LMFDB - Embedded newform 4030.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
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} - 18x^{6} + 12x^{5} + 98x^{4} - 18x^{3} - 173x^{2} - 48x + 32 $ x^8 - x^7 - 18x^6 + 12x^5 + 98x^4 - 18x^3 - 173x^2 - 48x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.14543$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.14543 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.14543 q^{6} -3.35025 q^{7} -1.00000 q^{8} +6.89376 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.14543 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.14543 q^{6} -3.35025 q^{7} -1.00000 q^{8} +6.89376 q^{9} +1.00000 q^{10} +4.63730 q^{11} -3.14543 q^{12} +1.00000 q^{13} +3.35025 q^{14} +3.14543 q^{15} +1.00000 q^{16} -6.47488 q^{17} -6.89376 q^{18} -1.14408 q^{19} -1.00000 q^{20} +10.5380 q^{21} -4.63730 q^{22} +5.14185 q^{23} +3.14543 q^{24} +1.00000 q^{25} -1.00000 q^{26} -12.2476 q^{27} -3.35025 q^{28} +7.69713 q^{29} -3.14543 q^{30} -1.00000 q^{31} -1.00000 q^{32} -14.5863 q^{33} +6.47488 q^{34} +3.35025 q^{35} +6.89376 q^{36} +7.39716 q^{37} +1.14408 q^{38} -3.14543 q^{39} +1.00000 q^{40} -8.55647 q^{41} -10.5380 q^{42} +3.35142 q^{43} +4.63730 q^{44} -6.89376 q^{45} -5.14185 q^{46} +8.04826 q^{47} -3.14543 q^{48} +4.22418 q^{49} -1.00000 q^{50} +20.3663 q^{51} +1.00000 q^{52} -0.862926 q^{53} +12.2476 q^{54} -4.63730 q^{55} +3.35025 q^{56} +3.59863 q^{57} -7.69713 q^{58} -14.0208 q^{59} +3.14543 q^{60} +13.8129 q^{61} +1.00000 q^{62} -23.0958 q^{63} +1.00000 q^{64} -1.00000 q^{65} +14.5863 q^{66} -13.7464 q^{67} -6.47488 q^{68} -16.1734 q^{69} -3.35025 q^{70} -11.2007 q^{71} -6.89376 q^{72} -7.01050 q^{73} -7.39716 q^{74} -3.14543 q^{75} -1.14408 q^{76} -15.5361 q^{77} +3.14543 q^{78} -13.8048 q^{79} -1.00000 q^{80} +17.8426 q^{81} +8.55647 q^{82} +3.45550 q^{83} +10.5380 q^{84} +6.47488 q^{85} -3.35142 q^{86} -24.2108 q^{87} -4.63730 q^{88} -4.22522 q^{89} +6.89376 q^{90} -3.35025 q^{91} +5.14185 q^{92} +3.14543 q^{93} -8.04826 q^{94} +1.14408 q^{95} +3.14543 q^{96} +12.1115 q^{97} -4.22418 q^{98} +31.9685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * 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
$3$	−3.14543	−1.81602	−0.908009	−	0.418951i	$-0.862398\pi$
$3$	−3.14543	−1.81602	−0.908009	+	0.418951i	$0.862398\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.14543	1.28412
$7$	−3.35025	−1.26628	−0.633138	−	0.774039i	$-0.718233\pi$
$7$	−3.35025	−1.26628	−0.633138	+	0.774039i	$0.718233\pi$
$8$	−1.00000	−0.353553
$9$	6.89376	2.29792
$10$	1.00000	0.316228
$11$	4.63730	1.39820	0.699100	−	0.715024i	$-0.253584\pi$
$11$	4.63730	1.39820	0.699100	+	0.715024i	$0.253584\pi$
$12$	−3.14543	−0.908009
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.35025	0.895392
$15$	3.14543	0.812148
$16$	1.00000	0.250000
$17$	−6.47488	−1.57039	−0.785194	−	0.619249i	$-0.787437\pi$
$17$	−6.47488	−1.57039	−0.785194	+	0.619249i	$0.787437\pi$
$18$	−6.89376	−1.62487
$19$	−1.14408	−0.262470	−0.131235	−	0.991351i	$-0.541894\pi$
$19$	−1.14408	−0.262470	−0.131235	+	0.991351i	$0.541894\pi$
$20$	−1.00000	−0.223607
$21$	10.5380	2.29958
$22$	−4.63730	−0.988677
$23$	5.14185	1.07215	0.536075	−	0.844170i	$-0.319906\pi$
$23$	5.14185	1.07215	0.536075	+	0.844170i	$0.319906\pi$
$24$	3.14543	0.642059
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−12.2476	−2.35704
$28$	−3.35025	−0.633138
$29$	7.69713	1.42932	0.714660	−	0.699472i	$-0.246581\pi$
$29$	7.69713	1.42932	0.714660	+	0.699472i	$0.246581\pi$
$30$	−3.14543	−0.574275
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−14.5863	−2.53916
$34$	6.47488	1.11043
$35$	3.35025	0.566296
$36$	6.89376	1.14896
$37$	7.39716	1.21609	0.608043	−	0.793904i	$-0.291955\pi$
$37$	7.39716	1.21609	0.608043	+	0.793904i	$0.291955\pi$
$38$	1.14408	0.185594
$39$	−3.14543	−0.503673
$40$	1.00000	0.158114
$41$	−8.55647	−1.33630	−0.668148	−	0.744028i	$-0.732913\pi$
$41$	−8.55647	−1.33630	−0.668148	+	0.744028i	$0.732913\pi$
$42$	−10.5380	−1.62605
$43$	3.35142	0.511087	0.255544	−	0.966798i	$-0.417746\pi$
$43$	3.35142	0.511087	0.255544	+	0.966798i	$0.417746\pi$
$44$	4.63730	0.699100
$45$	−6.89376	−1.02766
$46$	−5.14185	−0.758125
$47$	8.04826	1.17396	0.586980	−	0.809601i	$-0.300317\pi$
$47$	8.04826	1.17396	0.586980	+	0.809601i	$0.300317\pi$
$48$	−3.14543	−0.454004
$49$	4.22418	0.603454
$50$	−1.00000	−0.141421
$51$	20.3663	2.85185
$52$	1.00000	0.138675
$53$	−0.862926	−0.118532	−0.0592660	−	0.998242i	$-0.518876\pi$
$53$	−0.862926	−0.118532	−0.0592660	+	0.998242i	$0.518876\pi$
$54$	12.2476	1.66668
$55$	−4.63730	−0.625294
$56$	3.35025	0.447696
$57$	3.59863	0.476650
$58$	−7.69713	−1.01068
$59$	−14.0208	−1.82536	−0.912679	−	0.408676i	$-0.865991\pi$
$59$	−14.0208	−1.82536	−0.912679	+	0.408676i	$0.865991\pi$
$60$	3.14543	0.406074
$61$	13.8129	1.76856	0.884282	−	0.466953i	$-0.154648\pi$
$61$	13.8129	1.76856	0.884282	+	0.466953i	$0.154648\pi$
$62$	1.00000	0.127000
$63$	−23.0958	−2.90980
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	14.5863	1.79545
$67$	−13.7464	−1.67939	−0.839693	−	0.543062i	$-0.817265\pi$
$67$	−13.7464	−1.67939	−0.839693	+	0.543062i	$0.817265\pi$
$68$	−6.47488	−0.785194
$69$	−16.1734	−1.94704
$70$	−3.35025	−0.400432
$71$	−11.2007	−1.32928	−0.664640	−	0.747164i	$-0.731415\pi$
$71$	−11.2007	−1.32928	−0.664640	+	0.747164i	$0.731415\pi$
$72$	−6.89376	−0.812437
$73$	−7.01050	−0.820517	−0.410259	−	0.911969i	$-0.634562\pi$
$73$	−7.01050	−0.820517	−0.410259	+	0.911969i	$0.634562\pi$
$74$	−7.39716	−0.859902
$75$	−3.14543	−0.363204
$76$	−1.14408	−0.131235
$77$	−15.5361	−1.77051
$78$	3.14543	0.356150
$79$	−13.8048	−1.55316	−0.776581	−	0.630018i	$-0.783048\pi$
$79$	−13.8048	−1.55316	−0.776581	+	0.630018i	$0.783048\pi$
$80$	−1.00000	−0.111803
$81$	17.8426	1.98252
$82$	8.55647	0.944904
$83$	3.45550	0.379290	0.189645	−	0.981853i	$-0.439266\pi$
$83$	3.45550	0.379290	0.189645	+	0.981853i	$0.439266\pi$
$84$	10.5380	1.14979
$85$	6.47488	0.702299
$86$	−3.35142	−0.361393
$87$	−24.2108	−2.59567
$88$	−4.63730	−0.494338
$89$	−4.22522	−0.447873	−0.223936	−	0.974604i	$-0.571891\pi$
$89$	−4.22522	−0.447873	−0.223936	+	0.974604i	$0.571891\pi$
$90$	6.89376	0.726666
$91$	−3.35025	−0.351202
$92$	5.14185	0.536075
$93$	3.14543	0.326166
$94$	−8.04826	−0.830115
$95$	1.14408	0.117380
$96$	3.14543	0.321030
$97$	12.1115	1.22974	0.614870	−	0.788628i	$-0.289208\pi$
$97$	12.1115	1.22974	0.614870	+	0.788628i	$0.289208\pi$
$98$	−4.22418	−0.426706
$99$	31.9685	3.21295
$100$	1.00000	0.100000
$101$	10.2061	1.01554	0.507771	−	0.861492i	$-0.330470\pi$
$101$	10.2061	1.01554	0.507771	+	0.861492i	$0.330470\pi$
$102$	−20.3663	−2.01657
$103$	6.91460	0.681315	0.340658	−	0.940187i	$-0.389350\pi$
$103$	6.91460	0.681315	0.340658	+	0.940187i	$0.389350\pi$
$104$	−1.00000	−0.0980581
$105$	−10.5380	−1.02840
$106$	0.862926	0.0838148
$107$	−17.5948	−1.70096	−0.850478	−	0.526010i	$-0.823687\pi$
$107$	−17.5948	−1.70096	−0.850478	+	0.526010i	$0.823687\pi$
$108$	−12.2476	−1.17852
$109$	−14.3971	−1.37900	−0.689498	−	0.724287i	$-0.742169\pi$
$109$	−14.3971	−1.37900	−0.689498	+	0.724287i	$0.742169\pi$
$110$	4.63730	0.442150
$111$	−23.2673	−2.20843
$112$	−3.35025	−0.316569
$113$	−9.14511	−0.860300	−0.430150	−	0.902757i	$-0.641539\pi$
$113$	−9.14511	−0.860300	−0.430150	+	0.902757i	$0.641539\pi$
$114$	−3.59863	−0.337042
$115$	−5.14185	−0.479480
$116$	7.69713	0.714660
$117$	6.89376	0.637328
$118$	14.0208	1.29072
$119$	21.6925	1.98855
$120$	−3.14543	−0.287138
$121$	10.5046	0.954963
$122$	−13.8129	−1.25056
$123$	26.9138	2.42674
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	23.0958	2.05754
$127$	9.36671	0.831161	0.415581	−	0.909556i	$-0.363578\pi$
$127$	9.36671	0.831161	0.415581	+	0.909556i	$0.363578\pi$
$128$	−1.00000	−0.0883883
$129$	−10.5417	−0.928143
$130$	1.00000	0.0877058
$131$	−11.4980	−1.00458	−0.502291	−	0.864699i	$-0.667509\pi$
$131$	−11.4980	−1.00458	−0.502291	+	0.864699i	$0.667509\pi$
$132$	−14.5863	−1.26958
$133$	3.83295	0.332359
$134$	13.7464	1.18751
$135$	12.2476	1.05410
$136$	6.47488	0.555216
$137$	11.0646	0.945313	0.472656	−	0.881247i	$-0.343295\pi$
$137$	11.0646	0.945313	0.472656	+	0.881247i	$0.343295\pi$
$138$	16.1734	1.37677
$139$	12.2289	1.03724	0.518622	−	0.855004i	$-0.326445\pi$
$139$	12.2289	1.03724	0.518622	+	0.855004i	$0.326445\pi$
$140$	3.35025	0.283148
$141$	−25.3153	−2.13193
$142$	11.2007	0.939942
$143$	4.63730	0.387791
$144$	6.89376	0.574480
$145$	−7.69713	−0.639212
$146$	7.01050	0.580193
$147$	−13.2869	−1.09588
$148$	7.39716	0.608043
$149$	18.4227	1.50924	0.754622	−	0.656159i	$-0.227820\pi$
$149$	18.4227	1.50924	0.754622	+	0.656159i	$0.227820\pi$
$150$	3.14543	0.256824
$151$	1.15579	0.0940567	0.0470283	−	0.998894i	$-0.485025\pi$
$151$	1.15579	0.0940567	0.0470283	+	0.998894i	$0.485025\pi$
$152$	1.14408	0.0927971
$153$	−44.6363	−3.60863
$154$	15.5361	1.25194
$155$	1.00000	0.0803219
$156$	−3.14543	−0.251836
$157$	−13.9727	−1.11515	−0.557573	−	0.830128i	$-0.688267\pi$
$157$	−13.9727	−1.11515	−0.557573	+	0.830128i	$0.688267\pi$
$158$	13.8048	1.09825
$159$	2.71428	0.215256
$160$	1.00000	0.0790569
$161$	−17.2265	−1.35764
$162$	−17.8426	−1.40185
$163$	−9.84184	−0.770872	−0.385436	−	0.922735i	$-0.625949\pi$
$163$	−9.84184	−0.770872	−0.385436	+	0.922735i	$0.625949\pi$
$164$	−8.55647	−0.668148
$165$	14.5863	1.13555
$166$	−3.45550	−0.268198
$167$	−8.88827	−0.687795	−0.343898	−	0.939007i	$-0.611747\pi$
$167$	−8.88827	−0.687795	−0.343898	+	0.939007i	$0.611747\pi$
$168$	−10.5380	−0.813024
$169$	1.00000	0.0769231
$170$	−6.47488	−0.496601
$171$	−7.88701	−0.603135
$172$	3.35142	0.255544
$173$	18.5162	1.40776	0.703881	−	0.710318i	$-0.251449\pi$
$173$	18.5162	1.40776	0.703881	+	0.710318i	$0.251449\pi$
$174$	24.2108	1.83542
$175$	−3.35025	−0.253255
$176$	4.63730	0.349550
$177$	44.1017	3.31488
$178$	4.22522	0.316694
$179$	12.5807	0.940324	0.470162	−	0.882580i	$-0.344195\pi$
$179$	12.5807	0.940324	0.470162	+	0.882580i	$0.344195\pi$
$180$	−6.89376	−0.513830
$181$	−18.2020	−1.35294	−0.676471	−	0.736469i	$-0.736492\pi$
$181$	−18.2020	−1.35294	−0.676471	+	0.736469i	$0.736492\pi$
$182$	3.35025	0.248337
$183$	−43.4477	−3.21174
$184$	−5.14185	−0.379062
$185$	−7.39716	−0.543850
$186$	−3.14543	−0.230634
$187$	−30.0260	−2.19572
$188$	8.04826	0.586980
$189$	41.0324	2.98467
$190$	−1.14408	−0.0830003
$191$	4.19886	0.303819	0.151909	−	0.988394i	$-0.451458\pi$
$191$	4.19886	0.303819	0.151909	+	0.988394i	$0.451458\pi$
$192$	−3.14543	−0.227002
$193$	21.1562	1.52286	0.761429	−	0.648249i	$-0.224498\pi$
$193$	21.1562	1.52286	0.761429	+	0.648249i	$0.224498\pi$
$194$	−12.1115	−0.869558
$195$	3.14543	0.225249
$196$	4.22418	0.301727
$197$	16.0906	1.14641	0.573204	−	0.819413i	$-0.305700\pi$
$197$	16.0906	1.14641	0.573204	+	0.819413i	$0.305700\pi$
$198$	−31.9685	−2.27190
$199$	−24.1237	−1.71008	−0.855041	−	0.518561i	$-0.826468\pi$
$199$	−24.1237	−1.71008	−0.855041	+	0.518561i	$0.826468\pi$
$200$	−1.00000	−0.0707107
$201$	43.2383	3.04979
$202$	−10.2061	−0.718097
$203$	−25.7873	−1.80991
$204$	20.3663	1.42593
$205$	8.55647	0.597610
$206$	−6.91460	−0.481763
$207$	35.4467	2.46372
$208$	1.00000	0.0693375
$209$	−5.30545	−0.366985
$210$	10.5380	0.727191
$211$	−13.0587	−0.898997	−0.449499	−	0.893281i	$-0.648397\pi$
$211$	−13.0587	−0.898997	−0.449499	+	0.893281i	$0.648397\pi$
$212$	−0.862926	−0.0592660
$213$	35.2311	2.41399
$214$	17.5948	1.20276
$215$	−3.35142	−0.228565
$216$	12.2476	0.833341
$217$	3.35025	0.227430
$218$	14.3971	0.975098
$219$	22.0511	1.49007
$220$	−4.63730	−0.312647
$221$	−6.47488	−0.435548
$222$	23.2673	1.56160
$223$	17.7443	1.18825	0.594124	−	0.804373i	$-0.297499\pi$
$223$	17.7443	1.18825	0.594124	+	0.804373i	$0.297499\pi$
$224$	3.35025	0.223848
$225$	6.89376	0.459584
$226$	9.14511	0.608324
$227$	24.5278	1.62796	0.813982	−	0.580890i	$-0.197295\pi$
$227$	24.5278	1.62796	0.813982	+	0.580890i	$0.197295\pi$
$228$	3.59863	0.238325
$229$	4.74074	0.313277	0.156638	−	0.987656i	$-0.449934\pi$
$229$	4.74074	0.313277	0.156638	+	0.987656i	$0.449934\pi$
$230$	5.14185	0.339044
$231$	48.8679	3.21527
$232$	−7.69713	−0.505341
$233$	26.8253	1.75739	0.878693	−	0.477387i	$-0.158416\pi$
$233$	26.8253	1.75739	0.878693	+	0.477387i	$0.158416\pi$
$234$	−6.89376	−0.450659
$235$	−8.04826	−0.525011
$236$	−14.0208	−0.912679
$237$	43.4221	2.82057
$238$	−21.6925	−1.40611
$239$	−0.417155	−0.0269835	−0.0134918	−	0.999909i	$-0.504295\pi$
$239$	−0.417155	−0.0269835	−0.0134918	+	0.999909i	$0.504295\pi$
$240$	3.14543	0.203037
$241$	3.99975	0.257646	0.128823	−	0.991668i	$-0.458880\pi$
$241$	3.99975	0.257646	0.128823	+	0.991668i	$0.458880\pi$
$242$	−10.5046	−0.675261
$243$	−19.3802	−1.24324
$244$	13.8129	0.884282
$245$	−4.22418	−0.269873
$246$	−26.9138	−1.71596
$247$	−1.14408	−0.0727961
$248$	1.00000	0.0635001
$249$	−10.8690	−0.688797
$250$	1.00000	0.0632456
$251$	−5.85806	−0.369757	−0.184879	−	0.982761i	$-0.559189\pi$
$251$	−5.85806	−0.369757	−0.184879	+	0.982761i	$0.559189\pi$
$252$	−23.0958	−1.45490
$253$	23.8443	1.49908
$254$	−9.36671	−0.587720
$255$	−20.3663	−1.27539
$256$	1.00000	0.0625000
$257$	2.01499	0.125692	0.0628459	−	0.998023i	$-0.479982\pi$
$257$	2.01499	0.125692	0.0628459	+	0.998023i	$0.479982\pi$
$258$	10.5417	0.656296
$259$	−24.7823	−1.53990
$260$	−1.00000	−0.0620174
$261$	53.0621	3.28446
$262$	11.4980	0.710347
$263$	16.8755	1.04059	0.520294	−	0.853987i	$-0.325822\pi$
$263$	16.8755	1.04059	0.520294	+	0.853987i	$0.325822\pi$
$264$	14.5863	0.897727
$265$	0.862926	0.0530091
$266$	−3.83295	−0.235013
$267$	13.2902	0.813345
$268$	−13.7464	−0.839693
$269$	18.2442	1.11237	0.556185	−	0.831058i	$-0.312264\pi$
$269$	18.2442	1.11237	0.556185	+	0.831058i	$0.312264\pi$
$270$	−12.2476	−0.745363
$271$	−25.0245	−1.52013	−0.760066	−	0.649846i	$-0.774833\pi$
$271$	−25.0245	−1.52013	−0.760066	+	0.649846i	$0.774833\pi$
$272$	−6.47488	−0.392597
$273$	10.5380	0.637788
$274$	−11.0646	−0.668437
$275$	4.63730	0.279640
$276$	−16.1734	−0.973522
$277$	−5.31815	−0.319537	−0.159768	−	0.987155i	$-0.551075\pi$
$277$	−5.31815	−0.319537	−0.159768	+	0.987155i	$0.551075\pi$
$278$	−12.2289	−0.733442
$279$	−6.89376	−0.412719
$280$	−3.35025	−0.200216
$281$	−20.1345	−1.20112	−0.600561	−	0.799579i	$-0.705056\pi$
$281$	−20.1345	−1.20112	−0.600561	+	0.799579i	$0.705056\pi$
$282$	25.3153	1.50750
$283$	−10.8267	−0.643583	−0.321792	−	0.946811i	$-0.604285\pi$
$283$	−10.8267	−0.643583	−0.321792	+	0.946811i	$0.604285\pi$
$284$	−11.2007	−0.664640
$285$	−3.59863	−0.213164
$286$	−4.63730	−0.274210
$287$	28.6663	1.69212
$288$	−6.89376	−0.406219
$289$	24.9241	1.46612
$290$	7.69713	0.451991
$291$	−38.0960	−2.23323
$292$	−7.01050	−0.410259
$293$	6.89881	0.403033	0.201516	−	0.979485i	$-0.435413\pi$
$293$	6.89881	0.403033	0.201516	+	0.979485i	$0.435413\pi$
$294$	13.2869	0.774906
$295$	14.0208	0.816325
$296$	−7.39716	−0.429951
$297$	−56.7957	−3.29562
$298$	−18.4227	−1.06720
$299$	5.14185	0.297361
$300$	−3.14543	−0.181602
$301$	−11.2281	−0.647177
$302$	−1.15579	−0.0665081
$303$	−32.1025	−1.84424
$304$	−1.14408	−0.0656175
$305$	−13.8129	−0.790926
$306$	44.6363	2.55169
$307$	−25.5027	−1.45552	−0.727758	−	0.685834i	$-0.759437\pi$
$307$	−25.5027	−1.45552	−0.727758	+	0.685834i	$0.759437\pi$
$308$	−15.5361	−0.885253
$309$	−21.7494	−1.23728
$310$	−1.00000	−0.0567962
$311$	28.4500	1.61325	0.806625	−	0.591063i	$-0.201292\pi$
$311$	28.4500	1.61325	0.806625	+	0.591063i	$0.201292\pi$
$312$	3.14543	0.178075
$313$	−7.12373	−0.402657	−0.201329	−	0.979524i	$-0.564526\pi$
$313$	−7.12373	−0.402657	−0.201329	+	0.979524i	$0.564526\pi$
$314$	13.9727	0.788527
$315$	23.0958	1.30130
$316$	−13.8048	−0.776581
$317$	5.03241	0.282648	0.141324	−	0.989963i	$-0.454864\pi$
$317$	5.03241	0.282648	0.141324	+	0.989963i	$0.454864\pi$
$318$	−2.71428	−0.152209
$319$	35.6939	1.99848
$320$	−1.00000	−0.0559017
$321$	55.3434	3.08897
$322$	17.2265	0.959995
$323$	7.40778	0.412180
$324$	17.8426	0.991258
$325$	1.00000	0.0554700
$326$	9.84184	0.545089
$327$	45.2853	2.50428
$328$	8.55647	0.472452
$329$	−26.9637	−1.48656
$330$	−14.5863	−0.802952
$331$	8.31196	0.456866	0.228433	−	0.973560i	$-0.426640\pi$
$331$	8.31196	0.456866	0.228433	+	0.973560i	$0.426640\pi$
$332$	3.45550	0.189645
$333$	50.9942	2.79447
$334$	8.88827	0.486345
$335$	13.7464	0.751044
$336$	10.5380	0.574895
$337$	12.6077	0.686783	0.343392	−	0.939192i	$-0.388424\pi$
$337$	12.6077	0.686783	0.343392	+	0.939192i	$0.388424\pi$
$338$	−1.00000	−0.0543928
$339$	28.7654	1.56232
$340$	6.47488	0.351150
$341$	−4.63730	−0.251124
$342$	7.88701	0.426481
$343$	9.29970	0.502137
$344$	−3.35142	−0.180697
$345$	16.1734	0.870744
$346$	−18.5162	−0.995438
$347$	−22.7434	−1.22093	−0.610464	−	0.792044i	$-0.709017\pi$
$347$	−22.7434	−1.22093	−0.610464	+	0.792044i	$0.709017\pi$
$348$	−24.2108	−1.29784
$349$	−2.49993	−0.133818	−0.0669090	−	0.997759i	$-0.521314\pi$
$349$	−2.49993	−0.133818	−0.0669090	+	0.997759i	$0.521314\pi$
$350$	3.35025	0.179078
$351$	−12.2476	−0.653727
$352$	−4.63730	−0.247169
$353$	−17.1102	−0.910685	−0.455342	−	0.890316i	$-0.650483\pi$
$353$	−17.1102	−0.910685	−0.455342	+	0.890316i	$0.650483\pi$
$354$	−44.1017	−2.34398
$355$	11.2007	0.594472
$356$	−4.22522	−0.223936
$357$	−68.2322	−3.61123
$358$	−12.5807	−0.664910
$359$	22.8197	1.20438	0.602190	−	0.798353i	$-0.294295\pi$
$359$	22.8197	1.20438	0.602190	+	0.798353i	$0.294295\pi$
$360$	6.89376	0.363333
$361$	−17.6911	−0.931110
$362$	18.2020	0.956674
$363$	−33.0415	−1.73423
$364$	−3.35025	−0.175601
$365$	7.01050	0.366946
$366$	43.4477	2.27105
$367$	−1.16650	−0.0608909	−0.0304455	−	0.999536i	$-0.509693\pi$
$367$	−1.16650	−0.0608909	−0.0304455	+	0.999536i	$0.509693\pi$
$368$	5.14185	0.268038
$369$	−58.9862	−3.07070
$370$	7.39716	0.384560
$371$	2.89102	0.150094
$372$	3.14543	0.163083
$373$	2.08368	0.107889	0.0539446	−	0.998544i	$-0.482821\pi$
$373$	2.08368	0.107889	0.0539446	+	0.998544i	$0.482821\pi$
$374$	30.0260	1.55261
$375$	3.14543	0.162430
$376$	−8.04826	−0.415058
$377$	7.69713	0.396422
$378$	−41.0324	−2.11048
$379$	0.659422	0.0338722	0.0169361	−	0.999857i	$-0.494609\pi$
$379$	0.659422	0.0338722	0.0169361	+	0.999857i	$0.494609\pi$
$380$	1.14408	0.0586901
$381$	−29.4624	−1.50940
$382$	−4.19886	−0.214832
$383$	−7.85714	−0.401481	−0.200741	−	0.979644i	$-0.564335\pi$
$383$	−7.85714	−0.401481	−0.200741	+	0.979644i	$0.564335\pi$
$384$	3.14543	0.160515
$385$	15.5361	0.791795
$386$	−21.1562	−1.07682
$387$	23.1039	1.17444
$388$	12.1115	0.614870
$389$	12.1788	0.617490	0.308745	−	0.951145i	$-0.400091\pi$
$389$	12.1788	0.617490	0.308745	+	0.951145i	$0.400091\pi$
$390$	−3.14543	−0.159275
$391$	−33.2929	−1.68369
$392$	−4.22418	−0.213353
$393$	36.1661	1.82434
$394$	−16.0906	−0.810633
$395$	13.8048	0.694595
$396$	31.9685	1.60648
$397$	−0.463787	−0.0232768	−0.0116384	−	0.999932i	$-0.503705\pi$
$397$	−0.463787	−0.0232768	−0.0116384	+	0.999932i	$0.503705\pi$
$398$	24.1237	1.20921
$399$	−12.0563	−0.603570
$400$	1.00000	0.0500000
$401$	8.75695	0.437301	0.218651	−	0.975803i	$-0.429834\pi$
$401$	8.75695	0.437301	0.218651	+	0.975803i	$0.429834\pi$
$402$	−43.2383	−2.15653
$403$	−1.00000	−0.0498135
$404$	10.2061	0.507771
$405$	−17.8426	−0.886608
$406$	25.7873	1.27980
$407$	34.3029	1.70033
$408$	−20.3663	−1.00828
$409$	27.5546	1.36249	0.681244	−	0.732057i	$-0.261439\pi$
$409$	27.5546	1.36249	0.681244	+	0.732057i	$0.261439\pi$
$410$	−8.55647	−0.422574
$411$	−34.8030	−1.71670
$412$	6.91460	0.340658
$413$	46.9734	2.31141
$414$	−35.4467	−1.74211
$415$	−3.45550	−0.169624
$416$	−1.00000	−0.0490290
$417$	−38.4653	−1.88365
$418$	5.30545	0.259498
$419$	34.7063	1.69551	0.847757	−	0.530385i	$-0.177953\pi$
$419$	34.7063	1.69551	0.847757	+	0.530385i	$0.177953\pi$
$420$	−10.5380	−0.514201
$421$	36.3113	1.76970	0.884852	−	0.465873i	$-0.154260\pi$
$421$	36.3113	1.76970	0.884852	+	0.465873i	$0.154260\pi$
$422$	13.0587	0.635687
$423$	55.4828	2.69767
$424$	0.862926	0.0419074
$425$	−6.47488	−0.314078
$426$	−35.2311	−1.70695
$427$	−46.2768	−2.23949
$428$	−17.5948	−0.850478
$429$	−14.5863	−0.704235
$430$	3.35142	0.161620
$431$	−19.2993	−0.929615	−0.464807	−	0.885412i	$-0.653876\pi$
$431$	−19.2993	−0.929615	−0.464807	+	0.885412i	$0.653876\pi$
$432$	−12.2476	−0.589261
$433$	22.2064	1.06717	0.533585	−	0.845746i	$-0.320844\pi$
$433$	22.2064	1.06717	0.533585	+	0.845746i	$0.320844\pi$
$434$	−3.35025	−0.160817
$435$	24.2108	1.16082
$436$	−14.3971	−0.689498
$437$	−5.88269	−0.281407
$438$	−22.0511	−1.05364
$439$	26.5566	1.26748	0.633740	−	0.773546i	$-0.281519\pi$
$439$	26.5566	1.26748	0.633740	+	0.773546i	$0.281519\pi$
$440$	4.63730	0.221075
$441$	29.1205	1.38669
$442$	6.47488	0.307979
$443$	1.97318	0.0937487	0.0468744	−	0.998901i	$-0.485074\pi$
$443$	1.97318	0.0937487	0.0468744	+	0.998901i	$0.485074\pi$
$444$	−23.2673	−1.10422
$445$	4.22522	0.200295
$446$	−17.7443	−0.840219
$447$	−57.9473	−2.74081
$448$	−3.35025	−0.158284
$449$	−2.03683	−0.0961242	−0.0480621	−	0.998844i	$-0.515305\pi$
$449$	−2.03683	−0.0961242	−0.0480621	+	0.998844i	$0.515305\pi$
$450$	−6.89376	−0.324975
$451$	−39.6790	−1.86841
$452$	−9.14511	−0.430150
$453$	−3.63546	−0.170809
$454$	−24.5278	−1.15114
$455$	3.35025	0.157062
$456$	−3.59863	−0.168521
$457$	4.57196	0.213867	0.106934	−	0.994266i	$-0.465897\pi$
$457$	4.57196	0.213867	0.106934	+	0.994266i	$0.465897\pi$
$458$	−4.74074	−0.221520
$459$	79.3015	3.70148
$460$	−5.14185	−0.239740
$461$	−1.96968	−0.0917372	−0.0458686	−	0.998947i	$-0.514606\pi$
$461$	−1.96968	−0.0917372	−0.0458686	+	0.998947i	$0.514606\pi$
$462$	−48.8679	−2.27354
$463$	−19.3359	−0.898613	−0.449307	−	0.893378i	$-0.648329\pi$
$463$	−19.3359	−0.898613	−0.449307	+	0.893378i	$0.648329\pi$
$464$	7.69713	0.357330
$465$	−3.14543	−0.145866
$466$	−26.8253	−1.24266
$467$	−25.1131	−1.16210	−0.581049	−	0.813869i	$-0.697357\pi$
$467$	−25.1131	−1.16210	−0.581049	+	0.813869i	$0.697357\pi$
$468$	6.89376	0.318664
$469$	46.0538	2.12657
$470$	8.04826	0.371239
$471$	43.9503	2.02512
$472$	14.0208	0.645362
$473$	15.5416	0.714602
$474$	−43.4221	−1.99444
$475$	−1.14408	−0.0524940
$476$	21.6925	0.994273
$477$	−5.94881	−0.272377
$478$	0.417155	0.0190802
$479$	17.8450	0.815357	0.407678	−	0.913126i	$-0.366338\pi$
$479$	17.8450	0.815357	0.407678	+	0.913126i	$0.366338\pi$
$480$	−3.14543	−0.143569
$481$	7.39716	0.337281
$482$	−3.99975	−0.182183
$483$	54.1848	2.46549
$484$	10.5046	0.477482
$485$	−12.1115	−0.549957
$486$	19.3802	0.879102
$487$	−27.2416	−1.23443	−0.617216	−	0.786793i	$-0.711740\pi$
$487$	−27.2416	−1.23443	−0.617216	+	0.786793i	$0.711740\pi$
$488$	−13.8129	−0.625282
$489$	30.9569	1.39992
$490$	4.22418	0.190829
$491$	26.8803	1.21309	0.606545	−	0.795049i	$-0.292555\pi$
$491$	26.8803	1.21309	0.606545	+	0.795049i	$0.292555\pi$
$492$	26.9138	1.21337
$493$	−49.8380	−2.24459
$494$	1.14408	0.0514746
$495$	−31.9685	−1.43688
$496$	−1.00000	−0.0449013
$497$	37.5252	1.68323
$498$	10.8690	0.487053
$499$	21.1748	0.947913	0.473956	−	0.880548i	$-0.342825\pi$
$499$	21.1748	0.947913	0.473956	+	0.880548i	$0.342825\pi$
$500$	−1.00000	−0.0447214
$501$	27.9575	1.24905
$502$	5.85806	0.261458
$503$	29.7524	1.32659	0.663296	−	0.748357i	$-0.269157\pi$
$503$	29.7524	1.32659	0.663296	+	0.748357i	$0.269157\pi$
$504$	23.0958	1.02877
$505$	−10.2061	−0.454164
$506$	−23.8443	−1.06001
$507$	−3.14543	−0.139694
$508$	9.36671	0.415581
$509$	0.247817	0.0109843	0.00549214	−	0.999985i	$-0.498252\pi$
$509$	0.247817	0.0109843	0.00549214	+	0.999985i	$0.498252\pi$
$510$	20.3663	0.901835
$511$	23.4869	1.03900
$512$	−1.00000	−0.0441942
$513$	14.0122	0.618653
$514$	−2.01499	−0.0888775
$515$	−6.91460	−0.304693
$516$	−10.5417	−0.464072
$517$	37.3223	1.64143
$518$	24.7823	1.08887
$519$	−58.2415	−2.55652
$520$	1.00000	0.0438529
$521$	2.28502	0.100109	0.0500543	−	0.998746i	$-0.484061\pi$
$521$	2.28502	0.100109	0.0500543	+	0.998746i	$0.484061\pi$
$522$	−53.0621	−2.32247
$523$	−18.3920	−0.804226	−0.402113	−	0.915590i	$-0.631724\pi$
$523$	−18.3920	−0.804226	−0.402113	+	0.915590i	$0.631724\pi$
$524$	−11.4980	−0.502291
$525$	10.5380	0.459916
$526$	−16.8755	−0.735807
$527$	6.47488	0.282050
$528$	−14.5863	−0.634789
$529$	3.43865	0.149506
$530$	−0.862926	−0.0374831
$531$	−96.6564	−4.19453
$532$	3.83295	0.166180
$533$	−8.55647	−0.370622
$534$	−13.2902	−0.575122
$535$	17.5948	0.760691
$536$	13.7464	0.593753
$537$	−39.5717	−1.70765
$538$	−18.2442	−0.786565
$539$	19.5888	0.843749
$540$	12.2476	0.527051
$541$	−33.8446	−1.45509	−0.727547	−	0.686058i	$-0.759340\pi$
$541$	−33.8446	−1.45509	−0.727547	+	0.686058i	$0.759340\pi$
$542$	25.0245	1.07490
$543$	57.2531	2.45697
$544$	6.47488	0.277608
$545$	14.3971	0.616706
$546$	−10.5380	−0.450984
$547$	3.11771	0.133304	0.0666518	−	0.997776i	$-0.478768\pi$
$547$	3.11771	0.133304	0.0666518	+	0.997776i	$0.478768\pi$
$548$	11.0646	0.472656
$549$	95.2230	4.06402
$550$	−4.63730	−0.197735
$551$	−8.80613	−0.375154
$552$	16.1734	0.688384
$553$	46.2495	1.96673
$554$	5.31815	0.225947
$555$	23.2673	0.987641
$556$	12.2289	0.518622
$557$	1.12015	0.0474621	0.0237310	−	0.999718i	$-0.492445\pi$
$557$	1.12015	0.0474621	0.0237310	+	0.999718i	$0.492445\pi$
$558$	6.89376	0.291836
$559$	3.35142	0.141750
$560$	3.35025	0.141574
$561$	94.4448	3.98746
$562$	20.1345	0.849322
$563$	15.9666	0.672912	0.336456	−	0.941699i	$-0.390772\pi$
$563$	15.9666	0.672912	0.336456	+	0.941699i	$0.390772\pi$
$564$	−25.3153	−1.06597
$565$	9.14511	0.384738
$566$	10.8267	0.455082
$567$	−59.7773	−2.51041
$568$	11.2007	0.469971
$569$	9.07009	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$569$	9.07009	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$570$	3.59863	0.150730
$571$	8.59574	0.359721	0.179860	−	0.983692i	$-0.442435\pi$
$571$	8.59574	0.359721	0.179860	+	0.983692i	$0.442435\pi$
$572$	4.63730	0.193895
$573$	−13.2072	−0.551740
$574$	−28.6663	−1.19651
$575$	5.14185	0.214430
$576$	6.89376	0.287240
$577$	2.43524	0.101381	0.0506903	−	0.998714i	$-0.483858\pi$
$577$	2.43524	0.101381	0.0506903	+	0.998714i	$0.483858\pi$
$578$	−24.9241	−1.03670
$579$	−66.5455	−2.76554
$580$	−7.69713	−0.319606
$581$	−11.5768	−0.480286
$582$	38.0960	1.57913
$583$	−4.00165	−0.165731
$584$	7.01050	0.290097
$585$	−6.89376	−0.285022
$586$	−6.89881	−0.284987
$587$	−33.7578	−1.39333	−0.696666	−	0.717395i	$-0.745334\pi$
$587$	−33.7578	−1.39333	−0.696666	+	0.717395i	$0.745334\pi$
$588$	−13.2869	−0.547941
$589$	1.14408	0.0471410
$590$	−14.0208	−0.577229
$591$	−50.6119	−2.08190
$592$	7.39716	0.304021
$593$	−11.4196	−0.468947	−0.234474	−	0.972122i	$-0.575337\pi$
$593$	−11.4196	−0.468947	−0.234474	+	0.972122i	$0.575337\pi$
$594$	56.7957	2.33036
$595$	−21.6925	−0.889304
$596$	18.4227	0.754622
$597$	75.8794	3.10554
$598$	−5.14185	−0.210266
$599$	19.6285	0.801997	0.400999	−	0.916079i	$-0.368663\pi$
$599$	19.6285	0.801997	0.400999	+	0.916079i	$0.368663\pi$
$600$	3.14543	0.128412
$601$	−2.19622	−0.0895858	−0.0447929	−	0.998996i	$-0.514263\pi$
$601$	−2.19622	−0.0895858	−0.0447929	+	0.998996i	$0.514263\pi$
$602$	11.2281	0.457623
$603$	−94.7641	−3.85909
$604$	1.15579	0.0470283
$605$	−10.5046	−0.427073
$606$	32.1025	1.30408
$607$	−23.7768	−0.965070	−0.482535	−	0.875877i	$-0.660284\pi$
$607$	−23.7768	−0.965070	−0.482535	+	0.875877i	$0.660284\pi$
$608$	1.14408	0.0463986
$609$	81.1123	3.28684
$610$	13.8129	0.559269
$611$	8.04826	0.325598
$612$	−44.6363	−1.80431
$613$	14.9463	0.603676	0.301838	−	0.953359i	$-0.402400\pi$
$613$	14.9463	0.603676	0.301838	+	0.953359i	$0.402400\pi$
$614$	25.5027	1.02920
$615$	−26.9138	−1.08527
$616$	15.5361	0.625969
$617$	−0.151001	−0.00607906	−0.00303953	−	0.999995i	$-0.500968\pi$
$617$	−0.151001	−0.00607906	−0.00303953	+	0.999995i	$0.500968\pi$
$618$	21.7494	0.874889
$619$	20.5346	0.825354	0.412677	−	0.910877i	$-0.364594\pi$
$619$	20.5346	0.825354	0.412677	+	0.910877i	$0.364594\pi$
$620$	1.00000	0.0401610
$621$	−62.9752	−2.52711
$622$	−28.4500	−1.14074
$623$	14.1556	0.567130
$624$	−3.14543	−0.125918
$625$	1.00000	0.0400000
$626$	7.12373	0.284722
$627$	16.6879	0.666452
$628$	−13.9727	−0.557573
$629$	−47.8957	−1.90973
$630$	−23.0958	−0.920159
$631$	21.7549	0.866051	0.433025	−	0.901382i	$-0.357446\pi$
$631$	21.7549	0.866051	0.433025	+	0.901382i	$0.357446\pi$
$632$	13.8048	0.549125
$633$	41.0752	1.63259
$634$	−5.03241	−0.199862
$635$	−9.36671	−0.371707
$636$	2.71428	0.107628
$637$	4.22418	0.167368
$638$	−35.6939	−1.41314
$639$	−77.2150	−3.05458
$640$	1.00000	0.0395285
$641$	11.0391	0.436019	0.218009	−	0.975947i	$-0.430044\pi$
$641$	11.0391	0.436019	0.218009	+	0.975947i	$0.430044\pi$
$642$	−55.3434	−2.18423
$643$	−6.84902	−0.270099	−0.135049	−	0.990839i	$-0.543119\pi$
$643$	−6.84902	−0.270099	−0.135049	+	0.990839i	$0.543119\pi$
$644$	−17.2265	−0.678819
$645$	10.5417	0.415078
$646$	−7.40778	−0.291455
$647$	31.4649	1.23701	0.618507	−	0.785779i	$-0.287738\pi$
$647$	31.4649	1.23701	0.618507	+	0.785779i	$0.287738\pi$
$648$	−17.8426	−0.700925
$649$	−65.0189	−2.55222
$650$	−1.00000	−0.0392232
$651$	−10.5380	−0.413017
$652$	−9.84184	−0.385436
$653$	−8.68221	−0.339761	−0.169881	−	0.985465i	$-0.554338\pi$
$653$	−8.68221	−0.339761	−0.169881	+	0.985465i	$0.554338\pi$
$654$	−45.2853	−1.77079
$655$	11.4980	0.449263
$656$	−8.55647	−0.334074
$657$	−48.3287	−1.88548
$658$	26.9637	1.05115
$659$	10.4357	0.406517	0.203259	−	0.979125i	$-0.434847\pi$
$659$	10.4357	0.406517	0.203259	+	0.979125i	$0.434847\pi$
$660$	14.5863	0.567773
$661$	−35.0487	−1.36324	−0.681619	−	0.731708i	$-0.738724\pi$
$661$	−35.0487	−1.36324	−0.681619	+	0.731708i	$0.738724\pi$
$662$	−8.31196	−0.323053
$663$	20.3663	0.790962
$664$	−3.45550	−0.134099
$665$	−3.83295	−0.148636
$666$	−50.9942	−1.97599
$667$	39.5775	1.53245
$668$	−8.88827	−0.343898
$669$	−55.8137	−2.15788
$670$	−13.7464	−0.531068
$671$	64.0547	2.47281
$672$	−10.5380	−0.406512
$673$	41.7574	1.60963	0.804816	−	0.593525i	$-0.202264\pi$
$673$	41.7574	1.60963	0.804816	+	0.593525i	$0.202264\pi$
$674$	−12.6077	−0.485629
$675$	−12.2476	−0.471409
$676$	1.00000	0.0384615
$677$	−42.7456	−1.64285	−0.821424	−	0.570319i	$-0.806820\pi$
$677$	−42.7456	−1.64285	−0.821424	+	0.570319i	$0.806820\pi$
$678$	−28.7654	−1.10473
$679$	−40.5767	−1.55719
$680$	−6.47488	−0.248300
$681$	−77.1505	−2.95641
$682$	4.63730	0.177572
$683$	18.1344	0.693895	0.346947	−	0.937885i	$-0.387218\pi$
$683$	18.1344	0.693895	0.346947	+	0.937885i	$0.387218\pi$
$684$	−7.88701	−0.301567
$685$	−11.0646	−0.422757
$686$	−9.29970	−0.355064
$687$	−14.9117	−0.568916
$688$	3.35142	0.127772
$689$	−0.862926	−0.0328749
$690$	−16.1734	−0.615709
$691$	14.7567	0.561371	0.280686	−	0.959800i	$-0.409438\pi$
$691$	14.7567	0.561371	0.280686	+	0.959800i	$0.409438\pi$
$692$	18.5162	0.703881
$693$	−107.102	−4.06848
$694$	22.7434	0.863327
$695$	−12.2289	−0.463870
$696$	24.2108	0.917708
$697$	55.4021	2.09850
$698$	2.49993	0.0946236
$699$	−84.3773	−3.19144
$700$	−3.35025	−0.126628
$701$	−21.7155	−0.820182	−0.410091	−	0.912045i	$-0.634503\pi$
$701$	−21.7155	−0.820182	−0.410091	+	0.912045i	$0.634503\pi$
$702$	12.2476	0.462255
$703$	−8.46294	−0.319186
$704$	4.63730	0.174775
$705$	25.3153	0.953429
$706$	17.1102	0.643951
$707$	−34.1929	−1.28596
$708$	44.1017	1.65744
$709$	13.8191	0.518988	0.259494	−	0.965745i	$-0.416444\pi$
$709$	13.8191	0.518988	0.259494	+	0.965745i	$0.416444\pi$
$710$	−11.2007	−0.420355
$711$	−95.1670	−3.56904
$712$	4.22522	0.158347
$713$	−5.14185	−0.192564
$714$	68.2322	2.55353
$715$	−4.63730	−0.173425
$716$	12.5807	0.470162
$717$	1.31213	0.0490025
$718$	−22.8197	−0.851625
$719$	37.3578	1.39321	0.696605	−	0.717454i	$-0.254693\pi$
$719$	37.3578	1.39321	0.696605	+	0.717454i	$0.254693\pi$
$720$	−6.89376	−0.256915
$721$	−23.1656	−0.862733
$722$	17.6911	0.658394
$723$	−12.5809	−0.467890
$724$	−18.2020	−0.676471
$725$	7.69713	0.285864
$726$	33.0415	1.22629
$727$	2.58336	0.0958113	0.0479057	−	0.998852i	$-0.484745\pi$
$727$	2.58336	0.0958113	0.0479057	+	0.998852i	$0.484745\pi$
$728$	3.35025	0.124169
$729$	7.43110	0.275226
$730$	−7.01050	−0.259470
$731$	−21.7001	−0.802606
$732$	−43.4477	−1.60587
$733$	−37.8787	−1.39908	−0.699541	−	0.714592i	$-0.746612\pi$
$733$	−37.8787	−1.39908	−0.699541	+	0.714592i	$0.746612\pi$
$734$	1.16650	0.0430564
$735$	13.2869	0.490094
$736$	−5.14185	−0.189531
$737$	−63.7461	−2.34812
$738$	58.9862	2.17131
$739$	−18.7594	−0.690075	−0.345037	−	0.938589i	$-0.612134\pi$
$739$	−18.7594	−0.690075	−0.345037	+	0.938589i	$0.612134\pi$
$740$	−7.39716	−0.271925
$741$	3.59863	0.132199
$742$	−2.89102	−0.106133
$743$	−9.19550	−0.337350	−0.168675	−	0.985672i	$-0.553949\pi$
$743$	−9.19550	−0.337350	−0.168675	+	0.985672i	$0.553949\pi$
$744$	−3.14543	−0.115317
$745$	−18.4227	−0.674955
$746$	−2.08368	−0.0762891
$747$	23.8214	0.871578
$748$	−30.0260	−1.09786
$749$	58.9471	2.15388
$750$	−3.14543	−0.114855
$751$	−27.4712	−1.00244	−0.501218	−	0.865321i	$-0.667115\pi$
$751$	−27.4712	−1.00244	−0.501218	+	0.865321i	$0.667115\pi$
$752$	8.04826	0.293490
$753$	18.4261	0.671486
$754$	−7.69713	−0.280313
$755$	−1.15579	−0.0420634
$756$	41.0324	1.49233
$757$	−8.27527	−0.300770	−0.150385	−	0.988628i	$-0.548051\pi$
$757$	−8.27527	−0.300770	−0.150385	+	0.988628i	$0.548051\pi$
$758$	−0.659422	−0.0239513
$759$	−75.0008	−2.72236
$760$	−1.14408	−0.0415001
$761$	46.5567	1.68768	0.843840	−	0.536595i	$-0.180290\pi$
$761$	46.5567	1.68768	0.843840	+	0.536595i	$0.180290\pi$
$762$	29.4624	1.06731
$763$	48.2341	1.74619
$764$	4.19886	0.151909
$765$	44.6363	1.61383
$766$	7.85714	0.283890
$767$	−14.0208	−0.506263
$768$	−3.14543	−0.113501
$769$	−28.2043	−1.01707	−0.508536	−	0.861041i	$-0.669813\pi$
$769$	−28.2043	−1.01707	−0.508536	+	0.861041i	$0.669813\pi$
$770$	−15.5361	−0.559883
$771$	−6.33803	−0.228259
$772$	21.1562	0.761429
$773$	24.2511	0.872250	0.436125	−	0.899886i	$-0.356351\pi$
$773$	24.2511	0.872250	0.436125	+	0.899886i	$0.356351\pi$
$774$	−23.1039	−0.830453
$775$	−1.00000	−0.0359211
$776$	−12.1115	−0.434779
$777$	77.9512	2.79648
$778$	−12.1788	−0.436632
$779$	9.78929	0.350738
$780$	3.14543	0.112625
$781$	−51.9411	−1.85860
$782$	33.2929	1.19055
$783$	−94.2711	−3.36897
$784$	4.22418	0.150863
$785$	13.9727	0.498708
$786$	−36.1661	−1.29000
$787$	−36.7547	−1.31016	−0.655082	−	0.755558i	$-0.727366\pi$
$787$	−36.7547	−1.31016	−0.655082	+	0.755558i	$0.727366\pi$
$788$	16.0906	0.573204
$789$	−53.0808	−1.88973
$790$	−13.8048	−0.491153
$791$	30.6384	1.08938
$792$	−31.9685	−1.13595
$793$	13.8129	0.490511
$794$	0.463787	0.0164592
$795$	−2.71428	−0.0962655
$796$	−24.1237	−0.855041
$797$	−47.1893	−1.67153	−0.835765	−	0.549087i	$-0.814975\pi$
$797$	−47.1893	−1.67153	−0.835765	+	0.549087i	$0.814975\pi$
$798$	12.0563	0.426789
$799$	−52.1115	−1.84357
$800$	−1.00000	−0.0353553
$801$	−29.1277	−1.02918
$802$	−8.75695	−0.309219
$803$	−32.5098	−1.14725
$804$	43.2383	1.52490
$805$	17.2265	0.607154
$806$	1.00000	0.0352235
$807$	−57.3861	−2.02009
$808$	−10.2061	−0.359048
$809$	42.2283	1.48467	0.742334	−	0.670030i	$-0.233719\pi$
$809$	42.2283	1.48467	0.742334	+	0.670030i	$0.233719\pi$
$810$	17.8426	0.626926
$811$	43.7053	1.53470	0.767350	−	0.641228i	$-0.221575\pi$
$811$	43.7053	1.53470	0.767350	+	0.641228i	$0.221575\pi$
$812$	−25.7873	−0.904957
$813$	78.7130	2.76059
$814$	−34.3029	−1.20232
$815$	9.84184	0.344745
$816$	20.3663	0.712963
$817$	−3.83430	−0.134145
$818$	−27.5546	−0.963424
$819$	−23.0958	−0.807033
$820$	8.55647	0.298805
$821$	−0.257194	−0.00897615	−0.00448807	−	0.999990i	$-0.501429\pi$
$821$	−0.257194	−0.00897615	−0.00448807	+	0.999990i	$0.501429\pi$
$822$	34.8030	1.21389
$823$	24.1762	0.842730	0.421365	−	0.906891i	$-0.361551\pi$
$823$	24.1762	0.842730	0.421365	+	0.906891i	$0.361551\pi$
$824$	−6.91460	−0.240881
$825$	−14.5863	−0.507831
$826$	−46.9734	−1.63441
$827$	26.6918	0.928166	0.464083	−	0.885792i	$-0.346384\pi$
$827$	26.6918	0.928166	0.464083	+	0.885792i	$0.346384\pi$
$828$	35.4467	1.23186
$829$	−11.8850	−0.412784	−0.206392	−	0.978469i	$-0.566172\pi$
$829$	−11.8850	−0.412784	−0.206392	+	0.978469i	$0.566172\pi$
$830$	3.45550	0.119942
$831$	16.7279	0.580284
$832$	1.00000	0.0346688
$833$	−27.3510	−0.947657
$834$	38.4653	1.33194
$835$	8.88827	0.307591
$836$	−5.30545	−0.183493
$837$	12.2476	0.423338
$838$	−34.7063	−1.19891
$839$	−37.5264	−1.29555	−0.647777	−	0.761830i	$-0.724301\pi$
$839$	−37.5264	−1.29555	−0.647777	+	0.761830i	$0.724301\pi$
$840$	10.5380	0.363595
$841$	30.2458	1.04296
$842$	−36.3113	−1.25137
$843$	63.3317	2.18126
$844$	−13.0587	−0.449499
$845$	−1.00000	−0.0344010
$846$	−55.4828	−1.90754
$847$	−35.1930	−1.20925
$848$	−0.862926	−0.0296330
$849$	34.0548	1.16876
$850$	6.47488	0.222087
$851$	38.0351	1.30383
$852$	35.2311	1.20700
$853$	−24.8448	−0.850670	−0.425335	−	0.905036i	$-0.639844\pi$
$853$	−24.8448	−0.850670	−0.425335	+	0.905036i	$0.639844\pi$
$854$	46.2768	1.58356
$855$	7.88701	0.269730
$856$	17.5948	0.601379
$857$	−4.57395	−0.156243	−0.0781216	−	0.996944i	$-0.524892\pi$
$857$	−4.57395	−0.156243	−0.0781216	+	0.996944i	$0.524892\pi$
$858$	14.5863	0.497969
$859$	6.99385	0.238627	0.119314	−	0.992857i	$-0.461931\pi$
$859$	6.99385	0.238627	0.119314	+	0.992857i	$0.461931\pi$
$860$	−3.35142	−0.114283
$861$	−90.1680	−3.07292
$862$	19.2993	0.657337
$863$	3.57814	0.121801	0.0609006	−	0.998144i	$-0.480603\pi$
$863$	3.57814	0.121801	0.0609006	+	0.998144i	$0.480603\pi$
$864$	12.2476	0.416671
$865$	−18.5162	−0.629570
$866$	−22.2064	−0.754604
$867$	−78.3970	−2.66250
$868$	3.35025	0.113715
$869$	−64.0171	−2.17163
$870$	−24.2108	−0.820823
$871$	−13.7464	−0.465778
$872$	14.3971	0.487549
$873$	83.4940	2.82584
$874$	5.88269	0.198985
$875$	3.35025	0.113259
$876$	22.0511	0.745037
$877$	39.9563	1.34923	0.674614	−	0.738171i	$-0.264310\pi$
$877$	39.9563	1.34923	0.674614	+	0.738171i	$0.264310\pi$
$878$	−26.5566	−0.896243
$879$	−21.6998	−0.731915
$880$	−4.63730	−0.156324
$881$	46.8567	1.57864	0.789322	−	0.613980i	$-0.210432\pi$
$881$	46.8567	1.57864	0.789322	+	0.613980i	$0.210432\pi$
$882$	−29.1205	−0.980537
$883$	−17.4731	−0.588015	−0.294008	−	0.955803i	$-0.594989\pi$
$883$	−17.4731	−0.588015	−0.294008	+	0.955803i	$0.594989\pi$
$884$	−6.47488	−0.217774
$885$	−44.1017	−1.48246
$886$	−1.97318	−0.0662904
$887$	17.1448	0.575667	0.287833	−	0.957681i	$-0.407065\pi$
$887$	17.1448	0.575667	0.287833	+	0.957681i	$0.407065\pi$
$888$	23.2673	0.780799
$889$	−31.3808	−1.05248
$890$	−4.22522	−0.141630
$891$	82.7418	2.77195
$892$	17.7443	0.594124
$893$	−9.20786	−0.308129
$894$	57.9473	1.93805
$895$	−12.5807	−0.420526
$896$	3.35025	0.111924
$897$	−16.1734	−0.540013
$898$	2.03683	0.0679701
$899$	−7.69713	−0.256714
$900$	6.89376	0.229792
$901$	5.58734	0.186141
$902$	39.6790	1.32117
$903$	35.3173	1.17529
$904$	9.14511	0.304162
$905$	18.2020	0.605054
$906$	3.63546	0.120780
$907$	−29.7648	−0.988324	−0.494162	−	0.869370i	$-0.664525\pi$
$907$	−29.7648	−0.988324	−0.494162	+	0.869370i	$0.664525\pi$
$908$	24.5278	0.813982
$909$	70.3582	2.33363
$910$	−3.35025	−0.111060
$911$	−44.1293	−1.46207	−0.731035	−	0.682340i	$-0.760963\pi$
$911$	−44.1293	−1.46207	−0.731035	+	0.682340i	$0.760963\pi$
$912$	3.59863	0.119162
$913$	16.0242	0.530323
$914$	−4.57196	−0.151227
$915$	43.4477	1.43634
$916$	4.74074	0.156638
$917$	38.5211	1.27208
$918$	−79.3015	−2.61734
$919$	50.7499	1.67409	0.837043	−	0.547137i	$-0.184283\pi$
$919$	50.7499	1.67409	0.837043	+	0.547137i	$0.184283\pi$
$920$	5.14185	0.169522
$921$	80.2170	2.64324
$922$	1.96968	0.0648680
$923$	−11.2007	−0.368676
$924$	48.8679	1.60764
$925$	7.39716	0.243217
$926$	19.3359	0.635416
$927$	47.6676	1.56561
$928$	−7.69713	−0.252671
$929$	33.6454	1.10387	0.551935	−	0.833887i	$-0.313890\pi$
$929$	33.6454	1.10387	0.551935	+	0.833887i	$0.313890\pi$
$930$	3.14543	0.103143
$931$	−4.83280	−0.158389
$932$	26.8253	0.878693
$933$	−89.4876	−2.92969
$934$	25.1131	0.821727
$935$	30.0260	0.981955
$936$	−6.89376	−0.225330
$937$	40.6763	1.32884	0.664418	−	0.747362i	$-0.268680\pi$
$937$	40.6763	1.32884	0.664418	+	0.747362i	$0.268680\pi$
$938$	−46.0538	−1.50371
$939$	22.4072	0.731232
$940$	−8.04826	−0.262505
$941$	1.41925	0.0462663	0.0231332	−	0.999732i	$-0.492636\pi$
$941$	1.41925	0.0462663	0.0231332	+	0.999732i	$0.492636\pi$
$942$	−43.9503	−1.43198
$943$	−43.9961	−1.43271
$944$	−14.0208	−0.456340
$945$	−41.0324	−1.33478
$946$	−15.5416	−0.505300
$947$	−35.8530	−1.16506	−0.582532	−	0.812807i	$-0.697938\pi$
$947$	−35.8530	−1.16506	−0.582532	+	0.812807i	$0.697938\pi$
$948$	43.4221	1.41028
$949$	−7.01050	−0.227571
$950$	1.14408	0.0371189
$951$	−15.8291	−0.513294
$952$	−21.6925	−0.703057
$953$	13.0567	0.422948	0.211474	−	0.977384i	$-0.432174\pi$
$953$	13.0567	0.422948	0.211474	+	0.977384i	$0.432174\pi$
$954$	5.94881	0.192600
$955$	−4.19886	−0.135872
$956$	−0.417155	−0.0134918
$957$	−112.273	−3.62927
$958$	−17.8450	−0.576544
$959$	−37.0692	−1.19703
$960$	3.14543	0.101518
$961$	1.00000	0.0322581
$962$	−7.39716	−0.238494
$963$	−121.295	−3.90866
$964$	3.99975	0.128823
$965$	−21.1562	−0.681043
$966$	−54.1848	−1.74337
$967$	−33.5987	−1.08046	−0.540230	−	0.841517i	$-0.681663\pi$
$967$	−33.5987	−1.08046	−0.540230	+	0.841517i	$0.681663\pi$
$968$	−10.5046	−0.337631
$969$	−23.3007	−0.748526
$970$	12.1115	0.388878
$971$	6.04426	0.193970	0.0969848	−	0.995286i	$-0.469080\pi$
$971$	6.04426	0.193970	0.0969848	+	0.995286i	$0.469080\pi$
$972$	−19.3802	−0.621619
$973$	−40.9700	−1.31344
$974$	27.2416	0.872876
$975$	−3.14543	−0.100735
$976$	13.8129	0.442141
$977$	−27.1528	−0.868696	−0.434348	−	0.900745i	$-0.643021\pi$
$977$	−27.1528	−0.868696	−0.434348	+	0.900745i	$0.643021\pi$
$978$	−30.9569	−0.989891
$979$	−19.5936	−0.626216
$980$	−4.22418	−0.134936
$981$	−99.2505	−3.16882
$982$	−26.8803	−0.857784
$983$	−27.9519	−0.891526	−0.445763	−	0.895151i	$-0.647068\pi$
$983$	−27.9519	−0.891526	−0.445763	+	0.895151i	$0.647068\pi$
$984$	−26.9138	−0.857981
$985$	−16.0906	−0.512689
$986$	49.8380	1.58716
$987$	84.8126	2.69961
$988$	−1.14408	−0.0363980
$989$	17.2325	0.547962
$990$	31.9685	1.01602
$991$	−3.90237	−0.123963	−0.0619815	−	0.998077i	$-0.519742\pi$
$991$	−3.90237	−0.123963	−0.0619815	+	0.998077i	$0.519742\pi$
$992$	1.00000	0.0317500
$993$	−26.1447	−0.829677
$994$	−37.5252	−1.19023
$995$	24.1237	0.764771
$996$	−10.8690	−0.344399
$997$	12.2230	0.387106	0.193553	−	0.981090i	$-0.437999\pi$
$997$	12.2230	0.387106	0.193553	+	0.981090i	$0.437999\pi$
$998$	−21.1748	−0.670275
$999$	−90.5972	−2.86637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.k.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.k.1.1	✓	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.14543 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.14543 q^{6} -3.35025 q^{7} -1.00000 q^{8} +6.89376 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.14543 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.14543 q^{6} -3.35025 q^{7} -1.00000 q^{8} +6.89376 q^{9} +1.00000 q^{10} +4.63730 q^{11} -3.14543 q^{12} +1.00000 q^{13} +3.35025 q^{14} +3.14543 q^{15} +1.00000 q^{16} -6.47488 q^{17} -6.89376 q^{18} -1.14408 q^{19} -1.00000 q^{20} +10.5380 q^{21} -4.63730 q^{22} +5.14185 q^{23} +3.14543 q^{24} +1.00000 q^{25} -1.00000 q^{26} -12.2476 q^{27} -3.35025 q^{28} +7.69713 q^{29} -3.14543 q^{30} -1.00000 q^{31} -1.00000 q^{32} -14.5863 q^{33} +6.47488 q^{34} +3.35025 q^{35} +6.89376 q^{36} +7.39716 q^{37} +1.14408 q^{38} -3.14543 q^{39} +1.00000 q^{40} -8.55647 q^{41} -10.5380 q^{42} +3.35142 q^{43} +4.63730 q^{44} -6.89376 q^{45} -5.14185 q^{46} +8.04826 q^{47} -3.14543 q^{48} +4.22418 q^{49} -1.00000 q^{50} +20.3663 q^{51} +1.00000 q^{52} -0.862926 q^{53} +12.2476 q^{54} -4.63730 q^{55} +3.35025 q^{56} +3.59863 q^{57} -7.69713 q^{58} -14.0208 q^{59} +3.14543 q^{60} +13.8129 q^{61} +1.00000 q^{62} -23.0958 q^{63} +1.00000 q^{64} -1.00000 q^{65} +14.5863 q^{66} -13.7464 q^{67} -6.47488 q^{68} -16.1734 q^{69} -3.35025 q^{70} -11.2007 q^{71} -6.89376 q^{72} -7.01050 q^{73} -7.39716 q^{74} -3.14543 q^{75} -1.14408 q^{76} -15.5361 q^{77} +3.14543 q^{78} -13.8048 q^{79} -1.00000 q^{80} +17.8426 q^{81} +8.55647 q^{82} +3.45550 q^{83} +10.5380 q^{84} +6.47488 q^{85} -3.35142 q^{86} -24.2108 q^{87} -4.63730 q^{88} -4.22522 q^{89} +6.89376 q^{90} -3.35025 q^{91} +5.14185 q^{92} +3.14543 q^{93} -8.04826 q^{94} +1.14408 q^{95} +3.14543 q^{96} +12.1115 q^{97} -4.22418 q^{98} +31.9685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * q^99

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