LMFDB - Embedded newform 4140.3.d.c.2161.9

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			2161.9
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} +0.509961i q^{7} +O(q^{10})$	$q-2.23607i q^{5} +0.509961i q^{7} +17.4828i q^{11} +16.2803 q^{13} -10.8574i q^{17} +9.45266i q^{19} +(0.700637 - 22.9893i) q^{23} -5.00000 q^{25} -20.6961 q^{29} +11.5002 q^{31} +1.14031 q^{35} -62.7695i q^{37} -12.0760 q^{41} -37.7801i q^{43} +83.3025 q^{47} +48.7399 q^{49} +58.9194i q^{53} +39.0928 q^{55} -50.0025 q^{59} -38.6463i q^{61} -36.4038i q^{65} +82.8428i q^{67} -125.880 q^{71} +138.846 q^{73} -8.91557 q^{77} +111.640i q^{79} -40.8586i q^{83} -24.2780 q^{85} +7.92462i q^{89} +8.30230i q^{91} +21.1368 q^{95} +45.4089i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			0.509961i			0.0728516i	0.999336	+	0.0364258i	$0.0115973\pi$
$7$			0.509961i			0.0728516i	−0.999336	+	0.0364258i	$0.988403\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			17.4828i			1.58935i	0.607036	+	0.794674i	$0.292358\pi$
$11$			17.4828i			1.58935i	−0.607036	+	0.794674i	$0.707642\pi$
$12$		0			0
$13$	16.2803			1.25233			0.626164	−	0.779692i	$-0.284624\pi$
$13$	16.2803			1.25233			0.626164	+	0.779692i	$0.284624\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	10.8574i		−	0.638673i	−0.947641	−	0.319337i	$-0.896540\pi$
$17$		−	10.8574i		−	0.638673i	0.947641	−	0.319337i	$-0.103460\pi$
$18$		0			0
$19$			9.45266i			0.497508i	0.968567	+	0.248754i	$0.0800211\pi$
$19$			9.45266i			0.497508i	−0.968567	+	0.248754i	$0.919979\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.700637	−	22.9893i	0.0304625	−	0.999536i
$23$	0.700637	−	22.9893i	0.0304625	−	0.999536i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−20.6961			−0.713659			−0.356830	−	0.934169i	$-0.616142\pi$
$29$	−20.6961			−0.713659			−0.356830	+	0.934169i	$0.616142\pi$
$30$		0			0
$31$	11.5002			0.370974			0.185487	−	0.982647i	$-0.440614\pi$
$31$	11.5002			0.370974			0.185487	+	0.982647i	$0.440614\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.14031			0.0325802
$36$		0			0
$37$		−	62.7695i		−	1.69647i	−0.529618	−	0.848237i	$-0.677665\pi$
$37$		−	62.7695i		−	1.69647i	0.529618	−	0.848237i	$-0.322335\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.0760			−0.294536			−0.147268	−	0.989097i	$-0.547048\pi$
$41$	−12.0760			−0.294536			−0.147268	+	0.989097i	$0.547048\pi$
$42$		0			0
$43$		−	37.7801i		−	0.878607i	−0.898339	−	0.439304i	$-0.855225\pi$
$43$		−	37.7801i		−	0.878607i	0.898339	−	0.439304i	$-0.144775\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	83.3025			1.77239			0.886197	−	0.463309i	$-0.153338\pi$
$47$	83.3025			1.77239			0.886197	+	0.463309i	$0.153338\pi$
$48$		0			0
$49$	48.7399			0.994693
$50$		0			0
$51$		0			0
$52$		0			0
$53$			58.9194i			1.11169i	0.831287	+	0.555844i	$0.187605\pi$
$53$			58.9194i			1.11169i	−0.831287	+	0.555844i	$0.812395\pi$
$54$		0			0
$55$	39.0928			0.710778
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−50.0025			−0.847501			−0.423750	−	0.905779i	$-0.639287\pi$
$59$	−50.0025			−0.847501			−0.423750	+	0.905779i	$0.639287\pi$
$60$		0			0
$61$		−	38.6463i		−	0.633545i	−0.948501	−	0.316773i	$-0.897401\pi$
$61$		−	38.6463i		−	0.633545i	0.948501	−	0.316773i	$-0.102599\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	36.4038i		−	0.560058i
$66$		0			0
$67$			82.8428i			1.23646i	0.785997	+	0.618230i	$0.212150\pi$
$67$			82.8428i			1.23646i	−0.785997	+	0.618230i	$0.787850\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−125.880			−1.77296			−0.886481	−	0.462765i	$-0.846857\pi$
$71$	−125.880			−1.77296			−0.886481	+	0.462765i	$0.846857\pi$
$72$		0			0
$73$	138.846			1.90199			0.950997	−	0.309201i	$-0.100062\pi$
$73$	138.846			1.90199			0.950997	+	0.309201i	$0.100062\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.91557			−0.115787
$78$		0			0
$79$			111.640i			1.41317i	0.707629	+	0.706584i	$0.249765\pi$
$79$			111.640i			1.41317i	−0.707629	+	0.706584i	$0.750235\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	40.8586i		−	0.492273i	−0.969235	−	0.246136i	$-0.920839\pi$
$83$		−	40.8586i		−	0.492273i	0.969235	−	0.246136i	$-0.0791611\pi$
$84$		0			0
$85$	−24.2780			−0.285623
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.92462i			0.0890406i	0.999008	+	0.0445203i	$0.0141759\pi$
$89$			7.92462i			0.0890406i	−0.999008	+	0.0445203i	$0.985824\pi$
$90$		0			0
$91$			8.30230i			0.0912341i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	21.1368			0.222492
$96$		0			0
$97$			45.4089i			0.468133i	0.972221	+	0.234067i	$0.0752033\pi$
$97$			45.4089i			0.468133i	−0.972221	+	0.234067i	$0.924797\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−124.242			−1.23012			−0.615060	−	0.788480i	$-0.710868\pi$
$101$	−124.242			−1.23012			−0.615060	+	0.788480i	$0.710868\pi$
$102$		0			0
$103$			75.6613i			0.734576i	0.930107	+	0.367288i	$0.119714\pi$
$103$			75.6613i			0.734576i	−0.930107	+	0.367288i	$0.880286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			73.1471i			0.683618i	0.939769	+	0.341809i	$0.111040\pi$
$107$			73.1471i			0.683618i	−0.939769	+	0.341809i	$0.888960\pi$
$108$		0			0
$109$			16.4933i			0.151315i	0.997134	+	0.0756573i	$0.0241055\pi$
$109$			16.4933i			0.151315i	−0.997134	+	0.0756573i	$0.975894\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	27.7044i		−	0.245171i	−0.992458	−	0.122586i	$-0.960881\pi$
$113$		−	27.7044i		−	0.245171i	0.992458	−	0.122586i	$-0.0391186\pi$
$114$		0			0
$115$	−51.4057	−	1.56667i	−0.447006	−	0.0136232i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.53688			0.0465284
$120$		0			0
$121$	−184.650			−1.52603
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	247.508			1.94888			0.974440	−	0.224648i	$-0.0721231\pi$
$127$	247.508			1.94888			0.974440	+	0.224648i	$0.0721231\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	82.2249			0.627671			0.313836	−	0.949477i	$-0.398386\pi$
$131$	82.2249			0.627671			0.313836	+	0.949477i	$0.398386\pi$
$132$		0			0
$133$	−4.82049			−0.0362443
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	238.456i		−	1.74055i	−0.492562	−	0.870277i	$-0.663940\pi$
$137$		−	238.456i		−	1.74055i	0.492562	−	0.870277i	$-0.336060\pi$
$138$		0			0
$139$	12.6086			0.0907092			0.0453546	−	0.998971i	$-0.485558\pi$
$139$	12.6086			0.0907092			0.0453546	+	0.998971i	$0.485558\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			284.625i			1.99039i
$144$		0			0
$145$			46.2779i			0.319158i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			172.426i			1.15722i	0.815605	+	0.578610i	$0.196405\pi$
$149$			172.426i			1.15722i	−0.815605	+	0.578610i	$0.803595\pi$
$150$		0			0
$151$	272.970			1.80775			0.903875	−	0.427797i	$-0.140710\pi$
$151$	272.970			1.80775			0.903875	+	0.427797i	$0.140710\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	25.7152i		−	0.165905i
$156$		0			0
$157$		−	188.925i		−	1.20334i	−0.798744	−	0.601670i	$-0.794502\pi$
$157$		−	188.925i		−	1.20334i	0.798744	−	0.601670i	$-0.205498\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	11.7237	+	0.357298i	0.0728178	+	0.00221924i
$162$		0			0
$163$	−18.3070			−0.112313			−0.0561564	−	0.998422i	$-0.517885\pi$
$163$	−18.3070			−0.112313			−0.0561564	+	0.998422i	$0.517885\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	157.961			0.945873			0.472936	−	0.881097i	$-0.343194\pi$
$167$	157.961			0.945873			0.472936	+	0.881097i	$0.343194\pi$
$168$		0			0
$169$	96.0468			0.568324
$170$		0			0
$171$		0			0
$172$		0			0
$173$	55.4532			0.320539			0.160269	−	0.987073i	$-0.448764\pi$
$173$	55.4532			0.320539			0.160269	+	0.987073i	$0.448764\pi$
$174$		0			0
$175$		−	2.54981i		−	0.0145703i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	228.819			1.27832			0.639160	−	0.769074i	$-0.279282\pi$
$179$	228.819			1.27832			0.639160	+	0.769074i	$0.279282\pi$
$180$		0			0
$181$			47.8977i			0.264628i	0.991208	+	0.132314i	$0.0422408\pi$
$181$			47.8977i			0.264628i	−0.991208	+	0.132314i	$0.957759\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−140.357			−0.758686
$186$		0			0
$187$	189.819			1.01507
$188$		0			0
$189$		0			0
$190$		0			0
$191$			45.4655i			0.238039i	0.992892	+	0.119020i	$0.0379751\pi$
$191$			45.4655i			0.238039i	−0.992892	+	0.119020i	$0.962025\pi$
$192$		0			0
$193$	69.5085			0.360148			0.180074	−	0.983653i	$-0.442366\pi$
$193$	69.5085			0.360148			0.180074	+	0.983653i	$0.442366\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	94.6975			0.480698			0.240349	−	0.970687i	$-0.422738\pi$
$197$	94.6975			0.480698			0.240349	+	0.970687i	$0.422738\pi$
$198$		0			0
$199$			98.0744i			0.492836i	0.969164	+	0.246418i	$0.0792536\pi$
$199$			98.0744i			0.492836i	−0.969164	+	0.246418i	$0.920746\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	10.5542i		−	0.0519912i
$204$		0			0
$205$			27.0027i			0.131720i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−165.259			−0.790714
$210$		0			0
$211$	106.252			0.503566			0.251783	−	0.967784i	$-0.418983\pi$
$211$	106.252			0.503566			0.251783	+	0.967784i	$0.418983\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−84.4789			−0.392925
$216$		0			0
$217$			5.86466i			0.0270261i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	176.762i		−	0.799828i
$222$		0			0
$223$	−418.394			−1.87621			−0.938103	−	0.346355i	$-0.887419\pi$
$223$	−418.394			−1.87621			−0.938103	+	0.346355i	$0.887419\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	274.156i		−	1.20774i	−0.797085	−	0.603868i	$-0.793625\pi$
$227$		−	274.156i		−	1.20774i	0.797085	−	0.603868i	$-0.206375\pi$
$228$		0			0
$229$		−	213.326i		−	0.931554i	−0.884902	−	0.465777i	$-0.845775\pi$
$229$		−	213.326i		−	0.931554i	0.884902	−	0.465777i	$-0.154225\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	389.932			1.67353			0.836765	−	0.547562i	$-0.184444\pi$
$233$	389.932			1.67353			0.836765	+	0.547562i	$0.184444\pi$
$234$		0			0
$235$		−	186.270i		−	0.792639i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	425.586			1.78069			0.890347	−	0.455283i	$-0.150462\pi$
$239$	425.586			1.78069			0.890347	+	0.455283i	$0.150462\pi$
$240$		0			0
$241$		−	304.081i		−	1.26175i	−0.775886	−	0.630873i	$-0.782697\pi$
$241$		−	304.081i		−	1.26175i	0.775886	−	0.630873i	$-0.217303\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	108.986i		−	0.444840i
$246$		0			0
$247$			153.892i			0.623043i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			242.060i			0.964383i	0.876066	+	0.482191i	$0.160159\pi$
$251$			242.060i			0.964383i	−0.876066	+	0.482191i	$0.839841\pi$
$252$		0			0
$253$	401.919	+	12.2491i	1.58861	+	0.0484155i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−246.901			−0.960706			−0.480353	−	0.877075i	$-0.659491\pi$
$257$	−246.901			−0.960706			−0.480353	+	0.877075i	$0.659491\pi$
$258$		0			0
$259$	32.0100			0.123591
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	196.173i		−	0.745904i	−0.927851	−	0.372952i	$-0.878346\pi$
$263$		−	196.173i		−	0.745904i	0.927851	−	0.372952i	$-0.121654\pi$
$264$		0			0
$265$	131.748			0.497162
$266$		0			0
$267$		0			0
$268$		0			0
$269$	370.516			1.37738			0.688691	−	0.725055i	$-0.258186\pi$
$269$	370.516			1.37738			0.688691	+	0.725055i	$0.258186\pi$
$270$		0			0
$271$	−12.2009			−0.0450219			−0.0225109	−	0.999747i	$-0.507166\pi$
$271$	−12.2009			−0.0450219			−0.0225109	+	0.999747i	$0.507166\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	87.4142i		−	0.317870i
$276$		0			0
$277$	79.9789			0.288733			0.144366	−	0.989524i	$-0.453886\pi$
$277$	79.9789			0.288733			0.144366	+	0.989524i	$0.453886\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	136.426i		−	0.485503i	−0.970089	−	0.242751i	$-0.921950\pi$
$281$		−	136.426i		−	0.485503i	0.970089	−	0.242751i	$-0.0780499\pi$
$282$		0			0
$283$		−	494.551i		−	1.74753i	−0.486349	−	0.873764i	$-0.661672\pi$
$283$		−	494.551i		−	1.74753i	0.486349	−	0.873764i	$-0.338328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	6.15827i		−	0.0214574i
$288$		0			0
$289$	171.116			0.592096
$290$		0			0
$291$		0			0
$292$		0			0
$293$			350.121i			1.19495i	0.801886	+	0.597477i	$0.203830\pi$
$293$			350.121i			1.19495i	−0.801886	+	0.597477i	$0.796170\pi$
$294$		0			0
$295$			111.809i			0.379014i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.4066	−	374.272i	0.0381490	−	1.25175i
$300$		0			0
$301$	19.2664			0.0640079
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−86.4157			−0.283330
$306$		0			0
$307$	−105.546			−0.343798			−0.171899	−	0.985115i	$-0.554990\pi$
$307$	−105.546			−0.343798			−0.171899	+	0.985115i	$0.554990\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	321.362			1.03332			0.516659	−	0.856192i	$-0.327176\pi$
$311$	321.362			1.03332			0.516659	+	0.856192i	$0.327176\pi$
$312$		0			0
$313$		−	325.597i		−	1.04025i	−0.854092	−	0.520123i	$-0.825886\pi$
$313$		−	325.597i		−	1.04025i	0.854092	−	0.520123i	$-0.174114\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	411.021			1.29660			0.648298	−	0.761387i	$-0.275481\pi$
$317$	411.021			1.29660			0.648298	+	0.761387i	$0.275481\pi$
$318$		0			0
$319$		−	361.827i		−	1.13425i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	102.632			0.317745
$324$		0			0
$325$	−81.4013			−0.250466
$326$		0			0
$327$		0			0
$328$		0			0
$329$			42.4810i			0.129122i
$330$		0			0
$331$	−384.316			−1.16108			−0.580538	−	0.814233i	$-0.697158\pi$
$331$	−384.316			−1.16108			−0.580538	+	0.814233i	$0.697158\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	185.242			0.552962
$336$		0			0
$337$			567.852i			1.68502i	0.538679	+	0.842511i	$0.318923\pi$
$337$			567.852i			1.68502i	−0.538679	+	0.842511i	$0.681077\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			201.056i			0.589608i
$342$		0			0
$343$			49.8436i			0.145317i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−106.386			−0.306589			−0.153294	−	0.988181i	$-0.548988\pi$
$347$	−106.386			−0.306589			−0.153294	+	0.988181i	$0.548988\pi$
$348$		0			0
$349$	206.861			0.592725			0.296363	−	0.955075i	$-0.404226\pi$
$349$	206.861			0.592725			0.296363	+	0.955075i	$0.404226\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−419.825			−1.18931			−0.594654	−	0.803982i	$-0.702711\pi$
$353$	−419.825			−1.18931			−0.594654	+	0.803982i	$0.702711\pi$
$354$		0			0
$355$			281.477i			0.792893i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			596.809i			1.66242i	0.555957	+	0.831211i	$0.312352\pi$
$359$			596.809i			1.66242i	−0.555957	+	0.831211i	$0.687648\pi$
$360$		0			0
$361$	271.647			0.752486
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	310.468i		−	0.850597i
$366$		0			0
$367$			593.802i			1.61799i	0.587816	+	0.808995i	$0.299988\pi$
$367$			593.802i			1.61799i	−0.587816	+	0.808995i	$0.700012\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−30.0466			−0.0809882
$372$		0			0
$373$		−	110.539i		−	0.296351i	−0.988961	−	0.148176i	$-0.952660\pi$
$373$		−	110.539i		−	0.296351i	0.988961	−	0.148176i	$-0.0473401\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−336.938			−0.893735
$378$		0			0
$379$			47.0117i			0.124041i	0.998075	+	0.0620207i	$0.0197545\pi$
$379$			47.0117i			0.124041i	−0.998075	+	0.0620207i	$0.980246\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			172.313i			0.449904i	0.974370	+	0.224952i	$0.0722225\pi$
$383$			172.313i			0.449904i	−0.974370	+	0.224952i	$0.927778\pi$
$384$		0			0
$385$			19.9358i			0.0517813i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	239.714i		−	0.616232i	−0.951349	−	0.308116i	$-0.900302\pi$
$389$		−	239.714i		−	0.616232i	0.951349	−	0.308116i	$-0.0996985\pi$
$390$		0			0
$391$	−249.605	−	7.60713i	−0.638377	−	0.0194556i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	249.635			0.631988
$396$		0			0
$397$	−282.931			−0.712672			−0.356336	−	0.934358i	$-0.615974\pi$
$397$	−282.931			−0.712672			−0.356336	+	0.934358i	$0.615974\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			292.991i			0.730650i	0.930880	+	0.365325i	$0.119042\pi$
$401$			292.991i			0.730650i	−0.930880	+	0.365325i	$0.880958\pi$
$402$		0			0
$403$	187.226			0.464581
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1097.39			2.69629
$408$		0			0
$409$	−178.561			−0.436581			−0.218290	−	0.975884i	$-0.570048\pi$
$409$	−178.561			−0.436581			−0.218290	+	0.975884i	$0.570048\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	25.4994i		−	0.0617418i
$414$		0			0
$415$	−91.3627			−0.220151
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	250.428i		−	0.597681i	−0.954303	−	0.298841i	$-0.903400\pi$
$419$		−	250.428i		−	0.597681i	0.954303	−	0.298841i	$-0.0965999\pi$
$420$		0			0
$421$		−	81.4260i		−	0.193411i	−0.995313	−	0.0967054i	$-0.969170\pi$
$421$		−	81.4260i		−	0.193411i	0.995313	−	0.0967054i	$-0.0308305\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			54.2872i			0.127735i
$426$		0			0
$427$	19.7081			0.0461548
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	754.798i		−	1.75127i	−0.482973	−	0.875635i	$-0.660443\pi$
$431$		−	754.798i		−	1.75127i	0.482973	−	0.875635i	$-0.339557\pi$
$432$		0			0
$433$			601.513i			1.38918i	0.719408	+	0.694588i	$0.244413\pi$
$433$			601.513i			1.38918i	−0.719408	+	0.694588i	$0.755587\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	217.310	+	6.62288i	0.497277	+	0.0151553i
$438$		0			0
$439$	−385.967			−0.879196			−0.439598	−	0.898195i	$-0.644879\pi$
$439$	−385.967			−0.879196			−0.439598	+	0.898195i	$0.644879\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	548.212			1.23750			0.618749	−	0.785589i	$-0.287640\pi$
$443$	548.212			1.23750			0.618749	+	0.785589i	$0.287640\pi$
$444$		0			0
$445$	17.7200			0.0398202
$446$		0			0
$447$		0			0
$448$		0			0
$449$	117.642			0.262009			0.131005	−	0.991382i	$-0.458180\pi$
$449$	117.642			0.262009			0.131005	+	0.991382i	$0.458180\pi$
$450$		0			0
$451$		−	211.122i		−	0.468120i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	18.5645			0.0408011
$456$		0			0
$457$			381.925i			0.835722i	0.908511	+	0.417861i	$0.137220\pi$
$457$			381.925i			0.835722i	−0.908511	+	0.417861i	$0.862780\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	874.410			1.89677			0.948384	−	0.317125i	$-0.102717\pi$
$461$	874.410			1.89677			0.948384	+	0.317125i	$0.102717\pi$
$462$		0			0
$463$	−117.395			−0.253553			−0.126776	−	0.991931i	$-0.540463\pi$
$463$	−117.395			−0.253553			−0.126776	+	0.991931i	$0.540463\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	389.737i		−	0.834556i	−0.908779	−	0.417278i	$-0.862984\pi$
$467$		−	389.737i		−	0.834556i	0.908779	−	0.417278i	$-0.137016\pi$
$468$		0			0
$469$	−42.2466			−0.0900781
$470$		0			0
$471$		0			0
$472$		0			0
$473$	660.504			1.39641
$474$		0			0
$475$		−	47.2633i		−	0.0995016i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			786.509i			1.64198i	0.570941	+	0.820991i	$0.306578\pi$
$479$			786.509i			1.64198i	−0.570941	+	0.820991i	$0.693422\pi$
$480$		0			0
$481$		−	1021.90i		−	2.12454i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	101.537			0.209355
$486$		0			0
$487$	−575.548			−1.18182			−0.590911	−	0.806736i	$-0.701232\pi$
$487$	−575.548			−1.18182			−0.590911	+	0.806736i	$0.701232\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	608.629			1.23957			0.619785	−	0.784772i	$-0.287220\pi$
$491$	608.629			1.23957			0.619785	+	0.784772i	$0.287220\pi$
$492$		0			0
$493$			224.707i			0.455795i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	64.1941i		−	0.129163i
$498$		0			0
$499$	826.989			1.65729			0.828647	−	0.559772i	$-0.189111\pi$
$499$	826.989			1.65729			0.828647	+	0.559772i	$0.189111\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			583.264i			1.15957i	0.814770	+	0.579785i	$0.196863\pi$
$503$			583.264i			1.15957i	−0.814770	+	0.579785i	$0.803137\pi$
$504$		0			0
$505$			277.814i			0.550126i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−349.999			−0.687621			−0.343811	−	0.939039i	$-0.611718\pi$
$509$	−349.999			−0.687621			−0.343811	+	0.939039i	$0.611718\pi$
$510$		0			0
$511$			70.8058i			0.138563i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	169.184			0.328512
$516$		0			0
$517$			1456.36i			2.81695i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			269.304i			0.516899i	0.966025	+	0.258450i	$0.0832116\pi$
$521$			269.304i			0.516899i	−0.966025	+	0.258450i	$0.916788\pi$
$522$		0			0
$523$		−	970.386i		−	1.85542i	−0.373298	−	0.927711i	$-0.621773\pi$
$523$		−	970.386i		−	1.85542i	0.373298	−	0.927711i	$-0.378227\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	124.863i		−	0.236931i
$528$		0			0
$529$	−528.018	−	32.2144i	−0.998144	−	0.0608967i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−196.600			−0.368855
$534$		0			0
$535$	163.562			0.305723
$536$		0			0
$537$		0			0
$538$		0			0
$539$			852.112i			1.58091i
$540$		0			0
$541$	249.290			0.460794			0.230397	−	0.973097i	$-0.425997\pi$
$541$	249.290			0.460794			0.230397	+	0.973097i	$0.425997\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	36.8801			0.0676700
$546$		0			0
$547$	−235.742			−0.430972			−0.215486	−	0.976507i	$-0.569134\pi$
$547$	−235.742			−0.430972			−0.215486	+	0.976507i	$0.569134\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	195.633i		−	0.355051i
$552$		0			0
$553$	−56.9322			−0.102952
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	289.906i		−	0.520478i	−0.965544	−	0.260239i	$-0.916199\pi$
$557$		−	289.906i		−	0.520478i	0.965544	−	0.260239i	$-0.0838014\pi$
$558$		0			0
$559$		−	615.070i		−	1.10030i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	707.769i		−	1.25714i	−0.777754	−	0.628569i	$-0.783641\pi$
$563$		−	707.769i		−	1.25714i	0.777754	−	0.628569i	$-0.216359\pi$
$564$		0			0
$565$	−61.9488			−0.109644
$566$		0			0
$567$		0			0
$568$		0			0
$569$			764.923i			1.34433i	0.740402	+	0.672165i	$0.234635\pi$
$569$			764.923i			1.34433i	−0.740402	+	0.672165i	$0.765365\pi$
$570$		0			0
$571$			430.556i			0.754038i	0.926205	+	0.377019i	$0.123051\pi$
$571$			430.556i			0.754038i	−0.926205	+	0.377019i	$0.876949\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.50319	+	114.947i	−0.00609250	+	0.199907i
$576$		0			0
$577$	569.697			0.987343			0.493671	−	0.869648i	$-0.335655\pi$
$577$	569.697			0.987343			0.493671	+	0.869648i	$0.335655\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	20.8363			0.0358629
$582$		0			0
$583$	−1030.08			−1.76686
$584$		0			0
$585$		0			0
$586$		0			0
$587$	174.298			0.296930			0.148465	−	0.988918i	$-0.452567\pi$
$587$	174.298			0.296930			0.148465	+	0.988918i	$0.452567\pi$
$588$		0			0
$589$			108.707i			0.184563i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	59.7042			0.100682			0.0503408	−	0.998732i	$-0.483969\pi$
$593$	59.7042			0.100682			0.0503408	+	0.998732i	$0.483969\pi$
$594$		0			0
$595$		−	12.3808i		−	0.0208081i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	838.229			1.39938			0.699690	−	0.714446i	$-0.253321\pi$
$599$	838.229			1.39938			0.699690	+	0.714446i	$0.253321\pi$
$600$		0			0
$601$	630.011			1.04827			0.524135	−	0.851635i	$-0.324389\pi$
$601$	630.011			1.04827			0.524135	+	0.851635i	$0.324389\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			412.889i			0.682461i
$606$		0			0
$607$	−502.932			−0.828553			−0.414277	−	0.910151i	$-0.635965\pi$
$607$	−502.932			−0.828553			−0.414277	+	0.910151i	$0.635965\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1356.19			2.21962
$612$		0			0
$613$			600.773i			0.980054i	0.871707	+	0.490027i	$0.163013\pi$
$613$			600.773i			0.980054i	−0.871707	+	0.490027i	$0.836987\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			387.062i			0.627330i	0.949534	+	0.313665i	$0.101557\pi$
$617$			387.062i			0.627330i	−0.949534	+	0.313665i	$0.898443\pi$
$618$		0			0
$619$		−	740.727i		−	1.19665i	−0.801253	−	0.598326i	$-0.795833\pi$
$619$		−	740.727i		−	1.19665i	0.801253	−	0.598326i	$-0.204167\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.04125			−0.00648675
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−681.517			−1.08349
$630$		0			0
$631$			267.285i			0.423589i	0.977314	+	0.211794i	$0.0679307\pi$
$631$			267.285i			0.423589i	−0.977314	+	0.211794i	$0.932069\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	553.444i		−	0.871566i
$636$		0			0
$637$	793.499			1.24568
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	975.866i		−	1.52241i	−0.648510	−	0.761206i	$-0.724608\pi$
$641$		−	975.866i		−	1.52241i	0.648510	−	0.761206i	$-0.275392\pi$
$642$		0			0
$643$			81.6305i			0.126953i	0.997983	+	0.0634763i	$0.0202187\pi$
$643$			81.6305i			0.126953i	−0.997983	+	0.0634763i	$0.979781\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−219.408			−0.339116			−0.169558	−	0.985520i	$-0.554234\pi$
$647$	−219.408			−0.339116			−0.169558	+	0.985520i	$0.554234\pi$
$648$		0			0
$649$		−	874.186i		−	1.34697i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−356.004			−0.545183			−0.272591	−	0.962130i	$-0.587881\pi$
$653$	−356.004			−0.545183			−0.272591	+	0.962130i	$0.587881\pi$
$654$		0			0
$655$		−	183.861i		−	0.280703i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	586.421i		−	0.889865i	−0.895564	−	0.444933i	$-0.853228\pi$
$659$		−	586.421i		−	0.889865i	0.895564	−	0.444933i	$-0.146772\pi$
$660$		0			0
$661$		−	690.550i		−	1.04471i	−0.852730	−	0.522353i	$-0.825054\pi$
$661$		−	690.550i		−	1.04471i	0.852730	−	0.522353i	$-0.174946\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			10.7789i			0.0162089i
$666$		0			0
$667$	−14.5005	+	475.790i	−0.0217398	+	0.713328i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	675.647			1.00692
$672$		0			0
$673$	47.5400			0.0706389			0.0353194	−	0.999376i	$-0.488755\pi$
$673$	47.5400			0.0706389			0.0353194	+	0.999376i	$0.488755\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	704.319i		−	1.04035i	−0.854059	−	0.520176i	$-0.825866\pi$
$677$		−	704.319i		−	1.04035i	0.854059	−	0.520176i	$-0.174134\pi$
$678$		0			0
$679$	−23.1568			−0.0341042
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−201.340			−0.294787			−0.147393	−	0.989078i	$-0.547088\pi$
$683$	−201.340			−0.294787			−0.147393	+	0.989078i	$0.547088\pi$
$684$		0			0
$685$	−533.204			−0.778399
$686$		0			0
$687$		0			0
$688$		0			0
$689$			959.223i			1.39220i
$690$		0			0
$691$	437.477			0.633107			0.316554	−	0.948575i	$-0.397474\pi$
$691$	437.477			0.633107			0.316554	+	0.948575i	$0.397474\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	28.1936i		−	0.0405664i
$696$		0			0
$697$			131.114i			0.188112i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	159.865i		−	0.228053i	−0.993478	−	0.114027i	$-0.963625\pi$
$701$		−	159.865i		−	0.228053i	0.993478	−	0.114027i	$-0.0363749\pi$
$702$		0			0
$703$	593.338			0.844009
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	63.3586i		−	0.0896162i
$708$		0			0
$709$		−	744.339i		−	1.04984i	−0.851151	−	0.524922i	$-0.824095\pi$
$709$		−	744.339i		−	1.04984i	0.851151	−	0.524922i	$-0.175905\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.05747	−	264.382i	0.0113008	−	0.370802i
$714$		0			0
$715$	636.441			0.890127
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1236.66			1.71997			0.859986	−	0.510318i	$-0.170472\pi$
$719$	1236.66			1.71997			0.859986	+	0.510318i	$0.170472\pi$
$720$		0			0
$721$	−38.5843			−0.0535150
$722$		0			0
$723$		0			0
$724$		0			0
$725$	103.481			0.142732
$726$		0			0
$727$			1191.67i			1.63917i	0.572960	+	0.819584i	$0.305795\pi$
$727$			1191.67i			1.63917i	−0.572960	+	0.819584i	$0.694205\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−410.196			−0.561143
$732$		0			0
$733$			108.024i			0.147372i	0.997281	+	0.0736860i	$0.0234763\pi$
$733$			108.024i			0.147372i	−0.997281	+	0.0736860i	$0.976524\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1448.33			−1.96517
$738$		0			0
$739$	−1015.56			−1.37424			−0.687118	−	0.726546i	$-0.741125\pi$
$739$	−1015.56			−1.37424			−0.687118	+	0.726546i	$0.741125\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	821.741i		−	1.10598i	−0.833189	−	0.552988i	$-0.813487\pi$
$743$		−	821.741i		−	1.10598i	0.833189	−	0.552988i	$-0.186513\pi$
$744$		0			0
$745$	385.556			0.517524
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−37.3022			−0.0498026
$750$		0			0
$751$		−	34.6047i		−	0.0460782i	−0.999735	−	0.0230391i	$-0.992666\pi$
$751$		−	34.6047i		−	0.0460782i	0.999735	−	0.0230391i	$-0.00733422\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	610.380i		−	0.808450i
$756$		0			0
$757$			1231.28i			1.62653i	0.581893	+	0.813265i	$0.302312\pi$
$757$			1231.28i			1.62653i	−0.581893	+	0.813265i	$0.697688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−989.284			−1.29998			−0.649990	−	0.759943i	$-0.725227\pi$
$761$	−989.284			−1.29998			−0.649990	+	0.759943i	$0.725227\pi$
$762$		0			0
$763$	−8.41094			−0.0110235
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−814.054			−1.06135
$768$		0			0
$769$			556.547i			0.723728i	0.932231	+	0.361864i	$0.117859\pi$
$769$			556.547i			0.723728i	−0.932231	+	0.361864i	$0.882141\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	831.961i		−	1.07628i	−0.842857	−	0.538138i	$-0.819128\pi$
$773$		−	831.961i		−	1.07628i	0.842857	−	0.538138i	$-0.180872\pi$
$774$		0			0
$775$	−57.5010			−0.0741949
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	114.150i		−	0.146534i
$780$		0			0
$781$		−	2200.75i		−	2.81786i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−422.448			−0.538150
$786$		0			0
$787$			428.008i			0.543847i	0.962319	+	0.271924i	$0.0876598\pi$
$787$			428.008i			0.543847i	−0.962319	+	0.271924i	$0.912340\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14.1281			0.0178611
$792$		0			0
$793$		−	629.171i		−	0.793406i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	267.891i		−	0.336124i	−0.985776	−	0.168062i	$-0.946249\pi$
$797$		−	267.891i		−	0.336124i	0.985776	−	0.168062i	$-0.0537508\pi$
$798$		0			0
$799$		−	904.452i		−	1.13198i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			2427.41i			3.02293i
$804$		0			0
$805$	0.798942	−	26.2149i	0.000992475	−	0.0325651i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−822.021			−1.01610			−0.508048	−	0.861329i	$-0.669633\pi$
$809$	−822.021			−1.01610			−0.508048	+	0.861329i	$0.669633\pi$
$810$		0			0
$811$	526.347			0.649009			0.324505	−	0.945884i	$-0.394802\pi$
$811$	526.347			0.649009			0.324505	+	0.945884i	$0.394802\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			40.9356i			0.0502278i
$816$		0			0
$817$	357.122			0.437114
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1003.97			−1.22286			−0.611429	−	0.791299i	$-0.709405\pi$
$821$	−1003.97			−1.22286			−0.611429	+	0.791299i	$0.709405\pi$
$822$		0			0
$823$	1444.89			1.75564			0.877822	−	0.478988i	$-0.158996\pi$
$823$	1444.89			1.75564			0.877822	+	0.478988i	$0.158996\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			604.307i			0.730721i	0.930866	+	0.365361i	$0.119054\pi$
$827$			604.307i			0.730721i	−0.930866	+	0.365361i	$0.880946\pi$
$828$		0			0
$829$	−1570.61			−1.89458			−0.947290	−	0.320379i	$-0.896190\pi$
$829$	−1570.61			−1.89458			−0.947290	+	0.320379i	$0.896190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	529.191i		−	0.635284i
$834$		0			0
$835$		−	353.211i		−	0.423007i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1006.78i			1.19997i	0.800010	+	0.599986i	$0.204827\pi$
$839$			1006.78i			1.19997i	−0.800010	+	0.599986i	$0.795173\pi$
$840$		0			0
$841$	−412.670			−0.490690
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	214.767i		−	0.254162i
$846$		0			0
$847$		−	94.1641i		−	0.111174i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1443.03	−	43.9786i	−1.69569	−	0.0516788i
$852$		0			0
$853$	−852.227			−0.999093			−0.499547	−	0.866287i	$-0.666500\pi$
$853$	−852.227			−0.999093			−0.499547	+	0.866287i	$0.666500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1434.66			−1.67405			−0.837026	−	0.547164i	$-0.815707\pi$
$857$	−1434.66			−1.67405			−0.837026	+	0.547164i	$0.815707\pi$
$858$		0			0
$859$	638.255			0.743021			0.371510	−	0.928429i	$-0.378840\pi$
$859$	638.255			0.743021			0.371510	+	0.928429i	$0.378840\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1133.11			1.31299			0.656497	−	0.754328i	$-0.272037\pi$
$863$	1133.11			1.31299			0.656497	+	0.754328i	$0.272037\pi$
$864$		0			0
$865$		−	123.997i		−	0.143349i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1951.79			−2.24602
$870$		0			0
$871$			1348.70i			1.54845i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.70154			−0.00651604
$876$		0			0
$877$	707.476			0.806700			0.403350	−	0.915046i	$-0.367846\pi$
$877$	707.476			0.806700			0.403350	+	0.915046i	$0.367846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			915.626i			1.03930i	0.854378	+	0.519652i	$0.173938\pi$
$881$			915.626i			1.03930i	−0.854378	+	0.519652i	$0.826062\pi$
$882$		0			0
$883$	331.386			0.375295			0.187648	−	0.982236i	$-0.439914\pi$
$883$	331.386			0.375295			0.187648	+	0.982236i	$0.439914\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	977.356			1.10187			0.550934	−	0.834549i	$-0.314272\pi$
$887$	977.356			1.10187			0.550934	+	0.834549i	$0.314272\pi$
$888$		0			0
$889$			126.219i			0.141979i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			787.430i			0.881780i
$894$		0			0
$895$		−	511.655i		−	0.571682i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−238.010			−0.264749
$900$		0			0
$901$	639.714			0.710005
$902$		0			0
$903$		0			0
$904$		0			0
$905$	107.103			0.118345
$906$		0			0
$907$		−	149.222i		−	0.164523i	−0.996611	−	0.0822615i	$-0.973786\pi$
$907$		−	149.222i		−	0.164523i	0.996611	−	0.0822615i	$-0.0262143\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	513.869i		−	0.564071i	−0.959404	−	0.282036i	$-0.908990\pi$
$911$		−	513.869i		−	0.564071i	0.959404	−	0.282036i	$-0.0910097\pi$
$912$		0			0
$913$	714.325			0.782393
$914$		0			0
$915$		0			0
$916$		0			0
$917$			41.9315i			0.0457268i
$918$		0			0
$919$			1411.58i			1.53600i	0.640450	+	0.768000i	$0.278748\pi$
$919$			1411.58i			1.53600i	−0.640450	+	0.768000i	$0.721252\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2049.36			−2.22033
$924$		0			0
$925$			313.848i			0.339295i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	56.7413			0.0610779			0.0305389	−	0.999534i	$-0.490278\pi$
$929$	56.7413			0.0610779			0.0305389	+	0.999534i	$0.490278\pi$
$930$		0			0
$931$			460.722i			0.494868i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	424.448i		−	0.453955i
$936$		0			0
$937$		−	1421.63i		−	1.51722i	−0.651546	−	0.758609i	$-0.725879\pi$
$937$		−	1421.63i		−	1.51722i	0.651546	−	0.758609i	$-0.274121\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1728.77i			1.83716i	0.395231	+	0.918582i	$0.370665\pi$
$941$			1728.77i			1.83716i	−0.395231	+	0.918582i	$0.629335\pi$
$942$		0			0
$943$	−8.46087	+	277.618i	−0.00897229	+	0.294399i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	689.937			0.728550			0.364275	−	0.931292i	$-0.381317\pi$
$947$	689.937			0.728550			0.364275	+	0.931292i	$0.381317\pi$
$948$		0			0
$949$	2260.44			2.38192
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	96.8807i		−	0.101659i	−0.998707	−	0.0508293i	$-0.983814\pi$
$953$		−	96.8807i		−	0.101659i	0.998707	−	0.0508293i	$-0.0161864\pi$
$954$		0			0
$955$	101.664			0.106454
$956$		0			0
$957$		0			0
$958$		0			0
$959$	121.603			0.126802
$960$		0			0
$961$	−828.745			−0.862378
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	155.426i		−	0.161063i
$966$		0			0
$967$	1084.28			1.12128			0.560639	−	0.828060i	$-0.310556\pi$
$967$	1084.28			1.12128			0.560639	+	0.828060i	$0.310556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	719.516i		−	0.741005i	−0.928831	−	0.370503i	$-0.879185\pi$
$971$		−	719.516i		−	0.741005i	0.928831	−	0.370503i	$-0.120815\pi$
$972$		0			0
$973$			6.42989i			0.00660831i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1382.60i		−	1.41515i	−0.706640	−	0.707573i	$-0.749790\pi$
$977$		−	1382.60i		−	1.41515i	0.706640	−	0.707573i	$-0.250210\pi$
$978$		0			0
$979$	−138.545			−0.141517
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1150.57i			1.17047i	0.810863	+	0.585236i	$0.198998\pi$
$983$			1150.57i			1.17047i	−0.810863	+	0.585236i	$0.801002\pi$
$984$		0			0
$985$		−	211.750i		−	0.214975i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−868.539	−	26.4701i	−0.878199	−	0.0267646i
$990$		0			0
$991$	−1062.02			−1.07167			−0.535833	−	0.844324i	$-0.680002\pi$
$991$	−1062.02			−1.07167			−0.535833	+	0.844324i	$0.680002\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	219.301			0.220403
$996$		0			0
$997$	−1114.18			−1.11753			−0.558767	−	0.829325i	$-0.688725\pi$
$997$	−1114.18			−1.11753			−0.558767	+	0.829325i	$0.688725\pi$
$998$		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.2161.9		32
3.2	odd	2		1380.3.d.a.781.29	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.24		32
69.68	even	2		1380.3.d.a.781.20	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.20	✓	32	69.68	even	2
1380.3.d.a.781.29	yes	32	3.2	odd	2
4140.3.d.c.2161.9		32	1.1	even	1	trivial
4140.3.d.c.2161.24		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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} +0.509961i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} +0.509961i q^{7} +17.4828i q^{11} +16.2803 q^{13} -10.8574i q^{17} +9.45266i q^{19} +(0.700637 - 22.9893i) q^{23} -5.00000 q^{25} -20.6961 q^{29} +11.5002 q^{31} +1.14031 q^{35} -62.7695i q^{37} -12.0760 q^{41} -37.7801i q^{43} +83.3025 q^{47} +48.7399 q^{49} +58.9194i q^{53} +39.0928 q^{55} -50.0025 q^{59} -38.6463i q^{61} -36.4038i q^{65} +82.8428i q^{67} -125.880 q^{71} +138.846 q^{73} -8.91557 q^{77} +111.640i q^{79} -40.8586i q^{83} -24.2780 q^{85} +7.92462i q^{89} +8.30230i q^{91} +21.1368 q^{95} +45.4089i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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