LMFDB - Embedded newform 6045.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +10.0000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +6.00000 q^{17} -2.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +5.00000 q^{21} -4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -10.0000 q^{28} +7.00000 q^{29} -2.00000 q^{30} +1.00000 q^{31} +8.00000 q^{32} -2.00000 q^{33} -12.0000 q^{34} +5.00000 q^{35} +2.00000 q^{36} -6.00000 q^{37} -8.00000 q^{38} +1.00000 q^{39} +9.00000 q^{41} -10.0000 q^{42} +11.0000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -12.0000 q^{46} -8.00000 q^{47} +4.00000 q^{48} +18.0000 q^{49} -2.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} +2.00000 q^{54} -2.00000 q^{55} -4.00000 q^{57} -14.0000 q^{58} -7.00000 q^{59} +2.00000 q^{60} +2.00000 q^{61} -2.00000 q^{62} -5.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} +4.00000 q^{66} -3.00000 q^{67} +12.0000 q^{68} -6.00000 q^{69} -10.0000 q^{70} +2.00000 q^{73} +12.0000 q^{74} -1.00000 q^{75} +8.00000 q^{76} -10.0000 q^{77} -2.00000 q^{78} +6.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -18.0000 q^{82} +11.0000 q^{83} +10.0000 q^{84} -6.00000 q^{85} -22.0000 q^{86} -7.00000 q^{87} -14.0000 q^{89} +2.00000 q^{90} +5.00000 q^{91} +12.0000 q^{92} -1.00000 q^{93} +16.0000 q^{94} -4.00000 q^{95} -8.00000 q^{96} +1.00000 q^{97} -36.0000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00000	0.816497
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000	−0.577350
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	10.0000	2.67261
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−2.00000	−0.471405
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	5.00000	1.09109
$22$	−4.00000	−0.852803
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−10.0000	−1.88982
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	−2.00000	−0.365148
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	8.00000	1.41421
$33$	−2.00000	−0.348155
$34$	−12.0000	−2.05798
$35$	5.00000	0.845154
$36$	2.00000	0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−8.00000	−1.29777
$39$	1.00000	0.160128
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	−10.0000	−1.54303
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	4.00000	0.603023
$45$	−1.00000	−0.149071
$46$	−12.0000	−1.76930
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	4.00000	0.577350
$49$	18.0000	2.57143
$50$	−2.00000	−0.282843
$51$	−6.00000	−0.840168
$52$	−2.00000	−0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	2.00000	0.272166
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−14.0000	−1.83829
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	−5.00000	−0.629941
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	4.00000	0.492366
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	12.0000	1.45521
$69$	−6.00000	−0.722315
$70$	−10.0000	−1.19523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	12.0000	1.39497
$75$	−1.00000	−0.115470
$76$	8.00000	0.917663
$77$	−10.0000	−1.13961
$78$	−2.00000	−0.226455
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−18.0000	−1.98777
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	10.0000	1.09109
$85$	−6.00000	−0.650791
$86$	−22.0000	−2.37232
$87$	−7.00000	−0.750479
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	2.00000	0.210819
$91$	5.00000	0.524142
$92$	12.0000	1.25109
$93$	−1.00000	−0.103695
$94$	16.0000	1.65027
$95$	−4.00000	−0.410391
$96$	−8.00000	−0.816497
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	−36.0000	−3.63655
$99$	2.00000	0.201008
$100$	2.00000	0.200000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	12.0000	1.18818
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	20.0000	1.94257
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	−2.00000	−0.192450
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	4.00000	0.381385
$111$	6.00000	0.569495
$112$	20.0000	1.88982
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	8.00000	0.749269
$115$	−6.00000	−0.559503
$116$	14.0000	1.29987
$117$	−1.00000	−0.0924500
$118$	14.0000	1.28880
$119$	−30.0000	−2.75010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−4.00000	−0.362143
$123$	−9.00000	−0.811503
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	10.0000	0.890871
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	0	0
$129$	−11.0000	−0.968496
$130$	−2.00000	−0.175412
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−4.00000	−0.348155
$133$	−20.0000	−1.73422
$134$	6.00000	0.518321
$135$	1.00000	0.0860663
$136$	0	0
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	12.0000	1.02151
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	10.0000	0.845154
$141$	8.00000	0.673722
$142$	0	0
$143$	−2.00000	−0.167248
$144$	−4.00000	−0.333333
$145$	−7.00000	−0.581318
$146$	−4.00000	−0.331042
$147$	−18.0000	−1.48461
$148$	−12.0000	−0.986394
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	2.00000	0.163299
$151$	13.0000	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	20.0000	1.61165
$155$	−1.00000	−0.0803219
$156$	2.00000	0.160128
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−12.0000	−0.954669
$159$	10.0000	0.793052
$160$	−8.00000	−0.632456
$161$	−30.0000	−2.36433
$162$	−2.00000	−0.157135
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	18.0000	1.40556
$165$	2.00000	0.155700
$166$	−22.0000	−1.70753
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	12.0000	0.920358
$171$	4.00000	0.305888
$172$	22.0000	1.67748
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	14.0000	1.06134
$175$	−5.00000	−0.377964
$176$	−8.00000	−0.603023
$177$	7.00000	0.526152
$178$	28.0000	2.09869
$179$	23.0000	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$179$	23.0000	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$180$	−2.00000	−0.149071
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	−10.0000	−0.741249
$183$	−2.00000	−0.147844
$184$	0	0
$185$	6.00000	0.441129
$186$	2.00000	0.146647
$187$	12.0000	0.877527
$188$	−16.0000	−1.16692
$189$	5.00000	0.363696
$190$	8.00000	0.580381
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	8.00000	0.577350
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	−2.00000	−0.143592
$195$	−1.00000	−0.0716115
$196$	36.0000	2.57143
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	−4.00000	−0.284268
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	12.0000	0.844317
$203$	−35.0000	−2.45652
$204$	−12.0000	−0.840168
$205$	−9.00000	−0.628587
$206$	−20.0000	−1.39347
$207$	6.00000	0.417029
$208$	4.00000	0.277350
$209$	8.00000	0.553372
$210$	10.0000	0.690066
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	−20.0000	−1.37361
$213$	0	0
$214$	30.0000	2.05076
$215$	−11.0000	−0.750194
$216$	0	0
$217$	−5.00000	−0.339422
$218$	−4.00000	−0.270914
$219$	−2.00000	−0.135147
$220$	−4.00000	−0.269680
$221$	−6.00000	−0.403604
$222$	−12.0000	−0.805387
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−40.0000	−2.67261
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−8.00000	−0.529813
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	12.0000	0.791257
$231$	10.0000	0.657952
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	2.00000	0.130744
$235$	8.00000	0.521862
$236$	−14.0000	−0.911322
$237$	−6.00000	−0.389742
$238$	60.0000	3.88922
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	−4.00000	−0.258199
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	14.0000	0.899954
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	−18.0000	−1.14998
$246$	18.0000	1.14764
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−11.0000	−0.697097
$250$	2.00000	0.126491
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	−10.0000	−0.629941
$253$	12.0000	0.754434
$254$	6.00000	0.376473
$255$	6.00000	0.375735
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	22.0000	1.36966
$259$	30.0000	1.86411
$260$	2.00000	0.124035
$261$	7.00000	0.433289
$262$	−36.0000	−2.22409
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	40.0000	2.45256
$267$	14.0000	0.856786
$268$	−6.00000	−0.366508
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	−2.00000	−0.121716
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	−24.0000	−1.45521
$273$	−5.00000	−0.302614
$274$	44.0000	2.65814
$275$	2.00000	0.120605
$276$	−12.0000	−0.722315
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	−32.0000	−1.91923
$279$	1.00000	0.0598684
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	−16.0000	−0.952786
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	4.00000	0.236525
$287$	−45.0000	−2.65627
$288$	8.00000	0.471405
$289$	19.0000	1.11765
$290$	14.0000	0.822108
$291$	−1.00000	−0.0586210
$292$	4.00000	0.234082
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	36.0000	2.09956
$295$	7.00000	0.407556
$296$	0	0
$297$	−2.00000	−0.116052
$298$	−18.0000	−1.04271
$299$	−6.00000	−0.346989
$300$	−2.00000	−0.115470
$301$	−55.0000	−3.17015
$302$	−26.0000	−1.49613
$303$	6.00000	0.344691
$304$	−16.0000	−0.917663
$305$	−2.00000	−0.114520
$306$	−12.0000	−0.685994
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−20.0000	−1.13961
$309$	−10.0000	−0.568880
$310$	2.00000	0.113592
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	−5.00000	−0.282617	−0.141308	−	0.989966i	$-0.545131\pi$
$313$	−5.00000	−0.282617	−0.141308	+	0.989966i	$0.545131\pi$
$314$	12.0000	0.677199
$315$	5.00000	0.281718
$316$	12.0000	0.675053
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	−20.0000	−1.12154
$319$	14.0000	0.783850
$320$	8.00000	0.447214
$321$	15.0000	0.837218
$322$	60.0000	3.34367
$323$	24.0000	1.33540
$324$	2.00000	0.111111
$325$	−1.00000	−0.0554700
$326$	−8.00000	−0.443079
$327$	−2.00000	−0.110600
$328$	0	0
$329$	40.0000	2.20527
$330$	−4.00000	−0.220193
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	22.0000	1.20741
$333$	−6.00000	−0.328798
$334$	−24.0000	−1.31322
$335$	3.00000	0.163908
$336$	−20.0000	−1.09109
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−2.00000	−0.108786
$339$	3.00000	0.162938
$340$	−12.0000	−0.650791
$341$	2.00000	0.108306
$342$	−8.00000	−0.432590
$343$	−55.0000	−2.96972
$344$	0	0
$345$	6.00000	0.323029
$346$	2.00000	0.107521
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−14.0000	−0.750479
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	10.0000	0.534522
$351$	1.00000	0.0533761
$352$	16.0000	0.852803
$353$	5.00000	0.266123	0.133062	−	0.991108i	$-0.457519\pi$
$353$	5.00000	0.266123	0.133062	+	0.991108i	$0.457519\pi$
$354$	−14.0000	−0.744092
$355$	0	0
$356$	−28.0000	−1.48400
$357$	30.0000	1.58777
$358$	−46.0000	−2.43118
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	16.0000	0.840941
$363$	7.00000	0.367405
$364$	10.0000	0.524142
$365$	−2.00000	−0.104685
$366$	4.00000	0.209083
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−24.0000	−1.25109
$369$	9.00000	0.468521
$370$	−12.0000	−0.623850
$371$	50.0000	2.59587
$372$	−2.00000	−0.103695
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	−24.0000	−1.24101
$375$	1.00000	0.0516398
$376$	0	0
$377$	−7.00000	−0.360518
$378$	−10.0000	−0.514344
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−8.00000	−0.410391
$381$	3.00000	0.153695
$382$	−12.0000	−0.613973
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	0	0
$385$	10.0000	0.509647
$386$	26.0000	1.32337
$387$	11.0000	0.559161
$388$	2.00000	0.101535
$389$	25.0000	1.26755	0.633775	−	0.773517i	$-0.281504\pi$
$389$	25.0000	1.26755	0.633775	+	0.773517i	$0.281504\pi$
$390$	2.00000	0.101274
$391$	36.0000	1.82060
$392$	0	0
$393$	−18.0000	−0.907980
$394$	−18.0000	−0.906827
$395$	−6.00000	−0.301893
$396$	4.00000	0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	32.0000	1.60402
$399$	20.0000	1.00125
$400$	−4.00000	−0.200000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	−6.00000	−0.299253
$403$	−1.00000	−0.0498135
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	70.0000	3.47404
$407$	−12.0000	−0.594818
$408$	0	0
$409$	13.0000	0.642809	0.321404	−	0.946942i	$-0.395845\pi$
$409$	13.0000	0.642809	0.321404	+	0.946942i	$0.395845\pi$
$410$	18.0000	0.888957
$411$	22.0000	1.08518
$412$	20.0000	0.985329
$413$	35.0000	1.72224
$414$	−12.0000	−0.589768
$415$	−11.0000	−0.539969
$416$	−8.00000	−0.392232
$417$	−16.0000	−0.783523
$418$	−16.0000	−0.782586
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	−10.0000	−0.487950
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	22.0000	1.07094
$423$	−8.00000	−0.388973
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	−10.0000	−0.483934
$428$	−30.0000	−1.45010
$429$	2.00000	0.0965609
$430$	22.0000	1.06093
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	4.00000	0.192450
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	10.0000	0.480015
$435$	7.00000	0.335624
$436$	4.00000	0.191565
$437$	24.0000	1.14808
$438$	4.00000	0.191127
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	12.0000	0.570782
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	12.0000	0.569495
$445$	14.0000	0.663664
$446$	4.00000	0.189405
$447$	−9.00000	−0.425685
$448$	40.0000	1.88982
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−2.00000	−0.0942809
$451$	18.0000	0.847587
$452$	−6.00000	−0.282216
$453$	−13.0000	−0.610793
$454$	−36.0000	−1.68956
$455$	−5.00000	−0.234404
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	−52.0000	−2.42980
$459$	−6.00000	−0.280056
$460$	−12.0000	−0.559503
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−20.0000	−0.930484
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−28.0000	−1.29987
$465$	1.00000	0.0463739
$466$	−6.00000	−0.277945
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	−2.00000	−0.0924500
$469$	15.0000	0.692636
$470$	−16.0000	−0.738025
$471$	6.00000	0.276465
$472$	0	0
$473$	22.0000	1.01156
$474$	12.0000	0.551178
$475$	4.00000	0.183533
$476$	−60.0000	−2.75010
$477$	−10.0000	−0.457869
$478$	−36.0000	−1.64660
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	8.00000	0.365148
$481$	6.00000	0.273576
$482$	14.0000	0.637683
$483$	30.0000	1.36505
$484$	−14.0000	−0.636364
$485$	−1.00000	−0.0454077
$486$	2.00000	0.0907218
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	36.0000	1.62631
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	−18.0000	−0.811503
$493$	42.0000	1.89158
$494$	8.00000	0.359937
$495$	−2.00000	−0.0898933
$496$	−4.00000	−0.179605
$497$	0	0
$498$	22.0000	0.985844
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	−2.00000	−0.0894427
$501$	−12.0000	−0.536120
$502$	14.0000	0.624851
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−24.0000	−1.06693
$507$	−1.00000	−0.0444116
$508$	−6.00000	−0.266207
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	−12.0000	−0.531369
$511$	−10.0000	−0.442374
$512$	−32.0000	−1.41421
$513$	−4.00000	−0.176604
$514$	−36.0000	−1.58789
$515$	−10.0000	−0.440653
$516$	−22.0000	−0.968496
$517$	−16.0000	−0.703679
$518$	−60.0000	−2.63625
$519$	1.00000	0.0438951
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	−14.0000	−0.612763
$523$	29.0000	1.26808	0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000	1.26808	0.634041	+	0.773300i	$0.281395\pi$
$524$	36.0000	1.57267
$525$	5.00000	0.218218
$526$	8.00000	0.348817
$527$	6.00000	0.261364
$528$	8.00000	0.348155
$529$	13.0000	0.565217
$530$	−20.0000	−0.868744
$531$	−7.00000	−0.303774
$532$	−40.0000	−1.73422
$533$	−9.00000	−0.389833
$534$	−28.0000	−1.21168
$535$	15.0000	0.648507
$536$	0	0
$537$	−23.0000	−0.992523
$538$	60.0000	2.58678
$539$	36.0000	1.55063
$540$	2.00000	0.0860663
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	26.0000	1.11680
$543$	8.00000	0.343313
$544$	48.0000	2.05798
$545$	−2.00000	−0.0856706
$546$	10.0000	0.427960
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	−44.0000	−1.87959
$549$	2.00000	0.0853579
$550$	−4.00000	−0.170561
$551$	28.0000	1.19284
$552$	0	0
$553$	−30.0000	−1.27573
$554$	−10.0000	−0.424859
$555$	−6.00000	−0.254686
$556$	32.0000	1.35710
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−2.00000	−0.0846668
$559$	−11.0000	−0.465250
$560$	−20.0000	−0.845154
$561$	−12.0000	−0.506640
$562$	−6.00000	−0.253095
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	16.0000	0.673722
$565$	3.00000	0.126211
$566$	28.0000	1.17693
$567$	−5.00000	−0.209980
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−8.00000	−0.335083
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−4.00000	−0.167248
$573$	−6.00000	−0.250654
$574$	90.0000	3.75653
$575$	6.00000	0.250217
$576$	−8.00000	−0.333333
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−38.0000	−1.58059
$579$	13.0000	0.540262
$580$	−14.0000	−0.581318
$581$	−55.0000	−2.28178
$582$	2.00000	0.0829027
$583$	−20.0000	−0.828315
$584$	0	0
$585$	1.00000	0.0413449
$586$	48.0000	1.98286
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	−36.0000	−1.48461
$589$	4.00000	0.164817
$590$	−14.0000	−0.576371
$591$	−9.00000	−0.370211
$592$	24.0000	0.986394
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	4.00000	0.164122
$595$	30.0000	1.22988
$596$	18.0000	0.737309
$597$	16.0000	0.654836
$598$	12.0000	0.490716
$599$	−48.0000	−1.96123	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000	−1.96123	−0.980613	+	0.195952i	$0.937220\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	110.000	4.48327
$603$	−3.00000	−0.122169
$604$	26.0000	1.05792
$605$	7.00000	0.284590
$606$	−12.0000	−0.487467
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	32.0000	1.29777
$609$	35.0000	1.41827
$610$	4.00000	0.161955
$611$	8.00000	0.323645
$612$	12.0000	0.485071
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	40.0000	1.61427
$615$	9.00000	0.362915
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	20.0000	0.804518
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	−2.00000	−0.0803219
$621$	−6.00000	−0.240772
$622$	40.0000	1.60385
$623$	70.0000	2.80449
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	−8.00000	−0.319489
$628$	−12.0000	−0.478852
$629$	−36.0000	−1.43541
$630$	−10.0000	−0.398410
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	11.0000	0.437211
$634$	−8.00000	−0.317721
$635$	3.00000	0.119051
$636$	20.0000	0.793052
$637$	−18.0000	−0.713186
$638$	−28.0000	−1.10853
$639$	0	0
$640$	0	0
$641$	−43.0000	−1.69840	−0.849199	−	0.528073i	$-0.822915\pi$
$641$	−43.0000	−1.69840	−0.849199	+	0.528073i	$0.822915\pi$
$642$	−30.0000	−1.18401
$643$	30.0000	1.18308	0.591542	−	0.806274i	$-0.298519\pi$
$643$	30.0000	1.18308	0.591542	+	0.806274i	$0.298519\pi$
$644$	−60.0000	−2.36433
$645$	11.0000	0.433125
$646$	−48.0000	−1.88853
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−14.0000	−0.549548
$650$	2.00000	0.0784465
$651$	5.00000	0.195965
$652$	8.00000	0.313304
$653$	47.0000	1.83925	0.919626	−	0.392795i	$-0.128492\pi$
$653$	47.0000	1.83925	0.919626	+	0.392795i	$0.128492\pi$
$654$	4.00000	0.156412
$655$	−18.0000	−0.703318
$656$	−36.0000	−1.40556
$657$	2.00000	0.0780274
$658$	−80.0000	−3.11872
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	4.00000	0.155700
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−26.0000	−1.01052
$663$	6.00000	0.233021
$664$	0	0
$665$	20.0000	0.775567
$666$	12.0000	0.464991
$667$	42.0000	1.62625
$668$	24.0000	0.928588
$669$	2.00000	0.0773245
$670$	−6.00000	−0.231800
$671$	4.00000	0.154418
$672$	40.0000	1.54303
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−44.0000	−1.69482
$675$	−1.00000	−0.0384900
$676$	2.00000	0.0769231
$677$	24.0000	0.922395	0.461197	−	0.887298i	$-0.347420\pi$
$677$	24.0000	0.922395	0.461197	+	0.887298i	$0.347420\pi$
$678$	−6.00000	−0.230429
$679$	−5.00000	−0.191882
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−4.00000	−0.153168
$683$	38.0000	1.45403	0.727015	−	0.686622i	$-0.240907\pi$
$683$	38.0000	1.45403	0.727015	+	0.686622i	$0.240907\pi$
$684$	8.00000	0.305888
$685$	22.0000	0.840577
$686$	110.000	4.19982
$687$	−26.0000	−0.991962
$688$	−44.0000	−1.67748
$689$	10.0000	0.380970
$690$	−12.0000	−0.456832
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	−2.00000	−0.0760286
$693$	−10.0000	−0.379869
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	54.0000	2.04540
$698$	−28.0000	−1.05982
$699$	−3.00000	−0.113470
$700$	−10.0000	−0.377964
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	−16.0000	−0.603023
$705$	−8.00000	−0.301297
$706$	−10.0000	−0.376355
$707$	30.0000	1.12827
$708$	14.0000	0.526152
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	0	0
$713$	6.00000	0.224702
$714$	−60.0000	−2.24544
$715$	2.00000	0.0747958
$716$	46.0000	1.71910
$717$	−18.0000	−0.672222
$718$	−54.0000	−2.01526
$719$	25.0000	0.932343	0.466171	−	0.884694i	$-0.345633\pi$
$719$	25.0000	0.932343	0.466171	+	0.884694i	$0.345633\pi$
$720$	4.00000	0.149071
$721$	−50.0000	−1.86210
$722$	6.00000	0.223297
$723$	7.00000	0.260333
$724$	−16.0000	−0.594635
$725$	7.00000	0.259973
$726$	−14.0000	−0.519589
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	66.0000	2.44110
$732$	−4.00000	−0.147844
$733$	19.0000	0.701781	0.350891	−	0.936416i	$-0.385879\pi$
$733$	19.0000	0.701781	0.350891	+	0.936416i	$0.385879\pi$
$734$	−32.0000	−1.18114
$735$	18.0000	0.663940
$736$	48.0000	1.76930
$737$	−6.00000	−0.221013
$738$	−18.0000	−0.662589
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	12.0000	0.441129
$741$	4.00000	0.146944
$742$	−100.000	−3.67112
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	52.0000	1.90386
$747$	11.0000	0.402469
$748$	24.0000	0.877527
$749$	75.0000	2.74044
$750$	−2.00000	−0.0730297
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	32.0000	1.16692
$753$	7.00000	0.255094
$754$	14.0000	0.509850
$755$	−13.0000	−0.473118
$756$	10.0000	0.363696
$757$	49.0000	1.78094	0.890468	−	0.455047i	$-0.150377\pi$
$757$	49.0000	1.78094	0.890468	+	0.455047i	$0.150377\pi$
$758$	68.0000	2.46987
$759$	−12.0000	−0.435572
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−6.00000	−0.217357
$763$	−10.0000	−0.362024
$764$	12.0000	0.434145
$765$	−6.00000	−0.216930
$766$	−22.0000	−0.794892
$767$	7.00000	0.252755
$768$	−16.0000	−0.577350
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−20.0000	−0.720750
$771$	−18.0000	−0.648254
$772$	−26.0000	−0.935760
$773$	−45.0000	−1.61854	−0.809269	−	0.587439i	$-0.800136\pi$
$773$	−45.0000	−1.61854	−0.809269	+	0.587439i	$0.800136\pi$
$774$	−22.0000	−0.790774
$775$	1.00000	0.0359211
$776$	0	0
$777$	−30.0000	−1.07624
$778$	−50.0000	−1.79259
$779$	36.0000	1.28983
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	−72.0000	−2.57471
$783$	−7.00000	−0.250160
$784$	−72.0000	−2.57143
$785$	6.00000	0.214149
$786$	36.0000	1.28408
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	18.0000	0.641223
$789$	4.00000	0.142404
$790$	12.0000	0.426941
$791$	15.0000	0.533339
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	4.00000	0.141955
$795$	−10.0000	−0.354663
$796$	−32.0000	−1.13421
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−40.0000	−1.41598
$799$	−48.0000	−1.69812
$800$	8.00000	0.282843
$801$	−14.0000	−0.494666
$802$	72.0000	2.54241
$803$	4.00000	0.141157
$804$	6.00000	0.211604
$805$	30.0000	1.05736
$806$	2.00000	0.0704470
$807$	30.0000	1.05605
$808$	0	0
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	2.00000	0.0702728
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	−70.0000	−2.45652
$813$	13.0000	0.455930
$814$	24.0000	0.841200
$815$	−4.00000	−0.140114
$816$	24.0000	0.840168
$817$	44.0000	1.53937
$818$	−26.0000	−0.909069
$819$	5.00000	0.174714
$820$	−18.0000	−0.628587
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−44.0000	−1.53468
$823$	41.0000	1.42917	0.714585	−	0.699549i	$-0.246616\pi$
$823$	41.0000	1.42917	0.714585	+	0.699549i	$0.246616\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	−70.0000	−2.43561
$827$	43.0000	1.49526	0.747628	−	0.664117i	$-0.231193\pi$
$827$	43.0000	1.49526	0.747628	+	0.664117i	$0.231193\pi$
$828$	12.0000	0.417029
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	22.0000	0.763631
$831$	−5.00000	−0.173448
$832$	8.00000	0.277350
$833$	108.000	3.74198
$834$	32.0000	1.10807
$835$	−12.0000	−0.415277
$836$	16.0000	0.553372
$837$	−1.00000	−0.0345651
$838$	60.0000	2.07267
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	24.0000	0.827095
$843$	−3.00000	−0.103325
$844$	−22.0000	−0.757271
$845$	−1.00000	−0.0344010
$846$	16.0000	0.550091
$847$	35.0000	1.20261
$848$	40.0000	1.37361
$849$	14.0000	0.480479
$850$	−12.0000	−0.411597
$851$	−36.0000	−1.23406
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	20.0000	0.684386
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	−4.00000	−0.136558
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−22.0000	−0.750194
$861$	45.0000	1.53360
$862$	−6.00000	−0.204361
$863$	−25.0000	−0.851010	−0.425505	−	0.904956i	$-0.639903\pi$
$863$	−25.0000	−0.851010	−0.425505	+	0.904956i	$0.639903\pi$
$864$	−8.00000	−0.272166
$865$	1.00000	0.0340010
$866$	−38.0000	−1.29129
$867$	−19.0000	−0.645274
$868$	−10.0000	−0.339422
$869$	12.0000	0.407072
$870$	−14.0000	−0.474644
$871$	3.00000	0.101651
$872$	0	0
$873$	1.00000	0.0338449
$874$	−48.0000	−1.62362
$875$	5.00000	0.169031
$876$	−4.00000	−0.135147
$877$	−54.0000	−1.82345	−0.911725	−	0.410801i	$-0.865249\pi$
$877$	−54.0000	−1.82345	−0.911725	+	0.410801i	$0.865249\pi$
$878$	−34.0000	−1.14744
$879$	24.0000	0.809500
$880$	8.00000	0.269680
$881$	−39.0000	−1.31394	−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000	−1.31394	−0.656972	+	0.753915i	$0.728163\pi$
$882$	−36.0000	−1.21218
$883$	15.0000	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$883$	15.0000	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$884$	−12.0000	−0.403604
$885$	−7.00000	−0.235302
$886$	−30.0000	−1.00787
$887$	9.00000	0.302190	0.151095	−	0.988519i	$-0.451720\pi$
$887$	9.00000	0.302190	0.151095	+	0.988519i	$0.451720\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	−28.0000	−0.938562
$891$	2.00000	0.0670025
$892$	−4.00000	−0.133930
$893$	−32.0000	−1.07084
$894$	18.0000	0.602010
$895$	−23.0000	−0.768805
$896$	0	0
$897$	6.00000	0.200334
$898$	60.0000	2.00223
$899$	7.00000	0.233463
$900$	2.00000	0.0666667
$901$	−60.0000	−1.99889
$902$	−36.0000	−1.19867
$903$	55.0000	1.83029
$904$	0	0
$905$	8.00000	0.265929
$906$	26.0000	0.863792
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	36.0000	1.19470
$909$	−6.00000	−0.199007
$910$	10.0000	0.331497
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	16.0000	0.529813
$913$	22.0000	0.728094
$914$	−40.0000	−1.32308
$915$	2.00000	0.0661180
$916$	52.0000	1.71813
$917$	−90.0000	−2.97206
$918$	12.0000	0.396059
$919$	−41.0000	−1.35247	−0.676233	−	0.736688i	$-0.736389\pi$
$919$	−41.0000	−1.35247	−0.676233	+	0.736688i	$0.736389\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−24.0000	−0.790398
$923$	0	0
$924$	20.0000	0.657952
$925$	−6.00000	−0.197279
$926$	8.00000	0.262896
$927$	10.0000	0.328443
$928$	56.0000	1.83829
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	−2.00000	−0.0655826
$931$	72.0000	2.35970
$932$	6.00000	0.196537
$933$	20.0000	0.654771
$934$	−42.0000	−1.37428
$935$	−12.0000	−0.392442
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−30.0000	−0.979535
$939$	5.00000	0.163169
$940$	16.0000	0.521862
$941$	34.0000	1.10837	0.554184	−	0.832394i	$-0.313030\pi$
$941$	34.0000	1.10837	0.554184	+	0.832394i	$0.313030\pi$
$942$	−12.0000	−0.390981
$943$	54.0000	1.75848
$944$	28.0000	0.911322
$945$	−5.00000	−0.162650
$946$	−44.0000	−1.43056
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−12.0000	−0.389742
$949$	−2.00000	−0.0649227
$950$	−8.00000	−0.259554
$951$	−4.00000	−0.129709
$952$	0	0
$953$	56.0000	1.81402	0.907009	−	0.421111i	$-0.138360\pi$
$953$	56.0000	1.81402	0.907009	+	0.421111i	$0.138360\pi$
$954$	20.0000	0.647524
$955$	−6.00000	−0.194155
$956$	36.0000	1.16432
$957$	−14.0000	−0.452556
$958$	78.0000	2.52007
$959$	110.000	3.55209
$960$	−8.00000	−0.258199
$961$	1.00000	0.0322581
$962$	−12.0000	−0.386896
$963$	−15.0000	−0.483368
$964$	−14.0000	−0.450910
$965$	13.0000	0.418485
$966$	−60.0000	−1.93047
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	2.00000	0.0642161
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	−2.00000	−0.0641500
$973$	−80.0000	−2.56468
$974$	8.00000	0.256337
$975$	1.00000	0.0320256
$976$	−8.00000	−0.256074
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	8.00000	0.255812
$979$	−28.0000	−0.894884
$980$	−36.0000	−1.14998
$981$	2.00000	0.0638551
$982$	10.0000	0.319113
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	−84.0000	−2.67510
$987$	−40.0000	−1.27321
$988$	−8.00000	−0.254514
$989$	66.0000	2.09868
$990$	4.00000	0.127128
$991$	−26.0000	−0.825917	−0.412959	−	0.910750i	$-0.635505\pi$
$991$	−26.0000	−0.825917	−0.412959	+	0.910750i	$0.635505\pi$
$992$	8.00000	0.254000
$993$	−13.0000	−0.412543
$994$	0	0
$995$	16.0000	0.507234
$996$	−22.0000	−0.697097
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	−82.0000	−2.59566
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.a.1.1	✓	1	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more