LMFDB - Embedded newform 4032.2.c.o.2017.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2017,4032)]

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, 1, 0, 0]))

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

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

chi := DirichletCharacter("4032.2017");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1344)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2017.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4032.2017
Dual form			4032.2.c.o.2017.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-2.73205i q^{5} +1.00000 q^{7} +0.732051i q^{11} +4.00000i q^{13} +6.19615 q^{17} +3.46410i q^{19} -1.26795 q^{23} -2.46410 q^{25} -0.535898i q^{29} -2.00000 q^{31} -2.73205i q^{35} +2.00000i q^{37} +7.66025 q^{41} +4.92820i q^{43} +4.00000 q^{47} +1.00000 q^{49} +3.46410i q^{53} +2.00000 q^{55} +8.00000i q^{59} +12.3923i q^{61} +10.9282 q^{65} +14.3923i q^{67} -13.6603 q^{71} -15.4641 q^{73} +0.732051i q^{77} -8.39230 q^{79} -9.46410i q^{83} -16.9282i q^{85} -1.80385 q^{89} +4.00000i q^{91} +9.46410 q^{95} +18.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 4 q^{17} - 12 q^{23} + 4 q^{25} - 8 q^{31} - 4 q^{41} + 16 q^{47} + 4 q^{49} + 8 q^{55} + 16 q^{65} - 20 q^{71} - 48 q^{73} + 8 q^{79} - 28 q^{89} + 24 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 4 * q^17 - 12 * q^23 + 4 * q^25 - 8 * q^31 - 4 * q^41 + 16 * q^47 + 4 * q^49 + 8 * q^55 + 16 * q^65 - 20 * q^71 - 48 * q^73 + 8 * q^79 - 28 * q^89 + 24 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.73205i	− 1.22181i	−0.791704	−	0.610905i	$-0.790806\pi$
$5$	− 2.73205i	− 1.22181i	0.791704	−	0.610905i	$-0.209194\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.732051i	0.220722i	0.993892	+	0.110361i	$0.0352006\pi$
$11$	0.732051i	0.220722i	−0.993892	+	0.110361i	$0.964799\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.19615	1.50279	0.751394	−	0.659854i	$-0.229382\pi$
$17$	6.19615	1.50279	0.751394	+	0.659854i	$0.229382\pi$
$18$	0	0
$19$	3.46410i	0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.46410i	0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.535898i	− 0.0995138i	−0.998761	−	0.0497569i	$-0.984155\pi$
$29$	− 0.535898i	− 0.0995138i	0.998761	−	0.0497569i	$-0.0158447\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.73205i	− 0.461801i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.66025	1.19633	0.598165	−	0.801373i	$-0.295897\pi$
$41$	7.66025	1.19633	0.598165	+	0.801373i	$0.295897\pi$
$42$	0	0
$43$	4.92820i	0.751544i	0.926712	+	0.375772i	$0.122622\pi$
$43$	4.92820i	0.751544i	−0.926712	+	0.375772i	$0.877378\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.46410i	0.475831i	0.971286	+	0.237915i	$0.0764641\pi$
$53$	3.46410i	0.475831i	−0.971286	+	0.237915i	$0.923536\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	12.3923i	1.58667i	0.608784	+	0.793336i	$0.291658\pi$
$61$	12.3923i	1.58667i	−0.608784	+	0.793336i	$0.708342\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.9282	1.35548
$66$	0	0
$67$	14.3923i	1.75830i	0.476545	+	0.879150i	$0.341889\pi$
$67$	14.3923i	1.75830i	−0.476545	+	0.879150i	$0.658111\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.6603	−1.62117	−0.810587	−	0.585618i	$-0.800852\pi$
$71$	−13.6603	−1.62117	−0.810587	+	0.585618i	$0.800852\pi$
$72$	0	0
$73$	−15.4641	−1.80994	−0.904968	−	0.425479i	$-0.860105\pi$
$73$	−15.4641	−1.80994	−0.904968	+	0.425479i	$0.860105\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.732051i	0.0834249i
$78$	0	0
$79$	−8.39230	−0.944208	−0.472104	−	0.881543i	$-0.656505\pi$
$79$	−8.39230	−0.944208	−0.472104	+	0.881543i	$0.656505\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.46410i	− 1.03882i	−0.854525	−	0.519410i	$-0.826152\pi$
$83$	− 9.46410i	− 1.03882i	0.854525	−	0.519410i	$-0.173848\pi$
$84$	0	0
$85$	− 16.9282i	− 1.83612i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.80385	−0.191207	−0.0956037	−	0.995419i	$-0.530478\pi$
$89$	−1.80385	−0.191207	−0.0956037	+	0.995419i	$0.530478\pi$
$90$	0	0
$91$	4.00000i	0.419314i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.46410	0.970996
$96$	0	0
$97$	18.3923	1.86746	0.933728	−	0.357984i	$-0.116536\pi$
$97$	18.3923	1.86746	0.933728	+	0.357984i	$0.116536\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.7321i	− 1.46589i	−0.680286	−	0.732947i	$-0.738144\pi$
$101$	− 14.7321i	− 1.46589i	0.680286	−	0.732947i	$-0.261856\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.19615i	0.212310i	0.994350	+	0.106155i	$0.0338540\pi$
$107$	2.19615i	0.212310i	−0.994350	+	0.106155i	$0.966146\pi$
$108$	0	0
$109$	− 2.92820i	− 0.280471i	−0.990118	−	0.140236i	$-0.955214\pi$
$109$	− 2.92820i	− 0.280471i	0.990118	−	0.140236i	$-0.0447860\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.3923	1.35391	0.676957	−	0.736022i	$-0.263298\pi$
$113$	14.3923	1.35391	0.676957	+	0.736022i	$0.263298\pi$
$114$	0	0
$115$	3.46410i	0.323029i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.19615	0.568000
$120$	0	0
$121$	10.4641	0.951282
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	15.3205	1.35948	0.679738	−	0.733455i	$-0.262094\pi$
$127$	15.3205	1.35948	0.679738	+	0.733455i	$0.262094\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.3923i	1.43220i	0.697997	+	0.716101i	$0.254075\pi$
$131$	16.3923i	1.43220i	−0.697997	+	0.716101i	$0.745925\pi$
$132$	0	0
$133$	3.46410i	0.300376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	0	0
$139$	2.92820i	0.248367i	0.992259	+	0.124183i	$0.0396311\pi$
$139$	2.92820i	0.248367i	−0.992259	+	0.124183i	$0.960369\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.92820	−0.244869
$144$	0	0
$145$	−1.46410	−0.121587
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 18.3923i	− 1.50676i	−0.657588	−	0.753378i	$-0.728423\pi$
$149$	− 18.3923i	− 1.50676i	0.657588	−	0.753378i	$-0.271577\pi$
$150$	0	0
$151$	14.9282	1.21484	0.607420	−	0.794381i	$-0.292205\pi$
$151$	14.9282	1.21484	0.607420	+	0.794381i	$0.292205\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.46410i	0.438887i
$156$	0	0
$157$	− 12.3923i	− 0.989014i	−0.869174	−	0.494507i	$-0.835349\pi$
$157$	− 12.3923i	− 0.989014i	0.869174	−	0.494507i	$-0.164651\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.26795	−0.0999284
$162$	0	0
$163$	− 11.4641i	− 0.897938i	−0.893547	−	0.448969i	$-0.851791\pi$
$163$	− 11.4641i	− 0.897938i	0.893547	−	0.448969i	$-0.148209\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.39230	0.339887	0.169943	−	0.985454i	$-0.445642\pi$
$167$	4.39230	0.339887	0.169943	+	0.985454i	$0.445642\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 22.0526i	− 1.67663i	−0.545190	−	0.838313i	$-0.683542\pi$
$173$	− 22.0526i	− 1.67663i	0.545190	−	0.838313i	$-0.316458\pi$
$174$	0	0
$175$	−2.46410	−0.186269
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.5885i	0.791418i	0.918376	+	0.395709i	$0.129501\pi$
$179$	10.5885i	0.791418i	−0.918376	+	0.395709i	$0.870499\pi$
$180$	0	0
$181$	− 5.07180i	− 0.376984i	−0.982075	−	0.188492i	$-0.939640\pi$
$181$	− 5.07180i	− 0.376984i	0.982075	−	0.188492i	$-0.0603599\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.46410	0.401729
$186$	0	0
$187$	4.53590i	0.331698i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.2679	1.53889	0.769447	−	0.638710i	$-0.220532\pi$
$191$	21.2679	1.53889	0.769447	+	0.638710i	$0.220532\pi$
$192$	0	0
$193$	−3.60770	−0.259688	−0.129844	−	0.991534i	$-0.541448\pi$
$193$	−3.60770	−0.259688	−0.129844	+	0.991534i	$0.541448\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.7846i	1.05336i	0.850064	+	0.526680i	$0.176563\pi$
$197$	14.7846i	1.05336i	−0.850064	+	0.526680i	$0.823437\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.535898i	− 0.0376127i
$204$	0	0
$205$	− 20.9282i	− 1.46169i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.53590	−0.175412
$210$	0	0
$211$	− 27.8564i	− 1.91771i	−0.283889	−	0.958857i	$-0.591625\pi$
$211$	− 27.8564i	− 1.91771i	0.283889	−	0.958857i	$-0.408375\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.4641	0.918244
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	24.7846i	1.66719i
$222$	0	0
$223$	−18.9282	−1.26753	−0.633763	−	0.773527i	$-0.718491\pi$
$223$	−18.9282	−1.26753	−0.633763	+	0.773527i	$0.718491\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.3205i	1.54784i	0.633286	+	0.773918i	$0.281706\pi$
$227$	23.3205i	1.54784i	−0.633286	+	0.773918i	$0.718294\pi$
$228$	0	0
$229$	2.92820i	0.193501i	0.995309	+	0.0967506i	$0.0308449\pi$
$229$	2.92820i	0.193501i	−0.995309	+	0.0967506i	$0.969155\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.928203	0.0608086	0.0304043	−	0.999538i	$-0.490321\pi$
$233$	0.928203	0.0608086	0.0304043	+	0.999538i	$0.490321\pi$
$234$	0	0
$235$	− 10.9282i	− 0.712877i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.3397	0.668823	0.334411	−	0.942427i	$-0.391463\pi$
$239$	10.3397	0.668823	0.334411	+	0.942427i	$0.391463\pi$
$240$	0	0
$241$	−0.535898	−0.0345202	−0.0172601	−	0.999851i	$-0.505494\pi$
$241$	−0.535898	−0.0345202	−0.0172601	+	0.999851i	$0.505494\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.73205i	− 0.174544i
$246$	0	0
$247$	−13.8564	−0.881662
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.3205i	0.714544i	0.934000	+	0.357272i	$0.116293\pi$
$251$	11.3205i	0.714544i	−0.934000	+	0.357272i	$0.883707\pi$
$252$	0	0
$253$	− 0.928203i	− 0.0583556i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.1962	−0.885532	−0.442766	−	0.896637i	$-0.646003\pi$
$257$	−14.1962	−0.885532	−0.442766	+	0.896637i	$0.646003\pi$
$258$	0	0
$259$	2.00000i	0.124274i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.19615	0.258746	0.129373	−	0.991596i	$-0.458704\pi$
$263$	4.19615	0.258746	0.129373	+	0.991596i	$0.458704\pi$
$264$	0	0
$265$	9.46410	0.581375
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.7321i	1.38600i	0.720939	+	0.692999i	$0.243711\pi$
$269$	22.7321i	1.38600i	−0.720939	+	0.692999i	$0.756289\pi$
$270$	0	0
$271$	−28.9282	−1.75726	−0.878632	−	0.477500i	$-0.841543\pi$
$271$	−28.9282	−1.75726	−0.878632	+	0.477500i	$0.841543\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.80385i	− 0.108776i
$276$	0	0
$277$	− 3.07180i	− 0.184566i	−0.995733	−	0.0922832i	$-0.970583\pi$
$277$	− 3.07180i	− 0.184566i	0.995733	−	0.0922832i	$-0.0294165\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.0718	0.899108	0.449554	−	0.893253i	$-0.351583\pi$
$281$	15.0718	0.899108	0.449554	+	0.893253i	$0.351583\pi$
$282$	0	0
$283$	18.3923i	1.09331i	0.837358	+	0.546655i	$0.184099\pi$
$283$	18.3923i	1.09331i	−0.837358	+	0.546655i	$0.815901\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.66025	0.452170
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 24.1962i	− 1.41355i	−0.707436	−	0.706777i	$-0.750148\pi$
$293$	− 24.1962i	− 1.41355i	0.707436	−	0.706777i	$-0.249852\pi$
$294$	0	0
$295$	21.8564	1.27253
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 5.07180i	− 0.293310i
$300$	0	0
$301$	4.92820i	0.284057i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.8564	1.93861
$306$	0	0
$307$	− 9.32051i	− 0.531949i	−0.963980	−	0.265975i	$-0.914306\pi$
$307$	− 9.32051i	− 0.531949i	0.963980	−	0.265975i	$-0.0856938\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−23.7128	−1.34463	−0.672315	−	0.740265i	$-0.734700\pi$
$311$	−23.7128	−1.34463	−0.672315	+	0.740265i	$0.734700\pi$
$312$	0	0
$313$	12.9282	0.730745	0.365373	−	0.930861i	$-0.380942\pi$
$313$	12.9282	0.730745	0.365373	+	0.930861i	$0.380942\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.39230i	− 0.134365i	−0.997741	−	0.0671826i	$-0.978599\pi$
$317$	− 2.39230i	− 0.134365i	0.997741	−	0.0671826i	$-0.0214010\pi$
$318$	0	0
$319$	0.392305	0.0219649
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.4641i	1.19429i
$324$	0	0
$325$	− 9.85641i	− 0.546735i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	− 0.928203i	− 0.0510187i	−0.999675	−	0.0255093i	$-0.991879\pi$
$331$	− 0.928203i	− 0.0510187i	0.999675	−	0.0255093i	$-0.00812075\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	39.3205	2.14831
$336$	0	0
$337$	21.7128	1.18277	0.591386	−	0.806389i	$-0.298581\pi$
$337$	21.7128	1.18277	0.591386	+	0.806389i	$0.298581\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.46410i	− 0.0792855i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.7321i	0.898224i	0.893476	+	0.449112i	$0.148260\pi$
$347$	16.7321i	0.898224i	−0.893476	+	0.449112i	$0.851740\pi$
$348$	0	0
$349$	19.3205i	1.03420i	0.855924	+	0.517102i	$0.172989\pi$
$349$	19.3205i	1.03420i	−0.855924	+	0.517102i	$0.827011\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−35.6603	−1.89800	−0.949002	−	0.315270i	$-0.897905\pi$
$353$	−35.6603	−1.89800	−0.949002	+	0.315270i	$0.897905\pi$
$354$	0	0
$355$	37.3205i	1.98077i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.1962	−1.69925	−0.849624	−	0.527389i	$-0.823171\pi$
$359$	−32.1962	−1.69925	−0.849624	+	0.527389i	$0.823171\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	42.2487i	2.21140i
$366$	0	0
$367$	13.8564	0.723299	0.361649	−	0.932314i	$-0.382214\pi$
$367$	13.8564	0.723299	0.361649	+	0.932314i	$0.382214\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.46410i	0.179847i
$372$	0	0
$373$	− 12.0000i	− 0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$	− 12.0000i	− 0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.14359	0.110401
$378$	0	0
$379$	− 28.9282i	− 1.48594i	−0.669324	−	0.742971i	$-0.733416\pi$
$379$	− 28.9282i	− 1.48594i	0.669324	−	0.742971i	$-0.266584\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7.60770	−0.388735	−0.194368	−	0.980929i	$-0.562265\pi$
$383$	−7.60770	−0.388735	−0.194368	+	0.980929i	$0.562265\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 20.5359i	− 1.04121i	−0.853797	−	0.520606i	$-0.825706\pi$
$389$	− 20.5359i	− 1.04121i	0.853797	−	0.520606i	$-0.174294\pi$
$390$	0	0
$391$	−7.85641	−0.397316
$392$	0	0
$393$	0	0
$394$	0	0
$395$	22.9282i	1.15364i
$396$	0	0
$397$	19.3205i	0.969669i	0.874606	+	0.484834i	$0.161120\pi$
$397$	19.3205i	0.969669i	−0.874606	+	0.484834i	$0.838880\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.3205	−1.06470	−0.532348	−	0.846526i	$-0.678690\pi$
$401$	−21.3205	−1.06470	−0.532348	+	0.846526i	$0.678690\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.46410	−0.0725728
$408$	0	0
$409$	8.24871	0.407873	0.203936	−	0.978984i	$-0.434626\pi$
$409$	8.24871	0.407873	0.203936	+	0.978984i	$0.434626\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	−25.8564	−1.26924
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9.07180i	− 0.443186i	−0.975139	−	0.221593i	$-0.928874\pi$
$419$	− 9.07180i	− 0.443186i	0.975139	−	0.221593i	$-0.0711257\pi$
$420$	0	0
$421$	− 2.78461i	− 0.135714i	−0.997695	−	0.0678568i	$-0.978384\pi$
$421$	− 2.78461i	− 0.135714i	0.997695	−	0.0678568i	$-0.0216161\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.2679	−0.740604
$426$	0	0
$427$	12.3923i	0.599706i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.19615	0.202122	0.101061	−	0.994880i	$-0.467776\pi$
$431$	4.19615	0.202122	0.101061	+	0.994880i	$0.467776\pi$
$432$	0	0
$433$	−6.78461	−0.326048	−0.163024	−	0.986622i	$-0.552125\pi$
$433$	−6.78461	−0.326048	−0.163024	+	0.986622i	$0.552125\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.39230i	− 0.210112i
$438$	0	0
$439$	−13.8564	−0.661330	−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564	−0.661330	−0.330665	+	0.943748i	$0.607273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 28.0526i	− 1.33282i	−0.745587	−	0.666409i	$-0.767831\pi$
$443$	− 28.0526i	− 1.33282i	0.745587	−	0.666409i	$-0.232169\pi$
$444$	0	0
$445$	4.92820i	0.233619i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.4641	1.10734	0.553670	−	0.832736i	$-0.313227\pi$
$449$	23.4641	1.10734	0.553670	+	0.832736i	$0.313227\pi$
$450$	0	0
$451$	5.60770i	0.264056i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.9282	0.512322
$456$	0	0
$457$	30.7846	1.44004	0.720022	−	0.693952i	$-0.244132\pi$
$457$	30.7846	1.44004	0.720022	+	0.693952i	$0.244132\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.2679i	0.804249i	0.915585	+	0.402124i	$0.131728\pi$
$461$	17.2679i	0.804249i	−0.915585	+	0.402124i	$0.868272\pi$
$462$	0	0
$463$	41.1769	1.91365	0.956827	−	0.290659i	$-0.0938745\pi$
$463$	41.1769	1.91365	0.956827	+	0.290659i	$0.0938745\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 0.392305i	− 0.0181537i	−0.999959	−	0.00907685i	$-0.997111\pi$
$467$	− 0.392305i	− 0.0181537i	0.999959	−	0.00907685i	$-0.00288929\pi$
$468$	0	0
$469$	14.3923i	0.664575i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.60770	−0.165882
$474$	0	0
$475$	− 8.53590i	− 0.391654i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−38.6410	−1.76555	−0.882777	−	0.469793i	$-0.844329\pi$
$479$	−38.6410	−1.76555	−0.882777	+	0.469793i	$0.844329\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 50.2487i	− 2.28168i
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 2.87564i	− 0.129776i	−0.997893	−	0.0648880i	$-0.979331\pi$
$491$	− 2.87564i	− 0.129776i	0.997893	−	0.0648880i	$-0.0206690\pi$
$492$	0	0
$493$	− 3.32051i	− 0.149548i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.6603	−0.612746
$498$	0	0
$499$	15.8564i	0.709830i	0.934899	+	0.354915i	$0.115490\pi$
$499$	15.8564i	0.709830i	−0.934899	+	0.354915i	$0.884510\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.39230	0.195843	0.0979216	−	0.995194i	$-0.468781\pi$
$503$	4.39230	0.195843	0.0979216	+	0.995194i	$0.468781\pi$
$504$	0	0
$505$	−40.2487	−1.79104
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 0.588457i	− 0.0260829i	−0.999915	−	0.0130415i	$-0.995849\pi$
$509$	− 0.588457i	− 0.0260829i	0.999915	−	0.0130415i	$-0.00415134\pi$
$510$	0	0
$511$	−15.4641	−0.684092
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 38.2487i	− 1.68544i
$516$	0	0
$517$	2.92820i	0.128782i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.1962	−0.621945	−0.310972	−	0.950419i	$-0.600655\pi$
$521$	−14.1962	−0.621945	−0.310972	+	0.950419i	$0.600655\pi$
$522$	0	0
$523$	1.07180i	0.0468664i	0.999725	+	0.0234332i	$0.00745970\pi$
$523$	1.07180i	0.0468664i	−0.999725	+	0.0234332i	$0.992540\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.3923	−0.539817
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.6410i	1.32721i
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.732051i	0.0315317i
$540$	0	0
$541$	26.0000i	1.11783i	0.829226	+	0.558914i	$0.188782\pi$
$541$	26.0000i	1.11783i	−0.829226	+	0.558914i	$0.811218\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.00000	−0.342682
$546$	0	0
$547$	4.24871i	0.181662i	0.995866	+	0.0908309i	$0.0289523\pi$
$547$	4.24871i	0.181662i	−0.995866	+	0.0908309i	$0.971048\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.85641	0.0790856
$552$	0	0
$553$	−8.39230	−0.356877
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.4641i	1.33318i	0.745426	+	0.666588i	$0.232246\pi$
$557$	31.4641i	1.33318i	−0.745426	+	0.666588i	$0.767754\pi$
$558$	0	0
$559$	−19.7128	−0.833763
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 22.6410i	− 0.954205i	−0.878848	−	0.477103i	$-0.841687\pi$
$563$	− 22.6410i	− 0.954205i	0.878848	−	0.477103i	$-0.158313\pi$
$564$	0	0
$565$	− 39.3205i	− 1.65423i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.14359	−0.341397	−0.170699	−	0.985323i	$-0.554602\pi$
$569$	−8.14359	−0.341397	−0.170699	+	0.985323i	$0.554602\pi$
$570$	0	0
$571$	− 22.3923i	− 0.937089i	−0.883440	−	0.468544i	$-0.844779\pi$
$571$	− 22.3923i	− 0.937089i	0.883440	−	0.468544i	$-0.155221\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.12436	0.130295
$576$	0	0
$577$	−24.6410	−1.02582	−0.512909	−	0.858443i	$-0.671432\pi$
$577$	−24.6410	−1.02582	−0.512909	+	0.858443i	$0.671432\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9.46410i	− 0.392637i
$582$	0	0
$583$	−2.53590	−0.105026
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.32051i	− 0.302150i	−0.988522	−	0.151075i	$-0.951727\pi$
$587$	− 7.32051i	− 0.302150i	0.988522	−	0.151075i	$-0.0482734\pi$
$588$	0	0
$589$	− 6.92820i	− 0.285472i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.6603	0.643090	0.321545	−	0.946894i	$-0.395798\pi$
$593$	15.6603	0.643090	0.321545	+	0.946894i	$0.395798\pi$
$594$	0	0
$595$	− 16.9282i	− 0.693989i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.19615	0.334886	0.167443	−	0.985882i	$-0.446449\pi$
$599$	8.19615	0.334886	0.167443	+	0.985882i	$0.446449\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 28.5885i	− 1.16229i
$606$	0	0
$607$	−22.6410	−0.918970	−0.459485	−	0.888185i	$-0.651966\pi$
$607$	−22.6410	−0.918970	−0.459485	+	0.888185i	$0.651966\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	41.8564i	1.69056i	0.534320	+	0.845282i	$0.320568\pi$
$613$	41.8564i	1.69056i	−0.534320	+	0.845282i	$0.679432\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.4641	1.42773	0.713865	−	0.700283i	$-0.246943\pi$
$617$	35.4641	1.42773	0.713865	+	0.700283i	$0.246943\pi$
$618$	0	0
$619$	17.8564i	0.717710i	0.933393	+	0.358855i	$0.116833\pi$
$619$	17.8564i	0.717710i	−0.933393	+	0.358855i	$0.883167\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.80385	−0.0722696
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.3923i	0.494114i
$630$	0	0
$631$	−22.5359	−0.897140	−0.448570	−	0.893748i	$-0.648067\pi$
$631$	−22.5359	−0.897140	−0.448570	+	0.893748i	$0.648067\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 41.8564i	− 1.66102i
$636$	0	0
$637$	4.00000i	0.158486i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.5359	1.44308	0.721541	−	0.692372i	$-0.243434\pi$
$641$	36.5359	1.44308	0.721541	+	0.692372i	$0.243434\pi$
$642$	0	0
$643$	40.2487i	1.58725i	0.608404	+	0.793627i	$0.291810\pi$
$643$	40.2487i	1.58725i	−0.608404	+	0.793627i	$0.708190\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.1051	−1.26218	−0.631091	−	0.775709i	$-0.717393\pi$
$647$	−32.1051	−1.26218	−0.631091	+	0.775709i	$0.717393\pi$
$648$	0	0
$649$	−5.85641	−0.229884
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.14359i	0.162151i	0.996708	+	0.0810757i	$0.0258355\pi$
$653$	4.14359i	0.162151i	−0.996708	+	0.0810757i	$0.974164\pi$
$654$	0	0
$655$	44.7846	1.74988
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5.12436i	− 0.199617i	−0.995007	−	0.0998083i	$-0.968177\pi$
$659$	− 5.12436i	− 0.199617i	0.995007	−	0.0998083i	$-0.0318229\pi$
$660$	0	0
$661$	− 40.3923i	− 1.57108i	−0.618812	−	0.785539i	$-0.712386\pi$
$661$	− 40.3923i	− 1.57108i	0.618812	−	0.785539i	$-0.287614\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.46410	0.367002
$666$	0	0
$667$	0.679492i	0.0263100i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.07180	−0.350213
$672$	0	0
$673$	−37.1769	−1.43306	−0.716532	−	0.697554i	$-0.754272\pi$
$673$	−37.1769	−1.43306	−0.716532	+	0.697554i	$0.754272\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.7321i	− 1.33486i	−0.744672	−	0.667431i	$-0.767394\pi$
$677$	− 34.7321i	− 1.33486i	0.744672	−	0.667431i	$-0.232606\pi$
$678$	0	0
$679$	18.3923	0.705832
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 33.9090i	− 1.29749i	−0.761006	−	0.648745i	$-0.775294\pi$
$683$	− 33.9090i	− 1.29749i	0.761006	−	0.648745i	$-0.224706\pi$
$684$	0	0
$685$	9.46410i	0.361605i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13.8564	−0.527887
$690$	0	0
$691$	− 16.7846i	− 0.638517i	−0.947668	−	0.319258i	$-0.896566\pi$
$691$	− 16.7846i	− 0.638517i	0.947668	−	0.319258i	$-0.103434\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	47.4641	1.79783
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.7128i	0.820082i	0.912067	+	0.410041i	$0.134486\pi$
$701$	21.7128i	0.820082i	−0.912067	+	0.410041i	$0.865514\pi$
$702$	0	0
$703$	−6.92820	−0.261302
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 14.7321i	− 0.554056i
$708$	0	0
$709$	10.9282i	0.410417i	0.978718	+	0.205209i	$0.0657873\pi$
$709$	10.9282i	0.410417i	−0.978718	+	0.205209i	$0.934213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.53590	0.0949701
$714$	0	0
$715$	8.00000i	0.299183i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.5359	0.392923	0.196461	−	0.980512i	$-0.437055\pi$
$719$	10.5359	0.392923	0.196461	+	0.980512i	$0.437055\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.32051i	0.0490424i
$726$	0	0
$727$	44.6410	1.65564	0.827822	−	0.560991i	$-0.189580\pi$
$727$	44.6410	1.65564	0.827822	+	0.560991i	$0.189580\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30.5359i	1.12941i
$732$	0	0
$733$	33.0718i	1.22153i	0.791810	+	0.610767i	$0.209139\pi$
$733$	33.0718i	1.22153i	−0.791810	+	0.610767i	$0.790861\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.5359	−0.388095
$738$	0	0
$739$	− 6.39230i	− 0.235145i	−0.993064	−	0.117572i	$-0.962489\pi$
$739$	− 6.39230i	− 0.235145i	0.993064	−	0.117572i	$-0.0375112\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.1962	−1.47465	−0.737327	−	0.675536i	$-0.763912\pi$
$743$	−40.1962	−1.47465	−0.737327	+	0.675536i	$0.763912\pi$
$744$	0	0
$745$	−50.2487	−1.84097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.19615i	0.0802457i
$750$	0	0
$751$	18.9282	0.690700	0.345350	−	0.938474i	$-0.387760\pi$
$751$	18.9282	0.690700	0.345350	+	0.938474i	$0.387760\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 40.7846i	− 1.48430i
$756$	0	0
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.7321	−0.606536	−0.303268	−	0.952905i	$-0.598078\pi$
$761$	−16.7321	−0.606536	−0.303268	+	0.952905i	$0.598078\pi$
$762$	0	0
$763$	− 2.92820i	− 0.106008i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.0000	−1.15545
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 45.3731i	− 1.63196i	−0.578083	−	0.815978i	$-0.696199\pi$
$773$	− 45.3731i	− 1.63196i	0.578083	−	0.815978i	$-0.303801\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	0	0
$778$	0	0
$779$	26.5359i	0.950747i
$780$	0	0
$781$	− 10.0000i	− 0.357828i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33.8564	−1.20839
$786$	0	0
$787$	− 26.9282i	− 0.959887i	−0.877299	−	0.479943i	$-0.840657\pi$
$787$	− 26.9282i	− 0.959887i	0.877299	−	0.479943i	$-0.159343\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.3923	0.511731
$792$	0	0
$793$	−49.5692	−1.76025
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.6603i	0.767246i	0.923490	+	0.383623i	$0.125324\pi$
$797$	21.6603i	0.767246i	−0.923490	+	0.383623i	$0.874676\pi$
$798$	0	0
$799$	24.7846	0.876816
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 11.3205i	− 0.399492i
$804$	0	0
$805$	3.46410i	0.122094i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−41.3205	−1.45275	−0.726376	−	0.687298i	$-0.758797\pi$
$809$	−41.3205	−1.45275	−0.726376	+	0.687298i	$0.758797\pi$
$810$	0	0
$811$	− 20.7846i	− 0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$	− 20.7846i	− 0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−31.3205	−1.09711
$816$	0	0
$817$	−17.0718	−0.597267
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.6795i	0.651919i	0.945384	+	0.325959i	$0.105687\pi$
$821$	18.6795i	0.651919i	−0.945384	+	0.325959i	$0.894313\pi$
$822$	0	0
$823$	20.3923	0.710831	0.355416	−	0.934708i	$-0.384339\pi$
$823$	20.3923	0.710831	0.355416	+	0.934708i	$0.384339\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 46.9808i	− 1.63368i	−0.576863	−	0.816841i	$-0.695724\pi$
$827$	− 46.9808i	− 1.63368i	0.576863	−	0.816841i	$-0.304276\pi$
$828$	0	0
$829$	− 8.78461i	− 0.305102i	−0.988296	−	0.152551i	$-0.951251\pi$
$829$	− 8.78461i	− 0.305102i	0.988296	−	0.152551i	$-0.0487488\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.19615	0.214684
$834$	0	0
$835$	− 12.0000i	− 0.415277i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.1051	1.52268	0.761339	−	0.648354i	$-0.224542\pi$
$839$	44.1051	1.52268	0.761339	+	0.648354i	$0.224542\pi$
$840$	0	0
$841$	28.7128	0.990097
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.19615i	0.281956i
$846$	0	0
$847$	10.4641	0.359551
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2.53590i	− 0.0869295i
$852$	0	0
$853$	30.2487i	1.03570i	0.855473	+	0.517848i	$0.173267\pi$
$853$	30.2487i	1.03570i	−0.855473	+	0.517848i	$0.826733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.732051	−0.0250064	−0.0125032	−	0.999922i	$-0.503980\pi$
$857$	−0.732051	−0.0250064	−0.0125032	+	0.999922i	$0.503980\pi$
$858$	0	0
$859$	11.1769i	0.381351i	0.981653	+	0.190676i	$0.0610679\pi$
$859$	11.1769i	0.381351i	−0.981653	+	0.190676i	$0.938932\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.1244	−1.74029	−0.870147	−	0.492793i	$-0.835976\pi$
$863$	−51.1244	−1.74029	−0.870147	+	0.492793i	$0.835976\pi$
$864$	0	0
$865$	−60.2487	−2.04852
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 6.14359i	− 0.208407i
$870$	0	0
$871$	−57.5692	−1.95066
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 6.92820i	− 0.234216i
$876$	0	0
$877$	22.9282i	0.774230i	0.922031	+	0.387115i	$0.126528\pi$
$877$	22.9282i	0.774230i	−0.922031	+	0.387115i	$0.873472\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.1244	−1.11599	−0.557994	−	0.829845i	$-0.688429\pi$
$881$	−33.1244	−1.11599	−0.557994	+	0.829845i	$0.688429\pi$
$882$	0	0
$883$	− 0.928203i	− 0.0312365i	−0.999878	−	0.0156183i	$-0.995028\pi$
$883$	− 0.928203i	− 0.0312365i	0.999878	−	0.0156183i	$-0.00497165\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.2487	1.41857	0.709286	−	0.704920i	$-0.249017\pi$
$887$	42.2487	1.41857	0.709286	+	0.704920i	$0.249017\pi$
$888$	0	0
$889$	15.3205	0.513833
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13.8564i	0.463687i
$894$	0	0
$895$	28.9282	0.966963
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.07180i	0.0357464i
$900$	0	0
$901$	21.4641i	0.715073i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.8564	−0.460603
$906$	0	0
$907$	11.8564i	0.393686i	0.980435	+	0.196843i	$0.0630688\pi$
$907$	11.8564i	0.393686i	−0.980435	+	0.196843i	$0.936931\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.9090	1.05719	0.528596	−	0.848874i	$-0.322719\pi$
$911$	31.9090	1.05719	0.528596	+	0.848874i	$0.322719\pi$
$912$	0	0
$913$	6.92820	0.229290
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.3923i	0.541322i
$918$	0	0
$919$	2.92820	0.0965925	0.0482963	−	0.998833i	$-0.484621\pi$
$919$	2.92820	0.0965925	0.0482963	+	0.998833i	$0.484621\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 54.6410i	− 1.79853i
$924$	0	0
$925$	− 4.92820i	− 0.162038i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13.4115	0.440018	0.220009	−	0.975498i	$-0.429391\pi$
$929$	13.4115	0.440018	0.220009	+	0.975498i	$0.429391\pi$
$930$	0	0
$931$	3.46410i	0.113531i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.3923	0.405272
$936$	0	0
$937$	−39.8564	−1.30205	−0.651026	−	0.759055i	$-0.725661\pi$
$937$	−39.8564	−1.30205	−0.651026	+	0.759055i	$0.725661\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 43.2295i	− 1.40924i	−0.709585	−	0.704620i	$-0.751117\pi$
$941$	− 43.2295i	− 1.40924i	0.709585	−	0.704620i	$-0.248883\pi$
$942$	0	0
$943$	−9.71281	−0.316293
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26.5885i	− 0.864009i	−0.901872	−	0.432004i	$-0.857806\pi$
$947$	− 26.5885i	− 0.864009i	0.901872	−	0.432004i	$-0.142194\pi$
$948$	0	0
$949$	− 61.8564i	− 2.00794i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.7846	−0.478920	−0.239460	−	0.970906i	$-0.576970\pi$
$953$	−14.7846	−0.478920	−0.239460	+	0.970906i	$0.576970\pi$
$954$	0	0
$955$	− 58.1051i	− 1.88024i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.46410	−0.111862
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.85641i	0.317289i
$966$	0	0
$967$	−41.4641	−1.33340	−0.666698	−	0.745328i	$-0.732293\pi$
$967$	−41.4641	−1.33340	−0.666698	+	0.745328i	$0.732293\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 32.7846i	− 1.05211i	−0.850451	−	0.526054i	$-0.823671\pi$
$971$	− 32.7846i	− 1.05211i	0.850451	−	0.526054i	$-0.176329\pi$
$972$	0	0
$973$	2.92820i	0.0938739i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.4974	−1.35961	−0.679807	−	0.733392i	$-0.737936\pi$
$977$	−42.4974	−1.35961	−0.679807	+	0.733392i	$0.737936\pi$
$978$	0	0
$979$	− 1.32051i	− 0.0422036i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.7846	−1.17325	−0.586623	−	0.809860i	$-0.699543\pi$
$983$	−36.7846	−1.17325	−0.586623	+	0.809860i	$0.699543\pi$
$984$	0	0
$985$	40.3923	1.28701
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 6.24871i	− 0.198697i
$990$	0	0
$991$	−11.2154	−0.356269	−0.178134	−	0.984006i	$-0.557006\pi$
$991$	−11.2154	−0.356269	−0.178134	+	0.984006i	$0.557006\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 21.8564i	− 0.692895i
$996$	0	0
$997$	− 3.32051i	− 0.105162i	−0.998617	−	0.0525808i	$-0.983255\pi$
$997$	− 3.32051i	− 0.105162i	0.998617	−	0.0525808i	$-0.0167447\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.c.o.2017.1		4
3.2	odd	2		1344.2.c.h.673.2	yes	4
4.3	odd	2		4032.2.c.l.2017.1		4
8.3	odd	2		4032.2.c.l.2017.4		4
8.5	even	2	inner	4032.2.c.o.2017.4		4
12.11	even	2		1344.2.c.e.673.4	yes	4
24.5	odd	2		1344.2.c.h.673.3	yes	4
24.11	even	2		1344.2.c.e.673.1	✓	4
48.5	odd	4		5376.2.a.n.1.1		2
48.11	even	4		5376.2.a.z.1.1		2
48.29	odd	4		5376.2.a.bd.1.2		2
48.35	even	4		5376.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.2.c.e.673.1	✓	4	24.11	even	2
1344.2.c.e.673.4	yes	4	12.11	even	2
1344.2.c.h.673.2	yes	4	3.2	odd	2
1344.2.c.h.673.3	yes	4	24.5	odd	2
4032.2.c.l.2017.1		4	4.3	odd	2
4032.2.c.l.2017.4		4	8.3	odd	2
4032.2.c.o.2017.1		4	1.1	even	1	trivial
4032.2.c.o.2017.4		4	8.5	even	2	inner
5376.2.a.n.1.1		2	48.5	odd	4
5376.2.a.t.1.2		2	48.35	even	4
5376.2.a.z.1.1		2	48.11	even	4
5376.2.a.bd.1.2		2	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.73205i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-2.73205i q^{5} +1.00000 q^{7} +0.732051i q^{11} +4.00000i q^{13} +6.19615 q^{17} +3.46410i q^{19} -1.26795 q^{23} -2.46410 q^{25} -0.535898i q^{29} -2.00000 q^{31} -2.73205i q^{35} +2.00000i q^{37} +7.66025 q^{41} +4.92820i q^{43} +4.00000 q^{47} +1.00000 q^{49} +3.46410i q^{53} +2.00000 q^{55} +8.00000i q^{59} +12.3923i q^{61} +10.9282 q^{65} +14.3923i q^{67} -13.6603 q^{71} -15.4641 q^{73} +0.732051i q^{77} -8.39230 q^{79} -9.46410i q^{83} -16.9282i q^{85} -1.80385 q^{89} +4.00000i q^{91} +9.46410 q^{95} +18.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 4 q^{17} - 12 q^{23} + 4 q^{25} - 8 q^{31} - 4 q^{41} + 16 q^{47} + 4 q^{49} + 8 q^{55} + 16 q^{65} - 20 q^{71} - 48 q^{73} + 8 q^{79} - 28 q^{89} + 24 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 4 * q^17 - 12 * q^23 + 4 * q^25 - 8 * q^31 - 4 * q^41 + 16 * q^47 + 4 * q^49 + 8 * q^55 + 16 * q^65 - 20 * q^71 - 48 * q^73 + 8 * q^79 - 28 * q^89 + 24 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

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