LMFDB - Embedded newform 2268.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2268.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.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:	$18.1100711784$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.04547$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04547 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.04547 q^{5} -1.00000 q^{7} -4.77753 q^{11} -5.27492 q^{13} +0.418627 q^{17} +6.54983 q^{19} -8.71780 q^{23} +4.27492 q^{25} -3.88273 q^{29} -6.00000 q^{31} -3.04547 q^{35} -11.5498 q^{37} +6.92820 q^{41} -6.27492 q^{43} -3.46410 q^{47} +1.00000 q^{49} +10.8685 q^{53} -14.5498 q^{55} +8.71780 q^{59} -11.2749 q^{61} -16.0646 q^{65} +0.274917 q^{67} -8.24163 q^{71} -2.72508 q^{73} +4.77753 q^{77} -3.72508 q^{79} -0.837253 q^{83} +1.27492 q^{85} +16.0646 q^{89} +5.27492 q^{91} +19.9474 q^{95} +14.5498 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} - 6 q^{13} - 4 q^{19} + 2 q^{25} - 24 q^{31} - 16 q^{37} - 10 q^{43} + 4 q^{49} - 28 q^{55} - 30 q^{61} - 14 q^{67} - 26 q^{73} - 30 q^{79} - 10 q^{85} + 6 q^{91} + 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 6 * q^13 - 4 * q^19 + 2 * q^25 - 24 * q^31 - 16 * q^37 - 10 * q^43 + 4 * q^49 - 28 * q^55 - 30 * q^61 - 14 * q^67 - 26 * q^73 - 30 * q^79 - 10 * q^85 + 6 * q^91 + 28 * 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
$5$	3.04547	1.36198	0.680989	−	0.732294i	$-0.261550\pi$
$5$	3.04547	1.36198	0.680989	+	0.732294i	$0.261550\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.77753	−1.44048	−0.720239	−	0.693726i	$-0.755968\pi$
$11$	−4.77753	−1.44048	−0.720239	+	0.693726i	$0.755968\pi$
$12$	0	0
$13$	−5.27492	−1.46300	−0.731499	−	0.681842i	$-0.761179\pi$
$13$	−5.27492	−1.46300	−0.731499	+	0.681842i	$0.761179\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.418627	0.101532	0.0507659	−	0.998711i	$-0.483834\pi$
$17$	0.418627	0.101532	0.0507659	+	0.998711i	$0.483834\pi$
$18$	0	0
$19$	6.54983	1.50264	0.751318	−	0.659941i	$-0.229419\pi$
$19$	6.54983	1.50264	0.751318	+	0.659941i	$0.229419\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.71780	−1.81779	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	−8.71780	−1.81779	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	4.27492	0.854983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.88273	−0.721005	−0.360502	−	0.932758i	$-0.617395\pi$
$29$	−3.88273	−0.721005	−0.360502	+	0.932758i	$0.617395\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.04547	−0.514779
$36$	0	0
$37$	−11.5498	−1.89878	−0.949391	−	0.314098i	$-0.898298\pi$
$37$	−11.5498	−1.89878	−0.949391	+	0.314098i	$0.898298\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	−6.27492	−0.956916	−0.478458	−	0.878110i	$-0.658804\pi$
$43$	−6.27492	−0.956916	−0.478458	+	0.878110i	$0.658804\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.8685	1.49290	0.746450	−	0.665442i	$-0.231757\pi$
$53$	10.8685	1.49290	0.746450	+	0.665442i	$0.231757\pi$
$54$	0	0
$55$	−14.5498	−1.96190
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.71780	1.13496	0.567480	−	0.823387i	$-0.307918\pi$
$59$	8.71780	1.13496	0.567480	+	0.823387i	$0.307918\pi$
$60$	0	0
$61$	−11.2749	−1.44361	−0.721803	−	0.692099i	$-0.756686\pi$
$61$	−11.2749	−1.44361	−0.721803	+	0.692099i	$0.756686\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.0646	−1.99257
$66$	0	0
$67$	0.274917	0.0335865	0.0167932	−	0.999859i	$-0.494654\pi$
$67$	0.274917	0.0335865	0.0167932	+	0.999859i	$0.494654\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.24163	−0.978101	−0.489051	−	0.872255i	$-0.662657\pi$
$71$	−8.24163	−0.978101	−0.489051	+	0.872255i	$0.662657\pi$
$72$	0	0
$73$	−2.72508	−0.318947	−0.159473	−	0.987202i	$-0.550980\pi$
$73$	−2.72508	−0.318947	−0.159473	+	0.987202i	$0.550980\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.77753	0.544450
$78$	0	0
$79$	−3.72508	−0.419105	−0.209552	−	0.977797i	$-0.567201\pi$
$79$	−3.72508	−0.419105	−0.209552	+	0.977797i	$0.567201\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.837253	−0.0919005	−0.0459503	−	0.998944i	$-0.514632\pi$
$83$	−0.837253	−0.0919005	−0.0459503	+	0.998944i	$0.514632\pi$
$84$	0	0
$85$	1.27492	0.138284
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.0646	1.70285	0.851424	−	0.524479i	$-0.175740\pi$
$89$	16.0646	1.70285	0.851424	+	0.524479i	$0.175740\pi$
$90$	0	0
$91$	5.27492	0.552962
$92$	0	0
$93$	0	0
$94$	0	0
$95$	19.9474	2.04656
$96$	0	0
$97$	14.5498	1.47731	0.738656	−	0.674083i	$-0.235461\pi$
$97$	14.5498	1.47731	0.738656	+	0.674083i	$0.235461\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.25370	0.522762	0.261381	−	0.965236i	$-0.415822\pi$
$101$	5.25370	0.522762	0.261381	+	0.965236i	$0.415822\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.94027	−0.380920	−0.190460	−	0.981695i	$-0.560998\pi$
$107$	−3.94027	−0.380920	−0.190460	+	0.981695i	$0.560998\pi$
$108$	0	0
$109$	13.8248	1.32417	0.662086	−	0.749428i	$-0.269672\pi$
$109$	13.8248	1.32417	0.662086	+	0.749428i	$0.269672\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.49751	0.893450	0.446725	−	0.894671i	$-0.352590\pi$
$113$	9.49751	0.893450	0.446725	+	0.894671i	$0.352590\pi$
$114$	0	0
$115$	−26.5498	−2.47578
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.418627	−0.0383755
$120$	0	0
$121$	11.8248	1.07498
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.20822	−0.197509
$126$	0	0
$127$	−18.8248	−1.67043	−0.835213	−	0.549926i	$-0.814656\pi$
$127$	−18.8248	−1.67043	−0.835213	+	0.549926i	$0.814656\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.62685	−0.229509	−0.114754	−	0.993394i	$-0.536608\pi$
$131$	−2.62685	−0.229509	−0.114754	+	0.993394i	$0.536608\pi$
$132$	0	0
$133$	−6.54983	−0.567943
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.0575438	0.00491630	0.00245815	−	0.999997i	$-0.499218\pi$
$137$	0.0575438	0.00491630	0.00245815	+	0.999997i	$0.499218\pi$
$138$	0	0
$139$	2.54983	0.216274	0.108137	−	0.994136i	$-0.465511\pi$
$139$	2.54983	0.216274	0.108137	+	0.994136i	$0.465511\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	25.2011	2.10742
$144$	0	0
$145$	−11.8248	−0.981992
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.68439	0.219914	0.109957	−	0.993936i	$-0.464929\pi$
$149$	2.68439	0.219914	0.109957	+	0.993936i	$0.464929\pi$
$150$	0	0
$151$	−8.82475	−0.718148	−0.359074	−	0.933309i	$-0.616907\pi$
$151$	−8.82475	−0.718148	−0.359074	+	0.933309i	$0.616907\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.2728	−1.46771
$156$	0	0
$157$	−19.8248	−1.58219	−0.791094	−	0.611695i	$-0.790488\pi$
$157$	−19.8248	−1.58219	−0.791094	+	0.611695i	$0.790488\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.71780	0.687059
$162$	0	0
$163$	−18.2749	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$163$	−18.2749	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.837253	0.0647886	0.0323943	−	0.999475i	$-0.489687\pi$
$167$	0.837253	0.0647886	0.0323943	+	0.999475i	$0.489687\pi$
$168$	0	0
$169$	14.8248	1.14037
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.34683	−0.558569	−0.279285	−	0.960208i	$-0.590097\pi$
$173$	−7.34683	−0.558569	−0.279285	+	0.960208i	$0.590097\pi$
$174$	0	0
$175$	−4.27492	−0.323153
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.3923	−0.776757	−0.388379	−	0.921500i	$-0.626965\pi$
$179$	−10.3923	−0.776757	−0.388379	+	0.921500i	$0.626965\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−35.1747	−2.58610
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.40437	−0.535762	−0.267881	−	0.963452i	$-0.586323\pi$
$191$	−7.40437	−0.535762	−0.267881	+	0.963452i	$0.586323\pi$
$192$	0	0
$193$	−12.3746	−0.890742	−0.445371	−	0.895346i	$-0.646928\pi$
$193$	−12.3746	−0.890742	−0.445371	+	0.895346i	$0.646928\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.52165	−0.250907	−0.125453	−	0.992100i	$-0.540039\pi$
$197$	−3.52165	−0.250907	−0.125453	+	0.992100i	$0.540039\pi$
$198$	0	0
$199$	11.0997	0.786835	0.393417	−	0.919360i	$-0.371293\pi$
$199$	11.0997	0.786835	0.393417	+	0.919360i	$0.371293\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.88273	0.272514
$204$	0	0
$205$	21.0997	1.47366
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−31.2920	−2.16451
$210$	0	0
$211$	11.7251	0.807188	0.403594	−	0.914938i	$-0.367761\pi$
$211$	11.7251	0.807188	0.403594	+	0.914938i	$0.367761\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.1101	−1.30330
$216$	0	0
$217$	6.00000	0.407307
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.20822	−0.148541
$222$	0	0
$223$	0.549834	0.0368196	0.0184098	−	0.999831i	$-0.494140\pi$
$223$	0.549834	0.0368196	0.0184098	+	0.999831i	$0.494140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.8564	0.919682	0.459841	−	0.888001i	$-0.347906\pi$
$227$	13.8564	0.919682	0.459841	+	0.888001i	$0.347906\pi$
$228$	0	0
$229$	−13.2749	−0.877231	−0.438616	−	0.898675i	$-0.644531\pi$
$229$	−13.2749	−0.877231	−0.438616	+	0.898675i	$0.644531\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.46192	−0.488846	−0.244423	−	0.969669i	$-0.578599\pi$
$233$	−7.46192	−0.488846	−0.244423	+	0.969669i	$0.578599\pi$
$234$	0	0
$235$	−10.5498	−0.688195
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.2608	1.37525	0.687623	−	0.726068i	$-0.258654\pi$
$239$	21.2608	1.37525	0.687623	+	0.726068i	$0.258654\pi$
$240$	0	0
$241$	−3.27492	−0.210956	−0.105478	−	0.994422i	$-0.533637\pi$
$241$	−3.27492	−0.210956	−0.105478	+	0.994422i	$0.533637\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.04547	0.194568
$246$	0	0
$247$	−34.5498	−2.19835
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.2487	−1.53057	−0.765283	−	0.643695i	$-0.777401\pi$
$251$	−24.2487	−1.53057	−0.765283	+	0.643695i	$0.777401\pi$
$252$	0	0
$253$	41.6495	2.61848
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.20822	0.137745	0.0688725	−	0.997625i	$-0.478060\pi$
$257$	2.20822	0.137745	0.0688725	+	0.997625i	$0.478060\pi$
$258$	0	0
$259$	11.5498	0.717672
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.40437	0.456573	0.228287	−	0.973594i	$-0.426688\pi$
$263$	7.40437	0.456573	0.228287	+	0.973594i	$0.426688\pi$
$264$	0	0
$265$	33.0997	2.03330
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.7583	1.87537	0.937683	−	0.347492i	$-0.112967\pi$
$269$	30.7583	1.87537	0.937683	+	0.347492i	$0.112967\pi$
$270$	0	0
$271$	15.0997	0.917240	0.458620	−	0.888633i	$-0.348344\pi$
$271$	15.0997	0.917240	0.458620	+	0.888633i	$0.348344\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.4235	−1.23159
$276$	0	0
$277$	−16.8248	−1.01090	−0.505451	−	0.862855i	$-0.668674\pi$
$277$	−16.8248	−1.01090	−0.505451	+	0.862855i	$0.668674\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.03341	−0.359923	−0.179961	−	0.983674i	$-0.557597\pi$
$281$	−6.03341	−0.359923	−0.179961	+	0.983674i	$0.557597\pi$
$282$	0	0
$283$	−3.45017	−0.205091	−0.102546	−	0.994728i	$-0.532699\pi$
$283$	−3.45017	−0.205091	−0.102546	+	0.994728i	$0.532699\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−16.8248	−0.989691
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.67232	0.331381	0.165690	−	0.986178i	$-0.447015\pi$
$293$	5.67232	0.331381	0.165690	+	0.986178i	$0.447015\pi$
$294$	0	0
$295$	26.5498	1.54579
$296$	0	0
$297$	0	0
$298$	0	0
$299$	45.9857	2.65942
$300$	0	0
$301$	6.27492	0.361680
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−34.3375	−1.96616
$306$	0	0
$307$	20.5498	1.17284	0.586420	−	0.810007i	$-0.300537\pi$
$307$	20.5498	1.17284	0.586420	+	0.810007i	$0.300537\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.0668	−0.684246	−0.342123	−	0.939655i	$-0.611146\pi$
$311$	−12.0668	−0.684246	−0.342123	+	0.939655i	$0.611146\pi$
$312$	0	0
$313$	−5.82475	−0.329234	−0.164617	−	0.986358i	$-0.552639\pi$
$313$	−5.82475	−0.329234	−0.164617	+	0.986358i	$0.552639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.9210	1.68053	0.840266	−	0.542174i	$-0.182399\pi$
$317$	29.9210	1.68053	0.840266	+	0.542174i	$0.182399\pi$
$318$	0	0
$319$	18.5498	1.03859
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.74194	0.152565
$324$	0	0
$325$	−22.5498	−1.25084
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.46410	0.190982
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.837253	0.0457440
$336$	0	0
$337$	−0.824752	−0.0449271	−0.0224635	−	0.999748i	$-0.507151\pi$
$337$	−0.824752	−0.0449271	−0.0224635	+	0.999748i	$0.507151\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	28.6652	1.55231
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.7967	0.955376	0.477688	−	0.878530i	$-0.341475\pi$
$347$	17.7967	0.955376	0.477688	+	0.878530i	$0.341475\pi$
$348$	0	0
$349$	−32.1993	−1.72359	−0.861796	−	0.507256i	$-0.830660\pi$
$349$	−32.1993	−1.72359	−0.861796	+	0.507256i	$0.830660\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.7611	−0.838878	−0.419439	−	0.907783i	$-0.637773\pi$
$353$	−15.7611	−0.838878	−0.419439	+	0.907783i	$0.637773\pi$
$354$	0	0
$355$	−25.0997	−1.33215
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.1222	0.850896	0.425448	−	0.904983i	$-0.360117\pi$
$359$	16.1222	0.850896	0.425448	+	0.904983i	$0.360117\pi$
$360$	0	0
$361$	23.9003	1.25791
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.29917	−0.434398
$366$	0	0
$367$	−5.45017	−0.284496	−0.142248	−	0.989831i	$-0.545433\pi$
$367$	−5.45017	−0.284496	−0.142248	+	0.989831i	$0.545433\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.8685	−0.564263
$372$	0	0
$373$	24.2749	1.25691	0.628454	−	0.777847i	$-0.283688\pi$
$373$	24.2749	1.25691	0.628454	+	0.777847i	$0.283688\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.4811	1.05483
$378$	0	0
$379$	−13.3746	−0.687006	−0.343503	−	0.939152i	$-0.611614\pi$
$379$	−13.3746	−0.687006	−0.343503	+	0.939152i	$0.611614\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.57919	0.182888	0.0914440	−	0.995810i	$-0.470852\pi$
$383$	3.57919	0.182888	0.0914440	+	0.995810i	$0.470852\pi$
$384$	0	0
$385$	14.5498	0.741528
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.5074	−0.532746	−0.266373	−	0.963870i	$-0.585825\pi$
$389$	−10.5074	−0.532746	−0.266373	+	0.963870i	$0.585825\pi$
$390$	0	0
$391$	−3.64950	−0.184563
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.3446	−0.570811
$396$	0	0
$397$	−2.17525	−0.109173	−0.0545863	−	0.998509i	$-0.517384\pi$
$397$	−2.17525	−0.109173	−0.0545863	+	0.998509i	$0.517384\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.9331	−1.34497	−0.672487	−	0.740108i	$-0.734774\pi$
$401$	−26.9331	−1.34497	−0.672487	+	0.740108i	$0.734774\pi$
$402$	0	0
$403$	31.6495	1.57657
$404$	0	0
$405$	0	0
$406$	0	0
$407$	55.1796	2.73515
$408$	0	0
$409$	10.3746	0.512990	0.256495	−	0.966546i	$-0.417432\pi$
$409$	10.3746	0.512990	0.256495	+	0.966546i	$0.417432\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.71780	−0.428975
$414$	0	0
$415$	−2.54983	−0.125166
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−29.6175	−1.44691	−0.723455	−	0.690372i	$-0.757447\pi$
$419$	−29.6175	−1.44691	−0.723455	+	0.690372i	$0.757447\pi$
$420$	0	0
$421$	5.90033	0.287565	0.143782	−	0.989609i	$-0.454074\pi$
$421$	5.90033	0.287565	0.143782	+	0.989609i	$0.454074\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.78959	0.0868081
$426$	0	0
$427$	11.2749	0.545631
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.5934	−1.71447	−0.857236	−	0.514924i	$-0.827820\pi$
$431$	−35.5934	−1.71447	−0.857236	+	0.514924i	$0.827820\pi$
$432$	0	0
$433$	2.72508	0.130959	0.0654796	−	0.997854i	$-0.479142\pi$
$433$	2.72508	0.130959	0.0654796	+	0.997854i	$0.479142\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−57.1001	−2.73147
$438$	0	0
$439$	4.54983	0.217152	0.108576	−	0.994088i	$-0.465371\pi$
$439$	4.54983	0.217152	0.108576	+	0.994088i	$0.465371\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.7561	1.65131	0.825656	−	0.564174i	$-0.190805\pi$
$443$	34.7561	1.65131	0.825656	+	0.564174i	$0.190805\pi$
$444$	0	0
$445$	48.9244	2.31924
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.5430	−0.591940	−0.295970	−	0.955197i	$-0.595643\pi$
$449$	−12.5430	−0.591940	−0.295970	+	0.955197i	$0.595643\pi$
$450$	0	0
$451$	−33.0997	−1.55860
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.0646	0.753121
$456$	0	0
$457$	−8.09967	−0.378887	−0.189443	−	0.981892i	$-0.560668\pi$
$457$	−8.09967	−0.378887	−0.189443	+	0.981892i	$0.560668\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.9906	1.25708	0.628540	−	0.777777i	$-0.283653\pi$
$461$	26.9906	1.25708	0.628540	+	0.777777i	$0.283653\pi$
$462$	0	0
$463$	30.4743	1.41626	0.708129	−	0.706083i	$-0.249539\pi$
$463$	30.4743	1.41626	0.708129	+	0.706083i	$0.249539\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.837253	0.0387435	0.0193717	−	0.999812i	$-0.493833\pi$
$467$	0.837253	0.0387435	0.0193717	+	0.999812i	$0.493833\pi$
$468$	0	0
$469$	−0.274917	−0.0126945
$470$	0	0
$471$	0	0
$472$	0	0
$473$	29.9786	1.37842
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	0	0
$478$	0	0
$479$	41.6843	1.90460	0.952302	−	0.305156i	$-0.0987087\pi$
$479$	41.6843	1.90460	0.952302	+	0.305156i	$0.0987087\pi$
$480$	0	0
$481$	60.9244	2.77791
$482$	0	0
$483$	0	0
$484$	0	0
$485$	44.3112	2.01207
$486$	0	0
$487$	−3.17525	−0.143884	−0.0719421	−	0.997409i	$-0.522920\pi$
$487$	−3.17525	−0.143884	−0.0719421	+	0.997409i	$0.522920\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.5934	1.60631	0.803153	−	0.595773i	$-0.203154\pi$
$491$	35.5934	1.60631	0.803153	+	0.595773i	$0.203154\pi$
$492$	0	0
$493$	−1.62541	−0.0732050
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.24163	0.369688
$498$	0	0
$499$	−27.6495	−1.23776	−0.618881	−	0.785485i	$-0.712414\pi$
$499$	−27.6495	−1.23776	−0.618881	+	0.785485i	$0.712414\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.2011	−1.12366	−0.561830	−	0.827253i	$-0.689902\pi$
$503$	−25.2011	−1.12366	−0.561830	+	0.827253i	$0.689902\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−35.5934	−1.57765	−0.788824	−	0.614619i	$-0.789310\pi$
$509$	−35.5934	−1.57765	−0.788824	+	0.614619i	$0.789310\pi$
$510$	0	0
$511$	2.72508	0.120551
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.1819	−0.536799
$516$	0	0
$517$	16.5498	0.727861
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.88054	−0.345253	−0.172626	−	0.984987i	$-0.555225\pi$
$521$	−7.88054	−0.345253	−0.172626	+	0.984987i	$0.555225\pi$
$522$	0	0
$523$	−5.64950	−0.247036	−0.123518	−	0.992342i	$-0.539418\pi$
$523$	−5.64950	−0.247036	−0.123518	+	0.992342i	$0.539418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.51176	−0.109414
$528$	0	0
$529$	53.0000	2.30435
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−36.5457	−1.58297
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.77753	−0.205783
$540$	0	0
$541$	45.1993	1.94327	0.971636	−	0.236483i	$-0.0759947\pi$
$541$	45.1993	1.94327	0.971636	+	0.236483i	$0.0759947\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	42.1029	1.80349
$546$	0	0
$547$	−17.3746	−0.742884	−0.371442	−	0.928456i	$-0.621136\pi$
$547$	−17.3746	−0.742884	−0.371442	+	0.928456i	$0.621136\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−25.4312	−1.08341
$552$	0	0
$553$	3.72508	0.158407
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.6032	1.55093	0.775465	−	0.631391i	$-0.217516\pi$
$557$	36.6032	1.55093	0.775465	+	0.631391i	$0.217516\pi$
$558$	0	0
$559$	33.0997	1.39997
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.0860	−1.05725	−0.528624	−	0.848856i	$-0.677292\pi$
$563$	−25.0860	−1.05725	−0.528624	+	0.848856i	$0.677292\pi$
$564$	0	0
$565$	28.9244	1.21686
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.19615	0.217834	0.108917	−	0.994051i	$-0.465262\pi$
$569$	5.19615	0.217834	0.108917	+	0.994051i	$0.465262\pi$
$570$	0	0
$571$	−10.5498	−0.441497	−0.220748	−	0.975331i	$-0.570850\pi$
$571$	−10.5498	−0.441497	−0.220748	+	0.975331i	$0.570850\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−37.2679	−1.55418
$576$	0	0
$577$	15.2749	0.635903	0.317952	−	0.948107i	$-0.397005\pi$
$577$	15.2749	0.635903	0.317952	+	0.948107i	$0.397005\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.837253	0.0347351
$582$	0	0
$583$	−51.9244	−2.15049
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.9950	0.784008	0.392004	−	0.919963i	$-0.371782\pi$
$587$	18.9950	0.784008	0.392004	+	0.919963i	$0.371782\pi$
$588$	0	0
$589$	−39.2990	−1.61929
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.2898	−1.44918	−0.724590	−	0.689181i	$-0.757971\pi$
$593$	−35.2898	−1.44918	−0.724590	+	0.689181i	$0.757971\pi$
$594$	0	0
$595$	−1.27492	−0.0522665
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−37.0219	−1.51267	−0.756336	−	0.654183i	$-0.773013\pi$
$599$	−37.0219	−1.51267	−0.756336	+	0.654183i	$0.773013\pi$
$600$	0	0
$601$	11.8248	0.482342	0.241171	−	0.970483i	$-0.422469\pi$
$601$	11.8248	0.482342	0.241171	+	0.970483i	$0.422469\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	36.0120	1.46410
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.2728	0.739240
$612$	0	0
$613$	24.8248	1.00266	0.501331	−	0.865256i	$-0.332844\pi$
$613$	24.8248	1.00266	0.501331	+	0.865256i	$0.332844\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.0838	1.17087	0.585434	−	0.810720i	$-0.300924\pi$
$617$	29.0838	1.17087	0.585434	+	0.810720i	$0.300924\pi$
$618$	0	0
$619$	−24.5498	−0.986741	−0.493371	−	0.869819i	$-0.664235\pi$
$619$	−24.5498	−0.986741	−0.493371	+	0.869819i	$0.664235\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.0646	−0.643616
$624$	0	0
$625$	−28.0997	−1.12399
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.83507	−0.192787
$630$	0	0
$631$	42.5498	1.69388	0.846941	−	0.531687i	$-0.178442\pi$
$631$	42.5498	1.69388	0.846941	+	0.531687i	$0.178442\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−57.3303	−2.27508
$636$	0	0
$637$	−5.27492	−0.209000
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.0958	1.03072	0.515362	−	0.856973i	$-0.327657\pi$
$641$	26.0958	1.03072	0.515362	+	0.856973i	$0.327657\pi$
$642$	0	0
$643$	−35.6495	−1.40588	−0.702940	−	0.711250i	$-0.748130\pi$
$643$	−35.6495	−1.40588	−0.702940	+	0.711250i	$0.748130\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.7846	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	−41.6495	−1.63489
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.7007	−1.20141	−0.600706	−	0.799470i	$-0.705114\pi$
$653$	−30.7007	−1.20141	−0.600706	+	0.799470i	$0.705114\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13.4953	0.525703	0.262852	−	0.964836i	$-0.415337\pi$
$659$	13.4953	0.525703	0.262852	+	0.964836i	$0.415337\pi$
$660$	0	0
$661$	−32.9244	−1.28061	−0.640306	−	0.768120i	$-0.721192\pi$
$661$	−32.9244	−1.28061	−0.640306	+	0.768120i	$0.721192\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19.9474	−0.773525
$666$	0	0
$667$	33.8488	1.31063
$668$	0	0
$669$	0	0
$670$	0	0
$671$	53.8662	2.07948
$672$	0	0
$673$	32.0997	1.23735	0.618676	−	0.785647i	$-0.287670\pi$
$673$	32.0997	1.23735	0.618676	+	0.785647i	$0.287670\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.6652	−1.10169	−0.550846	−	0.834607i	$-0.685695\pi$
$677$	−28.6652	−1.10169	−0.550846	+	0.834607i	$0.685695\pi$
$678$	0	0
$679$	−14.5498	−0.558371
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.6296	1.01895	0.509476	−	0.860485i	$-0.329839\pi$
$683$	26.6296	1.01895	0.509476	+	0.860485i	$0.329839\pi$
$684$	0	0
$685$	0.175248	0.00669590
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−57.3303	−2.18411
$690$	0	0
$691$	33.6495	1.28009	0.640044	−	0.768338i	$-0.278916\pi$
$691$	33.6495	1.28009	0.640044	+	0.768338i	$0.278916\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.76546	0.294561
$696$	0	0
$697$	2.90033	0.109858
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−40.9046	−1.54495	−0.772473	−	0.635048i	$-0.780980\pi$
$701$	−40.9046	−1.54495	−0.772473	+	0.635048i	$0.780980\pi$
$702$	0	0
$703$	−75.6495	−2.85318
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.25370	−0.197586
$708$	0	0
$709$	−46.0997	−1.73131	−0.865655	−	0.500642i	$-0.833097\pi$
$709$	−46.0997	−1.73131	−0.865655	+	0.500642i	$0.833097\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	52.3068	1.95890
$714$	0	0
$715$	76.7492	2.87026
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.97586	−0.222862	−0.111431	−	0.993772i	$-0.535543\pi$
$719$	−5.97586	−0.222862	−0.111431	+	0.993772i	$0.535543\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−16.5983	−0.616447
$726$	0	0
$727$	20.5498	0.762151	0.381076	−	0.924544i	$-0.375554\pi$
$727$	20.5498	0.762151	0.381076	+	0.924544i	$0.375554\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.62685	−0.0971575
$732$	0	0
$733$	−37.4502	−1.38325	−0.691627	−	0.722255i	$-0.743106\pi$
$733$	−37.4502	−1.38325	−0.691627	+	0.722255i	$0.743106\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.31342	−0.0483806
$738$	0	0
$739$	15.7251	0.578457	0.289228	−	0.957260i	$-0.406601\pi$
$739$	15.7251	0.578457	0.289228	+	0.957260i	$0.406601\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.8208	−0.433664	−0.216832	−	0.976209i	$-0.569572\pi$
$743$	−11.8208	−0.433664	−0.216832	+	0.976209i	$0.569572\pi$
$744$	0	0
$745$	8.17525	0.299518
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.94027	0.143974
$750$	0	0
$751$	27.9244	1.01898	0.509488	−	0.860478i	$-0.329835\pi$
$751$	27.9244	1.01898	0.509488	+	0.860478i	$0.329835\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−26.8756	−0.978102
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.7150	0.859668	0.429834	−	0.902908i	$-0.358572\pi$
$761$	23.7150	0.859668	0.429834	+	0.902908i	$0.358572\pi$
$762$	0	0
$763$	−13.8248	−0.500490
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.9857	−1.66045
$768$	0	0
$769$	7.82475	0.282168	0.141084	−	0.989998i	$-0.454941\pi$
$769$	7.82475	0.282168	0.141084	+	0.989998i	$0.454941\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.2224	1.23089	0.615447	−	0.788178i	$-0.288976\pi$
$773$	34.2224	1.23089	0.615447	+	0.788178i	$0.288976\pi$
$774$	0	0
$775$	−25.6495	−0.921357
$776$	0	0
$777$	0	0
$778$	0	0
$779$	45.3786	1.62586
$780$	0	0
$781$	39.3746	1.40893
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−60.3758	−2.15490
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.49751	−0.337692
$792$	0	0
$793$	59.4743	2.11199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.71998	−0.167190	−0.0835952	−	0.996500i	$-0.526640\pi$
$797$	−4.71998	−0.167190	−0.0835952	+	0.996500i	$0.526640\pi$
$798$	0	0
$799$	−1.45017	−0.0513032
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.0192	0.459436
$804$	0	0
$805$	26.5498	0.935759
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−44.3687	−1.55992	−0.779960	−	0.625829i	$-0.784761\pi$
$809$	−44.3687	−1.55992	−0.779960	+	0.625829i	$0.784761\pi$
$810$	0	0
$811$	−0.350497	−0.0123076	−0.00615380	−	0.999981i	$-0.501959\pi$
$811$	−0.350497	−0.0123076	−0.00615380	+	0.999981i	$0.501959\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55.6558	−1.94954
$816$	0	0
$817$	−41.0997	−1.43790
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.6794	0.756617	0.378308	−	0.925680i	$-0.376506\pi$
$821$	21.6794	0.756617	0.378308	+	0.925680i	$0.376506\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.91613	0.344818	0.172409	−	0.985025i	$-0.444845\pi$
$827$	9.91613	0.344818	0.172409	+	0.985025i	$0.444845\pi$
$828$	0	0
$829$	−51.6495	−1.79386	−0.896931	−	0.442171i	$-0.854208\pi$
$829$	−51.6495	−1.79386	−0.896931	+	0.442171i	$0.854208\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.418627	0.0145046
$834$	0	0
$835$	2.54983	0.0882407
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.7994	−1.44308	−0.721538	−	0.692375i	$-0.756564\pi$
$839$	−41.7994	−1.44308	−0.721538	+	0.692375i	$0.756564\pi$
$840$	0	0
$841$	−13.9244	−0.480152
$842$	0	0
$843$	0	0
$844$	0	0
$845$	45.1484	1.55315
$846$	0	0
$847$	−11.8248	−0.406303
$848$	0	0
$849$	0	0
$850$	0	0
$851$	100.689	3.45158
$852$	0	0
$853$	47.0997	1.61266	0.806331	−	0.591465i	$-0.201450\pi$
$853$	47.0997	1.61266	0.806331	+	0.591465i	$0.201450\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.3616	−0.968814	−0.484407	−	0.874843i	$-0.660965\pi$
$857$	−28.3616	−0.968814	−0.484407	+	0.874843i	$0.660965\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.7923	−0.877980	−0.438990	−	0.898492i	$-0.644664\pi$
$863$	−25.7923	−0.877980	−0.438990	+	0.898492i	$0.644664\pi$
$864$	0	0
$865$	−22.3746	−0.760759
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.7967	0.603711
$870$	0	0
$871$	−1.45017	−0.0491370
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.20822	0.0746515
$876$	0	0
$877$	−13.9003	−0.469381	−0.234690	−	0.972070i	$-0.575408\pi$
$877$	−13.9003	−0.469381	−0.234690	+	0.972070i	$0.575408\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8.83289	0.297587	0.148794	−	0.988868i	$-0.452461\pi$
$881$	8.83289	0.297587	0.148794	+	0.988868i	$0.452461\pi$
$882$	0	0
$883$	1.37459	0.0462585	0.0231293	−	0.999732i	$-0.492637\pi$
$883$	1.37459	0.0462585	0.0231293	+	0.999732i	$0.492637\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.9139	−1.77667	−0.888337	−	0.459192i	$-0.848139\pi$
$887$	−52.9139	−1.77667	−0.888337	+	0.459192i	$0.848139\pi$
$888$	0	0
$889$	18.8248	0.631362
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−22.6893	−0.759268
$894$	0	0
$895$	−31.6495	−1.05793
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.2964	0.776977
$900$	0	0
$901$	4.54983	0.151577
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.2728	0.607410
$906$	0	0
$907$	6.82475	0.226612	0.113306	−	0.993560i	$-0.463856\pi$
$907$	6.82475	0.226612	0.113306	+	0.993560i	$0.463856\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.4980	1.24237	0.621183	−	0.783666i	$-0.286652\pi$
$911$	37.4980	1.24237	0.621183	+	0.783666i	$0.286652\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.62685	0.0867462
$918$	0	0
$919$	−20.8248	−0.686945	−0.343473	−	0.939163i	$-0.611603\pi$
$919$	−20.8248	−0.686945	−0.343473	+	0.939163i	$0.611603\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	43.4739	1.43096
$924$	0	0
$925$	−49.3746	−1.62343
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.57701	0.248593	0.124297	−	0.992245i	$-0.460333\pi$
$929$	7.57701	0.248593	0.124297	+	0.992245i	$0.460333\pi$
$930$	0	0
$931$	6.54983	0.214662
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.09095	−0.199195
$936$	0	0
$937$	2.72508	0.0890246	0.0445123	−	0.999009i	$-0.485827\pi$
$937$	2.72508	0.0890246	0.0445123	+	0.999009i	$0.485827\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	37.0794	1.20875	0.604377	−	0.796698i	$-0.293422\pi$
$941$	37.0794	1.20875	0.604377	+	0.796698i	$0.293422\pi$
$942$	0	0
$943$	−60.3987	−1.96685
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.1961	−1.43618	−0.718090	−	0.695951i	$-0.754983\pi$
$947$	−44.1961	−1.43618	−0.718090	+	0.695951i	$0.754983\pi$
$948$	0	0
$949$	14.3746	0.466619
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.0603	0.779388	0.389694	−	0.920944i	$-0.372581\pi$
$953$	24.0603	0.779388	0.389694	+	0.920944i	$0.372581\pi$
$954$	0	0
$955$	−22.5498	−0.729696
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.0575438	−0.00185819
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−37.6865	−1.21317
$966$	0	0
$967$	42.5498	1.36831	0.684155	−	0.729336i	$-0.260171\pi$
$967$	42.5498	1.36831	0.684155	+	0.729336i	$0.260171\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.3205	−0.555842	−0.277921	−	0.960604i	$-0.589645\pi$
$971$	−17.3205	−0.555842	−0.277921	+	0.960604i	$0.589645\pi$
$972$	0	0
$973$	−2.54983	−0.0817439
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.92820	0.221653	0.110826	−	0.993840i	$-0.464650\pi$
$977$	6.92820	0.221653	0.110826	+	0.993840i	$0.464650\pi$
$978$	0	0
$979$	−76.7492	−2.45291
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0.952341	0.0303750	0.0151875	−	0.999885i	$-0.495165\pi$
$983$	0.952341	0.0303750	0.0151875	+	0.999885i	$0.495165\pi$
$984$	0	0
$985$	−10.7251	−0.341730
$986$	0	0
$987$	0	0
$988$	0	0
$989$	54.7035	1.73947
$990$	0	0
$991$	−13.9244	−0.442324	−0.221162	−	0.975237i	$-0.570985\pi$
$991$	−13.9244	−0.442324	−0.221162	+	0.975237i	$0.570985\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.8038	1.07165
$996$	0	0
$997$	−9.07558	−0.287426	−0.143713	−	0.989619i	$-0.545904\pi$
$997$	−9.07558	−0.287426	−0.143713	+	0.989619i	$0.545904\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.a.k.1.4	yes	4
3.2	odd	2	inner	2268.2.a.k.1.1	✓	4
4.3	odd	2		9072.2.a.ch.1.4		4
9.2	odd	6		2268.2.j.r.757.4		8
9.4	even	3		2268.2.j.r.1513.1		8
9.5	odd	6		2268.2.j.r.1513.4		8
9.7	even	3		2268.2.j.r.757.1		8
12.11	even	2		9072.2.a.ch.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.a.k.1.1	✓	4	3.2	odd	2	inner
2268.2.a.k.1.4	yes	4	1.1	even	1	trivial
2268.2.j.r.757.1		8	9.7	even	3
2268.2.j.r.757.4		8	9.2	odd	6
2268.2.j.r.1513.1		8	9.4	even	3
2268.2.j.r.1513.4		8	9.5	odd	6
9072.2.a.ch.1.1		4	12.11	even	2
9072.2.a.ch.1.4		4	4.3	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04547 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.04547 q^{5} -1.00000 q^{7} -4.77753 q^{11} -5.27492 q^{13} +0.418627 q^{17} +6.54983 q^{19} -8.71780 q^{23} +4.27492 q^{25} -3.88273 q^{29} -6.00000 q^{31} -3.04547 q^{35} -11.5498 q^{37} +6.92820 q^{41} -6.27492 q^{43} -3.46410 q^{47} +1.00000 q^{49} +10.8685 q^{53} -14.5498 q^{55} +8.71780 q^{59} -11.2749 q^{61} -16.0646 q^{65} +0.274917 q^{67} -8.24163 q^{71} -2.72508 q^{73} +4.77753 q^{77} -3.72508 q^{79} -0.837253 q^{83} +1.27492 q^{85} +16.0646 q^{89} +5.27492 q^{91} +19.9474 q^{95} +14.5498 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} - 6 q^{13} - 4 q^{19} + 2 q^{25} - 24 q^{31} - 16 q^{37} - 10 q^{43} + 4 q^{49} - 28 q^{55} - 30 q^{61} - 14 q^{67} - 26 q^{73} - 30 q^{79} - 10 q^{85} + 6 q^{91} + 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 6 * q^13 - 4 * q^19 + 2 * q^25 - 24 * q^31 - 16 * q^37 - 10 * q^43 + 4 * q^49 - 28 * q^55 - 30 * q^61 - 14 * q^67 - 26 * q^73 - 30 * q^79 - 10 * q^85 + 6 * q^91 + 28 * q^97

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