LMFDB - Embedded newform 1155.2.a.v.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.22219$ of defining polynomial
Character	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.63640 q^{2} -1.00000 q^{3} +4.95063 q^{4} +1.00000 q^{5} -2.63640 q^{6} +1.00000 q^{7} +7.77906 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.63640 q^{2} -1.00000 q^{3} +4.95063 q^{4} +1.00000 q^{5} -2.63640 q^{6} +1.00000 q^{7} +7.77906 q^{8} +1.00000 q^{9} +2.63640 q^{10} +1.00000 q^{11} -4.95063 q^{12} -3.14265 q^{13} +2.63640 q^{14} -1.00000 q^{15} +10.6075 q^{16} +0.677821 q^{17} +2.63640 q^{18} +1.95063 q^{19} +4.95063 q^{20} -1.00000 q^{21} +2.63640 q^{22} +0.293777 q^{23} -7.77906 q^{24} +1.00000 q^{25} -8.28531 q^{26} -1.00000 q^{27} +4.95063 q^{28} -1.82047 q^{29} -2.63640 q^{30} -1.86984 q^{31} +12.4075 q^{32} -1.00000 q^{33} +1.78701 q^{34} +1.00000 q^{35} +4.95063 q^{36} -4.03142 q^{37} +5.14265 q^{38} +3.14265 q^{39} +7.77906 q^{40} -11.0439 q^{41} -2.63640 q^{42} +12.1217 q^{43} +4.95063 q^{44} +1.00000 q^{45} +0.774514 q^{46} +2.89825 q^{47} -10.6075 q^{48} +1.00000 q^{49} +2.63640 q^{50} -0.677821 q^{51} -15.5581 q^{52} +11.2234 q^{53} -2.63640 q^{54} +1.00000 q^{55} +7.77906 q^{56} -1.95063 q^{57} -4.79951 q^{58} -7.35360 q^{59} -4.95063 q^{60} -7.22344 q^{61} -4.92966 q^{62} +1.00000 q^{63} +11.4963 q^{64} -3.14265 q^{65} -2.63640 q^{66} -12.6694 q^{67} +3.35564 q^{68} -0.293777 q^{69} +2.63640 q^{70} -5.77111 q^{71} +7.77906 q^{72} +9.27281 q^{73} -10.6285 q^{74} -1.00000 q^{75} +9.65685 q^{76} +1.00000 q^{77} +8.28531 q^{78} -5.40297 q^{79} +10.6075 q^{80} +1.00000 q^{81} -29.1162 q^{82} -2.97903 q^{83} -4.95063 q^{84} +0.677821 q^{85} +31.9577 q^{86} +1.82047 q^{87} +7.77906 q^{88} +7.47733 q^{89} +2.63640 q^{90} -3.14265 q^{91} +1.45438 q^{92} +1.86984 q^{93} +7.64095 q^{94} +1.95063 q^{95} -12.4075 q^{96} -2.80798 q^{97} +2.63640 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - 4 q^{15} + 6 q^{16} + 6 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{20} - 4 q^{21} + 2 q^{22} + 10 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{30} - 8 q^{31} + 14 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{35} + 6 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{40} - 2 q^{42} + 6 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} - 6 q^{48} + 4 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} + 14 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 6 q^{57} + 20 q^{58} + 2 q^{59} - 6 q^{60} + 2 q^{61} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 4 q^{65} - 2 q^{66} - 12 q^{67} + 20 q^{68} - 10 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 20 q^{73} - 32 q^{74} - 4 q^{75} + 16 q^{76} + 4 q^{77} - 4 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 14 q^{83} - 6 q^{84} + 6 q^{85} + 40 q^{86} - 6 q^{87} + 6 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{91} + 40 q^{92} + 8 q^{93} + 4 q^{94} - 6 q^{95} - 14 q^{96} - 14 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 6 * q^12 + 4 * q^13 + 2 * q^14 - 4 * q^15 + 6 * q^16 + 6 * q^17 + 2 * q^18 - 6 * q^19 + 6 * q^20 - 4 * q^21 + 2 * q^22 + 10 * q^23 - 6 * q^24 + 4 * q^25 - 4 * q^27 + 6 * q^28 + 6 * q^29 - 2 * q^30 - 8 * q^31 + 14 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^35 + 6 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 + 6 * q^40 - 2 * q^42 + 6 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 - 6 * q^48 + 4 * q^49 + 2 * q^50 - 6 * q^51 - 12 * q^52 + 14 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 6 * q^57 + 20 * q^58 + 2 * q^59 - 6 * q^60 + 2 * q^61 + 20 * q^62 + 4 * q^63 - 2 * q^64 + 4 * q^65 - 2 * q^66 - 12 * q^67 + 20 * q^68 - 10 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 20 * q^73 - 32 * q^74 - 4 * q^75 + 16 * q^76 + 4 * q^77 - 4 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 14 * q^83 - 6 * q^84 + 6 * q^85 + 40 * q^86 - 6 * q^87 + 6 * q^88 - 6 * q^89 + 2 * q^90 + 4 * q^91 + 40 * q^92 + 8 * q^93 + 4 * q^94 - 6 * q^95 - 14 * q^96 - 14 * q^97 + 2 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.63640	1.86422	0.932110	−	0.362176i	$-0.117966\pi$
$2$	2.63640	1.86422	0.932110	+	0.362176i	$0.117966\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.95063	2.47532
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.63640	−1.07631
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	7.77906	2.75031
$9$	1.00000	0.333333
$10$	2.63640	0.833704
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−4.95063	−1.42912
$13$	−3.14265	−0.871615	−0.435808	−	0.900040i	$-0.643537\pi$
$13$	−3.14265	−0.871615	−0.435808	+	0.900040i	$0.643537\pi$
$14$	2.63640	0.704609
$15$	−1.00000	−0.258199
$16$	10.6075	2.65187
$17$	0.677821	0.164396	0.0821979	−	0.996616i	$-0.473806\pi$
$17$	0.677821	0.164396	0.0821979	+	0.996616i	$0.473806\pi$
$18$	2.63640	0.621407
$19$	1.95063	0.447505	0.223753	−	0.974646i	$-0.428169\pi$
$19$	1.95063	0.447505	0.223753	+	0.974646i	$0.428169\pi$
$20$	4.95063	1.10699
$21$	−1.00000	−0.218218
$22$	2.63640	0.562083
$23$	0.293777	0.0612567	0.0306283	−	0.999531i	$-0.490249\pi$
$23$	0.293777	0.0612567	0.0306283	+	0.999531i	$0.490249\pi$
$24$	−7.77906	−1.58789
$25$	1.00000	0.200000
$26$	−8.28531	−1.62488
$27$	−1.00000	−0.192450
$28$	4.95063	0.935581
$29$	−1.82047	−0.338054	−0.169027	−	0.985611i	$-0.554062\pi$
$29$	−1.82047	−0.338054	−0.169027	+	0.985611i	$0.554062\pi$
$30$	−2.63640	−0.481339
$31$	−1.86984	−0.335834	−0.167917	−	0.985801i	$-0.553704\pi$
$31$	−1.86984	−0.335834	−0.167917	+	0.985801i	$0.553704\pi$
$32$	12.4075	2.19336
$33$	−1.00000	−0.174078
$34$	1.78701	0.306470
$35$	1.00000	0.169031
$36$	4.95063	0.825105
$37$	−4.03142	−0.662761	−0.331381	−	0.943497i	$-0.607514\pi$
$37$	−4.03142	−0.662761	−0.331381	+	0.943497i	$0.607514\pi$
$38$	5.14265	0.834249
$39$	3.14265	0.503227
$40$	7.77906	1.22998
$41$	−11.0439	−1.72477	−0.862385	−	0.506253i	$-0.831030\pi$
$41$	−11.0439	−1.72477	−0.862385	+	0.506253i	$0.831030\pi$
$42$	−2.63640	−0.406806
$43$	12.1217	1.84854	0.924270	−	0.381740i	$-0.124675\pi$
$43$	12.1217	1.84854	0.924270	+	0.381740i	$0.124675\pi$
$44$	4.95063	0.746336
$45$	1.00000	0.149071
$46$	0.774514	0.114196
$47$	2.89825	0.422753	0.211376	−	0.977405i	$-0.432205\pi$
$47$	2.89825	0.422753	0.211376	+	0.977405i	$0.432205\pi$
$48$	−10.6075	−1.53106
$49$	1.00000	0.142857
$50$	2.63640	0.372844
$51$	−0.677821	−0.0949139
$52$	−15.5581	−2.15752
$53$	11.2234	1.54166	0.770829	−	0.637042i	$-0.219842\pi$
$53$	11.2234	1.54166	0.770829	+	0.637042i	$0.219842\pi$
$54$	−2.63640	−0.358769
$55$	1.00000	0.134840
$56$	7.77906	1.03952
$57$	−1.95063	−0.258367
$58$	−4.79951	−0.630206
$59$	−7.35360	−0.957357	−0.478678	−	0.877990i	$-0.658884\pi$
$59$	−7.35360	−0.957357	−0.478678	+	0.877990i	$0.658884\pi$
$60$	−4.95063	−0.639124
$61$	−7.22344	−0.924867	−0.462433	−	0.886654i	$-0.653024\pi$
$61$	−7.22344	−0.924867	−0.462433	+	0.886654i	$0.653024\pi$
$62$	−4.92966	−0.626068
$63$	1.00000	0.125988
$64$	11.4963	1.43703
$65$	−3.14265	−0.389798
$66$	−2.63640	−0.324519
$67$	−12.6694	−1.54781	−0.773904	−	0.633303i	$-0.781699\pi$
$67$	−12.6694	−1.54781	−0.773904	+	0.633303i	$0.781699\pi$
$68$	3.35564	0.406931
$69$	−0.293777	−0.0353666
$70$	2.63640	0.315111
$71$	−5.77111	−0.684904	−0.342452	−	0.939535i	$-0.611257\pi$
$71$	−5.77111	−0.684904	−0.342452	+	0.939535i	$0.611257\pi$
$72$	7.77906	0.916771
$73$	9.27281	1.08530	0.542650	−	0.839959i	$-0.317421\pi$
$73$	9.27281	1.08530	0.542650	+	0.839959i	$0.317421\pi$
$74$	−10.6285	−1.23553
$75$	−1.00000	−0.115470
$76$	9.65685	1.10772
$77$	1.00000	0.113961
$78$	8.28531	0.938126
$79$	−5.40297	−0.607881	−0.303941	−	0.952691i	$-0.598302\pi$
$79$	−5.40297	−0.607881	−0.303941	+	0.952691i	$0.598302\pi$
$80$	10.6075	1.18595
$81$	1.00000	0.111111
$82$	−29.1162	−3.21535
$83$	−2.97903	−0.326991	−0.163496	−	0.986544i	$-0.552277\pi$
$83$	−2.97903	−0.326991	−0.163496	+	0.986544i	$0.552277\pi$
$84$	−4.95063	−0.540158
$85$	0.677821	0.0735200
$86$	31.9577	3.44608
$87$	1.82047	0.195175
$88$	7.77906	0.829250
$89$	7.47733	0.792595	0.396298	−	0.918122i	$-0.370295\pi$
$89$	7.47733	0.792595	0.396298	+	0.918122i	$0.370295\pi$
$90$	2.63640	0.277901
$91$	−3.14265	−0.329440
$92$	1.45438	0.151630
$93$	1.86984	0.193894
$94$	7.64095	0.788104
$95$	1.95063	0.200131
$96$	−12.4075	−1.26634
$97$	−2.80798	−0.285107	−0.142553	−	0.989787i	$-0.545531\pi$
$97$	−2.80798	−0.285107	−0.142553	+	0.989787i	$0.545531\pi$
$98$	2.63640	0.266317
$99$	1.00000	0.100504
$100$	4.95063	0.495063
$101$	12.9138	1.28497	0.642483	−	0.766300i	$-0.277904\pi$
$101$	12.9138	1.28497	0.642483	+	0.766300i	$0.277904\pi$
$102$	−1.78701	−0.176940
$103$	−16.7092	−1.64641	−0.823205	−	0.567744i	$-0.807816\pi$
$103$	−16.7092	−1.64641	−0.823205	+	0.567744i	$0.807816\pi$
$104$	−24.4469	−2.39721
$105$	−1.00000	−0.0975900
$106$	29.5895	2.87399
$107$	−5.30423	−0.512779	−0.256390	−	0.966574i	$-0.582533\pi$
$107$	−5.30423	−0.512779	−0.256390	+	0.966574i	$0.582533\pi$
$108$	−4.95063	−0.476375
$109$	2.74270	0.262703	0.131352	−	0.991336i	$-0.458068\pi$
$109$	2.74270	0.262703	0.131352	+	0.991336i	$0.458068\pi$
$110$	2.63640	0.251371
$111$	4.03142	0.382645
$112$	10.6075	1.00231
$113$	2.06527	0.194285	0.0971423	−	0.995270i	$-0.469030\pi$
$113$	2.06527	0.194285	0.0971423	+	0.995270i	$0.469030\pi$
$114$	−5.14265	−0.481654
$115$	0.293777	0.0273948
$116$	−9.01250	−0.836789
$117$	−3.14265	−0.290538
$118$	−19.3871	−1.78472
$119$	0.677821	0.0621358
$120$	−7.77906	−0.710128
$121$	1.00000	0.0909091
$122$	−19.0439	−1.72416
$123$	11.0439	0.995796
$124$	−9.25690	−0.831295
$125$	1.00000	0.0894427
$126$	2.63640	0.234870
$127$	−3.35360	−0.297584	−0.148792	−	0.988869i	$-0.547538\pi$
$127$	−3.35360	−0.297584	−0.148792	+	0.988869i	$0.547538\pi$
$128$	5.49375	0.485584
$129$	−12.1217	−1.06725
$130$	−8.28531	−0.726669
$131$	−17.9327	−1.56679	−0.783393	−	0.621527i	$-0.786513\pi$
$131$	−17.9327	−1.56679	−0.783393	+	0.621527i	$0.786513\pi$
$132$	−4.95063	−0.430897
$133$	1.95063	0.169141
$134$	−33.4015	−2.88545
$135$	−1.00000	−0.0860663
$136$	5.27281	0.452140
$137$	4.12373	0.352314	0.176157	−	0.984362i	$-0.443633\pi$
$137$	4.12373	0.352314	0.176157	+	0.984362i	$0.443633\pi$
$138$	−0.774514	−0.0659310
$139$	−5.75559	−0.488183	−0.244091	−	0.969752i	$-0.578490\pi$
$139$	−5.75559	−0.488183	−0.244091	+	0.969752i	$0.578490\pi$
$140$	4.95063	0.418405
$141$	−2.89825	−0.244076
$142$	−15.2150	−1.27681
$143$	−3.14265	−0.262802
$144$	10.6075	0.883957
$145$	−1.82047	−0.151182
$146$	24.4469	2.02324
$147$	−1.00000	−0.0824786
$148$	−19.9581	−1.64054
$149$	−3.91717	−0.320907	−0.160453	−	0.987043i	$-0.551296\pi$
$149$	−3.91717	−0.320907	−0.160453	+	0.987043i	$0.551296\pi$
$150$	−2.63640	−0.215262
$151$	−20.7900	−1.69187	−0.845934	−	0.533287i	$-0.820956\pi$
$151$	−20.7900	−1.69187	−0.845934	+	0.533287i	$0.820956\pi$
$152$	15.1741	1.23078
$153$	0.677821	0.0547986
$154$	2.63640	0.212448
$155$	−1.86984	−0.150189
$156$	15.5581	1.24565
$157$	−0.423933	−0.0338336	−0.0169168	−	0.999857i	$-0.505385\pi$
$157$	−0.423933	−0.0338336	−0.0169168	+	0.999857i	$0.505385\pi$
$158$	−14.2444	−1.13322
$159$	−11.2234	−0.890077
$160$	12.4075	0.980900
$161$	0.293777	0.0231529
$162$	2.63640	0.207136
$163$	1.60953	0.126068	0.0630341	−	0.998011i	$-0.479922\pi$
$163$	1.60953	0.126068	0.0630341	+	0.998011i	$0.479922\pi$
$164$	−54.6743	−4.26935
$165$	−1.00000	−0.0778499
$166$	−7.85394	−0.609584
$167$	17.5581	1.35869	0.679344	−	0.733820i	$-0.262264\pi$
$167$	17.5581	1.35869	0.679344	+	0.733820i	$0.262264\pi$
$168$	−7.77906	−0.600167
$169$	−3.12373	−0.240287
$170$	1.78701	0.137057
$171$	1.95063	0.149168
$172$	60.0100	4.57572
$173$	25.7038	1.95422	0.977111	−	0.212729i	$-0.0682350\pi$
$173$	25.7038	1.95422	0.977111	+	0.212729i	$0.0682350\pi$
$174$	4.79951	0.363850
$175$	1.00000	0.0755929
$176$	10.6075	0.799569
$177$	7.35360	0.552730
$178$	19.7133	1.47757
$179$	−21.0848	−1.57595	−0.787976	−	0.615705i	$-0.788871\pi$
$179$	−21.0848	−1.57595	−0.787976	+	0.615705i	$0.788871\pi$
$180$	4.95063	0.368998
$181$	13.8025	1.02593	0.512967	−	0.858408i	$-0.328546\pi$
$181$	13.8025	1.02593	0.512967	+	0.858408i	$0.328546\pi$
$182$	−8.28531	−0.614148
$183$	7.22344	0.533972
$184$	2.28531	0.168475
$185$	−4.03142	−0.296396
$186$	4.92966	0.361461
$187$	0.677821	0.0495672
$188$	14.3481	1.04645
$189$	−1.00000	−0.0727393
$190$	5.14265	0.373087
$191$	−9.25690	−0.669806	−0.334903	−	0.942253i	$-0.608704\pi$
$191$	−9.25690	−0.669806	−0.334903	+	0.942253i	$0.608704\pi$
$192$	−11.4963	−0.829670
$193$	−5.94216	−0.427726	−0.213863	−	0.976864i	$-0.568605\pi$
$193$	−5.94216	−0.427726	−0.213863	+	0.976864i	$0.568605\pi$
$194$	−7.40297	−0.531502
$195$	3.14265	0.225050
$196$	4.95063	0.353616
$197$	−14.7322	−1.04962	−0.524812	−	0.851218i	$-0.675865\pi$
$197$	−14.7322	−1.04962	−0.524812	+	0.851218i	$0.675865\pi$
$198$	2.63640	0.187361
$199$	12.5706	0.891107	0.445554	−	0.895255i	$-0.353007\pi$
$199$	12.5706	0.891107	0.445554	+	0.895255i	$0.353007\pi$
$200$	7.77906	0.550062
$201$	12.6694	0.893627
$202$	34.0459	2.39546
$203$	−1.82047	−0.127772
$204$	−3.35564	−0.234942
$205$	−11.0439	−0.771340
$206$	−44.0523	−3.06927
$207$	0.293777	0.0204189
$208$	−33.3356	−2.31141
$209$	1.95063	0.134928
$210$	−2.63640	−0.181929
$211$	24.3640	1.67729	0.838645	−	0.544678i	$-0.183348\pi$
$211$	24.3640	1.67729	0.838645	+	0.544678i	$0.183348\pi$
$212$	55.5631	3.81609
$213$	5.77111	0.395430
$214$	−13.9841	−0.955933
$215$	12.1217	0.826692
$216$	−7.77906	−0.529298
$217$	−1.86984	−0.126933
$218$	7.23088	0.489737
$219$	−9.27281	−0.626598
$220$	4.95063	0.333771
$221$	−2.13016	−0.143290
$222$	10.6285	0.713335
$223$	−12.8080	−0.857686	−0.428843	−	0.903379i	$-0.641079\pi$
$223$	−12.8080	−0.857686	−0.428843	+	0.903379i	$0.641079\pi$
$224$	12.4075	0.829012
$225$	1.00000	0.0666667
$226$	5.44490	0.362189
$227$	−5.85189	−0.388404	−0.194202	−	0.980962i	$-0.562212\pi$
$227$	−5.85189	−0.388404	−0.194202	+	0.980962i	$0.562212\pi$
$228$	−9.65685	−0.639541
$229$	−0.799507	−0.0528329	−0.0264165	−	0.999651i	$-0.508410\pi$
$229$	−0.799507	−0.0528329	−0.0264165	+	0.999651i	$0.508410\pi$
$230$	0.774514	0.0510700
$231$	−1.00000	−0.0657952
$232$	−14.1616	−0.929753
$233$	−11.4121	−0.747629	−0.373814	−	0.927504i	$-0.621950\pi$
$233$	−11.4121	−0.747629	−0.373814	+	0.927504i	$0.621950\pi$
$234$	−8.28531	−0.541627
$235$	2.89825	0.189061
$236$	−36.4049	−2.36976
$237$	5.40297	0.350960
$238$	1.78701	0.115835
$239$	15.7626	1.01960	0.509800	−	0.860293i	$-0.329719\pi$
$239$	15.7626	1.01960	0.509800	+	0.860293i	$0.329719\pi$
$240$	−10.6075	−0.684710
$241$	−20.3840	−1.31305	−0.656526	−	0.754304i	$-0.727975\pi$
$241$	−20.3840	−1.31305	−0.656526	+	0.754304i	$0.727975\pi$
$242$	2.63640	0.169475
$243$	−1.00000	−0.0641500
$244$	−35.7606	−2.28934
$245$	1.00000	0.0638877
$246$	29.1162	1.85638
$247$	−6.13016	−0.390053
$248$	−14.5456	−0.923648
$249$	2.97903	0.188789
$250$	2.63640	0.166741
$251$	29.6619	1.87224	0.936120	−	0.351681i	$-0.114390\pi$
$251$	29.6619	1.87224	0.936120	+	0.351681i	$0.114390\pi$
$252$	4.95063	0.311860
$253$	0.293777	0.0184696
$254$	−8.84144	−0.554761
$255$	−0.677821	−0.0424468
$256$	−8.50875	−0.531797
$257$	−4.21195	−0.262734	−0.131367	−	0.991334i	$-0.541937\pi$
$257$	−4.21195	−0.262734	−0.131367	+	0.991334i	$0.541937\pi$
$258$	−31.9577	−1.98960
$259$	−4.03142	−0.250500
$260$	−15.5581	−0.964873
$261$	−1.82047	−0.112685
$262$	−47.2778	−2.92083
$263$	24.2653	1.49626	0.748132	−	0.663550i	$-0.230951\pi$
$263$	24.2653	1.49626	0.748132	+	0.663550i	$0.230951\pi$
$264$	−7.77906	−0.478768
$265$	11.2234	0.689450
$266$	5.14265	0.315316
$267$	−7.47733	−0.457605
$268$	−62.7213	−3.83131
$269$	−0.652436	−0.0397797	−0.0198898	−	0.999802i	$-0.506332\pi$
$269$	−0.652436	−0.0397797	−0.0198898	+	0.999802i	$0.506332\pi$
$270$	−2.63640	−0.160446
$271$	10.4394	0.634151	0.317076	−	0.948400i	$-0.397299\pi$
$271$	10.4394	0.634151	0.317076	+	0.948400i	$0.397299\pi$
$272$	7.18998	0.435956
$273$	3.14265	0.190202
$274$	10.8718	0.656791
$275$	1.00000	0.0603023
$276$	−1.45438	−0.0875434
$277$	27.2978	1.64017	0.820083	−	0.572245i	$-0.193927\pi$
$277$	27.2978	1.64017	0.820083	+	0.572245i	$0.193927\pi$
$278$	−15.1741	−0.910080
$279$	−1.86984	−0.111945
$280$	7.77906	0.464888
$281$	21.8434	1.30307	0.651535	−	0.758619i	$-0.274126\pi$
$281$	21.8434	1.30307	0.651535	+	0.758619i	$0.274126\pi$
$282$	−7.64095	−0.455012
$283$	−2.13016	−0.126625	−0.0633123	−	0.997994i	$-0.520166\pi$
$283$	−2.13016	−0.126625	−0.0633123	+	0.997994i	$0.520166\pi$
$284$	−28.5706	−1.69535
$285$	−1.95063	−0.115545
$286$	−8.28531	−0.489920
$287$	−11.0439	−0.651902
$288$	12.4075	0.731120
$289$	−16.5406	−0.972974
$290$	−4.79951	−0.281837
$291$	2.80798	0.164607
$292$	45.9063	2.68646
$293$	29.9496	1.74967	0.874837	−	0.484417i	$-0.160968\pi$
$293$	29.9496	1.74967	0.874837	+	0.484417i	$0.160968\pi$
$294$	−2.63640	−0.153758
$295$	−7.35360	−0.428143
$296$	−31.3606	−1.82280
$297$	−1.00000	−0.0580259
$298$	−10.3272	−0.598241
$299$	−0.923238	−0.0533923
$300$	−4.95063	−0.285825
$301$	12.1217	0.698682
$302$	−54.8109	−3.15401
$303$	−12.9138	−0.741876
$304$	20.6913	1.18673
$305$	−7.22344	−0.413613
$306$	1.78701	0.102157
$307$	5.41848	0.309249	0.154624	−	0.987973i	$-0.450583\pi$
$307$	5.41848	0.309249	0.154624	+	0.987973i	$0.450583\pi$
$308$	4.95063	0.282088
$309$	16.7092	0.950555
$310$	−4.92966	−0.279986
$311$	−12.3421	−0.699857	−0.349928	−	0.936776i	$-0.613794\pi$
$311$	−12.3421	−0.699857	−0.349928	+	0.936776i	$0.613794\pi$
$312$	24.4469	1.38403
$313$	−0.325195	−0.0183811	−0.00919057	−	0.999958i	$-0.502925\pi$
$313$	−0.325195	−0.0183811	−0.00919057	+	0.999958i	$0.502925\pi$
$314$	−1.11766	−0.0630732
$315$	1.00000	0.0563436
$316$	−26.7481	−1.50470
$317$	15.4185	0.865988	0.432994	−	0.901397i	$-0.357457\pi$
$317$	15.4185	0.865988	0.432994	+	0.901397i	$0.357457\pi$
$318$	−29.5895	−1.65930
$319$	−1.82047	−0.101927
$320$	11.4963	0.642660
$321$	5.30423	0.296053
$322$	0.774514	0.0431620
$323$	1.32218	0.0735680
$324$	4.95063	0.275035
$325$	−3.14265	−0.174323
$326$	4.24337	0.235019
$327$	−2.74270	−0.151672
$328$	−85.9113	−4.74366
$329$	2.89825	0.159785
$330$	−2.63640	−0.145129
$331$	−21.7123	−1.19341	−0.596707	−	0.802459i	$-0.703524\pi$
$331$	−21.7123	−1.19341	−0.596707	+	0.802459i	$0.703524\pi$
$332$	−14.7481	−0.809407
$333$	−4.03142	−0.220920
$334$	46.2903	2.53289
$335$	−12.6694	−0.692201
$336$	−10.6075	−0.578686
$337$	33.3945	1.81911	0.909557	−	0.415579i	$-0.136421\pi$
$337$	33.3945	1.81911	0.909557	+	0.415579i	$0.136421\pi$
$338$	−8.23542	−0.447948
$339$	−2.06527	−0.112170
$340$	3.35564	0.181985
$341$	−1.86984	−0.101258
$342$	5.14265	0.278083
$343$	1.00000	0.0539949
$344$	94.2953	5.08406
$345$	−0.293777	−0.0158164
$346$	67.7656	3.64310
$347$	−18.4878	−0.992476	−0.496238	−	0.868187i	$-0.665286\pi$
$347$	−18.4878	−0.992476	−0.496238	+	0.868187i	$0.665286\pi$
$348$	9.01250	0.483121
$349$	−7.81099	−0.418113	−0.209056	−	0.977904i	$-0.567039\pi$
$349$	−7.81099	−0.418113	−0.209056	+	0.977904i	$0.567039\pi$
$350$	2.63640	0.140922
$351$	3.14265	0.167742
$352$	12.4075	0.661323
$353$	22.0628	1.17429	0.587143	−	0.809483i	$-0.300253\pi$
$353$	22.0628	1.17429	0.587143	+	0.809483i	$0.300253\pi$
$354$	19.3871	1.03041
$355$	−5.77111	−0.306298
$356$	37.0175	1.96192
$357$	−0.677821	−0.0358741
$358$	−55.5881	−2.93792
$359$	36.4948	1.92612	0.963062	−	0.269281i	$-0.0867862\pi$
$359$	36.4948	1.92612	0.963062	+	0.269281i	$0.0867862\pi$
$360$	7.77906	0.409992
$361$	−15.1950	−0.799739
$362$	36.3890	1.91257
$363$	−1.00000	−0.0524864
$364$	−15.5581	−0.815467
$365$	9.27281	0.485361
$366$	19.0439	0.995441
$367$	−26.4070	−1.37843	−0.689217	−	0.724555i	$-0.742045\pi$
$367$	−26.4070	−1.37843	−0.689217	+	0.724555i	$0.742045\pi$
$368$	3.11623	0.162445
$369$	−11.0439	−0.574923
$370$	−10.6285	−0.552547
$371$	11.2234	0.582692
$372$	9.25690	0.479948
$373$	−3.36950	−0.174466	−0.0872331	−	0.996188i	$-0.527803\pi$
$373$	−3.36950	−0.174466	−0.0872331	+	0.996188i	$0.527803\pi$
$374$	1.78701	0.0924041
$375$	−1.00000	−0.0516398
$376$	22.5456	1.16270
$377$	5.72112	0.294653
$378$	−2.63640	−0.135602
$379$	19.7372	1.01383	0.506917	−	0.861995i	$-0.330785\pi$
$379$	19.7372	1.01383	0.506917	+	0.861995i	$0.330785\pi$
$380$	9.65685	0.495386
$381$	3.35360	0.171810
$382$	−24.4049	−1.24867
$383$	36.6399	1.87221	0.936105	−	0.351720i	$-0.114403\pi$
$383$	36.6399	1.87221	0.936105	+	0.351720i	$0.114403\pi$
$384$	−5.49375	−0.280352
$385$	1.00000	0.0509647
$386$	−15.6659	−0.797375
$387$	12.1217	0.616180
$388$	−13.9013	−0.705730
$389$	25.7961	1.30791	0.653957	−	0.756532i	$-0.273108\pi$
$389$	25.7961	1.30791	0.653957	+	0.756532i	$0.273108\pi$
$390$	8.28531	0.419543
$391$	0.199128	0.0100703
$392$	7.77906	0.392902
$393$	17.9327	0.904584
$394$	−38.8400	−1.95673
$395$	−5.40297	−0.271853
$396$	4.95063	0.248779
$397$	33.7984	1.69629	0.848147	−	0.529760i	$-0.177718\pi$
$397$	33.7984	1.69629	0.848147	+	0.529760i	$0.177718\pi$
$398$	33.1412	1.66122
$399$	−1.95063	−0.0976537
$400$	10.6075	0.530374
$401$	31.7352	1.58478	0.792390	−	0.610015i	$-0.208836\pi$
$401$	31.7352	1.58478	0.792390	+	0.610015i	$0.208836\pi$
$402$	33.4015	1.66592
$403$	5.87627	0.292718
$404$	63.9313	3.18070
$405$	1.00000	0.0496904
$406$	−4.79951	−0.238196
$407$	−4.03142	−0.199830
$408$	−5.27281	−0.261043
$409$	7.12714	0.352414	0.176207	−	0.984353i	$-0.443617\pi$
$409$	7.12714	0.352414	0.176207	+	0.984353i	$0.443617\pi$
$410$	−29.1162	−1.43795
$411$	−4.12373	−0.203409
$412$	−82.7213	−4.07538
$413$	−7.35360	−0.361847
$414$	0.774514	0.0380653
$415$	−2.97903	−0.146235
$416$	−38.9925	−1.91176
$417$	5.75559	0.281853
$418$	5.14265	0.251535
$419$	−2.77013	−0.135330	−0.0676649	−	0.997708i	$-0.521555\pi$
$419$	−2.77013	−0.135330	−0.0676649	+	0.997708i	$0.521555\pi$
$420$	−4.95063	−0.241566
$421$	2.58999	0.126228	0.0631142	−	0.998006i	$-0.479897\pi$
$421$	2.58999	0.126228	0.0631142	+	0.998006i	$0.479897\pi$
$422$	64.2335	3.12684
$423$	2.89825	0.140918
$424$	87.3078	4.24004
$425$	0.677821	0.0328792
$426$	15.2150	0.737168
$427$	−7.22344	−0.349567
$428$	−26.2593	−1.26929
$429$	3.14265	0.151729
$430$	31.9577	1.54114
$431$	−26.0878	−1.25661	−0.628303	−	0.777968i	$-0.716250\pi$
$431$	−26.0878	−1.25661	−0.628303	+	0.777968i	$0.716250\pi$
$432$	−10.6075	−0.510353
$433$	−23.0284	−1.10667	−0.553337	−	0.832957i	$-0.686646\pi$
$433$	−23.0284	−1.10667	−0.553337	+	0.832957i	$0.686646\pi$
$434$	−4.92966	−0.236631
$435$	1.82047	0.0872851
$436$	13.5781	0.650274
$437$	0.573050	0.0274127
$438$	−24.4469	−1.16812
$439$	−19.0778	−0.910532	−0.455266	−	0.890355i	$-0.650456\pi$
$439$	−19.0778	−0.910532	−0.455266	+	0.890355i	$0.650456\pi$
$440$	7.77906	0.370852
$441$	1.00000	0.0476190
$442$	−5.61596	−0.267124
$443$	11.9331	0.566957	0.283479	−	0.958979i	$-0.408511\pi$
$443$	11.9331	0.566957	0.283479	+	0.958979i	$0.408511\pi$
$444$	19.9581	0.947168
$445$	7.47733	0.354459
$446$	−33.7670	−1.59891
$447$	3.91717	0.185276
$448$	11.4963	0.543147
$449$	23.7042	1.11867	0.559335	−	0.828942i	$-0.311057\pi$
$449$	23.7042	1.11867	0.559335	+	0.828942i	$0.311057\pi$
$450$	2.63640	0.124281
$451$	−11.0439	−0.520038
$452$	10.2244	0.480916
$453$	20.7900	0.976801
$454$	−15.4280	−0.724070
$455$	−3.14265	−0.147330
$456$	−15.1741	−0.710591
$457$	−26.0449	−1.21833	−0.609164	−	0.793044i	$-0.708495\pi$
$457$	−26.0449	−1.21833	−0.609164	+	0.793044i	$0.708495\pi$
$458$	−2.10783	−0.0984922
$459$	−0.677821	−0.0316380
$460$	1.45438	0.0678108
$461$	37.4654	1.74494	0.872469	−	0.488669i	$-0.162518\pi$
$461$	37.4654	1.74494	0.872469	+	0.488669i	$0.162518\pi$
$462$	−2.63640	−0.122657
$463$	−18.9547	−0.880898	−0.440449	−	0.897778i	$-0.645181\pi$
$463$	−18.9547	−0.880898	−0.440449	+	0.897778i	$0.645181\pi$
$464$	−19.3107	−0.896475
$465$	1.86984	0.0867119
$466$	−30.0868	−1.39374
$467$	31.3447	1.45046	0.725231	−	0.688506i	$-0.241733\pi$
$467$	31.3447	1.45046	0.725231	+	0.688506i	$0.241733\pi$
$468$	−15.5581	−0.719174
$469$	−12.6694	−0.585016
$470$	7.64095	0.352451
$471$	0.423933	0.0195338
$472$	−57.2041	−2.63303
$473$	12.1217	0.557356
$474$	14.2444	0.654267
$475$	1.95063	0.0895011
$476$	3.35564	0.153806
$477$	11.2234	0.513886
$478$	41.5567	1.90076
$479$	1.13357	0.0517939	0.0258970	−	0.999665i	$-0.491756\pi$
$479$	1.13357	0.0517939	0.0258970	+	0.999665i	$0.491756\pi$
$480$	−12.4075	−0.566323
$481$	12.6694	0.577673
$482$	−53.7406	−2.44782
$483$	−0.293777	−0.0133673
$484$	4.95063	0.225029
$485$	−2.80798	−0.127504
$486$	−2.63640	−0.119590
$487$	12.7322	0.576951	0.288475	−	0.957487i	$-0.406852\pi$
$487$	12.7322	0.576951	0.288475	+	0.957487i	$0.406852\pi$
$488$	−56.1916	−2.54367
$489$	−1.60953	−0.0727855
$490$	2.63640	0.119101
$491$	10.5727	0.477137	0.238569	−	0.971126i	$-0.423322\pi$
$491$	10.5727	0.477137	0.238569	+	0.971126i	$0.423322\pi$
$492$	54.6743	2.46491
$493$	−1.23396	−0.0555746
$494$	−16.1616	−0.727144
$495$	1.00000	0.0449467
$496$	−19.8343	−0.890588
$497$	−5.77111	−0.258869
$498$	7.85394	0.351943
$499$	−28.0294	−1.25477	−0.627384	−	0.778710i	$-0.715874\pi$
$499$	−28.0294	−1.25477	−0.627384	+	0.778710i	$0.715874\pi$
$500$	4.95063	0.221399
$501$	−17.5581	−0.784439
$502$	78.2007	3.49027
$503$	10.2779	0.458268	0.229134	−	0.973395i	$-0.426411\pi$
$503$	10.2779	0.458268	0.229134	+	0.973395i	$0.426411\pi$
$504$	7.77906	0.346507
$505$	12.9138	0.574655
$506$	0.774514	0.0344314
$507$	3.12373	0.138730
$508$	−16.6024	−0.736613
$509$	17.3536	0.769185	0.384592	−	0.923087i	$-0.374342\pi$
$509$	17.3536	0.769185	0.384592	+	0.923087i	$0.374342\pi$
$510$	−1.78701	−0.0791302
$511$	9.27281	0.410205
$512$	−33.4200	−1.47697
$513$	−1.95063	−0.0861225
$514$	−11.1044	−0.489795
$515$	−16.7092	−0.736297
$516$	−60.0100	−2.64179
$517$	2.89825	0.127465
$518$	−10.6285	−0.466987
$519$	−25.7038	−1.12827
$520$	−24.4469	−1.07207
$521$	−18.1535	−0.795319	−0.397660	−	0.917533i	$-0.630177\pi$
$521$	−18.1535	−0.795319	−0.397660	+	0.917533i	$0.630177\pi$
$522$	−4.79951	−0.210069
$523$	27.6558	1.20930	0.604652	−	0.796490i	$-0.293312\pi$
$523$	27.6558	1.20930	0.604652	+	0.796490i	$0.293312\pi$
$524$	−88.7781	−3.87829
$525$	−1.00000	−0.0436436
$526$	63.9732	2.78936
$527$	−1.26742	−0.0552097
$528$	−10.6075	−0.461632
$529$	−22.9137	−0.996248
$530$	29.5895	1.28529
$531$	−7.35360	−0.319119
$532$	9.65685	0.418678
$533$	34.7072	1.50334
$534$	−19.7133	−0.853076
$535$	−5.30423	−0.229322
$536$	−98.5556	−4.25695
$537$	21.0848	0.909877
$538$	−1.72008	−0.0741581
$539$	1.00000	0.0430730
$540$	−4.95063	−0.213041
$541$	−34.6081	−1.48792	−0.743958	−	0.668226i	$-0.767054\pi$
$541$	−34.6081	−1.48792	−0.743958	+	0.668226i	$0.767054\pi$
$542$	27.5226	1.18220
$543$	−13.8025	−0.592323
$544$	8.41007	0.360579
$545$	2.74270	0.117484
$546$	8.28531	0.354578
$547$	−0.220424	−0.00942466	−0.00471233	−	0.999989i	$-0.501500\pi$
$547$	−0.220424	−0.00942466	−0.00471233	+	0.999989i	$0.501500\pi$
$548$	20.4151	0.872089
$549$	−7.22344	−0.308289
$550$	2.63640	0.112417
$551$	−3.55107	−0.151281
$552$	−2.28531	−0.0972691
$553$	−5.40297	−0.229758
$554$	71.9681	3.05763
$555$	4.03142	0.171124
$556$	−28.4938	−1.20841
$557$	−23.5990	−0.999922	−0.499961	−	0.866048i	$-0.666652\pi$
$557$	−23.5990	−0.999922	−0.499961	+	0.866048i	$0.666652\pi$
$558$	−4.92966	−0.208689
$559$	−38.0943	−1.61122
$560$	10.6075	0.448248
$561$	−0.677821	−0.0286176
$562$	57.5881	2.42921
$563$	24.0409	1.01320	0.506602	−	0.862180i	$-0.330902\pi$
$563$	24.0409	1.01320	0.506602	+	0.862180i	$0.330902\pi$
$564$	−14.3481	−0.604166
$565$	2.06527	0.0868868
$566$	−5.61596	−0.236056
$567$	1.00000	0.0419961
$568$	−44.8938	−1.88370
$569$	−35.0604	−1.46981	−0.734905	−	0.678170i	$-0.762773\pi$
$569$	−35.0604	−1.46981	−0.734905	+	0.678170i	$0.762773\pi$
$570$	−5.14265	−0.215402
$571$	1.68224	0.0703995	0.0351998	−	0.999380i	$-0.488793\pi$
$571$	1.68224	0.0703995	0.0351998	+	0.999380i	$0.488793\pi$
$572$	−15.5581	−0.650518
$573$	9.25690	0.386713
$574$	−29.1162	−1.21529
$575$	0.293777	0.0122513
$576$	11.4963	0.479010
$577$	12.3163	0.512736	0.256368	−	0.966579i	$-0.417474\pi$
$577$	12.3163	0.512736	0.256368	+	0.966579i	$0.417474\pi$
$578$	−43.6076	−1.81384
$579$	5.94216	0.246948
$580$	−9.01250	−0.374224
$581$	−2.97903	−0.123591
$582$	7.40297	0.306863
$583$	11.2234	0.464827
$584$	72.1337	2.98491
$585$	−3.14265	−0.129933
$586$	78.9593	3.26178
$587$	−7.71430	−0.318403	−0.159202	−	0.987246i	$-0.550892\pi$
$587$	−7.71430	−0.318403	−0.159202	+	0.987246i	$0.550892\pi$
$588$	−4.95063	−0.204161
$589$	−3.64737	−0.150287
$590$	−19.3871	−0.798153
$591$	14.7322	0.606001
$592$	−42.7632	−1.75756
$593$	37.2778	1.53082	0.765408	−	0.643545i	$-0.222537\pi$
$593$	37.2778	1.53082	0.765408	+	0.643545i	$0.222537\pi$
$594$	−2.63640	−0.108173
$595$	0.677821	0.0277880
$596$	−19.3925	−0.794346
$597$	−12.5706	−0.514481
$598$	−2.43403	−0.0995349
$599$	−29.5321	−1.20665	−0.603324	−	0.797496i	$-0.706157\pi$
$599$	−29.5321	−1.20665	−0.603324	+	0.797496i	$0.706157\pi$
$600$	−7.77906	−0.317579
$601$	−19.0619	−0.777550	−0.388775	−	0.921333i	$-0.627102\pi$
$601$	−19.0619	−0.777550	−0.388775	+	0.921333i	$0.627102\pi$
$602$	31.9577	1.30250
$603$	−12.6694	−0.515936
$604$	−102.924	−4.18791
$605$	1.00000	0.0406558
$606$	−34.0459	−1.38302
$607$	−38.7254	−1.57181	−0.785907	−	0.618345i	$-0.787804\pi$
$607$	−38.7254	−1.57181	−0.785907	+	0.618345i	$0.787804\pi$
$608$	24.2025	0.981540
$609$	1.82047	0.0737693
$610$	−19.0439	−0.771066
$611$	−9.10818	−0.368478
$612$	3.35564	0.135644
$613$	35.1801	1.42091	0.710456	−	0.703742i	$-0.248489\pi$
$613$	35.1801	1.42091	0.710456	+	0.703742i	$0.248489\pi$
$614$	14.2853	0.576508
$615$	11.0439	0.445334
$616$	7.77906	0.313427
$617$	−20.5097	−0.825690	−0.412845	−	0.910801i	$-0.635465\pi$
$617$	−20.5097	−0.825690	−0.412845	+	0.910801i	$0.635465\pi$
$618$	44.0523	1.77204
$619$	19.2150	0.772315	0.386157	−	0.922433i	$-0.373802\pi$
$619$	19.2150	0.772315	0.386157	+	0.922433i	$0.373802\pi$
$620$	−9.25690	−0.371766
$621$	−0.293777	−0.0117889
$622$	−32.5388	−1.30469
$623$	7.47733	0.299573
$624$	33.3356	1.33449
$625$	1.00000	0.0400000
$626$	−0.857347	−0.0342665
$627$	−1.95063	−0.0779007
$628$	−2.09874	−0.0837488
$629$	−2.73258	−0.108955
$630$	2.63640	0.105037
$631$	19.5716	0.779132	0.389566	−	0.920998i	$-0.372625\pi$
$631$	19.5716	0.779132	0.389566	+	0.920998i	$0.372625\pi$
$632$	−42.0300	−1.67186
$633$	−24.3640	−0.968384
$634$	40.6494	1.61439
$635$	−3.35360	−0.133083
$636$	−55.5631	−2.20322
$637$	−3.14265	−0.124516
$638$	−4.79951	−0.190014
$639$	−5.77111	−0.228301
$640$	5.49375	0.217160
$641$	−32.8370	−1.29698	−0.648491	−	0.761222i	$-0.724600\pi$
$641$	−32.8370	−1.29698	−0.648491	+	0.761222i	$0.724600\pi$
$642$	13.9841	0.551908
$643$	−4.12169	−0.162543	−0.0812717	−	0.996692i	$-0.525898\pi$
$643$	−4.12169	−0.162543	−0.0812717	+	0.996692i	$0.525898\pi$
$644$	1.45438	0.0573106
$645$	−12.1217	−0.477291
$646$	3.48580	0.137147
$647$	−13.7147	−0.539180	−0.269590	−	0.962975i	$-0.586888\pi$
$647$	−13.7147	−0.539180	−0.269590	+	0.962975i	$0.586888\pi$
$648$	7.77906	0.305590
$649$	−7.35360	−0.288654
$650$	−8.28531	−0.324976
$651$	1.86984	0.0732849
$652$	7.96819	0.312058
$653$	−36.3147	−1.42110	−0.710552	−	0.703645i	$-0.751555\pi$
$653$	−36.3147	−1.42110	−0.710552	+	0.703645i	$0.751555\pi$
$654$	−7.23088	−0.282750
$655$	−17.9327	−0.700688
$656$	−117.148	−4.57387
$657$	9.27281	0.361767
$658$	7.64095	0.297875
$659$	−9.70583	−0.378085	−0.189043	−	0.981969i	$-0.560538\pi$
$659$	−9.70583	−0.378085	−0.189043	+	0.981969i	$0.560538\pi$
$660$	−4.95063	−0.192703
$661$	−11.5108	−0.447718	−0.223859	−	0.974622i	$-0.571865\pi$
$661$	−11.5108	−0.447718	−0.223859	+	0.974622i	$0.571865\pi$
$662$	−57.2423	−2.22479
$663$	2.13016	0.0827284
$664$	−23.1741	−0.899328
$665$	1.95063	0.0756422
$666$	−10.6285	−0.411844
$667$	−0.534813	−0.0207080
$668$	86.9238	3.36318
$669$	12.8080	0.495185
$670$	−33.4015	−1.29041
$671$	−7.22344	−0.278858
$672$	−12.4075	−0.478630
$673$	48.4479	1.86753	0.933764	−	0.357888i	$-0.116503\pi$
$673$	48.4479	1.86753	0.933764	+	0.357888i	$0.116503\pi$
$674$	88.0414	3.39123
$675$	−1.00000	−0.0384900
$676$	−15.4644	−0.594786
$677$	−1.34717	−0.0517760	−0.0258880	−	0.999665i	$-0.508241\pi$
$677$	−1.34717	−0.0517760	−0.0258880	+	0.999665i	$0.508241\pi$
$678$	−5.44490	−0.209110
$679$	−2.80798	−0.107760
$680$	5.27281	0.202203
$681$	5.85189	0.224245
$682$	−4.92966	−0.188767
$683$	43.0844	1.64858	0.824290	−	0.566168i	$-0.191575\pi$
$683$	43.0844	1.64858	0.824290	+	0.566168i	$0.191575\pi$
$684$	9.65685	0.369239
$685$	4.12373	0.157560
$686$	2.63640	0.100658
$687$	0.799507	0.0305031
$688$	128.581	4.90209
$689$	−35.2714	−1.34373
$690$	−0.774514	−0.0294853
$691$	0.619365	0.0235617	0.0117809	−	0.999931i	$-0.496250\pi$
$691$	0.619365	0.0235617	0.0117809	+	0.999931i	$0.496250\pi$
$692$	127.250	4.83732
$693$	1.00000	0.0379869
$694$	−48.7413	−1.85019
$695$	−5.75559	−0.218322
$696$	14.1616	0.536793
$697$	−7.48580	−0.283545
$698$	−20.5929	−0.779454
$699$	11.4121	0.431644
$700$	4.95063	0.187116
$701$	−17.7846	−0.671714	−0.335857	−	0.941913i	$-0.609026\pi$
$701$	−17.7846	−0.671714	−0.335857	+	0.941913i	$0.609026\pi$
$702$	8.28531	0.312709
$703$	−7.86381	−0.296589
$704$	11.4963	0.433281
$705$	−2.89825	−0.109154
$706$	58.1666	2.18913
$707$	12.9138	0.485672
$708$	36.4049	1.36818
$709$	9.41001	0.353400	0.176700	−	0.984265i	$-0.443458\pi$
$709$	9.41001	0.353400	0.176700	+	0.984265i	$0.443458\pi$
$710$	−15.2150	−0.571008
$711$	−5.40297	−0.202627
$712$	58.1666	2.17988
$713$	−0.549316	−0.0205721
$714$	−1.78701	−0.0668772
$715$	−3.14265	−0.117529
$716$	−104.383	−3.90098
$717$	−15.7626	−0.588666
$718$	96.2151	3.59072
$719$	6.50573	0.242623	0.121311	−	0.992614i	$-0.461290\pi$
$719$	6.50573	0.242623	0.121311	+	0.992614i	$0.461290\pi$
$720$	10.6075	0.395318
$721$	−16.7092	−0.622285
$722$	−40.0603	−1.49089
$723$	20.3840	0.758091
$724$	68.3312	2.53951
$725$	−1.82047	−0.0676107
$726$	−2.63640	−0.0978462
$727$	36.0688	1.33772	0.668860	−	0.743389i	$-0.266783\pi$
$727$	36.0688	1.33772	0.668860	+	0.743389i	$0.266783\pi$
$728$	−24.4469	−0.906062
$729$	1.00000	0.0370370
$730$	24.4469	0.904820
$731$	8.21633	0.303892
$732$	35.7606	1.32175
$733$	−48.2903	−1.78364	−0.891822	−	0.452387i	$-0.850572\pi$
$733$	−48.2903	−1.78364	−0.891822	+	0.452387i	$0.850572\pi$
$734$	−69.6195	−2.56970
$735$	−1.00000	−0.0368856
$736$	3.64504	0.134358
$737$	−12.6694	−0.466682
$738$	−29.1162	−1.07178
$739$	5.52709	0.203317	0.101659	−	0.994819i	$-0.467585\pi$
$739$	5.52709	0.203317	0.101659	+	0.994819i	$0.467585\pi$
$740$	−19.9581	−0.733673
$741$	6.13016	0.225197
$742$	29.5895	1.08627
$743$	−44.2484	−1.62332	−0.811658	−	0.584133i	$-0.801434\pi$
$743$	−44.2484	−1.62332	−0.811658	+	0.584133i	$0.801434\pi$
$744$	14.5456	0.533268
$745$	−3.91717	−0.143514
$746$	−8.88337	−0.325243
$747$	−2.97903	−0.108997
$748$	3.35564	0.122694
$749$	−5.30423	−0.193812
$750$	−2.63640	−0.0962679
$751$	8.60408	0.313967	0.156984	−	0.987601i	$-0.449823\pi$
$751$	8.60408	0.313967	0.156984	+	0.987601i	$0.449823\pi$
$752$	30.7431	1.12109
$753$	−29.6619	−1.08094
$754$	15.0832	0.549297
$755$	−20.7900	−0.756627
$756$	−4.95063	−0.180053
$757$	32.0878	1.16625	0.583126	−	0.812382i	$-0.301829\pi$
$757$	32.0878	1.16625	0.583126	+	0.812382i	$0.301829\pi$
$758$	52.0354	1.89001
$759$	−0.293777	−0.0106634
$760$	15.1741	0.550421
$761$	41.5980	1.50793	0.753963	−	0.656917i	$-0.228140\pi$
$761$	41.5980	1.50793	0.753963	+	0.656917i	$0.228140\pi$
$762$	8.84144	0.320292
$763$	2.74270	0.0992925
$764$	−45.8275	−1.65798
$765$	0.677821	0.0245067
$766$	96.5975	3.49021
$767$	23.1098	0.834447
$768$	8.50875	0.307033
$769$	44.8535	1.61746	0.808729	−	0.588181i	$-0.200156\pi$
$769$	44.8535	1.61746	0.808729	+	0.588181i	$0.200156\pi$
$770$	2.63640	0.0950094
$771$	4.21195	0.151690
$772$	−29.4174	−1.05876
$773$	15.3516	0.552157	0.276078	−	0.961135i	$-0.410965\pi$
$773$	15.3516	0.552157	0.276078	+	0.961135i	$0.410965\pi$
$774$	31.9577	1.14869
$775$	−1.86984	−0.0671668
$776$	−21.8434	−0.784133
$777$	4.03142	0.144626
$778$	68.0090	2.43824
$779$	−21.5426	−0.771844
$780$	15.5581	0.557070
$781$	−5.77111	−0.206506
$782$	0.524982	0.0187733
$783$	1.82047	0.0650584
$784$	10.6075	0.378839
$785$	−0.423933	−0.0151308
$786$	47.2778	1.68634
$787$	−2.41995	−0.0862617	−0.0431309	−	0.999069i	$-0.513733\pi$
$787$	−2.41995	−0.0862617	−0.0431309	+	0.999069i	$0.513733\pi$
$788$	−72.9336	−2.59815
$789$	−24.2653	−0.863868
$790$	−14.2444	−0.506793
$791$	2.06527	0.0734327
$792$	7.77906	0.276417
$793$	22.7008	0.806128
$794$	89.1064	3.16227
$795$	−11.2234	−0.398054
$796$	62.2325	2.20577
$797$	−12.2035	−0.432270	−0.216135	−	0.976363i	$-0.569345\pi$
$797$	−12.2035	−0.432270	−0.216135	+	0.976363i	$0.569345\pi$
$798$	−5.14265	−0.182048
$799$	1.96449	0.0694987
$800$	12.4075	0.438672
$801$	7.47733	0.264198
$802$	83.6669	2.95438
$803$	9.27281	0.327230
$804$	62.7213	2.21201
$805$	0.293777	0.0103543
$806$	15.4922	0.545690
$807$	0.652436	0.0229668
$808$	100.457	3.53406
$809$	37.2997	1.31139	0.655695	−	0.755026i	$-0.272376\pi$
$809$	37.2997	1.31139	0.655695	+	0.755026i	$0.272376\pi$
$810$	2.63640	0.0926338
$811$	−35.0394	−1.23040	−0.615200	−	0.788371i	$-0.710925\pi$
$811$	−35.0394	−1.23040	−0.615200	+	0.788371i	$0.710925\pi$
$812$	−9.01250	−0.316277
$813$	−10.4394	−0.366127
$814$	−10.6285	−0.372527
$815$	1.60953	0.0563794
$816$	−7.18998	−0.251700
$817$	23.6449	0.827232
$818$	18.7900	0.656978
$819$	−3.14265	−0.109813
$820$	−54.6743	−1.90931
$821$	−31.6170	−1.10344	−0.551720	−	0.834030i	$-0.686028\pi$
$821$	−31.6170	−1.10344	−0.551720	+	0.834030i	$0.686028\pi$
$822$	−10.8718	−0.379198
$823$	11.7292	0.408853	0.204427	−	0.978882i	$-0.434467\pi$
$823$	11.7292	0.408853	0.204427	+	0.978882i	$0.434467\pi$
$824$	−129.982	−4.52814
$825$	−1.00000	−0.0348155
$826$	−19.3871	−0.674562
$827$	−6.43779	−0.223864	−0.111932	−	0.993716i	$-0.535704\pi$
$827$	−6.43779	−0.223864	−0.111932	+	0.993716i	$0.535704\pi$
$828$	1.45438	0.0505432
$829$	−29.5422	−1.02604	−0.513022	−	0.858376i	$-0.671474\pi$
$829$	−29.5422	−1.02604	−0.513022	+	0.858376i	$0.671474\pi$
$830$	−7.85394	−0.272614
$831$	−27.2978	−0.946950
$832$	−36.1287	−1.25254
$833$	0.677821	0.0234851
$834$	15.1741	0.525435
$835$	17.5581	0.607624
$836$	9.65685	0.333989
$837$	1.86984	0.0646312
$838$	−7.30319	−0.252285
$839$	22.5727	0.779295	0.389647	−	0.920964i	$-0.372597\pi$
$839$	22.5727	0.779295	0.389647	+	0.920964i	$0.372597\pi$
$840$	−7.77906	−0.268403
$841$	−25.6859	−0.885720
$842$	6.82826	0.235318
$843$	−21.8434	−0.752327
$844$	120.617	4.15182
$845$	−3.12373	−0.107460
$846$	7.64095	0.262701
$847$	1.00000	0.0343604
$848$	119.052	4.08828
$849$	2.13016	0.0731068
$850$	1.78701	0.0612940
$851$	−1.18434	−0.0405986
$852$	28.5706	0.978813
$853$	20.2698	0.694025	0.347012	−	0.937861i	$-0.387196\pi$
$853$	20.2698	0.694025	0.347012	+	0.937861i	$0.387196\pi$
$854$	−19.0439	−0.651669
$855$	1.95063	0.0667102
$856$	−41.2619	−1.41030
$857$	3.77753	0.129038	0.0645190	−	0.997916i	$-0.479449\pi$
$857$	3.77753	0.129038	0.0645190	+	0.997916i	$0.479449\pi$
$858$	8.28531	0.282856
$859$	−24.4783	−0.835189	−0.417594	−	0.908634i	$-0.637127\pi$
$859$	−24.4783	−0.835189	−0.417594	+	0.908634i	$0.637127\pi$
$860$	60.0100	2.04632
$861$	11.0439	0.376376
$862$	−68.7781	−2.34259
$863$	0.850304	0.0289447	0.0144723	−	0.999895i	$-0.495393\pi$
$863$	0.850304	0.0289447	0.0144723	+	0.999895i	$0.495393\pi$
$864$	−12.4075	−0.422112
$865$	25.7038	0.873955
$866$	−60.7122	−2.06308
$867$	16.5406	0.561747
$868$	−9.25690	−0.314200
$869$	−5.40297	−0.183283
$870$	4.79951	0.162719
$871$	39.8154	1.34909
$872$	21.3356	0.722516
$873$	−2.80798	−0.0950357
$874$	1.51079	0.0511033
$875$	1.00000	0.0338062
$876$	−45.9063	−1.55103
$877$	5.67175	0.191521	0.0957607	−	0.995404i	$-0.469472\pi$
$877$	5.67175	0.191521	0.0957607	+	0.995404i	$0.469472\pi$
$878$	−50.2967	−1.69743
$879$	−29.9496	−1.01017
$880$	10.6075	0.357578
$881$	−22.3630	−0.753430	−0.376715	−	0.926329i	$-0.622946\pi$
$881$	−22.3630	−0.753430	−0.376715	+	0.926329i	$0.622946\pi$
$882$	2.63640	0.0887724
$883$	0.780940	0.0262807	0.0131404	−	0.999914i	$-0.495817\pi$
$883$	0.780940	0.0262807	0.0131404	+	0.999914i	$0.495817\pi$
$884$	−10.5456	−0.354688
$885$	7.35360	0.247188
$886$	31.4604	1.05693
$887$	−48.3975	−1.62503	−0.812515	−	0.582941i	$-0.801902\pi$
$887$	−48.3975	−1.62503	−0.812515	+	0.582941i	$0.801902\pi$
$888$	31.3606	1.05239
$889$	−3.35360	−0.112476
$890$	19.7133	0.660790
$891$	1.00000	0.0335013
$892$	−63.4076	−2.12304
$893$	5.65341	0.189184
$894$	10.3272	0.345395
$895$	−21.0848	−0.704788
$896$	5.49375	0.183533
$897$	0.923238	0.0308260
$898$	62.4938	2.08545
$899$	3.40400	0.113530
$900$	4.95063	0.165021
$901$	7.60749	0.253442
$902$	−29.1162	−0.969464
$903$	−12.1217	−0.403384
$904$	16.0659	0.534344
$905$	13.8025	0.458811
$906$	54.8109	1.82097
$907$	−49.7042	−1.65040	−0.825200	−	0.564841i	$-0.808938\pi$
$907$	−49.7042	−1.65040	−0.825200	+	0.564841i	$0.808938\pi$
$908$	−28.9706	−0.961422
$909$	12.9138	0.428322
$910$	−8.28531	−0.274655
$911$	39.5022	1.30877	0.654383	−	0.756163i	$-0.272928\pi$
$911$	39.5022	1.30877	0.654383	+	0.756163i	$0.272928\pi$
$912$	−20.6913	−0.685157
$913$	−2.97903	−0.0985916
$914$	−68.6649	−2.27123
$915$	7.22344	0.238800
$916$	−3.95807	−0.130778
$917$	−17.9327	−0.592189
$918$	−1.78701	−0.0589801
$919$	−26.5357	−0.875334	−0.437667	−	0.899137i	$-0.644195\pi$
$919$	−26.5357	−0.875334	−0.437667	+	0.899137i	$0.644195\pi$
$920$	2.28531	0.0753443
$921$	−5.41848	−0.178545
$922$	98.7740	3.25295
$923$	18.1366	0.596973
$924$	−4.95063	−0.162864
$925$	−4.03142	−0.132552
$926$	−49.9722	−1.64219
$927$	−16.7092	−0.548803
$928$	−22.5876	−0.741473
$929$	12.6315	0.414426	0.207213	−	0.978296i	$-0.433561\pi$
$929$	12.6315	0.414426	0.207213	+	0.978296i	$0.433561\pi$
$930$	4.92966	0.161650
$931$	1.95063	0.0639293
$932$	−56.4969	−1.85062
$933$	12.3421	0.404063
$934$	82.6374	2.70398
$935$	0.677821	0.0221671
$936$	−24.4469	−0.799071
$937$	16.7396	0.546860	0.273430	−	0.961892i	$-0.411842\pi$
$937$	16.7396	0.546860	0.273430	+	0.961892i	$0.411842\pi$
$938$	−33.4015	−1.09060
$939$	0.325195	0.0106124
$940$	14.3481	0.467985
$941$	17.5098	0.570802	0.285401	−	0.958408i	$-0.407873\pi$
$941$	17.5098	0.570802	0.285401	+	0.958408i	$0.407873\pi$
$942$	1.11766	0.0364153
$943$	−3.24445	−0.105654
$944$	−78.0032	−2.53879
$945$	−1.00000	−0.0325300
$946$	31.9577	1.03903
$947$	−16.2458	−0.527918	−0.263959	−	0.964534i	$-0.585028\pi$
$947$	−16.2458	−0.527918	−0.263959	+	0.964534i	$0.585028\pi$
$948$	26.7481	0.868738
$949$	−29.1412	−0.945964
$950$	5.14265	0.166850
$951$	−15.4185	−0.499978
$952$	5.27281	0.170893
$953$	−43.4144	−1.40633	−0.703165	−	0.711027i	$-0.748230\pi$
$953$	−43.4144	−1.40633	−0.703165	+	0.711027i	$0.748230\pi$
$954$	29.5895	0.957997
$955$	−9.25690	−0.299546
$956$	78.0350	2.52383
$957$	1.82047	0.0588476
$958$	2.98854	0.0965553
$959$	4.12373	0.133162
$960$	−11.4963	−0.371040
$961$	−27.5037	−0.887216
$962$	33.4015	1.07691
$963$	−5.30423	−0.170926
$964$	−100.914	−3.25022
$965$	−5.94216	−0.191285
$966$	−0.774514	−0.0249196
$967$	−23.4834	−0.755174	−0.377587	−	0.925974i	$-0.623246\pi$
$967$	−23.4834	−0.755174	−0.377587	+	0.925974i	$0.623246\pi$
$968$	7.77906	0.250028
$969$	−1.32218	−0.0424745
$970$	−7.40297	−0.237695
$971$	2.46983	0.0792606	0.0396303	−	0.999214i	$-0.487382\pi$
$971$	2.46983	0.0792606	0.0396303	+	0.999214i	$0.487382\pi$
$972$	−4.95063	−0.158792
$973$	−5.75559	−0.184516
$974$	33.5672	1.07556
$975$	3.14265	0.100645
$976$	−76.6225	−2.45263
$977$	19.2734	0.616612	0.308306	−	0.951287i	$-0.400238\pi$
$977$	19.2734	0.616612	0.308306	+	0.951287i	$0.400238\pi$
$978$	−4.24337	−0.135688
$979$	7.47733	0.238976
$980$	4.95063	0.158142
$981$	2.74270	0.0875678
$982$	27.8738	0.889489
$983$	−46.2304	−1.47452	−0.737261	−	0.675608i	$-0.763881\pi$
$983$	−46.2304	−1.47452	−0.737261	+	0.675608i	$0.763881\pi$
$984$	85.9113	2.73875
$985$	−14.7322	−0.469407
$986$	−3.25321	−0.103603
$987$	−2.89825	−0.0922522
$988$	−30.3481	−0.965503
$989$	3.56107	0.113235
$990$	2.63640	0.0837904
$991$	−12.6487	−0.401800	−0.200900	−	0.979612i	$-0.564387\pi$
$991$	−12.6487	−0.401800	−0.200900	+	0.979612i	$0.564387\pi$
$992$	−23.2001	−0.736604
$993$	21.7123	0.689018
$994$	−15.2150	−0.482590
$995$	12.5706	0.398515
$996$	14.7481	0.467311
$997$	44.1436	1.39804	0.699021	−	0.715101i	$-0.253619\pi$
$997$	44.1436	1.39804	0.699021	+	0.715101i	$0.253619\pi$
$998$	−73.8968	−2.33916
$999$	4.03142	0.127548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.v.1.4	✓	4
3.2	odd	2		3465.2.a.bj.1.1		4
5.4	even	2		5775.2.a.by.1.1		4
7.6	odd	2		8085.2.a.bq.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.v.1.4	✓	4	1.1	even	1	trivial
3465.2.a.bj.1.1		4	3.2	odd	2
5775.2.a.by.1.1		4	5.4	even	2
8085.2.a.bq.1.4		4	7.6	odd	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:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

\(f(q)\)	\(=\)	\(q+2.63640 q^{2} -1.00000 q^{3} +4.95063 q^{4} +1.00000 q^{5} -2.63640 q^{6} +1.00000 q^{7} +7.77906 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.63640 q^{2} -1.00000 q^{3} +4.95063 q^{4} +1.00000 q^{5} -2.63640 q^{6} +1.00000 q^{7} +7.77906 q^{8} +1.00000 q^{9} +2.63640 q^{10} +1.00000 q^{11} -4.95063 q^{12} -3.14265 q^{13} +2.63640 q^{14} -1.00000 q^{15} +10.6075 q^{16} +0.677821 q^{17} +2.63640 q^{18} +1.95063 q^{19} +4.95063 q^{20} -1.00000 q^{21} +2.63640 q^{22} +0.293777 q^{23} -7.77906 q^{24} +1.00000 q^{25} -8.28531 q^{26} -1.00000 q^{27} +4.95063 q^{28} -1.82047 q^{29} -2.63640 q^{30} -1.86984 q^{31} +12.4075 q^{32} -1.00000 q^{33} +1.78701 q^{34} +1.00000 q^{35} +4.95063 q^{36} -4.03142 q^{37} +5.14265 q^{38} +3.14265 q^{39} +7.77906 q^{40} -11.0439 q^{41} -2.63640 q^{42} +12.1217 q^{43} +4.95063 q^{44} +1.00000 q^{45} +0.774514 q^{46} +2.89825 q^{47} -10.6075 q^{48} +1.00000 q^{49} +2.63640 q^{50} -0.677821 q^{51} -15.5581 q^{52} +11.2234 q^{53} -2.63640 q^{54} +1.00000 q^{55} +7.77906 q^{56} -1.95063 q^{57} -4.79951 q^{58} -7.35360 q^{59} -4.95063 q^{60} -7.22344 q^{61} -4.92966 q^{62} +1.00000 q^{63} +11.4963 q^{64} -3.14265 q^{65} -2.63640 q^{66} -12.6694 q^{67} +3.35564 q^{68} -0.293777 q^{69} +2.63640 q^{70} -5.77111 q^{71} +7.77906 q^{72} +9.27281 q^{73} -10.6285 q^{74} -1.00000 q^{75} +9.65685 q^{76} +1.00000 q^{77} +8.28531 q^{78} -5.40297 q^{79} +10.6075 q^{80} +1.00000 q^{81} -29.1162 q^{82} -2.97903 q^{83} -4.95063 q^{84} +0.677821 q^{85} +31.9577 q^{86} +1.82047 q^{87} +7.77906 q^{88} +7.47733 q^{89} +2.63640 q^{90} -3.14265 q^{91} +1.45438 q^{92} +1.86984 q^{93} +7.64095 q^{94} +1.95063 q^{95} -12.4075 q^{96} -2.80798 q^{97} +2.63640 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} + 4 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 6 q^{12} + 4 q^{13} + 2 q^{14} - 4 q^{15} + 6 q^{16} + 6 q^{17} + 2 q^{18} - 6 q^{19} + 6 q^{20} - 4 q^{21} + 2 q^{22} + 10 q^{23} - 6 q^{24} + 4 q^{25} - 4 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{30} - 8 q^{31} + 14 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{35} + 6 q^{36} + 12 q^{37} + 4 q^{38} - 4 q^{39} + 6 q^{40} - 2 q^{42} + 6 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{46} - 6 q^{48} + 4 q^{49} + 2 q^{50} - 6 q^{51} - 12 q^{52} + 14 q^{53} - 2 q^{54} + 4 q^{55} + 6 q^{56} + 6 q^{57} + 20 q^{58} + 2 q^{59} - 6 q^{60} + 2 q^{61} + 20 q^{62} + 4 q^{63} - 2 q^{64} + 4 q^{65} - 2 q^{66} - 12 q^{67} + 20 q^{68} - 10 q^{69} + 2 q^{70} + 4 q^{71} + 6 q^{72} + 20 q^{73} - 32 q^{74} - 4 q^{75} + 16 q^{76} + 4 q^{77} - 4 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 14 q^{83} - 6 q^{84} + 6 q^{85} + 40 q^{86} - 6 q^{87} + 6 q^{88} - 6 q^{89} + 2 q^{90} + 4 q^{91} + 40 q^{92} + 8 q^{93} + 4 q^{94} - 6 q^{95} - 14 q^{96} - 14 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 + 4 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 6 * q^12 + 4 * q^13 + 2 * q^14 - 4 * q^15 + 6 * q^16 + 6 * q^17 + 2 * q^18 - 6 * q^19 + 6 * q^20 - 4 * q^21 + 2 * q^22 + 10 * q^23 - 6 * q^24 + 4 * q^25 - 4 * q^27 + 6 * q^28 + 6 * q^29 - 2 * q^30 - 8 * q^31 + 14 * q^32 - 4 * q^33 - 16 * q^34 + 4 * q^35 + 6 * q^36 + 12 * q^37 + 4 * q^38 - 4 * q^39 + 6 * q^40 - 2 * q^42 + 6 * q^43 + 6 * q^44 + 4 * q^45 - 4 * q^46 - 6 * q^48 + 4 * q^49 + 2 * q^50 - 6 * q^51 - 12 * q^52 + 14 * q^53 - 2 * q^54 + 4 * q^55 + 6 * q^56 + 6 * q^57 + 20 * q^58 + 2 * q^59 - 6 * q^60 + 2 * q^61 + 20 * q^62 + 4 * q^63 - 2 * q^64 + 4 * q^65 - 2 * q^66 - 12 * q^67 + 20 * q^68 - 10 * q^69 + 2 * q^70 + 4 * q^71 + 6 * q^72 + 20 * q^73 - 32 * q^74 - 4 * q^75 + 16 * q^76 + 4 * q^77 - 4 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 14 * q^83 - 6 * q^84 + 6 * q^85 + 40 * q^86 - 6 * q^87 + 6 * q^88 - 6 * q^89 + 2 * q^90 + 4 * q^91 + 40 * q^92 + 8 * q^93 + 4 * q^94 - 6 * q^95 - 14 * q^96 - 14 * q^97 + 2 * q^98 + 4 * q^99

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