LMFDB - Embedded newform 8034.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 12x^{5} + 43x^{4} - 38x^{3} - 49x^{2} + 23x + 20 $ x^8 - x^7 - 12x^6 + 12x^5 + 43x^4 - 38x^3 - 49x^2 + 23x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.685988$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.08319 q^{5} +1.00000 q^{6} +1.60046 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.08319 q^{5} +1.00000 q^{6} +1.60046 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.08319 q^{10} +3.20602 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.60046 q^{14} -3.08319 q^{15} +1.00000 q^{16} -6.03219 q^{17} +1.00000 q^{18} +0.515751 q^{19} -3.08319 q^{20} +1.60046 q^{21} +3.20602 q^{22} -9.32744 q^{23} +1.00000 q^{24} +4.50604 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.60046 q^{28} -0.930227 q^{29} -3.08319 q^{30} -9.06510 q^{31} +1.00000 q^{32} +3.20602 q^{33} -6.03219 q^{34} -4.93452 q^{35} +1.00000 q^{36} -4.24261 q^{37} +0.515751 q^{38} +1.00000 q^{39} -3.08319 q^{40} -9.43642 q^{41} +1.60046 q^{42} +1.93529 q^{43} +3.20602 q^{44} -3.08319 q^{45} -9.32744 q^{46} +11.4692 q^{47} +1.00000 q^{48} -4.43852 q^{49} +4.50604 q^{50} -6.03219 q^{51} +1.00000 q^{52} +8.31988 q^{53} +1.00000 q^{54} -9.88476 q^{55} +1.60046 q^{56} +0.515751 q^{57} -0.930227 q^{58} +2.57458 q^{59} -3.08319 q^{60} -4.66778 q^{61} -9.06510 q^{62} +1.60046 q^{63} +1.00000 q^{64} -3.08319 q^{65} +3.20602 q^{66} +3.54555 q^{67} -6.03219 q^{68} -9.32744 q^{69} -4.93452 q^{70} -9.68568 q^{71} +1.00000 q^{72} +5.52290 q^{73} -4.24261 q^{74} +4.50604 q^{75} +0.515751 q^{76} +5.13111 q^{77} +1.00000 q^{78} +1.31123 q^{79} -3.08319 q^{80} +1.00000 q^{81} -9.43642 q^{82} -8.81992 q^{83} +1.60046 q^{84} +18.5984 q^{85} +1.93529 q^{86} -0.930227 q^{87} +3.20602 q^{88} -14.4996 q^{89} -3.08319 q^{90} +1.60046 q^{91} -9.32744 q^{92} -9.06510 q^{93} +11.4692 q^{94} -1.59016 q^{95} +1.00000 q^{96} +0.597695 q^{97} -4.43852 q^{98} +3.20602 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - 7 q^{11} + 8 q^{12} + 8 q^{13} - 6 q^{14} - 8 q^{15} + 8 q^{16} - 20 q^{17} + 8 q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21} - 7 q^{22} - 14 q^{23} + 8 q^{24} - 2 q^{25} + 8 q^{26} + 8 q^{27} - 6 q^{28} - 25 q^{29} - 8 q^{30} - 12 q^{31} + 8 q^{32} - 7 q^{33} - 20 q^{34} - 18 q^{35} + 8 q^{36} - 15 q^{37} - 12 q^{38} + 8 q^{39} - 8 q^{40} - 18 q^{41} - 6 q^{42} - 8 q^{43} - 7 q^{44} - 8 q^{45} - 14 q^{46} - 12 q^{47} + 8 q^{48} - 8 q^{49} - 2 q^{50} - 20 q^{51} + 8 q^{52} - 25 q^{53} + 8 q^{54} - 8 q^{55} - 6 q^{56} - 12 q^{57} - 25 q^{58} - 9 q^{59} - 8 q^{60} - 2 q^{61} - 12 q^{62} - 6 q^{63} + 8 q^{64} - 8 q^{65} - 7 q^{66} - 8 q^{67} - 20 q^{68} - 14 q^{69} - 18 q^{70} - 13 q^{71} + 8 q^{72} - 2 q^{73} - 15 q^{74} - 2 q^{75} - 12 q^{76} - 5 q^{77} + 8 q^{78} + q^{79} - 8 q^{80} + 8 q^{81} - 18 q^{82} - 6 q^{83} - 6 q^{84} + 5 q^{85} - 8 q^{86} - 25 q^{87} - 7 q^{88} - 17 q^{89} - 8 q^{90} - 6 q^{91} - 14 q^{92} - 12 q^{93} - 12 q^{94} + 10 q^{95} + 8 q^{96} + 19 q^{97} - 8 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - 7 * q^11 + 8 * q^12 + 8 * q^13 - 6 * q^14 - 8 * q^15 + 8 * q^16 - 20 * q^17 + 8 * q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 - 7 * q^22 - 14 * q^23 + 8 * q^24 - 2 * q^25 + 8 * q^26 + 8 * q^27 - 6 * q^28 - 25 * q^29 - 8 * q^30 - 12 * q^31 + 8 * q^32 - 7 * q^33 - 20 * q^34 - 18 * q^35 + 8 * q^36 - 15 * q^37 - 12 * q^38 + 8 * q^39 - 8 * q^40 - 18 * q^41 - 6 * q^42 - 8 * q^43 - 7 * q^44 - 8 * q^45 - 14 * q^46 - 12 * q^47 + 8 * q^48 - 8 * q^49 - 2 * q^50 - 20 * q^51 + 8 * q^52 - 25 * q^53 + 8 * q^54 - 8 * q^55 - 6 * q^56 - 12 * q^57 - 25 * q^58 - 9 * q^59 - 8 * q^60 - 2 * q^61 - 12 * q^62 - 6 * q^63 + 8 * q^64 - 8 * q^65 - 7 * q^66 - 8 * q^67 - 20 * q^68 - 14 * q^69 - 18 * q^70 - 13 * q^71 + 8 * q^72 - 2 * q^73 - 15 * q^74 - 2 * q^75 - 12 * q^76 - 5 * q^77 + 8 * q^78 + q^79 - 8 * q^80 + 8 * q^81 - 18 * q^82 - 6 * q^83 - 6 * q^84 + 5 * q^85 - 8 * q^86 - 25 * q^87 - 7 * q^88 - 17 * q^89 - 8 * q^90 - 6 * q^91 - 14 * q^92 - 12 * q^93 - 12 * q^94 + 10 * q^95 + 8 * q^96 + 19 * q^97 - 8 * q^98 - 7 * 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
$5$	−3.08319	−1.37884	−0.689422	−	0.724360i	$-0.742135\pi$
$5$	−3.08319	−1.37884	−0.689422	+	0.724360i	$0.742135\pi$
$6$	1.00000	0.408248
$7$	1.60046	0.604917	0.302459	−	0.953162i	$-0.402193\pi$
$7$	1.60046	0.604917	0.302459	+	0.953162i	$0.402193\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.08319	−0.974989
$11$	3.20602	0.966652	0.483326	−	0.875440i	$-0.339429\pi$
$11$	3.20602	0.966652	0.483326	+	0.875440i	$0.339429\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.60046	0.427741
$15$	−3.08319	−0.796075
$16$	1.00000	0.250000
$17$	−6.03219	−1.46302	−0.731511	−	0.681830i	$-0.761185\pi$
$17$	−6.03219	−1.46302	−0.731511	+	0.681830i	$0.761185\pi$
$18$	1.00000	0.235702
$19$	0.515751	0.118321	0.0591607	−	0.998248i	$-0.481158\pi$
$19$	0.515751	0.118321	0.0591607	+	0.998248i	$0.481158\pi$
$20$	−3.08319	−0.689422
$21$	1.60046	0.349249
$22$	3.20602	0.683526
$23$	−9.32744	−1.94491	−0.972453	−	0.233098i	$-0.925114\pi$
$23$	−9.32744	−1.94491	−0.972453	+	0.233098i	$0.925114\pi$
$24$	1.00000	0.204124
$25$	4.50604	0.901208
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	1.60046	0.302459
$29$	−0.930227	−0.172739	−0.0863694	−	0.996263i	$-0.527527\pi$
$29$	−0.930227	−0.172739	−0.0863694	+	0.996263i	$0.527527\pi$
$30$	−3.08319	−0.562910
$31$	−9.06510	−1.62814	−0.814070	−	0.580767i	$-0.802753\pi$
$31$	−9.06510	−1.62814	−0.814070	+	0.580767i	$0.802753\pi$
$32$	1.00000	0.176777
$33$	3.20602	0.558097
$34$	−6.03219	−1.03451
$35$	−4.93452	−0.834086
$36$	1.00000	0.166667
$37$	−4.24261	−0.697481	−0.348740	−	0.937219i	$-0.613390\pi$
$37$	−4.24261	−0.697481	−0.348740	+	0.937219i	$0.613390\pi$
$38$	0.515751	0.0836659
$39$	1.00000	0.160128
$40$	−3.08319	−0.487495
$41$	−9.43642	−1.47372	−0.736861	−	0.676044i	$-0.763693\pi$
$41$	−9.43642	−1.47372	−0.736861	+	0.676044i	$0.763693\pi$
$42$	1.60046	0.246957
$43$	1.93529	0.295129	0.147565	−	0.989052i	$-0.452857\pi$
$43$	1.93529	0.295129	0.147565	+	0.989052i	$0.452857\pi$
$44$	3.20602	0.483326
$45$	−3.08319	−0.459614
$46$	−9.32744	−1.37526
$47$	11.4692	1.67296	0.836480	−	0.547998i	$-0.184610\pi$
$47$	11.4692	1.67296	0.836480	+	0.547998i	$0.184610\pi$
$48$	1.00000	0.144338
$49$	−4.43852	−0.634075
$50$	4.50604	0.637250
$51$	−6.03219	−0.844676
$52$	1.00000	0.138675
$53$	8.31988	1.14282	0.571412	−	0.820664i	$-0.306396\pi$
$53$	8.31988	1.14282	0.571412	+	0.820664i	$0.306396\pi$
$54$	1.00000	0.136083
$55$	−9.88476	−1.33286
$56$	1.60046	0.213871
$57$	0.515751	0.0683129
$58$	−0.930227	−0.122145
$59$	2.57458	0.335182	0.167591	−	0.985857i	$-0.446401\pi$
$59$	2.57458	0.335182	0.167591	+	0.985857i	$0.446401\pi$
$60$	−3.08319	−0.398038
$61$	−4.66778	−0.597648	−0.298824	−	0.954308i	$-0.596594\pi$
$61$	−4.66778	−0.597648	−0.298824	+	0.954308i	$0.596594\pi$
$62$	−9.06510	−1.15127
$63$	1.60046	0.201639
$64$	1.00000	0.125000
$65$	−3.08319	−0.382422
$66$	3.20602	0.394634
$67$	3.54555	0.433158	0.216579	−	0.976265i	$-0.430510\pi$
$67$	3.54555	0.433158	0.216579	+	0.976265i	$0.430510\pi$
$68$	−6.03219	−0.731511
$69$	−9.32744	−1.12289
$70$	−4.93452	−0.589788
$71$	−9.68568	−1.14948	−0.574739	−	0.818337i	$-0.694897\pi$
$71$	−9.68568	−1.14948	−0.574739	+	0.818337i	$0.694897\pi$
$72$	1.00000	0.117851
$73$	5.52290	0.646406	0.323203	−	0.946330i	$-0.395240\pi$
$73$	5.52290	0.646406	0.323203	+	0.946330i	$0.395240\pi$
$74$	−4.24261	−0.493193
$75$	4.50604	0.520313
$76$	0.515751	0.0591607
$77$	5.13111	0.584745
$78$	1.00000	0.113228
$79$	1.31123	0.147525	0.0737627	−	0.997276i	$-0.476499\pi$
$79$	1.31123	0.147525	0.0737627	+	0.997276i	$0.476499\pi$
$80$	−3.08319	−0.344711
$81$	1.00000	0.111111
$82$	−9.43642	−1.04208
$83$	−8.81992	−0.968112	−0.484056	−	0.875037i	$-0.660837\pi$
$83$	−8.81992	−0.968112	−0.484056	+	0.875037i	$0.660837\pi$
$84$	1.60046	0.174625
$85$	18.5984	2.01728
$86$	1.93529	0.208688
$87$	−0.930227	−0.0997308
$88$	3.20602	0.341763
$89$	−14.4996	−1.53696	−0.768479	−	0.639875i	$-0.778986\pi$
$89$	−14.4996	−1.53696	−0.768479	+	0.639875i	$0.778986\pi$
$90$	−3.08319	−0.324996
$91$	1.60046	0.167774
$92$	−9.32744	−0.972453
$93$	−9.06510	−0.940007
$94$	11.4692	1.18296
$95$	−1.59016	−0.163147
$96$	1.00000	0.102062
$97$	0.597695	0.0606867	0.0303434	−	0.999540i	$-0.490340\pi$
$97$	0.597695	0.0606867	0.0303434	+	0.999540i	$0.490340\pi$
$98$	−4.43852	−0.448359
$99$	3.20602	0.322217
$100$	4.50604	0.450604
$101$	12.3470	1.22857	0.614285	−	0.789084i	$-0.289445\pi$
$101$	12.3470	1.22857	0.614285	+	0.789084i	$0.289445\pi$
$102$	−6.03219	−0.597276
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	−4.93452	−0.481560
$106$	8.31988	0.808098
$107$	−3.33782	−0.322680	−0.161340	−	0.986899i	$-0.551582\pi$
$107$	−3.33782	−0.322680	−0.161340	+	0.986899i	$0.551582\pi$
$108$	1.00000	0.0962250
$109$	−2.87285	−0.275169	−0.137585	−	0.990490i	$-0.543934\pi$
$109$	−2.87285	−0.275169	−0.137585	+	0.990490i	$0.543934\pi$
$110$	−9.88476	−0.942475
$111$	−4.24261	−0.402691
$112$	1.60046	0.151229
$113$	−20.0433	−1.88552	−0.942759	−	0.333476i	$-0.891778\pi$
$113$	−20.0433	−1.88552	−0.942759	+	0.333476i	$0.891778\pi$
$114$	0.515751	0.0483045
$115$	28.7583	2.68172
$116$	−0.930227	−0.0863694
$117$	1.00000	0.0924500
$118$	2.57458	0.237009
$119$	−9.65429	−0.885008
$120$	−3.08319	−0.281455
$121$	−0.721424	−0.0655840
$122$	−4.66778	−0.422601
$123$	−9.43642	−0.850854
$124$	−9.06510	−0.814070
$125$	1.52297	0.136218
$126$	1.60046	0.142580
$127$	7.17390	0.636581	0.318291	−	0.947993i	$-0.396891\pi$
$127$	7.17390	0.636581	0.318291	+	0.947993i	$0.396891\pi$
$128$	1.00000	0.0883883
$129$	1.93529	0.170393
$130$	−3.08319	−0.270413
$131$	1.31153	0.114588	0.0572942	−	0.998357i	$-0.481753\pi$
$131$	1.31153	0.114588	0.0572942	+	0.998357i	$0.481753\pi$
$132$	3.20602	0.279048
$133$	0.825439	0.0715747
$134$	3.54555	0.306289
$135$	−3.08319	−0.265358
$136$	−6.03219	−0.517256
$137$	−14.0487	−1.20026	−0.600130	−	0.799903i	$-0.704884\pi$
$137$	−14.0487	−1.20026	−0.600130	+	0.799903i	$0.704884\pi$
$138$	−9.32744	−0.794005
$139$	11.1725	0.947639	0.473819	−	0.880622i	$-0.342875\pi$
$139$	11.1725	0.947639	0.473819	+	0.880622i	$0.342875\pi$
$140$	−4.93452	−0.417043
$141$	11.4692	0.965884
$142$	−9.68568	−0.812804
$143$	3.20602	0.268101
$144$	1.00000	0.0833333
$145$	2.86806	0.238180
$146$	5.52290	0.457078
$147$	−4.43852	−0.366083
$148$	−4.24261	−0.348740
$149$	−1.25825	−0.103080	−0.0515399	−	0.998671i	$-0.516413\pi$
$149$	−1.25825	−0.103080	−0.0515399	+	0.998671i	$0.516413\pi$
$150$	4.50604	0.367917
$151$	−11.0981	−0.903151	−0.451575	−	0.892233i	$-0.649138\pi$
$151$	−11.0981	−0.903151	−0.451575	+	0.892233i	$0.649138\pi$
$152$	0.515751	0.0418329
$153$	−6.03219	−0.487674
$154$	5.13111	0.413477
$155$	27.9494	2.24495
$156$	1.00000	0.0800641
$157$	13.3491	1.06537	0.532687	−	0.846312i	$-0.321182\pi$
$157$	13.3491	1.06537	0.532687	+	0.846312i	$0.321182\pi$
$158$	1.31123	0.104316
$159$	8.31988	0.659809
$160$	−3.08319	−0.243747
$161$	−14.9282	−1.17651
$162$	1.00000	0.0785674
$163$	−4.80985	−0.376737	−0.188368	−	0.982098i	$-0.560320\pi$
$163$	−4.80985	−0.376737	−0.188368	+	0.982098i	$0.560320\pi$
$164$	−9.43642	−0.736861
$165$	−9.88476	−0.769528
$166$	−8.81992	−0.684559
$167$	−9.67519	−0.748689	−0.374345	−	0.927290i	$-0.622132\pi$
$167$	−9.67519	−0.748689	−0.374345	+	0.927290i	$0.622132\pi$
$168$	1.60046	0.123478
$169$	1.00000	0.0769231
$170$	18.5984	1.42643
$171$	0.515751	0.0394405
$172$	1.93529	0.147565
$173$	0.590239	0.0448750	0.0224375	−	0.999748i	$-0.492857\pi$
$173$	0.590239	0.0448750	0.0224375	+	0.999748i	$0.492857\pi$
$174$	−0.930227	−0.0705203
$175$	7.21174	0.545157
$176$	3.20602	0.241663
$177$	2.57458	0.193517
$178$	−14.4996	−1.08679
$179$	−17.8976	−1.33773	−0.668863	−	0.743386i	$-0.733219\pi$
$179$	−17.8976	−1.33773	−0.668863	+	0.743386i	$0.733219\pi$
$180$	−3.08319	−0.229807
$181$	25.7869	1.91673	0.958364	−	0.285551i	$-0.0921766\pi$
$181$	25.7869	1.91673	0.958364	+	0.285551i	$0.0921766\pi$
$182$	1.60046	0.118634
$183$	−4.66778	−0.345052
$184$	−9.32744	−0.687628
$185$	13.0808	0.961716
$186$	−9.06510	−0.664685
$187$	−19.3393	−1.41423
$188$	11.4692	0.836480
$189$	1.60046	0.116416
$190$	−1.59016	−0.115362
$191$	−6.33469	−0.458362	−0.229181	−	0.973384i	$-0.573605\pi$
$191$	−6.33469	−0.458362	−0.229181	+	0.973384i	$0.573605\pi$
$192$	1.00000	0.0721688
$193$	−24.2056	−1.74236	−0.871180	−	0.490964i	$-0.836645\pi$
$193$	−24.2056	−1.74236	−0.871180	+	0.490964i	$0.836645\pi$
$194$	0.597695	0.0429120
$195$	−3.08319	−0.220792
$196$	−4.43852	−0.317037
$197$	−10.9897	−0.782986	−0.391493	−	0.920181i	$-0.628041\pi$
$197$	−10.9897	−0.782986	−0.391493	+	0.920181i	$0.628041\pi$
$198$	3.20602	0.227842
$199$	−3.50716	−0.248616	−0.124308	−	0.992244i	$-0.539671\pi$
$199$	−3.50716	−0.248616	−0.124308	+	0.992244i	$0.539671\pi$
$200$	4.50604	0.318625
$201$	3.54555	0.250084
$202$	12.3470	0.868730
$203$	−1.48879	−0.104493
$204$	−6.03219	−0.422338
$205$	29.0943	2.03203
$206$	1.00000	0.0696733
$207$	−9.32744	−0.648302
$208$	1.00000	0.0693375
$209$	1.65351	0.114376
$210$	−4.93452	−0.340514
$211$	11.4572	0.788744	0.394372	−	0.918951i	$-0.370962\pi$
$211$	11.4572	0.788744	0.394372	+	0.918951i	$0.370962\pi$
$212$	8.31988	0.571412
$213$	−9.68568	−0.663652
$214$	−3.33782	−0.228169
$215$	−5.96686	−0.406937
$216$	1.00000	0.0680414
$217$	−14.5083	−0.984890
$218$	−2.87285	−0.194574
$219$	5.52290	0.373203
$220$	−9.88476	−0.666431
$221$	−6.03219	−0.405769
$222$	−4.24261	−0.284745
$223$	−10.4999	−0.703123	−0.351561	−	0.936165i	$-0.614349\pi$
$223$	−10.4999	−0.703123	−0.351561	+	0.936165i	$0.614349\pi$
$224$	1.60046	0.106935
$225$	4.50604	0.300403
$226$	−20.0433	−1.33326
$227$	19.1504	1.27106	0.635528	−	0.772078i	$-0.280782\pi$
$227$	19.1504	1.27106	0.635528	+	0.772078i	$0.280782\pi$
$228$	0.515751	0.0341564
$229$	21.6465	1.43044	0.715221	−	0.698898i	$-0.246326\pi$
$229$	21.6465	1.43044	0.715221	+	0.698898i	$0.246326\pi$
$230$	28.7583	1.89626
$231$	5.13111	0.337602
$232$	−0.930227	−0.0610724
$233$	0.925493	0.0606311	0.0303155	−	0.999540i	$-0.490349\pi$
$233$	0.925493	0.0606311	0.0303155	+	0.999540i	$0.490349\pi$
$234$	1.00000	0.0653720
$235$	−35.3618	−2.30675
$236$	2.57458	0.167591
$237$	1.31123	0.0851738
$238$	−9.65429	−0.625795
$239$	−0.913802	−0.0591089	−0.0295545	−	0.999563i	$-0.509409\pi$
$239$	−0.913802	−0.0591089	−0.0295545	+	0.999563i	$0.509409\pi$
$240$	−3.08319	−0.199019
$241$	−5.68811	−0.366403	−0.183202	−	0.983075i	$-0.558646\pi$
$241$	−5.68811	−0.366403	−0.183202	+	0.983075i	$0.558646\pi$
$242$	−0.721424	−0.0463749
$243$	1.00000	0.0641500
$244$	−4.66778	−0.298824
$245$	13.6848	0.874290
$246$	−9.43642	−0.601644
$247$	0.515751	0.0328165
$248$	−9.06510	−0.575634
$249$	−8.81992	−0.558940
$250$	1.52297	0.0963209
$251$	1.90315	0.120126	0.0600630	−	0.998195i	$-0.480870\pi$
$251$	1.90315	0.120126	0.0600630	+	0.998195i	$0.480870\pi$
$252$	1.60046	0.100820
$253$	−29.9040	−1.88005
$254$	7.17390	0.450131
$255$	18.5984	1.16468
$256$	1.00000	0.0625000
$257$	5.53215	0.345086	0.172543	−	0.985002i	$-0.444802\pi$
$257$	5.53215	0.345086	0.172543	+	0.985002i	$0.444802\pi$
$258$	1.93529	0.120486
$259$	−6.79013	−0.421918
$260$	−3.08319	−0.191211
$261$	−0.930227	−0.0575796
$262$	1.31153	0.0810263
$263$	−12.0045	−0.740228	−0.370114	−	0.928986i	$-0.620681\pi$
$263$	−12.0045	−0.740228	−0.370114	+	0.928986i	$0.620681\pi$
$264$	3.20602	0.197317
$265$	−25.6517	−1.57577
$266$	0.825439	0.0506109
$267$	−14.4996	−0.887363
$268$	3.54555	0.216579
$269$	−30.3177	−1.84851	−0.924253	−	0.381781i	$-0.875311\pi$
$269$	−30.3177	−1.84851	−0.924253	+	0.381781i	$0.875311\pi$
$270$	−3.08319	−0.187637
$271$	14.3143	0.869534	0.434767	−	0.900543i	$-0.356831\pi$
$271$	14.3143	0.869534	0.434767	+	0.900543i	$0.356831\pi$
$272$	−6.03219	−0.365755
$273$	1.60046	0.0968643
$274$	−14.0487	−0.848711
$275$	14.4465	0.871155
$276$	−9.32744	−0.561446
$277$	−19.5947	−1.17733	−0.588666	−	0.808377i	$-0.700347\pi$
$277$	−19.5947	−1.17733	−0.588666	+	0.808377i	$0.700347\pi$
$278$	11.1725	0.670082
$279$	−9.06510	−0.542713
$280$	−4.93452	−0.294894
$281$	22.5367	1.34443	0.672213	−	0.740358i	$-0.265344\pi$
$281$	22.5367	1.34443	0.672213	+	0.740358i	$0.265344\pi$
$282$	11.4692	0.682983
$283$	−20.8304	−1.23824	−0.619119	−	0.785297i	$-0.712510\pi$
$283$	−20.8304	−1.23824	−0.619119	+	0.785297i	$0.712510\pi$
$284$	−9.68568	−0.574739
$285$	−1.59016	−0.0941928
$286$	3.20602	0.189576
$287$	−15.1026	−0.891480
$288$	1.00000	0.0589256
$289$	19.3874	1.14043
$290$	2.86806	0.168419
$291$	0.597695	0.0350375
$292$	5.52290	0.323203
$293$	5.77150	0.337174	0.168587	−	0.985687i	$-0.446080\pi$
$293$	5.77150	0.337174	0.168587	+	0.985687i	$0.446080\pi$
$294$	−4.43852	−0.258860
$295$	−7.93792	−0.462163
$296$	−4.24261	−0.246597
$297$	3.20602	0.186032
$298$	−1.25825	−0.0728885
$299$	−9.32744	−0.539420
$300$	4.50604	0.260156
$301$	3.09736	0.178529
$302$	−11.0981	−0.638624
$303$	12.3470	0.709315
$304$	0.515751	0.0295803
$305$	14.3916	0.824063
$306$	−6.03219	−0.344838
$307$	−7.53337	−0.429952	−0.214976	−	0.976619i	$-0.568967\pi$
$307$	−7.53337	−0.429952	−0.214976	+	0.976619i	$0.568967\pi$
$308$	5.13111	0.292372
$309$	1.00000	0.0568880
$310$	27.9494	1.58742
$311$	24.0190	1.36200	0.680998	−	0.732286i	$-0.261546\pi$
$311$	24.0190	1.36200	0.680998	+	0.732286i	$0.261546\pi$
$312$	1.00000	0.0566139
$313$	28.9557	1.63667	0.818336	−	0.574741i	$-0.194897\pi$
$313$	28.9557	1.63667	0.818336	+	0.574741i	$0.194897\pi$
$314$	13.3491	0.753333
$315$	−4.93452	−0.278029
$316$	1.31123	0.0737627
$317$	10.0682	0.565487	0.282743	−	0.959196i	$-0.408756\pi$
$317$	10.0682	0.565487	0.282743	+	0.959196i	$0.408756\pi$
$318$	8.31988	0.466556
$319$	−2.98233	−0.166978
$320$	−3.08319	−0.172355
$321$	−3.33782	−0.186299
$322$	−14.9282	−0.831917
$323$	−3.11111	−0.173107
$324$	1.00000	0.0555556
$325$	4.50604	0.249950
$326$	−4.80985	−0.266393
$327$	−2.87285	−0.158869
$328$	−9.43642	−0.521039
$329$	18.3561	1.01200
$330$	−9.88476	−0.544138
$331$	9.89622	0.543945	0.271973	−	0.962305i	$-0.412324\pi$
$331$	9.89622	0.543945	0.271973	+	0.962305i	$0.412324\pi$
$332$	−8.81992	−0.484056
$333$	−4.24261	−0.232494
$334$	−9.67519	−0.529403
$335$	−10.9316	−0.597258
$336$	1.60046	0.0873123
$337$	13.3853	0.729143	0.364571	−	0.931175i	$-0.381216\pi$
$337$	13.3853	0.729143	0.364571	+	0.931175i	$0.381216\pi$
$338$	1.00000	0.0543928
$339$	−20.0433	−1.08860
$340$	18.5984	1.00864
$341$	−29.0629	−1.57384
$342$	0.515751	0.0278886
$343$	−18.3069	−0.988480
$344$	1.93529	0.104344
$345$	28.7583	1.54829
$346$	0.590239	0.0317314
$347$	−35.1884	−1.88901	−0.944507	−	0.328493i	$-0.893459\pi$
$347$	−35.1884	−1.88901	−0.944507	+	0.328493i	$0.893459\pi$
$348$	−0.930227	−0.0498654
$349$	35.1552	1.88182	0.940908	−	0.338661i	$-0.109974\pi$
$349$	35.1552	1.88182	0.940908	+	0.338661i	$0.109974\pi$
$350$	7.21174	0.385484
$351$	1.00000	0.0533761
$352$	3.20602	0.170882
$353$	−2.16211	−0.115078	−0.0575388	−	0.998343i	$-0.518325\pi$
$353$	−2.16211	−0.115078	−0.0575388	+	0.998343i	$0.518325\pi$
$354$	2.57458	0.136837
$355$	29.8628	1.58495
$356$	−14.4996	−0.768479
$357$	−9.65429	−0.510959
$358$	−17.8976	−0.945915
$359$	−3.76597	−0.198760	−0.0993801	−	0.995050i	$-0.531686\pi$
$359$	−3.76597	−0.198760	−0.0993801	+	0.995050i	$0.531686\pi$
$360$	−3.08319	−0.162498
$361$	−18.7340	−0.986000
$362$	25.7869	1.35533
$363$	−0.721424	−0.0378650
$364$	1.60046	0.0838870
$365$	−17.0281	−0.891293
$366$	−4.66778	−0.243989
$367$	−33.2216	−1.73415	−0.867077	−	0.498174i	$-0.834004\pi$
$367$	−33.2216	−1.73415	−0.867077	+	0.498174i	$0.834004\pi$
$368$	−9.32744	−0.486227
$369$	−9.43642	−0.491241
$370$	13.0808	0.680036
$371$	13.3156	0.691314
$372$	−9.06510	−0.470003
$373$	10.1443	0.525250	0.262625	−	0.964898i	$-0.415412\pi$
$373$	10.1443	0.525250	0.262625	+	0.964898i	$0.415412\pi$
$374$	−19.3393	−1.00001
$375$	1.52297	0.0786457
$376$	11.4692	0.591481
$377$	−0.930227	−0.0479091
$378$	1.60046	0.0823188
$379$	−9.24349	−0.474806	−0.237403	−	0.971411i	$-0.576296\pi$
$379$	−9.24349	−0.474806	−0.237403	+	0.971411i	$0.576296\pi$
$380$	−1.59016	−0.0815733
$381$	7.17390	0.367530
$382$	−6.33469	−0.324111
$383$	4.09951	0.209475	0.104738	−	0.994500i	$-0.466600\pi$
$383$	4.09951	0.209475	0.104738	+	0.994500i	$0.466600\pi$
$384$	1.00000	0.0510310
$385$	−15.8202	−0.806271
$386$	−24.2056	−1.23203
$387$	1.93529	0.0983764
$388$	0.597695	0.0303434
$389$	−26.5798	−1.34765	−0.673824	−	0.738892i	$-0.735349\pi$
$389$	−26.5798	−1.34765	−0.673824	+	0.738892i	$0.735349\pi$
$390$	−3.08319	−0.156123
$391$	56.2649	2.84544
$392$	−4.43852	−0.224179
$393$	1.31153	0.0661577
$394$	−10.9897	−0.553655
$395$	−4.04278	−0.203414
$396$	3.20602	0.161109
$397$	24.0973	1.20941	0.604705	−	0.796450i	$-0.293291\pi$
$397$	24.0973	1.20941	0.604705	+	0.796450i	$0.293291\pi$
$398$	−3.50716	−0.175798
$399$	0.825439	0.0413237
$400$	4.50604	0.225302
$401$	−16.6925	−0.833586	−0.416793	−	0.909001i	$-0.636846\pi$
$401$	−16.6925	−0.833586	−0.416793	+	0.909001i	$0.636846\pi$
$402$	3.54555	0.176836
$403$	−9.06510	−0.451565
$404$	12.3470	0.614285
$405$	−3.08319	−0.153205
$406$	−1.48879	−0.0738875
$407$	−13.6019	−0.674221
$408$	−6.03219	−0.298638
$409$	−0.799726	−0.0395439	−0.0197719	−	0.999805i	$-0.506294\pi$
$409$	−0.799726	−0.0395439	−0.0197719	+	0.999805i	$0.506294\pi$
$410$	29.0943	1.43686
$411$	−14.0487	−0.692970
$412$	1.00000	0.0492665
$413$	4.12052	0.202757
$414$	−9.32744	−0.458419
$415$	27.1935	1.33487
$416$	1.00000	0.0490290
$417$	11.1725	0.547119
$418$	1.65351	0.0808758
$419$	−1.41353	−0.0690557	−0.0345278	−	0.999404i	$-0.510993\pi$
$419$	−1.41353	−0.0690557	−0.0345278	+	0.999404i	$0.510993\pi$
$420$	−4.93452	−0.240780
$421$	−11.1500	−0.543417	−0.271709	−	0.962380i	$-0.587589\pi$
$421$	−11.1500	−0.543417	−0.271709	+	0.962380i	$0.587589\pi$
$422$	11.4572	0.557726
$423$	11.4692	0.557653
$424$	8.31988	0.404049
$425$	−27.1813	−1.31849
$426$	−9.68568	−0.469273
$427$	−7.47060	−0.361528
$428$	−3.33782	−0.161340
$429$	3.20602	0.154788
$430$	−5.96686	−0.287748
$431$	−9.02345	−0.434644	−0.217322	−	0.976100i	$-0.569732\pi$
$431$	−9.02345	−0.434644	−0.217322	+	0.976100i	$0.569732\pi$
$432$	1.00000	0.0481125
$433$	−24.7664	−1.19020	−0.595098	−	0.803653i	$-0.702887\pi$
$433$	−24.7664	−1.19020	−0.595098	+	0.803653i	$0.702887\pi$
$434$	−14.5083	−0.696422
$435$	2.86806	0.137513
$436$	−2.87285	−0.137585
$437$	−4.81064	−0.230124
$438$	5.52290	0.263894
$439$	−7.63904	−0.364591	−0.182296	−	0.983244i	$-0.558353\pi$
$439$	−7.63904	−0.364591	−0.182296	+	0.983244i	$0.558353\pi$
$440$	−9.88476	−0.471238
$441$	−4.43852	−0.211358
$442$	−6.03219	−0.286922
$443$	−25.4015	−1.20686	−0.603430	−	0.797416i	$-0.706200\pi$
$443$	−25.4015	−1.20686	−0.603430	+	0.797416i	$0.706200\pi$
$444$	−4.24261	−0.201345
$445$	44.7051	2.11922
$446$	−10.4999	−0.497183
$447$	−1.25825	−0.0595132
$448$	1.60046	0.0756147
$449$	−25.1607	−1.18741	−0.593703	−	0.804684i	$-0.702335\pi$
$449$	−25.1607	−1.18741	−0.593703	+	0.804684i	$0.702335\pi$
$450$	4.50604	0.212417
$451$	−30.2534	−1.42458
$452$	−20.0433	−0.942759
$453$	−11.0981	−0.521434
$454$	19.1504	0.898772
$455$	−4.93452	−0.231334
$456$	0.515751	0.0241523
$457$	−13.3907	−0.626392	−0.313196	−	0.949688i	$-0.601400\pi$
$457$	−13.3907	−0.626392	−0.313196	+	0.949688i	$0.601400\pi$
$458$	21.6465	1.01148
$459$	−6.03219	−0.281559
$460$	28.7583	1.34086
$461$	12.2323	0.569716	0.284858	−	0.958570i	$-0.408054\pi$
$461$	12.2323	0.569716	0.284858	+	0.958570i	$0.408054\pi$
$462$	5.13111	0.238721
$463$	−8.13507	−0.378069	−0.189034	−	0.981970i	$-0.560536\pi$
$463$	−8.13507	−0.378069	−0.189034	+	0.981970i	$0.560536\pi$
$464$	−0.930227	−0.0431847
$465$	27.9494	1.29612
$466$	0.925493	0.0428727
$467$	−10.6999	−0.495131	−0.247566	−	0.968871i	$-0.579631\pi$
$467$	−10.6999	−0.495131	−0.247566	+	0.968871i	$0.579631\pi$
$468$	1.00000	0.0462250
$469$	5.67452	0.262025
$470$	−35.3618	−1.63112
$471$	13.3491	0.615094
$472$	2.57458	0.118505
$473$	6.20459	0.285287
$474$	1.31123	0.0602270
$475$	2.32400	0.106632
$476$	−9.65429	−0.442504
$477$	8.31988	0.380941
$478$	−0.913802	−0.0417963
$479$	19.8964	0.909090	0.454545	−	0.890724i	$-0.349802\pi$
$479$	19.8964	0.909090	0.454545	+	0.890724i	$0.349802\pi$
$480$	−3.08319	−0.140728
$481$	−4.24261	−0.193446
$482$	−5.68811	−0.259086
$483$	−14.9282	−0.679257
$484$	−0.721424	−0.0327920
$485$	−1.84280	−0.0836775
$486$	1.00000	0.0453609
$487$	−23.7705	−1.07714	−0.538572	−	0.842580i	$-0.681036\pi$
$487$	−23.7705	−1.07714	−0.538572	+	0.842580i	$0.681036\pi$
$488$	−4.66778	−0.211300
$489$	−4.80985	−0.217509
$490$	13.6848	0.618216
$491$	−32.8194	−1.48112	−0.740559	−	0.671992i	$-0.765439\pi$
$491$	−32.8194	−1.48112	−0.740559	+	0.671992i	$0.765439\pi$
$492$	−9.43642	−0.425427
$493$	5.61131	0.252721
$494$	0.515751	0.0232047
$495$	−9.88476	−0.444287
$496$	−9.06510	−0.407035
$497$	−15.5016	−0.695340
$498$	−8.81992	−0.395230
$499$	−30.7484	−1.37649	−0.688244	−	0.725479i	$-0.741618\pi$
$499$	−30.7484	−1.37649	−0.688244	+	0.725479i	$0.741618\pi$
$500$	1.52297	0.0681092
$501$	−9.67519	−0.432256
$502$	1.90315	0.0849419
$503$	14.8130	0.660481	0.330240	−	0.943897i	$-0.392870\pi$
$503$	14.8130	0.660481	0.330240	+	0.943897i	$0.392870\pi$
$504$	1.60046	0.0712902
$505$	−38.0680	−1.69401
$506$	−29.9040	−1.32939
$507$	1.00000	0.0444116
$508$	7.17390	0.318291
$509$	22.9367	1.01665	0.508326	−	0.861165i	$-0.330265\pi$
$509$	22.9367	1.01665	0.508326	+	0.861165i	$0.330265\pi$
$510$	18.5984	0.823550
$511$	8.83919	0.391023
$512$	1.00000	0.0441942
$513$	0.515751	0.0227710
$514$	5.53215	0.244013
$515$	−3.08319	−0.135861
$516$	1.93529	0.0851964
$517$	36.7706	1.61717
$518$	−6.79013	−0.298341
$519$	0.590239	0.0259086
$520$	−3.08319	−0.135207
$521$	2.40904	0.105542	0.0527710	−	0.998607i	$-0.483195\pi$
$521$	2.40904	0.105542	0.0527710	+	0.998607i	$0.483195\pi$
$522$	−0.930227	−0.0407149
$523$	−31.2379	−1.36594	−0.682968	−	0.730448i	$-0.739311\pi$
$523$	−31.2379	−1.36594	−0.682968	+	0.730448i	$0.739311\pi$
$524$	1.31153	0.0572942
$525$	7.21174	0.314746
$526$	−12.0045	−0.523421
$527$	54.6824	2.38200
$528$	3.20602	0.139524
$529$	64.0012	2.78266
$530$	−25.6517	−1.11424
$531$	2.57458	0.111727
$532$	0.825439	0.0357873
$533$	−9.43642	−0.408737
$534$	−14.4996	−0.627461
$535$	10.2911	0.444925
$536$	3.54555	0.153145
$537$	−17.8976	−0.772337
$538$	−30.3177	−1.30709
$539$	−14.2300	−0.612930
$540$	−3.08319	−0.132679
$541$	29.9326	1.28690	0.643451	−	0.765487i	$-0.277502\pi$
$541$	29.9326	1.28690	0.643451	+	0.765487i	$0.277502\pi$
$542$	14.3143	0.614853
$543$	25.7869	1.10662
$544$	−6.03219	−0.258628
$545$	8.85753	0.379415
$546$	1.60046	0.0684934
$547$	13.7157	0.586439	0.293219	−	0.956045i	$-0.405273\pi$
$547$	13.7157	0.586439	0.293219	+	0.956045i	$0.405273\pi$
$548$	−14.0487	−0.600130
$549$	−4.66778	−0.199216
$550$	14.4465	0.615999
$551$	−0.479766	−0.0204387
$552$	−9.32744	−0.397002
$553$	2.09858	0.0892407
$554$	−19.5947	−0.832499
$555$	13.0808	0.555247
$556$	11.1725	0.473819
$557$	−44.7898	−1.89780	−0.948902	−	0.315572i	$-0.897804\pi$
$557$	−44.7898	−1.89780	−0.948902	+	0.315572i	$0.897804\pi$
$558$	−9.06510	−0.383756
$559$	1.93529	0.0818541
$560$	−4.93452	−0.208522
$561$	−19.3393	−0.816508
$562$	22.5367	0.950652
$563$	4.69173	0.197733	0.0988663	−	0.995101i	$-0.468478\pi$
$563$	4.69173	0.197733	0.0988663	+	0.995101i	$0.468478\pi$
$564$	11.4692	0.482942
$565$	61.7973	2.59983
$566$	−20.8304	−0.875566
$567$	1.60046	0.0672131
$568$	−9.68568	−0.406402
$569$	12.5194	0.524839	0.262419	−	0.964954i	$-0.415480\pi$
$569$	12.5194	0.524839	0.262419	+	0.964954i	$0.415480\pi$
$570$	−1.59016	−0.0666043
$571$	−44.3078	−1.85422	−0.927111	−	0.374786i	$-0.877716\pi$
$571$	−44.3078	−1.85422	−0.927111	+	0.374786i	$0.877716\pi$
$572$	3.20602	0.134051
$573$	−6.33469	−0.264636
$574$	−15.1026	−0.630372
$575$	−42.0298	−1.75277
$576$	1.00000	0.0416667
$577$	−6.49804	−0.270517	−0.135259	−	0.990810i	$-0.543187\pi$
$577$	−6.49804	−0.270517	−0.135259	+	0.990810i	$0.543187\pi$
$578$	19.3874	0.806408
$579$	−24.2056	−1.00595
$580$	2.86806	0.119090
$581$	−14.1159	−0.585628
$582$	0.597695	0.0247752
$583$	26.6737	1.10471
$584$	5.52290	0.228539
$585$	−3.08319	−0.127474
$586$	5.77150	0.238418
$587$	29.6712	1.22466	0.612330	−	0.790602i	$-0.290232\pi$
$587$	29.6712	1.22466	0.612330	+	0.790602i	$0.290232\pi$
$588$	−4.43852	−0.183042
$589$	−4.67533	−0.192644
$590$	−7.93792	−0.326799
$591$	−10.9897	−0.452057
$592$	−4.24261	−0.174370
$593$	3.61788	0.148569	0.0742843	−	0.997237i	$-0.476333\pi$
$593$	3.61788	0.148569	0.0742843	+	0.997237i	$0.476333\pi$
$594$	3.20602	0.131545
$595$	29.7660	1.22029
$596$	−1.25825	−0.0515399
$597$	−3.50716	−0.143539
$598$	−9.32744	−0.381428
$599$	15.0809	0.616188	0.308094	−	0.951356i	$-0.400309\pi$
$599$	15.0809	0.616188	0.308094	+	0.951356i	$0.400309\pi$
$600$	4.50604	0.183958
$601$	20.1548	0.822133	0.411067	−	0.911605i	$-0.365156\pi$
$601$	20.1548	0.822133	0.411067	+	0.911605i	$0.365156\pi$
$602$	3.09736	0.126239
$603$	3.54555	0.144386
$604$	−11.0981	−0.451575
$605$	2.22429	0.0904301
$606$	12.3470	0.501562
$607$	13.5756	0.551015	0.275508	−	0.961299i	$-0.411154\pi$
$607$	13.5756	0.551015	0.275508	+	0.961299i	$0.411154\pi$
$608$	0.515751	0.0209165
$609$	−1.48879	−0.0603289
$610$	14.3916	0.582700
$611$	11.4692	0.463996
$612$	−6.03219	−0.243837
$613$	11.7673	0.475279	0.237639	−	0.971353i	$-0.423626\pi$
$613$	11.7673	0.475279	0.237639	+	0.971353i	$0.423626\pi$
$614$	−7.53337	−0.304022
$615$	29.0943	1.17319
$616$	5.13111	0.206738
$617$	11.7577	0.473349	0.236674	−	0.971589i	$-0.423943\pi$
$617$	11.7577	0.473349	0.236674	+	0.971589i	$0.423943\pi$
$618$	1.00000	0.0402259
$619$	25.1290	1.01002	0.505011	−	0.863113i	$-0.331489\pi$
$619$	25.1290	1.01002	0.505011	+	0.863113i	$0.331489\pi$
$620$	27.9494	1.12247
$621$	−9.32744	−0.374297
$622$	24.0190	0.963076
$623$	−23.2061	−0.929733
$624$	1.00000	0.0400320
$625$	−27.2258	−1.08903
$626$	28.9557	1.15730
$627$	1.65351	0.0660348
$628$	13.3491	0.532687
$629$	25.5922	1.02043
$630$	−4.93452	−0.196596
$631$	19.5902	0.779873	0.389937	−	0.920842i	$-0.372497\pi$
$631$	19.5902	0.779873	0.389937	+	0.920842i	$0.372497\pi$
$632$	1.31123	0.0521581
$633$	11.4572	0.455381
$634$	10.0682	0.399859
$635$	−22.1185	−0.877745
$636$	8.31988	0.329905
$637$	−4.43852	−0.175861
$638$	−2.98233	−0.118072
$639$	−9.68568	−0.383160
$640$	−3.08319	−0.121874
$641$	−15.8901	−0.627622	−0.313811	−	0.949485i	$-0.601606\pi$
$641$	−15.8901	−0.627622	−0.313811	+	0.949485i	$0.601606\pi$
$642$	−3.33782	−0.131733
$643$	30.3379	1.19641	0.598205	−	0.801343i	$-0.295881\pi$
$643$	30.3379	1.19641	0.598205	+	0.801343i	$0.295881\pi$
$644$	−14.9282	−0.588254
$645$	−5.96686	−0.234945
$646$	−3.11111	−0.122405
$647$	17.8039	0.699942	0.349971	−	0.936761i	$-0.386191\pi$
$647$	17.8039	0.699942	0.349971	+	0.936761i	$0.386191\pi$
$648$	1.00000	0.0392837
$649$	8.25417	0.324004
$650$	4.50604	0.176741
$651$	−14.5083	−0.568626
$652$	−4.80985	−0.188368
$653$	16.0993	0.630016	0.315008	−	0.949089i	$-0.397993\pi$
$653$	16.0993	0.630016	0.315008	+	0.949089i	$0.397993\pi$
$654$	−2.87285	−0.112337
$655$	−4.04368	−0.158000
$656$	−9.43642	−0.368430
$657$	5.52290	0.215469
$658$	18.3561	0.715594
$659$	31.5994	1.23094	0.615469	−	0.788161i	$-0.288967\pi$
$659$	31.5994	1.23094	0.615469	+	0.788161i	$0.288967\pi$
$660$	−9.88476	−0.384764
$661$	31.5181	1.22591	0.612956	−	0.790117i	$-0.289980\pi$
$661$	31.5181	1.22591	0.612956	+	0.790117i	$0.289980\pi$
$662$	9.89622	0.384627
$663$	−6.03219	−0.234271
$664$	−8.81992	−0.342279
$665$	−2.54498	−0.0986903
$666$	−4.24261	−0.164398
$667$	8.67664	0.335961
$668$	−9.67519	−0.374345
$669$	−10.4999	−0.405948
$670$	−10.9316	−0.422325
$671$	−14.9650	−0.577718
$672$	1.60046	0.0617391
$673$	−7.57594	−0.292031	−0.146016	−	0.989282i	$-0.546645\pi$
$673$	−7.57594	−0.292031	−0.146016	+	0.989282i	$0.546645\pi$
$674$	13.3853	0.515582
$675$	4.50604	0.173438
$676$	1.00000	0.0384615
$677$	3.20858	0.123316	0.0616579	−	0.998097i	$-0.480361\pi$
$677$	3.20858	0.123316	0.0616579	+	0.998097i	$0.480361\pi$
$678$	−20.0433	−0.769759
$679$	0.956587	0.0367105
$680$	18.5984	0.713215
$681$	19.1504	0.733845
$682$	−29.0629	−1.11288
$683$	32.7710	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$683$	32.7710	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$684$	0.515751	0.0197202
$685$	43.3147	1.65497
$686$	−18.3069	−0.698961
$687$	21.6465	0.825867
$688$	1.93529	0.0737823
$689$	8.31988	0.316962
$690$	28.7583	1.09481
$691$	−36.6829	−1.39548	−0.697741	−	0.716350i	$-0.745811\pi$
$691$	−36.6829	−1.39548	−0.697741	+	0.716350i	$0.745811\pi$
$692$	0.590239	0.0224375
$693$	5.13111	0.194915
$694$	−35.1884	−1.33573
$695$	−34.4469	−1.30665
$696$	−0.930227	−0.0352602
$697$	56.9223	2.15609
$698$	35.1552	1.33065
$699$	0.925493	0.0350054
$700$	7.21174	0.272578
$701$	−11.2283	−0.424087	−0.212043	−	0.977260i	$-0.568012\pi$
$701$	−11.2283	−0.424087	−0.212043	+	0.977260i	$0.568012\pi$
$702$	1.00000	0.0377426
$703$	−2.18813	−0.0825269
$704$	3.20602	0.120831
$705$	−35.3618	−1.33180
$706$	−2.16211	−0.0813721
$707$	19.7609	0.743183
$708$	2.57458	0.0967587
$709$	−22.9972	−0.863679	−0.431839	−	0.901951i	$-0.642135\pi$
$709$	−22.9972	−0.863679	−0.431839	+	0.901951i	$0.642135\pi$
$710$	29.8628	1.12073
$711$	1.31123	0.0491751
$712$	−14.4996	−0.543397
$713$	84.5542	3.16658
$714$	−9.65429	−0.361303
$715$	−9.88476	−0.369669
$716$	−17.8976	−0.668863
$717$	−0.913802	−0.0341266
$718$	−3.76597	−0.140545
$719$	−30.5472	−1.13922	−0.569608	−	0.821916i	$-0.692905\pi$
$719$	−30.5472	−1.13922	−0.569608	+	0.821916i	$0.692905\pi$
$720$	−3.08319	−0.114904
$721$	1.60046	0.0596043
$722$	−18.7340	−0.697207
$723$	−5.68811	−0.211543
$724$	25.7869	0.958364
$725$	−4.19164	−0.155674
$726$	−0.721424	−0.0267746
$727$	−25.8305	−0.957999	−0.478999	−	0.877815i	$-0.659000\pi$
$727$	−25.8305	−0.957999	−0.478999	+	0.877815i	$0.659000\pi$
$728$	1.60046	0.0593170
$729$	1.00000	0.0370370
$730$	−17.0281	−0.630239
$731$	−11.6741	−0.431780
$732$	−4.66778	−0.172526
$733$	52.7701	1.94911	0.974555	−	0.224150i	$-0.0719606\pi$
$733$	52.7701	1.94911	0.974555	+	0.224150i	$0.0719606\pi$
$734$	−33.2216	−1.22623
$735$	13.6848	0.504771
$736$	−9.32744	−0.343814
$737$	11.3671	0.418713
$738$	−9.43642	−0.347360
$739$	20.7452	0.763126	0.381563	−	0.924343i	$-0.375386\pi$
$739$	20.7452	0.763126	0.381563	+	0.924343i	$0.375386\pi$
$740$	13.0808	0.480858
$741$	0.515751	0.0189466
$742$	13.3156	0.488833
$743$	−16.7055	−0.612865	−0.306433	−	0.951892i	$-0.599135\pi$
$743$	−16.7055	−0.612865	−0.306433	+	0.951892i	$0.599135\pi$
$744$	−9.06510	−0.332343
$745$	3.87942	0.142131
$746$	10.1443	0.371408
$747$	−8.81992	−0.322704
$748$	−19.3393	−0.707116
$749$	−5.34206	−0.195195
$750$	1.52297	0.0556109
$751$	24.8940	0.908396	0.454198	−	0.890901i	$-0.349926\pi$
$751$	24.8940	0.908396	0.454198	+	0.890901i	$0.349926\pi$
$752$	11.4692	0.418240
$753$	1.90315	0.0693548
$754$	−0.930227	−0.0338769
$755$	34.2175	1.24530
$756$	1.60046	0.0582082
$757$	32.1356	1.16799	0.583994	−	0.811758i	$-0.301489\pi$
$757$	32.1356	1.16799	0.583994	+	0.811758i	$0.301489\pi$
$758$	−9.24349	−0.335739
$759$	−29.9040	−1.08545
$760$	−1.59016	−0.0576810
$761$	45.5297	1.65045	0.825225	−	0.564804i	$-0.191048\pi$
$761$	45.5297	1.65045	0.825225	+	0.564804i	$0.191048\pi$
$762$	7.17390	0.259883
$763$	−4.59789	−0.166455
$764$	−6.33469	−0.229181
$765$	18.5984	0.672426
$766$	4.09951	0.148121
$767$	2.57458	0.0929628
$768$	1.00000	0.0360844
$769$	50.7255	1.82921	0.914604	−	0.404350i	$-0.132502\pi$
$769$	50.7255	1.82921	0.914604	+	0.404350i	$0.132502\pi$
$770$	−15.8202	−0.570120
$771$	5.53215	0.199235
$772$	−24.2056	−0.871180
$773$	43.5438	1.56616	0.783081	−	0.621920i	$-0.213647\pi$
$773$	43.5438	1.56616	0.783081	+	0.621920i	$0.213647\pi$
$774$	1.93529	0.0695626
$775$	−40.8477	−1.46729
$776$	0.597695	0.0214560
$777$	−6.79013	−0.243595
$778$	−26.5798	−0.952931
$779$	−4.86685	−0.174373
$780$	−3.08319	−0.110396
$781$	−31.0525	−1.11115
$782$	56.2649	2.01203
$783$	−0.930227	−0.0332436
$784$	−4.43852	−0.158519
$785$	−41.1578	−1.46898
$786$	1.31153	0.0467806
$787$	−24.7363	−0.881753	−0.440877	−	0.897568i	$-0.645332\pi$
$787$	−24.7363	−0.881753	−0.440877	+	0.897568i	$0.645332\pi$
$788$	−10.9897	−0.391493
$789$	−12.0045	−0.427371
$790$	−4.04278	−0.143836
$791$	−32.0786	−1.14058
$792$	3.20602	0.113921
$793$	−4.66778	−0.165758
$794$	24.0973	0.855182
$795$	−25.6517	−0.909774
$796$	−3.50716	−0.124308
$797$	−24.3709	−0.863263	−0.431631	−	0.902050i	$-0.642062\pi$
$797$	−24.3709	−0.863263	−0.431631	+	0.902050i	$0.642062\pi$
$798$	0.825439	0.0292202
$799$	−69.1846	−2.44758
$800$	4.50604	0.159313
$801$	−14.4996	−0.512319
$802$	−16.6925	−0.589434
$803$	17.7065	0.624850
$804$	3.54555	0.125042
$805$	46.0265	1.62222
$806$	−9.06510	−0.319304
$807$	−30.3177	−1.06724
$808$	12.3470	0.434365
$809$	48.0237	1.68842	0.844212	−	0.536009i	$-0.180069\pi$
$809$	48.0237	1.68842	0.844212	+	0.536009i	$0.180069\pi$
$810$	−3.08319	−0.108332
$811$	21.1444	0.742481	0.371240	−	0.928537i	$-0.378933\pi$
$811$	21.1444	0.742481	0.371240	+	0.928537i	$0.378933\pi$
$812$	−1.48879	−0.0522464
$813$	14.3143	0.502025
$814$	−13.6019	−0.476746
$815$	14.8297	0.519461
$816$	−6.03219	−0.211169
$817$	0.998128	0.0349201
$818$	−0.799726	−0.0279618
$819$	1.60046	0.0559246
$820$	29.0943	1.01602
$821$	32.8652	1.14700	0.573501	−	0.819205i	$-0.305585\pi$
$821$	32.8652	1.14700	0.573501	+	0.819205i	$0.305585\pi$
$822$	−14.0487	−0.490004
$823$	27.8558	0.970993	0.485496	−	0.874239i	$-0.338639\pi$
$823$	27.8558	0.970993	0.485496	+	0.874239i	$0.338639\pi$
$824$	1.00000	0.0348367
$825$	14.4465	0.502961
$826$	4.12052	0.143371
$827$	−21.1362	−0.734977	−0.367488	−	0.930028i	$-0.619782\pi$
$827$	−21.1362	−0.734977	−0.367488	+	0.930028i	$0.619782\pi$
$828$	−9.32744	−0.324151
$829$	43.7822	1.52062	0.760310	−	0.649560i	$-0.225047\pi$
$829$	43.7822	1.52062	0.760310	+	0.649560i	$0.225047\pi$
$830$	27.1935	0.943899
$831$	−19.5947	−0.679733
$832$	1.00000	0.0346688
$833$	26.7740	0.927665
$834$	11.1725	0.386872
$835$	29.8304	1.03232
$836$	1.65351	0.0571878
$837$	−9.06510	−0.313336
$838$	−1.41353	−0.0488297
$839$	−14.7168	−0.508081	−0.254041	−	0.967194i	$-0.581760\pi$
$839$	−14.7168	−0.508081	−0.254041	+	0.967194i	$0.581760\pi$
$840$	−4.93452	−0.170257
$841$	−28.1347	−0.970161
$842$	−11.1500	−0.384254
$843$	22.5367	0.776204
$844$	11.4572	0.394372
$845$	−3.08319	−0.106065
$846$	11.4692	0.394320
$847$	−1.15461	−0.0396729
$848$	8.31988	0.285706
$849$	−20.8304	−0.714897
$850$	−27.1813	−0.932311
$851$	39.5727	1.35653
$852$	−9.68568	−0.331826
$853$	−20.9964	−0.718901	−0.359451	−	0.933164i	$-0.617036\pi$
$853$	−20.9964	−0.718901	−0.359451	+	0.933164i	$0.617036\pi$
$854$	−7.47060	−0.255639
$855$	−1.59016	−0.0543822
$856$	−3.33782	−0.114084
$857$	36.4712	1.24583	0.622916	−	0.782288i	$-0.285948\pi$
$857$	36.4712	1.24583	0.622916	+	0.782288i	$0.285948\pi$
$858$	3.20602	0.109452
$859$	−29.1733	−0.995380	−0.497690	−	0.867355i	$-0.665818\pi$
$859$	−29.1733	−0.995380	−0.497690	+	0.867355i	$0.665818\pi$
$860$	−5.96686	−0.203468
$861$	−15.1026	−0.514696
$862$	−9.02345	−0.307340
$863$	42.0092	1.43001	0.715004	−	0.699120i	$-0.246425\pi$
$863$	42.0092	1.43001	0.715004	+	0.699120i	$0.246425\pi$
$864$	1.00000	0.0340207
$865$	−1.81982	−0.0618756
$866$	−24.7664	−0.841596
$867$	19.3874	0.658429
$868$	−14.5083	−0.492445
$869$	4.20385	0.142606
$870$	2.86806	0.0972365
$871$	3.54555	0.120137
$872$	−2.87285	−0.0972870
$873$	0.597695	0.0202289
$874$	−4.81064	−0.162722
$875$	2.43745	0.0824009
$876$	5.52290	0.186601
$877$	26.3908	0.891155	0.445577	−	0.895243i	$-0.352998\pi$
$877$	26.3908	0.891155	0.445577	+	0.895243i	$0.352998\pi$
$878$	−7.63904	−0.257805
$879$	5.77150	0.194668
$880$	−9.88476	−0.333215
$881$	−48.6394	−1.63870	−0.819352	−	0.573291i	$-0.805667\pi$
$881$	−48.6394	−1.63870	−0.819352	+	0.573291i	$0.805667\pi$
$882$	−4.43852	−0.149453
$883$	32.1207	1.08095	0.540473	−	0.841361i	$-0.318245\pi$
$883$	32.1207	1.08095	0.540473	+	0.841361i	$0.318245\pi$
$884$	−6.03219	−0.202885
$885$	−7.93792	−0.266830
$886$	−25.4015	−0.853379
$887$	−39.4412	−1.32430	−0.662152	−	0.749369i	$-0.730357\pi$
$887$	−39.4412	−1.32430	−0.662152	+	0.749369i	$0.730357\pi$
$888$	−4.24261	−0.142373
$889$	11.4816	0.385079
$890$	44.7051	1.49852
$891$	3.20602	0.107406
$892$	−10.4999	−0.351561
$893$	5.91527	0.197947
$894$	−1.25825	−0.0420822
$895$	55.1815	1.84451
$896$	1.60046	0.0534677
$897$	−9.32744	−0.311434
$898$	−25.1607	−0.839623
$899$	8.43260	0.281243
$900$	4.50604	0.150201
$901$	−50.1871	−1.67198
$902$	−30.2534	−1.00733
$903$	3.09736	0.103074
$904$	−20.0433	−0.666631
$905$	−79.5059	−2.64287
$906$	−11.0981	−0.368710
$907$	−0.292582	−0.00971502	−0.00485751	−	0.999988i	$-0.501546\pi$
$907$	−0.292582	−0.00971502	−0.00485751	+	0.999988i	$0.501546\pi$
$908$	19.1504	0.635528
$909$	12.3470	0.409523
$910$	−4.93452	−0.163578
$911$	9.35726	0.310020	0.155010	−	0.987913i	$-0.450459\pi$
$911$	9.35726	0.310020	0.155010	+	0.987913i	$0.450459\pi$
$912$	0.515751	0.0170782
$913$	−28.2769	−0.935828
$914$	−13.3907	−0.442926
$915$	14.3916	0.475773
$916$	21.6465	0.715221
$917$	2.09905	0.0693166
$918$	−6.03219	−0.199092
$919$	22.8347	0.753246	0.376623	−	0.926367i	$-0.377085\pi$
$919$	22.8347	0.753246	0.376623	+	0.926367i	$0.377085\pi$
$920$	28.7583	0.948131
$921$	−7.53337	−0.248233
$922$	12.2323	0.402850
$923$	−9.68568	−0.318808
$924$	5.13111	0.168801
$925$	−19.1174	−0.628575
$926$	−8.13507	−0.267335
$927$	1.00000	0.0328443
$928$	−0.930227	−0.0305362
$929$	−7.74533	−0.254116	−0.127058	−	0.991895i	$-0.540553\pi$
$929$	−7.74533	−0.254116	−0.127058	+	0.991895i	$0.540553\pi$
$930$	27.9494	0.916496
$931$	−2.28917	−0.0750246
$932$	0.925493	0.0303155
$933$	24.0190	0.786348
$934$	−10.6999	−0.350111
$935$	59.6268	1.95001
$936$	1.00000	0.0326860
$937$	−25.6888	−0.839217	−0.419609	−	0.907705i	$-0.637833\pi$
$937$	−25.6888	−0.839217	−0.419609	+	0.907705i	$0.637833\pi$
$938$	5.67452	0.185280
$939$	28.9557	0.944933
$940$	−35.3618	−1.15337
$941$	−38.8587	−1.26676	−0.633379	−	0.773842i	$-0.718332\pi$
$941$	−38.8587	−1.26676	−0.633379	+	0.773842i	$0.718332\pi$
$942$	13.3491	0.434937
$943$	88.0177	2.86625
$944$	2.57458	0.0837955
$945$	−4.93452	−0.160520
$946$	6.20459	0.201728
$947$	26.5540	0.862890	0.431445	−	0.902139i	$-0.358004\pi$
$947$	26.5540	0.862890	0.431445	+	0.902139i	$0.358004\pi$
$948$	1.31123	0.0425869
$949$	5.52290	0.179281
$950$	2.32400	0.0754004
$951$	10.0682	0.326484
$952$	−9.65429	−0.312897
$953$	22.2954	0.722218	0.361109	−	0.932524i	$-0.382398\pi$
$953$	22.2954	0.722218	0.361109	+	0.932524i	$0.382398\pi$
$954$	8.31988	0.269366
$955$	19.5310	0.632009
$956$	−0.913802	−0.0295545
$957$	−2.98233	−0.0964050
$958$	19.8964	0.642823
$959$	−22.4844	−0.726058
$960$	−3.08319	−0.0995094
$961$	51.1760	1.65084
$962$	−4.24261	−0.136787
$963$	−3.33782	−0.107560
$964$	−5.68811	−0.183202
$965$	74.6305	2.40244
$966$	−14.9282	−0.480307
$967$	30.6773	0.986515	0.493258	−	0.869883i	$-0.335806\pi$
$967$	30.6773	0.986515	0.493258	+	0.869883i	$0.335806\pi$
$968$	−0.721424	−0.0231875
$969$	−3.11111	−0.0999433
$970$	−1.84280	−0.0591689
$971$	−32.3639	−1.03861	−0.519303	−	0.854590i	$-0.673808\pi$
$971$	−32.3639	−1.03861	−0.519303	+	0.854590i	$0.673808\pi$
$972$	1.00000	0.0320750
$973$	17.8811	0.573243
$974$	−23.7705	−0.761656
$975$	4.50604	0.144309
$976$	−4.66778	−0.149412
$977$	41.0913	1.31463	0.657314	−	0.753617i	$-0.271693\pi$
$977$	41.0913	1.31463	0.657314	+	0.753617i	$0.271693\pi$
$978$	−4.80985	−0.153802
$979$	−46.4861	−1.48570
$980$	13.6848	0.437145
$981$	−2.87285	−0.0917231
$982$	−32.8194	−1.04731
$983$	14.5719	0.464772	0.232386	−	0.972624i	$-0.425347\pi$
$983$	14.5719	0.464772	0.232386	+	0.972624i	$0.425347\pi$
$984$	−9.43642	−0.300822
$985$	33.8834	1.07962
$986$	5.61131	0.178701
$987$	18.3561	0.584280
$988$	0.515751	0.0164082
$989$	−18.0513	−0.573999
$990$	−9.88476	−0.314158
$991$	−37.2659	−1.18379	−0.591895	−	0.806015i	$-0.701620\pi$
$991$	−37.2659	−1.18379	−0.591895	+	0.806015i	$0.701620\pi$
$992$	−9.06510	−0.287817
$993$	9.89622	0.314047
$994$	−15.5016	−0.491679
$995$	10.8132	0.342803
$996$	−8.81992	−0.279470
$997$	−12.6670	−0.401169	−0.200585	−	0.979676i	$-0.564284\pi$
$997$	−12.6670	−0.401169	−0.200585	+	0.979676i	$0.564284\pi$
$998$	−30.7484	−0.973324
$999$	−4.24261	−0.134230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.p.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.p.1.2	✓	8	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.08319 q^{5} +1.00000 q^{6} +1.60046 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.08319 q^{5} +1.00000 q^{6} +1.60046 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.08319 q^{10} +3.20602 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.60046 q^{14} -3.08319 q^{15} +1.00000 q^{16} -6.03219 q^{17} +1.00000 q^{18} +0.515751 q^{19} -3.08319 q^{20} +1.60046 q^{21} +3.20602 q^{22} -9.32744 q^{23} +1.00000 q^{24} +4.50604 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.60046 q^{28} -0.930227 q^{29} -3.08319 q^{30} -9.06510 q^{31} +1.00000 q^{32} +3.20602 q^{33} -6.03219 q^{34} -4.93452 q^{35} +1.00000 q^{36} -4.24261 q^{37} +0.515751 q^{38} +1.00000 q^{39} -3.08319 q^{40} -9.43642 q^{41} +1.60046 q^{42} +1.93529 q^{43} +3.20602 q^{44} -3.08319 q^{45} -9.32744 q^{46} +11.4692 q^{47} +1.00000 q^{48} -4.43852 q^{49} +4.50604 q^{50} -6.03219 q^{51} +1.00000 q^{52} +8.31988 q^{53} +1.00000 q^{54} -9.88476 q^{55} +1.60046 q^{56} +0.515751 q^{57} -0.930227 q^{58} +2.57458 q^{59} -3.08319 q^{60} -4.66778 q^{61} -9.06510 q^{62} +1.60046 q^{63} +1.00000 q^{64} -3.08319 q^{65} +3.20602 q^{66} +3.54555 q^{67} -6.03219 q^{68} -9.32744 q^{69} -4.93452 q^{70} -9.68568 q^{71} +1.00000 q^{72} +5.52290 q^{73} -4.24261 q^{74} +4.50604 q^{75} +0.515751 q^{76} +5.13111 q^{77} +1.00000 q^{78} +1.31123 q^{79} -3.08319 q^{80} +1.00000 q^{81} -9.43642 q^{82} -8.81992 q^{83} +1.60046 q^{84} +18.5984 q^{85} +1.93529 q^{86} -0.930227 q^{87} +3.20602 q^{88} -14.4996 q^{89} -3.08319 q^{90} +1.60046 q^{91} -9.32744 q^{92} -9.06510 q^{93} +11.4692 q^{94} -1.59016 q^{95} +1.00000 q^{96} +0.597695 q^{97} -4.43852 q^{98} +3.20602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - 7 q^{11} + 8 q^{12} + 8 q^{13} - 6 q^{14} - 8 q^{15} + 8 q^{16} - 20 q^{17} + 8 q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21} - 7 q^{22} - 14 q^{23} + 8 q^{24} - 2 q^{25} + 8 q^{26} + 8 q^{27} - 6 q^{28} - 25 q^{29} - 8 q^{30} - 12 q^{31} + 8 q^{32} - 7 q^{33} - 20 q^{34} - 18 q^{35} + 8 q^{36} - 15 q^{37} - 12 q^{38} + 8 q^{39} - 8 q^{40} - 18 q^{41} - 6 q^{42} - 8 q^{43} - 7 q^{44} - 8 q^{45} - 14 q^{46} - 12 q^{47} + 8 q^{48} - 8 q^{49} - 2 q^{50} - 20 q^{51} + 8 q^{52} - 25 q^{53} + 8 q^{54} - 8 q^{55} - 6 q^{56} - 12 q^{57} - 25 q^{58} - 9 q^{59} - 8 q^{60} - 2 q^{61} - 12 q^{62} - 6 q^{63} + 8 q^{64} - 8 q^{65} - 7 q^{66} - 8 q^{67} - 20 q^{68} - 14 q^{69} - 18 q^{70} - 13 q^{71} + 8 q^{72} - 2 q^{73} - 15 q^{74} - 2 q^{75} - 12 q^{76} - 5 q^{77} + 8 q^{78} + q^{79} - 8 q^{80} + 8 q^{81} - 18 q^{82} - 6 q^{83} - 6 q^{84} + 5 q^{85} - 8 q^{86} - 25 q^{87} - 7 q^{88} - 17 q^{89} - 8 q^{90} - 6 q^{91} - 14 q^{92} - 12 q^{93} - 12 q^{94} + 10 q^{95} + 8 q^{96} + 19 q^{97} - 8 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^5 + 8 * q^6 - 6 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - 7 * q^11 + 8 * q^12 + 8 * q^13 - 6 * q^14 - 8 * q^15 + 8 * q^16 - 20 * q^17 + 8 * q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 - 7 * q^22 - 14 * q^23 + 8 * q^24 - 2 * q^25 + 8 * q^26 + 8 * q^27 - 6 * q^28 - 25 * q^29 - 8 * q^30 - 12 * q^31 + 8 * q^32 - 7 * q^33 - 20 * q^34 - 18 * q^35 + 8 * q^36 - 15 * q^37 - 12 * q^38 + 8 * q^39 - 8 * q^40 - 18 * q^41 - 6 * q^42 - 8 * q^43 - 7 * q^44 - 8 * q^45 - 14 * q^46 - 12 * q^47 + 8 * q^48 - 8 * q^49 - 2 * q^50 - 20 * q^51 + 8 * q^52 - 25 * q^53 + 8 * q^54 - 8 * q^55 - 6 * q^56 - 12 * q^57 - 25 * q^58 - 9 * q^59 - 8 * q^60 - 2 * q^61 - 12 * q^62 - 6 * q^63 + 8 * q^64 - 8 * q^65 - 7 * q^66 - 8 * q^67 - 20 * q^68 - 14 * q^69 - 18 * q^70 - 13 * q^71 + 8 * q^72 - 2 * q^73 - 15 * q^74 - 2 * q^75 - 12 * q^76 - 5 * q^77 + 8 * q^78 + q^79 - 8 * q^80 + 8 * q^81 - 18 * q^82 - 6 * q^83 - 6 * q^84 + 5 * q^85 - 8 * q^86 - 25 * q^87 - 7 * q^88 - 17 * q^89 - 8 * q^90 - 6 * q^91 - 14 * q^92 - 12 * q^93 - 12 * q^94 + 10 * q^95 + 8 * q^96 + 19 * q^97 - 8 * q^98 - 7 * q^99

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