LMFDB - Embedded newform 8034.2.a.p.1.3

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			1.3
Root			$0.965971$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24957 q^{5} +1.00000 q^{6} +3.57309 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24957 q^{5} +1.00000 q^{6} +3.57309 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.24957 q^{10} -3.14452 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.57309 q^{14} -2.24957 q^{15} +1.00000 q^{16} -2.24225 q^{17} +1.00000 q^{18} -6.27884 q^{19} -2.24957 q^{20} +3.57309 q^{21} -3.14452 q^{22} -4.02900 q^{23} +1.00000 q^{24} +0.0605787 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.57309 q^{28} +3.37405 q^{29} -2.24957 q^{30} -4.73747 q^{31} +1.00000 q^{32} -3.14452 q^{33} -2.24225 q^{34} -8.03793 q^{35} +1.00000 q^{36} +1.30881 q^{37} -6.27884 q^{38} +1.00000 q^{39} -2.24957 q^{40} -1.29845 q^{41} +3.57309 q^{42} +2.09446 q^{43} -3.14452 q^{44} -2.24957 q^{45} -4.02900 q^{46} -9.07827 q^{47} +1.00000 q^{48} +5.76698 q^{49} +0.0605787 q^{50} -2.24225 q^{51} +1.00000 q^{52} -12.2551 q^{53} +1.00000 q^{54} +7.07382 q^{55} +3.57309 q^{56} -6.27884 q^{57} +3.37405 q^{58} -10.4567 q^{59} -2.24957 q^{60} +3.20542 q^{61} -4.73747 q^{62} +3.57309 q^{63} +1.00000 q^{64} -2.24957 q^{65} -3.14452 q^{66} -1.89011 q^{67} -2.24225 q^{68} -4.02900 q^{69} -8.03793 q^{70} +10.0652 q^{71} +1.00000 q^{72} -8.11073 q^{73} +1.30881 q^{74} +0.0605787 q^{75} -6.27884 q^{76} -11.2356 q^{77} +1.00000 q^{78} -4.60171 q^{79} -2.24957 q^{80} +1.00000 q^{81} -1.29845 q^{82} +13.9578 q^{83} +3.57309 q^{84} +5.04410 q^{85} +2.09446 q^{86} +3.37405 q^{87} -3.14452 q^{88} +6.22450 q^{89} -2.24957 q^{90} +3.57309 q^{91} -4.02900 q^{92} -4.73747 q^{93} -9.07827 q^{94} +14.1247 q^{95} +1.00000 q^{96} +0.109151 q^{97} +5.76698 q^{98} -3.14452 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.24957	−1.00604	−0.503020	−	0.864275i	$-0.667778\pi$
$5$	−2.24957	−1.00604	−0.503020	+	0.864275i	$0.667778\pi$
$6$	1.00000	0.408248
$7$	3.57309	1.35050	0.675251	−	0.737588i	$-0.264035\pi$
$7$	3.57309	1.35050	0.675251	+	0.737588i	$0.264035\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.24957	−0.711377
$11$	−3.14452	−0.948108	−0.474054	−	0.880496i	$-0.657210\pi$
$11$	−3.14452	−0.948108	−0.474054	+	0.880496i	$0.657210\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.57309	0.954949
$15$	−2.24957	−0.580837
$16$	1.00000	0.250000
$17$	−2.24225	−0.543825	−0.271912	−	0.962322i	$-0.587656\pi$
$17$	−2.24225	−0.543825	−0.271912	+	0.962322i	$0.587656\pi$
$18$	1.00000	0.235702
$19$	−6.27884	−1.44047	−0.720233	−	0.693732i	$-0.755965\pi$
$19$	−6.27884	−1.44047	−0.720233	+	0.693732i	$0.755965\pi$
$20$	−2.24957	−0.503020
$21$	3.57309	0.779712
$22$	−3.14452	−0.670414
$23$	−4.02900	−0.840106	−0.420053	−	0.907500i	$-0.637988\pi$
$23$	−4.02900	−0.840106	−0.420053	+	0.907500i	$0.637988\pi$
$24$	1.00000	0.204124
$25$	0.0605787	0.0121157
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	3.57309	0.675251
$29$	3.37405	0.626545	0.313272	−	0.949663i	$-0.398575\pi$
$29$	3.37405	0.626545	0.313272	+	0.949663i	$0.398575\pi$
$30$	−2.24957	−0.410714
$31$	−4.73747	−0.850874	−0.425437	−	0.904988i	$-0.639880\pi$
$31$	−4.73747	−0.850874	−0.425437	+	0.904988i	$0.639880\pi$
$32$	1.00000	0.176777
$33$	−3.14452	−0.547390
$34$	−2.24225	−0.384542
$35$	−8.03793	−1.35866
$36$	1.00000	0.166667
$37$	1.30881	0.215167	0.107583	−	0.994196i	$-0.465689\pi$
$37$	1.30881	0.215167	0.107583	+	0.994196i	$0.465689\pi$
$38$	−6.27884	−1.01856
$39$	1.00000	0.160128
$40$	−2.24957	−0.355689
$41$	−1.29845	−0.202784	−0.101392	−	0.994847i	$-0.532330\pi$
$41$	−1.29845	−0.202784	−0.101392	+	0.994847i	$0.532330\pi$
$42$	3.57309	0.551340
$43$	2.09446	0.319402	0.159701	−	0.987165i	$-0.448947\pi$
$43$	2.09446	0.319402	0.159701	+	0.987165i	$0.448947\pi$
$44$	−3.14452	−0.474054
$45$	−2.24957	−0.335347
$46$	−4.02900	−0.594044
$47$	−9.07827	−1.32420	−0.662101	−	0.749415i	$-0.730335\pi$
$47$	−9.07827	−1.32420	−0.662101	+	0.749415i	$0.730335\pi$
$48$	1.00000	0.144338
$49$	5.76698	0.823854
$50$	0.0605787	0.00856712
$51$	−2.24225	−0.313977
$52$	1.00000	0.138675
$53$	−12.2551	−1.68336	−0.841681	−	0.539975i	$-0.818434\pi$
$53$	−12.2551	−1.68336	−0.841681	+	0.539975i	$0.818434\pi$
$54$	1.00000	0.136083
$55$	7.07382	0.953834
$56$	3.57309	0.477474
$57$	−6.27884	−0.831653
$58$	3.37405	0.443034
$59$	−10.4567	−1.36135	−0.680675	−	0.732586i	$-0.738313\pi$
$59$	−10.4567	−1.36135	−0.680675	+	0.732586i	$0.738313\pi$
$60$	−2.24957	−0.290419
$61$	3.20542	0.410412	0.205206	−	0.978719i	$-0.434214\pi$
$61$	3.20542	0.410412	0.205206	+	0.978719i	$0.434214\pi$
$62$	−4.73747	−0.601659
$63$	3.57309	0.450167
$64$	1.00000	0.125000
$65$	−2.24957	−0.279025
$66$	−3.14452	−0.387063
$67$	−1.89011	−0.230914	−0.115457	−	0.993312i	$-0.536833\pi$
$67$	−1.89011	−0.230914	−0.115457	+	0.993312i	$0.536833\pi$
$68$	−2.24225	−0.271912
$69$	−4.02900	−0.485035
$70$	−8.03793	−0.960716
$71$	10.0652	1.19452	0.597260	−	0.802048i	$-0.296256\pi$
$71$	10.0652	1.19452	0.597260	+	0.802048i	$0.296256\pi$
$72$	1.00000	0.117851
$73$	−8.11073	−0.949289	−0.474645	−	0.880178i	$-0.657423\pi$
$73$	−8.11073	−0.949289	−0.474645	+	0.880178i	$0.657423\pi$
$74$	1.30881	0.152146
$75$	0.0605787	0.00699502
$76$	−6.27884	−0.720233
$77$	−11.2356	−1.28042
$78$	1.00000	0.113228
$79$	−4.60171	−0.517733	−0.258867	−	0.965913i	$-0.583349\pi$
$79$	−4.60171	−0.517733	−0.258867	+	0.965913i	$0.583349\pi$
$80$	−2.24957	−0.251510
$81$	1.00000	0.111111
$82$	−1.29845	−0.143390
$83$	13.9578	1.53207	0.766033	−	0.642801i	$-0.222228\pi$
$83$	13.9578	1.53207	0.766033	+	0.642801i	$0.222228\pi$
$84$	3.57309	0.389856
$85$	5.04410	0.547109
$86$	2.09446	0.225851
$87$	3.37405	0.361736
$88$	−3.14452	−0.335207
$89$	6.22450	0.659796	0.329898	−	0.944017i	$-0.392986\pi$
$89$	6.22450	0.659796	0.329898	+	0.944017i	$0.392986\pi$
$90$	−2.24957	−0.237126
$91$	3.57309	0.374562
$92$	−4.02900	−0.420053
$93$	−4.73747	−0.491252
$94$	−9.07827	−0.936352
$95$	14.1247	1.44917
$96$	1.00000	0.102062
$97$	0.109151	0.0110826	0.00554130	−	0.999985i	$-0.498236\pi$
$97$	0.109151	0.0110826	0.00554130	+	0.999985i	$0.498236\pi$
$98$	5.76698	0.582553
$99$	−3.14452	−0.316036
$100$	0.0605787	0.00605787
$101$	−15.9360	−1.58569	−0.792844	−	0.609425i	$-0.791400\pi$
$101$	−15.9360	−1.58569	−0.792844	+	0.609425i	$0.791400\pi$
$102$	−2.24225	−0.222016
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	−8.03793	−0.784422
$106$	−12.2551	−1.19032
$107$	−0.420266	−0.0406287	−0.0203143	−	0.999794i	$-0.506467\pi$
$107$	−0.420266	−0.0406287	−0.0203143	+	0.999794i	$0.506467\pi$
$108$	1.00000	0.0962250
$109$	−3.47914	−0.333242	−0.166621	−	0.986021i	$-0.553286\pi$
$109$	−3.47914	−0.333242	−0.166621	+	0.986021i	$0.553286\pi$
$110$	7.07382	0.674463
$111$	1.30881	0.124227
$112$	3.57309	0.337625
$113$	−9.36131	−0.880638	−0.440319	−	0.897841i	$-0.645135\pi$
$113$	−9.36131	−0.880638	−0.440319	+	0.897841i	$0.645135\pi$
$114$	−6.27884	−0.588068
$115$	9.06354	0.845179
$116$	3.37405	0.313272
$117$	1.00000	0.0924500
$118$	−10.4567	−0.962619
$119$	−8.01176	−0.734436
$120$	−2.24957	−0.205357
$121$	−1.11200	−0.101091
$122$	3.20542	0.290205
$123$	−1.29845	−0.117077
$124$	−4.73747	−0.425437
$125$	11.1116	0.993851
$126$	3.57309	0.318316
$127$	2.30558	0.204587	0.102294	−	0.994754i	$-0.467382\pi$
$127$	2.30558	0.204587	0.102294	+	0.994754i	$0.467382\pi$
$128$	1.00000	0.0883883
$129$	2.09446	0.184407
$130$	−2.24957	−0.197301
$131$	−7.68277	−0.671246	−0.335623	−	0.941996i	$-0.608947\pi$
$131$	−7.68277	−0.671246	−0.335623	+	0.941996i	$0.608947\pi$
$132$	−3.14452	−0.273695
$133$	−22.4349	−1.94535
$134$	−1.89011	−0.163281
$135$	−2.24957	−0.193612
$136$	−2.24225	−0.192271
$137$	5.69384	0.486458	0.243229	−	0.969969i	$-0.421793\pi$
$137$	5.69384	0.486458	0.243229	+	0.969969i	$0.421793\pi$
$138$	−4.02900	−0.342972
$139$	15.5772	1.32124	0.660621	−	0.750720i	$-0.270293\pi$
$139$	15.5772	1.32124	0.660621	+	0.750720i	$0.270293\pi$
$140$	−8.03793	−0.679329
$141$	−9.07827	−0.764528
$142$	10.0652	0.844653
$143$	−3.14452	−0.262958
$144$	1.00000	0.0833333
$145$	−7.59016	−0.630329
$146$	−8.11073	−0.671249
$147$	5.76698	0.475652
$148$	1.30881	0.107583
$149$	−4.84494	−0.396913	−0.198456	−	0.980110i	$-0.563593\pi$
$149$	−4.84494	−0.396913	−0.198456	+	0.980110i	$0.563593\pi$
$150$	0.0605787	0.00494623
$151$	−1.56151	−0.127074	−0.0635370	−	0.997979i	$-0.520238\pi$
$151$	−1.56151	−0.127074	−0.0635370	+	0.997979i	$0.520238\pi$
$152$	−6.27884	−0.509282
$153$	−2.24225	−0.181275
$154$	−11.2356	−0.905394
$155$	10.6573	0.856013
$156$	1.00000	0.0800641
$157$	−11.9321	−0.952288	−0.476144	−	0.879367i	$-0.657966\pi$
$157$	−11.9321	−0.952288	−0.476144	+	0.879367i	$0.657966\pi$
$158$	−4.60171	−0.366093
$159$	−12.2551	−0.971889
$160$	−2.24957	−0.177844
$161$	−14.3960	−1.13456
$162$	1.00000	0.0785674
$163$	9.20600	0.721069	0.360535	−	0.932746i	$-0.382594\pi$
$163$	9.20600	0.721069	0.360535	+	0.932746i	$0.382594\pi$
$164$	−1.29845	−0.101392
$165$	7.07382	0.550696
$166$	13.9578	1.08333
$167$	25.3409	1.96093	0.980467	−	0.196681i	$-0.0630165\pi$
$167$	25.3409	1.96093	0.980467	+	0.196681i	$0.0630165\pi$
$168$	3.57309	0.275670
$169$	1.00000	0.0769231
$170$	5.04410	0.386865
$171$	−6.27884	−0.480155
$172$	2.09446	0.159701
$173$	−8.35653	−0.635335	−0.317667	−	0.948202i	$-0.602900\pi$
$173$	−8.35653	−0.635335	−0.317667	+	0.948202i	$0.602900\pi$
$174$	3.37405	0.255786
$175$	0.216453	0.0163623
$176$	−3.14452	−0.237027
$177$	−10.4567	−0.785975
$178$	6.22450	0.466546
$179$	−10.3299	−0.772092	−0.386046	−	0.922480i	$-0.626159\pi$
$179$	−10.3299	−0.772092	−0.386046	+	0.922480i	$0.626159\pi$
$180$	−2.24957	−0.167673
$181$	16.8081	1.24934	0.624669	−	0.780890i	$-0.285234\pi$
$181$	16.8081	1.24934	0.624669	+	0.780890i	$0.285234\pi$
$182$	3.57309	0.264855
$183$	3.20542	0.236951
$184$	−4.02900	−0.297022
$185$	−2.94426	−0.216466
$186$	−4.73747	−0.347368
$187$	7.05079	0.515605
$188$	−9.07827	−0.662101
$189$	3.57309	0.259904
$190$	14.1247	1.02471
$191$	−13.2334	−0.957535	−0.478767	−	0.877942i	$-0.658916\pi$
$191$	−13.2334	−0.957535	−0.478767	+	0.877942i	$0.658916\pi$
$192$	1.00000	0.0721688
$193$	23.1230	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$193$	23.1230	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$194$	0.109151	0.00783658
$195$	−2.24957	−0.161095
$196$	5.76698	0.411927
$197$	−19.6407	−1.39934	−0.699671	−	0.714465i	$-0.746670\pi$
$197$	−19.6407	−1.39934	−0.699671	+	0.714465i	$0.746670\pi$
$198$	−3.14452	−0.223471
$199$	10.5365	0.746914	0.373457	−	0.927648i	$-0.378172\pi$
$199$	10.5365	0.746914	0.373457	+	0.927648i	$0.378172\pi$
$200$	0.0605787	0.00428356
$201$	−1.89011	−0.133318
$202$	−15.9360	−1.12125
$203$	12.0558	0.846149
$204$	−2.24225	−0.156989
$205$	2.92096	0.204009
$206$	1.00000	0.0696733
$207$	−4.02900	−0.280035
$208$	1.00000	0.0693375
$209$	19.7439	1.36572
$210$	−8.03793	−0.554670
$211$	−22.4883	−1.54816	−0.774081	−	0.633087i	$-0.781788\pi$
$211$	−22.4883	−1.54816	−0.774081	+	0.633087i	$0.781788\pi$
$212$	−12.2551	−0.841681
$213$	10.0652	0.689656
$214$	−0.420266	−0.0287288
$215$	−4.71163	−0.321331
$216$	1.00000	0.0680414
$217$	−16.9274	−1.14911
$218$	−3.47914	−0.235637
$219$	−8.11073	−0.548072
$220$	7.07382	0.476917
$221$	−2.24225	−0.150830
$222$	1.30881	0.0878415
$223$	8.63267	0.578086	0.289043	−	0.957316i	$-0.406663\pi$
$223$	8.63267	0.578086	0.289043	+	0.957316i	$0.406663\pi$
$224$	3.57309	0.238737
$225$	0.0605787	0.00403858
$226$	−9.36131	−0.622705
$227$	12.7730	0.847772	0.423886	−	0.905716i	$-0.360666\pi$
$227$	12.7730	0.847772	0.423886	+	0.905716i	$0.360666\pi$
$228$	−6.27884	−0.415827
$229$	−0.442618	−0.0292490	−0.0146245	−	0.999893i	$-0.504655\pi$
$229$	−0.442618	−0.0292490	−0.0146245	+	0.999893i	$0.504655\pi$
$230$	9.06354	0.597632
$231$	−11.2356	−0.739251
$232$	3.37405	0.221517
$233$	−13.0390	−0.854215	−0.427107	−	0.904201i	$-0.640467\pi$
$233$	−13.0390	−0.854215	−0.427107	+	0.904201i	$0.640467\pi$
$234$	1.00000	0.0653720
$235$	20.4222	1.33220
$236$	−10.4567	−0.680675
$237$	−4.60171	−0.298913
$238$	−8.01176	−0.519325
$239$	−27.1264	−1.75466	−0.877330	−	0.479888i	$-0.840677\pi$
$239$	−27.1264	−1.75466	−0.877330	+	0.479888i	$0.840677\pi$
$240$	−2.24957	−0.145209
$241$	21.5350	1.38719	0.693597	−	0.720363i	$-0.256025\pi$
$241$	21.5350	1.38719	0.693597	+	0.720363i	$0.256025\pi$
$242$	−1.11200	−0.0714823
$243$	1.00000	0.0641500
$244$	3.20542	0.205206
$245$	−12.9732	−0.828830
$246$	−1.29845	−0.0827862
$247$	−6.27884	−0.399513
$248$	−4.73747	−0.300829
$249$	13.9578	0.884539
$250$	11.1116	0.702759
$251$	−24.3162	−1.53482	−0.767411	−	0.641155i	$-0.778455\pi$
$251$	−24.3162	−1.53482	−0.767411	+	0.641155i	$0.778455\pi$
$252$	3.57309	0.225084
$253$	12.6693	0.796511
$254$	2.30558	0.144665
$255$	5.04410	0.315874
$256$	1.00000	0.0625000
$257$	−1.88981	−0.117883	−0.0589416	−	0.998261i	$-0.518773\pi$
$257$	−1.88981	−0.117883	−0.0589416	+	0.998261i	$0.518773\pi$
$258$	2.09446	0.130395
$259$	4.67649	0.290583
$260$	−2.24957	−0.139513
$261$	3.37405	0.208848
$262$	−7.68277	−0.474643
$263$	−8.71368	−0.537308	−0.268654	−	0.963237i	$-0.586579\pi$
$263$	−8.71368	−0.537308	−0.268654	+	0.963237i	$0.586579\pi$
$264$	−3.14452	−0.193532
$265$	27.5687	1.69353
$266$	−22.4349	−1.37557
$267$	6.22450	0.380933
$268$	−1.89011	−0.115457
$269$	−0.698163	−0.0425677	−0.0212839	−	0.999773i	$-0.506775\pi$
$269$	−0.698163	−0.0425677	−0.0212839	+	0.999773i	$0.506775\pi$
$270$	−2.24957	−0.136905
$271$	24.2795	1.47487	0.737437	−	0.675416i	$-0.236036\pi$
$271$	24.2795	1.47487	0.737437	+	0.675416i	$0.236036\pi$
$272$	−2.24225	−0.135956
$273$	3.57309	0.216253
$274$	5.69384	0.343978
$275$	−0.190491	−0.0114870
$276$	−4.02900	−0.242518
$277$	0.188794	0.0113435	0.00567175	−	0.999984i	$-0.498195\pi$
$277$	0.188794	0.0113435	0.00567175	+	0.999984i	$0.498195\pi$
$278$	15.5772	0.934259
$279$	−4.73747	−0.283625
$280$	−8.03793	−0.480358
$281$	−4.21764	−0.251603	−0.125802	−	0.992055i	$-0.540150\pi$
$281$	−4.21764	−0.251603	−0.125802	+	0.992055i	$0.540150\pi$
$282$	−9.07827	−0.540603
$283$	−5.69210	−0.338360	−0.169180	−	0.985585i	$-0.554112\pi$
$283$	−5.69210	−0.338360	−0.169180	+	0.985585i	$0.554112\pi$
$284$	10.0652	0.597260
$285$	14.1247	0.836676
$286$	−3.14452	−0.185939
$287$	−4.63948	−0.273860
$288$	1.00000	0.0589256
$289$	−11.9723	−0.704254
$290$	−7.59016	−0.445710
$291$	0.109151	0.00639854
$292$	−8.11073	−0.474645
$293$	14.2594	0.833043	0.416521	−	0.909126i	$-0.363249\pi$
$293$	14.2594	0.833043	0.416521	+	0.909126i	$0.363249\pi$
$294$	5.76698	0.336337
$295$	23.5232	1.36957
$296$	1.30881	0.0760730
$297$	−3.14452	−0.182463
$298$	−4.84494	−0.280660
$299$	−4.02900	−0.233003
$300$	0.0605787	0.00349751
$301$	7.48368	0.431352
$302$	−1.56151	−0.0898548
$303$	−15.9360	−0.915497
$304$	−6.27884	−0.360116
$305$	−7.21082	−0.412890
$306$	−2.24225	−0.128181
$307$	−1.85507	−0.105874	−0.0529372	−	0.998598i	$-0.516858\pi$
$307$	−1.85507	−0.105874	−0.0529372	+	0.998598i	$0.516858\pi$
$308$	−11.2356	−0.640211
$309$	1.00000	0.0568880
$310$	10.6573	0.605293
$311$	4.21294	0.238894	0.119447	−	0.992841i	$-0.461888\pi$
$311$	4.21294	0.238894	0.119447	+	0.992841i	$0.461888\pi$
$312$	1.00000	0.0566139
$313$	−5.79894	−0.327775	−0.163888	−	0.986479i	$-0.552403\pi$
$313$	−5.79894	−0.327775	−0.163888	+	0.986479i	$0.552403\pi$
$314$	−11.9321	−0.673370
$315$	−8.03793	−0.452886
$316$	−4.60171	−0.258867
$317$	−4.45496	−0.250215	−0.125108	−	0.992143i	$-0.539928\pi$
$317$	−4.45496	−0.250215	−0.125108	+	0.992143i	$0.539928\pi$
$318$	−12.2551	−0.687230
$319$	−10.6098	−0.594032
$320$	−2.24957	−0.125755
$321$	−0.420266	−0.0234570
$322$	−14.3960	−0.802258
$323$	14.0787	0.783361
$324$	1.00000	0.0555556
$325$	0.0605787	0.00336030
$326$	9.20600	0.509873
$327$	−3.47914	−0.192397
$328$	−1.29845	−0.0716949
$329$	−32.4375	−1.78834
$330$	7.07382	0.389401
$331$	−7.17142	−0.394177	−0.197088	−	0.980386i	$-0.563149\pi$
$331$	−7.17142	−0.394177	−0.197088	+	0.980386i	$0.563149\pi$
$332$	13.9578	0.766033
$333$	1.30881	0.0717223
$334$	25.3409	1.38659
$335$	4.25195	0.232309
$336$	3.57309	0.194928
$337$	26.9060	1.46566	0.732832	−	0.680409i	$-0.238198\pi$
$337$	26.9060	1.46566	0.732832	+	0.680409i	$0.238198\pi$
$338$	1.00000	0.0543928
$339$	−9.36131	−0.508437
$340$	5.04410	0.273555
$341$	14.8971	0.806721
$342$	−6.27884	−0.339521
$343$	−4.40570	−0.237885
$344$	2.09446	0.112926
$345$	9.06354	0.487965
$346$	−8.35653	−0.449250
$347$	−9.70451	−0.520966	−0.260483	−	0.965478i	$-0.583882\pi$
$347$	−9.70451	−0.520966	−0.260483	+	0.965478i	$0.583882\pi$
$348$	3.37405	0.180868
$349$	1.26625	0.0677808	0.0338904	−	0.999426i	$-0.489210\pi$
$349$	1.26625	0.0677808	0.0338904	+	0.999426i	$0.489210\pi$
$350$	0.216453	0.0115699
$351$	1.00000	0.0533761
$352$	−3.14452	−0.167603
$353$	14.8770	0.791821	0.395910	−	0.918289i	$-0.370429\pi$
$353$	14.8770	0.791821	0.395910	+	0.918289i	$0.370429\pi$
$354$	−10.4567	−0.555769
$355$	−22.6424	−1.20173
$356$	6.22450	0.329898
$357$	−8.01176	−0.424027
$358$	−10.3299	−0.545951
$359$	−5.35220	−0.282478	−0.141239	−	0.989975i	$-0.545109\pi$
$359$	−5.35220	−0.282478	−0.141239	+	0.989975i	$0.545109\pi$
$360$	−2.24957	−0.118563
$361$	20.4239	1.07494
$362$	16.8081	0.883415
$363$	−1.11200	−0.0583651
$364$	3.57309	0.187281
$365$	18.2457	0.955023
$366$	3.20542	0.167550
$367$	29.2763	1.52821	0.764105	−	0.645092i	$-0.223181\pi$
$367$	29.2763	1.52821	0.764105	+	0.645092i	$0.223181\pi$
$368$	−4.02900	−0.210026
$369$	−1.29845	−0.0675946
$370$	−2.94426	−0.153065
$371$	−43.7884	−2.27338
$372$	−4.73747	−0.245626
$373$	−26.7536	−1.38525	−0.692625	−	0.721298i	$-0.743546\pi$
$373$	−26.7536	−1.38525	−0.692625	+	0.721298i	$0.743546\pi$
$374$	7.05079	0.364588
$375$	11.1116	0.573800
$376$	−9.07827	−0.468176
$377$	3.37405	0.173772
$378$	3.57309	0.183780
$379$	−11.3636	−0.583709	−0.291854	−	0.956463i	$-0.594272\pi$
$379$	−11.3636	−0.583709	−0.291854	+	0.956463i	$0.594272\pi$
$380$	14.1247	0.724583
$381$	2.30558	0.118119
$382$	−13.2334	−0.677079
$383$	12.7182	0.649868	0.324934	−	0.945737i	$-0.394658\pi$
$383$	12.7182	0.649868	0.324934	+	0.945737i	$0.394658\pi$
$384$	1.00000	0.0510310
$385$	25.2754	1.28815
$386$	23.1230	1.17693
$387$	2.09446	0.106467
$388$	0.109151	0.00554130
$389$	−6.29223	−0.319029	−0.159514	−	0.987196i	$-0.550993\pi$
$389$	−6.29223	−0.319029	−0.159514	+	0.987196i	$0.550993\pi$
$390$	−2.24957	−0.113912
$391$	9.03403	0.456870
$392$	5.76698	0.291276
$393$	−7.68277	−0.387544
$394$	−19.6407	−0.989485
$395$	10.3519	0.520860
$396$	−3.14452	−0.158018
$397$	−32.2482	−1.61849	−0.809245	−	0.587471i	$-0.800124\pi$
$397$	−32.2482	−1.61849	−0.809245	+	0.587471i	$0.800124\pi$
$398$	10.5365	0.528148
$399$	−22.4349	−1.12315
$400$	0.0605787	0.00302893
$401$	9.54032	0.476421	0.238210	−	0.971214i	$-0.423439\pi$
$401$	9.54032	0.476421	0.238210	+	0.971214i	$0.423439\pi$
$402$	−1.89011	−0.0942703
$403$	−4.73747	−0.235990
$404$	−15.9360	−0.792844
$405$	−2.24957	−0.111782
$406$	12.0558	0.598318
$407$	−4.11557	−0.204001
$408$	−2.24225	−0.111008
$409$	21.0449	1.04060	0.520302	−	0.853982i	$-0.325819\pi$
$409$	21.0449	1.04060	0.520302	+	0.853982i	$0.325819\pi$
$410$	2.92096	0.144256
$411$	5.69384	0.280857
$412$	1.00000	0.0492665
$413$	−37.3628	−1.83850
$414$	−4.02900	−0.198015
$415$	−31.3991	−1.54132
$416$	1.00000	0.0490290
$417$	15.5772	0.762820
$418$	19.7439	0.965708
$419$	5.74914	0.280864	0.140432	−	0.990090i	$-0.455151\pi$
$419$	5.74914	0.280864	0.140432	+	0.990090i	$0.455151\pi$
$420$	−8.03793	−0.392211
$421$	−33.1625	−1.61624	−0.808122	−	0.589016i	$-0.799516\pi$
$421$	−33.1625	−1.61624	−0.808122	+	0.589016i	$0.799516\pi$
$422$	−22.4883	−1.09472
$423$	−9.07827	−0.441400
$424$	−12.2551	−0.595158
$425$	−0.135832	−0.00658884
$426$	10.0652	0.487661
$427$	11.4532	0.554261
$428$	−0.420266	−0.0203143
$429$	−3.14452	−0.151819
$430$	−4.71163	−0.227215
$431$	−31.5818	−1.52124	−0.760621	−	0.649197i	$-0.775105\pi$
$431$	−31.5818	−1.52124	−0.760621	+	0.649197i	$0.775105\pi$
$432$	1.00000	0.0481125
$433$	25.8770	1.24357	0.621784	−	0.783189i	$-0.286408\pi$
$433$	25.8770	1.24357	0.621784	+	0.783189i	$0.286408\pi$
$434$	−16.9274	−0.812541
$435$	−7.59016	−0.363920
$436$	−3.47914	−0.166621
$437$	25.2975	1.21014
$438$	−8.11073	−0.387546
$439$	20.9467	0.999730	0.499865	−	0.866103i	$-0.333383\pi$
$439$	20.9467	0.999730	0.499865	+	0.866103i	$0.333383\pi$
$440$	7.07382	0.337231
$441$	5.76698	0.274618
$442$	−2.24225	−0.106653
$443$	21.4385	1.01857	0.509287	−	0.860597i	$-0.329909\pi$
$443$	21.4385	1.01857	0.509287	+	0.860597i	$0.329909\pi$
$444$	1.30881	0.0621133
$445$	−14.0025	−0.663781
$446$	8.63267	0.408769
$447$	−4.84494	−0.229158
$448$	3.57309	0.168813
$449$	−5.54699	−0.261779	−0.130889	−	0.991397i	$-0.541783\pi$
$449$	−5.54699	−0.261779	−0.130889	+	0.991397i	$0.541783\pi$
$450$	0.0605787	0.00285571
$451$	4.08300	0.192261
$452$	−9.36131	−0.440319
$453$	−1.56151	−0.0733662
$454$	12.7730	0.599465
$455$	−8.03793	−0.376824
$456$	−6.27884	−0.294034
$457$	−25.8771	−1.21048	−0.605240	−	0.796043i	$-0.706923\pi$
$457$	−25.8771	−1.21048	−0.605240	+	0.796043i	$0.706923\pi$
$458$	−0.442618	−0.0206822
$459$	−2.24225	−0.104659
$460$	9.06354	0.422590
$461$	−7.88020	−0.367017	−0.183509	−	0.983018i	$-0.558746\pi$
$461$	−7.88020	−0.367017	−0.183509	+	0.983018i	$0.558746\pi$
$462$	−11.2356	−0.522730
$463$	−25.9117	−1.20422	−0.602109	−	0.798414i	$-0.705673\pi$
$463$	−25.9117	−1.20422	−0.602109	+	0.798414i	$0.705673\pi$
$464$	3.37405	0.156636
$465$	10.6573	0.494219
$466$	−13.0390	−0.604021
$467$	−22.8015	−1.05513	−0.527564	−	0.849515i	$-0.676895\pi$
$467$	−22.8015	−1.05513	−0.527564	+	0.849515i	$0.676895\pi$
$468$	1.00000	0.0462250
$469$	−6.75355	−0.311850
$470$	20.4222	0.942007
$471$	−11.9321	−0.549804
$472$	−10.4567	−0.481310
$473$	−6.58606	−0.302827
$474$	−4.60171	−0.211364
$475$	−0.380364	−0.0174523
$476$	−8.01176	−0.367218
$477$	−12.2551	−0.561121
$478$	−27.1264	−1.24073
$479$	−21.4789	−0.981398	−0.490699	−	0.871329i	$-0.663259\pi$
$479$	−21.4789	−0.981398	−0.490699	+	0.871329i	$0.663259\pi$
$480$	−2.24957	−0.102678
$481$	1.30881	0.0596766
$482$	21.5350	0.980894
$483$	−14.3960	−0.655041
$484$	−1.11200	−0.0505457
$485$	−0.245543	−0.0111495
$486$	1.00000	0.0453609
$487$	10.2461	0.464295	0.232148	−	0.972681i	$-0.425425\pi$
$487$	10.2461	0.464295	0.232148	+	0.972681i	$0.425425\pi$
$488$	3.20542	0.145102
$489$	9.20600	0.416310
$490$	−12.9732	−0.586071
$491$	31.2264	1.40923	0.704613	−	0.709592i	$-0.251121\pi$
$491$	31.2264	1.40923	0.704613	+	0.709592i	$0.251121\pi$
$492$	−1.29845	−0.0585387
$493$	−7.56545	−0.340731
$494$	−6.27884	−0.282499
$495$	7.07382	0.317945
$496$	−4.73747	−0.212719
$497$	35.9639	1.61320
$498$	13.9578	0.625463
$499$	−4.03630	−0.180690	−0.0903449	−	0.995911i	$-0.528797\pi$
$499$	−4.03630	−0.180690	−0.0903449	+	0.995911i	$0.528797\pi$
$500$	11.1116	0.496925
$501$	25.3409	1.13215
$502$	−24.3162	−1.08528
$503$	−27.1589	−1.21096	−0.605478	−	0.795862i	$-0.707018\pi$
$503$	−27.1589	−1.21096	−0.605478	+	0.795862i	$0.707018\pi$
$504$	3.57309	0.159158
$505$	35.8491	1.59526
$506$	12.6693	0.563218
$507$	1.00000	0.0444116
$508$	2.30558	0.102294
$509$	−7.59144	−0.336485	−0.168242	−	0.985746i	$-0.553809\pi$
$509$	−7.59144	−0.336485	−0.168242	+	0.985746i	$0.553809\pi$
$510$	5.04410	0.223357
$511$	−28.9804	−1.28202
$512$	1.00000	0.0441942
$513$	−6.27884	−0.277218
$514$	−1.88981	−0.0833560
$515$	−2.24957	−0.0991280
$516$	2.09446	0.0922033
$517$	28.5468	1.25549
$518$	4.67649	0.205473
$519$	−8.35653	−0.366811
$520$	−2.24957	−0.0986503
$521$	−19.4417	−0.851756	−0.425878	−	0.904780i	$-0.640035\pi$
$521$	−19.4417	−0.851756	−0.425878	+	0.904780i	$0.640035\pi$
$522$	3.37405	0.147678
$523$	−37.0423	−1.61975	−0.809873	−	0.586606i	$-0.800464\pi$
$523$	−37.0423	−1.61975	−0.809873	+	0.586606i	$0.800464\pi$
$524$	−7.68277	−0.335623
$525$	0.216453	0.00944679
$526$	−8.71368	−0.379934
$527$	10.6226	0.462727
$528$	−3.14452	−0.136848
$529$	−6.76712	−0.294223
$530$	27.5687	1.19751
$531$	−10.4567	−0.453783
$532$	−22.4349	−0.972675
$533$	−1.29845	−0.0562421
$534$	6.22450	0.269360
$535$	0.945419	0.0408740
$536$	−1.89011	−0.0816405
$537$	−10.3299	−0.445767
$538$	−0.698163	−0.0300999
$539$	−18.1344	−0.781103
$540$	−2.24957	−0.0968062
$541$	19.7654	0.849779	0.424889	−	0.905245i	$-0.360313\pi$
$541$	19.7654	0.849779	0.424889	+	0.905245i	$0.360313\pi$
$542$	24.2795	1.04289
$543$	16.8081	0.721305
$544$	−2.24225	−0.0961356
$545$	7.82659	0.335254
$546$	3.57309	0.152914
$547$	40.0405	1.71201	0.856003	−	0.516970i	$-0.172940\pi$
$547$	40.0405	1.71201	0.856003	+	0.516970i	$0.172940\pi$
$548$	5.69384	0.243229
$549$	3.20542	0.136804
$550$	−0.190491	−0.00812255
$551$	−21.1851	−0.902516
$552$	−4.02900	−0.171486
$553$	−16.4423	−0.699199
$554$	0.188794	0.00802107
$555$	−2.94426	−0.124977
$556$	15.5772	0.660621
$557$	34.6842	1.46962	0.734809	−	0.678274i	$-0.237272\pi$
$557$	34.6842	1.46962	0.734809	+	0.678274i	$0.237272\pi$
$558$	−4.73747	−0.200553
$559$	2.09446	0.0885861
$560$	−8.03793	−0.339664
$561$	7.05079	0.297685
$562$	−4.21764	−0.177911
$563$	2.02479	0.0853346	0.0426673	−	0.999089i	$-0.486414\pi$
$563$	2.02479	0.0853346	0.0426673	+	0.999089i	$0.486414\pi$
$564$	−9.07827	−0.382264
$565$	21.0590	0.885957
$566$	−5.69210	−0.239257
$567$	3.57309	0.150056
$568$	10.0652	0.422326
$569$	11.1802	0.468697	0.234348	−	0.972153i	$-0.424704\pi$
$569$	11.1802	0.468697	0.234348	+	0.972153i	$0.424704\pi$
$570$	14.1247	0.591619
$571$	2.73496	0.114454	0.0572272	−	0.998361i	$-0.481774\pi$
$571$	2.73496	0.114454	0.0572272	+	0.998361i	$0.481774\pi$
$572$	−3.14452	−0.131479
$573$	−13.2334	−0.552833
$574$	−4.63948	−0.193648
$575$	−0.244072	−0.0101785
$576$	1.00000	0.0416667
$577$	−14.9591	−0.622755	−0.311377	−	0.950286i	$-0.600790\pi$
$577$	−14.9591	−0.622755	−0.311377	+	0.950286i	$0.600790\pi$
$578$	−11.9723	−0.497983
$579$	23.1230	0.960958
$580$	−7.59016	−0.315164
$581$	49.8724	2.06906
$582$	0.109151	0.00452445
$583$	38.5363	1.59601
$584$	−8.11073	−0.335624
$585$	−2.24957	−0.0930084
$586$	14.2594	0.589050
$587$	9.36674	0.386607	0.193303	−	0.981139i	$-0.438080\pi$
$587$	9.36674	0.386607	0.193303	+	0.981139i	$0.438080\pi$
$588$	5.76698	0.237826
$589$	29.7458	1.22566
$590$	23.5232	0.968433
$591$	−19.6407	−0.807911
$592$	1.30881	0.0537917
$593$	−29.4054	−1.20753	−0.603767	−	0.797161i	$-0.706334\pi$
$593$	−29.4054	−1.20753	−0.603767	+	0.797161i	$0.706334\pi$
$594$	−3.14452	−0.129021
$595$	18.0230	0.738872
$596$	−4.84494	−0.198456
$597$	10.5365	0.431231
$598$	−4.02900	−0.164758
$599$	3.16462	0.129303	0.0646515	−	0.997908i	$-0.479406\pi$
$599$	3.16462	0.129303	0.0646515	+	0.997908i	$0.479406\pi$
$600$	0.0605787	0.00247311
$601$	29.6153	1.20803	0.604016	−	0.796972i	$-0.293566\pi$
$601$	29.6153	1.20803	0.604016	+	0.796972i	$0.293566\pi$
$602$	7.48368	0.305012
$603$	−1.89011	−0.0769714
$604$	−1.56151	−0.0635370
$605$	2.50154	0.101702
$606$	−15.9360	−0.647354
$607$	−39.9667	−1.62220	−0.811098	−	0.584910i	$-0.801130\pi$
$607$	−39.9667	−1.62220	−0.811098	+	0.584910i	$0.801130\pi$
$608$	−6.27884	−0.254641
$609$	12.0558	0.488525
$610$	−7.21082	−0.291958
$611$	−9.07827	−0.367267
$612$	−2.24225	−0.0906375
$613$	6.60089	0.266607	0.133304	−	0.991075i	$-0.457441\pi$
$613$	6.60089	0.266607	0.133304	+	0.991075i	$0.457441\pi$
$614$	−1.85507	−0.0748646
$615$	2.92096	0.117784
$616$	−11.2356	−0.452697
$617$	−37.5573	−1.51200	−0.756000	−	0.654571i	$-0.772849\pi$
$617$	−37.5573	−1.51200	−0.756000	+	0.654571i	$0.772849\pi$
$618$	1.00000	0.0402259
$619$	28.4054	1.14171	0.570855	−	0.821051i	$-0.306612\pi$
$619$	28.4054	1.14171	0.570855	+	0.821051i	$0.306612\pi$
$620$	10.6573	0.428007
$621$	−4.02900	−0.161678
$622$	4.21294	0.168923
$623$	22.2407	0.891055
$624$	1.00000	0.0400320
$625$	−25.2992	−1.01197
$626$	−5.79894	−0.231772
$627$	19.7439	0.788497
$628$	−11.9321	−0.476144
$629$	−2.93467	−0.117013
$630$	−8.03793	−0.320239
$631$	−30.0986	−1.19821	−0.599103	−	0.800672i	$-0.704476\pi$
$631$	−30.0986	−1.19821	−0.599103	+	0.800672i	$0.704476\pi$
$632$	−4.60171	−0.183046
$633$	−22.4883	−0.893831
$634$	−4.45496	−0.176929
$635$	−5.18657	−0.205823
$636$	−12.2551	−0.485945
$637$	5.76698	0.228496
$638$	−10.6098	−0.420044
$639$	10.0652	0.398173
$640$	−2.24957	−0.0889222
$641$	−10.7234	−0.423548	−0.211774	−	0.977319i	$-0.567924\pi$
$641$	−10.7234	−0.423548	−0.211774	+	0.977319i	$0.567924\pi$
$642$	−0.420266	−0.0165866
$643$	−3.75290	−0.148000	−0.0739999	−	0.997258i	$-0.523576\pi$
$643$	−3.75290	−0.148000	−0.0739999	+	0.997258i	$0.523576\pi$
$644$	−14.3960	−0.567282
$645$	−4.71163	−0.185520
$646$	14.0787	0.553920
$647$	−6.08970	−0.239411	−0.119705	−	0.992809i	$-0.538195\pi$
$647$	−6.08970	−0.239411	−0.119705	+	0.992809i	$0.538195\pi$
$648$	1.00000	0.0392837
$649$	32.8814	1.29071
$650$	0.0605787	0.00237609
$651$	−16.9274	−0.663437
$652$	9.20600	0.360535
$653$	−6.06090	−0.237181	−0.118591	−	0.992943i	$-0.537838\pi$
$653$	−6.06090	−0.237181	−0.118591	+	0.992943i	$0.537838\pi$
$654$	−3.47914	−0.136045
$655$	17.2829	0.675301
$656$	−1.29845	−0.0506960
$657$	−8.11073	−0.316430
$658$	−32.4375	−1.26454
$659$	32.3121	1.25870	0.629350	−	0.777122i	$-0.283321\pi$
$659$	32.3121	1.25870	0.629350	+	0.777122i	$0.283321\pi$
$660$	7.07382	0.275348
$661$	37.0379	1.44061	0.720304	−	0.693659i	$-0.244002\pi$
$661$	37.0379	1.44061	0.720304	+	0.693659i	$0.244002\pi$
$662$	−7.17142	−0.278725
$663$	−2.24225	−0.0870817
$664$	13.9578	0.541667
$665$	50.4689	1.95710
$666$	1.30881	0.0507153
$667$	−13.5940	−0.526364
$668$	25.3409	0.980467
$669$	8.63267	0.333758
$670$	4.25195	0.164267
$671$	−10.0795	−0.389114
$672$	3.57309	0.137835
$673$	5.46085	0.210500	0.105250	−	0.994446i	$-0.466436\pi$
$673$	5.46085	0.210500	0.105250	+	0.994446i	$0.466436\pi$
$674$	26.9060	1.03638
$675$	0.0605787	0.00233167
$676$	1.00000	0.0384615
$677$	−39.6394	−1.52347	−0.761733	−	0.647891i	$-0.775651\pi$
$677$	−39.6394	−1.52347	−0.761733	+	0.647891i	$0.775651\pi$
$678$	−9.36131	−0.359519
$679$	0.390006	0.0149671
$680$	5.04410	0.193432
$681$	12.7730	0.489461
$682$	14.8971	0.570438
$683$	−39.4834	−1.51079	−0.755394	−	0.655270i	$-0.772555\pi$
$683$	−39.4834	−1.51079	−0.755394	+	0.655270i	$0.772555\pi$
$684$	−6.27884	−0.240078
$685$	−12.8087	−0.489396
$686$	−4.40570	−0.168210
$687$	−0.442618	−0.0168869
$688$	2.09446	0.0798504
$689$	−12.2551	−0.466881
$690$	9.06354	0.345043
$691$	−15.9030	−0.604977	−0.302489	−	0.953153i	$-0.597817\pi$
$691$	−15.9030	−0.604977	−0.302489	+	0.953153i	$0.597817\pi$
$692$	−8.35653	−0.317667
$693$	−11.2356	−0.426807
$694$	−9.70451	−0.368378
$695$	−35.0421	−1.32922
$696$	3.37405	0.127893
$697$	2.91145	0.110279
$698$	1.26625	0.0479282
$699$	−13.0390	−0.493181
$700$	0.216453	0.00818116
$701$	27.6034	1.04257	0.521283	−	0.853384i	$-0.325454\pi$
$701$	27.6034	1.04257	0.521283	+	0.853384i	$0.325454\pi$
$702$	1.00000	0.0377426
$703$	−8.21781	−0.309941
$704$	−3.14452	−0.118513
$705$	20.4222	0.769145
$706$	14.8770	0.559902
$707$	−56.9406	−2.14147
$708$	−10.4567	−0.392988
$709$	38.4541	1.44417	0.722087	−	0.691802i	$-0.243183\pi$
$709$	38.4541	1.44417	0.722087	+	0.691802i	$0.243183\pi$
$710$	−22.6424	−0.849754
$711$	−4.60171	−0.172578
$712$	6.22450	0.233273
$713$	19.0873	0.714824
$714$	−8.01176	−0.299832
$715$	7.07382	0.264546
$716$	−10.3299	−0.386046
$717$	−27.1264	−1.01305
$718$	−5.35220	−0.199742
$719$	11.2586	0.419875	0.209938	−	0.977715i	$-0.432674\pi$
$719$	11.2586	0.419875	0.209938	+	0.977715i	$0.432674\pi$
$720$	−2.24957	−0.0838366
$721$	3.57309	0.133069
$722$	20.4239	0.760098
$723$	21.5350	0.800897
$724$	16.8081	0.624669
$725$	0.204395	0.00759105
$726$	−1.11200	−0.0412704
$727$	12.4796	0.462842	0.231421	−	0.972854i	$-0.425662\pi$
$727$	12.4796	0.462842	0.231421	+	0.972854i	$0.425662\pi$
$728$	3.57309	0.132428
$729$	1.00000	0.0370370
$730$	18.2457	0.675303
$731$	−4.69629	−0.173699
$732$	3.20542	0.118476
$733$	−21.1042	−0.779501	−0.389751	−	0.920920i	$-0.627439\pi$
$733$	−21.1042	−0.779501	−0.389751	+	0.920920i	$0.627439\pi$
$734$	29.2763	1.08061
$735$	−12.9732	−0.478525
$736$	−4.02900	−0.148511
$737$	5.94350	0.218932
$738$	−1.29845	−0.0477966
$739$	43.0341	1.58303	0.791517	−	0.611147i	$-0.209292\pi$
$739$	43.0341	1.58303	0.791517	+	0.611147i	$0.209292\pi$
$740$	−2.94426	−0.108233
$741$	−6.27884	−0.230659
$742$	−43.7884	−1.60752
$743$	15.9256	0.584252	0.292126	−	0.956380i	$-0.405637\pi$
$743$	15.9256	0.584252	0.292126	+	0.956380i	$0.405637\pi$
$744$	−4.73747	−0.173684
$745$	10.8990	0.399310
$746$	−26.7536	−0.979520
$747$	13.9578	0.510689
$748$	7.05079	0.257802
$749$	−1.50165	−0.0548691
$750$	11.1116	0.405738
$751$	17.5550	0.640590	0.320295	−	0.947318i	$-0.396218\pi$
$751$	17.5550	0.640590	0.320295	+	0.947318i	$0.396218\pi$
$752$	−9.07827	−0.331050
$753$	−24.3162	−0.886130
$754$	3.37405	0.122876
$755$	3.51273	0.127841
$756$	3.57309	0.129952
$757$	−38.2913	−1.39172	−0.695861	−	0.718177i	$-0.744977\pi$
$757$	−38.2913	−1.39172	−0.695861	+	0.718177i	$0.744977\pi$
$758$	−11.3636	−0.412744
$759$	12.6693	0.459866
$760$	14.1247	0.512357
$761$	22.4567	0.814054	0.407027	−	0.913416i	$-0.366566\pi$
$761$	22.4567	0.814054	0.407027	+	0.913416i	$0.366566\pi$
$762$	2.30558	0.0835224
$763$	−12.4313	−0.450043
$764$	−13.2334	−0.478767
$765$	5.04410	0.182370
$766$	12.7182	0.459526
$767$	−10.4567	−0.377570
$768$	1.00000	0.0360844
$769$	−39.8080	−1.43551	−0.717757	−	0.696294i	$-0.754831\pi$
$769$	−39.8080	−1.43551	−0.717757	+	0.696294i	$0.754831\pi$
$770$	25.2754	0.910863
$771$	−1.88981	−0.0680599
$772$	23.1230	0.832214
$773$	34.8297	1.25274	0.626369	−	0.779527i	$-0.284540\pi$
$773$	34.8297	1.25274	0.626369	+	0.779527i	$0.284540\pi$
$774$	2.09446	0.0752837
$775$	−0.286989	−0.0103090
$776$	0.109151	0.00391829
$777$	4.67649	0.167768
$778$	−6.29223	−0.225587
$779$	8.15277	0.292103
$780$	−2.24957	−0.0805476
$781$	−31.6502	−1.13253
$782$	9.03403	0.323056
$783$	3.37405	0.120579
$784$	5.76698	0.205964
$785$	26.8422	0.958040
$786$	−7.68277	−0.274035
$787$	29.3329	1.04560	0.522802	−	0.852454i	$-0.324887\pi$
$787$	29.3329	1.04560	0.522802	+	0.852454i	$0.324887\pi$
$788$	−19.6407	−0.699671
$789$	−8.71368	−0.310215
$790$	10.3519	0.368304
$791$	−33.4488	−1.18930
$792$	−3.14452	−0.111736
$793$	3.20542	0.113828
$794$	−32.2482	−1.14445
$795$	27.5687	0.977759
$796$	10.5365	0.373457
$797$	17.0643	0.604449	0.302225	−	0.953237i	$-0.402271\pi$
$797$	17.0643	0.604449	0.302225	+	0.953237i	$0.402271\pi$
$798$	−22.4349	−0.794186
$799$	20.3557	0.720134
$800$	0.0605787	0.00214178
$801$	6.22450	0.219932
$802$	9.54032	0.336880
$803$	25.5043	0.900029
$804$	−1.89011	−0.0666592
$805$	32.3849	1.14142
$806$	−4.73747	−0.166870
$807$	−0.698163	−0.0245765
$808$	−15.9360	−0.560625
$809$	−0.928638	−0.0326492	−0.0163246	−	0.999867i	$-0.505197\pi$
$809$	−0.928638	−0.0326492	−0.0163246	+	0.999867i	$0.505197\pi$
$810$	−2.24957	−0.0790419
$811$	−3.94929	−0.138678	−0.0693392	−	0.997593i	$-0.522089\pi$
$811$	−3.94929	−0.138678	−0.0693392	+	0.997593i	$0.522089\pi$
$812$	12.0558	0.423075
$813$	24.2795	0.851519
$814$	−4.11557	−0.144251
$815$	−20.7096	−0.725424
$816$	−2.24225	−0.0784944
$817$	−13.1508	−0.460087
$818$	21.0449	0.735818
$819$	3.57309	0.124854
$820$	2.92096	0.102004
$821$	19.6318	0.685155	0.342578	−	0.939490i	$-0.388700\pi$
$821$	19.6318	0.685155	0.342578	+	0.939490i	$0.388700\pi$
$822$	5.69384	0.198596
$823$	−17.3178	−0.603662	−0.301831	−	0.953362i	$-0.597598\pi$
$823$	−17.3178	−0.603662	−0.301831	+	0.953362i	$0.597598\pi$
$824$	1.00000	0.0348367
$825$	−0.190491	−0.00663204
$826$	−37.3628	−1.30002
$827$	46.0756	1.60221	0.801103	−	0.598527i	$-0.204247\pi$
$827$	46.0756	1.60221	0.801103	+	0.598527i	$0.204247\pi$
$828$	−4.02900	−0.140018
$829$	−32.8438	−1.14071	−0.570357	−	0.821397i	$-0.693195\pi$
$829$	−32.8438	−1.14071	−0.570357	+	0.821397i	$0.693195\pi$
$830$	−31.3991	−1.08988
$831$	0.188794	0.00654918
$832$	1.00000	0.0346688
$833$	−12.9310	−0.448032
$834$	15.5772	0.539395
$835$	−57.0061	−1.97278
$836$	19.7439	0.682858
$837$	−4.73747	−0.163751
$838$	5.74914	0.198601
$839$	33.3595	1.15170	0.575848	−	0.817556i	$-0.304672\pi$
$839$	33.3595	1.15170	0.575848	+	0.817556i	$0.304672\pi$
$840$	−8.03793	−0.277335
$841$	−17.6158	−0.607442
$842$	−33.1625	−1.14286
$843$	−4.21764	−0.145263
$844$	−22.4883	−0.774081
$845$	−2.24957	−0.0773877
$846$	−9.07827	−0.312117
$847$	−3.97329	−0.136524
$848$	−12.2551	−0.420840
$849$	−5.69210	−0.195352
$850$	−0.135832	−0.00465901
$851$	−5.27320	−0.180763
$852$	10.0652	0.344828
$853$	5.31035	0.181823	0.0909114	−	0.995859i	$-0.471022\pi$
$853$	5.31035	0.181823	0.0909114	+	0.995859i	$0.471022\pi$
$854$	11.4532	0.391922
$855$	14.1247	0.483055
$856$	−0.420266	−0.0143644
$857$	36.1874	1.23614	0.618068	−	0.786124i	$-0.287916\pi$
$857$	36.1874	1.23614	0.618068	+	0.786124i	$0.287916\pi$
$858$	−3.14452	−0.107352
$859$	5.61775	0.191675	0.0958375	−	0.995397i	$-0.469447\pi$
$859$	5.61775	0.191675	0.0958375	+	0.995397i	$0.469447\pi$
$860$	−4.71163	−0.160665
$861$	−4.63948	−0.158113
$862$	−31.5818	−1.07568
$863$	−16.3255	−0.555728	−0.277864	−	0.960620i	$-0.589626\pi$
$863$	−16.3255	−0.555728	−0.277864	+	0.960620i	$0.589626\pi$
$864$	1.00000	0.0340207
$865$	18.7986	0.639172
$866$	25.8770	0.879335
$867$	−11.9723	−0.406601
$868$	−16.9274	−0.574553
$869$	14.4702	0.490867
$870$	−7.59016	−0.257331
$871$	−1.89011	−0.0640441
$872$	−3.47914	−0.117819
$873$	0.109151	0.00369420
$874$	25.2975	0.855700
$875$	39.7027	1.34220
$876$	−8.11073	−0.274036
$877$	−1.15096	−0.0388652	−0.0194326	−	0.999811i	$-0.506186\pi$
$877$	−1.15096	−0.0388652	−0.0194326	+	0.999811i	$0.506186\pi$
$878$	20.9467	0.706916
$879$	14.2594	0.480958
$880$	7.07382	0.238459
$881$	−32.8519	−1.10681	−0.553404	−	0.832913i	$-0.686672\pi$
$881$	−32.8519	−1.10681	−0.553404	+	0.832913i	$0.686672\pi$
$882$	5.76698	0.194184
$883$	−40.6655	−1.36850	−0.684251	−	0.729246i	$-0.739871\pi$
$883$	−40.6655	−1.36850	−0.684251	+	0.729246i	$0.739871\pi$
$884$	−2.24225	−0.0754150
$885$	23.5232	0.790722
$886$	21.4385	0.720241
$887$	46.6307	1.56571	0.782853	−	0.622207i	$-0.213764\pi$
$887$	46.6307	1.56571	0.782853	+	0.622207i	$0.213764\pi$
$888$	1.30881	0.0439208
$889$	8.23805	0.276295
$890$	−14.0025	−0.469364
$891$	−3.14452	−0.105345
$892$	8.63267	0.289043
$893$	57.0010	1.90747
$894$	−4.84494	−0.162039
$895$	23.2378	0.776755
$896$	3.57309	0.119369
$897$	−4.02900	−0.134525
$898$	−5.54699	−0.185106
$899$	−15.9844	−0.533111
$900$	0.0605787	0.00201929
$901$	27.4789	0.915454
$902$	4.08300	0.135949
$903$	7.48368	0.249041
$904$	−9.36131	−0.311353
$905$	−37.8111	−1.25688
$906$	−1.56151	−0.0518777
$907$	−27.6595	−0.918420	−0.459210	−	0.888328i	$-0.651867\pi$
$907$	−27.6595	−0.918420	−0.459210	+	0.888328i	$0.651867\pi$
$908$	12.7730	0.423886
$909$	−15.9360	−0.528562
$910$	−8.03793	−0.266455
$911$	43.0998	1.42796	0.713980	−	0.700166i	$-0.246891\pi$
$911$	43.0998	1.42796	0.713980	+	0.700166i	$0.246891\pi$
$912$	−6.27884	−0.207913
$913$	−43.8905	−1.45256
$914$	−25.8771	−0.855939
$915$	−7.21082	−0.238382
$916$	−0.442618	−0.0146245
$917$	−27.4512	−0.906519
$918$	−2.24225	−0.0740052
$919$	3.10024	0.102268	0.0511338	−	0.998692i	$-0.483717\pi$
$919$	3.10024	0.102268	0.0511338	+	0.998692i	$0.483717\pi$
$920$	9.06354	0.298816
$921$	−1.85507	−0.0611267
$922$	−7.88020	−0.259521
$923$	10.0652	0.331300
$924$	−11.2356	−0.369626
$925$	0.0792859	0.00260690
$926$	−25.9117	−0.851511
$927$	1.00000	0.0328443
$928$	3.37405	0.110758
$929$	34.4122	1.12903	0.564513	−	0.825424i	$-0.309064\pi$
$929$	34.4122	1.12903	0.564513	+	0.825424i	$0.309064\pi$
$930$	10.6573	0.349466
$931$	−36.2100	−1.18673
$932$	−13.0390	−0.427107
$933$	4.21294	0.137925
$934$	−22.8015	−0.746088
$935$	−15.8613	−0.518719
$936$	1.00000	0.0326860
$937$	7.93375	0.259184	0.129592	−	0.991567i	$-0.458633\pi$
$937$	7.93375	0.259184	0.129592	+	0.991567i	$0.458633\pi$
$938$	−6.75355	−0.220511
$939$	−5.79894	−0.189241
$940$	20.4222	0.666100
$941$	−20.5392	−0.669558	−0.334779	−	0.942297i	$-0.608662\pi$
$941$	−20.5392	−0.669558	−0.334779	+	0.942297i	$0.608662\pi$
$942$	−11.9321	−0.388770
$943$	5.23146	0.170360
$944$	−10.4567	−0.340337
$945$	−8.03793	−0.261474
$946$	−6.58606	−0.214131
$947$	−19.2653	−0.626038	−0.313019	−	0.949747i	$-0.601340\pi$
$947$	−19.2653	−0.626038	−0.313019	+	0.949747i	$0.601340\pi$
$948$	−4.60171	−0.149457
$949$	−8.11073	−0.263285
$950$	−0.380364	−0.0123406
$951$	−4.45496	−0.144462
$952$	−8.01176	−0.259662
$953$	−24.8062	−0.803552	−0.401776	−	0.915738i	$-0.631607\pi$
$953$	−24.8062	−0.803552	−0.401776	+	0.915738i	$0.631607\pi$
$954$	−12.2551	−0.396772
$955$	29.7695	0.963318
$956$	−27.1264	−0.877330
$957$	−10.6098	−0.342965
$958$	−21.4789	−0.693953
$959$	20.3446	0.656962
$960$	−2.24957	−0.0726047
$961$	−8.55640	−0.276013
$962$	1.30881	0.0421977
$963$	−0.420266	−0.0135429
$964$	21.5350	0.693597
$965$	−52.0168	−1.67448
$966$	−14.3960	−0.463184
$967$	27.6603	0.889494	0.444747	−	0.895656i	$-0.353294\pi$
$967$	27.6603	0.889494	0.444747	+	0.895656i	$0.353294\pi$
$968$	−1.11200	−0.0357412
$969$	14.0787	0.452274
$970$	−0.245543	−0.00788391
$971$	21.5689	0.692179	0.346090	−	0.938201i	$-0.387509\pi$
$971$	21.5689	0.692179	0.346090	+	0.938201i	$0.387509\pi$
$972$	1.00000	0.0320750
$973$	55.6588	1.78434
$974$	10.2461	0.328306
$975$	0.0605787	0.00194007
$976$	3.20542	0.102603
$977$	44.6697	1.42911	0.714556	−	0.699579i	$-0.246629\pi$
$977$	44.6697	1.42911	0.714556	+	0.699579i	$0.246629\pi$
$978$	9.20600	0.294375
$979$	−19.5731	−0.625558
$980$	−12.9732	−0.414415
$981$	−3.47914	−0.111081
$982$	31.2264	0.996473
$983$	18.4080	0.587124	0.293562	−	0.955940i	$-0.405159\pi$
$983$	18.4080	0.587124	0.293562	+	0.955940i	$0.405159\pi$
$984$	−1.29845	−0.0413931
$985$	44.1832	1.40779
$986$	−7.56545	−0.240933
$987$	−32.4375	−1.03250
$988$	−6.27884	−0.199757
$989$	−8.43858	−0.268331
$990$	7.07382	0.224821
$991$	17.2681	0.548539	0.274269	−	0.961653i	$-0.411564\pi$
$991$	17.2681	0.548539	0.274269	+	0.961653i	$0.411564\pi$
$992$	−4.73747	−0.150415
$993$	−7.17142	−0.227578
$994$	35.9639	1.14070
$995$	−23.7027	−0.751425
$996$	13.9578	0.442269
$997$	−12.4784	−0.395196	−0.197598	−	0.980283i	$-0.563314\pi$
$997$	−12.4784	−0.395196	−0.197598	+	0.980283i	$0.563314\pi$
$998$	−4.03630	−0.127767
$999$	1.30881	0.0414089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.p.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.p.1.3	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24957 q^{5} +1.00000 q^{6} +3.57309 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24957 q^{5} +1.00000 q^{6} +3.57309 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.24957 q^{10} -3.14452 q^{11} +1.00000 q^{12} +1.00000 q^{13} +3.57309 q^{14} -2.24957 q^{15} +1.00000 q^{16} -2.24225 q^{17} +1.00000 q^{18} -6.27884 q^{19} -2.24957 q^{20} +3.57309 q^{21} -3.14452 q^{22} -4.02900 q^{23} +1.00000 q^{24} +0.0605787 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.57309 q^{28} +3.37405 q^{29} -2.24957 q^{30} -4.73747 q^{31} +1.00000 q^{32} -3.14452 q^{33} -2.24225 q^{34} -8.03793 q^{35} +1.00000 q^{36} +1.30881 q^{37} -6.27884 q^{38} +1.00000 q^{39} -2.24957 q^{40} -1.29845 q^{41} +3.57309 q^{42} +2.09446 q^{43} -3.14452 q^{44} -2.24957 q^{45} -4.02900 q^{46} -9.07827 q^{47} +1.00000 q^{48} +5.76698 q^{49} +0.0605787 q^{50} -2.24225 q^{51} +1.00000 q^{52} -12.2551 q^{53} +1.00000 q^{54} +7.07382 q^{55} +3.57309 q^{56} -6.27884 q^{57} +3.37405 q^{58} -10.4567 q^{59} -2.24957 q^{60} +3.20542 q^{61} -4.73747 q^{62} +3.57309 q^{63} +1.00000 q^{64} -2.24957 q^{65} -3.14452 q^{66} -1.89011 q^{67} -2.24225 q^{68} -4.02900 q^{69} -8.03793 q^{70} +10.0652 q^{71} +1.00000 q^{72} -8.11073 q^{73} +1.30881 q^{74} +0.0605787 q^{75} -6.27884 q^{76} -11.2356 q^{77} +1.00000 q^{78} -4.60171 q^{79} -2.24957 q^{80} +1.00000 q^{81} -1.29845 q^{82} +13.9578 q^{83} +3.57309 q^{84} +5.04410 q^{85} +2.09446 q^{86} +3.37405 q^{87} -3.14452 q^{88} +6.22450 q^{89} -2.24957 q^{90} +3.57309 q^{91} -4.02900 q^{92} -4.73747 q^{93} -9.07827 q^{94} +14.1247 q^{95} +1.00000 q^{96} +0.109151 q^{97} +5.76698 q^{98} -3.14452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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