LMFDB - Embedded newform 2340.2.u.g.73.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(73,2340)] 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, 0, 3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.73"); 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.u (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-2,0,0,0,0,0,-14,0,14,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.31678304256.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 $ x^8 - 2x^7 + 2x^6 + 8x^5 + 32x^4 - 20x^3 + 8x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.4
Root			$-0.285451 + 0.285451i$ of defining polynomial
Character	$\chi$	$=$	2340.73
Dual form			2340.2.u.g.577.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.21777 - 0.285451i) q^{5} -1.26613i q^{7} +(-4.21777 + 4.21777i) q^{11} +(2.28545 - 2.78868i) q^{13} +(1.93232 + 1.93232i) q^{17} +(-2.55158 + 2.55158i) q^{19} +(5.21777 - 5.21777i) q^{23} +(4.83704 - 1.26613i) q^{25} +7.16941i q^{29} +(2.78868 + 2.78868i) q^{31} +(-0.361419 - 2.80799i) q^{35} +5.83704i q^{37} +(1.33381 + 1.33381i) q^{41} +(5.65332 - 5.65332i) q^{43} -10.8048i q^{47} +5.39691 q^{49} +(9.27258 + 9.27258i) q^{53} +(-8.15009 + 10.5580i) q^{55} +(2.55158 + 2.55158i) q^{59} +13.8499 q^{61} +(4.27258 - 6.83704i) q^{65} +1.30477 q^{67} +(-8.72745 - 8.72745i) q^{71} -7.64360 q^{73} +(5.34026 + 5.34026i) q^{77} +0.954914i q^{79} -5.13078i q^{83} +(4.83704 + 3.73387i) q^{85} +(-5.31122 - 5.31122i) q^{89} +(-3.53083 - 2.89368i) q^{91} +(-4.93048 + 6.38719i) q^{95} +2.81956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 14 q^{11} + 14 q^{13} + 2 q^{19} + 22 q^{23} + 12 q^{25} - 6 q^{31} + 4 q^{35} + 8 q^{41} - 14 q^{43} - 24 q^{49} + 8 q^{53} - 30 q^{55} - 2 q^{59} - 12 q^{61} - 32 q^{65} + 20 q^{67} + 22 q^{71}+ \cdots + 12 q^{97}+O(q^{100}) $ 8 * q - 2 * q^5 - 14 * q^11 + 14 * q^13 + 2 * q^19 + 22 * q^23 + 12 * q^25 - 6 * q^31 + 4 * q^35 + 8 * q^41 - 14 * q^43 - 24 * q^49 + 8 * q^53 - 30 * q^55 - 2 * q^59 - 12 * q^61 - 32 * q^65 + 20 * q^67 + 22 * q^71 - 28 * q^73 - 8 * q^77 + 12 * q^85 - 4 * q^89 + 6 * q^95 + 12 * q^97

Character values

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

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

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
$5$	2.21777	−	0.285451i	0.991818	−	0.127658i
$5$	2.21777	−	0.285451i	0.991818	−	0.127658i
$6$		0			0
$7$		−	1.26613i		−	0.478553i	−0.970951	−	0.239277i	$-0.923090\pi$
$7$		−	1.26613i		−	0.478553i	0.970951	−	0.239277i	$-0.0769103\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.21777	+	4.21777i	−1.27171	+	1.27171i	−0.326514	+	0.945192i	$0.605874\pi$
$11$	−4.21777	+	4.21777i	−1.27171	+	1.27171i	−0.945192	+	0.326514i	$0.894126\pi$
$12$		0			0
$13$	2.28545	−	2.78868i	0.633870	−	0.773439i
$13$	2.28545	−	2.78868i	0.633870	−	0.773439i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.93232	+	1.93232i	0.468657	+	0.468657i	0.901479	−	0.432822i	$-0.142482\pi$
$17$	1.93232	+	1.93232i	0.468657	+	0.468657i	−0.432822	+	0.901479i	$0.642482\pi$
$18$		0			0
$19$	−2.55158	+	2.55158i	−0.585373	+	0.585373i	−0.936375	−	0.351001i	$-0.885841\pi$
$19$	−2.55158	+	2.55158i	−0.585373	+	0.585373i	0.351001	+	0.936375i	$0.385841\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.21777	−	5.21777i	1.08798	−	1.08798i	0.0922445	−	0.995736i	$-0.470596\pi$
$23$	5.21777	−	5.21777i	1.08798	−	1.08798i	0.995736	−	0.0922445i	$-0.0294041\pi$
$24$		0			0
$25$	4.83704	−	1.26613i	0.967407	−	0.253226i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.16941i			1.33133i	0.746252	+	0.665663i	$0.231851\pi$
$29$			7.16941i			1.33133i	−0.746252	+	0.665663i	$0.768149\pi$
$30$		0			0
$31$	2.78868	+	2.78868i	0.500861	+	0.500861i	0.911705	−	0.410844i	$-0.134766\pi$
$31$	2.78868	+	2.78868i	0.500861	+	0.500861i	−0.410844	+	0.911705i	$0.634766\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.361419	−	2.80799i	−0.0610910	−	0.474638i
$36$		0			0
$37$			5.83704i			0.959603i	0.877377	+	0.479801i	$0.159291\pi$
$37$			5.83704i			0.959603i	−0.877377	+	0.479801i	$0.840709\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.33381	+	1.33381i	0.208306	+	0.208306i	0.803547	−	0.595241i	$-0.202943\pi$
$41$	1.33381	+	1.33381i	0.208306	+	0.208306i	−0.595241	+	0.803547i	$0.702943\pi$
$42$		0			0
$43$	5.65332	−	5.65332i	0.862123	−	0.862123i	−0.129461	−	0.991584i	$-0.541325\pi$
$43$	5.65332	−	5.65332i	0.862123	−	0.862123i	0.991584	+	0.129461i	$0.0413248\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.8048i		−	1.57605i	−0.615644	−	0.788024i	$-0.711104\pi$
$47$		−	10.8048i		−	1.57605i	0.615644	−	0.788024i	$-0.288896\pi$
$48$		0			0
$49$	5.39691			0.770987
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.27258	+	9.27258i	1.27369	+	1.27369i	0.944139	+	0.329548i	$0.106896\pi$
$53$	9.27258	+	9.27258i	1.27369	+	1.27369i	0.329548	+	0.944139i	$0.393104\pi$
$54$		0			0
$55$	−8.15009	+	10.5580i	−1.09896	+	1.42364i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.55158	+	2.55158i	0.332188	+	0.332188i	0.853417	−	0.521229i	$-0.174526\pi$
$59$	2.55158	+	2.55158i	0.332188	+	0.332188i	−0.521229	+	0.853417i	$0.674526\pi$
$60$		0			0
$61$	13.8499			1.77330			0.886651	−	0.462439i	$-0.153026\pi$
$61$	13.8499			1.77330			0.886651	+	0.462439i	$0.153026\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.27258	−	6.83704i	0.529948	−	0.848030i
$66$		0			0
$67$	1.30477			0.159403			0.0797015	−	0.996819i	$-0.474603\pi$
$67$	1.30477			0.159403			0.0797015	+	0.996819i	$0.474603\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.72745	−	8.72745i	−1.03576	−	1.03576i	−0.999337	−	0.0364208i	$-0.988404\pi$
$71$	−8.72745	−	8.72745i	−1.03576	−	1.03576i	−0.0364208	−	0.999337i	$-0.511596\pi$
$72$		0			0
$73$	−7.64360			−0.894615			−0.447308	−	0.894380i	$-0.647617\pi$
$73$	−7.64360			−0.894615			−0.447308	+	0.894380i	$0.647617\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.34026	+	5.34026i	0.608579	+	0.608579i
$78$		0			0
$79$			0.954914i			0.107436i	0.998556	+	0.0537181i	$0.0171072\pi$
$79$			0.954914i			0.107436i	−0.998556	+	0.0537181i	$0.982893\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	5.13078i		−	0.563176i	−0.959535	−	0.281588i	$-0.909139\pi$
$83$		−	5.13078i		−	0.563176i	0.959535	−	0.281588i	$-0.0908611\pi$
$84$		0			0
$85$	4.83704	+	3.73387i	0.524650	+	0.404995i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.31122	−	5.31122i	−0.562988	−	0.562988i	0.367167	−	0.930155i	$-0.380328\pi$
$89$	−5.31122	−	5.31122i	−0.562988	−	0.562988i	−0.930155	+	0.367167i	$0.880328\pi$
$90$		0			0
$91$	−3.53083	−	2.89368i	−0.370132	−	0.303341i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.93048	+	6.38719i	−0.505857	+	0.655312i
$96$		0			0
$97$	2.81956			0.286283			0.143141	−	0.989702i	$-0.454280\pi$
$97$	2.81956			0.286283			0.143141	+	0.989702i	$0.454280\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.27716i			0.127082i	0.997979	+	0.0635412i	$0.0202394\pi$
$101$			1.27716i			0.127082i	−0.997979	+	0.0635412i	$0.979761\pi$
$102$		0			0
$103$	0.313060	−	0.313060i	0.0308467	−	0.0308467i	−0.691515	−	0.722362i	$-0.743057\pi$
$103$	0.313060	−	0.313060i	0.0308467	−	0.0308467i	0.722362	+	0.691515i	$0.243057\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.19016	+	7.19016i	−0.695099	+	0.695099i	−0.963349	−	0.268250i	$-0.913555\pi$
$107$	−7.19016	+	7.19016i	−0.695099	+	0.695099i	0.268250	+	0.963349i	$0.413555\pi$
$108$		0			0
$109$	3.07413	−	3.07413i	0.294448	−	0.294448i	−0.544387	−	0.838834i	$-0.683238\pi$
$109$	3.07413	−	3.07413i	0.294448	−	0.294448i	0.838834	+	0.544387i	$0.183238\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.06310	−	9.06310i	−0.852585	−	0.852585i	0.137866	−	0.990451i	$-0.455976\pi$
$113$	−9.06310	−	9.06310i	−0.852585	−	0.852585i	−0.990451	+	0.137866i	$0.955976\pi$
$114$		0			0
$115$	10.0824	−	13.0613i	0.940190	−	1.21797i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.44658	−	2.44658i	0.224277	−	0.224277i
$120$		0			0
$121$		−	24.5792i		−	2.23447i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.3660	−	4.18873i	0.927166	−	0.374652i
$126$		0			0
$127$	6.88396	+	6.88396i	0.610853	+	0.610853i	0.943168	−	0.332315i	$-0.107830\pi$
$127$	6.88396	+	6.88396i	0.610853	+	0.610853i	−0.332315	+	0.943168i	$0.607830\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.193437			−0.0169007			−0.00845036	−	0.999964i	$-0.502690\pi$
$131$	−0.193437			−0.0169007			−0.00845036	+	0.999964i	$0.502690\pi$
$132$		0			0
$133$	3.23064	+	3.23064i	0.280132	+	0.280132i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.03864i			0.345044i	0.985006	+	0.172522i	$0.0551916\pi$
$137$			4.03864i			0.345044i	−0.985006	+	0.172522i	$0.944808\pi$
$138$		0			0
$139$			2.57090i			0.218061i	0.994038	+	0.109031i	$0.0347746\pi$
$139$			2.57090i			0.218061i	−0.994038	+	0.109031i	$0.965225\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.12249	+	21.4015i	0.177491	+	1.78968i
$144$		0			0
$145$	2.04652	+	15.9001i	0.169954	+	1.32043i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.898264	+	0.898264i	−0.0735887	+	0.0735887i	−0.742943	−	0.669354i	$-0.766571\pi$
$149$	−0.898264	+	0.898264i	−0.0735887	+	0.0735887i	0.669354	+	0.742943i	$0.266571\pi$
$150$		0			0
$151$	−7.58564	+	7.58564i	−0.617311	+	0.617311i	−0.944841	−	0.327530i	$-0.893784\pi$
$151$	−7.58564	+	7.58564i	−0.617311	+	0.617311i	0.327530	+	0.944841i	$0.393784\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.98068	+	5.38862i	0.560702	+	0.432824i
$156$		0			0
$157$	−5.36142	+	5.36142i	−0.427888	+	0.427888i	−0.887908	−	0.460020i	$-0.847842\pi$
$157$	−5.36142	+	5.36142i	−0.427888	+	0.427888i	0.460020	+	0.887908i	$0.347842\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.60639	−	6.60639i	−0.520657	−	0.520657i
$162$		0			0
$163$	16.7883			1.31496			0.657479	−	0.753473i	$-0.271623\pi$
$163$	16.7883			1.31496			0.657479	+	0.753473i	$0.271623\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			17.2790i			1.33709i	0.743671	+	0.668546i	$0.233083\pi$
$167$			17.2790i			1.33709i	−0.743671	+	0.668546i	$0.766917\pi$
$168$		0			0
$169$	−2.55342	−	12.7468i	−0.196417	−	0.980520i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.52582	−	4.52582i	0.344091	−	0.344091i	−0.513812	−	0.857903i	$-0.671767\pi$
$173$	4.52582	−	4.52582i	0.344091	−	0.344091i	0.857903	+	0.513812i	$0.171767\pi$
$174$		0			0
$175$	−1.60309	−	6.12433i	−0.121182	−	0.462956i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.01935			0.524651			0.262325	−	0.964980i	$-0.415511\pi$
$179$	7.01935			0.524651			0.262325	+	0.964980i	$0.415511\pi$
$180$		0			0
$181$			3.02116i			0.224561i	0.993677	+	0.112281i	$0.0358155\pi$
$181$			3.02116i			0.224561i	−0.993677	+	0.112281i	$0.964184\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.66619	+	12.9452i	0.122501	+	0.951751i
$186$		0			0
$187$	−16.3002			−1.19199
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.916276			0.0662994			0.0331497	−	0.999450i	$-0.489446\pi$
$191$	0.916276			0.0662994			0.0331497	+	0.999450i	$0.489446\pi$
$192$		0			0
$193$	15.0580			1.08390			0.541949	−	0.840412i	$-0.317687\pi$
$193$	15.0580			1.08390			0.541949	+	0.840412i	$0.317687\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.56445			0.396451			0.198225	−	0.980156i	$-0.436482\pi$
$197$	5.56445			0.396451			0.198225	+	0.980156i	$0.436482\pi$
$198$		0			0
$199$	16.2616			1.15275			0.576375	−	0.817185i	$-0.304467\pi$
$199$	16.2616			1.15275			0.576375	+	0.817185i	$0.304467\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.07743			0.637111
$204$		0			0
$205$	3.33883	+	2.57735i	0.233194	+	0.180010i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	21.5240i		−	1.48885i
$210$		0			0
$211$	−19.4383			−1.33819			−0.669094	−	0.743177i	$-0.733318\pi$
$211$	−19.4383			−1.33819			−0.669094	+	0.743177i	$0.733318\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.9240	−	14.1515i	0.745013	−	0.965126i
$216$		0			0
$217$	3.53083	−	3.53083i	0.239689	−	0.239689i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.80485	−	0.972392i	0.659545	−	0.0654101i
$222$		0			0
$223$			3.60496i			0.241406i	0.992689	+	0.120703i	$0.0385149\pi$
$223$			3.60496i			0.241406i	−0.992689	+	0.120703i	$0.961485\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−24.4374			−1.62197			−0.810984	−	0.585068i	$-0.801068\pi$
$227$	−24.4374			−1.62197			−0.810984	+	0.585068i	$0.801068\pi$
$228$		0			0
$229$	9.46315	+	9.46315i	0.625343	+	0.625343i	0.946893	−	0.321550i	$-0.104204\pi$
$229$	9.46315	+	9.46315i	0.625343	+	0.625343i	−0.321550	+	0.946893i	$0.604204\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.80011	−	1.80011i	0.117929	−	0.117929i	−0.645679	−	0.763609i	$-0.723426\pi$
$233$	1.80011	−	1.80011i	0.117929	−	0.117929i	0.763609	+	0.645679i	$0.223426\pi$
$234$		0			0
$235$	−3.08426	−	23.9627i	−0.201195	−	1.56315i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.0999	−	13.0999i	0.847362	−	0.847362i	−0.142441	−	0.989803i	$-0.545495\pi$
$239$	13.0999	−	13.0999i	0.847362	−	0.847362i	0.989803	+	0.142441i	$0.0454953\pi$
$240$		0			0
$241$	−13.2371	+	13.2371i	−0.852676	+	0.852676i	−0.990462	−	0.137786i	$-0.956001\pi$
$241$	−13.2371	+	13.2371i	−0.852676	+	0.852676i	0.137786	+	0.990462i	$0.456001\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.9691	−	1.54055i	0.764679	−	0.0984224i
$246$		0			0
$247$	1.28402	+	12.9471i	0.0817002	+	0.823802i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.62612i			0.291998i	0.989285	+	0.145999i	$0.0466396\pi$
$251$			4.62612i			0.291998i	−0.989285	+	0.145999i	$0.953360\pi$
$252$		0			0
$253$			44.0148i			2.76718i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4.55342	+	4.55342i	0.284035	+	0.284035i	0.834716	−	0.550681i	$-0.185632\pi$
$257$	4.55342	+	4.55342i	0.284035	+	0.284035i	−0.550681	+	0.834716i	$0.685632\pi$
$258$		0			0
$259$	7.39046			0.459221
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.72100	+	2.72100i	0.167784	+	0.167784i	0.786005	−	0.618221i	$-0.212146\pi$
$263$	2.72100	+	2.72100i	0.167784	+	0.167784i	−0.618221	+	0.786005i	$0.712146\pi$
$264$		0			0
$265$	23.2114	+	17.9176i	1.42586	+	1.10067i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	11.6870i		−	0.712567i	−0.934378	−	0.356284i	$-0.884044\pi$
$269$		−	11.6870i		−	0.712567i	0.934378	−	0.356284i	$-0.115956\pi$
$270$		0			0
$271$	−13.9660	+	13.9660i	−0.848372	+	0.848372i	−0.989930	−	0.141558i	$-0.954789\pi$
$271$	−13.9660	+	13.9660i	−0.848372	+	0.848372i	0.141558	+	0.989930i	$0.454789\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.0613	+	25.7418i	−0.908228	+	1.55229i
$276$		0			0
$277$	1.09672	+	1.09672i	0.0658954	+	0.0658954i	0.739286	−	0.673391i	$-0.235163\pi$
$277$	1.09672	+	1.09672i	0.0658954	+	0.0658954i	−0.673391	+	0.739286i	$0.735163\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.62755	−	8.62755i	0.514677	−	0.514677i	−0.401279	−	0.915956i	$-0.631434\pi$
$281$	8.62755	−	8.62755i	0.514677	−	0.514677i	0.915956	+	0.401279i	$0.131434\pi$
$282$		0			0
$283$	7.50507	−	7.50507i	0.446130	−	0.446130i	−0.447936	−	0.894066i	$-0.647841\pi$
$283$	7.50507	−	7.50507i	0.446130	−	0.446130i	0.894066	+	0.447936i	$0.147841\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.68878	−	1.68878i	0.0996856	−	0.0996856i
$288$		0			0
$289$		−	9.53226i		−	0.560721i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−21.0341			−1.22882			−0.614411	−	0.788986i	$-0.710606\pi$
$293$	−21.0341			−1.22882			−0.614411	+	0.788986i	$0.710606\pi$
$294$		0			0
$295$	6.38719	+	4.93048i	0.371876	+	0.287064i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.62571	−	26.4756i	−0.151849	−	1.53113i
$300$		0			0
$301$	−7.15785	−	7.15785i	−0.412572	−	0.412572i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	30.7160	−	3.95348i	1.75879	−	0.226376i
$306$		0			0
$307$		−	13.7984i		−	0.787516i	−0.919214	−	0.393758i	$-0.871175\pi$
$307$		−	13.7984i		−	0.787516i	0.919214	−	0.393758i	$-0.128825\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			19.9743i			1.13264i	0.824187	+	0.566318i	$0.191633\pi$
$311$			19.9743i			1.13264i	−0.824187	+	0.566318i	$0.808367\pi$
$312$		0			0
$313$	−4.92587	−	4.92587i	−0.278427	−	0.278427i	0.554054	−	0.832481i	$-0.313080\pi$
$313$	−4.92587	−	4.92587i	−0.278427	−	0.278427i	−0.832481	+	0.554054i	$0.813080\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.7146			−0.657956			−0.328978	−	0.944338i	$-0.606704\pi$
$317$	−11.7146			−0.657956			−0.328978	+	0.944338i	$0.606704\pi$
$318$		0			0
$319$	−30.2390	−	30.2390i	−1.69306	−	1.69306i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9.86096			−0.548679
$324$		0			0
$325$	7.52398	−	16.3826i	0.417355	−	0.908744i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−13.6804			−0.754223
$330$		0			0
$331$	−4.94061	−	4.94061i	−0.271561	−	0.271561i	0.558168	−	0.829728i	$-0.311505\pi$
$331$	−4.94061	−	4.94061i	−0.271561	−	0.271561i	−0.829728	+	0.558168i	$0.811505\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.89368	−	0.372448i	0.158099	−	0.0203490i
$336$		0			0
$337$	−8.67909	−	8.67909i	−0.472780	−	0.472780i	0.430033	−	0.902813i	$-0.358502\pi$
$337$	−8.67909	−	8.67909i	−0.472780	−	0.472780i	−0.902813	+	0.430033i	$0.858502\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−23.5240			−1.27390
$342$		0			0
$343$		−	15.6961i		−	0.847511i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.0627	+	19.0627i	−1.02334	+	1.02334i	−0.0236177	+	0.999721i	$0.507518\pi$
$347$	−19.0627	+	19.0627i	−1.02334	+	1.02334i	−0.999721	+	0.0236177i	$0.992482\pi$
$348$		0			0
$349$	2.00645	+	2.00645i	0.107403	+	0.107403i	0.758766	−	0.651363i	$-0.225803\pi$
$349$	2.00645	+	2.00645i	0.107403	+	0.107403i	−0.651363	+	0.758766i	$0.725803\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.41162i			0.128358i	0.997938	+	0.0641788i	$0.0204428\pi$
$353$			2.41162i			0.128358i	−0.997938	+	0.0641788i	$0.979557\pi$
$354$		0			0
$355$	−21.8468	−	16.8642i	−1.15951	−	0.895061i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.92546	−	1.92546i	−0.101622	−	0.101622i	0.654468	−	0.756090i	$-0.272893\pi$
$359$	−1.92546	−	1.92546i	−0.101622	−	0.101622i	−0.756090	+	0.654468i	$0.772893\pi$
$360$		0			0
$361$			5.97884i			0.314676i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.9518	+	2.18187i	−0.887296	+	0.114205i
$366$		0			0
$367$	−16.8435	+	16.8435i	−0.879224	+	0.879224i	−0.993454	−	0.114230i	$-0.963560\pi$
$367$	−16.8435	+	16.8435i	−0.879224	+	0.879224i	0.114230	+	0.993454i	$0.463560\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11.7403	−	11.7403i	0.609527	−	0.609527i
$372$		0			0
$373$	−20.4388	−	20.4388i	−1.05828	−	1.05828i	−0.998193	−	0.0600904i	$-0.980861\pi$
$373$	−20.4388	−	20.4388i	−1.05828	−	1.05828i	−0.0600904	−	0.998193i	$-0.519139\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.9932	+	16.3853i	1.02970	+	0.843888i
$378$		0			0
$379$	−15.7841	+	15.7841i	−0.810775	+	0.810775i	−0.984750	−	0.173975i	$-0.944339\pi$
$379$	−15.7841	+	15.7841i	−0.810775	+	0.810775i	0.173975	+	0.984750i	$0.444339\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	21.7432i		−	1.11102i	−0.831508	−	0.555512i	$-0.812522\pi$
$383$		−	21.7432i		−	1.11102i	0.831508	−	0.555512i	$-0.187478\pi$
$384$		0			0
$385$	13.3679	+	10.3191i	0.681290	+	0.525910i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.9227			−0.908718			−0.454359	−	0.890819i	$-0.650132\pi$
$389$	−17.9227			−0.908718			−0.454359	+	0.890819i	$0.650132\pi$
$390$		0			0
$391$	20.1648			1.01978
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.272581	+	2.11778i	0.0137151	+	0.106557i
$396$		0			0
$397$		−	33.9632i		−	1.70457i	−0.523081	−	0.852283i	$-0.675218\pi$
$397$		−	33.9632i		−	1.70457i	0.523081	−	0.852283i	$-0.324782\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.0857	+	15.0857i	−0.753343	+	0.753343i	−0.975102	−	0.221758i	$-0.928820\pi$
$401$	−15.0857	+	15.0857i	−0.753343	+	0.753343i	0.221758	+	0.975102i	$0.428820\pi$
$402$		0			0
$403$	14.1501	−	1.40333i	0.704866	−	0.0699048i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.6193	−	24.6193i	−1.22033	−	1.22033i
$408$		0			0
$409$	11.7141	−	11.7141i	0.579227	−	0.579227i	−0.355463	−	0.934690i	$-0.615677\pi$
$409$	11.7141	−	11.7141i	0.579227	−	0.579227i	0.934690	+	0.355463i	$0.115677\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.23064	−	3.23064i	0.158970	−	0.158970i
$414$		0			0
$415$	−1.46459	−	11.3789i	−0.0718937	−	0.558568i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	13.2486i		−	0.647234i	−0.946188	−	0.323617i	$-0.895101\pi$
$419$		−	13.2486i		−	0.647234i	0.946188	−	0.323617i	$-0.104899\pi$
$420$		0			0
$421$	−6.17729	−	6.17729i	−0.301063	−	0.301063i	0.540367	−	0.841430i	$-0.318286\pi$
$421$	−6.17729	−	6.17729i	−0.301063	−	0.301063i	−0.841430	+	0.540367i	$0.818286\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.7933	+	6.90013i	0.572058	+	0.334706i
$426$		0			0
$427$		−	17.5358i		−	0.848619i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.25641	−	4.25641i	−0.205024	−	0.205024i	0.597124	−	0.802149i	$-0.296310\pi$
$431$	−4.25641	−	4.25641i	−0.205024	−	0.205024i	−0.802149	+	0.597124i	$0.796310\pi$
$432$		0			0
$433$	−27.2579	+	27.2579i	−1.30993	+	1.30993i	−0.388467	+	0.921463i	$0.626995\pi$
$433$	−27.2579	+	27.2579i	−1.30993	+	1.30993i	−0.921463	+	0.388467i	$0.873005\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			26.6272i			1.27375i
$438$		0			0
$439$	−24.2063			−1.15531			−0.577653	−	0.816283i	$-0.696031\pi$
$439$	−24.2063			−1.15531			−0.577653	+	0.816283i	$0.696031\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.84991	+	9.84991i	0.467983	+	0.467983i	0.901261	−	0.433277i	$-0.142643\pi$
$443$	9.84991	+	9.84991i	0.467983	+	0.467983i	−0.433277	+	0.901261i	$0.642643\pi$
$444$		0			0
$445$	−13.2952	−	10.2630i	−0.630252	−	0.486512i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.7067	−	12.7067i	−0.599666	−	0.599666i	0.340558	−	0.940224i	$-0.389384\pi$
$449$	−12.7067	−	12.7067i	−0.599666	−	0.599666i	−0.940224	+	0.340558i	$0.889384\pi$
$450$		0			0
$451$	−11.2514			−0.529809
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.65659	−	5.40965i	−0.405827	−	0.253608i
$456$		0			0
$457$	10.4226			0.487551			0.243775	−	0.969832i	$-0.421614\pi$
$457$	10.4226			0.487551			0.243775	+	0.969832i	$0.421614\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	23.2616	+	23.2616i	1.08340	+	1.08340i	0.996190	+	0.0872084i	$0.0277946\pi$
$461$	23.2616	+	23.2616i	1.08340	+	1.08340i	0.0872084	+	0.996190i	$0.472205\pi$
$462$		0			0
$463$	2.36644			0.109978			0.0549888	−	0.998487i	$-0.482488\pi$
$463$	2.36644			0.109978			0.0549888	+	0.998487i	$0.482488\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.6648	−	22.6648i	−1.04880	−	1.04880i	−0.998746	−	0.0500547i	$-0.984060\pi$
$467$	−22.6648	−	22.6648i	−1.04880	−	1.04880i	−0.0500547	−	0.998746i	$-0.515940\pi$
$468$		0			0
$469$		−	1.65201i		−	0.0762828i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			47.6888i			2.19274i
$474$		0			0
$475$	−9.11146	+	15.5727i	−0.418062	+	0.714526i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.74861	+	1.74861i	0.0798958	+	0.0798958i	0.745925	−	0.666030i	$-0.232008\pi$
$479$	1.74861	+	1.74861i	0.0798958	+	0.0798958i	−0.666030	+	0.745925i	$0.732008\pi$
$480$		0			0
$481$	16.2776	+	13.3403i	0.742195	+	0.608263i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.25314	−	0.804846i	0.283940	−	0.0365462i
$486$		0			0
$487$	−14.3822			−0.651720			−0.325860	−	0.945418i	$-0.605654\pi$
$487$	−14.3822			−0.651720			−0.325860	+	0.945418i	$0.605654\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			13.4770i			0.608206i	0.952639	+	0.304103i	$0.0983568\pi$
$491$			13.4770i			0.608206i	−0.952639	+	0.304103i	$0.901643\pi$
$492$		0			0
$493$	−13.8536	+	13.8536i	−0.623935	+	0.623935i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.0501	+	11.0501i	−0.495665	+	0.495665i
$498$		0			0
$499$	−19.7644	+	19.7644i	−0.884775	+	0.884775i	−0.994015	−	0.109241i	$-0.965158\pi$
$499$	−19.7644	+	19.7644i	−0.884775	+	0.884775i	0.109241	+	0.994015i	$0.465158\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.04181	+	6.04181i	0.269391	+	0.269391i	0.828855	−	0.559464i	$-0.188993\pi$
$503$	6.04181	+	6.04181i	0.269391	+	0.269391i	−0.559464	+	0.828855i	$0.688993\pi$
$504$		0			0
$505$	0.364567	+	2.83245i	0.0162230	+	0.126043i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.1677	−	13.1677i	0.583648	−	0.583648i	−0.352256	−	0.935904i	$-0.614585\pi$
$509$	13.1677	−	13.1677i	0.583648	−	0.583648i	0.935904	+	0.352256i	$0.114585\pi$
$510$		0			0
$511$			9.67781i			0.428121i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.604932	−	0.783659i	0.0266565	−	0.0345321i
$516$		0			0
$517$	45.5724	+	45.5724i	2.00427	+	2.00427i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.8720			−0.520121			−0.260061	−	0.965592i	$-0.583743\pi$
$521$	−11.8720			−0.520121			−0.260061	+	0.965592i	$0.583743\pi$
$522$		0			0
$523$	28.9954	+	28.9954i	1.26788	+	1.26788i	0.947180	+	0.320702i	$0.103919\pi$
$523$	28.9954	+	28.9954i	1.26788	+	1.26788i	0.320702	+	0.947180i	$0.396081\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			10.7772i			0.469464i
$528$		0			0
$529$		−	31.4503i		−	1.36740i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.76792	−	0.671206i	0.293151	−	0.0290732i
$534$		0			0
$535$	−13.8937	+	17.9986i	−0.600678	+	0.778147i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−22.7629	+	22.7629i	−0.980469	+	0.980469i
$540$		0			0
$541$	−4.93375	+	4.93375i	−0.212119	+	0.212119i	−0.805167	−	0.593048i	$-0.797924\pi$
$541$	−4.93375	+	4.93375i	−0.212119	+	0.212119i	0.593048	+	0.805167i	$0.297924\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.94020	−	7.69523i	0.254450	−	0.329627i
$546$		0			0
$547$	−19.6257	+	19.6257i	−0.839135	+	0.839135i	−0.988745	−	0.149610i	$-0.952198\pi$
$547$	−19.6257	+	19.6257i	−0.839135	+	0.839135i	0.149610	+	0.988745i	$0.452198\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.2934	−	18.2934i	−0.779323	−	0.779323i
$552$		0			0
$553$	1.20905			0.0514139
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	31.5792i		−	1.33805i	−0.743238	−	0.669027i	$-0.766711\pi$
$557$		−	31.5792i		−	1.33805i	0.743238	−	0.669027i	$-0.233289\pi$
$558$		0			0
$559$	−2.84489	−	28.6857i	−0.120326	−	1.21327i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.3660	−	21.3660i	0.900471	−	0.900471i	−0.0950060	−	0.995477i	$-0.530287\pi$
$563$	21.3660	−	21.3660i	0.900471	−	0.900471i	0.995477	+	0.0950060i	$0.0302870\pi$
$564$		0			0
$565$	−22.6870	−	17.5128i	−0.954448	−	0.736770i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−41.8436			−1.75418			−0.877088	−	0.480329i	$-0.840517\pi$
$569$	−41.8436			−1.75418			−0.877088	+	0.480329i	$0.840517\pi$
$570$		0			0
$571$			22.5809i			0.944983i	0.881335	+	0.472491i	$0.156645\pi$
$571$			22.5809i			0.944983i	−0.881335	+	0.472491i	$0.843355\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.6322	−	31.8449i	0.777015	−	1.32803i
$576$		0			0
$577$	−23.6436			−0.984296			−0.492148	−	0.870512i	$-0.663788\pi$
$577$	−23.6436			−0.984296			−0.492148	+	0.870512i	$0.663788\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.49624			−0.269510
$582$		0			0
$583$	−78.2193			−3.23951
$584$		0			0
$585$		0			0
$586$		0			0
$587$	37.8370			1.56170			0.780851	−	0.624718i	$-0.214786\pi$
$587$	37.8370			1.56170			0.780851	+	0.624718i	$0.214786\pi$
$588$		0			0
$589$	−14.2311			−0.586381
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.64547			−0.149701			−0.0748507	−	0.997195i	$-0.523848\pi$
$593$	−3.64547			−0.149701			−0.0748507	+	0.997195i	$0.523848\pi$
$594$		0			0
$595$	4.72757	−	6.12433i	0.193812	−	0.251073i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	25.0451i		−	1.02331i	−0.859190	−	0.511657i	$-0.829032\pi$
$599$		−	25.0451i		−	1.02331i	0.859190	−	0.511657i	$-0.170968\pi$
$600$		0			0
$601$	4.47132			0.182389			0.0911945	−	0.995833i	$-0.470932\pi$
$601$	4.47132			0.182389			0.0911945	+	0.995833i	$0.470932\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.01617	−	54.5111i	−0.285248	−	2.21619i
$606$		0			0
$607$	−3.24538	+	3.24538i	−0.131726	+	0.131726i	−0.769896	−	0.638170i	$-0.779692\pi$
$607$	−3.24538	+	3.24538i	−0.131726	+	0.131726i	0.638170	+	0.769896i	$0.279692\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−30.1312	−	24.6940i	−1.21898	−	0.999010i
$612$		0			0
$613$		−	47.3997i		−	1.91445i	−0.289337	−	0.957227i	$-0.593435\pi$
$613$		−	47.3997i		−	1.91445i	0.289337	−	0.957227i	$-0.406565\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.45402			−0.0585366			−0.0292683	−	0.999572i	$-0.509318\pi$
$617$	−1.45402			−0.0585366			−0.0292683	+	0.999572i	$0.509318\pi$
$618$		0			0
$619$	11.7229	+	11.7229i	0.471182	+	0.471182i	0.902297	−	0.431115i	$-0.141880\pi$
$619$	11.7229	+	11.7229i	0.471182	+	0.471182i	−0.431115	+	0.902297i	$0.641880\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.72471	+	6.72471i	−0.269420	+	0.269420i
$624$		0			0
$625$	21.7938	−	12.2487i	0.871753	−	0.489946i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.2790	+	11.2790i	−0.449724	+	0.449724i
$630$		0			0
$631$	21.2758	−	21.2758i	0.846975	−	0.846975i	−0.142780	−	0.989754i	$-0.545604\pi$
$631$	21.2758	−	21.2758i	0.846975	−	0.846975i	0.989754	+	0.142780i	$0.0456041\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.2321	+	13.3020i	0.683835	+	0.527875i
$636$		0			0
$637$	12.3344	−	15.0502i	0.488706	−	0.596312i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.2716i		−	0.721684i	−0.932627	−	0.360842i	$-0.882489\pi$
$641$		−	18.2716i		−	0.721684i	0.932627	−	0.360842i	$-0.117511\pi$
$642$		0			0
$643$		−	27.6308i		−	1.08965i	−0.838550	−	0.544825i	$-0.816596\pi$
$643$		−	27.6308i		−	1.08965i	0.838550	−	0.544825i	$-0.183404\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.6648	+	22.6648i	0.891045	+	0.891045i	0.994621	−	0.103577i	$-0.0330288\pi$
$647$	22.6648	+	22.6648i	0.891045	+	0.891045i	−0.103577	+	0.994621i	$0.533029\pi$
$648$		0			0
$649$	−21.5240			−0.844891
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.3338	−	17.3338i	−0.678324	−	0.678324i	0.281296	−	0.959621i	$-0.409236\pi$
$653$	−17.3338	−	17.3338i	−0.678324	−	0.678324i	−0.959621	+	0.281296i	$0.909236\pi$
$654$		0			0
$655$	−0.429000	+	0.0552170i	−0.0167624	+	0.00215751i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22.5065i		−	0.876730i	−0.898797	−	0.438365i	$-0.855558\pi$
$659$		−	22.5065i		−	0.876730i	0.898797	−	0.438365i	$-0.144442\pi$
$660$		0			0
$661$	−8.66806	+	8.66806i	−0.337148	+	0.337148i	−0.855293	−	0.518145i	$-0.826623\pi$
$661$	−8.66806	+	8.66806i	−0.337148	+	0.337148i	0.518145	+	0.855293i	$0.326623\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.08702	+	6.24264i	0.313601	+	0.242079i
$666$		0			0
$667$	37.4084	+	37.4084i	1.44846	+	1.44846i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−58.4159	+	58.4159i	−2.25512	+	2.25512i
$672$		0			0
$673$	9.10360	−	9.10360i	0.350918	−	0.350918i	−0.509533	−	0.860451i	$-0.670182\pi$
$673$	9.10360	−	9.10360i	0.350918	−	0.350918i	0.860451	+	0.509533i	$0.170182\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.29832	−	9.29832i	0.357364	−	0.357364i	−0.505477	−	0.862840i	$-0.668683\pi$
$677$	9.29832	−	9.29832i	0.357364	−	0.357364i	0.862840	+	0.505477i	$0.168683\pi$
$678$		0			0
$679$		−	3.56993i		−	0.137001i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.8177			−0.413927			−0.206964	−	0.978349i	$-0.566358\pi$
$683$	−10.8177			−0.413927			−0.206964	+	0.978349i	$0.566358\pi$
$684$		0			0
$685$	1.15283	+	8.95678i	0.0440475	+	0.342221i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	47.0503	−	4.66619i	1.79247	−	0.177768i
$690$		0			0
$691$	−0.121054	−	0.121054i	−0.00460512	−	0.00460512i	0.704800	−	0.709406i	$-0.251037\pi$
$691$	−0.121054	−	0.121054i	−0.00460512	−	0.00460512i	−0.709406	+	0.704800i	$0.751037\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.733868	+	5.70168i	0.0278372	+	0.216277i
$696$		0			0
$697$			5.15470i			0.195248i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.15470i		−	0.0436125i	−0.999762	−	0.0218063i	$-0.993058\pi$
$701$		−	1.15470i		−	0.0436125i	0.999762	−	0.0218063i	$-0.00694170\pi$
$702$		0			0
$703$	−14.8937	−	14.8937i	−0.561726	−	0.561726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.61706			0.0608156
$708$		0			0
$709$	−37.1464	−	37.1464i	−1.39506	−	1.39506i	−0.813503	−	0.581561i	$-0.802442\pi$
$709$	−37.1464	−	37.1464i	−1.39506	−	1.39506i	−0.581561	−	0.813503i	$-0.697558\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	29.1014			1.08985
$714$		0			0
$715$	10.8163	+	46.8578i	0.404506	+	1.75238i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.4383			0.501165			0.250582	−	0.968095i	$-0.419378\pi$
$719$	13.4383			0.501165			0.250582	+	0.968095i	$0.419378\pi$
$720$		0			0
$721$	−0.396375	−	0.396375i	−0.0147618	−	0.0147618i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.07743	+	34.6787i	0.337127	+	1.28793i
$726$		0			0
$727$	−28.7860	−	28.7860i	−1.06761	−	1.06761i	−0.997542	−	0.0700702i	$-0.977678\pi$
$727$	−28.7860	−	28.7860i	−1.06761	−	1.06761i	−0.0700702	−	0.997542i	$-0.522322\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	21.8481			0.808080
$732$		0			0
$733$			28.5996i			1.05635i	0.849136	+	0.528174i	$0.177123\pi$
$733$			28.5996i			1.05635i	−0.849136	+	0.528174i	$0.822877\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.50322	+	5.50322i	−0.202714	+	0.202714i
$738$		0			0
$739$	11.0484	+	11.0484i	0.406420	+	0.406420i	0.880488	−	0.474068i	$-0.157215\pi$
$739$	11.0484	+	11.0484i	0.406420	+	0.406420i	−0.474068	+	0.880488i	$0.657215\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.7882i			0.432466i	0.976342	+	0.216233i	$0.0693771\pi$
$743$			11.7882i			0.432466i	−0.976342	+	0.216233i	$0.930623\pi$
$744$		0			0
$745$	−1.73574	+	2.24856i	−0.0635924	+	0.0823808i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.10370	+	9.10370i	0.332642	+	0.332642i
$750$		0			0
$751$		−	30.3583i		−	1.10779i	−0.832587	−	0.553895i	$-0.813141\pi$
$751$		−	30.3583i		−	1.10779i	0.832587	−	0.553895i	$-0.186859\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−14.6579	+	18.9886i	−0.533456	+	0.691065i
$756$		0			0
$757$	19.1677	−	19.1677i	0.696662	−	0.696662i	−0.267027	−	0.963689i	$-0.586041\pi$
$757$	19.1677	−	19.1677i	0.696662	−	0.696662i	0.963689	+	0.267027i	$0.0860414\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.2165	+	12.2165i	−0.442847	+	0.442847i	−0.892968	−	0.450121i	$-0.851381\pi$
$761$	−12.2165	+	12.2165i	−0.442847	+	0.442847i	0.450121	+	0.892968i	$0.351381\pi$
$762$		0			0
$763$	−3.89225	−	3.89225i	−0.140909	−	0.140909i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.9471	−	1.28402i	0.467491	−	0.0463632i
$768$		0			0
$769$	−23.5875	+	23.5875i	−0.850586	+	0.850586i	−0.990205	−	0.139619i	$-0.955412\pi$
$769$	−23.5875	+	23.5875i	−0.850586	+	0.850586i	0.139619	+	0.990205i	$0.455412\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			31.8600i			1.14592i	0.819582	+	0.572962i	$0.194206\pi$
$773$			31.8600i			1.14592i	−0.819582	+	0.572962i	$0.805794\pi$
$774$		0			0
$775$	17.0198	+	9.95809i	0.611368	+	0.357705i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.80666			−0.243874
$780$		0			0
$781$	73.6208			2.63436
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.3600	+	13.4208i	−0.369764	+	0.479010i
$786$		0			0
$787$		−	28.3333i		−	1.00997i	−0.863127	−	0.504987i	$-0.831497\pi$
$787$		−	28.3333i		−	1.00997i	0.863127	−	0.504987i	$-0.168503\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.4751	+	11.4751i	−0.408007	+	0.408007i
$792$		0			0
$793$	31.6533	−	38.6230i	1.12404	−	1.37154i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.2610	+	21.2610i	0.753104	+	0.753104i	0.975057	−	0.221953i	$-0.0712433\pi$
$797$	21.2610	+	21.2610i	0.753104	+	0.753104i	−0.221953	+	0.975057i	$0.571243\pi$
$798$		0			0
$799$	20.8784	−	20.8784i	0.738626	−	0.738626i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.2390	−	32.2390i	1.13769	−	1.13769i
$804$		0			0
$805$	−16.5373	−	12.7657i	−0.582863	−	0.449931i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			22.8766i			0.804300i	0.915574	+	0.402150i	$0.131737\pi$
$809$			22.8766i			0.804300i	−0.915574	+	0.402150i	$0.868263\pi$
$810$		0			0
$811$	−20.7214	−	20.7214i	−0.727628	−	0.727628i	0.242519	−	0.970147i	$-0.422026\pi$
$811$	−20.7214	−	20.7214i	−0.727628	−	0.727628i	−0.970147	+	0.242519i	$0.922026\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	37.2326	−	4.79223i	1.30420	−	0.167865i
$816$		0			0
$817$			28.8498i			1.00933i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.2951	−	28.2951i	−0.987505	−	0.987505i	0.0124179	−	0.999923i	$-0.496047\pi$
$821$	−28.2951	−	28.2951i	−0.987505	−	0.987505i	−0.999923	+	0.0124179i	$0.996047\pi$
$822$		0			0
$823$	−13.8518	+	13.8518i	−0.482842	+	0.482842i	−0.906038	−	0.423196i	$-0.860908\pi$
$823$	−13.8518	+	13.8518i	−0.482842	+	0.482842i	0.423196	+	0.906038i	$0.360908\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			10.5533i			0.366975i	0.983022	+	0.183488i	$0.0587387\pi$
$827$			10.5533i			0.366975i	−0.983022	+	0.183488i	$0.941261\pi$
$828$		0			0
$829$	35.3056			1.22622			0.613108	−	0.789999i	$-0.289919\pi$
$829$	35.3056			1.22622			0.613108	+	0.789999i	$0.289919\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.4286	+	10.4286i	0.361328	+	0.361328i
$834$		0			0
$835$	4.93232	+	38.3210i	0.170690	+	1.32615i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.46456	+	2.46456i	0.0850860	+	0.0850860i	0.748369	−	0.663283i	$-0.230837\pi$
$839$	2.46456	+	2.46456i	0.0850860	+	0.0850860i	−0.663283	+	0.748369i	$0.730837\pi$
$840$		0			0
$841$	−22.4005			−0.772431
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.30150	−	27.5406i	−0.319981	−	0.947424i
$846$		0			0
$847$	−31.1205			−1.06931
$848$		0			0
$849$		0			0
$850$		0			0
$851$	30.4563	+	30.4563i	1.04403	+	1.04403i
$852$		0			0
$853$	26.8214			0.918348			0.459174	−	0.888346i	$-0.348145\pi$
$853$	26.8214			0.918348			0.459174	+	0.888346i	$0.348145\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.6207	−	20.6207i	−0.704390	−	0.704390i	0.260959	−	0.965350i	$-0.415961\pi$
$857$	−20.6207	−	20.6207i	−0.704390	−	0.704390i	−0.965350	+	0.260959i	$0.915961\pi$
$858$		0			0
$859$		−	15.3195i		−	0.522696i	−0.965245	−	0.261348i	$-0.915833\pi$
$859$		−	15.3195i		−	0.522696i	0.965245	−	0.261348i	$-0.0841670\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			19.7984i			0.673945i	0.941514	+	0.336973i	$0.109403\pi$
$863$			19.7984i			0.673945i	−0.941514	+	0.336973i	$0.890597\pi$
$864$		0			0
$865$	8.74533	−	11.3291i	0.297350	−	0.385202i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.02761	−	4.02761i	−0.136627	−	0.136627i
$870$		0			0
$871$	2.98199	−	3.63858i	0.101041	−	0.123289i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.30349	−	13.1248i	−0.179291	−	0.443698i
$876$		0			0
$877$	−36.4393			−1.23047			−0.615234	−	0.788345i	$-0.710938\pi$
$877$	−36.4393			−1.23047			−0.615234	+	0.788345i	$0.710938\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	39.4236i		−	1.32821i	−0.747637	−	0.664107i	$-0.768812\pi$
$881$		−	39.4236i		−	1.32821i	0.747637	−	0.664107i	$-0.231188\pi$
$882$		0			0
$883$	−23.9341	+	23.9341i	−0.805445	+	0.805445i	−0.983941	−	0.178495i	$-0.942877\pi$
$883$	−23.9341	+	23.9341i	−0.805445	+	0.805445i	0.178495	+	0.983941i	$0.442877\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.8887	+	28.8887i	−0.969987	+	0.969987i	−0.999563	−	0.0295751i	$-0.990585\pi$
$887$	−28.8887	+	28.8887i	−0.969987	+	0.969987i	0.0295751	+	0.999563i	$0.490585\pi$
$888$		0			0
$889$	8.71601	−	8.71601i	0.292326	−	0.292326i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	27.5695	+	27.5695i	0.922577	+	0.922577i
$894$		0			0
$895$	15.5673	−	2.00368i	0.520358	−	0.0669757i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−19.9932	+	19.9932i	−0.666810	+	0.666810i
$900$		0			0
$901$			35.8352i			1.19384i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.862394	+	6.70025i	0.0286669	+	0.222724i
$906$		0			0
$907$	22.9907	+	22.9907i	0.763394	+	0.763394i	0.976934	−	0.213540i	$-0.0684994\pi$
$907$	22.9907	+	22.9907i	0.763394	+	0.763394i	−0.213540	+	0.976934i	$0.568499\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−37.1427			−1.23059			−0.615296	−	0.788296i	$-0.710963\pi$
$911$	−37.1427			−1.23059			−0.615296	+	0.788296i	$0.710963\pi$
$912$		0			0
$913$	21.6404	+	21.6404i	0.716194	+	0.716194i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			0.244917i			0.00808789i
$918$		0			0
$919$		−	46.7810i		−	1.54316i	−0.636131	−	0.771581i	$-0.719466\pi$
$919$		−	46.7810i		−	1.54316i	0.636131	−	0.771581i	$-0.280534\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−44.2842	+	4.39186i	−1.45763	+	0.144560i
$924$		0			0
$925$	7.39046	+	28.2339i	0.242997	+	0.928326i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29.4287	−	29.4287i	0.965525	−	0.965525i	−0.0339004	−	0.999425i	$-0.510793\pi$
$929$	29.4287	−	29.4287i	0.965525	−	0.965525i	0.999425	+	0.0339004i	$0.0107929\pi$
$930$		0			0
$931$	−13.7707	+	13.7707i	−0.451315	+	0.451315i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−36.1501	+	4.65291i	−1.18224	+	0.152166i
$936$		0			0
$937$	−29.2450	+	29.2450i	−0.955392	+	0.955392i	−0.999047	−	0.0436550i	$-0.986100\pi$
$937$	−29.2450	+	29.2450i	−0.955392	+	0.955392i	0.0436550	+	0.999047i	$0.486100\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.1015	−	15.1015i	−0.492293	−	0.492293i	0.416735	−	0.909028i	$-0.363174\pi$
$941$	−15.1015	−	15.1015i	−0.492293	−	0.492293i	−0.909028	+	0.416735i	$0.863174\pi$
$942$		0			0
$943$	13.9190			0.453266
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	38.1180i		−	1.23867i	−0.785127	−	0.619335i	$-0.787402\pi$
$947$		−	38.1180i		−	1.23867i	0.785127	−	0.619335i	$-0.212598\pi$
$948$		0			0
$949$	−17.4691	+	21.3155i	−0.567070	+	0.691931i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	35.8177	−	35.8177i	1.16025	−	1.16025i	0.175830	−	0.984421i	$-0.443739\pi$
$953$	35.8177	−	35.8177i	1.16025	−	1.16025i	0.984421	−	0.175830i	$-0.0562609\pi$
$954$		0			0
$955$	2.03209	−	0.261552i	0.0657570	−	0.00846363i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.11345			0.165122
$960$		0			0
$961$		−	15.4466i		−	0.498277i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	33.3952	−	4.29832i	1.07503	−	0.138368i
$966$		0			0
$967$	−15.1852			−0.488322			−0.244161	−	0.969735i	$-0.578513\pi$
$967$	−15.1852			−0.488322			−0.244161	+	0.969735i	$0.578513\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	52.5959			1.68788			0.843941	−	0.536436i	$-0.180230\pi$
$971$	52.5959			1.68788			0.843941	+	0.536436i	$0.180230\pi$
$972$		0			0
$973$	3.25510			0.104354
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−40.6014			−1.29895			−0.649477	−	0.760381i	$-0.725012\pi$
$977$	−40.6014			−1.29895			−0.649477	+	0.760381i	$0.725012\pi$
$978$		0			0
$979$	44.8030			1.43191
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.2368			−0.390294			−0.195147	−	0.980774i	$-0.562518\pi$
$983$	−12.2368			−0.390294			−0.195147	+	0.980774i	$0.562518\pi$
$984$		0			0
$985$	12.3407	−	1.58838i	0.393207	−	0.0506100i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	58.9955i		−	1.87595i
$990$		0			0
$991$	−20.9218			−0.664603			−0.332302	−	0.943173i	$-0.607825\pi$
$991$	−20.9218			−0.664603			−0.332302	+	0.943173i	$0.607825\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	36.0644	−	4.64188i	1.14332	−	0.147158i
$996$		0			0
$997$	9.72732	−	9.72732i	0.308067	−	0.308067i	−0.536092	−	0.844159i	$-0.680100\pi$
$997$	9.72732	−	9.72732i	0.308067	−	0.308067i	0.844159	+	0.536092i	$0.180100\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.u.g.73.4		8
3.2	odd	2		260.2.m.c.73.2	yes	8
5.2	odd	4		2340.2.bp.g.1477.2		8
12.11	even	2		1040.2.bg.m.593.3		8
13.5	odd	4		2340.2.bp.g.1513.2		8
15.2	even	4		260.2.r.c.177.2	yes	8
15.8	even	4		1300.2.r.c.957.3		8
15.14	odd	2		1300.2.m.c.593.3		8
39.5	even	4		260.2.r.c.213.2	yes	8
60.47	odd	4		1040.2.cd.m.177.3		8
65.57	even	4	inner	2340.2.u.g.577.4		8
156.83	odd	4		1040.2.cd.m.993.3		8
195.44	even	4		1300.2.r.c.993.3		8
195.83	odd	4		1300.2.m.c.57.3		8
195.122	odd	4		260.2.m.c.57.2	✓	8
780.707	even	4		1040.2.bg.m.577.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.m.c.57.2	✓	8	195.122	odd	4
260.2.m.c.73.2	yes	8	3.2	odd	2
260.2.r.c.177.2	yes	8	15.2	even	4
260.2.r.c.213.2	yes	8	39.5	even	4
1040.2.bg.m.577.3		8	780.707	even	4
1040.2.bg.m.593.3		8	12.11	even	2
1040.2.cd.m.177.3		8	60.47	odd	4
1040.2.cd.m.993.3		8	156.83	odd	4
1300.2.m.c.57.3		8	195.83	odd	4
1300.2.m.c.593.3		8	15.14	odd	2
1300.2.r.c.957.3		8	15.8	even	4
1300.2.r.c.993.3		8	195.44	even	4
2340.2.u.g.73.4		8	1.1	even	1	trivial
2340.2.u.g.577.4		8	65.57	even	4	inner
2340.2.bp.g.1477.2		8	5.2	odd	4
2340.2.bp.g.1513.2		8	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.21777 - 0.285451i) q^{5} -1.26613i q^{7} +(-4.21777 + 4.21777i) q^{11} +(2.28545 - 2.78868i) q^{13} +(1.93232 + 1.93232i) q^{17} +(-2.55158 + 2.55158i) q^{19} +(5.21777 - 5.21777i) q^{23} +(4.83704 - 1.26613i) q^{25} +7.16941i q^{29} +(2.78868 + 2.78868i) q^{31} +(-0.361419 - 2.80799i) q^{35} +5.83704i q^{37} +(1.33381 + 1.33381i) q^{41} +(5.65332 - 5.65332i) q^{43} -10.8048i q^{47} +5.39691 q^{49} +(9.27258 + 9.27258i) q^{53} +(-8.15009 + 10.5580i) q^{55} +(2.55158 + 2.55158i) q^{59} +13.8499 q^{61} +(4.27258 - 6.83704i) q^{65} +1.30477 q^{67} +(-8.72745 - 8.72745i) q^{71} -7.64360 q^{73} +(5.34026 + 5.34026i) q^{77} +0.954914i q^{79} -5.13078i q^{83} +(4.83704 + 3.73387i) q^{85} +(-5.31122 - 5.31122i) q^{89} +(-3.53083 - 2.89368i) q^{91} +(-4.93048 + 6.38719i) q^{95} +2.81956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 14 q^{11} + 14 q^{13} + 2 q^{19} + 22 q^{23} + 12 q^{25} - 6 q^{31} + 4 q^{35} + 8 q^{41} - 14 q^{43} - 24 q^{49} + 8 q^{53} - 30 q^{55} - 2 q^{59} - 12 q^{61} - 32 q^{65} + 20 q^{67} + 22 q^{71}+ \cdots + 12 q^{97}+O(q^{100}) \) 8 * q - 2 * q^5 - 14 * q^11 + 14 * q^13 + 2 * q^19 + 22 * q^23 + 12 * q^25 - 6 * q^31 + 4 * q^35 + 8 * q^41 - 14 * q^43 - 24 * q^49 + 8 * q^53 - 30 * q^55 - 2 * q^59 - 12 * q^61 - 32 * q^65 + 20 * q^67 + 22 * q^71 - 28 * q^73 - 8 * q^77 + 12 * q^85 - 4 * q^89 + 6 * q^95 + 12 * q^97

Introduction

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