LMFDB - Embedded newform 2304.2.k.j.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(577,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2304.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.k (of order $4$, degree $2$, 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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			577.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.577
Dual form			2304.2.k.j.1729.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.27792i q^{7} +O(q^{10})$	$q-5.27792i q^{7} +(4.46410 + 4.46410i) q^{13} +(2.44949 + 2.44949i) q^{19} +5.00000i q^{25} +4.52004 q^{31} +(8.46410 - 8.46410i) q^{37} +(7.34847 - 7.34847i) q^{43} -20.8564 q^{49} +(-3.53590 - 3.53590i) q^{61} +(-11.3137 - 11.3137i) q^{67} -13.8564i q^{73} +15.0759 q^{79} +(23.5612 - 23.5612i) q^{91} -13.8564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} + 40 q^{37} - 56 q^{49} - 56 q^{61}+O(q^{100}) $ 8 * q + 8 * q^13 + 40 * q^37 - 56 * q^49 - 56 * q^61

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$		−	5.27792i		−	1.99487i	−0.0716124	−	0.997433i	$-0.522814\pi$
$7$		−	5.27792i		−	1.99487i	0.0716124	−	0.997433i	$-0.477186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$11$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$12$		0			0
$13$	4.46410	+	4.46410i	1.23812	+	1.23812i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	4.46410	+	4.46410i	1.23812	+	1.23812i	0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.44949	+	2.44949i	0.561951	+	0.561951i	0.929861	−	0.367910i	$-0.119927\pi$
$19$	2.44949	+	2.44949i	0.561951	+	0.561951i	−0.367910	+	0.929861i	$0.619927\pi$
$20$		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
$27$		0			0
$28$		0			0
$29$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$29$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	4.52004			0.811824			0.405912	−	0.913912i	$-0.366954\pi$
$31$	4.52004			0.811824			0.405912	+	0.913912i	$0.366954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.46410	−	8.46410i	1.39149	−	1.39149i	0.569495	−	0.821995i	$-0.307139\pi$
$37$	8.46410	−	8.46410i	1.39149	−	1.39149i	0.821995	−	0.569495i	$-0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	7.34847	−	7.34847i	1.12063	−	1.12063i	0.128984	−	0.991647i	$-0.458828\pi$
$43$	7.34847	−	7.34847i	1.12063	−	1.12063i	0.991647	−	0.128984i	$-0.0411717\pi$
$44$		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
$49$	−20.8564			−2.97949
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−3.53590	−	3.53590i	−0.452725	−	0.452725i	0.443533	−	0.896258i	$-0.353725\pi$
$61$	−3.53590	−	3.53590i	−0.452725	−	0.452725i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−11.3137	−	11.3137i	−1.38219	−	1.38219i	−0.840721	−	0.541468i	$-0.817869\pi$
$67$	−11.3137	−	11.3137i	−1.38219	−	1.38219i	−0.541468	−	0.840721i	$-0.682131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	13.8564i		−	1.62177i	−0.585206	−	0.810885i	$-0.698986\pi$
$73$		−	13.8564i		−	1.62177i	0.585206	−	0.810885i	$-0.301014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	15.0759			1.69617			0.848084	−	0.529861i	$-0.177756\pi$
$79$	15.0759			1.69617			0.848084	+	0.529861i	$0.177756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	23.5612	−	23.5612i	2.46988	−	2.46988i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.8564			−1.40690			−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−13.8564			−1.40690			−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		−	16.5916i		−	1.63482i	−0.576055	−	0.817411i	$-0.695409\pi$
$103$		−	16.5916i		−	1.63482i	0.576055	−	0.817411i	$-0.304591\pi$
$104$		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
$109$	9.39230	+	9.39230i	0.899620	+	0.899620i	0.995402	−	0.0957826i	$-0.0305354\pi$
$109$	9.39230	+	9.39230i	0.899620	+	0.899620i	−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			11.0000i			1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−6.79367			−0.602840			−0.301420	−	0.953491i	$-0.597461\pi$
$127$	−6.79367			−0.602840			−0.301420	+	0.953491i	$0.597461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$	12.9282	−	12.9282i	1.12102	−	1.12102i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−11.3137	+	11.3137i	−0.959616	+	0.959616i	−0.999216	−	0.0395994i	$-0.987392\pi$
$139$	−11.3137	+	11.3137i	−0.959616	+	0.959616i	0.0395994	+	0.999216i	$0.487392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		−	14.3180i		−	1.16518i	−0.812765	−	0.582591i	$-0.802039\pi$
$151$		−	14.3180i		−	1.16518i	0.812765	−	0.582591i	$-0.197961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17.3923	+	17.3923i	1.38806	+	1.38806i	0.829396	+	0.558661i	$0.188685\pi$
$157$	17.3923	+	17.3923i	1.38806	+	1.38806i	0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	17.1464	+	17.1464i	1.34301	+	1.34301i	0.893045	+	0.449966i	$0.148564\pi$
$163$	17.1464	+	17.1464i	1.34301	+	1.34301i	0.449966	+	0.893045i	$0.351436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			26.8564i			2.06588i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$	26.3896			1.99487
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$	16.4641	−	16.4641i	1.22377	−	1.22377i	0.257485	−	0.966282i	$-0.417106\pi$
$181$	16.4641	−	16.4641i	1.22377	−	1.22377i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		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
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		−	17.3495i		−	1.22987i	−0.788576	−	0.614937i	$-0.789182\pi$
$199$		−	17.3495i		−	1.22987i	0.788576	−	0.614937i	$-0.210818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	11.3137	+	11.3137i	0.778868	+	0.778868i	0.979638	−	0.200770i	$-0.0643445\pi$
$211$	11.3137	+	11.3137i	0.778868	+	0.778868i	−0.200770	+	0.979638i	$0.564345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	23.8564i		−	1.61948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−27.1475			−1.81793			−0.908964	−	0.416874i	$-0.863126\pi$
$223$	−27.1475			−1.81793			−0.908964	+	0.416874i	$0.863126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	−21.3923	+	21.3923i	−1.41364	+	1.41364i	−0.686743	+	0.726900i	$0.740960\pi$
$229$	−21.3923	+	21.3923i	−1.41364	+	1.41364i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		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
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$			21.8695i			1.39153i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−44.6728	−	44.6728i	−2.77583	−	2.77583i
$260$		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
$267$		0			0
$268$		0			0
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$	7.55154			0.458723			0.229362	−	0.973341i	$-0.426336\pi$
$271$	7.55154			0.458723			0.229362	+	0.973341i	$0.426336\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.60770	+	2.60770i	−0.156681	+	0.156681i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−2.60770	+	2.60770i	−0.156681	+	0.156681i	0.624413	+	0.781094i	$0.285338\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	22.6274	−	22.6274i	1.34506	−	1.34506i	0.454120	−	0.890941i	$-0.349954\pi$
$283$	22.6274	−	22.6274i	1.34506	−	1.34506i	0.890941	−	0.454120i	$-0.150046\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−38.7846	−	38.7846i	−2.23551	−	2.23551i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	11.3137	+	11.3137i	0.645707	+	0.645707i	0.951953	−	0.306245i	$-0.0990727\pi$
$307$	11.3137	+	11.3137i	0.645707	+	0.645707i	−0.306245	+	0.951953i	$0.599073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		−	22.0000i		−	1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$		−	22.0000i		−	1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−22.3205	+	22.3205i	−1.23812	+	1.23812i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−22.6274	+	22.6274i	−1.24372	+	1.24372i	−0.285267	+	0.958448i	$0.592082\pi$
$331$	−22.6274	+	22.6274i	−1.24372	+	1.24372i	−0.958448	+	0.285267i	$0.907918\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.8564			0.754807			0.377403	−	0.926049i	$-0.376817\pi$
$337$	13.8564			0.754807			0.377403	+	0.926049i	$0.376817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			73.1330i			3.94881i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	−10.3205	−	10.3205i	−0.552444	−	0.552444i	0.374701	−	0.927146i	$-0.377745\pi$
$349$	−10.3205	−	10.3205i	−0.552444	−	0.552444i	−0.927146	+	0.374701i	$0.877745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		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
$361$		−	7.00000i		−	0.368421i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−24.1160			−1.25884			−0.629421	−	0.777064i	$-0.716708\pi$
$367$	−24.1160			−1.25884			−0.629421	+	0.777064i	$0.716708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	15.5359	−	15.5359i	0.804418	−	0.804418i	−0.179364	−	0.983783i	$-0.557404\pi$
$373$	15.5359	−	15.5359i	0.804418	−	0.804418i	0.983783	+	0.179364i	$0.0574041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−26.9444	+	26.9444i	−1.38404	+	1.38404i	−0.546734	+	0.837307i	$0.684129\pi$
$379$	−26.9444	+	26.9444i	−1.38404	+	1.38404i	−0.837307	+	0.546734i	$0.815871\pi$
$380$		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
$387$		0			0
$388$		0			0
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.60770	+	6.60770i	0.331631	+	0.331631i	0.853206	−	0.521575i	$-0.174655\pi$
$397$	6.60770	+	6.60770i	0.331631	+	0.331631i	−0.521575	+	0.853206i	$0.674655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	20.1779	+	20.1779i	1.00513	+	1.00513i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			13.8564i			0.685155i	0.939490	+	0.342578i	$0.111300\pi$
$409$			13.8564i			0.685155i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	6.32051	−	6.32051i	0.308043	−	0.308043i	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	6.32051	−	6.32051i	0.308043	−	0.308043i	0.844150	+	0.536107i	$0.180106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−18.6622	+	18.6622i	−0.903126	+	0.903126i
$428$		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
$433$	41.5692			1.99769			0.998845	−	0.0480569i	$-0.0153029\pi$
$433$	41.5692			1.99769			0.998845	+	0.0480569i	$0.0153029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	2.24642i		−	0.107216i	−0.998562	−	0.0536078i	$-0.982928\pi$
$439$		−	2.24642i		−	0.107216i	0.998562	−	0.0536078i	$-0.0170721\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			41.5692i			1.94453i	0.233890	+	0.972263i	$0.424854\pi$
$457$			41.5692i			1.94453i	−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$	−12.8023			−0.594971			−0.297486	−	0.954726i	$-0.596148\pi$
$463$	−12.8023			−0.594971			−0.297486	+	0.954726i	$0.596148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$	−59.7128	+	59.7128i	−2.75728	+	2.75728i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−12.2474	+	12.2474i	−0.561951	+	0.561951i
$476$		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
$481$	75.5692			3.44566
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	28.6632i		−	1.29885i	−0.760424	−	0.649427i	$-0.775009\pi$
$487$		−	28.6632i		−	1.29885i	0.760424	−	0.649427i	$-0.224991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	22.6274	+	22.6274i	1.01294	+	1.01294i	0.999915	+	0.0130272i	$0.00414679\pi$
$499$	22.6274	+	22.6274i	1.01294	+	1.01294i	0.0130272	+	0.999915i	$0.495853\pi$
$500$		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
$507$		0			0
$508$		0			0
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$	−73.1330			−3.23521
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−31.8434	+	31.8434i	−1.39241	+	1.39241i	−0.572528	+	0.819885i	$0.694037\pi$
$523$	−31.8434	+	31.8434i	−1.39241	+	1.39241i	−0.819885	+	0.572528i	$0.805963\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	19.5359	+	19.5359i	0.839914	+	0.839914i	0.988847	−	0.148933i	$-0.0475840\pi$
$541$	19.5359	+	19.5359i	0.839914	+	0.839914i	−0.148933	+	0.988847i	$0.547584\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17.1464	−	17.1464i	−0.733128	−	0.733128i	0.238110	−	0.971238i	$-0.423472\pi$
$547$	−17.1464	−	17.1464i	−0.733128	−	0.733128i	−0.971238	+	0.238110i	$0.923472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	79.5692i		−	3.38363i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$	65.6086			2.77495
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	11.3137	−	11.3137i	0.473464	−	0.473464i	−0.429570	−	0.903034i	$-0.641335\pi$
$571$	11.3137	−	11.3137i	0.473464	−	0.473464i	0.903034	+	0.429570i	$0.141335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	13.8564			0.576850			0.288425	−	0.957503i	$-0.406868\pi$
$577$	13.8564			0.576850			0.288425	+	0.957503i	$0.406868\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$	11.0718	+	11.0718i	0.456205	+	0.456205i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		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
$601$			41.5692i			1.69564i	0.530281	+	0.847822i	$0.322086\pi$
$601$			41.5692i			1.69564i	−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−45.9855			−1.86649			−0.933247	−	0.359235i	$-0.883038\pi$
$607$	−45.9855			−1.86649			−0.933247	+	0.359235i	$0.883038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−19.2487	+	19.2487i	−0.777448	+	0.777448i	−0.979396	−	0.201948i	$-0.935273\pi$
$613$	−19.2487	+	19.2487i	−0.777448	+	0.777448i	0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−22.6274	+	22.6274i	−0.909473	+	0.909473i	−0.996230	−	0.0867568i	$-0.972350\pi$
$619$	−22.6274	+	22.6274i	−0.909473	+	0.909473i	0.0867568	+	0.996230i	$0.472350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			48.2591i			1.92117i	0.277994	+	0.960583i	$0.410330\pi$
$631$			48.2591i			1.92117i	−0.277994	+	0.960583i	$0.589670\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−93.1051	−	93.1051i	−3.68896	−	3.68896i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	22.0454	+	22.0454i	0.869386	+	0.869386i	0.992404	−	0.123018i	$-0.0392574\pi$
$643$	22.0454	+	22.0454i	0.869386	+	0.869386i	−0.123018	+	0.992404i	$0.539257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	−1.67949	+	1.67949i	−0.0653247	+	0.0653247i	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−1.67949	+	1.67949i	−0.0653247	+	0.0653247i	0.673690	+	0.739014i	$0.264708\pi$
$662$		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
$673$	13.8564			0.534125			0.267063	−	0.963679i	$-0.413947\pi$
$673$	13.8564			0.534125			0.267063	+	0.963679i	$0.413947\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$			73.1330i			2.80659i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−36.7423	−	36.7423i	−1.39774	−	1.39774i	−0.806471	−	0.591274i	$-0.798625\pi$
$691$	−36.7423	−	36.7423i	−1.39774	−	1.39774i	−0.591274	−	0.806471i	$-0.701375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$	41.4655			1.56390
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	35.2487	−	35.2487i	1.32379	−	1.32379i	0.413114	−	0.910679i	$-0.364441\pi$
$709$	35.2487	−	35.2487i	1.32379	−	1.32379i	0.910679	−	0.413114i	$-0.135559\pi$
$710$		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
$721$	−87.5692			−3.26125
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			9.06729i			0.336287i	0.985763	+	0.168144i	$0.0537772\pi$
$727$			9.06729i			0.336287i	−0.985763	+	0.168144i	$0.946223\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	14.6077	+	14.6077i	0.539548	+	0.539548i	0.923396	−	0.383849i	$-0.125402\pi$
$733$	14.6077	+	14.6077i	0.539548	+	0.539548i	−0.383849	+	0.923396i	$0.625402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−11.3137	−	11.3137i	−0.416181	−	0.416181i	0.467704	−	0.883885i	$-0.345081\pi$
$739$	−11.3137	−	11.3137i	−0.416181	−	0.416181i	−0.883885	+	0.467704i	$0.845081\pi$
$740$		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
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−49.0170			−1.78866			−0.894328	−	0.447412i	$-0.852346\pi$
$751$	−49.0170			−1.78866			−0.894328	+	0.447412i	$0.852346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−11.2487	+	11.2487i	−0.408841	+	0.408841i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	−11.2487	+	11.2487i	−0.408841	+	0.408841i	0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	49.5718	−	49.5718i	1.79462	−	1.79462i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$			22.6002i			0.811824i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.44949	−	2.44949i	−0.0873149	−	0.0873149i	0.662100	−	0.749415i	$-0.269665\pi$
$787$	−2.44949	−	2.44949i	−0.0873149	−	0.0873149i	−0.749415	+	0.662100i	$0.769665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	31.5692i		−	1.12106i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−7.34847	+	7.34847i	−0.258040	+	0.258040i	−0.824256	−	0.566217i	$-0.808407\pi$
$811$	−7.34847	+	7.34847i	−0.258040	+	0.258040i	0.566217	+	0.824256i	$0.308407\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	36.0000			1.25948
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		−	19.6231i		−	0.684019i	−0.939696	−	0.342010i	$-0.888892\pi$
$823$		−	19.6231i		−	0.684019i	0.939696	−	0.342010i	$-0.111108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−5.67949	−	5.67949i	−0.197257	−	0.197257i	0.601566	−	0.798823i	$-0.294544\pi$
$829$	−5.67949	−	5.67949i	−0.197257	−	0.197257i	−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		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
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	58.0571			1.99487
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−32.4641	+	32.4641i	−1.11155	+	1.11155i	−0.118609	+	0.992941i	$0.537843\pi$
$853$	−32.4641	+	32.4641i	−1.11155	+	1.11155i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−12.2474	+	12.2474i	−0.417878	+	0.417878i	−0.884472	−	0.466594i	$-0.845481\pi$
$859$	−12.2474	+	12.2474i	−0.417878	+	0.417878i	0.466594	+	0.884472i	$0.345481\pi$
$860$		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
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	101.011i		−	3.42263i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.24871	+	7.24871i	0.244772	+	0.244772i	0.818821	−	0.574049i	$-0.194628\pi$
$877$	7.24871	+	7.24871i	0.244772	+	0.244772i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	41.6413	+	41.6413i	1.40134	+	1.40134i	0.795855	+	0.605487i	$0.207022\pi$
$883$	41.6413	+	41.6413i	1.40134	+	1.40134i	0.605487	+	0.795855i	$0.292978\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$			35.8564i			1.20259i
$890$		0			0
$891$		0			0
$892$		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
$907$	31.8434	−	31.8434i	1.05734	−	1.05734i	0.0590889	−	0.998253i	$-0.481180\pi$
$907$	31.8434	−	31.8434i	1.05734	−	1.05734i	0.998253	−	0.0590889i	$-0.0188195\pi$
$908$		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
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	58.8150i		−	1.94013i	−0.242852	−	0.970063i	$-0.578083\pi$
$919$		−	58.8150i		−	1.94013i	0.242852	−	0.970063i	$-0.421917\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	42.3205	+	42.3205i	1.39149	+	1.39149i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−51.0876	−	51.0876i	−1.67433	−	1.67433i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			26.0000i			0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$			26.0000i			0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	61.8564	−	61.8564i	2.00794	−	2.00794i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−10.5692			−0.340943
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	55.7835i		−	1.79388i	−0.442157	−	0.896938i	$-0.645787\pi$
$967$		−	55.7835i		−	1.79388i	0.442157	−	0.896938i	$-0.354213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$	59.7128	+	59.7128i	1.91430	+	1.91430i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		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
$991$	0.730668			0.0232104			0.0116052	−	0.999933i	$-0.496306\pi$
$991$	0.730668			0.0232104			0.0116052	+	0.999933i	$0.496306\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	36.1769	−	36.1769i	1.14573	−	1.14573i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	36.1769	−	36.1769i	1.14573	−	1.14573i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.k.j.577.1	yes	8
3.2	odd	2	CM	2304.2.k.j.577.1	yes	8
4.3	odd	2	inner	2304.2.k.j.577.4	yes	8
8.3	odd	2		2304.2.k.g.577.4	yes	8
8.5	even	2		2304.2.k.g.577.1	✓	8
12.11	even	2	inner	2304.2.k.j.577.4	yes	8
16.3	odd	4		2304.2.k.g.1729.1	yes	8
16.5	even	4	inner	2304.2.k.j.1729.4	yes	8
16.11	odd	4	inner	2304.2.k.j.1729.1	yes	8
16.13	even	4		2304.2.k.g.1729.4	yes	8
24.5	odd	2		2304.2.k.g.577.1	✓	8
24.11	even	2		2304.2.k.g.577.4	yes	8
32.5	even	8		9216.2.a.bg.1.4		4
32.11	odd	8		9216.2.a.bg.1.1		4
32.21	even	8		9216.2.a.bf.1.4		4
32.27	odd	8		9216.2.a.bf.1.1		4
48.5	odd	4	inner	2304.2.k.j.1729.4	yes	8
48.11	even	4	inner	2304.2.k.j.1729.1	yes	8
48.29	odd	4		2304.2.k.g.1729.4	yes	8
48.35	even	4		2304.2.k.g.1729.1	yes	8
96.5	odd	8		9216.2.a.bg.1.4		4
96.11	even	8		9216.2.a.bg.1.1		4
96.53	odd	8		9216.2.a.bf.1.4		4
96.59	even	8		9216.2.a.bf.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2304.2.k.g.577.1	✓	8	8.5	even	2
2304.2.k.g.577.1	✓	8	24.5	odd	2
2304.2.k.g.577.4	yes	8	8.3	odd	2
2304.2.k.g.577.4	yes	8	24.11	even	2
2304.2.k.g.1729.1	yes	8	16.3	odd	4
2304.2.k.g.1729.1	yes	8	48.35	even	4
2304.2.k.g.1729.4	yes	8	16.13	even	4
2304.2.k.g.1729.4	yes	8	48.29	odd	4
2304.2.k.j.577.1	yes	8	1.1	even	1	trivial
2304.2.k.j.577.1	yes	8	3.2	odd	2	CM
2304.2.k.j.577.4	yes	8	4.3	odd	2	inner
2304.2.k.j.577.4	yes	8	12.11	even	2	inner
2304.2.k.j.1729.1	yes	8	16.11	odd	4	inner
2304.2.k.j.1729.1	yes	8	48.11	even	4	inner
2304.2.k.j.1729.4	yes	8	16.5	even	4	inner
2304.2.k.j.1729.4	yes	8	48.5	odd	4	inner
9216.2.a.bf.1.1		4	32.27	odd	8
9216.2.a.bf.1.1		4	96.59	even	8
9216.2.a.bf.1.4		4	32.21	even	8
9216.2.a.bf.1.4		4	96.53	odd	8
9216.2.a.bg.1.1		4	32.11	odd	8
9216.2.a.bg.1.1		4	96.11	even	8
9216.2.a.bg.1.4		4	32.5	even	8
9216.2.a.bg.1.4		4	96.5	odd	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-5.27792i q^{7} +O(q^{10})\)	\(q-5.27792i q^{7} +(4.46410 + 4.46410i) q^{13} +(2.44949 + 2.44949i) q^{19} +5.00000i q^{25} +4.52004 q^{31} +(8.46410 - 8.46410i) q^{37} +(7.34847 - 7.34847i) q^{43} -20.8564 q^{49} +(-3.53590 - 3.53590i) q^{61} +(-11.3137 - 11.3137i) q^{67} -13.8564i q^{73} +15.0759 q^{79} +(23.5612 - 23.5612i) q^{91} -13.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} + 40 q^{37} - 56 q^{49} - 56 q^{61}+O(q^{100}) \) 8 * q + 8 * q^13 + 40 * q^37 - 56 * q^49 - 56 * q^61

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