LMFDB - Embedded newform 2790.2.a.bf.1.1

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			1.1
Root			$4.53113$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.53113 q^{7} -1.00000 q^{8} -1.00000 q^{10} -6.53113 q^{11} +6.00000 q^{13} +4.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} +4.53113 q^{19} +1.00000 q^{20} +6.53113 q^{22} -6.53113 q^{23} +1.00000 q^{25} -6.00000 q^{26} -4.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.53113 q^{35} -7.06226 q^{37} -4.53113 q^{38} -1.00000 q^{40} -7.06226 q^{41} -8.53113 q^{43} -6.53113 q^{44} +6.53113 q^{46} +5.06226 q^{47} +13.5311 q^{49} -1.00000 q^{50} +6.00000 q^{52} +2.53113 q^{53} -6.53113 q^{55} +4.53113 q^{56} +9.06226 q^{59} +5.06226 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +5.06226 q^{67} +4.00000 q^{68} +4.53113 q^{70} +12.5311 q^{71} -8.53113 q^{73} +7.06226 q^{74} +4.53113 q^{76} +29.5934 q^{77} -8.53113 q^{79} +1.00000 q^{80} +7.06226 q^{82} +8.00000 q^{83} +4.00000 q^{85} +8.53113 q^{86} +6.53113 q^{88} +6.53113 q^{89} -27.1868 q^{91} -6.53113 q^{92} -5.06226 q^{94} +4.53113 q^{95} +16.1245 q^{97} -13.5311 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.53113	−1.71261	−0.856303	−	0.516474i	$-0.827244\pi$
$7$	−4.53113	−1.71261	−0.856303	+	0.516474i	$0.827244\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−6.53113	−1.96921	−0.984605	−	0.174796i	$-0.944074\pi$
$11$	−6.53113	−1.96921	−0.984605	+	0.174796i	$0.944074\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	4.53113	1.21100
$15$	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
$19$	4.53113	1.03951	0.519756	−	0.854315i	$-0.326023\pi$
$19$	4.53113	1.03951	0.519756	+	0.854315i	$0.326023\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	6.53113	1.39244
$23$	−6.53113	−1.36183	−0.680917	−	0.732360i	$-0.738419\pi$
$23$	−6.53113	−1.36183	−0.680917	+	0.732360i	$0.738419\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	−4.53113	−0.856303
$29$	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
$35$	−4.53113	−0.765901
$36$	0	0
$37$	−7.06226	−1.16103	−0.580514	−	0.814250i	$-0.697148\pi$
$37$	−7.06226	−1.16103	−0.580514	+	0.814250i	$0.697148\pi$
$38$	−4.53113	−0.735046
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−7.06226	−1.10294	−0.551470	−	0.834195i	$-0.685933\pi$
$41$	−7.06226	−1.10294	−0.551470	+	0.834195i	$0.685933\pi$
$42$	0	0
$43$	−8.53113	−1.30098	−0.650492	−	0.759513i	$-0.725437\pi$
$43$	−8.53113	−1.30098	−0.650492	+	0.759513i	$0.725437\pi$
$44$	−6.53113	−0.984605
$45$	0	0
$46$	6.53113	0.962962
$47$	5.06226	0.738406	0.369203	−	0.929349i	$-0.379631\pi$
$47$	5.06226	0.738406	0.369203	+	0.929349i	$0.379631\pi$
$48$	0	0
$49$	13.5311	1.93302
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	2.53113	0.347677	0.173839	−	0.984774i	$-0.444383\pi$
$53$	2.53113	0.347677	0.173839	+	0.984774i	$0.444383\pi$
$54$	0	0
$55$	−6.53113	−0.880657
$56$	4.53113	0.605498
$57$	0	0
$58$	0	0
$59$	9.06226	1.17981	0.589903	−	0.807474i	$-0.299166\pi$
$59$	9.06226	1.17981	0.589903	+	0.807474i	$0.299166\pi$
$60$	0	0
$61$	5.06226	0.648156	0.324078	−	0.946030i	$-0.394946\pi$
$61$	5.06226	0.648156	0.324078	+	0.946030i	$0.394946\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	5.06226	0.618453	0.309227	−	0.950988i	$-0.399930\pi$
$67$	5.06226	0.618453	0.309227	+	0.950988i	$0.399930\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	4.53113	0.541573
$71$	12.5311	1.48717	0.743586	−	0.668641i	$-0.233124\pi$
$71$	12.5311	1.48717	0.743586	+	0.668641i	$0.233124\pi$
$72$	0	0
$73$	−8.53113	−0.998493	−0.499247	−	0.866460i	$-0.666390\pi$
$73$	−8.53113	−0.998493	−0.499247	+	0.866460i	$0.666390\pi$
$74$	7.06226	0.820971
$75$	0	0
$76$	4.53113	0.519756
$77$	29.5934	3.37248
$78$	0	0
$79$	−8.53113	−0.959827	−0.479913	−	0.877316i	$-0.659332\pi$
$79$	−8.53113	−0.959827	−0.479913	+	0.877316i	$0.659332\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	7.06226	0.779896
$83$	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
$86$	8.53113	0.919935
$87$	0	0
$88$	6.53113	0.696221
$89$	6.53113	0.692298	0.346149	−	0.938180i	$-0.387489\pi$
$89$	6.53113	0.692298	0.346149	+	0.938180i	$0.387489\pi$
$90$	0	0
$91$	−27.1868	−2.84995
$92$	−6.53113	−0.680917
$93$	0	0
$94$	−5.06226	−0.522132
$95$	4.53113	0.464884
$96$	0	0
$97$	16.1245	1.63720	0.818598	−	0.574367i	$-0.194752\pi$
$97$	16.1245	1.63720	0.818598	+	0.574367i	$0.194752\pi$
$98$	−13.5311	−1.36685
$99$	0	0
$100$	1.00000	0.100000
$101$	9.46887	0.942188	0.471094	−	0.882083i	$-0.343859\pi$
$101$	9.46887	0.942188	0.471094	+	0.882083i	$0.343859\pi$
$102$	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
$106$	−2.53113	−0.245845
$107$	−9.59339	−0.927428	−0.463714	−	0.885985i	$-0.653483\pi$
$107$	−9.59339	−0.927428	−0.463714	+	0.885985i	$0.653483\pi$
$108$	0	0
$109$	15.0623	1.44270	0.721351	−	0.692569i	$-0.243521\pi$
$109$	15.0623	1.44270	0.721351	+	0.692569i	$0.243521\pi$
$110$	6.53113	0.622719
$111$	0	0
$112$	−4.53113	−0.428151
$113$	7.59339	0.714326	0.357163	−	0.934042i	$-0.383744\pi$
$113$	7.59339	0.714326	0.357163	+	0.934042i	$0.383744\pi$
$114$	0	0
$115$	−6.53113	−0.609031
$116$	0	0
$117$	0	0
$118$	−9.06226	−0.834248
$119$	−18.1245	−1.66147
$120$	0	0
$121$	31.6556	2.87779
$122$	−5.06226	−0.458315
$123$	0	0
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.06226	0.626674	0.313337	−	0.949642i	$-0.398553\pi$
$127$	7.06226	0.626674	0.313337	+	0.949642i	$0.398553\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.00000	−0.526235
$131$	9.06226	0.791773	0.395887	−	0.918299i	$-0.370437\pi$
$131$	9.06226	0.791773	0.395887	+	0.918299i	$0.370437\pi$
$132$	0	0
$133$	−20.5311	−1.78027
$134$	−5.06226	−0.437312
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−17.0623	−1.45773	−0.728864	−	0.684659i	$-0.759951\pi$
$137$	−17.0623	−1.45773	−0.728864	+	0.684659i	$0.759951\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	−4.53113	−0.382950
$141$	0	0
$142$	−12.5311	−1.05159
$143$	−39.1868	−3.27696
$144$	0	0
$145$	0	0
$146$	8.53113	0.706041
$147$	0	0
$148$	−7.06226	−0.580514
$149$	18.5311	1.51813	0.759065	−	0.651015i	$-0.225657\pi$
$149$	18.5311	1.51813	0.759065	+	0.651015i	$0.225657\pi$
$150$	0	0
$151$	14.1245	1.14944	0.574718	−	0.818351i	$-0.305112\pi$
$151$	14.1245	1.14944	0.574718	+	0.818351i	$0.305112\pi$
$152$	−4.53113	−0.367523
$153$	0	0
$154$	−29.5934	−2.38470
$155$	1.00000	0.0803219
$156$	0	0
$157$	−22.5311	−1.79818	−0.899090	−	0.437764i	$-0.855771\pi$
$157$	−22.5311	−1.79818	−0.899090	+	0.437764i	$0.855771\pi$
$158$	8.53113	0.678700
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	29.5934	2.33229
$162$	0	0
$163$	5.06226	0.396507	0.198253	−	0.980151i	$-0.436473\pi$
$163$	5.06226	0.396507	0.198253	+	0.980151i	$0.436473\pi$
$164$	−7.06226	−0.551470
$165$	0	0
$166$	−8.00000	−0.620920
$167$	23.5934	1.82571	0.912856	−	0.408283i	$-0.133872\pi$
$167$	23.5934	1.82571	0.912856	+	0.408283i	$0.133872\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−8.53113	−0.650492
$173$	−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
$175$	−4.53113	−0.342521
$176$	−6.53113	−0.492302
$177$	0	0
$178$	−6.53113	−0.489529
$179$	3.06226	0.228884	0.114442	−	0.993430i	$-0.463492\pi$
$179$	3.06226	0.228884	0.114442	+	0.993430i	$0.463492\pi$
$180$	0	0
$181$	2.40661	0.178882	0.0894411	−	0.995992i	$-0.471492\pi$
$181$	2.40661	0.178882	0.0894411	+	0.995992i	$0.471492\pi$
$182$	27.1868	2.01522
$183$	0	0
$184$	6.53113	0.481481
$185$	−7.06226	−0.519228
$186$	0	0
$187$	−26.1245	−1.91041
$188$	5.06226	0.369203
$189$	0	0
$190$	−4.53113	−0.328723
$191$	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
$196$	13.5311	0.966509
$197$	−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
$199$	−8.53113	−0.604756	−0.302378	−	0.953188i	$-0.597780\pi$
$199$	−8.53113	−0.604756	−0.302378	+	0.953188i	$0.597780\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−9.46887	−0.666227
$203$	0	0
$204$	0	0
$205$	−7.06226	−0.493249
$206$	0	0
$207$	0	0
$208$	6.00000	0.416025
$209$	−29.5934	−2.04702
$210$	0	0
$211$	−1.59339	−0.109693	−0.0548466	−	0.998495i	$-0.517467\pi$
$211$	−1.59339	−0.109693	−0.0548466	+	0.998495i	$0.517467\pi$
$212$	2.53113	0.173839
$213$	0	0
$214$	9.59339	0.655790
$215$	−8.53113	−0.581818
$216$	0	0
$217$	−4.53113	−0.307593
$218$	−15.0623	−1.02014
$219$	0	0
$220$	−6.53113	−0.440329
$221$	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	$0.478668\pi$
$224$	4.53113	0.302749
$225$	0	0
$226$	−7.59339	−0.505105
$227$	24.5311	1.62819	0.814094	−	0.580733i	$-0.197234\pi$
$227$	24.5311	1.62819	0.814094	+	0.580733i	$0.197234\pi$
$228$	0	0
$229$	−5.59339	−0.369621	−0.184811	−	0.982774i	$-0.559167\pi$
$229$	−5.59339	−0.369621	−0.184811	+	0.982774i	$0.559167\pi$
$230$	6.53113	0.430650
$231$	0	0
$232$	0	0
$233$	−1.46887	−0.0962289	−0.0481145	−	0.998842i	$-0.515321\pi$
$233$	−1.46887	−0.0962289	−0.0481145	+	0.998842i	$0.515321\pi$
$234$	0	0
$235$	5.06226	0.330225
$236$	9.06226	0.589903
$237$	0	0
$238$	18.1245	1.17484
$239$	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	$0.479482\pi$
$242$	−31.6556	−2.03490
$243$	0	0
$244$	5.06226	0.324078
$245$	13.5311	0.864472
$246$	0	0
$247$	27.1868	1.72985
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−3.06226	−0.193288	−0.0966440	−	0.995319i	$-0.530811\pi$
$251$	−3.06226	−0.193288	−0.0966440	+	0.995319i	$0.530811\pi$
$252$	0	0
$253$	42.6556	2.68174
$254$	−7.06226	−0.443125
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.6556	−0.789437	−0.394719	−	0.918802i	$-0.629158\pi$
$257$	−12.6556	−0.789437	−0.394719	+	0.918802i	$0.629158\pi$
$258$	0	0
$259$	32.0000	1.98838
$260$	6.00000	0.372104
$261$	0	0
$262$	−9.06226	−0.559868
$263$	23.0623	1.42208	0.711040	−	0.703152i	$-0.248225\pi$
$263$	23.0623	1.42208	0.711040	+	0.703152i	$0.248225\pi$
$264$	0	0
$265$	2.53113	0.155486
$266$	20.5311	1.25884
$267$	0	0
$268$	5.06226	0.309227
$269$	19.1868	1.16984	0.584919	−	0.811092i	$-0.301126\pi$
$269$	19.1868	1.16984	0.584919	+	0.811092i	$0.301126\pi$
$270$	0	0
$271$	−11.4689	−0.696684	−0.348342	−	0.937367i	$-0.613255\pi$
$271$	−11.4689	−0.696684	−0.348342	+	0.937367i	$0.613255\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	17.0623	1.03077
$275$	−6.53113	−0.393842
$276$	0	0
$277$	−15.0623	−0.905003	−0.452502	−	0.891764i	$-0.649468\pi$
$277$	−15.0623	−0.905003	−0.452502	+	0.891764i	$0.649468\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	4.53113	0.270787
$281$	−15.0623	−0.898539	−0.449269	−	0.893396i	$-0.648316\pi$
$281$	−15.0623	−0.898539	−0.449269	+	0.893396i	$0.648316\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	12.5311	0.743586
$285$	0	0
$286$	39.1868	2.31716
$287$	32.0000	1.88890
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	−8.53113	−0.499247
$293$	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
$295$	9.06226	0.527625
$296$	7.06226	0.410485
$297$	0	0
$298$	−18.5311	−1.07348
$299$	−39.1868	−2.26623
$300$	0	0
$301$	38.6556	2.22807
$302$	−14.1245	−0.812775
$303$	0	0
$304$	4.53113	0.259878
$305$	5.06226	0.289864
$306$	0	0
$307$	−30.1245	−1.71930	−0.859648	−	0.510886i	$-0.829317\pi$
$307$	−30.1245	−1.71930	−0.859648	+	0.510886i	$0.829317\pi$
$308$	29.5934	1.68624
$309$	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
$313$	10.9377	0.618238	0.309119	−	0.951023i	$-0.399966\pi$
$313$	10.9377	0.618238	0.309119	+	0.951023i	$0.399966\pi$
$314$	22.5311	1.27151
$315$	0	0
$316$	−8.53113	−0.479913
$317$	−12.9377	−0.726656	−0.363328	−	0.931661i	$-0.618360\pi$
$317$	−12.9377	−0.726656	−0.363328	+	0.931661i	$0.618360\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	0	0
$322$	−29.5934	−1.64917
$323$	18.1245	1.00848
$324$	0	0
$325$	6.00000	0.332820
$326$	−5.06226	−0.280373
$327$	0	0
$328$	7.06226	0.389948
$329$	−22.9377	−1.26460
$330$	0	0
$331$	12.9377	0.711123	0.355561	−	0.934653i	$-0.384290\pi$
$331$	12.9377	0.711123	0.355561	+	0.934653i	$0.384290\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	−23.5934	−1.29097
$335$	5.06226	0.276581
$336$	0	0
$337$	−19.1868	−1.04517	−0.522585	−	0.852587i	$-0.675032\pi$
$337$	−19.1868	−1.04517	−0.522585	+	0.852587i	$0.675032\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	4.00000	0.216930
$341$	−6.53113	−0.353680
$342$	0	0
$343$	−29.5934	−1.59789
$344$	8.53113	0.459968
$345$	0	0
$346$	14.0000	0.752645
$347$	−10.1245	−0.543512	−0.271756	−	0.962366i	$-0.587604\pi$
$347$	−10.1245	−0.543512	−0.271756	+	0.962366i	$0.587604\pi$
$348$	0	0
$349$	8.93774	0.478426	0.239213	−	0.970967i	$-0.423111\pi$
$349$	8.93774	0.478426	0.239213	+	0.970967i	$0.423111\pi$
$350$	4.53113	0.242199
$351$	0	0
$352$	6.53113	0.348110
$353$	6.93774	0.369259	0.184629	−	0.982808i	$-0.440892\pi$
$353$	6.93774	0.369259	0.184629	+	0.982808i	$0.440892\pi$
$354$	0	0
$355$	12.5311	0.665083
$356$	6.53113	0.346149
$357$	0	0
$358$	−3.06226	−0.161845
$359$	−3.46887	−0.183080	−0.0915400	−	0.995801i	$-0.529179\pi$
$359$	−3.46887	−0.183080	−0.0915400	+	0.995801i	$0.529179\pi$
$360$	0	0
$361$	1.53113	0.0805857
$362$	−2.40661	−0.126489
$363$	0	0
$364$	−27.1868	−1.42497
$365$	−8.53113	−0.446540
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−6.53113	−0.340459
$369$	0	0
$370$	7.06226	0.367149
$371$	−11.4689	−0.595434
$372$	0	0
$373$	−18.5311	−0.959505	−0.479753	−	0.877404i	$-0.659274\pi$
$373$	−18.5311	−0.959505	−0.479753	+	0.877404i	$0.659274\pi$
$374$	26.1245	1.35087
$375$	0	0
$376$	−5.06226	−0.261066
$377$	0	0
$378$	0	0
$379$	21.5934	1.10918	0.554589	−	0.832124i	$-0.312876\pi$
$379$	21.5934	1.10918	0.554589	+	0.832124i	$0.312876\pi$
$380$	4.53113	0.232442
$381$	0	0
$382$	−8.00000	−0.409316
$383$	3.06226	0.156474	0.0782370	−	0.996935i	$-0.475071\pi$
$383$	3.06226	0.156474	0.0782370	+	0.996935i	$0.475071\pi$
$384$	0	0
$385$	29.5934	1.50822
$386$	−14.0000	−0.712581
$387$	0	0
$388$	16.1245	0.818598
$389$	−10.9377	−0.554566	−0.277283	−	0.960788i	$-0.589434\pi$
$389$	−10.9377	−0.554566	−0.277283	+	0.960788i	$0.589434\pi$
$390$	0	0
$391$	−26.1245	−1.32117
$392$	−13.5311	−0.683425
$393$	0	0
$394$	6.00000	0.302276
$395$	−8.53113	−0.429248
$396$	0	0
$397$	37.7179	1.89301	0.946504	−	0.322693i	$-0.104588\pi$
$397$	37.7179	1.89301	0.946504	+	0.322693i	$0.104588\pi$
$398$	8.53113	0.427627
$399$	0	0
$400$	1.00000	0.0500000
$401$	−21.7179	−1.08454	−0.542270	−	0.840204i	$-0.682435\pi$
$401$	−21.7179	−1.08454	−0.542270	+	0.840204i	$0.682435\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	9.46887	0.471094
$405$	0	0
$406$	0	0
$407$	46.1245	2.28631
$408$	0	0
$409$	7.06226	0.349206	0.174603	−	0.984639i	$-0.444136\pi$
$409$	7.06226	0.349206	0.174603	+	0.984639i	$0.444136\pi$
$410$	7.06226	0.348780
$411$	0	0
$412$	0	0
$413$	−41.0623	−2.02054
$414$	0	0
$415$	8.00000	0.392705
$416$	−6.00000	−0.294174
$417$	0	0
$418$	29.5934	1.44746
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−26.2490	−1.27930	−0.639650	−	0.768667i	$-0.720921\pi$
$421$	−26.2490	−1.27930	−0.639650	+	0.768667i	$0.720921\pi$
$422$	1.59339	0.0775648
$423$	0	0
$424$	−2.53113	−0.122922
$425$	4.00000	0.194029
$426$	0	0
$427$	−22.9377	−1.11004
$428$	−9.59339	−0.463714
$429$	0	0
$430$	8.53113	0.411408
$431$	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
$433$	−25.5934	−1.22994	−0.614970	−	0.788551i	$-0.710832\pi$
$433$	−25.5934	−1.22994	−0.614970	+	0.788551i	$0.710832\pi$
$434$	4.53113	0.217501
$435$	0	0
$436$	15.0623	0.721351
$437$	−29.5934	−1.41564
$438$	0	0
$439$	−25.0623	−1.19616	−0.598078	−	0.801438i	$-0.704069\pi$
$439$	−25.0623	−1.19616	−0.598078	+	0.801438i	$0.704069\pi$
$440$	6.53113	0.311359
$441$	0	0
$442$	−24.0000	−1.14156
$443$	−14.4066	−0.684479	−0.342239	−	0.939613i	$-0.611185\pi$
$443$	−14.4066	−0.684479	−0.342239	+	0.939613i	$0.611185\pi$
$444$	0	0
$445$	6.53113	0.309605
$446$	−2.00000	−0.0947027
$447$	0	0
$448$	−4.53113	−0.214076
$449$	4.12452	0.194648	0.0973240	−	0.995253i	$-0.468972\pi$
$449$	4.12452	0.194648	0.0973240	+	0.995253i	$0.468972\pi$
$450$	0	0
$451$	46.1245	2.17192
$452$	7.59339	0.357163
$453$	0	0
$454$	−24.5311	−1.15130
$455$	−27.1868	−1.27454
$456$	0	0
$457$	14.9377	0.698758	0.349379	−	0.936981i	$-0.386393\pi$
$457$	14.9377	0.698758	0.349379	+	0.936981i	$0.386393\pi$
$458$	5.59339	0.261362
$459$	0	0
$460$	−6.53113	−0.304515
$461$	−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
$463$	13.1868	0.612841	0.306421	−	0.951896i	$-0.400869\pi$
$463$	13.1868	0.612841	0.306421	+	0.951896i	$0.400869\pi$
$464$	0	0
$465$	0	0
$466$	1.46887	0.0680441
$467$	22.1245	1.02380	0.511900	−	0.859045i	$-0.328942\pi$
$467$	22.1245	1.02380	0.511900	+	0.859045i	$0.328942\pi$
$468$	0	0
$469$	−22.9377	−1.05917
$470$	−5.06226	−0.233505
$471$	0	0
$472$	−9.06226	−0.417124
$473$	55.7179	2.56191
$474$	0	0
$475$	4.53113	0.207902
$476$	−18.1245	−0.830736
$477$	0	0
$478$	−8.00000	−0.365911
$479$	−1.34436	−0.0614252	−0.0307126	−	0.999528i	$-0.509778\pi$
$479$	−1.34436	−0.0614252	−0.0307126	+	0.999528i	$0.509778\pi$
$480$	0	0
$481$	−42.3735	−1.93207
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	31.6556	1.43889
$485$	16.1245	0.732177
$486$	0	0
$487$	23.0623	1.04505	0.522525	−	0.852624i	$-0.324990\pi$
$487$	23.0623	1.04505	0.522525	+	0.852624i	$0.324990\pi$
$488$	−5.06226	−0.229158
$489$	0	0
$490$	−13.5311	−0.611274
$491$	7.59339	0.342685	0.171342	−	0.985212i	$-0.445190\pi$
$491$	7.59339	0.342685	0.171342	+	0.985212i	$0.445190\pi$
$492$	0	0
$493$	0	0
$494$	−27.1868	−1.22319
$495$	0	0
$496$	1.00000	0.0449013
$497$	−56.7802	−2.54694
$498$	0	0
$499$	−7.06226	−0.316150	−0.158075	−	0.987427i	$-0.550529\pi$
$499$	−7.06226	−0.316150	−0.158075	+	0.987427i	$0.550529\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	3.06226	0.136675
$503$	−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
$505$	9.46887	0.421359
$506$	−42.6556	−1.89627
$507$	0	0
$508$	7.06226	0.313337
$509$	−38.1245	−1.68984	−0.844920	−	0.534893i	$-0.820352\pi$
$509$	−38.1245	−1.68984	−0.844920	+	0.534893i	$0.820352\pi$
$510$	0	0
$511$	38.6556	1.71003
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.6556	0.558217
$515$	0	0
$516$	0	0
$517$	−33.0623	−1.45408
$518$	−32.0000	−1.40600
$519$	0	0
$520$	−6.00000	−0.263117
$521$	12.1245	0.531185	0.265592	−	0.964085i	$-0.414432\pi$
$521$	12.1245	0.531185	0.265592	+	0.964085i	$0.414432\pi$
$522$	0	0
$523$	9.59339	0.419490	0.209745	−	0.977756i	$-0.432737\pi$
$523$	9.59339	0.419490	0.209745	+	0.977756i	$0.432737\pi$
$524$	9.06226	0.395887
$525$	0	0
$526$	−23.0623	−1.00556
$527$	4.00000	0.174243
$528$	0	0
$529$	19.6556	0.854593
$530$	−2.53113	−0.109945
$531$	0	0
$532$	−20.5311	−0.890137
$533$	−42.3735	−1.83540
$534$	0	0
$535$	−9.59339	−0.414758
$536$	−5.06226	−0.218656
$537$	0	0
$538$	−19.1868	−0.827201
$539$	−88.3735	−3.80652
$540$	0	0
$541$	−4.12452	−0.177327	−0.0886634	−	0.996062i	$-0.528260\pi$
$541$	−4.12452	−0.177327	−0.0886634	+	0.996062i	$0.528260\pi$
$542$	11.4689	0.492630
$543$	0	0
$544$	−4.00000	−0.171499
$545$	15.0623	0.645196
$546$	0	0
$547$	−7.18677	−0.307284	−0.153642	−	0.988127i	$-0.549100\pi$
$547$	−7.18677	−0.307284	−0.153642	+	0.988127i	$0.549100\pi$
$548$	−17.0623	−0.728864
$549$	0	0
$550$	6.53113	0.278488
$551$	0	0
$552$	0	0
$553$	38.6556	1.64381
$554$	15.0623	0.639934
$555$	0	0
$556$	6.00000	0.254457
$557$	26.7802	1.13471	0.567356	−	0.823473i	$-0.307966\pi$
$557$	26.7802	1.13471	0.567356	+	0.823473i	$0.307966\pi$
$558$	0	0
$559$	−51.1868	−2.16497
$560$	−4.53113	−0.191475
$561$	0	0
$562$	15.0623	0.635363
$563$	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
$565$	7.59339	0.319456
$566$	12.0000	0.504398
$567$	0	0
$568$	−12.5311	−0.525794
$569$	−9.46887	−0.396956	−0.198478	−	0.980105i	$-0.563600\pi$
$569$	−9.46887	−0.396956	−0.198478	+	0.980105i	$0.563600\pi$
$570$	0	0
$571$	36.1245	1.51176	0.755882	−	0.654708i	$-0.227208\pi$
$571$	36.1245	1.51176	0.755882	+	0.654708i	$0.227208\pi$
$572$	−39.1868	−1.63848
$573$	0	0
$574$	−32.0000	−1.33565
$575$	−6.53113	−0.272367
$576$	0	0
$577$	37.1868	1.54811	0.774053	−	0.633121i	$-0.218226\pi$
$577$	37.1868	1.54811	0.774053	+	0.633121i	$0.218226\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	−36.2490	−1.50386
$582$	0	0
$583$	−16.5311	−0.684649
$584$	8.53113	0.353021
$585$	0	0
$586$	−14.0000	−0.578335
$587$	44.2490	1.82635	0.913176	−	0.407564i	$-0.133622\pi$
$587$	44.2490	1.82635	0.913176	+	0.407564i	$0.133622\pi$
$588$	0	0
$589$	4.53113	0.186702
$590$	−9.06226	−0.373087
$591$	0	0
$592$	−7.06226	−0.290257
$593$	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
$595$	−18.1245	−0.743033
$596$	18.5311	0.759065
$597$	0	0
$598$	39.1868	1.60247
$599$	13.5934	0.555411	0.277705	−	0.960666i	$-0.410426\pi$
$599$	13.5934	0.555411	0.277705	+	0.960666i	$0.410426\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	−38.6556	−1.57549
$603$	0	0
$604$	14.1245	0.574718
$605$	31.6556	1.28698
$606$	0	0
$607$	−14.4066	−0.584746	−0.292373	−	0.956304i	$-0.594445\pi$
$607$	−14.4066	−0.584746	−0.292373	+	0.956304i	$0.594445\pi$
$608$	−4.53113	−0.183762
$609$	0	0
$610$	−5.06226	−0.204965
$611$	30.3735	1.22878
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	30.1245	1.21573
$615$	0	0
$616$	−29.5934	−1.19235
$617$	3.59339	0.144664	0.0723321	−	0.997381i	$-0.476956\pi$
$617$	3.59339	0.144664	0.0723321	+	0.997381i	$0.476956\pi$
$618$	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
$623$	−29.5934	−1.18563
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.9377	−0.437160
$627$	0	0
$628$	−22.5311	−0.899090
$629$	−28.2490	−1.12636
$630$	0	0
$631$	30.6556	1.22038	0.610191	−	0.792254i	$-0.291093\pi$
$631$	30.6556	1.22038	0.610191	+	0.792254i	$0.291093\pi$
$632$	8.53113	0.339350
$633$	0	0
$634$	12.9377	0.513823
$635$	7.06226	0.280257
$636$	0	0
$637$	81.1868	3.21674
$638$	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
$643$	−17.5934	−0.693815	−0.346908	−	0.937899i	$-0.612768\pi$
$643$	−17.5934	−0.693815	−0.346908	+	0.937899i	$0.612768\pi$
$644$	29.5934	1.16614
$645$	0	0
$646$	−18.1245	−0.713100
$647$	6.53113	0.256765	0.128383	−	0.991725i	$-0.459021\pi$
$647$	6.53113	0.256765	0.128383	+	0.991725i	$0.459021\pi$
$648$	0	0
$649$	−59.1868	−2.32328
$650$	−6.00000	−0.235339
$651$	0	0
$652$	5.06226	0.198253
$653$	−38.2490	−1.49680	−0.748400	−	0.663248i	$-0.769178\pi$
$653$	−38.2490	−1.49680	−0.748400	+	0.663248i	$0.769178\pi$
$654$	0	0
$655$	9.06226	0.354092
$656$	−7.06226	−0.275735
$657$	0	0
$658$	22.9377	0.894206
$659$	−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	$0.562301\pi$
$662$	−12.9377	−0.502840
$663$	0	0
$664$	−8.00000	−0.310460
$665$	−20.5311	−0.796163
$666$	0	0
$667$	0	0
$668$	23.5934	0.912856
$669$	0	0
$670$	−5.06226	−0.195572
$671$	−33.0623	−1.27635
$672$	0	0
$673$	−9.06226	−0.349324	−0.174662	−	0.984628i	$-0.555883\pi$
$673$	−9.06226	−0.349324	−0.174662	+	0.984628i	$0.555883\pi$
$674$	19.1868	0.739047
$675$	0	0
$676$	23.0000	0.884615
$677$	−27.5934	−1.06050	−0.530250	−	0.847841i	$-0.677902\pi$
$677$	−27.5934	−1.06050	−0.530250	+	0.847841i	$0.677902\pi$
$678$	0	0
$679$	−73.0623	−2.80387
$680$	−4.00000	−0.153393
$681$	0	0
$682$	6.53113	0.250090
$683$	−7.46887	−0.285788	−0.142894	−	0.989738i	$-0.545641\pi$
$683$	−7.46887	−0.285788	−0.142894	+	0.989738i	$0.545641\pi$
$684$	0	0
$685$	−17.0623	−0.651915
$686$	29.5934	1.12988
$687$	0	0
$688$	−8.53113	−0.325246
$689$	15.1868	0.578570
$690$	0	0
$691$	0.531129	0.0202051	0.0101025	−	0.999949i	$-0.496784\pi$
$691$	0.531129	0.0202051	0.0101025	+	0.999949i	$0.496784\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	10.1245	0.384321
$695$	6.00000	0.227593
$696$	0	0
$697$	−28.2490	−1.07001
$698$	−8.93774	−0.338299
$699$	0	0
$700$	−4.53113	−0.171261
$701$	13.4689	0.508712	0.254356	−	0.967111i	$-0.418136\pi$
$701$	13.4689	0.508712	0.254356	+	0.967111i	$0.418136\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	−6.53113	−0.246151
$705$	0	0
$706$	−6.93774	−0.261105
$707$	−42.9047	−1.61360
$708$	0	0
$709$	49.5934	1.86252	0.931259	−	0.364357i	$-0.118711\pi$
$709$	49.5934	1.86252	0.931259	+	0.364357i	$0.118711\pi$
$710$	−12.5311	−0.470285
$711$	0	0
$712$	−6.53113	−0.244764
$713$	−6.53113	−0.244593
$714$	0	0
$715$	−39.1868	−1.46550
$716$	3.06226	0.114442
$717$	0	0
$718$	3.46887	0.129457
$719$	−15.1868	−0.566371	−0.283186	−	0.959065i	$-0.591391\pi$
$719$	−15.1868	−0.566371	−0.283186	+	0.959065i	$0.591391\pi$
$720$	0	0
$721$	0	0
$722$	−1.53113	−0.0569827
$723$	0	0
$724$	2.40661	0.0894411
$725$	0	0
$726$	0	0
$727$	37.8424	1.40350	0.701749	−	0.712424i	$-0.252403\pi$
$727$	37.8424	1.40350	0.701749	+	0.712424i	$0.252403\pi$
$728$	27.1868	1.00761
$729$	0	0
$730$	8.53113	0.315751
$731$	−34.1245	−1.26214
$732$	0	0
$733$	−28.1245	−1.03880	−0.519401	−	0.854530i	$-0.673845\pi$
$733$	−28.1245	−1.03880	−0.519401	+	0.854530i	$0.673845\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	6.53113	0.240741
$737$	−33.0623	−1.21786
$738$	0	0
$739$	−21.1868	−0.779368	−0.389684	−	0.920949i	$-0.627416\pi$
$739$	−21.1868	−0.779368	−0.389684	+	0.920949i	$0.627416\pi$
$740$	−7.06226	−0.259614
$741$	0	0
$742$	11.4689	0.421036
$743$	28.6556	1.05127	0.525637	−	0.850709i	$-0.323827\pi$
$743$	28.6556	1.05127	0.525637	+	0.850709i	$0.323827\pi$
$744$	0	0
$745$	18.5311	0.678928
$746$	18.5311	0.678473
$747$	0	0
$748$	−26.1245	−0.955207
$749$	43.4689	1.58832
$750$	0	0
$751$	30.9377	1.12893	0.564467	−	0.825456i	$-0.309082\pi$
$751$	30.9377	1.12893	0.564467	+	0.825456i	$0.309082\pi$
$752$	5.06226	0.184602
$753$	0	0
$754$	0	0
$755$	14.1245	0.514044
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−21.5934	−0.784307
$759$	0	0
$760$	−4.53113	−0.164361
$761$	−11.5934	−0.420260	−0.210130	−	0.977673i	$-0.567389\pi$
$761$	−11.5934	−0.420260	−0.210130	+	0.977673i	$0.567389\pi$
$762$	0	0
$763$	−68.2490	−2.47078
$764$	8.00000	0.289430
$765$	0	0
$766$	−3.06226	−0.110644
$767$	54.3735	1.96331
$768$	0	0
$769$	−4.40661	−0.158907	−0.0794533	−	0.996839i	$-0.525317\pi$
$769$	−4.40661	−0.158907	−0.0794533	+	0.996839i	$0.525317\pi$
$770$	−29.5934	−1.06647
$771$	0	0
$772$	14.0000	0.503871
$773$	−22.5311	−0.810388	−0.405194	−	0.914231i	$-0.632796\pi$
$773$	−22.5311	−0.810388	−0.405194	+	0.914231i	$0.632796\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	−16.1245	−0.578836
$777$	0	0
$778$	10.9377	0.392137
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−81.8424	−2.92855
$782$	26.1245	0.934211
$783$	0	0
$784$	13.5311	0.483255
$785$	−22.5311	−0.804170
$786$	0	0
$787$	−36.5311	−1.30219	−0.651097	−	0.758994i	$-0.725691\pi$
$787$	−36.5311	−1.30219	−0.651097	+	0.758994i	$0.725691\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	8.53113	0.303524
$791$	−34.4066	−1.22336
$792$	0	0
$793$	30.3735	1.07860
$794$	−37.7179	−1.33856
$795$	0	0
$796$	−8.53113	−0.302378
$797$	−12.1245	−0.429472	−0.214736	−	0.976672i	$-0.568889\pi$
$797$	−12.1245	−0.429472	−0.214736	+	0.976672i	$0.568889\pi$
$798$	0	0
$799$	20.2490	0.716359
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	21.7179	0.766886
$803$	55.7179	1.96624
$804$	0	0
$805$	29.5934	1.04303
$806$	−6.00000	−0.211341
$807$	0	0
$808$	−9.46887	−0.333114
$809$	−15.5934	−0.548234	−0.274117	−	0.961696i	$-0.588386\pi$
$809$	−15.5934	−0.548234	−0.274117	+	0.961696i	$0.588386\pi$
$810$	0	0
$811$	32.7802	1.15107	0.575534	−	0.817778i	$-0.304794\pi$
$811$	32.7802	1.15107	0.575534	+	0.817778i	$0.304794\pi$
$812$	0	0
$813$	0	0
$814$	−46.1245	−1.61666
$815$	5.06226	0.177323
$816$	0	0
$817$	−38.6556	−1.35239
$818$	−7.06226	−0.246926
$819$	0	0
$820$	−7.06226	−0.246625
$821$	−42.1245	−1.47016	−0.735078	−	0.677983i	$-0.762854\pi$
$821$	−42.1245	−1.47016	−0.735078	+	0.677983i	$0.762854\pi$
$822$	0	0
$823$	40.1245	1.39865	0.699326	−	0.714803i	$-0.253483\pi$
$823$	40.1245	1.39865	0.699326	+	0.714803i	$0.253483\pi$
$824$	0	0
$825$	0	0
$826$	41.0623	1.42874
$827$	−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
$829$	30.6556	1.06471	0.532357	−	0.846520i	$-0.321306\pi$
$829$	30.6556	1.06471	0.532357	+	0.846520i	$0.321306\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	6.00000	0.208013
$833$	54.1245	1.87530
$834$	0	0
$835$	23.5934	0.816483
$836$	−29.5934	−1.02351
$837$	0	0
$838$	0	0
$839$	−33.8424	−1.16837	−0.584185	−	0.811621i	$-0.698586\pi$
$839$	−33.8424	−1.16837	−0.584185	+	0.811621i	$0.698586\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	26.2490	0.904601
$843$	0	0
$844$	−1.59339	−0.0548466
$845$	23.0000	0.791224
$846$	0	0
$847$	−143.436	−4.92851
$848$	2.53113	0.0869193
$849$	0	0
$850$	−4.00000	−0.137199
$851$	46.1245	1.58113
$852$	0	0
$853$	57.7179	1.97622	0.988112	−	0.153738i	$-0.0491311\pi$
$853$	57.7179	1.97622	0.988112	+	0.153738i	$0.0491311\pi$
$854$	22.9377	0.784913
$855$	0	0
$856$	9.59339	0.327895
$857$	−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
$859$	−23.0623	−0.786874	−0.393437	−	0.919352i	$-0.628714\pi$
$859$	−23.0623	−0.786874	−0.393437	+	0.919352i	$0.628714\pi$
$860$	−8.53113	−0.290909
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−33.4689	−1.13929	−0.569647	−	0.821890i	$-0.692920\pi$
$863$	−33.4689	−1.13929	−0.569647	+	0.821890i	$0.692920\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	25.5934	0.869699
$867$	0	0
$868$	−4.53113	−0.153797
$869$	55.7179	1.89010
$870$	0	0
$871$	30.3735	1.02917
$872$	−15.0623	−0.510072
$873$	0	0
$874$	29.5934	1.00101
$875$	−4.53113	−0.153180
$876$	0	0
$877$	11.8755	0.401007	0.200503	−	0.979693i	$-0.435742\pi$
$877$	11.8755	0.401007	0.200503	+	0.979693i	$0.435742\pi$
$878$	25.0623	0.845810
$879$	0	0
$880$	−6.53113	−0.220164
$881$	−3.87548	−0.130568	−0.0652842	−	0.997867i	$-0.520795\pi$
$881$	−3.87548	−0.130568	−0.0652842	+	0.997867i	$0.520795\pi$
$882$	0	0
$883$	−13.3444	−0.449073	−0.224537	−	0.974466i	$-0.572087\pi$
$883$	−13.3444	−0.449073	−0.224537	+	0.974466i	$0.572087\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	14.4066	0.484000
$887$	−23.1868	−0.778536	−0.389268	−	0.921125i	$-0.627272\pi$
$887$	−23.1868	−0.778536	−0.389268	+	0.921125i	$0.627272\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	−6.53113	−0.218924
$891$	0	0
$892$	2.00000	0.0669650
$893$	22.9377	0.767582
$894$	0	0
$895$	3.06226	0.102360
$896$	4.53113	0.151374
$897$	0	0
$898$	−4.12452	−0.137637
$899$	0	0
$900$	0	0
$901$	10.1245	0.337297
$902$	−46.1245	−1.53578
$903$	0	0
$904$	−7.59339	−0.252552
$905$	2.40661	0.0799985
$906$	0	0
$907$	32.2490	1.07081	0.535406	−	0.844595i	$-0.320159\pi$
$907$	32.2490	1.07081	0.535406	+	0.844595i	$0.320159\pi$
$908$	24.5311	0.814094
$909$	0	0
$910$	27.1868	0.901233
$911$	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
$913$	−52.2490	−1.72919
$914$	−14.9377	−0.494097
$915$	0	0
$916$	−5.59339	−0.184811
$917$	−41.0623	−1.35600
$918$	0	0
$919$	−25.0623	−0.826728	−0.413364	−	0.910566i	$-0.635646\pi$
$919$	−25.0623	−0.826728	−0.413364	+	0.910566i	$0.635646\pi$
$920$	6.53113	0.215325
$921$	0	0
$922$	24.0000	0.790398
$923$	75.1868	2.47480
$924$	0	0
$925$	−7.06226	−0.232206
$926$	−13.1868	−0.433344
$927$	0	0
$928$	0	0
$929$	51.8424	1.70089	0.850447	−	0.526060i	$-0.176331\pi$
$929$	51.8424	1.70089	0.850447	+	0.526060i	$0.176331\pi$
$930$	0	0
$931$	61.3113	2.00940
$932$	−1.46887	−0.0481145
$933$	0	0
$934$	−22.1245	−0.723936
$935$	−26.1245	−0.854363
$936$	0	0
$937$	−8.93774	−0.291983	−0.145992	−	0.989286i	$-0.546637\pi$
$937$	−8.93774	−0.291983	−0.145992	+	0.989286i	$0.546637\pi$
$938$	22.9377	0.748944
$939$	0	0
$940$	5.06226	0.165113
$941$	54.1245	1.76441	0.882204	−	0.470867i	$-0.156059\pi$
$941$	54.1245	1.76441	0.882204	+	0.470867i	$0.156059\pi$
$942$	0	0
$943$	46.1245	1.50202
$944$	9.06226	0.294951
$945$	0	0
$946$	−55.7179	−1.81155
$947$	−18.9377	−0.615394	−0.307697	−	0.951484i	$-0.599558\pi$
$947$	−18.9377	−0.615394	−0.307697	+	0.951484i	$0.599558\pi$
$948$	0	0
$949$	−51.1868	−1.66159
$950$	−4.53113	−0.147009
$951$	0	0
$952$	18.1245	0.587419
$953$	−21.8755	−0.708616	−0.354308	−	0.935129i	$-0.615284\pi$
$953$	−21.8755	−0.708616	−0.354308	+	0.935129i	$0.615284\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	8.00000	0.258738
$957$	0	0
$958$	1.34436	0.0434342
$959$	77.3113	2.49651
$960$	0	0
$961$	1.00000	0.0322581
$962$	42.3735	1.36618
$963$	0	0
$964$	2.00000	0.0644157
$965$	14.0000	0.450676
$966$	0	0
$967$	−0.937742	−0.0301558	−0.0150779	−	0.999886i	$-0.504800\pi$
$967$	−0.937742	−0.0301558	−0.0150779	+	0.999886i	$0.504800\pi$
$968$	−31.6556	−1.01745
$969$	0	0
$970$	−16.1245	−0.517727
$971$	−15.1868	−0.487367	−0.243683	−	0.969855i	$-0.578356\pi$
$971$	−15.1868	−0.487367	−0.243683	+	0.969855i	$0.578356\pi$
$972$	0	0
$973$	−27.1868	−0.871568
$974$	−23.0623	−0.738962
$975$	0	0
$976$	5.06226	0.162039
$977$	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
$979$	−42.6556	−1.36328
$980$	13.5311	0.432236
$981$	0	0
$982$	−7.59339	−0.242315
$983$	−0.937742	−0.0299093	−0.0149547	−	0.999888i	$-0.504760\pi$
$983$	−0.937742	−0.0299093	−0.0149547	+	0.999888i	$0.504760\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	27.1868	0.864926
$989$	55.7179	1.77173
$990$	0	0
$991$	−32.5311	−1.03339	−0.516693	−	0.856171i	$-0.672837\pi$
$991$	−32.5311	−1.03339	−0.516693	+	0.856171i	$0.672837\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	56.7802	1.80096
$995$	−8.53113	−0.270455
$996$	0	0
$997$	−2.24903	−0.0712275	−0.0356138	−	0.999366i	$-0.511339\pi$
$997$	−2.24903	−0.0712275	−0.0356138	+	0.999366i	$0.511339\pi$
$998$	7.06226	0.223552
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bf.1.1		2
3.2	odd	2		930.2.a.q.1.1	✓	2
12.11	even	2		7440.2.a.bd.1.2		2
15.2	even	4		4650.2.d.bg.3349.3		4
15.8	even	4		4650.2.d.bg.3349.2		4
15.14	odd	2		4650.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.1	✓	2	3.2	odd	2
2790.2.a.bf.1.1		2	1.1	even	1	trivial
4650.2.a.bz.1.2		2	15.14	odd	2
4650.2.d.bg.3349.2		4	15.8	even	4
4650.2.d.bg.3349.3		4	15.2	even	4
7440.2.a.bd.1.2		2	12.11	even	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.53113 q^{7} -1.00000 q^{8} -1.00000 q^{10} -6.53113 q^{11} +6.00000 q^{13} +4.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} +4.53113 q^{19} +1.00000 q^{20} +6.53113 q^{22} -6.53113 q^{23} +1.00000 q^{25} -6.00000 q^{26} -4.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.53113 q^{35} -7.06226 q^{37} -4.53113 q^{38} -1.00000 q^{40} -7.06226 q^{41} -8.53113 q^{43} -6.53113 q^{44} +6.53113 q^{46} +5.06226 q^{47} +13.5311 q^{49} -1.00000 q^{50} +6.00000 q^{52} +2.53113 q^{53} -6.53113 q^{55} +4.53113 q^{56} +9.06226 q^{59} +5.06226 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +5.06226 q^{67} +4.00000 q^{68} +4.53113 q^{70} +12.5311 q^{71} -8.53113 q^{73} +7.06226 q^{74} +4.53113 q^{76} +29.5934 q^{77} -8.53113 q^{79} +1.00000 q^{80} +7.06226 q^{82} +8.00000 q^{83} +4.00000 q^{85} +8.53113 q^{86} +6.53113 q^{88} +6.53113 q^{89} -27.1868 q^{91} -6.53113 q^{92} -5.06226 q^{94} +4.53113 q^{95} +16.1245 q^{97} -13.5311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 5 q^{11} + 12 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + q^{19} + 2 q^{20} + 5 q^{22} - 5 q^{23} + 2 q^{25} - 12 q^{26} - q^{28} + 2 q^{31}+ \cdots - 19 q^{98}+O(q^{100}) \) 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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