LMFDB - Embedded newform 3744.2.g.d.1873.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1873,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, 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("3744.1873");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.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:	$29.8959905168$
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:	$\Z[a_1, \ldots, a_{13}]$
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			1873.2
Root			$0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	3744.1873
Dual form			3744.2.g.d.1873.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	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
$9$	0	0
$10$	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
$15$	0	0
$16$	0	0
$17$	4.35114	1.05531	0.527653	−	0.849460i	$-0.323072\pi$
$17$	4.35114	1.05531	0.527653	+	0.849460i	$0.323072\pi$
$18$	0	0
$19$	− 6.02967i	− 1.38330i	−0.722232	−	0.691651i	$-0.756884\pi$
$19$	− 6.02967i	− 1.38330i	0.722232	−	0.691651i	$-0.243116\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.568158	0.118469	0.0592346	−	0.998244i	$-0.481134\pi$
$23$	0.568158	0.118469	0.0592346	+	0.998244i	$0.481134\pi$
$24$	0	0
$25$	−1.35114	−0.270228
$26$	0	0
$27$	0	0
$28$	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
$33$	0	0
$34$	0	0
$35$	7.04029i	1.19003i
$36$	0	0
$37$	− 9.60845i	− 1.57962i	−0.613352	−	0.789810i	$-0.710179\pi$
$37$	− 9.60845i	− 1.57962i	0.613352	−	0.789810i	$-0.289821\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.2093	−1.75060	−0.875299	−	0.483582i	$-0.839336\pi$
$41$	−11.2093	−1.75060	−0.875299	+	0.483582i	$0.839336\pi$
$42$	0	0
$43$	9.48746i	1.44682i	0.690417	+	0.723412i	$0.257427\pi$
$43$	9.48746i	1.44682i	−0.690417	+	0.723412i	$0.742573\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.95669	0.577142	0.288571	−	0.957458i	$-0.406820\pi$
$47$	3.95669	0.577142	0.288571	+	0.957458i	$0.406820\pi$
$48$	0	0
$49$	0.804226	0.114889
$50$	0	0
$51$	0	0
$52$	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
$57$	0	0
$58$	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	$-0.846686\pi$
$61$	− 7.23607i	− 0.926484i	0.886232	−	0.463242i	$-0.153314\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.52015	−0.312586
$66$	0	0
$67$	− 10.0297i	− 1.22532i	−0.790347	−	0.612660i	$-0.790100\pi$
$67$	− 10.0297i	− 1.22532i	0.790347	−	0.612660i	$-0.209900\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.63052	−0.193507	−0.0967536	−	0.995308i	$-0.530846\pi$
$71$	−1.63052	−0.193507	−0.0967536	+	0.995308i	$0.530846\pi$
$72$	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
$75$	0	0
$76$	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
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	−11.4182	−1.21033	−0.605164	−	0.796101i	$-0.706892\pi$
$89$	−11.4182	−1.21033	−0.605164	+	0.796101i	$0.706892\pi$
$90$	0	0
$91$	2.79360i	0.292849i
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	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
$105$	0	0
$106$	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	$0.0841035\pi$
$109$	5.45309i	0.522311i	−0.965297	+	0.261155i	$0.915896\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.6334	1.28252	0.641262	−	0.767322i	$-0.278411\pi$
$113$	13.6334	1.28252	0.641262	+	0.767322i	$0.278411\pi$
$114$	0	0
$115$	− 1.43184i	− 0.133520i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.1554	−1.11428
$120$	0	0
$121$	−17.2360	−1.56691
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	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
$135$	0	0
$136$	0	0
$137$	9.23054	0.788619	0.394309	−	0.918978i	$-0.370984\pi$
$137$	9.23054	0.788619	0.394309	+	0.918978i	$0.370984\pi$
$138$	0	0
$139$	− 11.7507i	− 0.996681i	−0.866982	−	0.498340i	$-0.833943\pi$
$139$	− 11.7507i	− 0.996681i	0.866982	−	0.498340i	$-0.166057\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.31375	0.444358
$144$	0	0
$145$	−6.77706	−0.562804
$146$	0	0
$147$	0	0
$148$	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
$153$	0	0
$154$	0	0
$155$	4.81015i	0.386360i
$156$	0	0
$157$	− 13.4781i	− 1.07567i	−0.843051	−	0.537833i	$-0.819243\pi$
$157$	− 13.4781i	− 1.07567i	0.843051	−	0.537833i	$-0.180757\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.58721	−0.125090
$162$	0	0
$163$	9.79349i	0.767085i	0.923523	+	0.383543i	$0.125296\pi$
$163$	9.79349i	0.767085i	−0.923523	+	0.383543i	$0.874704\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.6054	−1.28497	−0.642484	−	0.766299i	$-0.722096\pi$
$167$	−16.6054	−1.28497	−0.642484	+	0.766299i	$0.722096\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	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
$177$	0	0
$178$	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	$-0.916154\pi$
$181$	− 7.00592i	− 0.520746i	0.965508	−	0.260373i	$-0.0838455\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−24.2147	−1.78030
$186$	0	0
$187$	23.1209i	1.69077i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.3701	−1.03978	−0.519891	−	0.854232i	$-0.674028\pi$
$191$	−14.3701	−1.03978	−0.519891	+	0.854232i	$0.674028\pi$
$192$	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
$195$	0	0
$196$	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
$201$	0	0
$202$	0	0
$203$	7.51243i	0.527269i
$204$	0	0
$205$	28.2491i	1.97300i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.0402	2.21627
$210$	0	0
$211$	− 1.57035i	− 0.108107i	−0.998538	−	0.0540537i	$-0.982786\pi$
$211$	− 1.57035i	− 0.108107i	0.998538	−	0.0540537i	$-0.0172142\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23.9098	1.63063
$216$	0	0
$217$	5.33209	0.361966
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	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	$0.283094\pi$
$229$	23.5064i	1.55335i	−0.629904	+	0.776673i	$0.716906\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.47433	−0.227611	−0.113805	−	0.993503i	$-0.536304\pi$
$233$	−3.47433	−0.227611	−0.113805	+	0.993503i	$0.536304\pi$
$234$	0	0
$235$	− 9.97144i	− 0.650465i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.5820	−0.749177	−0.374589	−	0.927191i	$-0.622216\pi$
$239$	−11.5820	−0.749177	−0.374589	+	0.927191i	$0.622216\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	− 2.02677i	− 0.129485i
$246$	0	0
$247$	−6.02967	−0.383659
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	−11.3510	−0.708058	−0.354029	−	0.935235i	$-0.615189\pi$
$257$	−11.3510	−0.708058	−0.354029	+	0.935235i	$0.615189\pi$
$258$	0	0
$259$	26.8422i	1.66789i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.88712	0.239690	0.119845	−	0.992793i	$-0.461760\pi$
$263$	3.88712	0.239690	0.119845	+	0.992793i	$0.461760\pi$
$264$	0	0
$265$	20.3107	1.24768
$266$	0	0
$267$	0	0
$268$	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
$273$	0	0
$274$	0	0
$275$	− 7.17963i	− 0.432948i
$276$	0	0
$277$	4.26825i	0.256454i	0.991745	+	0.128227i	$0.0409286\pi$
$277$	4.26825i	0.256454i	−0.991745	+	0.128227i	$0.959071\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.9900	1.13285	0.566424	−	0.824114i	$-0.308327\pi$
$281$	18.9900	1.13285	0.566424	+	0.824114i	$0.308327\pi$
$282$	0	0
$283$	− 13.1957i	− 0.784401i	−0.919880	−	0.392200i	$-0.871714\pi$
$283$	− 13.1957i	− 0.784401i	0.919880	−	0.392200i	$-0.128286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.3144	1.84843
$288$	0	0
$289$	1.93243	0.113672
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	0	0
$299$	− 0.568158i	− 0.0328574i
$300$	0	0
$301$	− 26.5042i	− 1.52768i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.2360	−1.04419
$306$	0	0
$307$	− 24.7863i	− 1.41463i	−0.706900	−	0.707314i	$-0.749907\pi$
$307$	− 24.7863i	− 1.41463i	0.706900	−	0.707314i	$-0.250093\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.72956	−0.381598	−0.190799	−	0.981629i	$-0.561108\pi$
$311$	−6.72956	−0.381598	−0.190799	+	0.981629i	$0.561108\pi$
$312$	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
$315$	0	0
$316$	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
$321$	0	0
$322$	0	0
$323$	− 26.2360i	− 1.45981i
$324$	0	0
$325$	1.35114i	0.0749478i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.0534	−0.609395
$330$	0	0
$331$	− 1.44967i	− 0.0796811i	−0.999206	−	0.0398405i	$-0.987315\pi$
$331$	− 1.44967i	− 0.0796811i	0.999206	−	0.0398405i	$-0.0126850\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	− 10.1422i	− 0.549233i
$342$	0	0
$343$	17.3085	0.934573
$344$	0	0
$345$	0	0
$346$	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	$-0.945661\pi$
$349$	− 6.34741i	− 0.339769i	0.985464	−	0.169885i	$-0.0543395\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−32.1856	−1.71307	−0.856534	−	0.516090i	$-0.827387\pi$
$353$	−32.1856	−1.71307	−0.856534	+	0.516090i	$0.827387\pi$
$354$	0	0
$355$	4.10915i	0.218091i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.9222	1.20979	0.604894	−	0.796306i	$-0.293216\pi$
$359$	22.9222	1.20979	0.604894	+	0.796306i	$0.293216\pi$
$360$	0	0
$361$	−17.3570	−0.913524
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	0	0
$371$	− 22.5146i	− 1.16890i
$372$	0	0
$373$	23.2975i	1.20630i	0.797628	+	0.603149i	$0.206088\pi$
$373$	23.2975i	1.20630i	−0.797628	+	0.603149i	$0.793912\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.68915	−0.138498
$378$	0	0
$379$	9.48276i	0.487097i	0.969889	+	0.243548i	$0.0783114\pi$
$379$	9.48276i	0.487097i	−0.969889	+	0.243548i	$0.921689\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.68406	−0.290442	−0.145221	−	0.989399i	$-0.546389\pi$
$383$	−5.68406	−0.290442	−0.145221	+	0.989399i	$0.546389\pi$
$384$	0	0
$385$	−37.4104	−1.90661
$386$	0	0
$387$	0	0
$388$	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
$393$	0	0
$394$	0	0
$395$	23.1209i	1.16334i
$396$	0	0
$397$	− 4.61064i	− 0.231402i	−0.993284	−	0.115701i	$-0.963089\pi$
$397$	− 4.61064i	− 0.231402i	0.993284	−	0.115701i	$-0.0369114\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.4701	0.972290	0.486145	−	0.873878i	$-0.338403\pi$
$401$	19.4701	0.972290	0.486145	+	0.873878i	$0.338403\pi$
$402$	0	0
$403$	1.90868i	0.0950780i
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	17.3167i	0.852097i
$414$	0	0
$415$	−11.6216	−0.570481
$416$	0	0
$417$	0	0
$418$	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	$-0.728734\pi$
$421$	− 30.8897i	− 1.50547i	0.658322	−	0.752736i	$-0.271266\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.87901	−0.285174
$426$	0	0
$427$	20.2147i	0.978258i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.3709	0.788559	0.394279	−	0.918991i	$-0.370994\pi$
$431$	16.3709	0.788559	0.394279	+	0.918991i	$0.370994\pi$
$432$	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
$435$	0	0
$436$	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	0
$442$	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
$447$	0	0
$448$	0	0
$449$	21.8310	1.03027	0.515134	−	0.857110i	$-0.327742\pi$
$449$	21.8310	1.03027	0.515134	+	0.857110i	$0.327742\pi$
$450$	0	0
$451$	− 59.5634i	− 2.80473i
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	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
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	−50.4140	−2.31804
$474$	0	0
$475$	8.14694i	0.373807i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.4023	−1.84603	−0.923015	−	0.384764i	$-0.874283\pi$
$479$	−40.4023	−1.84603	−0.923015	+	0.384764i	$0.874283\pi$
$480$	0	0
$481$	−9.60845	−0.438108
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	4.55503	0.204321
$498$	0	0
$499$	− 16.9348i	− 0.758107i	−0.925375	−	0.379053i	$-0.876250\pi$
$499$	− 16.9348i	− 0.758107i	0.925375	−	0.379053i	$-0.123750\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.9809	−0.578792	−0.289396	−	0.957209i	$-0.593454\pi$
$503$	−12.9809	−0.578792	−0.289396	+	0.957209i	$0.593454\pi$
$504$	0	0
$505$	7.60845	0.338572
$506$	0	0
$507$	0	0
$508$	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
$513$	0	0
$514$	0	0
$515$	32.5585i	1.43470i
$516$	0	0
$517$	21.0249i	0.924672i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.4175	1.55167	0.775834	−	0.630937i	$-0.217330\pi$
$521$	35.4175	1.55167	0.775834	+	0.630937i	$0.217330\pi$
$522$	0	0
$523$	− 4.72122i	− 0.206445i	−0.994658	−	0.103222i	$-0.967085\pi$
$523$	− 4.72122i	− 0.206445i	0.994658	−	0.103222i	$-0.0329153\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.30493	−0.361768
$528$	0	0
$529$	−22.6772	−0.985965
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.2093i	0.485529i
$534$	0	0
$535$	27.6049	1.19346
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.27346i	0.184071i
$540$	0	0
$541$	− 26.4200i	− 1.13589i	−0.823068	−	0.567943i	$-0.807739\pi$
$541$	− 26.4200i	− 1.13589i	0.823068	−	0.567943i	$-0.192261\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.7426	0.588667
$546$	0	0
$547$	26.9939i	1.15417i	0.816683	+	0.577087i	$0.195811\pi$
$547$	26.9939i	1.15417i	−0.816683	+	0.577087i	$0.804189\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.2147	−0.690770
$552$	0	0
$553$	25.6297	1.08989
$554$	0	0
$555$	0	0
$556$	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
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	−2.17684	−0.0912577	−0.0456289	−	0.998958i	$-0.514529\pi$
$569$	−2.17684	−0.0912577	−0.0456289	+	0.998958i	$0.514529\pi$
$570$	0	0
$571$	4.05934i	0.169878i	0.996386	+	0.0849391i	$0.0270696\pi$
$571$	4.05934i	0.169878i	−0.996386	+	0.0849391i	$0.972930\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	12.8826i	0.534461i
$582$	0	0
$583$	−42.8254	−1.77365
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	−27.6050	−1.13360	−0.566801	−	0.823855i	$-0.691819\pi$
$593$	−27.6050	−1.13360	−0.566801	+	0.823855i	$0.691819\pi$
$594$	0	0
$595$	30.6333i	1.25584i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.5204	−0.756724	−0.378362	−	0.925658i	$-0.623513\pi$
$599$	−18.5204	−0.756724	−0.378362	+	0.925658i	$0.623513\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	0	0
$611$	− 3.95669i	− 0.160070i
$612$	0	0
$613$	− 37.3356i	− 1.50797i	−0.656891	−	0.753985i	$-0.728129\pi$
$613$	− 37.3356i	− 1.50797i	0.656891	−	0.753985i	$-0.271871\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.1630	−0.811731	−0.405865	−	0.913933i	$-0.633030\pi$
$617$	−20.1630	−0.811731	−0.405865	+	0.913933i	$0.633030\pi$
$618$	0	0
$619$	− 15.1066i	− 0.607187i	−0.952802	−	0.303594i	$-0.901813\pi$
$619$	− 15.1066i	− 0.607187i	0.952802	−	0.303594i	$-0.0981865\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	31.8979	1.27796
$624$	0	0
$625$	−29.9301	−1.19720
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	48.9857i	1.94394i
$636$	0	0
$637$	− 0.804226i	− 0.0318646i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.5386	1.44319	0.721593	−	0.692317i	$-0.243410\pi$
$641$	36.5386	1.44319	0.721593	+	0.692317i	$0.243410\pi$
$642$	0	0
$643$	− 29.1064i	− 1.14785i	−0.818910	−	0.573923i	$-0.805421\pi$
$643$	− 29.1064i	− 1.14785i	0.818910	−	0.573923i	$-0.194579\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.1906	−0.597206	−0.298603	−	0.954377i	$-0.596521\pi$
$647$	−15.1906	−0.597206	−0.298603	+	0.954377i	$0.596521\pi$
$648$	0	0
$649$	32.9382	1.29294
$650$	0	0
$651$	0	0
$652$	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
$657$	0	0
$658$	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	$0.214451\pi$
$661$	32.0806i	1.24779i	−0.781508	+	0.623895i	$0.785549\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	42.4507	1.64617
$666$	0	0
$667$	− 1.52786i	− 0.0591591i
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	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
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	8.05934	0.307037
$690$	0	0
$691$	− 12.4856i	− 0.474974i	−0.971391	−	0.237487i	$-0.923676\pi$
$691$	− 12.4856i	− 0.474974i	0.971391	−	0.237487i	$-0.0763237\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−29.6135	−1.12330
$696$	0	0
$697$	−48.7732	−1.84742
$698$	0	0
$699$	0	0
$700$	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
$705$	0	0
$706$	0	0
$707$	− 8.43403i	− 0.317195i
$708$	0	0
$709$	− 41.6463i	− 1.56406i	−0.623240	−	0.782030i	$-0.714184\pi$
$709$	− 41.6463i	− 1.56406i	0.623240	−	0.782030i	$-0.285816\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.08443	−0.0406122
$714$	0	0
$715$	− 13.3914i	− 0.500811i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.0401	1.23219	0.616093	−	0.787673i	$-0.288714\pi$
$719$	33.0401	1.23219	0.616093	+	0.787673i	$0.288714\pi$
$720$	0	0
$721$	36.0914	1.34411
$722$	0	0
$723$	0	0
$724$	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
$729$	0	0
$730$	0	0
$731$	41.2813i	1.52684i
$732$	0	0
$733$	29.3035i	1.08235i	0.840910	+	0.541175i	$0.182020\pi$
$733$	29.3035i	1.08235i	−0.840910	+	0.541175i	$0.817980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	53.2952	1.96315
$738$	0	0
$739$	42.2408i	1.55385i	0.629591	+	0.776926i	$0.283222\pi$
$739$	42.2408i	1.55385i	−0.629591	+	0.776926i	$0.716778\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19.3835	0.711112	0.355556	−	0.934655i	$-0.384291\pi$
$743$	19.3835	0.711112	0.355556	+	0.934655i	$0.384291\pi$
$744$	0	0
$745$	44.3746	1.62576
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	0	0
$755$	− 29.2967i	− 1.06622i
$756$	0	0
$757$	6.34174i	0.230495i	0.993337	+	0.115247i	$0.0367660\pi$
$757$	6.34174i	0.230495i	−0.993337	+	0.115247i	$0.963234\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.1133	−0.692856	−0.346428	−	0.938077i	$-0.612605\pi$
$761$	−19.1133	−0.692856	−0.346428	+	0.938077i	$0.612605\pi$
$762$	0	0
$763$	− 15.2338i	− 0.551499i
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	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
$777$	0	0
$778$	0	0
$779$	67.5884i	2.42161i
$780$	0	0
$781$	− 8.66418i	− 0.310029i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33.9667	−1.21232
$786$	0	0
$787$	− 8.62409i	− 0.307416i	−0.988116	−	0.153708i	$-0.950879\pi$
$787$	− 8.62409i	− 0.307416i	0.988116	−	0.153708i	$-0.0491214\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−38.0864	−1.35420
$792$	0	0
$793$	−7.23607	−0.256960
$794$	0	0
$795$	0	0
$796$	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
$801$	0	0
$802$	0	0
$803$	− 27.4956i	− 0.970298i
$804$	0	0
$805$	4.00000i	0.140981i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.4163	0.858431	0.429216	−	0.903202i	$-0.358790\pi$
$809$	24.4163	0.858431	0.429216	+	0.903202i	$0.358790\pi$
$810$	0	0
$811$	9.54312i	0.335104i	0.985863	+	0.167552i	$0.0535863\pi$
$811$	9.54312i	0.335104i	−0.985863	+	0.167552i	$0.946414\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.6810	0.864539
$816$	0	0
$817$	57.2063	2.00139
$818$	0	0
$819$	0	0
$820$	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
$825$	0	0
$826$	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	$-0.877996\pi$
$829$	− 21.5350i	− 0.747940i	0.927441	−	0.373970i	$-0.122004\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.49930	0.121244
$834$	0	0
$835$	41.8481i	1.44821i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.6245	1.64418	0.822090	−	0.569358i	$-0.192808\pi$
$839$	47.6245	1.64418	0.822090	+	0.569358i	$0.192808\pi$
$840$	0	0
$841$	21.7685	0.750636
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.52015i	0.0866957i
$846$	0	0
$847$	48.1504	1.65447
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5.45912i	− 0.187136i
$852$	0	0
$853$	− 15.3358i	− 0.525088i	−0.964920	−	0.262544i	$-0.915438\pi$
$853$	− 15.3358i	− 0.525088i	0.964920	−	0.262544i	$-0.0845616\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.0828	−0.515218	−0.257609	−	0.966249i	$-0.582935\pi$
$857$	−15.0828	−0.515218	−0.257609	+	0.966249i	$0.582935\pi$
$858$	0	0
$859$	− 15.0249i	− 0.512642i	−0.966592	−	0.256321i	$-0.917490\pi$
$859$	− 15.0249i	− 0.512642i	0.966592	−	0.256321i	$-0.0825104\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.32919	−0.0452460	−0.0226230	−	0.999744i	$-0.507202\pi$
$863$	−1.32919	−0.0452460	−0.0226230	+	0.999744i	$0.507202\pi$
$864$	0	0
$865$	40.8210	1.38795
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 48.7506i	− 1.65375i
$870$	0	0
$871$	−10.0297	−0.339842
$872$	0	0
$873$	0	0
$874$	0	0
$875$	25.6890i	0.868448i
$876$	0	0
$877$	44.4244i	1.50011i	0.661378	+	0.750053i	$0.269972\pi$
$877$	44.4244i	1.50011i	−0.661378	+	0.750053i	$0.730028\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.5136	1.23017	0.615087	−	0.788459i	$-0.289121\pi$
$881$	36.5136	1.23017	0.615087	+	0.788459i	$0.289121\pi$
$882$	0	0
$883$	− 6.77808i	− 0.228101i	−0.993475	−	0.114050i	$-0.963617\pi$
$883$	− 6.77808i	− 0.228101i	0.993475	−	0.114050i	$-0.0363825\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.6502	−1.63351	−0.816756	−	0.576983i	$-0.804230\pi$
$887$	−48.6502	−1.63351	−0.816756	+	0.576983i	$0.804230\pi$
$888$	0	0
$889$	54.3011	1.82120
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 23.8575i	− 0.798362i
$894$	0	0
$895$	−3.09462	−0.103442
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.13273i	0.171186i
$900$	0	0
$901$	35.0673i	1.16826i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.6560	−0.586904
$906$	0	0
$907$	36.5196i	1.21261i	0.795230	+	0.606307i	$0.207350\pi$
$907$	36.5196i	1.21261i	−0.795230	+	0.606307i	$0.792650\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.9142	−0.792312	−0.396156	−	0.918183i	$-0.629656\pi$
$911$	−23.9142	−0.792312	−0.396156	+	0.918183i	$0.629656\pi$
$912$	0	0
$913$	24.5042	0.810971
$914$	0	0
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	1.63052i	0.0536692i
$924$	0	0
$925$	12.9824i	0.426858i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.5521	−1.03519	−0.517595	−	0.855625i	$-0.673173\pi$
$929$	−31.5521	−1.03519	−0.517595	+	0.855625i	$0.673173\pi$
$930$	0	0
$931$	− 4.84922i	− 0.158927i
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	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
$945$	0	0
$946$	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
$951$	0	0
$952$	0	0
$953$	17.0359	0.551847	0.275924	−	0.961180i	$-0.411016\pi$
$953$	17.0359	0.551847	0.275924	+	0.961180i	$0.411016\pi$
$954$	0	0
$955$	36.2147i	1.17188i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−25.7865	−0.832689
$960$	0	0
$961$	−27.3570	−0.882482
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	37.9292	1.21346	0.606732	−	0.794907i	$-0.292480\pi$
$977$	37.9292	1.21346	0.606732	+	0.794907i	$0.292480\pi$
$978$	0	0
$979$	− 60.6735i	− 1.93913i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.3543	−0.330250	−0.165125	−	0.986273i	$-0.552803\pi$
$983$	−10.3543	−0.330250	−0.165125	+	0.986273i	$0.552803\pi$
$984$	0	0
$985$	−17.3497	−0.552809
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	0	0
$995$	38.7981i	1.22998i
$996$	0	0
$997$	− 24.7402i	− 0.783529i	−0.920066	−	0.391764i	$-0.871865\pi$
$997$	− 24.7402i	− 0.783529i	0.920066	−	0.391764i	$-0.128135\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.a.157.3	✓	8	24.11	even	2
312.2.g.a.157.4	yes	8	12.11	even	2
936.2.g.d.469.5		8	4.3	odd	2
936.2.g.d.469.6		8	8.3	odd	2
1248.2.g.a.625.1		8	24.5	odd	2
1248.2.g.a.625.8		8	3.2	odd	2
3744.2.g.d.1873.2		8	1.1	even	1	trivial
3744.2.g.d.1873.7		8	8.5	even	2	inner
9984.2.a.s.1.4		4	48.29	odd	4
9984.2.a.y.1.1		4	48.11	even	4
9984.2.a.bb.1.4		4	48.35	even	4
9984.2.a.bh.1.1		4	48.5	odd	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.g (of order \(2\), degree \(1\), not minimal)

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

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