LMFDB - Embedded newform 4140.3.d.c.2161.5

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			2161.5
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -3.01539i q^{7} +O(q^{10})$	$q-2.23607i q^{5} -3.01539i q^{7} -4.03301i q^{11} -1.33514 q^{13} -11.3894i q^{17} +18.6621i q^{19} +(-22.9711 - 1.15231i) q^{23} -5.00000 q^{25} +40.0389 q^{29} -37.8306 q^{31} -6.74262 q^{35} -3.61988i q^{37} -62.5713 q^{41} +62.6862i q^{43} -7.72339 q^{47} +39.9074 q^{49} -41.3222i q^{53} -9.01807 q^{55} +12.9729 q^{59} +43.9984i q^{61} +2.98547i q^{65} +65.9253i q^{67} -33.0858 q^{71} -33.5634 q^{73} -12.1611 q^{77} -88.0802i q^{79} +102.424i q^{83} -25.4676 q^{85} -150.583i q^{89} +4.02598i q^{91} +41.7298 q^{95} +35.2693i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	3.01539i		−	0.430770i	−0.976529	−	0.215385i	$-0.930899\pi$
$7$		−	3.01539i		−	0.430770i	0.976529	−	0.215385i	$-0.0691007\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.03301i		−	0.366637i	−0.983054	−	0.183318i	$-0.941316\pi$
$11$		−	4.03301i		−	0.366637i	0.983054	−	0.183318i	$-0.0586839\pi$
$12$		0			0
$13$	−1.33514			−0.102703			−0.0513516	−	0.998681i	$-0.516353\pi$
$13$	−1.33514			−0.102703			−0.0513516	+	0.998681i	$0.516353\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	11.3894i		−	0.669967i	−0.942224	−	0.334983i	$-0.891269\pi$
$17$		−	11.3894i		−	0.669967i	0.942224	−	0.334983i	$-0.108731\pi$
$18$		0			0
$19$			18.6621i			0.982217i	0.871098	+	0.491109i	$0.163408\pi$
$19$			18.6621i			0.982217i	−0.871098	+	0.491109i	$0.836592\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−22.9711	−	1.15231i	−0.998744	−	0.0501005i
$23$	−22.9711	−	1.15231i	−0.998744	−	0.0501005i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	40.0389			1.38065			0.690326	−	0.723498i	$-0.257467\pi$
$29$	40.0389			1.38065			0.690326	+	0.723498i	$0.257467\pi$
$30$		0			0
$31$	−37.8306			−1.22034			−0.610171	−	0.792270i	$-0.708899\pi$
$31$	−37.8306			−1.22034			−0.610171	+	0.792270i	$0.708899\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.74262			−0.192646
$36$		0			0
$37$		−	3.61988i		−	0.0978345i	−0.998803	−	0.0489173i	$-0.984423\pi$
$37$		−	3.61988i		−	0.0978345i	0.998803	−	0.0489173i	$-0.0155771\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−62.5713			−1.52613			−0.763065	−	0.646322i	$-0.776306\pi$
$41$	−62.5713			−1.52613			−0.763065	+	0.646322i	$0.776306\pi$
$42$		0			0
$43$			62.6862i			1.45782i	0.684610	+	0.728909i	$0.259972\pi$
$43$			62.6862i			1.45782i	−0.684610	+	0.728909i	$0.740028\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.72339			−0.164327			−0.0821637	−	0.996619i	$-0.526183\pi$
$47$	−7.72339			−0.164327			−0.0821637	+	0.996619i	$0.526183\pi$
$48$		0			0
$49$	39.9074			0.814437
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	41.3222i		−	0.779665i	−0.920886	−	0.389833i	$-0.872533\pi$
$53$		−	41.3222i		−	0.779665i	0.920886	−	0.389833i	$-0.127467\pi$
$54$		0			0
$55$	−9.01807			−0.163965
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.9729			0.219880			0.109940	−	0.993938i	$-0.464934\pi$
$59$	12.9729			0.219880			0.109940	+	0.993938i	$0.464934\pi$
$60$		0			0
$61$			43.9984i			0.721285i	0.932704	+	0.360642i	$0.117443\pi$
$61$			43.9984i			0.721285i	−0.932704	+	0.360642i	$0.882557\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			2.98547i			0.0459303i
$66$		0			0
$67$			65.9253i			0.983960i	0.870607	+	0.491980i	$0.163727\pi$
$67$			65.9253i			0.983960i	−0.870607	+	0.491980i	$0.836273\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−33.0858			−0.465997			−0.232999	−	0.972477i	$-0.574854\pi$
$71$	−33.0858			−0.465997			−0.232999	+	0.972477i	$0.574854\pi$
$72$		0			0
$73$	−33.5634			−0.459773			−0.229886	−	0.973217i	$-0.573835\pi$
$73$	−33.5634			−0.459773			−0.229886	+	0.973217i	$0.573835\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.1611			−0.157936
$78$		0			0
$79$		−	88.0802i		−	1.11494i	−0.830197	−	0.557469i	$-0.811772\pi$
$79$		−	88.0802i		−	1.11494i	0.830197	−	0.557469i	$-0.188228\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			102.424i			1.23402i	0.786956	+	0.617009i	$0.211656\pi$
$83$			102.424i			1.23402i	−0.786956	+	0.617009i	$0.788344\pi$
$84$		0			0
$85$	−25.4676			−0.299618
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	150.583i		−	1.69195i	−0.533224	−	0.845974i	$-0.679020\pi$
$89$		−	150.583i		−	1.69195i	0.533224	−	0.845974i	$-0.320980\pi$
$90$		0			0
$91$			4.02598i			0.0442415i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	41.7298			0.439261
$96$		0			0
$97$			35.2693i			0.363601i	0.983335	+	0.181800i	$0.0581925\pi$
$97$			35.2693i			0.363601i	−0.983335	+	0.181800i	$0.941808\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−20.3165			−0.201154			−0.100577	−	0.994929i	$-0.532069\pi$
$101$	−20.3165			−0.201154			−0.100577	+	0.994929i	$0.532069\pi$
$102$		0			0
$103$			154.474i			1.49975i	0.661579	+	0.749875i	$0.269887\pi$
$103$			154.474i			1.49975i	−0.661579	+	0.749875i	$0.730113\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	163.132i		−	1.52460i	−0.647223	−	0.762301i	$-0.724070\pi$
$107$		−	163.132i		−	1.52460i	0.647223	−	0.762301i	$-0.275930\pi$
$108$		0			0
$109$			115.236i			1.05721i	0.848869	+	0.528604i	$0.177284\pi$
$109$			115.236i			1.05721i	−0.848869	+	0.528604i	$0.822716\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			86.5285i			0.765739i	0.923803	+	0.382869i	$0.125064\pi$
$113$			86.5285i			0.765739i	−0.923803	+	0.382869i	$0.874936\pi$
$114$		0			0
$115$	−2.57665	+	51.3650i	−0.0224056	+	0.446652i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−34.3436			−0.288602
$120$		0			0
$121$	104.735			0.865577
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−154.729			−1.21834			−0.609168	−	0.793041i	$-0.708496\pi$
$127$	−154.729			−1.21834			−0.609168	+	0.793041i	$0.708496\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	201.324			1.53683			0.768414	−	0.639954i	$-0.221046\pi$
$131$	201.324			1.53683			0.768414	+	0.639954i	$0.221046\pi$
$132$		0			0
$133$	56.2736			0.423110
$134$		0			0
$135$		0			0
$136$		0			0
$137$			19.7355i			0.144055i	0.997403	+	0.0720275i	$0.0229469\pi$
$137$			19.7355i			0.144055i	−0.997403	+	0.0720275i	$0.977053\pi$
$138$		0			0
$139$	3.33436			0.0239882			0.0119941	−	0.999928i	$-0.496182\pi$
$139$	3.33436			0.0239882			0.0119941	+	0.999928i	$0.496182\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			5.38463i			0.0376548i
$144$		0			0
$145$		−	89.5298i		−	0.617447i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			165.423i			1.11022i	0.831776	+	0.555112i	$0.187325\pi$
$149$			165.423i			1.11022i	−0.831776	+	0.555112i	$0.812675\pi$
$150$		0			0
$151$	19.8645			0.131553			0.0657764	−	0.997834i	$-0.479048\pi$
$151$	19.8645			0.131553			0.0657764	+	0.997834i	$0.479048\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			84.5918i			0.545753i
$156$		0			0
$157$			40.3250i			0.256847i	0.991719	+	0.128423i	$0.0409917\pi$
$157$			40.3250i			0.256847i	−0.991719	+	0.128423i	$0.959008\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.47467	+	69.2669i	−0.0215818	+	0.430229i
$162$		0			0
$163$	152.671			0.936630			0.468315	−	0.883562i	$-0.344861\pi$
$163$	152.671			0.936630			0.468315	+	0.883562i	$0.344861\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−96.0563			−0.575188			−0.287594	−	0.957752i	$-0.592855\pi$
$167$	−96.0563			−0.575188			−0.287594	+	0.957752i	$0.592855\pi$
$168$		0			0
$169$	−167.217			−0.989452
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−56.9815			−0.329373			−0.164686	−	0.986346i	$-0.552661\pi$
$173$	−56.9815			−0.329373			−0.164686	+	0.986346i	$0.552661\pi$
$174$		0			0
$175$			15.0770i			0.0861541i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	78.2577			0.437194			0.218597	−	0.975815i	$-0.429852\pi$
$179$	78.2577			0.437194			0.218597	+	0.975815i	$0.429852\pi$
$180$		0			0
$181$			34.4195i			0.190163i	0.995469	+	0.0950815i	$0.0303112\pi$
$181$			34.4195i			0.190163i	−0.995469	+	0.0950815i	$0.969689\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.09429			−0.0437529
$186$		0			0
$187$	−45.9337			−0.245635
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	100.308i		−	0.525175i	−0.964908	−	0.262587i	$-0.915424\pi$
$191$		−	100.308i		−	0.525175i	0.964908	−	0.262587i	$-0.0845758\pi$
$192$		0			0
$193$	215.009			1.11403			0.557017	−	0.830501i	$-0.311946\pi$
$193$	215.009			1.11403			0.557017	+	0.830501i	$0.311946\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	180.788			0.917705			0.458852	−	0.888512i	$-0.348261\pi$
$197$	180.788			0.917705			0.458852	+	0.888512i	$0.348261\pi$
$198$		0			0
$199$		−	72.7714i		−	0.365685i	−0.983142	−	0.182843i	$-0.941470\pi$
$199$		−	72.7714i		−	0.365685i	0.983142	−	0.182843i	$-0.0585299\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	120.733i		−	0.594744i
$204$		0			0
$205$			139.914i			0.682506i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	75.2645			0.360117
$210$		0			0
$211$	129.390			0.613221			0.306610	−	0.951835i	$-0.400805\pi$
$211$	129.390			0.613221			0.306610	+	0.951835i	$0.400805\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	140.171			0.651956
$216$		0			0
$217$			114.074i			0.525687i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			15.2065i			0.0688077i
$222$		0			0
$223$	−38.1785			−0.171204			−0.0856020	−	0.996329i	$-0.527281\pi$
$223$	−38.1785			−0.171204			−0.0856020	+	0.996329i	$0.527281\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			373.195i			1.64403i	0.569466	+	0.822015i	$0.307150\pi$
$227$			373.195i			1.64403i	−0.569466	+	0.822015i	$0.692850\pi$
$228$		0			0
$229$			304.196i			1.32837i	0.747569	+	0.664184i	$0.231221\pi$
$229$			304.196i			1.32837i	−0.747569	+	0.664184i	$0.768779\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−249.734			−1.07182			−0.535910	−	0.844275i	$-0.680031\pi$
$233$	−249.734			−1.07182			−0.535910	+	0.844275i	$0.680031\pi$
$234$		0			0
$235$			17.2700i			0.0734895i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−351.779			−1.47188			−0.735938	−	0.677049i	$-0.763259\pi$
$239$	−351.779			−1.47188			−0.735938	+	0.677049i	$0.763259\pi$
$240$		0			0
$241$			77.3484i			0.320948i	0.987040	+	0.160474i	$0.0513022\pi$
$241$			77.3484i			0.320948i	−0.987040	+	0.160474i	$0.948698\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	89.2357i		−	0.364227i
$246$		0			0
$247$		−	24.9166i		−	0.100877i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			402.696i			1.60437i	0.597077	+	0.802184i	$0.296329\pi$
$251$			402.696i			1.60437i	−0.597077	+	0.802184i	$0.703671\pi$
$252$		0			0
$253$	−4.64728	+	92.6426i	−0.0183687	+	0.366176i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	133.068			0.517775			0.258887	−	0.965907i	$-0.416644\pi$
$257$	133.068			0.517775			0.258887	+	0.965907i	$0.416644\pi$
$258$		0			0
$259$	−10.9153			−0.0421442
$260$		0			0
$261$		0			0
$262$		0			0
$263$			129.767i			0.493411i	0.969091	+	0.246705i	$0.0793480\pi$
$263$			129.767i			0.493411i	−0.969091	+	0.246705i	$0.920652\pi$
$264$		0			0
$265$	−92.3994			−0.348677
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−308.726			−1.14768			−0.573840	−	0.818968i	$-0.694547\pi$
$269$	−308.726			−1.14768			−0.573840	+	0.818968i	$0.694547\pi$
$270$		0			0
$271$	479.177			1.76818			0.884090	−	0.467316i	$-0.154779\pi$
$271$	479.177			1.76818			0.884090	+	0.467316i	$0.154779\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			20.1650i			0.0733274i
$276$		0			0
$277$	−472.636			−1.70627			−0.853134	−	0.521692i	$-0.825301\pi$
$277$	−472.636			−1.70627			−0.853134	+	0.521692i	$0.825301\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	202.809i		−	0.721741i	−0.932616	−	0.360871i	$-0.882480\pi$
$281$		−	202.809i		−	0.721741i	0.932616	−	0.360871i	$-0.117520\pi$
$282$		0			0
$283$			178.345i			0.630196i	0.949059	+	0.315098i	$0.102037\pi$
$283$			178.345i			0.630196i	−0.949059	+	0.315098i	$0.897963\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			188.677i			0.657412i
$288$		0			0
$289$	159.281			0.551144
$290$		0			0
$291$		0			0
$292$		0			0
$293$			186.134i			0.635268i	0.948213	+	0.317634i	$0.102888\pi$
$293$			186.134i			0.635268i	−0.948213	+	0.317634i	$0.897112\pi$
$294$		0			0
$295$		−	29.0083i		−	0.0983333i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	30.6697	+	1.53850i	0.102574	+	0.00514548i
$300$		0			0
$301$	189.024			0.627985
$302$		0			0
$303$		0			0
$304$		0			0
$305$	98.3834			0.322568
$306$		0			0
$307$	−47.7961			−0.155688			−0.0778438	−	0.996966i	$-0.524804\pi$
$307$	−47.7961			−0.155688			−0.0778438	+	0.996966i	$0.524804\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	19.6841			0.0632928			0.0316464	−	0.999499i	$-0.489925\pi$
$311$	19.6841			0.0632928			0.0316464	+	0.999499i	$0.489925\pi$
$312$		0			0
$313$			241.396i			0.771232i	0.922659	+	0.385616i	$0.126011\pi$
$313$			241.396i			0.771232i	−0.922659	+	0.385616i	$0.873989\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	537.913			1.69689			0.848443	−	0.529286i	$-0.177540\pi$
$317$	537.913			1.69689			0.848443	+	0.529286i	$0.177540\pi$
$318$		0			0
$319$		−	161.477i		−	0.506198i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	212.551			0.658053
$324$		0			0
$325$	6.67571			0.0205406
$326$		0			0
$327$		0			0
$328$		0			0
$329$			23.2891i			0.0707874i
$330$		0			0
$331$	−606.195			−1.83140			−0.915702	−	0.401858i	$-0.868364\pi$
$331$	−606.195			−1.83140			−0.915702	+	0.401858i	$0.868364\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	147.413			0.440040
$336$		0			0
$337$		−	207.336i		−	0.615240i	−0.951509	−	0.307620i	$-0.900467\pi$
$337$		−	207.336i		−	0.615240i	0.951509	−	0.307620i	$-0.0995326\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			152.571i			0.447422i
$342$		0			0
$343$		−	268.091i		−	0.781606i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	350.545			1.01022			0.505108	−	0.863056i	$-0.331453\pi$
$347$	350.545			1.01022			0.505108	+	0.863056i	$0.331453\pi$
$348$		0			0
$349$	431.065			1.23514			0.617572	−	0.786515i	$-0.288117\pi$
$349$	431.065			1.23514			0.617572	+	0.786515i	$0.288117\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−180.996			−0.512737			−0.256368	−	0.966579i	$-0.582526\pi$
$353$	−180.996			−0.512737			−0.256368	+	0.966579i	$0.582526\pi$
$354$		0			0
$355$			73.9821i			0.208400i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			516.224i			1.43795i	0.695036	+	0.718975i	$0.255388\pi$
$359$			516.224i			1.43795i	−0.695036	+	0.718975i	$0.744612\pi$
$360$		0			0
$361$	12.7251			0.0352496
$362$		0			0
$363$		0			0
$364$		0			0
$365$			75.0501i			0.205617i
$366$		0			0
$367$		−	529.931i		−	1.44395i	−0.691917	−	0.721977i	$-0.743233\pi$
$367$		−	529.931i		−	1.44395i	0.691917	−	0.721977i	$-0.256767\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−124.603			−0.335857
$372$		0			0
$373$		−	286.913i		−	0.769204i	−0.923082	−	0.384602i	$-0.874339\pi$
$373$		−	286.913i		−	0.769204i	0.923082	−	0.384602i	$-0.125661\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−53.4576			−0.141797
$378$		0			0
$379$			353.984i			0.933996i	0.884258	+	0.466998i	$0.154664\pi$
$379$			353.984i			0.933996i	−0.884258	+	0.466998i	$0.845336\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			528.270i			1.37930i	0.724145	+	0.689648i	$0.242235\pi$
$383$			528.270i			1.37930i	−0.724145	+	0.689648i	$0.757765\pi$
$384$		0			0
$385$			27.1930i			0.0706313i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			367.888i			0.945727i	0.881136	+	0.472863i	$0.156780\pi$
$389$			367.888i			0.945727i	−0.881136	+	0.472863i	$0.843220\pi$
$390$		0			0
$391$	−13.1242	+	261.628i	−0.0335657	+	0.669126i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−196.953			−0.498616
$396$		0			0
$397$	−134.700			−0.339294			−0.169647	−	0.985505i	$-0.554263\pi$
$397$	−134.700			−0.339294			−0.169647	+	0.985505i	$0.554263\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	164.980i		−	0.411421i	−0.978613	−	0.205711i	$-0.934049\pi$
$401$		−	164.980i		−	0.411421i	0.978613	−	0.205711i	$-0.0659505\pi$
$402$		0			0
$403$	50.5092			0.125333
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.5990			−0.0358697
$408$		0			0
$409$	−528.596			−1.29241			−0.646206	−	0.763163i	$-0.723645\pi$
$409$	−528.596			−1.29241			−0.646206	+	0.763163i	$0.723645\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	39.1184i		−	0.0947177i
$414$		0			0
$415$	229.026			0.551870
$416$		0			0
$417$		0			0
$418$		0			0
$419$			593.960i			1.41757i	0.705427	+	0.708783i	$0.250755\pi$
$419$			593.960i			1.41757i	−0.705427	+	0.708783i	$0.749245\pi$
$420$		0			0
$421$		−	636.816i		−	1.51263i	−0.654210	−	0.756313i	$-0.726999\pi$
$421$		−	636.816i		−	1.51263i	0.654210	−	0.756313i	$-0.273001\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			56.9472i			0.133993i
$426$		0			0
$427$	132.672			0.310708
$428$		0			0
$429$		0			0
$430$		0			0
$431$			741.443i			1.72029i	0.510053	+	0.860143i	$0.329626\pi$
$431$			741.443i			1.72029i	−0.510053	+	0.860143i	$0.670374\pi$
$432$		0			0
$433$			118.796i			0.274356i	0.990546	+	0.137178i	$0.0438032\pi$
$433$			118.796i			0.274356i	−0.990546	+	0.137178i	$0.956197\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	21.5046	−	428.690i	0.0492096	−	0.980984i
$438$		0			0
$439$	317.334			0.722857			0.361429	−	0.932400i	$-0.382289\pi$
$439$	317.334			0.722857			0.361429	+	0.932400i	$0.382289\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−733.729			−1.65627			−0.828137	−	0.560526i	$-0.810599\pi$
$443$	−733.729			−1.65627			−0.828137	+	0.560526i	$0.810599\pi$
$444$		0			0
$445$	−336.715			−0.756662
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−89.4746			−0.199275			−0.0996377	−	0.995024i	$-0.531768\pi$
$449$	−89.4746			−0.199275			−0.0996377	+	0.995024i	$0.531768\pi$
$450$		0			0
$451$			252.351i			0.559536i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.00236			0.0197854
$456$		0			0
$457$			884.800i			1.93611i	0.250746	+	0.968053i	$0.419324\pi$
$457$			884.800i			1.93611i	−0.250746	+	0.968053i	$0.580676\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−444.736			−0.964720			−0.482360	−	0.875973i	$-0.660220\pi$
$461$	−444.736			−0.964720			−0.482360	+	0.875973i	$0.660220\pi$
$462$		0			0
$463$	277.681			0.599743			0.299871	−	0.953980i	$-0.403056\pi$
$463$	277.681			0.599743			0.299871	+	0.953980i	$0.403056\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			842.314i			1.80367i	0.432081	+	0.901835i	$0.357779\pi$
$467$			842.314i			1.80367i	−0.432081	+	0.901835i	$0.642221\pi$
$468$		0			0
$469$	198.791			0.423861
$470$		0			0
$471$		0			0
$472$		0			0
$473$	252.814			0.534490
$474$		0			0
$475$		−	93.3106i		−	0.196443i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			236.202i			0.493115i	0.969128	+	0.246557i	$0.0792994\pi$
$479$			236.202i			0.493115i	−0.969128	+	0.246557i	$0.920701\pi$
$480$		0			0
$481$			4.83305i			0.0100479i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	78.8645			0.162607
$486$		0			0
$487$	−691.421			−1.41976			−0.709878	−	0.704325i	$-0.751250\pi$
$487$	−691.421			−1.41976			−0.709878	+	0.704325i	$0.751250\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	306.151			0.623526			0.311763	−	0.950160i	$-0.399081\pi$
$491$	306.151			0.623526			0.311763	+	0.950160i	$0.399081\pi$
$492$		0			0
$493$		−	456.021i		−	0.924992i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			99.7667i			0.200738i
$498$		0			0
$499$	287.223			0.575598			0.287799	−	0.957691i	$-0.407077\pi$
$499$	287.223			0.575598			0.287799	+	0.957691i	$0.407077\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			330.079i			0.656222i	0.944639	+	0.328111i	$0.106412\pi$
$503$			330.079i			0.656222i	−0.944639	+	0.328111i	$0.893588\pi$
$504$		0			0
$505$			45.4291i			0.0899587i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−379.314			−0.745215			−0.372608	−	0.927989i	$-0.621536\pi$
$509$	−379.314			−0.745215			−0.372608	+	0.927989i	$0.621536\pi$
$510$		0			0
$511$			101.207i			0.198056i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	345.415			0.670709
$516$		0			0
$517$			31.1485i			0.0602485i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			139.030i			0.266853i	0.991059	+	0.133427i	$0.0425980\pi$
$521$			139.030i			0.266853i	−0.991059	+	0.133427i	$0.957402\pi$
$522$		0			0
$523$		−	704.955i		−	1.34791i	−0.738774	−	0.673954i	$-0.764595\pi$
$523$		−	704.955i		−	1.34791i	0.738774	−	0.673954i	$-0.235405\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			430.869i			0.817588i
$528$		0			0
$529$	526.344	+	52.9398i	0.994980	+	0.100075i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	83.5416			0.156738
$534$		0			0
$535$	−364.775			−0.681823
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	160.947i		−	0.298603i
$540$		0			0
$541$	556.137			1.02798			0.513990	−	0.857796i	$-0.328167\pi$
$541$	556.137			1.02798			0.513990	+	0.857796i	$0.328167\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	257.675			0.472798
$546$		0			0
$547$	−533.002			−0.974409			−0.487204	−	0.873288i	$-0.661983\pi$
$547$	−533.002			−0.974409			−0.487204	+	0.873288i	$0.661983\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			747.211i			1.35610i
$552$		0			0
$553$	−265.596			−0.480283
$554$		0			0
$555$		0			0
$556$		0			0
$557$			91.1976i			0.163730i	0.996643	+	0.0818650i	$0.0260876\pi$
$557$			91.1976i			0.163730i	−0.996643	+	0.0818650i	$0.973912\pi$
$558$		0			0
$559$		−	83.6949i		−	0.149723i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			160.420i			0.284937i	0.989799	+	0.142469i	$0.0455040\pi$
$563$			160.420i			0.284937i	−0.989799	+	0.142469i	$0.954496\pi$
$564$		0			0
$565$	193.484			0.342449
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	341.261i		−	0.599756i	−0.953978	−	0.299878i	$-0.903054\pi$
$569$		−	341.261i		−	0.599756i	0.953978	−	0.299878i	$-0.0969459\pi$
$570$		0			0
$571$		−	743.441i		−	1.30200i	−0.759079	−	0.650999i	$-0.774350\pi$
$571$		−	743.441i		−	1.30200i	0.759079	−	0.650999i	$-0.225650\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	114.856	+	5.76156i	0.199749	+	0.0100201i
$576$		0			0
$577$	−743.041			−1.28777			−0.643883	−	0.765124i	$-0.722678\pi$
$577$	−743.041			−1.28777			−0.643883	+	0.765124i	$0.722678\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	308.847			0.531579
$582$		0			0
$583$	−166.653			−0.285854
$584$		0			0
$585$		0			0
$586$		0			0
$587$	131.577			0.224151			0.112075	−	0.993700i	$-0.464250\pi$
$587$	131.577			0.224151			0.112075	+	0.993700i	$0.464250\pi$
$588$		0			0
$589$		−	705.999i		−	1.19864i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−553.887			−0.934042			−0.467021	−	0.884246i	$-0.654673\pi$
$593$	−553.887			−0.934042			−0.467021	+	0.884246i	$0.654673\pi$
$594$		0			0
$595$			76.7947i			0.129067i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−898.695			−1.50033			−0.750163	−	0.661253i	$-0.770025\pi$
$599$	−898.695			−1.50033			−0.750163	+	0.661253i	$0.770025\pi$
$600$		0			0
$601$	1094.79			1.82161			0.910804	−	0.412839i	$-0.135463\pi$
$601$	1094.79			1.82161			0.910804	+	0.412839i	$0.135463\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	234.194i		−	0.387098i
$606$		0			0
$607$	−448.466			−0.738824			−0.369412	−	0.929266i	$-0.620441\pi$
$607$	−448.466			−0.738824			−0.369412	+	0.929266i	$0.620441\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.3118			0.0168770
$612$		0			0
$613$			911.585i			1.48709i	0.668687	+	0.743544i	$0.266857\pi$
$613$			911.585i			1.48709i	−0.668687	+	0.743544i	$0.733143\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			239.385i			0.387982i	0.981003	+	0.193991i	$0.0621433\pi$
$617$			239.385i			0.387982i	−0.981003	+	0.193991i	$0.937857\pi$
$618$		0			0
$619$		−	367.754i		−	0.594109i	−0.954861	−	0.297055i	$-0.903996\pi$
$619$		−	367.754i		−	0.594109i	0.954861	−	0.297055i	$-0.0960043\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−454.068			−0.728841
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−41.2284			−0.0655459
$630$		0			0
$631$		−	447.846i		−	0.709740i	−0.934916	−	0.354870i	$-0.884525\pi$
$631$		−	447.846i		−	0.709740i	0.934916	−	0.354870i	$-0.115475\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			345.984i			0.544856i
$636$		0			0
$637$	−53.2820			−0.0836453
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	251.671i		−	0.392623i	−0.980542	−	0.196311i	$-0.937104\pi$
$641$		−	251.671i		−	0.392623i	0.980542	−	0.196311i	$-0.0628963\pi$
$642$		0			0
$643$			440.538i			0.685129i	0.939494	+	0.342565i	$0.111296\pi$
$643$			440.538i			0.685129i	−0.939494	+	0.342565i	$0.888704\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	44.9281			0.0694406			0.0347203	−	0.999397i	$-0.488946\pi$
$647$	44.9281			0.0694406			0.0347203	+	0.999397i	$0.488946\pi$
$648$		0			0
$649$		−	52.3198i		−	0.0806161i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	909.469			1.39275			0.696377	−	0.717676i	$-0.254794\pi$
$653$	909.469			1.39275			0.696377	+	0.717676i	$0.254794\pi$
$654$		0			0
$655$		−	450.175i		−	0.687290i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	42.4255i		−	0.0643787i	−0.999482	−	0.0321893i	$-0.989752\pi$
$659$		−	42.4255i		−	0.0643787i	0.999482	−	0.0321893i	$-0.0102480\pi$
$660$		0			0
$661$			355.110i			0.537231i	0.963248	+	0.268615i	$0.0865661\pi$
$661$			355.110i			0.537231i	−0.963248	+	0.268615i	$0.913434\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	125.832i		−	0.189221i
$666$		0			0
$667$	−919.739	−	46.1373i	−1.37892	−	0.0691714i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	177.446			0.264450
$672$		0			0
$673$	−105.461			−0.156703			−0.0783517	−	0.996926i	$-0.524966\pi$
$673$	−105.461			−0.156703			−0.0783517	+	0.996926i	$0.524966\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			322.625i			0.476551i	0.971198	+	0.238275i	$0.0765821\pi$
$677$			322.625i			0.476551i	−0.971198	+	0.238275i	$0.923418\pi$
$678$		0			0
$679$	106.351			0.156629
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.6400			0.0302195			0.0151098	−	0.999886i	$-0.495190\pi$
$683$	20.6400			0.0302195			0.0151098	+	0.999886i	$0.495190\pi$
$684$		0			0
$685$	44.1300			0.0644234
$686$		0			0
$687$		0			0
$688$		0			0
$689$			55.1710i			0.0800741i
$690$		0			0
$691$	−734.523			−1.06299			−0.531493	−	0.847063i	$-0.678369\pi$
$691$	−734.523			−1.06299			−0.531493	+	0.847063i	$0.678369\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.45587i		−	0.0107279i
$696$		0			0
$697$			712.652i			1.02246i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	819.017i		−	1.16835i	−0.811626	−	0.584177i	$-0.801417\pi$
$701$		−	819.017i		−	1.16835i	0.811626	−	0.584177i	$-0.198583\pi$
$702$		0			0
$703$	67.5546			0.0960947
$704$		0			0
$705$		0			0
$706$		0			0
$707$			61.2623i			0.0866511i
$708$		0			0
$709$		−	325.160i		−	0.458617i	−0.973354	−	0.229309i	$-0.926353\pi$
$709$		−	325.160i		−	0.458617i	0.973354	−	0.229309i	$-0.0736465\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	869.011	+	43.5926i	1.21881	+	0.0611397i
$714$		0			0
$715$	12.0404			0.0168397
$716$		0			0
$717$		0			0
$718$		0			0
$719$	121.949			0.169609			0.0848044	−	0.996398i	$-0.472973\pi$
$719$	121.949			0.169609			0.0848044	+	0.996398i	$0.472973\pi$
$720$		0			0
$721$	465.801			0.646048
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−200.195			−0.276131
$726$		0			0
$727$		−	80.4624i		−	0.110677i	−0.998468	−	0.0553386i	$-0.982376\pi$
$727$		−	80.4624i		−	0.110677i	0.998468	−	0.0553386i	$-0.0176238\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	713.961			0.976690
$732$		0			0
$733$			675.835i			0.922012i	0.887397	+	0.461006i	$0.152511\pi$
$733$			675.835i			0.922012i	−0.887397	+	0.461006i	$0.847489\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	265.877			0.360756
$738$		0			0
$739$	562.291			0.760881			0.380441	−	0.924805i	$-0.375772\pi$
$739$	562.291			0.760881			0.380441	+	0.924805i	$0.375772\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			619.562i			0.833865i	0.908937	+	0.416933i	$0.136895\pi$
$743$			619.562i			0.833865i	−0.908937	+	0.416933i	$0.863105\pi$
$744$		0			0
$745$	369.898			0.496507
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−491.908			−0.656753
$750$		0			0
$751$			1099.92i			1.46461i	0.680979	+	0.732303i	$0.261555\pi$
$751$			1099.92i			1.46461i	−0.680979	+	0.732303i	$0.738445\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	44.4183i		−	0.0588322i
$756$		0			0
$757$		−	642.260i		−	0.848428i	−0.905562	−	0.424214i	$-0.860550\pi$
$757$		−	642.260i		−	0.848428i	0.905562	−	0.424214i	$-0.139450\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	738.733			0.970740			0.485370	−	0.874309i	$-0.338685\pi$
$761$	738.733			0.970740			0.485370	+	0.874309i	$0.338685\pi$
$762$		0			0
$763$	347.481			0.455414
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−17.3207			−0.0225824
$768$		0			0
$769$			1469.39i			1.91078i	0.295344	+	0.955391i	$0.404566\pi$
$769$			1469.39i			1.91078i	−0.295344	+	0.955391i	$0.595434\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1161.08i		−	1.50205i	−0.660276	−	0.751023i	$-0.729561\pi$
$773$		−	1161.08i		−	1.50205i	0.660276	−	0.751023i	$-0.270439\pi$
$774$		0			0
$775$	189.153			0.244068
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1167.71i		−	1.49899i
$780$		0			0
$781$			133.435i			0.170852i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	90.1694			0.114865
$786$		0			0
$787$			579.573i			0.736433i	0.929740	+	0.368217i	$0.120032\pi$
$787$			579.573i			0.736433i	−0.929740	+	0.368217i	$0.879968\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	260.917			0.329858
$792$		0			0
$793$		−	58.7441i		−	0.0740783i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	788.823i		−	0.989741i	−0.868967	−	0.494870i	$-0.835216\pi$
$797$		−	788.823i		−	0.989741i	0.868967	−	0.494870i	$-0.164784\pi$
$798$		0			0
$799$			87.9651i			0.110094i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			135.361i			0.168570i
$804$		0			0
$805$	154.886	+	7.76960i	0.192404	+	0.00965168i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−981.006			−1.21262			−0.606308	−	0.795230i	$-0.707350\pi$
$809$	−981.006			−1.21262			−0.606308	+	0.795230i	$0.707350\pi$
$810$		0			0
$811$	1052.95			1.29833			0.649165	−	0.760647i	$-0.275118\pi$
$811$	1052.95			1.29833			0.649165	+	0.760647i	$0.275118\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	341.382i		−	0.418874i
$816$		0			0
$817$	−1169.86			−1.43189
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−93.5063			−0.113893			−0.0569466	−	0.998377i	$-0.518136\pi$
$821$	−93.5063			−0.113893			−0.0569466	+	0.998377i	$0.518136\pi$
$822$		0			0
$823$	−959.132			−1.16541			−0.582705	−	0.812684i	$-0.698006\pi$
$823$	−959.132			−1.16541			−0.582705	+	0.812684i	$0.698006\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	883.788i		−	1.06867i	−0.845274	−	0.534334i	$-0.820563\pi$
$827$		−	883.788i		−	1.06867i	0.845274	−	0.534334i	$-0.179437\pi$
$828$		0			0
$829$	−239.043			−0.288351			−0.144176	−	0.989552i	$-0.546053\pi$
$829$	−239.043			−0.288351			−0.144176	+	0.989552i	$0.546053\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	454.523i		−	0.545646i
$834$		0			0
$835$			214.788i			0.257232i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			683.240i			0.814350i	0.913350	+	0.407175i	$0.133486\pi$
$839$			683.240i			0.814350i	−0.913350	+	0.407175i	$0.866514\pi$
$840$		0			0
$841$	762.116			0.906202
$842$		0			0
$843$		0			0
$844$		0			0
$845$			373.909i			0.442496i
$846$		0			0
$847$		−	315.817i		−	0.372865i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.17123	+	83.1526i	−0.00490156	+	0.0977116i
$852$		0			0
$853$	−192.176			−0.225295			−0.112647	−	0.993635i	$-0.535933\pi$
$853$	−192.176			−0.225295			−0.112647	+	0.993635i	$0.535933\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−163.549			−0.190839			−0.0954197	−	0.995437i	$-0.530419\pi$
$857$	−163.549			−0.190839			−0.0954197	+	0.995437i	$0.530419\pi$
$858$		0			0
$859$	870.751			1.01368			0.506840	−	0.862040i	$-0.330813\pi$
$859$	870.751			1.01368			0.506840	+	0.862040i	$0.330813\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−989.203			−1.14624			−0.573119	−	0.819473i	$-0.694267\pi$
$863$	−989.203			−1.14624			−0.573119	+	0.819473i	$0.694267\pi$
$864$		0			0
$865$			127.414i			0.147300i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−355.228			−0.408778
$870$		0			0
$871$		−	88.0196i		−	0.101056i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	33.7131			0.0385293
$876$		0			0
$877$	−1318.01			−1.50287			−0.751433	−	0.659810i	$-0.770637\pi$
$877$	−1318.01			−1.50287			−0.751433	+	0.659810i	$0.770637\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	382.249i		−	0.433881i	−0.976185	−	0.216940i	$-0.930392\pi$
$881$		−	382.249i		−	0.433881i	0.976185	−	0.216940i	$-0.0696077\pi$
$882$		0			0
$883$	487.473			0.552065			0.276033	−	0.961148i	$-0.410980\pi$
$883$	487.473			0.552065			0.276033	+	0.961148i	$0.410980\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1351.49			1.52367			0.761835	−	0.647771i	$-0.224299\pi$
$887$	1351.49			1.52367			0.761835	+	0.647771i	$0.224299\pi$
$888$		0			0
$889$			466.567i			0.524823i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	144.135i		−	0.161405i
$894$		0			0
$895$		−	174.990i		−	0.195519i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1514.70			−1.68487
$900$		0			0
$901$	−470.637			−0.522350
$902$		0			0
$903$		0			0
$904$		0			0
$905$	76.9643			0.0850435
$906$		0			0
$907$			1015.42i			1.11953i	0.828651	+	0.559766i	$0.189109\pi$
$907$			1015.42i			1.11953i	−0.828651	+	0.559766i	$0.810891\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1136.70i		−	1.24775i	−0.781523	−	0.623877i	$-0.785557\pi$
$911$		−	1136.70i		−	1.24775i	0.781523	−	0.623877i	$-0.214443\pi$
$912$		0			0
$913$	413.075			0.452437
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	607.072i		−	0.662020i
$918$		0			0
$919$			20.3590i			0.0221535i	0.999939	+	0.0110767i	$0.00352591\pi$
$919$			20.3590i			0.0221535i	−0.999939	+	0.0110767i	$0.996474\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	44.1742			0.0478594
$924$		0			0
$925$			18.0994i			0.0195669i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−636.798			−0.685466			−0.342733	−	0.939433i	$-0.611353\pi$
$929$	−636.798			−0.685466			−0.342733	+	0.939433i	$0.611353\pi$
$930$		0			0
$931$			744.757i			0.799954i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			102.711i			0.109851i
$936$		0			0
$937$		−	635.821i		−	0.678571i	−0.940683	−	0.339285i	$-0.889815\pi$
$937$		−	635.821i		−	0.678571i	0.940683	−	0.339285i	$-0.110185\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	711.797i		−	0.756426i	−0.925719	−	0.378213i	$-0.876539\pi$
$941$		−	711.797i		−	0.756426i	0.925719	−	0.378213i	$-0.123461\pi$
$942$		0			0
$943$	1437.33	+	72.1017i	1.52421	+	0.0764599i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−309.176			−0.326480			−0.163240	−	0.986586i	$-0.552194\pi$
$947$	−309.176			−0.326480			−0.163240	+	0.986586i	$0.552194\pi$
$948$		0			0
$949$	44.8119			0.0472201
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	99.3662i		−	0.104267i	−0.998640	−	0.0521334i	$-0.983398\pi$
$953$		−	99.3662i		−	0.104267i	0.998640	−	0.0521334i	$-0.0166021\pi$
$954$		0			0
$955$	−224.296			−0.234865
$956$		0			0
$957$		0			0
$958$		0			0
$959$	59.5104			0.0620547
$960$		0			0
$961$	470.153			0.489233
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	480.774i		−	0.498211i
$966$		0			0
$967$	201.828			0.208716			0.104358	−	0.994540i	$-0.466721\pi$
$967$	201.828			0.208716			0.104358	+	0.994540i	$0.466721\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	78.9731i		−	0.0813318i	−0.999173	−	0.0406659i	$-0.987052\pi$
$971$		−	78.9731i		−	0.0813318i	0.999173	−	0.0406659i	$-0.0129479\pi$
$972$		0			0
$973$		−	10.0544i		−	0.0103334i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1764.09i		−	1.80562i	−0.430044	−	0.902808i	$-0.641502\pi$
$977$		−	1764.09i		−	1.80562i	0.430044	−	0.902808i	$-0.358498\pi$
$978$		0			0
$979$	−607.303			−0.620330
$980$		0			0
$981$		0			0
$982$		0			0
$983$			549.040i			0.558535i	0.960213	+	0.279268i	$0.0900917\pi$
$983$			549.040i			0.558535i	−0.960213	+	0.279268i	$0.909908\pi$
$984$		0			0
$985$		−	404.254i		−	0.410410i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	72.2340	−	1439.97i	0.0730375	−	1.45599i
$990$		0			0
$991$	894.623			0.902748			0.451374	−	0.892335i	$-0.350934\pi$
$991$	894.623			0.902748			0.451374	+	0.892335i	$0.350934\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−162.722			−0.163539
$996$		0			0
$997$	−760.191			−0.762479			−0.381239	−	0.924476i	$-0.624503\pi$
$997$	−760.191			−0.762479			−0.381239	+	0.924476i	$0.624503\pi$
$998$		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.2161.5		32
3.2	odd	2		1380.3.d.a.781.27	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.28		32
69.68	even	2		1380.3.d.a.781.22	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.22	✓	32	69.68	even	2
1380.3.d.a.781.27	yes	32	3.2	odd	2
4140.3.d.c.2161.5		32	1.1	even	1	trivial
4140.3.d.c.2161.28		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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -3.01539i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -3.01539i q^{7} -4.03301i q^{11} -1.33514 q^{13} -11.3894i q^{17} +18.6621i q^{19} +(-22.9711 - 1.15231i) q^{23} -5.00000 q^{25} +40.0389 q^{29} -37.8306 q^{31} -6.74262 q^{35} -3.61988i q^{37} -62.5713 q^{41} +62.6862i q^{43} -7.72339 q^{47} +39.9074 q^{49} -41.3222i q^{53} -9.01807 q^{55} +12.9729 q^{59} +43.9984i q^{61} +2.98547i q^{65} +65.9253i q^{67} -33.0858 q^{71} -33.5634 q^{73} -12.1611 q^{77} -88.0802i q^{79} +102.424i q^{83} -25.4676 q^{85} -150.583i q^{89} +4.02598i q^{91} +41.7298 q^{95} +35.2693i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more