LMFDB - Embedded newform 4030.2.a.o.1.7

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} - 3x^{7} - 12x^{6} + 40x^{5} + 24x^{4} - 118x^{3} + 25x^{2} + 30x + 4 $ x^8 - 3x^7 - 12x^6 + 40x^5 + 24x^4 - 118x^3 + 25x^2 + 30*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.51811$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.51811 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.51811 q^{6} -2.11544 q^{7} +1.00000 q^{8} +3.34088 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.51811 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.51811 q^{6} -2.11544 q^{7} +1.00000 q^{8} +3.34088 q^{9} +1.00000 q^{10} +3.54492 q^{11} +2.51811 q^{12} -1.00000 q^{13} -2.11544 q^{14} +2.51811 q^{15} +1.00000 q^{16} +3.06474 q^{17} +3.34088 q^{18} +0.843778 q^{19} +1.00000 q^{20} -5.32691 q^{21} +3.54492 q^{22} +3.25706 q^{23} +2.51811 q^{24} +1.00000 q^{25} -1.00000 q^{26} +0.858374 q^{27} -2.11544 q^{28} +5.23535 q^{29} +2.51811 q^{30} +1.00000 q^{31} +1.00000 q^{32} +8.92650 q^{33} +3.06474 q^{34} -2.11544 q^{35} +3.34088 q^{36} -0.656555 q^{37} +0.843778 q^{38} -2.51811 q^{39} +1.00000 q^{40} -1.82787 q^{41} -5.32691 q^{42} -9.90805 q^{43} +3.54492 q^{44} +3.34088 q^{45} +3.25706 q^{46} +4.31684 q^{47} +2.51811 q^{48} -2.52492 q^{49} +1.00000 q^{50} +7.71736 q^{51} -1.00000 q^{52} -4.79487 q^{53} +0.858374 q^{54} +3.54492 q^{55} -2.11544 q^{56} +2.12473 q^{57} +5.23535 q^{58} -3.38870 q^{59} +2.51811 q^{60} -2.27647 q^{61} +1.00000 q^{62} -7.06743 q^{63} +1.00000 q^{64} -1.00000 q^{65} +8.92650 q^{66} +15.9129 q^{67} +3.06474 q^{68} +8.20165 q^{69} -2.11544 q^{70} -6.65546 q^{71} +3.34088 q^{72} +8.16517 q^{73} -0.656555 q^{74} +2.51811 q^{75} +0.843778 q^{76} -7.49906 q^{77} -2.51811 q^{78} +7.45891 q^{79} +1.00000 q^{80} -7.86116 q^{81} -1.82787 q^{82} -3.39648 q^{83} -5.32691 q^{84} +3.06474 q^{85} -9.90805 q^{86} +13.1832 q^{87} +3.54492 q^{88} -6.49743 q^{89} +3.34088 q^{90} +2.11544 q^{91} +3.25706 q^{92} +2.51811 q^{93} +4.31684 q^{94} +0.843778 q^{95} +2.51811 q^{96} -1.06751 q^{97} -2.52492 q^{98} +11.8432 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * 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
$3$	2.51811	1.45383	0.726916	−	0.686727i	$-0.240953\pi$
$3$	2.51811	1.45383	0.726916	+	0.686727i	$0.240953\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.51811	1.02801
$7$	−2.11544	−0.799561	−0.399780	−	0.916611i	$-0.630914\pi$
$7$	−2.11544	−0.799561	−0.399780	+	0.916611i	$0.630914\pi$
$8$	1.00000	0.353553
$9$	3.34088	1.11363
$10$	1.00000	0.316228
$11$	3.54492	1.06883	0.534417	−	0.845221i	$-0.320531\pi$
$11$	3.54492	1.06883	0.534417	+	0.845221i	$0.320531\pi$
$12$	2.51811	0.726916
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.11544	−0.565375
$15$	2.51811	0.650173
$16$	1.00000	0.250000
$17$	3.06474	0.743310	0.371655	−	0.928371i	$-0.378790\pi$
$17$	3.06474	0.743310	0.371655	+	0.928371i	$0.378790\pi$
$18$	3.34088	0.787453
$19$	0.843778	0.193576	0.0967879	−	0.995305i	$-0.469143\pi$
$19$	0.843778	0.193576	0.0967879	+	0.995305i	$0.469143\pi$
$20$	1.00000	0.223607
$21$	−5.32691	−1.16243
$22$	3.54492	0.755780
$23$	3.25706	0.679145	0.339572	−	0.940580i	$-0.389718\pi$
$23$	3.25706	0.679145	0.339572	+	0.940580i	$0.389718\pi$
$24$	2.51811	0.514007
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0.858374	0.165194
$28$	−2.11544	−0.399780
$29$	5.23535	0.972180	0.486090	−	0.873909i	$-0.338423\pi$
$29$	5.23535	0.972180	0.486090	+	0.873909i	$0.338423\pi$
$30$	2.51811	0.459742
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	8.92650	1.55390
$34$	3.06474	0.525599
$35$	−2.11544	−0.357574
$36$	3.34088	0.556813
$37$	−0.656555	−0.107937	−0.0539685	−	0.998543i	$-0.517187\pi$
$37$	−0.656555	−0.107937	−0.0539685	+	0.998543i	$0.517187\pi$
$38$	0.843778	0.136879
$39$	−2.51811	−0.403220
$40$	1.00000	0.158114
$41$	−1.82787	−0.285465	−0.142733	−	0.989761i	$-0.545589\pi$
$41$	−1.82787	−0.285465	−0.142733	+	0.989761i	$0.545589\pi$
$42$	−5.32691	−0.821960
$43$	−9.90805	−1.51096	−0.755482	−	0.655170i	$-0.772597\pi$
$43$	−9.90805	−1.51096	−0.755482	+	0.655170i	$0.772597\pi$
$44$	3.54492	0.534417
$45$	3.34088	0.498029
$46$	3.25706	0.480228
$47$	4.31684	0.629676	0.314838	−	0.949145i	$-0.398050\pi$
$47$	4.31684	0.629676	0.314838	+	0.949145i	$0.398050\pi$
$48$	2.51811	0.363458
$49$	−2.52492	−0.360703
$50$	1.00000	0.141421
$51$	7.71736	1.08065
$52$	−1.00000	−0.138675
$53$	−4.79487	−0.658626	−0.329313	−	0.944221i	$-0.606817\pi$
$53$	−4.79487	−0.658626	−0.329313	+	0.944221i	$0.606817\pi$
$54$	0.858374	0.116810
$55$	3.54492	0.477997
$56$	−2.11544	−0.282687
$57$	2.12473	0.281427
$58$	5.23535	0.687435
$59$	−3.38870	−0.441171	−0.220586	−	0.975368i	$-0.570797\pi$
$59$	−3.38870	−0.441171	−0.220586	+	0.975368i	$0.570797\pi$
$60$	2.51811	0.325087
$61$	−2.27647	−0.291472	−0.145736	−	0.989324i	$-0.546555\pi$
$61$	−2.27647	−0.291472	−0.145736	+	0.989324i	$0.546555\pi$
$62$	1.00000	0.127000
$63$	−7.06743	−0.890412
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	8.92650	1.09878
$67$	15.9129	1.94407	0.972033	−	0.234846i	$-0.0754584\pi$
$67$	15.9129	1.94407	0.972033	+	0.234846i	$0.0754584\pi$
$68$	3.06474	0.371655
$69$	8.20165	0.987362
$70$	−2.11544	−0.252843
$71$	−6.65546	−0.789858	−0.394929	−	0.918712i	$-0.629231\pi$
$71$	−6.65546	−0.789858	−0.394929	+	0.918712i	$0.629231\pi$
$72$	3.34088	0.393726
$73$	8.16517	0.955661	0.477830	−	0.878452i	$-0.341423\pi$
$73$	8.16517	0.955661	0.477830	+	0.878452i	$0.341423\pi$
$74$	−0.656555	−0.0763230
$75$	2.51811	0.290766
$76$	0.843778	0.0967879
$77$	−7.49906	−0.854598
$78$	−2.51811	−0.285120
$79$	7.45891	0.839193	0.419596	−	0.907711i	$-0.362172\pi$
$79$	7.45891	0.839193	0.419596	+	0.907711i	$0.362172\pi$
$80$	1.00000	0.111803
$81$	−7.86116	−0.873462
$82$	−1.82787	−0.201854
$83$	−3.39648	−0.372813	−0.186406	−	0.982473i	$-0.559684\pi$
$83$	−3.39648	−0.372813	−0.186406	+	0.982473i	$0.559684\pi$
$84$	−5.32691	−0.581213
$85$	3.06474	0.332418
$86$	−9.90805	−1.06841
$87$	13.1832	1.41339
$88$	3.54492	0.377890
$89$	−6.49743	−0.688726	−0.344363	−	0.938836i	$-0.611905\pi$
$89$	−6.49743	−0.688726	−0.344363	+	0.938836i	$0.611905\pi$
$90$	3.34088	0.352160
$91$	2.11544	0.221758
$92$	3.25706	0.339572
$93$	2.51811	0.261116
$94$	4.31684	0.445248
$95$	0.843778	0.0865698
$96$	2.51811	0.257004
$97$	−1.06751	−0.108389	−0.0541945	−	0.998530i	$-0.517259\pi$
$97$	−1.06751	−0.108389	−0.0541945	+	0.998530i	$0.517259\pi$
$98$	−2.52492	−0.255056
$99$	11.8432	1.19028
$100$	1.00000	0.100000
$101$	15.0060	1.49315	0.746575	−	0.665302i	$-0.231697\pi$
$101$	15.0060	1.49315	0.746575	+	0.665302i	$0.231697\pi$
$102$	7.71736	0.764133
$103$	9.27883	0.914271	0.457135	−	0.889397i	$-0.348875\pi$
$103$	9.27883	0.914271	0.457135	+	0.889397i	$0.348875\pi$
$104$	−1.00000	−0.0980581
$105$	−5.32691	−0.519853
$106$	−4.79487	−0.465719
$107$	−18.6121	−1.79930	−0.899650	−	0.436611i	$-0.856179\pi$
$107$	−18.6121	−1.79930	−0.899650	+	0.436611i	$0.856179\pi$
$108$	0.858374	0.0825970
$109$	11.3632	1.08840	0.544199	−	0.838956i	$-0.316834\pi$
$109$	11.3632	1.08840	0.544199	+	0.838956i	$0.316834\pi$
$110$	3.54492	0.337995
$111$	−1.65328	−0.156922
$112$	−2.11544	−0.199890
$113$	−14.0437	−1.32112	−0.660561	−	0.750772i	$-0.729682\pi$
$113$	−14.0437	−1.32112	−0.660561	+	0.750772i	$0.729682\pi$
$114$	2.12473	0.198999
$115$	3.25706	0.303723
$116$	5.23535	0.486090
$117$	−3.34088	−0.308864
$118$	−3.38870	−0.311955
$119$	−6.48328	−0.594321
$120$	2.51811	0.229871
$121$	1.56647	0.142406
$122$	−2.27647	−0.206102
$123$	−4.60277	−0.415018
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	−7.06743	−0.629616
$127$	17.1118	1.51843	0.759213	−	0.650843i	$-0.225584\pi$
$127$	17.1118	1.51843	0.759213	+	0.650843i	$0.225584\pi$
$128$	1.00000	0.0883883
$129$	−24.9496	−2.19669
$130$	−1.00000	−0.0877058
$131$	−9.24764	−0.807970	−0.403985	−	0.914766i	$-0.632375\pi$
$131$	−9.24764	−0.807970	−0.403985	+	0.914766i	$0.632375\pi$
$132$	8.92650	0.776952
$133$	−1.78496	−0.154776
$134$	15.9129	1.37466
$135$	0.858374	0.0738770
$136$	3.06474	0.262800
$137$	7.26156	0.620397	0.310199	−	0.950672i	$-0.399604\pi$
$137$	7.26156	0.620397	0.310199	+	0.950672i	$0.399604\pi$
$138$	8.20165	0.698171
$139$	−3.60293	−0.305597	−0.152798	−	0.988257i	$-0.548829\pi$
$139$	−3.60293	−0.305597	−0.152798	+	0.988257i	$0.548829\pi$
$140$	−2.11544	−0.178787
$141$	10.8703	0.915442
$142$	−6.65546	−0.558514
$143$	−3.54492	−0.296441
$144$	3.34088	0.278407
$145$	5.23535	0.434772
$146$	8.16517	0.675754
$147$	−6.35803	−0.524401
$148$	−0.656555	−0.0539685
$149$	−12.3060	−1.00814	−0.504072	−	0.863662i	$-0.668165\pi$
$149$	−12.3060	−1.00814	−0.504072	+	0.863662i	$0.668165\pi$
$150$	2.51811	0.205603
$151$	−18.2834	−1.48788	−0.743942	−	0.668244i	$-0.767046\pi$
$151$	−18.2834	−1.48788	−0.743942	+	0.668244i	$0.767046\pi$
$152$	0.843778	0.0684394
$153$	10.2389	0.827769
$154$	−7.49906	−0.604292
$155$	1.00000	0.0803219
$156$	−2.51811	−0.201610
$157$	19.5611	1.56114	0.780571	−	0.625067i	$-0.214928\pi$
$157$	19.5611	1.56114	0.780571	+	0.625067i	$0.214928\pi$
$158$	7.45891	0.593399
$159$	−12.0740	−0.957531
$160$	1.00000	0.0790569
$161$	−6.89012	−0.543017
$162$	−7.86116	−0.617631
$163$	−10.0185	−0.784706	−0.392353	−	0.919815i	$-0.628339\pi$
$163$	−10.0185	−0.784706	−0.392353	+	0.919815i	$0.628339\pi$
$164$	−1.82787	−0.142733
$165$	8.92650	0.694927
$166$	−3.39648	−0.263618
$167$	16.5439	1.28020	0.640102	−	0.768290i	$-0.278892\pi$
$167$	16.5439	1.28020	0.640102	+	0.768290i	$0.278892\pi$
$168$	−5.32691	−0.410980
$169$	1.00000	0.0769231
$170$	3.06474	0.235055
$171$	2.81896	0.215571
$172$	−9.90805	−0.755482
$173$	−9.58330	−0.728604	−0.364302	−	0.931281i	$-0.618692\pi$
$173$	−9.58330	−0.728604	−0.364302	+	0.931281i	$0.618692\pi$
$174$	13.1832	0.999415
$175$	−2.11544	−0.159912
$176$	3.54492	0.267209
$177$	−8.53312	−0.641388
$178$	−6.49743	−0.487003
$179$	18.5473	1.38629	0.693146	−	0.720798i	$-0.256224\pi$
$179$	18.5473	1.38629	0.693146	+	0.720798i	$0.256224\pi$
$180$	3.34088	0.249014
$181$	3.84534	0.285822	0.142911	−	0.989736i	$-0.454354\pi$
$181$	3.84534	0.285822	0.142911	+	0.989736i	$0.454354\pi$
$182$	2.11544	0.156807
$183$	−5.73239	−0.423751
$184$	3.25706	0.240114
$185$	−0.656555	−0.0482709
$186$	2.51811	0.184637
$187$	10.8643	0.794475
$188$	4.31684	0.314838
$189$	−1.81584	−0.132083
$190$	0.843778	0.0612141
$191$	−8.23049	−0.595538	−0.297769	−	0.954638i	$-0.596242\pi$
$191$	−8.23049	−0.595538	−0.297769	+	0.954638i	$0.596242\pi$
$192$	2.51811	0.181729
$193$	3.37321	0.242809	0.121405	−	0.992603i	$-0.461260\pi$
$193$	3.37321	0.242809	0.121405	+	0.992603i	$0.461260\pi$
$194$	−1.06751	−0.0766426
$195$	−2.51811	−0.180326
$196$	−2.52492	−0.180351
$197$	7.50540	0.534738	0.267369	−	0.963594i	$-0.413846\pi$
$197$	7.50540	0.534738	0.267369	+	0.963594i	$0.413846\pi$
$198$	11.8432	0.841657
$199$	−20.2672	−1.43670	−0.718352	−	0.695680i	$-0.755103\pi$
$199$	−20.2672	−1.43670	−0.718352	+	0.695680i	$0.755103\pi$
$200$	1.00000	0.0707107
$201$	40.0703	2.82634
$202$	15.0060	1.05582
$203$	−11.0751	−0.777317
$204$	7.71736	0.540324
$205$	−1.82787	−0.127664
$206$	9.27883	0.646487
$207$	10.8815	0.756314
$208$	−1.00000	−0.0693375
$209$	2.99113	0.206900
$210$	−5.32691	−0.367592
$211$	−7.98397	−0.549639	−0.274820	−	0.961496i	$-0.588618\pi$
$211$	−7.98397	−0.549639	−0.274820	+	0.961496i	$0.588618\pi$
$212$	−4.79487	−0.329313
$213$	−16.7592	−1.14832
$214$	−18.6121	−1.27230
$215$	−9.90805	−0.675723
$216$	0.858374	0.0584049
$217$	−2.11544	−0.143605
$218$	11.3632	0.769613
$219$	20.5608	1.38937
$220$	3.54492	0.238999
$221$	−3.06474	−0.206157
$222$	−1.65328	−0.110961
$223$	14.0876	0.943378	0.471689	−	0.881765i	$-0.343645\pi$
$223$	14.0876	0.943378	0.471689	+	0.881765i	$0.343645\pi$
$224$	−2.11544	−0.141344
$225$	3.34088	0.222725
$226$	−14.0437	−0.934175
$227$	24.9423	1.65548	0.827740	−	0.561112i	$-0.189626\pi$
$227$	24.9423	1.65548	0.827740	+	0.561112i	$0.189626\pi$
$228$	2.12473	0.140713
$229$	−20.6437	−1.36418	−0.682088	−	0.731270i	$-0.738928\pi$
$229$	−20.6437	−1.36418	−0.682088	+	0.731270i	$0.738928\pi$
$230$	3.25706	0.214764
$231$	−18.8835	−1.24244
$232$	5.23535	0.343718
$233$	1.05137	0.0688777	0.0344389	−	0.999407i	$-0.489036\pi$
$233$	1.05137	0.0688777	0.0344389	+	0.999407i	$0.489036\pi$
$234$	−3.34088	−0.218400
$235$	4.31684	0.281599
$236$	−3.38870	−0.220586
$237$	18.7824	1.22004
$238$	−6.48328	−0.420248
$239$	0.511977	0.0331170	0.0165585	−	0.999863i	$-0.494729\pi$
$239$	0.511977	0.0331170	0.0165585	+	0.999863i	$0.494729\pi$
$240$	2.51811	0.162543
$241$	−25.2862	−1.62883	−0.814415	−	0.580283i	$-0.802942\pi$
$241$	−25.2862	−1.62883	−0.814415	+	0.580283i	$0.802942\pi$
$242$	1.56647	0.100696
$243$	−22.3704	−1.43506
$244$	−2.27647	−0.145736
$245$	−2.52492	−0.161311
$246$	−4.60277	−0.293462
$247$	−0.843778	−0.0536883
$248$	1.00000	0.0635001
$249$	−8.55272	−0.542007
$250$	1.00000	0.0632456
$251$	−15.3123	−0.966504	−0.483252	−	0.875481i	$-0.660544\pi$
$251$	−15.3123	−0.966504	−0.483252	+	0.875481i	$0.660544\pi$
$252$	−7.06743	−0.445206
$253$	11.5460	0.725893
$254$	17.1118	1.07369
$255$	7.71736	0.483280
$256$	1.00000	0.0625000
$257$	−18.2421	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$257$	−18.2421	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$258$	−24.9496	−1.55329
$259$	1.38890	0.0863022
$260$	−1.00000	−0.0620174
$261$	17.4907	1.08265
$262$	−9.24764	−0.571321
$263$	−12.2808	−0.757265	−0.378632	−	0.925547i	$-0.623606\pi$
$263$	−12.2808	−0.757265	−0.378632	+	0.925547i	$0.623606\pi$
$264$	8.92650	0.549388
$265$	−4.79487	−0.294546
$266$	−1.78496	−0.109443
$267$	−16.3613	−1.00129
$268$	15.9129	0.972033
$269$	−28.1113	−1.71397	−0.856987	−	0.515337i	$-0.827667\pi$
$269$	−28.1113	−1.71397	−0.856987	+	0.515337i	$0.827667\pi$
$270$	0.858374	0.0522390
$271$	−9.28658	−0.564120	−0.282060	−	0.959397i	$-0.591018\pi$
$271$	−9.28658	−0.564120	−0.282060	+	0.959397i	$0.591018\pi$
$272$	3.06474	0.185827
$273$	5.32691	0.322399
$274$	7.26156	0.438687
$275$	3.54492	0.213767
$276$	8.20165	0.493681
$277$	11.4100	0.685562	0.342781	−	0.939415i	$-0.388631\pi$
$277$	11.4100	0.685562	0.342781	+	0.939415i	$0.388631\pi$
$278$	−3.60293	−0.216090
$279$	3.34088	0.200013
$280$	−2.11544	−0.126422
$281$	−19.8612	−1.18482	−0.592411	−	0.805636i	$-0.701824\pi$
$281$	−19.8612	−1.18482	−0.592411	+	0.805636i	$0.701824\pi$
$282$	10.8703	0.647315
$283$	−19.3165	−1.14824	−0.574122	−	0.818770i	$-0.694657\pi$
$283$	−19.3165	−1.14824	−0.574122	+	0.818770i	$0.694657\pi$
$284$	−6.65546	−0.394929
$285$	2.12473	0.125858
$286$	−3.54492	−0.209616
$287$	3.86674	0.228247
$288$	3.34088	0.196863
$289$	−7.60734	−0.447491
$290$	5.23535	0.307430
$291$	−2.68810	−0.157579
$292$	8.16517	0.477830
$293$	12.4686	0.728422	0.364211	−	0.931317i	$-0.381339\pi$
$293$	12.4686	0.728422	0.364211	+	0.931317i	$0.381339\pi$
$294$	−6.35803	−0.370808
$295$	−3.38870	−0.197298
$296$	−0.656555	−0.0381615
$297$	3.04287	0.176565
$298$	−12.3060	−0.712865
$299$	−3.25706	−0.188361
$300$	2.51811	0.145383
$301$	20.9599	1.20811
$302$	−18.2834	−1.05209
$303$	37.7867	2.17079
$304$	0.843778	0.0483940
$305$	−2.27647	−0.130350
$306$	10.2389	0.585321
$307$	17.1643	0.979618	0.489809	−	0.871830i	$-0.337066\pi$
$307$	17.1643	0.979618	0.489809	+	0.871830i	$0.337066\pi$
$308$	−7.49906	−0.427299
$309$	23.3651	1.32920
$310$	1.00000	0.0567962
$311$	−1.86659	−0.105845	−0.0529224	−	0.998599i	$-0.516854\pi$
$311$	−1.86659	−0.105845	−0.0529224	+	0.998599i	$0.516854\pi$
$312$	−2.51811	−0.142560
$313$	−28.8715	−1.63191	−0.815957	−	0.578112i	$-0.803790\pi$
$313$	−28.8715	−1.63191	−0.815957	+	0.578112i	$0.803790\pi$
$314$	19.5611	1.10389
$315$	−7.06743	−0.398204
$316$	7.45891	0.419596
$317$	−5.30258	−0.297823	−0.148911	−	0.988851i	$-0.547577\pi$
$317$	−5.30258	−0.297823	−0.148911	+	0.988851i	$0.547577\pi$
$318$	−12.0740	−0.677077
$319$	18.5589	1.03910
$320$	1.00000	0.0559017
$321$	−46.8674	−2.61588
$322$	−6.89012	−0.383971
$323$	2.58596	0.143887
$324$	−7.86116	−0.436731
$325$	−1.00000	−0.0554700
$326$	−10.0185	−0.554871
$327$	28.6138	1.58235
$328$	−1.82787	−0.100927
$329$	−9.13201	−0.503464
$330$	8.92650	0.491388
$331$	−25.6652	−1.41069	−0.705343	−	0.708866i	$-0.749207\pi$
$331$	−25.6652	−1.41069	−0.705343	+	0.708866i	$0.749207\pi$
$332$	−3.39648	−0.186406
$333$	−2.19347	−0.120202
$334$	16.5439	0.905241
$335$	15.9129	0.869412
$336$	−5.32691	−0.290607
$337$	−16.6183	−0.905257	−0.452628	−	0.891699i	$-0.649514\pi$
$337$	−16.6183	−0.905257	−0.452628	+	0.891699i	$0.649514\pi$
$338$	1.00000	0.0543928
$339$	−35.3637	−1.92069
$340$	3.06474	0.166209
$341$	3.54492	0.191968
$342$	2.81896	0.152432
$343$	20.1494	1.08796
$344$	−9.90805	−0.534206
$345$	8.20165	0.441562
$346$	−9.58330	−0.515201
$347$	−29.4959	−1.58342	−0.791712	−	0.610895i	$-0.790810\pi$
$347$	−29.4959	−1.58342	−0.791712	+	0.610895i	$0.790810\pi$
$348$	13.1832	0.706693
$349$	2.12568	0.113785	0.0568926	−	0.998380i	$-0.481881\pi$
$349$	2.12568	0.113785	0.0568926	+	0.998380i	$0.481881\pi$
$350$	−2.11544	−0.113075
$351$	−0.858374	−0.0458166
$352$	3.54492	0.188945
$353$	−25.6406	−1.36471	−0.682357	−	0.731020i	$-0.739045\pi$
$353$	−25.6406	−1.36471	−0.682357	+	0.731020i	$0.739045\pi$
$354$	−8.53312	−0.453530
$355$	−6.65546	−0.353235
$356$	−6.49743	−0.344363
$357$	−16.3256	−0.864043
$358$	18.5473	0.980256
$359$	22.2396	1.17376	0.586882	−	0.809673i	$-0.300355\pi$
$359$	22.2396	1.17376	0.586882	+	0.809673i	$0.300355\pi$
$360$	3.34088	0.176080
$361$	−18.2880	−0.962528
$362$	3.84534	0.202107
$363$	3.94454	0.207035
$364$	2.11544	0.110879
$365$	8.16517	0.427384
$366$	−5.73239	−0.299637
$367$	−22.7696	−1.18856	−0.594281	−	0.804257i	$-0.702563\pi$
$367$	−22.7696	−1.18856	−0.594281	+	0.804257i	$0.702563\pi$
$368$	3.25706	0.169786
$369$	−6.10669	−0.317901
$370$	−0.656555	−0.0341327
$371$	10.1433	0.526611
$372$	2.51811	0.130558
$373$	−4.58361	−0.237330	−0.118665	−	0.992934i	$-0.537862\pi$
$373$	−4.58361	−0.237330	−0.118665	+	0.992934i	$0.537862\pi$
$374$	10.8643	0.561778
$375$	2.51811	0.130035
$376$	4.31684	0.222624
$377$	−5.23535	−0.269634
$378$	−1.81584	−0.0933965
$379$	12.1471	0.623955	0.311978	−	0.950089i	$-0.399009\pi$
$379$	12.1471	0.623955	0.311978	+	0.950089i	$0.399009\pi$
$380$	0.843778	0.0432849
$381$	43.0894	2.20753
$382$	−8.23049	−0.421109
$383$	−29.6790	−1.51653	−0.758264	−	0.651948i	$-0.773952\pi$
$383$	−29.6790	−1.51653	−0.758264	+	0.651948i	$0.773952\pi$
$384$	2.51811	0.128502
$385$	−7.49906	−0.382188
$386$	3.37321	0.171692
$387$	−33.1016	−1.68265
$388$	−1.06751	−0.0541945
$389$	10.0468	0.509391	0.254695	−	0.967021i	$-0.418025\pi$
$389$	10.0468	0.509391	0.254695	+	0.967021i	$0.418025\pi$
$390$	−2.51811	−0.127509
$391$	9.98207	0.504815
$392$	−2.52492	−0.127528
$393$	−23.2866	−1.17465
$394$	7.50540	0.378117
$395$	7.45891	0.375298
$396$	11.8432	0.595141
$397$	−24.7942	−1.24438	−0.622192	−	0.782865i	$-0.713758\pi$
$397$	−24.7942	−1.24438	−0.622192	+	0.782865i	$0.713758\pi$
$398$	−20.2672	−1.01590
$399$	−4.49472	−0.225018
$400$	1.00000	0.0500000
$401$	20.2223	1.00985	0.504926	−	0.863163i	$-0.331520\pi$
$401$	20.2223	1.00985	0.504926	+	0.863163i	$0.331520\pi$
$402$	40.0703	1.99853
$403$	−1.00000	−0.0498135
$404$	15.0060	0.746575
$405$	−7.86116	−0.390624
$406$	−11.0751	−0.549646
$407$	−2.32744	−0.115367
$408$	7.71736	0.382066
$409$	18.2534	0.902574	0.451287	−	0.892379i	$-0.350965\pi$
$409$	18.2534	0.902574	0.451287	+	0.892379i	$0.350965\pi$
$410$	−1.82787	−0.0902720
$411$	18.2854	0.901953
$412$	9.27883	0.457135
$413$	7.16858	0.352743
$414$	10.8815	0.534795
$415$	−3.39648	−0.166727
$416$	−1.00000	−0.0490290
$417$	−9.07258	−0.444286
$418$	2.99113	0.146301
$419$	13.9566	0.681823	0.340911	−	0.940095i	$-0.389264\pi$
$419$	13.9566	0.681823	0.340911	+	0.940095i	$0.389264\pi$
$420$	−5.32691	−0.259926
$421$	−8.12836	−0.396152	−0.198076	−	0.980187i	$-0.563469\pi$
$421$	−8.12836	−0.396152	−0.198076	+	0.980187i	$0.563469\pi$
$422$	−7.98397	−0.388654
$423$	14.4220	0.701224
$424$	−4.79487	−0.232859
$425$	3.06474	0.148662
$426$	−16.7592	−0.811985
$427$	4.81572	0.233049
$428$	−18.6121	−0.899650
$429$	−8.92650	−0.430976
$430$	−9.90805	−0.477809
$431$	2.93844	0.141539	0.0707697	−	0.997493i	$-0.477454\pi$
$431$	2.93844	0.141539	0.0707697	+	0.997493i	$0.477454\pi$
$432$	0.858374	0.0412985
$433$	24.5968	1.18205	0.591024	−	0.806654i	$-0.298724\pi$
$433$	24.5968	1.18205	0.591024	+	0.806654i	$0.298724\pi$
$434$	−2.11544	−0.101544
$435$	13.1832	0.632086
$436$	11.3632	0.544199
$437$	2.74824	0.131466
$438$	20.5608	0.982433
$439$	24.1951	1.15477	0.577386	−	0.816472i	$-0.304073\pi$
$439$	24.1951	1.15477	0.577386	+	0.816472i	$0.304073\pi$
$440$	3.54492	0.168998
$441$	−8.43546	−0.401688
$442$	−3.06474	−0.145775
$443$	1.42958	0.0679212	0.0339606	−	0.999423i	$-0.489188\pi$
$443$	1.42958	0.0679212	0.0339606	+	0.999423i	$0.489188\pi$
$444$	−1.65328	−0.0784612
$445$	−6.49743	−0.308008
$446$	14.0876	0.667069
$447$	−30.9878	−1.46567
$448$	−2.11544	−0.0999451
$449$	−2.45806	−0.116003	−0.0580015	−	0.998316i	$-0.518473\pi$
$449$	−2.45806	−0.116003	−0.0580015	+	0.998316i	$0.518473\pi$
$450$	3.34088	0.157491
$451$	−6.47965	−0.305115
$452$	−14.0437	−0.660561
$453$	−46.0397	−2.16313
$454$	24.9423	1.17060
$455$	2.11544	0.0991733
$456$	2.12473	0.0994994
$457$	11.4182	0.534120	0.267060	−	0.963680i	$-0.413948\pi$
$457$	11.4182	0.534120	0.267060	+	0.963680i	$0.413948\pi$
$458$	−20.6437	−0.964619
$459$	2.63070	0.122790
$460$	3.25706	0.151861
$461$	20.2730	0.944206	0.472103	−	0.881543i	$-0.343495\pi$
$461$	20.2730	0.944206	0.472103	+	0.881543i	$0.343495\pi$
$462$	−18.8835	−0.878538
$463$	−1.58584	−0.0737003	−0.0368501	−	0.999321i	$-0.511732\pi$
$463$	−1.58584	−0.0737003	−0.0368501	+	0.999321i	$0.511732\pi$
$464$	5.23535	0.243045
$465$	2.51811	0.116775
$466$	1.05137	0.0487039
$467$	−22.9265	−1.06091	−0.530455	−	0.847713i	$-0.677979\pi$
$467$	−22.9265	−1.06091	−0.530455	+	0.847713i	$0.677979\pi$
$468$	−3.34088	−0.154432
$469$	−33.6627	−1.55440
$470$	4.31684	0.199121
$471$	49.2569	2.26964
$472$	−3.38870	−0.155978
$473$	−35.1233	−1.61497
$474$	18.7824	0.862702
$475$	0.843778	0.0387152
$476$	−6.48328	−0.297161
$477$	−16.0191	−0.733463
$478$	0.511977	0.0234173
$479$	−21.8494	−0.998327	−0.499163	−	0.866508i	$-0.666359\pi$
$479$	−21.8494	−0.998327	−0.499163	+	0.866508i	$0.666359\pi$
$480$	2.51811	0.114935
$481$	0.656555	0.0299364
$482$	−25.2862	−1.15176
$483$	−17.3501	−0.789456
$484$	1.56647	0.0712032
$485$	−1.06751	−0.0484730
$486$	−22.3704	−1.01474
$487$	−9.23171	−0.418329	−0.209164	−	0.977881i	$-0.567074\pi$
$487$	−9.23171	−0.418329	−0.209164	+	0.977881i	$0.567074\pi$
$488$	−2.27647	−0.103051
$489$	−25.2276	−1.14083
$490$	−2.52492	−0.114064
$491$	6.63258	0.299324	0.149662	−	0.988737i	$-0.452181\pi$
$491$	6.63258	0.299324	0.149662	+	0.988737i	$0.452181\pi$
$492$	−4.60277	−0.207509
$493$	16.0450	0.722631
$494$	−0.843778	−0.0379633
$495$	11.8432	0.532310
$496$	1.00000	0.0449013
$497$	14.0792	0.631539
$498$	−8.55272	−0.383257
$499$	−8.28705	−0.370979	−0.185490	−	0.982646i	$-0.559387\pi$
$499$	−8.28705	−0.370979	−0.185490	+	0.982646i	$0.559387\pi$
$500$	1.00000	0.0447214
$501$	41.6593	1.86120
$502$	−15.3123	−0.683421
$503$	−2.26220	−0.100867	−0.0504333	−	0.998727i	$-0.516060\pi$
$503$	−2.26220	−0.100867	−0.0504333	+	0.998727i	$0.516060\pi$
$504$	−7.06743	−0.314808
$505$	15.0060	0.667757
$506$	11.5460	0.513284
$507$	2.51811	0.111833
$508$	17.1118	0.759213
$509$	−12.2566	−0.543264	−0.271632	−	0.962401i	$-0.587563\pi$
$509$	−12.2566	−0.543264	−0.271632	+	0.962401i	$0.587563\pi$
$510$	7.71736	0.341731
$511$	−17.2729	−0.764109
$512$	1.00000	0.0441942
$513$	0.724276	0.0319776
$514$	−18.2421	−0.804626
$515$	9.27883	0.408874
$516$	−24.9496	−1.09834
$517$	15.3029	0.673019
$518$	1.38890	0.0610249
$519$	−24.1318	−1.05927
$520$	−1.00000	−0.0438529
$521$	−35.3810	−1.55007	−0.775034	−	0.631919i	$-0.782267\pi$
$521$	−35.3810	−1.55007	−0.775034	+	0.631919i	$0.782267\pi$
$522$	17.4907	0.765546
$523$	0.330996	0.0144735	0.00723673	−	0.999974i	$-0.497696\pi$
$523$	0.330996	0.0144735	0.00723673	+	0.999974i	$0.497696\pi$
$524$	−9.24764	−0.403985
$525$	−5.32691	−0.232485
$526$	−12.2808	−0.535467
$527$	3.06474	0.133502
$528$	8.92650	0.388476
$529$	−12.3915	−0.538762
$530$	−4.79487	−0.208276
$531$	−11.3212	−0.491300
$532$	−1.78496	−0.0773878
$533$	1.82787	0.0791738
$534$	−16.3613	−0.708021
$535$	−18.6121	−0.804672
$536$	15.9129	0.687331
$537$	46.7042	2.01543
$538$	−28.1113	−1.21196
$539$	−8.95065	−0.385532
$540$	0.858374	0.0369385
$541$	29.3116	1.26020	0.630102	−	0.776512i	$-0.283013\pi$
$541$	29.3116	1.26020	0.630102	+	0.776512i	$0.283013\pi$
$542$	−9.28658	−0.398893
$543$	9.68299	0.415537
$544$	3.06474	0.131400
$545$	11.3632	0.486746
$546$	5.32691	0.227971
$547$	24.4218	1.04420	0.522099	−	0.852885i	$-0.325149\pi$
$547$	24.4218	1.04420	0.522099	+	0.852885i	$0.325149\pi$
$548$	7.26156	0.310199
$549$	−7.60540	−0.324591
$550$	3.54492	0.151156
$551$	4.41747	0.188191
$552$	8.20165	0.349085
$553$	−15.7789	−0.670985
$554$	11.4100	0.484766
$555$	−1.65328	−0.0701778
$556$	−3.60293	−0.152798
$557$	14.4838	0.613699	0.306849	−	0.951758i	$-0.400725\pi$
$557$	14.4838	0.613699	0.306849	+	0.951758i	$0.400725\pi$
$558$	3.34088	0.141431
$559$	9.90805	0.419066
$560$	−2.11544	−0.0893936
$561$	27.3575	1.15503
$562$	−19.8612	−0.837796
$563$	39.3355	1.65779	0.828896	−	0.559402i	$-0.188969\pi$
$563$	39.3355	1.65779	0.828896	+	0.559402i	$0.188969\pi$
$564$	10.8703	0.457721
$565$	−14.0437	−0.590824
$566$	−19.3165	−0.811932
$567$	16.6298	0.698386
$568$	−6.65546	−0.279257
$569$	14.0966	0.590961	0.295481	−	0.955349i	$-0.404520\pi$
$569$	14.0966	0.590961	0.295481	+	0.955349i	$0.404520\pi$
$570$	2.12473	0.0889949
$571$	17.9308	0.750381	0.375190	−	0.926948i	$-0.377577\pi$
$571$	17.9308	0.750381	0.375190	+	0.926948i	$0.377577\pi$
$572$	−3.54492	−0.148221
$573$	−20.7253	−0.865811
$574$	3.86674	0.161395
$575$	3.25706	0.135829
$576$	3.34088	0.139203
$577$	33.7328	1.40432	0.702158	−	0.712021i	$-0.252220\pi$
$577$	33.7328	1.40432	0.702158	+	0.712021i	$0.252220\pi$
$578$	−7.60734	−0.316424
$579$	8.49412	0.353004
$580$	5.23535	0.217386
$581$	7.18505	0.298086
$582$	−2.68810	−0.111425
$583$	−16.9974	−0.703962
$584$	8.16517	0.337877
$585$	−3.34088	−0.138128
$586$	12.4686	0.515072
$587$	−12.3345	−0.509100	−0.254550	−	0.967060i	$-0.581927\pi$
$587$	−12.3345	−0.509100	−0.254550	+	0.967060i	$0.581927\pi$
$588$	−6.35803	−0.262201
$589$	0.843778	0.0347673
$590$	−3.38870	−0.139511
$591$	18.8994	0.777418
$592$	−0.656555	−0.0269843
$593$	21.1935	0.870313	0.435156	−	0.900355i	$-0.356693\pi$
$593$	21.1935	0.870313	0.435156	+	0.900355i	$0.356693\pi$
$594$	3.04287	0.124850
$595$	−6.48328	−0.265788
$596$	−12.3060	−0.504072
$597$	−51.0350	−2.08873
$598$	−3.25706	−0.133191
$599$	34.8517	1.42400	0.712001	−	0.702178i	$-0.247789\pi$
$599$	34.8517	1.42400	0.712001	+	0.702178i	$0.247789\pi$
$600$	2.51811	0.102801
$601$	10.1852	0.415464	0.207732	−	0.978186i	$-0.433392\pi$
$601$	10.1852	0.415464	0.207732	+	0.978186i	$0.433392\pi$
$602$	20.9599	0.854260
$603$	53.1630	2.16496
$604$	−18.2834	−0.743942
$605$	1.56647	0.0636860
$606$	37.7867	1.53498
$607$	−6.73899	−0.273527	−0.136764	−	0.990604i	$-0.543670\pi$
$607$	−6.73899	−0.273527	−0.136764	+	0.990604i	$0.543670\pi$
$608$	0.843778	0.0342197
$609$	−27.8882	−1.13009
$610$	−2.27647	−0.0921714
$611$	−4.31684	−0.174641
$612$	10.2389	0.413885
$613$	−9.22420	−0.372562	−0.186281	−	0.982497i	$-0.559643\pi$
$613$	−9.22420	−0.372562	−0.186281	+	0.982497i	$0.559643\pi$
$614$	17.1643	0.692695
$615$	−4.60277	−0.185602
$616$	−7.49906	−0.302146
$617$	48.4193	1.94929	0.974644	−	0.223759i	$-0.0718329\pi$
$617$	48.4193	1.94929	0.974644	+	0.223759i	$0.0718329\pi$
$618$	23.3651	0.939883
$619$	31.2974	1.25795	0.628975	−	0.777426i	$-0.283475\pi$
$619$	31.2974	1.25795	0.628975	+	0.777426i	$0.283475\pi$
$620$	1.00000	0.0401610
$621$	2.79578	0.112191
$622$	−1.86659	−0.0748436
$623$	13.7449	0.550678
$624$	−2.51811	−0.100805
$625$	1.00000	0.0400000
$626$	−28.8715	−1.15394
$627$	7.53198	0.300798
$628$	19.5611	0.780571
$629$	−2.01217	−0.0802307
$630$	−7.06743	−0.281573
$631$	27.6548	1.10092	0.550460	−	0.834862i	$-0.314453\pi$
$631$	27.6548	1.10092	0.550460	+	0.834862i	$0.314453\pi$
$632$	7.45891	0.296699
$633$	−20.1045	−0.799083
$634$	−5.30258	−0.210592
$635$	17.1118	0.679060
$636$	−12.0740	−0.478766
$637$	2.52492	0.100041
$638$	18.5589	0.734754
$639$	−22.2351	−0.879607
$640$	1.00000	0.0395285
$641$	11.8670	0.468717	0.234358	−	0.972150i	$-0.424701\pi$
$641$	11.8670	0.468717	0.234358	+	0.972150i	$0.424701\pi$
$642$	−46.8674	−1.84971
$643$	−27.9254	−1.10127	−0.550635	−	0.834746i	$-0.685614\pi$
$643$	−27.9254	−1.10127	−0.550635	+	0.834746i	$0.685614\pi$
$644$	−6.89012	−0.271509
$645$	−24.9496	−0.982388
$646$	2.58596	0.101743
$647$	13.5860	0.534120	0.267060	−	0.963680i	$-0.413948\pi$
$647$	13.5860	0.534120	0.267060	+	0.963680i	$0.413948\pi$
$648$	−7.86116	−0.308816
$649$	−12.0127	−0.471539
$650$	−1.00000	−0.0392232
$651$	−5.32691	−0.208778
$652$	−10.0185	−0.392353
$653$	−34.0921	−1.33413	−0.667063	−	0.745001i	$-0.732449\pi$
$653$	−34.0921	−1.33413	−0.667063	+	0.745001i	$0.732449\pi$
$654$	28.6138	1.11889
$655$	−9.24764	−0.361335
$656$	−1.82787	−0.0713663
$657$	27.2788	1.06425
$658$	−9.13201	−0.356003
$659$	0.285723	0.0111302	0.00556510	−	0.999985i	$-0.498229\pi$
$659$	0.285723	0.0111302	0.00556510	+	0.999985i	$0.498229\pi$
$660$	8.92650	0.347464
$661$	39.3580	1.53085	0.765425	−	0.643525i	$-0.222529\pi$
$661$	39.3580	1.53085	0.765425	+	0.643525i	$0.222529\pi$
$662$	−25.6652	−0.997506
$663$	−7.71736	−0.299718
$664$	−3.39648	−0.131809
$665$	−1.78496	−0.0692178
$666$	−2.19347	−0.0849954
$667$	17.0519	0.660251
$668$	16.5439	0.640102
$669$	35.4742	1.37151
$670$	15.9129	0.614767
$671$	−8.06989	−0.311535
$672$	−5.32691	−0.205490
$673$	−7.81171	−0.301119	−0.150560	−	0.988601i	$-0.548108\pi$
$673$	−7.81171	−0.301119	−0.150560	+	0.988601i	$0.548108\pi$
$674$	−16.6183	−0.640113
$675$	0.858374	0.0330388
$676$	1.00000	0.0384615
$677$	8.53020	0.327842	0.163921	−	0.986473i	$-0.447586\pi$
$677$	8.53020	0.327842	0.163921	+	0.986473i	$0.447586\pi$
$678$	−35.3637	−1.35813
$679$	2.25825	0.0866635
$680$	3.06474	0.117528
$681$	62.8075	2.40679
$682$	3.54492	0.135742
$683$	1.98282	0.0758703	0.0379352	−	0.999280i	$-0.487922\pi$
$683$	1.98282	0.0758703	0.0379352	+	0.999280i	$0.487922\pi$
$684$	2.81896	0.107786
$685$	7.26156	0.277450
$686$	20.1494	0.769307
$687$	−51.9832	−1.98328
$688$	−9.90805	−0.377741
$689$	4.79487	0.182670
$690$	8.20165	0.312231
$691$	26.0028	0.989193	0.494597	−	0.869123i	$-0.335316\pi$
$691$	26.0028	0.989193	0.494597	+	0.869123i	$0.335316\pi$
$692$	−9.58330	−0.364302
$693$	−25.0535	−0.951703
$694$	−29.4959	−1.11965
$695$	−3.60293	−0.136667
$696$	13.1832	0.499708
$697$	−5.60195	−0.212189
$698$	2.12568	0.0804583
$699$	2.64747	0.100137
$700$	−2.11544	−0.0799561
$701$	−14.2454	−0.538043	−0.269021	−	0.963134i	$-0.586700\pi$
$701$	−14.2454	−0.538043	−0.269021	+	0.963134i	$0.586700\pi$
$702$	−0.858374	−0.0323972
$703$	−0.553987	−0.0208940
$704$	3.54492	0.133604
$705$	10.8703	0.409398
$706$	−25.6406	−0.964998
$707$	−31.7442	−1.19386
$708$	−8.53312	−0.320694
$709$	−36.7838	−1.38144	−0.690722	−	0.723121i	$-0.742707\pi$
$709$	−36.7838	−1.38144	−0.690722	+	0.723121i	$0.742707\pi$
$710$	−6.65546	−0.249775
$711$	24.9193	0.934547
$712$	−6.49743	−0.243502
$713$	3.25706	0.121978
$714$	−16.3256	−0.610971
$715$	−3.54492	−0.132573
$716$	18.5473	0.693146
$717$	1.28921	0.0481466
$718$	22.2396	0.829976
$719$	−5.52291	−0.205970	−0.102985	−	0.994683i	$-0.532839\pi$
$719$	−5.52291	−0.205970	−0.102985	+	0.994683i	$0.532839\pi$
$720$	3.34088	0.124507
$721$	−19.6288	−0.731015
$722$	−18.2880	−0.680610
$723$	−63.6735	−2.36804
$724$	3.84534	0.142911
$725$	5.23535	0.194436
$726$	3.94454	0.146396
$727$	25.9634	0.962931	0.481465	−	0.876465i	$-0.340105\pi$
$727$	25.9634	0.962931	0.481465	+	0.876465i	$0.340105\pi$
$728$	2.11544	0.0784034
$729$	−32.7476	−1.21288
$730$	8.16517	0.302206
$731$	−30.3656	−1.12311
$732$	−5.73239	−0.211875
$733$	−24.0564	−0.888545	−0.444273	−	0.895892i	$-0.646538\pi$
$733$	−24.0564	−0.888545	−0.444273	+	0.895892i	$0.646538\pi$
$734$	−22.7696	−0.840441
$735$	−6.35803	−0.234519
$736$	3.25706	0.120057
$737$	56.4098	2.07788
$738$	−6.10669	−0.224790
$739$	−46.0374	−1.69351	−0.846756	−	0.531981i	$-0.821448\pi$
$739$	−46.0374	−1.69351	−0.846756	+	0.531981i	$0.821448\pi$
$740$	−0.656555	−0.0241355
$741$	−2.12473	−0.0780537
$742$	10.1433	0.372370
$743$	−20.4244	−0.749298	−0.374649	−	0.927167i	$-0.622237\pi$
$743$	−20.4244	−0.749298	−0.374649	+	0.927167i	$0.622237\pi$
$744$	2.51811	0.0923184
$745$	−12.3060	−0.450855
$746$	−4.58361	−0.167818
$747$	−11.3472	−0.415174
$748$	10.8643	0.397237
$749$	39.3728	1.43865
$750$	2.51811	0.0919484
$751$	25.7836	0.940855	0.470428	−	0.882439i	$-0.344100\pi$
$751$	25.7836	0.940855	0.470428	+	0.882439i	$0.344100\pi$
$752$	4.31684	0.157419
$753$	−38.5581	−1.40513
$754$	−5.23535	−0.190660
$755$	−18.2834	−0.665402
$756$	−1.81584	−0.0660413
$757$	45.0038	1.63569	0.817845	−	0.575438i	$-0.195168\pi$
$757$	45.0038	1.63569	0.817845	+	0.575438i	$0.195168\pi$
$758$	12.1471	0.441203
$759$	29.0742	1.05533
$760$	0.843778	0.0306070
$761$	44.3451	1.60751	0.803755	−	0.594961i	$-0.202832\pi$
$761$	44.3451	1.60751	0.803755	+	0.594961i	$0.202832\pi$
$762$	43.0894	1.56096
$763$	−24.0382	−0.870240
$764$	−8.23049	−0.297769
$765$	10.2389	0.370190
$766$	−29.6790	−1.07235
$767$	3.38870	0.122359
$768$	2.51811	0.0908645
$769$	−35.6994	−1.28735	−0.643677	−	0.765298i	$-0.722592\pi$
$769$	−35.6994	−1.28735	−0.643677	+	0.765298i	$0.722592\pi$
$770$	−7.49906	−0.270247
$771$	−45.9357	−1.65433
$772$	3.37321	0.121405
$773$	35.8272	1.28862	0.644308	−	0.764766i	$-0.277145\pi$
$773$	35.8272	1.28862	0.644308	+	0.764766i	$0.277145\pi$
$774$	−33.1016	−1.18981
$775$	1.00000	0.0359211
$776$	−1.06751	−0.0383213
$777$	3.49741	0.125469
$778$	10.0468	0.360194
$779$	−1.54231	−0.0552591
$780$	−2.51811	−0.0901628
$781$	−23.5931	−0.844227
$782$	9.98207	0.356958
$783$	4.49389	0.160598
$784$	−2.52492	−0.0901757
$785$	19.5611	0.698164
$786$	−23.2866	−0.830604
$787$	−10.3602	−0.369302	−0.184651	−	0.982804i	$-0.559115\pi$
$787$	−10.3602	−0.369302	−0.184651	+	0.982804i	$0.559115\pi$
$788$	7.50540	0.267369
$789$	−30.9243	−1.10094
$790$	7.45891	0.265376
$791$	29.7086	1.05632
$792$	11.8432	0.420828
$793$	2.27647	0.0808397
$794$	−24.7942	−0.879912
$795$	−12.0740	−0.428221
$796$	−20.2672	−0.718352
$797$	29.2026	1.03441	0.517205	−	0.855862i	$-0.326973\pi$
$797$	29.2026	1.03441	0.517205	+	0.855862i	$0.326973\pi$
$798$	−4.49472	−0.159112
$799$	13.2300	0.468044
$800$	1.00000	0.0353553
$801$	−21.7071	−0.766984
$802$	20.2223	0.714073
$803$	28.9449	1.02144
$804$	40.0703	1.41317
$805$	−6.89012	−0.242845
$806$	−1.00000	−0.0352235
$807$	−70.7873	−2.49183
$808$	15.0060	0.527908
$809$	13.4120	0.471542	0.235771	−	0.971809i	$-0.424239\pi$
$809$	13.4120	0.471542	0.235771	+	0.971809i	$0.424239\pi$
$810$	−7.86116	−0.276213
$811$	4.73823	0.166382	0.0831909	−	0.996534i	$-0.473489\pi$
$811$	4.73823	0.166382	0.0831909	+	0.996534i	$0.473489\pi$
$812$	−11.0751	−0.388659
$813$	−23.3846	−0.820135
$814$	−2.32744	−0.0815767
$815$	−10.0185	−0.350931
$816$	7.71736	0.270162
$817$	−8.36019	−0.292486
$818$	18.2534	0.638216
$819$	7.06743	0.246956
$820$	−1.82787	−0.0638319
$821$	−4.43224	−0.154686	−0.0773432	−	0.997005i	$-0.524644\pi$
$821$	−4.43224	−0.154686	−0.0773432	+	0.997005i	$0.524644\pi$
$822$	18.2854	0.637777
$823$	27.5058	0.958793	0.479397	−	0.877598i	$-0.340856\pi$
$823$	27.5058	0.958793	0.479397	+	0.877598i	$0.340856\pi$
$824$	9.27883	0.323244
$825$	8.92650	0.310781
$826$	7.16858	0.249427
$827$	−12.1009	−0.420791	−0.210395	−	0.977616i	$-0.567475\pi$
$827$	−12.1009	−0.420791	−0.210395	+	0.977616i	$0.567475\pi$
$828$	10.8815	0.378157
$829$	−9.47380	−0.329039	−0.164519	−	0.986374i	$-0.552607\pi$
$829$	−9.47380	−0.329039	−0.164519	+	0.986374i	$0.552607\pi$
$830$	−3.39648	−0.117894
$831$	28.7317	0.996692
$832$	−1.00000	−0.0346688
$833$	−7.73824	−0.268114
$834$	−9.07258	−0.314158
$835$	16.5439	0.572525
$836$	2.99113	0.103450
$837$	0.858374	0.0296697
$838$	13.9566	0.482121
$839$	−16.4949	−0.569466	−0.284733	−	0.958607i	$-0.591905\pi$
$839$	−16.4949	−0.569466	−0.284733	+	0.958607i	$0.591905\pi$
$840$	−5.32691	−0.183796
$841$	−1.59109	−0.0548652
$842$	−8.12836	−0.280122
$843$	−50.0128	−1.72253
$844$	−7.98397	−0.274820
$845$	1.00000	0.0344010
$846$	14.4220	0.495840
$847$	−3.31377	−0.113862
$848$	−4.79487	−0.164656
$849$	−48.6410	−1.66935
$850$	3.06474	0.105120
$851$	−2.13844	−0.0733049
$852$	−16.7592	−0.574160
$853$	23.2364	0.795600	0.397800	−	0.917472i	$-0.369774\pi$
$853$	23.2364	0.795600	0.397800	+	0.917472i	$0.369774\pi$
$854$	4.81572	0.164791
$855$	2.81896	0.0964064
$856$	−18.6121	−0.636149
$857$	−39.9698	−1.36534	−0.682671	−	0.730726i	$-0.739182\pi$
$857$	−39.9698	−1.36534	−0.682671	+	0.730726i	$0.739182\pi$
$858$	−8.92650	−0.304746
$859$	−11.5793	−0.395079	−0.197540	−	0.980295i	$-0.563295\pi$
$859$	−11.5793	−0.395079	−0.197540	+	0.980295i	$0.563295\pi$
$860$	−9.90805	−0.337862
$861$	9.73688	0.331832
$862$	2.93844	0.100084
$863$	−20.1215	−0.684943	−0.342472	−	0.939528i	$-0.611264\pi$
$863$	−20.1215	−0.684943	−0.342472	+	0.939528i	$0.611264\pi$
$864$	0.858374	0.0292025
$865$	−9.58330	−0.325842
$866$	24.5968	0.835835
$867$	−19.1561	−0.650576
$868$	−2.11544	−0.0718027
$869$	26.4412	0.896958
$870$	13.1832	0.446952
$871$	−15.9129	−0.539187
$872$	11.3632	0.384807
$873$	−3.56641	−0.120705
$874$	2.74824	0.0929605
$875$	−2.11544	−0.0715149
$876$	20.5608	0.694685
$877$	−8.62324	−0.291186	−0.145593	−	0.989345i	$-0.546509\pi$
$877$	−8.62324	−0.291186	−0.145593	+	0.989345i	$0.546509\pi$
$878$	24.1951	0.816547
$879$	31.3972	1.05900
$880$	3.54492	0.119499
$881$	−20.1744	−0.679694	−0.339847	−	0.940481i	$-0.610375\pi$
$881$	−20.1744	−0.679694	−0.339847	+	0.940481i	$0.610375\pi$
$882$	−8.43546	−0.284037
$883$	−9.85470	−0.331637	−0.165819	−	0.986156i	$-0.553027\pi$
$883$	−9.85470	−0.331637	−0.165819	+	0.986156i	$0.553027\pi$
$884$	−3.06474	−0.103078
$885$	−8.53312	−0.286838
$886$	1.42958	0.0480276
$887$	−22.2270	−0.746311	−0.373156	−	0.927769i	$-0.621724\pi$
$887$	−22.2270	−0.746311	−0.373156	+	0.927769i	$0.621724\pi$
$888$	−1.65328	−0.0554804
$889$	−36.1989	−1.21407
$890$	−6.49743	−0.217794
$891$	−27.8672	−0.933586
$892$	14.0876	0.471689
$893$	3.64245	0.121890
$894$	−30.9878	−1.03639
$895$	18.5473	0.619968
$896$	−2.11544	−0.0706718
$897$	−8.20165	−0.273845
$898$	−2.45806	−0.0820265
$899$	5.23535	0.174609
$900$	3.34088	0.111363
$901$	−14.6950	−0.489563
$902$	−6.47965	−0.215749
$903$	52.7793	1.75638
$904$	−14.0437	−0.467087
$905$	3.84534	0.127823
$906$	−46.0397	−1.52957
$907$	5.62384	0.186736	0.0933682	−	0.995632i	$-0.470237\pi$
$907$	5.62384	0.186736	0.0933682	+	0.995632i	$0.470237\pi$
$908$	24.9423	0.827740
$909$	50.1331	1.66281
$910$	2.11544	0.0701261
$911$	24.0263	0.796028	0.398014	−	0.917379i	$-0.369700\pi$
$911$	24.0263	0.796028	0.398014	+	0.917379i	$0.369700\pi$
$912$	2.12473	0.0703567
$913$	−12.0403	−0.398475
$914$	11.4182	0.377680
$915$	−5.73239	−0.189507
$916$	−20.6437	−0.682088
$917$	19.5628	0.646021
$918$	2.63070	0.0868259
$919$	−13.2748	−0.437896	−0.218948	−	0.975737i	$-0.570262\pi$
$919$	−13.2748	−0.437896	−0.218948	+	0.975737i	$0.570262\pi$
$920$	3.25706	0.107382
$921$	43.2216	1.42420
$922$	20.2730	0.667654
$923$	6.65546	0.219067
$924$	−18.8835	−0.621221
$925$	−0.656555	−0.0215874
$926$	−1.58584	−0.0521140
$927$	30.9995	1.01816
$928$	5.23535	0.171859
$929$	−5.01974	−0.164692	−0.0823462	−	0.996604i	$-0.526241\pi$
$929$	−5.01974	−0.164692	−0.0823462	+	0.996604i	$0.526241\pi$
$930$	2.51811	0.0825721
$931$	−2.13047	−0.0698234
$932$	1.05137	0.0344389
$933$	−4.70029	−0.153881
$934$	−22.9265	−0.750177
$935$	10.8643	0.355300
$936$	−3.34088	−0.109200
$937$	−9.12721	−0.298173	−0.149086	−	0.988824i	$-0.547633\pi$
$937$	−9.12721	−0.298173	−0.149086	+	0.988824i	$0.547633\pi$
$938$	−33.6627	−1.09913
$939$	−72.7017	−2.37253
$940$	4.31684	0.140800
$941$	−6.31901	−0.205994	−0.102997	−	0.994682i	$-0.532843\pi$
$941$	−6.31901	−0.205994	−0.102997	+	0.994682i	$0.532843\pi$
$942$	49.2569	1.60488
$943$	−5.95348	−0.193872
$944$	−3.38870	−0.110293
$945$	−1.81584	−0.0590692
$946$	−35.1233	−1.14196
$947$	41.2320	1.33986	0.669930	−	0.742425i	$-0.266324\pi$
$947$	41.2320	1.33986	0.669930	+	0.742425i	$0.266324\pi$
$948$	18.7824	0.610022
$949$	−8.16517	−0.265053
$950$	0.843778	0.0273758
$951$	−13.3525	−0.432984
$952$	−6.48328	−0.210124
$953$	26.2936	0.851732	0.425866	−	0.904786i	$-0.359970\pi$
$953$	26.2936	0.851732	0.425866	+	0.904786i	$0.359970\pi$
$954$	−16.0191	−0.518637
$955$	−8.23049	−0.266332
$956$	0.511977	0.0165585
$957$	46.7334	1.51068
$958$	−21.8494	−0.705924
$959$	−15.3614	−0.496045
$960$	2.51811	0.0812717
$961$	1.00000	0.0322581
$962$	0.656555	0.0211682
$963$	−62.1808	−2.00375
$964$	−25.2862	−0.814415
$965$	3.37321	0.108588
$966$	−17.3501	−0.558230
$967$	−5.04159	−0.162127	−0.0810633	−	0.996709i	$-0.525832\pi$
$967$	−5.04159	−0.162127	−0.0810633	+	0.996709i	$0.525832\pi$
$968$	1.56647	0.0503482
$969$	6.51174	0.209187
$970$	−1.06751	−0.0342756
$971$	−59.5329	−1.91050	−0.955250	−	0.295798i	$-0.904414\pi$
$971$	−59.5329	−1.91050	−0.955250	+	0.295798i	$0.904414\pi$
$972$	−22.3704	−0.717531
$973$	7.62178	0.244343
$974$	−9.23171	−0.295803
$975$	−2.51811	−0.0806441
$976$	−2.27647	−0.0728679
$977$	14.8639	0.475540	0.237770	−	0.971322i	$-0.423584\pi$
$977$	14.8639	0.475540	0.237770	+	0.971322i	$0.423584\pi$
$978$	−25.2276	−0.806689
$979$	−23.0329	−0.736134
$980$	−2.52492	−0.0806556
$981$	37.9631	1.21207
$982$	6.63258	0.211654
$983$	46.5700	1.48535	0.742676	−	0.669651i	$-0.233556\pi$
$983$	46.5700	1.48535	0.742676	+	0.669651i	$0.233556\pi$
$984$	−4.60277	−0.146731
$985$	7.50540	0.239142
$986$	16.0450	0.510977
$987$	−22.9954	−0.731952
$988$	−0.843778	−0.0268441
$989$	−32.2712	−1.02616
$990$	11.8432	0.376400
$991$	25.4923	0.809789	0.404894	−	0.914363i	$-0.367308\pi$
$991$	25.4923	0.809789	0.404894	+	0.914363i	$0.367308\pi$
$992$	1.00000	0.0317500
$993$	−64.6278	−2.05090
$994$	14.0792	0.446566
$995$	−20.2672	−0.642513
$996$	−8.55272	−0.271003
$997$	20.5287	0.650151	0.325076	−	0.945688i	$-0.394610\pi$
$997$	20.5287	0.650151	0.325076	+	0.945688i	$0.394610\pi$
$998$	−8.28705	−0.262322
$999$	−0.563570	−0.0178306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.o.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.o.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.51811 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.51811 q^{6} -2.11544 q^{7} +1.00000 q^{8} +3.34088 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.51811 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.51811 q^{6} -2.11544 q^{7} +1.00000 q^{8} +3.34088 q^{9} +1.00000 q^{10} +3.54492 q^{11} +2.51811 q^{12} -1.00000 q^{13} -2.11544 q^{14} +2.51811 q^{15} +1.00000 q^{16} +3.06474 q^{17} +3.34088 q^{18} +0.843778 q^{19} +1.00000 q^{20} -5.32691 q^{21} +3.54492 q^{22} +3.25706 q^{23} +2.51811 q^{24} +1.00000 q^{25} -1.00000 q^{26} +0.858374 q^{27} -2.11544 q^{28} +5.23535 q^{29} +2.51811 q^{30} +1.00000 q^{31} +1.00000 q^{32} +8.92650 q^{33} +3.06474 q^{34} -2.11544 q^{35} +3.34088 q^{36} -0.656555 q^{37} +0.843778 q^{38} -2.51811 q^{39} +1.00000 q^{40} -1.82787 q^{41} -5.32691 q^{42} -9.90805 q^{43} +3.54492 q^{44} +3.34088 q^{45} +3.25706 q^{46} +4.31684 q^{47} +2.51811 q^{48} -2.52492 q^{49} +1.00000 q^{50} +7.71736 q^{51} -1.00000 q^{52} -4.79487 q^{53} +0.858374 q^{54} +3.54492 q^{55} -2.11544 q^{56} +2.12473 q^{57} +5.23535 q^{58} -3.38870 q^{59} +2.51811 q^{60} -2.27647 q^{61} +1.00000 q^{62} -7.06743 q^{63} +1.00000 q^{64} -1.00000 q^{65} +8.92650 q^{66} +15.9129 q^{67} +3.06474 q^{68} +8.20165 q^{69} -2.11544 q^{70} -6.65546 q^{71} +3.34088 q^{72} +8.16517 q^{73} -0.656555 q^{74} +2.51811 q^{75} +0.843778 q^{76} -7.49906 q^{77} -2.51811 q^{78} +7.45891 q^{79} +1.00000 q^{80} -7.86116 q^{81} -1.82787 q^{82} -3.39648 q^{83} -5.32691 q^{84} +3.06474 q^{85} -9.90805 q^{86} +13.1832 q^{87} +3.54492 q^{88} -6.49743 q^{89} +3.34088 q^{90} +2.11544 q^{91} +3.25706 q^{92} +2.51811 q^{93} +4.31684 q^{94} +0.843778 q^{95} +2.51811 q^{96} -1.06751 q^{97} -2.52492 q^{98} +11.8432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * q^99

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