LMFDB - Embedded newform 3744.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1,3744)]

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, 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("3744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.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:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	3744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{5} +2.35114 q^{7} +O(q^{10})$	$q-1.23607 q^{5} +2.35114 q^{7} +6.15537 q^{11} +1.00000 q^{13} +6.47214 q^{17} +5.25731 q^{19} -3.47214 q^{25} +4.00000 q^{29} +2.35114 q^{31} -2.90617 q^{35} -10.9443 q^{37} +5.23607 q^{41} -7.60845 q^{43} -9.06154 q^{47} -1.47214 q^{49} -4.94427 q^{53} -7.60845 q^{55} +9.06154 q^{59} +4.47214 q^{61} -1.23607 q^{65} -14.6619 q^{67} +6.15537 q^{71} +6.00000 q^{73} +14.4721 q^{77} +15.2169 q^{79} -3.24920 q^{83} -8.00000 q^{85} +15.7082 q^{89} +2.35114 q^{91} -6.49839 q^{95} +10.9443 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} + 4 q^{13} + 8 q^{17} + 4 q^{25} + 16 q^{29} - 8 q^{37} + 12 q^{41} + 12 q^{49} + 16 q^{53} + 4 q^{65} + 24 q^{73} + 40 q^{77} - 32 q^{85} + 36 q^{89} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^13 + 8 * q^17 + 4 * q^25 + 16 * q^29 - 8 * q^37 + 12 * q^41 + 12 * q^49 + 16 * q^53 + 4 * q^65 + 24 * q^73 + 40 * q^77 - 32 * q^85 + 36 * q^89 + 8 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	2.35114	0.888648	0.444324	−	0.895866i	$-0.353444\pi$
$7$	2.35114	0.888648	0.444324	+	0.895866i	$0.353444\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.15537	1.85591	0.927957	−	0.372689i	$-0.121564\pi$
$11$	6.15537	1.85591	0.927957	+	0.372689i	$0.121564\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	5.25731	1.20611	0.603055	−	0.797700i	$-0.293950\pi$
$19$	5.25731	1.20611	0.603055	+	0.797700i	$0.293950\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	2.35114	0.422277	0.211139	−	0.977456i	$-0.432283\pi$
$31$	2.35114	0.422277	0.211139	+	0.977456i	$0.432283\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.90617	−0.491232
$36$	0	0
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.23607	0.817736	0.408868	−	0.912593i	$-0.365924\pi$
$41$	5.23607	0.817736	0.408868	+	0.912593i	$0.365924\pi$
$42$	0	0
$43$	−7.60845	−1.16028	−0.580139	−	0.814517i	$-0.697002\pi$
$43$	−7.60845	−1.16028	−0.580139	+	0.814517i	$0.697002\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.06154	−1.32176	−0.660881	−	0.750491i	$-0.729817\pi$
$47$	−9.06154	−1.32176	−0.660881	+	0.750491i	$0.729817\pi$
$48$	0	0
$49$	−1.47214	−0.210305
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.94427	−0.679148	−0.339574	−	0.940579i	$-0.610283\pi$
$53$	−4.94427	−0.679148	−0.339574	+	0.940579i	$0.610283\pi$
$54$	0	0
$55$	−7.60845	−1.02592
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.06154	1.17971	0.589856	−	0.807509i	$-0.299185\pi$
$59$	9.06154	1.17971	0.589856	+	0.807509i	$0.299185\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−14.6619	−1.79123	−0.895617	−	0.444827i	$-0.853265\pi$
$67$	−14.6619	−1.79123	−0.895617	+	0.444827i	$0.853265\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.15537	0.730508	0.365254	−	0.930908i	$-0.380982\pi$
$71$	6.15537	0.730508	0.365254	+	0.930908i	$0.380982\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	14.4721	1.64925
$78$	0	0
$79$	15.2169	1.71204	0.856018	−	0.516946i	$-0.172931\pi$
$79$	15.2169	1.71204	0.856018	+	0.516946i	$0.172931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.24920	−0.356646	−0.178323	−	0.983972i	$-0.557067\pi$
$83$	−3.24920	−0.356646	−0.178323	+	0.983972i	$0.557067\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.7082	1.66507	0.832533	−	0.553975i	$-0.186890\pi$
$89$	15.7082	1.66507	0.832533	+	0.553975i	$0.186890\pi$
$90$	0	0
$91$	2.35114	0.246467
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.49839	−0.666721
$96$	0	0
$97$	10.9443	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$97$	10.9443	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.52786	0.550043	0.275022	−	0.961438i	$-0.411315\pi$
$101$	5.52786	0.550043	0.275022	+	0.961438i	$0.411315\pi$
$102$	0	0
$103$	−17.0130	−1.67634	−0.838171	−	0.545407i	$-0.816375\pi$
$103$	−17.0130	−1.67634	−0.838171	+	0.545407i	$0.816375\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.2169	−1.47107	−0.735537	−	0.677485i	$-0.763070\pi$
$107$	−15.2169	−1.47107	−0.735537	+	0.677485i	$0.763070\pi$
$108$	0	0
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	15.2169	1.39493
$120$	0	0
$121$	26.8885	2.44441
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	4.70228	0.417260	0.208630	−	0.977995i	$-0.433100\pi$
$127$	4.70228	0.417260	0.208630	+	0.977995i	$0.433100\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.40456	−0.821681	−0.410840	−	0.911707i	$-0.634765\pi$
$131$	−9.40456	−0.821681	−0.410840	+	0.911707i	$0.634765\pi$
$132$	0	0
$133$	12.3607	1.07181
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.291796	0.0249298	0.0124649	−	0.999922i	$-0.496032\pi$
$137$	0.291796	0.0249298	0.0124649	+	0.999922i	$0.496032\pi$
$138$	0	0
$139$	−2.90617	−0.246498	−0.123249	−	0.992376i	$-0.539331\pi$
$139$	−2.90617	−0.246498	−0.123249	+	0.992376i	$0.539331\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.15537	0.514738
$144$	0	0
$145$	−4.94427	−0.410599
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.2361	1.41203	0.706017	−	0.708195i	$-0.250490\pi$
$149$	17.2361	1.41203	0.706017	+	0.708195i	$0.250490\pi$
$150$	0	0
$151$	22.2703	1.81233	0.906167	−	0.422921i	$-0.138995\pi$
$151$	22.2703	1.81233	0.906167	+	0.422921i	$0.138995\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.90617	−0.233429
$156$	0	0
$157$	5.41641	0.432276	0.216138	−	0.976363i	$-0.430654\pi$
$157$	5.41641	0.432276	0.216138	+	0.976363i	$0.430654\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.95959	0.780096	0.390048	−	0.920795i	$-0.372458\pi$
$163$	9.95959	0.780096	0.390048	+	0.920795i	$0.372458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.06154	−0.701203	−0.350601	−	0.936525i	$-0.614023\pi$
$167$	−9.06154	−0.701203	−0.350601	+	0.936525i	$0.614023\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.4721	0.796182	0.398091	−	0.917346i	$-0.369673\pi$
$173$	10.4721	0.796182	0.398091	+	0.917346i	$0.369673\pi$
$174$	0	0
$175$	−8.16348	−0.617101
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.7153	−1.62308	−0.811539	−	0.584299i	$-0.801370\pi$
$179$	−21.7153	−1.62308	−0.811539	+	0.584299i	$0.801370\pi$
$180$	0	0
$181$	−16.4721	−1.22436	−0.612182	−	0.790717i	$-0.709708\pi$
$181$	−16.4721	−1.22436	−0.612182	+	0.790717i	$0.709708\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	13.5279	0.994588
$186$	0	0
$187$	39.8384	2.91327
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.49839	0.470207	0.235104	−	0.971970i	$-0.424457\pi$
$191$	6.49839	0.470207	0.235104	+	0.971970i	$0.424457\pi$
$192$	0	0
$193$	−2.94427	−0.211933	−0.105967	−	0.994370i	$-0.533794\pi$
$193$	−2.94427	−0.211933	−0.105967	+	0.994370i	$0.533794\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.1803	−1.01031	−0.505154	−	0.863029i	$-0.668564\pi$
$197$	−14.1803	−1.01031	−0.505154	+	0.863029i	$0.668564\pi$
$198$	0	0
$199$	−14.1068	−1.00001	−0.500004	−	0.866023i	$-0.666668\pi$
$199$	−14.1068	−1.00001	−0.500004	+	0.866023i	$0.666668\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.40456	0.660071
$204$	0	0
$205$	−6.47214	−0.452034
$206$	0	0
$207$	0	0
$208$	0	0
$209$	32.3607	2.23844
$210$	0	0
$211$	5.81234	0.400138	0.200069	−	0.979782i	$-0.435883\pi$
$211$	5.81234	0.400138	0.200069	+	0.979782i	$0.435883\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.40456	0.641386
$216$	0	0
$217$	5.52786	0.375256
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.47214	0.435363
$222$	0	0
$223$	−22.2703	−1.49133	−0.745666	−	0.666320i	$-0.767868\pi$
$223$	−22.2703	−1.49133	−0.745666	+	0.666320i	$0.767868\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	27.8707	1.84984	0.924921	−	0.380161i	$-0.124131\pi$
$227$	27.8707	1.84984	0.924921	+	0.380161i	$0.124131\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.41641	−0.485865	−0.242933	−	0.970043i	$-0.578109\pi$
$233$	−7.41641	−0.485865	−0.242933	+	0.970043i	$0.578109\pi$
$234$	0	0
$235$	11.2007	0.730652
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.8707	−1.80280	−0.901402	−	0.432984i	$-0.857461\pi$
$239$	−27.8707	−1.80280	−0.901402	+	0.432984i	$0.857461\pi$
$240$	0	0
$241$	7.88854	0.508146	0.254073	−	0.967185i	$-0.418230\pi$
$241$	7.88854	0.508146	0.254073	+	0.967185i	$0.418230\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.81966	0.116254
$246$	0	0
$247$	5.25731	0.334515
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.1231	1.14392	0.571959	−	0.820282i	$-0.306184\pi$
$251$	18.1231	1.14392	0.571959	+	0.820282i	$0.306184\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.5279	0.843845	0.421922	−	0.906632i	$-0.361355\pi$
$257$	13.5279	0.843845	0.421922	+	0.906632i	$0.361355\pi$
$258$	0	0
$259$	−25.7315	−1.59888
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.3107	0.759112	0.379556	−	0.925169i	$-0.376077\pi$
$263$	12.3107	0.759112	0.379556	+	0.925169i	$0.376077\pi$
$264$	0	0
$265$	6.11146	0.375424
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	2.35114	0.142822	0.0714108	−	0.997447i	$-0.477250\pi$
$271$	2.35114	0.142822	0.0714108	+	0.997447i	$0.477250\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.3723	−1.28880
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.291796	0.0174071	0.00870355	−	0.999962i	$-0.497230\pi$
$281$	0.291796	0.0174071	0.00870355	+	0.999962i	$0.497230\pi$
$282$	0	0
$283$	−14.1068	−0.838565	−0.419282	−	0.907856i	$-0.637718\pi$
$283$	−14.1068	−0.838565	−0.419282	+	0.907856i	$0.637718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.3107	0.726680
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.1246	1.11727	0.558636	−	0.829413i	$-0.311325\pi$
$293$	19.1246	1.11727	0.558636	+	0.829413i	$0.311325\pi$
$294$	0	0
$295$	−11.2007	−0.652129
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−17.8885	−1.03108
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.52786	−0.316525
$306$	0	0
$307$	−9.95959	−0.568424	−0.284212	−	0.958761i	$-0.591732\pi$
$307$	−9.95959	−0.568424	−0.284212	+	0.958761i	$0.591732\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	23.8885	1.35026	0.675130	−	0.737699i	$-0.264087\pi$
$313$	23.8885	1.35026	0.675130	+	0.737699i	$0.264087\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.5967	1.21299	0.606497	−	0.795086i	$-0.292574\pi$
$317$	21.5967	1.21299	0.606497	+	0.795086i	$0.292574\pi$
$318$	0	0
$319$	24.6215	1.37854
$320$	0	0
$321$	0	0
$322$	0	0
$323$	34.0260	1.89326
$324$	0	0
$325$	−3.47214	−0.192599
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−21.3050	−1.17458
$330$	0	0
$331$	−5.25731	−0.288968	−0.144484	−	0.989507i	$-0.546152\pi$
$331$	−5.25731	−0.288968	−0.144484	+	0.989507i	$0.546152\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.1231	0.990169
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.4721	0.783710
$342$	0	0
$343$	−19.9192	−1.07553
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.6215	1.32175	0.660875	−	0.750496i	$-0.270185\pi$
$347$	24.6215	1.32175	0.660875	+	0.750496i	$0.270185\pi$
$348$	0	0
$349$	−7.88854	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$349$	−7.88854	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.7639	−0.998703	−0.499352	−	0.866399i	$-0.666428\pi$
$353$	−18.7639	−0.998703	−0.499352	+	0.866399i	$0.666428\pi$
$354$	0	0
$355$	−7.60845	−0.403815
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.6538	−0.667840	−0.333920	−	0.942601i	$-0.608372\pi$
$359$	−12.6538	−0.667840	−0.333920	+	0.942601i	$0.608372\pi$
$360$	0	0
$361$	8.63932	0.454701
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.41641	−0.388193
$366$	0	0
$367$	−18.1231	−0.946017	−0.473008	−	0.881058i	$-0.656832\pi$
$367$	−18.1231	−0.946017	−0.473008	+	0.881058i	$0.656832\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.6247	−0.603523
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−9.95959	−0.511590	−0.255795	−	0.966731i	$-0.582337\pi$
$379$	−9.95959	−0.511590	−0.255795	+	0.966731i	$0.582337\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.9677	−0.611521	−0.305761	−	0.952108i	$-0.598911\pi$
$383$	−11.9677	−0.611521	−0.305761	+	0.952108i	$0.598911\pi$
$384$	0	0
$385$	−17.8885	−0.911685
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.4721	0.733766	0.366883	−	0.930267i	$-0.380425\pi$
$389$	14.4721	0.733766	0.366883	+	0.930267i	$0.380425\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.8091	−0.946390
$396$	0	0
$397$	24.8328	1.24632	0.623162	−	0.782093i	$-0.285848\pi$
$397$	24.8328	1.24632	0.623162	+	0.782093i	$0.285848\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.7082	1.58343	0.791716	−	0.610889i	$-0.209188\pi$
$401$	31.7082	1.58343	0.791716	+	0.610889i	$0.209188\pi$
$402$	0	0
$403$	2.35114	0.117119
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−67.3660	−3.33921
$408$	0	0
$409$	−11.8885	−0.587851	−0.293925	−	0.955828i	$-0.594962\pi$
$409$	−11.8885	−0.587851	−0.293925	+	0.955828i	$0.594962\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.3050	1.04835
$414$	0	0
$415$	4.01623	0.197149
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.0292	1.02735	0.513673	−	0.857986i	$-0.328284\pi$
$419$	21.0292	1.02735	0.513673	+	0.857986i	$0.328284\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−22.4721	−1.09006
$426$	0	0
$427$	10.5146	0.508838
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.6538	0.609510	0.304755	−	0.952431i	$-0.401425\pi$
$431$	12.6538	0.609510	0.304755	+	0.952431i	$0.401425\pi$
$432$	0	0
$433$	−9.41641	−0.452524	−0.226262	−	0.974067i	$-0.572651\pi$
$433$	−9.41641	−0.452524	−0.226262	+	0.974067i	$0.572651\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	26.4176	1.26084	0.630421	−	0.776253i	$-0.282882\pi$
$439$	26.4176	1.26084	0.630421	+	0.776253i	$0.282882\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.2169	−0.722977	−0.361488	−	0.932377i	$-0.617731\pi$
$443$	−15.2169	−0.722977	−0.361488	+	0.932377i	$0.617731\pi$
$444$	0	0
$445$	−19.4164	−0.920426
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.1803	−1.23553	−0.617763	−	0.786364i	$-0.711961\pi$
$449$	−26.1803	−1.23553	−0.617763	+	0.786364i	$0.711961\pi$
$450$	0	0
$451$	32.2299	1.51765
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.90617	−0.136243
$456$	0	0
$457$	26.9443	1.26040	0.630200	−	0.776433i	$-0.282973\pi$
$457$	26.9443	1.26040	0.630200	+	0.776433i	$0.282973\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.0689	−1.12100	−0.560500	−	0.828155i	$-0.689391\pi$
$461$	−24.0689	−1.12100	−0.560500	+	0.828155i	$0.689391\pi$
$462$	0	0
$463$	11.7557	0.546334	0.273167	−	0.961967i	$-0.411929\pi$
$463$	11.7557	0.546334	0.273167	+	0.961967i	$0.411929\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.49839	−0.300710	−0.150355	−	0.988632i	$-0.548042\pi$
$467$	−6.49839	−0.300710	−0.150355	+	0.988632i	$0.548042\pi$
$468$	0	0
$469$	−34.4721	−1.59178
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−46.8328	−2.15338
$474$	0	0
$475$	−18.2541	−0.837556
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.5599	−0.710951	−0.355476	−	0.934686i	$-0.615681\pi$
$479$	−15.5599	−0.710951	−0.355476	+	0.934686i	$0.615681\pi$
$480$	0	0
$481$	−10.9443	−0.499016
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.5279	−0.614269
$486$	0	0
$487$	32.7849	1.48563	0.742814	−	0.669498i	$-0.233491\pi$
$487$	32.7849	1.48563	0.742814	+	0.669498i	$0.233491\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−36.9322	−1.66673	−0.833363	−	0.552725i	$-0.813588\pi$
$491$	−36.9322	−1.66673	−0.833363	+	0.552725i	$0.813588\pi$
$492$	0	0
$493$	25.8885	1.16596
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.4721	0.649164
$498$	0	0
$499$	−11.0697	−0.495546	−0.247773	−	0.968818i	$-0.579699\pi$
$499$	−11.0697	−0.495546	−0.247773	+	0.968818i	$0.579699\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.1231	0.808068	0.404034	−	0.914744i	$-0.367608\pi$
$503$	18.1231	0.808068	0.404034	+	0.914744i	$0.367608\pi$
$504$	0	0
$505$	−6.83282	−0.304056
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.2918	−0.544824	−0.272412	−	0.962181i	$-0.587821\pi$
$509$	−12.2918	−0.544824	−0.272412	+	0.962181i	$0.587821\pi$
$510$	0	0
$511$	14.1068	0.624050
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.0292	0.926659
$516$	0	0
$517$	−55.7771	−2.45307
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.11146	−0.0925046	−0.0462523	−	0.998930i	$-0.514728\pi$
$521$	−2.11146	−0.0925046	−0.0462523	+	0.998930i	$0.514728\pi$
$522$	0	0
$523$	−2.90617	−0.127078	−0.0635390	−	0.997979i	$-0.520239\pi$
$523$	−2.90617	−0.127078	−0.0635390	+	0.997979i	$0.520239\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.2169	0.662859
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.23607	0.226799
$534$	0	0
$535$	18.8091	0.813190
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9.06154	−0.390308
$540$	0	0
$541$	10.9443	0.470531	0.235266	−	0.971931i	$-0.424404\pi$
$541$	10.9443	0.470531	0.235266	+	0.971931i	$0.424404\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.4721	0.791259
$546$	0	0
$547$	2.90617	0.124259	0.0621294	−	0.998068i	$-0.480211\pi$
$547$	2.90617	0.124259	0.0621294	+	0.998068i	$0.480211\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21.0292	0.895876
$552$	0	0
$553$	35.7771	1.52140
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.5967	−0.915084	−0.457542	−	0.889188i	$-0.651270\pi$
$557$	−21.5967	−0.915084	−0.457542	+	0.889188i	$0.651270\pi$
$558$	0	0
$559$	−7.60845	−0.321803
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.90617	−0.122480	−0.0612402	−	0.998123i	$-0.519506\pi$
$563$	−2.90617	−0.122480	−0.0612402	+	0.998123i	$0.519506\pi$
$564$	0	0
$565$	−4.94427	−0.208007
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.52786	0.231740	0.115870	−	0.993264i	$-0.463034\pi$
$569$	5.52786	0.231740	0.115870	+	0.993264i	$0.463034\pi$
$570$	0	0
$571$	44.5407	1.86397	0.931984	−	0.362499i	$-0.118076\pi$
$571$	44.5407	1.86397	0.931984	+	0.362499i	$0.118076\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.9443	1.28823	0.644113	−	0.764930i	$-0.277226\pi$
$577$	30.9443	1.28823	0.644113	+	0.764930i	$0.277226\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.63932	−0.316932
$582$	0	0
$583$	−30.4338	−1.26044
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.6538	−0.522277	−0.261138	−	0.965301i	$-0.584098\pi$
$587$	−12.6538	−0.522277	−0.261138	+	0.965301i	$0.584098\pi$
$588$	0	0
$589$	12.3607	0.509313
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−32.2918	−1.32607	−0.663033	−	0.748591i	$-0.730731\pi$
$593$	−32.2918	−1.32607	−0.663033	+	0.748591i	$0.730731\pi$
$594$	0	0
$595$	−18.8091	−0.771099
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.49839	−0.265517	−0.132759	−	0.991148i	$-0.542383\pi$
$599$	−6.49839	−0.265517	−0.132759	+	0.991148i	$0.542383\pi$
$600$	0	0
$601$	7.88854	0.321780	0.160890	−	0.986972i	$-0.448563\pi$
$601$	7.88854	0.321780	0.160890	+	0.986972i	$0.448563\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−33.2361	−1.35124
$606$	0	0
$607$	−8.71851	−0.353873	−0.176937	−	0.984222i	$-0.556619\pi$
$607$	−8.71851	−0.353873	−0.176937	+	0.984222i	$0.556619\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.06154	−0.366591
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.0689	1.45208	0.726039	−	0.687653i	$-0.241359\pi$
$617$	36.0689	1.45208	0.726039	+	0.687653i	$0.241359\pi$
$618$	0	0
$619$	−28.7687	−1.15631	−0.578156	−	0.815926i	$-0.696228\pi$
$619$	−28.7687	−1.15631	−0.578156	+	0.815926i	$0.696228\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	36.9322	1.47966
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−70.8328	−2.82429
$630$	0	0
$631$	−13.9758	−0.556369	−0.278184	−	0.960528i	$-0.589733\pi$
$631$	−13.9758	−0.556369	−0.278184	+	0.960528i	$0.589733\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.81234	−0.230656
$636$	0	0
$637$	−1.47214	−0.0583282
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.3607	0.962189	0.481095	−	0.876669i	$-0.340239\pi$
$641$	24.3607	0.962189	0.481095	+	0.876669i	$0.340239\pi$
$642$	0	0
$643$	6.36737	0.251105	0.125552	−	0.992087i	$-0.459930\pi$
$643$	6.36737	0.251105	0.125552	+	0.992087i	$0.459930\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	55.7771	2.18944
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.9443	1.28921	0.644604	−	0.764516i	$-0.277022\pi$
$653$	32.9443	1.28921	0.644604	+	0.764516i	$0.277022\pi$
$654$	0	0
$655$	11.6247	0.454214
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.81234	0.226417	0.113208	−	0.993571i	$-0.463887\pi$
$659$	5.81234	0.226417	0.113208	+	0.993571i	$0.463887\pi$
$660$	0	0
$661$	45.7771	1.78052	0.890261	−	0.455450i	$-0.150522\pi$
$661$	45.7771	1.78052	0.890261	+	0.455450i	$0.150522\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.2786	−0.592480
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27.5276	1.06269
$672$	0	0
$673$	17.4164	0.671353	0.335677	−	0.941977i	$-0.391035\pi$
$673$	17.4164	0.671353	0.335677	+	0.941977i	$0.391035\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.7771	−1.06756	−0.533780	−	0.845623i	$-0.679229\pi$
$677$	−27.7771	−1.06756	−0.533780	+	0.845623i	$0.679229\pi$
$678$	0	0
$679$	25.7315	0.987485
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9.06154	−0.346730	−0.173365	−	0.984858i	$-0.555464\pi$
$683$	−9.06154	−0.346730	−0.173365	+	0.984858i	$0.555464\pi$
$684$	0	0
$685$	−0.360680	−0.0137809
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.94427	−0.188362
$690$	0	0
$691$	15.7719	0.599993	0.299996	−	0.953940i	$-0.403015\pi$
$691$	15.7719	0.599993	0.299996	+	0.953940i	$0.403015\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.59222	0.136261
$696$	0	0
$697$	33.8885	1.28362
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.47214	0.244449	0.122225	−	0.992502i	$-0.460997\pi$
$701$	6.47214	0.244449	0.122225	+	0.992502i	$0.460997\pi$
$702$	0	0
$703$	−57.5374	−2.17007
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.9968	0.488795
$708$	0	0
$709$	25.7771	0.968079	0.484039	−	0.875046i	$-0.339169\pi$
$709$	25.7771	0.968079	0.484039	+	0.875046i	$0.339169\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−7.60845	−0.284540
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.81234	−0.216764	−0.108382	−	0.994109i	$-0.534567\pi$
$719$	−5.81234	−0.216764	−0.108382	+	0.994109i	$0.534567\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−13.8885	−0.515808
$726$	0	0
$727$	−1.11006	−0.0411698	−0.0205849	−	0.999788i	$-0.506553\pi$
$727$	−1.11006	−0.0411698	−0.0205849	+	0.999788i	$0.506553\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−49.2429	−1.82132
$732$	0	0
$733$	−22.9443	−0.847466	−0.423733	−	0.905787i	$-0.639281\pi$
$733$	−22.9443	−0.847466	−0.423733	+	0.905787i	$0.639281\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−90.2492	−3.32437
$738$	0	0
$739$	8.84953	0.325535	0.162768	−	0.986664i	$-0.447958\pi$
$739$	8.84953	0.325535	0.162768	+	0.986664i	$0.447958\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.46931	−0.200650	−0.100325	−	0.994955i	$-0.531988\pi$
$743$	−5.46931	−0.200650	−0.100325	+	0.994955i	$0.531988\pi$
$744$	0	0
$745$	−21.3050	−0.780553
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−35.7771	−1.30727
$750$	0	0
$751$	11.2007	0.408718	0.204359	−	0.978896i	$-0.434489\pi$
$751$	11.2007	0.408718	0.204359	+	0.978896i	$0.434489\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−27.5276	−1.00183
$756$	0	0
$757$	−18.5836	−0.675432	−0.337716	−	0.941248i	$-0.609654\pi$
$757$	−18.5836	−0.675432	−0.337716	+	0.941248i	$0.609654\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.6525	−0.748652	−0.374326	−	0.927297i	$-0.622126\pi$
$761$	−20.6525	−0.748652	−0.374326	+	0.927297i	$0.622126\pi$
$762$	0	0
$763$	−35.1361	−1.27201
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.06154	0.327193
$768$	0	0
$769$	−36.8328	−1.32823	−0.664113	−	0.747633i	$-0.731190\pi$
$769$	−36.8328	−1.32823	−0.664113	+	0.747633i	$0.731190\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−35.7082	−1.28433	−0.642167	−	0.766564i	$-0.721965\pi$
$773$	−35.7082	−1.28433	−0.642167	+	0.766564i	$0.721965\pi$
$774$	0	0
$775$	−8.16348	−0.293241
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27.5276	0.986280
$780$	0	0
$781$	37.8885	1.35576
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.69505	−0.238957
$786$	0	0
$787$	−46.2057	−1.64706	−0.823528	−	0.567275i	$-0.807998\pi$
$787$	−46.2057	−1.64706	−0.823528	+	0.567275i	$0.807998\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.40456	0.334388
$792$	0	0
$793$	4.47214	0.158810
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.4164	0.404390	0.202195	−	0.979345i	$-0.435193\pi$
$797$	11.4164	0.404390	0.202195	+	0.979345i	$0.435193\pi$
$798$	0	0
$799$	−58.6475	−2.07480
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.9322	1.30331
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.00000	−0.140633	−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000	−0.140633	−0.0703163	+	0.997525i	$0.522401\pi$
$810$	0	0
$811$	−49.7980	−1.74864	−0.874322	−	0.485347i	$-0.838693\pi$
$811$	−49.7980	−1.74864	−0.874322	+	0.485347i	$0.838693\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.3107	−0.431226
$816$	0	0
$817$	−40.0000	−1.39942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.5967	−1.31214	−0.656068	−	0.754702i	$-0.727781\pi$
$821$	−37.5967	−1.31214	−0.656068	+	0.754702i	$0.727781\pi$
$822$	0	0
$823$	22.8254	0.795642	0.397821	−	0.917463i	$-0.369767\pi$
$823$	22.8254	0.795642	0.397821	+	0.917463i	$0.369767\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.1814	1.39724	0.698622	−	0.715491i	$-0.253797\pi$
$827$	40.1814	1.39724	0.698622	+	0.715491i	$0.253797\pi$
$828$	0	0
$829$	0.472136	0.0163980	0.00819898	−	0.999966i	$-0.497390\pi$
$829$	0.472136	0.0163980	0.00819898	+	0.999966i	$0.497390\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.52786	−0.330121
$834$	0	0
$835$	11.2007	0.387615
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.5892	−1.26320	−0.631599	−	0.775295i	$-0.717601\pi$
$839$	−36.5892	−1.26320	−0.631599	+	0.775295i	$0.717601\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.23607	−0.0425220
$846$	0	0
$847$	63.2188	2.17222
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−20.8328	−0.713302	−0.356651	−	0.934238i	$-0.616081\pi$
$853$	−20.8328	−0.713302	−0.356651	+	0.934238i	$0.616081\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.3050	1.68423	0.842113	−	0.539302i	$-0.181312\pi$
$857$	49.3050	1.68423	0.842113	+	0.539302i	$0.181312\pi$
$858$	0	0
$859$	15.2169	0.519194	0.259597	−	0.965717i	$-0.416410\pi$
$859$	15.2169	0.519194	0.259597	+	0.965717i	$0.416410\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.9677	−0.407385	−0.203693	−	0.979035i	$-0.565294\pi$
$863$	−11.9677	−0.407385	−0.203693	+	0.979035i	$0.565294\pi$
$864$	0	0
$865$	−12.9443	−0.440118
$866$	0	0
$867$	0	0
$868$	0	0
$869$	93.6656	3.17739
$870$	0	0
$871$	−14.6619	−0.496799
$872$	0	0
$873$	0	0
$874$	0	0
$875$	24.6215	0.832358
$876$	0	0
$877$	30.9443	1.04491	0.522457	−	0.852666i	$-0.325016\pi$
$877$	30.9443	1.04491	0.522457	+	0.852666i	$0.325016\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.36068	−0.146915	−0.0734575	−	0.997298i	$-0.523403\pi$
$881$	−4.36068	−0.146915	−0.0734575	+	0.997298i	$0.523403\pi$
$882$	0	0
$883$	48.1329	1.61980	0.809900	−	0.586568i	$-0.199521\pi$
$883$	48.1329	1.61980	0.809900	+	0.586568i	$0.199521\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.7445	1.43522	0.717611	−	0.696445i	$-0.245236\pi$
$887$	42.7445	1.43522	0.717611	+	0.696445i	$0.245236\pi$
$888$	0	0
$889$	11.0557	0.370797
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−47.6393	−1.59419
$894$	0	0
$895$	26.8416	0.897215
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.40456	0.313660
$900$	0	0
$901$	−32.0000	−1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.3607	0.676812
$906$	0	0
$907$	−4.70228	−0.156137	−0.0780684	−	0.996948i	$-0.524875\pi$
$907$	−4.70228	−0.156137	−0.0780684	+	0.996948i	$0.524875\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.9322	1.22362	0.611809	−	0.791005i	$-0.290442\pi$
$911$	36.9322	1.22362	0.611809	+	0.791005i	$0.290442\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.1115	−0.730185
$918$	0	0
$919$	15.2169	0.501959	0.250980	−	0.967992i	$-0.419247\pi$
$919$	15.2169	0.501959	0.250980	+	0.967992i	$0.419247\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.15537	0.202606
$924$	0	0
$925$	38.0000	1.24943
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13.8197	−0.453408	−0.226704	−	0.973964i	$-0.572795\pi$
$929$	−13.8197	−0.453408	−0.226704	+	0.973964i	$0.572795\pi$
$930$	0	0
$931$	−7.73948	−0.253651
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−49.2429	−1.61042
$936$	0	0
$937$	27.5279	0.899296	0.449648	−	0.893206i	$-0.351549\pi$
$937$	27.5279	0.899296	0.449648	+	0.893206i	$0.351549\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.7082	−0.381677	−0.190838	−	0.981621i	$-0.561121\pi$
$941$	−11.7082	−0.381677	−0.190838	+	0.981621i	$0.561121\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.46931	0.177729	0.0888644	−	0.996044i	$-0.471676\pi$
$947$	5.46931	0.177729	0.0888644	+	0.996044i	$0.471676\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39.4164	−1.27682	−0.638411	−	0.769695i	$-0.720408\pi$
$953$	−39.4164	−1.27682	−0.638411	+	0.769695i	$0.720408\pi$
$954$	0	0
$955$	−8.03246	−0.259924
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.686054	0.0221538
$960$	0	0
$961$	−25.4721	−0.821682
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.63932	0.117154
$966$	0	0
$967$	−41.0795	−1.32103	−0.660513	−	0.750815i	$-0.729661\pi$
$967$	−41.0795	−1.32103	−0.660513	+	0.750815i	$0.729661\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.4338	−0.976667	−0.488334	−	0.872657i	$-0.662395\pi$
$971$	−30.4338	−0.976667	−0.488334	+	0.872657i	$0.662395\pi$
$972$	0	0
$973$	−6.83282	−0.219050
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.1246	−1.50765	−0.753825	−	0.657075i	$-0.771793\pi$
$977$	−47.1246	−1.50765	−0.753825	+	0.657075i	$0.771793\pi$
$978$	0	0
$979$	96.6898	3.09022
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−55.3983	−1.76693	−0.883466	−	0.468496i	$-0.844796\pi$
$983$	−55.3983	−1.76693	−0.883466	+	0.468496i	$0.844796\pi$
$984$	0	0
$985$	17.5279	0.558484
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	33.3400	1.05908	0.529540	−	0.848285i	$-0.322365\pi$
$991$	33.3400	1.05908	0.529540	+	0.848285i	$0.322365\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.4370	0.552791
$996$	0	0
$997$	4.11146	0.130211	0.0651056	−	0.997878i	$-0.479262\pi$
$997$	4.11146	0.130211	0.0651056	+	0.997878i	$0.479262\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.a.bf.1.2	yes	4
3.2	odd	2		3744.2.a.bb.1.4	yes	4
4.3	odd	2	inner	3744.2.a.bf.1.1	yes	4
8.3	odd	2		7488.2.a.cz.1.3		4
8.5	even	2		7488.2.a.cz.1.4		4
12.11	even	2		3744.2.a.bb.1.3	✓	4
24.5	odd	2		7488.2.a.dd.1.2		4
24.11	even	2		7488.2.a.dd.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3744.2.a.bb.1.3	✓	4	12.11	even	2
3744.2.a.bb.1.4	yes	4	3.2	odd	2
3744.2.a.bf.1.1	yes	4	4.3	odd	2	inner
3744.2.a.bf.1.2	yes	4	1.1	even	1	trivial
7488.2.a.cz.1.3		4	8.3	odd	2
7488.2.a.cz.1.4		4	8.5	even	2
7488.2.a.dd.1.1		4	24.11	even	2
7488.2.a.dd.1.2		4	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{5} +2.35114 q^{7} +O(q^{10})\)	\(q-1.23607 q^{5} +2.35114 q^{7} +6.15537 q^{11} +1.00000 q^{13} +6.47214 q^{17} +5.25731 q^{19} -3.47214 q^{25} +4.00000 q^{29} +2.35114 q^{31} -2.90617 q^{35} -10.9443 q^{37} +5.23607 q^{41} -7.60845 q^{43} -9.06154 q^{47} -1.47214 q^{49} -4.94427 q^{53} -7.60845 q^{55} +9.06154 q^{59} +4.47214 q^{61} -1.23607 q^{65} -14.6619 q^{67} +6.15537 q^{71} +6.00000 q^{73} +14.4721 q^{77} +15.2169 q^{79} -3.24920 q^{83} -8.00000 q^{85} +15.7082 q^{89} +2.35114 q^{91} -6.49839 q^{95} +10.9443 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} + 4 q^{13} + 8 q^{17} + 4 q^{25} + 16 q^{29} - 8 q^{37} + 12 q^{41} + 12 q^{49} + 16 q^{53} + 4 q^{65} + 24 q^{73} + 40 q^{77} - 32 q^{85} + 36 q^{89} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^13 + 8 * q^17 + 4 * q^25 + 16 * q^29 - 8 * q^37 + 12 * q^41 + 12 * q^49 + 16 * q^53 + 4 * q^65 + 24 * q^73 + 40 * q^77 - 32 * q^85 + 36 * q^89 + 8 * q^97

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