LMFDB - Embedded newform 4334.2.a.a.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
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} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} + \cdots + 4 $ x^15 - x^14 - 23x^13 + 19x^12 + 194x^11 - 124x^10 - 761x^9 + 353x^8 + 1417x^7 - 465x^6 - 1128x^5 + 288x^4 + 316x^3 - 79x^2 - 20*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.670430$ of defining polynomial
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.670430 q^{3} +1.00000 q^{4} +1.45227 q^{5} -0.670430 q^{6} +0.581735 q^{7} -1.00000 q^{8} -2.55052 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.670430 q^{3} +1.00000 q^{4} +1.45227 q^{5} -0.670430 q^{6} +0.581735 q^{7} -1.00000 q^{8} -2.55052 q^{9} -1.45227 q^{10} +1.00000 q^{11} +0.670430 q^{12} -1.74957 q^{13} -0.581735 q^{14} +0.973643 q^{15} +1.00000 q^{16} +0.878022 q^{17} +2.55052 q^{18} -1.44313 q^{19} +1.45227 q^{20} +0.390013 q^{21} -1.00000 q^{22} +1.79749 q^{23} -0.670430 q^{24} -2.89092 q^{25} +1.74957 q^{26} -3.72124 q^{27} +0.581735 q^{28} -4.72877 q^{29} -0.973643 q^{30} -7.21035 q^{31} -1.00000 q^{32} +0.670430 q^{33} -0.878022 q^{34} +0.844835 q^{35} -2.55052 q^{36} +9.47907 q^{37} +1.44313 q^{38} -1.17297 q^{39} -1.45227 q^{40} +2.09545 q^{41} -0.390013 q^{42} +3.72420 q^{43} +1.00000 q^{44} -3.70404 q^{45} -1.79749 q^{46} -7.30278 q^{47} +0.670430 q^{48} -6.66158 q^{49} +2.89092 q^{50} +0.588652 q^{51} -1.74957 q^{52} -5.86818 q^{53} +3.72124 q^{54} +1.45227 q^{55} -0.581735 q^{56} -0.967517 q^{57} +4.72877 q^{58} -2.37342 q^{59} +0.973643 q^{60} +3.41206 q^{61} +7.21035 q^{62} -1.48373 q^{63} +1.00000 q^{64} -2.54085 q^{65} -0.670430 q^{66} +2.44924 q^{67} +0.878022 q^{68} +1.20509 q^{69} -0.844835 q^{70} -4.12441 q^{71} +2.55052 q^{72} -8.26236 q^{73} -9.47907 q^{74} -1.93816 q^{75} -1.44313 q^{76} +0.581735 q^{77} +1.17297 q^{78} +11.2789 q^{79} +1.45227 q^{80} +5.15674 q^{81} -2.09545 q^{82} -13.1640 q^{83} +0.390013 q^{84} +1.27512 q^{85} -3.72420 q^{86} -3.17031 q^{87} -1.00000 q^{88} +10.9176 q^{89} +3.70404 q^{90} -1.01779 q^{91} +1.79749 q^{92} -4.83404 q^{93} +7.30278 q^{94} -2.09581 q^{95} -0.670430 q^{96} -17.5033 q^{97} +6.66158 q^{98} -2.55052 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9 + 7 * q^10 + 15 * q^11 - q^12 - q^13 - q^14 - 6 * q^15 + 15 * q^16 - 6 * q^17 - 2 * q^18 - 14 * q^19 - 7 * q^20 - 3 * q^21 - 15 * q^22 + 2 * q^23 + q^24 - 10 * q^25 + q^26 - 7 * q^27 + q^28 + 8 * q^29 + 6 * q^30 - 33 * q^31 - 15 * q^32 - q^33 + 6 * q^34 - 8 * q^35 + 2 * q^36 - 9 * q^37 + 14 * q^38 - 9 * q^39 + 7 * q^40 - 10 * q^41 + 3 * q^42 - 6 * q^43 + 15 * q^44 - 20 * q^45 - 2 * q^46 - q^47 - q^48 - 30 * q^49 + 10 * q^50 + 12 * q^51 - q^52 + 6 * q^53 + 7 * q^54 - 7 * q^55 - q^56 - 24 * q^57 - 8 * q^58 - 15 * q^59 - 6 * q^60 - 25 * q^61 + 33 * q^62 + 12 * q^63 + 15 * q^64 + 31 * q^65 + q^66 - 13 * q^67 - 6 * q^68 - 43 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 4 * q^73 + 9 * q^74 - 5 * q^75 - 14 * q^76 + q^77 + 9 * q^78 - 20 * q^79 - 7 * q^80 + 11 * q^81 + 10 * q^82 + q^83 - 3 * q^84 - q^85 + 6 * q^86 + 22 * q^87 - 15 * q^88 - 41 * q^89 + 20 * q^90 - 31 * q^91 + 2 * q^92 + 14 * q^93 + q^94 + 41 * q^95 + q^96 - 57 * q^97 + 30 * q^98 + 2 * 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
$3$	0.670430	0.387073	0.193536	−	0.981093i	$-0.438004\pi$
$3$	0.670430	0.387073	0.193536	+	0.981093i	$0.438004\pi$
$4$	1.00000	0.500000
$5$	1.45227	0.649474	0.324737	−	0.945804i	$-0.394724\pi$
$5$	1.45227	0.649474	0.324737	+	0.945804i	$0.394724\pi$
$6$	−0.670430	−0.273702
$7$	0.581735	0.219875	0.109938	−	0.993938i	$-0.464935\pi$
$7$	0.581735	0.219875	0.109938	+	0.993938i	$0.464935\pi$
$8$	−1.00000	−0.353553
$9$	−2.55052	−0.850175
$10$	−1.45227	−0.459247
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.670430	0.193536
$13$	−1.74957	−0.485244	−0.242622	−	0.970121i	$-0.578007\pi$
$13$	−1.74957	−0.485244	−0.242622	+	0.970121i	$0.578007\pi$
$14$	−0.581735	−0.155475
$15$	0.973643	0.251394
$16$	1.00000	0.250000
$17$	0.878022	0.212952	0.106476	−	0.994315i	$-0.466043\pi$
$17$	0.878022	0.212952	0.106476	+	0.994315i	$0.466043\pi$
$18$	2.55052	0.601164
$19$	−1.44313	−0.331077	−0.165538	−	0.986203i	$-0.552936\pi$
$19$	−1.44313	−0.331077	−0.165538	+	0.986203i	$0.552936\pi$
$20$	1.45227	0.324737
$21$	0.390013	0.0851077
$22$	−1.00000	−0.213201
$23$	1.79749	0.374803	0.187402	−	0.982283i	$-0.439993\pi$
$23$	1.79749	0.374803	0.187402	+	0.982283i	$0.439993\pi$
$24$	−0.670430	−0.136851
$25$	−2.89092	−0.578184
$26$	1.74957	0.343119
$27$	−3.72124	−0.716152
$28$	0.581735	0.109938
$29$	−4.72877	−0.878111	−0.439055	−	0.898460i	$-0.644687\pi$
$29$	−4.72877	−0.878111	−0.439055	+	0.898460i	$0.644687\pi$
$30$	−0.973643	−0.177762
$31$	−7.21035	−1.29502	−0.647509	−	0.762058i	$-0.724189\pi$
$31$	−7.21035	−1.29502	−0.647509	+	0.762058i	$0.724189\pi$
$32$	−1.00000	−0.176777
$33$	0.670430	0.116707
$34$	−0.878022	−0.150580
$35$	0.844835	0.142803
$36$	−2.55052	−0.425087
$37$	9.47907	1.55835	0.779175	−	0.626806i	$-0.215638\pi$
$37$	9.47907	1.55835	0.779175	+	0.626806i	$0.215638\pi$
$38$	1.44313	0.234107
$39$	−1.17297	−0.187825
$40$	−1.45227	−0.229624
$41$	2.09545	0.327254	0.163627	−	0.986522i	$-0.447681\pi$
$41$	2.09545	0.327254	0.163627	+	0.986522i	$0.447681\pi$
$42$	−0.390013	−0.0601803
$43$	3.72420	0.567935	0.283968	−	0.958834i	$-0.408349\pi$
$43$	3.72420	0.567935	0.283968	+	0.958834i	$0.408349\pi$
$44$	1.00000	0.150756
$45$	−3.70404	−0.552166
$46$	−1.79749	−0.265026
$47$	−7.30278	−1.06522	−0.532610	−	0.846361i	$-0.678789\pi$
$47$	−7.30278	−1.06522	−0.532610	+	0.846361i	$0.678789\pi$
$48$	0.670430	0.0967682
$49$	−6.66158	−0.951655
$50$	2.89092	0.408838
$51$	0.588652	0.0824278
$52$	−1.74957	−0.242622
$53$	−5.86818	−0.806056	−0.403028	−	0.915188i	$-0.632042\pi$
$53$	−5.86818	−0.806056	−0.403028	+	0.915188i	$0.632042\pi$
$54$	3.72124	0.506396
$55$	1.45227	0.195824
$56$	−0.581735	−0.0777376
$57$	−0.967517	−0.128151
$58$	4.72877	0.620918
$59$	−2.37342	−0.308993	−0.154497	−	0.987993i	$-0.549376\pi$
$59$	−2.37342	−0.308993	−0.154497	+	0.987993i	$0.549376\pi$
$60$	0.973643	0.125697
$61$	3.41206	0.436869	0.218435	−	0.975852i	$-0.429905\pi$
$61$	3.41206	0.436869	0.218435	+	0.975852i	$0.429905\pi$
$62$	7.21035	0.915716
$63$	−1.48373	−0.186932
$64$	1.00000	0.125000
$65$	−2.54085	−0.315153
$66$	−0.670430	−0.0825242
$67$	2.44924	0.299222	0.149611	−	0.988745i	$-0.452198\pi$
$67$	2.44924	0.299222	0.149611	+	0.988745i	$0.452198\pi$
$68$	0.878022	0.106476
$69$	1.20509	0.145076
$70$	−0.844835	−0.100977
$71$	−4.12441	−0.489477	−0.244739	−	0.969589i	$-0.578702\pi$
$71$	−4.12441	−0.489477	−0.244739	+	0.969589i	$0.578702\pi$
$72$	2.55052	0.300582
$73$	−8.26236	−0.967036	−0.483518	−	0.875334i	$-0.660641\pi$
$73$	−8.26236	−0.967036	−0.483518	+	0.875334i	$0.660641\pi$
$74$	−9.47907	−1.10192
$75$	−1.93816	−0.223799
$76$	−1.44313	−0.165538
$77$	0.581735	0.0662949
$78$	1.17297	0.132812
$79$	11.2789	1.26898	0.634489	−	0.772932i	$-0.281211\pi$
$79$	11.2789	1.26898	0.634489	+	0.772932i	$0.281211\pi$
$80$	1.45227	0.162368
$81$	5.15674	0.572971
$82$	−2.09545	−0.231404
$83$	−13.1640	−1.44494	−0.722469	−	0.691403i	$-0.756993\pi$
$83$	−13.1640	−1.44494	−0.722469	+	0.691403i	$0.756993\pi$
$84$	0.390013	0.0425539
$85$	1.27512	0.138306
$86$	−3.72420	−0.401591
$87$	−3.17031	−0.339893
$88$	−1.00000	−0.106600
$89$	10.9176	1.15726	0.578631	−	0.815589i	$-0.303587\pi$
$89$	10.9176	1.15726	0.578631	+	0.815589i	$0.303587\pi$
$90$	3.70404	0.390440
$91$	−1.01779	−0.106693
$92$	1.79749	0.187402
$93$	−4.83404	−0.501266
$94$	7.30278	0.753224
$95$	−2.09581	−0.215025
$96$	−0.670430	−0.0684255
$97$	−17.5033	−1.77719	−0.888596	−	0.458690i	$-0.848319\pi$
$97$	−17.5033	−1.77719	−0.888596	+	0.458690i	$0.848319\pi$
$98$	6.66158	0.672922
$99$	−2.55052	−0.256337
$100$	−2.89092	−0.289092
$101$	−15.8836	−1.58048	−0.790238	−	0.612801i	$-0.790043\pi$
$101$	−15.8836	−1.58048	−0.790238	+	0.612801i	$0.790043\pi$
$102$	−0.588652	−0.0582853
$103$	−5.48887	−0.540835	−0.270417	−	0.962743i	$-0.587162\pi$
$103$	−5.48887	−0.540835	−0.270417	+	0.962743i	$0.587162\pi$
$104$	1.74957	0.171560
$105$	0.566403	0.0552752
$106$	5.86818	0.569968
$107$	−9.85634	−0.952848	−0.476424	−	0.879216i	$-0.658067\pi$
$107$	−9.85634	−0.952848	−0.476424	+	0.879216i	$0.658067\pi$
$108$	−3.72124	−0.358076
$109$	1.41601	0.135629	0.0678146	−	0.997698i	$-0.478397\pi$
$109$	1.41601	0.135629	0.0678146	+	0.997698i	$0.478397\pi$
$110$	−1.45227	−0.138468
$111$	6.35506	0.603195
$112$	0.581735	0.0549688
$113$	4.18759	0.393936	0.196968	−	0.980410i	$-0.436891\pi$
$113$	4.18759	0.393936	0.196968	+	0.980410i	$0.436891\pi$
$114$	0.967517	0.0906163
$115$	2.61044	0.243425
$116$	−4.72877	−0.439055
$117$	4.46233	0.412542
$118$	2.37342	0.218491
$119$	0.510776	0.0468228
$120$	−0.973643	−0.0888811
$121$	1.00000	0.0909091
$122$	−3.41206	−0.308913
$123$	1.40485	0.126671
$124$	−7.21035	−0.647509
$125$	−11.4597	−1.02499
$126$	1.48373	0.132181
$127$	−11.2158	−0.995246	−0.497623	−	0.867394i	$-0.665794\pi$
$127$	−11.2158	−0.995246	−0.497623	+	0.867394i	$0.665794\pi$
$128$	−1.00000	−0.0883883
$129$	2.49681	0.219832
$130$	2.54085	0.222847
$131$	−4.42703	−0.386791	−0.193396	−	0.981121i	$-0.561950\pi$
$131$	−4.42703	−0.386791	−0.193396	+	0.981121i	$0.561950\pi$
$132$	0.670430	0.0583534
$133$	−0.839519	−0.0727955
$134$	−2.44924	−0.211582
$135$	−5.40423	−0.465122
$136$	−0.878022	−0.0752898
$137$	7.62121	0.651124	0.325562	−	0.945521i	$-0.394447\pi$
$137$	7.62121	0.651124	0.325562	+	0.945521i	$0.394447\pi$
$138$	−1.20509	−0.102584
$139$	2.70108	0.229102	0.114551	−	0.993417i	$-0.463457\pi$
$139$	2.70108	0.229102	0.114551	+	0.993417i	$0.463457\pi$
$140$	0.844835	0.0714016
$141$	−4.89600	−0.412318
$142$	4.12441	0.346113
$143$	−1.74957	−0.146307
$144$	−2.55052	−0.212544
$145$	−6.86744	−0.570310
$146$	8.26236	0.683798
$147$	−4.46613	−0.368360
$148$	9.47907	0.779175
$149$	8.17643	0.669839	0.334920	−	0.942247i	$-0.391291\pi$
$149$	8.17643	0.669839	0.334920	+	0.942247i	$0.391291\pi$
$150$	1.93816	0.158250
$151$	20.6190	1.67795	0.838977	−	0.544168i	$-0.183154\pi$
$151$	20.6190	1.67795	0.838977	+	0.544168i	$0.183154\pi$
$152$	1.44313	0.117053
$153$	−2.23942	−0.181046
$154$	−0.581735	−0.0468776
$155$	−10.4714	−0.841080
$156$	−1.17297	−0.0939124
$157$	−13.8852	−1.10816	−0.554079	−	0.832464i	$-0.686929\pi$
$157$	−13.8852	−1.10816	−0.554079	+	0.832464i	$0.686929\pi$
$158$	−11.2789	−0.897303
$159$	−3.93420	−0.312002
$160$	−1.45227	−0.114812
$161$	1.04567	0.0824100
$162$	−5.15674	−0.405152
$163$	−8.11776	−0.635832	−0.317916	−	0.948119i	$-0.602983\pi$
$163$	−8.11776	−0.635832	−0.317916	+	0.948119i	$0.602983\pi$
$164$	2.09545	0.163627
$165$	0.973643	0.0757980
$166$	13.1640	1.02173
$167$	−18.7088	−1.44773	−0.723863	−	0.689943i	$-0.757635\pi$
$167$	−18.7088	−1.44773	−0.723863	+	0.689943i	$0.757635\pi$
$168$	−0.390013	−0.0300901
$169$	−9.93900	−0.764538
$170$	−1.27512	−0.0977974
$171$	3.68074	0.281473
$172$	3.72420	0.283968
$173$	13.1071	0.996515	0.498257	−	0.867029i	$-0.333973\pi$
$173$	13.1071	0.996515	0.498257	+	0.867029i	$0.333973\pi$
$174$	3.17031	0.240341
$175$	−1.68175	−0.127128
$176$	1.00000	0.0753778
$177$	−1.59121	−0.119603
$178$	−10.9176	−0.818308
$179$	6.14776	0.459505	0.229753	−	0.973249i	$-0.426208\pi$
$179$	6.14776	0.459505	0.229753	+	0.973249i	$0.426208\pi$
$180$	−3.70404	−0.276083
$181$	−20.2627	−1.50611	−0.753056	−	0.657956i	$-0.771421\pi$
$181$	−20.2627	−1.50611	−0.753056	+	0.657956i	$0.771421\pi$
$182$	1.01779	0.0754434
$183$	2.28754	0.169100
$184$	−1.79749	−0.132513
$185$	13.7661	1.01211
$186$	4.83404	0.354449
$187$	0.878022	0.0642073
$188$	−7.30278	−0.532610
$189$	−2.16477	−0.157464
$190$	2.09581	0.152046
$191$	16.7647	1.21305	0.606527	−	0.795063i	$-0.292562\pi$
$191$	16.7647	1.21305	0.606527	+	0.795063i	$0.292562\pi$
$192$	0.670430	0.0483841
$193$	24.3605	1.75351	0.876755	−	0.480938i	$-0.159704\pi$
$193$	24.3605	1.75351	0.876755	+	0.480938i	$0.159704\pi$
$194$	17.5033	1.25667
$195$	−1.70346	−0.121987
$196$	−6.66158	−0.475827
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	2.55052	0.181258
$199$	−9.12252	−0.646678	−0.323339	−	0.946283i	$-0.604805\pi$
$199$	−9.12252	−0.646678	−0.323339	+	0.946283i	$0.604805\pi$
$200$	2.89092	0.204419
$201$	1.64204	0.115821
$202$	15.8836	1.11756
$203$	−2.75089	−0.193075
$204$	0.588652	0.0412139
$205$	3.04315	0.212543
$206$	5.48887	0.382428
$207$	−4.58455	−0.318648
$208$	−1.74957	−0.121311
$209$	−1.44313	−0.0998234
$210$	−0.566403	−0.0390855
$211$	2.65293	0.182635	0.0913177	−	0.995822i	$-0.470892\pi$
$211$	2.65293	0.182635	0.0913177	+	0.995822i	$0.470892\pi$
$212$	−5.86818	−0.403028
$213$	−2.76513	−0.189463
$214$	9.85634	0.673765
$215$	5.40853	0.368859
$216$	3.72124	0.253198
$217$	−4.19452	−0.284742
$218$	−1.41601	−0.0959044
$219$	−5.53933	−0.374313
$220$	1.45227	0.0979118
$221$	−1.53616	−0.103333
$222$	−6.35506	−0.426523
$223$	−20.2664	−1.35714	−0.678568	−	0.734537i	$-0.737399\pi$
$223$	−20.2664	−1.35714	−0.678568	+	0.734537i	$0.737399\pi$
$224$	−0.581735	−0.0388688
$225$	7.37336	0.491557
$226$	−4.18759	−0.278555
$227$	12.0399	0.799113	0.399557	−	0.916708i	$-0.369164\pi$
$227$	12.0399	0.799113	0.399557	+	0.916708i	$0.369164\pi$
$228$	−0.967517	−0.0640754
$229$	6.05609	0.400198	0.200099	−	0.979776i	$-0.435874\pi$
$229$	6.05609	0.400198	0.200099	+	0.979776i	$0.435874\pi$
$230$	−2.61044	−0.172127
$231$	0.390013	0.0256609
$232$	4.72877	0.310459
$233$	21.5358	1.41086	0.705429	−	0.708781i	$-0.250754\pi$
$233$	21.5358	1.41086	0.705429	+	0.708781i	$0.250754\pi$
$234$	−4.46233	−0.291711
$235$	−10.6056	−0.691832
$236$	−2.37342	−0.154497
$237$	7.56172	0.491187
$238$	−0.510776	−0.0331087
$239$	−0.970463	−0.0627741	−0.0313870	−	0.999507i	$-0.509992\pi$
$239$	−0.970463	−0.0627741	−0.0313870	+	0.999507i	$0.509992\pi$
$240$	0.973643	0.0628484
$241$	5.52821	0.356103	0.178052	−	0.984021i	$-0.443021\pi$
$241$	5.52821	0.356103	0.178052	+	0.984021i	$0.443021\pi$
$242$	−1.00000	−0.0642824
$243$	14.6209	0.937934
$244$	3.41206	0.218435
$245$	−9.67440	−0.618075
$246$	−1.40485	−0.0895701
$247$	2.52486	0.160653
$248$	7.21035	0.457858
$249$	−8.82555	−0.559296
$250$	11.4597	0.724777
$251$	−5.81521	−0.367053	−0.183526	−	0.983015i	$-0.558751\pi$
$251$	−5.81521	−0.367053	−0.183526	+	0.983015i	$0.558751\pi$
$252$	−1.48373	−0.0934662
$253$	1.79749	0.113007
$254$	11.2158	0.703745
$255$	0.854880	0.0535347
$256$	1.00000	0.0625000
$257$	−12.8393	−0.800894	−0.400447	−	0.916320i	$-0.631145\pi$
$257$	−12.8393	−0.800894	−0.400447	+	0.916320i	$0.631145\pi$
$258$	−2.49681	−0.155445
$259$	5.51431	0.342643
$260$	−2.54085	−0.157577
$261$	12.0608	0.746547
$262$	4.42703	0.273503
$263$	24.7083	1.52358	0.761788	−	0.647826i	$-0.224322\pi$
$263$	24.7083	1.52358	0.761788	+	0.647826i	$0.224322\pi$
$264$	−0.670430	−0.0412621
$265$	−8.52216	−0.523512
$266$	0.839519	0.0514742
$267$	7.31948	0.447945
$268$	2.44924	0.149611
$269$	14.2598	0.869436	0.434718	−	0.900567i	$-0.356848\pi$
$269$	14.2598	0.869436	0.434718	+	0.900567i	$0.356848\pi$
$270$	5.40423	0.328891
$271$	−1.52473	−0.0926208	−0.0463104	−	0.998927i	$-0.514746\pi$
$271$	−1.52473	−0.0926208	−0.0463104	+	0.998927i	$0.514746\pi$
$272$	0.878022	0.0532379
$273$	−0.682355	−0.0412980
$274$	−7.62121	−0.460414
$275$	−2.89092	−0.174329
$276$	1.20509	0.0725381
$277$	−12.3034	−0.739242	−0.369621	−	0.929183i	$-0.620513\pi$
$277$	−12.3034	−0.739242	−0.369621	+	0.929183i	$0.620513\pi$
$278$	−2.70108	−0.162000
$279$	18.3902	1.10099
$280$	−0.844835	−0.0504885
$281$	2.57833	0.153810	0.0769052	−	0.997038i	$-0.475496\pi$
$281$	2.57833	0.153810	0.0769052	+	0.997038i	$0.475496\pi$
$282$	4.89600	0.291553
$283$	18.5999	1.10565	0.552825	−	0.833297i	$-0.313550\pi$
$283$	18.5999	1.10565	0.552825	+	0.833297i	$0.313550\pi$
$284$	−4.12441	−0.244739
$285$	−1.40509	−0.0832305
$286$	1.74957	0.103454
$287$	1.21900	0.0719551
$288$	2.55052	0.150291
$289$	−16.2291	−0.954652
$290$	6.86744	0.403270
$291$	−11.7348	−0.687903
$292$	−8.26236	−0.483518
$293$	−10.5237	−0.614799	−0.307399	−	0.951581i	$-0.599459\pi$
$293$	−10.5237	−0.614799	−0.307399	+	0.951581i	$0.599459\pi$
$294$	4.46613	0.260470
$295$	−3.44684	−0.200683
$296$	−9.47907	−0.550960
$297$	−3.72124	−0.215928
$298$	−8.17643	−0.473648
$299$	−3.14484	−0.181871
$300$	−1.93816	−0.111900
$301$	2.16650	0.124875
$302$	−20.6190	−1.18649
$303$	−10.6488	−0.611759
$304$	−1.44313	−0.0827692
$305$	4.95522	0.283735
$306$	2.23942	0.128019
$307$	−24.9200	−1.42226	−0.711129	−	0.703061i	$-0.751816\pi$
$307$	−24.9200	−1.42226	−0.711129	+	0.703061i	$0.751816\pi$
$308$	0.581735	0.0331474
$309$	−3.67990	−0.209342
$310$	10.4714	0.594733
$311$	−4.58415	−0.259943	−0.129972	−	0.991518i	$-0.541489\pi$
$311$	−4.58415	−0.259943	−0.129972	+	0.991518i	$0.541489\pi$
$312$	1.17297	0.0664061
$313$	−9.44214	−0.533701	−0.266851	−	0.963738i	$-0.585983\pi$
$313$	−9.44214	−0.533701	−0.266851	+	0.963738i	$0.585983\pi$
$314$	13.8852	0.783586
$315$	−2.15477	−0.121408
$316$	11.2789	0.634489
$317$	−6.26478	−0.351865	−0.175933	−	0.984402i	$-0.556294\pi$
$317$	−6.26478	−0.351865	−0.175933	+	0.984402i	$0.556294\pi$
$318$	3.93420	0.220619
$319$	−4.72877	−0.264760
$320$	1.45227	0.0811842
$321$	−6.60798	−0.368822
$322$	−1.04567	−0.0582726
$323$	−1.26710	−0.0705033
$324$	5.15674	0.286486
$325$	5.05787	0.280560
$326$	8.11776	0.449601
$327$	0.949336	0.0524984
$328$	−2.09545	−0.115702
$329$	−4.24828	−0.234215
$330$	−0.973643	−0.0535973
$331$	−36.3314	−1.99695	−0.998477	−	0.0551670i	$-0.982431\pi$
$331$	−36.3314	−1.99695	−0.998477	+	0.0551670i	$0.982431\pi$
$332$	−13.1640	−0.722469
$333$	−24.1766	−1.32487
$334$	18.7088	1.02370
$335$	3.55695	0.194337
$336$	0.390013	0.0212769
$337$	−1.20970	−0.0658963	−0.0329481	−	0.999457i	$-0.510490\pi$
$337$	−1.20970	−0.0658963	−0.0329481	+	0.999457i	$0.510490\pi$
$338$	9.93900	0.540610
$339$	2.80749	0.152482
$340$	1.27512	0.0691532
$341$	−7.21035	−0.390463
$342$	−3.68074	−0.199031
$343$	−7.94742	−0.429121
$344$	−3.72420	−0.200795
$345$	1.75012	0.0942232
$346$	−13.1071	−0.704642
$347$	−27.4242	−1.47221	−0.736104	−	0.676869i	$-0.763336\pi$
$347$	−27.4242	−1.47221	−0.736104	+	0.676869i	$0.763336\pi$
$348$	−3.17031	−0.169946
$349$	−3.73862	−0.200124	−0.100062	−	0.994981i	$-0.531904\pi$
$349$	−3.73862	−0.200124	−0.100062	+	0.994981i	$0.531904\pi$
$350$	1.68175	0.0898933
$351$	6.51057	0.347509
$352$	−1.00000	−0.0533002
$353$	13.7047	0.729428	0.364714	−	0.931120i	$-0.381167\pi$
$353$	13.7047	0.729428	0.364714	+	0.931120i	$0.381167\pi$
$354$	1.59121	0.0845721
$355$	−5.98974	−0.317902
$356$	10.9176	0.578631
$357$	0.342440	0.0181238
$358$	−6.14776	−0.324919
$359$	2.09314	0.110472	0.0552359	−	0.998473i	$-0.482409\pi$
$359$	2.09314	0.110472	0.0552359	+	0.998473i	$0.482409\pi$
$360$	3.70404	0.195220
$361$	−16.9174	−0.890388
$362$	20.2627	1.06498
$363$	0.670430	0.0351884
$364$	−1.01779	−0.0533466
$365$	−11.9992	−0.628064
$366$	−2.28754	−0.119572
$367$	−20.2990	−1.05960	−0.529799	−	0.848123i	$-0.677733\pi$
$367$	−20.2990	−1.05960	−0.529799	+	0.848123i	$0.677733\pi$
$368$	1.79749	0.0937008
$369$	−5.34449	−0.278223
$370$	−13.7661	−0.715668
$371$	−3.41372	−0.177232
$372$	−4.83404	−0.250633
$373$	23.8442	1.23460	0.617302	−	0.786726i	$-0.288226\pi$
$373$	23.8442	1.23460	0.617302	+	0.786726i	$0.288226\pi$
$374$	−0.878022	−0.0454014
$375$	−7.68294	−0.396745
$376$	7.30278	0.376612
$377$	8.27333	0.426098
$378$	2.16477	0.111344
$379$	2.46301	0.126516	0.0632582	−	0.997997i	$-0.479851\pi$
$379$	2.46301	0.126516	0.0632582	+	0.997997i	$0.479851\pi$
$380$	−2.09581	−0.107513
$381$	−7.51944	−0.385233
$382$	−16.7647	−0.857759
$383$	−7.71307	−0.394120	−0.197060	−	0.980391i	$-0.563139\pi$
$383$	−7.71307	−0.394120	−0.197060	+	0.980391i	$0.563139\pi$
$384$	−0.670430	−0.0342127
$385$	0.844835	0.0430568
$386$	−24.3605	−1.23992
$387$	−9.49866	−0.482844
$388$	−17.5033	−0.888596
$389$	−25.4912	−1.29245	−0.646227	−	0.763145i	$-0.723654\pi$
$389$	−25.4912	−1.29245	−0.646227	+	0.763145i	$0.723654\pi$
$390$	1.70346	0.0862580
$391$	1.57824	0.0798150
$392$	6.66158	0.336461
$393$	−2.96801	−0.149716
$394$	−1.00000	−0.0503793
$395$	16.3800	0.824167
$396$	−2.55052	−0.128169
$397$	−5.90761	−0.296495	−0.148247	−	0.988950i	$-0.547363\pi$
$397$	−5.90761	−0.296495	−0.148247	+	0.988950i	$0.547363\pi$
$398$	9.12252	0.457270
$399$	−0.562839	−0.0281772
$400$	−2.89092	−0.144546
$401$	11.8176	0.590143	0.295071	−	0.955475i	$-0.404657\pi$
$401$	11.8176	0.590143	0.295071	+	0.955475i	$0.404657\pi$
$402$	−1.64204	−0.0818977
$403$	12.6150	0.628400
$404$	−15.8836	−0.790238
$405$	7.48897	0.372130
$406$	2.75089	0.136524
$407$	9.47907	0.469860
$408$	−0.588652	−0.0291426
$409$	−31.3317	−1.54925	−0.774626	−	0.632420i	$-0.782062\pi$
$409$	−31.3317	−1.54925	−0.774626	+	0.632420i	$0.782062\pi$
$410$	−3.04315	−0.150291
$411$	5.10949	0.252032
$412$	−5.48887	−0.270417
$413$	−1.38070	−0.0679400
$414$	4.58455	0.225318
$415$	−19.1177	−0.938449
$416$	1.74957	0.0857798
$417$	1.81088	0.0886793
$418$	1.44313	0.0705858
$419$	21.8166	1.06581	0.532905	−	0.846175i	$-0.321100\pi$
$419$	21.8166	1.06581	0.532905	+	0.846175i	$0.321100\pi$
$420$	0.566403	0.0276376
$421$	11.9342	0.581635	0.290818	−	0.956779i	$-0.406073\pi$
$421$	11.9342	0.581635	0.290818	+	0.956779i	$0.406073\pi$
$422$	−2.65293	−0.129143
$423$	18.6259	0.905623
$424$	5.86818	0.284984
$425$	−2.53829	−0.123125
$426$	2.76513	0.133971
$427$	1.98491	0.0960567
$428$	−9.85634	−0.476424
$429$	−1.17297	−0.0566313
$430$	−5.40853	−0.260823
$431$	20.4229	0.983739	0.491869	−	0.870669i	$-0.336314\pi$
$431$	20.4229	0.983739	0.491869	+	0.870669i	$0.336314\pi$
$432$	−3.72124	−0.179038
$433$	−21.6487	−1.04037	−0.520186	−	0.854053i	$-0.674137\pi$
$433$	−21.6487	−1.04037	−0.520186	+	0.854053i	$0.674137\pi$
$434$	4.19452	0.201343
$435$	−4.60414	−0.220751
$436$	1.41601	0.0678146
$437$	−2.59402	−0.124089
$438$	5.53933	0.264680
$439$	−28.0791	−1.34014	−0.670071	−	0.742297i	$-0.733736\pi$
$439$	−28.0791	−1.34014	−0.670071	+	0.742297i	$0.733736\pi$
$440$	−1.45227	−0.0692341
$441$	16.9905	0.809073
$442$	1.53616	0.0730678
$443$	33.1630	1.57562	0.787810	−	0.615918i	$-0.211215\pi$
$443$	33.1630	1.57562	0.787810	+	0.615918i	$0.211215\pi$
$444$	6.35506	0.301598
$445$	15.8553	0.751611
$446$	20.2664	0.959640
$447$	5.48172	0.259277
$448$	0.581735	0.0274844
$449$	−9.86370	−0.465497	−0.232748	−	0.972537i	$-0.574772\pi$
$449$	−9.86370	−0.465497	−0.232748	+	0.972537i	$0.574772\pi$
$450$	−7.37336	−0.347584
$451$	2.09545	0.0986708
$452$	4.18759	0.196968
$453$	13.8236	0.649490
$454$	−12.0399	−0.565058
$455$	−1.47810	−0.0692944
$456$	0.967517	0.0453081
$457$	27.8772	1.30404	0.652020	−	0.758202i	$-0.273922\pi$
$457$	27.8772	1.30404	0.652020	+	0.758202i	$0.273922\pi$
$458$	−6.05609	−0.282983
$459$	−3.26733	−0.152506
$460$	2.61044	0.121712
$461$	6.85531	0.319283	0.159642	−	0.987175i	$-0.448966\pi$
$461$	6.85531	0.319283	0.159642	+	0.987175i	$0.448966\pi$
$462$	−0.390013	−0.0181450
$463$	24.8941	1.15693	0.578463	−	0.815709i	$-0.303653\pi$
$463$	24.8941	1.15693	0.578463	+	0.815709i	$0.303653\pi$
$464$	−4.72877	−0.219528
$465$	−7.02031	−0.325559
$466$	−21.5358	−0.997627
$467$	24.4652	1.13211	0.566056	−	0.824367i	$-0.308469\pi$
$467$	24.4652	1.13211	0.566056	+	0.824367i	$0.308469\pi$
$468$	4.46233	0.206271
$469$	1.42481	0.0657916
$470$	10.6056	0.489199
$471$	−9.30904	−0.428938
$472$	2.37342	0.109246
$473$	3.72420	0.171239
$474$	−7.56172	−0.347321
$475$	4.17197	0.191423
$476$	0.510776	0.0234114
$477$	14.9669	0.685288
$478$	0.970463	0.0443880
$479$	6.97325	0.318616	0.159308	−	0.987229i	$-0.449074\pi$
$479$	6.97325	0.318616	0.159308	+	0.987229i	$0.449074\pi$
$480$	−0.973643	−0.0444405
$481$	−16.5843	−0.756180
$482$	−5.52821	−0.251803
$483$	0.701045	0.0318987
$484$	1.00000	0.0454545
$485$	−25.4195	−1.15424
$486$	−14.6209	−0.663220
$487$	20.1608	0.913574	0.456787	−	0.889576i	$-0.349000\pi$
$487$	20.1608	0.913574	0.456787	+	0.889576i	$0.349000\pi$
$488$	−3.41206	−0.154457
$489$	−5.44239	−0.246113
$490$	9.67440	0.437045
$491$	−10.0159	−0.452012	−0.226006	−	0.974126i	$-0.572567\pi$
$491$	−10.0159	−0.452012	−0.226006	+	0.974126i	$0.572567\pi$
$492$	1.40485	0.0633356
$493$	−4.15196	−0.186995
$494$	−2.52486	−0.113599
$495$	−3.70404	−0.166484
$496$	−7.21035	−0.323754
$497$	−2.39931	−0.107624
$498$	8.82555	0.395482
$499$	−14.6515	−0.655891	−0.327946	−	0.944697i	$-0.606356\pi$
$499$	−14.6515	−0.655891	−0.327946	+	0.944697i	$0.606356\pi$
$500$	−11.4597	−0.512494
$501$	−12.5429	−0.560376
$502$	5.81521	0.259546
$503$	30.5437	1.36188	0.680939	−	0.732340i	$-0.261572\pi$
$503$	30.5437	1.36188	0.680939	+	0.732340i	$0.261572\pi$
$504$	1.48373	0.0660906
$505$	−23.0672	−1.02648
$506$	−1.79749	−0.0799083
$507$	−6.66340	−0.295932
$508$	−11.2158	−0.497623
$509$	−27.1028	−1.20131	−0.600655	−	0.799509i	$-0.705093\pi$
$509$	−27.1028	−1.20131	−0.600655	+	0.799509i	$0.705093\pi$
$510$	−0.854880	−0.0378547
$511$	−4.80650	−0.212627
$512$	−1.00000	−0.0441942
$513$	5.37023	0.237101
$514$	12.8393	0.566317
$515$	−7.97131	−0.351258
$516$	2.49681	0.109916
$517$	−7.30278	−0.321176
$518$	−5.51431	−0.242285
$519$	8.78740	0.385724
$520$	2.54085	0.111423
$521$	−30.2126	−1.32364	−0.661819	−	0.749664i	$-0.730215\pi$
$521$	−30.2126	−1.32364	−0.661819	+	0.749664i	$0.730215\pi$
$522$	−12.0608	−0.527889
$523$	−25.5303	−1.11636	−0.558180	−	0.829720i	$-0.688500\pi$
$523$	−25.5303	−1.11636	−0.558180	+	0.829720i	$0.688500\pi$
$524$	−4.42703	−0.193396
$525$	−1.12750	−0.0492079
$526$	−24.7083	−1.07733
$527$	−6.33085	−0.275776
$528$	0.670430	0.0291767
$529$	−19.7690	−0.859522
$530$	8.52216	0.370179
$531$	6.05347	0.262698
$532$	−0.839519	−0.0363978
$533$	−3.66614	−0.158798
$534$	−7.31948	−0.316745
$535$	−14.3140	−0.618850
$536$	−2.44924	−0.105791
$537$	4.12164	0.177862
$538$	−14.2598	−0.614784
$539$	−6.66158	−0.286935
$540$	−5.40423	−0.232561
$541$	−40.3397	−1.73434	−0.867170	−	0.498012i	$-0.834063\pi$
$541$	−40.3397	−1.73434	−0.867170	+	0.498012i	$0.834063\pi$
$542$	1.52473	0.0654928
$543$	−13.5847	−0.582975
$544$	−0.878022	−0.0376449
$545$	2.05643	0.0880876
$546$	0.682355	0.0292021
$547$	25.3920	1.08568	0.542842	−	0.839835i	$-0.317348\pi$
$547$	25.3920	1.08568	0.542842	+	0.839835i	$0.317348\pi$
$548$	7.62121	0.325562
$549$	−8.70253	−0.371415
$550$	2.89092	0.123269
$551$	6.82423	0.290722
$552$	−1.20509	−0.0512922
$553$	6.56134	0.279017
$554$	12.3034	0.522723
$555$	9.22924	0.391759
$556$	2.70108	0.114551
$557$	31.1387	1.31939	0.659695	−	0.751533i	$-0.270685\pi$
$557$	31.1387	1.31939	0.659695	+	0.751533i	$0.270685\pi$
$558$	−18.3902	−0.778518
$559$	−6.51576	−0.275587
$560$	0.844835	0.0357008
$561$	0.588652	0.0248529
$562$	−2.57833	−0.108760
$563$	−9.46949	−0.399091	−0.199546	−	0.979889i	$-0.563947\pi$
$563$	−9.46949	−0.399091	−0.199546	+	0.979889i	$0.563947\pi$
$564$	−4.89600	−0.206159
$565$	6.08150	0.255851
$566$	−18.5999	−0.781813
$567$	2.99986	0.125982
$568$	4.12441	0.173056
$569$	−20.9254	−0.877239	−0.438620	−	0.898673i	$-0.644532\pi$
$569$	−20.9254	−0.877239	−0.438620	+	0.898673i	$0.644532\pi$
$570$	1.40509	0.0588529
$571$	9.73681	0.407473	0.203736	−	0.979026i	$-0.434691\pi$
$571$	9.73681	0.407473	0.203736	+	0.979026i	$0.434691\pi$
$572$	−1.74957	−0.0731533
$573$	11.2396	0.469540
$574$	−1.21900	−0.0508799
$575$	−5.19641	−0.216705
$576$	−2.55052	−0.106272
$577$	−23.0279	−0.958665	−0.479333	−	0.877633i	$-0.659121\pi$
$577$	−23.0279	−0.958665	−0.479333	+	0.877633i	$0.659121\pi$
$578$	16.2291	0.675041
$579$	16.3320	0.678736
$580$	−6.86744	−0.285155
$581$	−7.65797	−0.317706
$582$	11.7348	0.486421
$583$	−5.86818	−0.243035
$584$	8.26236	0.341899
$585$	6.48049	0.267935
$586$	10.5237	0.434728
$587$	38.8693	1.60431	0.802153	−	0.597118i	$-0.203688\pi$
$587$	38.8693	1.60431	0.802153	+	0.597118i	$0.203688\pi$
$588$	−4.46613	−0.184180
$589$	10.4055	0.428750
$590$	3.44684	0.141904
$591$	0.670430	0.0275778
$592$	9.47907	0.389588
$593$	−39.6982	−1.63021	−0.815105	−	0.579313i	$-0.803321\pi$
$593$	−39.6982	−1.63021	−0.815105	+	0.579313i	$0.803321\pi$
$594$	3.72124	0.152684
$595$	0.741783	0.0304102
$596$	8.17643	0.334920
$597$	−6.11601	−0.250312
$598$	3.14484	0.128602
$599$	−12.4375	−0.508183	−0.254092	−	0.967180i	$-0.581776\pi$
$599$	−12.4375	−0.508183	−0.254092	+	0.967180i	$0.581776\pi$
$600$	1.93816	0.0791250
$601$	28.5369	1.16405	0.582023	−	0.813172i	$-0.302261\pi$
$601$	28.5369	1.16405	0.582023	+	0.813172i	$0.302261\pi$
$602$	−2.16650	−0.0882999
$603$	−6.24685	−0.254391
$604$	20.6190	0.838977
$605$	1.45227	0.0590430
$606$	10.6488	0.432579
$607$	−14.0992	−0.572269	−0.286134	−	0.958190i	$-0.592370\pi$
$607$	−14.0992	−0.572269	−0.286134	+	0.958190i	$0.592370\pi$
$608$	1.44313	0.0585266
$609$	−1.84428	−0.0747340
$610$	−4.95522	−0.200631
$611$	12.7767	0.516892
$612$	−2.23942	−0.0905230
$613$	6.23306	0.251751	0.125875	−	0.992046i	$-0.459826\pi$
$613$	6.23306	0.251751	0.125875	+	0.992046i	$0.459826\pi$
$614$	24.9200	1.00569
$615$	2.04022	0.0822696
$616$	−0.581735	−0.0234388
$617$	−15.3672	−0.618661	−0.309330	−	0.950955i	$-0.600105\pi$
$617$	−15.3672	−0.618661	−0.309330	+	0.950955i	$0.600105\pi$
$618$	3.67990	0.148027
$619$	−43.7344	−1.75783	−0.878916	−	0.476976i	$-0.841733\pi$
$619$	−43.7344	−1.75783	−0.878916	+	0.476976i	$0.841733\pi$
$620$	−10.4714	−0.420540
$621$	−6.68890	−0.268416
$622$	4.58415	0.183807
$623$	6.35115	0.254453
$624$	−1.17297	−0.0469562
$625$	−2.18798	−0.0875190
$626$	9.44214	0.377384
$627$	−0.967517	−0.0386389
$628$	−13.8852	−0.554079
$629$	8.32284	0.331853
$630$	2.15477	0.0858481
$631$	20.2652	0.806745	0.403372	−	0.915036i	$-0.367838\pi$
$631$	20.2652	0.806745	0.403372	+	0.915036i	$0.367838\pi$
$632$	−11.2789	−0.448651
$633$	1.77860	0.0706932
$634$	6.26478	0.248806
$635$	−16.2884	−0.646386
$636$	−3.93420	−0.156001
$637$	11.6549	0.461785
$638$	4.72877	0.187214
$639$	10.5194	0.416141
$640$	−1.45227	−0.0574059
$641$	40.8509	1.61351	0.806757	−	0.590884i	$-0.201221\pi$
$641$	40.8509	1.61351	0.806757	+	0.590884i	$0.201221\pi$
$642$	6.60798	0.260796
$643$	42.9460	1.69363	0.846813	−	0.531891i	$-0.178518\pi$
$643$	42.9460	1.69363	0.846813	+	0.531891i	$0.178518\pi$
$644$	1.04567	0.0412050
$645$	3.62604	0.142775
$646$	1.26710	0.0498534
$647$	−18.6608	−0.733634	−0.366817	−	0.930293i	$-0.619552\pi$
$647$	−18.6608	−0.733634	−0.366817	+	0.930293i	$0.619552\pi$
$648$	−5.15674	−0.202576
$649$	−2.37342	−0.0931650
$650$	−5.05787	−0.198386
$651$	−2.81213	−0.110216
$652$	−8.11776	−0.317916
$653$	−4.60539	−0.180223	−0.0901114	−	0.995932i	$-0.528722\pi$
$653$	−4.60539	−0.180223	−0.0901114	+	0.995932i	$0.528722\pi$
$654$	−0.949336	−0.0371220
$655$	−6.42923	−0.251211
$656$	2.09545	0.0818135
$657$	21.0733	0.822149
$658$	4.24828	0.165615
$659$	−5.01451	−0.195338	−0.0976689	−	0.995219i	$-0.531139\pi$
$659$	−5.01451	−0.195338	−0.0976689	+	0.995219i	$0.531139\pi$
$660$	0.973643	0.0378990
$661$	−11.5506	−0.449265	−0.224632	−	0.974444i	$-0.572118\pi$
$661$	−11.5506	−0.449265	−0.224632	+	0.974444i	$0.572118\pi$
$662$	36.3314	1.41206
$663$	−1.02989	−0.0399976
$664$	13.1640	0.510863
$665$	−1.21921	−0.0472788
$666$	24.1766	0.936824
$667$	−8.49993	−0.329119
$668$	−18.7088	−0.723863
$669$	−13.5872	−0.525311
$670$	−3.55695	−0.137417
$671$	3.41206	0.131721
$672$	−0.390013	−0.0150451
$673$	−34.9431	−1.34696	−0.673479	−	0.739207i	$-0.735201\pi$
$673$	−34.9431	−1.34696	−0.673479	+	0.739207i	$0.735201\pi$
$674$	1.20970	0.0465957
$675$	10.7578	0.414068
$676$	−9.93900	−0.382269
$677$	−6.04807	−0.232446	−0.116223	−	0.993223i	$-0.537079\pi$
$677$	−6.04807	−0.232446	−0.116223	+	0.993223i	$0.537079\pi$
$678$	−2.80749	−0.107821
$679$	−10.1823	−0.390761
$680$	−1.27512	−0.0488987
$681$	8.07188	0.309315
$682$	7.21035	0.276099
$683$	34.1221	1.30565	0.652823	−	0.757510i	$-0.273584\pi$
$683$	34.1221	1.30565	0.652823	+	0.757510i	$0.273584\pi$
$684$	3.68074	0.140736
$685$	11.0680	0.422888
$686$	7.94742	0.303434
$687$	4.06019	0.154906
$688$	3.72420	0.141984
$689$	10.2668	0.391134
$690$	−1.75012	−0.0666258
$691$	16.4399	0.625403	0.312701	−	0.949851i	$-0.398766\pi$
$691$	16.4399	0.625403	0.312701	+	0.949851i	$0.398766\pi$
$692$	13.1071	0.498257
$693$	−1.48373	−0.0563622
$694$	27.4242	1.04101
$695$	3.92269	0.148796
$696$	3.17031	0.120170
$697$	1.83985	0.0696893
$698$	3.73862	0.141509
$699$	14.4383	0.546105
$700$	−1.68175	−0.0635642
$701$	−47.5629	−1.79642	−0.898212	−	0.439562i	$-0.855134\pi$
$701$	−47.5629	−1.79642	−0.898212	+	0.439562i	$0.855134\pi$
$702$	−6.51057	−0.245726
$703$	−13.6795	−0.515933
$704$	1.00000	0.0376889
$705$	−7.11030	−0.267789
$706$	−13.7047	−0.515783
$707$	−9.24004	−0.347507
$708$	−1.59121	−0.0598015
$709$	13.5331	0.508247	0.254123	−	0.967172i	$-0.418213\pi$
$709$	13.5331	0.508247	0.254123	+	0.967172i	$0.418213\pi$
$710$	5.98974	0.224791
$711$	−28.7671	−1.07885
$712$	−10.9176	−0.409154
$713$	−12.9606	−0.485377
$714$	−0.342440	−0.0128155
$715$	−2.54085	−0.0950223
$716$	6.14776	0.229753
$717$	−0.650628	−0.0242981
$718$	−2.09314	−0.0781154
$719$	1.59784	0.0595892	0.0297946	−	0.999556i	$-0.490515\pi$
$719$	1.59784	0.0595892	0.0297946	+	0.999556i	$0.490515\pi$
$720$	−3.70404	−0.138041
$721$	−3.19307	−0.118916
$722$	16.9174	0.629600
$723$	3.70628	0.137838
$724$	−20.2627	−0.753056
$725$	13.6705	0.507710
$726$	−0.670430	−0.0248820
$727$	−4.16812	−0.154587	−0.0772935	−	0.997008i	$-0.524628\pi$
$727$	−4.16812	−0.154587	−0.0772935	+	0.997008i	$0.524628\pi$
$728$	1.01779	0.0377217
$729$	−5.66791	−0.209922
$730$	11.9992	0.444108
$731$	3.26993	0.120943
$732$	2.28754	0.0845501
$733$	−7.02841	−0.259600	−0.129800	−	0.991540i	$-0.541434\pi$
$733$	−7.02841	−0.259600	−0.129800	+	0.991540i	$0.541434\pi$
$734$	20.2990	0.749249
$735$	−6.48601	−0.239240
$736$	−1.79749	−0.0662565
$737$	2.44924	0.0902189
$738$	5.34449	0.196733
$739$	37.5307	1.38059	0.690295	−	0.723528i	$-0.257481\pi$
$739$	37.5307	1.38059	0.690295	+	0.723528i	$0.257481\pi$
$740$	13.7661	0.506054
$741$	1.69274	0.0621844
$742$	3.41372	0.125322
$743$	40.4714	1.48475	0.742375	−	0.669985i	$-0.233699\pi$
$743$	40.4714	1.48475	0.742375	+	0.669985i	$0.233699\pi$
$744$	4.83404	0.177224
$745$	11.8744	0.435043
$746$	−23.8442	−0.872997
$747$	33.5751	1.22845
$748$	0.878022	0.0321037
$749$	−5.73378	−0.209508
$750$	7.68294	0.280541
$751$	−13.6803	−0.499202	−0.249601	−	0.968349i	$-0.580300\pi$
$751$	−13.6803	−0.499202	−0.249601	+	0.968349i	$0.580300\pi$
$752$	−7.30278	−0.266305
$753$	−3.89869	−0.142076
$754$	−8.27333	−0.301297
$755$	29.9443	1.08979
$756$	−2.16477	−0.0787321
$757$	−25.4181	−0.923838	−0.461919	−	0.886922i	$-0.652839\pi$
$757$	−25.4181	−0.923838	−0.461919	+	0.886922i	$0.652839\pi$
$758$	−2.46301	−0.0894606
$759$	1.20509	0.0437421
$760$	2.09581	0.0760230
$761$	32.9698	1.19515	0.597577	−	0.801811i	$-0.296130\pi$
$761$	32.9698	1.19515	0.597577	+	0.801811i	$0.296130\pi$
$762$	7.51944	0.272401
$763$	0.823743	0.0298215
$764$	16.7647	0.606527
$765$	−3.25223	−0.117585
$766$	7.71307	0.278685
$767$	4.15248	0.149937
$768$	0.670430	0.0241921
$769$	16.4058	0.591608	0.295804	−	0.955249i	$-0.404413\pi$
$769$	16.4058	0.591608	0.295804	+	0.955249i	$0.404413\pi$
$770$	−0.844835	−0.0304457
$771$	−8.60785	−0.310004
$772$	24.3605	0.876755
$773$	−10.9550	−0.394023	−0.197011	−	0.980401i	$-0.563124\pi$
$773$	−10.9550	−0.394023	−0.197011	+	0.980401i	$0.563124\pi$
$774$	9.49866	0.341422
$775$	20.8446	0.748759
$776$	17.5033	0.628333
$777$	3.69696	0.132628
$778$	25.4912	0.913904
$779$	−3.02400	−0.108346
$780$	−1.70346	−0.0609936
$781$	−4.12441	−0.147583
$782$	−1.57824	−0.0564377
$783$	17.5969	0.628861
$784$	−6.66158	−0.237914
$785$	−20.1650	−0.719719
$786$	2.96801	0.105865
$787$	22.3992	0.798445	0.399223	−	0.916854i	$-0.369280\pi$
$787$	22.3992	0.798445	0.399223	+	0.916854i	$0.369280\pi$
$788$	1.00000	0.0356235
$789$	16.5652	0.589735
$790$	−16.3800	−0.582774
$791$	2.43607	0.0866167
$792$	2.55052	0.0906289
$793$	−5.96964	−0.211988
$794$	5.90761	0.209653
$795$	−5.71351	−0.202637
$796$	−9.12252	−0.323339
$797$	−5.81433	−0.205954	−0.102977	−	0.994684i	$-0.532837\pi$
$797$	−5.81433	−0.205954	−0.102977	+	0.994684i	$0.532837\pi$
$798$	0.562839	0.0199243
$799$	−6.41200	−0.226840
$800$	2.89092	0.102209
$801$	−27.8456	−0.983875
$802$	−11.8176	−0.417294
$803$	−8.26236	−0.291572
$804$	1.64204	0.0579104
$805$	1.51858	0.0535231
$806$	−12.6150	−0.444346
$807$	9.56020	0.336535
$808$	15.8836	0.558782
$809$	−24.8672	−0.874286	−0.437143	−	0.899392i	$-0.644010\pi$
$809$	−24.8672	−0.874286	−0.437143	+	0.899392i	$0.644010\pi$
$810$	−7.48897	−0.263135
$811$	−33.8516	−1.18869	−0.594345	−	0.804210i	$-0.702589\pi$
$811$	−33.8516	−1.18869	−0.594345	+	0.804210i	$0.702589\pi$
$812$	−2.75089	−0.0965374
$813$	−1.02222	−0.0358510
$814$	−9.47907	−0.332241
$815$	−11.7892	−0.412956
$816$	0.588652	0.0206069
$817$	−5.37450	−0.188030
$818$	31.3317	1.09549
$819$	2.59589	0.0907078
$820$	3.04315	0.106271
$821$	−11.7921	−0.411546	−0.205773	−	0.978600i	$-0.565971\pi$
$821$	−11.7921	−0.411546	−0.205773	+	0.978600i	$0.565971\pi$
$822$	−5.10949	−0.178214
$823$	22.2348	0.775057	0.387528	−	0.921858i	$-0.373329\pi$
$823$	22.2348	0.775057	0.387528	+	0.921858i	$0.373329\pi$
$824$	5.48887	0.191214
$825$	−1.93816	−0.0674781
$826$	1.38070	0.0480408
$827$	49.6898	1.72788	0.863942	−	0.503592i	$-0.167988\pi$
$827$	49.6898	1.72788	0.863942	+	0.503592i	$0.167988\pi$
$828$	−4.58455	−0.159324
$829$	−2.30742	−0.0801401	−0.0400700	−	0.999197i	$-0.512758\pi$
$829$	−2.30742	−0.0801401	−0.0400700	+	0.999197i	$0.512758\pi$
$830$	19.1177	0.663584
$831$	−8.24860	−0.286141
$832$	−1.74957	−0.0606555
$833$	−5.84902	−0.202656
$834$	−1.81088	−0.0627058
$835$	−27.1701	−0.940260
$836$	−1.44313	−0.0499117
$837$	26.8314	0.927430
$838$	−21.8166	−0.753642
$839$	−23.0998	−0.797495	−0.398747	−	0.917061i	$-0.630555\pi$
$839$	−23.0998	−0.797495	−0.398747	+	0.917061i	$0.630555\pi$
$840$	−0.566403	−0.0195427
$841$	−6.63873	−0.228922
$842$	−11.9342	−0.411278
$843$	1.72859	0.0595358
$844$	2.65293	0.0913177
$845$	−14.4341	−0.496547
$846$	−18.6259	−0.640372
$847$	0.581735	0.0199887
$848$	−5.86818	−0.201514
$849$	12.4699	0.427967
$850$	2.53829	0.0870627
$851$	17.0386	0.584075
$852$	−2.76513	−0.0947317
$853$	21.8585	0.748422	0.374211	−	0.927344i	$-0.377914\pi$
$853$	21.8585	0.748422	0.374211	+	0.927344i	$0.377914\pi$
$854$	−1.98491	−0.0679223
$855$	5.34541	0.182809
$856$	9.85634	0.336883
$857$	23.5931	0.805924	0.402962	−	0.915217i	$-0.367981\pi$
$857$	23.5931	0.805924	0.402962	+	0.915217i	$0.367981\pi$
$858$	1.17297	0.0400444
$859$	−47.5269	−1.62160	−0.810799	−	0.585325i	$-0.800967\pi$
$859$	−47.5269	−1.62160	−0.810799	+	0.585325i	$0.800967\pi$
$860$	5.40853	0.184429
$861$	0.817252	0.0278519
$862$	−20.4229	−0.695608
$863$	−0.543606	−0.0185046	−0.00925229	−	0.999957i	$-0.502945\pi$
$863$	−0.543606	−0.0185046	−0.00925229	+	0.999957i	$0.502945\pi$
$864$	3.72124	0.126599
$865$	19.0350	0.647210
$866$	21.6487	0.735654
$867$	−10.8805	−0.369520
$868$	−4.19452	−0.142371
$869$	11.2789	0.382611
$870$	4.60414	0.156095
$871$	−4.28512	−0.145196
$872$	−1.41601	−0.0479522
$873$	44.6426	1.51092
$874$	2.59402	0.0877439
$875$	−6.66652	−0.225370
$876$	−5.53933	−0.187157
$877$	12.3806	0.418064	0.209032	−	0.977909i	$-0.432969\pi$
$877$	12.3806	0.418064	0.209032	+	0.977909i	$0.432969\pi$
$878$	28.0791	0.947623
$879$	−7.05538	−0.237972
$880$	1.45227	0.0489559
$881$	47.9940	1.61696	0.808479	−	0.588525i	$-0.200291\pi$
$881$	47.9940	1.61696	0.808479	+	0.588525i	$0.200291\pi$
$882$	−16.9905	−0.572101
$883$	43.6675	1.46953	0.734764	−	0.678323i	$-0.237293\pi$
$883$	43.6675	1.46953	0.734764	+	0.678323i	$0.237293\pi$
$884$	−1.53616	−0.0516667
$885$	−2.31087	−0.0776790
$886$	−33.1630	−1.11413
$887$	15.3029	0.513822	0.256911	−	0.966435i	$-0.417295\pi$
$887$	15.3029	0.513822	0.256911	+	0.966435i	$0.417295\pi$
$888$	−6.35506	−0.213262
$889$	−6.52465	−0.218830
$890$	−15.8553	−0.531470
$891$	5.15674	0.172757
$892$	−20.2664	−0.678568
$893$	10.5389	0.352669
$894$	−5.48172	−0.183336
$895$	8.92819	0.298436
$896$	−0.581735	−0.0194344
$897$	−2.10840	−0.0703974
$898$	9.86370	0.329156
$899$	34.0961	1.13717
$900$	7.37336	0.245779
$901$	−5.15239	−0.171651
$902$	−2.09545	−0.0697708
$903$	1.45248	0.0483357
$904$	−4.18759	−0.139277
$905$	−29.4268	−0.978180
$906$	−13.8236	−0.459259
$907$	−13.4133	−0.445383	−0.222691	−	0.974889i	$-0.571484\pi$
$907$	−13.4133	−0.445383	−0.222691	+	0.974889i	$0.571484\pi$
$908$	12.0399	0.399557
$909$	40.5114	1.34368
$910$	1.47810	0.0489985
$911$	21.9989	0.728855	0.364427	−	0.931232i	$-0.381265\pi$
$911$	21.9989	0.728855	0.364427	+	0.931232i	$0.381265\pi$
$912$	−0.967517	−0.0320377
$913$	−13.1640	−0.435665
$914$	−27.8772	−0.922095
$915$	3.32213	0.109826
$916$	6.05609	0.200099
$917$	−2.57536	−0.0850458
$918$	3.26733	0.107838
$919$	51.8312	1.70975	0.854877	−	0.518831i	$-0.173633\pi$
$919$	51.8312	1.70975	0.854877	+	0.518831i	$0.173633\pi$
$920$	−2.61044	−0.0860637
$921$	−16.7071	−0.550518
$922$	−6.85531	−0.225768
$923$	7.21595	0.237516
$924$	0.390013	0.0128305
$925$	−27.4033	−0.901013
$926$	−24.8941	−0.818070
$927$	13.9995	0.459804
$928$	4.72877	0.155229
$929$	43.7786	1.43633	0.718165	−	0.695873i	$-0.244982\pi$
$929$	43.7786	1.43633	0.718165	+	0.695873i	$0.244982\pi$
$930$	7.02031	0.230205
$931$	9.61353	0.315071
$932$	21.5358	0.705429
$933$	−3.07335	−0.100617
$934$	−24.4652	−0.800524
$935$	1.27512	0.0417010
$936$	−4.46233	−0.145856
$937$	58.8654	1.92305	0.961524	−	0.274722i	$-0.0885857\pi$
$937$	58.8654	1.92305	0.961524	+	0.274722i	$0.0885857\pi$
$938$	−1.42481	−0.0465217
$939$	−6.33029	−0.206581
$940$	−10.6056	−0.345916
$941$	10.2901	0.335447	0.167724	−	0.985834i	$-0.446358\pi$
$941$	10.2901	0.335447	0.167724	+	0.985834i	$0.446358\pi$
$942$	9.30904	0.303305
$943$	3.76656	0.122656
$944$	−2.37342	−0.0772484
$945$	−3.14383	−0.102269
$946$	−3.72420	−0.121084
$947$	−36.8445	−1.19728	−0.598642	−	0.801016i	$-0.704293\pi$
$947$	−36.8445	−1.19728	−0.598642	+	0.801016i	$0.704293\pi$
$948$	7.56172	0.245593
$949$	14.4556	0.469248
$950$	−4.17197	−0.135357
$951$	−4.20010	−0.136197
$952$	−0.510776	−0.0165544
$953$	17.3996	0.563629	0.281814	−	0.959469i	$-0.409064\pi$
$953$	17.3996	0.563629	0.281814	+	0.959469i	$0.409064\pi$
$954$	−14.9669	−0.484572
$955$	24.3469	0.787847
$956$	−0.970463	−0.0313870
$957$	−3.17031	−0.102482
$958$	−6.97325	−0.225295
$959$	4.43353	0.143166
$960$	0.973643	0.0314242
$961$	20.9892	0.677071
$962$	16.5843	0.534700
$963$	25.1388	0.810087
$964$	5.52821	0.178052
$965$	35.3780	1.13886
$966$	−0.701045	−0.0225558
$967$	−40.3722	−1.29828	−0.649141	−	0.760668i	$-0.724872\pi$
$967$	−40.3722	−1.29828	−0.649141	+	0.760668i	$0.724872\pi$
$968$	−1.00000	−0.0321412
$969$	−0.849501	−0.0272899
$970$	25.4195	0.816171
$971$	−10.6896	−0.343047	−0.171523	−	0.985180i	$-0.554869\pi$
$971$	−10.6896	−0.343047	−0.171523	+	0.985180i	$0.554869\pi$
$972$	14.6209	0.468967
$973$	1.57131	0.0503739
$974$	−20.1608	−0.645995
$975$	3.39095	0.108597
$976$	3.41206	0.109217
$977$	27.2821	0.872831	0.436415	−	0.899745i	$-0.356248\pi$
$977$	27.2821	0.872831	0.436415	+	0.899745i	$0.356248\pi$
$978$	5.44239	0.174028
$979$	10.9176	0.348928
$980$	−9.67440	−0.309037
$981$	−3.61157	−0.115309
$982$	10.0159	0.319621
$983$	−48.8256	−1.55729	−0.778647	−	0.627463i	$-0.784093\pi$
$983$	−48.8256	−1.55729	−0.778647	+	0.627463i	$0.784093\pi$
$984$	−1.40485	−0.0447850
$985$	1.45227	0.0462731
$986$	4.15196	0.132225
$987$	−2.84818	−0.0906585
$988$	2.52486	0.0803265
$989$	6.69422	0.212864
$990$	3.70404	0.117722
$991$	44.6020	1.41683	0.708414	−	0.705797i	$-0.249411\pi$
$991$	44.6020	1.41683	0.708414	+	0.705797i	$0.249411\pi$
$992$	7.21035	0.228929
$993$	−24.3577	−0.772967
$994$	2.39931	0.0761016
$995$	−13.2483	−0.420000
$996$	−8.82555	−0.279648
$997$	21.5525	0.682575	0.341287	−	0.939959i	$-0.389137\pi$
$997$	21.5525	0.682575	0.341287	+	0.939959i	$0.389137\pi$
$998$	14.6515	0.463785
$999$	−35.2739	−1.11602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.a.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.a.1.10	✓	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.670430 q^{3} +1.00000 q^{4} +1.45227 q^{5} -0.670430 q^{6} +0.581735 q^{7} -1.00000 q^{8} -2.55052 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.670430 q^{3} +1.00000 q^{4} +1.45227 q^{5} -0.670430 q^{6} +0.581735 q^{7} -1.00000 q^{8} -2.55052 q^{9} -1.45227 q^{10} +1.00000 q^{11} +0.670430 q^{12} -1.74957 q^{13} -0.581735 q^{14} +0.973643 q^{15} +1.00000 q^{16} +0.878022 q^{17} +2.55052 q^{18} -1.44313 q^{19} +1.45227 q^{20} +0.390013 q^{21} -1.00000 q^{22} +1.79749 q^{23} -0.670430 q^{24} -2.89092 q^{25} +1.74957 q^{26} -3.72124 q^{27} +0.581735 q^{28} -4.72877 q^{29} -0.973643 q^{30} -7.21035 q^{31} -1.00000 q^{32} +0.670430 q^{33} -0.878022 q^{34} +0.844835 q^{35} -2.55052 q^{36} +9.47907 q^{37} +1.44313 q^{38} -1.17297 q^{39} -1.45227 q^{40} +2.09545 q^{41} -0.390013 q^{42} +3.72420 q^{43} +1.00000 q^{44} -3.70404 q^{45} -1.79749 q^{46} -7.30278 q^{47} +0.670430 q^{48} -6.66158 q^{49} +2.89092 q^{50} +0.588652 q^{51} -1.74957 q^{52} -5.86818 q^{53} +3.72124 q^{54} +1.45227 q^{55} -0.581735 q^{56} -0.967517 q^{57} +4.72877 q^{58} -2.37342 q^{59} +0.973643 q^{60} +3.41206 q^{61} +7.21035 q^{62} -1.48373 q^{63} +1.00000 q^{64} -2.54085 q^{65} -0.670430 q^{66} +2.44924 q^{67} +0.878022 q^{68} +1.20509 q^{69} -0.844835 q^{70} -4.12441 q^{71} +2.55052 q^{72} -8.26236 q^{73} -9.47907 q^{74} -1.93816 q^{75} -1.44313 q^{76} +0.581735 q^{77} +1.17297 q^{78} +11.2789 q^{79} +1.45227 q^{80} +5.15674 q^{81} -2.09545 q^{82} -13.1640 q^{83} +0.390013 q^{84} +1.27512 q^{85} -3.72420 q^{86} -3.17031 q^{87} -1.00000 q^{88} +10.9176 q^{89} +3.70404 q^{90} -1.01779 q^{91} +1.79749 q^{92} -4.83404 q^{93} +7.30278 q^{94} -2.09581 q^{95} -0.670430 q^{96} -17.5033 q^{97} +6.66158 q^{98} -2.55052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9 + 7 * q^10 + 15 * q^11 - q^12 - q^13 - q^14 - 6 * q^15 + 15 * q^16 - 6 * q^17 - 2 * q^18 - 14 * q^19 - 7 * q^20 - 3 * q^21 - 15 * q^22 + 2 * q^23 + q^24 - 10 * q^25 + q^26 - 7 * q^27 + q^28 + 8 * q^29 + 6 * q^30 - 33 * q^31 - 15 * q^32 - q^33 + 6 * q^34 - 8 * q^35 + 2 * q^36 - 9 * q^37 + 14 * q^38 - 9 * q^39 + 7 * q^40 - 10 * q^41 + 3 * q^42 - 6 * q^43 + 15 * q^44 - 20 * q^45 - 2 * q^46 - q^47 - q^48 - 30 * q^49 + 10 * q^50 + 12 * q^51 - q^52 + 6 * q^53 + 7 * q^54 - 7 * q^55 - q^56 - 24 * q^57 - 8 * q^58 - 15 * q^59 - 6 * q^60 - 25 * q^61 + 33 * q^62 + 12 * q^63 + 15 * q^64 + 31 * q^65 + q^66 - 13 * q^67 - 6 * q^68 - 43 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 4 * q^73 + 9 * q^74 - 5 * q^75 - 14 * q^76 + q^77 + 9 * q^78 - 20 * q^79 - 7 * q^80 + 11 * q^81 + 10 * q^82 + q^83 - 3 * q^84 - q^85 + 6 * q^86 + 22 * q^87 - 15 * q^88 - 41 * q^89 + 20 * q^90 - 31 * q^91 + 2 * q^92 + 14 * q^93 + q^94 + 41 * q^95 + q^96 - 57 * q^97 + 30 * q^98 + 2 * q^99

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