LMFDB - Embedded newform 882.2.d.a.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,2,Mod(881,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("882.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	882.d (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:	$7.04280545828$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	882.881
Dual form			882.2.d.a.881.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -0.717439 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -0.717439 q^{5} +1.00000i q^{8} +0.717439i q^{10} -3.00000i q^{11} +2.44949i q^{13} +1.00000 q^{16} -5.91359 q^{17} -5.91359i q^{19} +0.717439 q^{20} -3.00000 q^{22} -4.24264i q^{23} -4.48528 q^{25} +2.44949 q^{26} -7.24264i q^{29} +9.08052i q^{31} -1.00000i q^{32} +5.91359i q^{34} -0.242641 q^{37} -5.91359 q^{38} -0.717439i q^{40} -11.8272 q^{41} -0.242641 q^{43} +3.00000i q^{44} -4.24264 q^{46} -5.91359 q^{47} +4.48528i q^{50} -2.44949i q^{52} -7.24264i q^{53} +2.15232i q^{55} -7.24264 q^{58} +8.06591 q^{59} +1.01461i q^{61} +9.08052 q^{62} -1.00000 q^{64} -1.75736i q^{65} -10.0000 q^{67} +5.91359 q^{68} +1.75736i q^{71} +1.43488i q^{73} +0.242641i q^{74} +5.91359i q^{76} -2.75736 q^{79} -0.717439 q^{80} +11.8272i q^{82} -6.63103 q^{83} +4.24264 q^{85} +0.242641i q^{86} +3.00000 q^{88} +10.3923 q^{89} +4.24264i q^{92} +5.91359i q^{94} +4.24264i q^{95} -13.5592i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} - 24 q^{22} + 32 q^{25} + 32 q^{37} + 32 q^{43} - 24 q^{58} - 8 q^{64} - 80 q^{67} - 56 q^{79} + 24 q^{88}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 - 24 * q^22 + 32 * q^25 + 32 * q^37 + 32 * q^43 - 24 * q^58 - 8 * q^64 - 80 * q^67 - 56 * q^79 + 24 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−0.717439	−0.320848	−0.160424	−	0.987048i	$-0.551286\pi$
$5$	−0.717439	−0.320848	−0.160424	+	0.987048i	$0.551286\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0.717439i	0.226874i
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	0	0
$13$	2.44949i	0.679366i	0.940540	+	0.339683i	$0.110320\pi$
$13$	2.44949i	0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.91359	−1.43426	−0.717128	−	0.696941i	$-0.754544\pi$
$17$	−5.91359	−1.43426	−0.717128	+	0.696941i	$0.754544\pi$
$18$	0	0
$19$	− 5.91359i	− 1.35667i	−0.734752	−	0.678335i	$-0.762702\pi$
$19$	− 5.91359i	− 1.35667i	0.734752	−	0.678335i	$-0.237298\pi$
$20$	0.717439	0.160424
$21$	0	0
$22$	−3.00000	−0.639602
$23$	− 4.24264i	− 0.884652i	−0.896854	−	0.442326i	$-0.854153\pi$
$23$	− 4.24264i	− 0.884652i	0.896854	−	0.442326i	$-0.145847\pi$
$24$	0	0
$25$	−4.48528	−0.897056
$26$	2.44949	0.480384
$27$	0	0
$28$	0	0
$29$	− 7.24264i	− 1.34492i	−0.740131	−	0.672462i	$-0.765237\pi$
$29$	− 7.24264i	− 1.34492i	0.740131	−	0.672462i	$-0.234763\pi$
$30$	0	0
$31$	9.08052i	1.63091i	0.578821	+	0.815455i	$0.303513\pi$
$31$	9.08052i	1.63091i	−0.578821	+	0.815455i	$0.696487\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	5.91359i	1.01417i
$35$	0	0
$36$	0	0
$37$	−0.242641	−0.0398899	−0.0199449	−	0.999801i	$-0.506349\pi$
$37$	−0.242641	−0.0398899	−0.0199449	+	0.999801i	$0.506349\pi$
$38$	−5.91359	−0.959311
$39$	0	0
$40$	− 0.717439i	− 0.113437i
$41$	−11.8272	−1.84710	−0.923548	−	0.383483i	$-0.874724\pi$
$41$	−11.8272	−1.84710	−0.923548	+	0.383483i	$0.874724\pi$
$42$	0	0
$43$	−0.242641	−0.0370024	−0.0185012	−	0.999829i	$-0.505889\pi$
$43$	−0.242641	−0.0370024	−0.0185012	+	0.999829i	$0.505889\pi$
$44$	3.00000i	0.452267i
$45$	0	0
$46$	−4.24264	−0.625543
$47$	−5.91359	−0.862586	−0.431293	−	0.902212i	$-0.641942\pi$
$47$	−5.91359	−0.862586	−0.431293	+	0.902212i	$0.641942\pi$
$48$	0	0
$49$	0	0
$50$	4.48528i	0.634315i
$51$	0	0
$52$	− 2.44949i	− 0.339683i
$53$	− 7.24264i	− 0.994853i	−0.867506	−	0.497427i	$-0.834278\pi$
$53$	− 7.24264i	− 0.994853i	0.867506	−	0.497427i	$-0.165722\pi$
$54$	0	0
$55$	2.15232i	0.290218i
$56$	0	0
$57$	0	0
$58$	−7.24264	−0.951005
$59$	8.06591	1.05009	0.525046	−	0.851074i	$-0.324048\pi$
$59$	8.06591	1.05009	0.525046	+	0.851074i	$0.324048\pi$
$60$	0	0
$61$	1.01461i	0.129908i	0.997888	+	0.0649539i	$0.0206900\pi$
$61$	1.01461i	0.129908i	−0.997888	+	0.0649539i	$0.979310\pi$
$62$	9.08052	1.15323
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 1.75736i	− 0.217974i
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	5.91359	0.717128
$69$	0	0
$70$	0	0
$71$	1.75736i	0.208560i	0.994548	+	0.104280i	$0.0332538\pi$
$71$	1.75736i	0.208560i	−0.994548	+	0.104280i	$0.966746\pi$
$72$	0	0
$73$	1.43488i	0.167940i	0.996468	+	0.0839699i	$0.0267600\pi$
$73$	1.43488i	0.167940i	−0.996468	+	0.0839699i	$0.973240\pi$
$74$	0.242641i	0.0282064i
$75$	0	0
$76$	5.91359i	0.678335i
$77$	0	0
$78$	0	0
$79$	−2.75736	−0.310227	−0.155114	−	0.987897i	$-0.549574\pi$
$79$	−2.75736	−0.310227	−0.155114	+	0.987897i	$0.549574\pi$
$80$	−0.717439	−0.0802121
$81$	0	0
$82$	11.8272i	1.30609i
$83$	−6.63103	−0.727850	−0.363925	−	0.931428i	$-0.618564\pi$
$83$	−6.63103	−0.727850	−0.363925	+	0.931428i	$0.618564\pi$
$84$	0	0
$85$	4.24264	0.460179
$86$	0.242641i	0.0261646i
$87$	0	0
$88$	3.00000	0.319801
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	0	0
$91$	0	0
$92$	4.24264i	0.442326i
$93$	0	0
$94$	5.91359i	0.609940i
$95$	4.24264i	0.435286i
$96$	0	0
$97$	− 13.5592i	− 1.37673i	−0.725364	−	0.688366i	$-0.758328\pi$
$97$	− 13.5592i	− 1.37673i	0.725364	−	0.688366i	$-0.241672\pi$
$98$	0	0
$99$	0	0
$100$	4.48528	0.448528
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 5.49333i	− 0.541273i	−0.962682	−	0.270637i	$-0.912766\pi$
$103$	− 5.49333i	− 0.541273i	0.962682	−	0.270637i	$-0.0872342\pi$
$104$	−2.44949	−0.240192
$105$	0	0
$106$	−7.24264	−0.703467
$107$	11.4853i	1.11032i	0.831742	+	0.555162i	$0.187344\pi$
$107$	11.4853i	1.11032i	−0.831742	+	0.555162i	$0.812656\pi$
$108$	0	0
$109$	18.4853	1.77057	0.885284	−	0.465050i	$-0.153964\pi$
$109$	18.4853	1.77057	0.885284	+	0.465050i	$0.153964\pi$
$110$	2.15232	0.205215
$111$	0	0
$112$	0	0
$113$	− 8.48528i	− 0.798228i	−0.916901	−	0.399114i	$-0.869318\pi$
$113$	− 8.48528i	− 0.798228i	0.916901	−	0.399114i	$-0.130682\pi$
$114$	0	0
$115$	3.04384i	0.283839i
$116$	7.24264i	0.672462i
$117$	0	0
$118$	− 8.06591i	− 0.742527i
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	1.01461	0.0918586
$123$	0	0
$124$	− 9.08052i	− 0.815455i
$125$	6.80511	0.608668
$126$	0	0
$127$	3.24264	0.287738	0.143869	−	0.989597i	$-0.454046\pi$
$127$	3.24264	0.287738	0.143869	+	0.989597i	$0.454046\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−1.75736	−0.154131
$131$	−5.19615	−0.453990	−0.226995	−	0.973896i	$-0.572890\pi$
$131$	−5.19615	−0.453990	−0.226995	+	0.973896i	$0.572890\pi$
$132$	0	0
$133$	0	0
$134$	10.0000i	0.863868i
$135$	0	0
$136$	− 5.91359i	− 0.507086i
$137$	− 2.48528i	− 0.212332i	−0.994348	−	0.106166i	$-0.966143\pi$
$137$	− 2.48528i	− 0.212332i	0.994348	−	0.106166i	$-0.0338575\pi$
$138$	0	0
$139$	0.594346i	0.0504118i	0.999682	+	0.0252059i	$0.00802413\pi$
$139$	0.594346i	0.0504118i	−0.999682	+	0.0252059i	$0.991976\pi$
$140$	0	0
$141$	0	0
$142$	1.75736	0.147474
$143$	7.34847	0.614510
$144$	0	0
$145$	5.19615i	0.431517i
$146$	1.43488	0.118751
$147$	0	0
$148$	0.242641	0.0199449
$149$	3.51472i	0.287937i	0.989582	+	0.143968i	$0.0459864\pi$
$149$	3.51472i	0.287937i	−0.989582	+	0.143968i	$0.954014\pi$
$150$	0	0
$151$	5.24264	0.426640	0.213320	−	0.976982i	$-0.431572\pi$
$151$	5.24264	0.426640	0.213320	+	0.976982i	$0.431572\pi$
$152$	5.91359	0.479656
$153$	0	0
$154$	0	0
$155$	− 6.51472i	− 0.523275i
$156$	0	0
$157$	− 14.6969i	− 1.17294i	−0.809970	−	0.586472i	$-0.800517\pi$
$157$	− 14.6969i	− 1.17294i	0.809970	−	0.586472i	$-0.199483\pi$
$158$	2.75736i	0.219364i
$159$	0	0
$160$	0.717439i	0.0567185i
$161$	0	0
$162$	0	0
$163$	−2.24264	−0.175657	−0.0878286	−	0.996136i	$-0.527993\pi$
$163$	−2.24264	−0.175657	−0.0878286	+	0.996136i	$0.527993\pi$
$164$	11.8272	0.923548
$165$	0	0
$166$	6.63103i	0.514668i
$167$	16.1318	1.24832	0.624159	−	0.781298i	$-0.285442\pi$
$167$	16.1318	1.24832	0.624159	+	0.781298i	$0.285442\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	− 4.24264i	− 0.325396i
$171$	0	0
$172$	0.242641	0.0185012
$173$	−20.7846	−1.58022	−0.790112	−	0.612962i	$-0.789978\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$174$	0	0
$175$	0	0
$176$	− 3.00000i	− 0.226134i
$177$	0	0
$178$	− 10.3923i	− 0.778936i
$179$	26.4853i	1.97960i	0.142454	+	0.989801i	$0.454501\pi$
$179$	26.4853i	1.97960i	−0.142454	+	0.989801i	$0.545499\pi$
$180$	0	0
$181$	− 11.8272i	− 0.879108i	−0.898216	−	0.439554i	$-0.855137\pi$
$181$	− 11.8272i	− 0.879108i	0.898216	−	0.439554i	$-0.144863\pi$
$182$	0	0
$183$	0	0
$184$	4.24264	0.312772
$185$	0.174080	0.0127986
$186$	0	0
$187$	17.7408i	1.29733i
$188$	5.91359	0.431293
$189$	0	0
$190$	4.24264	0.307794
$191$	− 8.48528i	− 0.613973i	−0.951714	−	0.306987i	$-0.900679\pi$
$191$	− 8.48528i	− 0.613973i	0.951714	−	0.306987i	$-0.0993207\pi$
$192$	0	0
$193$	9.48528	0.682765	0.341383	−	0.939924i	$-0.389105\pi$
$193$	9.48528	0.682765	0.341383	+	0.939924i	$0.389105\pi$
$194$	−13.5592	−0.973496
$195$	0	0
$196$	0	0
$197$	26.4853i	1.88700i	0.331375	+	0.943499i	$0.392487\pi$
$197$	26.4853i	1.88700i	−0.331375	+	0.943499i	$0.607513\pi$
$198$	0	0
$199$	− 23.0600i	− 1.63468i	−0.576155	−	0.817341i	$-0.695447\pi$
$199$	− 23.0600i	− 1.63468i	0.576155	−	0.817341i	$-0.304553\pi$
$200$	− 4.48528i	− 0.317157i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.48528	0.592638
$206$	−5.49333	−0.382738
$207$	0	0
$208$	2.44949i	0.169842i
$209$	−17.7408	−1.22716
$210$	0	0
$211$	−0.242641	−0.0167041	−0.00835204	−	0.999965i	$-0.502659\pi$
$211$	−0.242641	−0.0167041	−0.00835204	+	0.999965i	$0.502659\pi$
$212$	7.24264i	0.497427i
$213$	0	0
$214$	11.4853	0.785118
$215$	0.174080	0.0118721
$216$	0	0
$217$	0	0
$218$	− 18.4853i	− 1.25198i
$219$	0	0
$220$	− 2.15232i	− 0.145109i
$221$	− 14.4853i	− 0.974385i
$222$	0	0
$223$	− 2.15232i	− 0.144130i	−0.997400	−	0.0720649i	$-0.977041\pi$
$223$	− 2.15232i	− 0.144130i	0.997400	−	0.0720649i	$-0.0229589\pi$
$224$	0	0
$225$	0	0
$226$	−8.48528	−0.564433
$227$	15.5885	1.03464	0.517321	−	0.855791i	$-0.326929\pi$
$227$	15.5885	1.03464	0.517321	+	0.855791i	$0.326929\pi$
$228$	0	0
$229$	− 13.8564i	− 0.915657i	−0.889041	−	0.457829i	$-0.848627\pi$
$229$	− 13.8564i	− 0.915657i	0.889041	−	0.457829i	$-0.151373\pi$
$230$	3.04384	0.200705
$231$	0	0
$232$	7.24264	0.475503
$233$	− 18.7279i	− 1.22691i	−0.789731	−	0.613453i	$-0.789780\pi$
$233$	− 18.7279i	− 1.22691i	0.789731	−	0.613453i	$-0.210220\pi$
$234$	0	0
$235$	4.24264	0.276759
$236$	−8.06591	−0.525046
$237$	0	0
$238$	0	0
$239$	− 12.7279i	− 0.823301i	−0.911342	−	0.411650i	$-0.864952\pi$
$239$	− 12.7279i	− 0.823301i	0.911342	−	0.411650i	$-0.135048\pi$
$240$	0	0
$241$	− 7.22538i	− 0.465427i	−0.972545	−	0.232714i	$-0.925239\pi$
$241$	− 7.22538i	− 0.465427i	0.972545	−	0.232714i	$-0.0747605\pi$
$242$	− 2.00000i	− 0.128565i
$243$	0	0
$244$	− 1.01461i	− 0.0649539i
$245$	0	0
$246$	0	0
$247$	14.4853	0.921676
$248$	−9.08052	−0.576614
$249$	0	0
$250$	− 6.80511i	− 0.430393i
$251$	27.4156	1.73046	0.865230	−	0.501375i	$-0.167172\pi$
$251$	27.4156	1.73046	0.865230	+	0.501375i	$0.167172\pi$
$252$	0	0
$253$	−12.7279	−0.800198
$254$	− 3.24264i	− 0.203461i
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.30463	0.268516	0.134258	−	0.990946i	$-0.457135\pi$
$257$	4.30463	0.268516	0.134258	+	0.990946i	$0.457135\pi$
$258$	0	0
$259$	0	0
$260$	1.75736i	0.108987i
$261$	0	0
$262$	5.19615i	0.321019i
$263$	15.2132i	0.938086i	0.883175	+	0.469043i	$0.155401\pi$
$263$	15.2132i	0.938086i	−0.883175	+	0.469043i	$0.844599\pi$
$264$	0	0
$265$	5.19615i	0.319197i
$266$	0	0
$267$	0	0
$268$	10.0000	0.610847
$269$	−13.9795	−0.852345	−0.426173	−	0.904642i	$-0.640138\pi$
$269$	−13.9795	−0.852345	−0.426173	+	0.904642i	$0.640138\pi$
$270$	0	0
$271$	− 6.21076i	− 0.377277i	−0.982047	−	0.188639i	$-0.939593\pi$
$271$	− 6.21076i	− 0.377277i	0.982047	−	0.188639i	$-0.0604075\pi$
$272$	−5.91359	−0.358564
$273$	0	0
$274$	−2.48528	−0.150141
$275$	13.4558i	0.811418i
$276$	0	0
$277$	12.9706	0.779326	0.389663	−	0.920958i	$-0.372592\pi$
$277$	12.9706	0.779326	0.389663	+	0.920958i	$0.372592\pi$
$278$	0.594346	0.0356465
$279$	0	0
$280$	0	0
$281$	− 6.00000i	− 0.357930i	−0.983855	−	0.178965i	$-0.942725\pi$
$281$	− 6.00000i	− 0.357930i	0.983855	−	0.178965i	$-0.0572749\pi$
$282$	0	0
$283$	21.2049i	1.26050i	0.776393	+	0.630250i	$0.217047\pi$
$283$	21.2049i	1.26050i	−0.776393	+	0.630250i	$0.782953\pi$
$284$	− 1.75736i	− 0.104280i
$285$	0	0
$286$	− 7.34847i	− 0.434524i
$287$	0	0
$288$	0	0
$289$	17.9706	1.05709
$290$	5.19615	0.305129
$291$	0	0
$292$	− 1.43488i	− 0.0839699i
$293$	0.717439	0.0419132	0.0209566	−	0.999780i	$-0.493329\pi$
$293$	0.717439	0.0419132	0.0209566	+	0.999780i	$0.493329\pi$
$294$	0	0
$295$	−5.78680	−0.336920
$296$	− 0.242641i	− 0.0141032i
$297$	0	0
$298$	3.51472	0.203602
$299$	10.3923	0.601003
$300$	0	0
$301$	0	0
$302$	− 5.24264i	− 0.301680i
$303$	0	0
$304$	− 5.91359i	− 0.339168i
$305$	− 0.727922i	− 0.0416807i
$306$	0	0
$307$	9.97204i	0.569134i	0.958656	+	0.284567i	$0.0918499\pi$
$307$	9.97204i	0.569134i	−0.958656	+	0.284567i	$0.908150\pi$
$308$	0	0
$309$	0	0
$310$	−6.51472	−0.370011
$311$	−8.95743	−0.507929	−0.253965	−	0.967214i	$-0.581735\pi$
$311$	−8.95743	−0.507929	−0.253965	+	0.967214i	$0.581735\pi$
$312$	0	0
$313$	18.4582i	1.04332i	0.853154	+	0.521660i	$0.174687\pi$
$313$	18.4582i	1.04332i	−0.853154	+	0.521660i	$0.825313\pi$
$314$	−14.6969	−0.829396
$315$	0	0
$316$	2.75736	0.155114
$317$	− 1.24264i	− 0.0697937i	−0.999391	−	0.0348968i	$-0.988890\pi$
$317$	− 1.24264i	− 0.0697937i	0.999391	−	0.0348968i	$-0.0111103\pi$
$318$	0	0
$319$	−21.7279	−1.21653
$320$	0.717439	0.0401061
$321$	0	0
$322$	0	0
$323$	34.9706i	1.94581i
$324$	0	0
$325$	− 10.9867i	− 0.609430i
$326$	2.24264i	0.124208i
$327$	0	0
$328$	− 11.8272i	− 0.653047i
$329$	0	0
$330$	0	0
$331$	−33.4558	−1.83890	−0.919450	−	0.393208i	$-0.871365\pi$
$331$	−33.4558	−1.83890	−0.919450	+	0.393208i	$0.871365\pi$
$332$	6.63103	0.363925
$333$	0	0
$334$	− 16.1318i	− 0.882694i
$335$	7.17439	0.391979
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	− 7.00000i	− 0.380750i
$339$	0	0
$340$	−4.24264	−0.230089
$341$	27.2416	1.47521
$342$	0	0
$343$	0	0
$344$	− 0.242641i	− 0.0130823i
$345$	0	0
$346$	20.7846i	1.11739i
$347$	2.48528i	0.133417i	0.997773	+	0.0667084i	$0.0212497\pi$
$347$	2.48528i	0.133417i	−0.997773	+	0.0667084i	$0.978750\pi$
$348$	0	0
$349$	2.27541i	0.121800i	0.998144	+	0.0608999i	$0.0193971\pi$
$349$	2.27541i	0.121800i	−0.998144	+	0.0608999i	$0.980603\pi$
$350$	0	0
$351$	0	0
$352$	−3.00000	−0.159901
$353$	−8.95743	−0.476756	−0.238378	−	0.971172i	$-0.576616\pi$
$353$	−8.95743	−0.476756	−0.238378	+	0.971172i	$0.576616\pi$
$354$	0	0
$355$	− 1.26080i	− 0.0669162i
$356$	−10.3923	−0.550791
$357$	0	0
$358$	26.4853	1.39979
$359$	18.0000i	0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$	18.0000i	0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$	0	0
$361$	−15.9706	−0.840556
$362$	−11.8272	−0.621623
$363$	0	0
$364$	0	0
$365$	− 1.02944i	− 0.0538832i
$366$	0	0
$367$	15.4144i	0.804624i	0.915503	+	0.402312i	$0.131793\pi$
$367$	15.4144i	0.804624i	−0.915503	+	0.402312i	$0.868207\pi$
$368$	− 4.24264i	− 0.221163i
$369$	0	0
$370$	− 0.174080i	− 0.00904998i
$371$	0	0
$372$	0	0
$373$	−29.4558	−1.52517	−0.762583	−	0.646891i	$-0.776069\pi$
$373$	−29.4558	−1.52517	−0.762583	+	0.646891i	$0.776069\pi$
$374$	17.7408	0.917354
$375$	0	0
$376$	− 5.91359i	− 0.304970i
$377$	17.7408	0.913696
$378$	0	0
$379$	12.4853	0.641326	0.320663	−	0.947193i	$-0.396094\pi$
$379$	12.4853	0.641326	0.320663	+	0.947193i	$0.396094\pi$
$380$	− 4.24264i	− 0.217643i
$381$	0	0
$382$	−8.48528	−0.434145
$383$	−22.2195	−1.13536	−0.567681	−	0.823248i	$-0.692159\pi$
$383$	−22.2195	−1.13536	−0.567681	+	0.823248i	$0.692159\pi$
$384$	0	0
$385$	0	0
$386$	− 9.48528i	− 0.482788i
$387$	0	0
$388$	13.5592i	0.688366i
$389$	− 31.4558i	− 1.59487i	−0.603402	−	0.797437i	$-0.706188\pi$
$389$	− 31.4558i	− 1.59487i	0.603402	−	0.797437i	$-0.293812\pi$
$390$	0	0
$391$	25.0892i	1.26882i
$392$	0	0
$393$	0	0
$394$	26.4853	1.33431
$395$	1.97824	0.0995359
$396$	0	0
$397$	13.8564i	0.695433i	0.937600	+	0.347717i	$0.113043\pi$
$397$	13.8564i	0.695433i	−0.937600	+	0.347717i	$0.886957\pi$
$398$	−23.0600	−1.15589
$399$	0	0
$400$	−4.48528	−0.224264
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−22.2426	−1.10798
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.727922i	0.0360818i
$408$	0	0
$409$	− 14.9941i	− 0.741411i	−0.928750	−	0.370706i	$-0.879116\pi$
$409$	− 14.9941i	− 0.741411i	0.928750	−	0.370706i	$-0.120884\pi$
$410$	− 8.48528i	− 0.419058i
$411$	0	0
$412$	5.49333i	0.270637i
$413$	0	0
$414$	0	0
$415$	4.75736	0.233530
$416$	2.44949	0.120096
$417$	0	0
$418$	17.7408i	0.867730i
$419$	−23.6544	−1.15559	−0.577796	−	0.816181i	$-0.696087\pi$
$419$	−23.6544	−1.15559	−0.577796	+	0.816181i	$0.696087\pi$
$420$	0	0
$421$	−14.2426	−0.694144	−0.347072	−	0.937839i	$-0.612824\pi$
$421$	−14.2426	−0.694144	−0.347072	+	0.937839i	$0.612824\pi$
$422$	0.242641i	0.0118116i
$423$	0	0
$424$	7.24264	0.351734
$425$	26.5241	1.28661
$426$	0	0
$427$	0	0
$428$	− 11.4853i	− 0.555162i
$429$	0	0
$430$	− 0.174080i	− 0.00839488i
$431$	− 3.51472i	− 0.169298i	−0.996411	−	0.0846490i	$-0.973023\pi$
$431$	− 3.51472i	− 0.169298i	0.996411	−	0.0846490i	$-0.0269769\pi$
$432$	0	0
$433$	3.46410i	0.166474i	0.996530	+	0.0832370i	$0.0265259\pi$
$433$	3.46410i	0.166474i	−0.996530	+	0.0832370i	$0.973474\pi$
$434$	0	0
$435$	0	0
$436$	−18.4853	−0.885284
$437$	−25.0892	−1.20018
$438$	0	0
$439$	− 16.8493i	− 0.804171i	−0.915602	−	0.402086i	$-0.868285\pi$
$439$	− 16.8493i	− 0.804171i	0.915602	−	0.402086i	$-0.131715\pi$
$440$	−2.15232	−0.102608
$441$	0	0
$442$	−14.4853	−0.688995
$443$	16.4558i	0.781841i	0.920425	+	0.390920i	$0.127843\pi$
$443$	16.4558i	0.781841i	−0.920425	+	0.390920i	$0.872157\pi$
$444$	0	0
$445$	−7.45584	−0.353441
$446$	−2.15232	−0.101915
$447$	0	0
$448$	0	0
$449$	1.75736i	0.0829349i	0.999140	+	0.0414675i	$0.0132033\pi$
$449$	1.75736i	0.0829349i	−0.999140	+	0.0414675i	$0.986797\pi$
$450$	0	0
$451$	35.4815i	1.67076i
$452$	8.48528i	0.399114i
$453$	0	0
$454$	− 15.5885i	− 0.731603i
$455$	0	0
$456$	0	0
$457$	−23.0000	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$457$	−23.0000	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$458$	−13.8564	−0.647467
$459$	0	0
$460$	− 3.04384i	− 0.141920i
$461$	−32.6118	−1.51888	−0.759441	−	0.650576i	$-0.774528\pi$
$461$	−32.6118	−1.51888	−0.759441	+	0.650576i	$0.774528\pi$
$462$	0	0
$463$	−29.4558	−1.36893	−0.684465	−	0.729046i	$-0.739964\pi$
$463$	−29.4558	−1.36893	−0.684465	+	0.729046i	$0.739964\pi$
$464$	− 7.24264i	− 0.336231i
$465$	0	0
$466$	−18.7279	−0.867554
$467$	39.7862	1.84108	0.920542	−	0.390643i	$-0.127747\pi$
$467$	39.7862	1.84108	0.920542	+	0.390643i	$0.127747\pi$
$468$	0	0
$469$	0	0
$470$	− 4.24264i	− 0.195698i
$471$	0	0
$472$	8.06591i	0.371264i
$473$	0.727922i	0.0334699i
$474$	0	0
$475$	26.5241i	1.21701i
$476$	0	0
$477$	0	0
$478$	−12.7279	−0.582162
$479$	12.0013	0.548352	0.274176	−	0.961680i	$-0.411595\pi$
$479$	12.0013	0.548352	0.274176	+	0.961680i	$0.411595\pi$
$480$	0	0
$481$	− 0.594346i	− 0.0270998i
$482$	−7.22538	−0.329107
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	9.72792i	0.441722i
$486$	0	0
$487$	−14.2132	−0.644062	−0.322031	−	0.946729i	$-0.604365\pi$
$487$	−14.2132	−0.644062	−0.322031	+	0.946729i	$0.604365\pi$
$488$	−1.01461	−0.0459293
$489$	0	0
$490$	0	0
$491$	13.9706i	0.630483i	0.949012	+	0.315241i	$0.102085\pi$
$491$	13.9706i	0.630483i	−0.949012	+	0.315241i	$0.897915\pi$
$492$	0	0
$493$	42.8300i	1.92897i
$494$	− 14.4853i	− 0.651724i
$495$	0	0
$496$	9.08052i	0.407727i
$497$	0	0
$498$	0	0
$499$	31.9411	1.42988	0.714941	−	0.699185i	$-0.246454\pi$
$499$	31.9411	1.42988	0.714941	+	0.699185i	$0.246454\pi$
$500$	−6.80511	−0.304334
$501$	0	0
$502$	− 27.4156i	− 1.22362i
$503$	31.0028	1.38235	0.691174	−	0.722688i	$-0.257094\pi$
$503$	31.0028	1.38235	0.691174	+	0.722688i	$0.257094\pi$
$504$	0	0
$505$	0	0
$506$	12.7279i	0.565825i
$507$	0	0
$508$	−3.24264	−0.143869
$509$	17.1974	0.762262	0.381131	−	0.924521i	$-0.375535\pi$
$509$	17.1974	0.762262	0.381131	+	0.924521i	$0.375535\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	− 4.30463i	− 0.189869i
$515$	3.94113i	0.173667i
$516$	0	0
$517$	17.7408i	0.780238i
$518$	0	0
$519$	0	0
$520$	1.75736	0.0770653
$521$	−33.8726	−1.48399	−0.741993	−	0.670408i	$-0.766119\pi$
$521$	−33.8726	−1.48399	−0.741993	+	0.670408i	$0.766119\pi$
$522$	0	0
$523$	6.75412i	0.295337i	0.989037	+	0.147669i	$0.0471769\pi$
$523$	6.75412i	0.295337i	−0.989037	+	0.147669i	$0.952823\pi$
$524$	5.19615	0.226995
$525$	0	0
$526$	15.2132	0.663327
$527$	− 53.6985i	− 2.33914i
$528$	0	0
$529$	5.00000	0.217391
$530$	5.19615	0.225706
$531$	0	0
$532$	0	0
$533$	− 28.9706i	− 1.25485i
$534$	0	0
$535$	− 8.23999i	− 0.356246i
$536$	− 10.0000i	− 0.431934i
$537$	0	0
$538$	13.9795i	0.602699i
$539$	0	0
$540$	0	0
$541$	14.7279	0.633203	0.316601	−	0.948559i	$-0.397458\pi$
$541$	14.7279	0.633203	0.316601	+	0.948559i	$0.397458\pi$
$542$	−6.21076	−0.266775
$543$	0	0
$544$	5.91359i	0.253543i
$545$	−13.2621	−0.568084
$546$	0	0
$547$	−39.6985	−1.69738	−0.848692	−	0.528887i	$-0.822610\pi$
$547$	−39.6985	−1.69738	−0.848692	+	0.528887i	$0.822610\pi$
$548$	2.48528i	0.106166i
$549$	0	0
$550$	13.4558	0.573759
$551$	−42.8300	−1.82462
$552$	0	0
$553$	0	0
$554$	− 12.9706i	− 0.551066i
$555$	0	0
$556$	− 0.594346i	− 0.0252059i
$557$	9.72792i	0.412185i	0.978532	+	0.206093i	$0.0660748\pi$
$557$	9.72792i	0.412185i	−0.978532	+	0.206093i	$0.933925\pi$
$558$	0	0
$559$	− 0.594346i	− 0.0251382i
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	34.5900	1.45780	0.728898	−	0.684622i	$-0.240033\pi$
$563$	34.5900	1.45780	0.728898	+	0.684622i	$0.240033\pi$
$564$	0	0
$565$	6.08767i	0.256110i
$566$	21.2049	0.891307
$567$	0	0
$568$	−1.75736	−0.0737372
$569$	10.2426i	0.429394i	0.976681	+	0.214697i	$0.0688764\pi$
$569$	10.2426i	0.429394i	−0.976681	+	0.214697i	$0.931124\pi$
$570$	0	0
$571$	8.72792	0.365252	0.182626	−	0.983182i	$-0.441540\pi$
$571$	8.72792	0.365252	0.182626	+	0.983182i	$0.441540\pi$
$572$	−7.34847	−0.307255
$573$	0	0
$574$	0	0
$575$	19.0294i	0.793582i
$576$	0	0
$577$	− 10.6895i	− 0.445009i	−0.974932	−	0.222504i	$-0.928577\pi$
$577$	− 10.6895i	− 0.445009i	0.974932	−	0.222504i	$-0.0714232\pi$
$578$	− 17.9706i	− 0.747477i
$579$	0	0
$580$	− 5.19615i	− 0.215758i
$581$	0	0
$582$	0	0
$583$	−21.7279	−0.899879
$584$	−1.43488	−0.0593757
$585$	0	0
$586$	− 0.717439i	− 0.0296371i
$587$	−5.19615	−0.214468	−0.107234	−	0.994234i	$-0.534199\pi$
$587$	−5.19615	−0.214468	−0.107234	+	0.994234i	$0.534199\pi$
$588$	0	0
$589$	53.6985	2.21261
$590$	5.78680i	0.238239i
$591$	0	0
$592$	−0.242641	−0.00997247
$593$	−23.4803	−0.964220	−0.482110	−	0.876111i	$-0.660129\pi$
$593$	−23.4803	−0.964220	−0.482110	+	0.876111i	$0.660129\pi$
$594$	0	0
$595$	0	0
$596$	− 3.51472i	− 0.143968i
$597$	0	0
$598$	− 10.3923i	− 0.424973i
$599$	− 7.45584i	− 0.304638i	−0.988331	−	0.152319i	$-0.951326\pi$
$599$	− 7.45584i	− 0.304638i	0.988331	−	0.152319i	$-0.0486740\pi$
$600$	0	0
$601$	− 23.3572i	− 0.952760i	−0.879240	−	0.476380i	$-0.841949\pi$
$601$	− 23.3572i	− 0.952760i	0.879240	−	0.476380i	$-0.158051\pi$
$602$	0	0
$603$	0	0
$604$	−5.24264	−0.213320
$605$	−1.43488	−0.0583361
$606$	0	0
$607$	− 20.0672i	− 0.814501i	−0.913316	−	0.407251i	$-0.866488\pi$
$607$	− 20.0672i	− 0.814501i	0.913316	−	0.407251i	$-0.133512\pi$
$608$	−5.91359	−0.239828
$609$	0	0
$610$	−0.727922	−0.0294727
$611$	− 14.4853i	− 0.586012i
$612$	0	0
$613$	−37.2132	−1.50303	−0.751514	−	0.659718i	$-0.770676\pi$
$613$	−37.2132	−1.50303	−0.751514	+	0.659718i	$0.770676\pi$
$614$	9.97204	0.402439
$615$	0	0
$616$	0	0
$617$	17.6985i	0.712514i	0.934388	+	0.356257i	$0.115947\pi$
$617$	17.6985i	0.712514i	−0.934388	+	0.356257i	$0.884053\pi$
$618$	0	0
$619$	− 6.15978i	− 0.247582i	−0.992308	−	0.123791i	$-0.960495\pi$
$619$	− 6.15978i	− 0.247582i	0.992308	−	0.123791i	$-0.0395053\pi$
$620$	6.51472i	0.261637i
$621$	0	0
$622$	8.95743i	0.359160i
$623$	0	0
$624$	0	0
$625$	17.5442	0.701766
$626$	18.4582	0.737739
$627$	0	0
$628$	14.6969i	0.586472i
$629$	1.43488	0.0572123
$630$	0	0
$631$	24.7574	0.985575	0.492787	−	0.870150i	$-0.335978\pi$
$631$	24.7574	0.985575	0.492787	+	0.870150i	$0.335978\pi$
$632$	− 2.75736i	− 0.109682i
$633$	0	0
$634$	−1.24264	−0.0493516
$635$	−2.32640	−0.0923202
$636$	0	0
$637$	0	0
$638$	21.7279i	0.860217i
$639$	0	0
$640$	− 0.717439i	− 0.0283593i
$641$	− 17.6985i	− 0.699048i	−0.936927	−	0.349524i	$-0.886343\pi$
$641$	− 17.6985i	− 0.699048i	0.936927	−	0.349524i	$-0.113657\pi$
$642$	0	0
$643$	− 32.0174i	− 1.26264i	−0.775520	−	0.631322i	$-0.782512\pi$
$643$	− 32.0174i	− 1.26264i	0.775520	−	0.631322i	$-0.217488\pi$
$644$	0	0
$645$	0	0
$646$	34.9706	1.37590
$647$	−32.4377	−1.27526	−0.637629	−	0.770344i	$-0.720085\pi$
$647$	−32.4377	−1.27526	−0.637629	+	0.770344i	$0.720085\pi$
$648$	0	0
$649$	− 24.1977i	− 0.949844i
$650$	−10.9867	−0.430932
$651$	0	0
$652$	2.24264	0.0878286
$653$	− 19.2426i	− 0.753023i	−0.926412	−	0.376511i	$-0.877124\pi$
$653$	− 19.2426i	− 0.753023i	0.926412	−	0.376511i	$-0.122876\pi$
$654$	0	0
$655$	3.72792	0.145662
$656$	−11.8272	−0.461774
$657$	0	0
$658$	0	0
$659$	− 6.00000i	− 0.233727i	−0.993148	−	0.116863i	$-0.962716\pi$
$659$	− 6.00000i	− 0.233727i	0.993148	−	0.116863i	$-0.0372840\pi$
$660$	0	0
$661$	− 35.6556i	− 1.38684i	−0.720532	−	0.693422i	$-0.756102\pi$
$661$	− 35.6556i	− 1.38684i	0.720532	−	0.693422i	$-0.243898\pi$
$662$	33.4558i	1.30030i
$663$	0	0
$664$	− 6.63103i	− 0.257334i
$665$	0	0
$666$	0	0
$667$	−30.7279	−1.18979
$668$	−16.1318	−0.624159
$669$	0	0
$670$	− 7.17439i	− 0.277171i
$671$	3.04384	0.117506
$672$	0	0
$673$	17.9706	0.692714	0.346357	−	0.938103i	$-0.387419\pi$
$673$	17.9706	0.692714	0.346357	+	0.938103i	$0.387419\pi$
$674$	5.00000i	0.192593i
$675$	0	0
$676$	−7.00000	−0.269231
$677$	2.15232	0.0827203	0.0413601	−	0.999144i	$-0.486831\pi$
$677$	2.15232	0.0827203	0.0413601	+	0.999144i	$0.486831\pi$
$678$	0	0
$679$	0	0
$680$	4.24264i	0.162698i
$681$	0	0
$682$	− 27.2416i	− 1.04313i
$683$	− 7.97056i	− 0.304985i	−0.988305	−	0.152493i	$-0.951270\pi$
$683$	− 7.97056i	− 0.304985i	0.988305	−	0.152493i	$-0.0487300\pi$
$684$	0	0
$685$	1.78304i	0.0681264i
$686$	0	0
$687$	0	0
$688$	−0.242641	−0.00925059
$689$	17.7408	0.675870
$690$	0	0
$691$	28.5533i	1.08622i	0.839661	+	0.543110i	$0.182753\pi$
$691$	28.5533i	1.08622i	−0.839661	+	0.543110i	$0.817247\pi$
$692$	20.7846	0.790112
$693$	0	0
$694$	2.48528	0.0943400
$695$	− 0.426407i	− 0.0161745i
$696$	0	0
$697$	69.9411	2.64921
$698$	2.27541	0.0861255
$699$	0	0
$700$	0	0
$701$	− 20.6985i	− 0.781771i	−0.920439	−	0.390885i	$-0.872169\pi$
$701$	− 20.6985i	− 0.781771i	0.920439	−	0.390885i	$-0.127831\pi$
$702$	0	0
$703$	1.43488i	0.0541174i
$704$	3.00000i	0.113067i
$705$	0	0
$706$	8.95743i	0.337117i
$707$	0	0
$708$	0	0
$709$	26.9706	1.01290	0.506450	−	0.862269i	$-0.330957\pi$
$709$	26.9706	1.01290	0.506450	+	0.862269i	$0.330957\pi$
$710$	−1.26080	−0.0473169
$711$	0	0
$712$	10.3923i	0.389468i
$713$	38.5254	1.44279
$714$	0	0
$715$	−5.27208	−0.197165
$716$	− 26.4853i	− 0.989801i
$717$	0	0
$718$	18.0000	0.671754
$719$	−16.1318	−0.601615	−0.300808	−	0.953685i	$-0.597256\pi$
$719$	−16.1318	−0.601615	−0.300808	+	0.953685i	$0.597256\pi$
$720$	0	0
$721$	0	0
$722$	15.9706i	0.594363i
$723$	0	0
$724$	11.8272i	0.439554i
$725$	32.4853i	1.20647i
$726$	0	0
$727$	11.7041i	0.434081i	0.976163	+	0.217040i	$0.0696403\pi$
$727$	11.7041i	0.434081i	−0.976163	+	0.217040i	$0.930360\pi$
$728$	0	0
$729$	0	0
$730$	−1.02944	−0.0381012
$731$	1.43488	0.0530709
$732$	0	0
$733$	4.72490i	0.174518i	0.996186	+	0.0872591i	$0.0278108\pi$
$733$	4.72490i	0.174518i	−0.996186	+	0.0872591i	$0.972189\pi$
$734$	15.4144	0.568955
$735$	0	0
$736$	−4.24264	−0.156386
$737$	30.0000i	1.10506i
$738$	0	0
$739$	15.4558	0.568552	0.284276	−	0.958742i	$-0.408247\pi$
$739$	15.4558	0.568552	0.284276	+	0.958742i	$0.408247\pi$
$740$	−0.174080	−0.00639930
$741$	0	0
$742$	0	0
$743$	− 38.4853i	− 1.41189i	−0.708268	−	0.705944i	$-0.750523\pi$
$743$	− 38.4853i	− 1.41189i	0.708268	−	0.705944i	$-0.249477\pi$
$744$	0	0
$745$	− 2.52160i	− 0.0923841i
$746$	29.4558i	1.07846i
$747$	0	0
$748$	− 17.7408i	− 0.648667i
$749$	0	0
$750$	0	0
$751$	35.2426	1.28602	0.643011	−	0.765857i	$-0.277685\pi$
$751$	35.2426	1.28602	0.643011	+	0.765857i	$0.277685\pi$
$752$	−5.91359	−0.215646
$753$	0	0
$754$	− 17.7408i	− 0.646081i
$755$	−3.76127	−0.136887
$756$	0	0
$757$	−33.7574	−1.22693	−0.613466	−	0.789721i	$-0.710225\pi$
$757$	−33.7574	−1.22693	−0.613466	+	0.789721i	$0.710225\pi$
$758$	− 12.4853i	− 0.453486i
$759$	0	0
$760$	−4.24264	−0.153897
$761$	−29.5680	−1.07184	−0.535919	−	0.844270i	$-0.680035\pi$
$761$	−29.5680	−1.07184	−0.535919	+	0.844270i	$0.680035\pi$
$762$	0	0
$763$	0	0
$764$	8.48528i	0.306987i
$765$	0	0
$766$	22.2195i	0.802823i
$767$	19.7574i	0.713397i
$768$	0	0
$769$	− 9.84895i	− 0.355162i	−0.984106	−	0.177581i	$-0.943173\pi$
$769$	− 9.84895i	− 0.355162i	0.984106	−	0.177581i	$-0.0568272\pi$
$770$	0	0
$771$	0	0
$772$	−9.48528	−0.341383
$773$	16.1318	0.580221	0.290111	−	0.956993i	$-0.406308\pi$
$773$	16.1318	0.580221	0.290111	+	0.956993i	$0.406308\pi$
$774$	0	0
$775$	− 40.7287i	− 1.46302i
$776$	13.5592	0.486748
$777$	0	0
$778$	−31.4558	−1.12775
$779$	69.9411i	2.50590i
$780$	0	0
$781$	5.27208	0.188650
$782$	25.0892	0.897190
$783$	0	0
$784$	0	0
$785$	10.5442i	0.376337i
$786$	0	0
$787$	− 32.1915i	− 1.14750i	−0.819029	−	0.573752i	$-0.805487\pi$
$787$	− 32.1915i	− 1.14750i	0.819029	−	0.573752i	$-0.194513\pi$
$788$	− 26.4853i	− 0.943499i
$789$	0	0
$790$	− 1.97824i	− 0.0703825i
$791$	0	0
$792$	0	0
$793$	−2.48528	−0.0882549
$794$	13.8564	0.491745
$795$	0	0
$796$	23.0600i	0.817341i
$797$	−6.45695	−0.228717	−0.114358	−	0.993440i	$-0.536481\pi$
$797$	−6.45695	−0.228717	−0.114358	+	0.993440i	$0.536481\pi$
$798$	0	0
$799$	34.9706	1.23717
$800$	4.48528i	0.158579i
$801$	0	0
$802$	0	0
$803$	4.30463	0.151907
$804$	0	0
$805$	0	0
$806$	22.2426i	0.783464i
$807$	0	0
$808$	0	0
$809$	− 7.02944i	− 0.247142i	−0.992336	−	0.123571i	$-0.960565\pi$
$809$	− 7.02944i	− 0.247142i	0.992336	−	0.123571i	$-0.0394346\pi$
$810$	0	0
$811$	− 31.1769i	− 1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$	− 31.1769i	− 1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$	0	0
$813$	0	0
$814$	0.727922	0.0255137
$815$	1.60896	0.0563593
$816$	0	0
$817$	1.43488i	0.0502000i
$818$	−14.9941	−0.524257
$819$	0	0
$820$	−8.48528	−0.296319
$821$	− 40.7574i	− 1.42244i	−0.702969	−	0.711221i	$-0.748143\pi$
$821$	− 40.7574i	− 1.42244i	0.702969	−	0.711221i	$-0.251857\pi$
$822$	0	0
$823$	29.9411	1.04368	0.521841	−	0.853043i	$-0.325245\pi$
$823$	29.9411	1.04368	0.521841	+	0.853043i	$0.325245\pi$
$824$	5.49333	0.191369
$825$	0	0
$826$	0	0
$827$	− 37.9706i	− 1.32037i	−0.751105	−	0.660183i	$-0.770479\pi$
$827$	− 37.9706i	− 1.32037i	0.751105	−	0.660183i	$-0.229521\pi$
$828$	0	0
$829$	− 13.0880i	− 0.454564i	−0.973829	−	0.227282i	$-0.927016\pi$
$829$	− 13.0880i	− 0.454564i	0.973829	−	0.227282i	$-0.0729840\pi$
$830$	− 4.75736i	− 0.165130i
$831$	0	0
$832$	− 2.44949i	− 0.0849208i
$833$	0	0
$834$	0	0
$835$	−11.5736	−0.400521
$836$	17.7408	0.613578
$837$	0	0
$838$	23.6544i	0.817126i
$839$	−10.2182	−0.352772	−0.176386	−	0.984321i	$-0.556441\pi$
$839$	−10.2182	−0.352772	−0.176386	+	0.984321i	$0.556441\pi$
$840$	0	0
$841$	−23.4558	−0.808822
$842$	14.2426i	0.490834i
$843$	0	0
$844$	0.242641	0.00835204
$845$	−5.02207	−0.172765
$846$	0	0
$847$	0	0
$848$	− 7.24264i	− 0.248713i
$849$	0	0
$850$	− 26.5241i	− 0.909770i
$851$	1.02944i	0.0352887i
$852$	0	0
$853$	− 36.9164i	− 1.26399i	−0.774971	−	0.631997i	$-0.782235\pi$
$853$	− 36.9164i	− 1.26399i	0.774971	−	0.631997i	$-0.217765\pi$
$854$	0	0
$855$	0	0
$856$	−11.4853	−0.392559
$857$	33.8726	1.15707	0.578533	−	0.815659i	$-0.303625\pi$
$857$	33.8726	1.15707	0.578533	+	0.815659i	$0.303625\pi$
$858$	0	0
$859$	9.37769i	0.319963i	0.987120	+	0.159981i	$0.0511434\pi$
$859$	9.37769i	0.319963i	−0.987120	+	0.159981i	$0.948857\pi$
$860$	−0.174080	−0.00593607
$861$	0	0
$862$	−3.51472	−0.119712
$863$	− 33.5147i	− 1.14085i	−0.821348	−	0.570427i	$-0.806778\pi$
$863$	− 33.5147i	− 1.14085i	0.821348	−	0.570427i	$-0.193222\pi$
$864$	0	0
$865$	14.9117	0.507013
$866$	3.46410	0.117715
$867$	0	0
$868$	0	0
$869$	8.27208i	0.280611i
$870$	0	0
$871$	− 24.4949i	− 0.829978i
$872$	18.4853i	0.625991i
$873$	0	0
$874$	25.0892i	0.848656i
$875$	0	0
$876$	0	0
$877$	4.48528	0.151457	0.0757286	−	0.997128i	$-0.475872\pi$
$877$	4.48528	0.151457	0.0757286	+	0.997128i	$0.475872\pi$
$878$	−16.8493	−0.568635
$879$	0	0
$880$	2.15232i	0.0725546i
$881$	−19.0016	−0.640179	−0.320090	−	0.947387i	$-0.603713\pi$
$881$	−19.0016	−0.640179	−0.320090	+	0.947387i	$0.603713\pi$
$882$	0	0
$883$	41.4558	1.39510	0.697550	−	0.716536i	$-0.254273\pi$
$883$	41.4558	1.39510	0.697550	+	0.716536i	$0.254273\pi$
$884$	14.4853i	0.487193i
$885$	0	0
$886$	16.4558	0.552845
$887$	10.5664	0.354784	0.177392	−	0.984140i	$-0.443234\pi$
$887$	10.5664	0.354784	0.177392	+	0.984140i	$0.443234\pi$
$888$	0	0
$889$	0	0
$890$	7.45584i	0.249920i
$891$	0	0
$892$	2.15232i	0.0720649i
$893$	34.9706i	1.17025i
$894$	0	0
$895$	− 19.0016i	− 0.635153i
$896$	0	0
$897$	0	0
$898$	1.75736	0.0586438
$899$	65.7669	2.19345
$900$	0	0
$901$	42.8300i	1.42687i
$902$	35.4815	1.18141
$903$	0	0
$904$	8.48528	0.282216
$905$	8.48528i	0.282060i
$906$	0	0
$907$	−31.6985	−1.05253	−0.526265	−	0.850320i	$-0.676408\pi$
$907$	−31.6985	−1.05253	−0.526265	+	0.850320i	$0.676408\pi$
$908$	−15.5885	−0.517321
$909$	0	0
$910$	0	0
$911$	6.72792i	0.222906i	0.993770	+	0.111453i	$0.0355505\pi$
$911$	6.72792i	0.222906i	−0.993770	+	0.111453i	$0.964450\pi$
$912$	0	0
$913$	19.8931i	0.658365i
$914$	23.0000i	0.760772i
$915$	0	0
$916$	13.8564i	0.457829i
$917$	0	0
$918$	0	0
$919$	36.4853	1.20354	0.601769	−	0.798670i	$-0.294463\pi$
$919$	36.4853	1.20354	0.601769	+	0.798670i	$0.294463\pi$
$920$	−3.04384	−0.100352
$921$	0	0
$922$	32.6118i	1.07401i
$923$	−4.30463	−0.141689
$924$	0	0
$925$	1.08831	0.0357835
$926$	29.4558i	0.967979i
$927$	0	0
$928$	−7.24264	−0.237751
$929$	−31.0028	−1.01717	−0.508585	−	0.861012i	$-0.669831\pi$
$929$	−31.0028	−1.01717	−0.508585	+	0.861012i	$0.669831\pi$
$930$	0	0
$931$	0	0
$932$	18.7279i	0.613453i
$933$	0	0
$934$	− 39.7862i	− 1.30184i
$935$	− 12.7279i	− 0.416248i
$936$	0	0
$937$	35.1844i	1.14942i	0.818356	+	0.574712i	$0.194886\pi$
$937$	35.1844i	1.14942i	−0.818356	+	0.574712i	$0.805114\pi$
$938$	0	0
$939$	0	0
$940$	−4.24264	−0.138380
$941$	27.5897	0.899399	0.449700	−	0.893180i	$-0.351531\pi$
$941$	27.5897	0.899399	0.449700	+	0.893180i	$0.351531\pi$
$942$	0	0
$943$	50.1785i	1.63404i
$944$	8.06591	0.262523
$945$	0	0
$946$	0.727922	0.0236668
$947$	10.9706i	0.356495i	0.983986	+	0.178248i	$0.0570428\pi$
$947$	10.9706i	0.356495i	−0.983986	+	0.178248i	$0.942957\pi$
$948$	0	0
$949$	−3.51472	−0.114093
$950$	26.5241	0.860556
$951$	0	0
$952$	0	0
$953$	17.6985i	0.573310i	0.958034	+	0.286655i	$0.0925434\pi$
$953$	17.6985i	0.573310i	−0.958034	+	0.286655i	$0.907457\pi$
$954$	0	0
$955$	6.08767i	0.196992i
$956$	12.7279i	0.411650i
$957$	0	0
$958$	− 12.0013i	− 0.387743i
$959$	0	0
$960$	0	0
$961$	−51.4558	−1.65987
$962$	−0.594346	−0.0191625
$963$	0	0
$964$	7.22538i	0.232714i
$965$	−6.80511	−0.219064
$966$	0	0
$967$	47.7279	1.53483	0.767413	−	0.641153i	$-0.221544\pi$
$967$	47.7279	1.53483	0.767413	+	0.641153i	$0.221544\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	9.72792	0.312345
$971$	27.0675	0.868637	0.434318	−	0.900759i	$-0.356989\pi$
$971$	27.0675	0.868637	0.434318	+	0.900759i	$0.356989\pi$
$972$	0	0
$973$	0	0
$974$	14.2132i	0.455420i
$975$	0	0
$976$	1.01461i	0.0324769i
$977$	− 40.2426i	− 1.28748i	−0.765246	−	0.643738i	$-0.777383\pi$
$977$	− 40.2426i	− 1.28748i	0.765246	−	0.643738i	$-0.222617\pi$
$978$	0	0
$979$	− 31.1769i	− 0.996419i
$980$	0	0
$981$	0	0
$982$	13.9706	0.445819
$983$	−47.3087	−1.50891	−0.754457	−	0.656349i	$-0.772100\pi$
$983$	−47.3087	−1.50891	−0.754457	+	0.656349i	$0.772100\pi$
$984$	0	0
$985$	− 19.0016i	− 0.605440i
$986$	42.8300	1.36399
$987$	0	0
$988$	−14.4853	−0.460838
$989$	1.02944i	0.0327342i
$990$	0	0
$991$	52.2132	1.65861	0.829304	−	0.558798i	$-0.188737\pi$
$991$	52.2132	1.65861	0.829304	+	0.558798i	$0.188737\pi$
$992$	9.08052	0.288307
$993$	0	0
$994$	0	0
$995$	16.5442i	0.524485i
$996$	0	0
$997$	38.9456i	1.23342i	0.787190	+	0.616711i	$0.211535\pi$
$997$	38.9456i	1.23342i	−0.787190	+	0.616711i	$0.788465\pi$
$998$	− 31.9411i	− 1.01108i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.2.d.a.881.2		8
3.2	odd	2	inner	882.2.d.a.881.7		8
4.3	odd	2		7056.2.k.f.881.4		8
7.2	even	3		126.2.k.a.17.2	✓	8
7.3	odd	6		126.2.k.a.89.3	yes	8
7.4	even	3		882.2.k.a.215.4		8
7.5	odd	6		882.2.k.a.521.1		8
7.6	odd	2	inner	882.2.d.a.881.3		8
12.11	even	2		7056.2.k.f.881.5		8
21.2	odd	6		126.2.k.a.17.3	yes	8
21.5	even	6		882.2.k.a.521.4		8
21.11	odd	6		882.2.k.a.215.1		8
21.17	even	6		126.2.k.a.89.2	yes	8
21.20	even	2	inner	882.2.d.a.881.6		8
28.3	even	6		1008.2.bt.c.593.2		8
28.23	odd	6		1008.2.bt.c.17.3		8
28.27	even	2		7056.2.k.f.881.6		8
35.2	odd	12		3150.2.bp.b.899.2		8
35.3	even	12		3150.2.bp.e.1349.2		8
35.9	even	6		3150.2.bf.a.1151.3		8
35.17	even	12		3150.2.bp.b.1349.3		8
35.23	odd	12		3150.2.bp.e.899.3		8
35.24	odd	6		3150.2.bf.a.1601.1		8
63.2	odd	6		1134.2.t.e.1025.2		8
63.16	even	3		1134.2.t.e.1025.3		8
63.23	odd	6		1134.2.l.f.269.3		8
63.31	odd	6		1134.2.t.e.593.2		8
63.38	even	6		1134.2.l.f.215.4		8
63.52	odd	6		1134.2.l.f.215.1		8
63.58	even	3		1134.2.l.f.269.2		8
63.59	even	6		1134.2.t.e.593.3		8
84.23	even	6		1008.2.bt.c.17.2		8
84.59	odd	6		1008.2.bt.c.593.3		8
84.83	odd	2		7056.2.k.f.881.3		8
105.2	even	12		3150.2.bp.e.899.2		8
105.17	odd	12		3150.2.bp.e.1349.3		8
105.23	even	12		3150.2.bp.b.899.3		8
105.38	odd	12		3150.2.bp.b.1349.2		8
105.44	odd	6		3150.2.bf.a.1151.1		8
105.59	even	6		3150.2.bf.a.1601.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.k.a.17.2	✓	8	7.2	even	3
126.2.k.a.17.3	yes	8	21.2	odd	6
126.2.k.a.89.2	yes	8	21.17	even	6
126.2.k.a.89.3	yes	8	7.3	odd	6
882.2.d.a.881.2		8	1.1	even	1	trivial
882.2.d.a.881.3		8	7.6	odd	2	inner
882.2.d.a.881.6		8	21.20	even	2	inner
882.2.d.a.881.7		8	3.2	odd	2	inner
882.2.k.a.215.1		8	21.11	odd	6
882.2.k.a.215.4		8	7.4	even	3
882.2.k.a.521.1		8	7.5	odd	6
882.2.k.a.521.4		8	21.5	even	6
1008.2.bt.c.17.2		8	84.23	even	6
1008.2.bt.c.17.3		8	28.23	odd	6
1008.2.bt.c.593.2		8	28.3	even	6
1008.2.bt.c.593.3		8	84.59	odd	6
1134.2.l.f.215.1		8	63.52	odd	6
1134.2.l.f.215.4		8	63.38	even	6
1134.2.l.f.269.2		8	63.58	even	3
1134.2.l.f.269.3		8	63.23	odd	6
1134.2.t.e.593.2		8	63.31	odd	6
1134.2.t.e.593.3		8	63.59	even	6
1134.2.t.e.1025.2		8	63.2	odd	6
1134.2.t.e.1025.3		8	63.16	even	3
3150.2.bf.a.1151.1		8	105.44	odd	6
3150.2.bf.a.1151.3		8	35.9	even	6
3150.2.bf.a.1601.1		8	35.24	odd	6
3150.2.bf.a.1601.3		8	105.59	even	6
3150.2.bp.b.899.2		8	35.2	odd	12
3150.2.bp.b.899.3		8	105.23	even	12
3150.2.bp.b.1349.2		8	105.38	odd	12
3150.2.bp.b.1349.3		8	35.17	even	12
3150.2.bp.e.899.2		8	105.2	even	12
3150.2.bp.e.899.3		8	35.23	odd	12
3150.2.bp.e.1349.2		8	35.3	even	12
3150.2.bp.e.1349.3		8	105.17	odd	12
7056.2.k.f.881.3		8	84.83	odd	2
7056.2.k.f.881.4		8	4.3	odd	2
7056.2.k.f.881.5		8	12.11	even	2
7056.2.k.f.881.6		8	28.27	even	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.717439 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.717439 q^{5} +1.00000i q^{8} +0.717439i q^{10} -3.00000i q^{11} +2.44949i q^{13} +1.00000 q^{16} -5.91359 q^{17} -5.91359i q^{19} +0.717439 q^{20} -3.00000 q^{22} -4.24264i q^{23} -4.48528 q^{25} +2.44949 q^{26} -7.24264i q^{29} +9.08052i q^{31} -1.00000i q^{32} +5.91359i q^{34} -0.242641 q^{37} -5.91359 q^{38} -0.717439i q^{40} -11.8272 q^{41} -0.242641 q^{43} +3.00000i q^{44} -4.24264 q^{46} -5.91359 q^{47} +4.48528i q^{50} -2.44949i q^{52} -7.24264i q^{53} +2.15232i q^{55} -7.24264 q^{58} +8.06591 q^{59} +1.01461i q^{61} +9.08052 q^{62} -1.00000 q^{64} -1.75736i q^{65} -10.0000 q^{67} +5.91359 q^{68} +1.75736i q^{71} +1.43488i q^{73} +0.242641i q^{74} +5.91359i q^{76} -2.75736 q^{79} -0.717439 q^{80} +11.8272i q^{82} -6.63103 q^{83} +4.24264 q^{85} +0.242641i q^{86} +3.00000 q^{88} +10.3923 q^{89} +4.24264i q^{92} +5.91359i q^{94} +4.24264i q^{95} -13.5592i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} - 24 q^{22} + 32 q^{25} + 32 q^{37} + 32 q^{43} - 24 q^{58} - 8 q^{64} - 80 q^{67} - 56 q^{79} + 24 q^{88}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 - 24 * q^22 + 32 * q^25 + 32 * q^37 + 32 * q^43 - 24 * q^58 - 8 * q^64 - 80 * q^67 - 56 * q^79 + 24 * q^88

Introduction

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