LMFDB - Embedded newform 2160.3.c.q.1889.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,3,Mod(1889,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.1889"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.8
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.q.1889.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12464 + 4.52614i) q^{5} -4.02248i q^{7} +11.5580i q^{11} -13.5502i q^{13} +4.76977 q^{17} -8.28351 q^{19} +33.4085 q^{23} +(-15.9718 - 19.2328i) q^{25} -21.3299i q^{29} -51.2723 q^{31} +(18.2063 + 8.54633i) q^{35} +1.30139i q^{37} +0.277551i q^{41} +5.16141i q^{43} -49.9448 q^{47} +32.8197 q^{49} +14.5859 q^{53} +(-52.3133 - 24.5567i) q^{55} -56.4461i q^{59} +108.803 q^{61} +(61.3301 + 28.7893i) q^{65} +61.7275i q^{67} -63.6541i q^{71} +3.03261i q^{73} +46.4920 q^{77} -24.1416 q^{79} +54.1367 q^{83} +(-10.1341 + 21.5886i) q^{85} +19.2145i q^{89} -54.5054 q^{91} +(17.5995 - 37.4923i) q^{95} -179.628i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100}) $ 24 * q - 72 * q^25 + 72 * q^31 - 408 * q^49 + 168 * q^55 - 240 * q^61 + 312 * q^79 - 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.12464	+	4.52614i	−0.424928	+	0.905227i
$5$	−2.12464	+	4.52614i	−0.424928	+	0.905227i
$6$		0			0
$7$		−	4.02248i		−	0.574640i	−0.957835	−	0.287320i	$-0.907236\pi$
$7$		−	4.02248i		−	0.574640i	0.957835	−	0.287320i	$-0.0927643\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			11.5580i			1.05073i	0.850877	+	0.525366i	$0.176071\pi$
$11$			11.5580i			1.05073i	−0.850877	+	0.525366i	$0.823929\pi$
$12$		0			0
$13$		−	13.5502i		−	1.04232i	−0.853458	−	0.521162i	$-0.825499\pi$
$13$		−	13.5502i		−	1.04232i	0.853458	−	0.521162i	$-0.174501\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.76977			0.280575			0.140287	−	0.990111i	$-0.455197\pi$
$17$	4.76977			0.280575			0.140287	+	0.990111i	$0.455197\pi$
$18$		0			0
$19$	−8.28351			−0.435974			−0.217987	−	0.975952i	$-0.569949\pi$
$19$	−8.28351			−0.435974			−0.217987	+	0.975952i	$0.569949\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	33.4085			1.45254			0.726271	−	0.687409i	$-0.241252\pi$
$23$	33.4085			1.45254			0.726271	+	0.687409i	$0.241252\pi$
$24$		0			0
$25$	−15.9718	−	19.2328i	−0.638872	−	0.769313i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	21.3299i		−	0.735515i	−0.929922	−	0.367757i	$-0.880126\pi$
$29$		−	21.3299i		−	0.735515i	0.929922	−	0.367757i	$-0.119874\pi$
$30$		0			0
$31$	−51.2723			−1.65395			−0.826973	−	0.562242i	$-0.809939\pi$
$31$	−51.2723			−1.65395			−0.826973	+	0.562242i	$0.809939\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	18.2063	+	8.54633i	0.520180	+	0.244181i
$36$		0			0
$37$			1.30139i			0.0351728i	0.999845	+	0.0175864i	$0.00559822\pi$
$37$			1.30139i			0.0351728i	−0.999845	+	0.0175864i	$0.994402\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.277551i			0.00676953i	0.999994	+	0.00338477i	$0.00107741\pi$
$41$			0.277551i			0.00676953i	−0.999994	+	0.00338477i	$0.998923\pi$
$42$		0			0
$43$			5.16141i			0.120033i	0.998197	+	0.0600164i	$0.0191153\pi$
$43$			5.16141i			0.120033i	−0.998197	+	0.0600164i	$0.980885\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−49.9448			−1.06266			−0.531328	−	0.847166i	$-0.678307\pi$
$47$	−49.9448			−1.06266			−0.531328	+	0.847166i	$0.678307\pi$
$48$		0			0
$49$	32.8197			0.669789
$50$		0			0
$51$		0			0
$52$		0			0
$53$	14.5859			0.275206			0.137603	−	0.990487i	$-0.456060\pi$
$53$	14.5859			0.275206			0.137603	+	0.990487i	$0.456060\pi$
$54$		0			0
$55$	−52.3133	−	24.5567i	−0.951151	−	0.446485i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	56.4461i		−	0.956714i	−0.878166	−	0.478357i	$-0.841233\pi$
$59$		−	56.4461i		−	0.956714i	0.878166	−	0.478357i	$-0.158767\pi$
$60$		0			0
$61$	108.803			1.78366			0.891830	−	0.452370i	$-0.149421\pi$
$61$	108.803			1.78366			0.891830	+	0.452370i	$0.149421\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	61.3301	+	28.7893i	0.943540	+	0.442913i
$66$		0			0
$67$			61.7275i			0.921306i	0.887580	+	0.460653i	$0.152385\pi$
$67$			61.7275i			0.921306i	−0.887580	+	0.460653i	$0.847615\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	63.6541i		−	0.896537i	−0.893899	−	0.448268i	$-0.852041\pi$
$71$		−	63.6541i		−	0.896537i	0.893899	−	0.448268i	$-0.147959\pi$
$72$		0			0
$73$			3.03261i			0.0415427i	0.999784	+	0.0207713i	$0.00661220\pi$
$73$			3.03261i			0.0415427i	−0.999784	+	0.0207713i	$0.993388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	46.4920			0.603792
$78$		0			0
$79$	−24.1416			−0.305590			−0.152795	−	0.988258i	$-0.548827\pi$
$79$	−24.1416			−0.305590			−0.152795	+	0.988258i	$0.548827\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	54.1367			0.652249			0.326124	−	0.945327i	$-0.394257\pi$
$83$	54.1367			0.652249			0.326124	+	0.945327i	$0.394257\pi$
$84$		0			0
$85$	−10.1341	+	21.5886i	−0.119224	+	0.253984i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			19.2145i			0.215893i	0.994157	+	0.107946i	$0.0344275\pi$
$89$			19.2145i			0.215893i	−0.994157	+	0.107946i	$0.965572\pi$
$90$		0			0
$91$	−54.5054			−0.598961
$92$		0			0
$93$		0			0
$94$		0			0
$95$	17.5995	−	37.4923i	0.185258	−	0.394655i
$96$		0			0
$97$		−	179.628i		−	1.85183i	−0.377727	−	0.925917i	$-0.623294\pi$
$97$		−	179.628i		−	1.85183i	0.377727	−	0.925917i	$-0.376706\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	163.871i		−	1.62248i	−0.584711	−	0.811242i	$-0.698792\pi$
$101$		−	163.871i		−	1.62248i	0.584711	−	0.811242i	$-0.301208\pi$
$102$		0			0
$103$			124.295i			1.20675i	0.797458	+	0.603374i	$0.206177\pi$
$103$			124.295i			1.20675i	−0.797458	+	0.603374i	$0.793823\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	96.6934			0.903677			0.451838	−	0.892100i	$-0.350768\pi$
$107$	96.6934			0.903677			0.451838	+	0.892100i	$0.350768\pi$
$108$		0			0
$109$	187.754			1.72251			0.861256	−	0.508172i	$-0.169678\pi$
$109$	187.754			1.72251			0.861256	+	0.508172i	$0.169678\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	71.3349			0.631283			0.315641	−	0.948879i	$-0.397780\pi$
$113$	71.3349			0.631283			0.315641	+	0.948879i	$0.397780\pi$
$114$		0			0
$115$	−70.9810	+	151.211i	−0.617226	+	1.31488i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	19.1863i		−	0.161230i
$120$		0			0
$121$	−12.5884			−0.104037
$122$		0			0
$123$		0			0
$124$		0			0
$125$	120.985	−	31.4277i	0.967878	−	0.251421i
$126$		0			0
$127$		−	141.372i		−	1.11317i	−0.830792	−	0.556584i	$-0.812112\pi$
$127$		−	141.372i		−	1.11317i	0.830792	−	0.556584i	$-0.187888\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			216.320i			1.65130i	0.564186	+	0.825648i	$0.309190\pi$
$131$			216.320i			1.65130i	−0.564186	+	0.825648i	$0.690810\pi$
$132$		0			0
$133$			33.3202i			0.250528i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−118.321			−0.863657			−0.431828	−	0.901956i	$-0.642131\pi$
$137$	−118.321			−0.863657			−0.431828	+	0.901956i	$0.642131\pi$
$138$		0			0
$139$	−108.779			−0.782579			−0.391290	−	0.920268i	$-0.627971\pi$
$139$	−108.779			−0.782579			−0.391290	+	0.920268i	$0.627971\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	156.614			1.09520
$144$		0			0
$145$	96.5421	+	45.3184i	0.665808	+	0.312541i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			42.3938i			0.284522i	0.989829	+	0.142261i	$0.0454372\pi$
$149$			42.3938i			0.284522i	−0.989829	+	0.142261i	$0.954563\pi$
$150$		0			0
$151$	137.390			0.909869			0.454934	−	0.890525i	$-0.349663\pi$
$151$	137.390			0.909869			0.454934	+	0.890525i	$0.349663\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	108.935	−	232.065i	0.702808	−	1.49720i
$156$		0			0
$157$			80.8173i			0.514760i	0.966310	+	0.257380i	$0.0828592\pi$
$157$			80.8173i			0.514760i	−0.966310	+	0.257380i	$0.917141\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	134.385i		−	0.834688i
$162$		0			0
$163$		−	99.4938i		−	0.610391i	−0.952290	−	0.305196i	$-0.901278\pi$
$163$		−	99.4938i		−	0.610391i	0.952290	−	0.305196i	$-0.0987219\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	197.912			1.18510			0.592551	−	0.805533i	$-0.298121\pi$
$167$	197.912			1.18510			0.592551	+	0.805533i	$0.298121\pi$
$168$		0			0
$169$	−14.6082			−0.0864389
$170$		0			0
$171$		0			0
$172$		0			0
$173$	175.218			1.01282			0.506411	−	0.862292i	$-0.330972\pi$
$173$	175.218			1.01282			0.506411	+	0.862292i	$0.330972\pi$
$174$		0			0
$175$	−77.3636	+	64.2462i	−0.442078	+	0.367121i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.03418i		−	0.0113641i	−0.999984	−	0.00568207i	$-0.998191\pi$
$179$		−	2.03418i		−	0.0113641i	0.999984	−	0.00568207i	$-0.00180867\pi$
$180$		0			0
$181$	224.356			1.23953			0.619767	−	0.784786i	$-0.287227\pi$
$181$	224.356			1.23953			0.619767	+	0.784786i	$0.287227\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.89029	−	2.76500i	−0.0318394	−	0.0149459i
$186$		0			0
$187$			55.1293i			0.294809i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	145.691i		−	0.762780i	−0.924414	−	0.381390i	$-0.875446\pi$
$191$		−	145.691i		−	0.762780i	0.924414	−	0.381390i	$-0.124554\pi$
$192$		0			0
$193$		−	209.535i		−	1.08567i	−0.839838	−	0.542837i	$-0.817350\pi$
$193$		−	209.535i		−	1.08567i	0.839838	−	0.542837i	$-0.182650\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	297.757			1.51146			0.755729	−	0.654885i	$-0.227283\pi$
$197$	297.757			1.51146			0.755729	+	0.654885i	$0.227283\pi$
$198$		0			0
$199$	−25.0214			−0.125736			−0.0628678	−	0.998022i	$-0.520025\pi$
$199$	−25.0214			−0.125736			−0.0628678	+	0.998022i	$0.520025\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−85.7992			−0.422656
$204$		0			0
$205$	−1.25623	−	0.589696i	−0.00612796	−	0.00287657i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	95.7411i		−	0.458092i
$210$		0			0
$211$	196.797			0.932688			0.466344	−	0.884603i	$-0.345571\pi$
$211$	196.797			0.932688			0.466344	+	0.884603i	$0.345571\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−23.3612	−	10.9661i	−0.108657	−	0.0510053i
$216$		0			0
$217$			206.242i			0.950423i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	64.6314i		−	0.292450i
$222$		0			0
$223$			163.393i			0.732702i	0.930477	+	0.366351i	$0.119393\pi$
$223$			163.393i			0.732702i	−0.930477	+	0.366351i	$0.880607\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	89.7140			0.395216			0.197608	−	0.980281i	$-0.436683\pi$
$227$	89.7140			0.395216			0.197608	+	0.980281i	$0.436683\pi$
$228$		0			0
$229$	−273.661			−1.19503			−0.597514	−	0.801859i	$-0.703845\pi$
$229$	−273.661			−1.19503			−0.597514	+	0.801859i	$0.703845\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	57.6614			0.247474			0.123737	−	0.992315i	$-0.460512\pi$
$233$	57.6614			0.247474			0.123737	+	0.992315i	$0.460512\pi$
$234$		0			0
$235$	106.115	−	226.057i	0.451553	−	0.961945i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	131.371i		−	0.549670i	−0.961491	−	0.274835i	$-0.911377\pi$
$239$		−	131.371i		−	0.549670i	0.961491	−	0.274835i	$-0.0886233\pi$
$240$		0			0
$241$	−326.355			−1.35417			−0.677086	−	0.735904i	$-0.736757\pi$
$241$	−326.355			−1.35417			−0.677086	+	0.735904i	$0.736757\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−69.7300	+	148.546i	−0.284612	+	0.606311i
$246$		0			0
$247$			112.243i			0.454426i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			239.945i			0.955958i	0.878371	+	0.477979i	$0.158630\pi$
$251$			239.945i			0.955958i	−0.878371	+	0.477979i	$0.841370\pi$
$252$		0			0
$253$			386.137i			1.52623i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	218.132			0.848763			0.424381	−	0.905484i	$-0.360492\pi$
$257$	218.132			0.848763			0.424381	+	0.905484i	$0.360492\pi$
$258$		0			0
$259$	5.23483			0.0202117
$260$		0			0
$261$		0			0
$262$		0			0
$263$	179.288			0.681704			0.340852	−	0.940117i	$-0.389284\pi$
$263$	179.288			0.681704			0.340852	+	0.940117i	$0.389284\pi$
$264$		0			0
$265$	−30.9898	+	66.0178i	−0.116943	+	0.249124i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			168.744i			0.627300i	0.949539	+	0.313650i	$0.101552\pi$
$269$			168.744i			0.627300i	−0.949539	+	0.313650i	$0.898448\pi$
$270$		0			0
$271$	−156.629			−0.577966			−0.288983	−	0.957334i	$-0.593317\pi$
$271$	−156.629			−0.577966			−0.288983	+	0.957334i	$0.593317\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	222.294	−	184.603i	0.808341	−	0.671283i
$276$		0			0
$277$		−	475.405i		−	1.71626i	−0.513429	−	0.858132i	$-0.671625\pi$
$277$		−	475.405i		−	1.71626i	0.513429	−	0.858132i	$-0.328375\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	400.012i		−	1.42353i	−0.702418	−	0.711765i	$-0.747896\pi$
$281$		−	400.012i		−	1.42353i	0.702418	−	0.711765i	$-0.252104\pi$
$282$		0			0
$283$		−	205.228i		−	0.725189i	−0.931947	−	0.362594i	$-0.881891\pi$
$283$		−	205.228i		−	0.725189i	0.931947	−	0.362594i	$-0.118109\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.11644			0.00389004
$288$		0			0
$289$	−266.249			−0.921278
$290$		0			0
$291$		0			0
$292$		0			0
$293$	520.290			1.77574			0.887868	−	0.460099i	$-0.152186\pi$
$293$	520.290			1.77574			0.887868	+	0.460099i	$0.152186\pi$
$294$		0			0
$295$	255.483	+	119.928i	0.866043	+	0.406535i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	452.692i		−	1.51402i
$300$		0			0
$301$	20.7617			0.0689756
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−231.168	+	492.459i	−0.757928	+	1.61462i
$306$		0			0
$307$		−	527.621i		−	1.71863i	−0.511443	−	0.859317i	$-0.670889\pi$
$307$		−	527.621i		−	1.71863i	0.511443	−	0.859317i	$-0.329111\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			604.316i			1.94314i	0.236755	+	0.971569i	$0.423916\pi$
$311$			604.316i			1.94314i	−0.236755	+	0.971569i	$0.576084\pi$
$312$		0			0
$313$		−	36.8867i		−	0.117849i	−0.998262	−	0.0589244i	$-0.981233\pi$
$313$		−	36.8867i		−	0.117849i	0.998262	−	0.0589244i	$-0.0187671\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	416.238			1.31305			0.656526	−	0.754303i	$-0.272025\pi$
$317$	416.238			1.31305			0.656526	+	0.754303i	$0.272025\pi$
$318$		0			0
$319$	246.532			0.772828
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−39.5105			−0.122323
$324$		0			0
$325$	−260.609	+	216.421i	−0.801873	+	0.665911i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			200.902i			0.610645i
$330$		0			0
$331$	40.4935			0.122337			0.0611684	−	0.998127i	$-0.480517\pi$
$331$	40.4935			0.122337			0.0611684	+	0.998127i	$0.480517\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−279.387	−	131.149i	−0.833991	−	0.391489i
$336$		0			0
$337$		−	340.334i		−	1.00989i	−0.863151	−	0.504946i	$-0.831512\pi$
$337$		−	340.334i		−	1.00989i	0.863151	−	0.504946i	$-0.168488\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	592.608i		−	1.73785i
$342$		0			0
$343$		−	329.118i		−	0.959527i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	306.046			0.881976			0.440988	−	0.897513i	$-0.354628\pi$
$347$	306.046			0.881976			0.440988	+	0.897513i	$0.354628\pi$
$348$		0			0
$349$	−491.641			−1.40871			−0.704357	−	0.709846i	$-0.748765\pi$
$349$	−491.641			−1.40871			−0.704357	+	0.709846i	$0.748765\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−603.944			−1.71089			−0.855444	−	0.517895i	$-0.826716\pi$
$353$	−603.944			−1.71089			−0.855444	+	0.517895i	$0.826716\pi$
$354$		0			0
$355$	288.107	+	135.242i	0.811569	+	0.380964i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	253.931i		−	0.707328i	−0.935372	−	0.353664i	$-0.884936\pi$
$359$		−	253.931i		−	0.707328i	0.935372	−	0.353664i	$-0.115064\pi$
$360$		0			0
$361$	−292.384			−0.809927
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.7260	−	6.44322i	−0.0376055	−	0.0176527i
$366$		0			0
$367$		−	599.158i		−	1.63258i	−0.577641	−	0.816291i	$-0.696026\pi$
$367$		−	599.158i		−	1.63258i	0.577641	−	0.816291i	$-0.303974\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	58.6716i		−	0.158144i
$372$		0			0
$373$		−	24.9992i		−	0.0670220i	−0.999438	−	0.0335110i	$-0.989331\pi$
$373$		−	24.9992i		−	0.0670220i	0.999438	−	0.0335110i	$-0.0106689\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−289.025			−0.766644
$378$		0			0
$379$	−201.602			−0.531931			−0.265966	−	0.963983i	$-0.585691\pi$
$379$	−201.602			−0.531931			−0.265966	+	0.963983i	$0.585691\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	123.029			0.321226			0.160613	−	0.987017i	$-0.448653\pi$
$383$	123.029			0.321226			0.160613	+	0.987017i	$0.448653\pi$
$384$		0			0
$385$	−98.7788	+	210.429i	−0.256568	+	0.546569i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	720.820i		−	1.85301i	−0.376287	−	0.926503i	$-0.622799\pi$
$389$		−	720.820i		−	1.85301i	0.376287	−	0.926503i	$-0.377201\pi$
$390$		0			0
$391$	159.351			0.407547
$392$		0			0
$393$		0			0
$394$		0			0
$395$	51.2923	−	109.268i	0.129854	−	0.276629i
$396$		0			0
$397$			164.898i			0.415360i	0.978197	+	0.207680i	$0.0665913\pi$
$397$			164.898i			0.415360i	−0.978197	+	0.207680i	$0.933409\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			183.334i			0.457192i	0.973521	+	0.228596i	$0.0734134\pi$
$401$			183.334i			0.457192i	−0.973521	+	0.228596i	$0.926587\pi$
$402$		0			0
$403$			694.750i			1.72395i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−15.0416			−0.0369572
$408$		0			0
$409$	105.873			0.258857			0.129429	−	0.991589i	$-0.458686\pi$
$409$	105.873			0.258857			0.129429	+	0.991589i	$0.458686\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−227.053			−0.549766
$414$		0			0
$415$	−115.021	+	245.030i	−0.277159	+	0.590433i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			167.717i			0.400280i	0.979767	+	0.200140i	$0.0641397\pi$
$419$			167.717i			0.400280i	−0.979767	+	0.200140i	$0.935860\pi$
$420$		0			0
$421$	−38.9554			−0.0925307			−0.0462653	−	0.998929i	$-0.514732\pi$
$421$	−38.9554			−0.0925307			−0.0462653	+	0.998929i	$0.514732\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−76.1819	−	91.7363i	−0.179251	−	0.215850i
$426$		0			0
$427$		−	437.659i		−	1.02496i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			649.135i			1.50611i	0.657955	+	0.753057i	$0.271422\pi$
$431$			649.135i			1.50611i	−0.657955	+	0.753057i	$0.728578\pi$
$432$		0			0
$433$			805.930i			1.86127i	0.365950	+	0.930635i	$0.380744\pi$
$433$			805.930i			1.86127i	−0.365950	+	0.930635i	$0.619256\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−276.739			−0.633270
$438$		0			0
$439$	695.630			1.58458			0.792289	−	0.610146i	$-0.208889\pi$
$439$	695.630			1.58458			0.792289	+	0.610146i	$0.208889\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−633.721			−1.43052			−0.715261	−	0.698858i	$-0.753692\pi$
$443$	−633.721			−1.43052			−0.715261	+	0.698858i	$0.753692\pi$
$444$		0			0
$445$	−86.9673	−	40.8238i	−0.195432	−	0.0917390i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			525.671i			1.17076i	0.810760	+	0.585379i	$0.199054\pi$
$449$			525.671i			1.17076i	−0.810760	+	0.585379i	$0.800946\pi$
$450$		0			0
$451$	−3.20795			−0.00711296
$452$		0			0
$453$		0			0
$454$		0			0
$455$	115.804	−	246.699i	0.254515	−	0.542196i
$456$		0			0
$457$			119.625i			0.261762i	0.991398	+	0.130881i	$0.0417805\pi$
$457$			119.625i			0.261762i	−0.991398	+	0.130881i	$0.958219\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	799.662i		−	1.73463i	−0.497764	−	0.867313i	$-0.665845\pi$
$461$		−	799.662i		−	1.73463i	0.497764	−	0.867313i	$-0.334155\pi$
$462$		0			0
$463$		−	246.612i		−	0.532640i	−0.963885	−	0.266320i	$-0.914192\pi$
$463$		−	246.612i		−	0.532640i	0.963885	−	0.266320i	$-0.0858078\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15.9382			0.0341288			0.0170644	−	0.999854i	$-0.494568\pi$
$467$	15.9382			0.0341288			0.0170644	+	0.999854i	$0.494568\pi$
$468$		0			0
$469$	248.298			0.529419
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−59.6558			−0.126122
$474$		0			0
$475$	132.303	+	159.315i	0.278532	+	0.335401i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	106.512i		−	0.222363i	−0.993800	−	0.111182i	$-0.964536\pi$
$479$		−	106.512i		−	0.222363i	0.993800	−	0.111182i	$-0.0354635\pi$
$480$		0			0
$481$	17.6342			0.0366615
$482$		0			0
$483$		0			0
$484$		0			0
$485$	813.020	+	381.645i	1.67633	+	0.786897i
$486$		0			0
$487$		−	51.8946i		−	0.106560i	−0.998580	−	0.0532798i	$-0.983032\pi$
$487$		−	51.8946i		−	0.106560i	0.998580	−	0.0532798i	$-0.0169675\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			169.261i			0.344728i	0.985033	+	0.172364i	$0.0551405\pi$
$491$			169.261i			0.344728i	−0.985033	+	0.172364i	$0.944860\pi$
$492$		0			0
$493$		−	101.739i		−	0.206367i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−256.047			−0.515186
$498$		0			0
$499$	512.534			1.02712			0.513561	−	0.858053i	$-0.328326\pi$
$499$	512.534			1.02712			0.513561	+	0.858053i	$0.328326\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	451.842			0.898295			0.449148	−	0.893458i	$-0.351728\pi$
$503$	451.842			0.898295			0.449148	+	0.893458i	$0.351728\pi$
$504$		0			0
$505$	741.702	+	348.167i	1.46872	+	0.689439i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			847.482i			1.66499i	0.554029	+	0.832497i	$0.313090\pi$
$509$			847.482i			1.66499i	−0.554029	+	0.832497i	$0.686910\pi$
$510$		0			0
$511$	12.1986			0.0238721
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−562.576	−	264.082i	−1.09238	−	0.512781i
$516$		0			0
$517$		−	577.265i		−	1.11657i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			26.8677i			0.0515694i	0.999668	+	0.0257847i	$0.00820843\pi$
$521$			26.8677i			0.0515694i	−0.999668	+	0.0257847i	$0.991792\pi$
$522$		0			0
$523$			645.180i			1.23361i	0.787115	+	0.616806i	$0.211574\pi$
$523$			645.180i			1.23361i	−0.787115	+	0.616806i	$0.788426\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−244.557			−0.464056
$528$		0			0
$529$	587.125			1.10988
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.76087			0.00705604
$534$		0			0
$535$	−205.439	+	437.647i	−0.383998	+	0.818033i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			379.331i			0.703768i
$540$		0			0
$541$	−733.306			−1.35546			−0.677732	−	0.735309i	$-0.737037\pi$
$541$	−733.306			−1.35546			−0.677732	+	0.735309i	$0.737037\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−398.909	+	849.799i	−0.731944	+	1.55926i
$546$		0			0
$547$			396.610i			0.725063i	0.931971	+	0.362532i	$0.118088\pi$
$547$			396.610i			0.725063i	−0.931971	+	0.362532i	$0.881912\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			176.687i			0.320665i
$552$		0			0
$553$			97.1093i			0.175604i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	439.875			0.789722			0.394861	−	0.918741i	$-0.370793\pi$
$557$	439.875			0.789722			0.394861	+	0.918741i	$0.370793\pi$
$558$		0			0
$559$	69.9382			0.125113
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−631.764			−1.12214			−0.561069	−	0.827769i	$-0.689610\pi$
$563$	−631.764			−1.12214			−0.561069	+	0.827769i	$0.689610\pi$
$564$		0			0
$565$	−151.561	+	322.872i	−0.268250	+	0.571454i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			874.810i			1.53745i	0.639579	+	0.768726i	$0.279109\pi$
$569$			874.810i			1.53745i	−0.639579	+	0.768726i	$0.720891\pi$
$570$		0			0
$571$	167.804			0.293877			0.146938	−	0.989146i	$-0.453058\pi$
$571$	167.804			0.293877			0.146938	+	0.989146i	$0.453058\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−533.593	−	642.539i	−0.927988	−	1.11746i
$576$		0			0
$577$		−	706.060i		−	1.22367i	−0.790984	−	0.611837i	$-0.790431\pi$
$577$		−	706.060i		−	1.22367i	0.790984	−	0.611837i	$-0.209569\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	217.764i		−	0.374808i
$582$		0			0
$583$			168.585i			0.289168i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−540.407			−0.920626			−0.460313	−	0.887757i	$-0.652263\pi$
$587$	−540.407			−0.920626			−0.460313	+	0.887757i	$0.652263\pi$
$588$		0			0
$589$	424.714			0.721077
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−605.315			−1.02077			−0.510384	−	0.859947i	$-0.670497\pi$
$593$	−605.315			−1.02077			−0.510384	+	0.859947i	$0.670497\pi$
$594$		0			0
$595$	86.8399	+	40.7640i	0.145949	+	0.0685110i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			599.687i			1.00115i	0.865694	+	0.500574i	$0.166878\pi$
$599$			599.687i			1.00115i	−0.865694	+	0.500574i	$0.833122\pi$
$600$		0			0
$601$	304.427			0.506533			0.253267	−	0.967397i	$-0.418495\pi$
$601$	304.427			0.506533			0.253267	+	0.967397i	$0.418495\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	26.7459	−	56.9769i	0.0442081	−	0.0941767i
$606$		0			0
$607$		−	160.000i		−	0.263592i	−0.991277	−	0.131796i	$-0.957926\pi$
$607$		−	160.000i		−	0.263592i	0.991277	−	0.131796i	$-0.0420744\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			676.763i			1.10763i
$612$		0			0
$613$		−	18.0851i		−	0.0295026i	−0.999891	−	0.0147513i	$-0.995304\pi$
$613$		−	18.0851i		−	0.0295026i	0.999891	−	0.0147513i	$-0.00469565\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1058.31			−1.71525			−0.857625	−	0.514276i	$-0.828061\pi$
$617$	−1058.31			−1.71525			−0.857625	+	0.514276i	$0.828061\pi$
$618$		0			0
$619$	−981.467			−1.58557			−0.792784	−	0.609502i	$-0.791369\pi$
$619$	−981.467			−1.58557			−0.792784	+	0.609502i	$0.791369\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	77.2898			0.124061
$624$		0			0
$625$	−114.803	+	614.366i	−0.183685	+	0.982985i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			6.20736i			0.00986862i
$630$		0			0
$631$	−635.232			−1.00671			−0.503353	−	0.864081i	$-0.667900\pi$
$631$	−635.232			−1.00671			−0.503353	+	0.864081i	$0.667900\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	639.870	+	300.365i	1.00767	+	0.473016i
$636$		0			0
$637$		−	444.713i		−	0.698137i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	659.039i		−	1.02814i	−0.857748	−	0.514071i	$-0.828137\pi$
$641$		−	659.039i		−	1.02814i	0.857748	−	0.514071i	$-0.171863\pi$
$642$		0			0
$643$		−	75.0317i		−	0.116690i	−0.998296	−	0.0583450i	$-0.981418\pi$
$643$		−	75.0317i		−	0.116690i	0.998296	−	0.0583450i	$-0.0185823\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−913.249			−1.41151			−0.705756	−	0.708455i	$-0.749393\pi$
$647$	−913.249			−1.41151			−0.705756	+	0.708455i	$0.749393\pi$
$648$		0			0
$649$	652.407			1.00525
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1028.62			1.57522			0.787609	−	0.616176i	$-0.211319\pi$
$653$	1028.62			1.57522			0.787609	+	0.616176i	$0.211319\pi$
$654$		0			0
$655$	−979.092	−	459.602i	−1.49480	−	0.701682i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	755.002i		−	1.14568i	−0.819668	−	0.572839i	$-0.805842\pi$
$659$		−	755.002i		−	1.14568i	0.819668	−	0.572839i	$-0.194158\pi$
$660$		0			0
$661$	−57.9908			−0.0877319			−0.0438660	−	0.999037i	$-0.513967\pi$
$661$	−57.9908			−0.0877319			−0.0438660	+	0.999037i	$0.513967\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−150.812	−	70.7935i	−0.226785	−	0.106456i
$666$		0			0
$667$		−	712.600i		−	1.06837i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1257.55i			1.87415i
$672$		0			0
$673$		−	389.002i		−	0.578011i	−0.957327	−	0.289006i	$-0.906675\pi$
$673$		−	389.002i		−	0.578011i	0.957327	−	0.289006i	$-0.0933247\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−739.927			−1.09295			−0.546475	−	0.837475i	$-0.684031\pi$
$677$	−739.927			−1.09295			−0.546475	+	0.837475i	$0.684031\pi$
$678$		0			0
$679$	−722.550			−1.06414
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1098.80			1.60878			0.804392	−	0.594100i	$-0.202491\pi$
$683$	1098.80			1.60878			0.804392	+	0.594100i	$0.202491\pi$
$684$		0			0
$685$	251.390	−	535.537i	0.366992	−	0.781805i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	197.642i		−	0.286854i
$690$		0			0
$691$	−138.645			−0.200644			−0.100322	−	0.994955i	$-0.531987\pi$
$691$	−138.645			−0.200644			−0.100322	+	0.994955i	$0.531987\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	231.115	−	492.346i	0.332540	−	0.708412i
$696$		0			0
$697$			1.32385i			0.00189936i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	2.42379i		−	0.00345762i	−0.999999	−	0.00172881i	$-0.999450\pi$
$701$		−	2.42379i		−	0.00345762i	0.999999	−	0.00172881i	$-0.000550298\pi$
$702$		0			0
$703$		−	10.7801i		−	0.0153344i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−659.167			−0.932344
$708$		0			0
$709$	64.3983			0.0908298			0.0454149	−	0.998968i	$-0.485539\pi$
$709$	64.3983			0.0908298			0.0454149	+	0.998968i	$0.485539\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1712.93			−2.40242
$714$		0			0
$715$	−332.748	+	708.856i	−0.465382	+	0.991407i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	129.041i		−	0.179473i	−0.995966	−	0.0897363i	$-0.971398\pi$
$719$		−	129.041i		−	0.179473i	0.995966	−	0.0897363i	$-0.0286024\pi$
$720$		0			0
$721$	499.974			0.693446
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−410.235	+	340.677i	−0.565841	+	0.469900i
$726$		0			0
$727$		−	380.981i		−	0.524045i	−0.965062	−	0.262023i	$-0.915611\pi$
$727$		−	380.981i		−	0.524045i	0.965062	−	0.262023i	$-0.0843895\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			24.6188i			0.0336782i
$732$		0			0
$733$			155.618i			0.212303i	0.994350	+	0.106152i	$0.0338529\pi$
$733$			155.618i			0.212303i	−0.994350	+	0.106152i	$0.966147\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−713.449			−0.968045
$738$		0			0
$739$	569.361			0.770448			0.385224	−	0.922823i	$-0.374124\pi$
$739$	569.361			0.770448			0.385224	+	0.922823i	$0.374124\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1360.27			−1.83078			−0.915390	−	0.402569i	$-0.868117\pi$
$743$	−1360.27			−1.83078			−0.915390	+	0.402569i	$0.868117\pi$
$744$		0			0
$745$	−191.880	−	90.0715i	−0.257557	−	0.120901i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	388.947i		−	0.519289i
$750$		0			0
$751$	−1062.32			−1.41454			−0.707268	−	0.706946i	$-0.750072\pi$
$751$	−1062.32			−1.41454			−0.707268	+	0.706946i	$0.750072\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−291.905	+	621.847i	−0.386629	+	0.823638i
$756$		0			0
$757$		−	1289.84i		−	1.70389i	−0.523633	−	0.851944i	$-0.675424\pi$
$757$		−	1289.84i		−	1.70389i	0.523633	−	0.851944i	$-0.324576\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			88.5494i			0.116359i	0.998306	+	0.0581796i	$0.0185296\pi$
$761$			88.5494i			0.116359i	−0.998306	+	0.0581796i	$0.981470\pi$
$762$		0			0
$763$		−	755.236i		−	0.989824i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−764.857			−0.997206
$768$		0			0
$769$	−46.8810			−0.0609637			−0.0304818	−	0.999535i	$-0.509704\pi$
$769$	−46.8810			−0.0609637			−0.0304818	+	0.999535i	$0.509704\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−204.832			−0.264983			−0.132491	−	0.991184i	$-0.542298\pi$
$773$	−204.832			−0.264983			−0.132491	+	0.991184i	$0.542298\pi$
$774$		0			0
$775$	818.911	+	986.111i	1.05666	+	1.27240i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	2.29909i		−	0.00295134i
$780$		0			0
$781$	735.717			0.942020
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−365.790	−	171.708i	−0.465974	−	0.218736i
$786$		0			0
$787$		−	628.504i		−	0.798608i	−0.916819	−	0.399304i	$-0.869252\pi$
$787$		−	628.504i		−	0.798608i	0.916819	−	0.399304i	$-0.130748\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	286.943i		−	0.362760i
$792$		0			0
$793$		−	1474.31i		−	1.85915i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1005.12			1.26113			0.630566	−	0.776136i	$-0.282823\pi$
$797$	1005.12			1.26113			0.630566	+	0.776136i	$0.282823\pi$
$798$		0			0
$799$	−238.226			−0.298155
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−35.0511			−0.0436502
$804$		0			0
$805$	608.244	+	285.520i	0.755583	+	0.354683i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	20.1783i		−	0.0249423i	−0.999922	−	0.0124712i	$-0.996030\pi$
$809$		−	20.1783i		−	0.0249423i	0.999922	−	0.0124712i	$-0.00396980\pi$
$810$		0			0
$811$	−854.296			−1.05339			−0.526693	−	0.850055i	$-0.676568\pi$
$811$	−854.296			−1.05339			−0.526693	+	0.850055i	$0.676568\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	450.322	+	211.389i	0.552543	+	0.259373i
$816$		0			0
$817$		−	42.7546i		−	0.0523312i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			77.1306i			0.0939471i	0.998896	+	0.0469736i	$0.0149577\pi$
$821$			77.1306i			0.0939471i	−0.998896	+	0.0469736i	$0.985042\pi$
$822$		0			0
$823$			1548.24i			1.88122i	0.339493	+	0.940609i	$0.389745\pi$
$823$			1548.24i			1.88122i	−0.339493	+	0.940609i	$0.610255\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	28.6739			0.0346722			0.0173361	−	0.999850i	$-0.494481\pi$
$827$	28.6739			0.0346722			0.0173361	+	0.999850i	$0.494481\pi$
$828$		0			0
$829$	−467.578			−0.564026			−0.282013	−	0.959411i	$-0.591002\pi$
$829$	−467.578			−0.564026			−0.282013	+	0.959411i	$0.591002\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	156.542			0.187926
$834$		0			0
$835$	−420.492	+	895.777i	−0.503584	+	1.07279i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1277.59i		−	1.52275i	−0.648311	−	0.761376i	$-0.724524\pi$
$839$		−	1277.59i		−	1.52275i	0.648311	−	0.761376i	$-0.275476\pi$
$840$		0			0
$841$	386.034			0.459018
$842$		0			0
$843$		0			0
$844$		0			0
$845$	31.0371	−	66.1186i	0.0367303	−	0.0782468i
$846$		0			0
$847$			50.6367i			0.0597836i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			43.4776i			0.0510900i
$852$		0			0
$853$			1657.14i			1.94272i	0.237604	+	0.971362i	$0.423638\pi$
$853$			1657.14i			1.94272i	−0.237604	+	0.971362i	$0.576362\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−917.993			−1.07117			−0.535585	−	0.844481i	$-0.679909\pi$
$857$	−917.993			−1.07117			−0.535585	+	0.844481i	$0.679909\pi$
$858$		0			0
$859$	−1471.65			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$859$	−1471.65			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−847.452			−0.981984			−0.490992	−	0.871164i	$-0.663366\pi$
$863$	−847.452			−0.981984			−0.490992	+	0.871164i	$0.663366\pi$
$864$		0			0
$865$	−372.276	+	793.061i	−0.430376	+	0.916833i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	279.030i		−	0.321094i
$870$		0			0
$871$	836.420			0.960299
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−126.417	−	486.659i	−0.144477	−	0.556181i
$876$		0			0
$877$		−	960.589i		−	1.09531i	−0.836703	−	0.547656i	$-0.815520\pi$
$877$		−	960.589i		−	1.09531i	0.836703	−	0.547656i	$-0.184480\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1239.44i			1.40685i	0.710768	+	0.703427i	$0.248348\pi$
$881$			1239.44i			1.40685i	−0.710768	+	0.703427i	$0.751652\pi$
$882$		0			0
$883$			1081.77i			1.22511i	0.790430	+	0.612553i	$0.209857\pi$
$883$			1081.77i			1.22511i	−0.790430	+	0.612553i	$0.790143\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1493.65			1.68393			0.841967	−	0.539529i	$-0.181398\pi$
$887$	1493.65			1.68393			0.841967	+	0.539529i	$0.181398\pi$
$888$		0			0
$889$	−568.667			−0.639670
$890$		0			0
$891$		0			0
$892$		0			0
$893$	413.718			0.463291
$894$		0			0
$895$	9.20698	+	4.32191i	0.0102871	+	0.00482894i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1093.63i			1.21650i
$900$		0			0
$901$	69.5715			0.0772159
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−476.675	+	1015.46i	−0.526713	+	1.12206i
$906$		0			0
$907$		−	1331.28i		−	1.46779i	−0.679263	−	0.733895i	$-0.737701\pi$
$907$		−	1331.28i		−	1.46779i	0.679263	−	0.733895i	$-0.262299\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			926.885i			1.01744i	0.860933	+	0.508719i	$0.169881\pi$
$911$			926.885i			1.01744i	−0.860933	+	0.508719i	$0.830119\pi$
$912$		0			0
$913$			625.714i			0.685339i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	870.142			0.948900
$918$		0			0
$919$	130.701			0.142221			0.0711104	−	0.997468i	$-0.477346\pi$
$919$	130.701			0.142221			0.0711104	+	0.997468i	$0.477346\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−862.527			−0.934482
$924$		0			0
$925$	25.0295	−	20.7856i	0.0270589	−	0.0224709i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			514.471i			0.553790i	0.960900	+	0.276895i	$0.0893054\pi$
$929$			514.471i			0.553790i	−0.960900	+	0.276895i	$0.910695\pi$
$930$		0			0
$931$	−271.862			−0.292011
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−249.523	−	117.130i	−0.266869	−	0.125273i
$936$		0			0
$937$		−	1538.89i		−	1.64236i	−0.570670	−	0.821179i	$-0.693316\pi$
$937$		−	1538.89i		−	1.64236i	0.570670	−	0.821179i	$-0.306684\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	832.549i		−	0.884749i	−0.896830	−	0.442374i	$-0.854136\pi$
$941$		−	832.549i		−	0.884749i	0.896830	−	0.442374i	$-0.145864\pi$
$942$		0			0
$943$			9.27255i			0.00983303i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1196.52			−1.26349			−0.631743	−	0.775178i	$-0.717660\pi$
$947$	−1196.52			−1.26349			−0.631743	+	0.775178i	$0.717660\pi$
$948$		0			0
$949$	41.0926			0.0433009
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−405.006			−0.424980			−0.212490	−	0.977163i	$-0.568157\pi$
$953$	−405.006			−0.424980			−0.212490	+	0.977163i	$0.568157\pi$
$954$		0			0
$955$	659.417	+	309.541i	0.690489	+	0.324127i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			475.944i			0.496292i
$960$		0			0
$961$	1667.85			1.73553
$962$		0			0
$963$		0			0
$964$		0			0
$965$	948.383	+	445.187i	0.982781	+	0.461333i
$966$		0			0
$967$		−	1461.35i		−	1.51122i	−0.655019	−	0.755612i	$-0.727339\pi$
$967$		−	1461.35i		−	1.51122i	0.655019	−	0.755612i	$-0.272661\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			987.845i			1.01735i	0.860959	+	0.508674i	$0.169864\pi$
$971$			987.845i			1.01735i	−0.860959	+	0.508674i	$0.830136\pi$
$972$		0			0
$973$			437.559i			0.449701i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1032.47			−1.05677			−0.528386	−	0.849004i	$-0.677203\pi$
$977$	−1032.47			−1.05677			−0.528386	+	0.849004i	$0.677203\pi$
$978$		0			0
$979$	−222.082			−0.226845
$980$		0			0
$981$		0			0
$982$		0			0
$983$	254.697			0.259102			0.129551	−	0.991573i	$-0.458646\pi$
$983$	254.697			0.259102			0.129551	+	0.991573i	$0.458646\pi$
$984$		0			0
$985$	−632.627	+	1347.69i	−0.642261	+	1.36821i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			172.435i			0.174353i
$990$		0			0
$991$	275.074			0.277572			0.138786	−	0.990322i	$-0.455680\pi$
$991$	275.074			0.277572			0.138786	+	0.990322i	$0.455680\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	53.1615	−	113.250i	0.0534286	−	0.113819i
$996$		0			0
$997$		−	139.811i		−	0.140232i	−0.997539	−	0.0701161i	$-0.977663\pi$
$997$		−	139.811i		−	0.140232i	0.997539	−	0.0701161i	$-0.0223370\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.q.1889.8		24
3.2	odd	2	inner	2160.3.c.q.1889.17		24
4.3	odd	2		1080.3.c.c.809.8	yes	24
5.4	even	2	inner	2160.3.c.q.1889.18		24
12.11	even	2		1080.3.c.c.809.17	yes	24
15.14	odd	2	inner	2160.3.c.q.1889.7		24
20.19	odd	2		1080.3.c.c.809.18	yes	24
60.59	even	2		1080.3.c.c.809.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.3.c.c.809.7	✓	24	60.59	even	2
1080.3.c.c.809.8	yes	24	4.3	odd	2
1080.3.c.c.809.17	yes	24	12.11	even	2
1080.3.c.c.809.18	yes	24	20.19	odd	2
2160.3.c.q.1889.7		24	15.14	odd	2	inner
2160.3.c.q.1889.8		24	1.1	even	1	trivial
2160.3.c.q.1889.17		24	3.2	odd	2	inner
2160.3.c.q.1889.18		24	5.4	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.12464 + 4.52614i) q^{5} -4.02248i q^{7} +11.5580i q^{11} -13.5502i q^{13} +4.76977 q^{17} -8.28351 q^{19} +33.4085 q^{23} +(-15.9718 - 19.2328i) q^{25} -21.3299i q^{29} -51.2723 q^{31} +(18.2063 + 8.54633i) q^{35} +1.30139i q^{37} +0.277551i q^{41} +5.16141i q^{43} -49.9448 q^{47} +32.8197 q^{49} +14.5859 q^{53} +(-52.3133 - 24.5567i) q^{55} -56.4461i q^{59} +108.803 q^{61} +(61.3301 + 28.7893i) q^{65} +61.7275i q^{67} -63.6541i q^{71} +3.03261i q^{73} +46.4920 q^{77} -24.1416 q^{79} +54.1367 q^{83} +(-10.1341 + 21.5886i) q^{85} +19.2145i q^{89} -54.5054 q^{91} +(17.5995 - 37.4923i) q^{95} -179.628i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100}) \) 24 * q - 72 * q^25 + 72 * q^31 - 408 * q^49 + 168 * q^55 - 240 * q^61 + 312 * q^79 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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