LMFDB - Embedded newform 6762.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.70156 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.70156 q^{10} -1.00000 q^{12} +0.701562 q^{13} +2.70156 q^{15} +1.00000 q^{16} -1.00000 q^{18} -7.40312 q^{19} -2.70156 q^{20} -1.00000 q^{23} +1.00000 q^{24} +2.29844 q^{25} -0.701562 q^{26} -1.00000 q^{27} +6.70156 q^{29} -2.70156 q^{30} -6.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -2.70156 q^{37} +7.40312 q^{38} -0.701562 q^{39} +2.70156 q^{40} -1.29844 q^{41} -0.701562 q^{43} -2.70156 q^{45} +1.00000 q^{46} +2.70156 q^{47} -1.00000 q^{48} -2.29844 q^{50} +0.701562 q^{52} -3.40312 q^{53} +1.00000 q^{54} +7.40312 q^{57} -6.70156 q^{58} +13.4031 q^{59} +2.70156 q^{60} -10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -1.89531 q^{65} +4.00000 q^{67} +1.00000 q^{69} -5.40312 q^{71} -1.00000 q^{72} -7.40312 q^{73} +2.70156 q^{74} -2.29844 q^{75} -7.40312 q^{76} +0.701562 q^{78} -2.70156 q^{80} +1.00000 q^{81} +1.29844 q^{82} +15.4031 q^{83} +0.701562 q^{86} -6.70156 q^{87} +2.59688 q^{89} +2.70156 q^{90} -1.00000 q^{92} +6.00000 q^{93} -2.70156 q^{94} +20.0000 q^{95} +1.00000 q^{96} -18.1047 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - q^{10} - 2 q^{12} - 5 q^{13} - q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{19} + q^{20} - 2 q^{23} + 2 q^{24} + 11 q^{25} + 5 q^{26}+ \cdots - 17 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - q^10 - 2 * q^12 - 5 * q^13 - q^15 + 2 * q^16 - 2 * q^18 - 2 * q^19 + q^20 - 2 * q^23 + 2 * q^24 + 11 * q^25 + 5 * q^26 - 2 * q^27 + 7 * q^29 + q^30 - 12 * q^31 - 2 * q^32 + 2 * q^36 + q^37 + 2 * q^38 + 5 * q^39 - q^40 - 9 * q^41 + 5 * q^43 + q^45 + 2 * q^46 - q^47 - 2 * q^48 - 11 * q^50 - 5 * q^52 + 6 * q^53 + 2 * q^54 + 2 * q^57 - 7 * q^58 + 14 * q^59 - q^60 - 20 * q^61 + 12 * q^62 + 2 * q^64 - 23 * q^65 + 8 * q^67 + 2 * q^69 + 2 * q^71 - 2 * q^72 - 2 * q^73 - q^74 - 11 * q^75 - 2 * q^76 - 5 * q^78 + q^80 + 2 * q^81 + 9 * q^82 + 18 * q^83 - 5 * q^86 - 7 * q^87 + 18 * q^89 - q^90 - 2 * q^92 + 12 * q^93 + q^94 + 40 * q^95 + 2 * q^96 - 17 * q^97

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
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.70156	−1.20818	−0.604088	−	0.796918i	$-0.706462\pi$
$5$	−2.70156	−1.20818	−0.604088	+	0.796918i	$0.706462\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.70156	0.854309
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	0.701562	0.194578	0.0972892	−	0.995256i	$-0.468983\pi$
$13$	0.701562	0.194578	0.0972892	+	0.995256i	$0.468983\pi$
$14$	0	0
$15$	2.70156	0.697540
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−7.40312	−1.69839	−0.849197	−	0.528077i	$-0.822913\pi$
$19$	−7.40312	−1.69839	−0.849197	+	0.528077i	$0.822913\pi$
$20$	−2.70156	−0.604088
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	2.29844	0.459688
$26$	−0.701562	−0.137588
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.70156	1.24445	0.622224	−	0.782839i	$-0.286229\pi$
$29$	6.70156	1.24445	0.622224	+	0.782839i	$0.286229\pi$
$30$	−2.70156	−0.493236
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.70156	−0.444134	−0.222067	−	0.975031i	$-0.571280\pi$
$37$	−2.70156	−0.444134	−0.222067	+	0.975031i	$0.571280\pi$
$38$	7.40312	1.20095
$39$	−0.701562	−0.112340
$40$	2.70156	0.427154
$41$	−1.29844	−0.202782	−0.101391	−	0.994847i	$-0.532329\pi$
$41$	−1.29844	−0.202782	−0.101391	+	0.994847i	$0.532329\pi$
$42$	0	0
$43$	−0.701562	−0.106987	−0.0534936	−	0.998568i	$-0.517036\pi$
$43$	−0.701562	−0.106987	−0.0534936	+	0.998568i	$0.517036\pi$
$44$	0	0
$45$	−2.70156	−0.402725
$46$	1.00000	0.147442
$47$	2.70156	0.394063	0.197032	−	0.980397i	$-0.436870\pi$
$47$	2.70156	0.394063	0.197032	+	0.980397i	$0.436870\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−2.29844	−0.325048
$51$	0	0
$52$	0.701562	0.0972892
$53$	−3.40312	−0.467455	−0.233728	−	0.972302i	$-0.575092\pi$
$53$	−3.40312	−0.467455	−0.233728	+	0.972302i	$0.575092\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	7.40312	0.980568
$58$	−6.70156	−0.879958
$59$	13.4031	1.74494	0.872469	−	0.488669i	$-0.162518\pi$
$59$	13.4031	1.74494	0.872469	+	0.488669i	$0.162518\pi$
$60$	2.70156	0.348770
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.89531	−0.235085
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−5.40312	−0.641233	−0.320616	−	0.947209i	$-0.603890\pi$
$71$	−5.40312	−0.641233	−0.320616	+	0.947209i	$0.603890\pi$
$72$	−1.00000	−0.117851
$73$	−7.40312	−0.866470	−0.433235	−	0.901281i	$-0.642628\pi$
$73$	−7.40312	−0.866470	−0.433235	+	0.901281i	$0.642628\pi$
$74$	2.70156	0.314050
$75$	−2.29844	−0.265401
$76$	−7.40312	−0.849197
$77$	0	0
$78$	0.701562	0.0794363
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.70156	−0.302044
$81$	1.00000	0.111111
$82$	1.29844	0.143388
$83$	15.4031	1.69071	0.845356	−	0.534203i	$-0.179388\pi$
$83$	15.4031	1.69071	0.845356	+	0.534203i	$0.179388\pi$
$84$	0	0
$85$	0	0
$86$	0.701562	0.0756514
$87$	−6.70156	−0.718483
$88$	0	0
$89$	2.59688	0.275268	0.137634	−	0.990483i	$-0.456050\pi$
$89$	2.59688	0.275268	0.137634	+	0.990483i	$0.456050\pi$
$90$	2.70156	0.284770
$91$	0	0
$92$	−1.00000	−0.104257
$93$	6.00000	0.622171
$94$	−2.70156	−0.278645
$95$	20.0000	2.05196
$96$	1.00000	0.102062
$97$	−18.1047	−1.83825	−0.919126	−	0.393963i	$-0.871104\pi$
$97$	−18.1047	−1.83825	−0.919126	+	0.393963i	$0.871104\pi$
$98$	0	0
$99$	0	0
$100$	2.29844	0.229844
$101$	−17.4031	−1.73168	−0.865838	−	0.500325i	$-0.833214\pi$
$101$	−17.4031	−1.73168	−0.865838	+	0.500325i	$0.833214\pi$
$102$	0	0
$103$	−2.10469	−0.207381	−0.103690	−	0.994610i	$-0.533065\pi$
$103$	−2.10469	−0.207381	−0.103690	+	0.994610i	$0.533065\pi$
$104$	−0.701562	−0.0687938
$105$	0	0
$106$	3.40312	0.330541
$107$	−10.8062	−1.04468	−0.522340	−	0.852737i	$-0.674941\pi$
$107$	−10.8062	−1.04468	−0.522340	+	0.852737i	$0.674941\pi$
$108$	−1.00000	−0.0962250
$109$	9.29844	0.890629	0.445314	−	0.895374i	$-0.353092\pi$
$109$	9.29844	0.890629	0.445314	+	0.895374i	$0.353092\pi$
$110$	0	0
$111$	2.70156	0.256421
$112$	0	0
$113$	−10.7016	−1.00672	−0.503359	−	0.864077i	$-0.667903\pi$
$113$	−10.7016	−1.00672	−0.503359	+	0.864077i	$0.667903\pi$
$114$	−7.40312	−0.693366
$115$	2.70156	0.251922
$116$	6.70156	0.622224
$117$	0.701562	0.0648594
$118$	−13.4031	−1.23386
$119$	0	0
$120$	−2.70156	−0.246618
$121$	−11.0000	−1.00000
$122$	10.0000	0.905357
$123$	1.29844	0.117076
$124$	−6.00000	−0.538816
$125$	7.29844	0.652792
$126$	0	0
$127$	−2.10469	−0.186761	−0.0933804	−	0.995631i	$-0.529767\pi$
$127$	−2.10469	−0.186761	−0.0933804	+	0.995631i	$0.529767\pi$
$128$	−1.00000	−0.0883883
$129$	0.701562	0.0617691
$130$	1.89531	0.166230
$131$	−2.59688	−0.226890	−0.113445	−	0.993544i	$-0.536189\pi$
$131$	−2.59688	−0.226890	−0.113445	+	0.993544i	$0.536189\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	2.70156	0.232513
$136$	0	0
$137$	20.1047	1.71766	0.858830	−	0.512261i	$-0.171192\pi$
$137$	20.1047	1.71766	0.858830	+	0.512261i	$0.171192\pi$
$138$	−1.00000	−0.0851257
$139$	1.89531	0.160758	0.0803792	−	0.996764i	$-0.474387\pi$
$139$	1.89531	0.160758	0.0803792	+	0.996764i	$0.474387\pi$
$140$	0	0
$141$	−2.70156	−0.227513
$142$	5.40312	0.453420
$143$	0	0
$144$	1.00000	0.0833333
$145$	−18.1047	−1.50351
$146$	7.40312	0.612687
$147$	0	0
$148$	−2.70156	−0.222067
$149$	−4.59688	−0.376591	−0.188295	−	0.982112i	$-0.560296\pi$
$149$	−4.59688	−0.376591	−0.188295	+	0.982112i	$0.560296\pi$
$150$	2.29844	0.187667
$151$	−10.1047	−0.822308	−0.411154	−	0.911566i	$-0.634874\pi$
$151$	−10.1047	−0.822308	−0.411154	+	0.911566i	$0.634874\pi$
$152$	7.40312	0.600473
$153$	0	0
$154$	0	0
$155$	16.2094	1.30197
$156$	−0.701562	−0.0561699
$157$	0.596876	0.0476359	0.0238179	−	0.999716i	$-0.492418\pi$
$157$	0.596876	0.0476359	0.0238179	+	0.999716i	$0.492418\pi$
$158$	0	0
$159$	3.40312	0.269885
$160$	2.70156	0.213577
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−1.29844	−0.101391
$165$	0	0
$166$	−15.4031	−1.19551
$167$	8.59688	0.665246	0.332623	−	0.943060i	$-0.392066\pi$
$167$	8.59688	0.665246	0.332623	+	0.943060i	$0.392066\pi$
$168$	0	0
$169$	−12.5078	−0.962139
$170$	0	0
$171$	−7.40312	−0.566131
$172$	−0.701562	−0.0534936
$173$	−9.40312	−0.714906	−0.357453	−	0.933931i	$-0.616355\pi$
$173$	−9.40312	−0.714906	−0.357453	+	0.933931i	$0.616355\pi$
$174$	6.70156	0.508044
$175$	0	0
$176$	0	0
$177$	−13.4031	−1.00744
$178$	−2.59688	−0.194644
$179$	−0.701562	−0.0524372	−0.0262186	−	0.999656i	$-0.508347\pi$
$179$	−0.701562	−0.0524372	−0.0262186	+	0.999656i	$0.508347\pi$
$180$	−2.70156	−0.201363
$181$	8.80625	0.654563	0.327282	−	0.944927i	$-0.393867\pi$
$181$	8.80625	0.654563	0.327282	+	0.944927i	$0.393867\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	1.00000	0.0737210
$185$	7.29844	0.536592
$186$	−6.00000	−0.439941
$187$	0	0
$188$	2.70156	0.197032
$189$	0	0
$190$	−20.0000	−1.45095
$191$	−2.80625	−0.203053	−0.101527	−	0.994833i	$-0.532373\pi$
$191$	−2.80625	−0.203053	−0.101527	+	0.994833i	$0.532373\pi$
$192$	−1.00000	−0.0721688
$193$	12.1047	0.871314	0.435657	−	0.900113i	$-0.356516\pi$
$193$	12.1047	0.871314	0.435657	+	0.900113i	$0.356516\pi$
$194$	18.1047	1.29984
$195$	1.89531	0.135726
$196$	0	0
$197$	−25.5078	−1.81736	−0.908678	−	0.417497i	$-0.862907\pi$
$197$	−25.5078	−1.81736	−0.908678	+	0.417497i	$0.862907\pi$
$198$	0	0
$199$	18.1047	1.28341	0.641704	−	0.766953i	$-0.278228\pi$
$199$	18.1047	1.28341	0.641704	+	0.766953i	$0.278228\pi$
$200$	−2.29844	−0.162524
$201$	−4.00000	−0.282138
$202$	17.4031	1.22448
$203$	0	0
$204$	0	0
$205$	3.50781	0.244996
$206$	2.10469	0.146640
$207$	−1.00000	−0.0695048
$208$	0.701562	0.0486446
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−3.40312	−0.233728
$213$	5.40312	0.370216
$214$	10.8062	0.738700
$215$	1.89531	0.129259
$216$	1.00000	0.0680414
$217$	0	0
$218$	−9.29844	−0.629770
$219$	7.40312	0.500257
$220$	0	0
$221$	0	0
$222$	−2.70156	−0.181317
$223$	−4.80625	−0.321850	−0.160925	−	0.986967i	$-0.551448\pi$
$223$	−4.80625	−0.321850	−0.160925	+	0.986967i	$0.551448\pi$
$224$	0	0
$225$	2.29844	0.153229
$226$	10.7016	0.711857
$227$	−5.50781	−0.365566	−0.182783	−	0.983153i	$-0.558511\pi$
$227$	−5.50781	−0.365566	−0.182783	+	0.983153i	$0.558511\pi$
$228$	7.40312	0.490284
$229$	10.2094	0.674654	0.337327	−	0.941387i	$-0.390477\pi$
$229$	10.2094	0.674654	0.337327	+	0.941387i	$0.390477\pi$
$230$	−2.70156	−0.178136
$231$	0	0
$232$	−6.70156	−0.439979
$233$	3.19375	0.209230	0.104615	−	0.994513i	$-0.466639\pi$
$233$	3.19375	0.209230	0.104615	+	0.994513i	$0.466639\pi$
$234$	−0.701562	−0.0458626
$235$	−7.29844	−0.476098
$236$	13.4031	0.872469
$237$	0	0
$238$	0	0
$239$	6.80625	0.440260	0.220130	−	0.975471i	$-0.429352\pi$
$239$	6.80625	0.440260	0.220130	+	0.975471i	$0.429352\pi$
$240$	2.70156	0.174385
$241$	14.1047	0.908563	0.454281	−	0.890858i	$-0.349896\pi$
$241$	14.1047	0.908563	0.454281	+	0.890858i	$0.349896\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−1.29844	−0.0827854
$247$	−5.19375	−0.330470
$248$	6.00000	0.381000
$249$	−15.4031	−0.976133
$250$	−7.29844	−0.461594
$251$	30.9109	1.95108	0.975540	−	0.219820i	$-0.0705470\pi$
$251$	30.9109	1.95108	0.975540	+	0.219820i	$0.0705470\pi$
$252$	0	0
$253$	0	0
$254$	2.10469	0.132060
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.6125	1.72242	0.861210	−	0.508249i	$-0.169707\pi$
$257$	27.6125	1.72242	0.861210	+	0.508249i	$0.169707\pi$
$258$	−0.701562	−0.0436773
$259$	0	0
$260$	−1.89531	−0.117542
$261$	6.70156	0.414816
$262$	2.59688	0.160436
$263$	15.5078	0.956253	0.478126	−	0.878291i	$-0.341316\pi$
$263$	15.5078	0.956253	0.478126	+	0.878291i	$0.341316\pi$
$264$	0	0
$265$	9.19375	0.564768
$266$	0	0
$267$	−2.59688	−0.158926
$268$	4.00000	0.244339
$269$	−9.40312	−0.573319	−0.286659	−	0.958033i	$-0.592545\pi$
$269$	−9.40312	−0.573319	−0.286659	+	0.958033i	$0.592545\pi$
$270$	−2.70156	−0.164412
$271$	−4.59688	−0.279240	−0.139620	−	0.990205i	$-0.544588\pi$
$271$	−4.59688	−0.279240	−0.139620	+	0.990205i	$0.544588\pi$
$272$	0	0
$273$	0	0
$274$	−20.1047	−1.21457
$275$	0	0
$276$	1.00000	0.0601929
$277$	−0.596876	−0.0358628	−0.0179314	−	0.999839i	$-0.505708\pi$
$277$	−0.596876	−0.0358628	−0.0179314	+	0.999839i	$0.505708\pi$
$278$	−1.89531	−0.113673
$279$	−6.00000	−0.359211
$280$	0	0
$281$	10.7016	0.638402	0.319201	−	0.947687i	$-0.396586\pi$
$281$	10.7016	0.638402	0.319201	+	0.947687i	$0.396586\pi$
$282$	2.70156	0.160876
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	−5.40312	−0.320616
$285$	−20.0000	−1.18470
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	18.1047	1.06314
$291$	18.1047	1.06132
$292$	−7.40312	−0.433235
$293$	28.8062	1.68288	0.841440	−	0.540351i	$-0.181709\pi$
$293$	28.8062	1.68288	0.841440	+	0.540351i	$0.181709\pi$
$294$	0	0
$295$	−36.2094	−2.10819
$296$	2.70156	0.157025
$297$	0	0
$298$	4.59688	0.266290
$299$	−0.701562	−0.0405724
$300$	−2.29844	−0.132700
$301$	0	0
$302$	10.1047	0.581459
$303$	17.4031	0.999783
$304$	−7.40312	−0.424598
$305$	27.0156	1.54691
$306$	0	0
$307$	14.1047	0.804997	0.402498	−	0.915421i	$-0.368142\pi$
$307$	14.1047	0.804997	0.402498	+	0.915421i	$0.368142\pi$
$308$	0	0
$309$	2.10469	0.119731
$310$	−16.2094	−0.920631
$311$	−22.2094	−1.25938	−0.629689	−	0.776847i	$-0.716818\pi$
$311$	−22.2094	−1.25938	−0.629689	+	0.776847i	$0.716818\pi$
$312$	0.701562	0.0397181
$313$	17.4031	0.983683	0.491841	−	0.870685i	$-0.336324\pi$
$313$	17.4031	0.983683	0.491841	+	0.870685i	$0.336324\pi$
$314$	−0.596876	−0.0336836
$315$	0	0
$316$	0	0
$317$	1.29844	0.0729275	0.0364638	−	0.999335i	$-0.488391\pi$
$317$	1.29844	0.0729275	0.0364638	+	0.999335i	$0.488391\pi$
$318$	−3.40312	−0.190838
$319$	0	0
$320$	−2.70156	−0.151022
$321$	10.8062	0.603146
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.61250	0.0894452
$326$	12.0000	0.664619
$327$	−9.29844	−0.514205
$328$	1.29844	0.0716942
$329$	0	0
$330$	0	0
$331$	33.6125	1.84751	0.923755	−	0.382984i	$-0.125104\pi$
$331$	33.6125	1.84751	0.923755	+	0.382984i	$0.125104\pi$
$332$	15.4031	0.845356
$333$	−2.70156	−0.148045
$334$	−8.59688	−0.470400
$335$	−10.8062	−0.590408
$336$	0	0
$337$	4.59688	0.250408	0.125204	−	0.992131i	$-0.460041\pi$
$337$	4.59688	0.250408	0.125204	+	0.992131i	$0.460041\pi$
$338$	12.5078	0.680335
$339$	10.7016	0.581229
$340$	0	0
$341$	0	0
$342$	7.40312	0.400315
$343$	0	0
$344$	0.701562	0.0378257
$345$	−2.70156	−0.145447
$346$	9.40312	0.505515
$347$	−11.5078	−0.617772	−0.308886	−	0.951099i	$-0.599956\pi$
$347$	−11.5078	−0.617772	−0.308886	+	0.951099i	$0.599956\pi$
$348$	−6.70156	−0.359241
$349$	4.20937	0.225323	0.112661	−	0.993633i	$-0.464063\pi$
$349$	4.20937	0.225323	0.112661	+	0.993633i	$0.464063\pi$
$350$	0	0
$351$	−0.701562	−0.0374466
$352$	0	0
$353$	−24.3141	−1.29411	−0.647053	−	0.762445i	$-0.723999\pi$
$353$	−24.3141	−1.29411	−0.647053	+	0.762445i	$0.723999\pi$
$354$	13.4031	0.712368
$355$	14.5969	0.774722
$356$	2.59688	0.137634
$357$	0	0
$358$	0.701562	0.0370787
$359$	3.29844	0.174085	0.0870424	−	0.996205i	$-0.472258\pi$
$359$	3.29844	0.174085	0.0870424	+	0.996205i	$0.472258\pi$
$360$	2.70156	0.142385
$361$	35.8062	1.88454
$362$	−8.80625	−0.462846
$363$	11.0000	0.577350
$364$	0	0
$365$	20.0000	1.04685
$366$	−10.0000	−0.522708
$367$	28.9109	1.50914	0.754569	−	0.656220i	$-0.227846\pi$
$367$	28.9109	1.50914	0.754569	+	0.656220i	$0.227846\pi$
$368$	−1.00000	−0.0521286
$369$	−1.29844	−0.0675940
$370$	−7.29844	−0.379428
$371$	0	0
$372$	6.00000	0.311086
$373$	20.8062	1.07731	0.538653	−	0.842527i	$-0.318933\pi$
$373$	20.8062	1.07731	0.538653	+	0.842527i	$0.318933\pi$
$374$	0	0
$375$	−7.29844	−0.376890
$376$	−2.70156	−0.139322
$377$	4.70156	0.242143
$378$	0	0
$379$	−11.2984	−0.580362	−0.290181	−	0.956972i	$-0.593715\pi$
$379$	−11.2984	−0.580362	−0.290181	+	0.956972i	$0.593715\pi$
$380$	20.0000	1.02598
$381$	2.10469	0.107826
$382$	2.80625	0.143580
$383$	18.8062	0.960954	0.480477	−	0.877007i	$-0.340463\pi$
$383$	18.8062	0.960954	0.480477	+	0.877007i	$0.340463\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−12.1047	−0.616112
$387$	−0.701562	−0.0356624
$388$	−18.1047	−0.919126
$389$	11.1938	0.567546	0.283773	−	0.958892i	$-0.408414\pi$
$389$	11.1938	0.567546	0.283773	+	0.958892i	$0.408414\pi$
$390$	−1.89531	−0.0959729
$391$	0	0
$392$	0	0
$393$	2.59688	0.130995
$394$	25.5078	1.28506
$395$	0	0
$396$	0	0
$397$	32.2094	1.61654	0.808271	−	0.588811i	$-0.200404\pi$
$397$	32.2094	1.61654	0.808271	+	0.588811i	$0.200404\pi$
$398$	−18.1047	−0.907506
$399$	0	0
$400$	2.29844	0.114922
$401$	3.19375	0.159488	0.0797442	−	0.996815i	$-0.474590\pi$
$401$	3.19375	0.159488	0.0797442	+	0.996815i	$0.474590\pi$
$402$	4.00000	0.199502
$403$	−4.20937	−0.209684
$404$	−17.4031	−0.865838
$405$	−2.70156	−0.134242
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6.20937	−0.307034	−0.153517	−	0.988146i	$-0.549060\pi$
$409$	−6.20937	−0.307034	−0.153517	+	0.988146i	$0.549060\pi$
$410$	−3.50781	−0.173238
$411$	−20.1047	−0.991691
$412$	−2.10469	−0.103690
$413$	0	0
$414$	1.00000	0.0491473
$415$	−41.6125	−2.04268
$416$	−0.701562	−0.0343969
$417$	−1.89531	−0.0928139
$418$	0	0
$419$	15.4031	0.752492	0.376246	−	0.926520i	$-0.377215\pi$
$419$	15.4031	0.752492	0.376246	+	0.926520i	$0.377215\pi$
$420$	0	0
$421$	9.29844	0.453178	0.226589	−	0.973990i	$-0.427243\pi$
$421$	9.29844	0.453178	0.226589	+	0.973990i	$0.427243\pi$
$422$	12.0000	0.584151
$423$	2.70156	0.131354
$424$	3.40312	0.165270
$425$	0	0
$426$	−5.40312	−0.261782
$427$	0	0
$428$	−10.8062	−0.522340
$429$	0	0
$430$	−1.89531	−0.0914001
$431$	−14.1047	−0.679399	−0.339699	−	0.940534i	$-0.610325\pi$
$431$	−14.1047	−0.679399	−0.339699	+	0.940534i	$0.610325\pi$
$432$	−1.00000	−0.0481125
$433$	2.10469	0.101145	0.0505724	−	0.998720i	$-0.483895\pi$
$433$	2.10469	0.101145	0.0505724	+	0.998720i	$0.483895\pi$
$434$	0	0
$435$	18.1047	0.868053
$436$	9.29844	0.445314
$437$	7.40312	0.354139
$438$	−7.40312	−0.353735
$439$	−28.8062	−1.37485	−0.687424	−	0.726257i	$-0.741258\pi$
$439$	−28.8062	−1.37485	−0.687424	+	0.726257i	$0.741258\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.10469	−0.0999967	−0.0499983	−	0.998749i	$-0.515922\pi$
$443$	−2.10469	−0.0999967	−0.0499983	+	0.998749i	$0.515922\pi$
$444$	2.70156	0.128210
$445$	−7.01562	−0.332572
$446$	4.80625	0.227582
$447$	4.59688	0.217425
$448$	0	0
$449$	−23.6125	−1.11434	−0.557171	−	0.830398i	$-0.688113\pi$
$449$	−23.6125	−1.11434	−0.557171	+	0.830398i	$0.688113\pi$
$450$	−2.29844	−0.108349
$451$	0	0
$452$	−10.7016	−0.503359
$453$	10.1047	0.474760
$454$	5.50781	0.258494
$455$	0	0
$456$	−7.40312	−0.346683
$457$	22.2094	1.03891	0.519455	−	0.854498i	$-0.326135\pi$
$457$	22.2094	1.03891	0.519455	+	0.854498i	$0.326135\pi$
$458$	−10.2094	−0.477053
$459$	0	0
$460$	2.70156	0.125961
$461$	−14.5969	−0.679844	−0.339922	−	0.940454i	$-0.610401\pi$
$461$	−14.5969	−0.679844	−0.339922	+	0.940454i	$0.610401\pi$
$462$	0	0
$463$	6.10469	0.283709	0.141854	−	0.989888i	$-0.454694\pi$
$463$	6.10469	0.283709	0.141854	+	0.989888i	$0.454694\pi$
$464$	6.70156	0.311112
$465$	−16.2094	−0.751692
$466$	−3.19375	−0.147948
$467$	14.7016	0.680307	0.340154	−	0.940370i	$-0.389521\pi$
$467$	14.7016	0.680307	0.340154	+	0.940370i	$0.389521\pi$
$468$	0.701562	0.0324297
$469$	0	0
$470$	7.29844	0.336652
$471$	−0.596876	−0.0275026
$472$	−13.4031	−0.616929
$473$	0	0
$474$	0	0
$475$	−17.0156	−0.780730
$476$	0	0
$477$	−3.40312	−0.155818
$478$	−6.80625	−0.311311
$479$	5.40312	0.246875	0.123438	−	0.992352i	$-0.460608\pi$
$479$	5.40312	0.246875	0.123438	+	0.992352i	$0.460608\pi$
$480$	−2.70156	−0.123309
$481$	−1.89531	−0.0864189
$482$	−14.1047	−0.642451
$483$	0	0
$484$	−11.0000	−0.500000
$485$	48.9109	2.22093
$486$	1.00000	0.0453609
$487$	6.10469	0.276630	0.138315	−	0.990388i	$-0.455831\pi$
$487$	6.10469	0.276630	0.138315	+	0.990388i	$0.455831\pi$
$488$	10.0000	0.452679
$489$	12.0000	0.542659
$490$	0	0
$491$	33.6125	1.51691	0.758455	−	0.651725i	$-0.225954\pi$
$491$	33.6125	1.51691	0.758455	+	0.651725i	$0.225954\pi$
$492$	1.29844	0.0585381
$493$	0	0
$494$	5.19375	0.233678
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	15.4031	0.690231
$499$	−27.0156	−1.20939	−0.604693	−	0.796459i	$-0.706704\pi$
$499$	−27.0156	−1.20939	−0.604693	+	0.796459i	$0.706704\pi$
$500$	7.29844	0.326396
$501$	−8.59688	−0.384080
$502$	−30.9109	−1.37962
$503$	−1.40312	−0.0625622	−0.0312811	−	0.999511i	$-0.509959\pi$
$503$	−1.40312	−0.0625622	−0.0312811	+	0.999511i	$0.509959\pi$
$504$	0	0
$505$	47.0156	2.09217
$506$	0	0
$507$	12.5078	0.555491
$508$	−2.10469	−0.0933804
$509$	−10.8062	−0.478979	−0.239489	−	0.970899i	$-0.576980\pi$
$509$	−10.8062	−0.478979	−0.239489	+	0.970899i	$0.576980\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	7.40312	0.326856
$514$	−27.6125	−1.21794
$515$	5.68594	0.250552
$516$	0.701562	0.0308846
$517$	0	0
$518$	0	0
$519$	9.40312	0.412751
$520$	1.89531	0.0831150
$521$	−2.59688	−0.113771	−0.0568856	−	0.998381i	$-0.518117\pi$
$521$	−2.59688	−0.113771	−0.0568856	+	0.998381i	$0.518117\pi$
$522$	−6.70156	−0.293319
$523$	−38.4187	−1.67993	−0.839967	−	0.542637i	$-0.817426\pi$
$523$	−38.4187	−1.67993	−0.839967	+	0.542637i	$0.817426\pi$
$524$	−2.59688	−0.113445
$525$	0	0
$526$	−15.5078	−0.676173
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.19375	−0.399351
$531$	13.4031	0.581646
$532$	0	0
$533$	−0.910935	−0.0394570
$534$	2.59688	0.112378
$535$	29.1938	1.26216
$536$	−4.00000	−0.172774
$537$	0.701562	0.0302746
$538$	9.40312	0.405397
$539$	0	0
$540$	2.70156	0.116257
$541$	18.2094	0.782882	0.391441	−	0.920203i	$-0.371977\pi$
$541$	18.2094	0.782882	0.391441	+	0.920203i	$0.371977\pi$
$542$	4.59688	0.197453
$543$	−8.80625	−0.377912
$544$	0	0
$545$	−25.1203	−1.07604
$546$	0	0
$547$	24.2094	1.03512	0.517559	−	0.855648i	$-0.326841\pi$
$547$	24.2094	1.03512	0.517559	+	0.855648i	$0.326841\pi$
$548$	20.1047	0.858830
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−49.6125	−2.11356
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	0.596876	0.0253588
$555$	−7.29844	−0.309801
$556$	1.89531	0.0803792
$557$	8.59688	0.364261	0.182131	−	0.983274i	$-0.441701\pi$
$557$	8.59688	0.364261	0.182131	+	0.983274i	$0.441701\pi$
$558$	6.00000	0.254000
$559$	−0.492189	−0.0208174
$560$	0	0
$561$	0	0
$562$	−10.7016	−0.451418
$563$	25.7172	1.08385	0.541925	−	0.840427i	$-0.317696\pi$
$563$	25.7172	1.08385	0.541925	+	0.840427i	$0.317696\pi$
$564$	−2.70156	−0.113756
$565$	28.9109	1.21629
$566$	−10.0000	−0.420331
$567$	0	0
$568$	5.40312	0.226710
$569$	−8.10469	−0.339766	−0.169883	−	0.985464i	$-0.554339\pi$
$569$	−8.10469	−0.339766	−0.169883	+	0.985464i	$0.554339\pi$
$570$	20.0000	0.837708
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	2.80625	0.117233
$574$	0	0
$575$	−2.29844	−0.0958515
$576$	1.00000	0.0416667
$577$	−27.6125	−1.14952	−0.574762	−	0.818321i	$-0.694905\pi$
$577$	−27.6125	−1.14952	−0.574762	+	0.818321i	$0.694905\pi$
$578$	17.0000	0.707107
$579$	−12.1047	−0.503054
$580$	−18.1047	−0.751756
$581$	0	0
$582$	−18.1047	−0.750463
$583$	0	0
$584$	7.40312	0.306343
$585$	−1.89531	−0.0783616
$586$	−28.8062	−1.18998
$587$	26.8062	1.10641	0.553206	−	0.833044i	$-0.313404\pi$
$587$	26.8062	1.10641	0.553206	+	0.833044i	$0.313404\pi$
$588$	0	0
$589$	44.4187	1.83024
$590$	36.2094	1.49072
$591$	25.5078	1.04925
$592$	−2.70156	−0.111034
$593$	−2.91093	−0.119538	−0.0597689	−	0.998212i	$-0.519036\pi$
$593$	−2.91093	−0.119538	−0.0597689	+	0.998212i	$0.519036\pi$
$594$	0	0
$595$	0	0
$596$	−4.59688	−0.188295
$597$	−18.1047	−0.740975
$598$	0.701562	0.0286890
$599$	21.6125	0.883063	0.441531	−	0.897246i	$-0.354435\pi$
$599$	21.6125	0.883063	0.441531	+	0.897246i	$0.354435\pi$
$600$	2.29844	0.0938333
$601$	−12.5969	−0.513837	−0.256919	−	0.966433i	$-0.582707\pi$
$601$	−12.5969	−0.513837	−0.256919	+	0.966433i	$0.582707\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−10.1047	−0.411154
$605$	29.7172	1.20818
$606$	−17.4031	−0.706954
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	7.40312	0.300236
$609$	0	0
$610$	−27.0156	−1.09383
$611$	1.89531	0.0766762
$612$	0	0
$613$	32.3141	1.30515	0.652576	−	0.757723i	$-0.273688\pi$
$613$	32.3141	1.30515	0.652576	+	0.757723i	$0.273688\pi$
$614$	−14.1047	−0.569219
$615$	−3.50781	−0.141449
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−2.10469	−0.0846629
$619$	30.0000	1.20580	0.602901	−	0.797816i	$-0.294011\pi$
$619$	30.0000	1.20580	0.602901	+	0.797816i	$0.294011\pi$
$620$	16.2094	0.650984
$621$	1.00000	0.0401286
$622$	22.2094	0.890515
$623$	0	0
$624$	−0.701562	−0.0280850
$625$	−31.2094	−1.24837
$626$	−17.4031	−0.695569
$627$	0	0
$628$	0.596876	0.0238179
$629$	0	0
$630$	0	0
$631$	34.8062	1.38561	0.692807	−	0.721123i	$-0.256374\pi$
$631$	34.8062	1.38561	0.692807	+	0.721123i	$0.256374\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	−1.29844	−0.0515676
$635$	5.68594	0.225640
$636$	3.40312	0.134943
$637$	0	0
$638$	0	0
$639$	−5.40312	−0.213744
$640$	2.70156	0.106789
$641$	−5.29844	−0.209276	−0.104638	−	0.994510i	$-0.533368\pi$
$641$	−5.29844	−0.209276	−0.104638	+	0.994510i	$0.533368\pi$
$642$	−10.8062	−0.426489
$643$	−28.8062	−1.13601	−0.568004	−	0.823026i	$-0.692284\pi$
$643$	−28.8062	−1.13601	−0.568004	+	0.823026i	$0.692284\pi$
$644$	0	0
$645$	−1.89531	−0.0746279
$646$	0	0
$647$	−34.2094	−1.34491	−0.672455	−	0.740138i	$-0.734760\pi$
$647$	−34.2094	−1.34491	−0.672455	+	0.740138i	$0.734760\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−1.61250	−0.0632473
$651$	0	0
$652$	−12.0000	−0.469956
$653$	27.8953	1.09163	0.545814	−	0.837906i	$-0.316221\pi$
$653$	27.8953	1.09163	0.545814	+	0.837906i	$0.316221\pi$
$654$	9.29844	0.363598
$655$	7.01562	0.274123
$656$	−1.29844	−0.0506955
$657$	−7.40312	−0.288823
$658$	0	0
$659$	−25.6125	−0.997721	−0.498861	−	0.866682i	$-0.666248\pi$
$659$	−25.6125	−0.997721	−0.498861	+	0.866682i	$0.666248\pi$
$660$	0	0
$661$	−4.80625	−0.186941	−0.0934707	−	0.995622i	$-0.529796\pi$
$661$	−4.80625	−0.186941	−0.0934707	+	0.995622i	$0.529796\pi$
$662$	−33.6125	−1.30639
$663$	0	0
$664$	−15.4031	−0.597757
$665$	0	0
$666$	2.70156	0.104683
$667$	−6.70156	−0.259486
$668$	8.59688	0.332623
$669$	4.80625	0.185820
$670$	10.8062	0.417482
$671$	0	0
$672$	0	0
$673$	−14.7016	−0.566704	−0.283352	−	0.959016i	$-0.591446\pi$
$673$	−14.7016	−0.566704	−0.283352	+	0.959016i	$0.591446\pi$
$674$	−4.59688	−0.177065
$675$	−2.29844	−0.0884669
$676$	−12.5078	−0.481070
$677$	28.8062	1.10711	0.553557	−	0.832811i	$-0.313270\pi$
$677$	28.8062	1.10711	0.553557	+	0.832811i	$0.313270\pi$
$678$	−10.7016	−0.410991
$679$	0	0
$680$	0	0
$681$	5.50781	0.211060
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−7.40312	−0.283066
$685$	−54.3141	−2.07523
$686$	0	0
$687$	−10.2094	−0.389512
$688$	−0.701562	−0.0267468
$689$	−2.38750	−0.0909566
$690$	2.70156	0.102847
$691$	35.2984	1.34282	0.671408	−	0.741088i	$-0.265690\pi$
$691$	35.2984	1.34282	0.671408	+	0.741088i	$0.265690\pi$
$692$	−9.40312	−0.357453
$693$	0	0
$694$	11.5078	0.436831
$695$	−5.12031	−0.194224
$696$	6.70156	0.254022
$697$	0	0
$698$	−4.20937	−0.159327
$699$	−3.19375	−0.120799
$700$	0	0
$701$	27.4031	1.03500	0.517501	−	0.855683i	$-0.326862\pi$
$701$	27.4031	1.03500	0.517501	+	0.855683i	$0.326862\pi$
$702$	0.701562	0.0264788
$703$	20.0000	0.754314
$704$	0	0
$705$	7.29844	0.274875
$706$	24.3141	0.915072
$707$	0	0
$708$	−13.4031	−0.503720
$709$	−40.8062	−1.53251	−0.766255	−	0.642536i	$-0.777882\pi$
$709$	−40.8062	−1.53251	−0.766255	+	0.642536i	$0.777882\pi$
$710$	−14.5969	−0.547811
$711$	0	0
$712$	−2.59688	−0.0973220
$713$	6.00000	0.224702
$714$	0	0
$715$	0	0
$716$	−0.701562	−0.0262186
$717$	−6.80625	−0.254184
$718$	−3.29844	−0.123097
$719$	17.5078	0.652931	0.326466	−	0.945209i	$-0.394142\pi$
$719$	17.5078	0.652931	0.326466	+	0.945209i	$0.394142\pi$
$720$	−2.70156	−0.100681
$721$	0	0
$722$	−35.8062	−1.33257
$723$	−14.1047	−0.524559
$724$	8.80625	0.327282
$725$	15.4031	0.572058
$726$	−11.0000	−0.408248
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.0000	−0.740233
$731$	0	0
$732$	10.0000	0.369611
$733$	33.0156	1.21946	0.609730	−	0.792609i	$-0.291278\pi$
$733$	33.0156	1.21946	0.609730	+	0.792609i	$0.291278\pi$
$734$	−28.9109	−1.06712
$735$	0	0
$736$	1.00000	0.0368605
$737$	0	0
$738$	1.29844	0.0477962
$739$	−0.209373	−0.00770190	−0.00385095	−	0.999993i	$-0.501226\pi$
$739$	−0.209373	−0.00770190	−0.00385095	+	0.999993i	$0.501226\pi$
$740$	7.29844	0.268296
$741$	5.19375	0.190797
$742$	0	0
$743$	29.6125	1.08638	0.543189	−	0.839611i	$-0.317217\pi$
$743$	29.6125	1.08638	0.543189	+	0.839611i	$0.317217\pi$
$744$	−6.00000	−0.219971
$745$	12.4187	0.454988
$746$	−20.8062	−0.761771
$747$	15.4031	0.563571
$748$	0	0
$749$	0	0
$750$	7.29844	0.266501
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	2.70156	0.0985158
$753$	−30.9109	−1.12646
$754$	−4.70156	−0.171221
$755$	27.2984	0.993492
$756$	0	0
$757$	32.8062	1.19236	0.596182	−	0.802850i	$-0.296684\pi$
$757$	32.8062	1.19236	0.596182	+	0.802850i	$0.296684\pi$
$758$	11.2984	0.410378
$759$	0	0
$760$	−20.0000	−0.725476
$761$	35.6125	1.29095	0.645476	−	0.763781i	$-0.276659\pi$
$761$	35.6125	1.29095	0.645476	+	0.763781i	$0.276659\pi$
$762$	−2.10469	−0.0762447
$763$	0	0
$764$	−2.80625	−0.101527
$765$	0	0
$766$	−18.8062	−0.679497
$767$	9.40312	0.339527
$768$	−1.00000	−0.0360844
$769$	9.89531	0.356834	0.178417	−	0.983955i	$-0.442902\pi$
$769$	9.89531	0.356834	0.178417	+	0.983955i	$0.442902\pi$
$770$	0	0
$771$	−27.6125	−0.994440
$772$	12.1047	0.435657
$773$	25.7172	0.924983	0.462491	−	0.886624i	$-0.346956\pi$
$773$	25.7172	0.924983	0.462491	+	0.886624i	$0.346956\pi$
$774$	0.701562	0.0252171
$775$	−13.7906	−0.495374
$776$	18.1047	0.649920
$777$	0	0
$778$	−11.1938	−0.401315
$779$	9.61250	0.344403
$780$	1.89531	0.0678631
$781$	0	0
$782$	0	0
$783$	−6.70156	−0.239494
$784$	0	0
$785$	−1.61250	−0.0575525
$786$	−2.59688	−0.0926275
$787$	39.8219	1.41950	0.709748	−	0.704455i	$-0.248809\pi$
$787$	39.8219	1.41950	0.709748	+	0.704455i	$0.248809\pi$
$788$	−25.5078	−0.908678
$789$	−15.5078	−0.552093
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7.01562	−0.249132
$794$	−32.2094	−1.14307
$795$	−9.19375	−0.326069
$796$	18.1047	0.641704
$797$	25.2984	0.896117	0.448058	−	0.894004i	$-0.352116\pi$
$797$	25.2984	0.896117	0.448058	+	0.894004i	$0.352116\pi$
$798$	0	0
$799$	0	0
$800$	−2.29844	−0.0812621
$801$	2.59688	0.0917561
$802$	−3.19375	−0.112775
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	4.20937	0.148269
$807$	9.40312	0.331006
$808$	17.4031	0.612240
$809$	−8.59688	−0.302250	−0.151125	−	0.988515i	$-0.548290\pi$
$809$	−8.59688	−0.302250	−0.151125	+	0.988515i	$0.548290\pi$
$810$	2.70156	0.0949232
$811$	46.3141	1.62631	0.813153	−	0.582050i	$-0.197749\pi$
$811$	46.3141	1.62631	0.813153	+	0.582050i	$0.197749\pi$
$812$	0	0
$813$	4.59688	0.161219
$814$	0	0
$815$	32.4187	1.13558
$816$	0	0
$817$	5.19375	0.181706
$818$	6.20937	0.217106
$819$	0	0
$820$	3.50781	0.122498
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	20.1047	0.701231
$823$	−23.7172	−0.826729	−0.413365	−	0.910566i	$-0.635646\pi$
$823$	−23.7172	−0.826729	−0.413365	+	0.910566i	$0.635646\pi$
$824$	2.10469	0.0733202
$825$	0	0
$826$	0	0
$827$	31.0156	1.07852	0.539259	−	0.842140i	$-0.318704\pi$
$827$	31.0156	1.07852	0.539259	+	0.842140i	$0.318704\pi$
$828$	−1.00000	−0.0347524
$829$	−8.20937	−0.285123	−0.142562	−	0.989786i	$-0.545534\pi$
$829$	−8.20937	−0.285123	−0.142562	+	0.989786i	$0.545534\pi$
$830$	41.6125	1.44439
$831$	0.596876	0.0207054
$832$	0.701562	0.0243223
$833$	0	0
$834$	1.89531	0.0656293
$835$	−23.2250	−0.803734
$836$	0	0
$837$	6.00000	0.207390
$838$	−15.4031	−0.532092
$839$	6.59688	0.227749	0.113875	−	0.993495i	$-0.463674\pi$
$839$	6.59688	0.227749	0.113875	+	0.993495i	$0.463674\pi$
$840$	0	0
$841$	15.9109	0.548653
$842$	−9.29844	−0.320445
$843$	−10.7016	−0.368581
$844$	−12.0000	−0.413057
$845$	33.7906	1.16243
$846$	−2.70156	−0.0928816
$847$	0	0
$848$	−3.40312	−0.116864
$849$	−10.0000	−0.343199
$850$	0	0
$851$	2.70156	0.0926084
$852$	5.40312	0.185108
$853$	−21.8953	−0.749681	−0.374841	−	0.927089i	$-0.622303\pi$
$853$	−21.8953	−0.749681	−0.374841	+	0.927089i	$0.622303\pi$
$854$	0	0
$855$	20.0000	0.683986
$856$	10.8062	0.369350
$857$	−46.9109	−1.60245	−0.801224	−	0.598365i	$-0.795817\pi$
$857$	−46.9109	−1.60245	−0.801224	+	0.598365i	$0.795817\pi$
$858$	0	0
$859$	−40.9109	−1.39586	−0.697932	−	0.716164i	$-0.745896\pi$
$859$	−40.9109	−1.39586	−0.697932	+	0.716164i	$0.745896\pi$
$860$	1.89531	0.0646297
$861$	0	0
$862$	14.1047	0.480408
$863$	14.8062	0.504011	0.252005	−	0.967726i	$-0.418910\pi$
$863$	14.8062	0.504011	0.252005	+	0.967726i	$0.418910\pi$
$864$	1.00000	0.0340207
$865$	25.4031	0.863732
$866$	−2.10469	−0.0715202
$867$	17.0000	0.577350
$868$	0	0
$869$	0	0
$870$	−18.1047	−0.613806
$871$	2.80625	0.0950861
$872$	−9.29844	−0.314885
$873$	−18.1047	−0.612751
$874$	−7.40312	−0.250414
$875$	0	0
$876$	7.40312	0.250128
$877$	−9.79063	−0.330606	−0.165303	−	0.986243i	$-0.552860\pi$
$877$	−9.79063	−0.330606	−0.165303	+	0.986243i	$0.552860\pi$
$878$	28.8062	0.972164
$879$	−28.8062	−0.971611
$880$	0	0
$881$	−28.2094	−0.950398	−0.475199	−	0.879878i	$-0.657624\pi$
$881$	−28.2094	−0.950398	−0.475199	+	0.879878i	$0.657624\pi$
$882$	0	0
$883$	29.4031	0.989494	0.494747	−	0.869037i	$-0.335261\pi$
$883$	29.4031	0.989494	0.494747	+	0.869037i	$0.335261\pi$
$884$	0	0
$885$	36.2094	1.21716
$886$	2.10469	0.0707083
$887$	13.7906	0.463044	0.231522	−	0.972830i	$-0.425629\pi$
$887$	13.7906	0.463044	0.231522	+	0.972830i	$0.425629\pi$
$888$	−2.70156	−0.0906585
$889$	0	0
$890$	7.01562	0.235164
$891$	0	0
$892$	−4.80625	−0.160925
$893$	−20.0000	−0.669274
$894$	−4.59688	−0.153743
$895$	1.89531	0.0633533
$896$	0	0
$897$	0.701562	0.0234245
$898$	23.6125	0.787959
$899$	−40.2094	−1.34106
$900$	2.29844	0.0766146
$901$	0	0
$902$	0	0
$903$	0	0
$904$	10.7016	0.355929
$905$	−23.7906	−0.790827
$906$	−10.1047	−0.335706
$907$	−22.1047	−0.733974	−0.366987	−	0.930226i	$-0.619611\pi$
$907$	−22.1047	−0.733974	−0.366987	+	0.930226i	$0.619611\pi$
$908$	−5.50781	−0.182783
$909$	−17.4031	−0.577225
$910$	0	0
$911$	−4.70156	−0.155770	−0.0778849	−	0.996962i	$-0.524817\pi$
$911$	−4.70156	−0.155770	−0.0778849	+	0.996962i	$0.524817\pi$
$912$	7.40312	0.245142
$913$	0	0
$914$	−22.2094	−0.734621
$915$	−27.0156	−0.893109
$916$	10.2094	0.337327
$917$	0	0
$918$	0	0
$919$	26.8062	0.884257	0.442128	−	0.896952i	$-0.354224\pi$
$919$	26.8062	0.884257	0.442128	+	0.896952i	$0.354224\pi$
$920$	−2.70156	−0.0890679
$921$	−14.1047	−0.464765
$922$	14.5969	0.480723
$923$	−3.79063	−0.124770
$924$	0	0
$925$	−6.20937	−0.204163
$926$	−6.10469	−0.200612
$927$	−2.10469	−0.0691270
$928$	−6.70156	−0.219990
$929$	−17.2984	−0.567543	−0.283772	−	0.958892i	$-0.591586\pi$
$929$	−17.2984	−0.567543	−0.283772	+	0.958892i	$0.591586\pi$
$930$	16.2094	0.531526
$931$	0	0
$932$	3.19375	0.104615
$933$	22.2094	0.727102
$934$	−14.7016	−0.481050
$935$	0	0
$936$	−0.701562	−0.0229313
$937$	10.3141	0.336946	0.168473	−	0.985706i	$-0.446116\pi$
$937$	10.3141	0.336946	0.168473	+	0.985706i	$0.446116\pi$
$938$	0	0
$939$	−17.4031	−0.567929
$940$	−7.29844	−0.238049
$941$	−48.3141	−1.57499	−0.787497	−	0.616319i	$-0.788623\pi$
$941$	−48.3141	−1.57499	−0.787497	+	0.616319i	$0.788623\pi$
$942$	0.596876	0.0194473
$943$	1.29844	0.0422830
$944$	13.4031	0.436235
$945$	0	0
$946$	0	0
$947$	35.5078	1.15385	0.576924	−	0.816798i	$-0.304253\pi$
$947$	35.5078	1.15385	0.576924	+	0.816798i	$0.304253\pi$
$948$	0	0
$949$	−5.19375	−0.168596
$950$	17.0156	0.552060
$951$	−1.29844	−0.0421047
$952$	0	0
$953$	4.80625	0.155690	0.0778448	−	0.996965i	$-0.475196\pi$
$953$	4.80625	0.155690	0.0778448	+	0.996965i	$0.475196\pi$
$954$	3.40312	0.110180
$955$	7.58125	0.245324
$956$	6.80625	0.220130
$957$	0	0
$958$	−5.40312	−0.174567
$959$	0	0
$960$	2.70156	0.0871925
$961$	5.00000	0.161290
$962$	1.89531	0.0611074
$963$	−10.8062	−0.348226
$964$	14.1047	0.454281
$965$	−32.7016	−1.05270
$966$	0	0
$967$	−56.4187	−1.81430	−0.907152	−	0.420803i	$-0.861749\pi$
$967$	−56.4187	−1.81430	−0.907152	+	0.420803i	$0.861749\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−48.9109	−1.57044
$971$	20.5969	0.660985	0.330493	−	0.943809i	$-0.392785\pi$
$971$	20.5969	0.660985	0.330493	+	0.943809i	$0.392785\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−6.10469	−0.195607
$975$	−1.61250	−0.0516412
$976$	−10.0000	−0.320092
$977$	−46.7016	−1.49412	−0.747058	−	0.664759i	$-0.768534\pi$
$977$	−46.7016	−1.49412	−0.747058	+	0.664759i	$0.768534\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	0	0
$981$	9.29844	0.296876
$982$	−33.6125	−1.07262
$983$	−33.8219	−1.07875	−0.539375	−	0.842066i	$-0.681339\pi$
$983$	−33.8219	−1.07875	−0.539375	+	0.842066i	$0.681339\pi$
$984$	−1.29844	−0.0413927
$985$	68.9109	2.19568
$986$	0	0
$987$	0	0
$988$	−5.19375	−0.165235
$989$	0.701562	0.0223084
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	6.00000	0.190500
$993$	−33.6125	−1.06666
$994$	0	0
$995$	−48.9109	−1.55058
$996$	−15.4031	−0.488067
$997$	−61.4031	−1.94466	−0.972328	−	0.233619i	$-0.924943\pi$
$997$	−61.4031	−1.94466	−0.972328	+	0.233619i	$0.924943\pi$
$998$	27.0156	0.855165
$999$	2.70156	0.0854736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bq.1.1		2
7.6	odd	2		966.2.a.m.1.2	✓	2
21.20	even	2		2898.2.a.bc.1.1		2
28.27	even	2		7728.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.m.1.2	✓	2	7.6	odd	2
2898.2.a.bc.1.1		2	21.20	even	2
6762.2.a.bq.1.1		2	1.1	even	1	trivial
7728.2.a.z.1.2		2	28.27	even	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:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.70156 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.70156 q^{10} -1.00000 q^{12} +0.701562 q^{13} +2.70156 q^{15} +1.00000 q^{16} -1.00000 q^{18} -7.40312 q^{19} -2.70156 q^{20} -1.00000 q^{23} +1.00000 q^{24} +2.29844 q^{25} -0.701562 q^{26} -1.00000 q^{27} +6.70156 q^{29} -2.70156 q^{30} -6.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -2.70156 q^{37} +7.40312 q^{38} -0.701562 q^{39} +2.70156 q^{40} -1.29844 q^{41} -0.701562 q^{43} -2.70156 q^{45} +1.00000 q^{46} +2.70156 q^{47} -1.00000 q^{48} -2.29844 q^{50} +0.701562 q^{52} -3.40312 q^{53} +1.00000 q^{54} +7.40312 q^{57} -6.70156 q^{58} +13.4031 q^{59} +2.70156 q^{60} -10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -1.89531 q^{65} +4.00000 q^{67} +1.00000 q^{69} -5.40312 q^{71} -1.00000 q^{72} -7.40312 q^{73} +2.70156 q^{74} -2.29844 q^{75} -7.40312 q^{76} +0.701562 q^{78} -2.70156 q^{80} +1.00000 q^{81} +1.29844 q^{82} +15.4031 q^{83} +0.701562 q^{86} -6.70156 q^{87} +2.59688 q^{89} +2.70156 q^{90} -1.00000 q^{92} +6.00000 q^{93} -2.70156 q^{94} +20.0000 q^{95} +1.00000 q^{96} -18.1047 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} - q^{10} - 2 q^{12} - 5 q^{13} - q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{19} + q^{20} - 2 q^{23} + 2 q^{24} + 11 q^{25} + 5 q^{26}+ \cdots - 17 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 - 2 * q^8 + 2 * q^9 - q^10 - 2 * q^12 - 5 * q^13 - q^15 + 2 * q^16 - 2 * q^18 - 2 * q^19 + q^20 - 2 * q^23 + 2 * q^24 + 11 * q^25 + 5 * q^26 - 2 * q^27 + 7 * q^29 + q^30 - 12 * q^31 - 2 * q^32 + 2 * q^36 + q^37 + 2 * q^38 + 5 * q^39 - q^40 - 9 * q^41 + 5 * q^43 + q^45 + 2 * q^46 - q^47 - 2 * q^48 - 11 * q^50 - 5 * q^52 + 6 * q^53 + 2 * q^54 + 2 * q^57 - 7 * q^58 + 14 * q^59 - q^60 - 20 * q^61 + 12 * q^62 + 2 * q^64 - 23 * q^65 + 8 * q^67 + 2 * q^69 + 2 * q^71 - 2 * q^72 - 2 * q^73 - q^74 - 11 * q^75 - 2 * q^76 - 5 * q^78 + q^80 + 2 * q^81 + 9 * q^82 + 18 * q^83 - 5 * q^86 - 7 * q^87 + 18 * q^89 - q^90 - 2 * q^92 + 12 * q^93 + q^94 + 40 * q^95 + 2 * q^96 - 17 * q^97

Introduction

L-functions

Modular forms

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Groups

Database

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