LMFDB - Embedded newform 123.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [123,2,Mod(1,123)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(123, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("123.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 123 = 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	123.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:	$0.982159944862$
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	$\chi$	$=$	123.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -4.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +8.00000 q^{10} -3.00000 q^{11} +2.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} -4.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -8.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} -6.00000 q^{23} +11.0000 q^{25} +12.0000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +5.00000 q^{29} +8.00000 q^{30} +7.00000 q^{31} +8.00000 q^{32} -3.00000 q^{33} -6.00000 q^{34} +8.00000 q^{35} +2.00000 q^{36} -7.00000 q^{37} -6.00000 q^{39} +1.00000 q^{41} +4.00000 q^{42} -1.00000 q^{43} -6.00000 q^{44} -4.00000 q^{45} +12.0000 q^{46} +3.00000 q^{47} -4.00000 q^{48} -3.00000 q^{49} -22.0000 q^{50} +3.00000 q^{51} -12.0000 q^{52} -6.00000 q^{53} -2.00000 q^{54} +12.0000 q^{55} -10.0000 q^{58} -8.00000 q^{60} -3.00000 q^{61} -14.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +24.0000 q^{65} +6.00000 q^{66} -2.00000 q^{67} +6.00000 q^{68} -6.00000 q^{69} -16.0000 q^{70} -3.00000 q^{71} -11.0000 q^{73} +14.0000 q^{74} +11.0000 q^{75} +6.00000 q^{77} +12.0000 q^{78} +10.0000 q^{79} +16.0000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -16.0000 q^{83} -4.00000 q^{84} -12.0000 q^{85} +2.00000 q^{86} +5.00000 q^{87} -10.0000 q^{89} +8.00000 q^{90} +12.0000 q^{91} -12.0000 q^{92} +7.00000 q^{93} -6.00000 q^{94} +8.00000 q^{96} -12.0000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−2.00000	−0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	8.00000	2.52982
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	4.00000	1.06904
$15$	−4.00000	−1.03280
$16$	−4.00000	−1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.00000	−0.471405
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−8.00000	−1.78885
$21$	−2.00000	−0.436436
$22$	6.00000	1.27920
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	12.0000	2.35339
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	8.00000	1.46059
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	8.00000	1.41421
$33$	−3.00000	−0.522233
$34$	−6.00000	−1.02899
$35$	8.00000	1.35225
$36$	2.00000	0.333333
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	4.00000	0.617213
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−6.00000	−0.904534
$45$	−4.00000	−0.596285
$46$	12.0000	1.76930
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−4.00000	−0.577350
$49$	−3.00000	−0.428571
$50$	−22.0000	−3.11127
$51$	3.00000	0.420084
$52$	−12.0000	−1.66410
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−2.00000	−0.272166
$55$	12.0000	1.61808
$56$	0	0
$57$	0	0
$58$	−10.0000	−1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−8.00000	−1.03280
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	−14.0000	−1.77800
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	24.0000	2.97683
$66$	6.00000	0.738549
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	6.00000	0.727607
$69$	−6.00000	−0.722315
$70$	−16.0000	−1.91237
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	14.0000	1.62747
$75$	11.0000	1.27017
$76$	0	0
$77$	6.00000	0.683763
$78$	12.0000	1.35873
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	16.0000	1.78885
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	−4.00000	−0.436436
$85$	−12.0000	−1.30158
$86$	2.00000	0.215666
$87$	5.00000	0.536056
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	8.00000	0.843274
$91$	12.0000	1.25794
$92$	−12.0000	−1.25109
$93$	7.00000	0.725866
$94$	−6.00000	−0.618853
$95$	0	0
$96$	8.00000	0.816497
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	6.00000	0.606092
$99$	−3.00000	−0.301511
$100$	22.0000	2.20000
$101$	−13.0000	−1.29355	−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000	−1.29355	−0.646774	+	0.762682i	$0.723882\pi$
$102$	−6.00000	−0.594089
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	12.0000	1.16554
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	2.00000	0.192450
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−24.0000	−2.28831
$111$	−7.00000	−0.664411
$112$	8.00000	0.755929
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	10.0000	0.928477
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	6.00000	0.543214
$123$	1.00000	0.0901670
$124$	14.0000	1.25724
$125$	−24.0000	−2.14663
$126$	4.00000	0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	−48.0000	−4.20988
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	4.00000	0.345547
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	12.0000	1.02151
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	16.0000	1.35225
$141$	3.00000	0.252646
$142$	6.00000	0.503509
$143$	18.0000	1.50524
$144$	−4.00000	−0.333333
$145$	−20.0000	−1.66091
$146$	22.0000	1.82073
$147$	−3.00000	−0.247436
$148$	−14.0000	−1.15079
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−22.0000	−1.79629
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	−12.0000	−0.966988
$155$	−28.0000	−2.24901
$156$	−12.0000	−0.960769
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−20.0000	−1.59111
$159$	−6.00000	−0.475831
$160$	−32.0000	−2.52982
$161$	12.0000	0.945732
$162$	−2.00000	−0.157135
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	2.00000	0.156174
$165$	12.0000	0.934199
$166$	32.0000	2.48368
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	24.0000	1.84072
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	−10.0000	−0.758098
$175$	−22.0000	−1.66304
$176$	12.0000	0.904534
$177$	0	0
$178$	20.0000	1.49906
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	−8.00000	−0.596285
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	−24.0000	−1.77900
$183$	−3.00000	−0.221766
$184$	0	0
$185$	28.0000	2.05860
$186$	−14.0000	−1.02653
$187$	−9.00000	−0.658145
$188$	6.00000	0.437595
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−8.00000	−0.577350
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	24.0000	1.72310
$195$	24.0000	1.71868
$196$	−6.00000	−0.428571
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	6.00000	0.426401
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	26.0000	1.82935
$203$	−10.0000	−0.701862
$204$	6.00000	0.420084
$205$	−4.00000	−0.279372
$206$	2.00000	0.139347
$207$	−6.00000	−0.417029
$208$	24.0000	1.66410
$209$	0	0
$210$	−16.0000	−1.10410
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−12.0000	−0.824163
$213$	−3.00000	−0.205557
$214$	−36.0000	−2.46091
$215$	4.00000	0.272798
$216$	0	0
$217$	−14.0000	−0.950382
$218$	−20.0000	−1.35457
$219$	−11.0000	−0.743311
$220$	24.0000	1.61808
$221$	−18.0000	−1.21081
$222$	14.0000	0.939618
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−16.0000	−1.06904
$225$	11.0000	0.733333
$226$	−8.00000	−0.532152
$227$	23.0000	1.52656	0.763282	−	0.646066i	$-0.223587\pi$
$227$	23.0000	1.52656	0.763282	+	0.646066i	$0.223587\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	−48.0000	−3.16503
$231$	6.00000	0.394771
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	12.0000	0.784465
$235$	−12.0000	−0.782794
$236$	0	0
$237$	10.0000	0.649570
$238$	12.0000	0.777844
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	16.0000	1.03280
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	4.00000	0.257130
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	12.0000	0.766652
$246$	−2.00000	−0.127515
$247$	0	0
$248$	0	0
$249$	−16.0000	−1.01396
$250$	48.0000	3.03579
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	−4.00000	−0.251976
$253$	18.0000	1.13165
$254$	−16.0000	−1.00393
$255$	−12.0000	−0.751469
$256$	16.0000	1.00000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	2.00000	0.124515
$259$	14.0000	0.869918
$260$	48.0000	2.97683
$261$	5.00000	0.309492
$262$	−4.00000	−0.247121
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	8.00000	0.486864
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	−12.0000	−0.727607
$273$	12.0000	0.726273
$274$	34.0000	2.05402
$275$	−33.0000	−1.98997
$276$	−12.0000	−0.722315
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	0	0
$279$	7.00000	0.419079
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	−6.00000	−0.357295
$283$	−31.0000	−1.84276	−0.921379	−	0.388664i	$-0.872937\pi$
$283$	−31.0000	−1.84276	−0.921379	+	0.388664i	$0.872937\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−36.0000	−2.12872
$287$	−2.00000	−0.118056
$288$	8.00000	0.471405
$289$	−8.00000	−0.470588
$290$	40.0000	2.34888
$291$	−12.0000	−0.703452
$292$	−22.0000	−1.28745
$293$	29.0000	1.69420	0.847099	−	0.531435i	$-0.178347\pi$
$293$	29.0000	1.69420	0.847099	+	0.531435i	$0.178347\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	0	0
$297$	−3.00000	−0.174078
$298$	−20.0000	−1.15857
$299$	36.0000	2.08193
$300$	22.0000	1.27017
$301$	2.00000	0.115278
$302$	−24.0000	−1.38104
$303$	−13.0000	−0.746830
$304$	0	0
$305$	12.0000	0.687118
$306$	−6.00000	−0.342997
$307$	13.0000	0.741949	0.370975	−	0.928643i	$-0.379024\pi$
$307$	13.0000	0.741949	0.370975	+	0.928643i	$0.379024\pi$
$308$	12.0000	0.683763
$309$	−1.00000	−0.0568880
$310$	56.0000	3.18059
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	4.00000	0.225733
$315$	8.00000	0.450749
$316$	20.0000	1.12509
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	12.0000	0.672927
$319$	−15.0000	−0.839839
$320$	32.0000	1.78885
$321$	18.0000	1.00466
$322$	−24.0000	−1.33747
$323$	0	0
$324$	2.00000	0.111111
$325$	−66.0000	−3.66102
$326$	−38.0000	−2.10463
$327$	10.0000	0.553001
$328$	0	0
$329$	−6.00000	−0.330791
$330$	−24.0000	−1.32116
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−32.0000	−1.75623
$333$	−7.00000	−0.383598
$334$	−16.0000	−0.875481
$335$	8.00000	0.437087
$336$	8.00000	0.436436
$337$	−7.00000	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$338$	−46.0000	−2.50207
$339$	4.00000	0.217250
$340$	−24.0000	−1.30158
$341$	−21.0000	−1.13721
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	24.0000	1.29212
$346$	32.0000	1.72033
$347$	13.0000	0.697877	0.348938	−	0.937146i	$-0.386542\pi$
$347$	13.0000	0.697877	0.348938	+	0.937146i	$0.386542\pi$
$348$	10.0000	0.536056
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	44.0000	2.35190
$351$	−6.00000	−0.320256
$352$	−24.0000	−1.27920
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−20.0000	−1.06000
$357$	−6.00000	−0.317554
$358$	10.0000	0.528516
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−24.0000	−1.26141
$363$	−2.00000	−0.104973
$364$	24.0000	1.25794
$365$	44.0000	2.30307
$366$	6.00000	0.313625
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	24.0000	1.25109
$369$	1.00000	0.0520579
$370$	−56.0000	−2.91130
$371$	12.0000	0.623009
$372$	14.0000	0.725866
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	18.0000	0.930758
$375$	−24.0000	−1.23935
$376$	0	0
$377$	−30.0000	−1.54508
$378$	4.00000	0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	16.0000	0.818631
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	−28.0000	−1.42516
$387$	−1.00000	−0.0508329
$388$	−24.0000	−1.21842
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	−48.0000	−2.43057
$391$	−18.0000	−0.910299
$392$	0	0
$393$	2.00000	0.100887
$394$	24.0000	1.20910
$395$	−40.0000	−2.01262
$396$	−6.00000	−0.301511
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	40.0000	2.00502
$399$	0	0
$400$	−44.0000	−2.20000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	4.00000	0.199502
$403$	−42.0000	−2.09217
$404$	−26.0000	−1.29355
$405$	−4.00000	−0.198762
$406$	20.0000	0.992583
$407$	21.0000	1.04093
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	8.00000	0.395092
$411$	−17.0000	−0.838548
$412$	−2.00000	−0.0985329
$413$	0	0
$414$	12.0000	0.589768
$415$	64.0000	3.14164
$416$	−48.0000	−2.35339
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	16.0000	0.780720
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	16.0000	0.778868
$423$	3.00000	0.145865
$424$	0	0
$425$	33.0000	1.60074
$426$	6.00000	0.290701
$427$	6.00000	0.290360
$428$	36.0000	1.74013
$429$	18.0000	0.869048
$430$	−8.00000	−0.385794
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−4.00000	−0.192450
$433$	−21.0000	−1.00920	−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000	−1.00920	−0.504598	+	0.863355i	$0.668359\pi$
$434$	28.0000	1.34404
$435$	−20.0000	−0.958927
$436$	20.0000	0.957826
$437$	0	0
$438$	22.0000	1.05120
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	36.0000	1.71235
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−14.0000	−0.664411
$445$	40.0000	1.89618
$446$	32.0000	1.51524
$447$	10.0000	0.472984
$448$	16.0000	0.755929
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	−22.0000	−1.03709
$451$	−3.00000	−0.141264
$452$	8.00000	0.376288
$453$	12.0000	0.563809
$454$	−46.0000	−2.15889
$455$	−48.0000	−2.25027
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	40.0000	1.86908
$459$	3.00000	0.140028
$460$	48.0000	2.23801
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	−12.0000	−0.558291
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−20.0000	−0.928477
$465$	−28.0000	−1.29847
$466$	12.0000	0.555889
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	−12.0000	−0.554700
$469$	4.00000	0.184703
$470$	24.0000	1.10704
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	3.00000	0.137940
$474$	−20.0000	−0.918630
$475$	0	0
$476$	−12.0000	−0.550019
$477$	−6.00000	−0.274721
$478$	−40.0000	−1.82956
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	−32.0000	−1.46059
$481$	42.0000	1.91504
$482$	−14.0000	−0.637683
$483$	12.0000	0.546019
$484$	−4.00000	−0.181818
$485$	48.0000	2.17957
$486$	−2.00000	−0.0907218
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	0	0
$489$	19.0000	0.859210
$490$	−24.0000	−1.08421
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	2.00000	0.0901670
$493$	15.0000	0.675566
$494$	0	0
$495$	12.0000	0.539360
$496$	−28.0000	−1.25724
$497$	6.00000	0.269137
$498$	32.0000	1.43395
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−48.0000	−2.14663
$501$	8.00000	0.357414
$502$	36.0000	1.60676
$503$	−1.00000	−0.0445878	−0.0222939	−	0.999751i	$-0.507097\pi$
$503$	−1.00000	−0.0445878	−0.0222939	+	0.999751i	$0.507097\pi$
$504$	0	0
$505$	52.0000	2.31397
$506$	−36.0000	−1.60040
$507$	23.0000	1.02147
$508$	16.0000	0.709885
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	24.0000	1.06274
$511$	22.0000	0.973223
$512$	−32.0000	−1.41421
$513$	0	0
$514$	−6.00000	−0.264649
$515$	4.00000	0.176261
$516$	−2.00000	−0.0880451
$517$	−9.00000	−0.395820
$518$	−28.0000	−1.23025
$519$	−16.0000	−0.702322
$520$	0	0
$521$	7.00000	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$521$	7.00000	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$522$	−10.0000	−0.437688
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	4.00000	0.174741
$525$	−22.0000	−0.960159
$526$	−18.0000	−0.784837
$527$	21.0000	0.914774
$528$	12.0000	0.522233
$529$	13.0000	0.565217
$530$	−48.0000	−2.08499
$531$	0	0
$532$	0	0
$533$	−6.00000	−0.259889
$534$	20.0000	0.865485
$535$	−72.0000	−3.11283
$536$	0	0
$537$	−5.00000	−0.215766
$538$	0	0
$539$	9.00000	0.387657
$540$	−8.00000	−0.344265
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	6.00000	0.257722
$543$	12.0000	0.514969
$544$	24.0000	1.02899
$545$	−40.0000	−1.71341
$546$	−24.0000	−1.02711
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−34.0000	−1.45241
$549$	−3.00000	−0.128037
$550$	66.0000	2.81425
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	−46.0000	−1.95435
$555$	28.0000	1.18853
$556$	0	0
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−14.0000	−0.592667
$559$	6.00000	0.253773
$560$	−32.0000	−1.35225
$561$	−9.00000	−0.379980
$562$	26.0000	1.09674
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	6.00000	0.252646
$565$	−16.0000	−0.673125
$566$	62.0000	2.60605
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	36.0000	1.50524
$573$	−8.00000	−0.334205
$574$	4.00000	0.166957
$575$	−66.0000	−2.75239
$576$	−8.00000	−0.333333
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	16.0000	0.665512
$579$	14.0000	0.581820
$580$	−40.0000	−1.66091
$581$	32.0000	1.32758
$582$	24.0000	0.994832
$583$	18.0000	0.745484
$584$	0	0
$585$	24.0000	0.992278
$586$	−58.0000	−2.39596
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	0	0
$591$	−12.0000	−0.493614
$592$	28.0000	1.15079
$593$	−11.0000	−0.451716	−0.225858	−	0.974160i	$-0.572519\pi$
$593$	−11.0000	−0.451716	−0.225858	+	0.974160i	$0.572519\pi$
$594$	6.00000	0.246183
$595$	24.0000	0.983904
$596$	20.0000	0.819232
$597$	−20.0000	−0.818546
$598$	−72.0000	−2.94430
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−4.00000	−0.163028
$603$	−2.00000	−0.0814463
$604$	24.0000	0.976546
$605$	8.00000	0.325246
$606$	26.0000	1.05618
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	−10.0000	−0.405220
$610$	−24.0000	−0.971732
$611$	−18.0000	−0.728202
$612$	6.00000	0.242536
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−26.0000	−1.04927
$615$	−4.00000	−0.161296
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	2.00000	0.0804518
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	−56.0000	−2.24901
$621$	−6.00000	−0.240772
$622$	16.0000	0.641542
$623$	20.0000	0.801283
$624$	24.0000	0.960769
$625$	41.0000	1.64000
$626$	12.0000	0.479616
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−21.0000	−0.837325
$630$	−16.0000	−0.637455
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	−6.00000	−0.238290
$635$	−32.0000	−1.26988
$636$	−12.0000	−0.475831
$637$	18.0000	0.713186
$638$	30.0000	1.18771
$639$	−3.00000	−0.118678
$640$	0	0
$641$	7.00000	0.276483	0.138242	−	0.990399i	$-0.455855\pi$
$641$	7.00000	0.276483	0.138242	+	0.990399i	$0.455855\pi$
$642$	−36.0000	−1.42081
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	24.0000	0.945732
$645$	4.00000	0.157500
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	0	0
$650$	132.000	5.17747
$651$	−14.0000	−0.548703
$652$	38.0000	1.48819
$653$	−41.0000	−1.60445	−0.802227	−	0.597019i	$-0.796352\pi$
$653$	−41.0000	−1.60445	−0.802227	+	0.597019i	$0.796352\pi$
$654$	−20.0000	−0.782062
$655$	−8.00000	−0.312586
$656$	−4.00000	−0.156174
$657$	−11.0000	−0.429151
$658$	12.0000	0.467809
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	24.0000	0.934199
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	16.0000	0.621858
$663$	−18.0000	−0.699062
$664$	0	0
$665$	0	0
$666$	14.0000	0.542489
$667$	−30.0000	−1.16160
$668$	16.0000	0.619059
$669$	−16.0000	−0.618596
$670$	−16.0000	−0.618134
$671$	9.00000	0.347441
$672$	−16.0000	−0.617213
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	14.0000	0.539260
$675$	11.0000	0.423390
$676$	46.0000	1.76923
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	−8.00000	−0.307238
$679$	24.0000	0.921035
$680$	0	0
$681$	23.0000	0.881362
$682$	42.0000	1.60826
$683$	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
$685$	68.0000	2.59815
$686$	−40.0000	−1.52721
$687$	−20.0000	−0.763048
$688$	4.00000	0.152499
$689$	36.0000	1.37149
$690$	−48.0000	−1.82733
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−32.0000	−1.21646
$693$	6.00000	0.227921
$694$	−26.0000	−0.986947
$695$	0	0
$696$	0	0
$697$	3.00000	0.113633
$698$	10.0000	0.378506
$699$	−6.00000	−0.226941
$700$	−44.0000	−1.66304
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	12.0000	0.452911
$703$	0	0
$704$	24.0000	0.904534
$705$	−12.0000	−0.451946
$706$	−28.0000	−1.05379
$707$	26.0000	0.977831
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−24.0000	−0.900704
$711$	10.0000	0.375029
$712$	0	0
$713$	−42.0000	−1.57291
$714$	12.0000	0.449089
$715$	−72.0000	−2.69265
$716$	−10.0000	−0.373718
$717$	20.0000	0.746914
$718$	20.0000	0.746393
$719$	−35.0000	−1.30528	−0.652640	−	0.757668i	$-0.726339\pi$
$719$	−35.0000	−1.30528	−0.652640	+	0.757668i	$0.726339\pi$
$720$	16.0000	0.596285
$721$	2.00000	0.0744839
$722$	38.0000	1.41421
$723$	7.00000	0.260333
$724$	24.0000	0.891953
$725$	55.0000	2.04265
$726$	4.00000	0.148454
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−88.0000	−3.25703
$731$	−3.00000	−0.110959
$732$	−6.00000	−0.221766
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	−46.0000	−1.69789
$735$	12.0000	0.442627
$736$	−48.0000	−1.76930
$737$	6.00000	0.221013
$738$	−2.00000	−0.0736210
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	56.0000	2.05860
$741$	0	0
$742$	−24.0000	−0.881068
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	−40.0000	−1.46549
$746$	2.00000	0.0732252
$747$	−16.0000	−0.585409
$748$	−18.0000	−0.658145
$749$	−36.0000	−1.31541
$750$	48.0000	1.75271
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	−12.0000	−0.437595
$753$	−18.0000	−0.655956
$754$	60.0000	2.18507
$755$	−48.0000	−1.74690
$756$	−4.00000	−0.145479
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	40.0000	1.45287
$759$	18.0000	0.653359
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−16.0000	−0.579619
$763$	−20.0000	−0.724049
$764$	−16.0000	−0.578860
$765$	−12.0000	−0.433861
$766$	−18.0000	−0.650366
$767$	0	0
$768$	16.0000	0.577350
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	48.0000	1.72980
$771$	3.00000	0.108042
$772$	28.0000	1.00774
$773$	9.00000	0.323708	0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000	0.323708	0.161854	+	0.986815i	$0.448253\pi$
$774$	2.00000	0.0718885
$775$	77.0000	2.76592
$776$	0	0
$777$	14.0000	0.502247
$778$	0	0
$779$	0	0
$780$	48.0000	1.71868
$781$	9.00000	0.322045
$782$	36.0000	1.28736
$783$	5.00000	0.178685
$784$	12.0000	0.428571
$785$	8.00000	0.285532
$786$	−4.00000	−0.142675
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	−24.0000	−0.854965
$789$	9.00000	0.320408
$790$	80.0000	2.84627
$791$	−8.00000	−0.284447
$792$	0	0
$793$	18.0000	0.639199
$794$	44.0000	1.56150
$795$	24.0000	0.851192
$796$	−40.0000	−1.41776
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	88.0000	3.11127
$801$	−10.0000	−0.353333
$802$	16.0000	0.564980
$803$	33.0000	1.16454
$804$	−4.00000	−0.141069
$805$	−48.0000	−1.69178
$806$	84.0000	2.95877
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	8.00000	0.281091
$811$	37.0000	1.29925	0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000	1.29925	0.649623	+	0.760257i	$0.274927\pi$
$812$	−20.0000	−0.701862
$813$	−3.00000	−0.105215
$814$	−42.0000	−1.47210
$815$	−76.0000	−2.66216
$816$	−12.0000	−0.420084
$817$	0	0
$818$	10.0000	0.349642
$819$	12.0000	0.419314
$820$	−8.00000	−0.279372
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	34.0000	1.18589
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	0	0
$825$	−33.0000	−1.14891
$826$	0	0
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	−12.0000	−0.417029
$829$	45.0000	1.56291	0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000	1.56291	0.781457	+	0.623959i	$0.214477\pi$
$830$	−128.000	−4.44294
$831$	23.0000	0.797861
$832$	48.0000	1.66410
$833$	−9.00000	−0.311832
$834$	0	0
$835$	−32.0000	−1.10741
$836$	0	0
$837$	7.00000	0.241955
$838$	−60.0000	−2.07267
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	56.0000	1.92989
$843$	−13.0000	−0.447744
$844$	−16.0000	−0.550743
$845$	−92.0000	−3.16490
$846$	−6.00000	−0.206284
$847$	4.00000	0.137442
$848$	24.0000	0.824163
$849$	−31.0000	−1.06392
$850$	−66.0000	−2.26378
$851$	42.0000	1.43974
$852$	−6.00000	−0.205557
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−12.0000	−0.410632
$855$	0	0
$856$	0	0
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	−36.0000	−1.22902
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	8.00000	0.272798
$861$	−2.00000	−0.0681598
$862$	36.0000	1.22616
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	8.00000	0.272166
$865$	64.0000	2.17607
$866$	42.0000	1.42722
$867$	−8.00000	−0.271694
$868$	−28.0000	−0.950382
$869$	−30.0000	−1.01768
$870$	40.0000	1.35613
$871$	12.0000	0.406604
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	48.0000	1.62270
$876$	−22.0000	−0.743311
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	20.0000	0.674967
$879$	29.0000	0.978146
$880$	−48.0000	−1.61808
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	6.00000	0.202031
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	72.0000	2.41889
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−80.0000	−2.68161
$891$	−3.00000	−0.100504
$892$	−32.0000	−1.07144
$893$	0	0
$894$	−20.0000	−0.668900
$895$	20.0000	0.668526
$896$	0	0
$897$	36.0000	1.20201
$898$	−80.0000	−2.66963
$899$	35.0000	1.16732
$900$	22.0000	0.733333
$901$	−18.0000	−0.599667
$902$	6.00000	0.199778
$903$	2.00000	0.0665558
$904$	0	0
$905$	−48.0000	−1.59557
$906$	−24.0000	−0.797347
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	46.0000	1.52656
$909$	−13.0000	−0.431183
$910$	96.0000	3.18237
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	−56.0000	−1.85232
$915$	12.0000	0.396708
$916$	−40.0000	−1.32164
$917$	−4.00000	−0.132092
$918$	−6.00000	−0.198030
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	13.0000	0.428365
$922$	−44.0000	−1.44906
$923$	18.0000	0.592477
$924$	12.0000	0.394771
$925$	−77.0000	−2.53174
$926$	−68.0000	−2.23462
$927$	−1.00000	−0.0328443
$928$	40.0000	1.31306
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	56.0000	1.83631
$931$	0	0
$932$	−12.0000	−0.393073
$933$	−8.00000	−0.261908
$934$	−36.0000	−1.17796
$935$	36.0000	1.17733
$936$	0	0
$937$	−32.0000	−1.04539	−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000	−1.04539	−0.522697	+	0.852518i	$0.675074\pi$
$938$	−8.00000	−0.261209
$939$	−6.00000	−0.195803
$940$	−24.0000	−0.782794
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	4.00000	0.130327
$943$	−6.00000	−0.195387
$944$	0	0
$945$	8.00000	0.260240
$946$	−6.00000	−0.195077
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	20.0000	0.649570
$949$	66.0000	2.14245
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	12.0000	0.388514
$955$	32.0000	1.03550
$956$	40.0000	1.29369
$957$	−15.0000	−0.484881
$958$	30.0000	0.969256
$959$	34.0000	1.09792
$960$	32.0000	1.03280
$961$	18.0000	0.580645
$962$	−84.0000	−2.70827
$963$	18.0000	0.580042
$964$	14.0000	0.450910
$965$	−56.0000	−1.80270
$966$	−24.0000	−0.772187
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	0	0
$969$	0	0
$970$	−96.0000	−3.08237
$971$	−33.0000	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$971$	−33.0000	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	−46.0000	−1.47394
$975$	−66.0000	−2.11369
$976$	12.0000	0.384111
$977$	23.0000	0.735835	0.367918	−	0.929858i	$-0.380071\pi$
$977$	23.0000	0.735835	0.367918	+	0.929858i	$0.380071\pi$
$978$	−38.0000	−1.21511
$979$	30.0000	0.958804
$980$	24.0000	0.766652
$981$	10.0000	0.319275
$982$	16.0000	0.510581
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	−30.0000	−0.955395
$987$	−6.00000	−0.190982
$988$	0	0
$989$	6.00000	0.190789
$990$	−24.0000	−0.762770
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	56.0000	1.77800
$993$	−8.00000	−0.253872
$994$	−12.0000	−0.380617
$995$	80.0000	2.53617
$996$	−32.0000	−1.01396
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	−40.0000	−1.26618
$999$	−7.00000	−0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	123.2.a.a.1.1	✓	1
3.2	odd	2		369.2.a.b.1.1		1
4.3	odd	2		1968.2.a.a.1.1		1
5.4	even	2		3075.2.a.m.1.1		1
7.6	odd	2		6027.2.a.a.1.1		1
8.3	odd	2		7872.2.a.bj.1.1		1
8.5	even	2		7872.2.a.r.1.1		1
12.11	even	2		5904.2.a.v.1.1		1
15.14	odd	2		9225.2.a.d.1.1		1
41.40	even	2		5043.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.a.1.1	✓	1	1.1	even	1	trivial
369.2.a.b.1.1		1	3.2	odd	2
1968.2.a.a.1.1		1	4.3	odd	2
3075.2.a.m.1.1		1	5.4	even	2
5043.2.a.a.1.1		1	41.40	even	2
5904.2.a.v.1.1		1	12.11	even	2
6027.2.a.a.1.1		1	7.6	odd	2
7872.2.a.r.1.1		1	8.5	even	2
7872.2.a.bj.1.1		1	8.3	odd	2
9225.2.a.d.1.1		1	15.14	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:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	123.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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