LMFDB - Embedded newform 4140.3.d.b.2161.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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, 0, 0, 1]))

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

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

chi := DirichletCharacter("4140.2161");

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.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.11
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.b.2161.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +4.50512i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +4.50512i q^{7} -16.7256i q^{11} +3.28056 q^{13} -7.91035i q^{17} -5.02586i q^{19} +(9.32297 - 21.0258i) q^{23} -5.00000 q^{25} -46.6444 q^{29} -19.2951 q^{31} +10.0737 q^{35} +7.65403i q^{37} -44.7331 q^{41} +33.2928i q^{43} -34.0399 q^{47} +28.7039 q^{49} -55.1208i q^{53} -37.3995 q^{55} +41.8005 q^{59} +60.8160i q^{61} -7.33555i q^{65} +99.3338i q^{67} +75.5229 q^{71} -4.63517 q^{73} +75.3507 q^{77} -146.747i q^{79} +95.3465i q^{83} -17.6881 q^{85} -18.9550i q^{89} +14.7793i q^{91} -11.2382 q^{95} -39.5365i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) $ 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

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
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			4.50512i			0.643588i	0.946810	+	0.321794i	$0.104286\pi$
$7$			4.50512i			0.643588i	−0.946810	+	0.321794i	$0.895714\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	16.7256i		−	1.52051i	−0.649626	−	0.760254i	$-0.725075\pi$
$11$		−	16.7256i		−	1.52051i	0.649626	−	0.760254i	$-0.274925\pi$
$12$		0			0
$13$	3.28056			0.252351			0.126175	−	0.992008i	$-0.459730\pi$
$13$	3.28056			0.252351			0.126175	+	0.992008i	$0.459730\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.91035i		−	0.465315i	−0.972559	−	0.232657i	$-0.925258\pi$
$17$		−	7.91035i		−	0.465315i	0.972559	−	0.232657i	$-0.0747421\pi$
$18$		0			0
$19$		−	5.02586i		−	0.264519i	−0.991215	−	0.132259i	$-0.957777\pi$
$19$		−	5.02586i		−	0.264519i	0.991215	−	0.132259i	$-0.0422232\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	9.32297	−	21.0258i	0.405347	−	0.914163i
$23$	9.32297	−	21.0258i	0.405347	−	0.914163i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−46.6444			−1.60843			−0.804213	−	0.594341i	$-0.797413\pi$
$29$	−46.6444			−1.60843			−0.804213	+	0.594341i	$0.797413\pi$
$30$		0			0
$31$	−19.2951			−0.622423			−0.311211	−	0.950341i	$-0.600735\pi$
$31$	−19.2951			−0.622423			−0.311211	+	0.950341i	$0.600735\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.0737			0.287821
$36$		0			0
$37$			7.65403i			0.206866i	0.994636	+	0.103433i	$0.0329827\pi$
$37$			7.65403i			0.206866i	−0.994636	+	0.103433i	$0.967017\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−44.7331			−1.09105			−0.545525	−	0.838094i	$-0.683670\pi$
$41$	−44.7331			−1.09105			−0.545525	+	0.838094i	$0.683670\pi$
$42$		0			0
$43$			33.2928i			0.774251i	0.922027	+	0.387126i	$0.126532\pi$
$43$			33.2928i			0.774251i	−0.922027	+	0.387126i	$0.873468\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−34.0399			−0.724252			−0.362126	−	0.932129i	$-0.617949\pi$
$47$	−34.0399			−0.724252			−0.362126	+	0.932129i	$0.617949\pi$
$48$		0			0
$49$	28.7039			0.585794
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	55.1208i		−	1.04001i	−0.854162	−	0.520007i	$-0.825929\pi$
$53$		−	55.1208i		−	1.04001i	0.854162	−	0.520007i	$-0.174071\pi$
$54$		0			0
$55$	−37.3995			−0.679992
$56$		0			0
$57$		0			0
$58$		0			0
$59$	41.8005			0.708484			0.354242	−	0.935154i	$-0.384739\pi$
$59$	41.8005			0.708484			0.354242	+	0.935154i	$0.384739\pi$
$60$		0			0
$61$			60.8160i			0.996983i	0.866895	+	0.498492i	$0.166113\pi$
$61$			60.8160i			0.996983i	−0.866895	+	0.498492i	$0.833887\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7.33555i		−	0.112855i
$66$		0			0
$67$			99.3338i			1.48259i	0.671177	+	0.741297i	$0.265789\pi$
$67$			99.3338i			1.48259i	−0.671177	+	0.741297i	$0.734211\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	75.5229			1.06370			0.531852	−	0.846838i	$-0.321496\pi$
$71$	75.5229			1.06370			0.531852	+	0.846838i	$0.321496\pi$
$72$		0			0
$73$	−4.63517			−0.0634955			−0.0317477	−	0.999496i	$-0.510107\pi$
$73$	−4.63517			−0.0634955			−0.0317477	+	0.999496i	$0.510107\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	75.3507			0.978581
$78$		0			0
$79$		−	146.747i		−	1.85755i	−0.370640	−	0.928777i	$-0.620862\pi$
$79$		−	146.747i		−	1.85755i	0.370640	−	0.928777i	$-0.379138\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			95.3465i			1.14875i	0.818591	+	0.574377i	$0.194756\pi$
$83$			95.3465i			1.14875i	−0.818591	+	0.574377i	$0.805244\pi$
$84$		0			0
$85$	−17.6881			−0.208095
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.9550i		−	0.212977i	−0.994314	−	0.106489i	$-0.966039\pi$
$89$		−	18.9550i		−	0.212977i	0.994314	−	0.106489i	$-0.0339608\pi$
$90$		0			0
$91$			14.7793i			0.162410i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.2382			−0.118296
$96$		0			0
$97$		−	39.5365i		−	0.407593i	−0.979013	−	0.203796i	$-0.934672\pi$
$97$		−	39.5365i		−	0.407593i	0.979013	−	0.203796i	$-0.0653280\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−106.867			−1.05809			−0.529046	−	0.848593i	$-0.677450\pi$
$101$	−106.867			−1.05809			−0.529046	+	0.848593i	$0.677450\pi$
$102$		0			0
$103$			63.2367i			0.613949i	0.951718	+	0.306974i	$0.0993166\pi$
$103$			63.2367i			0.613949i	−0.951718	+	0.306974i	$0.900683\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			69.0952i			0.645749i	0.946442	+	0.322875i	$0.104649\pi$
$107$			69.0952i			0.645749i	−0.946442	+	0.322875i	$0.895351\pi$
$108$		0			0
$109$			73.7413i			0.676526i	0.941052	+	0.338263i	$0.109839\pi$
$109$			73.7413i			0.676526i	−0.941052	+	0.338263i	$0.890161\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.2883i			0.0910467i	0.998963	+	0.0455234i	$0.0144955\pi$
$113$			10.2883i			0.0910467i	−0.998963	+	0.0455234i	$0.985504\pi$
$114$		0			0
$115$	−47.0150	−	20.8468i	−0.408826	−	0.181276i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	35.6370			0.299471
$120$		0			0
$121$	−158.745			−1.31194
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	149.110			1.17410			0.587049	−	0.809551i	$-0.300290\pi$
$127$	149.110			1.17410			0.587049	+	0.809551i	$0.300290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−31.3444			−0.239271			−0.119635	−	0.992818i	$-0.538173\pi$
$131$	−31.3444			−0.239271			−0.119635	+	0.992818i	$0.538173\pi$
$132$		0			0
$133$	22.6421			0.170241
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	20.4938i		−	0.149590i	−0.997199	−	0.0747951i	$-0.976170\pi$
$137$		−	20.4938i		−	0.149590i	0.997199	−	0.0747951i	$-0.0238303\pi$
$138$		0			0
$139$	−3.55870			−0.0256021			−0.0128011	−	0.999918i	$-0.504075\pi$
$139$	−3.55870			−0.0256021			−0.0128011	+	0.999918i	$0.504075\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	54.8693i		−	0.383701i
$144$		0			0
$145$			104.300i			0.719310i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	43.7333i		−	0.293512i	−0.989173	−	0.146756i	$-0.953117\pi$
$149$		−	43.7333i		−	0.293512i	0.989173	−	0.146756i	$-0.0468833\pi$
$150$		0			0
$151$	−241.291			−1.59795			−0.798976	−	0.601363i	$-0.794625\pi$
$151$	−241.291			−1.59795			−0.798976	+	0.601363i	$0.794625\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			43.1452i			0.278356i
$156$		0			0
$157$		−	237.304i		−	1.51149i	−0.654865	−	0.755746i	$-0.727275\pi$
$157$		−	237.304i		−	1.51149i	0.654865	−	0.755746i	$-0.272725\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	94.7235	+	42.0011i	0.588344	+	0.260876i
$162$		0			0
$163$	−268.485			−1.64715			−0.823573	−	0.567211i	$-0.808023\pi$
$163$	−268.485			−1.64715			−0.823573	+	0.567211i	$0.808023\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−129.062			−0.772824			−0.386412	−	0.922326i	$-0.626286\pi$
$167$	−129.062			−0.772824			−0.386412	+	0.922326i	$0.626286\pi$
$168$		0			0
$169$	−158.238			−0.936319
$170$		0			0
$171$		0			0
$172$		0			0
$173$	261.404			1.51101			0.755504	−	0.655144i	$-0.227392\pi$
$173$	261.404			1.51101			0.755504	+	0.655144i	$0.227392\pi$
$174$		0			0
$175$		−	22.5256i		−	0.128718i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−212.837			−1.18903			−0.594517	−	0.804083i	$-0.702657\pi$
$179$	−212.837			−1.18903			−0.594517	+	0.804083i	$0.702657\pi$
$180$		0			0
$181$			142.181i			0.785530i	0.919639	+	0.392765i	$0.128481\pi$
$181$			142.181i			0.785530i	−0.919639	+	0.392765i	$0.871519\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	17.1149			0.0925132
$186$		0			0
$187$	−132.305			−0.707515
$188$		0			0
$189$		0			0
$190$		0			0
$191$			374.861i			1.96262i	0.192431	+	0.981310i	$0.438363\pi$
$191$			374.861i			1.96262i	−0.192431	+	0.981310i	$0.561637\pi$
$192$		0			0
$193$	62.3779			0.323202			0.161601	−	0.986856i	$-0.448334\pi$
$193$	62.3779			0.323202			0.161601	+	0.986856i	$0.448334\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−65.4646			−0.332308			−0.166154	−	0.986100i	$-0.553135\pi$
$197$	−65.4646			−0.332308			−0.166154	+	0.986100i	$0.553135\pi$
$198$		0			0
$199$			11.2001i			0.0562818i	0.999604	+	0.0281409i	$0.00895871\pi$
$199$			11.2001i			0.0562818i	−0.999604	+	0.0281409i	$0.991041\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	210.138i		−	1.03516i
$204$		0			0
$205$			100.026i			0.487933i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−84.0604			−0.402203
$210$		0			0
$211$	−122.214			−0.579214			−0.289607	−	0.957146i	$-0.593525\pi$
$211$	−122.214			−0.579214			−0.289607	+	0.957146i	$0.593525\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	74.4450			0.346256
$216$		0			0
$217$		−	86.9267i		−	0.400584i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	25.9504i		−	0.117423i
$222$		0			0
$223$	−56.4834			−0.253289			−0.126644	−	0.991948i	$-0.540421\pi$
$223$	−56.4834			−0.253289			−0.126644	+	0.991948i	$0.540421\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	50.9497i		−	0.224448i	−0.993683	−	0.112224i	$-0.964203\pi$
$227$		−	50.9497i		−	0.224448i	0.993683	−	0.112224i	$-0.0357974\pi$
$228$		0			0
$229$			250.713i			1.09482i	0.836865	+	0.547409i	$0.184386\pi$
$229$			250.713i			1.09482i	−0.836865	+	0.547409i	$0.815614\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	105.830			0.454206			0.227103	−	0.973871i	$-0.427075\pi$
$233$	105.830			0.454206			0.227103	+	0.973871i	$0.427075\pi$
$234$		0			0
$235$			76.1154i			0.323895i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	198.228			0.829405			0.414703	−	0.909957i	$-0.363886\pi$
$239$	198.228			0.829405			0.414703	+	0.909957i	$0.363886\pi$
$240$		0			0
$241$			141.253i			0.586111i	0.956095	+	0.293056i	$0.0946721\pi$
$241$			141.253i			0.586111i	−0.956095	+	0.293056i	$0.905328\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	64.1839i		−	0.261975i
$246$		0			0
$247$		−	16.4876i		−	0.0667515i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	313.146i		−	1.24759i	−0.781587	−	0.623797i	$-0.785589\pi$
$251$		−	313.146i		−	1.24759i	0.781587	−	0.623797i	$-0.214411\pi$
$252$		0			0
$253$	−351.668	−	155.932i	−1.38999	−	0.616333i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−220.699			−0.858751			−0.429376	−	0.903126i	$-0.641266\pi$
$257$	−220.699			−0.858751			−0.429376	+	0.903126i	$0.641266\pi$
$258$		0			0
$259$	−34.4823			−0.133136
$260$		0			0
$261$		0			0
$262$		0			0
$263$			23.3556i			0.0888047i	0.999014	+	0.0444023i	$0.0141384\pi$
$263$			23.3556i			0.0888047i	−0.999014	+	0.0444023i	$0.985862\pi$
$264$		0			0
$265$	−123.254			−0.465109
$266$		0			0
$267$		0			0
$268$		0			0
$269$	36.7299			0.136542			0.0682712	−	0.997667i	$-0.478252\pi$
$269$	36.7299			0.136542			0.0682712	+	0.997667i	$0.478252\pi$
$270$		0			0
$271$	−409.730			−1.51192			−0.755959	−	0.654619i	$-0.772829\pi$
$271$	−409.730			−1.51192			−0.755959	+	0.654619i	$0.772829\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			83.6279i			0.304102i
$276$		0			0
$277$	81.7757			0.295219			0.147610	−	0.989046i	$-0.452842\pi$
$277$	81.7757			0.295219			0.147610	+	0.989046i	$0.452842\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			162.029i			0.576617i	0.957538	+	0.288308i	$0.0930928\pi$
$281$			162.029i			0.576617i	−0.957538	+	0.288308i	$0.906907\pi$
$282$		0			0
$283$		−	390.052i		−	1.37828i	−0.724630	−	0.689138i	$-0.757989\pi$
$283$		−	390.052i		−	1.37828i	0.724630	−	0.689138i	$-0.242011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	201.528i		−	0.702187i
$288$		0			0
$289$	226.426			0.783482
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.07936i		−	0.0309876i	−0.999880	−	0.0154938i	$-0.995068\pi$
$293$		−	9.07936i		−	0.0309876i	0.999880	−	0.0154938i	$-0.00493203\pi$
$294$		0			0
$295$		−	93.4689i		−	0.316844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	30.5846	−	68.9762i	0.102290	−	0.230690i
$300$		0			0
$301$	−149.988			−0.498299
$302$		0			0
$303$		0			0
$304$		0			0
$305$	135.989			0.445864
$306$		0			0
$307$	−347.241			−1.13108			−0.565538	−	0.824722i	$-0.691332\pi$
$307$	−347.241			−1.13108			−0.565538	+	0.824722i	$0.691332\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−446.607			−1.43604			−0.718018	−	0.696024i	$-0.754951\pi$
$311$	−446.607			−1.43604			−0.718018	+	0.696024i	$0.754951\pi$
$312$		0			0
$313$			196.972i			0.629303i	0.949207	+	0.314652i	$0.101888\pi$
$313$			196.972i			0.629303i	−0.949207	+	0.314652i	$0.898112\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	267.299			0.843214			0.421607	−	0.906779i	$-0.361466\pi$
$317$	267.299			0.843214			0.421607	+	0.906779i	$0.361466\pi$
$318$		0			0
$319$			780.154i			2.44563i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−39.7563			−0.123085
$324$		0			0
$325$	−16.4028			−0.0504702
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	153.354i		−	0.466120i
$330$		0			0
$331$	−323.583			−0.977591			−0.488796	−	0.872398i	$-0.662564\pi$
$331$	−323.583			−0.977591			−0.488796	+	0.872398i	$0.662564\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	222.117			0.663036
$336$		0			0
$337$			429.028i			1.27308i	0.771243	+	0.636540i	$0.219635\pi$
$337$			429.028i			1.27308i	−0.771243	+	0.636540i	$0.780365\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			322.722i			0.946399i
$342$		0			0
$343$			350.065i			1.02060i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	28.9140			0.0833257			0.0416628	−	0.999132i	$-0.486734\pi$
$347$	28.9140			0.0833257			0.0416628	+	0.999132i	$0.486734\pi$
$348$		0			0
$349$	−459.116			−1.31552			−0.657760	−	0.753228i	$-0.728496\pi$
$349$	−459.116			−1.31552			−0.657760	+	0.753228i	$0.728496\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−497.975			−1.41069			−0.705347	−	0.708862i	$-0.749209\pi$
$353$	−497.975			−1.41069			−0.705347	+	0.708862i	$0.749209\pi$
$354$		0			0
$355$		−	168.874i		−	0.475702i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.5119i			0.0682782i	0.999417	+	0.0341391i	$0.0108689\pi$
$359$			24.5119i			0.0682782i	−0.999417	+	0.0341391i	$0.989131\pi$
$360$		0			0
$361$	335.741			0.930030
$362$		0			0
$363$		0			0
$364$		0			0
$365$			10.3646i			0.0283960i
$366$		0			0
$367$			506.076i			1.37895i	0.724308	+	0.689477i	$0.242159\pi$
$367$			506.076i			1.37895i	−0.724308	+	0.689477i	$0.757841\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	248.325			0.669341
$372$		0			0
$373$			141.697i			0.379885i	0.981795	+	0.189942i	$0.0608302\pi$
$373$			141.697i			0.379885i	−0.981795	+	0.189942i	$0.939170\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−153.020			−0.405888
$378$		0			0
$379$		−	488.878i		−	1.28992i	−0.764218	−	0.644958i	$-0.776875\pi$
$379$		−	488.878i		−	1.28992i	0.764218	−	0.644958i	$-0.223125\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			14.6483i			0.0382462i	0.999817	+	0.0191231i	$0.00608744\pi$
$383$			14.6483i			0.0382462i	−0.999817	+	0.0191231i	$0.993913\pi$
$384$		0			0
$385$		−	168.489i		−	0.437635i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			172.342i			0.443037i	0.975156	+	0.221519i	$0.0711014\pi$
$389$			172.342i			0.443037i	−0.975156	+	0.221519i	$0.928899\pi$
$390$		0			0
$391$	−166.321	−	73.7480i	−0.425374	−	0.188614i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−328.136			−0.830723
$396$		0			0
$397$	175.371			0.441742			0.220871	−	0.975303i	$-0.429110\pi$
$397$	175.371			0.441742			0.220871	+	0.975303i	$0.429110\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			449.214i			1.12024i	0.828413	+	0.560118i	$0.189244\pi$
$401$			449.214i			1.12024i	−0.828413	+	0.560118i	$0.810756\pi$
$402$		0			0
$403$	−63.2987			−0.157069
$404$		0			0
$405$		0			0
$406$		0			0
$407$	128.018			0.314541
$408$		0			0
$409$	−620.134			−1.51622			−0.758110	−	0.652127i	$-0.773877\pi$
$409$	−620.134			−1.51622			−0.758110	+	0.652127i	$0.773877\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			188.316i			0.455972i
$414$		0			0
$415$	213.201			0.513738
$416$		0			0
$417$		0			0
$418$		0			0
$419$			206.896i			0.493784i	0.969043	+	0.246892i	$0.0794093\pi$
$419$			206.896i			0.493784i	−0.969043	+	0.246892i	$0.920591\pi$
$420$		0			0
$421$		−	220.733i		−	0.524307i	−0.965026	−	0.262153i	$-0.915567\pi$
$421$		−	220.733i		−	0.524307i	0.965026	−	0.262153i	$-0.0844327\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			39.5517i			0.0930629i
$426$		0			0
$427$	−273.983			−0.641646
$428$		0			0
$429$		0			0
$430$		0			0
$431$			90.1264i			0.209110i	0.994519	+	0.104555i	$0.0333418\pi$
$431$			90.1264i			0.209110i	−0.994519	+	0.104555i	$0.966658\pi$
$432$		0			0
$433$			348.623i			0.805133i	0.915391	+	0.402567i	$0.131882\pi$
$433$			348.623i			0.805133i	−0.915391	+	0.402567i	$0.868118\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−105.672	−	46.8559i	−0.241813	−	0.107222i
$438$		0			0
$439$	−193.342			−0.440414			−0.220207	−	0.975453i	$-0.570673\pi$
$439$	−193.342			−0.440414			−0.220207	+	0.975453i	$0.570673\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	639.655			1.44392			0.721958	−	0.691937i	$-0.243243\pi$
$443$	639.655			1.44392			0.721958	+	0.691937i	$0.243243\pi$
$444$		0			0
$445$	−42.3846			−0.0952464
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−354.736			−0.790058			−0.395029	−	0.918669i	$-0.629265\pi$
$449$	−354.736			−0.790058			−0.395029	+	0.918669i	$0.629265\pi$
$450$		0			0
$451$			748.187i			1.65895i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	33.0475			0.0726319
$456$		0			0
$457$		−	592.163i		−	1.29576i	−0.761742	−	0.647881i	$-0.775655\pi$
$457$		−	592.163i		−	1.29576i	0.761742	−	0.647881i	$-0.224345\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	340.832			0.739332			0.369666	−	0.929165i	$-0.379472\pi$
$461$	340.832			0.739332			0.369666	+	0.929165i	$0.379472\pi$
$462$		0			0
$463$	−180.466			−0.389776			−0.194888	−	0.980826i	$-0.562434\pi$
$463$	−180.466			−0.389776			−0.194888	+	0.980826i	$0.562434\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	641.661i		−	1.37401i	−0.726654	−	0.687003i	$-0.758926\pi$
$467$		−	641.661i		−	1.37401i	0.726654	−	0.687003i	$-0.241074\pi$
$468$		0			0
$469$	−447.510			−0.954180
$470$		0			0
$471$		0			0
$472$		0			0
$473$	556.842			1.17725
$474$		0			0
$475$			25.1293i			0.0529038i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			6.67841i			0.0139424i	0.999976	+	0.00697120i	$0.00221902\pi$
$479$			6.67841i			0.0139424i	−0.999976	+	0.00697120i	$0.997781\pi$
$480$		0			0
$481$			25.1095i			0.0522027i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−88.4063			−0.182281
$486$		0			0
$487$	−315.161			−0.647147			−0.323574	−	0.946203i	$-0.604884\pi$
$487$	−315.161			−0.647147			−0.323574	+	0.946203i	$0.604884\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	27.8038			0.0566268			0.0283134	−	0.999599i	$-0.490986\pi$
$491$	27.8038			0.0566268			0.0283134	+	0.999599i	$0.490986\pi$
$492$		0			0
$493$			368.973i			0.748425i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			340.240i			0.684587i
$498$		0			0
$499$	210.058			0.420958			0.210479	−	0.977598i	$-0.432498\pi$
$499$	210.058			0.420958			0.210479	+	0.977598i	$0.432498\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			96.4559i			0.191761i	0.995393	+	0.0958807i	$0.0305667\pi$
$503$			96.4559i			0.191761i	−0.995393	+	0.0958807i	$0.969433\pi$
$504$		0			0
$505$			238.963i			0.473193i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	568.643			1.11718			0.558589	−	0.829445i	$-0.311343\pi$
$509$	568.643			1.11718			0.558589	+	0.829445i	$0.311343\pi$
$510$		0			0
$511$		−	20.8820i		−	0.0408649i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	141.402			0.274566
$516$		0			0
$517$			569.336i			1.10123i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	40.7079i		−	0.0781342i	−0.999237	−	0.0390671i	$-0.987561\pi$
$521$		−	40.7079i		−	0.0781342i	0.999237	−	0.0390671i	$-0.0124386\pi$
$522$		0			0
$523$			768.320i			1.46906i	0.678575	+	0.734531i	$0.262598\pi$
$523$			768.320i			1.46906i	−0.678575	+	0.734531i	$0.737402\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			152.631i			0.289622i
$528$		0			0
$529$	−355.164	−	392.045i	−0.671388	−	0.741106i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−146.750			−0.275327
$534$		0			0
$535$	154.501			0.288788
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	480.090i		−	0.890705i
$540$		0			0
$541$	713.405			1.31868			0.659340	−	0.751845i	$-0.270836\pi$
$541$	713.405			1.31868			0.659340	+	0.751845i	$0.270836\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	164.891			0.302552
$546$		0			0
$547$	−845.349			−1.54543			−0.772714	−	0.634755i	$-0.781101\pi$
$547$	−845.349			−1.54543			−0.772714	+	0.634755i	$0.781101\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			234.428i			0.425459i
$552$		0			0
$553$	661.111			1.19550
$554$		0			0
$555$		0			0
$556$		0			0
$557$			20.4984i			0.0368014i	0.999831	+	0.0184007i	$0.00585745\pi$
$557$			20.4984i			0.0368014i	−0.999831	+	0.0184007i	$0.994143\pi$
$558$		0			0
$559$			109.219i			0.195383i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	696.419i		−	1.23698i	−0.785794	−	0.618489i	$-0.787745\pi$
$563$		−	696.419i		−	1.23698i	0.785794	−	0.618489i	$-0.212255\pi$
$564$		0			0
$565$	23.0053			0.0407173
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	164.123i		−	0.288441i	−0.989546	−	0.144221i	$-0.953932\pi$
$569$		−	164.123i		−	0.288441i	0.989546	−	0.144221i	$-0.0460675\pi$
$570$		0			0
$571$		−	356.989i		−	0.625199i	−0.949885	−	0.312600i	$-0.898800\pi$
$571$		−	356.989i		−	0.625199i	0.949885	−	0.312600i	$-0.101200\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−46.6149	+	105.129i	−0.0810693	+	0.182833i
$576$		0			0
$577$	−955.102			−1.65529			−0.827645	−	0.561252i	$-0.810320\pi$
$577$	−955.102			−1.65529			−0.827645	+	0.561252i	$0.810320\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−429.547			−0.739324
$582$		0			0
$583$	−921.927			−1.58135
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−943.977			−1.60814			−0.804069	−	0.594536i	$-0.797336\pi$
$587$	−943.977			−1.60814			−0.804069	+	0.594536i	$0.797336\pi$
$588$		0			0
$589$			96.9745i			0.164643i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	282.505			0.476399			0.238199	−	0.971216i	$-0.423443\pi$
$593$	282.505			0.476399			0.238199	+	0.971216i	$0.423443\pi$
$594$		0			0
$595$		−	79.6869i		−	0.133928i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−872.279			−1.45622			−0.728112	−	0.685458i	$-0.759602\pi$
$599$	−872.279			−1.45622			−0.728112	+	0.685458i	$0.759602\pi$
$600$		0			0
$601$	421.219			0.700863			0.350432	−	0.936588i	$-0.386035\pi$
$601$	421.219			0.700863			0.350432	+	0.936588i	$0.386035\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			354.965i			0.586719i
$606$		0			0
$607$	−527.228			−0.868581			−0.434290	−	0.900773i	$-0.643001\pi$
$607$	−527.228			−0.868581			−0.434290	+	0.900773i	$0.643001\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−111.670			−0.182766
$612$		0			0
$613$		−	932.715i		−	1.52156i	−0.649011	−	0.760779i	$-0.724817\pi$
$613$		−	932.715i		−	1.52156i	0.649011	−	0.760779i	$-0.275183\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	666.640i		−	1.08045i	−0.841519	−	0.540227i	$-0.818338\pi$
$617$		−	666.640i		−	1.08045i	0.841519	−	0.540227i	$-0.181662\pi$
$618$		0			0
$619$		−	368.300i		−	0.594992i	−0.954723	−	0.297496i	$-0.903849\pi$
$619$		−	368.300i		−	0.594992i	0.954723	−	0.297496i	$-0.0961515\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	85.3944			0.137070
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	60.5461			0.0962577
$630$		0			0
$631$			668.239i			1.05902i	0.848305	+	0.529508i	$0.177624\pi$
$631$			668.239i			1.05902i	−0.848305	+	0.529508i	$0.822376\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	333.421i		−	0.525073i
$636$		0			0
$637$	94.1649			0.147826
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	7.09640i		−	0.0110708i	−0.999985	−	0.00553541i	$-0.998238\pi$
$641$		−	7.09640i		−	0.0110708i	0.999985	−	0.00553541i	$-0.00176199\pi$
$642$		0			0
$643$		−	349.594i		−	0.543692i	−0.962341	−	0.271846i	$-0.912366\pi$
$643$		−	349.594i		−	0.543692i	0.962341	−	0.271846i	$-0.0876342\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1161.51			−1.79523			−0.897613	−	0.440785i	$-0.854700\pi$
$647$	−1161.51			−1.79523			−0.897613	+	0.440785i	$0.854700\pi$
$648$		0			0
$649$		−	699.138i		−	1.07725i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−973.812			−1.49129			−0.745644	−	0.666344i	$-0.767858\pi$
$653$	−973.812			−1.49129			−0.745644	+	0.666344i	$0.767858\pi$
$654$		0			0
$655$			70.0883i			0.107005i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	643.261i		−	0.976117i	−0.872811	−	0.488059i	$-0.837705\pi$
$659$		−	643.261i		−	0.976117i	0.872811	−	0.488059i	$-0.162295\pi$
$660$		0			0
$661$		−	709.808i		−	1.07384i	−0.843633	−	0.536919i	$-0.819588\pi$
$661$		−	709.808i		−	1.07384i	0.843633	−	0.536919i	$-0.180412\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	50.6292i		−	0.0761342i
$666$		0			0
$667$	−434.864	+	980.733i	−0.651970	+	1.47036i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1017.18			1.51592
$672$		0			0
$673$	−171.463			−0.254774			−0.127387	−	0.991853i	$-0.540659\pi$
$673$	−171.463			−0.254774			−0.127387	+	0.991853i	$0.540659\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1311.57i		−	1.93732i	−0.248392	−	0.968660i	$-0.579902\pi$
$677$		−	1311.57i		−	1.93732i	0.248392	−	0.968660i	$-0.420098\pi$
$678$		0			0
$679$	178.117			0.262322
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1251.08			−1.83174			−0.915872	−	0.401470i	$-0.868499\pi$
$683$	−1251.08			−1.83174			−0.915872	+	0.401470i	$0.868499\pi$
$684$		0			0
$685$	−45.8256			−0.0668987
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	180.827i		−	0.262448i
$690$		0			0
$691$	458.895			0.664103			0.332052	−	0.943261i	$-0.392259\pi$
$691$	458.895			0.664103			0.332052	+	0.943261i	$0.392259\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			7.95748i			0.0114496i
$696$		0			0
$697$			353.854i			0.507682i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1065.87i			1.52050i	0.649629	+	0.760252i	$0.274924\pi$
$701$			1065.87i			1.52050i	−0.649629	+	0.760252i	$0.725076\pi$
$702$		0			0
$703$	38.4681			0.0547199
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	481.450i		−	0.680976i
$708$		0			0
$709$			1291.74i			1.82192i	0.412490	+	0.910962i	$0.364659\pi$
$709$			1291.74i			1.82192i	−0.412490	+	0.910962i	$0.635341\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−179.888	+	405.694i	−0.252297	+	0.568996i
$714$		0			0
$715$	−122.691			−0.171596
$716$		0			0
$717$		0			0
$718$		0			0
$719$	832.051			1.15723			0.578617	−	0.815599i	$-0.303593\pi$
$719$	832.051			1.15723			0.578617	+	0.815599i	$0.303593\pi$
$720$		0			0
$721$	−284.889			−0.395130
$722$		0			0
$723$		0			0
$724$		0			0
$725$	233.222			0.321685
$726$		0			0
$727$		−	254.392i		−	0.349920i	−0.984576	−	0.174960i	$-0.944020\pi$
$727$		−	254.392i		−	0.349920i	0.984576	−	0.174960i	$-0.0559796\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	263.358			0.360270
$732$		0			0
$733$		−	1009.28i		−	1.37691i	−0.725277	−	0.688457i	$-0.758288\pi$
$733$		−	1009.28i		−	1.37691i	0.725277	−	0.688457i	$-0.241712\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1661.42			2.25430
$738$		0			0
$739$	421.530			0.570407			0.285203	−	0.958467i	$-0.407939\pi$
$739$	421.530			0.570407			0.285203	+	0.958467i	$0.407939\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			232.509i			0.312932i	0.987683	+	0.156466i	$0.0500102\pi$
$743$			232.509i			0.312932i	−0.987683	+	0.156466i	$0.949990\pi$
$744$		0			0
$745$	−97.7907			−0.131263
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−311.282			−0.415596
$750$		0			0
$751$		−	497.992i		−	0.663105i	−0.943437	−	0.331552i	$-0.892428\pi$
$751$		−	497.992i		−	0.663105i	0.943437	−	0.331552i	$-0.107572\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			539.543i			0.714626i
$756$		0			0
$757$		−	75.3812i		−	0.0995788i	−0.998760	−	0.0497894i	$-0.984145\pi$
$757$		−	75.3812i		−	0.0995788i	0.998760	−	0.0497894i	$-0.0158550\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1074.06			1.41138			0.705688	−	0.708523i	$-0.250638\pi$
$761$	1074.06			1.41138			0.705688	+	0.708523i	$0.250638\pi$
$762$		0			0
$763$	−332.213			−0.435404
$764$		0			0
$765$		0			0
$766$		0			0
$767$	137.129			0.178786
$768$		0			0
$769$		−	107.714i		−	0.140070i	−0.997545	−	0.0700349i	$-0.977689\pi$
$769$		−	107.714i		−	0.140070i	0.997545	−	0.0700349i	$-0.0223111\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	796.138i		−	1.02993i	−0.857210	−	0.514966i	$-0.827804\pi$
$773$		−	796.138i		−	1.02993i	0.857210	−	0.514966i	$-0.172196\pi$
$774$		0			0
$775$	96.4755			0.124485
$776$		0			0
$777$		0			0
$778$		0			0
$779$			224.822i			0.288604i
$780$		0			0
$781$		−	1263.16i		−	1.61737i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−530.628			−0.675960
$786$		0			0
$787$			92.6042i			0.117667i	0.998268	+	0.0588337i	$0.0187382\pi$
$787$			92.6042i			0.117667i	−0.998268	+	0.0588337i	$0.981262\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−46.3499			−0.0585966
$792$		0			0
$793$			199.510i			0.251589i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	796.483i		−	0.999352i	−0.866213	−	0.499676i	$-0.833452\pi$
$797$		−	796.483i		−	0.999352i	0.866213	−	0.499676i	$-0.166548\pi$
$798$		0			0
$799$			269.267i			0.337005i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			77.5259i			0.0965454i
$804$		0			0
$805$	93.9172	−	211.808i	0.116667	−	0.263116i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	279.321			0.345268			0.172634	−	0.984986i	$-0.444772\pi$
$809$	279.321			0.345268			0.172634	+	0.984986i	$0.444772\pi$
$810$		0			0
$811$	359.059			0.442736			0.221368	−	0.975190i	$-0.428948\pi$
$811$	359.059			0.442736			0.221368	+	0.975190i	$0.428948\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			600.350i			0.736626i
$816$		0			0
$817$	167.325			0.204804
$818$		0			0
$819$		0			0
$820$		0			0
$821$	566.638			0.690180			0.345090	−	0.938570i	$-0.387848\pi$
$821$	566.638			0.690180			0.345090	+	0.938570i	$0.387848\pi$
$822$		0			0
$823$	−127.948			−0.155466			−0.0777329	−	0.996974i	$-0.524768\pi$
$823$	−127.948			−0.155466			−0.0777329	+	0.996974i	$0.524768\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.0636i			0.0145872i	0.999973	+	0.00729360i	$0.00232164\pi$
$827$			12.0636i			0.0145872i	−0.999973	+	0.00729360i	$0.997678\pi$
$828$		0			0
$829$	840.795			1.01423			0.507114	−	0.861879i	$-0.330712\pi$
$829$	840.795			1.01423			0.507114	+	0.861879i	$0.330712\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	227.058i		−	0.272579i
$834$		0			0
$835$			288.591i			0.345617i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	158.463i		−	0.188872i	−0.995531	−	0.0944358i	$-0.969895\pi$
$839$		−	158.463i		−	0.188872i	0.995531	−	0.0944358i	$-0.0301047\pi$
$840$		0			0
$841$	1334.70			1.58704
$842$		0			0
$843$		0			0
$844$		0			0
$845$			353.831i			0.418735i
$846$		0			0
$847$		−	715.166i		−	0.844351i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	160.932	+	71.3583i	0.189109	+	0.0838523i
$852$		0			0
$853$	450.469			0.528099			0.264050	−	0.964509i	$-0.414942\pi$
$853$	450.469			0.528099			0.264050	+	0.964509i	$0.414942\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−838.563			−0.978487			−0.489243	−	0.872147i	$-0.662727\pi$
$857$	−838.563			−0.978487			−0.489243	+	0.872147i	$0.662727\pi$
$858$		0			0
$859$	−743.346			−0.865362			−0.432681	−	0.901547i	$-0.642432\pi$
$859$	−743.346			−0.865362			−0.432681	+	0.901547i	$0.642432\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	920.734			1.06690			0.533450	−	0.845832i	$-0.320895\pi$
$863$	920.734			1.06690			0.533450	+	0.845832i	$0.320895\pi$
$864$		0			0
$865$		−	584.518i		−	0.675743i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2454.42			−2.82442
$870$		0			0
$871$			325.870i			0.374134i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−50.3687			−0.0575643
$876$		0			0
$877$	952.517			1.08611			0.543054	−	0.839698i	$-0.317268\pi$
$877$	952.517			1.08611			0.543054	+	0.839698i	$0.317268\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1233.48i			1.40009i	0.714099	+	0.700045i	$0.246837\pi$
$881$			1233.48i			1.40009i	−0.714099	+	0.700045i	$0.753163\pi$
$882$		0			0
$883$	−1645.24			−1.86324			−0.931622	−	0.363428i	$-0.881606\pi$
$883$	−1645.24			−1.86324			−0.931622	+	0.363428i	$0.881606\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1412.37			1.59230			0.796148	−	0.605102i	$-0.206868\pi$
$887$	1412.37			1.59230			0.796148	+	0.605102i	$0.206868\pi$
$888$		0			0
$889$			671.760i			0.755636i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			171.079i			0.191578i
$894$		0			0
$895$			475.918i			0.531752i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	900.008			1.00112
$900$		0			0
$901$	−436.025			−0.483934
$902$		0			0
$903$		0			0
$904$		0			0
$905$	317.926			0.351300
$906$		0			0
$907$			23.7746i			0.0262124i	0.999914	+	0.0131062i	$0.00417195\pi$
$907$			23.7746i			0.0262124i	−0.999914	+	0.0131062i	$0.995828\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			907.624i			0.996295i	0.867092	+	0.498147i	$0.165986\pi$
$911$			907.624i			0.996295i	−0.867092	+	0.498147i	$0.834014\pi$
$912$		0			0
$913$	1594.73			1.74669
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	141.210i		−	0.153992i
$918$		0			0
$919$			443.884i			0.483008i	0.970400	+	0.241504i	$0.0776407\pi$
$919$			443.884i			0.483008i	−0.970400	+	0.241504i	$0.922359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	247.757			0.268426
$924$		0			0
$925$		−	38.2702i		−	0.0413732i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−687.674			−0.740230			−0.370115	−	0.928986i	$-0.620682\pi$
$929$	−687.674			−0.740230			−0.370115	+	0.928986i	$0.620682\pi$
$930$		0			0
$931$		−	144.262i		−	0.154954i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			295.843i			0.316410i
$936$		0			0
$937$		−	335.833i		−	0.358413i	−0.983811	−	0.179207i	$-0.942647\pi$
$937$		−	335.833i		−	0.358413i	0.983811	−	0.179207i	$-0.0573530\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	0.00945708i		−	1.00500e-5i	−1.00000		5.02502e-6i	$-0.999998\pi$
$941$		−	0.00945708i		−	1.00500e-5i	1.00000		5.02502e-6i	$-1.59951e-6\pi$
$942$		0			0
$943$	−417.045	+	940.547i	−0.442254	+	0.997398i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1303.23			−1.37617			−0.688084	−	0.725631i	$-0.741548\pi$
$947$	−1303.23			−1.37617			−0.688084	+	0.725631i	$0.741548\pi$
$948$		0			0
$949$	−15.2060			−0.0160231
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1053.25i			1.10519i	0.833449	+	0.552597i	$0.186363\pi$
$953$			1053.25i			1.10519i	−0.833449	+	0.552597i	$0.813637\pi$
$954$		0			0
$955$	838.214			0.877711
$956$		0			0
$957$		0			0
$958$		0			0
$959$	92.3272			0.0962744
$960$		0			0
$961$	−588.699			−0.612590
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	139.481i		−	0.144540i
$966$		0			0
$967$	1599.40			1.65398			0.826990	−	0.562216i	$-0.190051\pi$
$967$	1599.40			1.65398			0.826990	+	0.562216i	$0.190051\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1031.90i			1.06272i	0.847146	+	0.531361i	$0.178319\pi$
$971$			1031.90i			1.06272i	−0.847146	+	0.531361i	$0.821681\pi$
$972$		0			0
$973$		−	16.0323i		−	0.0164772i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	443.622i		−	0.454066i	−0.973887	−	0.227033i	$-0.927097\pi$
$977$		−	443.622i		−	0.454066i	0.973887	−	0.227033i	$-0.0729025\pi$
$978$		0			0
$979$	−317.033			−0.323834
$980$		0			0
$981$		0			0
$982$		0			0
$983$			528.007i			0.537138i	0.963260	+	0.268569i	$0.0865507\pi$
$983$			528.007i			0.537138i	−0.963260	+	0.268569i	$0.913449\pi$
$984$		0			0
$985$			146.383i			0.148613i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	700.006	+	310.388i	0.707792	+	0.313840i
$990$		0			0
$991$	−1185.02			−1.19578			−0.597890	−	0.801578i	$-0.703994\pi$
$991$	−1185.02			−1.19578			−0.597890	+	0.801578i	$0.703994\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25.0441			0.0251700
$996$		0			0
$997$	1381.72			1.38588			0.692939	−	0.720996i	$-0.256315\pi$
$997$	1381.72			1.38588			0.692939	+	0.720996i	$0.256315\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.b.2161.11	yes	32
3.2	odd	2	inner	4140.3.d.b.2161.27	yes	32
23.22	odd	2	inner	4140.3.d.b.2161.22	yes	32
69.68	even	2	inner	4140.3.d.b.2161.6	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.d.b.2161.6	✓	32	69.68	even	2	inner
4140.3.d.b.2161.11	yes	32	1.1	even	1	trivial
4140.3.d.b.2161.22	yes	32	23.22	odd	2	inner
4140.3.d.b.2161.27	yes	32	3.2	odd	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +4.50512i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +4.50512i q^{7} -16.7256i q^{11} +3.28056 q^{13} -7.91035i q^{17} -5.02586i q^{19} +(9.32297 - 21.0258i) q^{23} -5.00000 q^{25} -46.6444 q^{29} -19.2951 q^{31} +10.0737 q^{35} +7.65403i q^{37} -44.7331 q^{41} +33.2928i q^{43} -34.0399 q^{47} +28.7039 q^{49} -55.1208i q^{53} -37.3995 q^{55} +41.8005 q^{59} +60.8160i q^{61} -7.33555i q^{65} +99.3338i q^{67} +75.5229 q^{71} -4.63517 q^{73} +75.3507 q^{77} -146.747i q^{79} +95.3465i q^{83} -17.6881 q^{85} -18.9550i q^{89} +14.7793i q^{91} -11.2382 q^{95} -39.5365i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 24 q^{13} - 160 q^{25} - 28 q^{31} - 260 q^{49} + 120 q^{55} - 296 q^{73} - 60 q^{85}+O(q^{100}) \) 32 * q - 24 * q^13 - 160 * q^25 - 28 * q^31 - 260 * q^49 + 120 * q^55 - 296 * q^73 - 60 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more