LMFDB - Embedded newform 4032.2.i.c.1889.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1889,4032)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4032.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.i (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.18
Character	$\chi$	$=$	4032.1889
Dual form			4032.2.i.c.1889.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.91870i q^{5} +(-2.03709 + 1.68827i) q^{7} +O(q^{10})$	$q+3.91870i q^{5} +(-2.03709 + 1.68827i) q^{7} +1.62598 q^{11} -2.54668 q^{13} +2.94004 q^{17} +2.72048 q^{19} +8.57805i q^{23} -10.3562 q^{25} -8.21126 q^{29} +8.08855i q^{31} +(-6.61582 - 7.98274i) q^{35} -8.05701i q^{37} +9.17811 q^{41} +9.05669i q^{43} -1.17360 q^{47} +(1.29949 - 6.87832i) q^{49} +2.44949 q^{53} +6.37173i q^{55} +1.45662i q^{59} -9.74827 q^{61} -9.97967i q^{65} +7.35618i q^{67} -7.71526i q^{71} +15.0398i q^{73} +(-3.31228 + 2.74510i) q^{77} +0.0913558 q^{79} -2.03273i q^{83} +11.5211i q^{85} +1.48342 q^{89} +(5.18782 - 4.29949i) q^{91} +10.6607i q^{95} -7.53884i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 80 q^{25} - 16 q^{49}+O(q^{100}) $ 48 * q - 80 * q^25 - 16 * q^49

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$5$			3.91870i			1.75249i	0.481862	+	0.876247i	$0.339961\pi$
$5$			3.91870i			1.75249i	−0.481862	+	0.876247i	$0.660039\pi$
$6$		0			0
$7$	−2.03709	+	1.68827i	−0.769948	+	0.638106i
$7$	−2.03709	+	1.68827i	−0.769948	+	0.638106i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.62598			0.490252			0.245126	−	0.969491i	$-0.421171\pi$
$11$	1.62598			0.490252			0.245126	+	0.969491i	$0.421171\pi$
$12$		0			0
$13$	−2.54668			−0.706322			−0.353161	−	0.935562i	$-0.614893\pi$
$13$	−2.54668			−0.706322			−0.353161	+	0.935562i	$0.614893\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.94004			0.713064			0.356532	−	0.934283i	$-0.383959\pi$
$17$	2.94004			0.713064			0.356532	+	0.934283i	$0.383959\pi$
$18$		0			0
$19$	2.72048			0.624122			0.312061	−	0.950062i	$-0.398981\pi$
$19$	2.72048			0.624122			0.312061	+	0.950062i	$0.398981\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.57805i			1.78865i	0.447421	+	0.894324i	$0.352343\pi$
$23$			8.57805i			1.78865i	−0.447421	+	0.894324i	$0.647657\pi$
$24$		0			0
$25$	−10.3562			−2.07124
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.21126			−1.52479			−0.762396	−	0.647111i	$-0.775977\pi$
$29$	−8.21126			−1.52479			−0.762396	+	0.647111i	$0.775977\pi$
$30$		0			0
$31$			8.08855i			1.45275i	0.687300	+	0.726374i	$0.258796\pi$
$31$			8.08855i			1.45275i	−0.687300	+	0.726374i	$0.741204\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.61582	−	7.98274i	−1.11828	−	1.34933i
$36$		0			0
$37$		−	8.05701i		−	1.32456i	−0.749254	−	0.662282i	$-0.769588\pi$
$37$		−	8.05701i		−	1.32456i	0.749254	−	0.662282i	$-0.230412\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9.17811			1.43338			0.716690	−	0.697392i	$-0.245656\pi$
$41$	9.17811			1.43338			0.716690	+	0.697392i	$0.245656\pi$
$42$		0			0
$43$			9.05669i			1.38113i	0.723269	+	0.690566i	$0.242639\pi$
$43$			9.05669i			1.38113i	−0.723269	+	0.690566i	$0.757361\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.17360			−0.171187			−0.0855934	−	0.996330i	$-0.527279\pi$
$47$	−1.17360			−0.171187			−0.0855934	+	0.996330i	$0.527279\pi$
$48$		0			0
$49$	1.29949	−	6.87832i	0.185641	−	0.982618i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.44949			0.336463			0.168232	−	0.985747i	$-0.446194\pi$
$53$	2.44949			0.336463			0.168232	+	0.985747i	$0.446194\pi$
$54$		0			0
$55$			6.37173i			0.859164i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.45662i			0.189635i	0.995495	+	0.0948177i	$0.0302268\pi$
$59$			1.45662i			0.189635i	−0.995495	+	0.0948177i	$0.969773\pi$
$60$		0			0
$61$	−9.74827			−1.24814			−0.624069	−	0.781369i	$-0.714522\pi$
$61$	−9.74827			−1.24814			−0.624069	+	0.781369i	$0.714522\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	9.97967i		−	1.23783i
$66$		0			0
$67$			7.35618i			0.898700i	0.893356	+	0.449350i	$0.148344\pi$
$67$			7.35618i			0.898700i	−0.893356	+	0.449350i	$0.851656\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.71526i		−	0.915633i	−0.889047	−	0.457817i	$-0.848632\pi$
$71$		−	7.71526i		−	0.915633i	0.889047	−	0.457817i	$-0.151368\pi$
$72$		0			0
$73$			15.0398i			1.76027i	0.474722	+	0.880136i	$0.342549\pi$
$73$			15.0398i			1.76027i	−0.474722	+	0.880136i	$0.657451\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.31228	+	2.74510i	−0.377469	+	0.312833i
$78$		0			0
$79$	0.0913558			0.0102783			0.00513916	−	0.999987i	$-0.498364\pi$
$79$	0.0913558			0.0102783			0.00513916	+	0.999987i	$0.498364\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.03273i		−	0.223121i	−0.993758	−	0.111561i	$-0.964415\pi$
$83$		−	2.03273i		−	0.223121i	0.993758	−	0.111561i	$-0.0355849\pi$
$84$		0			0
$85$			11.5211i			1.24964i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.48342			0.157242			0.0786211	−	0.996905i	$-0.474948\pi$
$89$	1.48342			0.157242			0.0786211	+	0.996905i	$0.474948\pi$
$90$		0			0
$91$	5.18782	−	4.29949i	0.543832	−	0.450709i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			10.6607i			1.09377i
$96$		0			0
$97$		−	7.53884i		−	0.765453i	−0.923862	−	0.382727i	$-0.874985\pi$
$97$		−	7.53884i		−	0.765453i	0.923862	−	0.382727i	$-0.125015\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	11.7561i		−	1.16977i	−0.811115	−	0.584887i	$-0.801139\pi$
$101$		−	11.7561i		−	1.16977i	0.811115	−	0.584887i	$-0.198861\pi$
$102$		0			0
$103$		−	14.0789i		−	1.38724i	−0.720342	−	0.693620i	$-0.756015\pi$
$103$		−	14.0789i		−	1.38724i	0.720342	−	0.693620i	$-0.243985\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.35369			−0.807582			−0.403791	−	0.914851i	$-0.632308\pi$
$107$	−8.35369			−0.807582			−0.403791	+	0.914851i	$0.632308\pi$
$108$		0			0
$109$			4.68427i			0.448671i	0.974512	+	0.224336i	$0.0720212\pi$
$109$			4.68427i			0.448671i	−0.974512	+	0.224336i	$0.927979\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	5.73703i		−	0.539694i	−0.962903	−	0.269847i	$-0.913027\pi$
$113$		−	5.73703i		−	0.539694i	0.962903	−	0.269847i	$-0.0869732\pi$
$114$		0			0
$115$	−33.6148			−3.13459
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.98913	+	4.96358i	−0.549022	+	0.455010i
$120$		0			0
$121$	−8.35618			−0.759653
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	20.9892i		−	1.87733i
$126$		0			0
$127$	1.12881			0.100166			0.0500828	−	0.998745i	$-0.484051\pi$
$127$	1.12881			0.100166			0.0500828	+	0.998745i	$0.484051\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	21.8456i		−	1.90866i	−0.298761	−	0.954328i	$-0.596573\pi$
$131$		−	21.8456i		−	1.90866i	0.298761	−	0.954328i	$-0.403427\pi$
$132$		0			0
$133$	−5.54187	+	4.59291i	−0.480541	+	0.398256i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	10.9703i		−	0.937260i	−0.883395	−	0.468630i	$-0.844748\pi$
$137$		−	10.9703i		−	0.937260i	0.883395	−	0.468630i	$-0.155252\pi$
$138$		0			0
$139$	−8.31568			−0.705327			−0.352663	−	0.935750i	$-0.614724\pi$
$139$	−8.31568			−0.705327			−0.352663	+	0.935750i	$0.614724\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.14086			−0.346276
$144$		0			0
$145$		−	32.1774i		−	2.67219i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.8438			−1.13413			−0.567065	−	0.823673i	$-0.691921\pi$
$149$	−13.8438			−1.13413			−0.567065	+	0.823673i	$0.691921\pi$
$150$		0			0
$151$	19.1508			1.55847			0.779233	−	0.626734i	$-0.215609\pi$
$151$	19.1508			1.55847			0.779233	+	0.626734i	$0.215609\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−31.6966			−2.54593
$156$		0			0
$157$	12.4192			0.991162			0.495581	−	0.868562i	$-0.334955\pi$
$157$	12.4192			0.991162			0.495581	+	0.868562i	$0.334955\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.4821	−	17.4743i	−1.14135	−	1.37717i
$162$		0			0
$163$			11.3562i			0.889485i	0.895659	+	0.444742i	$0.146705\pi$
$163$			11.3562i			0.889485i	−0.895659	+	0.444742i	$0.853295\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.66379			0.515660			0.257830	−	0.966190i	$-0.416993\pi$
$167$	6.66379			0.515660			0.257830	+	0.966190i	$0.416993\pi$
$168$		0			0
$169$	−6.51441			−0.501109
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	9.23315i		−	0.701984i	−0.936379	−	0.350992i	$-0.885844\pi$
$173$		−	9.23315i		−	0.701984i	0.936379	−	0.350992i	$-0.114156\pi$
$174$		0			0
$175$	21.0965	−	17.4840i	1.59474	−	1.32167i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.3520			−1.22221			−0.611103	−	0.791551i	$-0.709274\pi$
$179$	−16.3520			−1.22221			−0.611103	+	0.791551i	$0.709274\pi$
$180$		0			0
$181$	13.6304			1.01314			0.506571	−	0.862198i	$-0.330913\pi$
$181$	13.6304			1.01314			0.506571	+	0.862198i	$0.330913\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	31.5730			2.32129
$186$		0			0
$187$	4.78045			0.349581
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.4770i		−	0.975164i	−0.873077	−	0.487582i	$-0.837879\pi$
$191$		−	13.4770i		−	0.975164i	0.873077	−	0.487582i	$-0.162121\pi$
$192$		0			0
$193$	11.9552			0.860551			0.430275	−	0.902698i	$-0.358416\pi$
$193$	11.9552			0.860551			0.430275	+	0.902698i	$0.358416\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.90865			0.349727			0.174864	−	0.984593i	$-0.444052\pi$
$197$	4.90865			0.349727			0.174864	+	0.984593i	$0.444052\pi$
$198$		0			0
$199$			9.36692i			0.664004i	0.943279	+	0.332002i	$0.107724\pi$
$199$			9.36692i			0.664004i	−0.943279	+	0.332002i	$0.892276\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.7271	−	13.8628i	1.17401	−	0.972979i
$204$		0			0
$205$			35.9662i			2.51199i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.42346			0.305977
$210$		0			0
$211$		−	9.65567i		−	0.664724i	−0.943152	−	0.332362i	$-0.892155\pi$
$211$		−	9.65567i		−	0.664724i	0.943152	−	0.332362i	$-0.107845\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−35.4904			−2.42043
$216$		0			0
$217$	−13.6557	−	16.4771i	−0.927007	−	1.11854i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.48734			−0.503653
$222$		0			0
$223$		−	7.70721i		−	0.516113i	−0.966130	−	0.258056i	$-0.916918\pi$
$223$		−	7.70721i		−	0.516113i	0.966130	−	0.258056i	$-0.0830820\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			15.0314i			0.997668i	0.866698	+	0.498834i	$0.166238\pi$
$227$			15.0314i			0.997668i	−0.866698	+	0.498834i	$0.833762\pi$
$228$		0			0
$229$	−6.87735			−0.454468			−0.227234	−	0.973840i	$-0.572968\pi$
$229$	−6.87735			−0.454468			−0.227234	+	0.973840i	$0.572968\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.49461i			0.490988i	0.969398	+	0.245494i	$0.0789502\pi$
$233$			7.49461i			0.490988i	−0.969398	+	0.245494i	$0.921050\pi$
$234$		0			0
$235$		−	4.59897i		−	0.300004i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.40464i			0.349597i	0.984604	+	0.174799i	$0.0559275\pi$
$239$			5.40464i			0.349597i	−0.984604	+	0.174799i	$0.944073\pi$
$240$		0			0
$241$			11.6588i			0.751008i	0.926821	+	0.375504i	$0.122530\pi$
$241$			11.6588i			0.751008i	−0.926821	+	0.375504i	$0.877470\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	26.9541	+	5.09229i	1.72203	+	0.325335i
$246$		0			0
$247$	−6.92820			−0.440831
$248$		0			0
$249$		0			0
$250$		0			0
$251$			11.1840i			0.705930i	0.935637	+	0.352965i	$0.114827\pi$
$251$			11.1840i			0.705930i	−0.935637	+	0.352965i	$0.885173\pi$
$252$		0			0
$253$			13.9478i			0.876888i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.39665			−0.274256			−0.137128	−	0.990553i	$-0.543787\pi$
$257$	−4.39665			−0.274256			−0.137128	+	0.990553i	$0.543787\pi$
$258$		0			0
$259$	13.6024	+	16.4129i	0.845213	+	1.01985i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			12.6142i			0.777828i	0.921274	+	0.388914i	$0.127150\pi$
$263$			12.6142i			0.777828i	−0.921274	+	0.388914i	$0.872850\pi$
$264$		0			0
$265$			9.59881i			0.589650i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	19.4177i		−	1.18392i	−0.805967	−	0.591960i	$-0.798354\pi$
$269$		−	19.4177i		−	1.18392i	0.805967	−	0.591960i	$-0.201646\pi$
$270$		0			0
$271$			15.7387i			0.956055i	0.878345	+	0.478028i	$0.158648\pi$
$271$			15.7387i			0.956055i	−0.878345	+	0.478028i	$0.841352\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.8390			−1.01543
$276$		0			0
$277$		−	26.5977i		−	1.59810i	−0.601265	−	0.799050i	$-0.705336\pi$
$277$		−	26.5977i		−	1.59810i	0.601265	−	0.799050i	$-0.294664\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.75568i		−	0.224045i	−0.993706	−	0.112023i	$-0.964267\pi$
$281$		−	3.75568i		−	0.224045i	0.993706	−	0.112023i	$-0.0357329\pi$
$282$		0			0
$283$	−27.3591			−1.62633			−0.813164	−	0.582035i	$-0.802257\pi$
$283$	−27.3591			−1.62633			−0.813164	+	0.582035i	$0.802257\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−18.6966	+	15.4951i	−1.10363	+	0.914648i
$288$		0			0
$289$	−8.35618			−0.491540
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.9079i			1.45514i	0.686035	+	0.727569i	$0.259350\pi$
$293$			24.9079i			1.45514i	−0.686035	+	0.727569i	$0.740650\pi$
$294$		0			0
$295$	−5.70804			−0.332335
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	21.8456i		−	1.26336i
$300$		0			0
$301$	−15.2901	−	18.4493i	−0.881309	−	1.06340i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	38.2005i		−	2.18736i
$306$		0			0
$307$	−11.5424			−0.658762			−0.329381	−	0.944197i	$-0.606840\pi$
$307$	−11.5424			−0.658762			−0.329381	+	0.944197i	$0.606840\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.1322			1.36841			0.684207	−	0.729288i	$-0.260148\pi$
$311$	24.1322			1.36841			0.684207	+	0.729288i	$0.260148\pi$
$312$		0			0
$313$		−	16.6314i		−	0.940060i	−0.882650	−	0.470030i	$-0.844243\pi$
$313$		−	16.6314i		−	0.940060i	0.882650	−	0.470030i	$-0.155757\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.0012			−1.06721			−0.533607	−	0.845733i	$-0.679164\pi$
$317$	−19.0012			−1.06721			−0.533607	+	0.845733i	$0.679164\pi$
$318$		0			0
$319$	−13.3514			−0.747533
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.99832			0.445038
$324$		0			0
$325$	26.3739			1.46296
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.39073	−	1.98135i	0.131805	−	0.109235i
$330$		0			0
$331$			23.7691i			1.30647i	0.757157	+	0.653233i	$0.226588\pi$
$331$			23.7691i			1.30647i	−0.757157	+	0.653233i	$0.773412\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−28.8266			−1.57497
$336$		0			0
$337$	−1.19795			−0.0652563			−0.0326282	−	0.999468i	$-0.510388\pi$
$337$	−1.19795			−0.0652563			−0.0326282	+	0.999468i	$0.510388\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			13.1519i			0.712213i
$342$		0			0
$343$	8.96530	+	16.2057i	0.484080	+	0.875024i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.62598			0.0872873			0.0436437	−	0.999047i	$-0.486103\pi$
$347$	1.62598			0.0872873			0.0436437	+	0.999047i	$0.486103\pi$
$348$		0			0
$349$	−23.2544			−1.24478			−0.622391	−	0.782707i	$-0.713838\pi$
$349$	−23.2544			−1.24478			−0.622391	+	0.782707i	$0.713838\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	30.4476			1.62056			0.810280	−	0.586043i	$-0.199315\pi$
$353$	30.4476			1.62056			0.810280	+	0.586043i	$0.199315\pi$
$354$		0			0
$355$	30.2338			1.60464
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	7.71526i		−	0.407196i	−0.979055	−	0.203598i	$-0.934736\pi$
$359$		−	7.71526i		−	0.407196i	0.979055	−	0.203598i	$-0.0652636\pi$
$360$		0			0
$361$	−11.5990			−0.610472
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−58.9363			−3.08487
$366$		0			0
$367$			37.3905i			1.95177i	0.218292	+	0.975884i	$0.429952\pi$
$367$			37.3905i			1.95177i	−0.218292	+	0.975884i	$0.570048\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.98984	+	4.13540i	−0.259059	+	0.214699i
$372$		0			0
$373$		−	17.5169i		−	0.906991i	−0.891259	−	0.453495i	$-0.850177\pi$
$373$		−	17.5169i		−	0.906991i	0.891259	−	0.453495i	$-0.149823\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.9115			1.07699
$378$		0			0
$379$			25.6557i			1.31784i	0.752212	+	0.658922i	$0.228987\pi$
$379$			25.6557i			1.31784i	−0.752212	+	0.658922i	$0.771013\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	37.8376			1.93341			0.966706	−	0.255889i	$-0.0823681\pi$
$383$	37.8376			1.93341			0.966706	+	0.255889i	$0.0823681\pi$
$384$		0			0
$385$	−10.7572	−	12.9798i	−0.548238	−	0.661512i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	23.0374			1.16804			0.584021	−	0.811739i	$-0.301479\pi$
$389$	23.0374			1.16804			0.584021	+	0.811739i	$0.301479\pi$
$390$		0			0
$391$			25.2198i			1.27542i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			0.357996i			0.0180127i
$396$		0			0
$397$	−18.2753			−0.917210			−0.458605	−	0.888640i	$-0.651651\pi$
$397$	−18.2753			−0.917210			−0.458605	+	0.888640i	$0.651651\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			17.4743i			0.872624i	0.899795	+	0.436312i	$0.143716\pi$
$401$			17.4743i			0.872624i	−0.899795	+	0.436312i	$0.856284\pi$
$402$		0			0
$403$		−	20.5990i		−	1.02611i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.1006i		−	0.649371i
$408$		0			0
$409$			17.9145i			0.885814i	0.896567	+	0.442907i	$0.146053\pi$
$409$			17.9145i			0.885814i	−0.896567	+	0.442907i	$0.853947\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.45916	−	2.96726i	−0.121007	−	0.146009i
$414$		0			0
$415$	7.96566			0.391019
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	26.0509i		−	1.27267i	−0.771413	−	0.636335i	$-0.780450\pi$
$419$		−	26.0509i		−	1.27267i	0.771413	−	0.636335i	$-0.219550\pi$
$420$		0			0
$421$			1.11513i			0.0543480i	0.999631	+	0.0271740i	$0.00865082\pi$
$421$			1.11513i			0.0543480i	−0.999631	+	0.0271740i	$0.991349\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−30.4476			−1.47692
$426$		0			0
$427$	19.8581	−	16.4577i	0.961002	−	0.796445i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			22.5414i			1.08578i	0.839803	+	0.542890i	$0.182670\pi$
$431$			22.5414i			1.08578i	−0.839803	+	0.542890i	$0.817330\pi$
$432$		0			0
$433$			25.9596i			1.24754i	0.781608	+	0.623770i	$0.214400\pi$
$433$			25.9596i			1.24754i	−0.781608	+	0.623770i	$0.785600\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			23.3364i			1.11633i
$438$		0			0
$439$		−	18.0283i		−	0.860442i	−0.902724	−	0.430221i	$-0.858436\pi$
$439$		−	18.0283i		−	0.860442i	0.902724	−	0.430221i	$-0.141564\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.3350			0.491033			0.245516	−	0.969392i	$-0.421043\pi$
$443$	10.3350			0.491033			0.245516	+	0.969392i	$0.421043\pi$
$444$		0			0
$445$			5.81307i			0.275566i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			35.7155i			1.68552i	0.538292	+	0.842758i	$0.319070\pi$
$449$			35.7155i			1.68552i	−0.538292	+	0.842758i	$0.680930\pi$
$450$		0			0
$451$	14.9234			0.702718
$452$		0			0
$453$		0			0
$454$		0			0
$455$	16.8484	+	20.3295i	0.789864	+	0.953062i
$456$		0			0
$457$	13.1979			0.617374			0.308687	−	0.951164i	$-0.400110\pi$
$457$	13.1979			0.617374			0.308687	+	0.951164i	$0.400110\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			1.57150i			0.0731921i	0.999330	+	0.0365960i	$0.0116515\pi$
$461$			1.57150i			0.0731921i	−0.999330	+	0.0365960i	$0.988349\pi$
$462$		0			0
$463$	−38.0274			−1.76729			−0.883643	−	0.468162i	$-0.844916\pi$
$463$	−38.0274			−1.76729			−0.883643	+	0.468162i	$0.844916\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.9655i			0.738795i	0.929272	+	0.369397i	$0.120436\pi$
$467$			15.9655i			0.738795i	−0.929272	+	0.369397i	$0.879564\pi$
$468$		0			0
$469$	−12.4192	−	14.9852i	−0.573466	−	0.691953i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			14.7260i			0.677103i
$474$		0			0
$475$	−28.1738			−1.29270
$476$		0			0
$477$		0			0
$478$		0			0
$479$	25.8594			1.18154			0.590772	−	0.806838i	$-0.298823\pi$
$479$	25.8594			1.18154			0.590772	+	0.806838i	$0.298823\pi$
$480$		0			0
$481$			20.5186i			0.935570i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.5424			1.34145
$486$		0			0
$487$	1.63386			0.0740370			0.0370185	−	0.999315i	$-0.488214\pi$
$487$	1.63386			0.0740370			0.0370185	+	0.999315i	$0.488214\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.61687			−0.388874			−0.194437	−	0.980915i	$-0.562288\pi$
$491$	−8.61687			−0.388874			−0.194437	+	0.980915i	$0.562288\pi$
$492$		0			0
$493$	−24.1414			−1.08727
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.0255	+	15.7167i	0.584271	+	0.704990i
$498$		0			0
$499$		−	16.3680i		−	0.732734i	−0.930471	−	0.366367i	$-0.880602\pi$
$499$		−	16.3680i		−	0.732734i	0.930471	−	0.366367i	$-0.119398\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	32.5232			1.45014			0.725068	−	0.688677i	$-0.241808\pi$
$503$	32.5232			1.45014			0.725068	+	0.688677i	$0.241808\pi$
$504$		0			0
$505$	46.0685			2.05002
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	0.775695i		−	0.0343821i	−0.999852	−	0.0171910i	$-0.994528\pi$
$509$		−	0.775695i		−	0.0343821i	0.999852	−	0.0171910i	$-0.00547235\pi$
$510$		0			0
$511$	−25.3912	−	30.6374i	−1.12324	−	1.35532i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	55.1711			2.43113
$516$		0			0
$517$	−1.90825			−0.0839247
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.2749			−1.23875			−0.619374	−	0.785096i	$-0.712614\pi$
$521$	−28.2749			−1.23875			−0.619374	+	0.785096i	$0.712614\pi$
$522$		0			0
$523$	20.5186			0.897218			0.448609	−	0.893728i	$-0.351920\pi$
$523$	20.5186			0.897218			0.448609	+	0.893728i	$0.351920\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			23.7807i			1.03590i
$528$		0			0
$529$	−50.5830			−2.19926
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−23.3737			−1.01243
$534$		0			0
$535$		−	32.7356i		−	1.41528i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.11294	−	11.1840i	0.0910109	−	0.481730i
$540$		0			0
$541$		−	11.7038i		−	0.503187i	−0.967833	−	0.251593i	$-0.919045\pi$
$541$		−	11.7038i		−	0.503187i	0.967833	−	0.251593i	$-0.0809546\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−18.3562			−0.786294
$546$		0			0
$547$			42.0685i			1.79872i	0.437208	+	0.899360i	$0.355967\pi$
$547$			42.0685i			1.79872i	−0.437208	+	0.899360i	$0.644033\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−22.3386			−0.951655
$552$		0			0
$553$	−0.186100	+	0.154233i	−0.00791378	+	0.00655867i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.9162			0.928620			0.464310	−	0.885673i	$-0.346302\pi$
$557$	21.9162			0.928620			0.464310	+	0.885673i	$0.346302\pi$
$558$		0			0
$559$		−	23.0645i		−	0.975525i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			26.2690i			1.10711i	0.832814	+	0.553554i	$0.186729\pi$
$563$			26.2690i			1.10711i	−0.832814	+	0.553554i	$0.813271\pi$
$564$		0			0
$565$	22.4817			0.945811
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	40.4618i		−	1.69625i	−0.529798	−	0.848124i	$-0.677732\pi$
$569$		−	40.4618i		−	1.69625i	0.529798	−	0.848124i	$-0.322268\pi$
$570$		0			0
$571$			6.44074i			0.269537i	0.990877	+	0.134768i	$0.0430290\pi$
$571$			6.44074i			0.269537i	−0.990877	+	0.134768i	$0.956971\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	88.8358i		−	3.70471i
$576$		0			0
$577$			17.1376i			0.713450i	0.934210	+	0.356725i	$0.116107\pi$
$577$			17.1376i			0.713450i	−0.934210	+	0.356725i	$0.883893\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.43180	+	4.14086i	0.142375	+	0.171792i
$582$		0			0
$583$	3.98283			0.164952
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	19.2367i		−	0.793984i	−0.917822	−	0.396992i	$-0.870054\pi$
$587$		−	19.2367i		−	0.793984i	0.917822	−	0.396992i	$-0.129946\pi$
$588$		0			0
$589$			22.0048i			0.906691i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.0268037			−0.00110070			−0.000550349	−	1.00000i	$-0.500175\pi$
$593$	−0.0268037			−0.00110070			−0.000550349		1.00000i	$0.500175\pi$
$594$		0			0
$595$	−19.4508	−	23.4696i	−0.797403	−	0.962158i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			16.0460i			0.655624i	0.944743	+	0.327812i	$0.106311\pi$
$599$			16.0460i			0.655624i	−0.944743	+	0.327812i	$0.893689\pi$
$600$		0			0
$601$		−	22.5786i		−	0.921001i	−0.887660	−	0.460500i	$-0.847670\pi$
$601$		−	22.5786i		−	0.921001i	0.887660	−	0.460500i	$-0.152330\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	32.7453i		−	1.33129i
$606$		0			0
$607$			27.4523i			1.11425i	0.830427	+	0.557127i	$0.188096\pi$
$607$			27.4523i			1.11425i	−0.830427	+	0.557127i	$0.811904\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.98878			0.120913
$612$		0			0
$613$		−	14.0391i		−	0.567035i	−0.958967	−	0.283517i	$-0.908499\pi$
$613$		−	14.0391i		−	0.567035i	0.958967	−	0.283517i	$-0.0915013\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			19.4556i			0.783254i	0.920124	+	0.391627i	$0.128088\pi$
$617$			19.4556i			0.783254i	−0.920124	+	0.391627i	$0.871912\pi$
$618$		0			0
$619$	2.05997			0.0827971			0.0413985	−	0.999143i	$-0.486819\pi$
$619$	2.05997			0.0827971			0.0413985	+	0.999143i	$0.486819\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.02186	+	2.50441i	−0.121068	+	0.100337i
$624$		0			0
$625$	30.4696			1.21878
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	23.6879i		−	0.944499i
$630$		0			0
$631$	−43.9182			−1.74836			−0.874178	−	0.485606i	$-0.838599\pi$
$631$	−43.9182			−1.74836			−0.874178	+	0.485606i	$0.838599\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			4.42346i			0.175540i
$636$		0			0
$637$	−3.30938	+	17.5169i	−0.131122	+	0.694045i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			40.1986i			1.58775i	0.608081	+	0.793875i	$0.291940\pi$
$641$			40.1986i			1.58775i	−0.608081	+	0.793875i	$0.708060\pi$
$642$		0			0
$643$	39.8705			1.57234			0.786169	−	0.618011i	$-0.212061\pi$
$643$	39.8705			1.57234			0.786169	+	0.618011i	$0.212061\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−23.6879			−0.931268			−0.465634	−	0.884977i	$-0.654174\pi$
$647$	−23.6879			−0.931268			−0.465634	+	0.884977i	$0.654174\pi$
$648$		0			0
$649$			2.36843i			0.0929692i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.44949			−0.0958559			−0.0479280	−	0.998851i	$-0.515262\pi$
$653$	−2.44949			−0.0958559			−0.0479280	+	0.998851i	$0.515262\pi$
$654$		0			0
$655$	85.6061			3.34491
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.8282			−1.23985			−0.619925	−	0.784661i	$-0.712837\pi$
$659$	−31.8282			−1.23985			−0.619925	+	0.784661i	$0.712837\pi$
$660$		0			0
$661$	6.18031			0.240386			0.120193	−	0.992751i	$-0.461649\pi$
$661$	6.18031			0.240386			0.120193	+	0.992751i	$0.461649\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−17.9982	−	21.7169i	−0.697941	−	0.842146i
$666$		0			0
$667$		−	70.4366i		−	2.72731i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15.8505			−0.611903
$672$		0			0
$673$	−4.55413			−0.175549			−0.0877744	−	0.996140i	$-0.527975\pi$
$673$	−4.55413			−0.175549			−0.0877744	+	0.996140i	$0.527975\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			21.7649i			0.836495i	0.908333	+	0.418247i	$0.137355\pi$
$677$			21.7649i			0.836495i	−0.908333	+	0.418247i	$0.862645\pi$
$678$		0			0
$679$	12.7276	+	15.3573i	0.488440	+	0.589359i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.88749			−0.378334			−0.189167	−	0.981945i	$-0.560579\pi$
$683$	−9.88749			−0.378334			−0.189167	+	0.981945i	$0.560579\pi$
$684$		0			0
$685$	42.9895			1.64254
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.23807			−0.237652
$690$		0			0
$691$	−45.1572			−1.71786			−0.858931	−	0.512092i	$-0.828871\pi$
$691$	−45.1572			−1.71786			−0.858931	+	0.512092i	$0.828871\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	32.5866i		−	1.23608i
$696$		0			0
$697$	26.9840			1.02209
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.4490			−0.772349			−0.386175	−	0.922426i	$-0.626204\pi$
$701$	−20.4490			−0.772349			−0.386175	+	0.922426i	$0.626204\pi$
$702$		0			0
$703$		−	21.9190i		−	0.826689i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.8475	+	23.9482i	0.746440	+	0.900666i
$708$		0			0
$709$			11.4298i			0.429254i	0.976696	+	0.214627i	$0.0688535\pi$
$709$			11.4298i			0.429254i	−0.976696	+	0.214627i	$0.931146\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−69.3840			−2.59845
$714$		0			0
$715$		−	16.2268i		−	0.606847i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	34.4926			1.28636			0.643178	−	0.765717i	$-0.277616\pi$
$719$	34.4926			1.28636			0.643178	+	0.765717i	$0.277616\pi$
$720$		0			0
$721$	23.7691	+	28.6801i	0.885206	+	1.06810i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	85.0372			3.15820
$726$		0			0
$727$			28.4821i			1.05634i	0.849138	+	0.528172i	$0.177122\pi$
$727$			28.4821i			1.05634i	−0.849138	+	0.528172i	$0.822878\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			26.6270i			0.984836i
$732$		0			0
$733$	22.1574			0.818404			0.409202	−	0.912444i	$-0.365807\pi$
$733$	22.1574			0.818404			0.409202	+	0.912444i	$0.365807\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			11.9610i			0.440590i
$738$		0			0
$739$		−	26.1582i		−	0.962246i	−0.876653	−	0.481123i	$-0.840229\pi$
$739$		−	26.1582i		−	0.962246i	0.876653	−	0.481123i	$-0.159771\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	48.7619i		−	1.78890i	−0.447168	−	0.894450i	$-0.647568\pi$
$743$		−	48.7619i		−	1.78890i	0.447168	−	0.894450i	$-0.352432\pi$
$744$		0			0
$745$		−	54.2497i		−	1.98756i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.0172	−	14.1033i	0.621796	−	0.515323i
$750$		0			0
$751$	−16.8931			−0.616439			−0.308220	−	0.951315i	$-0.599733\pi$
$751$	−16.8931			−0.616439			−0.308220	+	0.951315i	$0.599733\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			75.0460i			2.73120i
$756$		0			0
$757$			14.8802i			0.540829i	0.962744	+	0.270415i	$0.0871608\pi$
$757$			14.8802i			0.540829i	−0.962744	+	0.270415i	$0.912839\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.78781			0.0648080			0.0324040	−	0.999475i	$-0.489684\pi$
$761$	1.78781			0.0648080			0.0324040	+	0.999475i	$0.489684\pi$
$762$		0			0
$763$	−7.90831	−	9.54228i	−0.286300	−	0.345454i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.70954i		−	0.133944i
$768$		0			0
$769$			11.6967i			0.421793i	0.977508	+	0.210897i	$0.0676383\pi$
$769$			11.6967i			0.421793i	−0.977508	+	0.210897i	$0.932362\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			32.5696i			1.17145i	0.810511	+	0.585723i	$0.199190\pi$
$773$			32.5696i			1.17145i	−0.810511	+	0.585723i	$0.800810\pi$
$774$		0			0
$775$		−	83.7665i		−	3.00898i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.9689			0.894603
$780$		0			0
$781$		−	12.5449i		−	0.448891i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			48.6672i			1.73701i
$786$		0			0
$787$	33.7690			1.20374			0.601868	−	0.798596i	$-0.294424\pi$
$787$	33.7690			1.20374			0.601868	+	0.798596i	$0.294424\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.68566	+	11.6869i	0.344382	+	0.415537i
$792$		0			0
$793$	24.8257			0.881588
$794$		0			0
$795$		0			0
$796$		0			0
$797$			14.2790i			0.505789i	0.967494	+	0.252894i	$0.0813825\pi$
$797$			14.2790i			0.505789i	−0.967494	+	0.252894i	$0.918617\pi$
$798$		0			0
$799$	−3.45042			−0.122067
$800$		0			0
$801$		0			0
$802$		0			0
$803$			24.4544i			0.862977i
$804$		0			0
$805$	68.4764	−	56.7508i	2.41348	−	2.00020i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	15.9799i		−	0.561823i	−0.959734	−	0.280911i	$-0.909363\pi$
$809$		−	15.9799i		−	0.561823i	0.959734	−	0.280911i	$-0.0906367\pi$
$810$		0			0
$811$	28.3280			0.994732			0.497366	−	0.867541i	$-0.334301\pi$
$811$	28.3280			0.994732			0.497366	+	0.867541i	$0.334301\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−44.5014			−1.55882
$816$		0			0
$817$			24.6386i			0.861995i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.5883			1.17224			0.586120	−	0.810225i	$-0.300655\pi$
$821$	33.5883			1.17224			0.586120	+	0.810225i	$0.300655\pi$
$822$		0			0
$823$	−9.45989			−0.329751			−0.164875	−	0.986314i	$-0.552722\pi$
$823$	−9.45989			−0.329751			−0.164875	+	0.986314i	$0.552722\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.8447			0.898709			0.449354	−	0.893354i	$-0.351654\pi$
$827$	25.8447			0.898709			0.449354	+	0.893354i	$0.351654\pi$
$828$		0			0
$829$	−17.9611			−0.623815			−0.311907	−	0.950113i	$-0.600968\pi$
$829$	−17.9611			−0.623815			−0.311907	+	0.950113i	$0.600968\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.82054	−	20.2225i	0.132374	−	0.700669i
$834$		0			0
$835$			26.1134i			0.903691i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	14.3254			0.494569			0.247285	−	0.968943i	$-0.420462\pi$
$839$	14.3254			0.494569			0.247285	+	0.968943i	$0.420462\pi$
$840$		0			0
$841$	38.4247			1.32499
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	25.5280i		−	0.878190i
$846$		0			0
$847$	17.0223	−	14.1075i	0.584893	−	0.484739i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	69.1135			2.36918
$852$		0			0
$853$	30.1418			1.03204			0.516018	−	0.856577i	$-0.327414\pi$
$853$	30.1418			1.03204			0.516018	+	0.856577i	$0.327414\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−33.7724			−1.15364			−0.576821	−	0.816870i	$-0.695707\pi$
$857$	−33.7724			−1.15364			−0.576821	+	0.816870i	$0.695707\pi$
$858$		0			0
$859$	−29.4190			−1.00376			−0.501882	−	0.864936i	$-0.667359\pi$
$859$	−29.4190			−1.00376			−0.501882	+	0.864936i	$0.667359\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	30.8915i		−	1.05156i	−0.850621	−	0.525780i	$-0.823774\pi$
$863$		−	30.8915i		−	1.05156i	0.850621	−	0.525780i	$-0.176226\pi$
$864$		0			0
$865$	36.1819			1.23022
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.148543			0.00503897
$870$		0			0
$871$		−	18.7338i		−	0.634772i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	35.4355	+	42.7570i	1.19794	+	1.44545i
$876$		0			0
$877$			19.7472i			0.666814i	0.942783	+	0.333407i	$0.108198\pi$
$877$			19.7472i			0.666814i	−0.942783	+	0.333407i	$0.891802\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.7188			1.13601			0.568007	−	0.823023i	$-0.307714\pi$
$881$	33.7188			1.13601			0.568007	+	0.823023i	$0.307714\pi$
$882$		0			0
$883$		−	25.2835i		−	0.850856i	−0.904992	−	0.425428i	$-0.860123\pi$
$883$		−	25.2835i		−	0.850856i	0.904992	−	0.425428i	$-0.139877\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−38.0134			−1.27636			−0.638182	−	0.769886i	$-0.720313\pi$
$887$	−38.0134			−1.27636			−0.638182	+	0.769886i	$0.720313\pi$
$888$		0			0
$889$	−2.29949	+	1.90573i	−0.0771223	+	0.0639163i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.19275			−0.106841
$894$		0			0
$895$		−	64.0786i		−	2.14191i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	66.4172i		−	2.21514i
$900$		0			0
$901$	7.20159			0.239920
$902$		0			0
$903$		0			0
$904$		0			0
$905$			53.4135i			1.77553i
$906$		0			0
$907$			25.9721i			0.862391i	0.902259	+	0.431195i	$0.141908\pi$
$907$			25.9721i			0.862391i	−0.902259	+	0.431195i	$0.858092\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			13.6256i			0.451435i	0.974193	+	0.225718i	$0.0724727\pi$
$911$			13.6256i			0.451435i	−0.974193	+	0.225718i	$0.927527\pi$
$912$		0			0
$913$		−	3.30519i		−	0.109386i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36.8812	+	44.5014i	1.21793	+	1.46957i
$918$		0			0
$919$	−5.11164			−0.168617			−0.0843087	−	0.996440i	$-0.526868\pi$
$919$	−5.11164			−0.168617			−0.0843087	+	0.996440i	$0.526868\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			19.6483i			0.646732i
$924$		0			0
$925$			83.4399i			2.74349i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	36.3276			1.19187			0.595935	−	0.803032i	$-0.296781\pi$
$929$	36.3276			1.19187			0.595935	+	0.803032i	$0.296781\pi$
$930$		0			0
$931$	3.53523	−	18.7124i	0.115863	−	0.613273i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			18.7331i			0.612639i
$936$		0			0
$937$			1.32103i			0.0431562i	0.999767	+	0.0215781i	$0.00686906\pi$
$937$			1.32103i			0.0431562i	−0.999767	+	0.0215781i	$0.993131\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			11.1360i			0.363024i	0.983389	+	0.181512i	$0.0580991\pi$
$941$			11.1360i			0.363024i	−0.983389	+	0.181512i	$0.941901\pi$
$942$		0			0
$943$			78.7303i			2.56381i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.5858			−1.09139			−0.545695	−	0.837984i	$-0.683734\pi$
$947$	−33.5858			−1.09139			−0.545695	+	0.837984i	$0.683734\pi$
$948$		0			0
$949$		−	38.3015i		−	1.24332i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			56.3855i			1.82651i	0.407392	+	0.913253i	$0.366438\pi$
$953$			56.3855i			1.82651i	−0.407392	+	0.913253i	$0.633562\pi$
$954$		0			0
$955$	52.8124			1.70897
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.5209	+	22.3476i	0.598071	+	0.721642i
$960$		0			0
$961$	−34.4247			−1.11047
$962$		0			0
$963$		0			0
$964$		0			0
$965$			46.8486i			1.50811i
$966$		0			0
$967$	−14.1305			−0.454405			−0.227203	−	0.973847i	$-0.572958\pi$
$967$	−14.1305			−0.454405			−0.227203	+	0.973847i	$0.572958\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			43.3331i			1.39063i	0.718707	+	0.695313i	$0.244734\pi$
$971$			43.3331i			1.39063i	−0.718707	+	0.695313i	$0.755266\pi$
$972$		0			0
$973$	16.9398	−	14.0391i	0.543065	−	0.450073i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	44.2007i		−	1.41411i	−0.707160	−	0.707053i	$-0.750024\pi$
$977$		−	44.2007i		−	1.41411i	0.707160	−	0.707053i	$-0.249976\pi$
$978$		0			0
$979$	2.41202			0.0770884
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.66379			−0.212542			−0.106271	−	0.994337i	$-0.533891\pi$
$983$	−6.66379			−0.212542			−0.106271	+	0.994337i	$0.533891\pi$
$984$		0			0
$985$			19.2355i			0.612895i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−77.6888			−2.47036
$990$		0			0
$991$	−24.8862			−0.790535			−0.395267	−	0.918566i	$-0.629348\pi$
$991$	−24.8862			−0.790535			−0.395267	+	0.918566i	$0.629348\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−36.7061			−1.16366
$996$		0			0
$997$	8.85125			0.280322			0.140161	−	0.990129i	$-0.455238\pi$
$997$	8.85125			0.280322			0.140161	+	0.990129i	$0.455238\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.i.c.1889.18	yes	48
3.2	odd	2	inner	4032.2.i.c.1889.41	yes	48
4.3	odd	2	inner	4032.2.i.c.1889.16	yes	48
7.6	odd	2	inner	4032.2.i.c.1889.33	yes	48
8.3	odd	2	inner	4032.2.i.c.1889.7	✓	48
8.5	even	2	inner	4032.2.i.c.1889.35	yes	48
12.11	even	2	inner	4032.2.i.c.1889.13	yes	48
21.20	even	2	inner	4032.2.i.c.1889.36	yes	48
24.5	odd	2	inner	4032.2.i.c.1889.34	yes	48
24.11	even	2	inner	4032.2.i.c.1889.32	yes	48
28.27	even	2	inner	4032.2.i.c.1889.31	yes	48
56.13	odd	2	inner	4032.2.i.c.1889.42	yes	48
56.27	even	2	inner	4032.2.i.c.1889.14	yes	48
84.83	odd	2	inner	4032.2.i.c.1889.8	yes	48
168.83	odd	2	inner	4032.2.i.c.1889.15	yes	48
168.125	even	2	inner	4032.2.i.c.1889.17	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.i.c.1889.7	✓	48	8.3	odd	2	inner
4032.2.i.c.1889.8	yes	48	84.83	odd	2	inner
4032.2.i.c.1889.13	yes	48	12.11	even	2	inner
4032.2.i.c.1889.14	yes	48	56.27	even	2	inner
4032.2.i.c.1889.15	yes	48	168.83	odd	2	inner
4032.2.i.c.1889.16	yes	48	4.3	odd	2	inner
4032.2.i.c.1889.17	yes	48	168.125	even	2	inner
4032.2.i.c.1889.18	yes	48	1.1	even	1	trivial
4032.2.i.c.1889.31	yes	48	28.27	even	2	inner
4032.2.i.c.1889.32	yes	48	24.11	even	2	inner
4032.2.i.c.1889.33	yes	48	7.6	odd	2	inner
4032.2.i.c.1889.34	yes	48	24.5	odd	2	inner
4032.2.i.c.1889.35	yes	48	8.5	even	2	inner
4032.2.i.c.1889.36	yes	48	21.20	even	2	inner
4032.2.i.c.1889.41	yes	48	3.2	odd	2	inner
4032.2.i.c.1889.42	yes	48	56.13	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.91870i q^{5} +(-2.03709 + 1.68827i) q^{7} +O(q^{10})\)	\(q+3.91870i q^{5} +(-2.03709 + 1.68827i) q^{7} +1.62598 q^{11} -2.54668 q^{13} +2.94004 q^{17} +2.72048 q^{19} +8.57805i q^{23} -10.3562 q^{25} -8.21126 q^{29} +8.08855i q^{31} +(-6.61582 - 7.98274i) q^{35} -8.05701i q^{37} +9.17811 q^{41} +9.05669i q^{43} -1.17360 q^{47} +(1.29949 - 6.87832i) q^{49} +2.44949 q^{53} +6.37173i q^{55} +1.45662i q^{59} -9.74827 q^{61} -9.97967i q^{65} +7.35618i q^{67} -7.71526i q^{71} +15.0398i q^{73} +(-3.31228 + 2.74510i) q^{77} +0.0913558 q^{79} -2.03273i q^{83} +11.5211i q^{85} +1.48342 q^{89} +(5.18782 - 4.29949i) q^{91} +10.6607i q^{95} -7.53884i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 80 q^{25} - 16 q^{49}+O(q^{100}) \) 48 * q - 80 * q^25 - 16 * q^49

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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