LMFDB - Embedded newform 2790.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2790,2,Mod(1,2790)] 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("2790.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,2,0,-1,-2,0,-2,-5,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.53113 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.53113 q^{11} +6.00000 q^{13} -3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.53113 q^{19} +1.00000 q^{20} -1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{25} -6.00000 q^{26} +3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +3.53113 q^{35} +9.06226 q^{37} +3.53113 q^{38} -1.00000 q^{40} +9.06226 q^{41} -0.468871 q^{43} +1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} +5.46887 q^{49} -1.00000 q^{50} +6.00000 q^{52} -5.53113 q^{53} +1.53113 q^{55} -3.53113 q^{56} -7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -11.0623 q^{67} +4.00000 q^{68} -3.53113 q^{70} +4.46887 q^{71} -0.468871 q^{73} -9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -0.468871 q^{79} +1.00000 q^{80} -9.06226 q^{82} +8.00000 q^{83} +4.00000 q^{85} +0.468871 q^{86} -1.53113 q^{88} -1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} +11.0623 q^{94} -3.53113 q^{95} -16.1245 q^{97} -5.46887 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 5 q^{11} + 12 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + q^{19} + 2 q^{20} + 5 q^{22} - 5 q^{23} + 2 q^{25} - 12 q^{26} - q^{28} + 2 q^{31}+ \cdots - 19 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - q^7 - 2 * q^8 - 2 * q^10 - 5 * q^11 + 12 * q^13 + q^14 + 2 * q^16 + 8 * q^17 + q^19 + 2 * q^20 + 5 * q^22 - 5 * q^23 + 2 * q^25 - 12 * q^26 - q^28 + 2 * q^31 - 2 * q^32 - 8 * q^34 - q^35 + 2 * q^37 - q^38 - 2 * q^40 + 2 * q^41 - 9 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 19 * q^49 - 2 * q^50 + 12 * q^52 - 3 * q^53 - 5 * q^55 + q^56 + 2 * q^59 - 6 * q^61 - 2 * q^62 + 2 * q^64 + 12 * q^65 - 6 * q^67 + 8 * q^68 + q^70 + 17 * q^71 - 9 * q^73 - 2 * q^74 + q^76 + 35 * q^77 - 9 * q^79 + 2 * q^80 - 2 * q^82 + 16 * q^83 + 8 * q^85 + 9 * q^86 + 5 * q^88 + 5 * q^89 - 6 * q^91 - 5 * q^92 + 6 * q^94 + q^95 - 19 * q^98

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.53113	1.33464	0.667321	−	0.744771i	$-0.267441\pi$
$7$	3.53113	1.33464	0.667321	+	0.744771i	$0.267441\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.53113	0.461653	0.230826	−	0.972995i	$-0.425857\pi$
$11$	1.53113	0.461653	0.230826	+	0.972995i	$0.425857\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	−3.53113	−0.943734
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−3.53113	−0.810097	−0.405048	−	0.914295i	$-0.632745\pi$
$19$	−3.53113	−0.810097	−0.405048	+	0.914295i	$0.632745\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.53113	−0.326438
$23$	1.53113	0.319262	0.159631	−	0.987177i	$-0.448969\pi$
$23$	1.53113	0.319262	0.159631	+	0.987177i	$0.448969\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	3.53113	0.667321
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	3.53113	0.596870
$36$	0	0
$37$	9.06226	1.48983	0.744913	−	0.667162i	$-0.232491\pi$
$37$	9.06226	1.48983	0.744913	+	0.667162i	$0.232491\pi$
$38$	3.53113	0.572825
$39$	0	0
$40$	−1.00000	−0.158114
$41$	9.06226	1.41529	0.707643	−	0.706570i	$-0.249758\pi$
$41$	9.06226	1.41529	0.707643	+	0.706570i	$0.249758\pi$
$42$	0	0
$43$	−0.468871	−0.0715022	−0.0357511	−	0.999361i	$-0.511382\pi$
$43$	−0.468871	−0.0715022	−0.0357511	+	0.999361i	$0.511382\pi$
$44$	1.53113	0.230826
$45$	0	0
$46$	−1.53113	−0.225753
$47$	−11.0623	−1.61360	−0.806798	−	0.590827i	$-0.798801\pi$
$47$	−11.0623	−1.61360	−0.806798	+	0.590827i	$0.798801\pi$
$48$	0	0
$49$	5.46887	0.781267
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	−5.53113	−0.759759	−0.379879	−	0.925036i	$-0.624035\pi$
$53$	−5.53113	−0.759759	−0.379879	+	0.925036i	$0.624035\pi$
$54$	0	0
$55$	1.53113	0.206457
$56$	−3.53113	−0.471867
$57$	0	0
$58$	0	0
$59$	−7.06226	−0.919428	−0.459714	−	0.888067i	$-0.652048\pi$
$59$	−7.06226	−0.919428	−0.459714	+	0.888067i	$0.652048\pi$
$60$	0	0
$61$	−11.0623	−1.41638	−0.708188	−	0.706023i	$-0.750487\pi$
$61$	−11.0623	−1.41638	−0.708188	+	0.706023i	$0.750487\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	−11.0623	−1.35147	−0.675735	−	0.737145i	$-0.736174\pi$
$67$	−11.0623	−1.35147	−0.675735	+	0.737145i	$0.736174\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−3.53113	−0.422051
$71$	4.46887	0.530357	0.265179	−	0.964199i	$-0.414569\pi$
$71$	4.46887	0.530357	0.265179	+	0.964199i	$0.414569\pi$
$72$	0	0
$73$	−0.468871	−0.0548772	−0.0274386	−	0.999623i	$-0.508735\pi$
$73$	−0.468871	−0.0548772	−0.0274386	+	0.999623i	$0.508735\pi$
$74$	−9.06226	−1.05347
$75$	0	0
$76$	−3.53113	−0.405048
$77$	5.40661	0.616141
$78$	0	0
$79$	−0.468871	−0.0527521	−0.0263761	−	0.999652i	$-0.508397\pi$
$79$	−0.468871	−0.0527521	−0.0263761	+	0.999652i	$0.508397\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.06226	−1.00076
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0.468871	0.0505597
$87$	0	0
$88$	−1.53113	−0.163219
$89$	−1.53113	−0.162299	−0.0811497	−	0.996702i	$-0.525859\pi$
$89$	−1.53113	−0.162299	−0.0811497	+	0.996702i	$0.525859\pi$
$90$	0	0
$91$	21.1868	2.22098
$92$	1.53113	0.159631
$93$	0	0
$94$	11.0623	1.14098
$95$	−3.53113	−0.362286
$96$	0	0
$97$	−16.1245	−1.63720	−0.818598	−	0.574367i	$-0.805248\pi$
$97$	−16.1245	−1.63720	−0.818598	+	0.574367i	$0.805248\pi$
$98$	−5.46887	−0.552439
$99$	0	0
$100$	1.00000	0.100000
$101$	17.5311	1.74441	0.872206	−	0.489138i	$-0.162689\pi$
$101$	17.5311	1.74441	0.872206	+	0.489138i	$0.162689\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	5.53113	0.537231
$107$	14.5934	1.41080	0.705398	−	0.708811i	$-0.250768\pi$
$107$	14.5934	1.41080	0.705398	+	0.708811i	$0.250768\pi$
$108$	0	0
$109$	−1.06226	−0.101746	−0.0508729	−	0.998705i	$-0.516200\pi$
$109$	−1.06226	−0.101746	−0.0508729	+	0.998705i	$0.516200\pi$
$110$	−1.53113	−0.145987
$111$	0	0
$112$	3.53113	0.333660
$113$	−16.5934	−1.56097	−0.780487	−	0.625172i	$-0.785029\pi$
$113$	−16.5934	−1.56097	−0.780487	+	0.625172i	$0.785029\pi$
$114$	0	0
$115$	1.53113	0.142779
$116$	0	0
$117$	0	0
$118$	7.06226	0.650134
$119$	14.1245	1.29479
$120$	0	0
$121$	−8.65564	−0.786877
$122$	11.0623	1.00153
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.06226	−0.804145	−0.402073	−	0.915608i	$-0.631710\pi$
$127$	−9.06226	−0.804145	−0.402073	+	0.915608i	$0.631710\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.00000	−0.526235
$131$	−7.06226	−0.617032	−0.308516	−	0.951219i	$-0.599832\pi$
$131$	−7.06226	−0.617032	−0.308516	+	0.951219i	$0.599832\pi$
$132$	0	0
$133$	−12.4689	−1.08119
$134$	11.0623	0.955634
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−0.937742	−0.0801167	−0.0400584	−	0.999197i	$-0.512754\pi$
$137$	−0.937742	−0.0801167	−0.0400584	+	0.999197i	$0.512754\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	3.53113	0.298435
$141$	0	0
$142$	−4.46887	−0.375019
$143$	9.18677	0.768237
$144$	0	0
$145$	0	0
$146$	0.468871	0.0388041
$147$	0	0
$148$	9.06226	0.744913
$149$	10.4689	0.857643	0.428822	−	0.903389i	$-0.358929\pi$
$149$	10.4689	0.857643	0.428822	+	0.903389i	$0.358929\pi$
$150$	0	0
$151$	−18.1245	−1.47495	−0.737476	−	0.675373i	$-0.763983\pi$
$151$	−18.1245	−1.47495	−0.737476	+	0.675373i	$0.763983\pi$
$152$	3.53113	0.286412
$153$	0	0
$154$	−5.40661	−0.435677
$155$	1.00000	0.0803219
$156$	0	0
$157$	−14.4689	−1.15474	−0.577371	−	0.816482i	$-0.695921\pi$
$157$	−14.4689	−1.15474	−0.577371	+	0.816482i	$0.695921\pi$
$158$	0.468871	0.0373014
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	5.40661	0.426101
$162$	0	0
$163$	−11.0623	−0.866463	−0.433231	−	0.901283i	$-0.642627\pi$
$163$	−11.0623	−0.866463	−0.433231	+	0.901283i	$0.642627\pi$
$164$	9.06226	0.707643
$165$	0	0
$166$	−8.00000	−0.620920
$167$	−0.593387	−0.0459176	−0.0229588	−	0.999736i	$-0.507309\pi$
$167$	−0.593387	−0.0459176	−0.0229588	+	0.999736i	$0.507309\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−0.468871	−0.0357511
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	3.53113	0.266928
$176$	1.53113	0.115413
$177$	0	0
$178$	1.53113	0.114763
$179$	−13.0623	−0.976319	−0.488159	−	0.872754i	$-0.662332\pi$
$179$	−13.0623	−0.976319	−0.488159	+	0.872754i	$0.662332\pi$
$180$	0	0
$181$	26.5934	1.97667	0.988335	−	0.152293i	$-0.0486657\pi$
$181$	26.5934	1.97667	0.988335	+	0.152293i	$0.0486657\pi$
$182$	−21.1868	−1.57047
$183$	0	0
$184$	−1.53113	−0.112876
$185$	9.06226	0.666270
$186$	0	0
$187$	6.12452	0.447869
$188$	−11.0623	−0.806798
$189$	0	0
$190$	3.53113	0.256175
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	16.1245	1.15767
$195$	0	0
$196$	5.46887	0.390634
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−0.468871	−0.0332374	−0.0166187	−	0.999862i	$-0.505290\pi$
$199$	−0.468871	−0.0332374	−0.0166187	+	0.999862i	$0.505290\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−17.5311	−1.23349
$203$	0	0
$204$	0	0
$205$	9.06226	0.632936
$206$	0	0
$207$	0	0
$208$	6.00000	0.416025
$209$	−5.40661	−0.373983
$210$	0	0
$211$	22.5934	1.55539	0.777696	−	0.628640i	$-0.216388\pi$
$211$	22.5934	1.55539	0.777696	+	0.628640i	$0.216388\pi$
$212$	−5.53113	−0.379879
$213$	0	0
$214$	−14.5934	−0.997583
$215$	−0.468871	−0.0319767
$216$	0	0
$217$	3.53113	0.239709
$218$	1.06226	0.0719452
$219$	0	0
$220$	1.53113	0.103229
$221$	24.0000	1.61441
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−3.53113	−0.235933
$225$	0	0
$226$	16.5934	1.10378
$227$	16.4689	1.09308	0.546539	−	0.837434i	$-0.315945\pi$
$227$	16.4689	1.09308	0.546539	+	0.837434i	$0.315945\pi$
$228$	0	0
$229$	18.5934	1.22869	0.614343	−	0.789039i	$-0.289421\pi$
$229$	18.5934	1.22869	0.614343	+	0.789039i	$0.289421\pi$
$230$	−1.53113	−0.100960
$231$	0	0
$232$	0	0
$233$	−9.53113	−0.624405	−0.312203	−	0.950016i	$-0.601067\pi$
$233$	−9.53113	−0.624405	−0.312203	+	0.950016i	$0.601067\pi$
$234$	0	0
$235$	−11.0623	−0.721622
$236$	−7.06226	−0.459714
$237$	0	0
$238$	−14.1245	−0.915556
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	8.65564	0.556406
$243$	0	0
$244$	−11.0623	−0.708188
$245$	5.46887	0.349393
$246$	0	0
$247$	−21.1868	−1.34808
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	13.0623	0.824482	0.412241	−	0.911075i	$-0.364746\pi$
$251$	13.0623	0.824482	0.412241	+	0.911075i	$0.364746\pi$
$252$	0	0
$253$	2.34436	0.147388
$254$	9.06226	0.568617
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.6556	1.72511	0.862556	−	0.505962i	$-0.168862\pi$
$257$	27.6556	1.72511	0.862556	+	0.505962i	$0.168862\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	6.00000	0.372104
$261$	0	0
$262$	7.06226	0.436308
$263$	6.93774	0.427800	0.213900	−	0.976856i	$-0.431383\pi$
$263$	6.93774	0.427800	0.213900	+	0.976856i	$0.431383\pi$
$264$	0	0
$265$	−5.53113	−0.339775
$266$	12.4689	0.764516
$267$	0	0
$268$	−11.0623	−0.675735
$269$	−29.1868	−1.77955	−0.889774	−	0.456400i	$-0.849138\pi$
$269$	−29.1868	−1.77955	−0.889774	+	0.456400i	$0.849138\pi$
$270$	0	0
$271$	−19.5311	−1.18643	−0.593216	−	0.805043i	$-0.702142\pi$
$271$	−19.5311	−1.18643	−0.593216	+	0.805043i	$0.702142\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	0.937742	0.0566511
$275$	1.53113	0.0923305
$276$	0	0
$277$	1.06226	0.0638249	0.0319124	−	0.999491i	$-0.489840\pi$
$277$	1.06226	0.0638249	0.0319124	+	0.999491i	$0.489840\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	−3.53113	−0.211025
$281$	1.06226	0.0633690	0.0316845	−	0.999498i	$-0.489913\pi$
$281$	1.06226	0.0633690	0.0316845	+	0.999498i	$0.489913\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	4.46887	0.265179
$285$	0	0
$286$	−9.18677	−0.543225
$287$	32.0000	1.88890
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	−0.468871	−0.0274386
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−7.06226	−0.411181
$296$	−9.06226	−0.526733
$297$	0	0
$298$	−10.4689	−0.606445
$299$	9.18677	0.531285
$300$	0	0
$301$	−1.65564	−0.0954298
$302$	18.1245	1.04295
$303$	0	0
$304$	−3.53113	−0.202524
$305$	−11.0623	−0.633423
$306$	0	0
$307$	2.12452	0.121253	0.0606263	−	0.998161i	$-0.480690\pi$
$307$	2.12452	0.121253	0.0606263	+	0.998161i	$0.480690\pi$
$308$	5.40661	0.308070
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	27.0623	1.52965	0.764825	−	0.644239i	$-0.222826\pi$
$313$	27.0623	1.52965	0.764825	+	0.644239i	$0.222826\pi$
$314$	14.4689	0.816526
$315$	0	0
$316$	−0.468871	−0.0263761
$317$	−29.0623	−1.63230	−0.816150	−	0.577841i	$-0.803895\pi$
$317$	−29.0623	−1.63230	−0.816150	+	0.577841i	$0.803895\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	−5.40661	−0.301299
$323$	−14.1245	−0.785909
$324$	0	0
$325$	6.00000	0.332820
$326$	11.0623	0.612682
$327$	0	0
$328$	−9.06226	−0.500379
$329$	−39.0623	−2.15357
$330$	0	0
$331$	29.0623	1.59741	0.798703	−	0.601725i	$-0.205520\pi$
$331$	29.0623	1.59741	0.798703	+	0.601725i	$0.205520\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	0.593387	0.0324687
$335$	−11.0623	−0.604396
$336$	0	0
$337$	29.1868	1.58990	0.794952	−	0.606672i	$-0.207496\pi$
$337$	29.1868	1.58990	0.794952	+	0.606672i	$0.207496\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	4.00000	0.216930
$341$	1.53113	0.0829153
$342$	0	0
$343$	−5.40661	−0.291930
$344$	0.468871	0.0252798
$345$	0	0
$346$	14.0000	0.752645
$347$	22.1245	1.18771	0.593853	−	0.804573i	$-0.297606\pi$
$347$	22.1245	1.18771	0.593853	+	0.804573i	$0.297606\pi$
$348$	0	0
$349$	25.0623	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$349$	25.0623	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$350$	−3.53113	−0.188747
$351$	0	0
$352$	−1.53113	−0.0816094
$353$	23.0623	1.22748	0.613740	−	0.789508i	$-0.289664\pi$
$353$	23.0623	1.22748	0.613740	+	0.789508i	$0.289664\pi$
$354$	0	0
$355$	4.46887	0.237183
$356$	−1.53113	−0.0811497
$357$	0	0
$358$	13.0623	0.690362
$359$	−11.5311	−0.608590	−0.304295	−	0.952578i	$-0.598421\pi$
$359$	−11.5311	−0.608590	−0.304295	+	0.952578i	$0.598421\pi$
$360$	0	0
$361$	−6.53113	−0.343744
$362$	−26.5934	−1.39772
$363$	0	0
$364$	21.1868	1.11049
$365$	−0.468871	−0.0245418
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	1.53113	0.0798156
$369$	0	0
$370$	−9.06226	−0.471124
$371$	−19.5311	−1.01401
$372$	0	0
$373$	−10.4689	−0.542058	−0.271029	−	0.962571i	$-0.587364\pi$
$373$	−10.4689	−0.542058	−0.271029	+	0.962571i	$0.587364\pi$
$374$	−6.12452	−0.316691
$375$	0	0
$376$	11.0623	0.570492
$377$	0	0
$378$	0	0
$379$	−2.59339	−0.133213	−0.0666067	−	0.997779i	$-0.521217\pi$
$379$	−2.59339	−0.133213	−0.0666067	+	0.997779i	$0.521217\pi$
$380$	−3.53113	−0.181143
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−13.0623	−0.667450	−0.333725	−	0.942670i	$-0.608306\pi$
$383$	−13.0623	−0.667450	−0.333725	+	0.942670i	$0.608306\pi$
$384$	0	0
$385$	5.40661	0.275547
$386$	−14.0000	−0.712581
$387$	0	0
$388$	−16.1245	−0.818598
$389$	−27.0623	−1.37211	−0.686055	−	0.727549i	$-0.740659\pi$
$389$	−27.0623	−1.37211	−0.686055	+	0.727549i	$0.740659\pi$
$390$	0	0
$391$	6.12452	0.309730
$392$	−5.46887	−0.276220
$393$	0	0
$394$	6.00000	0.302276
$395$	−0.468871	−0.0235915
$396$	0	0
$397$	−18.7179	−0.939425	−0.469712	−	0.882820i	$-0.655642\pi$
$397$	−18.7179	−0.939425	−0.469712	+	0.882820i	$0.655642\pi$
$398$	0.468871	0.0235024
$399$	0	0
$400$	1.00000	0.0500000
$401$	34.7179	1.73373	0.866865	−	0.498544i	$-0.166132\pi$
$401$	34.7179	1.73373	0.866865	+	0.498544i	$0.166132\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	17.5311	0.872206
$405$	0	0
$406$	0	0
$407$	13.8755	0.687782
$408$	0	0
$409$	−9.06226	−0.448100	−0.224050	−	0.974578i	$-0.571928\pi$
$409$	−9.06226	−0.448100	−0.224050	+	0.974578i	$0.571928\pi$
$410$	−9.06226	−0.447553
$411$	0	0
$412$	0	0
$413$	−24.9377	−1.22711
$414$	0	0
$415$	8.00000	0.392705
$416$	−6.00000	−0.294174
$417$	0	0
$418$	5.40661	0.264446
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	38.2490	1.86414	0.932072	−	0.362273i	$-0.117999\pi$
$421$	38.2490	1.86414	0.932072	+	0.362273i	$0.117999\pi$
$422$	−22.5934	−1.09983
$423$	0	0
$424$	5.53113	0.268615
$425$	4.00000	0.194029
$426$	0	0
$427$	−39.0623	−1.89036
$428$	14.5934	0.705398
$429$	0	0
$430$	0.468871	0.0226110
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−1.40661	−0.0675975	−0.0337988	−	0.999429i	$-0.510761\pi$
$433$	−1.40661	−0.0675975	−0.0337988	+	0.999429i	$0.510761\pi$
$434$	−3.53113	−0.169500
$435$	0	0
$436$	−1.06226	−0.0508729
$437$	−5.40661	−0.258633
$438$	0	0
$439$	−8.93774	−0.426575	−0.213288	−	0.976989i	$-0.568417\pi$
$439$	−8.93774	−0.426575	−0.213288	+	0.976989i	$0.568417\pi$
$440$	−1.53113	−0.0729937
$441$	0	0
$442$	−24.0000	−1.14156
$443$	−38.5934	−1.83363	−0.916814	−	0.399316i	$-0.869248\pi$
$443$	−38.5934	−1.83363	−0.916814	+	0.399316i	$0.869248\pi$
$444$	0	0
$445$	−1.53113	−0.0725825
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	3.53113	0.166830
$449$	−28.1245	−1.32728	−0.663639	−	0.748053i	$-0.730989\pi$
$449$	−28.1245	−1.32728	−0.663639	+	0.748053i	$0.730989\pi$
$450$	0	0
$451$	13.8755	0.653371
$452$	−16.5934	−0.780487
$453$	0	0
$454$	−16.4689	−0.772922
$455$	21.1868	0.993251
$456$	0	0
$457$	31.0623	1.45303	0.726516	−	0.687150i	$-0.241138\pi$
$457$	31.0623	1.45303	0.726516	+	0.687150i	$0.241138\pi$
$458$	−18.5934	−0.868812
$459$	0	0
$460$	1.53113	0.0713893
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	−35.1868	−1.63527	−0.817634	−	0.575738i	$-0.804715\pi$
$463$	−35.1868	−1.63527	−0.817634	+	0.575738i	$0.804715\pi$
$464$	0	0
$465$	0	0
$466$	9.53113	0.441521
$467$	−10.1245	−0.468507	−0.234253	−	0.972176i	$-0.575265\pi$
$467$	−10.1245	−0.468507	−0.234253	+	0.972176i	$0.575265\pi$
$468$	0	0
$469$	−39.0623	−1.80373
$470$	11.0623	0.510264
$471$	0	0
$472$	7.06226	0.325067
$473$	−0.717902	−0.0330092
$474$	0	0
$475$	−3.53113	−0.162019
$476$	14.1245	0.647396
$477$	0	0
$478$	−8.00000	−0.365911
$479$	−41.6556	−1.90329	−0.951647	−	0.307192i	$-0.900611\pi$
$479$	−41.6556	−1.90329	−0.951647	+	0.307192i	$0.900611\pi$
$480$	0	0
$481$	54.3735	2.47922
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−8.65564	−0.393438
$485$	−16.1245	−0.732177
$486$	0	0
$487$	6.93774	0.314379	0.157190	−	0.987568i	$-0.449757\pi$
$487$	6.93774	0.314379	0.157190	+	0.987568i	$0.449757\pi$
$488$	11.0623	0.500765
$489$	0	0
$490$	−5.46887	−0.247058
$491$	−16.5934	−0.748849	−0.374425	−	0.927257i	$-0.622160\pi$
$491$	−16.5934	−0.748849	−0.374425	+	0.927257i	$0.622160\pi$
$492$	0	0
$493$	0	0
$494$	21.1868	0.953238
$495$	0	0
$496$	1.00000	0.0449013
$497$	15.7802	0.707837
$498$	0	0
$499$	9.06226	0.405682	0.202841	−	0.979212i	$-0.434982\pi$
$499$	9.06226	0.405682	0.202841	+	0.979212i	$0.434982\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−13.0623	−0.582997
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	17.5311	0.780125
$506$	−2.34436	−0.104219
$507$	0	0
$508$	−9.06226	−0.402073
$509$	−5.87548	−0.260426	−0.130213	−	0.991486i	$-0.541566\pi$
$509$	−5.87548	−0.260426	−0.130213	+	0.991486i	$0.541566\pi$
$510$	0	0
$511$	−1.65564	−0.0732414
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−27.6556	−1.21984
$515$	0	0
$516$	0	0
$517$	−16.9377	−0.744921
$518$	−32.0000	−1.40600
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−20.1245	−0.881671	−0.440836	−	0.897588i	$-0.645318\pi$
$521$	−20.1245	−0.881671	−0.440836	+	0.897588i	$0.645318\pi$
$522$	0	0
$523$	−14.5934	−0.638124	−0.319062	−	0.947734i	$-0.603368\pi$
$523$	−14.5934	−0.638124	−0.319062	+	0.947734i	$0.603368\pi$
$524$	−7.06226	−0.308516
$525$	0	0
$526$	−6.93774	−0.302500
$527$	4.00000	0.174243
$528$	0	0
$529$	−20.6556	−0.898071
$530$	5.53113	0.240257
$531$	0	0
$532$	−12.4689	−0.540594
$533$	54.3735	2.35518
$534$	0	0
$535$	14.5934	0.630927
$536$	11.0623	0.477817
$537$	0	0
$538$	29.1868	1.25833
$539$	8.37355	0.360674
$540$	0	0
$541$	28.1245	1.20917	0.604584	−	0.796542i	$-0.293340\pi$
$541$	28.1245	1.20917	0.604584	+	0.796542i	$0.293340\pi$
$542$	19.5311	0.838934
$543$	0	0
$544$	−4.00000	−0.171499
$545$	−1.06226	−0.0455021
$546$	0	0
$547$	41.1868	1.76102	0.880510	−	0.474028i	$-0.157201\pi$
$547$	41.1868	1.76102	0.880510	+	0.474028i	$0.157201\pi$
$548$	−0.937742	−0.0400584
$549$	0	0
$550$	−1.53113	−0.0652876
$551$	0	0
$552$	0	0
$553$	−1.65564	−0.0704052
$554$	−1.06226	−0.0451310
$555$	0	0
$556$	6.00000	0.254457
$557$	−45.7802	−1.93977	−0.969884	−	0.243568i	$-0.921682\pi$
$557$	−45.7802	−1.93977	−0.969884	+	0.243568i	$0.921682\pi$
$558$	0	0
$559$	−2.81323	−0.118987
$560$	3.53113	0.149217
$561$	0	0
$562$	−1.06226	−0.0448086
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−16.5934	−0.698089
$566$	12.0000	0.504398
$567$	0	0
$568$	−4.46887	−0.187510
$569$	−17.5311	−0.734943	−0.367472	−	0.930035i	$-0.619776\pi$
$569$	−17.5311	−0.734943	−0.367472	+	0.930035i	$0.619776\pi$
$570$	0	0
$571$	3.87548	0.162184	0.0810920	−	0.996707i	$-0.474159\pi$
$571$	3.87548	0.162184	0.0810920	+	0.996707i	$0.474159\pi$
$572$	9.18677	0.384118
$573$	0	0
$574$	−32.0000	−1.33565
$575$	1.53113	0.0638525
$576$	0	0
$577$	−11.1868	−0.465711	−0.232856	−	0.972511i	$-0.574807\pi$
$577$	−11.1868	−0.465711	−0.232856	+	0.972511i	$0.574807\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	28.2490	1.17197
$582$	0	0
$583$	−8.46887	−0.350745
$584$	0.468871	0.0194020
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−20.2490	−0.835767	−0.417883	−	0.908501i	$-0.637228\pi$
$587$	−20.2490	−0.835767	−0.417883	+	0.908501i	$0.637228\pi$
$588$	0	0
$589$	−3.53113	−0.145498
$590$	7.06226	0.290749
$591$	0	0
$592$	9.06226	0.372456
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	14.1245	0.579049
$596$	10.4689	0.428822
$597$	0	0
$598$	−9.18677	−0.375675
$599$	−10.5934	−0.432834	−0.216417	−	0.976301i	$-0.569437\pi$
$599$	−10.5934	−0.432834	−0.216417	+	0.976301i	$0.569437\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	1.65564	0.0674790
$603$	0	0
$604$	−18.1245	−0.737476
$605$	−8.65564	−0.351902
$606$	0	0
$607$	−38.5934	−1.56646	−0.783229	−	0.621734i	$-0.786429\pi$
$607$	−38.5934	−1.56646	−0.783229	+	0.621734i	$0.786429\pi$
$608$	3.53113	0.143206
$609$	0	0
$610$	11.0623	0.447898
$611$	−66.3735	−2.68519
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	−2.12452	−0.0857385
$615$	0	0
$616$	−5.40661	−0.217839
$617$	−20.5934	−0.829059	−0.414529	−	0.910036i	$-0.636054\pi$
$617$	−20.5934	−0.829059	−0.414529	+	0.910036i	$0.636054\pi$
$618$	0	0
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−5.40661	−0.216611
$624$	0	0
$625$	1.00000	0.0400000
$626$	−27.0623	−1.08163
$627$	0	0
$628$	−14.4689	−0.577371
$629$	36.2490	1.44534
$630$	0	0
$631$	−9.65564	−0.384385	−0.192193	−	0.981357i	$-0.561560\pi$
$631$	−9.65564	−0.384385	−0.192193	+	0.981357i	$0.561560\pi$
$632$	0.468871	0.0186507
$633$	0	0
$634$	29.0623	1.15421
$635$	−9.06226	−0.359625
$636$	0	0
$637$	32.8132	1.30011
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	6.59339	0.260018	0.130009	−	0.991513i	$-0.458499\pi$
$643$	6.59339	0.260018	0.130009	+	0.991513i	$0.458499\pi$
$644$	5.40661	0.213050
$645$	0	0
$646$	14.1245	0.555722
$647$	−1.53113	−0.0601949	−0.0300974	−	0.999547i	$-0.509582\pi$
$647$	−1.53113	−0.0601949	−0.0300974	+	0.999547i	$0.509582\pi$
$648$	0	0
$649$	−10.8132	−0.424456
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−11.0623	−0.433231
$653$	26.2490	1.02720	0.513602	−	0.858029i	$-0.328311\pi$
$653$	26.2490	1.02720	0.513602	+	0.858029i	$0.328311\pi$
$654$	0	0
$655$	−7.06226	−0.275945
$656$	9.06226	0.353822
$657$	0	0
$658$	39.0623	1.52281
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−29.0623	−1.12954
$663$	0	0
$664$	−8.00000	−0.310460
$665$	−12.4689	−0.483522
$666$	0	0
$667$	0	0
$668$	−0.593387	−0.0229588
$669$	0	0
$670$	11.0623	0.427372
$671$	−16.9377	−0.653874
$672$	0	0
$673$	7.06226	0.272230	0.136115	−	0.990693i	$-0.456538\pi$
$673$	7.06226	0.272230	0.136115	+	0.990693i	$0.456538\pi$
$674$	−29.1868	−1.12423
$675$	0	0
$676$	23.0000	0.884615
$677$	−3.40661	−0.130927	−0.0654634	−	0.997855i	$-0.520853\pi$
$677$	−3.40661	−0.130927	−0.0654634	+	0.997855i	$0.520853\pi$
$678$	0	0
$679$	−56.9377	−2.18507
$680$	−4.00000	−0.153393
$681$	0	0
$682$	−1.53113	−0.0586300
$683$	−15.5311	−0.594282	−0.297141	−	0.954834i	$-0.596033\pi$
$683$	−15.5311	−0.594282	−0.297141	+	0.954834i	$0.596033\pi$
$684$	0	0
$685$	−0.937742	−0.0358293
$686$	5.40661	0.206425
$687$	0	0
$688$	−0.468871	−0.0178755
$689$	−33.1868	−1.26432
$690$	0	0
$691$	−7.53113	−0.286498	−0.143249	−	0.989687i	$-0.545755\pi$
$691$	−7.53113	−0.286498	−0.143249	+	0.989687i	$0.545755\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	−22.1245	−0.839835
$695$	6.00000	0.227593
$696$	0	0
$697$	36.2490	1.37303
$698$	−25.0623	−0.948620
$699$	0	0
$700$	3.53113	0.133464
$701$	21.5311	0.813220	0.406610	−	0.913602i	$-0.366711\pi$
$701$	21.5311	0.813220	0.406610	+	0.913602i	$0.366711\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	1.53113	0.0577066
$705$	0	0
$706$	−23.0623	−0.867960
$707$	61.9047	2.32816
$708$	0	0
$709$	25.4066	0.954165	0.477083	−	0.878858i	$-0.341694\pi$
$709$	25.4066	0.954165	0.477083	+	0.878858i	$0.341694\pi$
$710$	−4.46887	−0.167714
$711$	0	0
$712$	1.53113	0.0573815
$713$	1.53113	0.0573412
$714$	0	0
$715$	9.18677	0.343566
$716$	−13.0623	−0.488159
$717$	0	0
$718$	11.5311	0.430338
$719$	33.1868	1.23766	0.618829	−	0.785526i	$-0.287607\pi$
$719$	33.1868	1.23766	0.618829	+	0.785526i	$0.287607\pi$
$720$	0	0
$721$	0	0
$722$	6.53113	0.243063
$723$	0	0
$724$	26.5934	0.988335
$725$	0	0
$726$	0	0
$727$	−50.8424	−1.88564	−0.942820	−	0.333301i	$-0.891837\pi$
$727$	−50.8424	−1.88564	−0.942820	+	0.333301i	$0.891837\pi$
$728$	−21.1868	−0.785234
$729$	0	0
$730$	0.468871	0.0173537
$731$	−1.87548	−0.0693673
$732$	0	0
$733$	4.12452	0.152342	0.0761712	−	0.997095i	$-0.475730\pi$
$733$	4.12452	0.152342	0.0761712	+	0.997095i	$0.475730\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−1.53113	−0.0564382
$737$	−16.9377	−0.623910
$738$	0	0
$739$	27.1868	1.00008	0.500041	−	0.866002i	$-0.333318\pi$
$739$	27.1868	1.00008	0.500041	+	0.866002i	$0.333318\pi$
$740$	9.06226	0.333135
$741$	0	0
$742$	19.5311	0.717010
$743$	−11.6556	−0.427604	−0.213802	−	0.976877i	$-0.568585\pi$
$743$	−11.6556	−0.427604	−0.213802	+	0.976877i	$0.568585\pi$
$744$	0	0
$745$	10.4689	0.383550
$746$	10.4689	0.383293
$747$	0	0
$748$	6.12452	0.223934
$749$	51.5311	1.88291
$750$	0	0
$751$	47.0623	1.71733	0.858663	−	0.512540i	$-0.171296\pi$
$751$	47.0623	1.71733	0.858663	+	0.512540i	$0.171296\pi$
$752$	−11.0623	−0.403399
$753$	0	0
$754$	0	0
$755$	−18.1245	−0.659619
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	2.59339	0.0941960
$759$	0	0
$760$	3.53113	0.128088
$761$	12.5934	0.456510	0.228255	−	0.973601i	$-0.426698\pi$
$761$	12.5934	0.456510	0.228255	+	0.973601i	$0.426698\pi$
$762$	0	0
$763$	−3.75097	−0.135794
$764$	8.00000	0.289430
$765$	0	0
$766$	13.0623	0.471959
$767$	−42.3735	−1.53002
$768$	0	0
$769$	−28.5934	−1.03110	−0.515552	−	0.856858i	$-0.672413\pi$
$769$	−28.5934	−1.03110	−0.515552	+	0.856858i	$0.672413\pi$
$770$	−5.40661	−0.194841
$771$	0	0
$772$	14.0000	0.503871
$773$	−14.4689	−0.520409	−0.260205	−	0.965554i	$-0.583790\pi$
$773$	−14.4689	−0.520409	−0.260205	+	0.965554i	$0.583790\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	16.1245	0.578836
$777$	0	0
$778$	27.0623	0.970229
$779$	−32.0000	−1.14652
$780$	0	0
$781$	6.84242	0.244841
$782$	−6.12452	−0.219012
$783$	0	0
$784$	5.46887	0.195317
$785$	−14.4689	−0.516416
$786$	0	0
$787$	−28.4689	−1.01481	−0.507403	−	0.861709i	$-0.669394\pi$
$787$	−28.4689	−1.01481	−0.507403	+	0.861709i	$0.669394\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0.468871	0.0166817
$791$	−58.5934	−2.08334
$792$	0	0
$793$	−66.3735	−2.35699
$794$	18.7179	0.664273
$795$	0	0
$796$	−0.468871	−0.0166187
$797$	20.1245	0.712847	0.356423	−	0.934325i	$-0.383996\pi$
$797$	20.1245	0.712847	0.356423	+	0.934325i	$0.383996\pi$
$798$	0	0
$799$	−44.2490	−1.56542
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−34.7179	−1.22593
$803$	−0.717902	−0.0253342
$804$	0	0
$805$	5.40661	0.190558
$806$	−6.00000	−0.211341
$807$	0	0
$808$	−17.5311	−0.616743
$809$	8.59339	0.302127	0.151064	−	0.988524i	$-0.451730\pi$
$809$	8.59339	0.302127	0.151064	+	0.988524i	$0.451730\pi$
$810$	0	0
$811$	−39.7802	−1.39687	−0.698435	−	0.715673i	$-0.746120\pi$
$811$	−39.7802	−1.39687	−0.698435	+	0.715673i	$0.746120\pi$
$812$	0	0
$813$	0	0
$814$	−13.8755	−0.486335
$815$	−11.0623	−0.387494
$816$	0	0
$817$	1.65564	0.0579237
$818$	9.06226	0.316854
$819$	0	0
$820$	9.06226	0.316468
$821$	−9.87548	−0.344657	−0.172328	−	0.985040i	$-0.555129\pi$
$821$	−9.87548	−0.344657	−0.172328	+	0.985040i	$0.555129\pi$
$822$	0	0
$823$	7.87548	0.274522	0.137261	−	0.990535i	$-0.456170\pi$
$823$	7.87548	0.274522	0.137261	+	0.990535i	$0.456170\pi$
$824$	0	0
$825$	0	0
$826$	24.9377	0.867695
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−9.65564	−0.335354	−0.167677	−	0.985842i	$-0.553627\pi$
$829$	−9.65564	−0.335354	−0.167677	+	0.985842i	$0.553627\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	6.00000	0.208013
$833$	21.8755	0.757941
$834$	0	0
$835$	−0.593387	−0.0205350
$836$	−5.40661	−0.186992
$837$	0	0
$838$	0	0
$839$	54.8424	1.89337	0.946685	−	0.322160i	$-0.104409\pi$
$839$	54.8424	1.89337	0.946685	+	0.322160i	$0.104409\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−38.2490	−1.31815
$843$	0	0
$844$	22.5934	0.777696
$845$	23.0000	0.791224
$846$	0	0
$847$	−30.5642	−1.05020
$848$	−5.53113	−0.189940
$849$	0	0
$850$	−4.00000	−0.137199
$851$	13.8755	0.475645
$852$	0	0
$853$	1.28210	0.0438982	0.0219491	−	0.999759i	$-0.493013\pi$
$853$	1.28210	0.0438982	0.0219491	+	0.999759i	$0.493013\pi$
$854$	39.0623	1.33668
$855$	0	0
$856$	−14.5934	−0.498792
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	−6.93774	−0.236713	−0.118356	−	0.992971i	$-0.537763\pi$
$859$	−6.93774	−0.236713	−0.118356	+	0.992971i	$0.537763\pi$
$860$	−0.468871	−0.0159884
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−41.5311	−1.41374	−0.706868	−	0.707345i	$-0.749893\pi$
$863$	−41.5311	−1.41374	−0.706868	+	0.707345i	$0.749893\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	1.40661	0.0477987
$867$	0	0
$868$	3.53113	0.119854
$869$	−0.717902	−0.0243532
$870$	0	0
$871$	−66.3735	−2.24898
$872$	1.06226	0.0359726
$873$	0	0
$874$	5.40661	0.182881
$875$	3.53113	0.119374
$876$	0	0
$877$	44.1245	1.48998	0.744990	−	0.667076i	$-0.232454\pi$
$877$	44.1245	1.48998	0.744990	+	0.667076i	$0.232454\pi$
$878$	8.93774	0.301634
$879$	0	0
$880$	1.53113	0.0516143
$881$	−36.1245	−1.21707	−0.608533	−	0.793529i	$-0.708242\pi$
$881$	−36.1245	−1.21707	−0.608533	+	0.793529i	$0.708242\pi$
$882$	0	0
$883$	−53.6556	−1.80566	−0.902828	−	0.430002i	$-0.858513\pi$
$883$	−53.6556	−1.80566	−0.902828	+	0.430002i	$0.858513\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	38.5934	1.29657
$887$	25.1868	0.845689	0.422845	−	0.906202i	$-0.361032\pi$
$887$	25.1868	0.845689	0.422845	+	0.906202i	$0.361032\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	1.53113	0.0513236
$891$	0	0
$892$	2.00000	0.0669650
$893$	39.0623	1.30717
$894$	0	0
$895$	−13.0623	−0.436623
$896$	−3.53113	−0.117967
$897$	0	0
$898$	28.1245	0.938527
$899$	0	0
$900$	0	0
$901$	−22.1245	−0.737074
$902$	−13.8755	−0.462003
$903$	0	0
$904$	16.5934	0.551888
$905$	26.5934	0.883994
$906$	0	0
$907$	−32.2490	−1.07081	−0.535406	−	0.844595i	$-0.679841\pi$
$907$	−32.2490	−1.07081	−0.535406	+	0.844595i	$0.679841\pi$
$908$	16.4689	0.546539
$909$	0	0
$910$	−21.1868	−0.702335
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	12.2490	0.405384
$914$	−31.0623	−1.02745
$915$	0	0
$916$	18.5934	0.614343
$917$	−24.9377	−0.823517
$918$	0	0
$919$	−8.93774	−0.294829	−0.147414	−	0.989075i	$-0.547095\pi$
$919$	−8.93774	−0.294829	−0.147414	+	0.989075i	$0.547095\pi$
$920$	−1.53113	−0.0504798
$921$	0	0
$922$	24.0000	0.790398
$923$	26.8132	0.882568
$924$	0	0
$925$	9.06226	0.297965
$926$	35.1868	1.15631
$927$	0	0
$928$	0	0
$929$	−36.8424	−1.20876	−0.604380	−	0.796696i	$-0.706579\pi$
$929$	−36.8424	−1.20876	−0.604380	+	0.796696i	$0.706579\pi$
$930$	0	0
$931$	−19.3113	−0.632902
$932$	−9.53113	−0.312203
$933$	0	0
$934$	10.1245	0.331284
$935$	6.12452	0.200293
$936$	0	0
$937$	−25.0623	−0.818748	−0.409374	−	0.912367i	$-0.634253\pi$
$937$	−25.0623	−0.818748	−0.409374	+	0.912367i	$0.634253\pi$
$938$	39.0623	1.27543
$939$	0	0
$940$	−11.0623	−0.360811
$941$	21.8755	0.713120	0.356560	−	0.934272i	$-0.383949\pi$
$941$	21.8755	0.713120	0.356560	+	0.934272i	$0.383949\pi$
$942$	0	0
$943$	13.8755	0.451848
$944$	−7.06226	−0.229857
$945$	0	0
$946$	0.717902	0.0233410
$947$	−35.0623	−1.13937	−0.569685	−	0.821863i	$-0.692935\pi$
$947$	−35.0623	−1.13937	−0.569685	+	0.821863i	$0.692935\pi$
$948$	0	0
$949$	−2.81323	−0.0913212
$950$	3.53113	0.114565
$951$	0	0
$952$	−14.1245	−0.457778
$953$	−54.1245	−1.75327	−0.876633	−	0.481161i	$-0.840215\pi$
$953$	−54.1245	−1.75327	−0.876633	+	0.481161i	$0.840215\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	8.00000	0.258738
$957$	0	0
$958$	41.6556	1.34583
$959$	−3.31129	−0.106927
$960$	0	0
$961$	1.00000	0.0322581
$962$	−54.3735	−1.75307
$963$	0	0
$964$	2.00000	0.0644157
$965$	14.0000	0.450676
$966$	0	0
$967$	−17.0623	−0.548685	−0.274343	−	0.961632i	$-0.588460\pi$
$967$	−17.0623	−0.548685	−0.274343	+	0.961632i	$0.588460\pi$
$968$	8.65564	0.278203
$969$	0	0
$970$	16.1245	0.517727
$971$	33.1868	1.06501	0.532507	−	0.846426i	$-0.321250\pi$
$971$	33.1868	1.06501	0.532507	+	0.846426i	$0.321250\pi$
$972$	0	0
$973$	21.1868	0.679217
$974$	−6.93774	−0.222300
$975$	0	0
$976$	−11.0623	−0.354094
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	−2.34436	−0.0749259
$980$	5.46887	0.174697
$981$	0	0
$982$	16.5934	0.529516
$983$	−17.0623	−0.544202	−0.272101	−	0.962269i	$-0.587718\pi$
$983$	−17.0623	−0.544202	−0.272101	+	0.962269i	$0.587718\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	−21.1868	−0.674041
$989$	−0.717902	−0.0228280
$990$	0	0
$991$	−24.4689	−0.777279	−0.388640	−	0.921390i	$-0.627055\pi$
$991$	−24.4689	−0.777279	−0.388640	+	0.921390i	$0.627055\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−15.7802	−0.500516
$995$	−0.468871	−0.0148642
$996$	0	0
$997$	62.2490	1.97145	0.985723	−	0.168373i	$-0.0538514\pi$
$997$	62.2490	1.97145	0.985723	+	0.168373i	$0.0538514\pi$
$998$	−9.06226	−0.286861
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bf.1.2		2
3.2	odd	2		930.2.a.q.1.2	✓	2
12.11	even	2		7440.2.a.bd.1.1		2
15.2	even	4		4650.2.d.bg.3349.4		4
15.8	even	4		4650.2.d.bg.3349.1		4
15.14	odd	2		4650.2.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.2	✓	2	3.2	odd	2
2790.2.a.bf.1.2		2	1.1	even	1	trivial
4650.2.a.bz.1.1		2	15.14	odd	2
4650.2.d.bg.3349.1		4	15.8	even	4
4650.2.d.bg.3349.4		4	15.2	even	4
7440.2.a.bd.1.1		2	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.53113 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.53113 q^{11} +6.00000 q^{13} -3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.53113 q^{19} +1.00000 q^{20} -1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{25} -6.00000 q^{26} +3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +3.53113 q^{35} +9.06226 q^{37} +3.53113 q^{38} -1.00000 q^{40} +9.06226 q^{41} -0.468871 q^{43} +1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} +5.46887 q^{49} -1.00000 q^{50} +6.00000 q^{52} -5.53113 q^{53} +1.53113 q^{55} -3.53113 q^{56} -7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -11.0623 q^{67} +4.00000 q^{68} -3.53113 q^{70} +4.46887 q^{71} -0.468871 q^{73} -9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -0.468871 q^{79} +1.00000 q^{80} -9.06226 q^{82} +8.00000 q^{83} +4.00000 q^{85} +0.468871 q^{86} -1.53113 q^{88} -1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} +11.0623 q^{94} -3.53113 q^{95} -16.1245 q^{97} -5.46887 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 5 q^{11} + 12 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + q^{19} + 2 q^{20} + 5 q^{22} - 5 q^{23} + 2 q^{25} - 12 q^{26} - q^{28} + 2 q^{31}+ \cdots - 19 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - q^7 - 2 * q^8 - 2 * q^10 - 5 * q^11 + 12 * q^13 + q^14 + 2 * q^16 + 8 * q^17 + q^19 + 2 * q^20 + 5 * q^22 - 5 * q^23 + 2 * q^25 - 12 * q^26 - q^28 + 2 * q^31 - 2 * q^32 - 8 * q^34 - q^35 + 2 * q^37 - q^38 - 2 * q^40 + 2 * q^41 - 9 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 19 * q^49 - 2 * q^50 + 12 * q^52 - 3 * q^53 - 5 * q^55 + q^56 + 2 * q^59 - 6 * q^61 - 2 * q^62 + 2 * q^64 + 12 * q^65 - 6 * q^67 + 8 * q^68 + q^70 + 17 * q^71 - 9 * q^73 - 2 * q^74 + q^76 + 35 * q^77 - 9 * q^79 + 2 * q^80 - 2 * q^82 + 16 * q^83 + 8 * q^85 + 9 * q^86 + 5 * q^88 + 5 * q^89 - 6 * q^91 - 5 * q^92 + 6 * q^94 + q^95 - 19 * q^98

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