LMFDB - Embedded newform 2448.2.be.p.1585.1

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			1585.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2448.1585
Dual form			2448.2.be.p.1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.41421 + 2.41421i) q^{5} +(1.41421 + 1.41421i) q^{7} +O(q^{10})$	$q+(-2.41421 + 2.41421i) q^{5} +(1.41421 + 1.41421i) q^{7} +(-2.82843 - 2.82843i) q^{11} -1.17157 q^{13} +(-1.00000 - 4.00000i) q^{17} -2.82843i q^{19} +(2.58579 + 2.58579i) q^{23} -6.65685i q^{25} +(-3.24264 + 3.24264i) q^{29} +(2.58579 - 2.58579i) q^{31} -6.82843 q^{35} +(5.58579 - 5.58579i) q^{37} +(-8.65685 - 8.65685i) q^{41} +9.65685i q^{43} +10.8284 q^{47} -3.00000i q^{49} -6.82843i q^{53} +13.6569 q^{55} +2.82843i q^{59} +(3.24264 + 3.24264i) q^{61} +(2.82843 - 2.82843i) q^{65} +8.48528 q^{67} +(-0.242641 + 0.242641i) q^{71} +(1.00000 - 1.00000i) q^{73} -8.00000i q^{77} +(-4.24264 - 4.24264i) q^{79} +10.8284i q^{83} +(12.0711 + 7.24264i) q^{85} +16.9706 q^{89} +(-1.65685 - 1.65685i) q^{91} +(6.82843 + 6.82843i) q^{95} +(-5.48528 + 5.48528i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.41421	+	2.41421i	−1.07967	+	1.07967i	−0.0831305	+	0.996539i	$0.526492\pi$
$5$	−2.41421	+	2.41421i	−1.07967	+	1.07967i	−0.996539	+	0.0831305i	$0.973508\pi$
$6$		0			0
$7$	1.41421	+	1.41421i	0.534522	+	0.534522i	0.921915	−	0.387392i	$-0.126624\pi$
$7$	1.41421	+	1.41421i	0.534522	+	0.534522i	−0.387392	+	0.921915i	$0.626624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	0.137675	−	0.990478i	$-0.456037\pi$
$11$	−2.82843	−	2.82843i	−0.852803	−	0.852803i	−0.990478	+	0.137675i	$0.956037\pi$
$12$		0			0
$13$	−1.17157			−0.324936			−0.162468	−	0.986714i	$-0.551945\pi$
$13$	−1.17157			−0.324936			−0.162468	+	0.986714i	$0.551945\pi$
$14$		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	$-0.894823\pi$
$19$		−	2.82843i		−	0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.58579	+	2.58579i	0.539174	+	0.539174i	0.923286	−	0.384113i	$-0.125493\pi$
$23$	2.58579	+	2.58579i	0.539174	+	0.539174i	−0.384113	+	0.923286i	$0.625493\pi$
$24$		0			0
$25$		−	6.65685i		−	1.33137i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.24264	+	3.24264i	−0.602143	+	0.602143i	−0.940881	−	0.338738i	$-0.890000\pi$
$29$	−3.24264	+	3.24264i	−0.602143	+	0.602143i	0.338738	+	0.940881i	$0.390000\pi$
$30$		0			0
$31$	2.58579	−	2.58579i	0.464421	−	0.464421i	−0.435680	−	0.900101i	$-0.643492\pi$
$31$	2.58579	−	2.58579i	0.464421	−	0.464421i	0.900101	+	0.435680i	$0.143492\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.82843			−1.15421
$36$		0			0
$37$	5.58579	−	5.58579i	0.918298	−	0.918298i	−0.0786080	−	0.996906i	$-0.525048\pi$
$37$	5.58579	−	5.58579i	0.918298	−	0.918298i	0.996906	+	0.0786080i	$0.0250475\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.65685	−	8.65685i	−1.35197	−	1.35197i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	−8.65685	−	8.65685i	−1.35197	−	1.35197i	−0.468521	−	0.883452i	$-0.655213\pi$
$42$		0			0
$43$			9.65685i			1.47266i	0.676625	+	0.736328i	$0.263442\pi$
$43$			9.65685i			1.47266i	−0.676625	+	0.736328i	$0.736558\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10.8284			1.57949			0.789744	−	0.613436i	$-0.210213\pi$
$47$	10.8284			1.57949			0.789744	+	0.613436i	$0.210213\pi$
$48$		0			0
$49$		−	3.00000i		−	0.428571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.82843i		−	0.937957i	−0.883210	−	0.468978i	$-0.844622\pi$
$53$		−	6.82843i		−	0.937957i	0.883210	−	0.468978i	$-0.155378\pi$
$54$		0			0
$55$	13.6569			1.84149
$56$		0			0
$57$		0			0
$58$		0			0
$59$			2.82843i			0.368230i	0.982905	+	0.184115i	$0.0589419\pi$
$59$			2.82843i			0.368230i	−0.982905	+	0.184115i	$0.941058\pi$
$60$		0			0
$61$	3.24264	+	3.24264i	0.415178	+	0.415178i	0.883538	−	0.468360i	$-0.155155\pi$
$61$	3.24264	+	3.24264i	0.415178	+	0.415178i	−0.468360	+	0.883538i	$0.655155\pi$
$62$		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	$-0.326557\pi$
$67$	8.48528			1.03664			0.518321	+	0.855186i	$0.326557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.242641	+	0.242641i	−0.0287962	+	0.0287962i	−0.721358	−	0.692562i	$-0.756482\pi$
$71$	−0.242641	+	0.242641i	−0.0287962	+	0.0287962i	0.692562	+	0.721358i	$0.256482\pi$
$72$		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	0.426944	−	0.904278i	$-0.359590\pi$
$79$	−4.24264	−	4.24264i	−0.477334	−	0.477334i	−0.904278	+	0.426944i	$0.859590\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.8284i			1.18857i	0.804253	+	0.594287i	$0.202566\pi$
$83$			10.8284i			1.18857i	−0.804253	+	0.594287i	$0.797434\pi$
$84$		0			0
$85$	12.0711	+	7.24264i	1.30929	+	0.785575i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.9706			1.79888			0.899438	−	0.437048i	$-0.143976\pi$
$89$	16.9706			1.79888			0.899438	+	0.437048i	$0.143976\pi$
$90$		0			0
$91$	−1.65685	−	1.65685i	−0.173686	−	0.173686i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.82843	+	6.82843i	0.700582	+	0.700582i
$96$		0			0
$97$	−5.48528	+	5.48528i	−0.556946	+	0.556946i	−0.928437	−	0.371491i	$-0.878847\pi$
$97$	−5.48528	+	5.48528i	−0.556946	+	0.556946i	0.371491	+	0.928437i	$0.378847\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.4853			−1.24233			−0.621166	−	0.783679i	$-0.713341\pi$
$101$	−12.4853			−1.24233			−0.621166	+	0.783679i	$0.713341\pi$
$102$		0			0
$103$	10.8284			1.06696			0.533478	−	0.845814i	$-0.320885\pi$
$103$	10.8284			1.06696			0.533478	+	0.845814i	$0.320885\pi$
$104$		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
$109$	0.414214	+	0.414214i	0.0396745	+	0.0396745i	0.726666	−	0.686991i	$-0.241069\pi$
$109$	0.414214	+	0.414214i	0.0396745	+	0.0396745i	−0.686991	+	0.726666i	$0.741069\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.82843	+	5.82843i	0.548292	+	0.548292i	0.925947	−	0.377654i	$-0.123269\pi$
$113$	5.82843	+	5.82843i	0.548292	+	0.548292i	−0.377654	+	0.925947i	$0.623269\pi$
$114$		0			0
$115$	−12.4853			−1.16426
$116$		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
$127$		−	21.6569i		−	1.92174i	−0.277008	−	0.960868i	$-0.589343\pi$
$127$		−	21.6569i		−	1.92174i	0.277008	−	0.960868i	$-0.410657\pi$
$128$		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
$137$	23.3137			1.99182			0.995912	−	0.0903258i	$-0.0287908\pi$
$137$	23.3137			1.99182			0.995912	+	0.0903258i	$0.0287908\pi$
$138$		0			0
$139$	6.82843	−	6.82843i	0.579180	−	0.579180i	−0.355498	−	0.934677i	$-0.615689\pi$
$139$	6.82843	−	6.82843i	0.579180	−	0.579180i	0.934677	+	0.355498i	$0.115689\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.31371	+	3.31371i	0.277106	+	0.277106i
$144$		0			0
$145$		−	15.6569i		−	1.30023i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.31371			−0.435316			−0.217658	−	0.976025i	$-0.569842\pi$
$149$	−5.31371			−0.435316			−0.217658	+	0.976025i	$0.569842\pi$
$150$		0			0
$151$			5.65685i			0.460348i	0.973149	+	0.230174i	$0.0739296\pi$
$151$			5.65685i			0.460348i	−0.973149	+	0.230174i	$0.926070\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			12.4853i			1.00284i
$156$		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
$161$			7.31371i			0.576401i
$162$		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	−0.378447	−	0.925623i	$-0.623542\pi$
$167$	7.07107	−	7.07107i	0.547176	−	0.547176i	0.925623	+	0.378447i	$0.123542\pi$
$168$		0			0
$169$	−11.6274			−0.894417
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.7279	−	13.7279i	1.04371	−	1.04371i	0.0447148	−	0.999000i	$-0.485762\pi$
$173$	13.7279	−	13.7279i	1.04371	−	1.04371i	0.999000	−	0.0447148i	$-0.0142379\pi$
$174$		0			0
$175$	9.41421	−	9.41421i	0.711648	−	0.711648i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.9706i		−	0.969465i	−0.874662	−	0.484733i	$-0.838917\pi$
$179$		−	12.9706i		−	0.969465i	0.874662	−	0.484733i	$-0.161083\pi$
$180$		0			0
$181$	−4.07107	−	4.07107i	−0.302600	−	0.302600i	0.539430	−	0.842030i	$-0.318640\pi$
$181$	−4.07107	−	4.07107i	−0.302600	−	0.302600i	−0.842030	+	0.539430i	$0.818640\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			26.9706i			1.98292i
$186$		0			0
$187$	−8.48528	+	14.1421i	−0.620505	+	1.03418i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.1421			1.60215			0.801074	−	0.598565i	$-0.204262\pi$
$191$	22.1421			1.60215			0.801074	+	0.598565i	$0.204262\pi$
$192$		0			0
$193$	−17.1421	−	17.1421i	−1.23392	−	1.23392i	−0.962447	−	0.271471i	$-0.912490\pi$
$193$	−17.1421	−	17.1421i	−1.23392	−	1.23392i	−0.271471	−	0.962447i	$-0.587510\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.41421	−	8.41421i	−0.599488	−	0.599488i	0.340688	−	0.940176i	$-0.389340\pi$
$197$	−8.41421	−	8.41421i	−0.599488	−	0.599488i	−0.940176	+	0.340688i	$0.889340\pi$
$198$		0			0
$199$	5.41421	−	5.41421i	0.383803	−	0.383803i	−0.488667	−	0.872470i	$-0.662517\pi$
$199$	5.41421	−	5.41421i	0.383803	−	0.383803i	0.872470	+	0.488667i	$0.162517\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.17157			−0.643718
$204$		0			0
$205$	41.7990			2.91937
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000	+	8.00000i	−0.553372	+	0.553372i
$210$		0			0
$211$	−10.1421	−	10.1421i	−0.698213	−	0.698213i	0.265812	−	0.964025i	$-0.414360\pi$
$211$	−10.1421	−	10.1421i	−0.698213	−	0.698213i	−0.964025	+	0.265812i	$0.914360\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−23.3137	−	23.3137i	−1.58998	−	1.58998i
$216$		0			0
$217$	7.31371			0.496487
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.17157	+	4.68629i	0.0788085	+	0.315234i
$222$		0			0
$223$			26.8284i			1.79656i	0.439419	+	0.898282i	$0.355184\pi$
$223$			26.8284i			1.79656i	−0.439419	+	0.898282i	$0.644816\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.4853	−	12.4853i	−0.828677	−	0.828677i	0.158657	−	0.987334i	$-0.449284\pi$
$227$	−12.4853	−	12.4853i	−0.828677	−	0.828677i	−0.987334	+	0.158657i	$0.949284\pi$
$228$		0			0
$229$		−	13.3137i		−	0.879795i	−0.898048	−	0.439897i	$-0.855015\pi$
$229$		−	13.3137i		−	0.879795i	0.898048	−	0.439897i	$-0.144985\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.17157	−	8.17157i	0.535338	−	0.535338i	−0.386818	−	0.922156i	$-0.626426\pi$
$233$	8.17157	−	8.17157i	0.535338	−	0.535338i	0.922156	+	0.386818i	$0.126426\pi$
$234$		0			0
$235$	−26.1421	+	26.1421i	−1.70532	+	1.70532i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.7990			−0.763213			−0.381607	−	0.924325i	$-0.624629\pi$
$239$	−11.7990			−0.763213			−0.381607	+	0.924325i	$0.624629\pi$
$240$		0			0
$241$	−1.82843	+	1.82843i	−0.117779	+	0.117779i	−0.763540	−	0.645761i	$-0.776540\pi$
$241$	−1.82843	+	1.82843i	−0.117779	+	0.117779i	0.645761	+	0.763540i	$0.276540\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.24264	+	7.24264i	0.462715	+	0.462715i
$246$		0			0
$247$			3.31371i			0.210846i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.65685			0.104580			0.0522899	−	0.998632i	$-0.483348\pi$
$251$	1.65685			0.104580			0.0522899	+	0.998632i	$0.483348\pi$
$252$		0			0
$253$		−	14.6274i		−	0.919618i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	9.31371i		−	0.580973i	−0.956879	−	0.290487i	$-0.906183\pi$
$257$		−	9.31371i		−	0.580973i	0.956879	−	0.290487i	$-0.0938172\pi$
$258$		0			0
$259$	15.7990			0.981701
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.1716i		−	0.812194i	−0.913830	−	0.406097i	$-0.866890\pi$
$263$		−	13.1716i		−	0.812194i	0.913830	−	0.406097i	$-0.133110\pi$
$264$		0			0
$265$	16.4853	+	16.4853i	1.01268	+	1.01268i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.0711	+	12.0711i	−0.735986	+	0.735986i	−0.971799	−	0.235813i	$-0.924225\pi$
$269$	−12.0711	+	12.0711i	−0.735986	+	0.735986i	0.235813	+	0.971799i	$0.424225\pi$
$270$		0			0
$271$	−16.9706			−1.03089			−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706			−1.03089			−0.515444	+	0.856923i	$0.672373\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−18.8284	+	18.8284i	−1.13540	+	1.13540i
$276$		0			0
$277$	−18.0711	+	18.0711i	−1.08579	+	1.08579i	−0.0898279	+	0.995957i	$0.528632\pi$
$277$	−18.0711	+	18.0711i	−1.08579	+	1.08579i	−0.995957	+	0.0898279i	$0.971368\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	13.6569i		−	0.814700i	−0.913272	−	0.407350i	$-0.866453\pi$
$281$		−	13.6569i		−	0.814700i	0.913272	−	0.407350i	$-0.133547\pi$
$282$		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
$287$		−	24.4853i		−	1.44532i
$288$		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
$295$	−6.82843	−	6.82843i	−0.397566	−	0.397566i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.02944	−	3.02944i	−0.175197	−	0.175197i
$300$		0			0
$301$	−13.6569	+	13.6569i	−0.787168	+	0.787168i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−15.6569			−0.896509
$306$		0			0
$307$	24.2843			1.38598			0.692988	−	0.720949i	$-0.256294\pi$
$307$	24.2843			1.38598			0.692988	+	0.720949i	$0.256294\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.24264	+	4.24264i	−0.240578	+	0.240578i	−0.817089	−	0.576511i	$-0.804414\pi$
$311$	−4.24264	+	4.24264i	−0.240578	+	0.240578i	0.576511	+	0.817089i	$0.304414\pi$
$312$		0			0
$313$	13.4853	+	13.4853i	0.762233	+	0.762233i	0.976726	−	0.214492i	$-0.0688097\pi$
$313$	13.4853	+	13.4853i	0.762233	+	0.762233i	−0.214492	+	0.976726i	$0.568810\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.8995	+	18.8995i	1.06150	+	1.06150i	0.997981	+	0.0635209i	$0.0202329\pi$
$317$	18.8995	+	18.8995i	1.06150	+	1.06150i	0.0635209	+	0.997981i	$0.479767\pi$
$318$		0			0
$319$	18.3431			1.02702
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−11.3137	+	2.82843i	−0.629512	+	0.157378i
$324$		0			0
$325$			7.79899i			0.432610i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	15.3137	+	15.3137i	0.844272	+	0.844272i
$330$		0			0
$331$			0.686292i			0.0377220i	0.999822	+	0.0188610i	$0.00600400\pi$
$331$			0.686292i			0.0377220i	−0.999822	+	0.0188610i	$0.993996\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20.4853	+	20.4853i	−1.11923	+	1.11923i
$336$		0			0
$337$	−23.9706	+	23.9706i	−1.30576	+	1.30576i	−0.381314	+	0.924445i	$0.624528\pi$
$337$	−23.9706	+	23.9706i	−1.30576	+	1.30576i	−0.924445	+	0.381314i	$0.875472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.6274			−0.792119
$342$		0			0
$343$	14.1421	−	14.1421i	0.763604	−	0.763604i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.6569	+	17.6569i	0.947870	+	0.947870i	0.998707	−	0.0508373i	$-0.0161890\pi$
$347$	17.6569	+	17.6569i	0.947870	+	0.947870i	−0.0508373	+	0.998707i	$0.516189\pi$
$348$		0			0
$349$		−	22.8284i		−	1.22198i	−0.791639	−	0.610989i	$-0.790772\pi$
$349$		−	22.8284i		−	1.22198i	0.791639	−	0.610989i	$-0.209228\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.6274			−1.31079			−0.655393	−	0.755288i	$-0.727497\pi$
$353$	−24.6274			−1.31079			−0.655393	+	0.755288i	$0.727497\pi$
$354$		0			0
$355$		−	1.17157i		−	0.0621806i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			30.6274i			1.61645i	0.588872	+	0.808227i	$0.299572\pi$
$359$			30.6274i			1.61645i	−0.588872	+	0.808227i	$0.700428\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.82843i			0.252731i
$366$		0			0
$367$	−11.7574	−	11.7574i	−0.613729	−	0.613729i	0.330187	−	0.943916i	$-0.392888\pi$
$367$	−11.7574	−	11.7574i	−0.613729	−	0.613729i	−0.943916	+	0.330187i	$0.892888\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.65685	−	9.65685i	0.501359	−	0.501359i
$372$		0			0
$373$	10.1421			0.525140			0.262570	−	0.964913i	$-0.415430\pi$
$373$	10.1421			0.525140			0.262570	+	0.964913i	$0.415430\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.79899	−	3.79899i	0.195658	−	0.195658i
$378$		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	$-0.882328\pi$
$383$		−	14.1421i		−	0.722629i	0.932444	−	0.361315i	$-0.117672\pi$
$384$		0			0
$385$	19.3137	+	19.3137i	0.984318	+	0.984318i
$386$		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
$391$	7.75736	−	12.9289i	0.392307	−	0.653844i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.4853			1.03073
$396$		0			0
$397$	−8.07107	−	8.07107i	−0.405075	−	0.405075i	0.474942	−	0.880017i	$-0.342469\pi$
$397$	−8.07107	−	8.07107i	−0.405075	−	0.405075i	−0.880017	+	0.474942i	$0.842469\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.3431	+	11.3431i	0.566450	+	0.566450i	0.931132	−	0.364682i	$-0.118822\pi$
$401$	11.3431	+	11.3431i	0.566450	+	0.566450i	−0.364682	+	0.931132i	$0.618822\pi$
$402$		0			0
$403$	−3.02944	+	3.02944i	−0.150907	+	0.150907i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−31.5980			−1.56625
$408$		0			0
$409$	5.31371			0.262746			0.131373	−	0.991333i	$-0.458061\pi$
$409$	5.31371			0.262746			0.131373	+	0.991333i	$0.458061\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.00000	+	4.00000i	−0.196827	+	0.196827i
$414$		0			0
$415$	−26.1421	−	26.1421i	−1.28327	−	1.28327i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.17157	+	1.17157i	0.0572351	+	0.0572351i	0.735145	−	0.677910i	$-0.237114\pi$
$419$	1.17157	+	1.17157i	0.0572351	+	0.0572351i	−0.677910	+	0.735145i	$0.737114\pi$
$420$		0			0
$421$	18.1421			0.884194			0.442097	−	0.896967i	$-0.354235\pi$
$421$	18.1421			0.884194			0.442097	+	0.896967i	$0.354235\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−26.6274	+	6.65685i	−1.29162	+	0.322905i
$426$		0			0
$427$			9.17157i			0.443844i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.89949	+	9.89949i	0.476842	+	0.476842i	0.904120	−	0.427278i	$-0.140528\pi$
$431$	9.89949	+	9.89949i	0.476842	+	0.476842i	−0.427278	+	0.904120i	$0.640528\pi$
$432$		0			0
$433$			13.3137i			0.639816i	0.947449	+	0.319908i	$0.103652\pi$
$433$			13.3137i			0.639816i	−0.947449	+	0.319908i	$0.896348\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.31371	−	7.31371i	0.349862	−	0.349862i
$438$		0			0
$439$	3.07107	−	3.07107i	0.146574	−	0.146574i	−0.630012	−	0.776586i	$-0.716950\pi$
$439$	3.07107	−	3.07107i	0.146574	−	0.146574i	0.776586	+	0.630012i	$0.216950\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.65685			0.458811			0.229405	−	0.973331i	$-0.426322\pi$
$443$	9.65685			0.458811			0.229405	+	0.973331i	$0.426322\pi$
$444$		0			0
$445$	−40.9706	+	40.9706i	−1.94219	+	1.94219i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.3137	−	12.3137i	−0.581120	−	0.581120i	0.354091	−	0.935211i	$-0.384790\pi$
$449$	−12.3137	−	12.3137i	−0.581120	−	0.581120i	−0.935211	+	0.354091i	$0.884790\pi$
$450$		0			0
$451$			48.9706i			2.30593i
$452$		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
$461$			4.48528i			0.208900i	0.994530	+	0.104450i	$0.0333083\pi$
$461$			4.48528i			0.208900i	−0.994530	+	0.104450i	$0.966692\pi$
$462$		0			0
$463$	34.8284			1.61861			0.809307	−	0.587386i	$-0.199843\pi$
$463$	34.8284			1.61861			0.809307	+	0.587386i	$0.199843\pi$
$464$		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
$473$	27.3137	−	27.3137i	1.25589	−	1.25589i
$474$		0			0
$475$	−18.8284			−0.863907
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.72792	−	4.72792i	0.216024	−	0.216024i	−0.590796	−	0.806821i	$-0.701186\pi$
$479$	4.72792	−	4.72792i	0.216024	−	0.216024i	0.806821	+	0.590796i	$0.201186\pi$
$480$		0			0
$481$	−6.54416	+	6.54416i	−0.298388	+	0.298388i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	26.4853i		−	1.20263i
$486$		0			0
$487$	22.3848	+	22.3848i	1.01435	+	1.01435i	0.999896	+	0.0144555i	$0.00460150\pi$
$487$	22.3848	+	22.3848i	1.01435	+	1.01435i	0.0144555	+	0.999896i	$0.495399\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3.02944i			0.136717i	0.997661	+	0.0683583i	$0.0217761\pi$
$491$			3.02944i			0.136717i	−0.997661	+	0.0683583i	$0.978224\pi$
$492$		0			0
$493$	16.2132	+	9.72792i	0.730206	+	0.438124i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.686292			−0.0307844
$498$		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
$503$	−11.0711	−	11.0711i	−0.493635	−	0.493635i	0.415815	−	0.909449i	$-0.363496\pi$
$503$	−11.0711	−	11.0711i	−0.493635	−	0.493635i	−0.909449	+	0.415815i	$0.863496\pi$
$504$		0			0
$505$	30.1421	−	30.1421i	1.34131	−	1.34131i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.6274			1.09159			0.545796	−	0.837918i	$-0.316228\pi$
$509$	24.6274			1.09159			0.545796	+	0.837918i	$0.316228\pi$
$510$		0			0
$511$	2.82843			0.125122
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−26.1421	+	26.1421i	−1.15196	+	1.15196i
$516$		0			0
$517$	−30.6274	−	30.6274i	−1.34699	−	1.34699i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.9706	−	27.9706i	−1.22541	−	1.22541i	−0.965681	−	0.259732i	$-0.916366\pi$
$521$	−27.9706	−	27.9706i	−1.22541	−	1.22541i	−0.259732	−	0.965681i	$-0.583634\pi$
$522$		0			0
$523$	11.7990			0.515934			0.257967	−	0.966154i	$-0.416947\pi$
$523$	11.7990			0.515934			0.257967	+	0.966154i	$0.416947\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.9289	−	7.75736i	−0.563193	−	0.337916i
$528$		0			0
$529$		−	9.62742i		−	0.418583i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.1421	+	10.1421i	0.439305	+	0.439305i
$534$		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
$541$	11.2426	−	11.2426i	0.483359	−	0.483359i	−0.422844	−	0.906203i	$-0.638968\pi$
$541$	11.2426	−	11.2426i	0.483359	−	0.483359i	0.906203	+	0.422844i	$0.138968\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	−18.8284	+	18.8284i	−0.805045	+	0.805045i	−0.983879	−	0.178834i	$-0.942768\pi$
$547$	−18.8284	+	18.8284i	−0.805045	+	0.805045i	0.178834	+	0.983879i	$0.442768\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.17157	+	9.17157i	0.390722	+	0.390722i
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−36.4853			−1.54593			−0.772965	−	0.634448i	$-0.781227\pi$
$557$	−36.4853			−1.54593			−0.772965	+	0.634448i	$0.781227\pi$
$558$		0			0
$559$		−	11.3137i		−	0.478519i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.8284i			0.793524i	0.917922	+	0.396762i	$0.129866\pi$
$563$			18.8284i			0.793524i	−0.917922	+	0.396762i	$0.870134\pi$
$564$		0			0
$565$	−28.1421			−1.18395
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	0.686292i		−	0.0287708i	−0.999897	−	0.0143854i	$-0.995421\pi$
$569$		−	0.686292i		−	0.0287708i	0.999897	−	0.0143854i	$-0.00457918\pi$
$570$		0			0
$571$	−2.82843	−	2.82843i	−0.118366	−	0.118366i	0.645443	−	0.763809i	$-0.276673\pi$
$571$	−2.82843	−	2.82843i	−0.118366	−	0.118366i	−0.763809	+	0.645443i	$0.776673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	17.2132	−	17.2132i	0.717840	−	0.717840i
$576$		0			0
$577$	−12.6863			−0.528137			−0.264069	−	0.964504i	$-0.585065\pi$
$577$	−12.6863			−0.528137			−0.264069	+	0.964504i	$0.585065\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.3137	+	15.3137i	−0.635320	+	0.635320i
$582$		0			0
$583$	−19.3137	+	19.3137i	−0.799892	+	0.799892i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	43.5980i		−	1.79948i	−0.436425	−	0.899741i	$-0.643756\pi$
$587$		−	43.5980i		−	1.79948i	0.436425	−	0.899741i	$-0.356244\pi$
$588$		0			0
$589$	−7.31371	−	7.31371i	−0.301356	−	0.301356i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.97056i			0.204117i	0.994778	+	0.102058i	$0.0325428\pi$
$593$			4.97056i			0.204117i	−0.994778	+	0.102058i	$0.967457\pi$
$594$		0			0
$595$	6.82843	+	27.3137i	0.279938	+	1.11975i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.37258			0.0560822			0.0280411	−	0.999607i	$-0.491073\pi$
$599$	1.37258			0.0560822			0.0280411	+	0.999607i	$0.491073\pi$
$600$		0			0
$601$	−20.6569	−	20.6569i	−0.842611	−	0.842611i	0.146587	−	0.989198i	$-0.453171\pi$
$601$	−20.6569	−	20.6569i	−0.842611	−	0.842611i	−0.989198	+	0.146587i	$0.953171\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.0711	−	12.0711i	−0.490759	−	0.490759i
$606$		0			0
$607$	−21.8995	+	21.8995i	−0.888873	+	0.888873i	−0.994415	−	0.105542i	$-0.966342\pi$
$607$	−21.8995	+	21.8995i	−0.888873	+	0.888873i	0.105542	+	0.994415i	$0.466342\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.6863			−0.513232
$612$		0			0
$613$	−5.31371			−0.214619			−0.107309	−	0.994226i	$-0.534224\pi$
$613$	−5.31371			−0.214619			−0.107309	+	0.994226i	$0.534224\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.48528	+	9.48528i	−0.381863	+	0.381863i	−0.871773	−	0.489910i	$-0.837030\pi$
$617$	−9.48528	+	9.48528i	−0.381863	+	0.381863i	0.489910	+	0.871773i	$0.337030\pi$
$618$		0			0
$619$	−7.31371	−	7.31371i	−0.293963	−	0.293963i	0.544681	−	0.838643i	$-0.316651\pi$
$619$	−7.31371	−	7.31371i	−0.293963	−	0.293963i	−0.838643	+	0.544681i	$0.816651\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24.0000	+	24.0000i	0.961540	+	0.961540i
$624$		0			0
$625$	13.9706			0.558823
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.9289	−	16.7574i	−1.11360	−	0.668160i
$630$		0			0
$631$		−	24.4853i		−	0.974744i	−0.873195	−	0.487372i	$-0.837956\pi$
$631$		−	24.4853i		−	0.974744i	0.873195	−	0.487372i	$-0.162044\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	52.2843	+	52.2843i	2.07484	+	2.07484i
$636$		0			0
$637$			3.51472i			0.139258i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−32.3137	+	32.3137i	−1.27631	+	1.27631i	−0.333600	+	0.942715i	$0.608264\pi$
$641$	−32.3137	+	32.3137i	−1.27631	+	1.27631i	−0.942715	+	0.333600i	$0.891736\pi$
$642$		0			0
$643$	−0.485281	+	0.485281i	−0.0191376	+	0.0191376i	−0.716611	−	0.697473i	$-0.754308\pi$
$643$	−0.485281	+	0.485281i	−0.0191376	+	0.0191376i	0.697473	+	0.716611i	$0.254308\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.1421			0.555985			0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421			0.555985			0.277992	+	0.960583i	$0.410331\pi$
$648$		0			0
$649$	8.00000	−	8.00000i	0.314027	−	0.314027i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	33.0416	+	33.0416i	1.29302	+	1.29302i	0.932910	+	0.360108i	$0.117260\pi$
$653$	33.0416	+	33.0416i	1.29302	+	1.29302i	0.360108	+	0.932910i	$0.382740\pi$
$654$		0			0
$655$			57.9411i			2.26395i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−22.1421			−0.862535			−0.431268	−	0.902224i	$-0.641934\pi$
$659$	−22.1421			−0.862535			−0.431268	+	0.902224i	$0.641934\pi$
$660$		0			0
$661$			3.51472i			0.136707i	0.997661	+	0.0683534i	$0.0217745\pi$
$661$			3.51472i			0.136707i	−0.997661	+	0.0683534i	$0.978225\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			19.3137i			0.748953i
$666$		0			0
$667$	−16.7696			−0.649320
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	18.3431i		−	0.708129i
$672$		0			0
$673$	−23.1421	−	23.1421i	−0.892064	−	0.892064i	0.102653	−	0.994717i	$-0.467267\pi$
$673$	−23.1421	−	23.1421i	−0.892064	−	0.892064i	−0.994717	+	0.102653i	$0.967267\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.5563	−	16.5563i	0.636312	−	0.636312i	−0.313332	−	0.949644i	$-0.601445\pi$
$677$	16.5563	−	16.5563i	0.636312	−	0.636312i	0.949644	+	0.313332i	$0.101445\pi$
$678$		0			0
$679$	−15.5147			−0.595400
$680$		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
$685$	−56.2843	+	56.2843i	−2.15051	+	2.15051i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			8.00000i			0.304776i
$690$		0			0
$691$	−14.8284	−	14.8284i	−0.564100	−	0.564100i	0.366369	−	0.930469i	$-0.380601\pi$
$691$	−14.8284	−	14.8284i	−0.564100	−	0.564100i	−0.930469	+	0.366369i	$0.880601\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			32.9706i			1.25064i
$696$		0			0
$697$	−25.9706	+	43.2843i	−0.983705	+	1.63951i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.82843			0.257906			0.128953	−	0.991651i	$-0.458838\pi$
$701$	6.82843			0.257906			0.128953	+	0.991651i	$0.458838\pi$
$702$		0			0
$703$	−15.7990	−	15.7990i	−0.595870	−	0.595870i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.6569	−	17.6569i	−0.664054	−	0.664054i
$708$		0			0
$709$	21.2426	−	21.2426i	0.797784	−	0.797784i	−0.184962	−	0.982746i	$-0.559216\pi$
$709$	21.2426	−	21.2426i	0.797784	−	0.797784i	0.982746	+	0.184962i	$0.0592161\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.3726			0.500807
$714$		0			0
$715$	−16.0000			−0.598366
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.7574	−	19.7574i	0.736825	−	0.736825i	−0.235137	−	0.971962i	$-0.575554\pi$
$719$	19.7574	−	19.7574i	0.736825	−	0.736825i	0.971962	+	0.235137i	$0.0755539\pi$
$720$		0			0
$721$	15.3137	+	15.3137i	0.570312	+	0.570312i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	21.5858	+	21.5858i	0.801676	+	0.801676i
$726$		0			0
$727$	44.2843			1.64241			0.821206	−	0.570631i	$-0.193301\pi$
$727$	44.2843			1.64241			0.821206	+	0.570631i	$0.193301\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	38.6274	−	9.65685i	1.42869	−	0.357172i
$732$		0			0
$733$			35.1127i			1.29692i	0.761250	+	0.648459i	$0.224586\pi$
$733$			35.1127i			1.29692i	−0.761250	+	0.648459i	$0.775414\pi$
$734$		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	$0.118643\pi$
$739$			19.7990i			0.728318i	−0.931337	+	0.364159i	$0.881357\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−25.4142	+	25.4142i	−0.932357	+	0.932357i	−0.997853	−	0.0654958i	$-0.979137\pi$
$743$	−25.4142	+	25.4142i	−0.932357	+	0.932357i	0.0654958	+	0.997853i	$0.479137\pi$
$744$		0			0
$745$	12.8284	−	12.8284i	0.469997	−	0.469997i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.58579	+	6.58579i	−0.240319	+	0.240319i	−0.816982	−	0.576663i	$-0.804354\pi$
$751$	−6.58579	+	6.58579i	−0.240319	+	0.240319i	0.576663	+	0.816982i	$0.304354\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13.6569	−	13.6569i	−0.497024	−	0.497024i
$756$		0			0
$757$		−	14.6863i		−	0.533782i	−0.963727	−	0.266891i	$-0.914004\pi$
$757$		−	14.6863i		−	0.533782i	0.963727	−	0.266891i	$-0.0859964\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−39.3137			−1.42512			−0.712560	−	0.701611i	$-0.752465\pi$
$761$	−39.3137			−1.42512			−0.712560	+	0.701611i	$0.752465\pi$
$762$		0			0
$763$			1.17157i			0.0424138i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.31371i		−	0.119651i
$768$		0			0
$769$	−48.9706			−1.76592			−0.882962	−	0.469445i	$-0.844454\pi$
$769$	−48.9706			−1.76592			−0.882962	+	0.469445i	$0.844454\pi$
$770$		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
$775$	−17.2132	−	17.2132i	−0.618317	−	0.618317i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.4853	+	24.4853i	−0.877276	+	0.877276i
$780$		0			0
$781$	1.37258			0.0491149
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24.1421	−	24.1421i	0.861670	−	0.861670i
$786$		0			0
$787$	38.1421	−	38.1421i	1.35962	−	1.35962i	0.485238	−	0.874382i	$-0.338733\pi$
$787$	38.1421	−	38.1421i	1.35962	−	1.35962i	0.874382	−	0.485238i	$-0.161267\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			16.4853i			0.586149i
$792$		0			0
$793$	−3.79899	−	3.79899i	−0.134906	−	0.134906i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	12.4853i		−	0.442251i	−0.975245	−	0.221126i	$-0.929027\pi$
$797$		−	12.4853i		−	0.442251i	0.975245	−	0.221126i	$-0.0709731\pi$
$798$		0			0
$799$	−10.8284	−	43.3137i	−0.383082	−	1.53233i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.65685			−0.199626
$804$		0			0
$805$	−17.6569	−	17.6569i	−0.622322	−	0.622322i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.62742	+	5.62742i	0.197849	+	0.197849i	0.799077	−	0.601228i	$-0.205322\pi$
$809$	5.62742	+	5.62742i	0.197849	+	0.197849i	−0.601228	+	0.799077i	$0.705322\pi$
$810$		0			0
$811$	−15.7990	+	15.7990i	−0.554778	+	0.554778i	−0.927816	−	0.373038i	$-0.878316\pi$
$811$	−15.7990	+	15.7990i	−0.554778	+	0.554778i	0.373038	+	0.927816i	$0.378316\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	38.6274			1.35306
$816$		0			0
$817$	27.3137			0.955586
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.7574	+	26.7574i	−0.933838	+	0.933838i	−0.997943	−	0.0641049i	$-0.979581\pi$
$821$	−26.7574	+	26.7574i	−0.933838	+	0.933838i	0.0641049	+	0.997943i	$0.479581\pi$
$822$		0			0
$823$	−13.8995	−	13.8995i	−0.484506	−	0.484506i	0.422061	−	0.906567i	$-0.361307\pi$
$823$	−13.8995	−	13.8995i	−0.484506	−	0.484506i	−0.906567	+	0.422061i	$0.861307\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−26.8284	−	26.8284i	−0.932916	−	0.932916i	0.0649713	−	0.997887i	$-0.479304\pi$
$827$	−26.8284	−	26.8284i	−0.932916	−	0.932916i	−0.997887	+	0.0649713i	$0.979304\pi$
$828$		0			0
$829$	−30.6863			−1.06578			−0.532889	−	0.846185i	$-0.678894\pi$
$829$	−30.6863			−1.06578			−0.532889	+	0.846185i	$0.678894\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.0000	+	3.00000i	−0.415775	+	0.103944i
$834$		0			0
$835$			34.1421i			1.18154i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.8995	+	17.8995i	0.617959	+	0.617959i	0.945008	−	0.327048i	$-0.106054\pi$
$839$	17.8995	+	17.8995i	0.617959	+	0.617959i	−0.327048	+	0.945008i	$0.606054\pi$
$840$		0			0
$841$			7.97056i			0.274847i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.0711	−	28.0711i	0.965674	−	0.965674i
$846$		0			0
$847$	−7.07107	+	7.07107i	−0.242965	+	0.242965i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	28.8873			0.990244
$852$		0			0
$853$	9.10051	−	9.10051i	0.311595	−	0.311595i	−0.533932	−	0.845527i	$-0.679286\pi$
$853$	9.10051	−	9.10051i	0.311595	−	0.311595i	0.845527	+	0.533932i	$0.179286\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.2843	−	15.2843i	−0.522101	−	0.522101i	0.396105	−	0.918205i	$-0.370362\pi$
$857$	−15.2843	−	15.2843i	−0.522101	−	0.522101i	−0.918205	+	0.396105i	$0.870362\pi$
$858$		0			0
$859$		−	54.9117i		−	1.87356i	−0.349915	−	0.936781i	$-0.613790\pi$
$859$		−	54.9117i		−	1.87356i	0.349915	−	0.936781i	$-0.386210\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29.6569			−1.00953			−0.504766	−	0.863256i	$-0.668421\pi$
$863$	−29.6569			−1.00953			−0.504766	+	0.863256i	$0.668421\pi$
$864$		0			0
$865$			66.2843i			2.25373i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			24.0000i			0.814144i
$870$		0			0
$871$	−9.94113			−0.336842
$872$		0			0
$873$		0			0
$874$		0			0
$875$			11.3137i			0.382473i
$876$		0			0
$877$	2.75736	+	2.75736i	0.0931094	+	0.0931094i	0.752127	−	0.659018i	$-0.229028\pi$
$877$	2.75736	+	2.75736i	0.0931094	+	0.0931094i	−0.659018	+	0.752127i	$0.729028\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.7990	−	16.7990i	0.565972	−	0.565972i	−0.365025	−	0.930998i	$-0.618940\pi$
$881$	16.7990	−	16.7990i	0.565972	−	0.565972i	0.930998	+	0.365025i	$0.118940\pi$
$882$		0			0
$883$	−12.2010			−0.410597			−0.205298	−	0.978699i	$-0.565816\pi$
$883$	−12.2010			−0.410597			−0.205298	+	0.978699i	$0.565816\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.5563	−	11.5563i	0.388024	−	0.388024i	−0.485958	−	0.873982i	$-0.661529\pi$
$887$	11.5563	−	11.5563i	0.388024	−	0.388024i	0.873982	+	0.485958i	$0.161529\pi$
$888$		0			0
$889$	30.6274	−	30.6274i	1.02721	−	1.02721i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	30.6274i		−	1.02491i
$894$		0			0
$895$	31.3137	+	31.3137i	1.04670	+	1.04670i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			16.7696i			0.559296i
$900$		0			0
$901$	−27.3137	+	6.82843i	−0.909952	+	0.227488i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.6569			0.653416
$906$		0			0
$907$	0.686292	+	0.686292i	0.0227879	+	0.0227879i	0.718409	−	0.695621i	$-0.244871\pi$
$907$	0.686292	+	0.686292i	0.0227879	+	0.0227879i	−0.695621	+	0.718409i	$0.744871\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.2426	−	16.2426i	−0.538143	−	0.538143i	0.384840	−	0.922983i	$-0.374256\pi$
$911$	−16.2426	−	16.2426i	−0.538143	−	0.538143i	−0.922983	+	0.384840i	$0.874256\pi$
$912$		0			0
$913$	30.6274	−	30.6274i	1.01362	−	1.01362i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	33.9411			1.12083
$918$		0			0
$919$	58.9117			1.94332			0.971659	−	0.236388i	$-0.0759638\pi$
$919$	58.9117			1.94332			0.971659	+	0.236388i	$0.0759638\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.284271	−	0.284271i	0.00935690	−	0.00935690i
$924$		0			0
$925$	−37.1838	−	37.1838i	−1.22259	−	1.22259i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.9706	−	15.9706i	−0.523977	−	0.523977i	0.394793	−	0.918770i	$-0.370816\pi$
$929$	−15.9706	−	15.9706i	−0.523977	−	0.523977i	−0.918770	+	0.394793i	$0.870816\pi$
$930$		0			0
$931$	−8.48528			−0.278094
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−13.6569	−	54.6274i	−0.446627	−	1.78651i
$936$		0			0
$937$		−	16.6863i		−	0.545117i	−0.962139	−	0.272559i	$-0.912130\pi$
$937$		−	16.6863i		−	0.545117i	0.962139	−	0.272559i	$-0.0878699\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.10051	−	5.10051i	−0.166272	−	0.166272i	0.619067	−	0.785338i	$-0.287511\pi$
$941$	−5.10051	−	5.10051i	−0.166272	−	0.166272i	−0.785338	+	0.619067i	$0.787511\pi$
$942$		0			0
$943$		−	44.7696i		−	1.45790i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.14214	+	6.14214i	−0.199593	+	0.199593i	−0.799825	−	0.600233i	$-0.795075\pi$
$947$	−6.14214	+	6.14214i	−0.199593	+	0.199593i	0.600233	+	0.799825i	$0.295075\pi$
$948$		0			0
$949$	−1.17157	+	1.17157i	−0.0380309	+	0.0380309i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	36.6863			1.18839			0.594193	−	0.804323i	$-0.297472\pi$
$953$	36.6863			1.18839			0.594193	+	0.804323i	$0.297472\pi$
$954$		0			0
$955$	−53.4558	+	53.4558i	−1.72979	+	1.72979i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.9706	+	32.9706i	1.06467	+	1.06467i
$960$		0			0
$961$			17.6274i			0.568626i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	82.7696			2.66445
$966$		0			0
$967$			16.4853i			0.530131i	0.964230	+	0.265065i	$0.0853936\pi$
$967$			16.4853i			0.530131i	−0.964230	+	0.265065i	$0.914606\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	10.8284i		−	0.347501i	−0.984790	−	0.173750i	$-0.944411\pi$
$971$		−	10.8284i		−	0.347501i	0.984790	−	0.173750i	$-0.0555886\pi$
$972$		0			0
$973$	19.3137			0.619169
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	10.6274i		−	0.340001i	−0.985444	−	0.170001i	$-0.945623\pi$
$977$		−	10.6274i		−	0.340001i	0.985444	−	0.170001i	$-0.0543770\pi$
$978$		0			0
$979$	−48.0000	−	48.0000i	−1.53409	−	1.53409i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.07107	+	3.07107i	−0.0979519	+	0.0979519i	−0.754385	−	0.656433i	$-0.772065\pi$
$983$	−3.07107	+	3.07107i	−0.0979519	+	0.0979519i	0.656433	+	0.754385i	$0.272065\pi$
$984$		0			0
$985$	40.6274			1.29450
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.9706	+	24.9706i	−0.794018	+	0.794018i
$990$		0			0
$991$	33.8995	−	33.8995i	1.07685	−	1.07685i	0.0800632	−	0.996790i	$-0.474488\pi$
$991$	33.8995	−	33.8995i	1.07685	−	1.07685i	0.996790	−	0.0800632i	$-0.0255122\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			26.1421i			0.828761i
$996$		0			0
$997$	17.7279	+	17.7279i	0.561449	+	0.561449i	0.929719	−	0.368270i	$-0.120050\pi$
$997$	17.7279	+	17.7279i	0.561449	+	0.561449i	−0.368270	+	0.929719i	$0.620050\pi$
$998$		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.1585.1		4
3.2	odd	2		816.2.bd.c.769.1		4
4.3	odd	2		1224.2.w.g.361.1		4
12.11	even	2		408.2.v.b.361.2	yes	4
17.13	even	4	inner	2448.2.be.p.1441.1		4
51.47	odd	4		816.2.bd.c.625.1		4
68.47	odd	4		1224.2.w.g.217.1		4
204.47	even	4		408.2.v.b.217.2	✓	4
204.59	even	8		6936.2.a.v.1.2		2
204.179	even	8		6936.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.v.b.217.2	✓	4	204.47	even	4
408.2.v.b.361.2	yes	4	12.11	even	2
816.2.bd.c.625.1		4	51.47	odd	4
816.2.bd.c.769.1		4	3.2	odd	2
1224.2.w.g.217.1		4	68.47	odd	4
1224.2.w.g.361.1		4	4.3	odd	2
2448.2.be.p.1441.1		4	17.13	even	4	inner
2448.2.be.p.1585.1		4	1.1	even	1	trivial
6936.2.a.v.1.2		2	204.59	even	8
6936.2.a.w.1.1		2	204.179	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)

\(f(q)\)	\(=\)	\(q+(-2.41421 + 2.41421i) q^{5} +(1.41421 + 1.41421i) q^{7} +O(q^{10})\)	\(q+(-2.41421 + 2.41421i) q^{5} +(1.41421 + 1.41421i) q^{7} +(-2.82843 - 2.82843i) q^{11} -1.17157 q^{13} +(-1.00000 - 4.00000i) q^{17} -2.82843i q^{19} +(2.58579 + 2.58579i) q^{23} -6.65685i q^{25} +(-3.24264 + 3.24264i) q^{29} +(2.58579 - 2.58579i) q^{31} -6.82843 q^{35} +(5.58579 - 5.58579i) q^{37} +(-8.65685 - 8.65685i) q^{41} +9.65685i q^{43} +10.8284 q^{47} -3.00000i q^{49} -6.82843i q^{53} +13.6569 q^{55} +2.82843i q^{59} +(3.24264 + 3.24264i) q^{61} +(2.82843 - 2.82843i) q^{65} +8.48528 q^{67} +(-0.242641 + 0.242641i) q^{71} +(1.00000 - 1.00000i) q^{73} -8.00000i q^{77} +(-4.24264 - 4.24264i) q^{79} +10.8284i q^{83} +(12.0711 + 7.24264i) q^{85} +16.9706 q^{89} +(-1.65685 - 1.65685i) q^{91} +(6.82843 + 6.82843i) q^{95} +(-5.48528 + 5.48528i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 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

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