LMFDB - Embedded newform 1368.2.e.d.379.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(379,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("1368.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.e (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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.207360000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 $ x^8 + 6x^6 + 32x^4 + 24*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			379.2
Root			$1.14412 - 1.98168i$ of defining polynomial
Character	$\chi$	$=$	1368.379
Dual form			1368.2.e.d.379.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} -3.46410i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} -3.46410i q^{7} +2.82843 q^{8} +3.16228 q^{11} +4.47214 q^{13} +(-4.24264 + 2.44949i) q^{14} +(-2.00000 - 3.46410i) q^{16} -6.32456 q^{17} +(-2.00000 + 3.87298i) q^{19} +(-2.23607 - 3.87298i) q^{22} -5.47723i q^{23} +5.00000 q^{25} +(-3.16228 - 5.47723i) q^{26} +(6.00000 + 3.46410i) q^{28} -1.41421 q^{29} +8.94427 q^{31} +(-2.82843 + 4.89898i) q^{32} +(4.47214 + 7.74597i) q^{34} +4.47214 q^{37} +(6.15763 - 0.289123i) q^{38} +2.44949i q^{41} +(-3.16228 + 5.47723i) q^{44} +(-6.70820 + 3.87298i) q^{46} -5.47723i q^{47} -5.00000 q^{49} +(-3.53553 - 6.12372i) q^{50} +(-4.47214 + 7.74597i) q^{52} -4.24264 q^{53} -9.79796i q^{56} +(1.00000 + 1.73205i) q^{58} -4.89898i q^{59} -10.3923i q^{61} +(-6.32456 - 10.9545i) q^{62} +8.00000 q^{64} -7.74597i q^{67} +(6.32456 - 10.9545i) q^{68} -11.3137 q^{71} -12.0000 q^{73} +(-3.16228 - 5.47723i) q^{74} +(-4.70820 - 7.33708i) q^{76} -10.9545i q^{77} +4.47214 q^{79} +(3.00000 - 1.73205i) q^{82} +15.8114 q^{83} +8.94427 q^{88} +17.1464i q^{89} -15.4919i q^{91} +(9.48683 + 5.47723i) q^{92} +(-6.70820 + 3.87298i) q^{94} -15.4919i q^{97} +(3.53553 + 6.12372i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 16 q^{16} - 16 q^{19} + 40 q^{25} + 48 q^{28} - 40 q^{49} + 8 q^{58} + 64 q^{64} - 96 q^{73} + 16 q^{76} + 24 q^{82}+O(q^{100}) $ 8 * q - 8 * q^4 - 16 * q^16 - 16 * q^19 + 40 * q^25 + 48 * q^28 - 40 * q^49 + 8 * q^58 + 64 * q^64 - 96 * q^73 + 16 * q^76 + 24 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	−0.707107	−	1.22474i	−0.500000	−	0.866025i
$2$	−0.707107	−	1.22474i	−0.500000	−	0.866025i
$3$		0			0
$3$		0			0
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	3.46410i		−	1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$		−	3.46410i		−	1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$	2.82843			1.00000
$9$		0			0
$10$		0			0
$11$	3.16228			0.953463			0.476731	−	0.879049i	$-0.341821\pi$
$11$	3.16228			0.953463			0.476731	+	0.879049i	$0.341821\pi$
$12$		0			0
$13$	4.47214			1.24035			0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214			1.24035			0.620174	+	0.784465i	$0.287062\pi$
$14$	−4.24264	+	2.44949i	−1.13389	+	0.654654i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−6.32456			−1.53393			−0.766965	−	0.641689i	$-0.778234\pi$
$17$	−6.32456			−1.53393			−0.766965	+	0.641689i	$0.778234\pi$
$18$		0			0
$19$	−2.00000	+	3.87298i	−0.458831	+	0.888523i
$19$	−2.00000	+	3.87298i	−0.458831	+	0.888523i
$20$		0			0
$21$		0			0
$22$	−2.23607	−	3.87298i	−0.476731	−	0.825723i
$23$		−	5.47723i		−	1.14208i	−0.820922	−	0.571040i	$-0.806540\pi$
$23$		−	5.47723i		−	1.14208i	0.820922	−	0.571040i	$-0.193460\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$	−3.16228	−	5.47723i	−0.620174	−	1.07417i
$27$		0			0
$28$	6.00000	+	3.46410i	1.13389	+	0.654654i
$29$	−1.41421			−0.262613			−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421			−0.262613			−0.131306	+	0.991342i	$0.541917\pi$
$30$		0			0
$31$	8.94427			1.60644			0.803219	−	0.595683i	$-0.203119\pi$
$31$	8.94427			1.60644			0.803219	+	0.595683i	$0.203119\pi$
$32$	−2.82843	+	4.89898i	−0.500000	+	0.866025i
$33$		0			0
$34$	4.47214	+	7.74597i	0.766965	+	1.32842i
$35$		0			0
$36$		0			0
$37$	4.47214			0.735215			0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214			0.735215			0.367607	+	0.929981i	$0.380177\pi$
$38$	6.15763	−	0.289123i	0.998899	−	0.0469020i
$39$		0			0
$40$		0			0
$41$			2.44949i			0.382546i	0.981537	+	0.191273i	$0.0612616\pi$
$41$			2.44949i			0.382546i	−0.981537	+	0.191273i	$0.938738\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−3.16228	+	5.47723i	−0.476731	+	0.825723i
$45$		0			0
$46$	−6.70820	+	3.87298i	−0.989071	+	0.571040i
$47$		−	5.47723i		−	0.798935i	−0.916747	−	0.399468i	$-0.869195\pi$
$47$		−	5.47723i		−	0.798935i	0.916747	−	0.399468i	$-0.130805\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$	−3.53553	−	6.12372i	−0.500000	−	0.866025i
$51$		0			0
$52$	−4.47214	+	7.74597i	−0.620174	+	1.07417i
$53$	−4.24264			−0.582772			−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264			−0.582772			−0.291386	+	0.956606i	$0.594116\pi$
$54$		0			0
$55$		0			0
$56$		−	9.79796i		−	1.30931i
$57$		0			0
$58$	1.00000	+	1.73205i	0.131306	+	0.227429i
$59$		−	4.89898i		−	0.637793i	−0.947790	−	0.318896i	$-0.896688\pi$
$59$		−	4.89898i		−	0.637793i	0.947790	−	0.318896i	$-0.103312\pi$
$60$		0			0
$61$		−	10.3923i		−	1.33060i	−0.746577	−	0.665299i	$-0.768304\pi$
$61$		−	10.3923i		−	1.33060i	0.746577	−	0.665299i	$-0.231696\pi$
$62$	−6.32456	−	10.9545i	−0.803219	−	1.39122i
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$		−	7.74597i		−	0.946320i	−0.880976	−	0.473160i	$-0.843113\pi$
$67$		−	7.74597i		−	0.946320i	0.880976	−	0.473160i	$-0.156887\pi$
$68$	6.32456	−	10.9545i	0.766965	−	1.32842i
$69$		0			0
$70$		0			0
$71$	−11.3137			−1.34269			−0.671345	−	0.741145i	$-0.734283\pi$
$71$	−11.3137			−1.34269			−0.671345	+	0.741145i	$0.734283\pi$
$72$		0			0
$73$	−12.0000			−1.40449			−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000			−1.40449			−0.702247	+	0.711934i	$0.747820\pi$
$74$	−3.16228	−	5.47723i	−0.367607	−	0.636715i
$75$		0			0
$76$	−4.70820	−	7.33708i	−0.540068	−	0.841621i
$77$		−	10.9545i		−	1.24838i
$78$		0			0
$79$	4.47214			0.503155			0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214			0.503155			0.251577	+	0.967837i	$0.419051\pi$
$80$		0			0
$81$		0			0
$82$	3.00000	−	1.73205i	0.331295	−	0.191273i
$83$	15.8114			1.73553			0.867763	−	0.496979i	$-0.165557\pi$
$83$	15.8114			1.73553			0.867763	+	0.496979i	$0.165557\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	8.94427			0.953463
$89$			17.1464i			1.81752i	0.417322	+	0.908759i	$0.362969\pi$
$89$			17.1464i			1.81752i	−0.417322	+	0.908759i	$0.637031\pi$
$90$		0			0
$91$		−	15.4919i		−	1.62400i
$92$	9.48683	+	5.47723i	0.989071	+	0.571040i
$93$		0			0
$94$	−6.70820	+	3.87298i	−0.691898	+	0.399468i
$95$		0			0
$96$		0			0
$97$		−	15.4919i		−	1.57297i	−0.617611	−	0.786484i	$-0.711899\pi$
$97$		−	15.4919i		−	1.57297i	0.617611	−	0.786484i	$-0.288101\pi$
$98$	3.53553	+	6.12372i	0.357143	+	0.618590i
$99$		0			0
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$		−	10.9545i		−	1.09001i	−0.838433	−	0.545004i	$-0.816528\pi$
$101$		−	10.9545i		−	1.09001i	0.838433	−	0.545004i	$-0.183472\pi$
$102$		0			0
$103$	−4.47214			−0.440653			−0.220326	−	0.975426i	$-0.570712\pi$
$103$	−4.47214			−0.440653			−0.220326	+	0.975426i	$0.570712\pi$
$104$	12.6491			1.24035
$105$		0			0
$106$	3.00000	+	5.19615i	0.291386	+	0.504695i
$107$		−	14.6969i		−	1.42081i	−0.703795	−	0.710403i	$-0.748513\pi$
$107$		−	14.6969i		−	1.42081i	0.703795	−	0.710403i	$-0.251487\pi$
$108$		0			0
$109$	−4.47214			−0.428353			−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214			−0.428353			−0.214176	+	0.976795i	$0.568707\pi$
$110$		0			0
$111$		0			0
$112$	−12.0000	+	6.92820i	−1.13389	+	0.654654i
$113$		−	12.2474i		−	1.15214i	−0.817399	−	0.576072i	$-0.804585\pi$
$113$		−	12.2474i		−	1.15214i	0.817399	−	0.576072i	$-0.195415\pi$
$114$		0			0
$115$		0			0
$116$	1.41421	−	2.44949i	0.131306	−	0.227429i
$117$		0			0
$118$	−6.00000	+	3.46410i	−0.552345	+	0.318896i
$119$			21.9089i			2.00839i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	−12.7279	+	7.34847i	−1.15233	+	0.665299i
$123$		0			0
$124$	−8.94427	+	15.4919i	−0.803219	+	1.39122i
$125$		0			0
$126$		0			0
$127$	8.94427			0.793676			0.396838	−	0.917889i	$-0.370108\pi$
$127$	8.94427			0.793676			0.396838	+	0.917889i	$0.370108\pi$
$128$	−5.65685	−	9.79796i	−0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$	3.16228			0.276289			0.138145	−	0.990412i	$-0.455886\pi$
$131$	3.16228			0.276289			0.138145	+	0.990412i	$0.455886\pi$
$132$		0			0
$133$	13.4164	+	6.92820i	1.16335	+	0.600751i
$134$	−9.48683	+	5.47723i	−0.819538	+	0.473160i
$135$		0			0
$136$	−17.8885			−1.53393
$137$	−12.6491			−1.08069			−0.540343	−	0.841445i	$-0.681706\pi$
$137$	−12.6491			−1.08069			−0.540343	+	0.841445i	$0.681706\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$	8.00000	+	13.8564i	0.671345	+	1.16280i
$143$	14.1421			1.18262
$144$		0			0
$145$		0			0
$146$	8.48528	+	14.6969i	0.702247	+	1.21633i
$147$		0			0
$148$	−4.47214	+	7.74597i	−0.367607	+	0.636715i
$149$		−	21.9089i		−	1.79485i	−0.441170	−	0.897424i	$-0.645436\pi$
$149$		−	21.9089i		−	1.79485i	0.441170	−	0.897424i	$-0.354564\pi$
$150$		0			0
$151$	−4.47214			−0.363937			−0.181969	−	0.983304i	$-0.558247\pi$
$151$	−4.47214			−0.363937			−0.181969	+	0.983304i	$0.558247\pi$
$152$	−5.65685	+	10.9545i	−0.458831	+	0.888523i
$153$		0			0
$154$	−13.4164	+	7.74597i	−1.08112	+	0.624188i
$155$		0			0
$156$		0			0
$157$		−	3.46410i		−	0.276465i	−0.990400	−	0.138233i	$-0.955858\pi$
$157$		−	3.46410i		−	0.276465i	0.990400	−	0.138233i	$-0.0441422\pi$
$158$	−3.16228	−	5.47723i	−0.251577	−	0.435745i
$159$		0			0
$160$		0			0
$161$	−18.9737			−1.49533
$162$		0			0
$163$	−12.0000			−0.939913			−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000			−0.939913			−0.469956	+	0.882690i	$0.655730\pi$
$164$	−4.24264	−	2.44949i	−0.331295	−	0.191273i
$165$		0			0
$166$	−11.1803	−	19.3649i	−0.867763	−	1.50301i
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.24264			−0.322562			−0.161281	−	0.986909i	$-0.551563\pi$
$173$	−4.24264			−0.322562			−0.161281	+	0.986909i	$0.551563\pi$
$174$		0			0
$175$		−	17.3205i		−	1.30931i
$176$	−6.32456	−	10.9545i	−0.476731	−	0.825723i
$177$		0			0
$178$	21.0000	−	12.1244i	1.57402	−	0.908759i
$179$			19.5959i			1.46467i	0.680946	+	0.732334i	$0.261569\pi$
$179$			19.5959i			1.46467i	−0.680946	+	0.732334i	$0.738431\pi$
$180$		0			0
$181$	4.47214			0.332411			0.166206	−	0.986091i	$-0.446848\pi$
$181$	4.47214			0.332411			0.166206	+	0.986091i	$0.446848\pi$
$182$	−18.9737	+	10.9545i	−1.40642	+	0.811998i
$183$		0			0
$184$		−	15.4919i		−	1.14208i
$185$		0			0
$186$		0			0
$187$	−20.0000			−1.46254
$188$	9.48683	+	5.47723i	0.691898	+	0.399468i
$189$		0			0
$190$		0			0
$191$			16.4317i			1.18895i	0.804112	+	0.594477i	$0.202641\pi$
$191$			16.4317i			1.18895i	−0.804112	+	0.594477i	$0.797359\pi$
$192$		0			0
$193$			15.4919i			1.11513i	0.830132	+	0.557567i	$0.188265\pi$
$193$			15.4919i			1.11513i	−0.830132	+	0.557567i	$0.811735\pi$
$194$	−18.9737	+	10.9545i	−1.36223	+	0.786484i
$195$		0			0
$196$	5.00000	−	8.66025i	0.357143	−	0.618590i
$197$			21.9089i			1.56094i	0.625190	+	0.780472i	$0.285021\pi$
$197$			21.9089i			1.56094i	−0.625190	+	0.780472i	$0.714979\pi$
$198$		0			0
$199$			3.46410i			0.245564i	0.992434	+	0.122782i	$0.0391815\pi$
$199$			3.46410i			0.245564i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	14.1421			1.00000
$201$		0			0
$202$	−13.4164	+	7.74597i	−0.943975	+	0.545004i
$203$			4.89898i			0.343841i
$204$		0			0
$205$		0			0
$206$	3.16228	+	5.47723i	0.220326	+	0.381616i
$207$		0			0
$208$	−8.94427	−	15.4919i	−0.620174	−	1.07417i
$209$	−6.32456	+	12.2474i	−0.437479	+	0.847174i
$210$		0			0
$211$			23.2379i			1.59976i	0.600158	+	0.799882i	$0.295104\pi$
$211$			23.2379i			1.59976i	−0.600158	+	0.799882i	$0.704896\pi$
$212$	4.24264	−	7.34847i	0.291386	−	0.504695i
$213$		0			0
$214$	−18.0000	+	10.3923i	−1.23045	+	0.710403i
$215$		0			0
$216$		0			0
$217$		−	30.9839i		−	2.10332i
$218$	3.16228	+	5.47723i	0.214176	+	0.370965i
$219$		0			0
$220$		0			0
$221$	−28.2843			−1.90261
$222$		0			0
$223$	22.3607			1.49738			0.748691	−	0.662919i	$-0.230683\pi$
$223$	22.3607			1.49738			0.748691	+	0.662919i	$0.230683\pi$
$224$	16.9706	+	9.79796i	1.13389	+	0.654654i
$225$		0			0
$226$	−15.0000	+	8.66025i	−0.997785	+	0.576072i
$227$		−	14.6969i		−	0.975470i	−0.872992	−	0.487735i	$-0.837823\pi$
$227$		−	14.6969i		−	0.975470i	0.872992	−	0.487735i	$-0.162177\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−4.00000			−0.262613
$233$	6.32456			0.414335			0.207168	−	0.978305i	$-0.433575\pi$
$233$	6.32456			0.414335			0.207168	+	0.978305i	$0.433575\pi$
$234$		0			0
$235$		0			0
$236$	8.48528	+	4.89898i	0.552345	+	0.318896i
$237$		0			0
$238$	26.8328	−	15.4919i	1.73931	−	1.00419i
$239$			16.4317i			1.06288i	0.847097	+	0.531438i	$0.178348\pi$
$239$			16.4317i			1.06288i	−0.847097	+	0.531438i	$0.821652\pi$
$240$		0			0
$241$			15.4919i			0.997923i	0.866624	+	0.498962i	$0.166285\pi$
$241$			15.4919i			0.997923i	−0.866624	+	0.498962i	$0.833715\pi$
$242$	0.707107	+	1.22474i	0.0454545	+	0.0787296i
$243$		0			0
$244$	18.0000	+	10.3923i	1.15233	+	0.665299i
$245$		0			0
$246$		0			0
$247$	−8.94427	+	17.3205i	−0.569110	+	1.10208i
$248$	25.2982			1.60644
$249$		0			0
$250$		0			0
$251$	15.8114			0.998006			0.499003	−	0.866600i	$-0.333700\pi$
$251$	15.8114			0.998006			0.499003	+	0.866600i	$0.333700\pi$
$252$		0			0
$253$		−	17.3205i		−	1.08893i
$254$	−6.32456	−	10.9545i	−0.396838	−	0.687343i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$			2.44949i			0.152795i	0.997077	+	0.0763975i	$0.0243418\pi$
$257$			2.44949i			0.152795i	−0.997077	+	0.0763975i	$0.975658\pi$
$258$		0			0
$259$		−	15.4919i		−	0.962622i
$260$		0			0
$261$		0			0
$262$	−2.23607	−	3.87298i	−0.138145	−	0.239274i
$263$			16.4317i			1.01322i	0.862175	+	0.506610i	$0.169102\pi$
$263$			16.4317i			1.01322i	−0.862175	+	0.506610i	$0.830898\pi$
$264$		0			0
$265$		0			0
$266$	−1.00155	−	21.3307i	−0.0614091	−	1.30787i
$267$		0			0
$268$	13.4164	+	7.74597i	0.819538	+	0.473160i
$269$	21.2132			1.29339			0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132			1.29339			0.646696	+	0.762748i	$0.276150\pi$
$270$		0			0
$271$		−	10.3923i		−	0.631288i	−0.948878	−	0.315644i	$-0.897780\pi$
$271$		−	10.3923i		−	0.631288i	0.948878	−	0.315644i	$-0.102220\pi$
$272$	12.6491	+	21.9089i	0.766965	+	1.32842i
$273$		0			0
$274$	8.94427	+	15.4919i	0.540343	+	0.935902i
$275$	15.8114			0.953463
$276$		0			0
$277$			13.8564i			0.832551i	0.909239	+	0.416275i	$0.136665\pi$
$277$			13.8564i			0.832551i	−0.909239	+	0.416275i	$0.863335\pi$
$278$	−8.48528	−	14.6969i	−0.508913	−	0.881464i
$279$		0			0
$280$		0			0
$281$			7.34847i			0.438373i	0.975683	+	0.219186i	$0.0703403\pi$
$281$			7.34847i			0.438373i	−0.975683	+	0.219186i	$0.929660\pi$
$282$		0			0
$283$	20.0000			1.18888			0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000			1.18888			0.594438	+	0.804141i	$0.297374\pi$
$284$	11.3137	−	19.5959i	0.671345	−	1.16280i
$285$		0			0
$286$	−10.0000	−	17.3205i	−0.591312	−	1.02418i
$287$	8.48528			0.500870
$288$		0			0
$289$	23.0000			1.35294
$290$		0			0
$291$		0			0
$292$	12.0000	−	20.7846i	0.702247	−	1.21633i
$293$	−29.6985			−1.73500			−0.867502	−	0.497434i	$-0.834276\pi$
$293$	−29.6985			−1.73500			−0.867502	+	0.497434i	$0.834276\pi$
$294$		0			0
$295$		0			0
$296$	12.6491			0.735215
$297$		0			0
$298$	−26.8328	+	15.4919i	−1.55438	+	0.897424i
$299$		−	24.4949i		−	1.41658i
$300$		0			0
$301$		0			0
$302$	3.16228	+	5.47723i	0.181969	+	0.315179i
$303$		0			0
$304$	17.4164	−	0.817763i	0.998899	−	0.0469020i
$305$		0			0
$306$		0			0
$307$		−	15.4919i		−	0.884171i	−0.896973	−	0.442086i	$-0.854239\pi$
$307$		−	15.4919i		−	0.884171i	0.896973	−	0.442086i	$-0.145761\pi$
$308$	18.9737	+	10.9545i	1.08112	+	0.624188i
$309$		0			0
$310$		0			0
$311$		−	5.47723i		−	0.310585i	−0.987869	−	0.155292i	$-0.950368\pi$
$311$		−	5.47723i		−	0.310585i	0.987869	−	0.155292i	$-0.0496320\pi$
$312$		0			0
$313$	−20.0000			−1.13047			−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000			−1.13047			−0.565233	+	0.824931i	$0.691214\pi$
$314$	−4.24264	+	2.44949i	−0.239426	+	0.138233i
$315$		0			0
$316$	−4.47214	+	7.74597i	−0.251577	+	0.435745i
$317$	29.6985			1.66803			0.834017	−	0.551739i	$-0.186036\pi$
$317$	29.6985			1.66803			0.834017	+	0.551739i	$0.186036\pi$
$318$		0			0
$319$	−4.47214			−0.250392
$320$		0			0
$321$		0			0
$322$	13.4164	+	23.2379i	0.747667	+	1.29500i
$323$	12.6491	−	24.4949i	0.703815	−	1.36293i
$324$		0			0
$325$	22.3607			1.24035
$326$	8.48528	+	14.6969i	0.469956	+	0.813988i
$327$		0			0
$328$			6.92820i			0.382546i
$329$	−18.9737			−1.04605
$330$		0			0
$331$			15.4919i			0.851514i	0.904838	+	0.425757i	$0.139992\pi$
$331$			15.4919i			0.851514i	−0.904838	+	0.425757i	$0.860008\pi$
$332$	−15.8114	+	27.3861i	−0.867763	+	1.50301i
$333$		0			0
$334$	−14.0000	−	24.2487i	−0.766046	−	1.32683i
$335$		0			0
$336$		0			0
$337$			15.4919i			0.843899i	0.906619	+	0.421950i	$0.138654\pi$
$337$			15.4919i			0.843899i	−0.906619	+	0.421950i	$0.861346\pi$
$338$	−4.94975	−	8.57321i	−0.269231	−	0.466321i
$339$		0			0
$340$		0			0
$341$	28.2843			1.53168
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$		0			0
$346$	3.00000	+	5.19615i	0.161281	+	0.279347i
$347$	15.8114			0.848800			0.424400	−	0.905475i	$-0.360485\pi$
$347$	15.8114			0.848800			0.424400	+	0.905475i	$0.360485\pi$
$348$		0			0
$349$		−	13.8564i		−	0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$		−	13.8564i		−	0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$	−21.2132	+	12.2474i	−1.13389	+	0.654654i
$351$		0			0
$352$	−8.94427	+	15.4919i	−0.476731	+	0.825723i
$353$	−6.32456			−0.336622			−0.168311	−	0.985734i	$-0.553831\pi$
$353$	−6.32456			−0.336622			−0.168311	+	0.985734i	$0.553831\pi$
$354$		0			0
$355$		0			0
$356$	−29.6985	−	17.1464i	−1.57402	−	0.908759i
$357$		0			0
$358$	24.0000	−	13.8564i	1.26844	−	0.732334i
$359$		−	5.47723i		−	0.289077i	−0.989499	−	0.144538i	$-0.953830\pi$
$359$		−	5.47723i		−	0.289077i	0.989499	−	0.144538i	$-0.0461697\pi$
$360$		0			0
$361$	−11.0000	−	15.4919i	−0.578947	−	0.815365i
$362$	−3.16228	−	5.47723i	−0.166206	−	0.287877i
$363$		0			0
$364$	26.8328	+	15.4919i	1.40642	+	0.811998i
$365$		0			0
$366$		0			0
$367$		−	24.2487i		−	1.26577i	−0.774245	−	0.632886i	$-0.781870\pi$
$367$		−	24.2487i		−	1.26577i	0.774245	−	0.632886i	$-0.218130\pi$
$368$	−18.9737	+	10.9545i	−0.989071	+	0.571040i
$369$		0			0
$370$		0			0
$371$			14.6969i			0.763027i
$372$		0			0
$373$	−31.3050			−1.62091			−0.810454	−	0.585802i	$-0.800780\pi$
$373$	−31.3050			−1.62091			−0.810454	+	0.585802i	$0.800780\pi$
$374$	14.1421	+	24.4949i	0.731272	+	1.26660i
$375$		0			0
$376$		−	15.4919i		−	0.798935i
$377$	−6.32456			−0.325731
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	20.1246	−	11.6190i	1.02966	−	0.594477i
$383$	2.82843			0.144526			0.0722629	−	0.997386i	$-0.476978\pi$
$383$	2.82843			0.144526			0.0722629	+	0.997386i	$0.476978\pi$
$384$		0			0
$385$		0			0
$386$	18.9737	−	10.9545i	0.965734	−	0.557567i
$387$		0			0
$388$	26.8328	+	15.4919i	1.36223	+	0.786484i
$389$			10.9545i			0.555413i	0.960666	+	0.277706i	$0.0895742\pi$
$389$			10.9545i			0.555413i	−0.960666	+	0.277706i	$0.910426\pi$
$390$		0			0
$391$			34.6410i			1.75187i
$392$	−14.1421			−0.714286
$393$		0			0
$394$	26.8328	−	15.4919i	1.35182	−	0.780472i
$395$		0			0
$396$		0			0
$397$			27.7128i			1.39087i	0.718591	+	0.695433i	$0.244787\pi$
$397$			27.7128i			1.39087i	−0.718591	+	0.695433i	$0.755213\pi$
$398$	4.24264	−	2.44949i	0.212664	−	0.122782i
$399$		0			0
$400$	−10.0000	−	17.3205i	−0.500000	−	0.866025i
$401$			17.1464i			0.856252i	0.903719	+	0.428126i	$0.140826\pi$
$401$			17.1464i			0.856252i	−0.903719	+	0.428126i	$0.859174\pi$
$402$		0			0
$403$	40.0000			1.99254
$404$	18.9737	+	10.9545i	0.943975	+	0.545004i
$405$		0			0
$406$	6.00000	−	3.46410i	0.297775	−	0.171920i
$407$	14.1421			0.701000
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	4.47214	−	7.74597i	0.220326	−	0.381616i
$413$	−16.9706			−0.835067
$414$		0			0
$415$		0			0
$416$	−12.6491	+	21.9089i	−0.620174	+	1.07417i
$417$		0			0
$418$	19.4721	−	0.914287i	0.952413	−	0.0447193i
$419$	3.16228			0.154487			0.0772437	−	0.997012i	$-0.475388\pi$
$419$	3.16228			0.154487			0.0772437	+	0.997012i	$0.475388\pi$
$420$		0			0
$421$	−4.47214			−0.217959			−0.108979	−	0.994044i	$-0.534758\pi$
$421$	−4.47214			−0.217959			−0.108979	+	0.994044i	$0.534758\pi$
$422$	28.4605	−	16.4317i	1.38544	−	0.799882i
$423$		0			0
$424$	−12.0000			−0.582772
$425$	−31.6228			−1.53393
$426$		0			0
$427$	−36.0000			−1.74216
$428$	25.4558	+	14.6969i	1.23045	+	0.710403i
$429$		0			0
$430$		0			0
$431$	−25.4558			−1.22616			−0.613082	−	0.790019i	$-0.710071\pi$
$431$	−25.4558			−1.22616			−0.613082	+	0.790019i	$0.710071\pi$
$432$		0			0
$433$			30.9839i			1.48899i	0.667628	+	0.744495i	$0.267310\pi$
$433$			30.9839i			1.48899i	−0.667628	+	0.744495i	$0.732690\pi$
$434$	−37.9473	+	21.9089i	−1.82153	+	1.05166i
$435$		0			0
$436$	4.47214	−	7.74597i	0.214176	−	0.370965i
$437$	21.2132	+	10.9545i	1.01477	+	0.524022i
$438$		0			0
$439$	−22.3607			−1.06722			−0.533609	−	0.845732i	$-0.679164\pi$
$439$	−22.3607			−1.06722			−0.533609	+	0.845732i	$0.679164\pi$
$440$		0			0
$441$		0			0
$442$	20.0000	+	34.6410i	0.951303	+	1.64771i
$443$	15.8114			0.751222			0.375611	−	0.926777i	$-0.377433\pi$
$443$	15.8114			0.751222			0.375611	+	0.926777i	$0.377433\pi$
$444$		0			0
$445$		0			0
$446$	−15.8114	−	27.3861i	−0.748691	−	1.29677i
$447$		0			0
$448$		−	27.7128i		−	1.30931i
$449$		−	7.34847i		−	0.346796i	−0.984852	−	0.173398i	$-0.944525\pi$
$449$		−	7.34847i		−	0.346796i	0.984852	−	0.173398i	$-0.0554746\pi$
$450$		0			0
$451$			7.74597i			0.364743i
$452$	21.2132	+	12.2474i	0.997785	+	0.576072i
$453$		0			0
$454$	−18.0000	+	10.3923i	−0.844782	+	0.487735i
$455$		0			0
$456$		0			0
$457$	4.00000			0.187112			0.0935561	−	0.995614i	$-0.470177\pi$
$457$	4.00000			0.187112			0.0935561	+	0.995614i	$0.470177\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	10.9545i		−	0.510200i	−0.966915	−	0.255100i	$-0.917892\pi$
$461$		−	10.9545i		−	0.510200i	0.966915	−	0.255100i	$-0.0821083\pi$
$462$		0			0
$463$		−	3.46410i		−	0.160990i	−0.996755	−	0.0804952i	$-0.974350\pi$
$463$		−	3.46410i		−	0.160990i	0.996755	−	0.0804952i	$-0.0256502\pi$
$464$	2.82843	+	4.89898i	0.131306	+	0.227429i
$465$		0			0
$466$	−4.47214	−	7.74597i	−0.207168	−	0.358825i
$467$	−3.16228			−0.146333			−0.0731664	−	0.997320i	$-0.523310\pi$
$467$	−3.16228			−0.146333			−0.0731664	+	0.997320i	$0.523310\pi$
$468$		0			0
$469$	−26.8328			−1.23902
$470$		0			0
$471$		0			0
$472$		−	13.8564i		−	0.637793i
$473$		0			0
$474$		0			0
$475$	−10.0000	+	19.3649i	−0.458831	+	0.888523i
$476$	−37.9473	−	21.9089i	−1.73931	−	1.00419i
$477$		0			0
$478$	20.1246	−	11.6190i	0.920478	−	0.531438i
$479$			27.3861i			1.25130i	0.780102	+	0.625652i	$0.215167\pi$
$479$			27.3861i			1.25130i	−0.780102	+	0.625652i	$0.784833\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$	18.9737	−	10.9545i	0.864227	−	0.498962i
$483$		0			0
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$		0			0
$486$		0			0
$487$	−8.94427			−0.405304			−0.202652	−	0.979251i	$-0.564956\pi$
$487$	−8.94427			−0.405304			−0.202652	+	0.979251i	$0.564956\pi$
$488$		−	29.3939i		−	1.33060i
$489$		0			0
$490$		0			0
$491$	−22.1359			−0.998981			−0.499491	−	0.866319i	$-0.666479\pi$
$491$	−22.1359			−0.998981			−0.499491	+	0.866319i	$0.666479\pi$
$492$		0			0
$493$	8.94427			0.402830
$494$	27.5378	−	1.29300i	1.23898	−	0.0581747i
$495$		0			0
$496$	−17.8885	−	30.9839i	−0.803219	−	1.39122i
$497$			39.1918i			1.75799i
$498$		0			0
$499$	−8.00000			−0.358129			−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000			−0.358129			−0.179065	+	0.983837i	$0.557307\pi$
$500$		0			0
$501$		0			0
$502$	−11.1803	−	19.3649i	−0.499003	−	0.864299i
$503$		−	16.4317i		−	0.732652i	−0.930487	−	0.366326i	$-0.880615\pi$
$503$		−	16.4317i		−	0.732652i	0.930487	−	0.366326i	$-0.119385\pi$
$504$		0			0
$505$		0			0
$506$	−21.2132	+	12.2474i	−0.943042	+	0.544466i
$507$		0			0
$508$	−8.94427	+	15.4919i	−0.396838	+	0.687343i
$509$	−15.5563			−0.689523			−0.344762	−	0.938690i	$-0.612040\pi$
$509$	−15.5563			−0.689523			−0.344762	+	0.938690i	$0.612040\pi$
$510$		0			0
$511$			41.5692i			1.83891i
$512$	22.6274			1.00000
$513$		0			0
$514$	3.00000	−	1.73205i	0.132324	−	0.0763975i
$515$		0			0
$516$		0			0
$517$		−	17.3205i		−	0.761755i
$518$	−18.9737	+	10.9545i	−0.833655	+	0.481311i
$519$		0			0
$520$		0			0
$521$		−	26.9444i		−	1.18046i	−0.807237	−	0.590228i	$-0.799038\pi$
$521$		−	26.9444i		−	1.18046i	0.807237	−	0.590228i	$-0.200962\pi$
$522$		0			0
$523$			7.74597i			0.338707i	0.985555	+	0.169354i	$0.0541680\pi$
$523$			7.74597i			0.338707i	−0.985555	+	0.169354i	$0.945832\pi$
$524$	−3.16228	+	5.47723i	−0.138145	+	0.239274i
$525$		0			0
$526$	20.1246	−	11.6190i	0.877475	−	0.506610i
$527$	−56.5685			−2.46416
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$	−25.4164	+	16.3097i	−1.10194	+	0.707115i
$533$			10.9545i			0.474490i
$534$		0			0
$535$		0			0
$536$		−	21.9089i		−	0.946320i
$537$		0			0
$538$	−15.0000	−	25.9808i	−0.646696	−	1.12011i
$539$	−15.8114			−0.681045
$540$		0			0
$541$			17.3205i			0.744667i	0.928099	+	0.372333i	$0.121442\pi$
$541$			17.3205i			0.744667i	−0.928099	+	0.372333i	$0.878558\pi$
$542$	−12.7279	+	7.34847i	−0.546711	+	0.315644i
$543$		0			0
$544$	17.8885	−	30.9839i	0.766965	−	1.32842i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	12.6491	−	21.9089i	0.540343	−	0.935902i
$549$		0			0
$550$	−11.1803	−	19.3649i	−0.476731	−	0.825723i
$551$	2.82843	−	5.47723i	0.120495	−	0.233338i
$552$		0			0
$553$		−	15.4919i		−	0.658784i
$554$	16.9706	−	9.79796i	0.721010	−	0.416275i
$555$		0			0
$556$	−12.0000	+	20.7846i	−0.508913	+	0.881464i
$557$			32.8634i			1.39246i	0.717816	+	0.696232i	$0.245142\pi$
$557$			32.8634i			1.39246i	−0.717816	+	0.696232i	$0.754858\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	9.00000	−	5.19615i	0.379642	−	0.219186i
$563$			19.5959i			0.825869i	0.910761	+	0.412935i	$0.135496\pi$
$563$			19.5959i			0.825869i	−0.910761	+	0.412935i	$0.864504\pi$
$564$		0			0
$565$		0			0
$566$	−14.1421	−	24.4949i	−0.594438	−	1.02960i
$567$		0			0
$568$	−32.0000			−1.34269
$569$		−	7.34847i		−	0.308064i	−0.988066	−	0.154032i	$-0.950774\pi$
$569$		−	7.34847i		−	0.308064i	0.988066	−	0.154032i	$-0.0492259\pi$
$570$		0			0
$571$	24.0000			1.00437			0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000			1.00437			0.502184	+	0.864761i	$0.332530\pi$
$572$	−14.1421	+	24.4949i	−0.591312	+	1.02418i
$573$		0			0
$574$	−6.00000	−	10.3923i	−0.250435	−	0.433766i
$575$		−	27.3861i		−	1.14208i
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−16.2635	−	28.1691i	−0.676471	−	1.17168i
$579$		0			0
$580$		0			0
$581$		−	54.7723i		−	2.27234i
$582$		0			0
$583$	−13.4164			−0.555651
$584$	−33.9411			−1.40449
$585$		0			0
$586$	21.0000	+	36.3731i	0.867502	+	1.50256i
$587$	22.1359			0.913648			0.456824	−	0.889557i	$-0.348987\pi$
$587$	22.1359			0.913648			0.456824	+	0.889557i	$0.348987\pi$
$588$		0			0
$589$	−17.8885	+	34.6410i	−0.737085	+	1.42736i
$590$		0			0
$591$		0			0
$592$	−8.94427	−	15.4919i	−0.367607	−	0.636715i
$593$	31.6228			1.29859			0.649296	−	0.760536i	$-0.275064\pi$
$593$	31.6228			1.29859			0.649296	+	0.760536i	$0.275064\pi$
$594$		0			0
$595$		0			0
$596$	37.9473	+	21.9089i	1.55438	+	0.897424i
$597$		0			0
$598$	−30.0000	+	17.3205i	−1.22679	+	0.708288i
$599$	28.2843			1.15566			0.577832	−	0.816156i	$-0.303899\pi$
$599$	28.2843			1.15566			0.577832	+	0.816156i	$0.303899\pi$
$600$		0			0
$601$			30.9839i			1.26386i	0.775026	+	0.631929i	$0.217737\pi$
$601$			30.9839i			1.26386i	−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$	4.47214	−	7.74597i	0.181969	−	0.315179i
$605$		0			0
$606$		0			0
$607$	−8.94427			−0.363037			−0.181518	−	0.983388i	$-0.558101\pi$
$607$	−8.94427			−0.363037			−0.181518	+	0.983388i	$0.558101\pi$
$608$	−13.3168	−	20.7524i	−0.540068	−	0.841621i
$609$		0			0
$610$		0			0
$611$		−	24.4949i		−	0.990957i
$612$		0			0
$613$		−	24.2487i		−	0.979396i	−0.871892	−	0.489698i	$-0.837107\pi$
$613$		−	24.2487i		−	0.979396i	0.871892	−	0.489698i	$-0.162893\pi$
$614$	−18.9737	+	10.9545i	−0.765715	+	0.442086i
$615$		0			0
$616$		−	30.9839i		−	1.24838i
$617$	25.2982			1.01847			0.509234	−	0.860628i	$-0.329929\pi$
$617$	25.2982			1.01847			0.509234	+	0.860628i	$0.329929\pi$
$618$		0			0
$619$	32.0000			1.28619			0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000			1.28619			0.643094	+	0.765787i	$0.277650\pi$
$620$		0			0
$621$		0			0
$622$	−6.70820	+	3.87298i	−0.268974	+	0.155292i
$623$	59.3970			2.37969
$624$		0			0
$625$	25.0000			1.00000
$626$	14.1421	+	24.4949i	0.565233	+	0.979013i
$627$		0			0
$628$	6.00000	+	3.46410i	0.239426	+	0.138233i
$629$	−28.2843			−1.12777
$630$		0			0
$631$		−	17.3205i		−	0.689519i	−0.938691	−	0.344759i	$-0.887961\pi$
$631$		−	17.3205i		−	0.689519i	0.938691	−	0.344759i	$-0.112039\pi$
$632$	12.6491			0.503155
$633$		0			0
$634$	−21.0000	−	36.3731i	−0.834017	−	1.44456i
$635$		0			0
$636$		0			0
$637$	−22.3607			−0.885962
$638$	3.16228	+	5.47723i	0.125196	+	0.216845i
$639$		0			0
$640$		0			0
$641$			7.34847i			0.290247i	0.989414	+	0.145124i	$0.0463580\pi$
$641$			7.34847i			0.290247i	−0.989414	+	0.145124i	$0.953642\pi$
$642$		0			0
$643$	8.00000			0.315489			0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000			0.315489			0.157745	+	0.987480i	$0.449578\pi$
$644$	18.9737	−	32.8634i	0.747667	−	1.29500i
$645$		0			0
$646$	−38.9443	+	1.82857i	−1.53224	+	0.0719443i
$647$			5.47723i			0.215332i	0.994187	+	0.107666i	$0.0343377\pi$
$647$			5.47723i			0.215332i	−0.994187	+	0.107666i	$0.965662\pi$
$648$		0			0
$649$		−	15.4919i		−	0.608112i
$650$	−15.8114	−	27.3861i	−0.620174	−	1.07417i
$651$		0			0
$652$	12.0000	−	20.7846i	0.469956	−	0.813988i
$653$		−	32.8634i		−	1.28604i	−0.765848	−	0.643021i	$-0.777681\pi$
$653$		−	32.8634i		−	1.28604i	0.765848	−	0.643021i	$-0.222319\pi$
$654$		0			0
$655$		0			0
$656$	8.48528	−	4.89898i	0.331295	−	0.191273i
$657$		0			0
$658$	13.4164	+	23.2379i	0.523026	+	0.905908i
$659$		−	9.79796i		−	0.381674i	−0.981622	−	0.190837i	$-0.938880\pi$
$659$		−	9.79796i		−	0.381674i	0.981622	−	0.190837i	$-0.0611202\pi$
$660$		0			0
$661$	−31.3050			−1.21762			−0.608811	−	0.793315i	$-0.708353\pi$
$661$	−31.3050			−1.21762			−0.608811	+	0.793315i	$0.708353\pi$
$662$	18.9737	−	10.9545i	0.737432	−	0.425757i
$663$		0			0
$664$	44.7214			1.73553
$665$		0			0
$666$		0			0
$667$			7.74597i			0.299925i
$668$	−19.7990	+	34.2929i	−0.766046	+	1.32683i
$669$		0			0
$670$		0			0
$671$		−	32.8634i		−	1.26868i
$672$		0			0
$673$			46.4758i			1.79151i	0.444548	+	0.895755i	$0.353364\pi$
$673$			46.4758i			1.79151i	−0.444548	+	0.895755i	$0.646636\pi$
$674$	18.9737	−	10.9545i	0.730838	−	0.421950i
$675$		0			0
$676$	−7.00000	+	12.1244i	−0.269231	+	0.466321i
$677$	−29.6985			−1.14141			−0.570703	−	0.821157i	$-0.693329\pi$
$677$	−29.6985			−1.14141			−0.570703	+	0.821157i	$0.693329\pi$
$678$		0			0
$679$	−53.6656			−2.05950
$680$		0			0
$681$		0			0
$682$	−20.0000	−	34.6410i	−0.765840	−	1.32647i
$683$			44.0908i			1.68709i	0.537060	+	0.843544i	$0.319535\pi$
$683$			44.0908i			1.68709i	−0.537060	+	0.843544i	$0.680465\pi$
$684$		0			0
$685$		0			0
$686$	−8.48528	+	4.89898i	−0.323970	+	0.187044i
$687$		0			0
$688$		0			0
$689$	−18.9737			−0.722839
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$	4.24264	−	7.34847i	0.161281	−	0.279347i
$693$		0			0
$694$	−11.1803	−	19.3649i	−0.424400	−	0.735082i
$695$		0			0
$696$		0			0
$697$		−	15.4919i		−	0.586799i
$698$	−16.9706	+	9.79796i	−0.642345	+	0.370858i
$699$		0			0
$700$	30.0000	+	17.3205i	1.13389	+	0.654654i
$701$		−	21.9089i		−	0.827488i	−0.910393	−	0.413744i	$-0.864221\pi$
$701$		−	21.9089i		−	0.827488i	0.910393	−	0.413744i	$-0.135779\pi$
$702$		0			0
$703$	−8.94427	+	17.3205i	−0.337340	+	0.653255i
$704$	25.2982			0.953463
$705$		0			0
$706$	4.47214	+	7.74597i	0.168311	+	0.291523i
$707$	−37.9473			−1.42716
$708$		0			0
$709$			3.46410i			0.130097i	0.997882	+	0.0650485i	$0.0207202\pi$
$709$			3.46410i			0.130097i	−0.997882	+	0.0650485i	$0.979280\pi$
$710$		0			0
$711$		0			0
$712$			48.4974i			1.81752i
$713$		−	48.9898i		−	1.83468i
$714$		0			0
$715$		0			0
$716$	−33.9411	−	19.5959i	−1.26844	−	0.732334i
$717$		0			0
$718$	−6.70820	+	3.87298i	−0.250348	+	0.144538i
$719$			5.47723i			0.204266i	0.994771	+	0.102133i	$0.0325667\pi$
$719$			5.47723i			0.204266i	−0.994771	+	0.102133i	$0.967433\pi$
$720$		0			0
$721$			15.4919i			0.576950i
$722$	−11.1955	+	24.4266i	−0.416653	+	0.909066i
$723$		0			0
$724$	−4.47214	+	7.74597i	−0.166206	+	0.287877i
$725$	−7.07107			−0.262613
$726$		0			0
$727$		−	3.46410i		−	0.128476i	−0.997935	−	0.0642382i	$-0.979538\pi$
$727$		−	3.46410i		−	0.128476i	0.997935	−	0.0642382i	$-0.0204617\pi$
$728$		−	43.8178i		−	1.62400i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$			45.0333i			1.66334i	0.555267	+	0.831672i	$0.312616\pi$
$733$			45.0333i			1.66334i	−0.555267	+	0.831672i	$0.687384\pi$
$734$	−29.6985	+	17.1464i	−1.09619	+	0.632886i
$735$		0			0
$736$	26.8328	+	15.4919i	0.989071	+	0.571040i
$737$		−	24.4949i		−	0.902281i
$738$		0			0
$739$	−12.0000			−0.441427			−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$		0			0
$742$	18.0000	−	10.3923i	0.660801	−	0.381514i
$743$	−16.9706			−0.622590			−0.311295	−	0.950313i	$-0.600763\pi$
$743$	−16.9706			−0.622590			−0.311295	+	0.950313i	$0.600763\pi$
$744$		0			0
$745$		0			0
$746$	22.1359	+	38.3406i	0.810454	+	1.40375i
$747$		0			0
$748$	20.0000	−	34.6410i	0.731272	−	1.26660i
$749$	−50.9117			−1.86027
$750$		0			0
$751$	−31.3050			−1.14233			−0.571167	−	0.820834i	$-0.693509\pi$
$751$	−31.3050			−1.14233			−0.571167	+	0.820834i	$0.693509\pi$
$752$	−18.9737	+	10.9545i	−0.691898	+	0.399468i
$753$		0			0
$754$	4.47214	+	7.74597i	0.162866	+	0.282091i
$755$		0			0
$756$		0			0
$757$		−	13.8564i		−	0.503620i	−0.967777	−	0.251810i	$-0.918974\pi$
$757$		−	13.8564i		−	0.503620i	0.967777	−	0.251810i	$-0.0810257\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.6491			0.458530			0.229265	−	0.973364i	$-0.426368\pi$
$761$	12.6491			0.458530			0.229265	+	0.973364i	$0.426368\pi$
$762$		0			0
$763$			15.4919i			0.560846i
$764$	−28.4605	−	16.4317i	−1.02966	−	0.594477i
$765$		0			0
$766$	−2.00000	−	3.46410i	−0.0722629	−	0.125163i
$767$		−	21.9089i		−	0.791085i
$768$		0			0
$769$	−28.0000			−1.00971			−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000			−1.00971			−0.504853	+	0.863205i	$0.668453\pi$
$770$		0			0
$771$		0			0
$772$	−26.8328	−	15.4919i	−0.965734	−	0.557567i
$773$	24.0416			0.864717			0.432359	−	0.901702i	$-0.357681\pi$
$773$	24.0416			0.864717			0.432359	+	0.901702i	$0.357681\pi$
$774$		0			0
$775$	44.7214			1.60644
$776$		−	43.8178i		−	1.57297i
$777$		0			0
$778$	13.4164	−	7.74597i	0.481002	−	0.277706i
$779$	−9.48683	−	4.89898i	−0.339901	−	0.175524i
$780$		0			0
$781$	−35.7771			−1.28020
$782$	42.4264	−	24.4949i	1.51717	−	0.875936i
$783$		0			0
$784$	10.0000	+	17.3205i	0.357143	+	0.618590i
$785$		0			0
$786$		0			0
$787$			7.74597i			0.276114i	0.990424	+	0.138057i	$0.0440857\pi$
$787$			7.74597i			0.276114i	−0.990424	+	0.138057i	$0.955914\pi$
$788$	−37.9473	−	21.9089i	−1.35182	−	0.780472i
$789$		0			0
$790$		0			0
$791$	−42.4264			−1.50851
$792$		0			0
$793$		−	46.4758i		−	1.65040i
$794$	33.9411	−	19.5959i	1.20453	−	0.695433i
$795$		0			0
$796$	−6.00000	−	3.46410i	−0.212664	−	0.122782i
$797$	41.0122			1.45273			0.726363	−	0.687311i	$-0.241209\pi$
$797$	41.0122			1.45273			0.726363	+	0.687311i	$0.241209\pi$
$798$		0			0
$799$			34.6410i			1.22551i
$800$	−14.1421	+	24.4949i	−0.500000	+	0.866025i
$801$		0			0
$802$	21.0000	−	12.1244i	0.741536	−	0.428126i
$803$	−37.9473			−1.33913
$804$		0			0
$805$		0			0
$806$	−28.2843	−	48.9898i	−0.996271	−	1.72559i
$807$		0			0
$808$		−	30.9839i		−	1.09001i
$809$	31.6228			1.11180			0.555899	−	0.831250i	$-0.312374\pi$
$809$	31.6228			1.11180			0.555899	+	0.831250i	$0.312374\pi$
$810$		0			0
$811$		−	30.9839i		−	1.08799i	−0.839088	−	0.543995i	$-0.816911\pi$
$811$		−	30.9839i		−	1.08799i	0.839088	−	0.543995i	$-0.183089\pi$
$812$	−8.48528	−	4.89898i	−0.297775	−	0.171920i
$813$		0			0
$814$	−10.0000	−	17.3205i	−0.350500	−	0.607083i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	21.9089i		−	0.764626i	−0.924033	−	0.382313i	$-0.875128\pi$
$821$		−	21.9089i		−	0.764626i	0.924033	−	0.382313i	$-0.124872\pi$
$822$		0			0
$823$			10.3923i			0.362253i	0.983460	+	0.181126i	$0.0579743\pi$
$823$			10.3923i			0.362253i	−0.983460	+	0.181126i	$0.942026\pi$
$824$	−12.6491			−0.440653
$825$		0			0
$826$	12.0000	+	20.7846i	0.417533	+	0.723189i
$827$			48.9898i			1.70354i	0.523914	+	0.851771i	$0.324471\pi$
$827$			48.9898i			1.70354i	−0.523914	+	0.851771i	$0.675529\pi$
$828$		0			0
$829$	49.1935			1.70856			0.854280	−	0.519813i	$-0.173998\pi$
$829$	49.1935			1.70856			0.854280	+	0.519813i	$0.173998\pi$
$830$		0			0
$831$		0			0
$832$	35.7771			1.24035
$833$	31.6228			1.09566
$834$		0			0
$835$		0			0
$836$	−14.8886	−	23.2019i	−0.514935	−	0.802454i
$837$		0			0
$838$	−2.23607	−	3.87298i	−0.0772437	−	0.133790i
$839$	8.48528			0.292944			0.146472	−	0.989215i	$-0.453208\pi$
$839$	8.48528			0.292944			0.146472	+	0.989215i	$0.453208\pi$
$840$		0			0
$841$	−27.0000			−0.931034
$842$	3.16228	+	5.47723i	0.108979	+	0.188758i
$843$		0			0
$844$	−40.2492	−	23.2379i	−1.38544	−	0.799882i
$845$		0			0
$846$		0			0
$847$			3.46410i			0.119028i
$848$	8.48528	+	14.6969i	0.291386	+	0.504695i
$849$		0			0
$850$	22.3607	+	38.7298i	0.766965	+	1.32842i
$851$		−	24.4949i		−	0.839674i
$852$		0			0
$853$			27.7128i			0.948869i	0.880291	+	0.474434i	$0.157347\pi$
$853$			27.7128i			0.948869i	−0.880291	+	0.474434i	$0.842653\pi$
$854$	25.4558	+	44.0908i	0.871081	+	1.50876i
$855$		0			0
$856$		−	41.5692i		−	1.42081i
$857$			26.9444i			0.920403i	0.887815	+	0.460201i	$0.152223\pi$
$857$			26.9444i			0.920403i	−0.887815	+	0.460201i	$0.847777\pi$
$858$		0			0
$859$	−52.0000			−1.77422			−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000			−1.77422			−0.887109	+	0.461561i	$0.847290\pi$
$860$		0			0
$861$		0			0
$862$	18.0000	+	31.1769i	0.613082	+	1.06189i
$863$	−16.9706			−0.577685			−0.288842	−	0.957377i	$-0.593270\pi$
$863$	−16.9706			−0.577685			−0.288842	+	0.957377i	$0.593270\pi$
$864$		0			0
$865$		0			0
$866$	37.9473	−	21.9089i	1.28950	−	0.744495i
$867$		0			0
$868$	53.6656	+	30.9839i	1.82153	+	1.05166i
$869$	14.1421			0.479739
$870$		0			0
$871$		−	34.6410i		−	1.17377i
$872$	−12.6491			−0.428353
$873$		0			0
$874$	−1.58359	−	33.7267i	−0.0535658	−	1.14082i
$875$		0			0
$876$		0			0
$877$	−4.47214			−0.151013			−0.0755067	−	0.997145i	$-0.524057\pi$
$877$	−4.47214			−0.151013			−0.0755067	+	0.997145i	$0.524057\pi$
$878$	15.8114	+	27.3861i	0.533609	+	0.924237i
$879$		0			0
$880$		0			0
$881$	31.6228			1.06540			0.532699	−	0.846305i	$-0.321178\pi$
$881$	31.6228			1.06540			0.532699	+	0.846305i	$0.321178\pi$
$882$		0			0
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$	28.2843	−	48.9898i	0.951303	−	1.64771i
$885$		0			0
$886$	−11.1803	−	19.3649i	−0.375611	−	0.650577i
$887$	−53.7401			−1.80442			−0.902208	−	0.431301i	$-0.858055\pi$
$887$	−53.7401			−1.80442			−0.902208	+	0.431301i	$0.858055\pi$
$888$		0			0
$889$		−	30.9839i		−	1.03917i
$890$		0			0
$891$		0			0
$892$	−22.3607	+	38.7298i	−0.748691	+	1.29677i
$893$	21.2132	+	10.9545i	0.709873	+	0.366577i
$894$		0			0
$895$		0			0
$896$	−33.9411	+	19.5959i	−1.13389	+	0.654654i
$897$		0			0
$898$	−9.00000	+	5.19615i	−0.300334	+	0.173398i
$899$	−12.6491			−0.421871
$900$		0			0
$901$	26.8328			0.893931
$902$	9.48683	−	5.47723i	0.315877	−	0.182372i
$903$		0			0
$904$		−	34.6410i		−	1.15214i
$905$		0			0
$906$		0			0
$907$		−	23.2379i		−	0.771602i	−0.922582	−	0.385801i	$-0.873925\pi$
$907$		−	23.2379i		−	0.771602i	0.922582	−	0.385801i	$-0.126075\pi$
$908$	25.4558	+	14.6969i	0.844782	+	0.487735i
$909$		0			0
$910$		0			0
$911$	31.1127			1.03081			0.515405	−	0.856947i	$-0.327642\pi$
$911$	31.1127			1.03081			0.515405	+	0.856947i	$0.327642\pi$
$912$		0			0
$913$	50.0000			1.65476
$914$	−2.82843	−	4.89898i	−0.0935561	−	0.162044i
$915$		0			0
$916$		0			0
$917$		−	10.9545i		−	0.361748i
$918$		0			0
$919$		−	38.1051i		−	1.25697i	−0.777821	−	0.628486i	$-0.783675\pi$
$919$		−	38.1051i		−	1.25697i	0.777821	−	0.628486i	$-0.216325\pi$
$920$		0			0
$921$		0			0
$922$	−13.4164	+	7.74597i	−0.441846	+	0.255100i
$923$	−50.5964			−1.66540
$924$		0			0
$925$	22.3607			0.735215
$926$	−4.24264	+	2.44949i	−0.139422	+	0.0804952i
$927$		0			0
$928$	4.00000	−	6.92820i	0.131306	−	0.227429i
$929$	12.6491			0.415004			0.207502	−	0.978235i	$-0.433467\pi$
$929$	12.6491			0.415004			0.207502	+	0.978235i	$0.433467\pi$
$930$		0			0
$931$	10.0000	−	19.3649i	0.327737	−	0.634660i
$932$	−6.32456	+	10.9545i	−0.207168	+	0.358825i
$933$		0			0
$934$	2.23607	+	3.87298i	0.0731664	+	0.126728i
$935$		0			0
$936$		0			0
$937$	−6.00000			−0.196011			−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000			−0.196011			−0.0980057	+	0.995186i	$0.531246\pi$
$938$	18.9737	+	32.8634i	0.619512	+	1.07303i
$939$		0			0
$940$		0			0
$941$	35.3553			1.15255			0.576276	−	0.817255i	$-0.304506\pi$
$941$	35.3553			1.15255			0.576276	+	0.817255i	$0.304506\pi$
$942$		0			0
$943$	13.4164			0.436898
$944$	−16.9706	+	9.79796i	−0.552345	+	0.318896i
$945$		0			0
$946$		0			0
$947$	53.7587			1.74692			0.873462	−	0.486893i	$-0.161870\pi$
$947$	53.7587			1.74692			0.873462	+	0.486893i	$0.161870\pi$
$948$		0			0
$949$	−53.6656			−1.74206
$950$	30.7882	−	1.44562i	0.998899	−	0.0469020i
$951$		0			0
$952$			61.9677i			2.00839i
$953$			31.8434i			1.03151i	0.856737	+	0.515754i	$0.172488\pi$
$953$			31.8434i			1.03151i	−0.856737	+	0.515754i	$0.827512\pi$
$954$		0			0
$955$		0			0
$956$	−28.4605	−	16.4317i	−0.920478	−	0.531438i
$957$		0			0
$958$	33.5410	−	19.3649i	1.08366	−	0.625652i
$959$			43.8178i			1.41495i
$960$		0			0
$961$	49.0000			1.58065
$962$	−14.1421	−	24.4949i	−0.455961	−	0.789747i
$963$		0			0
$964$	−26.8328	−	15.4919i	−0.864227	−	0.498962i
$965$		0			0
$966$		0			0
$967$			31.1769i			1.00258i	0.865279	+	0.501291i	$0.167141\pi$
$967$			31.1769i			1.00258i	−0.865279	+	0.501291i	$0.832859\pi$
$968$	−2.82843			−0.0909091
$969$		0			0
$970$		0			0
$971$			14.6969i			0.471647i	0.971796	+	0.235824i	$0.0757788\pi$
$971$			14.6969i			0.471647i	−0.971796	+	0.235824i	$0.924221\pi$
$972$		0			0
$973$		−	41.5692i		−	1.33265i
$974$	6.32456	+	10.9545i	0.202652	+	0.351003i
$975$		0			0
$976$	−36.0000	+	20.7846i	−1.15233	+	0.665299i
$977$		−	36.7423i		−	1.17549i	−0.809046	−	0.587746i	$-0.800015\pi$
$977$		−	36.7423i		−	1.17549i	0.809046	−	0.587746i	$-0.199985\pi$
$978$		0			0
$979$			54.2218i			1.73294i
$980$		0			0
$981$		0			0
$982$	15.6525	+	27.1109i	0.499491	+	0.865143i
$983$	−50.9117			−1.62383			−0.811915	−	0.583775i	$-0.801575\pi$
$983$	−50.9117			−1.62383			−0.811915	+	0.583775i	$0.801575\pi$
$984$		0			0
$985$		0			0
$986$	−6.32456	−	10.9545i	−0.201415	−	0.348861i
$987$		0			0
$988$	−21.0557	−	32.8124i	−0.669872	−	1.04390i
$989$		0			0
$990$		0			0
$991$	−8.94427			−0.284124			−0.142062	−	0.989858i	$-0.545373\pi$
$991$	−8.94427			−0.284124			−0.142062	+	0.989858i	$0.545373\pi$
$992$	−25.2982	+	43.8178i	−0.803219	+	1.39122i
$993$		0			0
$994$	48.0000	−	27.7128i	1.52247	−	0.878997i
$995$		0			0
$996$		0			0
$997$		−	58.8897i		−	1.86506i	−0.361097	−	0.932528i	$-0.617598\pi$
$997$		−	58.8897i		−	1.86506i	0.361097	−	0.932528i	$-0.382402\pi$
$998$	5.65685	+	9.79796i	0.179065	+	0.310149i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.e.d.379.2	yes	8
3.2	odd	2	inner	1368.2.e.d.379.7	yes	8
4.3	odd	2		5472.2.e.d.5167.6		8
8.3	odd	2	inner	1368.2.e.d.379.6	yes	8
8.5	even	2		5472.2.e.d.5167.1		8
12.11	even	2		5472.2.e.d.5167.8		8
19.18	odd	2	inner	1368.2.e.d.379.8	yes	8
24.5	odd	2		5472.2.e.d.5167.3		8
24.11	even	2	inner	1368.2.e.d.379.3	yes	8
57.56	even	2	inner	1368.2.e.d.379.1	✓	8
76.75	even	2		5472.2.e.d.5167.5		8
152.37	odd	2		5472.2.e.d.5167.2		8
152.75	even	2	inner	1368.2.e.d.379.4	yes	8
228.227	odd	2		5472.2.e.d.5167.7		8
456.227	odd	2	inner	1368.2.e.d.379.5	yes	8
456.341	even	2		5472.2.e.d.5167.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.e.d.379.1	✓	8	57.56	even	2	inner
1368.2.e.d.379.2	yes	8	1.1	even	1	trivial
1368.2.e.d.379.3	yes	8	24.11	even	2	inner
1368.2.e.d.379.4	yes	8	152.75	even	2	inner
1368.2.e.d.379.5	yes	8	456.227	odd	2	inner
1368.2.e.d.379.6	yes	8	8.3	odd	2	inner
1368.2.e.d.379.7	yes	8	3.2	odd	2	inner
1368.2.e.d.379.8	yes	8	19.18	odd	2	inner
5472.2.e.d.5167.1		8	8.5	even	2
5472.2.e.d.5167.2		8	152.37	odd	2
5472.2.e.d.5167.3		8	24.5	odd	2
5472.2.e.d.5167.4		8	456.341	even	2
5472.2.e.d.5167.5		8	76.75	even	2
5472.2.e.d.5167.6		8	4.3	odd	2
5472.2.e.d.5167.7		8	228.227	odd	2
5472.2.e.d.5167.8		8	12.11	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} -3.46410i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} -3.46410i q^{7} +2.82843 q^{8} +3.16228 q^{11} +4.47214 q^{13} +(-4.24264 + 2.44949i) q^{14} +(-2.00000 - 3.46410i) q^{16} -6.32456 q^{17} +(-2.00000 + 3.87298i) q^{19} +(-2.23607 - 3.87298i) q^{22} -5.47723i q^{23} +5.00000 q^{25} +(-3.16228 - 5.47723i) q^{26} +(6.00000 + 3.46410i) q^{28} -1.41421 q^{29} +8.94427 q^{31} +(-2.82843 + 4.89898i) q^{32} +(4.47214 + 7.74597i) q^{34} +4.47214 q^{37} +(6.15763 - 0.289123i) q^{38} +2.44949i q^{41} +(-3.16228 + 5.47723i) q^{44} +(-6.70820 + 3.87298i) q^{46} -5.47723i q^{47} -5.00000 q^{49} +(-3.53553 - 6.12372i) q^{50} +(-4.47214 + 7.74597i) q^{52} -4.24264 q^{53} -9.79796i q^{56} +(1.00000 + 1.73205i) q^{58} -4.89898i q^{59} -10.3923i q^{61} +(-6.32456 - 10.9545i) q^{62} +8.00000 q^{64} -7.74597i q^{67} +(6.32456 - 10.9545i) q^{68} -11.3137 q^{71} -12.0000 q^{73} +(-3.16228 - 5.47723i) q^{74} +(-4.70820 - 7.33708i) q^{76} -10.9545i q^{77} +4.47214 q^{79} +(3.00000 - 1.73205i) q^{82} +15.8114 q^{83} +8.94427 q^{88} +17.1464i q^{89} -15.4919i q^{91} +(9.48683 + 5.47723i) q^{92} +(-6.70820 + 3.87298i) q^{94} -15.4919i q^{97} +(3.53553 + 6.12372i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 16 q^{16} - 16 q^{19} + 40 q^{25} + 48 q^{28} - 40 q^{49} + 8 q^{58} + 64 q^{64} - 96 q^{73} + 16 q^{76} + 24 q^{82}+O(q^{100}) \) 8 * q - 8 * q^4 - 16 * q^16 - 16 * q^19 + 40 * q^25 + 48 * q^28 - 40 * q^49 + 8 * q^58 + 64 * q^64 - 96 * q^73 + 16 * q^76 + 24 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more