LMFDB - Embedded newform 1248.2.g.a.625.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(625,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1248.625");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			625.1
Root			$0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1248.625
Dual form			1248.2.g.a.625.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.52015i q^{5} -2.79360 q^{7} -1.00000 q^{9} +5.31375i q^{11} +1.00000i q^{13} -2.52015 q^{15} -4.35114 q^{17} +6.02967i q^{19} +2.79360i q^{21} -0.568158 q^{23} -1.35114 q^{25} +1.00000i q^{27} -2.68915i q^{29} -1.90868 q^{31} +5.31375 q^{33} +7.04029i q^{35} +9.60845i q^{37} +1.00000 q^{39} +11.2093 q^{41} -9.48746i q^{43} +2.52015i q^{45} -3.95669 q^{47} +0.804226 q^{49} +4.35114i q^{51} +8.05934i q^{53} +13.3914 q^{55} +6.02967 q^{57} -6.19868i q^{59} +7.23607i q^{61} +2.79360 q^{63} +2.52015 q^{65} +10.0297i q^{67} +0.568158i q^{69} +1.63052 q^{71} -5.17442 q^{73} +1.35114i q^{75} -14.8445i q^{77} -9.17442 q^{79} +1.00000 q^{81} -4.61147i q^{83} +10.9655i q^{85} -2.68915 q^{87} +11.4182 q^{89} -2.79360i q^{91} +1.90868i q^{93} +15.1957 q^{95} -10.2514 q^{97} -5.31375i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 8 q^{9} + 4 q^{15} - 16 q^{17} + 8 q^{23} + 8 q^{25} + 4 q^{31} + 8 q^{39} + 36 q^{41} - 24 q^{47} - 24 q^{49} + 40 q^{55} + 12 q^{57} + 4 q^{63} - 4 q^{65} - 16 q^{71} + 32 q^{73} + 8 q^{81}+ \cdots - 56 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 8 * q^9 + 4 * q^15 - 16 * q^17 + 8 * q^23 + 8 * q^25 + 4 * q^31 + 8 * q^39 + 36 * q^41 - 24 * q^47 - 24 * q^49 + 40 * q^55 + 12 * q^57 + 4 * q^63 - 4 * q^65 - 16 * q^71 + 32 * q^73 + 8 * q^81 + 8 * q^87 + 60 * q^89 + 24 * q^95 - 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 2.52015i	− 1.12704i	−0.826101	−	0.563522i	$-0.809446\pi$
$5$	− 2.52015i	− 1.12704i	0.826101	−	0.563522i	$-0.190554\pi$
$6$	0	0
$7$	−2.79360	−1.05588	−0.527942	−	0.849281i	$-0.677036\pi$
$7$	−2.79360	−1.05588	−0.527942	+	0.849281i	$0.677036\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.31375i	1.60216i	0.598560	+	0.801078i	$0.295740\pi$
$11$	5.31375i	1.60216i	−0.598560	+	0.801078i	$0.704260\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−2.52015	−0.650699
$16$	0	0
$17$	−4.35114	−1.05531	−0.527653	−	0.849460i	$-0.676928\pi$
$17$	−4.35114	−1.05531	−0.527653	+	0.849460i	$0.676928\pi$
$18$	0	0
$19$	6.02967i	1.38330i	0.722232	+	0.691651i	$0.243116\pi$
$19$	6.02967i	1.38330i	−0.722232	+	0.691651i	$0.756884\pi$
$20$	0	0
$21$	2.79360i	0.609614i
$22$	0	0
$23$	−0.568158	−0.118469	−0.0592346	−	0.998244i	$-0.518866\pi$
$23$	−0.568158	−0.118469	−0.0592346	+	0.998244i	$0.518866\pi$
$24$	0	0
$25$	−1.35114	−0.270228
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 2.68915i	− 0.499363i	−0.968328	−	0.249682i	$-0.919674\pi$
$29$	− 2.68915i	− 0.499363i	0.968328	−	0.249682i	$-0.0803260\pi$
$30$	0	0
$31$	−1.90868	−0.342809	−0.171404	−	0.985201i	$-0.554830\pi$
$31$	−1.90868	−0.342809	−0.171404	+	0.985201i	$0.554830\pi$
$32$	0	0
$33$	5.31375	0.925005
$34$	0	0
$35$	7.04029i	1.19003i
$36$	0	0
$37$	9.60845i	1.57962i	0.613352	+	0.789810i	$0.289821\pi$
$37$	9.60845i	1.57962i	−0.613352	+	0.789810i	$0.710179\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	11.2093	1.75060	0.875299	−	0.483582i	$-0.160664\pi$
$41$	11.2093	1.75060	0.875299	+	0.483582i	$0.160664\pi$
$42$	0	0
$43$	− 9.48746i	− 1.44682i	−0.690417	−	0.723412i	$-0.742573\pi$
$43$	− 9.48746i	− 1.44682i	0.690417	−	0.723412i	$-0.257427\pi$
$44$	0	0
$45$	2.52015i	0.375681i
$46$	0	0
$47$	−3.95669	−0.577142	−0.288571	−	0.957458i	$-0.593180\pi$
$47$	−3.95669	−0.577142	−0.288571	+	0.957458i	$0.593180\pi$
$48$	0	0
$49$	0.804226	0.114889
$50$	0	0
$51$	4.35114i	0.609282i
$52$	0	0
$53$	8.05934i	1.10704i	0.832837	+	0.553518i	$0.186715\pi$
$53$	8.05934i	1.10704i	−0.832837	+	0.553518i	$0.813285\pi$
$54$	0	0
$55$	13.3914	1.80570
$56$	0	0
$57$	6.02967	0.798650
$58$	0	0
$59$	− 6.19868i	− 0.806999i	−0.914980	−	0.403500i	$-0.867794\pi$
$59$	− 6.19868i	− 0.806999i	0.914980	−	0.403500i	$-0.132206\pi$
$60$	0	0
$61$	7.23607i	0.926484i	0.886232	+	0.463242i	$0.153314\pi$
$61$	7.23607i	0.926484i	−0.886232	+	0.463242i	$0.846686\pi$
$62$	0	0
$63$	2.79360	0.351961
$64$	0	0
$65$	2.52015	0.312586
$66$	0	0
$67$	10.0297i	1.22532i	0.790347	+	0.612660i	$0.209900\pi$
$67$	10.0297i	1.22532i	−0.790347	+	0.612660i	$0.790100\pi$
$68$	0	0
$69$	0.568158i	0.0683982i
$70$	0	0
$71$	1.63052	0.193507	0.0967536	−	0.995308i	$-0.469154\pi$
$71$	1.63052	0.193507	0.0967536	+	0.995308i	$0.469154\pi$
$72$	0	0
$73$	−5.17442	−0.605620	−0.302810	−	0.953051i	$-0.597925\pi$
$73$	−5.17442	−0.605620	−0.302810	+	0.953051i	$0.597925\pi$
$74$	0	0
$75$	1.35114i	0.156016i
$76$	0	0
$77$	− 14.8445i	− 1.69169i
$78$	0	0
$79$	−9.17442	−1.03220	−0.516101	−	0.856528i	$-0.672617\pi$
$79$	−9.17442	−1.03220	−0.516101	+	0.856528i	$0.672617\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.61147i	− 0.506175i	−0.967443	−	0.253087i	$-0.918554\pi$
$83$	− 4.61147i	− 0.506175i	0.967443	−	0.253087i	$-0.0814460\pi$
$84$	0	0
$85$	10.9655i	1.18938i
$86$	0	0
$87$	−2.68915	−0.288307
$88$	0	0
$89$	11.4182	1.21033	0.605164	−	0.796101i	$-0.293108\pi$
$89$	11.4182	1.21033	0.605164	+	0.796101i	$0.293108\pi$
$90$	0	0
$91$	− 2.79360i	− 0.292849i
$92$	0	0
$93$	1.90868i	0.197921i
$94$	0	0
$95$	15.1957	1.55904
$96$	0	0
$97$	−10.2514	−1.04087	−0.520435	−	0.853901i	$-0.674230\pi$
$97$	−10.2514	−1.04087	−0.520435	+	0.853901i	$0.674230\pi$
$98$	0	0
$99$	− 5.31375i	− 0.534052i
$100$	0	0
$101$	3.01905i	0.300407i	0.988655	+	0.150203i	$0.0479929\pi$
$101$	3.01905i	0.300407i	−0.988655	+	0.150203i	$0.952007\pi$
$102$	0	0
$103$	−12.9193	−1.27298	−0.636488	−	0.771286i	$-0.719614\pi$
$103$	−12.9193	−1.27298	−0.636488	+	0.771286i	$0.719614\pi$
$104$	0	0
$105$	7.04029	0.687062
$106$	0	0
$107$	10.9537i	1.05893i	0.848331	+	0.529466i	$0.177608\pi$
$107$	10.9537i	1.05893i	−0.848331	+	0.529466i	$0.822392\pi$
$108$	0	0
$109$	− 5.45309i	− 0.522311i	−0.965297	−	0.261155i	$-0.915896\pi$
$109$	− 5.45309i	− 0.522311i	0.965297	−	0.261155i	$-0.0841035\pi$
$110$	0	0
$111$	9.60845	0.911994
$112$	0	0
$113$	−13.6334	−1.28252	−0.641262	−	0.767322i	$-0.721589\pi$
$113$	−13.6334	−1.28252	−0.641262	+	0.767322i	$0.721589\pi$
$114$	0	0
$115$	1.43184i	0.133520i
$116$	0	0
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	12.1554	1.11428
$120$	0	0
$121$	−17.2360	−1.56691
$122$	0	0
$123$	− 11.2093i	− 1.01071i
$124$	0	0
$125$	− 9.19566i	− 0.822485i
$126$	0	0
$127$	−19.4377	−1.72481	−0.862406	−	0.506217i	$-0.831044\pi$
$127$	−19.4377	−1.72481	−0.862406	+	0.506217i	$0.831044\pi$
$128$	0	0
$129$	−9.48746	−0.835324
$130$	0	0
$131$	12.7235i	1.11166i	0.831296	+	0.555830i	$0.187599\pi$
$131$	12.7235i	1.11166i	−0.831296	+	0.555830i	$0.812401\pi$
$132$	0	0
$133$	− 16.8445i	− 1.46061i
$134$	0	0
$135$	2.52015	0.216900
$136$	0	0
$137$	−9.23054	−0.788619	−0.394309	−	0.918978i	$-0.629016\pi$
$137$	−9.23054	−0.788619	−0.394309	+	0.918978i	$0.629016\pi$
$138$	0	0
$139$	11.7507i	0.996681i	0.866982	+	0.498340i	$0.166057\pi$
$139$	11.7507i	0.996681i	−0.866982	+	0.498340i	$0.833943\pi$
$140$	0	0
$141$	3.95669i	0.333213i
$142$	0	0
$143$	−5.31375	−0.444358
$144$	0	0
$145$	−6.77706	−0.562804
$146$	0	0
$147$	− 0.804226i	− 0.0663314i
$148$	0	0
$149$	17.6079i	1.44250i	0.692675	+	0.721249i	$0.256432\pi$
$149$	17.6079i	1.44250i	−0.692675	+	0.721249i	$0.743568\pi$
$150$	0	0
$151$	11.6250	0.946029	0.473014	−	0.881055i	$-0.343166\pi$
$151$	11.6250	0.946029	0.473014	+	0.881055i	$0.343166\pi$
$152$	0	0
$153$	4.35114	0.351769
$154$	0	0
$155$	4.81015i	0.386360i
$156$	0	0
$157$	13.4781i	1.07567i	0.843051	+	0.537833i	$0.180757\pi$
$157$	13.4781i	1.07567i	−0.843051	+	0.537833i	$0.819243\pi$
$158$	0	0
$159$	8.05934	0.639148
$160$	0	0
$161$	1.58721	0.125090
$162$	0	0
$163$	− 9.79349i	− 0.767085i	−0.923523	−	0.383543i	$-0.874704\pi$
$163$	− 9.79349i	− 0.767085i	0.923523	−	0.383543i	$-0.125296\pi$
$164$	0	0
$165$	− 13.3914i	− 1.04252i
$166$	0	0
$167$	16.6054	1.28497	0.642484	−	0.766299i	$-0.277904\pi$
$167$	16.6054	1.28497	0.642484	+	0.766299i	$0.277904\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 6.02967i	− 0.461101i
$172$	0	0
$173$	16.1979i	1.23150i	0.787942	+	0.615750i	$0.211147\pi$
$173$	16.1979i	1.23150i	−0.787942	+	0.615750i	$0.788853\pi$
$174$	0	0
$175$	3.77455	0.285329
$176$	0	0
$177$	−6.19868	−0.465921
$178$	0	0
$179$	− 1.22795i	− 0.0917816i	−0.998946	−	0.0458908i	$-0.985387\pi$
$179$	− 1.22795i	− 0.0917816i	0.998946	−	0.0458908i	$-0.0146126\pi$
$180$	0	0
$181$	7.00592i	0.520746i	0.965508	+	0.260373i	$0.0838455\pi$
$181$	7.00592i	0.520746i	−0.965508	+	0.260373i	$0.916154\pi$
$182$	0	0
$183$	7.23607	0.534906
$184$	0	0
$185$	24.2147	1.78030
$186$	0	0
$187$	− 23.1209i	− 1.69077i
$188$	0	0
$189$	− 2.79360i	− 0.203205i
$190$	0	0
$191$	14.3701	1.03978	0.519891	−	0.854232i	$-0.325972\pi$
$191$	14.3701	1.03978	0.519891	+	0.854232i	$0.325972\pi$
$192$	0	0
$193$	−3.67383	−0.264448	−0.132224	−	0.991220i	$-0.542212\pi$
$193$	−3.67383	−0.264448	−0.132224	+	0.991220i	$0.542212\pi$
$194$	0	0
$195$	− 2.52015i	− 0.180471i
$196$	0	0
$197$	− 6.88442i	− 0.490494i	−0.969461	−	0.245247i	$-0.921131\pi$
$197$	− 6.88442i	− 0.490494i	0.969461	−	0.245247i	$-0.0788691\pi$
$198$	0	0
$199$	−15.3952	−1.09133	−0.545667	−	0.838002i	$-0.683724\pi$
$199$	−15.3952	−1.09133	−0.545667	+	0.838002i	$0.683724\pi$
$200$	0	0
$201$	10.0297	0.707439
$202$	0	0
$203$	7.51243i	0.527269i
$204$	0	0
$205$	− 28.2491i	− 1.97300i
$206$	0	0
$207$	0.568158	0.0394897
$208$	0	0
$209$	−32.0402	−2.21627
$210$	0	0
$211$	1.57035i	0.108107i	0.998538	+	0.0540537i	$0.0172142\pi$
$211$	1.57035i	0.108107i	−0.998538	+	0.0540537i	$0.982786\pi$
$212$	0	0
$213$	− 1.63052i	− 0.111721i
$214$	0	0
$215$	−23.9098	−1.63063
$216$	0	0
$217$	5.33209	0.361966
$218$	0	0
$219$	5.17442i	0.349655i
$220$	0	0
$221$	− 4.35114i	− 0.292689i
$222$	0	0
$223$	−27.6571	−1.85205	−0.926027	−	0.377457i	$-0.876798\pi$
$223$	−27.6571	−1.85205	−0.926027	+	0.377457i	$0.876798\pi$
$224$	0	0
$225$	1.35114	0.0900761
$226$	0	0
$227$	− 22.9434i	− 1.52281i	−0.648276	−	0.761405i	$-0.724510\pi$
$227$	− 22.9434i	− 1.52281i	0.648276	−	0.761405i	$-0.275490\pi$
$228$	0	0
$229$	− 23.5064i	− 1.55335i	−0.629904	−	0.776673i	$-0.716906\pi$
$229$	− 23.5064i	− 1.55335i	0.629904	−	0.776673i	$-0.283094\pi$
$230$	0	0
$231$	−14.8445	−0.976698
$232$	0	0
$233$	3.47433	0.227611	0.113805	−	0.993503i	$-0.463696\pi$
$233$	3.47433	0.227611	0.113805	+	0.993503i	$0.463696\pi$
$234$	0	0
$235$	9.97144i	0.650465i
$236$	0	0
$237$	9.17442i	0.595942i
$238$	0	0
$239$	11.5820	0.749177	0.374589	−	0.927191i	$-0.377784\pi$
$239$	11.5820	0.749177	0.374589	+	0.927191i	$0.377784\pi$
$240$	0	0
$241$	10.0058	0.644531	0.322265	−	0.946649i	$-0.395556\pi$
$241$	10.0058	0.644531	0.322265	+	0.946649i	$0.395556\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 2.02677i	− 0.129485i
$246$	0	0
$247$	−6.02967	−0.383659
$248$	0	0
$249$	−4.61147	−0.292240
$250$	0	0
$251$	− 11.3723i	− 0.717811i	−0.933374	−	0.358906i	$-0.883150\pi$
$251$	− 11.3723i	− 0.717811i	0.933374	−	0.358906i	$-0.116850\pi$
$252$	0	0
$253$	− 3.01905i	− 0.189806i
$254$	0	0
$255$	10.9655	0.686687
$256$	0	0
$257$	11.3510	0.708058	0.354029	−	0.935235i	$-0.384811\pi$
$257$	11.3510	0.708058	0.354029	+	0.935235i	$0.384811\pi$
$258$	0	0
$259$	− 26.8422i	− 1.66789i
$260$	0	0
$261$	2.68915i	0.166454i
$262$	0	0
$263$	−3.88712	−0.239690	−0.119845	−	0.992793i	$-0.538240\pi$
$263$	−3.88712	−0.239690	−0.119845	+	0.992793i	$0.538240\pi$
$264$	0	0
$265$	20.3107	1.24768
$266$	0	0
$267$	− 11.4182i	− 0.698783i
$268$	0	0
$269$	− 25.2300i	− 1.53830i	−0.639067	−	0.769151i	$-0.720679\pi$
$269$	− 25.2300i	− 1.53830i	0.639067	−	0.769151i	$-0.279321\pi$
$270$	0	0
$271$	2.90649	0.176556	0.0882782	−	0.996096i	$-0.471864\pi$
$271$	2.90649	0.176556	0.0882782	+	0.996096i	$0.471864\pi$
$272$	0	0
$273$	−2.79360	−0.169077
$274$	0	0
$275$	− 7.17963i	− 0.432948i
$276$	0	0
$277$	− 4.26825i	− 0.256454i	−0.991745	−	0.128227i	$-0.959071\pi$
$277$	− 4.26825i	− 0.256454i	0.991745	−	0.128227i	$-0.0409286\pi$
$278$	0	0
$279$	1.90868	0.114270
$280$	0	0
$281$	−18.9900	−1.13285	−0.566424	−	0.824114i	$-0.691673\pi$
$281$	−18.9900	−1.13285	−0.566424	+	0.824114i	$0.691673\pi$
$282$	0	0
$283$	13.1957i	0.784401i	0.919880	+	0.392200i	$0.128286\pi$
$283$	13.1957i	0.784401i	−0.919880	+	0.392200i	$0.871714\pi$
$284$	0	0
$285$	− 15.1957i	− 0.900113i
$286$	0	0
$287$	−31.3144	−1.84843
$288$	0	0
$289$	1.93243	0.113672
$290$	0	0
$291$	10.2514i	0.600947i
$292$	0	0
$293$	− 12.0980i	− 0.706770i	−0.935478	−	0.353385i	$-0.885031\pi$
$293$	− 12.0980i	− 0.706770i	0.935478	−	0.353385i	$-0.114969\pi$
$294$	0	0
$295$	−15.6216	−0.909524
$296$	0	0
$297$	−5.31375	−0.308335
$298$	0	0
$299$	− 0.568158i	− 0.0328574i
$300$	0	0
$301$	26.5042i	1.52768i
$302$	0	0
$303$	3.01905	0.173440
$304$	0	0
$305$	18.2360	1.04419
$306$	0	0
$307$	24.7863i	1.41463i	0.706900	+	0.707314i	$0.250093\pi$
$307$	24.7863i	1.41463i	−0.706900	+	0.707314i	$0.749907\pi$
$308$	0	0
$309$	12.9193i	0.734953i
$310$	0	0
$311$	6.72956	0.381598	0.190799	−	0.981629i	$-0.438892\pi$
$311$	6.72956	0.381598	0.190799	+	0.981629i	$0.438892\pi$
$312$	0	0
$313$	11.4956	0.649768	0.324884	−	0.945754i	$-0.394675\pi$
$313$	11.4956	0.649768	0.324884	+	0.945754i	$0.394675\pi$
$314$	0	0
$315$	− 7.04029i	− 0.396676i
$316$	0	0
$317$	2.86756i	0.161058i	0.996752	+	0.0805291i	$0.0256610\pi$
$317$	2.86756i	0.161058i	−0.996752	+	0.0805291i	$0.974339\pi$
$318$	0	0
$319$	14.2895	0.800058
$320$	0	0
$321$	10.9537	0.611374
$322$	0	0
$323$	− 26.2360i	− 1.45981i
$324$	0	0
$325$	− 1.35114i	− 0.0749478i
$326$	0	0
$327$	−5.45309	−0.301556
$328$	0	0
$329$	11.0534	0.609395
$330$	0	0
$331$	1.44967i	0.0796811i	0.999206	+	0.0398405i	$0.0126850\pi$
$331$	1.44967i	0.0796811i	−0.999206	+	0.0398405i	$0.987315\pi$
$332$	0	0
$333$	− 9.60845i	− 0.526540i
$334$	0	0
$335$	25.2762	1.38099
$336$	0	0
$337$	−22.6774	−1.23532	−0.617660	−	0.786446i	$-0.711919\pi$
$337$	−22.6774	−1.23532	−0.617660	+	0.786446i	$0.711919\pi$
$338$	0	0
$339$	13.6334i	0.740466i
$340$	0	0
$341$	− 10.1422i	− 0.549233i
$342$	0	0
$343$	17.3085	0.934573
$344$	0	0
$345$	1.43184	0.0770878
$346$	0	0
$347$	− 33.7652i	− 1.81261i	−0.422621	−	0.906307i	$-0.638890\pi$
$347$	− 33.7652i	− 1.81261i	0.422621	−	0.906307i	$-0.361110\pi$
$348$	0	0
$349$	6.34741i	0.339769i	0.985464	+	0.169885i	$0.0543395\pi$
$349$	6.34741i	0.339769i	−0.985464	+	0.169885i	$0.945661\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	32.1856	1.71307	0.856534	−	0.516090i	$-0.172613\pi$
$353$	32.1856	1.71307	0.856534	+	0.516090i	$0.172613\pi$
$354$	0	0
$355$	− 4.10915i	− 0.218091i
$356$	0	0
$357$	− 12.1554i	− 0.643330i
$358$	0	0
$359$	−22.9222	−1.20979	−0.604894	−	0.796306i	$-0.706784\pi$
$359$	−22.9222	−1.20979	−0.604894	+	0.796306i	$0.706784\pi$
$360$	0	0
$361$	−17.3570	−0.913524
$362$	0	0
$363$	17.2360i	0.904653i
$364$	0	0
$365$	13.0403i	0.682560i
$366$	0	0
$367$	9.68093	0.505340	0.252670	−	0.967552i	$-0.418691\pi$
$367$	9.68093	0.505340	0.252670	+	0.967552i	$0.418691\pi$
$368$	0	0
$369$	−11.2093	−0.583533
$370$	0	0
$371$	− 22.5146i	− 1.16890i
$372$	0	0
$373$	− 23.2975i	− 1.20630i	−0.797628	−	0.603149i	$-0.793912\pi$
$373$	− 23.2975i	− 1.20630i	0.797628	−	0.603149i	$-0.206088\pi$
$374$	0	0
$375$	−9.19566	−0.474862
$376$	0	0
$377$	2.68915	0.138498
$378$	0	0
$379$	− 9.48276i	− 0.487097i	−0.969889	−	0.243548i	$-0.921689\pi$
$379$	− 9.48276i	− 0.487097i	0.969889	−	0.243548i	$-0.0783114\pi$
$380$	0	0
$381$	19.4377i	0.995821i
$382$	0	0
$383$	5.68406	0.290442	0.145221	−	0.989399i	$-0.453611\pi$
$383$	5.68406	0.290442	0.145221	+	0.989399i	$0.453611\pi$
$384$	0	0
$385$	−37.4104	−1.90661
$386$	0	0
$387$	9.48746i	0.482275i
$388$	0	0
$389$	1.79842i	0.0911834i	0.998960	+	0.0455917i	$0.0145173\pi$
$389$	1.79842i	0.0911834i	−0.998960	+	0.0455917i	$0.985483\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	12.7235	0.641817
$394$	0	0
$395$	23.1209i	1.16334i
$396$	0	0
$397$	4.61064i	0.231402i	0.993284	+	0.115701i	$0.0369114\pi$
$397$	4.61064i	0.231402i	−0.993284	+	0.115701i	$0.963089\pi$
$398$	0	0
$399$	−16.8445	−0.843281
$400$	0	0
$401$	−19.4701	−0.972290	−0.486145	−	0.873878i	$-0.661597\pi$
$401$	−19.4701	−0.972290	−0.486145	+	0.873878i	$0.661597\pi$
$402$	0	0
$403$	− 1.90868i	− 0.0950780i
$404$	0	0
$405$	− 2.52015i	− 0.125227i
$406$	0	0
$407$	−51.0569	−2.53080
$408$	0	0
$409$	34.5696	1.70936	0.854678	−	0.519159i	$-0.173755\pi$
$409$	34.5696	1.70936	0.854678	+	0.519159i	$0.173755\pi$
$410$	0	0
$411$	9.23054i	0.455309i
$412$	0	0
$413$	17.3167i	0.852097i
$414$	0	0
$415$	−11.6216	−0.570481
$416$	0	0
$417$	11.7507	0.575434
$418$	0	0
$419$	− 5.72235i	− 0.279555i	−0.990183	−	0.139778i	$-0.955361\pi$
$419$	− 5.72235i	− 0.279555i	0.990183	−	0.139778i	$-0.0446388\pi$
$420$	0	0
$421$	30.8897i	1.50547i	0.658322	+	0.752736i	$0.271266\pi$
$421$	30.8897i	1.50547i	−0.658322	+	0.752736i	$0.728734\pi$
$422$	0	0
$423$	3.95669	0.192381
$424$	0	0
$425$	5.87901	0.285174
$426$	0	0
$427$	− 20.2147i	− 0.978258i
$428$	0	0
$429$	5.31375i	0.256550i
$430$	0	0
$431$	−16.3709	−0.788559	−0.394279	−	0.918991i	$-0.629006\pi$
$431$	−16.3709	−0.788559	−0.394279	+	0.918991i	$0.629006\pi$
$432$	0	0
$433$	−9.17672	−0.441005	−0.220503	−	0.975386i	$-0.570770\pi$
$433$	−9.17672	−0.441005	−0.220503	+	0.975386i	$0.570770\pi$
$434$	0	0
$435$	6.77706i	0.324935i
$436$	0	0
$437$	− 3.42581i	− 0.163879i
$438$	0	0
$439$	19.1494	0.913953	0.456977	−	0.889479i	$-0.348932\pi$
$439$	19.1494	0.913953	0.456977	+	0.889479i	$0.348932\pi$
$440$	0	0
$441$	−0.804226	−0.0382965
$442$	0	0
$443$	5.89817i	0.280230i	0.990135	+	0.140115i	$0.0447473\pi$
$443$	5.89817i	0.280230i	−0.990135	+	0.140115i	$0.955253\pi$
$444$	0	0
$445$	− 28.7756i	− 1.36409i
$446$	0	0
$447$	17.6079	0.832827
$448$	0	0
$449$	−21.8310	−1.03027	−0.515134	−	0.857110i	$-0.672258\pi$
$449$	−21.8310	−1.03027	−0.515134	+	0.857110i	$0.672258\pi$
$450$	0	0
$451$	59.5634i	2.80473i
$452$	0	0
$453$	− 11.6250i	− 0.546190i
$454$	0	0
$455$	−7.04029	−0.330054
$456$	0	0
$457$	6.84244	0.320076	0.160038	−	0.987111i	$-0.448838\pi$
$457$	6.84244	0.320076	0.160038	+	0.987111i	$0.448838\pi$
$458$	0	0
$459$	− 4.35114i	− 0.203094i
$460$	0	0
$461$	− 22.2688i	− 1.03716i	−0.855029	−	0.518580i	$-0.826461\pi$
$461$	− 22.2688i	− 1.03716i	0.855029	−	0.518580i	$-0.173539\pi$
$462$	0	0
$463$	−26.4021	−1.22701	−0.613504	−	0.789692i	$-0.710240\pi$
$463$	−26.4021	−1.22701	−0.613504	+	0.789692i	$0.710240\pi$
$464$	0	0
$465$	4.81015	0.223065
$466$	0	0
$467$	11.3035i	0.523065i	0.965195	+	0.261532i	$0.0842278\pi$
$467$	11.3035i	0.523065i	−0.965195	+	0.261532i	$0.915772\pi$
$468$	0	0
$469$	− 28.0189i	− 1.29379i
$470$	0	0
$471$	13.4781	0.621036
$472$	0	0
$473$	50.4140	2.31804
$474$	0	0
$475$	− 8.14694i	− 0.373807i
$476$	0	0
$477$	− 8.05934i	− 0.369012i
$478$	0	0
$479$	40.4023	1.84603	0.923015	−	0.384764i	$-0.125717\pi$
$479$	40.4023	1.84603	0.923015	+	0.384764i	$0.125717\pi$
$480$	0	0
$481$	−9.60845	−0.438108
$482$	0	0
$483$	− 1.58721i	− 0.0722205i
$484$	0	0
$485$	25.8350i	1.17311i
$486$	0	0
$487$	36.4018	1.64952	0.824762	−	0.565480i	$-0.191309\pi$
$487$	36.4018	1.64952	0.824762	+	0.565480i	$0.191309\pi$
$488$	0	0
$489$	−9.79349	−0.442877
$490$	0	0
$491$	− 3.76382i	− 0.169859i	−0.996387	−	0.0849294i	$-0.972934\pi$
$491$	− 3.76382i	− 0.169859i	0.996387	−	0.0849294i	$-0.0270665\pi$
$492$	0	0
$493$	11.7009i	0.526981i
$494$	0	0
$495$	−13.3914	−0.601900
$496$	0	0
$497$	−4.55503	−0.204321
$498$	0	0
$499$	16.9348i	0.758107i	0.925375	+	0.379053i	$0.123750\pi$
$499$	16.9348i	0.758107i	−0.925375	+	0.379053i	$0.876250\pi$
$500$	0	0
$501$	− 16.6054i	− 0.741876i
$502$	0	0
$503$	12.9809	0.578792	0.289396	−	0.957209i	$-0.406546\pi$
$503$	12.9809	0.578792	0.289396	+	0.957209i	$0.406546\pi$
$504$	0	0
$505$	7.60845	0.338572
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	13.4857i	0.597741i	0.954294	+	0.298871i	$0.0966099\pi$
$509$	13.4857i	0.597741i	−0.954294	+	0.298871i	$0.903390\pi$
$510$	0	0
$511$	14.4553	0.639464
$512$	0	0
$513$	−6.02967	−0.266217
$514$	0	0
$515$	32.5585i	1.43470i
$516$	0	0
$517$	− 21.0249i	− 0.924672i
$518$	0	0
$519$	16.1979	0.711007
$520$	0	0
$521$	−35.4175	−1.55167	−0.775834	−	0.630937i	$-0.782670\pi$
$521$	−35.4175	−1.55167	−0.775834	+	0.630937i	$0.782670\pi$
$522$	0	0
$523$	4.72122i	0.206445i	0.994658	+	0.103222i	$0.0329153\pi$
$523$	4.72122i	0.206445i	−0.994658	+	0.103222i	$0.967085\pi$
$524$	0	0
$525$	− 3.77455i	− 0.164735i
$526$	0	0
$527$	8.30493	0.361768
$528$	0	0
$529$	−22.6772	−0.985965
$530$	0	0
$531$	6.19868i	0.269000i
$532$	0	0
$533$	11.2093i	0.485529i
$534$	0	0
$535$	27.6049	1.19346
$536$	0	0
$537$	−1.22795	−0.0529901
$538$	0	0
$539$	4.27346i	0.184071i
$540$	0	0
$541$	26.4200i	1.13589i	0.823068	+	0.567943i	$0.192261\pi$
$541$	26.4200i	1.13589i	−0.823068	+	0.567943i	$0.807739\pi$
$542$	0	0
$543$	7.00592	0.300653
$544$	0	0
$545$	−13.7426	−0.588667
$546$	0	0
$547$	− 26.9939i	− 1.15417i	−0.816683	−	0.577087i	$-0.804189\pi$
$547$	− 26.9939i	− 1.15417i	0.816683	−	0.577087i	$-0.195811\pi$
$548$	0	0
$549$	− 7.23607i	− 0.308828i
$550$	0	0
$551$	16.2147	0.690770
$552$	0	0
$553$	25.6297	1.08989
$554$	0	0
$555$	− 24.2147i	− 1.02786i
$556$	0	0
$557$	24.7636i	1.04927i	0.851329	+	0.524633i	$0.175797\pi$
$557$	24.7636i	1.04927i	−0.851329	+	0.524633i	$0.824203\pi$
$558$	0	0
$559$	9.48746	0.401277
$560$	0	0
$561$	−23.1209	−0.976164
$562$	0	0
$563$	− 19.7426i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$563$	− 19.7426i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$564$	0	0
$565$	34.3582i	1.44546i
$566$	0	0
$567$	−2.79360	−0.117320
$568$	0	0
$569$	2.17684	0.0912577	0.0456289	−	0.998958i	$-0.485471\pi$
$569$	2.17684	0.0912577	0.0456289	+	0.998958i	$0.485471\pi$
$570$	0	0
$571$	− 4.05934i	− 0.169878i	−0.996386	−	0.0849391i	$-0.972930\pi$
$571$	− 4.05934i	− 0.169878i	0.996386	−	0.0849391i	$-0.0270696\pi$
$572$	0	0
$573$	− 14.3701i	− 0.600319i
$574$	0	0
$575$	0.767662	0.0320137
$576$	0	0
$577$	−27.5489	−1.14687	−0.573437	−	0.819249i	$-0.694390\pi$
$577$	−27.5489	−1.14687	−0.573437	+	0.819249i	$0.694390\pi$
$578$	0	0
$579$	3.67383i	0.152679i
$580$	0	0
$581$	12.8826i	0.534461i
$582$	0	0
$583$	−42.8254	−1.77365
$584$	0	0
$585$	−2.52015	−0.104195
$586$	0	0
$587$	− 14.3268i	− 0.591329i	−0.955292	−	0.295664i	$-0.904459\pi$
$587$	− 14.3268i	− 0.591329i	0.955292	−	0.295664i	$-0.0955410\pi$
$588$	0	0
$589$	− 11.5087i	− 0.474208i
$590$	0	0
$591$	−6.88442	−0.283187
$592$	0	0
$593$	27.6050	1.13360	0.566801	−	0.823855i	$-0.308181\pi$
$593$	27.6050	1.13360	0.566801	+	0.823855i	$0.308181\pi$
$594$	0	0
$595$	− 30.6333i	− 1.25584i
$596$	0	0
$597$	15.3952i	0.630082i
$598$	0	0
$599$	18.5204	0.756724	0.378362	−	0.925658i	$-0.376487\pi$
$599$	18.5204	0.756724	0.378362	+	0.925658i	$0.376487\pi$
$600$	0	0
$601$	35.1640	1.43437	0.717185	−	0.696883i	$-0.245430\pi$
$601$	35.1640	1.43437	0.717185	+	0.696883i	$0.245430\pi$
$602$	0	0
$603$	− 10.0297i	− 0.408440i
$604$	0	0
$605$	43.4371i	1.76597i
$606$	0	0
$607$	35.5432	1.44265	0.721327	−	0.692594i	$-0.243532\pi$
$607$	35.5432	1.44265	0.721327	+	0.692594i	$0.243532\pi$
$608$	0	0
$609$	7.51243	0.304419
$610$	0	0
$611$	− 3.95669i	− 0.160070i
$612$	0	0
$613$	37.3356i	1.50797i	0.656891	+	0.753985i	$0.271871\pi$
$613$	37.3356i	1.50797i	−0.656891	+	0.753985i	$0.728129\pi$
$614$	0	0
$615$	−28.2491	−1.13911
$616$	0	0
$617$	20.1630	0.811731	0.405865	−	0.913933i	$-0.366970\pi$
$617$	20.1630	0.811731	0.405865	+	0.913933i	$0.366970\pi$
$618$	0	0
$619$	15.1066i	0.607187i	0.952802	+	0.303594i	$0.0981865\pi$
$619$	15.1066i	0.607187i	−0.952802	+	0.303594i	$0.901813\pi$
$620$	0	0
$621$	− 0.568158i	− 0.0227994i
$622$	0	0
$623$	−31.8979	−1.27796
$624$	0	0
$625$	−29.9301	−1.19720
$626$	0	0
$627$	32.0402i	1.27956i
$628$	0	0
$629$	− 41.8077i	− 1.66698i
$630$	0	0
$631$	−0.390213	−0.0155341	−0.00776706	−	0.999970i	$-0.502472\pi$
$631$	−0.390213	−0.0155341	−0.00776706	+	0.999970i	$0.502472\pi$
$632$	0	0
$633$	1.57035	0.0624158
$634$	0	0
$635$	48.9857i	1.94394i
$636$	0	0
$637$	0.804226i	0.0318646i
$638$	0	0
$639$	−1.63052	−0.0645024
$640$	0	0
$641$	−36.5386	−1.44319	−0.721593	−	0.692317i	$-0.756590\pi$
$641$	−36.5386	−1.44319	−0.721593	+	0.692317i	$0.756590\pi$
$642$	0	0
$643$	29.1064i	1.14785i	0.818910	+	0.573923i	$0.194579\pi$
$643$	29.1064i	1.14785i	−0.818910	+	0.573923i	$0.805421\pi$
$644$	0	0
$645$	23.9098i	0.941447i
$646$	0	0
$647$	15.1906	0.597206	0.298603	−	0.954377i	$-0.403479\pi$
$647$	15.1906	0.597206	0.298603	+	0.954377i	$0.403479\pi$
$648$	0	0
$649$	32.9382	1.29294
$650$	0	0
$651$	− 5.33209i	− 0.208981i
$652$	0	0
$653$	15.9205i	0.623016i	0.950243	+	0.311508i	$0.100834\pi$
$653$	15.9205i	0.623016i	−0.950243	+	0.311508i	$0.899166\pi$
$654$	0	0
$655$	32.0652	1.25289
$656$	0	0
$657$	5.17442	0.201873
$658$	0	0
$659$	− 7.56597i	− 0.294728i	−0.989082	−	0.147364i	$-0.952921\pi$
$659$	− 7.56597i	− 0.294728i	0.989082	−	0.147364i	$-0.0470789\pi$
$660$	0	0
$661$	− 32.0806i	− 1.24779i	−0.781508	−	0.623895i	$-0.785549\pi$
$661$	− 32.0806i	− 1.24779i	0.781508	−	0.623895i	$-0.214451\pi$
$662$	0	0
$663$	−4.35114	−0.168984
$664$	0	0
$665$	−42.4507	−1.64617
$666$	0	0
$667$	1.52786i	0.0591591i
$668$	0	0
$669$	27.6571i	1.06928i
$670$	0	0
$671$	−38.4507	−1.48437
$672$	0	0
$673$	19.0592	0.734679	0.367340	−	0.930087i	$-0.380269\pi$
$673$	19.0592	0.734679	0.367340	+	0.930087i	$0.380269\pi$
$674$	0	0
$675$	− 1.35114i	− 0.0520054i
$676$	0	0
$677$	− 4.40095i	− 0.169142i	−0.996417	−	0.0845711i	$-0.973048\pi$
$677$	− 4.40095i	− 0.169142i	0.996417	−	0.0845711i	$-0.0269520\pi$
$678$	0	0
$679$	28.6383	1.09904
$680$	0	0
$681$	−22.9434	−0.879195
$682$	0	0
$683$	33.8071i	1.29359i	0.762662	+	0.646797i	$0.223892\pi$
$683$	33.8071i	1.29359i	−0.762662	+	0.646797i	$0.776108\pi$
$684$	0	0
$685$	23.2623i	0.888808i
$686$	0	0
$687$	−23.5064	−0.896825
$688$	0	0
$689$	−8.05934	−0.307037
$690$	0	0
$691$	12.4856i	0.474974i	0.971391	+	0.237487i	$0.0763237\pi$
$691$	12.4856i	0.474974i	−0.971391	+	0.237487i	$0.923676\pi$
$692$	0	0
$693$	14.8445i	0.563897i
$694$	0	0
$695$	29.6135	1.12330
$696$	0	0
$697$	−48.7732	−1.84742
$698$	0	0
$699$	− 3.47433i	− 0.131411i
$700$	0	0
$701$	34.2016i	1.29178i	0.763432	+	0.645888i	$0.223513\pi$
$701$	34.2016i	1.29178i	−0.763432	+	0.645888i	$0.776487\pi$
$702$	0	0
$703$	−57.9358	−2.18509
$704$	0	0
$705$	9.97144	0.375546
$706$	0	0
$707$	− 8.43403i	− 0.317195i
$708$	0	0
$709$	41.6463i	1.56406i	0.623240	+	0.782030i	$0.285816\pi$
$709$	41.6463i	1.56406i	−0.623240	+	0.782030i	$0.714184\pi$
$710$	0	0
$711$	9.17442	0.344068
$712$	0	0
$713$	1.08443	0.0406122
$714$	0	0
$715$	13.3914i	0.500811i
$716$	0	0
$717$	− 11.5820i	− 0.432538i
$718$	0	0
$719$	−33.0401	−1.23219	−0.616093	−	0.787673i	$-0.711286\pi$
$719$	−33.0401	−1.23219	−0.616093	+	0.787673i	$0.711286\pi$
$720$	0	0
$721$	36.0914	1.34411
$722$	0	0
$723$	− 10.0058i	− 0.372120i
$724$	0	0
$725$	3.63342i	0.134942i
$726$	0	0
$727$	7.80551	0.289490	0.144745	−	0.989469i	$-0.453764\pi$
$727$	7.80551	0.289490	0.144745	+	0.989469i	$0.453764\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	41.2813i	1.52684i
$732$	0	0
$733$	− 29.3035i	− 1.08235i	−0.840910	−	0.541175i	$-0.817980\pi$
$733$	− 29.3035i	− 1.08235i	0.840910	−	0.541175i	$-0.182020\pi$
$734$	0	0
$735$	−2.02677	−0.0747585
$736$	0	0
$737$	−53.2952	−1.96315
$738$	0	0
$739$	− 42.2408i	− 1.55385i	−0.629591	−	0.776926i	$-0.716778\pi$
$739$	− 42.2408i	− 1.55385i	0.629591	−	0.776926i	$-0.283222\pi$
$740$	0	0
$741$	6.02967i	0.221506i
$742$	0	0
$743$	−19.3835	−0.711112	−0.355556	−	0.934655i	$-0.615709\pi$
$743$	−19.3835	−0.711112	−0.355556	+	0.934655i	$0.615709\pi$
$744$	0	0
$745$	44.3746	1.62576
$746$	0	0
$747$	4.61147i	0.168725i
$748$	0	0
$749$	− 30.6002i	− 1.11811i
$750$	0	0
$751$	−9.06886	−0.330927	−0.165464	−	0.986216i	$-0.552912\pi$
$751$	−9.06886	−0.330927	−0.165464	+	0.986216i	$0.552912\pi$
$752$	0	0
$753$	−11.3723	−0.414429
$754$	0	0
$755$	− 29.2967i	− 1.06622i
$756$	0	0
$757$	− 6.34174i	− 0.230495i	−0.993337	−	0.115247i	$-0.963234\pi$
$757$	− 6.34174i	− 0.230495i	0.993337	−	0.115247i	$-0.0367660\pi$
$758$	0	0
$759$	−3.01905	−0.109585
$760$	0	0
$761$	19.1133	0.692856	0.346428	−	0.938077i	$-0.387395\pi$
$761$	19.1133	0.692856	0.346428	+	0.938077i	$0.387395\pi$
$762$	0	0
$763$	15.2338i	0.551499i
$764$	0	0
$765$	− 10.9655i	− 0.396459i
$766$	0	0
$767$	6.19868	0.223821
$768$	0	0
$769$	−5.22795	−0.188525	−0.0942624	−	0.995547i	$-0.530049\pi$
$769$	−5.22795	−0.188525	−0.0942624	+	0.995547i	$0.530049\pi$
$770$	0	0
$771$	− 11.3510i	− 0.408797i
$772$	0	0
$773$	32.5414i	1.17043i	0.810877	+	0.585216i	$0.198990\pi$
$773$	32.5414i	1.17043i	−0.810877	+	0.585216i	$0.801010\pi$
$774$	0	0
$775$	2.57889	0.0926366
$776$	0	0
$777$	−26.8422	−0.962959
$778$	0	0
$779$	67.5884i	2.42161i
$780$	0	0
$781$	8.66418i	0.310029i
$782$	0	0
$783$	2.68915	0.0961025
$784$	0	0
$785$	33.9667	1.21232
$786$	0	0
$787$	8.62409i	0.307416i	0.988116	+	0.153708i	$0.0491214\pi$
$787$	8.62409i	0.307416i	−0.988116	+	0.153708i	$0.950879\pi$
$788$	0	0
$789$	3.88712i	0.138385i
$790$	0	0
$791$	38.0864	1.35420
$792$	0	0
$793$	−7.23607	−0.256960
$794$	0	0
$795$	− 20.3107i	− 0.720348i
$796$	0	0
$797$	− 27.6486i	− 0.979365i	−0.871901	−	0.489682i	$-0.837113\pi$
$797$	− 27.6486i	− 0.979365i	0.871901	−	0.489682i	$-0.162887\pi$
$798$	0	0
$799$	17.2161	0.609062
$800$	0	0
$801$	−11.4182	−0.403442
$802$	0	0
$803$	− 27.4956i	− 0.970298i
$804$	0	0
$805$	− 4.00000i	− 0.140981i
$806$	0	0
$807$	−25.2300	−0.888139
$808$	0	0
$809$	−24.4163	−0.858431	−0.429216	−	0.903202i	$-0.641210\pi$
$809$	−24.4163	−0.858431	−0.429216	+	0.903202i	$0.641210\pi$
$810$	0	0
$811$	− 9.54312i	− 0.335104i	−0.985863	−	0.167552i	$-0.946414\pi$
$811$	− 9.54312i	− 0.335104i	0.985863	−	0.167552i	$-0.0535863\pi$
$812$	0	0
$813$	− 2.90649i	− 0.101935i
$814$	0	0
$815$	−24.6810	−0.864539
$816$	0	0
$817$	57.2063	2.00139
$818$	0	0
$819$	2.79360i	0.0976164i
$820$	0	0
$821$	− 2.26876i	− 0.0791802i	−0.999216	−	0.0395901i	$-0.987395\pi$
$821$	− 2.26876i	− 0.0791802i	0.999216	−	0.0395901i	$-0.0126052\pi$
$822$	0	0
$823$	14.4047	0.502115	0.251058	−	0.967972i	$-0.419222\pi$
$823$	14.4047	0.502115	0.251058	+	0.967972i	$0.419222\pi$
$824$	0	0
$825$	−7.17963	−0.249963
$826$	0	0
$827$	15.2993i	0.532010i	0.963972	+	0.266005i	$0.0857038\pi$
$827$	15.2993i	0.532010i	−0.963972	+	0.266005i	$0.914296\pi$
$828$	0	0
$829$	21.5350i	0.747940i	0.927441	+	0.373970i	$0.122004\pi$
$829$	21.5350i	0.747940i	−0.927441	+	0.373970i	$0.877996\pi$
$830$	0	0
$831$	−4.26825	−0.148064
$832$	0	0
$833$	−3.49930	−0.121244
$834$	0	0
$835$	− 41.8481i	− 1.44821i
$836$	0	0
$837$	− 1.90868i	− 0.0659735i
$838$	0	0
$839$	−47.6245	−1.64418	−0.822090	−	0.569358i	$-0.807192\pi$
$839$	−47.6245	−1.64418	−0.822090	+	0.569358i	$0.807192\pi$
$840$	0	0
$841$	21.7685	0.750636
$842$	0	0
$843$	18.9900i	0.654050i
$844$	0	0
$845$	2.52015i	0.0866957i
$846$	0	0
$847$	48.1504	1.65447
$848$	0	0
$849$	13.1957	0.452874
$850$	0	0
$851$	− 5.45912i	− 0.187136i
$852$	0	0
$853$	15.3358i	0.525088i	0.964920	+	0.262544i	$0.0845616\pi$
$853$	15.3358i	0.525088i	−0.964920	+	0.262544i	$0.915438\pi$
$854$	0	0
$855$	−15.1957	−0.519681
$856$	0	0
$857$	15.0828	0.515218	0.257609	−	0.966249i	$-0.417065\pi$
$857$	15.0828	0.515218	0.257609	+	0.966249i	$0.417065\pi$
$858$	0	0
$859$	15.0249i	0.512642i	0.966592	+	0.256321i	$0.0825104\pi$
$859$	15.0249i	0.512642i	−0.966592	+	0.256321i	$0.917490\pi$
$860$	0	0
$861$	31.3144i	1.06719i
$862$	0	0
$863$	1.32919	0.0452460	0.0226230	−	0.999744i	$-0.492798\pi$
$863$	1.32919	0.0452460	0.0226230	+	0.999744i	$0.492798\pi$
$864$	0	0
$865$	40.8210	1.38795
$866$	0	0
$867$	− 1.93243i	− 0.0656287i
$868$	0	0
$869$	− 48.7506i	− 1.65375i
$870$	0	0
$871$	−10.0297	−0.339842
$872$	0	0
$873$	10.2514	0.346957
$874$	0	0
$875$	25.6890i	0.868448i
$876$	0	0
$877$	− 44.4244i	− 1.50011i	−0.661378	−	0.750053i	$-0.730028\pi$
$877$	− 44.4244i	− 1.50011i	0.661378	−	0.750053i	$-0.269972\pi$
$878$	0	0
$879$	−12.0980	−0.408054
$880$	0	0
$881$	−36.5136	−1.23017	−0.615087	−	0.788459i	$-0.710879\pi$
$881$	−36.5136	−1.23017	−0.615087	+	0.788459i	$0.710879\pi$
$882$	0	0
$883$	6.77808i	0.228101i	0.993475	+	0.114050i	$0.0363825\pi$
$883$	6.77808i	0.228101i	−0.993475	+	0.114050i	$0.963617\pi$
$884$	0	0
$885$	15.6216i	0.525114i
$886$	0	0
$887$	48.6502	1.63351	0.816756	−	0.576983i	$-0.195770\pi$
$887$	48.6502	1.63351	0.816756	+	0.576983i	$0.195770\pi$
$888$	0	0
$889$	54.3011	1.82120
$890$	0	0
$891$	5.31375i	0.178017i
$892$	0	0
$893$	− 23.8575i	− 0.798362i
$894$	0	0
$895$	−3.09462	−0.103442
$896$	0	0
$897$	−0.568158	−0.0189702
$898$	0	0
$899$	5.13273i	0.171186i
$900$	0	0
$901$	− 35.0673i	− 1.16826i
$902$	0	0
$903$	26.5042	0.882005
$904$	0	0
$905$	17.6560	0.586904
$906$	0	0
$907$	− 36.5196i	− 1.21261i	−0.795230	−	0.606307i	$-0.792650\pi$
$907$	− 36.5196i	− 1.21261i	0.795230	−	0.606307i	$-0.207350\pi$
$908$	0	0
$909$	− 3.01905i	− 0.100136i
$910$	0	0
$911$	23.9142	0.792312	0.396156	−	0.918183i	$-0.370344\pi$
$911$	23.9142	0.792312	0.396156	+	0.918183i	$0.370344\pi$
$912$	0	0
$913$	24.5042	0.810971
$914$	0	0
$915$	− 18.2360i	− 0.602862i
$916$	0	0
$917$	− 35.5445i	− 1.17378i
$918$	0	0
$919$	−18.4982	−0.610198	−0.305099	−	0.952321i	$-0.598690\pi$
$919$	−18.4982	−0.610198	−0.305099	+	0.952321i	$0.598690\pi$
$920$	0	0
$921$	24.7863	0.816736
$922$	0	0
$923$	1.63052i	0.0536692i
$924$	0	0
$925$	− 12.9824i	− 0.426858i
$926$	0	0
$927$	12.9193	0.424325
$928$	0	0
$929$	31.5521	1.03519	0.517595	−	0.855625i	$-0.326827\pi$
$929$	31.5521	1.03519	0.517595	+	0.855625i	$0.326827\pi$
$930$	0	0
$931$	4.84922i	0.158927i
$932$	0	0
$933$	− 6.72956i	− 0.220316i
$934$	0	0
$935$	−58.2680	−1.90557
$936$	0	0
$937$	−5.54201	−0.181050	−0.0905248	−	0.995894i	$-0.528854\pi$
$937$	−5.54201	−0.181050	−0.0905248	+	0.995894i	$0.528854\pi$
$938$	0	0
$939$	− 11.4956i	− 0.375144i
$940$	0	0
$941$	9.62480i	0.313760i	0.987618	+	0.156880i	$0.0501435\pi$
$941$	9.62480i	0.313760i	−0.987618	+	0.156880i	$0.949856\pi$
$942$	0	0
$943$	−6.36865	−0.207392
$944$	0	0
$945$	−7.04029	−0.229021
$946$	0	0
$947$	− 31.1231i	− 1.01137i	−0.862719	−	0.505683i	$-0.831241\pi$
$947$	− 31.1231i	− 1.01137i	0.862719	−	0.505683i	$-0.168759\pi$
$948$	0	0
$949$	− 5.17442i	− 0.167969i
$950$	0	0
$951$	2.86756	0.0929870
$952$	0	0
$953$	−17.0359	−0.551847	−0.275924	−	0.961180i	$-0.588984\pi$
$953$	−17.0359	−0.551847	−0.275924	+	0.961180i	$0.588984\pi$
$954$	0	0
$955$	− 36.2147i	− 1.17188i
$956$	0	0
$957$	− 14.2895i	− 0.461914i
$958$	0	0
$959$	25.7865	0.832689
$960$	0	0
$961$	−27.3570	−0.882482
$962$	0	0
$963$	− 10.9537i	− 0.352977i
$964$	0	0
$965$	9.25860i	0.298045i
$966$	0	0
$967$	−7.56565	−0.243295	−0.121647	−	0.992573i	$-0.538818\pi$
$967$	−7.56565	−0.243295	−0.121647	+	0.992573i	$0.538818\pi$
$968$	0	0
$969$	−26.2360	−0.842820
$970$	0	0
$971$	25.0676i	0.804457i	0.915539	+	0.402228i	$0.131764\pi$
$971$	25.0676i	0.804457i	−0.915539	+	0.402228i	$0.868236\pi$
$972$	0	0
$973$	− 32.8268i	− 1.05238i
$974$	0	0
$975$	−1.35114	−0.0432711
$976$	0	0
$977$	−37.9292	−1.21346	−0.606732	−	0.794907i	$-0.707520\pi$
$977$	−37.9292	−1.21346	−0.606732	+	0.794907i	$0.707520\pi$
$978$	0	0
$979$	60.6735i	1.93913i
$980$	0	0
$981$	5.45309i	0.174104i
$982$	0	0
$983$	10.3543	0.330250	0.165125	−	0.986273i	$-0.447197\pi$
$983$	10.3543	0.330250	0.165125	+	0.986273i	$0.447197\pi$
$984$	0	0
$985$	−17.3497	−0.552809
$986$	0	0
$987$	− 11.0534i	− 0.351834i
$988$	0	0
$989$	5.39038i	0.171404i
$990$	0	0
$991$	45.2348	1.43693	0.718466	−	0.695562i	$-0.244845\pi$
$991$	45.2348	1.43693	0.718466	+	0.695562i	$0.244845\pi$
$992$	0	0
$993$	1.44967	0.0460039
$994$	0	0
$995$	38.7981i	1.22998i
$996$	0	0
$997$	24.7402i	0.783529i	0.920066	+	0.391764i	$0.128135\pi$
$997$	24.7402i	0.783529i	−0.920066	+	0.391764i	$0.871865\pi$
$998$	0	0
$999$	−9.60845	−0.303998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.g.a.625.1		8
3.2	odd	2		3744.2.g.d.1873.7		8
4.3	odd	2		312.2.g.a.157.3	✓	8
8.3	odd	2		312.2.g.a.157.4	yes	8
8.5	even	2	inner	1248.2.g.a.625.8		8
12.11	even	2		936.2.g.d.469.6		8
16.3	odd	4		9984.2.a.y.1.1		4
16.5	even	4		9984.2.a.s.1.4		4
16.11	odd	4		9984.2.a.bb.1.4		4
16.13	even	4		9984.2.a.bh.1.1		4
24.5	odd	2		3744.2.g.d.1873.2		8
24.11	even	2		936.2.g.d.469.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.a.157.3	✓	8	4.3	odd	2
312.2.g.a.157.4	yes	8	8.3	odd	2
936.2.g.d.469.5		8	24.11	even	2
936.2.g.d.469.6		8	12.11	even	2
1248.2.g.a.625.1		8	1.1	even	1	trivial
1248.2.g.a.625.8		8	8.5	even	2	inner
3744.2.g.d.1873.2		8	24.5	odd	2
3744.2.g.d.1873.7		8	3.2	odd	2
9984.2.a.s.1.4		4	16.5	even	4
9984.2.a.y.1.1		4	16.3	odd	4
9984.2.a.bb.1.4		4	16.11	odd	4
9984.2.a.bh.1.1		4	16.13	even	4

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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.52015i q^{5} -2.79360 q^{7} -1.00000 q^{9} +5.31375i q^{11} +1.00000i q^{13} -2.52015 q^{15} -4.35114 q^{17} +6.02967i q^{19} +2.79360i q^{21} -0.568158 q^{23} -1.35114 q^{25} +1.00000i q^{27} -2.68915i q^{29} -1.90868 q^{31} +5.31375 q^{33} +7.04029i q^{35} +9.60845i q^{37} +1.00000 q^{39} +11.2093 q^{41} -9.48746i q^{43} +2.52015i q^{45} -3.95669 q^{47} +0.804226 q^{49} +4.35114i q^{51} +8.05934i q^{53} +13.3914 q^{55} +6.02967 q^{57} -6.19868i q^{59} +7.23607i q^{61} +2.79360 q^{63} +2.52015 q^{65} +10.0297i q^{67} +0.568158i q^{69} +1.63052 q^{71} -5.17442 q^{73} +1.35114i q^{75} -14.8445i q^{77} -9.17442 q^{79} +1.00000 q^{81} -4.61147i q^{83} +10.9655i q^{85} -2.68915 q^{87} +11.4182 q^{89} -2.79360i q^{91} +1.90868i q^{93} +15.1957 q^{95} -10.2514 q^{97} -5.31375i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 8 q^{9} + 4 q^{15} - 16 q^{17} + 8 q^{23} + 8 q^{25} + 4 q^{31} + 8 q^{39} + 36 q^{41} - 24 q^{47} - 24 q^{49} + 40 q^{55} + 12 q^{57} + 4 q^{63} - 4 q^{65} - 16 q^{71} + 32 q^{73} + 8 q^{81}+ \cdots - 56 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 8 * q^9 + 4 * q^15 - 16 * q^17 + 8 * q^23 + 8 * q^25 + 4 * q^31 + 8 * q^39 + 36 * q^41 - 24 * q^47 - 24 * q^49 + 40 * q^55 + 12 * q^57 + 4 * q^63 - 4 * q^65 - 16 * q^71 + 32 * q^73 + 8 * q^81 + 8 * q^87 + 60 * q^89 + 24 * q^95 - 56 * q^97

Introduction

L-functions

Modular forms

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Database

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