LMFDB - Embedded newform 2312.4.a.l.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 115 x^{12} + 662 x^{11} + 4615 x^{10} - 26180 x^{9} - 77800 x^{8} + 449840 x^{7} + \cdots + 6804162 $ x^14 - 6x^13 - 115x^12 + 662x^11 + 4615x^10 - 26180x^9 - 77800x^8 + 449840x^7 + 531871x^6 - 3469890x^5 - 680743x^4 + 11053578x^3 - 3766599x^2 - 11240100*x + 6804162
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{17}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.39279$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.38392 q^{3} -3.93156 q^{5} -36.3976 q^{7} +1.98658 q^{9} +O(q^{10})$	$q+5.38392 q^{3} -3.93156 q^{5} -36.3976 q^{7} +1.98658 q^{9} -24.2942 q^{11} -71.6546 q^{13} -21.1672 q^{15} -72.2853 q^{19} -195.962 q^{21} -128.525 q^{23} -109.543 q^{25} -134.670 q^{27} +116.052 q^{29} +49.8827 q^{31} -130.798 q^{33} +143.099 q^{35} +98.3236 q^{37} -385.783 q^{39} -277.944 q^{41} +344.162 q^{43} -7.81038 q^{45} +626.172 q^{47} +981.785 q^{49} +145.329 q^{53} +95.5140 q^{55} -389.178 q^{57} -454.009 q^{59} +253.423 q^{61} -72.3069 q^{63} +281.715 q^{65} -874.562 q^{67} -691.968 q^{69} +83.0135 q^{71} +390.139 q^{73} -589.770 q^{75} +884.249 q^{77} -657.385 q^{79} -778.691 q^{81} -163.049 q^{83} +624.815 q^{87} +418.761 q^{89} +2608.06 q^{91} +268.564 q^{93} +284.194 q^{95} -230.000 q^{97} -48.2624 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 98 q^{9}+O(q^{10}) $ 14 * q + 98 * q^9	$ 14 q + 98 q^{9} + 124 q^{13} + 60 q^{15} + 60 q^{19} + 148 q^{21} + 350 q^{25} + 116 q^{33} + 236 q^{35} + 436 q^{43} + 448 q^{47} + 1138 q^{49} + 1192 q^{53} - 1460 q^{55} - 692 q^{59} - 64 q^{67} + 2076 q^{69} + 1500 q^{77} - 166 q^{81} + 820 q^{83} - 1084 q^{87} - 476 q^{89} + 4372 q^{93}+O(q^{100}) $ 14 * q + 98 * q^9 + 124 * q^13 + 60 * q^15 + 60 * q^19 + 148 * q^21 + 350 * q^25 + 116 * q^33 + 236 * q^35 + 436 * q^43 + 448 * q^47 + 1138 * q^49 + 1192 * q^53 - 1460 * q^55 - 692 * q^59 - 64 * q^67 + 2076 * q^69 + 1500 * q^77 - 166 * q^81 + 820 * q^83 - 1084 * q^87 - 476 * q^89 + 4372 * q^93

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
$3$	5.38392	1.03614	0.518068	−	0.855340i	$-0.326651\pi$
$3$	5.38392	1.03614	0.518068	+	0.855340i	$0.326651\pi$
$4$	0	0
$5$	−3.93156	−0.351650	−0.175825	−	0.984421i	$-0.556259\pi$
$5$	−3.93156	−0.351650	−0.175825	+	0.984421i	$0.556259\pi$
$6$	0	0
$7$	−36.3976	−1.96529	−0.982643	−	0.185508i	$-0.940607\pi$
$7$	−36.3976	−1.96529	−0.982643	+	0.185508i	$0.940607\pi$
$8$	0	0
$9$	1.98658	0.0735772
$10$	0	0
$11$	−24.2942	−0.665906	−0.332953	−	0.942943i	$-0.608045\pi$
$11$	−24.2942	−0.665906	−0.332953	+	0.942943i	$0.608045\pi$
$12$	0	0
$13$	−71.6546	−1.52872	−0.764362	−	0.644787i	$-0.776946\pi$
$13$	−71.6546	−1.52872	−0.764362	+	0.644787i	$0.776946\pi$
$14$	0	0
$15$	−21.1672	−0.364357
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−72.2853	−0.872810	−0.436405	−	0.899750i	$-0.643749\pi$
$19$	−72.2853	−0.872810	−0.436405	+	0.899750i	$0.643749\pi$
$20$	0	0
$21$	−195.962	−2.03630
$22$	0	0
$23$	−128.525	−1.16519	−0.582594	−	0.812764i	$-0.697962\pi$
$23$	−128.525	−1.16519	−0.582594	+	0.812764i	$0.697962\pi$
$24$	0	0
$25$	−109.543	−0.876343
$26$	0	0
$27$	−134.670	−0.959900
$28$	0	0
$29$	116.052	0.743115	0.371557	−	0.928410i	$-0.378824\pi$
$29$	116.052	0.743115	0.371557	+	0.928410i	$0.378824\pi$
$30$	0	0
$31$	49.8827	0.289006	0.144503	−	0.989504i	$-0.453842\pi$
$31$	49.8827	0.289006	0.144503	+	0.989504i	$0.453842\pi$
$32$	0	0
$33$	−130.798	−0.689969
$34$	0	0
$35$	143.099	0.691092
$36$	0	0
$37$	98.3236	0.436873	0.218436	−	0.975851i	$-0.429904\pi$
$37$	98.3236	0.436873	0.218436	+	0.975851i	$0.429904\pi$
$38$	0	0
$39$	−385.783	−1.58397
$40$	0	0
$41$	−277.944	−1.05872	−0.529360	−	0.848397i	$-0.677568\pi$
$41$	−277.944	−1.05872	−0.529360	+	0.848397i	$0.677568\pi$
$42$	0	0
$43$	344.162	1.22056	0.610281	−	0.792185i	$-0.291057\pi$
$43$	344.162	1.22056	0.610281	+	0.792185i	$0.291057\pi$
$44$	0	0
$45$	−7.81038	−0.0258734
$46$	0	0
$47$	626.172	1.94333	0.971665	−	0.236362i	$-0.0759551\pi$
$47$	626.172	1.94333	0.971665	+	0.236362i	$0.0759551\pi$
$48$	0	0
$49$	981.785	2.86235
$50$	0	0
$51$	0	0
$52$	0	0
$53$	145.329	0.376651	0.188326	−	0.982107i	$-0.439694\pi$
$53$	145.329	0.376651	0.188326	+	0.982107i	$0.439694\pi$
$54$	0	0
$55$	95.5140	0.234166
$56$	0	0
$57$	−389.178	−0.904350
$58$	0	0
$59$	−454.009	−1.00181	−0.500906	−	0.865502i	$-0.667000\pi$
$59$	−454.009	−1.00181	−0.500906	+	0.865502i	$0.667000\pi$
$60$	0	0
$61$	253.423	0.531926	0.265963	−	0.963983i	$-0.414310\pi$
$61$	253.423	0.531926	0.265963	+	0.963983i	$0.414310\pi$
$62$	0	0
$63$	−72.3069	−0.144600
$64$	0	0
$65$	281.715	0.537575
$66$	0	0
$67$	−874.562	−1.59470	−0.797349	−	0.603518i	$-0.793765\pi$
$67$	−874.562	−1.59470	−0.797349	+	0.603518i	$0.793765\pi$
$68$	0	0
$69$	−691.968	−1.20729
$70$	0	0
$71$	83.0135	0.138759	0.0693795	−	0.997590i	$-0.477898\pi$
$71$	83.0135	0.138759	0.0693795	+	0.997590i	$0.477898\pi$
$72$	0	0
$73$	390.139	0.625511	0.312755	−	0.949834i	$-0.398748\pi$
$73$	390.139	0.625511	0.312755	+	0.949834i	$0.398748\pi$
$74$	0	0
$75$	−589.770	−0.908010
$76$	0	0
$77$	884.249	1.30870
$78$	0	0
$79$	−657.385	−0.936222	−0.468111	−	0.883670i	$-0.655065\pi$
$79$	−657.385	−0.936222	−0.468111	+	0.883670i	$0.655065\pi$
$80$	0	0
$81$	−778.691	−1.06816
$82$	0	0
$83$	−163.049	−0.215626	−0.107813	−	0.994171i	$-0.534385\pi$
$83$	−163.049	−0.215626	−0.107813	+	0.994171i	$0.534385\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	624.815	0.769968
$88$	0	0
$89$	418.761	0.498748	0.249374	−	0.968407i	$-0.419775\pi$
$89$	418.761	0.498748	0.249374	+	0.968407i	$0.419775\pi$
$90$	0	0
$91$	2608.06	3.00438
$92$	0	0
$93$	268.564	0.299450
$94$	0	0
$95$	284.194	0.306923
$96$	0	0
$97$	−230.000	−0.240752	−0.120376	−	0.992728i	$-0.538410\pi$
$97$	−230.000	−0.240752	−0.120376	+	0.992728i	$0.538410\pi$
$98$	0	0
$99$	−48.2624	−0.0489955
$100$	0	0
$101$	1060.55	1.04484	0.522420	−	0.852689i	$-0.325029\pi$
$101$	1060.55	1.04484	0.522420	+	0.852689i	$0.325029\pi$
$102$	0	0
$103$	671.076	0.641972	0.320986	−	0.947084i	$-0.395986\pi$
$103$	671.076	0.641972	0.320986	+	0.947084i	$0.395986\pi$
$104$	0	0
$105$	770.436	0.716065
$106$	0	0
$107$	247.680	0.223777	0.111889	−	0.993721i	$-0.464310\pi$
$107$	247.680	0.223777	0.111889	+	0.993721i	$0.464310\pi$
$108$	0	0
$109$	−720.616	−0.633234	−0.316617	−	0.948554i	$-0.602547\pi$
$109$	−720.616	−0.633234	−0.316617	+	0.948554i	$0.602547\pi$
$110$	0	0
$111$	529.366	0.452660
$112$	0	0
$113$	−295.560	−0.246053	−0.123026	−	0.992403i	$-0.539260\pi$
$113$	−295.560	−0.246053	−0.123026	+	0.992403i	$0.539260\pi$
$114$	0	0
$115$	505.304	0.409738
$116$	0	0
$117$	−142.348	−0.112479
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−740.793	−0.556569
$122$	0	0
$123$	−1496.43	−1.09698
$124$	0	0
$125$	922.120	0.659815
$126$	0	0
$127$	−450.551	−0.314803	−0.157401	−	0.987535i	$-0.550312\pi$
$127$	−450.551	−0.314803	−0.157401	+	0.987535i	$0.550312\pi$
$128$	0	0
$129$	1852.94	1.26467
$130$	0	0
$131$	−795.427	−0.530510	−0.265255	−	0.964178i	$-0.585456\pi$
$131$	−795.427	−0.530510	−0.265255	+	0.964178i	$0.585456\pi$
$132$	0	0
$133$	2631.01	1.71532
$134$	0	0
$135$	529.464	0.337548
$136$	0	0
$137$	−1480.25	−0.923109	−0.461555	−	0.887112i	$-0.652708\pi$
$137$	−1480.25	−0.923109	−0.461555	+	0.887112i	$0.652708\pi$
$138$	0	0
$139$	−3130.59	−1.91031	−0.955155	−	0.296108i	$-0.904311\pi$
$139$	−3130.59	−1.91031	−0.955155	+	0.296108i	$0.904311\pi$
$140$	0	0
$141$	3371.26	2.01355
$142$	0	0
$143$	1740.79	1.01799
$144$	0	0
$145$	−456.266	−0.261316
$146$	0	0
$147$	5285.85	2.96578
$148$	0	0
$149$	2885.80	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$149$	2885.80	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$150$	0	0
$151$	−1037.68	−0.559239	−0.279619	−	0.960111i	$-0.590208\pi$
$151$	−1037.68	−0.559239	−0.279619	+	0.960111i	$0.590208\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−196.117	−0.101629
$156$	0	0
$157$	−3420.64	−1.73883	−0.869415	−	0.494082i	$-0.835504\pi$
$157$	−3420.64	−1.73883	−0.869415	+	0.494082i	$0.835504\pi$
$158$	0	0
$159$	782.441	0.390262
$160$	0	0
$161$	4678.00	2.28993
$162$	0	0
$163$	81.2669	0.0390510	0.0195255	−	0.999809i	$-0.493784\pi$
$163$	81.2669	0.0390510	0.0195255	+	0.999809i	$0.493784\pi$
$164$	0	0
$165$	514.240	0.242627
$166$	0	0
$167$	1212.36	0.561767	0.280883	−	0.959742i	$-0.409373\pi$
$167$	1212.36	0.561767	0.280883	+	0.959742i	$0.409373\pi$
$168$	0	0
$169$	2937.38	1.33700
$170$	0	0
$171$	−143.601	−0.0642189
$172$	0	0
$173$	−1429.19	−0.628090	−0.314045	−	0.949408i	$-0.601684\pi$
$173$	−1429.19	−0.628090	−0.314045	+	0.949408i	$0.601684\pi$
$174$	0	0
$175$	3987.10	1.72226
$176$	0	0
$177$	−2444.35	−1.03801
$178$	0	0
$179$	1950.92	0.814629	0.407315	−	0.913288i	$-0.366465\pi$
$179$	1950.92	0.814629	0.407315	+	0.913288i	$0.366465\pi$
$180$	0	0
$181$	−2616.72	−1.07458	−0.537291	−	0.843397i	$-0.680552\pi$
$181$	−2616.72	−1.07458	−0.537291	+	0.843397i	$0.680552\pi$
$182$	0	0
$183$	1364.41	0.551148
$184$	0	0
$185$	−386.565	−0.153626
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4901.67	1.88648
$190$	0	0
$191$	2192.73	0.830683	0.415342	−	0.909665i	$-0.363662\pi$
$191$	2192.73	0.830683	0.415342	+	0.909665i	$0.363662\pi$
$192$	0	0
$193$	100.128	0.0373438	0.0186719	−	0.999826i	$-0.494056\pi$
$193$	100.128	0.0373438	0.0186719	+	0.999826i	$0.494056\pi$
$194$	0	0
$195$	1516.73	0.557001
$196$	0	0
$197$	801.112	0.289730	0.144865	−	0.989451i	$-0.453725\pi$
$197$	801.112	0.289730	0.144865	+	0.989451i	$0.453725\pi$
$198$	0	0
$199$	1117.64	0.398129	0.199064	−	0.979986i	$-0.436210\pi$
$199$	1117.64	0.398129	0.199064	+	0.979986i	$0.436210\pi$
$200$	0	0
$201$	−4708.57	−1.65232
$202$	0	0
$203$	−4224.02	−1.46043
$204$	0	0
$205$	1092.75	0.372298
$206$	0	0
$207$	−255.326	−0.0857312
$208$	0	0
$209$	1756.11	0.581210
$210$	0	0
$211$	−564.279	−0.184107	−0.0920535	−	0.995754i	$-0.529343\pi$
$211$	−564.279	−0.184107	−0.0920535	+	0.995754i	$0.529343\pi$
$212$	0	0
$213$	446.938	0.143773
$214$	0	0
$215$	−1353.09	−0.429210
$216$	0	0
$217$	−1815.61	−0.567980
$218$	0	0
$219$	2100.48	0.648114
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5051.79	−1.51701	−0.758505	−	0.651667i	$-0.774070\pi$
$223$	−5051.79	−1.51701	−0.758505	+	0.651667i	$0.774070\pi$
$224$	0	0
$225$	−217.616	−0.0644788
$226$	0	0
$227$	286.117	0.0836574	0.0418287	−	0.999125i	$-0.486682\pi$
$227$	286.117	0.0836574	0.0418287	+	0.999125i	$0.486682\pi$
$228$	0	0
$229$	−2265.18	−0.653657	−0.326828	−	0.945084i	$-0.605980\pi$
$229$	−2265.18	−0.653657	−0.326828	+	0.945084i	$0.605980\pi$
$230$	0	0
$231$	4760.73	1.35599
$232$	0	0
$233$	−5340.80	−1.50166	−0.750831	−	0.660494i	$-0.770347\pi$
$233$	−5340.80	−1.50166	−0.750831	+	0.660494i	$0.770347\pi$
$234$	0	0
$235$	−2461.83	−0.683371
$236$	0	0
$237$	−3539.31	−0.970053
$238$	0	0
$239$	7017.70	1.89932	0.949660	−	0.313284i	$-0.101429\pi$
$239$	7017.70	1.89932	0.949660	+	0.313284i	$0.101429\pi$
$240$	0	0
$241$	−4763.97	−1.27334	−0.636669	−	0.771137i	$-0.719688\pi$
$241$	−4763.97	−1.27334	−0.636669	+	0.771137i	$0.719688\pi$
$242$	0	0
$243$	−556.315	−0.146863
$244$	0	0
$245$	−3859.95	−1.00654
$246$	0	0
$247$	5179.58	1.33429
$248$	0	0
$249$	−877.843	−0.223418
$250$	0	0
$251$	−6592.92	−1.65793	−0.828967	−	0.559298i	$-0.811071\pi$
$251$	−6592.92	−1.65793	−0.828967	+	0.559298i	$0.811071\pi$
$252$	0	0
$253$	3122.41	0.775905
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6128.40	1.48747	0.743734	−	0.668476i	$-0.233053\pi$
$257$	6128.40	1.48747	0.743734	+	0.668476i	$0.233053\pi$
$258$	0	0
$259$	−3578.74	−0.858580
$260$	0	0
$261$	230.547	0.0546763
$262$	0	0
$263$	6142.19	1.44009	0.720045	−	0.693927i	$-0.244121\pi$
$263$	6142.19	1.44009	0.720045	+	0.693927i	$0.244121\pi$
$264$	0	0
$265$	−571.371	−0.132449
$266$	0	0
$267$	2254.57	0.516770
$268$	0	0
$269$	1176.28	0.266614	0.133307	−	0.991075i	$-0.457440\pi$
$269$	1176.28	0.266614	0.133307	+	0.991075i	$0.457440\pi$
$270$	0	0
$271$	−2498.30	−0.560004	−0.280002	−	0.959999i	$-0.590335\pi$
$271$	−2498.30	−0.560004	−0.280002	+	0.959999i	$0.590335\pi$
$272$	0	0
$273$	14041.6	3.11294
$274$	0	0
$275$	2661.25	0.583562
$276$	0	0
$277$	−5147.88	−1.11663	−0.558315	−	0.829629i	$-0.688552\pi$
$277$	−5147.88	−1.11663	−0.558315	+	0.829629i	$0.688552\pi$
$278$	0	0
$279$	99.0962	0.0212643
$280$	0	0
$281$	−3355.18	−0.712290	−0.356145	−	0.934431i	$-0.615909\pi$
$281$	−3355.18	−0.712290	−0.356145	+	0.934431i	$0.615909\pi$
$282$	0	0
$283$	−2612.83	−0.548822	−0.274411	−	0.961612i	$-0.588483\pi$
$283$	−2612.83	−0.548822	−0.274411	+	0.961612i	$0.588483\pi$
$284$	0	0
$285$	1530.08	0.318014
$286$	0	0
$287$	10116.5	2.08069
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−1238.30	−0.249452
$292$	0	0
$293$	1683.63	0.335695	0.167848	−	0.985813i	$-0.446318\pi$
$293$	1683.63	0.335695	0.167848	+	0.985813i	$0.446318\pi$
$294$	0	0
$295$	1784.96	0.352287
$296$	0	0
$297$	3271.70	0.639203
$298$	0	0
$299$	9209.41	1.78125
$300$	0	0
$301$	−12526.7	−2.39875
$302$	0	0
$303$	5709.92	1.08260
$304$	0	0
$305$	−996.348	−0.187052
$306$	0	0
$307$	5864.67	1.09028	0.545138	−	0.838346i	$-0.316477\pi$
$307$	5864.67	1.09028	0.545138	+	0.838346i	$0.316477\pi$
$308$	0	0
$309$	3613.02	0.665170
$310$	0	0
$311$	−6374.43	−1.16225	−0.581127	−	0.813813i	$-0.697388\pi$
$311$	−6374.43	−1.16225	−0.581127	+	0.813813i	$0.697388\pi$
$312$	0	0
$313$	3308.58	0.597482	0.298741	−	0.954334i	$-0.403433\pi$
$313$	3308.58	0.597482	0.298741	+	0.954334i	$0.403433\pi$
$314$	0	0
$315$	284.279	0.0508486
$316$	0	0
$317$	7435.37	1.31739	0.658694	−	0.752411i	$-0.271109\pi$
$317$	7435.37	1.31739	0.658694	+	0.752411i	$0.271109\pi$
$318$	0	0
$319$	−2819.39	−0.494845
$320$	0	0
$321$	1333.49	0.231863
$322$	0	0
$323$	0	0
$324$	0	0
$325$	7849.25	1.33969
$326$	0	0
$327$	−3879.74	−0.656116
$328$	0	0
$329$	−22791.1	−3.81920
$330$	0	0
$331$	953.626	0.158356	0.0791782	−	0.996860i	$-0.474770\pi$
$331$	953.626	0.158356	0.0791782	+	0.996860i	$0.474770\pi$
$332$	0	0
$333$	195.328	0.0321439
$334$	0	0
$335$	3438.40	0.560775
$336$	0	0
$337$	−9403.83	−1.52006	−0.760028	−	0.649890i	$-0.774815\pi$
$337$	−9403.83	−1.52006	−0.760028	+	0.649890i	$0.774815\pi$
$338$	0	0
$339$	−1591.27	−0.254944
$340$	0	0
$341$	−1211.86	−0.192451
$342$	0	0
$343$	−23250.2	−3.66004
$344$	0	0
$345$	2720.52	0.424544
$346$	0	0
$347$	−10350.8	−1.60133	−0.800664	−	0.599113i	$-0.795520\pi$
$347$	−10350.8	−1.60133	−0.800664	+	0.599113i	$0.795520\pi$
$348$	0	0
$349$	9266.42	1.42126	0.710630	−	0.703566i	$-0.248410\pi$
$349$	9266.42	1.42126	0.710630	+	0.703566i	$0.248410\pi$
$350$	0	0
$351$	9649.74	1.46742
$352$	0	0
$353$	−3315.92	−0.499968	−0.249984	−	0.968250i	$-0.580425\pi$
$353$	−3315.92	−0.499968	−0.249984	+	0.968250i	$0.580425\pi$
$354$	0	0
$355$	−326.373	−0.0487945
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5654.75	−0.831327	−0.415664	−	0.909518i	$-0.636451\pi$
$359$	−5654.75	−0.831327	−0.415664	+	0.909518i	$0.636451\pi$
$360$	0	0
$361$	−1633.83	−0.238203
$362$	0	0
$363$	−3988.37	−0.576681
$364$	0	0
$365$	−1533.85	−0.219961
$366$	0	0
$367$	12.7743	0.00181693	0.000908463	−	1.00000i	$-0.499711\pi$
$367$	12.7743	0.00181693	0.000908463		1.00000i	$0.499711\pi$
$368$	0	0
$369$	−552.159	−0.0778976
$370$	0	0
$371$	−5289.63	−0.740227
$372$	0	0
$373$	2955.39	0.410252	0.205126	−	0.978736i	$-0.434240\pi$
$373$	2955.39	0.410252	0.205126	+	0.978736i	$0.434240\pi$
$374$	0	0
$375$	4964.62	0.683658
$376$	0	0
$377$	−8315.66	−1.13602
$378$	0	0
$379$	4319.18	0.585387	0.292693	−	0.956206i	$-0.405448\pi$
$379$	4319.18	0.585387	0.292693	+	0.956206i	$0.405448\pi$
$380$	0	0
$381$	−2425.73	−0.326179
$382$	0	0
$383$	−5788.94	−0.772326	−0.386163	−	0.922431i	$-0.626200\pi$
$383$	−5788.94	−0.772326	−0.386163	+	0.922431i	$0.626200\pi$
$384$	0	0
$385$	−3476.48	−0.460202
$386$	0	0
$387$	683.706	0.0898055
$388$	0	0
$389$	−10329.9	−1.34640	−0.673198	−	0.739462i	$-0.735080\pi$
$389$	−10329.9	−1.34640	−0.673198	+	0.739462i	$0.735080\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4282.51	−0.549680
$394$	0	0
$395$	2584.55	0.329222
$396$	0	0
$397$	10928.4	1.38157	0.690785	−	0.723061i	$-0.257265\pi$
$397$	10928.4	1.38157	0.690785	+	0.723061i	$0.257265\pi$
$398$	0	0
$399$	14165.2	1.77731
$400$	0	0
$401$	6061.22	0.754820	0.377410	−	0.926046i	$-0.376815\pi$
$401$	6061.22	0.754820	0.377410	+	0.926046i	$0.376815\pi$
$402$	0	0
$403$	−3574.32	−0.441811
$404$	0	0
$405$	3061.47	0.375619
$406$	0	0
$407$	−2388.69	−0.290916
$408$	0	0
$409$	12598.8	1.52316	0.761580	−	0.648071i	$-0.224424\pi$
$409$	12598.8	1.52316	0.761580	+	0.648071i	$0.224424\pi$
$410$	0	0
$411$	−7969.53	−0.956466
$412$	0	0
$413$	16524.8	1.96885
$414$	0	0
$415$	641.038	0.0758248
$416$	0	0
$417$	−16854.8	−1.97934
$418$	0	0
$419$	2068.48	0.241174	0.120587	−	0.992703i	$-0.461522\pi$
$419$	2068.48	0.241174	0.120587	+	0.992703i	$0.461522\pi$
$420$	0	0
$421$	7194.58	0.832880	0.416440	−	0.909163i	$-0.363278\pi$
$421$	7194.58	0.832880	0.416440	+	0.909163i	$0.363278\pi$
$422$	0	0
$423$	1243.94	0.142985
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9223.99	−1.04539
$428$	0	0
$429$	9372.27	1.05477
$430$	0	0
$431$	13694.7	1.53051	0.765255	−	0.643728i	$-0.222613\pi$
$431$	13694.7	1.53051	0.765255	+	0.643728i	$0.222613\pi$
$432$	0	0
$433$	12504.4	1.38782	0.693909	−	0.720063i	$-0.255887\pi$
$433$	12504.4	1.38782	0.693909	+	0.720063i	$0.255887\pi$
$434$	0	0
$435$	−2456.50	−0.270759
$436$	0	0
$437$	9290.47	1.01699
$438$	0	0
$439$	−13970.1	−1.51881	−0.759406	−	0.650617i	$-0.774510\pi$
$439$	−13970.1	−1.51881	−0.759406	+	0.650617i	$0.774510\pi$
$440$	0	0
$441$	1950.40	0.210604
$442$	0	0
$443$	829.809	0.0889964	0.0444982	−	0.999009i	$-0.485831\pi$
$443$	829.809	0.0889964	0.0444982	+	0.999009i	$0.485831\pi$
$444$	0	0
$445$	−1646.38	−0.175384
$446$	0	0
$447$	15536.9	1.64401
$448$	0	0
$449$	−8396.70	−0.882550	−0.441275	−	0.897372i	$-0.645474\pi$
$449$	−8396.70	−0.882550	−0.441275	+	0.897372i	$0.645474\pi$
$450$	0	0
$451$	6752.41	0.705008
$452$	0	0
$453$	−5586.78	−0.579447
$454$	0	0
$455$	−10253.7	−1.05649
$456$	0	0
$457$	−4013.92	−0.410861	−0.205430	−	0.978672i	$-0.565859\pi$
$457$	−4013.92	−0.410861	−0.205430	+	0.978672i	$0.565859\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9576.45	0.967505	0.483753	−	0.875205i	$-0.339273\pi$
$461$	9576.45	0.967505	0.483753	+	0.875205i	$0.339273\pi$
$462$	0	0
$463$	−8366.78	−0.839822	−0.419911	−	0.907565i	$-0.637939\pi$
$463$	−8366.78	−0.839822	−0.419911	+	0.907565i	$0.637939\pi$
$464$	0	0
$465$	−1055.88	−0.105301
$466$	0	0
$467$	8324.94	0.824909	0.412454	−	0.910978i	$-0.364672\pi$
$467$	8324.94	0.824909	0.412454	+	0.910978i	$0.364672\pi$
$468$	0	0
$469$	31832.0	3.13404
$470$	0	0
$471$	−18416.4	−1.80166
$472$	0	0
$473$	−8361.12	−0.812779
$474$	0	0
$475$	7918.34	0.764881
$476$	0	0
$477$	288.709	0.0277129
$478$	0	0
$479$	−875.481	−0.0835109	−0.0417555	−	0.999128i	$-0.513295\pi$
$479$	−875.481	−0.0835109	−0.0417555	+	0.999128i	$0.513295\pi$
$480$	0	0
$481$	−7045.34	−0.667858
$482$	0	0
$483$	25186.0	2.37267
$484$	0	0
$485$	904.258	0.0846603
$486$	0	0
$487$	3961.92	0.368648	0.184324	−	0.982865i	$-0.440990\pi$
$487$	3961.92	0.368648	0.184324	+	0.982865i	$0.440990\pi$
$488$	0	0
$489$	437.534	0.0404621
$490$	0	0
$491$	5348.65	0.491611	0.245806	−	0.969319i	$-0.420947\pi$
$491$	5348.65	0.491611	0.245806	+	0.969319i	$0.420947\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	189.747	0.0172293
$496$	0	0
$497$	−3021.49	−0.272701
$498$	0	0
$499$	−3240.93	−0.290749	−0.145375	−	0.989377i	$-0.546439\pi$
$499$	−3240.93	−0.290749	−0.145375	+	0.989377i	$0.546439\pi$
$500$	0	0
$501$	6527.23	0.582067
$502$	0	0
$503$	−17198.8	−1.52457	−0.762283	−	0.647244i	$-0.775921\pi$
$503$	−17198.8	−1.52457	−0.762283	+	0.647244i	$0.775921\pi$
$504$	0	0
$505$	−4169.62	−0.367417
$506$	0	0
$507$	15814.6	1.38531
$508$	0	0
$509$	19542.7	1.70179	0.850897	−	0.525333i	$-0.176059\pi$
$509$	19542.7	1.70179	0.850897	+	0.525333i	$0.176059\pi$
$510$	0	0
$511$	−14200.1	−1.22931
$512$	0	0
$513$	9734.68	0.837810
$514$	0	0
$515$	−2638.38	−0.225749
$516$	0	0
$517$	−15212.3	−1.29408
$518$	0	0
$519$	−7694.67	−0.650787
$520$	0	0
$521$	19544.2	1.64347	0.821733	−	0.569872i	$-0.193007\pi$
$521$	19544.2	1.64347	0.821733	+	0.569872i	$0.193007\pi$
$522$	0	0
$523$	13396.9	1.12009	0.560044	−	0.828463i	$-0.310784\pi$
$523$	13396.9	1.12009	0.560044	+	0.828463i	$0.310784\pi$
$524$	0	0
$525$	21466.2	1.78450
$526$	0	0
$527$	0	0
$528$	0	0
$529$	4351.67	0.357662
$530$	0	0
$531$	−901.927	−0.0737106
$532$	0	0
$533$	19915.9	1.61849
$534$	0	0
$535$	−973.770	−0.0786911
$536$	0	0
$537$	10503.6	0.844066
$538$	0	0
$539$	−23851.7	−1.90605
$540$	0	0
$541$	10945.8	0.869868	0.434934	−	0.900462i	$-0.356772\pi$
$541$	10945.8	0.869868	0.434934	+	0.900462i	$0.356772\pi$
$542$	0	0
$543$	−14088.2	−1.11341
$544$	0	0
$545$	2833.15	0.222676
$546$	0	0
$547$	−12634.4	−0.987585	−0.493793	−	0.869580i	$-0.664390\pi$
$547$	−12634.4	−0.987585	−0.493793	+	0.869580i	$0.664390\pi$
$548$	0	0
$549$	503.446	0.0391376
$550$	0	0
$551$	−8388.86	−0.648598
$552$	0	0
$553$	23927.2	1.83994
$554$	0	0
$555$	−2081.24	−0.159178
$556$	0	0
$557$	−3440.59	−0.261728	−0.130864	−	0.991400i	$-0.541775\pi$
$557$	−3440.59	−0.261728	−0.130864	+	0.991400i	$0.541775\pi$
$558$	0	0
$559$	−24660.8	−1.86590
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11973.1	0.896278	0.448139	−	0.893964i	$-0.352087\pi$
$563$	11973.1	0.896278	0.448139	+	0.893964i	$0.352087\pi$
$564$	0	0
$565$	1162.01	0.0865243
$566$	0	0
$567$	28342.5	2.09925
$568$	0	0
$569$	12536.9	0.923680	0.461840	−	0.886963i	$-0.347189\pi$
$569$	12536.9	0.923680	0.461840	+	0.886963i	$0.347189\pi$
$570$	0	0
$571$	−3515.42	−0.257646	−0.128823	−	0.991668i	$-0.541120\pi$
$571$	−3515.42	−0.257646	−0.128823	+	0.991668i	$0.541120\pi$
$572$	0	0
$573$	11805.5	0.860701
$574$	0	0
$575$	14079.0	1.02110
$576$	0	0
$577$	15196.0	1.09639	0.548195	−	0.836350i	$-0.315315\pi$
$577$	15196.0	1.09639	0.548195	+	0.836350i	$0.315315\pi$
$578$	0	0
$579$	539.080	0.0386933
$580$	0	0
$581$	5934.60	0.423767
$582$	0	0
$583$	−3530.65	−0.250814
$584$	0	0
$585$	559.650	0.0395533
$586$	0	0
$587$	11409.2	0.802230	0.401115	−	0.916028i	$-0.368623\pi$
$587$	11409.2	0.802230	0.401115	+	0.916028i	$0.368623\pi$
$588$	0	0
$589$	−3605.79	−0.252248
$590$	0	0
$591$	4313.12	0.300200
$592$	0	0
$593$	3295.58	0.228218	0.114109	−	0.993468i	$-0.463599\pi$
$593$	3295.58	0.228218	0.114109	+	0.993468i	$0.463599\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6017.30	0.412515
$598$	0	0
$599$	1426.99	0.0973378	0.0486689	−	0.998815i	$-0.484502\pi$
$599$	1426.99	0.0973378	0.0486689	+	0.998815i	$0.484502\pi$
$600$	0	0
$601$	−28806.8	−1.95517	−0.977584	−	0.210548i	$-0.932475\pi$
$601$	−28806.8	−1.95517	−0.977584	+	0.210548i	$0.932475\pi$
$602$	0	0
$603$	−1737.39	−0.117333
$604$	0	0
$605$	2912.48	0.195717
$606$	0	0
$607$	3283.98	0.219592	0.109796	−	0.993954i	$-0.464980\pi$
$607$	3283.98	0.219592	0.109796	+	0.993954i	$0.464980\pi$
$608$	0	0
$609$	−22741.8	−1.51321
$610$	0	0
$611$	−44868.1	−2.97082
$612$	0	0
$613$	15397.0	1.01449	0.507244	−	0.861803i	$-0.330664\pi$
$613$	15397.0	1.01449	0.507244	+	0.861803i	$0.330664\pi$
$614$	0	0
$615$	5883.29	0.385752
$616$	0	0
$617$	29410.7	1.91901	0.959507	−	0.281684i	$-0.0908930\pi$
$617$	29410.7	1.91901	0.959507	+	0.281684i	$0.0908930\pi$
$618$	0	0
$619$	2604.90	0.169144	0.0845718	−	0.996417i	$-0.473048\pi$
$619$	2604.90	0.169144	0.0845718	+	0.996417i	$0.473048\pi$
$620$	0	0
$621$	17308.5	1.11846
$622$	0	0
$623$	−15241.9	−0.980182
$624$	0	0
$625$	10067.5	0.644319
$626$	0	0
$627$	9454.76	0.602212
$628$	0	0
$629$	0	0
$630$	0	0
$631$	6384.47	0.402792	0.201396	−	0.979510i	$-0.435452\pi$
$631$	6384.47	0.402792	0.201396	+	0.979510i	$0.435452\pi$
$632$	0	0
$633$	−3038.03	−0.190760
$634$	0	0
$635$	1771.37	0.110700
$636$	0	0
$637$	−70349.4	−4.37574
$638$	0	0
$639$	164.913	0.0102095
$640$	0	0
$641$	29758.9	1.83371	0.916854	−	0.399223i	$-0.130720\pi$
$641$	29758.9	1.83371	0.916854	+	0.399223i	$0.130720\pi$
$642$	0	0
$643$	15874.5	0.973605	0.486802	−	0.873512i	$-0.338163\pi$
$643$	15874.5	0.973605	0.486802	+	0.873512i	$0.338163\pi$
$644$	0	0
$645$	−7284.94	−0.444720
$646$	0	0
$647$	−19644.7	−1.19368	−0.596842	−	0.802359i	$-0.703578\pi$
$647$	−19644.7	−1.19368	−0.596842	+	0.802359i	$0.703578\pi$
$648$	0	0
$649$	11029.8	0.667113
$650$	0	0
$651$	−9775.10	−0.588504
$652$	0	0
$653$	5655.89	0.338947	0.169473	−	0.985535i	$-0.445793\pi$
$653$	5655.89	0.338947	0.169473	+	0.985535i	$0.445793\pi$
$654$	0	0
$655$	3127.27	0.186554
$656$	0	0
$657$	775.044	0.0460233
$658$	0	0
$659$	−22818.1	−1.34881	−0.674407	−	0.738360i	$-0.735600\pi$
$659$	−22818.1	−1.34881	−0.674407	+	0.738360i	$0.735600\pi$
$660$	0	0
$661$	−5851.00	−0.344293	−0.172146	−	0.985071i	$-0.555070\pi$
$661$	−5851.00	−0.344293	−0.172146	+	0.985071i	$0.555070\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10344.0	−0.603192
$666$	0	0
$667$	−14915.6	−0.865868
$668$	0	0
$669$	−27198.4	−1.57183
$670$	0	0
$671$	−6156.70	−0.354213
$672$	0	0
$673$	−12593.9	−0.721334	−0.360667	−	0.932695i	$-0.617451\pi$
$673$	−12593.9	−0.721334	−0.360667	+	0.932695i	$0.617451\pi$
$674$	0	0
$675$	14752.2	0.841201
$676$	0	0
$677$	25407.0	1.44235	0.721175	−	0.692752i	$-0.243602\pi$
$677$	25407.0	1.44235	0.721175	+	0.692752i	$0.243602\pi$
$678$	0	0
$679$	8371.43	0.473146
$680$	0	0
$681$	1540.43	0.0866804
$682$	0	0
$683$	−19925.5	−1.11629	−0.558146	−	0.829743i	$-0.688487\pi$
$683$	−19925.5	−1.11629	−0.558146	+	0.829743i	$0.688487\pi$
$684$	0	0
$685$	5819.68	0.324611
$686$	0	0
$687$	−12195.6	−0.677277
$688$	0	0
$689$	−10413.5	−0.575795
$690$	0	0
$691$	−26045.5	−1.43389	−0.716945	−	0.697130i	$-0.754460\pi$
$691$	−26045.5	−1.43389	−0.716945	+	0.697130i	$0.754460\pi$
$692$	0	0
$693$	1756.64	0.0962902
$694$	0	0
$695$	12308.1	0.671759
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−28754.4	−1.55593
$700$	0	0
$701$	−8521.04	−0.459109	−0.229554	−	0.973296i	$-0.573727\pi$
$701$	−8521.04	−0.459109	−0.229554	+	0.973296i	$0.573727\pi$
$702$	0	0
$703$	−7107.35	−0.381307
$704$	0	0
$705$	−13254.3	−0.708065
$706$	0	0
$707$	−38601.5	−2.05341
$708$	0	0
$709$	27767.4	1.47084	0.735420	−	0.677611i	$-0.236985\pi$
$709$	27767.4	1.47084	0.735420	+	0.677611i	$0.236985\pi$
$710$	0	0
$711$	−1305.95	−0.0688846
$712$	0	0
$713$	−6411.17	−0.336746
$714$	0	0
$715$	−6844.02	−0.357975
$716$	0	0
$717$	37782.7	1.96795
$718$	0	0
$719$	−28594.0	−1.48314	−0.741570	−	0.670876i	$-0.765918\pi$
$719$	−28594.0	−1.48314	−0.741570	+	0.670876i	$0.765918\pi$
$720$	0	0
$721$	−24425.6	−1.26166
$722$	0	0
$723$	−25648.9	−1.31935
$724$	0	0
$725$	−12712.7	−0.651223
$726$	0	0
$727$	22672.4	1.15663	0.578317	−	0.815812i	$-0.303710\pi$
$727$	22672.4	1.15663	0.578317	+	0.815812i	$0.303710\pi$
$728$	0	0
$729$	18029.5	0.915994
$730$	0	0
$731$	0	0
$732$	0	0
$733$	3241.83	0.163356	0.0816779	−	0.996659i	$-0.473972\pi$
$733$	3241.83	0.163356	0.0816779	+	0.996659i	$0.473972\pi$
$734$	0	0
$735$	−20781.7	−1.04292
$736$	0	0
$737$	21246.8	1.06192
$738$	0	0
$739$	8977.81	0.446893	0.223447	−	0.974716i	$-0.428269\pi$
$739$	8977.81	0.446893	0.223447	+	0.974716i	$0.428269\pi$
$740$	0	0
$741$	27886.4	1.38250
$742$	0	0
$743$	5317.32	0.262549	0.131274	−	0.991346i	$-0.458093\pi$
$743$	5317.32	0.262549	0.131274	+	0.991346i	$0.458093\pi$
$744$	0	0
$745$	−11345.7	−0.557953
$746$	0	0
$747$	−323.911	−0.0158652
$748$	0	0
$749$	−9014.96	−0.439786
$750$	0	0
$751$	18961.8	0.921339	0.460670	−	0.887572i	$-0.347609\pi$
$751$	18961.8	0.921339	0.460670	+	0.887572i	$0.347609\pi$
$752$	0	0
$753$	−35495.7	−1.71784
$754$	0	0
$755$	4079.70	0.196656
$756$	0	0
$757$	28601.5	1.37324	0.686619	−	0.727018i	$-0.259094\pi$
$757$	28601.5	1.37324	0.686619	+	0.727018i	$0.259094\pi$
$758$	0	0
$759$	16810.8	0.803943
$760$	0	0
$761$	23710.9	1.12946	0.564730	−	0.825276i	$-0.308980\pi$
$761$	23710.9	1.12946	0.564730	+	0.825276i	$0.308980\pi$
$762$	0	0
$763$	26228.7	1.24449
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32531.8	1.53149
$768$	0	0
$769$	3392.20	0.159071	0.0795357	−	0.996832i	$-0.474656\pi$
$769$	3392.20	0.159071	0.0795357	+	0.996832i	$0.474656\pi$
$770$	0	0
$771$	32994.8	1.54122
$772$	0	0
$773$	10444.4	0.485977	0.242988	−	0.970029i	$-0.421872\pi$
$773$	10444.4	0.485977	0.242988	+	0.970029i	$0.421872\pi$
$774$	0	0
$775$	−5464.29	−0.253269
$776$	0	0
$777$	−19267.7	−0.889606
$778$	0	0
$779$	20091.2	0.924061
$780$	0	0
$781$	−2016.74	−0.0924004
$782$	0	0
$783$	−15628.8	−0.713316
$784$	0	0
$785$	13448.4	0.611459
$786$	0	0
$787$	12690.3	0.574789	0.287394	−	0.957812i	$-0.407211\pi$
$787$	12690.3	0.574789	0.287394	+	0.957812i	$0.407211\pi$
$788$	0	0
$789$	33069.1	1.49213
$790$	0	0
$791$	10757.7	0.483564
$792$	0	0
$793$	−18158.9	−0.813168
$794$	0	0
$795$	−3076.21	−0.137235
$796$	0	0
$797$	−31949.2	−1.41995	−0.709974	−	0.704228i	$-0.751293\pi$
$797$	−31949.2	−1.41995	−0.709974	+	0.704228i	$0.751293\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	831.904	0.0366965
$802$	0	0
$803$	−9478.10	−0.416531
$804$	0	0
$805$	−18391.8	−0.805251
$806$	0	0
$807$	6333.02	0.276249
$808$	0	0
$809$	−18486.9	−0.803416	−0.401708	−	0.915768i	$-0.631583\pi$
$809$	−18486.9	−0.803416	−0.401708	+	0.915768i	$0.631583\pi$
$810$	0	0
$811$	−22297.4	−0.965434	−0.482717	−	0.875776i	$-0.660350\pi$
$811$	−22297.4	−0.965434	−0.482717	+	0.875776i	$0.660350\pi$
$812$	0	0
$813$	−13450.7	−0.580240
$814$	0	0
$815$	−319.506	−0.0137323
$816$	0	0
$817$	−24877.8	−1.06532
$818$	0	0
$819$	5181.12	0.221054
$820$	0	0
$821$	−18890.3	−0.803016	−0.401508	−	0.915855i	$-0.631514\pi$
$821$	−18890.3	−0.803016	−0.401508	+	0.915855i	$0.631514\pi$
$822$	0	0
$823$	−7979.34	−0.337962	−0.168981	−	0.985619i	$-0.554048\pi$
$823$	−7979.34	−0.337962	−0.168981	+	0.985619i	$0.554048\pi$
$824$	0	0
$825$	14328.0	0.604649
$826$	0	0
$827$	−4459.57	−0.187514	−0.0937571	−	0.995595i	$-0.529888\pi$
$827$	−4459.57	−0.187514	−0.0937571	+	0.995595i	$0.529888\pi$
$828$	0	0
$829$	−19686.3	−0.824769	−0.412384	−	0.911010i	$-0.635304\pi$
$829$	−19686.3	−0.824769	−0.412384	+	0.911010i	$0.635304\pi$
$830$	0	0
$831$	−27715.8	−1.15698
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4766.46	−0.197545
$836$	0	0
$837$	−6717.71	−0.277417
$838$	0	0
$839$	−28708.1	−1.18130	−0.590651	−	0.806927i	$-0.701129\pi$
$839$	−28708.1	−1.18130	−0.590651	+	0.806927i	$0.701129\pi$
$840$	0	0
$841$	−10920.9	−0.447780
$842$	0	0
$843$	−18064.0	−0.738029
$844$	0	0
$845$	−11548.5	−0.470154
$846$	0	0
$847$	26963.1	1.09382
$848$	0	0
$849$	−14067.3	−0.568654
$850$	0	0
$851$	−12637.0	−0.509039
$852$	0	0
$853$	7505.86	0.301285	0.150642	−	0.988588i	$-0.451866\pi$
$853$	7505.86	0.301285	0.150642	+	0.988588i	$0.451866\pi$
$854$	0	0
$855$	564.576	0.0225826
$856$	0	0
$857$	78.7833	0.00314024	0.00157012	−	0.999999i	$-0.499500\pi$
$857$	78.7833	0.00314024	0.00157012	+	0.999999i	$0.499500\pi$
$858$	0	0
$859$	−5441.53	−0.216138	−0.108069	−	0.994143i	$-0.534467\pi$
$859$	−5441.53	−0.216138	−0.108069	+	0.994143i	$0.534467\pi$
$860$	0	0
$861$	54466.3	2.15587
$862$	0	0
$863$	−7156.86	−0.282297	−0.141148	−	0.989988i	$-0.545079\pi$
$863$	−7156.86	−0.282297	−0.141148	+	0.989988i	$0.545079\pi$
$864$	0	0
$865$	5618.97	0.220868
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15970.6	0.623436
$870$	0	0
$871$	62666.4	2.43785
$872$	0	0
$873$	−456.914	−0.0177138
$874$	0	0
$875$	−33562.9	−1.29673
$876$	0	0
$877$	−14544.1	−0.559998	−0.279999	−	0.960000i	$-0.590334\pi$
$877$	−14544.1	−0.559998	−0.279999	+	0.960000i	$0.590334\pi$
$878$	0	0
$879$	9064.53	0.347826
$880$	0	0
$881$	−22447.1	−0.858413	−0.429207	−	0.903206i	$-0.641207\pi$
$881$	−22447.1	−0.858413	−0.429207	+	0.903206i	$0.641207\pi$
$882$	0	0
$883$	34760.2	1.32477	0.662385	−	0.749163i	$-0.269544\pi$
$883$	34760.2	1.32477	0.662385	+	0.749163i	$0.269544\pi$
$884$	0	0
$885$	9610.10	0.365017
$886$	0	0
$887$	5979.70	0.226357	0.113179	−	0.993575i	$-0.463897\pi$
$887$	5979.70	0.226357	0.113179	+	0.993575i	$0.463897\pi$
$888$	0	0
$889$	16399.0	0.618678
$890$	0	0
$891$	18917.7	0.711297
$892$	0	0
$893$	−45263.0	−1.69616
$894$	0	0
$895$	−7670.16	−0.286464
$896$	0	0
$897$	49582.7	1.84562
$898$	0	0
$899$	5788.99	0.214765
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−67442.5	−2.48543
$904$	0	0
$905$	10287.8	0.377876
$906$	0	0
$907$	37422.2	1.36999	0.684996	−	0.728547i	$-0.259804\pi$
$907$	37422.2	1.36999	0.684996	+	0.728547i	$0.259804\pi$
$908$	0	0
$909$	2106.87	0.0768764
$910$	0	0
$911$	25.4691	0.000926267 0	0.000463134	−	1.00000i	$-0.499853\pi$
$911$	25.4691	0.000926267 0	0.000463134		1.00000i	$0.499853\pi$
$912$	0	0
$913$	3961.14	0.143587
$914$	0	0
$915$	−5364.26	−0.193811
$916$	0	0
$917$	28951.6	1.04260
$918$	0	0
$919$	−11502.6	−0.412878	−0.206439	−	0.978459i	$-0.566188\pi$
$919$	−11502.6	−0.412878	−0.206439	+	0.978459i	$0.566188\pi$
$920$	0	0
$921$	31574.9	1.12967
$922$	0	0
$923$	−5948.30	−0.212124
$924$	0	0
$925$	−10770.6	−0.382850
$926$	0	0
$927$	1333.15	0.0472345
$928$	0	0
$929$	−37230.2	−1.31484	−0.657418	−	0.753526i	$-0.728351\pi$
$929$	−37230.2	−1.31484	−0.657418	+	0.753526i	$0.728351\pi$
$930$	0	0
$931$	−70968.7	−2.49829
$932$	0	0
$933$	−34319.4	−1.20425
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−72.1695	−0.00251619	−0.00125810	−	0.999999i	$-0.500400\pi$
$937$	−72.1695	−0.00251619	−0.00125810	+	0.999999i	$0.500400\pi$
$938$	0	0
$939$	17813.1	0.619072
$940$	0	0
$941$	9831.52	0.340594	0.170297	−	0.985393i	$-0.445527\pi$
$941$	9831.52	0.340594	0.170297	+	0.985393i	$0.445527\pi$
$942$	0	0
$943$	35722.7	1.23361
$944$	0	0
$945$	−19271.2	−0.663379
$946$	0	0
$947$	20552.0	0.705227	0.352614	−	0.935769i	$-0.385293\pi$
$947$	20552.0	0.705227	0.352614	+	0.935769i	$0.385293\pi$
$948$	0	0
$949$	−27955.2	−0.956233
$950$	0	0
$951$	40031.4	1.36499
$952$	0	0
$953$	−2360.48	−0.0802345	−0.0401172	−	0.999195i	$-0.512773\pi$
$953$	−2360.48	−0.0802345	−0.0401172	+	0.999195i	$0.512773\pi$
$954$	0	0
$955$	−8620.86	−0.292109
$956$	0	0
$957$	−15179.4	−0.512726
$958$	0	0
$959$	53877.4	1.81417
$960$	0	0
$961$	−27302.7	−0.916475
$962$	0	0
$963$	492.038	0.0164649
$964$	0	0
$965$	−393.659	−0.0131319
$966$	0	0
$967$	−9395.70	−0.312456	−0.156228	−	0.987721i	$-0.549934\pi$
$967$	−9395.70	−0.312456	−0.156228	+	0.987721i	$0.549934\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24081.5	−0.795893	−0.397947	−	0.917409i	$-0.630277\pi$
$971$	−24081.5	−0.795893	−0.397947	+	0.917409i	$0.630277\pi$
$972$	0	0
$973$	113946.	3.75430
$974$	0	0
$975$	42259.7	1.38810
$976$	0	0
$977$	−7503.29	−0.245703	−0.122851	−	0.992425i	$-0.539204\pi$
$977$	−7503.29	−0.245703	−0.122851	+	0.992425i	$0.539204\pi$
$978$	0	0
$979$	−10173.4	−0.332119
$980$	0	0
$981$	−1431.56	−0.0465916
$982$	0	0
$983$	−13201.0	−0.428328	−0.214164	−	0.976798i	$-0.568703\pi$
$983$	−13201.0	−0.428328	−0.214164	+	0.976798i	$0.568703\pi$
$984$	0	0
$985$	−3149.62	−0.101884
$986$	0	0
$987$	−122706.	−3.95721
$988$	0	0
$989$	−44233.3	−1.42218
$990$	0	0
$991$	38822.1	1.24442	0.622212	−	0.782849i	$-0.286234\pi$
$991$	38822.1	1.24442	0.622212	+	0.782849i	$0.286234\pi$
$992$	0	0
$993$	5134.24	0.164079
$994$	0	0
$995$	−4394.08	−0.140002
$996$	0	0
$997$	20727.4	0.658420	0.329210	−	0.944257i	$-0.393218\pi$
$997$	20727.4	0.658420	0.329210	+	0.944257i	$0.393218\pi$
$998$	0	0
$999$	−13241.3	−0.419354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.l.1.11		14
17.2	even	8		136.4.k.b.89.6	yes	14
17.9	even	8		136.4.k.b.81.6	✓	14
17.16	even	2	inner	2312.4.a.l.1.4		14
68.19	odd	8		272.4.o.f.225.2		14
68.43	odd	8		272.4.o.f.81.2		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.k.b.81.6	✓	14	17.9	even	8
136.4.k.b.89.6	yes	14	17.2	even	8
272.4.o.f.81.2		14	68.43	odd	8
272.4.o.f.225.2		14	68.19	odd	8
2312.4.a.l.1.4		14	17.16	even	2	inner
2312.4.a.l.1.11		14	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+5.38392 q^{3} -3.93156 q^{5} -36.3976 q^{7} +1.98658 q^{9} +O(q^{10})\)	\(q+5.38392 q^{3} -3.93156 q^{5} -36.3976 q^{7} +1.98658 q^{9} -24.2942 q^{11} -71.6546 q^{13} -21.1672 q^{15} -72.2853 q^{19} -195.962 q^{21} -128.525 q^{23} -109.543 q^{25} -134.670 q^{27} +116.052 q^{29} +49.8827 q^{31} -130.798 q^{33} +143.099 q^{35} +98.3236 q^{37} -385.783 q^{39} -277.944 q^{41} +344.162 q^{43} -7.81038 q^{45} +626.172 q^{47} +981.785 q^{49} +145.329 q^{53} +95.5140 q^{55} -389.178 q^{57} -454.009 q^{59} +253.423 q^{61} -72.3069 q^{63} +281.715 q^{65} -874.562 q^{67} -691.968 q^{69} +83.0135 q^{71} +390.139 q^{73} -589.770 q^{75} +884.249 q^{77} -657.385 q^{79} -778.691 q^{81} -163.049 q^{83} +624.815 q^{87} +418.761 q^{89} +2608.06 q^{91} +268.564 q^{93} +284.194 q^{95} -230.000 q^{97} -48.2624 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 98 q^{9}+O(q^{10}) \) 14 * q + 98 * q^9	\( 14 q + 98 q^{9} + 124 q^{13} + 60 q^{15} + 60 q^{19} + 148 q^{21} + 350 q^{25} + 116 q^{33} + 236 q^{35} + 436 q^{43} + 448 q^{47} + 1138 q^{49} + 1192 q^{53} - 1460 q^{55} - 692 q^{59} - 64 q^{67} + 2076 q^{69} + 1500 q^{77} - 166 q^{81} + 820 q^{83} - 1084 q^{87} - 476 q^{89} + 4372 q^{93}+O(q^{100}) \) 14 * q + 98 * q^9 + 124 * q^13 + 60 * q^15 + 60 * q^19 + 148 * q^21 + 350 * q^25 + 116 * q^33 + 236 * q^35 + 436 * q^43 + 448 * q^47 + 1138 * q^49 + 1192 * q^53 - 1460 * q^55 - 692 * q^59 - 64 * q^67 + 2076 * q^69 + 1500 * q^77 - 166 * q^81 + 820 * q^83 - 1084 * q^87 - 476 * q^89 + 4372 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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