LMFDB - Embedded newform 1024.2.e.g.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(257,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("1024.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.e (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:	$8.17668116698$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			769.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1024.769
Dual form			1024.2.e.g.257.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.414214 - 0.414214i) q^{3} +2.65685i q^{9} +O(q^{10})$	$q+(0.414214 - 0.414214i) q^{3} +2.65685i q^{9} +(4.41421 + 4.41421i) q^{11} -5.65685 q^{17} +(-5.24264 + 5.24264i) q^{19} -5.00000i q^{25} +(2.34315 + 2.34315i) q^{27} +3.65685 q^{33} +6.00000i q^{41} +(9.24264 + 9.24264i) q^{43} +7.00000 q^{49} +(-2.34315 + 2.34315i) q^{51} +4.34315i q^{57} +(10.0711 + 10.0711i) q^{59} +(-2.75736 + 2.75736i) q^{67} -16.9706i q^{73} +(-2.07107 - 2.07107i) q^{75} -6.02944 q^{81} +(7.58579 - 7.58579i) q^{83} -5.65685i q^{89} -16.9706 q^{97} +(-11.7279 + 11.7279i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3}+O(q^{10}) $ 4 * q - 4 * q^3	$ 4 q - 4 q^{3} + 12 q^{11} - 4 q^{19} + 32 q^{27} - 8 q^{33} + 20 q^{43} + 28 q^{49} - 32 q^{51} + 12 q^{59} - 28 q^{67} + 20 q^{75} - 92 q^{81} + 36 q^{83} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 12 * q^11 - 4 * q^19 + 32 * q^27 - 8 * q^33 + 20 * q^43 + 28 * q^49 - 32 * q^51 + 12 * q^59 - 28 * q^67 + 20 * q^75 - 92 * q^81 + 36 * q^83 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	0.414214	−	0.414214i	0.239146	−	0.239146i	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	0.414214	−	0.414214i	0.239146	−	0.239146i	0.816497	+	0.577350i	$0.195913\pi$
$4$		0			0
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$			2.65685i			0.885618i
$10$		0			0
$11$	4.41421	+	4.41421i	1.33094	+	1.33094i	0.904534	+	0.426401i	$0.140219\pi$
$11$	4.41421	+	4.41421i	1.33094	+	1.33094i	0.426401	+	0.904534i	$0.359781\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.65685			−1.37199			−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685			−1.37199			−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	−5.24264	+	5.24264i	−1.20274	+	1.20274i	−0.229416	+	0.973329i	$0.573682\pi$
$19$	−5.24264	+	5.24264i	−1.20274	+	1.20274i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$	2.34315	+	2.34315i	0.450939	+	0.450939i
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	3.65685			0.636577
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$	9.24264	+	9.24264i	1.40949	+	1.40949i	0.762493	+	0.646997i	$0.223975\pi$
$43$	9.24264	+	9.24264i	1.40949	+	1.40949i	0.646997	+	0.762493i	$0.276025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−2.34315	+	2.34315i	−0.328106	+	0.328106i
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			4.34315i			0.575264i
$58$		0			0
$59$	10.0711	+	10.0711i	1.31114	+	1.31114i	0.920575	+	0.390567i	$0.127721\pi$
$59$	10.0711	+	10.0711i	1.31114	+	1.31114i	0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$61$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.75736	+	2.75736i	−0.336865	+	0.336865i	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−2.75736	+	2.75736i	−0.336865	+	0.336865i	0.518321	+	0.855186i	$0.326557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	16.9706i		−	1.98625i	−0.117041	−	0.993127i	$-0.537341\pi$
$73$		−	16.9706i		−	1.98625i	0.117041	−	0.993127i	$-0.462659\pi$
$74$		0			0
$75$	−2.07107	−	2.07107i	−0.239146	−	0.239146i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−6.02944			−0.669937
$82$		0			0
$83$	7.58579	−	7.58579i	0.832648	−	0.832648i	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	7.58579	−	7.58579i	0.832648	−	0.832648i	0.987878	+	0.155230i	$0.0496119\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.9706			−1.72310			−0.861550	−	0.507673i	$-0.830506\pi$
$97$	−16.9706			−1.72310			−0.861550	+	0.507673i	$0.830506\pi$
$98$		0			0
$99$	−11.7279	+	11.7279i	−1.17870	+	1.17870i
$100$		0			0
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.89949	−	6.89949i	−0.666999	−	0.666999i	0.290021	−	0.957020i	$-0.406338\pi$
$107$	−6.89949	−	6.89949i	−0.666999	−	0.666999i	−0.957020	+	0.290021i	$0.906338\pi$
$108$		0			0
$109$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			27.9706i			2.54278i
$122$		0			0
$123$	2.48528	+	2.48528i	0.224090	+	0.224090i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	7.65685			0.674148
$130$		0			0
$131$	1.92893	−	1.92893i	0.168532	−	0.168532i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$	1.92893	−	1.92893i	0.168532	−	0.168532i	0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	−6.75736	−	6.75736i	−0.573152	−	0.573152i	0.359856	−	0.933008i	$-0.382826\pi$
$139$	−6.75736	−	6.75736i	−0.573152	−	0.573152i	−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.89949	−	2.89949i	0.239146	−	0.239146i
$148$		0			0
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	15.0294i		−	1.21506i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.7279	−	11.7279i	0.918602	−	0.918602i	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	11.7279	−	11.7279i	0.918602	−	0.918602i	0.996928	+	0.0783260i	$0.0249575\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$	−13.9289	−	13.9289i	−1.06517	−	1.06517i
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	8.34315			0.627109
$178$		0			0
$179$	18.8995	−	18.8995i	1.41261	−	1.41261i	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.8995	−	18.8995i	1.41261	−	1.41261i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−24.9706	−	24.9706i	−1.82603	−	1.82603i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−16.9706			−1.22157			−0.610784	−	0.791797i	$-0.709146\pi$
$193$	−16.9706			−1.22157			−0.610784	+	0.791797i	$0.709146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$			2.28427i			0.161120i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−46.2843			−3.20155
$210$		0			0
$211$	−19.7279	+	19.7279i	−1.35813	+	1.35813i	−0.481900	+	0.876226i	$0.660053\pi$
$211$	−19.7279	+	19.7279i	−1.35813	+	1.35813i	−0.876226	+	0.481900i	$0.839947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.02944	−	7.02944i	−0.475005	−	0.475005i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	13.2843			0.885618
$226$		0			0
$227$	16.4142	−	16.4142i	1.08945	−	1.08945i	0.0938647	−	0.995585i	$-0.470078\pi$
$227$	16.4142	−	16.4142i	1.08945	−	1.08945i	0.995585	−	0.0938647i	$-0.0299221\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.65685i			0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$			5.65685i			0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	16.9706			1.09317			0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706			1.09317			0.546585	+	0.837404i	$0.315928\pi$
$242$		0			0
$243$	−9.52691	+	9.52691i	−0.611152	+	0.611152i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		−	6.28427i		−	0.398250i
$250$		0			0
$251$	−12.5563	−	12.5563i	−0.792550	−	0.792550i	0.189358	−	0.981908i	$-0.439359\pi$
$251$	−12.5563	−	12.5563i	−0.792550	−	0.792550i	−0.981908	+	0.189358i	$0.939359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	30.0000			1.87135			0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000			1.87135			0.935674	+	0.352865i	$0.114792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.34315	−	2.34315i	−0.143398	−	0.143398i
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	22.0711	−	22.0711i	1.33094	−	1.33094i
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	28.2843i		−	1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$		−	28.2843i		−	1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$	−23.7279	−	23.7279i	−1.41048	−	1.41048i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	−23.7279	−	23.7279i	−1.41048	−	1.41048i	−0.653882	−	0.756596i	$-0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$	−7.02944	+	7.02944i	−0.412073	+	0.412073i
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			20.6863i			1.20034i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−21.2426	+	21.2426i	−1.21238	+	1.21238i	−0.242140	+	0.970241i	$0.577849\pi$
$307$	−21.2426	+	21.2426i	−1.21238	+	1.21238i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−5.71573			−0.319021
$322$		0			0
$323$	29.6569	−	29.6569i	1.65015	−	1.65015i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.272078	−	0.272078i	−0.0149548	−	0.0149548i	0.699590	−	0.714545i	$-0.253366\pi$
$331$	−0.272078	−	0.272078i	−0.0149548	−	0.0149548i	−0.714545	+	0.699590i	$0.753366\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	7.45584	−	7.45584i	0.404946	−	0.404946i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.3848	+	21.3848i	1.14799	+	1.14799i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	21.3848	+	21.3848i	1.14799	+	1.14799i	0.161048	+	0.986947i	$0.448512\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		−	35.9706i		−	1.89319i
$362$		0			0
$363$	11.5858	+	11.5858i	0.608096	+	0.608096i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−15.9411			−0.829862
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	14.7574	+	14.7574i	0.758035	+	0.758035i	0.975964	−	0.217930i	$-0.0699304\pi$
$379$	14.7574	+	14.7574i	0.758035	+	0.758035i	−0.217930	+	0.975964i	$0.569930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−24.5563	+	24.5563i	−1.24827	+	1.24827i
$388$		0			0
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	1.59798i		−	0.0806074i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	39.5980			1.97743			0.988714	−	0.149813i	$-0.0478671\pi$
$401$	39.5980			1.97743			0.988714	+	0.149813i	$0.0478671\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	22.0000i		−	1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$		−	22.0000i		−	1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$		0			0
$411$	−2.48528	−	2.48528i	−0.122590	−	0.122590i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.59798			−0.274134
$418$		0			0
$419$	−9.38478	+	9.38478i	−0.458476	+	0.458476i	−0.898155	−	0.439679i	$-0.855092\pi$
$419$	−9.38478	+	9.38478i	−0.458476	+	0.458476i	0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			28.2843i			1.37199i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	16.9706			0.815553			0.407777	−	0.913082i	$-0.366304\pi$
$433$	16.9706			0.815553			0.407777	+	0.913082i	$0.366304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$			18.5980i			0.885618i
$442$		0			0
$443$	19.5858	+	19.5858i	0.930549	+	0.930549i	0.997740	−	0.0671913i	$-0.0214038\pi$
$443$	19.5858	+	19.5858i	0.930549	+	0.930549i	−0.0671913	+	0.997740i	$0.521404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.65685			0.266963			0.133482	−	0.991051i	$-0.457384\pi$
$449$	5.65685			0.266963			0.133482	+	0.991051i	$0.457384\pi$
$450$		0			0
$451$	−26.4853	+	26.4853i	−1.24714	+	1.24714i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	26.0000i		−	1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$		−	26.0000i		−	1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$		0			0
$459$	−13.2548	−	13.2548i	−0.618683	−	0.618683i
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.556349	+	0.556349i	−0.0257448	+	0.0257448i	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−0.556349	+	0.556349i	−0.0257448	+	0.0257448i	0.694117	+	0.719862i	$0.255795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			81.5980i			3.75188i
$474$		0			0
$475$	26.2132	+	26.2132i	1.20274	+	1.20274i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		−	9.71573i		−	0.439360i
$490$		0			0
$491$	13.9289	+	13.9289i	0.628604	+	0.628604i	0.947717	−	0.319113i	$-0.103385\pi$
$491$	13.9289	+	13.9289i	0.628604	+	0.628604i	−0.319113	+	0.947717i	$0.603385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.2132	−	14.2132i	0.636270	−	0.636270i	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	14.2132	−	14.2132i	0.636270	−	0.636270i	0.949633	+	0.313363i	$0.101456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	5.38478	+	5.38478i	0.239146	+	0.239146i
$508$		0			0
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−24.5685			−1.08473
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6.00000i		−	0.262865i	−0.991325	−	0.131432i	$-0.958042\pi$
$521$		−	6.00000i		−	0.262865i	0.991325	−	0.131432i	$-0.0419576\pi$
$522$		0			0
$523$	31.7279	+	31.7279i	1.38737	+	1.38737i	0.830812	+	0.556553i	$0.187876\pi$
$523$	31.7279	+	31.7279i	1.38737	+	1.38737i	0.556553	+	0.830812i	$0.312124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−26.7574	+	26.7574i	−1.16117	+	1.16117i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	15.6569i		−	0.675643i
$538$		0			0
$539$	30.8995	+	30.8995i	1.33094	+	1.33094i
$540$		0			0
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−18.7574	+	18.7574i	−0.802007	+	0.802007i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$	−18.7574	+	18.7574i	−0.802007	+	0.802007i	0.181402	+	0.983409i	$0.441936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−20.6863			−0.873376
$562$		0			0
$563$	33.3848	−	33.3848i	1.40700	−	1.40700i	0.632175	−	0.774826i	$-0.282163\pi$
$563$	33.3848	−	33.3848i	1.40700	−	1.40700i	0.774826	−	0.632175i	$-0.217837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	42.0000i		−	1.76073i	−0.474295	−	0.880366i	$-0.657297\pi$
$569$		−	42.0000i		−	1.76073i	0.474295	−	0.880366i	$-0.342703\pi$
$570$		0			0
$571$	10.2132	+	10.2132i	0.427409	+	0.427409i	0.887745	−	0.460336i	$-0.152271\pi$
$571$	10.2132	+	10.2132i	0.427409	+	0.427409i	−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$	−7.02944	+	7.02944i	−0.292133	+	0.292133i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.0416	+	27.0416i	1.11613	+	1.11613i	0.992304	+	0.123823i	$0.0395156\pi$
$587$	27.0416	+	27.0416i	1.11613	+	1.11613i	0.123823	+	0.992304i	$0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			16.9706i			0.692244i	0.938190	+	0.346122i	$0.112502\pi$
$601$			16.9706i			0.692244i	−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−7.32590	−	7.32590i	−0.298334	−	0.298334i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	39.5980i		−	1.59415i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$		−	39.5980i		−	1.59415i	0.603877	−	0.797077i	$-0.293622\pi$
$618$		0			0
$619$	−34.2132	−	34.2132i	−1.37514	−	1.37514i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	−34.2132	−	34.2132i	−1.37514	−	1.37514i	−0.522514	−	0.852631i	$-0.675006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$	−19.1716	+	19.1716i	−0.765639	+	0.765639i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$			16.3431i			0.649582i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.2843			1.11716			0.558581	−	0.829450i	$-0.311346\pi$
$641$	28.2843			1.11716			0.558581	+	0.829450i	$0.311346\pi$
$642$		0			0
$643$	29.2426	−	29.2426i	1.15322	−	1.15322i	0.167313	−	0.985904i	$-0.446491\pi$
$643$	29.2426	−	29.2426i	1.15322	−	1.15322i	0.985904	−	0.167313i	$-0.0535092\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$			88.9117i			3.49009i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	45.0883			1.75906
$658$		0			0
$659$	−15.0416	+	15.0416i	−0.585939	+	0.585939i	−0.936529	−	0.350590i	$-0.885981\pi$
$659$	−15.0416	+	15.0416i	−0.585939	+	0.585939i	0.350590	+	0.936529i	$0.385981\pi$
$660$		0			0
$661$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$661$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	50.9117			1.96250			0.981251	−	0.192736i	$-0.0617360\pi$
$673$	50.9117			1.96250			0.981251	+	0.192736i	$0.0617360\pi$
$674$		0			0
$675$	11.7157	−	11.7157i	0.450939	−	0.450939i
$676$		0			0
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	13.5980i		−	0.521076i
$682$		0			0
$683$	36.5563	+	36.5563i	1.39879	+	1.39879i	0.803543	+	0.595247i	$0.202946\pi$
$683$	36.5563	+	36.5563i	1.39879	+	1.39879i	0.595247	+	0.803543i	$0.297054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−35.7279	+	35.7279i	−1.35915	+	1.35915i	−0.484193	+	0.874961i	$0.660887\pi$
$691$	−35.7279	+	35.7279i	−1.35915	+	1.35915i	−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	33.9411i		−	1.28561i
$698$		0			0
$699$	2.34315	+	2.34315i	0.0886259	+	0.0886259i
$700$		0			0
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	7.02944	−	7.02944i	0.261428	−	0.261428i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	10.1960i		−	0.377628i
$730$		0			0
$731$	−52.2843	−	52.2843i	−1.93380	−	1.93380i
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.3431			−0.896691
$738$		0			0
$739$	−38.2132	+	38.2132i	−1.40570	+	1.40570i	−0.625355	+	0.780340i	$0.715046\pi$
$739$	−38.2132	+	38.2132i	−1.40570	+	1.40570i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	20.1543	+	20.1543i	0.737408	+	0.737408i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−10.4020			−0.379071
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	54.0000i		−	1.95750i	−0.205061	−	0.978749i	$-0.565739\pi$
$761$		−	54.0000i		−	1.95750i	0.205061	−	0.978749i	$-0.434261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	50.9117			1.83592			0.917961	−	0.396670i	$-0.129834\pi$
$769$	50.9117			1.83592			0.917961	+	0.396670i	$0.129834\pi$
$770$		0			0
$771$	12.4264	−	12.4264i	0.447526	−	0.447526i
$772$		0			0
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−31.4558	−	31.4558i	−1.12702	−	1.12702i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	12.2721	−	12.2721i	0.437452	−	0.437452i	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	12.2721	−	12.2721i	0.437452	−	0.437452i	0.891154	+	0.453701i	$0.149897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	15.0294			0.531039
$802$		0			0
$803$	74.9117	−	74.9117i	2.64358	−	2.64358i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$		0			0
$811$	−2.21320	−	2.21320i	−0.0777161	−	0.0777161i	0.667180	−	0.744896i	$-0.267501\pi$
$811$	−2.21320	−	2.21320i	−0.0777161	−	0.0777161i	−0.744896	+	0.667180i	$0.767501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−96.9117			−3.39051
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$821$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		−	18.2843i		−	0.636577i
$826$		0			0
$827$	−17.1005	−	17.1005i	−0.594643	−	0.594643i	0.344239	−	0.938882i	$-0.388137\pi$
$827$	−17.1005	−	17.1005i	−0.594643	−	0.594643i	−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−39.5980			−1.37199
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$			29.0000i			1.00000i
$842$		0			0
$843$	−11.7157	−	11.7157i	−0.403511	−	0.403511i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.6569			−0.674621
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			54.0000i			1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$			54.0000i			1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$		0			0
$859$	−33.2426	−	33.2426i	−1.13422	−	1.13422i	−0.989467	−	0.144757i	$-0.953760\pi$
$859$	−33.2426	−	33.2426i	−1.13422	−	1.13422i	−0.144757	−	0.989467i	$-0.546240\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	6.21320	−	6.21320i	0.211011	−	0.211011i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	45.0883i		−	1.52601i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	28.6985	−	28.6985i	0.965781	−	0.965781i	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	28.6985	−	28.6985i	0.965781	−	0.965781i	0.999434	+	0.0336527i	$0.0107140\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−26.6152	−	26.6152i	−0.891644	−	0.891644i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−24.6985	−	24.6985i	−0.820100	−	0.820100i	0.166022	−	0.986122i	$-0.446908\pi$
$907$	−24.6985	−	24.6985i	−0.820100	−	0.820100i	−0.986122	+	0.166022i	$0.946908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	66.9706			2.21640
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$			17.5980i			0.579873i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.2843			0.927977			0.463988	−	0.885841i	$-0.346418\pi$
$929$	28.2843			0.927977			0.463988	+	0.885841i	$0.346418\pi$
$930$		0			0
$931$	−36.6985	+	36.6985i	−1.20274	+	1.20274i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			50.9117i			1.66321i	0.555366	+	0.831606i	$0.312578\pi$
$937$			50.9117i			1.66321i	−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	4.14214	+	4.14214i	0.135173	+	0.135173i
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.8701	+	11.8701i	−0.385725	+	0.385725i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	−11.8701	+	11.8701i	−0.385725	+	0.385725i	0.487435	+	0.873160i	$0.337933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	42.0000i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$		−	42.0000i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	18.3310	−	18.3310i	0.590707	−	0.590707i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		−	24.5685i		−	0.789255i
$970$		0			0
$971$	−11.4437	−	11.4437i	−0.367244	−	0.367244i	0.499227	−	0.866471i	$-0.333617\pi$
$971$	−11.4437	−	11.4437i	−0.367244	−	0.367244i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	62.2254			1.99077			0.995383	−	0.0959785i	$-0.0305980\pi$
$977$	62.2254			1.99077			0.995383	+	0.0959785i	$0.0305980\pi$
$978$		0			0
$979$	24.9706	−	24.9706i	0.798063	−	0.798063i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	−0.225397			−0.00715275
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.e.g.769.2		4
4.3	odd	2		1024.2.e.o.769.1		4
8.3	odd	2	CM	1024.2.e.g.769.2		4
8.5	even	2		1024.2.e.o.769.1		4
16.3	odd	4		1024.2.e.o.257.1		4
16.5	even	4		1024.2.e.o.257.1		4
16.11	odd	4	inner	1024.2.e.g.257.2		4
16.13	even	4	inner	1024.2.e.g.257.2		4
32.3	odd	8		512.2.a.a.1.2	✓	2
32.5	even	8		512.2.b.c.257.3		4
32.11	odd	8		512.2.b.c.257.3		4
32.13	even	8		512.2.a.a.1.2	✓	2
32.19	odd	8		512.2.a.f.1.1	yes	2
32.21	even	8		512.2.b.c.257.2		4
32.27	odd	8		512.2.b.c.257.2		4
32.29	even	8		512.2.a.f.1.1	yes	2
96.5	odd	8		4608.2.d.k.2305.4		4
96.11	even	8		4608.2.d.k.2305.4		4
96.29	odd	8		4608.2.a.i.1.1		2
96.35	even	8		4608.2.a.k.1.2		2
96.53	odd	8		4608.2.d.k.2305.1		4
96.59	even	8		4608.2.d.k.2305.1		4
96.77	odd	8		4608.2.a.k.1.2		2
96.83	even	8		4608.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.a.1.2	✓	2	32.3	odd	8
512.2.a.a.1.2	✓	2	32.13	even	8
512.2.a.f.1.1	yes	2	32.19	odd	8
512.2.a.f.1.1	yes	2	32.29	even	8
512.2.b.c.257.2		4	32.21	even	8
512.2.b.c.257.2		4	32.27	odd	8
512.2.b.c.257.3		4	32.5	even	8
512.2.b.c.257.3		4	32.11	odd	8
1024.2.e.g.257.2		4	16.11	odd	4	inner
1024.2.e.g.257.2		4	16.13	even	4	inner
1024.2.e.g.769.2		4	1.1	even	1	trivial
1024.2.e.g.769.2		4	8.3	odd	2	CM
1024.2.e.o.257.1		4	16.3	odd	4
1024.2.e.o.257.1		4	16.5	even	4
1024.2.e.o.769.1		4	4.3	odd	2
1024.2.e.o.769.1		4	8.5	even	2
4608.2.a.i.1.1		2	96.29	odd	8
4608.2.a.i.1.1		2	96.83	even	8
4608.2.a.k.1.2		2	96.35	even	8
4608.2.a.k.1.2		2	96.77	odd	8
4608.2.d.k.2305.1		4	96.53	odd	8
4608.2.d.k.2305.1		4	96.59	even	8
4608.2.d.k.2305.4		4	96.5	odd	8
4608.2.d.k.2305.4		4	96.11	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.e (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.414214 - 0.414214i) q^{3} +2.65685i q^{9} +O(q^{10})\)	\(q+(0.414214 - 0.414214i) q^{3} +2.65685i q^{9} +(4.41421 + 4.41421i) q^{11} -5.65685 q^{17} +(-5.24264 + 5.24264i) q^{19} -5.00000i q^{25} +(2.34315 + 2.34315i) q^{27} +3.65685 q^{33} +6.00000i q^{41} +(9.24264 + 9.24264i) q^{43} +7.00000 q^{49} +(-2.34315 + 2.34315i) q^{51} +4.34315i q^{57} +(10.0711 + 10.0711i) q^{59} +(-2.75736 + 2.75736i) q^{67} -16.9706i q^{73} +(-2.07107 - 2.07107i) q^{75} -6.02944 q^{81} +(7.58579 - 7.58579i) q^{83} -5.65685i q^{89} -16.9706 q^{97} +(-11.7279 + 11.7279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3}+O(q^{10}) \) 4 * q - 4 * q^3	\( 4 q - 4 q^{3} + 12 q^{11} - 4 q^{19} + 32 q^{27} - 8 q^{33} + 20 q^{43} + 28 q^{49} - 32 q^{51} + 12 q^{59} - 28 q^{67} + 20 q^{75} - 92 q^{81} + 36 q^{83} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 12 * q^11 - 4 * q^19 + 32 * q^27 - 8 * q^33 + 20 * q^43 + 28 * q^49 - 32 * q^51 + 12 * q^59 - 28 * q^67 + 20 * q^75 - 92 * q^81 + 36 * q^83 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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