LMFDB - Embedded newform 2340.2.bp.h.1513.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(1477,2340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2340.1477");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.bp (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:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 16x^{14} + 184x^{12} - 952x^{10} + 3559x^{8} - 6400x^{6} + 8200x^{4} - 2500x^{2} + 625 $ x^16 - 16x^14 + 184x^12 - 952x^10 + 3559x^8 - 6400x^6 + 8200x^4 - 2500*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1513.1
Root			$-2.66925 - 1.54109i$ of defining polynomial
Character	$\chi$	$=$	2340.1513
Dual form			2340.2.bp.h.1477.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.22422 - 0.229890i) q^{5} +2.73205 q^{7} +O(q^{10})$	$q+(-2.22422 - 0.229890i) q^{5} +2.73205 q^{7} +(-3.71025 - 3.71025i) q^{11} +(2.74513 - 2.33758i) q^{13} +(-2.25631 - 2.25631i) q^{17} +(0.443277 + 0.443277i) q^{19} +(0.459779 - 0.459779i) q^{23} +(4.89430 + 1.02265i) q^{25} +8.16470i q^{29} +(-4.03573 + 4.03573i) q^{31} +(-6.07668 - 0.628070i) q^{35} -7.78860 q^{37} +(4.17003 - 4.17003i) q^{41} +(-1.75821 + 1.75821i) q^{43} -2.05250 q^{47} +0.464102 q^{49} +(-4.44844 - 4.44844i) q^{53} +(7.39946 + 9.10536i) q^{55} +(-5.25081 + 5.25081i) q^{59} -13.3050 q^{61} +(-6.64315 + 4.56821i) q^{65} -11.1019i q^{67} +(6.17037 - 6.17037i) q^{71} -4.63351i q^{73} +(-10.1366 - 10.1366i) q^{77} -6.95436i q^{79} +1.37934 q^{83} +(4.49983 + 5.53723i) q^{85} +(1.19797 - 1.19797i) q^{89} +(7.49983 - 6.38638i) q^{91} +(-0.884040 - 1.08785i) q^{95} -9.56171i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7}+O(q^{10}) $ 16 * q + 16 * q^7	$ 16 q + 16 q^{7} + 8 q^{13} + 24 q^{25} - 32 q^{31} - 16 q^{37} + 16 q^{43} - 48 q^{49} + 8 q^{55} - 16 q^{61} - 16 q^{85} + 32 q^{91}+O(q^{100}) $ 16 * q + 16 * q^7 + 8 * q^13 + 24 * q^25 - 32 * q^31 - 16 * q^37 + 16 * q^43 - 48 * q^49 + 8 * q^55 - 16 * q^61 - 16 * q^85 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.22422	−	0.229890i	−0.994701	−	0.102810i
$5$	−2.22422	−	0.229890i	−0.994701	−	0.102810i
$6$		0			0
$7$	2.73205			1.03262			0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205			1.03262			0.516309	+	0.856402i	$0.327306\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.71025	−	3.71025i	−1.11868	−	1.11868i	−0.991935	−	0.126747i	$-0.959546\pi$
$11$	−3.71025	−	3.71025i	−1.11868	−	1.11868i	−0.126747	−	0.991935i	$-0.540454\pi$
$12$		0			0
$13$	2.74513	−	2.33758i	0.761362	−	0.648328i
$13$	2.74513	−	2.33758i	0.761362	−	0.648328i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.25631	−	2.25631i	−0.547236	−	0.547236i	0.378404	−	0.925640i	$-0.376473\pi$
$17$	−2.25631	−	2.25631i	−0.547236	−	0.547236i	−0.925640	+	0.378404i	$0.876473\pi$
$18$		0			0
$19$	0.443277	+	0.443277i	0.101695	+	0.101695i	0.756124	−	0.654429i	$-0.227091\pi$
$19$	0.443277	+	0.443277i	0.101695	+	0.101695i	−0.654429	+	0.756124i	$0.727091\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.459779	−	0.459779i	0.0958706	−	0.0958706i	−0.657545	−	0.753415i	$-0.728405\pi$
$23$	0.459779	−	0.459779i	0.0958706	−	0.0958706i	0.753415	+	0.657545i	$0.228405\pi$
$24$		0			0
$25$	4.89430	+	1.02265i	0.978860	+	0.204530i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.16470i			1.51615i	0.652169	+	0.758073i	$0.273859\pi$
$29$			8.16470i			1.51615i	−0.652169	+	0.758073i	$0.726141\pi$
$30$		0			0
$31$	−4.03573	+	4.03573i	−0.724838	+	0.724838i	−0.969587	−	0.244749i	$-0.921295\pi$
$31$	−4.03573	+	4.03573i	−0.724838	+	0.724838i	0.244749	+	0.969587i	$0.421295\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.07668	−	0.628070i	−1.02715	−	0.106163i
$36$		0			0
$37$	−7.78860			−1.28044			−0.640219	−	0.768192i	$-0.721156\pi$
$37$	−7.78860			−1.28044			−0.640219	+	0.768192i	$0.721156\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.17003	−	4.17003i	0.651249	−	0.651249i	−0.302045	−	0.953294i	$-0.597669\pi$
$41$	4.17003	−	4.17003i	0.651249	−	0.651249i	0.953294	+	0.302045i	$0.0976692\pi$
$42$		0			0
$43$	−1.75821	+	1.75821i	−0.268124	+	0.268124i	−0.828344	−	0.560220i	$-0.810717\pi$
$43$	−1.75821	+	1.75821i	−0.268124	+	0.268124i	0.560220	+	0.828344i	$0.310717\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.05250			−0.299388			−0.149694	−	0.988732i	$-0.547829\pi$
$47$	−2.05250			−0.299388			−0.149694	+	0.988732i	$0.547829\pi$
$48$		0			0
$49$	0.464102			0.0663002
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.44844	−	4.44844i	−0.611040	−	0.611040i	0.332177	−	0.943217i	$-0.392217\pi$
$53$	−4.44844	−	4.44844i	−0.611040	−	0.611040i	−0.943217	+	0.332177i	$0.892217\pi$
$54$		0			0
$55$	7.39946	+	9.10536i	0.997743	+	1.22777i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.25081	+	5.25081i	−0.683598	+	0.683598i	−0.960809	−	0.277211i	$-0.910590\pi$
$59$	−5.25081	+	5.25081i	−0.683598	+	0.683598i	0.277211	+	0.960809i	$0.410590\pi$
$60$		0			0
$61$	−13.3050			−1.70353			−0.851766	−	0.523922i	$-0.824468\pi$
$61$	−13.3050			−1.70353			−0.851766	+	0.523922i	$0.824468\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.64315	+	4.56821i	−0.823982	+	0.566617i
$66$		0			0
$67$		−	11.1019i		−	1.35631i	−0.734920	−	0.678154i	$-0.762780\pi$
$67$		−	11.1019i		−	1.35631i	0.734920	−	0.678154i	$-0.237220\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.17037	−	6.17037i	0.732288	−	0.732288i	−0.238784	−	0.971073i	$-0.576749\pi$
$71$	6.17037	−	6.17037i	0.732288	−	0.732288i	0.971073	+	0.238784i	$0.0767489\pi$
$72$		0			0
$73$		−	4.63351i		−	0.542311i	−0.962536	−	0.271156i	$-0.912594\pi$
$73$		−	4.63351i		−	0.542311i	0.962536	−	0.271156i	$-0.0874058\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.1366	−	10.1366i	−1.15517	−	1.15517i
$78$		0			0
$79$		−	6.95436i		−	0.782426i	−0.920300	−	0.391213i	$-0.872056\pi$
$79$		−	6.95436i		−	0.782426i	0.920300	−	0.391213i	$-0.127944\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.37934			0.151402			0.0757010	−	0.997131i	$-0.475881\pi$
$83$	1.37934			0.151402			0.0757010	+	0.997131i	$0.475881\pi$
$84$		0			0
$85$	4.49983	+	5.53723i	0.488075	+	0.600597i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.19797	−	1.19797i	0.126984	−	0.126984i	−0.640758	−	0.767743i	$-0.721380\pi$
$89$	1.19797	−	1.19797i	0.126984	−	0.126984i	0.767743	+	0.640758i	$0.221380\pi$
$90$		0			0
$91$	7.49983	−	6.38638i	0.786196	−	0.669475i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.884040	−	1.08785i	−0.0907006	−	0.111611i
$96$		0			0
$97$		−	9.56171i		−	0.970845i	−0.874280	−	0.485422i	$-0.838666\pi$
$97$		−	9.56171i		−	0.970845i	0.874280	−	0.485422i	$-0.161334\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	3.03625i		−	0.302118i	−0.988525	−	0.151059i	$-0.951732\pi$
$101$		−	3.03625i		−	0.302118i	0.988525	−	0.151059i	$-0.0482683\pi$
$102$		0			0
$103$	−10.1280	+	10.1280i	−0.997942	+	0.997942i	−0.999998	−	0.00205572i	$-0.999346\pi$
$103$	−10.1280	+	10.1280i	−0.997942	+	0.997942i	0.00205572	+	0.999998i	$0.499346\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.0806121	+	0.0806121i	−0.00779307	+	0.00779307i	−0.710993	−	0.703200i	$-0.751754\pi$
$107$	−0.0806121	+	0.0806121i	−0.00779307	+	0.00779307i	0.703200	+	0.710993i	$0.251754\pi$
$108$		0			0
$109$	−1.43370	−	1.43370i	−0.137324	−	0.137324i	0.635103	−	0.772427i	$-0.280958\pi$
$109$	−1.43370	−	1.43370i	−0.137324	−	0.137324i	−0.772427	+	0.635103i	$0.780958\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.97257	+	5.97257i	0.561853	+	0.561853i	0.929833	−	0.367981i	$-0.119951\pi$
$113$	5.97257	+	5.97257i	0.561853	+	0.561853i	−0.367981	+	0.929833i	$0.619951\pi$
$114$		0			0
$115$	−1.12835	+	0.916952i	−0.105219	+	0.0855062i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.16436	−	6.16436i	−0.565086	−	0.565086i
$120$		0			0
$121$			16.5319i			1.50290i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.6509	−	3.39975i	−0.952646	−	0.304083i
$126$		0			0
$127$	4.12835	+	4.12835i	0.366332	+	0.366332i	0.866138	−	0.499806i	$-0.166595\pi$
$127$	4.12835	+	4.12835i	0.366332	+	0.366332i	−0.499806	+	0.866138i	$0.666595\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.9104			−1.73958			−0.869792	−	0.493419i	$-0.835747\pi$
$131$	−19.9104			−1.73958			−0.869792	+	0.493419i	$0.835747\pi$
$132$		0			0
$133$	1.21106	+	1.21106i	0.105012	+	0.105012i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−19.6971			−1.68283			−0.841417	−	0.540387i	$-0.818278\pi$
$137$	−19.6971			−1.68283			−0.841417	+	0.540387i	$0.818278\pi$
$138$		0			0
$139$		−	6.75362i		−	0.572835i	−0.958105	−	0.286417i	$-0.907536\pi$
$139$		−	6.75362i		−	0.572835i	0.958105	−	0.286417i	$-0.0924644\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.8581	−	1.51211i	−1.57699	−	0.126449i
$144$		0			0
$145$	1.87698	−	18.1601i	0.155875	−	1.50811i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.90255	+	6.90255i	0.565479	+	0.565479i	0.930859	−	0.365380i	$-0.119061\pi$
$149$	6.90255	+	6.90255i	0.565479	+	0.565479i	−0.365380	+	0.930859i	$0.619061\pi$
$150$		0			0
$151$	−13.0354	−	13.0354i	−1.06080	−	1.06080i	−0.998028	−	0.0627770i	$-0.980004\pi$
$151$	−13.0354	−	13.0354i	−1.06080	−	1.06080i	−0.0627770	−	0.998028i	$-0.519996\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.90411	−	8.04857i	0.795518	−	0.646477i
$156$		0			0
$157$	−15.1654	+	15.1654i	−1.21033	+	1.21033i	−0.239414	+	0.970918i	$0.576955\pi$
$157$	−15.1654	+	15.1654i	−1.21033	+	1.21033i	−0.970918	+	0.239414i	$0.923045\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.25614	−	1.25614i	0.0989978	−	0.0989978i
$162$		0			0
$163$			7.96926i			0.624201i	0.950049	+	0.312100i	$0.101033\pi$
$163$			7.96926i			0.624201i	−0.950049	+	0.312100i	$0.898967\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.96106			−0.693428			−0.346714	−	0.937971i	$-0.612703\pi$
$167$	−8.96106			−0.693428			−0.346714	+	0.937971i	$0.612703\pi$
$168$		0			0
$169$	2.07145	−	12.8339i	0.159343	−	0.987223i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.5852	−	15.5852i	1.18492	−	1.18492i	0.206467	−	0.978454i	$-0.433803\pi$
$173$	15.5852	−	15.5852i	1.18492	−	1.18492i	0.978454	−	0.206467i	$-0.0661967\pi$
$174$		0			0
$175$	13.3715	+	2.79393i	1.01079	+	0.211201i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.0827			−1.05259			−0.526294	−	0.850303i	$-0.676419\pi$
$179$	−14.0827			−1.05259			−0.526294	+	0.850303i	$0.676419\pi$
$180$		0			0
$181$		−	10.3400i		−	0.768566i	−0.923215	−	0.384283i	$-0.874449\pi$
$181$		−	10.3400i		−	0.768566i	0.923215	−	0.384283i	$-0.125551\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	17.3236	+	1.79052i	1.27365	+	0.131642i
$186$		0			0
$187$			16.7430i			1.22437i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.6660			−1.20591			−0.602954	−	0.797776i	$-0.706010\pi$
$191$	−16.6660			−1.20591			−0.602954	+	0.797776i	$0.706010\pi$
$192$		0			0
$193$			2.93185i			0.211039i	0.994417	+	0.105520i	$0.0336506\pi$
$193$			2.93185i			0.211039i	−0.994417	+	0.105520i	$0.966349\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.3812i			1.02462i	0.858801	+	0.512310i	$0.171210\pi$
$197$			14.3812i			1.02462i	−0.858801	+	0.512310i	$0.828790\pi$
$198$		0			0
$199$	22.0446			1.56270			0.781351	−	0.624092i	$-0.214531\pi$
$199$	22.0446			1.56270			0.781351	+	0.624092i	$0.214531\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			22.3064i			1.56560i
$204$		0			0
$205$	−10.2337	+	8.31641i	−0.714753	+	0.580843i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.28934i		−	0.227528i
$210$		0			0
$211$	−7.76946			−0.534871			−0.267436	−	0.963576i	$-0.586176\pi$
$211$	−7.76946			−0.534871			−0.267436	+	0.963576i	$0.586176\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.31483	−	3.50644i	0.294269	−	0.239137i
$216$		0			0
$217$	−11.0258	+	11.0258i	−0.748481	+	0.748481i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−11.4682	−	0.919559i	−0.771433	−	0.0618562i
$222$		0			0
$223$	20.2710			1.35744			0.678722	−	0.734395i	$-0.262534\pi$
$223$	20.2710			1.35744			0.678722	+	0.734395i	$0.262534\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3.87959i		−	0.257498i	−0.991677	−	0.128749i	$-0.958904\pi$
$227$		−	3.87959i		−	0.257498i	0.991677	−	0.128749i	$-0.0410961\pi$
$228$		0			0
$229$	6.73171	−	6.73171i	0.444844	−	0.444844i	−0.448792	−	0.893636i	$-0.648146\pi$
$229$	6.73171	−	6.73171i	0.444844	−	0.444844i	0.893636	+	0.448792i	$0.148146\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.24548	−	9.24548i	0.605692	−	0.605692i	−0.336125	−	0.941817i	$-0.609117\pi$
$233$	9.24548	−	9.24548i	0.605692	−	0.605692i	0.941817	+	0.336125i	$0.109117\pi$
$234$		0			0
$235$	4.56521	+	0.471849i	0.297802	+	0.0307800i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−13.2543	−	13.2543i	−0.857349	−	0.857349i	0.133676	−	0.991025i	$-0.457322\pi$
$239$	−13.2543	−	13.2543i	−0.857349	−	0.857349i	−0.991025	+	0.133676i	$0.957322\pi$
$240$		0			0
$241$	7.38140	+	7.38140i	0.475477	+	0.475477i	0.903682	−	0.428204i	$-0.140854\pi$
$241$	7.38140	+	7.38140i	0.475477	+	0.475477i	−0.428204	+	0.903682i	$0.640854\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.03226	−	0.106692i	−0.0659489	−	0.00681631i
$246$		0			0
$247$	2.25305	+	0.180657i	0.143358	+	0.0114950i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	8.67664i		−	0.547665i	−0.961777	−	0.273832i	$-0.911709\pi$
$251$		−	8.67664i		−	0.547665i	0.961777	−	0.273832i	$-0.0882913\pi$
$252$		0			0
$253$	−3.41179			−0.214498
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.3174	+	16.3174i	1.01785	+	1.01785i	0.999838	+	0.0180122i	$0.00573376\pi$
$257$	16.3174	+	16.3174i	1.01785	+	1.01785i	0.0180122	+	0.999838i	$0.494266\pi$
$258$		0			0
$259$	−21.2789			−1.32220
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.443353	−	0.443353i	−0.0273383	−	0.0273383i	0.693305	−	0.720644i	$-0.256154\pi$
$263$	−0.443353	−	0.443353i	−0.0273383	−	0.0273383i	−0.720644	+	0.693305i	$0.756154\pi$
$264$		0			0
$265$	8.87165	+	10.9170i	0.544981	+	0.670623i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	1.81296i		−	0.110538i	−0.998471	−	0.0552690i	$-0.982398\pi$
$269$		−	1.81296i		−	0.110538i	0.998471	−	0.0552690i	$-0.0176016\pi$
$270$		0			0
$271$	−10.4542	−	10.4542i	−0.635047	−	0.635047i	0.314283	−	0.949329i	$-0.398236\pi$
$271$	−10.4542	−	10.4542i	−0.635047	−	0.635047i	−0.949329	+	0.314283i	$0.898236\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.3648	−	21.9534i	−0.866230	−	1.32384i
$276$		0			0
$277$	7.86040	+	7.86040i	0.472286	+	0.472286i	0.902654	−	0.430368i	$-0.141616\pi$
$277$	7.86040	+	7.86040i	0.472286	+	0.472286i	−0.430368	+	0.902654i	$0.641616\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0390	+	16.0390i	0.956805	+	0.956805i	0.999105	−	0.0423004i	$-0.0134687\pi$
$281$	16.0390	+	16.0390i	0.956805	+	0.956805i	−0.0423004	+	0.999105i	$0.513469\pi$
$282$		0			0
$283$	4.12835	−	4.12835i	0.245405	−	0.245405i	−0.573677	−	0.819082i	$-0.694483\pi$
$283$	4.12835	−	4.12835i	0.245405	−	0.245405i	0.819082	+	0.573677i	$0.194483\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.3927	−	11.3927i	0.672492	−	0.672492i
$288$		0			0
$289$		−	6.81811i		−	0.401066i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.92821i			0.521592i	0.965394	+	0.260796i	$0.0839849\pi$
$293$			8.92821i			0.521592i	−0.965394	+	0.260796i	$0.916015\pi$
$294$		0			0
$295$	12.8861	−	10.4718i	0.750256	−	0.609695i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.187383	−	2.33692i	0.0108366	−	0.135148i
$300$		0			0
$301$	−4.80351	+	4.80351i	−0.276869	+	0.276869i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	29.5933	+	3.05869i	1.69451	+	0.175140i
$306$		0			0
$307$	5.14419			0.293594			0.146797	−	0.989167i	$-0.453104\pi$
$307$	5.14419			0.293594			0.146797	+	0.989167i	$0.453104\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			25.6464i			1.45427i	0.686494	+	0.727136i	$0.259149\pi$
$311$			25.6464i			1.45427i	−0.686494	+	0.727136i	$0.740851\pi$
$312$		0			0
$313$	9.40721	+	9.40721i	0.531727	+	0.531727i	0.921086	−	0.389359i	$-0.127304\pi$
$313$	9.40721	+	9.40721i	0.531727	+	0.531727i	−0.389359	+	0.921086i	$0.627304\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7.00560i		−	0.393474i	−0.980456	−	0.196737i	$-0.936966\pi$
$317$		−	7.00560i		−	0.393474i	0.980456	−	0.196737i	$-0.0630344\pi$
$318$		0			0
$319$	30.2931	−	30.2931i	1.69609	−	1.69609i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.00034i		−	0.111302i
$324$		0			0
$325$	15.8260	−	8.63351i	0.877869	−	0.478901i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.60754			−0.309154
$330$		0			0
$331$	0.205726	−	0.205726i	0.0113077	−	0.0113077i	−0.701430	−	0.712738i	$-0.747455\pi$
$331$	0.205726	−	0.205726i	0.0113077	−	0.0113077i	0.712738	+	0.701430i	$0.247455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.55220	+	24.6930i	−0.139442	+	1.34912i
$336$		0			0
$337$	−3.96225	−	3.96225i	−0.215838	−	0.215838i	0.590904	−	0.806742i	$-0.298771\pi$
$337$	−3.96225	−	3.96225i	−0.215838	−	0.215838i	−0.806742	+	0.590904i	$0.798771\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	29.9471			1.62173
$342$		0			0
$343$	−17.8564			−0.964155
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.348607	−	0.348607i	0.0187142	−	0.0187142i	−0.697688	−	0.716402i	$-0.745788\pi$
$347$	0.348607	−	0.348607i	0.0187142	−	0.0187142i	0.716402	+	0.697688i	$0.245788\pi$
$348$		0			0
$349$	9.62527	−	9.62527i	0.515229	−	0.515229i	−0.400895	−	0.916124i	$-0.631301\pi$
$349$	9.62527	−	9.62527i	0.515229	−	0.515229i	0.916124	+	0.400895i	$0.131301\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	27.7787			1.47851			0.739255	−	0.673425i	$-0.235178\pi$
$353$	27.7787			1.47851			0.739255	+	0.673425i	$0.235178\pi$
$354$		0			0
$355$	−15.1428	+	12.3058i	−0.803694	+	0.653122i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.5633	−	18.5633i	0.979732	−	0.979732i	−0.0200665	−	0.999799i	$-0.506388\pi$
$359$	18.5633	−	18.5633i	0.979732	−	0.979732i	0.999799	+	0.0200665i	$0.00638781\pi$
$360$		0			0
$361$		−	18.6070i		−	0.979316i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.06520	+	10.3059i	−0.0557549	+	0.539437i
$366$		0			0
$367$	−22.6973	+	22.6973i	−1.18479	+	1.18479i	−0.206302	+	0.978488i	$0.566143\pi$
$367$	−22.6973	+	22.6973i	−1.18479	+	1.18479i	−0.978488	+	0.206302i	$0.933857\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.1534	−	12.1534i	−0.630971	−	0.630971i
$372$		0			0
$373$	24.7837	+	24.7837i	1.28325	+	1.28325i	0.938807	+	0.344443i	$0.111932\pi$
$373$	24.7837	+	24.7837i	1.28325	+	1.28325i	0.344443	+	0.938807i	$0.388068\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.0856	+	22.4131i	0.982960	+	1.15434i
$378$		0			0
$379$	−19.0615	−	19.0615i	−0.979126	−	0.979126i	0.0206610	−	0.999787i	$-0.493423\pi$
$379$	−19.0615	−	19.0615i	−0.979126	−	0.979126i	−0.999787	+	0.0206610i	$0.993423\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.45217			−0.482983			−0.241492	−	0.970403i	$-0.577637\pi$
$383$	−9.45217			−0.482983			−0.241492	+	0.970403i	$0.577637\pi$
$384$		0			0
$385$	20.2157	+	24.8763i	1.03029	+	1.26781i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	21.2829			1.07909			0.539544	−	0.841958i	$-0.318597\pi$
$389$	21.2829			1.07909			0.539544	+	0.841958i	$0.318597\pi$
$390$		0			0
$391$	−2.07481			−0.104928
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.59874	+	15.4680i	−0.0804411	+	0.778280i
$396$		0			0
$397$	−34.7791			−1.74551			−0.872756	−	0.488156i	$-0.837670\pi$
$397$	−34.7791			−1.74551			−0.872756	+	0.488156i	$0.837670\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.4397	+	16.4397i	0.820961	+	0.820961i	0.986246	−	0.165285i	$-0.0528543\pi$
$401$	16.4397	+	16.4397i	0.820961	+	0.820961i	−0.165285	+	0.986246i	$0.552854\pi$
$402$		0			0
$403$	−1.64476	+	20.5124i	−0.0819313	+	1.02180i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	28.8977	+	28.8977i	1.43240	+	1.43240i
$408$		0			0
$409$	−16.6140	−	16.6140i	−0.821510	−	0.821510i	0.164814	−	0.986325i	$-0.447298\pi$
$409$	−16.6140	−	16.6140i	−0.821510	−	0.821510i	−0.986325	+	0.164814i	$0.947298\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.3455	+	14.3455i	−0.705895	+	0.705895i
$414$		0			0
$415$	−3.06795	−	0.317096i	−0.150600	−	0.0155656i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			11.0464i			0.539653i	0.962909	+	0.269826i	$0.0869663\pi$
$419$			11.0464i			0.539653i	−0.962909	+	0.269826i	$0.913034\pi$
$420$		0			0
$421$	9.79650	−	9.79650i	0.477452	−	0.477452i	−0.426864	−	0.904316i	$-0.640382\pi$
$421$	9.79650	−	9.79650i	0.477452	−	0.477452i	0.904316	+	0.426864i	$0.140382\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.73565	−	13.3505i	−0.423741	−	0.647594i
$426$		0			0
$427$	−36.3500			−1.75910
$428$		0			0
$429$		0			0
$430$		0			0
$431$	24.9687	−	24.9687i	1.20270	−	1.20270i	0.229358	−	0.973342i	$-0.426337\pi$
$431$	24.9687	−	24.9687i	1.20270	−	1.20270i	0.973342	−	0.229358i	$-0.0736625\pi$
$432$		0			0
$433$	21.4489	−	21.4489i	1.03077	−	1.03077i	0.0312548	−	0.999511i	$-0.490050\pi$
$433$	21.4489	−	21.4489i	1.03077	−	1.03077i	0.999511	−	0.0312548i	$-0.00995034\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.407619			0.0194991
$438$		0			0
$439$	−12.2077			−0.582644			−0.291322	−	0.956625i	$-0.594095\pi$
$439$	−12.2077			−0.582644			−0.291322	+	0.956625i	$0.594095\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−27.2503	−	27.2503i	−1.29470	−	1.29470i	−0.931845	−	0.362858i	$-0.881801\pi$
$443$	−27.2503	−	27.2503i	−1.29470	−	1.29470i	−0.362858	−	0.931845i	$-0.618199\pi$
$444$		0			0
$445$	−2.93994	+	2.38914i	−0.139367	+	0.113256i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.764347	+	0.764347i	−0.0360718	+	0.0360718i	−0.724913	−	0.688841i	$-0.758120\pi$
$449$	−0.764347	+	0.764347i	−0.0360718	+	0.0360718i	0.688841	+	0.724913i	$0.258120\pi$
$450$		0			0
$451$	−30.9437			−1.45708
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−18.1494	+	12.4806i	−0.850858	+	0.585099i
$456$		0			0
$457$		−	2.36615i		−	0.110684i	−0.998467	−	0.0553419i	$-0.982375\pi$
$457$		−	2.36615i		−	0.110684i	0.998467	−	0.0553419i	$-0.0176249\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.95191	−	2.95191i	0.137484	−	0.137484i	−0.635015	−	0.772500i	$-0.719006\pi$
$461$	2.95191	−	2.95191i	0.137484	−	0.137484i	0.772500	+	0.635015i	$0.219006\pi$
$462$		0			0
$463$			14.6706i			0.681799i	0.940100	+	0.340900i	$0.110732\pi$
$463$			14.6706i			0.681799i	−0.940100	+	0.340900i	$0.889268\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.9419	−	24.9419i	−1.15417	−	1.15417i	−0.985708	−	0.168464i	$-0.946119\pi$
$467$	−24.9419	−	24.9419i	−1.15417	−	1.15417i	−0.168464	−	0.985708i	$-0.553881\pi$
$468$		0			0
$469$		−	30.3308i		−	1.40055i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13.0468			0.599891
$474$		0			0
$475$	1.71621	+	2.62285i	0.0787453	+	0.120345i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.4235	−	17.4235i	0.796099	−	0.796099i	−0.186379	−	0.982478i	$-0.559675\pi$
$479$	17.4235	−	17.4235i	0.796099	−	0.796099i	0.982478	+	0.186379i	$0.0596752\pi$
$480$		0			0
$481$	−21.3807	+	18.2065i	−0.974877	+	0.830144i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.19814	+	21.2673i	−0.0998124	+	0.965700i
$486$		0			0
$487$			34.9299i			1.58283i	0.611281	+	0.791413i	$0.290654\pi$
$487$			34.9299i			1.58283i	−0.611281	+	0.791413i	$0.709346\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			34.8553i			1.57300i	0.617591	+	0.786500i	$0.288109\pi$
$491$			34.8553i			1.57300i	−0.617591	+	0.786500i	$0.711891\pi$
$492$		0			0
$493$	18.4221	−	18.4221i	0.829690	−	0.829690i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.8578	−	16.8578i	0.756174	−	0.756174i
$498$		0			0
$499$	−22.0430	−	22.0430i	−0.986780	−	0.986780i	0.0131342	−	0.999914i	$-0.495819\pi$
$499$	−22.0430	−	22.0430i	−0.986780	−	0.986780i	−0.999914	+	0.0131342i	$0.995819\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10.6412	−	10.6412i	−0.474470	−	0.474470i	0.428888	−	0.903358i	$-0.358906\pi$
$503$	−10.6412	−	10.6412i	−0.474470	−	0.474470i	−0.903358	+	0.428888i	$0.858906\pi$
$504$		0			0
$505$	−0.698002	+	6.75328i	−0.0310607	+	0.300517i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.3501	+	20.3501i	0.902001	+	0.902001i	0.995609	−	0.0936080i	$-0.0298400\pi$
$509$	20.3501	+	20.3501i	0.902001	+	0.902001i	−0.0936080	+	0.995609i	$0.529840\pi$
$510$		0			0
$511$		−	12.6590i		−	0.560000i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.8552	−	20.1986i	1.09525	−	0.890056i
$516$		0			0
$517$	7.61530	+	7.61530i	0.334920	+	0.334920i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.9246			−0.872911			−0.436456	−	0.899726i	$-0.643766\pi$
$521$	−19.9246			−0.872911			−0.436456	+	0.899726i	$0.643766\pi$
$522$		0			0
$523$	−2.79926	−	2.79926i	−0.122403	−	0.122403i	0.643252	−	0.765655i	$-0.277585\pi$
$523$	−2.79926	−	2.79926i	−0.122403	−	0.122403i	−0.765655	+	0.643252i	$0.777585\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.2117			0.793315
$528$		0			0
$529$			22.5772i			0.981618i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.69949	−	21.1950i	0.0736132	−	0.918059i
$534$		0			0
$535$	0.197831	−	0.160767i	0.00855297	−	0.00695057i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.72193	−	1.72193i	−0.0741689	−	0.0741689i
$540$		0			0
$541$	27.5352	+	27.5352i	1.18383	+	1.18383i	0.978744	+	0.205088i	$0.0657480\pi$
$541$	27.5352	+	27.5352i	1.18383	+	1.18383i	0.205088	+	0.978744i	$0.434252\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.85928	+	3.51847i	0.122478	+	0.150714i
$546$		0			0
$547$	−4.46775	+	4.46775i	−0.191027	+	0.191027i	−0.796140	−	0.605113i	$-0.793128\pi$
$547$	−4.46775	+	4.46775i	−0.191027	+	0.191027i	0.605113	+	0.796140i	$0.293128\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.61922	+	3.61922i	−0.154184	+	0.154184i
$552$		0			0
$553$		−	18.9997i		−	0.807948i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22.0198			0.933011			0.466505	−	0.884518i	$-0.345513\pi$
$557$	22.0198			0.933011			0.466505	+	0.884518i	$0.345513\pi$
$558$		0			0
$559$	−0.716556	+	8.93644i	−0.0303071	+	0.377971i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.4543	+	28.4543i	−1.19921	+	1.19921i	−0.224802	+	0.974405i	$0.572173\pi$
$563$	−28.4543	+	28.4543i	−1.19921	+	1.19921i	−0.974405	+	0.224802i	$0.927827\pi$
$564$		0			0
$565$	−11.9113	−	14.6573i	−0.501111	−	0.616639i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.5309			0.693012			0.346506	−	0.938048i	$-0.387368\pi$
$569$	16.5309			0.693012			0.346506	+	0.938048i	$0.387368\pi$
$570$		0			0
$571$		−	29.2958i		−	1.22599i	−0.790086	−	0.612996i	$-0.789964\pi$
$571$		−	29.2958i		−	1.22599i	0.790086	−	0.612996i	$-0.210036\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.72049	−	1.78011i	0.113452	−	0.0742355i
$576$		0			0
$577$		−	17.1164i		−	0.712566i	−0.934378	−	0.356283i	$-0.884044\pi$
$577$		−	17.1164i		−	0.712566i	0.934378	−	0.356283i	$-0.115956\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.76842			0.156341
$582$		0			0
$583$			33.0096i			1.36712i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	15.6249i		−	0.644909i	−0.946585	−	0.322454i	$-0.895492\pi$
$587$		−	15.6249i		−	0.644909i	0.946585	−	0.322454i	$-0.104508\pi$
$588$		0			0
$589$	−3.57789			−0.147424
$590$		0			0
$591$		0			0
$592$		0			0
$593$			42.5033i			1.74540i	0.488254	+	0.872702i	$0.337634\pi$
$593$			42.5033i			1.74540i	−0.488254	+	0.872702i	$0.662366\pi$
$594$		0			0
$595$	12.2938	+	15.1280i	0.503995	+	0.620188i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			34.8695i			1.42473i	0.701810	+	0.712364i	$0.252375\pi$
$599$			34.8695i			1.42473i	−0.701810	+	0.712364i	$0.747625\pi$
$600$		0			0
$601$	−35.3949			−1.44379			−0.721895	−	0.692003i	$-0.756728\pi$
$601$	−35.3949			−1.44379			−0.721895	+	0.692003i	$0.756728\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.80052	−	36.7706i	0.154513	−	1.49494i
$606$		0			0
$607$	22.2111	−	22.2111i	0.901519	−	0.901519i	−0.0940487	−	0.995568i	$-0.529981\pi$
$607$	22.2111	−	22.2111i	0.901519	−	0.901519i	0.995568	+	0.0940487i	$0.0299809\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.63438	+	4.79788i	−0.227943	+	0.194102i
$612$		0			0
$613$	−11.6332			−0.469859			−0.234930	−	0.972012i	$-0.575486\pi$
$613$	−11.6332			−0.469859			−0.234930	+	0.972012i	$0.575486\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	33.9098i		−	1.36516i	−0.730813	−	0.682578i	$-0.760859\pi$
$617$		−	33.9098i		−	1.36516i	0.730813	−	0.682578i	$-0.239141\pi$
$618$		0			0
$619$	22.2166	−	22.2166i	0.892962	−	0.892962i	−0.101839	−	0.994801i	$-0.532473\pi$
$619$	22.2166	−	22.2166i	0.892962	−	0.892962i	0.994801	+	0.101839i	$0.0324726\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.27291	−	3.27291i	0.131126	−	0.131126i
$624$		0			0
$625$	22.9084	+	10.0103i	0.916335	+	0.400413i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.5735	+	17.5735i	0.700702	+	0.700702i
$630$		0			0
$631$	12.3304	+	12.3304i	0.490866	+	0.490866i	0.908579	−	0.417713i	$-0.137168\pi$
$631$	12.3304	+	12.3304i	0.490866	+	0.490866i	−0.417713	+	0.908579i	$0.637168\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.23329	−	10.1314i	−0.326728	−	0.402053i
$636$		0			0
$637$	1.27402	−	1.08487i	0.0504784	−	0.0429843i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	3.41771i		−	0.134991i	−0.997720	−	0.0674957i	$-0.978499\pi$
$641$		−	3.41771i		−	0.134991i	0.997720	−	0.0674957i	$-0.0215009\pi$
$642$		0			0
$643$	14.8260			0.584681			0.292340	−	0.956314i	$-0.405566\pi$
$643$	14.8260			0.584681			0.292340	+	0.956314i	$0.405566\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.7366	+	29.7366i	1.16907	+	1.16907i	0.982429	+	0.186638i	$0.0597591\pi$
$647$	29.7366	+	29.7366i	1.16907	+	1.16907i	0.186638	+	0.982429i	$0.440241\pi$
$648$		0			0
$649$	38.9636			1.52946
$650$		0			0
$651$		0			0
$652$		0			0
$653$	27.3474	+	27.3474i	1.07019	+	1.07019i	0.997344	+	0.0728416i	$0.0232068\pi$
$653$	27.3474	+	27.3474i	1.07019	+	1.07019i	0.0728416	+	0.997344i	$0.476793\pi$
$654$		0			0
$655$	44.2852	+	4.57721i	1.73037	+	0.178846i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	12.1293i		−	0.472491i	−0.971693	−	0.236245i	$-0.924083\pi$
$659$		−	12.1293i		−	0.472491i	0.971693	−	0.236245i	$-0.0759169\pi$
$660$		0			0
$661$	6.44954	+	6.44954i	0.250858	+	0.250858i	0.821322	−	0.570464i	$-0.193237\pi$
$661$	6.44954	+	6.44954i	0.250858	+	0.250858i	−0.570464	+	0.821322i	$0.693237\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.41524	−	2.97206i	−0.0936591	−	0.115252i
$666$		0			0
$667$	3.75396	+	3.75396i	0.145354	+	0.145354i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	49.3649	+	49.3649i	1.90571	+	1.90571i
$672$		0			0
$673$	−0.457094	+	0.457094i	−0.0176197	+	0.0176197i	−0.715862	−	0.698242i	$-0.753966\pi$
$673$	−0.457094	+	0.457094i	−0.0176197	+	0.0176197i	0.698242	+	0.715862i	$0.253966\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.5475	−	33.5475i	1.28933	−	1.28933i	0.354142	−	0.935192i	$-0.384773\pi$
$677$	33.5475	−	33.5475i	1.28933	−	1.28933i	0.935192	−	0.354142i	$-0.115227\pi$
$678$		0			0
$679$		−	26.1231i		−	1.00251i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	43.6290i		−	1.66942i	−0.550692	−	0.834709i	$-0.685636\pi$
$683$		−	43.6290i		−	1.66942i	0.550692	−	0.834709i	$-0.314364\pi$
$684$		0			0
$685$	43.8106	+	4.52815i	1.67392	+	0.173012i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22.6101	−	1.81296i	−0.861376	−	0.0690682i
$690$		0			0
$691$	−9.65769	+	9.65769i	−0.367396	+	0.367396i	−0.866527	−	0.499131i	$-0.833653\pi$
$691$	−9.65769	+	9.65769i	−0.367396	+	0.367396i	0.499131	+	0.866527i	$0.333653\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.55259	+	15.0215i	−0.0588930	+	0.569799i
$696$		0			0
$697$	−18.8178			−0.712774
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.28899i		−	0.0486846i	−0.999704	−	0.0243423i	$-0.992251\pi$
$701$		−	1.28899i		−	0.0486846i	0.999704	−	0.0243423i	$-0.00774917\pi$
$702$		0			0
$703$	−3.45251	−	3.45251i	−0.130214	−	0.130214i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	8.29518i		−	0.311972i
$708$		0			0
$709$	13.6715	−	13.6715i	0.513444	−	0.513444i	−0.402136	−	0.915580i	$-0.631732\pi$
$709$	13.6715	−	13.6715i	0.513444	−	0.513444i	0.915580	+	0.402136i	$0.131732\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.71109i			0.138981i
$714$		0			0
$715$	41.5969	+	7.69855i	1.55564	+	0.287909i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.21195			0.194373			0.0971864	−	0.995266i	$-0.469016\pi$
$719$	5.21195			0.194373			0.0971864	+	0.995266i	$0.469016\pi$
$720$		0			0
$721$	−27.6702	+	27.6702i	−1.03049	+	1.03049i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.34963	+	39.9605i	−0.310098	+	1.48410i
$726$		0			0
$727$	−6.03706	−	6.03706i	−0.223902	−	0.223902i	0.586237	−	0.810139i	$-0.300609\pi$
$727$	−6.03706	−	6.03706i	−0.223902	−	0.223902i	−0.810139	+	0.586237i	$0.800609\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.93412			0.293454
$732$		0			0
$733$	14.1301			0.521907			0.260953	−	0.965351i	$-0.415963\pi$
$733$	14.1301			0.521907			0.260953	+	0.965351i	$0.415963\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−41.1906	+	41.1906i	−1.51728	+	1.51728i
$738$		0			0
$739$	22.4164	−	22.4164i	0.824602	−	0.824602i	−0.162162	−	0.986764i	$-0.551847\pi$
$739$	22.4164	−	22.4164i	0.824602	−	0.824602i	0.986764	+	0.162162i	$0.0518467\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	34.3694			1.26089			0.630445	−	0.776234i	$-0.282872\pi$
$743$	34.3694			1.26089			0.630445	+	0.776234i	$0.282872\pi$
$744$		0			0
$745$	−13.7660	−	16.9396i	−0.504345	−	0.620619i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.220236	+	0.220236i	−0.00804726	+	0.00804726i
$750$		0			0
$751$		−	37.4753i		−	1.36749i	−0.729720	−	0.683746i	$-0.760350\pi$
$751$		−	37.4753i		−	1.36749i	0.729720	−	0.683746i	$-0.239650\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25.9969	+	31.9903i	0.946122	+	1.16424i
$756$		0			0
$757$	−20.1122	+	20.1122i	−0.730989	+	0.730989i	−0.970816	−	0.239827i	$-0.922909\pi$
$757$	−20.1122	+	20.1122i	−0.730989	+	0.730989i	0.239827	+	0.970816i	$0.422909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.9729	−	36.9729i	−1.34027	−	1.34027i	−0.895792	−	0.444473i	$-0.853391\pi$
$761$	−36.9729	−	36.9729i	−1.34027	−	1.34027i	−0.444473	−	0.895792i	$-0.646609\pi$
$762$		0			0
$763$	−3.91695	−	3.91695i	−0.141803	−	0.141803i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.13997	+	26.6883i	−0.0772697	+	0.963660i
$768$		0			0
$769$	15.1015	+	15.1015i	0.544574	+	0.544574i	0.924866	−	0.380292i	$-0.124177\pi$
$769$	15.1015	+	15.1015i	0.544574	+	0.544574i	−0.380292	+	0.924866i	$0.624177\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−32.1421			−1.15607			−0.578036	−	0.816011i	$-0.696181\pi$
$773$	−32.1421			−1.15607			−0.578036	+	0.816011i	$0.696181\pi$
$774$		0			0
$775$	−23.8792	+	15.6249i	−0.857766	+	0.561264i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.69696			0.132457
$780$		0			0
$781$	−45.7872			−1.63840
$782$		0			0
$783$		0			0
$784$		0			0
$785$	37.2176	−	30.2448i	1.32835	−	1.07948i
$786$		0			0
$787$	−4.41239			−0.157285			−0.0786423	−	0.996903i	$-0.525058\pi$
$787$	−4.41239			−0.157285			−0.0786423	+	0.996903i	$0.525058\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	16.3174	+	16.3174i	0.580179	+	0.580179i
$792$		0			0
$793$	−36.5240	+	31.1015i	−1.29700	+	1.10445i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.0040	−	11.0040i	−0.389781	−	0.389781i	0.484828	−	0.874609i	$-0.338882\pi$
$797$	−11.0040	−	11.0040i	−0.389781	−	0.389781i	−0.874609	+	0.484828i	$0.838882\pi$
$798$		0			0
$799$	4.63108	+	4.63108i	0.163836	+	0.163836i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.1915	+	17.1915i	−0.606674	+	0.606674i
$804$		0			0
$805$	−3.08271	+	2.50516i	−0.108651	+	0.0882952i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	29.1247i		−	1.02397i	−0.858994	−	0.511986i	$-0.828910\pi$
$809$		−	29.1247i		−	1.02397i	0.858994	−	0.511986i	$-0.171090\pi$
$810$		0			0
$811$	18.9557	−	18.9557i	0.665624	−	0.665624i	−0.291076	−	0.956700i	$-0.594013\pi$
$811$	18.9557	−	18.9557i	0.665624	−	0.665624i	0.956700	+	0.291076i	$0.0940131\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.83205	−	17.7254i	0.0641739	−	0.620893i
$816$		0			0
$817$	−1.55874			−0.0545335
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.2759	−	33.2759i	1.16134	−	1.16134i	0.177154	−	0.984183i	$-0.443311\pi$
$821$	33.2759	−	33.2759i	1.16134	−	1.16134i	0.984183	−	0.177154i	$-0.0566890\pi$
$822$		0			0
$823$	−16.2411	+	16.2411i	−0.566130	+	0.566130i	−0.931042	−	0.364912i	$-0.881099\pi$
$823$	−16.2411	+	16.2411i	−0.566130	+	0.566130i	0.364912	+	0.931042i	$0.381099\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−19.5342			−0.679269			−0.339635	−	0.940557i	$-0.610303\pi$
$827$	−19.5342			−0.679269			−0.339635	+	0.940557i	$0.610303\pi$
$828$		0			0
$829$	−2.12080			−0.0736583			−0.0368291	−	0.999322i	$-0.511726\pi$
$829$	−2.12080			−0.0736583			−0.0368291	+	0.999322i	$0.511726\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.04716	−	1.04716i	−0.0362819	−	0.0362819i
$834$		0			0
$835$	19.9314	+	2.06006i	0.689753	+	0.0712912i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	35.9943	−	35.9943i	1.24266	−	1.24266i	0.283768	−	0.958893i	$-0.408416\pi$
$839$	35.9943	−	35.9943i	1.24266	−	1.24266i	0.958893	−	0.283768i	$-0.0915845\pi$
$840$		0			0
$841$	−37.6623			−1.29870
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.55775	+	28.0692i	−0.259995	+	0.965610i
$846$		0			0
$847$			45.1660i			1.55192i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.58104	+	3.58104i	−0.122756	+	0.122756i
$852$		0			0
$853$		−	4.64604i		−	0.159077i	−0.996832	−	0.0795386i	$-0.974655\pi$
$853$		−	4.64604i		−	0.159077i	0.996832	−	0.0795386i	$-0.0253447\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.82846	+	9.82846i	0.335734	+	0.335734i	0.854759	−	0.519025i	$-0.173705\pi$
$857$	9.82846	+	9.82846i	0.335734	+	0.335734i	−0.519025	+	0.854759i	$0.673705\pi$
$858$		0			0
$859$			26.5316i			0.905245i	0.891702	+	0.452623i	$0.149512\pi$
$859$			26.5316i			0.905245i	−0.891702	+	0.452623i	$0.850488\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.4254			0.354885			0.177443	−	0.984131i	$-0.443218\pi$
$863$	10.4254			0.354885			0.177443	+	0.984131i	$0.443218\pi$
$864$		0			0
$865$	−38.2478	+	31.0820i	−1.30046	+	1.05682i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25.8024	+	25.8024i	−0.875287	+	0.875287i
$870$		0			0
$871$	−25.9515	−	30.4760i	−0.879331	−	1.03264i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−29.0988	−	9.28828i	−0.983719	−	0.314001i
$876$		0			0
$877$		−	40.4476i		−	1.36582i	−0.730504	−	0.682909i	$-0.760715\pi$
$877$		−	40.4476i		−	1.36582i	0.730504	−	0.682909i	$-0.239285\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17.1414i			0.577509i	0.957403	+	0.288755i	$0.0932411\pi$
$881$			17.1414i			0.577509i	−0.957403	+	0.288755i	$0.906759\pi$
$882$		0			0
$883$	−20.8713	+	20.8713i	−0.702375	+	0.702375i	−0.964920	−	0.262544i	$-0.915438\pi$
$883$	−20.8713	+	20.8713i	−0.702375	+	0.702375i	0.262544	+	0.964920i	$0.415438\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.5641	−	18.5641i	0.623322	−	0.623322i	−0.323057	−	0.946379i	$-0.604711\pi$
$887$	18.5641	−	18.5641i	0.623322	−	0.623322i	0.946379	+	0.323057i	$0.104711\pi$
$888$		0			0
$889$	11.2789	+	11.2789i	0.378281	+	0.378281i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.909827	−	0.909827i	−0.0304462	−	0.0304462i
$894$		0			0
$895$	31.3229	+	3.23746i	1.04701	+	0.108216i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32.9505	−	32.9505i	−1.09896	−	1.09896i
$900$		0			0
$901$			20.0741i			0.668766i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.37706	+	22.9984i	−0.0790161	+	0.764493i
$906$		0			0
$907$	2.44920	+	2.44920i	0.0813243	+	0.0813243i	0.746599	−	0.665275i	$-0.231685\pi$
$907$	2.44920	+	2.44920i	0.0813243	+	0.0813243i	−0.665275	+	0.746599i	$0.731685\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−47.0348			−1.55833			−0.779167	−	0.626816i	$-0.784358\pi$
$911$	−47.0348			−1.55833			−0.779167	+	0.626816i	$0.784358\pi$
$912$		0			0
$913$	−5.11769	−	5.11769i	−0.169371	−	0.169371i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−54.3963			−1.79633
$918$		0			0
$919$		−	30.2517i		−	0.997911i	−0.866628	−	0.498955i	$-0.833717\pi$
$919$		−	30.2517i		−	0.997911i	0.866628	−	0.498955i	$-0.166283\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.51473	−	31.3622i	0.0827734	−	1.03230i
$924$		0			0
$925$	−38.1198	−	7.96502i	−1.25337	−	0.261888i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.1760	−	13.1760i	−0.432290	−	0.432290i	0.457117	−	0.889407i	$-0.348882\pi$
$929$	−13.1760	−	13.1760i	−0.432290	−	0.432290i	−0.889407	+	0.457117i	$0.848882\pi$
$930$		0			0
$931$	0.205726	+	0.205726i	0.00674238	+	0.00674238i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.84903	−	37.2400i	0.125877	−	1.21788i
$936$		0			0
$937$	5.99206	−	5.99206i	0.195752	−	0.195752i	−0.602424	−	0.798176i	$-0.705798\pi$
$937$	5.99206	−	5.99206i	0.195752	−	0.195752i	0.798176	+	0.602424i	$0.205798\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.3026	−	21.3026i	0.694446	−	0.694446i	−0.268761	−	0.963207i	$-0.586614\pi$
$941$	21.3026	−	21.3026i	0.694446	−	0.694446i	0.963207	+	0.268761i	$0.0866141\pi$
$942$		0			0
$943$		−	3.83459i		−	0.124871i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−53.3066			−1.73223			−0.866116	−	0.499843i	$-0.833391\pi$
$947$	−53.3066			−1.73223			−0.866116	+	0.499843i	$0.833391\pi$
$948$		0			0
$949$	−10.8312	−	12.7196i	−0.351595	−	0.412895i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.5990	−	13.5990i	0.440515	−	0.440515i	−0.451670	−	0.892185i	$-0.649172\pi$
$953$	13.5990	−	13.5990i	0.440515	−	0.440515i	0.892185	+	0.451670i	$0.149172\pi$
$954$		0			0
$955$	37.0688	+	3.83134i	1.19952	+	0.123979i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−53.8134			−1.73772
$960$		0			0
$961$		−	1.57419i		−	0.0507804i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.674003	−	6.52109i	0.0216969	−	0.209921i
$966$		0			0
$967$			17.5954i			0.565830i	0.959145	+	0.282915i	$0.0913015\pi$
$967$			17.5954i			0.565830i	−0.959145	+	0.282915i	$0.908698\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−44.1478			−1.41677			−0.708385	−	0.705826i	$-0.750576\pi$
$971$	−44.1478			−1.41677			−0.708385	+	0.705826i	$0.750576\pi$
$972$		0			0
$973$		−	18.4512i		−	0.591519i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			25.0649i			0.801897i	0.916101	+	0.400949i	$0.131319\pi$
$977$			25.0649i			0.801897i	−0.916101	+	0.400949i	$0.868681\pi$
$978$		0			0
$979$	−8.88952			−0.284110
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	41.9377i		−	1.33761i	−0.743440	−	0.668803i	$-0.766807\pi$
$983$		−	41.9377i		−	1.33761i	0.743440	−	0.668803i	$-0.233193\pi$
$984$		0			0
$985$	3.30609	−	31.9870i	0.105341	−	1.01919i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.61677i			0.0514104i
$990$		0			0
$991$	54.5655			1.73333			0.866665	−	0.498890i	$-0.166259\pi$
$991$	54.5655			1.73333			0.866665	+	0.498890i	$0.166259\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−49.0321	−	5.06783i	−1.55442	−	0.160661i
$996$		0			0
$997$	33.8336	−	33.8336i	1.07152	−	1.07152i	0.0742824	−	0.997237i	$-0.476333\pi$
$997$	33.8336	−	33.8336i	1.07152	−	1.07152i	0.997237	−	0.0742824i	$-0.0236666\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bp.h.1513.1	yes	16
3.2	odd	2	inner	2340.2.bp.h.1513.8	yes	16
5.2	odd	4		2340.2.u.h.577.5	yes	16
13.8	odd	4		2340.2.u.h.73.5	yes	16
15.2	even	4		2340.2.u.h.577.4	yes	16
39.8	even	4		2340.2.u.h.73.4	✓	16
65.47	even	4	inner	2340.2.bp.h.1477.1	yes	16
195.47	odd	4	inner	2340.2.bp.h.1477.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.u.h.73.4	✓	16	39.8	even	4
2340.2.u.h.73.5	yes	16	13.8	odd	4
2340.2.u.h.577.4	yes	16	15.2	even	4
2340.2.u.h.577.5	yes	16	5.2	odd	4
2340.2.bp.h.1477.1	yes	16	65.47	even	4	inner
2340.2.bp.h.1477.8	yes	16	195.47	odd	4	inner
2340.2.bp.h.1513.1	yes	16	1.1	even	1	trivial
2340.2.bp.h.1513.8	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.bp (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.22422 - 0.229890i) q^{5} +2.73205 q^{7} +O(q^{10})\)	\(q+(-2.22422 - 0.229890i) q^{5} +2.73205 q^{7} +(-3.71025 - 3.71025i) q^{11} +(2.74513 - 2.33758i) q^{13} +(-2.25631 - 2.25631i) q^{17} +(0.443277 + 0.443277i) q^{19} +(0.459779 - 0.459779i) q^{23} +(4.89430 + 1.02265i) q^{25} +8.16470i q^{29} +(-4.03573 + 4.03573i) q^{31} +(-6.07668 - 0.628070i) q^{35} -7.78860 q^{37} +(4.17003 - 4.17003i) q^{41} +(-1.75821 + 1.75821i) q^{43} -2.05250 q^{47} +0.464102 q^{49} +(-4.44844 - 4.44844i) q^{53} +(7.39946 + 9.10536i) q^{55} +(-5.25081 + 5.25081i) q^{59} -13.3050 q^{61} +(-6.64315 + 4.56821i) q^{65} -11.1019i q^{67} +(6.17037 - 6.17037i) q^{71} -4.63351i q^{73} +(-10.1366 - 10.1366i) q^{77} -6.95436i q^{79} +1.37934 q^{83} +(4.49983 + 5.53723i) q^{85} +(1.19797 - 1.19797i) q^{89} +(7.49983 - 6.38638i) q^{91} +(-0.884040 - 1.08785i) q^{95} -9.56171i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 16 * q + 16 * q^7	\( 16 q + 16 q^{7} + 8 q^{13} + 24 q^{25} - 32 q^{31} - 16 q^{37} + 16 q^{43} - 48 q^{49} + 8 q^{55} - 16 q^{61} - 16 q^{85} + 32 q^{91}+O(q^{100}) \) 16 * q + 16 * q^7 + 8 * q^13 + 24 * q^25 - 32 * q^31 - 16 * q^37 + 16 * q^43 - 48 * q^49 + 8 * q^55 - 16 * q^61 - 16 * q^85 + 32 * q^91

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