LMFDB - Embedded newform 8022.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8022,2,Mod(1,8022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8022, 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("8022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8022 = 2 \cdot 3 \cdot 7 \cdot 191 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8022.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0559925015$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + \cdots + 4704 $ x^15 - 53x^13 - x^12 + 1068x^11 + 45x^10 - 10139x^9 - 615x^8 + 45390x^7 + 2130x^6 - 84842x^5 + 7822x^4 + 62828x^3 - 16144x^2 - 13616x + 4704
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.88066$ of defining polynomial
Character	$\chi$	$=$	8022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.88066 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.88066 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.88066 q^{10} -2.01167 q^{11} +1.00000 q^{12} +4.46496 q^{13} -1.00000 q^{14} -3.88066 q^{15} +1.00000 q^{16} +1.73476 q^{17} -1.00000 q^{18} +6.29040 q^{19} -3.88066 q^{20} +1.00000 q^{21} +2.01167 q^{22} +3.07078 q^{23} -1.00000 q^{24} +10.0595 q^{25} -4.46496 q^{26} +1.00000 q^{27} +1.00000 q^{28} -5.82268 q^{29} +3.88066 q^{30} +3.34602 q^{31} -1.00000 q^{32} -2.01167 q^{33} -1.73476 q^{34} -3.88066 q^{35} +1.00000 q^{36} -3.58791 q^{37} -6.29040 q^{38} +4.46496 q^{39} +3.88066 q^{40} -8.04967 q^{41} -1.00000 q^{42} +5.48422 q^{43} -2.01167 q^{44} -3.88066 q^{45} -3.07078 q^{46} -0.224077 q^{47} +1.00000 q^{48} +1.00000 q^{49} -10.0595 q^{50} +1.73476 q^{51} +4.46496 q^{52} +5.63333 q^{53} -1.00000 q^{54} +7.80659 q^{55} -1.00000 q^{56} +6.29040 q^{57} +5.82268 q^{58} +6.71729 q^{59} -3.88066 q^{60} +0.0358422 q^{61} -3.34602 q^{62} +1.00000 q^{63} +1.00000 q^{64} -17.3270 q^{65} +2.01167 q^{66} -8.81581 q^{67} +1.73476 q^{68} +3.07078 q^{69} +3.88066 q^{70} +2.99618 q^{71} -1.00000 q^{72} -12.6614 q^{73} +3.58791 q^{74} +10.0595 q^{75} +6.29040 q^{76} -2.01167 q^{77} -4.46496 q^{78} +8.53794 q^{79} -3.88066 q^{80} +1.00000 q^{81} +8.04967 q^{82} -4.06908 q^{83} +1.00000 q^{84} -6.73200 q^{85} -5.48422 q^{86} -5.82268 q^{87} +2.01167 q^{88} -3.76268 q^{89} +3.88066 q^{90} +4.46496 q^{91} +3.07078 q^{92} +3.34602 q^{93} +0.224077 q^{94} -24.4109 q^{95} -1.00000 q^{96} -10.9632 q^{97} -1.00000 q^{98} -2.01167 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9	$ 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) $ 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9 - 7 * q^11 + 15 * q^12 + 10 * q^13 - 15 * q^14 + 15 * q^16 - 3 * q^17 - 15 * q^18 + 12 * q^19 + 15 * q^21 + 7 * q^22 - q^23 - 15 * q^24 + 31 * q^25 - 10 * q^26 + 15 * q^27 + 15 * q^28 - 3 * q^29 + 19 * q^31 - 15 * q^32 - 7 * q^33 + 3 * q^34 + 15 * q^36 + 25 * q^37 - 12 * q^38 + 10 * q^39 + 8 * q^41 - 15 * q^42 + 25 * q^43 - 7 * q^44 + q^46 + 11 * q^47 + 15 * q^48 + 15 * q^49 - 31 * q^50 - 3 * q^51 + 10 * q^52 - 4 * q^53 - 15 * q^54 + 9 * q^55 - 15 * q^56 + 12 * q^57 + 3 * q^58 + 27 * q^61 - 19 * q^62 + 15 * q^63 + 15 * q^64 - 2 * q^65 + 7 * q^66 + 31 * q^67 - 3 * q^68 - q^69 - 16 * q^71 - 15 * q^72 + 26 * q^73 - 25 * q^74 + 31 * q^75 + 12 * q^76 - 7 * q^77 - 10 * q^78 + 32 * q^79 + 15 * q^81 - 8 * q^82 + 7 * q^83 + 15 * q^84 + 26 * q^85 - 25 * q^86 - 3 * q^87 + 7 * q^88 - 11 * q^89 + 10 * q^91 - q^92 + 19 * q^93 - 11 * q^94 - 8 * q^95 - 15 * q^96 + 30 * q^97 - 15 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.88066	−1.73548	−0.867741	−	0.497017i	$-0.834429\pi$
$5$	−3.88066	−1.73548	−0.867741	+	0.497017i	$0.834429\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.88066	1.22717
$11$	−2.01167	−0.606541	−0.303270	−	0.952905i	$-0.598079\pi$
$11$	−2.01167	−0.606541	−0.303270	+	0.952905i	$0.598079\pi$
$12$	1.00000	0.288675
$13$	4.46496	1.23836	0.619178	−	0.785250i	$-0.287466\pi$
$13$	4.46496	1.23836	0.619178	+	0.785250i	$0.287466\pi$
$14$	−1.00000	−0.267261
$15$	−3.88066	−1.00198
$16$	1.00000	0.250000
$17$	1.73476	0.420741	0.210370	−	0.977622i	$-0.432533\pi$
$17$	1.73476	0.420741	0.210370	+	0.977622i	$0.432533\pi$
$18$	−1.00000	−0.235702
$19$	6.29040	1.44312	0.721559	−	0.692353i	$-0.243426\pi$
$19$	6.29040	1.44312	0.721559	+	0.692353i	$0.243426\pi$
$20$	−3.88066	−0.867741
$21$	1.00000	0.218218
$22$	2.01167	0.428889
$23$	3.07078	0.640303	0.320151	−	0.947366i	$-0.396266\pi$
$23$	3.07078	0.640303	0.320151	+	0.947366i	$0.396266\pi$
$24$	−1.00000	−0.204124
$25$	10.0595	2.01190
$26$	−4.46496	−0.875650
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	−5.82268	−1.08125	−0.540623	−	0.841265i	$-0.681811\pi$
$29$	−5.82268	−1.08125	−0.540623	+	0.841265i	$0.681811\pi$
$30$	3.88066	0.708508
$31$	3.34602	0.600963	0.300481	−	0.953788i	$-0.402853\pi$
$31$	3.34602	0.600963	0.300481	+	0.953788i	$0.402853\pi$
$32$	−1.00000	−0.176777
$33$	−2.01167	−0.350187
$34$	−1.73476	−0.297509
$35$	−3.88066	−0.655950
$36$	1.00000	0.166667
$37$	−3.58791	−0.589848	−0.294924	−	0.955521i	$-0.595294\pi$
$37$	−3.58791	−0.589848	−0.294924	+	0.955521i	$0.595294\pi$
$38$	−6.29040	−1.02044
$39$	4.46496	0.714965
$40$	3.88066	0.613585
$41$	−8.04967	−1.25715	−0.628574	−	0.777750i	$-0.716361\pi$
$41$	−8.04967	−1.25715	−0.628574	+	0.777750i	$0.716361\pi$
$42$	−1.00000	−0.154303
$43$	5.48422	0.836336	0.418168	−	0.908370i	$-0.362672\pi$
$43$	5.48422	0.836336	0.418168	+	0.908370i	$0.362672\pi$
$44$	−2.01167	−0.303270
$45$	−3.88066	−0.578494
$46$	−3.07078	−0.452762
$47$	−0.224077	−0.0326851	−0.0163425	−	0.999866i	$-0.505202\pi$
$47$	−0.224077	−0.0326851	−0.0163425	+	0.999866i	$0.505202\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−10.0595	−1.42263
$51$	1.73476	0.242915
$52$	4.46496	0.619178
$53$	5.63333	0.773797	0.386899	−	0.922122i	$-0.373546\pi$
$53$	5.63333	0.773797	0.386899	+	0.922122i	$0.373546\pi$
$54$	−1.00000	−0.136083
$55$	7.80659	1.05264
$56$	−1.00000	−0.133631
$57$	6.29040	0.833184
$58$	5.82268	0.764556
$59$	6.71729	0.874517	0.437258	−	0.899336i	$-0.355950\pi$
$59$	6.71729	0.874517	0.437258	+	0.899336i	$0.355950\pi$
$60$	−3.88066	−0.500990
$61$	0.0358422	0.00458912	0.00229456	−	0.999997i	$-0.499270\pi$
$61$	0.0358422	0.00458912	0.00229456	+	0.999997i	$0.499270\pi$
$62$	−3.34602	−0.424945
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−17.3270	−2.14915
$66$	2.01167	0.247619
$67$	−8.81581	−1.07702	−0.538511	−	0.842618i	$-0.681013\pi$
$67$	−8.81581	−1.07702	−0.538511	+	0.842618i	$0.681013\pi$
$68$	1.73476	0.210370
$69$	3.07078	0.369679
$70$	3.88066	0.463827
$71$	2.99618	0.355582	0.177791	−	0.984068i	$-0.443105\pi$
$71$	2.99618	0.355582	0.177791	+	0.984068i	$0.443105\pi$
$72$	−1.00000	−0.117851
$73$	−12.6614	−1.48191	−0.740955	−	0.671554i	$-0.765627\pi$
$73$	−12.6614	−1.48191	−0.740955	+	0.671554i	$0.765627\pi$
$74$	3.58791	0.417085
$75$	10.0595	1.16157
$76$	6.29040	0.721559
$77$	−2.01167	−0.229251
$78$	−4.46496	−0.505557
$79$	8.53794	0.960594	0.480297	−	0.877106i	$-0.340529\pi$
$79$	8.53794	0.960594	0.480297	+	0.877106i	$0.340529\pi$
$80$	−3.88066	−0.433870
$81$	1.00000	0.111111
$82$	8.04967	0.888937
$83$	−4.06908	−0.446639	−0.223320	−	0.974745i	$-0.571689\pi$
$83$	−4.06908	−0.446639	−0.223320	+	0.974745i	$0.571689\pi$
$84$	1.00000	0.109109
$85$	−6.73200	−0.730188
$86$	−5.48422	−0.591379
$87$	−5.82268	−0.624257
$88$	2.01167	0.214445
$89$	−3.76268	−0.398844	−0.199422	−	0.979914i	$-0.563906\pi$
$89$	−3.76268	−0.398844	−0.199422	+	0.979914i	$0.563906\pi$
$90$	3.88066	0.409057
$91$	4.46496	0.468055
$92$	3.07078	0.320151
$93$	3.34602	0.346966
$94$	0.224077	0.0231118
$95$	−24.4109	−2.50450
$96$	−1.00000	−0.102062
$97$	−10.9632	−1.11315	−0.556574	−	0.830798i	$-0.687884\pi$
$97$	−10.9632	−1.11315	−0.556574	+	0.830798i	$0.687884\pi$
$98$	−1.00000	−0.101015
$99$	−2.01167	−0.202180
$100$	10.0595	1.00595
$101$	−11.3382	−1.12819	−0.564096	−	0.825709i	$-0.690775\pi$
$101$	−11.3382	−1.12819	−0.564096	+	0.825709i	$0.690775\pi$
$102$	−1.73476	−0.171767
$103$	1.02597	0.101092	0.0505458	−	0.998722i	$-0.483904\pi$
$103$	1.02597	0.101092	0.0505458	+	0.998722i	$0.483904\pi$
$104$	−4.46496	−0.437825
$105$	−3.88066	−0.378713
$106$	−5.63333	−0.547157
$107$	16.7663	1.62086	0.810429	−	0.585837i	$-0.199234\pi$
$107$	16.7663	1.62086	0.810429	+	0.585837i	$0.199234\pi$
$108$	1.00000	0.0962250
$109$	8.21536	0.786888	0.393444	−	0.919348i	$-0.371283\pi$
$109$	8.21536	0.786888	0.393444	+	0.919348i	$0.371283\pi$
$110$	−7.80659	−0.744329
$111$	−3.58791	−0.340549
$112$	1.00000	0.0944911
$113$	13.3870	1.25934	0.629670	−	0.776863i	$-0.283190\pi$
$113$	13.3870	1.25934	0.629670	+	0.776863i	$0.283190\pi$
$114$	−6.29040	−0.589150
$115$	−11.9167	−1.11123
$116$	−5.82268	−0.540623
$117$	4.46496	0.412785
$118$	−6.71729	−0.618377
$119$	1.73476	0.159025
$120$	3.88066	0.354254
$121$	−6.95319	−0.632108
$122$	−0.0358422	−0.00324500
$123$	−8.04967	−0.725814
$124$	3.34602	0.300481
$125$	−19.6341	−1.75613
$126$	−1.00000	−0.0890871
$127$	−11.1399	−0.988504	−0.494252	−	0.869319i	$-0.664558\pi$
$127$	−11.1399	−0.988504	−0.494252	+	0.869319i	$0.664558\pi$
$128$	−1.00000	−0.0883883
$129$	5.48422	0.482859
$130$	17.3270	1.51968
$131$	20.3822	1.78080	0.890400	−	0.455179i	$-0.150425\pi$
$131$	20.3822	1.78080	0.890400	+	0.455179i	$0.150425\pi$
$132$	−2.01167	−0.175093
$133$	6.29040	0.545447
$134$	8.81581	0.761570
$135$	−3.88066	−0.333994
$136$	−1.73476	−0.148754
$137$	−4.33493	−0.370358	−0.185179	−	0.982705i	$-0.559286\pi$
$137$	−4.33493	−0.370358	−0.185179	+	0.982705i	$0.559286\pi$
$138$	−3.07078	−0.261402
$139$	−10.9549	−0.929186	−0.464593	−	0.885524i	$-0.653799\pi$
$139$	−10.9549	−0.929186	−0.464593	+	0.885524i	$0.653799\pi$
$140$	−3.88066	−0.327975
$141$	−0.224077	−0.0188707
$142$	−2.99618	−0.251434
$143$	−8.98202	−0.751114
$144$	1.00000	0.0833333
$145$	22.5958	1.87648
$146$	12.6614	1.04787
$147$	1.00000	0.0824786
$148$	−3.58791	−0.294924
$149$	−4.22306	−0.345967	−0.172983	−	0.984925i	$-0.555341\pi$
$149$	−4.22306	−0.345967	−0.172983	+	0.984925i	$0.555341\pi$
$150$	−10.0595	−0.821354
$151$	3.14434	0.255883	0.127941	−	0.991782i	$-0.459163\pi$
$151$	3.14434	0.255883	0.127941	+	0.991782i	$0.459163\pi$
$152$	−6.29040	−0.510219
$153$	1.73476	0.140247
$154$	2.01167	0.162105
$155$	−12.9848	−1.04296
$156$	4.46496	0.357483
$157$	20.1494	1.60810	0.804049	−	0.594563i	$-0.202675\pi$
$157$	20.1494	1.60810	0.804049	+	0.594563i	$0.202675\pi$
$158$	−8.53794	−0.679242
$159$	5.63333	0.446752
$160$	3.88066	0.306793
$161$	3.07078	0.242012
$162$	−1.00000	−0.0785674
$163$	−11.1335	−0.872040	−0.436020	−	0.899937i	$-0.643612\pi$
$163$	−11.1335	−0.872040	−0.436020	+	0.899937i	$0.643612\pi$
$164$	−8.04967	−0.628574
$165$	7.80659	0.607742
$166$	4.06908	0.315822
$167$	14.9965	1.16046	0.580230	−	0.814453i	$-0.302963\pi$
$167$	14.9965	1.16046	0.580230	+	0.814453i	$0.302963\pi$
$168$	−1.00000	−0.0771517
$169$	6.93585	0.533527
$170$	6.73200	0.516321
$171$	6.29040	0.481039
$172$	5.48422	0.418168
$173$	5.88540	0.447458	0.223729	−	0.974651i	$-0.428177\pi$
$173$	5.88540	0.447458	0.223729	+	0.974651i	$0.428177\pi$
$174$	5.82268	0.441417
$175$	10.0595	0.760426
$176$	−2.01167	−0.151635
$177$	6.71729	0.504902
$178$	3.76268	0.282025
$179$	10.8061	0.807686	0.403843	−	0.914828i	$-0.367674\pi$
$179$	10.8061	0.807686	0.403843	+	0.914828i	$0.367674\pi$
$180$	−3.88066	−0.289247
$181$	−13.9117	−1.03405	−0.517023	−	0.855971i	$-0.672960\pi$
$181$	−13.9117	−1.03405	−0.517023	+	0.855971i	$0.672960\pi$
$182$	−4.46496	−0.330965
$183$	0.0358422	0.00264953
$184$	−3.07078	−0.226381
$185$	13.9234	1.02367
$186$	−3.34602	−0.245342
$187$	−3.48976	−0.255196
$188$	−0.224077	−0.0163425
$189$	1.00000	0.0727393
$190$	24.4109	1.77095
$191$	1.00000	0.0723575
$191$	1.00000	0.0723575
$192$	1.00000	0.0721688
$193$	−15.5868	−1.12196	−0.560982	−	0.827828i	$-0.689576\pi$
$193$	−15.5868	−1.12196	−0.560982	+	0.827828i	$0.689576\pi$
$194$	10.9632	0.787114
$195$	−17.3270	−1.24081
$196$	1.00000	0.0714286
$197$	2.40918	0.171647	0.0858233	−	0.996310i	$-0.472648\pi$
$197$	2.40918	0.171647	0.0858233	+	0.996310i	$0.472648\pi$
$198$	2.01167	0.142963
$199$	2.78185	0.197200	0.0986001	−	0.995127i	$-0.468564\pi$
$199$	2.78185	0.197200	0.0986001	+	0.995127i	$0.468564\pi$
$200$	−10.0595	−0.711313
$201$	−8.81581	−0.621819
$202$	11.3382	0.797753
$203$	−5.82268	−0.408672
$204$	1.73476	0.121457
$205$	31.2380	2.18176
$206$	−1.02597	−0.0714825
$207$	3.07078	0.213434
$208$	4.46496	0.309589
$209$	−12.6542	−0.875310
$210$	3.88066	0.267791
$211$	−4.69674	−0.323337	−0.161669	−	0.986845i	$-0.551688\pi$
$211$	−4.69674	−0.323337	−0.161669	+	0.986845i	$0.551688\pi$
$212$	5.63333	0.386899
$213$	2.99618	0.205295
$214$	−16.7663	−1.14612
$215$	−21.2824	−1.45145
$216$	−1.00000	−0.0680414
$217$	3.34602	0.227143
$218$	−8.21536	−0.556414
$219$	−12.6614	−0.855581
$220$	7.80659	0.526320
$221$	7.74562	0.521027
$222$	3.58791	0.240804
$223$	9.21803	0.617285	0.308642	−	0.951178i	$-0.400125\pi$
$223$	9.21803	0.617285	0.308642	+	0.951178i	$0.400125\pi$
$224$	−1.00000	−0.0668153
$225$	10.0595	0.670632
$226$	−13.3870	−0.890488
$227$	−4.06226	−0.269622	−0.134811	−	0.990871i	$-0.543043\pi$
$227$	−4.06226	−0.269622	−0.134811	+	0.990871i	$0.543043\pi$
$228$	6.29040	0.416592
$229$	5.16446	0.341277	0.170638	−	0.985334i	$-0.445417\pi$
$229$	5.16446	0.341277	0.170638	+	0.985334i	$0.445417\pi$
$230$	11.9167	0.785761
$231$	−2.01167	−0.132358
$232$	5.82268	0.382278
$233$	22.6263	1.48230	0.741148	−	0.671341i	$-0.234282\pi$
$233$	22.6263	1.48230	0.741148	+	0.671341i	$0.234282\pi$
$234$	−4.46496	−0.291883
$235$	0.869567	0.0567243
$236$	6.71729	0.437258
$237$	8.53794	0.554599
$238$	−1.73476	−0.112448
$239$	26.2182	1.69591	0.847956	−	0.530066i	$-0.177833\pi$
$239$	26.2182	1.69591	0.847956	+	0.530066i	$0.177833\pi$
$240$	−3.88066	−0.250495
$241$	2.59835	0.167374	0.0836871	−	0.996492i	$-0.473330\pi$
$241$	2.59835	0.167374	0.0836871	+	0.996492i	$0.473330\pi$
$242$	6.95319	0.446968
$243$	1.00000	0.0641500
$244$	0.0358422	0.00229456
$245$	−3.88066	−0.247926
$246$	8.04967	0.513228
$247$	28.0864	1.78709
$248$	−3.34602	−0.212472
$249$	−4.06908	−0.257867
$250$	19.6341	1.24177
$251$	−12.3603	−0.780175	−0.390088	−	0.920778i	$-0.627555\pi$
$251$	−12.3603	−0.780175	−0.390088	+	0.920778i	$0.627555\pi$
$252$	1.00000	0.0629941
$253$	−6.17740	−0.388370
$254$	11.1399	0.698978
$255$	−6.73200	−0.421574
$256$	1.00000	0.0625000
$257$	11.6547	0.726998	0.363499	−	0.931595i	$-0.381582\pi$
$257$	11.6547	0.726998	0.363499	+	0.931595i	$0.381582\pi$
$258$	−5.48422	−0.341433
$259$	−3.58791	−0.222942
$260$	−17.3270	−1.07457
$261$	−5.82268	−0.360415
$262$	−20.3822	−1.25922
$263$	−11.8105	−0.728264	−0.364132	−	0.931347i	$-0.618634\pi$
$263$	−11.8105	−0.728264	−0.364132	+	0.931347i	$0.618634\pi$
$264$	2.01167	0.123810
$265$	−21.8610	−1.34291
$266$	−6.29040	−0.385689
$267$	−3.76268	−0.230273
$268$	−8.81581	−0.538511
$269$	12.9205	0.787777	0.393889	−	0.919158i	$-0.371130\pi$
$269$	12.9205	0.787777	0.393889	+	0.919158i	$0.371130\pi$
$270$	3.88066	0.236169
$271$	24.6771	1.49903	0.749514	−	0.661988i	$-0.230287\pi$
$271$	24.6771	1.49903	0.749514	+	0.661988i	$0.230287\pi$
$272$	1.73476	0.105185
$273$	4.46496	0.270232
$274$	4.33493	0.261883
$275$	−20.2364	−1.22030
$276$	3.07078	0.184839
$277$	19.1102	1.14822	0.574110	−	0.818778i	$-0.305348\pi$
$277$	19.1102	1.14822	0.574110	+	0.818778i	$0.305348\pi$
$278$	10.9549	0.657034
$279$	3.34602	0.200321
$280$	3.88066	0.231914
$281$	27.2508	1.62565	0.812823	−	0.582511i	$-0.197930\pi$
$281$	27.2508	1.62565	0.812823	+	0.582511i	$0.197930\pi$
$282$	0.224077	0.0133436
$283$	7.06366	0.419891	0.209946	−	0.977713i	$-0.432671\pi$
$283$	7.06366	0.419891	0.209946	+	0.977713i	$0.432671\pi$
$284$	2.99618	0.177791
$285$	−24.4109	−1.44598
$286$	8.98202	0.531118
$287$	−8.04967	−0.475157
$288$	−1.00000	−0.0589256
$289$	−13.9906	−0.822977
$290$	−22.5958	−1.32687
$291$	−10.9632	−0.642676
$292$	−12.6614	−0.740955
$293$	−18.0744	−1.05592	−0.527960	−	0.849269i	$-0.677043\pi$
$293$	−18.0744	−1.05592	−0.527960	+	0.849269i	$0.677043\pi$
$294$	−1.00000	−0.0583212
$295$	−26.0675	−1.51771
$296$	3.58791	0.208543
$297$	−2.01167	−0.116729
$298$	4.22306	0.244635
$299$	13.7109	0.792923
$300$	10.0595	0.580785
$301$	5.48422	0.316105
$302$	−3.14434	−0.180936
$303$	−11.3382	−0.651362
$304$	6.29040	0.360779
$305$	−0.139091	−0.00796434
$306$	−1.73476	−0.0991695
$307$	4.64898	0.265331	0.132666	−	0.991161i	$-0.457646\pi$
$307$	4.64898	0.265331	0.132666	+	0.991161i	$0.457646\pi$
$308$	−2.01167	−0.114625
$309$	1.02597	0.0583652
$310$	12.9848	0.737484
$311$	−0.765606	−0.0434135	−0.0217068	−	0.999764i	$-0.506910\pi$
$311$	−0.765606	−0.0434135	−0.0217068	+	0.999764i	$0.506910\pi$
$312$	−4.46496	−0.252778
$313$	−3.53875	−0.200022	−0.100011	−	0.994986i	$-0.531888\pi$
$313$	−3.53875	−0.200022	−0.100011	+	0.994986i	$0.531888\pi$
$314$	−20.1494	−1.13710
$315$	−3.88066	−0.218650
$316$	8.53794	0.480297
$317$	−9.93810	−0.558179	−0.279090	−	0.960265i	$-0.590033\pi$
$317$	−9.93810	−0.558179	−0.279090	+	0.960265i	$0.590033\pi$
$318$	−5.63333	−0.315901
$319$	11.7133	0.655820
$320$	−3.88066	−0.216935
$321$	16.7663	0.935803
$322$	−3.07078	−0.171128
$323$	10.9123	0.607178
$324$	1.00000	0.0555556
$325$	44.9152	2.49145
$326$	11.1335	0.616625
$327$	8.21536	0.454310
$328$	8.04967	0.444469
$329$	−0.224077	−0.0123538
$330$	−7.80659	−0.429739
$331$	2.48943	0.136831	0.0684157	−	0.997657i	$-0.478206\pi$
$331$	2.48943	0.136831	0.0684157	+	0.997657i	$0.478206\pi$
$332$	−4.06908	−0.223320
$333$	−3.58791	−0.196616
$334$	−14.9965	−0.820569
$335$	34.2111	1.86915
$336$	1.00000	0.0545545
$337$	−6.13577	−0.334237	−0.167118	−	0.985937i	$-0.553446\pi$
$337$	−6.13577	−0.334237	−0.167118	+	0.985937i	$0.553446\pi$
$338$	−6.93585	−0.377260
$339$	13.3870	0.727080
$340$	−6.73200	−0.365094
$341$	−6.73108	−0.364509
$342$	−6.29040	−0.340146
$343$	1.00000	0.0539949
$344$	−5.48422	−0.295689
$345$	−11.9167	−0.641571
$346$	−5.88540	−0.316401
$347$	7.23784	0.388548	0.194274	−	0.980947i	$-0.437765\pi$
$347$	7.23784	0.388548	0.194274	+	0.980947i	$0.437765\pi$
$348$	−5.82268	−0.312129
$349$	21.5684	1.15453	0.577265	−	0.816557i	$-0.304120\pi$
$349$	21.5684	1.15453	0.577265	+	0.816557i	$0.304120\pi$
$350$	−10.0595	−0.537702
$351$	4.46496	0.238322
$352$	2.01167	0.107222
$353$	15.3360	0.816253	0.408127	−	0.912925i	$-0.366182\pi$
$353$	15.3360	0.816253	0.408127	+	0.912925i	$0.366182\pi$
$354$	−6.71729	−0.357020
$355$	−11.6272	−0.617105
$356$	−3.76268	−0.199422
$357$	1.73476	0.0918132
$358$	−10.8061	−0.571120
$359$	22.2927	1.17656	0.588281	−	0.808657i	$-0.299805\pi$
$359$	22.2927	1.17656	0.588281	+	0.808657i	$0.299805\pi$
$360$	3.88066	0.204528
$361$	20.5692	1.08259
$362$	13.9117	0.731182
$363$	−6.95319	−0.364948
$364$	4.46496	0.234027
$365$	49.1347	2.57183
$366$	−0.0358422	−0.00187350
$367$	−6.62455	−0.345799	−0.172899	−	0.984940i	$-0.555314\pi$
$367$	−6.62455	−0.345799	−0.172899	+	0.984940i	$0.555314\pi$
$368$	3.07078	0.160076
$369$	−8.04967	−0.419049
$370$	−13.9234	−0.723844
$371$	5.63333	0.292468
$372$	3.34602	0.173483
$373$	7.38189	0.382220	0.191110	−	0.981569i	$-0.438791\pi$
$373$	7.38189	0.382220	0.191110	+	0.981569i	$0.438791\pi$
$374$	3.48976	0.180451
$375$	−19.6341	−1.01390
$376$	0.224077	0.0115559
$377$	−25.9980	−1.33897
$378$	−1.00000	−0.0514344
$379$	23.4327	1.20366	0.601829	−	0.798625i	$-0.294439\pi$
$379$	23.4327	1.20366	0.601829	+	0.798625i	$0.294439\pi$
$380$	−24.4109	−1.25225
$381$	−11.1399	−0.570713
$382$	−1.00000	−0.0511645
$383$	−8.64882	−0.441934	−0.220967	−	0.975281i	$-0.570921\pi$
$383$	−8.64882	−0.441934	−0.220967	+	0.975281i	$0.570921\pi$
$384$	−1.00000	−0.0510310
$385$	7.80659	0.397861
$386$	15.5868	0.793348
$387$	5.48422	0.278779
$388$	−10.9632	−0.556574
$389$	−24.3180	−1.23297	−0.616486	−	0.787366i	$-0.711444\pi$
$389$	−24.3180	−1.23297	−0.616486	+	0.787366i	$0.711444\pi$
$390$	17.3270	0.877385
$391$	5.32707	0.269401
$392$	−1.00000	−0.0505076
$393$	20.3822	1.02815
$394$	−2.40918	−0.121373
$395$	−33.1328	−1.66709
$396$	−2.01167	−0.101090
$397$	−7.30542	−0.366649	−0.183324	−	0.983052i	$-0.558686\pi$
$397$	−7.30542	−0.366649	−0.183324	+	0.983052i	$0.558686\pi$
$398$	−2.78185	−0.139442
$399$	6.29040	0.314914
$400$	10.0595	0.502974
$401$	−23.0785	−1.15248	−0.576242	−	0.817279i	$-0.695481\pi$
$401$	−23.0785	−1.15248	−0.576242	+	0.817279i	$0.695481\pi$
$402$	8.81581	0.439693
$403$	14.9398	0.744206
$404$	−11.3382	−0.564096
$405$	−3.88066	−0.192831
$406$	5.82268	0.288975
$407$	7.21768	0.357767
$408$	−1.73476	−0.0858833
$409$	−2.25795	−0.111649	−0.0558243	−	0.998441i	$-0.517779\pi$
$409$	−2.25795	−0.111649	−0.0558243	+	0.998441i	$0.517779\pi$
$410$	−31.2380	−1.54273
$411$	−4.33493	−0.213826
$412$	1.02597	0.0505458
$413$	6.71729	0.330536
$414$	−3.07078	−0.150921
$415$	15.7907	0.775134
$416$	−4.46496	−0.218913
$417$	−10.9549	−0.536466
$418$	12.6542	0.618938
$419$	14.4799	0.707390	0.353695	−	0.935361i	$-0.384925\pi$
$419$	14.4799	0.707390	0.353695	+	0.935361i	$0.384925\pi$
$420$	−3.88066	−0.189357
$421$	25.6315	1.24920	0.624601	−	0.780944i	$-0.285262\pi$
$421$	25.6315	1.24920	0.624601	+	0.780944i	$0.285262\pi$
$422$	4.69674	0.228634
$423$	−0.224077	−0.0108950
$424$	−5.63333	−0.273579
$425$	17.4508	0.846487
$426$	−2.99618	−0.145166
$427$	0.0358422	0.00173453
$428$	16.7663	0.810429
$429$	−8.98202	−0.433656
$430$	21.2824	1.02633
$431$	−9.20429	−0.443355	−0.221677	−	0.975120i	$-0.571153\pi$
$431$	−9.20429	−0.443355	−0.221677	+	0.975120i	$0.571153\pi$
$432$	1.00000	0.0481125
$433$	8.64704	0.415550	0.207775	−	0.978177i	$-0.433378\pi$
$433$	8.64704	0.415550	0.207775	+	0.978177i	$0.433378\pi$
$434$	−3.34602	−0.160614
$435$	22.5958	1.08339
$436$	8.21536	0.393444
$437$	19.3165	0.924032
$438$	12.6614	0.604987
$439$	38.6137	1.84293	0.921466	−	0.388458i	$-0.126992\pi$
$439$	38.6137	1.84293	0.921466	+	0.388458i	$0.126992\pi$
$440$	−7.80659	−0.372165
$441$	1.00000	0.0476190
$442$	−7.74562	−0.368422
$443$	22.5589	1.07181	0.535903	−	0.844280i	$-0.319971\pi$
$443$	22.5589	1.07181	0.535903	+	0.844280i	$0.319971\pi$
$444$	−3.58791	−0.170274
$445$	14.6017	0.692186
$446$	−9.21803	−0.436486
$447$	−4.22306	−0.199744
$448$	1.00000	0.0472456
$449$	36.6078	1.72763	0.863815	−	0.503809i	$-0.168068\pi$
$449$	36.6078	1.72763	0.863815	+	0.503809i	$0.168068\pi$
$450$	−10.0595	−0.474209
$451$	16.1933	0.762511
$452$	13.3870	0.629670
$453$	3.14434	0.147734
$454$	4.06226	0.190651
$455$	−17.3270	−0.812301
$456$	−6.29040	−0.294575
$457$	−4.85377	−0.227050	−0.113525	−	0.993535i	$-0.536214\pi$
$457$	−4.85377	−0.227050	−0.113525	+	0.993535i	$0.536214\pi$
$458$	−5.16446	−0.241319
$459$	1.73476	0.0809716
$460$	−11.9167	−0.555617
$461$	−26.6375	−1.24063	−0.620315	−	0.784353i	$-0.712995\pi$
$461$	−26.6375	−1.24063	−0.620315	+	0.784353i	$0.712995\pi$
$462$	2.01167	0.0935913
$463$	−8.16950	−0.379669	−0.189834	−	0.981816i	$-0.560795\pi$
$463$	−8.16950	−0.379669	−0.189834	+	0.981816i	$0.560795\pi$
$464$	−5.82268	−0.270311
$465$	−12.9848	−0.602153
$466$	−22.6263	−1.04814
$467$	35.5021	1.64284	0.821422	−	0.570322i	$-0.193181\pi$
$467$	35.5021	1.64284	0.821422	+	0.570322i	$0.193181\pi$
$468$	4.46496	0.206393
$469$	−8.81581	−0.407076
$470$	−0.869567	−0.0401102
$471$	20.1494	0.928436
$472$	−6.71729	−0.309188
$473$	−11.0324	−0.507272
$474$	−8.53794	−0.392161
$475$	63.2782	2.90340
$476$	1.73476	0.0795125
$477$	5.63333	0.257932
$478$	−26.2182	−1.19919
$479$	−6.35702	−0.290459	−0.145230	−	0.989398i	$-0.546392\pi$
$479$	−6.35702	−0.290459	−0.145230	+	0.989398i	$0.546392\pi$
$480$	3.88066	0.177127
$481$	−16.0198	−0.730442
$482$	−2.59835	−0.118351
$483$	3.07078	0.139726
$484$	−6.95319	−0.316054
$485$	42.5445	1.93185
$486$	−1.00000	−0.0453609
$487$	−17.6438	−0.799516	−0.399758	−	0.916621i	$-0.630906\pi$
$487$	−17.6438	−0.799516	−0.399758	+	0.916621i	$0.630906\pi$
$488$	−0.0358422	−0.00162250
$489$	−11.1335	−0.503472
$490$	3.88066	0.175310
$491$	28.4637	1.28455	0.642274	−	0.766475i	$-0.277991\pi$
$491$	28.4637	1.28455	0.642274	+	0.766475i	$0.277991\pi$
$492$	−8.04967	−0.362907
$493$	−10.1010	−0.454924
$494$	−28.0864	−1.26367
$495$	7.80659	0.350880
$496$	3.34602	0.150241
$497$	2.99618	0.134397
$498$	4.06908	0.182340
$499$	36.5245	1.63506	0.817530	−	0.575887i	$-0.195343\pi$
$499$	36.5245	1.63506	0.817530	+	0.575887i	$0.195343\pi$
$500$	−19.6341	−0.878065
$501$	14.9965	0.669992
$502$	12.3603	0.551667
$503$	26.3771	1.17610	0.588048	−	0.808826i	$-0.299896\pi$
$503$	26.3771	1.17610	0.588048	+	0.808826i	$0.299896\pi$
$504$	−1.00000	−0.0445435
$505$	43.9996	1.95796
$506$	6.17740	0.274619
$507$	6.93585	0.308032
$508$	−11.1399	−0.494252
$509$	30.6016	1.35639	0.678195	−	0.734882i	$-0.262762\pi$
$509$	30.6016	1.35639	0.678195	+	0.734882i	$0.262762\pi$
$510$	6.73200	0.298098
$511$	−12.6614	−0.560109
$512$	−1.00000	−0.0441942
$513$	6.29040	0.277728
$514$	−11.6547	−0.514065
$515$	−3.98142	−0.175443
$516$	5.48422	0.241429
$517$	0.450770	0.0198248
$518$	3.58791	0.157643
$519$	5.88540	0.258340
$520$	17.3270	0.759838
$521$	−29.1244	−1.27596	−0.637981	−	0.770052i	$-0.720230\pi$
$521$	−29.1244	−1.27596	−0.637981	+	0.770052i	$0.720230\pi$
$522$	5.82268	0.254852
$523$	−3.90870	−0.170915	−0.0854577	−	0.996342i	$-0.527235\pi$
$523$	−3.90870	−0.170915	−0.0854577	+	0.996342i	$0.527235\pi$
$524$	20.3822	0.890400
$525$	10.0595	0.439032
$526$	11.8105	0.514960
$527$	5.80454	0.252850
$528$	−2.01167	−0.0875466
$529$	−13.5703	−0.590012
$530$	21.8610	0.949581
$531$	6.71729	0.291506
$532$	6.29040	0.272724
$533$	−35.9414	−1.55680
$534$	3.76268	0.162827
$535$	−65.0642	−2.81297
$536$	8.81581	0.380785
$537$	10.8061	0.466318
$538$	−12.9205	−0.557042
$539$	−2.01167	−0.0866487
$540$	−3.88066	−0.166997
$541$	21.9522	0.943797	0.471899	−	0.881653i	$-0.343569\pi$
$541$	21.9522	0.943797	0.471899	+	0.881653i	$0.343569\pi$
$542$	−24.6771	−1.05997
$543$	−13.9117	−0.597007
$544$	−1.73476	−0.0743772
$545$	−31.8810	−1.36563
$546$	−4.46496	−0.191083
$547$	−41.7393	−1.78465	−0.892323	−	0.451398i	$-0.850925\pi$
$547$	−41.7393	−1.78465	−0.892323	+	0.451398i	$0.850925\pi$
$548$	−4.33493	−0.185179
$549$	0.0358422	0.00152971
$550$	20.2364	0.862881
$551$	−36.6270	−1.56036
$552$	−3.07078	−0.130701
$553$	8.53794	0.363070
$554$	−19.1102	−0.811914
$555$	13.9234	0.591016
$556$	−10.9549	−0.464593
$557$	14.4480	0.612182	0.306091	−	0.952002i	$-0.400979\pi$
$557$	14.4480	0.612182	0.306091	+	0.952002i	$0.400979\pi$
$558$	−3.34602	−0.141648
$559$	24.4868	1.03568
$560$	−3.88066	−0.163988
$561$	−3.48976	−0.147338
$562$	−27.2508	−1.14951
$563$	9.05324	0.381549	0.190774	−	0.981634i	$-0.438900\pi$
$563$	9.05324	0.381549	0.190774	+	0.981634i	$0.438900\pi$
$564$	−0.224077	−0.00943536
$565$	−51.9502	−2.18556
$566$	−7.06366	−0.296908
$567$	1.00000	0.0419961
$568$	−2.99618	−0.125717
$569$	−11.6103	−0.486730	−0.243365	−	0.969935i	$-0.578251\pi$
$569$	−11.6103	−0.486730	−0.243365	+	0.969935i	$0.578251\pi$
$570$	24.4109	1.02246
$571$	−11.5258	−0.482339	−0.241170	−	0.970483i	$-0.577531\pi$
$571$	−11.5258	−0.482339	−0.241170	+	0.970483i	$0.577531\pi$
$572$	−8.98202	−0.375557
$573$	1.00000	0.0417756
$574$	8.04967	0.335987
$575$	30.8905	1.28822
$576$	1.00000	0.0416667
$577$	−12.8136	−0.533436	−0.266718	−	0.963775i	$-0.585939\pi$
$577$	−12.8136	−0.533436	−0.266718	+	0.963775i	$0.585939\pi$
$578$	13.9906	0.581933
$579$	−15.5868	−0.647766
$580$	22.5958	0.938241
$581$	−4.06908	−0.168814
$582$	10.9632	0.454441
$583$	−11.3324	−0.469340
$584$	12.6614	0.523934
$585$	−17.3270	−0.716382
$586$	18.0744	0.746648
$587$	−30.4680	−1.25755	−0.628775	−	0.777587i	$-0.716443\pi$
$587$	−30.4680	−1.25755	−0.628775	+	0.777587i	$0.716443\pi$
$588$	1.00000	0.0412393
$589$	21.0478	0.867260
$590$	26.0675	1.07318
$591$	2.40918	0.0991003
$592$	−3.58791	−0.147462
$593$	5.37549	0.220745	0.110372	−	0.993890i	$-0.464796\pi$
$593$	5.37549	0.220745	0.110372	+	0.993890i	$0.464796\pi$
$594$	2.01167	0.0825398
$595$	−6.73200	−0.275985
$596$	−4.22306	−0.172983
$597$	2.78185	0.113854
$598$	−13.7109	−0.560681
$599$	−14.6629	−0.599110	−0.299555	−	0.954079i	$-0.596838\pi$
$599$	−14.6629	−0.599110	−0.299555	+	0.954079i	$0.596838\pi$
$600$	−10.0595	−0.410677
$601$	10.5960	0.432218	0.216109	−	0.976369i	$-0.430663\pi$
$601$	10.5960	0.432218	0.216109	+	0.976369i	$0.430663\pi$
$602$	−5.48422	−0.223520
$603$	−8.81581	−0.359007
$604$	3.14434	0.127941
$605$	26.9829	1.09701
$606$	11.3382	0.460583
$607$	−6.47448	−0.262791	−0.131396	−	0.991330i	$-0.541946\pi$
$607$	−6.47448	−0.262791	−0.131396	+	0.991330i	$0.541946\pi$
$608$	−6.29040	−0.255110
$609$	−5.82268	−0.235947
$610$	0.139091	0.00563164
$611$	−1.00050	−0.0404758
$612$	1.73476	0.0701235
$613$	−17.5101	−0.707228	−0.353614	−	0.935391i	$-0.615047\pi$
$613$	−17.5101	−0.707228	−0.353614	+	0.935391i	$0.615047\pi$
$614$	−4.64898	−0.187618
$615$	31.2380	1.25964
$616$	2.01167	0.0810524
$617$	−25.6806	−1.03386	−0.516930	−	0.856028i	$-0.672925\pi$
$617$	−25.6806	−1.03386	−0.516930	+	0.856028i	$0.672925\pi$
$618$	−1.02597	−0.0412704
$619$	−0.437569	−0.0175874	−0.00879369	−	0.999961i	$-0.502799\pi$
$619$	−0.437569	−0.0175874	−0.00879369	+	0.999961i	$0.502799\pi$
$620$	−12.9848	−0.521480
$621$	3.07078	0.123226
$622$	0.765606	0.0306980
$623$	−3.76268	−0.150749
$624$	4.46496	0.178741
$625$	25.8958	1.03583
$626$	3.53875	0.141437
$627$	−12.6542	−0.505360
$628$	20.1494	0.804049
$629$	−6.22415	−0.248173
$630$	3.88066	0.154609
$631$	−2.83514	−0.112865	−0.0564326	−	0.998406i	$-0.517973\pi$
$631$	−2.83514	−0.112865	−0.0564326	+	0.998406i	$0.517973\pi$
$632$	−8.53794	−0.339621
$633$	−4.69674	−0.186679
$634$	9.93810	0.394692
$635$	43.2300	1.71553
$636$	5.63333	0.223376
$637$	4.46496	0.176908
$638$	−11.7133	−0.463734
$639$	2.99618	0.118527
$640$	3.88066	0.153396
$641$	−45.9690	−1.81567	−0.907833	−	0.419332i	$-0.862264\pi$
$641$	−45.9690	−1.81567	−0.907833	+	0.419332i	$0.862264\pi$
$642$	−16.7663	−0.661713
$643$	37.0227	1.46003	0.730016	−	0.683430i	$-0.239513\pi$
$643$	37.0227	1.46003	0.730016	+	0.683430i	$0.239513\pi$
$644$	3.07078	0.121006
$645$	−21.2824	−0.837992
$646$	−10.9123	−0.429340
$647$	45.3524	1.78299	0.891494	−	0.453032i	$-0.149658\pi$
$647$	45.3524	1.78299	0.891494	+	0.453032i	$0.149658\pi$
$648$	−1.00000	−0.0392837
$649$	−13.5130	−0.530430
$650$	−44.9152	−1.76172
$651$	3.34602	0.131141
$652$	−11.1335	−0.436020
$653$	−5.18979	−0.203092	−0.101546	−	0.994831i	$-0.532379\pi$
$653$	−5.18979	−0.203092	−0.101546	+	0.994831i	$0.532379\pi$
$654$	−8.21536	−0.321246
$655$	−79.0962	−3.09055
$656$	−8.04967	−0.314287
$657$	−12.6614	−0.493970
$658$	0.224077	0.00873545
$659$	7.55256	0.294206	0.147103	−	0.989121i	$-0.453005\pi$
$659$	7.55256	0.294206	0.147103	+	0.989121i	$0.453005\pi$
$660$	7.80659	0.303871
$661$	−23.8261	−0.926726	−0.463363	−	0.886169i	$-0.653357\pi$
$661$	−23.8261	−0.926726	−0.463363	+	0.886169i	$0.653357\pi$
$662$	−2.48943	−0.0967544
$663$	7.74562	0.300815
$664$	4.06908	0.157911
$665$	−24.4109	−0.946614
$666$	3.58791	0.139028
$667$	−17.8802	−0.692324
$668$	14.9965	0.580230
$669$	9.21803	0.356390
$670$	−34.2111	−1.32169
$671$	−0.0721027	−0.00278349
$672$	−1.00000	−0.0385758
$673$	−36.8509	−1.42050	−0.710249	−	0.703951i	$-0.751418\pi$
$673$	−36.8509	−1.42050	−0.710249	+	0.703951i	$0.751418\pi$
$674$	6.13577	0.236341
$675$	10.0595	0.387190
$676$	6.93585	0.266763
$677$	−10.0381	−0.385794	−0.192897	−	0.981219i	$-0.561788\pi$
$677$	−10.0381	−0.385794	−0.192897	+	0.981219i	$0.561788\pi$
$678$	−13.3870	−0.514123
$679$	−10.9632	−0.420730
$680$	6.73200	0.258160
$681$	−4.06226	−0.155666
$682$	6.73108	0.257747
$683$	−24.0584	−0.920569	−0.460284	−	0.887771i	$-0.652253\pi$
$683$	−24.0584	−0.920569	−0.460284	+	0.887771i	$0.652253\pi$
$684$	6.29040	0.240520
$685$	16.8224	0.642749
$686$	−1.00000	−0.0381802
$687$	5.16446	0.197036
$688$	5.48422	0.209084
$689$	25.1526	0.958237
$690$	11.9167	0.453659
$691$	34.9815	1.33076	0.665379	−	0.746505i	$-0.268270\pi$
$691$	34.9815	1.33076	0.665379	+	0.746505i	$0.268270\pi$
$692$	5.88540	0.223729
$693$	−2.01167	−0.0764170
$694$	−7.23784	−0.274745
$695$	42.5123	1.61259
$696$	5.82268	0.220708
$697$	−13.9642	−0.528933
$698$	−21.5684	−0.816376
$699$	22.6263	0.855804
$700$	10.0595	0.380213
$701$	−7.73414	−0.292114	−0.146057	−	0.989276i	$-0.546658\pi$
$701$	−7.73414	−0.292114	−0.146057	+	0.989276i	$0.546658\pi$
$702$	−4.46496	−0.168519
$703$	−22.5694	−0.851220
$704$	−2.01167	−0.0758176
$705$	0.869567	0.0327498
$706$	−15.3360	−0.577178
$707$	−11.3382	−0.426417
$708$	6.71729	0.252451
$709$	2.48368	0.0932764	0.0466382	−	0.998912i	$-0.485149\pi$
$709$	2.48368	0.0932764	0.0466382	+	0.998912i	$0.485149\pi$
$710$	11.6272	0.436359
$711$	8.53794	0.320198
$712$	3.76268	0.141013
$713$	10.2749	0.384798
$714$	−1.73476	−0.0649217
$715$	34.8561	1.30354
$716$	10.8061	0.403843
$717$	26.2182	0.979136
$718$	−22.2927	−0.831955
$719$	−18.1064	−0.675255	−0.337627	−	0.941280i	$-0.609624\pi$
$719$	−18.1064	−0.675255	−0.337627	+	0.941280i	$0.609624\pi$
$720$	−3.88066	−0.144623
$721$	1.02597	0.0382090
$722$	−20.5692	−0.765505
$723$	2.59835	0.0966336
$724$	−13.9117	−0.517023
$725$	−58.5732	−2.17535
$726$	6.95319	0.258057
$727$	13.4189	0.497680	0.248840	−	0.968545i	$-0.419951\pi$
$727$	13.4189	0.497680	0.248840	+	0.968545i	$0.419951\pi$
$728$	−4.46496	−0.165482
$729$	1.00000	0.0370370
$730$	−49.1347	−1.81856
$731$	9.51380	0.351880
$732$	0.0358422	0.00132477
$733$	39.3552	1.45362	0.726809	−	0.686839i	$-0.241002\pi$
$733$	39.3552	1.45362	0.726809	+	0.686839i	$0.241002\pi$
$734$	6.62455	0.244517
$735$	−3.88066	−0.143140
$736$	−3.07078	−0.113191
$737$	17.7345	0.653258
$738$	8.04967	0.296312
$739$	−41.3152	−1.51980	−0.759901	−	0.650039i	$-0.774753\pi$
$739$	−41.3152	−1.51980	−0.759901	+	0.650039i	$0.774753\pi$
$740$	13.9234	0.511835
$741$	28.0864	1.03178
$742$	−5.63333	−0.206806
$743$	−12.8367	−0.470933	−0.235466	−	0.971883i	$-0.575662\pi$
$743$	−12.8367	−0.470933	−0.235466	+	0.971883i	$0.575662\pi$
$744$	−3.34602	−0.122671
$745$	16.3882	0.600419
$746$	−7.38189	−0.270270
$747$	−4.06908	−0.148880
$748$	−3.48976	−0.127598
$749$	16.7663	0.612627
$750$	19.6341	0.716937
$751$	10.1776	0.371386	0.185693	−	0.982608i	$-0.440547\pi$
$751$	10.1776	0.371386	0.185693	+	0.982608i	$0.440547\pi$
$752$	−0.224077	−0.00817126
$753$	−12.3603	−0.450434
$754$	25.9980	0.946793
$755$	−12.2021	−0.444080
$756$	1.00000	0.0363696
$757$	−18.0837	−0.657263	−0.328631	−	0.944458i	$-0.606587\pi$
$757$	−18.0837	−0.657263	−0.328631	+	0.944458i	$0.606587\pi$
$758$	−23.4327	−0.851115
$759$	−6.17740	−0.224225
$760$	24.4109	0.885476
$761$	−6.77128	−0.245459	−0.122729	−	0.992440i	$-0.539165\pi$
$761$	−6.77128	−0.245459	−0.122729	+	0.992440i	$0.539165\pi$
$762$	11.1399	0.403555
$763$	8.21536	0.297416
$764$	1.00000	0.0361787
$765$	−6.73200	−0.243396
$766$	8.64882	0.312495
$767$	29.9924	1.08296
$768$	1.00000	0.0360844
$769$	10.1726	0.366833	0.183417	−	0.983035i	$-0.441284\pi$
$769$	10.1726	0.366833	0.183417	+	0.983035i	$0.441284\pi$
$770$	−7.80659	−0.281330
$771$	11.6547	0.419732
$772$	−15.5868	−0.560982
$773$	5.38499	0.193685	0.0968424	−	0.995300i	$-0.469126\pi$
$773$	5.38499	0.193685	0.0968424	+	0.995300i	$0.469126\pi$
$774$	−5.48422	−0.197126
$775$	33.6592	1.20908
$776$	10.9632	0.393557
$777$	−3.58791	−0.128715
$778$	24.3180	0.871843
$779$	−50.6357	−1.81421
$780$	−17.3270	−0.620405
$781$	−6.02733	−0.215675
$782$	−5.32707	−0.190496
$783$	−5.82268	−0.208086
$784$	1.00000	0.0357143
$785$	−78.1929	−2.79082
$786$	−20.3822	−0.727008
$787$	−47.5484	−1.69492	−0.847459	−	0.530861i	$-0.821868\pi$
$787$	−47.5484	−1.69492	−0.847459	+	0.530861i	$0.821868\pi$
$788$	2.40918	0.0858233
$789$	−11.8105	−0.420463
$790$	33.1328	1.17881
$791$	13.3870	0.475986
$792$	2.01167	0.0714815
$793$	0.160034	0.00568297
$794$	7.30542	0.259260
$795$	−21.8610	−0.775330
$796$	2.78185	0.0986001
$797$	23.3366	0.826626	0.413313	−	0.910589i	$-0.364371\pi$
$797$	23.3366	0.826626	0.413313	+	0.910589i	$0.364371\pi$
$798$	−6.29040	−0.222678
$799$	−0.388720	−0.0137519
$800$	−10.0595	−0.355657
$801$	−3.76268	−0.132948
$802$	23.0785	0.814929
$803$	25.4706	0.898839
$804$	−8.81581	−0.310910
$805$	−11.9167	−0.420007
$806$	−14.9398	−0.526233
$807$	12.9205	0.454823
$808$	11.3382	0.398876
$809$	−53.4675	−1.87982	−0.939908	−	0.341428i	$-0.889089\pi$
$809$	−53.4675	−1.87982	−0.939908	+	0.341428i	$0.889089\pi$
$810$	3.88066	0.136352
$811$	5.60153	0.196696	0.0983481	−	0.995152i	$-0.468644\pi$
$811$	5.60153	0.196696	0.0983481	+	0.995152i	$0.468644\pi$
$812$	−5.82268	−0.204336
$813$	24.6771	0.865465
$814$	−7.21768	−0.252979
$815$	43.2051	1.51341
$816$	1.73476	0.0607287
$817$	34.4979	1.20693
$818$	2.25795	0.0789475
$819$	4.46496	0.156018
$820$	31.2380	1.09088
$821$	−7.14646	−0.249413	−0.124707	−	0.992194i	$-0.539799\pi$
$821$	−7.14646	−0.249413	−0.124707	+	0.992194i	$0.539799\pi$
$822$	4.33493	0.151198
$823$	−1.28223	−0.0446958	−0.0223479	−	0.999750i	$-0.507114\pi$
$823$	−1.28223	−0.0446958	−0.0223479	+	0.999750i	$0.507114\pi$
$824$	−1.02597	−0.0357413
$825$	−20.2364	−0.704539
$826$	−6.71729	−0.233724
$827$	1.66038	0.0577372	0.0288686	−	0.999583i	$-0.490810\pi$
$827$	1.66038	0.0577372	0.0288686	+	0.999583i	$0.490810\pi$
$828$	3.07078	0.106717
$829$	56.9662	1.97852	0.989258	−	0.146177i	$-0.0466969\pi$
$829$	56.9662	1.97852	0.989258	+	0.146177i	$0.0466969\pi$
$830$	−15.7907	−0.548103
$831$	19.1102	0.662925
$832$	4.46496	0.154795
$833$	1.73476	0.0601058
$834$	10.9549	0.379339
$835$	−58.1961	−2.01396
$836$	−12.6542	−0.437655
$837$	3.34602	0.115655
$838$	−14.4799	−0.500200
$839$	39.0970	1.34978	0.674889	−	0.737919i	$-0.264191\pi$
$839$	39.0970	1.34978	0.674889	+	0.737919i	$0.264191\pi$
$840$	3.88066	0.133895
$841$	4.90365	0.169092
$842$	−25.6315	−0.883319
$843$	27.2508	0.938567
$844$	−4.69674	−0.161669
$845$	−26.9156	−0.925926
$846$	0.224077	0.00770394
$847$	−6.95319	−0.238914
$848$	5.63333	0.193449
$849$	7.06366	0.242424
$850$	−17.4508	−0.598557
$851$	−11.0177	−0.377681
$852$	2.99618	0.102648
$853$	1.62720	0.0557142	0.0278571	−	0.999612i	$-0.491132\pi$
$853$	1.62720	0.0557142	0.0278571	+	0.999612i	$0.491132\pi$
$854$	−0.0358422	−0.00122650
$855$	−24.4109	−0.834835
$856$	−16.7663	−0.573060
$857$	19.3211	0.659997	0.329998	−	0.943981i	$-0.392952\pi$
$857$	19.3211	0.659997	0.329998	+	0.943981i	$0.392952\pi$
$858$	8.98202	0.306641
$859$	52.7084	1.79839	0.899194	−	0.437550i	$-0.144154\pi$
$859$	52.7084	1.79839	0.899194	+	0.437550i	$0.144154\pi$
$860$	−21.2824	−0.725723
$861$	−8.04967	−0.274332
$862$	9.20429	0.313499
$863$	−22.4946	−0.765725	−0.382862	−	0.923805i	$-0.625062\pi$
$863$	−22.4946	−0.765725	−0.382862	+	0.923805i	$0.625062\pi$
$864$	−1.00000	−0.0340207
$865$	−22.8392	−0.776556
$866$	−8.64704	−0.293839
$867$	−13.9906	−0.475146
$868$	3.34602	0.113571
$869$	−17.1755	−0.582640
$870$	−22.5958	−0.766070
$871$	−39.3622	−1.33374
$872$	−8.21536	−0.278207
$873$	−10.9632	−0.371049
$874$	−19.3165	−0.653389
$875$	−19.6341	−0.663755
$876$	−12.6614	−0.427791
$877$	41.3338	1.39574	0.697872	−	0.716223i	$-0.254131\pi$
$877$	41.3338	1.39574	0.697872	+	0.716223i	$0.254131\pi$
$878$	−38.6137	−1.30315
$879$	−18.0744	−0.609636
$880$	7.80659	0.263160
$881$	−30.8008	−1.03770	−0.518852	−	0.854864i	$-0.673640\pi$
$881$	−30.8008	−1.03770	−0.518852	+	0.854864i	$0.673640\pi$
$882$	−1.00000	−0.0336718
$883$	0.111616	0.00375618	0.00187809	−	0.999998i	$-0.499402\pi$
$883$	0.111616	0.00375618	0.00187809	+	0.999998i	$0.499402\pi$
$884$	7.74562	0.260514
$885$	−26.0675	−0.876249
$886$	−22.5589	−0.757881
$887$	44.6701	1.49988	0.749938	−	0.661508i	$-0.230083\pi$
$887$	44.6701	1.49988	0.749938	+	0.661508i	$0.230083\pi$
$888$	3.58791	0.120402
$889$	−11.1399	−0.373619
$890$	−14.6017	−0.489450
$891$	−2.01167	−0.0673934
$892$	9.21803	0.308642
$893$	−1.40954	−0.0471684
$894$	4.22306	0.141240
$895$	−41.9348	−1.40172
$896$	−1.00000	−0.0334077
$897$	13.7109	0.457794
$898$	−36.6078	−1.22162
$899$	−19.4828	−0.649788
$900$	10.0595	0.335316
$901$	9.77246	0.325568
$902$	−16.1933	−0.539177
$903$	5.48422	0.182503
$904$	−13.3870	−0.445244
$905$	53.9864	1.79457
$906$	−3.14434	−0.104464
$907$	−19.5894	−0.650456	−0.325228	−	0.945636i	$-0.605441\pi$
$907$	−19.5894	−0.650456	−0.325228	+	0.945636i	$0.605441\pi$
$908$	−4.06226	−0.134811
$909$	−11.3382	−0.376064
$910$	17.3270	0.574383
$911$	−0.954684	−0.0316301	−0.0158150	−	0.999875i	$-0.505034\pi$
$911$	−0.954684	−0.0316301	−0.0158150	+	0.999875i	$0.505034\pi$
$912$	6.29040	0.208296
$913$	8.18563	0.270905
$914$	4.85377	0.160548
$915$	−0.139091	−0.00459822
$916$	5.16446	0.170638
$917$	20.3822	0.673079
$918$	−1.73476	−0.0572556
$919$	41.8118	1.37924	0.689622	−	0.724170i	$-0.257777\pi$
$919$	41.8118	1.37924	0.689622	+	0.724170i	$0.257777\pi$
$920$	11.9167	0.392880
$921$	4.64898	0.153189
$922$	26.6375	0.877258
$923$	13.3778	0.440337
$924$	−2.01167	−0.0661790
$925$	−36.0925	−1.18671
$926$	8.16950	0.268466
$927$	1.02597	0.0336972
$928$	5.82268	0.191139
$929$	−46.1151	−1.51299	−0.756494	−	0.654001i	$-0.773089\pi$
$929$	−46.1151	−1.51299	−0.756494	+	0.654001i	$0.773089\pi$
$930$	12.9848	0.425787
$931$	6.29040	0.206160
$932$	22.6263	0.741148
$933$	−0.765606	−0.0250648
$934$	−35.5021	−1.16167
$935$	13.5426	0.442889
$936$	−4.46496	−0.145942
$937$	−8.46666	−0.276594	−0.138297	−	0.990391i	$-0.544163\pi$
$937$	−8.46666	−0.276594	−0.138297	+	0.990391i	$0.544163\pi$
$938$	8.81581	0.287846
$939$	−3.53875	−0.115483
$940$	0.869567	0.0283622
$941$	10.7866	0.351633	0.175817	−	0.984423i	$-0.443743\pi$
$941$	10.7866	0.351633	0.175817	+	0.984423i	$0.443743\pi$
$942$	−20.1494	−0.656503
$943$	−24.7188	−0.804955
$944$	6.71729	0.218629
$945$	−3.88066	−0.126238
$946$	11.0324	0.358695
$947$	19.3530	0.628887	0.314444	−	0.949276i	$-0.398182\pi$
$947$	19.3530	0.628887	0.314444	+	0.949276i	$0.398182\pi$
$948$	8.53794	0.277300
$949$	−56.5328	−1.83513
$950$	−63.2782	−2.05302
$951$	−9.93810	−0.322265
$952$	−1.73476	−0.0562238
$953$	25.9315	0.840004	0.420002	−	0.907523i	$-0.362029\pi$
$953$	25.9315	0.840004	0.420002	+	0.907523i	$0.362029\pi$
$954$	−5.63333	−0.182386
$955$	−3.88066	−0.125575
$956$	26.2182	0.847956
$957$	11.7133	0.378638
$958$	6.35702	0.205386
$959$	−4.33493	−0.139982
$960$	−3.88066	−0.125248
$961$	−19.8042	−0.638844
$962$	16.0198	0.516501
$963$	16.7663	0.540286
$964$	2.59835	0.0836871
$965$	60.4871	1.94715
$966$	−3.07078	−0.0988009
$967$	−15.2309	−0.489792	−0.244896	−	0.969549i	$-0.578754\pi$
$967$	−15.2309	−0.489792	−0.244896	+	0.969549i	$0.578754\pi$
$968$	6.95319	0.223484
$969$	10.9123	0.350555
$970$	−42.5445	−1.36602
$971$	37.5990	1.20661	0.603304	−	0.797511i	$-0.293851\pi$
$971$	37.5990	1.20661	0.603304	+	0.797511i	$0.293851\pi$
$972$	1.00000	0.0320750
$973$	−10.9549	−0.351199
$974$	17.6438	0.565343
$975$	44.9152	1.43844
$976$	0.0358422	0.00114728
$977$	14.0219	0.448600	0.224300	−	0.974520i	$-0.427991\pi$
$977$	14.0219	0.448600	0.224300	+	0.974520i	$0.427991\pi$
$978$	11.1335	0.356009
$979$	7.56928	0.241915
$980$	−3.88066	−0.123963
$981$	8.21536	0.262296
$982$	−28.4637	−0.908312
$983$	48.9536	1.56138	0.780688	−	0.624921i	$-0.214869\pi$
$983$	48.9536	1.56138	0.780688	+	0.624921i	$0.214869\pi$
$984$	8.04967	0.256614
$985$	−9.34918	−0.297890
$986$	10.1010	0.321680
$987$	−0.224077	−0.00713246
$988$	28.0864	0.893547
$989$	16.8409	0.535508
$990$	−7.80659	−0.248110
$991$	30.3110	0.962860	0.481430	−	0.876485i	$-0.340118\pi$
$991$	30.3110	0.962860	0.481430	+	0.876485i	$0.340118\pi$
$992$	−3.34602	−0.106236
$993$	2.48943	0.0789996
$994$	−2.99618	−0.0950332
$995$	−10.7954	−0.342238
$996$	−4.06908	−0.128934
$997$	−4.02453	−0.127458	−0.0637291	−	0.997967i	$-0.520299\pi$
$997$	−4.02453	−0.127458	−0.0637291	+	0.997967i	$0.520299\pi$
$998$	−36.5245	−1.15616
$999$	−3.58791	−0.113516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8022.2.a.z.1.2	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8022.2.a.z.1.2	✓	15	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:	\( N \)	\(=\)	\( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.88066 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.88066 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.88066 q^{10} -2.01167 q^{11} +1.00000 q^{12} +4.46496 q^{13} -1.00000 q^{14} -3.88066 q^{15} +1.00000 q^{16} +1.73476 q^{17} -1.00000 q^{18} +6.29040 q^{19} -3.88066 q^{20} +1.00000 q^{21} +2.01167 q^{22} +3.07078 q^{23} -1.00000 q^{24} +10.0595 q^{25} -4.46496 q^{26} +1.00000 q^{27} +1.00000 q^{28} -5.82268 q^{29} +3.88066 q^{30} +3.34602 q^{31} -1.00000 q^{32} -2.01167 q^{33} -1.73476 q^{34} -3.88066 q^{35} +1.00000 q^{36} -3.58791 q^{37} -6.29040 q^{38} +4.46496 q^{39} +3.88066 q^{40} -8.04967 q^{41} -1.00000 q^{42} +5.48422 q^{43} -2.01167 q^{44} -3.88066 q^{45} -3.07078 q^{46} -0.224077 q^{47} +1.00000 q^{48} +1.00000 q^{49} -10.0595 q^{50} +1.73476 q^{51} +4.46496 q^{52} +5.63333 q^{53} -1.00000 q^{54} +7.80659 q^{55} -1.00000 q^{56} +6.29040 q^{57} +5.82268 q^{58} +6.71729 q^{59} -3.88066 q^{60} +0.0358422 q^{61} -3.34602 q^{62} +1.00000 q^{63} +1.00000 q^{64} -17.3270 q^{65} +2.01167 q^{66} -8.81581 q^{67} +1.73476 q^{68} +3.07078 q^{69} +3.88066 q^{70} +2.99618 q^{71} -1.00000 q^{72} -12.6614 q^{73} +3.58791 q^{74} +10.0595 q^{75} +6.29040 q^{76} -2.01167 q^{77} -4.46496 q^{78} +8.53794 q^{79} -3.88066 q^{80} +1.00000 q^{81} +8.04967 q^{82} -4.06908 q^{83} +1.00000 q^{84} -6.73200 q^{85} -5.48422 q^{86} -5.82268 q^{87} +2.01167 q^{88} -3.76268 q^{89} +3.88066 q^{90} +4.46496 q^{91} +3.07078 q^{92} +3.34602 q^{93} +0.224077 q^{94} -24.4109 q^{95} -1.00000 q^{96} -10.9632 q^{97} -1.00000 q^{98} -2.01167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9	\( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 + 15 * q^3 + 15 * q^4 - 15 * q^6 + 15 * q^7 - 15 * q^8 + 15 * q^9 - 7 * q^11 + 15 * q^12 + 10 * q^13 - 15 * q^14 + 15 * q^16 - 3 * q^17 - 15 * q^18 + 12 * q^19 + 15 * q^21 + 7 * q^22 - q^23 - 15 * q^24 + 31 * q^25 - 10 * q^26 + 15 * q^27 + 15 * q^28 - 3 * q^29 + 19 * q^31 - 15 * q^32 - 7 * q^33 + 3 * q^34 + 15 * q^36 + 25 * q^37 - 12 * q^38 + 10 * q^39 + 8 * q^41 - 15 * q^42 + 25 * q^43 - 7 * q^44 + q^46 + 11 * q^47 + 15 * q^48 + 15 * q^49 - 31 * q^50 - 3 * q^51 + 10 * q^52 - 4 * q^53 - 15 * q^54 + 9 * q^55 - 15 * q^56 + 12 * q^57 + 3 * q^58 + 27 * q^61 - 19 * q^62 + 15 * q^63 + 15 * q^64 - 2 * q^65 + 7 * q^66 + 31 * q^67 - 3 * q^68 - q^69 - 16 * q^71 - 15 * q^72 + 26 * q^73 - 25 * q^74 + 31 * q^75 + 12 * q^76 - 7 * q^77 - 10 * q^78 + 32 * q^79 + 15 * q^81 - 8 * q^82 + 7 * q^83 + 15 * q^84 + 26 * q^85 - 25 * q^86 - 3 * q^87 + 7 * q^88 - 11 * q^89 + 10 * q^91 - q^92 + 19 * q^93 - 11 * q^94 - 8 * q^95 - 15 * q^96 + 30 * q^97 - 15 * q^98 - 7 * q^99

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