LMFDB - Embedded newform 1872.2.c.j.1585.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(1585,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1872.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.c (of order $2$, degree $1$, not 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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1872.1585
Dual form			1872.2.c.j.1585.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{5} -0.438447i q^{7} +O(q^{10})$	$q+1.56155i q^{5} -0.438447i q^{7} +5.12311i q^{11} +(0.561553 - 3.56155i) q^{13} +3.56155 q^{17} -2.00000i q^{19} -3.12311 q^{23} +2.56155 q^{25} +5.12311 q^{29} +5.12311i q^{31} +0.684658 q^{35} +9.56155i q^{37} +8.00000i q^{41} -9.56155 q^{43} -7.56155i q^{47} +6.80776 q^{49} -12.2462 q^{53} -8.00000 q^{55} +10.0000i q^{59} +2.87689 q^{61} +(5.56155 + 0.876894i) q^{65} +9.12311i q^{67} +6.68466i q^{71} -9.36932i q^{73} +2.24621 q^{77} +11.1231 q^{79} +8.24621i q^{83} +5.56155i q^{85} +3.12311i q^{89} +(-1.56155 - 0.246211i) q^{91} +3.12311 q^{95} +6.24621i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 6 q^{13} + 6 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 22 q^{35} - 30 q^{43} - 14 q^{49} - 16 q^{53} - 32 q^{55} + 28 q^{61} + 14 q^{65} - 24 q^{77} + 28 q^{79} + 2 q^{91} - 4 q^{95}+O(q^{100}) $ 4 * q - 6 * q^13 + 6 * q^17 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 22 * q^35 - 30 * q^43 - 14 * q^49 - 16 * q^53 - 32 * q^55 + 28 * q^61 + 14 * q^65 - 24 * q^77 + 28 * q^79 + 2 * q^91 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$			1.56155i			0.698348i	0.937058	+	0.349174i	$0.113538\pi$
$5$			1.56155i			0.698348i	−0.937058	+	0.349174i	$0.886462\pi$
$6$		0			0
$7$		−	0.438447i		−	0.165717i	−0.996561	−	0.0828587i	$-0.973595\pi$
$7$		−	0.438447i		−	0.165717i	0.996561	−	0.0828587i	$-0.0264050\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.12311i			1.54467i	0.635213	+	0.772337i	$0.280912\pi$
$11$			5.12311i			1.54467i	−0.635213	+	0.772337i	$0.719088\pi$
$12$		0			0
$13$	0.561553	−	3.56155i	0.155747	−	0.987797i
$13$	0.561553	−	3.56155i	0.155747	−	0.987797i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.56155			0.863803			0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155			0.863803			0.431902	+	0.901921i	$0.357843\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.12311			−0.651213			−0.325606	−	0.945505i	$-0.605568\pi$
$23$	−3.12311			−0.651213			−0.325606	+	0.945505i	$0.605568\pi$
$24$		0			0
$25$	2.56155			0.512311
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.12311			0.951337			0.475668	−	0.879625i	$-0.342206\pi$
$29$	5.12311			0.951337			0.475668	+	0.879625i	$0.342206\pi$
$30$		0			0
$31$			5.12311i			0.920137i	0.887883	+	0.460068i	$0.152175\pi$
$31$			5.12311i			0.920137i	−0.887883	+	0.460068i	$0.847825\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.684658			0.115728
$36$		0			0
$37$			9.56155i			1.57191i	0.618284	+	0.785955i	$0.287828\pi$
$37$			9.56155i			1.57191i	−0.618284	+	0.785955i	$0.712172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	−9.56155			−1.45812			−0.729062	−	0.684448i	$-0.760043\pi$
$43$	−9.56155			−1.45812			−0.729062	+	0.684448i	$0.760043\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.56155i		−	1.10297i	−0.834186	−	0.551483i	$-0.814062\pi$
$47$		−	7.56155i		−	1.10297i	0.834186	−	0.551483i	$-0.185938\pi$
$48$		0			0
$49$	6.80776			0.972538
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−12.2462			−1.68215			−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462			−1.68215			−0.841073	+	0.540921i	$0.818076\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$	2.87689			0.368349			0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689			0.368349			0.184174	+	0.982894i	$0.441039\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.56155	+	0.876894i	0.689826	+	0.108765i
$66$		0			0
$67$			9.12311i			1.11456i	0.830323	+	0.557282i	$0.188156\pi$
$67$			9.12311i			1.11456i	−0.830323	+	0.557282i	$0.811844\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.68466i			0.793323i	0.917965	+	0.396662i	$0.129831\pi$
$71$			6.68466i			0.793323i	−0.917965	+	0.396662i	$0.870169\pi$
$72$		0			0
$73$		−	9.36932i		−	1.09660i	−0.836283	−	0.548298i	$-0.815276\pi$
$73$		−	9.36932i		−	1.09660i	0.836283	−	0.548298i	$-0.184724\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.24621			0.255980
$78$		0			0
$79$	11.1231			1.25145			0.625724	−	0.780045i	$-0.284804\pi$
$79$	11.1231			1.25145			0.625724	+	0.780045i	$0.284804\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.24621i			0.905139i	0.891729	+	0.452570i	$0.149493\pi$
$83$			8.24621i			0.905139i	−0.891729	+	0.452570i	$0.850507\pi$
$84$		0			0
$85$			5.56155i			0.603235i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.12311i			0.331049i	0.986206	+	0.165524i	$0.0529316\pi$
$89$			3.12311i			0.331049i	−0.986206	+	0.165524i	$0.947068\pi$
$90$		0			0
$91$	−1.56155	−	0.246211i	−0.163695	−	0.0258100i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.12311			0.320424
$96$		0			0
$97$			6.24621i			0.634207i	0.948391	+	0.317103i	$0.102710\pi$
$97$			6.24621i			0.634207i	−0.948391	+	0.317103i	$0.897290\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.3693			1.13129			0.565645	−	0.824649i	$-0.308627\pi$
$101$	11.3693			1.13129			0.565645	+	0.824649i	$0.308627\pi$
$102$		0			0
$103$	−3.12311			−0.307729			−0.153864	−	0.988092i	$-0.549172\pi$
$103$	−3.12311			−0.307729			−0.153864	+	0.988092i	$0.549172\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.00000			−0.386695			−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000			−0.386695			−0.193347	+	0.981130i	$0.561934\pi$
$108$		0			0
$109$		−	3.31534i		−	0.317552i	−0.987315	−	0.158776i	$-0.949245\pi$
$109$		−	3.31534i		−	0.317552i	0.987315	−	0.158776i	$-0.0507548\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$		−	4.87689i		−	0.454773i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	1.56155i		−	0.143147i
$120$		0			0
$121$	−15.2462			−1.38602
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.8078i			1.05612i
$126$		0			0
$127$	4.87689			0.432754			0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689			0.432754			0.216377	+	0.976310i	$0.430576\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.31534			−0.289663			−0.144831	−	0.989456i	$-0.546264\pi$
$131$	−3.31534			−0.289663			−0.144831	+	0.989456i	$0.546264\pi$
$132$		0			0
$133$	−0.876894			−0.0760364
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.12311i			0.266825i	0.991061	+	0.133412i	$0.0425935\pi$
$137$			3.12311i			0.266825i	−0.991061	+	0.133412i	$0.957406\pi$
$138$		0			0
$139$	7.80776			0.662246			0.331123	−	0.943588i	$-0.392573\pi$
$139$	7.80776			0.662246			0.331123	+	0.943588i	$0.392573\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.2462	+	2.87689i	1.52582	+	0.240578i
$144$		0			0
$145$			8.00000i			0.664364i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	7.12311i		−	0.583548i	−0.956487	−	0.291774i	$-0.905755\pi$
$149$		−	7.12311i		−	0.583548i	0.956487	−	0.291774i	$-0.0942454\pi$
$150$		0			0
$151$		−	11.5616i		−	0.940866i	−0.882436	−	0.470433i	$-0.844098\pi$
$151$		−	11.5616i		−	0.940866i	0.882436	−	0.470433i	$-0.155902\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	12.2462			0.977354			0.488677	−	0.872465i	$-0.337480\pi$
$157$	12.2462			0.977354			0.488677	+	0.872465i	$0.337480\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.36932i			0.107917i
$162$		0			0
$163$			20.2462i			1.58581i	0.609348	+	0.792903i	$0.291431\pi$
$163$			20.2462i			1.58581i	−0.609348	+	0.792903i	$0.708569\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.87689i		−	0.532150i	−0.963952	−	0.266075i	$-0.914273\pi$
$167$		−	6.87689i		−	0.532150i	0.963952	−	0.266075i	$-0.0857269\pi$
$168$		0			0
$169$	−12.3693	−	4.00000i	−0.951486	−	0.307692i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.24621			−0.322833			−0.161417	−	0.986886i	$-0.551606\pi$
$173$	−4.24621			−0.322833			−0.161417	+	0.986886i	$0.551606\pi$
$174$		0			0
$175$		−	1.12311i		−	0.0848988i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	17.5616			1.31261			0.656306	−	0.754495i	$-0.272118\pi$
$179$	17.5616			1.31261			0.656306	+	0.754495i	$0.272118\pi$
$180$		0			0
$181$	−8.24621			−0.612936			−0.306468	−	0.951881i	$-0.599147\pi$
$181$	−8.24621			−0.612936			−0.306468	+	0.951881i	$0.599147\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.9309			−1.09774
$186$		0			0
$187$			18.2462i			1.33430i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−22.2462			−1.60968			−0.804840	−	0.593492i	$-0.797749\pi$
$191$	−22.2462			−1.60968			−0.804840	+	0.593492i	$0.797749\pi$
$192$		0			0
$193$			23.6155i			1.69988i	0.526877	+	0.849941i	$0.323363\pi$
$193$			23.6155i			1.69988i	−0.526877	+	0.849941i	$0.676637\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.56155i		−	0.681232i	−0.940202	−	0.340616i	$-0.889364\pi$
$197$		−	9.56155i		−	0.681232i	0.940202	−	0.340616i	$-0.110636\pi$
$198$		0			0
$199$	14.2462			1.00989			0.504944	−	0.863152i	$-0.331513\pi$
$199$	14.2462			1.00989			0.504944	+	0.863152i	$0.331513\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.24621i		−	0.157653i
$204$		0			0
$205$	−12.4924			−0.872509
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.2462			0.708745
$210$		0			0
$211$	22.0540			1.51826			0.759129	−	0.650940i	$-0.225625\pi$
$211$	22.0540			1.51826			0.759129	+	0.650940i	$0.225625\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	14.9309i		−	1.01828i
$216$		0			0
$217$	2.24621			0.152483
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.00000	−	12.6847i	0.134535	−	0.853262i
$222$		0			0
$223$		−	27.1771i		−	1.81991i	−0.414705	−	0.909956i	$-0.636115\pi$
$223$		−	27.1771i		−	1.81991i	0.414705	−	0.909956i	$-0.363885\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3.36932i			0.223629i	0.993729	+	0.111815i	$0.0356663\pi$
$227$			3.36932i			0.223629i	−0.993729	+	0.111815i	$0.964334\pi$
$228$		0			0
$229$			20.6847i			1.36688i	0.730006	+	0.683440i	$0.239517\pi$
$229$			20.6847i			1.36688i	−0.730006	+	0.683440i	$0.760483\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.56155			0.233325			0.116663	−	0.993172i	$-0.462780\pi$
$233$	3.56155			0.233325			0.116663	+	0.993172i	$0.462780\pi$
$234$		0			0
$235$	11.8078			0.770254
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.93087i		−	0.577690i	−0.957376	−	0.288845i	$-0.906729\pi$
$239$		−	8.93087i		−	0.577690i	0.957376	−	0.288845i	$-0.0932712\pi$
$240$		0			0
$241$		−	27.1231i		−	1.74715i	−0.486687	−	0.873576i	$-0.661795\pi$
$241$		−	27.1231i		−	1.74715i	0.486687	−	0.873576i	$-0.338205\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			10.6307i			0.679169i
$246$		0			0
$247$	−7.12311	−	1.12311i	−0.453232	−	0.0714615i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.4924			1.54595			0.772974	−	0.634438i	$-0.218768\pi$
$251$	24.4924			1.54595			0.772974	+	0.634438i	$0.218768\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.930870			−0.0580661			−0.0290330	−	0.999578i	$-0.509243\pi$
$257$	−0.930870			−0.0580661			−0.0290330	+	0.999578i	$0.509243\pi$
$258$		0			0
$259$	4.19224			0.260493
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.49242			−0.277015			−0.138507	−	0.990361i	$-0.544230\pi$
$263$	−4.49242			−0.277015			−0.138507	+	0.990361i	$0.544230\pi$
$264$		0			0
$265$		−	19.1231i		−	1.17472i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.8769			−0.663176			−0.331588	−	0.943424i	$-0.607584\pi$
$269$	−10.8769			−0.663176			−0.331588	+	0.943424i	$0.607584\pi$
$270$		0			0
$271$		−	14.6847i		−	0.892029i	−0.895026	−	0.446015i	$-0.852843\pi$
$271$		−	14.6847i		−	0.892029i	0.895026	−	0.446015i	$-0.147157\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			13.1231i			0.791353i
$276$		0			0
$277$	7.36932			0.442779			0.221390	−	0.975185i	$-0.428941\pi$
$277$	7.36932			0.442779			0.221390	+	0.975185i	$0.428941\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9.36932i		−	0.558927i	−0.960156	−	0.279463i	$-0.909843\pi$
$281$		−	9.36932i		−	0.558927i	0.960156	−	0.279463i	$-0.0901565\pi$
$282$		0			0
$283$	−12.0000			−0.713326			−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000			−0.713326			−0.356663	+	0.934233i	$0.616086\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.50758			0.207046
$288$		0			0
$289$	−4.31534			−0.253844
$290$		0			0
$291$		0			0
$292$		0			0
$293$			12.6847i			0.741046i	0.928823	+	0.370523i	$0.120821\pi$
$293$			12.6847i			0.741046i	−0.928823	+	0.370523i	$0.879179\pi$
$294$		0			0
$295$	−15.6155			−0.909171
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.75379	+	11.1231i	−0.101424	+	0.643266i
$300$		0			0
$301$			4.19224i			0.241636i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			4.49242i			0.257235i
$306$		0			0
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	19.1231			1.08437			0.542186	−	0.840259i	$-0.317597\pi$
$311$	19.1231			1.08437			0.542186	+	0.840259i	$0.317597\pi$
$312$		0			0
$313$	4.43845			0.250876			0.125438	−	0.992101i	$-0.459966\pi$
$313$	4.43845			0.250876			0.125438	+	0.992101i	$0.459966\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.1231i		−	0.849398i	−0.905335	−	0.424699i	$-0.860380\pi$
$317$		−	15.1231i		−	0.849398i	0.905335	−	0.424699i	$-0.139620\pi$
$318$		0			0
$319$			26.2462i			1.46951i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	7.12311i		−	0.396340i
$324$		0			0
$325$	1.43845	−	9.12311i	0.0797907	−	0.506059i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.31534			−0.182781
$330$		0			0
$331$		−	0.246211i		−	0.0135330i	−0.999977	−	0.00676650i	$-0.997846\pi$
$331$		−	0.246211i		−	0.0135330i	0.999977	−	0.00676650i	$-0.00215386\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.2462			−0.778354
$336$		0			0
$337$	1.31534			0.0716512			0.0358256	−	0.999358i	$-0.488594\pi$
$337$	1.31534			0.0716512			0.0358256	+	0.999358i	$0.488594\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−26.2462			−1.42131
$342$		0			0
$343$		−	6.05398i		−	0.326884i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.56155			−0.0838285			−0.0419143	−	0.999121i	$-0.513346\pi$
$347$	−1.56155			−0.0838285			−0.0419143	+	0.999121i	$0.513346\pi$
$348$		0			0
$349$		−	10.9309i		−	0.585116i	−0.956248	−	0.292558i	$-0.905494\pi$
$349$		−	10.9309i		−	0.585116i	0.956248	−	0.292558i	$-0.0945065\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	7.61553i		−	0.405334i	−0.979248	−	0.202667i	$-0.935039\pi$
$353$		−	7.61553i		−	0.405334i	0.979248	−	0.202667i	$-0.0649608\pi$
$354$		0			0
$355$	−10.4384			−0.554015
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	13.1231i		−	0.692611i	−0.938122	−	0.346306i	$-0.887436\pi$
$359$		−	13.1231i		−	0.692611i	0.938122	−	0.346306i	$-0.112564\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.6307			0.765805
$366$		0			0
$367$	−28.4924			−1.48729			−0.743646	−	0.668573i	$-0.766905\pi$
$367$	−28.4924			−1.48729			−0.743646	+	0.668573i	$0.766905\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			5.36932i			0.278761i
$372$		0			0
$373$	−35.3693			−1.83135			−0.915677	−	0.401915i	$-0.868345\pi$
$373$	−35.3693			−1.83135			−0.915677	+	0.401915i	$0.868345\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.87689	−	18.2462i	0.148168	−	0.939728i
$378$		0			0
$379$		−	26.0000i		−	1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$		−	26.0000i		−	1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	29.8078i		−	1.52311i	−0.648103	−	0.761553i	$-0.724437\pi$
$383$		−	29.8078i		−	1.52311i	0.648103	−	0.761553i	$-0.275563\pi$
$384$		0			0
$385$			3.50758i			0.178763i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.7538			0.595941			0.297970	−	0.954575i	$-0.403690\pi$
$389$	11.7538			0.595941			0.297970	+	0.954575i	$0.403690\pi$
$390$		0			0
$391$	−11.1231			−0.562520
$392$		0			0
$393$		0			0
$394$		0			0
$395$			17.3693i			0.873945i
$396$		0			0
$397$		−	0.876894i		−	0.0440101i	−0.999758	−	0.0220050i	$-0.992995\pi$
$397$		−	0.876894i		−	0.0440101i	0.999758	−	0.0220050i	$-0.00700499\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			17.3693i			0.867382i	0.901062	+	0.433691i	$0.142789\pi$
$401$			17.3693i			0.867382i	−0.901062	+	0.433691i	$0.857211\pi$
$402$		0			0
$403$	18.2462	+	2.87689i	0.908909	+	0.143308i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−48.9848			−2.42809
$408$		0			0
$409$			25.3693i			1.25443i	0.778845	+	0.627216i	$0.215806\pi$
$409$			25.3693i			1.25443i	−0.778845	+	0.627216i	$0.784194\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.38447			0.215746
$414$		0			0
$415$	−12.8769			−0.632102
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−30.4384			−1.48702			−0.743508	−	0.668727i	$-0.766839\pi$
$419$	−30.4384			−1.48702			−0.743508	+	0.668727i	$0.766839\pi$
$420$		0			0
$421$		−	23.8078i		−	1.16032i	−0.814503	−	0.580160i	$-0.802990\pi$
$421$		−	23.8078i		−	1.16032i	0.814503	−	0.580160i	$-0.197010\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.12311			0.442536
$426$		0			0
$427$		−	1.26137i		−	0.0610418i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	37.4233i		−	1.80262i	−0.433178	−	0.901308i	$-0.642608\pi$
$431$		−	37.4233i		−	1.80262i	0.433178	−	0.901308i	$-0.357392\pi$
$432$		0			0
$433$	9.31534			0.447667			0.223833	−	0.974627i	$-0.428143\pi$
$433$	9.31534			0.447667			0.223833	+	0.974627i	$0.428143\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.24621i			0.298797i
$438$		0			0
$439$	37.8617			1.80704			0.903521	−	0.428544i	$-0.140973\pi$
$439$	37.8617			1.80704			0.903521	+	0.428544i	$0.140973\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.17708			−0.436016			−0.218008	−	0.975947i	$-0.569956\pi$
$443$	−9.17708			−0.436016			−0.218008	+	0.975947i	$0.569956\pi$
$444$		0			0
$445$	−4.87689			−0.231187
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.0000i		−	0.755087i	−0.925992	−	0.377543i	$-0.876769\pi$
$449$		−	16.0000i		−	0.755087i	0.925992	−	0.377543i	$-0.123231\pi$
$450$		0			0
$451$	−40.9848			−1.92990
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.384472	−	2.43845i	0.0180243	−	0.114316i
$456$		0			0
$457$		−	15.6155i		−	0.730464i	−0.930917	−	0.365232i	$-0.880990\pi$
$457$		−	15.6155i		−	0.730464i	0.930917	−	0.365232i	$-0.119010\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	9.17708i		−	0.427419i	−0.976897	−	0.213710i	$-0.931445\pi$
$461$		−	9.17708i		−	0.427419i	0.976897	−	0.213710i	$-0.0685546\pi$
$462$		0			0
$463$			1.61553i			0.0750800i	0.999295	+	0.0375400i	$0.0119522\pi$
$463$			1.61553i			0.0750800i	−0.999295	+	0.0375400i	$0.988048\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.4924			0.763178			0.381589	−	0.924332i	$-0.375377\pi$
$467$	16.4924			0.763178			0.381589	+	0.924332i	$0.375377\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	48.9848i		−	2.25233i
$474$		0			0
$475$		−	5.12311i		−	0.235064i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	29.8078i		−	1.36195i	−0.732306	−	0.680976i	$-0.761556\pi$
$479$		−	29.8078i		−	1.36195i	0.732306	−	0.680976i	$-0.238444\pi$
$480$		0			0
$481$	34.0540	+	5.36932i	1.55273	+	0.244820i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.75379			−0.442897
$486$		0			0
$487$			15.8617i			0.718764i	0.933191	+	0.359382i	$0.117012\pi$
$487$			15.8617i			0.718764i	−0.933191	+	0.359382i	$0.882988\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	31.8078			1.43546			0.717732	−	0.696319i	$-0.245180\pi$
$491$	31.8078			1.43546			0.717732	+	0.696319i	$0.245180\pi$
$492$		0			0
$493$	18.2462			0.821768
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.93087			0.131467
$498$		0			0
$499$			25.1231i			1.12466i	0.826911	+	0.562332i	$0.190096\pi$
$499$			25.1231i			1.12466i	−0.826911	+	0.562332i	$0.809904\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9.36932			0.417757			0.208879	−	0.977942i	$-0.433019\pi$
$503$	9.36932			0.417757			0.208879	+	0.977942i	$0.433019\pi$
$504$		0			0
$505$			17.7538i			0.790033i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	2.63068i		−	0.116603i	−0.998299	−	0.0583015i	$-0.981432\pi$
$509$		−	2.63068i		−	0.116603i	0.998299	−	0.0583015i	$-0.0185685\pi$
$510$		0			0
$511$	−4.10795			−0.181725
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	4.87689i		−	0.214902i
$516$		0			0
$517$	38.7386			1.70372
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.930870			−0.0407821			−0.0203911	−	0.999792i	$-0.506491\pi$
$521$	−0.930870			−0.0407821			−0.0203911	+	0.999792i	$0.506491\pi$
$522$		0			0
$523$	13.7538			0.601411			0.300706	−	0.953717i	$-0.402778\pi$
$523$	13.7538			0.601411			0.300706	+	0.953717i	$0.402778\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			18.2462i			0.794817i
$528$		0			0
$529$	−13.2462			−0.575922
$530$		0			0
$531$		0			0
$532$		0			0
$533$	28.4924	+	4.49242i	1.23414	+	0.194588i
$534$		0			0
$535$		−	6.24621i		−	0.270047i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			34.8769i			1.50225i
$540$		0			0
$541$			25.1771i			1.08245i	0.840879	+	0.541224i	$0.182039\pi$
$541$			25.1771i			1.08245i	−0.840879	+	0.541224i	$0.817961\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.17708			0.221762
$546$		0			0
$547$	−23.8078			−1.01795			−0.508973	−	0.860782i	$-0.669975\pi$
$547$	−23.8078			−1.01795			−0.508973	+	0.860782i	$0.669975\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.2462i		−	0.436503i
$552$		0			0
$553$		−	4.87689i		−	0.207387i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	18.9309i		−	0.802127i	−0.916050	−	0.401063i	$-0.868641\pi$
$557$		−	18.9309i		−	0.802127i	0.916050	−	0.401063i	$-0.131359\pi$
$558$		0			0
$559$	−5.36932	+	34.0540i	−0.227098	+	1.44033i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−26.9309			−1.13500			−0.567500	−	0.823373i	$-0.692089\pi$
$563$	−26.9309			−1.13500			−0.567500	+	0.823373i	$0.692089\pi$
$564$		0			0
$565$		−	15.6155i		−	0.656950i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.19224			0.0919033			0.0459517	−	0.998944i	$-0.485368\pi$
$569$	2.19224			0.0919033			0.0459517	+	0.998944i	$0.485368\pi$
$570$		0			0
$571$	14.0540			0.588141			0.294070	−	0.955784i	$-0.404990\pi$
$571$	14.0540			0.588141			0.294070	+	0.955784i	$0.404990\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.00000			−0.333623
$576$		0			0
$577$		−	29.8617i		−	1.24316i	−0.783351	−	0.621580i	$-0.786491\pi$
$577$		−	29.8617i		−	1.24316i	0.783351	−	0.621580i	$-0.213509\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.61553			0.149997
$582$		0			0
$583$		−	62.7386i		−	2.59837i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	9.50758i		−	0.392420i	−0.980562	−	0.196210i	$-0.937137\pi$
$587$		−	9.50758i		−	0.392420i	0.980562	−	0.196210i	$-0.0628634\pi$
$588$		0			0
$589$	10.2462			0.422188
$590$		0			0
$591$		0			0
$592$		0			0
$593$			25.7538i			1.05758i	0.848752	+	0.528791i	$0.177354\pi$
$593$			25.7538i			1.05758i	−0.848752	+	0.528791i	$0.822646\pi$
$594$		0			0
$595$	2.43845			0.0999666
$596$		0			0
$597$		0			0
$598$		0			0
$599$	41.3693			1.69030			0.845152	−	0.534526i	$-0.179510\pi$
$599$	41.3693			1.69030			0.845152	+	0.534526i	$0.179510\pi$
$600$		0			0
$601$	21.8078			0.889557			0.444779	−	0.895641i	$-0.353282\pi$
$601$	21.8078			0.889557			0.444779	+	0.895641i	$0.353282\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	23.8078i		−	0.967923i
$606$		0			0
$607$	7.61553			0.309105			0.154552	−	0.987985i	$-0.450606\pi$
$607$	7.61553			0.309105			0.154552	+	0.987985i	$0.450606\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−26.9309	−	4.24621i	−1.08951	−	0.171783i
$612$		0			0
$613$		−	18.6307i		−	0.752486i	−0.926521	−	0.376243i	$-0.877216\pi$
$613$		−	18.6307i		−	0.752486i	0.926521	−	0.376243i	$-0.122784\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.75379i			0.0706049i	0.999377	+	0.0353024i	$0.0112395\pi$
$617$			1.75379i			0.0706049i	−0.999377	+	0.0353024i	$0.988761\pi$
$618$		0			0
$619$		−	17.6155i		−	0.708028i	−0.935240	−	0.354014i	$-0.884817\pi$
$619$		−	17.6155i		−	0.708028i	0.935240	−	0.354014i	$-0.115183\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.36932			0.0548605
$624$		0			0
$625$	−5.63068			−0.225227
$626$		0			0
$627$		0			0
$628$		0			0
$629$			34.0540i			1.35782i
$630$		0			0
$631$		−	11.5616i		−	0.460258i	−0.973160	−	0.230129i	$-0.926085\pi$
$631$		−	11.5616i		−	0.460258i	0.973160	−	0.230129i	$-0.0739148\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			7.61553i			0.302213i
$636$		0			0
$637$	3.82292	−	24.2462i	0.151470	−	0.960670i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.75379			−0.148266			−0.0741329	−	0.997248i	$-0.523619\pi$
$641$	−3.75379			−0.148266			−0.0741329	+	0.997248i	$0.523619\pi$
$642$		0			0
$643$			25.1231i			0.990759i	0.868677	+	0.495379i	$0.164971\pi$
$643$			25.1231i			0.990759i	−0.868677	+	0.495379i	$0.835029\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.12311			0.122782			0.0613910	−	0.998114i	$-0.480446\pi$
$647$	3.12311			0.122782			0.0613910	+	0.998114i	$0.480446\pi$
$648$		0			0
$649$	−51.2311			−2.01099
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.6155			−0.532817			−0.266408	−	0.963860i	$-0.585837\pi$
$653$	−13.6155			−0.532817			−0.266408	+	0.963860i	$0.585837\pi$
$654$		0			0
$655$		−	5.17708i		−	0.202285i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.73863			0.262500			0.131250	−	0.991349i	$-0.458101\pi$
$659$	6.73863			0.262500			0.131250	+	0.991349i	$0.458101\pi$
$660$		0			0
$661$			32.8769i			1.27876i	0.768890	+	0.639381i	$0.220810\pi$
$661$			32.8769i			1.27876i	−0.768890	+	0.639381i	$0.779190\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.36932i		−	0.0530998i
$666$		0			0
$667$	−16.0000			−0.619522
$668$		0			0
$669$		0			0
$670$		0			0
$671$			14.7386i			0.568979i
$672$		0			0
$673$	−22.3002			−0.859609			−0.429805	−	0.902922i	$-0.641418\pi$
$673$	−22.3002			−0.859609			−0.429805	+	0.902922i	$0.641418\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.7386			1.10452			0.552258	−	0.833673i	$-0.313766\pi$
$677$	28.7386			1.10452			0.552258	+	0.833673i	$0.313766\pi$
$678$		0			0
$679$	2.73863			0.105099
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	12.2462i		−	0.468588i	−0.972166	−	0.234294i	$-0.924722\pi$
$683$		−	12.2462i		−	0.468588i	0.972166	−	0.234294i	$-0.0752779\pi$
$684$		0			0
$685$	−4.87689			−0.186337
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.87689	+	43.6155i	−0.261989	+	1.66162i
$690$		0			0
$691$			27.8617i			1.05991i	0.848026	+	0.529955i	$0.177791\pi$
$691$			27.8617i			1.05991i	−0.848026	+	0.529955i	$0.822209\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			12.1922i			0.462478i
$696$		0			0
$697$			28.4924i			1.07923i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−37.2311			−1.40620			−0.703099	−	0.711092i	$-0.748201\pi$
$701$	−37.2311			−1.40620			−0.703099	+	0.711092i	$0.748201\pi$
$702$		0			0
$703$	19.1231			0.721242
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	4.98485i		−	0.187474i
$708$		0			0
$709$			7.12311i			0.267514i	0.991014	+	0.133757i	$0.0427041\pi$
$709$			7.12311i			0.267514i	−0.991014	+	0.133757i	$0.957296\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	−4.49242	+	28.4924i	−0.168007	+	1.06556i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0000			−0.596699			−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000			−0.596699			−0.298350	+	0.954457i	$0.596436\pi$
$720$		0			0
$721$			1.36932i			0.0509960i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	13.1231			0.487380
$726$		0			0
$727$	−39.2311			−1.45500			−0.727500	−	0.686108i	$-0.759318\pi$
$727$	−39.2311			−1.45500			−0.727500	+	0.686108i	$0.759318\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.0540			−1.25953
$732$		0			0
$733$			4.30019i			0.158831i	0.996842	+	0.0794155i	$0.0253054\pi$
$733$			4.30019i			0.158831i	−0.996842	+	0.0794155i	$0.974695\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−46.7386			−1.72164
$738$		0			0
$739$		−	15.8617i		−	0.583484i	−0.956497	−	0.291742i	$-0.905765\pi$
$739$		−	15.8617i		−	0.583484i	0.956497	−	0.291742i	$-0.0942348\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7.94602i		−	0.291511i	−0.989321	−	0.145756i	$-0.953439\pi$
$743$		−	7.94602i		−	0.291511i	0.989321	−	0.145756i	$-0.0465613\pi$
$744$		0			0
$745$	11.1231			0.407519
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1.75379i			0.0640820i
$750$		0			0
$751$	3.50758			0.127993			0.0639967	−	0.997950i	$-0.479615\pi$
$751$	3.50758			0.127993			0.0639967	+	0.997950i	$0.479615\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.0540			0.657051
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	44.8769i		−	1.62679i	−0.581714	−	0.813393i	$-0.697618\pi$
$761$		−	44.8769i		−	1.62679i	0.581714	−	0.813393i	$-0.302382\pi$
$762$		0			0
$763$	−1.45360			−0.0526239
$764$		0			0
$765$		0			0
$766$		0			0
$767$	35.6155	+	5.61553i	1.28600	+	0.202765i
$768$		0			0
$769$			28.4924i			1.02746i	0.857951	+	0.513732i	$0.171737\pi$
$769$			28.4924i			1.02746i	−0.857951	+	0.513732i	$0.828263\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			14.0540i			0.505486i	0.967533	+	0.252743i	$0.0813328\pi$
$773$			14.0540i			0.505486i	−0.967533	+	0.252743i	$0.918667\pi$
$774$		0			0
$775$			13.1231i			0.471396i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.0000			0.573259
$780$		0			0
$781$	−34.2462			−1.22543
$782$		0			0
$783$		0			0
$784$		0			0
$785$			19.1231i			0.682533i
$786$		0			0
$787$		−	3.36932i		−	0.120103i	−0.998195	−	0.0600516i	$-0.980873\pi$
$787$		−	3.36932i		−	0.120103i	0.998195	−	0.0600516i	$-0.0191265\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.38447i			0.155894i
$792$		0			0
$793$	1.61553	−	10.2462i	0.0573691	−	0.363854i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.2462			−0.433783			−0.216892	−	0.976196i	$-0.569592\pi$
$797$	−12.2462			−0.433783			−0.216892	+	0.976196i	$0.569592\pi$
$798$		0			0
$799$		−	26.9309i		−	0.952746i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	48.0000			1.69388
$804$		0			0
$805$	−2.13826			−0.0753638
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−42.6847			−1.50071			−0.750356	−	0.661034i	$-0.770118\pi$
$809$	−42.6847			−1.50071			−0.750356	+	0.661034i	$0.770118\pi$
$810$		0			0
$811$			23.3693i			0.820608i	0.911949	+	0.410304i	$0.134577\pi$
$811$			23.3693i			0.820608i	−0.911949	+	0.410304i	$0.865423\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−31.6155			−1.10744
$816$		0			0
$817$			19.1231i			0.669033i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			42.5464i			1.48488i	0.669913	+	0.742440i	$0.266331\pi$
$821$			42.5464i			1.48488i	−0.669913	+	0.742440i	$0.733669\pi$
$822$		0			0
$823$	50.3542			1.75524			0.877618	−	0.479361i	$-0.159131\pi$
$823$	50.3542			1.75524			0.877618	+	0.479361i	$0.159131\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$		0			0
$829$	−6.87689			−0.238844			−0.119422	−	0.992844i	$-0.538104\pi$
$829$	−6.87689			−0.238844			−0.119422	+	0.992844i	$0.538104\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24.2462			0.840081
$834$		0			0
$835$	10.7386			0.371626
$836$		0			0
$837$		0			0
$838$		0			0
$839$			43.8617i			1.51428i	0.653255	+	0.757138i	$0.273403\pi$
$839$			43.8617i			1.51428i	−0.653255	+	0.757138i	$0.726597\pi$
$840$		0			0
$841$	−2.75379			−0.0949582
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.24621	−	19.3153i	0.214876	−	0.664468i
$846$		0			0
$847$			6.68466i			0.229688i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	29.8617i		−	1.02365i
$852$		0			0
$853$			4.68466i			0.160400i	0.996779	+	0.0801998i	$0.0255558\pi$
$853$			4.68466i			0.160400i	−0.996779	+	0.0801998i	$0.974444\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	50.4924			1.72479			0.862394	−	0.506237i	$-0.168964\pi$
$857$	50.4924			1.72479			0.862394	+	0.506237i	$0.168964\pi$
$858$		0			0
$859$	10.2462			0.349596			0.174798	−	0.984604i	$-0.444073\pi$
$859$	10.2462			0.349596			0.174798	+	0.984604i	$0.444073\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			48.0540i			1.63578i	0.575377	+	0.817888i	$0.304855\pi$
$863$			48.0540i			1.63578i	−0.575377	+	0.817888i	$0.695145\pi$
$864$		0			0
$865$		−	6.63068i		−	0.225450i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			56.9848i			1.93308i
$870$		0			0
$871$	32.4924	+	5.12311i	1.10096	+	0.173590i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5.17708			0.175017
$876$		0			0
$877$		−	7.42329i		−	0.250667i	−0.992115	−	0.125333i	$-0.960000\pi$
$877$		−	7.42329i		−	0.250667i	0.992115	−	0.125333i	$-0.0400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.7926			1.44172			0.720860	−	0.693081i	$-0.243747\pi$
$881$	42.7926			1.44172			0.720860	+	0.693081i	$0.243747\pi$
$882$		0			0
$883$	31.8078			1.07042			0.535208	−	0.844720i	$-0.320233\pi$
$883$	31.8078			1.07042			0.535208	+	0.844720i	$0.320233\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.26137			−0.176659			−0.0883297	−	0.996091i	$-0.528153\pi$
$887$	−5.26137			−0.176659			−0.0883297	+	0.996091i	$0.528153\pi$
$888$		0			0
$889$		−	2.13826i		−	0.0717150i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.1231			−0.506075
$894$		0			0
$895$			27.4233i			0.916660i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			26.2462i			0.875360i
$900$		0			0
$901$	−43.6155			−1.45304
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	12.8769i		−	0.428042i
$906$		0			0
$907$	22.4384			0.745056			0.372528	−	0.928021i	$-0.378491\pi$
$907$	22.4384			0.745056			0.372528	+	0.928021i	$0.378491\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.7538			0.853261			0.426631	−	0.904426i	$-0.359700\pi$
$911$	25.7538			0.853261			0.426631	+	0.904426i	$0.359700\pi$
$912$		0			0
$913$	−42.2462			−1.39815
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.45360i			0.0480022i
$918$		0			0
$919$	−1.75379			−0.0578522			−0.0289261	−	0.999582i	$-0.509209\pi$
$919$	−1.75379			−0.0578522			−0.0289261	+	0.999582i	$0.509209\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	23.8078	+	3.75379i	0.783642	+	0.123557i
$924$		0			0
$925$			24.4924i			0.805306i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			18.7386i			0.614795i	0.951581	+	0.307397i	$0.0994581\pi$
$929$			18.7386i			0.614795i	−0.951581	+	0.307397i	$0.900542\pi$
$930$		0			0
$931$		−	13.6155i		−	0.446231i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−28.4924			−0.931802
$936$		0			0
$937$	19.7538			0.645328			0.322664	−	0.946514i	$-0.395422\pi$
$937$	19.7538			0.645328			0.322664	+	0.946514i	$0.395422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	42.5464i		−	1.38697i	−0.720470	−	0.693486i	$-0.756074\pi$
$941$		−	42.5464i		−	1.38697i	0.720470	−	0.693486i	$-0.243926\pi$
$942$		0			0
$943$		−	24.9848i		−	0.813618i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	15.3693i		−	0.499436i	−0.968319	−	0.249718i	$-0.919662\pi$
$947$		−	15.3693i		−	0.499436i	0.968319	−	0.249718i	$-0.0803379\pi$
$948$		0			0
$949$	−33.3693	−	5.26137i	−1.08321	−	0.170791i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.0540			−0.649612			−0.324806	−	0.945781i	$-0.605299\pi$
$953$	−20.0540			−0.649612			−0.324806	+	0.945781i	$0.605299\pi$
$954$		0			0
$955$		−	34.7386i		−	1.12412i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.36932			0.0442175
$960$		0			0
$961$	4.75379			0.153348
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−36.8769			−1.18711
$966$		0			0
$967$			15.5616i			0.500426i	0.968191	+	0.250213i	$0.0805006\pi$
$967$			15.5616i			0.500426i	−0.968191	+	0.250213i	$0.919499\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−57.1771			−1.83490			−0.917450	−	0.397851i	$-0.869756\pi$
$971$	−57.1771			−1.83490			−0.917450	+	0.397851i	$0.869756\pi$
$972$		0			0
$973$		−	3.42329i		−	0.109746i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			47.2311i			1.51106i	0.655117	+	0.755528i	$0.272619\pi$
$977$			47.2311i			1.51106i	−0.655117	+	0.755528i	$0.727381\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		0			0
$981$		0			0
$982$		0			0
$983$			10.1922i			0.325082i	0.986702	+	0.162541i	$0.0519690\pi$
$983$			10.1922i			0.325082i	−0.986702	+	0.162541i	$0.948031\pi$
$984$		0			0
$985$	14.9309			0.475737
$986$		0			0
$987$		0			0
$988$		0			0
$989$	29.8617			0.949548
$990$		0			0
$991$	7.61553			0.241915			0.120958	−	0.992658i	$-0.461403\pi$
$991$	7.61553			0.241915			0.120958	+	0.992658i	$0.461403\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			22.2462i			0.705252i
$996$		0			0
$997$	−1.61553			−0.0511643			−0.0255821	−	0.999673i	$-0.508144\pi$
$997$	−1.61553			−0.0511643			−0.0255821	+	0.999673i	$0.508144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.c.j.1585.3		4
3.2	odd	2		208.2.f.b.129.1		4
4.3	odd	2		936.2.c.d.649.3		4
12.11	even	2		104.2.f.a.25.3	✓	4
13.12	even	2	inner	1872.2.c.j.1585.2		4
24.5	odd	2		832.2.f.f.129.4		4
24.11	even	2		832.2.f.i.129.2		4
39.5	even	4		2704.2.a.t.1.1		2
39.8	even	4		2704.2.a.s.1.1		2
39.38	odd	2		208.2.f.b.129.2		4
52.51	odd	2		936.2.c.d.649.2		4
60.23	odd	4		2600.2.f.b.649.3		4
60.47	odd	4		2600.2.f.a.649.2		4
60.59	even	2		2600.2.k.a.2001.1		4
156.11	odd	12		1352.2.i.g.529.1		4
156.23	even	6		1352.2.o.e.1161.2		8
156.35	even	6		1352.2.o.e.361.1		8
156.47	odd	4		1352.2.a.d.1.2		2
156.59	odd	12		1352.2.i.g.1329.1		4
156.71	odd	12		1352.2.i.h.1329.1		4
156.83	odd	4		1352.2.a.e.1.2		2
156.95	even	6		1352.2.o.e.361.2		8
156.107	even	6		1352.2.o.e.1161.1		8
156.119	odd	12		1352.2.i.h.529.1		4
156.155	even	2		104.2.f.a.25.4	yes	4
312.77	odd	2		832.2.f.f.129.3		4
312.155	even	2		832.2.f.i.129.1		4
780.467	odd	4		2600.2.f.b.649.2		4
780.623	odd	4		2600.2.f.a.649.3		4
780.779	even	2		2600.2.k.a.2001.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.f.a.25.3	✓	4	12.11	even	2
104.2.f.a.25.4	yes	4	156.155	even	2
208.2.f.b.129.1		4	3.2	odd	2
208.2.f.b.129.2		4	39.38	odd	2
832.2.f.f.129.3		4	312.77	odd	2
832.2.f.f.129.4		4	24.5	odd	2
832.2.f.i.129.1		4	312.155	even	2
832.2.f.i.129.2		4	24.11	even	2
936.2.c.d.649.2		4	52.51	odd	2
936.2.c.d.649.3		4	4.3	odd	2
1352.2.a.d.1.2		2	156.47	odd	4
1352.2.a.e.1.2		2	156.83	odd	4
1352.2.i.g.529.1		4	156.11	odd	12
1352.2.i.g.1329.1		4	156.59	odd	12
1352.2.i.h.529.1		4	156.119	odd	12
1352.2.i.h.1329.1		4	156.71	odd	12
1352.2.o.e.361.1		8	156.35	even	6
1352.2.o.e.361.2		8	156.95	even	6
1352.2.o.e.1161.1		8	156.107	even	6
1352.2.o.e.1161.2		8	156.23	even	6
1872.2.c.j.1585.2		4	13.12	even	2	inner
1872.2.c.j.1585.3		4	1.1	even	1	trivial
2600.2.f.a.649.2		4	60.47	odd	4
2600.2.f.a.649.3		4	780.623	odd	4
2600.2.f.b.649.2		4	780.467	odd	4
2600.2.f.b.649.3		4	60.23	odd	4
2600.2.k.a.2001.1		4	60.59	even	2
2600.2.k.a.2001.2		4	780.779	even	2
2704.2.a.s.1.1		2	39.8	even	4
2704.2.a.t.1.1		2	39.5	even	4

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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{5} -0.438447i q^{7} +O(q^{10})\)	\(q+1.56155i q^{5} -0.438447i q^{7} +5.12311i q^{11} +(0.561553 - 3.56155i) q^{13} +3.56155 q^{17} -2.00000i q^{19} -3.12311 q^{23} +2.56155 q^{25} +5.12311 q^{29} +5.12311i q^{31} +0.684658 q^{35} +9.56155i q^{37} +8.00000i q^{41} -9.56155 q^{43} -7.56155i q^{47} +6.80776 q^{49} -12.2462 q^{53} -8.00000 q^{55} +10.0000i q^{59} +2.87689 q^{61} +(5.56155 + 0.876894i) q^{65} +9.12311i q^{67} +6.68466i q^{71} -9.36932i q^{73} +2.24621 q^{77} +11.1231 q^{79} +8.24621i q^{83} +5.56155i q^{85} +3.12311i q^{89} +(-1.56155 - 0.246211i) q^{91} +3.12311 q^{95} +6.24621i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 6 q^{13} + 6 q^{17} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 22 q^{35} - 30 q^{43} - 14 q^{49} - 16 q^{53} - 32 q^{55} + 28 q^{61} + 14 q^{65} - 24 q^{77} + 28 q^{79} + 2 q^{91} - 4 q^{95}+O(q^{100}) \) 4 * q - 6 * q^13 + 6 * q^17 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 22 * q^35 - 30 * q^43 - 14 * q^49 - 16 * q^53 - 32 * q^55 + 28 * q^61 + 14 * q^65 - 24 * q^77 + 28 * q^79 + 2 * q^91 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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