LMFDB - Embedded newform 8034.2.a.bc.1.4

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			1.4
Root			$2.43337$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.43337 q^{5} -1.00000 q^{6} -2.53842 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.43337 q^{5} -1.00000 q^{6} -2.53842 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.43337 q^{10} +5.77661 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.53842 q^{14} +2.43337 q^{15} +1.00000 q^{16} -4.04922 q^{17} +1.00000 q^{18} -2.00627 q^{19} -2.43337 q^{20} +2.53842 q^{21} +5.77661 q^{22} +8.05359 q^{23} -1.00000 q^{24} +0.921312 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.53842 q^{28} +0.475455 q^{29} +2.43337 q^{30} -6.32058 q^{31} +1.00000 q^{32} -5.77661 q^{33} -4.04922 q^{34} +6.17693 q^{35} +1.00000 q^{36} -2.13431 q^{37} -2.00627 q^{38} +1.00000 q^{39} -2.43337 q^{40} +1.29937 q^{41} +2.53842 q^{42} -1.58896 q^{43} +5.77661 q^{44} -2.43337 q^{45} +8.05359 q^{46} +8.72962 q^{47} -1.00000 q^{48} -0.556418 q^{49} +0.921312 q^{50} +4.04922 q^{51} -1.00000 q^{52} +0.740185 q^{53} -1.00000 q^{54} -14.0566 q^{55} -2.53842 q^{56} +2.00627 q^{57} +0.475455 q^{58} -2.73337 q^{59} +2.43337 q^{60} -6.04981 q^{61} -6.32058 q^{62} -2.53842 q^{63} +1.00000 q^{64} +2.43337 q^{65} -5.77661 q^{66} -3.11178 q^{67} -4.04922 q^{68} -8.05359 q^{69} +6.17693 q^{70} +8.83959 q^{71} +1.00000 q^{72} -8.59636 q^{73} -2.13431 q^{74} -0.921312 q^{75} -2.00627 q^{76} -14.6635 q^{77} +1.00000 q^{78} -2.59198 q^{79} -2.43337 q^{80} +1.00000 q^{81} +1.29937 q^{82} -3.99096 q^{83} +2.53842 q^{84} +9.85327 q^{85} -1.58896 q^{86} -0.475455 q^{87} +5.77661 q^{88} -9.66240 q^{89} -2.43337 q^{90} +2.53842 q^{91} +8.05359 q^{92} +6.32058 q^{93} +8.72962 q^{94} +4.88201 q^{95} -1.00000 q^{96} +12.2300 q^{97} -0.556418 q^{98} +5.77661 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.43337	−1.08824	−0.544119	−	0.839008i	$-0.683136\pi$
$5$	−2.43337	−1.08824	−0.544119	+	0.839008i	$0.683136\pi$
$6$	−1.00000	−0.408248
$7$	−2.53842	−0.959433	−0.479717	−	0.877424i	$-0.659260\pi$
$7$	−2.53842	−0.959433	−0.479717	+	0.877424i	$0.659260\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.43337	−0.769501
$11$	5.77661	1.74171	0.870856	−	0.491538i	$-0.163565\pi$
$11$	5.77661	1.74171	0.870856	+	0.491538i	$0.163565\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.53842	−0.678422
$15$	2.43337	0.628295
$16$	1.00000	0.250000
$17$	−4.04922	−0.982081	−0.491040	−	0.871137i	$-0.663383\pi$
$17$	−4.04922	−0.982081	−0.491040	+	0.871137i	$0.663383\pi$
$18$	1.00000	0.235702
$19$	−2.00627	−0.460270	−0.230135	−	0.973159i	$-0.573917\pi$
$19$	−2.00627	−0.460270	−0.230135	+	0.973159i	$0.573917\pi$
$20$	−2.43337	−0.544119
$21$	2.53842	0.553929
$22$	5.77661	1.23158
$23$	8.05359	1.67929	0.839644	−	0.543136i	$-0.182763\pi$
$23$	8.05359	1.67929	0.839644	+	0.543136i	$0.182763\pi$
$24$	−1.00000	−0.204124
$25$	0.921312	0.184262
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−2.53842	−0.479717
$29$	0.475455	0.0882898	0.0441449	−	0.999025i	$-0.485944\pi$
$29$	0.475455	0.0882898	0.0441449	+	0.999025i	$0.485944\pi$
$30$	2.43337	0.444271
$31$	−6.32058	−1.13521	−0.567605	−	0.823301i	$-0.692130\pi$
$31$	−6.32058	−1.13521	−0.567605	+	0.823301i	$0.692130\pi$
$32$	1.00000	0.176777
$33$	−5.77661	−1.00558
$34$	−4.04922	−0.694436
$35$	6.17693	1.04409
$36$	1.00000	0.166667
$37$	−2.13431	−0.350879	−0.175439	−	0.984490i	$-0.556135\pi$
$37$	−2.13431	−0.350879	−0.175439	+	0.984490i	$0.556135\pi$
$38$	−2.00627	−0.325460
$39$	1.00000	0.160128
$40$	−2.43337	−0.384750
$41$	1.29937	0.202928	0.101464	−	0.994839i	$-0.467647\pi$
$41$	1.29937	0.202928	0.101464	+	0.994839i	$0.467647\pi$
$42$	2.53842	0.391687
$43$	−1.58896	−0.242314	−0.121157	−	0.992633i	$-0.538660\pi$
$43$	−1.58896	−0.242314	−0.121157	+	0.992633i	$0.538660\pi$
$44$	5.77661	0.870856
$45$	−2.43337	−0.362746
$46$	8.05359	1.18744
$47$	8.72962	1.27335	0.636673	−	0.771134i	$-0.280310\pi$
$47$	8.72962	1.27335	0.636673	+	0.771134i	$0.280310\pi$
$48$	−1.00000	−0.144338
$49$	−0.556418	−0.0794883
$50$	0.921312	0.130293
$51$	4.04922	0.567005
$52$	−1.00000	−0.138675
$53$	0.740185	0.101672	0.0508361	−	0.998707i	$-0.483811\pi$
$53$	0.740185	0.101672	0.0508361	+	0.998707i	$0.483811\pi$
$54$	−1.00000	−0.136083
$55$	−14.0566	−1.89540
$56$	−2.53842	−0.339211
$57$	2.00627	0.265737
$58$	0.475455	0.0624303
$59$	−2.73337	−0.355854	−0.177927	−	0.984044i	$-0.556939\pi$
$59$	−2.73337	−0.355854	−0.177927	+	0.984044i	$0.556939\pi$
$60$	2.43337	0.314147
$61$	−6.04981	−0.774599	−0.387300	−	0.921954i	$-0.626592\pi$
$61$	−6.04981	−0.774599	−0.387300	+	0.921954i	$0.626592\pi$
$62$	−6.32058	−0.802715
$63$	−2.53842	−0.319811
$64$	1.00000	0.125000
$65$	2.43337	0.301823
$66$	−5.77661	−0.711051
$67$	−3.11178	−0.380164	−0.190082	−	0.981768i	$-0.560875\pi$
$67$	−3.11178	−0.380164	−0.190082	+	0.981768i	$0.560875\pi$
$68$	−4.04922	−0.491040
$69$	−8.05359	−0.969538
$70$	6.17693	0.738284
$71$	8.83959	1.04907	0.524533	−	0.851390i	$-0.324240\pi$
$71$	8.83959	1.04907	0.524533	+	0.851390i	$0.324240\pi$
$72$	1.00000	0.117851
$73$	−8.59636	−1.00613	−0.503064	−	0.864249i	$-0.667794\pi$
$73$	−8.59636	−1.00613	−0.503064	+	0.864249i	$0.667794\pi$
$74$	−2.13431	−0.248109
$75$	−0.921312	−0.106384
$76$	−2.00627	−0.230135
$77$	−14.6635	−1.67106
$78$	1.00000	0.113228
$79$	−2.59198	−0.291620	−0.145810	−	0.989313i	$-0.546579\pi$
$79$	−2.59198	−0.291620	−0.145810	+	0.989313i	$0.546579\pi$
$80$	−2.43337	−0.272060
$81$	1.00000	0.111111
$82$	1.29937	0.143492
$83$	−3.99096	−0.438064	−0.219032	−	0.975718i	$-0.570290\pi$
$83$	−3.99096	−0.438064	−0.219032	+	0.975718i	$0.570290\pi$
$84$	2.53842	0.276964
$85$	9.85327	1.06874
$86$	−1.58896	−0.171342
$87$	−0.475455	−0.0509741
$88$	5.77661	0.615788
$89$	−9.66240	−1.02421	−0.512106	−	0.858922i	$-0.671135\pi$
$89$	−9.66240	−1.02421	−0.512106	+	0.858922i	$0.671135\pi$
$90$	−2.43337	−0.256500
$91$	2.53842	0.266099
$92$	8.05359	0.839644
$93$	6.32058	0.655414
$94$	8.72962	0.900391
$95$	4.88201	0.500884
$96$	−1.00000	−0.102062
$97$	12.2300	1.24176	0.620882	−	0.783904i	$-0.286774\pi$
$97$	12.2300	1.24176	0.620882	+	0.783904i	$0.286774\pi$
$98$	−0.556418	−0.0562067
$99$	5.77661	0.580571
$100$	0.921312	0.0921312
$101$	11.1817	1.11262	0.556311	−	0.830974i	$-0.312216\pi$
$101$	11.1817	1.11262	0.556311	+	0.830974i	$0.312216\pi$
$102$	4.04922	0.400933
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−6.17693	−0.602807
$106$	0.740185	0.0718931
$107$	10.8476	1.04868	0.524338	−	0.851510i	$-0.324313\pi$
$107$	10.8476	1.04868	0.524338	+	0.851510i	$0.324313\pi$
$108$	−1.00000	−0.0962250
$109$	13.9607	1.33719	0.668596	−	0.743626i	$-0.266896\pi$
$109$	13.9607	1.33719	0.668596	+	0.743626i	$0.266896\pi$
$110$	−14.0566	−1.34025
$111$	2.13431	0.202580
$112$	−2.53842	−0.239858
$113$	−9.77485	−0.919540	−0.459770	−	0.888038i	$-0.652068\pi$
$113$	−9.77485	−0.919540	−0.459770	+	0.888038i	$0.652068\pi$
$114$	2.00627	0.187905
$115$	−19.5974	−1.82747
$116$	0.475455	0.0441449
$117$	−1.00000	−0.0924500
$118$	−2.73337	−0.251627
$119$	10.2786	0.942241
$120$	2.43337	0.222136
$121$	22.3692	2.03356
$122$	−6.04981	−0.547724
$123$	−1.29937	−0.117161
$124$	−6.32058	−0.567605
$125$	9.92498	0.887717
$126$	−2.53842	−0.226141
$127$	−0.741066	−0.0657589	−0.0328795	−	0.999459i	$-0.510468\pi$
$127$	−0.741066	−0.0657589	−0.0328795	+	0.999459i	$0.510468\pi$
$128$	1.00000	0.0883883
$129$	1.58896	0.139900
$130$	2.43337	0.213421
$131$	12.8407	1.12189	0.560947	−	0.827851i	$-0.310437\pi$
$131$	12.8407	1.12189	0.560947	+	0.827851i	$0.310437\pi$
$132$	−5.77661	−0.502789
$133$	5.09276	0.441598
$134$	−3.11178	−0.268817
$135$	2.43337	0.209432
$136$	−4.04922	−0.347218
$137$	0.740211	0.0632405	0.0316202	−	0.999500i	$-0.489933\pi$
$137$	0.740211	0.0632405	0.0316202	+	0.999500i	$0.489933\pi$
$138$	−8.05359	−0.685567
$139$	7.89186	0.669379	0.334689	−	0.942329i	$-0.391369\pi$
$139$	7.89186	0.669379	0.334689	+	0.942329i	$0.391369\pi$
$140$	6.17693	0.522046
$141$	−8.72962	−0.735166
$142$	8.83959	0.741802
$143$	−5.77661	−0.483064
$144$	1.00000	0.0833333
$145$	−1.15696	−0.0960803
$146$	−8.59636	−0.711440
$147$	0.556418	0.0458926
$148$	−2.13431	−0.175439
$149$	−11.9282	−0.977194	−0.488597	−	0.872510i	$-0.662491\pi$
$149$	−11.9282	−0.977194	−0.488597	+	0.872510i	$0.662491\pi$
$150$	−0.921312	−0.0752248
$151$	22.9927	1.87112	0.935559	−	0.353171i	$-0.114897\pi$
$151$	22.9927	1.87112	0.935559	+	0.353171i	$0.114897\pi$
$152$	−2.00627	−0.162730
$153$	−4.04922	−0.327360
$154$	−14.6635	−1.18162
$155$	15.3803	1.23538
$156$	1.00000	0.0800641
$157$	7.41021	0.591399	0.295700	−	0.955281i	$-0.404447\pi$
$157$	7.41021	0.591399	0.295700	+	0.955281i	$0.404447\pi$
$158$	−2.59198	−0.206207
$159$	−0.740185	−0.0587005
$160$	−2.43337	−0.192375
$161$	−20.4434	−1.61117
$162$	1.00000	0.0785674
$163$	22.3303	1.74904	0.874520	−	0.484989i	$-0.161176\pi$
$163$	22.3303	1.74904	0.874520	+	0.484989i	$0.161176\pi$
$164$	1.29937	0.101464
$165$	14.0566	1.09431
$166$	−3.99096	−0.309758
$167$	−4.53339	−0.350804	−0.175402	−	0.984497i	$-0.556123\pi$
$167$	−4.53339	−0.350804	−0.175402	+	0.984497i	$0.556123\pi$
$168$	2.53842	0.195843
$169$	1.00000	0.0769231
$170$	9.85327	0.755712
$171$	−2.00627	−0.153423
$172$	−1.58896	−0.121157
$173$	−9.08335	−0.690595	−0.345297	−	0.938493i	$-0.612222\pi$
$173$	−9.08335	−0.690595	−0.345297	+	0.938493i	$0.612222\pi$
$174$	−0.475455	−0.0360441
$175$	−2.33868	−0.176788
$176$	5.77661	0.435428
$177$	2.73337	0.205453
$178$	−9.66240	−0.724228
$179$	−17.1575	−1.28241	−0.641207	−	0.767368i	$-0.721566\pi$
$179$	−17.1575	−1.28241	−0.641207	+	0.767368i	$0.721566\pi$
$180$	−2.43337	−0.181373
$181$	−10.9658	−0.815084	−0.407542	−	0.913187i	$-0.633614\pi$
$181$	−10.9658	−0.815084	−0.407542	+	0.913187i	$0.633614\pi$
$182$	2.53842	0.188160
$183$	6.04981	0.447215
$184$	8.05359	0.593718
$185$	5.19358	0.381840
$186$	6.32058	0.463448
$187$	−23.3908	−1.71050
$188$	8.72962	0.636673
$189$	2.53842	0.184643
$190$	4.88201	0.354178
$191$	19.7366	1.42809	0.714047	−	0.700098i	$-0.246860\pi$
$191$	19.7366	1.42809	0.714047	+	0.700098i	$0.246860\pi$
$192$	−1.00000	−0.0721688
$193$	−14.9310	−1.07475	−0.537377	−	0.843342i	$-0.680585\pi$
$193$	−14.9310	−1.07475	−0.537377	+	0.843342i	$0.680585\pi$
$194$	12.2300	0.878060
$195$	−2.43337	−0.174258
$196$	−0.556418	−0.0397442
$197$	−21.4303	−1.52685	−0.763424	−	0.645898i	$-0.776483\pi$
$197$	−21.4303	−1.52685	−0.763424	+	0.645898i	$0.776483\pi$
$198$	5.77661	0.410526
$199$	−4.79816	−0.340133	−0.170066	−	0.985433i	$-0.554398\pi$
$199$	−4.79816	−0.340133	−0.170066	+	0.985433i	$0.554398\pi$
$200$	0.921312	0.0651466
$201$	3.11178	0.219488
$202$	11.1817	0.786743
$203$	−1.20690	−0.0847081
$204$	4.04922	0.283502
$205$	−3.16186	−0.220834
$206$	−1.00000	−0.0696733
$207$	8.05359	0.559763
$208$	−1.00000	−0.0693375
$209$	−11.5894	−0.801658
$210$	−6.17693	−0.426249
$211$	15.9089	1.09521	0.547606	−	0.836736i	$-0.315539\pi$
$211$	15.9089	1.09521	0.547606	+	0.836736i	$0.315539\pi$
$212$	0.740185	0.0508361
$213$	−8.83959	−0.605679
$214$	10.8476	0.741526
$215$	3.86653	0.263695
$216$	−1.00000	−0.0680414
$217$	16.0443	1.08916
$218$	13.9607	0.945537
$219$	8.59636	0.580888
$220$	−14.0566	−0.947699
$221$	4.04922	0.272380
$222$	2.13431	0.143246
$223$	8.61008	0.576573	0.288287	−	0.957544i	$-0.406914\pi$
$223$	8.61008	0.576573	0.288287	+	0.957544i	$0.406914\pi$
$224$	−2.53842	−0.169605
$225$	0.921312	0.0614208
$226$	−9.77485	−0.650213
$227$	16.4584	1.09238	0.546190	−	0.837661i	$-0.316078\pi$
$227$	16.4584	1.09238	0.546190	+	0.837661i	$0.316078\pi$
$228$	2.00627	0.132869
$229$	16.3135	1.07803	0.539013	−	0.842297i	$-0.318797\pi$
$229$	16.3135	1.07803	0.539013	+	0.842297i	$0.318797\pi$
$230$	−19.5974	−1.29221
$231$	14.6635	0.964785
$232$	0.475455	0.0312151
$233$	9.69358	0.635047	0.317524	−	0.948250i	$-0.397149\pi$
$233$	9.69358	0.635047	0.317524	+	0.948250i	$0.397149\pi$
$234$	−1.00000	−0.0653720
$235$	−21.2424	−1.38570
$236$	−2.73337	−0.177927
$237$	2.59198	0.168367
$238$	10.2786	0.666265
$239$	14.6164	0.945454	0.472727	−	0.881209i	$-0.343270\pi$
$239$	14.6164	0.945454	0.472727	+	0.881209i	$0.343270\pi$
$240$	2.43337	0.157074
$241$	19.0765	1.22883	0.614413	−	0.788985i	$-0.289393\pi$
$241$	19.0765	1.22883	0.614413	+	0.788985i	$0.289393\pi$
$242$	22.3692	1.43795
$243$	−1.00000	−0.0641500
$244$	−6.04981	−0.387300
$245$	1.35397	0.0865022
$246$	−1.29937	−0.0828450
$247$	2.00627	0.127656
$248$	−6.32058	−0.401357
$249$	3.99096	0.252917
$250$	9.92498	0.627711
$251$	12.6704	0.799750	0.399875	−	0.916570i	$-0.369054\pi$
$251$	12.6704	0.799750	0.399875	+	0.916570i	$0.369054\pi$
$252$	−2.53842	−0.159906
$253$	46.5224	2.92484
$254$	−0.741066	−0.0464986
$255$	−9.85327	−0.617036
$256$	1.00000	0.0625000
$257$	26.2911	1.63999	0.819997	−	0.572368i	$-0.193975\pi$
$257$	26.2911	1.63999	0.819997	+	0.572368i	$0.193975\pi$
$258$	1.58896	0.0989241
$259$	5.41778	0.336644
$260$	2.43337	0.150911
$261$	0.475455	0.0294299
$262$	12.8407	0.793299
$263$	3.53155	0.217765	0.108882	−	0.994055i	$-0.465273\pi$
$263$	3.53155	0.217765	0.108882	+	0.994055i	$0.465273\pi$
$264$	−5.77661	−0.355526
$265$	−1.80115	−0.110644
$266$	5.09276	0.312257
$267$	9.66240	0.591329
$268$	−3.11178	−0.190082
$269$	17.9185	1.09251	0.546254	−	0.837619i	$-0.316053\pi$
$269$	17.9185	1.09251	0.546254	+	0.837619i	$0.316053\pi$
$270$	2.43337	0.148090
$271$	1.14289	0.0694255	0.0347127	−	0.999397i	$-0.488948\pi$
$271$	1.14289	0.0694255	0.0347127	+	0.999397i	$0.488948\pi$
$272$	−4.04922	−0.245520
$273$	−2.53842	−0.153632
$274$	0.740211	0.0447178
$275$	5.32206	0.320932
$276$	−8.05359	−0.484769
$277$	−23.1873	−1.39319	−0.696595	−	0.717465i	$-0.745302\pi$
$277$	−23.1873	−1.39319	−0.696595	+	0.717465i	$0.745302\pi$
$278$	7.89186	0.473322
$279$	−6.32058	−0.378403
$280$	6.17693	0.369142
$281$	30.2126	1.80234	0.901168	−	0.433471i	$-0.142711\pi$
$281$	30.2126	1.80234	0.901168	+	0.433471i	$0.142711\pi$
$282$	−8.72962	−0.519841
$283$	−11.4266	−0.679241	−0.339621	−	0.940562i	$-0.610299\pi$
$283$	−11.4266	−0.679241	−0.339621	+	0.940562i	$0.610299\pi$
$284$	8.83959	0.524533
$285$	−4.88201	−0.289185
$286$	−5.77661	−0.341578
$287$	−3.29836	−0.194696
$288$	1.00000	0.0589256
$289$	−0.603800	−0.0355176
$290$	−1.15696	−0.0679390
$291$	−12.2300	−0.716933
$292$	−8.59636	−0.503064
$293$	11.3285	0.661815	0.330908	−	0.943663i	$-0.392645\pi$
$293$	11.3285	0.661815	0.330908	+	0.943663i	$0.392645\pi$
$294$	0.556418	0.0324510
$295$	6.65131	0.387254
$296$	−2.13431	−0.124054
$297$	−5.77661	−0.335193
$298$	−11.9282	−0.690980
$299$	−8.05359	−0.465751
$300$	−0.921312	−0.0531920
$301$	4.03344	0.232484
$302$	22.9927	1.32308
$303$	−11.1817	−0.642373
$304$	−2.00627	−0.115068
$305$	14.7215	0.842948
$306$	−4.04922	−0.231479
$307$	19.6190	1.11972	0.559859	−	0.828588i	$-0.310855\pi$
$307$	19.6190	1.11972	0.559859	+	0.828588i	$0.310855\pi$
$308$	−14.6635	−0.835528
$309$	1.00000	0.0568880
$310$	15.3803	0.873545
$311$	−26.0380	−1.47648	−0.738239	−	0.674539i	$-0.764343\pi$
$311$	−26.0380	−1.47648	−0.738239	+	0.674539i	$0.764343\pi$
$312$	1.00000	0.0566139
$313$	1.14456	0.0646941	0.0323471	−	0.999477i	$-0.489702\pi$
$313$	1.14456	0.0646941	0.0323471	+	0.999477i	$0.489702\pi$
$314$	7.41021	0.418183
$315$	6.17693	0.348031
$316$	−2.59198	−0.145810
$317$	−24.8040	−1.39313	−0.696566	−	0.717493i	$-0.745290\pi$
$317$	−24.8040	−1.39313	−0.696566	+	0.717493i	$0.745290\pi$
$318$	−0.740185	−0.0415075
$319$	2.74652	0.153775
$320$	−2.43337	−0.136030
$321$	−10.8476	−0.605454
$322$	−20.4434	−1.13927
$323$	8.12384	0.452023
$324$	1.00000	0.0555556
$325$	−0.921312	−0.0511052
$326$	22.3303	1.23676
$327$	−13.9607	−0.772028
$328$	1.29937	0.0717459
$329$	−22.1594	−1.22169
$330$	14.0566	0.773793
$331$	26.1663	1.43823	0.719116	−	0.694890i	$-0.244547\pi$
$331$	26.1663	1.43823	0.719116	+	0.694890i	$0.244547\pi$
$332$	−3.99096	−0.219032
$333$	−2.13431	−0.116960
$334$	−4.53339	−0.248056
$335$	7.57212	0.413709
$336$	2.53842	0.138482
$337$	18.2750	0.995502	0.497751	−	0.867320i	$-0.334159\pi$
$337$	18.2750	0.995502	0.497751	+	0.867320i	$0.334159\pi$
$338$	1.00000	0.0543928
$339$	9.77485	0.530897
$340$	9.85327	0.534369
$341$	−36.5115	−1.97721
$342$	−2.00627	−0.108487
$343$	19.1814	1.03570
$344$	−1.58896	−0.0856708
$345$	19.5974	1.05509
$346$	−9.08335	−0.488324
$347$	28.6629	1.53870	0.769351	−	0.638826i	$-0.220580\pi$
$347$	28.6629	1.53870	0.769351	+	0.638826i	$0.220580\pi$
$348$	−0.475455	−0.0254871
$349$	−6.88183	−0.368376	−0.184188	−	0.982891i	$-0.558966\pi$
$349$	−6.88183	−0.368376	−0.184188	+	0.982891i	$0.558966\pi$
$350$	−2.33868	−0.125008
$351$	1.00000	0.0533761
$352$	5.77661	0.307894
$353$	−1.99704	−0.106292	−0.0531459	−	0.998587i	$-0.516925\pi$
$353$	−1.99704	−0.106292	−0.0531459	+	0.998587i	$0.516925\pi$
$354$	2.73337	0.145277
$355$	−21.5100	−1.14163
$356$	−9.66240	−0.512106
$357$	−10.2786	−0.544003
$358$	−17.1575	−0.906804
$359$	11.0783	0.584689	0.292345	−	0.956313i	$-0.405565\pi$
$359$	11.0783	0.584689	0.292345	+	0.956313i	$0.405565\pi$
$360$	−2.43337	−0.128250
$361$	−14.9749	−0.788151
$362$	−10.9658	−0.576351
$363$	−22.3692	−1.17408
$364$	2.53842	0.133049
$365$	20.9182	1.09491
$366$	6.04981	0.316229
$367$	−5.65883	−0.295388	−0.147694	−	0.989033i	$-0.547185\pi$
$367$	−5.65883	−0.295388	−0.147694	+	0.989033i	$0.547185\pi$
$368$	8.05359	0.419822
$369$	1.29937	0.0676427
$370$	5.19358	0.270001
$371$	−1.87890	−0.0975477
$372$	6.32058	0.327707
$373$	−15.2596	−0.790114	−0.395057	−	0.918657i	$-0.629275\pi$
$373$	−15.2596	−0.790114	−0.395057	+	0.918657i	$0.629275\pi$
$374$	−23.3908	−1.20951
$375$	−9.92498	−0.512524
$376$	8.72962	0.450196
$377$	−0.475455	−0.0244872
$378$	2.53842	0.130562
$379$	−31.9693	−1.64215	−0.821077	−	0.570818i	$-0.806626\pi$
$379$	−31.9693	−1.64215	−0.821077	+	0.570818i	$0.806626\pi$
$380$	4.88201	0.250442
$381$	0.741066	0.0379659
$382$	19.7366	1.00981
$383$	−23.7745	−1.21482	−0.607411	−	0.794388i	$-0.707792\pi$
$383$	−23.7745	−1.21482	−0.607411	+	0.794388i	$0.707792\pi$
$384$	−1.00000	−0.0510310
$385$	35.6817	1.81851
$386$	−14.9310	−0.759966
$387$	−1.58896	−0.0807712
$388$	12.2300	0.620882
$389$	25.7378	1.30496	0.652479	−	0.757807i	$-0.273729\pi$
$389$	25.7378	1.30496	0.652479	+	0.757807i	$0.273729\pi$
$390$	−2.43337	−0.123219
$391$	−32.6108	−1.64920
$392$	−0.556418	−0.0281034
$393$	−12.8407	−0.647726
$394$	−21.4303	−1.07964
$395$	6.30725	0.317352
$396$	5.77661	0.290285
$397$	34.3402	1.72348	0.861742	−	0.507346i	$-0.169374\pi$
$397$	34.3402	1.72348	0.861742	+	0.507346i	$0.169374\pi$
$398$	−4.79816	−0.240510
$399$	−5.09276	−0.254957
$400$	0.921312	0.0460656
$401$	21.4238	1.06985	0.534927	−	0.844898i	$-0.320339\pi$
$401$	21.4238	1.06985	0.534927	+	0.844898i	$0.320339\pi$
$402$	3.11178	0.155201
$403$	6.32058	0.314851
$404$	11.1817	0.556311
$405$	−2.43337	−0.120915
$406$	−1.20690	−0.0598977
$407$	−12.3291	−0.611130
$408$	4.04922	0.200466
$409$	30.7711	1.52153	0.760765	−	0.649027i	$-0.224824\pi$
$409$	30.7711	1.52153	0.760765	+	0.649027i	$0.224824\pi$
$410$	−3.16186	−0.156153
$411$	−0.740211	−0.0365119
$412$	−1.00000	−0.0492665
$413$	6.93844	0.341419
$414$	8.05359	0.395812
$415$	9.71149	0.476718
$416$	−1.00000	−0.0490290
$417$	−7.89186	−0.386466
$418$	−11.5894	−0.566858
$419$	−3.50389	−0.171176	−0.0855881	−	0.996331i	$-0.527277\pi$
$419$	−3.50389	−0.171176	−0.0855881	+	0.996331i	$0.527277\pi$
$420$	−6.17693	−0.301403
$421$	10.2446	0.499294	0.249647	−	0.968337i	$-0.419685\pi$
$421$	10.2446	0.499294	0.249647	+	0.968337i	$0.419685\pi$
$422$	15.9089	0.774432
$423$	8.72962	0.424449
$424$	0.740185	0.0359466
$425$	−3.73060	−0.180961
$426$	−8.83959	−0.428280
$427$	15.3570	0.743176
$428$	10.8476	0.524338
$429$	5.77661	0.278897
$430$	3.86653	0.186460
$431$	−21.0949	−1.01611	−0.508053	−	0.861326i	$-0.669635\pi$
$431$	−21.0949	−1.01611	−0.508053	+	0.861326i	$0.669635\pi$
$432$	−1.00000	−0.0481125
$433$	14.0426	0.674847	0.337423	−	0.941353i	$-0.390445\pi$
$433$	14.0426	0.674847	0.337423	+	0.941353i	$0.390445\pi$
$434$	16.0443	0.770151
$435$	1.15696	0.0554720
$436$	13.9607	0.668596
$437$	−16.1577	−0.772927
$438$	8.59636	0.410750
$439$	36.2834	1.73171	0.865857	−	0.500291i	$-0.166774\pi$
$439$	36.2834	1.73171	0.865857	+	0.500291i	$0.166774\pi$
$440$	−14.0566	−0.670124
$441$	−0.556418	−0.0264961
$442$	4.04922	0.192602
$443$	−20.4398	−0.971123	−0.485561	−	0.874203i	$-0.661385\pi$
$443$	−20.4398	−0.971123	−0.485561	+	0.874203i	$0.661385\pi$
$444$	2.13431	0.101290
$445$	23.5122	1.11459
$446$	8.61008	0.407699
$447$	11.9282	0.564183
$448$	−2.53842	−0.119929
$449$	8.93778	0.421800	0.210900	−	0.977508i	$-0.432361\pi$
$449$	8.93778	0.421800	0.210900	+	0.977508i	$0.432361\pi$
$450$	0.921312	0.0434311
$451$	7.50597	0.353442
$452$	−9.77485	−0.459770
$453$	−22.9927	−1.08029
$454$	16.4584	0.772429
$455$	−6.17693	−0.289579
$456$	2.00627	0.0939523
$457$	−6.45931	−0.302154	−0.151077	−	0.988522i	$-0.548274\pi$
$457$	−6.45931	−0.302154	−0.151077	+	0.988522i	$0.548274\pi$
$458$	16.3135	0.762280
$459$	4.04922	0.189002
$460$	−19.5974	−0.913733
$461$	19.3065	0.899193	0.449597	−	0.893232i	$-0.351568\pi$
$461$	19.3065	0.899193	0.449597	+	0.893232i	$0.351568\pi$
$462$	14.6635	0.682206
$463$	16.9074	0.785752	0.392876	−	0.919591i	$-0.371480\pi$
$463$	16.9074	0.785752	0.392876	+	0.919591i	$0.371480\pi$
$464$	0.475455	0.0220724
$465$	−15.3803	−0.713246
$466$	9.69358	0.449046
$467$	26.3793	1.22069	0.610345	−	0.792136i	$-0.291031\pi$
$467$	26.3793	1.22069	0.610345	+	0.792136i	$0.291031\pi$
$468$	−1.00000	−0.0462250
$469$	7.89900	0.364742
$470$	−21.2424	−0.979840
$471$	−7.41021	−0.341445
$472$	−2.73337	−0.125814
$473$	−9.17877	−0.422041
$474$	2.59198	0.119053
$475$	−1.84840	−0.0848105
$476$	10.2786	0.471120
$477$	0.740185	0.0338907
$478$	14.6164	0.668537
$479$	−20.7552	−0.948328	−0.474164	−	0.880437i	$-0.657250\pi$
$479$	−20.7552	−0.948328	−0.474164	+	0.880437i	$0.657250\pi$
$480$	2.43337	0.111068
$481$	2.13431	0.0973162
$482$	19.0765	0.868911
$483$	20.4434	0.930207
$484$	22.3692	1.01678
$485$	−29.7601	−1.35134
$486$	−1.00000	−0.0453609
$487$	−26.9324	−1.22042	−0.610211	−	0.792239i	$-0.708915\pi$
$487$	−26.9324	−1.22042	−0.610211	+	0.792239i	$0.708915\pi$
$488$	−6.04981	−0.273862
$489$	−22.3303	−1.00981
$490$	1.35397	0.0611663
$491$	13.5361	0.610876	0.305438	−	0.952212i	$-0.401197\pi$
$491$	13.5361	0.610876	0.305438	+	0.952212i	$0.401197\pi$
$492$	−1.29937	−0.0585803
$493$	−1.92522	−0.0867077
$494$	2.00627	0.0902664
$495$	−14.0566	−0.631799
$496$	−6.32058	−0.283802
$497$	−22.4386	−1.00651
$498$	3.99096	0.178839
$499$	5.86009	0.262334	0.131167	−	0.991360i	$-0.458128\pi$
$499$	5.86009	0.262334	0.131167	+	0.991360i	$0.458128\pi$
$500$	9.92498	0.443858
$501$	4.53339	0.202537
$502$	12.6704	0.565508
$503$	−15.7566	−0.702552	−0.351276	−	0.936272i	$-0.614252\pi$
$503$	−15.7566	−0.702552	−0.351276	+	0.936272i	$0.614252\pi$
$504$	−2.53842	−0.113070
$505$	−27.2093	−1.21080
$506$	46.5224	2.06817
$507$	−1.00000	−0.0444116
$508$	−0.741066	−0.0328795
$509$	−36.1402	−1.60189	−0.800943	−	0.598741i	$-0.795668\pi$
$509$	−36.1402	−1.60189	−0.800943	+	0.598741i	$0.795668\pi$
$510$	−9.85327	−0.436310
$511$	21.8212	0.965312
$512$	1.00000	0.0441942
$513$	2.00627	0.0885791
$514$	26.2911	1.15965
$515$	2.43337	0.107227
$516$	1.58896	0.0699499
$517$	50.4276	2.21780
$518$	5.41778	0.238044
$519$	9.08335	0.398715
$520$	2.43337	0.106711
$521$	−2.58432	−0.113221	−0.0566105	−	0.998396i	$-0.518029\pi$
$521$	−2.58432	−0.113221	−0.0566105	+	0.998396i	$0.518029\pi$
$522$	0.475455	0.0208101
$523$	0.904305	0.0395425	0.0197712	−	0.999805i	$-0.493706\pi$
$523$	0.904305	0.0395425	0.0197712	+	0.999805i	$0.493706\pi$
$524$	12.8407	0.560947
$525$	2.33868	0.102068
$526$	3.53155	0.153983
$527$	25.5934	1.11487
$528$	−5.77661	−0.251395
$529$	41.8603	1.82001
$530$	−1.80115	−0.0782369
$531$	−2.73337	−0.118618
$532$	5.09276	0.220799
$533$	−1.29937	−0.0562821
$534$	9.66240	0.418133
$535$	−26.3963	−1.14121
$536$	−3.11178	−0.134408
$537$	17.1575	0.740402
$538$	17.9185	0.772520
$539$	−3.21421	−0.138446
$540$	2.43337	0.104716
$541$	−42.6924	−1.83549	−0.917746	−	0.397168i	$-0.869993\pi$
$541$	−42.6924	−1.83549	−0.917746	+	0.397168i	$0.869993\pi$
$542$	1.14289	0.0490912
$543$	10.9658	0.470589
$544$	−4.04922	−0.173609
$545$	−33.9716	−1.45518
$546$	−2.53842	−0.108634
$547$	−31.1663	−1.33257	−0.666287	−	0.745695i	$-0.732117\pi$
$547$	−31.1663	−1.33257	−0.666287	+	0.745695i	$0.732117\pi$
$548$	0.740211	0.0316202
$549$	−6.04981	−0.258200
$550$	5.32206	0.226933
$551$	−0.953892	−0.0406371
$552$	−8.05359	−0.342783
$553$	6.57953	0.279790
$554$	−23.1873	−0.985134
$555$	−5.19358	−0.220455
$556$	7.89186	0.334689
$557$	17.3499	0.735138	0.367569	−	0.929996i	$-0.380190\pi$
$557$	17.3499	0.735138	0.367569	+	0.929996i	$0.380190\pi$
$558$	−6.32058	−0.267572
$559$	1.58896	0.0672057
$560$	6.17693	0.261023
$561$	23.3908	0.987559
$562$	30.2126	1.27444
$563$	−2.56647	−0.108164	−0.0540819	−	0.998537i	$-0.517223\pi$
$563$	−2.56647	−0.108164	−0.0540819	+	0.998537i	$0.517223\pi$
$564$	−8.72962	−0.367583
$565$	23.7859	1.00068
$566$	−11.4266	−0.480296
$567$	−2.53842	−0.106604
$568$	8.83959	0.370901
$569$	−36.2540	−1.51984	−0.759922	−	0.650014i	$-0.774763\pi$
$569$	−36.2540	−1.51984	−0.759922	+	0.650014i	$0.774763\pi$
$570$	−4.88201	−0.204485
$571$	−2.03912	−0.0853346	−0.0426673	−	0.999089i	$-0.513586\pi$
$571$	−2.03912	−0.0853346	−0.0426673	+	0.999089i	$0.513586\pi$
$572$	−5.77661	−0.241532
$573$	−19.7366	−0.824510
$574$	−3.29836	−0.137671
$575$	7.41987	0.309430
$576$	1.00000	0.0416667
$577$	27.1431	1.12998	0.564992	−	0.825097i	$-0.308879\pi$
$577$	27.1431	1.12998	0.564992	+	0.825097i	$0.308879\pi$
$578$	−0.603800	−0.0251148
$579$	14.9310	0.620509
$580$	−1.15696	−0.0480401
$581$	10.1307	0.420293
$582$	−12.2300	−0.506948
$583$	4.27576	0.177084
$584$	−8.59636	−0.355720
$585$	2.43337	0.100608
$586$	11.3285	0.467974
$587$	18.2302	0.752442	0.376221	−	0.926530i	$-0.377223\pi$
$587$	18.2302	0.752442	0.376221	+	0.926530i	$0.377223\pi$
$588$	0.556418	0.0229463
$589$	12.6808	0.522503
$590$	6.65131	0.273830
$591$	21.4303	0.881526
$592$	−2.13431	−0.0877196
$593$	4.48531	0.184190	0.0920948	−	0.995750i	$-0.470644\pi$
$593$	4.48531	0.184190	0.0920948	+	0.995750i	$0.470644\pi$
$594$	−5.77661	−0.237017
$595$	−25.0118	−1.02538
$596$	−11.9282	−0.488597
$597$	4.79816	0.196376
$598$	−8.05359	−0.329336
$599$	−6.04937	−0.247170	−0.123585	−	0.992334i	$-0.539439\pi$
$599$	−6.04937	−0.247170	−0.123585	+	0.992334i	$0.539439\pi$
$600$	−0.921312	−0.0376124
$601$	−13.5231	−0.551619	−0.275810	−	0.961212i	$-0.588946\pi$
$601$	−13.5231	−0.551619	−0.275810	+	0.961212i	$0.588946\pi$
$602$	4.03344	0.164391
$603$	−3.11178	−0.126721
$604$	22.9927	0.935559
$605$	−54.4326	−2.21300
$606$	−11.1817	−0.454226
$607$	7.52273	0.305338	0.152669	−	0.988277i	$-0.451213\pi$
$607$	7.52273	0.305338	0.152669	+	0.988277i	$0.451213\pi$
$608$	−2.00627	−0.0813651
$609$	1.20690	0.0489062
$610$	14.7215	0.596054
$611$	−8.72962	−0.353163
$612$	−4.04922	−0.163680
$613$	−25.2747	−1.02084	−0.510418	−	0.859926i	$-0.670509\pi$
$613$	−25.2747	−1.02084	−0.510418	+	0.859926i	$0.670509\pi$
$614$	19.6190	0.791760
$615$	3.16186	0.127499
$616$	−14.6635	−0.590808
$617$	11.9571	0.481373	0.240687	−	0.970603i	$-0.422627\pi$
$617$	11.9571	0.481373	0.240687	+	0.970603i	$0.422627\pi$
$618$	1.00000	0.0402259
$619$	20.8781	0.839161	0.419581	−	0.907718i	$-0.362177\pi$
$619$	20.8781	0.839161	0.419581	+	0.907718i	$0.362177\pi$
$620$	15.3803	0.617689
$621$	−8.05359	−0.323179
$622$	−26.0380	−1.04403
$623$	24.5272	0.982663
$624$	1.00000	0.0400320
$625$	−28.7577	−1.15031
$626$	1.14456	0.0457456
$627$	11.5894	0.462838
$628$	7.41021	0.295700
$629$	8.64230	0.344591
$630$	6.17693	0.246095
$631$	20.3175	0.808826	0.404413	−	0.914576i	$-0.367476\pi$
$631$	20.3175	0.808826	0.404413	+	0.914576i	$0.367476\pi$
$632$	−2.59198	−0.103103
$633$	−15.9089	−0.632321
$634$	−24.8040	−0.985093
$635$	1.80329	0.0715614
$636$	−0.740185	−0.0293502
$637$	0.556418	0.0220461
$638$	2.74652	0.108736
$639$	8.83959	0.349689
$640$	−2.43337	−0.0961876
$641$	35.7464	1.41190	0.705948	−	0.708263i	$-0.250521\pi$
$641$	35.7464	1.41190	0.705948	+	0.708263i	$0.250521\pi$
$642$	−10.8476	−0.428121
$643$	−31.5474	−1.24411	−0.622054	−	0.782974i	$-0.713702\pi$
$643$	−31.5474	−1.24411	−0.622054	+	0.782974i	$0.713702\pi$
$644$	−20.4434	−0.805583
$645$	−3.86653	−0.152244
$646$	8.12384	0.319628
$647$	−44.6583	−1.75570	−0.877849	−	0.478937i	$-0.841022\pi$
$647$	−44.6583	−1.75570	−0.877849	+	0.478937i	$0.841022\pi$
$648$	1.00000	0.0392837
$649$	−15.7896	−0.619796
$650$	−0.921312	−0.0361368
$651$	−16.0443	−0.628826
$652$	22.3303	0.874520
$653$	−16.2905	−0.637495	−0.318748	−	0.947840i	$-0.603262\pi$
$653$	−16.2905	−0.637495	−0.318748	+	0.947840i	$0.603262\pi$
$654$	−13.9607	−0.545906
$655$	−31.2462	−1.22089
$656$	1.29937	0.0507320
$657$	−8.59636	−0.335376
$658$	−22.1594	−0.863865
$659$	36.9170	1.43808	0.719040	−	0.694968i	$-0.244582\pi$
$659$	36.9170	1.43808	0.719040	+	0.694968i	$0.244582\pi$
$660$	14.0566	0.547154
$661$	16.0315	0.623552	0.311776	−	0.950156i	$-0.399076\pi$
$661$	16.0315	0.623552	0.311776	+	0.950156i	$0.399076\pi$
$662$	26.1663	1.01698
$663$	−4.04922	−0.157259
$664$	−3.99096	−0.154879
$665$	−12.3926	−0.480564
$666$	−2.13431	−0.0827029
$667$	3.82912	0.148264
$668$	−4.53339	−0.175402
$669$	−8.61008	−0.332885
$670$	7.57212	0.292537
$671$	−34.9474	−1.34913
$672$	2.53842	0.0979217
$673$	15.0154	0.578800	0.289400	−	0.957208i	$-0.406544\pi$
$673$	15.0154	0.578800	0.289400	+	0.957208i	$0.406544\pi$
$674$	18.2750	0.703926
$675$	−0.921312	−0.0354613
$676$	1.00000	0.0384615
$677$	29.3815	1.12922	0.564611	−	0.825357i	$-0.309026\pi$
$677$	29.3815	1.12922	0.564611	+	0.825357i	$0.309026\pi$
$678$	9.77485	0.375401
$679$	−31.0448	−1.19139
$680$	9.85327	0.377856
$681$	−16.4584	−0.630686
$682$	−36.5115	−1.39810
$683$	−8.35468	−0.319683	−0.159842	−	0.987143i	$-0.551098\pi$
$683$	−8.35468	−0.319683	−0.159842	+	0.987143i	$0.551098\pi$
$684$	−2.00627	−0.0767117
$685$	−1.80121	−0.0688207
$686$	19.1814	0.732348
$687$	−16.3135	−0.622399
$688$	−1.58896	−0.0605784
$689$	−0.740185	−0.0281988
$690$	19.5974	0.746060
$691$	−20.9229	−0.795943	−0.397971	−	0.917398i	$-0.630286\pi$
$691$	−20.9229	−0.795943	−0.397971	+	0.917398i	$0.630286\pi$
$692$	−9.08335	−0.345297
$693$	−14.6635	−0.557019
$694$	28.6629	1.08803
$695$	−19.2038	−0.728443
$696$	−0.475455	−0.0180221
$697$	−5.26145	−0.199292
$698$	−6.88183	−0.260481
$699$	−9.69358	−0.366645
$700$	−2.33868	−0.0883938
$701$	−15.3988	−0.581605	−0.290803	−	0.956783i	$-0.593922\pi$
$701$	−15.3988	−0.581605	−0.290803	+	0.956783i	$0.593922\pi$
$702$	1.00000	0.0377426
$703$	4.28201	0.161499
$704$	5.77661	0.217714
$705$	21.2424	0.800036
$706$	−1.99704	−0.0751597
$707$	−28.3839	−1.06749
$708$	2.73337	0.102726
$709$	31.4589	1.18146	0.590731	−	0.806868i	$-0.298839\pi$
$709$	31.4589	1.18146	0.590731	+	0.806868i	$0.298839\pi$
$710$	−21.5100	−0.807258
$711$	−2.59198	−0.0972067
$712$	−9.66240	−0.362114
$713$	−50.9033	−1.90635
$714$	−10.2786	−0.384668
$715$	14.0566	0.525689
$716$	−17.1575	−0.641207
$717$	−14.6164	−0.545858
$718$	11.0783	0.413438
$719$	−35.5732	−1.32666	−0.663328	−	0.748329i	$-0.730856\pi$
$719$	−35.5732	−1.32666	−0.663328	+	0.748329i	$0.730856\pi$
$720$	−2.43337	−0.0906865
$721$	2.53842	0.0945357
$722$	−14.9749	−0.557307
$723$	−19.0765	−0.709463
$724$	−10.9658	−0.407542
$725$	0.438043	0.0162685
$726$	−22.3692	−0.830198
$727$	−19.6665	−0.729390	−0.364695	−	0.931127i	$-0.618827\pi$
$727$	−19.6665	−0.729390	−0.364695	+	0.931127i	$0.618827\pi$
$728$	2.53842	0.0940801
$729$	1.00000	0.0370370
$730$	20.9182	0.774216
$731$	6.43404	0.237971
$732$	6.04981	0.223607
$733$	5.59025	0.206481	0.103240	−	0.994656i	$-0.467079\pi$
$733$	5.59025	0.206481	0.103240	+	0.994656i	$0.467079\pi$
$734$	−5.65883	−0.208871
$735$	−1.35397	−0.0499421
$736$	8.05359	0.296859
$737$	−17.9755	−0.662137
$738$	1.29937	0.0478306
$739$	−10.6477	−0.391683	−0.195841	−	0.980636i	$-0.562744\pi$
$739$	−10.6477	−0.391683	−0.195841	+	0.980636i	$0.562744\pi$
$740$	5.19358	0.190920
$741$	−2.00627	−0.0737022
$742$	−1.87890	−0.0689766
$743$	−30.2932	−1.11135	−0.555674	−	0.831400i	$-0.687540\pi$
$743$	−30.2932	−1.11135	−0.555674	+	0.831400i	$0.687540\pi$
$744$	6.32058	0.231724
$745$	29.0257	1.06342
$746$	−15.2596	−0.558695
$747$	−3.99096	−0.146021
$748$	−23.3908	−0.855251
$749$	−27.5358	−1.00614
$750$	−9.92498	−0.362409
$751$	12.2962	0.448695	0.224348	−	0.974509i	$-0.427975\pi$
$751$	12.2962	0.448695	0.224348	+	0.974509i	$0.427975\pi$
$752$	8.72962	0.318336
$753$	−12.6704	−0.461736
$754$	−0.475455	−0.0173150
$755$	−55.9498	−2.03622
$756$	2.53842	0.0923215
$757$	−16.9986	−0.617824	−0.308912	−	0.951091i	$-0.599965\pi$
$757$	−16.9986	−0.617824	−0.308912	+	0.951091i	$0.599965\pi$
$758$	−31.9693	−1.16118
$759$	−46.5224	−1.68866
$760$	4.88201	0.177089
$761$	−43.5302	−1.57797	−0.788984	−	0.614414i	$-0.789392\pi$
$761$	−43.5302	−1.57797	−0.788984	+	0.614414i	$0.789392\pi$
$762$	0.741066	0.0268460
$763$	−35.4381	−1.28295
$764$	19.7366	0.714047
$765$	9.85327	0.356246
$766$	−23.7745	−0.859009
$767$	2.73337	0.0986963
$768$	−1.00000	−0.0360844
$769$	22.4858	0.810858	0.405429	−	0.914126i	$-0.367122\pi$
$769$	22.4858	0.810858	0.405429	+	0.914126i	$0.367122\pi$
$770$	35.6817	1.28588
$771$	−26.2911	−0.946851
$772$	−14.9310	−0.537377
$773$	12.7123	0.457230	0.228615	−	0.973517i	$-0.426580\pi$
$773$	12.7123	0.457230	0.228615	+	0.973517i	$0.426580\pi$
$774$	−1.58896	−0.0571139
$775$	−5.82323	−0.209177
$776$	12.2300	0.439030
$777$	−5.41778	−0.194362
$778$	25.7378	0.922745
$779$	−2.60690	−0.0934017
$780$	−2.43337	−0.0871288
$781$	51.0629	1.82717
$782$	−32.6108	−1.16616
$783$	−0.475455	−0.0169914
$784$	−0.556418	−0.0198721
$785$	−18.0318	−0.643583
$786$	−12.8407	−0.458012
$787$	34.7227	1.23773	0.618866	−	0.785497i	$-0.287592\pi$
$787$	34.7227	1.23773	0.618866	+	0.785497i	$0.287592\pi$
$788$	−21.4303	−0.763424
$789$	−3.53155	−0.125727
$790$	6.30725	0.224402
$791$	24.8127	0.882237
$792$	5.77661	0.205263
$793$	6.04981	0.214835
$794$	34.3402	1.21869
$795$	1.80115	0.0638801
$796$	−4.79816	−0.170066
$797$	−46.3092	−1.64036	−0.820179	−	0.572108i	$-0.806126\pi$
$797$	−46.3092	−1.64036	−0.820179	+	0.572108i	$0.806126\pi$
$798$	−5.09276	−0.180282
$799$	−35.3482	−1.25053
$800$	0.921312	0.0325733
$801$	−9.66240	−0.341404
$802$	21.4238	0.756501
$803$	−49.6578	−1.75239
$804$	3.11178	0.109744
$805$	49.7464	1.75333
$806$	6.32058	0.222633
$807$	−17.9185	−0.630760
$808$	11.1817	0.393372
$809$	31.4300	1.10502	0.552510	−	0.833506i	$-0.313670\pi$
$809$	31.4300	1.10502	0.552510	+	0.833506i	$0.313670\pi$
$810$	−2.43337	−0.0855001
$811$	−22.5263	−0.791007	−0.395503	−	0.918465i	$-0.629430\pi$
$811$	−22.5263	−0.791007	−0.395503	+	0.918465i	$0.629430\pi$
$812$	−1.20690	−0.0423541
$813$	−1.14289	−0.0400828
$814$	−12.3291	−0.432134
$815$	−54.3379	−1.90337
$816$	4.04922	0.141751
$817$	3.18788	0.111530
$818$	30.7711	1.07588
$819$	2.53842	0.0886996
$820$	−3.16186	−0.110417
$821$	17.9755	0.627350	0.313675	−	0.949530i	$-0.398440\pi$
$821$	17.9755	0.627350	0.313675	+	0.949530i	$0.398440\pi$
$822$	−0.740211	−0.0258178
$823$	23.0890	0.804833	0.402417	−	0.915457i	$-0.368170\pi$
$823$	23.0890	0.804833	0.402417	+	0.915457i	$0.368170\pi$
$824$	−1.00000	−0.0348367
$825$	−5.32206	−0.185290
$826$	6.93844	0.241419
$827$	4.81901	0.167573	0.0837866	−	0.996484i	$-0.473299\pi$
$827$	4.81901	0.167573	0.0837866	+	0.996484i	$0.473299\pi$
$828$	8.05359	0.279881
$829$	14.9496	0.519222	0.259611	−	0.965713i	$-0.416406\pi$
$829$	14.9496	0.519222	0.259611	+	0.965713i	$0.416406\pi$
$830$	9.71149	0.337091
$831$	23.1873	0.804358
$832$	−1.00000	−0.0346688
$833$	2.25306	0.0780639
$834$	−7.89186	−0.273273
$835$	11.0314	0.381759
$836$	−11.5894	−0.400829
$837$	6.32058	0.218471
$838$	−3.50389	−0.121040
$839$	−14.0849	−0.486264	−0.243132	−	0.969993i	$-0.578175\pi$
$839$	−14.0849	−0.486264	−0.243132	+	0.969993i	$0.578175\pi$
$840$	−6.17693	−0.213124
$841$	−28.7739	−0.992205
$842$	10.2446	0.353054
$843$	−30.2126	−1.04058
$844$	15.9089	0.547606
$845$	−2.43337	−0.0837106
$846$	8.72962	0.300130
$847$	−56.7824	−1.95107
$848$	0.740185	0.0254181
$849$	11.4266	0.392160
$850$	−3.73060	−0.127958
$851$	−17.1889	−0.589227
$852$	−8.83959	−0.302839
$853$	14.4201	0.493736	0.246868	−	0.969049i	$-0.420599\pi$
$853$	14.4201	0.493736	0.246868	+	0.969049i	$0.420599\pi$
$854$	15.3570	0.525505
$855$	4.88201	0.166961
$856$	10.8476	0.370763
$857$	39.1070	1.33587	0.667935	−	0.744219i	$-0.267178\pi$
$857$	39.1070	1.33587	0.667935	+	0.744219i	$0.267178\pi$
$858$	5.77661	0.197210
$859$	21.0370	0.717774	0.358887	−	0.933381i	$-0.383156\pi$
$859$	21.0370	0.717774	0.358887	+	0.933381i	$0.383156\pi$
$860$	3.86653	0.131847
$861$	3.29836	0.112408
$862$	−21.0949	−0.718495
$863$	5.08200	0.172993	0.0864966	−	0.996252i	$-0.472433\pi$
$863$	5.08200	0.172993	0.0864966	+	0.996252i	$0.472433\pi$
$864$	−1.00000	−0.0340207
$865$	22.1032	0.751532
$866$	14.0426	0.477189
$867$	0.603800	0.0205061
$868$	16.0443	0.544579
$869$	−14.9728	−0.507918
$870$	1.15696	0.0392246
$871$	3.11178	0.105439
$872$	13.9607	0.472769
$873$	12.2300	0.413922
$874$	−16.1577	−0.546542
$875$	−25.1938	−0.851705
$876$	8.59636	0.290444
$877$	−2.14792	−0.0725301	−0.0362650	−	0.999342i	$-0.511546\pi$
$877$	−2.14792	−0.0725301	−0.0362650	+	0.999342i	$0.511546\pi$
$878$	36.2834	1.22451
$879$	−11.3285	−0.382099
$880$	−14.0566	−0.473849
$881$	32.0163	1.07866	0.539329	−	0.842095i	$-0.318678\pi$
$881$	32.0163	1.07866	0.539329	+	0.842095i	$0.318678\pi$
$882$	−0.556418	−0.0187356
$883$	−12.9803	−0.436820	−0.218410	−	0.975857i	$-0.570087\pi$
$883$	−12.9803	−0.436820	−0.218410	+	0.975857i	$0.570087\pi$
$884$	4.04922	0.136190
$885$	−6.65131	−0.223581
$886$	−20.4398	−0.686687
$887$	57.7282	1.93832	0.969162	−	0.246426i	$-0.0792564\pi$
$887$	57.7282	1.93832	0.969162	+	0.246426i	$0.0792564\pi$
$888$	2.13431	0.0716228
$889$	1.88114	0.0630913
$890$	23.5122	0.788132
$891$	5.77661	0.193524
$892$	8.61008	0.288287
$893$	−17.5140	−0.586083
$894$	11.9282	0.398938
$895$	41.7507	1.39557
$896$	−2.53842	−0.0848027
$897$	8.05359	0.268901
$898$	8.93778	0.298257
$899$	−3.00515	−0.100227
$900$	0.921312	0.0307104
$901$	−2.99717	−0.0998503
$902$	7.50597	0.249921
$903$	−4.03344	−0.134224
$904$	−9.77485	−0.325107
$905$	26.6840	0.887005
$906$	−22.9927	−0.763881
$907$	−20.2845	−0.673537	−0.336769	−	0.941587i	$-0.609334\pi$
$907$	−20.2845	−0.673537	−0.336769	+	0.941587i	$0.609334\pi$
$908$	16.4584	0.546190
$909$	11.1817	0.370874
$910$	−6.17693	−0.204763
$911$	40.8680	1.35402	0.677008	−	0.735976i	$-0.263276\pi$
$911$	40.8680	1.35402	0.677008	+	0.735976i	$0.263276\pi$
$912$	2.00627	0.0664343
$913$	−23.0542	−0.762982
$914$	−6.45931	−0.213655
$915$	−14.7215	−0.486676
$916$	16.3135	0.539013
$917$	−32.5950	−1.07638
$918$	4.04922	0.133644
$919$	−1.12941	−0.0372557	−0.0186279	−	0.999826i	$-0.505930\pi$
$919$	−1.12941	−0.0372557	−0.0186279	+	0.999826i	$0.505930\pi$
$920$	−19.5974	−0.646107
$921$	−19.6190	−0.646469
$922$	19.3065	0.635826
$923$	−8.83959	−0.290959
$924$	14.6635	0.482392
$925$	−1.96637	−0.0646538
$926$	16.9074	0.555611
$927$	−1.00000	−0.0328443
$928$	0.475455	0.0156076
$929$	10.3220	0.338652	0.169326	−	0.985560i	$-0.445841\pi$
$929$	10.3220	0.338652	0.169326	+	0.985560i	$0.445841\pi$
$930$	−15.3803	−0.504341
$931$	1.11633	0.0365861
$932$	9.69358	0.317524
$933$	26.0380	0.852445
$934$	26.3793	0.863158
$935$	56.9185	1.86143
$936$	−1.00000	−0.0326860
$937$	−21.1152	−0.689804	−0.344902	−	0.938639i	$-0.612088\pi$
$937$	−21.1152	−0.689804	−0.344902	+	0.938639i	$0.612088\pi$
$938$	7.89900	0.257912
$939$	−1.14456	−0.0373512
$940$	−21.2424	−0.692852
$941$	−33.5643	−1.09416	−0.547082	−	0.837079i	$-0.684262\pi$
$941$	−33.5643	−1.09416	−0.547082	+	0.837079i	$0.684262\pi$
$942$	−7.41021	−0.241438
$943$	10.4646	0.340775
$944$	−2.73337	−0.0889636
$945$	−6.17693	−0.200936
$946$	−9.17877	−0.298428
$947$	6.52226	0.211945	0.105973	−	0.994369i	$-0.466204\pi$
$947$	6.52226	0.211945	0.105973	+	0.994369i	$0.466204\pi$
$948$	2.59198	0.0841835
$949$	8.59636	0.279050
$950$	−1.84840	−0.0599701
$951$	24.8040	0.804325
$952$	10.2786	0.333132
$953$	−40.4482	−1.31025	−0.655124	−	0.755522i	$-0.727383\pi$
$953$	−40.4482	−1.31025	−0.655124	+	0.755522i	$0.727383\pi$
$954$	0.740185	0.0239644
$955$	−48.0267	−1.55411
$956$	14.6164	0.472727
$957$	−2.74652	−0.0887822
$958$	−20.7552	−0.670569
$959$	−1.87897	−0.0606750
$960$	2.43337	0.0785368
$961$	8.94975	0.288702
$962$	2.13431	0.0688130
$963$	10.8476	0.349559
$964$	19.0765	0.614413
$965$	36.3326	1.16959
$966$	20.4434	0.657755
$967$	−18.6062	−0.598334	−0.299167	−	0.954201i	$-0.596709\pi$
$967$	−18.6062	−0.598334	−0.299167	+	0.954201i	$0.596709\pi$
$968$	22.3692	0.718973
$969$	−8.12384	−0.260975
$970$	−29.7601	−0.955539
$971$	6.51579	0.209102	0.104551	−	0.994520i	$-0.466660\pi$
$971$	6.51579	0.209102	0.104551	+	0.994520i	$0.466660\pi$
$972$	−1.00000	−0.0320750
$973$	−20.0329	−0.642224
$974$	−26.9324	−0.862968
$975$	0.921312	0.0295056
$976$	−6.04981	−0.193650
$977$	−41.9897	−1.34337	−0.671684	−	0.740838i	$-0.734429\pi$
$977$	−41.9897	−1.34337	−0.671684	+	0.740838i	$0.734429\pi$
$978$	−22.3303	−0.714043
$979$	−55.8159	−1.78388
$980$	1.35397	0.0432511
$981$	13.9607	0.445731
$982$	13.5361	0.431954
$983$	−58.5253	−1.86667	−0.933334	−	0.359010i	$-0.883114\pi$
$983$	−58.5253	−1.86667	−0.933334	+	0.359010i	$0.883114\pi$
$984$	−1.29937	−0.0414225
$985$	52.1480	1.66157
$986$	−1.92522	−0.0613116
$987$	22.1594	0.705343
$988$	2.00627	0.0638280
$989$	−12.7968	−0.406914
$990$	−14.0566	−0.446750
$991$	−36.4884	−1.15909	−0.579546	−	0.814939i	$-0.696770\pi$
$991$	−36.4884	−1.15909	−0.579546	+	0.814939i	$0.696770\pi$
$992$	−6.32058	−0.200679
$993$	−26.1663	−0.830363
$994$	−22.4386	−0.711710
$995$	11.6757	0.370145
$996$	3.99096	0.126458
$997$	−13.1037	−0.414999	−0.207499	−	0.978235i	$-0.566532\pi$
$997$	−13.1037	−0.414999	−0.207499	+	0.978235i	$0.566532\pi$
$998$	5.86009	0.185498
$999$	2.13431	0.0675266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bc.1.4	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bc.1.4	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.43337 q^{5} -1.00000 q^{6} -2.53842 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.43337 q^{5} -1.00000 q^{6} -2.53842 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.43337 q^{10} +5.77661 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.53842 q^{14} +2.43337 q^{15} +1.00000 q^{16} -4.04922 q^{17} +1.00000 q^{18} -2.00627 q^{19} -2.43337 q^{20} +2.53842 q^{21} +5.77661 q^{22} +8.05359 q^{23} -1.00000 q^{24} +0.921312 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.53842 q^{28} +0.475455 q^{29} +2.43337 q^{30} -6.32058 q^{31} +1.00000 q^{32} -5.77661 q^{33} -4.04922 q^{34} +6.17693 q^{35} +1.00000 q^{36} -2.13431 q^{37} -2.00627 q^{38} +1.00000 q^{39} -2.43337 q^{40} +1.29937 q^{41} +2.53842 q^{42} -1.58896 q^{43} +5.77661 q^{44} -2.43337 q^{45} +8.05359 q^{46} +8.72962 q^{47} -1.00000 q^{48} -0.556418 q^{49} +0.921312 q^{50} +4.04922 q^{51} -1.00000 q^{52} +0.740185 q^{53} -1.00000 q^{54} -14.0566 q^{55} -2.53842 q^{56} +2.00627 q^{57} +0.475455 q^{58} -2.73337 q^{59} +2.43337 q^{60} -6.04981 q^{61} -6.32058 q^{62} -2.53842 q^{63} +1.00000 q^{64} +2.43337 q^{65} -5.77661 q^{66} -3.11178 q^{67} -4.04922 q^{68} -8.05359 q^{69} +6.17693 q^{70} +8.83959 q^{71} +1.00000 q^{72} -8.59636 q^{73} -2.13431 q^{74} -0.921312 q^{75} -2.00627 q^{76} -14.6635 q^{77} +1.00000 q^{78} -2.59198 q^{79} -2.43337 q^{80} +1.00000 q^{81} +1.29937 q^{82} -3.99096 q^{83} +2.53842 q^{84} +9.85327 q^{85} -1.58896 q^{86} -0.475455 q^{87} +5.77661 q^{88} -9.66240 q^{89} -2.43337 q^{90} +2.53842 q^{91} +8.05359 q^{92} +6.32058 q^{93} +8.72962 q^{94} +4.88201 q^{95} -1.00000 q^{96} +12.2300 q^{97} -0.556418 q^{98} +5.77661 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) \) 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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