LMFDB - Embedded newform 1248.4.m.a.337.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,4,Mod(337,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1248.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1248.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$73.6343836872$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.21
Character	$\chi$	$=$	1248.337
Dual form			1248.4.m.a.337.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -8.71484 q^{5} +16.6675i q^{7} -9.00000 q^{9} +35.9563 q^{11} +(46.8360 - 1.83976i) q^{13} +26.1445i q^{15} +23.1651 q^{17} -30.2303 q^{19} +50.0026 q^{21} -72.9013 q^{23} -49.0516 q^{25} +27.0000i q^{27} +119.929i q^{29} -96.3119i q^{31} -107.869i q^{33} -145.255i q^{35} -33.6307 q^{37} +(-5.51929 - 140.508i) q^{39} -60.2958i q^{41} +6.93666i q^{43} +78.4335 q^{45} -452.018i q^{47} +65.1938 q^{49} -69.4954i q^{51} +308.251i q^{53} -313.354 q^{55} +90.6910i q^{57} -92.9154 q^{59} +538.354i q^{61} -150.008i q^{63} +(-408.169 + 16.0332i) q^{65} -484.925 q^{67} +218.704i q^{69} -447.724i q^{71} +763.242i q^{73} +147.155i q^{75} +599.303i q^{77} -354.543 q^{79} +81.0000 q^{81} +232.957 q^{83} -201.880 q^{85} +359.788 q^{87} +1319.37i q^{89} +(30.6643 + 780.641i) q^{91} -288.936 q^{93} +263.453 q^{95} +1333.11i q^{97} -323.607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) $ 84 * q - 756 * q^9 - 104 * q^17 + 2188 * q^25 - 3396 * q^49 + 1616 * q^55 + 696 * q^65 - 3160 * q^79 + 6804 * q^81 + 2088 * q^87 - 2480 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−8.71484			−0.779479			−0.389739	−	0.920925i	$-0.627435\pi$
$5$	−8.71484			−0.779479			−0.389739	+	0.920925i	$0.627435\pi$
$6$		0			0
$7$			16.6675i			0.899962i	0.893038	+	0.449981i	$0.148569\pi$
$7$			16.6675i			0.899962i	−0.893038	+	0.449981i	$0.851431\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	35.9563			0.985568			0.492784	−	0.870152i	$-0.335979\pi$
$11$	35.9563			0.985568			0.492784	+	0.870152i	$0.335979\pi$
$12$		0			0
$13$	46.8360	−	1.83976i	0.999229	−	0.0392506i
$13$	46.8360	−	1.83976i	0.999229	−	0.0392506i
$14$		0			0
$15$			26.1445i			0.450032i
$16$		0			0
$17$	23.1651			0.330492			0.165246	−	0.986252i	$-0.447158\pi$
$17$	23.1651			0.330492			0.165246	+	0.986252i	$0.447158\pi$
$18$		0			0
$19$	−30.2303			−0.365017			−0.182508	−	0.983204i	$-0.558422\pi$
$19$	−30.2303			−0.365017			−0.182508	+	0.983204i	$0.558422\pi$
$20$		0			0
$21$	50.0026			0.519593
$22$		0			0
$23$	−72.9013			−0.660912			−0.330456	−	0.943821i	$-0.607203\pi$
$23$	−72.9013			−0.660912			−0.330456	+	0.943821i	$0.607203\pi$
$24$		0			0
$25$	−49.0516			−0.392413
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$			119.929i			0.767943i	0.923345	+	0.383971i	$0.125444\pi$
$29$			119.929i			0.767943i	−0.923345	+	0.383971i	$0.874556\pi$
$30$		0			0
$31$		−	96.3119i		−	0.558004i	−0.960290	−	0.279002i	$-0.909996\pi$
$31$		−	96.3119i		−	0.558004i	0.960290	−	0.279002i	$-0.0900037\pi$
$32$		0			0
$33$		−	107.869i		−	0.569018i
$34$		0			0
$35$		−	145.255i		−	0.701501i
$36$		0			0
$37$	−33.6307			−0.149428			−0.0747142	−	0.997205i	$-0.523804\pi$
$37$	−33.6307			−0.149428			−0.0747142	+	0.997205i	$0.523804\pi$
$38$		0			0
$39$	−5.51929	−	140.508i	−0.0226614	−	0.576905i
$40$		0			0
$41$		−	60.2958i		−	0.229674i	−0.993384	−	0.114837i	$-0.963365\pi$
$41$		−	60.2958i		−	0.229674i	0.993384	−	0.114837i	$-0.0366345\pi$
$42$		0			0
$43$			6.93666i			0.0246007i	0.999924	+	0.0123004i	$0.00391542\pi$
$43$			6.93666i			0.0246007i	−0.999924	+	0.0123004i	$0.996085\pi$
$44$		0			0
$45$	78.4335			0.259826
$46$		0			0
$47$		−	452.018i		−	1.40284i	−0.712747	−	0.701421i	$-0.752549\pi$
$47$		−	452.018i		−	1.40284i	0.712747	−	0.701421i	$-0.247451\pi$
$48$		0			0
$49$	65.1938			0.190069
$50$		0			0
$51$		−	69.4954i		−	0.190810i
$52$		0			0
$53$			308.251i			0.798897i	0.916756	+	0.399449i	$0.130798\pi$
$53$			308.251i			0.798897i	−0.916756	+	0.399449i	$0.869202\pi$
$54$		0			0
$55$	−313.354			−0.768229
$56$		0			0
$57$			90.6910i			0.210742i
$58$		0			0
$59$	−92.9154			−0.205026			−0.102513	−	0.994732i	$-0.532688\pi$
$59$	−92.9154			−0.205026			−0.102513	+	0.994732i	$0.532688\pi$
$60$		0			0
$61$			538.354i			1.12999i	0.825095	+	0.564993i	$0.191121\pi$
$61$			538.354i			1.12999i	−0.825095	+	0.564993i	$0.808879\pi$
$62$		0			0
$63$		−	150.008i		−	0.299987i
$64$		0			0
$65$	−408.169	+	16.0332i	−0.778878	+	0.0305950i
$66$		0			0
$67$	−484.925			−0.884224			−0.442112	−	0.896960i	$-0.645771\pi$
$67$	−484.925			−0.884224			−0.442112	+	0.896960i	$0.645771\pi$
$68$		0			0
$69$			218.704i			0.381578i
$70$		0			0
$71$		−	447.724i		−	0.748381i	−0.927352	−	0.374190i	$-0.877921\pi$
$71$		−	447.724i		−	0.748381i	0.927352	−	0.374190i	$-0.122079\pi$
$72$		0			0
$73$			763.242i			1.22371i	0.790971	+	0.611854i	$0.209576\pi$
$73$			763.242i			1.22371i	−0.790971	+	0.611854i	$0.790424\pi$
$74$		0			0
$75$			147.155i			0.226560i
$76$		0			0
$77$			599.303i			0.886973i
$78$		0			0
$79$	−354.543			−0.504927			−0.252464	−	0.967606i	$-0.581241\pi$
$79$	−354.543			−0.504927			−0.252464	+	0.967606i	$0.581241\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	232.957			0.308076			0.154038	−	0.988065i	$-0.450772\pi$
$83$	232.957			0.308076			0.154038	+	0.988065i	$0.450772\pi$
$84$		0			0
$85$	−201.880			−0.257612
$86$		0			0
$87$	359.788			0.443372
$88$		0			0
$89$			1319.37i			1.57138i	0.618622	+	0.785689i	$0.287691\pi$
$89$			1319.37i			1.57138i	−0.618622	+	0.785689i	$0.712309\pi$
$90$		0			0
$91$	30.6643	+	780.641i	0.0353241	+	0.899268i
$92$		0			0
$93$	−288.936			−0.322164
$94$		0			0
$95$	263.453			0.284523
$96$		0			0
$97$			1333.11i			1.39543i	0.716374	+	0.697716i	$0.245800\pi$
$97$			1333.11i			1.39543i	−0.716374	+	0.697716i	$0.754200\pi$
$98$		0			0
$99$	−323.607			−0.328523
$100$		0			0
$101$		−	834.686i		−	0.822321i	−0.911563	−	0.411160i	$-0.865124\pi$
$101$		−	834.686i		−	0.822321i	0.911563	−	0.411160i	$-0.134876\pi$
$102$		0			0
$103$	−81.2939			−0.0777682			−0.0388841	−	0.999244i	$-0.512380\pi$
$103$	−81.2939			−0.0777682			−0.0388841	+	0.999244i	$0.512380\pi$
$104$		0			0
$105$	−435.764			−0.405012
$106$		0			0
$107$			1329.88i			1.20154i	0.799423	+	0.600769i	$0.205139\pi$
$107$			1329.88i			1.20154i	−0.799423	+	0.600769i	$0.794861\pi$
$108$		0			0
$109$	−292.074			−0.256657			−0.128329	−	0.991732i	$-0.540961\pi$
$109$	−292.074			−0.256657			−0.128329	+	0.991732i	$0.540961\pi$
$110$		0			0
$111$			100.892i			0.0862726i
$112$		0			0
$113$	383.578			0.319328			0.159664	−	0.987171i	$-0.448959\pi$
$113$	383.578			0.319328			0.159664	+	0.987171i	$0.448959\pi$
$114$		0			0
$115$	635.323			0.515167
$116$		0			0
$117$	−421.524	+	16.5579i	−0.333076	+	0.0130835i
$118$		0			0
$119$			386.105i			0.297430i
$120$		0			0
$121$	−38.1413			−0.0286561
$122$		0			0
$123$	−180.887			−0.132602
$124$		0			0
$125$	1516.83			1.08536
$126$		0			0
$127$	−568.964			−0.397539			−0.198769	−	0.980046i	$-0.563694\pi$
$127$	−568.964			−0.397539			−0.198769	+	0.980046i	$0.563694\pi$
$128$		0			0
$129$	20.8100			0.0142032
$130$		0			0
$131$			1759.15i			1.17327i	0.809853	+	0.586633i	$0.199547\pi$
$131$			1759.15i			1.17327i	−0.809853	+	0.586633i	$0.800453\pi$
$132$		0			0
$133$		−	503.865i		−	0.328501i
$134$		0			0
$135$		−	235.301i		−	0.150011i
$136$		0			0
$137$		−	798.340i		−	0.497860i	−0.968521	−	0.248930i	$-0.919921\pi$
$137$		−	798.340i		−	0.497860i	0.968521	−	0.248930i	$-0.0800789\pi$
$138$		0			0
$139$			2438.56i			1.48803i	0.668162	+	0.744016i	$0.267081\pi$
$139$			2438.56i			1.48803i	−0.668162	+	0.744016i	$0.732919\pi$
$140$		0			0
$141$	−1356.05			−0.809931
$142$		0			0
$143$	1684.05	−	66.1511i	0.984808	−	0.0386841i
$144$		0			0
$145$		−	1045.17i		−	0.598595i
$146$		0			0
$147$		−	195.581i		−	0.109737i
$148$		0			0
$149$	1811.39			0.995938			0.497969	−	0.867195i	$-0.334079\pi$
$149$	1811.39			0.995938			0.497969	+	0.867195i	$0.334079\pi$
$150$		0			0
$151$		−	341.374i		−	0.183978i	−0.995760	−	0.0919888i	$-0.970678\pi$
$151$		−	341.374i		−	0.183978i	0.995760	−	0.0919888i	$-0.0293224\pi$
$152$		0			0
$153$	−208.486			−0.110164
$154$		0			0
$155$			839.343i			0.434953i
$156$		0			0
$157$			2602.82i			1.32310i	0.749899	+	0.661552i	$0.230102\pi$
$157$			2602.82i			1.32310i	−0.749899	+	0.661552i	$0.769898\pi$
$158$		0			0
$159$	924.753			0.461243
$160$		0			0
$161$		−	1215.08i		−	0.594795i
$162$		0			0
$163$	280.578			0.134825			0.0674127	−	0.997725i	$-0.478526\pi$
$163$	280.578			0.134825			0.0674127	+	0.997725i	$0.478526\pi$
$164$		0			0
$165$			940.061i			0.443537i
$166$		0			0
$167$		−	445.106i		−	0.206248i	−0.994669	−	0.103124i	$-0.967116\pi$
$167$		−	445.106i		−	0.206248i	0.994669	−	0.103124i	$-0.0328838\pi$
$168$		0			0
$169$	2190.23	−	172.334i	0.996919	−	0.0784408i
$170$		0			0
$171$	272.073			0.121672
$172$		0			0
$173$			4293.12i			1.88670i	0.331796	+	0.943351i	$0.392345\pi$
$173$			4293.12i			1.88670i	−0.331796	+	0.943351i	$0.607655\pi$
$174$		0			0
$175$		−	817.569i		−	0.353156i
$176$		0			0
$177$			278.746i			0.118372i
$178$		0			0
$179$		−	1254.41i		−	0.523792i	−0.965096	−	0.261896i	$-0.915652\pi$
$179$		−	1254.41i		−	0.523792i	0.965096	−	0.261896i	$-0.0843478\pi$
$180$		0			0
$181$		−	581.299i		−	0.238716i	−0.992851	−	0.119358i	$-0.961916\pi$
$181$		−	581.299i		−	0.238716i	0.992851	−	0.119358i	$-0.0380837\pi$
$182$		0			0
$183$	1615.06			0.652398
$184$		0			0
$185$	293.086			0.116476
$186$		0			0
$187$	832.933			0.325723
$188$		0			0
$189$	−450.023			−0.173198
$190$		0			0
$191$	3296.98			1.24901			0.624506	−	0.781020i	$-0.285300\pi$
$191$	3296.98			1.24901			0.624506	+	0.781020i	$0.285300\pi$
$192$		0			0
$193$			450.627i			0.168067i	0.996463	+	0.0840333i	$0.0267802\pi$
$193$			450.627i			0.168067i	−0.996463	+	0.0840333i	$0.973220\pi$
$194$		0			0
$195$	48.0997	+	1224.51i	0.0176640	+	0.449685i
$196$		0			0
$197$	4208.90			1.52219			0.761095	−	0.648640i	$-0.224662\pi$
$197$	4208.90			1.52219			0.761095	+	0.648640i	$0.224662\pi$
$198$		0			0
$199$	−4910.70			−1.74930			−0.874649	−	0.484756i	$-0.838908\pi$
$199$	−4910.70			−1.74930			−0.874649	+	0.484756i	$0.838908\pi$
$200$		0			0
$201$			1454.77i			0.510507i
$202$		0			0
$203$	−1998.93			−0.691119
$204$		0			0
$205$			525.468i			0.179026i
$206$		0			0
$207$	656.112			0.220304
$208$		0			0
$209$	−1086.97			−0.359749
$210$		0			0
$211$			3970.82i			1.29556i	0.761829	+	0.647778i	$0.224302\pi$
$211$			3970.82i			1.29556i	−0.761829	+	0.647778i	$0.775698\pi$
$212$		0			0
$213$	−1343.17			−0.432078
$214$		0			0
$215$		−	60.4519i		−	0.0191757i
$216$		0			0
$217$	1605.28			0.502182
$218$		0			0
$219$	2289.72			0.706508
$220$		0			0
$221$	1084.96	−	42.6183i	0.330238	−	0.0129720i
$222$		0			0
$223$		−	484.716i		−	0.145556i	−0.997348	−	0.0727780i	$-0.976814\pi$
$223$		−	484.716i		−	0.145556i	0.997348	−	0.0727780i	$-0.0231864\pi$
$224$		0			0
$225$	441.464			0.130804
$226$		0			0
$227$	−6155.05			−1.79967			−0.899835	−	0.436231i	$-0.856313\pi$
$227$	−6155.05			−1.79967			−0.899835	+	0.436231i	$0.856313\pi$
$228$		0			0
$229$	−5029.53			−1.45136			−0.725679	−	0.688034i	$-0.758474\pi$
$229$	−5029.53			−1.45136			−0.725679	+	0.688034i	$0.758474\pi$
$230$		0			0
$231$	1797.91			0.512094
$232$		0			0
$233$	−1563.57			−0.439627			−0.219813	−	0.975542i	$-0.570545\pi$
$233$	−1563.57			−0.439627			−0.219813	+	0.975542i	$0.570545\pi$
$234$		0			0
$235$			3939.26i			1.09349i
$236$		0			0
$237$			1063.63i			0.291520i
$238$		0			0
$239$			6996.70i			1.89364i	0.321770	+	0.946818i	$0.395722\pi$
$239$			6996.70i			1.89364i	−0.321770	+	0.946818i	$0.604278\pi$
$240$		0			0
$241$		−	3982.68i		−	1.06451i	−0.846584	−	0.532255i	$-0.821345\pi$
$241$		−	3982.68i		−	1.06451i	0.846584	−	0.532255i	$-0.178655\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−568.153			−0.148155
$246$		0			0
$247$	−1415.87	+	55.6166i	−0.364735	+	0.0143271i
$248$		0			0
$249$		−	698.871i		−	0.177868i
$250$		0			0
$251$			1286.97i			0.323636i	0.986821	+	0.161818i	$0.0517358\pi$
$251$			1286.97i			0.323636i	−0.986821	+	0.161818i	$0.948264\pi$
$252$		0			0
$253$	−2621.27			−0.651374
$254$		0			0
$255$			605.641i			0.148732i
$256$		0			0
$257$	6235.56			1.51348			0.756738	−	0.653718i	$-0.226792\pi$
$257$	6235.56			1.51348			0.756738	+	0.653718i	$0.226792\pi$
$258$		0			0
$259$		−	560.540i		−	0.134480i
$260$		0			0
$261$		−	1079.36i		−	0.255981i
$262$		0			0
$263$	−692.393			−0.162337			−0.0811687	−	0.996700i	$-0.525865\pi$
$263$	−692.393			−0.162337			−0.0811687	+	0.996700i	$0.525865\pi$
$264$		0			0
$265$		−	2686.36i		−	0.622723i
$266$		0			0
$267$	3958.10			0.907235
$268$		0			0
$269$			7517.66i			1.70394i	0.523591	+	0.851970i	$0.324592\pi$
$269$			7517.66i			1.70394i	−0.523591	+	0.851970i	$0.675408\pi$
$270$		0			0
$271$		−	3416.45i		−	0.765810i	−0.923788	−	0.382905i	$-0.874924\pi$
$271$		−	3416.45i		−	0.765810i	0.923788	−	0.382905i	$-0.125076\pi$
$272$		0			0
$273$	2341.92	−	91.9928i	0.519193	−	0.0203943i
$274$		0			0
$275$	−1763.72			−0.386749
$276$		0			0
$277$			5784.73i			1.25477i	0.778710	+	0.627385i	$0.215875\pi$
$277$			5784.73i			1.25477i	−0.778710	+	0.627385i	$0.784125\pi$
$278$		0			0
$279$			866.807i			0.186001i
$280$		0			0
$281$		−	3098.96i		−	0.657895i	−0.944348	−	0.328948i	$-0.893306\pi$
$281$		−	3098.96i		−	0.657895i	0.944348	−	0.328948i	$-0.106694\pi$
$282$		0			0
$283$		−	8198.04i		−	1.72199i	−0.508614	−	0.860994i	$-0.669842\pi$
$283$		−	8198.04i		−	1.72199i	0.508614	−	0.860994i	$-0.330158\pi$
$284$		0			0
$285$		−	790.358i		−	0.164269i
$286$		0			0
$287$	1004.98			0.206698
$288$		0			0
$289$	−4376.38			−0.890775
$290$		0			0
$291$	3999.33			0.805653
$292$		0			0
$293$	−2594.28			−0.517267			−0.258634	−	0.965975i	$-0.583272\pi$
$293$	−2594.28			−0.517267			−0.258634	+	0.965975i	$0.583272\pi$
$294$		0			0
$295$	809.743			0.159814
$296$		0			0
$297$			970.821i			0.189673i
$298$		0			0
$299$	−3414.41	+	134.121i	−0.660403	+	0.0259412i
$300$		0			0
$301$	−115.617			−0.0221397
$302$		0			0
$303$	−2504.06			−0.474767
$304$		0			0
$305$		−	4691.67i		−	0.880801i
$306$		0			0
$307$	2227.35			0.414076			0.207038	−	0.978333i	$-0.433618\pi$
$307$	2227.35			0.414076			0.207038	+	0.978333i	$0.433618\pi$
$308$		0			0
$309$			243.882i			0.0448995i
$310$		0			0
$311$	−6254.74			−1.14043			−0.570215	−	0.821496i	$-0.693140\pi$
$311$	−6254.74			−1.14043			−0.570215	+	0.821496i	$0.693140\pi$
$312$		0			0
$313$	−8081.71			−1.45944			−0.729720	−	0.683746i	$-0.760350\pi$
$313$	−8081.71			−1.45944			−0.729720	+	0.683746i	$0.760350\pi$
$314$		0			0
$315$			1307.29i			0.233834i
$316$		0			0
$317$	−607.880			−0.107703			−0.0538517	−	0.998549i	$-0.517150\pi$
$317$	−607.880			−0.107703			−0.0538517	+	0.998549i	$0.517150\pi$
$318$		0			0
$319$			4312.22i			0.756859i
$320$		0			0
$321$	3989.65			0.693708
$322$		0			0
$323$	−700.290			−0.120635
$324$		0			0
$325$	−2297.38	+	90.2433i	−0.392110	+	0.0154024i
$326$		0			0
$327$			876.223i			0.148181i
$328$		0			0
$329$	7534.02			1.26250
$330$		0			0
$331$	8021.66			1.33206			0.666028	−	0.745927i	$-0.267993\pi$
$331$	8021.66			1.33206			0.666028	+	0.745927i	$0.267993\pi$
$332$		0			0
$333$	302.676			0.0498095
$334$		0			0
$335$	4226.04			0.689234
$336$		0			0
$337$	−10370.6			−1.67633			−0.838164	−	0.545419i	$-0.816371\pi$
$337$	−10370.6			−1.67633			−0.838164	+	0.545419i	$0.816371\pi$
$338$		0			0
$339$		−	1150.74i		−	0.184364i
$340$		0			0
$341$		−	3463.02i		−	0.549951i
$342$		0			0
$343$			6803.58i			1.07102i
$344$		0			0
$345$		−	1905.97i		−	0.297432i
$346$		0			0
$347$		−	2078.35i		−	0.321532i	−0.986993	−	0.160766i	$-0.948604\pi$
$347$		−	2078.35i		−	0.321532i	0.986993	−	0.160766i	$-0.0513965\pi$
$348$		0			0
$349$	5847.40			0.896861			0.448430	−	0.893818i	$-0.351983\pi$
$349$	5847.40			0.896861			0.448430	+	0.893818i	$0.351983\pi$
$350$		0			0
$351$	49.6736	+	1264.57i	0.00755379	+	0.192302i
$352$		0			0
$353$		−	1046.41i		−	0.157776i	−0.996883	−	0.0788881i	$-0.974863\pi$
$353$		−	1046.41i		−	0.157776i	0.996883	−	0.0788881i	$-0.0251370\pi$
$354$		0			0
$355$			3901.84i			0.583347i
$356$		0			0
$357$	1158.32			0.171721
$358$		0			0
$359$			9018.15i			1.32579i	0.748711	+	0.662896i	$0.230673\pi$
$359$			9018.15i			1.32579i	−0.748711	+	0.662896i	$0.769327\pi$
$360$		0			0
$361$	−5945.13			−0.866763
$362$		0			0
$363$			114.424i			0.0165446i
$364$		0			0
$365$		−	6651.53i		−	0.953854i
$366$		0			0
$367$	248.518			0.0353475			0.0176737	−	0.999844i	$-0.494374\pi$
$367$	248.518			0.0353475			0.0176737	+	0.999844i	$0.494374\pi$
$368$		0			0
$369$			542.662i			0.0765579i
$370$		0			0
$371$	−5137.78			−0.718977
$372$		0			0
$373$			5757.08i			0.799170i	0.916696	+	0.399585i	$0.130846\pi$
$373$			5757.08i			0.799170i	−0.916696	+	0.399585i	$0.869154\pi$
$374$		0			0
$375$		−	4550.49i		−	0.626631i
$376$		0			0
$377$	220.642	+	5617.02i	0.0301422	+	0.767351i
$378$		0			0
$379$	6826.63			0.925226			0.462613	−	0.886560i	$-0.346912\pi$
$379$	6826.63			0.925226			0.462613	+	0.886560i	$0.346912\pi$
$380$		0			0
$381$			1706.89i			0.229519i
$382$		0			0
$383$			1577.59i			0.210472i	0.994447	+	0.105236i	$0.0335598\pi$
$383$			1577.59i			0.210472i	−0.994447	+	0.105236i	$0.966440\pi$
$384$		0			0
$385$		−	5222.83i		−	0.691377i
$386$		0			0
$387$		−	62.4299i		−	0.00820024i
$388$		0			0
$389$		−	4906.45i		−	0.639504i	−0.947501	−	0.319752i	$-0.896400\pi$
$389$		−	4906.45i		−	0.639504i	0.947501	−	0.319752i	$-0.103600\pi$
$390$		0			0
$391$	−1688.77			−0.218426
$392$		0			0
$393$	5277.46			0.677386
$394$		0			0
$395$	3089.79			0.393580
$396$		0			0
$397$	−807.657			−0.102104			−0.0510518	−	0.998696i	$-0.516257\pi$
$397$	−807.657			−0.102104			−0.0510518	+	0.998696i	$0.516257\pi$
$398$		0			0
$399$	−1511.59			−0.189660
$400$		0			0
$401$			6460.01i			0.804483i	0.915534	+	0.402241i	$0.131769\pi$
$401$			6460.01i			0.804483i	−0.915534	+	0.402241i	$0.868231\pi$
$402$		0			0
$403$	−177.191	−	4510.87i	−0.0219020	−	0.557574i
$404$		0			0
$405$	−705.902			−0.0866088
$406$		0			0
$407$	−1209.24			−0.147272
$408$		0			0
$409$			6999.02i			0.846159i	0.906093	+	0.423079i	$0.139051\pi$
$409$			6999.02i			0.846159i	−0.906093	+	0.423079i	$0.860949\pi$
$410$		0			0
$411$	−2395.02			−0.287440
$412$		0			0
$413$		−	1548.67i		−	0.184516i
$414$		0			0
$415$	−2030.18			−0.240139
$416$		0			0
$417$	7315.69			0.859115
$418$		0			0
$419$			12515.6i			1.45925i	0.683848	+	0.729624i	$0.260305\pi$
$419$			12515.6i			1.45925i	−0.683848	+	0.729624i	$0.739695\pi$
$420$		0			0
$421$	12863.5			1.48914			0.744571	−	0.667543i	$-0.232654\pi$
$421$	12863.5			1.48914			0.744571	+	0.667543i	$0.232654\pi$
$422$		0			0
$423$			4068.16i			0.467614i
$424$		0			0
$425$	−1136.29			−0.129689
$426$		0			0
$427$	−8973.03			−1.01694
$428$		0			0
$429$	−198.453	−	5052.16i	−0.0223343	−	0.568579i
$430$		0			0
$431$		−	13608.6i		−	1.52089i	−0.649401	−	0.760446i	$-0.724980\pi$
$431$		−	13608.6i		−	1.52089i	0.649401	−	0.760446i	$-0.275020\pi$
$432$		0			0
$433$	−12975.4			−1.44009			−0.720043	−	0.693929i	$-0.755878\pi$
$433$	−12975.4			−1.44009			−0.720043	+	0.693929i	$0.755878\pi$
$434$		0			0
$435$	−3135.50			−0.345599
$436$		0			0
$437$	2203.83			0.241244
$438$		0			0
$439$	−10025.0			−1.08991			−0.544953	−	0.838467i	$-0.683452\pi$
$439$	−10025.0			−1.08991			−0.544953	+	0.838467i	$0.683452\pi$
$440$		0			0
$441$	−586.744			−0.0633564
$442$		0			0
$443$		−	7965.70i		−	0.854315i	−0.904177	−	0.427158i	$-0.859515\pi$
$443$		−	7965.70i		−	0.854315i	0.904177	−	0.427158i	$-0.140485\pi$
$444$		0			0
$445$		−	11498.1i		−	1.22486i
$446$		0			0
$447$		−	5434.17i		−	0.575005i
$448$		0			0
$449$		−	2522.14i		−	0.265094i	−0.991177	−	0.132547i	$-0.957684\pi$
$449$		−	2522.14i		−	0.265094i	0.991177	−	0.132547i	$-0.0423155\pi$
$450$		0			0
$451$		−	2168.02i		−	0.226359i
$452$		0			0
$453$	−1024.12			−0.106219
$454$		0			0
$455$	−267.234	−	6803.16i	−0.0275343	−	0.700960i
$456$		0			0
$457$			9207.77i			0.942497i	0.882000	+	0.471249i	$0.156197\pi$
$457$			9207.77i			0.942497i	−0.882000	+	0.471249i	$0.843803\pi$
$458$		0			0
$459$			625.458i			0.0636033i
$460$		0			0
$461$	−4470.19			−0.451622			−0.225811	−	0.974171i	$-0.572503\pi$
$461$	−4470.19			−0.451622			−0.225811	+	0.974171i	$0.572503\pi$
$462$		0			0
$463$			8509.64i			0.854161i	0.904214	+	0.427081i	$0.140458\pi$
$463$			8509.64i			0.854161i	−0.904214	+	0.427081i	$0.859542\pi$
$464$		0			0
$465$	2518.03			0.251120
$466$		0			0
$467$		−	3571.08i		−	0.353854i	−0.984224	−	0.176927i	$-0.943384\pi$
$467$		−	3571.08i		−	0.353854i	0.984224	−	0.176927i	$-0.0566157\pi$
$468$		0			0
$469$		−	8082.49i		−	0.795767i
$470$		0			0
$471$	7808.45			0.763895
$472$		0			0
$473$			249.417i			0.0242457i
$474$		0			0
$475$	1482.85			0.143237
$476$		0			0
$477$		−	2774.26i		−	0.266299i
$478$		0			0
$479$		−	4738.57i		−	0.452006i	−0.974127	−	0.226003i	$-0.927434\pi$
$479$		−	4738.57i		−	0.452006i	0.974127	−	0.226003i	$-0.0725659\pi$
$480$		0			0
$481$	−1575.13	+	61.8725i	−0.149313	+	0.00586516i
$482$		0			0
$483$	−3645.25			−0.343405
$484$		0			0
$485$		−	11617.8i		−	1.08771i
$486$		0			0
$487$		−	20179.6i		−	1.87767i	−0.344371	−	0.938834i	$-0.611908\pi$
$487$		−	20179.6i		−	1.87767i	0.344371	−	0.938834i	$-0.388092\pi$
$488$		0			0
$489$		−	841.733i		−	0.0778415i
$490$		0			0
$491$		−	5526.37i		−	0.507946i	−0.967211	−	0.253973i	$-0.918263\pi$
$491$		−	5526.37i		−	0.507946i	0.967211	−	0.253973i	$-0.0817375\pi$
$492$		0			0
$493$			2778.18i			0.253799i
$494$		0			0
$495$	2820.18			0.256076
$496$		0			0
$497$	7462.45			0.673514
$498$		0			0
$499$	−11790.6			−1.05776			−0.528878	−	0.848698i	$-0.677387\pi$
$499$	−11790.6			−1.05776			−0.528878	+	0.848698i	$0.677387\pi$
$500$		0			0
$501$	−1335.32			−0.119077
$502$		0			0
$503$	17921.2			1.58860			0.794301	−	0.607525i	$-0.207837\pi$
$503$	17921.2			1.58860			0.794301	+	0.607525i	$0.207837\pi$
$504$		0			0
$505$			7274.16i			0.640982i
$506$		0			0
$507$	−517.003	−	6570.69i	−0.0452878	−	0.575571i
$508$		0			0
$509$	7918.98			0.689593			0.344796	−	0.938677i	$-0.387948\pi$
$509$	7918.98			0.689593			0.344796	+	0.938677i	$0.387948\pi$
$510$		0			0
$511$	−12721.3			−1.10129
$512$		0			0
$513$		−	816.219i		−	0.0702475i
$514$		0			0
$515$	708.463			0.0606187
$516$		0			0
$517$		−	16252.9i		−	1.38260i
$518$		0			0
$519$	12879.3			1.08929
$520$		0			0
$521$	−9351.85			−0.786395			−0.393198	−	0.919454i	$-0.628631\pi$
$521$	−9351.85			−0.786395			−0.393198	+	0.919454i	$0.628631\pi$
$522$		0			0
$523$			10017.4i			0.837531i	0.908094	+	0.418766i	$0.137537\pi$
$523$			10017.4i			0.837531i	−0.908094	+	0.418766i	$0.862463\pi$
$524$		0			0
$525$	−2452.71			−0.203895
$526$		0			0
$527$		−	2231.08i		−	0.184416i
$528$		0			0
$529$	−6852.40			−0.563195
$530$		0			0
$531$	836.239			0.0683421
$532$		0			0
$533$	−110.930	−	2824.02i	−0.00901484	−	0.229497i
$534$		0			0
$535$		−	11589.7i		−	0.936573i
$536$		0			0
$537$	−3763.22			−0.302412
$538$		0			0
$539$	2344.13			0.187326
$540$		0			0
$541$	−18747.9			−1.48990			−0.744951	−	0.667119i	$-0.767527\pi$
$541$	−18747.9			−1.48990			−0.744951	+	0.667119i	$0.767527\pi$
$542$		0			0
$543$	−1743.90			−0.137823
$544$		0			0
$545$	2545.38			0.200059
$546$		0			0
$547$		−	7024.44i		−	0.549074i	−0.961577	−	0.274537i	$-0.911475\pi$
$547$		−	7024.44i		−	0.549074i	0.961577	−	0.274537i	$-0.0885246\pi$
$548$		0			0
$549$		−	4845.19i		−	0.376662i
$550$		0			0
$551$		−	3625.51i		−	0.280312i
$552$		0			0
$553$		−	5909.36i		−	0.454415i
$554$		0			0
$555$		−	879.258i		−	0.0672476i
$556$		0			0
$557$	11378.9			0.865604			0.432802	−	0.901489i	$-0.357525\pi$
$557$	11378.9			0.865604			0.432802	+	0.901489i	$0.357525\pi$
$558$		0			0
$559$	12.7618	+	324.886i	0.000965593	+	0.0245818i
$560$		0			0
$561$		−	2498.80i		−	0.188056i
$562$		0			0
$563$		−	15789.5i		−	1.18197i	−0.806683	−	0.590985i	$-0.798739\pi$
$563$		−	15789.5i		−	1.18197i	0.806683	−	0.590985i	$-0.201261\pi$
$564$		0			0
$565$	−3342.82			−0.248909
$566$		0			0
$567$			1350.07i			0.0999957i
$568$		0			0
$569$	13400.2			0.987284			0.493642	−	0.869665i	$-0.335665\pi$
$569$	13400.2			0.987284			0.493642	+	0.869665i	$0.335665\pi$
$570$		0			0
$571$		−	22295.1i		−	1.63401i	−0.576630	−	0.817005i	$-0.695633\pi$
$571$		−	22295.1i		−	1.63401i	0.576630	−	0.817005i	$-0.304367\pi$
$572$		0			0
$573$		−	9890.95i		−	0.721117i
$574$		0			0
$575$	3575.93			0.259350
$576$		0			0
$577$			7646.84i			0.551719i	0.961198	+	0.275860i	$0.0889625\pi$
$577$			7646.84i			0.551719i	−0.961198	+	0.275860i	$0.911038\pi$
$578$		0			0
$579$	1351.88			0.0970332
$580$		0			0
$581$			3882.81i			0.277257i
$582$		0			0
$583$			11083.6i			0.787367i
$584$		0			0
$585$	3673.52	−	144.299i	0.259626	−	0.0101983i
$586$		0			0
$587$	7365.05			0.517867			0.258934	−	0.965895i	$-0.416629\pi$
$587$	7365.05			0.517867			0.258934	+	0.965895i	$0.416629\pi$
$588$		0			0
$589$			2911.54i			0.203681i
$590$		0			0
$591$		−	12626.7i		−	0.878837i
$592$		0			0
$593$		−	16147.1i		−	1.11818i	−0.829106	−	0.559092i	$-0.811150\pi$
$593$		−	16147.1i		−	1.11818i	0.829106	−	0.559092i	$-0.188850\pi$
$594$		0			0
$595$		−	3364.84i		−	0.231841i
$596$		0			0
$597$			14732.1i			1.00996i
$598$		0			0
$599$	6027.18			0.411125			0.205562	−	0.978644i	$-0.434098\pi$
$599$	6027.18			0.411125			0.205562	+	0.978644i	$0.434098\pi$
$600$		0			0
$601$	−2117.99			−0.143751			−0.0718757	−	0.997414i	$-0.522899\pi$
$601$	−2117.99			−0.143751			−0.0718757	+	0.997414i	$0.522899\pi$
$602$		0			0
$603$	4364.32			0.294741
$604$		0			0
$605$	332.395			0.0223369
$606$		0			0
$607$	−11132.7			−0.744418			−0.372209	−	0.928149i	$-0.621399\pi$
$607$	−11132.7			−0.744418			−0.372209	+	0.928149i	$0.621399\pi$
$608$		0			0
$609$			5996.78i			0.399018i
$610$		0			0
$611$	−831.605	−	21170.7i	−0.0550624	−	1.40176i
$612$		0			0
$613$	−17390.2			−1.14581			−0.572906	−	0.819621i	$-0.694184\pi$
$613$	−17390.2			−1.14581			−0.572906	+	0.819621i	$0.694184\pi$
$614$		0			0
$615$	1576.40			0.103361
$616$		0			0
$617$			17583.9i			1.14733i	0.819090	+	0.573665i	$0.194479\pi$
$617$			17583.9i			1.14733i	−0.819090	+	0.573665i	$0.805521\pi$
$618$		0			0
$619$	9088.88			0.590166			0.295083	−	0.955472i	$-0.404653\pi$
$619$	9088.88			0.590166			0.295083	+	0.955472i	$0.404653\pi$
$620$		0			0
$621$		−	1968.34i		−	0.127193i
$622$		0			0
$623$	−21990.6			−1.41418
$624$		0			0
$625$	−7087.49			−0.453599
$626$		0			0
$627$			3260.92i			0.207701i
$628$		0			0
$629$	−779.059			−0.0493849
$630$		0			0
$631$			8004.69i			0.505011i	0.967596	+	0.252505i	$0.0812545\pi$
$631$			8004.69i			0.505011i	−0.967596	+	0.252505i	$0.918745\pi$
$632$		0			0
$633$	11912.5			0.747990
$634$		0			0
$635$	4958.43			0.309873
$636$		0			0
$637$	3053.42	−	119.941i	0.189923	−	0.00746034i
$638$		0			0
$639$			4029.51i			0.249460i
$640$		0			0
$641$	31376.8			1.93340			0.966699	−	0.255917i	$-0.0823775\pi$
$641$	31376.8			1.93340			0.966699	+	0.255917i	$0.0823775\pi$
$642$		0			0
$643$	−24518.9			−1.50378			−0.751891	−	0.659288i	$-0.770858\pi$
$643$	−24518.9			−1.50378			−0.751891	+	0.659288i	$0.770858\pi$
$644$		0			0
$645$	−181.356			−0.0110711
$646$		0			0
$647$	19032.6			1.15649			0.578243	−	0.815864i	$-0.303738\pi$
$647$	19032.6			1.15649			0.578243	+	0.815864i	$0.303738\pi$
$648$		0			0
$649$	−3340.90			−0.202067
$650$		0			0
$651$		−	4815.84i		−	0.289935i
$652$		0			0
$653$		−	1630.31i		−	0.0977012i	−0.998806	−	0.0488506i	$-0.984444\pi$
$653$		−	1630.31i		−	0.0977012i	0.998806	−	0.0488506i	$-0.0155558\pi$
$654$		0			0
$655$		−	15330.7i		−	0.914536i
$656$		0			0
$657$		−	6869.17i		−	0.407903i
$658$		0			0
$659$		−	6216.87i		−	0.367489i	−0.982974	−	0.183744i	$-0.941178\pi$
$659$		−	6216.87i		−	0.367489i	0.982974	−	0.183744i	$-0.0588218\pi$
$660$		0			0
$661$	17029.9			1.00210			0.501050	−	0.865418i	$-0.332947\pi$
$661$	17029.9			1.00210			0.501050	+	0.865418i	$0.332947\pi$
$662$		0			0
$663$	−127.855	−	3254.89i	−0.00748940	−	0.190663i
$664$		0			0
$665$			4391.10i			0.256059i
$666$		0			0
$667$		−	8743.01i		−	0.507542i
$668$		0			0
$669$	−1454.15			−0.0840368
$670$		0			0
$671$			19357.2i			1.11368i
$672$		0			0
$673$	−8854.72			−0.507168			−0.253584	−	0.967313i	$-0.581609\pi$
$673$	−8854.72			−0.507168			−0.253584	+	0.967313i	$0.581609\pi$
$674$		0			0
$675$		−	1324.39i		−	0.0755199i
$676$		0			0
$677$		−	25869.2i		−	1.46859i	−0.678830	−	0.734295i	$-0.737513\pi$
$677$		−	25869.2i		−	1.46859i	0.678830	−	0.734295i	$-0.262487\pi$
$678$		0			0
$679$	−22219.7			−1.25584
$680$		0			0
$681$			18465.1i			1.03904i
$682$		0			0
$683$	14554.1			0.815370			0.407685	−	0.913123i	$-0.366336\pi$
$683$	14554.1			0.815370			0.407685	+	0.913123i	$0.366336\pi$
$684$		0			0
$685$			6957.41i			0.388071i
$686$		0			0
$687$			15088.6i			0.837942i
$688$		0			0
$689$	567.109	+	14437.3i	0.0313572	+	0.798281i
$690$		0			0
$691$	31889.4			1.75561			0.877807	−	0.479015i	$-0.159006\pi$
$691$	31889.4			1.75561			0.877807	+	0.479015i	$0.159006\pi$
$692$		0			0
$693$		−	5393.73i		−	0.295658i
$694$		0			0
$695$		−	21251.7i		−	1.15989i
$696$		0			0
$697$		−	1396.76i		−	0.0759054i
$698$		0			0
$699$			4690.72i			0.253819i
$700$		0			0
$701$			24149.2i			1.30115i	0.759443	+	0.650574i	$0.225471\pi$
$701$			24149.2i			1.30115i	−0.759443	+	0.650574i	$0.774529\pi$
$702$		0			0
$703$	1016.67			0.0545439
$704$		0			0
$705$	11817.8			0.631324
$706$		0			0
$707$	13912.2			0.740057
$708$		0			0
$709$	10731.6			0.568454			0.284227	−	0.958757i	$-0.408263\pi$
$709$	10731.6			0.568454			0.284227	+	0.958757i	$0.408263\pi$
$710$		0			0
$711$	3190.89			0.168309
$712$		0			0
$713$			7021.27i			0.368792i
$714$		0			0
$715$	−14676.2	+	576.496i	−0.767637	+	0.0301535i
$716$		0			0
$717$	20990.1			1.09329
$718$		0			0
$719$	21369.6			1.10842			0.554208	−	0.832378i	$-0.313021\pi$
$719$	21369.6			1.10842			0.554208	+	0.832378i	$0.313021\pi$
$720$		0			0
$721$		−	1354.97i		−	0.0699884i
$722$		0			0
$723$	−11948.0			−0.614596
$724$		0			0
$725$		−	5882.73i		−	0.301351i
$726$		0			0
$727$	−6216.16			−0.317118			−0.158559	−	0.987350i	$-0.550685\pi$
$727$	−6216.16			−0.317118			−0.158559	+	0.987350i	$0.550685\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			160.689i			0.00813035i
$732$		0			0
$733$	18224.5			0.918333			0.459166	−	0.888350i	$-0.348148\pi$
$733$	18224.5			0.918333			0.459166	+	0.888350i	$0.348148\pi$
$734$		0			0
$735$			1704.46i			0.0855373i
$736$		0			0
$737$	−17436.1			−0.871462
$738$		0			0
$739$	−2355.87			−0.117269			−0.0586346	−	0.998280i	$-0.518675\pi$
$739$	−2355.87			−0.117269			−0.0586346	+	0.998280i	$0.518675\pi$
$740$		0			0
$741$	166.850	+	4247.61i	0.00827177	+	0.210580i
$742$		0			0
$743$			16897.0i			0.834306i	0.908836	+	0.417153i	$0.136972\pi$
$743$			16897.0i			0.834306i	−0.908836	+	0.417153i	$0.863028\pi$
$744$		0			0
$745$	−15786.0			−0.776312
$746$		0			0
$747$	−2096.61			−0.102692
$748$		0			0
$749$	−22165.8			−1.08134
$750$		0			0
$751$	−6055.22			−0.294218			−0.147109	−	0.989120i	$-0.546997\pi$
$751$	−6055.22			−0.294218			−0.147109	+	0.989120i	$0.546997\pi$
$752$		0			0
$753$	3860.91			0.186852
$754$		0			0
$755$			2975.02i			0.143407i
$756$		0			0
$757$			14120.5i			0.677965i	0.940793	+	0.338982i	$0.110083\pi$
$757$			14120.5i			0.677965i	−0.940793	+	0.338982i	$0.889917\pi$
$758$		0			0
$759$			7863.80i			0.376071i
$760$		0			0
$761$			5880.45i			0.280113i	0.990143	+	0.140057i	$0.0447285\pi$
$761$			5880.45i			0.280113i	−0.990143	+	0.140057i	$0.955272\pi$
$762$		0			0
$763$		−	4868.15i		−	0.230982i
$764$		0			0
$765$	1816.92			0.0858706
$766$		0			0
$767$	−4351.79	+	170.942i	−0.204868	+	0.00804741i
$768$		0			0
$769$		−	36398.3i		−	1.70683i	−0.521229	−	0.853417i	$-0.674526\pi$
$769$		−	36398.3i		−	1.70683i	0.521229	−	0.853417i	$-0.325474\pi$
$770$		0			0
$771$		−	18706.7i		−	0.873806i
$772$		0			0
$773$	−7579.44			−0.352669			−0.176335	−	0.984330i	$-0.556424\pi$
$773$	−7579.44			−0.352669			−0.176335	+	0.984330i	$0.556424\pi$
$774$		0			0
$775$			4724.25i			0.218968i
$776$		0			0
$777$	−1681.62			−0.0776420
$778$		0			0
$779$			1822.76i			0.0838347i
$780$		0			0
$781$		−	16098.5i		−	0.737580i
$782$		0			0
$783$	−3238.09			−0.147791
$784$		0			0
$785$		−	22683.1i		−	1.03133i
$786$		0			0
$787$	−33424.2			−1.51390			−0.756952	−	0.653470i	$-0.773313\pi$
$787$	−33424.2			−1.51390			−0.756952	+	0.653470i	$0.773313\pi$
$788$		0			0
$789$			2077.18i			0.0937256i
$790$		0			0
$791$			6393.30i			0.287383i
$792$		0			0
$793$	990.443	+	25214.4i	0.0443527	+	1.12912i
$794$		0			0
$795$	−8059.08			−0.359529
$796$		0			0
$797$		−	12915.2i		−	0.574001i	−0.957930	−	0.287000i	$-0.907342\pi$
$797$		−	12915.2i		−	0.574001i	0.957930	−	0.287000i	$-0.0926581\pi$
$798$		0			0
$799$		−	10471.1i		−	0.463628i
$800$		0			0
$801$		−	11874.3i		−	0.523793i
$802$		0			0
$803$			27443.4i			1.20605i
$804$		0			0
$805$			10589.3i			0.463630i
$806$		0			0
$807$	22553.0			0.983770
$808$		0			0
$809$	−42970.1			−1.86743			−0.933713	−	0.358023i	$-0.883451\pi$
$809$	−42970.1			−1.86743			−0.933713	+	0.358023i	$0.883451\pi$
$810$		0			0
$811$	23953.2			1.03713			0.518564	−	0.855039i	$-0.326467\pi$
$811$	23953.2			1.03713			0.518564	+	0.855039i	$0.326467\pi$
$812$		0			0
$813$	−10249.3			−0.442140
$814$		0			0
$815$	−2445.19			−0.105094
$816$		0			0
$817$		−	209.698i		−	0.00897967i
$818$		0			0
$819$	−275.978	−	7025.77i	−0.0117747	−	0.299756i
$820$		0			0
$821$	10738.8			0.456501			0.228250	−	0.973602i	$-0.426700\pi$
$821$	10738.8			0.456501			0.228250	+	0.973602i	$0.426700\pi$
$822$		0			0
$823$	−14437.4			−0.611491			−0.305746	−	0.952113i	$-0.598906\pi$
$823$	−14437.4			−0.611491			−0.305746	+	0.952113i	$0.598906\pi$
$824$		0			0
$825$			5291.15i			0.223290i
$826$		0			0
$827$	−15364.2			−0.646031			−0.323015	−	0.946394i	$-0.604697\pi$
$827$	−15364.2			−0.646031			−0.323015	+	0.946394i	$0.604697\pi$
$828$		0			0
$829$		−	36608.6i		−	1.53374i	−0.641803	−	0.766870i	$-0.721813\pi$
$829$		−	36608.6i		−	1.53374i	0.641803	−	0.766870i	$-0.278187\pi$
$830$		0			0
$831$	17354.2			0.724441
$832$		0			0
$833$	1510.22			0.0628164
$834$		0			0
$835$			3879.03i			0.160766i
$836$		0			0
$837$	2600.42			0.107388
$838$		0			0
$839$			9733.80i			0.400534i	0.979741	+	0.200267i	$0.0641810\pi$
$839$			9733.80i			0.400534i	−0.979741	+	0.200267i	$0.935819\pi$
$840$		0			0
$841$	10005.9			0.410264
$842$		0			0
$843$	−9296.89			−0.379836
$844$		0			0
$845$	−19087.5	+	1501.87i	−0.777077	+	0.0611429i
$846$		0			0
$847$		−	635.721i		−	0.0257894i
$848$		0			0
$849$	−24594.1			−0.994191
$850$		0			0
$851$	2451.72			0.0987591
$852$		0			0
$853$	−7250.95			−0.291053			−0.145526	−	0.989354i	$-0.546488\pi$
$853$	−7250.95			−0.291053			−0.145526	+	0.989354i	$0.546488\pi$
$854$		0			0
$855$	−2371.07			−0.0948409
$856$		0			0
$857$	39519.2			1.57520			0.787602	−	0.616185i	$-0.211323\pi$
$857$	39519.2			1.57520			0.787602	+	0.616185i	$0.211323\pi$
$858$		0			0
$859$			46491.8i			1.84666i	0.384010	+	0.923329i	$0.374543\pi$
$859$			46491.8i			1.84666i	−0.384010	+	0.923329i	$0.625457\pi$
$860$		0			0
$861$		−	3014.95i		−	0.119337i
$862$		0			0
$863$			11505.9i			0.453842i	0.973913	+	0.226921i	$0.0728659\pi$
$863$			11505.9i			0.453842i	−0.973913	+	0.226921i	$0.927134\pi$
$864$		0			0
$865$		−	37413.8i		−	1.47064i
$866$		0			0
$867$			13129.1i			0.514289i
$868$		0			0
$869$	−12748.1			−0.497640
$870$		0			0
$871$	−22712.0	+	892.146i	−0.883542	+	0.0347063i
$872$		0			0
$873$		−	11998.0i		−	0.465144i
$874$		0			0
$875$			25281.8i			0.976779i
$876$		0			0
$877$	20615.5			0.793771			0.396886	−	0.917868i	$-0.370091\pi$
$877$	20615.5			0.793771			0.396886	+	0.917868i	$0.370091\pi$
$878$		0			0
$879$			7782.84i			0.298645i
$880$		0			0
$881$	−23657.2			−0.904688			−0.452344	−	0.891844i	$-0.649412\pi$
$881$	−23657.2			−0.904688			−0.452344	+	0.891844i	$0.649412\pi$
$882$		0			0
$883$		−	26112.2i		−	0.995182i	−0.867412	−	0.497591i	$-0.834218\pi$
$883$		−	26112.2i		−	0.995182i	0.867412	−	0.497591i	$-0.165782\pi$
$884$		0			0
$885$		−	2429.23i		−	0.0922685i
$886$		0			0
$887$	32404.5			1.22665			0.613323	−	0.789832i	$-0.289832\pi$
$887$	32404.5			1.22665			0.613323	+	0.789832i	$0.289832\pi$
$888$		0			0
$889$		−	9483.22i		−	0.357770i
$890$		0			0
$891$	2912.46			0.109508
$892$		0			0
$893$			13664.7i			0.512061i
$894$		0			0
$895$			10932.0i			0.408285i
$896$		0			0
$897$	402.363	+	10243.2i	0.0149772	+	0.381284i
$898$		0			0
$899$	11550.6			0.428515
$900$		0			0
$901$			7140.68i			0.264029i
$902$		0			0
$903$			346.851i			0.0127824i
$904$		0			0
$905$			5065.93i			0.186074i
$906$		0			0
$907$			31753.5i			1.16247i	0.813737	+	0.581233i	$0.197429\pi$
$907$			31753.5i			1.16247i	−0.813737	+	0.581233i	$0.802571\pi$
$908$		0			0
$909$			7512.18i			0.274107i
$910$		0			0
$911$	−45254.7			−1.64583			−0.822916	−	0.568163i	$-0.807654\pi$
$911$	−45254.7			−1.64583			−0.822916	+	0.568163i	$0.807654\pi$
$912$		0			0
$913$	8376.28			0.303630
$914$		0			0
$915$	−14075.0			−0.508531
$916$		0			0
$917$	−29320.7			−1.05589
$918$		0			0
$919$	−40149.3			−1.44113			−0.720567	−	0.693385i	$-0.756118\pi$
$919$	−40149.3			−1.44113			−0.720567	+	0.693385i	$0.756118\pi$
$920$		0			0
$921$		−	6682.04i		−	0.239067i
$922$		0			0
$923$	−823.705	−	20969.6i	−0.0293744	−	0.747804i
$924$		0			0
$925$	1649.64			0.0586376
$926$		0			0
$927$	731.645			0.0259227
$928$		0			0
$929$		−	39873.3i		−	1.40818i	−0.710110	−	0.704091i	$-0.751355\pi$
$929$		−	39873.3i		−	1.40818i	0.710110	−	0.704091i	$-0.248645\pi$
$930$		0			0
$931$	−1970.83			−0.0693784
$932$		0			0
$933$			18764.2i			0.658427i
$934$		0			0
$935$	−7258.88			−0.253894
$936$		0			0
$937$	9738.57			0.339536			0.169768	−	0.985484i	$-0.445698\pi$
$937$	9738.57			0.339536			0.169768	+	0.985484i	$0.445698\pi$
$938$		0			0
$939$			24245.1i			0.842609i
$940$		0			0
$941$	−22634.3			−0.784119			−0.392059	−	0.919940i	$-0.628237\pi$
$941$	−22634.3			−0.784119			−0.392059	+	0.919940i	$0.628237\pi$
$942$		0			0
$943$			4395.64i			0.151794i
$944$		0			0
$945$	3921.88			0.135004
$946$		0			0
$947$	27205.9			0.933550			0.466775	−	0.884376i	$-0.345416\pi$
$947$	27205.9			0.933550			0.466775	+	0.884376i	$0.345416\pi$
$948$		0			0
$949$	1404.18	+	35747.2i	0.0480313	+	1.22276i
$950$		0			0
$951$			1823.64i			0.0621826i
$952$		0			0
$953$	22624.9			0.769038			0.384519	−	0.923117i	$-0.374367\pi$
$953$	22624.9			0.769038			0.384519	+	0.923117i	$0.374367\pi$
$954$		0			0
$955$	−28732.7			−0.973578
$956$		0			0
$957$	12936.7			0.436973
$958$		0			0
$959$	13306.4			0.448055
$960$		0			0
$961$	20515.0			0.688631
$962$		0			0
$963$		−	11968.9i		−	0.400513i
$964$		0			0
$965$		−	3927.14i		−	0.131004i
$966$		0			0
$967$			50031.6i			1.66381i	0.554916	+	0.831907i	$0.312751\pi$
$967$			50031.6i			1.66381i	−0.554916	+	0.831907i	$0.687249\pi$
$968$		0			0
$969$			2100.87i			0.0696487i
$970$		0			0
$971$		−	6575.84i		−	0.217331i	−0.994078	−	0.108666i	$-0.965342\pi$
$971$		−	6575.84i		−	0.217331i	0.994078	−	0.108666i	$-0.0346578\pi$
$972$		0			0
$973$	−40644.8			−1.33917
$974$		0			0
$975$	270.730	+	6892.15i	0.00889261	+	0.226385i
$976$		0			0
$977$		−	34070.7i		−	1.11568i	−0.829949	−	0.557840i	$-0.811630\pi$
$977$		−	34070.7i		−	1.11568i	0.829949	−	0.557840i	$-0.188370\pi$
$978$		0			0
$979$			47439.6i			1.54870i
$980$		0			0
$981$	2628.67			0.0855524
$982$		0			0
$983$		−	10931.9i		−	0.354705i	−0.984147	−	0.177352i	$-0.943247\pi$
$983$		−	10931.9i		−	0.354705i	0.984147	−	0.177352i	$-0.0567532\pi$
$984$		0			0
$985$	−36679.9			−1.18652
$986$		0			0
$987$		−	22602.1i		−	0.728907i
$988$		0			0
$989$		−	505.692i		−	0.0162589i
$990$		0			0
$991$	4446.58			0.142533			0.0712666	−	0.997457i	$-0.477296\pi$
$991$	4446.58			0.142533			0.0712666	+	0.997457i	$0.477296\pi$
$992$		0			0
$993$		−	24065.0i		−	0.769063i
$994$		0			0
$995$	42796.0			1.36354
$996$		0			0
$997$			19510.8i			0.619774i	0.950773	+	0.309887i	$0.100291\pi$
$997$			19510.8i			0.619774i	−0.950773	+	0.309887i	$0.899709\pi$
$998$		0			0
$999$		−	908.029i		−	0.0287575i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.4.m.a.337.21		84
4.3	odd	2		312.4.m.a.181.3	✓	84
8.3	odd	2		312.4.m.a.181.81	yes	84
8.5	even	2	inner	1248.4.m.a.337.24		84
13.12	even	2	inner	1248.4.m.a.337.22		84
52.51	odd	2		312.4.m.a.181.82	yes	84
104.51	odd	2		312.4.m.a.181.4	yes	84
104.77	even	2	inner	1248.4.m.a.337.23		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.m.a.181.3	✓	84	4.3	odd	2
312.4.m.a.181.4	yes	84	104.51	odd	2
312.4.m.a.181.81	yes	84	8.3	odd	2
312.4.m.a.181.82	yes	84	52.51	odd	2
1248.4.m.a.337.21		84	1.1	even	1	trivial
1248.4.m.a.337.22		84	13.12	even	2	inner
1248.4.m.a.337.23		84	104.77	even	2	inner
1248.4.m.a.337.24		84	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1248.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -8.71484 q^{5} +16.6675i q^{7} -9.00000 q^{9} +35.9563 q^{11} +(46.8360 - 1.83976i) q^{13} +26.1445i q^{15} +23.1651 q^{17} -30.2303 q^{19} +50.0026 q^{21} -72.9013 q^{23} -49.0516 q^{25} +27.0000i q^{27} +119.929i q^{29} -96.3119i q^{31} -107.869i q^{33} -145.255i q^{35} -33.6307 q^{37} +(-5.51929 - 140.508i) q^{39} -60.2958i q^{41} +6.93666i q^{43} +78.4335 q^{45} -452.018i q^{47} +65.1938 q^{49} -69.4954i q^{51} +308.251i q^{53} -313.354 q^{55} +90.6910i q^{57} -92.9154 q^{59} +538.354i q^{61} -150.008i q^{63} +(-408.169 + 16.0332i) q^{65} -484.925 q^{67} +218.704i q^{69} -447.724i q^{71} +763.242i q^{73} +147.155i q^{75} +599.303i q^{77} -354.543 q^{79} +81.0000 q^{81} +232.957 q^{83} -201.880 q^{85} +359.788 q^{87} +1319.37i q^{89} +(30.6643 + 780.641i) q^{91} -288.936 q^{93} +263.453 q^{95} +1333.11i q^{97} -323.607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) \) 84 * q - 756 * q^9 - 104 * q^17 + 2188 * q^25 - 3396 * q^49 + 1616 * q^55 + 696 * q^65 - 3160 * q^79 + 6804 * q^81 + 2088 * q^87 - 2480 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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