LMFDB - Embedded newform 8450.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 650)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} +1.79129 q^{6} +2.79129 q^{7} -1.00000 q^{8} +0.208712 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} +1.79129 q^{6} +2.79129 q^{7} -1.00000 q^{8} +0.208712 q^{9} -0.791288 q^{11} -1.79129 q^{12} -2.79129 q^{14} +1.00000 q^{16} -3.79129 q^{17} -0.208712 q^{18} -5.00000 q^{19} -5.00000 q^{21} +0.791288 q^{22} +4.58258 q^{23} +1.79129 q^{24} +5.00000 q^{27} +2.79129 q^{28} +0.791288 q^{29} -0.417424 q^{31} -1.00000 q^{32} +1.41742 q^{33} +3.79129 q^{34} +0.208712 q^{36} +7.37386 q^{37} +5.00000 q^{38} +6.16515 q^{41} +5.00000 q^{42} -11.9564 q^{43} -0.791288 q^{44} -4.58258 q^{46} -3.79129 q^{47} -1.79129 q^{48} +0.791288 q^{49} +6.79129 q^{51} +4.58258 q^{53} -5.00000 q^{54} -2.79129 q^{56} +8.95644 q^{57} -0.791288 q^{58} -4.58258 q^{59} +0.582576 q^{61} +0.417424 q^{62} +0.582576 q^{63} +1.00000 q^{64} -1.41742 q^{66} +11.0000 q^{67} -3.79129 q^{68} -8.20871 q^{69} +1.58258 q^{71} -0.208712 q^{72} +11.1652 q^{73} -7.37386 q^{74} -5.00000 q^{76} -2.20871 q^{77} -7.95644 q^{79} -9.58258 q^{81} -6.16515 q^{82} +12.1652 q^{83} -5.00000 q^{84} +11.9564 q^{86} -1.41742 q^{87} +0.791288 q^{88} -14.3739 q^{89} +4.58258 q^{92} +0.747727 q^{93} +3.79129 q^{94} +1.79129 q^{96} -13.7913 q^{97} -0.791288 q^{98} -0.165151 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9	$ 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} - q^{14} + 2 q^{16} - 3 q^{17} - 5 q^{18} - 10 q^{19} - 10 q^{21} - 3 q^{22} - q^{24} + 10 q^{27} + q^{28} - 3 q^{29} - 10 q^{31} - 2 q^{32} + 12 q^{33} + 3 q^{34} + 5 q^{36} + q^{37} + 10 q^{38} - 6 q^{41} + 10 q^{42} - q^{43} + 3 q^{44} - 3 q^{47} + q^{48} - 3 q^{49} + 9 q^{51} - 10 q^{54} - q^{56} - 5 q^{57} + 3 q^{58} - 8 q^{61} + 10 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{66} + 22 q^{67} - 3 q^{68} - 21 q^{69} - 6 q^{71} - 5 q^{72} + 4 q^{73} - q^{74} - 10 q^{76} - 9 q^{77} + 7 q^{79} - 10 q^{81} + 6 q^{82} + 6 q^{83} - 10 q^{84} + q^{86} - 12 q^{87} - 3 q^{88} - 15 q^{89} - 26 q^{93} + 3 q^{94} - q^{96} - 23 q^{97} + 3 q^{98} + 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9 + 3 * q^11 + q^12 - q^14 + 2 * q^16 - 3 * q^17 - 5 * q^18 - 10 * q^19 - 10 * q^21 - 3 * q^22 - q^24 + 10 * q^27 + q^28 - 3 * q^29 - 10 * q^31 - 2 * q^32 + 12 * q^33 + 3 * q^34 + 5 * q^36 + q^37 + 10 * q^38 - 6 * q^41 + 10 * q^42 - q^43 + 3 * q^44 - 3 * q^47 + q^48 - 3 * q^49 + 9 * q^51 - 10 * q^54 - q^56 - 5 * q^57 + 3 * q^58 - 8 * q^61 + 10 * q^62 - 8 * q^63 + 2 * q^64 - 12 * q^66 + 22 * q^67 - 3 * q^68 - 21 * q^69 - 6 * q^71 - 5 * q^72 + 4 * q^73 - q^74 - 10 * q^76 - 9 * q^77 + 7 * q^79 - 10 * q^81 + 6 * q^82 + 6 * q^83 - 10 * q^84 + q^86 - 12 * q^87 - 3 * q^88 - 15 * q^89 - 26 * q^93 + 3 * q^94 - q^96 - 23 * q^97 + 3 * q^98 + 18 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.79129	−1.03420	−0.517100	−	0.855925i	$-0.672989\pi$
$3$	−1.79129	−1.03420	−0.517100	+	0.855925i	$0.672989\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.79129	0.731290
$7$	2.79129	1.05501	0.527504	−	0.849553i	$-0.323128\pi$
$7$	2.79129	1.05501	0.527504	+	0.849553i	$0.323128\pi$
$8$	−1.00000	−0.353553
$9$	0.208712	0.0695707
$10$	0	0
$11$	−0.791288	−0.238582	−0.119291	−	0.992859i	$-0.538062\pi$
$11$	−0.791288	−0.238582	−0.119291	+	0.992859i	$0.538062\pi$
$12$	−1.79129	−0.517100
$13$	0	0
$13$	0	0
$14$	−2.79129	−0.746003
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.79129	−0.919522	−0.459761	−	0.888043i	$-0.652065\pi$
$17$	−3.79129	−0.919522	−0.459761	+	0.888043i	$0.652065\pi$
$18$	−0.208712	−0.0491939
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	0.791288	0.168703
$23$	4.58258	0.955533	0.477767	−	0.878487i	$-0.341446\pi$
$23$	4.58258	0.955533	0.477767	+	0.878487i	$0.341446\pi$
$24$	1.79129	0.365645
$25$	0	0
$26$	0	0
$27$	5.00000	0.962250
$28$	2.79129	0.527504
$29$	0.791288	0.146938	0.0734692	−	0.997297i	$-0.476593\pi$
$29$	0.791288	0.146938	0.0734692	+	0.997297i	$0.476593\pi$
$30$	0	0
$31$	−0.417424	−0.0749716	−0.0374858	−	0.999297i	$-0.511935\pi$
$31$	−0.417424	−0.0749716	−0.0374858	+	0.999297i	$0.511935\pi$
$32$	−1.00000	−0.176777
$33$	1.41742	0.246742
$34$	3.79129	0.650201
$35$	0	0
$36$	0.208712	0.0347854
$37$	7.37386	1.21226	0.606128	−	0.795367i	$-0.292722\pi$
$37$	7.37386	1.21226	0.606128	+	0.795367i	$0.292722\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	0	0
$41$	6.16515	0.962835	0.481417	−	0.876491i	$-0.340122\pi$
$41$	6.16515	0.962835	0.481417	+	0.876491i	$0.340122\pi$
$42$	5.00000	0.771517
$43$	−11.9564	−1.82334	−0.911670	−	0.410923i	$-0.865206\pi$
$43$	−11.9564	−1.82334	−0.911670	+	0.410923i	$0.865206\pi$
$44$	−0.791288	−0.119291
$45$	0	0
$46$	−4.58258	−0.675664
$47$	−3.79129	−0.553016	−0.276508	−	0.961012i	$-0.589177\pi$
$47$	−3.79129	−0.553016	−0.276508	+	0.961012i	$0.589177\pi$
$48$	−1.79129	−0.258550
$49$	0.791288	0.113041
$50$	0	0
$51$	6.79129	0.950971
$52$	0	0
$53$	4.58258	0.629465	0.314733	−	0.949180i	$-0.398085\pi$
$53$	4.58258	0.629465	0.314733	+	0.949180i	$0.398085\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	−2.79129	−0.373002
$57$	8.95644	1.18631
$58$	−0.791288	−0.103901
$59$	−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
$61$	0.582576	0.0745912	0.0372956	−	0.999304i	$-0.488126\pi$
$61$	0.582576	0.0745912	0.0372956	+	0.999304i	$0.488126\pi$
$62$	0.417424	0.0530129
$63$	0.582576	0.0733976
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.41742	−0.174473
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	−3.79129	−0.459761
$69$	−8.20871	−0.988213
$70$	0	0
$71$	1.58258	0.187817	0.0939086	−	0.995581i	$-0.470064\pi$
$71$	1.58258	0.187817	0.0939086	+	0.995581i	$0.470064\pi$
$72$	−0.208712	−0.0245970
$73$	11.1652	1.30678	0.653391	−	0.757021i	$-0.273346\pi$
$73$	11.1652	1.30678	0.653391	+	0.757021i	$0.273346\pi$
$74$	−7.37386	−0.857194
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−2.20871	−0.251706
$78$	0	0
$79$	−7.95644	−0.895169	−0.447585	−	0.894242i	$-0.647716\pi$
$79$	−7.95644	−0.895169	−0.447585	+	0.894242i	$0.647716\pi$
$80$	0	0
$81$	−9.58258	−1.06473
$82$	−6.16515	−0.680827
$83$	12.1652	1.33530	0.667649	−	0.744476i	$-0.267301\pi$
$83$	12.1652	1.33530	0.667649	+	0.744476i	$0.267301\pi$
$84$	−5.00000	−0.545545
$85$	0	0
$86$	11.9564	1.28930
$87$	−1.41742	−0.151964
$88$	0.791288	0.0843516
$89$	−14.3739	−1.52363	−0.761813	−	0.647797i	$-0.775691\pi$
$89$	−14.3739	−1.52363	−0.761813	+	0.647797i	$0.775691\pi$
$90$	0	0
$91$	0	0
$92$	4.58258	0.477767
$93$	0.747727	0.0775357
$94$	3.79129	0.391041
$95$	0	0
$96$	1.79129	0.182823
$97$	−13.7913	−1.40029	−0.700147	−	0.713999i	$-0.746882\pi$
$97$	−13.7913	−1.40029	−0.700147	+	0.713999i	$0.746882\pi$
$98$	−0.791288	−0.0799321
$99$	−0.165151	−0.0165983
$100$	0	0
$101$	7.74773	0.770928	0.385464	−	0.922723i	$-0.374041\pi$
$101$	7.74773	0.770928	0.385464	+	0.922723i	$0.374041\pi$
$102$	−6.79129	−0.672438
$103$	−6.58258	−0.648600	−0.324300	−	0.945954i	$-0.605129\pi$
$103$	−6.58258	−0.648600	−0.324300	+	0.945954i	$0.605129\pi$
$104$	0	0
$105$	0	0
$106$	−4.58258	−0.445099
$107$	19.5826	1.89312	0.946560	−	0.322529i	$-0.104533\pi$
$107$	19.5826	1.89312	0.946560	+	0.322529i	$0.104533\pi$
$108$	5.00000	0.481125
$109$	−20.1652	−1.93147	−0.965736	−	0.259528i	$-0.916433\pi$
$109$	−20.1652	−1.93147	−0.965736	+	0.259528i	$0.916433\pi$
$110$	0	0
$111$	−13.2087	−1.25372
$112$	2.79129	0.263752
$113$	14.3739	1.35218	0.676090	−	0.736819i	$-0.263673\pi$
$113$	14.3739	1.35218	0.676090	+	0.736819i	$0.263673\pi$
$114$	−8.95644	−0.838847
$115$	0	0
$116$	0.791288	0.0734692
$117$	0	0
$118$	4.58258	0.421860
$119$	−10.5826	−0.970103
$120$	0	0
$121$	−10.3739	−0.943079
$122$	−0.582576	−0.0527439
$123$	−11.0436	−0.995764
$124$	−0.417424	−0.0374858
$125$	0	0
$126$	−0.582576	−0.0519000
$127$	−2.62614	−0.233032	−0.116516	−	0.993189i	$-0.537173\pi$
$127$	−2.62614	−0.233032	−0.116516	+	0.993189i	$0.537173\pi$
$128$	−1.00000	−0.0883883
$129$	21.4174	1.88570
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	1.41742	0.123371
$133$	−13.9564	−1.21018
$134$	−11.0000	−0.950255
$135$	0	0
$136$	3.79129	0.325100
$137$	−3.16515	−0.270417	−0.135209	−	0.990817i	$-0.543170\pi$
$137$	−3.16515	−0.270417	−0.135209	+	0.990817i	$0.543170\pi$
$138$	8.20871	0.698772
$139$	12.4174	1.05323	0.526616	−	0.850103i	$-0.323461\pi$
$139$	12.4174	1.05323	0.526616	+	0.850103i	$0.323461\pi$
$140$	0	0
$141$	6.79129	0.571930
$142$	−1.58258	−0.132807
$143$	0	0
$144$	0.208712	0.0173927
$145$	0	0
$146$	−11.1652	−0.924035
$147$	−1.41742	−0.116907
$148$	7.37386	0.606128
$149$	−18.9564	−1.55297	−0.776486	−	0.630134i	$-0.783000\pi$
$149$	−18.9564	−1.55297	−0.776486	+	0.630134i	$0.783000\pi$
$150$	0	0
$151$	−7.37386	−0.600077	−0.300038	−	0.953927i	$-0.596999\pi$
$151$	−7.37386	−0.600077	−0.300038	+	0.953927i	$0.596999\pi$
$152$	5.00000	0.405554
$153$	−0.791288	−0.0639718
$154$	2.20871	0.177983
$155$	0	0
$156$	0	0
$157$	13.1652	1.05069	0.525347	−	0.850888i	$-0.323936\pi$
$157$	13.1652	1.05069	0.525347	+	0.850888i	$0.323936\pi$
$158$	7.95644	0.632980
$159$	−8.20871	−0.650993
$160$	0	0
$161$	12.7913	1.00809
$162$	9.58258	0.752878
$163$	−18.3739	−1.43915	−0.719576	−	0.694414i	$-0.755664\pi$
$163$	−18.3739	−1.43915	−0.719576	+	0.694414i	$0.755664\pi$
$164$	6.16515	0.481417
$165$	0	0
$166$	−12.1652	−0.944199
$167$	−10.5826	−0.818904	−0.409452	−	0.912332i	$-0.634280\pi$
$167$	−10.5826	−0.818904	−0.409452	+	0.912332i	$0.634280\pi$
$168$	5.00000	0.385758
$169$	0	0
$170$	0	0
$171$	−1.04356	−0.0798031
$172$	−11.9564	−0.911670
$173$	−22.1216	−1.68187	−0.840937	−	0.541134i	$-0.817995\pi$
$173$	−22.1216	−1.68187	−0.840937	+	0.541134i	$0.817995\pi$
$174$	1.41742	0.107455
$175$	0	0
$176$	−0.791288	−0.0596456
$177$	8.20871	0.617005
$178$	14.3739	1.07737
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−6.37386	−0.473766	−0.236883	−	0.971538i	$-0.576126\pi$
$181$	−6.37386	−0.473766	−0.236883	+	0.971538i	$0.576126\pi$
$182$	0	0
$183$	−1.04356	−0.0771422
$184$	−4.58258	−0.337832
$185$	0	0
$186$	−0.747727	−0.0548260
$187$	3.00000	0.219382
$188$	−3.79129	−0.276508
$189$	13.9564	1.01518
$190$	0	0
$191$	25.7477	1.86304	0.931520	−	0.363690i	$-0.118483\pi$
$191$	25.7477	1.86304	0.931520	+	0.363690i	$0.118483\pi$
$192$	−1.79129	−0.129275
$193$	−4.95644	−0.356772	−0.178386	−	0.983961i	$-0.557088\pi$
$193$	−4.95644	−0.356772	−0.178386	+	0.983961i	$0.557088\pi$
$194$	13.7913	0.990157
$195$	0	0
$196$	0.791288	0.0565206
$197$	12.9564	0.923108	0.461554	−	0.887112i	$-0.347292\pi$
$197$	12.9564	0.923108	0.461554	+	0.887112i	$0.347292\pi$
$198$	0.165151	0.0117368
$199$	27.1216	1.92260	0.961299	−	0.275506i	$-0.0888455\pi$
$199$	27.1216	1.92260	0.961299	+	0.275506i	$0.0888455\pi$
$200$	0	0
$201$	−19.7042	−1.38982
$202$	−7.74773	−0.545128
$203$	2.20871	0.155021
$204$	6.79129	0.475485
$205$	0	0
$206$	6.58258	0.458630
$207$	0.956439	0.0664771
$208$	0	0
$209$	3.95644	0.273673
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	4.58258	0.314733
$213$	−2.83485	−0.194241
$214$	−19.5826	−1.33864
$215$	0	0
$216$	−5.00000	−0.340207
$217$	−1.16515	−0.0790956
$218$	20.1652	1.36576
$219$	−20.0000	−1.35147
$220$	0	0
$221$	0	0
$222$	13.2087	0.886511
$223$	−17.7477	−1.18848	−0.594238	−	0.804289i	$-0.702546\pi$
$223$	−17.7477	−1.18848	−0.594238	+	0.804289i	$0.702546\pi$
$224$	−2.79129	−0.186501
$225$	0	0
$226$	−14.3739	−0.956135
$227$	7.41742	0.492312	0.246156	−	0.969230i	$-0.420832\pi$
$227$	7.41742	0.492312	0.246156	+	0.969230i	$0.420832\pi$
$228$	8.95644	0.593155
$229$	−1.37386	−0.0907875	−0.0453937	−	0.998969i	$-0.514454\pi$
$229$	−1.37386	−0.0907875	−0.0453937	+	0.998969i	$0.514454\pi$
$230$	0	0
$231$	3.95644	0.260315
$232$	−0.791288	−0.0519506
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	0	0
$236$	−4.58258	−0.298300
$237$	14.2523	0.925785
$238$	10.5826	0.685966
$239$	−0.165151	−0.0106828	−0.00534138	−	0.999986i	$-0.501700\pi$
$239$	−0.165151	−0.0106828	−0.00534138	+	0.999986i	$0.501700\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	10.3739	0.666857
$243$	2.16515	0.138895
$244$	0.582576	0.0372956
$245$	0	0
$246$	11.0436	0.704112
$247$	0	0
$248$	0.417424	0.0265065
$249$	−21.7913	−1.38097
$250$	0	0
$251$	−6.62614	−0.418238	−0.209119	−	0.977890i	$-0.567060\pi$
$251$	−6.62614	−0.418238	−0.209119	+	0.977890i	$0.567060\pi$
$252$	0.582576	0.0366988
$253$	−3.62614	−0.227973
$254$	2.62614	0.164778
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.9129	1.42927	0.714633	−	0.699500i	$-0.246594\pi$
$257$	22.9129	1.42927	0.714633	+	0.699500i	$0.246594\pi$
$258$	−21.4174	−1.33339
$259$	20.5826	1.27894
$260$	0	0
$261$	0.165151	0.0102226
$262$	15.0000	0.926703
$263$	−12.9564	−0.798928	−0.399464	−	0.916749i	$-0.630804\pi$
$263$	−12.9564	−0.798928	−0.399464	+	0.916749i	$0.630804\pi$
$264$	−1.41742	−0.0872364
$265$	0	0
$266$	13.9564	0.855724
$267$	25.7477	1.57574
$268$	11.0000	0.671932
$269$	21.3303	1.30053	0.650266	−	0.759707i	$-0.274658\pi$
$269$	21.3303	1.30053	0.650266	+	0.759707i	$0.274658\pi$
$270$	0	0
$271$	7.16515	0.435252	0.217626	−	0.976032i	$-0.430169\pi$
$271$	7.16515	0.435252	0.217626	+	0.976032i	$0.430169\pi$
$272$	−3.79129	−0.229881
$273$	0	0
$274$	3.16515	0.191214
$275$	0	0
$276$	−8.20871	−0.494106
$277$	21.3739	1.28423	0.642115	−	0.766608i	$-0.278057\pi$
$277$	21.3739	1.28423	0.642115	+	0.766608i	$0.278057\pi$
$278$	−12.4174	−0.744748
$279$	−0.0871215	−0.00521583
$280$	0	0
$281$	−17.5390	−1.04629	−0.523145	−	0.852244i	$-0.675241\pi$
$281$	−17.5390	−1.04629	−0.523145	+	0.852244i	$0.675241\pi$
$282$	−6.79129	−0.404415
$283$	−19.5390	−1.16147	−0.580737	−	0.814091i	$-0.697236\pi$
$283$	−19.5390	−1.16147	−0.580737	+	0.814091i	$0.697236\pi$
$284$	1.58258	0.0939086
$285$	0	0
$286$	0	0
$287$	17.2087	1.01580
$288$	−0.208712	−0.0122985
$289$	−2.62614	−0.154479
$290$	0	0
$291$	24.7042	1.44818
$292$	11.1652	0.653391
$293$	20.3739	1.19025	0.595127	−	0.803632i	$-0.297102\pi$
$293$	20.3739	1.19025	0.595127	+	0.803632i	$0.297102\pi$
$294$	1.41742	0.0826659
$295$	0	0
$296$	−7.37386	−0.428597
$297$	−3.95644	−0.229576
$298$	18.9564	1.09812
$299$	0	0
$300$	0	0
$301$	−33.3739	−1.92364
$302$	7.37386	0.424318
$303$	−13.8784	−0.797294
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0.791288	0.0452349
$307$	5.16515	0.294791	0.147395	−	0.989078i	$-0.452911\pi$
$307$	5.16515	0.294791	0.147395	+	0.989078i	$0.452911\pi$
$308$	−2.20871	−0.125853
$309$	11.7913	0.670783
$310$	0	0
$311$	−3.62614	−0.205619	−0.102810	−	0.994701i	$-0.532783\pi$
$311$	−3.62614	−0.205619	−0.102810	+	0.994701i	$0.532783\pi$
$312$	0	0
$313$	18.3739	1.03855	0.519276	−	0.854607i	$-0.326202\pi$
$313$	18.3739	1.03855	0.519276	+	0.854607i	$0.326202\pi$
$314$	−13.1652	−0.742952
$315$	0	0
$316$	−7.95644	−0.447585
$317$	−5.53901	−0.311102	−0.155551	−	0.987828i	$-0.549715\pi$
$317$	−5.53901	−0.311102	−0.155551	+	0.987828i	$0.549715\pi$
$318$	8.20871	0.460322
$319$	−0.626136	−0.0350569
$320$	0	0
$321$	−35.0780	−1.95786
$322$	−12.7913	−0.712831
$323$	18.9564	1.05476
$324$	−9.58258	−0.532365
$325$	0	0
$326$	18.3739	1.01763
$327$	36.1216	1.99753
$328$	−6.16515	−0.340414
$329$	−10.5826	−0.583436
$330$	0	0
$331$	10.9564	0.602220	0.301110	−	0.953589i	$-0.402643\pi$
$331$	10.9564	0.602220	0.301110	+	0.953589i	$0.402643\pi$
$332$	12.1652	0.667649
$333$	1.53901	0.0843375
$334$	10.5826	0.579053
$335$	0	0
$336$	−5.00000	−0.272772
$337$	−27.7477	−1.51152	−0.755758	−	0.654852i	$-0.772731\pi$
$337$	−27.7477	−1.51152	−0.755758	+	0.654852i	$0.772731\pi$
$338$	0	0
$339$	−25.7477	−1.39842
$340$	0	0
$341$	0.330303	0.0178869
$342$	1.04356	0.0564293
$343$	−17.3303	−0.935748
$344$	11.9564	0.644648
$345$	0	0
$346$	22.1216	1.18926
$347$	−19.4174	−1.04238	−0.521191	−	0.853440i	$-0.674512\pi$
$347$	−19.4174	−1.04238	−0.521191	+	0.853440i	$0.674512\pi$
$348$	−1.41742	−0.0759819
$349$	−4.04356	−0.216447	−0.108223	−	0.994127i	$-0.534516\pi$
$349$	−4.04356	−0.216447	−0.108223	+	0.994127i	$0.534516\pi$
$350$	0	0
$351$	0	0
$352$	0.791288	0.0421758
$353$	−32.3739	−1.72309	−0.861543	−	0.507684i	$-0.830502\pi$
$353$	−32.3739	−1.72309	−0.861543	+	0.507684i	$0.830502\pi$
$354$	−8.20871	−0.436288
$355$	0	0
$356$	−14.3739	−0.761813
$357$	18.9564	1.00328
$358$	−6.00000	−0.317110
$359$	29.2087	1.54158	0.770788	−	0.637091i	$-0.219863\pi$
$359$	29.2087	1.54158	0.770788	+	0.637091i	$0.219863\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	6.37386	0.335003
$363$	18.5826	0.975332
$364$	0	0
$365$	0	0
$366$	1.04356	0.0545478
$367$	2.74773	0.143430	0.0717151	−	0.997425i	$-0.477153\pi$
$367$	2.74773	0.143430	0.0717151	+	0.997425i	$0.477153\pi$
$368$	4.58258	0.238883
$369$	1.28674	0.0669851
$370$	0	0
$371$	12.7913	0.664091
$372$	0.747727	0.0387678
$373$	−5.16515	−0.267441	−0.133721	−	0.991019i	$-0.542693\pi$
$373$	−5.16515	−0.267441	−0.133721	+	0.991019i	$0.542693\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	3.79129	0.195521
$377$	0	0
$378$	−13.9564	−0.717842
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	0	0
$381$	4.70417	0.241002
$382$	−25.7477	−1.31737
$383$	−25.7477	−1.31565	−0.657824	−	0.753172i	$-0.728523\pi$
$383$	−25.7477	−1.31565	−0.657824	+	0.753172i	$0.728523\pi$
$384$	1.79129	0.0914113
$385$	0	0
$386$	4.95644	0.252276
$387$	−2.49545	−0.126851
$388$	−13.7913	−0.700147
$389$	30.3303	1.53781	0.768904	−	0.639365i	$-0.220803\pi$
$389$	30.3303	1.53781	0.768904	+	0.639365i	$0.220803\pi$
$390$	0	0
$391$	−17.3739	−0.878634
$392$	−0.791288	−0.0399661
$393$	26.8693	1.35538
$394$	−12.9564	−0.652736
$395$	0	0
$396$	−0.165151	−0.00829917
$397$	−10.7913	−0.541599	−0.270800	−	0.962636i	$-0.587288\pi$
$397$	−10.7913	−0.541599	−0.270800	+	0.962636i	$0.587288\pi$
$398$	−27.1216	−1.35948
$399$	25.0000	1.25157
$400$	0	0
$401$	6.62614	0.330893	0.165447	−	0.986219i	$-0.447093\pi$
$401$	6.62614	0.330893	0.165447	+	0.986219i	$0.447093\pi$
$402$	19.7042	0.982754
$403$	0	0
$404$	7.74773	0.385464
$405$	0	0
$406$	−2.20871	−0.109617
$407$	−5.83485	−0.289223
$408$	−6.79129	−0.336219
$409$	−32.4955	−1.60680	−0.803398	−	0.595442i	$-0.796977\pi$
$409$	−32.4955	−1.60680	−0.803398	+	0.595442i	$0.796977\pi$
$410$	0	0
$411$	5.66970	0.279666
$412$	−6.58258	−0.324300
$413$	−12.7913	−0.629418
$414$	−0.956439	−0.0470064
$415$	0	0
$416$	0	0
$417$	−22.2432	−1.08925
$418$	−3.95644	−0.193516
$419$	−35.5390	−1.73619	−0.868097	−	0.496394i	$-0.834657\pi$
$419$	−35.5390	−1.73619	−0.868097	+	0.496394i	$0.834657\pi$
$420$	0	0
$421$	−23.9564	−1.16757	−0.583783	−	0.811910i	$-0.698428\pi$
$421$	−23.9564	−1.16757	−0.583783	+	0.811910i	$0.698428\pi$
$422$	−5.00000	−0.243396
$423$	−0.791288	−0.0384737
$424$	−4.58258	−0.222550
$425$	0	0
$426$	2.83485	0.137349
$427$	1.62614	0.0786943
$428$	19.5826	0.946560
$429$	0	0
$430$	0	0
$431$	2.20871	0.106390	0.0531950	−	0.998584i	$-0.483060\pi$
$431$	2.20871	0.106390	0.0531950	+	0.998584i	$0.483060\pi$
$432$	5.00000	0.240563
$433$	−21.4174	−1.02926	−0.514628	−	0.857414i	$-0.672070\pi$
$433$	−21.4174	−1.02926	−0.514628	+	0.857414i	$0.672070\pi$
$434$	1.16515	0.0559291
$435$	0	0
$436$	−20.1652	−0.965736
$437$	−22.9129	−1.09607
$438$	20.0000	0.955637
$439$	−8.41742	−0.401742	−0.200871	−	0.979618i	$-0.564377\pi$
$439$	−8.41742	−0.401742	−0.200871	+	0.979618i	$0.564377\pi$
$440$	0	0
$441$	0.165151	0.00786435
$442$	0	0
$443$	−24.9564	−1.18572	−0.592858	−	0.805307i	$-0.702001\pi$
$443$	−24.9564	−1.18572	−0.592858	+	0.805307i	$0.702001\pi$
$444$	−13.2087	−0.626858
$445$	0	0
$446$	17.7477	0.840379
$447$	33.9564	1.60608
$448$	2.79129	0.131876
$449$	−18.6261	−0.879022	−0.439511	−	0.898237i	$-0.644848\pi$
$449$	−18.6261	−0.879022	−0.439511	+	0.898237i	$0.644848\pi$
$450$	0	0
$451$	−4.87841	−0.229715
$452$	14.3739	0.676090
$453$	13.2087	0.620599
$454$	−7.41742	−0.348117
$455$	0	0
$456$	−8.95644	−0.419424
$457$	−6.37386	−0.298157	−0.149078	−	0.988825i	$-0.547631\pi$
$457$	−6.37386	−0.298157	−0.149078	+	0.988825i	$0.547631\pi$
$458$	1.37386	0.0641964
$459$	−18.9564	−0.884811
$460$	0	0
$461$	−33.7913	−1.57382	−0.786909	−	0.617070i	$-0.788320\pi$
$461$	−33.7913	−1.57382	−0.786909	+	0.617070i	$0.788320\pi$
$462$	−3.95644	−0.184070
$463$	−4.95644	−0.230345	−0.115173	−	0.993345i	$-0.536742\pi$
$463$	−4.95644	−0.230345	−0.115173	+	0.993345i	$0.536742\pi$
$464$	0.791288	0.0367346
$465$	0	0
$466$	−15.0000	−0.694862
$467$	−5.37386	−0.248673	−0.124336	−	0.992240i	$-0.539680\pi$
$467$	−5.37386	−0.248673	−0.124336	+	0.992240i	$0.539680\pi$
$468$	0	0
$469$	30.7042	1.41779
$470$	0	0
$471$	−23.5826	−1.08663
$472$	4.58258	0.210930
$473$	9.46099	0.435017
$474$	−14.2523	−0.654629
$475$	0	0
$476$	−10.5826	−0.485052
$477$	0.956439	0.0437923
$478$	0.165151	0.00755385
$479$	3.95644	0.180774	0.0903872	−	0.995907i	$-0.471190\pi$
$479$	3.95644	0.180774	0.0903872	+	0.995907i	$0.471190\pi$
$480$	0	0
$481$	0	0
$482$	−7.00000	−0.318841
$483$	−22.9129	−1.04257
$484$	−10.3739	−0.471539
$485$	0	0
$486$	−2.16515	−0.0982133
$487$	−20.1216	−0.911796	−0.455898	−	0.890032i	$-0.650682\pi$
$487$	−20.1216	−0.911796	−0.455898	+	0.890032i	$0.650682\pi$
$488$	−0.582576	−0.0263720
$489$	32.9129	1.48837
$490$	0	0
$491$	−10.9129	−0.492491	−0.246246	−	0.969207i	$-0.579197\pi$
$491$	−10.9129	−0.492491	−0.246246	+	0.969207i	$0.579197\pi$
$492$	−11.0436	−0.497882
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	−0.417424	−0.0187429
$497$	4.41742	0.198149
$498$	21.7913	0.976491
$499$	17.7477	0.794497	0.397249	−	0.917711i	$-0.369965\pi$
$499$	17.7477	0.794497	0.397249	+	0.917711i	$0.369965\pi$
$500$	0	0
$501$	18.9564	0.846911
$502$	6.62614	0.295739
$503$	−1.91288	−0.0852910	−0.0426455	−	0.999090i	$-0.513579\pi$
$503$	−1.91288	−0.0852910	−0.0426455	+	0.999090i	$0.513579\pi$
$504$	−0.582576	−0.0259500
$505$	0	0
$506$	3.62614	0.161201
$507$	0	0
$508$	−2.62614	−0.116516
$509$	19.9129	0.882623	0.441311	−	0.897354i	$-0.354513\pi$
$509$	19.9129	0.882623	0.441311	+	0.897354i	$0.354513\pi$
$510$	0	0
$511$	31.1652	1.37867
$512$	−1.00000	−0.0441942
$513$	−25.0000	−1.10378
$514$	−22.9129	−1.01064
$515$	0	0
$516$	21.4174	0.942850
$517$	3.00000	0.131940
$518$	−20.5826	−0.904346
$519$	39.6261	1.73939
$520$	0	0
$521$	−0.330303	−0.0144708	−0.00723541	−	0.999974i	$-0.502303\pi$
$521$	−0.330303	−0.0144708	−0.00723541	+	0.999974i	$0.502303\pi$
$522$	−0.165151	−0.00722848
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	12.9564	0.564928
$527$	1.58258	0.0689381
$528$	1.41742	0.0616855
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	−0.956439	−0.0415059
$532$	−13.9564	−0.605088
$533$	0	0
$534$	−25.7477	−1.11421
$535$	0	0
$536$	−11.0000	−0.475128
$537$	−10.7477	−0.463799
$538$	−21.3303	−0.919615
$539$	−0.626136	−0.0269696
$540$	0	0
$541$	17.9129	0.770135	0.385067	−	0.922888i	$-0.374178\pi$
$541$	17.9129	0.770135	0.385067	+	0.922888i	$0.374178\pi$
$542$	−7.16515	−0.307770
$543$	11.4174	0.489969
$544$	3.79129	0.162550
$545$	0	0
$546$	0	0
$547$	−4.66970	−0.199662	−0.0998309	−	0.995004i	$-0.531830\pi$
$547$	−4.66970	−0.199662	−0.0998309	+	0.995004i	$0.531830\pi$
$548$	−3.16515	−0.135209
$549$	0.121591	0.00518936
$550$	0	0
$551$	−3.95644	−0.168550
$552$	8.20871	0.349386
$553$	−22.2087	−0.944411
$554$	−21.3739	−0.908088
$555$	0	0
$556$	12.4174	0.526616
$557$	46.7477	1.98076	0.990382	−	0.138357i	$-0.0441822\pi$
$557$	46.7477	1.98076	0.990382	+	0.138357i	$0.0441822\pi$
$558$	0.0871215	0.00368815
$559$	0	0
$560$	0	0
$561$	−5.37386	−0.226885
$562$	17.5390	0.739839
$563$	35.7042	1.50475	0.752376	−	0.658734i	$-0.228908\pi$
$563$	35.7042	1.50475	0.752376	+	0.658734i	$0.228908\pi$
$564$	6.79129	0.285965
$565$	0	0
$566$	19.5390	0.821286
$567$	−26.7477	−1.12330
$568$	−1.58258	−0.0664034
$569$	20.3739	0.854117	0.427058	−	0.904224i	$-0.359550\pi$
$569$	20.3739	0.854117	0.427058	+	0.904224i	$0.359550\pi$
$570$	0	0
$571$	7.37386	0.308587	0.154293	−	0.988025i	$-0.450690\pi$
$571$	7.37386	0.308587	0.154293	+	0.988025i	$0.450690\pi$
$572$	0	0
$573$	−46.1216	−1.92676
$574$	−17.2087	−0.718278
$575$	0	0
$576$	0.208712	0.00869634
$577$	−29.7477	−1.23841	−0.619207	−	0.785228i	$-0.712546\pi$
$577$	−29.7477	−1.23841	−0.619207	+	0.785228i	$0.712546\pi$
$578$	2.62614	0.109233
$579$	8.87841	0.368974
$580$	0	0
$581$	33.9564	1.40875
$582$	−24.7042	−1.02402
$583$	−3.62614	−0.150179
$584$	−11.1652	−0.462017
$585$	0	0
$586$	−20.3739	−0.841637
$587$	−37.2867	−1.53899	−0.769494	−	0.638654i	$-0.779492\pi$
$587$	−37.2867	−1.53899	−0.769494	+	0.638654i	$0.779492\pi$
$588$	−1.41742	−0.0584536
$589$	2.08712	0.0859983
$590$	0	0
$591$	−23.2087	−0.954679
$592$	7.37386	0.303064
$593$	−7.58258	−0.311379	−0.155690	−	0.987806i	$-0.549760\pi$
$593$	−7.58258	−0.311379	−0.155690	+	0.987806i	$0.549760\pi$
$594$	3.95644	0.162335
$595$	0	0
$596$	−18.9564	−0.776486
$597$	−48.5826	−1.98835
$598$	0	0
$599$	−19.7477	−0.806870	−0.403435	−	0.915008i	$-0.632184\pi$
$599$	−19.7477	−0.806870	−0.403435	+	0.915008i	$0.632184\pi$
$600$	0	0
$601$	9.74773	0.397618	0.198809	−	0.980038i	$-0.436293\pi$
$601$	9.74773	0.397618	0.198809	+	0.980038i	$0.436293\pi$
$602$	33.3739	1.36022
$603$	2.29583	0.0934936
$604$	−7.37386	−0.300038
$605$	0	0
$606$	13.8784	0.563772
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	5.00000	0.202777
$609$	−3.95644	−0.160323
$610$	0	0
$611$	0	0
$612$	−0.791288	−0.0319859
$613$	0.417424	0.0168596	0.00842980	−	0.999964i	$-0.497317\pi$
$613$	0.417424	0.0168596	0.00842980	+	0.999964i	$0.497317\pi$
$614$	−5.16515	−0.208449
$615$	0	0
$616$	2.20871	0.0889915
$617$	11.8348	0.476453	0.238227	−	0.971210i	$-0.423434\pi$
$617$	11.8348	0.476453	0.238227	+	0.971210i	$0.423434\pi$
$618$	−11.7913	−0.474315
$619$	−35.7913	−1.43857	−0.719287	−	0.694713i	$-0.755531\pi$
$619$	−35.7913	−1.43857	−0.719287	+	0.694713i	$0.755531\pi$
$620$	0	0
$621$	22.9129	0.919462
$622$	3.62614	0.145395
$623$	−40.1216	−1.60744
$624$	0	0
$625$	0	0
$626$	−18.3739	−0.734367
$627$	−7.08712	−0.283032
$628$	13.1652	0.525347
$629$	−27.9564	−1.11470
$630$	0	0
$631$	3.37386	0.134311	0.0671557	−	0.997743i	$-0.478608\pi$
$631$	3.37386	0.134311	0.0671557	+	0.997743i	$0.478608\pi$
$632$	7.95644	0.316490
$633$	−8.95644	−0.355987
$634$	5.53901	0.219982
$635$	0	0
$636$	−8.20871	−0.325497
$637$	0	0
$638$	0.626136	0.0247890
$639$	0.330303	0.0130666
$640$	0	0
$641$	0.791288	0.0312540	0.0156270	−	0.999878i	$-0.495026\pi$
$641$	0.791288	0.0312540	0.0156270	+	0.999878i	$0.495026\pi$
$642$	35.0780	1.38442
$643$	−29.7477	−1.17314	−0.586568	−	0.809900i	$-0.699521\pi$
$643$	−29.7477	−1.17314	−0.586568	+	0.809900i	$0.699521\pi$
$644$	12.7913	0.504047
$645$	0	0
$646$	−18.9564	−0.745831
$647$	−23.0436	−0.905936	−0.452968	−	0.891527i	$-0.649635\pi$
$647$	−23.0436	−0.905936	−0.452968	+	0.891527i	$0.649635\pi$
$648$	9.58258	0.376439
$649$	3.62614	0.142338
$650$	0	0
$651$	2.08712	0.0818007
$652$	−18.3739	−0.719576
$653$	40.5826	1.58812	0.794059	−	0.607840i	$-0.207964\pi$
$653$	40.5826	1.58812	0.794059	+	0.607840i	$0.207964\pi$
$654$	−36.1216	−1.41247
$655$	0	0
$656$	6.16515	0.240709
$657$	2.33030	0.0909138
$658$	10.5826	0.412552
$659$	−33.4955	−1.30480	−0.652399	−	0.757876i	$-0.726237\pi$
$659$	−33.4955	−1.30480	−0.652399	+	0.757876i	$0.726237\pi$
$660$	0	0
$661$	−14.9564	−0.581738	−0.290869	−	0.956763i	$-0.593944\pi$
$661$	−14.9564	−0.581738	−0.290869	+	0.956763i	$0.593944\pi$
$662$	−10.9564	−0.425834
$663$	0	0
$664$	−12.1652	−0.472099
$665$	0	0
$666$	−1.53901	−0.0596356
$667$	3.62614	0.140405
$668$	−10.5826	−0.409452
$669$	31.7913	1.22912
$670$	0	0
$671$	−0.460985	−0.0177961
$672$	5.00000	0.192879
$673$	−21.9129	−0.844679	−0.422340	−	0.906438i	$-0.638791\pi$
$673$	−21.9129	−0.844679	−0.422340	+	0.906438i	$0.638791\pi$
$674$	27.7477	1.06880
$675$	0	0
$676$	0	0
$677$	31.7477	1.22016	0.610082	−	0.792338i	$-0.291136\pi$
$677$	31.7477	1.22016	0.610082	+	0.792338i	$0.291136\pi$
$678$	25.7477	0.988836
$679$	−38.4955	−1.47732
$680$	0	0
$681$	−13.2867	−0.509149
$682$	−0.330303	−0.0126479
$683$	16.9129	0.647153	0.323577	−	0.946202i	$-0.395115\pi$
$683$	16.9129	0.647153	0.323577	+	0.946202i	$0.395115\pi$
$684$	−1.04356	−0.0399015
$685$	0	0
$686$	17.3303	0.661674
$687$	2.46099	0.0938924
$688$	−11.9564	−0.455835
$689$	0	0
$690$	0	0
$691$	−30.7477	−1.16970	−0.584849	−	0.811142i	$-0.698846\pi$
$691$	−30.7477	−1.16970	−0.584849	+	0.811142i	$0.698846\pi$
$692$	−22.1216	−0.840937
$693$	−0.460985	−0.0175114
$694$	19.4174	0.737075
$695$	0	0
$696$	1.41742	0.0537273
$697$	−23.3739	−0.885348
$698$	4.04356	0.153051
$699$	−26.8693	−1.01629
$700$	0	0
$701$	46.1216	1.74199	0.870994	−	0.491293i	$-0.163476\pi$
$701$	46.1216	1.74199	0.870994	+	0.491293i	$0.163476\pi$
$702$	0	0
$703$	−36.8693	−1.39055
$704$	−0.791288	−0.0298228
$705$	0	0
$706$	32.3739	1.21841
$707$	21.6261	0.813335
$708$	8.20871	0.308502
$709$	6.20871	0.233173	0.116587	−	0.993181i	$-0.462805\pi$
$709$	6.20871	0.233173	0.116587	+	0.993181i	$0.462805\pi$
$710$	0	0
$711$	−1.66061	−0.0622776
$712$	14.3739	0.538683
$713$	−1.91288	−0.0716379
$714$	−18.9564	−0.709427
$715$	0	0
$716$	6.00000	0.224231
$717$	0.295834	0.0110481
$718$	−29.2087	−1.09006
$719$	−20.5390	−0.765976	−0.382988	−	0.923753i	$-0.625105\pi$
$719$	−20.5390	−0.765976	−0.382988	+	0.923753i	$0.625105\pi$
$720$	0	0
$721$	−18.3739	−0.684278
$722$	−6.00000	−0.223297
$723$	−12.5390	−0.466331
$724$	−6.37386	−0.236883
$725$	0	0
$726$	−18.5826	−0.689664
$727$	−4.83485	−0.179315	−0.0896573	−	0.995973i	$-0.528577\pi$
$727$	−4.83485	−0.179315	−0.0896573	+	0.995973i	$0.528577\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	0	0
$731$	45.3303	1.67660
$732$	−1.04356	−0.0385711
$733$	5.62614	0.207806	0.103903	−	0.994587i	$-0.466867\pi$
$733$	5.62614	0.207806	0.103903	+	0.994587i	$0.466867\pi$
$734$	−2.74773	−0.101420
$735$	0	0
$736$	−4.58258	−0.168916
$737$	−8.70417	−0.320622
$738$	−1.28674	−0.0473656
$739$	−20.6261	−0.758745	−0.379372	−	0.925244i	$-0.623860\pi$
$739$	−20.6261	−0.758745	−0.379372	+	0.925244i	$0.623860\pi$
$740$	0	0
$741$	0	0
$742$	−12.7913	−0.469583
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	−0.747727	−0.0274130
$745$	0	0
$746$	5.16515	0.189110
$747$	2.53901	0.0928977
$748$	3.00000	0.109691
$749$	54.6606	1.99726
$750$	0	0
$751$	−52.4955	−1.91559	−0.957793	−	0.287458i	$-0.907190\pi$
$751$	−52.4955	−1.91559	−0.957793	+	0.287458i	$0.907190\pi$
$752$	−3.79129	−0.138254
$753$	11.8693	0.432542
$754$	0	0
$755$	0	0
$756$	13.9564	0.507591
$757$	20.2523	0.736081	0.368041	−	0.929810i	$-0.380029\pi$
$757$	20.2523	0.736081	0.368041	+	0.929810i	$0.380029\pi$
$758$	17.0000	0.617468
$759$	6.49545	0.235770
$760$	0	0
$761$	−34.9129	−1.26559	−0.632795	−	0.774319i	$-0.718093\pi$
$761$	−34.9129	−1.26559	−0.632795	+	0.774319i	$0.718093\pi$
$762$	−4.70417	−0.170414
$763$	−56.2867	−2.03772
$764$	25.7477	0.931520
$765$	0	0
$766$	25.7477	0.930303
$767$	0	0
$768$	−1.79129	−0.0646375
$769$	−9.25227	−0.333645	−0.166823	−	0.985987i	$-0.553351\pi$
$769$	−9.25227	−0.333645	−0.166823	+	0.985987i	$0.553351\pi$
$770$	0	0
$771$	−41.0436	−1.47815
$772$	−4.95644	−0.178386
$773$	−36.6606	−1.31859	−0.659295	−	0.751884i	$-0.729145\pi$
$773$	−36.6606	−1.31859	−0.659295	+	0.751884i	$0.729145\pi$
$774$	2.49545	0.0896972
$775$	0	0
$776$	13.7913	0.495078
$777$	−36.8693	−1.32268
$778$	−30.3303	−1.08739
$779$	−30.8258	−1.10445
$780$	0	0
$781$	−1.25227	−0.0448098
$782$	17.3739	0.621288
$783$	3.95644	0.141392
$784$	0.791288	0.0282603
$785$	0	0
$786$	−26.8693	−0.958397
$787$	11.4610	0.408540	0.204270	−	0.978915i	$-0.434518\pi$
$787$	11.4610	0.408540	0.204270	+	0.978915i	$0.434518\pi$
$788$	12.9564	0.461554
$789$	23.2087	0.826252
$790$	0	0
$791$	40.1216	1.42656
$792$	0.165151	0.00586840
$793$	0	0
$794$	10.7913	0.382968
$795$	0	0
$796$	27.1216	0.961299
$797$	20.0780	0.711200	0.355600	−	0.934638i	$-0.384276\pi$
$797$	20.0780	0.711200	0.355600	+	0.934638i	$0.384276\pi$
$798$	−25.0000	−0.884990
$799$	14.3739	0.508511
$800$	0	0
$801$	−3.00000	−0.106000
$802$	−6.62614	−0.233977
$803$	−8.83485	−0.311775
$804$	−19.7042	−0.694912
$805$	0	0
$806$	0	0
$807$	−38.2087	−1.34501
$808$	−7.74773	−0.272564
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−31.7042	−1.11328	−0.556642	−	0.830753i	$-0.687910\pi$
$811$	−31.7042	−1.11328	−0.556642	+	0.830753i	$0.687910\pi$
$812$	2.20871	0.0775106
$813$	−12.8348	−0.450138
$814$	5.83485	0.204511
$815$	0	0
$816$	6.79129	0.237743
$817$	59.7822	2.09151
$818$	32.4955	1.13618
$819$	0	0
$820$	0	0
$821$	34.9129	1.21847	0.609234	−	0.792991i	$-0.291477\pi$
$821$	34.9129	1.21847	0.609234	+	0.792991i	$0.291477\pi$
$822$	−5.66970	−0.197753
$823$	−48.7477	−1.69924	−0.849619	−	0.527396i	$-0.823168\pi$
$823$	−48.7477	−1.69924	−0.849619	+	0.527396i	$0.823168\pi$
$824$	6.58258	0.229315
$825$	0	0
$826$	12.7913	0.445066
$827$	−8.66970	−0.301475	−0.150737	−	0.988574i	$-0.548165\pi$
$827$	−8.66970	−0.301475	−0.150737	+	0.988574i	$0.548165\pi$
$828$	0.956439	0.0332386
$829$	15.2867	0.530930	0.265465	−	0.964120i	$-0.414474\pi$
$829$	15.2867	0.530930	0.265465	+	0.964120i	$0.414474\pi$
$830$	0	0
$831$	−38.2867	−1.32815
$832$	0	0
$833$	−3.00000	−0.103944
$834$	22.2432	0.770218
$835$	0	0
$836$	3.95644	0.136836
$837$	−2.08712	−0.0721415
$838$	35.5390	1.22767
$839$	−44.7042	−1.54336	−0.771680	−	0.636011i	$-0.780583\pi$
$839$	−44.7042	−1.54336	−0.771680	+	0.636011i	$0.780583\pi$
$840$	0	0
$841$	−28.3739	−0.978409
$842$	23.9564	0.825593
$843$	31.4174	1.08207
$844$	5.00000	0.172107
$845$	0	0
$846$	0.791288	0.0272050
$847$	−28.9564	−0.994955
$848$	4.58258	0.157366
$849$	35.0000	1.20120
$850$	0	0
$851$	33.7913	1.15835
$852$	−2.83485	−0.0971203
$853$	9.12159	0.312317	0.156159	−	0.987732i	$-0.450089\pi$
$853$	9.12159	0.312317	0.156159	+	0.987732i	$0.450089\pi$
$854$	−1.62614	−0.0556452
$855$	0	0
$856$	−19.5826	−0.669319
$857$	5.83485	0.199315	0.0996573	−	0.995022i	$-0.468225\pi$
$857$	5.83485	0.199315	0.0996573	+	0.995022i	$0.468225\pi$
$858$	0	0
$859$	−38.2867	−1.30633	−0.653163	−	0.757217i	$-0.726559\pi$
$859$	−38.2867	−1.30633	−0.653163	+	0.757217i	$0.726559\pi$
$860$	0	0
$861$	−30.8258	−1.05054
$862$	−2.20871	−0.0752290
$863$	−13.4174	−0.456734	−0.228367	−	0.973575i	$-0.573339\pi$
$863$	−13.4174	−0.456734	−0.228367	+	0.973575i	$0.573339\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	21.4174	0.727794
$867$	4.70417	0.159762
$868$	−1.16515	−0.0395478
$869$	6.29583	0.213572
$870$	0	0
$871$	0	0
$872$	20.1652	0.682878
$873$	−2.87841	−0.0974194
$874$	22.9129	0.775040
$875$	0	0
$876$	−20.0000	−0.675737
$877$	45.7477	1.54479	0.772395	−	0.635142i	$-0.219058\pi$
$877$	45.7477	1.54479	0.772395	+	0.635142i	$0.219058\pi$
$878$	8.41742	0.284074
$879$	−36.4955	−1.23096
$880$	0	0
$881$	−27.9564	−0.941876	−0.470938	−	0.882166i	$-0.656085\pi$
$881$	−27.9564	−0.941876	−0.470938	+	0.882166i	$0.656085\pi$
$882$	−0.165151	−0.00556094
$883$	−2.95644	−0.0994921	−0.0497461	−	0.998762i	$-0.515841\pi$
$883$	−2.95644	−0.0994921	−0.0497461	+	0.998762i	$0.515841\pi$
$884$	0	0
$885$	0	0
$886$	24.9564	0.838428
$887$	−32.8348	−1.10249	−0.551243	−	0.834345i	$-0.685846\pi$
$887$	−32.8348	−1.10249	−0.551243	+	0.834345i	$0.685846\pi$
$888$	13.2087	0.443255
$889$	−7.33030	−0.245850
$890$	0	0
$891$	7.58258	0.254026
$892$	−17.7477	−0.594238
$893$	18.9564	0.634353
$894$	−33.9564	−1.13567
$895$	0	0
$896$	−2.79129	−0.0932504
$897$	0	0
$898$	18.6261	0.621562
$899$	−0.330303	−0.0110162
$900$	0	0
$901$	−17.3739	−0.578807
$902$	4.87841	0.162433
$903$	59.7822	1.98943
$904$	−14.3739	−0.478068
$905$	0	0
$906$	−13.2087	−0.438830
$907$	22.4955	0.746949	0.373475	−	0.927640i	$-0.378166\pi$
$907$	22.4955	0.746949	0.373475	+	0.927640i	$0.378166\pi$
$908$	7.41742	0.246156
$909$	1.61704	0.0536340
$910$	0	0
$911$	8.66970	0.287240	0.143620	−	0.989633i	$-0.454126\pi$
$911$	8.66970	0.287240	0.143620	+	0.989633i	$0.454126\pi$
$912$	8.95644	0.296577
$913$	−9.62614	−0.318579
$914$	6.37386	0.210829
$915$	0	0
$916$	−1.37386	−0.0453937
$917$	−41.8693	−1.38265
$918$	18.9564	0.625656
$919$	−17.2523	−0.569100	−0.284550	−	0.958661i	$-0.591844\pi$
$919$	−17.2523	−0.569100	−0.284550	+	0.958661i	$0.591844\pi$
$920$	0	0
$921$	−9.25227	−0.304873
$922$	33.7913	1.11286
$923$	0	0
$924$	3.95644	0.130157
$925$	0	0
$926$	4.95644	0.162879
$927$	−1.37386	−0.0451236
$928$	−0.791288	−0.0259753
$929$	−4.74773	−0.155768	−0.0778839	−	0.996962i	$-0.524816\pi$
$929$	−4.74773	−0.155768	−0.0778839	+	0.996962i	$0.524816\pi$
$930$	0	0
$931$	−3.95644	−0.129667
$932$	15.0000	0.491341
$933$	6.49545	0.212652
$934$	5.37386	0.175838
$935$	0	0
$936$	0	0
$937$	−46.3739	−1.51497	−0.757484	−	0.652854i	$-0.773572\pi$
$937$	−46.3739	−1.51497	−0.757484	+	0.652854i	$0.773572\pi$
$938$	−30.7042	−1.00253
$939$	−32.9129	−1.07407
$940$	0	0
$941$	33.6261	1.09618	0.548090	−	0.836419i	$-0.315355\pi$
$941$	33.6261	1.09618	0.548090	+	0.836419i	$0.315355\pi$
$942$	23.5826	0.768362
$943$	28.2523	0.920021
$944$	−4.58258	−0.149150
$945$	0	0
$946$	−9.46099	−0.307603
$947$	−32.2087	−1.04664	−0.523321	−	0.852135i	$-0.675307\pi$
$947$	−32.2087	−1.04664	−0.523321	+	0.852135i	$0.675307\pi$
$948$	14.2523	0.462892
$949$	0	0
$950$	0	0
$951$	9.92197	0.321742
$952$	10.5826	0.342983
$953$	24.4955	0.793486	0.396743	−	0.917930i	$-0.370140\pi$
$953$	24.4955	0.793486	0.396743	+	0.917930i	$0.370140\pi$
$954$	−0.956439	−0.0309659
$955$	0	0
$956$	−0.165151	−0.00534138
$957$	1.12159	0.0362559
$958$	−3.95644	−0.127827
$959$	−8.83485	−0.285292
$960$	0	0
$961$	−30.8258	−0.994379
$962$	0	0
$963$	4.08712	0.131706
$964$	7.00000	0.225455
$965$	0	0
$966$	22.9129	0.737210
$967$	−20.7477	−0.667202	−0.333601	−	0.942714i	$-0.608264\pi$
$967$	−20.7477	−0.667202	−0.333601	+	0.942714i	$0.608264\pi$
$968$	10.3739	0.333429
$969$	−33.9564	−1.09084
$970$	0	0
$971$	−36.4955	−1.17119	−0.585597	−	0.810602i	$-0.699140\pi$
$971$	−36.4955	−1.17119	−0.585597	+	0.810602i	$0.699140\pi$
$972$	2.16515	0.0694473
$973$	34.6606	1.11117
$974$	20.1216	0.644737
$975$	0	0
$976$	0.582576	0.0186478
$977$	−1.41742	−0.0453474	−0.0226737	−	0.999743i	$-0.507218\pi$
$977$	−1.41742	−0.0453474	−0.0226737	+	0.999743i	$0.507218\pi$
$978$	−32.9129	−1.05244
$979$	11.3739	0.363510
$980$	0	0
$981$	−4.20871	−0.134374
$982$	10.9129	0.348244
$983$	−50.5390	−1.61194	−0.805972	−	0.591953i	$-0.798357\pi$
$983$	−50.5390	−1.61194	−0.805972	+	0.591953i	$0.798357\pi$
$984$	11.0436	0.352056
$985$	0	0
$986$	3.00000	0.0955395
$987$	18.9564	0.603390
$988$	0	0
$989$	−54.7913	−1.74226
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0.417424	0.0132532
$993$	−19.6261	−0.622817
$994$	−4.41742	−0.140112
$995$	0	0
$996$	−21.7913	−0.690483
$997$	−36.1216	−1.14398	−0.571991	−	0.820260i	$-0.693829\pi$
$997$	−36.1216	−1.14398	−0.571991	+	0.820260i	$0.693829\pi$
$998$	−17.7477	−0.561794
$999$	36.8693	1.16649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.be.1.1		2
5.4	even	2		8450.2.a.bh.1.2		2
13.4	even	6		650.2.e.d.601.2	yes	4
13.10	even	6		650.2.e.d.451.2	✓	4
13.12	even	2		8450.2.a.bk.1.1		2
65.4	even	6		650.2.e.i.601.1	yes	4
65.17	odd	12		650.2.o.f.549.4		8
65.23	odd	12		650.2.o.f.399.4		8
65.43	odd	12		650.2.o.f.549.1		8
65.49	even	6		650.2.e.i.451.1	yes	4
65.62	odd	12		650.2.o.f.399.1		8
65.64	even	2		8450.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.e.d.451.2	✓	4	13.10	even	6
650.2.e.d.601.2	yes	4	13.4	even	6
650.2.e.i.451.1	yes	4	65.49	even	6
650.2.e.i.601.1	yes	4	65.4	even	6
650.2.o.f.399.1		8	65.62	odd	12
650.2.o.f.399.4		8	65.23	odd	12
650.2.o.f.549.1		8	65.43	odd	12
650.2.o.f.549.4		8	65.17	odd	12
8450.2.a.bb.1.2		2	65.64	even	2
8450.2.a.be.1.1		2	1.1	even	1	trivial
8450.2.a.bh.1.2		2	5.4	even	2
8450.2.a.bk.1.1		2	13.12	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} +1.79129 q^{6} +2.79129 q^{7} -1.00000 q^{8} +0.208712 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.79129 q^{3} +1.00000 q^{4} +1.79129 q^{6} +2.79129 q^{7} -1.00000 q^{8} +0.208712 q^{9} -0.791288 q^{11} -1.79129 q^{12} -2.79129 q^{14} +1.00000 q^{16} -3.79129 q^{17} -0.208712 q^{18} -5.00000 q^{19} -5.00000 q^{21} +0.791288 q^{22} +4.58258 q^{23} +1.79129 q^{24} +5.00000 q^{27} +2.79129 q^{28} +0.791288 q^{29} -0.417424 q^{31} -1.00000 q^{32} +1.41742 q^{33} +3.79129 q^{34} +0.208712 q^{36} +7.37386 q^{37} +5.00000 q^{38} +6.16515 q^{41} +5.00000 q^{42} -11.9564 q^{43} -0.791288 q^{44} -4.58258 q^{46} -3.79129 q^{47} -1.79129 q^{48} +0.791288 q^{49} +6.79129 q^{51} +4.58258 q^{53} -5.00000 q^{54} -2.79129 q^{56} +8.95644 q^{57} -0.791288 q^{58} -4.58258 q^{59} +0.582576 q^{61} +0.417424 q^{62} +0.582576 q^{63} +1.00000 q^{64} -1.41742 q^{66} +11.0000 q^{67} -3.79129 q^{68} -8.20871 q^{69} +1.58258 q^{71} -0.208712 q^{72} +11.1652 q^{73} -7.37386 q^{74} -5.00000 q^{76} -2.20871 q^{77} -7.95644 q^{79} -9.58258 q^{81} -6.16515 q^{82} +12.1652 q^{83} -5.00000 q^{84} +11.9564 q^{86} -1.41742 q^{87} +0.791288 q^{88} -14.3739 q^{89} +4.58258 q^{92} +0.747727 q^{93} +3.79129 q^{94} +1.79129 q^{96} -13.7913 q^{97} -0.791288 q^{98} -0.165151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9	\( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} - q^{14} + 2 q^{16} - 3 q^{17} - 5 q^{18} - 10 q^{19} - 10 q^{21} - 3 q^{22} - q^{24} + 10 q^{27} + q^{28} - 3 q^{29} - 10 q^{31} - 2 q^{32} + 12 q^{33} + 3 q^{34} + 5 q^{36} + q^{37} + 10 q^{38} - 6 q^{41} + 10 q^{42} - q^{43} + 3 q^{44} - 3 q^{47} + q^{48} - 3 q^{49} + 9 q^{51} - 10 q^{54} - q^{56} - 5 q^{57} + 3 q^{58} - 8 q^{61} + 10 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{66} + 22 q^{67} - 3 q^{68} - 21 q^{69} - 6 q^{71} - 5 q^{72} + 4 q^{73} - q^{74} - 10 q^{76} - 9 q^{77} + 7 q^{79} - 10 q^{81} + 6 q^{82} + 6 q^{83} - 10 q^{84} + q^{86} - 12 q^{87} - 3 q^{88} - 15 q^{89} - 26 q^{93} + 3 q^{94} - q^{96} - 23 q^{97} + 3 q^{98} + 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + 5 * q^9 + 3 * q^11 + q^12 - q^14 + 2 * q^16 - 3 * q^17 - 5 * q^18 - 10 * q^19 - 10 * q^21 - 3 * q^22 - q^24 + 10 * q^27 + q^28 - 3 * q^29 - 10 * q^31 - 2 * q^32 + 12 * q^33 + 3 * q^34 + 5 * q^36 + q^37 + 10 * q^38 - 6 * q^41 + 10 * q^42 - q^43 + 3 * q^44 - 3 * q^47 + q^48 - 3 * q^49 + 9 * q^51 - 10 * q^54 - q^56 - 5 * q^57 + 3 * q^58 - 8 * q^61 + 10 * q^62 - 8 * q^63 + 2 * q^64 - 12 * q^66 + 22 * q^67 - 3 * q^68 - 21 * q^69 - 6 * q^71 - 5 * q^72 + 4 * q^73 - q^74 - 10 * q^76 - 9 * q^77 + 7 * q^79 - 10 * q^81 + 6 * q^82 + 6 * q^83 - 10 * q^84 + q^86 - 12 * q^87 - 3 * q^88 - 15 * q^89 - 26 * q^93 + 3 * q^94 - q^96 - 23 * q^97 + 3 * q^98 + 18 * q^99

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