LMFDB - Embedded newform 2312.4.a.n.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:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + \cdots - 176969301147 $ x^18 - 294x^16 - 14x^15 + 34371x^14 + 2670x^13 - 2054705x^12 - 160284x^11 + 67981059x^10 + 2824200x^9 - 1279285428x^8 + 12108606x^7 + 13538185208x^6 - 853720902x^5 - 75816227907x^4 + 7747530606x^3 + 195817930515x^2 - 12873873186x - 176969301147
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.18564$ 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)

$f(q)$	$=$	$q+1.18564 q^{3} -19.4950 q^{5} -3.40994 q^{7} -25.5943 q^{9} +O(q^{10})$	$q+1.18564 q^{3} -19.4950 q^{5} -3.40994 q^{7} -25.5943 q^{9} -18.9542 q^{11} +37.8880 q^{13} -23.1141 q^{15} +39.8848 q^{19} -4.04295 q^{21} +137.453 q^{23} +255.057 q^{25} -62.3578 q^{27} -16.9187 q^{29} -228.196 q^{31} -22.4728 q^{33} +66.4769 q^{35} +142.447 q^{37} +44.9215 q^{39} +189.952 q^{41} +312.727 q^{43} +498.961 q^{45} -110.416 q^{47} -331.372 q^{49} +43.6534 q^{53} +369.513 q^{55} +47.2889 q^{57} +776.906 q^{59} -40.0560 q^{61} +87.2748 q^{63} -738.629 q^{65} +929.861 q^{67} +162.970 q^{69} +916.206 q^{71} -1158.21 q^{73} +302.405 q^{75} +64.6327 q^{77} -737.298 q^{79} +617.111 q^{81} -1244.81 q^{83} -20.0595 q^{87} +724.256 q^{89} -129.196 q^{91} -270.558 q^{93} -777.556 q^{95} -1308.82 q^{97} +485.119 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) $ 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9	$ 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) $ 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9 - 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 - 153 * q^23 + 306 * q^25 - 768 * q^27 - 51 * q^29 - 303 * q^31 + 525 * q^33 - 255 * q^35 - 717 * q^37 + 216 * q^39 + 393 * q^41 - 390 * q^43 - 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 - 810 * q^57 - 204 * q^59 - 534 * q^61 - 2556 * q^63 + 2127 * q^65 - 405 * q^67 + 2547 * q^69 + 426 * q^71 - 1149 * q^73 - 2226 * q^75 - 357 * q^77 - 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 - 6090 * q^91 + 606 * q^93 - 2109 * q^95 - 2349 * q^97 - 1428 * 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
$3$	1.18564	0.228176	0.114088	−	0.993471i	$-0.463605\pi$
$3$	1.18564	0.228176	0.114088	+	0.993471i	$0.463605\pi$
$4$	0	0
$5$	−19.4950	−1.74369	−0.871845	−	0.489782i	$-0.837076\pi$
$5$	−19.4950	−1.74369	−0.871845	+	0.489782i	$0.837076\pi$
$6$	0	0
$7$	−3.40994	−0.184119	−0.0920597	−	0.995753i	$-0.529345\pi$
$7$	−3.40994	−0.184119	−0.0920597	+	0.995753i	$0.529345\pi$
$8$	0	0
$9$	−25.5943	−0.947936
$10$	0	0
$11$	−18.9542	−0.519537	−0.259769	−	0.965671i	$-0.583646\pi$
$11$	−18.9542	−0.519537	−0.259769	+	0.965671i	$0.583646\pi$
$12$	0	0
$13$	37.8880	0.808326	0.404163	−	0.914687i	$-0.367563\pi$
$13$	37.8880	0.808326	0.404163	+	0.914687i	$0.367563\pi$
$14$	0	0
$15$	−23.1141	−0.397869
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	39.8848	0.481589	0.240795	−	0.970576i	$-0.422592\pi$
$19$	39.8848	0.481589	0.240795	+	0.970576i	$0.422592\pi$
$20$	0	0
$21$	−4.04295	−0.0420117
$22$	0	0
$23$	137.453	1.24613	0.623065	−	0.782170i	$-0.285887\pi$
$23$	137.453	1.24613	0.623065	+	0.782170i	$0.285887\pi$
$24$	0	0
$25$	255.057	2.04046
$26$	0	0
$27$	−62.3578	−0.444473
$28$	0	0
$29$	−16.9187	−0.108336	−0.0541678	−	0.998532i	$-0.517251\pi$
$29$	−16.9187	−0.108336	−0.0541678	+	0.998532i	$0.517251\pi$
$30$	0	0
$31$	−228.196	−1.32210	−0.661052	−	0.750340i	$-0.729890\pi$
$31$	−228.196	−1.32210	−0.661052	+	0.750340i	$0.729890\pi$
$32$	0	0
$33$	−22.4728	−0.118546
$34$	0	0
$35$	66.4769	0.321047
$36$	0	0
$37$	142.447	0.632922	0.316461	−	0.948605i	$-0.397505\pi$
$37$	142.447	0.632922	0.316461	+	0.948605i	$0.397505\pi$
$38$	0	0
$39$	44.9215	0.184441
$40$	0	0
$41$	189.952	0.723550	0.361775	−	0.932265i	$-0.382171\pi$
$41$	189.952	0.723550	0.361775	+	0.932265i	$0.382171\pi$
$42$	0	0
$43$	312.727	1.10908	0.554539	−	0.832158i	$-0.312895\pi$
$43$	312.727	1.10908	0.554539	+	0.832158i	$0.312895\pi$
$44$	0	0
$45$	498.961	1.65291
$46$	0	0
$47$	−110.416	−0.342677	−0.171339	−	0.985212i	$-0.554809\pi$
$47$	−110.416	−0.342677	−0.171339	+	0.985212i	$0.554809\pi$
$48$	0	0
$49$	−331.372	−0.966100
$50$	0	0
$51$	0	0
$52$	0	0
$53$	43.6534	0.113137	0.0565685	−	0.998399i	$-0.481984\pi$
$53$	43.6534	0.113137	0.0565685	+	0.998399i	$0.481984\pi$
$54$	0	0
$55$	369.513	0.905912
$56$	0	0
$57$	47.2889	0.109887
$58$	0	0
$59$	776.906	1.71432	0.857158	−	0.515054i	$-0.172228\pi$
$59$	776.906	1.71432	0.857158	+	0.515054i	$0.172228\pi$
$60$	0	0
$61$	−40.0560	−0.0840761	−0.0420380	−	0.999116i	$-0.513385\pi$
$61$	−40.0560	−0.0840761	−0.0420380	+	0.999116i	$0.513385\pi$
$62$	0	0
$63$	87.2748	0.174533
$64$	0	0
$65$	−738.629	−1.40947
$66$	0	0
$67$	929.861	1.69553	0.847765	−	0.530372i	$-0.177948\pi$
$67$	929.861	1.69553	0.847765	+	0.530372i	$0.177948\pi$
$68$	0	0
$69$	162.970	0.284337
$70$	0	0
$71$	916.206	1.53146	0.765730	−	0.643163i	$-0.222378\pi$
$71$	916.206	1.53146	0.765730	+	0.643163i	$0.222378\pi$
$72$	0	0
$73$	−1158.21	−1.85696	−0.928478	−	0.371388i	$-0.878882\pi$
$73$	−1158.21	−1.85696	−0.928478	+	0.371388i	$0.878882\pi$
$74$	0	0
$75$	302.405	0.465583
$76$	0	0
$77$	64.6327	0.0956569
$78$	0	0
$79$	−737.298	−1.05003	−0.525016	−	0.851093i	$-0.675941\pi$
$79$	−737.298	−1.05003	−0.525016	+	0.851093i	$0.675941\pi$
$80$	0	0
$81$	617.111	0.846517
$82$	0	0
$83$	−1244.81	−1.64621	−0.823105	−	0.567889i	$-0.807760\pi$
$83$	−1244.81	−1.64621	−0.823105	+	0.567889i	$0.807760\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−20.0595	−0.0247196
$88$	0	0
$89$	724.256	0.862596	0.431298	−	0.902210i	$-0.358056\pi$
$89$	724.256	0.862596	0.431298	+	0.902210i	$0.358056\pi$
$90$	0	0
$91$	−129.196	−0.148829
$92$	0	0
$93$	−270.558	−0.301673
$94$	0	0
$95$	−777.556	−0.839742
$96$	0	0
$97$	−1308.82	−1.37001	−0.685004	−	0.728539i	$-0.740200\pi$
$97$	−1308.82	−1.37001	−0.685004	+	0.728539i	$0.740200\pi$
$98$	0	0
$99$	485.119	0.492488
$100$	0	0
$101$	−725.363	−0.714617	−0.357309	−	0.933986i	$-0.616306\pi$
$101$	−725.363	−0.714617	−0.357309	+	0.933986i	$0.616306\pi$
$102$	0	0
$103$	619.726	0.592849	0.296424	−	0.955056i	$-0.404206\pi$
$103$	619.726	0.592849	0.296424	+	0.955056i	$0.404206\pi$
$104$	0	0
$105$	78.8176	0.0732553
$106$	0	0
$107$	−349.710	−0.315960	−0.157980	−	0.987442i	$-0.550498\pi$
$107$	−349.710	−0.315960	−0.157980	+	0.987442i	$0.550498\pi$
$108$	0	0
$109$	642.941	0.564978	0.282489	−	0.959271i	$-0.408840\pi$
$109$	642.941	0.564978	0.282489	+	0.959271i	$0.408840\pi$
$110$	0	0
$111$	168.891	0.144418
$112$	0	0
$113$	1684.64	1.40246	0.701229	−	0.712936i	$-0.252635\pi$
$113$	1684.64	1.40246	0.701229	+	0.712936i	$0.252635\pi$
$114$	0	0
$115$	−2679.66	−2.17287
$116$	0	0
$117$	−969.716	−0.766241
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−971.738	−0.730081
$122$	0	0
$123$	225.215	0.165097
$124$	0	0
$125$	−2535.47	−1.81423
$126$	0	0
$127$	1298.12	0.907001	0.453501	−	0.891256i	$-0.350175\pi$
$127$	1298.12	0.907001	0.453501	+	0.891256i	$0.350175\pi$
$128$	0	0
$129$	370.781	0.253065
$130$	0	0
$131$	324.996	0.216756	0.108378	−	0.994110i	$-0.465434\pi$
$131$	324.996	0.216756	0.108378	+	0.994110i	$0.465434\pi$
$132$	0	0
$133$	−136.005	−0.0886699
$134$	0	0
$135$	1215.67	0.775023
$136$	0	0
$137$	−1164.64	−0.726291	−0.363145	−	0.931733i	$-0.618297\pi$
$137$	−1164.64	−0.726291	−0.363145	+	0.931733i	$0.618297\pi$
$138$	0	0
$139$	−3184.95	−1.94348	−0.971739	−	0.236057i	$-0.924145\pi$
$139$	−3184.95	−1.94348	−0.971739	+	0.236057i	$0.924145\pi$
$140$	0	0
$141$	−130.913	−0.0781908
$142$	0	0
$143$	−718.137	−0.419956
$144$	0	0
$145$	329.832	0.188904
$146$	0	0
$147$	−392.888	−0.220441
$148$	0	0
$149$	281.605	0.154832	0.0774160	−	0.996999i	$-0.475333\pi$
$149$	281.605	0.154832	0.0774160	+	0.996999i	$0.475333\pi$
$150$	0	0
$151$	2728.61	1.47054	0.735269	−	0.677775i	$-0.237056\pi$
$151$	2728.61	1.47054	0.735269	+	0.677775i	$0.237056\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4448.69	2.30534
$156$	0	0
$157$	−3443.03	−1.75021	−0.875107	−	0.483929i	$-0.839209\pi$
$157$	−3443.03	−1.75021	−0.875107	+	0.483929i	$0.839209\pi$
$158$	0	0
$159$	51.7572	0.0258152
$160$	0	0
$161$	−468.707	−0.229437
$162$	0	0
$163$	−318.662	−0.153126	−0.0765631	−	0.997065i	$-0.524395\pi$
$163$	−318.662	−0.153126	−0.0765631	+	0.997065i	$0.524395\pi$
$164$	0	0
$165$	438.109	0.206708
$166$	0	0
$167$	2346.36	1.08723	0.543613	−	0.839336i	$-0.317056\pi$
$167$	2346.36	1.08723	0.543613	+	0.839336i	$0.317056\pi$
$168$	0	0
$169$	−761.499	−0.346608
$170$	0	0
$171$	−1020.82	−0.456516
$172$	0	0
$173$	−1860.96	−0.817838	−0.408919	−	0.912571i	$-0.634094\pi$
$173$	−1860.96	−0.817838	−0.408919	+	0.912571i	$0.634094\pi$
$174$	0	0
$175$	−869.728	−0.375687
$176$	0	0
$177$	921.130	0.391166
$178$	0	0
$179$	−1044.30	−0.436061	−0.218031	−	0.975942i	$-0.569963\pi$
$179$	−1044.30	−0.436061	−0.218031	+	0.975942i	$0.569963\pi$
$180$	0	0
$181$	−2078.75	−0.853661	−0.426830	−	0.904332i	$-0.640370\pi$
$181$	−2078.75	−0.853661	−0.426830	+	0.904332i	$0.640370\pi$
$182$	0	0
$183$	−47.4919	−0.0191842
$184$	0	0
$185$	−2777.01	−1.10362
$186$	0	0
$187$	0	0
$188$	0	0
$189$	212.636	0.0818360
$190$	0	0
$191$	−3197.87	−1.21147	−0.605733	−	0.795668i	$-0.707120\pi$
$191$	−3197.87	−1.21147	−0.605733	+	0.795668i	$0.707120\pi$
$192$	0	0
$193$	−2366.72	−0.882696	−0.441348	−	0.897336i	$-0.645500\pi$
$193$	−2366.72	−0.882696	−0.441348	+	0.897336i	$0.645500\pi$
$194$	0	0
$195$	−875.747	−0.321608
$196$	0	0
$197$	3182.09	1.15084	0.575418	−	0.817860i	$-0.304839\pi$
$197$	3182.09	1.15084	0.575418	+	0.817860i	$0.304839\pi$
$198$	0	0
$199$	−1715.27	−0.611016	−0.305508	−	0.952189i	$-0.598826\pi$
$199$	−1715.27	−0.611016	−0.305508	+	0.952189i	$0.598826\pi$
$200$	0	0
$201$	1102.48	0.386880
$202$	0	0
$203$	57.6919	0.0199467
$204$	0	0
$205$	−3703.13	−1.26165
$206$	0	0
$207$	−3518.02	−1.18125
$208$	0	0
$209$	−755.984	−0.250204
$210$	0	0
$211$	−808.797	−0.263886	−0.131943	−	0.991257i	$-0.542122\pi$
$211$	−808.797	−0.263886	−0.131943	+	0.991257i	$0.542122\pi$
$212$	0	0
$213$	1086.29	0.349443
$214$	0	0
$215$	−6096.62	−1.93389
$216$	0	0
$217$	778.134	0.243425
$218$	0	0
$219$	−1373.21	−0.423713
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1613.96	0.484657	0.242328	−	0.970194i	$-0.422089\pi$
$223$	1613.96	0.484657	0.242328	+	0.970194i	$0.422089\pi$
$224$	0	0
$225$	−6527.99	−1.93422
$226$	0	0
$227$	−2817.47	−0.823799	−0.411899	−	0.911229i	$-0.635134\pi$
$227$	−2817.47	−0.823799	−0.411899	+	0.911229i	$0.635134\pi$
$228$	0	0
$229$	−3234.86	−0.933473	−0.466737	−	0.884396i	$-0.654570\pi$
$229$	−3234.86	−0.933473	−0.466737	+	0.884396i	$0.654570\pi$
$230$	0	0
$231$	76.6310	0.0218266
$232$	0	0
$233$	3442.67	0.967970	0.483985	−	0.875076i	$-0.339189\pi$
$233$	3442.67	0.967970	0.483985	+	0.875076i	$0.339189\pi$
$234$	0	0
$235$	2152.57	0.597523
$236$	0	0
$237$	−874.169	−0.239592
$238$	0	0
$239$	−3210.75	−0.868980	−0.434490	−	0.900677i	$-0.643071\pi$
$239$	−3210.75	−0.868980	−0.434490	+	0.900677i	$0.643071\pi$
$240$	0	0
$241$	−6655.71	−1.77897	−0.889486	−	0.456962i	$-0.848937\pi$
$241$	−6655.71	−1.77897	−0.889486	+	0.456962i	$0.848937\pi$
$242$	0	0
$243$	2415.33	0.637628
$244$	0	0
$245$	6460.12	1.68458
$246$	0	0
$247$	1511.15	0.389281
$248$	0	0
$249$	−1475.89	−0.375626
$250$	0	0
$251$	−2302.88	−0.579108	−0.289554	−	0.957162i	$-0.593507\pi$
$251$	−2302.88	−0.579108	−0.289554	+	0.957162i	$0.593507\pi$
$252$	0	0
$253$	−2605.32	−0.647411
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5818.48	1.41225	0.706123	−	0.708089i	$-0.250443\pi$
$257$	5818.48	1.41225	0.706123	+	0.708089i	$0.250443\pi$
$258$	0	0
$259$	−485.735	−0.116533
$260$	0	0
$261$	433.023	0.102695
$262$	0	0
$263$	4795.55	1.12436	0.562180	−	0.827015i	$-0.309963\pi$
$263$	4795.55	1.12436	0.562180	+	0.827015i	$0.309963\pi$
$264$	0	0
$265$	−851.026	−0.197276
$266$	0	0
$267$	858.706	0.196824
$268$	0	0
$269$	−1227.66	−0.278259	−0.139129	−	0.990274i	$-0.544430\pi$
$269$	−1227.66	−0.278259	−0.139129	+	0.990274i	$0.544430\pi$
$270$	0	0
$271$	1121.30	0.251345	0.125672	−	0.992072i	$-0.459891\pi$
$271$	1121.30	0.251345	0.125672	+	0.992072i	$0.459891\pi$
$272$	0	0
$273$	−153.179	−0.0339591
$274$	0	0
$275$	−4834.40	−1.06009
$276$	0	0
$277$	−39.0024	−0.00846002	−0.00423001	−	0.999991i	$-0.501346\pi$
$277$	−39.0024	−0.00846002	−0.00423001	+	0.999991i	$0.501346\pi$
$278$	0	0
$279$	5840.51	1.25327
$280$	0	0
$281$	7853.13	1.66718	0.833592	−	0.552381i	$-0.186281\pi$
$281$	7853.13	1.66718	0.833592	+	0.552381i	$0.186281\pi$
$282$	0	0
$283$	7335.68	1.54085	0.770425	−	0.637530i	$-0.220044\pi$
$283$	7335.68	1.54085	0.770425	+	0.637530i	$0.220044\pi$
$284$	0	0
$285$	−921.900	−0.191609
$286$	0	0
$287$	−647.726	−0.133220
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−1551.79	−0.312603
$292$	0	0
$293$	1308.91	0.260981	0.130491	−	0.991450i	$-0.458345\pi$
$293$	1308.91	0.260981	0.130491	+	0.991450i	$0.458345\pi$
$294$	0	0
$295$	−15145.8	−2.98923
$296$	0	0
$297$	1181.94	0.230920
$298$	0	0
$299$	5207.83	1.00728
$300$	0	0
$301$	−1066.38	−0.204203
$302$	0	0
$303$	−860.018	−0.163059
$304$	0	0
$305$	780.893	0.146603
$306$	0	0
$307$	3534.22	0.657031	0.328516	−	0.944499i	$-0.393452\pi$
$307$	3534.22	0.657031	0.328516	+	0.944499i	$0.393452\pi$
$308$	0	0
$309$	734.771	0.135274
$310$	0	0
$311$	−9833.81	−1.79300	−0.896502	−	0.443039i	$-0.853900\pi$
$311$	−9833.81	−1.79300	−0.896502	+	0.443039i	$0.853900\pi$
$312$	0	0
$313$	6801.24	1.22821	0.614103	−	0.789226i	$-0.289518\pi$
$313$	6801.24	1.22821	0.614103	+	0.789226i	$0.289518\pi$
$314$	0	0
$315$	−1701.43	−0.304332
$316$	0	0
$317$	−2247.46	−0.398202	−0.199101	−	0.979979i	$-0.563802\pi$
$317$	−2247.46	−0.398202	−0.199101	+	0.979979i	$0.563802\pi$
$318$	0	0
$319$	320.681	0.0562844
$320$	0	0
$321$	−414.629	−0.0720945
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9663.60	1.64935
$326$	0	0
$327$	762.296	0.128915
$328$	0	0
$329$	376.512	0.0630935
$330$	0	0
$331$	−7207.48	−1.19685	−0.598427	−	0.801177i	$-0.704207\pi$
$331$	−7207.48	−1.19685	−0.598427	+	0.801177i	$0.704207\pi$
$332$	0	0
$333$	−3645.82	−0.599970
$334$	0	0
$335$	−18127.7	−2.95648
$336$	0	0
$337$	−1124.92	−0.181835	−0.0909176	−	0.995858i	$-0.528980\pi$
$337$	−1124.92	−0.181835	−0.0909176	+	0.995858i	$0.528980\pi$
$338$	0	0
$339$	1997.38	0.320008
$340$	0	0
$341$	4325.28	0.686882
$342$	0	0
$343$	2299.57	0.361997
$344$	0	0
$345$	−3177.11	−0.495796
$346$	0	0
$347$	−9053.64	−1.40065	−0.700324	−	0.713825i	$-0.746961\pi$
$347$	−9053.64	−1.40065	−0.700324	+	0.713825i	$0.746961\pi$
$348$	0	0
$349$	−7232.88	−1.10936	−0.554681	−	0.832063i	$-0.687160\pi$
$349$	−7232.88	−1.10936	−0.554681	+	0.832063i	$0.687160\pi$
$350$	0	0
$351$	−2362.61	−0.359279
$352$	0	0
$353$	−759.240	−0.114477	−0.0572383	−	0.998361i	$-0.518229\pi$
$353$	−759.240	−0.114477	−0.0572383	+	0.998361i	$0.518229\pi$
$354$	0	0
$355$	−17861.5	−2.67039
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7036.39	1.03445	0.517223	−	0.855851i	$-0.326966\pi$
$359$	7036.39	1.03445	0.517223	+	0.855851i	$0.326966\pi$
$360$	0	0
$361$	−5268.20	−0.768072
$362$	0	0
$363$	−1152.13	−0.166587
$364$	0	0
$365$	22579.3	3.23796
$366$	0	0
$367$	−12694.9	−1.80563	−0.902817	−	0.430026i	$-0.858505\pi$
$367$	−12694.9	−1.80563	−0.902817	+	0.430026i	$0.858505\pi$
$368$	0	0
$369$	−4861.69	−0.685879
$370$	0	0
$371$	−148.856	−0.0208307
$372$	0	0
$373$	11107.7	1.54192	0.770958	−	0.636886i	$-0.219778\pi$
$373$	11107.7	1.54192	0.770958	+	0.636886i	$0.219778\pi$
$374$	0	0
$375$	−3006.15	−0.413964
$376$	0	0
$377$	−641.017	−0.0875705
$378$	0	0
$379$	−8334.59	−1.12960	−0.564801	−	0.825227i	$-0.691047\pi$
$379$	−8334.59	−1.12960	−0.564801	+	0.825227i	$0.691047\pi$
$380$	0	0
$381$	1539.10	0.206956
$382$	0	0
$383$	11168.7	1.49007	0.745033	−	0.667028i	$-0.232434\pi$
$383$	11168.7	1.49007	0.745033	+	0.667028i	$0.232434\pi$
$384$	0	0
$385$	−1260.02	−0.166796
$386$	0	0
$387$	−8004.01	−1.05133
$388$	0	0
$389$	−7878.46	−1.02687	−0.513437	−	0.858127i	$-0.671628\pi$
$389$	−7878.46	−1.02687	−0.513437	+	0.858127i	$0.671628\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	385.328	0.0494586
$394$	0	0
$395$	14373.7	1.83093
$396$	0	0
$397$	5925.60	0.749112	0.374556	−	0.927204i	$-0.377795\pi$
$397$	5925.60	0.749112	0.374556	+	0.927204i	$0.377795\pi$
$398$	0	0
$399$	−161.252	−0.0202324
$400$	0	0
$401$	6827.12	0.850199	0.425100	−	0.905147i	$-0.360239\pi$
$401$	6827.12	0.850199	0.425100	+	0.905147i	$0.360239\pi$
$402$	0	0
$403$	−8645.89	−1.06869
$404$	0	0
$405$	−12030.6	−1.47606
$406$	0	0
$407$	−2699.97	−0.328827
$408$	0	0
$409$	−1632.17	−0.197324	−0.0986619	−	0.995121i	$-0.531456\pi$
$409$	−1632.17	−0.197324	−0.0986619	+	0.995121i	$0.531456\pi$
$410$	0	0
$411$	−1380.84	−0.165722
$412$	0	0
$413$	−2649.20	−0.315639
$414$	0	0
$415$	24267.6	2.87048
$416$	0	0
$417$	−3776.19	−0.443456
$418$	0	0
$419$	7101.70	0.828021	0.414011	−	0.910272i	$-0.364128\pi$
$419$	7101.70	0.828021	0.414011	+	0.910272i	$0.364128\pi$
$420$	0	0
$421$	−4682.52	−0.542072	−0.271036	−	0.962569i	$-0.587366\pi$
$421$	−4682.52	−0.542072	−0.271036	+	0.962569i	$0.587366\pi$
$422$	0	0
$423$	2826.02	0.324836
$424$	0	0
$425$	0	0
$426$	0	0
$427$	136.588	0.0154800
$428$	0	0
$429$	−851.451	−0.0958239
$430$	0	0
$431$	−17413.2	−1.94609	−0.973043	−	0.230623i	$-0.925924\pi$
$431$	−17413.2	−1.94609	−0.973043	+	0.230623i	$0.925924\pi$
$432$	0	0
$433$	17056.5	1.89304	0.946518	−	0.322651i	$-0.104574\pi$
$433$	17056.5	1.89304	0.946518	+	0.322651i	$0.104574\pi$
$434$	0	0
$435$	391.061	0.0431033
$436$	0	0
$437$	5482.30	0.600123
$438$	0	0
$439$	−1831.05	−0.199069	−0.0995346	−	0.995034i	$-0.531735\pi$
$439$	−1831.05	−0.199069	−0.0995346	+	0.995034i	$0.531735\pi$
$440$	0	0
$441$	8481.23	0.915801
$442$	0	0
$443$	11049.5	1.18506	0.592528	−	0.805550i	$-0.298130\pi$
$443$	11049.5	1.18506	0.592528	+	0.805550i	$0.298130\pi$
$444$	0	0
$445$	−14119.4	−1.50410
$446$	0	0
$447$	333.882	0.0353290
$448$	0	0
$449$	−4496.88	−0.472652	−0.236326	−	0.971674i	$-0.575943\pi$
$449$	−4496.88	−0.472652	−0.236326	+	0.971674i	$0.575943\pi$
$450$	0	0
$451$	−3600.40	−0.375911
$452$	0	0
$453$	3235.15	0.335542
$454$	0	0
$455$	2518.68	0.259511
$456$	0	0
$457$	−8200.19	−0.839363	−0.419682	−	0.907671i	$-0.637858\pi$
$457$	−8200.19	−0.839363	−0.419682	+	0.907671i	$0.637858\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16052.2	1.62175	0.810873	−	0.585223i	$-0.198993\pi$
$461$	16052.2	1.62175	0.810873	+	0.585223i	$0.198993\pi$
$462$	0	0
$463$	8912.52	0.894600	0.447300	−	0.894384i	$-0.352386\pi$
$463$	8912.52	0.894600	0.447300	+	0.894384i	$0.352386\pi$
$464$	0	0
$465$	5274.54	0.526024
$466$	0	0
$467$	−8069.38	−0.799585	−0.399793	−	0.916606i	$-0.630918\pi$
$467$	−8069.38	−0.799585	−0.399793	+	0.916606i	$0.630918\pi$
$468$	0	0
$469$	−3170.77	−0.312180
$470$	0	0
$471$	−4082.19	−0.399357
$472$	0	0
$473$	−5927.49	−0.576207
$474$	0	0
$475$	10172.9	0.982661
$476$	0	0
$477$	−1117.28	−0.107247
$478$	0	0
$479$	7851.19	0.748914	0.374457	−	0.927244i	$-0.377829\pi$
$479$	7851.19	0.748914	0.374457	+	0.927244i	$0.377829\pi$
$480$	0	0
$481$	5397.03	0.511608
$482$	0	0
$483$	−555.718	−0.0523520
$484$	0	0
$485$	25515.6	2.38887
$486$	0	0
$487$	4546.30	0.423024	0.211512	−	0.977375i	$-0.432161\pi$
$487$	4546.30	0.423024	0.211512	+	0.977375i	$0.432161\pi$
$488$	0	0
$489$	−377.818	−0.0349397
$490$	0	0
$491$	−14586.2	−1.34067	−0.670333	−	0.742060i	$-0.733849\pi$
$491$	−14586.2	−1.34067	−0.670333	+	0.742060i	$0.733849\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−9457.42	−0.858746
$496$	0	0
$497$	−3124.20	−0.281971
$498$	0	0
$499$	−6740.92	−0.604739	−0.302370	−	0.953191i	$-0.597778\pi$
$499$	−6740.92	−0.604739	−0.302370	+	0.953191i	$0.597778\pi$
$500$	0	0
$501$	2781.93	0.248079
$502$	0	0
$503$	4238.78	0.375741	0.187871	−	0.982194i	$-0.439841\pi$
$503$	4238.78	0.375741	0.187871	+	0.982194i	$0.439841\pi$
$504$	0	0
$505$	14141.0	1.24607
$506$	0	0
$507$	−902.862	−0.0790878
$508$	0	0
$509$	−8324.26	−0.724885	−0.362442	−	0.932006i	$-0.618057\pi$
$509$	−8324.26	−0.724885	−0.362442	+	0.932006i	$0.618057\pi$
$510$	0	0
$511$	3949.41	0.341902
$512$	0	0
$513$	−2487.13	−0.214053
$514$	0	0
$515$	−12081.6	−1.03374
$516$	0	0
$517$	2092.85	0.178034
$518$	0	0
$519$	−2206.42	−0.186611
$520$	0	0
$521$	−16965.0	−1.42659	−0.713294	−	0.700865i	$-0.752797\pi$
$521$	−16965.0	−1.42659	−0.713294	+	0.700865i	$0.752797\pi$
$522$	0	0
$523$	−12324.2	−1.03040	−0.515199	−	0.857071i	$-0.672282\pi$
$523$	−12324.2	−1.03040	−0.515199	+	0.857071i	$0.672282\pi$
$524$	0	0
$525$	−1031.18	−0.0857229
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6726.42	0.552842
$530$	0	0
$531$	−19884.3	−1.62506
$532$	0	0
$533$	7196.91	0.584865
$534$	0	0
$535$	6817.61	0.550936
$536$	0	0
$537$	−1238.17	−0.0994989
$538$	0	0
$539$	6280.90	0.501925
$540$	0	0
$541$	−5678.55	−0.451275	−0.225637	−	0.974211i	$-0.572446\pi$
$541$	−5678.55	−0.451275	−0.225637	+	0.974211i	$0.572446\pi$
$542$	0	0
$543$	−2464.65	−0.194785
$544$	0	0
$545$	−12534.2	−0.985146
$546$	0	0
$547$	−4762.92	−0.372300	−0.186150	−	0.982521i	$-0.559601\pi$
$547$	−4762.92	−0.372300	−0.186150	+	0.982521i	$0.559601\pi$
$548$	0	0
$549$	1025.20	0.0796987
$550$	0	0
$551$	−674.800	−0.0521732
$552$	0	0
$553$	2514.14	0.193331
$554$	0	0
$555$	−3292.53	−0.251820
$556$	0	0
$557$	17850.0	1.35786	0.678932	−	0.734201i	$-0.262443\pi$
$557$	17850.0	1.35786	0.678932	+	0.734201i	$0.262443\pi$
$558$	0	0
$559$	11848.6	0.896497
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9782.51	0.732298	0.366149	−	0.930556i	$-0.380676\pi$
$563$	9782.51	0.732298	0.366149	+	0.930556i	$0.380676\pi$
$564$	0	0
$565$	−32842.2	−2.44545
$566$	0	0
$567$	−2104.31	−0.155860
$568$	0	0
$569$	−4728.44	−0.348377	−0.174189	−	0.984712i	$-0.555730\pi$
$569$	−4728.44	−0.348377	−0.174189	+	0.984712i	$0.555730\pi$
$570$	0	0
$571$	6230.73	0.456652	0.228326	−	0.973585i	$-0.426675\pi$
$571$	6230.73	0.456652	0.228326	+	0.973585i	$0.426675\pi$
$572$	0	0
$573$	−3791.52	−0.276428
$574$	0	0
$575$	35058.4	2.54267
$576$	0	0
$577$	−9996.35	−0.721236	−0.360618	−	0.932714i	$-0.617434\pi$
$577$	−9996.35	−0.721236	−0.360618	+	0.932714i	$0.617434\pi$
$578$	0	0
$579$	−2806.08	−0.201410
$580$	0	0
$581$	4244.72	0.303099
$582$	0	0
$583$	−827.416	−0.0587789
$584$	0	0
$585$	18904.7	1.33609
$586$	0	0
$587$	−13160.3	−0.925352	−0.462676	−	0.886527i	$-0.653111\pi$
$587$	−13160.3	−0.925352	−0.462676	+	0.886527i	$0.653111\pi$
$588$	0	0
$589$	−9101.55	−0.636711
$590$	0	0
$591$	3772.81	0.262593
$592$	0	0
$593$	−7253.62	−0.502311	−0.251156	−	0.967947i	$-0.580811\pi$
$593$	−7253.62	−0.502311	−0.251156	+	0.967947i	$0.580811\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2033.69	−0.139419
$598$	0	0
$599$	10524.3	0.717880	0.358940	−	0.933361i	$-0.383138\pi$
$599$	10524.3	0.717880	0.358940	+	0.933361i	$0.383138\pi$
$600$	0	0
$601$	6043.58	0.410188	0.205094	−	0.978742i	$-0.434250\pi$
$601$	6043.58	0.410188	0.205094	+	0.978742i	$0.434250\pi$
$602$	0	0
$603$	−23799.1	−1.60725
$604$	0	0
$605$	18944.1	1.27304
$606$	0	0
$607$	−5894.86	−0.394176	−0.197088	−	0.980386i	$-0.563149\pi$
$607$	−5894.86	−0.394176	−0.197088	+	0.980386i	$0.563149\pi$
$608$	0	0
$609$	68.4017	0.00455136
$610$	0	0
$611$	−4183.44	−0.276995
$612$	0	0
$613$	−12052.4	−0.794117	−0.397058	−	0.917793i	$-0.629969\pi$
$613$	−12052.4	−0.794117	−0.397058	+	0.917793i	$0.629969\pi$
$614$	0	0
$615$	−4390.57	−0.287878
$616$	0	0
$617$	−4296.88	−0.280366	−0.140183	−	0.990126i	$-0.544769\pi$
$617$	−4296.88	−0.280366	−0.140183	+	0.990126i	$0.544769\pi$
$618$	0	0
$619$	−14018.0	−0.910225	−0.455113	−	0.890434i	$-0.650401\pi$
$619$	−14018.0	−0.910225	−0.455113	+	0.890434i	$0.650401\pi$
$620$	0	0
$621$	−8571.29	−0.553871
$622$	0	0
$623$	−2469.67	−0.158821
$624$	0	0
$625$	17546.9	1.12300
$626$	0	0
$627$	−896.324	−0.0570905
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−21427.0	−1.35181	−0.675907	−	0.736987i	$-0.736248\pi$
$631$	−21427.0	−1.35181	−0.675907	+	0.736987i	$0.736248\pi$
$632$	0	0
$633$	−958.941	−0.0602125
$634$	0	0
$635$	−25306.8	−1.58153
$636$	0	0
$637$	−12555.0	−0.780924
$638$	0	0
$639$	−23449.6	−1.45172
$640$	0	0
$641$	11840.0	0.729565	0.364783	−	0.931093i	$-0.381143\pi$
$641$	11840.0	0.729565	0.364783	+	0.931093i	$0.381143\pi$
$642$	0	0
$643$	−11773.3	−0.722076	−0.361038	−	0.932551i	$-0.617578\pi$
$643$	−11773.3	−0.722076	−0.361038	+	0.932551i	$0.617578\pi$
$644$	0	0
$645$	−7228.39	−0.441268
$646$	0	0
$647$	−24623.0	−1.49618	−0.748090	−	0.663597i	$-0.769029\pi$
$647$	−24623.0	−1.49618	−0.748090	+	0.663597i	$0.769029\pi$
$648$	0	0
$649$	−14725.6	−0.890651
$650$	0	0
$651$	922.586	0.0555438
$652$	0	0
$653$	−10159.0	−0.608810	−0.304405	−	0.952543i	$-0.598458\pi$
$653$	−10159.0	−0.608810	−0.304405	+	0.952543i	$0.598458\pi$
$654$	0	0
$655$	−6335.81	−0.377955
$656$	0	0
$657$	29643.4	1.76027
$658$	0	0
$659$	−33561.7	−1.98388	−0.991940	−	0.126708i	$-0.959559\pi$
$659$	−33561.7	−1.98388	−0.991940	+	0.126708i	$0.959559\pi$
$660$	0	0
$661$	20341.1	1.19694	0.598471	−	0.801145i	$-0.295775\pi$
$661$	20341.1	1.19694	0.598471	+	0.801145i	$0.295775\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2651.42	0.154613
$666$	0	0
$667$	−2325.54	−0.135000
$668$	0	0
$669$	1913.57	0.110587
$670$	0	0
$671$	759.229	0.0436806
$672$	0	0
$673$	17775.7	1.01813	0.509067	−	0.860727i	$-0.329990\pi$
$673$	17775.7	1.01813	0.509067	+	0.860727i	$0.329990\pi$
$674$	0	0
$675$	−15904.8	−0.906926
$676$	0	0
$677$	24013.0	1.36321	0.681606	−	0.731720i	$-0.261282\pi$
$677$	24013.0	1.36321	0.681606	+	0.731720i	$0.261282\pi$
$678$	0	0
$679$	4463.00	0.252245
$680$	0	0
$681$	−3340.51	−0.187971
$682$	0	0
$683$	−33490.3	−1.87624	−0.938118	−	0.346316i	$-0.887432\pi$
$683$	−33490.3	−1.87624	−0.938118	+	0.346316i	$0.887432\pi$
$684$	0	0
$685$	22704.7	1.26643
$686$	0	0
$687$	−3835.37	−0.212996
$688$	0	0
$689$	1653.94	0.0914516
$690$	0	0
$691$	−14322.4	−0.788496	−0.394248	−	0.919004i	$-0.628995\pi$
$691$	−14322.4	−0.788496	−0.394248	+	0.919004i	$0.628995\pi$
$692$	0	0
$693$	−1654.23	−0.0906765
$694$	0	0
$695$	62090.7	3.38882
$696$	0	0
$697$	0	0
$698$	0	0
$699$	4081.76	0.220868
$700$	0	0
$701$	3897.92	0.210018	0.105009	−	0.994471i	$-0.466513\pi$
$701$	3897.92	0.210018	0.105009	+	0.994471i	$0.466513\pi$
$702$	0	0
$703$	5681.46	0.304809
$704$	0	0
$705$	2552.16	0.136341
$706$	0	0
$707$	2473.44	0.131575
$708$	0	0
$709$	31719.1	1.68017	0.840083	−	0.542459i	$-0.182506\pi$
$709$	31719.1	1.68017	0.840083	+	0.542459i	$0.182506\pi$
$710$	0	0
$711$	18870.6	0.995362
$712$	0	0
$713$	−31366.3	−1.64751
$714$	0	0
$715$	14000.1	0.732273
$716$	0	0
$717$	−3806.79	−0.198281
$718$	0	0
$719$	25933.4	1.34514	0.672568	−	0.740036i	$-0.265191\pi$
$719$	25933.4	1.34514	0.672568	+	0.740036i	$0.265191\pi$
$720$	0	0
$721$	−2113.23	−0.109155
$722$	0	0
$723$	−7891.27	−0.405919
$724$	0	0
$725$	−4315.24	−0.221054
$726$	0	0
$727$	−3744.21	−0.191011	−0.0955055	−	0.995429i	$-0.530447\pi$
$727$	−3744.21	−0.191011	−0.0955055	+	0.995429i	$0.530447\pi$
$728$	0	0
$729$	−13798.3	−0.701026
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−33073.7	−1.66658	−0.833291	−	0.552834i	$-0.813546\pi$
$733$	−33073.7	−1.66658	−0.833291	+	0.552834i	$0.813546\pi$
$734$	0	0
$735$	7659.37	0.384381
$736$	0	0
$737$	−17624.8	−0.880891
$738$	0	0
$739$	−204.991	−0.0102039	−0.00510197	−	0.999987i	$-0.501624\pi$
$739$	−204.991	−0.0102039	−0.00510197	+	0.999987i	$0.501624\pi$
$740$	0	0
$741$	1791.68	0.0888247
$742$	0	0
$743$	−31.6576	−0.00156313	−0.000781564	−	1.00000i	$-0.500249\pi$
$743$	−31.6576	−0.00156313	−0.000781564		1.00000i	$0.500249\pi$
$744$	0	0
$745$	−5489.90	−0.269979
$746$	0	0
$747$	31859.9	1.56050
$748$	0	0
$749$	1192.49	0.0581743
$750$	0	0
$751$	11142.5	0.541407	0.270703	−	0.962663i	$-0.412744\pi$
$751$	11142.5	0.541407	0.270703	+	0.962663i	$0.412744\pi$
$752$	0	0
$753$	−2730.38	−0.132139
$754$	0	0
$755$	−53194.4	−2.56416
$756$	0	0
$757$	11700.5	0.561771	0.280886	−	0.959741i	$-0.409372\pi$
$757$	11700.5	0.561771	0.280886	+	0.959741i	$0.409372\pi$
$758$	0	0
$759$	−3088.97	−0.147724
$760$	0	0
$761$	14725.8	0.701460	0.350730	−	0.936477i	$-0.385933\pi$
$761$	14725.8	0.701460	0.350730	+	0.936477i	$0.385933\pi$
$762$	0	0
$763$	−2192.39	−0.104023
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29435.4	1.38573
$768$	0	0
$769$	−30720.8	−1.44060	−0.720300	−	0.693663i	$-0.755996\pi$
$769$	−30720.8	−1.44060	−0.720300	+	0.693663i	$0.755996\pi$
$770$	0	0
$771$	6898.62	0.322241
$772$	0	0
$773$	−16317.4	−0.759245	−0.379622	−	0.925142i	$-0.623946\pi$
$773$	−16317.4	−0.759245	−0.379622	+	0.925142i	$0.623946\pi$
$774$	0	0
$775$	−58203.0	−2.69769
$776$	0	0
$777$	−575.906	−0.0265901
$778$	0	0
$779$	7576.21	0.348454
$780$	0	0
$781$	−17366.0	−0.795650
$782$	0	0
$783$	1055.02	0.0481522
$784$	0	0
$785$	67122.0	3.05183
$786$	0	0
$787$	−4235.81	−0.191856	−0.0959278	−	0.995388i	$-0.530582\pi$
$787$	−4235.81	−0.191856	−0.0959278	+	0.995388i	$0.530582\pi$
$788$	0	0
$789$	5685.79	0.256552
$790$	0	0
$791$	−5744.52	−0.258220
$792$	0	0
$793$	−1517.64	−0.0679609
$794$	0	0
$795$	−1009.01	−0.0450137
$796$	0	0
$797$	−13490.4	−0.599566	−0.299783	−	0.954007i	$-0.596914\pi$
$797$	−13490.4	−0.599566	−0.299783	+	0.954007i	$0.596914\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−18536.8	−0.817685
$802$	0	0
$803$	21952.9	0.964758
$804$	0	0
$805$	9137.47	0.400067
$806$	0	0
$807$	−1455.56	−0.0634920
$808$	0	0
$809$	11737.7	0.510104	0.255052	−	0.966927i	$-0.417907\pi$
$809$	11737.7	0.510104	0.255052	+	0.966927i	$0.417907\pi$
$810$	0	0
$811$	−21782.5	−0.943143	−0.471571	−	0.881828i	$-0.656313\pi$
$811$	−21782.5	−0.943143	−0.471571	+	0.881828i	$0.656313\pi$
$812$	0	0
$813$	1329.46	0.0573509
$814$	0	0
$815$	6212.34	0.267004
$816$	0	0
$817$	12473.0	0.534120
$818$	0	0
$819$	3306.67	0.141080
$820$	0	0
$821$	−26891.4	−1.14314	−0.571568	−	0.820555i	$-0.693665\pi$
$821$	−26891.4	−1.14314	−0.571568	+	0.820555i	$0.693665\pi$
$822$	0	0
$823$	3300.25	0.139781	0.0698904	−	0.997555i	$-0.477735\pi$
$823$	3300.25	0.139781	0.0698904	+	0.997555i	$0.477735\pi$
$824$	0	0
$825$	−5731.85	−0.241888
$826$	0	0
$827$	−19211.6	−0.807804	−0.403902	−	0.914802i	$-0.632346\pi$
$827$	−19211.6	−0.807804	−0.403902	+	0.914802i	$0.632346\pi$
$828$	0	0
$829$	21443.2	0.898377	0.449189	−	0.893437i	$-0.351713\pi$
$829$	21443.2	0.898377	0.449189	+	0.893437i	$0.351713\pi$
$830$	0	0
$831$	−46.2427	−0.00193038
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−45742.4	−1.89578
$836$	0	0
$837$	14229.8	0.587639
$838$	0	0
$839$	15884.8	0.653640	0.326820	−	0.945087i	$-0.394023\pi$
$839$	15884.8	0.653640	0.326820	+	0.945087i	$0.394023\pi$
$840$	0	0
$841$	−24102.8	−0.988263
$842$	0	0
$843$	9310.98	0.380412
$844$	0	0
$845$	14845.5	0.604378
$846$	0	0
$847$	3313.57	0.134422
$848$	0	0
$849$	8697.46	0.351585
$850$	0	0
$851$	19579.8	0.788704
$852$	0	0
$853$	−6492.40	−0.260604	−0.130302	−	0.991474i	$-0.541595\pi$
$853$	−6492.40	−0.260604	−0.130302	+	0.991474i	$0.541595\pi$
$854$	0	0
$855$	19901.0	0.796022
$856$	0	0
$857$	−1583.10	−0.0631010	−0.0315505	−	0.999502i	$-0.510045\pi$
$857$	−1583.10	−0.0631010	−0.0315505	+	0.999502i	$0.510045\pi$
$858$	0	0
$859$	−13925.0	−0.553103	−0.276551	−	0.960999i	$-0.589192\pi$
$859$	−13925.0	−0.553103	−0.276551	+	0.960999i	$0.589192\pi$
$860$	0	0
$861$	−767.969	−0.0303976
$862$	0	0
$863$	−20235.8	−0.798188	−0.399094	−	0.916910i	$-0.630675\pi$
$863$	−20235.8	−0.798188	−0.399094	+	0.916910i	$0.630675\pi$
$864$	0	0
$865$	36279.5	1.42606
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13974.9	0.545530
$870$	0	0
$871$	35230.6	1.37054
$872$	0	0
$873$	33498.3	1.29868
$874$	0	0
$875$	8645.78	0.334035
$876$	0	0
$877$	10883.0	0.419033	0.209516	−	0.977805i	$-0.432811\pi$
$877$	10883.0	0.419033	0.209516	+	0.977805i	$0.432811\pi$
$878$	0	0
$879$	1551.90	0.0595498
$880$	0	0
$881$	−46079.8	−1.76217	−0.881083	−	0.472961i	$-0.843185\pi$
$881$	−46079.8	−1.76217	−0.881083	+	0.472961i	$0.843185\pi$
$882$	0	0
$883$	26837.6	1.02283	0.511414	−	0.859335i	$-0.329122\pi$
$883$	26837.6	1.02283	0.511414	+	0.859335i	$0.329122\pi$
$884$	0	0
$885$	−17957.5	−0.682072
$886$	0	0
$887$	31685.2	1.19942	0.599710	−	0.800217i	$-0.295282\pi$
$887$	31685.2	1.19942	0.599710	+	0.800217i	$0.295282\pi$
$888$	0	0
$889$	−4426.50	−0.166997
$890$	0	0
$891$	−11696.9	−0.439797
$892$	0	0
$893$	−4403.92	−0.165030
$894$	0	0
$895$	20358.8	0.760356
$896$	0	0
$897$	6174.61	0.229837
$898$	0	0
$899$	3860.79	0.143231
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1264.34	−0.0465942
$904$	0	0
$905$	40525.4	1.48852
$906$	0	0
$907$	46173.4	1.69037	0.845183	−	0.534477i	$-0.179491\pi$
$907$	46173.4	1.69037	0.845183	+	0.534477i	$0.179491\pi$
$908$	0	0
$909$	18565.1	0.677411
$910$	0	0
$911$	19101.7	0.694697	0.347348	−	0.937736i	$-0.387082\pi$
$911$	19101.7	0.694697	0.347348	+	0.937736i	$0.387082\pi$
$912$	0	0
$913$	23594.4	0.855267
$914$	0	0
$915$	925.857	0.0334512
$916$	0	0
$917$	−1108.22	−0.0399090
$918$	0	0
$919$	48505.0	1.74106	0.870530	−	0.492116i	$-0.163776\pi$
$919$	48505.0	1.74106	0.870530	+	0.492116i	$0.163776\pi$
$920$	0	0
$921$	4190.31	0.149919
$922$	0	0
$923$	34713.2	1.23792
$924$	0	0
$925$	36332.1	1.29145
$926$	0	0
$927$	−15861.4	−0.561982
$928$	0	0
$929$	39619.7	1.39922	0.699612	−	0.714523i	$-0.253356\pi$
$929$	39619.7	1.39922	0.699612	+	0.714523i	$0.253356\pi$
$930$	0	0
$931$	−13216.7	−0.465263
$932$	0	0
$933$	−11659.3	−0.409121
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10398.1	−0.362529	−0.181265	−	0.983434i	$-0.558019\pi$
$937$	−10398.1	−0.362529	−0.181265	+	0.983434i	$0.558019\pi$
$938$	0	0
$939$	8063.81	0.280248
$940$	0	0
$941$	52003.6	1.80156	0.900781	−	0.434273i	$-0.142995\pi$
$941$	52003.6	1.80156	0.900781	+	0.434273i	$0.142995\pi$
$942$	0	0
$943$	26109.6	0.901638
$944$	0	0
$945$	−4145.35	−0.142697
$946$	0	0
$947$	29931.1	1.02707	0.513533	−	0.858070i	$-0.328337\pi$
$947$	29931.1	1.02707	0.513533	+	0.858070i	$0.328337\pi$
$948$	0	0
$949$	−43882.1	−1.50103
$950$	0	0
$951$	−2664.68	−0.0908603
$952$	0	0
$953$	−5046.69	−0.171541	−0.0857704	−	0.996315i	$-0.527335\pi$
$953$	−5046.69	−0.171541	−0.0857704	+	0.996315i	$0.527335\pi$
$954$	0	0
$955$	62342.7	2.11242
$956$	0	0
$957$	380.212	0.0128428
$958$	0	0
$959$	3971.35	0.133724
$960$	0	0
$961$	22282.4	0.747958
$962$	0	0
$963$	8950.56	0.299510
$964$	0	0
$965$	46139.3	1.53915
$966$	0	0
$967$	−36104.5	−1.20066	−0.600332	−	0.799751i	$-0.704965\pi$
$967$	−36104.5	−1.20066	−0.600332	+	0.799751i	$0.704965\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−44506.2	−1.47093	−0.735465	−	0.677563i	$-0.763036\pi$
$971$	−44506.2	−1.47093	−0.735465	+	0.677563i	$0.763036\pi$
$972$	0	0
$973$	10860.5	0.357832
$974$	0	0
$975$	11457.5	0.376343
$976$	0	0
$977$	15598.9	0.510800	0.255400	−	0.966835i	$-0.417793\pi$
$977$	15598.9	0.510800	0.255400	+	0.966835i	$0.417793\pi$
$978$	0	0
$979$	−13727.7	−0.448151
$980$	0	0
$981$	−16455.6	−0.535563
$982$	0	0
$983$	33083.6	1.07345	0.536727	−	0.843756i	$-0.319661\pi$
$983$	33083.6	1.07345	0.536727	+	0.843756i	$0.319661\pi$
$984$	0	0
$985$	−62035.0	−2.00670
$986$	0	0
$987$	446.407	0.0143964
$988$	0	0
$989$	42985.3	1.38206
$990$	0	0
$991$	−32338.7	−1.03660	−0.518301	−	0.855198i	$-0.673435\pi$
$991$	−32338.7	−1.03660	−0.518301	+	0.855198i	$0.673435\pi$
$992$	0	0
$993$	−8545.46	−0.273094
$994$	0	0
$995$	33439.3	1.06542
$996$	0	0
$997$	43521.5	1.38249	0.691244	−	0.722621i	$-0.257063\pi$
$997$	43521.5	1.38249	0.691244	+	0.722621i	$0.257063\pi$
$998$	0	0
$999$	−8882.67	−0.281317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.n.1.11	✓	18
17.16	even	2		2312.4.a.q.1.8	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.n.1.11	✓	18	1.1	even	1	trivial
2312.4.a.q.1.8	yes	18	17.16	even	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18564 q^{3} -19.4950 q^{5} -3.40994 q^{7} -25.5943 q^{9} +O(q^{10})\)	\(q+1.18564 q^{3} -19.4950 q^{5} -3.40994 q^{7} -25.5943 q^{9} -18.9542 q^{11} +37.8880 q^{13} -23.1141 q^{15} +39.8848 q^{19} -4.04295 q^{21} +137.453 q^{23} +255.057 q^{25} -62.3578 q^{27} -16.9187 q^{29} -228.196 q^{31} -22.4728 q^{33} +66.4769 q^{35} +142.447 q^{37} +44.9215 q^{39} +189.952 q^{41} +312.727 q^{43} +498.961 q^{45} -110.416 q^{47} -331.372 q^{49} +43.6534 q^{53} +369.513 q^{55} +47.2889 q^{57} +776.906 q^{59} -40.0560 q^{61} +87.2748 q^{63} -738.629 q^{65} +929.861 q^{67} +162.970 q^{69} +916.206 q^{71} -1158.21 q^{73} +302.405 q^{75} +64.6327 q^{77} -737.298 q^{79} +617.111 q^{81} -1244.81 q^{83} -20.0595 q^{87} +724.256 q^{89} -129.196 q^{91} -270.558 q^{93} -777.556 q^{95} -1308.82 q^{97} +485.119 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) \) 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9} - 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} - 153 q^{23} + 306 q^{25} - 768 q^{27} - 51 q^{29} - 303 q^{31} + 525 q^{33} - 255 q^{35} - 717 q^{37} + 216 q^{39} + 393 q^{41} - 390 q^{43} - 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} - 810 q^{57} - 204 q^{59} - 534 q^{61} - 2556 q^{63} + 2127 q^{65} - 405 q^{67} + 2547 q^{69} + 426 q^{71} - 1149 q^{73} - 2226 q^{75} - 357 q^{77} - 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} - 6090 q^{91} + 606 q^{93} - 2109 q^{95} - 2349 q^{97} - 1428 q^{99}+O(q^{100}) \) 18 * q - 18 * q^3 - 51 * q^7 + 120 * q^9 - 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 - 153 * q^23 + 306 * q^25 - 768 * q^27 - 51 * q^29 - 303 * q^31 + 525 * q^33 - 255 * q^35 - 717 * q^37 + 216 * q^39 + 393 * q^41 - 390 * q^43 - 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 - 810 * q^57 - 204 * q^59 - 534 * q^61 - 2556 * q^63 + 2127 * q^65 - 405 * q^67 + 2547 * q^69 + 426 * q^71 - 1149 * q^73 - 2226 * q^75 - 357 * q^77 - 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 - 6090 * q^91 + 606 * q^93 - 2109 * q^95 - 2349 * q^97 - 1428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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