LMFDB - Embedded newform 2790.2.a.bh.1.2

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			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.46410 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{10} -0.732051 q^{11} -4.73205 q^{13} +3.46410 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.46410 q^{19} +1.00000 q^{20} -0.732051 q^{22} +7.46410 q^{23} +1.00000 q^{25} -4.73205 q^{26} +3.46410 q^{28} +2.73205 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +3.46410 q^{35} -3.26795 q^{37} -1.46410 q^{38} +1.00000 q^{40} +9.46410 q^{41} +6.73205 q^{43} -0.732051 q^{44} +7.46410 q^{46} -3.46410 q^{47} +5.00000 q^{49} +1.00000 q^{50} -4.73205 q^{52} +8.73205 q^{53} -0.732051 q^{55} +3.46410 q^{56} +2.73205 q^{58} +1.46410 q^{59} -15.1244 q^{61} -1.00000 q^{62} +1.00000 q^{64} -4.73205 q^{65} -12.3923 q^{67} +4.00000 q^{68} +3.46410 q^{70} -13.8564 q^{71} +16.3923 q^{73} -3.26795 q^{74} -1.46410 q^{76} -2.53590 q^{77} +10.5359 q^{79} +1.00000 q^{80} +9.46410 q^{82} +1.26795 q^{83} +4.00000 q^{85} +6.73205 q^{86} -0.732051 q^{88} +12.9282 q^{89} -16.3923 q^{91} +7.46410 q^{92} -3.46410 q^{94} -1.46410 q^{95} +4.00000 q^{97} +5.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−0.732051	−0.220722	−0.110361	−	0.993892i	$-0.535201\pi$
$11$	−0.732051	−0.220722	−0.110361	+	0.993892i	$0.535201\pi$
$12$	0	0
$13$	−4.73205	−1.31243	−0.656217	−	0.754572i	$-0.727845\pi$
$13$	−4.73205	−1.31243	−0.656217	+	0.754572i	$0.727845\pi$
$14$	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
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−0.732051	−0.156074
$23$	7.46410	1.55637	0.778186	−	0.628033i	$-0.216140\pi$
$23$	7.46410	1.55637	0.778186	+	0.628033i	$0.216140\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.73205	−0.928032
$27$	0	0
$28$	3.46410	0.654654
$29$	2.73205	0.507329	0.253665	−	0.967292i	$-0.418364\pi$
$29$	2.73205	0.507329	0.253665	+	0.967292i	$0.418364\pi$
$30$	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
$37$	−3.26795	−0.537248	−0.268624	−	0.963245i	$-0.586569\pi$
$37$	−3.26795	−0.537248	−0.268624	+	0.963245i	$0.586569\pi$
$38$	−1.46410	−0.237509
$39$	0	0
$40$	1.00000	0.158114
$41$	9.46410	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$41$	9.46410	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$42$	0	0
$43$	6.73205	1.02663	0.513314	−	0.858201i	$-0.328418\pi$
$43$	6.73205	1.02663	0.513314	+	0.858201i	$0.328418\pi$
$44$	−0.732051	−0.110361
$45$	0	0
$46$	7.46410	1.10052
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	1.00000	0.141421
$51$	0	0
$52$	−4.73205	−0.656217
$53$	8.73205	1.19944	0.599720	−	0.800210i	$-0.295279\pi$
$53$	8.73205	1.19944	0.599720	+	0.800210i	$0.295279\pi$
$54$	0	0
$55$	−0.732051	−0.0987097
$56$	3.46410	0.462910
$57$	0	0
$58$	2.73205	0.358736
$59$	1.46410	0.190610	0.0953049	−	0.995448i	$-0.469617\pi$
$59$	1.46410	0.190610	0.0953049	+	0.995448i	$0.469617\pi$
$60$	0	0
$61$	−15.1244	−1.93648	−0.968238	−	0.250032i	$-0.919559\pi$
$61$	−15.1244	−1.93648	−0.968238	+	0.250032i	$0.919559\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.73205	−0.586939
$66$	0	0
$67$	−12.3923	−1.51396	−0.756980	−	0.653437i	$-0.773326\pi$
$67$	−12.3923	−1.51396	−0.756980	+	0.653437i	$0.773326\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	3.46410	0.414039
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	16.3923	1.91857	0.959287	−	0.282433i	$-0.0911414\pi$
$73$	16.3923	1.91857	0.959287	+	0.282433i	$0.0911414\pi$
$74$	−3.26795	−0.379891
$75$	0	0
$76$	−1.46410	−0.167944
$77$	−2.53590	−0.288992
$78$	0	0
$79$	10.5359	1.18538	0.592691	−	0.805430i	$-0.298066\pi$
$79$	10.5359	1.18538	0.592691	+	0.805430i	$0.298066\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	9.46410	1.04514
$83$	1.26795	0.139176	0.0695878	−	0.997576i	$-0.477832\pi$
$83$	1.26795	0.139176	0.0695878	+	0.997576i	$0.477832\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	6.73205	0.725936
$87$	0	0
$88$	−0.732051	−0.0780369
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	−16.3923	−1.71838
$92$	7.46410	0.778186
$93$	0	0
$94$	−3.46410	−0.357295
$95$	−1.46410	−0.150214
$96$	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
$101$	3.07180	0.305655	0.152828	−	0.988253i	$-0.451162\pi$
$101$	3.07180	0.305655	0.152828	+	0.988253i	$0.451162\pi$
$102$	0	0
$103$	−10.9282	−1.07679	−0.538394	−	0.842693i	$-0.680969\pi$
$103$	−10.9282	−1.07679	−0.538394	+	0.842693i	$0.680969\pi$
$104$	−4.73205	−0.464016
$105$	0	0
$106$	8.73205	0.848132
$107$	−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
$109$	−8.53590	−0.817591	−0.408795	−	0.912626i	$-0.634051\pi$
$109$	−8.53590	−0.817591	−0.408795	+	0.912626i	$0.634051\pi$
$110$	−0.732051	−0.0697983
$111$	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
$115$	7.46410	0.696031
$116$	2.73205	0.253665
$117$	0	0
$118$	1.46410	0.134781
$119$	13.8564	1.27021
$120$	0	0
$121$	−10.4641	−0.951282
$122$	−15.1244	−1.36929
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.8564	1.05209	0.526043	−	0.850458i	$-0.323675\pi$
$127$	11.8564	1.05209	0.526043	+	0.850458i	$0.323675\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.73205	−0.415028
$131$	8.39230	0.733239	0.366620	−	0.930371i	$-0.380515\pi$
$131$	8.39230	0.733239	0.366620	+	0.930371i	$0.380515\pi$
$132$	0	0
$133$	−5.07180	−0.439781
$134$	−12.3923	−1.07053
$135$	0	0
$136$	4.00000	0.342997
$137$	−20.7846	−1.77575	−0.887875	−	0.460086i	$-0.847819\pi$
$137$	−20.7846	−1.77575	−0.887875	+	0.460086i	$0.847819\pi$
$138$	0	0
$139$	−3.66025	−0.310459	−0.155229	−	0.987878i	$-0.549612\pi$
$139$	−3.66025	−0.310459	−0.155229	+	0.987878i	$0.549612\pi$
$140$	3.46410	0.292770
$141$	0	0
$142$	−13.8564	−1.16280
$143$	3.46410	0.289683
$144$	0	0
$145$	2.73205	0.226884
$146$	16.3923	1.35664
$147$	0	0
$148$	−3.26795	−0.268624
$149$	−21.3205	−1.74664	−0.873322	−	0.487143i	$-0.838039\pi$
$149$	−21.3205	−1.74664	−0.873322	+	0.487143i	$0.838039\pi$
$150$	0	0
$151$	−5.46410	−0.444662	−0.222331	−	0.974971i	$-0.571367\pi$
$151$	−5.46410	−0.444662	−0.222331	+	0.974971i	$0.571367\pi$
$152$	−1.46410	−0.118754
$153$	0	0
$154$	−2.53590	−0.204349
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	7.85641	0.627009	0.313505	−	0.949587i	$-0.398497\pi$
$157$	7.85641	0.627009	0.313505	+	0.949587i	$0.398497\pi$
$158$	10.5359	0.838191
$159$	0	0
$160$	1.00000	0.0790569
$161$	25.8564	2.03777
$162$	0	0
$163$	−2.53590	−0.198627	−0.0993134	−	0.995056i	$-0.531665\pi$
$163$	−2.53590	−0.198627	−0.0993134	+	0.995056i	$0.531665\pi$
$164$	9.46410	0.739022
$165$	0	0
$166$	1.26795	0.0984119
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	9.39230	0.722485
$170$	4.00000	0.306786
$171$	0	0
$172$	6.73205	0.513314
$173$	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
$176$	−0.732051	−0.0551804
$177$	0	0
$178$	12.9282	0.969010
$179$	11.2679	0.842206	0.421103	−	0.907013i	$-0.361643\pi$
$179$	11.2679	0.842206	0.421103	+	0.907013i	$0.361643\pi$
$180$	0	0
$181$	15.1244	1.12418	0.562092	−	0.827075i	$-0.309997\pi$
$181$	15.1244	1.12418	0.562092	+	0.827075i	$0.309997\pi$
$182$	−16.3923	−1.21508
$183$	0	0
$184$	7.46410	0.550261
$185$	−3.26795	−0.240264
$186$	0	0
$187$	−2.92820	−0.214131
$188$	−3.46410	−0.252646
$189$	0	0
$190$	−1.46410	−0.106217
$191$	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
$193$	−11.8564	−0.853443	−0.426721	−	0.904383i	$-0.640332\pi$
$193$	−11.8564	−0.853443	−0.426721	+	0.904383i	$0.640332\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	5.00000	0.357143
$197$	−24.7321	−1.76209	−0.881043	−	0.473037i	$-0.843158\pi$
$197$	−24.7321	−1.76209	−0.881043	+	0.473037i	$0.843158\pi$
$198$	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
$202$	3.07180	0.216131
$203$	9.46410	0.664250
$204$	0	0
$205$	9.46410	0.661002
$206$	−10.9282	−0.761404
$207$	0	0
$208$	−4.73205	−0.328109
$209$	1.07180	0.0741377
$210$	0	0
$211$	13.0718	0.899900	0.449950	−	0.893054i	$-0.351442\pi$
$211$	13.0718	0.899900	0.449950	+	0.893054i	$0.351442\pi$
$212$	8.73205	0.599720
$213$	0	0
$214$	−4.00000	−0.273434
$215$	6.73205	0.459122
$216$	0	0
$217$	−3.46410	−0.235159
$218$	−8.53590	−0.578124
$219$	0	0
$220$	−0.732051	−0.0493549
$221$	−18.9282	−1.27325
$222$	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
$227$	6.53590	0.433803	0.216901	−	0.976194i	$-0.430405\pi$
$227$	6.53590	0.433803	0.216901	+	0.976194i	$0.430405\pi$
$228$	0	0
$229$	14.0526	0.928619	0.464310	−	0.885673i	$-0.346302\pi$
$229$	14.0526	0.928619	0.464310	+	0.885673i	$0.346302\pi$
$230$	7.46410	0.492168
$231$	0	0
$232$	2.73205	0.179368
$233$	16.7846	1.09960	0.549798	−	0.835298i	$-0.314705\pi$
$233$	16.7846	1.09960	0.549798	+	0.835298i	$0.314705\pi$
$234$	0	0
$235$	−3.46410	−0.225973
$236$	1.46410	0.0953049
$237$	0	0
$238$	13.8564	0.898177
$239$	2.53590	0.164034	0.0820168	−	0.996631i	$-0.473864\pi$
$239$	2.53590	0.164034	0.0820168	+	0.996631i	$0.473864\pi$
$240$	0	0
$241$	28.2487	1.81966	0.909830	−	0.414982i	$-0.136212\pi$
$241$	28.2487	1.81966	0.909830	+	0.414982i	$0.136212\pi$
$242$	−10.4641	−0.672658
$243$	0	0
$244$	−15.1244	−0.968238
$245$	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
$251$	−30.9808	−1.95549	−0.977744	−	0.209801i	$-0.932718\pi$
$251$	−30.9808	−1.95549	−0.977744	+	0.209801i	$0.932718\pi$
$252$	0	0
$253$	−5.46410	−0.343525
$254$	11.8564	0.743937
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.07180	−0.191613	−0.0958067	−	0.995400i	$-0.530543\pi$
$257$	−3.07180	−0.191613	−0.0958067	+	0.995400i	$0.530543\pi$
$258$	0	0
$259$	−11.3205	−0.703422
$260$	−4.73205	−0.293469
$261$	0	0
$262$	8.39230	0.518478
$263$	14.7846	0.911658	0.455829	−	0.890067i	$-0.349343\pi$
$263$	14.7846	0.911658	0.455829	+	0.890067i	$0.349343\pi$
$264$	0	0
$265$	8.73205	0.536406
$266$	−5.07180	−0.310972
$267$	0	0
$268$	−12.3923	−0.756980
$269$	17.6603	1.07676	0.538382	−	0.842701i	$-0.319036\pi$
$269$	17.6603	1.07676	0.538382	+	0.842701i	$0.319036\pi$
$270$	0	0
$271$	1.46410	0.0889378	0.0444689	−	0.999011i	$-0.485840\pi$
$271$	1.46410	0.0889378	0.0444689	+	0.999011i	$0.485840\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−20.7846	−1.25564
$275$	−0.732051	−0.0441443
$276$	0	0
$277$	−2.87564	−0.172781	−0.0863904	−	0.996261i	$-0.527533\pi$
$277$	−2.87564	−0.172781	−0.0863904	+	0.996261i	$0.527533\pi$
$278$	−3.66025	−0.219527
$279$	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
$283$	2.53590	0.150744	0.0753718	−	0.997156i	$-0.475986\pi$
$283$	2.53590	0.150744	0.0753718	+	0.997156i	$0.475986\pi$
$284$	−13.8564	−0.822226
$285$	0	0
$286$	3.46410	0.204837
$287$	32.7846	1.93521
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	2.73205	0.160432
$291$	0	0
$292$	16.3923	0.959287
$293$	−17.3205	−1.01187	−0.505937	−	0.862570i	$-0.668853\pi$
$293$	−17.3205	−1.01187	−0.505937	+	0.862570i	$0.668853\pi$
$294$	0	0
$295$	1.46410	0.0852433
$296$	−3.26795	−0.189946
$297$	0	0
$298$	−21.3205	−1.23506
$299$	−35.3205	−2.04264
$300$	0	0
$301$	23.3205	1.34417
$302$	−5.46410	−0.314424
$303$	0	0
$304$	−1.46410	−0.0839720
$305$	−15.1244	−0.866018
$306$	0	0
$307$	16.7846	0.957948	0.478974	−	0.877829i	$-0.341009\pi$
$307$	16.7846	0.957948	0.478974	+	0.877829i	$0.341009\pi$
$308$	−2.53590	−0.144496
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	−26.9282	−1.52696	−0.763479	−	0.645832i	$-0.776510\pi$
$311$	−26.9282	−1.52696	−0.763479	+	0.645832i	$0.776510\pi$
$312$	0	0
$313$	−12.7846	−0.722629	−0.361314	−	0.932444i	$-0.617672\pi$
$313$	−12.7846	−0.722629	−0.361314	+	0.932444i	$0.617672\pi$
$314$	7.85641	0.443363
$315$	0	0
$316$	10.5359	0.592691
$317$	−16.5359	−0.928749	−0.464374	−	0.885639i	$-0.653721\pi$
$317$	−16.5359	−0.928749	−0.464374	+	0.885639i	$0.653721\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	25.8564	1.44092
$323$	−5.85641	−0.325859
$324$	0	0
$325$	−4.73205	−0.262487
$326$	−2.53590	−0.140450
$327$	0	0
$328$	9.46410	0.522568
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−16.7321	−0.919677	−0.459838	−	0.888003i	$-0.652093\pi$
$331$	−16.7321	−0.919677	−0.459838	+	0.888003i	$0.652093\pi$
$332$	1.26795	0.0695878
$333$	0	0
$334$	17.3205	0.947736
$335$	−12.3923	−0.677064
$336$	0	0
$337$	−32.3923	−1.76452	−0.882261	−	0.470761i	$-0.843979\pi$
$337$	−32.3923	−1.76452	−0.882261	+	0.470761i	$0.843979\pi$
$338$	9.39230	0.510874
$339$	0	0
$340$	4.00000	0.216930
$341$	0.732051	0.0396428
$342$	0	0
$343$	−6.92820	−0.374088
$344$	6.73205	0.362968
$345$	0	0
$346$	2.00000	0.107521
$347$	−18.0526	−0.969112	−0.484556	−	0.874760i	$-0.661019\pi$
$347$	−18.0526	−0.969112	−0.484556	+	0.874760i	$0.661019\pi$
$348$	0	0
$349$	−35.4641	−1.89835	−0.949175	−	0.314749i	$-0.898080\pi$
$349$	−35.4641	−1.89835	−0.949175	+	0.314749i	$0.898080\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	−0.732051	−0.0390184
$353$	−4.39230	−0.233779	−0.116889	−	0.993145i	$-0.537292\pi$
$353$	−4.39230	−0.233779	−0.116889	+	0.993145i	$0.537292\pi$
$354$	0	0
$355$	−13.8564	−0.735422
$356$	12.9282	0.685193
$357$	0	0
$358$	11.2679	0.595530
$359$	19.0718	1.00657	0.503285	−	0.864120i	$-0.332124\pi$
$359$	19.0718	1.00657	0.503285	+	0.864120i	$0.332124\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	15.1244	0.794918
$363$	0	0
$364$	−16.3923	−0.859190
$365$	16.3923	0.858012
$366$	0	0
$367$	−35.4641	−1.85121	−0.925606	−	0.378490i	$-0.876444\pi$
$367$	−35.4641	−1.85121	−0.925606	+	0.378490i	$0.876444\pi$
$368$	7.46410	0.389093
$369$	0	0
$370$	−3.26795	−0.169893
$371$	30.2487	1.57043
$372$	0	0
$373$	−25.3205	−1.31105	−0.655523	−	0.755175i	$-0.727552\pi$
$373$	−25.3205	−1.31105	−0.655523	+	0.755175i	$0.727552\pi$
$374$	−2.92820	−0.151414
$375$	0	0
$376$	−3.46410	−0.178647
$377$	−12.9282	−0.665836
$378$	0	0
$379$	27.3205	1.40336	0.701680	−	0.712492i	$-0.252434\pi$
$379$	27.3205	1.40336	0.701680	+	0.712492i	$0.252434\pi$
$380$	−1.46410	−0.0751068
$381$	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
$385$	−2.53590	−0.129241
$386$	−11.8564	−0.603475
$387$	0	0
$388$	4.00000	0.203069
$389$	−15.8038	−0.801287	−0.400643	−	0.916234i	$-0.631213\pi$
$389$	−15.8038	−0.801287	−0.400643	+	0.916234i	$0.631213\pi$
$390$	0	0
$391$	29.8564	1.50990
$392$	5.00000	0.252538
$393$	0	0
$394$	−24.7321	−1.24598
$395$	10.5359	0.530119
$396$	0	0
$397$	−32.2487	−1.61852	−0.809258	−	0.587453i	$-0.800131\pi$
$397$	−32.2487	−1.61852	−0.809258	+	0.587453i	$0.800131\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	0.535898	0.0267615	0.0133807	−	0.999910i	$-0.495741\pi$
$401$	0.535898	0.0267615	0.0133807	+	0.999910i	$0.495741\pi$
$402$	0	0
$403$	4.73205	0.235720
$404$	3.07180	0.152828
$405$	0	0
$406$	9.46410	0.469695
$407$	2.39230	0.118582
$408$	0	0
$409$	−15.4641	−0.764651	−0.382325	−	0.924028i	$-0.624877\pi$
$409$	−15.4641	−0.764651	−0.382325	+	0.924028i	$0.624877\pi$
$410$	9.46410	0.467399
$411$	0	0
$412$	−10.9282	−0.538394
$413$	5.07180	0.249567
$414$	0	0
$415$	1.26795	0.0622412
$416$	−4.73205	−0.232008
$417$	0	0
$418$	1.07180	0.0524233
$419$	−16.3923	−0.800816	−0.400408	−	0.916337i	$-0.631132\pi$
$419$	−16.3923	−0.800816	−0.400408	+	0.916337i	$0.631132\pi$
$420$	0	0
$421$	−16.2487	−0.791914	−0.395957	−	0.918269i	$-0.629587\pi$
$421$	−16.2487	−0.791914	−0.395957	+	0.918269i	$0.629587\pi$
$422$	13.0718	0.636325
$423$	0	0
$424$	8.73205	0.424066
$425$	4.00000	0.194029
$426$	0	0
$427$	−52.3923	−2.53544
$428$	−4.00000	−0.193347
$429$	0	0
$430$	6.73205	0.324648
$431$	14.7846	0.712150	0.356075	−	0.934457i	$-0.384115\pi$
$431$	14.7846	0.712150	0.356075	+	0.934457i	$0.384115\pi$
$432$	0	0
$433$	−24.7846	−1.19107	−0.595536	−	0.803328i	$-0.703060\pi$
$433$	−24.7846	−1.19107	−0.595536	+	0.803328i	$0.703060\pi$
$434$	−3.46410	−0.166282
$435$	0	0
$436$	−8.53590	−0.408795
$437$	−10.9282	−0.522767
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	−0.732051	−0.0348992
$441$	0	0
$442$	−18.9282	−0.900323
$443$	21.0718	1.00115	0.500576	−	0.865693i	$-0.333122\pi$
$443$	21.0718	1.00115	0.500576	+	0.865693i	$0.333122\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	10.0000	0.473514
$447$	0	0
$448$	3.46410	0.163663
$449$	5.32051	0.251090	0.125545	−	0.992088i	$-0.459932\pi$
$449$	5.32051	0.251090	0.125545	+	0.992088i	$0.459932\pi$
$450$	0	0
$451$	−6.92820	−0.326236
$452$	−8.00000	−0.376288
$453$	0	0
$454$	6.53590	0.306745
$455$	−16.3923	−0.768483
$456$	0	0
$457$	1.07180	0.0501365	0.0250683	−	0.999686i	$-0.492020\pi$
$457$	1.07180	0.0501365	0.0250683	+	0.999686i	$0.492020\pi$
$458$	14.0526	0.656633
$459$	0	0
$460$	7.46410	0.348016
$461$	30.7321	1.43133	0.715667	−	0.698441i	$-0.246123\pi$
$461$	30.7321	1.43133	0.715667	+	0.698441i	$0.246123\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	2.73205	0.126832
$465$	0	0
$466$	16.7846	0.777532
$467$	−33.8564	−1.56669	−0.783344	−	0.621589i	$-0.786488\pi$
$467$	−33.8564	−1.56669	−0.783344	+	0.621589i	$0.786488\pi$
$468$	0	0
$469$	−42.9282	−1.98224
$470$	−3.46410	−0.159787
$471$	0	0
$472$	1.46410	0.0673907
$473$	−4.92820	−0.226599
$474$	0	0
$475$	−1.46410	−0.0671776
$476$	13.8564	0.635107
$477$	0	0
$478$	2.53590	0.115989
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	15.4641	0.705102
$482$	28.2487	1.28669
$483$	0	0
$484$	−10.4641	−0.475641
$485$	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	$0.572752\pi$
$488$	−15.1244	−0.684647
$489$	0	0
$490$	5.00000	0.225877
$491$	−15.2679	−0.689033	−0.344516	−	0.938780i	$-0.611957\pi$
$491$	−15.2679	−0.689033	−0.344516	+	0.938780i	$0.611957\pi$
$492$	0	0
$493$	10.9282	0.492182
$494$	6.92820	0.311715
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−48.0000	−2.15309
$498$	0	0
$499$	−5.12436	−0.229398	−0.114699	−	0.993400i	$-0.536590\pi$
$499$	−5.12436	−0.229398	−0.114699	+	0.993400i	$0.536590\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−30.9808	−1.38274
$503$	38.6410	1.72292	0.861459	−	0.507827i	$-0.169551\pi$
$503$	38.6410	1.72292	0.861459	+	0.507827i	$0.169551\pi$
$504$	0	0
$505$	3.07180	0.136693
$506$	−5.46410	−0.242909
$507$	0	0
$508$	11.8564	0.526043
$509$	29.6603	1.31467	0.657334	−	0.753600i	$-0.271684\pi$
$509$	29.6603	1.31467	0.657334	+	0.753600i	$0.271684\pi$
$510$	0	0
$511$	56.7846	2.51200
$512$	1.00000	0.0441942
$513$	0	0
$514$	−3.07180	−0.135491
$515$	−10.9282	−0.481554
$516$	0	0
$517$	2.53590	0.111529
$518$	−11.3205	−0.497395
$519$	0	0
$520$	−4.73205	−0.207514
$521$	8.92820	0.391152	0.195576	−	0.980689i	$-0.437342\pi$
$521$	8.92820	0.391152	0.195576	+	0.980689i	$0.437342\pi$
$522$	0	0
$523$	−19.8038	−0.865962	−0.432981	−	0.901403i	$-0.642538\pi$
$523$	−19.8038	−0.865962	−0.432981	+	0.901403i	$0.642538\pi$
$524$	8.39230	0.366620
$525$	0	0
$526$	14.7846	0.644640
$527$	−4.00000	−0.174243
$528$	0	0
$529$	32.7128	1.42230
$530$	8.73205	0.379296
$531$	0	0
$532$	−5.07180	−0.219890
$533$	−44.7846	−1.93984
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−12.3923	−0.535266
$537$	0	0
$538$	17.6603	0.761388
$539$	−3.66025	−0.157658
$540$	0	0
$541$	13.3205	0.572693	0.286347	−	0.958126i	$-0.407559\pi$
$541$	13.3205	0.572693	0.286347	+	0.958126i	$0.407559\pi$
$542$	1.46410	0.0628885
$543$	0	0
$544$	4.00000	0.171499
$545$	−8.53590	−0.365638
$546$	0	0
$547$	3.32051	0.141975	0.0709873	−	0.997477i	$-0.477385\pi$
$547$	3.32051	0.141975	0.0709873	+	0.997477i	$0.477385\pi$
$548$	−20.7846	−0.887875
$549$	0	0
$550$	−0.732051	−0.0312148
$551$	−4.00000	−0.170406
$552$	0	0
$553$	36.4974	1.55203
$554$	−2.87564	−0.122174
$555$	0	0
$556$	−3.66025	−0.155229
$557$	−2.87564	−0.121845	−0.0609225	−	0.998143i	$-0.519404\pi$
$557$	−2.87564	−0.121845	−0.0609225	+	0.998143i	$0.519404\pi$
$558$	0	0
$559$	−31.8564	−1.34738
$560$	3.46410	0.146385
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−28.3923	−1.19659	−0.598296	−	0.801275i	$-0.704156\pi$
$563$	−28.3923	−1.19659	−0.598296	+	0.801275i	$0.704156\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	2.53590	0.106592
$567$	0	0
$568$	−13.8564	−0.581402
$569$	5.32051	0.223047	0.111524	−	0.993762i	$-0.464427\pi$
$569$	5.32051	0.223047	0.111524	+	0.993762i	$0.464427\pi$
$570$	0	0
$571$	−2.58846	−0.108324	−0.0541618	−	0.998532i	$-0.517249\pi$
$571$	−2.58846	−0.108324	−0.0541618	+	0.998532i	$0.517249\pi$
$572$	3.46410	0.144841
$573$	0	0
$574$	32.7846	1.36840
$575$	7.46410	0.311275
$576$	0	0
$577$	−5.07180	−0.211142	−0.105571	−	0.994412i	$-0.533667\pi$
$577$	−5.07180	−0.211142	−0.105571	+	0.994412i	$0.533667\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	2.73205	0.113442
$581$	4.39230	0.182224
$582$	0	0
$583$	−6.39230	−0.264742
$584$	16.3923	0.678318
$585$	0	0
$586$	−17.3205	−0.715504
$587$	−14.0526	−0.580011	−0.290006	−	0.957025i	$-0.593657\pi$
$587$	−14.0526	−0.580011	−0.290006	+	0.957025i	$0.593657\pi$
$588$	0	0
$589$	1.46410	0.0603273
$590$	1.46410	0.0602761
$591$	0	0
$592$	−3.26795	−0.134312
$593$	−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
$596$	−21.3205	−0.873322
$597$	0	0
$598$	−35.3205	−1.44436
$599$	−0.143594	−0.00586707	−0.00293354	−	0.999996i	$-0.500934\pi$
$599$	−0.143594	−0.00586707	−0.00293354	+	0.999996i	$0.500934\pi$
$600$	0	0
$601$	15.0718	0.614791	0.307396	−	0.951582i	$-0.400542\pi$
$601$	15.0718	0.614791	0.307396	+	0.951582i	$0.400542\pi$
$602$	23.3205	0.950473
$603$	0	0
$604$	−5.46410	−0.222331
$605$	−10.4641	−0.425426
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	−1.46410	−0.0593772
$609$	0	0
$610$	−15.1244	−0.612367
$611$	16.3923	0.663162
$612$	0	0
$613$	−14.5885	−0.589222	−0.294611	−	0.955617i	$-0.595190\pi$
$613$	−14.5885	−0.589222	−0.294611	+	0.955617i	$0.595190\pi$
$614$	16.7846	0.677372
$615$	0	0
$616$	−2.53590	−0.102174
$617$	−8.14359	−0.327849	−0.163924	−	0.986473i	$-0.552415\pi$
$617$	−8.14359	−0.327849	−0.163924	+	0.986473i	$0.552415\pi$
$618$	0	0
$619$	10.1962	0.409818	0.204909	−	0.978781i	$-0.434310\pi$
$619$	10.1962	0.409818	0.204909	+	0.978781i	$0.434310\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	−26.9282	−1.07972
$623$	44.7846	1.79426
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.7846	−0.510976
$627$	0	0
$628$	7.85641	0.313505
$629$	−13.0718	−0.521207
$630$	0	0
$631$	−20.3923	−0.811805	−0.405902	−	0.913916i	$-0.633043\pi$
$631$	−20.3923	−0.811805	−0.405902	+	0.913916i	$0.633043\pi$
$632$	10.5359	0.419096
$633$	0	0
$634$	−16.5359	−0.656724
$635$	11.8564	0.470507
$636$	0	0
$637$	−23.6603	−0.937453
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	1.00000	0.0395285
$641$	−3.46410	−0.136824	−0.0684119	−	0.997657i	$-0.521793\pi$
$641$	−3.46410	−0.136824	−0.0684119	+	0.997657i	$0.521793\pi$
$642$	0	0
$643$	45.6603	1.80066	0.900332	−	0.435203i	$-0.143323\pi$
$643$	45.6603	1.80066	0.900332	+	0.435203i	$0.143323\pi$
$644$	25.8564	1.01889
$645$	0	0
$646$	−5.85641	−0.230417
$647$	−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
$649$	−1.07180	−0.0420717
$650$	−4.73205	−0.185606
$651$	0	0
$652$	−2.53590	−0.0993134
$653$	−0.928203	−0.0363234	−0.0181617	−	0.999835i	$-0.505781\pi$
$653$	−0.928203	−0.0363234	−0.0181617	+	0.999835i	$0.505781\pi$
$654$	0	0
$655$	8.39230	0.327914
$656$	9.46410	0.369511
$657$	0	0
$658$	−12.0000	−0.467809
$659$	−14.5359	−0.566238	−0.283119	−	0.959085i	$-0.591369\pi$
$659$	−14.5359	−0.566238	−0.283119	+	0.959085i	$0.591369\pi$
$660$	0	0
$661$	−16.2487	−0.632002	−0.316001	−	0.948759i	$-0.602340\pi$
$661$	−16.2487	−0.632002	−0.316001	+	0.948759i	$0.602340\pi$
$662$	−16.7321	−0.650310
$663$	0	0
$664$	1.26795	0.0492060
$665$	−5.07180	−0.196676
$666$	0	0
$667$	20.3923	0.789593
$668$	17.3205	0.670151
$669$	0	0
$670$	−12.3923	−0.478757
$671$	11.0718	0.427422
$672$	0	0
$673$	−4.78461	−0.184433	−0.0922166	−	0.995739i	$-0.529395\pi$
$673$	−4.78461	−0.184433	−0.0922166	+	0.995739i	$0.529395\pi$
$674$	−32.3923	−1.24770
$675$	0	0
$676$	9.39230	0.361242
$677$	−28.0526	−1.07815	−0.539074	−	0.842259i	$-0.681226\pi$
$677$	−28.0526	−1.07815	−0.539074	+	0.842259i	$0.681226\pi$
$678$	0	0
$679$	13.8564	0.531760
$680$	4.00000	0.153393
$681$	0	0
$682$	0.732051	0.0280317
$683$	39.7128	1.51957	0.759784	−	0.650175i	$-0.225305\pi$
$683$	39.7128	1.51957	0.759784	+	0.650175i	$0.225305\pi$
$684$	0	0
$685$	−20.7846	−0.794139
$686$	−6.92820	−0.264520
$687$	0	0
$688$	6.73205	0.256657
$689$	−41.3205	−1.57419
$690$	0	0
$691$	36.7846	1.39935	0.699676	−	0.714460i	$-0.253328\pi$
$691$	36.7846	1.39935	0.699676	+	0.714460i	$0.253328\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−18.0526	−0.685266
$695$	−3.66025	−0.138841
$696$	0	0
$697$	37.8564	1.43391
$698$	−35.4641	−1.34234
$699$	0	0
$700$	3.46410	0.130931
$701$	−27.0718	−1.02249	−0.511244	−	0.859436i	$-0.670815\pi$
$701$	−27.0718	−1.02249	−0.511244	+	0.859436i	$0.670815\pi$
$702$	0	0
$703$	4.78461	0.180455
$704$	−0.732051	−0.0275902
$705$	0	0
$706$	−4.39230	−0.165307
$707$	10.6410	0.400197
$708$	0	0
$709$	30.0526	1.12865	0.564324	−	0.825554i	$-0.309137\pi$
$709$	30.0526	1.12865	0.564324	+	0.825554i	$0.309137\pi$
$710$	−13.8564	−0.520022
$711$	0	0
$712$	12.9282	0.484505
$713$	−7.46410	−0.279533
$714$	0	0
$715$	3.46410	0.129550
$716$	11.2679	0.421103
$717$	0	0
$718$	19.0718	0.711753
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	0	0
$721$	−37.8564	−1.40985
$722$	−16.8564	−0.627330
$723$	0	0
$724$	15.1244	0.562092
$725$	2.73205	0.101466
$726$	0	0
$727$	−4.78461	−0.177451	−0.0887257	−	0.996056i	$-0.528279\pi$
$727$	−4.78461	−0.177451	−0.0887257	+	0.996056i	$0.528279\pi$
$728$	−16.3923	−0.607539
$729$	0	0
$730$	16.3923	0.606706
$731$	26.9282	0.995976
$732$	0	0
$733$	−38.3923	−1.41805	−0.709026	−	0.705182i	$-0.750865\pi$
$733$	−38.3923	−1.41805	−0.709026	+	0.705182i	$0.750865\pi$
$734$	−35.4641	−1.30900
$735$	0	0
$736$	7.46410	0.275130
$737$	9.07180	0.334164
$738$	0	0
$739$	34.1962	1.25793	0.628963	−	0.777435i	$-0.283480\pi$
$739$	34.1962	1.25793	0.628963	+	0.777435i	$0.283480\pi$
$740$	−3.26795	−0.120132
$741$	0	0
$742$	30.2487	1.11047
$743$	−33.7128	−1.23680	−0.618402	−	0.785862i	$-0.712219\pi$
$743$	−33.7128	−1.23680	−0.618402	+	0.785862i	$0.712219\pi$
$744$	0	0
$745$	−21.3205	−0.781123
$746$	−25.3205	−0.927050
$747$	0	0
$748$	−2.92820	−0.107066
$749$	−13.8564	−0.506302
$750$	0	0
$751$	19.8564	0.724571	0.362285	−	0.932067i	$-0.381997\pi$
$751$	19.8564	0.724571	0.362285	+	0.932067i	$0.381997\pi$
$752$	−3.46410	−0.126323
$753$	0	0
$754$	−12.9282	−0.470817
$755$	−5.46410	−0.198859
$756$	0	0
$757$	−44.4449	−1.61538	−0.807688	−	0.589610i	$-0.799281\pi$
$757$	−44.4449	−1.61538	−0.807688	+	0.589610i	$0.799281\pi$
$758$	27.3205	0.992326
$759$	0	0
$760$	−1.46410	−0.0531085
$761$	−7.46410	−0.270573	−0.135287	−	0.990806i	$-0.543196\pi$
$761$	−7.46410	−0.270573	−0.135287	+	0.990806i	$0.543196\pi$
$762$	0	0
$763$	−29.5692	−1.07048
$764$	2.00000	0.0723575
$765$	0	0
$766$	2.00000	0.0722629
$767$	−6.92820	−0.250163
$768$	0	0
$769$	4.67949	0.168747	0.0843734	−	0.996434i	$-0.473111\pi$
$769$	4.67949	0.168747	0.0843734	+	0.996434i	$0.473111\pi$
$770$	−2.53590	−0.0913874
$771$	0	0
$772$	−11.8564	−0.426721
$773$	2.48334	0.0893195	0.0446598	−	0.999002i	$-0.485780\pi$
$773$	2.48334	0.0893195	0.0446598	+	0.999002i	$0.485780\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	4.00000	0.143592
$777$	0	0
$778$	−15.8038	−0.566595
$779$	−13.8564	−0.496457
$780$	0	0
$781$	10.1436	0.362966
$782$	29.8564	1.06766
$783$	0	0
$784$	5.00000	0.178571
$785$	7.85641	0.280407
$786$	0	0
$787$	9.66025	0.344351	0.172175	−	0.985066i	$-0.444920\pi$
$787$	9.66025	0.344351	0.172175	+	0.985066i	$0.444920\pi$
$788$	−24.7321	−0.881043
$789$	0	0
$790$	10.5359	0.374850
$791$	−27.7128	−0.985354
$792$	0	0
$793$	71.5692	2.54150
$794$	−32.2487	−1.14446
$795$	0	0
$796$	−20.0000	−0.708881
$797$	36.8372	1.30484	0.652420	−	0.757858i	$-0.273754\pi$
$797$	36.8372	1.30484	0.652420	+	0.757858i	$0.273754\pi$
$798$	0	0
$799$	−13.8564	−0.490204
$800$	1.00000	0.0353553
$801$	0	0
$802$	0.535898	0.0189232
$803$	−12.0000	−0.423471
$804$	0	0
$805$	25.8564	0.911319
$806$	4.73205	0.166679
$807$	0	0
$808$	3.07180	0.108065
$809$	−32.2487	−1.13380	−0.566902	−	0.823785i	$-0.691858\pi$
$809$	−32.2487	−1.13380	−0.566902	+	0.823785i	$0.691858\pi$
$810$	0	0
$811$	4.39230	0.154235	0.0771173	−	0.997022i	$-0.475428\pi$
$811$	4.39230	0.154235	0.0771173	+	0.997022i	$0.475428\pi$
$812$	9.46410	0.332125
$813$	0	0
$814$	2.39230	0.0838502
$815$	−2.53590	−0.0888286
$816$	0	0
$817$	−9.85641	−0.344832
$818$	−15.4641	−0.540690
$819$	0	0
$820$	9.46410	0.330501
$821$	35.5167	1.23954	0.619770	−	0.784784i	$-0.287226\pi$
$821$	35.5167	1.23954	0.619770	+	0.784784i	$0.287226\pi$
$822$	0	0
$823$	−24.1436	−0.841593	−0.420796	−	0.907155i	$-0.638249\pi$
$823$	−24.1436	−0.841593	−0.420796	+	0.907155i	$0.638249\pi$
$824$	−10.9282	−0.380702
$825$	0	0
$826$	5.07180	0.176470
$827$	−11.5167	−0.400474	−0.200237	−	0.979748i	$-0.564171\pi$
$827$	−11.5167	−0.400474	−0.200237	+	0.979748i	$0.564171\pi$
$828$	0	0
$829$	−0.875644	−0.0304124	−0.0152062	−	0.999884i	$-0.504840\pi$
$829$	−0.875644	−0.0304124	−0.0152062	+	0.999884i	$0.504840\pi$
$830$	1.26795	0.0440112
$831$	0	0
$832$	−4.73205	−0.164054
$833$	20.0000	0.692959
$834$	0	0
$835$	17.3205	0.599401
$836$	1.07180	0.0370689
$837$	0	0
$838$	−16.3923	−0.566263
$839$	46.7846	1.61518	0.807592	−	0.589742i	$-0.200770\pi$
$839$	46.7846	1.61518	0.807592	+	0.589742i	$0.200770\pi$
$840$	0	0
$841$	−21.5359	−0.742617
$842$	−16.2487	−0.559968
$843$	0	0
$844$	13.0718	0.449950
$845$	9.39230	0.323105
$846$	0	0
$847$	−36.2487	−1.24552
$848$	8.73205	0.299860
$849$	0	0
$850$	4.00000	0.137199
$851$	−24.3923	−0.836157
$852$	0	0
$853$	−24.1436	−0.826661	−0.413330	−	0.910581i	$-0.635635\pi$
$853$	−24.1436	−0.826661	−0.413330	+	0.910581i	$0.635635\pi$
$854$	−52.3923	−1.79283
$855$	0	0
$856$	−4.00000	−0.136717
$857$	11.0718	0.378205	0.189103	−	0.981957i	$-0.439442\pi$
$857$	11.0718	0.378205	0.189103	+	0.981957i	$0.439442\pi$
$858$	0	0
$859$	20.4449	0.697570	0.348785	−	0.937203i	$-0.386594\pi$
$859$	20.4449	0.697570	0.348785	+	0.937203i	$0.386594\pi$
$860$	6.73205	0.229561
$861$	0	0
$862$	14.7846	0.503566
$863$	31.8564	1.08440	0.542202	−	0.840248i	$-0.317591\pi$
$863$	31.8564	1.08440	0.542202	+	0.840248i	$0.317591\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	−24.7846	−0.842215
$867$	0	0
$868$	−3.46410	−0.117579
$869$	−7.71281	−0.261639
$870$	0	0
$871$	58.6410	1.98698
$872$	−8.53590	−0.289062
$873$	0	0
$874$	−10.9282	−0.369652
$875$	3.46410	0.117108
$876$	0	0
$877$	45.3205	1.53036	0.765182	−	0.643814i	$-0.222649\pi$
$877$	45.3205	1.53036	0.765182	+	0.643814i	$0.222649\pi$
$878$	22.0000	0.742464
$879$	0	0
$880$	−0.732051	−0.0246774
$881$	−35.5692	−1.19836	−0.599179	−	0.800615i	$-0.704506\pi$
$881$	−35.5692	−1.19836	−0.599179	+	0.800615i	$0.704506\pi$
$882$	0	0
$883$	−2.33975	−0.0787387	−0.0393694	−	0.999225i	$-0.512535\pi$
$883$	−2.33975	−0.0787387	−0.0393694	+	0.999225i	$0.512535\pi$
$884$	−18.9282	−0.636624
$885$	0	0
$886$	21.0718	0.707921
$887$	50.1051	1.68237	0.841183	−	0.540751i	$-0.181860\pi$
$887$	50.1051	1.68237	0.841183	+	0.540751i	$0.181860\pi$
$888$	0	0
$889$	41.0718	1.37750
$890$	12.9282	0.433354
$891$	0	0
$892$	10.0000	0.334825
$893$	5.07180	0.169721
$894$	0	0
$895$	11.2679	0.376646
$896$	3.46410	0.115728
$897$	0	0
$898$	5.32051	0.177548
$899$	−2.73205	−0.0911190
$900$	0	0
$901$	34.9282	1.16363
$902$	−6.92820	−0.230684
$903$	0	0
$904$	−8.00000	−0.266076
$905$	15.1244	0.502751
$906$	0	0
$907$	−9.85641	−0.327277	−0.163638	−	0.986520i	$-0.552323\pi$
$907$	−9.85641	−0.327277	−0.163638	+	0.986520i	$0.552323\pi$
$908$	6.53590	0.216901
$909$	0	0
$910$	−16.3923	−0.543400
$911$	30.9282	1.02470	0.512349	−	0.858778i	$-0.328776\pi$
$911$	30.9282	1.02470	0.512349	+	0.858778i	$0.328776\pi$
$912$	0	0
$913$	−0.928203	−0.0307190
$914$	1.07180	0.0354519
$915$	0	0
$916$	14.0526	0.464310
$917$	29.0718	0.960035
$918$	0	0
$919$	−40.7846	−1.34536	−0.672680	−	0.739933i	$-0.734857\pi$
$919$	−40.7846	−1.34536	−0.672680	+	0.739933i	$0.734857\pi$
$920$	7.46410	0.246084
$921$	0	0
$922$	30.7321	1.01211
$923$	65.5692	2.15824
$924$	0	0
$925$	−3.26795	−0.107450
$926$	10.3923	0.341512
$927$	0	0
$928$	2.73205	0.0896840
$929$	22.3923	0.734668	0.367334	−	0.930089i	$-0.380271\pi$
$929$	22.3923	0.734668	0.367334	+	0.930089i	$0.380271\pi$
$930$	0	0
$931$	−7.32051	−0.239920
$932$	16.7846	0.549798
$933$	0	0
$934$	−33.8564	−1.10782
$935$	−2.92820	−0.0957625
$936$	0	0
$937$	−32.9282	−1.07572	−0.537859	−	0.843035i	$-0.680767\pi$
$937$	−32.9282	−1.07572	−0.537859	+	0.843035i	$0.680767\pi$
$938$	−42.9282	−1.40166
$939$	0	0
$940$	−3.46410	−0.112987
$941$	23.4115	0.763194	0.381597	−	0.924329i	$-0.375374\pi$
$941$	23.4115	0.763194	0.381597	+	0.924329i	$0.375374\pi$
$942$	0	0
$943$	70.6410	2.30039
$944$	1.46410	0.0476524
$945$	0	0
$946$	−4.92820	−0.160230
$947$	−0.196152	−0.00637410	−0.00318705	−	0.999995i	$-0.501014\pi$
$947$	−0.196152	−0.00637410	−0.00318705	+	0.999995i	$0.501014\pi$
$948$	0	0
$949$	−77.5692	−2.51800
$950$	−1.46410	−0.0475017
$951$	0	0
$952$	13.8564	0.449089
$953$	−17.0718	−0.553010	−0.276505	−	0.961013i	$-0.589176\pi$
$953$	−17.0718	−0.553010	−0.276505	+	0.961013i	$0.589176\pi$
$954$	0	0
$955$	2.00000	0.0647185
$956$	2.53590	0.0820168
$957$	0	0
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	1.00000	0.0322581
$962$	15.4641	0.498583
$963$	0	0
$964$	28.2487	0.909830
$965$	−11.8564	−0.381671
$966$	0	0
$967$	−39.8564	−1.28170	−0.640848	−	0.767668i	$-0.721417\pi$
$967$	−39.8564	−1.28170	−0.640848	+	0.767668i	$0.721417\pi$
$968$	−10.4641	−0.336329
$969$	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
$973$	−12.6795	−0.406486
$974$	−10.0000	−0.320421
$975$	0	0
$976$	−15.1244	−0.484119
$977$	−1.21539	−0.0388838	−0.0194419	−	0.999811i	$-0.506189\pi$
$977$	−1.21539	−0.0388838	−0.0194419	+	0.999811i	$0.506189\pi$
$978$	0	0
$979$	−9.46410	−0.302474
$980$	5.00000	0.159719
$981$	0	0
$982$	−15.2679	−0.487220
$983$	−17.7128	−0.564951	−0.282475	−	0.959275i	$-0.591156\pi$
$983$	−17.7128	−0.564951	−0.282475	+	0.959275i	$0.591156\pi$
$984$	0	0
$985$	−24.7321	−0.788029
$986$	10.9282	0.348025
$987$	0	0
$988$	6.92820	0.220416
$989$	50.2487	1.59782
$990$	0	0
$991$	26.9282	0.855403	0.427701	−	0.903920i	$-0.359324\pi$
$991$	26.9282	0.855403	0.427701	+	0.903920i	$0.359324\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−48.0000	−1.52247
$995$	−20.0000	−0.634043
$996$	0	0
$997$	39.1769	1.24075	0.620373	−	0.784307i	$-0.286981\pi$
$997$	39.1769	1.24075	0.620373	+	0.784307i	$0.286981\pi$
$998$	−5.12436	−0.162209
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bh.1.2		2
3.2	odd	2		310.2.a.c.1.1	✓	2
12.11	even	2		2480.2.a.s.1.2		2
15.2	even	4		1550.2.b.h.249.2		4
15.8	even	4		1550.2.b.h.249.3		4
15.14	odd	2		1550.2.a.j.1.2		2
24.5	odd	2		9920.2.a.bt.1.2		2
24.11	even	2		9920.2.a.bl.1.1		2
93.92	even	2		9610.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.a.c.1.1	✓	2	3.2	odd	2
1550.2.a.j.1.2		2	15.14	odd	2
1550.2.b.h.249.2		4	15.2	even	4
1550.2.b.h.249.3		4	15.8	even	4
2480.2.a.s.1.2		2	12.11	even	2
2790.2.a.bh.1.2		2	1.1	even	1	trivial
9610.2.a.j.1.2		2	93.92	even	2
9920.2.a.bl.1.1		2	24.11	even	2
9920.2.a.bt.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.46410 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.46410 q^{7} +1.00000 q^{8} +1.00000 q^{10} -0.732051 q^{11} -4.73205 q^{13} +3.46410 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.46410 q^{19} +1.00000 q^{20} -0.732051 q^{22} +7.46410 q^{23} +1.00000 q^{25} -4.73205 q^{26} +3.46410 q^{28} +2.73205 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +3.46410 q^{35} -3.26795 q^{37} -1.46410 q^{38} +1.00000 q^{40} +9.46410 q^{41} +6.73205 q^{43} -0.732051 q^{44} +7.46410 q^{46} -3.46410 q^{47} +5.00000 q^{49} +1.00000 q^{50} -4.73205 q^{52} +8.73205 q^{53} -0.732051 q^{55} +3.46410 q^{56} +2.73205 q^{58} +1.46410 q^{59} -15.1244 q^{61} -1.00000 q^{62} +1.00000 q^{64} -4.73205 q^{65} -12.3923 q^{67} +4.00000 q^{68} +3.46410 q^{70} -13.8564 q^{71} +16.3923 q^{73} -3.26795 q^{74} -1.46410 q^{76} -2.53590 q^{77} +10.5359 q^{79} +1.00000 q^{80} +9.46410 q^{82} +1.26795 q^{83} +4.00000 q^{85} +6.73205 q^{86} -0.732051 q^{88} +12.9282 q^{89} -16.3923 q^{91} +7.46410 q^{92} -3.46410 q^{94} -1.46410 q^{95} +4.00000 q^{97} +5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 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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