LMFDB - Embedded newform 2700.2.j.j.593.1

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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.33973862400.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 44x^{4} + 100 $ x^8 + 44*x^4 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$-1.79576 - 1.79576i$ of defining polynomial
Character	$\chi$	$=$	2700.593
Dual form			2700.2.j.j.1457.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.224745 + 0.224745i) q^{7} +O(q^{10})$	$q+(-0.224745 + 0.224745i) q^{7} -2.78434i q^{11} +(1.22474 + 1.22474i) q^{13} +(2.78434 + 2.78434i) q^{17} +3.89898i q^{19} +(-6.19445 + 6.19445i) q^{23} -2.78434 q^{29} +0.449490 q^{31} +(-3.67423 + 3.67423i) q^{37} -9.60455i q^{41} +(-0.550510 - 0.550510i) q^{43} +6.89898i q^{49} +(6.19445 - 6.19445i) q^{53} +12.3889 q^{59} +12.7980 q^{61} +(-8.67423 + 8.67423i) q^{67} +9.60455i q^{71} +(6.22474 + 6.22474i) q^{73} +(0.625766 + 0.625766i) q^{77} +9.24745i q^{79} +(3.41011 - 3.41011i) q^{83} -12.3889 q^{89} -0.550510 q^{91} +(4.77526 - 4.77526i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} - 16 q^{31} - 24 q^{43} + 24 q^{61} - 40 q^{67} + 40 q^{73} - 24 q^{91} + 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 16 * q^31 - 24 * q^43 + 24 * q^61 - 40 * q^67 + 40 * q^73 - 24 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$7$	−0.224745	+	0.224745i	−0.0849456	+	0.0849456i	−0.748303	−	0.663357i	$-0.769131\pi$
$7$	−0.224745	+	0.224745i	−0.0849456	+	0.0849456i	0.663357	+	0.748303i	$0.269131\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.78434i		−	0.839510i	−0.907637	−	0.419755i	$-0.862116\pi$
$11$		−	2.78434i		−	0.839510i	0.907637	−	0.419755i	$-0.137884\pi$
$12$		0			0
$13$	1.22474	+	1.22474i	0.339683	+	0.339683i	0.856248	−	0.516565i	$-0.172790\pi$
$13$	1.22474	+	1.22474i	0.339683	+	0.339683i	−0.516565	+	0.856248i	$0.672790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.78434	+	2.78434i	0.675302	+	0.675302i	0.958933	−	0.283632i	$-0.0915393\pi$
$17$	2.78434	+	2.78434i	0.675302	+	0.675302i	−0.283632	+	0.958933i	$0.591539\pi$
$18$		0			0
$19$			3.89898i			0.894487i	0.894412	+	0.447244i	$0.147594\pi$
$19$			3.89898i			0.894487i	−0.894412	+	0.447244i	$0.852406\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.19445	+	6.19445i	−1.29163	+	1.29163i	−0.357854	+	0.933778i	$0.616491\pi$
$23$	−6.19445	+	6.19445i	−1.29163	+	1.29163i	−0.933778	+	0.357854i	$0.883509\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.78434			−0.517039			−0.258520	−	0.966006i	$-0.583235\pi$
$29$	−2.78434			−0.517039			−0.258520	+	0.966006i	$0.583235\pi$
$30$		0			0
$31$	0.449490			0.0807307			0.0403654	−	0.999185i	$-0.487148\pi$
$31$	0.449490			0.0807307			0.0403654	+	0.999185i	$0.487148\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.67423	+	3.67423i	−0.604040	+	0.604040i	−0.941382	−	0.337342i	$-0.890472\pi$
$37$	−3.67423	+	3.67423i	−0.604040	+	0.604040i	0.337342	+	0.941382i	$0.390472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.60455i		−	1.49998i	−0.661450	−	0.749990i	$-0.730058\pi$
$41$		−	9.60455i		−	1.49998i	0.661450	−	0.749990i	$-0.269942\pi$
$42$		0			0
$43$	−0.550510	−	0.550510i	−0.0839520	−	0.0839520i	0.663884	−	0.747836i	$-0.268907\pi$
$43$	−0.550510	−	0.550510i	−0.0839520	−	0.0839520i	−0.747836	+	0.663884i	$0.768907\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$			6.89898i			0.985568i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.19445	−	6.19445i	0.850873	−	0.850873i	−0.139368	−	0.990241i	$-0.544507\pi$
$53$	6.19445	−	6.19445i	0.850873	−	0.850873i	0.990241	+	0.139368i	$0.0445071\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.3889			1.61290			0.806448	−	0.591305i	$-0.201387\pi$
$59$	12.3889			1.61290			0.806448	+	0.591305i	$0.201387\pi$
$60$		0			0
$61$	12.7980			1.63861			0.819305	−	0.573357i	$-0.194359\pi$
$61$	12.7980			1.63861			0.819305	+	0.573357i	$0.194359\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−8.67423	+	8.67423i	−1.05973	+	1.05973i	−0.0616272	+	0.998099i	$0.519629\pi$
$67$	−8.67423	+	8.67423i	−1.05973	+	1.05973i	−0.998099	+	0.0616272i	$0.980371\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.60455i			1.13985i	0.821696	+	0.569925i	$0.193028\pi$
$71$			9.60455i			1.13985i	−0.821696	+	0.569925i	$0.806972\pi$
$72$		0			0
$73$	6.22474	+	6.22474i	0.728551	+	0.728551i	0.970331	−	0.241780i	$-0.0777312\pi$
$73$	6.22474	+	6.22474i	0.728551	+	0.728551i	−0.241780	+	0.970331i	$0.577731\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.625766	+	0.625766i	0.0713127	+	0.0713127i
$78$		0			0
$79$			9.24745i			1.04042i	0.854039	+	0.520210i	$0.174146\pi$
$79$			9.24745i			1.04042i	−0.854039	+	0.520210i	$0.825854\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.41011	−	3.41011i	0.374308	−	0.374308i	−0.494736	−	0.869044i	$-0.664735\pi$
$83$	3.41011	−	3.41011i	0.374308	−	0.374308i	0.869044	+	0.494736i	$0.164735\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.3889			−1.31322			−0.656610	−	0.754230i	$-0.728010\pi$
$89$	−12.3889			−1.31322			−0.656610	+	0.754230i	$0.728010\pi$
$90$		0			0
$91$	−0.550510			−0.0577092
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.77526	−	4.77526i	0.484854	−	0.484854i	−0.421824	−	0.906678i	$-0.638610\pi$
$97$	4.77526	−	4.77526i	0.484854	−	0.484854i	0.906678	+	0.421824i	$0.138610\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			17.9576i			1.78685i	0.449217	+	0.893423i	$0.351703\pi$
$101$			17.9576i			1.78685i	−0.449217	+	0.893423i	$0.648297\pi$
$102$		0			0
$103$	6.22474	+	6.22474i	0.613342	+	0.613342i	0.943815	−	0.330473i	$-0.107208\pi$
$103$	6.22474	+	6.22474i	0.613342	+	0.613342i	−0.330473	+	0.943815i	$0.607208\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.97879	−	8.97879i	−0.868012	−	0.868012i	0.124240	−	0.992252i	$-0.460351\pi$
$107$	−8.97879	−	8.97879i	−0.868012	−	0.868012i	−0.992252	+	0.124240i	$0.960351\pi$
$108$		0			0
$109$			11.3485i			1.08699i	0.839414	+	0.543493i	$0.182899\pi$
$109$			11.3485i			1.08699i	−0.839414	+	0.543493i	$0.817101\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.625766	+	0.625766i	−0.0588671	+	0.0588671i	−0.735927	−	0.677060i	$-0.763254\pi$
$113$	−0.625766	+	0.625766i	−0.0588671	+	0.0588671i	0.677060	+	0.735927i	$0.263254\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.25153			−0.114728
$120$		0			0
$121$	3.24745			0.295223
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−9.89898	+	9.89898i	−0.878392	+	0.878392i	−0.993368	−	0.114976i	$-0.963321\pi$
$127$	−9.89898	+	9.89898i	−0.878392	+	0.878392i	0.114976	+	0.993368i	$0.463321\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			5.56868i			0.486538i	0.969959	+	0.243269i	$0.0782198\pi$
$131$			5.56868i			0.486538i	−0.969959	+	0.243269i	$0.921780\pi$
$132$		0			0
$133$	−0.876276	−	0.876276i	−0.0759827	−	0.0759827i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.41011	−	3.41011i	−0.291345	−	0.291345i	0.546266	−	0.837611i	$-0.316049\pi$
$137$	−3.41011	−	3.41011i	−0.291345	−	0.291345i	−0.837611	+	0.546266i	$0.816049\pi$
$138$		0			0
$139$			19.2474i			1.63255i	0.577665	+	0.816274i	$0.303964\pi$
$139$			19.2474i			1.63255i	−0.577665	+	0.816274i	$0.696036\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.41011	−	3.41011i	0.285167	−	0.285167i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.82021			−0.558734			−0.279367	−	0.960184i	$-0.590125\pi$
$149$	−6.82021			−0.558734			−0.279367	+	0.960184i	$0.590125\pi$
$150$		0			0
$151$	10.5505			0.858588			0.429294	−	0.903165i	$-0.358762\pi$
$151$	10.5505			0.858588			0.429294	+	0.903165i	$0.358762\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.3485	+	13.3485i	−1.06532	+	1.06532i	−0.0676121	+	0.997712i	$0.521538\pi$
$157$	−13.3485	+	13.3485i	−1.06532	+	1.06532i	−0.997712	+	0.0676121i	$0.978462\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.78434i		−	0.219437i
$162$		0			0
$163$	−3.77526	−	3.77526i	−0.295701	−	0.295701i	0.543626	−	0.839327i	$-0.317051\pi$
$163$	−3.77526	−	3.77526i	−0.295701	−	0.295701i	−0.839327	+	0.543626i	$0.817051\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.60455	−	9.60455i	−0.743223	−	0.743223i	0.229974	−	0.973197i	$-0.426136\pi$
$167$	−9.60455	−	9.60455i	−0.743223	−	0.743223i	−0.973197	+	0.229974i	$0.926136\pi$
$168$		0			0
$169$		−	10.0000i		−	0.769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.56868	−	5.56868i	0.423379	−	0.423379i	−0.462986	−	0.886365i	$-0.653222\pi$
$173$	5.56868	−	5.56868i	0.423379	−	0.423379i	0.886365	+	0.462986i	$0.153222\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.60455			0.717878			0.358939	−	0.933361i	$-0.383139\pi$
$179$	9.60455			0.717878			0.358939	+	0.933361i	$0.383139\pi$
$180$		0			0
$181$	11.2474			0.836016			0.418008	−	0.908443i	$-0.362728\pi$
$181$	11.2474			0.836016			0.418008	+	0.908443i	$0.362728\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7.75255	−	7.75255i	0.566923	−	0.566923i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.3889i		−	0.896429i	−0.893926	−	0.448214i	$-0.852060\pi$
$191$		−	12.3889i		−	0.896429i	0.893926	−	0.448214i	$-0.147940\pi$
$192$		0			0
$193$	11.5732	+	11.5732i	0.833058	+	0.833058i	0.987934	−	0.154876i	$-0.0494977\pi$
$193$	11.5732	+	11.5732i	0.833058	+	0.833058i	−0.154876	+	0.987934i	$0.549498\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.82021	+	6.82021i	0.485920	+	0.485920i	0.907016	−	0.421096i	$-0.138354\pi$
$197$	6.82021	+	6.82021i	0.485920	+	0.485920i	−0.421096	+	0.907016i	$0.638354\pi$
$198$		0			0
$199$			9.24745i			0.655534i	0.944759	+	0.327767i	$0.106296\pi$
$199$			9.24745i			0.655534i	−0.944759	+	0.327767i	$0.893704\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.625766	−	0.625766i	0.0439202	−	0.0439202i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.8561			0.750931
$210$		0			0
$211$	−13.2474			−0.911992			−0.455996	−	0.889982i	$-0.650717\pi$
$211$	−13.2474			−0.911992			−0.455996	+	0.889982i	$0.650717\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.101021	+	0.101021i	−0.00685772	+	0.00685772i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			6.82021i			0.458777i
$222$		0			0
$223$	−12.7980	−	12.7980i	−0.857015	−	0.857015i	0.133971	−	0.990985i	$-0.457227\pi$
$223$	−12.7980	−	12.7980i	−0.857015	−	0.857015i	−0.990985	+	0.133971i	$0.957227\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.7990	+	15.7990i	1.04862	+	1.04862i	0.998756	+	0.0498603i	$0.0158776\pi$
$227$	15.7990	+	15.7990i	1.04862	+	1.04862i	0.0498603	+	0.998756i	$0.484122\pi$
$228$		0			0
$229$			1.34847i			0.0891094i	0.999007	+	0.0445547i	$0.0141869\pi$
$229$			1.34847i			0.0891094i	−0.999007	+	0.0445547i	$0.985813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−15.1732	+	15.1732i	−0.994032	+	0.994032i	−0.999982	−	0.00595067i	$-0.998106\pi$
$233$	−15.1732	+	15.1732i	−0.994032	+	0.994032i	0.00595067	+	0.999982i	$0.498106\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.35302			−0.540312			−0.270156	−	0.962817i	$-0.587075\pi$
$239$	−8.35302			−0.540312			−0.270156	+	0.962817i	$0.587075\pi$
$240$		0			0
$241$	19.0000			1.22390			0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000			1.22390			0.611949	+	0.790897i	$0.290386\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.77526	+	4.77526i	−0.303842	+	0.303842i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	21.9934i		−	1.38821i	−0.719872	−	0.694107i	$-0.755799\pi$
$251$		−	21.9934i		−	1.38821i	0.719872	−	0.694107i	$-0.244201\pi$
$252$		0			0
$253$	17.2474	+	17.2474i	1.08434	+	1.08434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	20.7419	+	20.7419i	1.29385	+	1.29385i	0.932389	+	0.361456i	$0.117720\pi$
$257$	20.7419	+	20.7419i	1.29385	+	1.29385i	0.361456	+	0.932389i	$0.382280\pi$
$258$		0			0
$259$		−	1.65153i		−	0.102621i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.3889	+	12.3889i	−0.763932	+	0.763932i	−0.977031	−	0.213099i	$-0.931644\pi$
$263$	−12.3889	+	12.3889i	−0.763932	+	0.763932i	0.213099	+	0.977031i	$0.431644\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.3889			0.755364			0.377682	−	0.925935i	$-0.376721\pi$
$269$	12.3889			0.755364			0.377682	+	0.925935i	$0.376721\pi$
$270$		0			0
$271$	−21.0000			−1.27566			−0.637830	−	0.770178i	$-0.720168\pi$
$271$	−21.0000			−1.27566			−0.637830	+	0.770178i	$0.720168\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	13.8990	−	13.8990i	0.835109	−	0.835109i	−0.153102	−	0.988210i	$-0.548926\pi$
$277$	13.8990	−	13.8990i	0.835109	−	0.835109i	0.988210	+	0.153102i	$0.0489262\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	23.5263i		−	1.40346i	−0.712444	−	0.701729i	$-0.752412\pi$
$281$		−	23.5263i		−	1.40346i	0.712444	−	0.701729i	$-0.247588\pi$
$282$		0			0
$283$	−2.79796	−	2.79796i	−0.166321	−	0.166321i	0.619039	−	0.785360i	$-0.287522\pi$
$283$	−2.79796	−	2.79796i	−0.166321	−	0.166321i	−0.785360	+	0.619039i	$0.787522\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.15857	+	2.15857i	0.127417	+	0.127417i
$288$		0			0
$289$		−	1.49490i		−	0.0879351i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.2303	−	10.2303i	0.597662	−	0.597662i	−0.342028	−	0.939690i	$-0.611114\pi$
$293$	10.2303	−	10.2303i	0.597662	−	0.597662i	0.939690	+	0.342028i	$0.111114\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−15.1732			−0.877491
$300$		0			0
$301$	0.247449			0.0142627
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−13.3485	+	13.3485i	−0.761837	+	0.761837i	−0.976654	−	0.214817i	$-0.931085\pi$
$307$	−13.3485	+	13.3485i	−0.761837	+	0.761837i	0.214817	+	0.976654i	$0.431085\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.25153i			0.0709679i	0.999370	+	0.0354839i	$0.0112973\pi$
$311$			1.25153i			0.0709679i	−0.999370	+	0.0354839i	$0.988703\pi$
$312$		0			0
$313$	−13.7753	−	13.7753i	−0.778623	−	0.778623i	0.200973	−	0.979597i	$-0.435590\pi$
$313$	−13.7753	−	13.7753i	−0.778623	−	0.778623i	−0.979597	+	0.200973i	$0.935590\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−21.9934	−	21.9934i	−1.23528	−	1.23528i	−0.961910	−	0.273365i	$-0.911863\pi$
$317$	−21.9934	−	21.9934i	−1.23528	−	1.23528i	−0.273365	−	0.961910i	$-0.588137\pi$
$318$		0			0
$319$			7.75255i			0.434060i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.8561	+	10.8561i	−0.604049	+	0.604049i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.75255			0.371154			0.185577	−	0.982630i	$-0.440585\pi$
$331$	6.75255			0.371154			0.185577	+	0.982630i	$0.440585\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.42679	+	6.42679i	−0.350089	+	0.350089i	−0.860143	−	0.510053i	$-0.829626\pi$
$337$	−6.42679	+	6.42679i	−0.350089	+	0.350089i	0.510053	+	0.860143i	$0.329626\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1.25153i		−	0.0677743i
$342$		0			0
$343$	−3.12372	−	3.12372i	−0.168665	−	0.168665i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.625766	−	0.625766i	−0.0335929	−	0.0335929i	0.690111	−	0.723704i	$-0.257562\pi$
$347$	−0.625766	−	0.625766i	−0.0335929	−	0.0335929i	−0.723704	+	0.690111i	$0.757562\pi$
$348$		0			0
$349$		−	13.0000i		−	0.695874i	−0.937518	−	0.347937i	$-0.886882\pi$
$349$		−	13.0000i		−	0.695874i	0.937518	−	0.347937i	$-0.113118\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.1732	+	15.1732i	−0.807590	+	0.807590i	−0.984269	−	0.176679i	$-0.943465\pi$
$353$	−15.1732	+	15.1732i	−0.807590	+	0.807590i	0.176679	+	0.984269i	$0.443465\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.03587			−0.213005			−0.106503	−	0.994312i	$-0.533965\pi$
$359$	−4.03587			−0.213005			−0.106503	+	0.994312i	$0.533965\pi$
$360$		0			0
$361$	3.79796			0.199893
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.5732	−	18.5732i	0.969514	−	0.969514i	−0.0300350	−	0.999549i	$-0.509562\pi$
$367$	18.5732	−	18.5732i	0.969514	−	0.969514i	0.999549	+	0.0300350i	$0.00956186\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			2.78434i			0.144556i
$372$		0			0
$373$	4.32577	+	4.32577i	0.223980	+	0.223980i	0.810172	−	0.586192i	$-0.199374\pi$
$373$	4.32577	+	4.32577i	0.223980	+	0.223980i	−0.586192	+	0.810172i	$0.699374\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.41011	−	3.41011i	−0.175629	−	0.175629i
$378$		0			0
$379$		−	13.0000i		−	0.667765i	−0.942615	−	0.333883i	$-0.891641\pi$
$379$		−	13.0000i		−	0.667765i	0.942615	−	0.333883i	$-0.108359\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	23.5263	−	23.5263i	1.20214	−	1.20214i	0.228620	−	0.973516i	$-0.426579\pi$
$383$	23.5263	−	23.5263i	1.20214	−	1.20214i	0.973516	−	0.228620i	$-0.0734212\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.7060			0.847030			0.423515	−	0.905889i	$-0.360796\pi$
$389$	16.7060			0.847030			0.423515	+	0.905889i	$0.360796\pi$
$390$		0			0
$391$	−34.4949			−1.74448
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	16.1464	−	16.1464i	0.810366	−	0.810366i	−0.174323	−	0.984689i	$-0.555774\pi$
$397$	16.1464	−	16.1464i	0.810366	−	0.810366i	0.984689	+	0.174323i	$0.0557736\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.53281i		−	0.0765448i	−0.999267	−	0.0382724i	$-0.987815\pi$
$401$		−	1.53281i		−	0.0765448i	0.999267	−	0.0382724i	$-0.0121855\pi$
$402$		0			0
$403$	0.550510	+	0.550510i	0.0274229	+	0.0274229i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.2303	+	10.2303i	0.507098	+	0.507098i
$408$		0			0
$409$		−	27.4949i		−	1.35954i	−0.733428	−	0.679768i	$-0.762081\pi$
$409$		−	27.4949i		−	1.35954i	0.733428	−	0.679768i	$-0.237919\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.78434	+	2.78434i	−0.137008	+	0.137008i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−30.3465			−1.48252			−0.741261	−	0.671217i	$-0.765772\pi$
$419$	−30.3465			−1.48252			−0.741261	+	0.671217i	$0.765772\pi$
$420$		0			0
$421$	11.2474			0.548167			0.274084	−	0.961706i	$-0.411626\pi$
$421$	11.2474			0.548167			0.274084	+	0.961706i	$0.411626\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.87628	+	2.87628i	−0.139193	+	0.139193i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			20.7419i			0.999103i	0.866284	+	0.499551i	$0.166502\pi$
$431$			20.7419i			0.999103i	−0.866284	+	0.499551i	$0.833498\pi$
$432$		0			0
$433$	−2.79796	−	2.79796i	−0.134461	−	0.134461i	0.636673	−	0.771134i	$-0.280310\pi$
$433$	−2.79796	−	2.79796i	−0.134461	−	0.134461i	−0.771134	+	0.636673i	$0.780310\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.1520	−	24.1520i	−1.15535	−	1.15535i
$438$		0			0
$439$		−	13.1464i		−	0.627445i	−0.949515	−	0.313722i	$-0.898424\pi$
$439$		−	13.1464i		−	0.627445i	0.949515	−	0.313722i	$-0.101576\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.82021	−	6.82021i	0.324038	−	0.324038i	−0.526276	−	0.850314i	$-0.676412\pi$
$443$	6.82021	−	6.82021i	0.324038	−	0.324038i	0.850314	+	0.526276i	$0.176412\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.3889			0.584668			0.292334	−	0.956316i	$-0.405568\pi$
$449$	12.3889			0.584668			0.292334	+	0.956316i	$0.405568\pi$
$450$		0			0
$451$	−26.7423			−1.25925
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	22.3485	−	22.3485i	1.04542	−	1.04542i	0.0464990	−	0.998918i	$-0.485194\pi$
$457$	22.3485	−	22.3485i	1.04542	−	1.04542i	0.998918	−	0.0464990i	$-0.0148064\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			29.0949i			1.35509i	0.735483	+	0.677543i	$0.236955\pi$
$461$			29.0949i			1.35509i	−0.735483	+	0.677543i	$0.763045\pi$
$462$		0			0
$463$	−1.02270	−	1.02270i	−0.0475291	−	0.0475291i	0.682943	−	0.730472i	$-0.260700\pi$
$463$	−1.02270	−	1.02270i	−0.0475291	−	0.0475291i	−0.730472	+	0.682943i	$0.760700\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.4248	+	16.4248i	0.760048	+	0.760048i	0.976331	−	0.216283i	$-0.0693934\pi$
$467$	16.4248	+	16.4248i	0.760048	+	0.760048i	−0.216283	+	0.976331i	$0.569393\pi$
$468$		0			0
$469$		−	3.89898i		−	0.180038i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.53281	+	1.53281i	−0.0704786	+	0.0704786i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.03587			0.184404			0.0922019	−	0.995740i	$-0.470609\pi$
$479$	4.03587			0.184404			0.0922019	+	0.995740i	$0.470609\pi$
$480$		0			0
$481$	−9.00000			−0.410365
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−18.6742	+	18.6742i	−0.846210	+	0.846210i	−0.989658	−	0.143448i	$-0.954181\pi$
$487$	−18.6742	+	18.6742i	−0.846210	+	0.846210i	0.143448	+	0.989658i	$0.454181\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			39.9510i			1.80296i	0.432816	+	0.901482i	$0.357520\pi$
$491$			39.9510i			1.80296i	−0.432816	+	0.901482i	$0.642480\pi$
$492$		0			0
$493$	−7.75255	−	7.75255i	−0.349157	−	0.349157i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.15857	−	2.15857i	−0.0968253	−	0.0968253i
$498$		0			0
$499$		−	0.898979i		−	0.0402438i	−0.999798	−	0.0201219i	$-0.993595\pi$
$499$		−	0.898979i		−	0.0402438i	0.999798	−	0.0201219i	$-0.00640544\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.5833	−	18.5833i	0.828590	−	0.828590i	−0.158732	−	0.987322i	$-0.550740\pi$
$503$	18.5833	−	18.5833i	0.828590	−	0.828590i	0.987322	+	0.158732i	$0.0507404\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.4248			−0.728015			−0.364007	−	0.931396i	$-0.618592\pi$
$509$	−16.4248			−0.728015			−0.364007	+	0.931396i	$0.618592\pi$
$510$		0			0
$511$	−2.79796			−0.123774
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	15.1732i		−	0.664751i	−0.943147	−	0.332376i	$-0.892150\pi$
$521$		−	15.1732i		−	0.664751i	0.943147	−	0.332376i	$-0.107850\pi$
$522$		0			0
$523$	−27.9217	−	27.9217i	−1.22093	−	1.22093i	−0.967302	−	0.253628i	$-0.918376\pi$
$523$	−27.9217	−	27.9217i	−1.22093	−	1.22093i	−0.253628	−	0.967302i	$-0.581624\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.25153	+	1.25153i	0.0545176	+	0.0545176i
$528$		0			0
$529$		−	53.7423i		−	2.33662i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.7631	−	11.7631i	0.509518	−	0.509518i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	19.2091			0.827395
$540$		0			0
$541$	10.5505			0.453602			0.226801	−	0.973941i	$-0.427173\pi$
$541$	10.5505			0.453602			0.226801	+	0.973941i	$0.427173\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	23.5732	−	23.5732i	1.00792	−	1.00792i	0.00794945	−	0.999968i	$-0.497470\pi$
$547$	23.5732	−	23.5732i	1.00792	−	1.00792i	0.999968	−	0.00794945i	$-0.00253042\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.8561i		−	0.462485i
$552$		0			0
$553$	−2.07832	−	2.07832i	−0.0883790	−	0.0883790i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.5263	−	23.5263i	−0.996839	−	0.996839i	0.00315559	−	0.999995i	$-0.498996\pi$
$557$	−23.5263	−	23.5263i	−0.996839	−	0.996839i	−0.999995	+	0.00315559i	$0.998996\pi$
$558$		0			0
$559$		−	1.34847i		−	0.0570342i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.35302	−	8.35302i	0.352038	−	0.352038i	−0.508829	−	0.860867i	$-0.669922\pi$
$563$	8.35302	−	8.35302i	0.352038	−	0.352038i	0.860867	+	0.508829i	$0.169922\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.9576			0.752821			0.376410	−	0.926453i	$-0.377158\pi$
$569$	17.9576			0.752821			0.376410	+	0.926453i	$0.377158\pi$
$570$		0			0
$571$	−9.44949			−0.395449			−0.197724	−	0.980258i	$-0.563355\pi$
$571$	−9.44949			−0.395449			−0.197724	+	0.980258i	$0.563355\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	22.0227	−	22.0227i	0.916817	−	0.916817i	−0.0799794	−	0.996797i	$-0.525485\pi$
$577$	22.0227	−	22.0227i	0.916817	−	0.916817i	0.996797	+	0.0799794i	$0.0254854\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.53281i			0.0635916i
$582$		0			0
$583$	−17.2474	−	17.2474i	−0.714316	−	0.714316i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.2303	−	10.2303i	−0.422250	−	0.422250i	0.463727	−	0.885978i	$-0.346512\pi$
$587$	−10.2303	−	10.2303i	−0.422250	−	0.422250i	−0.885978	+	0.463727i	$0.846512\pi$
$588$		0			0
$589$			1.75255i			0.0722126i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.97879	−	8.97879i	0.368715	−	0.368715i	−0.498294	−	0.867008i	$-0.666040\pi$
$593$	8.97879	−	8.97879i	0.368715	−	0.368715i	0.867008	+	0.498294i	$0.166040\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	34.3823			1.40482			0.702412	−	0.711770i	$-0.252106\pi$
$599$	34.3823			1.40482			0.702412	+	0.711770i	$0.252106\pi$
$600$		0			0
$601$	−14.0454			−0.572924			−0.286462	−	0.958092i	$-0.592479\pi$
$601$	−14.0454			−0.572924			−0.286462	+	0.958092i	$0.592479\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3.67423	+	3.67423i	−0.149133	+	0.149133i	−0.777730	−	0.628598i	$-0.783629\pi$
$607$	−3.67423	+	3.67423i	−0.149133	+	0.149133i	0.628598	+	0.777730i	$0.283629\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	31.2247	+	31.2247i	1.26116	+	1.26116i	0.950533	+	0.310622i	$0.100537\pi$
$613$	31.2247	+	31.2247i	1.26116	+	1.26116i	0.310622	+	0.950533i	$0.399463\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11.7631	+	11.7631i	0.473566	+	0.473566i	0.903066	−	0.429501i	$-0.141310\pi$
$617$	11.7631	+	11.7631i	0.473566	+	0.473566i	−0.429501	+	0.903066i	$0.641310\pi$
$618$		0			0
$619$			1.65153i			0.0663806i	0.999449	+	0.0331903i	$0.0105667\pi$
$619$			1.65153i			0.0663806i	−0.999449	+	0.0331903i	$0.989433\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.78434	−	2.78434i	0.111552	−	0.111552i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.4606			−0.815819
$630$		0			0
$631$	−35.4949			−1.41303			−0.706515	−	0.707698i	$-0.749734\pi$
$631$	−35.4949			−1.41303			−0.706515	+	0.707698i	$0.749734\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8.44949	+	8.44949i	−0.334781	+	0.334781i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			8.07175i			0.318815i	0.987213	+	0.159407i	$0.0509583\pi$
$641$			8.07175i			0.318815i	−0.987213	+	0.159407i	$0.949042\pi$
$642$		0			0
$643$	12.5505	+	12.5505i	0.494944	+	0.494944i	0.909860	−	0.414916i	$-0.136189\pi$
$643$	12.5505	+	12.5505i	0.494944	+	0.494944i	−0.414916	+	0.909860i	$0.636189\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13.0147	−	13.0147i	−0.511659	−	0.511659i	0.403375	−	0.915035i	$-0.367837\pi$
$647$	−13.0147	−	13.0147i	−0.511659	−	0.511659i	−0.915035	+	0.403375i	$0.867837\pi$
$648$		0			0
$649$		−	34.4949i		−	1.35404i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.9934	+	21.9934i	−0.860670	+	0.860670i	−0.991416	−	0.130746i	$-0.958263\pi$
$653$	−21.9934	+	21.9934i	−0.860670	+	0.860670i	0.130746	+	0.991416i	$0.458263\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−49.5556			−1.93041			−0.965206	−	0.261492i	$-0.915785\pi$
$659$	−49.5556			−1.93041			−0.965206	+	0.261492i	$0.915785\pi$
$660$		0			0
$661$	−1.00000			−0.0388955			−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000			−0.0388955			−0.0194477	+	0.999811i	$0.506191\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.2474	−	17.2474i	0.667824	−	0.667824i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	35.6339i		−	1.37563i
$672$		0			0
$673$	−18.2702	−	18.2702i	−0.704263	−	0.704263i	0.261060	−	0.965323i	$-0.415928\pi$
$673$	−18.2702	−	18.2702i	−0.704263	−	0.704263i	−0.965323	+	0.261060i	$0.915928\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11.1374	+	11.1374i	0.428044	+	0.428044i	0.887961	−	0.459918i	$-0.152121\pi$
$677$	11.1374	+	11.1374i	0.428044	+	0.428044i	−0.459918	+	0.887961i	$0.652121\pi$
$678$		0			0
$679$			2.14643i			0.0823724i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−20.1161	+	20.1161i	−0.769723	+	0.769723i	−0.978058	−	0.208335i	$-0.933196\pi$
$683$	−20.1161	+	20.1161i	−0.769723	+	0.769723i	0.208335	+	0.978058i	$0.433196\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.1732			0.578054
$690$		0			0
$691$	−0.247449			−0.00941339			−0.00470670	−	0.999989i	$-0.501498\pi$
$691$	−0.247449			−0.00941339			−0.00470670	+	0.999989i	$0.501498\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	26.7423	−	26.7423i	1.01294	−	1.01294i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			17.6763i			0.667625i	0.942640	+	0.333812i	$0.108335\pi$
$701$			17.6763i			0.667625i	−0.942640	+	0.333812i	$0.891665\pi$
$702$		0			0
$703$	−14.3258	−	14.3258i	−0.540306	−	0.540306i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.03587	−	4.03587i	−0.151785	−	0.151785i
$708$		0			0
$709$		−	8.34847i		−	0.313533i	−0.987636	−	0.156767i	$-0.949893\pi$
$709$		−	8.34847i		−	0.313533i	0.987636	−	0.156767i	$-0.0501071\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.78434	+	2.78434i	−0.104274	+	0.104274i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−49.8369			−1.85860			−0.929300	−	0.369325i	$-0.879589\pi$
$719$	−49.8369			−1.85860			−0.929300	+	0.369325i	$0.879589\pi$
$720$		0			0
$721$	−2.79796			−0.104201
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	13.8990	−	13.8990i	0.515485	−	0.515485i	−0.400717	−	0.916202i	$-0.631239\pi$
$727$	13.8990	−	13.8990i	0.515485	−	0.515485i	0.916202	+	0.400717i	$0.131239\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	3.06562i		−	0.113386i
$732$		0			0
$733$	2.55051	+	2.55051i	0.0942052	+	0.0942052i	0.752639	−	0.658434i	$-0.228781\pi$
$733$	2.55051	+	2.55051i	0.0942052	+	0.0942052i	−0.658434	+	0.752639i	$0.728781\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.1520	+	24.1520i	0.889651	+	0.889651i
$738$		0			0
$739$		−	26.2474i		−	0.965528i	−0.875750	−	0.482764i	$-0.839633\pi$
$739$		−	26.2474i		−	0.965528i	0.875750	−	0.482764i	$-0.160367\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.3823	+	34.3823i	−1.26137	+	1.26137i	−0.310934	+	0.950431i	$0.600642\pi$
$743$	−34.3823	+	34.3823i	−1.26137	+	1.26137i	−0.950431	+	0.310934i	$0.899358\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.03587			0.147468
$750$		0			0
$751$	9.00000			0.328415			0.164207	−	0.986426i	$-0.447493\pi$
$751$	9.00000			0.328415			0.164207	+	0.986426i	$0.447493\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−19.7196	+	19.7196i	−0.716723	+	0.716723i	−0.967933	−	0.251210i	$-0.919172\pi$
$757$	−19.7196	+	19.7196i	−0.716723	+	0.716723i	0.251210	+	0.967933i	$0.419172\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	23.2450i		−	0.842630i	−0.906914	−	0.421315i	$-0.861569\pi$
$761$		−	23.2450i		−	0.842630i	0.906914	−	0.421315i	$-0.138431\pi$
$762$		0			0
$763$	−2.55051	−	2.55051i	−0.0923347	−	0.0923347i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	15.1732	+	15.1732i	0.547874	+	0.547874i
$768$		0			0
$769$		−	29.7423i		−	1.07254i	−0.844048	−	0.536268i	$-0.819834\pi$
$769$		−	29.7423i		−	1.07254i	0.844048	−	0.536268i	$-0.180166\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.41011	−	3.41011i	0.122653	−	0.122653i	−0.643116	−	0.765769i	$-0.722359\pi$
$773$	3.41011	−	3.41011i	0.122653	−	0.122653i	0.765769	+	0.643116i	$0.222359\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	37.4480			1.34171
$780$		0			0
$781$	26.7423			0.956916
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−17.4722	+	17.4722i	−0.622816	+	0.622816i	−0.946251	−	0.323434i	$-0.895163\pi$
$787$	−17.4722	+	17.4722i	−0.622816	+	0.622816i	0.323434	+	0.946251i	$0.395163\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	0.281275i		−	0.0100010i
$792$		0			0
$793$	15.6742	+	15.6742i	0.556608	+	0.556608i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.9934	+	21.9934i	0.779048	+	0.779048i	0.979669	−	0.200621i	$-0.0642960\pi$
$797$	21.9934	+	21.9934i	0.779048	+	0.779048i	−0.200621	+	0.979669i	$0.564296\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.3318	−	17.3318i	0.611626	−	0.611626i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.25153			0.0440015			0.0220008	−	0.999758i	$-0.492996\pi$
$809$	1.25153			0.0440015			0.0220008	+	0.999758i	$0.492996\pi$
$810$		0			0
$811$	−19.5505			−0.686511			−0.343256	−	0.939242i	$-0.611530\pi$
$811$	−19.5505			−0.686511			−0.343256	+	0.939242i	$0.611530\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.14643	−	2.14643i	0.0750940	−	0.0750940i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	17.9576i		−	0.626724i	−0.949634	−	0.313362i	$-0.898545\pi$
$821$		−	17.9576i		−	0.626724i	0.949634	−	0.313362i	$-0.101455\pi$
$822$		0			0
$823$	−31.0227	−	31.0227i	−1.08138	−	1.08138i	−0.996381	−	0.0850028i	$-0.972910\pi$
$823$	−31.0227	−	31.0227i	−1.08138	−	1.08138i	−0.0850028	−	0.996381i	$-0.527090\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.82021	−	6.82021i	−0.237162	−	0.237162i	0.578512	−	0.815674i	$-0.303634\pi$
$827$	−6.82021	−	6.82021i	−0.237162	−	0.237162i	−0.815674	+	0.578512i	$0.803634\pi$
$828$		0			0
$829$			49.2474i			1.71043i	0.518270	+	0.855217i	$0.326576\pi$
$829$			49.2474i			1.71043i	−0.518270	+	0.855217i	$0.673424\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−19.2091	+	19.2091i	−0.665556	+	0.665556i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.7778			0.855424			0.427712	−	0.903915i	$-0.359320\pi$
$839$	24.7778			0.855424			0.427712	+	0.903915i	$0.359320\pi$
$840$		0			0
$841$	−21.2474			−0.732671
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.729847	+	0.729847i	−0.0250779	+	0.0250779i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	45.5197i		−	1.56040i
$852$		0			0
$853$	3.47219	+	3.47219i	0.118886	+	0.118886i	0.764047	−	0.645161i	$-0.223210\pi$
$853$	3.47219	+	3.47219i	0.118886	+	0.118886i	−0.645161	+	0.764047i	$0.723210\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.41011	−	3.41011i	−0.116487	−	0.116487i	0.646460	−	0.762947i	$-0.276249\pi$
$857$	−3.41011	−	3.41011i	−0.116487	−	0.116487i	−0.762947	+	0.646460i	$0.776249\pi$
$858$		0			0
$859$		−	5.24745i		−	0.179041i	−0.995985	−	0.0895203i	$-0.971467\pi$
$859$		−	5.24745i		−	0.179041i	0.995985	−	0.0895203i	$-0.0285334\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.8561	−	10.8561i	0.369545	−	0.369545i	−0.497766	−	0.867311i	$-0.665846\pi$
$863$	10.8561	−	10.8561i	0.369545	−	0.369545i	0.867311	+	0.497766i	$0.165846\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	25.7480			0.873443
$870$		0			0
$871$	−21.2474			−0.719942
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10.2247	+	10.2247i	−0.345265	+	0.345265i	−0.858342	−	0.513077i	$-0.828505\pi$
$877$	−10.2247	+	10.2247i	−0.345265	+	0.345265i	0.513077	+	0.858342i	$0.328505\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	2.78434i		−	0.0938068i	−0.998899	−	0.0469034i	$-0.985065\pi$
$881$		−	2.78434i		−	0.0938068i	0.998899	−	0.0469034i	$-0.0149353\pi$
$882$		0			0
$883$	−22.9217	−	22.9217i	−0.771376	−	0.771376i	0.206971	−	0.978347i	$-0.433639\pi$
$883$	−22.9217	−	22.9217i	−0.771376	−	0.771376i	−0.978347	+	0.206971i	$0.933639\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.6404	+	13.6404i	0.458001	+	0.458001i	0.897999	−	0.439998i	$-0.145021\pi$
$887$	13.6404	+	13.6404i	0.458001	+	0.458001i	−0.439998	+	0.897999i	$0.645021\pi$
$888$		0			0
$889$		−	4.44949i		−	0.149231i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.25153			−0.0417409
$900$		0			0
$901$	34.4949			1.14919
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−13.6742	+	13.6742i	−0.454046	+	0.454046i	−0.896695	−	0.442649i	$-0.854039\pi$
$907$	−13.6742	+	13.6742i	−0.454046	+	0.454046i	0.442649	+	0.896695i	$0.354039\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			4.31715i			0.143034i	0.997439	+	0.0715168i	$0.0227839\pi$
$911$			4.31715i			0.143034i	−0.997439	+	0.0715168i	$0.977216\pi$
$912$		0			0
$913$	−9.49490	−	9.49490i	−0.314235	−	0.314235i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.25153	−	1.25153i	−0.0413292	−	0.0413292i
$918$		0			0
$919$		−	18.6515i		−	0.615257i	−0.951507	−	0.307629i	$-0.900465\pi$
$919$		−	18.6515i		−	0.615257i	0.951507	−	0.307629i	$-0.0995354\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.7631	+	11.7631i	−0.387188	+	0.387188i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−56.6571			−1.85886			−0.929429	−	0.369001i	$-0.879700\pi$
$929$	−56.6571			−1.85886			−0.929429	+	0.369001i	$0.879700\pi$
$930$		0			0
$931$	−26.8990			−0.881578
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	9.77526	−	9.77526i	0.319344	−	0.319344i	−0.529171	−	0.848515i	$-0.677497\pi$
$937$	9.77526	−	9.77526i	0.319344	−	0.319344i	0.848515	+	0.529171i	$0.177497\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	15.1732i		−	0.494633i	−0.968935	−	0.247317i	$-0.920451\pi$
$941$		−	15.1732i		−	0.494633i	0.968935	−	0.247317i	$-0.0795488\pi$
$942$		0			0
$943$	59.4949	+	59.4949i	1.93742	+	1.93742i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.4248	−	16.4248i	−0.533733	−	0.533733i	0.387948	−	0.921681i	$-0.373184\pi$
$947$	−16.4248	−	16.4248i	−0.533733	−	0.533733i	−0.921681	+	0.387948i	$0.873184\pi$
$948$		0			0
$949$			15.2474i			0.494953i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.5263	−	23.5263i	0.762090	−	0.762090i	−0.214610	−	0.976700i	$-0.568848\pi$
$953$	23.5263	−	23.5263i	0.762090	−	0.762090i	0.976700	+	0.214610i	$0.0688480\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.53281			0.0494969
$960$		0			0
$961$	−30.7980			−0.993483
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	31.3258	−	31.3258i	1.00737	−	1.00737i	0.00739605	−	0.999973i	$-0.497646\pi$
$967$	31.3258	−	31.3258i	1.00737	−	1.00737i	0.999973	−	0.00739605i	$-0.00235426\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	2.50306i		−	0.0803272i	−0.999193	−	0.0401636i	$-0.987212\pi$
$971$		−	2.50306i		−	0.0803272i	0.999193	−	0.0401636i	$-0.0127879\pi$
$972$		0			0
$973$	−4.32577	−	4.32577i	−0.138678	−	0.138678i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33.7566	−	33.7566i	−1.07997	−	1.07997i	−0.996511	−	0.0834572i	$-0.973404\pi$
$977$	−33.7566	−	33.7566i	−1.07997	−	1.07997i	−0.0834572	−	0.996511i	$-0.526596\pi$
$978$		0			0
$979$			34.4949i			1.10246i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.6848	+	25.6848i	−0.819219	+	0.819219i	−0.985995	−	0.166776i	$-0.946664\pi$
$983$	−25.6848	+	25.6848i	−0.819219	+	0.819219i	0.166776	+	0.985995i	$0.446664\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.82021			0.216870
$990$		0			0
$991$	35.0454			1.11325			0.556627	−	0.830763i	$-0.312095\pi$
$991$	35.0454			1.11325			0.556627	+	0.830763i	$0.312095\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.101021	−	0.101021i	0.00319935	−	0.00319935i	−0.705505	−	0.708705i	$-0.749280\pi$
$997$	0.101021	−	0.101021i	0.00319935	−	0.00319935i	0.708705	+	0.705505i	$0.249280\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.j.j.593.1		8
3.2	odd	2	inner	2700.2.j.j.593.2		8
5.2	odd	4	inner	2700.2.j.j.1457.1		8
5.3	odd	4		540.2.j.a.377.2	yes	8
5.4	even	2		540.2.j.a.53.3	yes	8
15.2	even	4	inner	2700.2.j.j.1457.2		8
15.8	even	4		540.2.j.a.377.3	yes	8
15.14	odd	2		540.2.j.a.53.2	✓	8
20.3	even	4		2160.2.w.e.1457.2		8
20.19	odd	2		2160.2.w.e.593.3		8
45.4	even	6		1620.2.x.d.593.1		16
45.13	odd	12		1620.2.x.d.917.1		16
45.14	odd	6		1620.2.x.d.593.4		16
45.23	even	12		1620.2.x.d.917.4		16
45.29	odd	6		1620.2.x.d.53.1		16
45.34	even	6		1620.2.x.d.53.4		16
45.38	even	12		1620.2.x.d.377.1		16
45.43	odd	12		1620.2.x.d.377.4		16
60.23	odd	4		2160.2.w.e.1457.3		8
60.59	even	2		2160.2.w.e.593.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.j.a.53.2	✓	8	15.14	odd	2
540.2.j.a.53.3	yes	8	5.4	even	2
540.2.j.a.377.2	yes	8	5.3	odd	4
540.2.j.a.377.3	yes	8	15.8	even	4
1620.2.x.d.53.1		16	45.29	odd	6
1620.2.x.d.53.4		16	45.34	even	6
1620.2.x.d.377.1		16	45.38	even	12
1620.2.x.d.377.4		16	45.43	odd	12
1620.2.x.d.593.1		16	45.4	even	6
1620.2.x.d.593.4		16	45.14	odd	6
1620.2.x.d.917.1		16	45.13	odd	12
1620.2.x.d.917.4		16	45.23	even	12
2160.2.w.e.593.2		8	60.59	even	2
2160.2.w.e.593.3		8	20.19	odd	2
2160.2.w.e.1457.2		8	20.3	even	4
2160.2.w.e.1457.3		8	60.23	odd	4
2700.2.j.j.593.1		8	1.1	even	1	trivial
2700.2.j.j.593.2		8	3.2	odd	2	inner
2700.2.j.j.1457.1		8	5.2	odd	4	inner
2700.2.j.j.1457.2		8	15.2	even	4	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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.224745 + 0.224745i) q^{7} +O(q^{10})\)	\(q+(-0.224745 + 0.224745i) q^{7} -2.78434i q^{11} +(1.22474 + 1.22474i) q^{13} +(2.78434 + 2.78434i) q^{17} +3.89898i q^{19} +(-6.19445 + 6.19445i) q^{23} -2.78434 q^{29} +0.449490 q^{31} +(-3.67423 + 3.67423i) q^{37} -9.60455i q^{41} +(-0.550510 - 0.550510i) q^{43} +6.89898i q^{49} +(6.19445 - 6.19445i) q^{53} +12.3889 q^{59} +12.7980 q^{61} +(-8.67423 + 8.67423i) q^{67} +9.60455i q^{71} +(6.22474 + 6.22474i) q^{73} +(0.625766 + 0.625766i) q^{77} +9.24745i q^{79} +(3.41011 - 3.41011i) q^{83} -12.3889 q^{89} -0.550510 q^{91} +(4.77526 - 4.77526i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} - 16 q^{31} - 24 q^{43} + 24 q^{61} - 40 q^{67} + 40 q^{73} - 24 q^{91} + 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 16 * q^31 - 24 * q^43 + 24 * q^61 - 40 * q^67 + 40 * q^73 - 24 * q^91 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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