LMFDB - Embedded newform 8013.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8013,2,Mod(1,8013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8013, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.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:	$63.9841271397$
Analytic rank:	$1$
Dimension:	$116$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72179 q^{2} -1.00000 q^{3} +5.40814 q^{4} +2.52515 q^{5} +2.72179 q^{6} +2.50598 q^{7} -9.27623 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.72179 q^{2} -1.00000 q^{3} +5.40814 q^{4} +2.52515 q^{5} +2.72179 q^{6} +2.50598 q^{7} -9.27623 q^{8} +1.00000 q^{9} -6.87293 q^{10} -2.56170 q^{11} -5.40814 q^{12} +0.00898741 q^{13} -6.82075 q^{14} -2.52515 q^{15} +14.4317 q^{16} -3.23473 q^{17} -2.72179 q^{18} +3.79073 q^{19} +13.6564 q^{20} -2.50598 q^{21} +6.97241 q^{22} -6.52425 q^{23} +9.27623 q^{24} +1.37638 q^{25} -0.0244618 q^{26} -1.00000 q^{27} +13.5527 q^{28} -8.10042 q^{29} +6.87293 q^{30} +6.15328 q^{31} -20.7275 q^{32} +2.56170 q^{33} +8.80426 q^{34} +6.32798 q^{35} +5.40814 q^{36} -8.98686 q^{37} -10.3176 q^{38} -0.00898741 q^{39} -23.4239 q^{40} -4.09476 q^{41} +6.82075 q^{42} -2.64014 q^{43} -13.8540 q^{44} +2.52515 q^{45} +17.7576 q^{46} +3.00910 q^{47} -14.4317 q^{48} -0.720059 q^{49} -3.74623 q^{50} +3.23473 q^{51} +0.0486051 q^{52} +7.63121 q^{53} +2.72179 q^{54} -6.46868 q^{55} -23.2460 q^{56} -3.79073 q^{57} +22.0476 q^{58} +14.7425 q^{59} -13.6564 q^{60} +7.79775 q^{61} -16.7479 q^{62} +2.50598 q^{63} +27.5525 q^{64} +0.0226945 q^{65} -6.97241 q^{66} -9.86850 q^{67} -17.4939 q^{68} +6.52425 q^{69} -17.2234 q^{70} +10.3072 q^{71} -9.27623 q^{72} +15.2781 q^{73} +24.4603 q^{74} -1.37638 q^{75} +20.5008 q^{76} -6.41957 q^{77} +0.0244618 q^{78} +7.82538 q^{79} +36.4421 q^{80} +1.00000 q^{81} +11.1451 q^{82} -10.6038 q^{83} -13.5527 q^{84} -8.16819 q^{85} +7.18592 q^{86} +8.10042 q^{87} +23.7629 q^{88} +7.94221 q^{89} -6.87293 q^{90} +0.0225223 q^{91} -35.2840 q^{92} -6.15328 q^{93} -8.19015 q^{94} +9.57215 q^{95} +20.7275 q^{96} +8.26956 q^{97} +1.95985 q^{98} -2.56170 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9}+O(q^{10}) $ 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18} + 3 q^{19} - 54 q^{20} + 33 q^{21} - 22 q^{22} - 58 q^{23} + 45 q^{24} + 126 q^{25} - 21 q^{26} - 116 q^{27} - 77 q^{28} - 38 q^{29} - 3 q^{30} + 17 q^{31} - 106 q^{32} + 57 q^{33} + 35 q^{34} - 72 q^{35} + 116 q^{36} - 41 q^{37} - 45 q^{38} - 6 q^{39} + 5 q^{40} - 39 q^{41} + 9 q^{42} - 118 q^{43} - 103 q^{44} - 20 q^{45} - 8 q^{46} - 65 q^{47} - 112 q^{48} + 165 q^{49} - 72 q^{50} + 30 q^{51} - 10 q^{52} - 58 q^{53} + 16 q^{54} + 14 q^{55} - 23 q^{56} - 3 q^{57} - 27 q^{58} - 75 q^{59} + 54 q^{60} + 45 q^{61} - 73 q^{62} - 33 q^{63} + 111 q^{64} - 86 q^{65} + 22 q^{66} - 127 q^{67} - 94 q^{68} + 58 q^{69} - 7 q^{70} - 61 q^{71} - 45 q^{72} + 15 q^{73} - 51 q^{74} - 126 q^{75} + 96 q^{76} - 57 q^{77} + 21 q^{78} + 7 q^{79} - 144 q^{80} + 116 q^{81} - 37 q^{82} - 194 q^{83} + 77 q^{84} + 3 q^{85} - 57 q^{86} + 38 q^{87} - 42 q^{88} - 56 q^{89} + 3 q^{90} - 39 q^{91} - 138 q^{92} - 17 q^{93} + 51 q^{94} - 127 q^{95} + 106 q^{96} + 57 q^{97} - 105 q^{98} - 57 q^{99}+O(q^{100}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9 + 3 * q^10 - 57 * q^11 - 116 * q^12 + 6 * q^13 - 9 * q^14 + 20 * q^15 + 112 * q^16 - 30 * q^17 - 16 * q^18 + 3 * q^19 - 54 * q^20 + 33 * q^21 - 22 * q^22 - 58 * q^23 + 45 * q^24 + 126 * q^25 - 21 * q^26 - 116 * q^27 - 77 * q^28 - 38 * q^29 - 3 * q^30 + 17 * q^31 - 106 * q^32 + 57 * q^33 + 35 * q^34 - 72 * q^35 + 116 * q^36 - 41 * q^37 - 45 * q^38 - 6 * q^39 + 5 * q^40 - 39 * q^41 + 9 * q^42 - 118 * q^43 - 103 * q^44 - 20 * q^45 - 8 * q^46 - 65 * q^47 - 112 * q^48 + 165 * q^49 - 72 * q^50 + 30 * q^51 - 10 * q^52 - 58 * q^53 + 16 * q^54 + 14 * q^55 - 23 * q^56 - 3 * q^57 - 27 * q^58 - 75 * q^59 + 54 * q^60 + 45 * q^61 - 73 * q^62 - 33 * q^63 + 111 * q^64 - 86 * q^65 + 22 * q^66 - 127 * q^67 - 94 * q^68 + 58 * q^69 - 7 * q^70 - 61 * q^71 - 45 * q^72 + 15 * q^73 - 51 * q^74 - 126 * q^75 + 96 * q^76 - 57 * q^77 + 21 * q^78 + 7 * q^79 - 144 * q^80 + 116 * q^81 - 37 * q^82 - 194 * q^83 + 77 * q^84 + 3 * q^85 - 57 * q^86 + 38 * q^87 - 42 * q^88 - 56 * q^89 + 3 * q^90 - 39 * q^91 - 138 * q^92 - 17 * q^93 + 51 * q^94 - 127 * q^95 + 106 * q^96 + 57 * q^97 - 105 * q^98 - 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.72179	−1.92460	−0.962298	−	0.271998i	$-0.912315\pi$
$2$	−2.72179	−1.92460	−0.962298	+	0.271998i	$0.912315\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.40814	2.70407
$5$	2.52515	1.12928	0.564641	−	0.825337i	$-0.309015\pi$
$5$	2.52515	1.12928	0.564641	+	0.825337i	$0.309015\pi$
$6$	2.72179	1.11117
$7$	2.50598	0.947172	0.473586	−	0.880748i	$-0.342959\pi$
$7$	2.50598	0.947172	0.473586	+	0.880748i	$0.342959\pi$
$8$	−9.27623	−3.27964
$9$	1.00000	0.333333
$10$	−6.87293	−2.17341
$11$	−2.56170	−0.772382	−0.386191	−	0.922419i	$-0.626209\pi$
$11$	−2.56170	−0.772382	−0.386191	+	0.922419i	$0.626209\pi$
$12$	−5.40814	−1.56119
$13$	0.00898741	0.00249266	0.00124633	−	0.999999i	$-0.499603\pi$
$13$	0.00898741	0.00249266	0.00124633	+	0.999999i	$0.499603\pi$
$14$	−6.82075	−1.82292
$15$	−2.52515	−0.651991
$16$	14.4317	3.60791
$17$	−3.23473	−0.784538	−0.392269	−	0.919851i	$-0.628310\pi$
$17$	−3.23473	−0.784538	−0.392269	+	0.919851i	$0.628310\pi$
$18$	−2.72179	−0.641532
$19$	3.79073	0.869652	0.434826	−	0.900514i	$-0.356810\pi$
$19$	3.79073	0.869652	0.434826	+	0.900514i	$0.356810\pi$
$20$	13.6564	3.05365
$21$	−2.50598	−0.546850
$22$	6.97241	1.48652
$23$	−6.52425	−1.36040	−0.680200	−	0.733026i	$-0.738107\pi$
$23$	−6.52425	−1.36040	−0.680200	+	0.733026i	$0.738107\pi$
$24$	9.27623	1.89350
$25$	1.37638	0.275277
$26$	−0.0244618	−0.00479736
$27$	−1.00000	−0.192450
$28$	13.5527	2.56122
$29$	−8.10042	−1.50421	−0.752105	−	0.659043i	$-0.770962\pi$
$29$	−8.10042	−1.50421	−0.752105	+	0.659043i	$0.770962\pi$
$30$	6.87293	1.25482
$31$	6.15328	1.10516	0.552581	−	0.833459i	$-0.313643\pi$
$31$	6.15328	1.10516	0.552581	+	0.833459i	$0.313643\pi$
$32$	−20.7275	−3.66413
$33$	2.56170	0.445935
$34$	8.80426	1.50992
$35$	6.32798	1.06962
$36$	5.40814	0.901356
$37$	−8.98686	−1.47743	−0.738716	−	0.674017i	$-0.764567\pi$
$37$	−8.98686	−1.47743	−0.738716	+	0.674017i	$0.764567\pi$
$38$	−10.3176	−1.67373
$39$	−0.00898741	−0.00143914
$40$	−23.4239	−3.70364
$41$	−4.09476	−0.639494	−0.319747	−	0.947503i	$-0.603598\pi$
$41$	−4.09476	−0.639494	−0.319747	+	0.947503i	$0.603598\pi$
$42$	6.82075	1.05246
$43$	−2.64014	−0.402618	−0.201309	−	0.979528i	$-0.564520\pi$
$43$	−2.64014	−0.402618	−0.201309	+	0.979528i	$0.564520\pi$
$44$	−13.8540	−2.08857
$45$	2.52515	0.376427
$46$	17.7576	2.61822
$47$	3.00910	0.438923	0.219461	−	0.975621i	$-0.429570\pi$
$47$	3.00910	0.438923	0.219461	+	0.975621i	$0.429570\pi$
$48$	−14.4317	−2.08303
$49$	−0.720059	−0.102866
$50$	−3.74623	−0.529796
$51$	3.23473	0.452953
$52$	0.0486051	0.00674032
$53$	7.63121	1.04823	0.524114	−	0.851648i	$-0.324397\pi$
$53$	7.63121	1.04823	0.524114	+	0.851648i	$0.324397\pi$
$54$	2.72179	0.370389
$55$	−6.46868	−0.872236
$56$	−23.2460	−3.10638
$57$	−3.79073	−0.502094
$58$	22.0476	2.89500
$59$	14.7425	1.91932	0.959658	−	0.281171i	$-0.0907226\pi$
$59$	14.7425	1.91932	0.959658	+	0.281171i	$0.0907226\pi$
$60$	−13.6564	−1.76303
$61$	7.79775	0.998399	0.499200	−	0.866487i	$-0.333627\pi$
$61$	7.79775	0.998399	0.499200	+	0.866487i	$0.333627\pi$
$62$	−16.7479	−2.12699
$63$	2.50598	0.315724
$64$	27.5525	3.44406
$65$	0.0226945	0.00281491
$66$	−6.97241	−0.858244
$67$	−9.86850	−1.20563	−0.602814	−	0.797881i	$-0.705954\pi$
$67$	−9.86850	−1.20563	−0.602814	+	0.797881i	$0.705954\pi$
$68$	−17.4939	−2.12144
$69$	6.52425	0.785428
$70$	−17.2234	−2.05859
$71$	10.3072	1.22324	0.611618	−	0.791153i	$-0.290519\pi$
$71$	10.3072	1.22324	0.611618	+	0.791153i	$0.290519\pi$
$72$	−9.27623	−1.09321
$73$	15.2781	1.78817	0.894086	−	0.447895i	$-0.147826\pi$
$73$	15.2781	1.78817	0.894086	+	0.447895i	$0.147826\pi$
$74$	24.4603	2.84346
$75$	−1.37638	−0.158931
$76$	20.5008	2.35160
$77$	−6.41957	−0.731578
$78$	0.0244618	0.00276976
$79$	7.82538	0.880424	0.440212	−	0.897894i	$-0.354903\pi$
$79$	7.82538	0.880424	0.440212	+	0.897894i	$0.354903\pi$
$80$	36.4421	4.07435
$81$	1.00000	0.111111
$82$	11.1451	1.23077
$83$	−10.6038	−1.16391	−0.581957	−	0.813220i	$-0.697713\pi$
$83$	−10.6038	−1.16391	−0.581957	+	0.813220i	$0.697713\pi$
$84$	−13.5527	−1.47872
$85$	−8.16819	−0.885964
$86$	7.18592	0.774877
$87$	8.10042	0.868456
$88$	23.7629	2.53313
$89$	7.94221	0.841873	0.420936	−	0.907090i	$-0.361702\pi$
$89$	7.94221	0.841873	0.420936	+	0.907090i	$0.361702\pi$
$90$	−6.87293	−0.724470
$91$	0.0225223	0.00236098
$92$	−35.2840	−3.67862
$93$	−6.15328	−0.638066
$94$	−8.19015	−0.844749
$95$	9.57215	0.982082
$96$	20.7275	2.11549
$97$	8.26956	0.839647	0.419824	−	0.907606i	$-0.362092\pi$
$97$	8.26956	0.839647	0.419824	+	0.907606i	$0.362092\pi$
$98$	1.95985	0.197974
$99$	−2.56170	−0.257461
$100$	7.44367	0.744367
$101$	4.54551	0.452295	0.226147	−	0.974093i	$-0.427387\pi$
$101$	4.54551	0.452295	0.226147	+	0.974093i	$0.427387\pi$
$102$	−8.80426	−0.871752
$103$	−19.1423	−1.88614	−0.943072	−	0.332589i	$-0.892078\pi$
$103$	−19.1423	−1.88614	−0.943072	+	0.332589i	$0.892078\pi$
$104$	−0.0833692	−0.00817502
$105$	−6.32798	−0.617547
$106$	−20.7706	−2.01741
$107$	14.2842	1.38090	0.690452	−	0.723378i	$-0.257412\pi$
$107$	14.2842	1.38090	0.690452	+	0.723378i	$0.257412\pi$
$108$	−5.40814	−0.520398
$109$	−7.79780	−0.746894	−0.373447	−	0.927652i	$-0.621824\pi$
$109$	−7.79780	−0.746894	−0.373447	+	0.927652i	$0.621824\pi$
$110$	17.6064	1.67870
$111$	8.98686	0.852995
$112$	36.1655	3.41731
$113$	−10.6535	−1.00220	−0.501098	−	0.865390i	$-0.667071\pi$
$113$	−10.6535	−1.00220	−0.501098	+	0.865390i	$0.667071\pi$
$114$	10.3176	0.966327
$115$	−16.4747	−1.53628
$116$	−43.8082	−4.06749
$117$	0.00898741	0.000830886 0
$118$	−40.1261	−3.69391
$119$	−8.10618	−0.743092
$120$	23.4239	2.13830
$121$	−4.43769	−0.403427
$122$	−21.2238	−1.92152
$123$	4.09476	0.369212
$124$	33.2778	2.98843
$125$	−9.15018	−0.818417
$126$	−6.82075	−0.607641
$127$	2.69640	0.239267	0.119633	−	0.992818i	$-0.461828\pi$
$127$	2.69640	0.239267	0.119633	+	0.992818i	$0.461828\pi$
$128$	−33.5371	−2.96429
$129$	2.64014	0.232452
$130$	−0.0617698	−0.00541757
$131$	−6.52381	−0.569988	−0.284994	−	0.958529i	$-0.591992\pi$
$131$	−6.52381	−0.569988	−0.284994	+	0.958529i	$0.591992\pi$
$132$	13.8540	1.20584
$133$	9.49949	0.823710
$134$	26.8600	2.32035
$135$	−2.52515	−0.217330
$136$	30.0061	2.57300
$137$	13.9536	1.19214	0.596069	−	0.802933i	$-0.296729\pi$
$137$	13.9536	1.19214	0.596069	+	0.802933i	$0.296729\pi$
$138$	−17.7576	−1.51163
$139$	−18.1020	−1.53539	−0.767696	−	0.640814i	$-0.778597\pi$
$139$	−18.1020	−1.53539	−0.767696	+	0.640814i	$0.778597\pi$
$140$	34.2226	2.89233
$141$	−3.00910	−0.253412
$142$	−28.0539	−2.35423
$143$	−0.0230230	−0.00192528
$144$	14.4317	1.20264
$145$	−20.4548	−1.69868
$146$	−41.5839	−3.44151
$147$	0.720059	0.0593894
$148$	−48.6022	−3.99507
$149$	−10.0819	−0.825941	−0.412971	−	0.910744i	$-0.635509\pi$
$149$	−10.0819	−0.825941	−0.412971	+	0.910744i	$0.635509\pi$
$150$	3.74623	0.305878
$151$	−9.24131	−0.752047	−0.376024	−	0.926610i	$-0.622709\pi$
$151$	−9.24131	−0.752047	−0.376024	+	0.926610i	$0.622709\pi$
$152$	−35.1636	−2.85215
$153$	−3.23473	−0.261513
$154$	17.4727	1.40799
$155$	15.5380	1.24804
$156$	−0.0486051	−0.00389152
$157$	9.12763	0.728465	0.364232	−	0.931308i	$-0.381331\pi$
$157$	9.12763	0.728465	0.364232	+	0.931308i	$0.381331\pi$
$158$	−21.2990	−1.69446
$159$	−7.63121	−0.605195
$160$	−52.3400	−4.13784
$161$	−16.3497	−1.28853
$162$	−2.72179	−0.213844
$163$	1.58935	0.124487	0.0622436	−	0.998061i	$-0.480174\pi$
$163$	1.58935	0.124487	0.0622436	+	0.998061i	$0.480174\pi$
$164$	−22.1450	−1.72923
$165$	6.46868	0.503586
$166$	28.8612	2.24006
$167$	−1.69481	−0.131148	−0.0655740	−	0.997848i	$-0.520888\pi$
$167$	−1.69481	−0.131148	−0.0655740	+	0.997848i	$0.520888\pi$
$168$	23.2460	1.79347
$169$	−12.9999	−0.999994
$170$	22.2321	1.70512
$171$	3.79073	0.289884
$172$	−14.2783	−1.08871
$173$	−13.4851	−1.02525	−0.512627	−	0.858611i	$-0.671328\pi$
$173$	−13.4851	−1.02525	−0.512627	+	0.858611i	$0.671328\pi$
$174$	−22.0476	−1.67143
$175$	3.44919	0.260734
$176$	−36.9696	−2.78669
$177$	−14.7425	−1.10812
$178$	−21.6170	−1.62026
$179$	2.36174	0.176525	0.0882624	−	0.996097i	$-0.471869\pi$
$179$	2.36174	0.176525	0.0882624	+	0.996097i	$0.471869\pi$
$180$	13.6564	1.01788
$181$	1.71203	0.127254	0.0636271	−	0.997974i	$-0.479733\pi$
$181$	1.71203	0.127254	0.0636271	+	0.997974i	$0.479733\pi$
$182$	−0.0613009	−0.00454392
$183$	−7.79775	−0.576426
$184$	60.5204	4.46163
$185$	−22.6932	−1.66844
$186$	16.7479	1.22802
$187$	8.28642	0.605963
$188$	16.2736	1.18688
$189$	−2.50598	−0.182283
$190$	−26.0534	−1.89011
$191$	−12.6718	−0.916901	−0.458450	−	0.888720i	$-0.651595\pi$
$191$	−12.6718	−0.916901	−0.458450	+	0.888720i	$0.651595\pi$
$192$	−27.5525	−1.98843
$193$	9.04369	0.650979	0.325489	−	0.945546i	$-0.394471\pi$
$193$	9.04369	0.650979	0.325489	+	0.945546i	$0.394471\pi$
$194$	−22.5080	−1.61598
$195$	−0.0226945	−0.00162519
$196$	−3.89417	−0.278155
$197$	−21.2056	−1.51083	−0.755417	−	0.655245i	$-0.772565\pi$
$197$	−21.2056	−1.51083	−0.755417	+	0.655245i	$0.772565\pi$
$198$	6.97241	0.495507
$199$	−23.7970	−1.68692	−0.843462	−	0.537189i	$-0.819486\pi$
$199$	−23.7970	−1.68692	−0.843462	+	0.537189i	$0.819486\pi$
$200$	−12.7676	−0.902809
$201$	9.86850	0.696070
$202$	−12.3719	−0.870485
$203$	−20.2995	−1.42475
$204$	17.4939	1.22482
$205$	−10.3399	−0.722169
$206$	52.1012	3.63006
$207$	−6.52425	−0.453467
$208$	0.129703	0.00899330
$209$	−9.71070	−0.671703
$210$	17.2234	1.18853
$211$	0.0540010	0.00371758	0.00185879	−	0.999998i	$-0.499408\pi$
$211$	0.0540010	0.00371758	0.00185879	+	0.999998i	$0.499408\pi$
$212$	41.2706	2.83448
$213$	−10.3072	−0.706235
$214$	−38.8785	−2.65768
$215$	−6.66676	−0.454669
$216$	9.27623	0.631167
$217$	15.4200	1.04678
$218$	21.2240	1.43747
$219$	−15.2781	−1.03240
$220$	−34.9835	−2.35859
$221$	−0.0290719	−0.00195558
$222$	−24.4603	−1.64167
$223$	17.0202	1.13976	0.569878	−	0.821729i	$-0.306990\pi$
$223$	17.0202	1.13976	0.569878	+	0.821729i	$0.306990\pi$
$224$	−51.9427	−3.47057
$225$	1.37638	0.0917589
$226$	28.9966	1.92882
$227$	−7.30529	−0.484869	−0.242435	−	0.970168i	$-0.577946\pi$
$227$	−7.30529	−0.484869	−0.242435	+	0.970168i	$0.577946\pi$
$228$	−20.5008	−1.35770
$229$	4.09113	0.270350	0.135175	−	0.990822i	$-0.456840\pi$
$229$	4.09113	0.270350	0.135175	+	0.990822i	$0.456840\pi$
$230$	44.8407	2.95671
$231$	6.41957	0.422377
$232$	75.1413	4.93327
$233$	−19.0021	−1.24487	−0.622435	−	0.782671i	$-0.713857\pi$
$233$	−19.0021	−1.24487	−0.622435	+	0.782671i	$0.713857\pi$
$234$	−0.0244618	−0.00159912
$235$	7.59844	0.495668
$236$	79.7297	5.18996
$237$	−7.82538	−0.508313
$238$	22.0633	1.43015
$239$	22.3477	1.44555	0.722776	−	0.691082i	$-0.242866\pi$
$239$	22.3477	1.44555	0.722776	+	0.691082i	$0.242866\pi$
$240$	−36.4421	−2.35233
$241$	25.1073	1.61730	0.808651	−	0.588288i	$-0.200198\pi$
$241$	25.1073	1.61730	0.808651	+	0.588288i	$0.200198\pi$
$242$	12.0785	0.776433
$243$	−1.00000	−0.0641500
$244$	42.1713	2.69974
$245$	−1.81826	−0.116164
$246$	−11.1451	−0.710584
$247$	0.0340688	0.00216774
$248$	−57.0793	−3.62454
$249$	10.6038	0.671986
$250$	24.9048	1.57512
$251$	−14.5639	−0.919264	−0.459632	−	0.888110i	$-0.652019\pi$
$251$	−14.5639	−0.919264	−0.459632	+	0.888110i	$0.652019\pi$
$252$	13.5527	0.853739
$253$	16.7132	1.05075
$254$	−7.33902	−0.460491
$255$	8.16819	0.511512
$256$	36.1760	2.26100
$257$	−2.58712	−0.161380	−0.0806901	−	0.996739i	$-0.525712\pi$
$257$	−2.58712	−0.161380	−0.0806901	+	0.996739i	$0.525712\pi$
$258$	−7.18592	−0.447376
$259$	−22.5209	−1.39938
$260$	0.122735	0.00761171
$261$	−8.10042	−0.501403
$262$	17.7564	1.09700
$263$	−3.15240	−0.194385	−0.0971926	−	0.995266i	$-0.530986\pi$
$263$	−3.15240	−0.194385	−0.0971926	+	0.995266i	$0.530986\pi$
$264$	−23.7629	−1.46251
$265$	19.2700	1.18374
$266$	−25.8556	−1.58531
$267$	−7.94221	−0.486056
$268$	−53.3702	−3.26010
$269$	8.08202	0.492770	0.246385	−	0.969172i	$-0.420757\pi$
$269$	8.08202	0.492770	0.246385	+	0.969172i	$0.420757\pi$
$270$	6.87293	0.418273
$271$	−17.4109	−1.05764	−0.528818	−	0.848735i	$-0.677365\pi$
$271$	−17.4109	−1.05764	−0.528818	+	0.848735i	$0.677365\pi$
$272$	−46.6826	−2.83055
$273$	−0.0225223	−0.00136311
$274$	−37.9788	−2.29438
$275$	−3.52588	−0.212619
$276$	35.2840	2.12385
$277$	−7.01665	−0.421590	−0.210795	−	0.977530i	$-0.567605\pi$
$277$	−7.01665	−0.421590	−0.210795	+	0.977530i	$0.567605\pi$
$278$	49.2699	2.95501
$279$	6.15328	0.368388
$280$	−58.6998	−3.50798
$281$	−4.80692	−0.286757	−0.143378	−	0.989668i	$-0.545797\pi$
$281$	−4.80692	−0.286757	−0.143378	+	0.989668i	$0.545797\pi$
$282$	8.19015	0.487716
$283$	−30.7205	−1.82614	−0.913072	−	0.407798i	$-0.866297\pi$
$283$	−30.7205	−1.82614	−0.913072	+	0.407798i	$0.866297\pi$
$284$	55.7425	3.30771
$285$	−9.57215	−0.567005
$286$	0.0626638	0.00370539
$287$	−10.2614	−0.605711
$288$	−20.7275	−1.22138
$289$	−6.53650	−0.384500
$290$	55.6736	3.26927
$291$	−8.26956	−0.484770
$292$	82.6263	4.83534
$293$	−6.92345	−0.404472	−0.202236	−	0.979337i	$-0.564821\pi$
$293$	−6.92345	−0.404472	−0.202236	+	0.979337i	$0.564821\pi$
$294$	−1.95985	−0.114301
$295$	37.2271	2.16745
$296$	83.3642	4.84544
$297$	2.56170	0.148645
$298$	27.4408	1.58960
$299$	−0.0586361	−0.00339101
$300$	−7.44367	−0.429760
$301$	−6.61615	−0.381349
$302$	25.1529	1.44739
$303$	−4.54551	−0.261133
$304$	54.7064	3.13763
$305$	19.6905	1.12747
$306$	8.80426	0.503306
$307$	1.57390	0.0898273	0.0449136	−	0.998991i	$-0.485699\pi$
$307$	1.57390	0.0898273	0.0449136	+	0.998991i	$0.485699\pi$
$308$	−34.7179	−1.97824
$309$	19.1423	1.08897
$310$	−42.2911	−2.40197
$311$	−10.1026	−0.572865	−0.286433	−	0.958100i	$-0.592469\pi$
$311$	−10.1026	−0.572865	−0.286433	+	0.958100i	$0.592469\pi$
$312$	0.0833692	0.00471985
$313$	−1.66158	−0.0939181	−0.0469590	−	0.998897i	$-0.514953\pi$
$313$	−1.66158	−0.0939181	−0.0469590	+	0.998897i	$0.514953\pi$
$314$	−24.8435	−1.40200
$315$	6.32798	0.356541
$316$	42.3207	2.38072
$317$	−17.9676	−1.00916	−0.504580	−	0.863365i	$-0.668352\pi$
$317$	−17.9676	−1.00916	−0.504580	+	0.863365i	$0.668352\pi$
$318$	20.7706	1.16475
$319$	20.7509	1.16182
$320$	69.5742	3.88932
$321$	−14.2842	−0.797266
$322$	44.5003	2.47991
$323$	−12.2620	−0.682275
$324$	5.40814	0.300452
$325$	0.0123701	0.000686171 0
$326$	−4.32587	−0.239588
$327$	7.79780	0.431219
$328$	37.9839	2.09731
$329$	7.54076	0.415735
$330$	−17.6064	−0.969199
$331$	−11.2722	−0.619575	−0.309788	−	0.950806i	$-0.600258\pi$
$331$	−11.2722	−0.619575	−0.309788	+	0.950806i	$0.600258\pi$
$332$	−57.3465	−3.14730
$333$	−8.98686	−0.492477
$334$	4.61290	0.252407
$335$	−24.9194	−1.36149
$336$	−36.1655	−1.97299
$337$	−14.3189	−0.779999	−0.389999	−	0.920815i	$-0.627525\pi$
$337$	−14.3189	−0.779999	−0.389999	+	0.920815i	$0.627525\pi$
$338$	35.3830	1.92458
$339$	10.6535	0.578619
$340$	−44.1747	−2.39571
$341$	−15.7629	−0.853607
$342$	−10.3176	−0.557909
$343$	−19.3463	−1.04460
$344$	24.4906	1.32044
$345$	16.4747	0.886969
$346$	36.7036	1.97320
$347$	−0.584727	−0.0313898	−0.0156949	−	0.999877i	$-0.504996\pi$
$347$	−0.584727	−0.0313898	−0.0156949	+	0.999877i	$0.504996\pi$
$348$	43.8082	2.34836
$349$	1.50402	0.0805083	0.0402541	−	0.999189i	$-0.487183\pi$
$349$	1.50402	0.0805083	0.0402541	+	0.999189i	$0.487183\pi$
$350$	−9.38797	−0.501808
$351$	−0.00898741	−0.000479712 0
$352$	53.0976	2.83011
$353$	−16.3046	−0.867807	−0.433904	−	0.900959i	$-0.642864\pi$
$353$	−16.3046	−0.867807	−0.433904	+	0.900959i	$0.642864\pi$
$354$	40.1261	2.13268
$355$	26.0271	1.38138
$356$	42.9526	2.27648
$357$	8.10618	0.429025
$358$	−6.42816	−0.339739
$359$	−21.2309	−1.12052	−0.560262	−	0.828316i	$-0.689299\pi$
$359$	−21.2309	−1.12052	−0.560262	+	0.828316i	$0.689299\pi$
$360$	−23.4239	−1.23455
$361$	−4.63040	−0.243705
$362$	−4.65979	−0.244913
$363$	4.43769	0.232918
$364$	0.121803	0.00638424
$365$	38.5796	2.01935
$366$	21.2238	1.10939
$367$	29.1069	1.51937	0.759685	−	0.650291i	$-0.225353\pi$
$367$	29.1069	1.51937	0.759685	+	0.650291i	$0.225353\pi$
$368$	−94.1558	−4.90821
$369$	−4.09476	−0.213165
$370$	61.7660	3.21106
$371$	19.1237	0.992852
$372$	−33.2778	−1.72537
$373$	9.46520	0.490089	0.245045	−	0.969512i	$-0.421197\pi$
$373$	9.46520	0.490089	0.245045	+	0.969512i	$0.421197\pi$
$374$	−22.5539	−1.16623
$375$	9.15018	0.472513
$376$	−27.9131	−1.43951
$377$	−0.0728018	−0.00374948
$378$	6.82075	0.350822
$379$	−16.8735	−0.866735	−0.433367	−	0.901217i	$-0.642675\pi$
$379$	−16.8735	−0.866735	−0.433367	+	0.901217i	$0.642675\pi$
$380$	51.7675	2.65562
$381$	−2.69640	−0.138141
$382$	34.4900	1.76466
$383$	21.6284	1.10516	0.552580	−	0.833460i	$-0.313643\pi$
$383$	21.6284	1.10516	0.552580	+	0.833460i	$0.313643\pi$
$384$	33.5371	1.71144
$385$	−16.2104	−0.826158
$386$	−24.6150	−1.25287
$387$	−2.64014	−0.134206
$388$	44.7229	2.27046
$389$	3.88704	0.197081	0.0985405	−	0.995133i	$-0.468583\pi$
$389$	3.88704	0.197081	0.0985405	+	0.995133i	$0.468583\pi$
$390$	0.0617698	0.00312783
$391$	21.1042	1.06729
$392$	6.67943	0.337362
$393$	6.52381	0.329083
$394$	57.7170	2.90774
$395$	19.7602	0.994246
$396$	−13.8540	−0.696191
$397$	−34.9958	−1.75639	−0.878196	−	0.478302i	$-0.841253\pi$
$397$	−34.9958	−1.75639	−0.878196	+	0.478302i	$0.841253\pi$
$398$	64.7704	3.24665
$399$	−9.49949	−0.475569
$400$	19.8635	0.993175
$401$	12.0922	0.603857	0.301928	−	0.953331i	$-0.402370\pi$
$401$	12.0922	0.603857	0.301928	+	0.953331i	$0.402370\pi$
$402$	−26.8600	−1.33965
$403$	0.0553021	0.00275479
$404$	24.5827	1.22304
$405$	2.52515	0.125476
$406$	55.2510	2.74206
$407$	23.0216	1.14114
$408$	−30.0061	−1.48552
$409$	8.05810	0.398448	0.199224	−	0.979954i	$-0.436158\pi$
$409$	8.05810	0.398448	0.199224	+	0.979954i	$0.436158\pi$
$410$	28.1430	1.38988
$411$	−13.9536	−0.688281
$412$	−103.524	−5.10026
$413$	36.9445	1.81792
$414$	17.7576	0.872740
$415$	−26.7761	−1.31439
$416$	−0.186286	−0.00913343
$417$	18.1020	0.886459
$418$	26.4305	1.29276
$419$	−8.65585	−0.422866	−0.211433	−	0.977393i	$-0.567813\pi$
$419$	−8.65585	−0.422866	−0.211433	+	0.977393i	$0.567813\pi$
$420$	−34.2226	−1.66989
$421$	7.39750	0.360532	0.180266	−	0.983618i	$-0.442304\pi$
$421$	7.39750	0.360532	0.180266	+	0.983618i	$0.442304\pi$
$422$	−0.146979	−0.00715484
$423$	3.00910	0.146308
$424$	−70.7889	−3.43781
$425$	−4.45223	−0.215965
$426$	28.0539	1.35922
$427$	19.5410	0.945656
$428$	77.2508	3.73406
$429$	0.0230230	0.00111156
$430$	18.1455	0.875055
$431$	−3.19963	−0.154121	−0.0770605	−	0.997026i	$-0.524553\pi$
$431$	−3.19963	−0.154121	−0.0770605	+	0.997026i	$0.524553\pi$
$432$	−14.4317	−0.694343
$433$	−20.4711	−0.983779	−0.491890	−	0.870657i	$-0.663694\pi$
$433$	−20.4711	−0.983779	−0.491890	+	0.870657i	$0.663694\pi$
$434$	−41.9700	−2.01463
$435$	20.4548	0.980732
$436$	−42.1716	−2.01965
$437$	−24.7316	−1.18308
$438$	41.5839	1.98696
$439$	−17.4778	−0.834169	−0.417085	−	0.908868i	$-0.636948\pi$
$439$	−17.4778	−0.834169	−0.417085	+	0.908868i	$0.636948\pi$
$440$	60.0049	2.86062
$441$	−0.720059	−0.0342885
$442$	0.0791275	0.00376371
$443$	−19.4787	−0.925462	−0.462731	−	0.886499i	$-0.653130\pi$
$443$	−19.4787	−0.925462	−0.462731	+	0.886499i	$0.653130\pi$
$444$	48.6022	2.30656
$445$	20.0553	0.950711
$446$	−46.3254	−2.19357
$447$	10.0819	0.476857
$448$	69.0460	3.26212
$449$	5.39102	0.254418	0.127209	−	0.991876i	$-0.459398\pi$
$449$	5.39102	0.254418	0.127209	+	0.991876i	$0.459398\pi$
$450$	−3.74623	−0.176599
$451$	10.4895	0.493933
$452$	−57.6156	−2.71001
$453$	9.24131	0.434195
$454$	19.8835	0.933177
$455$	0.0568721	0.00266621
$456$	35.1636	1.64669
$457$	−29.3444	−1.37267	−0.686337	−	0.727284i	$-0.740782\pi$
$457$	−29.3444	−1.37267	−0.686337	+	0.727284i	$0.740782\pi$
$458$	−11.1352	−0.520314
$459$	3.23473	0.150984
$460$	−89.0975	−4.15419
$461$	22.1480	1.03153	0.515766	−	0.856729i	$-0.327507\pi$
$461$	22.1480	1.03153	0.515766	+	0.856729i	$0.327507\pi$
$462$	−17.4727	−0.812905
$463$	42.8284	1.99041	0.995203	−	0.0978322i	$-0.0311908\pi$
$463$	42.8284	1.99041	0.995203	+	0.0978322i	$0.0311908\pi$
$464$	−116.903	−5.42706
$465$	−15.5380	−0.720556
$466$	51.7198	2.39587
$467$	−5.38484	−0.249181	−0.124590	−	0.992208i	$-0.539762\pi$
$467$	−5.38484	−0.249181	−0.124590	+	0.992208i	$0.539762\pi$
$468$	0.0486051	0.00224677
$469$	−24.7303	−1.14194
$470$	−20.6814	−0.953960
$471$	−9.12763	−0.420579
$472$	−136.755	−6.29467
$473$	6.76326	0.310975
$474$	21.2990	0.978296
$475$	5.21749	0.239395
$476$	−43.8393	−2.00937
$477$	7.63121	0.349409
$478$	−60.8257	−2.78210
$479$	−37.1420	−1.69706	−0.848530	−	0.529147i	$-0.822512\pi$
$479$	−37.1420	−1.69706	−0.848530	+	0.529147i	$0.822512\pi$
$480$	52.3400	2.38898
$481$	−0.0807686	−0.00368273
$482$	−68.3368	−3.11265
$483$	16.3497	0.743935
$484$	−23.9996	−1.09089
$485$	20.8819	0.948198
$486$	2.72179	0.123463
$487$	−27.7231	−1.25625	−0.628126	−	0.778112i	$-0.716178\pi$
$487$	−27.7231	−1.25625	−0.628126	+	0.778112i	$0.716178\pi$
$488$	−72.3337	−3.27439
$489$	−1.58935	−0.0718727
$490$	4.94891	0.223569
$491$	23.6462	1.06714	0.533570	−	0.845756i	$-0.320850\pi$
$491$	23.6462	1.06714	0.533570	+	0.845756i	$0.320850\pi$
$492$	22.1450	0.998374
$493$	26.2027	1.18011
$494$	−0.0927280	−0.00417203
$495$	−6.46868	−0.290745
$496$	88.8021	3.98733
$497$	25.8296	1.15861
$498$	−28.8612	−1.29330
$499$	28.0641	1.25632	0.628161	−	0.778084i	$-0.283808\pi$
$499$	28.0641	1.25632	0.628161	+	0.778084i	$0.283808\pi$
$500$	−49.4854	−2.21305
$501$	1.69481	0.0757184
$502$	39.6398	1.76921
$503$	−41.0190	−1.82895	−0.914473	−	0.404647i	$-0.867394\pi$
$503$	−41.0190	−1.82895	−0.914473	+	0.404647i	$0.867394\pi$
$504$	−23.2460	−1.03546
$505$	11.4781	0.510768
$506$	−45.4897	−2.02227
$507$	12.9999	0.577347
$508$	14.5825	0.646993
$509$	41.7365	1.84994	0.924969	−	0.380042i	$-0.124090\pi$
$509$	41.7365	1.84994	0.924969	+	0.380042i	$0.124090\pi$
$510$	−22.2321	−0.984453
$511$	38.2868	1.69371
$512$	−31.3892	−1.38722
$513$	−3.79073	−0.167365
$514$	7.04160	0.310592
$515$	−48.3371	−2.12999
$516$	14.2783	0.628565
$517$	−7.70842	−0.339016
$518$	61.2972	2.69324
$519$	13.4851	0.591931
$520$	−0.210520	−0.00923190
$521$	−33.1305	−1.45147	−0.725737	−	0.687972i	$-0.758501\pi$
$521$	−33.1305	−1.45147	−0.725737	+	0.687972i	$0.758501\pi$
$522$	22.0476	0.964999
$523$	14.2584	0.623476	0.311738	−	0.950168i	$-0.399089\pi$
$523$	14.2584	0.623476	0.311738	+	0.950168i	$0.399089\pi$
$524$	−35.2817	−1.54129
$525$	−3.44919	−0.150535
$526$	8.58016	0.374113
$527$	−19.9042	−0.867042
$528$	36.9696	1.60889
$529$	19.5659	0.850690
$530$	−52.4488	−2.27823
$531$	14.7425	0.639772
$532$	51.3745	2.22737
$533$	−0.0368013	−0.00159404
$534$	21.6170	0.935460
$535$	36.0697	1.55943
$536$	91.5424	3.95403
$537$	−2.36174	−0.101917
$538$	−21.9976	−0.948382
$539$	1.84457	0.0794514
$540$	−13.6564	−0.587676
$541$	29.1284	1.25233	0.626163	−	0.779692i	$-0.284624\pi$
$541$	29.1284	1.25233	0.626163	+	0.779692i	$0.284624\pi$
$542$	47.3888	2.03552
$543$	−1.71203	−0.0734703
$544$	67.0478	2.87465
$545$	−19.6906	−0.843453
$546$	0.0613009	0.00262343
$547$	−3.88636	−0.166169	−0.0830843	−	0.996543i	$-0.526477\pi$
$547$	−3.88636	−0.166169	−0.0830843	+	0.996543i	$0.526477\pi$
$548$	75.4630	3.22362
$549$	7.79775	0.332800
$550$	9.59671	0.409205
$551$	−30.7065	−1.30814
$552$	−60.5204	−2.57592
$553$	19.6102	0.833912
$554$	19.0978	0.811390
$555$	22.6932	0.963272
$556$	−97.8981	−4.15181
$557$	−28.5133	−1.20815	−0.604073	−	0.796929i	$-0.706456\pi$
$557$	−28.5133	−1.20815	−0.604073	+	0.796929i	$0.706456\pi$
$558$	−16.7479	−0.708997
$559$	−0.0237281	−0.00100359
$560$	91.3232	3.85911
$561$	−8.28642	−0.349853
$562$	13.0834	0.551891
$563$	−29.9051	−1.26035	−0.630175	−	0.776453i	$-0.717017\pi$
$563$	−29.9051	−1.26035	−0.630175	+	0.776453i	$0.717017\pi$
$564$	−16.2736	−0.685244
$565$	−26.9017	−1.13176
$566$	83.6147	3.51459
$567$	2.50598	0.105241
$568$	−95.6116	−4.01177
$569$	−23.8466	−0.999701	−0.499851	−	0.866112i	$-0.666612\pi$
$569$	−23.8466	−0.999701	−0.499851	+	0.866112i	$0.666612\pi$
$570$	26.0534	1.09126
$571$	−35.7122	−1.49451	−0.747254	−	0.664538i	$-0.768628\pi$
$571$	−35.7122	−1.49451	−0.747254	+	0.664538i	$0.768628\pi$
$572$	−0.124512	−0.00520610
$573$	12.6718	0.529373
$574$	27.9293	1.16575
$575$	−8.97987	−0.374487
$576$	27.5525	1.14802
$577$	22.1907	0.923813	0.461906	−	0.886929i	$-0.347166\pi$
$577$	22.1907	0.923813	0.461906	+	0.886929i	$0.347166\pi$
$578$	17.7910	0.740007
$579$	−9.04369	−0.375843
$580$	−110.622	−4.59334
$581$	−26.5728	−1.10243
$582$	22.5080	0.932987
$583$	−19.5489	−0.809632
$584$	−141.724	−5.86456
$585$	0.0226945	0.000938304 0
$586$	18.8442	0.778446
$587$	0.672144	0.0277423	0.0138712	−	0.999904i	$-0.495585\pi$
$587$	0.672144	0.0277423	0.0138712	+	0.999904i	$0.495585\pi$
$588$	3.89417	0.160593
$589$	23.3254	0.961107
$590$	−101.324	−4.17146
$591$	21.2056	0.872280
$592$	−129.695	−5.33044
$593$	16.6713	0.684607	0.342304	−	0.939589i	$-0.388793\pi$
$593$	16.6713	0.684607	0.342304	+	0.939589i	$0.388793\pi$
$594$	−6.97241	−0.286081
$595$	−20.4693	−0.839160
$596$	−54.5243	−2.23340
$597$	23.7970	0.973946
$598$	0.159595	0.00652633
$599$	34.1618	1.39581	0.697907	−	0.716188i	$-0.254115\pi$
$599$	34.1618	1.39581	0.697907	+	0.716188i	$0.254115\pi$
$600$	12.7676	0.521237
$601$	47.3171	1.93010	0.965052	−	0.262060i	$-0.0844017\pi$
$601$	47.3171	1.93010	0.965052	+	0.262060i	$0.0844017\pi$
$602$	18.0078	0.733942
$603$	−9.86850	−0.401876
$604$	−49.9783	−2.03359
$605$	−11.2058	−0.455582
$606$	12.3719	0.502575
$607$	−6.45291	−0.261915	−0.130958	−	0.991388i	$-0.541805\pi$
$607$	−6.45291	−0.261915	−0.130958	+	0.991388i	$0.541805\pi$
$608$	−78.5722	−3.18652
$609$	20.2995	0.822577
$610$	−53.5934	−2.16993
$611$	0.0270440	0.00109408
$612$	−17.4939	−0.707148
$613$	−19.1541	−0.773625	−0.386813	−	0.922158i	$-0.626424\pi$
$613$	−19.1541	−0.773625	−0.386813	+	0.922158i	$0.626424\pi$
$614$	−4.28383	−0.172881
$615$	10.3399	0.416944
$616$	59.5494	2.39931
$617$	17.6867	0.712038	0.356019	−	0.934479i	$-0.384134\pi$
$617$	17.6867	0.712038	0.356019	+	0.934479i	$0.384134\pi$
$618$	−52.1012	−2.09582
$619$	5.85378	0.235283	0.117642	−	0.993056i	$-0.462467\pi$
$619$	5.85378	0.235283	0.117642	+	0.993056i	$0.462467\pi$
$620$	84.0314	3.37478
$621$	6.52425	0.261809
$622$	27.4971	1.10253
$623$	19.9030	0.797398
$624$	−0.129703	−0.00519228
$625$	−29.9875	−1.19950
$626$	4.52247	0.180754
$627$	9.71070	0.387808
$628$	49.3635	1.96982
$629$	29.0701	1.15910
$630$	−17.2234	−0.686198
$631$	−6.50682	−0.259032	−0.129516	−	0.991577i	$-0.541342\pi$
$631$	−6.50682	−0.259032	−0.129516	+	0.991577i	$0.541342\pi$
$632$	−72.5899	−2.88747
$633$	−0.0540010	−0.00214635
$634$	48.9039	1.94222
$635$	6.80881	0.270199
$636$	−41.2706	−1.63649
$637$	−0.00647146	−0.000256409 0
$638$	−56.4794	−2.23604
$639$	10.3072	0.407745
$640$	−84.6863	−3.34752
$641$	−12.2774	−0.484927	−0.242464	−	0.970161i	$-0.577955\pi$
$641$	−12.2774	−0.484927	−0.242464	+	0.970161i	$0.577955\pi$
$642$	38.8785	1.53441
$643$	−35.3402	−1.39368	−0.696841	−	0.717226i	$-0.745412\pi$
$643$	−35.3402	−1.39368	−0.696841	+	0.717226i	$0.745412\pi$
$644$	−88.4211	−3.48428
$645$	6.66676	0.262504
$646$	33.3745	1.31310
$647$	−34.9218	−1.37292	−0.686459	−	0.727169i	$-0.740836\pi$
$647$	−34.9218	−1.37292	−0.686459	+	0.727169i	$0.740836\pi$
$648$	−9.27623	−0.364405
$649$	−37.7660	−1.48244
$650$	−0.0336689	−0.00132060
$651$	−15.4200	−0.604358
$652$	8.59540	0.336622
$653$	46.9414	1.83696	0.918480	−	0.395467i	$-0.129417\pi$
$653$	46.9414	1.83696	0.918480	+	0.395467i	$0.129417\pi$
$654$	−21.2240	−0.829923
$655$	−16.4736	−0.643677
$656$	−59.0942	−2.30724
$657$	15.2781	0.596057
$658$	−20.5244	−0.800123
$659$	−29.6517	−1.15507	−0.577533	−	0.816367i	$-0.695984\pi$
$659$	−29.6517	−1.15507	−0.577533	+	0.816367i	$0.695984\pi$
$660$	34.9835	1.36173
$661$	−32.0522	−1.24669	−0.623343	−	0.781949i	$-0.714226\pi$
$661$	−32.0522	−1.24669	−0.623343	+	0.781949i	$0.714226\pi$
$662$	30.6805	1.19243
$663$	0.0290719	0.00112906
$664$	98.3628	3.81722
$665$	23.9876	0.930200
$666$	24.4603	0.947819
$667$	52.8492	2.04633
$668$	−9.16574	−0.354633
$669$	−17.0202	−0.658038
$670$	67.8255	2.62033
$671$	−19.9755	−0.771145
$672$	51.9427	2.00373
$673$	20.2900	0.782121	0.391060	−	0.920365i	$-0.372108\pi$
$673$	20.2900	0.782121	0.391060	+	0.920365i	$0.372108\pi$
$674$	38.9730	1.50118
$675$	−1.37638	−0.0529770
$676$	−70.3053	−2.70405
$677$	−50.6689	−1.94736	−0.973682	−	0.227911i	$-0.926810\pi$
$677$	−50.6689	−1.94736	−0.973682	+	0.227911i	$0.926810\pi$
$678$	−28.9966	−1.11361
$679$	20.7234	0.795290
$680$	75.7699	2.90564
$681$	7.30529	0.279939
$682$	42.9032	1.64285
$683$	10.5830	0.404946	0.202473	−	0.979288i	$-0.435102\pi$
$683$	10.5830	0.404946	0.202473	+	0.979288i	$0.435102\pi$
$684$	20.5008	0.783866
$685$	35.2350	1.34626
$686$	52.6566	2.01044
$687$	−4.09113	−0.156086
$688$	−38.1017	−1.45261
$689$	0.0685848	0.00261287
$690$	−44.8407	−1.70706
$691$	−36.5629	−1.39092	−0.695458	−	0.718566i	$-0.744799\pi$
$691$	−36.5629	−1.39092	−0.695458	+	0.718566i	$0.744799\pi$
$692$	−72.9294	−2.77236
$693$	−6.41957	−0.243859
$694$	1.59150	0.0604127
$695$	−45.7103	−1.73389
$696$	−75.1413	−2.84822
$697$	13.2455	0.501707
$698$	−4.09362	−0.154946
$699$	19.0021	0.718726
$700$	18.6537	0.705043
$701$	16.9132	0.638805	0.319402	−	0.947619i	$-0.396518\pi$
$701$	16.9132	0.638805	0.319402	+	0.947619i	$0.396518\pi$
$702$	0.0244618	0.000923252 0
$703$	−34.0667	−1.28485
$704$	−70.5812	−2.66013
$705$	−7.59844	−0.286174
$706$	44.3777	1.67018
$707$	11.3910	0.428401
$708$	−79.7297	−2.99642
$709$	−7.82711	−0.293953	−0.146977	−	0.989140i	$-0.546954\pi$
$709$	−7.82711	−0.293953	−0.146977	+	0.989140i	$0.546954\pi$
$710$	−70.8404	−2.65859
$711$	7.82538	0.293475
$712$	−73.6738	−2.76104
$713$	−40.1456	−1.50346
$714$	−22.0633	−0.825699
$715$	−0.0581366	−0.00217419
$716$	12.7726	0.477335
$717$	−22.3477	−0.834590
$718$	57.7860	2.15655
$719$	24.2547	0.904547	0.452274	−	0.891879i	$-0.350613\pi$
$719$	24.2547	0.904547	0.452274	+	0.891879i	$0.350613\pi$
$720$	36.4421	1.35812
$721$	−47.9702	−1.78650
$722$	12.6030	0.469034
$723$	−25.1073	−0.933750
$724$	9.25890	0.344104
$725$	−11.1493	−0.414074
$726$	−12.0785	−0.448274
$727$	−46.4606	−1.72313	−0.861564	−	0.507649i	$-0.830515\pi$
$727$	−46.4606	−1.72313	−0.861564	+	0.507649i	$0.830515\pi$
$728$	−0.208922	−0.00774315
$729$	1.00000	0.0370370
$730$	−105.006	−3.88643
$731$	8.54016	0.315869
$732$	−42.1713	−1.55870
$733$	14.3395	0.529641	0.264820	−	0.964298i	$-0.414687\pi$
$733$	14.3395	0.529641	0.264820	+	0.964298i	$0.414687\pi$
$734$	−79.2230	−2.92417
$735$	1.81826	0.0670674
$736$	135.231	4.98469
$737$	25.2801	0.931206
$738$	11.1451	0.410256
$739$	−17.4087	−0.640391	−0.320196	−	0.947351i	$-0.603749\pi$
$739$	−17.4087	−0.640391	−0.320196	+	0.947351i	$0.603749\pi$
$740$	−122.728	−4.51156
$741$	−0.0340688	−0.00125155
$742$	−52.0506	−1.91084
$743$	18.2556	0.669731	0.334866	−	0.942266i	$-0.391309\pi$
$743$	18.2556	0.669731	0.334866	+	0.942266i	$0.391309\pi$
$744$	57.0793	2.09263
$745$	−25.4583	−0.932720
$746$	−25.7623	−0.943224
$747$	−10.6038	−0.387971
$748$	44.8141	1.63856
$749$	35.7959	1.30795
$750$	−24.9048	−0.909396
$751$	−41.6406	−1.51949	−0.759743	−	0.650223i	$-0.774675\pi$
$751$	−41.6406	−1.51949	−0.759743	+	0.650223i	$0.774675\pi$
$752$	43.4264	1.58360
$753$	14.5639	0.530737
$754$	0.198151	0.00721624
$755$	−23.3357	−0.849273
$756$	−13.5527	−0.492906
$757$	25.1082	0.912571	0.456286	−	0.889833i	$-0.349180\pi$
$757$	25.1082	0.912571	0.456286	+	0.889833i	$0.349180\pi$
$758$	45.9262	1.66811
$759$	−16.7132	−0.606650
$760$	−88.7934	−3.22088
$761$	47.1014	1.70742	0.853712	−	0.520745i	$-0.174346\pi$
$761$	47.1014	1.70742	0.853712	+	0.520745i	$0.174346\pi$
$762$	7.33902	0.265865
$763$	−19.5411	−0.707437
$764$	−68.5309	−2.47936
$765$	−8.16819	−0.295321
$766$	−58.8680	−2.12699
$767$	0.132497	0.00478420
$768$	−36.1760	−1.30539
$769$	22.5017	0.811433	0.405717	−	0.913999i	$-0.367022\pi$
$769$	22.5017	0.811433	0.405717	+	0.913999i	$0.367022\pi$
$770$	44.1212	1.59002
$771$	2.58712	0.0931729
$772$	48.9095	1.76029
$773$	42.0679	1.51308	0.756539	−	0.653949i	$-0.226889\pi$
$773$	42.0679	1.51308	0.756539	+	0.653949i	$0.226889\pi$
$774$	7.18592	0.258292
$775$	8.46928	0.304226
$776$	−76.7103	−2.75374
$777$	22.5209	0.807933
$778$	−10.5797	−0.379301
$779$	−15.5221	−0.556137
$780$	−0.122735	−0.00439462
$781$	−26.4039	−0.944805
$782$	−57.4412	−2.05409
$783$	8.10042	0.289485
$784$	−10.3916	−0.371130
$785$	23.0486	0.822642
$786$	−17.7564	−0.633351
$787$	4.26341	0.151974	0.0759871	−	0.997109i	$-0.475789\pi$
$787$	4.26341	0.151974	0.0759871	+	0.997109i	$0.475789\pi$
$788$	−114.683	−4.08540
$789$	3.15240	0.112228
$790$	−53.7832	−1.91352
$791$	−26.6975	−0.949252
$792$	23.7629	0.844378
$793$	0.0700815	0.00248867
$794$	95.2513	3.38034
$795$	−19.2700	−0.683435
$796$	−128.697	−4.56156
$797$	19.5551	0.692679	0.346339	−	0.938109i	$-0.387425\pi$
$797$	19.5551	0.692679	0.346339	+	0.938109i	$0.387425\pi$
$798$	25.8556	0.915278
$799$	−9.73365	−0.344352
$800$	−28.5290	−1.00865
$801$	7.94221	0.280624
$802$	−32.9125	−1.16218
$803$	−39.1380	−1.38115
$804$	53.3702	1.88222
$805$	−41.2853	−1.45512
$806$	−0.150521	−0.00530186
$807$	−8.08202	−0.284501
$808$	−42.1651	−1.48336
$809$	27.9305	0.981984	0.490992	−	0.871164i	$-0.336634\pi$
$809$	27.9305	0.981984	0.490992	+	0.871164i	$0.336634\pi$
$810$	−6.87293	−0.241490
$811$	10.1825	0.357557	0.178778	−	0.983889i	$-0.442786\pi$
$811$	10.1825	0.357557	0.178778	+	0.983889i	$0.442786\pi$
$812$	−109.782	−3.85261
$813$	17.4109	0.610627
$814$	−62.6601	−2.19623
$815$	4.01334	0.140581
$816$	46.6826	1.63422
$817$	−10.0081	−0.350138
$818$	−21.9325	−0.766850
$819$	0.0225223	0.000786992 0
$820$	−55.9195	−1.95279
$821$	22.5442	0.786799	0.393400	−	0.919368i	$-0.371299\pi$
$821$	22.5442	0.786799	0.393400	+	0.919368i	$0.371299\pi$
$822$	37.9788	1.32466
$823$	19.9032	0.693782	0.346891	−	0.937905i	$-0.387237\pi$
$823$	19.9032	0.693782	0.346891	+	0.937905i	$0.387237\pi$
$824$	177.568	6.18587
$825$	3.52588	0.122755
$826$	−100.555	−3.49876
$827$	−5.82618	−0.202596	−0.101298	−	0.994856i	$-0.532300\pi$
$827$	−5.82618	−0.202596	−0.101298	+	0.994856i	$0.532300\pi$
$828$	−35.2840	−1.22621
$829$	−18.7305	−0.650538	−0.325269	−	0.945622i	$-0.605455\pi$
$829$	−18.7305	−0.650538	−0.325269	+	0.945622i	$0.605455\pi$
$830$	72.8788	2.52966
$831$	7.01665	0.243405
$832$	0.247625	0.00858487
$833$	2.32920	0.0807019
$834$	−49.2699	−1.70608
$835$	−4.27964	−0.148103
$836$	−52.5168	−1.81633
$837$	−6.15328	−0.212689
$838$	23.5594	0.813846
$839$	13.9342	0.481061	0.240531	−	0.970642i	$-0.422679\pi$
$839$	13.9342	0.481061	0.240531	+	0.970642i	$0.422679\pi$
$840$	58.6998	2.02533
$841$	36.6168	1.26265
$842$	−20.1344	−0.693878
$843$	4.80692	0.165559
$844$	0.292044	0.0100526
$845$	−32.8267	−1.12927
$846$	−8.19015	−0.281583
$847$	−11.1208	−0.382114
$848$	110.131	3.78192
$849$	30.7205	1.05432
$850$	12.1180	0.415645
$851$	58.6326	2.00990
$852$	−55.7425	−1.90971
$853$	12.4302	0.425603	0.212802	−	0.977095i	$-0.431741\pi$
$853$	12.4302	0.425603	0.212802	+	0.977095i	$0.431741\pi$
$854$	−53.1865	−1.82000
$855$	9.57215	0.327361
$856$	−132.503	−4.52887
$857$	−30.1810	−1.03096	−0.515481	−	0.856901i	$-0.672387\pi$
$857$	−30.1810	−1.03096	−0.515481	+	0.856901i	$0.672387\pi$
$858$	−0.0626638	−0.00213931
$859$	−16.0584	−0.547907	−0.273954	−	0.961743i	$-0.588331\pi$
$859$	−16.0584	−0.547907	−0.273954	+	0.961743i	$0.588331\pi$
$860$	−36.0548	−1.22946
$861$	10.2614	0.349707
$862$	8.70873	0.296621
$863$	−54.0899	−1.84124	−0.920620	−	0.390460i	$-0.872316\pi$
$863$	−54.0899	−1.84124	−0.920620	+	0.390460i	$0.872316\pi$
$864$	20.7275	0.705163
$865$	−34.0520	−1.15780
$866$	55.7181	1.89338
$867$	6.53650	0.221991
$868$	83.3935	2.83056
$869$	−20.0463	−0.680023
$870$	−55.6736	−1.88751
$871$	−0.0886922	−0.00300522
$872$	72.3342	2.44954
$873$	8.26956	0.279882
$874$	67.3143	2.27694
$875$	−22.9302	−0.775181
$876$	−82.6263	−2.79168
$877$	27.8199	0.939410	0.469705	−	0.882823i	$-0.344360\pi$
$877$	27.8199	0.939410	0.469705	+	0.882823i	$0.344360\pi$
$878$	47.5709	1.60544
$879$	6.92345	0.233522
$880$	−93.3537	−3.14695
$881$	34.1497	1.15053	0.575266	−	0.817967i	$-0.304899\pi$
$881$	34.1497	1.15053	0.575266	+	0.817967i	$0.304899\pi$
$882$	1.95985	0.0659915
$883$	−37.1696	−1.25086	−0.625428	−	0.780282i	$-0.715075\pi$
$883$	−37.1696	−1.25086	−0.625428	+	0.780282i	$0.715075\pi$
$884$	−0.157225	−0.00528803
$885$	−37.2271	−1.25138
$886$	53.0170	1.78114
$887$	−40.7429	−1.36801	−0.684006	−	0.729476i	$-0.739764\pi$
$887$	−40.7429	−1.36801	−0.684006	+	0.729476i	$0.739764\pi$
$888$	−83.3642	−2.79752
$889$	6.75712	0.226627
$890$	−54.5862	−1.82973
$891$	−2.56170	−0.0858202
$892$	92.0475	3.08198
$893$	11.4067	0.381710
$894$	−27.4408	−0.917757
$895$	5.96375	0.199346
$896$	−84.0434	−2.80769
$897$	0.0586361	0.00195780
$898$	−14.6732	−0.489652
$899$	−49.8442	−1.66240
$900$	7.44367	0.248122
$901$	−24.6849	−0.822375
$902$	−28.5503	−0.950622
$903$	6.61615	0.220172
$904$	98.8242	3.28685
$905$	4.32314	0.143706
$906$	−25.1529	−0.835649
$907$	−46.3779	−1.53995	−0.769976	−	0.638073i	$-0.779732\pi$
$907$	−46.3779	−1.53995	−0.769976	+	0.638073i	$0.779732\pi$
$908$	−39.5080	−1.31112
$909$	4.54551	0.150765
$910$	−0.154794	−0.00513137
$911$	52.9838	1.75543	0.877716	−	0.479182i	$-0.159067\pi$
$911$	52.9838	1.75543	0.877716	+	0.479182i	$0.159067\pi$
$912$	−54.7064	−1.81151
$913$	27.1636	0.898985
$914$	79.8693	2.64184
$915$	−19.6905	−0.650947
$916$	22.1254	0.731044
$917$	−16.3486	−0.539877
$918$	−8.80426	−0.290584
$919$	−43.6697	−1.44053	−0.720265	−	0.693699i	$-0.755980\pi$
$919$	−43.6697	−1.44053	−0.720265	+	0.693699i	$0.755980\pi$
$920$	152.823	5.03843
$921$	−1.57390	−0.0518618
$922$	−60.2821	−1.98528
$923$	0.0926347	0.00304911
$924$	34.7179	1.14214
$925$	−12.3694	−0.406702
$926$	−116.570	−3.83073
$927$	−19.1423	−0.628715
$928$	167.901	5.51163
$929$	9.20676	0.302064	0.151032	−	0.988529i	$-0.451740\pi$
$929$	9.20676	0.302064	0.151032	+	0.988529i	$0.451740\pi$
$930$	42.2911	1.38678
$931$	−2.72954	−0.0894572
$932$	−102.766	−3.36621
$933$	10.1026	0.330744
$934$	14.6564	0.479572
$935$	20.9244	0.684303
$936$	−0.0833692	−0.00272501
$937$	−59.9993	−1.96009	−0.980047	−	0.198768i	$-0.936306\pi$
$937$	−59.9993	−1.96009	−0.980047	+	0.198768i	$0.936306\pi$
$938$	67.3106	2.19777
$939$	1.66158	0.0542236
$940$	41.0934	1.34032
$941$	−32.8631	−1.07131	−0.535654	−	0.844438i	$-0.679935\pi$
$941$	−32.8631	−1.07131	−0.535654	+	0.844438i	$0.679935\pi$
$942$	24.8435	0.809445
$943$	26.7152	0.869968
$944$	212.759	6.92473
$945$	−6.32798	−0.205849
$946$	−18.4082	−0.598501
$947$	31.9849	1.03937	0.519685	−	0.854358i	$-0.326049\pi$
$947$	31.9849	1.03937	0.519685	+	0.854358i	$0.326049\pi$
$948$	−42.3207	−1.37451
$949$	0.137311	0.00445730
$950$	−14.2009	−0.460738
$951$	17.9676	0.582638
$952$	75.1947	2.43708
$953$	−54.1353	−1.75361	−0.876807	−	0.480842i	$-0.840331\pi$
$953$	−54.1353	−1.75361	−0.876807	+	0.480842i	$0.840331\pi$
$954$	−20.7706	−0.672472
$955$	−31.9983	−1.03544
$956$	120.859	3.90887
$957$	−20.7509	−0.670780
$958$	101.093	3.26615
$959$	34.9675	1.12916
$960$	−69.5742	−2.24550
$961$	6.86291	0.221384
$962$	0.219835	0.00708777
$963$	14.2842	0.460302
$964$	135.784	4.37330
$965$	22.8367	0.735138
$966$	−44.5003	−1.43177
$967$	−34.7130	−1.11630	−0.558148	−	0.829742i	$-0.688488\pi$
$967$	−34.7130	−1.11630	−0.558148	+	0.829742i	$0.688488\pi$
$968$	41.1650	1.32309
$969$	12.2620	0.393912
$970$	−56.8361	−1.82490
$971$	16.7071	0.536158	0.268079	−	0.963397i	$-0.413611\pi$
$971$	16.7071	0.536158	0.268079	+	0.963397i	$0.413611\pi$
$972$	−5.40814	−0.173466
$973$	−45.3633	−1.45428
$974$	75.4564	2.41778
$975$	−0.0123701	−0.000396161 0
$976$	112.534	3.60214
$977$	26.2235	0.838963	0.419482	−	0.907764i	$-0.362212\pi$
$977$	26.2235	0.838963	0.419482	+	0.907764i	$0.362212\pi$
$978$	4.32587	0.138326
$979$	−20.3456	−0.650247
$980$	−9.83338	−0.314116
$981$	−7.79780	−0.248965
$982$	−64.3601	−2.05381
$983$	25.4377	0.811336	0.405668	−	0.914020i	$-0.367039\pi$
$983$	25.4377	0.811336	0.405668	+	0.914020i	$0.367039\pi$
$984$	−37.9839	−1.21088
$985$	−53.5472	−1.70616
$986$	−71.3182	−2.27123
$987$	−7.54076	−0.240025
$988$	0.184249	0.00586173
$989$	17.2250	0.547722
$990$	17.6064	0.559567
$991$	32.1914	1.02259	0.511297	−	0.859404i	$-0.329165\pi$
$991$	32.1914	1.02259	0.511297	+	0.859404i	$0.329165\pi$
$992$	−127.542	−4.04946
$993$	11.2722	0.357712
$994$	−70.3026	−2.22986
$995$	−60.0910	−1.90501
$996$	57.3465	1.81709
$997$	−18.8373	−0.596585	−0.298292	−	0.954475i	$-0.596417\pi$
$997$	−18.8373	−0.596585	−0.298292	+	0.954475i	$0.596417\pi$
$998$	−76.3846	−2.41791
$999$	8.98686	0.284332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.c.1.5	✓	116

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.c.1.5	✓	116	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:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72179 q^{2} -1.00000 q^{3} +5.40814 q^{4} +2.52515 q^{5} +2.72179 q^{6} +2.50598 q^{7} -9.27623 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.72179 q^{2} -1.00000 q^{3} +5.40814 q^{4} +2.52515 q^{5} +2.72179 q^{6} +2.50598 q^{7} -9.27623 q^{8} +1.00000 q^{9} -6.87293 q^{10} -2.56170 q^{11} -5.40814 q^{12} +0.00898741 q^{13} -6.82075 q^{14} -2.52515 q^{15} +14.4317 q^{16} -3.23473 q^{17} -2.72179 q^{18} +3.79073 q^{19} +13.6564 q^{20} -2.50598 q^{21} +6.97241 q^{22} -6.52425 q^{23} +9.27623 q^{24} +1.37638 q^{25} -0.0244618 q^{26} -1.00000 q^{27} +13.5527 q^{28} -8.10042 q^{29} +6.87293 q^{30} +6.15328 q^{31} -20.7275 q^{32} +2.56170 q^{33} +8.80426 q^{34} +6.32798 q^{35} +5.40814 q^{36} -8.98686 q^{37} -10.3176 q^{38} -0.00898741 q^{39} -23.4239 q^{40} -4.09476 q^{41} +6.82075 q^{42} -2.64014 q^{43} -13.8540 q^{44} +2.52515 q^{45} +17.7576 q^{46} +3.00910 q^{47} -14.4317 q^{48} -0.720059 q^{49} -3.74623 q^{50} +3.23473 q^{51} +0.0486051 q^{52} +7.63121 q^{53} +2.72179 q^{54} -6.46868 q^{55} -23.2460 q^{56} -3.79073 q^{57} +22.0476 q^{58} +14.7425 q^{59} -13.6564 q^{60} +7.79775 q^{61} -16.7479 q^{62} +2.50598 q^{63} +27.5525 q^{64} +0.0226945 q^{65} -6.97241 q^{66} -9.86850 q^{67} -17.4939 q^{68} +6.52425 q^{69} -17.2234 q^{70} +10.3072 q^{71} -9.27623 q^{72} +15.2781 q^{73} +24.4603 q^{74} -1.37638 q^{75} +20.5008 q^{76} -6.41957 q^{77} +0.0244618 q^{78} +7.82538 q^{79} +36.4421 q^{80} +1.00000 q^{81} +11.1451 q^{82} -10.6038 q^{83} -13.5527 q^{84} -8.16819 q^{85} +7.18592 q^{86} +8.10042 q^{87} +23.7629 q^{88} +7.94221 q^{89} -6.87293 q^{90} +0.0225223 q^{91} -35.2840 q^{92} -6.15328 q^{93} -8.19015 q^{94} +9.57215 q^{95} +20.7275 q^{96} +8.26956 q^{97} +1.95985 q^{98} -2.56170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9}+O(q^{10}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9	\( 116 q - 16 q^{2} - 116 q^{3} + 116 q^{4} - 20 q^{5} + 16 q^{6} - 33 q^{7} - 45 q^{8} + 116 q^{9} + 3 q^{10} - 57 q^{11} - 116 q^{12} + 6 q^{13} - 9 q^{14} + 20 q^{15} + 112 q^{16} - 30 q^{17} - 16 q^{18} + 3 q^{19} - 54 q^{20} + 33 q^{21} - 22 q^{22} - 58 q^{23} + 45 q^{24} + 126 q^{25} - 21 q^{26} - 116 q^{27} - 77 q^{28} - 38 q^{29} - 3 q^{30} + 17 q^{31} - 106 q^{32} + 57 q^{33} + 35 q^{34} - 72 q^{35} + 116 q^{36} - 41 q^{37} - 45 q^{38} - 6 q^{39} + 5 q^{40} - 39 q^{41} + 9 q^{42} - 118 q^{43} - 103 q^{44} - 20 q^{45} - 8 q^{46} - 65 q^{47} - 112 q^{48} + 165 q^{49} - 72 q^{50} + 30 q^{51} - 10 q^{52} - 58 q^{53} + 16 q^{54} + 14 q^{55} - 23 q^{56} - 3 q^{57} - 27 q^{58} - 75 q^{59} + 54 q^{60} + 45 q^{61} - 73 q^{62} - 33 q^{63} + 111 q^{64} - 86 q^{65} + 22 q^{66} - 127 q^{67} - 94 q^{68} + 58 q^{69} - 7 q^{70} - 61 q^{71} - 45 q^{72} + 15 q^{73} - 51 q^{74} - 126 q^{75} + 96 q^{76} - 57 q^{77} + 21 q^{78} + 7 q^{79} - 144 q^{80} + 116 q^{81} - 37 q^{82} - 194 q^{83} + 77 q^{84} + 3 q^{85} - 57 q^{86} + 38 q^{87} - 42 q^{88} - 56 q^{89} + 3 q^{90} - 39 q^{91} - 138 q^{92} - 17 q^{93} + 51 q^{94} - 127 q^{95} + 106 q^{96} + 57 q^{97} - 105 q^{98} - 57 q^{99}+O(q^{100}) \) 116 * q - 16 * q^2 - 116 * q^3 + 116 * q^4 - 20 * q^5 + 16 * q^6 - 33 * q^7 - 45 * q^8 + 116 * q^9 + 3 * q^10 - 57 * q^11 - 116 * q^12 + 6 * q^13 - 9 * q^14 + 20 * q^15 + 112 * q^16 - 30 * q^17 - 16 * q^18 + 3 * q^19 - 54 * q^20 + 33 * q^21 - 22 * q^22 - 58 * q^23 + 45 * q^24 + 126 * q^25 - 21 * q^26 - 116 * q^27 - 77 * q^28 - 38 * q^29 - 3 * q^30 + 17 * q^31 - 106 * q^32 + 57 * q^33 + 35 * q^34 - 72 * q^35 + 116 * q^36 - 41 * q^37 - 45 * q^38 - 6 * q^39 + 5 * q^40 - 39 * q^41 + 9 * q^42 - 118 * q^43 - 103 * q^44 - 20 * q^45 - 8 * q^46 - 65 * q^47 - 112 * q^48 + 165 * q^49 - 72 * q^50 + 30 * q^51 - 10 * q^52 - 58 * q^53 + 16 * q^54 + 14 * q^55 - 23 * q^56 - 3 * q^57 - 27 * q^58 - 75 * q^59 + 54 * q^60 + 45 * q^61 - 73 * q^62 - 33 * q^63 + 111 * q^64 - 86 * q^65 + 22 * q^66 - 127 * q^67 - 94 * q^68 + 58 * q^69 - 7 * q^70 - 61 * q^71 - 45 * q^72 + 15 * q^73 - 51 * q^74 - 126 * q^75 + 96 * q^76 - 57 * q^77 + 21 * q^78 + 7 * q^79 - 144 * q^80 + 116 * q^81 - 37 * q^82 - 194 * q^83 + 77 * q^84 + 3 * q^85 - 57 * q^86 + 38 * q^87 - 42 * q^88 - 56 * q^89 + 3 * q^90 - 39 * q^91 - 138 * q^92 - 17 * q^93 + 51 * q^94 - 127 * q^95 + 106 * q^96 + 57 * q^97 - 105 * q^98 - 57 * q^99

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