LMFDB - Embedded newform 2340.2.bi.d.1061.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(161,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.161"); 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.bi (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.58498535041007616.52
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 $ x^12 - 12x^9 + 72x^6 - 324*x^3 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1061.3
Root			$-1.44857 + 0.949550i$ of defining polynomial
Character	$\chi$	$=$	2340.1061
Dual form			2340.2.bi.d.161.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +(2.39812 - 2.39812i) q^{7} +(1.91323 + 1.91323i) q^{11} +(2.69240 + 2.39812i) q^{13} +1.97724 q^{17} +(2.00000 + 2.00000i) q^{19} -7.21792 q^{23} -1.00000i q^{25} -2.41225i q^{29} +(4.79624 + 4.79624i) q^{31} +3.39145i q^{35} +(0.692400 - 0.692400i) q^{37} +(-4.86968 + 4.86968i) q^{41} -11.0039i q^{43} +(7.21792 + 7.21792i) q^{47} -4.50196i q^{49} -0.0188362i q^{53} -2.70572 q^{55} -6.91340 q^{61} +(-3.59954 + 0.208088i) q^{65} +(3.38480 + 3.38480i) q^{67} +(3.90931 - 3.90931i) q^{71} +(3.09052 - 3.09052i) q^{73} +9.17632 q^{77} +15.6830 q^{79} +(0.563027 - 0.563027i) q^{83} +(-1.39812 + 1.39812i) q^{85} +(9.11232 + 9.11232i) q^{89} +(12.2077 - 0.705720i) q^{91} -2.82843 q^{95} +(-0.103840 - 0.103840i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) $ 12 * q + 12 * q^13 + 24 * q^19 - 12 * q^37 - 24 * q^55 - 12 * q^73 + 24 * q^79 + 12 * q^85 + 72 * q^91 + 36 * q^97

Character values

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

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

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
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$6$		0			0
$7$	2.39812	−	2.39812i	0.906404	−	0.906404i	−0.0895758	−	0.995980i	$-0.528551\pi$
$7$	2.39812	−	2.39812i	0.906404	−	0.906404i	0.995980	+	0.0895758i	$0.0285511\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.91323	+	1.91323i	0.576861	+	0.576861i	0.934037	−	0.357176i	$-0.116260\pi$
$11$	1.91323	+	1.91323i	0.576861	+	0.576861i	−0.357176	+	0.934037i	$0.616260\pi$
$12$		0			0
$13$	2.69240	+	2.39812i	0.746738	+	0.665119i
$13$	2.69240	+	2.39812i	0.746738	+	0.665119i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.97724			0.479551			0.239776	−	0.970828i	$-0.422926\pi$
$17$	1.97724			0.479551			0.239776	+	0.970828i	$0.422926\pi$
$18$		0			0
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	0.898272	−	0.439440i	$-0.144823\pi$
$19$	2.00000	+	2.00000i	0.458831	+	0.458831i	−0.439440	+	0.898272i	$0.644823\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.21792			−1.50504			−0.752520	−	0.658569i	$-0.771162\pi$
$23$	−7.21792			−1.50504			−0.752520	+	0.658569i	$0.771162\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	2.41225i		−	0.447944i	−0.974596	−	0.223972i	$-0.928098\pi$
$29$		−	2.41225i		−	0.447944i	0.974596	−	0.223972i	$-0.0719024\pi$
$30$		0			0
$31$	4.79624	+	4.79624i	0.861430	+	0.861430i	0.991504	−	0.130074i	$-0.0415215\pi$
$31$	4.79624	+	4.79624i	0.861430	+	0.861430i	−0.130074	+	0.991504i	$0.541522\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.39145i			0.573260i
$36$		0			0
$37$	0.692400	−	0.692400i	0.113830	−	0.113830i	−0.647898	−	0.761727i	$-0.724352\pi$
$37$	0.692400	−	0.692400i	0.113830	−	0.113830i	0.761727	+	0.647898i	$0.224352\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.86968	+	4.86968i	−0.760515	+	0.760515i	−0.976415	−	0.215900i	$-0.930731\pi$
$41$	−4.86968	+	4.86968i	−0.760515	+	0.760515i	0.215900	+	0.976415i	$0.430731\pi$
$42$		0			0
$43$		−	11.0039i		−	1.67808i	−0.544068	−	0.839041i	$-0.683117\pi$
$43$		−	11.0039i		−	1.67808i	0.544068	−	0.839041i	$-0.316883\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.21792	+	7.21792i	1.05284	+	1.05284i	0.998524	+	0.0543181i	$0.0172985\pi$
$47$	7.21792	+	7.21792i	1.05284	+	1.05284i	0.0543181	+	0.998524i	$0.482702\pi$
$48$		0			0
$49$		−	4.50196i		−	0.643137i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.0188362i		−	0.00258735i	−0.999999	−	0.00129368i	$-0.999588\pi$
$53$		−	0.0188362i		−	0.00258735i	0.999999	−	0.00129368i	$-0.000411790\pi$
$54$		0			0
$55$	−2.70572			−0.364839
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−6.91340			−0.885170			−0.442585	−	0.896727i	$-0.645939\pi$
$61$	−6.91340			−0.885170			−0.442585	+	0.896727i	$0.645939\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.59954	+	0.208088i	−0.446468	+	0.0258101i
$66$		0			0
$67$	3.38480	+	3.38480i	0.413519	+	0.413519i	0.882963	−	0.469443i	$-0.155545\pi$
$67$	3.38480	+	3.38480i	0.413519	+	0.413519i	−0.469443	+	0.882963i	$0.655545\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.90931	−	3.90931i	0.463950	−	0.463950i	−0.435998	−	0.899948i	$-0.643605\pi$
$71$	3.90931	−	3.90931i	0.463950	−	0.463950i	0.899948	+	0.435998i	$0.143605\pi$
$72$		0			0
$73$	3.09052	−	3.09052i	0.361718	−	0.361718i	−0.502727	−	0.864445i	$-0.667670\pi$
$73$	3.09052	−	3.09052i	0.361718	−	0.361718i	0.864445	+	0.502727i	$0.167670\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.17632			1.04574
$78$		0			0
$79$	15.6830			1.76448			0.882238	−	0.470804i	$-0.156036\pi$
$79$	15.6830			1.76448			0.882238	+	0.470804i	$0.156036\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.563027	−	0.563027i	0.0618002	−	0.0618002i	−0.675531	−	0.737331i	$-0.736086\pi$
$83$	0.563027	−	0.563027i	0.0618002	−	0.0618002i	0.737331	+	0.675531i	$0.236086\pi$
$84$		0			0
$85$	−1.39812	+	1.39812i	−0.151647	+	0.151647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.11232	+	9.11232i	0.965904	+	0.965904i	0.999438	−	0.0335340i	$-0.0106762\pi$
$89$	9.11232	+	9.11232i	0.965904	+	0.965904i	−0.0335340	+	0.999438i	$0.510676\pi$
$90$		0			0
$91$	12.2077	−	0.705720i	1.27971	−	0.0739795i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.82843			−0.290191
$96$		0			0
$97$	−0.103840	−	0.103840i	−0.0105433	−	0.0105433i	0.701815	−	0.712359i	$-0.252373\pi$
$97$	−0.103840	−	0.103840i	−0.0105433	−	0.0105433i	−0.712359	+	0.701815i	$0.752373\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.3193			1.12631			0.563154	−	0.826352i	$-0.309588\pi$
$101$	11.3193			1.12631			0.563154	+	0.826352i	$0.309588\pi$
$102$		0			0
$103$		−	2.02664i		−	0.199691i	−0.995003	−	0.0998453i	$-0.968165\pi$
$103$		−	2.02664i		−	0.199691i	0.995003	−	0.0998453i	$-0.0318348\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.38949i		−	0.424348i	−0.977232	−	0.212174i	$-0.931946\pi$
$107$		−	4.38949i		−	0.424348i	0.977232	−	0.212174i	$-0.0680544\pi$
$108$		0			0
$109$	−5.11716	−	5.11716i	−0.490135	−	0.490135i	0.418214	−	0.908349i	$-0.362656\pi$
$109$	−5.11716	−	5.11716i	−0.490135	−	0.490135i	−0.908349	+	0.418214i	$0.862656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.8520i			1.39716i	0.715532	+	0.698580i	$0.246184\pi$
$113$			14.8520i			1.39716i	−0.715532	+	0.698580i	$0.753816\pi$
$114$		0			0
$115$	5.10384	−	5.10384i	0.475936	−	0.475936i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.74166	−	4.74166i	0.434667	−	0.434667i
$120$		0			0
$121$		−	3.67908i		−	0.334462i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$			11.3848i			1.01024i	0.863050	+	0.505119i	$0.168551\pi$
$127$			11.3848i			1.01024i	−0.863050	+	0.505119i	$0.831449\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			16.3942i			1.43237i	0.697910	+	0.716186i	$0.254114\pi$
$131$			16.3942i			1.43237i	−0.697910	+	0.716186i	$0.745886\pi$
$132$		0			0
$133$	9.59248			0.831774
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.1704	−	12.1704i	−1.03979	−	1.03979i	−0.999175	−	0.0406159i	$-0.987068\pi$
$137$	−12.1704	−	12.1704i	−1.03979	−	1.03979i	−0.0406159	−	0.999175i	$-0.512932\pi$
$138$		0			0
$139$	11.6830			0.990939			0.495470	−	0.868625i	$-0.334996\pi$
$139$	11.6830			0.990939			0.495470	+	0.868625i	$0.334996\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.563027	+	9.73935i	0.0470827	+	0.814445i
$144$		0			0
$145$	1.70572	+	1.70572i	0.141652	+	0.141652i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.04321	−	1.04321i	0.0854631	−	0.0854631i	−0.663083	−	0.748546i	$-0.730752\pi$
$149$	1.04321	−	1.04321i	0.0854631	−	0.0854631i	0.748546	+	0.663083i	$0.230752\pi$
$150$		0			0
$151$	−8.11716	+	8.11716i	−0.660565	+	0.660565i	−0.955513	−	0.294948i	$-0.904698\pi$
$151$	−8.11716	+	8.11716i	−0.660565	+	0.660565i	0.294948	+	0.955513i	$0.404698\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.78291			−0.544816
$156$		0			0
$157$	23.1850			1.85036			0.925181	−	0.379527i	$-0.123913\pi$
$157$	23.1850			1.85036			0.925181	+	0.379527i	$0.123913\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−17.3094	+	17.3094i	−1.36417	+	1.36417i
$162$		0			0
$163$	−5.99060	+	5.99060i	−0.469220	+	0.469220i	−0.901662	−	0.432442i	$-0.857652\pi$
$163$	−5.99060	+	5.99060i	−0.469220	+	0.469220i	0.432442	+	0.901662i	$0.357652\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.38949	−	4.38949i	−0.339669	−	0.339669i	0.516574	−	0.856243i	$-0.327207\pi$
$167$	−4.38949	−	4.38949i	−0.339669	−	0.339669i	−0.856243	+	0.516574i	$0.827207\pi$
$168$		0			0
$169$	1.49804	+	12.9134i	0.115234	+	0.993338i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.41225			−0.183400			−0.0917001	−	0.995787i	$-0.529230\pi$
$173$	−2.41225			−0.183400			−0.0917001	+	0.995787i	$0.529230\pi$
$174$		0			0
$175$	−2.39812	−	2.39812i	−0.181281	−	0.181281i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.65882			−0.348216			−0.174108	−	0.984727i	$-0.555704\pi$
$179$	−4.65882			−0.348216			−0.174108	+	0.984727i	$0.555704\pi$
$180$		0			0
$181$		−	19.8640i		−	1.47648i	−0.674537	−	0.738241i	$-0.735657\pi$
$181$		−	19.8640i		−	1.47648i	0.674537	−	0.738241i	$-0.264343\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			0.979202i			0.0719924i
$186$		0			0
$187$	3.78292	+	3.78292i	0.276635	+	0.276635i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.3962i			1.11403i	0.830502	+	0.557015i	$0.188053\pi$
$191$			15.3962i			1.11403i	−0.830502	+	0.557015i	$0.811947\pi$
$192$		0			0
$193$	8.28488	−	8.28488i	0.596359	−	0.596359i	−0.342983	−	0.939342i	$-0.611437\pi$
$193$	8.28488	−	8.28488i	0.596359	−	0.596359i	0.939342	+	0.342983i	$0.111437\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.2890	−	14.2890i	1.01805	−	1.01805i	0.0182141	−	0.999834i	$-0.494202\pi$
$197$	14.2890	−	14.2890i	1.01805	−	1.01805i	0.999834	−	0.0182141i	$-0.00579804\pi$
$198$		0			0
$199$			2.58856i			0.183498i	0.995782	+	0.0917491i	$0.0292458\pi$
$199$			2.58856i			0.183498i	−0.995782	+	0.0917491i	$0.970754\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.78487	−	5.78487i	−0.406018	−	0.406018i
$204$		0			0
$205$		−	6.88676i		−	0.480992i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			7.65293i			0.529364i
$210$		0			0
$211$	−21.7735			−1.49895			−0.749476	−	0.662032i	$-0.769694\pi$
$211$	−21.7735			−1.49895			−0.749476	+	0.662032i	$0.769694\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.78095	+	7.78095i	0.530656	+	0.530656i
$216$		0			0
$217$	23.0039			1.56161
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.32352	+	4.74166i	0.358099	+	0.318959i
$222$		0			0
$223$	12.2077	+	12.2077i	0.817487	+	0.817487i	0.985743	−	0.168257i	$-0.0538137\pi$
$223$	12.2077	+	12.2077i	0.817487	+	0.817487i	−0.168257	+	0.985743i	$0.553814\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.48920	−	6.48920i	0.430704	−	0.430704i	−0.458164	−	0.888868i	$-0.651493\pi$
$227$	6.48920	−	6.48920i	0.430704	−	0.430704i	0.888868	+	0.458164i	$0.151493\pi$
$228$		0			0
$229$	−3.79624	+	3.79624i	−0.250863	+	0.250863i	−0.821324	−	0.570462i	$-0.806764\pi$
$229$	−3.79624	+	3.79624i	−0.250863	+	0.250863i	0.570462	+	0.821324i	$0.306764\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	22.0699			1.44585			0.722925	−	0.690927i	$-0.242797\pi$
$233$	22.0699			1.44585			0.722925	+	0.690927i	$0.242797\pi$
$234$		0			0
$235$	−10.2077			−0.665876
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.86771	+	5.86771i	−0.379551	+	0.379551i	−0.870940	−	0.491389i	$-0.836489\pi$
$239$	−5.86771	+	5.86771i	−0.379551	+	0.379551i	0.491389	+	0.870940i	$0.336489\pi$
$240$		0			0
$241$	8.29428	−	8.29428i	0.534282	−	0.534282i	−0.387562	−	0.921844i	$-0.626683\pi$
$241$	8.29428	−	8.29428i	0.534282	−	0.534282i	0.921844	+	0.387562i	$0.126683\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.18337	+	3.18337i	0.203378	+	0.203378i
$246$		0			0
$247$	0.588561	+	10.1810i	0.0374492	+	0.647804i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.1025			−0.953261			−0.476631	−	0.879104i	$-0.658142\pi$
$251$	−15.1025			−0.953261			−0.476631	+	0.879104i	$0.658142\pi$
$252$		0			0
$253$	−13.8096	−	13.8096i	−0.868200	−	0.868200i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.23283			−0.575928			−0.287964	−	0.957641i	$-0.592978\pi$
$257$	−9.23283			−0.575928			−0.287964	+	0.957641i	$0.592978\pi$
$258$		0			0
$259$		−	3.32092i		−	0.206352i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.3942i		−	1.01091i	−0.862852	−	0.505456i	$-0.831324\pi$
$263$		−	16.3942i		−	1.01091i	0.862852	−	0.505456i	$-0.168676\pi$
$264$		0			0
$265$	0.0133192	+	0.0133192i	0.000818192	+	0.000818192i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	15.2737i		−	0.931256i	−0.884981	−	0.465628i	$-0.845829\pi$
$269$		−	15.2737i		−	0.931256i	0.884981	−	0.465628i	$-0.154171\pi$
$270$		0			0
$271$	−1.29428	+	1.29428i	−0.0786219	+	0.0786219i	−0.745324	−	0.666702i	$-0.767705\pi$
$271$	−1.29428	+	1.29428i	−0.0786219	+	0.0786219i	0.666702	+	0.745324i	$0.267705\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.91323	−	1.91323i	0.115372	−	0.115372i
$276$		0			0
$277$		−	23.0306i		−	1.38377i	−0.722007	−	0.691886i	$-0.756780\pi$
$277$		−	23.0306i		−	1.38377i	0.722007	−	0.691886i	$-0.243220\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−15.1402	−	15.1402i	−0.903187	−	0.903187i	0.0925235	−	0.995711i	$-0.470507\pi$
$281$	−15.1402	−	15.1402i	−0.903187	−	0.903187i	−0.995711	+	0.0925235i	$0.970507\pi$
$282$		0			0
$283$			26.2077i			1.55788i	0.627095	+	0.778942i	$0.284244\pi$
$283$			26.2077i			1.55788i	−0.627095	+	0.778942i	$0.715756\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			23.3561i			1.37867i
$288$		0			0
$289$	−13.0905			−0.770031
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.91092	−	6.91092i	−0.403740	−	0.403740i	0.475808	−	0.879549i	$-0.342156\pi$
$293$	−6.91092	−	6.91092i	−0.403740	−	0.403740i	−0.879549	+	0.475808i	$0.842156\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.4335	−	17.3094i	−1.12387	−	1.00103i
$300$		0			0
$301$	−26.3887	−	26.3887i	−1.52102	−	1.52102i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.88851	−	4.88851i	0.279915	−	0.279915i
$306$		0			0
$307$	−0.779001	+	0.779001i	−0.0444600	+	0.0444600i	−0.728987	−	0.684527i	$-0.760009\pi$
$307$	−0.779001	+	0.779001i	−0.0444600	+	0.0444600i	0.684527	+	0.728987i	$0.260009\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−18.0966			−1.02616			−0.513082	−	0.858339i	$-0.671497\pi$
$311$	−18.0966			−1.02616			−0.513082	+	0.858339i	$0.671497\pi$
$312$		0			0
$313$	−8.41536			−0.475664			−0.237832	−	0.971306i	$-0.576437\pi$
$313$	−8.41536			−0.475664			−0.237832	+	0.971306i	$0.576437\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.6408	−	18.6408i	1.04697	−	1.04697i	0.0481310	−	0.998841i	$-0.484674\pi$
$317$	18.6408	−	18.6408i	1.04697	−	1.04697i	0.998841	−	0.0481310i	$-0.0153265\pi$
$318$		0			0
$319$	4.61520	−	4.61520i	0.258402	−	0.258402i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.95448	+	3.95448i	0.220033	+	0.220033i
$324$		0			0
$325$	2.39812	−	2.69240i	0.133024	−	0.149348i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	34.6189			1.90860
$330$		0			0
$331$	−12.7696	−	12.7696i	−0.701881	−	0.701881i	0.262933	−	0.964814i	$-0.415310\pi$
$331$	−12.7696	−	12.7696i	−0.701881	−	0.701881i	−0.964814	+	0.262933i	$0.915310\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.78683			−0.261533
$336$		0			0
$337$			9.41144i			0.512674i	0.966588	+	0.256337i	$0.0825156\pi$
$337$			9.41144i			0.512674i	−0.966588	+	0.256337i	$0.917484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			18.3526i			0.993852i
$342$		0			0
$343$	5.99060	+	5.99060i	0.323462	+	0.323462i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	27.5290i		−	1.47783i	−0.673797	−	0.738916i	$-0.735338\pi$
$347$		−	27.5290i		−	1.47783i	0.673797	−	0.738916i	$-0.264662\pi$
$348$		0			0
$349$	−9.79624	+	9.79624i	−0.524381	+	0.524381i	−0.918891	−	0.394511i	$-0.870914\pi$
$349$	−9.79624	+	9.79624i	−0.524381	+	0.524381i	0.394511	+	0.918891i	$0.370914\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.6655	−	15.6655i	0.833792	−	0.833792i	−0.154241	−	0.988033i	$-0.549293\pi$
$353$	15.6655	−	15.6655i	0.833792	−	0.833792i	0.988033	+	0.154241i	$0.0492933\pi$
$354$		0			0
$355$			5.52860i			0.293428i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.52688	−	6.52688i	−0.344475	−	0.344475i	0.513571	−	0.858047i	$-0.328322\pi$
$359$	−6.52688	−	6.52688i	−0.344475	−	0.344475i	−0.858047	+	0.513571i	$0.828322\pi$
$360$		0			0
$361$		−	11.0000i		−	0.578947i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.37066i			0.228771i
$366$		0			0
$367$	−12.6152			−0.658508			−0.329254	−	0.944241i	$-0.606797\pi$
$367$	−12.6152			−0.658508			−0.329254	+	0.944241i	$0.606797\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.0451715	−	0.0451715i	−0.00234519	−	0.00234519i
$372$		0			0
$373$	−15.9812			−0.827475			−0.413738	−	0.910396i	$-0.635777\pi$
$373$	−15.9812			−0.827475			−0.413738	+	0.910396i	$0.635777\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.78487	−	6.49475i	0.297936	−	0.334497i
$378$		0			0
$379$	17.9812	+	17.9812i	0.923632	+	0.923632i	0.997284	−	0.0736518i	$-0.0234653\pi$
$379$	17.9812	+	17.9812i	0.923632	+	0.923632i	−0.0736518	+	0.997284i	$0.523465\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.8728	−	13.8728i	0.708868	−	0.708868i	−0.257430	−	0.966297i	$-0.582876\pi$
$383$	13.8728	−	13.8728i	0.708868	−	0.708868i	0.966297	+	0.257430i	$0.0828755\pi$
$384$		0			0
$385$	−6.48864	+	6.48864i	−0.330692	+	0.330692i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.00358			−0.0508836			−0.0254418	−	0.999676i	$-0.508099\pi$
$389$	−1.00358			−0.0508836			−0.0254418	+	0.999676i	$0.508099\pi$
$390$		0			0
$391$	−14.2716			−0.721744
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.0896	+	11.0896i	−0.557976	+	0.557976i
$396$		0			0
$397$	−5.69632	+	5.69632i	−0.285890	+	0.285890i	−0.835453	−	0.549563i	$-0.814794\pi$
$397$	−5.69632	+	5.69632i	−0.285890	+	0.285890i	0.549563	+	0.835453i	$0.314794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.24068	+	5.24068i	0.261707	+	0.261707i	0.825747	−	0.564040i	$-0.190754\pi$
$401$	5.24068	+	5.24068i	0.261707	+	0.261707i	−0.564040	+	0.825747i	$0.690754\pi$
$402$		0			0
$403$	1.41144	+	24.4154i	0.0703088	+	1.21622i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.64945			0.131328
$408$		0			0
$409$	−18.6830	−	18.6830i	−0.923815	−	0.923815i	0.0734816	−	0.997297i	$-0.476589\pi$
$409$	−18.6830	−	18.6830i	−0.923815	−	0.923815i	−0.997297	+	0.0734816i	$0.976589\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			0.796240i			0.0390859i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	9.27996i		−	0.453356i	−0.973970	−	0.226678i	$-0.927214\pi$
$419$		−	9.27996i		−	0.453356i	0.973970	−	0.226678i	$-0.0727865\pi$
$420$		0			0
$421$	1.84952	+	1.84952i	0.0901399	+	0.0901399i	0.750739	−	0.660599i	$-0.229698\pi$
$421$	1.84952	+	1.84952i	0.0901399	+	0.0901399i	−0.660599	+	0.750739i	$0.729698\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.97724i		−	0.0959102i
$426$		0			0
$427$	−16.5792	+	16.5792i	−0.802322	+	0.802322i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−17.3546	+	17.3546i	−0.835942	+	0.835942i	−0.988322	−	0.152380i	$-0.951306\pi$
$431$	−17.3546	+	17.3546i	−0.835942	+	0.835942i	0.152380	+	0.988322i	$0.451306\pi$
$432$		0			0
$433$			23.8534i			1.14632i	0.819442	+	0.573162i	$0.194283\pi$
$433$			23.8534i			1.14632i	−0.819442	+	0.573162i	$0.805717\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.4358	−	14.4358i	−0.690560	−	0.690560i
$438$		0			0
$439$			21.9095i			1.04568i	0.852430	+	0.522841i	$0.175128\pi$
$439$			21.9095i			1.04568i	−0.852430	+	0.522841i	$0.824872\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	13.7071i		−	0.651245i	−0.945500	−	0.325623i	$-0.894426\pi$
$443$		−	13.7071i		−	0.651245i	0.945500	−	0.325623i	$-0.105574\pi$
$444$		0			0
$445$	−12.8868			−0.610891
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.6821	+	20.6821i	0.976047	+	0.976047i	0.999720	−	0.0236732i	$-0.00753613\pi$
$449$	20.6821	+	20.6821i	0.976047	+	0.976047i	−0.0236732	+	0.999720i	$0.507536\pi$
$450$		0			0
$451$	−18.6336			−0.877424
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.13311	+	9.13115i	−0.381286	+	0.428075i
$456$		0			0
$457$	−18.1038	−	18.1038i	−0.846862	−	0.846862i	0.142878	−	0.989740i	$-0.454364\pi$
$457$	−18.1038	−	18.1038i	−0.846862	−	0.846862i	−0.989740	+	0.142878i	$0.954364\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−26.6703	+	26.6703i	−1.24216	+	1.24216i	−0.283055	+	0.959104i	$0.591348\pi$
$461$	−26.6703	+	26.6703i	−1.24216	+	1.24216i	−0.959104	+	0.283055i	$0.908652\pi$
$462$		0			0
$463$	8.04388	−	8.04388i	0.373831	−	0.373831i	−0.495040	−	0.868870i	$-0.664846\pi$
$463$	8.04388	−	8.04388i	0.373831	−	0.373831i	0.868870	+	0.495040i	$0.164846\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.50770			−0.439964			−0.219982	−	0.975504i	$-0.570600\pi$
$467$	−9.50770			−0.439964			−0.219982	+	0.975504i	$0.570600\pi$
$468$		0			0
$469$	16.2343			0.749631
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.0531	−	21.0531i	0.968021	−	0.968021i
$474$		0			0
$475$	2.00000	−	2.00000i	0.0917663	−	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.3900	−	22.3900i	−1.02302	−	1.02302i	−0.999729	−	0.0232953i	$-0.992584\pi$
$479$	−22.3900	−	22.3900i	−1.02302	−	1.02302i	−0.0232953	−	0.999729i	$-0.507416\pi$
$480$		0			0
$481$	3.52468	−	0.203760i	0.160712	−	0.00929065i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.146852			0.00666818
$486$		0			0
$487$	16.7868	+	16.7868i	0.760684	+	0.760684i	0.976446	−	0.215762i	$-0.0692234\pi$
$487$	16.7868	+	16.7868i	0.760684	+	0.760684i	−0.215762	+	0.976446i	$0.569223\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−31.4591			−1.41973			−0.709864	−	0.704339i	$-0.751244\pi$
$491$	−31.4591			−1.41973			−0.709864	+	0.704339i	$0.751244\pi$
$492$		0			0
$493$		−	4.76960i		−	0.214812i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	18.7500i		−	0.841052i
$498$		0			0
$499$	−25.8640	−	25.8640i	−1.15783	−	1.15783i	−0.984941	−	0.172893i	$-0.944689\pi$
$499$	−25.8640	−	25.8640i	−1.15783	−	1.15783i	−0.172893	−	0.984941i	$-0.555311\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			37.0256i			1.65089i	0.564483	+	0.825445i	$0.309076\pi$
$503$			37.0256i			1.65089i	−0.564483	+	0.825445i	$0.690924\pi$
$504$		0			0
$505$	−8.00392	+	8.00392i	−0.356170	+	0.356170i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	26.3389	−	26.3389i	1.16745	−	1.16745i	0.184647	−	0.982805i	$-0.440886\pi$
$509$	26.3389	−	26.3389i	1.16745	−	1.16745i	0.982805	−	0.184647i	$-0.0591141\pi$
$510$		0			0
$511$		−	14.8229i		−	0.655726i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.43305	+	1.43305i	0.0631477	+	0.0631477i
$516$		0			0
$517$			27.6191i			1.21469i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.60579i			0.420837i	0.977611	+	0.210419i	$0.0674827\pi$
$521$			9.60579i			0.420837i	−0.977611	+	0.210419i	$0.932517\pi$
$522$		0			0
$523$	−36.9773			−1.61690			−0.808452	−	0.588562i	$-0.799694\pi$
$523$	−36.9773			−1.61690			−0.808452	+	0.588562i	$0.799694\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.48332	+	9.48332i	0.413100	+	0.413100i
$528$		0			0
$529$	29.0984			1.26515
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.7892	+	1.43305i	−1.07374	+	0.0620723i
$534$		0			0
$535$	3.10384	+	3.10384i	0.134191	+	0.134191i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.61330	−	8.61330i	0.371001	−	0.371001i
$540$		0			0
$541$	−26.9173	+	26.9173i	−1.15727	+	1.15727i	−0.172205	+	0.985061i	$0.555089\pi$
$541$	−26.9173	+	26.9173i	−1.15727	+	1.15727i	−0.985061	+	0.172205i	$0.944911\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.23676			0.309989
$546$		0			0
$547$	9.46472			0.404682			0.202341	−	0.979315i	$-0.435145\pi$
$547$	9.46472			0.404682			0.202341	+	0.979315i	$0.435145\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.82450	−	4.82450i	0.205531	−	0.205531i
$552$		0			0
$553$	37.6097	−	37.6097i	1.59933	−	1.59933i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.78487	+	5.78487i	0.245113	+	0.245113i	0.818961	−	0.573849i	$-0.194550\pi$
$557$	5.78487	+	5.78487i	0.245113	+	0.245113i	−0.573849	+	0.818961i	$0.694550\pi$
$558$		0			0
$559$	26.3887	−	29.6270i	1.11612	−	1.25309i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.2985			−0.518318			−0.259159	−	0.965835i	$-0.583445\pi$
$563$	−12.2985			−0.518318			−0.259159	+	0.965835i	$0.583445\pi$
$564$		0			0
$565$	−10.5020	−	10.5020i	−0.441821	−	0.441821i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.5426			−0.945036			−0.472518	−	0.881321i	$-0.656655\pi$
$569$	−22.5426			−0.945036			−0.472518	+	0.881321i	$0.656655\pi$
$570$		0			0
$571$			33.9252i			1.41972i	0.704341	+	0.709862i	$0.251243\pi$
$571$			33.9252i			1.41972i	−0.704341	+	0.709862i	$0.748757\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			7.21792i			0.301008i
$576$		0			0
$577$	−6.90008	−	6.90008i	−0.287254	−	0.287254i	0.548739	−	0.835993i	$-0.315108\pi$
$577$	−6.90008	−	6.90008i	−0.287254	−	0.287254i	−0.835993	+	0.548739i	$0.815108\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	2.70041i		−	0.112032i
$582$		0			0
$583$	0.0360380	−	0.0360380i	0.00149254	−	0.00149254i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.84299	−	9.84299i	0.406264	−	0.406264i	−0.474170	−	0.880433i	$-0.657252\pi$
$587$	9.84299	−	9.84299i	0.406264	−	0.406264i	0.880433	+	0.474170i	$0.157252\pi$
$588$		0			0
$589$			19.1850i			0.790503i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.24656	−	2.24656i	−0.0922553	−	0.0922553i	0.659473	−	0.751728i	$-0.270780\pi$
$593$	−2.24656	−	2.24656i	−0.0922553	−	0.0922553i	−0.751728	+	0.659473i	$0.770780\pi$
$594$		0			0
$595$			6.70572i			0.274908i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	42.2718i		−	1.72718i	−0.504195	−	0.863590i	$-0.668211\pi$
$599$		−	42.2718i		−	1.72718i	0.504195	−	0.863590i	$-0.331789\pi$
$600$		0			0
$601$	−42.5059			−1.73385			−0.866925	−	0.498438i	$-0.833907\pi$
$601$	−42.5059			−1.73385			−0.866925	+	0.498438i	$0.833907\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.60150	+	2.60150i	0.105766	+	0.105766i
$606$		0			0
$607$	5.17712			0.210133			0.105066	−	0.994465i	$-0.466494\pi$
$607$	5.17712			0.210133			0.105066	+	0.994465i	$0.466494\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.12409	+	36.7430i	0.0859316	+	1.48646i
$612$		0			0
$613$	−29.2622	−	29.2622i	−1.18189	−	1.18189i	−0.979255	−	0.202633i	$-0.935050\pi$
$613$	−29.2622	−	29.2622i	−1.18189	−	1.18189i	−0.202633	−	0.979255i	$-0.564950\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.7944	−	29.7944i	1.19948	−	1.19948i	0.225153	−	0.974323i	$-0.427712\pi$
$617$	29.7944	−	29.7944i	1.19948	−	1.19948i	0.974323	−	0.225153i	$-0.0722881\pi$
$618$		0			0
$619$	−14.2982	+	14.2982i	−0.574693	+	0.574693i	−0.933436	−	0.358743i	$-0.883205\pi$
$619$	−14.2982	+	14.2982i	−0.574693	+	0.574693i	0.358743	+	0.933436i	$0.383205\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	43.7049			1.75100
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.36904	−	1.36904i	0.0545873	−	0.0545873i
$630$		0			0
$631$	1.79232	−	1.79232i	0.0713512	−	0.0713512i	−0.670531	−	0.741882i	$-0.733934\pi$
$631$	1.79232	−	1.79232i	0.0713512	−	0.0713512i	0.741882	+	0.670531i	$0.233934\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.05027	−	8.05027i	−0.319465	−	0.319465i
$636$		0			0
$637$	10.7962	−	12.1211i	0.427763	−	0.480255i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−32.3346			−1.27714			−0.638571	−	0.769563i	$-0.720474\pi$
$641$	−32.3346			−1.27714			−0.638571	+	0.769563i	$0.720474\pi$
$642$		0			0
$643$	12.3793	+	12.3793i	0.488193	+	0.488193i	0.907736	−	0.419543i	$-0.137810\pi$
$643$	12.3793	+	12.3793i	0.488193	+	0.488193i	−0.419543	+	0.907736i	$0.637810\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.5961			0.416575			0.208287	−	0.978068i	$-0.433211\pi$
$647$	10.5961			0.416575			0.208287	+	0.978068i	$0.433211\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	25.2957i		−	0.989897i	−0.868922	−	0.494949i	$-0.835187\pi$
$653$		−	25.2957i		−	0.989897i	0.868922	−	0.494949i	$-0.164813\pi$
$654$		0			0
$655$	−11.5925	−	11.5925i	−0.452956	−	0.452956i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	9.53599i		−	0.371469i	−0.982600	−	0.185735i	$-0.940533\pi$
$659$		−	9.53599i		−	0.371469i	0.982600	−	0.185735i	$-0.0594665\pi$
$660$		0			0
$661$	−3.64184	+	3.64184i	−0.141651	+	0.141651i	−0.774376	−	0.632725i	$-0.781936\pi$
$661$	−3.64184	+	3.64184i	−0.141651	+	0.141651i	0.632725	+	0.774376i	$0.281936\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.78291	+	6.78291i	−0.263030	+	0.263030i
$666$		0			0
$667$			17.4114i			0.674174i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13.2269	−	13.2269i	−0.510620	−	0.510620i
$672$		0			0
$673$			37.8002i			1.45709i	0.684998	+	0.728545i	$0.259803\pi$
$673$			37.8002i			1.45709i	−0.684998	+	0.728545i	$0.740197\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	37.0178i		−	1.42271i	−0.702832	−	0.711355i	$-0.748082\pi$
$677$		−	37.0178i		−	1.42271i	0.702832	−	0.711355i	$-0.251918\pi$
$678$		0			0
$679$	−0.498040			−0.0191130
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.68516	−	3.68516i	−0.141009	−	0.141009i	0.633079	−	0.774087i	$-0.281791\pi$
$683$	−3.68516	−	3.68516i	−0.141009	−	0.141009i	−0.774087	+	0.633079i	$0.781791\pi$
$684$		0			0
$685$	17.2116			0.657621
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.0451715	−	0.0507146i	0.00172090	−	0.00193207i
$690$		0			0
$691$	−23.2943	−	23.2943i	−0.886156	−	0.886156i	0.107995	−	0.994151i	$-0.465557\pi$
$691$	−23.2943	−	23.2943i	−0.886156	−	0.886156i	−0.994151	+	0.107995i	$0.965557\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.26113	+	8.26113i	−0.313362	+	0.313362i
$696$		0			0
$697$	−9.62852	+	9.62852i	−0.364706	+	0.364706i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	25.6647			0.969344			0.484672	−	0.874696i	$-0.338939\pi$
$701$	25.6647			0.969344			0.484672	+	0.874696i	$0.338939\pi$
$702$		0			0
$703$	2.76960			0.104458
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27.1449	−	27.1449i	1.02089	−	1.02089i
$708$		0			0
$709$	−20.9173	+	20.9173i	−0.785566	+	0.785566i	−0.980764	−	0.195198i	$-0.937465\pi$
$709$	−20.9173	+	20.9173i	−0.785566	+	0.785566i	0.195198	+	0.980764i	$0.437465\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−34.6189	−	34.6189i	−1.29649	−	1.29649i
$714$		0			0
$715$	−7.28488	−	6.48864i	−0.272439	−	0.242661i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.95252			0.184698			0.0923489	−	0.995727i	$-0.470562\pi$
$719$	4.95252			0.184698			0.0923489	+	0.995727i	$0.470562\pi$
$720$		0			0
$721$	−4.86012	−	4.86012i	−0.181000	−	0.181000i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.41225			−0.0895888
$726$		0			0
$727$		−	10.6418i		−	0.394684i	−0.980335	−	0.197342i	$-0.936769\pi$
$727$		−	10.6418i		−	0.394684i	0.980335	−	0.197342i	$-0.0632309\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	21.7574i		−	0.804726i
$732$		0			0
$733$	−5.33424	−	5.33424i	−0.197025	−	0.197025i	0.601699	−	0.798723i	$-0.294491\pi$
$733$	−5.33424	−	5.33424i	−0.197025	−	0.197025i	−0.798723	+	0.601699i	$0.794491\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.9518i			0.477087i
$738$		0			0
$739$	−16.3887	+	16.3887i	−0.602869	+	0.602869i	−0.941073	−	0.338204i	$-0.890181\pi$
$739$	−16.3887	+	16.3887i	−0.602869	+	0.602869i	0.338204	+	0.941073i	$0.390181\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−30.2538	+	30.2538i	−1.10990	+	1.10990i	−0.116740	+	0.993162i	$0.537245\pi$
$743$	−30.2538	+	30.2538i	−1.10990	+	1.10990i	−0.993162	+	0.116740i	$0.962755\pi$
$744$		0			0
$745$			1.47532i			0.0540516i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.5265	−	10.5265i	−0.384631	−	0.384631i
$750$		0			0
$751$		−	3.91732i		−	0.142945i	−0.997443	−	0.0714725i	$-0.977230\pi$
$751$		−	3.91732i		−	0.142945i	0.997443	−	0.0714725i	$-0.0227698\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	11.4794i		−	0.417778i
$756$		0			0
$757$	7.98120			0.290082			0.145041	−	0.989426i	$-0.453669\pi$
$757$	7.98120			0.290082			0.145041	+	0.989426i	$0.453669\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.70041	+	2.70041i	0.0978899	+	0.0978899i	0.754356	−	0.656466i	$-0.227949\pi$
$761$	2.70041	+	2.70041i	0.0978899	+	0.0978899i	−0.656466	+	0.754356i	$0.727949\pi$
$762$		0			0
$763$	−24.5431			−0.888521
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−36.0039	−	36.0039i	−1.29833	−	1.29833i	−0.929492	−	0.368843i	$-0.879754\pi$
$769$	−36.0039	−	36.0039i	−1.29833	−	1.29833i	−0.368843	−	0.929492i	$-0.620246\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−35.9239	+	35.9239i	−1.29209	+	1.29209i	−0.358603	+	0.933490i	$0.616747\pi$
$773$	−35.9239	+	35.9239i	−1.29209	+	1.29209i	−0.933490	+	0.358603i	$0.883253\pi$
$774$		0			0
$775$	4.79624	−	4.79624i	0.172286	−	0.172286i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19.4787			−0.697897
$780$		0			0
$781$	14.9588			0.535269
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.3942	+	16.3942i	−0.585136	+	0.585136i
$786$		0			0
$787$	32.5964	−	32.5964i	1.16194	−	1.16194i	0.177885	−	0.984051i	$-0.443075\pi$
$787$	32.5964	−	32.5964i	1.16194	−	1.16194i	0.984051	−	0.177885i	$-0.0569255\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	35.6169	+	35.6169i	1.26639	+	1.26639i
$792$		0			0
$793$	−18.6136	−	16.5792i	−0.660990	−	0.588743i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.5286			−1.08138			−0.540690	−	0.841222i	$-0.681837\pi$
$797$	−30.5286			−1.08138			−0.540690	+	0.841222i	$0.681837\pi$
$798$		0			0
$799$	14.2716	+	14.2716i	0.504892	+	0.504892i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.8258			0.417322
$804$		0			0
$805$		−	24.4792i		−	0.862780i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			29.5062i			1.03738i	0.854961	+	0.518692i	$0.173581\pi$
$809$			29.5062i			1.03738i	−0.854961	+	0.518692i	$0.826419\pi$
$810$		0			0
$811$	−14.9773	−	14.9773i	−0.525923	−	0.525923i	0.393431	−	0.919354i	$-0.371288\pi$
$811$	−14.9773	−	14.9773i	−0.525923	−	0.525923i	−0.919354	+	0.393431i	$0.871288\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	8.47199i		−	0.296761i
$816$		0			0
$817$	22.0078	−	22.0078i	0.769957	−	0.769957i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.2977	−	23.2977i	0.813094	−	0.813094i	−0.172002	−	0.985097i	$-0.555024\pi$
$821$	23.2977	−	23.2977i	0.813094	−	0.813094i	0.985097	+	0.172002i	$0.0550237\pi$
$822$		0			0
$823$			6.96944i			0.242939i	0.992595	+	0.121470i	$0.0387607\pi$
$823$			6.96944i			0.242939i	−0.992595	+	0.121470i	$0.961239\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.1646	−	15.1646i	−0.527323	−	0.527323i	0.392450	−	0.919773i	$-0.371628\pi$
$827$	−15.1646	−	15.1646i	−0.527323	−	0.527323i	−0.919773	+	0.392450i	$0.871628\pi$
$828$		0			0
$829$			12.5886i			0.437219i	0.975812	+	0.218609i	$0.0701521\pi$
$829$			12.5886i			0.437219i	−0.975812	+	0.218609i	$0.929848\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	8.90146i		−	0.308417i
$834$		0			0
$835$	6.20768			0.214826
$836$		0			0
$837$		0			0
$838$		0			0
$839$	37.4398	+	37.4398i	1.29257	+	1.29257i	0.933195	+	0.359371i	$0.117009\pi$
$839$	37.4398	+	37.4398i	1.29257	+	1.29257i	0.359371	+	0.933195i	$0.382991\pi$
$840$		0			0
$841$	23.1810			0.799346
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.1904	−	8.07188i	−0.350561	−	0.277681i
$846$		0			0
$847$	−8.82288	−	8.82288i	−0.303158	−	0.303158i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.99769	+	4.99769i	−0.171319	+	0.171319i
$852$		0			0
$853$	2.90008	−	2.90008i	0.0992968	−	0.0992968i	−0.655713	−	0.755010i	$-0.727632\pi$
$853$	2.90008	−	2.90008i	0.0992968	−	0.0992968i	0.755010	+	0.655713i	$0.227632\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.5175			1.04246			0.521230	−	0.853416i	$-0.325473\pi$
$857$	30.5175			1.04246			0.521230	+	0.853416i	$0.325473\pi$
$858$		0			0
$859$	−30.9212			−1.05502			−0.527510	−	0.849549i	$-0.676874\pi$
$859$	−30.9212			−1.05502			−0.527510	+	0.849549i	$0.676874\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.20070	−	9.20070i	0.313196	−	0.313196i	−0.532951	−	0.846146i	$-0.678917\pi$
$863$	9.20070	−	9.20070i	0.313196	−	0.313196i	0.846146	+	0.532951i	$0.178917\pi$
$864$		0			0
$865$	1.70572	−	1.70572i	0.0579962	−	0.0579962i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	30.0052	+	30.0052i	1.01786	+	1.01786i
$870$		0			0
$871$	0.996080	+	17.2304i	0.0337509	+	0.583830i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.39145			0.114652
$876$		0			0
$877$	27.5059	+	27.5059i	0.928808	+	0.928808i	0.997629	−	0.0688214i	$-0.0219239\pi$
$877$	27.5059	+	27.5059i	0.928808	+	0.928808i	−0.0688214	+	0.997629i	$0.521924\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−42.9706			−1.44772			−0.723858	−	0.689949i	$-0.757633\pi$
$881$	−42.9706			−1.44772			−0.723858	+	0.689949i	$0.757633\pi$
$882$		0			0
$883$		−	20.8495i		−	0.701642i	−0.936443	−	0.350821i	$-0.885903\pi$
$883$		−	20.8495i		−	0.701642i	0.936443	−	0.350821i	$-0.114097\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	10.6227i		−	0.356674i	−0.983969	−	0.178337i	$-0.942928\pi$
$887$		−	10.6227i		−	0.356674i	0.983969	−	0.178337i	$-0.0570718\pi$
$888$		0			0
$889$	27.3021	+	27.3021i	0.915684	+	0.915684i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			28.8717i			0.966154i
$894$		0			0
$895$	3.29428	−	3.29428i	0.110116	−	0.110116i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.5697	−	11.5697i	0.385872	−	0.385872i
$900$		0			0
$901$		−	0.0372437i		−	0.00124077i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	14.0460	+	14.0460i	0.466905	+	0.466905i
$906$		0			0
$907$			43.9279i			1.45860i	0.684193	+	0.729301i	$0.260155\pi$
$907$			43.9279i			1.45860i	−0.684193	+	0.729301i	$0.739845\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			37.2289i			1.23345i	0.787179	+	0.616725i	$0.211541\pi$
$911$			37.2289i			1.23345i	−0.787179	+	0.616725i	$0.788459\pi$
$912$		0			0
$913$	2.15440			0.0713003
$914$		0			0
$915$		0			0
$916$		0			0
$917$	39.3154	+	39.3154i	1.29831	+	1.29831i
$918$		0			0
$919$	49.2222			1.62369			0.811845	−	0.583873i	$-0.198463\pi$
$919$	49.2222			1.62369			0.811845	+	0.583873i	$0.198463\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.9004	−	1.15043i	0.655030	−	0.0378670i
$924$		0			0
$925$	−0.692400	−	0.692400i	−0.0227660	−	0.0227660i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.9545	−	28.9545i	0.949967	−	0.949967i	−0.0488396	−	0.998807i	$-0.515552\pi$
$929$	28.9545	−	28.9545i	0.949967	−	0.949967i	0.998807	+	0.0488396i	$0.0155523\pi$
$930$		0			0
$931$	9.00392	−	9.00392i	0.295092	−	0.295092i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.34986			−0.174959
$936$		0			0
$937$	−14.0345			−0.458486			−0.229243	−	0.973369i	$-0.573625\pi$
$937$	−14.0345			−0.458486			−0.229243	+	0.973369i	$0.573625\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	33.5757	−	33.5757i	1.09454	−	1.09454i	0.0994976	−	0.995038i	$-0.468276\pi$
$941$	33.5757	−	33.5757i	1.09454	−	1.09454i	0.995038	−	0.0994976i	$-0.0317236\pi$
$942$		0			0
$943$	35.1489	−	35.1489i	1.14461	−	1.14461i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.8643	−	25.8643i	−0.840476	−	0.840476i	0.148445	−	0.988921i	$-0.452573\pi$
$947$	−25.8643	−	25.8643i	−0.840476	−	0.840476i	−0.988921	+	0.148445i	$0.952573\pi$
$948$		0			0
$949$	15.7324	−	0.909479i	0.510694	−	0.0295230i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−53.1560			−1.72189			−0.860947	−	0.508695i	$-0.830128\pi$
$953$	−53.1560			−1.72189			−0.860947	+	0.508695i	$0.830128\pi$
$954$		0			0
$955$	−10.8868	−	10.8868i	−0.352287	−	0.352287i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−58.3723			−1.88494
$960$		0			0
$961$			15.0078i			0.484124i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			11.7166i			0.377170i
$966$		0			0
$967$	14.9773	+	14.9773i	0.481637	+	0.481637i	0.905654	−	0.424017i	$-0.139380\pi$
$967$	14.9773	+	14.9773i	0.481637	+	0.481637i	−0.424017	+	0.905654i	$0.639380\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			28.9243i			0.928226i	0.885776	+	0.464113i	$0.153627\pi$
$971$			28.9243i			0.928226i	−0.885776	+	0.464113i	$0.846373\pi$
$972$		0			0
$973$	28.0172	−	28.0172i	0.898191	−	0.898191i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.5886	−	11.5886i	0.370751	−	0.370751i	−0.496999	−	0.867751i	$-0.665565\pi$
$977$	11.5886	−	11.5886i	0.370751	−	0.370751i	0.867751	+	0.496999i	$0.165565\pi$
$978$		0			0
$979$			34.8680i			1.11438i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.0134	+	22.0134i	0.702119	+	0.702119i	0.964865	−	0.262746i	$-0.0846281\pi$
$983$	22.0134	+	22.0134i	0.702119	+	0.702119i	−0.262746	+	0.964865i	$0.584628\pi$
$984$		0			0
$985$			20.2077i			0.643870i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			79.4254i			2.52558i
$990$		0			0
$991$	53.0568			1.68541			0.842703	−	0.538379i	$-0.180963\pi$
$991$	53.0568			1.68541			0.842703	+	0.538379i	$0.180963\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.83039	−	1.83039i	−0.0580272	−	0.0580272i
$996$		0			0
$997$	3.37696			0.106949			0.0534747	−	0.998569i	$-0.482970\pi$
$997$	3.37696			0.106949			0.0534747	+	0.998569i	$0.482970\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.bi.d.1061.3	yes	12
3.2	odd	2	inner	2340.2.bi.d.1061.6	yes	12
13.5	odd	4	inner	2340.2.bi.d.161.6	yes	12
39.5	even	4	inner	2340.2.bi.d.161.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.bi.d.161.3	✓	12	39.5	even	4	inner
2340.2.bi.d.161.6	yes	12	13.5	odd	4	inner
2340.2.bi.d.1061.3	yes	12	1.1	even	1	trivial
2340.2.bi.d.1061.6	yes	12	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +(2.39812 - 2.39812i) q^{7} +(1.91323 + 1.91323i) q^{11} +(2.69240 + 2.39812i) q^{13} +1.97724 q^{17} +(2.00000 + 2.00000i) q^{19} -7.21792 q^{23} -1.00000i q^{25} -2.41225i q^{29} +(4.79624 + 4.79624i) q^{31} +3.39145i q^{35} +(0.692400 - 0.692400i) q^{37} +(-4.86968 + 4.86968i) q^{41} -11.0039i q^{43} +(7.21792 + 7.21792i) q^{47} -4.50196i q^{49} -0.0188362i q^{53} -2.70572 q^{55} -6.91340 q^{61} +(-3.59954 + 0.208088i) q^{65} +(3.38480 + 3.38480i) q^{67} +(3.90931 - 3.90931i) q^{71} +(3.09052 - 3.09052i) q^{73} +9.17632 q^{77} +15.6830 q^{79} +(0.563027 - 0.563027i) q^{83} +(-1.39812 + 1.39812i) q^{85} +(9.11232 + 9.11232i) q^{89} +(12.2077 - 0.705720i) q^{91} -2.82843 q^{95} +(-0.103840 - 0.103840i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{13} + 24 q^{19} - 12 q^{37} - 24 q^{55} - 12 q^{73} + 24 q^{79} + 12 q^{85} + 72 q^{91} + 36 q^{97}+O(q^{100}) \) 12 * q + 12 * q^13 + 24 * q^19 - 12 * q^37 - 24 * q^55 - 12 * q^73 + 24 * q^79 + 12 * q^85 + 72 * q^91 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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