LMFDB - Embedded newform 5775.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$46.1136071673$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79129 q^{2} +1.00000 q^{3} +5.79129 q^{4} +2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.79129 q^{2} +1.00000 q^{3} +5.79129 q^{4} +2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} -1.00000 q^{11} +5.79129 q^{12} -1.00000 q^{13} -2.79129 q^{14} +17.9564 q^{16} +1.58258 q^{17} +2.79129 q^{18} +2.58258 q^{19} -1.00000 q^{21} -2.79129 q^{22} -3.58258 q^{23} +10.5826 q^{24} -2.79129 q^{26} +1.00000 q^{27} -5.79129 q^{28} +10.1652 q^{29} -5.58258 q^{31} +28.9564 q^{32} -1.00000 q^{33} +4.41742 q^{34} +5.79129 q^{36} -1.00000 q^{37} +7.20871 q^{38} -1.00000 q^{39} +7.16515 q^{41} -2.79129 q^{42} +7.58258 q^{43} -5.79129 q^{44} -10.0000 q^{46} -10.5826 q^{47} +17.9564 q^{48} +1.00000 q^{49} +1.58258 q^{51} -5.79129 q^{52} +0.417424 q^{53} +2.79129 q^{54} -10.5826 q^{56} +2.58258 q^{57} +28.3739 q^{58} -4.58258 q^{59} +10.0000 q^{61} -15.5826 q^{62} -1.00000 q^{63} +44.9129 q^{64} -2.79129 q^{66} +0.582576 q^{67} +9.16515 q^{68} -3.58258 q^{69} -7.16515 q^{71} +10.5826 q^{72} -7.00000 q^{73} -2.79129 q^{74} +14.9564 q^{76} +1.00000 q^{77} -2.79129 q^{78} -11.1652 q^{79} +1.00000 q^{81} +20.0000 q^{82} +2.41742 q^{83} -5.79129 q^{84} +21.1652 q^{86} +10.1652 q^{87} -10.5826 q^{88} -9.16515 q^{89} +1.00000 q^{91} -20.7477 q^{92} -5.58258 q^{93} -29.5390 q^{94} +28.9564 q^{96} +11.5826 q^{97} +2.79129 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} + 7 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 + 7 * q^4 + q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} + 7 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{11} + 7 q^{12} - 2 q^{13} - q^{14} + 13 q^{16} - 6 q^{17} + q^{18} - 4 q^{19} - 2 q^{21} - q^{22} + 2 q^{23} + 12 q^{24} - q^{26} + 2 q^{27} - 7 q^{28} + 2 q^{29} - 2 q^{31} + 35 q^{32} - 2 q^{33} + 18 q^{34} + 7 q^{36} - 2 q^{37} + 19 q^{38} - 2 q^{39} - 4 q^{41} - q^{42} + 6 q^{43} - 7 q^{44} - 20 q^{46} - 12 q^{47} + 13 q^{48} + 2 q^{49} - 6 q^{51} - 7 q^{52} + 10 q^{53} + q^{54} - 12 q^{56} - 4 q^{57} + 43 q^{58} + 20 q^{61} - 22 q^{62} - 2 q^{63} + 44 q^{64} - q^{66} - 8 q^{67} + 2 q^{69} + 4 q^{71} + 12 q^{72} - 14 q^{73} - q^{74} + 7 q^{76} + 2 q^{77} - q^{78} - 4 q^{79} + 2 q^{81} + 40 q^{82} + 14 q^{83} - 7 q^{84} + 24 q^{86} + 2 q^{87} - 12 q^{88} + 2 q^{91} - 14 q^{92} - 2 q^{93} - 27 q^{94} + 35 q^{96} + 14 q^{97} + q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 + 7 * q^4 + q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^11 + 7 * q^12 - 2 * q^13 - q^14 + 13 * q^16 - 6 * q^17 + q^18 - 4 * q^19 - 2 * q^21 - q^22 + 2 * q^23 + 12 * q^24 - q^26 + 2 * q^27 - 7 * q^28 + 2 * q^29 - 2 * q^31 + 35 * q^32 - 2 * q^33 + 18 * q^34 + 7 * q^36 - 2 * q^37 + 19 * q^38 - 2 * q^39 - 4 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 20 * q^46 - 12 * q^47 + 13 * q^48 + 2 * q^49 - 6 * q^51 - 7 * q^52 + 10 * q^53 + q^54 - 12 * q^56 - 4 * q^57 + 43 * q^58 + 20 * q^61 - 22 * q^62 - 2 * q^63 + 44 * q^64 - q^66 - 8 * q^67 + 2 * q^69 + 4 * q^71 + 12 * q^72 - 14 * q^73 - q^74 + 7 * q^76 + 2 * q^77 - q^78 - 4 * q^79 + 2 * q^81 + 40 * q^82 + 14 * q^83 - 7 * q^84 + 24 * q^86 + 2 * q^87 - 12 * q^88 + 2 * q^91 - 14 * q^92 - 2 * q^93 - 27 * q^94 + 35 * q^96 + 14 * q^97 + q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.79129	1.97374	0.986869	−	0.161521i	$-0.0516399\pi$
$2$	2.79129	1.97374	0.986869	+	0.161521i	$0.0516399\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.79129	2.89564
$5$	0	0
$5$	0	0
$6$	2.79129	1.13954
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	10.5826	3.74151
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	5.79129	1.67180
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−2.79129	−0.746003
$15$	0	0
$16$	17.9564	4.48911
$17$	1.58258	0.383831	0.191915	−	0.981411i	$-0.438530\pi$
$17$	1.58258	0.383831	0.191915	+	0.981411i	$0.438530\pi$
$18$	2.79129	0.657913
$19$	2.58258	0.592483	0.296242	−	0.955113i	$-0.404267\pi$
$19$	2.58258	0.592483	0.296242	+	0.955113i	$0.404267\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−2.79129	−0.595105
$23$	−3.58258	−0.747019	−0.373509	−	0.927626i	$-0.621846\pi$
$23$	−3.58258	−0.747019	−0.373509	+	0.927626i	$0.621846\pi$
$24$	10.5826	2.16016
$25$	0	0
$26$	−2.79129	−0.547417
$27$	1.00000	0.192450
$28$	−5.79129	−1.09445
$29$	10.1652	1.88762	0.943811	−	0.330487i	$-0.107213\pi$
$29$	10.1652	1.88762	0.943811	+	0.330487i	$0.107213\pi$
$30$	0	0
$31$	−5.58258	−1.00266	−0.501330	−	0.865256i	$-0.667156\pi$
$31$	−5.58258	−1.00266	−0.501330	+	0.865256i	$0.667156\pi$
$32$	28.9564	5.11882
$33$	−1.00000	−0.174078
$34$	4.41742	0.757582
$35$	0	0
$36$	5.79129	0.965215
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	7.20871	1.16941
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.16515	1.11901	0.559504	−	0.828827i	$-0.310992\pi$
$41$	7.16515	1.11901	0.559504	+	0.828827i	$0.310992\pi$
$42$	−2.79129	−0.430705
$43$	7.58258	1.15633	0.578166	−	0.815919i	$-0.303769\pi$
$43$	7.58258	1.15633	0.578166	+	0.815919i	$0.303769\pi$
$44$	−5.79129	−0.873069
$45$	0	0
$46$	−10.0000	−1.47442
$47$	−10.5826	−1.54363	−0.771814	−	0.635849i	$-0.780650\pi$
$47$	−10.5826	−1.54363	−0.771814	+	0.635849i	$0.780650\pi$
$48$	17.9564	2.59179
$49$	1.00000	0.142857
$50$	0	0
$51$	1.58258	0.221605
$52$	−5.79129	−0.803107
$53$	0.417424	0.0573376	0.0286688	−	0.999589i	$-0.490873\pi$
$53$	0.417424	0.0573376	0.0286688	+	0.999589i	$0.490873\pi$
$54$	2.79129	0.379846
$55$	0	0
$56$	−10.5826	−1.41416
$57$	2.58258	0.342071
$58$	28.3739	3.72567
$59$	−4.58258	−0.596601	−0.298300	−	0.954472i	$-0.596420\pi$
$59$	−4.58258	−0.596601	−0.298300	+	0.954472i	$0.596420\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−15.5826	−1.97899
$63$	−1.00000	−0.125988
$64$	44.9129	5.61411
$65$	0	0
$66$	−2.79129	−0.343584
$67$	0.582576	0.0711729	0.0355865	−	0.999367i	$-0.488670\pi$
$67$	0.582576	0.0711729	0.0355865	+	0.999367i	$0.488670\pi$
$68$	9.16515	1.11144
$69$	−3.58258	−0.431291
$70$	0	0
$71$	−7.16515	−0.850347	−0.425174	−	0.905112i	$-0.639787\pi$
$71$	−7.16515	−0.850347	−0.425174	+	0.905112i	$0.639787\pi$
$72$	10.5826	1.24717
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−2.79129	−0.324481
$75$	0	0
$76$	14.9564	1.71562
$77$	1.00000	0.113961
$78$	−2.79129	−0.316051
$79$	−11.1652	−1.25618	−0.628089	−	0.778142i	$-0.716163\pi$
$79$	−11.1652	−1.25618	−0.628089	+	0.778142i	$0.716163\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	20.0000	2.20863
$83$	2.41742	0.265347	0.132673	−	0.991160i	$-0.457644\pi$
$83$	2.41742	0.265347	0.132673	+	0.991160i	$0.457644\pi$
$84$	−5.79129	−0.631881
$85$	0	0
$86$	21.1652	2.28230
$87$	10.1652	1.08982
$88$	−10.5826	−1.12811
$89$	−9.16515	−0.971504	−0.485752	−	0.874097i	$-0.661454\pi$
$89$	−9.16515	−0.971504	−0.485752	+	0.874097i	$0.661454\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−20.7477	−2.16310
$93$	−5.58258	−0.578886
$94$	−29.5390	−3.04672
$95$	0	0
$96$	28.9564	2.95535
$97$	11.5826	1.17603	0.588016	−	0.808849i	$-0.299909\pi$
$97$	11.5826	1.17603	0.588016	+	0.808849i	$0.299909\pi$
$98$	2.79129	0.281963
$99$	−1.00000	−0.100504
$100$	0	0
$101$	2.41742	0.240543	0.120271	−	0.992741i	$-0.461624\pi$
$101$	2.41742	0.240543	0.120271	+	0.992741i	$0.461624\pi$
$102$	4.41742	0.437390
$103$	−17.1652	−1.69133	−0.845666	−	0.533712i	$-0.820797\pi$
$103$	−17.1652	−1.69133	−0.845666	+	0.533712i	$0.820797\pi$
$104$	−10.5826	−1.03771
$105$	0	0
$106$	1.16515	0.113170
$107$	3.41742	0.330375	0.165187	−	0.986262i	$-0.447177\pi$
$107$	3.41742	0.330375	0.165187	+	0.986262i	$0.447177\pi$
$108$	5.79129	0.557267
$109$	−5.58258	−0.534714	−0.267357	−	0.963598i	$-0.586150\pi$
$109$	−5.58258	−0.534714	−0.267357	+	0.963598i	$0.586150\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	−17.9564	−1.69672
$113$	9.16515	0.862185	0.431092	−	0.902308i	$-0.358128\pi$
$113$	9.16515	0.862185	0.431092	+	0.902308i	$0.358128\pi$
$114$	7.20871	0.675158
$115$	0	0
$116$	58.8693	5.46588
$117$	−1.00000	−0.0924500
$118$	−12.7913	−1.17753
$119$	−1.58258	−0.145074
$120$	0	0
$121$	1.00000	0.0909091
$122$	27.9129	2.52711
$123$	7.16515	0.646060
$124$	−32.3303	−2.90335
$125$	0	0
$126$	−2.79129	−0.248668
$127$	2.41742	0.214512	0.107256	−	0.994231i	$-0.465794\pi$
$127$	2.41742	0.214512	0.107256	+	0.994231i	$0.465794\pi$
$128$	67.4519	5.96196
$129$	7.58258	0.667609
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	−5.79129	−0.504067
$133$	−2.58258	−0.223938
$134$	1.62614	0.140477
$135$	0	0
$136$	16.7477	1.43611
$137$	2.41742	0.206534	0.103267	−	0.994654i	$-0.467070\pi$
$137$	2.41742	0.206534	0.103267	+	0.994654i	$0.467070\pi$
$138$	−10.0000	−0.851257
$139$	−7.16515	−0.607740	−0.303870	−	0.952713i	$-0.598279\pi$
$139$	−7.16515	−0.607740	−0.303870	+	0.952713i	$0.598279\pi$
$140$	0	0
$141$	−10.5826	−0.891214
$142$	−20.0000	−1.67836
$143$	1.00000	0.0836242
$144$	17.9564	1.49637
$145$	0	0
$146$	−19.5390	−1.61706
$147$	1.00000	0.0824786
$148$	−5.79129	−0.476041
$149$	−12.1652	−0.996608	−0.498304	−	0.867002i	$-0.666044\pi$
$149$	−12.1652	−0.996608	−0.498304	+	0.867002i	$0.666044\pi$
$150$	0	0
$151$	−5.58258	−0.454304	−0.227152	−	0.973859i	$-0.572941\pi$
$151$	−5.58258	−0.454304	−0.227152	+	0.973859i	$0.572941\pi$
$152$	27.3303	2.21678
$153$	1.58258	0.127944
$154$	2.79129	0.224928
$155$	0	0
$156$	−5.79129	−0.463674
$157$	−0.834849	−0.0666282	−0.0333141	−	0.999445i	$-0.510606\pi$
$157$	−0.834849	−0.0666282	−0.0333141	+	0.999445i	$0.510606\pi$
$158$	−31.1652	−2.47937
$159$	0.417424	0.0331039
$160$	0	0
$161$	3.58258	0.282347
$162$	2.79129	0.219304
$163$	0.582576	0.0456309	0.0228154	−	0.999740i	$-0.492737\pi$
$163$	0.582576	0.0456309	0.0228154	+	0.999740i	$0.492737\pi$
$164$	41.4955	3.24025
$165$	0	0
$166$	6.74773	0.523725
$167$	−22.7477	−1.76027	−0.880136	−	0.474722i	$-0.842549\pi$
$167$	−22.7477	−1.76027	−0.880136	+	0.474722i	$0.842549\pi$
$168$	−10.5826	−0.816463
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.58258	0.197494
$172$	43.9129	3.34833
$173$	−11.1652	−0.848871	−0.424435	−	0.905458i	$-0.639527\pi$
$173$	−11.1652	−0.848871	−0.424435	+	0.905458i	$0.639527\pi$
$174$	28.3739	2.15102
$175$	0	0
$176$	−17.9564	−1.35352
$177$	−4.58258	−0.344447
$178$	−25.5826	−1.91750
$179$	−22.3303	−1.66905	−0.834523	−	0.550974i	$-0.814256\pi$
$179$	−22.3303	−1.66905	−0.834523	+	0.550974i	$0.814256\pi$
$180$	0	0
$181$	3.58258	0.266291	0.133145	−	0.991097i	$-0.457492\pi$
$181$	3.58258	0.266291	0.133145	+	0.991097i	$0.457492\pi$
$182$	2.79129	0.206904
$183$	10.0000	0.739221
$184$	−37.9129	−2.79497
$185$	0	0
$186$	−15.5826	−1.14257
$187$	−1.58258	−0.115729
$188$	−61.2867	−4.46980
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	2.41742	0.174919	0.0874593	−	0.996168i	$-0.472125\pi$
$191$	2.41742	0.174919	0.0874593	+	0.996168i	$0.472125\pi$
$192$	44.9129	3.24131
$193$	11.5826	0.833732	0.416866	−	0.908968i	$-0.363128\pi$
$193$	11.5826	0.833732	0.416866	+	0.908968i	$0.363128\pi$
$194$	32.3303	2.32118
$195$	0	0
$196$	5.79129	0.413663
$197$	13.1652	0.937978	0.468989	−	0.883204i	$-0.344618\pi$
$197$	13.1652	0.937978	0.468989	+	0.883204i	$0.344618\pi$
$198$	−2.79129	−0.198368
$199$	−0.417424	−0.0295904	−0.0147952	−	0.999891i	$-0.504710\pi$
$199$	−0.417424	−0.0295904	−0.0147952	+	0.999891i	$0.504710\pi$
$200$	0	0
$201$	0.582576	0.0410917
$202$	6.74773	0.474768
$203$	−10.1652	−0.713454
$204$	9.16515	0.641689
$205$	0	0
$206$	−47.9129	−3.33825
$207$	−3.58258	−0.249006
$208$	−17.9564	−1.24506
$209$	−2.58258	−0.178640
$210$	0	0
$211$	5.16515	0.355584	0.177792	−	0.984068i	$-0.443105\pi$
$211$	5.16515	0.355584	0.177792	+	0.984068i	$0.443105\pi$
$212$	2.41742	0.166029
$213$	−7.16515	−0.490948
$214$	9.53901	0.652074
$215$	0	0
$216$	10.5826	0.720053
$217$	5.58258	0.378970
$218$	−15.5826	−1.05539
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−1.58258	−0.106456
$222$	−2.79129	−0.187339
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	−28.9564	−1.93473
$225$	0	0
$226$	25.5826	1.70173
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	14.9564	0.990514
$229$	−26.7477	−1.76754	−0.883770	−	0.467922i	$-0.845003\pi$
$229$	−26.7477	−1.76754	−0.883770	+	0.467922i	$0.845003\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	107.573	7.06255
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−2.79129	−0.182472
$235$	0	0
$236$	−26.5390	−1.72754
$237$	−11.1652	−0.725255
$238$	−4.41742	−0.286339
$239$	7.41742	0.479793	0.239897	−	0.970798i	$-0.422886\pi$
$239$	7.41742	0.479793	0.239897	+	0.970798i	$0.422886\pi$
$240$	0	0
$241$	8.16515	0.525964	0.262982	−	0.964801i	$-0.415294\pi$
$241$	8.16515	0.525964	0.262982	+	0.964801i	$0.415294\pi$
$242$	2.79129	0.179431
$243$	1.00000	0.0641500
$244$	57.9129	3.70749
$245$	0	0
$246$	20.0000	1.27515
$247$	−2.58258	−0.164325
$248$	−59.0780	−3.75146
$249$	2.41742	0.153198
$250$	0	0
$251$	16.5826	1.04668	0.523341	−	0.852123i	$-0.324685\pi$
$251$	16.5826	1.04668	0.523341	+	0.852123i	$0.324685\pi$
$252$	−5.79129	−0.364817
$253$	3.58258	0.225235
$254$	6.74773	0.423390
$255$	0	0
$256$	98.4519	6.15324
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	21.1652	1.31768
$259$	1.00000	0.0621370
$260$	0	0
$261$	10.1652	0.629207
$262$	−44.6606	−2.75914
$263$	−22.9129	−1.41287	−0.706434	−	0.707779i	$-0.749697\pi$
$263$	−22.9129	−1.41287	−0.706434	+	0.707779i	$0.749697\pi$
$264$	−10.5826	−0.651313
$265$	0	0
$266$	−7.20871	−0.441995
$267$	−9.16515	−0.560898
$268$	3.37386	0.206092
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	14.5826	0.885828	0.442914	−	0.896564i	$-0.353945\pi$
$271$	14.5826	0.885828	0.442914	+	0.896564i	$0.353945\pi$
$272$	28.4174	1.72306
$273$	1.00000	0.0605228
$274$	6.74773	0.407645
$275$	0	0
$276$	−20.7477	−1.24887
$277$	0.834849	0.0501612	0.0250806	−	0.999685i	$-0.492016\pi$
$277$	0.834849	0.0501612	0.0250806	+	0.999685i	$0.492016\pi$
$278$	−20.0000	−1.19952
$279$	−5.58258	−0.334220
$280$	0	0
$281$	9.33030	0.556599	0.278300	−	0.960494i	$-0.410229\pi$
$281$	9.33030	0.556599	0.278300	+	0.960494i	$0.410229\pi$
$282$	−29.5390	−1.75902
$283$	−0.252273	−0.0149961	−0.00749803	−	0.999972i	$-0.502387\pi$
$283$	−0.252273	−0.0149961	−0.00749803	+	0.999972i	$0.502387\pi$
$284$	−41.4955	−2.46230
$285$	0	0
$286$	2.79129	0.165052
$287$	−7.16515	−0.422946
$288$	28.9564	1.70627
$289$	−14.4955	−0.852674
$290$	0	0
$291$	11.5826	0.678983
$292$	−40.5390	−2.37237
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	2.79129	0.162791
$295$	0	0
$296$	−10.5826	−0.615100
$297$	−1.00000	−0.0580259
$298$	−33.9564	−1.96704
$299$	3.58258	0.207186
$300$	0	0
$301$	−7.58258	−0.437052
$302$	−15.5826	−0.896676
$303$	2.41742	0.138877
$304$	46.3739	2.65972
$305$	0	0
$306$	4.41742	0.252527
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	5.79129	0.329989
$309$	−17.1652	−0.976491
$310$	0	0
$311$	−22.3303	−1.26624	−0.633118	−	0.774056i	$-0.718225\pi$
$311$	−22.3303	−1.26624	−0.633118	+	0.774056i	$0.718225\pi$
$312$	−10.5826	−0.599120
$313$	−10.4174	−0.588828	−0.294414	−	0.955678i	$-0.595124\pi$
$313$	−10.4174	−0.588828	−0.294414	+	0.955678i	$0.595124\pi$
$314$	−2.33030	−0.131507
$315$	0	0
$316$	−64.6606	−3.63744
$317$	31.5826	1.77385	0.886927	−	0.461909i	$-0.152835\pi$
$317$	31.5826	1.77385	0.886927	+	0.461909i	$0.152835\pi$
$318$	1.16515	0.0653384
$319$	−10.1652	−0.569139
$320$	0	0
$321$	3.41742	0.190742
$322$	10.0000	0.557278
$323$	4.08712	0.227414
$324$	5.79129	0.321738
$325$	0	0
$326$	1.62614	0.0900634
$327$	−5.58258	−0.308717
$328$	75.8258	4.18678
$329$	10.5826	0.583436
$330$	0	0
$331$	15.1652	0.833552	0.416776	−	0.909009i	$-0.363160\pi$
$331$	15.1652	0.833552	0.416776	+	0.909009i	$0.363160\pi$
$332$	14.0000	0.768350
$333$	−1.00000	−0.0547997
$334$	−63.4955	−3.47432
$335$	0	0
$336$	−17.9564	−0.979604
$337$	−8.41742	−0.458526	−0.229263	−	0.973364i	$-0.573632\pi$
$337$	−8.41742	−0.458526	−0.229263	+	0.973364i	$0.573632\pi$
$338$	−33.4955	−1.82191
$339$	9.16515	0.497783
$340$	0	0
$341$	5.58258	0.302313
$342$	7.20871	0.389803
$343$	−1.00000	−0.0539949
$344$	80.2432	4.32642
$345$	0	0
$346$	−31.1652	−1.67545
$347$	10.3303	0.554560	0.277280	−	0.960789i	$-0.410567\pi$
$347$	10.3303	0.554560	0.277280	+	0.960789i	$0.410567\pi$
$348$	58.8693	3.15573
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−28.9564	−1.54338
$353$	−5.83485	−0.310558	−0.155279	−	0.987871i	$-0.549628\pi$
$353$	−5.83485	−0.310558	−0.155279	+	0.987871i	$0.549628\pi$
$354$	−12.7913	−0.679849
$355$	0	0
$356$	−53.0780	−2.81313
$357$	−1.58258	−0.0837588
$358$	−62.3303	−3.29426
$359$	−27.1652	−1.43372	−0.716861	−	0.697216i	$-0.754422\pi$
$359$	−27.1652	−1.43372	−0.716861	+	0.697216i	$0.754422\pi$
$360$	0	0
$361$	−12.3303	−0.648963
$362$	10.0000	0.525588
$363$	1.00000	0.0524864
$364$	5.79129	0.303546
$365$	0	0
$366$	27.9129	1.45903
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−64.3303	−3.35345
$369$	7.16515	0.373003
$370$	0	0
$371$	−0.417424	−0.0216716
$372$	−32.3303	−1.67625
$373$	7.25227	0.375508	0.187754	−	0.982216i	$-0.439879\pi$
$373$	7.25227	0.375508	0.187754	+	0.982216i	$0.439879\pi$
$374$	−4.41742	−0.228420
$375$	0	0
$376$	−111.991	−5.77549
$377$	−10.1652	−0.523532
$378$	−2.79129	−0.143568
$379$	−3.41742	−0.175541	−0.0877706	−	0.996141i	$-0.527974\pi$
$379$	−3.41742	−0.175541	−0.0877706	+	0.996141i	$0.527974\pi$
$380$	0	0
$381$	2.41742	0.123848
$382$	6.74773	0.345244
$383$	26.3303	1.34542	0.672708	−	0.739908i	$-0.265131\pi$
$383$	26.3303	1.34542	0.672708	+	0.739908i	$0.265131\pi$
$384$	67.4519	3.44214
$385$	0	0
$386$	32.3303	1.64557
$387$	7.58258	0.385444
$388$	67.0780	3.40537
$389$	10.3303	0.523767	0.261884	−	0.965099i	$-0.415656\pi$
$389$	10.3303	0.523767	0.261884	+	0.965099i	$0.415656\pi$
$390$	0	0
$391$	−5.66970	−0.286729
$392$	10.5826	0.534501
$393$	−16.0000	−0.807093
$394$	36.7477	1.85132
$395$	0	0
$396$	−5.79129	−0.291023
$397$	22.4174	1.12510	0.562549	−	0.826764i	$-0.309821\pi$
$397$	22.4174	1.12510	0.562549	+	0.826764i	$0.309821\pi$
$398$	−1.16515	−0.0584038
$399$	−2.58258	−0.129290
$400$	0	0
$401$	−13.9129	−0.694776	−0.347388	−	0.937721i	$-0.612931\pi$
$401$	−13.9129	−0.694776	−0.347388	+	0.937721i	$0.612931\pi$
$402$	1.62614	0.0811043
$403$	5.58258	0.278088
$404$	14.0000	0.696526
$405$	0	0
$406$	−28.3739	−1.40817
$407$	1.00000	0.0495682
$408$	16.7477	0.829136
$409$	−28.3303	−1.40084	−0.700422	−	0.713729i	$-0.747005\pi$
$409$	−28.3303	−1.40084	−0.700422	+	0.713729i	$0.747005\pi$
$410$	0	0
$411$	2.41742	0.119243
$412$	−99.4083	−4.89750
$413$	4.58258	0.225494
$414$	−10.0000	−0.491473
$415$	0	0
$416$	−28.9564	−1.41971
$417$	−7.16515	−0.350879
$418$	−7.20871	−0.352590
$419$	−6.58258	−0.321580	−0.160790	−	0.986989i	$-0.551404\pi$
$419$	−6.58258	−0.321580	−0.160790	+	0.986989i	$0.551404\pi$
$420$	0	0
$421$	39.6606	1.93294	0.966470	−	0.256780i	$-0.0826617\pi$
$421$	39.6606	1.93294	0.966470	+	0.256780i	$0.0826617\pi$
$422$	14.4174	0.701829
$423$	−10.5826	−0.514542
$424$	4.41742	0.214529
$425$	0	0
$426$	−20.0000	−0.969003
$427$	−10.0000	−0.483934
$428$	19.7913	0.956648
$429$	1.00000	0.0482805
$430$	0	0
$431$	−9.74773	−0.469531	−0.234766	−	0.972052i	$-0.575432\pi$
$431$	−9.74773	−0.469531	−0.234766	+	0.972052i	$0.575432\pi$
$432$	17.9564	0.863930
$433$	−7.16515	−0.344335	−0.172168	−	0.985068i	$-0.555077\pi$
$433$	−7.16515	−0.344335	−0.172168	+	0.985068i	$0.555077\pi$
$434$	15.5826	0.747988
$435$	0	0
$436$	−32.3303	−1.54834
$437$	−9.25227	−0.442596
$438$	−19.5390	−0.933610
$439$	26.5826	1.26872	0.634359	−	0.773039i	$-0.281264\pi$
$439$	26.5826	1.26872	0.634359	+	0.773039i	$0.281264\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−4.41742	−0.210115
$443$	4.83485	0.229711	0.114855	−	0.993382i	$-0.463360\pi$
$443$	4.83485	0.229711	0.114855	+	0.993382i	$0.463360\pi$
$444$	−5.79129	−0.274842
$445$	0	0
$446$	−16.7477	−0.793028
$447$	−12.1652	−0.575392
$448$	−44.9129	−2.12193
$449$	18.3303	0.865060	0.432530	−	0.901619i	$-0.357621\pi$
$449$	18.3303	0.865060	0.432530	+	0.901619i	$0.357621\pi$
$450$	0	0
$451$	−7.16515	−0.337394
$452$	53.0780	2.49658
$453$	−5.58258	−0.262292
$454$	61.4083	2.88204
$455$	0	0
$456$	27.3303	1.27986
$457$	−25.9129	−1.21215	−0.606077	−	0.795406i	$-0.707258\pi$
$457$	−25.9129	−1.21215	−0.606077	+	0.795406i	$0.707258\pi$
$458$	−74.6606	−3.48866
$459$	1.58258	0.0738683
$460$	0	0
$461$	−18.3303	−0.853727	−0.426864	−	0.904316i	$-0.640382\pi$
$461$	−18.3303	−0.853727	−0.426864	+	0.904316i	$0.640382\pi$
$462$	2.79129	0.129862
$463$	0.582576	0.0270746	0.0135373	−	0.999908i	$-0.495691\pi$
$463$	0.582576	0.0270746	0.0135373	+	0.999908i	$0.495691\pi$
$464$	182.530	8.47374
$465$	0	0
$466$	39.0780	1.81025
$467$	29.4174	1.36128	0.680638	−	0.732620i	$-0.261703\pi$
$467$	29.4174	1.36128	0.680638	+	0.732620i	$0.261703\pi$
$468$	−5.79129	−0.267702
$469$	−0.582576	−0.0269008
$470$	0	0
$471$	−0.834849	−0.0384678
$472$	−48.4955	−2.23218
$473$	−7.58258	−0.348647
$474$	−31.1652	−1.43146
$475$	0	0
$476$	−9.16515	−0.420084
$477$	0.417424	0.0191125
$478$	20.7042	0.946987
$479$	−6.41742	−0.293220	−0.146610	−	0.989194i	$-0.546836\pi$
$479$	−6.41742	−0.293220	−0.146610	+	0.989194i	$0.546836\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	22.7913	1.03811
$483$	3.58258	0.163013
$484$	5.79129	0.263240
$485$	0	0
$486$	2.79129	0.126615
$487$	26.3303	1.19314	0.596570	−	0.802561i	$-0.296530\pi$
$487$	26.3303	1.19314	0.596570	+	0.802561i	$0.296530\pi$
$488$	105.826	4.79051
$489$	0.582576	0.0263450
$490$	0	0
$491$	22.9129	1.03404	0.517022	−	0.855972i	$-0.327041\pi$
$491$	22.9129	1.03404	0.517022	+	0.855972i	$0.327041\pi$
$492$	41.4955	1.87076
$493$	16.0871	0.724528
$494$	−7.20871	−0.324335
$495$	0	0
$496$	−100.243	−4.50105
$497$	7.16515	0.321401
$498$	6.74773	0.302373
$499$	14.2523	0.638019	0.319010	−	0.947751i	$-0.396650\pi$
$499$	14.2523	0.638019	0.319010	+	0.947751i	$0.396650\pi$
$500$	0	0
$501$	−22.7477	−1.01629
$502$	46.2867	2.06588
$503$	−26.7477	−1.19262	−0.596311	−	0.802753i	$-0.703368\pi$
$503$	−26.7477	−1.19262	−0.596311	+	0.802753i	$0.703368\pi$
$504$	−10.5826	−0.471385
$505$	0	0
$506$	10.0000	0.444554
$507$	−12.0000	−0.532939
$508$	14.0000	0.621150
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	139.904	6.18293
$513$	2.58258	0.114024
$514$	−53.0345	−2.33925
$515$	0	0
$516$	43.9129	1.93316
$517$	10.5826	0.465421
$518$	2.79129	0.122642
$519$	−11.1652	−0.490096
$520$	0	0
$521$	34.1652	1.49680	0.748401	−	0.663246i	$-0.230822\pi$
$521$	34.1652	1.49680	0.748401	+	0.663246i	$0.230822\pi$
$522$	28.3739	1.24189
$523$	−24.5826	−1.07492	−0.537460	−	0.843289i	$-0.680616\pi$
$523$	−24.5826	−1.07492	−0.537460	+	0.843289i	$0.680616\pi$
$524$	−92.6606	−4.04790
$525$	0	0
$526$	−63.9564	−2.78863
$527$	−8.83485	−0.384852
$528$	−17.9564	−0.781454
$529$	−10.1652	−0.441963
$530$	0	0
$531$	−4.58258	−0.198867
$532$	−14.9564	−0.648444
$533$	−7.16515	−0.310357
$534$	−25.5826	−1.10707
$535$	0	0
$536$	6.16515	0.266294
$537$	−22.3303	−0.963624
$538$	27.9129	1.20341
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−18.3303	−0.788081	−0.394041	−	0.919093i	$-0.628923\pi$
$541$	−18.3303	−0.788081	−0.394041	+	0.919093i	$0.628923\pi$
$542$	40.7042	1.74839
$543$	3.58258	0.153743
$544$	45.8258	1.96476
$545$	0	0
$546$	2.79129	0.119456
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	14.0000	0.598050
$549$	10.0000	0.426790
$550$	0	0
$551$	26.2523	1.11838
$552$	−37.9129	−1.61368
$553$	11.1652	0.474791
$554$	2.33030	0.0990051
$555$	0	0
$556$	−41.4955	−1.75980
$557$	27.3303	1.15802	0.579011	−	0.815320i	$-0.303439\pi$
$557$	27.3303	1.15802	0.579011	+	0.815320i	$0.303439\pi$
$558$	−15.5826	−0.659663
$559$	−7.58258	−0.320709
$560$	0	0
$561$	−1.58258	−0.0668164
$562$	26.0436	1.09858
$563$	28.4174	1.19765	0.598826	−	0.800879i	$-0.295634\pi$
$563$	28.4174	1.19765	0.598826	+	0.800879i	$0.295634\pi$
$564$	−61.2867	−2.58064
$565$	0	0
$566$	−0.704166	−0.0295983
$567$	−1.00000	−0.0419961
$568$	−75.8258	−3.18158
$569$	46.6606	1.95611	0.978057	−	0.208337i	$-0.0668050\pi$
$569$	46.6606	1.95611	0.978057	+	0.208337i	$0.0668050\pi$
$570$	0	0
$571$	47.1652	1.97380	0.986900	−	0.161333i	$-0.0515792\pi$
$571$	47.1652	1.97380	0.986900	+	0.161333i	$0.0515792\pi$
$572$	5.79129	0.242146
$573$	2.41742	0.100989
$574$	−20.0000	−0.834784
$575$	0	0
$576$	44.9129	1.87137
$577$	−23.9129	−0.995506	−0.497753	−	0.867319i	$-0.665841\pi$
$577$	−23.9129	−0.995506	−0.497753	+	0.867319i	$0.665841\pi$
$578$	−40.4610	−1.68296
$579$	11.5826	0.481355
$580$	0	0
$581$	−2.41742	−0.100292
$582$	32.3303	1.34013
$583$	−0.417424	−0.0172879
$584$	−74.0780	−3.06537
$585$	0	0
$586$	0	0
$587$	−10.2523	−0.423157	−0.211578	−	0.977361i	$-0.567860\pi$
$587$	−10.2523	−0.423157	−0.211578	+	0.977361i	$0.567860\pi$
$588$	5.79129	0.238829
$589$	−14.4174	−0.594060
$590$	0	0
$591$	13.1652	0.541542
$592$	−17.9564	−0.738005
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	−2.79129	−0.114528
$595$	0	0
$596$	−70.4519	−2.88582
$597$	−0.417424	−0.0170840
$598$	10.0000	0.408930
$599$	−11.1652	−0.456196	−0.228098	−	0.973638i	$-0.573251\pi$
$599$	−11.1652	−0.456196	−0.228098	+	0.973638i	$0.573251\pi$
$600$	0	0
$601$	−30.4955	−1.24394	−0.621968	−	0.783043i	$-0.713667\pi$
$601$	−30.4955	−1.24394	−0.621968	+	0.783043i	$0.713667\pi$
$602$	−21.1652	−0.862627
$603$	0.582576	0.0237243
$604$	−32.3303	−1.31550
$605$	0	0
$606$	6.74773	0.274108
$607$	−5.74773	−0.233293	−0.116647	−	0.993173i	$-0.537214\pi$
$607$	−5.74773	−0.233293	−0.116647	+	0.993173i	$0.537214\pi$
$608$	74.7822	3.03282
$609$	−10.1652	−0.411913
$610$	0	0
$611$	10.5826	0.428125
$612$	9.16515	0.370479
$613$	−0.747727	−0.0302004	−0.0151002	−	0.999886i	$-0.504807\pi$
$613$	−0.747727	−0.0302004	−0.0151002	+	0.999886i	$0.504807\pi$
$614$	0	0
$615$	0	0
$616$	10.5826	0.426384
$617$	−21.1652	−0.852077	−0.426038	−	0.904705i	$-0.640091\pi$
$617$	−21.1652	−0.852077	−0.426038	+	0.904705i	$0.640091\pi$
$618$	−47.9129	−1.92734
$619$	35.0780	1.40991	0.704953	−	0.709254i	$-0.250968\pi$
$619$	35.0780	1.40991	0.704953	+	0.709254i	$0.250968\pi$
$620$	0	0
$621$	−3.58258	−0.143764
$622$	−62.3303	−2.49922
$623$	9.16515	0.367194
$624$	−17.9564	−0.718833
$625$	0	0
$626$	−29.0780	−1.16219
$627$	−2.58258	−0.103138
$628$	−4.83485	−0.192931
$629$	−1.58258	−0.0631014
$630$	0	0
$631$	4.83485	0.192472	0.0962361	−	0.995359i	$-0.469320\pi$
$631$	4.83485	0.192472	0.0962361	+	0.995359i	$0.469320\pi$
$632$	−118.156	−4.70000
$633$	5.16515	0.205296
$634$	88.1561	3.50112
$635$	0	0
$636$	2.41742	0.0958571
$637$	−1.00000	−0.0396214
$638$	−28.3739	−1.12333
$639$	−7.16515	−0.283449
$640$	0	0
$641$	34.4174	1.35941	0.679703	−	0.733487i	$-0.262109\pi$
$641$	34.4174	1.35941	0.679703	+	0.733487i	$0.262109\pi$
$642$	9.53901	0.376475
$643$	44.2432	1.74478	0.872390	−	0.488810i	$-0.162569\pi$
$643$	44.2432	1.74478	0.872390	+	0.488810i	$0.162569\pi$
$644$	20.7477	0.817575
$645$	0	0
$646$	11.4083	0.448855
$647$	−34.9129	−1.37257	−0.686283	−	0.727334i	$-0.740759\pi$
$647$	−34.9129	−1.37257	−0.686283	+	0.727334i	$0.740759\pi$
$648$	10.5826	0.415723
$649$	4.58258	0.179882
$650$	0	0
$651$	5.58258	0.218798
$652$	3.37386	0.132131
$653$	−6.33030	−0.247724	−0.123862	−	0.992299i	$-0.539528\pi$
$653$	−6.33030	−0.247724	−0.123862	+	0.992299i	$0.539528\pi$
$654$	−15.5826	−0.609327
$655$	0	0
$656$	128.661	5.02335
$657$	−7.00000	−0.273096
$658$	29.5390	1.15155
$659$	−19.4174	−0.756395	−0.378198	−	0.925725i	$-0.623456\pi$
$659$	−19.4174	−0.756395	−0.378198	+	0.925725i	$0.623456\pi$
$660$	0	0
$661$	25.0780	0.975422	0.487711	−	0.873005i	$-0.337832\pi$
$661$	25.0780	0.975422	0.487711	+	0.873005i	$0.337832\pi$
$662$	42.3303	1.64521
$663$	−1.58258	−0.0614621
$664$	25.5826	0.992796
$665$	0	0
$666$	−2.79129	−0.108160
$667$	−36.4174	−1.41009
$668$	−131.739	−5.09712
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−10.0000	−0.386046
$672$	−28.9564	−1.11702
$673$	−38.7477	−1.49362	−0.746808	−	0.665040i	$-0.768414\pi$
$673$	−38.7477	−1.49362	−0.746808	+	0.665040i	$0.768414\pi$
$674$	−23.4955	−0.905011
$675$	0	0
$676$	−69.4955	−2.67290
$677$	26.8348	1.03135	0.515674	−	0.856785i	$-0.327542\pi$
$677$	26.8348	1.03135	0.515674	+	0.856785i	$0.327542\pi$
$678$	25.5826	0.982493
$679$	−11.5826	−0.444498
$680$	0	0
$681$	22.0000	0.843042
$682$	15.5826	0.596688
$683$	−31.0780	−1.18917	−0.594584	−	0.804034i	$-0.702683\pi$
$683$	−31.0780	−1.18917	−0.594584	+	0.804034i	$0.702683\pi$
$684$	14.9564	0.571874
$685$	0	0
$686$	−2.79129	−0.106572
$687$	−26.7477	−1.02049
$688$	136.156	5.19090
$689$	−0.417424	−0.0159026
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	−64.6606	−2.45803
$693$	1.00000	0.0379869
$694$	28.8348	1.09456
$695$	0	0
$696$	107.573	4.07756
$697$	11.3394	0.429510
$698$	−41.8693	−1.58478
$699$	14.0000	0.529529
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	−2.79129	−0.105350
$703$	−2.58258	−0.0974037
$704$	−44.9129	−1.69272
$705$	0	0
$706$	−16.2867	−0.612960
$707$	−2.41742	−0.0909166
$708$	−26.5390	−0.997397
$709$	−45.6606	−1.71482	−0.857410	−	0.514634i	$-0.827928\pi$
$709$	−45.6606	−1.71482	−0.857410	+	0.514634i	$0.827928\pi$
$710$	0	0
$711$	−11.1652	−0.418726
$712$	−96.9909	−3.63489
$713$	20.0000	0.749006
$714$	−4.41742	−0.165318
$715$	0	0
$716$	−129.321	−4.83296
$717$	7.41742	0.277009
$718$	−75.8258	−2.82979
$719$	50.0780	1.86760	0.933798	−	0.357801i	$-0.116474\pi$
$719$	50.0780	1.86760	0.933798	+	0.357801i	$0.116474\pi$
$720$	0	0
$721$	17.1652	0.639264
$722$	−34.4174	−1.28088
$723$	8.16515	0.303665
$724$	20.7477	0.771083
$725$	0	0
$726$	2.79129	0.103594
$727$	−29.9129	−1.10941	−0.554704	−	0.832048i	$-0.687168\pi$
$727$	−29.9129	−1.10941	−0.554704	+	0.832048i	$0.687168\pi$
$728$	10.5826	0.392216
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	57.9129	2.14052
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−61.4083	−2.26662
$735$	0	0
$736$	−103.739	−3.82386
$737$	−0.582576	−0.0214595
$738$	20.0000	0.736210
$739$	13.9129	0.511794	0.255897	−	0.966704i	$-0.417629\pi$
$739$	13.9129	0.511794	0.255897	+	0.966704i	$0.417629\pi$
$740$	0	0
$741$	−2.58258	−0.0948733
$742$	−1.16515	−0.0427741
$743$	−29.2432	−1.07283	−0.536414	−	0.843955i	$-0.680221\pi$
$743$	−29.2432	−1.07283	−0.536414	+	0.843955i	$0.680221\pi$
$744$	−59.0780	−2.16591
$745$	0	0
$746$	20.2432	0.741156
$747$	2.41742	0.0884489
$748$	−9.16515	−0.335111
$749$	−3.41742	−0.124870
$750$	0	0
$751$	36.9129	1.34697	0.673485	−	0.739201i	$-0.264797\pi$
$751$	36.9129	1.34697	0.673485	+	0.739201i	$0.264797\pi$
$752$	−190.025	−6.92951
$753$	16.5826	0.604303
$754$	−28.3739	−1.03332
$755$	0	0
$756$	−5.79129	−0.210627
$757$	9.33030	0.339116	0.169558	−	0.985520i	$-0.445766\pi$
$757$	9.33030	0.339116	0.169558	+	0.985520i	$0.445766\pi$
$758$	−9.53901	−0.346473
$759$	3.58258	0.130039
$760$	0	0
$761$	5.66970	0.205526	0.102763	−	0.994706i	$-0.467232\pi$
$761$	5.66970	0.205526	0.102763	+	0.994706i	$0.467232\pi$
$762$	6.74773	0.244444
$763$	5.58258	0.202103
$764$	14.0000	0.506502
$765$	0	0
$766$	73.4955	2.65550
$767$	4.58258	0.165467
$768$	98.4519	3.55258
$769$	−48.4955	−1.74879	−0.874395	−	0.485214i	$-0.838742\pi$
$769$	−48.4955	−1.74879	−0.874395	+	0.485214i	$0.838742\pi$
$770$	0	0
$771$	−19.0000	−0.684268
$772$	67.0780	2.41419
$773$	12.1652	0.437550	0.218775	−	0.975775i	$-0.429794\pi$
$773$	12.1652	0.437550	0.218775	+	0.975775i	$0.429794\pi$
$774$	21.1652	0.760766
$775$	0	0
$776$	122.573	4.40013
$777$	1.00000	0.0358748
$778$	28.8348	1.03378
$779$	18.5045	0.662994
$780$	0	0
$781$	7.16515	0.256389
$782$	−15.8258	−0.565928
$783$	10.1652	0.363273
$784$	17.9564	0.641301
$785$	0	0
$786$	−44.6606	−1.59299
$787$	−29.4174	−1.04862	−0.524309	−	0.851528i	$-0.675676\pi$
$787$	−29.4174	−1.04862	−0.524309	+	0.851528i	$0.675676\pi$
$788$	76.2432	2.71605
$789$	−22.9129	−0.815720
$790$	0	0
$791$	−9.16515	−0.325875
$792$	−10.5826	−0.376035
$793$	−10.0000	−0.355110
$794$	62.5735	2.22065
$795$	0	0
$796$	−2.41742	−0.0856833
$797$	−2.49545	−0.0883935	−0.0441968	−	0.999023i	$-0.514073\pi$
$797$	−2.49545	−0.0883935	−0.0441968	+	0.999023i	$0.514073\pi$
$798$	−7.20871	−0.255186
$799$	−16.7477	−0.592492
$800$	0	0
$801$	−9.16515	−0.323835
$802$	−38.8348	−1.37131
$803$	7.00000	0.247025
$804$	3.37386	0.118987
$805$	0	0
$806$	15.5826	0.548873
$807$	10.0000	0.352017
$808$	25.5826	0.899992
$809$	−27.3303	−0.960882	−0.480441	−	0.877027i	$-0.659523\pi$
$809$	−27.3303	−0.960882	−0.480441	+	0.877027i	$0.659523\pi$
$810$	0	0
$811$	−29.7477	−1.04458	−0.522292	−	0.852767i	$-0.674923\pi$
$811$	−29.7477	−1.04458	−0.522292	+	0.852767i	$0.674923\pi$
$812$	−58.8693	−2.06591
$813$	14.5826	0.511433
$814$	2.79129	0.0978346
$815$	0	0
$816$	28.4174	0.994809
$817$	19.5826	0.685108
$818$	−79.0780	−2.76490
$819$	1.00000	0.0349428
$820$	0	0
$821$	−47.0000	−1.64031	−0.820156	−	0.572140i	$-0.806113\pi$
$821$	−47.0000	−1.64031	−0.820156	+	0.572140i	$0.806113\pi$
$822$	6.74773	0.235354
$823$	21.4174	0.746564	0.373282	−	0.927718i	$-0.378232\pi$
$823$	21.4174	0.746564	0.373282	+	0.927718i	$0.378232\pi$
$824$	−181.652	−6.32813
$825$	0	0
$826$	12.7913	0.445066
$827$	36.9129	1.28359	0.641793	−	0.766878i	$-0.278191\pi$
$827$	36.9129	1.28359	0.641793	+	0.766878i	$0.278191\pi$
$828$	−20.7477	−0.721033
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	0.834849	0.0289606
$832$	−44.9129	−1.55707
$833$	1.58258	0.0548330
$834$	−20.0000	−0.692543
$835$	0	0
$836$	−14.9564	−0.517279
$837$	−5.58258	−0.192962
$838$	−18.3739	−0.634715
$839$	52.9129	1.82676	0.913378	−	0.407113i	$-0.133465\pi$
$839$	52.9129	1.82676	0.913378	+	0.407113i	$0.133465\pi$
$840$	0	0
$841$	74.3303	2.56311
$842$	110.704	3.81512
$843$	9.33030	0.321353
$844$	29.9129	1.02964
$845$	0	0
$846$	−29.5390	−1.01557
$847$	−1.00000	−0.0343604
$848$	7.49545	0.257395
$849$	−0.252273	−0.00865798
$850$	0	0
$851$	3.58258	0.122809
$852$	−41.4955	−1.42161
$853$	1.16515	0.0398940	0.0199470	−	0.999801i	$-0.493650\pi$
$853$	1.16515	0.0398940	0.0199470	+	0.999801i	$0.493650\pi$
$854$	−27.9129	−0.955159
$855$	0	0
$856$	36.1652	1.23610
$857$	44.3303	1.51429	0.757147	−	0.653244i	$-0.226593\pi$
$857$	44.3303	1.51429	0.757147	+	0.653244i	$0.226593\pi$
$858$	2.79129	0.0952930
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	−7.16515	−0.244188
$862$	−27.2087	−0.926732
$863$	−23.5826	−0.802760	−0.401380	−	0.915912i	$-0.631469\pi$
$863$	−23.5826	−0.802760	−0.401380	+	0.915912i	$0.631469\pi$
$864$	28.9564	0.985118
$865$	0	0
$866$	−20.0000	−0.679628
$867$	−14.4955	−0.492291
$868$	32.3303	1.09736
$869$	11.1652	0.378752
$870$	0	0
$871$	−0.582576	−0.0197398
$872$	−59.0780	−2.00063
$873$	11.5826	0.392011
$874$	−25.8258	−0.873569
$875$	0	0
$876$	−40.5390	−1.36969
$877$	45.4955	1.53627	0.768136	−	0.640287i	$-0.221184\pi$
$877$	45.4955	1.53627	0.768136	+	0.640287i	$0.221184\pi$
$878$	74.1996	2.50412
$879$	0	0
$880$	0	0
$881$	35.6606	1.20144	0.600718	−	0.799461i	$-0.294881\pi$
$881$	35.6606	1.20144	0.600718	+	0.799461i	$0.294881\pi$
$882$	2.79129	0.0939876
$883$	28.2523	0.950765	0.475382	−	0.879779i	$-0.342310\pi$
$883$	28.2523	0.950765	0.475382	+	0.879779i	$0.342310\pi$
$884$	−9.16515	−0.308257
$885$	0	0
$886$	13.4955	0.453389
$887$	−11.2523	−0.377814	−0.188907	−	0.981995i	$-0.560495\pi$
$887$	−11.2523	−0.377814	−0.188907	+	0.981995i	$0.560495\pi$
$888$	−10.5826	−0.355128
$889$	−2.41742	−0.0810778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−34.7477	−1.16344
$893$	−27.3303	−0.914574
$894$	−33.9564	−1.13567
$895$	0	0
$896$	−67.4519	−2.25341
$897$	3.58258	0.119619
$898$	51.1652	1.70740
$899$	−56.7477	−1.89264
$900$	0	0
$901$	0.660606	0.0220080
$902$	−20.0000	−0.665927
$903$	−7.58258	−0.252332
$904$	96.9909	3.22587
$905$	0	0
$906$	−15.5826	−0.517696
$907$	−5.66970	−0.188259	−0.0941296	−	0.995560i	$-0.530007\pi$
$907$	−5.66970	−0.188259	−0.0941296	+	0.995560i	$0.530007\pi$
$908$	127.408	4.22819
$909$	2.41742	0.0801809
$910$	0	0
$911$	51.4955	1.70612	0.853060	−	0.521812i	$-0.174744\pi$
$911$	51.4955	1.70612	0.853060	+	0.521812i	$0.174744\pi$
$912$	46.3739	1.53559
$913$	−2.41742	−0.0800051
$914$	−72.3303	−2.39247
$915$	0	0
$916$	−154.904	−5.11817
$917$	16.0000	0.528367
$918$	4.41742	0.145797
$919$	−53.9129	−1.77842	−0.889211	−	0.457498i	$-0.848746\pi$
$919$	−53.9129	−1.77842	−0.889211	+	0.457498i	$0.848746\pi$
$920$	0	0
$921$	0	0
$922$	−51.1652	−1.68503
$923$	7.16515	0.235844
$924$	5.79129	0.190519
$925$	0	0
$926$	1.62614	0.0534382
$927$	−17.1652	−0.563778
$928$	294.347	9.66240
$929$	15.3303	0.502971	0.251485	−	0.967861i	$-0.419081\pi$
$929$	15.3303	0.502971	0.251485	+	0.967861i	$0.419081\pi$
$930$	0	0
$931$	2.58258	0.0846405
$932$	81.0780	2.65580
$933$	−22.3303	−0.731061
$934$	82.1125	2.68680
$935$	0	0
$936$	−10.5826	−0.345902
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	−1.62614	−0.0530952
$939$	−10.4174	−0.339960
$940$	0	0
$941$	16.8348	0.548800	0.274400	−	0.961616i	$-0.411521\pi$
$941$	16.8348	0.548800	0.274400	+	0.961616i	$0.411521\pi$
$942$	−2.33030	−0.0759254
$943$	−25.6697	−0.835920
$944$	−82.2867	−2.67821
$945$	0	0
$946$	−21.1652	−0.688138
$947$	−26.8348	−0.872015	−0.436008	−	0.899943i	$-0.643608\pi$
$947$	−26.8348	−0.872015	−0.436008	+	0.899943i	$0.643608\pi$
$948$	−64.6606	−2.10008
$949$	7.00000	0.227230
$950$	0	0
$951$	31.5826	1.02414
$952$	−16.7477	−0.542797
$953$	9.83485	0.318582	0.159291	−	0.987232i	$-0.449079\pi$
$953$	9.83485	0.318582	0.159291	+	0.987232i	$0.449079\pi$
$954$	1.16515	0.0377232
$955$	0	0
$956$	42.9564	1.38931
$957$	−10.1652	−0.328593
$958$	−17.9129	−0.578739
$959$	−2.41742	−0.0780627
$960$	0	0
$961$	0.165151	0.00532746
$962$	2.79129	0.0899947
$963$	3.41742	0.110125
$964$	47.2867	1.52300
$965$	0	0
$966$	10.0000	0.321745
$967$	−5.16515	−0.166100	−0.0830500	−	0.996545i	$-0.526466\pi$
$967$	−5.16515	−0.166100	−0.0830500	+	0.996545i	$0.526466\pi$
$968$	10.5826	0.340137
$969$	4.08712	0.131297
$970$	0	0
$971$	−3.41742	−0.109670	−0.0548352	−	0.998495i	$-0.517463\pi$
$971$	−3.41742	−0.109670	−0.0548352	+	0.998495i	$0.517463\pi$
$972$	5.79129	0.185756
$973$	7.16515	0.229704
$974$	73.4955	2.35495
$975$	0	0
$976$	179.564	5.74772
$977$	−50.7477	−1.62356	−0.811782	−	0.583961i	$-0.801502\pi$
$977$	−50.7477	−1.62356	−0.811782	+	0.583961i	$0.801502\pi$
$978$	1.62614	0.0519981
$979$	9.16515	0.292920
$980$	0	0
$981$	−5.58258	−0.178238
$982$	63.9564	2.04093
$983$	23.1652	0.738854	0.369427	−	0.929260i	$-0.379554\pi$
$983$	23.1652	0.738854	0.369427	+	0.929260i	$0.379554\pi$
$984$	75.8258	2.41724
$985$	0	0
$986$	44.9038	1.43003
$987$	10.5826	0.336847
$988$	−14.9564	−0.475828
$989$	−27.1652	−0.863802
$990$	0	0
$991$	−47.7477	−1.51676	−0.758378	−	0.651815i	$-0.774008\pi$
$991$	−47.7477	−1.51676	−0.758378	+	0.651815i	$0.774008\pi$
$992$	−161.652	−5.13244
$993$	15.1652	0.481252
$994$	20.0000	0.634361
$995$	0	0
$996$	14.0000	0.443607
$997$	62.6606	1.98448	0.992241	−	0.124332i	$-0.0396789\pi$
$997$	62.6606	1.98448	0.992241	+	0.124332i	$0.0396789\pi$
$998$	39.7822	1.25928
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bn.1.2		2
5.4	even	2		231.2.a.b.1.1	✓	2
15.14	odd	2		693.2.a.j.1.2		2
20.19	odd	2		3696.2.a.bl.1.1		2
35.34	odd	2		1617.2.a.o.1.1		2
55.54	odd	2		2541.2.a.z.1.2		2
105.104	even	2		4851.2.a.ba.1.2		2
165.164	even	2		7623.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.b.1.1	✓	2	5.4	even	2
693.2.a.j.1.2		2	15.14	odd	2
1617.2.a.o.1.1		2	35.34	odd	2
2541.2.a.z.1.2		2	55.54	odd	2
3696.2.a.bl.1.1		2	20.19	odd	2
4851.2.a.ba.1.2		2	105.104	even	2
5775.2.a.bn.1.2		2	1.1	even	1	trivial
7623.2.a.bf.1.1		2	165.164	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79129 q^{2} +1.00000 q^{3} +5.79129 q^{4} +2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.79129 q^{2} +1.00000 q^{3} +5.79129 q^{4} +2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} -1.00000 q^{11} +5.79129 q^{12} -1.00000 q^{13} -2.79129 q^{14} +17.9564 q^{16} +1.58258 q^{17} +2.79129 q^{18} +2.58258 q^{19} -1.00000 q^{21} -2.79129 q^{22} -3.58258 q^{23} +10.5826 q^{24} -2.79129 q^{26} +1.00000 q^{27} -5.79129 q^{28} +10.1652 q^{29} -5.58258 q^{31} +28.9564 q^{32} -1.00000 q^{33} +4.41742 q^{34} +5.79129 q^{36} -1.00000 q^{37} +7.20871 q^{38} -1.00000 q^{39} +7.16515 q^{41} -2.79129 q^{42} +7.58258 q^{43} -5.79129 q^{44} -10.0000 q^{46} -10.5826 q^{47} +17.9564 q^{48} +1.00000 q^{49} +1.58258 q^{51} -5.79129 q^{52} +0.417424 q^{53} +2.79129 q^{54} -10.5826 q^{56} +2.58258 q^{57} +28.3739 q^{58} -4.58258 q^{59} +10.0000 q^{61} -15.5826 q^{62} -1.00000 q^{63} +44.9129 q^{64} -2.79129 q^{66} +0.582576 q^{67} +9.16515 q^{68} -3.58258 q^{69} -7.16515 q^{71} +10.5826 q^{72} -7.00000 q^{73} -2.79129 q^{74} +14.9564 q^{76} +1.00000 q^{77} -2.79129 q^{78} -11.1652 q^{79} +1.00000 q^{81} +20.0000 q^{82} +2.41742 q^{83} -5.79129 q^{84} +21.1652 q^{86} +10.1652 q^{87} -10.5826 q^{88} -9.16515 q^{89} +1.00000 q^{91} -20.7477 q^{92} -5.58258 q^{93} -29.5390 q^{94} +28.9564 q^{96} +11.5826 q^{97} +2.79129 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} + 7 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 + 7 * q^4 + q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} + 7 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{11} + 7 q^{12} - 2 q^{13} - q^{14} + 13 q^{16} - 6 q^{17} + q^{18} - 4 q^{19} - 2 q^{21} - q^{22} + 2 q^{23} + 12 q^{24} - q^{26} + 2 q^{27} - 7 q^{28} + 2 q^{29} - 2 q^{31} + 35 q^{32} - 2 q^{33} + 18 q^{34} + 7 q^{36} - 2 q^{37} + 19 q^{38} - 2 q^{39} - 4 q^{41} - q^{42} + 6 q^{43} - 7 q^{44} - 20 q^{46} - 12 q^{47} + 13 q^{48} + 2 q^{49} - 6 q^{51} - 7 q^{52} + 10 q^{53} + q^{54} - 12 q^{56} - 4 q^{57} + 43 q^{58} + 20 q^{61} - 22 q^{62} - 2 q^{63} + 44 q^{64} - q^{66} - 8 q^{67} + 2 q^{69} + 4 q^{71} + 12 q^{72} - 14 q^{73} - q^{74} + 7 q^{76} + 2 q^{77} - q^{78} - 4 q^{79} + 2 q^{81} + 40 q^{82} + 14 q^{83} - 7 q^{84} + 24 q^{86} + 2 q^{87} - 12 q^{88} + 2 q^{91} - 14 q^{92} - 2 q^{93} - 27 q^{94} + 35 q^{96} + 14 q^{97} + q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 + 7 * q^4 + q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^11 + 7 * q^12 - 2 * q^13 - q^14 + 13 * q^16 - 6 * q^17 + q^18 - 4 * q^19 - 2 * q^21 - q^22 + 2 * q^23 + 12 * q^24 - q^26 + 2 * q^27 - 7 * q^28 + 2 * q^29 - 2 * q^31 + 35 * q^32 - 2 * q^33 + 18 * q^34 + 7 * q^36 - 2 * q^37 + 19 * q^38 - 2 * q^39 - 4 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 20 * q^46 - 12 * q^47 + 13 * q^48 + 2 * q^49 - 6 * q^51 - 7 * q^52 + 10 * q^53 + q^54 - 12 * q^56 - 4 * q^57 + 43 * q^58 + 20 * q^61 - 22 * q^62 - 2 * q^63 + 44 * q^64 - q^66 - 8 * q^67 + 2 * q^69 + 4 * q^71 + 12 * q^72 - 14 * q^73 - q^74 + 7 * q^76 + 2 * q^77 - q^78 - 4 * q^79 + 2 * q^81 + 40 * q^82 + 14 * q^83 - 7 * q^84 + 24 * q^86 + 2 * q^87 - 12 * q^88 + 2 * q^91 - 14 * q^92 - 2 * q^93 - 27 * q^94 + 35 * q^96 + 14 * q^97 + q^98 - 2 * q^99

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