LMFDB - Embedded newform 5472.2.e.d.5167.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(5167,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, 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("5472.5167");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.e (of order $2$, degree $1$, not 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:	$43.6941399860$
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:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5167.6
Root			$1.14412 - 1.98168i$ of defining polynomial
Character	$\chi$	$=$	5472.5167
Dual form			5472.2.e.d.5167.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410i q^{7} +O(q^{10})$	$q+3.46410i q^{7} -3.16228 q^{11} +4.47214 q^{13} -6.32456 q^{17} +(2.00000 - 3.87298i) q^{19} +5.47723i q^{23} +5.00000 q^{25} -1.41421 q^{29} -8.94427 q^{31} +4.47214 q^{37} +2.44949i q^{41} +5.47723i q^{47} -5.00000 q^{49} -4.24264 q^{53} +4.89898i q^{59} -10.3923i q^{61} +7.74597i q^{67} +11.3137 q^{71} -12.0000 q^{73} -10.9545i q^{77} -4.47214 q^{79} -15.8114 q^{83} +17.1464i q^{89} +15.4919i q^{91} -15.4919i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{19} + 40 q^{25} - 40 q^{49} - 96 q^{73}+O(q^{100}) $ 8 * q + 16 * q^19 + 40 * q^25 - 40 * q^49 - 96 * q^73

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$			3.46410i			1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$			3.46410i			1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.16228			−0.953463			−0.476731	−	0.879049i	$-0.658179\pi$
$11$	−3.16228			−0.953463			−0.476731	+	0.879049i	$0.658179\pi$
$12$		0			0
$13$	4.47214			1.24035			0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214			1.24035			0.620174	+	0.784465i	$0.287062\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−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
$22$		0			0
$23$			5.47723i			1.14208i	0.820922	+	0.571040i	$0.193460\pi$
$23$			5.47723i			1.14208i	−0.820922	+	0.571040i	$0.806540\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−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	$-0.796881\pi$
$31$	−8.94427			−1.60644			−0.803219	+	0.595683i	$0.796881\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		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	$0.380177\pi$
$38$		0			0
$39$		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	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.47723i			0.798935i	0.916747	+	0.399468i	$0.130805\pi$
$47$			5.47723i			0.798935i	−0.916747	+	0.399468i	$0.869195\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−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
$56$		0			0
$57$		0			0
$58$		0			0
$59$			4.89898i			0.637793i	0.947790	+	0.318896i	$0.103312\pi$
$59$			4.89898i			0.637793i	−0.947790	+	0.318896i	$0.896688\pi$
$60$		0			0
$61$		−	10.3923i		−	1.33060i	−0.746577	−	0.665299i	$-0.768304\pi$
$61$		−	10.3923i		−	1.33060i	0.746577	−	0.665299i	$-0.231696\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			7.74597i			0.946320i	0.880976	+	0.473160i	$0.156887\pi$
$67$			7.74597i			0.946320i	−0.880976	+	0.473160i	$0.843113\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.3137			1.34269			0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137			1.34269			0.671345	+	0.741145i	$0.265717\pi$
$72$		0			0
$73$	−12.0000			−1.40449			−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000			−1.40449			−0.702247	+	0.711934i	$0.747820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	10.9545i		−	1.24838i
$78$		0			0
$79$	−4.47214			−0.503155			−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214			−0.503155			−0.251577	+	0.967837i	$0.580949\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.8114			−1.73553			−0.867763	−	0.496979i	$-0.834443\pi$
$83$	−15.8114			−1.73553			−0.867763	+	0.496979i	$0.834443\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			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
$92$		0			0
$93$		0			0
$94$		0			0
$95$		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	$-0.288101\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	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	$-0.429288\pi$
$103$	4.47214			0.440653			0.220326	+	0.975426i	$0.429288\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.6969i			1.42081i	0.703795	+	0.710403i	$0.251487\pi$
$107$			14.6969i			1.42081i	−0.703795	+	0.710403i	$0.748513\pi$
$108$		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
$112$		0			0
$113$		−	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
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	21.9089i		−	2.00839i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.94427			−0.793676			−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427			−0.793676			−0.396838	+	0.917889i	$0.629892\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.16228			−0.276289			−0.138145	−	0.990412i	$-0.544114\pi$
$131$	−3.16228			−0.276289			−0.138145	+	0.990412i	$0.544114\pi$
$132$		0			0
$133$	13.4164	+	6.92820i	1.16335	+	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−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	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.1421			−1.18262
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	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	$-0.441753\pi$
$151$	4.47214			0.363937			0.181969	+	0.983304i	$0.441753\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		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	$-0.0441422\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.9737			−1.49533
$162$		0			0
$163$	12.0000			0.939913			0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000			0.939913			0.469956	+	0.882690i	$0.344270\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−19.7990			−1.53209			−0.766046	−	0.642786i	$-0.777779\pi$
$167$	−19.7990			−1.53209			−0.766046	+	0.642786i	$0.777779\pi$
$168$		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
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.5959i		−	1.46467i	−0.680946	−	0.732334i	$-0.738431\pi$
$179$		−	19.5959i		−	1.46467i	0.680946	−	0.732334i	$-0.261569\pi$
$180$		0			0
$181$	4.47214			0.332411			0.166206	−	0.986091i	$-0.446848\pi$
$181$	4.47214			0.332411			0.166206	+	0.986091i	$0.446848\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	20.0000			1.46254
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	16.4317i		−	1.18895i	−0.804112	−	0.594477i	$-0.797359\pi$
$191$		−	16.4317i		−	1.18895i	0.804112	−	0.594477i	$-0.202641\pi$
$192$		0			0
$193$			15.4919i			1.11513i	0.830132	+	0.557567i	$0.188265\pi$
$193$			15.4919i			1.11513i	−0.830132	+	0.557567i	$0.811735\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			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	$-0.960818\pi$
$199$		−	3.46410i		−	0.245564i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	4.89898i		−	0.343841i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.32456	+	12.2474i	−0.437479	+	0.847174i
$210$		0			0
$211$		−	23.2379i		−	1.59976i	−0.600158	−	0.799882i	$-0.704896\pi$
$211$		−	23.2379i		−	1.59976i	0.600158	−	0.799882i	$-0.295104\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	30.9839i		−	2.10332i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−28.2843			−1.90261
$222$		0			0
$223$	−22.3607			−1.49738			−0.748691	−	0.662919i	$-0.769317\pi$
$223$	−22.3607			−1.49738			−0.748691	+	0.662919i	$0.769317\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			14.6969i			0.975470i	0.872992	+	0.487735i	$0.162177\pi$
$227$			14.6969i			0.975470i	−0.872992	+	0.487735i	$0.837823\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	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
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	16.4317i		−	1.06288i	−0.847097	−	0.531438i	$-0.821652\pi$
$239$		−	16.4317i		−	1.06288i	0.847097	−	0.531438i	$-0.178348\pi$
$240$		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			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.94427	−	17.3205i	0.569110	−	1.10208i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.8114			−0.998006			−0.499003	−	0.866600i	$-0.666300\pi$
$251$	−15.8114			−0.998006			−0.499003	+	0.866600i	$0.666300\pi$
$252$		0			0
$253$		−	17.3205i		−	1.08893i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			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
$262$		0			0
$263$		−	16.4317i		−	1.01322i	−0.862175	−	0.506610i	$-0.830898\pi$
$263$		−	16.4317i		−	1.01322i	0.862175	−	0.506610i	$-0.169102\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	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	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−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	$0.863335\pi$
$278$		0			0
$279$		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	$-0.702626\pi$
$283$	−20.0000			−1.18888			−0.594438	+	0.804141i	$0.702626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.48528			−0.500870
$288$		0			0
$289$	23.0000			1.35294
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−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
$296$		0			0
$297$		0			0
$298$		0			0
$299$			24.4949i			1.41658i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			15.4919i			0.884171i	0.896973	+	0.442086i	$0.145761\pi$
$307$			15.4919i			0.884171i	−0.896973	+	0.442086i	$0.854239\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			5.47723i			0.310585i	0.987869	+	0.155292i	$0.0496320\pi$
$311$			5.47723i			0.310585i	−0.987869	+	0.155292i	$0.950368\pi$
$312$		0			0
$313$	−20.0000			−1.13047			−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000			−1.13047			−0.565233	+	0.824931i	$0.691214\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	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
$322$		0			0
$323$	−12.6491	+	24.4949i	−0.703815	+	1.36293i
$324$		0			0
$325$	22.3607			1.24035
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−18.9737			−1.04605
$330$		0			0
$331$		−	15.4919i		−	0.851514i	−0.904838	−	0.425757i	$-0.860008\pi$
$331$		−	15.4919i		−	0.851514i	0.904838	−	0.425757i	$-0.139992\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		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	$0.861346\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	28.2843			1.53168
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.8114			−0.848800			−0.424400	−	0.905475i	$-0.639515\pi$
$347$	−15.8114			−0.848800			−0.424400	+	0.905475i	$0.639515\pi$
$348$		0			0
$349$		−	13.8564i		−	0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$		−	13.8564i		−	0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−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
$356$		0			0
$357$		0			0
$358$		0			0
$359$			5.47723i			0.289077i	0.989499	+	0.144538i	$0.0461697\pi$
$359$			5.47723i			0.289077i	−0.989499	+	0.144538i	$0.953830\pi$
$360$		0			0
$361$	−11.0000	−	15.4919i	−0.578947	−	0.815365i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$			24.2487i			1.26577i	0.774245	+	0.632886i	$0.218130\pi$
$367$			24.2487i			1.26577i	−0.774245	+	0.632886i	$0.781870\pi$
$368$		0			0
$369$		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	$0.800780\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−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
$382$		0			0
$383$	−2.82843			−0.144526			−0.0722629	−	0.997386i	$-0.523022\pi$
$383$	−2.82843			−0.144526			−0.0722629	+	0.997386i	$0.523022\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			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
$392$		0			0
$393$		0			0
$394$		0			0
$395$		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	$0.755213\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			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
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−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
$412$		0			0
$413$	−16.9706			−0.835067
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.16228			−0.154487			−0.0772437	−	0.997012i	$-0.524612\pi$
$419$	−3.16228			−0.154487			−0.0772437	+	0.997012i	$0.524612\pi$
$420$		0			0
$421$	−4.47214			−0.217959			−0.108979	−	0.994044i	$-0.534758\pi$
$421$	−4.47214			−0.217959			−0.108979	+	0.994044i	$0.534758\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−31.6228			−1.53393
$426$		0			0
$427$	36.0000			1.74216
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.4558			1.22616			0.613082	−	0.790019i	$-0.289929\pi$
$431$	25.4558			1.22616			0.613082	+	0.790019i	$0.289929\pi$
$432$		0			0
$433$			30.9839i			1.48899i	0.667628	+	0.744495i	$0.267310\pi$
$433$			30.9839i			1.48899i	−0.667628	+	0.744495i	$0.732690\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.2132	+	10.9545i	1.01477	+	0.524022i
$438$		0			0
$439$	22.3607			1.06722			0.533609	−	0.845732i	$-0.320836\pi$
$439$	22.3607			1.06722			0.533609	+	0.845732i	$0.320836\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−15.8114			−0.751222			−0.375611	−	0.926777i	$-0.622567\pi$
$443$	−15.8114			−0.751222			−0.375611	+	0.926777i	$0.622567\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	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
$452$		0			0
$453$		0			0
$454$		0			0
$455$		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	$0.0256502\pi$
$463$			3.46410i			0.160990i	−0.996755	+	0.0804952i	$0.974350\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.16228			0.146333			0.0731664	−	0.997320i	$-0.476690\pi$
$467$	3.16228			0.146333			0.0731664	+	0.997320i	$0.476690\pi$
$468$		0			0
$469$	−26.8328			−1.23902
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	10.0000	−	19.3649i	0.458831	−	0.888523i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	27.3861i		−	1.25130i	−0.780102	−	0.625652i	$-0.784833\pi$
$479$		−	27.3861i		−	1.25130i	0.780102	−	0.625652i	$-0.215167\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	8.94427			0.405304			0.202652	−	0.979251i	$-0.435044\pi$
$487$	8.94427			0.405304			0.202652	+	0.979251i	$0.435044\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.1359			0.998981			0.499491	−	0.866319i	$-0.333521\pi$
$491$	22.1359			0.998981			0.499491	+	0.866319i	$0.333521\pi$
$492$		0			0
$493$	8.94427			0.402830
$494$		0			0
$495$		0			0
$496$		0			0
$497$			39.1918i			1.75799i
$498$		0			0
$499$	8.00000			0.358129			0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000			0.358129			0.179065	+	0.983837i	$0.442693\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			16.4317i			0.732652i	0.930487	+	0.366326i	$0.119385\pi$
$503$			16.4317i			0.732652i	−0.930487	+	0.366326i	$0.880615\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−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
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	17.3205i		−	0.761755i
$518$		0			0
$519$		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	$-0.945832\pi$
$523$		−	7.74597i		−	0.338707i	0.985555	−	0.169354i	$-0.0541680\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	56.5685			2.46416
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$			10.9545i			0.474490i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	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	$0.878558\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.82843	+	5.47723i	−0.120495	+	0.233338i
$552$		0			0
$553$		−	15.4919i		−	0.658784i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			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
$562$		0			0
$563$		−	19.5959i		−	0.825869i	−0.910761	−	0.412935i	$-0.864504\pi$
$563$		−	19.5959i		−	0.825869i	0.910761	−	0.412935i	$-0.135496\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	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	$-0.667470\pi$
$571$	−24.0000			−1.00437			−0.502184	+	0.864761i	$0.667470\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			27.3861i			1.14208i
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	54.7723i		−	2.27234i
$582$		0			0
$583$	13.4164			0.555651
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.1359			−0.913648			−0.456824	−	0.889557i	$-0.651013\pi$
$587$	−22.1359			−0.913648			−0.456824	+	0.889557i	$0.651013\pi$
$588$		0			0
$589$	−17.8885	+	34.6410i	−0.737085	+	1.42736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	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
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−28.2843			−1.15566			−0.577832	−	0.816156i	$-0.696101\pi$
$599$	−28.2843			−1.15566			−0.577832	+	0.816156i	$0.696101\pi$
$600$		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
$604$		0			0
$605$		0			0
$606$		0			0
$607$	8.94427			0.363037			0.181518	−	0.983388i	$-0.441899\pi$
$607$	8.94427			0.363037			0.181518	+	0.983388i	$0.441899\pi$
$608$		0			0
$609$		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	$-0.162893\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	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	$-0.722350\pi$
$619$	−32.0000			−1.28619			−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−59.3970			−2.37969
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−28.2843			−1.12777
$630$		0			0
$631$			17.3205i			0.689519i	0.938691	+	0.344759i	$0.112039\pi$
$631$			17.3205i			0.689519i	−0.938691	+	0.344759i	$0.887961\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−22.3607			−0.885962
$638$		0			0
$639$		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	$-0.550422\pi$
$643$	−8.00000			−0.315489			−0.157745	+	0.987480i	$0.550422\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	5.47723i		−	0.215332i	−0.994187	−	0.107666i	$-0.965662\pi$
$647$		−	5.47723i		−	0.215332i	0.994187	−	0.107666i	$-0.0343377\pi$
$648$		0			0
$649$		−	15.4919i		−	0.608112i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	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
$656$		0			0
$657$		0			0
$658$		0			0
$659$			9.79796i			0.381674i	0.981622	+	0.190837i	$0.0611202\pi$
$659$			9.79796i			0.381674i	−0.981622	+	0.190837i	$0.938880\pi$
$660$		0			0
$661$	−31.3050			−1.21762			−0.608811	−	0.793315i	$-0.708353\pi$
$661$	−31.3050			−1.21762			−0.608811	+	0.793315i	$0.708353\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	7.74597i		−	0.299925i
$668$		0			0
$669$		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	$0.646636\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−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
$682$		0			0
$683$		−	44.0908i		−	1.68709i	−0.537060	−	0.843544i	$-0.680465\pi$
$683$		−	44.0908i		−	1.68709i	0.537060	−	0.843544i	$-0.319535\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.9737			−0.722839
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	15.4919i		−	0.586799i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	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
$704$		0			0
$705$		0			0
$706$		0			0
$707$	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
$712$		0			0
$713$		−	48.9898i		−	1.83468i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	5.47723i		−	0.204266i	−0.994771	−	0.102133i	$-0.967433\pi$
$719$		−	5.47723i		−	0.204266i	0.994771	−	0.102133i	$-0.0325667\pi$
$720$		0			0
$721$			15.4919i			0.576950i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.07107			−0.262613
$726$		0			0
$727$			3.46410i			0.128476i	0.997935	+	0.0642382i	$0.0204617\pi$
$727$			3.46410i			0.128476i	−0.997935	+	0.0642382i	$0.979538\pi$
$728$		0			0
$729$		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	$0.687384\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	24.4949i		−	0.902281i
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.9706			0.622590			0.311295	−	0.950313i	$-0.399237\pi$
$743$	16.9706			0.622590			0.311295	+	0.950313i	$0.399237\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−50.9117			−1.86027
$750$		0			0
$751$	31.3050			1.14233			0.571167	−	0.820834i	$-0.306491\pi$
$751$	31.3050			1.14233			0.571167	+	0.820834i	$0.306491\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		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
$764$		0			0
$765$		0			0
$766$		0			0
$767$			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
$772$		0			0
$773$	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
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.48683	+	4.89898i	0.339901	+	0.175524i
$780$		0			0
$781$	−35.7771			−1.28020
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	7.74597i		−	0.276114i	−0.990424	−	0.138057i	$-0.955914\pi$
$787$		−	7.74597i		−	0.276114i	0.990424	−	0.138057i	$-0.0440857\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	42.4264			1.50851
$792$		0			0
$793$		−	46.4758i		−	1.65040i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	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
$800$		0			0
$801$		0			0
$802$		0			0
$803$	37.9473			1.33913
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	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	$0.183089\pi$
$811$			30.9839i			1.08799i	−0.839088	+	0.543995i	$0.816911\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		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	$-0.942026\pi$
$823$		−	10.3923i		−	0.362253i	0.983460	−	0.181126i	$-0.0579743\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.9898i		−	1.70354i	−0.523914	−	0.851771i	$-0.675529\pi$
$827$		−	48.9898i		−	1.70354i	0.523914	−	0.851771i	$-0.324471\pi$
$828$		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
$832$		0			0
$833$	31.6228			1.09566
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.48528			−0.292944			−0.146472	−	0.989215i	$-0.546792\pi$
$839$	−8.48528			−0.292944			−0.146472	+	0.989215i	$0.546792\pi$
$840$		0			0
$841$	−27.0000			−0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	3.46410i		−	0.119028i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			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	$0.842653\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			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	$-0.152710\pi$
$859$	52.0000			1.77422			0.887109	+	0.461561i	$0.152710\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16.9706			0.577685			0.288842	−	0.957377i	$-0.406730\pi$
$863$	16.9706			0.577685			0.288842	+	0.957377i	$0.406730\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.1421			0.479739
$870$		0			0
$871$			34.6410i			1.17377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		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	$0.524057\pi$
$878$		0			0
$879$		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	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	53.7401			1.80442			0.902208	−	0.431301i	$-0.141945\pi$
$887$	53.7401			1.80442			0.902208	+	0.431301i	$0.141945\pi$
$888$		0			0
$889$		−	30.9839i		−	1.03917i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21.2132	+	10.9545i	0.709873	+	0.366577i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.6491			0.421871
$900$		0			0
$901$	26.8328			0.893931
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			23.2379i			0.771602i	0.922582	+	0.385801i	$0.126075\pi$
$907$			23.2379i			0.771602i	−0.922582	+	0.385801i	$0.873925\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−31.1127			−1.03081			−0.515405	−	0.856947i	$-0.672358\pi$
$911$	−31.1127			−1.03081			−0.515405	+	0.856947i	$0.672358\pi$
$912$		0			0
$913$	50.0000			1.65476
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	10.9545i		−	0.361748i
$918$		0			0
$919$			38.1051i			1.25697i	0.777821	+	0.628486i	$0.216325\pi$
$919$			38.1051i			1.25697i	−0.777821	+	0.628486i	$0.783675\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	50.5964			1.66540
$924$		0			0
$925$	22.3607			0.735215
$926$		0			0
$927$		0			0
$928$		0			0
$929$	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
$932$		0			0
$933$		0			0
$934$		0			0
$935$		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	$0.531246\pi$
$938$		0			0
$939$		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
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−53.7587			−1.74692			−0.873462	−	0.486893i	$-0.838130\pi$
$947$	−53.7587			−1.74692			−0.873462	+	0.486893i	$0.838130\pi$
$948$		0			0
$949$	−53.6656			−1.74206
$950$		0			0
$951$		0			0
$952$		0			0
$953$			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
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	43.8178i		−	1.41495i
$960$		0			0
$961$	49.0000			1.58065
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	31.1769i		−	1.00258i	−0.865279	−	0.501291i	$-0.832859\pi$
$967$		−	31.1769i		−	1.00258i	0.865279	−	0.501291i	$-0.167141\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	14.6969i		−	0.471647i	−0.971796	−	0.235824i	$-0.924221\pi$
$971$		−	14.6969i		−	0.471647i	0.971796	−	0.235824i	$-0.0757788\pi$
$972$		0			0
$973$		−	41.5692i		−	1.33265i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	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
$982$		0			0
$983$	50.9117			1.62383			0.811915	−	0.583775i	$-0.198425\pi$
$983$	50.9117			1.62383			0.811915	+	0.583775i	$0.198425\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	8.94427			0.284124			0.142062	−	0.989858i	$-0.454627\pi$
$991$	8.94427			0.284124			0.142062	+	0.989858i	$0.454627\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		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	$-0.382402\pi$
$998$		0			0
$999$		0			0

Twists

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

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

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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.46410i q^{7} +O(q^{10})\)	\(q+3.46410i q^{7} -3.16228 q^{11} +4.47214 q^{13} -6.32456 q^{17} +(2.00000 - 3.87298i) q^{19} +5.47723i q^{23} +5.00000 q^{25} -1.41421 q^{29} -8.94427 q^{31} +4.47214 q^{37} +2.44949i q^{41} +5.47723i q^{47} -5.00000 q^{49} -4.24264 q^{53} +4.89898i q^{59} -10.3923i q^{61} +7.74597i q^{67} +11.3137 q^{71} -12.0000 q^{73} -10.9545i q^{77} -4.47214 q^{79} -15.8114 q^{83} +17.1464i q^{89} +15.4919i q^{91} -15.4919i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{19} + 40 q^{25} - 40 q^{49} - 96 q^{73}+O(q^{100}) \) 8 * q + 16 * q^19 + 40 * q^25 - 40 * q^49 - 96 * q^73

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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