LMFDB - Embedded newform 4140.3.d.c.2161.8

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:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.8
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -1.24316i q^{7} +O(q^{10})$	$q-2.23607i q^{5} -1.24316i q^{7} -0.645254i q^{11} +4.44571 q^{13} +24.6738i q^{17} -35.0975i q^{19} +(21.9604 - 6.83676i) q^{23} -5.00000 q^{25} +27.8373 q^{29} -2.38638 q^{31} -2.77979 q^{35} -23.0724i q^{37} -12.6215 q^{41} +30.4211i q^{43} -47.1500 q^{47} +47.4546 q^{49} +52.3772i q^{53} -1.44283 q^{55} -64.7943 q^{59} -26.0902i q^{61} -9.94092i q^{65} -5.24177i q^{67} +53.8322 q^{71} +39.3733 q^{73} -0.802155 q^{77} -83.2069i q^{79} +110.729i q^{83} +55.1723 q^{85} -8.64323i q^{89} -5.52674i q^{91} -78.4805 q^{95} -64.9425i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) $ 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

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
$7$		−	1.24316i		−	0.177594i	−0.996050	−	0.0887972i	$-0.971698\pi$
$7$		−	1.24316i		−	0.177594i	0.996050	−	0.0887972i	$-0.0283023\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.645254i		−	0.0586595i	−0.999570	−	0.0293297i	$-0.990663\pi$
$11$		−	0.645254i		−	0.0586595i	0.999570	−	0.0293297i	$-0.00933728\pi$
$12$		0			0
$13$	4.44571			0.341978			0.170989	−	0.985273i	$-0.445304\pi$
$13$	4.44571			0.341978			0.170989	+	0.985273i	$0.445304\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			24.6738i			1.45140i	0.688011	+	0.725700i	$0.258484\pi$
$17$			24.6738i			1.45140i	−0.688011	+	0.725700i	$0.741516\pi$
$18$		0			0
$19$		−	35.0975i		−	1.84724i	−0.383310	−	0.923620i	$-0.625216\pi$
$19$		−	35.0975i		−	1.84724i	0.383310	−	0.923620i	$-0.374784\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.9604	−	6.83676i	0.954800	−	0.297250i
$23$	21.9604	−	6.83676i	0.954800	−	0.297250i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	27.8373			0.959908			0.479954	−	0.877294i	$-0.340653\pi$
$29$	27.8373			0.959908			0.479954	+	0.877294i	$0.340653\pi$
$30$		0			0
$31$	−2.38638			−0.0769800			−0.0384900	−	0.999259i	$-0.512255\pi$
$31$	−2.38638			−0.0769800			−0.0384900	+	0.999259i	$0.512255\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.77979			−0.0794226
$36$		0			0
$37$		−	23.0724i		−	0.623578i	−0.950151	−	0.311789i	$-0.899072\pi$
$37$		−	23.0724i		−	0.623578i	0.950151	−	0.311789i	$-0.100928\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.6215			−0.307840			−0.153920	−	0.988083i	$-0.549190\pi$
$41$	−12.6215			−0.307840			−0.153920	+	0.988083i	$0.549190\pi$
$42$		0			0
$43$			30.4211i			0.707467i	0.935346	+	0.353734i	$0.115088\pi$
$43$			30.4211i			0.707467i	−0.935346	+	0.353734i	$0.884912\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−47.1500			−1.00319			−0.501596	−	0.865102i	$-0.667253\pi$
$47$	−47.1500			−1.00319			−0.501596	+	0.865102i	$0.667253\pi$
$48$		0			0
$49$	47.4546			0.968460
$50$		0			0
$51$		0			0
$52$		0			0
$53$			52.3772i			0.988249i	0.869391	+	0.494124i	$0.164511\pi$
$53$			52.3772i			0.988249i	−0.869391	+	0.494124i	$0.835489\pi$
$54$		0			0
$55$	−1.44283			−0.0262333
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−64.7943			−1.09821			−0.549104	−	0.835754i	$-0.685031\pi$
$59$	−64.7943			−1.09821			−0.549104	+	0.835754i	$0.685031\pi$
$60$		0			0
$61$		−	26.0902i		−	0.427709i	−0.976866	−	0.213854i	$-0.931398\pi$
$61$		−	26.0902i		−	0.427709i	0.976866	−	0.213854i	$-0.0686018\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	9.94092i		−	0.152937i
$66$		0			0
$67$		−	5.24177i		−	0.0782353i	−0.999235	−	0.0391177i	$-0.987545\pi$
$67$		−	5.24177i		−	0.0782353i	0.999235	−	0.0391177i	$-0.0124547\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	53.8322			0.758200			0.379100	−	0.925356i	$-0.376234\pi$
$71$	53.8322			0.758200			0.379100	+	0.925356i	$0.376234\pi$
$72$		0			0
$73$	39.3733			0.539361			0.269680	−	0.962950i	$-0.413082\pi$
$73$	39.3733			0.539361			0.269680	+	0.962950i	$0.413082\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.802155			−0.0104176
$78$		0			0
$79$		−	83.2069i		−	1.05325i	−0.850097	−	0.526626i	$-0.823457\pi$
$79$		−	83.2069i		−	1.05325i	0.850097	−	0.526626i	$-0.176543\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			110.729i			1.33408i	0.745022	+	0.667040i	$0.232439\pi$
$83$			110.729i			1.33408i	−0.745022	+	0.667040i	$0.767561\pi$
$84$		0			0
$85$	55.1723			0.649086
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.64323i		−	0.0971150i	−0.998820	−	0.0485575i	$-0.984538\pi$
$89$		−	8.64323i		−	0.0971150i	0.998820	−	0.0485575i	$-0.0154624\pi$
$90$		0			0
$91$		−	5.52674i		−	0.0607334i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−78.4805			−0.826110
$96$		0			0
$97$		−	64.9425i		−	0.669511i	−0.942305	−	0.334755i	$-0.891346\pi$
$97$		−	64.9425i		−	0.669511i	0.942305	−	0.334755i	$-0.108654\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	176.714			1.74965			0.874823	−	0.484442i	$-0.160977\pi$
$101$	176.714			1.74965			0.874823	+	0.484442i	$0.160977\pi$
$102$		0			0
$103$		−	42.7395i		−	0.414947i	−0.978241	−	0.207473i	$-0.933476\pi$
$103$		−	42.7395i		−	0.414947i	0.978241	−	0.207473i	$-0.0665240\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	137.408i		−	1.28419i	−0.766627	−	0.642093i	$-0.778066\pi$
$107$		−	137.408i		−	1.28419i	0.766627	−	0.642093i	$-0.221934\pi$
$108$		0			0
$109$		−	24.1903i		−	0.221929i	−0.993824	−	0.110965i	$-0.964606\pi$
$109$		−	24.1903i		−	0.221929i	0.993824	−	0.110965i	$-0.0353941\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			87.7092i			0.776187i	0.921620	+	0.388094i	$0.126866\pi$
$113$			87.7092i			0.776187i	−0.921620	+	0.388094i	$0.873134\pi$
$114$		0			0
$115$	−15.2875	−	49.1049i	−0.132934	−	0.426999i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	30.6735			0.257761
$120$		0			0
$121$	120.584			0.996559
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	114.723			0.903333			0.451666	−	0.892187i	$-0.350830\pi$
$127$	114.723			0.903333			0.451666	+	0.892187i	$0.350830\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−201.730			−1.53992			−0.769961	−	0.638091i	$-0.779725\pi$
$131$	−201.730			−1.53992			−0.769961	+	0.638091i	$0.779725\pi$
$132$		0			0
$133$	−43.6319			−0.328059
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.3300i		−	0.0754014i	−0.999289	−	0.0377007i	$-0.987997\pi$
$137$		−	10.3300i		−	0.0754014i	0.999289	−	0.0377007i	$-0.0120033\pi$
$138$		0			0
$139$	244.636			1.75997			0.879985	−	0.475002i	$-0.157553\pi$
$139$	244.636			1.75997			0.879985	+	0.475002i	$0.157553\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.86862i		−	0.0200603i
$144$		0			0
$145$		−	62.2462i		−	0.429284i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	70.3387i		−	0.472072i	−0.971744	−	0.236036i	$-0.924152\pi$
$149$		−	70.3387i		−	0.472072i	0.971744	−	0.236036i	$-0.0758484\pi$
$150$		0			0
$151$	−239.373			−1.58525			−0.792625	−	0.609709i	$-0.791286\pi$
$151$	−239.373			−1.58525			−0.792625	+	0.609709i	$0.791286\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			5.33611i			0.0344265i
$156$		0			0
$157$			10.0365i			0.0639268i	0.999489	+	0.0319634i	$0.0101760\pi$
$157$			10.0365i			0.0639268i	−0.999489	+	0.0319634i	$0.989824\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.49919	−	27.3003i	−0.0527900	−	0.169567i
$162$		0			0
$163$	−134.923			−0.827751			−0.413876	−	0.910333i	$-0.635825\pi$
$163$	−134.923			−0.827751			−0.413876	+	0.910333i	$0.635825\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	38.0772			0.228007			0.114004	−	0.993480i	$-0.463632\pi$
$167$	38.0772			0.228007			0.114004	+	0.993480i	$0.463632\pi$
$168$		0			0
$169$	−149.236			−0.883051
$170$		0			0
$171$		0			0
$172$		0			0
$173$	101.735			0.588064			0.294032	−	0.955796i	$-0.405003\pi$
$173$	101.735			0.588064			0.294032	+	0.955796i	$0.405003\pi$
$174$		0			0
$175$			6.21580i			0.0355189i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	267.620			1.49509			0.747543	−	0.664213i	$-0.231233\pi$
$179$	267.620			1.49509			0.747543	+	0.664213i	$0.231233\pi$
$180$		0			0
$181$		−	192.444i		−	1.06323i	−0.846987	−	0.531614i	$-0.821586\pi$
$181$		−	192.444i		−	1.06323i	0.846987	−	0.531614i	$-0.178414\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−51.5914			−0.278872
$186$		0			0
$187$	15.9209			0.0851384
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	174.564i		−	0.913947i	−0.889480	−	0.456974i	$-0.848933\pi$
$191$		−	174.564i		−	0.913947i	0.889480	−	0.456974i	$-0.151067\pi$
$192$		0			0
$193$	−193.243			−1.00126			−0.500630	−	0.865661i	$-0.666898\pi$
$193$	−193.243			−1.00126			−0.500630	+	0.865661i	$0.666898\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−54.1642			−0.274945			−0.137473	−	0.990506i	$-0.543898\pi$
$197$	−54.1642			−0.274945			−0.137473	+	0.990506i	$0.543898\pi$
$198$		0			0
$199$		−	176.608i		−	0.887475i	−0.896157	−	0.443737i	$-0.853652\pi$
$199$		−	176.608i		−	0.887475i	0.896157	−	0.443737i	$-0.146348\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	34.6063i		−	0.170474i
$204$		0			0
$205$			28.2224i			0.137670i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−22.6468			−0.108358
$210$		0			0
$211$	62.6260			0.296806			0.148403	−	0.988927i	$-0.452587\pi$
$211$	62.6260			0.296806			0.148403	+	0.988927i	$0.452587\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	68.0236			0.316389
$216$		0			0
$217$			2.96665i			0.0136712i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			109.693i			0.496347i
$222$		0			0
$223$	122.876			0.551015			0.275507	−	0.961299i	$-0.411154\pi$
$223$	122.876			0.551015			0.275507	+	0.961299i	$0.411154\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	83.1198i		−	0.366166i	−0.983097	−	0.183083i	$-0.941392\pi$
$227$		−	83.1198i		−	0.366166i	0.983097	−	0.183083i	$-0.0586078\pi$
$228$		0			0
$229$			28.5713i			0.124765i	0.998052	+	0.0623826i	$0.0198699\pi$
$229$			28.5713i			0.124765i	−0.998052	+	0.0623826i	$0.980130\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−141.131			−0.605714			−0.302857	−	0.953036i	$-0.597940\pi$
$233$	−141.131			−0.605714			−0.302857	+	0.953036i	$0.597940\pi$
$234$		0			0
$235$			105.431i			0.448641i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	91.5351			0.382992			0.191496	−	0.981493i	$-0.438666\pi$
$239$	91.5351			0.382992			0.191496	+	0.981493i	$0.438666\pi$
$240$		0			0
$241$		−	200.005i		−	0.829896i	−0.909845	−	0.414948i	$-0.863800\pi$
$241$		−	200.005i		−	0.829896i	0.909845	−	0.414948i	$-0.136200\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	106.112i		−	0.433109i
$246$		0			0
$247$		−	156.034i		−	0.631715i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	180.365i		−	0.718584i	−0.933225	−	0.359292i	$-0.883018\pi$
$251$		−	180.365i		−	0.718584i	0.933225	−	0.359292i	$-0.116982\pi$
$252$		0			0
$253$	−4.41145	−	14.1700i	−0.0174365	−	0.0560080i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−278.898			−1.08521			−0.542604	−	0.839989i	$-0.682562\pi$
$257$	−278.898			−1.08521			−0.542604	+	0.839989i	$0.682562\pi$
$258$		0			0
$259$	−28.6827			−0.110744
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	344.941i		−	1.31156i	−0.754951	−	0.655781i	$-0.772340\pi$
$263$		−	344.941i		−	1.31156i	0.754951	−	0.655781i	$-0.227660\pi$
$264$		0			0
$265$	117.119			0.441958
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−260.499			−0.968397			−0.484199	−	0.874958i	$-0.660889\pi$
$269$	−260.499			−0.968397			−0.484199	+	0.874958i	$0.660889\pi$
$270$		0			0
$271$	−96.5451			−0.356255			−0.178128	−	0.984007i	$-0.557004\pi$
$271$	−96.5451			−0.356255			−0.178128	+	0.984007i	$0.557004\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			3.22627i			0.0117319i
$276$		0			0
$277$	245.186			0.885148			0.442574	−	0.896732i	$-0.354065\pi$
$277$	245.186			0.885148			0.442574	+	0.896732i	$0.354065\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.1366i		−	0.0716606i	−0.999358	−	0.0358303i	$-0.988592\pi$
$281$		−	20.1366i		−	0.0716606i	0.999358	−	0.0358303i	$-0.0114076\pi$
$282$		0			0
$283$		−	316.431i		−	1.11813i	−0.829124	−	0.559065i	$-0.811160\pi$
$283$		−	316.431i		−	1.11813i	0.829124	−	0.559065i	$-0.188840\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			15.6905i			0.0546707i
$288$		0			0
$289$	−319.797			−1.10656
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	467.587i		−	1.59586i	−0.602749	−	0.797931i	$-0.705928\pi$
$293$		−	467.587i		−	1.59586i	0.602749	−	0.797931i	$-0.294072\pi$
$294$		0			0
$295$			144.884i			0.491134i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	97.6296	−	30.3943i	0.326520	−	0.101653i
$300$		0			0
$301$	37.8183			0.125642
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−58.3395			−0.191277
$306$		0			0
$307$	440.208			1.43390			0.716951	−	0.697124i	$-0.245537\pi$
$307$	440.208			1.43390			0.716951	+	0.697124i	$0.245537\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.06739			0.0227247			0.0113624	−	0.999935i	$-0.496383\pi$
$311$	7.06739			0.0227247			0.0113624	+	0.999935i	$0.496383\pi$
$312$		0			0
$313$		−	228.455i		−	0.729887i	−0.931030	−	0.364944i	$-0.881088\pi$
$313$		−	228.455i		−	0.729887i	0.931030	−	0.364944i	$-0.118912\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−146.513			−0.462187			−0.231094	−	0.972931i	$-0.574230\pi$
$317$	−146.513			−0.462187			−0.231094	+	0.972931i	$0.574230\pi$
$318$		0			0
$319$		−	17.9622i		−	0.0563077i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	865.990			2.68108
$324$		0			0
$325$	−22.2286			−0.0683956
$326$		0			0
$327$		0			0
$328$		0			0
$329$			58.6150i			0.178161i
$330$		0			0
$331$	147.841			0.446650			0.223325	−	0.974744i	$-0.428309\pi$
$331$	147.841			0.446650			0.223325	+	0.974744i	$0.428309\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.7209			−0.0349879
$336$		0			0
$337$			148.877i			0.441771i	0.975300	+	0.220886i	$0.0708947\pi$
$337$			148.877i			0.441771i	−0.975300	+	0.220886i	$0.929105\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.53982i			0.00451561i
$342$		0			0
$343$		−	119.908i		−	0.349587i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	202.020			0.582190			0.291095	−	0.956694i	$-0.405980\pi$
$347$	202.020			0.582190			0.291095	+	0.956694i	$0.405980\pi$
$348$		0			0
$349$	286.634			0.821301			0.410651	−	0.911793i	$-0.365302\pi$
$349$	286.634			0.821301			0.410651	+	0.911793i	$0.365302\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	97.4876			0.276169			0.138084	−	0.990420i	$-0.455905\pi$
$353$	97.4876			0.276169			0.138084	+	0.990420i	$0.455905\pi$
$354$		0			0
$355$		−	120.373i		−	0.339078i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	587.903i		−	1.63761i	−0.574069	−	0.818807i	$-0.694636\pi$
$359$		−	587.903i		−	1.63761i	0.574069	−	0.818807i	$-0.305364\pi$
$360$		0			0
$361$	−870.838			−2.41229
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	88.0415i		−	0.241209i
$366$		0			0
$367$			388.731i			1.05921i	0.848244	+	0.529606i	$0.177660\pi$
$367$			388.731i			1.05921i	−0.848244	+	0.529606i	$0.822340\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	65.1132			0.175507
$372$		0			0
$373$		−	300.106i		−	0.804575i	−0.915513	−	0.402287i	$-0.868215\pi$
$373$		−	300.106i		−	0.804575i	0.915513	−	0.402287i	$-0.131785\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	123.757			0.328267
$378$		0			0
$379$			644.658i			1.70094i	0.526021	+	0.850472i	$0.323683\pi$
$379$			644.658i			1.70094i	−0.526021	+	0.850472i	$0.676317\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	397.378i		−	1.03754i	−0.854914	−	0.518770i	$-0.826390\pi$
$383$		−	397.378i		−	1.03754i	0.854914	−	0.518770i	$-0.173610\pi$
$384$		0			0
$385$			1.79367i			0.00465889i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			571.886i			1.47014i	0.677989	+	0.735072i	$0.262852\pi$
$389$			571.886i			1.47014i	−0.677989	+	0.735072i	$0.737148\pi$
$390$		0			0
$391$	168.689	+	541.847i	0.431429	+	1.38580i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−186.056			−0.471028
$396$		0			0
$397$	614.917			1.54891			0.774454	−	0.632630i	$-0.218025\pi$
$397$	614.917			1.54891			0.774454	+	0.632630i	$0.218025\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	312.234i		−	0.778637i	−0.921103	−	0.389319i	$-0.872711\pi$
$401$		−	312.234i		−	0.778637i	0.921103	−	0.389319i	$-0.127289\pi$
$402$		0			0
$403$	−10.6092			−0.0263255
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.8875			−0.0365787
$408$		0			0
$409$	−291.617			−0.713000			−0.356500	−	0.934295i	$-0.616030\pi$
$409$	−291.617			−0.713000			−0.356500	+	0.934295i	$0.616030\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			80.5497i			0.195036i
$414$		0			0
$415$	247.597			0.596619
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	326.333i		−	0.778839i	−0.921061	−	0.389419i	$-0.872676\pi$
$419$		−	326.333i		−	0.778839i	0.921061	−	0.389419i	$-0.127324\pi$
$420$		0			0
$421$		−	220.340i		−	0.523372i	−0.965153	−	0.261686i	$-0.915722\pi$
$421$		−	220.340i		−	0.523372i	0.965153	−	0.261686i	$-0.0842785\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	123.369i		−	0.290280i
$426$		0			0
$427$	−32.4344			−0.0759587
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	50.5056i		−	0.117182i	−0.998282	−	0.0585912i	$-0.981339\pi$
$431$		−	50.5056i		−	0.117182i	0.998282	−	0.0585912i	$-0.0186608\pi$
$432$		0			0
$433$		−	734.667i		−	1.69669i	−0.529444	−	0.848345i	$-0.677599\pi$
$433$		−	734.667i		−	1.69669i	0.529444	−	0.848345i	$-0.322401\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−239.953	−	770.756i	−0.549092	−	1.76374i
$438$		0			0
$439$	−519.867			−1.18421			−0.592103	−	0.805862i	$-0.701702\pi$
$439$	−519.867			−1.18421			−0.592103	+	0.805862i	$0.701702\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	309.417			0.698458			0.349229	−	0.937037i	$-0.386444\pi$
$443$	309.417			0.698458			0.349229	+	0.937037i	$0.386444\pi$
$444$		0			0
$445$	−19.3269			−0.0434311
$446$		0			0
$447$		0			0
$448$		0			0
$449$	586.186			1.30554			0.652768	−	0.757558i	$-0.273607\pi$
$449$	586.186			1.30554			0.652768	+	0.757558i	$0.273607\pi$
$450$		0			0
$451$			8.14405i			0.0180578i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.3582			−0.0271608
$456$		0			0
$457$			557.443i			1.21979i	0.792483	+	0.609893i	$0.208788\pi$
$457$			557.443i			1.21979i	−0.792483	+	0.609893i	$0.791212\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	302.840			0.656919			0.328460	−	0.944518i	$-0.393470\pi$
$461$	302.840			0.656919			0.328460	+	0.944518i	$0.393470\pi$
$462$		0			0
$463$	−115.612			−0.249701			−0.124850	−	0.992176i	$-0.539845\pi$
$463$	−115.612			−0.249701			−0.124850	+	0.992176i	$0.539845\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	506.237i		−	1.08402i	−0.840372	−	0.542010i	$-0.817664\pi$
$467$		−	506.237i		−	1.08402i	0.840372	−	0.542010i	$-0.182336\pi$
$468$		0			0
$469$	−6.51636			−0.0138942
$470$		0			0
$471$		0			0
$472$		0			0
$473$	19.6293			0.0414997
$474$		0			0
$475$			175.488i			0.369448i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	751.363i		−	1.56861i	−0.620377	−	0.784304i	$-0.713020\pi$
$479$		−	751.363i		−	1.56861i	0.620377	−	0.784304i	$-0.286980\pi$
$480$		0			0
$481$		−	102.573i		−	0.213250i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−145.216			−0.299414
$486$		0			0
$487$	−247.149			−0.507493			−0.253747	−	0.967271i	$-0.581663\pi$
$487$	−247.149			−0.507493			−0.253747	+	0.967271i	$0.581663\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−135.860			−0.276701			−0.138350	−	0.990383i	$-0.544180\pi$
$491$	−135.860			−0.276701			−0.138350	+	0.990383i	$0.544180\pi$
$492$		0			0
$493$			686.853i			1.39321i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	66.9221i		−	0.134652i
$498$		0			0
$499$	−896.475			−1.79654			−0.898272	−	0.439440i	$-0.855177\pi$
$499$	−896.475			−1.79654			−0.898272	+	0.439440i	$0.855177\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	612.150i		−	1.21700i	−0.793554	−	0.608499i	$-0.791772\pi$
$503$		−	612.150i		−	1.21700i	0.793554	−	0.608499i	$-0.208228\pi$
$504$		0			0
$505$		−	395.145i		−	0.782466i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	530.292			1.04183			0.520915	−	0.853608i	$-0.325591\pi$
$509$	530.292			1.04183			0.520915	+	0.853608i	$0.325591\pi$
$510$		0			0
$511$		−	48.9474i		−	0.0957874i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−95.5684			−0.185570
$516$		0			0
$517$			30.4237i			0.0588467i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			703.652i			1.35058i	0.737552	+	0.675290i	$0.235982\pi$
$521$			703.652i			1.35058i	−0.737552	+	0.675290i	$0.764018\pi$
$522$		0			0
$523$			329.442i			0.629909i	0.949107	+	0.314954i	$0.101989\pi$
$523$			329.442i			0.629909i	−0.949107	+	0.314954i	$0.898011\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	58.8811i		−	0.111729i
$528$		0			0
$529$	435.517	−	300.276i	0.823284	−	0.567629i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−56.1114			−0.105275
$534$		0			0
$535$	−307.253			−0.574305
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	30.6203i		−	0.0568094i
$540$		0			0
$541$	−169.968			−0.314174			−0.157087	−	0.987585i	$-0.550210\pi$
$541$	−169.968			−0.314174			−0.157087	+	0.987585i	$0.550210\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−54.0912			−0.0992499
$546$		0			0
$547$	246.547			0.450725			0.225363	−	0.974275i	$-0.427643\pi$
$547$	246.547			0.450725			0.225363	+	0.974275i	$0.427643\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	977.022i		−	1.77318i
$552$		0			0
$553$	−103.439			−0.187052
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	592.630i		−	1.06397i	−0.846755	−	0.531984i	$-0.821447\pi$
$557$		−	592.630i		−	1.06397i	0.846755	−	0.531984i	$-0.178553\pi$
$558$		0			0
$559$			135.244i			0.241938i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	342.931i		−	0.609114i	−0.952494	−	0.304557i	$-0.901492\pi$
$563$		−	342.931i		−	0.609114i	0.952494	−	0.304557i	$-0.0985085\pi$
$564$		0			0
$565$	196.124			0.347122
$566$		0			0
$567$		0			0
$568$		0			0
$569$			1073.82i			1.88720i	0.331090	+	0.943599i	$0.392584\pi$
$569$			1073.82i			1.88720i	−0.331090	+	0.943599i	$0.607416\pi$
$570$		0			0
$571$			6.10928i			0.0106993i	0.999986	+	0.00534963i	$0.00170285\pi$
$571$			6.10928i			0.0106993i	−0.999986	+	0.00534963i	$0.998297\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−109.802	+	34.1838i	−0.190960	+	0.0594501i
$576$		0			0
$577$	925.791			1.60449			0.802245	−	0.596995i	$-0.203639\pi$
$577$	925.791			1.60449			0.802245	+	0.596995i	$0.203639\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	137.653			0.236925
$582$		0			0
$583$	33.7966			0.0579701
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−374.547			−0.638071			−0.319035	−	0.947743i	$-0.603359\pi$
$587$	−374.547			−0.638071			−0.319035	+	0.947743i	$0.603359\pi$
$588$		0			0
$589$			83.7561i			0.142200i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−593.889			−1.00150			−0.500749	−	0.865592i	$-0.666942\pi$
$593$	−593.889			−1.00150			−0.500749	+	0.865592i	$0.666942\pi$
$594$		0			0
$595$		−	68.5881i		−	0.115274i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	986.438			1.64681			0.823404	−	0.567456i	$-0.192072\pi$
$599$	986.438			1.64681			0.823404	+	0.567456i	$0.192072\pi$
$600$		0			0
$601$	−425.487			−0.707965			−0.353983	−	0.935252i	$-0.615173\pi$
$601$	−425.487			−0.707965			−0.353983	+	0.935252i	$0.615173\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	269.633i		−	0.445675i
$606$		0			0
$607$	1056.53			1.74057			0.870287	−	0.492545i	$-0.163933\pi$
$607$	1056.53			1.74057			0.870287	+	0.492545i	$0.163933\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−209.615			−0.343069
$612$		0			0
$613$			1030.50i			1.68107i	0.541754	+	0.840537i	$0.317760\pi$
$613$			1030.50i			1.68107i	−0.541754	+	0.840537i	$0.682240\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			115.498i			0.187193i	0.995610	+	0.0935967i	$0.0298364\pi$
$617$			115.498i			0.187193i	−0.995610	+	0.0935967i	$0.970164\pi$
$618$		0			0
$619$		−	726.045i		−	1.17293i	−0.809974	−	0.586466i	$-0.800519\pi$
$619$		−	726.045i		−	1.17293i	0.809974	−	0.586466i	$-0.199481\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.7449			−0.0172471
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	569.283			0.905061
$630$		0			0
$631$			226.600i			0.359113i	0.983748	+	0.179557i	$0.0574663\pi$
$631$			226.600i			0.359113i	−0.983748	+	0.179557i	$0.942534\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	256.529i		−	0.403983i
$636$		0			0
$637$	210.969			0.331192
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	670.496i		−	1.04602i	−0.852328	−	0.523008i	$-0.824810\pi$
$641$		−	670.496i		−	1.04602i	0.852328	−	0.523008i	$-0.175190\pi$
$642$		0			0
$643$		−	19.1937i		−	0.0298503i	−0.999889	−	0.0149251i	$-0.995249\pi$
$643$		−	19.1937i		−	0.0298503i	0.999889	−	0.0149251i	$-0.00475099\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−610.380			−0.943400			−0.471700	−	0.881759i	$-0.656359\pi$
$647$	−610.380			−0.943400			−0.471700	+	0.881759i	$0.656359\pi$
$648$		0			0
$649$			41.8088i			0.0644203i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	345.483			0.529071			0.264535	−	0.964376i	$-0.414781\pi$
$653$	345.483			0.529071			0.264535	+	0.964376i	$0.414781\pi$
$654$		0			0
$655$			451.082i			0.688674i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	576.064i		−	0.874149i	−0.899425	−	0.437074i	$-0.856015\pi$
$659$		−	576.064i		−	0.874149i	0.899425	−	0.437074i	$-0.143985\pi$
$660$		0			0
$661$		−	919.436i		−	1.39098i	−0.718537	−	0.695489i	$-0.755188\pi$
$661$		−	919.436i		−	1.39098i	0.718537	−	0.695489i	$-0.244812\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			97.5639i			0.146713i
$666$		0			0
$667$	611.319	−	190.317i	0.916520	−	0.285333i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.8348			−0.0250892
$672$		0			0
$673$	1166.59			1.73342			0.866711	−	0.498811i	$-0.166230\pi$
$673$	1166.59			1.73342			0.866711	+	0.498811i	$0.166230\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			766.112i			1.13163i	0.824533	+	0.565814i	$0.191438\pi$
$677$			766.112i			1.13163i	−0.824533	+	0.565814i	$0.808562\pi$
$678$		0			0
$679$	−80.7340			−0.118901
$680$		0			0
$681$		0			0
$682$		0			0
$683$	325.142			0.476050			0.238025	−	0.971259i	$-0.423500\pi$
$683$	325.142			0.476050			0.238025	+	0.971259i	$0.423500\pi$
$684$		0			0
$685$	−23.0986			−0.0337205
$686$		0			0
$687$		0			0
$688$		0			0
$689$			232.854i			0.337959i
$690$		0			0
$691$	−688.292			−0.996081			−0.498040	−	0.867154i	$-0.665947\pi$
$691$	−688.292			−0.996081			−0.498040	+	0.867154i	$0.665947\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	547.022i		−	0.787082i
$696$		0			0
$697$		−	311.419i		−	0.446800i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			371.835i			0.530436i	0.964189	+	0.265218i	$0.0854439\pi$
$701$			371.835i			0.530436i	−0.964189	+	0.265218i	$0.914556\pi$
$702$		0			0
$703$	−809.784			−1.15190
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	219.684i		−	0.310727i
$708$		0			0
$709$			273.162i			0.385278i	0.981270	+	0.192639i	$0.0617046\pi$
$709$			273.162i			0.385278i	−0.981270	+	0.192639i	$0.938295\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−52.4058	+	16.3151i	−0.0735005	+	0.0228823i
$714$		0			0
$715$	−6.41442			−0.00897122
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1338.95			1.86224			0.931118	−	0.364718i	$-0.118834\pi$
$719$	1338.95			1.86224			0.931118	+	0.364718i	$0.118834\pi$
$720$		0			0
$721$	−53.1321			−0.0736922
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−139.187			−0.191982
$726$		0			0
$727$			715.094i			0.983624i	0.870702	+	0.491812i	$0.163665\pi$
$727$			715.094i			0.983624i	−0.870702	+	0.491812i	$0.836335\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−750.604			−1.02682
$732$		0			0
$733$			344.396i			0.469844i	0.972014	+	0.234922i	$0.0754835\pi$
$733$			344.396i			0.469844i	−0.972014	+	0.234922i	$0.924516\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.38227			−0.00458924
$738$		0			0
$739$	211.174			0.285756			0.142878	−	0.989740i	$-0.454364\pi$
$739$	211.174			0.285756			0.142878	+	0.989740i	$0.454364\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			835.061i			1.12391i	0.827169	+	0.561953i	$0.189950\pi$
$743$			835.061i			1.12391i	−0.827169	+	0.561953i	$0.810050\pi$
$744$		0			0
$745$	−157.282			−0.211117
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−170.820			−0.228064
$750$		0			0
$751$			401.865i			0.535106i	0.963543	+	0.267553i	$0.0862151\pi$
$751$			401.865i			0.535106i	−0.963543	+	0.267553i	$0.913785\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			535.254i			0.708946i
$756$		0			0
$757$		−	213.695i		−	0.282292i	−0.989989	−	0.141146i	$-0.954921\pi$
$757$		−	213.695i		−	0.282292i	0.989989	−	0.141146i	$-0.0450787\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.9452			−0.0262092			−0.0131046	−	0.999914i	$-0.504171\pi$
$761$	−19.9452			−0.0262092			−0.0131046	+	0.999914i	$0.504171\pi$
$762$		0			0
$763$	−30.0724			−0.0394134
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−288.057			−0.375563
$768$		0			0
$769$		−	1434.95i		−	1.86600i	−0.359876	−	0.933000i	$-0.617181\pi$
$769$		−	1434.95i		−	1.86600i	0.359876	−	0.933000i	$-0.382819\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1152.17i			1.49051i	0.666779	+	0.745256i	$0.267673\pi$
$773$			1152.17i			1.49051i	−0.666779	+	0.745256i	$0.732327\pi$
$774$		0			0
$775$	11.9319			0.0153960
$776$		0			0
$777$		0			0
$778$		0			0
$779$			442.982i			0.568655i
$780$		0			0
$781$		−	34.7355i		−	0.0444756i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.4423			0.0285889
$786$		0			0
$787$		−	648.255i		−	0.823704i	−0.911251	−	0.411852i	$-0.864882\pi$
$787$		−	648.255i		−	0.823704i	0.911251	−	0.411852i	$-0.135118\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	109.037			0.137846
$792$		0			0
$793$		−	115.990i		−	0.146267i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			861.490i			1.08092i	0.841371	+	0.540458i	$0.181749\pi$
$797$			861.490i			1.08092i	−0.841371	+	0.540458i	$0.818251\pi$
$798$		0			0
$799$		−	1163.37i		−	1.45603i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	25.4058i		−	0.0316386i
$804$		0			0
$805$	−61.0453	+	19.0048i	−0.0758327	+	0.0236084i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−581.630			−0.718949			−0.359475	−	0.933155i	$-0.617044\pi$
$809$	−581.630			−0.718949			−0.359475	+	0.933155i	$0.617044\pi$
$810$		0			0
$811$	427.760			0.527447			0.263724	−	0.964598i	$-0.415049\pi$
$811$	427.760			0.527447			0.263724	+	0.964598i	$0.415049\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			301.698i			0.370182i
$816$		0			0
$817$	1067.71			1.30686
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−571.710			−0.696358			−0.348179	−	0.937428i	$-0.613200\pi$
$821$	−571.710			−0.696358			−0.348179	+	0.937428i	$0.613200\pi$
$822$		0			0
$823$	−1018.86			−1.23798			−0.618990	−	0.785399i	$-0.712458\pi$
$823$	−1018.86			−1.23798			−0.618990	+	0.785399i	$0.712458\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	555.453i		−	0.671648i	−0.941925	−	0.335824i	$-0.890985\pi$
$827$		−	555.453i		−	0.671648i	0.941925	−	0.335824i	$-0.109015\pi$
$828$		0			0
$829$	−899.797			−1.08540			−0.542700	−	0.839926i	$-0.682598\pi$
$829$	−899.797			−1.08540			−0.542700	+	0.839926i	$0.682598\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1170.88i			1.40562i
$834$		0			0
$835$		−	85.1432i		−	0.101968i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			63.4558i			0.0756326i	0.999285	+	0.0378163i	$0.0120402\pi$
$839$			63.4558i			0.0756326i	−0.999285	+	0.0378163i	$0.987960\pi$
$840$		0			0
$841$	−66.0832			−0.0785770
$842$		0			0
$843$		0			0
$844$		0			0
$845$			333.701i			0.394912i
$846$		0			0
$847$		−	149.905i		−	0.176983i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−157.740	−	506.678i	−0.185359	−	0.595392i
$852$		0			0
$853$	−387.087			−0.453794			−0.226897	−	0.973919i	$-0.572858\pi$
$853$	−387.087			−0.453794			−0.226897	+	0.973919i	$0.572858\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1286.10			−1.50070			−0.750348	−	0.661043i	$-0.770114\pi$
$857$	−1286.10			−1.50070			−0.750348	+	0.661043i	$0.770114\pi$
$858$		0			0
$859$	−145.590			−0.169488			−0.0847438	−	0.996403i	$-0.527007\pi$
$859$	−145.590			−0.169488			−0.0847438	+	0.996403i	$0.527007\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−586.464			−0.679565			−0.339782	−	0.940504i	$-0.610353\pi$
$863$	−586.464			−0.679565			−0.339782	+	0.940504i	$0.610353\pi$
$864$		0			0
$865$		−	227.487i		−	0.262990i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−53.6896			−0.0617832
$870$		0			0
$871$		−	23.3034i		−	0.0267548i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	13.8990			0.0158845
$876$		0			0
$877$	−215.488			−0.245710			−0.122855	−	0.992425i	$-0.539205\pi$
$877$	−215.488			−0.245710			−0.122855	+	0.992425i	$0.539205\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			244.305i			0.277305i	0.990341	+	0.138652i	$0.0442770\pi$
$881$			244.305i			0.277305i	−0.990341	+	0.138652i	$0.955723\pi$
$882$		0			0
$883$	806.275			0.913109			0.456554	−	0.889696i	$-0.349083\pi$
$883$	806.275			0.913109			0.456554	+	0.889696i	$0.349083\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	173.061			0.195108			0.0975540	−	0.995230i	$-0.468898\pi$
$887$	173.061			0.195108			0.0975540	+	0.995230i	$0.468898\pi$
$888$		0			0
$889$		−	142.619i		−	0.160427i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1654.85i			1.85313i
$894$		0			0
$895$		−	598.418i		−	0.668623i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−66.4304			−0.0738937
$900$		0			0
$901$	−1292.34			−1.43434
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−430.319			−0.475490
$906$		0			0
$907$			1723.01i			1.89967i	0.312744	+	0.949837i	$0.398752\pi$
$907$			1723.01i			1.89967i	−0.312744	+	0.949837i	$0.601248\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	862.103i		−	0.946326i	−0.880975	−	0.473163i	$-0.843112\pi$
$911$		−	862.103i		−	0.946326i	0.880975	−	0.473163i	$-0.156888\pi$
$912$		0			0
$913$	71.4481			0.0782564
$914$		0			0
$915$		0			0
$916$		0			0
$917$			250.783i			0.273482i
$918$		0			0
$919$		−	1163.55i		−	1.26611i	−0.774109	−	0.633053i	$-0.781802\pi$
$919$		−	1163.55i		−	1.26611i	0.774109	−	0.633053i	$-0.218198\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	239.323			0.259288
$924$		0			0
$925$			115.362i			0.124716i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	985.061			1.06035			0.530173	−	0.847890i	$-0.322127\pi$
$929$	985.061			1.06035			0.530173	+	0.847890i	$0.322127\pi$
$930$		0			0
$931$		−	1665.54i		−	1.78898i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	35.6002i		−	0.0380751i
$936$		0			0
$937$		−	221.368i		−	0.236251i	−0.992999	−	0.118126i	$-0.962311\pi$
$937$		−	221.368i		−	0.236251i	0.992999	−	0.118126i	$-0.0376886\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			117.679i			0.125057i	0.998043	+	0.0625285i	$0.0199164\pi$
$941$			117.679i			0.125057i	−0.998043	+	0.0625285i	$0.980084\pi$
$942$		0			0
$943$	−277.172	+	86.2898i	−0.293926	+	0.0915056i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−982.718			−1.03772			−0.518859	−	0.854860i	$-0.673643\pi$
$947$	−982.718			−1.03772			−0.518859	+	0.854860i	$0.673643\pi$
$948$		0			0
$949$	175.043			0.184450
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	642.653i		−	0.674347i	−0.941443	−	0.337173i	$-0.890529\pi$
$953$		−	642.653i		−	0.674347i	0.941443	−	0.337173i	$-0.109471\pi$
$954$		0			0
$955$	−390.337			−0.408730
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.8418			−0.0133909
$960$		0			0
$961$	−955.305			−0.994074
$962$		0			0
$963$		0			0
$964$		0			0
$965$			432.105i			0.447777i
$966$		0			0
$967$	1822.91			1.88512			0.942559	−	0.334039i	$-0.108412\pi$
$967$	1822.91			1.88512			0.942559	+	0.334039i	$0.108412\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1490.41i			1.53492i	0.641095	+	0.767461i	$0.278480\pi$
$971$			1490.41i			1.53492i	−0.641095	+	0.767461i	$0.721520\pi$
$972$		0			0
$973$		−	304.122i		−	0.312561i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			476.626i			0.487847i	0.969795	+	0.243923i	$0.0784345\pi$
$977$			476.626i			0.487847i	−0.969795	+	0.243923i	$0.921565\pi$
$978$		0			0
$979$	−5.57708			−0.00569671
$980$		0			0
$981$		0			0
$982$		0			0
$983$			518.857i			0.527830i	0.964546	+	0.263915i	$0.0850139\pi$
$983$			518.857i			0.527830i	−0.964546	+	0.263915i	$0.914986\pi$
$984$		0			0
$985$			121.115i			0.122959i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	207.982	+	668.059i	0.210295	+	0.675489i
$990$		0			0
$991$	−349.963			−0.353141			−0.176571	−	0.984288i	$-0.556500\pi$
$991$	−349.963			−0.353141			−0.176571	+	0.984288i	$0.556500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−394.906			−0.396891
$996$		0			0
$997$	39.3423			0.0394607			0.0197303	−	0.999805i	$-0.493719\pi$
$997$	39.3423			0.0394607			0.0197303	+	0.999805i	$0.493719\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.c.2161.8		32
3.2	odd	2		1380.3.d.a.781.28	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.25		32
69.68	even	2		1380.3.d.a.781.21	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.21	✓	32	69.68	even	2
1380.3.d.a.781.28	yes	32	3.2	odd	2
4140.3.d.c.2161.8		32	1.1	even	1	trivial
4140.3.d.c.2161.25		32	23.22	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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -1.24316i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -1.24316i q^{7} -0.645254i q^{11} +4.44571 q^{13} +24.6738i q^{17} -35.0975i q^{19} +(21.9604 - 6.83676i) q^{23} -5.00000 q^{25} +27.8373 q^{29} -2.38638 q^{31} -2.77979 q^{35} -23.0724i q^{37} -12.6215 q^{41} +30.4211i q^{43} -47.1500 q^{47} +47.4546 q^{49} +52.3772i q^{53} -1.44283 q^{55} -64.7943 q^{59} -26.0902i q^{61} -9.94092i q^{65} -5.24177i q^{67} +53.8322 q^{71} +39.3733 q^{73} -0.802155 q^{77} -83.2069i q^{79} +110.729i q^{83} +55.1723 q^{85} -8.64323i q^{89} -5.52674i q^{91} -78.4805 q^{95} -64.9425i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) \) 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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