LMFDB - Embedded newform 6036.2.a.h.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.64052 q^{5} +2.86228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.64052 q^{5} +2.86228 q^{7} +1.00000 q^{9} -3.28758 q^{11} -4.30362 q^{13} -3.64052 q^{15} +6.11147 q^{17} -0.135042 q^{19} +2.86228 q^{21} +2.55641 q^{23} +8.25341 q^{25} +1.00000 q^{27} +2.37199 q^{29} -10.0917 q^{31} -3.28758 q^{33} -10.4202 q^{35} +3.35365 q^{37} -4.30362 q^{39} +9.70499 q^{41} -10.1736 q^{43} -3.64052 q^{45} +4.02094 q^{47} +1.19264 q^{49} +6.11147 q^{51} +1.81779 q^{53} +11.9685 q^{55} -0.135042 q^{57} -3.96537 q^{59} +1.10697 q^{61} +2.86228 q^{63} +15.6674 q^{65} -15.1209 q^{67} +2.55641 q^{69} +1.58601 q^{71} -11.6466 q^{73} +8.25341 q^{75} -9.40998 q^{77} +16.6826 q^{79} +1.00000 q^{81} -5.41422 q^{83} -22.2490 q^{85} +2.37199 q^{87} +15.8071 q^{89} -12.3182 q^{91} -10.0917 q^{93} +0.491622 q^{95} -2.19887 q^{97} -3.28758 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.64052	−1.62809	−0.814046	−	0.580801i	$-0.802739\pi$
$5$	−3.64052	−1.62809	−0.814046	+	0.580801i	$0.802739\pi$
$6$	0	0
$7$	2.86228	1.08184	0.540920	−	0.841074i	$-0.318076\pi$
$7$	2.86228	1.08184	0.540920	+	0.841074i	$0.318076\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.28758	−0.991243	−0.495622	−	0.868539i	$-0.665060\pi$
$11$	−3.28758	−0.991243	−0.495622	+	0.868539i	$0.665060\pi$
$12$	0	0
$13$	−4.30362	−1.19361	−0.596805	−	0.802386i	$-0.703563\pi$
$13$	−4.30362	−1.19361	−0.596805	+	0.802386i	$0.703563\pi$
$14$	0	0
$15$	−3.64052	−0.939979
$16$	0	0
$17$	6.11147	1.48225	0.741125	−	0.671367i	$-0.234292\pi$
$17$	6.11147	1.48225	0.741125	+	0.671367i	$0.234292\pi$
$18$	0	0
$19$	−0.135042	−0.0309807	−0.0154903	−	0.999880i	$-0.504931\pi$
$19$	−0.135042	−0.0309807	−0.0154903	+	0.999880i	$0.504931\pi$
$20$	0	0
$21$	2.86228	0.624601
$22$	0	0
$23$	2.55641	0.533048	0.266524	−	0.963828i	$-0.414125\pi$
$23$	2.55641	0.533048	0.266524	+	0.963828i	$0.414125\pi$
$24$	0	0
$25$	8.25341	1.65068
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.37199	0.440468	0.220234	−	0.975447i	$-0.429318\pi$
$29$	2.37199	0.440468	0.220234	+	0.975447i	$0.429318\pi$
$30$	0	0
$31$	−10.0917	−1.81253	−0.906264	−	0.422713i	$-0.861078\pi$
$31$	−10.0917	−1.81253	−0.906264	+	0.422713i	$0.861078\pi$
$32$	0	0
$33$	−3.28758	−0.572295
$34$	0	0
$35$	−10.4202	−1.76133
$36$	0	0
$37$	3.35365	0.551337	0.275668	−	0.961253i	$-0.411101\pi$
$37$	3.35365	0.551337	0.275668	+	0.961253i	$0.411101\pi$
$38$	0	0
$39$	−4.30362	−0.689131
$40$	0	0
$41$	9.70499	1.51567	0.757833	−	0.652449i	$-0.226258\pi$
$41$	9.70499	1.51567	0.757833	+	0.652449i	$0.226258\pi$
$42$	0	0
$43$	−10.1736	−1.55146	−0.775728	−	0.631067i	$-0.782617\pi$
$43$	−10.1736	−1.55146	−0.775728	+	0.631067i	$0.782617\pi$
$44$	0	0
$45$	−3.64052	−0.542697
$46$	0	0
$47$	4.02094	0.586514	0.293257	−	0.956034i	$-0.405261\pi$
$47$	4.02094	0.586514	0.293257	+	0.956034i	$0.405261\pi$
$48$	0	0
$49$	1.19264	0.170378
$50$	0	0
$51$	6.11147	0.855778
$52$	0	0
$53$	1.81779	0.249692	0.124846	−	0.992176i	$-0.460156\pi$
$53$	1.81779	0.249692	0.124846	+	0.992176i	$0.460156\pi$
$54$	0	0
$55$	11.9685	1.61383
$56$	0	0
$57$	−0.135042	−0.0178867
$58$	0	0
$59$	−3.96537	−0.516247	−0.258124	−	0.966112i	$-0.583104\pi$
$59$	−3.96537	−0.516247	−0.258124	+	0.966112i	$0.583104\pi$
$60$	0	0
$61$	1.10697	0.141733	0.0708666	−	0.997486i	$-0.477424\pi$
$61$	1.10697	0.141733	0.0708666	+	0.997486i	$0.477424\pi$
$62$	0	0
$63$	2.86228	0.360613
$64$	0	0
$65$	15.6674	1.94331
$66$	0	0
$67$	−15.1209	−1.84731	−0.923653	−	0.383229i	$-0.874812\pi$
$67$	−15.1209	−1.84731	−0.923653	+	0.383229i	$0.874812\pi$
$68$	0	0
$69$	2.55641	0.307755
$70$	0	0
$71$	1.58601	0.188225	0.0941123	−	0.995562i	$-0.469999\pi$
$71$	1.58601	0.188225	0.0941123	+	0.995562i	$0.469999\pi$
$72$	0	0
$73$	−11.6466	−1.36313	−0.681567	−	0.731756i	$-0.738701\pi$
$73$	−11.6466	−1.36313	−0.681567	+	0.731756i	$0.738701\pi$
$74$	0	0
$75$	8.25341	0.953021
$76$	0	0
$77$	−9.40998	−1.07237
$78$	0	0
$79$	16.6826	1.87694	0.938469	−	0.345364i	$-0.112244\pi$
$79$	16.6826	1.87694	0.938469	+	0.345364i	$0.112244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.41422	−0.594288	−0.297144	−	0.954833i	$-0.596034\pi$
$83$	−5.41422	−0.594288	−0.297144	+	0.954833i	$0.596034\pi$
$84$	0	0
$85$	−22.2490	−2.41324
$86$	0	0
$87$	2.37199	0.254304
$88$	0	0
$89$	15.8071	1.67555	0.837774	−	0.546017i	$-0.183857\pi$
$89$	15.8071	1.67555	0.837774	+	0.546017i	$0.183857\pi$
$90$	0	0
$91$	−12.3182	−1.29130
$92$	0	0
$93$	−10.0917	−1.04646
$94$	0	0
$95$	0.491622	0.0504394
$96$	0	0
$97$	−2.19887	−0.223262	−0.111631	−	0.993750i	$-0.535607\pi$
$97$	−2.19887	−0.223262	−0.111631	+	0.993750i	$0.535607\pi$
$98$	0	0
$99$	−3.28758	−0.330414
$100$	0	0
$101$	9.25034	0.920443	0.460222	−	0.887804i	$-0.347770\pi$
$101$	9.25034	0.920443	0.460222	+	0.887804i	$0.347770\pi$
$102$	0	0
$103$	15.0022	1.47821	0.739107	−	0.673589i	$-0.235248\pi$
$103$	15.0022	1.47821	0.739107	+	0.673589i	$0.235248\pi$
$104$	0	0
$105$	−10.4202	−1.01691
$106$	0	0
$107$	14.8129	1.43202	0.716009	−	0.698091i	$-0.245967\pi$
$107$	14.8129	1.43202	0.716009	+	0.698091i	$0.245967\pi$
$108$	0	0
$109$	15.1079	1.44707	0.723536	−	0.690287i	$-0.242516\pi$
$109$	15.1079	1.44707	0.723536	+	0.690287i	$0.242516\pi$
$110$	0	0
$111$	3.35365	0.318314
$112$	0	0
$113$	1.06412	0.100104	0.0500520	−	0.998747i	$-0.484061\pi$
$113$	1.06412	0.100104	0.0500520	+	0.998747i	$0.484061\pi$
$114$	0	0
$115$	−9.30666	−0.867850
$116$	0	0
$117$	−4.30362	−0.397870
$118$	0	0
$119$	17.4927	1.60356
$120$	0	0
$121$	−0.191805	−0.0174368
$122$	0	0
$123$	9.70499	0.875070
$124$	0	0
$125$	−11.8441	−1.05937
$126$	0	0
$127$	18.1780	1.61303	0.806517	−	0.591210i	$-0.201350\pi$
$127$	18.1780	1.61303	0.806517	+	0.591210i	$0.201350\pi$
$128$	0	0
$129$	−10.1736	−0.895734
$130$	0	0
$131$	−1.96530	−0.171709	−0.0858547	−	0.996308i	$-0.527362\pi$
$131$	−1.96530	−0.171709	−0.0858547	+	0.996308i	$0.527362\pi$
$132$	0	0
$133$	−0.386527	−0.0335161
$134$	0	0
$135$	−3.64052	−0.313326
$136$	0	0
$137$	17.8328	1.52356	0.761780	−	0.647836i	$-0.224326\pi$
$137$	17.8328	1.52356	0.761780	+	0.647836i	$0.224326\pi$
$138$	0	0
$139$	12.9396	1.09752	0.548762	−	0.835979i	$-0.315099\pi$
$139$	12.9396	1.09752	0.548762	+	0.835979i	$0.315099\pi$
$140$	0	0
$141$	4.02094	0.338624
$142$	0	0
$143$	14.1485	1.18316
$144$	0	0
$145$	−8.63529	−0.717122
$146$	0	0
$147$	1.19264	0.0983677
$148$	0	0
$149$	−20.2324	−1.65750	−0.828752	−	0.559616i	$-0.810949\pi$
$149$	−20.2324	−1.65750	−0.828752	+	0.559616i	$0.810949\pi$
$150$	0	0
$151$	3.11737	0.253688	0.126844	−	0.991923i	$-0.459515\pi$
$151$	3.11737	0.253688	0.126844	+	0.991923i	$0.459515\pi$
$152$	0	0
$153$	6.11147	0.494083
$154$	0	0
$155$	36.7392	2.95096
$156$	0	0
$157$	9.71309	0.775189	0.387594	−	0.921830i	$-0.373306\pi$
$157$	9.71309	0.775189	0.387594	+	0.921830i	$0.373306\pi$
$158$	0	0
$159$	1.81779	0.144160
$160$	0	0
$161$	7.31715	0.576672
$162$	0	0
$163$	−11.8308	−0.926658	−0.463329	−	0.886186i	$-0.653345\pi$
$163$	−11.8308	−0.926658	−0.463329	+	0.886186i	$0.653345\pi$
$164$	0	0
$165$	11.9685	0.931748
$166$	0	0
$167$	6.70008	0.518468	0.259234	−	0.965815i	$-0.416530\pi$
$167$	6.70008	0.518468	0.259234	+	0.965815i	$0.416530\pi$
$168$	0	0
$169$	5.52117	0.424706
$170$	0	0
$171$	−0.135042	−0.0103269
$172$	0	0
$173$	25.3435	1.92683	0.963415	−	0.268015i	$-0.0863675\pi$
$173$	25.3435	1.92683	0.963415	+	0.268015i	$0.0863675\pi$
$174$	0	0
$175$	23.6236	1.78577
$176$	0	0
$177$	−3.96537	−0.298055
$178$	0	0
$179$	15.9462	1.19188	0.595939	−	0.803030i	$-0.296780\pi$
$179$	15.9462	1.19188	0.595939	+	0.803030i	$0.296780\pi$
$180$	0	0
$181$	12.3006	0.914294	0.457147	−	0.889391i	$-0.348871\pi$
$181$	12.3006	0.914294	0.457147	+	0.889391i	$0.348871\pi$
$182$	0	0
$183$	1.10697	0.0818297
$184$	0	0
$185$	−12.2090	−0.897626
$186$	0	0
$187$	−20.0920	−1.46927
$188$	0	0
$189$	2.86228	0.208200
$190$	0	0
$191$	−2.73072	−0.197588	−0.0987940	−	0.995108i	$-0.531498\pi$
$191$	−2.73072	−0.197588	−0.0987940	+	0.995108i	$0.531498\pi$
$192$	0	0
$193$	−18.4977	−1.33149	−0.665747	−	0.746178i	$-0.731887\pi$
$193$	−18.4977	−1.33149	−0.665747	+	0.746178i	$0.731887\pi$
$194$	0	0
$195$	15.6674	1.12197
$196$	0	0
$197$	−1.15807	−0.0825089	−0.0412545	−	0.999149i	$-0.513135\pi$
$197$	−1.15807	−0.0825089	−0.0412545	+	0.999149i	$0.513135\pi$
$198$	0	0
$199$	−1.68890	−0.119723	−0.0598614	−	0.998207i	$-0.519066\pi$
$199$	−1.68890	−0.119723	−0.0598614	+	0.998207i	$0.519066\pi$
$200$	0	0
$201$	−15.1209	−1.06654
$202$	0	0
$203$	6.78930	0.476516
$204$	0	0
$205$	−35.3313	−2.46764
$206$	0	0
$207$	2.55641	0.177683
$208$	0	0
$209$	0.443961	0.0307094
$210$	0	0
$211$	−5.92456	−0.407864	−0.203932	−	0.978985i	$-0.565372\pi$
$211$	−5.92456	−0.407864	−0.203932	+	0.978985i	$0.565372\pi$
$212$	0	0
$213$	1.58601	0.108671
$214$	0	0
$215$	37.0372	2.52591
$216$	0	0
$217$	−28.8853	−1.96086
$218$	0	0
$219$	−11.6466	−0.787006
$220$	0	0
$221$	−26.3015	−1.76923
$222$	0	0
$223$	−24.4950	−1.64031	−0.820153	−	0.572145i	$-0.806112\pi$
$223$	−24.4950	−1.64031	−0.820153	+	0.572145i	$0.806112\pi$
$224$	0	0
$225$	8.25341	0.550227
$226$	0	0
$227$	−8.95090	−0.594092	−0.297046	−	0.954863i	$-0.596001\pi$
$227$	−8.95090	−0.594092	−0.297046	+	0.954863i	$0.596001\pi$
$228$	0	0
$229$	23.4521	1.54976	0.774880	−	0.632109i	$-0.217810\pi$
$229$	23.4521	1.54976	0.774880	+	0.632109i	$0.217810\pi$
$230$	0	0
$231$	−9.40998	−0.619131
$232$	0	0
$233$	−0.754123	−0.0494042	−0.0247021	−	0.999695i	$-0.507864\pi$
$233$	−0.754123	−0.0494042	−0.0247021	+	0.999695i	$0.507864\pi$
$234$	0	0
$235$	−14.6383	−0.954899
$236$	0	0
$237$	16.6826	1.08365
$238$	0	0
$239$	0.223949	0.0144861	0.00724303	−	0.999974i	$-0.497694\pi$
$239$	0.223949	0.0144861	0.00724303	+	0.999974i	$0.497694\pi$
$240$	0	0
$241$	−8.09293	−0.521312	−0.260656	−	0.965432i	$-0.583939\pi$
$241$	−8.09293	−0.521312	−0.260656	+	0.965432i	$0.583939\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.34185	−0.277391
$246$	0	0
$247$	0.581169	0.0369789
$248$	0	0
$249$	−5.41422	−0.343112
$250$	0	0
$251$	3.94011	0.248698	0.124349	−	0.992239i	$-0.460316\pi$
$251$	3.94011	0.248698	0.124349	+	0.992239i	$0.460316\pi$
$252$	0	0
$253$	−8.40440	−0.528380
$254$	0	0
$255$	−22.2490	−1.39328
$256$	0	0
$257$	3.84186	0.239649	0.119824	−	0.992795i	$-0.461767\pi$
$257$	3.84186	0.239649	0.119824	+	0.992795i	$0.461767\pi$
$258$	0	0
$259$	9.59908	0.596458
$260$	0	0
$261$	2.37199	0.146823
$262$	0	0
$263$	4.68245	0.288732	0.144366	−	0.989524i	$-0.453886\pi$
$263$	4.68245	0.288732	0.144366	+	0.989524i	$0.453886\pi$
$264$	0	0
$265$	−6.61770	−0.406522
$266$	0	0
$267$	15.8071	0.967378
$268$	0	0
$269$	6.53308	0.398329	0.199165	−	0.979966i	$-0.436177\pi$
$269$	6.53308	0.398329	0.199165	+	0.979966i	$0.436177\pi$
$270$	0	0
$271$	3.11279	0.189089	0.0945444	−	0.995521i	$-0.469861\pi$
$271$	3.11279	0.189089	0.0945444	+	0.995521i	$0.469861\pi$
$272$	0	0
$273$	−12.3182	−0.745530
$274$	0	0
$275$	−27.1338	−1.63623
$276$	0	0
$277$	12.7980	0.768956	0.384478	−	0.923134i	$-0.374381\pi$
$277$	12.7980	0.768956	0.384478	+	0.923134i	$0.374381\pi$
$278$	0	0
$279$	−10.0917	−0.604176
$280$	0	0
$281$	−0.544674	−0.0324925	−0.0162463	−	0.999868i	$-0.505172\pi$
$281$	−0.544674	−0.0324925	−0.0162463	+	0.999868i	$0.505172\pi$
$282$	0	0
$283$	30.5014	1.81312	0.906560	−	0.422078i	$-0.138699\pi$
$283$	30.5014	1.81312	0.906560	+	0.422078i	$0.138699\pi$
$284$	0	0
$285$	0.491622	0.0291212
$286$	0	0
$287$	27.7784	1.63971
$288$	0	0
$289$	20.3501	1.19707
$290$	0	0
$291$	−2.19887	−0.128900
$292$	0	0
$293$	−11.0640	−0.646369	−0.323184	−	0.946336i	$-0.604753\pi$
$293$	−11.0640	−0.646369	−0.323184	+	0.946336i	$0.604753\pi$
$294$	0	0
$295$	14.4360	0.840498
$296$	0	0
$297$	−3.28758	−0.190765
$298$	0	0
$299$	−11.0018	−0.636251
$300$	0	0
$301$	−29.1196	−1.67843
$302$	0	0
$303$	9.25034	0.531418
$304$	0	0
$305$	−4.02995	−0.230754
$306$	0	0
$307$	0.706571	0.0403261	0.0201631	−	0.999797i	$-0.493581\pi$
$307$	0.706571	0.0403261	0.0201631	+	0.999797i	$0.493581\pi$
$308$	0	0
$309$	15.0022	0.853447
$310$	0	0
$311$	−30.4096	−1.72437	−0.862185	−	0.506594i	$-0.830904\pi$
$311$	−30.4096	−1.72437	−0.862185	+	0.506594i	$0.830904\pi$
$312$	0	0
$313$	16.5723	0.936722	0.468361	−	0.883537i	$-0.344845\pi$
$313$	16.5723	0.936722	0.468361	+	0.883537i	$0.344845\pi$
$314$	0	0
$315$	−10.4202	−0.587111
$316$	0	0
$317$	21.0721	1.18353	0.591765	−	0.806111i	$-0.298431\pi$
$317$	21.0721	1.18353	0.591765	+	0.806111i	$0.298431\pi$
$318$	0	0
$319$	−7.79811	−0.436611
$320$	0	0
$321$	14.8129	0.826776
$322$	0	0
$323$	−0.825304	−0.0459211
$324$	0	0
$325$	−35.5196	−1.97027
$326$	0	0
$327$	15.1079	0.835467
$328$	0	0
$329$	11.5091	0.634515
$330$	0	0
$331$	−25.1394	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$331$	−25.1394	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$332$	0	0
$333$	3.35365	0.183779
$334$	0	0
$335$	55.0478	3.00758
$336$	0	0
$337$	5.98769	0.326170	0.163085	−	0.986612i	$-0.447855\pi$
$337$	5.98769	0.326170	0.163085	+	0.986612i	$0.447855\pi$
$338$	0	0
$339$	1.06412	0.0577951
$340$	0	0
$341$	33.1774	1.79666
$342$	0	0
$343$	−16.6223	−0.897518
$344$	0	0
$345$	−9.30666	−0.501054
$346$	0	0
$347$	35.2283	1.89116	0.945578	−	0.325395i	$-0.105497\pi$
$347$	35.2283	1.89116	0.945578	+	0.325395i	$0.105497\pi$
$348$	0	0
$349$	−0.986748	−0.0528194	−0.0264097	−	0.999651i	$-0.508407\pi$
$349$	−0.986748	−0.0528194	−0.0264097	+	0.999651i	$0.508407\pi$
$350$	0	0
$351$	−4.30362	−0.229710
$352$	0	0
$353$	−28.9746	−1.54216	−0.771082	−	0.636736i	$-0.780284\pi$
$353$	−28.9746	−1.54216	−0.771082	+	0.636736i	$0.780284\pi$
$354$	0	0
$355$	−5.77390	−0.306447
$356$	0	0
$357$	17.4927	0.925814
$358$	0	0
$359$	2.17489	0.114786	0.0573932	−	0.998352i	$-0.481721\pi$
$359$	2.17489	0.114786	0.0573932	+	0.998352i	$0.481721\pi$
$360$	0	0
$361$	−18.9818	−0.999040
$362$	0	0
$363$	−0.191805	−0.0100671
$364$	0	0
$365$	42.3998	2.21931
$366$	0	0
$367$	27.2229	1.42102	0.710512	−	0.703686i	$-0.248464\pi$
$367$	27.2229	1.42102	0.710512	+	0.703686i	$0.248464\pi$
$368$	0	0
$369$	9.70499	0.505222
$370$	0	0
$371$	5.20302	0.270127
$372$	0	0
$373$	32.6376	1.68991	0.844955	−	0.534837i	$-0.179627\pi$
$373$	32.6376	1.68991	0.844955	+	0.534837i	$0.179627\pi$
$374$	0	0
$375$	−11.8441	−0.611627
$376$	0	0
$377$	−10.2082	−0.525747
$378$	0	0
$379$	21.4096	1.09974	0.549869	−	0.835251i	$-0.314678\pi$
$379$	21.4096	1.09974	0.549869	+	0.835251i	$0.314678\pi$
$380$	0	0
$381$	18.1780	0.931286
$382$	0	0
$383$	29.5010	1.50743	0.753715	−	0.657202i	$-0.228260\pi$
$383$	29.5010	1.50743	0.753715	+	0.657202i	$0.228260\pi$
$384$	0	0
$385$	34.2572	1.74591
$386$	0	0
$387$	−10.1736	−0.517152
$388$	0	0
$389$	−15.2317	−0.772275	−0.386138	−	0.922441i	$-0.626191\pi$
$389$	−15.2317	−0.772275	−0.386138	+	0.922441i	$0.626191\pi$
$390$	0	0
$391$	15.6234	0.790110
$392$	0	0
$393$	−1.96530	−0.0991364
$394$	0	0
$395$	−60.7333	−3.05583
$396$	0	0
$397$	1.82650	0.0916696	0.0458348	−	0.998949i	$-0.485405\pi$
$397$	1.82650	0.0916696	0.0458348	+	0.998949i	$0.485405\pi$
$398$	0	0
$399$	−0.386527	−0.0193506
$400$	0	0
$401$	−3.20055	−0.159828	−0.0799140	−	0.996802i	$-0.525465\pi$
$401$	−3.20055	−0.159828	−0.0799140	+	0.996802i	$0.525465\pi$
$402$	0	0
$403$	43.4310	2.16345
$404$	0	0
$405$	−3.64052	−0.180899
$406$	0	0
$407$	−11.0254	−0.546509
$408$	0	0
$409$	7.59810	0.375702	0.187851	−	0.982198i	$-0.439848\pi$
$409$	7.59810	0.375702	0.187851	+	0.982198i	$0.439848\pi$
$410$	0	0
$411$	17.8328	0.879628
$412$	0	0
$413$	−11.3500	−0.558497
$414$	0	0
$415$	19.7106	0.967554
$416$	0	0
$417$	12.9396	0.633656
$418$	0	0
$419$	−5.98137	−0.292209	−0.146104	−	0.989269i	$-0.546674\pi$
$419$	−5.98137	−0.292209	−0.146104	+	0.989269i	$0.546674\pi$
$420$	0	0
$421$	22.7719	1.10983	0.554917	−	0.831906i	$-0.312750\pi$
$421$	22.7719	1.10983	0.554917	+	0.831906i	$0.312750\pi$
$422$	0	0
$423$	4.02094	0.195505
$424$	0	0
$425$	50.4405	2.44672
$426$	0	0
$427$	3.16846	0.153333
$428$	0	0
$429$	14.1485	0.683097
$430$	0	0
$431$	−0.794805	−0.0382844	−0.0191422	−	0.999817i	$-0.506094\pi$
$431$	−0.794805	−0.0382844	−0.0191422	+	0.999817i	$0.506094\pi$
$432$	0	0
$433$	1.73000	0.0831384	0.0415692	−	0.999136i	$-0.486764\pi$
$433$	1.73000	0.0831384	0.0415692	+	0.999136i	$0.486764\pi$
$434$	0	0
$435$	−8.63529	−0.414030
$436$	0	0
$437$	−0.345221	−0.0165142
$438$	0	0
$439$	−19.4798	−0.929722	−0.464861	−	0.885384i	$-0.653896\pi$
$439$	−19.4798	−0.929722	−0.464861	+	0.885384i	$0.653896\pi$
$440$	0	0
$441$	1.19264	0.0567926
$442$	0	0
$443$	−27.6154	−1.31205	−0.656023	−	0.754741i	$-0.727762\pi$
$443$	−27.6154	−1.31205	−0.656023	+	0.754741i	$0.727762\pi$
$444$	0	0
$445$	−57.5461	−2.72795
$446$	0	0
$447$	−20.2324	−0.956960
$448$	0	0
$449$	6.00086	0.283198	0.141599	−	0.989924i	$-0.454776\pi$
$449$	6.00086	0.283198	0.141599	+	0.989924i	$0.454776\pi$
$450$	0	0
$451$	−31.9060	−1.50239
$452$	0	0
$453$	3.11737	0.146467
$454$	0	0
$455$	44.8446	2.10235
$456$	0	0
$457$	8.38628	0.392294	0.196147	−	0.980575i	$-0.437157\pi$
$457$	8.38628	0.392294	0.196147	+	0.980575i	$0.437157\pi$
$458$	0	0
$459$	6.11147	0.285259
$460$	0	0
$461$	18.6418	0.868234	0.434117	−	0.900857i	$-0.357060\pi$
$461$	18.6418	0.868234	0.434117	+	0.900857i	$0.357060\pi$
$462$	0	0
$463$	−7.27649	−0.338167	−0.169083	−	0.985602i	$-0.554081\pi$
$463$	−7.27649	−0.338167	−0.169083	+	0.985602i	$0.554081\pi$
$464$	0	0
$465$	36.7392	1.70374
$466$	0	0
$467$	−16.9155	−0.782758	−0.391379	−	0.920230i	$-0.628002\pi$
$467$	−16.9155	−0.782758	−0.391379	+	0.920230i	$0.628002\pi$
$468$	0	0
$469$	−43.2801	−1.99849
$470$	0	0
$471$	9.71309	0.447555
$472$	0	0
$473$	33.4465	1.53787
$474$	0	0
$475$	−1.11455	−0.0511392
$476$	0	0
$477$	1.81779	0.0832308
$478$	0	0
$479$	24.9135	1.13833	0.569164	−	0.822224i	$-0.307267\pi$
$479$	24.9135	1.13833	0.569164	+	0.822224i	$0.307267\pi$
$480$	0	0
$481$	−14.4328	−0.658081
$482$	0	0
$483$	7.31715	0.332942
$484$	0	0
$485$	8.00504	0.363490
$486$	0	0
$487$	22.2421	1.00789	0.503943	−	0.863737i	$-0.331882\pi$
$487$	22.2421	1.00789	0.503943	+	0.863737i	$0.331882\pi$
$488$	0	0
$489$	−11.8308	−0.535006
$490$	0	0
$491$	−26.9595	−1.21667	−0.608333	−	0.793682i	$-0.708161\pi$
$491$	−26.9595	−1.21667	−0.608333	+	0.793682i	$0.708161\pi$
$492$	0	0
$493$	14.4964	0.652883
$494$	0	0
$495$	11.9685	0.537945
$496$	0	0
$497$	4.53960	0.203629
$498$	0	0
$499$	−0.736164	−0.0329552	−0.0164776	−	0.999864i	$-0.505245\pi$
$499$	−0.736164	−0.0329552	−0.0164776	+	0.999864i	$0.505245\pi$
$500$	0	0
$501$	6.70008	0.299337
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−33.6761	−1.49857
$506$	0	0
$507$	5.52117	0.245204
$508$	0	0
$509$	−22.6017	−1.00180	−0.500902	−	0.865504i	$-0.666998\pi$
$509$	−22.6017	−1.00180	−0.500902	+	0.865504i	$0.666998\pi$
$510$	0	0
$511$	−33.3359	−1.47469
$512$	0	0
$513$	−0.135042	−0.00596224
$514$	0	0
$515$	−54.6159	−2.40667
$516$	0	0
$517$	−13.2192	−0.581378
$518$	0	0
$519$	25.3435	1.11246
$520$	0	0
$521$	−2.83435	−0.124175	−0.0620875	−	0.998071i	$-0.519776\pi$
$521$	−2.83435	−0.124175	−0.0620875	+	0.998071i	$0.519776\pi$
$522$	0	0
$523$	45.3147	1.98147	0.990736	−	0.135803i	$-0.0433614\pi$
$523$	45.3147	1.98147	0.990736	+	0.135803i	$0.0433614\pi$
$524$	0	0
$525$	23.6236	1.03102
$526$	0	0
$527$	−61.6753	−2.68662
$528$	0	0
$529$	−16.4648	−0.715860
$530$	0	0
$531$	−3.96537	−0.172082
$532$	0	0
$533$	−41.7666	−1.80911
$534$	0	0
$535$	−53.9268	−2.33146
$536$	0	0
$537$	15.9462	0.688131
$538$	0	0
$539$	−3.92092	−0.168886
$540$	0	0
$541$	37.2382	1.60100	0.800498	−	0.599335i	$-0.204568\pi$
$541$	37.2382	1.60100	0.800498	+	0.599335i	$0.204568\pi$
$542$	0	0
$543$	12.3006	0.527868
$544$	0	0
$545$	−55.0006	−2.35597
$546$	0	0
$547$	21.0341	0.899353	0.449676	−	0.893192i	$-0.351539\pi$
$547$	21.0341	0.899353	0.449676	+	0.893192i	$0.351539\pi$
$548$	0	0
$549$	1.10697	0.0472444
$550$	0	0
$551$	−0.320318	−0.0136460
$552$	0	0
$553$	47.7502	2.03055
$554$	0	0
$555$	−12.2090	−0.518245
$556$	0	0
$557$	−5.87668	−0.249003	−0.124501	−	0.992219i	$-0.539733\pi$
$557$	−5.87668	−0.249003	−0.124501	+	0.992219i	$0.539733\pi$
$558$	0	0
$559$	43.7833	1.85183
$560$	0	0
$561$	−20.0920	−0.848284
$562$	0	0
$563$	15.2324	0.641969	0.320985	−	0.947084i	$-0.395986\pi$
$563$	15.2324	0.641969	0.320985	+	0.947084i	$0.395986\pi$
$564$	0	0
$565$	−3.87396	−0.162979
$566$	0	0
$567$	2.86228	0.120204
$568$	0	0
$569$	−41.7962	−1.75219	−0.876093	−	0.482142i	$-0.839859\pi$
$569$	−41.7962	−1.75219	−0.876093	+	0.482142i	$0.839859\pi$
$570$	0	0
$571$	−9.35200	−0.391369	−0.195684	−	0.980667i	$-0.562693\pi$
$571$	−9.35200	−0.391369	−0.195684	+	0.980667i	$0.562693\pi$
$572$	0	0
$573$	−2.73072	−0.114077
$574$	0	0
$575$	21.0991	0.879892
$576$	0	0
$577$	34.3693	1.43081	0.715407	−	0.698708i	$-0.246241\pi$
$577$	34.3693	1.43081	0.715407	+	0.698708i	$0.246241\pi$
$578$	0	0
$579$	−18.4977	−0.768738
$580$	0	0
$581$	−15.4970	−0.642924
$582$	0	0
$583$	−5.97613	−0.247506
$584$	0	0
$585$	15.6674	0.647769
$586$	0	0
$587$	5.34234	0.220502	0.110251	−	0.993904i	$-0.464835\pi$
$587$	5.34234	0.220502	0.110251	+	0.993904i	$0.464835\pi$
$588$	0	0
$589$	1.36280	0.0561533
$590$	0	0
$591$	−1.15807	−0.0476366
$592$	0	0
$593$	0.255919	0.0105093	0.00525466	−	0.999986i	$-0.498327\pi$
$593$	0.255919	0.0105093	0.00525466	+	0.999986i	$0.498327\pi$
$594$	0	0
$595$	−63.6828	−2.61074
$596$	0	0
$597$	−1.68890	−0.0691220
$598$	0	0
$599$	1.81810	0.0742855	0.0371427	−	0.999310i	$-0.488174\pi$
$599$	1.81810	0.0742855	0.0371427	+	0.999310i	$0.488174\pi$
$600$	0	0
$601$	−18.2567	−0.744704	−0.372352	−	0.928091i	$-0.621449\pi$
$601$	−18.2567	−0.744704	−0.372352	+	0.928091i	$0.621449\pi$
$602$	0	0
$603$	−15.1209	−0.615769
$604$	0	0
$605$	0.698270	0.0283887
$606$	0	0
$607$	37.5511	1.52415	0.762076	−	0.647487i	$-0.224180\pi$
$607$	37.5511	1.52415	0.762076	+	0.647487i	$0.224180\pi$
$608$	0	0
$609$	6.78930	0.275116
$610$	0	0
$611$	−17.3046	−0.700070
$612$	0	0
$613$	−42.0129	−1.69688	−0.848442	−	0.529288i	$-0.822459\pi$
$613$	−42.0129	−1.69688	−0.848442	+	0.529288i	$0.822459\pi$
$614$	0	0
$615$	−35.3313	−1.42469
$616$	0	0
$617$	5.58273	0.224752	0.112376	−	0.993666i	$-0.464154\pi$
$617$	5.58273	0.224752	0.112376	+	0.993666i	$0.464154\pi$
$618$	0	0
$619$	−1.29002	−0.0518504	−0.0259252	−	0.999664i	$-0.508253\pi$
$619$	−1.29002	−0.0518504	−0.0259252	+	0.999664i	$0.508253\pi$
$620$	0	0
$621$	2.55641	0.102585
$622$	0	0
$623$	45.2443	1.81267
$624$	0	0
$625$	1.85169	0.0740678
$626$	0	0
$627$	0.443961	0.0177301
$628$	0	0
$629$	20.4957	0.817219
$630$	0	0
$631$	3.26335	0.129912	0.0649559	−	0.997888i	$-0.479309\pi$
$631$	3.26335	0.129912	0.0649559	+	0.997888i	$0.479309\pi$
$632$	0	0
$633$	−5.92456	−0.235480
$634$	0	0
$635$	−66.1774	−2.62617
$636$	0	0
$637$	−5.13269	−0.203365
$638$	0	0
$639$	1.58601	0.0627415
$640$	0	0
$641$	−1.34296	−0.0530439	−0.0265219	−	0.999648i	$-0.508443\pi$
$641$	−1.34296	−0.0530439	−0.0265219	+	0.999648i	$0.508443\pi$
$642$	0	0
$643$	−15.7228	−0.620047	−0.310024	−	0.950729i	$-0.600337\pi$
$643$	−15.7228	−0.620047	−0.310024	+	0.950729i	$0.600337\pi$
$644$	0	0
$645$	37.0372	1.45834
$646$	0	0
$647$	14.0001	0.550401	0.275201	−	0.961387i	$-0.411256\pi$
$647$	14.0001	0.550401	0.275201	+	0.961387i	$0.411256\pi$
$648$	0	0
$649$	13.0365	0.511727
$650$	0	0
$651$	−28.8853	−1.13211
$652$	0	0
$653$	−30.4744	−1.19256	−0.596278	−	0.802778i	$-0.703355\pi$
$653$	−30.4744	−1.19256	−0.596278	+	0.802778i	$0.703355\pi$
$654$	0	0
$655$	7.15473	0.279558
$656$	0	0
$657$	−11.6466	−0.454378
$658$	0	0
$659$	11.7148	0.456343	0.228171	−	0.973621i	$-0.426725\pi$
$659$	11.7148	0.456343	0.228171	+	0.973621i	$0.426725\pi$
$660$	0	0
$661$	21.2955	0.828301	0.414150	−	0.910209i	$-0.364079\pi$
$661$	21.2955	0.828301	0.414150	+	0.910209i	$0.364079\pi$
$662$	0	0
$663$	−26.3015	−1.02147
$664$	0	0
$665$	1.40716	0.0545673
$666$	0	0
$667$	6.06377	0.234790
$668$	0	0
$669$	−24.4950	−0.947031
$670$	0	0
$671$	−3.63926	−0.140492
$672$	0	0
$673$	−22.0033	−0.848165	−0.424082	−	0.905624i	$-0.639403\pi$
$673$	−22.0033	−0.848165	−0.424082	+	0.905624i	$0.639403\pi$
$674$	0	0
$675$	8.25341	0.317674
$676$	0	0
$677$	−26.4760	−1.01756	−0.508778	−	0.860898i	$-0.669903\pi$
$677$	−26.4760	−1.01756	−0.508778	+	0.860898i	$0.669903\pi$
$678$	0	0
$679$	−6.29379	−0.241533
$680$	0	0
$681$	−8.95090	−0.342999
$682$	0	0
$683$	−6.24910	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$683$	−6.24910	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$684$	0	0
$685$	−64.9207	−2.48049
$686$	0	0
$687$	23.4521	0.894754
$688$	0	0
$689$	−7.82308	−0.298036
$690$	0	0
$691$	−27.2028	−1.03484	−0.517422	−	0.855730i	$-0.673108\pi$
$691$	−27.2028	−1.03484	−0.517422	+	0.855730i	$0.673108\pi$
$692$	0	0
$693$	−9.40998	−0.357456
$694$	0	0
$695$	−47.1070	−1.78687
$696$	0	0
$697$	59.3118	2.24660
$698$	0	0
$699$	−0.754123	−0.0285235
$700$	0	0
$701$	−22.5382	−0.851256	−0.425628	−	0.904898i	$-0.639947\pi$
$701$	−22.5382	−0.851256	−0.425628	+	0.904898i	$0.639947\pi$
$702$	0	0
$703$	−0.452883	−0.0170808
$704$	0	0
$705$	−14.6383	−0.551311
$706$	0	0
$707$	26.4771	0.995772
$708$	0	0
$709$	34.5113	1.29610	0.648049	−	0.761599i	$-0.275585\pi$
$709$	34.5113	1.29610	0.648049	+	0.761599i	$0.275585\pi$
$710$	0	0
$711$	16.6826	0.625646
$712$	0	0
$713$	−25.7986	−0.966163
$714$	0	0
$715$	−51.5080	−1.92629
$716$	0	0
$717$	0.223949	0.00836353
$718$	0	0
$719$	−41.6706	−1.55405	−0.777025	−	0.629469i	$-0.783272\pi$
$719$	−41.6706	−1.55405	−0.777025	+	0.629469i	$0.783272\pi$
$720$	0	0
$721$	42.9406	1.59919
$722$	0	0
$723$	−8.09293	−0.300979
$724$	0	0
$725$	19.5770	0.727072
$726$	0	0
$727$	−9.76283	−0.362083	−0.181042	−	0.983475i	$-0.557947\pi$
$727$	−9.76283	−0.362083	−0.181042	+	0.983475i	$0.557947\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−62.1756	−2.29965
$732$	0	0
$733$	−12.0110	−0.443637	−0.221818	−	0.975088i	$-0.571199\pi$
$733$	−12.0110	−0.443637	−0.221818	+	0.975088i	$0.571199\pi$
$734$	0	0
$735$	−4.34185	−0.160152
$736$	0	0
$737$	49.7111	1.83113
$738$	0	0
$739$	1.64927	0.0606696	0.0303348	−	0.999540i	$-0.490343\pi$
$739$	1.64927	0.0606696	0.0303348	+	0.999540i	$0.490343\pi$
$740$	0	0
$741$	0.581169	0.0213498
$742$	0	0
$743$	−14.6716	−0.538247	−0.269124	−	0.963106i	$-0.586734\pi$
$743$	−14.6716	−0.538247	−0.269124	+	0.963106i	$0.586734\pi$
$744$	0	0
$745$	73.6566	2.69857
$746$	0	0
$747$	−5.41422	−0.198096
$748$	0	0
$749$	42.3987	1.54921
$750$	0	0
$751$	−41.7546	−1.52365	−0.761823	−	0.647785i	$-0.775695\pi$
$751$	−41.7546	−1.52365	−0.761823	+	0.647785i	$0.775695\pi$
$752$	0	0
$753$	3.94011	0.143586
$754$	0	0
$755$	−11.3489	−0.413027
$756$	0	0
$757$	13.2891	0.483000	0.241500	−	0.970401i	$-0.422361\pi$
$757$	13.2891	0.483000	0.241500	+	0.970401i	$0.422361\pi$
$758$	0	0
$759$	−8.40440	−0.305060
$760$	0	0
$761$	27.8527	1.00966	0.504829	−	0.863219i	$-0.331555\pi$
$761$	27.8527	1.00966	0.504829	+	0.863219i	$0.331555\pi$
$762$	0	0
$763$	43.2430	1.56550
$764$	0	0
$765$	−22.2490	−0.804413
$766$	0	0
$767$	17.0655	0.616198
$768$	0	0
$769$	−35.0642	−1.26445	−0.632223	−	0.774786i	$-0.717857\pi$
$769$	−35.0642	−1.26445	−0.632223	+	0.774786i	$0.717857\pi$
$770$	0	0
$771$	3.84186	0.138361
$772$	0	0
$773$	15.6828	0.564072	0.282036	−	0.959404i	$-0.408990\pi$
$773$	15.6828	0.564072	0.282036	+	0.959404i	$0.408990\pi$
$774$	0	0
$775$	−83.2911	−2.99191
$776$	0	0
$777$	9.59908	0.344365
$778$	0	0
$779$	−1.31058	−0.0469564
$780$	0	0
$781$	−5.21413	−0.186576
$782$	0	0
$783$	2.37199	0.0847680
$784$	0	0
$785$	−35.3607	−1.26208
$786$	0	0
$787$	20.3090	0.723939	0.361970	−	0.932190i	$-0.382104\pi$
$787$	20.3090	0.723939	0.361970	+	0.932190i	$0.382104\pi$
$788$	0	0
$789$	4.68245	0.166700
$790$	0	0
$791$	3.04581	0.108297
$792$	0	0
$793$	−4.76399	−0.169174
$794$	0	0
$795$	−6.61770	−0.234706
$796$	0	0
$797$	−49.5982	−1.75686	−0.878430	−	0.477871i	$-0.841409\pi$
$797$	−49.5982	−1.75686	−0.878430	+	0.477871i	$0.841409\pi$
$798$	0	0
$799$	24.5739	0.869361
$800$	0	0
$801$	15.8071	0.558516
$802$	0	0
$803$	38.2892	1.35120
$804$	0	0
$805$	−26.6383	−0.938875
$806$	0	0
$807$	6.53308	0.229975
$808$	0	0
$809$	20.5964	0.724130	0.362065	−	0.932153i	$-0.382072\pi$
$809$	20.5964	0.724130	0.362065	+	0.932153i	$0.382072\pi$
$810$	0	0
$811$	36.0949	1.26746	0.633732	−	0.773553i	$-0.281522\pi$
$811$	36.0949	1.26746	0.633732	+	0.773553i	$0.281522\pi$
$812$	0	0
$813$	3.11279	0.109170
$814$	0	0
$815$	43.0702	1.50868
$816$	0	0
$817$	1.37386	0.0480652
$818$	0	0
$819$	−12.3182	−0.430432
$820$	0	0
$821$	0.546656	0.0190784	0.00953921	−	0.999955i	$-0.496964\pi$
$821$	0.546656	0.0190784	0.00953921	+	0.999955i	$0.496964\pi$
$822$	0	0
$823$	−42.3707	−1.47695	−0.738475	−	0.674281i	$-0.764454\pi$
$823$	−42.3707	−1.47695	−0.738475	+	0.674281i	$0.764454\pi$
$824$	0	0
$825$	−27.1338	−0.944676
$826$	0	0
$827$	20.3339	0.707079	0.353539	−	0.935420i	$-0.384978\pi$
$827$	20.3339	0.707079	0.353539	+	0.935420i	$0.384978\pi$
$828$	0	0
$829$	38.2574	1.32873	0.664367	−	0.747407i	$-0.268701\pi$
$829$	38.2574	1.32873	0.664367	+	0.747407i	$0.268701\pi$
$830$	0	0
$831$	12.7980	0.443957
$832$	0	0
$833$	7.28882	0.252543
$834$	0	0
$835$	−24.3918	−0.844112
$836$	0	0
$837$	−10.0917	−0.348821
$838$	0	0
$839$	−21.0559	−0.726930	−0.363465	−	0.931608i	$-0.618406\pi$
$839$	−21.0559	−0.726930	−0.363465	+	0.931608i	$0.618406\pi$
$840$	0	0
$841$	−23.3737	−0.805988
$842$	0	0
$843$	−0.544674	−0.0187596
$844$	0	0
$845$	−20.1000	−0.691460
$846$	0	0
$847$	−0.548999	−0.0188638
$848$	0	0
$849$	30.5014	1.04680
$850$	0	0
$851$	8.57329	0.293889
$852$	0	0
$853$	39.5154	1.35298	0.676490	−	0.736452i	$-0.263500\pi$
$853$	39.5154	1.35298	0.676490	+	0.736452i	$0.263500\pi$
$854$	0	0
$855$	0.491622	0.0168131
$856$	0	0
$857$	32.1377	1.09780	0.548901	−	0.835887i	$-0.315046\pi$
$857$	32.1377	1.09780	0.548901	+	0.835887i	$0.315046\pi$
$858$	0	0
$859$	−39.5768	−1.35034	−0.675171	−	0.737661i	$-0.735930\pi$
$859$	−39.5768	−1.35034	−0.675171	+	0.737661i	$0.735930\pi$
$860$	0	0
$861$	27.7784	0.946686
$862$	0	0
$863$	−2.81070	−0.0956772	−0.0478386	−	0.998855i	$-0.515233\pi$
$863$	−2.81070	−0.0956772	−0.0478386	+	0.998855i	$0.515233\pi$
$864$	0	0
$865$	−92.2636	−3.13705
$866$	0	0
$867$	20.3501	0.691126
$868$	0	0
$869$	−54.8454	−1.86050
$870$	0	0
$871$	65.0745	2.20496
$872$	0	0
$873$	−2.19887	−0.0744206
$874$	0	0
$875$	−33.9011	−1.14607
$876$	0	0
$877$	−17.3681	−0.586481	−0.293240	−	0.956039i	$-0.594734\pi$
$877$	−17.3681	−0.586481	−0.293240	+	0.956039i	$0.594734\pi$
$878$	0	0
$879$	−11.0640	−0.373181
$880$	0	0
$881$	20.4208	0.687996	0.343998	−	0.938970i	$-0.388219\pi$
$881$	20.4208	0.687996	0.343998	+	0.938970i	$0.388219\pi$
$882$	0	0
$883$	−48.6656	−1.63773	−0.818864	−	0.573988i	$-0.805395\pi$
$883$	−48.6656	−1.63773	−0.818864	+	0.573988i	$0.805395\pi$
$884$	0	0
$885$	14.4360	0.485261
$886$	0	0
$887$	−34.3090	−1.15198	−0.575992	−	0.817455i	$-0.695384\pi$
$887$	−34.3090	−1.15198	−0.575992	+	0.817455i	$0.695384\pi$
$888$	0	0
$889$	52.0305	1.74505
$890$	0	0
$891$	−3.28758	−0.110138
$892$	0	0
$893$	−0.542994	−0.0181706
$894$	0	0
$895$	−58.0526	−1.94049
$896$	0	0
$897$	−11.0018	−0.367340
$898$	0	0
$899$	−23.9375	−0.798360
$900$	0	0
$901$	11.1094	0.370107
$902$	0	0
$903$	−29.1196	−0.969041
$904$	0	0
$905$	−44.7805	−1.48855
$906$	0	0
$907$	−34.7172	−1.15276	−0.576382	−	0.817180i	$-0.695536\pi$
$907$	−34.7172	−1.15276	−0.576382	+	0.817180i	$0.695536\pi$
$908$	0	0
$909$	9.25034	0.306814
$910$	0	0
$911$	32.2255	1.06768	0.533839	−	0.845586i	$-0.320749\pi$
$911$	32.2255	1.06768	0.533839	+	0.845586i	$0.320749\pi$
$912$	0	0
$913$	17.7997	0.589084
$914$	0	0
$915$	−4.02995	−0.133226
$916$	0	0
$917$	−5.62525	−0.185762
$918$	0	0
$919$	−40.9663	−1.35135	−0.675677	−	0.737198i	$-0.736149\pi$
$919$	−40.9663	−1.35135	−0.675677	+	0.737198i	$0.736149\pi$
$920$	0	0
$921$	0.706571	0.0232823
$922$	0	0
$923$	−6.82558	−0.224667
$924$	0	0
$925$	27.6790	0.910081
$926$	0	0
$927$	15.0022	0.492738
$928$	0	0
$929$	−48.7127	−1.59821	−0.799106	−	0.601190i	$-0.794694\pi$
$929$	−48.7127	−1.59821	−0.799106	+	0.601190i	$0.794694\pi$
$930$	0	0
$931$	−0.161057	−0.00527842
$932$	0	0
$933$	−30.4096	−0.995565
$934$	0	0
$935$	73.1453	2.39211
$936$	0	0
$937$	−35.2939	−1.15300	−0.576501	−	0.817096i	$-0.695582\pi$
$937$	−35.2939	−1.15300	−0.576501	+	0.817096i	$0.695582\pi$
$938$	0	0
$939$	16.5723	0.540816
$940$	0	0
$941$	−56.3588	−1.83724	−0.918621	−	0.395139i	$-0.870697\pi$
$941$	−56.3588	−1.83724	−0.918621	+	0.395139i	$0.870697\pi$
$942$	0	0
$943$	24.8099	0.807922
$944$	0	0
$945$	−10.4202	−0.338969
$946$	0	0
$947$	34.9159	1.13461	0.567307	−	0.823506i	$-0.307985\pi$
$947$	34.9159	1.13461	0.567307	+	0.823506i	$0.307985\pi$
$948$	0	0
$949$	50.1227	1.62705
$950$	0	0
$951$	21.0721	0.683311
$952$	0	0
$953$	−3.57623	−0.115845	−0.0579227	−	0.998321i	$-0.518448\pi$
$953$	−3.57623	−0.115845	−0.0579227	+	0.998321i	$0.518448\pi$
$954$	0	0
$955$	9.94125	0.321691
$956$	0	0
$957$	−7.79811	−0.252077
$958$	0	0
$959$	51.0425	1.64825
$960$	0	0
$961$	70.8429	2.28525
$962$	0	0
$963$	14.8129	0.477340
$964$	0	0
$965$	67.3413	2.16779
$966$	0	0
$967$	−27.5802	−0.886920	−0.443460	−	0.896294i	$-0.646249\pi$
$967$	−27.5802	−0.886920	−0.443460	+	0.896294i	$0.646249\pi$
$968$	0	0
$969$	−0.825304	−0.0265126
$970$	0	0
$971$	−25.0143	−0.802746	−0.401373	−	0.915915i	$-0.631467\pi$
$971$	−25.0143	−0.802746	−0.401373	+	0.915915i	$0.631467\pi$
$972$	0	0
$973$	37.0368	1.18734
$974$	0	0
$975$	−35.5196	−1.13754
$976$	0	0
$977$	40.9339	1.30959	0.654796	−	0.755806i	$-0.272755\pi$
$977$	40.9339	1.30959	0.654796	+	0.755806i	$0.272755\pi$
$978$	0	0
$979$	−51.9671	−1.66088
$980$	0	0
$981$	15.1079	0.482357
$982$	0	0
$983$	−19.7175	−0.628891	−0.314445	−	0.949276i	$-0.601818\pi$
$983$	−19.7175	−0.628891	−0.314445	+	0.949276i	$0.601818\pi$
$984$	0	0
$985$	4.21597	0.134332
$986$	0	0
$987$	11.5091	0.366337
$988$	0	0
$989$	−26.0078	−0.827000
$990$	0	0
$991$	−35.8852	−1.13993	−0.569965	−	0.821669i	$-0.693043\pi$
$991$	−35.8852	−1.13993	−0.569965	+	0.821669i	$0.693043\pi$
$992$	0	0
$993$	−25.1394	−0.797773
$994$	0	0
$995$	6.14847	0.194920
$996$	0	0
$997$	−17.6854	−0.560102	−0.280051	−	0.959985i	$-0.590351\pi$
$997$	−17.6854	−0.560102	−0.280051	+	0.959985i	$0.590351\pi$
$998$	0	0
$999$	3.35365	0.106105

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.1	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.64052 q^{5} +2.86228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.64052 q^{5} +2.86228 q^{7} +1.00000 q^{9} -3.28758 q^{11} -4.30362 q^{13} -3.64052 q^{15} +6.11147 q^{17} -0.135042 q^{19} +2.86228 q^{21} +2.55641 q^{23} +8.25341 q^{25} +1.00000 q^{27} +2.37199 q^{29} -10.0917 q^{31} -3.28758 q^{33} -10.4202 q^{35} +3.35365 q^{37} -4.30362 q^{39} +9.70499 q^{41} -10.1736 q^{43} -3.64052 q^{45} +4.02094 q^{47} +1.19264 q^{49} +6.11147 q^{51} +1.81779 q^{53} +11.9685 q^{55} -0.135042 q^{57} -3.96537 q^{59} +1.10697 q^{61} +2.86228 q^{63} +15.6674 q^{65} -15.1209 q^{67} +2.55641 q^{69} +1.58601 q^{71} -11.6466 q^{73} +8.25341 q^{75} -9.40998 q^{77} +16.6826 q^{79} +1.00000 q^{81} -5.41422 q^{83} -22.2490 q^{85} +2.37199 q^{87} +15.8071 q^{89} -12.3182 q^{91} -10.0917 q^{93} +0.491622 q^{95} -2.19887 q^{97} -3.28758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more