LMFDB - Embedded newform 5070.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.464102 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -0.535898 q^{19} -1.00000 q^{20} +2.00000 q^{21} -0.464102 q^{22} -0.267949 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +3.73205 q^{29} -1.00000 q^{30} -1.73205 q^{31} -1.00000 q^{32} -0.464102 q^{33} -4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +1.19615 q^{37} +0.535898 q^{38} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} +1.92820 q^{43} +0.464102 q^{44} -1.00000 q^{45} +0.267949 q^{46} -10.4641 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} -12.9282 q^{53} +1.00000 q^{54} -0.464102 q^{55} +2.00000 q^{56} +0.535898 q^{57} -3.73205 q^{58} +1.53590 q^{59} +1.00000 q^{60} +10.3923 q^{61} +1.73205 q^{62} -2.00000 q^{63} +1.00000 q^{64} +0.464102 q^{66} -4.53590 q^{67} +4.00000 q^{68} +0.267949 q^{69} -2.00000 q^{70} -8.39230 q^{71} -1.00000 q^{72} +2.00000 q^{73} -1.19615 q^{74} -1.00000 q^{75} -0.535898 q^{76} -0.928203 q^{77} -0.0717968 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.92820 q^{83} +2.00000 q^{84} -4.00000 q^{85} -1.92820 q^{86} -3.73205 q^{87} -0.464102 q^{88} +7.46410 q^{89} +1.00000 q^{90} -0.267949 q^{92} +1.73205 q^{93} +10.4641 q^{94} +0.535898 q^{95} +1.00000 q^{96} -7.46410 q^{97} +3.00000 q^{98} +0.464102 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} + 4 q^{14} + 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{20} + 4 q^{21} + 6 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{27} - 4 q^{28} + 4 q^{29} - 2 q^{30} - 2 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 2 q^{36} - 8 q^{37} + 8 q^{38} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 10 q^{43} - 6 q^{44} - 2 q^{45} + 4 q^{46} - 14 q^{47} - 2 q^{48} - 6 q^{49} - 2 q^{50} - 8 q^{51} - 12 q^{53} + 2 q^{54} + 6 q^{55} + 4 q^{56} + 8 q^{57} - 4 q^{58} + 10 q^{59} + 2 q^{60} - 4 q^{63} + 2 q^{64} - 6 q^{66} - 16 q^{67} + 8 q^{68} + 4 q^{69} - 4 q^{70} + 4 q^{71} - 2 q^{72} + 4 q^{73} + 8 q^{74} - 2 q^{75} - 8 q^{76} + 12 q^{77} - 14 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} - 4 q^{83} + 4 q^{84} - 8 q^{85} + 10 q^{86} - 4 q^{87} + 6 q^{88} + 8 q^{89} + 2 q^{90} - 4 q^{92} + 14 q^{94} + 8 q^{95} + 2 q^{96} - 8 q^{97} + 6 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 + 4 * q^14 + 2 * q^15 + 2 * q^16 + 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^20 + 4 * q^21 + 6 * q^22 - 4 * q^23 + 2 * q^24 + 2 * q^25 - 2 * q^27 - 4 * q^28 + 4 * q^29 - 2 * q^30 - 2 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 2 * q^36 - 8 * q^37 + 8 * q^38 + 2 * q^40 + 4 * q^41 - 4 * q^42 - 10 * q^43 - 6 * q^44 - 2 * q^45 + 4 * q^46 - 14 * q^47 - 2 * q^48 - 6 * q^49 - 2 * q^50 - 8 * q^51 - 12 * q^53 + 2 * q^54 + 6 * q^55 + 4 * q^56 + 8 * q^57 - 4 * q^58 + 10 * q^59 + 2 * q^60 - 4 * q^63 + 2 * q^64 - 6 * q^66 - 16 * q^67 + 8 * q^68 + 4 * q^69 - 4 * q^70 + 4 * q^71 - 2 * q^72 + 4 * q^73 + 8 * q^74 - 2 * q^75 - 8 * q^76 + 12 * q^77 - 14 * q^79 - 2 * q^80 + 2 * q^81 - 4 * q^82 - 4 * q^83 + 4 * q^84 - 8 * q^85 + 10 * q^86 - 4 * q^87 + 6 * q^88 + 8 * q^89 + 2 * q^90 - 4 * q^92 + 14 * q^94 + 8 * q^95 + 2 * q^96 - 8 * q^97 + 6 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0.464102	0.139932	0.0699660	−	0.997549i	$-0.477711\pi$
$11$	0.464102	0.139932	0.0699660	+	0.997549i	$0.477711\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	−0.535898	−0.122944	−0.0614718	−	0.998109i	$-0.519579\pi$
$19$	−0.535898	−0.122944	−0.0614718	+	0.998109i	$0.519579\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	−0.464102	−0.0989468
$23$	−0.267949	−0.0558713	−0.0279356	−	0.999610i	$-0.508893\pi$
$23$	−0.267949	−0.0558713	−0.0279356	+	0.999610i	$0.508893\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	3.73205	0.693024	0.346512	−	0.938045i	$-0.387366\pi$
$29$	3.73205	0.693024	0.346512	+	0.938045i	$0.387366\pi$
$30$	−1.00000	−0.182574
$31$	−1.73205	−0.311086	−0.155543	−	0.987829i	$-0.549713\pi$
$31$	−1.73205	−0.311086	−0.155543	+	0.987829i	$0.549713\pi$
$32$	−1.00000	−0.176777
$33$	−0.464102	−0.0807897
$34$	−4.00000	−0.685994
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	1.19615	0.196646	0.0983231	−	0.995155i	$-0.468652\pi$
$37$	1.19615	0.196646	0.0983231	+	0.995155i	$0.468652\pi$
$38$	0.535898	0.0869342
$39$	0	0
$40$	1.00000	0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	1.92820	0.294048	0.147024	−	0.989133i	$-0.453031\pi$
$43$	1.92820	0.294048	0.147024	+	0.989133i	$0.453031\pi$
$44$	0.464102	0.0699660
$45$	−1.00000	−0.149071
$46$	0.267949	0.0395070
$47$	−10.4641	−1.52635	−0.763173	−	0.646194i	$-0.776360\pi$
$47$	−10.4641	−1.52635	−0.763173	+	0.646194i	$0.776360\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−12.9282	−1.77583	−0.887913	−	0.460012i	$-0.847845\pi$
$53$	−12.9282	−1.77583	−0.887913	+	0.460012i	$0.847845\pi$
$54$	1.00000	0.136083
$55$	−0.464102	−0.0625794
$56$	2.00000	0.267261
$57$	0.535898	0.0709815
$58$	−3.73205	−0.490042
$59$	1.53590	0.199957	0.0999785	−	0.994990i	$-0.468123\pi$
$59$	1.53590	0.199957	0.0999785	+	0.994990i	$0.468123\pi$
$60$	1.00000	0.129099
$61$	10.3923	1.33060	0.665299	−	0.746577i	$-0.268304\pi$
$61$	10.3923	1.33060	0.665299	+	0.746577i	$0.268304\pi$
$62$	1.73205	0.219971
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0.464102	0.0571270
$67$	−4.53590	−0.554148	−0.277074	−	0.960849i	$-0.589365\pi$
$67$	−4.53590	−0.554148	−0.277074	+	0.960849i	$0.589365\pi$
$68$	4.00000	0.485071
$69$	0.267949	0.0322573
$70$	−2.00000	−0.239046
$71$	−8.39230	−0.995983	−0.497992	−	0.867182i	$-0.665929\pi$
$71$	−8.39230	−0.995983	−0.497992	+	0.867182i	$0.665929\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−1.19615	−0.139050
$75$	−1.00000	−0.115470
$76$	−0.535898	−0.0614718
$77$	−0.928203	−0.105779
$78$	0	0
$79$	−0.0717968	−0.00807777	−0.00403888	−	0.999992i	$-0.501286\pi$
$79$	−0.0717968	−0.00807777	−0.00403888	+	0.999992i	$0.501286\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	4.92820	0.540941	0.270470	−	0.962728i	$-0.412821\pi$
$83$	4.92820	0.540941	0.270470	+	0.962728i	$0.412821\pi$
$84$	2.00000	0.218218
$85$	−4.00000	−0.433861
$86$	−1.92820	−0.207924
$87$	−3.73205	−0.400118
$88$	−0.464102	−0.0494734
$89$	7.46410	0.791193	0.395597	−	0.918424i	$-0.370538\pi$
$89$	7.46410	0.791193	0.395597	+	0.918424i	$0.370538\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−0.267949	−0.0279356
$93$	1.73205	0.179605
$94$	10.4641	1.07929
$95$	0.535898	0.0549820
$96$	1.00000	0.102062
$97$	−7.46410	−0.757865	−0.378932	−	0.925424i	$-0.623709\pi$
$97$	−7.46410	−0.757865	−0.378932	+	0.925424i	$0.623709\pi$
$98$	3.00000	0.303046
$99$	0.464102	0.0466440
$100$	1.00000	0.100000
$101$	10.9282	1.08740	0.543698	−	0.839281i	$-0.317024\pi$
$101$	10.9282	1.08740	0.543698	+	0.839281i	$0.317024\pi$
$102$	4.00000	0.396059
$103$	−15.8564	−1.56238	−0.781189	−	0.624295i	$-0.785387\pi$
$103$	−15.8564	−1.56238	−0.781189	+	0.624295i	$0.785387\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	12.9282	1.25570
$107$	19.8564	1.91959	0.959796	−	0.280700i	$-0.0905665\pi$
$107$	19.8564	1.91959	0.959796	+	0.280700i	$0.0905665\pi$
$108$	−1.00000	−0.0962250
$109$	−11.8564	−1.13564	−0.567819	−	0.823154i	$-0.692213\pi$
$109$	−11.8564	−1.13564	−0.567819	+	0.823154i	$0.692213\pi$
$110$	0.464102	0.0442504
$111$	−1.19615	−0.113534
$112$	−2.00000	−0.188982
$113$	11.1962	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$113$	11.1962	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$114$	−0.535898	−0.0501915
$115$	0.267949	0.0249864
$116$	3.73205	0.346512
$117$	0	0
$118$	−1.53590	−0.141391
$119$	−8.00000	−0.733359
$120$	−1.00000	−0.0912871
$121$	−10.7846	−0.980419
$122$	−10.3923	−0.940875
$123$	−2.00000	−0.180334
$124$	−1.73205	−0.155543
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	−8.92820	−0.792250	−0.396125	−	0.918197i	$-0.629645\pi$
$127$	−8.92820	−0.792250	−0.396125	+	0.918197i	$0.629645\pi$
$128$	−1.00000	−0.0883883
$129$	−1.92820	−0.169769
$130$	0	0
$131$	−1.33975	−0.117054	−0.0585271	−	0.998286i	$-0.518640\pi$
$131$	−1.33975	−0.117054	−0.0585271	+	0.998286i	$0.518640\pi$
$132$	−0.464102	−0.0403949
$133$	1.07180	0.0929366
$134$	4.53590	0.391842
$135$	1.00000	0.0860663
$136$	−4.00000	−0.342997
$137$	4.46410	0.381394	0.190697	−	0.981649i	$-0.438925\pi$
$137$	4.46410	0.381394	0.190697	+	0.981649i	$0.438925\pi$
$138$	−0.267949	−0.0228093
$139$	0.928203	0.0787292	0.0393646	−	0.999225i	$-0.487467\pi$
$139$	0.928203	0.0787292	0.0393646	+	0.999225i	$0.487467\pi$
$140$	2.00000	0.169031
$141$	10.4641	0.881236
$142$	8.39230	0.704267
$143$	0	0
$144$	1.00000	0.0833333
$145$	−3.73205	−0.309930
$146$	−2.00000	−0.165521
$147$	3.00000	0.247436
$148$	1.19615	0.0983231
$149$	20.4641	1.67648	0.838242	−	0.545298i	$-0.183584\pi$
$149$	20.4641	1.67648	0.838242	+	0.545298i	$0.183584\pi$
$150$	1.00000	0.0816497
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	0.535898	0.0434671
$153$	4.00000	0.323381
$154$	0.928203	0.0747967
$155$	1.73205	0.139122
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	0.0717968	0.00571184
$159$	12.9282	1.02527
$160$	1.00000	0.0790569
$161$	0.535898	0.0422347
$162$	−1.00000	−0.0785674
$163$	−23.0526	−1.80562	−0.902808	−	0.430044i	$-0.858498\pi$
$163$	−23.0526	−1.80562	−0.902808	+	0.430044i	$0.858498\pi$
$164$	2.00000	0.156174
$165$	0.464102	0.0361303
$166$	−4.92820	−0.382503
$167$	18.3205	1.41768	0.708842	−	0.705368i	$-0.249218\pi$
$167$	18.3205	1.41768	0.708842	+	0.705368i	$0.249218\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	4.00000	0.306786
$171$	−0.535898	−0.0409812
$172$	1.92820	0.147024
$173$	2.92820	0.222627	0.111314	−	0.993785i	$-0.464494\pi$
$173$	2.92820	0.222627	0.111314	+	0.993785i	$0.464494\pi$
$174$	3.73205	0.282926
$175$	−2.00000	−0.151186
$176$	0.464102	0.0349830
$177$	−1.53590	−0.115445
$178$	−7.46410	−0.559458
$179$	16.2679	1.21592	0.607962	−	0.793966i	$-0.291987\pi$
$179$	16.2679	1.21592	0.607962	+	0.793966i	$0.291987\pi$
$180$	−1.00000	−0.0745356
$181$	−10.9282	−0.812287	−0.406143	−	0.913809i	$-0.633127\pi$
$181$	−10.9282	−0.812287	−0.406143	+	0.913809i	$0.633127\pi$
$182$	0	0
$183$	−10.3923	−0.768221
$184$	0.267949	0.0197535
$185$	−1.19615	−0.0879429
$186$	−1.73205	−0.127000
$187$	1.85641	0.135754
$188$	−10.4641	−0.763173
$189$	2.00000	0.145479
$190$	−0.535898	−0.0388782
$191$	14.5359	1.05178	0.525890	−	0.850552i	$-0.323732\pi$
$191$	14.5359	1.05178	0.525890	+	0.850552i	$0.323732\pi$
$192$	−1.00000	−0.0721688
$193$	23.3205	1.67865	0.839323	−	0.543632i	$-0.182951\pi$
$193$	23.3205	1.67865	0.839323	+	0.543632i	$0.182951\pi$
$194$	7.46410	0.535891
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−16.3923	−1.16790	−0.583952	−	0.811788i	$-0.698494\pi$
$197$	−16.3923	−1.16790	−0.583952	+	0.811788i	$0.698494\pi$
$198$	−0.464102	−0.0329823
$199$	18.9282	1.34178	0.670892	−	0.741555i	$-0.265911\pi$
$199$	18.9282	1.34178	0.670892	+	0.741555i	$0.265911\pi$
$200$	−1.00000	−0.0707107
$201$	4.53590	0.319938
$202$	−10.9282	−0.768906
$203$	−7.46410	−0.523877
$204$	−4.00000	−0.280056
$205$	−2.00000	−0.139686
$206$	15.8564	1.10477
$207$	−0.267949	−0.0186238
$208$	0	0
$209$	−0.248711	−0.0172037
$210$	2.00000	0.138013
$211$	23.3205	1.60545	0.802725	−	0.596349i	$-0.203383\pi$
$211$	23.3205	1.60545	0.802725	+	0.596349i	$0.203383\pi$
$212$	−12.9282	−0.887913
$213$	8.39230	0.575031
$214$	−19.8564	−1.35736
$215$	−1.92820	−0.131502
$216$	1.00000	0.0680414
$217$	3.46410	0.235159
$218$	11.8564	0.803017
$219$	−2.00000	−0.135147
$220$	−0.464102	−0.0312897
$221$	0	0
$222$	1.19615	0.0802805
$223$	−27.4641	−1.83913	−0.919566	−	0.392935i	$-0.871460\pi$
$223$	−27.4641	−1.83913	−0.919566	+	0.392935i	$0.871460\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−11.1962	−0.744757
$227$	−4.39230	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$227$	−4.39230	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$228$	0.535898	0.0354907
$229$	19.8564	1.31215	0.656074	−	0.754696i	$-0.272216\pi$
$229$	19.8564	1.31215	0.656074	+	0.754696i	$0.272216\pi$
$230$	−0.267949	−0.0176680
$231$	0.928203	0.0610713
$232$	−3.73205	−0.245021
$233$	18.1244	1.18737	0.593683	−	0.804699i	$-0.297673\pi$
$233$	18.1244	1.18737	0.593683	+	0.804699i	$0.297673\pi$
$234$	0	0
$235$	10.4641	0.682603
$236$	1.53590	0.0999785
$237$	0.0717968	0.00466370
$238$	8.00000	0.518563
$239$	−4.39230	−0.284115	−0.142057	−	0.989858i	$-0.545372\pi$
$239$	−4.39230	−0.284115	−0.142057	+	0.989858i	$0.545372\pi$
$240$	1.00000	0.0645497
$241$	14.2679	0.919079	0.459540	−	0.888157i	$-0.348014\pi$
$241$	14.2679	0.919079	0.459540	+	0.888157i	$0.348014\pi$
$242$	10.7846	0.693261
$243$	−1.00000	−0.0641500
$244$	10.3923	0.665299
$245$	3.00000	0.191663
$246$	2.00000	0.127515
$247$	0	0
$248$	1.73205	0.109985
$249$	−4.92820	−0.312312
$250$	1.00000	0.0632456
$251$	−12.2679	−0.774346	−0.387173	−	0.922007i	$-0.626548\pi$
$251$	−12.2679	−0.774346	−0.387173	+	0.922007i	$0.626548\pi$
$252$	−2.00000	−0.125988
$253$	−0.124356	−0.00781817
$254$	8.92820	0.560205
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	22.6603	1.41351	0.706754	−	0.707459i	$-0.250159\pi$
$257$	22.6603	1.41351	0.706754	+	0.707459i	$0.250159\pi$
$258$	1.92820	0.120045
$259$	−2.39230	−0.148651
$260$	0	0
$261$	3.73205	0.231008
$262$	1.33975	0.0827698
$263$	−18.1244	−1.11760	−0.558798	−	0.829304i	$-0.688737\pi$
$263$	−18.1244	−1.11760	−0.558798	+	0.829304i	$0.688737\pi$
$264$	0.464102	0.0285635
$265$	12.9282	0.794173
$266$	−1.07180	−0.0657161
$267$	−7.46410	−0.456796
$268$	−4.53590	−0.277074
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−1.00000	−0.0608581
$271$	9.19615	0.558626	0.279313	−	0.960200i	$-0.409893\pi$
$271$	9.19615	0.558626	0.279313	+	0.960200i	$0.409893\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−4.46410	−0.269686
$275$	0.464102	0.0279864
$276$	0.267949	0.0161286
$277$	−9.92820	−0.596528	−0.298264	−	0.954483i	$-0.596408\pi$
$277$	−9.92820	−0.596528	−0.298264	+	0.954483i	$0.596408\pi$
$278$	−0.928203	−0.0556699
$279$	−1.73205	−0.103695
$280$	−2.00000	−0.119523
$281$	4.92820	0.293992	0.146996	−	0.989137i	$-0.453040\pi$
$281$	4.92820	0.293992	0.146996	+	0.989137i	$0.453040\pi$
$282$	−10.4641	−0.623128
$283$	−3.92820	−0.233507	−0.116754	−	0.993161i	$-0.537249\pi$
$283$	−3.92820	−0.233507	−0.116754	+	0.993161i	$0.537249\pi$
$284$	−8.39230	−0.497992
$285$	−0.535898	−0.0317439
$286$	0	0
$287$	−4.00000	−0.236113
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	3.73205	0.219154
$291$	7.46410	0.437553
$292$	2.00000	0.117041
$293$	−4.14359	−0.242071	−0.121036	−	0.992648i	$-0.538622\pi$
$293$	−4.14359	−0.242071	−0.121036	+	0.992648i	$0.538622\pi$
$294$	−3.00000	−0.174964
$295$	−1.53590	−0.0894235
$296$	−1.19615	−0.0695249
$297$	−0.464102	−0.0269299
$298$	−20.4641	−1.18545
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−3.85641	−0.222280
$302$	−10.3923	−0.598010
$303$	−10.9282	−0.627809
$304$	−0.535898	−0.0307359
$305$	−10.3923	−0.595062
$306$	−4.00000	−0.228665
$307$	12.5359	0.715462	0.357731	−	0.933825i	$-0.383551\pi$
$307$	12.5359	0.715462	0.357731	+	0.933825i	$0.383551\pi$
$308$	−0.928203	−0.0528893
$309$	15.8564	0.902039
$310$	−1.73205	−0.0983739
$311$	7.60770	0.431393	0.215696	−	0.976460i	$-0.430798\pi$
$311$	7.60770	0.431393	0.215696	+	0.976460i	$0.430798\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	5.00000	0.282166
$315$	2.00000	0.112687
$316$	−0.0717968	−0.00403888
$317$	−21.4641	−1.20554	−0.602772	−	0.797913i	$-0.705937\pi$
$317$	−21.4641	−1.20554	−0.602772	+	0.797913i	$0.705937\pi$
$318$	−12.9282	−0.724978
$319$	1.73205	0.0969762
$320$	−1.00000	−0.0559017
$321$	−19.8564	−1.10828
$322$	−0.535898	−0.0298644
$323$	−2.14359	−0.119273
$324$	1.00000	0.0555556
$325$	0	0
$326$	23.0526	1.27676
$327$	11.8564	0.655661
$328$	−2.00000	−0.110432
$329$	20.9282	1.15381
$330$	−0.464102	−0.0255480
$331$	−24.7846	−1.36229	−0.681143	−	0.732151i	$-0.738517\pi$
$331$	−24.7846	−1.36229	−0.681143	+	0.732151i	$0.738517\pi$
$332$	4.92820	0.270470
$333$	1.19615	0.0655487
$334$	−18.3205	−1.00245
$335$	4.53590	0.247823
$336$	2.00000	0.109109
$337$	25.3205	1.37930	0.689648	−	0.724145i	$-0.257765\pi$
$337$	25.3205	1.37930	0.689648	+	0.724145i	$0.257765\pi$
$338$	0	0
$339$	−11.1962	−0.608092
$340$	−4.00000	−0.216930
$341$	−0.803848	−0.0435308
$342$	0.535898	0.0289781
$343$	20.0000	1.07990
$344$	−1.92820	−0.103962
$345$	−0.267949	−0.0144259
$346$	−2.92820	−0.157421
$347$	22.3923	1.20208	0.601041	−	0.799218i	$-0.294753\pi$
$347$	22.3923	1.20208	0.601041	+	0.799218i	$0.294753\pi$
$348$	−3.73205	−0.200059
$349$	−14.5359	−0.778089	−0.389044	−	0.921219i	$-0.627195\pi$
$349$	−14.5359	−0.778089	−0.389044	+	0.921219i	$0.627195\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−0.464102	−0.0247367
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	1.53590	0.0816321
$355$	8.39230	0.445417
$356$	7.46410	0.395597
$357$	8.00000	0.423405
$358$	−16.2679	−0.859788
$359$	18.9282	0.998992	0.499496	−	0.866316i	$-0.333518\pi$
$359$	18.9282	0.998992	0.499496	+	0.866316i	$0.333518\pi$
$360$	1.00000	0.0527046
$361$	−18.7128	−0.984885
$362$	10.9282	0.574374
$363$	10.7846	0.566045
$364$	0	0
$365$	−2.00000	−0.104685
$366$	10.3923	0.543214
$367$	36.3923	1.89966	0.949831	−	0.312762i	$-0.101254\pi$
$367$	36.3923	1.89966	0.949831	+	0.312762i	$0.101254\pi$
$368$	−0.267949	−0.0139678
$369$	2.00000	0.104116
$370$	1.19615	0.0621850
$371$	25.8564	1.34240
$372$	1.73205	0.0898027
$373$	25.7846	1.33508	0.667538	−	0.744576i	$-0.267348\pi$
$373$	25.7846	1.33508	0.667538	+	0.744576i	$0.267348\pi$
$374$	−1.85641	−0.0959925
$375$	1.00000	0.0516398
$376$	10.4641	0.539645
$377$	0	0
$378$	−2.00000	−0.102869
$379$	0.143594	0.00737590	0.00368795	−	0.999993i	$-0.498826\pi$
$379$	0.143594	0.00737590	0.00368795	+	0.999993i	$0.498826\pi$
$380$	0.535898	0.0274910
$381$	8.92820	0.457406
$382$	−14.5359	−0.743721
$383$	−4.60770	−0.235442	−0.117721	−	0.993047i	$-0.537559\pi$
$383$	−4.60770	−0.235442	−0.117721	+	0.993047i	$0.537559\pi$
$384$	1.00000	0.0510310
$385$	0.928203	0.0473056
$386$	−23.3205	−1.18698
$387$	1.92820	0.0980161
$388$	−7.46410	−0.378932
$389$	20.2679	1.02763	0.513813	−	0.857902i	$-0.328233\pi$
$389$	20.2679	1.02763	0.513813	+	0.857902i	$0.328233\pi$
$390$	0	0
$391$	−1.07180	−0.0542031
$392$	3.00000	0.151523
$393$	1.33975	0.0675812
$394$	16.3923	0.825832
$395$	0.0717968	0.00361249
$396$	0.464102	0.0233220
$397$	−12.1244	−0.608504	−0.304252	−	0.952592i	$-0.598406\pi$
$397$	−12.1244	−0.608504	−0.304252	+	0.952592i	$0.598406\pi$
$398$	−18.9282	−0.948785
$399$	−1.07180	−0.0536570
$400$	1.00000	0.0500000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	−4.53590	−0.226230
$403$	0	0
$404$	10.9282	0.543698
$405$	−1.00000	−0.0496904
$406$	7.46410	0.370437
$407$	0.555136	0.0275171
$408$	4.00000	0.198030
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	2.00000	0.0987730
$411$	−4.46410	−0.220198
$412$	−15.8564	−0.781189
$413$	−3.07180	−0.151153
$414$	0.267949	0.0131690
$415$	−4.92820	−0.241916
$416$	0	0
$417$	−0.928203	−0.0454543
$418$	0.248711	0.0121649
$419$	1.60770	0.0785410	0.0392705	−	0.999229i	$-0.487497\pi$
$419$	1.60770	0.0785410	0.0392705	+	0.999229i	$0.487497\pi$
$420$	−2.00000	−0.0975900
$421$	16.3923	0.798912	0.399456	−	0.916752i	$-0.369199\pi$
$421$	16.3923	0.798912	0.399456	+	0.916752i	$0.369199\pi$
$422$	−23.3205	−1.13522
$423$	−10.4641	−0.508782
$424$	12.9282	0.627849
$425$	4.00000	0.194029
$426$	−8.39230	−0.406608
$427$	−20.7846	−1.00584
$428$	19.8564	0.959796
$429$	0	0
$430$	1.92820	0.0929862
$431$	7.60770	0.366450	0.183225	−	0.983071i	$-0.441346\pi$
$431$	7.60770	0.366450	0.183225	+	0.983071i	$0.441346\pi$
$432$	−1.00000	−0.0481125
$433$	15.3205	0.736257	0.368128	−	0.929775i	$-0.379999\pi$
$433$	15.3205	0.736257	0.368128	+	0.929775i	$0.379999\pi$
$434$	−3.46410	−0.166282
$435$	3.73205	0.178938
$436$	−11.8564	−0.567819
$437$	0.143594	0.00686901
$438$	2.00000	0.0955637
$439$	−9.85641	−0.470421	−0.235210	−	0.971945i	$-0.575578\pi$
$439$	−9.85641	−0.470421	−0.235210	+	0.971945i	$0.575578\pi$
$440$	0.464102	0.0221252
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−4.39230	−0.208685	−0.104342	−	0.994541i	$-0.533274\pi$
$443$	−4.39230	−0.208685	−0.104342	+	0.994541i	$0.533274\pi$
$444$	−1.19615	−0.0567669
$445$	−7.46410	−0.353832
$446$	27.4641	1.30046
$447$	−20.4641	−0.967919
$448$	−2.00000	−0.0944911
$449$	39.7128	1.87416	0.937082	−	0.349110i	$-0.113516\pi$
$449$	39.7128	1.87416	0.937082	+	0.349110i	$0.113516\pi$
$450$	−1.00000	−0.0471405
$451$	0.928203	0.0437074
$452$	11.1962	0.526623
$453$	−10.3923	−0.488273
$454$	4.39230	0.206141
$455$	0	0
$456$	−0.535898	−0.0250957
$457$	31.4641	1.47183	0.735914	−	0.677075i	$-0.236753\pi$
$457$	31.4641	1.47183	0.735914	+	0.677075i	$0.236753\pi$
$458$	−19.8564	−0.927829
$459$	−4.00000	−0.186704
$460$	0.267949	0.0124932
$461$	6.46410	0.301063	0.150532	−	0.988605i	$-0.451901\pi$
$461$	6.46410	0.301063	0.150532	+	0.988605i	$0.451901\pi$
$462$	−0.928203	−0.0431839
$463$	20.9282	0.972616	0.486308	−	0.873787i	$-0.338343\pi$
$463$	20.9282	0.972616	0.486308	+	0.873787i	$0.338343\pi$
$464$	3.73205	0.173256
$465$	−1.73205	−0.0803219
$466$	−18.1244	−0.839595
$467$	−11.8564	−0.548649	−0.274325	−	0.961637i	$-0.588454\pi$
$467$	−11.8564	−0.548649	−0.274325	+	0.961637i	$0.588454\pi$
$468$	0	0
$469$	9.07180	0.418897
$470$	−10.4641	−0.482673
$471$	5.00000	0.230388
$472$	−1.53590	−0.0706955
$473$	0.894882	0.0411467
$474$	−0.0717968	−0.00329773
$475$	−0.535898	−0.0245887
$476$	−8.00000	−0.366679
$477$	−12.9282	−0.591942
$478$	4.39230	0.200899
$479$	1.46410	0.0668965	0.0334483	−	0.999440i	$-0.489351\pi$
$479$	1.46410	0.0668965	0.0334483	+	0.999440i	$0.489351\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−14.2679	−0.649887
$483$	−0.535898	−0.0243842
$484$	−10.7846	−0.490210
$485$	7.46410	0.338927
$486$	1.00000	0.0453609
$487$	39.1769	1.77528	0.887638	−	0.460542i	$-0.152345\pi$
$487$	39.1769	1.77528	0.887638	+	0.460542i	$0.152345\pi$
$488$	−10.3923	−0.470438
$489$	23.0526	1.04247
$490$	−3.00000	−0.135526
$491$	−17.3205	−0.781664	−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205	−0.781664	−0.390832	+	0.920462i	$0.627813\pi$
$492$	−2.00000	−0.0901670
$493$	14.9282	0.672332
$494$	0	0
$495$	−0.464102	−0.0208598
$496$	−1.73205	−0.0777714
$497$	16.7846	0.752893
$498$	4.92820	0.220838
$499$	−13.4641	−0.602736	−0.301368	−	0.953508i	$-0.597443\pi$
$499$	−13.4641	−0.602736	−0.301368	+	0.953508i	$0.597443\pi$
$500$	−1.00000	−0.0447214
$501$	−18.3205	−0.818500
$502$	12.2679	0.547545
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	2.00000	0.0890871
$505$	−10.9282	−0.486299
$506$	0.124356	0.00552828
$507$	0	0
$508$	−8.92820	−0.396125
$509$	−19.3923	−0.859549	−0.429774	−	0.902936i	$-0.641407\pi$
$509$	−19.3923	−0.859549	−0.429774	+	0.902936i	$0.641407\pi$
$510$	−4.00000	−0.177123
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	0.535898	0.0236605
$514$	−22.6603	−0.999501
$515$	15.8564	0.698717
$516$	−1.92820	−0.0848844
$517$	−4.85641	−0.213585
$518$	2.39230	0.105112
$519$	−2.92820	−0.128534
$520$	0	0
$521$	17.3205	0.758825	0.379413	−	0.925228i	$-0.376126\pi$
$521$	17.3205	0.758825	0.379413	+	0.925228i	$0.376126\pi$
$522$	−3.73205	−0.163347
$523$	−29.7846	−1.30239	−0.651195	−	0.758910i	$-0.725732\pi$
$523$	−29.7846	−1.30239	−0.651195	+	0.758910i	$0.725732\pi$
$524$	−1.33975	−0.0585271
$525$	2.00000	0.0872872
$526$	18.1244	0.790259
$527$	−6.92820	−0.301797
$528$	−0.464102	−0.0201974
$529$	−22.9282	−0.996878
$530$	−12.9282	−0.561565
$531$	1.53590	0.0666523
$532$	1.07180	0.0464683
$533$	0	0
$534$	7.46410	0.323003
$535$	−19.8564	−0.858467
$536$	4.53590	0.195921
$537$	−16.2679	−0.702014
$538$	−12.0000	−0.517357
$539$	−1.39230	−0.0599708
$540$	1.00000	0.0430331
$541$	13.0718	0.562000	0.281000	−	0.959708i	$-0.409334\pi$
$541$	13.0718	0.562000	0.281000	+	0.959708i	$0.409334\pi$
$542$	−9.19615	−0.395009
$543$	10.9282	0.468974
$544$	−4.00000	−0.171499
$545$	11.8564	0.507873
$546$	0	0
$547$	9.07180	0.387882	0.193941	−	0.981013i	$-0.437873\pi$
$547$	9.07180	0.387882	0.193941	+	0.981013i	$0.437873\pi$
$548$	4.46410	0.190697
$549$	10.3923	0.443533
$550$	−0.464102	−0.0197894
$551$	−2.00000	−0.0852029
$552$	−0.267949	−0.0114047
$553$	0.143594	0.00610622
$554$	9.92820	0.421809
$555$	1.19615	0.0507738
$556$	0.928203	0.0393646
$557$	37.7128	1.59794	0.798972	−	0.601369i	$-0.205378\pi$
$557$	37.7128	1.59794	0.798972	+	0.601369i	$0.205378\pi$
$558$	1.73205	0.0733236
$559$	0	0
$560$	2.00000	0.0845154
$561$	−1.85641	−0.0783775
$562$	−4.92820	−0.207884
$563$	39.3205	1.65716	0.828581	−	0.559869i	$-0.189149\pi$
$563$	39.3205	1.65716	0.828581	+	0.559869i	$0.189149\pi$
$564$	10.4641	0.440618
$565$	−11.1962	−0.471026
$566$	3.92820	0.165115
$567$	−2.00000	−0.0839921
$568$	8.39230	0.352133
$569$	5.32051	0.223047	0.111524	−	0.993762i	$-0.464427\pi$
$569$	5.32051	0.223047	0.111524	+	0.993762i	$0.464427\pi$
$570$	0.535898	0.0224463
$571$	−45.1769	−1.89060	−0.945298	−	0.326209i	$-0.894229\pi$
$571$	−45.1769	−1.89060	−0.945298	+	0.326209i	$0.894229\pi$
$572$	0	0
$573$	−14.5359	−0.607246
$574$	4.00000	0.166957
$575$	−0.267949	−0.0111743
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	1.00000	0.0415945
$579$	−23.3205	−0.969167
$580$	−3.73205	−0.154965
$581$	−9.85641	−0.408913
$582$	−7.46410	−0.309397
$583$	−6.00000	−0.248495
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	4.14359	0.171170
$587$	18.3923	0.759132	0.379566	−	0.925165i	$-0.376073\pi$
$587$	18.3923	0.759132	0.379566	+	0.925165i	$0.376073\pi$
$588$	3.00000	0.123718
$589$	0.928203	0.0382459
$590$	1.53590	0.0632319
$591$	16.3923	0.674289
$592$	1.19615	0.0491616
$593$	−31.1051	−1.27733	−0.638667	−	0.769483i	$-0.720514\pi$
$593$	−31.1051	−1.27733	−0.638667	+	0.769483i	$0.720514\pi$
$594$	0.464102	0.0190423
$595$	8.00000	0.327968
$596$	20.4641	0.838242
$597$	−18.9282	−0.774680
$598$	0	0
$599$	10.3923	0.424618	0.212309	−	0.977203i	$-0.431902\pi$
$599$	10.3923	0.424618	0.212309	+	0.977203i	$0.431902\pi$
$600$	1.00000	0.0408248
$601$	21.7846	0.888613	0.444306	−	0.895875i	$-0.353450\pi$
$601$	21.7846	0.888613	0.444306	+	0.895875i	$0.353450\pi$
$602$	3.85641	0.157175
$603$	−4.53590	−0.184716
$604$	10.3923	0.422857
$605$	10.7846	0.438457
$606$	10.9282	0.443928
$607$	43.1769	1.75250	0.876248	−	0.481860i	$-0.160038\pi$
$607$	43.1769	1.75250	0.876248	+	0.481860i	$0.160038\pi$
$608$	0.535898	0.0217335
$609$	7.46410	0.302461
$610$	10.3923	0.420772
$611$	0	0
$612$	4.00000	0.161690
$613$	−0.947441	−0.0382668	−0.0191334	−	0.999817i	$-0.506091\pi$
$613$	−0.947441	−0.0382668	−0.0191334	+	0.999817i	$0.506091\pi$
$614$	−12.5359	−0.505908
$615$	2.00000	0.0806478
$616$	0.928203	0.0373984
$617$	15.5359	0.625452	0.312726	−	0.949843i	$-0.398758\pi$
$617$	15.5359	0.625452	0.312726	+	0.949843i	$0.398758\pi$
$618$	−15.8564	−0.637838
$619$	24.2487	0.974638	0.487319	−	0.873224i	$-0.337975\pi$
$619$	24.2487	0.974638	0.487319	+	0.873224i	$0.337975\pi$
$620$	1.73205	0.0695608
$621$	0.267949	0.0107524
$622$	−7.60770	−0.305041
$623$	−14.9282	−0.598086
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.0000	1.11911
$627$	0.248711	0.00993257
$628$	−5.00000	−0.199522
$629$	4.78461	0.190775
$630$	−2.00000	−0.0796819
$631$	24.5359	0.976759	0.488379	−	0.872631i	$-0.337588\pi$
$631$	24.5359	0.976759	0.488379	+	0.872631i	$0.337588\pi$
$632$	0.0717968	0.00285592
$633$	−23.3205	−0.926907
$634$	21.4641	0.852448
$635$	8.92820	0.354305
$636$	12.9282	0.512637
$637$	0	0
$638$	−1.73205	−0.0685725
$639$	−8.39230	−0.331994
$640$	1.00000	0.0395285
$641$	−27.8564	−1.10026	−0.550131	−	0.835078i	$-0.685422\pi$
$641$	−27.8564	−1.10026	−0.550131	+	0.835078i	$0.685422\pi$
$642$	19.8564	0.783670
$643$	27.4641	1.08308	0.541539	−	0.840675i	$-0.317842\pi$
$643$	27.4641	1.08308	0.541539	+	0.840675i	$0.317842\pi$
$644$	0.535898	0.0211174
$645$	1.92820	0.0759229
$646$	2.14359	0.0843386
$647$	−21.3205	−0.838196	−0.419098	−	0.907941i	$-0.637654\pi$
$647$	−21.3205	−0.838196	−0.419098	+	0.907941i	$0.637654\pi$
$648$	−1.00000	−0.0392837
$649$	0.712813	0.0279804
$650$	0	0
$651$	−3.46410	−0.135769
$652$	−23.0526	−0.902808
$653$	−44.2487	−1.73159	−0.865793	−	0.500402i	$-0.833185\pi$
$653$	−44.2487	−1.73159	−0.865793	+	0.500402i	$0.833185\pi$
$654$	−11.8564	−0.463622
$655$	1.33975	0.0523482
$656$	2.00000	0.0780869
$657$	2.00000	0.0780274
$658$	−20.9282	−0.815866
$659$	−3.73205	−0.145380	−0.0726900	−	0.997355i	$-0.523158\pi$
$659$	−3.73205	−0.145380	−0.0726900	+	0.997355i	$0.523158\pi$
$660$	0.464102	0.0180651
$661$	43.3205	1.68497	0.842486	−	0.538718i	$-0.181091\pi$
$661$	43.3205	1.68497	0.842486	+	0.538718i	$0.181091\pi$
$662$	24.7846	0.963281
$663$	0	0
$664$	−4.92820	−0.191251
$665$	−1.07180	−0.0415625
$666$	−1.19615	−0.0463500
$667$	−1.00000	−0.0387202
$668$	18.3205	0.708842
$669$	27.4641	1.06182
$670$	−4.53590	−0.175237
$671$	4.82309	0.186193
$672$	−2.00000	−0.0771517
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	−25.3205	−0.975310
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	11.6077	0.446120	0.223060	−	0.974805i	$-0.428395\pi$
$677$	11.6077	0.446120	0.223060	+	0.974805i	$0.428395\pi$
$678$	11.1962	0.429986
$679$	14.9282	0.572892
$680$	4.00000	0.153393
$681$	4.39230	0.168313
$682$	0.803848	0.0307809
$683$	0.784610	0.0300223	0.0150111	−	0.999887i	$-0.495222\pi$
$683$	0.784610	0.0300223	0.0150111	+	0.999887i	$0.495222\pi$
$684$	−0.535898	−0.0204906
$685$	−4.46410	−0.170565
$686$	−20.0000	−0.763604
$687$	−19.8564	−0.757569
$688$	1.92820	0.0735121
$689$	0	0
$690$	0.267949	0.0102007
$691$	35.1769	1.33819	0.669096	−	0.743176i	$-0.266681\pi$
$691$	35.1769	1.33819	0.669096	+	0.743176i	$0.266681\pi$
$692$	2.92820	0.111314
$693$	−0.928203	−0.0352595
$694$	−22.3923	−0.850000
$695$	−0.928203	−0.0352088
$696$	3.73205	0.141463
$697$	8.00000	0.303022
$698$	14.5359	0.550192
$699$	−18.1244	−0.685526
$700$	−2.00000	−0.0755929
$701$	3.73205	0.140958	0.0704788	−	0.997513i	$-0.477547\pi$
$701$	3.73205	0.140958	0.0704788	+	0.997513i	$0.477547\pi$
$702$	0	0
$703$	−0.641016	−0.0241764
$704$	0.464102	0.0174915
$705$	−10.4641	−0.394101
$706$	−2.00000	−0.0752710
$707$	−21.8564	−0.821995
$708$	−1.53590	−0.0577226
$709$	−9.07180	−0.340698	−0.170349	−	0.985384i	$-0.554490\pi$
$709$	−9.07180	−0.340698	−0.170349	+	0.985384i	$0.554490\pi$
$710$	−8.39230	−0.314958
$711$	−0.0717968	−0.00269259
$712$	−7.46410	−0.279729
$713$	0.464102	0.0173807
$714$	−8.00000	−0.299392
$715$	0	0
$716$	16.2679	0.607962
$717$	4.39230	0.164034
$718$	−18.9282	−0.706394
$719$	−34.6410	−1.29189	−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410	−1.29189	−0.645946	+	0.763383i	$0.723537\pi$
$720$	−1.00000	−0.0372678
$721$	31.7128	1.18105
$722$	18.7128	0.696419
$723$	−14.2679	−0.530631
$724$	−10.9282	−0.406143
$725$	3.73205	0.138605
$726$	−10.7846	−0.400254
$727$	−23.7128	−0.879460	−0.439730	−	0.898130i	$-0.644926\pi$
$727$	−23.7128	−0.879460	−0.439730	+	0.898130i	$0.644926\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	7.71281	0.285269
$732$	−10.3923	−0.384111
$733$	37.0718	1.36928	0.684639	−	0.728882i	$-0.259960\pi$
$733$	37.0718	1.36928	0.684639	+	0.728882i	$0.259960\pi$
$734$	−36.3923	−1.34326
$735$	−3.00000	−0.110657
$736$	0.267949	0.00987674
$737$	−2.10512	−0.0775430
$738$	−2.00000	−0.0736210
$739$	−15.3205	−0.563574	−0.281787	−	0.959477i	$-0.590927\pi$
$739$	−15.3205	−0.563574	−0.281787	+	0.959477i	$0.590927\pi$
$740$	−1.19615	−0.0439714
$741$	0	0
$742$	−25.8564	−0.949219
$743$	−33.5359	−1.23031	−0.615156	−	0.788405i	$-0.710907\pi$
$743$	−33.5359	−1.23031	−0.615156	+	0.788405i	$0.710907\pi$
$744$	−1.73205	−0.0635001
$745$	−20.4641	−0.749747
$746$	−25.7846	−0.944042
$747$	4.92820	0.180314
$748$	1.85641	0.0678769
$749$	−39.7128	−1.45107
$750$	−1.00000	−0.0365148
$751$	−27.9282	−1.01911	−0.509557	−	0.860437i	$-0.670191\pi$
$751$	−27.9282	−1.01911	−0.509557	+	0.860437i	$0.670191\pi$
$752$	−10.4641	−0.381587
$753$	12.2679	0.447069
$754$	0	0
$755$	−10.3923	−0.378215
$756$	2.00000	0.0727393
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−0.143594	−0.00521555
$759$	0.124356	0.00451382
$760$	−0.535898	−0.0194391
$761$	18.9282	0.686147	0.343073	−	0.939309i	$-0.388532\pi$
$761$	18.9282	0.686147	0.343073	+	0.939309i	$0.388532\pi$
$762$	−8.92820	−0.323435
$763$	23.7128	0.858461
$764$	14.5359	0.525890
$765$	−4.00000	−0.144620
$766$	4.60770	0.166483
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	19.5885	0.706378	0.353189	−	0.935552i	$-0.385097\pi$
$769$	19.5885	0.706378	0.353189	+	0.935552i	$0.385097\pi$
$770$	−0.928203	−0.0334501
$771$	−22.6603	−0.816089
$772$	23.3205	0.839323
$773$	27.7128	0.996761	0.498380	−	0.866959i	$-0.333928\pi$
$773$	27.7128	0.996761	0.498380	+	0.866959i	$0.333928\pi$
$774$	−1.92820	−0.0693078
$775$	−1.73205	−0.0622171
$776$	7.46410	0.267946
$777$	2.39230	0.0858235
$778$	−20.2679	−0.726641
$779$	−1.07180	−0.0384011
$780$	0	0
$781$	−3.89488	−0.139370
$782$	1.07180	0.0383274
$783$	−3.73205	−0.133373
$784$	−3.00000	−0.107143
$785$	5.00000	0.178458
$786$	−1.33975	−0.0477872
$787$	−1.73205	−0.0617409	−0.0308705	−	0.999523i	$-0.509828\pi$
$787$	−1.73205	−0.0617409	−0.0308705	+	0.999523i	$0.509828\pi$
$788$	−16.3923	−0.583952
$789$	18.1244	0.645244
$790$	−0.0717968	−0.00255441
$791$	−22.3923	−0.796179
$792$	−0.464102	−0.0164911
$793$	0	0
$794$	12.1244	0.430277
$795$	−12.9282	−0.458516
$796$	18.9282	0.670892
$797$	37.8564	1.34094	0.670471	−	0.741935i	$-0.266092\pi$
$797$	37.8564	1.34094	0.670471	+	0.741935i	$0.266092\pi$
$798$	1.07180	0.0379412
$799$	−41.8564	−1.48077
$800$	−1.00000	−0.0353553
$801$	7.46410	0.263731
$802$	−32.0000	−1.12996
$803$	0.928203	0.0327556
$804$	4.53590	0.159969
$805$	−0.535898	−0.0188879
$806$	0	0
$807$	−12.0000	−0.422420
$808$	−10.9282	−0.384453
$809$	−1.21539	−0.0427308	−0.0213654	−	0.999772i	$-0.506801\pi$
$809$	−1.21539	−0.0427308	−0.0213654	+	0.999772i	$0.506801\pi$
$810$	1.00000	0.0351364
$811$	0.784610	0.0275514	0.0137757	−	0.999905i	$-0.495615\pi$
$811$	0.784610	0.0275514	0.0137757	+	0.999905i	$0.495615\pi$
$812$	−7.46410	−0.261939
$813$	−9.19615	−0.322523
$814$	−0.555136	−0.0194575
$815$	23.0526	0.807496
$816$	−4.00000	−0.140028
$817$	−1.03332	−0.0361513
$818$	4.00000	0.139857
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	35.3923	1.23520	0.617600	−	0.786492i	$-0.288105\pi$
$821$	35.3923	1.23520	0.617600	+	0.786492i	$0.288105\pi$
$822$	4.46410	0.155703
$823$	23.1769	0.807896	0.403948	−	0.914782i	$-0.367638\pi$
$823$	23.1769	0.807896	0.403948	+	0.914782i	$0.367638\pi$
$824$	15.8564	0.552384
$825$	−0.464102	−0.0161579
$826$	3.07180	0.106881
$827$	−17.3205	−0.602293	−0.301147	−	0.953578i	$-0.597369\pi$
$827$	−17.3205	−0.602293	−0.301147	+	0.953578i	$0.597369\pi$
$828$	−0.267949	−0.00931188
$829$	−23.4641	−0.814942	−0.407471	−	0.913218i	$-0.633589\pi$
$829$	−23.4641	−0.814942	−0.407471	+	0.913218i	$0.633589\pi$
$830$	4.92820	0.171060
$831$	9.92820	0.344406
$832$	0	0
$833$	−12.0000	−0.415775
$834$	0.928203	0.0321410
$835$	−18.3205	−0.634007
$836$	−0.248711	−0.00860186
$837$	1.73205	0.0598684
$838$	−1.60770	−0.0555369
$839$	39.5692	1.36608	0.683041	−	0.730380i	$-0.260657\pi$
$839$	39.5692	1.36608	0.683041	+	0.730380i	$0.260657\pi$
$840$	2.00000	0.0690066
$841$	−15.0718	−0.519717
$842$	−16.3923	−0.564916
$843$	−4.92820	−0.169736
$844$	23.3205	0.802725
$845$	0	0
$846$	10.4641	0.359763
$847$	21.5692	0.741127
$848$	−12.9282	−0.443956
$849$	3.92820	0.134816
$850$	−4.00000	−0.137199
$851$	−0.320508	−0.0109869
$852$	8.39230	0.287516
$853$	−35.8372	−1.22704	−0.613521	−	0.789679i	$-0.710247\pi$
$853$	−35.8372	−1.22704	−0.613521	+	0.789679i	$0.710247\pi$
$854$	20.7846	0.711235
$855$	0.535898	0.0183273
$856$	−19.8564	−0.678678
$857$	20.5167	0.700836	0.350418	−	0.936593i	$-0.386040\pi$
$857$	20.5167	0.700836	0.350418	+	0.936593i	$0.386040\pi$
$858$	0	0
$859$	27.1769	0.927264	0.463632	−	0.886028i	$-0.346546\pi$
$859$	27.1769	0.927264	0.463632	+	0.886028i	$0.346546\pi$
$860$	−1.92820	−0.0657512
$861$	4.00000	0.136320
$862$	−7.60770	−0.259119
$863$	−42.4641	−1.44549	−0.722747	−	0.691112i	$-0.757121\pi$
$863$	−42.4641	−1.44549	−0.722747	+	0.691112i	$0.757121\pi$
$864$	1.00000	0.0340207
$865$	−2.92820	−0.0995619
$866$	−15.3205	−0.520612
$867$	1.00000	0.0339618
$868$	3.46410	0.117579
$869$	−0.0333210	−0.00113034
$870$	−3.73205	−0.126528
$871$	0	0
$872$	11.8564	0.401509
$873$	−7.46410	−0.252622
$874$	−0.143594	−0.00485712
$875$	2.00000	0.0676123
$876$	−2.00000	−0.0675737
$877$	28.1244	0.949692	0.474846	−	0.880069i	$-0.342504\pi$
$877$	28.1244	0.949692	0.474846	+	0.880069i	$0.342504\pi$
$878$	9.85641	0.332638
$879$	4.14359	0.139760
$880$	−0.464102	−0.0156449
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	3.00000	0.101015
$883$	−16.0718	−0.540859	−0.270430	−	0.962740i	$-0.587166\pi$
$883$	−16.0718	−0.540859	−0.270430	+	0.962740i	$0.587166\pi$
$884$	0	0
$885$	1.53590	0.0516287
$886$	4.39230	0.147562
$887$	−10.1244	−0.339943	−0.169971	−	0.985449i	$-0.554368\pi$
$887$	−10.1244	−0.339943	−0.169971	+	0.985449i	$0.554368\pi$
$888$	1.19615	0.0401402
$889$	17.8564	0.598885
$890$	7.46410	0.250197
$891$	0.464102	0.0155480
$892$	−27.4641	−0.919566
$893$	5.60770	0.187654
$894$	20.4641	0.684422
$895$	−16.2679	−0.543778
$896$	2.00000	0.0668153
$897$	0	0
$898$	−39.7128	−1.32523
$899$	−6.46410	−0.215590
$900$	1.00000	0.0333333
$901$	−51.7128	−1.72280
$902$	−0.928203	−0.0309058
$903$	3.85641	0.128333
$904$	−11.1962	−0.372378
$905$	10.9282	0.363266
$906$	10.3923	0.345261
$907$	−6.85641	−0.227663	−0.113832	−	0.993500i	$-0.536312\pi$
$907$	−6.85641	−0.227663	−0.113832	+	0.993500i	$0.536312\pi$
$908$	−4.39230	−0.145764
$909$	10.9282	0.362466
$910$	0	0
$911$	−24.2487	−0.803396	−0.401698	−	0.915772i	$-0.631580\pi$
$911$	−24.2487	−0.803396	−0.401698	+	0.915772i	$0.631580\pi$
$912$	0.535898	0.0177454
$913$	2.28719	0.0756948
$914$	−31.4641	−1.04074
$915$	10.3923	0.343559
$916$	19.8564	0.656074
$917$	2.67949	0.0884846
$918$	4.00000	0.132020
$919$	−54.6410	−1.80244	−0.901220	−	0.433361i	$-0.857328\pi$
$919$	−54.6410	−1.80244	−0.901220	+	0.433361i	$0.857328\pi$
$920$	−0.267949	−0.00883402
$921$	−12.5359	−0.413072
$922$	−6.46410	−0.212884
$923$	0	0
$924$	0.928203	0.0305356
$925$	1.19615	0.0393292
$926$	−20.9282	−0.687743
$927$	−15.8564	−0.520793
$928$	−3.73205	−0.122511
$929$	33.8564	1.11079	0.555396	−	0.831586i	$-0.312567\pi$
$929$	33.8564	1.11079	0.555396	+	0.831586i	$0.312567\pi$
$930$	1.73205	0.0567962
$931$	1.60770	0.0526901
$932$	18.1244	0.593683
$933$	−7.60770	−0.249065
$934$	11.8564	0.387953
$935$	−1.85641	−0.0607110
$936$	0	0
$937$	44.6410	1.45836	0.729179	−	0.684323i	$-0.239902\pi$
$937$	44.6410	1.45836	0.729179	+	0.684323i	$0.239902\pi$
$938$	−9.07180	−0.296205
$939$	28.0000	0.913745
$940$	10.4641	0.341301
$941$	−26.7846	−0.873153	−0.436577	−	0.899667i	$-0.643809\pi$
$941$	−26.7846	−0.873153	−0.436577	+	0.899667i	$0.643809\pi$
$942$	−5.00000	−0.162909
$943$	−0.535898	−0.0174513
$944$	1.53590	0.0499892
$945$	−2.00000	−0.0650600
$946$	−0.894882	−0.0290951
$947$	−2.53590	−0.0824056	−0.0412028	−	0.999151i	$-0.513119\pi$
$947$	−2.53590	−0.0824056	−0.0412028	+	0.999151i	$0.513119\pi$
$948$	0.0717968	0.00233185
$949$	0	0
$950$	0.535898	0.0173868
$951$	21.4641	0.696021
$952$	8.00000	0.259281
$953$	15.7321	0.509611	0.254806	−	0.966992i	$-0.417989\pi$
$953$	15.7321	0.509611	0.254806	+	0.966992i	$0.417989\pi$
$954$	12.9282	0.418566
$955$	−14.5359	−0.470371
$956$	−4.39230	−0.142057
$957$	−1.73205	−0.0559893
$958$	−1.46410	−0.0473030
$959$	−8.92820	−0.288307
$960$	1.00000	0.0322749
$961$	−28.0000	−0.903226
$962$	0	0
$963$	19.8564	0.639864
$964$	14.2679	0.459540
$965$	−23.3205	−0.750714
$966$	0.535898	0.0172422
$967$	34.5359	1.11060	0.555300	−	0.831650i	$-0.312604\pi$
$967$	34.5359	1.11060	0.555300	+	0.831650i	$0.312604\pi$
$968$	10.7846	0.346630
$969$	2.14359	0.0688621
$970$	−7.46410	−0.239658
$971$	11.4641	0.367901	0.183950	−	0.982936i	$-0.441111\pi$
$971$	11.4641	0.367901	0.183950	+	0.982936i	$0.441111\pi$
$972$	−1.00000	−0.0320750
$973$	−1.85641	−0.0595137
$974$	−39.1769	−1.25531
$975$	0	0
$976$	10.3923	0.332650
$977$	33.3923	1.06831	0.534157	−	0.845385i	$-0.320629\pi$
$977$	33.3923	1.06831	0.534157	+	0.845385i	$0.320629\pi$
$978$	−23.0526	−0.737140
$979$	3.46410	0.110713
$980$	3.00000	0.0958315
$981$	−11.8564	−0.378546
$982$	17.3205	0.552720
$983$	−57.3923	−1.83053	−0.915265	−	0.402852i	$-0.868019\pi$
$983$	−57.3923	−1.83053	−0.915265	+	0.402852i	$0.868019\pi$
$984$	2.00000	0.0637577
$985$	16.3923	0.522302
$986$	−14.9282	−0.475411
$987$	−20.9282	−0.666152
$988$	0	0
$989$	−0.516660	−0.0164288
$990$	0.464102	0.0147501
$991$	−25.1436	−0.798713	−0.399356	−	0.916796i	$-0.630766\pi$
$991$	−25.1436	−0.798713	−0.399356	+	0.916796i	$0.630766\pi$
$992$	1.73205	0.0549927
$993$	24.7846	0.786516
$994$	−16.7846	−0.532375
$995$	−18.9282	−0.600064
$996$	−4.92820	−0.156156
$997$	−47.5692	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$997$	−47.5692	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$998$	13.4641	0.426199
$999$	−1.19615	−0.0378446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.y.1.2		2
13.5	odd	4		5070.2.b.o.1351.4		4
13.6	odd	12		390.2.bb.b.361.2	yes	4
13.8	odd	4		5070.2.b.o.1351.1		4
13.11	odd	12		390.2.bb.b.121.2	✓	4
13.12	even	2		5070.2.a.bg.1.1		2
39.11	even	12		1170.2.bs.e.901.1		4
39.32	even	12		1170.2.bs.e.361.1		4
65.19	odd	12		1950.2.bc.b.751.1		4
65.24	odd	12		1950.2.bc.b.901.1		4
65.32	even	12		1950.2.y.f.49.2		4
65.37	even	12		1950.2.y.c.199.1		4
65.58	even	12		1950.2.y.c.49.1		4
65.63	even	12		1950.2.y.f.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.bb.b.121.2	✓	4	13.11	odd	12
390.2.bb.b.361.2	yes	4	13.6	odd	12
1170.2.bs.e.361.1		4	39.32	even	12
1170.2.bs.e.901.1		4	39.11	even	12
1950.2.y.c.49.1		4	65.58	even	12
1950.2.y.c.199.1		4	65.37	even	12
1950.2.y.f.49.2		4	65.32	even	12
1950.2.y.f.199.2		4	65.63	even	12
1950.2.bc.b.751.1		4	65.19	odd	12
1950.2.bc.b.901.1		4	65.24	odd	12
5070.2.a.y.1.2		2	1.1	even	1	trivial
5070.2.a.bg.1.1		2	13.12	even	2
5070.2.b.o.1351.1		4	13.8	odd	4
5070.2.b.o.1351.4		4	13.5	odd	4

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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.464102 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -0.535898 q^{19} -1.00000 q^{20} +2.00000 q^{21} -0.464102 q^{22} -0.267949 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +3.73205 q^{29} -1.00000 q^{30} -1.73205 q^{31} -1.00000 q^{32} -0.464102 q^{33} -4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +1.19615 q^{37} +0.535898 q^{38} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} +1.92820 q^{43} +0.464102 q^{44} -1.00000 q^{45} +0.267949 q^{46} -10.4641 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} -12.9282 q^{53} +1.00000 q^{54} -0.464102 q^{55} +2.00000 q^{56} +0.535898 q^{57} -3.73205 q^{58} +1.53590 q^{59} +1.00000 q^{60} +10.3923 q^{61} +1.73205 q^{62} -2.00000 q^{63} +1.00000 q^{64} +0.464102 q^{66} -4.53590 q^{67} +4.00000 q^{68} +0.267949 q^{69} -2.00000 q^{70} -8.39230 q^{71} -1.00000 q^{72} +2.00000 q^{73} -1.19615 q^{74} -1.00000 q^{75} -0.535898 q^{76} -0.928203 q^{77} -0.0717968 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.92820 q^{83} +2.00000 q^{84} -4.00000 q^{85} -1.92820 q^{86} -3.73205 q^{87} -0.464102 q^{88} +7.46410 q^{89} +1.00000 q^{90} -0.267949 q^{92} +1.73205 q^{93} +10.4641 q^{94} +0.535898 q^{95} +1.00000 q^{96} -7.46410 q^{97} +3.00000 q^{98} +0.464102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} + 4 q^{14} + 2 q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{20} + 4 q^{21} + 6 q^{22} - 4 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{27} - 4 q^{28} + 4 q^{29} - 2 q^{30} - 2 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 2 q^{36} - 8 q^{37} + 8 q^{38} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 10 q^{43} - 6 q^{44} - 2 q^{45} + 4 q^{46} - 14 q^{47} - 2 q^{48} - 6 q^{49} - 2 q^{50} - 8 q^{51} - 12 q^{53} + 2 q^{54} + 6 q^{55} + 4 q^{56} + 8 q^{57} - 4 q^{58} + 10 q^{59} + 2 q^{60} - 4 q^{63} + 2 q^{64} - 6 q^{66} - 16 q^{67} + 8 q^{68} + 4 q^{69} - 4 q^{70} + 4 q^{71} - 2 q^{72} + 4 q^{73} + 8 q^{74} - 2 q^{75} - 8 q^{76} + 12 q^{77} - 14 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} - 4 q^{83} + 4 q^{84} - 8 q^{85} + 10 q^{86} - 4 q^{87} + 6 q^{88} + 8 q^{89} + 2 q^{90} - 4 q^{92} + 14 q^{94} + 8 q^{95} + 2 q^{96} - 8 q^{97} + 6 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 + 4 * q^14 + 2 * q^15 + 2 * q^16 + 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^20 + 4 * q^21 + 6 * q^22 - 4 * q^23 + 2 * q^24 + 2 * q^25 - 2 * q^27 - 4 * q^28 + 4 * q^29 - 2 * q^30 - 2 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 2 * q^36 - 8 * q^37 + 8 * q^38 + 2 * q^40 + 4 * q^41 - 4 * q^42 - 10 * q^43 - 6 * q^44 - 2 * q^45 + 4 * q^46 - 14 * q^47 - 2 * q^48 - 6 * q^49 - 2 * q^50 - 8 * q^51 - 12 * q^53 + 2 * q^54 + 6 * q^55 + 4 * q^56 + 8 * q^57 - 4 * q^58 + 10 * q^59 + 2 * q^60 - 4 * q^63 + 2 * q^64 - 6 * q^66 - 16 * q^67 + 8 * q^68 + 4 * q^69 - 4 * q^70 + 4 * q^71 - 2 * q^72 + 4 * q^73 + 8 * q^74 - 2 * q^75 - 8 * q^76 + 12 * q^77 - 14 * q^79 - 2 * q^80 + 2 * q^81 - 4 * q^82 - 4 * q^83 + 4 * q^84 - 8 * q^85 + 10 * q^86 - 4 * q^87 + 6 * q^88 + 8 * q^89 + 2 * q^90 - 4 * q^92 + 14 * q^94 + 8 * q^95 + 2 * q^96 - 8 * q^97 + 6 * q^98 - 6 * q^99

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