LMFDB - Embedded newform 2160.3.c.q.1889.24

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			1889.24
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.q.1889.23

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.67464 + 1.77419i) q^{5} -9.67132i q^{7} -8.58075i q^{11} -19.0519i q^{13} +26.2977 q^{17} -26.6206 q^{19} +25.8661 q^{23} +(18.7045 + 16.5874i) q^{25} -18.3616i q^{29} +7.56234 q^{31} +(17.1588 - 45.2099i) q^{35} +47.0280i q^{37} +0.602357i q^{41} -62.2825i q^{43} -79.2020 q^{47} -44.5345 q^{49} -80.9629 q^{53} +(15.2239 - 40.1119i) q^{55} -15.4109i q^{59} -62.7576 q^{61} +(33.8017 - 89.0605i) q^{65} -48.3536i q^{67} +99.6423i q^{71} +60.7117i q^{73} -82.9872 q^{77} -60.8209 q^{79} +77.0753 q^{83} +(122.932 + 46.6572i) q^{85} +28.7841i q^{89} -184.257 q^{91} +(-124.442 - 47.2300i) q^{95} +23.3070i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100})$ 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
$5$	4.67464	+	1.77419i	0.934928	+	0.354839i
$5$	4.67464	+	1.77419i	0.934928	+	0.354839i
$6$		0			0
$7$		−	9.67132i		−	1.38162i	−0.723037	−	0.690809i	$-0.757255\pi$
$7$		−	9.67132i		−	1.38162i	0.723037	−	0.690809i	$-0.242745\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	8.58075i		−	0.780068i	−0.920801	−	0.390034i	$-0.872463\pi$
$11$		−	8.58075i		−	0.780068i	0.920801	−	0.390034i	$-0.127537\pi$
$12$		0			0
$13$		−	19.0519i		−	1.46553i	−0.680483	−	0.732763i	$-0.738230\pi$
$13$		−	19.0519i		−	1.46553i	0.680483	−	0.732763i	$-0.261770\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.2977			1.54692			0.773462	−	0.633843i	$-0.218523\pi$
$17$	26.2977			1.54692			0.773462	+	0.633843i	$0.218523\pi$
$18$		0			0
$19$	−26.6206			−1.40108			−0.700541	−	0.713612i	$-0.747058\pi$
$19$	−26.6206			−1.40108			−0.700541	+	0.713612i	$0.747058\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	25.8661			1.12462			0.562308	−	0.826928i	$-0.309914\pi$
$23$	25.8661			1.12462			0.562308	+	0.826928i	$0.309914\pi$
$24$		0			0
$25$	18.7045	+	16.5874i	0.748179	+	0.663497i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	18.3616i		−	0.633158i	−0.948566	−	0.316579i	$-0.897466\pi$
$29$		−	18.3616i		−	0.633158i	0.948566	−	0.316579i	$-0.102534\pi$
$30$		0			0
$31$	7.56234			0.243946			0.121973	−	0.992533i	$-0.461078\pi$
$31$	7.56234			0.243946			0.121973	+	0.992533i	$0.461078\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	17.1588	−	45.2099i	0.490251	−	1.29171i
$36$		0			0
$37$			47.0280i			1.27103i	0.772090	+	0.635513i	$0.219211\pi$
$37$			47.0280i			1.27103i	−0.772090	+	0.635513i	$0.780789\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.602357i			0.0146916i	0.999973	+	0.00734582i	$0.00233827\pi$
$41$			0.602357i			0.0146916i	−0.999973	+	0.00734582i	$0.997662\pi$
$42$		0			0
$43$		−	62.2825i		−	1.44843i	−0.689574	−	0.724215i	$-0.742202\pi$
$43$		−	62.2825i		−	1.44843i	0.689574	−	0.724215i	$-0.257798\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−79.2020			−1.68515			−0.842574	−	0.538580i	$-0.818961\pi$
$47$	−79.2020			−1.68515			−0.842574	+	0.538580i	$0.818961\pi$
$48$		0			0
$49$	−44.5345			−0.908868
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−80.9629			−1.52760			−0.763801	−	0.645452i	$-0.776669\pi$
$53$	−80.9629			−1.52760			−0.763801	+	0.645452i	$0.776669\pi$
$54$		0			0
$55$	15.2239	−	40.1119i	0.276798	−	0.729307i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	15.4109i		−	0.261202i	−0.991435	−	0.130601i	$-0.958309\pi$
$59$		−	15.4109i		−	0.261202i	0.991435	−	0.130601i	$-0.0416907\pi$
$60$		0			0
$61$	−62.7576			−1.02881			−0.514407	−	0.857546i	$-0.671988\pi$
$61$	−62.7576			−1.02881			−0.514407	+	0.857546i	$0.671988\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	33.8017	−	89.0605i	0.520025	−	1.37016i
$66$		0			0
$67$		−	48.3536i		−	0.721696i	−0.932625	−	0.360848i	$-0.882487\pi$
$67$		−	48.3536i		−	0.721696i	0.932625	−	0.360848i	$-0.117513\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			99.6423i			1.40341i	0.712466	+	0.701706i	$0.247578\pi$
$71$			99.6423i			1.40341i	−0.712466	+	0.701706i	$0.752422\pi$
$72$		0			0
$73$			60.7117i			0.831667i	0.909441	+	0.415834i	$0.136510\pi$
$73$			60.7117i			0.831667i	−0.909441	+	0.415834i	$0.863490\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−82.9872			−1.07776
$78$		0			0
$79$	−60.8209			−0.769884			−0.384942	−	0.922941i	$-0.625779\pi$
$79$	−60.8209			−0.769884			−0.384942	+	0.922941i	$0.625779\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	77.0753			0.928618			0.464309	−	0.885673i	$-0.346303\pi$
$83$	77.0753			0.928618			0.464309	+	0.885673i	$0.346303\pi$
$84$		0			0
$85$	122.932	+	46.6572i	1.44626	+	0.548908i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			28.7841i			0.323417i	0.986839	+	0.161708i	$0.0517004\pi$
$89$			28.7841i			0.323417i	−0.986839	+	0.161708i	$0.948300\pi$
$90$		0			0
$91$	−184.257			−2.02480
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−124.442	−	47.2300i	−1.30991	−	0.497158i
$96$		0			0
$97$			23.3070i			0.240279i	0.992757	+	0.120139i	$0.0383342\pi$
$97$			23.3070i			0.240279i	−0.992757	+	0.120139i	$0.961666\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	114.136i		−	1.13006i	−0.825071	−	0.565029i	$-0.808865\pi$
$101$		−	114.136i		−	1.13006i	0.825071	−	0.565029i	$-0.191135\pi$
$102$		0			0
$103$		−	166.463i		−	1.61614i	−0.589085	−	0.808071i	$-0.700511\pi$
$103$		−	166.463i		−	1.61614i	0.589085	−	0.808071i	$-0.299489\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.42346			−0.0506865			−0.0253433	−	0.999679i	$-0.508068\pi$
$107$	−5.42346			−0.0506865			−0.0253433	+	0.999679i	$0.508068\pi$
$108$		0			0
$109$	201.133			1.84525			0.922627	−	0.385694i	$-0.126038\pi$
$109$	201.133			1.84525			0.922627	+	0.385694i	$0.126038\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−54.7095			−0.484155			−0.242077	−	0.970257i	$-0.577829\pi$
$113$	−54.7095			−0.484155			−0.242077	+	0.970257i	$0.577829\pi$
$114$		0			0
$115$	120.915	+	45.8915i	1.05143	+	0.399057i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	254.334i		−	2.13726i
$120$		0			0
$121$	47.3708			0.391494
$122$		0			0
$123$		0			0
$124$		0			0
$125$	58.0074	+	110.726i	0.464059	+	0.885804i
$126$		0			0
$127$			52.7864i			0.415641i	0.978167	+	0.207820i	$0.0666370\pi$
$127$			52.7864i			0.415641i	−0.978167	+	0.207820i	$0.933363\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	162.244i		−	1.23850i	−0.785193	−	0.619251i	$-0.787436\pi$
$131$		−	162.244i		−	1.23850i	0.785193	−	0.619251i	$-0.212564\pi$
$132$		0			0
$133$			257.456i			1.93576i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−104.273			−0.761115			−0.380557	−	0.924757i	$-0.624268\pi$
$137$	−104.273			−0.761115			−0.380557	+	0.924757i	$0.624268\pi$
$138$		0			0
$139$	135.011			0.971303			0.485652	−	0.874152i	$-0.338582\pi$
$139$	135.011			0.971303			0.485652	+	0.874152i	$0.338582\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−163.479			−1.14321
$144$		0			0
$145$	32.5770	−	85.8337i	0.224669	−	0.591957i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	286.649i		−	1.92382i	−0.273374	−	0.961908i	$-0.588140\pi$
$149$		−	286.649i		−	1.92382i	0.273374	−	0.961908i	$-0.411860\pi$
$150$		0			0
$151$	2.67570			0.0177199			0.00885994	−	0.999961i	$-0.497180\pi$
$151$	2.67570			0.0177199			0.00885994	+	0.999961i	$0.497180\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	35.3512	+	13.4170i	0.228072	+	0.0865616i
$156$		0			0
$157$			290.335i			1.84927i	0.380860	+	0.924633i	$0.375628\pi$
$157$			290.335i			1.84927i	−0.380860	+	0.924633i	$0.624372\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	250.160i		−	1.55379i
$162$		0			0
$163$			187.526i			1.15047i	0.817989	+	0.575234i	$0.195089\pi$
$163$			187.526i			1.15047i	−0.817989	+	0.575234i	$0.804911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−78.3652			−0.469253			−0.234626	−	0.972086i	$-0.575387\pi$
$167$	−78.3652			−0.469253			−0.234626	+	0.972086i	$0.575387\pi$
$168$		0			0
$169$	−193.973			−1.14777
$170$		0			0
$171$		0			0
$172$		0			0
$173$	227.871			1.31717			0.658586	−	0.752505i	$-0.271155\pi$
$173$	227.871			1.31717			0.658586	+	0.752505i	$0.271155\pi$
$174$		0			0
$175$	160.422	−	180.897i	0.916699	−	1.03370i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	89.2278i		−	0.498479i	−0.968442	−	0.249240i	$-0.919819\pi$
$179$		−	89.2278i		−	0.498479i	0.968442	−	0.249240i	$-0.0801807\pi$
$180$		0			0
$181$	−24.5898			−0.135856			−0.0679278	−	0.997690i	$-0.521639\pi$
$181$	−24.5898			−0.135856			−0.0679278	+	0.997690i	$0.521639\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−83.4367	+	219.839i	−0.451009	+	1.18832i
$186$		0			0
$187$		−	225.654i		−	1.20671i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	28.7155i		−	0.150343i	−0.997171	−	0.0751714i	$-0.976050\pi$
$191$		−	28.7155i		−	0.150343i	0.997171	−	0.0751714i	$-0.0239504\pi$
$192$		0			0
$193$		−	224.729i		−	1.16440i	−0.813046	−	0.582200i	$-0.802192\pi$
$193$		−	224.729i		−	1.16440i	0.813046	−	0.582200i	$-0.197808\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−109.880			−0.557766			−0.278883	−	0.960325i	$-0.589964\pi$
$197$	−109.880			−0.557766			−0.278883	+	0.960325i	$0.589964\pi$
$198$		0			0
$199$	−84.1079			−0.422653			−0.211326	−	0.977416i	$-0.567778\pi$
$199$	−84.1079			−0.422653			−0.211326	+	0.977416i	$0.567778\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−177.581			−0.874782
$204$		0			0
$205$	−1.06870	+	2.81580i	−0.00521316	+	0.0137356i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			228.424i			1.09294i
$210$		0			0
$211$	−58.6451			−0.277939			−0.138970	−	0.990297i	$-0.544379\pi$
$211$	−58.6451			−0.277939			−0.138970	+	0.990297i	$0.544379\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	110.501	−	291.148i	0.513959	−	1.35418i
$216$		0			0
$217$		−	73.1378i		−	0.337041i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	501.020i		−	2.26706i
$222$		0			0
$223$			207.329i			0.929727i	0.885382	+	0.464864i	$0.153897\pi$
$223$			207.329i			0.929727i	−0.885382	+	0.464864i	$0.846103\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	279.237			1.23012			0.615060	−	0.788480i	$-0.289132\pi$
$227$	279.237			1.23012			0.615060	+	0.788480i	$0.289132\pi$
$228$		0			0
$229$	133.801			0.584283			0.292142	−	0.956375i	$-0.405632\pi$
$229$	133.801			0.584283			0.292142	+	0.956375i	$0.405632\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.8561			0.0508843			0.0254422	−	0.999676i	$-0.491901\pi$
$233$	11.8561			0.0508843			0.0254422	+	0.999676i	$0.491901\pi$
$234$		0			0
$235$	−370.241	−	140.520i	−1.57549	−	0.597955i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			145.655i			0.609434i	0.952443	+	0.304717i	$0.0985619\pi$
$239$			145.655i			0.609434i	−0.952443	+	0.304717i	$0.901438\pi$
$240$		0			0
$241$	−104.021			−0.431621			−0.215810	−	0.976435i	$-0.569239\pi$
$241$	−104.021			−0.431621			−0.215810	+	0.976435i	$0.569239\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−208.183	−	79.0128i	−0.849725	−	0.322501i
$246$		0			0
$247$			507.171i			2.05332i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4.36967i		−	0.0174090i	−0.999962	−	0.00870452i	$-0.997229\pi$
$251$		−	4.36967i		−	0.0174090i	0.999962	−	0.00870452i	$-0.00277077\pi$
$252$		0			0
$253$		−	221.951i		−	0.877276i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	240.157			0.934463			0.467231	−	0.884135i	$-0.345252\pi$
$257$	240.157			0.934463			0.467231	+	0.884135i	$0.345252\pi$
$258$		0			0
$259$	454.823			1.75607
$260$		0			0
$261$		0			0
$262$		0			0
$263$	164.037			0.623715			0.311858	−	0.950129i	$-0.399049\pi$
$263$	164.037			0.623715			0.311858	+	0.950129i	$0.399049\pi$
$264$		0			0
$265$	−378.472	−	143.644i	−1.42820	−	0.542052i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	414.194i		−	1.53976i	−0.638192	−	0.769878i	$-0.720317\pi$
$269$		−	414.194i		−	1.53976i	0.638192	−	0.769878i	$-0.279683\pi$
$270$		0			0
$271$	−350.857			−1.29468			−0.647338	−	0.762203i	$-0.724118\pi$
$271$	−350.857			−1.29468			−0.647338	+	0.762203i	$0.724118\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	142.332	−	160.498i	0.517572	−	0.583631i
$276$		0			0
$277$		−	334.661i		−	1.20816i	−0.796923	−	0.604081i	$-0.793540\pi$
$277$		−	334.661i		−	1.20816i	0.796923	−	0.604081i	$-0.206460\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	452.747i		−	1.61120i	−0.592459	−	0.805600i	$-0.701843\pi$
$281$		−	452.747i		−	1.61120i	0.592459	−	0.805600i	$-0.298157\pi$
$282$		0			0
$283$		−	177.203i		−	0.626160i	−0.949727	−	0.313080i	$-0.898639\pi$
$283$		−	177.203i		−	0.626160i	0.949727	−	0.313080i	$-0.101361\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.82559			0.0202982
$288$		0			0
$289$	402.569			1.39297
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−371.342			−1.26738			−0.633690	−	0.773587i	$-0.718461\pi$
$293$	−371.342			−1.26738			−0.633690	+	0.773587i	$0.718461\pi$
$294$		0			0
$295$	27.3419	−	72.0405i	0.0926845	−	0.244205i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	492.798i		−	1.64815i
$300$		0			0
$301$	−602.354			−2.00118
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−293.369	−	111.344i	−0.961866	−	0.365063i
$306$		0			0
$307$			350.668i			1.14224i	0.820866	+	0.571120i	$0.193491\pi$
$307$			350.668i			1.14224i	−0.820866	+	0.571120i	$0.806509\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			314.727i			1.01198i	0.862538	+	0.505992i	$0.168874\pi$
$311$			314.727i			1.01198i	−0.862538	+	0.505992i	$0.831126\pi$
$312$		0			0
$313$		−	82.9434i		−	0.264995i	−0.991183	−	0.132497i	$-0.957700\pi$
$313$		−	82.9434i		−	0.264995i	0.991183	−	0.132497i	$-0.0422996\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	339.930			1.07234			0.536168	−	0.844111i	$-0.319871\pi$
$317$	339.930			1.07234			0.536168	+	0.844111i	$0.319871\pi$
$318$		0			0
$319$	−157.556			−0.493906
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−700.060			−2.16737
$324$		0			0
$325$	316.021	−	356.355i	0.972372	−	1.09648i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			765.988i			2.32823i
$330$		0			0
$331$	418.232			1.26354			0.631771	−	0.775155i	$-0.282328\pi$
$331$	418.232			1.26354			0.631771	+	0.775155i	$0.282328\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	85.7886	−	226.036i	0.256085	−	0.674733i
$336$		0			0
$337$			150.481i			0.446530i	0.974758	+	0.223265i	$0.0716715\pi$
$337$			150.481i			0.446530i	−0.974758	+	0.223265i	$0.928328\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	64.8905i		−	0.190295i
$342$		0			0
$343$		−	43.1872i		−	0.125910i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−576.039			−1.66005			−0.830027	−	0.557724i	$-0.811675\pi$
$347$	−576.039			−1.66005			−0.830027	+	0.557724i	$0.811675\pi$
$348$		0			0
$349$	−205.529			−0.588908			−0.294454	−	0.955666i	$-0.595138\pi$
$349$	−205.529			−0.588908			−0.294454	+	0.955666i	$0.595138\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	414.970			1.17555			0.587777	−	0.809023i	$-0.300003\pi$
$353$	414.970			1.17555			0.587777	+	0.809023i	$0.300003\pi$
$354$		0			0
$355$	−176.785	+	465.792i	−0.497985	+	1.31209i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	292.348i		−	0.814340i	−0.913352	−	0.407170i	$-0.866516\pi$
$359$		−	292.348i		−	0.814340i	0.913352	−	0.407170i	$-0.133484\pi$
$360$		0			0
$361$	347.655			0.963033
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−107.714	+	283.805i	−0.295108	+	0.777549i
$366$		0			0
$367$			130.948i			0.356807i	0.983957	+	0.178403i	$0.0570932\pi$
$367$			130.948i			0.356807i	−0.983957	+	0.178403i	$0.942907\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			783.019i			2.11056i
$372$		0			0
$373$		−	667.901i		−	1.79062i	−0.445443	−	0.895310i	$-0.646954\pi$
$373$		−	667.901i		−	1.79062i	0.445443	−	0.895310i	$-0.353046\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−349.822			−0.927910
$378$		0			0
$379$	136.623			0.360482			0.180241	−	0.983622i	$-0.442312\pi$
$379$	136.623			0.360482			0.180241	+	0.983622i	$0.442312\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	291.572			0.761285			0.380642	−	0.924722i	$-0.375703\pi$
$383$	291.572			0.761285			0.380642	+	0.924722i	$0.375703\pi$
$384$		0			0
$385$	−387.935	−	147.235i	−1.00762	−	0.382429i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			28.2735i			0.0726826i	0.999339	+	0.0363413i	$0.0115703\pi$
$389$			28.2735i			0.0726826i	−0.999339	+	0.0363413i	$0.988430\pi$
$390$		0			0
$391$	680.220			1.73969
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−284.315	−	107.908i	−0.719786	−	0.273185i
$396$		0			0
$397$		−	21.2201i		−	0.0534510i	−0.999643	−	0.0267255i	$-0.991492\pi$
$397$		−	21.2201i		−	0.0534510i	0.999643	−	0.0267255i	$-0.00850801\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			154.148i			0.384409i	0.981355	+	0.192205i	$0.0615637\pi$
$401$			154.148i			0.384409i	−0.981355	+	0.192205i	$0.938436\pi$
$402$		0			0
$403$		−	144.077i		−	0.357510i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	403.535			0.991487
$408$		0			0
$409$	−458.331			−1.12061			−0.560307	−	0.828285i	$-0.689317\pi$
$409$	−458.331			−1.12061			−0.560307	+	0.828285i	$0.689317\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−149.044			−0.360881
$414$		0			0
$415$	360.299	+	136.746i	0.868190	+	0.329509i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			102.481i			0.244586i	0.992494	+	0.122293i	$0.0390247\pi$
$419$			102.481i			0.244586i	−0.992494	+	0.122293i	$0.960975\pi$
$420$		0			0
$421$	−116.487			−0.276691			−0.138346	−	0.990384i	$-0.544178\pi$
$421$	−116.487			−0.276691			−0.138346	+	0.990384i	$0.544178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	491.885	+	436.211i	1.15738	+	1.02638i
$426$		0			0
$427$			606.949i			1.42143i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			146.831i			0.340675i	0.985386	+	0.170338i	$0.0544858\pi$
$431$			146.831i			0.340675i	−0.985386	+	0.170338i	$0.945514\pi$
$432$		0			0
$433$		−	7.56030i		−	0.0174603i	−0.999962	−	0.00873013i	$-0.997221\pi$
$433$		−	7.56030i		−	0.0174603i	0.999962	−	0.00873013i	$-0.00277892\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−688.572			−1.57568
$438$		0			0
$439$	−221.140			−0.503735			−0.251868	−	0.967762i	$-0.581045\pi$
$439$	−221.140			−0.503735			−0.251868	+	0.967762i	$0.581045\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−54.3553			−0.122698			−0.0613491	−	0.998116i	$-0.519540\pi$
$443$	−54.3553			−0.122698			−0.0613491	+	0.998116i	$0.519540\pi$
$444$		0			0
$445$	−51.0685	+	134.555i	−0.114761	+	0.302371i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			342.268i			0.762289i	0.924515	+	0.381144i	$0.124470\pi$
$449$			342.268i			0.762289i	−0.924515	+	0.381144i	$0.875530\pi$
$450$		0			0
$451$	5.16868			0.0114605
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−861.333	−	326.907i	−1.89304	−	0.718476i
$456$		0			0
$457$			577.506i			1.26369i	0.775095	+	0.631844i	$0.217702\pi$
$457$			577.506i			1.26369i	−0.775095	+	0.631844i	$0.782298\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	776.198i		−	1.68373i	−0.539690	−	0.841864i	$-0.681459\pi$
$461$		−	776.198i		−	1.68373i	0.539690	−	0.841864i	$-0.318541\pi$
$462$		0			0
$463$		−	526.497i		−	1.13714i	−0.822634	−	0.568571i	$-0.807496\pi$
$463$		−	526.497i		−	1.13714i	0.822634	−	0.568571i	$-0.192504\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	106.009			0.227000			0.113500	−	0.993538i	$-0.463794\pi$
$467$	106.009			0.227000			0.113500	+	0.993538i	$0.463794\pi$
$468$		0			0
$469$	−467.643			−0.997108
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−534.430			−1.12987
$474$		0			0
$475$	−497.924	−	441.567i	−1.04826	−	0.929614i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			106.127i			0.221560i	0.993845	+	0.110780i	$0.0353349\pi$
$479$			106.127i			0.221560i	−0.993845	+	0.110780i	$0.964665\pi$
$480$		0			0
$481$	895.970			1.86272
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−41.3512	+	108.952i	−0.0852602	+	0.224643i
$486$		0			0
$487$			461.009i			0.946631i	0.880893	+	0.473315i	$0.156943\pi$
$487$			461.009i			0.946631i	−0.880893	+	0.473315i	$0.843057\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			710.511i			1.44707i	0.690288	+	0.723534i	$0.257484\pi$
$491$			710.511i			1.44707i	−0.690288	+	0.723534i	$0.742516\pi$
$492$		0			0
$493$		−	482.867i		−	0.979447i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	963.673			1.93898
$498$		0			0
$499$	−103.124			−0.206662			−0.103331	−	0.994647i	$-0.532950\pi$
$499$	−103.124			−0.206662			−0.103331	+	0.994647i	$0.532950\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	862.704			1.71512			0.857558	−	0.514387i	$-0.171980\pi$
$503$	862.704			1.71512			0.857558	+	0.514387i	$0.171980\pi$
$504$		0			0
$505$	202.499	−	533.544i	0.400988	−	1.05652i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	443.439i		−	0.871196i	−0.900141	−	0.435598i	$-0.856537\pi$
$509$		−	443.439i		−	0.871196i	0.900141	−	0.435598i	$-0.143463\pi$
$510$		0			0
$511$	587.163			1.14905
$512$		0			0
$513$		0			0
$514$		0			0
$515$	295.337	−	778.153i	0.573470	−	1.51098i
$516$		0			0
$517$			679.612i			1.31453i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			602.433i			1.15630i	0.815930	+	0.578151i	$0.196226\pi$
$521$			602.433i			1.15630i	−0.815930	+	0.578151i	$0.803774\pi$
$522$		0			0
$523$			810.326i			1.54938i	0.632340	+	0.774691i	$0.282094\pi$
$523$			810.326i			1.54938i	−0.632340	+	0.774691i	$0.717906\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	198.872			0.377367
$528$		0			0
$529$	140.058			0.264759
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.4760			0.0215310
$534$		0			0
$535$	−25.3527	−	9.62226i	−0.0473882	−	0.0179855i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			382.139i			0.708978i
$540$		0			0
$541$	877.470			1.62194			0.810971	−	0.585087i	$-0.198940\pi$
$541$	877.470			1.62194			0.810971	+	0.585087i	$0.198940\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	940.222	+	356.848i	1.72518	+	0.654767i
$546$		0			0
$547$			627.998i			1.14808i	0.818828	+	0.574039i	$0.194624\pi$
$547$			627.998i			1.14808i	−0.818828	+	0.574039i	$0.805376\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			488.796i			0.887107i
$552$		0			0
$553$			588.218i			1.06369i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	195.072			0.350219			0.175109	−	0.984549i	$-0.443972\pi$
$557$	195.072			0.350219			0.175109	+	0.984549i	$0.443972\pi$
$558$		0			0
$559$	−1186.60			−2.12271
$560$		0			0
$561$		0			0
$562$		0			0
$563$	831.859			1.47755			0.738774	−	0.673953i	$-0.235405\pi$
$563$	831.859			1.47755			0.738774	+	0.673953i	$0.235405\pi$
$564$		0			0
$565$	−255.747	−	97.0652i	−0.452650	−	0.171797i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	555.389i		−	0.976078i	−0.872822	−	0.488039i	$-0.837712\pi$
$569$		−	555.389i		−	0.976078i	0.872822	−	0.488039i	$-0.162288\pi$
$570$		0			0
$571$	−453.650			−0.794484			−0.397242	−	0.917714i	$-0.630033\pi$
$571$	−453.650			−0.794484			−0.397242	+	0.917714i	$0.630033\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	483.813	+	429.053i	0.841414	+	0.746178i
$576$		0			0
$577$			103.410i			0.179220i	0.995977	+	0.0896098i	$0.0285620\pi$
$577$			103.410i			0.179220i	−0.995977	+	0.0896098i	$0.971438\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	745.420i		−	1.28299i
$582$		0			0
$583$			694.722i			1.19163i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	851.682			1.45091			0.725453	−	0.688272i	$-0.241630\pi$
$587$	851.682			1.45091			0.725453	+	0.688272i	$0.241630\pi$
$588$		0			0
$589$	−201.314			−0.341789
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1032.74			1.74155			0.870775	−	0.491681i	$-0.163617\pi$
$593$	1032.74			1.74155			0.870775	+	0.491681i	$0.163617\pi$
$594$		0			0
$595$	451.237	−	1188.92i	0.758381	−	1.99818i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	710.247i		−	1.18572i	−0.805305	−	0.592861i	$-0.797998\pi$
$599$		−	710.247i		−	1.18572i	0.805305	−	0.592861i	$-0.202002\pi$
$600$		0			0
$601$	647.056			1.07663			0.538316	−	0.842743i	$-0.319061\pi$
$601$	647.056			1.07663			0.538316	+	0.842743i	$0.319061\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	221.441	+	84.0449i	0.366019	+	0.138917i
$606$		0			0
$607$		−	498.590i		−	0.821401i	−0.911770	−	0.410700i	$-0.865284\pi$
$607$		−	498.590i		−	0.821401i	0.911770	−	0.410700i	$-0.134716\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1508.94i			2.46963i
$612$		0			0
$613$			374.575i			0.611053i	0.952184	+	0.305526i	$0.0988325\pi$
$613$			374.575i			0.611053i	−0.952184	+	0.305526i	$0.901168\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−170.764			−0.276765			−0.138383	−	0.990379i	$-0.544190\pi$
$617$	−170.764			−0.276765			−0.138383	+	0.990379i	$0.544190\pi$
$618$		0			0
$619$	531.705			0.858974			0.429487	−	0.903073i	$-0.358694\pi$
$619$	531.705			0.858974			0.429487	+	0.903073i	$0.358694\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	278.380			0.446838
$624$		0			0
$625$	74.7152	+	620.518i	0.119544	+	0.992829i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1236.73i			1.96618i
$630$		0			0
$631$	671.104			1.06356			0.531778	−	0.846884i	$-0.321524\pi$
$631$	671.104			1.06356			0.531778	+	0.846884i	$0.321524\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−93.6532	+	246.757i	−0.147485	+	0.388594i
$636$		0			0
$637$			848.465i			1.33197i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			828.196i			1.29204i	0.763321	+	0.646019i	$0.223567\pi$
$641$			828.196i			1.29204i	−0.763321	+	0.646019i	$0.776433\pi$
$642$		0			0
$643$		−	730.562i		−	1.13618i	−0.822968	−	0.568088i	$-0.807683\pi$
$643$		−	730.562i		−	1.13618i	0.822968	−	0.568088i	$-0.192317\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−936.478			−1.44742			−0.723708	−	0.690106i	$-0.757564\pi$
$647$	−936.478			−1.44742			−0.723708	+	0.690106i	$0.757564\pi$
$648$		0			0
$649$	−132.237			−0.203755
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1006.97			−1.54207			−0.771037	−	0.636791i	$-0.780262\pi$
$653$	−1006.97			−1.54207			−0.771037	+	0.636791i	$0.780262\pi$
$654$		0			0
$655$	287.852	−	758.430i	0.439468	−	1.15791i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			183.595i			0.278597i	0.990250	+	0.139299i	$0.0444848\pi$
$659$			183.595i			0.278597i	−0.990250	+	0.139299i	$0.955515\pi$
$660$		0			0
$661$	−157.268			−0.237924			−0.118962	−	0.992899i	$-0.537957\pi$
$661$	−157.268			−0.237924			−0.118962	+	0.992899i	$0.537957\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−456.777	+	1203.51i	−0.686883	+	1.80980i
$666$		0			0
$667$		−	474.943i		−	0.712059i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			538.507i			0.802544i
$672$		0			0
$673$			145.482i			0.216169i	0.994142	+	0.108085i	$0.0344717\pi$
$673$			145.482i			0.216169i	−0.994142	+	0.108085i	$0.965528\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−50.4636			−0.0745400			−0.0372700	−	0.999305i	$-0.511866\pi$
$677$	−50.4636			−0.0745400			−0.0372700	+	0.999305i	$0.511866\pi$
$678$		0			0
$679$	225.410			0.331974
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−292.599			−0.428403			−0.214202	−	0.976789i	$-0.568715\pi$
$683$	−292.599			−0.428403			−0.214202	+	0.976789i	$0.568715\pi$
$684$		0			0
$685$	−487.437	−	185.000i	−0.711587	−	0.270073i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1542.49i			2.23874i
$690$		0			0
$691$	−312.011			−0.451535			−0.225768	−	0.974181i	$-0.572489\pi$
$691$	−312.011			−0.451535			−0.225768	+	0.974181i	$0.572489\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	631.128	+	239.536i	0.908098	+	0.344656i
$696$		0			0
$697$			15.8406i			0.0227268i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1192.92i		−	1.70174i	−0.525377	−	0.850869i	$-0.676076\pi$
$701$		−	1192.92i		−	1.70174i	0.525377	−	0.850869i	$-0.323924\pi$
$702$		0			0
$703$		−	1251.91i		−	1.78081i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1103.84			−1.56131
$708$		0			0
$709$	147.979			0.208715			0.104358	−	0.994540i	$-0.466721\pi$
$709$	147.979			0.208715			0.104358	+	0.994540i	$0.466721\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	195.609			0.274346
$714$		0			0
$715$	−764.205	−	290.043i	−1.06882	−	0.405655i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			396.045i			0.550827i	0.961326	+	0.275413i	$0.0888148\pi$
$719$			396.045i			0.550827i	−0.961326	+	0.275413i	$0.911185\pi$
$720$		0			0
$721$	−1609.91			−2.23289
$722$		0			0
$723$		0			0
$724$		0			0
$725$	304.571	−	343.444i	0.420098	−	0.473716i
$726$		0			0
$727$			195.196i			0.268496i	0.990948	+	0.134248i	$0.0428619\pi$
$727$			195.196i			0.268496i	−0.990948	+	0.134248i	$0.957138\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1637.89i		−	2.24061i
$732$		0			0
$733$		−	587.367i		−	0.801319i	−0.916227	−	0.400659i	$-0.868781\pi$
$733$		−	587.367i		−	0.801319i	0.916227	−	0.400659i	$-0.131219\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−414.910			−0.562972
$738$		0			0
$739$	265.498			0.359267			0.179634	−	0.983734i	$-0.442509\pi$
$739$	265.498			0.359267			0.179634	+	0.983734i	$0.442509\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1345.90			−1.81144			−0.905718	−	0.423881i	$-0.860667\pi$
$743$	−1345.90			−1.81144			−0.905718	+	0.423881i	$0.860667\pi$
$744$		0			0
$745$	508.570	−	1339.98i	0.682644	−	1.79863i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			52.4520i			0.0700294i
$750$		0			0
$751$	−355.090			−0.472823			−0.236411	−	0.971653i	$-0.575971\pi$
$751$	−355.090			−0.472823			−0.236411	+	0.971653i	$0.575971\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.5079	+	4.74721i	0.0165668	+	0.00628770i
$756$		0			0
$757$			978.475i			1.29257i	0.763096	+	0.646285i	$0.223678\pi$
$757$			978.475i			1.29257i	−0.763096	+	0.646285i	$0.776322\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			334.505i			0.439560i	0.975549	+	0.219780i	$0.0705339\pi$
$761$			334.505i			0.439560i	−0.975549	+	0.219780i	$0.929466\pi$
$762$		0			0
$763$		−	1945.22i		−	2.54943i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−293.607			−0.382799
$768$		0			0
$769$	117.430			0.152705			0.0763527	−	0.997081i	$-0.475673\pi$
$769$	117.430			0.152705			0.0763527	+	0.997081i	$0.475673\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	553.474			0.716007			0.358004	−	0.933720i	$-0.383458\pi$
$773$	553.474			0.716007			0.358004	+	0.933720i	$0.383458\pi$
$774$		0			0
$775$	141.450	+	125.440i	0.182516	+	0.161858i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	16.0351i		−	0.0205842i
$780$		0			0
$781$	855.005			1.09476
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−515.110	+	1357.21i	−0.656191	+	1.72893i
$786$		0			0
$787$			731.043i			0.928898i	0.885600	+	0.464449i	$0.153748\pi$
$787$			731.043i			0.928898i	−0.885600	+	0.464449i	$0.846252\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			529.113i			0.668917i
$792$		0			0
$793$			1195.65i			1.50775i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1108.67			1.39105			0.695527	−	0.718500i	$-0.255171\pi$
$797$	1108.67			1.39105			0.695527	+	0.718500i	$0.255171\pi$
$798$		0			0
$799$	−2082.83			−2.60680
$800$		0			0
$801$		0			0
$802$		0			0
$803$	520.952			0.648757
$804$		0			0
$805$	443.832	−	1169.41i	0.551344	−	1.45268i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1154.29i			1.42681i	0.700753	+	0.713404i	$0.252848\pi$
$809$			1154.29i			1.42681i	−0.700753	+	0.713404i	$0.747152\pi$
$810$		0			0
$811$	−1231.92			−1.51901			−0.759507	−	0.650499i	$-0.774560\pi$
$811$	−1231.92			−1.51901			−0.759507	+	0.650499i	$0.774560\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−332.707	+	876.617i	−0.408230	+	1.07560i
$816$		0			0
$817$			1658.00i			2.02937i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1050.87i		−	1.27999i	−0.768380	−	0.639994i	$-0.778937\pi$
$821$		−	1050.87i		−	1.27999i	0.768380	−	0.639994i	$-0.221063\pi$
$822$		0			0
$823$			438.283i			0.532543i	0.963898	+	0.266272i	$0.0857918\pi$
$823$			438.283i			0.532543i	−0.963898	+	0.266272i	$0.914208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	281.785			0.340732			0.170366	−	0.985381i	$-0.445505\pi$
$827$	281.785			0.340732			0.170366	+	0.985381i	$0.445505\pi$
$828$		0			0
$829$	264.448			0.318996			0.159498	−	0.987198i	$-0.449012\pi$
$829$	264.448			0.318996			0.159498	+	0.987198i	$0.449012\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1171.16			−1.40595
$834$		0			0
$835$	−366.329	−	139.035i	−0.438717	−	0.166509i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1593.36i			1.89912i	0.313585	+	0.949560i	$0.398470\pi$
$839$			1593.36i			1.89912i	−0.313585	+	0.949560i	$0.601530\pi$
$840$		0			0
$841$	503.853			0.599111
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−906.754	−	344.145i	−1.07308	−	0.407273i
$846$		0			0
$847$		−	458.138i		−	0.540895i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1216.43i			1.42942i
$852$		0			0
$853$			316.402i			0.370928i	0.982651	+	0.185464i	$0.0593788\pi$
$853$			316.402i			0.370928i	−0.982651	+	0.185464i	$0.940621\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	223.873			0.261229			0.130614	−	0.991433i	$-0.458305\pi$
$857$	223.873			0.261229			0.130614	+	0.991433i	$0.458305\pi$
$858$		0			0
$859$	487.293			0.567280			0.283640	−	0.958931i	$-0.408458\pi$
$859$	487.293			0.567280			0.283640	+	0.958931i	$0.408458\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−722.801			−0.837545			−0.418772	−	0.908091i	$-0.637539\pi$
$863$	−722.801			−0.837545			−0.418772	+	0.908091i	$0.637539\pi$
$864$		0			0
$865$	1065.21	+	404.287i	1.23146	+	0.467384i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			521.888i			0.600562i
$870$		0			0
$871$	−921.226			−1.05766
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1070.86	−	561.008i	1.22384	−	0.641153i
$876$		0			0
$877$			127.733i			0.145648i	0.997345	+	0.0728239i	$0.0232011\pi$
$877$			127.733i			0.145648i	−0.997345	+	0.0728239i	$0.976799\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	476.607i		−	0.540984i	−0.962722	−	0.270492i	$-0.912814\pi$
$881$		−	476.607i		−	0.540984i	0.962722	−	0.270492i	$-0.0871863\pi$
$882$		0			0
$883$			42.5119i			0.0481449i	0.999710	+	0.0240724i	$0.00766324\pi$
$883$			42.5119i			0.0481449i	−0.999710	+	0.0240724i	$0.992337\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.2543			0.0126881			0.00634405	−	0.999980i	$-0.497981\pi$
$887$	11.2543			0.0126881			0.00634405	+	0.999980i	$0.497981\pi$
$888$		0			0
$889$	510.514			0.574257
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2108.40			2.36103
$894$		0			0
$895$	158.307	−	417.108i	0.176880	−	0.466042i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	138.856i		−	0.154457i
$900$		0			0
$901$	−2129.14			−2.36308
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−114.949	−	43.6271i	−0.127015	−	0.0482068i
$906$		0			0
$907$			1492.02i			1.64500i	0.568762	+	0.822502i	$0.307423\pi$
$907$			1492.02i			1.64500i	−0.568762	+	0.822502i	$0.692577\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	202.505i		−	0.222289i	−0.993804	−	0.111144i	$-0.964548\pi$
$911$		−	202.505i		−	0.222289i	0.993804	−	0.111144i	$-0.0354516\pi$
$912$		0			0
$913$		−	661.363i		−	0.724385i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1569.11			−1.71114
$918$		0			0
$919$	1675.22			1.82287			0.911437	−	0.411440i	$-0.134974\pi$
$919$	1675.22			1.82287			0.911437	+	0.411440i	$0.134974\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1898.37			2.05674
$924$		0			0
$925$	−780.073	+	879.634i	−0.843322	+	0.950955i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	249.474i		−	0.268540i	−0.990945	−	0.134270i	$-0.957131\pi$
$929$		−	249.474i		−	0.268540i	0.990945	−	0.134270i	$-0.0428690\pi$
$930$		0			0
$931$	1185.53			1.27340
$932$		0			0
$933$		0			0
$934$		0			0
$935$	400.353	−	1054.85i	0.428186	−	1.12818i
$936$		0			0
$937$			669.921i			0.714964i	0.933920	+	0.357482i	$0.116365\pi$
$937$			669.921i			0.714964i	−0.933920	+	0.357482i	$0.883635\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1505.28i			1.59966i	0.600226	+	0.799830i	$0.295077\pi$
$941$			1505.28i			1.59966i	−0.600226	+	0.799830i	$0.704923\pi$
$942$		0			0
$943$			15.5807i			0.0165224i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1647.39			1.73959			0.869794	−	0.493415i	$-0.164252\pi$
$947$	1647.39			1.73959			0.869794	+	0.493415i	$0.164252\pi$
$948$		0			0
$949$	1156.67			1.21883
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1785.17			1.87321			0.936605	−	0.350386i	$-0.113950\pi$
$953$	1785.17			1.87321			0.936605	+	0.350386i	$0.113950\pi$
$954$		0			0
$955$	50.9468	−	134.234i	0.0533474	−	0.140560i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1008.46i			1.05157i
$960$		0			0
$961$	−903.811			−0.940490
$962$		0			0
$963$		0			0
$964$		0			0
$965$	398.713	−	1050.53i	0.413174	−	1.08863i
$966$		0			0
$967$		−	371.768i		−	0.384455i	−0.981350	−	0.192228i	$-0.938429\pi$
$967$		−	371.768i		−	0.384455i	0.981350	−	0.192228i	$-0.0615712\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	564.620i		−	0.581483i	−0.956802	−	0.290741i	$-0.906098\pi$
$971$		−	564.620i		−	0.581483i	0.956802	−	0.290741i	$-0.0939019\pi$
$972$		0			0
$973$		−	1305.74i		−	1.34197i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	485.194			0.496616			0.248308	−	0.968681i	$-0.420125\pi$
$977$	485.194			0.496616			0.248308	+	0.968681i	$0.420125\pi$
$978$		0			0
$979$	246.989			0.252287
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−959.728			−0.976325			−0.488163	−	0.872753i	$-0.662333\pi$
$983$	−959.728			−0.976325			−0.488163	+	0.872753i	$0.662333\pi$
$984$		0			0
$985$	−513.649	−	194.948i	−0.521471	−	0.197917i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1611.01i		−	1.62893i
$990$		0			0
$991$	184.184			0.185857			0.0929286	−	0.995673i	$-0.470377\pi$
$991$	184.184			0.185857			0.0929286	+	0.995673i	$0.470377\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−393.174	−	149.224i	−0.395150	−	0.149973i
$996$		0			0
$997$		−	655.370i		−	0.657342i	−0.944445	−	0.328671i	$-0.893399\pi$
$997$		−	655.370i		−	0.657342i	0.944445	−	0.328671i	$-0.106601\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.q.1889.24		24
3.2	odd	2	inner	2160.3.c.q.1889.1		24
4.3	odd	2		1080.3.c.c.809.24	yes	24
5.4	even	2	inner	2160.3.c.q.1889.2		24
12.11	even	2		1080.3.c.c.809.1	✓	24
15.14	odd	2	inner	2160.3.c.q.1889.23		24
20.19	odd	2		1080.3.c.c.809.2	yes	24
60.59	even	2		1080.3.c.c.809.23	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.3.c.c.809.1	✓	24	12.11	even	2
1080.3.c.c.809.2	yes	24	20.19	odd	2
1080.3.c.c.809.23	yes	24	60.59	even	2
1080.3.c.c.809.24	yes	24	4.3	odd	2
2160.3.c.q.1889.1		24	3.2	odd	2	inner
2160.3.c.q.1889.2		24	5.4	even	2	inner
2160.3.c.q.1889.23		24	15.14	odd	2	inner
2160.3.c.q.1889.24		24	1.1	even	1	trivial

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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2160.3.c.q.1889.24

Level	$2160$
Weight	$3$
Character	2160.1889
Analytic conductor	$58.856$
Analytic rank	$0$
Dimension	$24$
Inner twists	$4$