LMFDB - Embedded newform 2415.2.a.d.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{39} +3.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +4.00000 q^{52} -1.00000 q^{54} -2.00000 q^{55} +3.00000 q^{56} +6.00000 q^{58} +2.00000 q^{59} -1.00000 q^{60} -14.0000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} +2.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} +3.00000 q^{72} +8.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -2.00000 q^{77} +4.00000 q^{78} +10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} -2.00000 q^{85} +8.00000 q^{86} -6.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} -1.00000 q^{92} -2.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−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
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−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	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	3.00000	0.474342
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−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
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	4.00000	0.554700
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	−2.00000	−0.269680
$56$	3.00000	0.400892
$57$	0	0
$58$	6.00000	0.787839
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	−1.00000	−0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	2.00000	0.254000
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	2.00000	0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	2.00000	0.242536
$69$	1.00000	0.120386
$70$	−1.00000	−0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−4.00000	−0.464991
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.00000	−0.227921
$78$	4.00000	0.452911
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	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
$87$	−6.00000	−0.643268
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	−4.00000	−0.419314
$92$	−1.00000	−0.104257
$93$	−2.00000	−0.207390
$94$	8.00000	0.825137
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	−2.00000	−0.201008
$100$	−1.00000	−0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	2.00000	0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−12.0000	−1.17670
$105$	1.00000	0.0975900
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−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
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	6.00000	0.557086
$117$	−4.00000	−0.369800
$118$	−2.00000	−0.184115
$119$	−2.00000	−0.183340
$120$	3.00000	0.273861
$121$	−7.00000	−0.636364
$122$	14.0000	1.26750
$123$	−2.00000	−0.180334
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	3.00000	0.265165
$129$	−8.00000	−0.704361
$130$	4.00000	0.350823
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	12.0000	1.03664
$135$	1.00000	0.0860663
$136$	−6.00000	−0.514496
$137$	−20.0000	−1.70872	−0.854358	−	0.519685i	$-0.826049\pi$
$137$	−20.0000	−1.70872	−0.854358	+	0.519685i	$0.826049\pi$
$138$	−1.00000	−0.0851257
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−1.00000	−0.0845154
$141$	−8.00000	−0.673722
$142$	0	0
$143$	8.00000	0.668994
$144$	−1.00000	−0.0833333
$145$	−6.00000	−0.498273
$146$	−8.00000	−0.662085
$147$	1.00000	0.0824786
$148$	−4.00000	−0.328798
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−1.00000	−0.0816497
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	2.00000	0.161165
$155$	−2.00000	−0.160644
$156$	4.00000	0.320256
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	−5.00000	−0.395285
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	2.00000	0.156174
$165$	−2.00000	−0.155700
$166$	−12.0000	−0.931381
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	3.00000	0.231455
$169$	3.00000	0.230769
$170$	2.00000	0.153393
$171$	0	0
$172$	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	$0.548590\pi$
$174$	6.00000	0.454859
$175$	1.00000	0.0755929
$176$	2.00000	0.150756
$177$	2.00000	0.150329
$178$	−6.00000	−0.449719
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.00000	−0.0745356
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	4.00000	0.296500
$183$	−14.0000	−1.03491
$184$	3.00000	0.221163
$185$	4.00000	0.294086
$186$	2.00000	0.146647
$187$	4.00000	0.292509
$188$	8.00000	0.583460
$189$	1.00000	0.0727393
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	7.00000	0.505181
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	14.0000	1.00514
$195$	−4.00000	−0.286446
$196$	−1.00000	−0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	2.00000	0.142134
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	3.00000	0.212132
$201$	−12.0000	−0.846415
$202$	−18.0000	−1.26648
$203$	−6.00000	−0.421117
$204$	2.00000	0.140028
$205$	−2.00000	−0.139686
$206$	−8.00000	−0.557386
$207$	1.00000	0.0695048
$208$	4.00000	0.277350
$209$	0	0
$210$	−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	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−8.00000	−0.545595
$216$	3.00000	0.204124
$217$	−2.00000	−0.135769
$218$	−2.00000	−0.135457
$219$	8.00000	0.540590
$220$	2.00000	0.134840
$221$	8.00000	0.538138
$222$	−4.00000	−0.268462
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−5.00000	−0.334077
$225$	1.00000	0.0666667
$226$	8.00000	0.532152
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−1.00000	−0.0659380
$231$	−2.00000	−0.131590
$232$	−18.0000	−1.18176
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	4.00000	0.261488
$235$	−8.00000	−0.521862
$236$	−2.00000	−0.130189
$237$	10.0000	0.649570
$238$	2.00000	0.129641
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−1.00000	−0.0645497
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	1.00000	0.0638877
$246$	2.00000	0.127515
$247$	0	0
$248$	−6.00000	−0.381000
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	−1.00000	−0.0629941
$253$	−2.00000	−0.125739
$254$	16.0000	1.00393
$255$	−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
$260$	4.00000	0.248069
$261$	−6.00000	−0.371391
$262$	6.00000	0.370681
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	12.0000	0.733017
$269$	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
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	2.00000	0.121268
$273$	−4.00000	−0.242091
$274$	20.0000	1.20824
$275$	−2.00000	−0.120605
$276$	−1.00000	−0.0601929
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	10.0000	0.599760
$279$	−2.00000	−0.119737
$280$	3.00000	0.179284
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	−8.00000	−0.473050
$287$	−2.00000	−0.118056
$288$	−5.00000	−0.294628
$289$	−13.0000	−0.764706
$290$	6.00000	0.352332
$291$	−14.0000	−0.820695
$292$	−8.00000	−0.468165
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	−1.00000	−0.0583212
$295$	2.00000	0.116445
$296$	12.0000	0.697486
$297$	−2.00000	−0.116052
$298$	−2.00000	−0.115857
$299$	−4.00000	−0.231326
$300$	−1.00000	−0.0577350
$301$	−8.00000	−0.461112
$302$	20.0000	1.15087
$303$	18.0000	1.03407
$304$	0	0
$305$	−14.0000	−0.801638
$306$	2.00000	0.114332
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	2.00000	0.113961
$309$	8.00000	0.455104
$310$	2.00000	0.113592
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	−12.0000	−0.679366
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−10.0000	−0.564333
$315$	1.00000	0.0563436
$316$	−10.0000	−0.562544
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	7.00000	0.391312
$321$	4.00000	0.223258
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	−4.00000	−0.221880
$326$	20.0000	1.10770
$327$	2.00000	0.110600
$328$	−6.00000	−0.331295
$329$	−8.00000	−0.441054
$330$	2.00000	0.110096
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−12.0000	−0.658586
$333$	4.00000	0.219199
$334$	16.0000	0.875481
$335$	−12.0000	−0.655630
$336$	−1.00000	−0.0545545
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	−3.00000	−0.163178
$339$	−8.00000	−0.434500
$340$	2.00000	0.108465
$341$	4.00000	0.216612
$342$	0	0
$343$	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	$0.0828835\pi$
$348$	6.00000	0.321634
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	−1.00000	−0.0534522
$351$	−4.00000	−0.213504
$352$	10.0000	0.533002
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−2.00000	−0.105851
$358$	−4.00000	−0.211407
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	3.00000	0.158114
$361$	−19.0000	−1.00000
$362$	26.0000	1.36653
$363$	−7.00000	−0.367405
$364$	4.00000	0.209657
$365$	8.00000	0.418739
$366$	14.0000	0.731792
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	−4.00000	−0.207950
$371$	0	0
$372$	2.00000	0.103695
$373$	28.0000	1.44979	0.724893	−	0.688862i	$-0.241889\pi$
$373$	28.0000	1.44979	0.724893	+	0.688862i	$0.241889\pi$
$374$	−4.00000	−0.206835
$375$	1.00000	0.0516398
$376$	−24.0000	−1.23771
$377$	24.0000	1.23606
$378$	−1.00000	−0.0514344
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	18.0000	0.920960
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	3.00000	0.153093
$385$	−2.00000	−0.101929
$386$	−22.0000	−1.11977
$387$	−8.00000	−0.406663
$388$	14.0000	0.710742
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	4.00000	0.202548
$391$	−2.00000	−0.101144
$392$	3.00000	0.151523
$393$	−6.00000	−0.302660
$394$	−18.0000	−0.906827
$395$	10.0000	0.503155
$396$	2.00000	0.100504
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	12.0000	0.598506
$403$	8.00000	0.398508
$404$	−18.0000	−0.895533
$405$	1.00000	0.0496904
$406$	6.00000	0.297775
$407$	−8.00000	−0.396545
$408$	−6.00000	−0.297044
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	2.00000	0.0987730
$411$	−20.0000	−0.986527
$412$	−8.00000	−0.394132
$413$	2.00000	0.0984136
$414$	−1.00000	−0.0491473
$415$	12.0000	0.589057
$416$	20.0000	0.980581
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−1.00000	−0.0487950
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−8.00000	−0.389434
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−14.0000	−0.677507
$428$	−4.00000	−0.193347
$429$	8.00000	0.386244
$430$	8.00000	0.385794
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	2.00000	0.0960031
$435$	−6.00000	−0.287678
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	−8.00000	−0.382255
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	−6.00000	−0.286039
$441$	1.00000	0.0476190
$442$	−8.00000	−0.380521
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−4.00000	−0.189832
$445$	6.00000	0.284427
$446$	−16.0000	−0.757622
$447$	2.00000	0.0945968
$448$	7.00000	0.330719
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−1.00000	−0.0471405
$451$	4.00000	0.188353
$452$	8.00000	0.376288
$453$	−20.0000	−0.939682
$454$	−4.00000	−0.187729
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	14.0000	0.654177
$459$	−2.00000	−0.0933520
$460$	−1.00000	−0.0466252
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	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	$0.559519\pi$
$464$	6.00000	0.278543
$465$	−2.00000	−0.0927478
$466$	−6.00000	−0.277945
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	4.00000	0.184900
$469$	−12.0000	−0.554109
$470$	8.00000	0.369012
$471$	10.0000	0.460776
$472$	6.00000	0.276172
$473$	16.0000	0.735681
$474$	−10.0000	−0.459315
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	−8.00000	−0.365911
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	−5.00000	−0.228218
$481$	−16.0000	−0.729537
$482$	6.00000	0.273293
$483$	1.00000	0.0455016
$484$	7.00000	0.318182
$485$	−14.0000	−0.635707
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−42.0000	−1.90125
$489$	−20.0000	−0.904431
$490$	−1.00000	−0.0451754
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	2.00000	0.0901670
$493$	12.0000	0.540453
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	2.00000	0.0898027
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−1.00000	−0.0447214
$501$	−16.0000	−0.714827
$502$	−16.0000	−0.714115
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	3.00000	0.133631
$505$	18.0000	0.800989
$506$	2.00000	0.0889108
$507$	3.00000	0.133235
$508$	16.0000	0.709885
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	2.00000	0.0885615
$511$	8.00000	0.353899
$512$	11.0000	0.486136
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	8.00000	0.352180
$517$	16.0000	0.703679
$518$	−4.00000	−0.175750
$519$	−4.00000	−0.175581
$520$	−12.0000	−0.526235
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	6.00000	0.262613
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	6.00000	0.262111
$525$	1.00000	0.0436436
$526$	−20.0000	−0.872041
$527$	4.00000	0.174243
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	8.00000	0.346518
$534$	−6.00000	−0.259645
$535$	4.00000	0.172935
$536$	−36.0000	−1.55496
$537$	4.00000	0.172613
$538$	−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	$0.486311\pi$
$542$	10.0000	0.429537
$543$	−26.0000	−1.11577
$544$	10.0000	0.428746
$545$	2.00000	0.0856706
$546$	4.00000	0.171184
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	20.0000	0.854358
$549$	−14.0000	−0.597505
$550$	2.00000	0.0852803
$551$	0	0
$552$	3.00000	0.127688
$553$	10.0000	0.425243
$554$	−10.0000	−0.424859
$555$	4.00000	0.169791
$556$	10.0000	0.424094
$557$	40.0000	1.69485	0.847427	−	0.530912i	$-0.178150\pi$
$557$	40.0000	1.69485	0.847427	+	0.530912i	$0.178150\pi$
$558$	2.00000	0.0846668
$559$	32.0000	1.35346
$560$	−1.00000	−0.0422577
$561$	4.00000	0.168880
$562$	18.0000	0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	8.00000	0.336861
$565$	−8.00000	−0.336563
$566$	−4.00000	−0.168133
$567$	1.00000	0.0419961
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	−8.00000	−0.334497
$573$	−18.0000	−0.751961
$574$	2.00000	0.0834784
$575$	1.00000	0.0417029
$576$	7.00000	0.291667
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	13.0000	0.540729
$579$	22.0000	0.914289
$580$	6.00000	0.249136
$581$	12.0000	0.497844
$582$	14.0000	0.580319
$583$	0	0
$584$	24.0000	0.993127
$585$	−4.00000	−0.165380
$586$	10.0000	0.413096
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	−2.00000	−0.0823387
$591$	18.0000	0.740421
$592$	−4.00000	−0.164399
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	2.00000	0.0820610
$595$	−2.00000	−0.0819920
$596$	−2.00000	−0.0819232
$597$	−20.0000	−0.818546
$598$	4.00000	0.163572
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	3.00000	0.122474
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	8.00000	0.326056
$603$	−12.0000	−0.488678
$604$	20.0000	0.813788
$605$	−7.00000	−0.284590
$606$	−18.0000	−0.731200
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	14.0000	0.566843
$611$	32.0000	1.29458
$612$	2.00000	0.0808452
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	20.0000	0.807134
$615$	−2.00000	−0.0806478
$616$	−6.00000	−0.241747
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	−8.00000	−0.321807
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	2.00000	0.0803219
$621$	1.00000	0.0401286
$622$	6.00000	0.240578
$623$	6.00000	0.240385
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−8.00000	−0.318981
$630$	−1.00000	−0.0398410
$631$	10.0000	0.398094	0.199047	−	0.979990i	$-0.436215\pi$
$631$	10.0000	0.398094	0.199047	+	0.979990i	$0.436215\pi$
$632$	30.0000	1.19334
$633$	8.00000	0.317971
$634$	−10.0000	−0.397151
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−4.00000	−0.158486
$638$	−12.0000	−0.475085
$639$	0	0
$640$	3.00000	0.118585
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−4.00000	−0.157867
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−1.00000	−0.0394055
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−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
$649$	−4.00000	−0.157014
$650$	4.00000	0.156893
$651$	−2.00000	−0.0783862
$652$	20.0000	0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−2.00000	−0.0782062
$655$	−6.00000	−0.234439
$656$	2.00000	0.0780869
$657$	8.00000	0.312110
$658$	8.00000	0.311872
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	2.00000	0.0778499
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	4.00000	0.155464
$663$	8.00000	0.310694
$664$	36.0000	1.39707
$665$	0	0
$666$	−4.00000	−0.154997
$667$	−6.00000	−0.232321
$668$	16.0000	0.619059
$669$	16.0000	0.618596
$670$	12.0000	0.463600
$671$	28.0000	1.08093
$672$	−5.00000	−0.192879
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	−3.00000	−0.115385
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	8.00000	0.307238
$679$	−14.0000	−0.537271
$680$	−6.00000	−0.230089
$681$	4.00000	0.153280
$682$	−4.00000	−0.153168
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	−20.0000	−0.764161
$686$	−1.00000	−0.0381802
$687$	−14.0000	−0.534133
$688$	8.00000	0.304997
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	4.00000	0.152057
$693$	−2.00000	−0.0759737
$694$	−36.0000	−1.36654
$695$	−10.0000	−0.379322
$696$	−18.0000	−0.682288
$697$	4.00000	0.151511
$698$	26.0000	0.984115
$699$	6.00000	0.226941
$700$	−1.00000	−0.0377964
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	4.00000	0.150970
$703$	0	0
$704$	−14.0000	−0.527645
$705$	−8.00000	−0.301297
$706$	4.00000	0.150542
$707$	18.0000	0.676960
$708$	−2.00000	−0.0751646
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	18.0000	0.674579
$713$	−2.00000	−0.0749006
$714$	2.00000	0.0748481
$715$	8.00000	0.299183
$716$	−4.00000	−0.149487
$717$	8.00000	0.298765
$718$	−10.0000	−0.373197
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−1.00000	−0.0372678
$721$	8.00000	0.297936
$722$	19.0000	0.707107
$723$	−6.00000	−0.223142
$724$	26.0000	0.966282
$725$	−6.00000	−0.222834
$726$	7.00000	0.259794
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	−12.0000	−0.444750
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	16.0000	0.591781
$732$	14.0000	0.517455
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	−32.0000	−1.18114
$735$	1.00000	0.0368856
$736$	−5.00000	−0.184302
$737$	24.0000	0.884051
$738$	2.00000	0.0736210
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	−6.00000	−0.219971
$745$	2.00000	0.0732743
$746$	−28.0000	−1.02515
$747$	12.0000	0.439057
$748$	−4.00000	−0.146254
$749$	4.00000	0.146157
$750$	−1.00000	−0.0365148
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	8.00000	0.291730
$753$	16.0000	0.583072
$754$	−24.0000	−0.874028
$755$	−20.0000	−0.727875
$756$	−1.00000	−0.0363696
$757$	24.0000	0.872295	0.436147	−	0.899875i	$-0.356343\pi$
$757$	24.0000	0.872295	0.436147	+	0.899875i	$0.356343\pi$
$758$	−26.0000	−0.944363
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	16.0000	0.579619
$763$	2.00000	0.0724049
$764$	18.0000	0.651217
$765$	−2.00000	−0.0723102
$766$	−24.0000	−0.867155
$767$	−8.00000	−0.288863
$768$	−17.0000	−0.613435
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	−22.0000	−0.791797
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	8.00000	0.287554
$775$	−2.00000	−0.0718421
$776$	−42.0000	−1.50771
$777$	4.00000	0.143499
$778$	6.00000	0.215110
$779$	0	0
$780$	4.00000	0.143223
$781$	0	0
$782$	2.00000	0.0715199
$783$	−6.00000	−0.214423
$784$	−1.00000	−0.0357143
$785$	10.0000	0.356915
$786$	6.00000	0.214013
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−18.0000	−0.641223
$789$	20.0000	0.712019
$790$	−10.0000	−0.355784
$791$	−8.00000	−0.284447
$792$	−6.00000	−0.213201
$793$	56.0000	1.98862
$794$	0	0
$795$	0	0
$796$	20.0000	0.708881
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	−5.00000	−0.176777
$801$	6.00000	0.212000
$802$	14.0000	0.494357
$803$	−16.0000	−0.564628
$804$	12.0000	0.423207
$805$	1.00000	0.0352454
$806$	−8.00000	−0.281788
$807$	10.0000	0.352017
$808$	54.0000	1.89971
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	6.00000	0.210559
$813$	−10.0000	−0.350715
$814$	8.00000	0.280400
$815$	−20.0000	−0.700569
$816$	2.00000	0.0700140
$817$	0	0
$818$	−34.0000	−1.18878
$819$	−4.00000	−0.139771
$820$	2.00000	0.0698430
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	20.0000	0.697580
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	24.0000	0.836080
$825$	−2.00000	−0.0696311
$826$	−2.00000	−0.0695889
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−1.00000	−0.0347524
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	−12.0000	−0.416526
$831$	10.0000	0.346896
$832$	−28.0000	−0.970725
$833$	−2.00000	−0.0692959
$834$	10.0000	0.346272
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	12.0000	0.414533
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	3.00000	0.103510
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	−18.0000	−0.619953
$844$	−8.00000	−0.275371
$845$	3.00000	0.103203
$846$	8.00000	0.275046
$847$	−7.00000	−0.240523
$848$	0	0
$849$	4.00000	0.137280
$850$	2.00000	0.0685994
$851$	4.00000	0.137118
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	12.0000	0.410152
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	−8.00000	−0.273115
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	8.00000	0.272798
$861$	−2.00000	−0.0681598
$862$	−10.0000	−0.340601
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−5.00000	−0.170103
$865$	−4.00000	−0.136004
$866$	−2.00000	−0.0679628
$867$	−13.0000	−0.441503
$868$	2.00000	0.0678844
$869$	−20.0000	−0.678454
$870$	6.00000	0.203419
$871$	48.0000	1.62642
$872$	6.00000	0.203186
$873$	−14.0000	−0.473828
$874$	0	0
$875$	1.00000	0.0338062
$876$	−8.00000	−0.270295
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	14.0000	0.472477
$879$	−10.0000	−0.337292
$880$	2.00000	0.0674200
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−1.00000	−0.0336718
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−8.00000	−0.269069
$885$	2.00000	0.0672293
$886$	−4.00000	−0.134383
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	12.0000	0.402694
$889$	−16.0000	−0.536623
$890$	−6.00000	−0.201120
$891$	−2.00000	−0.0670025
$892$	−16.0000	−0.535720
$893$	0	0
$894$	−2.00000	−0.0668900
$895$	4.00000	0.133705
$896$	3.00000	0.100223
$897$	−4.00000	−0.133556
$898$	10.0000	0.333704
$899$	12.0000	0.400222
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	−4.00000	−0.133185
$903$	−8.00000	−0.266223
$904$	−24.0000	−0.798228
$905$	−26.0000	−0.864269
$906$	20.0000	0.664455
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−4.00000	−0.132745
$909$	18.0000	0.597022
$910$	4.00000	0.132599
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	20.0000	0.661541
$915$	−14.0000	−0.462826
$916$	14.0000	0.462573
$917$	−6.00000	−0.198137
$918$	2.00000	0.0660098
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	3.00000	0.0989071
$921$	−20.0000	−0.659022
$922$	−18.0000	−0.592798
$923$	0	0
$924$	2.00000	0.0657952
$925$	4.00000	0.131519
$926$	8.00000	0.262896
$927$	8.00000	0.262754
$928$	30.0000	0.984798
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	2.00000	0.0655826
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−6.00000	−0.196431
$934$	−20.0000	−0.654420
$935$	4.00000	0.130814
$936$	−12.0000	−0.392232
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	12.0000	0.391814
$939$	−26.0000	−0.848478
$940$	8.00000	0.260931
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−10.0000	−0.325818
$943$	−2.00000	−0.0651290
$944$	−2.00000	−0.0650945
$945$	1.00000	0.0325300
$946$	−16.0000	−0.520205
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	−10.0000	−0.324785
$949$	−32.0000	−1.03876
$950$	0	0
$951$	10.0000	0.324272
$952$	−6.00000	−0.194461
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	−8.00000	−0.258738
$957$	12.0000	0.387905
$958$	−28.0000	−0.904639
$959$	−20.0000	−0.645834
$960$	7.00000	0.225924
$961$	−27.0000	−0.870968
$962$	16.0000	0.515861
$963$	4.00000	0.128898
$964$	6.00000	0.193247
$965$	22.0000	0.708205
$966$	−1.00000	−0.0321745
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	14.0000	0.449513
$971$	56.0000	1.79713	0.898563	−	0.438845i	$-0.144612\pi$
$971$	56.0000	1.79713	0.898563	+	0.438845i	$0.144612\pi$
$972$	−1.00000	−0.0320750
$973$	−10.0000	−0.320585
$974$	8.00000	0.256337
$975$	−4.00000	−0.128103
$976$	14.0000	0.448129
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	20.0000	0.639529
$979$	−12.0000	−0.383522
$980$	−1.00000	−0.0319438
$981$	2.00000	0.0638551
$982$	12.0000	0.382935
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	−6.00000	−0.191273
$985$	18.0000	0.573528
$986$	−12.0000	−0.382158
$987$	−8.00000	−0.254643
$988$	0	0
$989$	−8.00000	−0.254385
$990$	2.00000	0.0635642
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	10.0000	0.317500
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−20.0000	−0.634043
$996$	−12.0000	−0.380235
$997$	12.0000	0.380044	0.190022	−	0.981780i	$-0.439144\pi$
$997$	12.0000	0.380044	0.190022	+	0.981780i	$0.439144\pi$
$998$	24.0000	0.759707
$999$	4.00000	0.126554

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.d.1.1	✓	1	1.1	even	1	trivial
7245.2.a.m.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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