LMFDB - Embedded newform 5577.2.a.bf.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5577,2,Mod(1,5577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5577, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5577.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5577 = 3 \cdot 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5577.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:	$44.5325692073$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 5 x^{12} + 90 x^{11} - 84 x^{10} - 450 x^{9} + 761 x^{8} + 782 x^{7} - 2061 x^{6} + \cdots + 64 $ x^14 - 6x^13 - 5x^12 + 90x^11 - 84x^10 - 450x^9 + 761x^8 + 782x^7 - 2061x^6 - 92x^5 + 1921x^4 - 400x^3 - 635x^2 + 112*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.81001$ of defining polynomial
Character	$\chi$	$=$	5577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81001 q^{2} +1.00000 q^{3} +1.27615 q^{4} -2.99723 q^{5} -1.81001 q^{6} -4.54162 q^{7} +1.31019 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.81001 q^{2} +1.00000 q^{3} +1.27615 q^{4} -2.99723 q^{5} -1.81001 q^{6} -4.54162 q^{7} +1.31019 q^{8} +1.00000 q^{9} +5.42503 q^{10} -1.00000 q^{11} +1.27615 q^{12} +8.22038 q^{14} -2.99723 q^{15} -4.92374 q^{16} +5.59175 q^{17} -1.81001 q^{18} -0.429586 q^{19} -3.82491 q^{20} -4.54162 q^{21} +1.81001 q^{22} -7.72067 q^{23} +1.31019 q^{24} +3.98341 q^{25} +1.00000 q^{27} -5.79576 q^{28} +4.56940 q^{29} +5.42503 q^{30} -0.104718 q^{31} +6.29167 q^{32} -1.00000 q^{33} -10.1211 q^{34} +13.6123 q^{35} +1.27615 q^{36} +8.29366 q^{37} +0.777556 q^{38} -3.92693 q^{40} -10.5956 q^{41} +8.22038 q^{42} +2.28838 q^{43} -1.27615 q^{44} -2.99723 q^{45} +13.9745 q^{46} +9.86720 q^{47} -4.92374 q^{48} +13.6263 q^{49} -7.21002 q^{50} +5.59175 q^{51} -10.6258 q^{53} -1.81001 q^{54} +2.99723 q^{55} -5.95036 q^{56} -0.429586 q^{57} -8.27068 q^{58} +8.83165 q^{59} -3.82491 q^{60} -5.46149 q^{61} +0.189541 q^{62} -4.54162 q^{63} -1.54051 q^{64} +1.81001 q^{66} -6.76105 q^{67} +7.13589 q^{68} -7.72067 q^{69} -24.6384 q^{70} +2.50552 q^{71} +1.31019 q^{72} +13.2630 q^{73} -15.0116 q^{74} +3.98341 q^{75} -0.548215 q^{76} +4.54162 q^{77} +7.11430 q^{79} +14.7576 q^{80} +1.00000 q^{81} +19.1781 q^{82} +0.999903 q^{83} -5.79576 q^{84} -16.7598 q^{85} -4.14200 q^{86} +4.56940 q^{87} -1.31019 q^{88} -7.25837 q^{89} +5.42503 q^{90} -9.85270 q^{92} -0.104718 q^{93} -17.8598 q^{94} +1.28757 q^{95} +6.29167 q^{96} +9.47227 q^{97} -24.6637 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 6 q^{2} + 14 q^{3} + 18 q^{4} - 12 q^{5} - 6 q^{6} - 12 q^{7} - 12 q^{8} + 14 q^{9}+O(q^{10}) $ 14 * q - 6 * q^2 + 14 * q^3 + 18 * q^4 - 12 * q^5 - 6 * q^6 - 12 * q^7 - 12 * q^8 + 14 * q^9	$ 14 q - 6 q^{2} + 14 q^{3} + 18 q^{4} - 12 q^{5} - 6 q^{6} - 12 q^{7} - 12 q^{8} + 14 q^{9} - 14 q^{11} + 18 q^{12} - 2 q^{14} - 12 q^{15} + 22 q^{16} + 2 q^{17} - 6 q^{18} + 2 q^{19} - 44 q^{20} - 12 q^{21} + 6 q^{22} + 2 q^{23} - 12 q^{24} + 20 q^{25} + 14 q^{27} - 24 q^{28} - 28 q^{31} - 30 q^{32} - 14 q^{33} + 16 q^{34} - 2 q^{35} + 18 q^{36} - 10 q^{38} + 10 q^{40} - 40 q^{41} - 2 q^{42} - 2 q^{43} - 18 q^{44} - 12 q^{45} - 32 q^{46} - 48 q^{47} + 22 q^{48} + 10 q^{49} + 2 q^{51} + 8 q^{53} - 6 q^{54} + 12 q^{55} - 10 q^{56} + 2 q^{57} + 16 q^{58} - 40 q^{59} - 44 q^{60} + 4 q^{61} - 6 q^{62} - 12 q^{63} + 16 q^{64} + 6 q^{66} - 40 q^{67} + 22 q^{68} + 2 q^{69} + 40 q^{70} - 36 q^{71} - 12 q^{72} - 10 q^{73} - 48 q^{74} + 20 q^{75} + 4 q^{76} + 12 q^{77} - 24 q^{79} - 68 q^{80} + 14 q^{81} + 46 q^{82} - 12 q^{83} - 24 q^{84} - 34 q^{85} - 48 q^{86} + 12 q^{88} - 20 q^{89} + 36 q^{92} - 28 q^{93} - 50 q^{94} - 60 q^{95} - 30 q^{96} + 16 q^{97} - 44 q^{98} - 14 q^{99}+O(q^{100}) $ 14 * q - 6 * q^2 + 14 * q^3 + 18 * q^4 - 12 * q^5 - 6 * q^6 - 12 * q^7 - 12 * q^8 + 14 * q^9 - 14 * q^11 + 18 * q^12 - 2 * q^14 - 12 * q^15 + 22 * q^16 + 2 * q^17 - 6 * q^18 + 2 * q^19 - 44 * q^20 - 12 * q^21 + 6 * q^22 + 2 * q^23 - 12 * q^24 + 20 * q^25 + 14 * q^27 - 24 * q^28 - 28 * q^31 - 30 * q^32 - 14 * q^33 + 16 * q^34 - 2 * q^35 + 18 * q^36 - 10 * q^38 + 10 * q^40 - 40 * q^41 - 2 * q^42 - 2 * q^43 - 18 * q^44 - 12 * q^45 - 32 * q^46 - 48 * q^47 + 22 * q^48 + 10 * q^49 + 2 * q^51 + 8 * q^53 - 6 * q^54 + 12 * q^55 - 10 * q^56 + 2 * q^57 + 16 * q^58 - 40 * q^59 - 44 * q^60 + 4 * q^61 - 6 * q^62 - 12 * q^63 + 16 * q^64 + 6 * q^66 - 40 * q^67 + 22 * q^68 + 2 * q^69 + 40 * q^70 - 36 * q^71 - 12 * q^72 - 10 * q^73 - 48 * q^74 + 20 * q^75 + 4 * q^76 + 12 * q^77 - 24 * q^79 - 68 * q^80 + 14 * q^81 + 46 * q^82 - 12 * q^83 - 24 * q^84 - 34 * q^85 - 48 * q^86 + 12 * q^88 - 20 * q^89 + 36 * q^92 - 28 * q^93 - 50 * q^94 - 60 * q^95 - 30 * q^96 + 16 * q^97 - 44 * q^98 - 14 * 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 $

$2$	−1.81001	−1.27987	−0.639936	−	0.768428i	$-0.721039\pi$
$2$	−1.81001	−1.27987	−0.639936	+	0.768428i	$0.721039\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.27615	0.638073
$5$	−2.99723	−1.34040	−0.670202	−	0.742179i	$-0.733792\pi$
$5$	−2.99723	−1.34040	−0.670202	+	0.742179i	$0.733792\pi$
$6$	−1.81001	−0.738935
$7$	−4.54162	−1.71657	−0.858285	−	0.513174i	$-0.828470\pi$
$7$	−4.54162	−1.71657	−0.858285	+	0.513174i	$0.828470\pi$
$8$	1.31019	0.463220
$9$	1.00000	0.333333
$10$	5.42503	1.71555
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.27615	0.368392
$13$	0	0
$13$	0	0
$14$	8.22038	2.19699
$15$	−2.99723	−0.773882
$16$	−4.92374	−1.23094
$17$	5.59175	1.35620	0.678100	−	0.734970i	$-0.262804\pi$
$17$	5.59175	1.35620	0.678100	+	0.734970i	$0.262804\pi$
$18$	−1.81001	−0.426624
$19$	−0.429586	−0.0985538	−0.0492769	−	0.998785i	$-0.515692\pi$
$19$	−0.429586	−0.0985538	−0.0492769	+	0.998785i	$0.515692\pi$
$20$	−3.82491	−0.855275
$21$	−4.54162	−0.991062
$22$	1.81001	0.385896
$23$	−7.72067	−1.60987	−0.804935	−	0.593363i	$-0.797800\pi$
$23$	−7.72067	−1.60987	−0.804935	+	0.593363i	$0.797800\pi$
$24$	1.31019	0.267440
$25$	3.98341	0.796682
$26$	0	0
$27$	1.00000	0.192450
$28$	−5.79576	−1.09530
$29$	4.56940	0.848517	0.424259	−	0.905541i	$-0.360535\pi$
$29$	4.56940	0.848517	0.424259	+	0.905541i	$0.360535\pi$
$30$	5.42503	0.990471
$31$	−0.104718	−0.0188080	−0.00940398	−	0.999956i	$-0.502993\pi$
$31$	−0.104718	−0.0188080	−0.00940398	+	0.999956i	$0.502993\pi$
$32$	6.29167	1.11222
$33$	−1.00000	−0.174078
$34$	−10.1211	−1.73576
$35$	13.6123	2.30090
$36$	1.27615	0.212691
$37$	8.29366	1.36347	0.681734	−	0.731600i	$-0.261226\pi$
$37$	8.29366	1.36347	0.681734	+	0.731600i	$0.261226\pi$
$38$	0.777556	0.126136
$39$	0	0
$40$	−3.92693	−0.620902
$41$	−10.5956	−1.65475	−0.827376	−	0.561648i	$-0.810167\pi$
$41$	−10.5956	−1.65475	−0.827376	+	0.561648i	$0.810167\pi$
$42$	8.22038	1.26843
$43$	2.28838	0.348975	0.174488	−	0.984659i	$-0.444173\pi$
$43$	2.28838	0.348975	0.174488	+	0.984659i	$0.444173\pi$
$44$	−1.27615	−0.192386
$45$	−2.99723	−0.446801
$46$	13.9745	2.06043
$47$	9.86720	1.43928	0.719639	−	0.694348i	$-0.244307\pi$
$47$	9.86720	1.43928	0.719639	+	0.694348i	$0.244307\pi$
$48$	−4.92374	−0.710681
$49$	13.6263	1.94661
$50$	−7.21002	−1.01965
$51$	5.59175	0.783002
$52$	0	0
$53$	−10.6258	−1.45956	−0.729780	−	0.683682i	$-0.760378\pi$
$53$	−10.6258	−1.45956	−0.729780	+	0.683682i	$0.760378\pi$
$54$	−1.81001	−0.246312
$55$	2.99723	0.404147
$56$	−5.95036	−0.795150
$57$	−0.429586	−0.0569001
$58$	−8.27068	−1.08599
$59$	8.83165	1.14978	0.574892	−	0.818229i	$-0.305044\pi$
$59$	8.83165	1.14978	0.574892	+	0.818229i	$0.305044\pi$
$60$	−3.82491	−0.493793
$61$	−5.46149	−0.699272	−0.349636	−	0.936886i	$-0.613695\pi$
$61$	−5.46149	−0.699272	−0.349636	+	0.936886i	$0.613695\pi$
$62$	0.189541	0.0240718
$63$	−4.54162	−0.572190
$64$	−1.54051	−0.192564
$65$	0	0
$66$	1.81001	0.222797
$67$	−6.76105	−0.825994	−0.412997	−	0.910732i	$-0.635518\pi$
$67$	−6.76105	−0.825994	−0.412997	+	0.910732i	$0.635518\pi$
$68$	7.13589	0.865354
$69$	−7.72067	−0.929459
$70$	−24.6384	−2.94485
$71$	2.50552	0.297351	0.148675	−	0.988886i	$-0.452499\pi$
$71$	2.50552	0.297351	0.148675	+	0.988886i	$0.452499\pi$
$72$	1.31019	0.154407
$73$	13.2630	1.55232	0.776160	−	0.630536i	$-0.217165\pi$
$73$	13.2630	1.55232	0.776160	+	0.630536i	$0.217165\pi$
$74$	−15.0116	−1.74507
$75$	3.98341	0.459965
$76$	−0.548215	−0.0628845
$77$	4.54162	0.517565
$78$	0	0
$79$	7.11430	0.800421	0.400210	−	0.916423i	$-0.368937\pi$
$79$	7.11430	0.800421	0.400210	+	0.916423i	$0.368937\pi$
$80$	14.7576	1.64995
$81$	1.00000	0.111111
$82$	19.1781	2.11787
$83$	0.999903	0.109754	0.0548768	−	0.998493i	$-0.482523\pi$
$83$	0.999903	0.109754	0.0548768	+	0.998493i	$0.482523\pi$
$84$	−5.79576	−0.632370
$85$	−16.7598	−1.81785
$86$	−4.14200	−0.446644
$87$	4.56940	0.489892
$88$	−1.31019	−0.139666
$89$	−7.25837	−0.769386	−0.384693	−	0.923045i	$-0.625693\pi$
$89$	−7.25837	−0.769386	−0.384693	+	0.923045i	$0.625693\pi$
$90$	5.42503	0.571849
$91$	0	0
$92$	−9.85270	−1.02721
$93$	−0.104718	−0.0108588
$94$	−17.8598	−1.84209
$95$	1.28757	0.132102
$96$	6.29167	0.642141
$97$	9.47227	0.961764	0.480882	−	0.876785i	$-0.340317\pi$
$97$	9.47227	0.961764	0.480882	+	0.876785i	$0.340317\pi$
$98$	−24.6637	−2.49141
$99$	−1.00000	−0.100504
$100$	5.08341	0.508341
$101$	2.64259	0.262948	0.131474	−	0.991320i	$-0.458029\pi$
$101$	2.64259	0.262948	0.131474	+	0.991320i	$0.458029\pi$
$102$	−10.1211	−1.00214
$103$	14.4302	1.42185	0.710927	−	0.703265i	$-0.248275\pi$
$103$	14.4302	1.42185	0.710927	+	0.703265i	$0.248275\pi$
$104$	0	0
$105$	13.6123	1.32842
$106$	19.2328	1.86805
$107$	13.5851	1.31332	0.656661	−	0.754186i	$-0.271968\pi$
$107$	13.5851	1.31332	0.656661	+	0.754186i	$0.271968\pi$
$108$	1.27615	0.122797
$109$	0.777084	0.0744312	0.0372156	−	0.999307i	$-0.488151\pi$
$109$	0.777084	0.0744312	0.0372156	+	0.999307i	$0.488151\pi$
$110$	−5.42503	−0.517256
$111$	8.29366	0.787199
$112$	22.3617	2.11299
$113$	−10.8816	−1.02366	−0.511828	−	0.859088i	$-0.671031\pi$
$113$	−10.8816	−1.02366	−0.511828	+	0.859088i	$0.671031\pi$
$114$	0.777556	0.0728248
$115$	23.1406	2.15788
$116$	5.83123	0.541416
$117$	0	0
$118$	−15.9854	−1.47158
$119$	−25.3956	−2.32801
$120$	−3.92693	−0.358478
$121$	1.00000	0.0909091
$122$	9.88536	0.894978
$123$	−10.5956	−0.955372
$124$	−0.133636	−0.0120009
$125$	3.04696	0.272528
$126$	8.22038	0.732330
$127$	−7.75630	−0.688260	−0.344130	−	0.938922i	$-0.611826\pi$
$127$	−7.75630	−0.688260	−0.344130	+	0.938922i	$0.611826\pi$
$128$	−9.79499	−0.865763
$129$	2.28838	0.201481
$130$	0	0
$131$	4.07520	0.356052	0.178026	−	0.984026i	$-0.443029\pi$
$131$	4.07520	0.356052	0.178026	+	0.984026i	$0.443029\pi$
$132$	−1.27615	−0.111074
$133$	1.95101	0.169174
$134$	12.2376	1.05717
$135$	−2.99723	−0.257961
$136$	7.32623	0.628219
$137$	3.69680	0.315839	0.157920	−	0.987452i	$-0.449521\pi$
$137$	3.69680	0.315839	0.157920	+	0.987452i	$0.449521\pi$
$138$	13.9745	1.18959
$139$	0.566920	0.0480855	0.0240428	−	0.999711i	$-0.492346\pi$
$139$	0.566920	0.0480855	0.0240428	+	0.999711i	$0.492346\pi$
$140$	17.3713	1.46814
$141$	9.86720	0.830968
$142$	−4.53503	−0.380571
$143$	0	0
$144$	−4.92374	−0.410312
$145$	−13.6956	−1.13736
$146$	−24.0062	−1.98677
$147$	13.6263	1.12388
$148$	10.5839	0.869992
$149$	0.155025	0.0127002	0.00635009	−	0.999980i	$-0.497979\pi$
$149$	0.155025	0.0127002	0.00635009	+	0.999980i	$0.497979\pi$
$150$	−7.21002	−0.588696
$151$	11.5828	0.942594	0.471297	−	0.881975i	$-0.343786\pi$
$151$	11.5828	0.942594	0.471297	+	0.881975i	$0.343786\pi$
$152$	−0.562837	−0.0456521
$153$	5.59175	0.452066
$154$	−8.22038	−0.662417
$155$	0.313865	0.0252103
$156$	0	0
$157$	−10.1019	−0.806219	−0.403110	−	0.915152i	$-0.632071\pi$
$157$	−10.1019	−0.806219	−0.403110	+	0.915152i	$0.632071\pi$
$158$	−12.8770	−1.02444
$159$	−10.6258	−0.842677
$160$	−18.8576	−1.49082
$161$	35.0643	2.76345
$162$	−1.81001	−0.142208
$163$	11.0607	0.866339	0.433170	−	0.901312i	$-0.357395\pi$
$163$	11.0607	0.866339	0.433170	+	0.901312i	$0.357395\pi$
$164$	−13.5215	−1.05585
$165$	2.99723	0.233334
$166$	−1.80984	−0.140471
$167$	−23.7250	−1.83590	−0.917949	−	0.396698i	$-0.870156\pi$
$167$	−23.7250	−1.83590	−0.917949	+	0.396698i	$0.870156\pi$
$168$	−5.95036	−0.459080
$169$	0	0
$170$	30.3354	2.32662
$171$	−0.429586	−0.0328513
$172$	2.92031	0.222672
$173$	−13.3655	−1.01616	−0.508080	−	0.861310i	$-0.669644\pi$
$173$	−13.3655	−1.01616	−0.508080	+	0.861310i	$0.669644\pi$
$174$	−8.27068	−0.626999
$175$	−18.0911	−1.36756
$176$	4.92374	0.371141
$177$	8.83165	0.663828
$178$	13.1377	0.984716
$179$	−17.1585	−1.28249	−0.641243	−	0.767338i	$-0.721581\pi$
$179$	−17.1585	−1.28249	−0.641243	+	0.767338i	$0.721581\pi$
$180$	−3.82491	−0.285092
$181$	7.27163	0.540496	0.270248	−	0.962791i	$-0.412894\pi$
$181$	7.27163	0.540496	0.270248	+	0.962791i	$0.412894\pi$
$182$	0	0
$183$	−5.46149	−0.403725
$184$	−10.1155	−0.745725
$185$	−24.8580	−1.82760
$186$	0.189541	0.0138979
$187$	−5.59175	−0.408909
$188$	12.5920	0.918365
$189$	−4.54162	−0.330354
$190$	−2.33052	−0.169074
$191$	−11.9166	−0.862253	−0.431126	−	0.902292i	$-0.641884\pi$
$191$	−11.9166	−0.862253	−0.431126	+	0.902292i	$0.641884\pi$
$192$	−1.54051	−0.111177
$193$	7.47020	0.537717	0.268858	−	0.963180i	$-0.413354\pi$
$193$	7.47020	0.537717	0.268858	+	0.963180i	$0.413354\pi$
$194$	−17.1449	−1.23093
$195$	0	0
$196$	17.3891	1.24208
$197$	−3.32844	−0.237142	−0.118571	−	0.992946i	$-0.537831\pi$
$197$	−3.32844	−0.237142	−0.118571	+	0.992946i	$0.537831\pi$
$198$	1.81001	0.128632
$199$	1.90310	0.134907	0.0674535	−	0.997722i	$-0.478513\pi$
$199$	1.90310	0.134907	0.0674535	+	0.997722i	$0.478513\pi$
$200$	5.21900	0.369039
$201$	−6.76105	−0.476888
$202$	−4.78313	−0.336540
$203$	−20.7525	−1.45654
$204$	7.13589	0.499612
$205$	31.7574	2.21804
$206$	−26.1189	−1.81979
$207$	−7.72067	−0.536623
$208$	0	0
$209$	0.429586	0.0297151
$210$	−24.6384	−1.70021
$211$	−1.11069	−0.0764631	−0.0382316	−	0.999269i	$-0.512172\pi$
$211$	−1.11069	−0.0764631	−0.0382316	+	0.999269i	$0.512172\pi$
$212$	−13.5600	−0.931306
$213$	2.50552	0.171676
$214$	−24.5892	−1.68088
$215$	−6.85882	−0.467768
$216$	1.31019	0.0891468
$217$	0.475590	0.0322852
$218$	−1.40653	−0.0952624
$219$	13.2630	0.896232
$220$	3.82491	0.257875
$221$	0	0
$222$	−15.0116	−1.00751
$223$	−21.7416	−1.45592	−0.727961	−	0.685618i	$-0.759532\pi$
$223$	−21.7416	−1.45592	−0.727961	+	0.685618i	$0.759532\pi$
$224$	−28.5743	−1.90920
$225$	3.98341	0.265561
$226$	19.6958	1.31015
$227$	−27.5677	−1.82973	−0.914866	−	0.403757i	$-0.867704\pi$
$227$	−27.5677	−1.82973	−0.914866	+	0.403757i	$0.867704\pi$
$228$	−0.548215	−0.0363064
$229$	17.5599	1.16039	0.580197	−	0.814476i	$-0.302976\pi$
$229$	17.5599	1.16039	0.580197	+	0.814476i	$0.302976\pi$
$230$	−41.8849	−2.76181
$231$	4.54162	0.298816
$232$	5.98676	0.393050
$233$	−5.10960	−0.334741	−0.167371	−	0.985894i	$-0.553528\pi$
$233$	−5.10960	−0.334741	−0.167371	+	0.985894i	$0.553528\pi$
$234$	0	0
$235$	−29.5743	−1.92921
$236$	11.2705	0.733646
$237$	7.11430	0.462123
$238$	45.9663	2.97956
$239$	−11.8494	−0.766475	−0.383238	−	0.923650i	$-0.625191\pi$
$239$	−11.8494	−0.766475	−0.383238	+	0.923650i	$0.625191\pi$
$240$	14.7576	0.952600
$241$	−0.628349	−0.0404755	−0.0202378	−	0.999795i	$-0.506442\pi$
$241$	−0.628349	−0.0404755	−0.0202378	+	0.999795i	$0.506442\pi$
$242$	−1.81001	−0.116352
$243$	1.00000	0.0641500
$244$	−6.96965	−0.446186
$245$	−40.8411	−2.60924
$246$	19.1781	1.22275
$247$	0	0
$248$	−0.137200	−0.00871223
$249$	0.999903	0.0633663
$250$	−5.51503	−0.348801
$251$	−12.8077	−0.808418	−0.404209	−	0.914667i	$-0.632453\pi$
$251$	−12.8077	−0.808418	−0.404209	+	0.914667i	$0.632453\pi$
$252$	−5.79576	−0.365099
$253$	7.72067	0.485394
$254$	14.0390	0.880885
$255$	−16.7598	−1.04954
$256$	20.8101	1.30063
$257$	13.1560	0.820649	0.410325	−	0.911939i	$-0.365415\pi$
$257$	13.1560	0.820649	0.410325	+	0.911939i	$0.365415\pi$
$258$	−4.14200	−0.257870
$259$	−37.6666	−2.34049
$260$	0	0
$261$	4.56940	0.282839
$262$	−7.37617	−0.455701
$263$	−8.70410	−0.536718	−0.268359	−	0.963319i	$-0.586481\pi$
$263$	−8.70410	−0.536718	−0.268359	+	0.963319i	$0.586481\pi$
$264$	−1.31019	−0.0806363
$265$	31.8479	1.95640
$266$	−3.53136	−0.216522
$267$	−7.25837	−0.444205
$268$	−8.62808	−0.527044
$269$	10.1179	0.616896	0.308448	−	0.951241i	$-0.400190\pi$
$269$	10.1179	0.616896	0.308448	+	0.951241i	$0.400190\pi$
$270$	5.42503	0.330157
$271$	−28.9152	−1.75648	−0.878238	−	0.478224i	$-0.841281\pi$
$271$	−28.9152	−1.75648	−0.878238	+	0.478224i	$0.841281\pi$
$272$	−27.5324	−1.66939
$273$	0	0
$274$	−6.69126	−0.404234
$275$	−3.98341	−0.240209
$276$	−9.85270	−0.593063
$277$	−9.29036	−0.558204	−0.279102	−	0.960262i	$-0.590037\pi$
$277$	−9.29036	−0.558204	−0.279102	+	0.960262i	$0.590037\pi$
$278$	−1.02613	−0.0615433
$279$	−0.104718	−0.00626932
$280$	17.8346	1.06582
$281$	−1.40995	−0.0841105	−0.0420553	−	0.999115i	$-0.513391\pi$
$281$	−1.40995	−0.0841105	−0.0420553	+	0.999115i	$0.513391\pi$
$282$	−17.8598	−1.06353
$283$	−3.21205	−0.190937	−0.0954684	−	0.995432i	$-0.530435\pi$
$283$	−3.21205	−0.190937	−0.0954684	+	0.995432i	$0.530435\pi$
$284$	3.19741	0.189732
$285$	1.28757	0.0762691
$286$	0	0
$287$	48.1211	2.84050
$288$	6.29167	0.370740
$289$	14.2677	0.839277
$290$	24.7892	1.45567
$291$	9.47227	0.555274
$292$	16.9256	0.990493
$293$	−21.7378	−1.26994	−0.634969	−	0.772538i	$-0.718987\pi$
$293$	−21.7378	−1.26994	−0.634969	+	0.772538i	$0.718987\pi$
$294$	−24.6637	−1.43842
$295$	−26.4705	−1.54117
$296$	10.8662	0.631586
$297$	−1.00000	−0.0580259
$298$	−0.280598	−0.0162546
$299$	0	0
$300$	5.08341	0.293491
$301$	−10.3930	−0.599040
$302$	−20.9650	−1.20640
$303$	2.64259	0.151813
$304$	2.11517	0.121313
$305$	16.3693	0.937306
$306$	−10.1211	−0.578587
$307$	−15.2363	−0.869579	−0.434789	−	0.900532i	$-0.643177\pi$
$307$	−15.2363	−0.869579	−0.434789	+	0.900532i	$0.643177\pi$
$308$	5.79576	0.330244
$309$	14.4302	0.820908
$310$	−0.568100	−0.0322659
$311$	−7.66303	−0.434530	−0.217265	−	0.976113i	$-0.569714\pi$
$311$	−7.66303	−0.434530	−0.217265	+	0.976113i	$0.569714\pi$
$312$	0	0
$313$	13.9879	0.790642	0.395321	−	0.918543i	$-0.370633\pi$
$313$	13.9879	0.790642	0.395321	+	0.918543i	$0.370633\pi$
$314$	18.2846	1.03186
$315$	13.6123	0.766965
$316$	9.07888	0.510727
$317$	−2.11260	−0.118655	−0.0593277	−	0.998239i	$-0.518896\pi$
$317$	−2.11260	−0.118655	−0.0593277	+	0.998239i	$0.518896\pi$
$318$	19.2328	1.07852
$319$	−4.56940	−0.255838
$320$	4.61728	0.258114
$321$	13.5851	0.758247
$322$	−63.4668	−3.53687
$323$	−2.40214	−0.133659
$324$	1.27615	0.0708970
$325$	0	0
$326$	−20.0200	−1.10880
$327$	0.777084	0.0429729
$328$	−13.8822	−0.766515
$329$	−44.8130	−2.47062
$330$	−5.42503	−0.298638
$331$	15.2543	0.838454	0.419227	−	0.907882i	$-0.362301\pi$
$331$	15.2543	0.838454	0.419227	+	0.907882i	$0.362301\pi$
$332$	1.27602	0.0700308
$333$	8.29366	0.454490
$334$	42.9426	2.34972
$335$	20.2644	1.10716
$336$	22.3617	1.21993
$337$	−19.7832	−1.07766	−0.538831	−	0.842414i	$-0.681134\pi$
$337$	−19.7832	−1.07766	−0.538831	+	0.842414i	$0.681134\pi$
$338$	0	0
$339$	−10.8816	−0.591008
$340$	−21.3879	−1.15992
$341$	0.104718	0.00567081
$342$	0.777556	0.0420454
$343$	−30.0940	−1.62492
$344$	2.99821	0.161652
$345$	23.1406	1.24585
$346$	24.1917	1.30055
$347$	5.23479	0.281018	0.140509	−	0.990079i	$-0.455126\pi$
$347$	5.23479	0.281018	0.140509	+	0.990079i	$0.455126\pi$
$348$	5.83123	0.312587
$349$	−20.9482	−1.12133	−0.560664	−	0.828043i	$-0.689454\pi$
$349$	−20.9482	−1.12133	−0.560664	+	0.828043i	$0.689454\pi$
$350$	32.7452	1.75030
$351$	0	0
$352$	−6.29167	−0.335347
$353$	−14.2790	−0.759993	−0.379996	−	0.924988i	$-0.624075\pi$
$353$	−14.2790	−0.759993	−0.379996	+	0.924988i	$0.624075\pi$
$354$	−15.9854	−0.849615
$355$	−7.50964	−0.398570
$356$	−9.26274	−0.490924
$357$	−25.3956	−1.34408
$358$	31.0571	1.64142
$359$	−17.5026	−0.923751	−0.461875	−	0.886945i	$-0.652823\pi$
$359$	−17.5026	−0.923751	−0.461875	+	0.886945i	$0.652823\pi$
$360$	−3.92693	−0.206967
$361$	−18.8155	−0.990287
$362$	−13.1617	−0.691766
$363$	1.00000	0.0524864
$364$	0	0
$365$	−39.7524	−2.08073
$366$	9.88536	0.516716
$367$	9.18212	0.479303	0.239651	−	0.970859i	$-0.422967\pi$
$367$	9.18212	0.479303	0.239651	+	0.970859i	$0.422967\pi$
$368$	38.0146	1.98165
$369$	−10.5956	−0.551584
$370$	44.9933	2.33909
$371$	48.2581	2.50544
$372$	−0.133636	−0.00692870
$373$	12.4239	0.643287	0.321643	−	0.946861i	$-0.395765\pi$
$373$	12.4239	0.643287	0.321643	+	0.946861i	$0.395765\pi$
$374$	10.1211	0.523352
$375$	3.04696	0.157344
$376$	12.9279	0.666703
$377$	0	0
$378$	8.22038	0.422811
$379$	30.1527	1.54884	0.774420	−	0.632672i	$-0.218042\pi$
$379$	30.1527	1.54884	0.774420	+	0.632672i	$0.218042\pi$
$380$	1.64313	0.0842906
$381$	−7.75630	−0.397367
$382$	21.5691	1.10357
$383$	−8.92987	−0.456295	−0.228148	−	0.973627i	$-0.573267\pi$
$383$	−8.92987	−0.456295	−0.228148	+	0.973627i	$0.573267\pi$
$384$	−9.79499	−0.499848
$385$	−13.6123	−0.693746
$386$	−13.5212	−0.688209
$387$	2.28838	0.116325
$388$	12.0880	0.613675
$389$	−2.58520	−0.131075	−0.0655373	−	0.997850i	$-0.520876\pi$
$389$	−2.58520	−0.131075	−0.0655373	+	0.997850i	$0.520876\pi$
$390$	0	0
$391$	−43.1721	−2.18331
$392$	17.8529	0.901709
$393$	4.07520	0.205567
$394$	6.02452	0.303511
$395$	−21.3232	−1.07289
$396$	−1.27615	−0.0641287
$397$	−18.5419	−0.930593	−0.465297	−	0.885155i	$-0.654052\pi$
$397$	−18.5419	−0.930593	−0.465297	+	0.885155i	$0.654052\pi$
$398$	−3.44463	−0.172664
$399$	1.95101	0.0976729
$400$	−19.6133	−0.980665
$401$	21.1628	1.05682	0.528411	−	0.848989i	$-0.322788\pi$
$401$	21.1628	1.05682	0.528411	+	0.848989i	$0.322788\pi$
$402$	12.2376	0.610355
$403$	0	0
$404$	3.37234	0.167780
$405$	−2.99723	−0.148934
$406$	37.5622	1.86418
$407$	−8.29366	−0.411101
$408$	7.32623	0.362702
$409$	−17.1558	−0.848299	−0.424150	−	0.905592i	$-0.639427\pi$
$409$	−17.1558	−0.848299	−0.424150	+	0.905592i	$0.639427\pi$
$410$	−57.4814	−2.83880
$411$	3.69680	0.182350
$412$	18.4151	0.907247
$413$	−40.1100	−1.97368
$414$	13.9745	0.686809
$415$	−2.99694	−0.147114
$416$	0	0
$417$	0.566920	0.0277622
$418$	−0.777556	−0.0380315
$419$	−15.7242	−0.768176	−0.384088	−	0.923296i	$-0.625484\pi$
$419$	−15.7242	−0.768176	−0.384088	+	0.923296i	$0.625484\pi$
$420$	17.3713	0.847631
$421$	14.8298	0.722758	0.361379	−	0.932419i	$-0.382306\pi$
$421$	14.8298	0.722758	0.361379	+	0.932419i	$0.382306\pi$
$422$	2.01036	0.0978630
$423$	9.86720	0.479760
$424$	−13.9217	−0.676098
$425$	22.2742	1.08046
$426$	−4.53503	−0.219723
$427$	24.8040	1.20035
$428$	17.3366	0.837996
$429$	0	0
$430$	12.4146	0.598683
$431$	38.8705	1.87233	0.936164	−	0.351563i	$-0.114350\pi$
$431$	38.8705	1.87233	0.936164	+	0.351563i	$0.114350\pi$
$432$	−4.92374	−0.236894
$433$	31.8976	1.53290	0.766451	−	0.642302i	$-0.222021\pi$
$433$	31.8976	1.53290	0.766451	+	0.642302i	$0.222021\pi$
$434$	−0.860824	−0.0413209
$435$	−13.6956	−0.656652
$436$	0.991673	0.0474925
$437$	3.31669	0.158659
$438$	−24.0062	−1.14706
$439$	22.0899	1.05429	0.527147	−	0.849774i	$-0.323262\pi$
$439$	22.0899	1.05429	0.527147	+	0.849774i	$0.323262\pi$
$440$	3.92693	0.187209
$441$	13.6263	0.648870
$442$	0	0
$443$	39.1634	1.86071	0.930355	−	0.366660i	$-0.119499\pi$
$443$	39.1634	1.86071	0.930355	+	0.366660i	$0.119499\pi$
$444$	10.5839	0.502290
$445$	21.7550	1.03129
$446$	39.3525	1.86340
$447$	0.155025	0.00733245
$448$	6.99641	0.330549
$449$	30.3755	1.43351	0.716755	−	0.697325i	$-0.245627\pi$
$449$	30.3755	1.43351	0.716755	+	0.697325i	$0.245627\pi$
$450$	−7.21002	−0.339884
$451$	10.5956	0.498927
$452$	−13.8865	−0.653167
$453$	11.5828	0.544207
$454$	49.8979	2.34182
$455$	0	0
$456$	−0.562837	−0.0263573
$457$	30.3717	1.42073	0.710364	−	0.703835i	$-0.248531\pi$
$457$	30.3717	1.42073	0.710364	+	0.703835i	$0.248531\pi$
$458$	−31.7837	−1.48516
$459$	5.59175	0.261001
$460$	29.5308	1.37688
$461$	0.597729	0.0278390	0.0139195	−	0.999903i	$-0.495569\pi$
$461$	0.597729	0.0278390	0.0139195	+	0.999903i	$0.495569\pi$
$462$	−8.22038	−0.382447
$463$	−37.7369	−1.75378	−0.876892	−	0.480687i	$-0.840387\pi$
$463$	−37.7369	−1.75378	−0.876892	+	0.480687i	$0.840387\pi$
$464$	−22.4986	−1.04447
$465$	0.313865	0.0145552
$466$	9.24845	0.428426
$467$	−8.43576	−0.390360	−0.195180	−	0.980767i	$-0.562529\pi$
$467$	−8.43576	−0.390360	−0.195180	+	0.980767i	$0.562529\pi$
$468$	0	0
$469$	30.7061	1.41787
$470$	53.5299	2.46915
$471$	−10.1019	−0.465471
$472$	11.5711	0.532603
$473$	−2.28838	−0.105220
$474$	−12.8770	−0.591459
$475$	−1.71122	−0.0785161
$476$	−32.4085	−1.48544
$477$	−10.6258	−0.486520
$478$	21.4476	0.980991
$479$	−16.0997	−0.735613	−0.367807	−	0.929902i	$-0.619891\pi$
$479$	−16.0997	−0.735613	−0.367807	+	0.929902i	$0.619891\pi$
$480$	−18.8576	−0.860728
$481$	0	0
$482$	1.13732	0.0518035
$483$	35.0643	1.59548
$484$	1.27615	0.0580066
$485$	−28.3906	−1.28915
$486$	−1.81001	−0.0821038
$487$	−31.1833	−1.41305	−0.706525	−	0.707688i	$-0.749738\pi$
$487$	−31.1833	−1.41305	−0.706525	+	0.707688i	$0.749738\pi$
$488$	−7.15556	−0.323917
$489$	11.0607	0.500181
$490$	73.9229	3.33950
$491$	−6.04904	−0.272989	−0.136495	−	0.990641i	$-0.543584\pi$
$491$	−6.04904	−0.272989	−0.136495	+	0.990641i	$0.543584\pi$
$492$	−13.5215	−0.609597
$493$	25.5510	1.15076
$494$	0	0
$495$	2.99723	0.134716
$496$	0.515606	0.0231514
$497$	−11.3791	−0.510423
$498$	−1.80984	−0.0811007
$499$	9.70687	0.434539	0.217270	−	0.976112i	$-0.430285\pi$
$499$	9.70687	0.434539	0.217270	+	0.976112i	$0.430285\pi$
$500$	3.88836	0.173893
$501$	−23.7250	−1.05996
$502$	23.1822	1.03467
$503$	−6.67187	−0.297484	−0.148742	−	0.988876i	$-0.547522\pi$
$503$	−6.67187	−0.297484	−0.148742	+	0.988876i	$0.547522\pi$
$504$	−5.95036	−0.265050
$505$	−7.92047	−0.352456
$506$	−13.9745	−0.621243
$507$	0	0
$508$	−9.89817	−0.439160
$509$	−35.0462	−1.55340	−0.776698	−	0.629873i	$-0.783107\pi$
$509$	−35.0462	−1.55340	−0.776698	+	0.629873i	$0.783107\pi$
$510$	30.3354	1.34328
$511$	−60.2356	−2.66466
$512$	−18.0765	−0.798877
$513$	−0.429586	−0.0189667
$514$	−23.8125	−1.05033
$515$	−43.2508	−1.90586
$516$	2.92031	0.128560
$517$	−9.86720	−0.433959
$518$	68.1770	2.99553
$519$	−13.3655	−0.586680
$520$	0	0
$521$	−25.2873	−1.10786	−0.553928	−	0.832564i	$-0.686872\pi$
$521$	−25.2873	−1.10786	−0.553928	+	0.832564i	$0.686872\pi$
$522$	−8.27068	−0.361998
$523$	−25.5247	−1.11612	−0.558059	−	0.829801i	$-0.688454\pi$
$523$	−25.5247	−1.11612	−0.558059	+	0.829801i	$0.688454\pi$
$524$	5.20055	0.227187
$525$	−18.0911	−0.789561
$526$	15.7545	0.686930
$527$	−0.585559	−0.0255073
$528$	4.92374	0.214278
$529$	36.6087	1.59168
$530$	−57.6451	−2.50394
$531$	8.83165	0.383261
$532$	2.48978	0.107946
$533$	0	0
$534$	13.1377	0.568526
$535$	−40.7178	−1.76038
$536$	−8.85822	−0.382617
$537$	−17.1585	−0.740443
$538$	−18.3134	−0.789549
$539$	−13.6263	−0.586925
$540$	−3.82491	−0.164598
$541$	25.0265	1.07597	0.537987	−	0.842953i	$-0.319185\pi$
$541$	25.0265	1.07597	0.537987	+	0.842953i	$0.319185\pi$
$542$	52.3369	2.24806
$543$	7.27163	0.312056
$544$	35.1815	1.50839
$545$	−2.32910	−0.0997678
$546$	0	0
$547$	−30.3554	−1.29790	−0.648951	−	0.760830i	$-0.724792\pi$
$547$	−30.3554	−1.29790	−0.648951	+	0.760830i	$0.724792\pi$
$548$	4.71766	0.201528
$549$	−5.46149	−0.233091
$550$	7.21002	0.307436
$551$	−1.96295	−0.0836246
$552$	−10.1155	−0.430544
$553$	−32.3104	−1.37398
$554$	16.8157	0.714429
$555$	−24.8580	−1.05516
$556$	0.723472	0.0306821
$557$	−3.13681	−0.132911	−0.0664554	−	0.997789i	$-0.521169\pi$
$557$	−3.13681	−0.132911	−0.0664554	+	0.997789i	$0.521169\pi$
$558$	0.189541	0.00802393
$559$	0	0
$560$	−67.0234	−2.83225
$561$	−5.59175	−0.236084
$562$	2.55203	0.107651
$563$	−4.15226	−0.174997	−0.0874985	−	0.996165i	$-0.527887\pi$
$563$	−4.15226	−0.174997	−0.0874985	+	0.996165i	$0.527887\pi$
$564$	12.5920	0.530218
$565$	32.6147	1.37211
$566$	5.81386	0.244375
$567$	−4.54162	−0.190730
$568$	3.28270	0.137739
$569$	12.5584	0.526475	0.263238	−	0.964731i	$-0.415210\pi$
$569$	12.5584	0.526475	0.263238	+	0.964731i	$0.415210\pi$
$570$	−2.33052	−0.0976147
$571$	−11.4736	−0.480154	−0.240077	−	0.970754i	$-0.577173\pi$
$571$	−11.4736	−0.480154	−0.240077	+	0.970754i	$0.577173\pi$
$572$	0	0
$573$	−11.9166	−0.497822
$574$	−87.0997	−3.63547
$575$	−30.7546	−1.28255
$576$	−1.54051	−0.0641880
$577$	31.6119	1.31602	0.658010	−	0.753009i	$-0.271398\pi$
$577$	31.6119	1.31602	0.658010	+	0.753009i	$0.271398\pi$
$578$	−25.8247	−1.07417
$579$	7.47020	0.310451
$580$	−17.4775	−0.725716
$581$	−4.54117	−0.188400
$582$	−17.1449	−0.710680
$583$	10.6258	0.440074
$584$	17.3770	0.719066
$585$	0	0
$586$	39.3457	1.62536
$587$	−29.8286	−1.23116	−0.615579	−	0.788075i	$-0.711078\pi$
$587$	−29.8286	−1.23116	−0.615579	+	0.788075i	$0.711078\pi$
$588$	17.3891	0.717115
$589$	0.0449855	0.00185360
$590$	47.9120	1.97251
$591$	−3.32844	−0.136914
$592$	−40.8358	−1.67834
$593$	5.64786	0.231930	0.115965	−	0.993253i	$-0.463004\pi$
$593$	5.64786	0.231930	0.115965	+	0.993253i	$0.463004\pi$
$594$	1.81001	0.0742657
$595$	76.1165	3.12047
$596$	0.197835	0.00810364
$597$	1.90310	0.0778886
$598$	0	0
$599$	10.6802	0.436382	0.218191	−	0.975906i	$-0.429984\pi$
$599$	10.6802	0.436382	0.218191	+	0.975906i	$0.429984\pi$
$600$	5.21900	0.213065
$601$	−8.23162	−0.335775	−0.167887	−	0.985806i	$-0.553695\pi$
$601$	−8.23162	−0.335775	−0.167887	+	0.985806i	$0.553695\pi$
$602$	18.8114	0.766695
$603$	−6.76105	−0.275331
$604$	14.7813	0.601444
$605$	−2.99723	−0.121855
$606$	−4.78313	−0.194301
$607$	−12.8508	−0.521598	−0.260799	−	0.965393i	$-0.583986\pi$
$607$	−12.8508	−0.521598	−0.260799	+	0.965393i	$0.583986\pi$
$608$	−2.70281	−0.109614
$609$	−20.7525	−0.840933
$610$	−29.6287	−1.19963
$611$	0	0
$612$	7.13589	0.288451
$613$	−30.4198	−1.22864	−0.614321	−	0.789056i	$-0.710570\pi$
$613$	−30.4198	−1.22864	−0.614321	+	0.789056i	$0.710570\pi$
$614$	27.5778	1.11295
$615$	31.7574	1.28058
$616$	5.95036	0.239747
$617$	12.8886	0.518876	0.259438	−	0.965760i	$-0.416463\pi$
$617$	12.8886	0.518876	0.259438	+	0.965760i	$0.416463\pi$
$618$	−26.1189	−1.05066
$619$	24.0307	0.965876	0.482938	−	0.875655i	$-0.339570\pi$
$619$	24.0307	0.965876	0.482938	+	0.875655i	$0.339570\pi$
$620$	0.400538	0.0160860
$621$	−7.72067	−0.309820
$622$	13.8702	0.556143
$623$	32.9647	1.32070
$624$	0	0
$625$	−29.0495	−1.16198
$626$	−25.3182	−1.01192
$627$	0.429586	0.0171560
$628$	−12.8915	−0.514427
$629$	46.3761	1.84914
$630$	−24.6384	−0.981617
$631$	9.24422	0.368007	0.184003	−	0.982926i	$-0.441094\pi$
$631$	9.24422	0.368007	0.184003	+	0.982926i	$0.441094\pi$
$632$	9.32104	0.370771
$633$	−1.11069	−0.0441460
$634$	3.82383	0.151864
$635$	23.2474	0.922546
$636$	−13.5600	−0.537690
$637$	0	0
$638$	8.27068	0.327439
$639$	2.50552	0.0991169
$640$	29.3579	1.16047
$641$	15.5746	0.615159	0.307579	−	0.951522i	$-0.400481\pi$
$641$	15.5746	0.615159	0.307579	+	0.951522i	$0.400481\pi$
$642$	−24.5892	−0.970459
$643$	−8.49472	−0.334999	−0.167500	−	0.985872i	$-0.553569\pi$
$643$	−8.49472	−0.334999	−0.167500	+	0.985872i	$0.553569\pi$
$644$	44.7472	1.76329
$645$	−6.85882	−0.270066
$646$	4.34790	0.171066
$647$	−21.4881	−0.844783	−0.422391	−	0.906414i	$-0.638809\pi$
$647$	−21.4881	−0.844783	−0.422391	+	0.906414i	$0.638809\pi$
$648$	1.31019	0.0514689
$649$	−8.83165	−0.346673
$650$	0	0
$651$	0.475590	0.0186399
$652$	14.1150	0.552788
$653$	−36.4897	−1.42795	−0.713976	−	0.700170i	$-0.753107\pi$
$653$	−36.4897	−1.42795	−0.713976	+	0.700170i	$0.753107\pi$
$654$	−1.40653	−0.0549998
$655$	−12.2143	−0.477254
$656$	52.1699	2.03689
$657$	13.2630	0.517440
$658$	81.1121	3.16208
$659$	−19.0057	−0.740356	−0.370178	−	0.928961i	$-0.620703\pi$
$659$	−19.0057	−0.740356	−0.370178	+	0.928961i	$0.620703\pi$
$660$	3.82491	0.148884
$661$	27.4944	1.06941	0.534704	−	0.845039i	$-0.320423\pi$
$661$	27.4944	1.06941	0.534704	+	0.845039i	$0.320423\pi$
$662$	−27.6105	−1.07311
$663$	0	0
$664$	1.31006	0.0508401
$665$	−5.84765	−0.226762
$666$	−15.0116	−0.581689
$667$	−35.2788	−1.36600
$668$	−30.2766	−1.17144
$669$	−21.7416	−0.840577
$670$	−36.6789	−1.41703
$671$	5.46149	0.210838
$672$	−28.5743	−1.10228
$673$	−27.2729	−1.05129	−0.525647	−	0.850703i	$-0.676177\pi$
$673$	−27.2729	−1.05129	−0.525647	+	0.850703i	$0.676177\pi$
$674$	35.8079	1.37927
$675$	3.98341	0.153322
$676$	0	0
$677$	37.4349	1.43874	0.719370	−	0.694627i	$-0.244431\pi$
$677$	37.4349	1.43874	0.719370	+	0.694627i	$0.244431\pi$
$678$	19.6958	0.756414
$679$	−43.0194	−1.65093
$680$	−21.9584	−0.842067
$681$	−27.5677	−1.05640
$682$	−0.189541	−0.00725792
$683$	9.17617	0.351116	0.175558	−	0.984469i	$-0.443827\pi$
$683$	9.17617	0.351116	0.175558	+	0.984469i	$0.443827\pi$
$684$	−0.548215	−0.0209615
$685$	−11.0802	−0.423352
$686$	54.4704	2.07969
$687$	17.5599	0.669954
$688$	−11.2674	−0.429566
$689$	0	0
$690$	−41.8849	−1.59453
$691$	26.4798	1.00734	0.503670	−	0.863896i	$-0.331983\pi$
$691$	26.4798	1.00734	0.503670	+	0.863896i	$0.331983\pi$
$692$	−17.0563	−0.648384
$693$	4.54162	0.172522
$694$	−9.47503	−0.359667
$695$	−1.69919	−0.0644540
$696$	5.98676	0.226928
$697$	−59.2479	−2.24417
$698$	37.9164	1.43516
$699$	−5.10960	−0.193263
$700$	−23.0869	−0.872603
$701$	−17.2177	−0.650304	−0.325152	−	0.945662i	$-0.605416\pi$
$701$	−17.2177	−0.650304	−0.325152	+	0.945662i	$0.605416\pi$
$702$	0	0
$703$	−3.56284	−0.134375
$704$	1.54051	0.0580602
$705$	−29.5743	−1.11383
$706$	25.8451	0.972694
$707$	−12.0016	−0.451368
$708$	11.2705	0.423571
$709$	−47.1650	−1.77132	−0.885660	−	0.464335i	$-0.846293\pi$
$709$	−47.1650	−1.77132	−0.885660	+	0.464335i	$0.846293\pi$
$710$	13.5925	0.510119
$711$	7.11430	0.266807
$712$	−9.50981	−0.356395
$713$	0.808495	0.0302784
$714$	45.9663	1.72025
$715$	0	0
$716$	−21.8967	−0.818319
$717$	−11.8494	−0.442525
$718$	31.6799	1.18228
$719$	12.1810	0.454273	0.227137	−	0.973863i	$-0.427064\pi$
$719$	12.1810	0.454273	0.227137	+	0.973863i	$0.427064\pi$
$720$	14.7576	0.549984
$721$	−65.5366	−2.44071
$722$	34.0562	1.26744
$723$	−0.628349	−0.0233685
$724$	9.27967	0.344876
$725$	18.2018	0.675998
$726$	−1.81001	−0.0671759
$727$	−20.6572	−0.766131	−0.383066	−	0.923721i	$-0.625132\pi$
$727$	−20.6572	−0.766131	−0.383066	+	0.923721i	$0.625132\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	71.9523	2.66307
$731$	12.7961	0.473280
$732$	−6.96965	−0.257606
$733$	12.2521	0.452542	0.226271	−	0.974064i	$-0.427346\pi$
$733$	12.2521	0.452542	0.226271	+	0.974064i	$0.427346\pi$
$734$	−16.6197	−0.613446
$735$	−40.8411	−1.50645
$736$	−48.5759	−1.79053
$737$	6.76105	0.249046
$738$	19.1781	0.705957
$739$	−45.9946	−1.69194	−0.845970	−	0.533231i	$-0.820978\pi$
$739$	−45.9946	−1.69194	−0.845970	+	0.533231i	$0.820978\pi$
$740$	−31.7225	−1.16614
$741$	0	0
$742$	−87.3478	−3.20664
$743$	4.54254	0.166650	0.0833248	−	0.996522i	$-0.473446\pi$
$743$	4.54254	0.166650	0.0833248	+	0.996522i	$0.473446\pi$
$744$	−0.137200	−0.00503001
$745$	−0.464647	−0.0170234
$746$	−22.4875	−0.823325
$747$	0.999903	0.0365845
$748$	−7.13589	−0.260914
$749$	−61.6984	−2.25441
$750$	−5.51503	−0.201380
$751$	−2.34656	−0.0856272	−0.0428136	−	0.999083i	$-0.513632\pi$
$751$	−2.34656	−0.0856272	−0.0428136	+	0.999083i	$0.513632\pi$
$752$	−48.5836	−1.77166
$753$	−12.8077	−0.466740
$754$	0	0
$755$	−34.7163	−1.26346
$756$	−5.79576	−0.210790
$757$	36.3932	1.32274	0.661368	−	0.750062i	$-0.269976\pi$
$757$	36.3932	1.32274	0.661368	+	0.750062i	$0.269976\pi$
$758$	−54.5768	−1.98232
$759$	7.72067	0.280242
$760$	1.68695	0.0611923
$761$	−31.9781	−1.15920	−0.579602	−	0.814900i	$-0.696792\pi$
$761$	−31.9781	−1.15920	−0.579602	+	0.814900i	$0.696792\pi$
$762$	14.0390	0.508579
$763$	−3.52922	−0.127766
$764$	−15.2073	−0.550180
$765$	−16.7598	−0.605952
$766$	16.1632	0.584000
$767$	0	0
$768$	20.8101	0.750919
$769$	25.2070	0.908986	0.454493	−	0.890750i	$-0.349820\pi$
$769$	25.2070	0.908986	0.454493	+	0.890750i	$0.349820\pi$
$770$	24.6384	0.887906
$771$	13.1560	0.473802
$772$	9.53307	0.343103
$773$	18.5858	0.668485	0.334243	−	0.942487i	$-0.391520\pi$
$773$	18.5858	0.668485	0.334243	+	0.942487i	$0.391520\pi$
$774$	−4.14200	−0.148881
$775$	−0.417136	−0.0149840
$776$	12.4104	0.445508
$777$	−37.6666	−1.35128
$778$	4.67924	0.167759
$779$	4.55172	0.163082
$780$	0	0
$781$	−2.50552	−0.0896546
$782$	78.1420	2.79435
$783$	4.56940	0.163297
$784$	−67.0922	−2.39615
$785$	30.2778	1.08066
$786$	−7.37617	−0.263099
$787$	3.03672	0.108247	0.0541236	−	0.998534i	$-0.482763\pi$
$787$	3.03672	0.108247	0.0541236	+	0.998534i	$0.482763\pi$
$788$	−4.24758	−0.151314
$789$	−8.70410	−0.309874
$790$	38.5953	1.37316
$791$	49.4201	1.75718
$792$	−1.31019	−0.0465554
$793$	0	0
$794$	33.5611	1.19104
$795$	31.8479	1.12953
$796$	2.42863	0.0860805
$797$	−2.47849	−0.0877926	−0.0438963	−	0.999036i	$-0.513977\pi$
$797$	−2.47849	−0.0877926	−0.0438963	+	0.999036i	$0.513977\pi$
$798$	−3.53136	−0.125009
$799$	55.1749	1.95195
$800$	25.0623	0.886086
$801$	−7.25837	−0.256462
$802$	−38.3050	−1.35260
$803$	−13.2630	−0.468042
$804$	−8.62808	−0.304289
$805$	−105.096	−3.70414
$806$	0	0
$807$	10.1179	0.356165
$808$	3.46229	0.121803
$809$	24.4940	0.861162	0.430581	−	0.902552i	$-0.358309\pi$
$809$	24.4940	0.861162	0.430581	+	0.902552i	$0.358309\pi$
$810$	5.42503	0.190616
$811$	−31.4384	−1.10395	−0.551976	−	0.833860i	$-0.686126\pi$
$811$	−31.4384	−1.10395	−0.551976	+	0.833860i	$0.686126\pi$
$812$	−26.4832	−0.929378
$813$	−28.9152	−1.01410
$814$	15.0116	0.526157
$815$	−33.1514	−1.16124
$816$	−27.5324	−0.963825
$817$	−0.983058	−0.0343928
$818$	31.0522	1.08571
$819$	0	0
$820$	40.5271	1.41527
$821$	11.9697	0.417745	0.208872	−	0.977943i	$-0.433021\pi$
$821$	11.9697	0.417745	0.208872	+	0.977943i	$0.433021\pi$
$822$	−6.69126	−0.233385
$823$	4.93635	0.172070	0.0860352	−	0.996292i	$-0.472580\pi$
$823$	4.93635	0.172070	0.0860352	+	0.996292i	$0.472580\pi$
$824$	18.9063	0.658632
$825$	−3.98341	−0.138685
$826$	72.5996	2.52606
$827$	48.5995	1.68997	0.844984	−	0.534791i	$-0.179610\pi$
$827$	48.5995	1.68997	0.844984	+	0.534791i	$0.179610\pi$
$828$	−9.85270	−0.342405
$829$	25.4622	0.884339	0.442169	−	0.896932i	$-0.354209\pi$
$829$	25.4622	0.884339	0.442169	+	0.896932i	$0.354209\pi$
$830$	5.42451	0.188287
$831$	−9.29036	−0.322279
$832$	0	0
$833$	76.1947	2.63999
$834$	−1.02613	−0.0355320
$835$	71.1095	2.46085
$836$	0.548215	0.0189604
$837$	−0.104718	−0.00361959
$838$	28.4610	0.983167
$839$	18.9650	0.654744	0.327372	−	0.944896i	$-0.393837\pi$
$839$	18.9650	0.654744	0.327372	+	0.944896i	$0.393837\pi$
$840$	17.8346	0.615352
$841$	−8.12055	−0.280019
$842$	−26.8420	−0.925038
$843$	−1.40995	−0.0485612
$844$	−1.41740	−0.0487890
$845$	0	0
$846$	−17.8598	−0.614031
$847$	−4.54162	−0.156052
$848$	52.3185	1.79662
$849$	−3.21205	−0.110237
$850$	−40.3167	−1.38285
$851$	−64.0326	−2.19501
$852$	3.19741	0.109542
$853$	−9.05734	−0.310117	−0.155059	−	0.987905i	$-0.549557\pi$
$853$	−9.05734	−0.310117	−0.155059	+	0.987905i	$0.549557\pi$
$854$	−44.8955	−1.53629
$855$	1.28757	0.0440340
$856$	17.7990	0.608358
$857$	−32.6253	−1.11446	−0.557230	−	0.830358i	$-0.688136\pi$
$857$	−32.6253	−1.11446	−0.557230	+	0.830358i	$0.688136\pi$
$858$	0	0
$859$	−30.6821	−1.04686	−0.523429	−	0.852069i	$-0.675348\pi$
$859$	−30.6821	−1.04686	−0.523429	+	0.852069i	$0.675348\pi$
$860$	−8.75286	−0.298470
$861$	48.1211	1.63996
$862$	−70.3562	−2.39634
$863$	−26.7121	−0.909290	−0.454645	−	0.890673i	$-0.650234\pi$
$863$	−26.7121	−0.909290	−0.454645	+	0.890673i	$0.650234\pi$
$864$	6.29167	0.214047
$865$	40.0595	1.36206
$866$	−57.7351	−1.96192
$867$	14.2677	0.484557
$868$	0.606923	0.0206003
$869$	−7.11430	−0.241336
$870$	24.7892	0.840431
$871$	0	0
$872$	1.01812	0.0344780
$873$	9.47227	0.320588
$874$	−6.00325	−0.203063
$875$	−13.8381	−0.467813
$876$	16.9256	0.571862
$877$	−12.2054	−0.412147	−0.206073	−	0.978537i	$-0.566069\pi$
$877$	−12.2054	−0.412147	−0.206073	+	0.978537i	$0.566069\pi$
$878$	−39.9830	−1.34936
$879$	−21.7378	−0.733199
$880$	−14.7576	−0.497479
$881$	33.4767	1.12786	0.563929	−	0.825823i	$-0.309289\pi$
$881$	33.4767	1.12786	0.563929	+	0.825823i	$0.309289\pi$
$882$	−24.6637	−0.830471
$883$	−13.2020	−0.444284	−0.222142	−	0.975014i	$-0.571305\pi$
$883$	−13.2020	−0.444284	−0.222142	+	0.975014i	$0.571305\pi$
$884$	0	0
$885$	−26.4705	−0.889797
$886$	−70.8863	−2.38147
$887$	0.465829	0.0156410	0.00782051	−	0.999969i	$-0.497511\pi$
$887$	0.465829	0.0156410	0.00782051	+	0.999969i	$0.497511\pi$
$888$	10.8662	0.364647
$889$	35.2261	1.18145
$890$	−39.3769	−1.31992
$891$	−1.00000	−0.0335013
$892$	−27.7454	−0.928985
$893$	−4.23881	−0.141846
$894$	−0.280598	−0.00938460
$895$	51.4280	1.71905
$896$	44.4851	1.48614
$897$	0	0
$898$	−54.9801	−1.83471
$899$	−0.478500	−0.0159589
$900$	5.08341	0.169447
$901$	−59.4166	−1.97945
$902$	−19.1781	−0.638562
$903$	−10.3930	−0.345856
$904$	−14.2569	−0.474178
$905$	−21.7948	−0.724483
$906$	−20.9650	−0.696515
$907$	−59.1052	−1.96255	−0.981277	−	0.192600i	$-0.938308\pi$
$907$	−59.1052	−1.96255	−0.981277	+	0.192600i	$0.938308\pi$
$908$	−35.1804	−1.16750
$909$	2.64259	0.0876493
$910$	0	0
$911$	−19.9000	−0.659318	−0.329659	−	0.944100i	$-0.606934\pi$
$911$	−19.9000	−0.659318	−0.329659	+	0.944100i	$0.606934\pi$
$912$	2.11517	0.0700403
$913$	−0.999903	−0.0330920
$914$	−54.9731	−1.81835
$915$	16.3693	0.541154
$916$	22.4091	0.740416
$917$	−18.5080	−0.611188
$918$	−10.1211	−0.334048
$919$	1.02617	0.0338503	0.0169251	−	0.999857i	$-0.494612\pi$
$919$	1.02617	0.0338503	0.0169251	+	0.999857i	$0.494612\pi$
$920$	30.3185	0.999572
$921$	−15.2363	−0.502052
$922$	−1.08190	−0.0356304
$923$	0	0
$924$	5.79576	0.190667
$925$	33.0370	1.08625
$926$	68.3043	2.24462
$927$	14.4302	0.473952
$928$	28.7492	0.943738
$929$	26.4873	0.869019	0.434509	−	0.900667i	$-0.356922\pi$
$929$	26.4873	0.869019	0.434509	+	0.900667i	$0.356922\pi$
$930$	−0.568100	−0.0186287
$931$	−5.85365	−0.191846
$932$	−6.52060	−0.213589
$933$	−7.66303	−0.250876
$934$	15.2688	0.499611
$935$	16.7598	0.548104
$936$	0	0
$937$	5.64595	0.184445	0.0922225	−	0.995738i	$-0.470603\pi$
$937$	5.64595	0.184445	0.0922225	+	0.995738i	$0.470603\pi$
$938$	−55.5784	−1.81470
$939$	13.9879	0.456477
$940$	−37.7411	−1.23098
$941$	−46.4041	−1.51273	−0.756366	−	0.654149i	$-0.773027\pi$
$941$	−46.4041	−1.51273	−0.756366	+	0.654149i	$0.773027\pi$
$942$	18.2846	0.595743
$943$	81.8050	2.66394
$944$	−43.4848	−1.41531
$945$	13.6123	0.442808
$946$	4.14200	0.134668
$947$	−36.7172	−1.19315	−0.596574	−	0.802558i	$-0.703472\pi$
$947$	−36.7172	−1.19315	−0.596574	+	0.802558i	$0.703472\pi$
$948$	9.07888	0.294868
$949$	0	0
$950$	3.09733	0.100491
$951$	−2.11260	−0.0685057
$952$	−33.2729	−1.07838
$953$	−14.3637	−0.465285	−0.232642	−	0.972562i	$-0.574737\pi$
$953$	−14.3637	−0.465285	−0.232642	+	0.972562i	$0.574737\pi$
$954$	19.2328	0.622683
$955$	35.7167	1.15577
$956$	−15.1216	−0.489067
$957$	−4.56940	−0.147708
$958$	29.1406	0.941491
$959$	−16.7895	−0.542160
$960$	4.61728	0.149022
$961$	−30.9890	−0.999646
$962$	0	0
$963$	13.5851	0.437774
$964$	−0.801865	−0.0258263
$965$	−22.3899	−0.720758
$966$	−63.4668	−2.04201
$967$	−44.4755	−1.43023	−0.715117	−	0.699005i	$-0.753627\pi$
$967$	−44.4755	−1.43023	−0.715117	+	0.699005i	$0.753627\pi$
$968$	1.31019	0.0421109
$969$	−2.40214	−0.0771678
$970$	51.3874	1.64995
$971$	26.8276	0.860938	0.430469	−	0.902605i	$-0.358348\pi$
$971$	26.8276	0.860938	0.430469	+	0.902605i	$0.358348\pi$
$972$	1.27615	0.0409324
$973$	−2.57473	−0.0825421
$974$	56.4422	1.80852
$975$	0	0
$976$	26.8910	0.860758
$977$	7.97638	0.255187	0.127594	−	0.991827i	$-0.459275\pi$
$977$	7.97638	0.255187	0.127594	+	0.991827i	$0.459275\pi$
$978$	−20.0200	−0.640168
$979$	7.25837	0.231979
$980$	−52.1192	−1.66489
$981$	0.777084	0.0248104
$982$	10.9488	0.349392
$983$	−2.96567	−0.0945900	−0.0472950	−	0.998881i	$-0.515060\pi$
$983$	−2.96567	−0.0945900	−0.0472950	+	0.998881i	$0.515060\pi$
$984$	−13.8822	−0.442548
$985$	9.97612	0.317866
$986$	−46.2476	−1.47282
$987$	−44.8130	−1.42641
$988$	0	0
$989$	−17.6678	−0.561805
$990$	−5.42503	−0.172419
$991$	−22.3604	−0.710301	−0.355150	−	0.934809i	$-0.615570\pi$
$991$	−22.3604	−0.710301	−0.355150	+	0.934809i	$0.615570\pi$
$992$	−0.658853	−0.0209186
$993$	15.2543	0.484081
$994$	20.5963	0.653276
$995$	−5.70403	−0.180830
$996$	1.27602	0.0404323
$997$	−51.2433	−1.62289	−0.811445	−	0.584428i	$-0.801319\pi$
$997$	−51.2433	−1.62289	−0.811445	+	0.584428i	$0.801319\pi$
$998$	−17.5696	−0.556155
$999$	8.29366	0.262400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.bf.1.5		14
13.6	odd	12		429.2.s.b.166.11	✓	28
13.11	odd	12		429.2.s.b.199.11	yes	28
13.12	even	2		5577.2.a.bg.1.10		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.s.b.166.11	✓	28	13.6	odd	12
429.2.s.b.199.11	yes	28	13.11	odd	12
5577.2.a.bf.1.5		14	1.1	even	1	trivial
5577.2.a.bg.1.10		14	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:	\( N \)	\(=\)	\( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5577.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81001 q^{2} +1.00000 q^{3} +1.27615 q^{4} -2.99723 q^{5} -1.81001 q^{6} -4.54162 q^{7} +1.31019 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.81001 q^{2} +1.00000 q^{3} +1.27615 q^{4} -2.99723 q^{5} -1.81001 q^{6} -4.54162 q^{7} +1.31019 q^{8} +1.00000 q^{9} +5.42503 q^{10} -1.00000 q^{11} +1.27615 q^{12} +8.22038 q^{14} -2.99723 q^{15} -4.92374 q^{16} +5.59175 q^{17} -1.81001 q^{18} -0.429586 q^{19} -3.82491 q^{20} -4.54162 q^{21} +1.81001 q^{22} -7.72067 q^{23} +1.31019 q^{24} +3.98341 q^{25} +1.00000 q^{27} -5.79576 q^{28} +4.56940 q^{29} +5.42503 q^{30} -0.104718 q^{31} +6.29167 q^{32} -1.00000 q^{33} -10.1211 q^{34} +13.6123 q^{35} +1.27615 q^{36} +8.29366 q^{37} +0.777556 q^{38} -3.92693 q^{40} -10.5956 q^{41} +8.22038 q^{42} +2.28838 q^{43} -1.27615 q^{44} -2.99723 q^{45} +13.9745 q^{46} +9.86720 q^{47} -4.92374 q^{48} +13.6263 q^{49} -7.21002 q^{50} +5.59175 q^{51} -10.6258 q^{53} -1.81001 q^{54} +2.99723 q^{55} -5.95036 q^{56} -0.429586 q^{57} -8.27068 q^{58} +8.83165 q^{59} -3.82491 q^{60} -5.46149 q^{61} +0.189541 q^{62} -4.54162 q^{63} -1.54051 q^{64} +1.81001 q^{66} -6.76105 q^{67} +7.13589 q^{68} -7.72067 q^{69} -24.6384 q^{70} +2.50552 q^{71} +1.31019 q^{72} +13.2630 q^{73} -15.0116 q^{74} +3.98341 q^{75} -0.548215 q^{76} +4.54162 q^{77} +7.11430 q^{79} +14.7576 q^{80} +1.00000 q^{81} +19.1781 q^{82} +0.999903 q^{83} -5.79576 q^{84} -16.7598 q^{85} -4.14200 q^{86} +4.56940 q^{87} -1.31019 q^{88} -7.25837 q^{89} +5.42503 q^{90} -9.85270 q^{92} -0.104718 q^{93} -17.8598 q^{94} +1.28757 q^{95} +6.29167 q^{96} +9.47227 q^{97} -24.6637 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 6 q^{2} + 14 q^{3} + 18 q^{4} - 12 q^{5} - 6 q^{6} - 12 q^{7} - 12 q^{8} + 14 q^{9}+O(q^{10}) \) 14 * q - 6 * q^2 + 14 * q^3 + 18 * q^4 - 12 * q^5 - 6 * q^6 - 12 * q^7 - 12 * q^8 + 14 * q^9	\( 14 q - 6 q^{2} + 14 q^{3} + 18 q^{4} - 12 q^{5} - 6 q^{6} - 12 q^{7} - 12 q^{8} + 14 q^{9} - 14 q^{11} + 18 q^{12} - 2 q^{14} - 12 q^{15} + 22 q^{16} + 2 q^{17} - 6 q^{18} + 2 q^{19} - 44 q^{20} - 12 q^{21} + 6 q^{22} + 2 q^{23} - 12 q^{24} + 20 q^{25} + 14 q^{27} - 24 q^{28} - 28 q^{31} - 30 q^{32} - 14 q^{33} + 16 q^{34} - 2 q^{35} + 18 q^{36} - 10 q^{38} + 10 q^{40} - 40 q^{41} - 2 q^{42} - 2 q^{43} - 18 q^{44} - 12 q^{45} - 32 q^{46} - 48 q^{47} + 22 q^{48} + 10 q^{49} + 2 q^{51} + 8 q^{53} - 6 q^{54} + 12 q^{55} - 10 q^{56} + 2 q^{57} + 16 q^{58} - 40 q^{59} - 44 q^{60} + 4 q^{61} - 6 q^{62} - 12 q^{63} + 16 q^{64} + 6 q^{66} - 40 q^{67} + 22 q^{68} + 2 q^{69} + 40 q^{70} - 36 q^{71} - 12 q^{72} - 10 q^{73} - 48 q^{74} + 20 q^{75} + 4 q^{76} + 12 q^{77} - 24 q^{79} - 68 q^{80} + 14 q^{81} + 46 q^{82} - 12 q^{83} - 24 q^{84} - 34 q^{85} - 48 q^{86} + 12 q^{88} - 20 q^{89} + 36 q^{92} - 28 q^{93} - 50 q^{94} - 60 q^{95} - 30 q^{96} + 16 q^{97} - 44 q^{98} - 14 q^{99}+O(q^{100}) \) 14 * q - 6 * q^2 + 14 * q^3 + 18 * q^4 - 12 * q^5 - 6 * q^6 - 12 * q^7 - 12 * q^8 + 14 * q^9 - 14 * q^11 + 18 * q^12 - 2 * q^14 - 12 * q^15 + 22 * q^16 + 2 * q^17 - 6 * q^18 + 2 * q^19 - 44 * q^20 - 12 * q^21 + 6 * q^22 + 2 * q^23 - 12 * q^24 + 20 * q^25 + 14 * q^27 - 24 * q^28 - 28 * q^31 - 30 * q^32 - 14 * q^33 + 16 * q^34 - 2 * q^35 + 18 * q^36 - 10 * q^38 + 10 * q^40 - 40 * q^41 - 2 * q^42 - 2 * q^43 - 18 * q^44 - 12 * q^45 - 32 * q^46 - 48 * q^47 + 22 * q^48 + 10 * q^49 + 2 * q^51 + 8 * q^53 - 6 * q^54 + 12 * q^55 - 10 * q^56 + 2 * q^57 + 16 * q^58 - 40 * q^59 - 44 * q^60 + 4 * q^61 - 6 * q^62 - 12 * q^63 + 16 * q^64 + 6 * q^66 - 40 * q^67 + 22 * q^68 + 2 * q^69 + 40 * q^70 - 36 * q^71 - 12 * q^72 - 10 * q^73 - 48 * q^74 + 20 * q^75 + 4 * q^76 + 12 * q^77 - 24 * q^79 - 68 * q^80 + 14 * q^81 + 46 * q^82 - 12 * q^83 - 24 * q^84 - 34 * q^85 - 48 * q^86 + 12 * q^88 - 20 * q^89 + 36 * q^92 - 28 * q^93 - 50 * q^94 - 60 * q^95 - 30 * q^96 + 16 * q^97 - 44 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

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