LMFDB - Embedded newform 8030.2.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8030,2,Mod(1,8030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8030, 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("8030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8030 = 2 \cdot 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8030.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:	$64.1198728231$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 7 x^{14} - 6 x^{13} + 136 x^{12} - 149 x^{11} - 876 x^{10} + 1631 x^{9} + 2142 x^{8} + \cdots - 32 $ x^15 - 7x^14 - 6x^13 + 136x^12 - 149x^11 - 876x^10 + 1631x^9 + 2142x^8 - 5473x^7 - 1914x^6 + 7517x^5 + 392x^4 - 3966x^3 - 79x^2 + 491x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.29384$ of defining polynomial
Character	$\chi$	$=$	8030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.29384 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.29384 q^{6} -4.27001 q^{7} +1.00000 q^{8} -1.32597 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.29384 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.29384 q^{6} -4.27001 q^{7} +1.00000 q^{8} -1.32597 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.29384 q^{12} -0.887514 q^{13} -4.27001 q^{14} +1.29384 q^{15} +1.00000 q^{16} -7.55436 q^{17} -1.32597 q^{18} -0.196436 q^{19} -1.00000 q^{20} +5.52472 q^{21} -1.00000 q^{22} -4.62098 q^{23} -1.29384 q^{24} +1.00000 q^{25} -0.887514 q^{26} +5.59713 q^{27} -4.27001 q^{28} -8.14596 q^{29} +1.29384 q^{30} -9.85588 q^{31} +1.00000 q^{32} +1.29384 q^{33} -7.55436 q^{34} +4.27001 q^{35} -1.32597 q^{36} -0.128879 q^{37} -0.196436 q^{38} +1.14830 q^{39} -1.00000 q^{40} +4.66318 q^{41} +5.52472 q^{42} -1.93222 q^{43} -1.00000 q^{44} +1.32597 q^{45} -4.62098 q^{46} -6.59566 q^{47} -1.29384 q^{48} +11.2330 q^{49} +1.00000 q^{50} +9.77416 q^{51} -0.887514 q^{52} +1.85194 q^{53} +5.59713 q^{54} +1.00000 q^{55} -4.27001 q^{56} +0.254158 q^{57} -8.14596 q^{58} +8.26418 q^{59} +1.29384 q^{60} -4.02649 q^{61} -9.85588 q^{62} +5.66190 q^{63} +1.00000 q^{64} +0.887514 q^{65} +1.29384 q^{66} -8.72028 q^{67} -7.55436 q^{68} +5.97882 q^{69} +4.27001 q^{70} +2.02927 q^{71} -1.32597 q^{72} -1.00000 q^{73} -0.128879 q^{74} -1.29384 q^{75} -0.196436 q^{76} +4.27001 q^{77} +1.14830 q^{78} +12.7324 q^{79} -1.00000 q^{80} -3.26390 q^{81} +4.66318 q^{82} +0.592684 q^{83} +5.52472 q^{84} +7.55436 q^{85} -1.93222 q^{86} +10.5396 q^{87} -1.00000 q^{88} -15.4886 q^{89} +1.32597 q^{90} +3.78969 q^{91} -4.62098 q^{92} +12.7520 q^{93} -6.59566 q^{94} +0.196436 q^{95} -1.29384 q^{96} -4.73382 q^{97} +11.2330 q^{98} +1.32597 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 + 7 * q^3 + 15 * q^4 - 15 * q^5 + 7 * q^6 + 3 * q^7 + 15 * q^8 + 16 * q^9	\( 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9} - 15 q^{10} - 15 q^{11} + 7 q^{12} - q^{13} + 3 q^{14} - 7 q^{15} + 15 q^{16} + 2 q^{17} + 16 q^{18} + 23 q^{19} - 15 q^{20} + 20 q^{21} - 15 q^{22} + 7 q^{24} + 15 q^{25} - q^{26} + 19 q^{27} + 3 q^{28} + 23 q^{29} - 7 q^{30} + 9 q^{31} + 15 q^{32} - 7 q^{33} + 2 q^{34} - 3 q^{35} + 16 q^{36} + 11 q^{37} + 23 q^{38} + 7 q^{39} - 15 q^{40} + 27 q^{41} + 20 q^{42} + 7 q^{43} - 15 q^{44} - 16 q^{45} - 18 q^{47} + 7 q^{48} + 16 q^{49} + 15 q^{50} + 21 q^{51} - q^{52} - 19 q^{53} + 19 q^{54} + 15 q^{55} + 3 q^{56} + 11 q^{57} + 23 q^{58} + 2 q^{59} - 7 q^{60} + 31 q^{61} + 9 q^{62} + 20 q^{63} + 15 q^{64} + q^{65} - 7 q^{66} + 49 q^{67} + 2 q^{68} + 33 q^{69} - 3 q^{70} + 32 q^{71} + 16 q^{72} - 15 q^{73} + 11 q^{74} + 7 q^{75} + 23 q^{76} - 3 q^{77} + 7 q^{78} + 36 q^{79} - 15 q^{80} + 23 q^{81} + 27 q^{82} + 33 q^{83} + 20 q^{84} - 2 q^{85} + 7 q^{86} + 29 q^{87} - 15 q^{88} + 6 q^{89} - 16 q^{90} + 33 q^{91} + 20 q^{93} - 18 q^{94} - 23 q^{95} + 7 q^{96} + 30 q^{97} + 16 q^{98} - 16 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 + 7 * q^3 + 15 * q^4 - 15 * q^5 + 7 * q^6 + 3 * q^7 + 15 * q^8 + 16 * q^9 - 15 * q^10 - 15 * q^11 + 7 * q^12 - q^13 + 3 * q^14 - 7 * q^15 + 15 * q^16 + 2 * q^17 + 16 * q^18 + 23 * q^19 - 15 * q^20 + 20 * q^21 - 15 * q^22 + 7 * q^24 + 15 * q^25 - q^26 + 19 * q^27 + 3 * q^28 + 23 * q^29 - 7 * q^30 + 9 * q^31 + 15 * q^32 - 7 * q^33 + 2 * q^34 - 3 * q^35 + 16 * q^36 + 11 * q^37 + 23 * q^38 + 7 * q^39 - 15 * q^40 + 27 * q^41 + 20 * q^42 + 7 * q^43 - 15 * q^44 - 16 * q^45 - 18 * q^47 + 7 * q^48 + 16 * q^49 + 15 * q^50 + 21 * q^51 - q^52 - 19 * q^53 + 19 * q^54 + 15 * q^55 + 3 * q^56 + 11 * q^57 + 23 * q^58 + 2 * q^59 - 7 * q^60 + 31 * q^61 + 9 * q^62 + 20 * q^63 + 15 * q^64 + q^65 - 7 * q^66 + 49 * q^67 + 2 * q^68 + 33 * q^69 - 3 * q^70 + 32 * q^71 + 16 * q^72 - 15 * q^73 + 11 * q^74 + 7 * q^75 + 23 * q^76 - 3 * q^77 + 7 * q^78 + 36 * q^79 - 15 * q^80 + 23 * q^81 + 27 * q^82 + 33 * q^83 + 20 * q^84 - 2 * q^85 + 7 * q^86 + 29 * q^87 - 15 * q^88 + 6 * q^89 - 16 * q^90 + 33 * q^91 + 20 * q^93 - 18 * q^94 - 23 * q^95 + 7 * q^96 + 30 * q^97 + 16 * q^98 - 16 * 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
$3$	−1.29384	−0.747001	−0.373500	−	0.927630i	$-0.621843\pi$
$3$	−1.29384	−0.747001	−0.373500	+	0.927630i	$0.621843\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.29384	−0.528209
$7$	−4.27001	−1.61391	−0.806956	−	0.590612i	$-0.798887\pi$
$7$	−4.27001	−1.61391	−0.806956	+	0.590612i	$0.798887\pi$
$8$	1.00000	0.353553
$9$	−1.32597	−0.441990
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.29384	−0.373500
$13$	−0.887514	−0.246152	−0.123076	−	0.992397i	$-0.539276\pi$
$13$	−0.887514	−0.246152	−0.123076	+	0.992397i	$0.539276\pi$
$14$	−4.27001	−1.14121
$15$	1.29384	0.334069
$16$	1.00000	0.250000
$17$	−7.55436	−1.83220	−0.916100	−	0.400949i	$-0.868681\pi$
$17$	−7.55436	−1.83220	−0.916100	+	0.400949i	$0.868681\pi$
$18$	−1.32597	−0.312534
$19$	−0.196436	−0.0450656	−0.0225328	−	0.999746i	$-0.507173\pi$
$19$	−0.196436	−0.0450656	−0.0225328	+	0.999746i	$0.507173\pi$
$20$	−1.00000	−0.223607
$21$	5.52472	1.20559
$22$	−1.00000	−0.213201
$23$	−4.62098	−0.963541	−0.481770	−	0.876298i	$-0.660006\pi$
$23$	−4.62098	−0.963541	−0.481770	+	0.876298i	$0.660006\pi$
$24$	−1.29384	−0.264105
$25$	1.00000	0.200000
$26$	−0.887514	−0.174056
$27$	5.59713	1.07717
$28$	−4.27001	−0.806956
$29$	−8.14596	−1.51267	−0.756333	−	0.654187i	$-0.773011\pi$
$29$	−8.14596	−1.51267	−0.756333	+	0.654187i	$0.773011\pi$
$30$	1.29384	0.236222
$31$	−9.85588	−1.77017	−0.885084	−	0.465430i	$-0.845900\pi$
$31$	−9.85588	−1.77017	−0.885084	+	0.465430i	$0.845900\pi$
$32$	1.00000	0.176777
$33$	1.29384	0.225229
$34$	−7.55436	−1.29556
$35$	4.27001	0.721763
$36$	−1.32597	−0.220995
$37$	−0.128879	−0.0211876	−0.0105938	−	0.999944i	$-0.503372\pi$
$37$	−0.128879	−0.0211876	−0.0105938	+	0.999944i	$0.503372\pi$
$38$	−0.196436	−0.0318662
$39$	1.14830	0.183876
$40$	−1.00000	−0.158114
$41$	4.66318	0.728266	0.364133	−	0.931347i	$-0.381365\pi$
$41$	4.66318	0.728266	0.364133	+	0.931347i	$0.381365\pi$
$42$	5.52472	0.852483
$43$	−1.93222	−0.294662	−0.147331	−	0.989087i	$-0.547068\pi$
$43$	−1.93222	−0.294662	−0.147331	+	0.989087i	$0.547068\pi$
$44$	−1.00000	−0.150756
$45$	1.32597	0.197664
$46$	−4.62098	−0.681326
$47$	−6.59566	−0.962076	−0.481038	−	0.876700i	$-0.659740\pi$
$47$	−6.59566	−0.962076	−0.481038	+	0.876700i	$0.659740\pi$
$48$	−1.29384	−0.186750
$49$	11.2330	1.60471
$50$	1.00000	0.141421
$51$	9.77416	1.36866
$52$	−0.887514	−0.123076
$53$	1.85194	0.254383	0.127192	−	0.991878i	$-0.459404\pi$
$53$	1.85194	0.254383	0.127192	+	0.991878i	$0.459404\pi$
$54$	5.59713	0.761673
$55$	1.00000	0.134840
$56$	−4.27001	−0.570604
$57$	0.254158	0.0336641
$58$	−8.14596	−1.06962
$59$	8.26418	1.07590	0.537952	−	0.842975i	$-0.319198\pi$
$59$	8.26418	1.07590	0.537952	+	0.842975i	$0.319198\pi$
$60$	1.29384	0.167034
$61$	−4.02649	−0.515539	−0.257770	−	0.966206i	$-0.582988\pi$
$61$	−4.02649	−0.515539	−0.257770	+	0.966206i	$0.582988\pi$
$62$	−9.85588	−1.25170
$63$	5.66190	0.713332
$64$	1.00000	0.125000
$65$	0.887514	0.110083
$66$	1.29384	0.159261
$67$	−8.72028	−1.06535	−0.532676	−	0.846319i	$-0.678813\pi$
$67$	−8.72028	−1.06535	−0.532676	+	0.846319i	$0.678813\pi$
$68$	−7.55436	−0.916100
$69$	5.97882	0.719766
$70$	4.27001	0.510364
$71$	2.02927	0.240830	0.120415	−	0.992724i	$-0.461577\pi$
$71$	2.02927	0.240830	0.120415	+	0.992724i	$0.461577\pi$
$72$	−1.32597	−0.156267
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	−0.128879	−0.0149819
$75$	−1.29384	−0.149400
$76$	−0.196436	−0.0225328
$77$	4.27001	0.486613
$78$	1.14830	0.130020
$79$	12.7324	1.43250	0.716252	−	0.697842i	$-0.245856\pi$
$79$	12.7324	1.43250	0.716252	+	0.697842i	$0.245856\pi$
$80$	−1.00000	−0.111803
$81$	−3.26390	−0.362656
$82$	4.66318	0.514962
$83$	0.592684	0.0650555	0.0325278	−	0.999471i	$-0.489644\pi$
$83$	0.592684	0.0650555	0.0325278	+	0.999471i	$0.489644\pi$
$84$	5.52472	0.602797
$85$	7.55436	0.819385
$86$	−1.93222	−0.208357
$87$	10.5396	1.12996
$88$	−1.00000	−0.106600
$89$	−15.4886	−1.64179	−0.820893	−	0.571083i	$-0.806524\pi$
$89$	−15.4886	−1.64179	−0.820893	+	0.571083i	$0.806524\pi$
$90$	1.32597	0.139769
$91$	3.78969	0.397268
$92$	−4.62098	−0.481770
$93$	12.7520	1.32232
$94$	−6.59566	−0.680290
$95$	0.196436	0.0201540
$96$	−1.29384	−0.132052
$97$	−4.73382	−0.480646	−0.240323	−	0.970693i	$-0.577253\pi$
$97$	−4.73382	−0.480646	−0.240323	+	0.970693i	$0.577253\pi$
$98$	11.2330	1.13470
$99$	1.32597	0.133265
$100$	1.00000	0.100000
$101$	−13.9216	−1.38525	−0.692624	−	0.721299i	$-0.743546\pi$
$101$	−13.9216	−1.38525	−0.692624	+	0.721299i	$0.743546\pi$
$102$	9.77416	0.967786
$103$	6.98722	0.688471	0.344236	−	0.938883i	$-0.388138\pi$
$103$	6.98722	0.688471	0.344236	+	0.938883i	$0.388138\pi$
$104$	−0.887514	−0.0870280
$105$	−5.52472	−0.539158
$106$	1.85194	0.179876
$107$	−16.8546	−1.62940	−0.814699	−	0.579884i	$-0.803098\pi$
$107$	−16.8546	−1.62940	−0.814699	+	0.579884i	$0.803098\pi$
$108$	5.59713	0.538584
$109$	6.04853	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$109$	6.04853	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$110$	1.00000	0.0953463
$111$	0.166749	0.0158271
$112$	−4.27001	−0.403478
$113$	13.0397	1.22667	0.613335	−	0.789823i	$-0.289828\pi$
$113$	13.0397	1.22667	0.613335	+	0.789823i	$0.289828\pi$
$114$	0.254158	0.0238041
$115$	4.62098	0.430908
$116$	−8.14596	−0.756333
$117$	1.17682	0.108797
$118$	8.26418	0.760779
$119$	32.2572	2.95701
$120$	1.29384	0.118111
$121$	1.00000	0.0909091
$122$	−4.02649	−0.364541
$123$	−6.03342	−0.544015
$124$	−9.85588	−0.885084
$125$	−1.00000	−0.0894427
$126$	5.66190	0.504402
$127$	1.30463	0.115767	0.0578837	−	0.998323i	$-0.481565\pi$
$127$	1.30463	0.115767	0.0578837	+	0.998323i	$0.481565\pi$
$128$	1.00000	0.0883883
$129$	2.50000	0.220112
$130$	0.887514	0.0778402
$131$	15.6409	1.36655	0.683276	−	0.730160i	$-0.260555\pi$
$131$	15.6409	1.36655	0.683276	+	0.730160i	$0.260555\pi$
$132$	1.29384	0.112615
$133$	0.838786	0.0727319
$134$	−8.72028	−0.753317
$135$	−5.59713	−0.481724
$136$	−7.55436	−0.647781
$137$	−19.4455	−1.66134	−0.830669	−	0.556767i	$-0.812042\pi$
$137$	−19.4455	−1.66134	−0.830669	+	0.556767i	$0.812042\pi$
$138$	5.97882	0.508951
$139$	10.6371	0.902226	0.451113	−	0.892467i	$-0.351027\pi$
$139$	10.6371	0.902226	0.451113	+	0.892467i	$0.351027\pi$
$140$	4.27001	0.360882
$141$	8.53375	0.718671
$142$	2.02927	0.170293
$143$	0.887514	0.0742177
$144$	−1.32597	−0.110497
$145$	8.14596	0.676485
$146$	−1.00000	−0.0827606
$147$	−14.5337	−1.19872
$148$	−0.128879	−0.0105938
$149$	−11.1216	−0.911119	−0.455559	−	0.890205i	$-0.650561\pi$
$149$	−11.1216	−0.911119	−0.455559	+	0.890205i	$0.650561\pi$
$150$	−1.29384	−0.105642
$151$	−20.3784	−1.65837	−0.829184	−	0.558976i	$-0.811194\pi$
$151$	−20.3784	−1.65837	−0.829184	+	0.558976i	$0.811194\pi$
$152$	−0.196436	−0.0159331
$153$	10.0168	0.809814
$154$	4.27001	0.344087
$155$	9.85588	0.791644
$156$	1.14830	0.0919380
$157$	8.36617	0.667693	0.333846	−	0.942627i	$-0.391653\pi$
$157$	8.36617	0.667693	0.333846	+	0.942627i	$0.391653\pi$
$158$	12.7324	1.01293
$159$	−2.39612	−0.190025
$160$	−1.00000	−0.0790569
$161$	19.7316	1.55507
$162$	−3.26390	−0.256436
$163$	−18.3223	−1.43512	−0.717558	−	0.696499i	$-0.754740\pi$
$163$	−18.3223	−1.43512	−0.717558	+	0.696499i	$0.754740\pi$
$164$	4.66318	0.364133
$165$	−1.29384	−0.100726
$166$	0.592684	0.0460012
$167$	23.4175	1.81210	0.906051	−	0.423168i	$-0.139082\pi$
$167$	23.4175	1.81210	0.906051	+	0.423168i	$0.139082\pi$
$168$	5.52472	0.426242
$169$	−12.2123	−0.939409
$170$	7.55436	0.579393
$171$	0.260469	0.0199185
$172$	−1.93222	−0.147331
$173$	−12.5369	−0.953160	−0.476580	−	0.879131i	$-0.658124\pi$
$173$	−12.5369	−0.953160	−0.476580	+	0.879131i	$0.658124\pi$
$174$	10.5396	0.799005
$175$	−4.27001	−0.322782
$176$	−1.00000	−0.0753778
$177$	−10.6926	−0.803702
$178$	−15.4886	−1.16092
$179$	−14.2507	−1.06515	−0.532574	−	0.846383i	$-0.678775\pi$
$179$	−14.2507	−1.06515	−0.532574	+	0.846383i	$0.678775\pi$
$180$	1.32597	0.0988319
$181$	17.3914	1.29269	0.646346	−	0.763044i	$-0.276296\pi$
$181$	17.3914	1.29269	0.646346	+	0.763044i	$0.276296\pi$
$182$	3.78969	0.280911
$183$	5.20965	0.385108
$184$	−4.62098	−0.340663
$185$	0.128879	0.00947536
$186$	12.7520	0.935020
$187$	7.55436	0.552429
$188$	−6.59566	−0.481038
$189$	−23.8998	−1.73845
$190$	0.196436	0.0142510
$191$	8.35812	0.604772	0.302386	−	0.953186i	$-0.402217\pi$
$191$	8.35812	0.604772	0.302386	+	0.953186i	$0.402217\pi$
$192$	−1.29384	−0.0933751
$193$	15.3099	1.10203	0.551014	−	0.834496i	$-0.314241\pi$
$193$	15.3099	1.10203	0.551014	+	0.834496i	$0.314241\pi$
$194$	−4.73382	−0.339868
$195$	−1.14830	−0.0822318
$196$	11.2330	0.802356
$197$	−1.74352	−0.124220	−0.0621102	−	0.998069i	$-0.519783\pi$
$197$	−1.74352	−0.124220	−0.0621102	+	0.998069i	$0.519783\pi$
$198$	1.32597	0.0942325
$199$	21.9807	1.55817	0.779084	−	0.626920i	$-0.215685\pi$
$199$	21.9807	1.55817	0.779084	+	0.626920i	$0.215685\pi$
$200$	1.00000	0.0707107
$201$	11.2827	0.795819
$202$	−13.9216	−0.979519
$203$	34.7833	2.44131
$204$	9.77416	0.684328
$205$	−4.66318	−0.325690
$206$	6.98722	0.486823
$207$	6.12727	0.425875
$208$	−0.887514	−0.0615381
$209$	0.196436	0.0135878
$210$	−5.52472	−0.381242
$211$	−1.48005	−0.101891	−0.0509453	−	0.998701i	$-0.516223\pi$
$211$	−1.48005	−0.101891	−0.0509453	+	0.998701i	$0.516223\pi$
$212$	1.85194	0.127192
$213$	−2.62556	−0.179901
$214$	−16.8546	−1.15216
$215$	1.93222	0.131777
$216$	5.59713	0.380836
$217$	42.0847	2.85690
$218$	6.04853	0.409658
$219$	1.29384	0.0874299
$220$	1.00000	0.0674200
$221$	6.70460	0.451000
$222$	0.166749	0.0111915
$223$	−19.8739	−1.33085	−0.665427	−	0.746463i	$-0.731751\pi$
$223$	−19.8739	−1.33085	−0.665427	+	0.746463i	$0.731751\pi$
$224$	−4.27001	−0.285302
$225$	−1.32597	−0.0883979
$226$	13.0397	0.867386
$227$	−8.49117	−0.563579	−0.281790	−	0.959476i	$-0.590928\pi$
$227$	−8.49117	−0.563579	−0.281790	+	0.959476i	$0.590928\pi$
$228$	0.254158	0.0168320
$229$	−3.78347	−0.250019	−0.125009	−	0.992156i	$-0.539896\pi$
$229$	−3.78347	−0.250019	−0.125009	+	0.992156i	$0.539896\pi$
$230$	4.62098	0.304698
$231$	−5.52472	−0.363500
$232$	−8.14596	−0.534808
$233$	−19.1862	−1.25693	−0.628467	−	0.777837i	$-0.716317\pi$
$233$	−19.1862	−1.25693	−0.628467	+	0.777837i	$0.716317\pi$
$234$	1.17682	0.0769309
$235$	6.59566	0.430253
$236$	8.26418	0.537952
$237$	−16.4737	−1.07008
$238$	32.2572	2.09092
$239$	25.8422	1.67160	0.835798	−	0.549037i	$-0.185005\pi$
$239$	25.8422	1.67160	0.835798	+	0.549037i	$0.185005\pi$
$240$	1.29384	0.0835172
$241$	−1.39400	−0.0897956	−0.0448978	−	0.998992i	$-0.514296\pi$
$241$	−1.39400	−0.0897956	−0.0448978	+	0.998992i	$0.514296\pi$
$242$	1.00000	0.0642824
$243$	−12.5684	−0.806263
$244$	−4.02649	−0.257770
$245$	−11.2330	−0.717649
$246$	−6.03342	−0.384677
$247$	0.174340	0.0110930
$248$	−9.85588	−0.625849
$249$	−0.766841	−0.0485966
$250$	−1.00000	−0.0632456
$251$	4.15141	0.262035	0.131017	−	0.991380i	$-0.458176\pi$
$251$	4.15141	0.262035	0.131017	+	0.991380i	$0.458176\pi$
$252$	5.66190	0.356666
$253$	4.62098	0.290518
$254$	1.30463	0.0818599
$255$	−9.77416	−0.612082
$256$	1.00000	0.0625000
$257$	29.6399	1.84889	0.924444	−	0.381318i	$-0.124530\pi$
$257$	29.6399	1.84889	0.924444	+	0.381318i	$0.124530\pi$
$258$	2.50000	0.155643
$259$	0.550314	0.0341948
$260$	0.887514	0.0550413
$261$	10.8013	0.668583
$262$	15.6409	0.966298
$263$	7.86541	0.485002	0.242501	−	0.970151i	$-0.422032\pi$
$263$	7.86541	0.485002	0.242501	+	0.970151i	$0.422032\pi$
$264$	1.29384	0.0796306
$265$	−1.85194	−0.113764
$266$	0.838786	0.0514292
$267$	20.0398	1.22642
$268$	−8.72028	−0.532676
$269$	15.4023	0.939096	0.469548	−	0.882907i	$-0.344417\pi$
$269$	15.4023	0.939096	0.469548	+	0.882907i	$0.344417\pi$
$270$	−5.59713	−0.340630
$271$	4.86545	0.295555	0.147777	−	0.989021i	$-0.452788\pi$
$271$	4.86545	0.295555	0.147777	+	0.989021i	$0.452788\pi$
$272$	−7.55436	−0.458050
$273$	−4.90327	−0.296760
$274$	−19.4455	−1.17474
$275$	−1.00000	−0.0603023
$276$	5.97882	0.359883
$277$	15.2680	0.917367	0.458683	−	0.888600i	$-0.348321\pi$
$277$	15.2680	0.917367	0.458683	+	0.888600i	$0.348321\pi$
$278$	10.6371	0.637970
$279$	13.0686	0.782396
$280$	4.27001	0.255182
$281$	−16.7984	−1.00211	−0.501055	−	0.865415i	$-0.667055\pi$
$281$	−16.7984	−1.00211	−0.501055	+	0.865415i	$0.667055\pi$
$282$	8.53375	0.508177
$283$	18.3336	1.08982	0.544910	−	0.838494i	$-0.316564\pi$
$283$	18.3336	1.08982	0.544910	+	0.838494i	$0.316564\pi$
$284$	2.02927	0.120415
$285$	−0.254158	−0.0150550
$286$	0.887514	0.0524798
$287$	−19.9118	−1.17536
$288$	−1.32597	−0.0781335
$289$	40.0683	2.35696
$290$	8.14596	0.478347
$291$	6.12482	0.359043
$292$	−1.00000	−0.0585206
$293$	−0.714296	−0.0417296	−0.0208648	−	0.999782i	$-0.506642\pi$
$293$	−0.714296	−0.0417296	−0.0208648	+	0.999782i	$0.506642\pi$
$294$	−14.5337	−0.847624
$295$	−8.26418	−0.481159
$296$	−0.128879	−0.00749093
$297$	−5.59713	−0.324778
$298$	−11.1216	−0.644258
$299$	4.10118	0.237178
$300$	−1.29384	−0.0747001
$301$	8.25062	0.475558
$302$	−20.3784	−1.17264
$303$	18.0123	1.03478
$304$	−0.196436	−0.0112664
$305$	4.02649	0.230556
$306$	10.0168	0.572625
$307$	−19.6535	−1.12168	−0.560842	−	0.827923i	$-0.689522\pi$
$307$	−19.6535	−1.12168	−0.560842	+	0.827923i	$0.689522\pi$
$308$	4.27001	0.243306
$309$	−9.04037	−0.514289
$310$	9.85588	0.559777
$311$	−0.758585	−0.0430154	−0.0215077	−	0.999769i	$-0.506847\pi$
$311$	−0.758585	−0.0430154	−0.0215077	+	0.999769i	$0.506847\pi$
$312$	1.14830	0.0650100
$313$	−25.9344	−1.46590	−0.732949	−	0.680284i	$-0.761857\pi$
$313$	−25.9344	−1.46590	−0.732949	+	0.680284i	$0.761857\pi$
$314$	8.36617	0.472130
$315$	−5.66190	−0.319012
$316$	12.7324	0.716252
$317$	−31.3946	−1.76330	−0.881650	−	0.471905i	$-0.843567\pi$
$317$	−31.3946	−1.76330	−0.881650	+	0.471905i	$0.843567\pi$
$318$	−2.39612	−0.134368
$319$	8.14596	0.456086
$320$	−1.00000	−0.0559017
$321$	21.8072	1.21716
$322$	19.7316	1.09960
$323$	1.48395	0.0825693
$324$	−3.26390	−0.181328
$325$	−0.887514	−0.0492304
$326$	−18.3223	−1.01478
$327$	−7.82586	−0.432771
$328$	4.66318	0.257481
$329$	28.1635	1.55271
$330$	−1.29384	−0.0712237
$331$	15.6771	0.861693	0.430847	−	0.902425i	$-0.358215\pi$
$331$	15.6771	0.861693	0.430847	+	0.902425i	$0.358215\pi$
$332$	0.592684	0.0325278
$333$	0.170889	0.00936468
$334$	23.4175	1.28135
$335$	8.72028	0.476440
$336$	5.52472	0.301398
$337$	22.1901	1.20877	0.604386	−	0.796691i	$-0.293418\pi$
$337$	22.1901	1.20877	0.604386	+	0.796691i	$0.293418\pi$
$338$	−12.2123	−0.664263
$339$	−16.8713	−0.916323
$340$	7.55436	0.409693
$341$	9.85588	0.533726
$342$	0.260469	0.0140845
$343$	−18.0749	−0.975951
$344$	−1.93222	−0.104179
$345$	−5.97882	−0.321889
$346$	−12.5369	−0.673986
$347$	−9.12308	−0.489753	−0.244876	−	0.969554i	$-0.578747\pi$
$347$	−9.12308	−0.489753	−0.244876	+	0.969554i	$0.578747\pi$
$348$	10.5396	0.564982
$349$	24.1035	1.29023	0.645116	−	0.764085i	$-0.276809\pi$
$349$	24.1035	1.29023	0.645116	+	0.764085i	$0.276809\pi$
$350$	−4.27001	−0.228242
$351$	−4.96753	−0.265147
$352$	−1.00000	−0.0533002
$353$	0.450792	0.0239933	0.0119966	−	0.999928i	$-0.496181\pi$
$353$	0.450792	0.0239933	0.0119966	+	0.999928i	$0.496181\pi$
$354$	−10.6926	−0.568303
$355$	−2.02927	−0.107703
$356$	−15.4886	−0.820893
$357$	−41.7357	−2.20889
$358$	−14.2507	−0.753174
$359$	28.5097	1.50468	0.752342	−	0.658773i	$-0.228924\pi$
$359$	28.5097	1.50468	0.752342	+	0.658773i	$0.228924\pi$
$360$	1.32597	0.0698847
$361$	−18.9614	−0.997969
$362$	17.3914	0.914071
$363$	−1.29384	−0.0679092
$364$	3.78969	0.198634
$365$	1.00000	0.0523424
$366$	5.20965	0.272313
$367$	−16.5120	−0.861918	−0.430959	−	0.902371i	$-0.641825\pi$
$367$	−16.5120	−0.861918	−0.430959	+	0.902371i	$0.641825\pi$
$368$	−4.62098	−0.240885
$369$	−6.18322	−0.321886
$370$	0.128879	0.00670009
$371$	−7.90780	−0.410553
$372$	12.7520	0.661159
$373$	−22.9766	−1.18968	−0.594842	−	0.803842i	$-0.702785\pi$
$373$	−22.9766	−1.18968	−0.594842	+	0.803842i	$0.702785\pi$
$374$	7.55436	0.390627
$375$	1.29384	0.0668138
$376$	−6.59566	−0.340145
$377$	7.22965	0.372346
$378$	−23.8998	−1.22927
$379$	−34.1745	−1.75543	−0.877714	−	0.479185i	$-0.840932\pi$
$379$	−34.1745	−1.75543	−0.877714	+	0.479185i	$0.840932\pi$
$380$	0.196436	0.0100770
$381$	−1.68799	−0.0864783
$382$	8.35812	0.427639
$383$	−28.3315	−1.44767	−0.723835	−	0.689973i	$-0.757622\pi$
$383$	−28.3315	−1.44767	−0.723835	+	0.689973i	$0.757622\pi$
$384$	−1.29384	−0.0660262
$385$	−4.27001	−0.217620
$386$	15.3099	0.779252
$387$	2.56207	0.130237
$388$	−4.73382	−0.240323
$389$	15.2748	0.774462	0.387231	−	0.921983i	$-0.373432\pi$
$389$	15.2748	0.774462	0.387231	+	0.921983i	$0.373432\pi$
$390$	−1.14830	−0.0581467
$391$	34.9085	1.76540
$392$	11.2330	0.567351
$393$	−20.2369	−1.02082
$394$	−1.74352	−0.0878371
$395$	−12.7324	−0.640635
$396$	1.32597	0.0666324
$397$	8.94998	0.449187	0.224593	−	0.974453i	$-0.427895\pi$
$397$	8.94998	0.449187	0.224593	+	0.974453i	$0.427895\pi$
$398$	21.9807	1.10179
$399$	−1.08526	−0.0543308
$400$	1.00000	0.0500000
$401$	−13.6495	−0.681621	−0.340811	−	0.940132i	$-0.610702\pi$
$401$	−13.6495	−0.681621	−0.340811	+	0.940132i	$0.610702\pi$
$402$	11.2827	0.562729
$403$	8.74724	0.435731
$404$	−13.9216	−0.692624
$405$	3.26390	0.162185
$406$	34.7833	1.72627
$407$	0.128879	0.00638829
$408$	9.77416	0.483893
$409$	−40.3810	−1.99671	−0.998356	−	0.0573195i	$-0.981745\pi$
$409$	−40.3810	−1.99671	−0.998356	+	0.0573195i	$0.981745\pi$
$410$	−4.66318	−0.230298
$411$	25.1594	1.24102
$412$	6.98722	0.344236
$413$	−35.2881	−1.73641
$414$	6.12727	0.301139
$415$	−0.592684	−0.0290937
$416$	−0.887514	−0.0435140
$417$	−13.7627	−0.673964
$418$	0.196436	0.00960802
$419$	−18.0354	−0.881085	−0.440542	−	0.897732i	$-0.645214\pi$
$419$	−18.0354	−0.881085	−0.440542	+	0.897732i	$0.645214\pi$
$420$	−5.52472	−0.269579
$421$	26.0929	1.27169	0.635845	−	0.771817i	$-0.280652\pi$
$421$	26.0929	1.27169	0.635845	+	0.771817i	$0.280652\pi$
$422$	−1.48005	−0.0720476
$423$	8.74564	0.425227
$424$	1.85194	0.0899381
$425$	−7.55436	−0.366440
$426$	−2.62556	−0.127209
$427$	17.1931	0.832035
$428$	−16.8546	−0.814699
$429$	−1.14830	−0.0554407
$430$	1.93222	0.0931802
$431$	19.8376	0.955545	0.477773	−	0.878483i	$-0.341444\pi$
$431$	19.8376	0.955545	0.477773	+	0.878483i	$0.341444\pi$
$432$	5.59713	0.269292
$433$	13.7770	0.662080	0.331040	−	0.943617i	$-0.392600\pi$
$433$	13.7770	0.662080	0.331040	+	0.943617i	$0.392600\pi$
$434$	42.0847	2.02013
$435$	−10.5396	−0.505335
$436$	6.04853	0.289672
$437$	0.907729	0.0434225
$438$	1.29384	0.0618222
$439$	34.8097	1.66138	0.830688	−	0.556739i	$-0.187948\pi$
$439$	34.8097	1.66138	0.830688	+	0.556739i	$0.187948\pi$
$440$	1.00000	0.0476731
$441$	−14.8946	−0.709266
$442$	6.70460	0.318905
$443$	−22.3474	−1.06175	−0.530877	−	0.847449i	$-0.678138\pi$
$443$	−22.3474	−1.06175	−0.530877	+	0.847449i	$0.678138\pi$
$444$	0.166749	0.00791356
$445$	15.4886	0.734229
$446$	−19.8739	−0.941056
$447$	14.3896	0.680607
$448$	−4.27001	−0.201739
$449$	−17.3283	−0.817774	−0.408887	−	0.912585i	$-0.634083\pi$
$449$	−17.3283	−0.817774	−0.408887	+	0.912585i	$0.634083\pi$
$450$	−1.32597	−0.0625068
$451$	−4.66318	−0.219580
$452$	13.0397	0.613335
$453$	26.3664	1.23880
$454$	−8.49117	−0.398511
$455$	−3.78969	−0.177664
$456$	0.254158	0.0119020
$457$	−16.5825	−0.775698	−0.387849	−	0.921723i	$-0.626782\pi$
$457$	−16.5825	−0.775698	−0.387849	+	0.921723i	$0.626782\pi$
$458$	−3.78347	−0.176790
$459$	−42.2827	−1.97359
$460$	4.62098	0.215454
$461$	8.77633	0.408754	0.204377	−	0.978892i	$-0.434483\pi$
$461$	8.77633	0.408754	0.204377	+	0.978892i	$0.434483\pi$
$462$	−5.52472	−0.257033
$463$	11.7645	0.546741	0.273370	−	0.961909i	$-0.411862\pi$
$463$	11.7645	0.546741	0.273370	+	0.961909i	$0.411862\pi$
$464$	−8.14596	−0.378167
$465$	−12.7520	−0.591359
$466$	−19.1862	−0.888786
$467$	−37.0970	−1.71664	−0.858322	−	0.513111i	$-0.828493\pi$
$467$	−37.0970	−1.71664	−0.858322	+	0.513111i	$0.828493\pi$
$468$	1.17682	0.0543984
$469$	37.2357	1.71938
$470$	6.59566	0.304235
$471$	−10.8245	−0.498767
$472$	8.26418	0.380390
$473$	1.93222	0.0888438
$474$	−16.4737	−0.756662
$475$	−0.196436	−0.00901312
$476$	32.2572	1.47851
$477$	−2.45561	−0.112435
$478$	25.8422	1.18200
$479$	11.6255	0.531181	0.265590	−	0.964086i	$-0.414433\pi$
$479$	11.6255	0.531181	0.265590	+	0.964086i	$0.414433\pi$
$480$	1.29384	0.0590556
$481$	0.114382	0.00521536
$482$	−1.39400	−0.0634951
$483$	−25.5296	−1.16164
$484$	1.00000	0.0454545
$485$	4.73382	0.214952
$486$	−12.5684	−0.570114
$487$	13.5372	0.613428	0.306714	−	0.951802i	$-0.400770\pi$
$487$	13.5372	0.613428	0.306714	+	0.951802i	$0.400770\pi$
$488$	−4.02649	−0.182271
$489$	23.7062	1.07203
$490$	−11.2330	−0.507454
$491$	−23.6807	−1.06869	−0.534347	−	0.845265i	$-0.679442\pi$
$491$	−23.6807	−1.06869	−0.534347	+	0.845265i	$0.679442\pi$
$492$	−6.03342	−0.272008
$493$	61.5375	2.77151
$494$	0.174340	0.00784394
$495$	−1.32597	−0.0595979
$496$	−9.85588	−0.442542
$497$	−8.66501	−0.388679
$498$	−0.766841	−0.0343630
$499$	18.8542	0.844030	0.422015	−	0.906589i	$-0.361323\pi$
$499$	18.8542	0.844030	0.422015	+	0.906589i	$0.361323\pi$
$500$	−1.00000	−0.0447214
$501$	−30.2986	−1.35364
$502$	4.15141	0.185287
$503$	31.0490	1.38441	0.692203	−	0.721703i	$-0.256640\pi$
$503$	31.0490	1.38441	0.692203	+	0.721703i	$0.256640\pi$
$504$	5.66190	0.252201
$505$	13.9216	0.619502
$506$	4.62098	0.205428
$507$	15.8008	0.701739
$508$	1.30463	0.0578837
$509$	−29.0629	−1.28819	−0.644095	−	0.764946i	$-0.722766\pi$
$509$	−29.0629	−1.28819	−0.644095	+	0.764946i	$0.722766\pi$
$510$	−9.77416	−0.432807
$511$	4.27001	0.188894
$512$	1.00000	0.0441942
$513$	−1.09948	−0.0485432
$514$	29.6399	1.30736
$515$	−6.98722	−0.307894
$516$	2.50000	0.110056
$517$	6.59566	0.290077
$518$	0.550314	0.0241794
$519$	16.2207	0.712011
$520$	0.887514	0.0389201
$521$	42.4031	1.85771	0.928856	−	0.370441i	$-0.120793\pi$
$521$	42.4031	1.85771	0.928856	+	0.370441i	$0.120793\pi$
$522$	10.8013	0.472759
$523$	−26.8768	−1.17524	−0.587621	−	0.809136i	$-0.699935\pi$
$523$	−26.8768	−1.17524	−0.587621	+	0.809136i	$0.699935\pi$
$524$	15.6409	0.683276
$525$	5.52472	0.241119
$526$	7.86541	0.342948
$527$	74.4549	3.24331
$528$	1.29384	0.0563073
$529$	−1.64656	−0.0715897
$530$	−1.85194	−0.0804431
$531$	−10.9580	−0.475539
$532$	0.838786	0.0363660
$533$	−4.13864	−0.179264
$534$	20.0398	0.867206
$535$	16.8546	0.728689
$536$	−8.72028	−0.376659
$537$	18.4382	0.795667
$538$	15.4023	0.664041
$539$	−11.2330	−0.483839
$540$	−5.59713	−0.240862
$541$	−37.3347	−1.60514	−0.802572	−	0.596556i	$-0.796535\pi$
$541$	−37.3347	−1.60514	−0.802572	+	0.596556i	$0.796535\pi$
$542$	4.86545	0.208989
$543$	−22.5017	−0.965642
$544$	−7.55436	−0.323890
$545$	−6.04853	−0.259091
$546$	−4.90327	−0.209841
$547$	21.2620	0.909097	0.454548	−	0.890722i	$-0.349801\pi$
$547$	21.2620	0.909097	0.454548	+	0.890722i	$0.349801\pi$
$548$	−19.4455	−0.830669
$549$	5.33900	0.227863
$550$	−1.00000	−0.0426401
$551$	1.60016	0.0681692
$552$	5.97882	0.254476
$553$	−54.3673	−2.31193
$554$	15.2680	0.648676
$555$	−0.166749	−0.00707810
$556$	10.6371	0.451113
$557$	34.2218	1.45002	0.725012	−	0.688736i	$-0.241834\pi$
$557$	34.2218	1.45002	0.725012	+	0.688736i	$0.241834\pi$
$558$	13.0686	0.553238
$559$	1.71488	0.0725316
$560$	4.27001	0.180441
$561$	−9.77416	−0.412665
$562$	−16.7984	−0.708599
$563$	40.9348	1.72520	0.862599	−	0.505889i	$-0.168835\pi$
$563$	40.9348	1.72520	0.862599	+	0.505889i	$0.168835\pi$
$564$	8.53375	0.359336
$565$	−13.0397	−0.548583
$566$	18.3336	0.770620
$567$	13.9369	0.585294
$568$	2.02927	0.0851464
$569$	7.48746	0.313891	0.156945	−	0.987607i	$-0.449835\pi$
$569$	7.48746	0.313891	0.156945	+	0.987607i	$0.449835\pi$
$570$	−0.254158	−0.0106455
$571$	−30.0290	−1.25667	−0.628336	−	0.777942i	$-0.716264\pi$
$571$	−30.0290	−1.25667	−0.628336	+	0.777942i	$0.716264\pi$
$572$	0.887514	0.0371088
$573$	−10.8141	−0.451765
$574$	−19.9118	−0.831103
$575$	−4.62098	−0.192708
$576$	−1.32597	−0.0552487
$577$	−34.2917	−1.42758	−0.713792	−	0.700358i	$-0.753024\pi$
$577$	−34.2917	−1.42758	−0.713792	+	0.700358i	$0.753024\pi$
$578$	40.0683	1.66662
$579$	−19.8086	−0.823216
$580$	8.14596	0.338242
$581$	−2.53077	−0.104994
$582$	6.12482	0.253882
$583$	−1.85194	−0.0766995
$584$	−1.00000	−0.0413803
$585$	−1.17682	−0.0486554
$586$	−0.714296	−0.0295073
$587$	−43.2939	−1.78693	−0.893465	−	0.449134i	$-0.851733\pi$
$587$	−43.2939	−1.78693	−0.893465	+	0.449134i	$0.851733\pi$
$588$	−14.5337	−0.599360
$589$	1.93605	0.0797738
$590$	−8.26418	−0.340231
$591$	2.25584	0.0927928
$592$	−0.128879	−0.00529689
$593$	−14.9420	−0.613595	−0.306798	−	0.951775i	$-0.599257\pi$
$593$	−14.9420	−0.613595	−0.306798	+	0.951775i	$0.599257\pi$
$594$	−5.59713	−0.229653
$595$	−32.2572	−1.32242
$596$	−11.1216	−0.455559
$597$	−28.4395	−1.16395
$598$	4.10118	0.167710
$599$	−5.92151	−0.241947	−0.120973	−	0.992656i	$-0.538602\pi$
$599$	−5.92151	−0.241947	−0.120973	+	0.992656i	$0.538602\pi$
$600$	−1.29384	−0.0528209
$601$	−16.5530	−0.675212	−0.337606	−	0.941288i	$-0.609617\pi$
$601$	−16.5530	−0.675212	−0.337606	+	0.941288i	$0.609617\pi$
$602$	8.25062	0.336270
$603$	11.5628	0.470874
$604$	−20.3784	−0.829184
$605$	−1.00000	−0.0406558
$606$	18.0123	0.731701
$607$	6.10436	0.247768	0.123884	−	0.992297i	$-0.460465\pi$
$607$	6.10436	0.247768	0.123884	+	0.992297i	$0.460465\pi$
$608$	−0.196436	−0.00796655
$609$	−45.0042	−1.82366
$610$	4.02649	0.163028
$611$	5.85374	0.236817
$612$	10.0168	0.404907
$613$	−36.5737	−1.47720	−0.738600	−	0.674144i	$-0.764513\pi$
$613$	−36.5737	−1.47720	−0.738600	+	0.674144i	$0.764513\pi$
$614$	−19.6535	−0.793150
$615$	6.03342	0.243291
$616$	4.27001	0.172044
$617$	−20.9337	−0.842758	−0.421379	−	0.906885i	$-0.638454\pi$
$617$	−20.9337	−0.842758	−0.421379	+	0.906885i	$0.638454\pi$
$618$	−9.04037	−0.363657
$619$	42.0647	1.69072	0.845362	−	0.534193i	$-0.179385\pi$
$619$	42.0647	1.69072	0.845362	+	0.534193i	$0.179385\pi$
$620$	9.85588	0.395822
$621$	−25.8642	−1.03789
$622$	−0.758585	−0.0304165
$623$	66.1363	2.64970
$624$	1.14830	0.0459690
$625$	1.00000	0.0400000
$626$	−25.9344	−1.03655
$627$	−0.254158	−0.0101501
$628$	8.36617	0.333846
$629$	0.973597	0.0388199
$630$	−5.66190	−0.225575
$631$	−28.9672	−1.15316	−0.576582	−	0.817039i	$-0.695614\pi$
$631$	−28.9672	−1.15316	−0.576582	+	0.817039i	$0.695614\pi$
$632$	12.7324	0.506466
$633$	1.91495	0.0761124
$634$	−31.3946	−1.24684
$635$	−1.30463	−0.0517727
$636$	−2.39612	−0.0950124
$637$	−9.96943	−0.395003
$638$	8.14596	0.322502
$639$	−2.69075	−0.106445
$640$	−1.00000	−0.0395285
$641$	−21.9647	−0.867554	−0.433777	−	0.901020i	$-0.642819\pi$
$641$	−21.9647	−0.867554	−0.433777	+	0.901020i	$0.642819\pi$
$642$	21.8072	0.860663
$643$	39.4944	1.55751	0.778753	−	0.627331i	$-0.215853\pi$
$643$	39.4944	1.55751	0.778753	+	0.627331i	$0.215853\pi$
$644$	19.7316	0.777535
$645$	−2.50000	−0.0984373
$646$	1.48395	0.0583853
$647$	−14.2307	−0.559465	−0.279732	−	0.960078i	$-0.590246\pi$
$647$	−14.2307	−0.559465	−0.279732	+	0.960078i	$0.590246\pi$
$648$	−3.26390	−0.128218
$649$	−8.26418	−0.324397
$650$	−0.887514	−0.0348112
$651$	−54.4510	−2.13410
$652$	−18.3223	−0.717558
$653$	−18.8000	−0.735700	−0.367850	−	0.929885i	$-0.619906\pi$
$653$	−18.8000	−0.735700	−0.367850	+	0.929885i	$0.619906\pi$
$654$	−7.82586	−0.306015
$655$	−15.6409	−0.611141
$656$	4.66318	0.182066
$657$	1.32597	0.0517310
$658$	28.1635	1.09793
$659$	5.70298	0.222157	0.111078	−	0.993812i	$-0.464570\pi$
$659$	5.70298	0.222157	0.111078	+	0.993812i	$0.464570\pi$
$660$	−1.29384	−0.0503628
$661$	19.7169	0.766899	0.383449	−	0.923562i	$-0.374736\pi$
$661$	19.7169	0.766899	0.383449	+	0.923562i	$0.374736\pi$
$662$	15.6771	0.609309
$663$	−8.67471	−0.336898
$664$	0.592684	0.0230006
$665$	−0.838786	−0.0325267
$666$	0.170889	0.00662183
$667$	37.6423	1.45752
$668$	23.4175	0.906051
$669$	25.7137	0.994150
$670$	8.72028	0.336894
$671$	4.02649	0.155441
$672$	5.52472	0.213121
$673$	−5.26204	−0.202837	−0.101418	−	0.994844i	$-0.532338\pi$
$673$	−5.26204	−0.202837	−0.101418	+	0.994844i	$0.532338\pi$
$674$	22.1901	0.854732
$675$	5.59713	0.215434
$676$	−12.2123	−0.469705
$677$	2.88517	0.110886	0.0554431	−	0.998462i	$-0.482343\pi$
$677$	2.88517	0.110886	0.0554431	+	0.998462i	$0.482343\pi$
$678$	−16.8713	−0.647938
$679$	20.2134	0.775721
$680$	7.55436	0.289696
$681$	10.9863	0.420994
$682$	9.85588	0.377401
$683$	−3.04471	−0.116503	−0.0582513	−	0.998302i	$-0.518552\pi$
$683$	−3.04471	−0.116503	−0.0582513	+	0.998302i	$0.518552\pi$
$684$	0.260469	0.00995927
$685$	19.4455	0.742973
$686$	−18.0749	−0.690101
$687$	4.89522	0.186764
$688$	−1.93222	−0.0736654
$689$	−1.64362	−0.0626171
$690$	−5.97882	−0.227610
$691$	−24.2446	−0.922307	−0.461154	−	0.887320i	$-0.652564\pi$
$691$	−24.2446	−0.922307	−0.461154	+	0.887320i	$0.652564\pi$
$692$	−12.5369	−0.476580
$693$	−5.66190	−0.215078
$694$	−9.12308	−0.346308
$695$	−10.6371	−0.403488
$696$	10.5396	0.399502
$697$	−35.2273	−1.33433
$698$	24.1035	0.912332
$699$	24.8240	0.938930
$700$	−4.27001	−0.161391
$701$	−25.9068	−0.978488	−0.489244	−	0.872147i	$-0.662727\pi$
$701$	−25.9068	−0.978488	−0.489244	+	0.872147i	$0.662727\pi$
$702$	−4.96753	−0.187487
$703$	0.0253165	0.000954830 0
$704$	−1.00000	−0.0376889
$705$	−8.53375	−0.321400
$706$	0.450792	0.0169658
$707$	59.4453	2.23567
$708$	−10.6926	−0.401851
$709$	−19.2342	−0.722354	−0.361177	−	0.932497i	$-0.617625\pi$
$709$	−19.2342	−0.722354	−0.361177	+	0.932497i	$0.617625\pi$
$710$	−2.02927	−0.0761572
$711$	−16.8827	−0.633152
$712$	−15.4886	−0.580459
$713$	45.5438	1.70563
$714$	−41.7357	−1.56192
$715$	−0.887514	−0.0331912
$716$	−14.2507	−0.532574
$717$	−33.4358	−1.24868
$718$	28.5097	1.06397
$719$	51.0135	1.90248	0.951242	−	0.308446i	$-0.0998090\pi$
$719$	51.0135	1.90248	0.951242	+	0.308446i	$0.0998090\pi$
$720$	1.32597	0.0494159
$721$	−29.8355	−1.11113
$722$	−18.9614	−0.705671
$723$	1.80362	0.0670774
$724$	17.3914	0.646346
$725$	−8.14596	−0.302533
$726$	−1.29384	−0.0480190
$727$	13.0464	0.483865	0.241932	−	0.970293i	$-0.422219\pi$
$727$	13.0464	0.483865	0.241932	+	0.970293i	$0.422219\pi$
$728$	3.78969	0.140455
$729$	26.0533	0.964935
$730$	1.00000	0.0370117
$731$	14.5967	0.539879
$732$	5.20965	0.192554
$733$	−13.7649	−0.508417	−0.254208	−	0.967149i	$-0.581815\pi$
$733$	−13.7649	−0.508417	−0.254208	+	0.967149i	$0.581815\pi$
$734$	−16.5120	−0.609468
$735$	14.5337	0.536084
$736$	−4.62098	−0.170332
$737$	8.72028	0.321216
$738$	−6.18322	−0.227608
$739$	19.3092	0.710299	0.355149	−	0.934810i	$-0.384430\pi$
$739$	19.3092	0.710299	0.355149	+	0.934810i	$0.384430\pi$
$740$	0.128879	0.00473768
$741$	−0.225569	−0.00828648
$742$	−7.90780	−0.290304
$743$	−26.4120	−0.968961	−0.484480	−	0.874802i	$-0.660991\pi$
$743$	−26.4120	−0.968961	−0.484480	+	0.874802i	$0.660991\pi$
$744$	12.7520	0.467510
$745$	11.1216	0.407465
$746$	−22.9766	−0.841234
$747$	−0.785881	−0.0287539
$748$	7.55436	0.276215
$749$	71.9694	2.62970
$750$	1.29384	0.0472445
$751$	−32.7066	−1.19348	−0.596741	−	0.802434i	$-0.703538\pi$
$751$	−32.7066	−1.19348	−0.596741	+	0.802434i	$0.703538\pi$
$752$	−6.59566	−0.240519
$753$	−5.37128	−0.195740
$754$	7.22965	0.263288
$755$	20.3784	0.741645
$756$	−23.8998	−0.869227
$757$	12.8023	0.465308	0.232654	−	0.972560i	$-0.425259\pi$
$757$	12.8023	0.465308	0.232654	+	0.972560i	$0.425259\pi$
$758$	−34.1745	−1.24127
$759$	−5.97882	−0.217018
$760$	0.196436	0.00712550
$761$	−35.2622	−1.27825	−0.639126	−	0.769102i	$-0.720704\pi$
$761$	−35.2622	−1.27825	−0.639126	+	0.769102i	$0.720704\pi$
$762$	−1.68799	−0.0611494
$763$	−25.8273	−0.935011
$764$	8.35812	0.302386
$765$	−10.0168	−0.362160
$766$	−28.3315	−1.02366
$767$	−7.33458	−0.264836
$768$	−1.29384	−0.0466876
$769$	−29.1384	−1.05076	−0.525379	−	0.850869i	$-0.676076\pi$
$769$	−29.1384	−1.05076	−0.525379	+	0.850869i	$0.676076\pi$
$770$	−4.27001	−0.153880
$771$	−38.3494	−1.38112
$772$	15.3099	0.551014
$773$	18.1561	0.653029	0.326514	−	0.945192i	$-0.394126\pi$
$773$	18.1561	0.653029	0.326514	+	0.945192i	$0.394126\pi$
$774$	2.56207	0.0920917
$775$	−9.85588	−0.354034
$776$	−4.73382	−0.169934
$777$	−0.712020	−0.0255436
$778$	15.2748	0.547627
$779$	−0.916018	−0.0328197
$780$	−1.14830	−0.0411159
$781$	−2.02927	−0.0726131
$782$	34.9085	1.24833
$783$	−45.5940	−1.62939
$784$	11.2330	0.401178
$785$	−8.36617	−0.298601
$786$	−20.2369	−0.721826
$787$	19.7798	0.705072	0.352536	−	0.935798i	$-0.385319\pi$
$787$	19.7798	0.705072	0.352536	+	0.935798i	$0.385319\pi$
$788$	−1.74352	−0.0621102
$789$	−10.1766	−0.362297
$790$	−12.7324	−0.452997
$791$	−55.6795	−1.97974
$792$	1.32597	0.0471162
$793$	3.57357	0.126901
$794$	8.94998	0.317623
$795$	2.39612	0.0849816
$796$	21.9807	0.779084
$797$	−33.1946	−1.17581	−0.587907	−	0.808929i	$-0.700048\pi$
$797$	−33.1946	−1.17581	−0.587907	+	0.808929i	$0.700048\pi$
$798$	−1.08526	−0.0384177
$799$	49.8260	1.76272
$800$	1.00000	0.0353553
$801$	20.5374	0.725652
$802$	−13.6495	−0.481979
$803$	1.00000	0.0352892
$804$	11.2827	0.397909
$805$	−19.7316	−0.695448
$806$	8.74724	0.308108
$807$	−19.9282	−0.701505
$808$	−13.9216	−0.489759
$809$	26.7982	0.942175	0.471088	−	0.882086i	$-0.343862\pi$
$809$	26.7982	0.942175	0.471088	+	0.882086i	$0.343862\pi$
$810$	3.26390	0.114682
$811$	−3.48289	−0.122301	−0.0611504	−	0.998129i	$-0.519477\pi$
$811$	−3.48289	−0.122301	−0.0611504	+	0.998129i	$0.519477\pi$
$812$	34.7833	1.22065
$813$	−6.29513	−0.220780
$814$	0.128879	0.00451720
$815$	18.3223	0.641803
$816$	9.77416	0.342164
$817$	0.379559	0.0132791
$818$	−40.3810	−1.41189
$819$	−5.02502	−0.175588
$820$	−4.66318	−0.162845
$821$	47.6326	1.66239	0.831194	−	0.555982i	$-0.187658\pi$
$821$	47.6326	1.66239	0.831194	+	0.555982i	$0.187658\pi$
$822$	25.1594	0.877534
$823$	15.5285	0.541288	0.270644	−	0.962679i	$-0.412763\pi$
$823$	15.5285	0.541288	0.270644	+	0.962679i	$0.412763\pi$
$824$	6.98722	0.243411
$825$	1.29384	0.0450459
$826$	−35.2881	−1.22783
$827$	22.8777	0.795537	0.397768	−	0.917486i	$-0.369785\pi$
$827$	22.8777	0.795537	0.397768	+	0.917486i	$0.369785\pi$
$828$	6.12727	0.212937
$829$	−23.3330	−0.810388	−0.405194	−	0.914231i	$-0.632796\pi$
$829$	−23.3330	−0.810388	−0.405194	+	0.914231i	$0.632796\pi$
$830$	−0.592684	−0.0205724
$831$	−19.7544	−0.685274
$832$	−0.887514	−0.0307690
$833$	−84.8579	−2.94015
$834$	−13.7627	−0.476564
$835$	−23.4175	−0.810397
$836$	0.196436	0.00679390
$837$	−55.1646	−1.90677
$838$	−18.0354	−0.623021
$839$	−40.4777	−1.39745	−0.698723	−	0.715392i	$-0.746248\pi$
$839$	−40.4777	−1.39745	−0.698723	+	0.715392i	$0.746248\pi$
$840$	−5.52472	−0.190621
$841$	37.3566	1.28816
$842$	26.0929	0.899221
$843$	21.7345	0.748577
$844$	−1.48005	−0.0509453
$845$	12.2123	0.420117
$846$	8.74564	0.300681
$847$	−4.27001	−0.146719
$848$	1.85194	0.0635959
$849$	−23.7208	−0.814097
$850$	−7.55436	−0.259112
$851$	0.595546	0.0204151
$852$	−2.62556	−0.0899503
$853$	−21.9096	−0.750169	−0.375084	−	0.926991i	$-0.622386\pi$
$853$	−21.9096	−0.750169	−0.375084	+	0.926991i	$0.622386\pi$
$854$	17.1931	0.588337
$855$	−0.260469	−0.00890784
$856$	−16.8546	−0.576079
$857$	−15.3377	−0.523926	−0.261963	−	0.965078i	$-0.584370\pi$
$857$	−15.3377	−0.523926	−0.261963	+	0.965078i	$0.584370\pi$
$858$	−1.14830	−0.0392025
$859$	10.7800	0.367808	0.183904	−	0.982944i	$-0.441126\pi$
$859$	10.7800	0.367808	0.183904	+	0.982944i	$0.441126\pi$
$860$	1.93222	0.0658883
$861$	25.7628	0.877992
$862$	19.8376	0.675673
$863$	−38.1247	−1.29778	−0.648890	−	0.760882i	$-0.724766\pi$
$863$	−38.1247	−1.29778	−0.648890	+	0.760882i	$0.724766\pi$
$864$	5.59713	0.190418
$865$	12.5369	0.426266
$866$	13.7770	0.468161
$867$	−51.8422	−1.76065
$868$	42.0847	1.42845
$869$	−12.7324	−0.431916
$870$	−10.5396	−0.357326
$871$	7.73937	0.262239
$872$	6.04853	0.204829
$873$	6.27689	0.212441
$874$	0.907729	0.0307044
$875$	4.27001	0.144353
$876$	1.29384	0.0437149
$877$	−0.457848	−0.0154604	−0.00773021	−	0.999970i	$-0.502461\pi$
$877$	−0.457848	−0.0154604	−0.00773021	+	0.999970i	$0.502461\pi$
$878$	34.8097	1.17477
$879$	0.924187	0.0311720
$880$	1.00000	0.0337100
$881$	−7.22137	−0.243294	−0.121647	−	0.992573i	$-0.538818\pi$
$881$	−7.22137	−0.243294	−0.121647	+	0.992573i	$0.538818\pi$
$882$	−14.8946	−0.501527
$883$	−10.3902	−0.349659	−0.174829	−	0.984599i	$-0.555937\pi$
$883$	−10.3902	−0.349659	−0.174829	+	0.984599i	$0.555937\pi$
$884$	6.70460	0.225500
$885$	10.6926	0.359426
$886$	−22.3474	−0.750774
$887$	4.95575	0.166398	0.0831988	−	0.996533i	$-0.473486\pi$
$887$	4.95575	0.166398	0.0831988	+	0.996533i	$0.473486\pi$
$888$	0.166749	0.00559573
$889$	−5.57079	−0.186838
$890$	15.4886	0.519178
$891$	3.26390	0.109345
$892$	−19.8739	−0.665427
$893$	1.29563	0.0433565
$894$	14.3896	0.481262
$895$	14.2507	0.476349
$896$	−4.27001	−0.142651
$897$	−5.30629	−0.177172
$898$	−17.3283	−0.578253
$899$	80.2856	2.67767
$900$	−1.32597	−0.0441990
$901$	−13.9902	−0.466082
$902$	−4.66318	−0.155267
$903$	−10.6750	−0.355242
$904$	13.0397	0.433693
$905$	−17.3914	−0.578110
$906$	26.3664	0.875966
$907$	6.25681	0.207754	0.103877	−	0.994590i	$-0.466875\pi$
$907$	6.25681	0.207754	0.103877	+	0.994590i	$0.466875\pi$
$908$	−8.49117	−0.281790
$909$	18.4596	0.612265
$910$	−3.78969	−0.125627
$911$	24.7621	0.820406	0.410203	−	0.911994i	$-0.365458\pi$
$911$	24.7621	0.820406	0.410203	+	0.911994i	$0.365458\pi$
$912$	0.254158	0.00841601
$913$	−0.592684	−0.0196150
$914$	−16.5825	−0.548501
$915$	−5.20965	−0.172226
$916$	−3.78347	−0.125009
$917$	−66.7868	−2.20549
$918$	−42.2827	−1.39554
$919$	−3.66404	−0.120866	−0.0604328	−	0.998172i	$-0.519248\pi$
$919$	−3.66404	−0.120866	−0.0604328	+	0.998172i	$0.519248\pi$
$920$	4.62098	0.152349
$921$	25.4285	0.837898
$922$	8.77633	0.289033
$923$	−1.80101	−0.0592809
$924$	−5.52472	−0.181750
$925$	−0.128879	−0.00423751
$926$	11.7645	0.386604
$927$	−9.26483	−0.304297
$928$	−8.14596	−0.267404
$929$	−37.9217	−1.24417	−0.622086	−	0.782949i	$-0.713714\pi$
$929$	−37.9217	−1.24417	−0.622086	+	0.782949i	$0.713714\pi$
$930$	−12.7520	−0.418154
$931$	−2.20657	−0.0723173
$932$	−19.1862	−0.628467
$933$	0.981490	0.0321326
$934$	−37.0970	−1.21385
$935$	−7.55436	−0.247054
$936$	1.17682	0.0384654
$937$	37.8810	1.23752	0.618759	−	0.785581i	$-0.287636\pi$
$937$	37.8810	1.23752	0.618759	+	0.785581i	$0.287636\pi$
$938$	37.2357	1.21579
$939$	33.5550	1.09503
$940$	6.59566	0.215127
$941$	11.0895	0.361508	0.180754	−	0.983528i	$-0.442146\pi$
$941$	11.0895	0.361508	0.180754	+	0.983528i	$0.442146\pi$
$942$	−10.8245	−0.352682
$943$	−21.5484	−0.701713
$944$	8.26418	0.268976
$945$	23.8998	0.777460
$946$	1.93222	0.0628220
$947$	−7.40904	−0.240761	−0.120381	−	0.992728i	$-0.538412\pi$
$947$	−7.40904	−0.240761	−0.120381	+	0.992728i	$0.538412\pi$
$948$	−16.4737	−0.535041
$949$	0.887514	0.0288099
$950$	−0.196436	−0.00637324
$951$	40.6198	1.31719
$952$	32.2572	1.04546
$953$	24.9119	0.806975	0.403488	−	0.914985i	$-0.367798\pi$
$953$	24.9119	0.806975	0.403488	+	0.914985i	$0.367798\pi$
$954$	−2.45561	−0.0795034
$955$	−8.35812	−0.270462
$956$	25.8422	0.835798
$957$	−10.5396	−0.340697
$958$	11.6255	0.375602
$959$	83.0323	2.68125
$960$	1.29384	0.0417586
$961$	66.1384	2.13350
$962$	0.114382	0.00368782
$963$	22.3487	0.720177
$964$	−1.39400	−0.0448978
$965$	−15.3099	−0.492842
$966$	−25.5296	−0.821402
$967$	−33.5183	−1.07788	−0.538938	−	0.842345i	$-0.681174\pi$
$967$	−33.5183	−1.07788	−0.538938	+	0.842345i	$0.681174\pi$
$968$	1.00000	0.0321412
$969$	−1.92000	−0.0616793
$970$	4.73382	0.151994
$971$	−18.5654	−0.595792	−0.297896	−	0.954598i	$-0.596285\pi$
$971$	−18.5654	−0.595792	−0.297896	+	0.954598i	$0.596285\pi$
$972$	−12.5684	−0.403132
$973$	−45.4205	−1.45611
$974$	13.5372	0.433759
$975$	1.14830	0.0367752
$976$	−4.02649	−0.128885
$977$	−14.6630	−0.469112	−0.234556	−	0.972103i	$-0.575364\pi$
$977$	−14.6630	−0.469112	−0.234556	+	0.972103i	$0.575364\pi$
$978$	23.7062	0.758042
$979$	15.4886	0.495017
$980$	−11.2330	−0.358824
$981$	−8.02016	−0.256064
$982$	−23.6807	−0.755681
$983$	18.6775	0.595720	0.297860	−	0.954610i	$-0.403727\pi$
$983$	18.6775	0.595720	0.297860	+	0.954610i	$0.403727\pi$
$984$	−6.03342	−0.192338
$985$	1.74352	0.0555531
$986$	61.5375	1.95975
$987$	−36.4392	−1.15987
$988$	0.174340	0.00554650
$989$	8.92877	0.283918
$990$	−1.32597	−0.0421420
$991$	9.09972	0.289062	0.144531	−	0.989500i	$-0.453833\pi$
$991$	9.09972	0.289062	0.144531	+	0.989500i	$0.453833\pi$
$992$	−9.85588	−0.312925
$993$	−20.2838	−0.643686
$994$	−8.66501	−0.274838
$995$	−21.9807	−0.696833
$996$	−0.766841	−0.0242983
$997$	−50.7994	−1.60883	−0.804416	−	0.594066i	$-0.797522\pi$
$997$	−50.7994	−1.60883	−0.804416	+	0.594066i	$0.797522\pi$
$998$	18.8542	0.596819
$999$	−0.721351	−0.0228225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8030.2.a.bg.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8030.2.a.bg.1.4	✓	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.29384 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.29384 q^{6} -4.27001 q^{7} +1.00000 q^{8} -1.32597 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.29384 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.29384 q^{6} -4.27001 q^{7} +1.00000 q^{8} -1.32597 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.29384 q^{12} -0.887514 q^{13} -4.27001 q^{14} +1.29384 q^{15} +1.00000 q^{16} -7.55436 q^{17} -1.32597 q^{18} -0.196436 q^{19} -1.00000 q^{20} +5.52472 q^{21} -1.00000 q^{22} -4.62098 q^{23} -1.29384 q^{24} +1.00000 q^{25} -0.887514 q^{26} +5.59713 q^{27} -4.27001 q^{28} -8.14596 q^{29} +1.29384 q^{30} -9.85588 q^{31} +1.00000 q^{32} +1.29384 q^{33} -7.55436 q^{34} +4.27001 q^{35} -1.32597 q^{36} -0.128879 q^{37} -0.196436 q^{38} +1.14830 q^{39} -1.00000 q^{40} +4.66318 q^{41} +5.52472 q^{42} -1.93222 q^{43} -1.00000 q^{44} +1.32597 q^{45} -4.62098 q^{46} -6.59566 q^{47} -1.29384 q^{48} +11.2330 q^{49} +1.00000 q^{50} +9.77416 q^{51} -0.887514 q^{52} +1.85194 q^{53} +5.59713 q^{54} +1.00000 q^{55} -4.27001 q^{56} +0.254158 q^{57} -8.14596 q^{58} +8.26418 q^{59} +1.29384 q^{60} -4.02649 q^{61} -9.85588 q^{62} +5.66190 q^{63} +1.00000 q^{64} +0.887514 q^{65} +1.29384 q^{66} -8.72028 q^{67} -7.55436 q^{68} +5.97882 q^{69} +4.27001 q^{70} +2.02927 q^{71} -1.32597 q^{72} -1.00000 q^{73} -0.128879 q^{74} -1.29384 q^{75} -0.196436 q^{76} +4.27001 q^{77} +1.14830 q^{78} +12.7324 q^{79} -1.00000 q^{80} -3.26390 q^{81} +4.66318 q^{82} +0.592684 q^{83} +5.52472 q^{84} +7.55436 q^{85} -1.93222 q^{86} +10.5396 q^{87} -1.00000 q^{88} -15.4886 q^{89} +1.32597 q^{90} +3.78969 q^{91} -4.62098 q^{92} +12.7520 q^{93} -6.59566 q^{94} +0.196436 q^{95} -1.29384 q^{96} -4.73382 q^{97} +11.2330 q^{98} +1.32597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 + 7 * q^3 + 15 * q^4 - 15 * q^5 + 7 * q^6 + 3 * q^7 + 15 * q^8 + 16 * q^9	\( 15 q + 15 q^{2} + 7 q^{3} + 15 q^{4} - 15 q^{5} + 7 q^{6} + 3 q^{7} + 15 q^{8} + 16 q^{9} - 15 q^{10} - 15 q^{11} + 7 q^{12} - q^{13} + 3 q^{14} - 7 q^{15} + 15 q^{16} + 2 q^{17} + 16 q^{18} + 23 q^{19} - 15 q^{20} + 20 q^{21} - 15 q^{22} + 7 q^{24} + 15 q^{25} - q^{26} + 19 q^{27} + 3 q^{28} + 23 q^{29} - 7 q^{30} + 9 q^{31} + 15 q^{32} - 7 q^{33} + 2 q^{34} - 3 q^{35} + 16 q^{36} + 11 q^{37} + 23 q^{38} + 7 q^{39} - 15 q^{40} + 27 q^{41} + 20 q^{42} + 7 q^{43} - 15 q^{44} - 16 q^{45} - 18 q^{47} + 7 q^{48} + 16 q^{49} + 15 q^{50} + 21 q^{51} - q^{52} - 19 q^{53} + 19 q^{54} + 15 q^{55} + 3 q^{56} + 11 q^{57} + 23 q^{58} + 2 q^{59} - 7 q^{60} + 31 q^{61} + 9 q^{62} + 20 q^{63} + 15 q^{64} + q^{65} - 7 q^{66} + 49 q^{67} + 2 q^{68} + 33 q^{69} - 3 q^{70} + 32 q^{71} + 16 q^{72} - 15 q^{73} + 11 q^{74} + 7 q^{75} + 23 q^{76} - 3 q^{77} + 7 q^{78} + 36 q^{79} - 15 q^{80} + 23 q^{81} + 27 q^{82} + 33 q^{83} + 20 q^{84} - 2 q^{85} + 7 q^{86} + 29 q^{87} - 15 q^{88} + 6 q^{89} - 16 q^{90} + 33 q^{91} + 20 q^{93} - 18 q^{94} - 23 q^{95} + 7 q^{96} + 30 q^{97} + 16 q^{98} - 16 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 + 7 * q^3 + 15 * q^4 - 15 * q^5 + 7 * q^6 + 3 * q^7 + 15 * q^8 + 16 * q^9 - 15 * q^10 - 15 * q^11 + 7 * q^12 - q^13 + 3 * q^14 - 7 * q^15 + 15 * q^16 + 2 * q^17 + 16 * q^18 + 23 * q^19 - 15 * q^20 + 20 * q^21 - 15 * q^22 + 7 * q^24 + 15 * q^25 - q^26 + 19 * q^27 + 3 * q^28 + 23 * q^29 - 7 * q^30 + 9 * q^31 + 15 * q^32 - 7 * q^33 + 2 * q^34 - 3 * q^35 + 16 * q^36 + 11 * q^37 + 23 * q^38 + 7 * q^39 - 15 * q^40 + 27 * q^41 + 20 * q^42 + 7 * q^43 - 15 * q^44 - 16 * q^45 - 18 * q^47 + 7 * q^48 + 16 * q^49 + 15 * q^50 + 21 * q^51 - q^52 - 19 * q^53 + 19 * q^54 + 15 * q^55 + 3 * q^56 + 11 * q^57 + 23 * q^58 + 2 * q^59 - 7 * q^60 + 31 * q^61 + 9 * q^62 + 20 * q^63 + 15 * q^64 + q^65 - 7 * q^66 + 49 * q^67 + 2 * q^68 + 33 * q^69 - 3 * q^70 + 32 * q^71 + 16 * q^72 - 15 * q^73 + 11 * q^74 + 7 * q^75 + 23 * q^76 - 3 * q^77 + 7 * q^78 + 36 * q^79 - 15 * q^80 + 23 * q^81 + 27 * q^82 + 33 * q^83 + 20 * q^84 - 2 * q^85 + 7 * q^86 + 29 * q^87 - 15 * q^88 + 6 * q^89 - 16 * q^90 + 33 * q^91 + 20 * q^93 - 18 * q^94 - 23 * q^95 + 7 * q^96 + 30 * q^97 + 16 * q^98 - 16 * q^99

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