LMFDB - Embedded newform 1080.2.d.d.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(109,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("1080.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.d (of order $2$, degree $1$, 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:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			109.4
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1080.109
Dual form			1080.2.d.d.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(-0.792893 + 2.09077i) q^{5} +0.717439i q^{7} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +(-0.792893 + 2.09077i) q^{5} +0.717439i q^{7} +2.82843 q^{8} +(-1.12132 + 2.95680i) q^{10} +3.16693i q^{11} +1.01461i q^{14} +4.00000 q^{16} +(-1.58579 + 4.18154i) q^{20} +4.47871i q^{22} +(-3.74264 - 3.31552i) q^{25} +1.43488i q^{28} +10.3923i q^{29} -0.757359 q^{31} +5.65685 q^{32} +(-1.50000 - 0.568852i) q^{35} +(-2.24264 + 5.91359i) q^{40} +6.33386i q^{44} +6.48528 q^{49} +(-5.29289 - 4.68885i) q^{50} +10.0711 q^{53} +(-6.62132 - 2.51104i) q^{55} +2.02922i q^{56} +14.6969i q^{58} -10.3923i q^{59} -1.07107 q^{62} +8.00000 q^{64} +(-2.12132 - 0.804479i) q^{70} -17.0233i q^{73} -2.27208 q^{77} -10.0000 q^{79} +(-3.17157 + 8.36308i) q^{80} +12.1716 q^{83} +8.95743i q^{88} -8.06591i q^{97} +9.17157 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 6 q^{5}+O(q^{10}) $ 4 * q + 8 * q^4 - 6 * q^5	$ 4 q + 8 q^{4} - 6 q^{5} + 4 q^{10} + 16 q^{16} - 12 q^{20} + 2 q^{25} - 20 q^{31} - 6 q^{35} + 8 q^{40} - 8 q^{49} - 24 q^{50} + 12 q^{53} - 18 q^{55} + 24 q^{62} + 32 q^{64} - 60 q^{77} - 40 q^{79} - 24 q^{80} + 60 q^{83} + 48 q^{98}+O(q^{100}) $ 4 * q + 8 * q^4 - 6 * q^5 + 4 * q^10 + 16 * q^16 - 12 * q^20 + 2 * q^25 - 20 * q^31 - 6 * q^35 + 8 * q^40 - 8 * q^49 - 24 * q^50 + 12 * q^53 - 18 * q^55 + 24 * q^62 + 32 * q^64 - 60 * q^77 - 40 * q^79 - 24 * q^80 + 60 * q^83 + 48 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.41421			1.00000
$2$	1.41421			1.00000
$3$		0			0
$3$		0			0
$4$	2.00000			1.00000
$5$	−0.792893	+	2.09077i	−0.354593	+	0.935021i
$5$	−0.792893	+	2.09077i	−0.354593	+	0.935021i
$6$		0			0
$7$			0.717439i			0.271166i	0.990766	+	0.135583i	$0.0432908\pi$
$7$			0.717439i			0.271166i	−0.990766	+	0.135583i	$0.956709\pi$
$8$	2.82843			1.00000
$9$		0			0
$10$	−1.12132	+	2.95680i	−0.354593	+	0.935021i
$11$			3.16693i			0.954865i	0.878668	+	0.477432i	$0.158432\pi$
$11$			3.16693i			0.954865i	−0.878668	+	0.477432i	$0.841568\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$			1.01461i			0.271166i
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−1.58579	+	4.18154i	−0.354593	+	0.935021i
$21$		0			0
$22$			4.47871i			0.954865i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.74264	−	3.31552i	−0.748528	−	0.663103i
$26$		0			0
$27$		0			0
$28$			1.43488i			0.271166i
$29$			10.3923i			1.92980i	0.262613	+	0.964901i	$0.415416\pi$
$29$			10.3923i			1.92980i	−0.262613	+	0.964901i	$0.584584\pi$
$30$		0			0
$31$	−0.757359			−0.136026			−0.0680129	−	0.997684i	$-0.521666\pi$
$31$	−0.757359			−0.136026			−0.0680129	+	0.997684i	$0.521666\pi$
$32$	5.65685			1.00000
$33$		0			0
$34$		0			0
$35$	−1.50000	−	0.568852i	−0.253546	−	0.0961536i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$	−2.24264	+	5.91359i	−0.354593	+	0.935021i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$			6.33386i			0.954865i
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	6.48528			0.926469
$50$	−5.29289	−	4.68885i	−0.748528	−	0.663103i
$51$		0			0
$52$		0			0
$53$	10.0711			1.38337			0.691684	−	0.722200i	$-0.256869\pi$
$53$	10.0711			1.38337			0.691684	+	0.722200i	$0.256869\pi$
$54$		0			0
$55$	−6.62132	−	2.51104i	−0.892819	−	0.338588i
$56$			2.02922i			0.271166i
$57$		0			0
$58$			14.6969i			1.92980i
$59$		−	10.3923i		−	1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$		−	10.3923i		−	1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−1.07107			−0.136026
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$	−2.12132	−	0.804479i	−0.253546	−	0.0961536i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	17.0233i		−	1.99243i	−0.0869195	−	0.996215i	$-0.527702\pi$
$73$		−	17.0233i		−	1.99243i	0.0869195	−	0.996215i	$-0.472298\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.27208			−0.258927
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$	−3.17157	+	8.36308i	−0.354593	+	0.935021i
$81$		0			0
$82$		0			0
$83$	12.1716			1.33600			0.668002	−	0.744160i	$-0.267150\pi$
$83$	12.1716			1.33600			0.668002	+	0.744160i	$0.267150\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$			8.95743i			0.954865i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	8.06591i		−	0.818969i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$		−	8.06591i		−	0.818969i	0.912317	−	0.409484i	$-0.134291\pi$
$98$	9.17157			0.926469
$99$		0			0
$100$	−7.48528	−	6.63103i	−0.748528	−	0.663103i
$101$			18.8785i			1.87848i	0.343263	+	0.939239i	$0.388468\pi$
$101$			18.8785i			1.87848i	−0.343263	+	0.939239i	$0.611532\pi$
$102$		0			0
$103$		−	14.6969i		−	1.44813i	−0.689730	−	0.724066i	$-0.742271\pi$
$103$		−	14.6969i		−	1.44813i	0.689730	−	0.724066i	$-0.257729\pi$
$104$		0			0
$105$		0			0
$106$	14.2426			1.38337
$107$	−20.6569			−1.99697			−0.998487	−	0.0549930i	$-0.982486\pi$
$107$	−20.6569			−1.99697			−0.998487	+	0.0549930i	$0.982486\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−9.36396	−	3.55114i	−0.892819	−	0.338588i
$111$		0			0
$112$			2.86976i			0.271166i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$			20.7846i			1.92980i
$117$		0			0
$118$		−	14.6969i		−	1.35296i
$119$		0			0
$120$		0			0
$121$	0.970563			0.0882330
$122$		0			0
$123$		0			0
$124$	−1.51472			−0.136026
$125$	9.89949	−	5.19615i	0.885438	−	0.464758i
$126$		0			0
$127$		−	21.5020i		−	1.90800i	−0.299809	−	0.953999i	$-0.596923\pi$
$127$		−	21.5020i		−	1.90800i	0.299809	−	0.953999i	$-0.403077\pi$
$128$	11.3137			1.00000
$129$		0			0
$130$		0			0
$131$		−	17.8639i		−	1.56077i	−0.625297	−	0.780387i	$-0.715022\pi$
$131$		−	17.8639i		−	1.56077i	0.625297	−	0.780387i	$-0.284978\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	−3.00000	−	1.13770i	−0.253546	−	0.0961536i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−21.7279	−	8.23999i	−1.80441	−	0.684294i
$146$		−	24.0746i		−	1.99243i
$147$		0			0
$148$		0			0
$149$			14.5738i			1.19394i	0.802265	+	0.596968i	$0.203628\pi$
$149$			14.5738i			1.19394i	−0.802265	+	0.596968i	$0.796372\pi$
$150$		0			0
$151$	20.2132			1.64493			0.822464	−	0.568818i	$-0.192599\pi$
$151$	20.2132			1.64493			0.822464	+	0.568818i	$0.192599\pi$
$152$		0			0
$153$		0			0
$154$	−3.21320			−0.258927
$155$	0.600505	−	1.58346i	0.0482337	−	0.127187i
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−14.1421			−1.12509
$159$		0			0
$160$	−4.48528	+	11.8272i	−0.354593	+	0.935021i
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	17.2132			1.33600
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	24.8995			1.89307			0.946537	−	0.322596i	$-0.104555\pi$
$173$	24.8995			1.89307			0.946537	+	0.322596i	$0.104555\pi$
$174$		0			0
$175$	2.37868	−	2.68512i	0.179811	−	0.202976i
$176$			12.6677i			0.954865i
$177$		0			0
$178$		0			0
$179$		−	21.9223i		−	1.63855i	−0.573400	−	0.819275i	$-0.694376\pi$
$179$		−	21.9223i		−	1.63855i	0.573400	−	0.819275i	$-0.305624\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		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
$193$			27.4156i			1.97342i	0.162488	+	0.986710i	$0.448048\pi$
$193$			27.4156i			1.97342i	−0.162488	+	0.986710i	$0.551952\pi$
$194$		−	11.4069i		−	0.818969i
$195$		0			0
$196$	12.9706			0.926469
$197$	−28.0711			−1.99998			−0.999990	−	0.00438501i	$-0.998604\pi$
$197$	−28.0711			−1.99998			−0.999990	+	0.00438501i	$0.998604\pi$
$198$		0			0
$199$	−14.2132			−1.00755			−0.503774	−	0.863836i	$-0.668055\pi$
$199$	−14.2132			−1.00755			−0.503774	+	0.863836i	$0.668055\pi$
$200$	−10.5858	−	9.37769i	−0.748528	−	0.663103i
$201$		0			0
$202$			26.6982i			1.87848i
$203$	−7.45584			−0.523298
$204$		0			0
$205$		0			0
$206$		−	20.7846i		−	1.44813i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	20.1421			1.38337
$213$		0			0
$214$	−29.2132			−1.99697
$215$		0			0
$216$		0			0
$217$		−	0.543359i		−	0.0368856i
$218$		0			0
$219$		0			0
$220$	−13.2426	−	5.02207i	−0.892819	−	0.338588i
$221$		0			0
$222$		0			0
$223$			14.6969i			0.984180i	0.870544	+	0.492090i	$0.163767\pi$
$223$			14.6969i			0.984180i	−0.870544	+	0.492090i	$0.836233\pi$
$224$			4.05845i			0.271166i
$225$		0			0
$226$		0			0
$227$	−28.2843			−1.87729			−0.938647	−	0.344881i	$-0.887919\pi$
$227$	−28.2843			−1.87729			−0.938647	+	0.344881i	$0.887919\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$			29.3939i			1.92980i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		−	20.7846i		−	1.35296i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$	1.37258			0.0882330
$243$		0			0
$244$		0			0
$245$	−5.14214	+	13.5592i	−0.328519	+	0.866268i
$246$		0			0
$247$		0			0
$248$	−2.14214			−0.136026
$249$		0			0
$250$	14.0000	−	7.34847i	0.885438	−	0.464758i
$251$		−	31.1769i		−	1.96787i	−0.178529	−	0.983935i	$-0.557134\pi$
$251$		−	31.1769i		−	1.96787i	0.178529	−	0.983935i	$-0.442866\pi$
$252$		0			0
$253$		0			0
$254$		−	30.4085i		−	1.90800i
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		−	25.2633i		−	1.56077i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−7.98528	+	21.0563i	−0.490532	+	1.29348i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	10.3923i		−	0.633630i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$		−	10.3923i		−	0.633630i	0.948487	−	0.316815i	$-0.102613\pi$
$270$		0			0
$271$	32.2132			1.95681			0.978406	−	0.206691i	$-0.0662693\pi$
$271$	32.2132			1.95681			0.978406	+	0.206691i	$0.0662693\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.5000	−	11.8527i	0.633174	−	0.714743i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$	−4.24264	−	1.60896i	−0.253546	−	0.0961536i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$	−30.7279	−	11.6531i	−1.80441	−	0.684294i
$291$		0			0
$292$		−	34.0467i		−	1.99243i
$293$	−14.1421			−0.826192			−0.413096	−	0.910687i	$-0.635553\pi$
$293$	−14.1421			−0.826192			−0.413096	+	0.910687i	$0.635553\pi$
$294$		0			0
$295$	21.7279	+	8.23999i	1.26505	+	0.479751i
$296$		0			0
$297$		0			0
$298$			20.6105i			1.19394i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	28.5858			1.64493
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−4.54416			−0.258927
$309$		0			0
$310$	0.849242	−	2.23936i	0.0482337	−	0.127187i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	24.5459i		−	1.38742i	−0.720257	−	0.693708i	$-0.755976\pi$
$313$		−	24.5459i		−	1.38742i	0.720257	−	0.693708i	$-0.244024\pi$
$314$		0			0
$315$		0			0
$316$	−20.0000			−1.12509
$317$	0.556349			0.0312477			0.0156238	−	0.999878i	$-0.495027\pi$
$317$	0.556349			0.0312477			0.0156238	+	0.999878i	$0.495027\pi$
$318$		0			0
$319$	−32.9117			−1.84270
$320$	−6.34315	+	16.7262i	−0.354593	+	0.935021i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	24.3431			1.33600
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			29.3939i			1.60119i	0.599208	+	0.800593i	$0.295482\pi$
$337$			29.3939i			1.60119i	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−18.3848			−1.00000
$339$		0			0
$340$		0			0
$341$		−	2.39850i		−	0.129886i
$342$		0			0
$343$			9.67487i			0.522394i
$344$		0			0
$345$		0			0
$346$	35.2132			1.89307
$347$	−6.85786			−0.368149			−0.184075	−	0.982912i	$-0.558929\pi$
$347$	−6.85786			−0.368149			−0.184075	+	0.982912i	$0.558929\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	3.36396	−	3.79733i	0.179811	−	0.202976i
$351$		0			0
$352$			17.9149i			0.954865i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		−	31.0028i		−	1.63855i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	35.5919	+	13.4977i	1.86296	+	0.706501i
$366$		0			0
$367$			30.4595i			1.58997i	0.606628	+	0.794986i	$0.292522\pi$
$367$			30.4595i			1.58997i	−0.606628	+	0.794986i	$0.707478\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			7.22538i			0.375123i
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	1.80152	−	4.75039i	0.0918137	−	0.242102i
$386$			38.7716i			1.97342i
$387$		0			0
$388$		−	16.1318i		−	0.818969i
$389$			14.8200i			0.751405i	0.926740	+	0.375703i	$0.122599\pi$
$389$			14.8200i			0.751405i	−0.926740	+	0.375703i	$0.877401\pi$
$390$		0			0
$391$		0			0
$392$	18.3431			0.926469
$393$		0			0
$394$	−39.6985			−1.99998
$395$	7.92893	−	20.9077i	0.398948	−	1.05198i
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	−20.1005			−1.00755
$399$		0			0
$400$	−14.9706	−	13.2621i	−0.748528	−	0.663103i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$			37.7570i			1.87848i
$405$		0			0
$406$	−10.5442			−0.523298
$407$		0			0
$408$		0			0
$409$	38.9411			1.92551			0.962757	−	0.270367i	$-0.0871450\pi$
$409$	38.9411			1.92551			0.962757	+	0.270367i	$0.0871450\pi$
$410$		0			0
$411$		0			0
$412$		−	29.3939i		−	1.44813i
$413$	7.45584			0.366878
$414$		0			0
$415$	−9.65076	+	25.4480i	−0.473737	+	1.24919i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			10.3923i			0.507697i	0.967244	+	0.253849i	$0.0816965\pi$
$419$			10.3923i			0.507697i	−0.967244	+	0.253849i	$0.918303\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	28.4853			1.38337
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−41.3137			−1.99697
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$			31.7203i			1.52438i	0.647354	+	0.762190i	$0.275876\pi$
$433$			31.7203i			1.52438i	−0.647354	+	0.762190i	$0.724124\pi$
$434$		−	0.768426i		−	0.0368856i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−38.2132			−1.82382			−0.911908	−	0.410394i	$-0.865391\pi$
$439$	−38.2132			−1.82382			−0.911908	+	0.410394i	$0.865391\pi$
$440$	−18.7279	−	7.10228i	−0.892819	−	0.338588i
$441$		0			0
$442$		0			0
$443$	28.2843			1.34383			0.671913	−	0.740630i	$-0.265473\pi$
$443$	28.2843			1.34383			0.671913	+	0.740630i	$0.265473\pi$
$444$		0			0
$445$		0			0
$446$			20.7846i			0.984180i
$447$		0			0
$448$			5.73951i			0.271166i
$449$		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
$454$	−40.0000			−1.87729
$455$		0			0
$456$		0			0
$457$		−	23.1110i		−	1.08109i	−0.841316	−	0.540544i	$-0.818219\pi$
$457$		−	23.1110i		−	1.08109i	0.841316	−	0.540544i	$-0.181781\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			1.90613i			0.0887774i	0.999014	+	0.0443887i	$0.0141340\pi$
$461$			1.90613i			0.0887774i	−0.999014	+	0.0443887i	$0.985866\pi$
$462$		0			0
$463$		−	5.37023i		−	0.249576i	−0.992183	−	0.124788i	$-0.960175\pi$
$463$		−	5.37023i		−	0.249576i	0.992183	−	0.124788i	$-0.0398251\pi$
$464$			41.5692i			1.92980i
$465$		0			0
$466$		0			0
$467$	4.79899			0.222071			0.111035	−	0.993816i	$-0.464583\pi$
$467$	4.79899			0.222071			0.111035	+	0.993816i	$0.464583\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	29.3939i		−	1.35296i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		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
$482$	14.1421			0.644157
$483$		0			0
$484$	1.94113			0.0882330
$485$	16.8640	+	6.39540i	0.765753	+	0.290400i
$486$		0			0
$487$			44.0908i			1.99795i	0.0453143	+	0.998973i	$0.485571\pi$
$487$			44.0908i			1.99795i	−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$		0			0
$490$	−7.27208	+	19.1757i	−0.328519	+	0.866268i
$491$			38.6485i			1.74418i	0.489344	+	0.872091i	$0.337236\pi$
$491$			38.6485i			1.74418i	−0.489344	+	0.872091i	$0.662764\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−3.02944			−0.136026
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	19.7990	−	10.3923i	0.885438	−	0.464758i
$501$		0			0
$502$		−	44.0908i		−	1.96787i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−39.4706	−	14.9686i	−1.75642	−	0.666095i
$506$		0			0
$507$		0			0
$508$		−	43.0041i		−	1.90800i
$509$		−	24.9662i		−	1.10661i	−0.832980	−	0.553303i	$-0.813367\pi$
$509$		−	24.9662i		−	1.10661i	0.832980	−	0.553303i	$-0.186633\pi$
$510$		0			0
$511$	12.2132			0.540280
$512$	22.6274			1.00000
$513$		0			0
$514$		0			0
$515$	30.7279	+	11.6531i	1.35403	+	0.513497i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		−	35.7277i		−	1.56077i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$	−11.2929	+	29.7781i	−0.490532	+	1.29348i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	16.3787	−	43.1887i	0.708112	−	1.86721i
$536$		0			0
$537$		0			0
$538$		−	14.6969i		−	0.633630i
$539$			20.5384i			0.884653i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	45.5563			1.95681
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	14.8492	−	16.7622i	0.633174	−	0.714743i
$551$		0			0
$552$		0			0
$553$		−	7.17439i		−	0.305086i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−46.0711			−1.95209			−0.976047	−	0.217560i	$-0.930190\pi$
$557$	−46.0711			−1.95209			−0.976047	+	0.217560i	$0.930190\pi$
$558$		0			0
$559$		0			0
$560$	−6.00000	−	2.27541i	−0.253546	−	0.0961536i
$561$		0			0
$562$		0			0
$563$	−47.1421			−1.98680			−0.993402	−	0.114684i	$-0.963415\pi$
$563$	−47.1421			−1.98680			−0.993402	+	0.114684i	$0.963415\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			29.3939i			1.22368i	0.790980	+	0.611842i	$0.209571\pi$
$577$			29.3939i			1.22368i	−0.790980	+	0.611842i	$0.790429\pi$
$578$	24.0416			1.00000
$579$		0			0
$580$	−43.4558	−	16.4800i	−1.80441	−	0.684294i
$581$			8.73236i			0.362279i
$582$		0			0
$583$			31.8944i			1.32093i
$584$		−	48.1493i		−	1.99243i
$585$		0			0
$586$	−20.0000			−0.826192
$587$	−37.6274			−1.55305			−0.776525	−	0.630087i	$-0.783019\pi$
$587$	−37.6274			−1.55305			−0.776525	+	0.630087i	$0.783019\pi$
$588$		0			0
$589$		0			0
$590$	30.7279	+	11.6531i	1.26505	+	0.479751i
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$			29.1477i			1.19394i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−43.4264			−1.77140			−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−43.4264			−1.77140			−0.885700	+	0.464258i	$0.846321\pi$
$602$		0			0
$603$		0			0
$604$	40.4264			1.64493
$605$	−0.769553	+	2.02922i	−0.0312868	+	0.0824997i
$606$		0			0
$607$		−	44.0908i		−	1.78959i	−0.446476	−	0.894795i	$-0.647321\pi$
$607$		−	44.0908i		−	1.78959i	0.446476	−	0.894795i	$-0.352679\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$	−6.42641			−0.258927
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$	1.20101	−	3.16693i	0.0482337	−	0.127187i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	3.01472	+	24.8176i	0.120589	+	0.992703i
$626$		−	34.7131i		−	1.38742i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	29.2426			1.16413			0.582066	−	0.813142i	$-0.302245\pi$
$631$	29.2426			1.16413			0.582066	+	0.813142i	$0.302245\pi$
$632$	−28.2843			−1.12509
$633$		0			0
$634$	0.786797			0.0312477
$635$	44.9558	+	17.0488i	1.78402	+	0.676562i
$636$		0			0
$637$		0			0
$638$	−46.5442			−1.84270
$639$		0			0
$640$	−8.97056	+	23.6544i	−0.354593	+	0.935021i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	32.9117			1.29190
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.04163			−0.353826			−0.176913	−	0.984226i	$-0.556611\pi$
$653$	−9.04163			−0.353826			−0.176913	+	0.984226i	$0.556611\pi$
$654$		0			0
$655$	37.3492	+	14.1641i	1.45936	+	0.553439i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			51.3162i			1.99900i	0.0316976	+	0.999498i	$0.489909\pi$
$659$			51.3162i			1.99900i	−0.0316976	+	0.999498i	$0.510091\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	34.4264			1.33600
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			9.84895i			0.379649i	0.981818	+	0.189824i	$0.0607919\pi$
$673$			9.84895i			0.379649i	−0.981818	+	0.189824i	$0.939208\pi$
$674$			41.5692i			1.60119i
$675$		0			0
$676$	−26.0000			−1.00000
$677$	2.82843			0.108705			0.0543526	−	0.998522i	$-0.482690\pi$
$677$	2.82843			0.108705			0.0543526	+	0.998522i	$0.482690\pi$
$678$		0			0
$679$	5.78680			0.222077
$680$		0			0
$681$		0			0
$682$		−	3.39200i		−	0.129886i
$683$	5.65685			0.216454			0.108227	−	0.994126i	$-0.465483\pi$
$683$	5.65685			0.216454			0.108227	+	0.994126i	$0.465483\pi$
$684$		0			0
$685$		0			0
$686$			13.6823i			0.522394i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	49.7990			1.89307
$693$		0			0
$694$	−9.69848			−0.368149
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	4.75736	−	5.37023i	0.179811	−	0.202976i
$701$			12.7908i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$701$			12.7908i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$702$		0			0
$703$		0			0
$704$			25.3354i			0.954865i
$705$		0			0
$706$		0			0
$707$	−13.5442			−0.509380
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	43.8446i		−	1.63855i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	10.5442			0.392685
$722$	26.8701			1.00000
$723$		0			0
$724$		0			0
$725$	34.4558	−	38.8947i	1.27966	−	1.44451i
$726$		0			0
$727$			28.6764i			1.06355i	0.846886	+	0.531775i	$0.178475\pi$
$727$			28.6764i			1.06355i	−0.846886	+	0.531775i	$0.821525\pi$
$728$		0			0
$729$		0			0
$730$	50.3345	+	19.0886i	1.86296	+	0.706501i
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$			43.0762i			1.58997i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$			10.2182i			0.375123i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−30.4706	−	11.5555i	−1.11636	−	0.423361i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	14.8200i		−	0.541512i
$750$		0			0
$751$	−51.6690			−1.88543			−0.942715	−	0.333599i	$-0.891737\pi$
$751$	−51.6690			−1.88543			−0.942715	+	0.333599i	$0.891737\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0269	+	42.2612i	−0.583279	+	1.53804i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	55.4264			1.99873			0.999364	−	0.0356685i	$-0.0113561\pi$
$769$	55.4264			1.99873			0.999364	+	0.0356685i	$0.0113561\pi$
$770$	2.54773	−	6.71807i	0.0918137	−	0.242102i
$771$		0			0
$772$			54.8313i			1.97342i
$773$	−19.7990			−0.712120			−0.356060	−	0.934463i	$-0.615880\pi$
$773$	−19.7990			−0.712120			−0.356060	+	0.934463i	$0.615880\pi$
$774$		0			0
$775$	2.83452	+	2.51104i	0.101819	+	0.0901991i
$776$		−	22.8138i		−	0.818969i
$777$		0			0
$778$			20.9587i			0.751405i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	25.9411			0.926469
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−56.1421			−1.99998
$789$		0			0
$790$	11.2132	−	29.5680i	0.398948	−	1.05198i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−28.4264			−1.00755
$797$	−41.8701			−1.48311			−0.741557	−	0.670890i	$-0.765912\pi$
$797$	−41.8701			−1.48311			−0.741557	+	0.670890i	$0.765912\pi$
$798$		0			0
$799$		0			0
$800$	−21.1716	−	18.7554i	−0.748528	−	0.663103i
$801$		0			0
$802$		0			0
$803$	53.9117			1.90250
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$			53.3964i			1.87848i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−14.9117			−0.523298
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	55.0711			1.92551
$819$		0			0
$820$		0			0
$821$		−	31.1769i		−	1.08808i	−0.839059	−	0.544041i	$-0.816894\pi$
$821$		−	31.1769i		−	1.08808i	0.839059	−	0.544041i	$-0.183106\pi$
$822$		0			0
$823$		−	56.9836i		−	1.98632i	−0.116747	−	0.993162i	$-0.537247\pi$
$823$		−	56.9836i		−	1.98632i	0.116747	−	0.993162i	$-0.462753\pi$
$824$		−	41.5692i		−	1.44813i
$825$		0			0
$826$	10.5442			0.366878
$827$	56.5685			1.96708			0.983540	−	0.180688i	$-0.0578324\pi$
$827$	56.5685			1.96708			0.983540	+	0.180688i	$0.0578324\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−13.6482	+	35.9889i	−0.473737	+	1.24919i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$			14.6969i			0.507697i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−79.0000			−2.72414
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.3076	−	27.1800i	0.354593	−	0.935021i
$846$		0			0
$847$			0.696320i			0.0239258i
$848$	40.2843			1.38337
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−58.4264			−1.99697
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−19.7426	+	52.0591i	−0.671270	+	1.77006i
$866$			44.8592i			1.52438i
$867$		0			0
$868$		−	1.08672i		−	0.0368856i
$869$		−	31.6693i		−	1.07431i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.72792	+	7.10228i	0.126027	+	0.240101i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−54.0416			−1.82382
$879$		0			0
$880$	−26.4853	−	10.0441i	−0.892819	−	0.338588i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	40.0000			1.34383
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	15.4264			0.517385
$890$		0			0
$891$		0			0
$892$			29.3939i			0.984180i
$893$		0			0
$894$		0			0
$895$	45.8345	+	17.3821i	1.53208	+	0.581018i
$896$			8.11689i			0.271166i
$897$		0			0
$898$		0			0
$899$		−	7.87071i		−	0.262503i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	−56.5685			−1.87729
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$			38.5465i			1.27570i
$914$		−	32.6839i		−	1.08109i
$915$		0			0
$916$		0			0
$917$	12.8162			0.423229
$918$		0			0
$919$	−54.6985			−1.80434			−0.902168	−	0.431384i	$-0.858025\pi$
$919$	−54.6985			−1.80434			−0.902168	+	0.431384i	$0.858025\pi$
$920$		0			0
$921$		0			0
$922$			2.69568i			0.0887774i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		−	7.59466i		−	0.249576i
$927$		0			0
$928$			58.7878i			1.92980i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	6.78680			0.222071
$935$		0			0
$936$		0			0
$937$		−	60.0274i		−	1.96101i	−0.196492	−	0.980505i	$-0.562955\pi$
$937$		−	60.0274i		−	1.96101i	0.196492	−	0.980505i	$-0.437045\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	60.6939i		−	1.97856i	−0.146017	−	0.989282i	$-0.546646\pi$
$941$		−	60.6939i		−	1.97856i	0.146017	−	0.989282i	$-0.453354\pi$
$942$		0			0
$943$		0			0
$944$		−	41.5692i		−	1.35296i
$945$		0			0
$946$		0			0
$947$	−49.2843			−1.60152			−0.800762	−	0.598983i	$-0.795572\pi$
$947$	−49.2843			−1.60152			−0.800762	+	0.598983i	$0.795572\pi$
$948$		0			0
$949$		0			0
$950$		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
$961$	−30.4264			−0.981497
$962$		0			0
$963$		0			0
$964$	20.0000			0.644157
$965$	−57.3198	−	21.7377i	−1.84519	−	0.699761i
$966$		0			0
$967$		−	51.2441i		−	1.64790i	−0.566663	−	0.823949i	$-0.691766\pi$
$967$		−	51.2441i		−	1.64790i	0.566663	−	0.823949i	$-0.308234\pi$
$968$	2.74517			0.0882330
$969$		0			0
$970$	23.8492	+	9.04447i	0.765753	+	0.290400i
$971$			55.6208i			1.78496i	0.451090	+	0.892479i	$0.351035\pi$
$971$			55.6208i			1.78496i	−0.451090	+	0.892479i	$0.648965\pi$
$972$		0			0
$973$		0			0
$974$			62.3538i			1.99795i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−10.2843	+	27.1185i	−0.328519	+	0.866268i
$981$		0			0
$982$			54.6572i			1.74418i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	22.2574	−	58.6902i	0.709178	−	1.87002i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	50.2132			1.59507			0.797537	−	0.603269i	$-0.206136\pi$
$991$	50.2132			1.59507			0.797537	+	0.603269i	$0.206136\pi$
$992$	−4.28427			−0.136026
$993$		0			0
$994$		0			0
$995$	11.2696	−	29.7165i	0.357269	−	0.942078i
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.d.d.109.4	yes	4
3.2	odd	2		1080.2.d.e.109.1	yes	4
4.3	odd	2		4320.2.d.a.3889.4		4
5.4	even	2		1080.2.d.e.109.2	yes	4
8.3	odd	2		4320.2.d.f.3889.1		4
8.5	even	2		1080.2.d.e.109.1	yes	4
12.11	even	2		4320.2.d.f.3889.1		4
15.14	odd	2	inner	1080.2.d.d.109.3	✓	4
20.19	odd	2		4320.2.d.f.3889.2		4
24.5	odd	2	CM	1080.2.d.d.109.4	yes	4
24.11	even	2		4320.2.d.a.3889.4		4
40.19	odd	2		4320.2.d.a.3889.3		4
40.29	even	2	inner	1080.2.d.d.109.3	✓	4
60.59	even	2		4320.2.d.a.3889.3		4
120.29	odd	2		1080.2.d.e.109.2	yes	4
120.59	even	2		4320.2.d.f.3889.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.d.d.109.3	✓	4	15.14	odd	2	inner
1080.2.d.d.109.3	✓	4	40.29	even	2	inner
1080.2.d.d.109.4	yes	4	1.1	even	1	trivial
1080.2.d.d.109.4	yes	4	24.5	odd	2	CM
1080.2.d.e.109.1	yes	4	3.2	odd	2
1080.2.d.e.109.1	yes	4	8.5	even	2
1080.2.d.e.109.2	yes	4	5.4	even	2
1080.2.d.e.109.2	yes	4	120.29	odd	2
4320.2.d.a.3889.3		4	40.19	odd	2
4320.2.d.a.3889.3		4	60.59	even	2
4320.2.d.a.3889.4		4	4.3	odd	2
4320.2.d.a.3889.4		4	24.11	even	2
4320.2.d.f.3889.1		4	8.3	odd	2
4320.2.d.f.3889.1		4	12.11	even	2
4320.2.d.f.3889.2		4	20.19	odd	2
4320.2.d.f.3889.2		4	120.59	even	2

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-0.792893 + 2.09077i) q^{5} +0.717439i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-0.792893 + 2.09077i) q^{5} +0.717439i q^{7} +2.82843 q^{8} +(-1.12132 + 2.95680i) q^{10} +3.16693i q^{11} +1.01461i q^{14} +4.00000 q^{16} +(-1.58579 + 4.18154i) q^{20} +4.47871i q^{22} +(-3.74264 - 3.31552i) q^{25} +1.43488i q^{28} +10.3923i q^{29} -0.757359 q^{31} +5.65685 q^{32} +(-1.50000 - 0.568852i) q^{35} +(-2.24264 + 5.91359i) q^{40} +6.33386i q^{44} +6.48528 q^{49} +(-5.29289 - 4.68885i) q^{50} +10.0711 q^{53} +(-6.62132 - 2.51104i) q^{55} +2.02922i q^{56} +14.6969i q^{58} -10.3923i q^{59} -1.07107 q^{62} +8.00000 q^{64} +(-2.12132 - 0.804479i) q^{70} -17.0233i q^{73} -2.27208 q^{77} -10.0000 q^{79} +(-3.17157 + 8.36308i) q^{80} +12.1716 q^{83} +8.95743i q^{88} -8.06591i q^{97} +9.17157 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 6 q^{5}+O(q^{10}) \) 4 * q + 8 * q^4 - 6 * q^5	\( 4 q + 8 q^{4} - 6 q^{5} + 4 q^{10} + 16 q^{16} - 12 q^{20} + 2 q^{25} - 20 q^{31} - 6 q^{35} + 8 q^{40} - 8 q^{49} - 24 q^{50} + 12 q^{53} - 18 q^{55} + 24 q^{62} + 32 q^{64} - 60 q^{77} - 40 q^{79} - 24 q^{80} + 60 q^{83} + 48 q^{98}+O(q^{100}) \) 4 * q + 8 * q^4 - 6 * q^5 + 4 * q^10 + 16 * q^16 - 12 * q^20 + 2 * q^25 - 20 * q^31 - 6 * q^35 + 8 * q^40 - 8 * q^49 - 24 * q^50 + 12 * q^53 - 18 * q^55 + 24 * q^62 + 32 * q^64 - 60 * q^77 - 40 * q^79 - 24 * q^80 + 60 * q^83 + 48 * q^98

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