0.789033\pi$
$758$	0	0
$759$	−3.62639	−0.131630
$760$	0	0
$761$	−17.9239	−0.649739	−0.324870	−	0.945759i	$-0.605320\pi$
$761$	−17.9239	−0.649739	−0.324870	+	0.945759i	$0.605320\pi$
$762$	0	0
$763$	−75.7652	−2.74289
$764$	0	0
$765$	1.92111	0.0694578
$766$	0	0
$767$	3.23177	0.116693
$768$	0	0
$769$	−31.3505	−1.13053	−0.565264	−	0.824910i	$-0.691226\pi$
$769$	−31.3505	−1.13053	−0.565264	+	0.824910i	$0.691226\pi$
$770$	0	0
$771$	−42.0901	−1.51584
$772$	0	0
$773$	−9.21456	−0.331425	−0.165712	−	0.986174i	$-0.552992\pi$
$773$	−9.21456	−0.331425	−0.165712	+	0.986174i	$0.552992\pi$
$774$	0	0
$775$	43.6769	1.56892
$776$	0	0
$777$	−80.0321	−2.87114
$778$	0	0
$779$	30.4954	1.09261
$780$	0	0
$781$	3.03028	0.108432
$782$	0	0
$783$	−2.11530	−0.0755945
$784$	0	0
$785$	−16.0986	−0.574585
$786$	0	0
$787$	1.87853	0.0669625	0.0334813	−	0.999439i	$-0.489341\pi$
$787$	1.87853	0.0669625	0.0334813	+	0.999439i	$0.489341\pi$
$788$	0	0
$789$	43.5929	1.55195
$790$	0	0
$791$	−6.81902	−0.242456
$792$	0	0
$793$	38.0861	1.35248
$794$	0	0
$795$	10.7880	0.382612
$796$	0	0
$797$	−22.3282	−0.790905	−0.395452	−	0.918486i	$-0.629412\pi$
$797$	−22.3282	−0.790905	−0.395452	+	0.918486i	$0.629412\pi$
$798$	0	0
$799$	10.6913	0.378230
$800$	0	0
$801$	−30.0916	−1.06323
$802$	0	0
$803$	−9.41554	−0.332267
$804$	0	0
$805$	−4.80304	−0.169285
$806$	0	0
$807$	−17.6328	−0.620704
$808$	0	0
$809$	4.11252	0.144588	0.0722942	−	0.997383i	$-0.476968\pi$
$809$	4.11252	0.144588	0.0722942	+	0.997383i	$0.476968\pi$
$810$	0	0
$811$	4.98960	0.175208	0.0876042	−	0.996155i	$-0.472079\pi$
$811$	4.98960	0.175208	0.0876042	+	0.996155i	$0.472079\pi$
$812$	0	0
$813$	7.95560	0.279015
$814$	0	0
$815$	−8.42570	−0.295139
$816$	0	0
$817$	−23.3275	−0.816126
$818$	0	0
$819$	42.5871	1.48811
$820$	0	0
$821$	−39.4527	−1.37691	−0.688455	−	0.725279i	$-0.741711\pi$
$821$	−39.4527	−1.37691	−0.688455	+	0.725279i	$0.741711\pi$
$822$	0	0
$823$	2.61869	0.0912819	0.0456410	−	0.998958i	$-0.485467\pi$
$823$	2.61869	0.0912819	0.0456410	+	0.998958i	$0.485467\pi$
$824$	0	0
$825$	10.2842	0.358050
$826$	0	0
$827$	16.7544	0.582608	0.291304	−	0.956630i	$-0.405911\pi$
$827$	16.7544	0.582608	0.291304	+	0.956630i	$0.405911\pi$
$828$	0	0
$829$	25.0348	0.869495	0.434748	−	0.900552i	$-0.356838\pi$
$829$	25.0348	0.869495	0.434748	+	0.900552i	$0.356838\pi$
$830$	0	0
$831$	−78.6263	−2.72752
$832$	0	0
$833$	−13.4671	−0.466609
$834$	0	0
$835$	9.15869	0.316949
$836$	0	0
$837$	−2.02899	−0.0701323
$838$	0	0
$839$	17.3197	0.597941	0.298971	−	0.954262i	$-0.403357\pi$
$839$	17.3197	0.597941	0.298971	+	0.954262i	$0.403357\pi$
$840$	0	0
$841$	70.4954	2.43088
$842$	0	0
$843$	−51.8640	−1.78629
$844$	0	0
$845$	1.68555	0.0579848
$846$	0	0
$847$	−45.8814	−1.57650
$848$	0	0
$849$	−12.0337	−0.412995
$850$	0	0
$851$	11.7107	0.401439
$852$	0	0
$853$	−51.3757	−1.75907	−0.879535	−	0.475833i	$-0.842147\pi$
$853$	−51.3757	−1.75907	−0.879535	+	0.475833i	$0.842147\pi$
$854$	0	0
$855$	11.4847	0.392770
$856$	0	0
$857$	−6.72178	−0.229612	−0.114806	−	0.993388i	$-0.536625\pi$
$857$	−6.72178	−0.229612	−0.114806	+	0.993388i	$0.536625\pi$
$858$	0	0
$859$	14.2503	0.486212	0.243106	−	0.970000i	$-0.421834\pi$
$859$	14.2503	0.486212	0.243106	+	0.970000i	$0.421834\pi$
$860$	0	0
$861$	56.1164	1.91244
$862$	0	0
$863$	−49.6186	−1.68904	−0.844518	−	0.535527i	$-0.820113\pi$
$863$	−49.6186	−1.68904	−0.844518	+	0.535527i	$0.820113\pi$
$864$	0	0
$865$	−14.4624	−0.491735
$866$	0	0
$867$	−2.43162	−0.0825822
$868$	0	0
$869$	1.77631	0.0602571
$870$	0	0
$871$	27.6883	0.938183
$872$	0	0
$873$	−25.2608	−0.854950
$874$	0	0
$875$	28.5402	0.964834
$876$	0	0
$877$	45.0620	1.52164	0.760818	−	0.648965i	$-0.224798\pi$
$877$	45.0620	1.52164	0.760818	+	0.648965i	$0.224798\pi$
$878$	0	0
$879$	34.1212	1.15088
$880$	0	0
$881$	58.1095	1.95776	0.978879	−	0.204443i	$-0.0655382\pi$
$881$	58.1095	1.95776	0.978879	+	0.204443i	$0.0655382\pi$
$882$	0	0
$883$	−6.35261	−0.213782	−0.106891	−	0.994271i	$-0.534090\pi$
$883$	−6.35261	−0.213782	−0.106891	+	0.994271i	$0.534090\pi$
$884$	0	0
$885$	1.60376	0.0539098
$886$	0	0
$887$	19.4738	0.653865	0.326932	−	0.945048i	$-0.393985\pi$
$887$	19.4738	0.653865	0.326932	+	0.945048i	$0.393985\pi$
$888$	0	0
$889$	−29.5539	−0.991205
$890$	0	0
$891$	−8.57362	−0.287227
$892$	0	0
$893$	63.9145	2.13882
$894$	0	0
$895$	11.9164	0.398321
$896$	0	0
$897$	−12.6497	−0.422363
$898$	0	0
$899$	95.4361	3.18297
$900$	0	0
$901$	−6.72672	−0.224100
$902$	0	0
$903$	−42.9263	−1.42850
$904$	0	0
$905$	−0.767811	−0.0255229
$906$	0	0
$907$	−29.5814	−0.982234	−0.491117	−	0.871094i	$-0.663411\pi$
$907$	−29.5814	−0.982234	−0.491117	+	0.871094i	$0.663411\pi$
$908$	0	0
$909$	−1.67540	−0.0555695
$910$	0	0
$911$	−41.5301	−1.37595	−0.687977	−	0.725732i	$-0.741501\pi$
$911$	−41.5301	−1.37595	−0.687977	+	0.725732i	$0.741501\pi$
$912$	0	0
$913$	−6.96423	−0.230482
$914$	0	0
$915$	18.9001	0.624819
$916$	0	0
$917$	−71.2647	−2.35337
$918$	0	0
$919$	19.7037	0.649965	0.324983	−	0.945720i	$-0.394642\pi$
$919$	19.7037	0.649965	0.324983	+	0.945720i	$0.394642\pi$
$920$	0	0
$921$	20.6816	0.681483
$922$	0	0
$923$	10.5704	0.347928
$924$	0	0
$925$	−33.2109	−1.09197
$926$	0	0
$927$	1.78643	0.0586739
$928$	0	0
$929$	−28.8855	−0.947701	−0.473851	−	0.880605i	$-0.657136\pi$
$929$	−28.8855	−0.947701	−0.473851	+	0.880605i	$0.657136\pi$
$930$	0	0
$931$	−80.5090	−2.63858
$932$	0	0
$933$	73.8172	2.41667
$934$	0	0
$935$	0.611049	0.0199835
$936$	0	0
$937$	−6.81226	−0.222547	−0.111273	−	0.993790i	$-0.535493\pi$
$937$	−6.81226	−0.222547	−0.111273	+	0.993790i	$0.535493\pi$
$938$	0	0
$939$	−2.69363	−0.0879032
$940$	0	0
$941$	−45.0627	−1.46900	−0.734500	−	0.678608i	$-0.762583\pi$
$941$	−45.0627	−1.46900	−0.734500	+	0.678608i	$0.762583\pi$
$942$	0	0
$943$	−8.21125	−0.267395
$944$	0	0
$945$	−0.632763	−0.0205838
$946$	0	0
$947$	26.6042	0.864519	0.432260	−	0.901749i	$-0.357716\pi$
$947$	26.6042	0.864519	0.432260	+	0.901749i	$0.357716\pi$
$948$	0	0
$949$	−32.8437	−1.06615
$950$	0	0
$951$	38.4362	1.24638
$952$	0	0
$953$	27.0787	0.877164	0.438582	−	0.898691i	$-0.355481\pi$
$953$	27.0787	0.877164	0.438582	+	0.898691i	$0.355481\pi$
$954$	0	0
$955$	2.12691	0.0688252
$956$	0	0
$957$	22.4715	0.726399
$958$	0	0
$959$	74.4780	2.40502
$960$	0	0
$961$	60.5424	1.95298
$962$	0	0
$963$	−3.54614	−0.114273
$964$	0	0
$965$	−5.68238	−0.182922
$966$	0	0
$967$	17.9677	0.577804	0.288902	−	0.957359i	$-0.406710\pi$
$967$	17.9677	0.577804	0.288902	+	0.957359i	$0.406710\pi$
$968$	0	0
$969$	−14.5367	−0.466986
$970$	0	0
$971$	−41.9029	−1.34473	−0.672364	−	0.740221i	$-0.734721\pi$
$971$	−41.9029	−1.34473	−0.672364	+	0.740221i	$0.734721\pi$
$972$	0	0
$973$	−58.6974	−1.88175
$974$	0	0
$975$	35.8739	1.14888
$976$	0	0
$977$	2.37095	0.0758535	0.0379267	−	0.999281i	$-0.487925\pi$
$977$	2.37095	0.0758535	0.0379267	+	0.999281i	$0.487925\pi$
$978$	0	0
$979$	−9.57126	−0.305899
$980$	0	0
$981$	−48.7810	−1.55746
$982$	0	0
$983$	2.01321	0.0642113	0.0321056	−	0.999484i	$-0.489779\pi$
$983$	2.01321	0.0642113	0.0321056	+	0.999484i	$0.489779\pi$
$984$	0	0
$985$	13.5023	0.430220
$986$	0	0
$987$	117.613	3.74366
$988$	0	0
$989$	6.28121	0.199731
$990$	0	0
$991$	−9.09875	−0.289031	−0.144516	−	0.989503i	$-0.546162\pi$
$991$	−9.09875	−0.289031	−0.144516	+	0.989503i	$0.546162\pi$
$992$	0	0
$993$	33.6607	1.06819
$994$	0	0
$995$	5.85285	0.185548
$996$	0	0
$997$	58.7618	1.86100	0.930502	−	0.366286i	$-0.119371\pi$
$997$	58.7618	1.86100	0.930502	+	0.366286i	$0.119371\pi$
$998$	0	0
$999$	1.54280	0.0488120

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.2	✓	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43162 q^{3} -0.659543 q^{5} +4.52406 q^{7} +2.91279 q^{9} +O(q^{10})\)	\(q-2.43162 q^{3} -0.659543 q^{5} +4.52406 q^{7} +2.91279 q^{9} +0.926475 q^{11} +3.23177 q^{13} +1.60376 q^{15} -1.00000 q^{17} -5.97818 q^{19} -11.0008 q^{21} +1.60970 q^{23} -4.56500 q^{25} +0.212065 q^{27} -9.97474 q^{29} -9.56778 q^{31} -2.25284 q^{33} -2.98381 q^{35} +7.27511 q^{37} -7.85845 q^{39} -5.10111 q^{41} +3.90210 q^{43} -1.92111 q^{45} -10.6913 q^{47} +13.4671 q^{49} +2.43162 q^{51} +6.72672 q^{53} -0.611049 q^{55} +14.5367 q^{57} +1.00000 q^{59} +11.7849 q^{61} +13.1776 q^{63} -2.13149 q^{65} +8.56754 q^{67} -3.91418 q^{69} +3.27076 q^{71} -10.1628 q^{73} +11.1004 q^{75} +4.19143 q^{77} +1.91728 q^{79} -9.25403 q^{81} -7.51691 q^{83} +0.659543 q^{85} +24.2548 q^{87} -10.3308 q^{89} +14.6207 q^{91} +23.2652 q^{93} +3.94287 q^{95} -8.67239 q^{97} +2.69862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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