LMFDB - Embedded newform 2340.2.u.g.577.4

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			577.4
Root			$-0.285451 - 0.285451i$ of defining polynomial
Character	$\chi$	$=$	2340.577
Dual form			2340.2.u.g.73.4

$q$-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 $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

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

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
$5$	2.21777	+	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	$0.0769103\pi$
$7$			1.26613i			0.478553i	−0.970951	+	0.239277i	$0.923090\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.21777	−	4.21777i	−1.27171	−	1.27171i	−0.945192	−	0.326514i	$-0.894126\pi$
$11$	−4.21777	−	4.21777i	−1.27171	−	1.27171i	−0.326514	−	0.945192i	$-0.605874\pi$
$12$		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	−0.432822	−	0.901479i	$-0.642482\pi$
$17$	1.93232	−	1.93232i	0.468657	−	0.468657i	0.901479	+	0.432822i	$0.142482\pi$
$18$		0			0
$19$	−2.55158	−	2.55158i	−0.585373	−	0.585373i	0.351001	−	0.936375i	$-0.385841\pi$
$19$	−2.55158	−	2.55158i	−0.585373	−	0.585373i	−0.936375	+	0.351001i	$0.885841\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.21777	+	5.21777i	1.08798	+	1.08798i	0.995736	+	0.0922445i	$0.0294041\pi$
$23$	5.21777	+	5.21777i	1.08798	+	1.08798i	0.0922445	+	0.995736i	$0.470596\pi$
$24$		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	$-0.768149\pi$
$29$		−	7.16941i		−	1.33133i	0.746252	−	0.665663i	$-0.231851\pi$
$30$		0			0
$31$	2.78868	−	2.78868i	0.500861	−	0.500861i	−0.410844	−	0.911705i	$-0.634766\pi$
$31$	2.78868	−	2.78868i	0.500861	−	0.500861i	0.911705	+	0.410844i	$0.134766\pi$
$32$		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	$-0.840709\pi$
$37$		−	5.83704i		−	0.959603i	0.877377	−	0.479801i	$-0.159291\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.33381	−	1.33381i	0.208306	−	0.208306i	−0.595241	−	0.803547i	$-0.702943\pi$
$41$	1.33381	−	1.33381i	0.208306	−	0.208306i	0.803547	+	0.595241i	$0.202943\pi$
$42$		0			0
$43$	5.65332	+	5.65332i	0.862123	+	0.862123i	0.991584	−	0.129461i	$-0.0413248\pi$
$43$	5.65332	+	5.65332i	0.862123	+	0.862123i	−0.129461	+	0.991584i	$0.541325\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.8048i			1.57605i	0.615644	+	0.788024i	$0.288896\pi$
$47$			10.8048i			1.57605i	−0.615644	+	0.788024i	$0.711104\pi$
$48$		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	0.329548	−	0.944139i	$-0.393104\pi$
$53$	9.27258	−	9.27258i	1.27369	−	1.27369i	0.944139	−	0.329548i	$-0.106896\pi$
$54$		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	−0.521229	−	0.853417i	$-0.674526\pi$
$59$	2.55158	−	2.55158i	0.332188	−	0.332188i	0.853417	+	0.521229i	$0.174526\pi$
$60$		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	−0.0364208	+	0.999337i	$0.511596\pi$
$71$	−8.72745	+	8.72745i	−1.03576	+	1.03576i	−0.999337	+	0.0364208i	$0.988404\pi$
$72$		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	$-0.982893\pi$
$79$		−	0.954914i		−	0.107436i	0.998556	−	0.0537181i	$-0.0171072\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.13078i			0.563176i	0.959535	+	0.281588i	$0.0908611\pi$
$83$			5.13078i			0.563176i	−0.959535	+	0.281588i	$0.909139\pi$
$84$		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	−0.930155	−	0.367167i	$-0.880328\pi$
$89$	−5.31122	+	5.31122i	−0.562988	+	0.562988i	0.367167	+	0.930155i	$0.380328\pi$
$90$		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	$-0.979761\pi$
$101$		−	1.27716i		−	0.127082i	0.997979	−	0.0635412i	$-0.0202394\pi$
$102$		0			0
$103$	0.313060	+	0.313060i	0.0308467	+	0.0308467i	0.722362	−	0.691515i	$-0.243057\pi$
$103$	0.313060	+	0.313060i	0.0308467	+	0.0308467i	−0.691515	+	0.722362i	$0.743057\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.19016	−	7.19016i	−0.695099	−	0.695099i	0.268250	−	0.963349i	$-0.413555\pi$
$107$	−7.19016	−	7.19016i	−0.695099	−	0.695099i	−0.963349	+	0.268250i	$0.913555\pi$
$108$		0			0
$109$	3.07413	+	3.07413i	0.294448	+	0.294448i	0.838834	−	0.544387i	$-0.183238\pi$
$109$	3.07413	+	3.07413i	0.294448	+	0.294448i	−0.544387	+	0.838834i	$0.683238\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.06310	+	9.06310i	−0.852585	+	0.852585i	−0.990451	−	0.137866i	$-0.955976\pi$
$113$	−9.06310	+	9.06310i	−0.852585	+	0.852585i	0.137866	+	0.990451i	$0.455976\pi$
$114$		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	−0.332315	−	0.943168i	$-0.607830\pi$
$127$	6.88396	−	6.88396i	0.610853	−	0.610853i	0.943168	+	0.332315i	$0.107830\pi$
$128$		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	$-0.944808\pi$
$137$		−	4.03864i		−	0.345044i	0.985006	−	0.172522i	$-0.0551916\pi$
$138$		0			0
$139$		−	2.57090i		−	0.218061i	−0.994038	−	0.109031i	$-0.965225\pi$
$139$		−	2.57090i		−	0.218061i	0.994038	−	0.109031i	$-0.0347746\pi$
$140$		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	0.669354	−	0.742943i	$-0.266571\pi$
$149$	−0.898264	−	0.898264i	−0.0735887	−	0.0735887i	−0.742943	+	0.669354i	$0.766571\pi$
$150$		0			0
$151$	−7.58564	−	7.58564i	−0.617311	−	0.617311i	0.327530	−	0.944841i	$-0.393784\pi$
$151$	−7.58564	−	7.58564i	−0.617311	−	0.617311i	−0.944841	+	0.327530i	$0.893784\pi$
$152$		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	0.460020	−	0.887908i	$-0.347842\pi$
$157$	−5.36142	−	5.36142i	−0.427888	−	0.427888i	−0.887908	+	0.460020i	$0.847842\pi$
$158$		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	$-0.766917\pi$
$167$		−	17.2790i		−	1.33709i	0.743671	−	0.668546i	$-0.233083\pi$
$168$		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	0.857903	−	0.513812i	$-0.171767\pi$
$173$	4.52582	+	4.52582i	0.344091	+	0.344091i	−0.513812	+	0.857903i	$0.671767\pi$
$174$		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	$-0.964184\pi$
$181$		−	3.02116i		−	0.224561i	0.993677	−	0.112281i	$-0.0358155\pi$
$182$		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	$-0.961485\pi$
$223$		−	3.60496i		−	0.241406i	0.992689	−	0.120703i	$-0.0385149\pi$
$224$		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	−0.321550	−	0.946893i	$-0.604204\pi$
$229$	9.46315	−	9.46315i	0.625343	−	0.625343i	0.946893	+	0.321550i	$0.104204\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.80011	+	1.80011i	0.117929	+	0.117929i	0.763609	−	0.645679i	$-0.223426\pi$
$233$	1.80011	+	1.80011i	0.117929	+	0.117929i	−0.645679	+	0.763609i	$0.723426\pi$
$234$		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	0.989803	−	0.142441i	$-0.0454953\pi$
$239$	13.0999	+	13.0999i	0.847362	+	0.847362i	−0.142441	+	0.989803i	$0.545495\pi$
$240$		0			0
$241$	−13.2371	−	13.2371i	−0.852676	−	0.852676i	0.137786	−	0.990462i	$-0.456001\pi$
$241$	−13.2371	−	13.2371i	−0.852676	−	0.852676i	−0.990462	+	0.137786i	$0.956001\pi$
$242$		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	$-0.953360\pi$
$251$		−	4.62612i		−	0.291998i	0.989285	−	0.145999i	$-0.0466396\pi$
$252$		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	−0.550681	−	0.834716i	$-0.685632\pi$
$257$	4.55342	−	4.55342i	0.284035	−	0.284035i	0.834716	+	0.550681i	$0.185632\pi$
$258$		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	−0.618221	−	0.786005i	$-0.712146\pi$
$263$	2.72100	−	2.72100i	0.167784	−	0.167784i	0.786005	+	0.618221i	$0.212146\pi$
$264$		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	$0.115956\pi$
$269$			11.6870i			0.712567i	−0.934378	+	0.356284i	$0.884044\pi$
$270$		0			0
$271$	−13.9660	−	13.9660i	−0.848372	−	0.848372i	0.141558	−	0.989930i	$-0.454789\pi$
$271$	−13.9660	−	13.9660i	−0.848372	−	0.848372i	−0.989930	+	0.141558i	$0.954789\pi$
$272$		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	−0.673391	−	0.739286i	$-0.735163\pi$
$277$	1.09672	−	1.09672i	0.0658954	−	0.0658954i	0.739286	+	0.673391i	$0.235163\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.62755	+	8.62755i	0.514677	+	0.514677i	0.915956	−	0.401279i	$-0.131434\pi$
$281$	8.62755	+	8.62755i	0.514677	+	0.514677i	−0.401279	+	0.915956i	$0.631434\pi$
$282$		0			0
$283$	7.50507	+	7.50507i	0.446130	+	0.446130i	0.894066	−	0.447936i	$-0.147841\pi$
$283$	7.50507	+	7.50507i	0.446130	+	0.446130i	−0.447936	+	0.894066i	$0.647841\pi$
$284$		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	$0.128825\pi$
$307$			13.7984i			0.787516i	−0.919214	+	0.393758i	$0.871175\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	19.9743i		−	1.13264i	−0.824187	−	0.566318i	$-0.808367\pi$
$311$		−	19.9743i		−	1.13264i	0.824187	−	0.566318i	$-0.191633\pi$
$312$		0			0
$313$	−4.92587	+	4.92587i	−0.278427	+	0.278427i	−0.832481	−	0.554054i	$-0.813080\pi$
$313$	−4.92587	+	4.92587i	−0.278427	+	0.278427i	0.554054	+	0.832481i	$0.313080\pi$
$314$		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	−0.829728	−	0.558168i	$-0.811505\pi$
$331$	−4.94061	+	4.94061i	−0.271561	+	0.271561i	0.558168	+	0.829728i	$0.311505\pi$
$332$		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	−0.902813	−	0.430033i	$-0.858502\pi$
$337$	−8.67909	+	8.67909i	−0.472780	+	0.472780i	0.430033	+	0.902813i	$0.358502\pi$
$338$		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	−0.999721	−	0.0236177i	$-0.992482\pi$
$347$	−19.0627	−	19.0627i	−1.02334	−	1.02334i	−0.0236177	−	0.999721i	$-0.507518\pi$
$348$		0			0
$349$	2.00645	−	2.00645i	0.107403	−	0.107403i	−0.651363	−	0.758766i	$-0.725803\pi$
$349$	2.00645	−	2.00645i	0.107403	−	0.107403i	0.758766	+	0.651363i	$0.225803\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.41162i		−	0.128358i	−0.997938	−	0.0641788i	$-0.979557\pi$
$353$		−	2.41162i		−	0.128358i	0.997938	−	0.0641788i	$-0.0204428\pi$
$354$		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	−0.756090	−	0.654468i	$-0.772893\pi$
$359$	−1.92546	+	1.92546i	−0.101622	+	0.101622i	0.654468	+	0.756090i	$0.272893\pi$
$360$		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	0.114230	−	0.993454i	$-0.463560\pi$
$367$	−16.8435	−	16.8435i	−0.879224	−	0.879224i	−0.993454	+	0.114230i	$0.963560\pi$
$368$		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	−0.0600904	+	0.998193i	$0.519139\pi$
$373$	−20.4388	+	20.4388i	−1.05828	+	1.05828i	−0.998193	+	0.0600904i	$0.980861\pi$
$374$		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	0.173975	−	0.984750i	$-0.444339\pi$
$379$	−15.7841	−	15.7841i	−0.810775	−	0.810775i	−0.984750	+	0.173975i	$0.944339\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			21.7432i			1.11102i	0.831508	+	0.555512i	$0.187478\pi$
$383$			21.7432i			1.11102i	−0.831508	+	0.555512i	$0.812522\pi$
$384$		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	$0.324782\pi$
$397$			33.9632i			1.70457i	−0.523081	+	0.852283i	$0.675218\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.0857	−	15.0857i	−0.753343	−	0.753343i	0.221758	−	0.975102i	$-0.428820\pi$
$401$	−15.0857	−	15.0857i	−0.753343	−	0.753343i	−0.975102	+	0.221758i	$0.928820\pi$
$402$		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	0.934690	−	0.355463i	$-0.115677\pi$
$409$	11.7141	+	11.7141i	0.579227	+	0.579227i	−0.355463	+	0.934690i	$0.615677\pi$
$410$		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	$0.104899\pi$
$419$			13.2486i			0.647234i	−0.946188	+	0.323617i	$0.895101\pi$
$420$		0			0
$421$	−6.17729	+	6.17729i	−0.301063	+	0.301063i	−0.841430	−	0.540367i	$-0.818286\pi$
$421$	−6.17729	+	6.17729i	−0.301063	+	0.301063i	0.540367	+	0.841430i	$0.318286\pi$
$422$		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	−0.802149	−	0.597124i	$-0.796310\pi$
$431$	−4.25641	+	4.25641i	−0.205024	+	0.205024i	0.597124	+	0.802149i	$0.296310\pi$
$432$		0			0
$433$	−27.2579	−	27.2579i	−1.30993	−	1.30993i	−0.921463	−	0.388467i	$-0.873005\pi$
$433$	−27.2579	−	27.2579i	−1.30993	−	1.30993i	−0.388467	−	0.921463i	$-0.626995\pi$
$434$		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	−0.433277	−	0.901261i	$-0.642643\pi$
$443$	9.84991	−	9.84991i	0.467983	−	0.467983i	0.901261	+	0.433277i	$0.142643\pi$
$444$		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	−0.940224	−	0.340558i	$-0.889384\pi$
$449$	−12.7067	+	12.7067i	−0.599666	+	0.599666i	0.340558	+	0.940224i	$0.389384\pi$
$450$		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	0.0872084	−	0.996190i	$-0.472205\pi$
$461$	23.2616	−	23.2616i	1.08340	−	1.08340i	0.996190	−	0.0872084i	$-0.0277946\pi$
$462$		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	−0.0500547	+	0.998746i	$0.515940\pi$
$467$	−22.6648	+	22.6648i	−1.04880	+	1.04880i	−0.998746	+	0.0500547i	$0.984060\pi$
$468$		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	−0.666030	−	0.745925i	$-0.732008\pi$
$479$	1.74861	−	1.74861i	0.0798958	−	0.0798958i	0.745925	+	0.666030i	$0.232008\pi$
$480$		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	$-0.901643\pi$
$491$		−	13.4770i		−	0.608206i	0.952639	−	0.304103i	$-0.0983568\pi$
$492$		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	0.109241	−	0.994015i	$-0.465158\pi$
$499$	−19.7644	−	19.7644i	−0.884775	−	0.884775i	−0.994015	+	0.109241i	$0.965158\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.04181	−	6.04181i	0.269391	−	0.269391i	−0.559464	−	0.828855i	$-0.688993\pi$
$503$	6.04181	−	6.04181i	0.269391	−	0.269391i	0.828855	+	0.559464i	$0.188993\pi$
$504$		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	0.935904	−	0.352256i	$-0.114585\pi$
$509$	13.1677	+	13.1677i	0.583648	+	0.583648i	−0.352256	+	0.935904i	$0.614585\pi$
$510$		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	0.320702	−	0.947180i	$-0.396081\pi$
$523$	28.9954	−	28.9954i	1.26788	−	1.26788i	0.947180	−	0.320702i	$-0.103919\pi$
$524$		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	0.593048	−	0.805167i	$-0.297924\pi$
$541$	−4.93375	−	4.93375i	−0.212119	−	0.212119i	−0.805167	+	0.593048i	$0.797924\pi$
$542$		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	0.149610	−	0.988745i	$-0.452198\pi$
$547$	−19.6257	−	19.6257i	−0.839135	−	0.839135i	−0.988745	+	0.149610i	$0.952198\pi$
$548$		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	$0.233289\pi$
$557$			31.5792i			1.33805i	−0.743238	+	0.669027i	$0.766711\pi$
$558$		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	0.995477	−	0.0950060i	$-0.0302870\pi$
$563$	21.3660	+	21.3660i	0.900471	+	0.900471i	−0.0950060	+	0.995477i	$0.530287\pi$
$564$		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	$-0.843355\pi$
$571$		−	22.5809i		−	0.944983i	0.881335	−	0.472491i	$-0.156645\pi$
$572$		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	$0.170968\pi$
$599$			25.0451i			1.02331i	−0.859190	+	0.511657i	$0.829032\pi$
$600$		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	0.638170	−	0.769896i	$-0.279692\pi$
$607$	−3.24538	−	3.24538i	−0.131726	−	0.131726i	−0.769896	+	0.638170i	$0.779692\pi$
$608$		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	$0.406565\pi$
$613$			47.3997i			1.91445i	−0.289337	+	0.957227i	$0.593435\pi$
$614$		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	−0.431115	−	0.902297i	$-0.641880\pi$
$619$	11.7229	−	11.7229i	0.471182	−	0.471182i	0.902297	+	0.431115i	$0.141880\pi$
$620$		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	0.989754	−	0.142780i	$-0.0456041\pi$
$631$	21.2758	+	21.2758i	0.846975	+	0.846975i	−0.142780	+	0.989754i	$0.545604\pi$
$632$		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	$0.117511\pi$
$641$			18.2716i			0.721684i	−0.932627	+	0.360842i	$0.882489\pi$
$642$		0			0
$643$			27.6308i			1.08965i	0.838550	+	0.544825i	$0.183404\pi$
$643$			27.6308i			1.08965i	−0.838550	+	0.544825i	$0.816596\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	22.6648	−	22.6648i	0.891045	−	0.891045i	−0.103577	−	0.994621i	$-0.533029\pi$
$647$	22.6648	−	22.6648i	0.891045	−	0.891045i	0.994621	+	0.103577i	$0.0330288\pi$
$648$		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	−0.959621	−	0.281296i	$-0.909236\pi$
$653$	−17.3338	+	17.3338i	−0.678324	+	0.678324i	0.281296	+	0.959621i	$0.409236\pi$
$654$		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	$0.144442\pi$
$659$			22.5065i			0.876730i	−0.898797	+	0.438365i	$0.855558\pi$
$660$		0			0
$661$	−8.66806	−	8.66806i	−0.337148	−	0.337148i	0.518145	−	0.855293i	$-0.326623\pi$
$661$	−8.66806	−	8.66806i	−0.337148	−	0.337148i	−0.855293	+	0.518145i	$0.826623\pi$
$662$		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	0.860451	−	0.509533i	$-0.170182\pi$
$673$	9.10360	+	9.10360i	0.350918	+	0.350918i	−0.509533	+	0.860451i	$0.670182\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.29832	+	9.29832i	0.357364	+	0.357364i	0.862840	−	0.505477i	$-0.168683\pi$
$677$	9.29832	+	9.29832i	0.357364	+	0.357364i	−0.505477	+	0.862840i	$0.668683\pi$
$678$		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	−0.709406	−	0.704800i	$-0.751037\pi$
$691$	−0.121054	+	0.121054i	−0.00460512	+	0.00460512i	0.704800	+	0.709406i	$0.251037\pi$
$692$		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	$0.00694170\pi$
$701$			1.15470i			0.0436125i	−0.999762	+	0.0218063i	$0.993058\pi$
$702$		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	−0.581561	+	0.813503i	$0.697558\pi$
$709$	−37.1464	+	37.1464i	−1.39506	+	1.39506i	−0.813503	+	0.581561i	$0.802442\pi$
$710$		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	−0.0700702	+	0.997542i	$0.522322\pi$
$727$	−28.7860	+	28.7860i	−1.06761	+	1.06761i	−0.997542	+	0.0700702i	$0.977678\pi$
$728$		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	$-0.822877\pi$
$733$		−	28.5996i		−	1.05635i	0.849136	−	0.528174i	$-0.177123\pi$
$734$		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	−0.474068	−	0.880488i	$-0.657215\pi$
$739$	11.0484	−	11.0484i	0.406420	−	0.406420i	0.880488	+	0.474068i	$0.157215\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11.7882i		−	0.432466i	−0.976342	−	0.216233i	$-0.930623\pi$
$743$		−	11.7882i		−	0.432466i	0.976342	−	0.216233i	$-0.0693771\pi$
$744$		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	$0.186859\pi$
$751$			30.3583i			1.10779i	−0.832587	+	0.553895i	$0.813141\pi$
$752$		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	0.963689	−	0.267027i	$-0.0860414\pi$
$757$	19.1677	+	19.1677i	0.696662	+	0.696662i	−0.267027	+	0.963689i	$0.586041\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.2165	−	12.2165i	−0.442847	−	0.442847i	0.450121	−	0.892968i	$-0.351381\pi$
$761$	−12.2165	−	12.2165i	−0.442847	−	0.442847i	−0.892968	+	0.450121i	$0.851381\pi$
$762$		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	0.139619	−	0.990205i	$-0.455412\pi$
$769$	−23.5875	−	23.5875i	−0.850586	−	0.850586i	−0.990205	+	0.139619i	$0.955412\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	31.8600i		−	1.14592i	−0.819582	−	0.572962i	$-0.805794\pi$
$773$		−	31.8600i		−	1.14592i	0.819582	−	0.572962i	$-0.194206\pi$
$774$		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	$0.168503\pi$
$787$			28.3333i			1.00997i	−0.863127	+	0.504987i	$0.831497\pi$
$788$		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	−0.221953	−	0.975057i	$-0.571243\pi$
$797$	21.2610	−	21.2610i	0.753104	−	0.753104i	0.975057	+	0.221953i	$0.0712433\pi$
$798$		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	$-0.868263\pi$
$809$		−	22.8766i		−	0.804300i	0.915574	−	0.402150i	$-0.131737\pi$
$810$		0			0
$811$	−20.7214	+	20.7214i	−0.727628	+	0.727628i	−0.970147	−	0.242519i	$-0.922026\pi$
$811$	−20.7214	+	20.7214i	−0.727628	+	0.727628i	0.242519	+	0.970147i	$0.422026\pi$
$812$		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	−0.999923	−	0.0124179i	$-0.996047\pi$
$821$	−28.2951	+	28.2951i	−0.987505	+	0.987505i	0.0124179	+	0.999923i	$0.496047\pi$
$822$		0			0
$823$	−13.8518	−	13.8518i	−0.482842	−	0.482842i	0.423196	−	0.906038i	$-0.360908\pi$
$823$	−13.8518	−	13.8518i	−0.482842	−	0.482842i	−0.906038	+	0.423196i	$0.860908\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	10.5533i		−	0.366975i	−0.983022	−	0.183488i	$-0.941261\pi$
$827$		−	10.5533i		−	0.366975i	0.983022	−	0.183488i	$-0.0587387\pi$
$828$		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	−0.663283	−	0.748369i	$-0.730837\pi$
$839$	2.46456	−	2.46456i	0.0850860	−	0.0850860i	0.748369	+	0.663283i	$0.230837\pi$
$840$		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	−0.965350	−	0.260959i	$-0.915961\pi$
$857$	−20.6207	+	20.6207i	−0.704390	+	0.704390i	0.260959	+	0.965350i	$0.415961\pi$
$858$		0			0
$859$			15.3195i			0.522696i	0.965245	+	0.261348i	$0.0841670\pi$
$859$			15.3195i			0.522696i	−0.965245	+	0.261348i	$0.915833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	19.7984i		−	0.673945i	−0.941514	−	0.336973i	$-0.890597\pi$
$863$		−	19.7984i		−	0.673945i	0.941514	−	0.336973i	$-0.109403\pi$
$864$		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	$0.231188\pi$
$881$			39.4236i			1.32821i	−0.747637	+	0.664107i	$0.768812\pi$
$882$		0			0
$883$	−23.9341	−	23.9341i	−0.805445	−	0.805445i	0.178495	−	0.983941i	$-0.442877\pi$
$883$	−23.9341	−	23.9341i	−0.805445	−	0.805445i	−0.983941	+	0.178495i	$0.942877\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.8887	−	28.8887i	−0.969987	−	0.969987i	0.0295751	−	0.999563i	$-0.490585\pi$
$887$	−28.8887	−	28.8887i	−0.969987	−	0.969987i	−0.999563	+	0.0295751i	$0.990585\pi$
$888$		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	−0.213540	−	0.976934i	$-0.568499\pi$
$907$	22.9907	−	22.9907i	0.763394	−	0.763394i	0.976934	+	0.213540i	$0.0684994\pi$
$908$		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	$0.280534\pi$
$919$			46.7810i			1.54316i	−0.636131	+	0.771581i	$0.719466\pi$
$920$		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	0.999425	−	0.0339004i	$-0.0107929\pi$
$929$	29.4287	+	29.4287i	0.965525	+	0.965525i	−0.0339004	+	0.999425i	$0.510793\pi$
$930$		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	0.0436550	−	0.999047i	$-0.486100\pi$
$937$	−29.2450	−	29.2450i	−0.955392	−	0.955392i	−0.999047	+	0.0436550i	$0.986100\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.1015	+	15.1015i	−0.492293	+	0.492293i	−0.909028	−	0.416735i	$-0.863174\pi$
$941$	−15.1015	+	15.1015i	−0.492293	+	0.492293i	0.416735	+	0.909028i	$0.363174\pi$
$942$		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	$0.212598\pi$
$947$			38.1180i			1.23867i	−0.785127	+	0.619335i	$0.787402\pi$
$948$		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	0.984421	+	0.175830i	$0.0562609\pi$
$953$	35.8177	+	35.8177i	1.16025	+	1.16025i	0.175830	+	0.984421i	$0.443739\pi$
$954$		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	0.844159	−	0.536092i	$-0.180100\pi$
$997$	9.72732	+	9.72732i	0.308067	+	0.308067i	−0.536092	+	0.844159i	$0.680100\pi$
$998$		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.577.4		8
3.2	odd	2		260.2.m.c.57.2	✓	8
5.3	odd	4		2340.2.bp.g.1513.2		8
12.11	even	2		1040.2.bg.m.577.3		8
13.8	odd	4		2340.2.bp.g.1477.2		8
15.2	even	4		1300.2.r.c.993.3		8
15.8	even	4		260.2.r.c.213.2	yes	8
15.14	odd	2		1300.2.m.c.57.3		8
39.8	even	4		260.2.r.c.177.2	yes	8
60.23	odd	4		1040.2.cd.m.993.3		8
65.8	even	4	inner	2340.2.u.g.73.4		8
156.47	odd	4		1040.2.cd.m.177.3		8
195.8	odd	4		260.2.m.c.73.2	yes	8
195.47	odd	4		1300.2.m.c.593.3		8
195.164	even	4		1300.2.r.c.957.3		8
780.203	even	4		1040.2.bg.m.593.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.m.c.57.2	✓	8	3.2	odd	2
260.2.m.c.73.2	yes	8	195.8	odd	4
260.2.r.c.177.2	yes	8	39.8	even	4
260.2.r.c.213.2	yes	8	15.8	even	4
1040.2.bg.m.577.3		8	12.11	even	2
1040.2.bg.m.593.3		8	780.203	even	4
1040.2.cd.m.177.3		8	156.47	odd	4
1040.2.cd.m.993.3		8	60.23	odd	4
1300.2.m.c.57.3		8	15.14	odd	2
1300.2.m.c.593.3		8	195.47	odd	4
1300.2.r.c.957.3		8	195.164	even	4
1300.2.r.c.993.3		8	15.2	even	4
2340.2.u.g.73.4		8	65.8	even	4	inner
2340.2.u.g.577.4		8	1.1	even	1	trivial
2340.2.bp.g.1477.2		8	13.8	odd	4
2340.2.bp.g.1513.2		8	5.3	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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