LMFDB - Embedded newform 2448.2.be.p.1585.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,2,Mod(1441,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 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("2448.1441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.be (of order $4$, degree $2$, 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:	$19.5473784148$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1585.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2448.1585
Dual form			2448.2.be.p.1441.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.414214 - 0.414214i) q^{5} +(-1.41421 - 1.41421i) q^{7} +O(q^{10})$	$q+(0.414214 - 0.414214i) q^{5} +(-1.41421 - 1.41421i) q^{7} +(2.82843 + 2.82843i) q^{11} -6.82843 q^{13} +(-1.00000 - 4.00000i) q^{17} +2.82843i q^{19} +(5.41421 + 5.41421i) q^{23} +4.65685i q^{25} +(5.24264 - 5.24264i) q^{29} +(5.41421 - 5.41421i) q^{31} -1.17157 q^{35} +(8.41421 - 8.41421i) q^{37} +(2.65685 + 2.65685i) q^{41} -1.65685i q^{43} +5.17157 q^{47} -3.00000i q^{49} -1.17157i q^{53} +2.34315 q^{55} -2.82843i q^{59} +(-5.24264 - 5.24264i) q^{61} +(-2.82843 + 2.82843i) q^{65} -8.48528 q^{67} +(8.24264 - 8.24264i) q^{71} +(1.00000 - 1.00000i) q^{73} -8.00000i q^{77} +(4.24264 + 4.24264i) q^{79} +5.17157i q^{83} +(-2.07107 - 1.24264i) q^{85} -16.9706 q^{89} +(9.65685 + 9.65685i) q^{91} +(1.17157 + 1.17157i) q^{95} +(11.4853 - 11.4853i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} - 16 q^{13} - 4 q^{17} + 16 q^{23} + 4 q^{29} + 16 q^{31} - 16 q^{35} + 28 q^{37} - 12 q^{41} + 32 q^{47} + 32 q^{55} - 4 q^{61} + 16 q^{71} + 4 q^{73} + 20 q^{85} + 16 q^{91} + 16 q^{95} + 12 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 16 * q^13 - 4 * q^17 + 16 * q^23 + 4 * q^29 + 16 * q^31 - 16 * q^35 + 28 * q^37 - 12 * q^41 + 32 * q^47 + 32 * q^55 - 4 * q^61 + 16 * q^71 + 4 * q^73 + 20 * q^85 + 16 * q^91 + 16 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	0.414214	−	0.414214i	0.185242	−	0.185242i	−0.608394	−	0.793635i	$-0.708186\pi$
$5$	0.414214	−	0.414214i	0.185242	−	0.185242i	0.793635	+	0.608394i	$0.208186\pi$
$6$		0			0
$7$	−1.41421	−	1.41421i	−0.534522	−	0.534522i	0.387392	−	0.921915i	$-0.373376\pi$
$7$	−1.41421	−	1.41421i	−0.534522	−	0.534522i	−0.921915	+	0.387392i	$0.873376\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.82843	+	2.82843i	0.852803	+	0.852803i	0.990478	−	0.137675i	$-0.0439628\pi$
$11$	2.82843	+	2.82843i	0.852803	+	0.852803i	−0.137675	+	0.990478i	$0.543963\pi$
$12$		0			0
$13$	−6.82843			−1.89386			−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843			−1.89386			−0.946932	+	0.321433i	$0.895836\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$18$		0			0
$19$			2.82843i			0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$			2.82843i			0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.41421	+	5.41421i	1.12894	+	1.12894i	0.990350	+	0.138592i	$0.0442577\pi$
$23$	5.41421	+	5.41421i	1.12894	+	1.12894i	0.138592	+	0.990350i	$0.455742\pi$
$24$		0			0
$25$			4.65685i			0.931371i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.24264	−	5.24264i	0.973534	−	0.973534i	−0.0261248	−	0.999659i	$-0.508317\pi$
$29$	5.24264	−	5.24264i	0.973534	−	0.973534i	0.999659	+	0.0261248i	$0.00831671\pi$
$30$		0			0
$31$	5.41421	−	5.41421i	0.972421	−	0.972421i	−0.0272083	−	0.999630i	$-0.508662\pi$
$31$	5.41421	−	5.41421i	0.972421	−	0.972421i	0.999630	+	0.0272083i	$0.00866175\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.17157			−0.198032
$36$		0			0
$37$	8.41421	−	8.41421i	1.38329	−	1.38329i	0.544578	−	0.838710i	$-0.316690\pi$
$37$	8.41421	−	8.41421i	1.38329	−	1.38329i	0.838710	−	0.544578i	$-0.183310\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.65685	+	2.65685i	0.414931	+	0.414931i	0.883452	−	0.468521i	$-0.155213\pi$
$41$	2.65685	+	2.65685i	0.414931	+	0.414931i	−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	1.65685i		−	0.252668i	−0.991988	−	0.126334i	$-0.959679\pi$
$43$		−	1.65685i		−	0.252668i	0.991988	−	0.126334i	$-0.0403211\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.17157			0.754351			0.377176	−	0.926142i	$-0.376895\pi$
$47$	5.17157			0.754351			0.377176	+	0.926142i	$0.376895\pi$
$48$		0			0
$49$		−	3.00000i		−	0.428571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.17157i		−	0.160928i	−0.996758	−	0.0804640i	$-0.974360\pi$
$53$		−	1.17157i		−	0.160928i	0.996758	−	0.0804640i	$-0.0256402\pi$
$54$		0			0
$55$	2.34315			0.315950
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	2.82843i		−	0.368230i	−0.982905	−	0.184115i	$-0.941058\pi$
$59$		−	2.82843i		−	0.368230i	0.982905	−	0.184115i	$-0.0589419\pi$
$60$		0			0
$61$	−5.24264	−	5.24264i	−0.671251	−	0.671251i	0.286753	−	0.958005i	$-0.407424\pi$
$61$	−5.24264	−	5.24264i	−0.671251	−	0.671251i	−0.958005	+	0.286753i	$0.907424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.82843	+	2.82843i	−0.350823	+	0.350823i
$66$		0			0
$67$	−8.48528			−1.03664			−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528			−1.03664			−0.518321	+	0.855186i	$0.673443\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.24264	−	8.24264i	0.978221	−	0.978221i	−0.0215464	−	0.999768i	$-0.506859\pi$
$71$	8.24264	−	8.24264i	0.978221	−	0.978221i	0.999768	+	0.0215464i	$0.00685895\pi$
$72$		0			0
$73$	1.00000	−	1.00000i	0.117041	−	0.117041i	−0.646160	−	0.763202i	$-0.723626\pi$
$73$	1.00000	−	1.00000i	0.117041	−	0.117041i	0.763202	+	0.646160i	$0.223626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	8.00000i		−	0.911685i
$78$		0			0
$79$	4.24264	+	4.24264i	0.477334	+	0.477334i	0.904278	−	0.426944i	$-0.140410\pi$
$79$	4.24264	+	4.24264i	0.477334	+	0.477334i	−0.426944	+	0.904278i	$0.640410\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.17157i			0.567654i	0.958876	+	0.283827i	$0.0916041\pi$
$83$			5.17157i			0.567654i	−0.958876	+	0.283827i	$0.908396\pi$
$84$		0			0
$85$	−2.07107	−	1.24264i	−0.224639	−	0.134783i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.9706			−1.79888			−0.899438	−	0.437048i	$-0.856024\pi$
$89$	−16.9706			−1.79888			−0.899438	+	0.437048i	$0.856024\pi$
$90$		0			0
$91$	9.65685	+	9.65685i	1.01231	+	1.01231i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.17157	+	1.17157i	0.120201	+	0.120201i
$96$		0			0
$97$	11.4853	−	11.4853i	1.16615	−	1.16615i	0.183050	−	0.983104i	$-0.441403\pi$
$97$	11.4853	−	11.4853i	1.16615	−	1.16615i	0.983104	−	0.183050i	$-0.0585970\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.48528			0.446302			0.223151	−	0.974784i	$-0.428366\pi$
$101$	4.48528			0.446302			0.223151	+	0.974784i	$0.428366\pi$
$102$		0			0
$103$	5.17157			0.509570			0.254785	−	0.966998i	$-0.417995\pi$
$103$	5.17157			0.509570			0.254785	+	0.966998i	$0.417995\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−2.41421	−	2.41421i	−0.231240	−	0.231240i	0.581970	−	0.813210i	$-0.302282\pi$
$109$	−2.41421	−	2.41421i	−0.231240	−	0.231240i	−0.813210	+	0.581970i	$0.802282\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.171573	+	0.171573i	0.0161402	+	0.0161402i	0.715131	−	0.698991i	$-0.246367\pi$
$113$	0.171573	+	0.171573i	0.0161402	+	0.0161402i	−0.698991	+	0.715131i	$0.746367\pi$
$114$		0			0
$115$	4.48528			0.418255
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.24264	+	7.07107i	−0.388922	+	0.648204i
$120$		0			0
$121$			5.00000i			0.454545i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.00000	+	4.00000i	0.357771	+	0.357771i
$126$		0			0
$127$		−	10.3431i		−	0.917806i	−0.888486	−	0.458903i	$-0.848243\pi$
$127$		−	10.3431i		−	0.917806i	0.888486	−	0.458903i	$-0.151757\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000	−	12.0000i	1.04844	−	1.04844i	0.0496797	−	0.998765i	$-0.484180\pi$
$131$	12.0000	−	12.0000i	1.04844	−	1.04844i	0.998765	−	0.0496797i	$-0.0158200\pi$
$132$		0			0
$133$	4.00000	−	4.00000i	0.346844	−	0.346844i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.686292			0.0586338			0.0293169	−	0.999570i	$-0.490667\pi$
$137$	0.686292			0.0586338			0.0293169	+	0.999570i	$0.490667\pi$
$138$		0			0
$139$	1.17157	−	1.17157i	0.0993715	−	0.0993715i	−0.655673	−	0.755045i	$-0.727615\pi$
$139$	1.17157	−	1.17157i	0.0993715	−	0.0993715i	0.755045	+	0.655673i	$0.227615\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.3137	−	19.3137i	−1.61509	−	1.61509i
$144$		0			0
$145$		−	4.34315i		−	0.360679i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.3137			1.41839			0.709197	−	0.705010i	$-0.249058\pi$
$149$	17.3137			1.41839			0.709197	+	0.705010i	$0.249058\pi$
$150$		0			0
$151$		−	5.65685i		−	0.460348i	−0.973149	−	0.230174i	$-0.926070\pi$
$151$		−	5.65685i		−	0.460348i	0.973149	−	0.230174i	$-0.0739296\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4.48528i		−	0.360266i
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	15.3137i		−	1.20689i
$162$		0			0
$163$	−8.00000	−	8.00000i	−0.626608	−	0.626608i	0.320605	−	0.947213i	$-0.396114\pi$
$163$	−8.00000	−	8.00000i	−0.626608	−	0.626608i	−0.947213	+	0.320605i	$0.896114\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.07107	+	7.07107i	−0.547176	+	0.547176i	−0.925623	−	0.378447i	$-0.876458\pi$
$167$	−7.07107	+	7.07107i	−0.547176	+	0.547176i	0.378447	+	0.925623i	$0.376458\pi$
$168$		0			0
$169$	33.6274			2.58672
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.7279	+	11.7279i	−0.891657	+	0.891657i	−0.994679	−	0.103022i	$-0.967149\pi$
$173$	−11.7279	+	11.7279i	−0.891657	+	0.891657i	0.103022	+	0.994679i	$0.467149\pi$
$174$		0			0
$175$	6.58579	−	6.58579i	0.497839	−	0.497839i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			20.9706i			1.56741i	0.621131	+	0.783707i	$0.286673\pi$
$179$			20.9706i			1.56741i	−0.621131	+	0.783707i	$0.713327\pi$
$180$		0			0
$181$	10.0711	+	10.0711i	0.748577	+	0.748577i	0.974212	−	0.225635i	$-0.0724458\pi$
$181$	10.0711	+	10.0711i	0.748577	+	0.748577i	−0.225635	+	0.974212i	$0.572446\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.97056i		−	0.512486i
$186$		0			0
$187$	8.48528	−	14.1421i	0.620505	−	1.03418i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.14214			−0.444429			−0.222215	−	0.974998i	$-0.571329\pi$
$191$	−6.14214			−0.444429			−0.222215	+	0.974998i	$0.571329\pi$
$192$		0			0
$193$	11.1421	+	11.1421i	0.802028	+	0.802028i	0.983412	−	0.181384i	$-0.0580577\pi$
$193$	11.1421	+	11.1421i	0.802028	+	0.802028i	−0.181384	+	0.983412i	$0.558058\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.58579	−	5.58579i	−0.397971	−	0.397971i	0.479546	−	0.877517i	$-0.340801\pi$
$197$	−5.58579	−	5.58579i	−0.397971	−	0.397971i	−0.877517	+	0.479546i	$0.840801\pi$
$198$		0			0
$199$	2.58579	−	2.58579i	0.183302	−	0.183302i	−0.609491	−	0.792793i	$-0.708626\pi$
$199$	2.58579	−	2.58579i	0.183302	−	0.183302i	0.792793	+	0.609491i	$0.208626\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.8284			−1.04075
$204$		0			0
$205$	2.20101			0.153725
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000	+	8.00000i	−0.553372	+	0.553372i
$210$		0			0
$211$	18.1421	+	18.1421i	1.24896	+	1.24896i	0.956181	+	0.292775i	$0.0945786\pi$
$211$	18.1421	+	18.1421i	1.24896	+	1.24896i	0.292775	+	0.956181i	$0.405421\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.686292	−	0.686292i	−0.0468047	−	0.0468047i
$216$		0			0
$217$	−15.3137			−1.03956
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.82843	+	27.3137i	0.459330	+	1.83732i
$222$		0			0
$223$			21.1716i			1.41775i	0.705332	+	0.708877i	$0.250798\pi$
$223$			21.1716i			1.41775i	−0.705332	+	0.708877i	$0.749202\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.48528	+	4.48528i	0.297699	+	0.297699i	0.840112	−	0.542413i	$-0.182489\pi$
$227$	4.48528	+	4.48528i	0.297699	+	0.297699i	−0.542413	+	0.840112i	$0.682489\pi$
$228$		0			0
$229$			9.31371i			0.615467i	0.951473	+	0.307734i	$0.0995706\pi$
$229$			9.31371i			0.615467i	−0.951473	+	0.307734i	$0.900429\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.8284	−	13.8284i	0.905930	−	0.905930i	−0.0900104	−	0.995941i	$-0.528690\pi$
$233$	13.8284	−	13.8284i	0.905930	−	0.905930i	0.995941	+	0.0900104i	$0.0286900\pi$
$234$		0			0
$235$	2.14214	−	2.14214i	0.139738	−	0.139738i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	27.7990			1.79817			0.899084	−	0.437777i	$-0.144234\pi$
$239$	27.7990			1.79817			0.899084	+	0.437777i	$0.144234\pi$
$240$		0			0
$241$	3.82843	−	3.82843i	0.246611	−	0.246611i	−0.572967	−	0.819578i	$-0.694208\pi$
$241$	3.82843	−	3.82843i	0.246611	−	0.246611i	0.819578	+	0.572967i	$0.194208\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.24264	−	1.24264i	−0.0793894	−	0.0793894i
$246$		0			0
$247$		−	19.3137i		−	1.22890i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.65685			−0.609535			−0.304768	−	0.952427i	$-0.598579\pi$
$251$	−9.65685			−0.609535			−0.304768	+	0.952427i	$0.598579\pi$
$252$		0			0
$253$			30.6274i			1.92553i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			13.3137i			0.830486i	0.909710	+	0.415243i	$0.136304\pi$
$257$			13.3137i			0.830486i	−0.909710	+	0.415243i	$0.863696\pi$
$258$		0			0
$259$	−23.7990			−1.47880
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	18.8284i		−	1.16101i	−0.814256	−	0.580505i	$-0.802855\pi$
$263$		−	18.8284i		−	1.16101i	0.814256	−	0.580505i	$-0.197145\pi$
$264$		0			0
$265$	−0.485281	−	0.485281i	−0.0298106	−	0.0298106i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.07107	−	2.07107i	0.126275	−	0.126275i	−0.641145	−	0.767420i	$-0.721540\pi$
$269$	2.07107	−	2.07107i	0.126275	−	0.126275i	0.767420	+	0.641145i	$0.221540\pi$
$270$		0			0
$271$	16.9706			1.03089			0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706			1.03089			0.515444	+	0.856923i	$0.327627\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.1716	+	13.1716i	−0.794276	+	0.794276i
$276$		0			0
$277$	−3.92893	+	3.92893i	−0.236067	+	0.236067i	−0.815219	−	0.579153i	$-0.803384\pi$
$277$	−3.92893	+	3.92893i	−0.236067	+	0.236067i	0.579153	+	0.815219i	$0.303384\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2.34315i		−	0.139780i	−0.997555	−	0.0698902i	$-0.977735\pi$
$281$		−	2.34315i		−	0.139780i	0.997555	−	0.0698902i	$-0.0222649\pi$
$282$		0			0
$283$	−20.0000	−	20.0000i	−1.18888	−	1.18888i	−0.977378	−	0.211498i	$-0.932166\pi$
$283$	−20.0000	−	20.0000i	−1.18888	−	1.18888i	−0.211498	−	0.977378i	$-0.567834\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	7.51472i		−	0.443580i
$288$		0			0
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.00000			0.116841			0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000			0.116841			0.0584206	+	0.998292i	$0.481394\pi$
$294$		0			0
$295$	−1.17157	−	1.17157i	−0.0682116	−	0.0682116i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−36.9706	−	36.9706i	−2.13806	−	2.13806i
$300$		0			0
$301$	−2.34315	+	2.34315i	−0.135057	+	0.135057i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.34315			−0.248688
$306$		0			0
$307$	−32.2843			−1.84256			−0.921280	−	0.388899i	$-0.872855\pi$
$307$	−32.2843			−1.84256			−0.921280	+	0.388899i	$0.872855\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.24264	−	4.24264i	0.240578	−	0.240578i	−0.576511	−	0.817089i	$-0.695586\pi$
$311$	4.24264	−	4.24264i	0.240578	−	0.240578i	0.817089	+	0.576511i	$0.195586\pi$
$312$		0			0
$313$	−3.48528	−	3.48528i	−0.197000	−	0.197000i	0.601713	−	0.798713i	$-0.294485\pi$
$313$	−3.48528	−	3.48528i	−0.197000	−	0.197000i	−0.798713	+	0.601713i	$0.794485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.899495	−	0.899495i	−0.0505207	−	0.0505207i	0.681395	−	0.731916i	$-0.261374\pi$
$317$	−0.899495	−	0.899495i	−0.0505207	−	0.0505207i	−0.731916	+	0.681395i	$0.761374\pi$
$318$		0			0
$319$	29.6569			1.66047
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.3137	−	2.82843i	0.629512	−	0.157378i
$324$		0			0
$325$		−	31.7990i		−	1.76389i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.31371	−	7.31371i	−0.403218	−	0.403218i
$330$		0			0
$331$			23.3137i			1.28144i	0.767776	+	0.640719i	$0.221363\pi$
$331$			23.3137i			1.28144i	−0.767776	+	0.640719i	$0.778637\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.51472	+	3.51472i	−0.192030	+	0.192030i
$336$		0			0
$337$	9.97056	−	9.97056i	0.543131	−	0.543131i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	9.97056	−	9.97056i	0.543131	−	0.543131i	0.924445	+	0.381314i	$0.124528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.6274			1.65857
$342$		0			0
$343$	−14.1421	+	14.1421i	−0.763604	+	0.763604i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.34315	+	6.34315i	0.340518	+	0.340518i	0.856562	−	0.516044i	$-0.172596\pi$
$347$	6.34315	+	6.34315i	0.340518	+	0.340518i	−0.516044	+	0.856562i	$0.672596\pi$
$348$		0			0
$349$		−	17.1716i		−	0.919173i	−0.888133	−	0.459587i	$-0.847998\pi$
$349$		−	17.1716i		−	0.919173i	0.888133	−	0.459587i	$-0.152002\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.6274			1.09789			0.548943	−	0.835860i	$-0.315030\pi$
$353$	20.6274			1.09789			0.548943	+	0.835860i	$0.315030\pi$
$354$		0			0
$355$		−	6.82843i		−	0.362415i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	14.6274i		−	0.772006i	−0.922498	−	0.386003i	$-0.873856\pi$
$359$		−	14.6274i		−	0.772006i	0.922498	−	0.386003i	$-0.126144\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	0.828427i		−	0.0433619i
$366$		0			0
$367$	−20.2426	−	20.2426i	−1.05666	−	1.05666i	−0.998296	−	0.0583617i	$-0.981412\pi$
$367$	−20.2426	−	20.2426i	−1.05666	−	1.05666i	−0.0583617	−	0.998296i	$-0.518588\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.65685	+	1.65685i	−0.0860196	+	0.0860196i
$372$		0			0
$373$	−18.1421			−0.939364			−0.469682	−	0.882836i	$-0.655631\pi$
$373$	−18.1421			−0.939364			−0.469682	+	0.882836i	$0.655631\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−35.7990	+	35.7990i	−1.84374	+	1.84374i
$378$		0			0
$379$	−8.00000	+	8.00000i	−0.410932	+	0.410932i	−0.882063	−	0.471131i	$-0.843846\pi$
$379$	−8.00000	+	8.00000i	−0.410932	+	0.410932i	0.471131	+	0.882063i	$0.343846\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			14.1421i			0.722629i	0.932444	+	0.361315i	$0.117672\pi$
$383$			14.1421i			0.722629i	−0.932444	+	0.361315i	$0.882328\pi$
$384$		0			0
$385$	−3.31371	−	3.31371i	−0.168882	−	0.168882i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	2.00000i		−	0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$		−	2.00000i		−	0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$		0			0
$391$	16.2426	−	27.0711i	0.821426	−	1.36904i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.51472			0.176845
$396$		0			0
$397$	6.07107	+	6.07107i	0.304698	+	0.304698i	0.842849	−	0.538151i	$-0.180877\pi$
$397$	6.07107	+	6.07107i	0.304698	+	0.304698i	−0.538151	+	0.842849i	$0.680877\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.6569	+	22.6569i	1.13143	+	1.13143i	0.989940	+	0.141490i	$0.0451892\pi$
$401$	22.6569	+	22.6569i	1.13143	+	1.13143i	0.141490	+	0.989940i	$0.454811\pi$
$402$		0			0
$403$	−36.9706	+	36.9706i	−1.84163	+	1.84163i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	47.5980			2.35934
$408$		0			0
$409$	−17.3137			−0.856108			−0.428054	−	0.903753i	$-0.640801\pi$
$409$	−17.3137			−0.856108			−0.428054	+	0.903753i	$0.640801\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.00000	+	4.00000i	−0.196827	+	0.196827i
$414$		0			0
$415$	2.14214	+	2.14214i	0.105153	+	0.105153i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.82843	+	6.82843i	0.333590	+	0.333590i	0.853948	−	0.520358i	$-0.174201\pi$
$419$	6.82843	+	6.82843i	0.333590	+	0.333590i	−0.520358	+	0.853948i	$0.674201\pi$
$420$		0			0
$421$	−10.1421			−0.494297			−0.247149	−	0.968978i	$-0.579494\pi$
$421$	−10.1421			−0.494297			−0.247149	+	0.968978i	$0.579494\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	18.6274	−	4.65685i	0.903562	−	0.225891i
$426$		0			0
$427$			14.8284i			0.717598i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.89949	−	9.89949i	−0.476842	−	0.476842i	0.427278	−	0.904120i	$-0.359472\pi$
$431$	−9.89949	−	9.89949i	−0.476842	−	0.476842i	−0.904120	+	0.427278i	$0.859472\pi$
$432$		0			0
$433$		−	9.31371i		−	0.447588i	−0.974636	−	0.223794i	$-0.928156\pi$
$433$		−	9.31371i		−	0.447588i	0.974636	−	0.223794i	$-0.0718443\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−15.3137	+	15.3137i	−0.732554	+	0.732554i
$438$		0			0
$439$	−11.0711	+	11.0711i	−0.528393	+	0.528393i	−0.920093	−	0.391700i	$-0.871887\pi$
$439$	−11.0711	+	11.0711i	−0.528393	+	0.528393i	0.391700	+	0.920093i	$0.371887\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.65685			−0.0787195			−0.0393598	−	0.999225i	$-0.512532\pi$
$443$	−1.65685			−0.0787195			−0.0393598	+	0.999225i	$0.512532\pi$
$444$		0			0
$445$	−7.02944	+	7.02944i	−0.333227	+	0.333227i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.3137	+	10.3137i	0.486734	+	0.486734i	0.907274	−	0.420540i	$-0.138159\pi$
$449$	10.3137	+	10.3137i	0.486734	+	0.486734i	−0.420540	+	0.907274i	$0.638159\pi$
$450$		0			0
$451$			15.0294i			0.707709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.00000			0.375046
$456$		0			0
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	12.4853i		−	0.581498i	−0.956799	−	0.290749i	$-0.906096\pi$
$461$		−	12.4853i		−	0.581498i	0.956799	−	0.290749i	$-0.0939044\pi$
$462$		0			0
$463$	29.1716			1.35572			0.677859	−	0.735192i	$-0.262908\pi$
$463$	29.1716			1.35572			0.677859	+	0.735192i	$0.262908\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	4.00000i		−	0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$		−	4.00000i		−	0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$		0			0
$469$	12.0000	+	12.0000i	0.554109	+	0.554109i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.68629	−	4.68629i	0.215476	−	0.215476i
$474$		0			0
$475$	−13.1716			−0.604353
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−20.7279	+	20.7279i	−0.947083	+	0.947083i	−0.998669	−	0.0515856i	$-0.983572\pi$
$479$	−20.7279	+	20.7279i	−0.947083	+	0.947083i	0.0515856	+	0.998669i	$0.483572\pi$
$480$		0			0
$481$	−57.4558	+	57.4558i	−2.61976	+	2.61976i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	9.51472i		−	0.432041i
$486$		0			0
$487$	−14.3848	−	14.3848i	−0.651836	−	0.651836i	0.301599	−	0.953435i	$-0.402480\pi$
$487$	−14.3848	−	14.3848i	−0.651836	−	0.651836i	−0.953435	+	0.301599i	$0.902480\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			36.9706i			1.66846i	0.551418	+	0.834229i	$0.314087\pi$
$491$			36.9706i			1.66846i	−0.551418	+	0.834229i	$0.685913\pi$
$492$		0			0
$493$	−26.2132	−	15.7279i	−1.18058	−	0.708350i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−23.3137			−1.04576
$498$		0			0
$499$	8.00000	+	8.00000i	0.358129	+	0.358129i	0.863123	−	0.504994i	$-0.168505\pi$
$499$	8.00000	+	8.00000i	0.358129	+	0.358129i	−0.504994	+	0.863123i	$0.668505\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.07107	+	3.07107i	0.136932	+	0.136932i	0.772250	−	0.635318i	$-0.219131\pi$
$503$	3.07107	+	3.07107i	0.136932	+	0.136932i	−0.635318	+	0.772250i	$0.719131\pi$
$504$		0			0
$505$	1.85786	−	1.85786i	0.0826739	−	0.0826739i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−20.6274			−0.914294			−0.457147	−	0.889391i	$-0.651129\pi$
$509$	−20.6274			−0.914294			−0.457147	+	0.889391i	$0.651129\pi$
$510$		0			0
$511$	−2.82843			−0.125122
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.14214	−	2.14214i	0.0943938	−	0.0943938i
$516$		0			0
$517$	14.6274	+	14.6274i	0.643313	+	0.643313i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.97056	+	5.97056i	0.261575	+	0.261575i	0.825694	−	0.564119i	$-0.190784\pi$
$521$	5.97056	+	5.97056i	0.261575	+	0.261575i	−0.564119	+	0.825694i	$0.690784\pi$
$522$		0			0
$523$	−27.7990			−1.21556			−0.607782	−	0.794104i	$-0.707941\pi$
$523$	−27.7990			−1.21556			−0.607782	+	0.794104i	$0.707941\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27.0711	−	16.2426i	−1.17923	−	0.707541i
$528$		0			0
$529$			35.6274i			1.54902i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18.1421	−	18.1421i	−0.785823	−	0.785823i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.48528	−	8.48528i	0.365487	−	0.365487i
$540$		0			0
$541$	2.75736	−	2.75736i	0.118548	−	0.118548i	−0.645344	−	0.763892i	$-0.723286\pi$
$541$	2.75736	−	2.75736i	0.118548	−	0.118548i	0.763892	+	0.645344i	$0.223286\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	−13.1716	+	13.1716i	−0.563176	+	0.563176i	−0.930208	−	0.367032i	$-0.880374\pi$
$547$	−13.1716	+	13.1716i	−0.563176	+	0.563176i	0.367032	+	0.930208i	$0.380374\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.8284	+	14.8284i	0.631712	+	0.631712i
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5147			−0.826865			−0.413433	−	0.910535i	$-0.635670\pi$
$557$	−19.5147			−0.826865			−0.413433	+	0.910535i	$0.635670\pi$
$558$		0			0
$559$			11.3137i			0.478519i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			13.1716i			0.555116i	0.960709	+	0.277558i	$0.0895250\pi$
$563$			13.1716i			0.555116i	−0.960709	+	0.277558i	$0.910475\pi$
$564$		0			0
$565$	0.142136			0.00597969
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	23.3137i		−	0.977362i	−0.872463	−	0.488681i	$-0.837478\pi$
$569$		−	23.3137i		−	0.977362i	0.872463	−	0.488681i	$-0.162522\pi$
$570$		0			0
$571$	2.82843	+	2.82843i	0.118366	+	0.118366i	0.763809	−	0.645443i	$-0.223327\pi$
$571$	2.82843	+	2.82843i	0.118366	+	0.118366i	−0.645443	+	0.763809i	$0.723327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−25.2132	+	25.2132i	−1.05146	+	1.05146i
$576$		0			0
$577$	−35.3137			−1.47013			−0.735064	−	0.677997i	$-0.762848\pi$
$577$	−35.3137			−1.47013			−0.735064	+	0.677997i	$0.762848\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.31371	−	7.31371i	0.303424	−	0.303424i
$582$		0			0
$583$	3.31371	−	3.31371i	0.137240	−	0.137240i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			35.5980i			1.46929i	0.678454	+	0.734643i	$0.262650\pi$
$587$			35.5980i			1.46929i	−0.678454	+	0.734643i	$0.737350\pi$
$588$		0			0
$589$	15.3137	+	15.3137i	0.630990	+	0.630990i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	28.9706i		−	1.18968i	−0.803845	−	0.594839i	$-0.797216\pi$
$593$		−	28.9706i		−	1.18968i	0.803845	−	0.594839i	$-0.202784\pi$
$594$		0			0
$595$	1.17157	+	4.68629i	0.0480298	+	0.192119i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	46.6274			1.90514			0.952572	−	0.304312i	$-0.0984267\pi$
$599$	46.6274			1.90514			0.952572	+	0.304312i	$0.0984267\pi$
$600$		0			0
$601$	−9.34315	−	9.34315i	−0.381115	−	0.381115i	0.490389	−	0.871504i	$-0.336855\pi$
$601$	−9.34315	−	9.34315i	−0.381115	−	0.381115i	−0.871504	+	0.490389i	$0.836855\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.07107	+	2.07107i	0.0842009	+	0.0842009i
$606$		0			0
$607$	−2.10051	+	2.10051i	−0.0852569	+	0.0852569i	−0.748449	−	0.663192i	$-0.769201\pi$
$607$	−2.10051	+	2.10051i	−0.0852569	+	0.0852569i	0.663192	+	0.748449i	$0.269201\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−35.3137			−1.42864
$612$		0			0
$613$	17.3137			0.699294			0.349647	−	0.936881i	$-0.386302\pi$
$613$	17.3137			0.699294			0.349647	+	0.936881i	$0.386302\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.48528	−	7.48528i	0.301346	−	0.301346i	−0.540194	−	0.841540i	$-0.681649\pi$
$617$	7.48528	−	7.48528i	0.301346	−	0.301346i	0.841540	+	0.540194i	$0.181649\pi$
$618$		0			0
$619$	15.3137	+	15.3137i	0.615510	+	0.615510i	0.944376	−	0.328867i	$-0.106667\pi$
$619$	15.3137	+	15.3137i	0.615510	+	0.615510i	−0.328867	+	0.944376i	$0.606667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.0000	+	24.0000i	0.961540	+	0.961540i
$624$		0			0
$625$	−19.9706			−0.798823
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−42.0711	−	25.2426i	−1.67748	−	1.00649i
$630$		0			0
$631$		−	7.51472i		−	0.299156i	−0.988750	−	0.149578i	$-0.952208\pi$
$631$		−	7.51472i		−	0.299156i	0.988750	−	0.149578i	$-0.0477915\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.28427	−	4.28427i	−0.170016	−	0.170016i
$636$		0			0
$637$			20.4853i			0.811656i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.68629	+	9.68629i	−0.382586	+	0.382586i	−0.872033	−	0.489447i	$-0.837199\pi$
$641$	−9.68629	+	9.68629i	−0.382586	+	0.382586i	0.489447	+	0.872033i	$0.337199\pi$
$642$		0			0
$643$	16.4853	−	16.4853i	0.650116	−	0.650116i	−0.302905	−	0.953021i	$-0.597956\pi$
$643$	16.4853	−	16.4853i	0.650116	−	0.650116i	0.953021	+	0.302905i	$0.0979564\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14.1421			−0.555985			−0.277992	−	0.960583i	$-0.589669\pi$
$647$	−14.1421			−0.555985			−0.277992	+	0.960583i	$0.589669\pi$
$648$		0			0
$649$	8.00000	−	8.00000i	0.314027	−	0.314027i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.0416	−	15.0416i	−0.588624	−	0.588624i	0.348634	−	0.937259i	$-0.386646\pi$
$653$	−15.0416	−	15.0416i	−0.588624	−	0.588624i	−0.937259	+	0.348634i	$0.886646\pi$
$654$		0			0
$655$		−	9.94113i		−	0.388432i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.14214			0.239264			0.119632	−	0.992818i	$-0.461829\pi$
$659$	6.14214			0.239264			0.119632	+	0.992818i	$0.461829\pi$
$660$		0			0
$661$			20.4853i			0.796785i	0.917215	+	0.398393i	$0.130432\pi$
$661$			20.4853i			0.796785i	−0.917215	+	0.398393i	$0.869568\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	3.31371i		−	0.128500i
$666$		0			0
$667$	56.7696			2.19813
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	29.6569i		−	1.14489i
$672$		0			0
$673$	5.14214	+	5.14214i	0.198215	+	0.198215i	0.799234	−	0.601020i	$-0.205239\pi$
$673$	5.14214	+	5.14214i	0.198215	+	0.198215i	−0.601020	+	0.799234i	$0.705239\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.5563	+	14.5563i	−0.559446	+	0.559446i	−0.929150	−	0.369704i	$-0.879459\pi$
$677$	−14.5563	+	14.5563i	−0.559446	+	0.559446i	0.369704	+	0.929150i	$0.379459\pi$
$678$		0			0
$679$	−32.4853			−1.24667
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.00000	+	8.00000i	−0.306111	+	0.306111i	−0.843399	−	0.537288i	$-0.819449\pi$
$683$	−8.00000	+	8.00000i	−0.306111	+	0.306111i	0.537288	+	0.843399i	$0.319449\pi$
$684$		0			0
$685$	0.284271	−	0.284271i	0.0108614	−	0.0108614i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			8.00000i			0.304776i
$690$		0			0
$691$	−9.17157	−	9.17157i	−0.348903	−	0.348903i	0.510798	−	0.859701i	$-0.329350\pi$
$691$	−9.17157	−	9.17157i	−0.348903	−	0.348903i	−0.859701	+	0.510798i	$0.829350\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	0.970563i		−	0.0368155i
$696$		0			0
$697$	7.97056	−	13.2843i	0.301907	−	0.503178i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.17157			0.0442497			0.0221248	−	0.999755i	$-0.492957\pi$
$701$	1.17157			0.0442497			0.0221248	+	0.999755i	$0.492957\pi$
$702$		0			0
$703$	23.7990	+	23.7990i	0.897596	+	0.897596i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.34315	−	6.34315i	−0.238559	−	0.238559i
$708$		0			0
$709$	12.7574	−	12.7574i	0.479113	−	0.479113i	−0.425735	−	0.904848i	$-0.639984\pi$
$709$	12.7574	−	12.7574i	0.479113	−	0.479113i	0.904848	+	0.425735i	$0.139984\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	58.6274			2.19561
$714$		0			0
$715$	−16.0000			−0.598366
$716$		0			0
$717$		0			0
$718$		0			0
$719$	28.2426	−	28.2426i	1.05327	−	1.05327i	0.0547740	−	0.998499i	$-0.482556\pi$
$719$	28.2426	−	28.2426i	1.05327	−	1.05327i	0.998499	−	0.0547740i	$-0.0174438\pi$
$720$		0			0
$721$	−7.31371	−	7.31371i	−0.272377	−	0.272377i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.4142	+	24.4142i	0.906721	+	0.906721i
$726$		0			0
$727$	−12.2843			−0.455598			−0.227799	−	0.973708i	$-0.573153\pi$
$727$	−12.2843			−0.455598			−0.227799	+	0.973708i	$0.573153\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.62742	+	1.65685i	−0.245124	+	0.0612810i
$732$		0			0
$733$		−	27.1127i		−	1.00143i	−0.865612	−	0.500715i	$-0.833070\pi$
$733$		−	27.1127i		−	1.00143i	0.865612	−	0.500715i	$-0.166930\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.0000	−	24.0000i	−0.884051	−	0.884051i
$738$		0			0
$739$		−	19.7990i		−	0.728318i	−0.931337	−	0.364159i	$-0.881357\pi$
$739$		−	19.7990i		−	0.728318i	0.931337	−	0.364159i	$-0.118643\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.5858	+	22.5858i	−0.828592	+	0.828592i	−0.987322	−	0.158730i	$-0.949260\pi$
$743$	−22.5858	+	22.5858i	−0.828592	+	0.828592i	0.158730	+	0.987322i	$0.449260\pi$
$744$		0			0
$745$	7.17157	−	7.17157i	0.262746	−	0.262746i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−9.41421	+	9.41421i	−0.343530	+	0.343530i	−0.857693	−	0.514163i	$-0.828103\pi$
$751$	−9.41421	+	9.41421i	−0.343530	+	0.343530i	0.514163	+	0.857693i	$0.328103\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.34315	−	2.34315i	−0.0852758	−	0.0852758i
$756$		0			0
$757$		−	37.3137i		−	1.35619i	−0.734974	−	0.678095i	$-0.762806\pi$
$757$		−	37.3137i		−	1.35619i	0.734974	−	0.678095i	$-0.237194\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.6863			−0.604878			−0.302439	−	0.953169i	$-0.597801\pi$
$761$	−16.6863			−0.604878			−0.302439	+	0.953169i	$0.597801\pi$
$762$		0			0
$763$			6.82843i			0.247206i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			19.3137i			0.697378i
$768$		0			0
$769$	−15.0294			−0.541975			−0.270988	−	0.962583i	$-0.587350\pi$
$769$	−15.0294			−0.541975			−0.270988	+	0.962583i	$0.587350\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	38.0000i		−	1.36677i	−0.730061	−	0.683383i	$-0.760508\pi$
$773$		−	38.0000i		−	1.36677i	0.730061	−	0.683383i	$-0.239492\pi$
$774$		0			0
$775$	25.2132	+	25.2132i	0.905685	+	0.905685i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.51472	+	7.51472i	−0.269243	+	0.269243i
$780$		0			0
$781$	46.6274			1.66846
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.14214	+	4.14214i	−0.147839	+	0.147839i
$786$		0			0
$787$	9.85786	−	9.85786i	0.351395	−	0.351395i	−0.509233	−	0.860628i	$-0.670071\pi$
$787$	9.85786	−	9.85786i	0.351395	−	0.351395i	0.860628	+	0.509233i	$0.170071\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	0.485281i		−	0.0172546i
$792$		0			0
$793$	35.7990	+	35.7990i	1.27126	+	1.27126i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			4.48528i			0.158877i	0.996840	+	0.0794384i	$0.0253127\pi$
$797$			4.48528i			0.158877i	−0.996840	+	0.0794384i	$0.974687\pi$
$798$		0			0
$799$	−5.17157	−	20.6863i	−0.182957	−	0.731828i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5.65685			0.199626
$804$		0			0
$805$	−6.34315	−	6.34315i	−0.223567	−	0.223567i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.6274	−	39.6274i	−1.39323	−	1.39323i	−0.817982	−	0.575244i	$-0.804907\pi$
$809$	−39.6274	−	39.6274i	−1.39323	−	1.39323i	−0.575244	−	0.817982i	$-0.695093\pi$
$810$		0			0
$811$	23.7990	−	23.7990i	0.835696	−	0.835696i	−0.152593	−	0.988289i	$-0.548763\pi$
$811$	23.7990	−	23.7990i	0.835696	−	0.835696i	0.988289	+	0.152593i	$0.0487625\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.62742			−0.232148
$816$		0			0
$817$	4.68629			0.163953
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−35.2426	+	35.2426i	−1.22998	+	1.22998i	−0.266005	+	0.963972i	$0.585704\pi$
$821$	−35.2426	+	35.2426i	−1.22998	+	1.22998i	−0.963972	+	0.266005i	$0.914296\pi$
$822$		0			0
$823$	5.89949	+	5.89949i	0.205643	+	0.205643i	0.802413	−	0.596769i	$-0.203549\pi$
$823$	5.89949	+	5.89949i	0.205643	+	0.205643i	−0.596769	+	0.802413i	$0.703549\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.1716	−	21.1716i	−0.736208	−	0.736208i	0.235634	−	0.971842i	$-0.424283\pi$
$827$	−21.1716	−	21.1716i	−0.736208	−	0.736208i	−0.971842	+	0.235634i	$0.924283\pi$
$828$		0			0
$829$	−53.3137			−1.85166			−0.925831	−	0.377938i	$-0.876633\pi$
$829$	−53.3137			−1.85166			−0.925831	+	0.377938i	$0.876633\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.0000	+	3.00000i	−0.415775	+	0.103944i
$834$		0			0
$835$			5.85786i			0.202720i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.89949	−	1.89949i	−0.0655778	−	0.0655778i	0.673557	−	0.739135i	$-0.264765\pi$
$839$	−1.89949	−	1.89949i	−0.0655778	−	0.0655778i	−0.739135	+	0.673557i	$0.764765\pi$
$840$		0			0
$841$		−	25.9706i		−	0.895537i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.9289	−	13.9289i	0.479170	−	0.479170i
$846$		0			0
$847$	7.07107	−	7.07107i	0.242965	−	0.242965i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	91.1127			3.12330
$852$		0			0
$853$	28.8995	−	28.8995i	0.989500	−	0.989500i	−0.0104456	−	0.999945i	$-0.503325\pi$
$853$	28.8995	−	28.8995i	0.989500	−	0.989500i	0.999945	+	0.0104456i	$0.00332499\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.2843	+	41.2843i	1.41024	+	1.41024i	0.758066	+	0.652178i	$0.226144\pi$
$857$	41.2843	+	41.2843i	1.41024	+	1.41024i	0.652178	+	0.758066i	$0.273856\pi$
$858$		0			0
$859$			46.9117i			1.60061i	0.599596	+	0.800303i	$0.295328\pi$
$859$			46.9117i			1.60061i	−0.599596	+	0.800303i	$0.704672\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.3431			−0.624408			−0.312204	−	0.950015i	$-0.601067\pi$
$863$	−18.3431			−0.624408			−0.312204	+	0.950015i	$0.601067\pi$
$864$		0			0
$865$			9.71573i			0.330345i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			24.0000i			0.814144i
$870$		0			0
$871$	57.9411			1.96326
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	11.3137i		−	0.382473i
$876$		0			0
$877$	11.2426	+	11.2426i	0.379637	+	0.379637i	0.870971	−	0.491334i	$-0.163491\pi$
$877$	11.2426	+	11.2426i	0.379637	+	0.379637i	−0.491334	+	0.870971i	$0.663491\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22.7990	+	22.7990i	−0.768117	+	0.768117i	−0.977775	−	0.209657i	$-0.932765\pi$
$881$	−22.7990	+	22.7990i	−0.768117	+	0.768117i	0.209657	+	0.977775i	$0.432765\pi$
$882$		0			0
$883$	−51.7990			−1.74317			−0.871587	−	0.490240i	$-0.836909\pi$
$883$	−51.7990			−1.74317			−0.871587	+	0.490240i	$0.836909\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.5563	+	19.5563i	−0.656638	+	0.656638i	−0.954583	−	0.297945i	$-0.903699\pi$
$887$	−19.5563	+	19.5563i	−0.656638	+	0.656638i	0.297945	+	0.954583i	$0.403699\pi$
$888$		0			0
$889$	−14.6274	+	14.6274i	−0.490588	+	0.490588i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			14.6274i			0.489488i
$894$		0			0
$895$	8.68629	+	8.68629i	0.290351	+	0.290351i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	56.7696i		−	1.89337i
$900$		0			0
$901$	−4.68629	+	1.17157i	−0.156123	+	0.0390308i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.34315			0.277336
$906$		0			0
$907$	23.3137	+	23.3137i	0.774119	+	0.774119i	0.978824	−	0.204705i	$-0.0656234\pi$
$907$	23.3137	+	23.3137i	0.774119	+	0.774119i	−0.204705	+	0.978824i	$0.565623\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.75736	−	7.75736i	−0.257013	−	0.257013i	0.566825	−	0.823838i	$-0.308171\pi$
$911$	−7.75736	−	7.75736i	−0.257013	−	0.257013i	−0.823838	+	0.566825i	$0.808171\pi$
$912$		0			0
$913$	−14.6274	+	14.6274i	−0.484097	+	0.484097i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−33.9411			−1.12083
$918$		0			0
$919$	−42.9117			−1.41553			−0.707763	−	0.706450i	$-0.750296\pi$
$919$	−42.9117			−1.41553			−0.707763	+	0.706450i	$0.750296\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−56.2843	+	56.2843i	−1.85262	+	1.85262i
$924$		0			0
$925$	39.1838	+	39.1838i	1.28835	+	1.28835i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.9706	+	17.9706i	0.589595	+	0.589595i	0.937522	−	0.347927i	$-0.113114\pi$
$929$	17.9706	+	17.9706i	0.589595	+	0.589595i	−0.347927	+	0.937522i	$0.613114\pi$
$930$		0			0
$931$	8.48528			0.278094
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.34315	−	9.37258i	−0.0766291	−	0.306516i
$936$		0			0
$937$		−	39.3137i		−	1.28432i	−0.766569	−	0.642161i	$-0.778038\pi$
$937$		−	39.3137i		−	1.28432i	0.766569	−	0.642161i	$-0.221962\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−24.8995	−	24.8995i	−0.811700	−	0.811700i	0.173188	−	0.984889i	$-0.444593\pi$
$941$	−24.8995	−	24.8995i	−0.811700	−	0.811700i	−0.984889	+	0.173188i	$0.944593\pi$
$942$		0			0
$943$			28.7696i			0.936866i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.1421	−	22.1421i	0.719523	−	0.719523i	−0.248985	−	0.968507i	$-0.580097\pi$
$947$	22.1421	−	22.1421i	0.719523	−	0.719523i	0.968507	+	0.248985i	$0.0800969\pi$
$948$		0			0
$949$	−6.82843	+	6.82843i	−0.221660	+	0.221660i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	59.3137			1.92136			0.960680	−	0.277659i	$-0.0895585\pi$
$953$	59.3137			1.92136			0.960680	+	0.277659i	$0.0895585\pi$
$954$		0			0
$955$	−2.54416	+	2.54416i	−0.0823270	+	0.0823270i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.970563	−	0.970563i	−0.0313411	−	0.0313411i
$960$		0			0
$961$		−	27.6274i		−	0.891207i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.23045			0.297139
$966$		0			0
$967$		−	0.485281i		−	0.0156056i	−0.999970	−	0.00780280i	$-0.997516\pi$
$967$		−	0.485281i		−	0.0156056i	0.999970	−	0.00780280i	$-0.00248373\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	5.17157i		−	0.165964i	−0.996551	−	0.0829818i	$-0.973556\pi$
$971$		−	5.17157i		−	0.165964i	0.996551	−	0.0829818i	$-0.0264443\pi$
$972$		0			0
$973$	−3.31371			−0.106233
$974$		0			0
$975$		0			0
$976$		0			0
$977$			34.6274i			1.10783i	0.832573	+	0.553915i	$0.186867\pi$
$977$			34.6274i			1.10783i	−0.832573	+	0.553915i	$0.813133\pi$
$978$		0			0
$979$	−48.0000	−	48.0000i	−1.53409	−	1.53409i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.0711	−	11.0711i	0.353112	−	0.353112i	−0.508154	−	0.861266i	$-0.669672\pi$
$983$	11.0711	−	11.0711i	0.353112	−	0.353112i	0.861266	+	0.508154i	$0.169672\pi$
$984$		0			0
$985$	−4.62742			−0.147442
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.97056	−	8.97056i	0.285247	−	0.285247i
$990$		0			0
$991$	14.1005	−	14.1005i	0.447917	−	0.447917i	−0.446744	−	0.894662i	$-0.647417\pi$
$991$	14.1005	−	14.1005i	0.447917	−	0.447917i	0.894662	+	0.446744i	$0.147417\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	2.14214i		−	0.0679103i
$996$		0			0
$997$	−7.72792	−	7.72792i	−0.244746	−	0.244746i	0.574064	−	0.818810i	$-0.305366\pi$
$997$	−7.72792	−	7.72792i	−0.244746	−	0.244746i	−0.818810	+	0.574064i	$0.805366\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.be.p.1585.2		4
3.2	odd	2		816.2.bd.c.769.2		4
4.3	odd	2		1224.2.w.g.361.2		4
12.11	even	2		408.2.v.b.361.1	yes	4
17.13	even	4	inner	2448.2.be.p.1441.2		4
51.47	odd	4		816.2.bd.c.625.2		4
68.47	odd	4		1224.2.w.g.217.2		4
204.47	even	4		408.2.v.b.217.1	✓	4
204.59	even	8		6936.2.a.w.1.2		2
204.179	even	8		6936.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.v.b.217.1	✓	4	204.47	even	4
408.2.v.b.361.1	yes	4	12.11	even	2
816.2.bd.c.625.2		4	51.47	odd	4
816.2.bd.c.769.2		4	3.2	odd	2
1224.2.w.g.217.2		4	68.47	odd	4
1224.2.w.g.361.2		4	4.3	odd	2
2448.2.be.p.1441.2		4	17.13	even	4	inner
2448.2.be.p.1585.2		4	1.1	even	1	trivial
6936.2.a.v.1.1		2	204.179	even	8
6936.2.a.w.1.2		2	204.59	even	8

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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.be (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.414214 - 0.414214i) q^{5} +(-1.41421 - 1.41421i) q^{7} +O(q^{10})\)	\(q+(0.414214 - 0.414214i) q^{5} +(-1.41421 - 1.41421i) q^{7} +(2.82843 + 2.82843i) q^{11} -6.82843 q^{13} +(-1.00000 - 4.00000i) q^{17} +2.82843i q^{19} +(5.41421 + 5.41421i) q^{23} +4.65685i q^{25} +(5.24264 - 5.24264i) q^{29} +(5.41421 - 5.41421i) q^{31} -1.17157 q^{35} +(8.41421 - 8.41421i) q^{37} +(2.65685 + 2.65685i) q^{41} -1.65685i q^{43} +5.17157 q^{47} -3.00000i q^{49} -1.17157i q^{53} +2.34315 q^{55} -2.82843i q^{59} +(-5.24264 - 5.24264i) q^{61} +(-2.82843 + 2.82843i) q^{65} -8.48528 q^{67} +(8.24264 - 8.24264i) q^{71} +(1.00000 - 1.00000i) q^{73} -8.00000i q^{77} +(4.24264 + 4.24264i) q^{79} +5.17157i q^{83} +(-2.07107 - 1.24264i) q^{85} -16.9706 q^{89} +(9.65685 + 9.65685i) q^{91} +(1.17157 + 1.17157i) q^{95} +(11.4853 - 11.4853i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} - 16 q^{13} - 4 q^{17} + 16 q^{23} + 4 q^{29} + 16 q^{31} - 16 q^{35} + 28 q^{37} - 12 q^{41} + 32 q^{47} + 32 q^{55} - 4 q^{61} + 16 q^{71} + 4 q^{73} + 20 q^{85} + 16 q^{91} + 16 q^{95} + 12 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 16 * q^13 - 4 * q^17 + 16 * q^23 + 4 * q^29 + 16 * q^31 - 16 * q^35 + 28 * q^37 - 12 * q^41 + 32 * q^47 + 32 * q^55 - 4 * q^61 + 16 * q^71 + 4 * q^73 + 20 * q^85 + 16 * q^91 + 16 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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