LMFDB - Embedded newform 6045.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 15x^{10} - x^{9} + 81x^{8} + 9x^{7} - 192x^{6} - 27x^{5} + 197x^{4} + 28x^{3} - 82x^{2} - 10x + 10 $ x^12 - 15x^10 - x^9 + 81x^8 + 9x^7 - 192x^6 - 27x^5 + 197x^4 + 28x^3 - 82x^2 - 10*x + 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51257$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51257 q^{2} -1.00000 q^{3} +4.31301 q^{4} -1.00000 q^{5} +2.51257 q^{6} -1.14239 q^{7} -5.81161 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.51257 q^{2} -1.00000 q^{3} +4.31301 q^{4} -1.00000 q^{5} +2.51257 q^{6} -1.14239 q^{7} -5.81161 q^{8} +1.00000 q^{9} +2.51257 q^{10} -0.654597 q^{11} -4.31301 q^{12} +1.00000 q^{13} +2.87033 q^{14} +1.00000 q^{15} +5.97606 q^{16} +3.63894 q^{17} -2.51257 q^{18} -8.27624 q^{19} -4.31301 q^{20} +1.14239 q^{21} +1.64472 q^{22} +7.99619 q^{23} +5.81161 q^{24} +1.00000 q^{25} -2.51257 q^{26} -1.00000 q^{27} -4.92713 q^{28} -6.82759 q^{29} -2.51257 q^{30} +1.00000 q^{31} -3.39205 q^{32} +0.654597 q^{33} -9.14310 q^{34} +1.14239 q^{35} +4.31301 q^{36} -6.60111 q^{37} +20.7946 q^{38} -1.00000 q^{39} +5.81161 q^{40} +7.16025 q^{41} -2.87033 q^{42} +3.63290 q^{43} -2.82329 q^{44} -1.00000 q^{45} -20.0910 q^{46} +3.87618 q^{47} -5.97606 q^{48} -5.69495 q^{49} -2.51257 q^{50} -3.63894 q^{51} +4.31301 q^{52} +0.397732 q^{53} +2.51257 q^{54} +0.654597 q^{55} +6.63911 q^{56} +8.27624 q^{57} +17.1548 q^{58} -3.13260 q^{59} +4.31301 q^{60} +4.55654 q^{61} -2.51257 q^{62} -1.14239 q^{63} -3.42935 q^{64} -1.00000 q^{65} -1.64472 q^{66} -6.66527 q^{67} +15.6948 q^{68} -7.99619 q^{69} -2.87033 q^{70} +4.12629 q^{71} -5.81161 q^{72} -7.37463 q^{73} +16.5858 q^{74} -1.00000 q^{75} -35.6955 q^{76} +0.747803 q^{77} +2.51257 q^{78} +2.64541 q^{79} -5.97606 q^{80} +1.00000 q^{81} -17.9906 q^{82} -0.00770923 q^{83} +4.92713 q^{84} -3.63894 q^{85} -9.12793 q^{86} +6.82759 q^{87} +3.80426 q^{88} +2.37701 q^{89} +2.51257 q^{90} -1.14239 q^{91} +34.4877 q^{92} -1.00000 q^{93} -9.73918 q^{94} +8.27624 q^{95} +3.39205 q^{96} +8.60832 q^{97} +14.3090 q^{98} -0.654597 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9	$ 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9} - 6 q^{11} - 6 q^{12} + 12 q^{13} + 5 q^{14} + 12 q^{15} - 6 q^{16} + 5 q^{17} - 24 q^{19} - 6 q^{20} + 7 q^{21} + q^{22} + 13 q^{23} + 3 q^{24} + 12 q^{25} - 12 q^{27} - 10 q^{28} + 3 q^{29} + 12 q^{31} - 6 q^{32} + 6 q^{33} - 15 q^{34} + 7 q^{35} + 6 q^{36} - 9 q^{37} + 16 q^{38} - 12 q^{39} + 3 q^{40} - 5 q^{41} - 5 q^{42} - 8 q^{43} + 5 q^{44} - 12 q^{45} - 2 q^{46} + 17 q^{47} + 6 q^{48} - 7 q^{49} - 5 q^{51} + 6 q^{52} + 16 q^{53} + 6 q^{55} + 17 q^{56} + 24 q^{57} + 36 q^{58} - 18 q^{59} + 6 q^{60} - 24 q^{61} - 7 q^{63} - 21 q^{64} - 12 q^{65} - q^{66} - 20 q^{67} + 23 q^{68} - 13 q^{69} - 5 q^{70} - 9 q^{71} - 3 q^{72} - 15 q^{73} + 10 q^{74} - 12 q^{75} - 30 q^{76} + 4 q^{77} - 25 q^{79} + 6 q^{80} + 12 q^{81} - 11 q^{82} - 13 q^{83} + 10 q^{84} - 5 q^{85} - 30 q^{86} - 3 q^{87} + 3 q^{88} + 35 q^{89} - 7 q^{91} + 5 q^{92} - 12 q^{93} + 4 q^{94} + 24 q^{95} + 6 q^{96} - 11 q^{97} + 16 q^{98} - 6 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9 - 6 * q^11 - 6 * q^12 + 12 * q^13 + 5 * q^14 + 12 * q^15 - 6 * q^16 + 5 * q^17 - 24 * q^19 - 6 * q^20 + 7 * q^21 + q^22 + 13 * q^23 + 3 * q^24 + 12 * q^25 - 12 * q^27 - 10 * q^28 + 3 * q^29 + 12 * q^31 - 6 * q^32 + 6 * q^33 - 15 * q^34 + 7 * q^35 + 6 * q^36 - 9 * q^37 + 16 * q^38 - 12 * q^39 + 3 * q^40 - 5 * q^41 - 5 * q^42 - 8 * q^43 + 5 * q^44 - 12 * q^45 - 2 * q^46 + 17 * q^47 + 6 * q^48 - 7 * q^49 - 5 * q^51 + 6 * q^52 + 16 * q^53 + 6 * q^55 + 17 * q^56 + 24 * q^57 + 36 * q^58 - 18 * q^59 + 6 * q^60 - 24 * q^61 - 7 * q^63 - 21 * q^64 - 12 * q^65 - q^66 - 20 * q^67 + 23 * q^68 - 13 * q^69 - 5 * q^70 - 9 * q^71 - 3 * q^72 - 15 * q^73 + 10 * q^74 - 12 * q^75 - 30 * q^76 + 4 * q^77 - 25 * q^79 + 6 * q^80 + 12 * q^81 - 11 * q^82 - 13 * q^83 + 10 * q^84 - 5 * q^85 - 30 * q^86 - 3 * q^87 + 3 * q^88 + 35 * q^89 - 7 * q^91 + 5 * q^92 - 12 * q^93 + 4 * q^94 + 24 * q^95 + 6 * q^96 - 11 * q^97 + 16 * q^98 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.51257	−1.77666	−0.888328	−	0.459209i	$-0.848133\pi$
$2$	−2.51257	−1.77666	−0.888328	+	0.459209i	$0.848133\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.31301	2.15651
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.51257	1.02575
$7$	−1.14239	−0.431782	−0.215891	−	0.976418i	$-0.569266\pi$
$7$	−1.14239	−0.431782	−0.215891	+	0.976418i	$0.569266\pi$
$8$	−5.81161	−2.05471
$9$	1.00000	0.333333
$10$	2.51257	0.794545
$11$	−0.654597	−0.197368	−0.0986842	−	0.995119i	$-0.531463\pi$
$11$	−0.654597	−0.197368	−0.0986842	+	0.995119i	$0.531463\pi$
$12$	−4.31301	−1.24506
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.87033	0.767127
$15$	1.00000	0.258199
$16$	5.97606	1.49401
$17$	3.63894	0.882573	0.441287	−	0.897366i	$-0.354522\pi$
$17$	3.63894	0.882573	0.441287	+	0.897366i	$0.354522\pi$
$18$	−2.51257	−0.592219
$19$	−8.27624	−1.89870	−0.949350	−	0.314221i	$-0.898257\pi$
$19$	−8.27624	−1.89870	−0.949350	+	0.314221i	$0.898257\pi$
$20$	−4.31301	−0.964419
$21$	1.14239	0.249289
$22$	1.64472	0.350656
$23$	7.99619	1.66732	0.833661	−	0.552277i	$-0.186241\pi$
$23$	7.99619	1.66732	0.833661	+	0.552277i	$0.186241\pi$
$24$	5.81161	1.18629
$25$	1.00000	0.200000
$26$	−2.51257	−0.492756
$27$	−1.00000	−0.192450
$28$	−4.92713	−0.931140
$29$	−6.82759	−1.26785	−0.633926	−	0.773394i	$-0.718558\pi$
$29$	−6.82759	−1.26785	−0.633926	+	0.773394i	$0.718558\pi$
$30$	−2.51257	−0.458731
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.39205	−0.599635
$33$	0.654597	0.113951
$34$	−9.14310	−1.56803
$35$	1.14239	0.193099
$36$	4.31301	0.718836
$37$	−6.60111	−1.08522	−0.542608	−	0.839986i	$-0.682563\pi$
$37$	−6.60111	−1.08522	−0.542608	+	0.839986i	$0.682563\pi$
$38$	20.7946	3.37334
$39$	−1.00000	−0.160128
$40$	5.81161	0.918896
$41$	7.16025	1.11824	0.559121	−	0.829086i	$-0.311139\pi$
$41$	7.16025	1.11824	0.559121	+	0.829086i	$0.311139\pi$
$42$	−2.87033	−0.442901
$43$	3.63290	0.554013	0.277006	−	0.960868i	$-0.410658\pi$
$43$	3.63290	0.554013	0.277006	+	0.960868i	$0.410658\pi$
$44$	−2.82329	−0.425626
$45$	−1.00000	−0.149071
$46$	−20.0910	−2.96226
$47$	3.87618	0.565399	0.282699	−	0.959209i	$-0.408770\pi$
$47$	3.87618	0.565399	0.282699	+	0.959209i	$0.408770\pi$
$48$	−5.97606	−0.862570
$49$	−5.69495	−0.813565
$50$	−2.51257	−0.355331
$51$	−3.63894	−0.509554
$52$	4.31301	0.598107
$53$	0.397732	0.0546327	0.0273163	−	0.999627i	$-0.491304\pi$
$53$	0.397732	0.0546327	0.0273163	+	0.999627i	$0.491304\pi$
$54$	2.51257	0.341918
$55$	0.654597	0.0882659
$56$	6.63911	0.887188
$57$	8.27624	1.09621
$58$	17.1548	2.25254
$59$	−3.13260	−0.407830	−0.203915	−	0.978989i	$-0.565367\pi$
$59$	−3.13260	−0.407830	−0.203915	+	0.978989i	$0.565367\pi$
$60$	4.31301	0.556808
$61$	4.55654	0.583405	0.291703	−	0.956509i	$-0.405778\pi$
$61$	4.55654	0.583405	0.291703	+	0.956509i	$0.405778\pi$
$62$	−2.51257	−0.319097
$63$	−1.14239	−0.143927
$64$	−3.42935	−0.428669
$65$	−1.00000	−0.124035
$66$	−1.64472	−0.202451
$67$	−6.66527	−0.814292	−0.407146	−	0.913363i	$-0.633476\pi$
$67$	−6.66527	−0.814292	−0.407146	+	0.913363i	$0.633476\pi$
$68$	15.6948	1.90328
$69$	−7.99619	−0.962629
$70$	−2.87033	−0.343070
$71$	4.12629	0.489701	0.244850	−	0.969561i	$-0.421261\pi$
$71$	4.12629	0.489701	0.244850	+	0.969561i	$0.421261\pi$
$72$	−5.81161	−0.684905
$73$	−7.37463	−0.863136	−0.431568	−	0.902080i	$-0.642039\pi$
$73$	−7.37463	−0.863136	−0.431568	+	0.902080i	$0.642039\pi$
$74$	16.5858	1.92806
$75$	−1.00000	−0.115470
$76$	−35.6955	−4.09456
$77$	0.747803	0.0852201
$78$	2.51257	0.284493
$79$	2.64541	0.297631	0.148816	−	0.988865i	$-0.452454\pi$
$79$	2.64541	0.297631	0.148816	+	0.988865i	$0.452454\pi$
$80$	−5.97606	−0.668144
$81$	1.00000	0.111111
$82$	−17.9906	−1.98673
$83$	−0.00770923	−0.000846198 0	−0.000423099	−	1.00000i	$-0.500135\pi$
$83$	−0.00770923	−0.000846198 0	−0.000423099		1.00000i	$0.500135\pi$
$84$	4.92713	0.537594
$85$	−3.63894	−0.394699
$86$	−9.12793	−0.984290
$87$	6.82759	0.731995
$88$	3.80426	0.405536
$89$	2.37701	0.251962	0.125981	−	0.992033i	$-0.459792\pi$
$89$	2.37701	0.251962	0.125981	+	0.992033i	$0.459792\pi$
$90$	2.51257	0.264848
$91$	−1.14239	−0.119755
$92$	34.4877	3.59559
$93$	−1.00000	−0.103695
$94$	−9.73918	−1.00452
$95$	8.27624	0.849124
$96$	3.39205	0.346200
$97$	8.60832	0.874042	0.437021	−	0.899451i	$-0.356034\pi$
$97$	8.60832	0.874042	0.437021	+	0.899451i	$0.356034\pi$
$98$	14.3090	1.44542
$99$	−0.654597	−0.0657895
$100$	4.31301	0.431301
$101$	16.9467	1.68626	0.843130	−	0.537710i	$-0.180710\pi$
$101$	16.9467	1.68626	0.843130	+	0.537710i	$0.180710\pi$
$102$	9.14310	0.905302
$103$	−0.347836	−0.0342733	−0.0171367	−	0.999853i	$-0.505455\pi$
$103$	−0.347836	−0.0342733	−0.0171367	+	0.999853i	$0.505455\pi$
$104$	−5.81161	−0.569875
$105$	−1.14239	−0.111486
$106$	−0.999330	−0.0970635
$107$	11.8845	1.14892	0.574460	−	0.818532i	$-0.305212\pi$
$107$	11.8845	1.14892	0.574460	+	0.818532i	$0.305212\pi$
$108$	−4.31301	−0.415020
$109$	−1.11573	−0.106868	−0.0534339	−	0.998571i	$-0.517017\pi$
$109$	−1.11573	−0.106868	−0.0534339	+	0.998571i	$0.517017\pi$
$110$	−1.64472	−0.156818
$111$	6.60111	0.626550
$112$	−6.82697	−0.645088
$113$	−9.53151	−0.896649	−0.448324	−	0.893871i	$-0.647979\pi$
$113$	−9.53151	−0.896649	−0.448324	+	0.893871i	$0.647979\pi$
$114$	−20.7946	−1.94760
$115$	−7.99619	−0.745649
$116$	−29.4475	−2.73413
$117$	1.00000	0.0924500
$118$	7.87088	0.724573
$119$	−4.15708	−0.381079
$120$	−5.81161	−0.530525
$121$	−10.5715	−0.961046
$122$	−11.4486	−1.03651
$123$	−7.16025	−0.645618
$124$	4.31301	0.387320
$125$	−1.00000	−0.0894427
$126$	2.87033	0.255709
$127$	17.8173	1.58103	0.790513	−	0.612445i	$-0.209814\pi$
$127$	17.8173	1.58103	0.790513	+	0.612445i	$0.209814\pi$
$128$	15.4006	1.36123
$129$	−3.63290	−0.319859
$130$	2.51257	0.220367
$131$	−20.2748	−1.77142	−0.885709	−	0.464241i	$-0.846327\pi$
$131$	−20.2748	−1.77142	−0.885709	+	0.464241i	$0.846327\pi$
$132$	2.82329	0.245736
$133$	9.45466	0.819823
$134$	16.7470	1.44672
$135$	1.00000	0.0860663
$136$	−21.1481	−1.81344
$137$	9.85927	0.842334	0.421167	−	0.906983i	$-0.361621\pi$
$137$	9.85927	0.842334	0.421167	+	0.906983i	$0.361621\pi$
$138$	20.0910	1.71026
$139$	0.301784	0.0255970	0.0127985	−	0.999918i	$-0.495926\pi$
$139$	0.301784	0.0255970	0.0127985	+	0.999918i	$0.495926\pi$
$140$	4.92713	0.416418
$141$	−3.87618	−0.326433
$142$	−10.3676	−0.870030
$143$	−0.654597	−0.0547402
$144$	5.97606	0.498005
$145$	6.82759	0.567001
$146$	18.5293	1.53350
$147$	5.69495	0.469712
$148$	−28.4707	−2.34028
$149$	−23.4573	−1.92170	−0.960849	−	0.277073i	$-0.910636\pi$
$149$	−23.4573	−1.92170	−0.960849	+	0.277073i	$0.910636\pi$
$150$	2.51257	0.205151
$151$	−7.03795	−0.572740	−0.286370	−	0.958119i	$-0.592449\pi$
$151$	−7.03795	−0.572740	−0.286370	+	0.958119i	$0.592449\pi$
$152$	48.0983	3.90129
$153$	3.63894	0.294191
$154$	−1.87891	−0.151407
$155$	−1.00000	−0.0803219
$156$	−4.31301	−0.345317
$157$	21.8823	1.74640	0.873198	−	0.487366i	$-0.162042\pi$
$157$	21.8823	1.74640	0.873198	+	0.487366i	$0.162042\pi$
$158$	−6.64677	−0.528789
$159$	−0.397732	−0.0315422
$160$	3.39205	0.268165
$161$	−9.13474	−0.719919
$162$	−2.51257	−0.197406
$163$	−7.80984	−0.611714	−0.305857	−	0.952078i	$-0.598943\pi$
$163$	−7.80984	−0.611714	−0.305857	+	0.952078i	$0.598943\pi$
$164$	30.8822	2.41150
$165$	−0.654597	−0.0509603
$166$	0.0193700	0.00150340
$167$	−24.5505	−1.89978	−0.949889	−	0.312587i	$-0.898805\pi$
$167$	−24.5505	−1.89978	−0.949889	+	0.312587i	$0.898805\pi$
$168$	−6.63911	−0.512218
$169$	1.00000	0.0769231
$170$	9.14310	0.701244
$171$	−8.27624	−0.632900
$172$	15.6688	1.19473
$173$	−11.2159	−0.852732	−0.426366	−	0.904551i	$-0.640206\pi$
$173$	−11.2159	−0.852732	−0.426366	+	0.904551i	$0.640206\pi$
$174$	−17.1548	−1.30050
$175$	−1.14239	−0.0863563
$176$	−3.91191	−0.294871
$177$	3.13260	0.235461
$178$	−5.97240	−0.447651
$179$	6.68788	0.499876	0.249938	−	0.968262i	$-0.419590\pi$
$179$	6.68788	0.499876	0.249938	+	0.968262i	$0.419590\pi$
$180$	−4.31301	−0.321473
$181$	−12.0477	−0.895495	−0.447748	−	0.894160i	$-0.647774\pi$
$181$	−12.0477	−0.895495	−0.447748	+	0.894160i	$0.647774\pi$
$182$	2.87033	0.212763
$183$	−4.55654	−0.336829
$184$	−46.4708	−3.42587
$185$	6.60111	0.485324
$186$	2.51257	0.184231
$187$	−2.38204	−0.174192
$188$	16.7180	1.21929
$189$	1.14239	0.0830964
$190$	−20.7946	−1.50860
$191$	−0.704196	−0.0509538	−0.0254769	−	0.999675i	$-0.508110\pi$
$191$	−0.704196	−0.0509538	−0.0254769	+	0.999675i	$0.508110\pi$
$192$	3.42935	0.247492
$193$	−0.512892	−0.0369188	−0.0184594	−	0.999830i	$-0.505876\pi$
$193$	−0.512892	−0.0369188	−0.0184594	+	0.999830i	$0.505876\pi$
$194$	−21.6290	−1.55287
$195$	1.00000	0.0716115
$196$	−24.5624	−1.75446
$197$	23.0203	1.64013	0.820063	−	0.572274i	$-0.193939\pi$
$197$	23.0203	1.64013	0.820063	+	0.572274i	$0.193939\pi$
$198$	1.64472	0.116885
$199$	−16.1227	−1.14291	−0.571454	−	0.820634i	$-0.693620\pi$
$199$	−16.1227	−1.14291	−0.571454	+	0.820634i	$0.693620\pi$
$200$	−5.81161	−0.410943
$201$	6.66527	0.470132
$202$	−42.5798	−2.99590
$203$	7.79975	0.547435
$204$	−15.6948	−1.09886
$205$	−7.16025	−0.500093
$206$	0.873964	0.0608919
$207$	7.99619	0.555774
$208$	5.97606	0.414365
$209$	5.41760	0.374743
$210$	2.87033	0.198071
$211$	17.2280	1.18603	0.593013	−	0.805193i	$-0.297938\pi$
$211$	17.2280	1.18603	0.593013	+	0.805193i	$0.297938\pi$
$212$	1.71542	0.117816
$213$	−4.12629	−0.282729
$214$	−29.8607	−2.04124
$215$	−3.63290	−0.247762
$216$	5.81161	0.395430
$217$	−1.14239	−0.0775502
$218$	2.80336	0.189867
$219$	7.37463	0.498332
$220$	2.82329	0.190346
$221$	3.63894	0.244782
$222$	−16.5858	−1.11316
$223$	17.9952	1.20505	0.602525	−	0.798100i	$-0.294161\pi$
$223$	17.9952	1.20505	0.602525	+	0.798100i	$0.294161\pi$
$224$	3.87503	0.258911
$225$	1.00000	0.0666667
$226$	23.9486	1.59304
$227$	9.35116	0.620659	0.310329	−	0.950629i	$-0.399561\pi$
$227$	9.35116	0.620659	0.310329	+	0.950629i	$0.399561\pi$
$228$	35.6955	2.36399
$229$	−16.3436	−1.08002	−0.540008	−	0.841660i	$-0.681579\pi$
$229$	−16.3436	−1.08002	−0.540008	+	0.841660i	$0.681579\pi$
$230$	20.0910	1.32476
$231$	−0.747803	−0.0492018
$232$	39.6793	2.60507
$233$	26.8381	1.75822	0.879110	−	0.476618i	$-0.158138\pi$
$233$	26.8381	1.75822	0.879110	+	0.476618i	$0.158138\pi$
$234$	−2.51257	−0.164252
$235$	−3.87618	−0.252854
$236$	−13.5109	−0.879488
$237$	−2.64541	−0.171838
$238$	10.4450	0.677046
$239$	21.3764	1.38272	0.691362	−	0.722508i	$-0.257011\pi$
$239$	21.3764	1.38272	0.691362	+	0.722508i	$0.257011\pi$
$240$	5.97606	0.385753
$241$	14.3861	0.926689	0.463344	−	0.886178i	$-0.346649\pi$
$241$	14.3861	0.926689	0.463344	+	0.886178i	$0.346649\pi$
$242$	26.5617	1.70745
$243$	−1.00000	−0.0641500
$244$	19.6524	1.25812
$245$	5.69495	0.363837
$246$	17.9906	1.14704
$247$	−8.27624	−0.526605
$248$	−5.81161	−0.369038
$249$	0.00770923	0.000488553 0
$250$	2.51257	0.158909
$251$	−13.5782	−0.857049	−0.428524	−	0.903530i	$-0.640966\pi$
$251$	−13.5782	−0.857049	−0.428524	+	0.903530i	$0.640966\pi$
$252$	−4.92713	−0.310380
$253$	−5.23429	−0.329077
$254$	−44.7671	−2.80894
$255$	3.63894	0.227879
$256$	−31.8364	−1.98977
$257$	28.5701	1.78216	0.891078	−	0.453849i	$-0.149950\pi$
$257$	28.5701	1.78216	0.891078	+	0.453849i	$0.149950\pi$
$258$	9.12793	0.568280
$259$	7.54102	0.468576
$260$	−4.31301	−0.267482
$261$	−6.82759	−0.422617
$262$	50.9419	3.14720
$263$	−24.7543	−1.52641	−0.763206	−	0.646155i	$-0.776376\pi$
$263$	−24.7543	−1.52641	−0.763206	+	0.646155i	$0.776376\pi$
$264$	−3.80426	−0.234136
$265$	−0.397732	−0.0244325
$266$	−23.7555	−1.45654
$267$	−2.37701	−0.145471
$268$	−28.7474	−1.75603
$269$	31.9229	1.94637	0.973186	−	0.230020i	$-0.0738791\pi$
$269$	31.9229	1.94637	0.973186	+	0.230020i	$0.0738791\pi$
$270$	−2.51257	−0.152910
$271$	−9.19807	−0.558743	−0.279371	−	0.960183i	$-0.590126\pi$
$271$	−9.19807	−0.558743	−0.279371	+	0.960183i	$0.590126\pi$
$272$	21.7465	1.31858
$273$	1.14239	0.0691404
$274$	−24.7721	−1.49654
$275$	−0.654597	−0.0394737
$276$	−34.4877	−2.07591
$277$	−22.0031	−1.32204	−0.661020	−	0.750368i	$-0.729876\pi$
$277$	−22.0031	−1.32204	−0.661020	+	0.750368i	$0.729876\pi$
$278$	−0.758254	−0.0454771
$279$	1.00000	0.0598684
$280$	−6.63911	−0.396762
$281$	31.0026	1.84946	0.924729	−	0.380626i	$-0.124292\pi$
$281$	31.0026	1.84946	0.924729	+	0.380626i	$0.124292\pi$
$282$	9.73918	0.579960
$283$	23.7506	1.41183	0.705913	−	0.708299i	$-0.250537\pi$
$283$	23.7506	1.41183	0.705913	+	0.708299i	$0.250537\pi$
$284$	17.7968	1.05604
$285$	−8.27624	−0.490242
$286$	1.64472	0.0972545
$287$	−8.17977	−0.482837
$288$	−3.39205	−0.199878
$289$	−3.75809	−0.221064
$290$	−17.1548	−1.00737
$291$	−8.60832	−0.504628
$292$	−31.8069	−1.86136
$293$	−15.3130	−0.894595	−0.447297	−	0.894385i	$-0.647613\pi$
$293$	−15.3130	−0.894595	−0.447297	+	0.894385i	$0.647613\pi$
$294$	−14.3090	−0.834516
$295$	3.13260	0.182387
$296$	38.3631	2.22981
$297$	0.654597	0.0379836
$298$	58.9382	3.41420
$299$	7.99619	0.462432
$300$	−4.31301	−0.249012
$301$	−4.15018	−0.239212
$302$	17.6833	1.01756
$303$	−16.9467	−0.973562
$304$	−49.4593	−2.83668
$305$	−4.55654	−0.260907
$306$	−9.14310	−0.522676
$307$	9.68617	0.552819	0.276410	−	0.961040i	$-0.410855\pi$
$307$	9.68617	0.552819	0.276410	+	0.961040i	$0.410855\pi$
$308$	3.22528	0.183778
$309$	0.347836	0.0197877
$310$	2.51257	0.142704
$311$	−30.2809	−1.71707	−0.858536	−	0.512753i	$-0.828626\pi$
$311$	−30.2809	−1.71707	−0.858536	+	0.512753i	$0.828626\pi$
$312$	5.81161	0.329018
$313$	14.0953	0.796716	0.398358	−	0.917230i	$-0.369580\pi$
$313$	14.0953	0.796716	0.398358	+	0.917230i	$0.369580\pi$
$314$	−54.9808	−3.10274
$315$	1.14239	0.0643662
$316$	11.4097	0.641844
$317$	−25.4201	−1.42773	−0.713866	−	0.700282i	$-0.753058\pi$
$317$	−25.4201	−1.42773	−0.713866	+	0.700282i	$0.753058\pi$
$318$	0.999330	0.0560396
$319$	4.46932	0.250234
$320$	3.42935	0.191707
$321$	−11.8845	−0.663330
$322$	22.9517	1.27905
$323$	−30.1168	−1.67574
$324$	4.31301	0.239612
$325$	1.00000	0.0554700
$326$	19.6228	1.08680
$327$	1.11573	0.0617001
$328$	−41.6126	−2.29767
$329$	−4.42810	−0.244129
$330$	1.64472	0.0905390
$331$	2.46611	0.135550	0.0677748	−	0.997701i	$-0.478410\pi$
$331$	2.46611	0.135550	0.0677748	+	0.997701i	$0.478410\pi$
$332$	−0.0332500	−0.00182483
$333$	−6.60111	−0.361739
$334$	61.6850	3.37525
$335$	6.66527	0.364162
$336$	6.82697	0.372442
$337$	19.8032	1.07875	0.539373	−	0.842067i	$-0.318661\pi$
$337$	19.8032	1.07875	0.539373	+	0.842067i	$0.318661\pi$
$338$	−2.51257	−0.136666
$339$	9.53151	0.517680
$340$	−15.6948	−0.851171
$341$	−0.654597	−0.0354484
$342$	20.7946	1.12445
$343$	14.5025	0.783064
$344$	−21.1130	−1.13834
$345$	7.99619	0.430501
$346$	28.1808	1.51501
$347$	−14.8861	−0.799126	−0.399563	−	0.916706i	$-0.630838\pi$
$347$	−14.8861	−0.799126	−0.399563	+	0.916706i	$0.630838\pi$
$348$	29.4475	1.57855
$349$	−3.73308	−0.199827	−0.0999135	−	0.994996i	$-0.531857\pi$
$349$	−3.73308	−0.199827	−0.0999135	+	0.994996i	$0.531857\pi$
$350$	2.87033	0.153425
$351$	−1.00000	−0.0533761
$352$	2.22043	0.118349
$353$	−23.0671	−1.22774	−0.613869	−	0.789408i	$-0.710388\pi$
$353$	−23.0671	−1.22774	−0.613869	+	0.789408i	$0.710388\pi$
$354$	−7.87088	−0.418333
$355$	−4.12629	−0.219001
$356$	10.2521	0.543359
$357$	4.15708	0.220016
$358$	−16.8038	−0.888107
$359$	2.87601	0.151790	0.0758951	−	0.997116i	$-0.475819\pi$
$359$	2.87601	0.151790	0.0758951	+	0.997116i	$0.475819\pi$
$360$	5.81161	0.306299
$361$	49.4961	2.60506
$362$	30.2706	1.59099
$363$	10.5715	0.554860
$364$	−4.92713	−0.258252
$365$	7.37463	0.386006
$366$	11.4486	0.598430
$367$	−24.3383	−1.27045	−0.635225	−	0.772327i	$-0.719093\pi$
$367$	−24.3383	−1.27045	−0.635225	+	0.772327i	$0.719093\pi$
$368$	47.7857	2.49100
$369$	7.16025	0.372748
$370$	−16.5858	−0.862253
$371$	−0.454364	−0.0235894
$372$	−4.31301	−0.223619
$373$	−16.7108	−0.865252	−0.432626	−	0.901573i	$-0.642413\pi$
$373$	−16.7108	−0.865252	−0.432626	+	0.901573i	$0.642413\pi$
$374$	5.98505	0.309480
$375$	1.00000	0.0516398
$376$	−22.5268	−1.16173
$377$	−6.82759	−0.351639
$378$	−2.87033	−0.147634
$379$	−18.2110	−0.935438	−0.467719	−	0.883877i	$-0.654924\pi$
$379$	−18.2110	−0.935438	−0.467719	+	0.883877i	$0.654924\pi$
$380$	35.6955	1.83114
$381$	−17.8173	−0.912806
$382$	1.76934	0.0905274
$383$	−11.4541	−0.585277	−0.292639	−	0.956223i	$-0.594533\pi$
$383$	−11.4541	−0.585277	−0.292639	+	0.956223i	$0.594533\pi$
$384$	−15.4006	−0.785908
$385$	−0.747803	−0.0381116
$386$	1.28868	0.0655919
$387$	3.63290	0.184671
$388$	37.1278	1.88488
$389$	18.8185	0.954138	0.477069	−	0.878866i	$-0.341699\pi$
$389$	18.8185	0.954138	0.477069	+	0.878866i	$0.341699\pi$
$390$	−2.51257	−0.127229
$391$	29.0977	1.47153
$392$	33.0969	1.67164
$393$	20.2748	1.02273
$394$	−57.8400	−2.91394
$395$	−2.64541	−0.133105
$396$	−2.82329	−0.141875
$397$	−7.85174	−0.394068	−0.197034	−	0.980397i	$-0.563131\pi$
$397$	−7.85174	−0.394068	−0.197034	+	0.980397i	$0.563131\pi$
$398$	40.5094	2.03055
$399$	−9.45466	−0.473325
$400$	5.97606	0.298803
$401$	24.2561	1.21129	0.605647	−	0.795734i	$-0.292914\pi$
$401$	24.2561	1.21129	0.605647	+	0.795734i	$0.292914\pi$
$402$	−16.7470	−0.835262
$403$	1.00000	0.0498135
$404$	73.0913	3.63643
$405$	−1.00000	−0.0496904
$406$	−19.5974	−0.972604
$407$	4.32107	0.214188
$408$	21.1481	1.04699
$409$	26.9614	1.33316	0.666578	−	0.745435i	$-0.267758\pi$
$409$	26.9614	1.33316	0.666578	+	0.745435i	$0.267758\pi$
$410$	17.9906	0.888494
$411$	−9.85927	−0.486322
$412$	−1.50022	−0.0739107
$413$	3.57864	0.176093
$414$	−20.0910	−0.987419
$415$	0.00770923	0.000378431 0
$416$	−3.39205	−0.166309
$417$	−0.301784	−0.0147784
$418$	−13.6121	−0.665790
$419$	−18.7922	−0.918060	−0.459030	−	0.888421i	$-0.651803\pi$
$419$	−18.7922	−0.918060	−0.459030	+	0.888421i	$0.651803\pi$
$420$	−4.92713	−0.240419
$421$	3.38663	0.165054	0.0825271	−	0.996589i	$-0.473701\pi$
$421$	3.38663	0.165054	0.0825271	+	0.996589i	$0.473701\pi$
$422$	−43.2866	−2.10716
$423$	3.87618	0.188466
$424$	−2.31146	−0.112255
$425$	3.63894	0.176515
$426$	10.3676	0.502312
$427$	−5.20533	−0.251904
$428$	51.2581	2.47766
$429$	0.654597	0.0316043
$430$	9.12793	0.440188
$431$	30.5332	1.47073	0.735366	−	0.677670i	$-0.237010\pi$
$431$	30.5332	1.47073	0.735366	+	0.677670i	$0.237010\pi$
$432$	−5.97606	−0.287523
$433$	−11.7591	−0.565105	−0.282552	−	0.959252i	$-0.591181\pi$
$433$	−11.7591	−0.565105	−0.282552	+	0.959252i	$0.591181\pi$
$434$	2.87033	0.137780
$435$	−6.82759	−0.327358
$436$	−4.81217	−0.230461
$437$	−66.1784	−3.16574
$438$	−18.5293	−0.885364
$439$	−20.4264	−0.974901	−0.487451	−	0.873151i	$-0.662073\pi$
$439$	−20.4264	−0.974901	−0.487451	+	0.873151i	$0.662073\pi$
$440$	−3.80426	−0.181361
$441$	−5.69495	−0.271188
$442$	−9.14310	−0.434893
$443$	−17.1540	−0.815012	−0.407506	−	0.913203i	$-0.633601\pi$
$443$	−17.1540	−0.815012	−0.407506	+	0.913203i	$0.633601\pi$
$444$	28.4707	1.35116
$445$	−2.37701	−0.112681
$446$	−45.2143	−2.14096
$447$	23.4573	1.10949
$448$	3.91764	0.185091
$449$	−19.2802	−0.909890	−0.454945	−	0.890520i	$-0.650341\pi$
$449$	−19.2802	−0.909890	−0.454945	+	0.890520i	$0.650341\pi$
$450$	−2.51257	−0.118444
$451$	−4.68708	−0.220706
$452$	−41.1095	−1.93363
$453$	7.03795	0.330672
$454$	−23.4955	−1.10270
$455$	1.14239	0.0535559
$456$	−48.0983	−2.25241
$457$	−4.32058	−0.202108	−0.101054	−	0.994881i	$-0.532221\pi$
$457$	−4.32058	−0.202108	−0.101054	+	0.994881i	$0.532221\pi$
$458$	41.0645	1.91882
$459$	−3.63894	−0.169851
$460$	−34.4877	−1.60800
$461$	17.4359	0.812071	0.406036	−	0.913857i	$-0.366911\pi$
$461$	17.4359	0.812071	0.406036	+	0.913857i	$0.366911\pi$
$462$	1.87891	0.0874147
$463$	−11.0473	−0.513410	−0.256705	−	0.966490i	$-0.582637\pi$
$463$	−11.0473	−0.513410	−0.256705	+	0.966490i	$0.582637\pi$
$464$	−40.8021	−1.89419
$465$	1.00000	0.0463739
$466$	−67.4326	−3.12375
$467$	−27.1553	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$467$	−27.1553	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$468$	4.31301	0.199369
$469$	7.61431	0.351596
$470$	9.73918	0.449235
$471$	−21.8823	−1.00828
$472$	18.2054	0.837974
$473$	−2.37809	−0.109345
$474$	6.64677	0.305296
$475$	−8.27624	−0.379740
$476$	−17.9295	−0.821799
$477$	0.397732	0.0182109
$478$	−53.7097	−2.45663
$479$	−7.96072	−0.363735	−0.181867	−	0.983323i	$-0.558214\pi$
$479$	−7.96072	−0.363735	−0.181867	+	0.983323i	$0.558214\pi$
$480$	−3.39205	−0.154825
$481$	−6.60111	−0.300985
$482$	−36.1460	−1.64641
$483$	9.13474	0.415645
$484$	−45.5950	−2.07250
$485$	−8.60832	−0.390883
$486$	2.51257	0.113973
$487$	−0.203485	−0.00922077	−0.00461039	−	0.999989i	$-0.501468\pi$
$487$	−0.203485	−0.00922077	−0.00461039	+	0.999989i	$0.501468\pi$
$488$	−26.4808	−1.19873
$489$	7.80984	0.353173
$490$	−14.3090	−0.646414
$491$	−25.6886	−1.15931	−0.579655	−	0.814862i	$-0.696813\pi$
$491$	−25.6886	−1.15931	−0.579655	+	0.814862i	$0.696813\pi$
$492$	−30.8822	−1.39228
$493$	−24.8452	−1.11897
$494$	20.7946	0.935595
$495$	0.654597	0.0294220
$496$	5.97606	0.268333
$497$	−4.71382	−0.211444
$498$	−0.0193700	−0.000867990 0
$499$	−15.0804	−0.675090	−0.337545	−	0.941309i	$-0.609596\pi$
$499$	−15.0804	−0.675090	−0.337545	+	0.941309i	$0.609596\pi$
$500$	−4.31301	−0.192884
$501$	24.5505	1.09684
$502$	34.1162	1.52268
$503$	7.01530	0.312797	0.156398	−	0.987694i	$-0.450012\pi$
$503$	7.01530	0.312797	0.156398	+	0.987694i	$0.450012\pi$
$504$	6.63911	0.295729
$505$	−16.9467	−0.754118
$506$	13.1515	0.584656
$507$	−1.00000	−0.0444116
$508$	76.8461	3.40949
$509$	7.17163	0.317877	0.158938	−	0.987288i	$-0.449193\pi$
$509$	7.17163	0.317877	0.158938	+	0.987288i	$0.449193\pi$
$510$	−9.14310	−0.404863
$511$	8.42468	0.372686
$512$	49.1900	2.17391
$513$	8.27624	0.365405
$514$	−71.7845	−3.16628
$515$	0.347836	0.0153275
$516$	−15.6688	−0.689779
$517$	−2.53734	−0.111592
$518$	−18.9474	−0.832499
$519$	11.2159	0.492325
$520$	5.81161	0.254856
$521$	5.56533	0.243821	0.121911	−	0.992541i	$-0.461098\pi$
$521$	5.56533	0.243821	0.121911	+	0.992541i	$0.461098\pi$
$522$	17.1548	0.750846
$523$	−6.80377	−0.297508	−0.148754	−	0.988874i	$-0.547526\pi$
$523$	−6.80377	−0.297508	−0.148754	+	0.988874i	$0.547526\pi$
$524$	−87.4455	−3.82007
$525$	1.14239	0.0498578
$526$	62.1968	2.71191
$527$	3.63894	0.158515
$528$	3.91191	0.170244
$529$	40.9391	1.77996
$530$	0.999330	0.0434081
$531$	−3.13260	−0.135943
$532$	40.7781	1.76795
$533$	7.16025	0.310145
$534$	5.97240	0.258451
$535$	−11.8845	−0.513813
$536$	38.7359	1.67314
$537$	−6.68788	−0.288603
$538$	−80.2085	−3.45803
$539$	3.72790	0.160572
$540$	4.31301	0.185603
$541$	−29.2164	−1.25611	−0.628056	−	0.778168i	$-0.716149\pi$
$541$	−29.2164	−1.25611	−0.628056	+	0.778168i	$0.716149\pi$
$542$	23.1108	0.992694
$543$	12.0477	0.517014
$544$	−12.3435	−0.529222
$545$	1.11573	0.0477927
$546$	−2.87033	−0.122839
$547$	−9.52734	−0.407359	−0.203680	−	0.979038i	$-0.565290\pi$
$547$	−9.52734	−0.407359	−0.203680	+	0.979038i	$0.565290\pi$
$548$	42.5232	1.81650
$549$	4.55654	0.194468
$550$	1.64472	0.0701312
$551$	56.5068	2.40727
$552$	46.4708	1.97793
$553$	−3.02208	−0.128512
$554$	55.2844	2.34881
$555$	−6.60111	−0.280202
$556$	1.30160	0.0552001
$557$	−12.4338	−0.526839	−0.263420	−	0.964681i	$-0.584850\pi$
$557$	−12.4338	−0.526839	−0.263420	+	0.964681i	$0.584850\pi$
$558$	−2.51257	−0.106366
$559$	3.63290	0.153655
$560$	6.82697	0.288492
$561$	2.38204	0.100570
$562$	−77.8961	−3.28585
$563$	−42.1876	−1.77800	−0.888999	−	0.457910i	$-0.848598\pi$
$563$	−42.1876	−1.77800	−0.888999	+	0.457910i	$0.848598\pi$
$564$	−16.7180	−0.703955
$565$	9.53151	0.400994
$566$	−59.6750	−2.50833
$567$	−1.14239	−0.0479757
$568$	−23.9804	−1.00620
$569$	−41.8012	−1.75240	−0.876199	−	0.481950i	$-0.839929\pi$
$569$	−41.8012	−1.75240	−0.876199	+	0.481950i	$0.839929\pi$
$570$	20.7946	0.870992
$571$	−4.55328	−0.190549	−0.0952744	−	0.995451i	$-0.530373\pi$
$571$	−4.55328	−0.190549	−0.0952744	+	0.995451i	$0.530373\pi$
$572$	−2.82329	−0.118048
$573$	0.704196	0.0294182
$574$	20.5523	0.857834
$575$	7.99619	0.333464
$576$	−3.42935	−0.142890
$577$	10.5237	0.438105	0.219053	−	0.975713i	$-0.429703\pi$
$577$	10.5237	0.438105	0.219053	+	0.975713i	$0.429703\pi$
$578$	9.44248	0.392755
$579$	0.512892	0.0213151
$580$	29.4475	1.22274
$581$	0.00880692	0.000365373 0
$582$	21.6290	0.896551
$583$	−0.260354	−0.0107828
$584$	42.8585	1.77350
$585$	−1.00000	−0.0413449
$586$	38.4750	1.58939
$587$	33.9675	1.40199	0.700994	−	0.713168i	$-0.252740\pi$
$587$	33.9675	1.40199	0.700994	+	0.713168i	$0.252740\pi$
$588$	24.5624	1.01294
$589$	−8.27624	−0.341017
$590$	−7.87088	−0.324039
$591$	−23.0203	−0.946927
$592$	−39.4486	−1.62133
$593$	−3.35044	−0.137586	−0.0687931	−	0.997631i	$-0.521915\pi$
$593$	−3.35044	−0.137586	−0.0687931	+	0.997631i	$0.521915\pi$
$594$	−1.64472	−0.0674838
$595$	4.15708	0.170424
$596$	−101.172	−4.14415
$597$	16.1227	0.659858
$598$	−20.0910	−0.821582
$599$	−26.7457	−1.09280	−0.546400	−	0.837524i	$-0.684002\pi$
$599$	−26.7457	−1.09280	−0.546400	+	0.837524i	$0.684002\pi$
$600$	5.81161	0.237258
$601$	−17.0499	−0.695480	−0.347740	−	0.937591i	$-0.613051\pi$
$601$	−17.0499	−0.695480	−0.347740	+	0.937591i	$0.613051\pi$
$602$	10.4276	0.424998
$603$	−6.66527	−0.271431
$604$	−30.3548	−1.23512
$605$	10.5715	0.429793
$606$	42.5798	1.72969
$607$	10.5740	0.429184	0.214592	−	0.976704i	$-0.431158\pi$
$607$	10.5740	0.429184	0.214592	+	0.976704i	$0.431158\pi$
$608$	28.0734	1.13853
$609$	−7.79975	−0.316062
$610$	11.4486	0.463542
$611$	3.87618	0.156813
$612$	15.6948	0.634425
$613$	−8.28401	−0.334588	−0.167294	−	0.985907i	$-0.553503\pi$
$613$	−8.28401	−0.334588	−0.167294	+	0.985907i	$0.553503\pi$
$614$	−24.3372	−0.982169
$615$	7.16025	0.288729
$616$	−4.34594	−0.175103
$617$	−33.2342	−1.33796	−0.668980	−	0.743280i	$-0.733269\pi$
$617$	−33.2342	−1.33796	−0.668980	+	0.743280i	$0.733269\pi$
$618$	−0.873964	−0.0351560
$619$	−4.76162	−0.191386	−0.0956929	−	0.995411i	$-0.530507\pi$
$619$	−4.76162	−0.191386	−0.0956929	+	0.995411i	$0.530507\pi$
$620$	−4.31301	−0.173215
$621$	−7.99619	−0.320876
$622$	76.0829	3.05065
$623$	−2.71546	−0.108793
$624$	−5.97606	−0.239234
$625$	1.00000	0.0400000
$626$	−35.4156	−1.41549
$627$	−5.41760	−0.216358
$628$	94.3785	3.76611
$629$	−24.0211	−0.957783
$630$	−2.87033	−0.114357
$631$	−42.2443	−1.68172	−0.840860	−	0.541252i	$-0.817950\pi$
$631$	−42.2443	−1.68172	−0.840860	+	0.541252i	$0.817950\pi$
$632$	−15.3741	−0.611548
$633$	−17.2280	−0.684752
$634$	63.8697	2.53659
$635$	−17.8173	−0.707057
$636$	−1.71542	−0.0680210
$637$	−5.69495	−0.225642
$638$	−11.2295	−0.444580
$639$	4.12629	0.163234
$640$	−15.4006	−0.608762
$641$	−2.82403	−0.111542	−0.0557712	−	0.998444i	$-0.517762\pi$
$641$	−2.82403	−0.111542	−0.0557712	+	0.998444i	$0.517762\pi$
$642$	29.8607	1.17851
$643$	−28.3380	−1.11754	−0.558772	−	0.829322i	$-0.688727\pi$
$643$	−28.3380	−1.11754	−0.558772	+	0.829322i	$0.688727\pi$
$644$	−39.3983	−1.55251
$645$	3.63290	0.143045
$646$	75.6705	2.97722
$647$	−40.4453	−1.59007	−0.795034	−	0.606565i	$-0.792547\pi$
$647$	−40.4453	−1.59007	−0.795034	+	0.606565i	$0.792547\pi$
$648$	−5.81161	−0.228302
$649$	2.05059	0.0804927
$650$	−2.51257	−0.0985511
$651$	1.14239	0.0447737
$652$	−33.6839	−1.31916
$653$	40.2907	1.57670	0.788349	−	0.615228i	$-0.210936\pi$
$653$	40.2907	1.57670	0.788349	+	0.615228i	$0.210936\pi$
$654$	−2.80336	−0.109620
$655$	20.2748	0.792202
$656$	42.7901	1.67067
$657$	−7.37463	−0.287712
$658$	11.1259	0.433733
$659$	−29.4446	−1.14700	−0.573499	−	0.819206i	$-0.694414\pi$
$659$	−29.4446	−1.14700	−0.573499	+	0.819206i	$0.694414\pi$
$660$	−2.82329	−0.109896
$661$	15.2487	0.593107	0.296553	−	0.955016i	$-0.404163\pi$
$661$	15.2487	0.593107	0.296553	+	0.955016i	$0.404163\pi$
$662$	−6.19627	−0.240825
$663$	−3.63894	−0.141325
$664$	0.0448031	0.00173870
$665$	−9.45466	−0.366636
$666$	16.5858	0.642685
$667$	−54.5947	−2.11392
$668$	−105.887	−4.09689
$669$	−17.9952	−0.695736
$670$	−16.7470	−0.646991
$671$	−2.98270	−0.115146
$672$	−3.87503	−0.149483
$673$	−39.0239	−1.50426	−0.752130	−	0.659015i	$-0.770973\pi$
$673$	−39.0239	−1.50426	−0.752130	+	0.659015i	$0.770973\pi$
$674$	−49.7568	−1.91656
$675$	−1.00000	−0.0384900
$676$	4.31301	0.165885
$677$	21.9177	0.842365	0.421182	−	0.906976i	$-0.361615\pi$
$677$	21.9177	0.842365	0.421182	+	0.906976i	$0.361615\pi$
$678$	−23.9486	−0.919740
$679$	−9.83402	−0.377395
$680$	21.1481	0.810993
$681$	−9.35116	−0.358337
$682$	1.64472	0.0629797
$683$	−10.8514	−0.415217	−0.207609	−	0.978212i	$-0.566568\pi$
$683$	−10.8514	−0.415217	−0.207609	+	0.978212i	$0.566568\pi$
$684$	−35.6955	−1.36485
$685$	−9.85927	−0.376703
$686$	−36.4387	−1.39123
$687$	16.3436	0.623548
$688$	21.7104	0.827703
$689$	0.397732	0.0151524
$690$	−20.0910	−0.764851
$691$	−27.4876	−1.04568	−0.522838	−	0.852432i	$-0.675127\pi$
$691$	−27.4876	−1.04568	−0.522838	+	0.852432i	$0.675127\pi$
$692$	−48.3745	−1.83892
$693$	0.747803	0.0284067
$694$	37.4023	1.41977
$695$	−0.301784	−0.0114473
$696$	−39.6793	−1.50404
$697$	26.0557	0.986931
$698$	9.37962	0.355024
$699$	−26.8381	−1.01511
$700$	−4.92713	−0.186228
$701$	15.8439	0.598417	0.299209	−	0.954188i	$-0.403277\pi$
$701$	15.8439	0.598417	0.299209	+	0.954188i	$0.403277\pi$
$702$	2.51257	0.0948309
$703$	54.6324	2.06050
$704$	2.24484	0.0846057
$705$	3.87618	0.145985
$706$	57.9577	2.18127
$707$	−19.3597	−0.728096
$708$	13.5109	0.507772
$709$	−18.3180	−0.687946	−0.343973	−	0.938980i	$-0.611773\pi$
$709$	−18.3180	−0.687946	−0.343973	+	0.938980i	$0.611773\pi$
$710$	10.3676	0.389089
$711$	2.64541	0.0992105
$712$	−13.8143	−0.517711
$713$	7.99619	0.299460
$714$	−10.4450	−0.390893
$715$	0.654597	0.0244805
$716$	28.8449	1.07799
$717$	−21.3764	−0.798317
$718$	−7.22619	−0.269679
$719$	−41.4172	−1.54460	−0.772301	−	0.635257i	$-0.780894\pi$
$719$	−41.4172	−1.54460	−0.772301	+	0.635257i	$0.780894\pi$
$720$	−5.97606	−0.222715
$721$	0.397364	0.0147986
$722$	−124.363	−4.62830
$723$	−14.3861	−0.535024
$724$	−51.9617	−1.93114
$725$	−6.82759	−0.253570
$726$	−26.5617	−0.985795
$727$	−13.1228	−0.486699	−0.243350	−	0.969939i	$-0.578246\pi$
$727$	−13.1228	−0.486699	−0.243350	+	0.969939i	$0.578246\pi$
$728$	6.63911	0.246062
$729$	1.00000	0.0370370
$730$	−18.5293	−0.685800
$731$	13.2199	0.488957
$732$	−19.6524	−0.726375
$733$	−17.3246	−0.639897	−0.319949	−	0.947435i	$-0.603666\pi$
$733$	−17.3246	−0.639897	−0.319949	+	0.947435i	$0.603666\pi$
$734$	61.1517	2.25715
$735$	−5.69495	−0.210062
$736$	−27.1235	−0.999785
$737$	4.36306	0.160716
$738$	−17.9906	−0.662244
$739$	−8.47802	−0.311869	−0.155935	−	0.987767i	$-0.549839\pi$
$739$	−8.47802	−0.311869	−0.155935	+	0.987767i	$0.549839\pi$
$740$	28.4707	1.04660
$741$	8.27624	0.304035
$742$	1.14162	0.0419102
$743$	34.4242	1.26290	0.631451	−	0.775416i	$-0.282460\pi$
$743$	34.4242	1.26290	0.631451	+	0.775416i	$0.282460\pi$
$744$	5.81161	0.213064
$745$	23.4573	0.859409
$746$	41.9871	1.53726
$747$	−0.00770923	−0.000282066 0
$748$	−10.2738	−0.375647
$749$	−13.5767	−0.496083
$750$	−2.51257	−0.0917461
$751$	−19.2287	−0.701664	−0.350832	−	0.936438i	$-0.614101\pi$
$751$	−19.2287	−0.701664	−0.350832	+	0.936438i	$0.614101\pi$
$752$	23.1643	0.844714
$753$	13.5782	0.494817
$754$	17.1548	0.624741
$755$	7.03795	0.256137
$756$	4.92713	0.179198
$757$	−41.5800	−1.51125	−0.755625	−	0.655005i	$-0.772667\pi$
$757$	−41.5800	−1.51125	−0.755625	+	0.655005i	$0.772667\pi$
$758$	45.7565	1.66195
$759$	5.23429	0.189993
$760$	−48.0983	−1.74471
$761$	16.7849	0.608454	0.304227	−	0.952600i	$-0.401602\pi$
$761$	16.7849	0.608454	0.304227	+	0.952600i	$0.401602\pi$
$762$	44.7671	1.62174
$763$	1.27460	0.0461435
$764$	−3.03721	−0.109882
$765$	−3.63894	−0.131566
$766$	28.7792	1.03984
$767$	−3.13260	−0.113112
$768$	31.8364	1.14880
$769$	−47.2043	−1.70223	−0.851116	−	0.524978i	$-0.824073\pi$
$769$	−47.2043	−1.70223	−0.851116	+	0.524978i	$0.824073\pi$
$770$	1.87891	0.0677112
$771$	−28.5701	−1.02893
$772$	−2.21211	−0.0796155
$773$	−27.9882	−1.00667	−0.503333	−	0.864093i	$-0.667893\pi$
$773$	−27.9882	−1.00667	−0.503333	+	0.864093i	$0.667893\pi$
$774$	−9.12793	−0.328097
$775$	1.00000	0.0359211
$776$	−50.0282	−1.79591
$777$	−7.54102	−0.270533
$778$	−47.2829	−1.69517
$779$	−59.2599	−2.12321
$780$	4.31301	0.154431
$781$	−2.70106	−0.0966515
$782$	−73.1100	−2.61441
$783$	6.82759	0.243998
$784$	−34.0334	−1.21548
$785$	−21.8823	−0.781012
$786$	−50.9419	−1.81704
$787$	3.13078	0.111600	0.0558001	−	0.998442i	$-0.482229\pi$
$787$	3.13078	0.111600	0.0558001	+	0.998442i	$0.482229\pi$
$788$	99.2867	3.53694
$789$	24.7543	0.881275
$790$	6.64677	0.236482
$791$	10.8887	0.387156
$792$	3.80426	0.135179
$793$	4.55654	0.161808
$794$	19.7281	0.700123
$795$	0.397732	0.0141061
$796$	−69.5374	−2.46469
$797$	−12.8577	−0.455442	−0.227721	−	0.973726i	$-0.573127\pi$
$797$	−12.8577	−0.455442	−0.227721	+	0.973726i	$0.573127\pi$
$798$	23.7555	0.840936
$799$	14.1052	0.499006
$800$	−3.39205	−0.119927
$801$	2.37701	0.0839875
$802$	−60.9453	−2.15205
$803$	4.82742	0.170356
$804$	28.7474	1.01384
$805$	9.13474	0.321957
$806$	−2.51257	−0.0885015
$807$	−31.9229	−1.12374
$808$	−98.4876	−3.46478
$809$	14.6527	0.515161	0.257581	−	0.966257i	$-0.417075\pi$
$809$	14.6527	0.515161	0.257581	+	0.966257i	$0.417075\pi$
$810$	2.51257	0.0882827
$811$	−27.7271	−0.973629	−0.486814	−	0.873505i	$-0.661841\pi$
$811$	−27.7271	−0.973629	−0.486814	+	0.873505i	$0.661841\pi$
$812$	33.6404	1.18055
$813$	9.19807	0.322590
$814$	−10.8570	−0.380538
$815$	7.80984	0.273567
$816$	−21.7465	−0.761281
$817$	−30.0668	−1.05190
$818$	−67.7425	−2.36856
$819$	−1.14239	−0.0399182
$820$	−30.8822	−1.07845
$821$	6.87026	0.239774	0.119887	−	0.992788i	$-0.461747\pi$
$821$	6.87026	0.239774	0.119887	+	0.992788i	$0.461747\pi$
$822$	24.7721	0.864027
$823$	−21.3736	−0.745035	−0.372518	−	0.928025i	$-0.621505\pi$
$823$	−21.3736	−0.745035	−0.372518	+	0.928025i	$0.621505\pi$
$824$	2.02149	0.0704219
$825$	0.654597	0.0227902
$826$	−8.99159	−0.312857
$827$	−11.6658	−0.405659	−0.202829	−	0.979214i	$-0.565014\pi$
$827$	−11.6658	−0.405659	−0.202829	+	0.979214i	$0.565014\pi$
$828$	34.4877	1.19853
$829$	18.2601	0.634198	0.317099	−	0.948392i	$-0.397291\pi$
$829$	18.2601	0.634198	0.317099	+	0.948392i	$0.397291\pi$
$830$	−0.0193700	−0.000672342 0
$831$	22.0031	0.763280
$832$	−3.42935	−0.118891
$833$	−20.7236	−0.718031
$834$	0.758254	0.0262562
$835$	24.5505	0.849607
$836$	23.3662	0.808137
$837$	−1.00000	−0.0345651
$838$	47.2168	1.63108
$839$	−18.7878	−0.648627	−0.324313	−	0.945950i	$-0.605133\pi$
$839$	−18.7878	−0.648627	−0.324313	+	0.945950i	$0.605133\pi$
$840$	6.63911	0.229071
$841$	17.6160	0.607449
$842$	−8.50915	−0.293245
$843$	−31.0026	−1.06778
$844$	74.3047	2.55767
$845$	−1.00000	−0.0344010
$846$	−9.73918	−0.334840
$847$	12.0767	0.414962
$848$	2.37687	0.0816220
$849$	−23.7506	−0.815118
$850$	−9.14310	−0.313606
$851$	−52.7838	−1.80940
$852$	−17.7968	−0.609707
$853$	−36.0042	−1.23276	−0.616381	−	0.787448i	$-0.711402\pi$
$853$	−36.0042	−1.23276	−0.616381	+	0.787448i	$0.711402\pi$
$854$	13.0788	0.447546
$855$	8.27624	0.283041
$856$	−69.0683	−2.36070
$857$	−3.81976	−0.130481	−0.0652403	−	0.997870i	$-0.520781\pi$
$857$	−3.81976	−0.130481	−0.0652403	+	0.997870i	$0.520781\pi$
$858$	−1.64472	−0.0561499
$859$	20.1107	0.686170	0.343085	−	0.939304i	$-0.388528\pi$
$859$	20.1107	0.686170	0.343085	+	0.939304i	$0.388528\pi$
$860$	−15.6688	−0.534300
$861$	8.17977	0.278766
$862$	−76.7168	−2.61298
$863$	−12.6915	−0.432024	−0.216012	−	0.976391i	$-0.569305\pi$
$863$	−12.6915	−0.432024	−0.216012	+	0.976391i	$0.569305\pi$
$864$	3.39205	0.115400
$865$	11.2159	0.381353
$866$	29.5455	1.00400
$867$	3.75809	0.127632
$868$	−4.92713	−0.167238
$869$	−1.73168	−0.0587431
$870$	17.1548	0.581603
$871$	−6.66527	−0.225844
$872$	6.48420	0.219583
$873$	8.60832	0.291347
$874$	166.278	5.62444
$875$	1.14239	0.0386197
$876$	31.8069	1.07466
$877$	−5.66531	−0.191304	−0.0956520	−	0.995415i	$-0.530494\pi$
$877$	−5.66531	−0.191304	−0.0956520	+	0.995415i	$0.530494\pi$
$878$	51.3229	1.73206
$879$	15.3130	0.516495
$880$	3.91191	0.131870
$881$	9.20071	0.309980	0.154990	−	0.987916i	$-0.450466\pi$
$881$	9.20071	0.309980	0.154990	+	0.987916i	$0.450466\pi$
$882$	14.3090	0.481808
$883$	−50.7928	−1.70931	−0.854656	−	0.519194i	$-0.826232\pi$
$883$	−50.7928	−1.70931	−0.854656	+	0.519194i	$0.826232\pi$
$884$	15.6948	0.527874
$885$	−3.13260	−0.105301
$886$	43.1007	1.44800
$887$	17.5397	0.588927	0.294464	−	0.955663i	$-0.404859\pi$
$887$	17.5397	0.588927	0.294464	+	0.955663i	$0.404859\pi$
$888$	−38.3631	−1.28738
$889$	−20.3542	−0.682658
$890$	5.97240	0.200195
$891$	−0.654597	−0.0219298
$892$	77.6137	2.59870
$893$	−32.0802	−1.07352
$894$	−58.9382	−1.97119
$895$	−6.68788	−0.223551
$896$	−17.5934	−0.587755
$897$	−7.99619	−0.266985
$898$	48.4429	1.61656
$899$	−6.82759	−0.227713
$900$	4.31301	0.143767
$901$	1.44732	0.0482174
$902$	11.7766	0.392118
$903$	4.15018	0.138109
$904$	55.3934	1.84236
$905$	12.0477	0.400478
$906$	−17.6833	−0.587490
$907$	44.4338	1.47540	0.737700	−	0.675129i	$-0.235912\pi$
$907$	44.4338	1.47540	0.737700	+	0.675129i	$0.235912\pi$
$908$	40.3317	1.33845
$909$	16.9467	0.562087
$910$	−2.87033	−0.0951504
$911$	−56.6542	−1.87704	−0.938518	−	0.345230i	$-0.887801\pi$
$911$	−56.6542	−1.87704	−0.938518	+	0.345230i	$0.887801\pi$
$912$	49.4593	1.63776
$913$	0.00504644	0.000167013 0
$914$	10.8558	0.359077
$915$	4.55654	0.150635
$916$	−70.4902	−2.32906
$917$	23.1617	0.764865
$918$	9.14310	0.301767
$919$	−7.47585	−0.246606	−0.123303	−	0.992369i	$-0.539349\pi$
$919$	−7.47585	−0.246606	−0.123303	+	0.992369i	$0.539349\pi$
$920$	46.4708	1.53210
$921$	−9.68617	−0.319170
$922$	−43.8090	−1.44277
$923$	4.12629	0.135819
$924$	−3.22528	−0.106104
$925$	−6.60111	−0.217043
$926$	27.7571	0.912154
$927$	−0.347836	−0.0114244
$928$	23.1595	0.760249
$929$	10.9827	0.360332	0.180166	−	0.983636i	$-0.442337\pi$
$929$	10.9827	0.360332	0.180166	+	0.983636i	$0.442337\pi$
$930$	−2.51257	−0.0823905
$931$	47.1328	1.54471
$932$	115.753	3.79161
$933$	30.2809	0.991352
$934$	68.2297	2.23254
$935$	2.38204	0.0779011
$936$	−5.81161	−0.189958
$937$	−29.9811	−0.979441	−0.489720	−	0.871880i	$-0.662901\pi$
$937$	−29.9811	−0.979441	−0.489720	+	0.871880i	$0.662901\pi$
$938$	−19.1315	−0.624665
$939$	−14.0953	−0.459984
$940$	−16.7180	−0.545282
$941$	49.7065	1.62039	0.810193	−	0.586164i	$-0.199363\pi$
$941$	49.7065	1.62039	0.810193	+	0.586164i	$0.199363\pi$
$942$	54.9808	1.79137
$943$	57.2547	1.86447
$944$	−18.7206	−0.609304
$945$	−1.14239	−0.0371618
$946$	5.97512	0.194268
$947$	−21.8516	−0.710083	−0.355041	−	0.934851i	$-0.615533\pi$
$947$	−21.8516	−0.710083	−0.355041	+	0.934851i	$0.615533\pi$
$948$	−11.4097	−0.370569
$949$	−7.37463	−0.239391
$950$	20.7946	0.674667
$951$	25.4201	0.824302
$952$	24.1593	0.783008
$953$	35.0458	1.13524	0.567622	−	0.823290i	$-0.307864\pi$
$953$	35.0458	1.13524	0.567622	+	0.823290i	$0.307864\pi$
$954$	−0.999330	−0.0323545
$955$	0.704196	0.0227872
$956$	92.1967	2.98186
$957$	−4.46932	−0.144473
$958$	20.0019	0.646231
$959$	−11.2631	−0.363704
$960$	−3.42935	−0.110682
$961$	1.00000	0.0322581
$962$	16.5858	0.534747
$963$	11.8845	0.382974
$964$	62.0473	1.99841
$965$	0.512892	0.0165106
$966$	−22.9517	−0.738459
$967$	−51.7944	−1.66560	−0.832798	−	0.553577i	$-0.813262\pi$
$967$	−51.7944	−1.66560	−0.832798	+	0.553577i	$0.813262\pi$
$968$	61.4375	1.97467
$969$	30.1168	0.967490
$970$	21.6290	0.694465
$971$	−2.98051	−0.0956490	−0.0478245	−	0.998856i	$-0.515229\pi$
$971$	−2.98051	−0.0956490	−0.0478245	+	0.998856i	$0.515229\pi$
$972$	−4.31301	−0.138340
$973$	−0.344754	−0.0110523
$974$	0.511270	0.0163821
$975$	−1.00000	−0.0320256
$976$	27.2302	0.871616
$977$	6.81762	0.218115	0.109057	−	0.994035i	$-0.465217\pi$
$977$	6.81762	0.218115	0.109057	+	0.994035i	$0.465217\pi$
$978$	−19.6228	−0.627467
$979$	−1.55598	−0.0497295
$980$	24.5624	0.784617
$981$	−1.11573	−0.0356226
$982$	64.5444	2.05969
$983$	−5.03506	−0.160594	−0.0802968	−	0.996771i	$-0.525587\pi$
$983$	−5.03506	−0.160594	−0.0802968	+	0.996771i	$0.525587\pi$
$984$	41.6126	1.32656
$985$	−23.0203	−0.733486
$986$	62.4254	1.98803
$987$	4.42810	0.140948
$988$	−35.6955	−1.13563
$989$	29.0494	0.923717
$990$	−1.64472	−0.0522727
$991$	−34.3819	−1.09218	−0.546088	−	0.837728i	$-0.683884\pi$
$991$	−34.3819	−1.09218	−0.546088	+	0.837728i	$0.683884\pi$
$992$	−3.39205	−0.107698
$993$	−2.46611	−0.0782596
$994$	11.8438	0.375663
$995$	16.1227	0.511124
$996$	0.0332500	0.00105357
$997$	25.4172	0.804969	0.402485	−	0.915427i	$-0.368147\pi$
$997$	25.4172	0.804969	0.402485	+	0.915427i	$0.368147\pi$
$998$	37.8905	1.19940
$999$	6.60111	0.208850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.z.1.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.z.1.1	✓	12	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51257 q^{2} -1.00000 q^{3} +4.31301 q^{4} -1.00000 q^{5} +2.51257 q^{6} -1.14239 q^{7} -5.81161 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.51257 q^{2} -1.00000 q^{3} +4.31301 q^{4} -1.00000 q^{5} +2.51257 q^{6} -1.14239 q^{7} -5.81161 q^{8} +1.00000 q^{9} +2.51257 q^{10} -0.654597 q^{11} -4.31301 q^{12} +1.00000 q^{13} +2.87033 q^{14} +1.00000 q^{15} +5.97606 q^{16} +3.63894 q^{17} -2.51257 q^{18} -8.27624 q^{19} -4.31301 q^{20} +1.14239 q^{21} +1.64472 q^{22} +7.99619 q^{23} +5.81161 q^{24} +1.00000 q^{25} -2.51257 q^{26} -1.00000 q^{27} -4.92713 q^{28} -6.82759 q^{29} -2.51257 q^{30} +1.00000 q^{31} -3.39205 q^{32} +0.654597 q^{33} -9.14310 q^{34} +1.14239 q^{35} +4.31301 q^{36} -6.60111 q^{37} +20.7946 q^{38} -1.00000 q^{39} +5.81161 q^{40} +7.16025 q^{41} -2.87033 q^{42} +3.63290 q^{43} -2.82329 q^{44} -1.00000 q^{45} -20.0910 q^{46} +3.87618 q^{47} -5.97606 q^{48} -5.69495 q^{49} -2.51257 q^{50} -3.63894 q^{51} +4.31301 q^{52} +0.397732 q^{53} +2.51257 q^{54} +0.654597 q^{55} +6.63911 q^{56} +8.27624 q^{57} +17.1548 q^{58} -3.13260 q^{59} +4.31301 q^{60} +4.55654 q^{61} -2.51257 q^{62} -1.14239 q^{63} -3.42935 q^{64} -1.00000 q^{65} -1.64472 q^{66} -6.66527 q^{67} +15.6948 q^{68} -7.99619 q^{69} -2.87033 q^{70} +4.12629 q^{71} -5.81161 q^{72} -7.37463 q^{73} +16.5858 q^{74} -1.00000 q^{75} -35.6955 q^{76} +0.747803 q^{77} +2.51257 q^{78} +2.64541 q^{79} -5.97606 q^{80} +1.00000 q^{81} -17.9906 q^{82} -0.00770923 q^{83} +4.92713 q^{84} -3.63894 q^{85} -9.12793 q^{86} +6.82759 q^{87} +3.80426 q^{88} +2.37701 q^{89} +2.51257 q^{90} -1.14239 q^{91} +34.4877 q^{92} -1.00000 q^{93} -9.73918 q^{94} +8.27624 q^{95} +3.39205 q^{96} +8.60832 q^{97} +14.3090 q^{98} -0.654597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9	\( 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} - 7 q^{7} - 3 q^{8} + 12 q^{9} - 6 q^{11} - 6 q^{12} + 12 q^{13} + 5 q^{14} + 12 q^{15} - 6 q^{16} + 5 q^{17} - 24 q^{19} - 6 q^{20} + 7 q^{21} + q^{22} + 13 q^{23} + 3 q^{24} + 12 q^{25} - 12 q^{27} - 10 q^{28} + 3 q^{29} + 12 q^{31} - 6 q^{32} + 6 q^{33} - 15 q^{34} + 7 q^{35} + 6 q^{36} - 9 q^{37} + 16 q^{38} - 12 q^{39} + 3 q^{40} - 5 q^{41} - 5 q^{42} - 8 q^{43} + 5 q^{44} - 12 q^{45} - 2 q^{46} + 17 q^{47} + 6 q^{48} - 7 q^{49} - 5 q^{51} + 6 q^{52} + 16 q^{53} + 6 q^{55} + 17 q^{56} + 24 q^{57} + 36 q^{58} - 18 q^{59} + 6 q^{60} - 24 q^{61} - 7 q^{63} - 21 q^{64} - 12 q^{65} - q^{66} - 20 q^{67} + 23 q^{68} - 13 q^{69} - 5 q^{70} - 9 q^{71} - 3 q^{72} - 15 q^{73} + 10 q^{74} - 12 q^{75} - 30 q^{76} + 4 q^{77} - 25 q^{79} + 6 q^{80} + 12 q^{81} - 11 q^{82} - 13 q^{83} + 10 q^{84} - 5 q^{85} - 30 q^{86} - 3 q^{87} + 3 q^{88} + 35 q^{89} - 7 q^{91} + 5 q^{92} - 12 q^{93} + 4 q^{94} + 24 q^{95} + 6 q^{96} - 11 q^{97} + 16 q^{98} - 6 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 + 6 * q^4 - 12 * q^5 - 7 * q^7 - 3 * q^8 + 12 * q^9 - 6 * q^11 - 6 * q^12 + 12 * q^13 + 5 * q^14 + 12 * q^15 - 6 * q^16 + 5 * q^17 - 24 * q^19 - 6 * q^20 + 7 * q^21 + q^22 + 13 * q^23 + 3 * q^24 + 12 * q^25 - 12 * q^27 - 10 * q^28 + 3 * q^29 + 12 * q^31 - 6 * q^32 + 6 * q^33 - 15 * q^34 + 7 * q^35 + 6 * q^36 - 9 * q^37 + 16 * q^38 - 12 * q^39 + 3 * q^40 - 5 * q^41 - 5 * q^42 - 8 * q^43 + 5 * q^44 - 12 * q^45 - 2 * q^46 + 17 * q^47 + 6 * q^48 - 7 * q^49 - 5 * q^51 + 6 * q^52 + 16 * q^53 + 6 * q^55 + 17 * q^56 + 24 * q^57 + 36 * q^58 - 18 * q^59 + 6 * q^60 - 24 * q^61 - 7 * q^63 - 21 * q^64 - 12 * q^65 - q^66 - 20 * q^67 + 23 * q^68 - 13 * q^69 - 5 * q^70 - 9 * q^71 - 3 * q^72 - 15 * q^73 + 10 * q^74 - 12 * q^75 - 30 * q^76 + 4 * q^77 - 25 * q^79 + 6 * q^80 + 12 * q^81 - 11 * q^82 - 13 * q^83 + 10 * q^84 - 5 * q^85 - 30 * q^86 - 3 * q^87 + 3 * q^88 + 35 * q^89 - 7 * q^91 + 5 * q^92 - 12 * q^93 + 4 * q^94 + 24 * q^95 + 6 * q^96 - 11 * q^97 + 16 * q^98 - 6 * q^99

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