LMFDB - Embedded newform 4140.3.d.c.2161.2

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			2161.2
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -10.9556i q^{7} +O(q^{10})$	$q-2.23607i q^{5} -10.9556i q^{7} +15.6383i q^{11} +4.14100 q^{13} +22.0418i q^{17} -9.79095i q^{19} +(-6.20242 - 22.1479i) q^{23} -5.00000 q^{25} +46.1239 q^{29} +22.4774 q^{31} -24.4975 q^{35} +63.2681i q^{37} -48.0421 q^{41} -26.9062i q^{43} +42.7574 q^{47} -71.0258 q^{49} +48.2132i q^{53} +34.9683 q^{55} +11.2111 q^{59} +56.7333i q^{61} -9.25956i q^{65} -83.8024i q^{67} -29.8994 q^{71} -134.730 q^{73} +171.328 q^{77} +100.665i q^{79} +118.834i q^{83} +49.2870 q^{85} -2.18729i q^{89} -45.3673i q^{91} -21.8932 q^{95} +67.6333i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	10.9556i		−	1.56509i	−0.622594	−	0.782545i	$-0.713921\pi$
$7$		−	10.9556i		−	1.56509i	0.622594	−	0.782545i	$-0.286079\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			15.6383i			1.42167i	0.703361	+	0.710833i	$0.251682\pi$
$11$			15.6383i			1.42167i	−0.703361	+	0.710833i	$0.748318\pi$
$12$		0			0
$13$	4.14100			0.318539			0.159269	−	0.987235i	$-0.449086\pi$
$13$	4.14100			0.318539			0.159269	+	0.987235i	$0.449086\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			22.0418i			1.29658i	0.761394	+	0.648289i	$0.224515\pi$
$17$			22.0418i			1.29658i	−0.761394	+	0.648289i	$0.775485\pi$
$18$		0			0
$19$		−	9.79095i		−	0.515313i	−0.966237	−	0.257657i	$-0.917050\pi$
$19$		−	9.79095i		−	0.515313i	0.966237	−	0.257657i	$-0.0829503\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.20242	−	22.1479i	−0.269671	−	0.962953i
$23$	−6.20242	−	22.1479i	−0.269671	−	0.962953i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	46.1239			1.59048			0.795240	−	0.606295i	$-0.207345\pi$
$29$	46.1239			1.59048			0.795240	+	0.606295i	$0.207345\pi$
$30$		0			0
$31$	22.4774			0.725078			0.362539	−	0.931969i	$-0.381910\pi$
$31$	22.4774			0.725078			0.362539	+	0.931969i	$0.381910\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−24.4975			−0.699930
$36$		0			0
$37$			63.2681i			1.70995i	0.518671	+	0.854974i	$0.326427\pi$
$37$			63.2681i			1.70995i	−0.518671	+	0.854974i	$0.673573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−48.0421			−1.17176			−0.585879	−	0.810398i	$-0.699251\pi$
$41$	−48.0421			−1.17176			−0.585879	+	0.810398i	$0.699251\pi$
$42$		0			0
$43$		−	26.9062i		−	0.625725i	−0.949798	−	0.312863i	$-0.898712\pi$
$43$		−	26.9062i		−	0.625725i	0.949798	−	0.312863i	$-0.101288\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42.7574			0.909732			0.454866	−	0.890560i	$-0.349687\pi$
$47$	42.7574			0.909732			0.454866	+	0.890560i	$0.349687\pi$
$48$		0			0
$49$	−71.0258			−1.44951
$50$		0			0
$51$		0			0
$52$		0			0
$53$			48.2132i			0.909683i	0.890572	+	0.454842i	$0.150304\pi$
$53$			48.2132i			0.909683i	−0.890572	+	0.454842i	$0.849696\pi$
$54$		0			0
$55$	34.9683			0.635788
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11.2111			0.190019			0.0950095	−	0.995476i	$-0.469712\pi$
$59$	11.2111			0.190019			0.0950095	+	0.995476i	$0.469712\pi$
$60$		0			0
$61$			56.7333i			0.930054i	0.885297	+	0.465027i	$0.153955\pi$
$61$			56.7333i			0.930054i	−0.885297	+	0.465027i	$0.846045\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	9.25956i		−	0.142455i
$66$		0			0
$67$		−	83.8024i		−	1.25078i	−0.780312	−	0.625391i	$-0.784940\pi$
$67$		−	83.8024i		−	1.25078i	0.780312	−	0.625391i	$-0.215060\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−29.8994			−0.421118			−0.210559	−	0.977581i	$-0.567528\pi$
$71$	−29.8994			−0.421118			−0.210559	+	0.977581i	$0.567528\pi$
$72$		0			0
$73$	−134.730			−1.84562			−0.922808	−	0.385261i	$-0.874111\pi$
$73$	−134.730			−1.84562			−0.922808	+	0.385261i	$0.874111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	171.328			2.22503
$78$		0			0
$79$			100.665i			1.27424i	0.770766	+	0.637118i	$0.219874\pi$
$79$			100.665i			1.27424i	−0.770766	+	0.637118i	$0.780126\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			118.834i			1.43174i	0.698236	+	0.715868i	$0.253969\pi$
$83$			118.834i			1.43174i	−0.698236	+	0.715868i	$0.746031\pi$
$84$		0			0
$85$	49.2870			0.579847
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.18729i		−	0.0245763i	−0.999924	−	0.0122881i	$-0.996088\pi$
$89$		−	2.18729i		−	0.0245763i	0.999924	−	0.0122881i	$-0.00391153\pi$
$90$		0			0
$91$		−	45.3673i		−	0.498541i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−21.8932			−0.230455
$96$		0			0
$97$			67.6333i			0.697250i	0.937262	+	0.348625i	$0.113351\pi$
$97$			67.6333i			0.697250i	−0.937262	+	0.348625i	$0.886649\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	36.2157			0.358571			0.179285	−	0.983797i	$-0.442621\pi$
$101$	36.2157			0.358571			0.179285	+	0.983797i	$0.442621\pi$
$102$		0			0
$103$			85.3665i			0.828801i	0.910095	+	0.414400i	$0.136009\pi$
$103$			85.3665i			0.828801i	−0.910095	+	0.414400i	$0.863991\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			139.482i			1.30357i	0.758404	+	0.651785i	$0.225980\pi$
$107$			139.482i			1.30357i	−0.758404	+	0.651785i	$0.774020\pi$
$108$		0			0
$109$			8.40310i			0.0770927i	0.999257	+	0.0385463i	$0.0122727\pi$
$109$			8.40310i			0.0770927i	−0.999257	+	0.0385463i	$0.987727\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			35.4404i			0.313632i	0.987628	+	0.156816i	$0.0501230\pi$
$113$			35.4404i			0.313632i	−0.987628	+	0.156816i	$0.949877\pi$
$114$		0			0
$115$	−49.5242	+	13.8690i	−0.430646	+	0.120600i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	241.482			2.02926
$120$		0			0
$121$	−123.557			−1.02113
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	13.3112			0.104812			0.0524062	−	0.998626i	$-0.483311\pi$
$127$	13.3112			0.104812			0.0524062	+	0.998626i	$0.483311\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	173.536			1.32471			0.662353	−	0.749192i	$-0.269558\pi$
$131$	173.536			1.32471			0.662353	+	0.749192i	$0.269558\pi$
$132$		0			0
$133$	−107.266			−0.806512
$134$		0			0
$135$		0			0
$136$		0			0
$137$			155.519i			1.13517i	0.823314	+	0.567587i	$0.192123\pi$
$137$			155.519i			1.13517i	−0.823314	+	0.567587i	$0.807877\pi$
$138$		0			0
$139$	25.6263			0.184362			0.0921810	−	0.995742i	$-0.470616\pi$
$139$	25.6263			0.184362			0.0921810	+	0.995742i	$0.470616\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			64.7583i			0.452855i
$144$		0			0
$145$		−	103.136i		−	0.711284i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	158.719i		−	1.06523i	−0.846357	−	0.532615i	$-0.821209\pi$
$149$		−	158.719i		−	1.06523i	0.846357	−	0.532615i	$-0.178791\pi$
$150$		0			0
$151$	−81.4734			−0.539559			−0.269779	−	0.962922i	$-0.586951\pi$
$151$	−81.4734			−0.539559			−0.269779	+	0.962922i	$0.586951\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	50.2611i		−	0.324265i
$156$		0			0
$157$		−	196.147i		−	1.24935i	−0.780887	−	0.624673i	$-0.785232\pi$
$157$		−	196.147i		−	1.24935i	0.780887	−	0.624673i	$-0.214768\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−242.644	+	67.9515i	−1.50711	+	0.422059i
$162$		0			0
$163$	279.106			1.71231			0.856154	−	0.516720i	$-0.172847\pi$
$163$	279.106			1.71231			0.856154	+	0.516720i	$0.172847\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	93.7380			0.561305			0.280653	−	0.959809i	$-0.409449\pi$
$167$	93.7380			0.561305			0.280653	+	0.959809i	$0.409449\pi$
$168$		0			0
$169$	−151.852			−0.898533
$170$		0			0
$171$		0			0
$172$		0			0
$173$	309.333			1.78805			0.894025	−	0.448018i	$-0.147870\pi$
$173$	309.333			1.78805			0.894025	+	0.448018i	$0.147870\pi$
$174$		0			0
$175$			54.7782i			0.313018i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−44.0710			−0.246207			−0.123103	−	0.992394i	$-0.539285\pi$
$179$	−44.0710			−0.246207			−0.123103	+	0.992394i	$0.539285\pi$
$180$		0			0
$181$		−	297.831i		−	1.64548i	−0.568420	−	0.822739i	$-0.692445\pi$
$181$		−	297.831i		−	1.64548i	0.568420	−	0.822739i	$-0.307555\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	141.472			0.764712
$186$		0			0
$187$	−344.697			−1.84330
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	325.077i		−	1.70197i	−0.525187	−	0.850987i	$-0.676004\pi$
$191$		−	325.077i		−	1.70197i	0.525187	−	0.850987i	$-0.323996\pi$
$192$		0			0
$193$	310.873			1.61074			0.805370	−	0.592772i	$-0.201966\pi$
$193$	310.873			1.61074			0.805370	+	0.592772i	$0.201966\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	342.889			1.74056			0.870278	−	0.492561i	$-0.163939\pi$
$197$	342.889			1.74056			0.870278	+	0.492561i	$0.163939\pi$
$198$		0			0
$199$			145.639i			0.731854i	0.930644	+	0.365927i	$0.119248\pi$
$199$			145.639i			0.731854i	−0.930644	+	0.365927i	$0.880752\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	505.316i		−	2.48924i
$204$		0			0
$205$			107.425i			0.524027i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	153.114			0.732603
$210$		0			0
$211$	−306.927			−1.45463			−0.727315	−	0.686304i	$-0.759232\pi$
$211$	−306.927			−1.45463			−0.727315	+	0.686304i	$0.759232\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−60.1641			−0.279833
$216$		0			0
$217$		−	246.254i		−	1.13481i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			91.2752i			0.413010i
$222$		0			0
$223$	270.216			1.21173			0.605865	−	0.795567i	$-0.292827\pi$
$223$	270.216			1.21173			0.605865	+	0.795567i	$0.292827\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	224.771i		−	0.990182i	−0.868841	−	0.495091i	$-0.835135\pi$
$227$		−	224.771i		−	0.990182i	0.868841	−	0.495091i	$-0.164865\pi$
$228$		0			0
$229$			175.242i			0.765247i	0.923904	+	0.382623i	$0.124979\pi$
$229$			175.242i			0.765247i	−0.923904	+	0.382623i	$0.875021\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	426.831			1.83189			0.915946	−	0.401301i	$-0.131442\pi$
$233$	426.831			1.83189			0.915946	+	0.401301i	$0.131442\pi$
$234$		0			0
$235$		−	95.6084i		−	0.406844i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	242.185			1.01333			0.506663	−	0.862144i	$-0.330879\pi$
$239$	242.185			1.01333			0.506663	+	0.862144i	$0.330879\pi$
$240$		0			0
$241$		−	139.437i		−	0.578575i	−0.957242	−	0.289288i	$-0.906582\pi$
$241$		−	139.437i		−	0.578575i	0.957242	−	0.289288i	$-0.0934184\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			158.819i			0.648239i
$246$		0			0
$247$		−	40.5443i		−	0.164147i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			62.2349i			0.247948i	0.992285	+	0.123974i	$0.0395639\pi$
$251$			62.2349i			0.247948i	−0.992285	+	0.123974i	$0.960436\pi$
$252$		0			0
$253$	346.356	−	96.9955i	1.36900	−	0.383381i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	75.8351			0.295078			0.147539	−	0.989056i	$-0.452865\pi$
$257$	75.8351			0.295078			0.147539	+	0.989056i	$0.452865\pi$
$258$		0			0
$259$	693.142			2.67622
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	35.7658i		−	0.135992i	−0.997686	−	0.0679958i	$-0.978340\pi$
$263$		−	35.7658i		−	0.135992i	0.997686	−	0.0679958i	$-0.0216604\pi$
$264$		0			0
$265$	107.808			0.406823
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−134.370			−0.499516			−0.249758	−	0.968308i	$-0.580351\pi$
$269$	−134.370			−0.499516			−0.249758	+	0.968308i	$0.580351\pi$
$270$		0			0
$271$	−31.5527			−0.116431			−0.0582153	−	0.998304i	$-0.518541\pi$
$271$	−31.5527			−0.116431			−0.0582153	+	0.998304i	$0.518541\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	78.1916i		−	0.284333i
$276$		0			0
$277$	283.999			1.02527			0.512633	−	0.858608i	$-0.328670\pi$
$277$	283.999			1.02527			0.512633	+	0.858608i	$0.328670\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	312.962i		−	1.11374i	−0.830598	−	0.556872i	$-0.812001\pi$
$281$		−	312.962i		−	1.11374i	0.830598	−	0.556872i	$-0.187999\pi$
$282$		0			0
$283$		−	107.000i		−	0.378090i	−0.981968	−	0.189045i	$-0.939461\pi$
$283$		−	107.000i		−	0.378090i	0.981968	−	0.189045i	$-0.0605393\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			526.332i			1.83391i
$288$		0			0
$289$	−196.842			−0.681115
$290$		0			0
$291$		0			0
$292$		0			0
$293$			514.198i			1.75494i	0.479630	+	0.877471i	$0.340771\pi$
$293$			514.198i			1.75494i	−0.479630	+	0.877471i	$0.659229\pi$
$294$		0			0
$295$		−	25.0688i		−	0.0849791i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−25.6842	−	91.7145i	−0.0859005	−	0.306738i
$300$		0			0
$301$	−294.774			−0.979317
$302$		0			0
$303$		0			0
$304$		0			0
$305$	126.859			0.415933
$306$		0			0
$307$	429.667			1.39957			0.699784	−	0.714355i	$-0.253280\pi$
$307$	429.667			1.39957			0.699784	+	0.714355i	$0.253280\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−78.6548			−0.252909			−0.126455	−	0.991972i	$-0.540360\pi$
$311$	−78.6548			−0.252909			−0.126455	+	0.991972i	$0.540360\pi$
$312$		0			0
$313$			251.012i			0.801955i	0.916088	+	0.400978i	$0.131329\pi$
$313$			251.012i			0.801955i	−0.916088	+	0.400978i	$0.868671\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−544.328			−1.71712			−0.858561	−	0.512711i	$-0.828641\pi$
$317$	−544.328			−1.71712			−0.858561	+	0.512711i	$0.828641\pi$
$318$		0			0
$319$			721.300i			2.26113i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	215.810			0.668144
$324$		0			0
$325$	−20.7050			−0.0637077
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	468.434i		−	1.42381i
$330$		0			0
$331$	−48.0170			−0.145066			−0.0725332	−	0.997366i	$-0.523108\pi$
$331$	−48.0170			−0.145066			−0.0725332	+	0.997366i	$0.523108\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−187.388			−0.559367
$336$		0			0
$337$		−	269.752i		−	0.800450i	−0.916417	−	0.400225i	$-0.868932\pi$
$337$		−	269.752i		−	0.800450i	0.916417	−	0.400225i	$-0.131068\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			351.509i			1.03082i
$342$		0			0
$343$			241.307i			0.703519i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−328.476			−0.946617			−0.473309	−	0.880897i	$-0.656940\pi$
$347$	−328.476			−0.946617			−0.473309	+	0.880897i	$0.656940\pi$
$348$		0			0
$349$	−537.518			−1.54017			−0.770083	−	0.637944i	$-0.779785\pi$
$349$	−537.518			−1.54017			−0.770083	+	0.637944i	$0.779785\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−225.238			−0.638067			−0.319033	−	0.947743i	$-0.603358\pi$
$353$	−225.238			−0.638067			−0.319033	+	0.947743i	$0.603358\pi$
$354$		0			0
$355$			66.8570i			0.188330i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			467.736i			1.30289i	0.758698	+	0.651443i	$0.225836\pi$
$359$			467.736i			1.30289i	−0.758698	+	0.651443i	$0.774164\pi$
$360$		0			0
$361$	265.137			0.734452
$362$		0			0
$363$		0			0
$364$		0			0
$365$			301.265i			0.825384i
$366$		0			0
$367$			64.4715i			0.175672i	0.996135	+	0.0878358i	$0.0279951\pi$
$367$			64.4715i			0.175672i	−0.996135	+	0.0878358i	$0.972005\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	528.206			1.42374
$372$		0			0
$373$		−	115.957i		−	0.310877i	−0.987846	−	0.155439i	$-0.950321\pi$
$373$		−	115.957i		−	0.310877i	0.987846	−	0.155439i	$-0.0496791\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	190.999			0.506629
$378$		0			0
$379$			73.9571i			0.195137i	0.995229	+	0.0975687i	$0.0311066\pi$
$379$			73.9571i			0.195137i	−0.995229	+	0.0975687i	$0.968893\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			66.3787i			0.173312i	0.996238	+	0.0866562i	$0.0276182\pi$
$383$			66.3787i			0.173312i	−0.996238	+	0.0866562i	$0.972382\pi$
$384$		0			0
$385$		−	383.100i		−	0.995066i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			137.632i			0.353809i	0.984228	+	0.176904i	$0.0566083\pi$
$389$			137.632i			0.353809i	−0.984228	+	0.176904i	$0.943392\pi$
$390$		0			0
$391$	488.180	−	136.713i	1.24854	−	0.349649i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	225.093			0.569855
$396$		0			0
$397$	277.148			0.698107			0.349053	−	0.937103i	$-0.386503\pi$
$397$	277.148			0.698107			0.349053	+	0.937103i	$0.386503\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			418.621i			1.04394i	0.852963	+	0.521972i	$0.174803\pi$
$401$			418.621i			1.04394i	−0.852963	+	0.521972i	$0.825197\pi$
$402$		0			0
$403$	93.0791			0.230965
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−989.407			−2.43097
$408$		0			0
$409$	129.820			0.317407			0.158704	−	0.987326i	$-0.449269\pi$
$409$	129.820			0.317407			0.158704	+	0.987326i	$0.449269\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	122.825i		−	0.297397i
$414$		0			0
$415$	265.721			0.640292
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	293.288i		−	0.699971i	−0.936755	−	0.349986i	$-0.886186\pi$
$419$		−	293.288i		−	0.699971i	0.936755	−	0.349986i	$-0.113814\pi$
$420$		0			0
$421$			111.846i			0.265667i	0.991138	+	0.132833i	$0.0424075\pi$
$421$			111.846i			0.265667i	−0.991138	+	0.132833i	$0.957592\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	110.209i		−	0.259316i
$426$		0			0
$427$	621.549			1.45562
$428$		0			0
$429$		0			0
$430$		0			0
$431$			121.083i			0.280935i	0.990085	+	0.140467i	$0.0448605\pi$
$431$			121.083i			0.280935i	−0.990085	+	0.140467i	$0.955140\pi$
$432$		0			0
$433$			462.818i			1.06886i	0.845211	+	0.534432i	$0.179474\pi$
$433$			462.818i			1.06886i	−0.845211	+	0.534432i	$0.820526\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−216.849	+	60.7276i	−0.496222	+	0.138965i
$438$		0			0
$439$	730.162			1.66324			0.831620	−	0.555345i	$-0.187414\pi$
$439$	730.162			1.66324			0.831620	+	0.555345i	$0.187414\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	753.296			1.70044			0.850221	−	0.526426i	$-0.176468\pi$
$443$	753.296			1.70044			0.850221	+	0.526426i	$0.176468\pi$
$444$		0			0
$445$	−4.89093			−0.0109908
$446$		0			0
$447$		0			0
$448$		0			0
$449$	31.6902			0.0705795			0.0352898	−	0.999377i	$-0.488765\pi$
$449$	31.6902			0.0705795			0.0352898	+	0.999377i	$0.488765\pi$
$450$		0			0
$451$		−	751.298i		−	1.66585i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−101.444			−0.222955
$456$		0			0
$457$			410.263i			0.897732i	0.893599	+	0.448866i	$0.148172\pi$
$457$			410.263i			0.897732i	−0.893599	+	0.448866i	$0.851828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	166.556			0.361292			0.180646	−	0.983548i	$-0.442181\pi$
$461$	166.556			0.361292			0.180646	+	0.983548i	$0.442181\pi$
$462$		0			0
$463$	220.267			0.475738			0.237869	−	0.971297i	$-0.423551\pi$
$463$	220.267			0.475738			0.237869	+	0.971297i	$0.423551\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			763.901i			1.63576i	0.575387	+	0.817882i	$0.304852\pi$
$467$			763.901i			1.63576i	−0.575387	+	0.817882i	$0.695148\pi$
$468$		0			0
$469$	−918.108			−1.95759
$470$		0			0
$471$		0			0
$472$		0			0
$473$	420.768			0.889572
$474$		0			0
$475$			48.9548i			0.103063i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	76.7152i		−	0.160157i	−0.996789	−	0.0800785i	$-0.974483\pi$
$479$		−	76.7152i		−	0.160157i	0.996789	−	0.0800785i	$-0.0255171\pi$
$480$		0			0
$481$			261.993i			0.544684i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	151.233			0.311820
$486$		0			0
$487$	−121.432			−0.249347			−0.124674	−	0.992198i	$-0.539788\pi$
$487$	−121.432			−0.249347			−0.124674	+	0.992198i	$0.539788\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−493.477			−1.00505			−0.502523	−	0.864564i	$-0.667595\pi$
$491$	−493.477			−1.00505			−0.502523	+	0.864564i	$0.667595\pi$
$492$		0			0
$493$			1016.65i			2.06218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			327.566i			0.659087i
$498$		0			0
$499$	469.248			0.940377			0.470188	−	0.882566i	$-0.344186\pi$
$499$	469.248			0.940377			0.470188	+	0.882566i	$0.344186\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			275.437i			0.547588i	0.961788	+	0.273794i	$0.0882787\pi$
$503$			275.437i			0.547588i	−0.961788	+	0.273794i	$0.911721\pi$
$504$		0			0
$505$		−	80.9807i		−	0.160358i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	371.271			0.729412			0.364706	−	0.931123i	$-0.381169\pi$
$509$	371.271			0.729412			0.364706	+	0.931123i	$0.381169\pi$
$510$		0			0
$511$			1476.05i			2.88855i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	190.885			0.370651
$516$		0			0
$517$			668.654i			1.29333i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	862.959i		−	1.65635i	−0.560469	−	0.828176i	$-0.689379\pi$
$521$		−	862.959i		−	1.65635i	0.560469	−	0.828176i	$-0.310621\pi$
$522$		0			0
$523$			106.860i			0.204321i	0.994768	+	0.102160i	$0.0325755\pi$
$523$			106.860i			0.204321i	−0.994768	+	0.102160i	$0.967424\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			495.444i			0.940121i
$528$		0			0
$529$	−452.060	+	274.741i	−0.854556	+	0.519360i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−198.942			−0.373250
$534$		0			0
$535$	311.891			0.582974
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1110.72i		−	2.06071i
$540$		0			0
$541$	−150.515			−0.278216			−0.139108	−	0.990277i	$-0.544424\pi$
$541$	−150.515			−0.278216			−0.139108	+	0.990277i	$0.544424\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	18.7899			0.0344769
$546$		0			0
$547$	−525.905			−0.961435			−0.480717	−	0.876876i	$-0.659624\pi$
$547$	−525.905			−0.961435			−0.480717	+	0.876876i	$0.659624\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	451.597i		−	0.819595i
$552$		0			0
$553$	1102.84			1.99429
$554$		0			0
$555$		0			0
$556$		0			0
$557$			190.022i			0.341152i	0.985345	+	0.170576i	$0.0545629\pi$
$557$			190.022i			0.341152i	−0.985345	+	0.170576i	$0.945437\pi$
$558$		0			0
$559$		−	111.419i		−	0.199318i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	125.444i		−	0.222814i	−0.993775	−	0.111407i	$-0.964464\pi$
$563$		−	125.444i		−	0.222814i	0.993775	−	0.111407i	$-0.0355357\pi$
$564$		0			0
$565$	79.2472			0.140261
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	802.343i		−	1.41009i	−0.709161	−	0.705047i	$-0.750926\pi$
$569$		−	802.343i		−	1.41009i	0.709161	−	0.705047i	$-0.249074\pi$
$570$		0			0
$571$		−	804.407i		−	1.40877i	−0.709819	−	0.704384i	$-0.751223\pi$
$571$		−	804.407i		−	1.40877i	0.709819	−	0.704384i	$-0.248777\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	31.0121	+	110.740i	0.0539341	+	0.192591i
$576$		0			0
$577$	188.872			0.327334			0.163667	−	0.986516i	$-0.447668\pi$
$577$	188.872			0.327334			0.163667	+	0.986516i	$0.447668\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1301.90			2.24080
$582$		0			0
$583$	−753.974			−1.29327
$584$		0			0
$585$		0			0
$586$		0			0
$587$	625.806			1.06611			0.533055	−	0.846081i	$-0.321044\pi$
$587$	625.806			1.06611			0.533055	+	0.846081i	$0.321044\pi$
$588$		0			0
$589$		−	220.075i		−	0.373642i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−830.618			−1.40070			−0.700352	−	0.713797i	$-0.746974\pi$
$593$	−830.618			−1.40070			−0.700352	+	0.713797i	$0.746974\pi$
$594$		0			0
$595$		−	539.970i		−	0.907513i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	591.359			0.987244			0.493622	−	0.869676i	$-0.335673\pi$
$599$	591.359			0.987244			0.493622	+	0.869676i	$0.335673\pi$
$600$		0			0
$601$	481.738			0.801561			0.400780	−	0.916174i	$-0.368739\pi$
$601$	481.738			0.801561			0.400780	+	0.916174i	$0.368739\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			276.282i			0.456664i
$606$		0			0
$607$	−641.293			−1.05650			−0.528248	−	0.849090i	$-0.677151\pi$
$607$	−641.293			−1.05650			−0.528248	+	0.849090i	$0.677151\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	177.058			0.289785
$612$		0			0
$613$		−	944.634i		−	1.54100i	−0.637439	−	0.770501i	$-0.720006\pi$
$613$		−	944.634i		−	1.54100i	0.637439	−	0.770501i	$-0.279994\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			100.309i			0.162576i	0.996691	+	0.0812878i	$0.0259033\pi$
$617$			100.309i			0.162576i	−0.996691	+	0.0812878i	$0.974097\pi$
$618$		0			0
$619$			197.941i			0.319775i	0.987135	+	0.159887i	$0.0511131\pi$
$619$			197.941i			0.319775i	−0.987135	+	0.159887i	$0.948887\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.9631			−0.0384641
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1394.54			−2.21708
$630$		0			0
$631$		−	604.447i		−	0.957920i	−0.877837	−	0.478960i	$-0.841014\pi$
$631$		−	604.447i		−	0.957920i	0.877837	−	0.478960i	$-0.158986\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	29.7647i		−	0.0468735i
$636$		0			0
$637$	−294.118			−0.461724
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	138.982i		−	0.216821i	−0.994106	−	0.108410i	$-0.965424\pi$
$641$		−	138.982i		−	0.216821i	0.994106	−	0.108410i	$-0.0345760\pi$
$642$		0			0
$643$			967.966i			1.50539i	0.658369	+	0.752695i	$0.271247\pi$
$643$			967.966i			1.50539i	−0.658369	+	0.752695i	$0.728753\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	98.1923			0.151765			0.0758827	−	0.997117i	$-0.475823\pi$
$647$	98.1923			0.151765			0.0758827	+	0.997117i	$0.475823\pi$
$648$		0			0
$649$			175.323i			0.270143i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.12387			0.00784666			0.00392333	−	0.999992i	$-0.498751\pi$
$653$	5.12387			0.00784666			0.00392333	+	0.999992i	$0.498751\pi$
$654$		0			0
$655$		−	388.039i		−	0.592426i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			950.020i			1.44161i	0.693139	+	0.720804i	$0.256227\pi$
$659$			950.020i			1.44161i	−0.693139	+	0.720804i	$0.743773\pi$
$660$		0			0
$661$		−	548.308i		−	0.829513i	−0.909932	−	0.414756i	$-0.863867\pi$
$661$		−	548.308i		−	0.829513i	0.909932	−	0.414756i	$-0.136133\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			239.854i			0.360683i
$666$		0			0
$667$	−286.080	−	1021.55i	−0.428905	−	1.53156i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−887.213			−1.32223
$672$		0			0
$673$	−1297.44			−1.92785			−0.963924	−	0.266178i	$-0.914239\pi$
$673$	−1297.44			−1.92785			−0.963924	+	0.266178i	$0.914239\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			716.739i			1.05870i	0.848404	+	0.529350i	$0.177564\pi$
$677$			716.739i			1.05870i	−0.848404	+	0.529350i	$0.822436\pi$
$678$		0			0
$679$	740.965			1.09126
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−139.569			−0.204347			−0.102173	−	0.994767i	$-0.532580\pi$
$683$	−139.569			−0.204347			−0.102173	+	0.994767i	$0.532580\pi$
$684$		0			0
$685$	347.750			0.507665
$686$		0			0
$687$		0			0
$688$		0			0
$689$			199.651i			0.289769i
$690$		0			0
$691$	1003.08			1.45163			0.725817	−	0.687888i	$-0.241462\pi$
$691$	1003.08			1.45163			0.725817	+	0.687888i	$0.241462\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	57.3022i		−	0.0824492i
$696$		0			0
$697$		−	1058.94i		−	1.51928i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	790.112i		−	1.12712i	−0.826075	−	0.563561i	$-0.809431\pi$
$701$		−	790.112i		−	1.12712i	0.826075	−	0.563561i	$-0.190569\pi$
$702$		0			0
$703$	619.455			0.881159
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	396.765i		−	0.561196i
$708$		0			0
$709$			1309.66i			1.84719i	0.383367	+	0.923596i	$0.374765\pi$
$709$			1309.66i			1.84719i	−0.383367	+	0.923596i	$0.625235\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−139.415	−	497.828i	−0.195532	−	0.698216i
$714$		0			0
$715$	144.804			0.202523
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−83.2696			−0.115813			−0.0579065	−	0.998322i	$-0.518443\pi$
$719$	−83.2696			−0.115813			−0.0579065	+	0.998322i	$0.518443\pi$
$720$		0			0
$721$	935.244			1.29715
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−230.619			−0.318096
$726$		0			0
$727$		−	801.697i		−	1.10275i	−0.834258	−	0.551374i	$-0.814104\pi$
$727$		−	801.697i		−	1.10275i	0.834258	−	0.551374i	$-0.185896\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	593.062			0.811302
$732$		0			0
$733$		−	1183.49i		−	1.61459i	−0.590150	−	0.807294i	$-0.700931\pi$
$733$		−	1183.49i		−	1.61459i	0.590150	−	0.807294i	$-0.299069\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1310.53			1.77819
$738$		0			0
$739$	−131.897			−0.178481			−0.0892404	−	0.996010i	$-0.528444\pi$
$739$	−131.897			−0.178481			−0.0892404	+	0.996010i	$0.528444\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			926.182i			1.24654i	0.782005	+	0.623272i	$0.214197\pi$
$743$			926.182i			1.24654i	−0.782005	+	0.623272i	$0.785803\pi$
$744$		0			0
$745$	−354.907			−0.476386
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1528.11			2.04021
$750$		0			0
$751$			453.210i			0.603475i	0.953391	+	0.301737i	$0.0975666\pi$
$751$			453.210i			0.603475i	−0.953391	+	0.301737i	$0.902433\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			182.180i			0.241298i
$756$		0			0
$757$			292.660i			0.386605i	0.981139	+	0.193302i	$0.0619198\pi$
$757$			292.660i			0.386605i	−0.981139	+	0.193302i	$0.938080\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−558.472			−0.733866			−0.366933	−	0.930247i	$-0.619592\pi$
$761$	−558.472			−0.733866			−0.366933	+	0.930247i	$0.619592\pi$
$762$		0			0
$763$	92.0613			0.120657
$764$		0			0
$765$		0			0
$766$		0			0
$767$	46.4253			0.0605284
$768$		0			0
$769$		−	600.327i		−	0.780659i	−0.920675	−	0.390329i	$-0.872361\pi$
$769$		−	600.327i		−	0.780659i	0.920675	−	0.390329i	$-0.127639\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1488.43i		−	1.92553i	−0.270342	−	0.962764i	$-0.587137\pi$
$773$		−	1488.43i		−	1.92553i	0.270342	−	0.962764i	$-0.412863\pi$
$774$		0			0
$775$	−112.387			−0.145016
$776$		0			0
$777$		0			0
$778$		0			0
$779$			470.378i			0.603823i
$780$		0			0
$781$		−	467.576i		−	0.598688i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−438.598			−0.558724
$786$		0			0
$787$		−	493.323i		−	0.626840i	−0.949615	−	0.313420i	$-0.898525\pi$
$787$		−	493.323i		−	0.626840i	0.949615	−	0.313420i	$-0.101475\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	388.272			0.490863
$792$		0			0
$793$			234.933i			0.296258i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			870.248i			1.09190i	0.837816	+	0.545952i	$0.183832\pi$
$797$			870.248i			1.09190i	−0.837816	+	0.545952i	$0.816168\pi$
$798$		0			0
$799$			942.451i			1.17954i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	2106.95i		−	2.62385i
$804$		0			0
$805$	151.944	+	542.569i	0.188750	+	0.673999i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−109.777			−0.135695			−0.0678476	−	0.997696i	$-0.521613\pi$
$809$	−109.777			−0.135695			−0.0678476	+	0.997696i	$0.521613\pi$
$810$		0			0
$811$	1383.17			1.70551			0.852754	−	0.522313i	$-0.174931\pi$
$811$	1383.17			1.70551			0.852754	+	0.522313i	$0.174931\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	624.101i		−	0.765768i
$816$		0			0
$817$	−263.437			−0.322445
$818$		0			0
$819$		0			0
$820$		0			0
$821$	293.940			0.358027			0.179013	−	0.983847i	$-0.442709\pi$
$821$	293.940			0.358027			0.179013	+	0.983847i	$0.442709\pi$
$822$		0			0
$823$	−1607.53			−1.95326			−0.976628	−	0.214936i	$-0.931046\pi$
$823$	−1607.53			−1.95326			−0.976628	+	0.214936i	$0.931046\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1453.75i			1.75786i	0.476954	+	0.878928i	$0.341741\pi$
$827$			1453.75i			1.75786i	−0.476954	+	0.878928i	$0.658259\pi$
$828$		0			0
$829$	−822.523			−0.992187			−0.496093	−	0.868269i	$-0.665233\pi$
$829$	−822.523			−0.992187			−0.496093	+	0.868269i	$0.665233\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1565.54i		−	1.87940i
$834$		0			0
$835$		−	209.605i		−	0.251023i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	635.154i		−	0.757036i	−0.925594	−	0.378518i	$-0.876434\pi$
$839$		−	635.154i		−	0.757036i	0.925594	−	0.378518i	$-0.123566\pi$
$840$		0			0
$841$	1286.41			1.52962
$842$		0			0
$843$		0			0
$844$		0			0
$845$			339.552i			0.401836i
$846$		0			0
$847$			1353.64i			1.59816i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1401.26	−	392.415i	1.64660	−	0.461123i
$852$		0			0
$853$	−863.435			−1.01223			−0.506117	−	0.862465i	$-0.668919\pi$
$853$	−863.435			−1.01223			−0.506117	+	0.862465i	$0.668919\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	529.779			0.618179			0.309089	−	0.951033i	$-0.399976\pi$
$857$	529.779			0.618179			0.309089	+	0.951033i	$0.399976\pi$
$858$		0			0
$859$	−530.245			−0.617282			−0.308641	−	0.951179i	$-0.599874\pi$
$859$	−530.245			−0.617282			−0.308641	+	0.951179i	$0.599874\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−82.0388			−0.0950624			−0.0475312	−	0.998870i	$-0.515135\pi$
$863$	−82.0388			−0.0950624			−0.0475312	+	0.998870i	$0.515135\pi$
$864$		0			0
$865$		−	691.689i		−	0.799640i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1574.23			−1.81154
$870$		0			0
$871$		−	347.026i		−	0.398422i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	122.488			0.139986
$876$		0			0
$877$	−616.126			−0.702538			−0.351269	−	0.936275i	$-0.614250\pi$
$877$	−616.126			−0.702538			−0.351269	+	0.936275i	$0.614250\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			314.173i			0.356609i	0.983975	+	0.178305i	$0.0570613\pi$
$881$			314.173i			0.356609i	−0.983975	+	0.178305i	$0.942939\pi$
$882$		0			0
$883$	−1073.46			−1.21569			−0.607847	−	0.794054i	$-0.707967\pi$
$883$	−1073.46			−1.21569			−0.607847	+	0.794054i	$0.707967\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1155.76			−1.30300			−0.651499	−	0.758650i	$-0.725860\pi$
$887$	−1155.76			−1.30300			−0.651499	+	0.758650i	$0.725860\pi$
$888$		0			0
$889$		−	145.832i		−	0.164041i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	418.636i		−	0.468797i
$894$		0			0
$895$			98.5458i			0.110107i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1036.75			1.15322
$900$		0			0
$901$	−1062.71			−1.17948
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−665.971			−0.735880
$906$		0			0
$907$		−	486.355i		−	0.536224i	−0.963388	−	0.268112i	$-0.913600\pi$
$907$		−	486.355i		−	0.536224i	0.963388	−	0.268112i	$-0.0863998\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1191.75i			1.30818i	0.756417	+	0.654090i	$0.226948\pi$
$911$			1191.75i			1.30818i	−0.756417	+	0.654090i	$0.773052\pi$
$912$		0			0
$913$	−1858.37			−2.03545
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1901.20i		−	2.07328i
$918$		0			0
$919$		−	205.998i		−	0.224154i	−0.993700	−	0.112077i	$-0.964250\pi$
$919$		−	205.998i		−	0.224154i	0.993700	−	0.112077i	$-0.0357504\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−123.813			−0.134142
$924$		0			0
$925$		−	316.340i		−	0.341990i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1351.59			−1.45489			−0.727445	−	0.686166i	$-0.759292\pi$
$929$	−1351.59			−1.45489			−0.727445	+	0.686166i	$0.759292\pi$
$930$		0			0
$931$			695.410i			0.746950i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			770.766i			0.824349i
$936$		0			0
$937$			1153.39i			1.23094i	0.788161	+	0.615469i	$0.211034\pi$
$937$			1153.39i			1.23094i	−0.788161	+	0.615469i	$0.788966\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1153.58i		−	1.22590i	−0.790120	−	0.612952i	$-0.789982\pi$
$941$		−	1153.58i		−	1.22590i	0.790120	−	0.612952i	$-0.210018\pi$
$942$		0			0
$943$	297.978	+	1064.03i	0.315989	+	1.12835i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−726.208			−0.766851			−0.383426	−	0.923572i	$-0.625256\pi$
$947$	−726.208			−0.766851			−0.383426	+	0.923572i	$0.625256\pi$
$948$		0			0
$949$	−557.917			−0.587900
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1260.34i			1.32250i	0.750165	+	0.661251i	$0.229974\pi$
$953$			1260.34i			1.32250i	−0.750165	+	0.661251i	$0.770026\pi$
$954$		0			0
$955$	−726.894			−0.761146
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1703.81			1.77665
$960$		0			0
$961$	−455.765			−0.474261
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	695.133i		−	0.720345i
$966$		0			0
$967$	1800.75			1.86221			0.931103	−	0.364757i	$-0.118848\pi$
$967$	1800.75			1.86221			0.931103	+	0.364757i	$0.118848\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	5.23250i		−	0.00538878i	−0.999996	−	0.00269439i	$-0.999142\pi$
$971$		−	5.23250i		−	0.00538878i	0.999996	−	0.00269439i	$-0.000857652\pi$
$972$		0			0
$973$		−	280.752i		−	0.288543i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1419.78i			1.45320i	0.687058	+	0.726602i	$0.258902\pi$
$977$			1419.78i			1.45320i	−0.687058	+	0.726602i	$0.741098\pi$
$978$		0			0
$979$	34.2055			0.0349392
$980$		0			0
$981$		0			0
$982$		0			0
$983$			407.188i			0.414230i	0.978317	+	0.207115i	$0.0664074\pi$
$983$			407.188i			0.414230i	−0.978317	+	0.207115i	$0.933593\pi$
$984$		0			0
$985$		−	766.724i		−	0.778400i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−595.916	+	166.884i	−0.602544	+	0.168740i
$990$		0			0
$991$	633.181			0.638932			0.319466	−	0.947598i	$-0.396497\pi$
$991$	633.181			0.638932			0.319466	+	0.947598i	$0.396497\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	325.659			0.327295
$996$		0			0
$997$	254.868			0.255635			0.127817	−	0.991798i	$-0.459203\pi$
$997$	254.868			0.255635			0.127817	+	0.991798i	$0.459203\pi$
$998$		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.2161.2		32
3.2	odd	2		1380.3.d.a.781.10	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.31		32
69.68	even	2		1380.3.d.a.781.7	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.7	✓	32	69.68	even	2
1380.3.d.a.781.10	yes	32	3.2	odd	2
4140.3.d.c.2161.2		32	1.1	even	1	trivial
4140.3.d.c.2161.31		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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -10.9556i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -10.9556i q^{7} +15.6383i q^{11} +4.14100 q^{13} +22.0418i q^{17} -9.79095i q^{19} +(-6.20242 - 22.1479i) q^{23} -5.00000 q^{25} +46.1239 q^{29} +22.4774 q^{31} -24.4975 q^{35} +63.2681i q^{37} -48.0421 q^{41} -26.9062i q^{43} +42.7574 q^{47} -71.0258 q^{49} +48.2132i q^{53} +34.9683 q^{55} +11.2111 q^{59} +56.7333i q^{61} -9.25956i q^{65} -83.8024i q^{67} -29.8994 q^{71} -134.730 q^{73} +171.328 q^{77} +100.665i q^{79} +118.834i q^{83} +49.2870 q^{85} -2.18729i q^{89} -45.3673i q^{91} -21.8932 q^{95} +67.6333i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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