LMFDB - Embedded newform 4030.2.a.f.1.2

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.4418197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 12x^{2} + 3x - 1 $ x^6 - 7x^4 - x^3 + 12x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.191750$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.56820 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.56820 q^{6} -1.49195 q^{7} +1.00000 q^{8} -0.540753 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.56820 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.56820 q^{6} -1.49195 q^{7} +1.00000 q^{8} -0.540753 q^{9} -1.00000 q^{10} -1.02486 q^{11} -1.56820 q^{12} +1.00000 q^{13} -1.49195 q^{14} +1.56820 q^{15} +1.00000 q^{16} +1.43401 q^{17} -0.540753 q^{18} +2.94238 q^{19} -1.00000 q^{20} +2.33968 q^{21} -1.02486 q^{22} +1.14116 q^{23} -1.56820 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.55260 q^{27} -1.49195 q^{28} +6.02560 q^{29} +1.56820 q^{30} -1.00000 q^{31} +1.00000 q^{32} +1.60718 q^{33} +1.43401 q^{34} +1.49195 q^{35} -0.540753 q^{36} -6.22717 q^{37} +2.94238 q^{38} -1.56820 q^{39} -1.00000 q^{40} +3.45260 q^{41} +2.33968 q^{42} +4.32637 q^{43} -1.02486 q^{44} +0.540753 q^{45} +1.14116 q^{46} -4.37159 q^{47} -1.56820 q^{48} -4.77407 q^{49} +1.00000 q^{50} -2.24882 q^{51} +1.00000 q^{52} -9.04674 q^{53} +5.55260 q^{54} +1.02486 q^{55} -1.49195 q^{56} -4.61423 q^{57} +6.02560 q^{58} -8.83881 q^{59} +1.56820 q^{60} -11.6584 q^{61} -1.00000 q^{62} +0.806779 q^{63} +1.00000 q^{64} -1.00000 q^{65} +1.60718 q^{66} +9.32431 q^{67} +1.43401 q^{68} -1.78957 q^{69} +1.49195 q^{70} -15.9341 q^{71} -0.540753 q^{72} -4.75697 q^{73} -6.22717 q^{74} -1.56820 q^{75} +2.94238 q^{76} +1.52904 q^{77} -1.56820 q^{78} -5.97259 q^{79} -1.00000 q^{80} -7.08533 q^{81} +3.45260 q^{82} +0.949922 q^{83} +2.33968 q^{84} -1.43401 q^{85} +4.32637 q^{86} -9.44933 q^{87} -1.02486 q^{88} -10.5946 q^{89} +0.540753 q^{90} -1.49195 q^{91} +1.14116 q^{92} +1.56820 q^{93} -4.37159 q^{94} -2.94238 q^{95} -1.56820 q^{96} +13.4934 q^{97} -4.77407 q^{98} +0.554194 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 3 q^{3} + 6 q^{4} - 6 q^{5} - 3 q^{6} - 2 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - 3 * q^3 + 6 * q^4 - 6 * q^5 - 3 * q^6 - 2 * q^7 + 6 * q^8 - q^9	\( 6 q + 6 q^{2} - 3 q^{3} + 6 q^{4} - 6 q^{5} - 3 q^{6} - 2 q^{7} + 6 q^{8} - q^{9} - 6 q^{10} - 4 q^{11} - 3 q^{12} + 6 q^{13} - 2 q^{14} + 3 q^{15} + 6 q^{16} - 8 q^{17} - q^{18} - 9 q^{19} - 6 q^{20} - 5 q^{21} - 4 q^{22} - 7 q^{23} - 3 q^{24} + 6 q^{25} + 6 q^{26} + 9 q^{27} - 2 q^{28} - 14 q^{29} + 3 q^{30} - 6 q^{31} + 6 q^{32} - 6 q^{33} - 8 q^{34} + 2 q^{35} - q^{36} - 9 q^{38} - 3 q^{39} - 6 q^{40} + 2 q^{41} - 5 q^{42} - 7 q^{43} - 4 q^{44} + q^{45} - 7 q^{46} - 8 q^{47} - 3 q^{48} - 14 q^{49} + 6 q^{50} - 5 q^{51} + 6 q^{52} - 24 q^{53} + 9 q^{54} + 4 q^{55} - 2 q^{56} - 15 q^{57} - 14 q^{58} - 5 q^{59} + 3 q^{60} - 5 q^{61} - 6 q^{62} - 19 q^{63} + 6 q^{64} - 6 q^{65} - 6 q^{66} - 12 q^{67} - 8 q^{68} + 2 q^{70} - 10 q^{71} - q^{72} + 5 q^{73} - 3 q^{75} - 9 q^{76} - q^{77} - 3 q^{78} - 16 q^{79} - 6 q^{80} - 10 q^{81} + 2 q^{82} - 22 q^{83} - 5 q^{84} + 8 q^{85} - 7 q^{86} - 31 q^{87} - 4 q^{88} + 14 q^{89} + q^{90} - 2 q^{91} - 7 q^{92} + 3 q^{93} - 8 q^{94} + 9 q^{95} - 3 q^{96} - 9 q^{97} - 14 q^{98} - 21 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 3 * q^3 + 6 * q^4 - 6 * q^5 - 3 * q^6 - 2 * q^7 + 6 * q^8 - q^9 - 6 * q^10 - 4 * q^11 - 3 * q^12 + 6 * q^13 - 2 * q^14 + 3 * q^15 + 6 * q^16 - 8 * q^17 - q^18 - 9 * q^19 - 6 * q^20 - 5 * q^21 - 4 * q^22 - 7 * q^23 - 3 * q^24 + 6 * q^25 + 6 * q^26 + 9 * q^27 - 2 * q^28 - 14 * q^29 + 3 * q^30 - 6 * q^31 + 6 * q^32 - 6 * q^33 - 8 * q^34 + 2 * q^35 - q^36 - 9 * q^38 - 3 * q^39 - 6 * q^40 + 2 * q^41 - 5 * q^42 - 7 * q^43 - 4 * q^44 + q^45 - 7 * q^46 - 8 * q^47 - 3 * q^48 - 14 * q^49 + 6 * q^50 - 5 * q^51 + 6 * q^52 - 24 * q^53 + 9 * q^54 + 4 * q^55 - 2 * q^56 - 15 * q^57 - 14 * q^58 - 5 * q^59 + 3 * q^60 - 5 * q^61 - 6 * q^62 - 19 * q^63 + 6 * q^64 - 6 * q^65 - 6 * q^66 - 12 * q^67 - 8 * q^68 + 2 * q^70 - 10 * q^71 - q^72 + 5 * q^73 - 3 * q^75 - 9 * q^76 - q^77 - 3 * q^78 - 16 * q^79 - 6 * q^80 - 10 * q^81 + 2 * q^82 - 22 * q^83 - 5 * q^84 + 8 * q^85 - 7 * q^86 - 31 * q^87 - 4 * q^88 + 14 * q^89 + q^90 - 2 * q^91 - 7 * q^92 + 3 * q^93 - 8 * q^94 + 9 * q^95 - 3 * q^96 - 9 * q^97 - 14 * q^98 - 21 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.56820	−0.905400	−0.452700	−	0.891663i	$-0.649539\pi$
$3$	−1.56820	−0.905400	−0.452700	+	0.891663i	$0.649539\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.56820	−0.640214
$7$	−1.49195	−0.563906	−0.281953	−	0.959428i	$-0.590982\pi$
$7$	−1.49195	−0.563906	−0.281953	+	0.959428i	$0.590982\pi$
$8$	1.00000	0.353553
$9$	−0.540753	−0.180251
$10$	−1.00000	−0.316228
$11$	−1.02486	−0.309006	−0.154503	−	0.987992i	$-0.549378\pi$
$11$	−1.02486	−0.309006	−0.154503	+	0.987992i	$0.549378\pi$
$12$	−1.56820	−0.452700
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.49195	−0.398742
$15$	1.56820	0.404907
$16$	1.00000	0.250000
$17$	1.43401	0.347799	0.173900	−	0.984763i	$-0.444363\pi$
$17$	1.43401	0.347799	0.173900	+	0.984763i	$0.444363\pi$
$18$	−0.540753	−0.127457
$19$	2.94238	0.675027	0.337514	−	0.941321i	$-0.390414\pi$
$19$	2.94238	0.675027	0.337514	+	0.941321i	$0.390414\pi$
$20$	−1.00000	−0.223607
$21$	2.33968	0.510560
$22$	−1.02486	−0.218500
$23$	1.14116	0.237949	0.118974	−	0.992897i	$-0.462039\pi$
$23$	1.14116	0.237949	0.118974	+	0.992897i	$0.462039\pi$
$24$	−1.56820	−0.320107
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	5.55260	1.06860
$28$	−1.49195	−0.281953
$29$	6.02560	1.11893	0.559463	−	0.828856i	$-0.311008\pi$
$29$	6.02560	1.11893	0.559463	+	0.828856i	$0.311008\pi$
$30$	1.56820	0.286313
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	1.60718	0.279774
$34$	1.43401	0.245931
$35$	1.49195	0.252186
$36$	−0.540753	−0.0901255
$37$	−6.22717	−1.02374	−0.511870	−	0.859063i	$-0.671047\pi$
$37$	−6.22717	−1.02374	−0.511870	+	0.859063i	$0.671047\pi$
$38$	2.94238	0.477316
$39$	−1.56820	−0.251113
$40$	−1.00000	−0.158114
$41$	3.45260	0.539205	0.269603	−	0.962972i	$-0.413108\pi$
$41$	3.45260	0.539205	0.269603	+	0.962972i	$0.413108\pi$
$42$	2.33968	0.361021
$43$	4.32637	0.659765	0.329883	−	0.944022i	$-0.392991\pi$
$43$	4.32637	0.659765	0.329883	+	0.944022i	$0.392991\pi$
$44$	−1.02486	−0.154503
$45$	0.540753	0.0806107
$46$	1.14116	0.168255
$47$	−4.37159	−0.637662	−0.318831	−	0.947812i	$-0.603290\pi$
$47$	−4.37159	−0.637662	−0.318831	+	0.947812i	$0.603290\pi$
$48$	−1.56820	−0.226350
$49$	−4.77407	−0.682010
$50$	1.00000	0.141421
$51$	−2.24882	−0.314897
$52$	1.00000	0.138675
$53$	−9.04674	−1.24266	−0.621332	−	0.783547i	$-0.713408\pi$
$53$	−9.04674	−1.24266	−0.621332	+	0.783547i	$0.713408\pi$
$54$	5.55260	0.755614
$55$	1.02486	0.138192
$56$	−1.49195	−0.199371
$57$	−4.61423	−0.611169
$58$	6.02560	0.791200
$59$	−8.83881	−1.15071	−0.575357	−	0.817902i	$-0.695137\pi$
$59$	−8.83881	−1.15071	−0.575357	+	0.817902i	$0.695137\pi$
$60$	1.56820	0.202454
$61$	−11.6584	−1.49271	−0.746353	−	0.665550i	$-0.768197\pi$
$61$	−11.6584	−1.49271	−0.746353	+	0.665550i	$0.768197\pi$
$62$	−1.00000	−0.127000
$63$	0.806779	0.101645
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	1.60718	0.197830
$67$	9.32431	1.13915	0.569573	−	0.821941i	$-0.307109\pi$
$67$	9.32431	1.13915	0.569573	+	0.821941i	$0.307109\pi$
$68$	1.43401	0.173900
$69$	−1.78957	−0.215439
$70$	1.49195	0.178323
$71$	−15.9341	−1.89103	−0.945513	−	0.325583i	$-0.894439\pi$
$71$	−15.9341	−1.89103	−0.945513	+	0.325583i	$0.894439\pi$
$72$	−0.540753	−0.0637284
$73$	−4.75697	−0.556761	−0.278381	−	0.960471i	$-0.589798\pi$
$73$	−4.75697	−0.556761	−0.278381	+	0.960471i	$0.589798\pi$
$74$	−6.22717	−0.723893
$75$	−1.56820	−0.181080
$76$	2.94238	0.337514
$77$	1.52904	0.174250
$78$	−1.56820	−0.177564
$79$	−5.97259	−0.671969	−0.335984	−	0.941868i	$-0.609069\pi$
$79$	−5.97259	−0.671969	−0.335984	+	0.941868i	$0.609069\pi$
$80$	−1.00000	−0.111803
$81$	−7.08533	−0.787258
$82$	3.45260	0.381276
$83$	0.949922	0.104267	0.0521337	−	0.998640i	$-0.483398\pi$
$83$	0.949922	0.104267	0.0521337	+	0.998640i	$0.483398\pi$
$84$	2.33968	0.255280
$85$	−1.43401	−0.155541
$86$	4.32637	0.466525
$87$	−9.44933	−1.01307
$88$	−1.02486	−0.109250
$89$	−10.5946	−1.12303	−0.561513	−	0.827468i	$-0.689781\pi$
$89$	−10.5946	−1.12303	−0.561513	+	0.827468i	$0.689781\pi$
$90$	0.540753	0.0570004
$91$	−1.49195	−0.156399
$92$	1.14116	0.118974
$93$	1.56820	0.162615
$94$	−4.37159	−0.450895
$95$	−2.94238	−0.301881
$96$	−1.56820	−0.160054
$97$	13.4934	1.37005	0.685024	−	0.728520i	$-0.259792\pi$
$97$	13.4934	1.37005	0.685024	+	0.728520i	$0.259792\pi$
$98$	−4.77407	−0.482254
$99$	0.554194	0.0556986
$100$	1.00000	0.100000
$101$	7.27999	0.724386	0.362193	−	0.932103i	$-0.382028\pi$
$101$	7.27999	0.724386	0.362193	+	0.932103i	$0.382028\pi$
$102$	−2.24882	−0.222666
$103$	−11.8444	−1.16706	−0.583531	−	0.812091i	$-0.698329\pi$
$103$	−11.8444	−1.16706	−0.583531	+	0.812091i	$0.698329\pi$
$104$	1.00000	0.0980581
$105$	−2.33968	−0.228329
$106$	−9.04674	−0.878697
$107$	−13.9269	−1.34636	−0.673182	−	0.739477i	$-0.735073\pi$
$107$	−13.9269	−1.34636	−0.673182	+	0.739477i	$0.735073\pi$
$108$	5.55260	0.534300
$109$	0.767912	0.0735527	0.0367763	−	0.999324i	$-0.488291\pi$
$109$	0.767912	0.0735527	0.0367763	+	0.999324i	$0.488291\pi$
$110$	1.02486	0.0977162
$111$	9.76543	0.926894
$112$	−1.49195	−0.140976
$113$	16.9743	1.59680	0.798402	−	0.602124i	$-0.205679\pi$
$113$	16.9743	1.59680	0.798402	+	0.602124i	$0.205679\pi$
$114$	−4.61423	−0.432162
$115$	−1.14116	−0.106414
$116$	6.02560	0.559463
$117$	−0.540753	−0.0499927
$118$	−8.83881	−0.813678
$119$	−2.13948	−0.196126
$120$	1.56820	0.143156
$121$	−9.94967	−0.904515
$122$	−11.6584	−1.05550
$123$	−5.41436	−0.488196
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0.806779	0.0718736
$127$	−18.8774	−1.67510	−0.837549	−	0.546362i	$-0.816012\pi$
$127$	−18.8774	−1.67510	−0.837549	+	0.546362i	$0.816012\pi$
$128$	1.00000	0.0883883
$129$	−6.78461	−0.597351
$130$	−1.00000	−0.0877058
$131$	7.58378	0.662598	0.331299	−	0.943526i	$-0.392513\pi$
$131$	7.58378	0.662598	0.331299	+	0.943526i	$0.392513\pi$
$132$	1.60718	0.139887
$133$	−4.38989	−0.380652
$134$	9.32431	0.805498
$135$	−5.55260	−0.477892
$136$	1.43401	0.122966
$137$	15.2258	1.30083	0.650414	−	0.759580i	$-0.274595\pi$
$137$	15.2258	1.30083	0.650414	+	0.759580i	$0.274595\pi$
$138$	−1.78957	−0.152338
$139$	−8.98962	−0.762490	−0.381245	−	0.924474i	$-0.624505\pi$
$139$	−8.98962	−0.762490	−0.381245	+	0.924474i	$0.624505\pi$
$140$	1.49195	0.126093
$141$	6.85552	0.577339
$142$	−15.9341	−1.33716
$143$	−1.02486	−0.0857028
$144$	−0.540753	−0.0450628
$145$	−6.02560	−0.500399
$146$	−4.75697	−0.393690
$147$	7.48669	0.617492
$148$	−6.22717	−0.511870
$149$	11.4923	0.941482	0.470741	−	0.882271i	$-0.343987\pi$
$149$	11.4923	0.941482	0.470741	+	0.882271i	$0.343987\pi$
$150$	−1.56820	−0.128043
$151$	6.86593	0.558742	0.279371	−	0.960183i	$-0.409874\pi$
$151$	6.86593	0.558742	0.279371	+	0.960183i	$0.409874\pi$
$152$	2.94238	0.238658
$153$	−0.775447	−0.0626912
$154$	1.52904	0.123213
$155$	1.00000	0.0803219
$156$	−1.56820	−0.125556
$157$	−6.46147	−0.515681	−0.257841	−	0.966187i	$-0.583011\pi$
$157$	−6.46147	−0.515681	−0.257841	+	0.966187i	$0.583011\pi$
$158$	−5.97259	−0.475154
$159$	14.1871	1.12511
$160$	−1.00000	−0.0790569
$161$	−1.70256	−0.134181
$162$	−7.08533	−0.556676
$163$	−4.82792	−0.378152	−0.189076	−	0.981962i	$-0.560549\pi$
$163$	−4.82792	−0.378152	−0.189076	+	0.981962i	$0.560549\pi$
$164$	3.45260	0.269603
$165$	−1.60718	−0.125119
$166$	0.949922	0.0737282
$167$	−13.2027	−1.02165	−0.510826	−	0.859684i	$-0.670661\pi$
$167$	−13.2027	−1.02165	−0.510826	+	0.859684i	$0.670661\pi$
$168$	2.33968	0.180510
$169$	1.00000	0.0769231
$170$	−1.43401	−0.109984
$171$	−1.59110	−0.121674
$172$	4.32637	0.329883
$173$	−6.54788	−0.497826	−0.248913	−	0.968526i	$-0.580073\pi$
$173$	−6.54788	−0.497826	−0.248913	+	0.968526i	$0.580073\pi$
$174$	−9.44933	−0.716352
$175$	−1.49195	−0.112781
$176$	−1.02486	−0.0772515
$177$	13.8610	1.04186
$178$	−10.5946	−0.794099
$179$	−19.2959	−1.44224	−0.721120	−	0.692810i	$-0.756372\pi$
$179$	−19.2959	−1.44224	−0.721120	+	0.692810i	$0.756372\pi$
$180$	0.540753	0.0403054
$181$	10.1329	0.753176	0.376588	−	0.926381i	$-0.377097\pi$
$181$	10.1329	0.753176	0.376588	+	0.926381i	$0.377097\pi$
$182$	−1.49195	−0.110591
$183$	18.2827	1.35150
$184$	1.14116	0.0841277
$185$	6.22717	0.457830
$186$	1.56820	0.114986
$187$	−1.46966	−0.107472
$188$	−4.37159	−0.318831
$189$	−8.28423	−0.602589
$190$	−2.94238	−0.213462
$191$	−24.1076	−1.74436	−0.872181	−	0.489184i	$-0.837295\pi$
$191$	−24.1076	−1.74436	−0.872181	+	0.489184i	$0.837295\pi$
$192$	−1.56820	−0.113175
$193$	21.8594	1.57348	0.786739	−	0.617286i	$-0.211768\pi$
$193$	21.8594	1.57348	0.786739	+	0.617286i	$0.211768\pi$
$194$	13.4934	0.968771
$195$	1.56820	0.112301
$196$	−4.77407	−0.341005
$197$	−17.8184	−1.26951	−0.634753	−	0.772715i	$-0.718898\pi$
$197$	−17.8184	−1.26951	−0.634753	+	0.772715i	$0.718898\pi$
$198$	0.554194	0.0393849
$199$	−11.2098	−0.794640	−0.397320	−	0.917680i	$-0.630060\pi$
$199$	−11.2098	−0.794640	−0.397320	+	0.917680i	$0.630060\pi$
$200$	1.00000	0.0707107
$201$	−14.6224	−1.03138
$202$	7.27999	0.512219
$203$	−8.98992	−0.630968
$204$	−2.24882	−0.157449
$205$	−3.45260	−0.241140
$206$	−11.8444	−0.825237
$207$	−0.617088	−0.0428906
$208$	1.00000	0.0693375
$209$	−3.01551	−0.208587
$210$	−2.33968	−0.161453
$211$	6.83383	0.470460	0.235230	−	0.971940i	$-0.424416\pi$
$211$	6.83383	0.470460	0.235230	+	0.971940i	$0.424416\pi$
$212$	−9.04674	−0.621332
$213$	24.9878	1.71214
$214$	−13.9269	−0.952023
$215$	−4.32637	−0.295056
$216$	5.55260	0.377807
$217$	1.49195	0.101280
$218$	0.767912	0.0520096
$219$	7.45988	0.504092
$220$	1.02486	0.0690958
$221$	1.43401	0.0964622
$222$	9.76543	0.655413
$223$	15.5580	1.04184	0.520922	−	0.853604i	$-0.325588\pi$
$223$	15.5580	1.04184	0.520922	+	0.853604i	$0.325588\pi$
$224$	−1.49195	−0.0996854
$225$	−0.540753	−0.0360502
$226$	16.9743	1.12911
$227$	7.57399	0.502703	0.251352	−	0.967896i	$-0.419125\pi$
$227$	7.57399	0.502703	0.251352	+	0.967896i	$0.419125\pi$
$228$	−4.61423	−0.305585
$229$	−16.5673	−1.09480	−0.547399	−	0.836872i	$-0.684382\pi$
$229$	−16.5673	−1.09480	−0.547399	+	0.836872i	$0.684382\pi$
$230$	−1.14116	−0.0752461
$231$	−2.39784	−0.157766
$232$	6.02560	0.395600
$233$	2.48371	0.162713	0.0813567	−	0.996685i	$-0.474075\pi$
$233$	2.48371	0.162713	0.0813567	+	0.996685i	$0.474075\pi$
$234$	−0.540753	−0.0353501
$235$	4.37159	0.285171
$236$	−8.83881	−0.575357
$237$	9.36621	0.608400
$238$	−2.13948	−0.138682
$239$	0.510101	0.0329957	0.0164979	−	0.999864i	$-0.494748\pi$
$239$	0.510101	0.0329957	0.0164979	+	0.999864i	$0.494748\pi$
$240$	1.56820	0.101227
$241$	−2.23208	−0.143781	−0.0718904	−	0.997413i	$-0.522903\pi$
$241$	−2.23208	−0.143781	−0.0718904	+	0.997413i	$0.522903\pi$
$242$	−9.94967	−0.639589
$243$	−5.54661	−0.355815
$244$	−11.6584	−0.746353
$245$	4.77407	0.305004
$246$	−5.41436	−0.345207
$247$	2.94238	0.187219
$248$	−1.00000	−0.0635001
$249$	−1.48967	−0.0944037
$250$	−1.00000	−0.0632456
$251$	10.1588	0.641216	0.320608	−	0.947212i	$-0.396113\pi$
$251$	10.1588	0.641216	0.320608	+	0.947212i	$0.396113\pi$
$252$	0.806779	0.0508223
$253$	−1.16953	−0.0735276
$254$	−18.8774	−1.18447
$255$	2.24882	0.140826
$256$	1.00000	0.0625000
$257$	−8.18897	−0.510814	−0.255407	−	0.966834i	$-0.582209\pi$
$257$	−8.18897	−0.510814	−0.255407	+	0.966834i	$0.582209\pi$
$258$	−6.78461	−0.422391
$259$	9.29065	0.577293
$260$	−1.00000	−0.0620174
$261$	−3.25836	−0.201688
$262$	7.58378	0.468528
$263$	−18.0255	−1.11150	−0.555749	−	0.831350i	$-0.687569\pi$
$263$	−18.0255	−1.11150	−0.555749	+	0.831350i	$0.687569\pi$
$264$	1.60718	0.0989150
$265$	9.04674	0.555737
$266$	−4.38989	−0.269161
$267$	16.6144	1.01679
$268$	9.32431	0.569573
$269$	3.35767	0.204721	0.102360	−	0.994747i	$-0.467361\pi$
$269$	3.35767	0.204721	0.102360	+	0.994747i	$0.467361\pi$
$270$	−5.55260	−0.337921
$271$	24.7308	1.50229	0.751145	−	0.660137i	$-0.229502\pi$
$271$	24.7308	1.50229	0.751145	+	0.660137i	$0.229502\pi$
$272$	1.43401	0.0869498
$273$	2.33968	0.141604
$274$	15.2258	0.919825
$275$	−1.02486	−0.0618012
$276$	−1.78957	−0.107719
$277$	24.0705	1.44625	0.723127	−	0.690716i	$-0.242704\pi$
$277$	24.0705	1.44625	0.723127	+	0.690716i	$0.242704\pi$
$278$	−8.98962	−0.539162
$279$	0.540753	0.0323741
$280$	1.49195	0.0891613
$281$	−26.2186	−1.56407	−0.782034	−	0.623236i	$-0.785818\pi$
$281$	−26.2186	−1.56407	−0.782034	+	0.623236i	$0.785818\pi$
$282$	6.85552	0.408240
$283$	−22.8318	−1.35721	−0.678606	−	0.734502i	$-0.737416\pi$
$283$	−22.8318	−1.35721	−0.678606	+	0.734502i	$0.737416\pi$
$284$	−15.9341	−0.945513
$285$	4.61423	0.273323
$286$	−1.02486	−0.0606010
$287$	−5.15112	−0.304061
$288$	−0.540753	−0.0318642
$289$	−14.9436	−0.879036
$290$	−6.02560	−0.353835
$291$	−21.1604	−1.24044
$292$	−4.75697	−0.278381
$293$	−8.43198	−0.492601	−0.246301	−	0.969193i	$-0.579215\pi$
$293$	−8.43198	−0.492601	−0.246301	+	0.969193i	$0.579215\pi$
$294$	7.48669	0.436633
$295$	8.83881	0.514615
$296$	−6.22717	−0.361947
$297$	−5.69062	−0.330203
$298$	11.4923	0.665729
$299$	1.14116	0.0659952
$300$	−1.56820	−0.0905400
$301$	−6.45475	−0.372046
$302$	6.86593	0.395090
$303$	−11.4165	−0.655859
$304$	2.94238	0.168757
$305$	11.6584	0.667559
$306$	−0.775447	−0.0443294
$307$	−17.5432	−1.00124	−0.500620	−	0.865667i	$-0.666895\pi$
$307$	−17.5432	−1.00124	−0.500620	+	0.865667i	$0.666895\pi$
$308$	1.52904	0.0871251
$309$	18.5743	1.05666
$310$	1.00000	0.0567962
$311$	28.8255	1.63454	0.817272	−	0.576252i	$-0.195485\pi$
$311$	28.8255	1.63454	0.817272	+	0.576252i	$0.195485\pi$
$312$	−1.56820	−0.0887818
$313$	13.5453	0.765624	0.382812	−	0.923826i	$-0.374956\pi$
$313$	13.5453	0.765624	0.382812	+	0.923826i	$0.374956\pi$
$314$	−6.46147	−0.364642
$315$	−0.806779	−0.0454569
$316$	−5.97259	−0.335984
$317$	−14.0492	−0.789084	−0.394542	−	0.918878i	$-0.629097\pi$
$317$	−14.0492	−0.789084	−0.394542	+	0.918878i	$0.629097\pi$
$318$	14.1871	0.795572
$319$	−6.17537	−0.345754
$320$	−1.00000	−0.0559017
$321$	21.8401	1.21900
$322$	−1.70256	−0.0948802
$323$	4.21940	0.234774
$324$	−7.08533	−0.393629
$325$	1.00000	0.0554700
$326$	−4.82792	−0.267394
$327$	−1.20424	−0.0665946
$328$	3.45260	0.190638
$329$	6.52221	0.359581
$330$	−1.60718	−0.0884722
$331$	15.3162	0.841856	0.420928	−	0.907094i	$-0.361704\pi$
$331$	15.3162	0.841856	0.420928	+	0.907094i	$0.361704\pi$
$332$	0.949922	0.0521337
$333$	3.36736	0.184530
$334$	−13.2027	−0.722418
$335$	−9.32431	−0.509441
$336$	2.33968	0.127640
$337$	−6.07422	−0.330884	−0.165442	−	0.986220i	$-0.552905\pi$
$337$	−6.07422	−0.330884	−0.165442	+	0.986220i	$0.552905\pi$
$338$	1.00000	0.0543928
$339$	−26.6190	−1.44575
$340$	−1.43401	−0.0777703
$341$	1.02486	0.0554991
$342$	−1.59110	−0.0860368
$343$	17.5664	0.948495
$344$	4.32637	0.233262
$345$	1.78957	0.0963472
$346$	−6.54788	−0.352016
$347$	−1.19043	−0.0639056	−0.0319528	−	0.999489i	$-0.510173\pi$
$347$	−1.19043	−0.0639056	−0.0319528	+	0.999489i	$0.510173\pi$
$348$	−9.44933	−0.506537
$349$	−15.7136	−0.841128	−0.420564	−	0.907263i	$-0.638168\pi$
$349$	−15.7136	−0.841128	−0.420564	+	0.907263i	$0.638168\pi$
$350$	−1.49195	−0.0797483
$351$	5.55260	0.296376
$352$	−1.02486	−0.0546250
$353$	17.3692	0.924471	0.462235	−	0.886757i	$-0.347047\pi$
$353$	17.3692	0.924471	0.462235	+	0.886757i	$0.347047\pi$
$354$	13.8610	0.736704
$355$	15.9341	0.845693
$356$	−10.5946	−0.561513
$357$	3.35513	0.177572
$358$	−19.2959	−1.01982
$359$	−32.8369	−1.73307	−0.866534	−	0.499119i	$-0.833657\pi$
$359$	−32.8369	−1.73307	−0.866534	+	0.499119i	$0.833657\pi$
$360$	0.540753	0.0285002
$361$	−10.3424	−0.544338
$362$	10.1329	0.532576
$363$	15.6031	0.818948
$364$	−1.49195	−0.0781997
$365$	4.75697	0.248991
$366$	18.2827	0.955652
$367$	20.4606	1.06804	0.534018	−	0.845473i	$-0.320682\pi$
$367$	20.4606	1.06804	0.534018	+	0.845473i	$0.320682\pi$
$368$	1.14116	0.0594872
$369$	−1.86700	−0.0971923
$370$	6.22717	0.323735
$371$	13.4973	0.700746
$372$	1.56820	0.0813073
$373$	17.8481	0.924140	0.462070	−	0.886844i	$-0.347107\pi$
$373$	17.8481	0.924140	0.462070	+	0.886844i	$0.347107\pi$
$374$	−1.46966	−0.0759942
$375$	1.56820	0.0809814
$376$	−4.37159	−0.225447
$377$	6.02560	0.310334
$378$	−8.28423	−0.426095
$379$	−11.1753	−0.574036	−0.287018	−	0.957925i	$-0.592664\pi$
$379$	−11.1753	−0.574036	−0.287018	+	0.957925i	$0.592664\pi$
$380$	−2.94238	−0.150941
$381$	29.6035	1.51663
$382$	−24.1076	−1.23345
$383$	15.0592	0.769490	0.384745	−	0.923023i	$-0.374289\pi$
$383$	15.0592	0.769490	0.384745	+	0.923023i	$0.374289\pi$
$384$	−1.56820	−0.0800268
$385$	−1.52904	−0.0779270
$386$	21.8594	1.11262
$387$	−2.33950	−0.118923
$388$	13.4934	0.685024
$389$	−23.8515	−1.20932	−0.604659	−	0.796485i	$-0.706690\pi$
$389$	−23.8515	−1.20932	−0.604659	+	0.796485i	$0.706690\pi$
$390$	1.56820	0.0794088
$391$	1.63644	0.0827585
$392$	−4.77407	−0.241127
$393$	−11.8929	−0.599916
$394$	−17.8184	−0.897676
$395$	5.97259	0.300514
$396$	0.554194	0.0278493
$397$	−35.1081	−1.76202	−0.881011	−	0.473095i	$-0.843137\pi$
$397$	−35.1081	−1.76202	−0.881011	+	0.473095i	$0.843137\pi$
$398$	−11.2098	−0.561896
$399$	6.88422	0.344642
$400$	1.00000	0.0500000
$401$	2.85667	0.142655	0.0713275	−	0.997453i	$-0.477276\pi$
$401$	2.85667	0.142655	0.0713275	+	0.997453i	$0.477276\pi$
$402$	−14.6224	−0.729297
$403$	−1.00000	−0.0498135
$404$	7.27999	0.362193
$405$	7.08533	0.352073
$406$	−8.98992	−0.446162
$407$	6.38195	0.316342
$408$	−2.24882	−0.111333
$409$	−0.398169	−0.0196882	−0.00984410	−	0.999952i	$-0.503134\pi$
$409$	−0.398169	−0.0196882	−0.00984410	+	0.999952i	$0.503134\pi$
$410$	−3.45260	−0.170512
$411$	−23.8771	−1.17777
$412$	−11.8444	−0.583531
$413$	13.1871	0.648895
$414$	−0.617088	−0.0303282
$415$	−0.949922	−0.0466298
$416$	1.00000	0.0490290
$417$	14.0975	0.690358
$418$	−3.01551	−0.147493
$419$	−26.6342	−1.30116	−0.650582	−	0.759436i	$-0.725475\pi$
$419$	−26.6342	−1.30116	−0.650582	+	0.759436i	$0.725475\pi$
$420$	−2.33968	−0.114165
$421$	−7.83612	−0.381909	−0.190955	−	0.981599i	$-0.561158\pi$
$421$	−7.83612	−0.381909	−0.190955	+	0.981599i	$0.561158\pi$
$422$	6.83383	0.332666
$423$	2.36395	0.114939
$424$	−9.04674	−0.439348
$425$	1.43401	0.0695599
$426$	24.9878	1.21066
$427$	17.3938	0.841746
$428$	−13.9269	−0.673182
$429$	1.60718	0.0775953
$430$	−4.32637	−0.208636
$431$	16.9633	0.817095	0.408548	−	0.912737i	$-0.366035\pi$
$431$	16.9633	0.817095	0.408548	+	0.912737i	$0.366035\pi$
$432$	5.55260	0.267150
$433$	22.0858	1.06138	0.530689	−	0.847567i	$-0.321933\pi$
$433$	22.0858	1.06138	0.530689	+	0.847567i	$0.321933\pi$
$434$	1.49195	0.0716161
$435$	9.44933	0.453061
$436$	0.767912	0.0367763
$437$	3.35773	0.160622
$438$	7.45988	0.356447
$439$	−34.8308	−1.66238	−0.831192	−	0.555985i	$-0.812341\pi$
$439$	−34.8308	−1.66238	−0.831192	+	0.555985i	$0.812341\pi$
$440$	1.02486	0.0488581
$441$	2.58159	0.122933
$442$	1.43401	0.0682090
$443$	−7.79547	−0.370374	−0.185187	−	0.982703i	$-0.559289\pi$
$443$	−7.79547	−0.370374	−0.185187	+	0.982703i	$0.559289\pi$
$444$	9.76543	0.463447
$445$	10.5946	0.502233
$446$	15.5580	0.736694
$447$	−18.0221	−0.852418
$448$	−1.49195	−0.0704882
$449$	33.8963	1.59967	0.799834	−	0.600222i	$-0.204921\pi$
$449$	33.8963	1.59967	0.799834	+	0.600222i	$0.204921\pi$
$450$	−0.540753	−0.0254914
$451$	−3.53842	−0.166618
$452$	16.9743	0.798402
$453$	−10.7671	−0.505885
$454$	7.57399	0.355465
$455$	1.49195	0.0699439
$456$	−4.61423	−0.216081
$457$	24.5958	1.15054	0.575272	−	0.817962i	$-0.304896\pi$
$457$	24.5958	1.15054	0.575272	+	0.817962i	$0.304896\pi$
$458$	−16.5673	−0.774139
$459$	7.96251	0.371658
$460$	−1.14116	−0.0532070
$461$	9.93793	0.462855	0.231428	−	0.972852i	$-0.425660\pi$
$461$	9.93793	0.462855	0.231428	+	0.972852i	$0.425660\pi$
$462$	−2.39784	−0.111557
$463$	15.4200	0.716627	0.358313	−	0.933601i	$-0.383352\pi$
$463$	15.4200	0.716627	0.358313	+	0.933601i	$0.383352\pi$
$464$	6.02560	0.279731
$465$	−1.56820	−0.0727235
$466$	2.48371	0.115056
$467$	2.51737	0.116490	0.0582450	−	0.998302i	$-0.481450\pi$
$467$	2.51737	0.116490	0.0582450	+	0.998302i	$0.481450\pi$
$468$	−0.540753	−0.0249963
$469$	−13.9114	−0.642371
$470$	4.37159	0.201646
$471$	10.1329	0.466898
$472$	−8.83881	−0.406839
$473$	−4.43391	−0.203871
$474$	9.36621	0.430204
$475$	2.94238	0.135005
$476$	−2.13948	−0.0980630
$477$	4.89205	0.223992
$478$	0.510101	0.0233315
$479$	−12.2690	−0.560584	−0.280292	−	0.959915i	$-0.590431\pi$
$479$	−12.2690	−0.560584	−0.280292	+	0.959915i	$0.590431\pi$
$480$	1.56820	0.0715781
$481$	−6.22717	−0.283934
$482$	−2.23208	−0.101668
$483$	2.66996	0.121487
$484$	−9.94967	−0.452258
$485$	−13.4934	−0.612704
$486$	−5.54661	−0.251600
$487$	−42.7409	−1.93677	−0.968387	−	0.249453i	$-0.919749\pi$
$487$	−42.7409	−1.93677	−0.968387	+	0.249453i	$0.919749\pi$
$488$	−11.6584	−0.527751
$489$	7.57113	0.342379
$490$	4.77407	0.215671
$491$	−40.2387	−1.81595	−0.907973	−	0.419029i	$-0.862371\pi$
$491$	−40.2387	−1.81595	−0.907973	+	0.419029i	$0.862371\pi$
$492$	−5.41436	−0.244098
$493$	8.64079	0.389161
$494$	2.94238	0.132384
$495$	−0.554194	−0.0249092
$496$	−1.00000	−0.0449013
$497$	23.7729	1.06636
$498$	−1.48967	−0.0667535
$499$	−33.2894	−1.49024	−0.745118	−	0.666933i	$-0.767607\pi$
$499$	−33.2894	−1.49024	−0.745118	+	0.666933i	$0.767607\pi$
$500$	−1.00000	−0.0447214
$501$	20.7044	0.925004
$502$	10.1588	0.453408
$503$	9.02216	0.402278	0.201139	−	0.979563i	$-0.435536\pi$
$503$	9.02216	0.402278	0.201139	+	0.979563i	$0.435536\pi$
$504$	0.806779	0.0359368
$505$	−7.27999	−0.323955
$506$	−1.16953	−0.0519919
$507$	−1.56820	−0.0696461
$508$	−18.8774	−0.837549
$509$	11.7742	0.521881	0.260941	−	0.965355i	$-0.415967\pi$
$509$	11.7742	0.521881	0.260941	+	0.965355i	$0.415967\pi$
$510$	2.24882	0.0995793
$511$	7.09719	0.313961
$512$	1.00000	0.0441942
$513$	16.3378	0.721333
$514$	−8.18897	−0.361200
$515$	11.8444	0.521926
$516$	−6.78461	−0.298676
$517$	4.48025	0.197041
$518$	9.29065	0.408208
$519$	10.2684	0.450732
$520$	−1.00000	−0.0438529
$521$	27.6437	1.21109	0.605547	−	0.795809i	$-0.292954\pi$
$521$	27.6437	1.21109	0.605547	+	0.795809i	$0.292954\pi$
$522$	−3.25836	−0.142615
$523$	27.4075	1.19844	0.599222	−	0.800583i	$-0.295477\pi$
$523$	27.4075	1.19844	0.599222	+	0.800583i	$0.295477\pi$
$524$	7.58378	0.331299
$525$	2.33968	0.102112
$526$	−18.0255	−0.785948
$527$	−1.43401	−0.0624666
$528$	1.60718	0.0699435
$529$	−21.6977	−0.943380
$530$	9.04674	0.392965
$531$	4.77961	0.207418
$532$	−4.38989	−0.190326
$533$	3.45260	0.149549
$534$	16.6144	0.718978
$535$	13.9269	0.602112
$536$	9.32431	0.402749
$537$	30.2597	1.30580
$538$	3.35767	0.144759
$539$	4.89274	0.210745
$540$	−5.55260	−0.238946
$541$	38.6880	1.66333	0.831663	−	0.555281i	$-0.187389\pi$
$541$	38.6880	1.66333	0.831663	+	0.555281i	$0.187389\pi$
$542$	24.7308	1.06228
$543$	−15.8905	−0.681925
$544$	1.43401	0.0614828
$545$	−0.767912	−0.0328937
$546$	2.33968	0.100129
$547$	23.4131	1.00107	0.500537	−	0.865715i	$-0.333136\pi$
$547$	23.4131	1.00107	0.500537	+	0.865715i	$0.333136\pi$
$548$	15.2258	0.650414
$549$	6.30432	0.269062
$550$	−1.02486	−0.0437000
$551$	17.7296	0.755305
$552$	−1.78957	−0.0761692
$553$	8.91083	0.378927
$554$	24.0705	1.02266
$555$	−9.76543	−0.414520
$556$	−8.98962	−0.381245
$557$	26.9462	1.14175	0.570873	−	0.821039i	$-0.306605\pi$
$557$	26.9462	1.14175	0.570873	+	0.821039i	$0.306605\pi$
$558$	0.540753	0.0228919
$559$	4.32637	0.182986
$560$	1.49195	0.0630466
$561$	2.30471	0.0973051
$562$	−26.2186	−1.10596
$563$	−15.6819	−0.660914	−0.330457	−	0.943821i	$-0.607203\pi$
$563$	−15.6819	−0.660914	−0.330457	+	0.943821i	$0.607203\pi$
$564$	6.85552	0.288669
$565$	−16.9743	−0.714113
$566$	−22.8318	−0.959694
$567$	10.5710	0.443940
$568$	−15.9341	−0.668579
$569$	−38.1947	−1.60121	−0.800603	−	0.599196i	$-0.795487\pi$
$569$	−38.1947	−1.60121	−0.800603	+	0.599196i	$0.795487\pi$
$570$	4.61423	0.193269
$571$	33.8053	1.41471	0.707355	−	0.706859i	$-0.249888\pi$
$571$	33.8053	1.41471	0.707355	+	0.706859i	$0.249888\pi$
$572$	−1.02486	−0.0428514
$573$	37.8054	1.57934
$574$	−5.15112	−0.215004
$575$	1.14116	0.0475898
$576$	−0.540753	−0.0225314
$577$	32.6580	1.35957	0.679786	−	0.733410i	$-0.262073\pi$
$577$	32.6580	1.35957	0.679786	+	0.733410i	$0.262073\pi$
$578$	−14.9436	−0.621572
$579$	−34.2800	−1.42463
$580$	−6.02560	−0.250199
$581$	−1.41724	−0.0587970
$582$	−21.1604	−0.877125
$583$	9.27160	0.383991
$584$	−4.75697	−0.196845
$585$	0.540753	0.0223574
$586$	−8.43198	−0.348322
$587$	−11.7864	−0.486476	−0.243238	−	0.969967i	$-0.578210\pi$
$587$	−11.7864	−0.486476	−0.243238	+	0.969967i	$0.578210\pi$
$588$	7.48669	0.308746
$589$	−2.94238	−0.121238
$590$	8.83881	0.363888
$591$	27.9427	1.14941
$592$	−6.22717	−0.255935
$593$	−48.6819	−1.99912	−0.999562	−	0.0295887i	$-0.990580\pi$
$593$	−48.6819	−1.99912	−0.999562	+	0.0295887i	$0.990580\pi$
$594$	−5.69062	−0.233489
$595$	2.13948	0.0877102
$596$	11.4923	0.470741
$597$	17.5792	0.719467
$598$	1.14116	0.0466656
$599$	−26.6505	−1.08891	−0.544455	−	0.838790i	$-0.683264\pi$
$599$	−26.6505	−1.08891	−0.544455	+	0.838790i	$0.683264\pi$
$600$	−1.56820	−0.0640214
$601$	18.5276	0.755758	0.377879	−	0.925855i	$-0.376654\pi$
$601$	18.5276	0.755758	0.377879	+	0.925855i	$0.376654\pi$
$602$	−6.45475	−0.263076
$603$	−5.04215	−0.205332
$604$	6.86593	0.279371
$605$	9.94967	0.404512
$606$	−11.4165	−0.463763
$607$	46.2812	1.87850	0.939248	−	0.343240i	$-0.111524\pi$
$607$	46.2812	1.87850	0.939248	+	0.343240i	$0.111524\pi$
$608$	2.94238	0.119329
$609$	14.0980	0.571279
$610$	11.6584	0.472035
$611$	−4.37159	−0.176856
$612$	−0.775447	−0.0313456
$613$	15.7832	0.637477	0.318738	−	0.947843i	$-0.396741\pi$
$613$	15.7832	0.637477	0.318738	+	0.947843i	$0.396741\pi$
$614$	−17.5432	−0.707984
$615$	5.41436	0.218328
$616$	1.52904	0.0616067
$617$	−23.8985	−0.962116	−0.481058	−	0.876689i	$-0.659747\pi$
$617$	−23.8985	−0.962116	−0.481058	+	0.876689i	$0.659747\pi$
$618$	18.5743	0.747170
$619$	−18.4921	−0.743260	−0.371630	−	0.928381i	$-0.621201\pi$
$619$	−18.4921	−0.743260	−0.371630	+	0.928381i	$0.621201\pi$
$620$	1.00000	0.0401610
$621$	6.33643	0.254272
$622$	28.8255	1.15580
$623$	15.8067	0.633281
$624$	−1.56820	−0.0627782
$625$	1.00000	0.0400000
$626$	13.5453	0.541378
$627$	4.72892	0.188855
$628$	−6.46147	−0.257841
$629$	−8.92984	−0.356056
$630$	−0.806779	−0.0321429
$631$	−49.5013	−1.97061	−0.985307	−	0.170791i	$-0.945368\pi$
$631$	−49.5013	−1.97061	−0.985307	+	0.170791i	$0.945368\pi$
$632$	−5.97259	−0.237577
$633$	−10.7168	−0.425955
$634$	−14.0492	−0.557966
$635$	18.8774	0.749127
$636$	14.1871	0.562554
$637$	−4.77407	−0.189156
$638$	−6.17537	−0.244485
$639$	8.61640	0.340860
$640$	−1.00000	−0.0395285
$641$	−29.8726	−1.17990	−0.589949	−	0.807440i	$-0.700852\pi$
$641$	−29.8726	−1.17990	−0.589949	+	0.807440i	$0.700852\pi$
$642$	21.8401	0.861962
$643$	−6.08378	−0.239921	−0.119961	−	0.992779i	$-0.538277\pi$
$643$	−6.08378	−0.239921	−0.119961	+	0.992779i	$0.538277\pi$
$644$	−1.70256	−0.0670904
$645$	6.78461	0.267144
$646$	4.21940	0.166010
$647$	30.3037	1.19136	0.595681	−	0.803221i	$-0.296882\pi$
$647$	30.3037	1.19136	0.595681	+	0.803221i	$0.296882\pi$
$648$	−7.08533	−0.278338
$649$	9.05851	0.355577
$650$	1.00000	0.0392232
$651$	−2.33968	−0.0916993
$652$	−4.82792	−0.189076
$653$	−38.9720	−1.52509	−0.762547	−	0.646933i	$-0.776051\pi$
$653$	−38.9720	−1.52509	−0.762547	+	0.646933i	$0.776051\pi$
$654$	−1.20424	−0.0470895
$655$	−7.58378	−0.296323
$656$	3.45260	0.134801
$657$	2.57235	0.100357
$658$	6.52221	0.254262
$659$	−17.4836	−0.681064	−0.340532	−	0.940233i	$-0.610607\pi$
$659$	−17.4836	−0.681064	−0.340532	+	0.940233i	$0.610607\pi$
$660$	−1.60718	−0.0625593
$661$	13.3233	0.518218	0.259109	−	0.965848i	$-0.416571\pi$
$661$	13.3233	0.518218	0.259109	+	0.965848i	$0.416571\pi$
$662$	15.3162	0.595282
$663$	−2.24882	−0.0873368
$664$	0.949922	0.0368641
$665$	4.38989	0.170233
$666$	3.36736	0.130483
$667$	6.87619	0.266247
$668$	−13.2027	−0.510826
$669$	−24.3981	−0.943285
$670$	−9.32431	−0.360229
$671$	11.9482	0.461255
$672$	2.33968	0.0902552
$673$	6.19072	0.238635	0.119317	−	0.992856i	$-0.461929\pi$
$673$	6.19072	0.238635	0.119317	+	0.992856i	$0.461929\pi$
$674$	−6.07422	−0.233970
$675$	5.55260	0.213720
$676$	1.00000	0.0384615
$677$	−10.2505	−0.393957	−0.196979	−	0.980408i	$-0.563113\pi$
$677$	−10.2505	−0.393957	−0.196979	+	0.980408i	$0.563113\pi$
$678$	−26.6190	−1.02230
$679$	−20.1316	−0.772578
$680$	−1.43401	−0.0549919
$681$	−11.8775	−0.455147
$682$	1.02486	0.0392438
$683$	24.2058	0.926210	0.463105	−	0.886303i	$-0.346735\pi$
$683$	24.2058	0.926210	0.463105	+	0.886303i	$0.346735\pi$
$684$	−1.59110	−0.0608372
$685$	−15.2258	−0.581748
$686$	17.5664	0.670688
$687$	25.9808	0.991229
$688$	4.32637	0.164941
$689$	−9.04674	−0.344653
$690$	1.78957	0.0681278
$691$	−30.6935	−1.16763	−0.583817	−	0.811885i	$-0.698441\pi$
$691$	−30.6935	−1.16763	−0.583817	+	0.811885i	$0.698441\pi$
$692$	−6.54788	−0.248913
$693$	−0.826833	−0.0314088
$694$	−1.19043	−0.0451881
$695$	8.98962	0.340996
$696$	−9.44933	−0.358176
$697$	4.95107	0.187535
$698$	−15.7136	−0.594767
$699$	−3.89496	−0.147321
$700$	−1.49195	−0.0563906
$701$	−14.5471	−0.549437	−0.274718	−	0.961525i	$-0.588585\pi$
$701$	−14.5471	−0.549437	−0.274718	+	0.961525i	$0.588585\pi$
$702$	5.55260	0.209570
$703$	−18.3227	−0.691052
$704$	−1.02486	−0.0386257
$705$	−6.85552	−0.258194
$706$	17.3692	0.653700
$707$	−10.8614	−0.408486
$708$	13.8610	0.520928
$709$	10.3988	0.390535	0.195268	−	0.980750i	$-0.437442\pi$
$709$	10.3988	0.390535	0.195268	+	0.980750i	$0.437442\pi$
$710$	15.9341	0.597995
$711$	3.22970	0.121123
$712$	−10.5946	−0.397050
$713$	−1.14116	−0.0427369
$714$	3.35513	0.125563
$715$	1.02486	0.0383275
$716$	−19.2959	−0.721120
$717$	−0.799940	−0.0298743
$718$	−32.8369	−1.22546
$719$	26.4883	0.987847	0.493923	−	0.869505i	$-0.335562\pi$
$719$	26.4883	0.987847	0.493923	+	0.869505i	$0.335562\pi$
$720$	0.540753	0.0201527
$721$	17.6713	0.658113
$722$	−10.3424	−0.384905
$723$	3.50034	0.130179
$724$	10.1329	0.376588
$725$	6.02560	0.223785
$726$	15.6031	0.579084
$727$	6.15292	0.228199	0.114100	−	0.993469i	$-0.463602\pi$
$727$	6.15292	0.228199	0.114100	+	0.993469i	$0.463602\pi$
$728$	−1.49195	−0.0552955
$729$	29.9542	1.10941
$730$	4.75697	0.176063
$731$	6.20407	0.229466
$732$	18.2827	0.675748
$733$	30.9399	1.14279	0.571396	−	0.820675i	$-0.306402\pi$
$733$	30.9399	1.14279	0.571396	+	0.820675i	$0.306402\pi$
$734$	20.4606	0.755215
$735$	−7.48669	−0.276151
$736$	1.14116	0.0420638
$737$	−9.55608	−0.352003
$738$	−1.86700	−0.0687254
$739$	−22.3962	−0.823859	−0.411929	−	0.911216i	$-0.635145\pi$
$739$	−22.3962	−0.823859	−0.411929	+	0.911216i	$0.635145\pi$
$740$	6.22717	0.228915
$741$	−4.61423	−0.169508
$742$	13.4973	0.495502
$743$	−15.4041	−0.565123	−0.282561	−	0.959249i	$-0.591184\pi$
$743$	−15.4041	−0.565123	−0.282561	+	0.959249i	$0.591184\pi$
$744$	1.56820	0.0574929
$745$	−11.4923	−0.421044
$746$	17.8481	0.653465
$747$	−0.513673	−0.0187943
$748$	−1.46966	−0.0537360
$749$	20.7783	0.759223
$750$	1.56820	0.0572625
$751$	−47.3545	−1.72799	−0.863995	−	0.503500i	$-0.832045\pi$
$751$	−47.3545	−1.72799	−0.863995	+	0.503500i	$0.832045\pi$
$752$	−4.37159	−0.159415
$753$	−15.9310	−0.580557
$754$	6.02560	0.219439
$755$	−6.86593	−0.249877
$756$	−8.28423	−0.301295
$757$	−2.62160	−0.0952836	−0.0476418	−	0.998864i	$-0.515171\pi$
$757$	−2.62160	−0.0952836	−0.0476418	+	0.998864i	$0.515171\pi$
$758$	−11.1753	−0.405904
$759$	1.83405	0.0665719
$760$	−2.94238	−0.106731
$761$	32.2958	1.17072	0.585360	−	0.810773i	$-0.300953\pi$
$761$	32.2958	1.17072	0.585360	+	0.810773i	$0.300953\pi$
$762$	29.6035	1.07242
$763$	−1.14569	−0.0414768
$764$	−24.1076	−0.872181
$765$	0.775447	0.0280364
$766$	15.0592	0.544112
$767$	−8.83881	−0.319151
$768$	−1.56820	−0.0565875
$769$	−24.2679	−0.875122	−0.437561	−	0.899189i	$-0.644158\pi$
$769$	−24.2679	−0.875122	−0.437561	+	0.899189i	$0.644158\pi$
$770$	−1.52904	−0.0551027
$771$	12.8419	0.462491
$772$	21.8594	0.786739
$773$	32.8392	1.18114	0.590572	−	0.806985i	$-0.298902\pi$
$773$	32.8392	1.18114	0.590572	+	0.806985i	$0.298902\pi$
$774$	−2.33950	−0.0840916
$775$	−1.00000	−0.0359211
$776$	13.4934	0.484385
$777$	−14.5696	−0.522681
$778$	−23.8515	−0.855117
$779$	10.1588	0.363978
$780$	1.56820	0.0561505
$781$	16.3301	0.584338
$782$	1.63644	0.0585191
$783$	33.4578	1.19568
$784$	−4.77407	−0.170503
$785$	6.46147	0.230620
$786$	−11.8929	−0.424205
$787$	37.1132	1.32294	0.661471	−	0.749971i	$-0.269933\pi$
$787$	37.1132	1.32294	0.661471	+	0.749971i	$0.269933\pi$
$788$	−17.8184	−0.634753
$789$	28.2675	1.00635
$790$	5.97259	0.212495
$791$	−25.3248	−0.900448
$792$	0.554194	0.0196924
$793$	−11.6584	−0.414002
$794$	−35.1081	−1.24594
$795$	−14.1871	−0.503164
$796$	−11.2098	−0.397320
$797$	−1.57835	−0.0559082	−0.0279541	−	0.999609i	$-0.508899\pi$
$797$	−1.57835	−0.0559082	−0.0279541	+	0.999609i	$0.508899\pi$
$798$	6.88422	0.243699
$799$	−6.26891	−0.221778
$800$	1.00000	0.0353553
$801$	5.72907	0.202427
$802$	2.85667	0.100872
$803$	4.87521	0.172043
$804$	−14.6224	−0.515691
$805$	1.70256	0.0600075
$806$	−1.00000	−0.0352235
$807$	−5.26549	−0.185354
$808$	7.27999	0.256109
$809$	47.5883	1.67312	0.836558	−	0.547878i	$-0.184564\pi$
$809$	47.5883	1.67312	0.836558	+	0.547878i	$0.184564\pi$
$810$	7.08533	0.248953
$811$	44.4009	1.55913	0.779564	−	0.626323i	$-0.215441\pi$
$811$	44.4009	1.55913	0.779564	+	0.626323i	$0.215441\pi$
$812$	−8.98992	−0.315484
$813$	−38.7828	−1.36017
$814$	6.38195	0.223687
$815$	4.82792	0.169115
$816$	−2.24882	−0.0787244
$817$	12.7298	0.445360
$818$	−0.398169	−0.0139217
$819$	0.806779	0.0281912
$820$	−3.45260	−0.120570
$821$	40.7178	1.42106	0.710530	−	0.703667i	$-0.248455\pi$
$821$	40.7178	1.42106	0.710530	+	0.703667i	$0.248455\pi$
$822$	−23.8771	−0.832809
$823$	−37.6026	−1.31074	−0.655371	−	0.755307i	$-0.727488\pi$
$823$	−37.6026	−1.31074	−0.655371	+	0.755307i	$0.727488\pi$
$824$	−11.8444	−0.412619
$825$	1.60718	0.0559548
$826$	13.1871	0.458838
$827$	5.13543	0.178576	0.0892882	−	0.996006i	$-0.471541\pi$
$827$	5.13543	0.178576	0.0892882	+	0.996006i	$0.471541\pi$
$828$	−0.617088	−0.0214453
$829$	−1.37801	−0.0478601	−0.0239301	−	0.999714i	$-0.507618\pi$
$829$	−1.37801	−0.0478601	−0.0239301	+	0.999714i	$0.507618\pi$
$830$	−0.949922	−0.0329723
$831$	−37.7472	−1.30944
$832$	1.00000	0.0346688
$833$	−6.84608	−0.237203
$834$	14.0975	0.488157
$835$	13.2027	0.456897
$836$	−3.01551	−0.104294
$837$	−5.55260	−0.191926
$838$	−26.6342	−0.920062
$839$	14.6557	0.505972	0.252986	−	0.967470i	$-0.418587\pi$
$839$	14.6557	0.505972	0.252986	+	0.967470i	$0.418587\pi$
$840$	−2.33968	−0.0807267
$841$	7.30782	0.251994
$842$	−7.83612	−0.270051
$843$	41.1159	1.41611
$844$	6.83383	0.235230
$845$	−1.00000	−0.0344010
$846$	2.36395	0.0812743
$847$	14.8445	0.510062
$848$	−9.04674	−0.310666
$849$	35.8049	1.22882
$850$	1.43401	0.0491862
$851$	−7.10621	−0.243598
$852$	24.9878	0.856068
$853$	−11.6801	−0.399919	−0.199960	−	0.979804i	$-0.564081\pi$
$853$	−11.6801	−0.399919	−0.199960	+	0.979804i	$0.564081\pi$
$854$	17.3938	0.595204
$855$	1.59110	0.0544144
$856$	−13.9269	−0.476012
$857$	−13.6112	−0.464951	−0.232475	−	0.972602i	$-0.574683\pi$
$857$	−13.6112	−0.464951	−0.232475	+	0.972602i	$0.574683\pi$
$858$	1.60718	0.0548682
$859$	−49.8511	−1.70090	−0.850448	−	0.526059i	$-0.823669\pi$
$859$	−49.8511	−1.70090	−0.850448	+	0.526059i	$0.823669\pi$
$860$	−4.32637	−0.147528
$861$	8.07798	0.275297
$862$	16.9633	0.577774
$863$	25.2856	0.860731	0.430365	−	0.902655i	$-0.358385\pi$
$863$	25.2856	0.860731	0.430365	+	0.902655i	$0.358385\pi$
$864$	5.55260	0.188903
$865$	6.54788	0.222635
$866$	22.0858	0.750508
$867$	23.4345	0.795879
$868$	1.49195	0.0506402
$869$	6.12105	0.207642
$870$	9.44933	0.320362
$871$	9.32431	0.315942
$872$	0.767912	0.0260048
$873$	−7.29661	−0.246953
$874$	3.35773	0.113577
$875$	1.49195	0.0504373
$876$	7.45988	0.252046
$877$	41.9250	1.41571	0.707853	−	0.706360i	$-0.249664\pi$
$877$	41.9250	1.41571	0.707853	+	0.706360i	$0.249664\pi$
$878$	−34.8308	−1.17548
$879$	13.2230	0.446001
$880$	1.02486	0.0345479
$881$	−20.5535	−0.692467	−0.346233	−	0.938148i	$-0.612539\pi$
$881$	−20.5535	−0.692467	−0.346233	+	0.938148i	$0.612539\pi$
$882$	2.58159	0.0869268
$883$	25.1676	0.846958	0.423479	−	0.905906i	$-0.360809\pi$
$883$	25.1676	0.846958	0.423479	+	0.905906i	$0.360809\pi$
$884$	1.43401	0.0482311
$885$	−13.8610	−0.465932
$886$	−7.79547	−0.261894
$887$	−7.81956	−0.262555	−0.131278	−	0.991346i	$-0.541908\pi$
$887$	−7.81956	−0.262555	−0.131278	+	0.991346i	$0.541908\pi$
$888$	9.76543	0.327706
$889$	28.1642	0.944598
$890$	10.5946	0.355132
$891$	7.26144	0.243267
$892$	15.5580	0.520922
$893$	−12.8629	−0.430439
$894$	−18.0221	−0.602751
$895$	19.2959	0.644990
$896$	−1.49195	−0.0498427
$897$	−1.78957	−0.0597520
$898$	33.8963	1.13114
$899$	−6.02560	−0.200965
$900$	−0.540753	−0.0180251
$901$	−12.9731	−0.432198
$902$	−3.53842	−0.117816
$903$	10.1223	0.336850
$904$	16.9743	0.564556
$905$	−10.1329	−0.336831
$906$	−10.7671	−0.357715
$907$	−11.2322	−0.372960	−0.186480	−	0.982459i	$-0.559708\pi$
$907$	−11.2322	−0.372960	−0.186480	+	0.982459i	$0.559708\pi$
$908$	7.57399	0.251352
$909$	−3.93668	−0.130571
$910$	1.49195	0.0494578
$911$	−44.2229	−1.46517	−0.732585	−	0.680675i	$-0.761687\pi$
$911$	−44.2229	−1.46517	−0.732585	+	0.680675i	$0.761687\pi$
$912$	−4.61423	−0.152792
$913$	−0.973533	−0.0322193
$914$	24.5958	0.813558
$915$	−18.2827	−0.604407
$916$	−16.5673	−0.547399
$917$	−11.3147	−0.373643
$918$	7.96251	0.262802
$919$	34.1044	1.12500	0.562500	−	0.826797i	$-0.309840\pi$
$919$	34.1044	1.12500	0.562500	+	0.826797i	$0.309840\pi$
$920$	−1.14116	−0.0376230
$921$	27.5111	0.906523
$922$	9.93793	0.327288
$923$	−15.9341	−0.524476
$924$	−2.39784	−0.0788830
$925$	−6.22717	−0.204748
$926$	15.4200	0.506732
$927$	6.40489	0.210364
$928$	6.02560	0.197800
$929$	42.1831	1.38398	0.691991	−	0.721906i	$-0.256734\pi$
$929$	42.1831	1.38398	0.691991	+	0.721906i	$0.256734\pi$
$930$	−1.56820	−0.0514233
$931$	−14.0471	−0.460375
$932$	2.48371	0.0813567
$933$	−45.2041	−1.47992
$934$	2.51737	0.0823708
$935$	1.46966	0.0480629
$936$	−0.540753	−0.0176751
$937$	52.6460	1.71987	0.859935	−	0.510404i	$-0.170504\pi$
$937$	52.6460	1.71987	0.859935	+	0.510404i	$0.170504\pi$
$938$	−13.9114	−0.454225
$939$	−21.2417	−0.693196
$940$	4.37159	0.142585
$941$	27.7967	0.906145	0.453072	−	0.891474i	$-0.350328\pi$
$941$	27.7967	0.906145	0.453072	+	0.891474i	$0.350328\pi$
$942$	10.1329	0.330147
$943$	3.93998	0.128303
$944$	−8.83881	−0.287679
$945$	8.28423	0.269486
$946$	−4.43391	−0.144159
$947$	−13.2517	−0.430623	−0.215312	−	0.976545i	$-0.569077\pi$
$947$	−13.2517	−0.430623	−0.215312	+	0.976545i	$0.569077\pi$
$948$	9.36621	0.304200
$949$	−4.75697	−0.154418
$950$	2.94238	0.0954633
$951$	22.0320	0.714436
$952$	−2.13948	−0.0693410
$953$	44.8351	1.45235	0.726176	−	0.687509i	$-0.241296\pi$
$953$	44.8351	1.45235	0.726176	+	0.687509i	$0.241296\pi$
$954$	4.89205	0.158386
$955$	24.1076	0.780102
$956$	0.510101	0.0164979
$957$	9.68421	0.313046
$958$	−12.2690	−0.396393
$959$	−22.7162	−0.733545
$960$	1.56820	0.0506134
$961$	1.00000	0.0322581
$962$	−6.22717	−0.200772
$963$	7.53102	0.242684
$964$	−2.23208	−0.0718904
$965$	−21.8594	−0.703680
$966$	2.66996	0.0859045
$967$	11.6511	0.374675	0.187337	−	0.982296i	$-0.440014\pi$
$967$	11.6511	0.374675	0.187337	+	0.982296i	$0.440014\pi$
$968$	−9.94967	−0.319794
$969$	−6.61686	−0.212564
$970$	−13.4934	−0.433247
$971$	−19.2068	−0.616376	−0.308188	−	0.951325i	$-0.599723\pi$
$971$	−19.2068	−0.616376	−0.308188	+	0.951325i	$0.599723\pi$
$972$	−5.54661	−0.177908
$973$	13.4121	0.429973
$974$	−42.7409	−1.36951
$975$	−1.56820	−0.0502225
$976$	−11.6584	−0.373177
$977$	−31.8983	−1.02052	−0.510258	−	0.860021i	$-0.670450\pi$
$977$	−31.8983	−1.02052	−0.510258	+	0.860021i	$0.670450\pi$
$978$	7.57113	0.242098
$979$	10.8580	0.347022
$980$	4.77407	0.152502
$981$	−0.415251	−0.0132579
$982$	−40.2387	−1.28407
$983$	40.0632	1.27782	0.638908	−	0.769283i	$-0.279386\pi$
$983$	40.0632	1.27782	0.638908	+	0.769283i	$0.279386\pi$
$984$	−5.41436	−0.172603
$985$	17.8184	0.567740
$986$	8.64079	0.275179
$987$	−10.2281	−0.325565
$988$	2.94238	0.0936094
$989$	4.93710	0.156991
$990$	−0.554194	−0.0176135
$991$	−13.0190	−0.413563	−0.206781	−	0.978387i	$-0.566299\pi$
$991$	−13.0190	−0.413563	−0.206781	+	0.978387i	$0.566299\pi$
$992$	−1.00000	−0.0317500
$993$	−24.0189	−0.762217
$994$	23.7729	0.754031
$995$	11.2098	0.355374
$996$	−1.48967	−0.0472019
$997$	−41.8375	−1.32501	−0.662504	−	0.749058i	$-0.730506\pi$
$997$	−41.8375	−1.32501	−0.662504	+	0.749058i	$0.730506\pi$
$998$	−33.2894	−1.05376
$999$	−34.5770	−1.09397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.f.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.f.1.2	✓	6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.56820 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.56820 q^{6} -1.49195 q^{7} +1.00000 q^{8} -0.540753 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.56820 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.56820 q^{6} -1.49195 q^{7} +1.00000 q^{8} -0.540753 q^{9} -1.00000 q^{10} -1.02486 q^{11} -1.56820 q^{12} +1.00000 q^{13} -1.49195 q^{14} +1.56820 q^{15} +1.00000 q^{16} +1.43401 q^{17} -0.540753 q^{18} +2.94238 q^{19} -1.00000 q^{20} +2.33968 q^{21} -1.02486 q^{22} +1.14116 q^{23} -1.56820 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.55260 q^{27} -1.49195 q^{28} +6.02560 q^{29} +1.56820 q^{30} -1.00000 q^{31} +1.00000 q^{32} +1.60718 q^{33} +1.43401 q^{34} +1.49195 q^{35} -0.540753 q^{36} -6.22717 q^{37} +2.94238 q^{38} -1.56820 q^{39} -1.00000 q^{40} +3.45260 q^{41} +2.33968 q^{42} +4.32637 q^{43} -1.02486 q^{44} +0.540753 q^{45} +1.14116 q^{46} -4.37159 q^{47} -1.56820 q^{48} -4.77407 q^{49} +1.00000 q^{50} -2.24882 q^{51} +1.00000 q^{52} -9.04674 q^{53} +5.55260 q^{54} +1.02486 q^{55} -1.49195 q^{56} -4.61423 q^{57} +6.02560 q^{58} -8.83881 q^{59} +1.56820 q^{60} -11.6584 q^{61} -1.00000 q^{62} +0.806779 q^{63} +1.00000 q^{64} -1.00000 q^{65} +1.60718 q^{66} +9.32431 q^{67} +1.43401 q^{68} -1.78957 q^{69} +1.49195 q^{70} -15.9341 q^{71} -0.540753 q^{72} -4.75697 q^{73} -6.22717 q^{74} -1.56820 q^{75} +2.94238 q^{76} +1.52904 q^{77} -1.56820 q^{78} -5.97259 q^{79} -1.00000 q^{80} -7.08533 q^{81} +3.45260 q^{82} +0.949922 q^{83} +2.33968 q^{84} -1.43401 q^{85} +4.32637 q^{86} -9.44933 q^{87} -1.02486 q^{88} -10.5946 q^{89} +0.540753 q^{90} -1.49195 q^{91} +1.14116 q^{92} +1.56820 q^{93} -4.37159 q^{94} -2.94238 q^{95} -1.56820 q^{96} +13.4934 q^{97} -4.77407 q^{98} +0.554194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 3 q^{3} + 6 q^{4} - 6 q^{5} - 3 q^{6} - 2 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - 3 * q^3 + 6 * q^4 - 6 * q^5 - 3 * q^6 - 2 * q^7 + 6 * q^8 - q^9	\( 6 q + 6 q^{2} - 3 q^{3} + 6 q^{4} - 6 q^{5} - 3 q^{6} - 2 q^{7} + 6 q^{8} - q^{9} - 6 q^{10} - 4 q^{11} - 3 q^{12} + 6 q^{13} - 2 q^{14} + 3 q^{15} + 6 q^{16} - 8 q^{17} - q^{18} - 9 q^{19} - 6 q^{20} - 5 q^{21} - 4 q^{22} - 7 q^{23} - 3 q^{24} + 6 q^{25} + 6 q^{26} + 9 q^{27} - 2 q^{28} - 14 q^{29} + 3 q^{30} - 6 q^{31} + 6 q^{32} - 6 q^{33} - 8 q^{34} + 2 q^{35} - q^{36} - 9 q^{38} - 3 q^{39} - 6 q^{40} + 2 q^{41} - 5 q^{42} - 7 q^{43} - 4 q^{44} + q^{45} - 7 q^{46} - 8 q^{47} - 3 q^{48} - 14 q^{49} + 6 q^{50} - 5 q^{51} + 6 q^{52} - 24 q^{53} + 9 q^{54} + 4 q^{55} - 2 q^{56} - 15 q^{57} - 14 q^{58} - 5 q^{59} + 3 q^{60} - 5 q^{61} - 6 q^{62} - 19 q^{63} + 6 q^{64} - 6 q^{65} - 6 q^{66} - 12 q^{67} - 8 q^{68} + 2 q^{70} - 10 q^{71} - q^{72} + 5 q^{73} - 3 q^{75} - 9 q^{76} - q^{77} - 3 q^{78} - 16 q^{79} - 6 q^{80} - 10 q^{81} + 2 q^{82} - 22 q^{83} - 5 q^{84} + 8 q^{85} - 7 q^{86} - 31 q^{87} - 4 q^{88} + 14 q^{89} + q^{90} - 2 q^{91} - 7 q^{92} + 3 q^{93} - 8 q^{94} + 9 q^{95} - 3 q^{96} - 9 q^{97} - 14 q^{98} - 21 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 - 3 * q^3 + 6 * q^4 - 6 * q^5 - 3 * q^6 - 2 * q^7 + 6 * q^8 - q^9 - 6 * q^10 - 4 * q^11 - 3 * q^12 + 6 * q^13 - 2 * q^14 + 3 * q^15 + 6 * q^16 - 8 * q^17 - q^18 - 9 * q^19 - 6 * q^20 - 5 * q^21 - 4 * q^22 - 7 * q^23 - 3 * q^24 + 6 * q^25 + 6 * q^26 + 9 * q^27 - 2 * q^28 - 14 * q^29 + 3 * q^30 - 6 * q^31 + 6 * q^32 - 6 * q^33 - 8 * q^34 + 2 * q^35 - q^36 - 9 * q^38 - 3 * q^39 - 6 * q^40 + 2 * q^41 - 5 * q^42 - 7 * q^43 - 4 * q^44 + q^45 - 7 * q^46 - 8 * q^47 - 3 * q^48 - 14 * q^49 + 6 * q^50 - 5 * q^51 + 6 * q^52 - 24 * q^53 + 9 * q^54 + 4 * q^55 - 2 * q^56 - 15 * q^57 - 14 * q^58 - 5 * q^59 + 3 * q^60 - 5 * q^61 - 6 * q^62 - 19 * q^63 + 6 * q^64 - 6 * q^65 - 6 * q^66 - 12 * q^67 - 8 * q^68 + 2 * q^70 - 10 * q^71 - q^72 + 5 * q^73 - 3 * q^75 - 9 * q^76 - q^77 - 3 * q^78 - 16 * q^79 - 6 * q^80 - 10 * q^81 + 2 * q^82 - 22 * q^83 - 5 * q^84 + 8 * q^85 - 7 * q^86 - 31 * q^87 - 4 * q^88 + 14 * q^89 + q^90 - 2 * q^91 - 7 * q^92 + 3 * q^93 - 8 * q^94 + 9 * q^95 - 3 * q^96 - 9 * q^97 - 14 * q^98 - 21 * q^99

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