LMFDB - Embedded newform 4032.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, 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("4032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.46410 q^{5} -1.00000 q^{7} +3.46410 q^{11} -2.00000 q^{13} +3.46410 q^{17} -4.00000 q^{19} +3.46410 q^{23} +7.00000 q^{25} +4.00000 q^{31} -3.46410 q^{35} -2.00000 q^{37} +10.3923 q^{41} -4.00000 q^{43} -6.92820 q^{47} +1.00000 q^{49} +6.92820 q^{53} +12.0000 q^{55} -6.92820 q^{59} +10.0000 q^{61} -6.92820 q^{65} -4.00000 q^{67} +10.3923 q^{71} +14.0000 q^{73} -3.46410 q^{77} -8.00000 q^{79} +12.0000 q^{85} -3.46410 q^{89} +2.00000 q^{91} -13.8564 q^{95} +14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^7	$ 2 q - 2 q^{7} - 4 q^{13} - 8 q^{19} + 14 q^{25} + 8 q^{31} - 4 q^{37} - 8 q^{43} + 2 q^{49} + 24 q^{55} + 20 q^{61} - 8 q^{67} + 28 q^{73} - 16 q^{79} + 24 q^{85} + 4 q^{91} + 28 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 4 * q^13 - 8 * q^19 + 14 * q^25 + 8 * q^31 - 4 * q^37 - 8 * q^43 + 2 * q^49 + 24 * q^55 + 20 * q^61 - 8 * q^67 + 28 * q^73 - 16 * q^79 + 24 * q^85 + 4 * 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
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.3923	1.62301	0.811503	−	0.584349i	$-0.198650\pi$
$41$	10.3923	1.62301	0.811503	+	0.584349i	$0.198650\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.92820	−0.859338
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−13.8564	−1.42164
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.46410	−0.344691	−0.172345	−	0.985037i	$-0.555135\pi$
$101$	−3.46410	−0.344691	−0.172345	+	0.985037i	$0.555135\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.3205	1.67444	0.837218	−	0.546869i	$-0.184180\pi$
$107$	17.3205	1.67444	0.837218	+	0.546869i	$0.184180\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.8564	−1.21064	−0.605320	−	0.795982i	$-0.706955\pi$
$131$	−13.8564	−1.21064	−0.605320	+	0.795982i	$0.706955\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.92820	−0.579365
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.92820	−0.567581	−0.283790	−	0.958886i	$-0.591592\pi$
$149$	−6.92820	−0.567581	−0.283790	+	0.958886i	$0.591592\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.8564	1.11297
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.7846	1.60836	0.804181	−	0.594385i	$-0.202604\pi$
$167$	20.7846	1.60836	0.804181	+	0.594385i	$0.202604\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.3205	−1.31685	−0.658427	−	0.752645i	$-0.728778\pi$
$173$	−17.3205	−1.31685	−0.658427	+	0.752645i	$0.728778\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.3205	−1.29460	−0.647298	−	0.762237i	$-0.724101\pi$
$179$	−17.3205	−1.29460	−0.647298	+	0.762237i	$0.724101\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.92820	−0.509372
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.2487	1.75458	0.877288	−	0.479965i	$-0.159351\pi$
$191$	24.2487	1.75458	0.877288	+	0.479965i	$0.159351\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.7846	−1.48084	−0.740421	−	0.672143i	$-0.765374\pi$
$197$	−20.7846	−1.48084	−0.740421	+	0.672143i	$0.765374\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	36.0000	2.51435
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.8564	−0.958468
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.8564	−0.944999
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.92820	0.453882	0.226941	−	0.973909i	$-0.427128\pi$
$233$	6.92820	0.453882	0.226941	+	0.973909i	$0.427128\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.3923	−0.672222	−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923	−0.672222	−0.336111	+	0.941822i	$0.609112\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.46410	0.221313
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.7846	1.31191	0.655956	−	0.754799i	$-0.272265\pi$
$251$	20.7846	1.31191	0.655956	+	0.754799i	$0.272265\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.46410	−0.216085	−0.108042	−	0.994146i	$-0.534458\pi$
$257$	−3.46410	−0.216085	−0.108042	+	0.994146i	$0.534458\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.3205	1.06803	0.534014	−	0.845476i	$-0.320683\pi$
$263$	17.3205	1.06803	0.534014	+	0.845476i	$0.320683\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.3205	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$269$	17.3205	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.2487	1.46225
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.7846	−1.23991	−0.619953	−	0.784639i	$-0.712848\pi$
$281$	−20.7846	−1.23991	−0.619953	+	0.784639i	$0.712848\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.3923	−0.613438
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.3923	0.607125	0.303562	−	0.952812i	$-0.401824\pi$
$293$	10.3923	0.607125	0.303562	+	0.952812i	$0.401824\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.92820	−0.400668
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	34.6410	1.98354
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−34.6410	−1.96431	−0.982156	−	0.188069i	$-0.939777\pi$
$311$	−34.6410	−1.96431	−0.982156	+	0.188069i	$0.939777\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.92820	−0.389127	−0.194563	−	0.980890i	$-0.562329\pi$
$317$	−6.92820	−0.389127	−0.194563	+	0.980890i	$0.562329\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.8564	−0.770991
$324$	0	0
$325$	−14.0000	−0.776580
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.92820	0.381964
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.8564	−0.757056
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.8564	0.750366
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.3205	−0.929814	−0.464907	−	0.885360i	$-0.653912\pi$
$347$	−17.3205	−0.929814	−0.464907	+	0.885360i	$0.653912\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.46410	0.184376	0.0921878	−	0.995742i	$-0.470614\pi$
$353$	3.46410	0.184376	0.0921878	+	0.995742i	$0.470614\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.2487	1.27980	0.639899	−	0.768459i	$-0.278976\pi$
$359$	24.2487	1.27980	0.639899	+	0.768459i	$0.278976\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	48.4974	2.53847
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.92820	−0.359694
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.8564	0.708029	0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564	0.708029	0.354015	+	0.935240i	$0.384816\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.8564	−0.702548	−0.351274	−	0.936273i	$-0.614251\pi$
$389$	−13.8564	−0.702548	−0.351274	+	0.936273i	$0.614251\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−27.7128	−1.39438
$396$	0	0
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.92820	0.345978	0.172989	−	0.984924i	$-0.444657\pi$
$401$	6.92820	0.345978	0.172989	+	0.984924i	$0.444657\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.92820	−0.343418
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.7846	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$419$	20.7846	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.2487	1.17624
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.46410	−0.166860	−0.0834300	−	0.996514i	$-0.526587\pi$
$431$	−3.46410	−0.166860	−0.0834300	+	0.996514i	$0.526587\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13.8564	−0.662842
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.46410	−0.164584	−0.0822922	−	0.996608i	$-0.526224\pi$
$443$	−3.46410	−0.164584	−0.0822922	+	0.996608i	$0.526224\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	41.5692	1.96177	0.980886	−	0.194581i	$-0.0623348\pi$
$449$	41.5692	1.96177	0.980886	+	0.194581i	$0.0623348\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.1769	−1.45205	−0.726027	−	0.687666i	$-0.758635\pi$
$461$	−31.1769	−1.45205	−0.726027	+	0.687666i	$0.758635\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.92820	0.320599	0.160300	−	0.987068i	$-0.448754\pi$
$467$	6.92820	0.320599	0.160300	+	0.987068i	$0.448754\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−13.8564	−0.637118
$474$	0	0
$475$	−28.0000	−1.28473
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.92820	0.316558	0.158279	−	0.987394i	$-0.449406\pi$
$479$	6.92820	0.316558	0.158279	+	0.987394i	$0.449406\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	48.4974	2.20215
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.3923	0.468998	0.234499	−	0.972116i	$-0.424655\pi$
$491$	10.3923	0.468998	0.234499	+	0.972116i	$0.424655\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.3923	−0.466159
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.46410	0.153544	0.0767718	−	0.997049i	$-0.475539\pi$
$509$	3.46410	0.153544	0.0767718	+	0.997049i	$0.475539\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.8564	0.610586
$516$	0	0
$517$	−24.0000	−1.05552
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.46410	0.151765	0.0758825	−	0.997117i	$-0.475823\pi$
$521$	3.46410	0.151765	0.0758825	+	0.997117i	$0.475823\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.8564	0.603595
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−20.7846	−0.900281
$534$	0	0
$535$	60.0000	2.59403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.46410	0.149209
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.92820	−0.296772
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.92820	0.290445	0.145223	−	0.989399i	$-0.453610\pi$
$569$	6.92820	0.290445	0.145223	+	0.989399i	$0.453610\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.2487	1.01124
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.7846	−0.857873	−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846	−0.857873	−0.428936	+	0.903335i	$0.641112\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24.2487	−0.995775	−0.497888	−	0.867242i	$-0.665891\pi$
$593$	−24.2487	−0.995775	−0.497888	+	0.867242i	$0.665891\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.0333	−1.84001	−0.920006	−	0.391905i	$-0.871816\pi$
$599$	−45.0333	−1.84001	−0.920006	+	0.391905i	$0.871816\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.46410	0.140836
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.8564	0.560570
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.7846	−0.836757	−0.418378	−	0.908273i	$-0.637401\pi$
$617$	−20.7846	−0.836757	−0.418378	+	0.908273i	$0.637401\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.46410	0.138786
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.92820	−0.276246
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27.7128	−1.09975
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−48.4974	−1.91553	−0.957767	−	0.287547i	$-0.907160\pi$
$641$	−48.4974	−1.91553	−0.957767	+	0.287547i	$0.907160\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.92820	0.272376	0.136188	−	0.990683i	$-0.456515\pi$
$647$	6.92820	0.272376	0.136188	+	0.990683i	$0.456515\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.7128	−1.08449	−0.542243	−	0.840222i	$-0.682425\pi$
$653$	−27.7128	−1.08449	−0.542243	+	0.840222i	$0.682425\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.3923	−0.404827	−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923	−0.404827	−0.202413	+	0.979300i	$0.564878\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.8564	0.537328
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34.6410	1.33730
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.2487	0.931954	0.465977	−	0.884797i	$-0.345703\pi$
$677$	24.2487	0.931954	0.465977	+	0.884797i	$0.345703\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.2487	0.927851	0.463926	−	0.885874i	$-0.346441\pi$
$683$	24.2487	0.927851	0.463926	+	0.885874i	$0.346441\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13.8564	−0.527887
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−55.4256	−2.10241
$696$	0	0
$697$	36.0000	1.36360
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.46410	0.130281
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.8564	0.518927
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.7128	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.8564	−0.512498
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.8564	−0.510407
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.3923	0.381257	0.190628	−	0.981662i	$-0.438947\pi$
$743$	10.3923	0.381257	0.190628	+	0.981662i	$0.438947\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.3205	−0.632878
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−27.7128	−1.00857
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.1051	1.38131	0.690655	−	0.723185i	$-0.257322\pi$
$761$	38.1051	1.38131	0.690655	+	0.723185i	$0.257322\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.8564	0.500326
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−45.0333	−1.61974	−0.809868	−	0.586612i	$-0.800461\pi$
$773$	−45.0333	−1.61974	−0.809868	+	0.586612i	$0.800461\pi$
$774$	0	0
$775$	28.0000	1.00579
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−41.5692	−1.48937
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	34.6410	1.23639
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.1769	−1.10434	−0.552171	−	0.833731i	$-0.686201\pi$
$797$	−31.1769	−1.10434	−0.552171	+	0.833731i	$0.686201\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.4974	1.71144
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.7128	−0.974331	−0.487165	−	0.873310i	$-0.661969\pi$
$809$	−27.7128	−0.974331	−0.487165	+	0.873310i	$0.661969\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	69.2820	2.42684
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.92820	−0.241796	−0.120898	−	0.992665i	$-0.538577\pi$
$821$	−6.92820	−0.241796	−0.120898	+	0.992665i	$0.538577\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	72.0000	2.49166
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.7846	−0.717564	−0.358782	−	0.933421i	$-0.616808\pi$
$839$	−20.7846	−0.717564	−0.358782	+	0.933421i	$0.616808\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31.1769	−1.07252
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.92820	−0.237496
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.3205	−0.591657	−0.295829	−	0.955241i	$-0.595596\pi$
$857$	−17.3205	−0.591657	−0.295829	+	0.955241i	$0.595596\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.1051	−1.29711	−0.648557	−	0.761166i	$-0.724627\pi$
$863$	−38.1051	−1.29711	−0.648557	+	0.761166i	$0.724627\pi$
$864$	0	0
$865$	−60.0000	−2.04006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−27.7128	−0.940093
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−51.9615	−1.75063	−0.875314	−	0.483555i	$-0.839345\pi$
$881$	−51.9615	−1.75063	−0.875314	+	0.483555i	$0.839345\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.92820	−0.232626	−0.116313	−	0.993213i	$-0.537108\pi$
$887$	−6.92820	−0.232626	−0.116313	+	0.993213i	$0.537108\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.7128	0.927374
$894$	0	0
$895$	−60.0000	−2.00558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.92820	−0.230301
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.1769	−1.03294	−0.516469	−	0.856306i	$-0.672754\pi$
$911$	−31.1769	−1.03294	−0.516469	+	0.856306i	$0.672754\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.8564	0.457579
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20.7846	−0.684134
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.0333	−1.47750	−0.738748	−	0.673982i	$-0.764582\pi$
$929$	−45.0333	−1.47750	−0.738748	+	0.673982i	$0.764582\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	0	0
$934$	0	0
$935$	41.5692	1.35946
$936$	0	0
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.3205	0.564632	0.282316	−	0.959321i	$-0.408897\pi$
$941$	17.3205	0.564632	0.282316	+	0.959321i	$0.408897\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.2487	−0.787977	−0.393989	−	0.919115i	$-0.628905\pi$
$947$	−24.2487	−0.787977	−0.393989	+	0.919115i	$0.628905\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−41.5692	−1.34656	−0.673280	−	0.739388i	$-0.735115\pi$
$953$	−41.5692	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$954$	0	0
$955$	84.0000	2.71818
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.92820	0.223723
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	48.4974	1.56119
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.7128	0.889346	0.444673	−	0.895693i	$-0.353320\pi$
$971$	27.7128	0.889346	0.444673	+	0.895693i	$0.353320\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.6410	1.10826	0.554132	−	0.832429i	$-0.313050\pi$
$977$	34.6410	1.10826	0.554132	+	0.832429i	$0.313050\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.8564	−0.441951	−0.220975	−	0.975279i	$-0.570924\pi$
$983$	−13.8564	−0.441951	−0.220975	+	0.975279i	$0.570924\pi$
$984$	0	0
$985$	−72.0000	−2.29411
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.8564	−0.440608
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	55.4256	1.75711
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.a.bq.1.2		2
3.2	odd	2	inner	4032.2.a.bq.1.1		2
4.3	odd	2		4032.2.a.bt.1.2		2
8.3	odd	2		63.2.a.b.1.2	yes	2
8.5	even	2		1008.2.a.n.1.1		2
12.11	even	2		4032.2.a.bt.1.1		2
24.5	odd	2		1008.2.a.n.1.2		2
24.11	even	2		63.2.a.b.1.1	✓	2
40.3	even	4		1575.2.d.i.1324.1		4
40.19	odd	2		1575.2.a.q.1.1		2
40.27	even	4		1575.2.d.i.1324.4		4
56.3	even	6		441.2.e.i.226.1		4
56.11	odd	6		441.2.e.j.226.1		4
56.13	odd	2		7056.2.a.cm.1.2		2
56.19	even	6		441.2.e.i.361.1		4
56.27	even	2		441.2.a.g.1.2		2
56.51	odd	6		441.2.e.j.361.1		4
72.11	even	6		567.2.f.j.190.2		4
72.43	odd	6		567.2.f.j.190.1		4
72.59	even	6		567.2.f.j.379.2		4
72.67	odd	6		567.2.f.j.379.1		4
88.43	even	2		7623.2.a.bi.1.1		2
120.59	even	2		1575.2.a.q.1.2		2
120.83	odd	4		1575.2.d.i.1324.3		4
120.107	odd	4		1575.2.d.i.1324.2		4
168.11	even	6		441.2.e.j.226.2		4
168.59	odd	6		441.2.e.i.226.2		4
168.83	odd	2		441.2.a.g.1.1		2
168.107	even	6		441.2.e.j.361.2		4
168.125	even	2		7056.2.a.cm.1.1		2
168.131	odd	6		441.2.e.i.361.2		4
264.131	odd	2		7623.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.a.b.1.1	✓	2	24.11	even	2
63.2.a.b.1.2	yes	2	8.3	odd	2
441.2.a.g.1.1		2	168.83	odd	2
441.2.a.g.1.2		2	56.27	even	2
441.2.e.i.226.1		4	56.3	even	6
441.2.e.i.226.2		4	168.59	odd	6
441.2.e.i.361.1		4	56.19	even	6
441.2.e.i.361.2		4	168.131	odd	6
441.2.e.j.226.1		4	56.11	odd	6
441.2.e.j.226.2		4	168.11	even	6
441.2.e.j.361.1		4	56.51	odd	6
441.2.e.j.361.2		4	168.107	even	6
567.2.f.j.190.1		4	72.43	odd	6
567.2.f.j.190.2		4	72.11	even	6
567.2.f.j.379.1		4	72.67	odd	6
567.2.f.j.379.2		4	72.59	even	6
1008.2.a.n.1.1		2	8.5	even	2
1008.2.a.n.1.2		2	24.5	odd	2
1575.2.a.q.1.1		2	40.19	odd	2
1575.2.a.q.1.2		2	120.59	even	2
1575.2.d.i.1324.1		4	40.3	even	4
1575.2.d.i.1324.2		4	120.107	odd	4
1575.2.d.i.1324.3		4	120.83	odd	4
1575.2.d.i.1324.4		4	40.27	even	4
4032.2.a.bq.1.1		2	3.2	odd	2	inner
4032.2.a.bq.1.2		2	1.1	even	1	trivial
4032.2.a.bt.1.1		2	12.11	even	2
4032.2.a.bt.1.2		2	4.3	odd	2
7056.2.a.cm.1.1		2	168.125	even	2
7056.2.a.cm.1.2		2	56.13	odd	2
7623.2.a.bi.1.1		2	88.43	even	2
7623.2.a.bi.1.2		2	264.131	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.a (trivial)

\(f(q)\)	\(=\)	\(q+3.46410 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.46410 q^{5} -1.00000 q^{7} +3.46410 q^{11} -2.00000 q^{13} +3.46410 q^{17} -4.00000 q^{19} +3.46410 q^{23} +7.00000 q^{25} +4.00000 q^{31} -3.46410 q^{35} -2.00000 q^{37} +10.3923 q^{41} -4.00000 q^{43} -6.92820 q^{47} +1.00000 q^{49} +6.92820 q^{53} +12.0000 q^{55} -6.92820 q^{59} +10.0000 q^{61} -6.92820 q^{65} -4.00000 q^{67} +10.3923 q^{71} +14.0000 q^{73} -3.46410 q^{77} -8.00000 q^{79} +12.0000 q^{85} -3.46410 q^{89} +2.00000 q^{91} -13.8564 q^{95} +14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^7	\( 2 q - 2 q^{7} - 4 q^{13} - 8 q^{19} + 14 q^{25} + 8 q^{31} - 4 q^{37} - 8 q^{43} + 2 q^{49} + 24 q^{55} + 20 q^{61} - 8 q^{67} + 28 q^{73} - 16 q^{79} + 24 q^{85} + 4 q^{91} + 28 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 4 * q^13 - 8 * q^19 + 14 * q^25 + 8 * q^31 - 4 * q^37 - 8 * q^43 + 2 * q^49 + 24 * q^55 + 20 * q^61 - 8 * q^67 + 28 * q^73 - 16 * q^79 + 24 * q^85 + 4 * q^91 + 28 * q^97

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