LMFDB - Embedded newform 4140.3.c.a.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(4049,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("4140.4049");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.c (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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Character	$\chi$	$=$	4140.4049
Dual form			4140.3.c.a.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.95880 + 0.640555i) q^{5} -5.90914i q^{7} +O(q^{10})$	$q+(-4.95880 + 0.640555i) q^{5} -5.90914i q^{7} +19.5889i q^{11} -22.3573i q^{13} -14.4049 q^{17} +13.9303 q^{19} -4.79583 q^{23} +(24.1794 - 6.35277i) q^{25} +49.1668i q^{29} +12.8357 q^{31} +(3.78513 + 29.3023i) q^{35} -9.21290i q^{37} -72.5557i q^{41} -42.3352i q^{43} -11.8340 q^{47} +14.0820 q^{49} -67.3121 q^{53} +(-12.5478 - 97.1375i) q^{55} -103.370i q^{59} -4.23266 q^{61} +(14.3211 + 110.865i) q^{65} -35.0159i q^{67} +2.21240i q^{71} +75.7778i q^{73} +115.754 q^{77} -13.9649 q^{79} +42.0724 q^{83} +(71.4312 - 9.22716i) q^{85} +40.4432i q^{89} -132.113 q^{91} +(-69.0777 + 8.92314i) q^{95} +146.897i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q+O(q^{10}) $ 88 * q	$ 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) $ 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$	−4.95880	+	0.640555i	−0.991760	+	0.128111i
$5$	−4.95880	+	0.640555i	−0.991760	+	0.128111i
$6$		0			0
$7$		−	5.90914i		−	0.844163i	−0.906558	−	0.422082i	$-0.861300\pi$
$7$		−	5.90914i		−	0.844163i	0.906558	−	0.422082i	$-0.138700\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			19.5889i			1.78081i	0.455168	+	0.890406i	$0.349579\pi$
$11$			19.5889i			1.78081i	−0.455168	+	0.890406i	$0.650421\pi$
$12$		0			0
$13$		−	22.3573i		−	1.71979i	−0.510467	−	0.859897i	$-0.670528\pi$
$13$		−	22.3573i		−	1.71979i	0.510467	−	0.859897i	$-0.329472\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−14.4049			−0.847350			−0.423675	−	0.905814i	$-0.639260\pi$
$17$	−14.4049			−0.847350			−0.423675	+	0.905814i	$0.639260\pi$
$18$		0			0
$19$	13.9303			0.733175			0.366588	−	0.930383i	$-0.380526\pi$
$19$	13.9303			0.733175			0.366588	+	0.930383i	$0.380526\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.79583			−0.208514
$23$	−4.79583			−0.208514
$24$		0			0
$25$	24.1794	−	6.35277i	0.967175	−	0.254111i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			49.1668i			1.69541i	0.530471	+	0.847703i	$0.322015\pi$
$29$			49.1668i			1.69541i	−0.530471	+	0.847703i	$0.677985\pi$
$30$		0			0
$31$	12.8357			0.414056			0.207028	−	0.978335i	$-0.433621\pi$
$31$	12.8357			0.414056			0.207028	+	0.978335i	$0.433621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.78513	+	29.3023i	0.108147	+	0.837207i
$36$		0			0
$37$		−	9.21290i		−	0.248997i	−0.992220	−	0.124499i	$-0.960268\pi$
$37$		−	9.21290i		−	0.248997i	0.992220	−	0.124499i	$-0.0397323\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	72.5557i		−	1.76965i	−0.465923	−	0.884825i	$-0.654278\pi$
$41$		−	72.5557i		−	1.76965i	0.465923	−	0.884825i	$-0.345722\pi$
$42$		0			0
$43$		−	42.3352i		−	0.984539i	−0.870443	−	0.492270i	$-0.836167\pi$
$43$		−	42.3352i		−	0.984539i	0.870443	−	0.492270i	$-0.163833\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.8340			−0.251788			−0.125894	−	0.992044i	$-0.540180\pi$
$47$	−11.8340			−0.251788			−0.125894	+	0.992044i	$0.540180\pi$
$48$		0			0
$49$	14.0820			0.287388
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−67.3121			−1.27004			−0.635019	−	0.772496i	$-0.719008\pi$
$53$	−67.3121			−1.27004			−0.635019	+	0.772496i	$0.719008\pi$
$54$		0			0
$55$	−12.5478	−	97.1375i	−0.228141	−	1.76614i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	103.370i		−	1.75204i	−0.482279	−	0.876018i	$-0.660191\pi$
$59$		−	103.370i		−	1.75204i	0.482279	−	0.876018i	$-0.339809\pi$
$60$		0			0
$61$	−4.23266			−0.0693879			−0.0346940	−	0.999398i	$-0.511046\pi$
$61$	−4.23266			−0.0693879			−0.0346940	+	0.999398i	$0.511046\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14.3211	+	110.865i	0.220325	+	1.70562i
$66$		0			0
$67$		−	35.0159i		−	0.522625i	−0.965254	−	0.261313i	$-0.915845\pi$
$67$		−	35.0159i		−	0.522625i	0.965254	−	0.261313i	$-0.0841554\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			2.21240i			0.0311606i	0.999879	+	0.0155803i	$0.00495956\pi$
$71$			2.21240i			0.0311606i	−0.999879	+	0.0155803i	$0.995040\pi$
$72$		0			0
$73$			75.7778i			1.03805i	0.854758	+	0.519026i	$0.173705\pi$
$73$			75.7778i			1.03805i	−0.854758	+	0.519026i	$0.826295\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	115.754			1.50330
$78$		0			0
$79$	−13.9649			−0.176771			−0.0883855	−	0.996086i	$-0.528171\pi$
$79$	−13.9649			−0.176771			−0.0883855	+	0.996086i	$0.528171\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	42.0724			0.506897			0.253448	−	0.967349i	$-0.418435\pi$
$83$	42.0724			0.506897			0.253448	+	0.967349i	$0.418435\pi$
$84$		0			0
$85$	71.4312	−	9.22716i	0.840367	−	0.108555i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			40.4432i			0.454418i	0.973846	+	0.227209i	$0.0729601\pi$
$89$			40.4432i			0.454418i	−0.973846	+	0.227209i	$0.927040\pi$
$90$		0			0
$91$	−132.113			−1.45179
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−69.0777	+	8.92314i	−0.727134	+	0.0939278i
$96$		0			0
$97$			146.897i			1.51441i	0.653179	+	0.757203i	$0.273435\pi$
$97$			146.897i			1.51441i	−0.653179	+	0.757203i	$0.726565\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			95.1177i			0.941760i	0.882197	+	0.470880i	$0.156063\pi$
$101$			95.1177i			0.941760i	−0.882197	+	0.470880i	$0.843937\pi$
$102$		0			0
$103$			19.9392i			0.193585i	0.995305	+	0.0967923i	$0.0308583\pi$
$103$			19.9392i			0.193585i	−0.995305	+	0.0967923i	$0.969142\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−32.3011			−0.301880			−0.150940	−	0.988543i	$-0.548230\pi$
$107$	−32.3011			−0.301880			−0.150940	+	0.988543i	$0.548230\pi$
$108$		0			0
$109$	57.6293			0.528709			0.264355	−	0.964426i	$-0.414841\pi$
$109$	57.6293			0.528709			0.264355	+	0.964426i	$0.414841\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	211.992			1.87603			0.938016	−	0.346593i	$-0.112661\pi$
$113$	211.992			1.87603			0.938016	+	0.346593i	$0.112661\pi$
$114$		0			0
$115$	23.7816	−	3.07199i	0.206796	−	0.0267130i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			85.1209i			0.715302i
$120$		0			0
$121$	−262.726			−2.17129
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−115.831	+	46.9903i	−0.926651	+	0.375922i
$126$		0			0
$127$		−	142.578i		−	1.12266i	−0.827591	−	0.561332i	$-0.810289\pi$
$127$		−	142.578i		−	1.12266i	0.827591	−	0.561332i	$-0.189711\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			62.0381i			0.473573i	0.971562	+	0.236787i	$0.0760943\pi$
$131$			62.0381i			0.473573i	−0.971562	+	0.236787i	$0.923906\pi$
$132$		0			0
$133$		−	82.3163i		−	0.618920i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	47.8750			0.349453			0.174726	−	0.984617i	$-0.444096\pi$
$137$	47.8750			0.349453			0.174726	+	0.984617i	$0.444096\pi$
$138$		0			0
$139$	−82.2903			−0.592017			−0.296008	−	0.955185i	$-0.595656\pi$
$139$	−82.2903			−0.592017			−0.296008	+	0.955185i	$0.595656\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	437.956			3.06263
$144$		0			0
$145$	−31.4940	−	243.808i	−0.217200	−	1.68144i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	227.320i		−	1.52564i	−0.646614	−	0.762818i	$-0.723815\pi$
$149$		−	227.320i		−	1.52564i	0.646614	−	0.762818i	$-0.276185\pi$
$150$		0			0
$151$	−267.412			−1.77094			−0.885470	−	0.464697i	$-0.846163\pi$
$151$	−267.412			−1.77094			−0.885470	+	0.464697i	$0.846163\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−63.6498	+	8.22199i	−0.410644	+	0.0530451i
$156$		0			0
$157$			143.492i			0.913964i	0.889476	+	0.456982i	$0.151070\pi$
$157$			143.492i			0.913964i	−0.889476	+	0.456982i	$0.848930\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			28.3393i			0.176020i
$162$		0			0
$163$			254.844i			1.56346i	0.623617	+	0.781730i	$0.285663\pi$
$163$			254.844i			1.56346i	−0.623617	+	0.781730i	$0.714337\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−213.030			−1.27563			−0.637814	−	0.770191i	$-0.720161\pi$
$167$	−213.030			−1.27563			−0.637814	+	0.770191i	$0.720161\pi$
$168$		0			0
$169$	−330.850			−1.95769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−216.009			−1.24861			−0.624304	−	0.781181i	$-0.714617\pi$
$173$	−216.009			−1.24861			−0.624304	+	0.781181i	$0.714617\pi$
$174$		0			0
$175$	−37.5394	−	142.879i	−0.214511	−	0.816454i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			234.878i			1.31217i	0.754688	+	0.656084i	$0.227788\pi$
$179$			234.878i			1.31217i	−0.754688	+	0.656084i	$0.772212\pi$
$180$		0			0
$181$	253.227			1.39904			0.699521	−	0.714612i	$-0.253397\pi$
$181$	253.227			1.39904			0.699521	+	0.714612i	$0.253397\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.90137	+	45.6849i	0.0318993	+	0.246946i
$186$		0			0
$187$		−	282.177i		−	1.50897i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	248.163i		−	1.29928i	−0.760240	−	0.649642i	$-0.774919\pi$
$191$		−	248.163i		−	1.29928i	0.760240	−	0.649642i	$-0.225081\pi$
$192$		0			0
$193$			182.807i			0.947187i	0.880743	+	0.473594i	$0.157043\pi$
$193$			182.807i			0.947187i	−0.880743	+	0.473594i	$0.842957\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−82.4216			−0.418384			−0.209192	−	0.977875i	$-0.567083\pi$
$197$	−82.4216			−0.418384			−0.209192	+	0.977875i	$0.567083\pi$
$198$		0			0
$199$	−142.206			−0.714601			−0.357300	−	0.933990i	$-0.616303\pi$
$199$	−142.206			−0.714601			−0.357300	+	0.933990i	$0.616303\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	290.534			1.43120
$204$		0			0
$205$	46.4759	+	359.789i	0.226712	+	1.75507i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			272.880i			1.30565i
$210$		0			0
$211$	132.527			0.628092			0.314046	−	0.949408i	$-0.398315\pi$
$211$	132.527			0.628092			0.314046	+	0.949408i	$0.398315\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	27.1180	+	209.932i	0.126130	+	0.976426i
$216$		0			0
$217$		−	75.8481i		−	0.349531i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			322.056i			1.45727i
$222$		0			0
$223$			380.648i			1.70694i	0.521142	+	0.853470i	$0.325506\pi$
$223$			380.648i			1.70694i	−0.521142	+	0.853470i	$0.674494\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−251.579			−1.10828			−0.554140	−	0.832424i	$-0.686953\pi$
$227$	−251.579			−1.10828			−0.554140	+	0.832424i	$0.686953\pi$
$228$		0			0
$229$	253.497			1.10698			0.553488	−	0.832857i	$-0.313297\pi$
$229$	253.497			1.10698			0.553488	+	0.832857i	$0.313297\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−450.703			−1.93435			−0.967174	−	0.254113i	$-0.918216\pi$
$233$	−450.703			−1.93435			−0.967174	+	0.254113i	$0.918216\pi$
$234$		0			0
$235$	58.6825	−	7.58034i	0.249713	−	0.0322567i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	261.558i		−	1.09439i	−0.837007	−	0.547193i	$-0.815696\pi$
$239$		−	261.558i		−	1.09439i	0.837007	−	0.547193i	$-0.184304\pi$
$240$		0			0
$241$	−34.0989			−0.141489			−0.0707446	−	0.997494i	$-0.522538\pi$
$241$	−34.0989			−0.141489			−0.0707446	+	0.997494i	$0.522538\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−69.8299	+	9.02030i	−0.285020	+	0.0368176i
$246$		0			0
$247$		−	311.445i		−	1.26091i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			406.762i			1.62057i	0.586038	+	0.810284i	$0.300687\pi$
$251$			406.762i			1.62057i	−0.586038	+	0.810284i	$0.699313\pi$
$252$		0			0
$253$		−	93.9452i		−	0.371325i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−302.141			−1.17565			−0.587824	−	0.808989i	$-0.700015\pi$
$257$	−302.141			−1.17565			−0.587824	+	0.808989i	$0.700015\pi$
$258$		0			0
$259$	−54.4404			−0.210195
$260$		0			0
$261$		0			0
$262$		0			0
$263$	304.566			1.15805			0.579023	−	0.815311i	$-0.303434\pi$
$263$	304.566			1.15805			0.579023	+	0.815311i	$0.303434\pi$
$264$		0			0
$265$	333.787	−	43.1171i	1.25957	−	0.162706i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			399.211i			1.48406i	0.670369	+	0.742028i	$0.266136\pi$
$269$			399.211i			1.48406i	−0.670369	+	0.742028i	$0.733864\pi$
$270$		0			0
$271$	−39.3310			−0.145133			−0.0725663	−	0.997364i	$-0.523119\pi$
$271$	−39.3310			−0.145133			−0.0725663	+	0.997364i	$0.523119\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	124.444	+	473.648i	0.452523	+	1.72236i
$276$		0			0
$277$			20.7593i			0.0749435i	0.999298	+	0.0374717i	$0.0119304\pi$
$277$			20.7593i			0.0749435i	−0.999298	+	0.0374717i	$0.988070\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	121.543i		−	0.432538i	−0.976334	−	0.216269i	$-0.930611\pi$
$281$		−	121.543i		−	0.432538i	0.976334	−	0.216269i	$-0.0693888\pi$
$282$		0			0
$283$			50.7738i			0.179413i	0.995968	+	0.0897063i	$0.0285928\pi$
$283$			50.7738i			0.179413i	−0.995968	+	0.0897063i	$0.971407\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−428.742			−1.49387
$288$		0			0
$289$	−81.4976			−0.281999
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−22.4898			−0.0767569			−0.0383785	−	0.999263i	$-0.512219\pi$
$293$	−22.4898			−0.0767569			−0.0383785	+	0.999263i	$0.512219\pi$
$294$		0			0
$295$	66.2142	+	512.591i	0.224455	+	1.73760i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			107.222i			0.358602i
$300$		0			0
$301$	−250.165			−0.831112
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.9889	−	2.71125i	0.0688162	−	0.00888936i
$306$		0			0
$307$			454.697i			1.48110i	0.672002	+	0.740550i	$0.265435\pi$
$307$			454.697i			1.48110i	−0.672002	+	0.740550i	$0.734565\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			36.5558i			0.117543i	0.998271	+	0.0587714i	$0.0187183\pi$
$311$			36.5558i			0.117543i	−0.998271	+	0.0587714i	$0.981282\pi$
$312$		0			0
$313$			468.274i			1.49608i	0.663652	+	0.748041i	$0.269005\pi$
$313$			468.274i			1.49608i	−0.663652	+	0.748041i	$0.730995\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−129.480			−0.408453			−0.204226	−	0.978924i	$-0.565468\pi$
$317$	−129.480			−0.408453			−0.204226	+	0.978924i	$0.565468\pi$
$318$		0			0
$319$	−963.124			−3.01920
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−200.666			−0.621256
$324$		0			0
$325$	−142.031	−	540.586i	−0.437018	−	1.66334i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			69.9289i			0.212550i
$330$		0			0
$331$	−219.279			−0.662475			−0.331238	−	0.943547i	$-0.607466\pi$
$331$	−219.279			−0.662475			−0.331238	+	0.943547i	$0.607466\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	22.4296	+	173.637i	0.0669540	+	0.518319i
$336$		0			0
$337$			100.695i			0.298798i	0.988777	+	0.149399i	$0.0477338\pi$
$337$			100.695i			0.298798i	−0.988777	+	0.149399i	$0.952266\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			251.438i			0.737355i
$342$		0			0
$343$		−	372.761i		−	1.08677i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	624.121			1.79862			0.899309	−	0.437314i	$-0.144070\pi$
$347$	624.121			1.79862			0.899309	+	0.437314i	$0.144070\pi$
$348$		0			0
$349$	368.673			1.05637			0.528185	−	0.849129i	$-0.322873\pi$
$349$	368.673			1.05637			0.528185	+	0.849129i	$0.322873\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−343.180			−0.972181			−0.486091	−	0.873908i	$-0.661577\pi$
$353$	−343.180			−0.972181			−0.486091	+	0.873908i	$0.661577\pi$
$354$		0			0
$355$	−1.41716	−	10.9708i	−0.00399201	−	0.0309038i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			69.5292i			0.193675i	0.995300	+	0.0968373i	$0.0308727\pi$
$359$			69.5292i			0.193675i	−0.995300	+	0.0968373i	$0.969127\pi$
$360$		0			0
$361$	−166.946			−0.462454
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−48.5398	−	375.767i	−0.132986	−	1.02950i
$366$		0			0
$367$		−	159.675i		−	0.435081i	−0.976051	−	0.217541i	$-0.930197\pi$
$367$		−	159.675i		−	0.435081i	0.976051	−	0.217541i	$-0.0698034\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			397.757i			1.07212i
$372$		0			0
$373$			95.4895i			0.256004i	0.991774	+	0.128002i	$0.0408564\pi$
$373$			95.4895i			0.256004i	−0.991774	+	0.128002i	$0.959144\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1099.24			2.91575
$378$		0			0
$379$	−171.588			−0.452738			−0.226369	−	0.974042i	$-0.572686\pi$
$379$	−171.588			−0.452738			−0.226369	+	0.974042i	$0.572686\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−588.416			−1.53633			−0.768167	−	0.640249i	$-0.778831\pi$
$383$	−588.416			−1.53633			−0.768167	+	0.640249i	$0.778831\pi$
$384$		0			0
$385$	−574.000	+	74.1466i	−1.49091	+	0.192589i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	494.635i		−	1.27156i	−0.771872	−	0.635778i	$-0.780679\pi$
$389$		−	494.635i		−	1.27156i	0.771872	−	0.635778i	$-0.219321\pi$
$390$		0			0
$391$	69.0837			0.176685
$392$		0			0
$393$		0			0
$394$		0			0
$395$	69.2492	−	8.94529i	0.175314	−	0.0226463i
$396$		0			0
$397$		−	13.7733i		−	0.0346935i	−0.999850	−	0.0173467i	$-0.994478\pi$
$397$		−	13.7733i		−	0.0346935i	0.999850	−	0.0173467i	$-0.00552192\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	533.141i		−	1.32953i	−0.747053	−	0.664764i	$-0.768532\pi$
$401$		−	533.141i		−	1.32953i	0.747053	−	0.664764i	$-0.231468\pi$
$402$		0			0
$403$		−	286.972i		−	0.712090i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	180.471			0.443417
$408$		0			0
$409$	−676.598			−1.65427			−0.827137	−	0.562000i	$-0.810032\pi$
$409$	−676.598			−1.65427			−0.827137	+	0.562000i	$0.810032\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−610.829			−1.47900
$414$		0			0
$415$	−208.629	+	26.9497i	−0.502720	+	0.0649390i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			748.423i			1.78621i	0.449847	+	0.893106i	$0.351479\pi$
$419$			748.423i			1.78621i	−0.449847	+	0.893106i	$0.648521\pi$
$420$		0			0
$421$	−726.098			−1.72470			−0.862349	−	0.506315i	$-0.831007\pi$
$421$	−726.098			−1.72470			−0.862349	+	0.506315i	$0.831007\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−348.303	+	91.5112i	−0.819536	+	0.215321i
$426$		0			0
$427$			25.0114i			0.0585748i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			464.627i			1.07802i	0.842299	+	0.539010i	$0.181202\pi$
$431$			464.627i			1.07802i	−0.842299	+	0.539010i	$0.818798\pi$
$432$		0			0
$433$		−	125.037i		−	0.288769i	−0.989522	−	0.144385i	$-0.953880\pi$
$433$		−	125.037i		−	0.288769i	0.989522	−	0.144385i	$-0.0461202\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−66.8075			−0.152878
$438$		0			0
$439$	−116.543			−0.265474			−0.132737	−	0.991151i	$-0.542377\pi$
$439$	−116.543			−0.265474			−0.132737	+	0.991151i	$0.542377\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−177.209			−0.400021			−0.200011	−	0.979794i	$-0.564098\pi$
$443$	−177.209			−0.400021			−0.200011	+	0.979794i	$0.564098\pi$
$444$		0			0
$445$	−25.9061	−	200.550i	−0.0582159	−	0.450674i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			365.279i			0.813540i	0.913531	+	0.406770i	$0.133345\pi$
$449$			365.279i			0.813540i	−0.913531	+	0.406770i	$0.866655\pi$
$450$		0			0
$451$	1421.29			3.15141
$452$		0			0
$453$		0			0
$454$		0			0
$455$	655.120	−	84.6254i	1.43982	−	0.185990i
$456$		0			0
$457$		−	277.379i		−	0.606956i	−0.952838	−	0.303478i	$-0.901852\pi$
$457$		−	277.379i		−	0.606956i	0.952838	−	0.303478i	$-0.0981479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			671.712i			1.45708i	0.685006	+	0.728538i	$0.259800\pi$
$461$			671.712i			1.45708i	−0.685006	+	0.728538i	$0.740200\pi$
$462$		0			0
$463$			200.512i			0.433070i	0.976275	+	0.216535i	$0.0694756\pi$
$463$			200.512i			0.433070i	−0.976275	+	0.216535i	$0.930524\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	304.362			0.651738			0.325869	−	0.945415i	$-0.394343\pi$
$467$	304.362			0.651738			0.325869	+	0.945415i	$0.394343\pi$
$468$		0			0
$469$	−206.914			−0.441181
$470$		0			0
$471$		0			0
$472$		0			0
$473$	829.301			1.75328
$474$		0			0
$475$	336.827	−	88.4961i	0.709109	−	0.186308i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			374.271i			0.781359i	0.920527	+	0.390680i	$0.127760\pi$
$479$			374.271i			0.781359i	−0.920527	+	0.390680i	$0.872240\pi$
$480$		0			0
$481$	−205.976			−0.428224
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−94.0959	−	728.435i	−0.194012	−	1.50193i
$486$		0			0
$487$		−	282.771i		−	0.580638i	−0.956930	−	0.290319i	$-0.906239\pi$
$487$		−	282.771i		−	0.580638i	0.956930	−	0.290319i	$-0.0937614\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			592.184i			1.20608i	0.797712	+	0.603038i	$0.206043\pi$
$491$			592.184i			1.20608i	−0.797712	+	0.603038i	$0.793957\pi$
$492$		0			0
$493$		−	708.245i		−	1.43660i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.0734			0.0263046
$498$		0			0
$499$	−578.506			−1.15933			−0.579666	−	0.814854i	$-0.696817\pi$
$499$	−578.506			−1.15933			−0.579666	+	0.814854i	$0.696817\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	115.106			0.228838			0.114419	−	0.993433i	$-0.463499\pi$
$503$	115.106			0.228838			0.114419	+	0.993433i	$0.463499\pi$
$504$		0			0
$505$	−60.9281	−	471.670i	−0.120650	−	0.934000i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	591.657i		−	1.16239i	−0.813764	−	0.581196i	$-0.802585\pi$
$509$		−	591.657i		−	1.16239i	0.813764	−	0.581196i	$-0.197415\pi$
$510$		0			0
$511$	447.782			0.876286
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.7722	−	98.8746i	−0.0248003	−	0.191990i
$516$		0			0
$517$		−	231.816i		−	0.448386i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	80.5454i		−	0.154598i	−0.997008	−	0.0772989i	$-0.975370\pi$
$521$		−	80.5454i		−	0.154598i	0.997008	−	0.0772989i	$-0.0246296\pi$
$522$		0			0
$523$			452.610i			0.865411i	0.901535	+	0.432706i	$0.142441\pi$
$523$			452.610i			0.865411i	−0.901535	+	0.432706i	$0.857559\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−184.898			−0.350850
$528$		0			0
$529$	23.0000			0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1622.15			−3.04343
$534$		0			0
$535$	160.175	−	20.6906i	0.299392	−	0.0386741i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			275.852i			0.511784i
$540$		0			0
$541$	−421.295			−0.778735			−0.389367	−	0.921083i	$-0.627306\pi$
$541$	−421.295			−0.778735			−0.389367	+	0.921083i	$0.627306\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−285.772	+	36.9147i	−0.524353	+	0.0677335i
$546$		0			0
$547$		−	382.500i		−	0.699269i	−0.936886	−	0.349635i	$-0.886306\pi$
$547$		−	382.500i		−	0.699269i	0.936886	−	0.349635i	$-0.113694\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			684.910i			1.24303i
$552$		0			0
$553$			82.5207i			0.149224i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	118.346			0.212470			0.106235	−	0.994341i	$-0.466120\pi$
$557$	118.346			0.212470			0.106235	+	0.994341i	$0.466120\pi$
$558$		0			0
$559$	−946.501			−1.69320
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−940.323			−1.67020			−0.835101	−	0.550097i	$-0.814591\pi$
$563$	−940.323			−1.67020			−0.835101	+	0.550097i	$0.814591\pi$
$564$		0			0
$565$	−1051.22	+	135.792i	−1.86057	+	0.240340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	392.662i		−	0.690091i	−0.938586	−	0.345045i	$-0.887864\pi$
$569$		−	392.662i		−	0.690091i	0.938586	−	0.345045i	$-0.112136\pi$
$570$		0			0
$571$	376.829			0.659945			0.329973	−	0.943991i	$-0.392960\pi$
$571$	376.829			0.659945			0.329973	+	0.943991i	$0.392960\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−115.960	+	30.4668i	−0.201670	+	0.0529857i
$576$		0			0
$577$			661.675i			1.14675i	0.819293	+	0.573375i	$0.194366\pi$
$577$			661.675i			1.14675i	−0.819293	+	0.573375i	$0.805634\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	248.612i		−	0.427904i
$582$		0			0
$583$		−	1318.57i		−	2.26170i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	157.471			0.268264			0.134132	−	0.990963i	$-0.457175\pi$
$587$	157.471			0.268264			0.134132	+	0.990963i	$0.457175\pi$
$588$		0			0
$589$	178.806			0.303575
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.5853			−0.0212231			−0.0106116	−	0.999944i	$-0.503378\pi$
$593$	−12.5853			−0.0212231			−0.0106116	+	0.999944i	$0.503378\pi$
$594$		0			0
$595$	−54.5246	−	422.097i	−0.0916380	−	0.709407i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	227.955i		−	0.380559i	−0.981730	−	0.190280i	$-0.939061\pi$
$599$		−	227.955i		−	0.380559i	0.981730	−	0.190280i	$-0.0609394\pi$
$600$		0			0
$601$	677.995			1.12811			0.564055	−	0.825737i	$-0.309240\pi$
$601$	677.995			1.12811			0.564055	+	0.825737i	$0.309240\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1302.80	−	168.290i	2.15340	−	0.278166i
$606$		0			0
$607$		−	286.643i		−	0.472230i	−0.971725	−	0.236115i	$-0.924126\pi$
$607$		−	286.643i		−	0.472230i	0.971725	−	0.236115i	$-0.0758742\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			264.577i			0.433023i
$612$		0			0
$613$			331.956i			0.541527i	0.962646	+	0.270763i	$0.0872761\pi$
$613$			331.956i			0.541527i	−0.962646	+	0.270763i	$0.912724\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	642.962			1.04208			0.521039	−	0.853533i	$-0.325545\pi$
$617$	642.962			1.04208			0.521039	+	0.853533i	$0.325545\pi$
$618$		0			0
$619$	−959.585			−1.55022			−0.775109	−	0.631827i	$-0.782305\pi$
$619$	−959.585			−1.55022			−0.775109	+	0.631827i	$0.782305\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	238.985			0.383603
$624$		0			0
$625$	544.285	−	307.212i	0.870856	−	0.491539i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			132.711i			0.210988i
$630$		0			0
$631$	−261.520			−0.414454			−0.207227	−	0.978293i	$-0.566444\pi$
$631$	−261.520			−0.414454			−0.207227	+	0.978293i	$0.566444\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	91.3293	+	707.018i	0.143826	+	1.11341i
$636$		0			0
$637$		−	314.836i		−	0.494248i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			458.579i			0.715411i	0.933834	+	0.357706i	$0.116441\pi$
$641$			458.579i			0.715411i	−0.933834	+	0.357706i	$0.883559\pi$
$642$		0			0
$643$			860.107i			1.33765i	0.743421	+	0.668823i	$0.233202\pi$
$643$			860.107i			1.33765i	−0.743421	+	0.668823i	$0.766798\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−388.441			−0.600372			−0.300186	−	0.953881i	$-0.597049\pi$
$647$	−388.441			−0.600372			−0.300186	+	0.953881i	$0.597049\pi$
$648$		0			0
$649$	2024.91			3.12004
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.8405			0.0487603			0.0243802	−	0.999703i	$-0.492239\pi$
$653$	31.8405			0.0487603			0.0243802	+	0.999703i	$0.492239\pi$
$654$		0			0
$655$	−39.7388	−	307.635i	−0.0606699	−	0.469671i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	102.043i		−	0.154845i	−0.996998	−	0.0774226i	$-0.975331\pi$
$659$		−	102.043i		−	0.154845i	0.996998	−	0.0774226i	$-0.0246691\pi$
$660$		0			0
$661$	403.551			0.610516			0.305258	−	0.952270i	$-0.401257\pi$
$661$	403.551			0.610516			0.305258	+	0.952270i	$0.401257\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	52.7281	+	408.190i	0.0792904	+	0.613820i
$666$		0			0
$667$		−	235.796i		−	0.353517i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	82.9133i		−	0.123567i
$672$		0			0
$673$			827.757i			1.22995i	0.788546	+	0.614975i	$0.210834\pi$
$673$			827.757i			1.22995i	−0.788546	+	0.614975i	$0.789166\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−861.871			−1.27307			−0.636537	−	0.771246i	$-0.719634\pi$
$677$	−861.871			−1.27307			−0.636537	+	0.771246i	$0.719634\pi$
$678$		0			0
$679$	868.038			1.27841
$680$		0			0
$681$		0			0
$682$		0			0
$683$	479.387			0.701884			0.350942	−	0.936397i	$-0.385861\pi$
$683$	479.387			0.701884			0.350942	+	0.936397i	$0.385861\pi$
$684$		0			0
$685$	−237.403	+	30.6666i	−0.346573	+	0.0447687i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1504.92i			2.18421i
$690$		0			0
$691$	1302.48			1.88492			0.942459	−	0.334323i	$-0.108508\pi$
$691$	1302.48			1.88492			0.942459	+	0.334323i	$0.108508\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	408.061	−	52.7115i	0.587138	−	0.0758438i
$696$		0			0
$697$			1045.16i			1.49951i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	924.576i		−	1.31894i	−0.751731	−	0.659469i	$-0.770781\pi$
$701$		−	924.576i		−	1.31894i	0.751731	−	0.659469i	$-0.229219\pi$
$702$		0			0
$703$		−	128.339i		−	0.182559i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	562.064			0.794999
$708$		0			0
$709$	847.605			1.19549			0.597747	−	0.801685i	$-0.296063\pi$
$709$	847.605			1.19549			0.597747	+	0.801685i	$0.296063\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−61.5580			−0.0863366
$714$		0			0
$715$	−2171.74	+	280.535i	−3.03739	+	0.392356i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			478.624i			0.665681i	0.942983	+	0.332840i	$0.108007\pi$
$719$			478.624i			0.665681i	−0.942983	+	0.332840i	$0.891993\pi$
$720$		0			0
$721$	117.824			0.163417
$722$		0			0
$723$		0			0
$724$		0			0
$725$	312.345	+	1188.82i	0.430821	+	1.63975i
$726$		0			0
$727$			162.220i			0.223136i	0.993757	+	0.111568i	$0.0355873\pi$
$727$			162.220i			0.223136i	−0.993757	+	0.111568i	$0.964413\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			609.836i			0.834249i
$732$		0			0
$733$			341.345i			0.465682i	0.972515	+	0.232841i	$0.0748022\pi$
$733$			341.345i			0.465682i	−0.972515	+	0.232841i	$0.925198\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	685.924			0.930697
$738$		0			0
$739$	409.414			0.554011			0.277005	−	0.960868i	$-0.410658\pi$
$739$	409.414			0.554011			0.277005	+	0.960868i	$0.410658\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	102.519			0.137979			0.0689896	−	0.997617i	$-0.478022\pi$
$743$	102.519			0.137979			0.0689896	+	0.997617i	$0.478022\pi$
$744$		0			0
$745$	145.611	+	1127.23i	0.195451	+	1.51306i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			190.872i			0.254836i
$750$		0			0
$751$	−1087.66			−1.44829			−0.724144	−	0.689649i	$-0.757765\pi$
$751$	−1087.66			−1.44829			−0.724144	+	0.689649i	$0.757765\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1326.04	−	171.292i	1.75635	−	0.226877i
$756$		0			0
$757$		−	1205.50i		−	1.59247i	−0.604989	−	0.796234i	$-0.706823\pi$
$757$		−	1205.50i		−	1.59247i	0.604989	−	0.796234i	$-0.293177\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1000.54i			1.31477i	0.753553	+	0.657387i	$0.228338\pi$
$761$			1000.54i			1.31477i	−0.753553	+	0.657387i	$0.771662\pi$
$762$		0			0
$763$		−	340.540i		−	0.446317i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2311.08			−3.01314
$768$		0			0
$769$	829.770			1.07902			0.539512	−	0.841978i	$-0.318609\pi$
$769$	829.770			1.07902			0.539512	+	0.841978i	$0.318609\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1389.90			1.79807			0.899033	−	0.437882i	$-0.144271\pi$
$773$	1389.90			1.79807			0.899033	+	0.437882i	$0.144271\pi$
$774$		0			0
$775$	310.360	−	81.5424i	0.400464	−	0.105216i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1010.72i		−	1.29746i
$780$		0			0
$781$	−43.3385			−0.0554911
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−91.9147	−	711.550i	−0.117089	−	0.906433i
$786$		0			0
$787$			1249.53i			1.58771i	0.608107	+	0.793855i	$0.291929\pi$
$787$			1249.53i			1.58771i	−0.608107	+	0.793855i	$0.708071\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1252.69i		−	1.58368i
$792$		0			0
$793$			94.6311i			0.119333i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1450.21			1.81959			0.909794	−	0.415060i	$-0.136239\pi$
$797$	1450.21			1.81959			0.909794	+	0.415060i	$0.136239\pi$
$798$		0			0
$799$	170.468			0.213352
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1484.41			−1.84857
$804$		0			0
$805$	−18.1529	−	140.529i	−0.0225501	−	0.174570i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			640.396i			0.791589i	0.918339	+	0.395795i	$0.129531\pi$
$809$			640.396i			0.791589i	−0.918339	+	0.395795i	$0.870469\pi$
$810$		0			0
$811$	−1509.18			−1.86089			−0.930443	−	0.366437i	$-0.880578\pi$
$811$	−1509.18			−1.86089			−0.930443	+	0.366437i	$0.880578\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−163.242	−	1263.72i	−0.200296	−	1.55058i
$816$		0			0
$817$		−	589.743i		−	0.721840i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	751.427i		−	0.915258i	−0.889143	−	0.457629i	$-0.848699\pi$
$821$		−	751.427i		−	0.915258i	0.889143	−	0.457629i	$-0.151301\pi$
$822$		0			0
$823$			484.001i			0.588093i	0.955791	+	0.294047i	$0.0950021\pi$
$823$			484.001i			0.588093i	−0.955791	+	0.294047i	$0.904998\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	643.203			0.777754			0.388877	−	0.921290i	$-0.372863\pi$
$827$	643.203			0.777754			0.388877	+	0.921290i	$0.372863\pi$
$828$		0			0
$829$	−764.147			−0.921770			−0.460885	−	0.887460i	$-0.652468\pi$
$829$	−764.147			−0.921770			−0.460885	+	0.887460i	$0.652468\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−202.851			−0.243518
$834$		0			0
$835$	1056.37	−	136.457i	1.26512	−	0.163422i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	857.572i		−	1.02214i	−0.859540	−	0.511068i	$-0.829250\pi$
$839$		−	857.572i		−	1.02214i	0.859540	−	0.511068i	$-0.170750\pi$
$840$		0			0
$841$	−1576.37			−1.87440
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1640.62	−	211.928i	1.94156	−	0.250802i
$846$		0			0
$847$			1552.49i			1.83292i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			44.1835i			0.0519195i
$852$		0			0
$853$			492.532i			0.577411i	0.957418	+	0.288706i	$0.0932249\pi$
$853$			492.532i			0.577411i	−0.957418	+	0.288706i	$0.906775\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1090.91			1.27294			0.636468	−	0.771303i	$-0.280395\pi$
$857$	1090.91			1.27294			0.636468	+	0.771303i	$0.280395\pi$
$858$		0			0
$859$	364.170			0.423947			0.211973	−	0.977275i	$-0.432011\pi$
$859$	364.170			0.423947			0.211973	+	0.977275i	$0.432011\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1512.66			−1.75279			−0.876397	−	0.481589i	$-0.840060\pi$
$863$	−1512.66			−1.75279			−0.876397	+	0.481589i	$0.840060\pi$
$864$		0			0
$865$	1071.15	−	138.366i	1.23832	−	0.159960i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	273.558i		−	0.314796i
$870$		0			0
$871$	−782.862			−0.898808
$872$		0			0
$873$		0			0
$874$		0			0
$875$	277.673	+	684.464i	0.317340	+	0.782245i
$876$		0			0
$877$			314.957i			0.359130i	0.983746	+	0.179565i	$0.0574690\pi$
$877$			314.957i			0.359130i	−0.983746	+	0.179565i	$0.942531\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1287.76i		−	1.46171i	−0.682534	−	0.730854i	$-0.739122\pi$
$881$		−	1287.76i		−	1.46171i	0.682534	−	0.730854i	$-0.260878\pi$
$882$		0			0
$883$		−	1155.81i		−	1.30896i	−0.756079	−	0.654480i	$-0.772888\pi$
$883$		−	1155.81i		−	1.30896i	0.756079	−	0.654480i	$-0.227112\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−545.081			−0.614522			−0.307261	−	0.951625i	$-0.599412\pi$
$887$	−545.081			−0.614522			−0.307261	+	0.951625i	$0.599412\pi$
$888$		0			0
$889$	−842.516			−0.947712
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−164.852			−0.184604
$894$		0			0
$895$	−150.452	−	1164.71i	−0.168103	−	1.30135i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			631.091i			0.701992i
$900$		0			0
$901$	969.627			1.07617
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1255.70	+	162.206i	−1.38751	+	0.179233i
$906$		0			0
$907$		−	1624.98i		−	1.79160i	−0.444460	−	0.895799i	$-0.646604\pi$
$907$		−	1624.98i		−	1.79160i	0.444460	−	0.895799i	$-0.353396\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1367.89i			1.50152i	0.660573	+	0.750762i	$0.270313\pi$
$911$			1367.89i			1.50152i	−0.660573	+	0.750762i	$0.729687\pi$
$912$		0			0
$913$			824.153i			0.902687i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	366.592			0.399773
$918$		0			0
$919$	−671.531			−0.730719			−0.365359	−	0.930867i	$-0.619054\pi$
$919$	−671.531			−0.730719			−0.365359	+	0.930867i	$0.619054\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	49.4633			0.0535898
$924$		0			0
$925$	−58.5274	−	222.762i	−0.0632729	−	0.240824i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1541.76i		−	1.65959i	−0.558070	−	0.829794i	$-0.688458\pi$
$929$		−	1541.76i		−	1.65959i	0.558070	−	0.829794i	$-0.311542\pi$
$930$		0			0
$931$	196.167			0.210706
$932$		0			0
$933$		0			0
$934$		0			0
$935$	180.750	+	1399.26i	0.193316	+	1.49654i
$936$		0			0
$937$			1026.68i			1.09571i	0.836574	+	0.547854i	$0.184555\pi$
$937$			1026.68i			1.09571i	−0.836574	+	0.547854i	$0.815445\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	984.359i		−	1.04608i	−0.852309	−	0.523039i	$-0.824798\pi$
$941$		−	984.359i		−	1.04608i	0.852309	−	0.523039i	$-0.175202\pi$
$942$		0			0
$943$			347.965i			0.368998i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.0421			0.0190518			0.00952591	−	0.999955i	$-0.496968\pi$
$947$	18.0421			0.0190518			0.00952591	+	0.999955i	$0.496968\pi$
$948$		0			0
$949$	1694.19			1.78524
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1639.01			−1.71984			−0.859922	−	0.510426i	$-0.829488\pi$
$953$	−1639.01			−1.71984			−0.859922	+	0.510426i	$0.829488\pi$
$954$		0			0
$955$	158.962	+	1230.59i	0.166453	+	1.28858i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	282.901i		−	0.294995i
$960$		0			0
$961$	−796.244			−0.828558
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−117.098	−	906.504i	−0.121345	−	0.939382i
$966$		0			0
$967$		−	867.749i		−	0.897362i	−0.893692	−	0.448681i	$-0.851894\pi$
$967$		−	867.749i		−	0.897362i	0.893692	−	0.448681i	$-0.148106\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	984.099i		−	1.01349i	−0.862096	−	0.506745i	$-0.830848\pi$
$971$		−	984.099i		−	1.01349i	0.862096	−	0.506745i	$-0.169152\pi$
$972$		0			0
$973$			486.265i			0.499759i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−26.2062			−0.0268231			−0.0134115	−	0.999910i	$-0.504269\pi$
$977$	−26.2062			−0.0268231			−0.0134115	+	0.999910i	$0.504269\pi$
$978$		0			0
$979$	−792.239			−0.809233
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−603.724			−0.614164			−0.307082	−	0.951683i	$-0.599353\pi$
$983$	−603.724			−0.614164			−0.307082	+	0.951683i	$0.599353\pi$
$984$		0			0
$985$	408.712	−	52.7956i	0.414936	−	0.0535995i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			203.032i			0.205291i
$990$		0			0
$991$	642.257			0.648090			0.324045	−	0.946042i	$-0.394957\pi$
$991$	642.257			0.648090			0.324045	+	0.946042i	$0.394957\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	705.169	−	91.0904i	0.708712	−	0.0915482i
$996$		0			0
$997$		−	952.460i		−	0.955326i	−0.878543	−	0.477663i	$-0.841484\pi$
$997$		−	952.460i		−	0.955326i	0.878543	−	0.477663i	$-0.158516\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.c.a.4049.4	yes	88
3.2	odd	2	inner	4140.3.c.a.4049.85	yes	88
5.4	even	2	inner	4140.3.c.a.4049.86	yes	88
15.14	odd	2	inner	4140.3.c.a.4049.3	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.c.a.4049.3	✓	88	15.14	odd	2	inner
4140.3.c.a.4049.4	yes	88	1.1	even	1	trivial
4140.3.c.a.4049.85	yes	88	3.2	odd	2	inner
4140.3.c.a.4049.86	yes	88	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.95880 + 0.640555i) q^{5} -5.90914i q^{7} +O(q^{10})\)	\(q+(-4.95880 + 0.640555i) q^{5} -5.90914i q^{7} +19.5889i q^{11} -22.3573i q^{13} -14.4049 q^{17} +13.9303 q^{19} -4.79583 q^{23} +(24.1794 - 6.35277i) q^{25} +49.1668i q^{29} +12.8357 q^{31} +(3.78513 + 29.3023i) q^{35} -9.21290i q^{37} -72.5557i q^{41} -42.3352i q^{43} -11.8340 q^{47} +14.0820 q^{49} -67.3121 q^{53} +(-12.5478 - 97.1375i) q^{55} -103.370i q^{59} -4.23266 q^{61} +(14.3211 + 110.865i) q^{65} -35.0159i q^{67} +2.21240i q^{71} +75.7778i q^{73} +115.754 q^{77} -13.9649 q^{79} +42.0724 q^{83} +(71.4312 - 9.22716i) q^{85} +40.4432i q^{89} -132.113 q^{91} +(-69.0777 + 8.92314i) q^{95} +146.897i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q+O(q^{10}) \) 88 * q	\( 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) \) 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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