LMFDB - Embedded newform 3744.2.a.bb.1.1

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:	$1$
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			1.1
Root			$-1.17557$ 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)

$f(q)$	$=$	$q-3.23607 q^{5} -3.80423 q^{7} +O(q^{10})$	$q-3.23607 q^{5} -3.80423 q^{7} +1.45309 q^{11} +1.00000 q^{13} +2.47214 q^{17} +8.50651 q^{19} +5.47214 q^{25} -4.00000 q^{29} -3.80423 q^{31} +12.3107 q^{35} +6.94427 q^{37} -0.763932 q^{41} -4.70228 q^{43} +10.8576 q^{47} +7.47214 q^{49} -12.9443 q^{53} -4.70228 q^{55} -10.8576 q^{59} -4.47214 q^{61} -3.23607 q^{65} +6.71040 q^{67} +1.45309 q^{71} +6.00000 q^{73} -5.52786 q^{77} +9.40456 q^{79} -13.7638 q^{83} -8.00000 q^{85} -2.29180 q^{89} -3.80423 q^{91} -27.5276 q^{95} -6.94427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−3.80423	−1.43786	−0.718931	−	0.695081i	$-0.755368\pi$
$7$	−3.80423	−1.43786	−0.718931	+	0.695081i	$0.755368\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.45309	0.438122	0.219061	−	0.975711i	$-0.429701\pi$
$11$	1.45309	0.438122	0.219061	+	0.975711i	$0.429701\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	8.50651	1.95153	0.975763	−	0.218828i	$-0.0702234\pi$
$19$	8.50651	1.95153	0.975763	+	0.218828i	$0.0702234\pi$
$20$	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
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−3.80423	−0.683259	−0.341630	−	0.939835i	$-0.610979\pi$
$31$	−3.80423	−0.683259	−0.341630	+	0.939835i	$0.610979\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.3107	2.08089
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.763932	−0.119306	−0.0596531	−	0.998219i	$-0.518999\pi$
$41$	−0.763932	−0.119306	−0.0596531	+	0.998219i	$0.518999\pi$
$42$	0	0
$43$	−4.70228	−0.717091	−0.358546	−	0.933512i	$-0.616727\pi$
$43$	−4.70228	−0.717091	−0.358546	+	0.933512i	$0.616727\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8576	1.58375	0.791875	−	0.610683i	$-0.209105\pi$
$47$	10.8576	1.58375	0.791875	+	0.610683i	$0.209105\pi$
$48$	0	0
$49$	7.47214	1.06745
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.9443	−1.77803	−0.889016	−	0.457876i	$-0.848610\pi$
$53$	−12.9443	−1.77803	−0.889016	+	0.457876i	$0.848610\pi$
$54$	0	0
$55$	−4.70228	−0.634056
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.8576	−1.41355	−0.706773	−	0.707441i	$-0.749850\pi$
$59$	−10.8576	−1.41355	−0.706773	+	0.707441i	$0.749850\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.23607	−0.401385
$66$	0	0
$67$	6.71040	0.819805	0.409903	−	0.912129i	$-0.365563\pi$
$67$	6.71040	0.819805	0.409903	+	0.912129i	$0.365563\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.45309	0.172449	0.0862247	−	0.996276i	$-0.472520\pi$
$71$	1.45309	0.172449	0.0862247	+	0.996276i	$0.472520\pi$
$72$	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
$77$	−5.52786	−0.629959
$78$	0	0
$79$	9.40456	1.05810	0.529048	−	0.848592i	$-0.322549\pi$
$79$	9.40456	1.05810	0.529048	+	0.848592i	$0.322549\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.7638	−1.51078	−0.755388	−	0.655278i	$-0.772551\pi$
$83$	−13.7638	−1.51078	−0.755388	+	0.655278i	$0.772551\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.29180	−0.242930	−0.121465	−	0.992596i	$-0.538759\pi$
$89$	−2.29180	−0.242930	−0.121465	+	0.992596i	$0.538759\pi$
$90$	0	0
$91$	−3.80423	−0.398791
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−27.5276	−2.82428
$96$	0	0
$97$	−6.94427	−0.705084	−0.352542	−	0.935796i	$-0.614683\pi$
$97$	−6.94427	−0.705084	−0.352542	+	0.935796i	$0.614683\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.4721	−1.44003	−0.720016	−	0.693958i	$-0.755865\pi$
$101$	−14.4721	−1.44003	−0.720016	+	0.693958i	$0.755865\pi$
$102$	0	0
$103$	10.5146	1.03604	0.518018	−	0.855370i	$-0.326670\pi$
$103$	10.5146	1.03604	0.518018	+	0.855370i	$0.326670\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.40456	0.909174	0.454587	−	0.890702i	$-0.349787\pi$
$107$	9.40456	0.909174	0.454587	+	0.890702i	$0.349787\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.40456	−0.862115
$120$	0	0
$121$	−8.88854	−0.808049
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−7.60845	−0.675141	−0.337570	−	0.941300i	$-0.609605\pi$
$127$	−7.60845	−0.675141	−0.337570	+	0.941300i	$0.609605\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.2169	−1.32951	−0.664754	−	0.747063i	$-0.731463\pi$
$131$	−15.2169	−1.32951	−0.664754	+	0.747063i	$0.731463\pi$
$132$	0	0
$133$	−32.3607	−2.80603
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.7082	−1.17117	−0.585585	−	0.810611i	$-0.699135\pi$
$137$	−13.7082	−1.17117	−0.585585	+	0.810611i	$0.699135\pi$
$138$	0	0
$139$	−12.3107	−1.04418	−0.522091	−	0.852890i	$-0.674848\pi$
$139$	−12.3107	−1.04418	−0.522091	+	0.852890i	$0.674848\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.45309	0.121513
$144$	0	0
$145$	12.9443	1.07496
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.7639	−1.04566	−0.522831	−	0.852436i	$-0.675124\pi$
$149$	−12.7639	−1.04566	−0.522831	+	0.852436i	$0.675124\pi$
$150$	0	0
$151$	−2.00811	−0.163418	−0.0817090	−	0.996656i	$-0.526038\pi$
$151$	−2.00811	−0.163418	−0.0817090	+	0.996656i	$0.526038\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.3107	0.988822
$156$	0	0
$157$	−21.4164	−1.70922	−0.854608	−	0.519274i	$-0.826202\pi$
$157$	−21.4164	−1.70922	−0.854608	+	0.519274i	$0.826202\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.898056	0.0703412	0.0351706	−	0.999381i	$-0.488803\pi$
$163$	0.898056	0.0703412	0.0351706	+	0.999381i	$0.488803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.8576	0.840190	0.420095	−	0.907480i	$-0.361997\pi$
$167$	10.8576	0.840190	0.420095	+	0.907480i	$0.361997\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.52786	−0.116161	−0.0580807	−	0.998312i	$-0.518498\pi$
$173$	−1.52786	−0.116161	−0.0580807	+	0.998312i	$0.518498\pi$
$174$	0	0
$175$	−20.8172	−1.57364
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.1231	−1.35458	−0.677291	−	0.735715i	$-0.736846\pi$
$179$	−18.1231	−1.35458	−0.677291	+	0.735715i	$0.736846\pi$
$180$	0	0
$181$	−7.52786	−0.559542	−0.279771	−	0.960067i	$-0.590258\pi$
$181$	−7.52786	−0.559542	−0.279771	+	0.960067i	$0.590258\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−22.4721	−1.65218
$186$	0	0
$187$	3.59222	0.262689
$188$	0	0
$189$	0	0
$190$	0	0
$191$	27.5276	1.99183	0.995915	−	0.0902956i	$-0.0287812\pi$
$191$	27.5276	1.99183	0.995915	+	0.0902956i	$0.0287812\pi$
$192$	0	0
$193$	14.9443	1.07571	0.537856	−	0.843037i	$-0.319234\pi$
$193$	14.9443	1.07571	0.537856	+	0.843037i	$0.319234\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.18034	−0.582825	−0.291413	−	0.956597i	$-0.594125\pi$
$197$	−8.18034	−0.582825	−0.291413	+	0.956597i	$0.594125\pi$
$198$	0	0
$199$	22.8254	1.61805	0.809023	−	0.587777i	$-0.199997\pi$
$199$	22.8254	1.61805	0.809023	+	0.587777i	$0.199997\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.2169	1.06802
$204$	0	0
$205$	2.47214	0.172661
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.3607	0.855006
$210$	0	0
$211$	24.6215	1.69501	0.847506	−	0.530786i	$-0.178103\pi$
$211$	24.6215	1.69501	0.847506	+	0.530786i	$0.178103\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.2169	1.03778
$216$	0	0
$217$	14.4721	0.982433
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.47214	0.166294
$222$	0	0
$223$	2.00811	0.134473	0.0672366	−	0.997737i	$-0.478582\pi$
$223$	2.00811	0.134473	0.0672366	+	0.997737i	$0.478582\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.5762	1.29932	0.649658	−	0.760227i	$-0.274912\pi$
$227$	19.5762	1.29932	0.649658	+	0.760227i	$0.274912\pi$
$228$	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
$233$	−19.4164	−1.27201	−0.636006	−	0.771684i	$-0.719414\pi$
$233$	−19.4164	−1.27201	−0.636006	+	0.771684i	$0.719414\pi$
$234$	0	0
$235$	−35.1361	−2.29203
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.5762	−1.26628	−0.633138	−	0.774039i	$-0.718234\pi$
$239$	−19.5762	−1.26628	−0.633138	+	0.774039i	$0.718234\pi$
$240$	0	0
$241$	−27.8885	−1.79646	−0.898230	−	0.439527i	$-0.855146\pi$
$241$	−27.8885	−1.79646	−0.898230	+	0.439527i	$0.855146\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−24.1803	−1.54483
$246$	0	0
$247$	8.50651	0.541256
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−21.7153	−1.37066	−0.685329	−	0.728234i	$-0.740341\pi$
$251$	−21.7153	−1.37066	−0.685329	+	0.728234i	$0.740341\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.4721	−1.40177	−0.700887	−	0.713273i	$-0.747212\pi$
$257$	−22.4721	−1.40177	−0.700887	+	0.713273i	$0.747212\pi$
$258$	0	0
$259$	−26.4176	−1.64151
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.90617	0.179202	0.0896011	−	0.995978i	$-0.471441\pi$
$263$	2.90617	0.179202	0.0896011	+	0.995978i	$0.471441\pi$
$264$	0	0
$265$	41.8885	2.57319
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	−3.80423	−0.231090	−0.115545	−	0.993302i	$-0.536861\pi$
$271$	−3.80423	−0.231090	−0.115545	+	0.993302i	$0.536861\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.95148	0.479492
$276$	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
$281$	−13.7082	−0.817763	−0.408881	−	0.912588i	$-0.634081\pi$
$281$	−13.7082	−0.817763	−0.408881	+	0.912588i	$0.634081\pi$
$282$	0	0
$283$	22.8254	1.35683	0.678413	−	0.734680i	$-0.262668\pi$
$283$	22.8254	1.35683	0.678413	+	0.734680i	$0.262668\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.90617	0.171546
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.1246	1.23411	0.617056	−	0.786919i	$-0.288325\pi$
$293$	21.1246	1.23411	0.617056	+	0.786919i	$0.288325\pi$
$294$	0	0
$295$	35.1361	2.04570
$296$	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
$305$	14.4721	0.828672
$306$	0	0
$307$	−0.898056	−0.0512548	−0.0256274	−	0.999672i	$-0.508158\pi$
$307$	−0.898056	−0.0512548	−0.0256274	+	0.999672i	$0.508158\pi$
$308$	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
$313$	−11.8885	−0.671980	−0.335990	−	0.941866i	$-0.609071\pi$
$313$	−11.8885	−0.671980	−0.335990	+	0.941866i	$0.609071\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.5967	1.54999	0.774994	−	0.631969i	$-0.217753\pi$
$317$	27.5967	1.54999	0.774994	+	0.631969i	$0.217753\pi$
$318$	0	0
$319$	−5.81234	−0.325429
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.0292	1.17010
$324$	0	0
$325$	5.47214	0.303539
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−41.3050	−2.27722
$330$	0	0
$331$	−8.50651	−0.467560	−0.233780	−	0.972290i	$-0.575110\pi$
$331$	−8.50651	−0.467560	−0.233780	+	0.972290i	$0.575110\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−21.7153	−1.18643
$336$	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
$341$	−5.52786	−0.299351
$342$	0	0
$343$	−1.79611	−0.0969809
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.81234	0.312023	0.156011	−	0.987755i	$-0.450136\pi$
$347$	5.81234	0.312023	0.156011	+	0.987755i	$0.450136\pi$
$348$	0	0
$349$	27.8885	1.49284	0.746420	−	0.665475i	$-0.231771\pi$
$349$	27.8885	1.49284	0.746420	+	0.665475i	$0.231771\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	23.2361	1.23673	0.618366	−	0.785891i	$-0.287795\pi$
$353$	23.2361	1.23673	0.618366	+	0.785891i	$0.287795\pi$
$354$	0	0
$355$	−4.70228	−0.249571
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.9807	−1.52954	−0.764772	−	0.644301i	$-0.777148\pi$
$359$	−28.9807	−1.52954	−0.764772	+	0.644301i	$0.777148\pi$
$360$	0	0
$361$	53.3607	2.80846
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.4164	−1.01630
$366$	0	0
$367$	−21.7153	−1.13353	−0.566765	−	0.823880i	$-0.691805\pi$
$367$	−21.7153	−1.13353	−0.566765	+	0.823880i	$0.691805\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	49.2429	2.55657
$372$	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
$379$	−0.898056	−0.0461300	−0.0230650	−	0.999734i	$-0.507342\pi$
$379$	−0.898056	−0.0461300	−0.0230650	+	0.999734i	$0.507342\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.1684	1.18385	0.591925	−	0.805993i	$-0.298368\pi$
$383$	23.1684	1.18385	0.591925	+	0.805993i	$0.298368\pi$
$384$	0	0
$385$	17.8885	0.911685
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.52786	−0.280274	−0.140137	−	0.990132i	$-0.544754\pi$
$389$	−5.52786	−0.280274	−0.140137	+	0.990132i	$0.544754\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−30.4338	−1.53129
$396$	0	0
$397$	−28.8328	−1.44708	−0.723539	−	0.690284i	$-0.757486\pi$
$397$	−28.8328	−1.44708	−0.723539	+	0.690284i	$0.757486\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.2918	−0.913449	−0.456724	−	0.889608i	$-0.650977\pi$
$401$	−18.2918	−0.913449	−0.456724	+	0.889608i	$0.650977\pi$
$402$	0	0
$403$	−3.80423	−0.189502
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.0906	0.500173
$408$	0	0
$409$	23.8885	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$409$	23.8885	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	41.3050	2.03248
$414$	0	0
$415$	44.5407	2.18641
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−34.0260	−1.66228	−0.831140	−	0.556063i	$-0.812311\pi$
$419$	−34.0260	−1.66228	−0.831140	+	0.556063i	$0.812311\pi$
$420$	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
$425$	13.5279	0.656198
$426$	0	0
$427$	17.0130	0.823318
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.9807	1.39595	0.697976	−	0.716121i	$-0.254084\pi$
$431$	28.9807	1.39595	0.697976	+	0.716121i	$0.254084\pi$
$432$	0	0
$433$	17.4164	0.836979	0.418490	−	0.908222i	$-0.362560\pi$
$433$	17.4164	0.836979	0.418490	+	0.908222i	$0.362560\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−25.7315	−1.22810	−0.614049	−	0.789268i	$-0.710460\pi$
$439$	−25.7315	−1.22810	−0.614049	+	0.789268i	$0.710460\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.40456	0.446824	0.223412	−	0.974724i	$-0.428280\pi$
$443$	9.40456	0.446824	0.223412	+	0.974724i	$0.428280\pi$
$444$	0	0
$445$	7.41641	0.351571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.81966	0.180261	0.0901305	−	0.995930i	$-0.471272\pi$
$449$	3.81966	0.180261	0.0901305	+	0.995930i	$0.471272\pi$
$450$	0	0
$451$	−1.11006	−0.0522706
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.3107	0.577136
$456$	0	0
$457$	9.05573	0.423609	0.211805	−	0.977312i	$-0.432066\pi$
$457$	9.05573	0.423609	0.211805	+	0.977312i	$0.432066\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−34.0689	−1.58675	−0.793373	−	0.608736i	$-0.791677\pi$
$461$	−34.0689	−1.58675	−0.793373	+	0.608736i	$0.791677\pi$
$462$	0	0
$463$	−19.0211	−0.883987	−0.441993	−	0.897018i	$-0.645729\pi$
$463$	−19.0211	−0.883987	−0.441993	+	0.897018i	$0.645729\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.5276	−1.27383	−0.636914	−	0.770935i	$-0.719789\pi$
$467$	−27.5276	−1.27383	−0.636914	+	0.770935i	$0.719789\pi$
$468$	0	0
$469$	−25.5279	−1.17877
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.83282	−0.314173
$474$	0	0
$475$	46.5488	2.13580
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.6700	−0.761671	−0.380836	−	0.924643i	$-0.624364\pi$
$479$	−16.6700	−0.761671	−0.380836	+	0.924643i	$0.624364\pi$
$480$	0	0
$481$	6.94427	0.316632
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22.4721	1.02041
$486$	0	0
$487$	15.0049	0.679937	0.339969	−	0.940437i	$-0.389584\pi$
$487$	15.0049	0.679937	0.339969	+	0.940437i	$0.389584\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.71851	−0.393461	−0.196730	−	0.980458i	$-0.563032\pi$
$491$	−8.71851	−0.393461	−0.196730	+	0.980458i	$0.563032\pi$
$492$	0	0
$493$	−9.88854	−0.445358
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.52786	−0.247959
$498$	0	0
$499$	−33.1280	−1.48301	−0.741506	−	0.670946i	$-0.765888\pi$
$499$	−33.1280	−1.48301	−0.741506	+	0.670946i	$0.765888\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.7153	−0.968237	−0.484119	−	0.875002i	$-0.660860\pi$
$503$	−21.7153	−0.968237	−0.484119	+	0.875002i	$0.660860\pi$
$504$	0	0
$505$	46.8328	2.08403
$506$	0	0
$507$	0	0
$508$	0	0
$509$	25.7082	1.13950	0.569748	−	0.821819i	$-0.307041\pi$
$509$	25.7082	1.13950	0.569748	+	0.821819i	$0.307041\pi$
$510$	0	0
$511$	−22.8254	−1.00973
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−34.0260	−1.49937
$516$	0	0
$517$	15.7771	0.693876
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.8885	1.65993	0.829964	−	0.557818i	$-0.188361\pi$
$521$	37.8885	1.65993	0.829964	+	0.557818i	$0.188361\pi$
$522$	0	0
$523$	−12.3107	−0.538311	−0.269155	−	0.963097i	$-0.586745\pi$
$523$	−12.3107	−0.538311	−0.269155	+	0.963097i	$0.586745\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.40456	−0.409669
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.763932	−0.0330896
$534$	0	0
$535$	−30.4338	−1.31577
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.8576	0.467672
$540$	0	0
$541$	−6.94427	−0.298558	−0.149279	−	0.988795i	$-0.547695\pi$
$541$	−6.94427	−0.298558	−0.149279	+	0.988795i	$0.547695\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.52786	−0.408129
$546$	0	0
$547$	12.3107	0.526369	0.263184	−	0.964746i	$-0.415227\pi$
$547$	12.3107	0.526369	0.263184	+	0.964746i	$0.415227\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.0260	−1.44956
$552$	0	0
$553$	−35.7771	−1.52140
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.5967	−1.16931	−0.584656	−	0.811281i	$-0.698770\pi$
$557$	−27.5967	−1.16931	−0.584656	+	0.811281i	$0.698770\pi$
$558$	0	0
$559$	−4.70228	−0.198885
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.3107	0.518836	0.259418	−	0.965765i	$-0.416469\pi$
$563$	12.3107	0.518836	0.259418	+	0.965765i	$0.416469\pi$
$564$	0	0
$565$	12.9443	0.544570
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.4721	−0.606704	−0.303352	−	0.952879i	$-0.598106\pi$
$569$	−14.4721	−0.606704	−0.303352	+	0.952879i	$0.598106\pi$
$570$	0	0
$571$	−4.01623	−0.168074	−0.0840370	−	0.996463i	$-0.526781\pi$
$571$	−4.01623	−0.168074	−0.0840370	+	0.996463i	$0.526781\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.0557	0.543517	0.271759	−	0.962365i	$-0.412395\pi$
$577$	13.0557	0.543517	0.271759	+	0.962365i	$0.412395\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	52.3607	2.17229
$582$	0	0
$583$	−18.8091	−0.778994
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.9807	−1.19616	−0.598081	−	0.801435i	$-0.704070\pi$
$587$	−28.9807	−1.19616	−0.598081	+	0.801435i	$0.704070\pi$
$588$	0	0
$589$	−32.3607	−1.33340
$590$	0	0
$591$	0	0
$592$	0	0
$593$	45.7082	1.87701	0.938505	−	0.345264i	$-0.112211\pi$
$593$	45.7082	1.87701	0.938505	+	0.345264i	$0.112211\pi$
$594$	0	0
$595$	30.4338	1.24766
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.5276	−1.12475	−0.562374	−	0.826883i	$-0.690112\pi$
$599$	−27.5276	−1.12475	−0.562374	+	0.826883i	$0.690112\pi$
$600$	0	0
$601$	−27.8885	−1.13760	−0.568799	−	0.822477i	$-0.692592\pi$
$601$	−27.8885	−1.13760	−0.568799	+	0.822477i	$0.692592\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	28.7639	1.16942
$606$	0	0
$607$	−36.9322	−1.49903	−0.749516	−	0.661986i	$-0.769714\pi$
$607$	−36.9322	−1.49903	−0.749516	+	0.661986i	$0.769714\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.8576	0.439253
$612$	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
$617$	22.0689	0.888460	0.444230	−	0.895913i	$-0.353477\pi$
$617$	22.0689	0.888460	0.444230	+	0.895913i	$0.353477\pi$
$618$	0	0
$619$	29.5358	1.18714	0.593571	−	0.804782i	$-0.297718\pi$
$619$	29.5358	1.18714	0.593571	+	0.804782i	$0.297718\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.71851	0.349300
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.1672	0.684500
$630$	0	0
$631$	−45.4387	−1.80889	−0.904443	−	0.426594i	$-0.859713\pi$
$631$	−45.4387	−1.80889	−0.904443	+	0.426594i	$0.859713\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.6215	0.977073
$636$	0	0
$637$	7.47214	0.296057
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.3607	0.804199	0.402099	−	0.915596i	$-0.368281\pi$
$641$	20.3607	0.804199	0.402099	+	0.915596i	$0.368281\pi$
$642$	0	0
$643$	40.7364	1.60649	0.803244	−	0.595650i	$-0.203106\pi$
$643$	40.7364	1.60649	0.803244	+	0.595650i	$0.203106\pi$
$644$	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
$649$	−15.7771	−0.619305
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.0557	−0.589176	−0.294588	−	0.955624i	$-0.595182\pi$
$653$	−15.0557	−0.589176	−0.294588	+	0.955624i	$0.595182\pi$
$654$	0	0
$655$	49.2429	1.92408
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.6215	−0.959116	−0.479558	−	0.877510i	$-0.659203\pi$
$659$	−24.6215	−0.959116	−0.479558	+	0.877510i	$0.659203\pi$
$660$	0	0
$661$	−25.7771	−1.00261	−0.501306	−	0.865270i	$-0.667147\pi$
$661$	−25.7771	−1.00261	−0.501306	+	0.865270i	$0.667147\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	104.721	4.06092
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.49839	−0.250868
$672$	0	0
$673$	−9.41641	−0.362976	−0.181488	−	0.983393i	$-0.558091\pi$
$673$	−9.41641	−0.362976	−0.181488	+	0.983393i	$0.558091\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.7771	−1.68249	−0.841245	−	0.540654i	$-0.818177\pi$
$677$	−43.7771	−1.68249	−0.841245	+	0.540654i	$0.818177\pi$
$678$	0	0
$679$	26.4176	1.01381
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.8576	0.415456	0.207728	−	0.978187i	$-0.433393\pi$
$683$	10.8576	0.415456	0.207728	+	0.978187i	$0.433393\pi$
$684$	0	0
$685$	44.3607	1.69493
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.9443	−0.493137
$690$	0	0
$691$	25.5195	0.970808	0.485404	−	0.874290i	$-0.338672\pi$
$691$	25.5195	0.970808	0.485404	+	0.874290i	$0.338672\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	39.8384	1.51116
$696$	0	0
$697$	−1.88854	−0.0715337
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.47214	0.0933713	0.0466856	−	0.998910i	$-0.485134\pi$
$701$	2.47214	0.0933713	0.0466856	+	0.998910i	$0.485134\pi$
$702$	0	0
$703$	59.0715	2.22792
$704$	0	0
$705$	0	0
$706$	0	0
$707$	55.0553	2.07057
$708$	0	0
$709$	−45.7771	−1.71919	−0.859597	−	0.510972i	$-0.829286\pi$
$709$	−45.7771	−1.71919	−0.859597	+	0.510972i	$0.829286\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−4.70228	−0.175855
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.6215	0.918226	0.459113	−	0.888378i	$-0.348167\pi$
$719$	24.6215	0.918226	0.459113	+	0.888378i	$0.348167\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.8885	−0.812920
$726$	0	0
$727$	−32.2299	−1.19534	−0.597671	−	0.801742i	$-0.703907\pi$
$727$	−32.2299	−1.19534	−0.597671	+	0.801742i	$0.703907\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.6247	−0.429954
$732$	0	0
$733$	−5.05573	−0.186738	−0.0933688	−	0.995632i	$-0.529764\pi$
$733$	−5.05573	−0.186738	−0.0933688	+	0.995632i	$0.529764\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.75078	0.359174
$738$	0	0
$739$	−31.3319	−1.15256	−0.576281	−	0.817252i	$-0.695496\pi$
$739$	−31.3319	−1.15256	−0.576281	+	0.817252i	$0.695496\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	50.6960	1.85986	0.929928	−	0.367741i	$-0.119869\pi$
$743$	50.6960	1.85986	0.929928	+	0.367741i	$0.119869\pi$
$744$	0	0
$745$	41.3050	1.51330
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−35.7771	−1.30727
$750$	0	0
$751$	−35.1361	−1.28213	−0.641067	−	0.767485i	$-0.721508\pi$
$751$	−35.1361	−1.28213	−0.641067	+	0.767485i	$0.721508\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.49839	0.236501
$756$	0	0
$757$	−45.4164	−1.65069	−0.825344	−	0.564631i	$-0.809019\pi$
$757$	−45.4164	−1.65069	−0.825344	+	0.564631i	$0.809019\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.6525	−0.386152	−0.193076	−	0.981184i	$-0.561846\pi$
$761$	−10.6525	−0.386152	−0.193076	+	0.981184i	$0.561846\pi$
$762$	0	0
$763$	−11.2007	−0.405492
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.8576	−0.392047
$768$	0	0
$769$	16.8328	0.607007	0.303503	−	0.952830i	$-0.401844\pi$
$769$	16.8328	0.607007	0.303503	+	0.952830i	$0.401844\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.2918	0.801780	0.400890	−	0.916126i	$-0.368701\pi$
$773$	22.2918	0.801780	0.400890	+	0.916126i	$0.368701\pi$
$774$	0	0
$775$	−20.8172	−0.747777
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.49839	−0.232829
$780$	0	0
$781$	2.11146	0.0755538
$782$	0	0
$783$	0	0
$784$	0	0
$785$	69.3050	2.47360
$786$	0	0
$787$	−44.3287	−1.58015	−0.790073	−	0.613013i	$-0.789957\pi$
$787$	−44.3287	−1.58015	−0.790073	+	0.613013i	$0.789957\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.2169	0.541051
$792$	0	0
$793$	−4.47214	−0.158810
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.4164	0.546077	0.273039	−	0.962003i	$-0.411971\pi$
$797$	15.4164	0.546077	0.273039	+	0.962003i	$0.411971\pi$
$798$	0	0
$799$	26.8416	0.949587
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.71851	0.307670
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	−4.49028	−0.157675	−0.0788375	−	0.996887i	$-0.525121\pi$
$811$	−4.49028	−0.157675	−0.0788375	+	0.996887i	$0.525121\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.90617	−0.101799
$816$	0	0
$817$	−40.0000	−1.39942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.5967	−0.404729	−0.202365	−	0.979310i	$-0.564863\pi$
$821$	−11.5967	−0.404729	−0.202365	+	0.979310i	$0.564863\pi$
$822$	0	0
$823$	14.1068	0.491734	0.245867	−	0.969304i	$-0.420927\pi$
$823$	14.1068	0.491734	0.245867	+	0.969304i	$0.420927\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.4823	0.781787	0.390894	−	0.920436i	$-0.372166\pi$
$827$	22.4823	0.781787	0.390894	+	0.920436i	$0.372166\pi$
$828$	0	0
$829$	−8.47214	−0.294249	−0.147125	−	0.989118i	$-0.547002\pi$
$829$	−8.47214	−0.294249	−0.147125	+	0.989118i	$0.547002\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.4721	0.640022
$834$	0	0
$835$	−35.1361	−1.21593
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.3560	0.599197	0.299599	−	0.954065i	$-0.403147\pi$
$839$	17.3560	0.599197	0.299599	+	0.954065i	$0.403147\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.23607	−0.111324
$846$	0	0
$847$	33.8140	1.16186
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	32.8328	1.12417	0.562087	−	0.827078i	$-0.309999\pi$
$853$	32.8328	1.12417	0.562087	+	0.827078i	$0.309999\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.3050	0.454488	0.227244	−	0.973838i	$-0.427028\pi$
$857$	13.3050	0.454488	0.227244	+	0.973838i	$0.427028\pi$
$858$	0	0
$859$	9.40456	0.320880	0.160440	−	0.987046i	$-0.448709\pi$
$859$	9.40456	0.320880	0.160440	+	0.987046i	$0.448709\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.1684	0.788661	0.394330	−	0.918969i	$-0.370977\pi$
$863$	23.1684	0.788661	0.394330	+	0.918969i	$0.370977\pi$
$864$	0	0
$865$	4.94427	0.168110
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.6656	0.463575
$870$	0	0
$871$	6.71040	0.227373
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.81234	0.196493
$876$	0	0
$877$	13.0557	0.440861	0.220430	−	0.975403i	$-0.429254\pi$
$877$	13.0557	0.440861	0.220430	+	0.975403i	$0.429254\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40.3607	−1.35979	−0.679893	−	0.733311i	$-0.737974\pi$
$881$	−40.3607	−1.35979	−0.679893	+	0.733311i	$0.737974\pi$
$882$	0	0
$883$	−43.8546	−1.47582	−0.737912	−	0.674896i	$-0.764188\pi$
$883$	−43.8546	−1.47582	−0.737912	+	0.674896i	$0.764188\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.9030	−0.533969	−0.266985	−	0.963701i	$-0.586027\pi$
$887$	−15.9030	−0.533969	−0.266985	+	0.963701i	$0.586027\pi$
$888$	0	0
$889$	28.9443	0.970760
$890$	0	0
$891$	0	0
$892$	0	0
$893$	92.3607	3.09073
$894$	0	0
$895$	58.6475	1.96037
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.2169	0.507512
$900$	0	0
$901$	−32.0000	−1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.3607	0.809776
$906$	0	0
$907$	7.60845	0.252635	0.126317	−	0.991990i	$-0.459684\pi$
$907$	7.60845	0.252635	0.126317	+	0.991990i	$0.459684\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.71851	0.288857	0.144429	−	0.989515i	$-0.453866\pi$
$911$	8.71851	0.288857	0.144429	+	0.989515i	$0.453866\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	57.8885	1.91165
$918$	0	0
$919$	9.40456	0.310228	0.155114	−	0.987897i	$-0.450426\pi$
$919$	9.40456	0.310228	0.155114	+	0.987897i	$0.450426\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.45309	0.0478289
$924$	0	0
$925$	38.0000	1.24943
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.1803	1.18704	0.593519	−	0.804820i	$-0.297738\pi$
$929$	36.1803	1.18704	0.593519	+	0.804820i	$0.297738\pi$
$930$	0	0
$931$	63.5618	2.08315
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.6247	−0.380168
$936$	0	0
$937$	36.4721	1.19149	0.595746	−	0.803173i	$-0.296856\pi$
$937$	36.4721	1.19149	0.595746	+	0.803173i	$0.296856\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.70820	−0.0556859	−0.0278429	−	0.999612i	$-0.508864\pi$
$941$	−1.70820	−0.0556859	−0.0278429	+	0.999612i	$0.508864\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50.6960	−1.64740	−0.823700	−	0.567026i	$-0.808094\pi$
$947$	−50.6960	−1.64740	−0.823700	+	0.567026i	$0.808094\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	12.5836	0.407623	0.203811	−	0.979010i	$-0.434667\pi$
$953$	12.5836	0.407623	0.203811	+	0.979010i	$0.434667\pi$
$954$	0	0
$955$	−89.0813	−2.88260
$956$	0	0
$957$	0	0
$958$	0	0
$959$	52.1491	1.68398
$960$	0	0
$961$	−16.5279	−0.533157
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−48.3607	−1.55679
$966$	0	0
$967$	32.4419	1.04326	0.521631	−	0.853171i	$-0.325324\pi$
$967$	32.4419	1.04326	0.521631	+	0.853171i	$0.325324\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18.8091	0.603614	0.301807	−	0.953369i	$-0.402410\pi$
$971$	18.8091	0.603614	0.301807	+	0.953369i	$0.402410\pi$
$972$	0	0
$973$	46.8328	1.50139
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.87539	0.219963	0.109982	−	0.993934i	$-0.464921\pi$
$977$	6.87539	0.219963	0.109982	+	0.993934i	$0.464921\pi$
$978$	0	0
$979$	−3.33017	−0.106433
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.0778	−0.417116	−0.208558	−	0.978010i	$-0.566877\pi$
$983$	−13.0778	−0.417116	−0.208558	+	0.978010i	$0.566877\pi$
$984$	0	0
$985$	26.4721	0.843472
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	31.1199	0.988555	0.494278	−	0.869304i	$-0.335433\pi$
$991$	31.1199	0.988555	0.494278	+	0.869304i	$0.335433\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−73.8644	−2.34166
$996$	0	0
$997$	39.8885	1.26328	0.631641	−	0.775261i	$-0.282382\pi$
$997$	39.8885	1.26328	0.631641	+	0.775261i	$0.282382\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3744.2.a.bb.1.1	✓	4	1.1	even	1	trivial
3744.2.a.bb.1.2	yes	4	4.3	odd	2	inner
3744.2.a.bf.1.3	yes	4	3.2	odd	2
3744.2.a.bf.1.4	yes	4	12.11	even	2
7488.2.a.cz.1.1		4	24.5	odd	2
7488.2.a.cz.1.2		4	24.11	even	2
7488.2.a.dd.1.3		4	8.5	even	2
7488.2.a.dd.1.4		4	8.3	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{5} -3.80423 q^{7} +O(q^{10})\)	\(q-3.23607 q^{5} -3.80423 q^{7} +1.45309 q^{11} +1.00000 q^{13} +2.47214 q^{17} +8.50651 q^{19} +5.47214 q^{25} -4.00000 q^{29} -3.80423 q^{31} +12.3107 q^{35} +6.94427 q^{37} -0.763932 q^{41} -4.70228 q^{43} +10.8576 q^{47} +7.47214 q^{49} -12.9443 q^{53} -4.70228 q^{55} -10.8576 q^{59} -4.47214 q^{61} -3.23607 q^{65} +6.71040 q^{67} +1.45309 q^{71} +6.00000 q^{73} -5.52786 q^{77} +9.40456 q^{79} -13.7638 q^{83} -8.00000 q^{85} -2.29180 q^{89} -3.80423 q^{91} -27.5276 q^{95} -6.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 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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