LMFDB - Embedded newform 5225.2.a.y.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,5,4,17,0,-1,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 $ x^15 - 5x^14 - 11x^13 + 87x^12 - 4x^11 - 545x^10 + 431x^9 + 1480x^8 - 1763x^7 - 1609x^6 + 2516x^5 + 391x^4 - 1081x^3 - 45x^2 + 153x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.28989$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.28989 q^{2} +3.30063 q^{3} -0.336174 q^{4} +4.25747 q^{6} +0.563541 q^{7} -3.01342 q^{8} +7.89418 q^{9} +1.00000 q^{11} -1.10959 q^{12} -1.69967 q^{13} +0.726907 q^{14} -3.21464 q^{16} -0.0949420 q^{17} +10.1826 q^{18} +1.00000 q^{19} +1.86004 q^{21} +1.28989 q^{22} +0.625186 q^{23} -9.94618 q^{24} -2.19239 q^{26} +16.1539 q^{27} -0.189448 q^{28} +6.14313 q^{29} +3.11276 q^{31} +1.88029 q^{32} +3.30063 q^{33} -0.122465 q^{34} -2.65382 q^{36} +2.91877 q^{37} +1.28989 q^{38} -5.60998 q^{39} +0.562373 q^{41} +2.39925 q^{42} +10.9758 q^{43} -0.336174 q^{44} +0.806423 q^{46} +8.90408 q^{47} -10.6103 q^{48} -6.68242 q^{49} -0.313369 q^{51} +0.571385 q^{52} -0.489328 q^{53} +20.8368 q^{54} -1.69818 q^{56} +3.30063 q^{57} +7.92399 q^{58} -11.5958 q^{59} +13.8576 q^{61} +4.01513 q^{62} +4.44869 q^{63} +8.85465 q^{64} +4.25747 q^{66} -2.05662 q^{67} +0.0319171 q^{68} +2.06351 q^{69} +3.06337 q^{71} -23.7884 q^{72} -5.61322 q^{73} +3.76490 q^{74} -0.336174 q^{76} +0.563541 q^{77} -7.23628 q^{78} -5.37103 q^{79} +29.6355 q^{81} +0.725401 q^{82} +2.47369 q^{83} -0.625298 q^{84} +14.1576 q^{86} +20.2762 q^{87} -3.01342 q^{88} -14.0050 q^{89} -0.957832 q^{91} -0.210171 q^{92} +10.2741 q^{93} +11.4853 q^{94} +6.20615 q^{96} +6.13801 q^{97} -8.61961 q^{98} +7.89418 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * q^99

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 $

$2$	1.28989	0.912093	0.456046	−	0.889956i	$-0.349265\pi$
$2$	1.28989	0.912093	0.456046	+	0.889956i	$0.349265\pi$
$3$	3.30063	1.90562	0.952811	−	0.303565i	$-0.0981771\pi$
$3$	3.30063	1.90562	0.952811	+	0.303565i	$0.0981771\pi$
$4$	−0.336174	−0.168087
$5$	0	0
$5$	0	0
$6$	4.25747	1.73810
$7$	0.563541	0.212998	0.106499	−	0.994313i	$-0.466036\pi$
$7$	0.563541	0.212998	0.106499	+	0.994313i	$0.466036\pi$
$8$	−3.01342	−1.06540
$9$	7.89418	2.63139
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.10959	−0.320310
$13$	−1.69967	−0.471403	−0.235702	−	0.971825i	$-0.575739\pi$
$13$	−1.69967	−0.471403	−0.235702	+	0.971825i	$0.575739\pi$
$14$	0.726907	0.194274
$15$	0	0
$16$	−3.21464	−0.803660
$17$	−0.0949420	−0.0230268	−0.0115134	−	0.999934i	$-0.503665\pi$
$17$	−0.0949420	−0.0230268	−0.0115134	+	0.999934i	$0.503665\pi$
$18$	10.1826	2.40007
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.86004	0.405894
$22$	1.28989	0.275006
$23$	0.625186	0.130360	0.0651801	−	0.997874i	$-0.479238\pi$
$23$	0.625186	0.130360	0.0651801	+	0.997874i	$0.479238\pi$
$24$	−9.94618	−2.03026
$25$	0	0
$26$	−2.19239	−0.429963
$27$	16.1539	3.10882
$28$	−0.189448	−0.0358023
$29$	6.14313	1.14075	0.570376	−	0.821384i	$-0.306798\pi$
$29$	6.14313	1.14075	0.570376	+	0.821384i	$0.306798\pi$
$30$	0	0
$31$	3.11276	0.559069	0.279534	−	0.960136i	$-0.409820\pi$
$31$	3.11276	0.559069	0.279534	+	0.960136i	$0.409820\pi$
$32$	1.88029	0.332392
$33$	3.30063	0.574566
$34$	−0.122465	−0.0210026
$35$	0	0
$36$	−2.65382	−0.442303
$37$	2.91877	0.479842	0.239921	−	0.970792i	$-0.422878\pi$
$37$	2.91877	0.479842	0.239921	+	0.970792i	$0.422878\pi$
$38$	1.28989	0.209248
$39$	−5.60998	−0.898316
$40$	0	0
$41$	0.562373	0.0878279	0.0439139	−	0.999035i	$-0.486017\pi$
$41$	0.562373	0.0878279	0.0439139	+	0.999035i	$0.486017\pi$
$42$	2.39925	0.370213
$43$	10.9758	1.67379	0.836895	−	0.547364i	$-0.184369\pi$
$43$	10.9758	1.67379	0.836895	+	0.547364i	$0.184369\pi$
$44$	−0.336174	−0.0506802
$45$	0	0
$46$	0.806423	0.118901
$47$	8.90408	1.29879	0.649396	−	0.760450i	$-0.275022\pi$
$47$	8.90408	1.29879	0.649396	+	0.760450i	$0.275022\pi$
$48$	−10.6103	−1.53147
$49$	−6.68242	−0.954632
$50$	0	0
$51$	−0.313369	−0.0438804
$52$	0.571385	0.0792368
$53$	−0.489328	−0.0672144	−0.0336072	−	0.999435i	$-0.510700\pi$
$53$	−0.489328	−0.0672144	−0.0336072	+	0.999435i	$0.510700\pi$
$54$	20.8368	2.83553
$55$	0	0
$56$	−1.69818	−0.226929
$57$	3.30063	0.437179
$58$	7.92399	1.04047
$59$	−11.5958	−1.50965	−0.754824	−	0.655927i	$-0.772278\pi$
$59$	−11.5958	−1.50965	−0.754824	+	0.655927i	$0.772278\pi$
$60$	0	0
$61$	13.8576	1.77428	0.887139	−	0.461502i	$-0.152689\pi$
$61$	13.8576	1.77428	0.887139	+	0.461502i	$0.152689\pi$
$62$	4.01513	0.509922
$63$	4.44869	0.560482
$64$	8.85465	1.10683
$65$	0	0
$66$	4.25747	0.524058
$67$	−2.05662	−0.251257	−0.125628	−	0.992077i	$-0.540095\pi$
$67$	−2.05662	−0.251257	−0.125628	+	0.992077i	$0.540095\pi$
$68$	0.0319171	0.00387051
$69$	2.06351	0.248417
$70$	0	0
$71$	3.06337	0.363556	0.181778	−	0.983340i	$-0.441815\pi$
$71$	3.06337	0.363556	0.181778	+	0.983340i	$0.441815\pi$
$72$	−23.7884	−2.80349
$73$	−5.61322	−0.656978	−0.328489	−	0.944508i	$-0.606539\pi$
$73$	−5.61322	−0.656978	−0.328489	+	0.944508i	$0.606539\pi$
$74$	3.76490	0.437661
$75$	0	0
$76$	−0.336174	−0.0385618
$77$	0.563541	0.0642214
$78$	−7.23628	−0.819347
$79$	−5.37103	−0.604288	−0.302144	−	0.953262i	$-0.597702\pi$
$79$	−5.37103	−0.604288	−0.302144	+	0.953262i	$0.597702\pi$
$80$	0	0
$81$	29.6355	3.29283
$82$	0.725401	0.0801071
$83$	2.47369	0.271522	0.135761	−	0.990742i	$-0.456652\pi$
$83$	2.47369	0.271522	0.135761	+	0.990742i	$0.456652\pi$
$84$	−0.625298	−0.0682256
$85$	0	0
$86$	14.1576	1.52665
$87$	20.2762	2.17384
$88$	−3.01342	−0.321231
$89$	−14.0050	−1.48452	−0.742262	−	0.670110i	$-0.766247\pi$
$89$	−14.0050	−1.48452	−0.742262	+	0.670110i	$0.766247\pi$
$90$	0	0
$91$	−0.957832	−0.100408
$92$	−0.210171	−0.0219119
$93$	10.2741	1.06537
$94$	11.4853	1.18462
$95$	0	0
$96$	6.20615	0.633413
$97$	6.13801	0.623221	0.311610	−	0.950210i	$-0.399132\pi$
$97$	6.13801	0.623221	0.311610	+	0.950210i	$0.399132\pi$
$98$	−8.61961	−0.870712
$99$	7.89418	0.793395
$100$	0	0
$101$	12.4177	1.23561	0.617804	−	0.786332i	$-0.288023\pi$
$101$	12.4177	1.23561	0.617804	+	0.786332i	$0.288023\pi$
$102$	−0.404212	−0.0400230
$103$	−10.7035	−1.05465	−0.527324	−	0.849665i	$-0.676804\pi$
$103$	−10.7035	−1.05465	−0.527324	+	0.849665i	$0.676804\pi$
$104$	5.12181	0.502234
$105$	0	0
$106$	−0.631181	−0.0613057
$107$	2.53215	0.244792	0.122396	−	0.992481i	$-0.460942\pi$
$107$	2.53215	0.244792	0.122396	+	0.992481i	$0.460942\pi$
$108$	−5.43052	−0.522552
$109$	−10.2842	−0.985046	−0.492523	−	0.870299i	$-0.663925\pi$
$109$	−10.2842	−0.985046	−0.492523	+	0.870299i	$0.663925\pi$
$110$	0	0
$111$	9.63378	0.914398
$112$	−1.81158	−0.171178
$113$	−17.0132	−1.60047	−0.800236	−	0.599686i	$-0.795292\pi$
$113$	−17.0132	−1.60047	−0.800236	+	0.599686i	$0.795292\pi$
$114$	4.25747	0.398748
$115$	0	0
$116$	−2.06516	−0.191746
$117$	−13.4175	−1.24045
$118$	−14.9574	−1.37694
$119$	−0.0535037	−0.00490468
$120$	0	0
$121$	1.00000	0.0909091
$122$	17.8748	1.61831
$123$	1.85619	0.167367
$124$	−1.04643	−0.0939723
$125$	0	0
$126$	5.73834	0.511212
$127$	−5.76757	−0.511789	−0.255894	−	0.966705i	$-0.582370\pi$
$127$	−5.76757	−0.511789	−0.255894	+	0.966705i	$0.582370\pi$
$128$	7.66098	0.677141
$129$	36.2270	3.18961
$130$	0	0
$131$	−6.29905	−0.550351	−0.275175	−	0.961394i	$-0.588736\pi$
$131$	−6.29905	−0.550351	−0.275175	+	0.961394i	$0.588736\pi$
$132$	−1.10959	−0.0965772
$133$	0.563541	0.0488652
$134$	−2.65283	−0.229169
$135$	0	0
$136$	0.286100	0.0245329
$137$	9.89969	0.845787	0.422894	−	0.906179i	$-0.361014\pi$
$137$	9.89969	0.845787	0.422894	+	0.906179i	$0.361014\pi$
$138$	2.66171	0.226579
$139$	21.4383	1.81838	0.909188	−	0.416387i	$-0.136704\pi$
$139$	21.4383	1.81838	0.909188	+	0.416387i	$0.136704\pi$
$140$	0	0
$141$	29.3891	2.47501
$142$	3.95143	0.331596
$143$	−1.69967	−0.142133
$144$	−25.3769	−2.11474
$145$	0	0
$146$	−7.24046	−0.599225
$147$	−22.0562	−1.81917
$148$	−0.981214	−0.0806553
$149$	−5.05856	−0.414413	−0.207207	−	0.978297i	$-0.566437\pi$
$149$	−5.05856	−0.414413	−0.207207	+	0.978297i	$0.566437\pi$
$150$	0	0
$151$	−13.4546	−1.09492	−0.547460	−	0.836832i	$-0.684405\pi$
$151$	−13.4546	−1.09492	−0.547460	+	0.836832i	$0.684405\pi$
$152$	−3.01342	−0.244420
$153$	−0.749489	−0.0605926
$154$	0.726907	0.0585759
$155$	0	0
$156$	1.88593	0.150995
$157$	2.56715	0.204881	0.102440	−	0.994739i	$-0.467335\pi$
$157$	2.56715	0.204881	0.102440	+	0.994739i	$0.467335\pi$
$158$	−6.92806	−0.551167
$159$	−1.61509	−0.128085
$160$	0	0
$161$	0.352317	0.0277665
$162$	38.2266	3.00337
$163$	−14.3323	−1.12259	−0.561294	−	0.827616i	$-0.689696\pi$
$163$	−14.3323	−1.12259	−0.561294	+	0.827616i	$0.689696\pi$
$164$	−0.189055	−0.0147627
$165$	0	0
$166$	3.19079	0.247654
$167$	−16.9732	−1.31342	−0.656711	−	0.754142i	$-0.728053\pi$
$167$	−16.9732	−1.31342	−0.656711	+	0.754142i	$0.728053\pi$
$168$	−5.60508	−0.432441
$169$	−10.1111	−0.777779
$170$	0	0
$171$	7.89418	0.603683
$172$	−3.68977	−0.281343
$173$	−24.9706	−1.89848	−0.949241	−	0.314549i	$-0.898147\pi$
$173$	−24.9706	−1.89848	−0.949241	+	0.314549i	$0.898147\pi$
$174$	26.1542	1.98274
$175$	0	0
$176$	−3.21464	−0.242312
$177$	−38.2736	−2.87682
$178$	−18.0649	−1.35402
$179$	2.25582	0.168608	0.0843040	−	0.996440i	$-0.473133\pi$
$179$	2.25582	0.168608	0.0843040	+	0.996440i	$0.473133\pi$
$180$	0	0
$181$	5.14529	0.382446	0.191223	−	0.981547i	$-0.438755\pi$
$181$	5.14529	0.382446	0.191223	+	0.981547i	$0.438755\pi$
$182$	−1.23550	−0.0915814
$183$	45.7387	3.38110
$184$	−1.88394	−0.138886
$185$	0	0
$186$	13.2525	0.971719
$187$	−0.0949420	−0.00694285
$188$	−2.99332	−0.218310
$189$	9.10337	0.662172
$190$	0	0
$191$	−1.23491	−0.0893549	−0.0446774	−	0.999001i	$-0.514226\pi$
$191$	−1.23491	−0.0893549	−0.0446774	+	0.999001i	$0.514226\pi$
$192$	29.2260	2.10920
$193$	−6.39165	−0.460081	−0.230041	−	0.973181i	$-0.573886\pi$
$193$	−6.39165	−0.460081	−0.230041	+	0.973181i	$0.573886\pi$
$194$	7.91738	0.568435
$195$	0	0
$196$	2.24646	0.160461
$197$	5.23212	0.372773	0.186387	−	0.982476i	$-0.440322\pi$
$197$	5.23212	0.372773	0.186387	+	0.982476i	$0.440322\pi$
$198$	10.1826	0.723649
$199$	−19.1059	−1.35438	−0.677192	−	0.735807i	$-0.736803\pi$
$199$	−19.1059	−1.35438	−0.677192	+	0.735807i	$0.736803\pi$
$200$	0	0
$201$	−6.78816	−0.478800
$202$	16.0175	1.12699
$203$	3.46190	0.242978
$204$	0.105347	0.00737573
$205$	0	0
$206$	−13.8064	−0.961936
$207$	4.93532	0.343029
$208$	5.46382	0.378848
$209$	1.00000	0.0691714
$210$	0	0
$211$	−3.42015	−0.235453	−0.117726	−	0.993046i	$-0.537561\pi$
$211$	−3.42015	−0.235453	−0.117726	+	0.993046i	$0.537561\pi$
$212$	0.164499	0.0112979
$213$	10.1111	0.692799
$214$	3.26621	0.223273
$215$	0	0
$216$	−48.6784	−3.31214
$217$	1.75417	0.119081
$218$	−13.2655	−0.898453
$219$	−18.5272	−1.25195
$220$	0	0
$221$	0.161370	0.0108549
$222$	12.4265	0.834015
$223$	−10.6147	−0.710813	−0.355406	−	0.934712i	$-0.615657\pi$
$223$	−10.6147	−0.710813	−0.355406	+	0.934712i	$0.615657\pi$
$224$	1.05962	0.0707989
$225$	0	0
$226$	−21.9453	−1.45978
$227$	7.18967	0.477195	0.238598	−	0.971119i	$-0.423312\pi$
$227$	7.18967	0.477195	0.238598	+	0.971119i	$0.423312\pi$
$228$	−1.10959	−0.0734843
$229$	−21.1961	−1.40067	−0.700337	−	0.713812i	$-0.746967\pi$
$229$	−21.1961	−1.40067	−0.700337	+	0.713812i	$0.746967\pi$
$230$	0	0
$231$	1.86004	0.122382
$232$	−18.5118	−1.21536
$233$	26.7869	1.75487	0.877433	−	0.479699i	$-0.159254\pi$
$233$	26.7869	1.75487	0.877433	+	0.479699i	$0.159254\pi$
$234$	−17.3071	−1.13140
$235$	0	0
$236$	3.89822	0.253752
$237$	−17.7278	−1.15154
$238$	−0.0690141	−0.00447352
$239$	5.29366	0.342419	0.171209	−	0.985235i	$-0.445233\pi$
$239$	5.29366	0.342419	0.171209	+	0.985235i	$0.445233\pi$
$240$	0	0
$241$	1.54532	0.0995429	0.0497714	−	0.998761i	$-0.484151\pi$
$241$	1.54532	0.0995429	0.0497714	+	0.998761i	$0.484151\pi$
$242$	1.28989	0.0829175
$243$	49.3542	3.16608
$244$	−4.65856	−0.298233
$245$	0	0
$246$	2.39428	0.152654
$247$	−1.69967	−0.108147
$248$	−9.38005	−0.595634
$249$	8.16473	0.517419
$250$	0	0
$251$	−23.7169	−1.49700	−0.748500	−	0.663135i	$-0.769226\pi$
$251$	−23.7169	−1.49700	−0.748500	+	0.663135i	$0.769226\pi$
$252$	−1.49553	−0.0942098
$253$	0.625186	0.0393051
$254$	−7.43955	−0.466799
$255$	0	0
$256$	−7.82746	−0.489216
$257$	−2.55273	−0.159235	−0.0796174	−	0.996825i	$-0.525370\pi$
$257$	−2.55273	−0.159235	−0.0796174	+	0.996825i	$0.525370\pi$
$258$	46.7290	2.90922
$259$	1.64484	0.102206
$260$	0	0
$261$	48.4950	3.00176
$262$	−8.12511	−0.501971
$263$	−4.06910	−0.250911	−0.125456	−	0.992099i	$-0.540039\pi$
$263$	−4.06910	−0.250911	−0.125456	+	0.992099i	$0.540039\pi$
$264$	−9.94618	−0.612145
$265$	0	0
$266$	0.726907	0.0445696
$267$	−46.2252	−2.82894
$268$	0.691384	0.0422330
$269$	0.181743	0.0110811	0.00554055	−	0.999985i	$-0.498236\pi$
$269$	0.181743	0.0110811	0.00554055	+	0.999985i	$0.498236\pi$
$270$	0	0
$271$	−4.15652	−0.252491	−0.126245	−	0.991999i	$-0.540293\pi$
$271$	−4.15652	−0.252491	−0.126245	+	0.991999i	$0.540293\pi$
$272$	0.305204	0.0185057
$273$	−3.16145	−0.191340
$274$	12.7695	0.771436
$275$	0	0
$276$	−0.693698	−0.0417557
$277$	31.6346	1.90074	0.950368	−	0.311127i	$-0.100706\pi$
$277$	31.6346	1.90074	0.950368	+	0.311127i	$0.100706\pi$
$278$	27.6532	1.65853
$279$	24.5727	1.47113
$280$	0	0
$281$	1.38352	0.0825337	0.0412668	−	0.999148i	$-0.486861\pi$
$281$	1.38352	0.0825337	0.0412668	+	0.999148i	$0.486861\pi$
$282$	37.9088	2.25744
$283$	25.1918	1.49750	0.748749	−	0.662854i	$-0.230655\pi$
$283$	25.1918	1.49750	0.748749	+	0.662854i	$0.230655\pi$
$284$	−1.02983	−0.0611090
$285$	0	0
$286$	−2.19239	−0.129639
$287$	0.316920	0.0187072
$288$	14.8434	0.874653
$289$	−16.9910	−0.999470
$290$	0	0
$291$	20.2593	1.18762
$292$	1.88702	0.110430
$293$	11.8433	0.691895	0.345947	−	0.938254i	$-0.387558\pi$
$293$	11.8433	0.691895	0.345947	+	0.938254i	$0.387558\pi$
$294$	−28.4502	−1.65925
$295$	0	0
$296$	−8.79546	−0.511226
$297$	16.1539	0.937343
$298$	−6.52500	−0.377983
$299$	−1.06261	−0.0614522
$300$	0	0
$301$	6.18529	0.356514
$302$	−17.3550	−0.998668
$303$	40.9863	2.35460
$304$	−3.21464	−0.184372
$305$	0	0
$306$	−0.966761	−0.0552661
$307$	−12.4995	−0.713385	−0.356692	−	0.934222i	$-0.616096\pi$
$307$	−12.4995	−0.713385	−0.356692	+	0.934222i	$0.616096\pi$
$308$	−0.189448	−0.0107948
$309$	−35.3283	−2.00976
$310$	0	0
$311$	−4.01056	−0.227418	−0.113709	−	0.993514i	$-0.536273\pi$
$311$	−4.01056	−0.227418	−0.113709	+	0.993514i	$0.536273\pi$
$312$	16.9052	0.957069
$313$	19.8467	1.12180	0.560901	−	0.827883i	$-0.310455\pi$
$313$	19.8467	1.12180	0.560901	+	0.827883i	$0.310455\pi$
$314$	3.31135	0.186870
$315$	0	0
$316$	1.80560	0.101573
$317$	−16.2627	−0.913406	−0.456703	−	0.889619i	$-0.650970\pi$
$317$	−16.2627	−0.913406	−0.456703	+	0.889619i	$0.650970\pi$
$318$	−2.08330	−0.116825
$319$	6.14313	0.343949
$320$	0	0
$321$	8.35770	0.466481
$322$	0.454452	0.0253256
$323$	−0.0949420	−0.00528272
$324$	−9.96269	−0.553483
$325$	0	0
$326$	−18.4871	−1.02390
$327$	−33.9443	−1.87712
$328$	−1.69466	−0.0935721
$329$	5.01781	0.276641
$330$	0	0
$331$	17.4946	0.961593	0.480796	−	0.876832i	$-0.340348\pi$
$331$	17.4946	0.961593	0.480796	+	0.876832i	$0.340348\pi$
$332$	−0.831590	−0.0456394
$333$	23.0413	1.26265
$334$	−21.8936	−1.19796
$335$	0	0
$336$	−5.97936	−0.326201
$337$	−34.0435	−1.85447	−0.927234	−	0.374482i	$-0.877821\pi$
$337$	−34.0435	−1.85447	−0.927234	+	0.374482i	$0.877821\pi$
$338$	−13.0423	−0.709407
$339$	−56.1545	−3.04989
$340$	0	0
$341$	3.11276	0.168566
$342$	10.1826	0.550615
$343$	−7.71060	−0.416333
$344$	−33.0746	−1.78326
$345$	0	0
$346$	−32.2095	−1.73159
$347$	18.8146	1.01002	0.505011	−	0.863113i	$-0.331488\pi$
$347$	18.8146	1.01002	0.505011	+	0.863113i	$0.331488\pi$
$348$	−6.81635	−0.365394
$349$	26.8633	1.43796	0.718981	−	0.695030i	$-0.244609\pi$
$349$	26.8633	1.43796	0.718981	+	0.695030i	$0.244609\pi$
$350$	0	0
$351$	−27.4562	−1.46551
$352$	1.88029	0.100220
$353$	−8.92277	−0.474911	−0.237456	−	0.971398i	$-0.576313\pi$
$353$	−8.92277	−0.474911	−0.237456	+	0.971398i	$0.576313\pi$
$354$	−49.3688	−2.62392
$355$	0	0
$356$	4.70811	0.249529
$357$	−0.176596	−0.00934645
$358$	2.90977	0.153786
$359$	4.83871	0.255377	0.127689	−	0.991814i	$-0.459244\pi$
$359$	4.83871	0.255377	0.127689	+	0.991814i	$0.459244\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.63688	0.348826
$363$	3.30063	0.173238
$364$	0.321998	0.0168773
$365$	0	0
$366$	58.9981	3.08388
$367$	17.7897	0.928614	0.464307	−	0.885674i	$-0.346303\pi$
$367$	17.7897	0.928614	0.464307	+	0.885674i	$0.346303\pi$
$368$	−2.00975	−0.104765
$369$	4.43947	0.231110
$370$	0	0
$371$	−0.275756	−0.0143165
$372$	−3.45388	−0.179076
$373$	6.12381	0.317079	0.158539	−	0.987353i	$-0.449322\pi$
$373$	6.12381	0.317079	0.158539	+	0.987353i	$0.449322\pi$
$374$	−0.122465	−0.00633252
$375$	0	0
$376$	−26.8317	−1.38374
$377$	−10.4413	−0.537753
$378$	11.7424	0.603963
$379$	−24.5784	−1.26251	−0.631255	−	0.775575i	$-0.717460\pi$
$379$	−24.5784	−1.26251	−0.631255	+	0.775575i	$0.717460\pi$
$380$	0	0
$381$	−19.0366	−0.975276
$382$	−1.59290	−0.0814999
$383$	−29.3567	−1.50006	−0.750029	−	0.661405i	$-0.769960\pi$
$383$	−29.3567	−1.50006	−0.750029	+	0.661405i	$0.769960\pi$
$384$	25.2861	1.29037
$385$	0	0
$386$	−8.24455	−0.419637
$387$	86.6447	4.40440
$388$	−2.06344	−0.104755
$389$	23.3372	1.18324	0.591621	−	0.806216i	$-0.298488\pi$
$389$	23.3372	1.18324	0.591621	+	0.806216i	$0.298488\pi$
$390$	0	0
$391$	−0.0593564	−0.00300178
$392$	20.1369	1.01707
$393$	−20.7909	−1.04876
$394$	6.74888	0.340004
$395$	0	0
$396$	−2.65382	−0.133359
$397$	−14.4695	−0.726202	−0.363101	−	0.931750i	$-0.618282\pi$
$397$	−14.4695	−0.726202	−0.363101	+	0.931750i	$0.618282\pi$
$398$	−24.6446	−1.23532
$399$	1.86004	0.0931185
$400$	0	0
$401$	−12.7693	−0.637668	−0.318834	−	0.947811i	$-0.603291\pi$
$401$	−12.7693	−0.637668	−0.318834	+	0.947811i	$0.603291\pi$
$402$	−8.75600	−0.436710
$403$	−5.29066	−0.263547
$404$	−4.17451	−0.207690
$405$	0	0
$406$	4.46549	0.221618
$407$	2.91877	0.144678
$408$	0.944311	0.0467504
$409$	−23.0952	−1.14198	−0.570991	−	0.820956i	$-0.693441\pi$
$409$	−23.0952	−1.14198	−0.570991	+	0.820956i	$0.693441\pi$
$410$	0	0
$411$	32.6752	1.61175
$412$	3.59824	0.177273
$413$	−6.53472	−0.321552
$414$	6.36604	0.312874
$415$	0	0
$416$	−3.19587	−0.156690
$417$	70.7601	3.46513
$418$	1.28989	0.0630908
$419$	−2.04650	−0.0999783	−0.0499891	−	0.998750i	$-0.515919\pi$
$419$	−2.04650	−0.0999783	−0.0499891	+	0.998750i	$0.515919\pi$
$420$	0	0
$421$	4.36047	0.212516	0.106258	−	0.994339i	$-0.466113\pi$
$421$	4.36047	0.212516	0.106258	+	0.994339i	$0.466113\pi$
$422$	−4.41163	−0.214755
$423$	70.2903	3.41763
$424$	1.47455	0.0716104
$425$	0	0
$426$	13.0422	0.631897
$427$	7.80930	0.377918
$428$	−0.851244	−0.0411464
$429$	−5.60998	−0.270852
$430$	0	0
$431$	15.2001	0.732165	0.366083	−	0.930582i	$-0.380699\pi$
$431$	15.2001	0.732165	0.366083	+	0.930582i	$0.380699\pi$
$432$	−51.9289	−2.49843
$433$	14.5469	0.699081	0.349541	−	0.936921i	$-0.386338\pi$
$433$	14.5469	0.699081	0.349541	+	0.936921i	$0.386338\pi$
$434$	2.26269	0.108613
$435$	0	0
$436$	3.45728	0.165574
$437$	0.625186	0.0299067
$438$	−23.8981	−1.14189
$439$	34.3065	1.63736	0.818680	−	0.574250i	$-0.194706\pi$
$439$	34.3065	1.63736	0.818680	+	0.574250i	$0.194706\pi$
$440$	0	0
$441$	−52.7522	−2.51201
$442$	0.208150	0.00990069
$443$	1.56134	0.0741813	0.0370907	−	0.999312i	$-0.488191\pi$
$443$	1.56134	0.0741813	0.0370907	+	0.999312i	$0.488191\pi$
$444$	−3.23863	−0.153699
$445$	0	0
$446$	−13.6918	−0.648327
$447$	−16.6964	−0.789715
$448$	4.98996	0.235753
$449$	−20.7918	−0.981228	−0.490614	−	0.871377i	$-0.663227\pi$
$449$	−20.7918	−0.981228	−0.490614	+	0.871377i	$0.663227\pi$
$450$	0	0
$451$	0.562373	0.0264811
$452$	5.71942	0.269019
$453$	−44.4087	−2.08650
$454$	9.27391	0.435246
$455$	0	0
$456$	−9.94618	−0.465773
$457$	3.00835	0.140725	0.0703624	−	0.997521i	$-0.477584\pi$
$457$	3.00835	0.140725	0.0703624	+	0.997521i	$0.477584\pi$
$458$	−27.3407	−1.27755
$459$	−1.53368	−0.0715862
$460$	0	0
$461$	16.5782	0.772122	0.386061	−	0.922473i	$-0.373835\pi$
$461$	16.5782	0.772122	0.386061	+	0.922473i	$0.373835\pi$
$462$	2.39925	0.111623
$463$	28.9063	1.34339	0.671694	−	0.740828i	$-0.265567\pi$
$463$	28.9063	1.34339	0.671694	+	0.740828i	$0.265567\pi$
$464$	−19.7479	−0.916775
$465$	0	0
$466$	34.5522	1.60060
$467$	7.97738	0.369149	0.184574	−	0.982819i	$-0.440909\pi$
$467$	7.97738	0.369149	0.184574	+	0.982819i	$0.440909\pi$
$468$	4.51061	0.208503
$469$	−1.15899	−0.0535172
$470$	0	0
$471$	8.47321	0.390425
$472$	34.9430	1.60838
$473$	10.9758	0.504667
$474$	−22.8670	−1.05031
$475$	0	0
$476$	0.0179866	0.000824413 0
$477$	−3.86284	−0.176867
$478$	6.82826	0.312317
$479$	−2.05076	−0.0937014	−0.0468507	−	0.998902i	$-0.514919\pi$
$479$	−2.05076	−0.0937014	−0.0468507	+	0.998902i	$0.514919\pi$
$480$	0	0
$481$	−4.96093	−0.226199
$482$	1.99330	0.0907923
$483$	1.16287	0.0529124
$484$	−0.336174	−0.0152807
$485$	0	0
$486$	63.6617	2.88775
$487$	19.7907	0.896804	0.448402	−	0.893832i	$-0.351993\pi$
$487$	19.7907	0.896804	0.448402	+	0.893832i	$0.351993\pi$
$488$	−41.7586	−1.89032
$489$	−47.3055	−2.13923
$490$	0	0
$491$	−5.00033	−0.225662	−0.112831	−	0.993614i	$-0.535992\pi$
$491$	−5.00033	−0.225662	−0.112831	+	0.993614i	$0.535992\pi$
$492$	−0.624002	−0.0281322
$493$	−0.583242	−0.0262679
$494$	−2.19239	−0.0986403
$495$	0	0
$496$	−10.0064	−0.449301
$497$	1.72634	0.0774367
$498$	10.5316	0.471934
$499$	31.6046	1.41481	0.707407	−	0.706806i	$-0.249865\pi$
$499$	31.6046	1.41481	0.707407	+	0.706806i	$0.249865\pi$
$500$	0	0
$501$	−56.0222	−2.50289
$502$	−30.5923	−1.36540
$503$	−19.0877	−0.851080	−0.425540	−	0.904940i	$-0.639916\pi$
$503$	−19.0877	−0.851080	−0.425540	+	0.904940i	$0.639916\pi$
$504$	−13.4058	−0.597140
$505$	0	0
$506$	0.806423	0.0358499
$507$	−33.3731	−1.48215
$508$	1.93891	0.0860251
$509$	42.8558	1.89955	0.949775	−	0.312934i	$-0.101312\pi$
$509$	42.8558	1.89955	0.949775	+	0.312934i	$0.101312\pi$
$510$	0	0
$511$	−3.16328	−0.139935
$512$	−25.4185	−1.12335
$513$	16.1539	0.713211
$514$	−3.29275	−0.145237
$515$	0	0
$516$	−12.1786	−0.536132
$517$	8.90408	0.391601
$518$	2.12167	0.0932210
$519$	−82.4189	−3.61779
$520$	0	0
$521$	−1.32945	−0.0582443	−0.0291222	−	0.999576i	$-0.509271\pi$
$521$	−1.32945	−0.0582443	−0.0291222	+	0.999576i	$0.509271\pi$
$522$	62.5534	2.73789
$523$	19.3395	0.845656	0.422828	−	0.906210i	$-0.361037\pi$
$523$	19.3395	0.845656	0.422828	+	0.906210i	$0.361037\pi$
$524$	2.11758	0.0925069
$525$	0	0
$526$	−5.24871	−0.228854
$527$	−0.295532	−0.0128736
$528$	−10.6103	−0.461756
$529$	−22.6091	−0.983006
$530$	0	0
$531$	−91.5395	−3.97248
$532$	−0.189448	−0.00821361
$533$	−0.955847	−0.0414023
$534$	−59.6257	−2.58025
$535$	0	0
$536$	6.19746	0.267690
$537$	7.44563	0.321303
$538$	0.234430	0.0101070
$539$	−6.68242	−0.287832
$540$	0	0
$541$	−13.6158	−0.585390	−0.292695	−	0.956206i	$-0.594552\pi$
$541$	−13.6158	−0.585390	−0.292695	+	0.956206i	$0.594552\pi$
$542$	−5.36147	−0.230295
$543$	16.9827	0.728798
$544$	−0.178519	−0.00765393
$545$	0	0
$546$	−4.07794	−0.174520
$547$	−35.4126	−1.51413	−0.757066	−	0.653339i	$-0.773368\pi$
$547$	−35.4126	−1.51413	−0.757066	+	0.653339i	$0.773368\pi$
$548$	−3.32802	−0.142166
$549$	109.394	4.66882
$550$	0	0
$551$	6.14313	0.261706
$552$	−6.21821	−0.264665
$553$	−3.02679	−0.128712
$554$	40.8052	1.73365
$555$	0	0
$556$	−7.20702	−0.305646
$557$	−2.52845	−0.107134	−0.0535669	−	0.998564i	$-0.517059\pi$
$557$	−2.52845	−0.107134	−0.0535669	+	0.998564i	$0.517059\pi$
$558$	31.6962	1.34181
$559$	−18.6552	−0.789030
$560$	0	0
$561$	−0.313369	−0.0132304
$562$	1.78459	0.0752784
$563$	−22.8087	−0.961274	−0.480637	−	0.876920i	$-0.659594\pi$
$563$	−22.8087	−0.961274	−0.480637	+	0.876920i	$0.659594\pi$
$564$	−9.87986	−0.416017
$565$	0	0
$566$	32.4948	1.36586
$567$	16.7008	0.701368
$568$	−9.23122	−0.387334
$569$	−14.2866	−0.598926	−0.299463	−	0.954108i	$-0.596807\pi$
$569$	−14.2866	−0.598926	−0.299463	+	0.954108i	$0.596807\pi$
$570$	0	0
$571$	−11.3152	−0.473527	−0.236764	−	0.971567i	$-0.576087\pi$
$571$	−11.3152	−0.473527	−0.236764	+	0.971567i	$0.576087\pi$
$572$	0.571385	0.0238908
$573$	−4.07598	−0.170277
$574$	0.408793	0.0170627
$575$	0	0
$576$	69.9002	2.91251
$577$	−43.3748	−1.80572	−0.902858	−	0.429938i	$-0.858535\pi$
$577$	−43.3748	−1.80572	−0.902858	+	0.429938i	$0.858535\pi$
$578$	−21.9166	−0.911609
$579$	−21.0965	−0.876740
$580$	0	0
$581$	1.39402	0.0578338
$582$	26.1324	1.08322
$583$	−0.489328	−0.0202659
$584$	16.9150	0.699946
$585$	0	0
$586$	15.2766	0.631072
$587$	−21.8206	−0.900633	−0.450317	−	0.892869i	$-0.648689\pi$
$587$	−21.8206	−0.900633	−0.450317	+	0.892869i	$0.648689\pi$
$588$	7.41473	0.305779
$589$	3.11276	0.128259
$590$	0	0
$591$	17.2693	0.710365
$592$	−9.38278	−0.385630
$593$	−40.6744	−1.67030	−0.835149	−	0.550025i	$-0.814618\pi$
$593$	−40.6744	−1.67030	−0.835149	+	0.550025i	$0.814618\pi$
$594$	20.8368	0.854944
$595$	0	0
$596$	1.70056	0.0696575
$597$	−63.0616	−2.58094
$598$	−1.37065	−0.0560501
$599$	−33.5105	−1.36920	−0.684601	−	0.728918i	$-0.740024\pi$
$599$	−33.5105	−1.36920	−0.684601	+	0.728918i	$0.740024\pi$
$600$	0	0
$601$	8.85454	0.361184	0.180592	−	0.983558i	$-0.442199\pi$
$601$	8.85454	0.361184	0.180592	+	0.983558i	$0.442199\pi$
$602$	7.97837	0.325174
$603$	−16.2353	−0.661154
$604$	4.52309	0.184042
$605$	0	0
$606$	52.8680	2.14761
$607$	−38.7846	−1.57422	−0.787109	−	0.616814i	$-0.788423\pi$
$607$	−38.7846	−1.57422	−0.787109	+	0.616814i	$0.788423\pi$
$608$	1.88029	0.0762559
$609$	11.4265	0.463024
$610$	0	0
$611$	−15.1340	−0.612255
$612$	0.251959	0.0101848
$613$	−11.4604	−0.462883	−0.231441	−	0.972849i	$-0.574344\pi$
$613$	−11.4604	−0.462883	−0.231441	+	0.972849i	$0.574344\pi$
$614$	−16.1230	−0.650673
$615$	0	0
$616$	−1.69818	−0.0684217
$617$	44.6630	1.79806	0.899032	−	0.437884i	$-0.144272\pi$
$617$	44.6630	1.79806	0.899032	+	0.437884i	$0.144272\pi$
$618$	−45.5698	−1.83309
$619$	−10.0641	−0.404508	−0.202254	−	0.979333i	$-0.564827\pi$
$619$	−10.0641	−0.404508	−0.202254	+	0.979333i	$0.564827\pi$
$620$	0	0
$621$	10.0992	0.405266
$622$	−5.17320	−0.207427
$623$	−7.89237	−0.316201
$624$	18.0341	0.721940
$625$	0	0
$626$	25.6001	1.02319
$627$	3.30063	0.131815
$628$	−0.863009	−0.0344378
$629$	−0.277114	−0.0110492
$630$	0	0
$631$	−32.9805	−1.31293	−0.656466	−	0.754356i	$-0.727949\pi$
$631$	−32.9805	−1.31293	−0.656466	+	0.754356i	$0.727949\pi$
$632$	16.1851	0.643811
$633$	−11.2887	−0.448684
$634$	−20.9772	−0.833110
$635$	0	0
$636$	0.542952	0.0215295
$637$	11.3579	0.450016
$638$	7.92399	0.313714
$639$	24.1828	0.956658
$640$	0	0
$641$	−15.4514	−0.610292	−0.305146	−	0.952306i	$-0.598705\pi$
$641$	−15.4514	−0.610292	−0.305146	+	0.952306i	$0.598705\pi$
$642$	10.7805	0.425474
$643$	26.4799	1.04427	0.522133	−	0.852864i	$-0.325136\pi$
$643$	26.4799	1.04427	0.522133	+	0.852864i	$0.325136\pi$
$644$	−0.118440	−0.00466719
$645$	0	0
$646$	−0.122465	−0.00481833
$647$	40.9548	1.61010	0.805050	−	0.593207i	$-0.202138\pi$
$647$	40.9548	1.61010	0.805050	+	0.593207i	$0.202138\pi$
$648$	−89.3041	−3.50820
$649$	−11.5958	−0.455176
$650$	0	0
$651$	5.78986	0.226923
$652$	4.81813	0.188693
$653$	−23.3051	−0.911997	−0.455999	−	0.889980i	$-0.650718\pi$
$653$	−23.3051	−0.911997	−0.455999	+	0.889980i	$0.650718\pi$
$654$	−43.7846	−1.71211
$655$	0	0
$656$	−1.80782	−0.0705837
$657$	−44.3118	−1.72877
$658$	6.47244	0.252322
$659$	10.8517	0.422723	0.211361	−	0.977408i	$-0.432210\pi$
$659$	10.8517	0.422723	0.211361	+	0.977408i	$0.432210\pi$
$660$	0	0
$661$	8.94285	0.347837	0.173918	−	0.984760i	$-0.444357\pi$
$661$	8.94285	0.347837	0.173918	+	0.984760i	$0.444357\pi$
$662$	22.5662	0.877062
$663$	0.532623	0.0206854
$664$	−7.45425	−0.289281
$665$	0	0
$666$	29.7208	1.15166
$667$	3.84060	0.148709
$668$	5.70594	0.220769
$669$	−35.0352	−1.35454
$670$	0	0
$671$	13.8576	0.534965
$672$	3.49742	0.134916
$673$	−8.94979	−0.344989	−0.172494	−	0.985010i	$-0.555183\pi$
$673$	−8.94979	−0.344989	−0.172494	+	0.985010i	$0.555183\pi$
$674$	−43.9125	−1.69145
$675$	0	0
$676$	3.39910	0.130735
$677$	8.30060	0.319018	0.159509	−	0.987196i	$-0.449009\pi$
$677$	8.30060	0.319018	0.159509	+	0.987196i	$0.449009\pi$
$678$	−72.4333	−2.78178
$679$	3.45902	0.132745
$680$	0	0
$681$	23.7305	0.909353
$682$	4.01513	0.153747
$683$	37.6902	1.44217	0.721087	−	0.692844i	$-0.243643\pi$
$683$	37.6902	1.44217	0.721087	+	0.692844i	$0.243643\pi$
$684$	−2.65382	−0.101471
$685$	0	0
$686$	−9.94585	−0.379734
$687$	−69.9604	−2.66916
$688$	−35.2831	−1.34516
$689$	0.831695	0.0316851
$690$	0	0
$691$	−49.0286	−1.86514	−0.932568	−	0.360994i	$-0.882437\pi$
$691$	−49.0286	−1.86514	−0.932568	+	0.360994i	$0.882437\pi$
$692$	8.39449	0.319111
$693$	4.44869	0.168992
$694$	24.2689	0.921234
$695$	0	0
$696$	−61.1007	−2.31602
$697$	−0.0533928	−0.00202240
$698$	34.6509	1.31155
$699$	88.4136	3.34411
$700$	0	0
$701$	−8.98928	−0.339520	−0.169760	−	0.985485i	$-0.554299\pi$
$701$	−8.98928	−0.339520	−0.169760	+	0.985485i	$0.554299\pi$
$702$	−35.4156	−1.33668
$703$	2.91877	0.110083
$704$	8.85465	0.333722
$705$	0	0
$706$	−11.5094	−0.433163
$707$	6.99788	0.263182
$708$	12.8666	0.483556
$709$	22.4894	0.844607	0.422303	−	0.906455i	$-0.361222\pi$
$709$	22.4894	0.844607	0.422303	+	0.906455i	$0.361222\pi$
$710$	0	0
$711$	−42.3999	−1.59012
$712$	42.2028	1.58162
$713$	1.94605	0.0728803
$714$	−0.227790	−0.00852483
$715$	0	0
$716$	−0.758349	−0.0283408
$717$	17.4724	0.652520
$718$	6.24142	0.232928
$719$	−27.3255	−1.01907	−0.509534	−	0.860450i	$-0.670182\pi$
$719$	−27.3255	−1.01907	−0.509534	+	0.860450i	$0.670182\pi$
$720$	0	0
$721$	−6.03186	−0.224638
$722$	1.28989	0.0480049
$723$	5.10054	0.189691
$724$	−1.72971	−0.0642843
$725$	0	0
$726$	4.25747	0.158009
$727$	45.7801	1.69789	0.848945	−	0.528482i	$-0.177238\pi$
$727$	45.7801	1.69789	0.848945	+	0.528482i	$0.177238\pi$
$728$	2.88635	0.106975
$729$	73.9938	2.74051
$730$	0	0
$731$	−1.04206	−0.0385421
$732$	−15.3762	−0.568320
$733$	19.9594	0.737219	0.368610	−	0.929584i	$-0.379834\pi$
$733$	19.9594	0.737219	0.368610	+	0.929584i	$0.379834\pi$
$734$	22.9468	0.846982
$735$	0	0
$736$	1.17553	0.0433306
$737$	−2.05662	−0.0757567
$738$	5.72644	0.210793
$739$	22.5149	0.828226	0.414113	−	0.910226i	$-0.364092\pi$
$739$	22.5149	0.828226	0.414113	+	0.910226i	$0.364092\pi$
$740$	0	0
$741$	−5.60998	−0.206088
$742$	−0.355696	−0.0130580
$743$	26.2969	0.964739	0.482370	−	0.875968i	$-0.339776\pi$
$743$	26.2969	0.964739	0.482370	+	0.875968i	$0.339776\pi$
$744$	−30.9601	−1.13505
$745$	0	0
$746$	7.89906	0.289205
$747$	19.5277	0.714482
$748$	0.0319171	0.00116700
$749$	1.42697	0.0521404
$750$	0	0
$751$	−27.3472	−0.997914	−0.498957	−	0.866627i	$-0.666283\pi$
$751$	−27.3472	−0.997914	−0.498957	+	0.866627i	$0.666283\pi$
$752$	−28.6234	−1.04379
$753$	−78.2809	−2.85272
$754$	−13.4681	−0.490481
$755$	0	0
$756$	−3.06032	−0.111303
$757$	23.2243	0.844102	0.422051	−	0.906572i	$-0.361310\pi$
$757$	23.2243	0.844102	0.422051	+	0.906572i	$0.361310\pi$
$758$	−31.7036	−1.15153
$759$	2.06351	0.0749006
$760$	0	0
$761$	−38.8028	−1.40660	−0.703300	−	0.710893i	$-0.748291\pi$
$761$	−38.8028	−1.40660	−0.703300	+	0.710893i	$0.748291\pi$
$762$	−24.5552	−0.889542
$763$	−5.79555	−0.209813
$764$	0.415145	0.0150194
$765$	0	0
$766$	−37.8670	−1.36819
$767$	19.7090	0.711653
$768$	−25.8356	−0.932261
$769$	2.29406	0.0827261	0.0413630	−	0.999144i	$-0.486830\pi$
$769$	2.29406	0.0827261	0.0413630	+	0.999144i	$0.486830\pi$
$770$	0	0
$771$	−8.42561	−0.303441
$772$	2.14871	0.0773337
$773$	−52.2893	−1.88071	−0.940357	−	0.340189i	$-0.889509\pi$
$773$	−52.2893	−1.88071	−0.940357	+	0.340189i	$0.889509\pi$
$774$	111.762	4.01722
$775$	0	0
$776$	−18.4964	−0.663982
$777$	5.42902	0.194765
$778$	30.1025	1.07923
$779$	0.562373	0.0201491
$780$	0	0
$781$	3.06337	0.109616
$782$	−0.0765634	−0.00273790
$783$	99.2354	3.54638
$784$	21.4816	0.767199
$785$	0	0
$786$	−26.8180	−0.956567
$787$	−9.08106	−0.323705	−0.161852	−	0.986815i	$-0.551747\pi$
$787$	−9.08106	−0.323705	−0.161852	+	0.986815i	$0.551747\pi$
$788$	−1.75891	−0.0626584
$789$	−13.4306	−0.478142
$790$	0	0
$791$	−9.58765	−0.340898
$792$	−23.7884	−0.845285
$793$	−23.5532	−0.836400
$794$	−18.6641	−0.662363
$795$	0	0
$796$	6.42292	0.227654
$797$	−16.3422	−0.578870	−0.289435	−	0.957198i	$-0.593467\pi$
$797$	−16.3422	−0.578870	−0.289435	+	0.957198i	$0.593467\pi$
$798$	2.39925	0.0849327
$799$	−0.845371	−0.0299071
$800$	0	0
$801$	−110.558	−3.90636
$802$	−16.4710	−0.581612
$803$	−5.61322	−0.198086
$804$	2.28200	0.0804801
$805$	0	0
$806$	−6.82439	−0.240379
$807$	0.599868	0.0211164
$808$	−37.4197	−1.31642
$809$	−3.54010	−0.124463	−0.0622317	−	0.998062i	$-0.519822\pi$
$809$	−3.54010	−0.124463	−0.0622317	+	0.998062i	$0.519822\pi$
$810$	0	0
$811$	−32.9605	−1.15740	−0.578699	−	0.815541i	$-0.696439\pi$
$811$	−32.9605	−1.15740	−0.578699	+	0.815541i	$0.696439\pi$
$812$	−1.16380	−0.0408415
$813$	−13.7192	−0.481152
$814$	3.76490	0.131960
$815$	0	0
$816$	1.00737	0.0352649
$817$	10.9758	0.383994
$818$	−29.7903	−1.04159
$819$	−7.56129	−0.264213
$820$	0	0
$821$	18.7416	0.654086	0.327043	−	0.945010i	$-0.393948\pi$
$821$	18.7416	0.654086	0.327043	+	0.945010i	$0.393948\pi$
$822$	42.1476	1.47007
$823$	22.5046	0.784462	0.392231	−	0.919867i	$-0.371703\pi$
$823$	22.5046	0.784462	0.392231	+	0.919867i	$0.371703\pi$
$824$	32.2541	1.12362
$825$	0	0
$826$	−8.42909	−0.293286
$827$	11.9403	0.415205	0.207603	−	0.978213i	$-0.433434\pi$
$827$	11.9403	0.415205	0.207603	+	0.978213i	$0.433434\pi$
$828$	−1.65913	−0.0576587
$829$	39.5586	1.37393	0.686964	−	0.726691i	$-0.258943\pi$
$829$	39.5586	1.37393	0.686964	+	0.726691i	$0.258943\pi$
$830$	0	0
$831$	104.414	3.62208
$832$	−15.0500	−0.521764
$833$	0.634443	0.0219821
$834$	91.2729	3.16052
$835$	0	0
$836$	−0.336174	−0.0116268
$837$	50.2832	1.73804
$838$	−2.63977	−0.0911894
$839$	−34.8294	−1.20244	−0.601222	−	0.799082i	$-0.705319\pi$
$839$	−34.8294	−1.20244	−0.601222	+	0.799082i	$0.705319\pi$
$840$	0	0
$841$	8.73807	0.301313
$842$	5.62454	0.193835
$843$	4.56648	0.157278
$844$	1.14977	0.0395766
$845$	0	0
$846$	90.6671	3.11720
$847$	0.563541	0.0193635
$848$	1.57301	0.0540175
$849$	83.1489	2.85366
$850$	0	0
$851$	1.82477	0.0625523
$852$	−3.39908	−0.116451
$853$	55.1856	1.88952	0.944759	−	0.327765i	$-0.106295\pi$
$853$	55.1856	1.88952	0.944759	+	0.327765i	$0.106295\pi$
$854$	10.0732	0.344697
$855$	0	0
$856$	−7.63043	−0.260803
$857$	21.4134	0.731468	0.365734	−	0.930719i	$-0.380818\pi$
$857$	21.4134	0.731468	0.365734	+	0.930719i	$0.380818\pi$
$858$	−7.23628	−0.247042
$859$	−36.5639	−1.24754	−0.623772	−	0.781606i	$-0.714401\pi$
$859$	−36.5639	−1.24754	−0.623772	+	0.781606i	$0.714401\pi$
$860$	0	0
$861$	1.04604	0.0356488
$862$	19.6066	0.667802
$863$	52.7117	1.79433	0.897163	−	0.441699i	$-0.145624\pi$
$863$	52.7117	1.79433	0.897163	+	0.441699i	$0.145624\pi$
$864$	30.3740	1.03334
$865$	0	0
$866$	18.7640	0.637627
$867$	−56.0810	−1.90461
$868$	−0.589706	−0.0200159
$869$	−5.37103	−0.182200
$870$	0	0
$871$	3.49558	0.118443
$872$	30.9905	1.04947
$873$	48.4545	1.63994
$874$	0.806423	0.0272777
$875$	0	0
$876$	6.22836	0.210437
$877$	18.5720	0.627133	0.313566	−	0.949566i	$-0.398476\pi$
$877$	18.5720	0.627133	0.313566	+	0.949566i	$0.398476\pi$
$878$	44.2517	1.49342
$879$	39.0905	1.31849
$880$	0	0
$881$	−32.2045	−1.08500	−0.542499	−	0.840056i	$-0.682522\pi$
$881$	−32.2045	−1.08500	−0.542499	+	0.840056i	$0.682522\pi$
$882$	−68.0448	−2.29119
$883$	−41.2458	−1.38803	−0.694016	−	0.719960i	$-0.744160\pi$
$883$	−41.2458	−1.38803	−0.694016	+	0.719960i	$0.744160\pi$
$884$	−0.0542484	−0.00182457
$885$	0	0
$886$	2.01396	0.0676602
$887$	−24.4058	−0.819468	−0.409734	−	0.912205i	$-0.634379\pi$
$887$	−24.4058	−0.819468	−0.409734	+	0.912205i	$0.634379\pi$
$888$	−29.0306	−0.974203
$889$	−3.25026	−0.109010
$890$	0	0
$891$	29.6355	0.992826
$892$	3.56839	0.119478
$893$	8.90408	0.297964
$894$	−21.5366	−0.720293
$895$	0	0
$896$	4.31727	0.144230
$897$	−3.50728	−0.117105
$898$	−26.8193	−0.894970
$899$	19.1221	0.637758
$900$	0	0
$901$	0.0464578	0.00154773
$902$	0.725401	0.0241532
$903$	20.4154	0.679381
$904$	51.2680	1.70515
$905$	0	0
$906$	−57.2825	−1.90308
$907$	47.4499	1.57555	0.787774	−	0.615965i	$-0.211234\pi$
$907$	47.4499	1.57555	0.787774	+	0.615965i	$0.211234\pi$
$908$	−2.41698	−0.0802104
$909$	98.0276	3.25137
$910$	0	0
$911$	50.2159	1.66373	0.831863	−	0.554981i	$-0.187275\pi$
$911$	50.2159	1.66373	0.831863	+	0.554981i	$0.187275\pi$
$912$	−10.6103	−0.351343
$913$	2.47369	0.0818671
$914$	3.88046	0.128354
$915$	0	0
$916$	7.12557	0.235435
$917$	−3.54977	−0.117224
$918$	−1.97829	−0.0652932
$919$	−0.561735	−0.0185299	−0.00926497	−	0.999957i	$-0.502949\pi$
$919$	−0.561735	−0.0185299	−0.00926497	+	0.999957i	$0.502949\pi$
$920$	0	0
$921$	−41.2563	−1.35944
$922$	21.3841	0.704247
$923$	−5.20672	−0.171381
$924$	−0.625298	−0.0205708
$925$	0	0
$926$	37.2860	1.22529
$927$	−84.4953	−2.77519
$928$	11.5509	0.379176
$929$	24.7806	0.813025	0.406513	−	0.913645i	$-0.366745\pi$
$929$	24.7806	0.813025	0.406513	+	0.913645i	$0.366745\pi$
$930$	0	0
$931$	−6.68242	−0.219008
$932$	−9.00506	−0.294971
$933$	−13.2374	−0.433373
$934$	10.2900	0.336698
$935$	0	0
$936$	40.4324	1.32158
$937$	31.7278	1.03650	0.518251	−	0.855229i	$-0.326583\pi$
$937$	31.7278	1.03650	0.518251	+	0.855229i	$0.326583\pi$
$938$	−1.49497	−0.0488127
$939$	65.5067	2.13773
$940$	0	0
$941$	−24.1062	−0.785839	−0.392920	−	0.919573i	$-0.628535\pi$
$941$	−24.1062	−0.785839	−0.392920	+	0.919573i	$0.628535\pi$
$942$	10.9295	0.356104
$943$	0.351587	0.0114493
$944$	37.2764	1.21324
$945$	0	0
$946$	14.1576	0.460303
$947$	−52.7844	−1.71526	−0.857631	−	0.514266i	$-0.828064\pi$
$947$	−52.7844	−1.71526	−0.857631	+	0.514266i	$0.828064\pi$
$948$	5.95963	0.193560
$949$	9.54061	0.309701
$950$	0	0
$951$	−53.6773	−1.74060
$952$	0.161229	0.00522546
$953$	−43.2711	−1.40169	−0.700845	−	0.713314i	$-0.747193\pi$
$953$	−43.2711	−1.40169	−0.700845	+	0.713314i	$0.747193\pi$
$954$	−4.98265	−0.161319
$955$	0	0
$956$	−1.77959	−0.0575562
$957$	20.2762	0.655437
$958$	−2.64526	−0.0854644
$959$	5.57888	0.180151
$960$	0	0
$961$	−21.3107	−0.687442
$962$	−6.39908	−0.206315
$963$	19.9893	0.644145
$964$	−0.519497	−0.0167319
$965$	0	0
$966$	1.49998	0.0482610
$967$	−28.7928	−0.925915	−0.462958	−	0.886380i	$-0.653212\pi$
$967$	−28.7928	−0.925915	−0.462958	+	0.886380i	$0.653212\pi$
$968$	−3.01342	−0.0968549
$969$	−0.313369	−0.0100669
$970$	0	0
$971$	27.3569	0.877923	0.438962	−	0.898506i	$-0.355346\pi$
$971$	27.3569	0.877923	0.438962	+	0.898506i	$0.355346\pi$
$972$	−16.5916	−0.532177
$973$	12.0814	0.387311
$974$	25.5280	0.817968
$975$	0	0
$976$	−44.5470	−1.42592
$977$	48.1900	1.54174	0.770868	−	0.636995i	$-0.219823\pi$
$977$	48.1900	1.54174	0.770868	+	0.636995i	$0.219823\pi$
$978$	−61.0191	−1.95117
$979$	−14.0050	−0.447601
$980$	0	0
$981$	−81.1852	−2.59204
$982$	−6.44990	−0.205824
$983$	−38.6466	−1.23264	−0.616318	−	0.787498i	$-0.711376\pi$
$983$	−38.6466	−1.23264	−0.616318	+	0.787498i	$0.711376\pi$
$984$	−5.59346	−0.178313
$985$	0	0
$986$	−0.752320	−0.0239587
$987$	16.5619	0.527172
$988$	0.571385	0.0181782
$989$	6.86189	0.218196
$990$	0	0
$991$	4.47927	0.142289	0.0711444	−	0.997466i	$-0.477335\pi$
$991$	4.47927	0.142289	0.0711444	+	0.997466i	$0.477335\pi$
$992$	5.85290	0.185830
$993$	57.7434	1.83243
$994$	2.22679	0.0706295
$995$	0	0
$996$	−2.74477	−0.0869715
$997$	4.58855	0.145321	0.0726605	−	0.997357i	$-0.476851\pi$
$997$	4.58855	0.145321	0.0726605	+	0.997357i	$0.476851\pi$
$998$	40.7665	1.29044
$999$	47.1494	1.49174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.y.1.10	yes	15
5.4	even	2		5225.2.a.r.1.6	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.r.1.6	✓	15	5.4	even	2
5225.2.a.y.1.10	yes	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.28989 q^{2} +3.30063 q^{3} -0.336174 q^{4} +4.25747 q^{6} +0.563541 q^{7} -3.01342 q^{8} +7.89418 q^{9} +1.00000 q^{11} -1.10959 q^{12} -1.69967 q^{13} +0.726907 q^{14} -3.21464 q^{16} -0.0949420 q^{17} +10.1826 q^{18} +1.00000 q^{19} +1.86004 q^{21} +1.28989 q^{22} +0.625186 q^{23} -9.94618 q^{24} -2.19239 q^{26} +16.1539 q^{27} -0.189448 q^{28} +6.14313 q^{29} +3.11276 q^{31} +1.88029 q^{32} +3.30063 q^{33} -0.122465 q^{34} -2.65382 q^{36} +2.91877 q^{37} +1.28989 q^{38} -5.60998 q^{39} +0.562373 q^{41} +2.39925 q^{42} +10.9758 q^{43} -0.336174 q^{44} +0.806423 q^{46} +8.90408 q^{47} -10.6103 q^{48} -6.68242 q^{49} -0.313369 q^{51} +0.571385 q^{52} -0.489328 q^{53} +20.8368 q^{54} -1.69818 q^{56} +3.30063 q^{57} +7.92399 q^{58} -11.5958 q^{59} +13.8576 q^{61} +4.01513 q^{62} +4.44869 q^{63} +8.85465 q^{64} +4.25747 q^{66} -2.05662 q^{67} +0.0319171 q^{68} +2.06351 q^{69} +3.06337 q^{71} -23.7884 q^{72} -5.61322 q^{73} +3.76490 q^{74} -0.336174 q^{76} +0.563541 q^{77} -7.23628 q^{78} -5.37103 q^{79} +29.6355 q^{81} +0.725401 q^{82} +2.47369 q^{83} -0.625298 q^{84} +14.1576 q^{86} +20.2762 q^{87} -3.01342 q^{88} -14.0050 q^{89} -0.957832 q^{91} -0.210171 q^{92} +10.2741 q^{93} +11.4853 q^{94} +6.20615 q^{96} +6.13801 q^{97} -8.61961 q^{98} +7.89418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 21 * q^7 + 9 * q^8 + 15 * q^9 + 15 * q^11 + 11 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 17 * q^17 + 22 * q^18 + 15 * q^19 + 6 * q^21 + 5 * q^22 + 26 * q^23 + q^24 + 3 * q^26 + q^27 + 46 * q^28 + 9 * q^29 + 14 * q^31 + 18 * q^32 + 4 * q^33 - 13 * q^34 + 12 * q^36 + 9 * q^37 + 5 * q^38 - 22 * q^39 + 4 * q^41 - 6 * q^42 + 28 * q^43 + 17 * q^44 + 27 * q^46 + 14 * q^47 - 4 * q^48 + 32 * q^49 - 40 * q^51 + 14 * q^52 + 3 * q^53 - 39 * q^54 + 34 * q^56 + 4 * q^57 + 26 * q^58 + q^59 + 2 * q^61 - 3 * q^62 + 45 * q^63 + 5 * q^64 - q^66 + 37 * q^67 + 26 * q^68 - 7 * q^69 - 7 * q^71 + 16 * q^72 + 42 * q^73 - 43 * q^74 + 17 * q^76 + 21 * q^77 - 64 * q^78 - 10 * q^79 + 31 * q^81 + 22 * q^82 + 14 * q^83 - 32 * q^84 + 37 * q^86 + 29 * q^87 + 9 * q^88 + 15 * q^89 - 22 * q^91 + 26 * q^92 - 18 * q^93 - 44 * q^94 + 71 * q^96 + 8 * q^97 - 10 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more