LMFDB - Embedded newform 855.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,2,Mod(1,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,2,0,8,4,0,4,12,0,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.14243 q^{2} +2.59002 q^{4} +1.00000 q^{5} +3.10482 q^{7} +1.26409 q^{8} +2.14243 q^{10} +1.10482 q^{11} +1.77353 q^{13} +6.65187 q^{14} -2.47182 q^{16} -7.75669 q^{17} +1.00000 q^{19} +2.59002 q^{20} +2.36700 q^{22} +6.65187 q^{23} +1.00000 q^{25} +3.79966 q^{26} +8.04156 q^{28} -7.75669 q^{29} +6.57664 q^{31} -7.82389 q^{32} -16.6182 q^{34} +3.10482 q^{35} -1.40652 q^{37} +2.14243 q^{38} +1.26409 q^{40} -2.81995 q^{41} +3.10482 q^{43} +2.86151 q^{44} +14.2512 q^{46} -1.46492 q^{47} +2.63990 q^{49} +2.14243 q^{50} +4.59348 q^{52} +2.59348 q^{53} +1.10482 q^{55} +3.92477 q^{56} -16.6182 q^{58} +5.38969 q^{59} +4.07523 q^{61} +14.0900 q^{62} -11.8185 q^{64} +1.77353 q^{65} -15.8151 q^{67} -20.0900 q^{68} +6.65187 q^{70} -7.14638 q^{71} +0.243310 q^{73} -3.01339 q^{74} +2.59002 q^{76} +3.43026 q^{77} -9.38969 q^{79} -2.47182 q^{80} -6.04156 q^{82} -8.86151 q^{83} -7.75669 q^{85} +6.65187 q^{86} +1.39659 q^{88} -0.813048 q^{89} +5.50648 q^{91} +17.2285 q^{92} -3.13849 q^{94} +1.00000 q^{95} +3.19689 q^{97} +5.65581 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29}+ \cdots - 38 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

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 $

$2$	2.14243	1.51493	0.757465	−	0.652876i	$-0.226438\pi$
$2$	2.14243	1.51493	0.757465	+	0.652876i	$0.226438\pi$
$3$	0	0
$3$	0	0
$4$	2.59002	1.29501
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.10482	1.17351	0.586756	−	0.809764i	$-0.300405\pi$
$7$	3.10482	1.17351	0.586756	+	0.809764i	$0.300405\pi$
$8$	1.26409	0.446923
$9$	0	0
$10$	2.14243	0.677497
$11$	1.10482	0.333115	0.166558	−	0.986032i	$-0.446735\pi$
$11$	1.10482	0.333115	0.166558	+	0.986032i	$0.446735\pi$
$12$	0	0
$13$	1.77353	0.491888	0.245944	−	0.969284i	$-0.420902\pi$
$13$	1.77353	0.491888	0.245944	+	0.969284i	$0.420902\pi$
$14$	6.65187	1.77779
$15$	0	0
$16$	−2.47182	−0.617955
$17$	−7.75669	−1.88127	−0.940637	−	0.339415i	$-0.889771\pi$
$17$	−7.75669	−1.88127	−0.940637	+	0.339415i	$0.889771\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.59002	0.579147
$21$	0	0
$22$	2.36700	0.504647
$23$	6.65187	1.38701	0.693505	−	0.720451i	$-0.256065\pi$
$23$	6.65187	1.38701	0.693505	+	0.720451i	$0.256065\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.79966	0.745175
$27$	0	0
$28$	8.04156	1.51971
$29$	−7.75669	−1.44038	−0.720191	−	0.693776i	$-0.755946\pi$
$29$	−7.75669	−1.44038	−0.720191	+	0.693776i	$0.755946\pi$
$30$	0	0
$31$	6.57664	1.18120	0.590600	−	0.806965i	$-0.298891\pi$
$31$	6.57664	1.18120	0.590600	+	0.806965i	$0.298891\pi$
$32$	−7.82389	−1.38308
$33$	0	0
$34$	−16.6182	−2.85000
$35$	3.10482	0.524810
$36$	0	0
$37$	−1.40652	−0.231231	−0.115616	−	0.993294i	$-0.536884\pi$
$37$	−1.40652	−0.231231	−0.115616	+	0.993294i	$0.536884\pi$
$38$	2.14243	0.347549
$39$	0	0
$40$	1.26409	0.199870
$41$	−2.81995	−0.440402	−0.220201	−	0.975454i	$-0.570671\pi$
$41$	−2.81995	−0.440402	−0.220201	+	0.975454i	$0.570671\pi$
$42$	0	0
$43$	3.10482	0.473480	0.236740	−	0.971573i	$-0.423921\pi$
$43$	3.10482	0.473480	0.236740	+	0.971573i	$0.423921\pi$
$44$	2.86151	0.431389
$45$	0	0
$46$	14.2512	2.10122
$47$	−1.46492	−0.213680	−0.106840	−	0.994276i	$-0.534073\pi$
$47$	−1.46492	−0.213680	−0.106840	+	0.994276i	$0.534073\pi$
$48$	0	0
$49$	2.63990	0.377129
$50$	2.14243	0.302986
$51$	0	0
$52$	4.59348	0.637001
$53$	2.59348	0.356241	0.178121	−	0.984009i	$-0.442998\pi$
$53$	2.59348	0.356241	0.178121	+	0.984009i	$0.442998\pi$
$54$	0	0
$55$	1.10482	0.148974
$56$	3.92477	0.524469
$57$	0	0
$58$	−16.6182	−2.18208
$59$	5.38969	0.701678	0.350839	−	0.936436i	$-0.385897\pi$
$59$	5.38969	0.701678	0.350839	+	0.936436i	$0.385897\pi$
$60$	0	0
$61$	4.07523	0.521780	0.260890	−	0.965369i	$-0.415984\pi$
$61$	4.07523	0.521780	0.260890	+	0.965369i	$0.415984\pi$
$62$	14.0900	1.78943
$63$	0	0
$64$	−11.8185	−1.47732
$65$	1.77353	0.219979
$66$	0	0
$67$	−15.8151	−1.93212	−0.966060	−	0.258318i	$-0.916832\pi$
$67$	−15.8151	−1.93212	−0.966060	+	0.258318i	$0.916832\pi$
$68$	−20.0900	−2.43627
$69$	0	0
$70$	6.65187	0.795051
$71$	−7.14638	−0.848119	−0.424059	−	0.905634i	$-0.639395\pi$
$71$	−7.14638	−0.848119	−0.424059	+	0.905634i	$0.639395\pi$
$72$	0	0
$73$	0.243310	0.0284773	0.0142387	−	0.999899i	$-0.495468\pi$
$73$	0.243310	0.0284773	0.0142387	+	0.999899i	$0.495468\pi$
$74$	−3.01339	−0.350299
$75$	0	0
$76$	2.59002	0.297096
$77$	3.43026	0.390915
$78$	0	0
$79$	−9.38969	−1.05642	−0.528211	−	0.849113i	$-0.677137\pi$
$79$	−9.38969	−1.05642	−0.528211	+	0.849113i	$0.677137\pi$
$80$	−2.47182	−0.276358
$81$	0	0
$82$	−6.04156	−0.667178
$83$	−8.86151	−0.972677	−0.486338	−	0.873771i	$-0.661668\pi$
$83$	−8.86151	−0.972677	−0.486338	+	0.873771i	$0.661668\pi$
$84$	0	0
$85$	−7.75669	−0.841331
$86$	6.65187	0.717290
$87$	0	0
$88$	1.39659	0.148877
$89$	−0.813048	−0.0861829	−0.0430914	−	0.999071i	$-0.513721\pi$
$89$	−0.813048	−0.0861829	−0.0430914	+	0.999071i	$0.513721\pi$
$90$	0	0
$91$	5.50648	0.577236
$92$	17.2285	1.79620
$93$	0	0
$94$	−3.13849	−0.323711
$95$	1.00000	0.102598
$96$	0	0
$97$	3.19689	0.324595	0.162297	−	0.986742i	$-0.448110\pi$
$97$	3.19689	0.324595	0.162297	+	0.986742i	$0.448110\pi$
$98$	5.65581	0.571323
$99$	0	0
$100$	2.59002	0.259002
$101$	3.01887	0.300389	0.150195	−	0.988656i	$-0.452010\pi$
$101$	3.01887	0.300389	0.150195	+	0.988656i	$0.452010\pi$
$102$	0	0
$103$	6.05839	0.596951	0.298476	−	0.954417i	$-0.403522\pi$
$103$	6.05839	0.596951	0.298476	+	0.954417i	$0.403522\pi$
$104$	2.24190	0.219836
$105$	0	0
$106$	5.55635	0.539681
$107$	−20.3848	−1.97068	−0.985338	−	0.170616i	$-0.945424\pi$
$107$	−20.3848	−1.97068	−0.985338	+	0.170616i	$0.945424\pi$
$108$	0	0
$109$	4.72710	0.452774	0.226387	−	0.974037i	$-0.427309\pi$
$109$	4.72710	0.452774	0.226387	+	0.974037i	$0.427309\pi$
$110$	2.36700	0.225685
$111$	0	0
$112$	−7.67456	−0.725177
$113$	7.98316	0.750993	0.375496	−	0.926824i	$-0.377472\pi$
$113$	7.98316	0.750993	0.375496	+	0.926824i	$0.377472\pi$
$114$	0	0
$115$	6.65187	0.620290
$116$	−20.0900	−1.86531
$117$	0	0
$118$	11.5471	1.06299
$119$	−24.0831	−2.20770
$120$	0	0
$121$	−9.77938	−0.889034
$122$	8.73091	0.790460
$123$	0	0
$124$	17.0337	1.52967
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.63503	0.411293	0.205646	−	0.978626i	$-0.434070\pi$
$127$	4.63503	0.411293	0.205646	+	0.978626i	$0.434070\pi$
$128$	−9.67265	−0.854950
$129$	0	0
$130$	3.79966	0.333252
$131$	−15.5134	−1.35541	−0.677705	−	0.735334i	$-0.737025\pi$
$131$	−15.5134	−1.35541	−0.677705	+	0.735334i	$0.737025\pi$
$132$	0	0
$133$	3.10482	0.269222
$134$	−33.8828	−2.92703
$135$	0	0
$136$	−9.80515	−0.840785
$137$	0.813048	0.0694634	0.0347317	−	0.999397i	$-0.488942\pi$
$137$	0.813048	0.0694634	0.0347317	+	0.999397i	$0.488942\pi$
$138$	0	0
$139$	−12.7687	−1.08302	−0.541512	−	0.840693i	$-0.682148\pi$
$139$	−12.7687	−1.08302	−0.541512	+	0.840693i	$0.682148\pi$
$140$	8.04156	0.679636
$141$	0	0
$142$	−15.3106	−1.28484
$143$	1.95942	0.163855
$144$	0	0
$145$	−7.75669	−0.644158
$146$	0.521277	0.0431412
$147$	0	0
$148$	−3.64293	−0.299447
$149$	13.7982	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$149$	13.7982	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$150$	0	0
$151$	5.02269	0.408740	0.204370	−	0.978894i	$-0.434485\pi$
$151$	5.02269	0.408740	0.204370	+	0.978894i	$0.434485\pi$
$152$	1.26409	0.102531
$153$	0	0
$154$	7.34911	0.592208
$155$	6.57664	0.528248
$156$	0	0
$157$	7.18695	0.573581	0.286791	−	0.957993i	$-0.407412\pi$
$157$	7.18695	0.573581	0.286791	+	0.957993i	$0.407412\pi$
$158$	−20.1168	−1.60041
$159$	0	0
$160$	−7.82389	−0.618533
$161$	20.6529	1.62767
$162$	0	0
$163$	7.25528	0.568277	0.284139	−	0.958783i	$-0.408292\pi$
$163$	7.25528	0.568277	0.284139	+	0.958783i	$0.408292\pi$
$164$	−7.30374	−0.570326
$165$	0	0
$166$	−18.9852	−1.47354
$167$	7.33129	0.567312	0.283656	−	0.958926i	$-0.408453\pi$
$167$	7.33129	0.567312	0.283656	+	0.958926i	$0.408453\pi$
$168$	0	0
$169$	−9.85461	−0.758047
$170$	−16.6182	−1.27456
$171$	0	0
$172$	8.04156	0.613163
$173$	5.77353	0.438953	0.219477	−	0.975618i	$-0.429565\pi$
$173$	5.77353	0.438953	0.219477	+	0.975618i	$0.429565\pi$
$174$	0	0
$175$	3.10482	0.234702
$176$	−2.73091	−0.205850
$177$	0	0
$178$	−1.74190	−0.130561
$179$	4.48379	0.335134	0.167567	−	0.985861i	$-0.446409\pi$
$179$	4.48379	0.335134	0.167567	+	0.985861i	$0.446409\pi$
$180$	0	0
$181$	−2.73400	−0.203217	−0.101608	−	0.994824i	$-0.532399\pi$
$181$	−2.73400	−0.203217	−0.101608	+	0.994824i	$0.532399\pi$
$182$	11.7973	0.874471
$183$	0	0
$184$	8.40856	0.619887
$185$	−1.40652	−0.103410
$186$	0	0
$187$	−8.56974	−0.626681
$188$	−3.79418	−0.276719
$189$	0	0
$190$	2.14243	0.155429
$191$	17.7230	1.28239	0.641196	−	0.767377i	$-0.278438\pi$
$191$	17.7230	1.28239	0.641196	+	0.767377i	$0.278438\pi$
$192$	0	0
$193$	−13.5371	−0.974423	−0.487212	−	0.873284i	$-0.661986\pi$
$193$	−13.5371	−0.974423	−0.487212	+	0.873284i	$0.661986\pi$
$194$	6.84912	0.491738
$195$	0	0
$196$	6.83741	0.488386
$197$	21.5134	1.53276	0.766382	−	0.642385i	$-0.222055\pi$
$197$	21.5134	1.53276	0.766382	+	0.642385i	$0.222055\pi$
$198$	0	0
$199$	7.09410	0.502888	0.251444	−	0.967872i	$-0.419095\pi$
$199$	7.09410	0.502888	0.251444	+	0.967872i	$0.419095\pi$
$200$	1.26409	0.0893846
$201$	0	0
$202$	6.46774	0.455068
$203$	−24.0831	−1.69030
$204$	0	0
$205$	−2.81995	−0.196954
$206$	12.9797	0.904339
$207$	0	0
$208$	−4.38384	−0.303964
$209$	1.10482	0.0764219
$210$	0	0
$211$	11.0604	0.761431	0.380716	−	0.924692i	$-0.375678\pi$
$211$	11.0604	0.761431	0.380716	+	0.924692i	$0.375678\pi$
$212$	6.71717	0.461337
$213$	0	0
$214$	−43.6731	−2.98543
$215$	3.10482	0.211747
$216$	0	0
$217$	20.4193	1.38615
$218$	10.1275	0.685921
$219$	0	0
$220$	2.86151	0.192923
$221$	−13.7567	−0.925375
$222$	0	0
$223$	−12.6350	−0.846104	−0.423052	−	0.906105i	$-0.639041\pi$
$223$	−12.6350	−0.846104	−0.423052	+	0.906105i	$0.639041\pi$
$224$	−24.2918	−1.62306
$225$	0	0
$226$	17.1034	1.13770
$227$	6.62813	0.439925	0.219962	−	0.975508i	$-0.429407\pi$
$227$	6.62813	0.439925	0.219962	+	0.975508i	$0.429407\pi$
$228$	0	0
$229$	−0.0752308	−0.00497139	−0.00248570	−	0.999997i	$-0.500791\pi$
$229$	−0.0752308	−0.00497139	−0.00248570	+	0.999997i	$0.500791\pi$
$230$	14.2512	0.939696
$231$	0	0
$232$	−9.80515	−0.643740
$233$	−9.51338	−0.623242	−0.311621	−	0.950206i	$-0.600872\pi$
$233$	−9.51338	−0.623242	−0.311621	+	0.950206i	$0.600872\pi$
$234$	0	0
$235$	−1.46492	−0.0955607
$236$	13.9594	0.908681
$237$	0	0
$238$	−51.5965	−3.34450
$239$	2.92984	0.189515	0.0947577	−	0.995500i	$-0.469792\pi$
$239$	2.92984	0.189515	0.0947577	+	0.995500i	$0.469792\pi$
$240$	0	0
$241$	−9.42743	−0.607274	−0.303637	−	0.952788i	$-0.598201\pi$
$241$	−9.42743	−0.607274	−0.303637	+	0.952788i	$0.598201\pi$
$242$	−20.9517	−1.34682
$243$	0	0
$244$	10.5549	0.675711
$245$	2.63990	0.168657
$246$	0	0
$247$	1.77353	0.112847
$248$	8.31346	0.527905
$249$	0	0
$250$	2.14243	0.135499
$251$	14.9298	0.942363	0.471181	−	0.882036i	$-0.343828\pi$
$251$	14.9298	0.942363	0.471181	+	0.882036i	$0.343828\pi$
$252$	0	0
$253$	7.34911	0.462035
$254$	9.93026	0.623080
$255$	0	0
$256$	2.91405	0.182128
$257$	15.1295	0.943755	0.471877	−	0.881664i	$-0.343576\pi$
$257$	15.1295	0.943755	0.471877	+	0.881664i	$0.343576\pi$
$258$	0	0
$259$	−4.36700	−0.271352
$260$	4.59348	0.284875
$261$	0	0
$262$	−33.2364	−2.05335
$263$	15.2216	0.938605	0.469302	−	0.883038i	$-0.344505\pi$
$263$	15.2216	0.938605	0.469302	+	0.883038i	$0.344505\pi$
$264$	0	0
$265$	2.59348	0.159316
$266$	6.65187	0.407852
$267$	0	0
$268$	−40.9615	−2.50212
$269$	−11.5203	−0.702404	−0.351202	−	0.936300i	$-0.614227\pi$
$269$	−11.5203	−0.702404	−0.351202	+	0.936300i	$0.614227\pi$
$270$	0	0
$271$	2.16118	0.131282	0.0656411	−	0.997843i	$-0.479091\pi$
$271$	2.16118	0.131282	0.0656411	+	0.997843i	$0.479091\pi$
$272$	19.1731	1.16254
$273$	0	0
$274$	1.74190	0.105232
$275$	1.10482	0.0666231
$276$	0	0
$277$	23.4205	1.40720	0.703602	−	0.710595i	$-0.251574\pi$
$277$	23.4205	1.40720	0.703602	+	0.710595i	$0.251574\pi$
$278$	−27.3560	−1.64070
$279$	0	0
$280$	3.92477	0.234550
$281$	−25.6824	−1.53209	−0.766043	−	0.642789i	$-0.777777\pi$
$281$	−25.6824	−1.53209	−0.766043	+	0.642789i	$0.777777\pi$
$282$	0	0
$283$	7.38587	0.439045	0.219522	−	0.975607i	$-0.429550\pi$
$283$	7.38587	0.439045	0.219522	+	0.975607i	$0.429550\pi$
$284$	−18.5093	−1.09832
$285$	0	0
$286$	4.19794	0.248229
$287$	−8.75543	−0.516817
$288$	0	0
$289$	43.1662	2.53919
$290$	−16.6182	−0.975854
$291$	0	0
$292$	0.630180	0.0368785
$293$	24.1522	1.41099	0.705494	−	0.708716i	$-0.250725\pi$
$293$	24.1522	1.41099	0.705494	+	0.708716i	$0.250725\pi$
$294$	0	0
$295$	5.38969	0.313800
$296$	−1.77797	−0.103343
$297$	0	0
$298$	29.5618	1.71247
$299$	11.7973	0.682253
$300$	0	0
$301$	9.63990	0.555635
$302$	10.7608	0.619213
$303$	0	0
$304$	−2.47182	−0.141769
$305$	4.07523	0.233347
$306$	0	0
$307$	4.38482	0.250255	0.125127	−	0.992141i	$-0.460066\pi$
$307$	4.38482	0.250255	0.125127	+	0.992141i	$0.460066\pi$
$308$	8.88447	0.506239
$309$	0	0
$310$	14.0900	0.800259
$311$	15.7123	0.890963	0.445481	−	0.895291i	$-0.353033\pi$
$311$	15.7123	0.890963	0.445481	+	0.895291i	$0.353033\pi$
$312$	0	0
$313$	24.7794	1.40061	0.700307	−	0.713842i	$-0.253047\pi$
$313$	24.7794	1.40061	0.700307	+	0.713842i	$0.253047\pi$
$314$	15.3976	0.868935
$315$	0	0
$316$	−24.3195	−1.36808
$317$	27.3798	1.53780	0.768900	−	0.639369i	$-0.220804\pi$
$317$	27.3798	1.53780	0.768900	+	0.639369i	$0.220804\pi$
$318$	0	0
$319$	−8.56974	−0.479813
$320$	−11.8185	−0.660676
$321$	0	0
$322$	44.2474	2.46581
$323$	−7.75669	−0.431594
$324$	0	0
$325$	1.77353	0.0983775
$326$	15.5440	0.860900
$327$	0	0
$328$	−3.56467	−0.196826
$329$	−4.54831	−0.250756
$330$	0	0
$331$	4.70033	0.258354	0.129177	−	0.991622i	$-0.458767\pi$
$331$	4.70033	0.258354	0.129177	+	0.991622i	$0.458767\pi$
$332$	−22.9515	−1.25963
$333$	0	0
$334$	15.7068	0.859439
$335$	−15.8151	−0.864070
$336$	0	0
$337$	−20.6360	−1.12412	−0.562058	−	0.827098i	$-0.689990\pi$
$337$	−20.6360	−1.12412	−0.562058	+	0.827098i	$0.689990\pi$
$338$	−21.1128	−1.14839
$339$	0	0
$340$	−20.0900	−1.08953
$341$	7.26600	0.393476
$342$	0	0
$343$	−13.5373	−0.730947
$344$	3.92477	0.211609
$345$	0	0
$346$	12.3694	0.664983
$347$	−30.0148	−1.61128	−0.805639	−	0.592407i	$-0.798178\pi$
$347$	−30.0148	−1.61128	−0.805639	+	0.592407i	$0.798178\pi$
$348$	0	0
$349$	−14.7340	−0.788693	−0.394347	−	0.918962i	$-0.629029\pi$
$349$	−14.7340	−0.788693	−0.394347	+	0.918962i	$0.629029\pi$
$350$	6.65187	0.355557
$351$	0	0
$352$	−8.64399	−0.460726
$353$	29.6302	1.57705	0.788527	−	0.615000i	$-0.210844\pi$
$353$	29.6302	1.57705	0.788527	+	0.615000i	$0.210844\pi$
$354$	0	0
$355$	−7.14638	−0.379290
$356$	−2.10581	−0.111608
$357$	0	0
$358$	9.60623	0.507705
$359$	−21.7577	−1.14833	−0.574163	−	0.818741i	$-0.694672\pi$
$359$	−21.7577	−1.14833	−0.574163	+	0.818741i	$0.694672\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.85742	−0.307859
$363$	0	0
$364$	14.2619	0.747527
$365$	0.243310	0.0127355
$366$	0	0
$367$	30.0347	1.56780	0.783898	−	0.620889i	$-0.213228\pi$
$367$	30.0347	1.56780	0.783898	+	0.620889i	$0.213228\pi$
$368$	−16.4422	−0.857111
$369$	0	0
$370$	−3.01339	−0.156658
$371$	8.05227	0.418053
$372$	0	0
$373$	−37.3055	−1.93161	−0.965803	−	0.259277i	$-0.916516\pi$
$373$	−37.3055	−1.93161	−0.965803	+	0.259277i	$0.916516\pi$
$374$	−18.3601	−0.949378
$375$	0	0
$376$	−1.85179	−0.0954987
$377$	−13.7567	−0.708506
$378$	0	0
$379$	−14.7794	−0.759166	−0.379583	−	0.925158i	$-0.623932\pi$
$379$	−14.7794	−0.759166	−0.379583	+	0.925158i	$0.623932\pi$
$380$	2.59002	0.132865
$381$	0	0
$382$	37.9704	1.94273
$383$	29.0020	1.48193	0.740967	−	0.671541i	$-0.234367\pi$
$383$	29.0020	1.48193	0.740967	+	0.671541i	$0.234367\pi$
$384$	0	0
$385$	3.43026	0.174822
$386$	−29.0024	−1.47618
$387$	0	0
$388$	8.28001	0.420354
$389$	−21.1987	−1.07481	−0.537407	−	0.843323i	$-0.680596\pi$
$389$	−21.1987	−1.07481	−0.537407	+	0.843323i	$0.680596\pi$
$390$	0	0
$391$	−51.5965	−2.60935
$392$	3.33707	0.168548
$393$	0	0
$394$	46.0910	2.32203
$395$	−9.38969	−0.472446
$396$	0	0
$397$	10.3856	0.521238	0.260619	−	0.965442i	$-0.416073\pi$
$397$	10.3856	0.521238	0.260619	+	0.965442i	$0.416073\pi$
$398$	15.1987	0.761840
$399$	0	0
$400$	−2.47182	−0.123591
$401$	−25.6638	−1.28159	−0.640796	−	0.767712i	$-0.721395\pi$
$401$	−25.6638	−1.28159	−0.640796	+	0.767712i	$0.721395\pi$
$402$	0	0
$403$	11.6638	0.581017
$404$	7.81896	0.389008
$405$	0	0
$406$	−51.5965	−2.56069
$407$	−1.55395	−0.0770266
$408$	0	0
$409$	5.71611	0.282644	0.141322	−	0.989964i	$-0.454865\pi$
$409$	5.71611	0.282644	0.141322	+	0.989964i	$0.454865\pi$
$410$	−6.04156	−0.298371
$411$	0	0
$412$	15.6914	0.773059
$413$	16.7340	0.823427
$414$	0	0
$415$	−8.86151	−0.434994
$416$	−13.8759	−0.680321
$417$	0	0
$418$	2.36700	0.115774
$419$	−15.4303	−0.753818	−0.376909	−	0.926250i	$-0.623013\pi$
$419$	−15.4303	−0.753818	−0.376909	+	0.926250i	$0.623013\pi$
$420$	0	0
$421$	−1.18005	−0.0575121	−0.0287561	−	0.999586i	$-0.509155\pi$
$421$	−1.18005	−0.0575121	−0.0287561	+	0.999586i	$0.509155\pi$
$422$	23.6962	1.15352
$423$	0	0
$424$	3.27839	0.159213
$425$	−7.75669	−0.376255
$426$	0	0
$427$	12.6529	0.612315
$428$	−52.7972	−2.55205
$429$	0	0
$430$	6.65187	0.320782
$431$	14.0255	0.675585	0.337792	−	0.941221i	$-0.390320\pi$
$431$	14.0255	0.675585	0.337792	+	0.941221i	$0.390320\pi$
$432$	0	0
$433$	25.5894	1.22975	0.614874	−	0.788625i	$-0.289207\pi$
$433$	25.5894	1.22975	0.614874	+	0.788625i	$0.289207\pi$
$434$	43.7470	2.09992
$435$	0	0
$436$	12.2433	0.586348
$437$	6.65187	0.318202
$438$	0	0
$439$	−21.9732	−1.04873	−0.524363	−	0.851495i	$-0.675696\pi$
$439$	−21.9732	−1.04873	−0.524363	+	0.851495i	$0.675696\pi$
$440$	1.39659	0.0665798
$441$	0	0
$442$	−29.4728	−1.40188
$443$	19.3721	0.920395	0.460197	−	0.887817i	$-0.347779\pi$
$443$	19.3721	0.920395	0.460197	+	0.887817i	$0.347779\pi$
$444$	0	0
$445$	−0.813048	−0.0385422
$446$	−27.0697	−1.28179
$447$	0	0
$448$	−36.6944	−1.73365
$449$	19.1841	0.905355	0.452677	−	0.891674i	$-0.350469\pi$
$449$	19.1841	0.905355	0.452677	+	0.891674i	$0.350469\pi$
$450$	0	0
$451$	−3.11553	−0.146705
$452$	20.6766	0.972545
$453$	0	0
$454$	14.2003	0.666455
$455$	5.50648	0.258148
$456$	0	0
$457$	8.36010	0.391069	0.195534	−	0.980697i	$-0.437356\pi$
$457$	8.36010	0.391069	0.195534	+	0.980697i	$0.437356\pi$
$458$	−0.161177	−0.00753131
$459$	0	0
$460$	17.2285	0.803283
$461$	25.0268	1.16561	0.582806	−	0.812611i	$-0.301955\pi$
$461$	25.0268	1.16561	0.582806	+	0.812611i	$0.301955\pi$
$462$	0	0
$463$	32.3749	1.50459	0.752294	−	0.658827i	$-0.228947\pi$
$463$	32.3749	1.50459	0.752294	+	0.658827i	$0.228947\pi$
$464$	19.1731	0.890091
$465$	0	0
$466$	−20.3818	−0.944168
$467$	−15.2216	−0.704372	−0.352186	−	0.935930i	$-0.614562\pi$
$467$	−15.2216	−0.704372	−0.352186	+	0.935930i	$0.614562\pi$
$468$	0	0
$469$	−49.1030	−2.26736
$470$	−3.13849	−0.144768
$471$	0	0
$472$	6.81305	0.313596
$473$	3.43026	0.157724
$474$	0	0
$475$	1.00000	0.0458831
$476$	−62.3759	−2.85899
$477$	0	0
$478$	6.27698	0.287103
$479$	20.0485	0.916038	0.458019	−	0.888943i	$-0.348559\pi$
$479$	20.0485	0.916038	0.458019	+	0.888943i	$0.348559\pi$
$480$	0	0
$481$	−2.49451	−0.113740
$482$	−20.1977	−0.919978
$483$	0	0
$484$	−25.3288	−1.15131
$485$	3.19689	0.145163
$486$	0	0
$487$	−31.6508	−1.43424	−0.717118	−	0.696952i	$-0.754539\pi$
$487$	−31.6508	−1.43424	−0.717118	+	0.696952i	$0.754539\pi$
$488$	5.15146	0.233195
$489$	0	0
$490$	5.65581	0.255504
$491$	−24.8033	−1.11936	−0.559679	−	0.828710i	$-0.689076\pi$
$491$	−24.8033	−1.11936	−0.559679	+	0.828710i	$0.689076\pi$
$492$	0	0
$493$	60.1662	2.70975
$494$	3.79966	0.170955
$495$	0	0
$496$	−16.2563	−0.729928
$497$	−22.1882	−0.995277
$498$	0	0
$499$	−33.0237	−1.47834	−0.739171	−	0.673518i	$-0.764783\pi$
$499$	−33.0237	−1.47834	−0.739171	+	0.673518i	$0.764783\pi$
$500$	2.59002	0.115829
$501$	0	0
$502$	31.9862	1.42761
$503$	3.54396	0.158017	0.0790087	−	0.996874i	$-0.474825\pi$
$503$	3.54396	0.158017	0.0790087	+	0.996874i	$0.474825\pi$
$504$	0	0
$505$	3.01887	0.134338
$506$	15.7450	0.699950
$507$	0	0
$508$	12.0049	0.532629
$509$	2.11679	0.0938250	0.0469125	−	0.998899i	$-0.485062\pi$
$509$	2.11679	0.0938250	0.0469125	+	0.998899i	$0.485062\pi$
$510$	0	0
$511$	0.755435	0.0334185
$512$	25.5885	1.13086
$513$	0	0
$514$	32.4140	1.42972
$515$	6.05839	0.266965
$516$	0	0
$517$	−1.61847	−0.0711802
$518$	−9.35601	−0.411080
$519$	0	0
$520$	2.24190	0.0983136
$521$	8.60748	0.377101	0.188550	−	0.982064i	$-0.439621\pi$
$521$	8.60748	0.377101	0.188550	+	0.982064i	$0.439621\pi$
$522$	0	0
$523$	36.8349	1.61068	0.805340	−	0.592814i	$-0.201983\pi$
$523$	36.8349	1.61068	0.805340	+	0.592814i	$0.201983\pi$
$524$	−40.1800	−1.75527
$525$	0	0
$526$	32.6113	1.42192
$527$	−51.0130	−2.22216
$528$	0	0
$529$	21.2474	0.923799
$530$	5.55635	0.241353
$531$	0	0
$532$	8.04156	0.348646
$533$	−5.00125	−0.216628
$534$	0	0
$535$	−20.3848	−0.881313
$536$	−19.9917	−0.863509
$537$	0	0
$538$	−24.6814	−1.06409
$539$	2.91661	0.125627
$540$	0	0
$541$	−32.0079	−1.37613	−0.688063	−	0.725651i	$-0.741539\pi$
$541$	−32.0079	−1.37613	−0.688063	+	0.725651i	$0.741539\pi$
$542$	4.63018	0.198883
$543$	0	0
$544$	60.6875	2.60196
$545$	4.72710	0.202487
$546$	0	0
$547$	17.6240	0.753550	0.376775	−	0.926305i	$-0.377033\pi$
$547$	17.6240	0.753550	0.376775	+	0.926305i	$0.377033\pi$
$548$	2.10581	0.0899559
$549$	0	0
$550$	2.36700	0.100929
$551$	−7.75669	−0.330446
$552$	0	0
$553$	−29.1533	−1.23972
$554$	50.1769	2.13181
$555$	0	0
$556$	−33.0711	−1.40253
$557$	34.6865	1.46972	0.734858	−	0.678221i	$-0.237249\pi$
$557$	34.6865	1.46972	0.734858	+	0.678221i	$0.237249\pi$
$558$	0	0
$559$	5.50648	0.232899
$560$	−7.67456	−0.324309
$561$	0	0
$562$	−55.0229	−2.32100
$563$	−13.5073	−0.569263	−0.284632	−	0.958637i	$-0.591871\pi$
$563$	−13.5073	−0.569263	−0.284632	+	0.958637i	$0.591871\pi$
$564$	0	0
$565$	7.98316	0.335854
$566$	15.8238	0.665122
$567$	0	0
$568$	−9.03366	−0.379044
$569$	−32.3588	−1.35655	−0.678276	−	0.734807i	$-0.737273\pi$
$569$	−32.3588	−1.35655	−0.678276	+	0.734807i	$0.737273\pi$
$570$	0	0
$571$	−8.93293	−0.373831	−0.186916	−	0.982376i	$-0.559849\pi$
$571$	−8.93293	−0.373831	−0.186916	+	0.982376i	$0.559849\pi$
$572$	5.07496	0.212195
$573$	0	0
$574$	−18.7579	−0.782941
$575$	6.65187	0.277402
$576$	0	0
$577$	14.9059	0.620541	0.310270	−	0.950648i	$-0.399580\pi$
$577$	14.9059	0.620541	0.310270	+	0.950648i	$0.399580\pi$
$578$	92.4808	3.84669
$579$	0	0
$580$	−20.0900	−0.834193
$581$	−27.5134	−1.14145
$582$	0	0
$583$	2.86532	0.118669
$584$	0.307566	0.0127272
$585$	0	0
$586$	51.7446	2.13755
$587$	−7.37207	−0.304278	−0.152139	−	0.988359i	$-0.548616\pi$
$587$	−7.37207	−0.304278	−0.152139	+	0.988359i	$0.548616\pi$
$588$	0	0
$589$	6.57664	0.270986
$590$	11.5471	0.475385
$591$	0	0
$592$	3.47668	0.142890
$593$	17.6853	0.726247	0.363124	−	0.931741i	$-0.381710\pi$
$593$	17.6853	0.726247	0.363124	+	0.931741i	$0.381710\pi$
$594$	0	0
$595$	−24.0831	−0.987312
$596$	35.7378	1.46388
$597$	0	0
$598$	25.2749	1.03357
$599$	−5.30657	−0.216821	−0.108410	−	0.994106i	$-0.534576\pi$
$599$	−5.30657	−0.216821	−0.108410	+	0.994106i	$0.534576\pi$
$600$	0	0
$601$	43.7875	1.78613	0.893065	−	0.449927i	$-0.148550\pi$
$601$	43.7875	1.78613	0.893065	+	0.449927i	$0.148550\pi$
$602$	20.6529	0.841747
$603$	0	0
$604$	13.0089	0.529324
$605$	−9.77938	−0.397588
$606$	0	0
$607$	−7.24535	−0.294080	−0.147040	−	0.989131i	$-0.546975\pi$
$607$	−7.24535	−0.294080	−0.147040	+	0.989131i	$0.546975\pi$
$608$	−7.82389	−0.317301
$609$	0	0
$610$	8.73091	0.353504
$611$	−2.59807	−0.105107
$612$	0	0
$613$	20.8962	0.843988	0.421994	−	0.906599i	$-0.361330\pi$
$613$	20.8962	0.843988	0.421994	+	0.906599i	$0.361330\pi$
$614$	9.39419	0.379119
$615$	0	0
$616$	4.33616	0.174709
$617$	22.0494	0.887677	0.443839	−	0.896107i	$-0.353616\pi$
$617$	22.0494	0.887677	0.443839	+	0.896107i	$0.353616\pi$
$618$	0	0
$619$	−0.0484607	−0.00194780	−0.000973900	−	1.00000i	$-0.500310\pi$
$619$	−0.0484607	−0.00194780	−0.000973900		1.00000i	$0.500310\pi$
$620$	17.0337	0.684088
$621$	0	0
$622$	33.6626	1.34975
$623$	−2.52437	−0.101137
$624$	0	0
$625$	1.00000	0.0400000
$626$	53.0882	2.12183
$627$	0	0
$628$	18.6144	0.742795
$629$	10.9100	0.435009
$630$	0	0
$631$	−32.6182	−1.29851	−0.649255	−	0.760571i	$-0.724919\pi$
$631$	−32.6182	−1.29851	−0.649255	+	0.760571i	$0.724919\pi$
$632$	−11.8694	−0.472140
$633$	0	0
$634$	58.6593	2.32966
$635$	4.63503	0.183936
$636$	0	0
$637$	4.68193	0.185505
$638$	−18.3601	−0.726883
$639$	0	0
$640$	−9.67265	−0.382345
$641$	13.4136	0.529806	0.264903	−	0.964275i	$-0.414660\pi$
$641$	13.4136	0.529806	0.264903	+	0.964275i	$0.414660\pi$
$642$	0	0
$643$	23.8052	0.938783	0.469392	−	0.882990i	$-0.344473\pi$
$643$	23.8052	0.938783	0.469392	+	0.882990i	$0.344473\pi$
$644$	53.4914	2.10786
$645$	0	0
$646$	−16.6182	−0.653834
$647$	48.4580	1.90508	0.952540	−	0.304412i	$-0.0984601\pi$
$647$	48.4580	1.90508	0.952540	+	0.304412i	$0.0984601\pi$
$648$	0	0
$649$	5.95463	0.233740
$650$	3.79966	0.149035
$651$	0	0
$652$	18.7914	0.735926
$653$	−22.2351	−0.870128	−0.435064	−	0.900399i	$-0.643274\pi$
$653$	−22.2351	−0.870128	−0.435064	+	0.900399i	$0.643274\pi$
$654$	0	0
$655$	−15.5134	−0.606158
$656$	6.97041	0.272149
$657$	0	0
$658$	−9.74445	−0.379878
$659$	10.7072	0.417095	0.208547	−	0.978012i	$-0.433126\pi$
$659$	10.7072	0.417095	0.208547	+	0.978012i	$0.433126\pi$
$660$	0	0
$661$	−44.7980	−1.74244	−0.871220	−	0.490893i	$-0.836670\pi$
$661$	−44.7980	−1.74244	−0.871220	+	0.490893i	$0.836670\pi$
$662$	10.0702	0.391388
$663$	0	0
$664$	−11.2017	−0.434712
$665$	3.10482	0.120400
$666$	0	0
$667$	−51.5965	−1.99782
$668$	18.9882	0.734677
$669$	0	0
$670$	−33.8828	−1.30901
$671$	4.50239	0.173813
$672$	0	0
$673$	−43.2772	−1.66821	−0.834106	−	0.551604i	$-0.814016\pi$
$673$	−43.2772	−1.66821	−0.834106	+	0.551604i	$0.814016\pi$
$674$	−44.2113	−1.70296
$675$	0	0
$676$	−25.5237	−0.981680
$677$	−11.5330	−0.443251	−0.221625	−	0.975132i	$-0.571136\pi$
$677$	−11.5330	−0.443251	−0.221625	+	0.975132i	$0.571136\pi$
$678$	0	0
$679$	9.92575	0.380915
$680$	−9.80515	−0.376010
$681$	0	0
$682$	15.5669	0.596088
$683$	−0.916090	−0.0350532	−0.0175266	−	0.999846i	$-0.505579\pi$
$683$	−0.916090	−0.0350532	−0.0175266	+	0.999846i	$0.505579\pi$
$684$	0	0
$685$	0.813048	0.0310650
$686$	−29.0028	−1.10733
$687$	0	0
$688$	−7.67456	−0.292590
$689$	4.59960	0.175231
$690$	0	0
$691$	14.8278	0.564077	0.282039	−	0.959403i	$-0.408989\pi$
$691$	14.8278	0.564077	0.282039	+	0.959403i	$0.408989\pi$
$692$	14.9536	0.568450
$693$	0	0
$694$	−64.3047	−2.44097
$695$	−12.7687	−0.484343
$696$	0	0
$697$	21.8735	0.828517
$698$	−31.5666	−1.19481
$699$	0	0
$700$	8.04156	0.303942
$701$	−36.1722	−1.36620	−0.683102	−	0.730323i	$-0.739369\pi$
$701$	−36.1722	−1.36620	−0.683102	+	0.730323i	$0.739369\pi$
$702$	0	0
$703$	−1.40652	−0.0530481
$704$	−13.0573	−0.492117
$705$	0	0
$706$	63.4807	2.38913
$707$	9.37305	0.352510
$708$	0	0
$709$	10.4331	0.391823	0.195911	−	0.980622i	$-0.437234\pi$
$709$	10.4331	0.391823	0.195911	+	0.980622i	$0.437234\pi$
$710$	−15.3106	−0.574598
$711$	0	0
$712$	−1.02777	−0.0385171
$713$	43.7470	1.63834
$714$	0	0
$715$	1.95942	0.0732783
$716$	11.6131	0.434003
$717$	0	0
$718$	−46.6144	−1.73963
$719$	−7.73373	−0.288420	−0.144210	−	0.989547i	$-0.546064\pi$
$719$	−7.73373	−0.288420	−0.144210	+	0.989547i	$0.546064\pi$
$720$	0	0
$721$	18.8102	0.700529
$722$	2.14243	0.0797331
$723$	0	0
$724$	−7.08114	−0.263168
$725$	−7.75669	−0.288076
$726$	0	0
$727$	−39.5241	−1.46587	−0.732934	−	0.680300i	$-0.761849\pi$
$727$	−39.5241	−1.46587	−0.732934	+	0.680300i	$0.761849\pi$
$728$	6.96068	0.257980
$729$	0	0
$730$	0.521277	0.0192933
$731$	−24.0831	−0.890746
$732$	0	0
$733$	50.7498	1.87449	0.937243	−	0.348677i	$-0.113369\pi$
$733$	50.7498	1.87449	0.937243	+	0.348677i	$0.113369\pi$
$734$	64.3473	2.37510
$735$	0	0
$736$	−52.0435	−1.91835
$737$	−17.4728	−0.643619
$738$	0	0
$739$	0.419276	0.0154233	0.00771165	−	0.999970i	$-0.497545\pi$
$739$	0.419276	0.0154233	0.00771165	+	0.999970i	$0.497545\pi$
$740$	−3.64293	−0.133917
$741$	0	0
$742$	17.2515	0.633321
$743$	−19.5511	−0.717259	−0.358630	−	0.933480i	$-0.616756\pi$
$743$	−19.5511	−0.717259	−0.358630	+	0.933480i	$0.616756\pi$
$744$	0	0
$745$	13.7982	0.505529
$746$	−79.9246	−2.92625
$747$	0	0
$748$	−22.1958	−0.811560
$749$	−63.2912	−2.31261
$750$	0	0
$751$	8.76768	0.319937	0.159969	−	0.987122i	$-0.448861\pi$
$751$	8.76768	0.319937	0.159969	+	0.987122i	$0.448861\pi$
$752$	3.62102	0.132045
$753$	0	0
$754$	−29.4728	−1.07334
$755$	5.02269	0.182794
$756$	0	0
$757$	−48.0535	−1.74653	−0.873267	−	0.487241i	$-0.838003\pi$
$757$	−48.0535	−1.74653	−0.873267	+	0.487241i	$0.838003\pi$
$758$	−31.6638	−1.15008
$759$	0	0
$760$	1.26409	0.0458533
$761$	−29.3947	−1.06556	−0.532779	−	0.846254i	$-0.678852\pi$
$761$	−29.3947	−1.06556	−0.532779	+	0.846254i	$0.678852\pi$
$762$	0	0
$763$	14.6768	0.531336
$764$	45.9031	1.66071
$765$	0	0
$766$	62.1350	2.24503
$767$	9.55875	0.345146
$768$	0	0
$769$	−13.3790	−0.482458	−0.241229	−	0.970468i	$-0.577551\pi$
$769$	−13.3790	−0.482458	−0.241229	+	0.970468i	$0.577551\pi$
$770$	7.34911	0.264844
$771$	0	0
$772$	−35.0615	−1.26189
$773$	−0.133625	−0.00480617	−0.00240309	−	0.999997i	$-0.500765\pi$
$773$	−0.133625	−0.00480617	−0.00240309	+	0.999997i	$0.500765\pi$
$774$	0	0
$775$	6.57664	0.236240
$776$	4.04115	0.145069
$777$	0	0
$778$	−45.4167	−1.62827
$779$	−2.81995	−0.101035
$780$	0	0
$781$	−7.89545	−0.282522
$782$	−110.542	−3.95298
$783$	0	0
$784$	−6.52536	−0.233049
$785$	7.18695	0.256513
$786$	0	0
$787$	−4.84060	−0.172549	−0.0862744	−	0.996271i	$-0.527496\pi$
$787$	−4.84060	−0.172549	−0.0862744	+	0.996271i	$0.527496\pi$
$788$	55.7202	1.98495
$789$	0	0
$790$	−20.1168	−0.715723
$791$	24.7863	0.881299
$792$	0	0
$793$	7.22753	0.256657
$794$	22.2505	0.789640
$795$	0	0
$796$	18.3739	0.651246
$797$	−11.6993	−0.414410	−0.207205	−	0.978298i	$-0.566437\pi$
$797$	−11.6993	−0.414410	−0.207205	+	0.978298i	$0.566437\pi$
$798$	0	0
$799$	11.3629	0.401991
$800$	−7.82389	−0.276616
$801$	0	0
$802$	−54.9831	−1.94152
$803$	0.268814	0.00948624
$804$	0	0
$805$	20.6529	0.727917
$806$	24.9890	0.880200
$807$	0	0
$808$	3.81613	0.134251
$809$	33.1987	1.16720	0.583601	−	0.812040i	$-0.301643\pi$
$809$	33.1987	1.16720	0.583601	+	0.812040i	$0.301643\pi$
$810$	0	0
$811$	−26.5766	−0.933232	−0.466616	−	0.884460i	$-0.654527\pi$
$811$	−26.5766	−0.933232	−0.466616	+	0.884460i	$0.654527\pi$
$812$	−62.3759	−2.18896
$813$	0	0
$814$	−3.32924	−0.116690
$815$	7.25528	0.254141
$816$	0	0
$817$	3.10482	0.108624
$818$	12.2464	0.428185
$819$	0	0
$820$	−7.30374	−0.255058
$821$	−25.5807	−0.892773	−0.446387	−	0.894840i	$-0.647289\pi$
$821$	−25.5807	−0.892773	−0.446387	+	0.894840i	$0.647289\pi$
$822$	0	0
$823$	12.9783	0.452395	0.226198	−	0.974081i	$-0.427371\pi$
$823$	12.9783	0.452395	0.226198	+	0.974081i	$0.427371\pi$
$824$	7.65835	0.266791
$825$	0	0
$826$	35.8515	1.24743
$827$	12.5167	0.435248	0.217624	−	0.976033i	$-0.430169\pi$
$827$	12.5167	0.435248	0.217624	+	0.976033i	$0.430169\pi$
$828$	0	0
$829$	−19.4391	−0.675149	−0.337574	−	0.941299i	$-0.609606\pi$
$829$	−19.4391	−0.675149	−0.337574	+	0.941299i	$0.609606\pi$
$830$	−18.9852	−0.658986
$831$	0	0
$832$	−20.9605	−0.726674
$833$	−20.4769	−0.709482
$834$	0	0
$835$	7.33129	0.253710
$836$	2.86151	0.0989673
$837$	0	0
$838$	−33.0583	−1.14198
$839$	43.9972	1.51895	0.759475	−	0.650536i	$-0.225456\pi$
$839$	43.9972	1.51895	0.759475	+	0.650536i	$0.225456\pi$
$840$	0	0
$841$	31.1662	1.07470
$842$	−2.52818	−0.0871268
$843$	0	0
$844$	28.6468	0.986063
$845$	−9.85461	−0.339009
$846$	0	0
$847$	−30.3632	−1.04329
$848$	−6.41061	−0.220141
$849$	0	0
$850$	−16.6182	−0.569999
$851$	−9.35601	−0.320720
$852$	0	0
$853$	3.30374	0.113118	0.0565590	−	0.998399i	$-0.481987\pi$
$853$	3.30374	0.113118	0.0565590	+	0.998399i	$0.481987\pi$
$854$	27.1079	0.927614
$855$	0	0
$856$	−25.7682	−0.880740
$857$	−6.02374	−0.205767	−0.102883	−	0.994693i	$-0.532807\pi$
$857$	−6.02374	−0.205767	−0.102883	+	0.994693i	$0.532807\pi$
$858$	0	0
$859$	−36.0158	−1.22884	−0.614421	−	0.788978i	$-0.710610\pi$
$859$	−36.0158	−1.22884	−0.614421	+	0.788978i	$0.710610\pi$
$860$	8.04156	0.274215
$861$	0	0
$862$	30.0487	1.02346
$863$	−29.9250	−1.01866	−0.509329	−	0.860572i	$-0.670106\pi$
$863$	−29.9250	−1.01866	−0.509329	+	0.860572i	$0.670106\pi$
$864$	0	0
$865$	5.77353	0.196306
$866$	54.8236	1.86298
$867$	0	0
$868$	52.8864	1.79508
$869$	−10.3739	−0.351911
$870$	0	0
$871$	−28.0485	−0.950386
$872$	5.97548	0.202355
$873$	0	0
$874$	14.2512	0.482054
$875$	3.10482	0.104962
$876$	0	0
$877$	0.618980	0.0209015	0.0104507	−	0.999945i	$-0.496673\pi$
$877$	0.618980	0.0209015	0.0104507	+	0.999945i	$0.496673\pi$
$878$	−47.0762	−1.58874
$879$	0	0
$880$	−2.73091	−0.0920591
$881$	21.2423	0.715672	0.357836	−	0.933784i	$-0.383515\pi$
$881$	21.2423	0.715672	0.357836	+	0.933784i	$0.383515\pi$
$882$	0	0
$883$	−48.8971	−1.64552	−0.822760	−	0.568389i	$-0.807567\pi$
$883$	−48.8971	−1.64552	−0.822760	+	0.568389i	$0.807567\pi$
$884$	−35.6302	−1.19837
$885$	0	0
$886$	41.5034	1.39433
$887$	−58.2797	−1.95684	−0.978421	−	0.206622i	$-0.933753\pi$
$887$	−58.2797	−1.95684	−0.978421	+	0.206622i	$0.933753\pi$
$888$	0	0
$889$	14.3909	0.482657
$890$	−1.74190	−0.0583887
$891$	0	0
$892$	−32.7251	−1.09572
$893$	−1.46492	−0.0490216
$894$	0	0
$895$	4.48379	0.149877
$896$	−30.0318	−1.00329
$897$	0	0
$898$	41.1007	1.37155
$899$	−51.0130	−1.70138
$900$	0	0
$901$	−20.1168	−0.670187
$902$	−6.67483	−0.222247
$903$	0	0
$904$	10.0914	0.335636
$905$	−2.73400	−0.0908814
$906$	0	0
$907$	−36.5154	−1.21247	−0.606237	−	0.795284i	$-0.707322\pi$
$907$	−36.5154	−1.21247	−0.606237	+	0.795284i	$0.707322\pi$
$908$	17.1670	0.569708
$909$	0	0
$910$	11.7973	0.391076
$911$	−6.62735	−0.219574	−0.109787	−	0.993955i	$-0.535017\pi$
$911$	−6.62735	−0.219574	−0.109787	+	0.993955i	$0.535017\pi$
$912$	0	0
$913$	−9.79036	−0.324014
$914$	17.9110	0.592442
$915$	0	0
$916$	−0.194850	−0.00643802
$917$	−48.1662	−1.59059
$918$	0	0
$919$	7.91688	0.261154	0.130577	−	0.991438i	$-0.458317\pi$
$919$	7.91688	0.261154	0.130577	+	0.991438i	$0.458317\pi$
$920$	8.40856	0.277222
$921$	0	0
$922$	53.6182	1.76582
$923$	−12.6743	−0.417179
$924$	0	0
$925$	−1.40652	−0.0462462
$926$	69.3611	2.27935
$927$	0	0
$928$	60.6875	1.99217
$929$	−35.8597	−1.17652	−0.588259	−	0.808673i	$-0.700186\pi$
$929$	−35.8597	−1.17652	−0.588259	+	0.808673i	$0.700186\pi$
$930$	0	0
$931$	2.63990	0.0865192
$932$	−24.6399	−0.807106
$933$	0	0
$934$	−32.6113	−1.06707
$935$	−8.56974	−0.280260
$936$	0	0
$937$	−11.4420	−0.373793	−0.186896	−	0.982380i	$-0.559843\pi$
$937$	−11.4420	−0.373793	−0.186896	+	0.982380i	$0.559843\pi$
$938$	−105.200	−3.43490
$939$	0	0
$940$	−3.79418	−0.123752
$941$	0.360100	0.0117389	0.00586945	−	0.999983i	$-0.498132\pi$
$941$	0.360100	0.0117389	0.00586945	+	0.999983i	$0.498132\pi$
$942$	0	0
$943$	−18.7579	−0.610843
$944$	−13.3223	−0.433605
$945$	0	0
$946$	7.34911	0.238940
$947$	−2.63807	−0.0857256	−0.0428628	−	0.999081i	$-0.513648\pi$
$947$	−2.63807	−0.0857256	−0.0428628	+	0.999081i	$0.513648\pi$
$948$	0	0
$949$	0.431517	0.0140076
$950$	2.14243	0.0695097
$951$	0	0
$952$	−30.4432	−0.986670
$953$	−10.4430	−0.338282	−0.169141	−	0.985592i	$-0.554099\pi$
$953$	−10.4430	−0.338282	−0.169141	+	0.985592i	$0.554099\pi$
$954$	0	0
$955$	17.7230	0.573503
$956$	7.58835	0.245425
$957$	0	0
$958$	42.9525	1.38773
$959$	2.52437	0.0815160
$960$	0	0
$961$	12.2522	0.395232
$962$	−5.34432	−0.172308
$963$	0	0
$964$	−24.4173	−0.786428
$965$	−13.5371	−0.435775
$966$	0	0
$967$	33.2732	1.06999	0.534996	−	0.844854i	$-0.320313\pi$
$967$	33.2732	1.06999	0.534996	+	0.844854i	$0.320313\pi$
$968$	−12.3620	−0.397330
$969$	0	0
$970$	6.84912	0.219912
$971$	−23.5657	−0.756258	−0.378129	−	0.925753i	$-0.623432\pi$
$971$	−23.5657	−0.756258	−0.378129	+	0.925753i	$0.623432\pi$
$972$	0	0
$973$	−39.6444	−1.27094
$974$	−67.8098	−2.17277
$975$	0	0
$976$	−10.0732	−0.322437
$977$	−0.402439	−0.0128752	−0.00643759	−	0.999979i	$-0.502049\pi$
$977$	−0.402439	−0.0128752	−0.00643759	+	0.999979i	$0.502049\pi$
$978$	0	0
$979$	−0.898271	−0.0287089
$980$	6.83741	0.218413
$981$	0	0
$982$	−53.1395	−1.69575
$983$	−11.4927	−0.366561	−0.183281	−	0.983061i	$-0.558672\pi$
$983$	−11.4927	−0.366561	−0.183281	+	0.983061i	$0.558672\pi$
$984$	0	0
$985$	21.5134	0.685473
$986$	128.902	4.10508
$987$	0	0
$988$	4.59348	0.146138
$989$	20.6529	0.656723
$990$	0	0
$991$	6.05761	0.192426	0.0962132	−	0.995361i	$-0.469327\pi$
$991$	6.05761	0.192426	0.0962132	+	0.995361i	$0.469327\pi$
$992$	−51.4549	−1.63370
$993$	0	0
$994$	−47.5368	−1.50777
$995$	7.09410	0.224898
$996$	0	0
$997$	−48.1086	−1.52362	−0.761808	−	0.647803i	$-0.775688\pi$
$997$	−48.1086	−1.52362	−0.761808	+	0.647803i	$0.775688\pi$
$998$	−70.7510	−2.23959
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.m.1.3		4
3.2	odd	2		95.2.a.b.1.2	✓	4
5.4	even	2		4275.2.a.bo.1.2		4
12.11	even	2		1520.2.a.t.1.4		4
15.2	even	4		475.2.b.e.324.2		8
15.8	even	4		475.2.b.e.324.7		8
15.14	odd	2		475.2.a.i.1.3		4
21.20	even	2		4655.2.a.y.1.2		4
24.5	odd	2		6080.2.a.cc.1.4		4
24.11	even	2		6080.2.a.ch.1.1		4
57.56	even	2		1805.2.a.p.1.3		4
60.59	even	2		7600.2.a.cf.1.1		4
285.284	even	2		9025.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.2	✓	4	3.2	odd	2
475.2.a.i.1.3		4	15.14	odd	2
475.2.b.e.324.2		8	15.2	even	4
475.2.b.e.324.7		8	15.8	even	4
855.2.a.m.1.3		4	1.1	even	1	trivial
1520.2.a.t.1.4		4	12.11	even	2
1805.2.a.p.1.3		4	57.56	even	2
4275.2.a.bo.1.2		4	5.4	even	2
4655.2.a.y.1.2		4	21.20	even	2
6080.2.a.cc.1.4		4	24.5	odd	2
6080.2.a.ch.1.1		4	24.11	even	2
7600.2.a.cf.1.1		4	60.59	even	2
9025.2.a.bf.1.2		4	285.284	even	2

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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14243 q^{2} +2.59002 q^{4} +1.00000 q^{5} +3.10482 q^{7} +1.26409 q^{8} +2.14243 q^{10} +1.10482 q^{11} +1.77353 q^{13} +6.65187 q^{14} -2.47182 q^{16} -7.75669 q^{17} +1.00000 q^{19} +2.59002 q^{20} +2.36700 q^{22} +6.65187 q^{23} +1.00000 q^{25} +3.79966 q^{26} +8.04156 q^{28} -7.75669 q^{29} +6.57664 q^{31} -7.82389 q^{32} -16.6182 q^{34} +3.10482 q^{35} -1.40652 q^{37} +2.14243 q^{38} +1.26409 q^{40} -2.81995 q^{41} +3.10482 q^{43} +2.86151 q^{44} +14.2512 q^{46} -1.46492 q^{47} +2.63990 q^{49} +2.14243 q^{50} +4.59348 q^{52} +2.59348 q^{53} +1.10482 q^{55} +3.92477 q^{56} -16.6182 q^{58} +5.38969 q^{59} +4.07523 q^{61} +14.0900 q^{62} -11.8185 q^{64} +1.77353 q^{65} -15.8151 q^{67} -20.0900 q^{68} +6.65187 q^{70} -7.14638 q^{71} +0.243310 q^{73} -3.01339 q^{74} +2.59002 q^{76} +3.43026 q^{77} -9.38969 q^{79} -2.47182 q^{80} -6.04156 q^{82} -8.86151 q^{83} -7.75669 q^{85} +6.65187 q^{86} +1.39659 q^{88} -0.813048 q^{89} +5.50648 q^{91} +17.2285 q^{92} -3.13849 q^{94} +1.00000 q^{95} +3.19689 q^{97} +5.65581 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29}+ \cdots - 38 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

Introduction

L-functions

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