LMFDB - Embedded newform 6036.2.a.h.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.56884 q^{5} +1.63329 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.56884 q^{5} +1.63329 q^{7} +1.00000 q^{9} +3.76318 q^{11} -1.49974 q^{13} +1.56884 q^{15} -7.60445 q^{17} +7.25611 q^{19} +1.63329 q^{21} +5.83805 q^{23} -2.53874 q^{25} +1.00000 q^{27} +7.42558 q^{29} +6.49320 q^{31} +3.76318 q^{33} +2.56238 q^{35} -4.26124 q^{37} -1.49974 q^{39} +5.80942 q^{41} +0.704806 q^{43} +1.56884 q^{45} -12.6749 q^{47} -4.33235 q^{49} -7.60445 q^{51} +1.19925 q^{53} +5.90383 q^{55} +7.25611 q^{57} -5.25868 q^{59} -10.2312 q^{61} +1.63329 q^{63} -2.35285 q^{65} +15.1051 q^{67} +5.83805 q^{69} +7.00364 q^{71} -6.66022 q^{73} -2.53874 q^{75} +6.14637 q^{77} +1.98822 q^{79} +1.00000 q^{81} +0.798286 q^{83} -11.9302 q^{85} +7.42558 q^{87} +3.12564 q^{89} -2.44951 q^{91} +6.49320 q^{93} +11.3837 q^{95} +4.65214 q^{97} +3.76318 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.56884	0.701607	0.350804	−	0.936449i	$-0.385908\pi$
$5$	1.56884	0.701607	0.350804	+	0.936449i	$0.385908\pi$
$6$	0	0
$7$	1.63329	0.617327	0.308663	−	0.951171i	$-0.400118\pi$
$7$	1.63329	0.617327	0.308663	+	0.951171i	$0.400118\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.76318	1.13464	0.567320	−	0.823497i	$-0.307980\pi$
$11$	3.76318	1.13464	0.567320	+	0.823497i	$0.307980\pi$
$12$	0	0
$13$	−1.49974	−0.415952	−0.207976	−	0.978134i	$-0.566688\pi$
$13$	−1.49974	−0.415952	−0.207976	+	0.978134i	$0.566688\pi$
$14$	0	0
$15$	1.56884	0.405073
$16$	0	0
$17$	−7.60445	−1.84435	−0.922175	−	0.386773i	$-0.873590\pi$
$17$	−7.60445	−1.84435	−0.922175	+	0.386773i	$0.873590\pi$
$18$	0	0
$19$	7.25611	1.66466	0.832332	−	0.554277i	$-0.187005\pi$
$19$	7.25611	1.66466	0.832332	+	0.554277i	$0.187005\pi$
$20$	0	0
$21$	1.63329	0.356414
$22$	0	0
$23$	5.83805	1.21732	0.608659	−	0.793432i	$-0.291708\pi$
$23$	5.83805	1.21732	0.608659	+	0.793432i	$0.291708\pi$
$24$	0	0
$25$	−2.53874	−0.507747
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.42558	1.37890	0.689448	−	0.724335i	$-0.257853\pi$
$29$	7.42558	1.37890	0.689448	+	0.724335i	$0.257853\pi$
$30$	0	0
$31$	6.49320	1.16621	0.583106	−	0.812396i	$-0.301837\pi$
$31$	6.49320	1.16621	0.583106	+	0.812396i	$0.301837\pi$
$32$	0	0
$33$	3.76318	0.655085
$34$	0	0
$35$	2.56238	0.433121
$36$	0	0
$37$	−4.26124	−0.700543	−0.350271	−	0.936648i	$-0.613911\pi$
$37$	−4.26124	−0.700543	−0.350271	+	0.936648i	$0.613911\pi$
$38$	0	0
$39$	−1.49974	−0.240150
$40$	0	0
$41$	5.80942	0.907279	0.453640	−	0.891185i	$-0.350125\pi$
$41$	5.80942	0.907279	0.453640	+	0.891185i	$0.350125\pi$
$42$	0	0
$43$	0.704806	0.107482	0.0537409	−	0.998555i	$-0.482885\pi$
$43$	0.704806	0.107482	0.0537409	+	0.998555i	$0.482885\pi$
$44$	0	0
$45$	1.56884	0.233869
$46$	0	0
$47$	−12.6749	−1.84882	−0.924412	−	0.381394i	$-0.875444\pi$
$47$	−12.6749	−1.84882	−0.924412	+	0.381394i	$0.875444\pi$
$48$	0	0
$49$	−4.33235	−0.618908
$50$	0	0
$51$	−7.60445	−1.06484
$52$	0	0
$53$	1.19925	0.164729	0.0823646	−	0.996602i	$-0.473753\pi$
$53$	1.19925	0.164729	0.0823646	+	0.996602i	$0.473753\pi$
$54$	0	0
$55$	5.90383	0.796072
$56$	0	0
$57$	7.25611	0.961095
$58$	0	0
$59$	−5.25868	−0.684622	−0.342311	−	0.939587i	$-0.611210\pi$
$59$	−5.25868	−0.684622	−0.342311	+	0.939587i	$0.611210\pi$
$60$	0	0
$61$	−10.2312	−1.30997	−0.654983	−	0.755643i	$-0.727324\pi$
$61$	−10.2312	−1.30997	−0.654983	+	0.755643i	$0.727324\pi$
$62$	0	0
$63$	1.63329	0.205776
$64$	0	0
$65$	−2.35285	−0.291835
$66$	0	0
$67$	15.1051	1.84538	0.922692	−	0.385537i	$-0.125984\pi$
$67$	15.1051	1.84538	0.922692	+	0.385537i	$0.125984\pi$
$68$	0	0
$69$	5.83805	0.702818
$70$	0	0
$71$	7.00364	0.831180	0.415590	−	0.909552i	$-0.363575\pi$
$71$	7.00364	0.831180	0.415590	+	0.909552i	$0.363575\pi$
$72$	0	0
$73$	−6.66022	−0.779520	−0.389760	−	0.920916i	$-0.627442\pi$
$73$	−6.66022	−0.779520	−0.389760	+	0.920916i	$0.627442\pi$
$74$	0	0
$75$	−2.53874	−0.293148
$76$	0	0
$77$	6.14637	0.700444
$78$	0	0
$79$	1.98822	0.223692	0.111846	−	0.993726i	$-0.464324\pi$
$79$	1.98822	0.223692	0.111846	+	0.993726i	$0.464324\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.798286	0.0876233	0.0438117	−	0.999040i	$-0.486050\pi$
$83$	0.798286	0.0876233	0.0438117	+	0.999040i	$0.486050\pi$
$84$	0	0
$85$	−11.9302	−1.29401
$86$	0	0
$87$	7.42558	0.796106
$88$	0	0
$89$	3.12564	0.331317	0.165658	−	0.986183i	$-0.447025\pi$
$89$	3.12564	0.331317	0.165658	+	0.986183i	$0.447025\pi$
$90$	0	0
$91$	−2.44951	−0.256778
$92$	0	0
$93$	6.49320	0.673313
$94$	0	0
$95$	11.3837	1.16794
$96$	0	0
$97$	4.65214	0.472353	0.236177	−	0.971710i	$-0.424106\pi$
$97$	4.65214	0.472353	0.236177	+	0.971710i	$0.424106\pi$
$98$	0	0
$99$	3.76318	0.378213
$100$	0	0
$101$	8.07795	0.803786	0.401893	−	0.915687i	$-0.368352\pi$
$101$	8.07795	0.803786	0.401893	+	0.915687i	$0.368352\pi$
$102$	0	0
$103$	19.0473	1.87679	0.938394	−	0.345568i	$-0.112314\pi$
$103$	19.0473	1.87679	0.938394	+	0.345568i	$0.112314\pi$
$104$	0	0
$105$	2.56238	0.250063
$106$	0	0
$107$	16.2926	1.57507	0.787534	−	0.616271i	$-0.211357\pi$
$107$	16.2926	1.57507	0.787534	+	0.616271i	$0.211357\pi$
$108$	0	0
$109$	−11.8618	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$109$	−11.8618	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$110$	0	0
$111$	−4.26124	−0.404459
$112$	0	0
$113$	−13.9771	−1.31485	−0.657426	−	0.753519i	$-0.728355\pi$
$113$	−13.9771	−1.31485	−0.657426	+	0.753519i	$0.728355\pi$
$114$	0	0
$115$	9.15897	0.854079
$116$	0	0
$117$	−1.49974	−0.138651
$118$	0	0
$119$	−12.4203	−1.13857
$120$	0	0
$121$	3.16149	0.287408
$122$	0	0
$123$	5.80942	0.523818
$124$	0	0
$125$	−11.8271	−1.05785
$126$	0	0
$127$	11.4784	1.01854	0.509272	−	0.860605i	$-0.329915\pi$
$127$	11.4784	1.01854	0.509272	+	0.860605i	$0.329915\pi$
$128$	0	0
$129$	0.704806	0.0620547
$130$	0	0
$131$	17.7993	1.55514	0.777568	−	0.628799i	$-0.216453\pi$
$131$	17.7993	1.55514	0.777568	+	0.628799i	$0.216453\pi$
$132$	0	0
$133$	11.8513	1.02764
$134$	0	0
$135$	1.56884	0.135024
$136$	0	0
$137$	2.61261	0.223211	0.111605	−	0.993753i	$-0.464401\pi$
$137$	2.61261	0.223211	0.111605	+	0.993753i	$0.464401\pi$
$138$	0	0
$139$	6.80555	0.577239	0.288620	−	0.957444i	$-0.406804\pi$
$139$	6.80555	0.577239	0.288620	+	0.957444i	$0.406804\pi$
$140$	0	0
$141$	−12.6749	−1.06742
$142$	0	0
$143$	−5.64377	−0.471955
$144$	0	0
$145$	11.6496	0.967443
$146$	0	0
$147$	−4.33235	−0.357326
$148$	0	0
$149$	7.10240	0.581852	0.290926	−	0.956746i	$-0.406037\pi$
$149$	7.10240	0.581852	0.290926	+	0.956746i	$0.406037\pi$
$150$	0	0
$151$	−14.7380	−1.19937	−0.599683	−	0.800238i	$-0.704706\pi$
$151$	−14.7380	−1.19937	−0.599683	+	0.800238i	$0.704706\pi$
$152$	0	0
$153$	−7.60445	−0.614783
$154$	0	0
$155$	10.1868	0.818224
$156$	0	0
$157$	−5.44517	−0.434572	−0.217286	−	0.976108i	$-0.569720\pi$
$157$	−5.44517	−0.434572	−0.217286	+	0.976108i	$0.569720\pi$
$158$	0	0
$159$	1.19925	0.0951065
$160$	0	0
$161$	9.53524	0.751483
$162$	0	0
$163$	−9.35521	−0.732757	−0.366378	−	0.930466i	$-0.619402\pi$
$163$	−9.35521	−0.732757	−0.366378	+	0.930466i	$0.619402\pi$
$164$	0	0
$165$	5.90383	0.459612
$166$	0	0
$167$	−18.6598	−1.44394	−0.721970	−	0.691924i	$-0.756763\pi$
$167$	−18.6598	−1.44394	−0.721970	+	0.691924i	$0.756763\pi$
$168$	0	0
$169$	−10.7508	−0.826984
$170$	0	0
$171$	7.25611	0.554888
$172$	0	0
$173$	0.787031	0.0598369	0.0299184	−	0.999552i	$-0.490475\pi$
$173$	0.787031	0.0598369	0.0299184	+	0.999552i	$0.490475\pi$
$174$	0	0
$175$	−4.14650	−0.313446
$176$	0	0
$177$	−5.25868	−0.395266
$178$	0	0
$179$	−8.83257	−0.660177	−0.330089	−	0.943950i	$-0.607079\pi$
$179$	−8.83257	−0.660177	−0.330089	+	0.943950i	$0.607079\pi$
$180$	0	0
$181$	10.3323	0.767997	0.383998	−	0.923334i	$-0.374547\pi$
$181$	10.3323	0.767997	0.383998	+	0.923334i	$0.374547\pi$
$182$	0	0
$183$	−10.2312	−0.756310
$184$	0	0
$185$	−6.68520	−0.491506
$186$	0	0
$187$	−28.6169	−2.09267
$188$	0	0
$189$	1.63329	0.118805
$190$	0	0
$191$	2.88273	0.208587	0.104294	−	0.994547i	$-0.466742\pi$
$191$	2.88273	0.208587	0.104294	+	0.994547i	$0.466742\pi$
$192$	0	0
$193$	6.32110	0.455003	0.227501	−	0.973778i	$-0.426944\pi$
$193$	6.32110	0.455003	0.227501	+	0.973778i	$0.426944\pi$
$194$	0	0
$195$	−2.35285	−0.168491
$196$	0	0
$197$	−21.5158	−1.53294	−0.766469	−	0.642282i	$-0.777988\pi$
$197$	−21.5158	−1.53294	−0.766469	+	0.642282i	$0.777988\pi$
$198$	0	0
$199$	−20.7287	−1.46942	−0.734709	−	0.678382i	$-0.762681\pi$
$199$	−20.7287	−1.46942	−0.734709	+	0.678382i	$0.762681\pi$
$200$	0	0
$201$	15.1051	1.06543
$202$	0	0
$203$	12.1281	0.851229
$204$	0	0
$205$	9.11406	0.636554
$206$	0	0
$207$	5.83805	0.405772
$208$	0	0
$209$	27.3060	1.88880
$210$	0	0
$211$	−4.49028	−0.309124	−0.154562	−	0.987983i	$-0.549397\pi$
$211$	−4.49028	−0.309124	−0.154562	+	0.987983i	$0.549397\pi$
$212$	0	0
$213$	7.00364	0.479882
$214$	0	0
$215$	1.10573	0.0754101
$216$	0	0
$217$	10.6053	0.719935
$218$	0	0
$219$	−6.66022	−0.450056
$220$	0	0
$221$	11.4047	0.767161
$222$	0	0
$223$	−15.6458	−1.04772	−0.523861	−	0.851804i	$-0.675509\pi$
$223$	−15.6458	−1.04772	−0.523861	+	0.851804i	$0.675509\pi$
$224$	0	0
$225$	−2.53874	−0.169249
$226$	0	0
$227$	10.3909	0.689667	0.344833	−	0.938664i	$-0.387935\pi$
$227$	10.3909	0.689667	0.344833	+	0.938664i	$0.387935\pi$
$228$	0	0
$229$	2.33676	0.154417	0.0772086	−	0.997015i	$-0.475399\pi$
$229$	2.33676	0.154417	0.0772086	+	0.997015i	$0.475399\pi$
$230$	0	0
$231$	6.14637	0.404401
$232$	0	0
$233$	−23.7034	−1.55286	−0.776431	−	0.630202i	$-0.782972\pi$
$233$	−23.7034	−1.55286	−0.776431	+	0.630202i	$0.782972\pi$
$234$	0	0
$235$	−19.8849	−1.29715
$236$	0	0
$237$	1.98822	0.129149
$238$	0	0
$239$	−4.34830	−0.281268	−0.140634	−	0.990062i	$-0.544914\pi$
$239$	−4.34830	−0.281268	−0.140634	+	0.990062i	$0.544914\pi$
$240$	0	0
$241$	12.3513	0.795616	0.397808	−	0.917469i	$-0.369771\pi$
$241$	12.3513	0.795616	0.397808	+	0.917469i	$0.369771\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.79678	−0.434230
$246$	0	0
$247$	−10.8822	−0.692420
$248$	0	0
$249$	0.798286	0.0505893
$250$	0	0
$251$	17.0681	1.07733	0.538664	−	0.842521i	$-0.318929\pi$
$251$	17.0681	1.07733	0.538664	+	0.842521i	$0.318929\pi$
$252$	0	0
$253$	21.9696	1.38122
$254$	0	0
$255$	−11.9302	−0.747097
$256$	0	0
$257$	−30.8167	−1.92229	−0.961147	−	0.276036i	$-0.910979\pi$
$257$	−30.8167	−1.92229	−0.961147	+	0.276036i	$0.910979\pi$
$258$	0	0
$259$	−6.95985	−0.432464
$260$	0	0
$261$	7.42558	0.459632
$262$	0	0
$263$	−28.3036	−1.74527	−0.872637	−	0.488370i	$-0.837592\pi$
$263$	−28.3036	−1.74527	−0.872637	+	0.488370i	$0.837592\pi$
$264$	0	0
$265$	1.88143	0.115575
$266$	0	0
$267$	3.12564	0.191286
$268$	0	0
$269$	27.0855	1.65143	0.825715	−	0.564087i	$-0.190772\pi$
$269$	27.0855	1.65143	0.825715	+	0.564087i	$0.190772\pi$
$270$	0	0
$271$	8.72254	0.529857	0.264928	−	0.964268i	$-0.414652\pi$
$271$	8.72254	0.529857	0.264928	+	0.964268i	$0.414652\pi$
$272$	0	0
$273$	−2.44951	−0.148251
$274$	0	0
$275$	−9.55371	−0.576110
$276$	0	0
$277$	7.36864	0.442738	0.221369	−	0.975190i	$-0.428947\pi$
$277$	7.36864	0.442738	0.221369	+	0.975190i	$0.428947\pi$
$278$	0	0
$279$	6.49320	0.388738
$280$	0	0
$281$	9.29084	0.554245	0.277123	−	0.960835i	$-0.410619\pi$
$281$	9.29084	0.554245	0.277123	+	0.960835i	$0.410619\pi$
$282$	0	0
$283$	26.2886	1.56269	0.781347	−	0.624097i	$-0.214533\pi$
$283$	26.2886	1.56269	0.781347	+	0.624097i	$0.214533\pi$
$284$	0	0
$285$	11.3837	0.674311
$286$	0	0
$287$	9.48849	0.560088
$288$	0	0
$289$	40.8277	2.40163
$290$	0	0
$291$	4.65214	0.272713
$292$	0	0
$293$	21.9738	1.28372	0.641861	−	0.766821i	$-0.278163\pi$
$293$	21.9738	1.28372	0.641861	+	0.766821i	$0.278163\pi$
$294$	0	0
$295$	−8.25003	−0.480336
$296$	0	0
$297$	3.76318	0.218362
$298$	0	0
$299$	−8.75553	−0.506345
$300$	0	0
$301$	1.15115	0.0663515
$302$	0	0
$303$	8.07795	0.464066
$304$	0	0
$305$	−16.0511	−0.919082
$306$	0	0
$307$	7.61970	0.434879	0.217440	−	0.976074i	$-0.430229\pi$
$307$	7.61970	0.434879	0.217440	+	0.976074i	$0.430229\pi$
$308$	0	0
$309$	19.0473	1.08356
$310$	0	0
$311$	19.8702	1.12674	0.563369	−	0.826206i	$-0.309505\pi$
$311$	19.8702	1.12674	0.563369	+	0.826206i	$0.309505\pi$
$312$	0	0
$313$	−17.3373	−0.979962	−0.489981	−	0.871733i	$-0.662996\pi$
$313$	−17.3373	−0.979962	−0.489981	+	0.871733i	$0.662996\pi$
$314$	0	0
$315$	2.56238	0.144374
$316$	0	0
$317$	26.4642	1.48638	0.743190	−	0.669081i	$-0.233312\pi$
$317$	26.4642	1.48638	0.743190	+	0.669081i	$0.233312\pi$
$318$	0	0
$319$	27.9438	1.56455
$320$	0	0
$321$	16.2926	0.909366
$322$	0	0
$323$	−55.1787	−3.07022
$324$	0	0
$325$	3.80743	0.211198
$326$	0	0
$327$	−11.8618	−0.655957
$328$	0	0
$329$	−20.7018	−1.14133
$330$	0	0
$331$	−1.65818	−0.0911417	−0.0455708	−	0.998961i	$-0.514511\pi$
$331$	−1.65818	−0.0911417	−0.0455708	+	0.998961i	$0.514511\pi$
$332$	0	0
$333$	−4.26124	−0.233514
$334$	0	0
$335$	23.6975	1.29474
$336$	0	0
$337$	−30.7470	−1.67490	−0.837448	−	0.546517i	$-0.815954\pi$
$337$	−30.7470	−1.67490	−0.837448	+	0.546517i	$0.815954\pi$
$338$	0	0
$339$	−13.9771	−0.759131
$340$	0	0
$341$	24.4350	1.32323
$342$	0	0
$343$	−18.5091	−0.999395
$344$	0	0
$345$	9.15897	0.493103
$346$	0	0
$347$	−9.88852	−0.530844	−0.265422	−	0.964132i	$-0.585511\pi$
$347$	−9.88852	−0.530844	−0.265422	+	0.964132i	$0.585511\pi$
$348$	0	0
$349$	−23.6002	−1.26329	−0.631645	−	0.775258i	$-0.717620\pi$
$349$	−23.6002	−1.26329	−0.631645	+	0.775258i	$0.717620\pi$
$350$	0	0
$351$	−1.49974	−0.0800499
$352$	0	0
$353$	8.06576	0.429297	0.214648	−	0.976691i	$-0.431139\pi$
$353$	8.06576	0.429297	0.214648	+	0.976691i	$0.431139\pi$
$354$	0	0
$355$	10.9876	0.583162
$356$	0	0
$357$	−12.4203	−0.657352
$358$	0	0
$359$	0.273268	0.0144225	0.00721126	−	0.999974i	$-0.497705\pi$
$359$	0.273268	0.0144225	0.00721126	+	0.999974i	$0.497705\pi$
$360$	0	0
$361$	33.6511	1.77111
$362$	0	0
$363$	3.16149	0.165935
$364$	0	0
$365$	−10.4488	−0.546917
$366$	0	0
$367$	−18.4431	−0.962721	−0.481361	−	0.876523i	$-0.659857\pi$
$367$	−18.4431	−0.962721	−0.481361	+	0.876523i	$0.659857\pi$
$368$	0	0
$369$	5.80942	0.302426
$370$	0	0
$371$	1.95872	0.101692
$372$	0	0
$373$	28.5121	1.47630	0.738152	−	0.674635i	$-0.235699\pi$
$373$	28.5121	1.47630	0.738152	+	0.674635i	$0.235699\pi$
$374$	0	0
$375$	−11.8271	−0.610748
$376$	0	0
$377$	−11.1364	−0.573554
$378$	0	0
$379$	4.94522	0.254019	0.127009	−	0.991902i	$-0.459462\pi$
$379$	4.94522	0.254019	0.127009	+	0.991902i	$0.459462\pi$
$380$	0	0
$381$	11.4784	0.588057
$382$	0	0
$383$	36.8367	1.88227	0.941134	−	0.338034i	$-0.109762\pi$
$383$	36.8367	1.88227	0.941134	+	0.338034i	$0.109762\pi$
$384$	0	0
$385$	9.64268	0.491436
$386$	0	0
$387$	0.704806	0.0358273
$388$	0	0
$389$	−7.94294	−0.402723	−0.201362	−	0.979517i	$-0.564537\pi$
$389$	−7.94294	−0.402723	−0.201362	+	0.979517i	$0.564537\pi$
$390$	0	0
$391$	−44.3951	−2.24516
$392$	0	0
$393$	17.7993	0.897858
$394$	0	0
$395$	3.11920	0.156944
$396$	0	0
$397$	−31.4075	−1.57630	−0.788148	−	0.615485i	$-0.788960\pi$
$397$	−31.4075	−1.57630	−0.788148	+	0.615485i	$0.788960\pi$
$398$	0	0
$399$	11.8513	0.593310
$400$	0	0
$401$	−1.93441	−0.0965997	−0.0482999	−	0.998833i	$-0.515380\pi$
$401$	−1.93441	−0.0965997	−0.0482999	+	0.998833i	$0.515380\pi$
$402$	0	0
$403$	−9.73808	−0.485088
$404$	0	0
$405$	1.56884	0.0779564
$406$	0	0
$407$	−16.0358	−0.794864
$408$	0	0
$409$	−22.1079	−1.09317	−0.546583	−	0.837405i	$-0.684072\pi$
$409$	−22.1079	−1.09317	−0.546583	+	0.837405i	$0.684072\pi$
$410$	0	0
$411$	2.61261	0.128871
$412$	0	0
$413$	−8.58896	−0.422635
$414$	0	0
$415$	1.25238	0.0614772
$416$	0	0
$417$	6.80555	0.333269
$418$	0	0
$419$	−1.45783	−0.0712199	−0.0356099	−	0.999366i	$-0.511337\pi$
$419$	−1.45783	−0.0712199	−0.0356099	+	0.999366i	$0.511337\pi$
$420$	0	0
$421$	27.2821	1.32965	0.664824	−	0.747000i	$-0.268506\pi$
$421$	27.2821	1.32965	0.664824	+	0.747000i	$0.268506\pi$
$422$	0	0
$423$	−12.6749	−0.616275
$424$	0	0
$425$	19.3057	0.936463
$426$	0	0
$427$	−16.7105	−0.808678
$428$	0	0
$429$	−5.64377	−0.272484
$430$	0	0
$431$	14.5830	0.702439	0.351219	−	0.936293i	$-0.385767\pi$
$431$	14.5830	0.702439	0.351219	+	0.936293i	$0.385767\pi$
$432$	0	0
$433$	7.97506	0.383257	0.191628	−	0.981468i	$-0.438623\pi$
$433$	7.97506	0.383257	0.191628	+	0.981468i	$0.438623\pi$
$434$	0	0
$435$	11.6496	0.558554
$436$	0	0
$437$	42.3615	2.02643
$438$	0	0
$439$	−2.56568	−0.122453	−0.0612266	−	0.998124i	$-0.519501\pi$
$439$	−2.56568	−0.122453	−0.0612266	+	0.998124i	$0.519501\pi$
$440$	0	0
$441$	−4.33235	−0.206303
$442$	0	0
$443$	22.5418	1.07099	0.535497	−	0.844537i	$-0.320124\pi$
$443$	22.5418	1.07099	0.535497	+	0.844537i	$0.320124\pi$
$444$	0	0
$445$	4.90363	0.232454
$446$	0	0
$447$	7.10240	0.335932
$448$	0	0
$449$	0.145653	0.00687377	0.00343689	−	0.999994i	$-0.498906\pi$
$449$	0.145653	0.00687377	0.00343689	+	0.999994i	$0.498906\pi$
$450$	0	0
$451$	21.8619	1.02944
$452$	0	0
$453$	−14.7380	−0.692454
$454$	0	0
$455$	−3.84289	−0.180157
$456$	0	0
$457$	−28.6997	−1.34252	−0.671258	−	0.741224i	$-0.734246\pi$
$457$	−28.6997	−1.34252	−0.671258	+	0.741224i	$0.734246\pi$
$458$	0	0
$459$	−7.60445	−0.354945
$460$	0	0
$461$	33.4778	1.55922	0.779608	−	0.626268i	$-0.215419\pi$
$461$	33.4778	1.55922	0.779608	+	0.626268i	$0.215419\pi$
$462$	0	0
$463$	−40.0570	−1.86161	−0.930804	−	0.365518i	$-0.880892\pi$
$463$	−40.0570	−1.86161	−0.930804	+	0.365518i	$0.880892\pi$
$464$	0	0
$465$	10.1868	0.472402
$466$	0	0
$467$	20.7146	0.958557	0.479279	−	0.877663i	$-0.340898\pi$
$467$	20.7146	0.958557	0.479279	+	0.877663i	$0.340898\pi$
$468$	0	0
$469$	24.6711	1.13921
$470$	0	0
$471$	−5.44517	−0.250900
$472$	0	0
$473$	2.65231	0.121953
$474$	0	0
$475$	−18.4213	−0.845229
$476$	0	0
$477$	1.19925	0.0549097
$478$	0	0
$479$	28.2691	1.29165	0.645823	−	0.763487i	$-0.276514\pi$
$479$	28.2691	1.29165	0.645823	+	0.763487i	$0.276514\pi$
$480$	0	0
$481$	6.39072	0.291392
$482$	0	0
$483$	9.53524	0.433869
$484$	0	0
$485$	7.29847	0.331406
$486$	0	0
$487$	−9.39464	−0.425712	−0.212856	−	0.977084i	$-0.568276\pi$
$487$	−9.39464	−0.425712	−0.212856	+	0.977084i	$0.568276\pi$
$488$	0	0
$489$	−9.35521	−0.423057
$490$	0	0
$491$	36.2928	1.63787	0.818936	−	0.573886i	$-0.194565\pi$
$491$	36.2928	1.63787	0.818936	+	0.573886i	$0.194565\pi$
$492$	0	0
$493$	−56.4674	−2.54317
$494$	0	0
$495$	5.90383	0.265357
$496$	0	0
$497$	11.4390	0.513110
$498$	0	0
$499$	6.18581	0.276915	0.138458	−	0.990368i	$-0.455786\pi$
$499$	6.18581	0.276915	0.138458	+	0.990368i	$0.455786\pi$
$500$	0	0
$501$	−18.6598	−0.833659
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	12.6730	0.563942
$506$	0	0
$507$	−10.7508	−0.477460
$508$	0	0
$509$	−17.5761	−0.779048	−0.389524	−	0.921016i	$-0.627360\pi$
$509$	−17.5761	−0.779048	−0.389524	+	0.921016i	$0.627360\pi$
$510$	0	0
$511$	−10.8781	−0.481219
$512$	0	0
$513$	7.25611	0.320365
$514$	0	0
$515$	29.8822	1.31677
$516$	0	0
$517$	−47.6979	−2.09775
$518$	0	0
$519$	0.787031	0.0345468
$520$	0	0
$521$	7.83633	0.343316	0.171658	−	0.985157i	$-0.445088\pi$
$521$	7.83633	0.343316	0.171658	+	0.985157i	$0.445088\pi$
$522$	0	0
$523$	28.1857	1.23247	0.616237	−	0.787560i	$-0.288656\pi$
$523$	28.1857	1.23247	0.616237	+	0.787560i	$0.288656\pi$
$524$	0	0
$525$	−4.14650	−0.180968
$526$	0	0
$527$	−49.3772	−2.15090
$528$	0	0
$529$	11.0828	0.481861
$530$	0	0
$531$	−5.25868	−0.228207
$532$	0	0
$533$	−8.71259	−0.377384
$534$	0	0
$535$	25.5606	1.10508
$536$	0	0
$537$	−8.83257	−0.381154
$538$	0	0
$539$	−16.3034	−0.702237
$540$	0	0
$541$	−8.11709	−0.348981	−0.174491	−	0.984659i	$-0.555828\pi$
$541$	−8.11709	−0.348981	−0.174491	+	0.984659i	$0.555828\pi$
$542$	0	0
$543$	10.3323	0.443403
$544$	0	0
$545$	−18.6092	−0.797132
$546$	0	0
$547$	−25.5780	−1.09364	−0.546818	−	0.837252i	$-0.684161\pi$
$547$	−25.5780	−1.09364	−0.546818	+	0.837252i	$0.684161\pi$
$548$	0	0
$549$	−10.2312	−0.436656
$550$	0	0
$551$	53.8808	2.29540
$552$	0	0
$553$	3.24735	0.138091
$554$	0	0
$555$	−6.68520	−0.283771
$556$	0	0
$557$	9.54420	0.404401	0.202200	−	0.979344i	$-0.435191\pi$
$557$	9.54420	0.404401	0.202200	+	0.979344i	$0.435191\pi$
$558$	0	0
$559$	−1.05702	−0.0447073
$560$	0	0
$561$	−28.6169	−1.20821
$562$	0	0
$563$	−21.5818	−0.909564	−0.454782	−	0.890603i	$-0.650283\pi$
$563$	−21.5818	−0.909564	−0.454782	+	0.890603i	$0.650283\pi$
$564$	0	0
$565$	−21.9278	−0.922511
$566$	0	0
$567$	1.63329	0.0685919
$568$	0	0
$569$	−20.3038	−0.851180	−0.425590	−	0.904916i	$-0.639933\pi$
$569$	−20.3038	−0.851180	−0.425590	+	0.904916i	$0.639933\pi$
$570$	0	0
$571$	−39.1620	−1.63888	−0.819439	−	0.573166i	$-0.805715\pi$
$571$	−39.1620	−1.63888	−0.819439	+	0.573166i	$0.805715\pi$
$572$	0	0
$573$	2.88273	0.120428
$574$	0	0
$575$	−14.8213	−0.618089
$576$	0	0
$577$	27.3853	1.14006	0.570032	−	0.821623i	$-0.306931\pi$
$577$	27.3853	1.14006	0.570032	+	0.821623i	$0.306931\pi$
$578$	0	0
$579$	6.32110	0.262696
$580$	0	0
$581$	1.30384	0.0540922
$582$	0	0
$583$	4.51298	0.186908
$584$	0	0
$585$	−2.35285	−0.0972783
$586$	0	0
$587$	−27.0651	−1.11710	−0.558549	−	0.829472i	$-0.688642\pi$
$587$	−27.0651	−1.11710	−0.558549	+	0.829472i	$0.688642\pi$
$588$	0	0
$589$	47.1153	1.94135
$590$	0	0
$591$	−21.5158	−0.885042
$592$	0	0
$593$	13.0695	0.536701	0.268350	−	0.963321i	$-0.413522\pi$
$593$	13.0695	0.536701	0.268350	+	0.963321i	$0.413522\pi$
$594$	0	0
$595$	−19.4855	−0.798827
$596$	0	0
$597$	−20.7287	−0.848369
$598$	0	0
$599$	15.3169	0.625832	0.312916	−	0.949781i	$-0.398694\pi$
$599$	15.3169	0.625832	0.312916	+	0.949781i	$0.398694\pi$
$600$	0	0
$601$	−35.6393	−1.45376	−0.726879	−	0.686766i	$-0.759030\pi$
$601$	−35.6393	−1.45376	−0.726879	+	0.686766i	$0.759030\pi$
$602$	0	0
$603$	15.1051	0.615128
$604$	0	0
$605$	4.95987	0.201648
$606$	0	0
$607$	7.61659	0.309148	0.154574	−	0.987981i	$-0.450599\pi$
$607$	7.61659	0.309148	0.154574	+	0.987981i	$0.450599\pi$
$608$	0	0
$609$	12.1281	0.491457
$610$	0	0
$611$	19.0090	0.769022
$612$	0	0
$613$	13.6662	0.551973	0.275987	−	0.961161i	$-0.410995\pi$
$613$	13.6662	0.551973	0.275987	+	0.961161i	$0.410995\pi$
$614$	0	0
$615$	9.11406	0.367514
$616$	0	0
$617$	5.65828	0.227794	0.113897	−	0.993493i	$-0.463667\pi$
$617$	5.65828	0.227794	0.113897	+	0.993493i	$0.463667\pi$
$618$	0	0
$619$	−24.0090	−0.965003	−0.482501	−	0.875895i	$-0.660272\pi$
$619$	−24.0090	−0.965003	−0.482501	+	0.875895i	$0.660272\pi$
$620$	0	0
$621$	5.83805	0.234273
$622$	0	0
$623$	5.10508	0.204531
$624$	0	0
$625$	−5.86115	−0.234446
$626$	0	0
$627$	27.3060	1.09050
$628$	0	0
$629$	32.4044	1.29205
$630$	0	0
$631$	−5.42902	−0.216126	−0.108063	−	0.994144i	$-0.534465\pi$
$631$	−5.42902	−0.216126	−0.108063	+	0.994144i	$0.534465\pi$
$632$	0	0
$633$	−4.49028	−0.178473
$634$	0	0
$635$	18.0078	0.714619
$636$	0	0
$637$	6.49738	0.257436
$638$	0	0
$639$	7.00364	0.277060
$640$	0	0
$641$	−25.7998	−1.01903	−0.509515	−	0.860462i	$-0.670175\pi$
$641$	−25.7998	−1.01903	−0.509515	+	0.860462i	$0.670175\pi$
$642$	0	0
$643$	50.4177	1.98828	0.994140	−	0.108097i	$-0.0344758\pi$
$643$	50.4177	1.98828	0.994140	+	0.108097i	$0.0344758\pi$
$644$	0	0
$645$	1.10573	0.0435380
$646$	0	0
$647$	19.3312	0.759990	0.379995	−	0.924989i	$-0.375926\pi$
$647$	19.3312	0.759990	0.379995	+	0.924989i	$0.375926\pi$
$648$	0	0
$649$	−19.7893	−0.776799
$650$	0	0
$651$	10.6053	0.415654
$652$	0	0
$653$	−6.59412	−0.258048	−0.129024	−	0.991641i	$-0.541184\pi$
$653$	−6.59412	−0.258048	−0.129024	+	0.991641i	$0.541184\pi$
$654$	0	0
$655$	27.9243	1.09109
$656$	0	0
$657$	−6.66022	−0.259840
$658$	0	0
$659$	−36.8744	−1.43642	−0.718212	−	0.695824i	$-0.755039\pi$
$659$	−36.8744	−1.43642	−0.718212	+	0.695824i	$0.755039\pi$
$660$	0	0
$661$	−13.0611	−0.508018	−0.254009	−	0.967202i	$-0.581749\pi$
$661$	−13.0611	−0.508018	−0.254009	+	0.967202i	$0.581749\pi$
$662$	0	0
$663$	11.4047	0.442920
$664$	0	0
$665$	18.5929	0.721001
$666$	0	0
$667$	43.3509	1.67855
$668$	0	0
$669$	−15.6458	−0.604903
$670$	0	0
$671$	−38.5017	−1.48634
$672$	0	0
$673$	−30.2792	−1.16718	−0.583588	−	0.812050i	$-0.698352\pi$
$673$	−30.2792	−1.16718	−0.583588	+	0.812050i	$0.698352\pi$
$674$	0	0
$675$	−2.53874	−0.0977160
$676$	0	0
$677$	28.0357	1.07750	0.538750	−	0.842466i	$-0.318897\pi$
$677$	28.0357	1.07750	0.538750	+	0.842466i	$0.318897\pi$
$678$	0	0
$679$	7.59831	0.291596
$680$	0	0
$681$	10.3909	0.398179
$682$	0	0
$683$	−32.1635	−1.23070	−0.615351	−	0.788254i	$-0.710986\pi$
$683$	−32.1635	−1.23070	−0.615351	+	0.788254i	$0.710986\pi$
$684$	0	0
$685$	4.09878	0.156606
$686$	0	0
$687$	2.33676	0.0891528
$688$	0	0
$689$	−1.79855	−0.0685194
$690$	0	0
$691$	−33.2177	−1.26366	−0.631831	−	0.775106i	$-0.717696\pi$
$691$	−33.2177	−1.26366	−0.631831	+	0.775106i	$0.717696\pi$
$692$	0	0
$693$	6.14637	0.233481
$694$	0	0
$695$	10.6768	0.404995
$696$	0	0
$697$	−44.1775	−1.67334
$698$	0	0
$699$	−23.7034	−0.896546
$700$	0	0
$701$	−5.37269	−0.202924	−0.101462	−	0.994839i	$-0.532352\pi$
$701$	−5.37269	−0.202924	−0.101462	+	0.994839i	$0.532352\pi$
$702$	0	0
$703$	−30.9200	−1.16617
$704$	0	0
$705$	−19.8849	−0.748909
$706$	0	0
$707$	13.1937	0.496199
$708$	0	0
$709$	−12.7486	−0.478784	−0.239392	−	0.970923i	$-0.576948\pi$
$709$	−12.7486	−0.478784	−0.239392	+	0.970923i	$0.576948\pi$
$710$	0	0
$711$	1.98822	0.0745641
$712$	0	0
$713$	37.9076	1.41965
$714$	0	0
$715$	−8.85418	−0.331127
$716$	0	0
$717$	−4.34830	−0.162390
$718$	0	0
$719$	−23.5410	−0.877932	−0.438966	−	0.898504i	$-0.644655\pi$
$719$	−23.5410	−0.877932	−0.438966	+	0.898504i	$0.644655\pi$
$720$	0	0
$721$	31.1098	1.15859
$722$	0	0
$723$	12.3513	0.459349
$724$	0	0
$725$	−18.8516	−0.700130
$726$	0	0
$727$	−1.94077	−0.0719792	−0.0359896	−	0.999352i	$-0.511458\pi$
$727$	−1.94077	−0.0719792	−0.0359896	+	0.999352i	$0.511458\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.35966	−0.198234
$732$	0	0
$733$	−37.8358	−1.39750	−0.698749	−	0.715367i	$-0.746260\pi$
$733$	−37.8358	−1.39750	−0.698749	+	0.715367i	$0.746260\pi$
$734$	0	0
$735$	−6.79678	−0.250703
$736$	0	0
$737$	56.8432	2.09385
$738$	0	0
$739$	−23.8057	−0.875706	−0.437853	−	0.899046i	$-0.644261\pi$
$739$	−23.8057	−0.875706	−0.437853	+	0.899046i	$0.644261\pi$
$740$	0	0
$741$	−10.8822	−0.399769
$742$	0	0
$743$	−38.4856	−1.41190	−0.705949	−	0.708263i	$-0.749479\pi$
$743$	−38.4856	−1.41190	−0.705949	+	0.708263i	$0.749479\pi$
$744$	0	0
$745$	11.1425	0.408231
$746$	0	0
$747$	0.798286	0.0292078
$748$	0	0
$749$	26.6106	0.972332
$750$	0	0
$751$	8.81177	0.321546	0.160773	−	0.986991i	$-0.448601\pi$
$751$	8.81177	0.321546	0.160773	+	0.986991i	$0.448601\pi$
$752$	0	0
$753$	17.0681	0.621995
$754$	0	0
$755$	−23.1217	−0.841483
$756$	0	0
$757$	−39.8876	−1.44974	−0.724869	−	0.688886i	$-0.758100\pi$
$757$	−39.8876	−1.44974	−0.724869	+	0.688886i	$0.758100\pi$
$758$	0	0
$759$	21.9696	0.797446
$760$	0	0
$761$	−1.05887	−0.0383839	−0.0191920	−	0.999816i	$-0.506109\pi$
$761$	−1.05887	−0.0383839	−0.0191920	+	0.999816i	$0.506109\pi$
$762$	0	0
$763$	−19.3737	−0.701376
$764$	0	0
$765$	−11.9302	−0.431337
$766$	0	0
$767$	7.88663	0.284770
$768$	0	0
$769$	5.40978	0.195082	0.0975408	−	0.995232i	$-0.468902\pi$
$769$	5.40978	0.195082	0.0975408	+	0.995232i	$0.468902\pi$
$770$	0	0
$771$	−30.8167	−1.10984
$772$	0	0
$773$	27.7799	0.999174	0.499587	−	0.866264i	$-0.333485\pi$
$773$	27.7799	0.999174	0.499587	+	0.866264i	$0.333485\pi$
$774$	0	0
$775$	−16.4845	−0.592141
$776$	0	0
$777$	−6.95985	−0.249683
$778$	0	0
$779$	42.1538	1.51032
$780$	0	0
$781$	26.3559	0.943090
$782$	0	0
$783$	7.42558	0.265369
$784$	0	0
$785$	−8.54261	−0.304899
$786$	0	0
$787$	−11.9613	−0.426373	−0.213187	−	0.977011i	$-0.568384\pi$
$787$	−11.9613	−0.426373	−0.213187	+	0.977011i	$0.568384\pi$
$788$	0	0
$789$	−28.3036	−1.00763
$790$	0	0
$791$	−22.8287	−0.811694
$792$	0	0
$793$	15.3440	0.544883
$794$	0	0
$795$	1.88143	0.0667274
$796$	0	0
$797$	33.3418	1.18103	0.590514	−	0.807027i	$-0.298925\pi$
$797$	33.3418	1.18103	0.590514	+	0.807027i	$0.298925\pi$
$798$	0	0
$799$	96.3857	3.40988
$800$	0	0
$801$	3.12564	0.110439
$802$	0	0
$803$	−25.0636	−0.884475
$804$	0	0
$805$	14.9593	0.527246
$806$	0	0
$807$	27.0855	0.953454
$808$	0	0
$809$	−28.5635	−1.00424	−0.502119	−	0.864799i	$-0.667446\pi$
$809$	−28.5635	−1.00424	−0.502119	+	0.864799i	$0.667446\pi$
$810$	0	0
$811$	36.3738	1.27726	0.638629	−	0.769515i	$-0.279502\pi$
$811$	36.3738	1.27726	0.638629	+	0.769515i	$0.279502\pi$
$812$	0	0
$813$	8.72254	0.305913
$814$	0	0
$815$	−14.6768	−0.514108
$816$	0	0
$817$	5.11415	0.178921
$818$	0	0
$819$	−2.44951	−0.0855927
$820$	0	0
$821$	−14.7152	−0.513565	−0.256783	−	0.966469i	$-0.582662\pi$
$821$	−14.7152	−0.513565	−0.256783	+	0.966469i	$0.582662\pi$
$822$	0	0
$823$	5.54170	0.193172	0.0965858	−	0.995325i	$-0.469208\pi$
$823$	5.54170	0.193172	0.0965858	+	0.995325i	$0.469208\pi$
$824$	0	0
$825$	−9.55371	−0.332617
$826$	0	0
$827$	20.1402	0.700343	0.350171	−	0.936686i	$-0.386123\pi$
$827$	20.1402	0.700343	0.350171	+	0.936686i	$0.386123\pi$
$828$	0	0
$829$	1.23079	0.0427471	0.0213735	−	0.999772i	$-0.493196\pi$
$829$	1.23079	0.0427471	0.0213735	+	0.999772i	$0.493196\pi$
$830$	0	0
$831$	7.36864	0.255615
$832$	0	0
$833$	32.9452	1.14148
$834$	0	0
$835$	−29.2743	−1.01308
$836$	0	0
$837$	6.49320	0.224438
$838$	0	0
$839$	47.1283	1.62705	0.813524	−	0.581531i	$-0.197546\pi$
$839$	47.1283	1.62705	0.813524	+	0.581531i	$0.197546\pi$
$840$	0	0
$841$	26.1392	0.901353
$842$	0	0
$843$	9.29084	0.319994
$844$	0	0
$845$	−16.8663	−0.580218
$846$	0	0
$847$	5.16364	0.177425
$848$	0	0
$849$	26.2886	0.902222
$850$	0	0
$851$	−24.8773	−0.852783
$852$	0	0
$853$	17.8453	0.611012	0.305506	−	0.952190i	$-0.401174\pi$
$853$	17.8453	0.611012	0.305506	+	0.952190i	$0.401174\pi$
$854$	0	0
$855$	11.3837	0.389314
$856$	0	0
$857$	−7.21709	−0.246531	−0.123265	−	0.992374i	$-0.539337\pi$
$857$	−7.21709	−0.246531	−0.123265	+	0.992374i	$0.539337\pi$
$858$	0	0
$859$	−38.7568	−1.32236	−0.661182	−	0.750226i	$-0.729945\pi$
$859$	−38.7568	−1.32236	−0.661182	+	0.750226i	$0.729945\pi$
$860$	0	0
$861$	9.48849	0.323367
$862$	0	0
$863$	−11.4580	−0.390035	−0.195017	−	0.980800i	$-0.562476\pi$
$863$	−11.4580	−0.390035	−0.195017	+	0.980800i	$0.562476\pi$
$864$	0	0
$865$	1.23473	0.0419820
$866$	0	0
$867$	40.8277	1.38658
$868$	0	0
$869$	7.48202	0.253810
$870$	0	0
$871$	−22.6537	−0.767591
$872$	0	0
$873$	4.65214	0.157451
$874$	0	0
$875$	−19.3171	−0.653037
$876$	0	0
$877$	40.5097	1.36792	0.683958	−	0.729521i	$-0.260257\pi$
$877$	40.5097	1.36792	0.683958	+	0.729521i	$0.260257\pi$
$878$	0	0
$879$	21.9738	0.741157
$880$	0	0
$881$	−38.7955	−1.30705	−0.653527	−	0.756903i	$-0.726711\pi$
$881$	−38.7955	−1.30705	−0.653527	+	0.756903i	$0.726711\pi$
$882$	0	0
$883$	−15.2459	−0.513065	−0.256533	−	0.966536i	$-0.582580\pi$
$883$	−15.2459	−0.513065	−0.256533	+	0.966536i	$0.582580\pi$
$884$	0	0
$885$	−8.25003	−0.277322
$886$	0	0
$887$	−37.1707	−1.24807	−0.624035	−	0.781396i	$-0.714508\pi$
$887$	−37.1707	−1.24807	−0.624035	+	0.781396i	$0.714508\pi$
$888$	0	0
$889$	18.7476	0.628775
$890$	0	0
$891$	3.76318	0.126071
$892$	0	0
$893$	−91.9705	−3.07767
$894$	0	0
$895$	−13.8569	−0.463185
$896$	0	0
$897$	−8.75553	−0.292338
$898$	0	0
$899$	48.2158	1.60809
$900$	0	0
$901$	−9.11961	−0.303818
$902$	0	0
$903$	1.15115	0.0383080
$904$	0	0
$905$	16.2098	0.538832
$906$	0	0
$907$	−44.2704	−1.46997	−0.734986	−	0.678082i	$-0.762811\pi$
$907$	−44.2704	−1.46997	−0.734986	+	0.678082i	$0.762811\pi$
$908$	0	0
$909$	8.07795	0.267929
$910$	0	0
$911$	−18.9861	−0.629038	−0.314519	−	0.949251i	$-0.601843\pi$
$911$	−18.9861	−0.629038	−0.314519	+	0.949251i	$0.601843\pi$
$912$	0	0
$913$	3.00409	0.0994209
$914$	0	0
$915$	−16.0511	−0.530632
$916$	0	0
$917$	29.0715	0.960027
$918$	0	0
$919$	−26.9981	−0.890584	−0.445292	−	0.895385i	$-0.646900\pi$
$919$	−26.9981	−0.890584	−0.445292	+	0.895385i	$0.646900\pi$
$920$	0	0
$921$	7.61970	0.251078
$922$	0	0
$923$	−10.5036	−0.345731
$924$	0	0
$925$	10.8181	0.355699
$926$	0	0
$927$	19.0473	0.625596
$928$	0	0
$929$	42.2636	1.38662	0.693312	−	0.720638i	$-0.256151\pi$
$929$	42.2636	1.38662	0.693312	+	0.720638i	$0.256151\pi$
$930$	0	0
$931$	−31.4360	−1.03027
$932$	0	0
$933$	19.8702	0.650522
$934$	0	0
$935$	−44.8954	−1.46824
$936$	0	0
$937$	3.49025	0.114022	0.0570108	−	0.998374i	$-0.481843\pi$
$937$	3.49025	0.114022	0.0570108	+	0.998374i	$0.481843\pi$
$938$	0	0
$939$	−17.3373	−0.565781
$940$	0	0
$941$	−28.1245	−0.916832	−0.458416	−	0.888738i	$-0.651583\pi$
$941$	−28.1245	−0.916832	−0.458416	+	0.888738i	$0.651583\pi$
$942$	0	0
$943$	33.9157	1.10445
$944$	0	0
$945$	2.56238	0.0833542
$946$	0	0
$947$	−7.73205	−0.251258	−0.125629	−	0.992077i	$-0.540095\pi$
$947$	−7.73205	−0.251258	−0.125629	+	0.992077i	$0.540095\pi$
$948$	0	0
$949$	9.98857	0.324243
$950$	0	0
$951$	26.4642	0.858162
$952$	0	0
$953$	−15.6331	−0.506405	−0.253203	−	0.967413i	$-0.581484\pi$
$953$	−15.6331	−0.506405	−0.253203	+	0.967413i	$0.581484\pi$
$954$	0	0
$955$	4.52255	0.146346
$956$	0	0
$957$	27.9438	0.903293
$958$	0	0
$959$	4.26716	0.137794
$960$	0	0
$961$	11.1616	0.360053
$962$	0	0
$963$	16.2926	0.525023
$964$	0	0
$965$	9.91680	0.319233
$966$	0	0
$967$	16.1778	0.520242	0.260121	−	0.965576i	$-0.416237\pi$
$967$	16.1778	0.520242	0.260121	+	0.965576i	$0.416237\pi$
$968$	0	0
$969$	−55.1787	−1.77260
$970$	0	0
$971$	45.5466	1.46166	0.730829	−	0.682560i	$-0.239134\pi$
$971$	45.5466	1.46166	0.730829	+	0.682560i	$0.239134\pi$
$972$	0	0
$973$	11.1155	0.356345
$974$	0	0
$975$	3.80743	0.121935
$976$	0	0
$977$	−31.2601	−1.00010	−0.500049	−	0.865997i	$-0.666685\pi$
$977$	−31.2601	−1.00010	−0.500049	+	0.865997i	$0.666685\pi$
$978$	0	0
$979$	11.7623	0.375925
$980$	0	0
$981$	−11.8618	−0.378717
$982$	0	0
$983$	37.0087	1.18039	0.590197	−	0.807259i	$-0.299050\pi$
$983$	37.0087	1.18039	0.590197	+	0.807259i	$0.299050\pi$
$984$	0	0
$985$	−33.7549	−1.07552
$986$	0	0
$987$	−20.7018	−0.658947
$988$	0	0
$989$	4.11469	0.130840
$990$	0	0
$991$	−33.5090	−1.06445	−0.532225	−	0.846603i	$-0.678644\pi$
$991$	−33.5090	−1.06445	−0.532225	+	0.846603i	$0.678644\pi$
$992$	0	0
$993$	−1.65818	−0.0526207
$994$	0	0
$995$	−32.5200	−1.03095
$996$	0	0
$997$	−47.9084	−1.51727	−0.758637	−	0.651514i	$-0.774134\pi$
$997$	−47.9084	−1.51727	−0.758637	+	0.651514i	$0.774134\pi$
$998$	0	0
$999$	−4.26124	−0.134820

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.13	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.13	✓	24	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.56884 q^{5} +1.63329 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.56884 q^{5} +1.63329 q^{7} +1.00000 q^{9} +3.76318 q^{11} -1.49974 q^{13} +1.56884 q^{15} -7.60445 q^{17} +7.25611 q^{19} +1.63329 q^{21} +5.83805 q^{23} -2.53874 q^{25} +1.00000 q^{27} +7.42558 q^{29} +6.49320 q^{31} +3.76318 q^{33} +2.56238 q^{35} -4.26124 q^{37} -1.49974 q^{39} +5.80942 q^{41} +0.704806 q^{43} +1.56884 q^{45} -12.6749 q^{47} -4.33235 q^{49} -7.60445 q^{51} +1.19925 q^{53} +5.90383 q^{55} +7.25611 q^{57} -5.25868 q^{59} -10.2312 q^{61} +1.63329 q^{63} -2.35285 q^{65} +15.1051 q^{67} +5.83805 q^{69} +7.00364 q^{71} -6.66022 q^{73} -2.53874 q^{75} +6.14637 q^{77} +1.98822 q^{79} +1.00000 q^{81} +0.798286 q^{83} -11.9302 q^{85} +7.42558 q^{87} +3.12564 q^{89} -2.44951 q^{91} +6.49320 q^{93} +11.3837 q^{95} +4.65214 q^{97} +3.76318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Introduction

L-functions

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Database

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