LMFDB - Embedded newform 8034.2.a.bc.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
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} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + \cdots + 2048 $ x^15 - x^14 - 48x^13 + 44x^12 + 872x^11 - 707x^10 - 7580x^9 + 5112x^8 + 33191x^7 - 16428x^6 - 71361x^5 + 21747x^4 + 65434x^3 - 11840x^2 - 17600*x + 2048
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.97610$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.97610 q^{5} -1.00000 q^{6} +4.65524 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.97610 q^{5} -1.00000 q^{6} +4.65524 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.97610 q^{10} +2.35768 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.65524 q^{14} -1.97610 q^{15} +1.00000 q^{16} -1.40007 q^{17} +1.00000 q^{18} -4.22562 q^{19} +1.97610 q^{20} -4.65524 q^{21} +2.35768 q^{22} -1.68831 q^{23} -1.00000 q^{24} -1.09504 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.65524 q^{28} +0.920725 q^{29} -1.97610 q^{30} -4.23306 q^{31} +1.00000 q^{32} -2.35768 q^{33} -1.40007 q^{34} +9.19921 q^{35} +1.00000 q^{36} +1.22354 q^{37} -4.22562 q^{38} +1.00000 q^{39} +1.97610 q^{40} -8.28255 q^{41} -4.65524 q^{42} +9.40003 q^{43} +2.35768 q^{44} +1.97610 q^{45} -1.68831 q^{46} +13.2833 q^{47} -1.00000 q^{48} +14.6713 q^{49} -1.09504 q^{50} +1.40007 q^{51} -1.00000 q^{52} +11.7500 q^{53} -1.00000 q^{54} +4.65900 q^{55} +4.65524 q^{56} +4.22562 q^{57} +0.920725 q^{58} +4.69835 q^{59} -1.97610 q^{60} -1.74167 q^{61} -4.23306 q^{62} +4.65524 q^{63} +1.00000 q^{64} -1.97610 q^{65} -2.35768 q^{66} +10.0174 q^{67} -1.40007 q^{68} +1.68831 q^{69} +9.19921 q^{70} +6.96979 q^{71} +1.00000 q^{72} +11.5943 q^{73} +1.22354 q^{74} +1.09504 q^{75} -4.22562 q^{76} +10.9756 q^{77} +1.00000 q^{78} -4.59328 q^{79} +1.97610 q^{80} +1.00000 q^{81} -8.28255 q^{82} -2.82344 q^{83} -4.65524 q^{84} -2.76667 q^{85} +9.40003 q^{86} -0.920725 q^{87} +2.35768 q^{88} +3.99760 q^{89} +1.97610 q^{90} -4.65524 q^{91} -1.68831 q^{92} +4.23306 q^{93} +13.2833 q^{94} -8.35023 q^{95} -1.00000 q^{96} -11.8358 q^{97} +14.6713 q^{98} +2.35768 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9} - q^{10} + 3 q^{11} - 15 q^{12} - 15 q^{13} + 5 q^{14} + q^{15} + 15 q^{16} - 2 q^{17} + 15 q^{18} + 8 q^{19} - q^{20} - 5 q^{21} + 3 q^{22} + 3 q^{23} - 15 q^{24} + 22 q^{25} - 15 q^{26} - 15 q^{27} + 5 q^{28} + 26 q^{29} + q^{30} + 15 q^{32} - 3 q^{33} - 2 q^{34} - 8 q^{35} + 15 q^{36} + 25 q^{37} + 8 q^{38} + 15 q^{39} - q^{40} - q^{41} - 5 q^{42} + 10 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 3 q^{47} - 15 q^{48} + 32 q^{49} + 22 q^{50} + 2 q^{51} - 15 q^{52} + 13 q^{53} - 15 q^{54} - 2 q^{55} + 5 q^{56} - 8 q^{57} + 26 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 5 q^{63} + 15 q^{64} + q^{65} - 3 q^{66} + 29 q^{67} - 2 q^{68} - 3 q^{69} - 8 q^{70} + 18 q^{71} + 15 q^{72} + 23 q^{73} + 25 q^{74} - 22 q^{75} + 8 q^{76} + 17 q^{77} + 15 q^{78} + 27 q^{79} - q^{80} + 15 q^{81} - q^{82} + 7 q^{83} - 5 q^{84} + 43 q^{85} + 10 q^{86} - 26 q^{87} + 3 q^{88} + 35 q^{89} - q^{90} - 5 q^{91} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 15 q^{96} + 19 q^{97} + 32 q^{98} + 3 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9 - q^10 + 3 * q^11 - 15 * q^12 - 15 * q^13 + 5 * q^14 + q^15 + 15 * q^16 - 2 * q^17 + 15 * q^18 + 8 * q^19 - q^20 - 5 * q^21 + 3 * q^22 + 3 * q^23 - 15 * q^24 + 22 * q^25 - 15 * q^26 - 15 * q^27 + 5 * q^28 + 26 * q^29 + q^30 + 15 * q^32 - 3 * q^33 - 2 * q^34 - 8 * q^35 + 15 * q^36 + 25 * q^37 + 8 * q^38 + 15 * q^39 - q^40 - q^41 - 5 * q^42 + 10 * q^43 + 3 * q^44 - q^45 + 3 * q^46 - 3 * q^47 - 15 * q^48 + 32 * q^49 + 22 * q^50 + 2 * q^51 - 15 * q^52 + 13 * q^53 - 15 * q^54 - 2 * q^55 + 5 * q^56 - 8 * q^57 + 26 * q^58 + 28 * q^59 + q^60 + 22 * q^61 + 5 * q^63 + 15 * q^64 + q^65 - 3 * q^66 + 29 * q^67 - 2 * q^68 - 3 * q^69 - 8 * q^70 + 18 * q^71 + 15 * q^72 + 23 * q^73 + 25 * q^74 - 22 * q^75 + 8 * q^76 + 17 * q^77 + 15 * q^78 + 27 * q^79 - q^80 + 15 * q^81 - q^82 + 7 * q^83 - 5 * q^84 + 43 * q^85 + 10 * q^86 - 26 * q^87 + 3 * q^88 + 35 * q^89 - q^90 - 5 * q^91 + 3 * q^92 - 3 * q^94 + 6 * q^95 - 15 * q^96 + 19 * q^97 + 32 * q^98 + 3 * 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
$5$	1.97610	0.883738	0.441869	−	0.897080i	$-0.354316\pi$
$5$	1.97610	0.883738	0.441869	+	0.897080i	$0.354316\pi$
$6$	−1.00000	−0.408248
$7$	4.65524	1.75952	0.879758	−	0.475422i	$-0.157705\pi$
$7$	4.65524	1.75952	0.879758	+	0.475422i	$0.157705\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.97610	0.624897
$11$	2.35768	0.710867	0.355433	−	0.934702i	$-0.384333\pi$
$11$	2.35768	0.710867	0.355433	+	0.934702i	$0.384333\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.65524	1.24417
$15$	−1.97610	−0.510226
$16$	1.00000	0.250000
$17$	−1.40007	−0.339566	−0.169783	−	0.985481i	$-0.554307\pi$
$17$	−1.40007	−0.339566	−0.169783	+	0.985481i	$0.554307\pi$
$18$	1.00000	0.235702
$19$	−4.22562	−0.969423	−0.484712	−	0.874674i	$-0.661075\pi$
$19$	−4.22562	−0.969423	−0.484712	+	0.874674i	$0.661075\pi$
$20$	1.97610	0.441869
$21$	−4.65524	−1.01586
$22$	2.35768	0.502659
$23$	−1.68831	−0.352036	−0.176018	−	0.984387i	$-0.556322\pi$
$23$	−1.68831	−0.352036	−0.176018	+	0.984387i	$0.556322\pi$
$24$	−1.00000	−0.204124
$25$	−1.09504	−0.219008
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	4.65524	0.879758
$29$	0.920725	0.170974	0.0854872	−	0.996339i	$-0.472755\pi$
$29$	0.920725	0.170974	0.0854872	+	0.996339i	$0.472755\pi$
$30$	−1.97610	−0.360784
$31$	−4.23306	−0.760280	−0.380140	−	0.924929i	$-0.624124\pi$
$31$	−4.23306	−0.760280	−0.380140	+	0.924929i	$0.624124\pi$
$32$	1.00000	0.176777
$33$	−2.35768	−0.410419
$34$	−1.40007	−0.240110
$35$	9.19921	1.55495
$36$	1.00000	0.166667
$37$	1.22354	0.201149	0.100575	−	0.994930i	$-0.467932\pi$
$37$	1.22354	0.201149	0.100575	+	0.994930i	$0.467932\pi$
$38$	−4.22562	−0.685486
$39$	1.00000	0.160128
$40$	1.97610	0.312448
$41$	−8.28255	−1.29352	−0.646759	−	0.762695i	$-0.723876\pi$
$41$	−8.28255	−1.29352	−0.646759	+	0.762695i	$0.723876\pi$
$42$	−4.65524	−0.718319
$43$	9.40003	1.43349	0.716746	−	0.697335i	$-0.245631\pi$
$43$	9.40003	1.43349	0.716746	+	0.697335i	$0.245631\pi$
$44$	2.35768	0.355433
$45$	1.97610	0.294579
$46$	−1.68831	−0.248927
$47$	13.2833	1.93757	0.968783	−	0.247909i	$-0.0797434\pi$
$47$	13.2833	1.93757	0.968783	+	0.247909i	$0.0797434\pi$
$48$	−1.00000	−0.144338
$49$	14.6713	2.09590
$50$	−1.09504	−0.154862
$51$	1.40007	0.196049
$52$	−1.00000	−0.138675
$53$	11.7500	1.61399	0.806996	−	0.590556i	$-0.201092\pi$
$53$	11.7500	1.61399	0.806996	+	0.590556i	$0.201092\pi$
$54$	−1.00000	−0.136083
$55$	4.65900	0.628220
$56$	4.65524	0.622083
$57$	4.22562	0.559697
$58$	0.920725	0.120897
$59$	4.69835	0.611673	0.305837	−	0.952084i	$-0.401064\pi$
$59$	4.69835	0.611673	0.305837	+	0.952084i	$0.401064\pi$
$60$	−1.97610	−0.255113
$61$	−1.74167	−0.222998	−0.111499	−	0.993765i	$-0.535565\pi$
$61$	−1.74167	−0.222998	−0.111499	+	0.993765i	$0.535565\pi$
$62$	−4.23306	−0.537599
$63$	4.65524	0.586505
$64$	1.00000	0.125000
$65$	−1.97610	−0.245105
$66$	−2.35768	−0.290210
$67$	10.0174	1.22383	0.611913	−	0.790925i	$-0.290400\pi$
$67$	10.0174	1.22383	0.611913	+	0.790925i	$0.290400\pi$
$68$	−1.40007	−0.169783
$69$	1.68831	0.203248
$70$	9.19921	1.09952
$71$	6.96979	0.827162	0.413581	−	0.910467i	$-0.364278\pi$
$71$	6.96979	0.827162	0.413581	+	0.910467i	$0.364278\pi$
$72$	1.00000	0.117851
$73$	11.5943	1.35701	0.678505	−	0.734595i	$-0.262628\pi$
$73$	11.5943	1.35701	0.678505	+	0.734595i	$0.262628\pi$
$74$	1.22354	0.142234
$75$	1.09504	0.126444
$76$	−4.22562	−0.484712
$77$	10.9756	1.25078
$78$	1.00000	0.113228
$79$	−4.59328	−0.516785	−0.258392	−	0.966040i	$-0.583193\pi$
$79$	−4.59328	−0.516785	−0.258392	+	0.966040i	$0.583193\pi$
$80$	1.97610	0.220934
$81$	1.00000	0.111111
$82$	−8.28255	−0.914655
$83$	−2.82344	−0.309913	−0.154956	−	0.987921i	$-0.549524\pi$
$83$	−2.82344	−0.309913	−0.154956	+	0.987921i	$0.549524\pi$
$84$	−4.65524	−0.507928
$85$	−2.76667	−0.300087
$86$	9.40003	1.01363
$87$	−0.920725	−0.0987121
$88$	2.35768	0.251329
$89$	3.99760	0.423745	0.211873	−	0.977297i	$-0.432044\pi$
$89$	3.99760	0.423745	0.211873	+	0.977297i	$0.432044\pi$
$90$	1.97610	0.208299
$91$	−4.65524	−0.488002
$92$	−1.68831	−0.176018
$93$	4.23306	0.438948
$94$	13.2833	1.37007
$95$	−8.35023	−0.856716
$96$	−1.00000	−0.102062
$97$	−11.8358	−1.20174	−0.600872	−	0.799345i	$-0.705180\pi$
$97$	−11.8358	−1.20174	−0.600872	+	0.799345i	$0.705180\pi$
$98$	14.6713	1.48202
$99$	2.35768	0.236956
$100$	−1.09504	−0.109504
$101$	−7.65739	−0.761939	−0.380969	−	0.924588i	$-0.624410\pi$
$101$	−7.65739	−0.761939	−0.380969	+	0.924588i	$0.624410\pi$
$102$	1.40007	0.138627
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−9.19921	−0.897751
$106$	11.7500	1.14127
$107$	−2.19049	−0.211763	−0.105882	−	0.994379i	$-0.533766\pi$
$107$	−2.19049	−0.211763	−0.105882	+	0.994379i	$0.533766\pi$
$108$	−1.00000	−0.0962250
$109$	15.4443	1.47930	0.739649	−	0.672993i	$-0.234991\pi$
$109$	15.4443	1.47930	0.739649	+	0.672993i	$0.234991\pi$
$110$	4.65900	0.444218
$111$	−1.22354	−0.116134
$112$	4.65524	0.439879
$113$	16.2250	1.52632	0.763159	−	0.646211i	$-0.223647\pi$
$113$	16.2250	1.52632	0.763159	+	0.646211i	$0.223647\pi$
$114$	4.22562	0.395765
$115$	−3.33626	−0.311108
$116$	0.920725	0.0854872
$117$	−1.00000	−0.0924500
$118$	4.69835	0.432518
$119$	−6.51765	−0.597472
$120$	−1.97610	−0.180392
$121$	−5.44135	−0.494668
$122$	−1.74167	−0.157684
$123$	8.28255	0.746813
$124$	−4.23306	−0.380140
$125$	−12.0444	−1.07728
$126$	4.65524	0.414722
$127$	12.1773	1.08056	0.540279	−	0.841486i	$-0.318319\pi$
$127$	12.1773	1.08056	0.540279	+	0.841486i	$0.318319\pi$
$128$	1.00000	0.0883883
$129$	−9.40003	−0.827627
$130$	−1.97610	−0.173315
$131$	−11.5772	−1.01151	−0.505754	−	0.862678i	$-0.668786\pi$
$131$	−11.5772	−1.01151	−0.505754	+	0.862678i	$0.668786\pi$
$132$	−2.35768	−0.205210
$133$	−19.6713	−1.70572
$134$	10.0174	0.865375
$135$	−1.97610	−0.170075
$136$	−1.40007	−0.120055
$137$	5.67371	0.484738	0.242369	−	0.970184i	$-0.422076\pi$
$137$	5.67371	0.484738	0.242369	+	0.970184i	$0.422076\pi$
$138$	1.68831	0.143718
$139$	−19.7870	−1.67831	−0.839154	−	0.543894i	$-0.816949\pi$
$139$	−19.7870	−1.67831	−0.839154	+	0.543894i	$0.816949\pi$
$140$	9.19921	0.777475
$141$	−13.2833	−1.11865
$142$	6.96979	0.584892
$143$	−2.35768	−0.197159
$144$	1.00000	0.0833333
$145$	1.81944	0.151096
$146$	11.5943	0.959552
$147$	−14.6713	−1.21007
$148$	1.22354	0.100575
$149$	−19.8323	−1.62473	−0.812364	−	0.583151i	$-0.801820\pi$
$149$	−19.8323	−1.62473	−0.812364	+	0.583151i	$0.801820\pi$
$150$	1.09504	0.0894096
$151$	−16.6789	−1.35731	−0.678656	−	0.734456i	$-0.737437\pi$
$151$	−16.6789	−1.35731	−0.678656	+	0.734456i	$0.737437\pi$
$152$	−4.22562	−0.342743
$153$	−1.40007	−0.113189
$154$	10.9756	0.884436
$155$	−8.36493	−0.671888
$156$	1.00000	0.0800641
$157$	7.72650	0.616641	0.308321	−	0.951282i	$-0.400233\pi$
$157$	7.72650	0.616641	0.308321	+	0.951282i	$0.400233\pi$
$158$	−4.59328	−0.365422
$159$	−11.7500	−0.931839
$160$	1.97610	0.156224
$161$	−7.85948	−0.619414
$162$	1.00000	0.0785674
$163$	19.9507	1.56266	0.781330	−	0.624118i	$-0.214542\pi$
$163$	19.9507	1.56266	0.781330	+	0.624118i	$0.214542\pi$
$164$	−8.28255	−0.646759
$165$	−4.65900	−0.362703
$166$	−2.82344	−0.219141
$167$	15.6969	1.21466	0.607330	−	0.794449i	$-0.292240\pi$
$167$	15.6969	1.21466	0.607330	+	0.794449i	$0.292240\pi$
$168$	−4.65524	−0.359160
$169$	1.00000	0.0769231
$170$	−2.76667	−0.212194
$171$	−4.22562	−0.323141
$172$	9.40003	0.716746
$173$	−8.94426	−0.680019	−0.340010	−	0.940422i	$-0.610430\pi$
$173$	−8.94426	−0.680019	−0.340010	+	0.940422i	$0.610430\pi$
$174$	−0.920725	−0.0698000
$175$	−5.09767	−0.385348
$176$	2.35768	0.177717
$177$	−4.69835	−0.353150
$178$	3.99760	0.299633
$179$	22.5180	1.68307	0.841535	−	0.540202i	$-0.181652\pi$
$179$	22.5180	1.68307	0.841535	+	0.540202i	$0.181652\pi$
$180$	1.97610	0.147290
$181$	−4.81147	−0.357634	−0.178817	−	0.983882i	$-0.557227\pi$
$181$	−4.81147	−0.357634	−0.178817	+	0.983882i	$0.557227\pi$
$182$	−4.65524	−0.345069
$183$	1.74167	0.128748
$184$	−1.68831	−0.124464
$185$	2.41784	0.177763
$186$	4.23306	0.310383
$187$	−3.30091	−0.241386
$188$	13.2833	0.968783
$189$	−4.65524	−0.338619
$190$	−8.35023	−0.605790
$191$	7.01017	0.507238	0.253619	−	0.967304i	$-0.418379\pi$
$191$	7.01017	0.507238	0.253619	+	0.967304i	$0.418379\pi$
$192$	−1.00000	−0.0721688
$193$	12.1251	0.872781	0.436391	−	0.899757i	$-0.356257\pi$
$193$	12.1251	0.872781	0.436391	+	0.899757i	$0.356257\pi$
$194$	−11.8358	−0.849761
$195$	1.97610	0.141511
$196$	14.6713	1.04795
$197$	−17.9383	−1.27805	−0.639026	−	0.769185i	$-0.720662\pi$
$197$	−17.9383	−1.27805	−0.639026	+	0.769185i	$0.720662\pi$
$198$	2.35768	0.167553
$199$	−11.7186	−0.830710	−0.415355	−	0.909659i	$-0.636343\pi$
$199$	−11.7186	−0.830710	−0.415355	+	0.909659i	$0.636343\pi$
$200$	−1.09504	−0.0774310
$201$	−10.0174	−0.706576
$202$	−7.65739	−0.538772
$203$	4.28620	0.300832
$204$	1.40007	0.0980243
$205$	−16.3671	−1.14313
$206$	−1.00000	−0.0696733
$207$	−1.68831	−0.117345
$208$	−1.00000	−0.0693375
$209$	−9.96265	−0.689131
$210$	−9.19921	−0.634806
$211$	15.2946	1.05293	0.526463	−	0.850198i	$-0.323518\pi$
$211$	15.2946	1.05293	0.526463	+	0.850198i	$0.323518\pi$
$212$	11.7500	0.806996
$213$	−6.96979	−0.477562
$214$	−2.19049	−0.149739
$215$	18.5754	1.26683
$216$	−1.00000	−0.0680414
$217$	−19.7059	−1.33772
$218$	15.4443	1.04602
$219$	−11.5943	−0.783471
$220$	4.65900	0.314110
$221$	1.40007	0.0941787
$222$	−1.22354	−0.0821189
$223$	14.1192	0.945489	0.472744	−	0.881200i	$-0.343263\pi$
$223$	14.1192	0.945489	0.472744	+	0.881200i	$0.343263\pi$
$224$	4.65524	0.311041
$225$	−1.09504	−0.0730026
$226$	16.2250	1.07927
$227$	−1.43072	−0.0949605	−0.0474803	−	0.998872i	$-0.515119\pi$
$227$	−1.43072	−0.0949605	−0.0474803	+	0.998872i	$0.515119\pi$
$228$	4.22562	0.279848
$229$	25.0949	1.65832	0.829158	−	0.559015i	$-0.188820\pi$
$229$	25.0949	1.65832	0.829158	+	0.559015i	$0.188820\pi$
$230$	−3.33626	−0.219986
$231$	−10.9756	−0.722139
$232$	0.920725	0.0604486
$233$	−0.642463	−0.0420892	−0.0210446	−	0.999779i	$-0.506699\pi$
$233$	−0.642463	−0.0420892	−0.0210446	+	0.999779i	$0.506699\pi$
$234$	−1.00000	−0.0653720
$235$	26.2491	1.71230
$236$	4.69835	0.305837
$237$	4.59328	0.298366
$238$	−6.51765	−0.422477
$239$	−8.20641	−0.530829	−0.265414	−	0.964134i	$-0.585509\pi$
$239$	−8.20641	−0.530829	−0.265414	+	0.964134i	$0.585509\pi$
$240$	−1.97610	−0.127557
$241$	6.48254	0.417577	0.208789	−	0.977961i	$-0.433048\pi$
$241$	6.48254	0.417577	0.208789	+	0.977961i	$0.433048\pi$
$242$	−5.44135	−0.349783
$243$	−1.00000	−0.0641500
$244$	−1.74167	−0.111499
$245$	28.9919	1.85222
$246$	8.28255	0.528076
$247$	4.22562	0.268870
$248$	−4.23306	−0.268799
$249$	2.82344	0.178928
$250$	−12.0444	−0.761754
$251$	−29.4254	−1.85732	−0.928658	−	0.370936i	$-0.879037\pi$
$251$	−29.4254	−1.85732	−0.928658	+	0.370936i	$0.879037\pi$
$252$	4.65524	0.293253
$253$	−3.98049	−0.250251
$254$	12.1773	0.764070
$255$	2.76667	0.173256
$256$	1.00000	0.0625000
$257$	2.86093	0.178460	0.0892300	−	0.996011i	$-0.471559\pi$
$257$	2.86093	0.178460	0.0892300	+	0.996011i	$0.471559\pi$
$258$	−9.40003	−0.585221
$259$	5.69589	0.353925
$260$	−1.97610	−0.122552
$261$	0.920725	0.0569915
$262$	−11.5772	−0.715244
$263$	−3.22757	−0.199021	−0.0995103	−	0.995037i	$-0.531728\pi$
$263$	−3.22757	−0.199021	−0.0995103	+	0.995037i	$0.531728\pi$
$264$	−2.35768	−0.145105
$265$	23.2192	1.42635
$266$	−19.6713	−1.20612
$267$	−3.99760	−0.244649
$268$	10.0174	0.611913
$269$	−2.39878	−0.146256	−0.0731281	−	0.997323i	$-0.523298\pi$
$269$	−2.39878	−0.146256	−0.0731281	+	0.997323i	$0.523298\pi$
$270$	−1.97610	−0.120261
$271$	0.749348	0.0455197	0.0227598	−	0.999741i	$-0.492755\pi$
$271$	0.749348	0.0455197	0.0227598	+	0.999741i	$0.492755\pi$
$272$	−1.40007	−0.0848916
$273$	4.65524	0.281748
$274$	5.67371	0.342761
$275$	−2.58175	−0.155685
$276$	1.68831	0.101624
$277$	21.0829	1.26675	0.633374	−	0.773846i	$-0.281669\pi$
$277$	21.0829	1.26675	0.633374	+	0.773846i	$0.281669\pi$
$278$	−19.7870	−1.18674
$279$	−4.23306	−0.253427
$280$	9.19921	0.549758
$281$	−22.8003	−1.36015	−0.680077	−	0.733141i	$-0.738054\pi$
$281$	−22.8003	−1.36015	−0.680077	+	0.733141i	$0.738054\pi$
$282$	−13.2833	−0.791008
$283$	−31.8124	−1.89105	−0.945525	−	0.325550i	$-0.894451\pi$
$283$	−31.8124	−1.89105	−0.945525	+	0.325550i	$0.894451\pi$
$284$	6.96979	0.413581
$285$	8.35023	0.494625
$286$	−2.35768	−0.139412
$287$	−38.5573	−2.27596
$288$	1.00000	0.0589256
$289$	−15.0398	−0.884695
$290$	1.81944	0.106841
$291$	11.8358	0.693827
$292$	11.5943	0.678505
$293$	23.9481	1.39906	0.699530	−	0.714603i	$-0.253393\pi$
$293$	23.9481	1.39906	0.699530	+	0.714603i	$0.253393\pi$
$294$	−14.6713	−0.855646
$295$	9.28440	0.540559
$296$	1.22354	0.0711170
$297$	−2.35768	−0.136806
$298$	−19.8323	−1.14886
$299$	1.68831	0.0976373
$300$	1.09504	0.0632221
$301$	43.7594	2.52225
$302$	−16.6789	−0.959765
$303$	7.65739	0.439906
$304$	−4.22562	−0.242356
$305$	−3.44172	−0.197072
$306$	−1.40007	−0.0800365
$307$	−19.9043	−1.13600	−0.567999	−	0.823029i	$-0.692282\pi$
$307$	−19.9043	−1.13600	−0.567999	+	0.823029i	$0.692282\pi$
$308$	10.9756	0.625391
$309$	1.00000	0.0568880
$310$	−8.36493	−0.475096
$311$	−3.07394	−0.174307	−0.0871536	−	0.996195i	$-0.527777\pi$
$311$	−3.07394	−0.174307	−0.0871536	+	0.996195i	$0.527777\pi$
$312$	1.00000	0.0566139
$313$	17.0139	0.961681	0.480841	−	0.876808i	$-0.340332\pi$
$313$	17.0139	0.961681	0.480841	+	0.876808i	$0.340332\pi$
$314$	7.72650	0.436031
$315$	9.19921	0.518317
$316$	−4.59328	−0.258392
$317$	−15.5410	−0.872867	−0.436434	−	0.899736i	$-0.643759\pi$
$317$	−15.5410	−0.872867	−0.436434	+	0.899736i	$0.643759\pi$
$318$	−11.7500	−0.658910
$319$	2.17077	0.121540
$320$	1.97610	0.110467
$321$	2.19049	0.122261
$322$	−7.85948	−0.437991
$323$	5.91615	0.329183
$324$	1.00000	0.0555556
$325$	1.09504	0.0607419
$326$	19.9507	1.10497
$327$	−15.4443	−0.854073
$328$	−8.28255	−0.457327
$329$	61.8369	3.40918
$330$	−4.65900	−0.256470
$331$	−15.1150	−0.830794	−0.415397	−	0.909640i	$-0.636357\pi$
$331$	−15.1150	−0.830794	−0.415397	+	0.909640i	$0.636357\pi$
$332$	−2.82344	−0.154956
$333$	1.22354	0.0670498
$334$	15.6969	0.858895
$335$	19.7954	1.08154
$336$	−4.65524	−0.253964
$337$	−3.85788	−0.210152	−0.105076	−	0.994464i	$-0.533509\pi$
$337$	−3.85788	−0.210152	−0.105076	+	0.994464i	$0.533509\pi$
$338$	1.00000	0.0543928
$339$	−16.2250	−0.881220
$340$	−2.76667	−0.150044
$341$	−9.98019	−0.540458
$342$	−4.22562	−0.228495
$343$	35.7116	1.92825
$344$	9.40003	0.506816
$345$	3.33626	0.179618
$346$	−8.94426	−0.480846
$347$	−0.410196	−0.0220205	−0.0110102	−	0.999939i	$-0.503505\pi$
$347$	−0.410196	−0.0220205	−0.0110102	+	0.999939i	$0.503505\pi$
$348$	−0.920725	−0.0493560
$349$	24.9299	1.33447	0.667233	−	0.744849i	$-0.267478\pi$
$349$	24.9299	1.33447	0.667233	+	0.744849i	$0.267478\pi$
$350$	−5.09767	−0.272482
$351$	1.00000	0.0533761
$352$	2.35768	0.125665
$353$	−1.62571	−0.0865279	−0.0432639	−	0.999064i	$-0.513776\pi$
$353$	−1.62571	−0.0865279	−0.0432639	+	0.999064i	$0.513776\pi$
$354$	−4.69835	−0.249715
$355$	13.7730	0.730994
$356$	3.99760	0.211873
$357$	6.51765	0.344951
$358$	22.5180	1.19011
$359$	−3.25356	−0.171716	−0.0858582	−	0.996307i	$-0.527363\pi$
$359$	−3.25356	−0.171716	−0.0858582	+	0.996307i	$0.527363\pi$
$360$	1.97610	0.104149
$361$	−1.14415	−0.0602184
$362$	−4.81147	−0.252885
$363$	5.44135	0.285597
$364$	−4.65524	−0.244001
$365$	22.9115	1.19924
$366$	1.74167	0.0910387
$367$	−15.5365	−0.810998	−0.405499	−	0.914095i	$-0.632902\pi$
$367$	−15.5365	−0.810998	−0.405499	+	0.914095i	$0.632902\pi$
$368$	−1.68831	−0.0880091
$369$	−8.28255	−0.431172
$370$	2.41784	0.125698
$371$	54.6993	2.83985
$372$	4.23306	0.219474
$373$	−12.8180	−0.663692	−0.331846	−	0.943334i	$-0.607671\pi$
$373$	−12.8180	−0.663692	−0.331846	+	0.943334i	$0.607671\pi$
$374$	−3.30091	−0.170686
$375$	12.0444	0.621970
$376$	13.2833	0.685033
$377$	−0.920725	−0.0474198
$378$	−4.65524	−0.239440
$379$	−26.0379	−1.33748	−0.668739	−	0.743497i	$-0.733166\pi$
$379$	−26.0379	−1.33748	−0.668739	+	0.743497i	$0.733166\pi$
$380$	−8.35023	−0.428358
$381$	−12.1773	−0.623860
$382$	7.01017	0.358672
$383$	−37.9053	−1.93687	−0.968436	−	0.249262i	$-0.919812\pi$
$383$	−37.9053	−1.93687	−0.968436	+	0.249262i	$0.919812\pi$
$384$	−1.00000	−0.0510310
$385$	21.6888	1.10536
$386$	12.1251	0.617150
$387$	9.40003	0.477831
$388$	−11.8358	−0.600872
$389$	−25.9319	−1.31480	−0.657399	−	0.753542i	$-0.728344\pi$
$389$	−25.9319	−1.31480	−0.657399	+	0.753542i	$0.728344\pi$
$390$	1.97610	0.100064
$391$	2.36374	0.119540
$392$	14.6713	0.741011
$393$	11.5772	0.583995
$394$	−17.9383	−0.903719
$395$	−9.07677	−0.456702
$396$	2.35768	0.118478
$397$	13.6737	0.686264	0.343132	−	0.939287i	$-0.388512\pi$
$397$	13.6737	0.686264	0.343132	+	0.939287i	$0.388512\pi$
$398$	−11.7186	−0.587401
$399$	19.6713	0.984795
$400$	−1.09504	−0.0547520
$401$	−1.62541	−0.0811692	−0.0405846	−	0.999176i	$-0.512922\pi$
$401$	−1.62541	−0.0811692	−0.0405846	+	0.999176i	$0.512922\pi$
$402$	−10.0174	−0.499624
$403$	4.23306	0.210864
$404$	−7.65739	−0.380969
$405$	1.97610	0.0981931
$406$	4.28620	0.212720
$407$	2.88472	0.142990
$408$	1.40007	0.0693137
$409$	12.6629	0.626142	0.313071	−	0.949730i	$-0.398642\pi$
$409$	12.6629	0.626142	0.313071	+	0.949730i	$0.398642\pi$
$410$	−16.3671	−0.808315
$411$	−5.67371	−0.279863
$412$	−1.00000	−0.0492665
$413$	21.8720	1.07625
$414$	−1.68831	−0.0829758
$415$	−5.57939	−0.273882
$416$	−1.00000	−0.0490290
$417$	19.7870	0.968971
$418$	−9.96265	−0.487289
$419$	−26.5504	−1.29707	−0.648535	−	0.761185i	$-0.724618\pi$
$419$	−26.5504	−1.29707	−0.648535	+	0.761185i	$0.724618\pi$
$420$	−9.19921	−0.448875
$421$	−31.5279	−1.53658	−0.768288	−	0.640104i	$-0.778891\pi$
$421$	−31.5279	−1.53658	−0.768288	+	0.640104i	$0.778891\pi$
$422$	15.2946	0.744531
$423$	13.2833	0.645856
$424$	11.7500	0.570633
$425$	1.53313	0.0743677
$426$	−6.96979	−0.337688
$427$	−8.10791	−0.392369
$428$	−2.19049	−0.105882
$429$	2.35768	0.113830
$430$	18.5754	0.895784
$431$	−0.0732573	−0.00352868	−0.00176434	−	0.999998i	$-0.500562\pi$
$431$	−0.0732573	−0.00352868	−0.00176434	+	0.999998i	$0.500562\pi$
$432$	−1.00000	−0.0481125
$433$	−28.0122	−1.34618	−0.673091	−	0.739560i	$-0.735034\pi$
$433$	−28.0122	−1.34618	−0.673091	+	0.739560i	$0.735034\pi$
$434$	−19.7059	−0.945914
$435$	−1.81944	−0.0872356
$436$	15.4443	0.739649
$437$	7.13414	0.341272
$438$	−11.5943	−0.553997
$439$	35.9419	1.71542	0.857708	−	0.514137i	$-0.171888\pi$
$439$	35.9419	1.71542	0.857708	+	0.514137i	$0.171888\pi$
$440$	4.65900	0.222109
$441$	14.6713	0.698632
$442$	1.40007	0.0665944
$443$	7.72845	0.367190	0.183595	−	0.983002i	$-0.441227\pi$
$443$	7.72845	0.367190	0.183595	+	0.983002i	$0.441227\pi$
$444$	−1.22354	−0.0580668
$445$	7.89965	0.374480
$446$	14.1192	0.668562
$447$	19.8323	0.938037
$448$	4.65524	0.219939
$449$	−40.5691	−1.91458	−0.957288	−	0.289135i	$-0.906632\pi$
$449$	−40.5691	−1.91458	−0.957288	+	0.289135i	$0.906632\pi$
$450$	−1.09504	−0.0516207
$451$	−19.5276	−0.919519
$452$	16.2250	0.763159
$453$	16.6789	0.783645
$454$	−1.43072	−0.0671472
$455$	−9.19921	−0.431266
$456$	4.22562	0.197883
$457$	13.7808	0.644638	0.322319	−	0.946631i	$-0.395538\pi$
$457$	13.7808	0.644638	0.322319	+	0.946631i	$0.395538\pi$
$458$	25.0949	1.17261
$459$	1.40007	0.0653495
$460$	−3.33626	−0.155554
$461$	−15.1040	−0.703465	−0.351732	−	0.936101i	$-0.614407\pi$
$461$	−15.1040	−0.703465	−0.351732	+	0.936101i	$0.614407\pi$
$462$	−10.9756	−0.510629
$463$	−19.0640	−0.885980	−0.442990	−	0.896527i	$-0.646082\pi$
$463$	−19.0640	−0.885980	−0.442990	+	0.896527i	$0.646082\pi$
$464$	0.920725	0.0427436
$465$	8.36493	0.387914
$466$	−0.642463	−0.0297615
$467$	−30.1782	−1.39648	−0.698240	−	0.715864i	$-0.746033\pi$
$467$	−30.1782	−1.39648	−0.698240	+	0.715864i	$0.746033\pi$
$468$	−1.00000	−0.0462250
$469$	46.6336	2.15334
$470$	26.2491	1.21078
$471$	−7.72650	−0.356018
$472$	4.69835	0.216259
$473$	22.1623	1.01902
$474$	4.59328	0.210976
$475$	4.62722	0.212311
$476$	−6.51765	−0.298736
$477$	11.7500	0.537998
$478$	−8.20641	−0.375353
$479$	16.5702	0.757111	0.378556	−	0.925578i	$-0.376421\pi$
$479$	16.5702	0.757111	0.378556	+	0.925578i	$0.376421\pi$
$480$	−1.97610	−0.0901961
$481$	−1.22354	−0.0557888
$482$	6.48254	0.295272
$483$	7.85948	0.357619
$484$	−5.44135	−0.247334
$485$	−23.3887	−1.06203
$486$	−1.00000	−0.0453609
$487$	2.30340	0.104377	0.0521884	−	0.998637i	$-0.483380\pi$
$487$	2.30340	0.104377	0.0521884	+	0.998637i	$0.483380\pi$
$488$	−1.74167	−0.0788418
$489$	−19.9507	−0.902202
$490$	28.9919	1.30972
$491$	33.2881	1.50227	0.751134	−	0.660149i	$-0.229507\pi$
$491$	33.2881	1.50227	0.751134	+	0.660149i	$0.229507\pi$
$492$	8.28255	0.373406
$493$	−1.28908	−0.0580571
$494$	4.22562	0.190120
$495$	4.65900	0.209407
$496$	−4.23306	−0.190070
$497$	32.4461	1.45541
$498$	2.82344	0.126521
$499$	16.7316	0.749009	0.374504	−	0.927225i	$-0.377813\pi$
$499$	16.7316	0.749009	0.374504	+	0.927225i	$0.377813\pi$
$500$	−12.0444	−0.538642
$501$	−15.6969	−0.701285
$502$	−29.4254	−1.31332
$503$	5.43676	0.242413	0.121206	−	0.992627i	$-0.461324\pi$
$503$	5.43676	0.242413	0.121206	+	0.992627i	$0.461324\pi$
$504$	4.65524	0.207361
$505$	−15.1317	−0.673354
$506$	−3.98049	−0.176954
$507$	−1.00000	−0.0444116
$508$	12.1773	0.540279
$509$	−34.1782	−1.51492	−0.757461	−	0.652880i	$-0.773561\pi$
$509$	−34.1782	−1.51492	−0.757461	+	0.652880i	$0.773561\pi$
$510$	2.76667	0.122510
$511$	53.9743	2.38768
$512$	1.00000	0.0441942
$513$	4.22562	0.186566
$514$	2.86093	0.126190
$515$	−1.97610	−0.0870773
$516$	−9.40003	−0.413813
$517$	31.3177	1.37735
$518$	5.69589	0.250263
$519$	8.94426	0.392609
$520$	−1.97610	−0.0866576
$521$	25.9293	1.13599	0.567993	−	0.823034i	$-0.307720\pi$
$521$	25.9293	1.13599	0.567993	+	0.823034i	$0.307720\pi$
$522$	0.920725	0.0402990
$523$	6.35658	0.277954	0.138977	−	0.990296i	$-0.455619\pi$
$523$	6.35658	0.277954	0.138977	+	0.990296i	$0.455619\pi$
$524$	−11.5772	−0.505754
$525$	5.09767	0.222481
$526$	−3.22757	−0.140729
$527$	5.92657	0.258165
$528$	−2.35768	−0.102605
$529$	−20.1496	−0.876070
$530$	23.2192	1.00858
$531$	4.69835	0.203891
$532$	−19.6713	−0.852858
$533$	8.28255	0.358757
$534$	−3.99760	−0.172993
$535$	−4.32863	−0.187143
$536$	10.0174	0.432688
$537$	−22.5180	−0.971721
$538$	−2.39878	−0.103419
$539$	34.5901	1.48990
$540$	−1.97610	−0.0850377
$541$	−17.6478	−0.758737	−0.379369	−	0.925246i	$-0.623859\pi$
$541$	−17.6478	−0.758737	−0.379369	+	0.925246i	$0.623859\pi$
$542$	0.749348	0.0321873
$543$	4.81147	0.206480
$544$	−1.40007	−0.0600274
$545$	30.5195	1.30731
$546$	4.65524	0.199226
$547$	−14.6376	−0.625859	−0.312929	−	0.949776i	$-0.601310\pi$
$547$	−14.6376	−0.625859	−0.312929	+	0.949776i	$0.601310\pi$
$548$	5.67371	0.242369
$549$	−1.74167	−0.0743328
$550$	−2.58175	−0.110086
$551$	−3.89063	−0.165747
$552$	1.68831	0.0718591
$553$	−21.3828	−0.909291
$554$	21.0829	0.895727
$555$	−2.41784	−0.102632
$556$	−19.7870	−0.839154
$557$	−30.4552	−1.29043	−0.645213	−	0.764002i	$-0.723232\pi$
$557$	−30.4552	−1.29043	−0.645213	+	0.764002i	$0.723232\pi$
$558$	−4.23306	−0.179200
$559$	−9.40003	−0.397579
$560$	9.19921	0.388738
$561$	3.30091	0.139364
$562$	−22.8003	−0.961774
$563$	−6.82548	−0.287660	−0.143830	−	0.989602i	$-0.545942\pi$
$563$	−6.82548	−0.287660	−0.143830	+	0.989602i	$0.545942\pi$
$564$	−13.2833	−0.559327
$565$	32.0622	1.34886
$566$	−31.8124	−1.33717
$567$	4.65524	0.195502
$568$	6.96979	0.292446
$569$	2.03472	0.0853001	0.0426501	−	0.999090i	$-0.486420\pi$
$569$	2.03472	0.0853001	0.0426501	+	0.999090i	$0.486420\pi$
$570$	8.35023	0.349753
$571$	9.03720	0.378195	0.189098	−	0.981958i	$-0.439444\pi$
$571$	9.03720	0.378195	0.189098	+	0.981958i	$0.439444\pi$
$572$	−2.35768	−0.0985795
$573$	−7.01017	−0.292854
$574$	−38.5573	−1.60935
$575$	1.84876	0.0770987
$576$	1.00000	0.0416667
$577$	13.3567	0.556046	0.278023	−	0.960574i	$-0.410321\pi$
$577$	13.3567	0.556046	0.278023	+	0.960574i	$0.410321\pi$
$578$	−15.0398	−0.625574
$579$	−12.1251	−0.503901
$580$	1.81944	0.0755482
$581$	−13.1438	−0.545296
$582$	11.8358	0.490610
$583$	27.7028	1.14733
$584$	11.5943	0.479776
$585$	−1.97610	−0.0817016
$586$	23.9481	0.989285
$587$	−41.3212	−1.70551	−0.852755	−	0.522311i	$-0.825070\pi$
$587$	−41.3212	−1.70551	−0.852755	+	0.522311i	$0.825070\pi$
$588$	−14.6713	−0.605033
$589$	17.8873	0.737033
$590$	9.28440	0.382233
$591$	17.9383	0.737883
$592$	1.22354	0.0502873
$593$	9.26916	0.380639	0.190319	−	0.981722i	$-0.439048\pi$
$593$	9.26916	0.380639	0.190319	+	0.981722i	$0.439048\pi$
$594$	−2.35768	−0.0967367
$595$	−12.8795	−0.528009
$596$	−19.8323	−0.812364
$597$	11.7186	0.479611
$598$	1.68831	0.0690400
$599$	2.06596	0.0844128	0.0422064	−	0.999109i	$-0.486561\pi$
$599$	2.06596	0.0844128	0.0422064	+	0.999109i	$0.486561\pi$
$600$	1.09504	0.0447048
$601$	41.7134	1.70152	0.850762	−	0.525552i	$-0.176141\pi$
$601$	41.7134	1.70152	0.850762	+	0.525552i	$0.176141\pi$
$602$	43.7594	1.78350
$603$	10.0174	0.407942
$604$	−16.6789	−0.678656
$605$	−10.7526	−0.437157
$606$	7.65739	0.311060
$607$	4.01768	0.163073	0.0815364	−	0.996670i	$-0.474017\pi$
$607$	4.01768	0.163073	0.0815364	+	0.996670i	$0.474017\pi$
$608$	−4.22562	−0.171371
$609$	−4.28620	−0.173685
$610$	−3.44172	−0.139351
$611$	−13.2833	−0.537384
$612$	−1.40007	−0.0565944
$613$	−33.7009	−1.36117	−0.680583	−	0.732671i	$-0.738274\pi$
$613$	−33.7009	−1.36117	−0.680583	+	0.732671i	$0.738274\pi$
$614$	−19.9043	−0.803272
$615$	16.3671	0.659986
$616$	10.9756	0.442218
$617$	20.1016	0.809259	0.404630	−	0.914481i	$-0.367400\pi$
$617$	20.1016	0.809259	0.404630	+	0.914481i	$0.367400\pi$
$618$	1.00000	0.0402259
$619$	9.78828	0.393424	0.196712	−	0.980461i	$-0.436974\pi$
$619$	9.78828	0.393424	0.196712	+	0.980461i	$0.436974\pi$
$620$	−8.36493	−0.335944
$621$	1.68831	0.0677494
$622$	−3.07394	−0.123254
$623$	18.6098	0.745586
$624$	1.00000	0.0400320
$625$	−18.3257	−0.733028
$626$	17.0139	0.680011
$627$	9.96265	0.397870
$628$	7.72650	0.308321
$629$	−1.71304	−0.0683035
$630$	9.19921	0.366505
$631$	−26.1972	−1.04289	−0.521447	−	0.853284i	$-0.674608\pi$
$631$	−26.1972	−1.04289	−0.521447	+	0.853284i	$0.674608\pi$
$632$	−4.59328	−0.182711
$633$	−15.2946	−0.607907
$634$	−15.5410	−0.617210
$635$	24.0635	0.954929
$636$	−11.7500	−0.465920
$637$	−14.6713	−0.581297
$638$	2.17077	0.0859418
$639$	6.96979	0.275721
$640$	1.97610	0.0781121
$641$	−17.6597	−0.697516	−0.348758	−	0.937213i	$-0.613397\pi$
$641$	−17.6597	−0.697516	−0.348758	+	0.937213i	$0.613397\pi$
$642$	2.19049	0.0864519
$643$	−41.3631	−1.63120	−0.815601	−	0.578615i	$-0.803593\pi$
$643$	−41.3631	−1.63120	−0.815601	+	0.578615i	$0.803593\pi$
$644$	−7.85948	−0.309707
$645$	−18.5754	−0.731405
$646$	5.91615	0.232768
$647$	−48.1061	−1.89124	−0.945622	−	0.325267i	$-0.894546\pi$
$647$	−48.1061	−1.89124	−0.945622	+	0.325267i	$0.894546\pi$
$648$	1.00000	0.0392837
$649$	11.0772	0.434818
$650$	1.09504	0.0429510
$651$	19.7059	0.772335
$652$	19.9507	0.781330
$653$	−20.4563	−0.800517	−0.400259	−	0.916402i	$-0.631080\pi$
$653$	−20.4563	−0.800517	−0.400259	+	0.916402i	$0.631080\pi$
$654$	−15.4443	−0.603921
$655$	−22.8778	−0.893908
$656$	−8.28255	−0.323379
$657$	11.5943	0.452337
$658$	61.8369	2.41065
$659$	−12.7352	−0.496092	−0.248046	−	0.968748i	$-0.579788\pi$
$659$	−12.7352	−0.496092	−0.248046	+	0.968748i	$0.579788\pi$
$660$	−4.65900	−0.181351
$661$	−0.422485	−0.0164328	−0.00821639	−	0.999966i	$-0.502615\pi$
$661$	−0.422485	−0.0164328	−0.00821639	+	0.999966i	$0.502615\pi$
$662$	−15.1150	−0.587460
$663$	−1.40007	−0.0543741
$664$	−2.82344	−0.109571
$665$	−38.8723	−1.50740
$666$	1.22354	0.0474113
$667$	−1.55447	−0.0601892
$668$	15.6969	0.607330
$669$	−14.1192	−0.545878
$670$	19.7954	0.764764
$671$	−4.10631	−0.158522
$672$	−4.65524	−0.179580
$673$	13.2581	0.511064	0.255532	−	0.966801i	$-0.417749\pi$
$673$	13.2581	0.511064	0.255532	+	0.966801i	$0.417749\pi$
$674$	−3.85788	−0.148600
$675$	1.09504	0.0421481
$676$	1.00000	0.0384615
$677$	44.2611	1.70109	0.850545	−	0.525902i	$-0.176272\pi$
$677$	44.2611	1.70109	0.850545	+	0.525902i	$0.176272\pi$
$678$	−16.2250	−0.623117
$679$	−55.0985	−2.11449
$680$	−2.76667	−0.106097
$681$	1.43072	0.0548255
$682$	−9.98019	−0.382161
$683$	11.9060	0.455571	0.227786	−	0.973711i	$-0.426851\pi$
$683$	11.9060	0.455571	0.227786	+	0.973711i	$0.426851\pi$
$684$	−4.22562	−0.161571
$685$	11.2118	0.428381
$686$	35.7116	1.36348
$687$	−25.0949	−0.957429
$688$	9.40003	0.358373
$689$	−11.7500	−0.447641
$690$	3.33626	0.127009
$691$	17.3712	0.660831	0.330415	−	0.943836i	$-0.392811\pi$
$691$	17.3712	0.660831	0.330415	+	0.943836i	$0.392811\pi$
$692$	−8.94426	−0.340010
$693$	10.9756	0.416927
$694$	−0.410196	−0.0155708
$695$	−39.1009	−1.48318
$696$	−0.920725	−0.0349000
$697$	11.5961	0.439235
$698$	24.9299	0.943610
$699$	0.642463	0.0243002
$700$	−5.09767	−0.192674
$701$	49.4216	1.86663	0.933315	−	0.359059i	$-0.116902\pi$
$701$	49.4216	1.86663	0.933315	+	0.359059i	$0.116902\pi$
$702$	1.00000	0.0377426
$703$	−5.17023	−0.194999
$704$	2.35768	0.0888584
$705$	−26.2491	−0.988597
$706$	−1.62571	−0.0611844
$707$	−35.6470	−1.34064
$708$	−4.69835	−0.176575
$709$	2.66753	0.100181	0.0500907	−	0.998745i	$-0.484049\pi$
$709$	2.66753	0.100181	0.0500907	+	0.998745i	$0.484049\pi$
$710$	13.7730	0.516891
$711$	−4.59328	−0.172262
$712$	3.99760	0.149817
$713$	7.14670	0.267646
$714$	6.51765	0.243917
$715$	−4.65900	−0.174237
$716$	22.5180	0.841535
$717$	8.20641	0.306474
$718$	−3.25356	−0.121422
$719$	−32.1454	−1.19882	−0.599410	−	0.800442i	$-0.704598\pi$
$719$	−32.1454	−1.19882	−0.599410	+	0.800442i	$0.704598\pi$
$720$	1.97610	0.0736448
$721$	−4.65524	−0.173370
$722$	−1.14415	−0.0425808
$723$	−6.48254	−0.241088
$724$	−4.81147	−0.178817
$725$	−1.00823	−0.0374447
$726$	5.44135	0.201947
$727$	−12.3347	−0.457469	−0.228735	−	0.973489i	$-0.573459\pi$
$727$	−12.3347	−0.457469	−0.228735	+	0.973489i	$0.573459\pi$
$728$	−4.65524	−0.172535
$729$	1.00000	0.0370370
$730$	22.9115	0.847992
$731$	−13.1607	−0.486765
$732$	1.74167	0.0643741
$733$	9.97138	0.368301	0.184151	−	0.982898i	$-0.441047\pi$
$733$	9.97138	0.368301	0.184151	+	0.982898i	$0.441047\pi$
$734$	−15.5365	−0.573462
$735$	−28.9919	−1.06938
$736$	−1.68831	−0.0622318
$737$	23.6179	0.869977
$738$	−8.28255	−0.304885
$739$	21.2278	0.780877	0.390439	−	0.920629i	$-0.372323\pi$
$739$	21.2278	0.780877	0.390439	+	0.920629i	$0.372323\pi$
$740$	2.41784	0.0888816
$741$	−4.22562	−0.155232
$742$	54.6993	2.00807
$743$	16.2868	0.597505	0.298752	−	0.954331i	$-0.403430\pi$
$743$	16.2868	0.597505	0.298752	+	0.954331i	$0.403430\pi$
$744$	4.23306	0.155191
$745$	−39.1906	−1.43583
$746$	−12.8180	−0.469301
$747$	−2.82344	−0.103304
$748$	−3.30091	−0.120693
$749$	−10.1973	−0.372600
$750$	12.0444	0.439799
$751$	28.2516	1.03092	0.515458	−	0.856915i	$-0.327622\pi$
$751$	28.2516	1.03092	0.515458	+	0.856915i	$0.327622\pi$
$752$	13.2833	0.484392
$753$	29.4254	1.07232
$754$	−0.920725	−0.0335308
$755$	−32.9592	−1.19951
$756$	−4.65524	−0.169309
$757$	42.2881	1.53699	0.768494	−	0.639857i	$-0.221006\pi$
$757$	42.2881	1.53699	0.768494	+	0.639857i	$0.221006\pi$
$758$	−26.0379	−0.945740
$759$	3.98049	0.144482
$760$	−8.35023	−0.302895
$761$	−31.0413	−1.12525	−0.562623	−	0.826714i	$-0.690208\pi$
$761$	−31.0413	−1.12525	−0.562623	+	0.826714i	$0.690208\pi$
$762$	−12.1773	−0.441136
$763$	71.8971	2.60285
$764$	7.01017	0.253619
$765$	−2.76667	−0.100029
$766$	−37.9053	−1.36958
$767$	−4.69835	−0.169648
$768$	−1.00000	−0.0360844
$769$	−52.1811	−1.88170	−0.940850	−	0.338824i	$-0.889971\pi$
$769$	−52.1811	−1.88170	−0.940850	+	0.338824i	$0.889971\pi$
$770$	21.6888	0.781609
$771$	−2.86093	−0.103034
$772$	12.1251	0.436391
$773$	29.9114	1.07584	0.537918	−	0.842997i	$-0.319211\pi$
$773$	29.9114	1.07584	0.537918	+	0.842997i	$0.319211\pi$
$774$	9.40003	0.337877
$775$	4.63536	0.166507
$776$	−11.8358	−0.424881
$777$	−5.69589	−0.204339
$778$	−25.9319	−0.929703
$779$	34.9989	1.25397
$780$	1.97610	0.0707556
$781$	16.4325	0.588002
$782$	2.36374	0.0845273
$783$	−0.920725	−0.0329040
$784$	14.6713	0.523974
$785$	15.2683	0.544949
$786$	11.5772	0.412946
$787$	21.5388	0.767776	0.383888	−	0.923380i	$-0.374585\pi$
$787$	21.5388	0.767776	0.383888	+	0.923380i	$0.374585\pi$
$788$	−17.9383	−0.639026
$789$	3.22757	0.114905
$790$	−9.07677	−0.322937
$791$	75.5312	2.68558
$792$	2.35768	0.0837765
$793$	1.74167	0.0618486
$794$	13.6737	0.485262
$795$	−23.2192	−0.823501
$796$	−11.7186	−0.415355
$797$	39.2712	1.39106	0.695528	−	0.718499i	$-0.255170\pi$
$797$	39.2712	1.39106	0.695528	+	0.718499i	$0.255170\pi$
$798$	19.6713	0.696355
$799$	−18.5975	−0.657932
$800$	−1.09504	−0.0387155
$801$	3.99760	0.141248
$802$	−1.62541	−0.0573953
$803$	27.3356	0.964654
$804$	−10.0174	−0.353288
$805$	−15.5311	−0.547399
$806$	4.23306	0.149103
$807$	2.39878	0.0844410
$808$	−7.65739	−0.269386
$809$	44.9800	1.58141	0.790706	−	0.612196i	$-0.209714\pi$
$809$	44.9800	1.58141	0.790706	+	0.612196i	$0.209714\pi$
$810$	1.97610	0.0694330
$811$	16.1384	0.566696	0.283348	−	0.959017i	$-0.408555\pi$
$811$	16.1384	0.566696	0.283348	+	0.959017i	$0.408555\pi$
$812$	4.28620	0.150416
$813$	−0.749348	−0.0262808
$814$	2.88472	0.101109
$815$	39.4245	1.38098
$816$	1.40007	0.0490122
$817$	−39.7210	−1.38966
$818$	12.6629	0.442749
$819$	−4.65524	−0.162667
$820$	−16.3671	−0.571565
$821$	1.55416	0.0542404	0.0271202	−	0.999632i	$-0.491366\pi$
$821$	1.55416	0.0542404	0.0271202	+	0.999632i	$0.491366\pi$
$822$	−5.67371	−0.197893
$823$	−51.6454	−1.80025	−0.900123	−	0.435636i	$-0.856523\pi$
$823$	−51.6454	−1.80025	−0.900123	+	0.435636i	$0.856523\pi$
$824$	−1.00000	−0.0348367
$825$	2.58175	0.0898850
$826$	21.8720	0.761023
$827$	19.0442	0.662233	0.331117	−	0.943590i	$-0.392575\pi$
$827$	19.0442	0.662233	0.331117	+	0.943590i	$0.392575\pi$
$828$	−1.68831	−0.0586727
$829$	−26.9045	−0.934431	−0.467215	−	0.884144i	$-0.654743\pi$
$829$	−26.9045	−0.934431	−0.467215	+	0.884144i	$0.654743\pi$
$830$	−5.57939	−0.193664
$831$	−21.0829	−0.731358
$832$	−1.00000	−0.0346688
$833$	−20.5408	−0.711695
$834$	19.7870	0.685166
$835$	31.0185	1.07344
$836$	−9.96265	−0.344565
$837$	4.23306	0.146316
$838$	−26.5504	−0.917167
$839$	−41.9742	−1.44911	−0.724555	−	0.689217i	$-0.757955\pi$
$839$	−41.9742	−1.44911	−0.724555	+	0.689217i	$0.757955\pi$
$840$	−9.19921	−0.317403
$841$	−28.1523	−0.970768
$842$	−31.5279	−1.08652
$843$	22.8003	0.785285
$844$	15.2946	0.526463
$845$	1.97610	0.0679798
$846$	13.2833	0.456689
$847$	−25.3308	−0.870377
$848$	11.7500	0.403498
$849$	31.8124	1.09180
$850$	1.53313	0.0525859
$851$	−2.06572	−0.0708119
$852$	−6.96979	−0.238781
$853$	−2.02827	−0.0694467	−0.0347234	−	0.999397i	$-0.511055\pi$
$853$	−2.02827	−0.0694467	−0.0347234	+	0.999397i	$0.511055\pi$
$854$	−8.10791	−0.277447
$855$	−8.35023	−0.285572
$856$	−2.19049	−0.0748695
$857$	−10.4055	−0.355446	−0.177723	−	0.984081i	$-0.556873\pi$
$857$	−10.4055	−0.355446	−0.177723	+	0.984081i	$0.556873\pi$
$858$	2.35768	0.0804898
$859$	29.9199	1.02085	0.510427	−	0.859921i	$-0.329487\pi$
$859$	29.9199	1.02085	0.510427	+	0.859921i	$0.329487\pi$
$860$	18.5754	0.633415
$861$	38.5573	1.31403
$862$	−0.0732573	−0.00249515
$863$	1.86961	0.0636422	0.0318211	−	0.999494i	$-0.489869\pi$
$863$	1.86961	0.0636422	0.0318211	+	0.999494i	$0.489869\pi$
$864$	−1.00000	−0.0340207
$865$	−17.6747	−0.600959
$866$	−28.0122	−0.951894
$867$	15.0398	0.510779
$868$	−19.7059	−0.668862
$869$	−10.8295	−0.367365
$870$	−1.81944	−0.0616849
$871$	−10.0174	−0.339428
$872$	15.4443	0.523011
$873$	−11.8358	−0.400581
$874$	7.13414	0.241316
$875$	−56.0695	−1.89550
$876$	−11.5943	−0.391735
$877$	−9.92590	−0.335174	−0.167587	−	0.985857i	$-0.553597\pi$
$877$	−9.92590	−0.335174	−0.167587	+	0.985857i	$0.553597\pi$
$878$	35.9419	1.21298
$879$	−23.9481	−0.807748
$880$	4.65900	0.157055
$881$	−24.5217	−0.826158	−0.413079	−	0.910695i	$-0.635547\pi$
$881$	−24.5217	−0.826158	−0.413079	+	0.910695i	$0.635547\pi$
$882$	14.6713	0.494007
$883$	27.9675	0.941180	0.470590	−	0.882352i	$-0.344041\pi$
$883$	27.9675	0.941180	0.470590	+	0.882352i	$0.344041\pi$
$884$	1.40007	0.0470894
$885$	−9.28440	−0.312092
$886$	7.72845	0.259642
$887$	34.0164	1.14216	0.571080	−	0.820895i	$-0.306525\pi$
$887$	34.0164	1.14216	0.571080	+	0.820895i	$0.306525\pi$
$888$	−1.22354	−0.0410594
$889$	56.6881	1.90126
$890$	7.89965	0.264797
$891$	2.35768	0.0789852
$892$	14.1192	0.472744
$893$	−56.1301	−1.87832
$894$	19.8323	0.663292
$895$	44.4977	1.48739
$896$	4.65524	0.155521
$897$	−1.68831	−0.0563709
$898$	−40.5691	−1.35381
$899$	−3.89748	−0.129988
$900$	−1.09504	−0.0365013
$901$	−16.4509	−0.548057
$902$	−19.5276	−0.650198
$903$	−43.7594	−1.45622
$904$	16.2250	0.539635
$905$	−9.50794	−0.316054
$906$	16.6789	0.554121
$907$	−15.4638	−0.513466	−0.256733	−	0.966482i	$-0.582646\pi$
$907$	−15.4638	−0.513466	−0.256733	+	0.966482i	$0.582646\pi$
$908$	−1.43072	−0.0474803
$909$	−7.65739	−0.253980
$910$	−9.19921	−0.304951
$911$	−5.47641	−0.181442	−0.0907208	−	0.995876i	$-0.528917\pi$
$911$	−5.47641	−0.181442	−0.0907208	+	0.995876i	$0.528917\pi$
$912$	4.22562	0.139924
$913$	−6.65676	−0.220307
$914$	13.7808	0.455828
$915$	3.44172	0.113780
$916$	25.0949	0.829158
$917$	−53.8948	−1.77976
$918$	1.40007	0.0462091
$919$	−24.7430	−0.816195	−0.408098	−	0.912938i	$-0.633808\pi$
$919$	−24.7430	−0.816195	−0.408098	+	0.912938i	$0.633808\pi$
$920$	−3.33626	−0.109993
$921$	19.9043	0.655869
$922$	−15.1040	−0.497425
$923$	−6.96979	−0.229414
$924$	−10.9756	−0.361070
$925$	−1.33983	−0.0440533
$926$	−19.0640	−0.626482
$927$	−1.00000	−0.0328443
$928$	0.920725	0.0302243
$929$	−51.9988	−1.70602	−0.853012	−	0.521891i	$-0.825227\pi$
$929$	−51.9988	−1.70602	−0.853012	+	0.521891i	$0.825227\pi$
$930$	8.36493	0.274297
$931$	−61.9952	−2.03181
$932$	−0.642463	−0.0210446
$933$	3.07394	0.100636
$934$	−30.1782	−0.987460
$935$	−6.52292	−0.213322
$936$	−1.00000	−0.0326860
$937$	−47.7692	−1.56055	−0.780276	−	0.625435i	$-0.784921\pi$
$937$	−47.7692	−1.56055	−0.780276	+	0.625435i	$0.784921\pi$
$938$	46.6336	1.52264
$939$	−17.0139	−0.555227
$940$	26.2491	0.856150
$941$	49.4890	1.61330	0.806648	−	0.591032i	$-0.201279\pi$
$941$	49.4890	1.61330	0.806648	+	0.591032i	$0.201279\pi$
$942$	−7.72650	−0.251743
$943$	13.9835	0.455365
$944$	4.69835	0.152918
$945$	−9.19921	−0.299250
$946$	22.1623	0.720557
$947$	9.12725	0.296596	0.148298	−	0.988943i	$-0.452621\pi$
$947$	9.12725	0.296596	0.148298	+	0.988943i	$0.452621\pi$
$948$	4.59328	0.149183
$949$	−11.5943	−0.376367
$950$	4.62722	0.150127
$951$	15.5410	0.503950
$952$	−6.51765	−0.211238
$953$	29.1733	0.945016	0.472508	−	0.881326i	$-0.343349\pi$
$953$	29.1733	0.945016	0.472508	+	0.881326i	$0.343349\pi$
$954$	11.7500	0.380422
$955$	13.8528	0.448265
$956$	−8.20641	−0.265414
$957$	−2.17077	−0.0701712
$958$	16.5702	0.535359
$959$	26.4125	0.852903
$960$	−1.97610	−0.0637783
$961$	−13.0812	−0.421975
$962$	−1.22354	−0.0394486
$963$	−2.19049	−0.0705877
$964$	6.48254	0.208789
$965$	23.9603	0.771310
$966$	7.85948	0.252875
$967$	−30.1231	−0.968693	−0.484346	−	0.874876i	$-0.660943\pi$
$967$	−30.1231	−0.968693	−0.484346	+	0.874876i	$0.660943\pi$
$968$	−5.44135	−0.174892
$969$	−5.91615	−0.190054
$970$	−23.3887	−0.750966
$971$	−21.9567	−0.704622	−0.352311	−	0.935883i	$-0.614604\pi$
$971$	−21.9567	−0.704622	−0.352311	+	0.935883i	$0.614604\pi$
$972$	−1.00000	−0.0320750
$973$	−92.1130	−2.95301
$974$	2.30340	0.0738056
$975$	−1.09504	−0.0350693
$976$	−1.74167	−0.0557496
$977$	36.3325	1.16238	0.581190	−	0.813768i	$-0.302587\pi$
$977$	36.3325	1.16238	0.581190	+	0.813768i	$0.302587\pi$
$978$	−19.9507	−0.637953
$979$	9.42507	0.301226
$980$	28.9919	0.926111
$981$	15.4443	0.493100
$982$	33.2881	1.06226
$983$	−60.1263	−1.91773	−0.958866	−	0.283858i	$-0.908385\pi$
$983$	−60.1263	−1.91773	−0.958866	+	0.283858i	$0.908385\pi$
$984$	8.28255	0.264038
$985$	−35.4478	−1.12946
$986$	−1.28908	−0.0410526
$987$	−61.8369	−1.96829
$988$	4.22562	0.134435
$989$	−15.8701	−0.504641
$990$	4.65900	0.148073
$991$	43.3331	1.37652	0.688261	−	0.725463i	$-0.258374\pi$
$991$	43.3331	1.37652	0.688261	+	0.725463i	$0.258374\pi$
$992$	−4.23306	−0.134400
$993$	15.1150	0.479659
$994$	32.4461	1.02913
$995$	−23.1571	−0.734130
$996$	2.82344	0.0894641
$997$	4.68320	0.148318	0.0741592	−	0.997246i	$-0.476373\pi$
$997$	4.68320	0.148318	0.0741592	+	0.997246i	$0.476373\pi$
$998$	16.7316	0.529629
$999$	−1.22354	−0.0387112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bc.1.12	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bc.1.12	✓	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.97610 q^{5} -1.00000 q^{6} +4.65524 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.97610 q^{5} -1.00000 q^{6} +4.65524 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.97610 q^{10} +2.35768 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.65524 q^{14} -1.97610 q^{15} +1.00000 q^{16} -1.40007 q^{17} +1.00000 q^{18} -4.22562 q^{19} +1.97610 q^{20} -4.65524 q^{21} +2.35768 q^{22} -1.68831 q^{23} -1.00000 q^{24} -1.09504 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.65524 q^{28} +0.920725 q^{29} -1.97610 q^{30} -4.23306 q^{31} +1.00000 q^{32} -2.35768 q^{33} -1.40007 q^{34} +9.19921 q^{35} +1.00000 q^{36} +1.22354 q^{37} -4.22562 q^{38} +1.00000 q^{39} +1.97610 q^{40} -8.28255 q^{41} -4.65524 q^{42} +9.40003 q^{43} +2.35768 q^{44} +1.97610 q^{45} -1.68831 q^{46} +13.2833 q^{47} -1.00000 q^{48} +14.6713 q^{49} -1.09504 q^{50} +1.40007 q^{51} -1.00000 q^{52} +11.7500 q^{53} -1.00000 q^{54} +4.65900 q^{55} +4.65524 q^{56} +4.22562 q^{57} +0.920725 q^{58} +4.69835 q^{59} -1.97610 q^{60} -1.74167 q^{61} -4.23306 q^{62} +4.65524 q^{63} +1.00000 q^{64} -1.97610 q^{65} -2.35768 q^{66} +10.0174 q^{67} -1.40007 q^{68} +1.68831 q^{69} +9.19921 q^{70} +6.96979 q^{71} +1.00000 q^{72} +11.5943 q^{73} +1.22354 q^{74} +1.09504 q^{75} -4.22562 q^{76} +10.9756 q^{77} +1.00000 q^{78} -4.59328 q^{79} +1.97610 q^{80} +1.00000 q^{81} -8.28255 q^{82} -2.82344 q^{83} -4.65524 q^{84} -2.76667 q^{85} +9.40003 q^{86} -0.920725 q^{87} +2.35768 q^{88} +3.99760 q^{89} +1.97610 q^{90} -4.65524 q^{91} -1.68831 q^{92} +4.23306 q^{93} +13.2833 q^{94} -8.35023 q^{95} -1.00000 q^{96} -11.8358 q^{97} +14.6713 q^{98} +2.35768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9} - q^{10} + 3 q^{11} - 15 q^{12} - 15 q^{13} + 5 q^{14} + q^{15} + 15 q^{16} - 2 q^{17} + 15 q^{18} + 8 q^{19} - q^{20} - 5 q^{21} + 3 q^{22} + 3 q^{23} - 15 q^{24} + 22 q^{25} - 15 q^{26} - 15 q^{27} + 5 q^{28} + 26 q^{29} + q^{30} + 15 q^{32} - 3 q^{33} - 2 q^{34} - 8 q^{35} + 15 q^{36} + 25 q^{37} + 8 q^{38} + 15 q^{39} - q^{40} - q^{41} - 5 q^{42} + 10 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 3 q^{47} - 15 q^{48} + 32 q^{49} + 22 q^{50} + 2 q^{51} - 15 q^{52} + 13 q^{53} - 15 q^{54} - 2 q^{55} + 5 q^{56} - 8 q^{57} + 26 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 5 q^{63} + 15 q^{64} + q^{65} - 3 q^{66} + 29 q^{67} - 2 q^{68} - 3 q^{69} - 8 q^{70} + 18 q^{71} + 15 q^{72} + 23 q^{73} + 25 q^{74} - 22 q^{75} + 8 q^{76} + 17 q^{77} + 15 q^{78} + 27 q^{79} - q^{80} + 15 q^{81} - q^{82} + 7 q^{83} - 5 q^{84} + 43 q^{85} + 10 q^{86} - 26 q^{87} + 3 q^{88} + 35 q^{89} - q^{90} - 5 q^{91} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 15 q^{96} + 19 q^{97} + 32 q^{98} + 3 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9 - q^10 + 3 * q^11 - 15 * q^12 - 15 * q^13 + 5 * q^14 + q^15 + 15 * q^16 - 2 * q^17 + 15 * q^18 + 8 * q^19 - q^20 - 5 * q^21 + 3 * q^22 + 3 * q^23 - 15 * q^24 + 22 * q^25 - 15 * q^26 - 15 * q^27 + 5 * q^28 + 26 * q^29 + q^30 + 15 * q^32 - 3 * q^33 - 2 * q^34 - 8 * q^35 + 15 * q^36 + 25 * q^37 + 8 * q^38 + 15 * q^39 - q^40 - q^41 - 5 * q^42 + 10 * q^43 + 3 * q^44 - q^45 + 3 * q^46 - 3 * q^47 - 15 * q^48 + 32 * q^49 + 22 * q^50 + 2 * q^51 - 15 * q^52 + 13 * q^53 - 15 * q^54 - 2 * q^55 + 5 * q^56 - 8 * q^57 + 26 * q^58 + 28 * q^59 + q^60 + 22 * q^61 + 5 * q^63 + 15 * q^64 + q^65 - 3 * q^66 + 29 * q^67 - 2 * q^68 - 3 * q^69 - 8 * q^70 + 18 * q^71 + 15 * q^72 + 23 * q^73 + 25 * q^74 - 22 * q^75 + 8 * q^76 + 17 * q^77 + 15 * q^78 + 27 * q^79 - q^80 + 15 * q^81 - q^82 + 7 * q^83 - 5 * q^84 + 43 * q^85 + 10 * q^86 - 26 * q^87 + 3 * q^88 + 35 * q^89 - q^90 - 5 * q^91 + 3 * q^92 - 3 * q^94 + 6 * q^95 - 15 * q^96 + 19 * q^97 + 32 * q^98 + 3 * q^99

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