LMFDB - Embedded newform 2700.2.j.b.1457.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(593,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.j (of order $4$, degree $2$, 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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2700.1457
Dual form			2700.2.j.b.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+3.00000i q^{11} +(-1.22474 + 1.22474i) q^{13} +2.00000i q^{19} +(-3.67423 - 3.67423i) q^{23} -6.00000 q^{29} -2.00000 q^{31} +(1.22474 + 1.22474i) q^{37} +6.00000i q^{41} +(2.44949 - 2.44949i) q^{43} +(-3.67423 + 3.67423i) q^{47} -7.00000i q^{49} +(-7.34847 - 7.34847i) q^{53} -3.00000 q^{59} -1.00000 q^{61} +(-4.89898 - 4.89898i) q^{67} +15.0000i q^{71} +(4.89898 - 4.89898i) q^{73} +4.00000i q^{79} +(-7.34847 - 7.34847i) q^{83} -12.0000 q^{89} +(3.67423 + 3.67423i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{29} - 8 q^{31} - 12 q^{59} - 4 q^{61} - 48 q^{89}+O(q^{100}) $ 4 * q - 24 * q^29 - 8 * q^31 - 12 * q^59 - 4 * q^61 - 48 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$	−1.22474	+	1.22474i	−0.339683	+	0.339683i	−0.856248	−	0.516565i	$-0.827210\pi$
$13$	−1.22474	+	1.22474i	−0.339683	+	0.339683i	0.516565	+	0.856248i	$0.327210\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.67423	−	3.67423i	−0.766131	−	0.766131i	0.211292	−	0.977423i	$-0.432233\pi$
$23$	−3.67423	−	3.67423i	−0.766131	−	0.766131i	−0.977423	+	0.211292i	$0.932233\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.22474	+	1.22474i	0.201347	+	0.201347i	0.800577	−	0.599230i	$-0.204527\pi$
$37$	1.22474	+	1.22474i	0.201347	+	0.201347i	−0.599230	+	0.800577i	$0.704527\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	−0.495222	−	0.868766i	$-0.664913\pi$
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	0.868766	+	0.495222i	$0.164913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.67423	+	3.67423i	−0.535942	+	0.535942i	−0.922335	−	0.386392i	$-0.873721\pi$
$47$	−3.67423	+	3.67423i	−0.535942	+	0.535942i	0.386392	+	0.922335i	$0.373721\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.34847	−	7.34847i	−1.00939	−	1.00939i	−0.999955	−	0.00943438i	$-0.996997\pi$
$53$	−7.34847	−	7.34847i	−1.00939	−	1.00939i	−0.00943438	−	0.999955i	$-0.503003\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000			−0.390567			−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000			−0.390567			−0.195283	+	0.980747i	$0.562563\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.89898	−	4.89898i	−0.598506	−	0.598506i	0.341409	−	0.939915i	$-0.389096\pi$
$67$	−4.89898	−	4.89898i	−0.598506	−	0.598506i	−0.939915	+	0.341409i	$0.889096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			15.0000i			1.78017i	0.455792	+	0.890086i	$0.349356\pi$
$71$			15.0000i			1.78017i	−0.455792	+	0.890086i	$0.650644\pi$
$72$		0			0
$73$	4.89898	−	4.89898i	0.573382	−	0.573382i	−0.359690	−	0.933072i	$-0.617117\pi$
$73$	4.89898	−	4.89898i	0.573382	−	0.573382i	0.933072	+	0.359690i	$0.117117\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.34847	−	7.34847i	−0.806599	−	0.806599i	0.177518	−	0.984118i	$-0.443193\pi$
$83$	−7.34847	−	7.34847i	−0.806599	−	0.806599i	−0.984118	+	0.177518i	$0.943193\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.0000			−1.27200			−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000			−1.27200			−0.635999	+	0.771690i	$0.719412\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.67423	+	3.67423i	0.373062	+	0.373062i	0.868591	−	0.495529i	$-0.165026\pi$
$97$	3.67423	+	3.67423i	0.373062	+	0.373062i	−0.495529	+	0.868591i	$0.665026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12.0000i			1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$			12.0000i			1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$		0			0
$103$	−12.2474	+	12.2474i	−1.20678	+	1.20678i	−0.234712	+	0.972065i	$0.575415\pi$
$103$	−12.2474	+	12.2474i	−1.20678	+	1.20678i	−0.972065	+	0.234712i	$0.924585\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.67423	+	3.67423i	−0.355202	+	0.355202i	−0.862041	−	0.506839i	$-0.830814\pi$
$107$	−3.67423	+	3.67423i	−0.355202	+	0.355202i	0.506839	+	0.862041i	$0.330814\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.34847	−	7.34847i	−0.691286	−	0.691286i	0.271229	−	0.962515i	$-0.412570\pi$
$113$	−7.34847	−	7.34847i	−0.691286	−	0.691286i	−0.962515	+	0.271229i	$0.912570\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	0.953491	−	0.301420i	$-0.0974607\pi$
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	−0.301420	+	0.953491i	$0.597461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			15.0000i			1.31056i	0.755388	+	0.655278i	$0.227449\pi$
$131$			15.0000i			1.31056i	−0.755388	+	0.655278i	$0.772551\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.34847	−	7.34847i	0.627822	−	0.627822i	−0.319698	−	0.947520i	$-0.603581\pi$
$137$	7.34847	−	7.34847i	0.627822	−	0.627822i	0.947520	+	0.319698i	$0.103581\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.67423	−	3.67423i	−0.307255	−	0.307255i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−9.79796	−	9.79796i	−0.781962	−	0.781962i	0.198199	−	0.980162i	$-0.436491\pi$
$157$	−9.79796	−	9.79796i	−0.781962	−	0.781962i	−0.980162	+	0.198199i	$0.936491\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	12.2474	−	12.2474i	0.959294	−	0.959294i	−0.0399091	−	0.999203i	$-0.512707\pi$
$163$	12.2474	−	12.2474i	0.959294	−	0.959294i	0.999203	+	0.0399091i	$0.0127068\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.0227	+	11.0227i	−0.852962	+	0.852962i	−0.990497	−	0.137535i	$-0.956082\pi$
$167$	−11.0227	+	11.0227i	−0.852962	+	0.852962i	0.137535	+	0.990497i	$0.456082\pi$
$168$		0			0
$169$			10.0000i			0.769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.6969	−	14.6969i	−1.11739	−	1.11739i	−0.992123	−	0.125264i	$-0.960022\pi$
$173$	−14.6969	−	14.6969i	−1.11739	−	1.11739i	−0.125264	−	0.992123i	$-0.539978\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000			0.224231			0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000			0.224231			0.112115	+	0.993695i	$0.464237\pi$
$180$		0			0
$181$	1.00000			0.0743294			0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000			0.0743294			0.0371647	+	0.999309i	$0.488167\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			12.0000i			0.868290i	0.900843	+	0.434145i	$0.142949\pi$
$191$			12.0000i			0.868290i	−0.900843	+	0.434145i	$0.857051\pi$
$192$		0			0
$193$	−14.6969	+	14.6969i	−1.05791	+	1.05791i	−0.0596919	+	0.998217i	$0.519012\pi$
$193$	−14.6969	+	14.6969i	−1.05791	+	1.05791i	−0.998217	+	0.0596919i	$0.980988\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			10.0000i			0.708881i	0.935079	+	0.354441i	$0.115329\pi$
$199$			10.0000i			0.708881i	−0.935079	+	0.354441i	$0.884671\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	12.2474	−	12.2474i	0.820150	−	0.820150i	−0.165979	−	0.986129i	$-0.553079\pi$
$223$	12.2474	−	12.2474i	0.820150	−	0.820150i	0.986129	+	0.165979i	$0.0530785\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.0227	+	11.0227i	−0.731603	+	0.731603i	−0.970937	−	0.239335i	$-0.923071\pi$
$227$	−11.0227	+	11.0227i	−0.731603	+	0.731603i	0.239335	+	0.970937i	$0.423071\pi$
$228$		0			0
$229$		−	25.0000i		−	1.65205i	−0.563636	−	0.826023i	$-0.690598\pi$
$229$		−	25.0000i		−	1.65205i	0.563636	−	0.826023i	$-0.309402\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.00000			0.582162			0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000			0.582162			0.291081	+	0.956698i	$0.405985\pi$
$240$		0			0
$241$	−13.0000			−0.837404			−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000			−0.837404			−0.418702	+	0.908124i	$0.637515\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.44949	−	2.44949i	−0.155857	−	0.155857i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			9.00000i			0.568075i	0.958813	+	0.284037i	$0.0916740\pi$
$251$			9.00000i			0.568075i	−0.958813	+	0.284037i	$0.908326\pi$
$252$		0			0
$253$	11.0227	−	11.0227i	0.692991	−	0.692991i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.6969	−	14.6969i	0.916770	−	0.916770i	−0.0800232	−	0.996793i	$-0.525499\pi$
$257$	14.6969	−	14.6969i	0.916770	−	0.916770i	0.996793	+	0.0800232i	$0.0254994\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.67423	+	3.67423i	0.226563	+	0.226563i	0.811255	−	0.584692i	$-0.198785\pi$
$263$	3.67423	+	3.67423i	0.226563	+	0.226563i	−0.584692	+	0.811255i	$0.698785\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.0000			−0.731653			−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000			−0.731653			−0.365826	+	0.930683i	$0.619214\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	9.79796	+	9.79796i	0.588702	+	0.588702i	0.937280	−	0.348578i	$-0.113335\pi$
$277$	9.79796	+	9.79796i	0.588702	+	0.588702i	−0.348578	+	0.937280i	$0.613335\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	−17.1464	+	17.1464i	−1.01925	+	1.01925i	−0.0194383	+	0.999811i	$0.506188\pi$
$283$	−17.1464	+	17.1464i	−1.01925	+	1.01925i	−0.999811	+	0.0194383i	$0.993812\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.00000			0.520483
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.2474	+	12.2474i	0.698999	+	0.698999i	0.964195	−	0.265196i	$-0.0854366\pi$
$307$	12.2474	+	12.2474i	0.698999	+	0.698999i	−0.265196	+	0.964195i	$0.585437\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.00000i			0.170114i	0.996376	+	0.0850572i	$0.0271073\pi$
$311$			3.00000i			0.170114i	−0.996376	+	0.0850572i	$0.972893\pi$
$312$		0			0
$313$	4.89898	−	4.89898i	0.276907	−	0.276907i	−0.554966	−	0.831873i	$-0.687269\pi$
$313$	4.89898	−	4.89898i	0.276907	−	0.276907i	0.831873	+	0.554966i	$0.187269\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.34847	−	7.34847i	0.412731	−	0.412731i	−0.469958	−	0.882689i	$-0.655731\pi$
$317$	7.34847	−	7.34847i	0.412731	−	0.412731i	0.882689	+	0.469958i	$0.155731\pi$
$318$		0			0
$319$		−	18.0000i		−	1.00781i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	14.0000			0.769510			0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000			0.769510			0.384755	+	0.923019i	$0.374286\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−24.4949	−	24.4949i	−1.33432	−	1.33432i	−0.901460	−	0.432862i	$-0.857504\pi$
$337$	−24.4949	−	24.4949i	−1.33432	−	1.33432i	−0.432862	−	0.901460i	$-0.642496\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.00000i		−	0.324918i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.34847	+	7.34847i	−0.394486	+	0.394486i	−0.876283	−	0.481797i	$-0.839984\pi$
$347$	−7.34847	+	7.34847i	−0.394486	+	0.394486i	0.481797	+	0.876283i	$0.339984\pi$
$348$		0			0
$349$			14.0000i			0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$			14.0000i			0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.0454	+	22.0454i	1.17336	+	1.17336i	0.981404	+	0.191955i	$0.0614827\pi$
$353$	22.0454	+	22.0454i	1.17336	+	1.17336i	0.191955	+	0.981404i	$0.438517\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	21.0000			1.10834			0.554169	−	0.832404i	$-0.313036\pi$
$359$	21.0000			1.10834			0.554169	+	0.832404i	$0.313036\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9.79796	−	9.79796i	−0.511449	−	0.511449i	0.403521	−	0.914970i	$-0.367786\pi$
$367$	−9.79796	−	9.79796i	−0.511449	−	0.511449i	−0.914970	+	0.403521i	$0.867786\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−14.6969	+	14.6969i	−0.760979	+	0.760979i	−0.976499	−	0.215521i	$-0.930855\pi$
$373$	−14.6969	+	14.6969i	−0.760979	+	0.760979i	0.215521	+	0.976499i	$0.430855\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.34847	−	7.34847i	0.378465	−	0.378465i
$378$		0			0
$379$			26.0000i			1.33553i	0.744372	+	0.667765i	$0.232749\pi$
$379$			26.0000i			1.33553i	−0.744372	+	0.667765i	$0.767251\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.67423	+	3.67423i	0.187745	+	0.187745i	0.794720	−	0.606976i	$-0.207617\pi$
$383$	3.67423	+	3.67423i	0.187745	+	0.187745i	−0.606976	+	0.794720i	$0.707617\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.4722	+	13.4722i	0.676150	+	0.676150i	0.959127	−	0.282977i	$-0.0913219\pi$
$397$	13.4722	+	13.4722i	0.676150	+	0.676150i	−0.282977	+	0.959127i	$0.591322\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	36.0000i		−	1.79775i	−0.438201	−	0.898877i	$-0.644384\pi$
$401$		−	36.0000i		−	1.79775i	0.438201	−	0.898877i	$-0.355616\pi$
$402$		0			0
$403$	2.44949	−	2.44949i	0.122018	−	0.122018i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.67423	+	3.67423i	−0.182125	+	0.182125i
$408$		0			0
$409$		−	25.0000i		−	1.23617i	−0.786111	−	0.618085i	$-0.787909\pi$
$409$		−	25.0000i		−	1.23617i	0.786111	−	0.618085i	$-0.212091\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$	−5.00000			−0.243685			−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000			−0.243685			−0.121843	+	0.992549i	$0.538880\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	21.0000i		−	1.01153i	−0.862670	−	0.505767i	$-0.831209\pi$
$431$		−	21.0000i		−	1.01153i	0.862670	−	0.505767i	$-0.168791\pi$
$432$		0			0
$433$	15.9217	−	15.9217i	0.765147	−	0.765147i	−0.212101	−	0.977248i	$-0.568030\pi$
$433$	15.9217	−	15.9217i	0.765147	−	0.765147i	0.977248	+	0.212101i	$0.0680304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.34847	−	7.34847i	0.351525	−	0.351525i
$438$		0			0
$439$			4.00000i			0.190910i	0.995434	+	0.0954548i	$0.0304305\pi$
$439$			4.00000i			0.190910i	−0.995434	+	0.0954548i	$0.969569\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.3712	+	18.3712i	0.872841	+	0.872841i	0.992781	−	0.119940i	$-0.0382703\pi$
$443$	18.3712	+	18.3712i	0.872841	+	0.872841i	−0.119940	+	0.992781i	$0.538270\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	−18.0000			−0.847587
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.7196	+	25.7196i	1.20311	+	1.20311i	0.973214	+	0.229900i	$0.0738399\pi$
$457$	25.7196	+	25.7196i	1.20311	+	1.20311i	0.229900	+	0.973214i	$0.426160\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	12.0000i		−	0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$		−	12.0000i		−	0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$		0			0
$463$	−17.1464	+	17.1464i	−0.796862	+	0.796862i	−0.982599	−	0.185737i	$-0.940533\pi$
$463$	−17.1464	+	17.1464i	−0.796862	+	0.796862i	0.185737	+	0.982599i	$0.440533\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.0227	+	11.0227i	−0.510070	+	0.510070i	−0.914548	−	0.404478i	$-0.867453\pi$
$467$	−11.0227	+	11.0227i	−0.510070	+	0.510070i	0.404478	+	0.914548i	$0.367453\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.34847	+	7.34847i	0.337883	+	0.337883i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	−3.00000			−0.136788
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−4.89898	−	4.89898i	−0.221994	−	0.221994i	0.587344	−	0.809338i	$-0.300174\pi$
$487$	−4.89898	−	4.89898i	−0.221994	−	0.221994i	−0.809338	+	0.587344i	$0.800174\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	12.0000i		−	0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$		−	12.0000i		−	0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22.0454	+	22.0454i	0.982956	+	0.982956i	0.999857	−	0.0169010i	$-0.00538002\pi$
$503$	22.0454	+	22.0454i	0.982956	+	0.982956i	−0.0169010	+	0.999857i	$0.505380\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−11.0227	−	11.0227i	−0.484778	−	0.484778i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	36.0000i		−	1.57719i	−0.614914	−	0.788594i	$-0.710809\pi$
$521$		−	36.0000i		−	1.57719i	0.614914	−	0.788594i	$-0.289191\pi$
$522$		0			0
$523$	14.6969	−	14.6969i	0.642652	−	0.642652i	−0.308554	−	0.951207i	$-0.599845\pi$
$523$	14.6969	−	14.6969i	0.642652	−	0.642652i	0.951207	+	0.308554i	$0.0998452\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			4.00000i			0.173913i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.34847	−	7.34847i	−0.318298	−	0.318298i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21.0000			0.904534
$540$		0			0
$541$	11.0000			0.472927			0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000			0.472927			0.236463	+	0.971640i	$0.424012\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−19.5959	−	19.5959i	−0.837861	−	0.837861i	0.150716	−	0.988577i	$-0.451842\pi$
$547$	−19.5959	−	19.5959i	−0.837861	−	0.837861i	−0.988577	+	0.150716i	$0.951842\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	12.0000i		−	0.511217i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.3939	−	29.3939i	1.24546	−	1.24546i	0.287754	−	0.957704i	$-0.407091\pi$
$557$	29.3939	−	29.3939i	1.24546	−	1.24546i	0.957704	−	0.287754i	$-0.0929086\pi$
$558$		0			0
$559$			6.00000i			0.253773i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.3712	+	18.3712i	0.774253	+	0.774253i	0.978847	−	0.204594i	$-0.0655875\pi$
$563$	18.3712	+	18.3712i	0.774253	+	0.774253i	−0.204594	+	0.978847i	$0.565587\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−3.67423	−	3.67423i	−0.152960	−	0.152960i	0.626478	−	0.779439i	$-0.284496\pi$
$577$	−3.67423	−	3.67423i	−0.152960	−	0.152960i	−0.779439	+	0.626478i	$0.784496\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	22.0454	−	22.0454i	0.913027	−	0.913027i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.34847	−	7.34847i	0.303304	−	0.303304i	−0.539001	−	0.842305i	$-0.681198\pi$
$587$	7.34847	−	7.34847i	0.303304	−	0.303304i	0.842305	+	0.539001i	$0.181198\pi$
$588$		0			0
$589$		−	4.00000i		−	0.164817i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	29.3939	+	29.3939i	1.20706	+	1.20706i	0.971974	+	0.235088i	$0.0755377\pi$
$593$	29.3939	+	29.3939i	1.20706	+	1.20706i	0.235088	+	0.971974i	$0.424462\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−5.00000			−0.203954			−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000			−0.203954			−0.101977	+	0.994787i	$0.532517\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−2.44949	−	2.44949i	−0.0994217	−	0.0994217i	0.655646	−	0.755068i	$-0.272396\pi$
$607$	−2.44949	−	2.44949i	−0.0994217	−	0.0994217i	−0.755068	+	0.655646i	$0.772396\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	9.00000i		−	0.364101i
$612$		0			0
$613$	6.12372	−	6.12372i	0.247335	−	0.247335i	−0.572541	−	0.819876i	$-0.694042\pi$
$613$	6.12372	−	6.12372i	0.247335	−	0.247335i	0.819876	+	0.572541i	$0.194042\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.34847	−	7.34847i	0.295838	−	0.295838i	−0.543543	−	0.839381i	$-0.682918\pi$
$617$	7.34847	−	7.34847i	0.295838	−	0.295838i	0.839381	+	0.543543i	$0.182918\pi$
$618$		0			0
$619$			10.0000i			0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$			10.0000i			0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−10.0000			−0.398094			−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000			−0.398094			−0.199047	+	0.979990i	$0.563785\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	8.57321	+	8.57321i	0.339683	+	0.339683i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	48.0000i		−	1.89589i	−0.318440	−	0.947943i	$-0.603159\pi$
$641$		−	48.0000i		−	1.89589i	0.318440	−	0.947943i	$-0.396841\pi$
$642$		0			0
$643$	−22.0454	+	22.0454i	−0.869386	+	0.869386i	−0.992404	−	0.123018i	$-0.960743\pi$
$643$	−22.0454	+	22.0454i	−0.869386	+	0.869386i	0.123018	+	0.992404i	$0.460743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−25.7196	+	25.7196i	−1.01114	+	1.01114i	−0.0112063	+	0.999937i	$0.503567\pi$
$647$	−25.7196	+	25.7196i	−1.01114	+	1.01114i	−0.999937	+	0.0112063i	$0.996433\pi$
$648$		0			0
$649$		−	9.00000i		−	0.353281i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	14.6969	+	14.6969i	0.575136	+	0.575136i	0.933559	−	0.358423i	$-0.116686\pi$
$653$	14.6969	+	14.6969i	0.575136	+	0.575136i	−0.358423	+	0.933559i	$0.616686\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	25.0000			0.972387			0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000			0.972387			0.486194	+	0.873851i	$0.338385\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	22.0454	+	22.0454i	0.853602	+	0.853602i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	3.00000i		−	0.115814i
$672$		0			0
$673$	−20.8207	+	20.8207i	−0.802578	+	0.802578i	−0.983498	−	0.180920i	$-0.942092\pi$
$673$	−20.8207	+	20.8207i	−0.802578	+	0.802578i	0.180920	+	0.983498i	$0.442092\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.34847	−	7.34847i	0.282425	−	0.282425i	−0.551651	−	0.834075i	$-0.686002\pi$
$677$	7.34847	−	7.34847i	0.282425	−	0.282425i	0.834075	+	0.551651i	$0.186002\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3.67423	+	3.67423i	0.140591	+	0.140591i	0.773899	−	0.633309i	$-0.218304\pi$
$683$	3.67423	+	3.67423i	0.140591	+	0.140591i	−0.633309	+	0.773899i	$0.718304\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.0000			0.685745
$690$		0			0
$691$	46.0000			1.74992			0.874961	−	0.484193i	$-0.160887\pi$
$691$	46.0000			1.74992			0.874961	+	0.484193i	$0.160887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			12.0000i			0.453234i	0.973984	+	0.226617i	$0.0727665\pi$
$701$			12.0000i			0.453234i	−0.973984	+	0.226617i	$0.927233\pi$
$702$		0			0
$703$	−2.44949	+	2.44949i	−0.0923843	+	0.0923843i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	1.00000i		−	0.0375558i	−0.999824	−	0.0187779i	$-0.994022\pi$
$709$		−	1.00000i		−	0.0375558i	0.999824	−	0.0187779i	$-0.00597754\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	7.34847	+	7.34847i	0.275202	+	0.275202i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.00000			−0.111881			−0.0559406	−	0.998434i	$-0.517816\pi$
$719$	−3.00000			−0.111881			−0.0559406	+	0.998434i	$0.517816\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.89898	+	4.89898i	0.181693	+	0.181693i	0.792093	−	0.610400i	$-0.208991\pi$
$727$	4.89898	+	4.89898i	0.181693	+	0.181693i	−0.610400	+	0.792093i	$0.708991\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−3.67423	+	3.67423i	−0.135711	+	0.135711i	−0.771699	−	0.635988i	$-0.780593\pi$
$733$	−3.67423	+	3.67423i	−0.135711	+	0.135711i	0.635988	+	0.771699i	$0.280593\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.6969	−	14.6969i	0.541369	−	0.541369i
$738$		0			0
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.3712	+	18.3712i	0.673973	+	0.673973i	0.958630	−	0.284657i	$-0.0918796\pi$
$743$	18.3712	+	18.3712i	0.673973	+	0.673973i	−0.284657	+	0.958630i	$0.591880\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−20.0000			−0.729810			−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000			−0.729810			−0.364905	+	0.931045i	$0.618899\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−18.3712	−	18.3712i	−0.667712	−	0.667712i	0.289474	−	0.957186i	$-0.406520\pi$
$757$	−18.3712	−	18.3712i	−0.667712	−	0.667712i	−0.957186	+	0.289474i	$0.906520\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	42.0000i		−	1.52250i	−0.648459	−	0.761249i	$-0.724586\pi$
$761$		−	42.0000i		−	1.52250i	0.648459	−	0.761249i	$-0.275414\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.67423	−	3.67423i	0.132669	−	0.132669i
$768$		0			0
$769$		−	2.00000i		−	0.0721218i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$		−	2.00000i		−	0.0721218i	0.999350	−	0.0360609i	$-0.0114810\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.0454	−	22.0454i	−0.792918	−	0.792918i	0.189049	−	0.981968i	$-0.439459\pi$
$773$	−22.0454	−	22.0454i	−0.792918	−	0.792918i	−0.981968	+	0.189049i	$0.939459\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	−45.0000			−1.61023
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	14.6969	+	14.6969i	0.523889	+	0.523889i	0.918744	−	0.394854i	$-0.129205\pi$
$787$	14.6969	+	14.6969i	0.523889	+	0.523889i	−0.394854	+	0.918744i	$0.629205\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.22474	−	1.22474i	0.0434920	−	0.0434920i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.0454	+	22.0454i	−0.780888	+	0.780888i	−0.979981	−	0.199092i	$-0.936201\pi$
$797$	−22.0454	+	22.0454i	−0.780888	+	0.780888i	0.199092	+	0.979981i	$0.436201\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.6969	+	14.6969i	0.518644	+	0.518644i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4.89898	+	4.89898i	0.171394	+	0.171394i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			6.00000i			0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$			6.00000i			0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$		0			0
$823$	12.2474	−	12.2474i	0.426919	−	0.426919i	−0.460658	−	0.887578i	$-0.652387\pi$
$823$	12.2474	−	12.2474i	0.426919	−	0.426919i	0.887578	+	0.460658i	$0.152387\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.0227	+	11.0227i	−0.383297	+	0.383297i	−0.872289	−	0.488992i	$-0.837365\pi$
$827$	−11.0227	+	11.0227i	−0.383297	+	0.383297i	0.488992	+	0.872289i	$0.337365\pi$
$828$		0			0
$829$			17.0000i			0.590434i	0.955430	+	0.295217i	$0.0953920\pi$
$829$			17.0000i			0.590434i	−0.955430	+	0.295217i	$0.904608\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.0000			0.517858			0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000			0.517858			0.258929	+	0.965896i	$0.416631\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	9.00000i		−	0.308516i
$852$		0			0
$853$	−1.22474	+	1.22474i	−0.0419345	+	0.0419345i	−0.727763	−	0.685829i	$-0.759440\pi$
$853$	−1.22474	+	1.22474i	−0.0419345	+	0.0419345i	0.685829	+	0.727763i	$0.259440\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.6969	−	14.6969i	0.502038	−	0.502038i	−0.410033	−	0.912071i	$-0.634483\pi$
$857$	14.6969	−	14.6969i	0.502038	−	0.502038i	0.912071	+	0.410033i	$0.134483\pi$
$858$		0			0
$859$		−	2.00000i		−	0.0682391i	−0.999418	−	0.0341196i	$-0.989137\pi$
$859$		−	2.00000i		−	0.0682391i	0.999418	−	0.0341196i	$-0.0108627\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.34847	−	7.34847i	−0.250145	−	0.250145i	0.570885	−	0.821030i	$-0.306600\pi$
$863$	−7.34847	−	7.34847i	−0.250145	−	0.250145i	−0.821030	+	0.570885i	$0.806600\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.0000			−0.407072
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−33.0681	−	33.0681i	−1.11663	−	1.11663i	−0.992232	−	0.124398i	$-0.960300\pi$
$877$	−33.0681	−	33.0681i	−1.11663	−	1.11663i	−0.124398	−	0.992232i	$-0.539700\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	24.0000i		−	0.808581i	−0.914631	−	0.404290i	$-0.867519\pi$
$881$		−	24.0000i		−	0.808581i	0.914631	−	0.404290i	$-0.132481\pi$
$882$		0			0
$883$	−14.6969	+	14.6969i	−0.494591	+	0.494591i	−0.909749	−	0.415158i	$-0.863726\pi$
$883$	−14.6969	+	14.6969i	−0.494591	+	0.494591i	0.415158	+	0.909749i	$0.363726\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.0227	−	11.0227i	0.370106	−	0.370106i	−0.497410	−	0.867516i	$-0.665715\pi$
$887$	11.0227	−	11.0227i	0.370106	−	0.370106i	0.867516	+	0.497410i	$0.165715\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.34847	−	7.34847i	−0.245907	−	0.245907i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−31.8434	−	31.8434i	−1.05734	−	1.05734i	−0.998253	−	0.0590889i	$-0.981180\pi$
$907$	−31.8434	−	31.8434i	−1.05734	−	1.05734i	−0.0590889	−	0.998253i	$-0.518820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			15.0000i			0.496972i	0.968635	+	0.248486i	$0.0799330\pi$
$911$			15.0000i			0.496972i	−0.968635	+	0.248486i	$0.920067\pi$
$912$		0			0
$913$	22.0454	−	22.0454i	0.729597	−	0.729597i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	46.0000i		−	1.51740i	−0.651440	−	0.758700i	$-0.725835\pi$
$919$		−	46.0000i		−	1.51740i	0.651440	−	0.758700i	$-0.274165\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18.3712	−	18.3712i	−0.604695	−	0.604695i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$	14.0000			0.458831
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	11.0227	+	11.0227i	0.360096	+	0.360096i	0.863848	−	0.503752i	$-0.168048\pi$
$937$	11.0227	+	11.0227i	0.360096	+	0.360096i	−0.503752	+	0.863848i	$0.668048\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			36.0000i			1.17357i	0.809744	+	0.586783i	$0.199606\pi$
$941$			36.0000i			1.17357i	−0.809744	+	0.586783i	$0.800394\pi$
$942$		0			0
$943$	22.0454	−	22.0454i	0.717897	−	0.717897i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22.0454	+	22.0454i	−0.716379	+	0.716379i	−0.967862	−	0.251482i	$-0.919082\pi$
$947$	−22.0454	+	22.0454i	−0.716379	+	0.716379i	0.251482	+	0.967862i	$0.419082\pi$
$948$		0			0
$949$			12.0000i			0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.3939	+	29.3939i	0.952161	+	0.952161i	0.998907	−	0.0467458i	$-0.0148851\pi$
$953$	29.3939	+	29.3939i	0.952161	+	0.952161i	−0.0467458	+	0.998907i	$0.514885\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	34.2929	+	34.2929i	1.10278	+	1.10278i	0.994073	+	0.108710i	$0.0346721\pi$
$967$	34.2929	+	34.2929i	1.10278	+	1.10278i	0.108710	+	0.994073i	$0.465328\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			3.00000i			0.0962746i	0.998841	+	0.0481373i	$0.0153285\pi$
$971$			3.00000i			0.0962746i	−0.998841	+	0.0481373i	$0.984672\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.6969	−	14.6969i	0.470197	−	0.470197i	−0.431782	−	0.901978i	$-0.642115\pi$
$977$	14.6969	−	14.6969i	0.470197	−	0.470197i	0.901978	+	0.431782i	$0.142115\pi$
$978$		0			0
$979$		−	36.0000i		−	1.15056i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.67423	−	3.67423i	−0.117190	−	0.117190i	0.646080	−	0.763270i	$-0.276407\pi$
$983$	−3.67423	−	3.67423i	−0.117190	−	0.117190i	−0.763270	+	0.646080i	$0.776407\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	33.0681	+	33.0681i	1.04728	+	1.04728i	0.998826	+	0.0484521i	$0.0154288\pi$
$997$	33.0681	+	33.0681i	1.04728	+	1.04728i	0.0484521	+	0.998826i	$0.484571\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.j.b.1457.1	yes	4
3.2	odd	2		2700.2.j.h.1457.1	yes	4
5.2	odd	4		2700.2.j.h.593.2	yes	4
5.3	odd	4		2700.2.j.h.593.1	yes	4
5.4	even	2	inner	2700.2.j.b.1457.2	yes	4
15.2	even	4	inner	2700.2.j.b.593.2	yes	4
15.8	even	4	inner	2700.2.j.b.593.1	✓	4
15.14	odd	2		2700.2.j.h.1457.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2700.2.j.b.593.1	✓	4	15.8	even	4	inner
2700.2.j.b.593.2	yes	4	15.2	even	4	inner
2700.2.j.b.1457.1	yes	4	1.1	even	1	trivial
2700.2.j.b.1457.2	yes	4	5.4	even	2	inner
2700.2.j.h.593.1	yes	4	5.3	odd	4
2700.2.j.h.593.2	yes	4	5.2	odd	4
2700.2.j.h.1457.1	yes	4	3.2	odd	2
2700.2.j.h.1457.2	yes	4	15.14	odd	2

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+3.00000i q^{11} +(-1.22474 + 1.22474i) q^{13} +2.00000i q^{19} +(-3.67423 - 3.67423i) q^{23} -6.00000 q^{29} -2.00000 q^{31} +(1.22474 + 1.22474i) q^{37} +6.00000i q^{41} +(2.44949 - 2.44949i) q^{43} +(-3.67423 + 3.67423i) q^{47} -7.00000i q^{49} +(-7.34847 - 7.34847i) q^{53} -3.00000 q^{59} -1.00000 q^{61} +(-4.89898 - 4.89898i) q^{67} +15.0000i q^{71} +(4.89898 - 4.89898i) q^{73} +4.00000i q^{79} +(-7.34847 - 7.34847i) q^{83} -12.0000 q^{89} +(3.67423 + 3.67423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{29} - 8 q^{31} - 12 q^{59} - 4 q^{61} - 48 q^{89}+O(q^{100}) \) 4 * q - 24 * q^29 - 8 * q^31 - 12 * q^59 - 4 * q^61 - 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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