LMFDB - Embedded newform 2415.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,2,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, 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("2415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2415.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:	$19.2838720881$
Analytic rank:	$1$
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	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} -2.00000 q^{31} +5.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} -1.00000 q^{35} -1.00000 q^{36} -4.00000 q^{37} -4.00000 q^{38} -3.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -12.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} -6.00000 q^{61} -2.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} -4.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{77} -2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} +1.00000 q^{84} -2.00000 q^{85} -8.00000 q^{86} +2.00000 q^{87} +6.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +1.00000 q^{92} -2.00000 q^{93} -4.00000 q^{95} +5.00000 q^{96} +14.0000 q^{97} +1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−1.00000	−0.267261
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−2.00000	−0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	1.00000	0.182574
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.00000	0.883883
$33$	−2.00000	−0.348155
$34$	−2.00000	−0.342997
$35$	−1.00000	−0.169031
$36$	−1.00000	−0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−1.00000	−0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	2.00000	0.301511
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	1.00000	0.136083
$55$	−2.00000	−0.269680
$56$	3.00000	0.400892
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	−1.00000	−0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.00000	−0.254000
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	−1.00000	−0.120386
$70$	−1.00000	−0.119523
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−3.00000	−0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−4.00000	−0.464991
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	2.00000	0.227921
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	1.00000	0.109109
$85$	−2.00000	−0.216930
$86$	−8.00000	−0.862662
$87$	2.00000	0.214423
$88$	6.00000	0.639602
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−4.00000	−0.410391
$96$	5.00000	0.510310
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	1.00000	0.101015
$99$	−2.00000	−0.201008
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−2.00000	−0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	−12.0000	−1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−2.00000	−0.190693
$111$	−4.00000	−0.379663
$112$	1.00000	0.0944911
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	−4.00000	−0.374634
$115$	−1.00000	−0.0932505
$116$	−2.00000	−0.185695
$117$	0	0
$118$	6.00000	0.552345
$119$	2.00000	0.183340
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	−6.00000	−0.543214
$123$	−10.0000	−0.901670
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−3.00000	−0.265165
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	2.00000	0.174078
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	6.00000	0.514496
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−1.00000	−0.0851257
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−4.00000	−0.335673
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	2.00000	0.166091
$146$	4.00000	0.331042
$147$	1.00000	0.0824786
$148$	4.00000	0.328798
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	12.0000	0.973329
$153$	−2.00000	−0.161690
$154$	2.00000	0.161165
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−2.00000	−0.159111
$159$	−12.0000	−0.951662
$160$	5.00000	0.395285
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	10.0000	0.780869
$165$	−2.00000	−0.155700
$166$	12.0000	0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	3.00000	0.231455
$169$	−13.0000	−1.00000
$170$	−2.00000	−0.153393
$171$	−4.00000	−0.305888
$172$	8.00000	0.609994
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	2.00000	0.151620
$175$	−1.00000	−0.0755929
$176$	2.00000	0.150756
$177$	6.00000	0.450988
$178$	−10.0000	−0.749532
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−1.00000	−0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	3.00000	0.221163
$185$	−4.00000	−0.294086
$186$	−2.00000	−0.146647
$187$	4.00000	0.292509
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	−4.00000	−0.290191
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	7.00000	0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−2.00000	−0.142134
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−3.00000	−0.212132
$201$	−4.00000	−0.282138
$202$	2.00000	0.140720
$203$	−2.00000	−0.140372
$204$	2.00000	0.140028
$205$	−10.0000	−0.698430
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	8.00000	0.553372
$210$	−1.00000	−0.0690066
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	12.0000	0.824163
$213$	−4.00000	−0.274075
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	−3.00000	−0.204124
$217$	2.00000	0.135769
$218$	2.00000	0.135457
$219$	4.00000	0.270295
$220$	2.00000	0.134840
$221$	0	0
$222$	−4.00000	−0.268462
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−5.00000	−0.334077
$225$	1.00000	0.0666667
$226$	−20.0000	−1.33038
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	4.00000	0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−1.00000	−0.0659380
$231$	2.00000	0.131590
$232$	−6.00000	−0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	−2.00000	−0.129914
$238$	2.00000	0.129641
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−1.00000	−0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	1.00000	0.0638877
$246$	−10.0000	−0.637577
$247$	0	0
$248$	6.00000	0.381000
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	1.00000	0.0629941
$253$	2.00000	0.125739
$254$	0	0
$255$	−2.00000	−0.125245
$256$	−17.0000	−1.06250
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	−8.00000	−0.498058
$259$	4.00000	0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	−10.0000	−0.617802
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	6.00000	0.369274
$265$	−12.0000	−0.737154
$266$	4.00000	0.245256
$267$	−10.0000	−0.611990
$268$	4.00000	0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	1.00000	0.0608581
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	8.00000	0.483298
$275$	−2.00000	−0.120605
$276$	1.00000	0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−2.00000	−0.119952
$279$	−2.00000	−0.119737
$280$	3.00000	0.179284
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	4.00000	0.237356
$285$	−4.00000	−0.236940
$286$	0	0
$287$	10.0000	0.590281
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	14.0000	0.820695
$292$	−4.00000	−0.234082
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	1.00000	0.0583212
$295$	6.00000	0.349334
$296$	12.0000	0.697486
$297$	−2.00000	−0.116052
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.00000	0.461112
$302$	20.0000	1.15087
$303$	2.00000	0.114897
$304$	4.00000	0.229416
$305$	−6.00000	−0.343559
$306$	−2.00000	−0.114332
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−2.00000	−0.113961
$309$	0	0
$310$	−2.00000	−0.113592
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−2.00000	−0.112867
$315$	−1.00000	−0.0563436
$316$	2.00000	0.112509
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−12.0000	−0.672927
$319$	−4.00000	−0.223957
$320$	7.00000	0.391312
$321$	12.0000	0.669775
$322$	1.00000	0.0557278
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	2.00000	0.110600
$328$	30.0000	1.65647
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−12.0000	−0.658586
$333$	−4.00000	−0.219199
$334$	8.00000	0.437741
$335$	−4.00000	−0.218543
$336$	1.00000	0.0545545
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	−13.0000	−0.707107
$339$	−20.0000	−1.08625
$340$	2.00000	0.108465
$341$	4.00000	0.216612
$342$	−4.00000	−0.216295
$343$	−1.00000	−0.0539949
$344$	24.0000	1.29399
$345$	−1.00000	−0.0538382
$346$	−4.00000	−0.215041
$347$	36.0000	1.93258	0.966291	−	0.257454i	$-0.0828835\pi$
$347$	36.0000	1.93258	0.966291	+	0.257454i	$0.0828835\pi$
$348$	−2.00000	−0.107211
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−10.0000	−0.533002
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	6.00000	0.318896
$355$	−4.00000	−0.212298
$356$	10.0000	0.529999
$357$	2.00000	0.105851
$358$	24.0000	1.26844
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	−7.00000	−0.367405
$364$	0	0
$365$	4.00000	0.209370
$366$	−6.00000	−0.313625
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	1.00000	0.0521286
$369$	−10.0000	−0.520579
$370$	−4.00000	−0.207950
$371$	12.0000	0.623009
$372$	2.00000	0.103695
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	4.00000	0.206835
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	6.00000	0.306987
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−3.00000	−0.153093
$385$	2.00000	0.101929
$386$	6.00000	0.305392
$387$	−8.00000	−0.406663
$388$	−14.0000	−0.710742
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	−3.00000	−0.151523
$393$	−10.0000	−0.504433
$394$	−18.0000	−0.906827
$395$	−2.00000	−0.100631
$396$	2.00000	0.100504
$397$	−12.0000	−0.602263	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000	−0.602263	−0.301131	+	0.953583i	$0.597364\pi$
$398$	24.0000	1.20301
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	1.00000	0.0496904
$406$	−2.00000	−0.0992583
$407$	8.00000	0.396545
$408$	6.00000	0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−10.0000	−0.493865
$411$	8.00000	0.394611
$412$	0	0
$413$	−6.00000	−0.295241
$414$	−1.00000	−0.0491473
$415$	12.0000	0.589057
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	8.00000	0.391293
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	1.00000	0.0487950
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	36.0000	1.74831
$425$	−2.00000	−0.0970143
$426$	−4.00000	−0.193801
$427$	6.00000	0.290360
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	−1.00000	−0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	2.00000	0.0960031
$435$	2.00000	0.0958927
$436$	−2.00000	−0.0957826
$437$	4.00000	0.191346
$438$	4.00000	0.191127
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	6.00000	0.286039
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	4.00000	0.189832
$445$	−10.0000	−0.474045
$446$	−24.0000	−1.13643
$447$	−6.00000	−0.283790
$448$	−7.00000	−0.330719
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	1.00000	0.0471405
$451$	20.0000	0.941763
$452$	20.0000	0.940721
$453$	20.0000	0.939682
$454$	−12.0000	−0.563188
$455$	0	0
$456$	12.0000	0.561951
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	−6.00000	−0.280362
$459$	−2.00000	−0.0933520
$460$	1.00000	0.0466252
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	2.00000	0.0930484
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−2.00000	−0.0928477
$465$	−2.00000	−0.0927478
$466$	10.0000	0.463241
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	−18.0000	−0.828517
$473$	16.0000	0.735681
$474$	−2.00000	−0.0918630
$475$	−4.00000	−0.183533
$476$	−2.00000	−0.0916698
$477$	−12.0000	−0.549442
$478$	20.0000	0.914779
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	5.00000	0.228218
$481$	0	0
$482$	−14.0000	−0.637683
$483$	1.00000	0.0455016
$484$	7.00000	0.318182
$485$	14.0000	0.635707
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	18.0000	0.814822
$489$	−4.00000	−0.180886
$490$	1.00000	0.0451754
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	10.0000	0.450835
$493$	−4.00000	−0.180151
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	2.00000	0.0898027
$497$	4.00000	0.179425
$498$	12.0000	0.537733
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	0	0
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	3.00000	0.133631
$505$	2.00000	0.0889988
$506$	2.00000	0.0889108
$507$	−13.0000	−0.577350
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	−2.00000	−0.0885615
$511$	−4.00000	−0.176950
$512$	−11.0000	−0.486136
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	8.00000	0.352180
$517$	0	0
$518$	4.00000	0.175750
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	2.00000	0.0875376
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	10.0000	0.436852
$525$	−1.00000	−0.0436436
$526$	−12.0000	−0.523225
$527$	4.00000	0.174243
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	−12.0000	−0.521247
$531$	6.00000	0.260378
$532$	−4.00000	−0.173422
$533$	0	0
$534$	−10.0000	−0.432742
$535$	12.0000	0.518805
$536$	12.0000	0.518321
$537$	24.0000	1.03568
$538$	10.0000	0.431131
$539$	−2.00000	−0.0861461
$540$	−1.00000	−0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−2.00000	−0.0859074
$543$	−2.00000	−0.0858282
$544$	−10.0000	−0.428746
$545$	2.00000	0.0856706
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−8.00000	−0.341743
$549$	−6.00000	−0.256074
$550$	−2.00000	−0.0852803
$551$	−8.00000	−0.340811
$552$	3.00000	0.127688
$553$	2.00000	0.0850487
$554$	2.00000	0.0849719
$555$	−4.00000	−0.169791
$556$	2.00000	0.0848189
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	−2.00000	−0.0846668
$559$	0	0
$560$	1.00000	0.0422577
$561$	4.00000	0.168880
$562$	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	$0.774121\pi$
$564$	0	0
$565$	−20.0000	−0.841406
$566$	12.0000	0.504398
$567$	−1.00000	−0.0419961
$568$	12.0000	0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−4.00000	−0.167542
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	10.0000	0.417392
$575$	−1.00000	−0.0417029
$576$	7.00000	0.291667
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	−13.0000	−0.540729
$579$	6.00000	0.249351
$580$	−2.00000	−0.0830455
$581$	−12.0000	−0.497844
$582$	14.0000	0.580319
$583$	24.0000	0.993978
$584$	−12.0000	−0.496564
$585$	0	0
$586$	14.0000	0.578335
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	−1.00000	−0.0412393
$589$	8.00000	0.329634
$590$	6.00000	0.247016
$591$	−18.0000	−0.740421
$592$	4.00000	0.164399
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	−2.00000	−0.0820610
$595$	2.00000	0.0819920
$596$	6.00000	0.245770
$597$	24.0000	0.982255
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−3.00000	−0.122474
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	−20.0000	−0.813788
$605$	−7.00000	−0.284590
$606$	2.00000	0.0812444
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−20.0000	−0.811107
$609$	−2.00000	−0.0810441
$610$	−6.00000	−0.242933
$611$	0	0
$612$	2.00000	0.0808452
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−4.00000	−0.161427
$615$	−10.0000	−0.403239
$616$	−6.00000	−0.241747
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	2.00000	0.0803219
$621$	−1.00000	−0.0401286
$622$	−18.0000	−0.721734
$623$	10.0000	0.400642
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	8.00000	0.319489
$628$	2.00000	0.0798087
$629$	8.00000	0.318981
$630$	−1.00000	−0.0398410
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	6.00000	0.238667
$633$	8.00000	0.317971
$634$	30.0000	1.19145
$635$	0	0
$636$	12.0000	0.475831
$637$	0	0
$638$	−4.00000	−0.158362
$639$	−4.00000	−0.158238
$640$	−3.00000	−0.118585
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	12.0000	0.473602
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−1.00000	−0.0394055
$645$	−8.00000	−0.315000
$646$	8.00000	0.314756
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−3.00000	−0.117851
$649$	−12.0000	−0.471041
$650$	0	0
$651$	2.00000	0.0783862
$652$	4.00000	0.156652
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	2.00000	0.0782062
$655$	−10.0000	−0.390732
$656$	10.0000	0.390434
$657$	4.00000	0.156055
$658$	0	0
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	2.00000	0.0778499
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−36.0000	−1.39707
$665$	4.00000	0.155113
$666$	−4.00000	−0.154997
$667$	−2.00000	−0.0774403
$668$	−8.00000	−0.309529
$669$	−24.0000	−0.927894
$670$	−4.00000	−0.154533
$671$	12.0000	0.463255
$672$	−5.00000	−0.192879
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	16.0000	0.616297
$675$	1.00000	0.0384900
$676$	13.0000	0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−20.0000	−0.768095
$679$	−14.0000	−0.537271
$680$	6.00000	0.230089
$681$	−12.0000	−0.459841
$682$	4.00000	0.153168
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	4.00000	0.152944
$685$	8.00000	0.305664
$686$	−1.00000	−0.0381802
$687$	−6.00000	−0.228914
$688$	8.00000	0.304997
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	−50.0000	−1.90209	−0.951045	−	0.309053i	$-0.899988\pi$
$691$	−50.0000	−1.90209	−0.951045	+	0.309053i	$0.899988\pi$
$692$	4.00000	0.152057
$693$	2.00000	0.0759737
$694$	36.0000	1.36654
$695$	−2.00000	−0.0758643
$696$	−6.00000	−0.227429
$697$	20.0000	0.757554
$698$	−10.0000	−0.378506
$699$	10.0000	0.378235
$700$	1.00000	0.0377964
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	−14.0000	−0.527645
$705$	0	0
$706$	−12.0000	−0.451626
$707$	−2.00000	−0.0752177
$708$	−6.00000	−0.225494
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	−4.00000	−0.150117
$711$	−2.00000	−0.0750059
$712$	30.0000	1.12430
$713$	2.00000	0.0749006
$714$	2.00000	0.0748481
$715$	0	0
$716$	−24.0000	−0.896922
$717$	20.0000	0.746914
$718$	26.0000	0.970311
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−14.0000	−0.520666
$724$	2.00000	0.0743294
$725$	2.00000	0.0742781
$726$	−7.00000	−0.259794
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	16.0000	0.591781
$732$	6.00000	0.221766
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−8.00000	−0.295285
$735$	1.00000	0.0368856
$736$	−5.00000	−0.184302
$737$	8.00000	0.294684
$738$	−10.0000	−0.368105
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	12.0000	0.440534
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	6.00000	0.219971
$745$	−6.00000	−0.219823
$746$	−20.0000	−0.732252
$747$	12.0000	0.439057
$748$	−4.00000	−0.146254
$749$	−12.0000	−0.438470
$750$	1.00000	0.0365148
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	20.0000	0.727875
$756$	1.00000	0.0363696
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	−2.00000	−0.0726433
$759$	2.00000	0.0725954
$760$	12.0000	0.435286
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−6.00000	−0.217072
$765$	−2.00000	−0.0723102
$766$	−16.0000	−0.578103
$767$	0	0
$768$	−17.0000	−0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	−6.00000	−0.215945
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	−8.00000	−0.287554
$775$	−2.00000	−0.0718421
$776$	−42.0000	−1.50771
$777$	4.00000	0.143499
$778$	34.0000	1.21896
$779$	40.0000	1.43315
$780$	0	0
$781$	8.00000	0.286263
$782$	2.00000	0.0715199
$783$	2.00000	0.0714742
$784$	−1.00000	−0.0357143
$785$	−2.00000	−0.0713831
$786$	−10.0000	−0.356688
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	18.0000	0.641223
$789$	−12.0000	−0.427211
$790$	−2.00000	−0.0711568
$791$	20.0000	0.711118
$792$	6.00000	0.213201
$793$	0	0
$794$	−12.0000	−0.425864
$795$	−12.0000	−0.425596
$796$	−24.0000	−0.850657
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	5.00000	0.176777
$801$	−10.0000	−0.353333
$802$	2.00000	0.0706225
$803$	−8.00000	−0.282314
$804$	4.00000	0.141069
$805$	1.00000	0.0352454
$806$	0	0
$807$	10.0000	0.352017
$808$	−6.00000	−0.211079
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	1.00000	0.0351364
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	2.00000	0.0701862
$813$	−2.00000	−0.0701431
$814$	8.00000	0.280400
$815$	−4.00000	−0.140114
$816$	2.00000	0.0700140
$817$	32.0000	1.11954
$818$	10.0000	0.349642
$819$	0	0
$820$	10.0000	0.349215
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	8.00000	0.279032
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	−6.00000	−0.208767
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	1.00000	0.0347524
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	12.0000	0.416526
$831$	2.00000	0.0693792
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	−2.00000	−0.0692543
$835$	8.00000	0.276851
$836$	−8.00000	−0.276686
$837$	−2.00000	−0.0691301
$838$	20.0000	0.690889
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	3.00000	0.103510
$841$	−25.0000	−0.862069
$842$	−34.0000	−1.17172
$843$	6.00000	0.206651
$844$	−8.00000	−0.275371
$845$	−13.0000	−0.447214
$846$	0	0
$847$	7.00000	0.240523
$848$	12.0000	0.412082
$849$	12.0000	0.411839
$850$	−2.00000	−0.0685994
$851$	4.00000	0.137118
$852$	4.00000	0.137038
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	6.00000	0.205316
$855$	−4.00000	−0.136797
$856$	−36.0000	−1.23045
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	8.00000	0.272798
$861$	10.0000	0.340799
$862$	−30.0000	−1.02180
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	5.00000	0.170103
$865$	−4.00000	−0.136004
$866$	14.0000	0.475739
$867$	−13.0000	−0.441503
$868$	−2.00000	−0.0678844
$869$	4.00000	0.135691
$870$	2.00000	0.0678064
$871$	0	0
$872$	−6.00000	−0.203186
$873$	14.0000	0.473828
$874$	4.00000	0.135302
$875$	−1.00000	−0.0338062
$876$	−4.00000	−0.135147
$877$	54.0000	1.82345	0.911725	−	0.410801i	$-0.134751\pi$
$877$	54.0000	1.82345	0.911725	+	0.410801i	$0.134751\pi$
$878$	−22.0000	−0.742464
$879$	14.0000	0.472208
$880$	2.00000	0.0674200
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	1.00000	0.0336718
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	12.0000	0.403148
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	12.0000	0.402694
$889$	0	0
$890$	−10.0000	−0.335201
$891$	−2.00000	−0.0670025
$892$	24.0000	0.803579
$893$	0	0
$894$	−6.00000	−0.200670
$895$	24.0000	0.802232
$896$	3.00000	0.100223
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−4.00000	−0.133407
$900$	−1.00000	−0.0333333
$901$	24.0000	0.799556
$902$	20.0000	0.665927
$903$	8.00000	0.266223
$904$	60.0000	1.99557
$905$	−2.00000	−0.0664822
$906$	20.0000	0.664455
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	12.0000	0.398234
$909$	2.00000	0.0663358
$910$	0	0
$911$	10.0000	0.331315	0.165657	−	0.986183i	$-0.447025\pi$
$911$	10.0000	0.331315	0.165657	+	0.986183i	$0.447025\pi$
$912$	4.00000	0.132453
$913$	−24.0000	−0.794284
$914$	−12.0000	−0.396925
$915$	−6.00000	−0.198354
$916$	6.00000	0.198246
$917$	10.0000	0.330229
$918$	−2.00000	−0.0660098
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	3.00000	0.0989071
$921$	−4.00000	−0.131804
$922$	26.0000	0.856264
$923$	0	0
$924$	−2.00000	−0.0657952
$925$	−4.00000	−0.131519
$926$	−8.00000	−0.262896
$927$	0	0
$928$	10.0000	0.328266
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	−2.00000	−0.0655826
$931$	−4.00000	−0.131095
$932$	−10.0000	−0.327561
$933$	−18.0000	−0.589294
$934$	20.0000	0.654420
$935$	4.00000	0.130814
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	4.00000	0.130605
$939$	2.00000	0.0652675
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−2.00000	−0.0651635
$943$	10.0000	0.325645
$944$	−6.00000	−0.195283
$945$	−1.00000	−0.0325300
$946$	16.0000	0.520205
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	2.00000	0.0649570
$949$	0	0
$950$	−4.00000	−0.129777
$951$	30.0000	0.972817
$952$	−6.00000	−0.194461
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	−12.0000	−0.388514
$955$	6.00000	0.194155
$956$	−20.0000	−0.646846
$957$	−4.00000	−0.129302
$958$	4.00000	0.129234
$959$	−8.00000	−0.258333
$960$	7.00000	0.225924
$961$	−27.0000	−0.870968
$962$	0	0
$963$	12.0000	0.386695
$964$	14.0000	0.450910
$965$	6.00000	0.193147
$966$	1.00000	0.0321745
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	21.0000	0.674966
$969$	8.00000	0.256997
$970$	14.0000	0.449513
$971$	−56.0000	−1.79713	−0.898563	−	0.438845i	$-0.855388\pi$
$971$	−56.0000	−1.79713	−0.898563	+	0.438845i	$0.855388\pi$
$972$	−1.00000	−0.0320750
$973$	2.00000	0.0641171
$974$	24.0000	0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	−4.00000	−0.127906
$979$	20.0000	0.639203
$980$	−1.00000	−0.0319438
$981$	2.00000	0.0638551
$982$	40.0000	1.27645
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	30.0000	0.956365
$985$	−18.0000	−0.573528
$986$	−4.00000	−0.127386
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	−2.00000	−0.0635642
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	−10.0000	−0.317500
$993$	−12.0000	−0.380808
$994$	4.00000	0.126872
$995$	24.0000	0.760851
$996$	−12.0000	−0.380235
$997$	16.0000	0.506725	0.253363	−	0.967371i	$-0.418463\pi$
$997$	16.0000	0.506725	0.253363	+	0.967371i	$0.418463\pi$
$998$	−32.0000	−1.01294
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.2.a.i.1.1	✓	1
3.2	odd	2		7245.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.i.1.1	✓	1	1.1	even	1	trivial
7245.2.a.g.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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