LMFDB - Embedded newform 37.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [37,2,Mod(1,37)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("37.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(37, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	37.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$0.295446487479$
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	$\chi$	$=$	37.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} -5.00000 q^{11} -6.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +6.00000 q^{15} -4.00000 q^{16} -12.0000 q^{18} -4.00000 q^{20} +3.00000 q^{21} +10.0000 q^{22} +2.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} -9.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -12.0000 q^{30} -4.00000 q^{31} +8.00000 q^{32} +15.0000 q^{33} +2.00000 q^{35} +12.0000 q^{36} -1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -6.00000 q^{42} +2.00000 q^{43} -10.0000 q^{44} -12.0000 q^{45} -4.00000 q^{46} -9.00000 q^{47} +12.0000 q^{48} -6.00000 q^{49} +2.00000 q^{50} -4.00000 q^{52} +1.00000 q^{53} +18.0000 q^{54} +10.0000 q^{55} -12.0000 q^{58} +8.00000 q^{59} +12.0000 q^{60} -8.00000 q^{61} +8.00000 q^{62} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -30.0000 q^{66} +8.00000 q^{67} -6.00000 q^{69} -4.00000 q^{70} +9.00000 q^{71} -1.00000 q^{73} +2.00000 q^{74} +3.00000 q^{75} +5.00000 q^{77} -12.0000 q^{78} +4.00000 q^{79} +8.00000 q^{80} +9.00000 q^{81} +18.0000 q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} -18.0000 q^{87} +4.00000 q^{89} +24.0000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +12.0000 q^{93} +18.0000 q^{94} -24.0000 q^{96} +4.00000 q^{97} +12.0000 q^{98} -30.0000 q^{99} +O(q^{100})\)

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	$0.750000\pi$
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	2.00000	1.00000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	6.00000	2.44949
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	4.00000	1.26491
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−6.00000	−1.73205
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	6.00000	1.54919
$16$	−4.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−12.0000	−2.82843
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−4.00000	−0.894427
$21$	3.00000	0.654654
$22$	10.0000	2.13201
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	−9.00000	−1.73205
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−12.0000	−2.19089
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	8.00000	1.41421
$33$	15.0000	2.61116
$34$	0	0
$35$	2.00000	0.338062
$36$	12.0000	2.00000
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−6.00000	−0.925820
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−10.0000	−1.50756
$45$	−12.0000	−1.78885
$46$	−4.00000	−0.589768
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	12.0000	1.73205
$49$	−6.00000	−0.857143
$50$	2.00000	0.282843
$51$	0	0
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	18.0000	2.44949
$55$	10.0000	1.34840
$56$	0	0
$57$	0	0
$58$	−12.0000	−1.57568
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	12.0000	1.54919
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	8.00000	1.01600
$63$	−6.00000	−0.755929
$64$	−8.00000	−1.00000
$65$	4.00000	0.496139
$66$	−30.0000	−3.69274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	−4.00000	−0.478091
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	2.00000	0.232495
$75$	3.00000	0.346410
$76$	0	0
$77$	5.00000	0.569803
$78$	−12.0000	−1.35873
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	8.00000	0.894427
$81$	9.00000	1.00000
$82$	18.0000	1.98777
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	−4.00000	−0.431331
$87$	−18.0000	−1.92980
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	24.0000	2.52982
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	12.0000	1.24434
$94$	18.0000	1.85656
$95$	0	0
$96$	−24.0000	−2.44949
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	12.0000	1.21218
$99$	−30.0000	−3.01511
$100$	−2.00000	−0.200000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−18.0000	−1.73205
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−20.0000	−1.90693
$111$	3.00000	0.284747
$112$	4.00000	0.377964
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	12.0000	1.11417
$117$	−12.0000	−1.10940
$118$	−16.0000	−1.47292
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	16.0000	1.44857
$123$	27.0000	2.43451
$124$	−8.00000	−0.718421
$125$	12.0000	1.07331
$126$	12.0000	1.06904
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	−8.00000	−0.701646
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	30.0000	2.61116
$133$	0	0
$134$	−16.0000	−1.38219
$135$	18.0000	1.54919
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	12.0000	1.02151
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	4.00000	0.338062
$141$	27.0000	2.27381
$142$	−18.0000	−1.51053
$143$	10.0000	0.836242
$144$	−24.0000	−2.00000
$145$	−12.0000	−0.996546
$146$	2.00000	0.165521
$147$	18.0000	1.48461
$148$	−2.00000	−0.164399
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	−6.00000	−0.489898
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	−10.0000	−0.805823
$155$	8.00000	0.642575
$156$	12.0000	0.960769
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	−8.00000	−0.636446
$159$	−3.00000	−0.237915
$160$	−16.0000	−1.26491
$161$	−2.00000	−0.157622
$162$	−18.0000	−1.41421
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	−18.0000	−1.40556
$165$	−30.0000	−2.33550
$166$	30.0000	2.32845
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	36.0000	2.72915
$175$	1.00000	0.0755929
$176$	20.0000	1.50756
$177$	−24.0000	−1.80395
$178$	−8.00000	−0.599625
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−24.0000	−1.78885
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	−4.00000	−0.296500
$183$	24.0000	1.77413
$184$	0	0
$185$	2.00000	0.147043
$186$	−24.0000	−1.75977
$187$	0	0
$188$	−18.0000	−1.31278
$189$	9.00000	0.654654
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	24.0000	1.73205
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	−8.00000	−0.574367
$195$	−12.0000	−0.859338
$196$	−12.0000	−0.857143
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	60.0000	4.26401
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	−6.00000	−0.422159
$203$	−6.00000	−0.421117
$204$	0	0
$205$	18.0000	1.25717
$206$	−36.0000	−2.50824
$207$	12.0000	0.834058
$208$	8.00000	0.554700
$209$	0	0
$210$	12.0000	0.828079
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	2.00000	0.137361
$213$	−27.0000	−1.85001
$214$	24.0000	1.64061
$215$	−4.00000	−0.272798
$216$	0	0
$217$	4.00000	0.271538
$218$	32.0000	2.16731
$219$	3.00000	0.202721
$220$	20.0000	1.34840
$221$	0	0
$222$	−6.00000	−0.402694
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	−8.00000	−0.534522
$225$	−6.00000	−0.400000
$226$	36.0000	2.39468
$227$	−16.0000	−1.06196	−0.530979	−	0.847385i	$-0.678176\pi$
$227$	−16.0000	−1.06196	−0.530979	+	0.847385i	$0.678176\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	8.00000	0.527504
$231$	−15.0000	−0.986928
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	24.0000	1.56893
$235$	18.0000	1.17419
$236$	16.0000	1.04151
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−24.0000	−1.54919
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−28.0000	−1.79991
$243$	0	0
$244$	−16.0000	−1.02430
$245$	12.0000	0.766652
$246$	−54.0000	−3.44291
$247$	0	0
$248$	0	0
$249$	45.0000	2.85176
$250$	−24.0000	−1.51789
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−12.0000	−0.755929
$253$	−10.0000	−0.628695
$254$	−2.00000	−0.125491
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	12.0000	0.747087
$259$	1.00000	0.0621370
$260$	8.00000	0.496139
$261$	36.0000	2.22834
$262$	24.0000	1.48272
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−12.0000	−0.734388
$268$	16.0000	0.977356
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−36.0000	−2.19089
$271$	−31.0000	−1.88312	−0.941558	−	0.336851i	$-0.890638\pi$
$271$	−31.0000	−1.88312	−0.941558	+	0.336851i	$0.890638\pi$
$272$	0	0
$273$	−6.00000	−0.363137
$274$	12.0000	0.724947
$275$	5.00000	0.301511
$276$	−12.0000	−0.722315
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−8.00000	−0.479808
$279$	−24.0000	−1.43684
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	−54.0000	−3.21565
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	18.0000	1.06810
$285$	0	0
$286$	−20.0000	−1.18262
$287$	9.00000	0.531253
$288$	48.0000	2.82843
$289$	−17.0000	−1.00000
$290$	24.0000	1.40933
$291$	−12.0000	−0.703452
$292$	−2.00000	−0.117041
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−36.0000	−2.09956
$295$	−16.0000	−0.931556
$296$	0	0
$297$	45.0000	2.61116
$298$	10.0000	0.579284
$299$	−4.00000	−0.231326
$300$	6.00000	0.346410
$301$	−2.00000	−0.115278
$302$	−32.0000	−1.84139
$303$	−9.00000	−0.517036
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	−17.0000	−0.970241	−0.485121	−	0.874447i	$-0.661224\pi$
$307$	−17.0000	−0.970241	−0.485121	+	0.874447i	$0.661224\pi$
$308$	10.0000	0.569803
$309$	−54.0000	−3.07195
$310$	−16.0000	−0.908739
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−46.0000	−2.59593
$315$	12.0000	0.676123
$316$	8.00000	0.450035
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	6.00000	0.336463
$319$	−30.0000	−1.67968
$320$	16.0000	0.894427
$321$	36.0000	2.00932
$322$	4.00000	0.222911
$323$	0	0
$324$	18.0000	1.00000
$325$	2.00000	0.110940
$326$	36.0000	1.99386
$327$	48.0000	2.65441
$328$	0	0
$329$	9.00000	0.496186
$330$	60.0000	3.30289
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	−30.0000	−1.64646
$333$	−6.00000	−0.328798
$334$	24.0000	1.31322
$335$	−16.0000	−0.874173
$336$	−12.0000	−0.654654
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	18.0000	0.979071
$339$	54.0000	2.93288
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	12.0000	0.646058
$346$	−18.0000	−0.967686
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	−36.0000	−1.92980
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−2.00000	−0.106904
$351$	18.0000	0.960769
$352$	−40.0000	−2.13201
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	48.0000	2.55117
$355$	−18.0000	−0.955341
$356$	8.00000	0.423999
$357$	0	0
$358$	−36.0000	−1.90266
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−10.0000	−0.525588
$363$	−42.0000	−2.20443
$364$	4.00000	0.209657
$365$	2.00000	0.104685
$366$	−48.0000	−2.50900
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−8.00000	−0.417029
$369$	−54.0000	−2.81113
$370$	−4.00000	−0.207950
$371$	−1.00000	−0.0519174
$372$	24.0000	1.24434
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	−36.0000	−1.85903
$376$	0	0
$377$	−12.0000	−0.618031
$378$	−18.0000	−0.925820
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	−3.00000	−0.153695
$382$	8.00000	0.409316
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	0	0
$385$	−10.0000	−0.509647
$386$	52.0000	2.64673
$387$	12.0000	0.609994
$388$	8.00000	0.406138
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	24.0000	1.21529
$391$	0	0
$392$	0	0
$393$	36.0000	1.81596
$394$	−6.00000	−0.302276
$395$	−8.00000	−0.402524
$396$	−60.0000	−3.01511
$397$	−5.00000	−0.250943	−0.125471	−	0.992097i	$-0.540044\pi$
$397$	−5.00000	−0.250943	−0.125471	+	0.992097i	$0.540044\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	4.00000	0.200000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	48.0000	2.39402
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	−18.0000	−0.894427
$406$	12.0000	0.595550
$407$	5.00000	0.247841
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	−36.0000	−1.77791
$411$	18.0000	0.887875
$412$	36.0000	1.77359
$413$	−8.00000	−0.393654
$414$	−24.0000	−1.17954
$415$	30.0000	1.47264
$416$	−16.0000	−0.784465
$417$	−12.0000	−0.587643
$418$	0	0
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	−12.0000	−0.585540
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	26.0000	1.26566
$423$	−54.0000	−2.62557
$424$	0	0
$425$	0	0
$426$	54.0000	2.61631
$427$	8.00000	0.387147
$428$	−24.0000	−1.16008
$429$	−30.0000	−1.44841
$430$	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	$0.757018\pi$
$432$	36.0000	1.73205
$433$	9.00000	0.432512	0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000	0.432512	0.216256	+	0.976337i	$0.430615\pi$
$434$	−8.00000	−0.384012
$435$	36.0000	1.72607
$436$	−32.0000	−1.53252
$437$	0	0
$438$	−6.00000	−0.286691
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	1.00000	0.0475114	0.0237557	−	0.999718i	$-0.492438\pi$
$443$	1.00000	0.0475114	0.0237557	+	0.999718i	$0.492438\pi$
$444$	6.00000	0.284747
$445$	−8.00000	−0.379236
$446$	34.0000	1.60995
$447$	15.0000	0.709476
$448$	8.00000	0.377964
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	12.0000	0.565685
$451$	45.0000	2.11897
$452$	−36.0000	−1.69330
$453$	−48.0000	−2.25524
$454$	32.0000	1.50183
$455$	−4.00000	−0.187523
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	−8.00000	−0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	30.0000	1.39573
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−24.0000	−1.11417
$465$	−24.0000	−1.11297
$466$	−12.0000	−0.555889
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	−24.0000	−1.10940
$469$	−8.00000	−0.369406
$470$	−36.0000	−1.66056
$471$	−69.0000	−3.17935
$472$	0	0
$473$	−10.0000	−0.459800
$474$	24.0000	1.10236
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	12.0000	0.548867
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	48.0000	2.19089
$481$	2.00000	0.0911922
$482$	−28.0000	−1.27537
$483$	6.00000	0.273009
$484$	28.0000	1.27273
$485$	−8.00000	−0.363261
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	54.0000	2.44196
$490$	−24.0000	−1.08421
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	54.0000	2.43451
$493$	0	0
$494$	0	0
$495$	60.0000	2.69680
$496$	16.0000	0.718421
$497$	−9.00000	−0.403705
$498$	−90.0000	−4.03300
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	24.0000	1.07331
$501$	36.0000	1.60836
$502$	4.00000	0.178529
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	20.0000	0.889108
$507$	27.0000	1.19911
$508$	2.00000	0.0887357
$509$	−31.0000	−1.37405	−0.687025	−	0.726633i	$-0.741084\pi$
$509$	−31.0000	−1.37405	−0.687025	+	0.726633i	$0.741084\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	−32.0000	−1.41421
$513$	0	0
$514$	0	0
$515$	−36.0000	−1.58635
$516$	−12.0000	−0.528271
$517$	45.0000	1.97910
$518$	−2.00000	−0.0878750
$519$	−27.0000	−1.18517
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	−72.0000	−3.15135
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−24.0000	−1.04844
$525$	−3.00000	−0.130931
$526$	−38.0000	−1.65688
$527$	0	0
$528$	−60.0000	−2.61116
$529$	−19.0000	−0.826087
$530$	4.00000	0.173749
$531$	48.0000	2.08302
$532$	0	0
$533$	18.0000	0.779667
$534$	24.0000	1.03858
$535$	24.0000	1.03761
$536$	0	0
$537$	−54.0000	−2.33027
$538$	12.0000	0.517357
$539$	30.0000	1.29219
$540$	36.0000	1.54919
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	62.0000	2.66313
$543$	−15.0000	−0.643712
$544$	0	0
$545$	32.0000	1.37073
$546$	12.0000	0.513553
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	−48.0000	−2.04859
$550$	−10.0000	−0.426401
$551$	0	0
$552$	0	0
$553$	−4.00000	−0.170097
$554$	−24.0000	−1.01966
$555$	−6.00000	−0.254686
$556$	8.00000	0.339276
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	48.0000	2.03200
$559$	−4.00000	−0.169182
$560$	−8.00000	−0.338062
$561$	0	0
$562$	−24.0000	−1.01238
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	54.0000	2.27381
$565$	36.0000	1.51453
$566$	−8.00000	−0.336265
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	20.0000	0.836242
$573$	12.0000	0.501307
$574$	−18.0000	−0.751305
$575$	−2.00000	−0.0834058
$576$	−48.0000	−2.00000
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	34.0000	1.41421
$579$	78.0000	3.24157
$580$	−24.0000	−0.996546
$581$	15.0000	0.622305
$582$	24.0000	0.994832
$583$	−5.00000	−0.207079
$584$	0	0
$585$	24.0000	0.992278
$586$	4.00000	0.165238
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	36.0000	1.48461
$589$	0	0
$590$	32.0000	1.31742
$591$	−9.00000	−0.370211
$592$	4.00000	0.164399
$593$	−5.00000	−0.205325	−0.102663	−	0.994716i	$-0.532736\pi$
$593$	−5.00000	−0.205325	−0.102663	+	0.994716i	$0.532736\pi$
$594$	−90.0000	−3.69274
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−6.00000	−0.245564
$598$	8.00000	0.327144
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	4.00000	0.163028
$603$	48.0000	1.95471
$604$	32.0000	1.30206
$605$	−28.0000	−1.13836
$606$	18.0000	0.731200
$607$	−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
$609$	18.0000	0.729397
$610$	−32.0000	−1.29564
$611$	18.0000	0.728202
$612$	0	0
$613$	15.0000	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$613$	15.0000	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$614$	34.0000	1.37213
$615$	−54.0000	−2.17749
$616$	0	0
$617$	17.0000	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$617$	17.0000	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$618$	108.000	4.34440
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	16.0000	0.642575
$621$	−18.0000	−0.722315
$622$	0	0
$623$	−4.00000	−0.160257
$624$	−24.0000	−0.960769
$625$	−19.0000	−0.760000
$626$	−44.0000	−1.75859
$627$	0	0
$628$	46.0000	1.83560
$629$	0	0
$630$	−24.0000	−0.956183
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	39.0000	1.55011
$634$	−44.0000	−1.74746
$635$	−2.00000	−0.0793676
$636$	−6.00000	−0.237915
$637$	12.0000	0.475457
$638$	60.0000	2.37542
$639$	54.0000	2.13621
$640$	0	0
$641$	−1.00000	−0.0394976	−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000	−0.0394976	−0.0197488	+	0.999805i	$0.506287\pi$
$642$	−72.0000	−2.84161
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	−4.00000	−0.157622
$645$	12.0000	0.472500
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	−4.00000	−0.156893
$651$	−12.0000	−0.470317
$652$	−36.0000	−1.40987
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−96.0000	−3.75390
$655$	24.0000	0.937758
$656$	36.0000	1.40556
$657$	−6.00000	−0.234082
$658$	−18.0000	−0.701713
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	−60.0000	−2.33550
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	0	0
$665$	0	0
$666$	12.0000	0.464991
$667$	12.0000	0.464642
$668$	−24.0000	−0.928588
$669$	51.0000	1.97177
$670$	32.0000	1.23627
$671$	40.0000	1.54418
$672$	24.0000	0.925820
$673$	27.0000	1.04077	0.520387	−	0.853931i	$-0.325788\pi$
$673$	27.0000	1.04077	0.520387	+	0.853931i	$0.325788\pi$
$674$	50.0000	1.92593
$675$	9.00000	0.346410
$676$	−18.0000	−0.692308
$677$	−11.0000	−0.422764	−0.211382	−	0.977403i	$-0.567796\pi$
$677$	−11.0000	−0.422764	−0.211382	+	0.977403i	$0.567796\pi$
$678$	−108.000	−4.14772
$679$	−4.00000	−0.153506
$680$	0	0
$681$	48.0000	1.83936
$682$	−40.0000	−1.53168
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	−26.0000	−0.992685
$687$	−21.0000	−0.801200
$688$	−8.00000	−0.304997
$689$	−2.00000	−0.0761939
$690$	−24.0000	−0.913664
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	18.0000	0.684257
$693$	30.0000	1.13961
$694$	20.0000	0.759190
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0	0
$698$	−12.0000	−0.454207
$699$	−18.0000	−0.680823
$700$	2.00000	0.0755929
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	−36.0000	−1.35873
$703$	0	0
$704$	40.0000	1.50756
$705$	−54.0000	−2.03376
$706$	−16.0000	−0.602168
$707$	−3.00000	−0.112827
$708$	−48.0000	−1.80395
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	36.0000	1.35106
$711$	24.0000	0.900070
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−20.0000	−0.747958
$716$	36.0000	1.34538
$717$	18.0000	0.672222
$718$	30.0000	1.11959
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	48.0000	1.78885
$721$	−18.0000	−0.670355
$722$	38.0000	1.41421
$723$	−42.0000	−1.56200
$724$	10.0000	0.371647
$725$	−6.00000	−0.222834
$726$	84.0000	3.11753
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−4.00000	−0.148047
$731$	0	0
$732$	48.0000	1.77413
$733$	7.00000	0.258551	0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000	0.258551	0.129275	+	0.991609i	$0.458735\pi$
$734$	−16.0000	−0.590571
$735$	−36.0000	−1.32788
$736$	16.0000	0.589768
$737$	−40.0000	−1.47342
$738$	108.000	3.97553
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	2.00000	0.0734223
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	38.0000	1.39128
$747$	−90.0000	−3.29293
$748$	0	0
$749$	12.0000	0.438470
$750$	72.0000	2.62907
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	36.0000	1.31278
$753$	6.00000	0.218652
$754$	24.0000	0.874028
$755$	−32.0000	−1.16460
$756$	18.0000	0.654654
$757$	−50.0000	−1.81728	−0.908640	−	0.417579i	$-0.862879\pi$
$757$	−50.0000	−1.81728	−0.908640	+	0.417579i	$0.862879\pi$
$758$	−30.0000	−1.08965
$759$	30.0000	1.08893
$760$	0	0
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	6.00000	0.217357
$763$	16.0000	0.579239
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−40.0000	−1.44526
$767$	−16.0000	−0.577727
$768$	−48.0000	−1.73205
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	20.0000	0.720750
$771$	0	0
$772$	−52.0000	−1.87152
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	−24.0000	−0.862662
$775$	4.00000	0.143684
$776$	0	0
$777$	−3.00000	−0.107624
$778$	−8.00000	−0.286814
$779$	0	0
$780$	−24.0000	−0.859338
$781$	−45.0000	−1.61023
$782$	0	0
$783$	−54.0000	−1.92980
$784$	24.0000	0.857143
$785$	−46.0000	−1.64181
$786$	−72.0000	−2.56815
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	6.00000	0.213741
$789$	−57.0000	−2.02925
$790$	16.0000	0.569254
$791$	18.0000	0.640006
$792$	0	0
$793$	16.0000	0.568177
$794$	10.0000	0.354887
$795$	6.00000	0.212798
$796$	4.00000	0.141776
$797$	52.0000	1.84193	0.920967	−	0.389640i	$-0.127401\pi$
$797$	52.0000	1.84193	0.920967	+	0.389640i	$0.127401\pi$
$798$	0	0
$799$	0	0
$800$	−8.00000	−0.282843
$801$	24.0000	0.847998
$802$	−36.0000	−1.27120
$803$	5.00000	0.176446
$804$	−48.0000	−1.69283
$805$	4.00000	0.140981
$806$	−16.0000	−0.563576
$807$	18.0000	0.633630
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	36.0000	1.26491
$811$	47.0000	1.65039	0.825197	−	0.564846i	$-0.191064\pi$
$811$	47.0000	1.65039	0.825197	+	0.564846i	$0.191064\pi$
$812$	−12.0000	−0.421117
$813$	93.0000	3.26165
$814$	−10.0000	−0.350500
$815$	36.0000	1.26102
$816$	0	0
$817$	0	0
$818$	−40.0000	−1.39857
$819$	12.0000	0.419314
$820$	36.0000	1.25717
$821$	−47.0000	−1.64031	−0.820156	−	0.572140i	$-0.806113\pi$
$821$	−47.0000	−1.64031	−0.820156	+	0.572140i	$0.806113\pi$
$822$	−36.0000	−1.25564
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	−15.0000	−0.522233
$826$	16.0000	0.556711
$827$	22.0000	0.765015	0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000	0.765015	0.382507	+	0.923952i	$0.375061\pi$
$828$	24.0000	0.834058
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	−60.0000	−2.08263
$831$	−36.0000	−1.24883
$832$	16.0000	0.554700
$833$	0	0
$834$	24.0000	0.831052
$835$	24.0000	0.830554
$836$	0	0
$837$	36.0000	1.24434
$838$	−14.0000	−0.483622
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	48.0000	1.65419
$843$	−36.0000	−1.23991
$844$	−26.0000	−0.894957
$845$	18.0000	0.619219
$846$	108.000	3.71312
$847$	−14.0000	−0.481046
$848$	−4.00000	−0.137361
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	−54.0000	−1.85001
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	0	0
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	60.0000	2.04837
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−8.00000	−0.272798
$861$	−27.0000	−0.920158
$862$	60.0000	2.04361
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−72.0000	−2.44949
$865$	−18.0000	−0.612018
$866$	−18.0000	−0.611665
$867$	51.0000	1.73205
$868$	8.00000	0.271538
$869$	−20.0000	−0.678454
$870$	−72.0000	−2.44103
$871$	−16.0000	−0.542139
$872$	0	0
$873$	24.0000	0.812277
$874$	0	0
$875$	−12.0000	−0.405674
$876$	6.00000	0.202721
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	−56.0000	−1.88991
$879$	6.00000	0.202375
$880$	−40.0000	−1.34840
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	72.0000	2.42437
$883$	48.0000	1.61533	0.807664	−	0.589643i	$-0.200731\pi$
$883$	48.0000	1.61533	0.807664	+	0.589643i	$0.200731\pi$
$884$	0	0
$885$	48.0000	1.61350
$886$	−2.00000	−0.0671913
$887$	25.0000	0.839418	0.419709	−	0.907659i	$-0.362132\pi$
$887$	25.0000	0.839418	0.419709	+	0.907659i	$0.362132\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	16.0000	0.536321
$891$	−45.0000	−1.50756
$892$	−34.0000	−1.13840
$893$	0	0
$894$	−30.0000	−1.00335
$895$	−36.0000	−1.20335
$896$	0	0
$897$	12.0000	0.400668
$898$	−72.0000	−2.40267
$899$	−24.0000	−0.800445
$900$	−12.0000	−0.400000
$901$	0	0
$902$	−90.0000	−2.99667
$903$	6.00000	0.199667
$904$	0	0
$905$	−10.0000	−0.332411
$906$	96.0000	3.18939
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−32.0000	−1.06196
$909$	18.0000	0.597022
$910$	8.00000	0.265197
$911$	26.0000	0.861418	0.430709	−	0.902491i	$-0.358263\pi$
$911$	26.0000	0.861418	0.430709	+	0.902491i	$0.358263\pi$
$912$	0	0
$913$	75.0000	2.48214
$914$	−36.0000	−1.19077
$915$	−48.0000	−1.58683
$916$	14.0000	0.462573
$917$	12.0000	0.396275
$918$	0	0
$919$	−58.0000	−1.91324	−0.956622	−	0.291333i	$-0.905901\pi$
$919$	−58.0000	−1.91324	−0.956622	+	0.291333i	$0.905901\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	−60.0000	−1.97599
$923$	−18.0000	−0.592477
$924$	−30.0000	−0.986928
$925$	1.00000	0.0328798
$926$	44.0000	1.44593
$927$	108.000	3.54719
$928$	48.0000	1.57568
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	48.0000	1.57398
$931$	0	0
$932$	12.0000	0.393073
$933$	0	0
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	16.0000	0.522419
$939$	−66.0000	−2.15383
$940$	36.0000	1.17419
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	138.000	4.49628
$943$	−18.0000	−0.586161
$944$	−32.0000	−1.04151
$945$	−18.0000	−0.585540
$946$	20.0000	0.650256
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−24.0000	−0.779484
$949$	2.00000	0.0649227
$950$	0	0
$951$	−66.0000	−2.14020
$952$	0	0
$953$	61.0000	1.97598	0.987992	−	0.154506i	$-0.0493785\pi$
$953$	61.0000	1.97598	0.987992	+	0.154506i	$0.0493785\pi$
$954$	−12.0000	−0.388514
$955$	8.00000	0.258874
$956$	−12.0000	−0.388108
$957$	90.0000	2.90929
$958$	−28.0000	−0.904639
$959$	6.00000	0.193750
$960$	−48.0000	−1.54919
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	−72.0000	−2.32017
$964$	28.0000	0.901819
$965$	52.0000	1.67394
$966$	−12.0000	−0.386094
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	0	0
$969$	0	0
$970$	16.0000	0.513729
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	48.0000	1.53802
$975$	−6.00000	−0.192154
$976$	32.0000	1.02430
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	−108.000	−3.45346
$979$	−20.0000	−0.639203
$980$	24.0000	0.766652
$981$	−96.0000	−3.06504
$982$	56.0000	1.78703
$983$	9.00000	0.287055	0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000	0.287055	0.143528	+	0.989646i	$0.454155\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−27.0000	−0.859419
$988$	0	0
$989$	4.00000	0.127193
$990$	−120.000	−3.81385
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	−32.0000	−1.01600
$993$	6.00000	0.190404
$994$	18.0000	0.570925
$995$	−4.00000	−0.126809
$996$	90.0000	2.85176
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−24.0000	−0.759707
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.2.a.a.1.1	✓	1
3.2	odd	2		333.2.a.d.1.1		1
4.3	odd	2		592.2.a.e.1.1		1
5.2	odd	4		925.2.b.b.149.1		2
5.3	odd	4		925.2.b.b.149.2		2
5.4	even	2		925.2.a.e.1.1		1
7.6	odd	2		1813.2.a.a.1.1		1
8.3	odd	2		2368.2.a.b.1.1		1
8.5	even	2		2368.2.a.q.1.1		1
11.10	odd	2		4477.2.a.b.1.1		1
12.11	even	2		5328.2.a.r.1.1		1
13.12	even	2		6253.2.a.c.1.1		1
15.14	odd	2		8325.2.a.e.1.1		1
37.6	odd	4		1369.2.b.c.1368.2		2
37.31	odd	4		1369.2.b.c.1368.1		2
37.36	even	2		1369.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.a.a.1.1	✓	1	1.1	even	1	trivial
333.2.a.d.1.1		1	3.2	odd	2
592.2.a.e.1.1		1	4.3	odd	2
925.2.a.e.1.1		1	5.4	even	2
925.2.b.b.149.1		2	5.2	odd	4
925.2.b.b.149.2		2	5.3	odd	4
1369.2.a.e.1.1		1	37.36	even	2
1369.2.b.c.1368.1		2	37.31	odd	4
1369.2.b.c.1368.2		2	37.6	odd	4
1813.2.a.a.1.1		1	7.6	odd	2
2368.2.a.b.1.1		1	8.3	odd	2
2368.2.a.q.1.1		1	8.5	even	2
4477.2.a.b.1.1		1	11.10	odd	2
5328.2.a.r.1.1		1	12.11	even	2
6253.2.a.c.1.1		1	13.12	even	2
8325.2.a.e.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	37.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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