LMFDB - Embedded newform 2790.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.73205 q^{11} -1.26795 q^{13} -3.46410 q^{14} +1.00000 q^{16} +4.00000 q^{17} +5.46410 q^{19} +1.00000 q^{20} +2.73205 q^{22} +0.535898 q^{23} +1.00000 q^{25} -1.26795 q^{26} -3.46410 q^{28} -0.732051 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -3.46410 q^{35} -6.73205 q^{37} +5.46410 q^{38} +1.00000 q^{40} +2.53590 q^{41} +3.26795 q^{43} +2.73205 q^{44} +0.535898 q^{46} +3.46410 q^{47} +5.00000 q^{49} +1.00000 q^{50} -1.26795 q^{52} +5.26795 q^{53} +2.73205 q^{55} -3.46410 q^{56} -0.732051 q^{58} -5.46410 q^{59} +9.12436 q^{61} -1.00000 q^{62} +1.00000 q^{64} -1.26795 q^{65} +8.39230 q^{67} +4.00000 q^{68} -3.46410 q^{70} +13.8564 q^{71} -4.39230 q^{73} -6.73205 q^{74} +5.46410 q^{76} -9.46410 q^{77} +17.4641 q^{79} +1.00000 q^{80} +2.53590 q^{82} +4.73205 q^{83} +4.00000 q^{85} +3.26795 q^{86} +2.73205 q^{88} -0.928203 q^{89} +4.39230 q^{91} +0.535898 q^{92} +3.46410 q^{94} +5.46410 q^{95} +4.00000 q^{97} +5.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} - 6 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{34} - 10 q^{37} + 4 q^{38} + 2 q^{40} + 12 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 10 q^{49} + 2 q^{50} - 6 q^{52} + 14 q^{53} + 2 q^{55} + 2 q^{58} - 4 q^{59} - 6 q^{61} - 2 q^{62} + 2 q^{64} - 6 q^{65} - 4 q^{67} + 8 q^{68} + 12 q^{73} - 10 q^{74} + 4 q^{76} - 12 q^{77} + 28 q^{79} + 2 q^{80} + 12 q^{82} + 6 q^{83} + 8 q^{85} + 10 q^{86} + 2 q^{88} + 12 q^{89} - 12 q^{91} + 8 q^{92} + 4 q^{95} + 8 q^{97} + 10 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 - 6 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 + 2 * q^20 + 2 * q^22 + 8 * q^23 + 2 * q^25 - 6 * q^26 + 2 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^34 - 10 * q^37 + 4 * q^38 + 2 * q^40 + 12 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 10 * q^49 + 2 * q^50 - 6 * q^52 + 14 * q^53 + 2 * q^55 + 2 * q^58 - 4 * q^59 - 6 * q^61 - 2 * q^62 + 2 * q^64 - 6 * q^65 - 4 * q^67 + 8 * q^68 + 12 * q^73 - 10 * q^74 + 4 * q^76 - 12 * q^77 + 28 * q^79 + 2 * q^80 + 12 * q^82 + 6 * q^83 + 8 * q^85 + 10 * q^86 + 2 * q^88 + 12 * q^89 - 12 * q^91 + 8 * q^92 + 4 * q^95 + 8 * q^97 + 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	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
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	−1.26795	−0.351666	−0.175833	−	0.984420i	$-0.556262\pi$
$13$	−1.26795	−0.351666	−0.175833	+	0.984420i	$0.556262\pi$
$14$	−3.46410	−0.925820
$15$	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
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	2.73205	0.582475
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.26795	−0.248665
$27$	0	0
$28$	−3.46410	−0.654654
$29$	−0.732051	−0.135938	−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051	−0.135938	−0.0679692	+	0.997687i	$0.521652\pi$
$30$	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
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−6.73205	−1.10674	−0.553371	−	0.832935i	$-0.686659\pi$
$37$	−6.73205	−1.10674	−0.553371	+	0.832935i	$0.686659\pi$
$38$	5.46410	0.886394
$39$	0	0
$40$	1.00000	0.158114
$41$	2.53590	0.396041	0.198020	−	0.980198i	$-0.436549\pi$
$41$	2.53590	0.396041	0.198020	+	0.980198i	$0.436549\pi$
$42$	0	0
$43$	3.26795	0.498358	0.249179	−	0.968458i	$-0.419839\pi$
$43$	3.26795	0.498358	0.249179	+	0.968458i	$0.419839\pi$
$44$	2.73205	0.411872
$45$	0	0
$46$	0.535898	0.0790139
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.26795	−0.175833
$53$	5.26795	0.723608	0.361804	−	0.932254i	$-0.382161\pi$
$53$	5.26795	0.723608	0.361804	+	0.932254i	$0.382161\pi$
$54$	0	0
$55$	2.73205	0.368390
$56$	−3.46410	−0.462910
$57$	0	0
$58$	−0.732051	−0.0961230
$59$	−5.46410	−0.711365	−0.355683	−	0.934607i	$-0.615752\pi$
$59$	−5.46410	−0.711365	−0.355683	+	0.934607i	$0.615752\pi$
$60$	0	0
$61$	9.12436	1.16825	0.584127	−	0.811662i	$-0.301437\pi$
$61$	9.12436	1.16825	0.584127	+	0.811662i	$0.301437\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.26795	−0.157270
$66$	0	0
$67$	8.39230	1.02528	0.512642	−	0.858603i	$-0.328667\pi$
$67$	8.39230	1.02528	0.512642	+	0.858603i	$0.328667\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	−3.46410	−0.414039
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	0	0
$73$	−4.39230	−0.514080	−0.257040	−	0.966401i	$-0.582747\pi$
$73$	−4.39230	−0.514080	−0.257040	+	0.966401i	$0.582747\pi$
$74$	−6.73205	−0.782585
$75$	0	0
$76$	5.46410	0.626775
$77$	−9.46410	−1.07853
$78$	0	0
$79$	17.4641	1.96486	0.982432	−	0.186618i	$-0.0597528\pi$
$79$	17.4641	1.96486	0.982432	+	0.186618i	$0.0597528\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	2.53590	0.280043
$83$	4.73205	0.519410	0.259705	−	0.965688i	$-0.416375\pi$
$83$	4.73205	0.519410	0.259705	+	0.965688i	$0.416375\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	3.26795	0.352392
$87$	0	0
$88$	2.73205	0.291238
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	4.39230	0.460439
$92$	0.535898	0.0558713
$93$	0	0
$94$	3.46410	0.357295
$95$	5.46410	0.560605
$96$	0	0
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	5.00000	0.505076
$99$	0	0
$100$	1.00000	0.100000
$101$	16.9282	1.68442	0.842210	−	0.539150i	$-0.181255\pi$
$101$	16.9282	1.68442	0.842210	+	0.539150i	$0.181255\pi$
$102$	0	0
$103$	2.92820	0.288524	0.144262	−	0.989539i	$-0.453919\pi$
$103$	2.92820	0.288524	0.144262	+	0.989539i	$0.453919\pi$
$104$	−1.26795	−0.124333
$105$	0	0
$106$	5.26795	0.511668
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−15.4641	−1.48119	−0.740596	−	0.671950i	$-0.765457\pi$
$109$	−15.4641	−1.48119	−0.740596	+	0.671950i	$0.765457\pi$
$110$	2.73205	0.260491
$111$	0	0
$112$	−3.46410	−0.327327
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	0.535898	0.0499728
$116$	−0.732051	−0.0679692
$117$	0	0
$118$	−5.46410	−0.503011
$119$	−13.8564	−1.27021
$120$	0	0
$121$	−3.53590	−0.321445
$122$	9.12436	0.826080
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.8564	−1.40703	−0.703514	−	0.710681i	$-0.748387\pi$
$127$	−15.8564	−1.40703	−0.703514	+	0.710681i	$0.748387\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.26795	−0.111207
$131$	−12.3923	−1.08272	−0.541360	−	0.840791i	$-0.682091\pi$
$131$	−12.3923	−1.08272	−0.541360	+	0.840791i	$0.682091\pi$
$132$	0	0
$133$	−18.9282	−1.64128
$134$	8.39230	0.724985
$135$	0	0
$136$	4.00000	0.342997
$137$	20.7846	1.77575	0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846	1.77575	0.887875	+	0.460086i	$0.152181\pi$
$138$	0	0
$139$	13.6603	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$139$	13.6603	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$140$	−3.46410	−0.292770
$141$	0	0
$142$	13.8564	1.16280
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−0.732051	−0.0607935
$146$	−4.39230	−0.363510
$147$	0	0
$148$	−6.73205	−0.553371
$149$	13.3205	1.09126	0.545629	−	0.838027i	$-0.316291\pi$
$149$	13.3205	1.09126	0.545629	+	0.838027i	$0.316291\pi$
$150$	0	0
$151$	1.46410	0.119147	0.0595734	−	0.998224i	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	5.46410	0.443197
$153$	0	0
$154$	−9.46410	−0.762639
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−19.8564	−1.58471	−0.792357	−	0.610058i	$-0.791146\pi$
$157$	−19.8564	−1.58471	−0.792357	+	0.610058i	$0.791146\pi$
$158$	17.4641	1.38937
$159$	0	0
$160$	1.00000	0.0790569
$161$	−1.85641	−0.146305
$162$	0	0
$163$	−9.46410	−0.741286	−0.370643	−	0.928775i	$-0.620863\pi$
$163$	−9.46410	−0.741286	−0.370643	+	0.928775i	$0.620863\pi$
$164$	2.53590	0.198020
$165$	0	0
$166$	4.73205	0.367278
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	−11.3923	−0.876331
$170$	4.00000	0.306786
$171$	0	0
$172$	3.26795	0.249179
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−3.46410	−0.261861
$176$	2.73205	0.205936
$177$	0	0
$178$	−0.928203	−0.0695718
$179$	14.7321	1.10113	0.550563	−	0.834794i	$-0.314413\pi$
$179$	14.7321	1.10113	0.550563	+	0.834794i	$0.314413\pi$
$180$	0	0
$181$	−9.12436	−0.678208	−0.339104	−	0.940749i	$-0.610124\pi$
$181$	−9.12436	−0.678208	−0.339104	+	0.940749i	$0.610124\pi$
$182$	4.39230	0.325579
$183$	0	0
$184$	0.535898	0.0395070
$185$	−6.73205	−0.494950
$186$	0	0
$187$	10.9282	0.799149
$188$	3.46410	0.252646
$189$	0	0
$190$	5.46410	0.396408
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	0	0
$193$	15.8564	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$193$	15.8564	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	5.00000	0.357143
$197$	−21.2679	−1.51528	−0.757639	−	0.652673i	$-0.773647\pi$
$197$	−21.2679	−1.51528	−0.757639	+	0.652673i	$0.773647\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	16.9282	1.19106
$203$	2.53590	0.177985
$204$	0	0
$205$	2.53590	0.177115
$206$	2.92820	0.204018
$207$	0	0
$208$	−1.26795	−0.0879165
$209$	14.9282	1.03261
$210$	0	0
$211$	26.9282	1.85381	0.926907	−	0.375291i	$-0.122457\pi$
$211$	26.9282	1.85381	0.926907	+	0.375291i	$0.122457\pi$
$212$	5.26795	0.361804
$213$	0	0
$214$	−4.00000	−0.273434
$215$	3.26795	0.222872
$216$	0	0
$217$	3.46410	0.235159
$218$	−15.4641	−1.04736
$219$	0	0
$220$	2.73205	0.184195
$221$	−5.07180	−0.341166
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	−3.46410	−0.231455
$225$	0	0
$226$	−8.00000	−0.532152
$227$	13.4641	0.893644	0.446822	−	0.894623i	$-0.352556\pi$
$227$	13.4641	0.893644	0.446822	+	0.894623i	$0.352556\pi$
$228$	0	0
$229$	−24.0526	−1.58944	−0.794719	−	0.606978i	$-0.792382\pi$
$229$	−24.0526	−1.58944	−0.794719	+	0.606978i	$0.792382\pi$
$230$	0.535898	0.0353361
$231$	0	0
$232$	−0.732051	−0.0480615
$233$	−24.7846	−1.62369	−0.811847	−	0.583870i	$-0.801538\pi$
$233$	−24.7846	−1.62369	−0.811847	+	0.583870i	$0.801538\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	−5.46410	−0.355683
$237$	0	0
$238$	−13.8564	−0.898177
$239$	9.46410	0.612182	0.306091	−	0.952002i	$-0.400979\pi$
$239$	9.46410	0.612182	0.306091	+	0.952002i	$0.400979\pi$
$240$	0	0
$241$	−20.2487	−1.30433	−0.652167	−	0.758075i	$-0.726140\pi$
$241$	−20.2487	−1.30433	−0.652167	+	0.758075i	$0.726140\pi$
$242$	−3.53590	−0.227296
$243$	0	0
$244$	9.12436	0.584127
$245$	5.00000	0.319438
$246$	0	0
$247$	−6.92820	−0.440831
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	20.9808	1.32429	0.662147	−	0.749374i	$-0.269645\pi$
$251$	20.9808	1.32429	0.662147	+	0.749374i	$0.269645\pi$
$252$	0	0
$253$	1.46410	0.0920473
$254$	−15.8564	−0.994919
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.9282	−1.05595	−0.527976	−	0.849259i	$-0.677049\pi$
$257$	−16.9282	−1.05595	−0.527976	+	0.849259i	$0.677049\pi$
$258$	0	0
$259$	23.3205	1.44907
$260$	−1.26795	−0.0786349
$261$	0	0
$262$	−12.3923	−0.765599
$263$	−26.7846	−1.65161	−0.825805	−	0.563956i	$-0.809279\pi$
$263$	−26.7846	−1.65161	−0.825805	+	0.563956i	$0.809279\pi$
$264$	0	0
$265$	5.26795	0.323608
$266$	−18.9282	−1.16056
$267$	0	0
$268$	8.39230	0.512642
$269$	0.339746	0.0207147	0.0103573	−	0.999946i	$-0.496703\pi$
$269$	0.339746	0.0207147	0.0103573	+	0.999946i	$0.496703\pi$
$270$	0	0
$271$	−5.46410	−0.331921	−0.165960	−	0.986132i	$-0.553072\pi$
$271$	−5.46410	−0.331921	−0.165960	+	0.986132i	$0.553072\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	20.7846	1.25564
$275$	2.73205	0.164749
$276$	0	0
$277$	−27.1244	−1.62974	−0.814872	−	0.579641i	$-0.803193\pi$
$277$	−27.1244	−1.62974	−0.814872	+	0.579641i	$0.803193\pi$
$278$	13.6603	0.819288
$279$	0	0
$280$	−3.46410	−0.207020
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	9.46410	0.562582	0.281291	−	0.959622i	$-0.409237\pi$
$283$	9.46410	0.562582	0.281291	+	0.959622i	$0.409237\pi$
$284$	13.8564	0.822226
$285$	0	0
$286$	−3.46410	−0.204837
$287$	−8.78461	−0.518539
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−0.732051	−0.0429875
$291$	0	0
$292$	−4.39230	−0.257040
$293$	17.3205	1.01187	0.505937	−	0.862570i	$-0.331147\pi$
$293$	17.3205	1.01187	0.505937	+	0.862570i	$0.331147\pi$
$294$	0	0
$295$	−5.46410	−0.318132
$296$	−6.73205	−0.391293
$297$	0	0
$298$	13.3205	0.771636
$299$	−0.679492	−0.0392960
$300$	0	0
$301$	−11.3205	−0.652503
$302$	1.46410	0.0842496
$303$	0	0
$304$	5.46410	0.313388
$305$	9.12436	0.522459
$306$	0	0
$307$	−24.7846	−1.41453	−0.707266	−	0.706947i	$-0.750072\pi$
$307$	−24.7846	−1.41453	−0.707266	+	0.706947i	$0.750072\pi$
$308$	−9.46410	−0.539267
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	−13.0718	−0.741234	−0.370617	−	0.928786i	$-0.620854\pi$
$311$	−13.0718	−0.741234	−0.370617	+	0.928786i	$0.620854\pi$
$312$	0	0
$313$	28.7846	1.62700	0.813501	−	0.581563i	$-0.197559\pi$
$313$	28.7846	1.62700	0.813501	+	0.581563i	$0.197559\pi$
$314$	−19.8564	−1.12056
$315$	0	0
$316$	17.4641	0.982432
$317$	−23.4641	−1.31788	−0.658938	−	0.752198i	$-0.728994\pi$
$317$	−23.4641	−1.31788	−0.658938	+	0.752198i	$0.728994\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	−1.85641	−0.103453
$323$	21.8564	1.21612
$324$	0	0
$325$	−1.26795	−0.0703332
$326$	−9.46410	−0.524168
$327$	0	0
$328$	2.53590	0.140022
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−13.2679	−0.729272	−0.364636	−	0.931150i	$-0.618807\pi$
$331$	−13.2679	−0.729272	−0.364636	+	0.931150i	$0.618807\pi$
$332$	4.73205	0.259705
$333$	0	0
$334$	−17.3205	−0.947736
$335$	8.39230	0.458521
$336$	0	0
$337$	−11.6077	−0.632311	−0.316156	−	0.948707i	$-0.602392\pi$
$337$	−11.6077	−0.632311	−0.316156	+	0.948707i	$0.602392\pi$
$338$	−11.3923	−0.619660
$339$	0	0
$340$	4.00000	0.216930
$341$	−2.73205	−0.147949
$342$	0	0
$343$	6.92820	0.374088
$344$	3.26795	0.176196
$345$	0	0
$346$	2.00000	0.107521
$347$	20.0526	1.07648	0.538239	−	0.842792i	$-0.319090\pi$
$347$	20.0526	1.07648	0.538239	+	0.842792i	$0.319090\pi$
$348$	0	0
$349$	−28.5359	−1.52749	−0.763746	−	0.645517i	$-0.776642\pi$
$349$	−28.5359	−1.52749	−0.763746	+	0.645517i	$0.776642\pi$
$350$	−3.46410	−0.185164
$351$	0	0
$352$	2.73205	0.145619
$353$	16.3923	0.872474	0.436237	−	0.899832i	$-0.356311\pi$
$353$	16.3923	0.872474	0.436237	+	0.899832i	$0.356311\pi$
$354$	0	0
$355$	13.8564	0.735422
$356$	−0.928203	−0.0491947
$357$	0	0
$358$	14.7321	0.778613
$359$	32.9282	1.73788	0.868942	−	0.494914i	$-0.164800\pi$
$359$	32.9282	1.73788	0.868942	+	0.494914i	$0.164800\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−9.12436	−0.479565
$363$	0	0
$364$	4.39230	0.230219
$365$	−4.39230	−0.229904
$366$	0	0
$367$	−28.5359	−1.48956	−0.744781	−	0.667309i	$-0.767446\pi$
$367$	−28.5359	−1.48956	−0.744781	+	0.667309i	$0.767446\pi$
$368$	0.535898	0.0279356
$369$	0	0
$370$	−6.73205	−0.349983
$371$	−18.2487	−0.947426
$372$	0	0
$373$	9.32051	0.482598	0.241299	−	0.970451i	$-0.422427\pi$
$373$	9.32051	0.482598	0.241299	+	0.970451i	$0.422427\pi$
$374$	10.9282	0.565084
$375$	0	0
$376$	3.46410	0.178647
$377$	0.928203	0.0478049
$378$	0	0
$379$	−7.32051	−0.376029	−0.188015	−	0.982166i	$-0.560205\pi$
$379$	−7.32051	−0.376029	−0.188015	+	0.982166i	$0.560205\pi$
$380$	5.46410	0.280302
$381$	0	0
$382$	2.00000	0.102329
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	−9.46410	−0.482335
$386$	15.8564	0.807070
$387$	0	0
$388$	4.00000	0.203069
$389$	−26.1962	−1.32820	−0.664099	−	0.747645i	$-0.731185\pi$
$389$	−26.1962	−1.32820	−0.664099	+	0.747645i	$0.731185\pi$
$390$	0	0
$391$	2.14359	0.108406
$392$	5.00000	0.252538
$393$	0	0
$394$	−21.2679	−1.07146
$395$	17.4641	0.878714
$396$	0	0
$397$	16.2487	0.815499	0.407750	−	0.913094i	$-0.366314\pi$
$397$	16.2487	0.815499	0.407750	+	0.913094i	$0.366314\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	7.46410	0.372739	0.186370	−	0.982480i	$-0.440328\pi$
$401$	7.46410	0.372739	0.186370	+	0.982480i	$0.440328\pi$
$402$	0	0
$403$	1.26795	0.0631610
$404$	16.9282	0.842210
$405$	0	0
$406$	2.53590	0.125855
$407$	−18.3923	−0.911673
$408$	0	0
$409$	−8.53590	−0.422073	−0.211037	−	0.977478i	$-0.567684\pi$
$409$	−8.53590	−0.422073	−0.211037	+	0.977478i	$0.567684\pi$
$410$	2.53590	0.125239
$411$	0	0
$412$	2.92820	0.144262
$413$	18.9282	0.931396
$414$	0	0
$415$	4.73205	0.232287
$416$	−1.26795	−0.0621663
$417$	0	0
$418$	14.9282	0.730162
$419$	4.39230	0.214578	0.107289	−	0.994228i	$-0.465783\pi$
$419$	4.39230	0.214578	0.107289	+	0.994228i	$0.465783\pi$
$420$	0	0
$421$	32.2487	1.57171	0.785853	−	0.618413i	$-0.212224\pi$
$421$	32.2487	1.57171	0.785853	+	0.618413i	$0.212224\pi$
$422$	26.9282	1.31084
$423$	0	0
$424$	5.26795	0.255834
$425$	4.00000	0.194029
$426$	0	0
$427$	−31.6077	−1.52960
$428$	−4.00000	−0.193347
$429$	0	0
$430$	3.26795	0.157595
$431$	−26.7846	−1.29017	−0.645085	−	0.764111i	$-0.723178\pi$
$431$	−26.7846	−1.29017	−0.645085	+	0.764111i	$0.723178\pi$
$432$	0	0
$433$	16.7846	0.806617	0.403308	−	0.915064i	$-0.367860\pi$
$433$	16.7846	0.806617	0.403308	+	0.915064i	$0.367860\pi$
$434$	3.46410	0.166282
$435$	0	0
$436$	−15.4641	−0.740596
$437$	2.92820	0.140075
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	2.73205	0.130245
$441$	0	0
$442$	−5.07180	−0.241241
$443$	34.9282	1.65949	0.829745	−	0.558143i	$-0.188486\pi$
$443$	34.9282	1.65949	0.829745	+	0.558143i	$0.188486\pi$
$444$	0	0
$445$	−0.928203	−0.0440011
$446$	10.0000	0.473514
$447$	0	0
$448$	−3.46410	−0.163663
$449$	−29.3205	−1.38372	−0.691860	−	0.722032i	$-0.743209\pi$
$449$	−29.3205	−1.38372	−0.691860	+	0.722032i	$0.743209\pi$
$450$	0	0
$451$	6.92820	0.326236
$452$	−8.00000	−0.376288
$453$	0	0
$454$	13.4641	0.631902
$455$	4.39230	0.205914
$456$	0	0
$457$	14.9282	0.698312	0.349156	−	0.937065i	$-0.386468\pi$
$457$	14.9282	0.698312	0.349156	+	0.937065i	$0.386468\pi$
$458$	−24.0526	−1.12390
$459$	0	0
$460$	0.535898	0.0249864
$461$	27.2679	1.27000	0.634998	−	0.772514i	$-0.281001\pi$
$461$	27.2679	1.27000	0.634998	+	0.772514i	$0.281001\pi$
$462$	0	0
$463$	−10.3923	−0.482971	−0.241486	−	0.970404i	$-0.577635\pi$
$463$	−10.3923	−0.482971	−0.241486	+	0.970404i	$0.577635\pi$
$464$	−0.732051	−0.0339846
$465$	0	0
$466$	−24.7846	−1.14812
$467$	−6.14359	−0.284292	−0.142146	−	0.989846i	$-0.545400\pi$
$467$	−6.14359	−0.284292	−0.142146	+	0.989846i	$0.545400\pi$
$468$	0	0
$469$	−29.0718	−1.34241
$470$	3.46410	0.159787
$471$	0	0
$472$	−5.46410	−0.251506
$473$	8.92820	0.410519
$474$	0	0
$475$	5.46410	0.250710
$476$	−13.8564	−0.635107
$477$	0	0
$478$	9.46410	0.432878
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	8.53590	0.389203
$482$	−20.2487	−0.922304
$483$	0	0
$484$	−3.53590	−0.160723
$485$	4.00000	0.181631
$486$	0	0
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	9.12436	0.413040
$489$	0	0
$490$	5.00000	0.225877
$491$	−18.7321	−0.845366	−0.422683	−	0.906278i	$-0.638912\pi$
$491$	−18.7321	−0.845366	−0.422683	+	0.906278i	$0.638912\pi$
$492$	0	0
$493$	−2.92820	−0.131880
$494$	−6.92820	−0.311715
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−48.0000	−2.15309
$498$	0	0
$499$	19.1244	0.856124	0.428062	−	0.903749i	$-0.359197\pi$
$499$	19.1244	0.856124	0.428062	+	0.903749i	$0.359197\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	20.9808	0.936417
$503$	−30.6410	−1.36622	−0.683108	−	0.730318i	$-0.739372\pi$
$503$	−30.6410	−1.36622	−0.683108	+	0.730318i	$0.739372\pi$
$504$	0	0
$505$	16.9282	0.753295
$506$	1.46410	0.0650873
$507$	0	0
$508$	−15.8564	−0.703514
$509$	12.3397	0.546950	0.273475	−	0.961879i	$-0.411827\pi$
$509$	12.3397	0.546950	0.273475	+	0.961879i	$0.411827\pi$
$510$	0	0
$511$	15.2154	0.673089
$512$	1.00000	0.0441942
$513$	0	0
$514$	−16.9282	−0.746671
$515$	2.92820	0.129032
$516$	0	0
$517$	9.46410	0.416231
$518$	23.3205	1.02464
$519$	0	0
$520$	−1.26795	−0.0556033
$521$	−4.92820	−0.215909	−0.107954	−	0.994156i	$-0.534430\pi$
$521$	−4.92820	−0.215909	−0.107954	+	0.994156i	$0.534430\pi$
$522$	0	0
$523$	−30.1962	−1.32039	−0.660193	−	0.751096i	$-0.729525\pi$
$523$	−30.1962	−1.32039	−0.660193	+	0.751096i	$0.729525\pi$
$524$	−12.3923	−0.541360
$525$	0	0
$526$	−26.7846	−1.16786
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−22.7128	−0.987514
$530$	5.26795	0.228825
$531$	0	0
$532$	−18.9282	−0.820642
$533$	−3.21539	−0.139274
$534$	0	0
$535$	−4.00000	−0.172935
$536$	8.39230	0.362492
$537$	0	0
$538$	0.339746	0.0146475
$539$	13.6603	0.588389
$540$	0	0
$541$	−21.3205	−0.916640	−0.458320	−	0.888787i	$-0.651549\pi$
$541$	−21.3205	−0.916640	−0.458320	+	0.888787i	$0.651549\pi$
$542$	−5.46410	−0.234703
$543$	0	0
$544$	4.00000	0.171499
$545$	−15.4641	−0.662409
$546$	0	0
$547$	−31.3205	−1.33917	−0.669584	−	0.742736i	$-0.733528\pi$
$547$	−31.3205	−1.33917	−0.669584	+	0.742736i	$0.733528\pi$
$548$	20.7846	0.887875
$549$	0	0
$550$	2.73205	0.116495
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−60.4974	−2.57261
$554$	−27.1244	−1.15240
$555$	0	0
$556$	13.6603	0.579324
$557$	−27.1244	−1.14930	−0.574648	−	0.818401i	$-0.694861\pi$
$557$	−27.1244	−1.14930	−0.574648	+	0.818401i	$0.694861\pi$
$558$	0	0
$559$	−4.14359	−0.175255
$560$	−3.46410	−0.146385
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−7.60770	−0.320626	−0.160313	−	0.987066i	$-0.551250\pi$
$563$	−7.60770	−0.320626	−0.160313	+	0.987066i	$0.551250\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	9.46410	0.397806
$567$	0	0
$568$	13.8564	0.581402
$569$	−29.3205	−1.22918	−0.614590	−	0.788847i	$-0.710678\pi$
$569$	−29.3205	−1.22918	−0.614590	+	0.788847i	$0.710678\pi$
$570$	0	0
$571$	28.5885	1.19639	0.598195	−	0.801351i	$-0.295885\pi$
$571$	28.5885	1.19639	0.598195	+	0.801351i	$0.295885\pi$
$572$	−3.46410	−0.144841
$573$	0	0
$574$	−8.78461	−0.366663
$575$	0.535898	0.0223485
$576$	0	0
$577$	−18.9282	−0.787991	−0.393996	−	0.919112i	$-0.628908\pi$
$577$	−18.9282	−0.787991	−0.393996	+	0.919112i	$0.628908\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−0.732051	−0.0303968
$581$	−16.3923	−0.680067
$582$	0	0
$583$	14.3923	0.596068
$584$	−4.39230	−0.181755
$585$	0	0
$586$	17.3205	0.715504
$587$	24.0526	0.992755	0.496378	−	0.868107i	$-0.334663\pi$
$587$	24.0526	0.992755	0.496378	+	0.868107i	$0.334663\pi$
$588$	0	0
$589$	−5.46410	−0.225144
$590$	−5.46410	−0.224954
$591$	0	0
$592$	−6.73205	−0.276686
$593$	−8.00000	−0.328521	−0.164260	−	0.986417i	$-0.552524\pi$
$593$	−8.00000	−0.328521	−0.164260	+	0.986417i	$0.552524\pi$
$594$	0	0
$595$	−13.8564	−0.568057
$596$	13.3205	0.545629
$597$	0	0
$598$	−0.679492	−0.0277865
$599$	−27.8564	−1.13818	−0.569091	−	0.822275i	$-0.692705\pi$
$599$	−27.8564	−1.13818	−0.569091	+	0.822275i	$0.692705\pi$
$600$	0	0
$601$	28.9282	1.18001	0.590003	−	0.807401i	$-0.299127\pi$
$601$	28.9282	1.18001	0.590003	+	0.807401i	$0.299127\pi$
$602$	−11.3205	−0.461389
$603$	0	0
$604$	1.46410	0.0595734
$605$	−3.53590	−0.143755
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	5.46410	0.221599
$609$	0	0
$610$	9.12436	0.369434
$611$	−4.39230	−0.177694
$612$	0	0
$613$	16.5885	0.670001	0.335001	−	0.942218i	$-0.391263\pi$
$613$	16.5885	0.670001	0.335001	+	0.942218i	$0.391263\pi$
$614$	−24.7846	−1.00023
$615$	0	0
$616$	−9.46410	−0.381320
$617$	−35.8564	−1.44352	−0.721762	−	0.692141i	$-0.756668\pi$
$617$	−35.8564	−1.44352	−0.721762	+	0.692141i	$0.756668\pi$
$618$	0	0
$619$	−0.196152	−0.00788403	−0.00394202	−	0.999992i	$-0.501255\pi$
$619$	−0.196152	−0.00788403	−0.00394202	+	0.999992i	$0.501255\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	−13.0718	−0.524131
$623$	3.21539	0.128822
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.7846	1.15046
$627$	0	0
$628$	−19.8564	−0.792357
$629$	−26.9282	−1.07370
$630$	0	0
$631$	0.392305	0.0156174	0.00780870	−	0.999970i	$-0.497514\pi$
$631$	0.392305	0.0156174	0.00780870	+	0.999970i	$0.497514\pi$
$632$	17.4641	0.694685
$633$	0	0
$634$	−23.4641	−0.931879
$635$	−15.8564	−0.629242
$636$	0	0
$637$	−6.33975	−0.251190
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	1.00000	0.0395285
$641$	3.46410	0.136824	0.0684119	−	0.997657i	$-0.478207\pi$
$641$	3.46410	0.136824	0.0684119	+	0.997657i	$0.478207\pi$
$642$	0	0
$643$	28.3397	1.11761	0.558805	−	0.829299i	$-0.311260\pi$
$643$	28.3397	1.11761	0.558805	+	0.829299i	$0.311260\pi$
$644$	−1.85641	−0.0731527
$645$	0	0
$646$	21.8564	0.859929
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	−14.9282	−0.585983
$650$	−1.26795	−0.0497331
$651$	0	0
$652$	−9.46410	−0.370643
$653$	12.9282	0.505920	0.252960	−	0.967477i	$-0.418596\pi$
$653$	12.9282	0.505920	0.252960	+	0.967477i	$0.418596\pi$
$654$	0	0
$655$	−12.3923	−0.484207
$656$	2.53590	0.0990102
$657$	0	0
$658$	−12.0000	−0.467809
$659$	−21.4641	−0.836123	−0.418061	−	0.908419i	$-0.637290\pi$
$659$	−21.4641	−0.836123	−0.418061	+	0.908419i	$0.637290\pi$
$660$	0	0
$661$	32.2487	1.25433	0.627165	−	0.778887i	$-0.284215\pi$
$661$	32.2487	1.25433	0.627165	+	0.778887i	$0.284215\pi$
$662$	−13.2679	−0.515673
$663$	0	0
$664$	4.73205	0.183639
$665$	−18.9282	−0.734004
$666$	0	0
$667$	−0.392305	−0.0151901
$668$	−17.3205	−0.670151
$669$	0	0
$670$	8.39230	0.324223
$671$	24.9282	0.962343
$672$	0	0
$673$	36.7846	1.41794	0.708971	−	0.705237i	$-0.249160\pi$
$673$	36.7846	1.41794	0.708971	+	0.705237i	$0.249160\pi$
$674$	−11.6077	−0.447112
$675$	0	0
$676$	−11.3923	−0.438166
$677$	10.0526	0.386351	0.193176	−	0.981164i	$-0.438121\pi$
$677$	10.0526	0.386351	0.193176	+	0.981164i	$0.438121\pi$
$678$	0	0
$679$	−13.8564	−0.531760
$680$	4.00000	0.153393
$681$	0	0
$682$	−2.73205	−0.104616
$683$	−15.7128	−0.601234	−0.300617	−	0.953745i	$-0.597193\pi$
$683$	−15.7128	−0.601234	−0.300617	+	0.953745i	$0.597193\pi$
$684$	0	0
$685$	20.7846	0.794139
$686$	6.92820	0.264520
$687$	0	0
$688$	3.26795	0.124589
$689$	−6.67949	−0.254468
$690$	0	0
$691$	−4.78461	−0.182015	−0.0910076	−	0.995850i	$-0.529009\pi$
$691$	−4.78461	−0.182015	−0.0910076	+	0.995850i	$0.529009\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	20.0526	0.761185
$695$	13.6603	0.518163
$696$	0	0
$697$	10.1436	0.384216
$698$	−28.5359	−1.08010
$699$	0	0
$700$	−3.46410	−0.130931
$701$	−40.9282	−1.54584	−0.772918	−	0.634505i	$-0.781204\pi$
$701$	−40.9282	−1.54584	−0.772918	+	0.634505i	$0.781204\pi$
$702$	0	0
$703$	−36.7846	−1.38736
$704$	2.73205	0.102968
$705$	0	0
$706$	16.3923	0.616933
$707$	−58.6410	−2.20542
$708$	0	0
$709$	−8.05256	−0.302420	−0.151210	−	0.988502i	$-0.548317\pi$
$709$	−8.05256	−0.302420	−0.151210	+	0.988502i	$0.548317\pi$
$710$	13.8564	0.520022
$711$	0	0
$712$	−0.928203	−0.0347859
$713$	−0.535898	−0.0200696
$714$	0	0
$715$	−3.46410	−0.129550
$716$	14.7321	0.550563
$717$	0	0
$718$	32.9282	1.22887
$719$	25.8564	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$719$	25.8564	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$720$	0	0
$721$	−10.1436	−0.377767
$722$	10.8564	0.404034
$723$	0	0
$724$	−9.12436	−0.339104
$725$	−0.732051	−0.0271877
$726$	0	0
$727$	36.7846	1.36427	0.682133	−	0.731228i	$-0.261053\pi$
$727$	36.7846	1.36427	0.682133	+	0.731228i	$0.261053\pi$
$728$	4.39230	0.162790
$729$	0	0
$730$	−4.39230	−0.162566
$731$	13.0718	0.483478
$732$	0	0
$733$	−17.6077	−0.650355	−0.325178	−	0.945653i	$-0.605424\pi$
$733$	−17.6077	−0.650355	−0.325178	+	0.945653i	$0.605424\pi$
$734$	−28.5359	−1.05328
$735$	0	0
$736$	0.535898	0.0197535
$737$	22.9282	0.844571
$738$	0	0
$739$	23.8038	0.875639	0.437819	−	0.899063i	$-0.355751\pi$
$739$	23.8038	0.875639	0.437819	+	0.899063i	$0.355751\pi$
$740$	−6.73205	−0.247475
$741$	0	0
$742$	−18.2487	−0.669931
$743$	21.7128	0.796566	0.398283	−	0.917263i	$-0.369606\pi$
$743$	21.7128	0.796566	0.398283	+	0.917263i	$0.369606\pi$
$744$	0	0
$745$	13.3205	0.488026
$746$	9.32051	0.341248
$747$	0	0
$748$	10.9282	0.399575
$749$	13.8564	0.506302
$750$	0	0
$751$	−7.85641	−0.286684	−0.143342	−	0.989673i	$-0.545785\pi$
$751$	−7.85641	−0.286684	−0.143342	+	0.989673i	$0.545785\pi$
$752$	3.46410	0.126323
$753$	0	0
$754$	0.928203	0.0338032
$755$	1.46410	0.0532841
$756$	0	0
$757$	14.4449	0.525008	0.262504	−	0.964931i	$-0.415452\pi$
$757$	14.4449	0.525008	0.262504	+	0.964931i	$0.415452\pi$
$758$	−7.32051	−0.265893
$759$	0	0
$760$	5.46410	0.198204
$761$	−0.535898	−0.0194263	−0.00971315	−	0.999953i	$-0.503092\pi$
$761$	−0.535898	−0.0194263	−0.00971315	+	0.999953i	$0.503092\pi$
$762$	0	0
$763$	53.5692	1.93934
$764$	2.00000	0.0723575
$765$	0	0
$766$	2.00000	0.0722629
$767$	6.92820	0.250163
$768$	0	0
$769$	39.3205	1.41793	0.708967	−	0.705242i	$-0.249162\pi$
$769$	39.3205	1.41793	0.708967	+	0.705242i	$0.249162\pi$
$770$	−9.46410	−0.341063
$771$	0	0
$772$	15.8564	0.570685
$773$	47.5167	1.70906	0.854528	−	0.519406i	$-0.173847\pi$
$773$	47.5167	1.70906	0.854528	+	0.519406i	$0.173847\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	4.00000	0.143592
$777$	0	0
$778$	−26.1962	−0.939178
$779$	13.8564	0.496457
$780$	0	0
$781$	37.8564	1.35461
$782$	2.14359	0.0766547
$783$	0	0
$784$	5.00000	0.178571
$785$	−19.8564	−0.708706
$786$	0	0
$787$	−7.66025	−0.273059	−0.136529	−	0.990636i	$-0.543595\pi$
$787$	−7.66025	−0.273059	−0.136529	+	0.990636i	$0.543595\pi$
$788$	−21.2679	−0.757639
$789$	0	0
$790$	17.4641	0.621345
$791$	27.7128	0.985354
$792$	0	0
$793$	−11.5692	−0.410835
$794$	16.2487	0.576645
$795$	0	0
$796$	−20.0000	−0.708881
$797$	−42.8372	−1.51737	−0.758685	−	0.651457i	$-0.774158\pi$
$797$	−42.8372	−1.51737	−0.758685	+	0.651457i	$0.774158\pi$
$798$	0	0
$799$	13.8564	0.490204
$800$	1.00000	0.0353553
$801$	0	0
$802$	7.46410	0.263567
$803$	−12.0000	−0.423471
$804$	0	0
$805$	−1.85641	−0.0654297
$806$	1.26795	0.0446616
$807$	0	0
$808$	16.9282	0.595532
$809$	16.2487	0.571274	0.285637	−	0.958338i	$-0.407795\pi$
$809$	16.2487	0.571274	0.285637	+	0.958338i	$0.407795\pi$
$810$	0	0
$811$	−16.3923	−0.575612	−0.287806	−	0.957689i	$-0.592926\pi$
$811$	−16.3923	−0.575612	−0.287806	+	0.957689i	$0.592926\pi$
$812$	2.53590	0.0889926
$813$	0	0
$814$	−18.3923	−0.644650
$815$	−9.46410	−0.331513
$816$	0	0
$817$	17.8564	0.624717
$818$	−8.53590	−0.298451
$819$	0	0
$820$	2.53590	0.0885574
$821$	−9.51666	−0.332134	−0.166067	−	0.986114i	$-0.553107\pi$
$821$	−9.51666	−0.332134	−0.166067	+	0.986114i	$0.553107\pi$
$822$	0	0
$823$	−51.8564	−1.80760	−0.903800	−	0.427954i	$-0.859234\pi$
$823$	−51.8564	−1.80760	−0.903800	+	0.427954i	$0.859234\pi$
$824$	2.92820	0.102009
$825$	0	0
$826$	18.9282	0.658596
$827$	33.5167	1.16549	0.582744	−	0.812656i	$-0.301979\pi$
$827$	33.5167	1.16549	0.582744	+	0.812656i	$0.301979\pi$
$828$	0	0
$829$	−25.1244	−0.872605	−0.436302	−	0.899800i	$-0.643712\pi$
$829$	−25.1244	−0.872605	−0.436302	+	0.899800i	$0.643712\pi$
$830$	4.73205	0.164252
$831$	0	0
$832$	−1.26795	−0.0439582
$833$	20.0000	0.692959
$834$	0	0
$835$	−17.3205	−0.599401
$836$	14.9282	0.516303
$837$	0	0
$838$	4.39230	0.151730
$839$	5.21539	0.180055	0.0900276	−	0.995939i	$-0.471304\pi$
$839$	5.21539	0.180055	0.0900276	+	0.995939i	$0.471304\pi$
$840$	0	0
$841$	−28.4641	−0.981521
$842$	32.2487	1.11136
$843$	0	0
$844$	26.9282	0.926907
$845$	−11.3923	−0.391907
$846$	0	0
$847$	12.2487	0.420871
$848$	5.26795	0.180902
$849$	0	0
$850$	4.00000	0.137199
$851$	−3.60770	−0.123670
$852$	0	0
$853$	−51.8564	−1.77553	−0.887765	−	0.460297i	$-0.847743\pi$
$853$	−51.8564	−1.77553	−0.887765	+	0.460297i	$0.847743\pi$
$854$	−31.6077	−1.08159
$855$	0	0
$856$	−4.00000	−0.136717
$857$	24.9282	0.851531	0.425766	−	0.904833i	$-0.360005\pi$
$857$	24.9282	0.851531	0.425766	+	0.904833i	$0.360005\pi$
$858$	0	0
$859$	−38.4449	−1.31172	−0.655861	−	0.754882i	$-0.727694\pi$
$859$	−38.4449	−1.31172	−0.655861	+	0.754882i	$0.727694\pi$
$860$	3.26795	0.111436
$861$	0	0
$862$	−26.7846	−0.912287
$863$	4.14359	0.141050	0.0705248	−	0.997510i	$-0.477533\pi$
$863$	4.14359	0.141050	0.0705248	+	0.997510i	$0.477533\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	16.7846	0.570364
$867$	0	0
$868$	3.46410	0.117579
$869$	47.7128	1.61855
$870$	0	0
$871$	−10.6410	−0.360557
$872$	−15.4641	−0.523681
$873$	0	0
$874$	2.92820	0.0990480
$875$	−3.46410	−0.117108
$876$	0	0
$877$	10.6795	0.360621	0.180310	−	0.983610i	$-0.442290\pi$
$877$	10.6795	0.360621	0.180310	+	0.983610i	$0.442290\pi$
$878$	22.0000	0.742464
$879$	0	0
$880$	2.73205	0.0920974
$881$	47.5692	1.60265	0.801324	−	0.598231i	$-0.204129\pi$
$881$	47.5692	1.60265	0.801324	+	0.598231i	$0.204129\pi$
$882$	0	0
$883$	−19.6603	−0.661620	−0.330810	−	0.943697i	$-0.607322\pi$
$883$	−19.6603	−0.661620	−0.330810	+	0.943697i	$0.607322\pi$
$884$	−5.07180	−0.170583
$885$	0	0
$886$	34.9282	1.17344
$887$	−26.1051	−0.876524	−0.438262	−	0.898847i	$-0.644406\pi$
$887$	−26.1051	−0.876524	−0.438262	+	0.898847i	$0.644406\pi$
$888$	0	0
$889$	54.9282	1.84223
$890$	−0.928203	−0.0311134
$891$	0	0
$892$	10.0000	0.334825
$893$	18.9282	0.633408
$894$	0	0
$895$	14.7321	0.492438
$896$	−3.46410	−0.115728
$897$	0	0
$898$	−29.3205	−0.978438
$899$	0.732051	0.0244153
$900$	0	0
$901$	21.0718	0.702003
$902$	6.92820	0.230684
$903$	0	0
$904$	−8.00000	−0.266076
$905$	−9.12436	−0.303304
$906$	0	0
$907$	17.8564	0.592912	0.296456	−	0.955046i	$-0.404195\pi$
$907$	17.8564	0.592912	0.296456	+	0.955046i	$0.404195\pi$
$908$	13.4641	0.446822
$909$	0	0
$910$	4.39230	0.145603
$911$	17.0718	0.565614	0.282807	−	0.959177i	$-0.408734\pi$
$911$	17.0718	0.565614	0.282807	+	0.959177i	$0.408734\pi$
$912$	0	0
$913$	12.9282	0.427861
$914$	14.9282	0.493781
$915$	0	0
$916$	−24.0526	−0.794719
$917$	42.9282	1.41761
$918$	0	0
$919$	0.784610	0.0258819	0.0129409	−	0.999916i	$-0.495881\pi$
$919$	0.784610	0.0258819	0.0129409	+	0.999916i	$0.495881\pi$
$920$	0.535898	0.0176680
$921$	0	0
$922$	27.2679	0.898022
$923$	−17.5692	−0.578298
$924$	0	0
$925$	−6.73205	−0.221348
$926$	−10.3923	−0.341512
$927$	0	0
$928$	−0.732051	−0.0240307
$929$	1.60770	0.0527468	0.0263734	−	0.999652i	$-0.491604\pi$
$929$	1.60770	0.0527468	0.0263734	+	0.999652i	$0.491604\pi$
$930$	0	0
$931$	27.3205	0.895393
$932$	−24.7846	−0.811847
$933$	0	0
$934$	−6.14359	−0.201025
$935$	10.9282	0.357390
$936$	0	0
$937$	−19.0718	−0.623048	−0.311524	−	0.950238i	$-0.600840\pi$
$937$	−19.0718	−0.623048	−0.311524	+	0.950238i	$0.600840\pi$
$938$	−29.0718	−0.949228
$939$	0	0
$940$	3.46410	0.112987
$941$	54.5885	1.77953	0.889766	−	0.456416i	$-0.150867\pi$
$941$	54.5885	1.77953	0.889766	+	0.456416i	$0.150867\pi$
$942$	0	0
$943$	1.35898	0.0442546
$944$	−5.46410	−0.177841
$945$	0	0
$946$	8.92820	0.290281
$947$	10.1962	0.331330	0.165665	−	0.986182i	$-0.447023\pi$
$947$	10.1962	0.331330	0.165665	+	0.986182i	$0.447023\pi$
$948$	0	0
$949$	5.56922	0.180785
$950$	5.46410	0.177279
$951$	0	0
$952$	−13.8564	−0.449089
$953$	−30.9282	−1.00186	−0.500931	−	0.865487i	$-0.667009\pi$
$953$	−30.9282	−1.00186	−0.500931	+	0.865487i	$0.667009\pi$
$954$	0	0
$955$	2.00000	0.0647185
$956$	9.46410	0.306091
$957$	0	0
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	1.00000	0.0322581
$962$	8.53590	0.275208
$963$	0	0
$964$	−20.2487	−0.652167
$965$	15.8564	0.510436
$966$	0	0
$967$	−12.1436	−0.390512	−0.195256	−	0.980752i	$-0.562554\pi$
$967$	−12.1436	−0.390512	−0.195256	+	0.980752i	$0.562554\pi$
$968$	−3.53590	−0.113648
$969$	0	0
$970$	4.00000	0.128432
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	−47.3205	−1.51703
$974$	−10.0000	−0.320421
$975$	0	0
$976$	9.12436	0.292064
$977$	−42.7846	−1.36880	−0.684400	−	0.729106i	$-0.739936\pi$
$977$	−42.7846	−1.36880	−0.684400	+	0.729106i	$0.739936\pi$
$978$	0	0
$979$	−2.53590	−0.0810477
$980$	5.00000	0.159719
$981$	0	0
$982$	−18.7321	−0.597764
$983$	37.7128	1.20285	0.601426	−	0.798929i	$-0.294599\pi$
$983$	37.7128	1.20285	0.601426	+	0.798929i	$0.294599\pi$
$984$	0	0
$985$	−21.2679	−0.677653
$986$	−2.92820	−0.0932530
$987$	0	0
$988$	−6.92820	−0.220416
$989$	1.75129	0.0556877
$990$	0	0
$991$	13.0718	0.415239	0.207620	−	0.978210i	$-0.433428\pi$
$991$	13.0718	0.415239	0.207620	+	0.978210i	$0.433428\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−48.0000	−1.52247
$995$	−20.0000	−0.634043
$996$	0	0
$997$	−23.1769	−0.734020	−0.367010	−	0.930217i	$-0.619619\pi$
$997$	−23.1769	−0.734020	−0.367010	+	0.930217i	$0.619619\pi$
$998$	19.1244	0.605371
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bh.1.1		2
3.2	odd	2		310.2.a.c.1.2	✓	2
12.11	even	2		2480.2.a.s.1.1		2
15.2	even	4		1550.2.b.h.249.1		4
15.8	even	4		1550.2.b.h.249.4		4
15.14	odd	2		1550.2.a.j.1.1		2
24.5	odd	2		9920.2.a.bt.1.1		2
24.11	even	2		9920.2.a.bl.1.2		2
93.92	even	2		9610.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.a.c.1.2	✓	2	3.2	odd	2
1550.2.a.j.1.1		2	15.14	odd	2
1550.2.b.h.249.1		4	15.2	even	4
1550.2.b.h.249.4		4	15.8	even	4
2480.2.a.s.1.1		2	12.11	even	2
2790.2.a.bh.1.1		2	1.1	even	1	trivial
9610.2.a.j.1.1		2	93.92	even	2
9920.2.a.bl.1.2		2	24.11	even	2
9920.2.a.bt.1.1		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.73205 q^{11} -1.26795 q^{13} -3.46410 q^{14} +1.00000 q^{16} +4.00000 q^{17} +5.46410 q^{19} +1.00000 q^{20} +2.73205 q^{22} +0.535898 q^{23} +1.00000 q^{25} -1.26795 q^{26} -3.46410 q^{28} -0.732051 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -3.46410 q^{35} -6.73205 q^{37} +5.46410 q^{38} +1.00000 q^{40} +2.53590 q^{41} +3.26795 q^{43} +2.73205 q^{44} +0.535898 q^{46} +3.46410 q^{47} +5.00000 q^{49} +1.00000 q^{50} -1.26795 q^{52} +5.26795 q^{53} +2.73205 q^{55} -3.46410 q^{56} -0.732051 q^{58} -5.46410 q^{59} +9.12436 q^{61} -1.00000 q^{62} +1.00000 q^{64} -1.26795 q^{65} +8.39230 q^{67} +4.00000 q^{68} -3.46410 q^{70} +13.8564 q^{71} -4.39230 q^{73} -6.73205 q^{74} +5.46410 q^{76} -9.46410 q^{77} +17.4641 q^{79} +1.00000 q^{80} +2.53590 q^{82} +4.73205 q^{83} +4.00000 q^{85} +3.26795 q^{86} +2.73205 q^{88} -0.928203 q^{89} +4.39230 q^{91} +0.535898 q^{92} +3.46410 q^{94} +5.46410 q^{95} +4.00000 q^{97} +5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} - 6 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{34} - 10 q^{37} + 4 q^{38} + 2 q^{40} + 12 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 10 q^{49} + 2 q^{50} - 6 q^{52} + 14 q^{53} + 2 q^{55} + 2 q^{58} - 4 q^{59} - 6 q^{61} - 2 q^{62} + 2 q^{64} - 6 q^{65} - 4 q^{67} + 8 q^{68} + 12 q^{73} - 10 q^{74} + 4 q^{76} - 12 q^{77} + 28 q^{79} + 2 q^{80} + 12 q^{82} + 6 q^{83} + 8 q^{85} + 10 q^{86} + 2 q^{88} + 12 q^{89} - 12 q^{91} + 8 q^{92} + 4 q^{95} + 8 q^{97} + 10 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 - 6 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 + 2 * q^20 + 2 * q^22 + 8 * q^23 + 2 * q^25 - 6 * q^26 + 2 * q^29 - 2 * q^31 + 2 * q^32 + 8 * q^34 - 10 * q^37 + 4 * q^38 + 2 * q^40 + 12 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 10 * q^49 + 2 * q^50 - 6 * q^52 + 14 * q^53 + 2 * q^55 + 2 * q^58 - 4 * q^59 - 6 * q^61 - 2 * q^62 + 2 * q^64 - 6 * q^65 - 4 * q^67 + 8 * q^68 + 12 * q^73 - 10 * q^74 + 4 * q^76 - 12 * q^77 + 28 * q^79 + 2 * q^80 + 12 * q^82 + 6 * q^83 + 8 * q^85 + 10 * q^86 + 2 * q^88 + 12 * q^89 - 12 * q^91 + 8 * q^92 + 4 * q^95 + 8 * q^97 + 10 * q^98

Introduction

L-functions

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