LMFDB - Embedded newform 6045.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 7x^{7} + 22x^{6} + 14x^{5} - 52x^{4} - 5x^{3} + 41x^{2} - 4x - 5 $ x^9 - 3x^8 - 7x^7 + 22x^6 + 14x^5 - 52x^4 - 5x^3 + 41x^2 - 4x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.12407$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12407 q^{2} +1.00000 q^{3} +2.51167 q^{4} +1.00000 q^{5} -2.12407 q^{6} +2.73139 q^{7} -1.08683 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.12407 q^{2} +1.00000 q^{3} +2.51167 q^{4} +1.00000 q^{5} -2.12407 q^{6} +2.73139 q^{7} -1.08683 q^{8} +1.00000 q^{9} -2.12407 q^{10} -1.46063 q^{11} +2.51167 q^{12} +1.00000 q^{13} -5.80167 q^{14} +1.00000 q^{15} -2.71484 q^{16} -5.70068 q^{17} -2.12407 q^{18} -1.04377 q^{19} +2.51167 q^{20} +2.73139 q^{21} +3.10248 q^{22} -8.86066 q^{23} -1.08683 q^{24} +1.00000 q^{25} -2.12407 q^{26} +1.00000 q^{27} +6.86037 q^{28} -2.85192 q^{29} -2.12407 q^{30} +1.00000 q^{31} +7.94018 q^{32} -1.46063 q^{33} +12.1086 q^{34} +2.73139 q^{35} +2.51167 q^{36} +8.41324 q^{37} +2.21704 q^{38} +1.00000 q^{39} -1.08683 q^{40} +5.66728 q^{41} -5.80167 q^{42} +2.57534 q^{43} -3.66863 q^{44} +1.00000 q^{45} +18.8207 q^{46} -8.04770 q^{47} -2.71484 q^{48} +0.460513 q^{49} -2.12407 q^{50} -5.70068 q^{51} +2.51167 q^{52} +6.47549 q^{53} -2.12407 q^{54} -1.46063 q^{55} -2.96857 q^{56} -1.04377 q^{57} +6.05767 q^{58} -4.11628 q^{59} +2.51167 q^{60} -12.5622 q^{61} -2.12407 q^{62} +2.73139 q^{63} -11.4358 q^{64} +1.00000 q^{65} +3.10248 q^{66} -1.65069 q^{67} -14.3182 q^{68} -8.86066 q^{69} -5.80167 q^{70} -13.3271 q^{71} -1.08683 q^{72} -13.6038 q^{73} -17.8703 q^{74} +1.00000 q^{75} -2.62161 q^{76} -3.98956 q^{77} -2.12407 q^{78} -3.70559 q^{79} -2.71484 q^{80} +1.00000 q^{81} -12.0377 q^{82} +13.0471 q^{83} +6.86037 q^{84} -5.70068 q^{85} -5.47019 q^{86} -2.85192 q^{87} +1.58746 q^{88} +4.06255 q^{89} -2.12407 q^{90} +2.73139 q^{91} -22.2551 q^{92} +1.00000 q^{93} +17.0939 q^{94} -1.04377 q^{95} +7.94018 q^{96} -12.2970 q^{97} -0.978163 q^{98} -1.46063 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 9 q^{3} + 8 q^{4} + 9 q^{5} - 6 q^{6} - 12 q^{7} - 21 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q - 6 * q^2 + 9 * q^3 + 8 * q^4 + 9 * q^5 - 6 * q^6 - 12 * q^7 - 21 * q^8 + 9 * q^9	\( 9 q - 6 q^{2} + 9 q^{3} + 8 q^{4} + 9 q^{5} - 6 q^{6} - 12 q^{7} - 21 q^{8} + 9 q^{9} - 6 q^{10} - 10 q^{11} + 8 q^{12} + 9 q^{13} - 9 q^{14} + 9 q^{15} + 30 q^{16} - 17 q^{17} - 6 q^{18} + 6 q^{19} + 8 q^{20} - 12 q^{21} + 15 q^{22} - 23 q^{23} - 21 q^{24} + 9 q^{25} - 6 q^{26} + 9 q^{27} + 2 q^{28} - 4 q^{29} - 6 q^{30} + 9 q^{31} - 38 q^{32} - 10 q^{33} + 15 q^{34} - 12 q^{35} + 8 q^{36} + 11 q^{37} - 16 q^{38} + 9 q^{39} - 21 q^{40} - 8 q^{41} - 9 q^{42} - 15 q^{43} - q^{44} + 9 q^{45} + 26 q^{46} - 33 q^{47} + 30 q^{48} + 15 q^{49} - 6 q^{50} - 17 q^{51} + 8 q^{52} - 38 q^{53} - 6 q^{54} - 10 q^{55} + 37 q^{56} + 6 q^{57} - 26 q^{58} - 25 q^{59} + 8 q^{60} - 10 q^{61} - 6 q^{62} - 12 q^{63} + 47 q^{64} + 9 q^{65} + 15 q^{66} - 19 q^{67} + 7 q^{68} - 23 q^{69} - 9 q^{70} - 43 q^{71} - 21 q^{72} + q^{73} + 4 q^{74} + 9 q^{75} - 26 q^{76} + 2 q^{77} - 6 q^{78} - 9 q^{79} + 30 q^{80} + 9 q^{81} + 15 q^{82} - 12 q^{83} + 2 q^{84} - 17 q^{85} - 30 q^{86} - 4 q^{87} + q^{88} - 5 q^{89} - 6 q^{90} - 12 q^{91} - 57 q^{92} + 9 q^{93} + 40 q^{94} + 6 q^{95} - 38 q^{96} - 4 q^{97} + 34 q^{98} - 10 q^{99}+O(q^{100}) \) 9 * q - 6 * q^2 + 9 * q^3 + 8 * q^4 + 9 * q^5 - 6 * q^6 - 12 * q^7 - 21 * q^8 + 9 * q^9 - 6 * q^10 - 10 * q^11 + 8 * q^12 + 9 * q^13 - 9 * q^14 + 9 * q^15 + 30 * q^16 - 17 * q^17 - 6 * q^18 + 6 * q^19 + 8 * q^20 - 12 * q^21 + 15 * q^22 - 23 * q^23 - 21 * q^24 + 9 * q^25 - 6 * q^26 + 9 * q^27 + 2 * q^28 - 4 * q^29 - 6 * q^30 + 9 * q^31 - 38 * q^32 - 10 * q^33 + 15 * q^34 - 12 * q^35 + 8 * q^36 + 11 * q^37 - 16 * q^38 + 9 * q^39 - 21 * q^40 - 8 * q^41 - 9 * q^42 - 15 * q^43 - q^44 + 9 * q^45 + 26 * q^46 - 33 * q^47 + 30 * q^48 + 15 * q^49 - 6 * q^50 - 17 * q^51 + 8 * q^52 - 38 * q^53 - 6 * q^54 - 10 * q^55 + 37 * q^56 + 6 * q^57 - 26 * q^58 - 25 * q^59 + 8 * q^60 - 10 * q^61 - 6 * q^62 - 12 * q^63 + 47 * q^64 + 9 * q^65 + 15 * q^66 - 19 * q^67 + 7 * q^68 - 23 * q^69 - 9 * q^70 - 43 * q^71 - 21 * q^72 + q^73 + 4 * q^74 + 9 * q^75 - 26 * q^76 + 2 * q^77 - 6 * q^78 - 9 * q^79 + 30 * q^80 + 9 * q^81 + 15 * q^82 - 12 * q^83 + 2 * q^84 - 17 * q^85 - 30 * q^86 - 4 * q^87 + q^88 - 5 * q^89 - 6 * q^90 - 12 * q^91 - 57 * q^92 + 9 * q^93 + 40 * q^94 + 6 * q^95 - 38 * q^96 - 4 * q^97 + 34 * q^98 - 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.12407	−1.50194	−0.750972	−	0.660334i	$-0.770415\pi$
$2$	−2.12407	−1.50194	−0.750972	+	0.660334i	$0.770415\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.51167	1.25584
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.12407	−0.867148
$7$	2.73139	1.03237	0.516185	−	0.856477i	$-0.327352\pi$
$7$	2.73139	1.03237	0.516185	+	0.856477i	$0.327352\pi$
$8$	−1.08683	−0.384253
$9$	1.00000	0.333333
$10$	−2.12407	−0.671690
$11$	−1.46063	−0.440397	−0.220198	−	0.975455i	$-0.570670\pi$
$11$	−1.46063	−0.440397	−0.220198	+	0.975455i	$0.570670\pi$
$12$	2.51167	0.725058
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.80167	−1.55056
$15$	1.00000	0.258199
$16$	−2.71484	−0.678710
$17$	−5.70068	−1.38262	−0.691309	−	0.722560i	$-0.742965\pi$
$17$	−5.70068	−1.38262	−0.691309	+	0.722560i	$0.742965\pi$
$18$	−2.12407	−0.500648
$19$	−1.04377	−0.239457	−0.119728	−	0.992807i	$-0.538202\pi$
$19$	−1.04377	−0.239457	−0.119728	+	0.992807i	$0.538202\pi$
$20$	2.51167	0.561627
$21$	2.73139	0.596039
$22$	3.10248	0.661452
$23$	−8.86066	−1.84758	−0.923788	−	0.382905i	$-0.874924\pi$
$23$	−8.86066	−1.84758	−0.923788	+	0.382905i	$0.874924\pi$
$24$	−1.08683	−0.221849
$25$	1.00000	0.200000
$26$	−2.12407	−0.416564
$27$	1.00000	0.192450
$28$	6.86037	1.29649
$29$	−2.85192	−0.529588	−0.264794	−	0.964305i	$-0.585304\pi$
$29$	−2.85192	−0.529588	−0.264794	+	0.964305i	$0.585304\pi$
$30$	−2.12407	−0.387800
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	7.94018	1.40364
$33$	−1.46063	−0.254263
$34$	12.1086	2.07661
$35$	2.73139	0.461690
$36$	2.51167	0.418612
$37$	8.41324	1.38313	0.691564	−	0.722315i	$-0.256922\pi$
$37$	8.41324	1.38313	0.691564	+	0.722315i	$0.256922\pi$
$38$	2.21704	0.359651
$39$	1.00000	0.160128
$40$	−1.08683	−0.171843
$41$	5.66728	0.885080	0.442540	−	0.896749i	$-0.354077\pi$
$41$	5.66728	0.885080	0.442540	+	0.896749i	$0.354077\pi$
$42$	−5.80167	−0.895218
$43$	2.57534	0.392735	0.196368	−	0.980530i	$-0.437085\pi$
$43$	2.57534	0.392735	0.196368	+	0.980530i	$0.437085\pi$
$44$	−3.66863	−0.553067
$45$	1.00000	0.149071
$46$	18.8207	2.77496
$47$	−8.04770	−1.17388	−0.586939	−	0.809631i	$-0.699667\pi$
$47$	−8.04770	−1.17388	−0.586939	+	0.809631i	$0.699667\pi$
$48$	−2.71484	−0.391853
$49$	0.460513	0.0657876
$50$	−2.12407	−0.300389
$51$	−5.70068	−0.798254
$52$	2.51167	0.348307
$53$	6.47549	0.889476	0.444738	−	0.895661i	$-0.353297\pi$
$53$	6.47549	0.889476	0.444738	+	0.895661i	$0.353297\pi$
$54$	−2.12407	−0.289049
$55$	−1.46063	−0.196951
$56$	−2.96857	−0.396692
$57$	−1.04377	−0.138251
$58$	6.05767	0.795412
$59$	−4.11628	−0.535894	−0.267947	−	0.963434i	$-0.586345\pi$
$59$	−4.11628	−0.535894	−0.267947	+	0.963434i	$0.586345\pi$
$60$	2.51167	0.324256
$61$	−12.5622	−1.60843	−0.804214	−	0.594340i	$-0.797413\pi$
$61$	−12.5622	−1.60843	−0.804214	+	0.594340i	$0.797413\pi$
$62$	−2.12407	−0.269757
$63$	2.73139	0.344123
$64$	−11.4358	−1.42948
$65$	1.00000	0.124035
$66$	3.10248	0.381889
$67$	−1.65069	−0.201664	−0.100832	−	0.994903i	$-0.532151\pi$
$67$	−1.65069	−0.201664	−0.100832	+	0.994903i	$0.532151\pi$
$68$	−14.3182	−1.73634
$69$	−8.86066	−1.06670
$70$	−5.80167	−0.693433
$71$	−13.3271	−1.58163	−0.790816	−	0.612054i	$-0.790344\pi$
$71$	−13.3271	−1.58163	−0.790816	+	0.612054i	$0.790344\pi$
$72$	−1.08683	−0.128084
$73$	−13.6038	−1.59220	−0.796101	−	0.605163i	$-0.793108\pi$
$73$	−13.6038	−1.59220	−0.796101	+	0.605163i	$0.793108\pi$
$74$	−17.8703	−2.07738
$75$	1.00000	0.115470
$76$	−2.62161	−0.300719
$77$	−3.98956	−0.454653
$78$	−2.12407	−0.240504
$79$	−3.70559	−0.416912	−0.208456	−	0.978032i	$-0.566844\pi$
$79$	−3.70559	−0.416912	−0.208456	+	0.978032i	$0.566844\pi$
$80$	−2.71484	−0.303528
$81$	1.00000	0.111111
$82$	−12.0377	−1.32934
$83$	13.0471	1.43210	0.716052	−	0.698047i	$-0.245947\pi$
$83$	13.0471	1.43210	0.716052	+	0.698047i	$0.245947\pi$
$84$	6.86037	0.748528
$85$	−5.70068	−0.618325
$86$	−5.47019	−0.589866
$87$	−2.85192	−0.305758
$88$	1.58746	0.169224
$89$	4.06255	0.430629	0.215315	−	0.976545i	$-0.430922\pi$
$89$	4.06255	0.430629	0.215315	+	0.976545i	$0.430922\pi$
$90$	−2.12407	−0.223897
$91$	2.73139	0.286328
$92$	−22.2551	−2.32025
$93$	1.00000	0.103695
$94$	17.0939	1.76310
$95$	−1.04377	−0.107088
$96$	7.94018	0.810391
$97$	−12.2970	−1.24858	−0.624288	−	0.781194i	$-0.714611\pi$
$97$	−12.2970	−1.24858	−0.624288	+	0.781194i	$0.714611\pi$
$98$	−0.978163	−0.0988094
$99$	−1.46063	−0.146799
$100$	2.51167	0.251167
$101$	14.1971	1.41266	0.706332	−	0.707881i	$-0.250349\pi$
$101$	14.1971	1.41266	0.706332	+	0.707881i	$0.250349\pi$
$102$	12.1086	1.19893
$103$	2.66526	0.262616	0.131308	−	0.991342i	$-0.458082\pi$
$103$	2.66526	0.262616	0.131308	+	0.991342i	$0.458082\pi$
$104$	−1.08683	−0.106573
$105$	2.73139	0.266557
$106$	−13.7544	−1.33594
$107$	−17.5943	−1.70090	−0.850451	−	0.526055i	$-0.823671\pi$
$107$	−17.5943	−1.70090	−0.850451	+	0.526055i	$0.823671\pi$
$108$	2.51167	0.241686
$109$	0.914767	0.0876187	0.0438094	−	0.999040i	$-0.486051\pi$
$109$	0.914767	0.0876187	0.0438094	+	0.999040i	$0.486051\pi$
$110$	3.10248	0.295810
$111$	8.41324	0.798549
$112$	−7.41530	−0.700680
$113$	−15.5715	−1.46484	−0.732419	−	0.680854i	$-0.761609\pi$
$113$	−15.5715	−1.46484	−0.732419	+	0.680854i	$0.761609\pi$
$114$	2.21704	0.207645
$115$	−8.86066	−0.826261
$116$	−7.16309	−0.665076
$117$	1.00000	0.0924500
$118$	8.74326	0.804882
$119$	−15.5708	−1.42737
$120$	−1.08683	−0.0992138
$121$	−8.86656	−0.806051
$122$	26.6830	2.41577
$123$	5.66728	0.511001
$124$	2.51167	0.225555
$125$	1.00000	0.0894427
$126$	−5.80167	−0.516854
$127$	−6.41482	−0.569223	−0.284611	−	0.958643i	$-0.591865\pi$
$127$	−6.41482	−0.569223	−0.284611	+	0.958643i	$0.591865\pi$
$128$	8.41012	0.743356
$129$	2.57534	0.226746
$130$	−2.12407	−0.186293
$131$	−9.02445	−0.788470	−0.394235	−	0.919010i	$-0.628990\pi$
$131$	−9.02445	−0.788470	−0.394235	+	0.919010i	$0.628990\pi$
$132$	−3.66863	−0.319313
$133$	−2.85094	−0.247208
$134$	3.50619	0.302889
$135$	1.00000	0.0860663
$136$	6.19568	0.531275
$137$	12.9105	1.10302	0.551511	−	0.834168i	$-0.314051\pi$
$137$	12.9105	1.10302	0.551511	+	0.834168i	$0.314051\pi$
$138$	18.8207	1.60212
$139$	4.79954	0.407092	0.203546	−	0.979065i	$-0.434753\pi$
$139$	4.79954	0.407092	0.203546	+	0.979065i	$0.434753\pi$
$140$	6.86037	0.579807
$141$	−8.04770	−0.677738
$142$	28.3076	2.37552
$143$	−1.46063	−0.122144
$144$	−2.71484	−0.226237
$145$	−2.85192	−0.236839
$146$	28.8954	2.39140
$147$	0.460513	0.0379825
$148$	21.1313	1.73698
$149$	−14.2805	−1.16990	−0.584952	−	0.811068i	$-0.698887\pi$
$149$	−14.2805	−1.16990	−0.584952	+	0.811068i	$0.698887\pi$
$150$	−2.12407	−0.173430
$151$	16.4847	1.34151	0.670753	−	0.741681i	$-0.265971\pi$
$151$	16.4847	1.34151	0.670753	+	0.741681i	$0.265971\pi$
$152$	1.13440	0.0920121
$153$	−5.70068	−0.460872
$154$	8.47410	0.682863
$155$	1.00000	0.0803219
$156$	2.51167	0.201095
$157$	20.7352	1.65485	0.827423	−	0.561579i	$-0.189806\pi$
$157$	20.7352	1.65485	0.827423	+	0.561579i	$0.189806\pi$
$158$	7.87094	0.626178
$159$	6.47549	0.513539
$160$	7.94018	0.627726
$161$	−24.2020	−1.90738
$162$	−2.12407	−0.166883
$163$	−2.19990	−0.172310	−0.0861548	−	0.996282i	$-0.527458\pi$
$163$	−2.19990	−0.172310	−0.0861548	+	0.996282i	$0.527458\pi$
$164$	14.2344	1.11152
$165$	−1.46063	−0.113710
$166$	−27.7129	−2.15094
$167$	−14.4811	−1.12058	−0.560292	−	0.828295i	$-0.689311\pi$
$167$	−14.4811	−1.12058	−0.560292	+	0.828295i	$0.689311\pi$
$168$	−2.96857	−0.229030
$169$	1.00000	0.0769231
$170$	12.1086	0.928690
$171$	−1.04377	−0.0798190
$172$	6.46841	0.493211
$173$	10.8229	0.822849	0.411424	−	0.911444i	$-0.365031\pi$
$173$	10.8229	0.822849	0.411424	+	0.911444i	$0.365031\pi$
$174$	6.05767	0.459231
$175$	2.73139	0.206474
$176$	3.96538	0.298902
$177$	−4.11628	−0.309398
$178$	−8.62913	−0.646781
$179$	−3.91489	−0.292613	−0.146306	−	0.989239i	$-0.546738\pi$
$179$	−3.91489	−0.292613	−0.146306	+	0.989239i	$0.546738\pi$
$180$	2.51167	0.187209
$181$	8.36495	0.621762	0.310881	−	0.950449i	$-0.399376\pi$
$181$	8.36495	0.621762	0.310881	+	0.950449i	$0.399376\pi$
$182$	−5.80167	−0.430049
$183$	−12.5622	−0.928626
$184$	9.63005	0.709937
$185$	8.41324	0.618554
$186$	−2.12407	−0.155744
$187$	8.32659	0.608900
$188$	−20.2132	−1.47420
$189$	2.73139	0.198680
$190$	2.21704	0.160841
$191$	−22.4259	−1.62268	−0.811340	−	0.584575i	$-0.801261\pi$
$191$	−22.4259	−1.62268	−0.811340	+	0.584575i	$0.801261\pi$
$192$	−11.4358	−0.825309
$193$	−8.92501	−0.642437	−0.321218	−	0.947005i	$-0.604092\pi$
$193$	−8.92501	−0.642437	−0.321218	+	0.947005i	$0.604092\pi$
$194$	26.1198	1.87529
$195$	1.00000	0.0716115
$196$	1.15666	0.0826186
$197$	−18.8794	−1.34510	−0.672551	−	0.740051i	$-0.734801\pi$
$197$	−18.8794	−1.34510	−0.672551	+	0.740051i	$0.734801\pi$
$198$	3.10248	0.220484
$199$	2.31495	0.164102	0.0820512	−	0.996628i	$-0.473853\pi$
$199$	2.31495	0.164102	0.0820512	+	0.996628i	$0.473853\pi$
$200$	−1.08683	−0.0768507
$201$	−1.65069	−0.116431
$202$	−30.1556	−2.12174
$203$	−7.78971	−0.546731
$204$	−14.3182	−1.00248
$205$	5.66728	0.395820
$206$	−5.66121	−0.394435
$207$	−8.86066	−0.615859
$208$	−2.71484	−0.188240
$209$	1.52456	0.105456
$210$	−5.80167	−0.400353
$211$	17.3655	1.19549	0.597744	−	0.801687i	$-0.296064\pi$
$211$	17.3655	1.19549	0.597744	+	0.801687i	$0.296064\pi$
$212$	16.2643	1.11704
$213$	−13.3271	−0.913156
$214$	37.3714	2.55466
$215$	2.57534	0.175636
$216$	−1.08683	−0.0739496
$217$	2.73139	0.185419
$218$	−1.94303	−0.131598
$219$	−13.6038	−0.919259
$220$	−3.66863	−0.247339
$221$	−5.70068	−0.383469
$222$	−17.8703	−1.19938
$223$	−11.6517	−0.780252	−0.390126	−	0.920761i	$-0.627569\pi$
$223$	−11.6517	−0.780252	−0.390126	+	0.920761i	$0.627569\pi$
$224$	21.6878	1.44907
$225$	1.00000	0.0666667
$226$	33.0749	2.20011
$227$	−19.3862	−1.28671	−0.643354	−	0.765569i	$-0.722458\pi$
$227$	−19.3862	−1.28671	−0.643354	+	0.765569i	$0.722458\pi$
$228$	−2.62161	−0.173620
$229$	24.2816	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$229$	24.2816	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$230$	18.8207	1.24100
$231$	−3.98956	−0.262494
$232$	3.09956	0.203496
$233$	19.4169	1.27204	0.636021	−	0.771672i	$-0.280579\pi$
$233$	19.4169	1.27204	0.636021	+	0.771672i	$0.280579\pi$
$234$	−2.12407	−0.138855
$235$	−8.04770	−0.524974
$236$	−10.3387	−0.672995
$237$	−3.70559	−0.240704
$238$	33.0735	2.14383
$239$	−16.2244	−1.04947	−0.524735	−	0.851266i	$-0.675835\pi$
$239$	−16.2244	−1.04947	−0.524735	+	0.851266i	$0.675835\pi$
$240$	−2.71484	−0.175242
$241$	−0.502727	−0.0323835	−0.0161917	−	0.999869i	$-0.505154\pi$
$241$	−0.502727	−0.0323835	−0.0161917	+	0.999869i	$0.505154\pi$
$242$	18.8332	1.21064
$243$	1.00000	0.0641500
$244$	−31.5522	−2.01992
$245$	0.460513	0.0294211
$246$	−12.0377	−0.767496
$247$	−1.04377	−0.0664134
$248$	−1.08683	−0.0690139
$249$	13.0471	0.826826
$250$	−2.12407	−0.134338
$251$	7.98307	0.503887	0.251944	−	0.967742i	$-0.418930\pi$
$251$	7.98307	0.503887	0.251944	+	0.967742i	$0.418930\pi$
$252$	6.86037	0.432163
$253$	12.9422	0.813667
$254$	13.6255	0.854941
$255$	−5.70068	−0.356990
$256$	5.00795	0.312997
$257$	7.67339	0.478653	0.239326	−	0.970939i	$-0.423073\pi$
$257$	7.67339	0.478653	0.239326	+	0.970939i	$0.423073\pi$
$258$	−5.47019	−0.340559
$259$	22.9799	1.42790
$260$	2.51167	0.155767
$261$	−2.85192	−0.176529
$262$	19.1686	1.18424
$263$	−27.3288	−1.68517	−0.842584	−	0.538565i	$-0.818967\pi$
$263$	−27.3288	−1.68517	−0.842584	+	0.538565i	$0.818967\pi$
$264$	1.58746	0.0977015
$265$	6.47549	0.397786
$266$	6.05560	0.371293
$267$	4.06255	0.248624
$268$	−4.14601	−0.253258
$269$	14.3008	0.871938	0.435969	−	0.899962i	$-0.356406\pi$
$269$	14.3008	0.871938	0.435969	+	0.899962i	$0.356406\pi$
$270$	−2.12407	−0.129267
$271$	0.115509	0.00701664	0.00350832	−	0.999994i	$-0.498883\pi$
$271$	0.115509	0.00701664	0.00350832	+	0.999994i	$0.498883\pi$
$272$	15.4764	0.938396
$273$	2.73139	0.165311
$274$	−27.4229	−1.65668
$275$	−1.46063	−0.0880794
$276$	−22.2551	−1.33960
$277$	−1.58788	−0.0954068	−0.0477034	−	0.998862i	$-0.515190\pi$
$277$	−1.58788	−0.0954068	−0.0477034	+	0.998862i	$0.515190\pi$
$278$	−10.1946	−0.611430
$279$	1.00000	0.0598684
$280$	−2.96857	−0.177406
$281$	−15.2954	−0.912445	−0.456222	−	0.889866i	$-0.650798\pi$
$281$	−15.2954	−0.912445	−0.456222	+	0.889866i	$0.650798\pi$
$282$	17.0939	1.01793
$283$	0.431571	0.0256542	0.0128271	−	0.999918i	$-0.495917\pi$
$283$	0.431571	0.0256542	0.0128271	+	0.999918i	$0.495917\pi$
$284$	−33.4733	−1.98627
$285$	−1.04377	−0.0618275
$286$	3.10248	0.183454
$287$	15.4796	0.913730
$288$	7.94018	0.467879
$289$	15.4977	0.911630
$290$	6.05767	0.355719
$291$	−12.2970	−0.720866
$292$	−34.1683	−1.99955
$293$	−2.72723	−0.159327	−0.0796633	−	0.996822i	$-0.525384\pi$
$293$	−2.72723	−0.159327	−0.0796633	+	0.996822i	$0.525384\pi$
$294$	−0.978163	−0.0570476
$295$	−4.11628	−0.239659
$296$	−9.14378	−0.531471
$297$	−1.46063	−0.0847544
$298$	30.3328	1.75713
$299$	−8.86066	−0.512425
$300$	2.51167	0.145012
$301$	7.03426	0.405448
$302$	−35.0147	−2.01487
$303$	14.1971	0.815602
$304$	2.83366	0.162522
$305$	−12.5622	−0.719311
$306$	12.1086	0.692205
$307$	3.19633	0.182424	0.0912121	−	0.995831i	$-0.470926\pi$
$307$	3.19633	0.182424	0.0912121	+	0.995831i	$0.470926\pi$
$308$	−10.0205	−0.570970
$309$	2.66526	0.151622
$310$	−2.12407	−0.120639
$311$	3.19961	0.181433	0.0907166	−	0.995877i	$-0.471084\pi$
$311$	3.19961	0.181433	0.0907166	+	0.995877i	$0.471084\pi$
$312$	−1.08683	−0.0615298
$313$	1.88982	0.106819	0.0534094	−	0.998573i	$-0.482991\pi$
$313$	1.88982	0.106819	0.0534094	+	0.998573i	$0.482991\pi$
$314$	−44.0429	−2.48549
$315$	2.73139	0.153897
$316$	−9.30724	−0.523573
$317$	−16.0683	−0.902484	−0.451242	−	0.892402i	$-0.649019\pi$
$317$	−16.0683	−0.902484	−0.451242	+	0.892402i	$0.649019\pi$
$318$	−13.7544	−0.771308
$319$	4.16560	0.233229
$320$	−11.4358	−0.639281
$321$	−17.5943	−0.982016
$322$	51.4067	2.86478
$323$	5.95019	0.331077
$324$	2.51167	0.139537
$325$	1.00000	0.0554700
$326$	4.67274	0.258799
$327$	0.914767	0.0505867
$328$	−6.15938	−0.340095
$329$	−21.9814	−1.21188
$330$	3.10248	0.170786
$331$	−16.5894	−0.911834	−0.455917	−	0.890022i	$-0.650689\pi$
$331$	−16.5894	−0.911834	−0.455917	+	0.890022i	$0.650689\pi$
$332$	32.7700	1.79849
$333$	8.41324	0.461043
$334$	30.7589	1.68305
$335$	−1.65069	−0.0901871
$336$	−7.41530	−0.404538
$337$	−27.1428	−1.47856	−0.739282	−	0.673396i	$-0.764835\pi$
$337$	−27.1428	−1.47856	−0.739282	+	0.673396i	$0.764835\pi$
$338$	−2.12407	−0.115534
$339$	−15.5715	−0.845725
$340$	−14.3182	−0.776516
$341$	−1.46063	−0.0790976
$342$	2.21704	0.119884
$343$	−17.8619	−0.964453
$344$	−2.79896	−0.150910
$345$	−8.86066	−0.477042
$346$	−22.9886	−1.23587
$347$	17.0921	0.917551	0.458775	−	0.888552i	$-0.348288\pi$
$347$	17.0921	0.917551	0.458775	+	0.888552i	$0.348288\pi$
$348$	−7.16309	−0.383982
$349$	18.3971	0.984774	0.492387	−	0.870376i	$-0.336124\pi$
$349$	18.3971	0.984774	0.492387	+	0.870376i	$0.336124\pi$
$350$	−5.80167	−0.310112
$351$	1.00000	0.0533761
$352$	−11.5977	−0.618158
$353$	28.5014	1.51697	0.758487	−	0.651688i	$-0.225939\pi$
$353$	28.5014	1.51697	0.758487	+	0.651688i	$0.225939\pi$
$354$	8.74326	0.464699
$355$	−13.3271	−0.707327
$356$	10.2038	0.540800
$357$	−15.5708	−0.824094
$358$	8.31550	0.439488
$359$	29.3984	1.55159	0.775794	−	0.630987i	$-0.217350\pi$
$359$	29.3984	1.55159	0.775794	+	0.630987i	$0.217350\pi$
$360$	−1.08683	−0.0572811
$361$	−17.9105	−0.942660
$362$	−17.7677	−0.933852
$363$	−8.86656	−0.465374
$364$	6.86037	0.359581
$365$	−13.6038	−0.712055
$366$	26.6830	1.39475
$367$	14.9579	0.780797	0.390398	−	0.920646i	$-0.372337\pi$
$367$	14.9579	0.780797	0.390398	+	0.920646i	$0.372337\pi$
$368$	24.0553	1.25397
$369$	5.66728	0.295027
$370$	−17.8703	−0.929033
$371$	17.6871	0.918269
$372$	2.51167	0.130224
$373$	−7.71080	−0.399250	−0.199625	−	0.979872i	$-0.563972\pi$
$373$	−7.71080	−0.399250	−0.199625	+	0.979872i	$0.563972\pi$
$374$	−17.6863	−0.914535
$375$	1.00000	0.0516398
$376$	8.74650	0.451066
$377$	−2.85192	−0.146881
$378$	−5.80167	−0.298406
$379$	30.9198	1.58824	0.794121	−	0.607759i	$-0.207932\pi$
$379$	30.9198	1.58824	0.794121	+	0.607759i	$0.207932\pi$
$380$	−2.62161	−0.134486
$381$	−6.41482	−0.328641
$382$	47.6342	2.43718
$383$	−20.2864	−1.03659	−0.518294	−	0.855203i	$-0.673433\pi$
$383$	−20.2864	−1.03659	−0.518294	+	0.855203i	$0.673433\pi$
$384$	8.41012	0.429177
$385$	−3.98956	−0.203327
$386$	18.9574	0.964904
$387$	2.57534	0.130912
$388$	−30.8862	−1.56801
$389$	−14.3534	−0.727745	−0.363873	−	0.931449i	$-0.618546\pi$
$389$	−14.3534	−0.727745	−0.363873	+	0.931449i	$0.618546\pi$
$390$	−2.12407	−0.107556
$391$	50.5118	2.55449
$392$	−0.500501	−0.0252791
$393$	−9.02445	−0.455223
$394$	40.1012	2.02027
$395$	−3.70559	−0.186449
$396$	−3.66863	−0.184356
$397$	−27.3526	−1.37279	−0.686394	−	0.727229i	$-0.740808\pi$
$397$	−27.3526	−1.37279	−0.686394	+	0.727229i	$0.740808\pi$
$398$	−4.91712	−0.246473
$399$	−2.85094	−0.142726
$400$	−2.71484	−0.135742
$401$	10.6353	0.531101	0.265550	−	0.964097i	$-0.414446\pi$
$401$	10.6353	0.531101	0.265550	+	0.964097i	$0.414446\pi$
$402$	3.50619	0.174873
$403$	1.00000	0.0498135
$404$	35.6585	1.77408
$405$	1.00000	0.0496904
$406$	16.5459	0.821159
$407$	−12.2886	−0.609125
$408$	6.19568	0.306732
$409$	19.1997	0.949363	0.474681	−	0.880158i	$-0.342563\pi$
$409$	19.1997	0.949363	0.474681	+	0.880158i	$0.342563\pi$
$410$	−12.0377	−0.594499
$411$	12.9105	0.636830
$412$	6.69428	0.329803
$413$	−11.2432	−0.553240
$414$	18.8207	0.924985
$415$	13.0471	0.640456
$416$	7.94018	0.389299
$417$	4.79954	0.235035
$418$	−3.23827	−0.158389
$419$	10.0497	0.490962	0.245481	−	0.969401i	$-0.421054\pi$
$419$	10.0497	0.490962	0.245481	+	0.969401i	$0.421054\pi$
$420$	6.86037	0.334752
$421$	−39.9866	−1.94883	−0.974413	−	0.224764i	$-0.927839\pi$
$421$	−39.9866	−1.94883	−0.974413	+	0.224764i	$0.927839\pi$
$422$	−36.8855	−1.79556
$423$	−8.04770	−0.391292
$424$	−7.03777	−0.341784
$425$	−5.70068	−0.276523
$426$	28.3076	1.37151
$427$	−34.3124	−1.66049
$428$	−44.1911	−2.13606
$429$	−1.46063	−0.0705199
$430$	−5.47019	−0.263796
$431$	−17.0955	−0.823459	−0.411729	−	0.911306i	$-0.635075\pi$
$431$	−17.0955	−0.823459	−0.411729	+	0.911306i	$0.635075\pi$
$432$	−2.71484	−0.130618
$433$	−31.9442	−1.53514	−0.767570	−	0.640965i	$-0.778535\pi$
$433$	−31.9442	−1.53514	−0.767570	+	0.640965i	$0.778535\pi$
$434$	−5.80167	−0.278489
$435$	−2.85192	−0.136739
$436$	2.29760	0.110035
$437$	9.24848	0.442415
$438$	28.8954	1.38068
$439$	12.5222	0.597652	0.298826	−	0.954308i	$-0.403405\pi$
$439$	12.5222	0.597652	0.298826	+	0.954308i	$0.403405\pi$
$440$	1.58746	0.0756793
$441$	0.460513	0.0219292
$442$	12.1086	0.575949
$443$	35.9629	1.70865	0.854325	−	0.519740i	$-0.173971\pi$
$443$	35.9629	1.70865	0.854325	+	0.519740i	$0.173971\pi$
$444$	21.1313	1.00285
$445$	4.06255	0.192583
$446$	24.7489	1.17190
$447$	−14.2805	−0.675445
$448$	−31.2357	−1.47575
$449$	4.64674	0.219293	0.109647	−	0.993971i	$-0.465028\pi$
$449$	4.64674	0.219293	0.109647	+	0.993971i	$0.465028\pi$
$450$	−2.12407	−0.100130
$451$	−8.27780	−0.389787
$452$	−39.1104	−1.83960
$453$	16.4847	0.774519
$454$	41.1777	1.93257
$455$	2.73139	0.128050
$456$	1.13440	0.0531232
$457$	−12.1820	−0.569848	−0.284924	−	0.958550i	$-0.591968\pi$
$457$	−12.1820	−0.569848	−0.284924	+	0.958550i	$0.591968\pi$
$458$	−51.5759	−2.40998
$459$	−5.70068	−0.266085
$460$	−22.2551	−1.03765
$461$	−30.8084	−1.43489	−0.717446	−	0.696614i	$-0.754689\pi$
$461$	−30.8084	−1.43489	−0.717446	+	0.696614i	$0.754689\pi$
$462$	8.47410	0.394251
$463$	12.2206	0.567938	0.283969	−	0.958833i	$-0.408349\pi$
$463$	12.2206	0.567938	0.283969	+	0.958833i	$0.408349\pi$
$464$	7.74250	0.359437
$465$	1.00000	0.0463739
$466$	−41.2428	−1.91054
$467$	−15.0445	−0.696178	−0.348089	−	0.937462i	$-0.613169\pi$
$467$	−15.0445	−0.696178	−0.348089	+	0.937462i	$0.613169\pi$
$468$	2.51167	0.116102
$469$	−4.50870	−0.208192
$470$	17.0939	0.788481
$471$	20.7352	0.955426
$472$	4.47370	0.205919
$473$	−3.76162	−0.172959
$474$	7.87094	0.361524
$475$	−1.04377	−0.0478914
$476$	−39.1088	−1.79255
$477$	6.47549	0.296492
$478$	34.4618	1.57624
$479$	25.6885	1.17374	0.586868	−	0.809682i	$-0.300361\pi$
$479$	25.6885	1.17374	0.586868	+	0.809682i	$0.300361\pi$
$480$	7.94018	0.362418
$481$	8.41324	0.383611
$482$	1.06783	0.0486382
$483$	−24.2020	−1.10123
$484$	−22.2699	−1.01227
$485$	−12.2970	−0.558380
$486$	−2.12407	−0.0963498
$487$	−36.9360	−1.67373	−0.836865	−	0.547409i	$-0.815614\pi$
$487$	−36.9360	−1.67373	−0.836865	+	0.547409i	$0.815614\pi$
$488$	13.6530	0.618044
$489$	−2.19990	−0.0994830
$490$	−0.978163	−0.0441889
$491$	11.3345	0.511518	0.255759	−	0.966741i	$-0.417675\pi$
$491$	11.3345	0.511518	0.255759	+	0.966741i	$0.417675\pi$
$492$	14.2344	0.641734
$493$	16.2579	0.732217
$494$	2.21704	0.0997492
$495$	−1.46063	−0.0656505
$496$	−2.71484	−0.121900
$497$	−36.4015	−1.63283
$498$	−27.7129	−1.24185
$499$	21.1298	0.945899	0.472950	−	0.881089i	$-0.343189\pi$
$499$	21.1298	0.945899	0.472950	+	0.881089i	$0.343189\pi$
$500$	2.51167	0.112325
$501$	−14.4811	−0.646969
$502$	−16.9566	−0.756810
$503$	−25.9966	−1.15913	−0.579566	−	0.814925i	$-0.696778\pi$
$503$	−25.9966	−1.15913	−0.579566	+	0.814925i	$0.696778\pi$
$504$	−2.96857	−0.132231
$505$	14.1971	0.631762
$506$	−27.4901	−1.22208
$507$	1.00000	0.0444116
$508$	−16.1119	−0.714851
$509$	39.2680	1.74053	0.870263	−	0.492588i	$-0.163949\pi$
$509$	39.2680	1.74053	0.870263	+	0.492588i	$0.163949\pi$
$510$	12.1086	0.536179
$511$	−37.1573	−1.64374
$512$	−27.4575	−1.21346
$513$	−1.04377	−0.0460835
$514$	−16.2988	−0.718910
$515$	2.66526	0.117446
$516$	6.46841	0.284756
$517$	11.7547	0.516972
$518$	−48.8109	−2.14463
$519$	10.8229	0.475072
$520$	−1.08683	−0.0476608
$521$	1.31856	0.0577670	0.0288835	−	0.999583i	$-0.490805\pi$
$521$	1.31856	0.0577670	0.0288835	+	0.999583i	$0.490805\pi$
$522$	6.05767	0.265137
$523$	−2.12301	−0.0928326	−0.0464163	−	0.998922i	$-0.514780\pi$
$523$	−2.12301	−0.0928326	−0.0464163	+	0.998922i	$0.514780\pi$
$524$	−22.6665	−0.990190
$525$	2.73139	0.119208
$526$	58.0484	2.53103
$527$	−5.70068	−0.248325
$528$	3.96538	0.172571
$529$	55.5113	2.41354
$530$	−13.7544	−0.597452
$531$	−4.11628	−0.178631
$532$	−7.16064	−0.310453
$533$	5.66728	0.245477
$534$	−8.62913	−0.373419
$535$	−17.5943	−0.760666
$536$	1.79403	0.0774902
$537$	−3.91489	−0.168940
$538$	−30.3760	−1.30960
$539$	−0.672640	−0.0289727
$540$	2.51167	0.108085
$541$	37.4429	1.60980	0.804899	−	0.593412i	$-0.202219\pi$
$541$	37.4429	1.60980	0.804899	+	0.593412i	$0.202219\pi$
$542$	−0.245348	−0.0105386
$543$	8.36495	0.358974
$544$	−45.2644	−1.94069
$545$	0.914767	0.0391843
$546$	−5.80167	−0.248289
$547$	−24.1601	−1.03301	−0.516506	−	0.856284i	$-0.672768\pi$
$547$	−24.1601	−1.03301	−0.516506	+	0.856284i	$0.672768\pi$
$548$	32.4271	1.38522
$549$	−12.5622	−0.536143
$550$	3.10248	0.132290
$551$	2.97674	0.126813
$552$	9.63005	0.409882
$553$	−10.1214	−0.430407
$554$	3.37278	0.143296
$555$	8.41324	0.357122
$556$	12.0549	0.511241
$557$	38.1553	1.61669	0.808346	−	0.588707i	$-0.200363\pi$
$557$	38.1553	1.61669	0.808346	+	0.588707i	$0.200363\pi$
$558$	−2.12407	−0.0899191
$559$	2.57534	0.108925
$560$	−7.41530	−0.313354
$561$	8.32659	0.351549
$562$	32.4884	1.37044
$563$	4.10975	0.173205	0.0866027	−	0.996243i	$-0.472399\pi$
$563$	4.10975	0.173205	0.0866027	+	0.996243i	$0.472399\pi$
$564$	−20.2132	−0.851129
$565$	−15.5715	−0.655096
$566$	−0.916686	−0.0385312
$567$	2.73139	0.114708
$568$	14.4843	0.607747
$569$	−6.88012	−0.288430	−0.144215	−	0.989546i	$-0.546066\pi$
$569$	−6.88012	−0.288430	−0.144215	+	0.989546i	$0.546066\pi$
$570$	2.21704	0.0928615
$571$	18.3033	0.765971	0.382985	−	0.923754i	$-0.374896\pi$
$571$	18.3033	0.765971	0.382985	+	0.923754i	$0.374896\pi$
$572$	−3.66863	−0.153393
$573$	−22.4259	−0.936855
$574$	−32.8797	−1.37237
$575$	−8.86066	−0.369515
$576$	−11.4358	−0.476492
$577$	−33.1812	−1.38135	−0.690676	−	0.723164i	$-0.742687\pi$
$577$	−33.1812	−1.38135	−0.690676	+	0.723164i	$0.742687\pi$
$578$	−32.9182	−1.36922
$579$	−8.92501	−0.370911
$580$	−7.16309	−0.297431
$581$	35.6367	1.47846
$582$	26.1198	1.08270
$583$	−9.45830	−0.391723
$584$	14.7850	0.611809
$585$	1.00000	0.0413449
$586$	5.79283	0.239300
$587$	−12.9976	−0.536470	−0.268235	−	0.963353i	$-0.586440\pi$
$587$	−12.9976	−0.536470	−0.268235	+	0.963353i	$0.586440\pi$
$588$	1.15666	0.0476999
$589$	−1.04377	−0.0430077
$590$	8.74326	0.359954
$591$	−18.8794	−0.776595
$592$	−22.8406	−0.938743
$593$	−23.1751	−0.951689	−0.475844	−	0.879529i	$-0.657857\pi$
$593$	−23.1751	−0.951689	−0.475844	+	0.879529i	$0.657857\pi$
$594$	3.10248	0.127296
$595$	−15.5708	−0.638340
$596$	−35.8680	−1.46921
$597$	2.31495	0.0947446
$598$	18.8207	0.769634
$599$	6.79556	0.277659	0.138830	−	0.990316i	$-0.455666\pi$
$599$	6.79556	0.277659	0.138830	+	0.990316i	$0.455666\pi$
$600$	−1.08683	−0.0443697
$601$	14.4587	0.589784	0.294892	−	0.955531i	$-0.404716\pi$
$601$	14.4587	0.589784	0.294892	+	0.955531i	$0.404716\pi$
$602$	−14.9413	−0.608960
$603$	−1.65069	−0.0672215
$604$	41.4042	1.68471
$605$	−8.86656	−0.360477
$606$	−30.1556	−1.22499
$607$	25.6419	1.04077	0.520385	−	0.853931i	$-0.325788\pi$
$607$	25.6419	1.04077	0.520385	+	0.853931i	$0.325788\pi$
$608$	−8.28771	−0.336111
$609$	−7.78971	−0.315655
$610$	26.6830	1.08036
$611$	−8.04770	−0.325575
$612$	−14.3182	−0.578781
$613$	15.7320	0.635409	0.317704	−	0.948190i	$-0.397088\pi$
$613$	15.7320	0.635409	0.317704	+	0.948190i	$0.397088\pi$
$614$	−6.78923	−0.273991
$615$	5.66728	0.228527
$616$	4.33598	0.174702
$617$	18.2918	0.736399	0.368199	−	0.929747i	$-0.379974\pi$
$617$	18.2918	0.736399	0.368199	+	0.929747i	$0.379974\pi$
$618$	−5.66121	−0.227727
$619$	1.79616	0.0721939	0.0360969	−	0.999348i	$-0.488507\pi$
$619$	1.79616	0.0721939	0.0360969	+	0.999348i	$0.488507\pi$
$620$	2.51167	0.100871
$621$	−8.86066	−0.355566
$622$	−6.79620	−0.272503
$623$	11.0964	0.444569
$624$	−2.71484	−0.108681
$625$	1.00000	0.0400000
$626$	−4.01410	−0.160436
$627$	1.52456	0.0608851
$628$	52.0800	2.07822
$629$	−47.9612	−1.91234
$630$	−5.80167	−0.231144
$631$	30.4412	1.21184	0.605922	−	0.795524i	$-0.292804\pi$
$631$	30.4412	1.21184	0.605922	+	0.795524i	$0.292804\pi$
$632$	4.02736	0.160200
$633$	17.3655	0.690215
$634$	34.1301	1.35548
$635$	−6.41482	−0.254564
$636$	16.2643	0.644922
$637$	0.460513	0.0182462
$638$	−8.84803	−0.350297
$639$	−13.3271	−0.527211
$640$	8.41012	0.332439
$641$	−16.1964	−0.639720	−0.319860	−	0.947465i	$-0.603636\pi$
$641$	−16.1964	−0.639720	−0.319860	+	0.947465i	$0.603636\pi$
$642$	37.3714	1.47493
$643$	−14.9874	−0.591044	−0.295522	−	0.955336i	$-0.595494\pi$
$643$	−14.9874	−0.591044	−0.295522	+	0.955336i	$0.595494\pi$
$644$	−60.7874	−2.39536
$645$	2.57534	0.101404
$646$	−12.6386	−0.497260
$647$	−1.03097	−0.0405317	−0.0202658	−	0.999795i	$-0.506451\pi$
$647$	−1.03097	−0.0405317	−0.0202658	+	0.999795i	$0.506451\pi$
$648$	−1.08683	−0.0426948
$649$	6.01236	0.236006
$650$	−2.12407	−0.0833129
$651$	2.73139	0.107052
$652$	−5.52544	−0.216393
$653$	7.71798	0.302028	0.151014	−	0.988532i	$-0.451746\pi$
$653$	7.71798	0.302028	0.151014	+	0.988532i	$0.451746\pi$
$654$	−1.94303	−0.0759784
$655$	−9.02445	−0.352614
$656$	−15.3858	−0.600713
$657$	−13.6038	−0.530734
$658$	46.6901	1.82017
$659$	20.3520	0.792803	0.396401	−	0.918077i	$-0.370259\pi$
$659$	20.3520	0.792803	0.396401	+	0.918077i	$0.370259\pi$
$660$	−3.66863	−0.142801
$661$	−17.9003	−0.696241	−0.348121	−	0.937450i	$-0.613180\pi$
$661$	−17.9003	−0.696241	−0.348121	+	0.937450i	$0.613180\pi$
$662$	35.2370	1.36952
$663$	−5.70068	−0.221396
$664$	−14.1800	−0.550291
$665$	−2.85094	−0.110555
$666$	−17.8703	−0.692460
$667$	25.2699	0.978454
$668$	−36.3719	−1.40727
$669$	−11.6517	−0.450479
$670$	3.50619	0.135456
$671$	18.3488	0.708347
$672$	21.6878	0.836623
$673$	16.7546	0.645843	0.322921	−	0.946426i	$-0.395335\pi$
$673$	16.7546	0.645843	0.322921	+	0.946426i	$0.395335\pi$
$674$	57.6533	2.22072
$675$	1.00000	0.0384900
$676$	2.51167	0.0966029
$677$	−26.8054	−1.03021	−0.515107	−	0.857126i	$-0.672248\pi$
$677$	−26.8054	−1.03021	−0.515107	+	0.857126i	$0.672248\pi$
$678$	33.0749	1.27023
$679$	−33.5881	−1.28899
$680$	6.19568	0.237593
$681$	−19.3862	−0.742882
$682$	3.10248	0.118800
$683$	−35.4545	−1.35663	−0.678315	−	0.734771i	$-0.737290\pi$
$683$	−35.4545	−1.35663	−0.678315	+	0.734771i	$0.737290\pi$
$684$	−2.62161	−0.100240
$685$	12.9105	0.493286
$686$	37.9400	1.44855
$687$	24.2816	0.926403
$688$	−6.99163	−0.266553
$689$	6.47549	0.246696
$690$	18.8207	0.716491
$691$	−34.6486	−1.31809	−0.659047	−	0.752102i	$-0.729040\pi$
$691$	−34.6486	−1.31809	−0.659047	+	0.752102i	$0.729040\pi$
$692$	27.1836	1.03336
$693$	−3.98956	−0.151551
$694$	−36.3048	−1.37811
$695$	4.79954	0.182057
$696$	3.09956	0.117488
$697$	−32.3073	−1.22373
$698$	−39.0767	−1.47908
$699$	19.4169	0.734413
$700$	6.86037	0.259298
$701$	36.5549	1.38066	0.690330	−	0.723494i	$-0.257465\pi$
$701$	36.5549	1.38066	0.690330	+	0.723494i	$0.257465\pi$
$702$	−2.12407	−0.0801679
$703$	−8.78147	−0.331200
$704$	16.7035	0.629537
$705$	−8.04770	−0.303094
$706$	−60.5389	−2.27841
$707$	38.7779	1.45839
$708$	−10.3387	−0.388554
$709$	10.8258	0.406573	0.203287	−	0.979119i	$-0.434838\pi$
$709$	10.8258	0.406573	0.203287	+	0.979119i	$0.434838\pi$
$710$	28.3076	1.06237
$711$	−3.70559	−0.138971
$712$	−4.41531	−0.165471
$713$	−8.86066	−0.331834
$714$	33.0735	1.23774
$715$	−1.46063	−0.0546245
$716$	−9.83293	−0.367474
$717$	−16.2244	−0.605911
$718$	−62.4442	−2.33040
$719$	25.6899	0.958072	0.479036	−	0.877795i	$-0.340986\pi$
$719$	25.6899	0.958072	0.479036	+	0.877795i	$0.340986\pi$
$720$	−2.71484	−0.101176
$721$	7.27989	0.271117
$722$	38.0433	1.41582
$723$	−0.502727	−0.0186966
$724$	21.0100	0.780831
$725$	−2.85192	−0.105918
$726$	18.8332	0.698965
$727$	−13.4601	−0.499208	−0.249604	−	0.968348i	$-0.580300\pi$
$727$	−13.4601	−0.499208	−0.249604	+	0.968348i	$0.580300\pi$
$728$	−2.96857	−0.110022
$729$	1.00000	0.0370370
$730$	28.8954	1.06947
$731$	−14.6812	−0.543002
$732$	−31.5522	−1.16620
$733$	−32.1123	−1.18610	−0.593048	−	0.805167i	$-0.702076\pi$
$733$	−32.1123	−1.18610	−0.593048	+	0.805167i	$0.702076\pi$
$734$	−31.7717	−1.17271
$735$	0.460513	0.0169863
$736$	−70.3552	−2.59333
$737$	2.41106	0.0888124
$738$	−12.0377	−0.443114
$739$	−36.9619	−1.35967	−0.679833	−	0.733367i	$-0.737948\pi$
$739$	−36.9619	−1.35967	−0.679833	+	0.733367i	$0.737948\pi$
$740$	21.1313	0.776803
$741$	−1.04377	−0.0383438
$742$	−37.5687	−1.37919
$743$	−22.5694	−0.827990	−0.413995	−	0.910279i	$-0.635867\pi$
$743$	−22.5694	−0.827990	−0.413995	+	0.910279i	$0.635867\pi$
$744$	−1.08683	−0.0398452
$745$	−14.2805	−0.523197
$746$	16.3783	0.599652
$747$	13.0471	0.477368
$748$	20.9137	0.764680
$749$	−48.0569	−1.75596
$750$	−2.12407	−0.0775601
$751$	−5.88649	−0.214801	−0.107401	−	0.994216i	$-0.534253\pi$
$751$	−5.88649	−0.214801	−0.107401	+	0.994216i	$0.534253\pi$
$752$	21.8482	0.796722
$753$	7.98307	0.290919
$754$	6.05767	0.220607
$755$	16.4847	0.599940
$756$	6.86037	0.249509
$757$	−36.3358	−1.32065	−0.660324	−	0.750981i	$-0.729581\pi$
$757$	−36.3358	−1.32065	−0.660324	+	0.750981i	$0.729581\pi$
$758$	−65.6758	−2.38545
$759$	12.9422	0.469771
$760$	1.13440	0.0411491
$761$	−10.3283	−0.374401	−0.187200	−	0.982322i	$-0.559941\pi$
$761$	−10.3283	−0.374401	−0.187200	+	0.982322i	$0.559941\pi$
$762$	13.6255	0.493600
$763$	2.49859	0.0904550
$764$	−56.3265	−2.03782
$765$	−5.70068	−0.206108
$766$	43.0898	1.55690
$767$	−4.11628	−0.148630
$768$	5.00795	0.180709
$769$	−23.2666	−0.839016	−0.419508	−	0.907752i	$-0.637797\pi$
$769$	−23.2666	−0.839016	−0.419508	+	0.907752i	$0.637797\pi$
$770$	8.47410	0.305386
$771$	7.67339	0.276350
$772$	−22.4167	−0.806796
$773$	−20.1846	−0.725990	−0.362995	−	0.931791i	$-0.618246\pi$
$773$	−20.1846	−0.725990	−0.362995	+	0.931791i	$0.618246\pi$
$774$	−5.47019	−0.196622
$775$	1.00000	0.0359211
$776$	13.3648	0.479770
$777$	22.9799	0.824398
$778$	30.4876	1.09303
$779$	−5.91533	−0.211939
$780$	2.51167	0.0899324
$781$	19.4659	0.696546
$782$	−107.291	−3.83670
$783$	−2.85192	−0.101919
$784$	−1.25022	−0.0446507
$785$	20.7352	0.740070
$786$	19.1686	0.683720
$787$	44.3408	1.58058	0.790289	−	0.612734i	$-0.209930\pi$
$787$	44.3408	1.58058	0.790289	+	0.612734i	$0.209930\pi$
$788$	−47.4189	−1.68923
$789$	−27.3288	−0.972932
$790$	7.87094	0.280035
$791$	−42.5318	−1.51226
$792$	1.58746	0.0564080
$793$	−12.5622	−0.446098
$794$	58.0989	2.06185
$795$	6.47549	0.229662
$796$	5.81440	0.206086
$797$	−1.73863	−0.0615853	−0.0307927	−	0.999526i	$-0.509803\pi$
$797$	−1.73863	−0.0615853	−0.0307927	+	0.999526i	$0.509803\pi$
$798$	6.05560	0.214366
$799$	45.8773	1.62302
$800$	7.94018	0.280728
$801$	4.06255	0.143543
$802$	−22.5901	−0.797684
$803$	19.8701	0.701201
$804$	−4.14601	−0.146218
$805$	−24.2020	−0.853007
$806$	−2.12407	−0.0748172
$807$	14.3008	0.503413
$808$	−15.4299	−0.542821
$809$	−2.33089	−0.0819496	−0.0409748	−	0.999160i	$-0.513046\pi$
$809$	−2.33089	−0.0819496	−0.0409748	+	0.999160i	$0.513046\pi$
$810$	−2.12407	−0.0746322
$811$	−31.4009	−1.10263	−0.551317	−	0.834296i	$-0.685875\pi$
$811$	−31.4009	−1.10263	−0.551317	+	0.834296i	$0.685875\pi$
$812$	−19.5652	−0.686605
$813$	0.115509	0.00405106
$814$	26.1019	0.914872
$815$	−2.19990	−0.0770592
$816$	15.4764	0.541783
$817$	−2.68805	−0.0940431
$818$	−40.7814	−1.42589
$819$	2.73139	0.0954426
$820$	14.2344	0.497085
$821$	16.8129	0.586774	0.293387	−	0.955994i	$-0.405218\pi$
$821$	16.8129	0.586774	0.293387	+	0.955994i	$0.405218\pi$
$822$	−27.4229	−0.956483
$823$	−40.0481	−1.39599	−0.697994	−	0.716104i	$-0.745924\pi$
$823$	−40.0481	−1.39599	−0.697994	+	0.716104i	$0.745924\pi$
$824$	−2.89670	−0.100911
$825$	−1.46063	−0.0508527
$826$	23.8813	0.830936
$827$	42.4316	1.47549	0.737746	−	0.675078i	$-0.235890\pi$
$827$	42.4316	1.47549	0.737746	+	0.675078i	$0.235890\pi$
$828$	−22.2551	−0.773418
$829$	−6.37380	−0.221371	−0.110686	−	0.993855i	$-0.535305\pi$
$829$	−6.37380	−0.221371	−0.110686	+	0.993855i	$0.535305\pi$
$830$	−27.7129	−0.961930
$831$	−1.58788	−0.0550831
$832$	−11.4358	−0.396465
$833$	−2.62524	−0.0909591
$834$	−10.1946	−0.353009
$835$	−14.4811	−0.501140
$836$	3.82920	0.132436
$837$	1.00000	0.0345651
$838$	−21.3464	−0.737398
$839$	−10.0504	−0.346977	−0.173488	−	0.984836i	$-0.555504\pi$
$839$	−10.0504	−0.346977	−0.173488	+	0.984836i	$0.555504\pi$
$840$	−2.96857	−0.102425
$841$	−20.8666	−0.719537
$842$	84.9343	2.92703
$843$	−15.2954	−0.526800
$844$	43.6164	1.50134
$845$	1.00000	0.0344010
$846$	17.0939	0.587699
$847$	−24.2181	−0.832142
$848$	−17.5799	−0.603697
$849$	0.431571	0.0148115
$850$	12.1086	0.415323
$851$	−74.5469	−2.55543
$852$	−33.4733	−1.14678
$853$	44.0202	1.50722	0.753611	−	0.657320i	$-0.228310\pi$
$853$	44.0202	1.50722	0.753611	+	0.657320i	$0.228310\pi$
$854$	72.8819	2.49397
$855$	−1.04377	−0.0356961
$856$	19.1220	0.653577
$857$	−35.9637	−1.22850	−0.614249	−	0.789112i	$-0.710541\pi$
$857$	−35.9637	−1.22850	−0.614249	+	0.789112i	$0.710541\pi$
$858$	3.10248	0.105917
$859$	−4.27555	−0.145880	−0.0729399	−	0.997336i	$-0.523238\pi$
$859$	−4.27555	−0.145880	−0.0729399	+	0.997336i	$0.523238\pi$
$860$	6.46841	0.220571
$861$	15.4796	0.527542
$862$	36.3119	1.23679
$863$	−30.9507	−1.05358	−0.526788	−	0.849997i	$-0.676604\pi$
$863$	−30.9507	−1.05358	−0.526788	+	0.849997i	$0.676604\pi$
$864$	7.94018	0.270130
$865$	10.8229	0.367989
$866$	67.8517	2.30570
$867$	15.4977	0.526330
$868$	6.86037	0.232856
$869$	5.41251	0.183607
$870$	6.05767	0.205374
$871$	−1.65069	−0.0559317
$872$	−0.994198	−0.0336678
$873$	−12.2970	−0.416192
$874$	−19.6444	−0.664482
$875$	2.73139	0.0923380
$876$	−34.1683	−1.15444
$877$	9.96657	0.336547	0.168274	−	0.985740i	$-0.446181\pi$
$877$	9.96657	0.336547	0.168274	+	0.985740i	$0.446181\pi$
$878$	−26.5980	−0.897639
$879$	−2.72723	−0.0919872
$880$	3.96538	0.133673
$881$	−52.6241	−1.77295	−0.886476	−	0.462775i	$-0.846854\pi$
$881$	−52.6241	−1.77295	−0.886476	+	0.462775i	$0.846854\pi$
$882$	−0.978163	−0.0329365
$883$	−2.13051	−0.0716975	−0.0358488	−	0.999357i	$-0.511413\pi$
$883$	−2.13051	−0.0716975	−0.0358488	+	0.999357i	$0.511413\pi$
$884$	−14.3182	−0.481575
$885$	−4.11628	−0.138367
$886$	−76.3877	−2.56630
$887$	11.4660	0.384992	0.192496	−	0.981298i	$-0.438342\pi$
$887$	11.4660	0.384992	0.192496	+	0.981298i	$0.438342\pi$
$888$	−9.14378	−0.306845
$889$	−17.5214	−0.587648
$890$	−8.62913	−0.289249
$891$	−1.46063	−0.0489330
$892$	−29.2652	−0.979870
$893$	8.39993	0.281093
$894$	30.3328	1.01448
$895$	−3.91489	−0.130860
$896$	22.9713	0.767419
$897$	−8.86066	−0.295849
$898$	−9.87001	−0.329366
$899$	−2.85192	−0.0951168
$900$	2.51167	0.0837225
$901$	−36.9147	−1.22981
$902$	17.5826	0.585438
$903$	7.03426	0.234085
$904$	16.9236	0.562869
$905$	8.36495	0.278060
$906$	−35.0147	−1.16328
$907$	−40.8328	−1.35583	−0.677915	−	0.735140i	$-0.737116\pi$
$907$	−40.8328	−1.35583	−0.677915	+	0.735140i	$0.737116\pi$
$908$	−48.6919	−1.61590
$909$	14.1971	0.470888
$910$	−5.80167	−0.192324
$911$	−23.6909	−0.784915	−0.392458	−	0.919770i	$-0.628375\pi$
$911$	−23.6909	−0.784915	−0.392458	+	0.919770i	$0.628375\pi$
$912$	2.83366	0.0938320
$913$	−19.0570	−0.630694
$914$	25.8753	0.855880
$915$	−12.5622	−0.415294
$916$	60.9876	2.01509
$917$	−24.6493	−0.813993
$918$	12.1086	0.399645
$919$	43.8370	1.44605	0.723025	−	0.690822i	$-0.242751\pi$
$919$	43.8370	1.44605	0.723025	+	0.690822i	$0.242751\pi$
$920$	9.63005	0.317494
$921$	3.19633	0.105323
$922$	65.4393	2.15513
$923$	−13.3271	−0.438666
$924$	−10.0205	−0.329649
$925$	8.41324	0.276626
$926$	−25.9574	−0.853012
$927$	2.66526	0.0875388
$928$	−22.6447	−0.743350
$929$	−52.9023	−1.73567	−0.867834	−	0.496853i	$-0.834489\pi$
$929$	−52.9023	−1.73567	−0.867834	+	0.496853i	$0.834489\pi$
$930$	−2.12407	−0.0696510
$931$	−0.480669	−0.0157533
$932$	48.7688	1.59748
$933$	3.19961	0.104751
$934$	31.9556	1.04562
$935$	8.32659	0.272309
$936$	−1.08683	−0.0355242
$937$	39.8023	1.30028	0.650142	−	0.759813i	$-0.274709\pi$
$937$	39.8023	1.30028	0.650142	+	0.759813i	$0.274709\pi$
$938$	9.57679	0.312693
$939$	1.88982	0.0616718
$940$	−20.2132	−0.659282
$941$	50.4450	1.64446	0.822230	−	0.569155i	$-0.192730\pi$
$941$	50.4450	1.64446	0.822230	+	0.569155i	$0.192730\pi$
$942$	−44.0429	−1.43500
$943$	−50.2158	−1.63525
$944$	11.1750	0.363716
$945$	2.73139	0.0888523
$946$	7.98994	0.259775
$947$	−0.326652	−0.0106148	−0.00530739	−	0.999986i	$-0.501689\pi$
$947$	−0.326652	−0.0106148	−0.00530739	+	0.999986i	$0.501689\pi$
$948$	−9.30724	−0.302285
$949$	−13.6038	−0.441598
$950$	2.21704	0.0719302
$951$	−16.0683	−0.521049
$952$	16.9228	0.548473
$953$	11.3263	0.366896	0.183448	−	0.983029i	$-0.441274\pi$
$953$	11.3263	0.366896	0.183448	+	0.983029i	$0.441274\pi$
$954$	−13.7544	−0.445315
$955$	−22.4259	−0.725685
$956$	−40.7504	−1.31796
$957$	4.16560	0.134655
$958$	−54.5641	−1.76289
$959$	35.2638	1.13873
$960$	−11.4358	−0.369089
$961$	1.00000	0.0322581
$962$	−17.8703	−0.576162
$963$	−17.5943	−0.566967
$964$	−1.26269	−0.0406684
$965$	−8.92501	−0.287306
$966$	51.4067	1.65398
$967$	17.1772	0.552382	0.276191	−	0.961103i	$-0.410928\pi$
$967$	17.1772	0.552382	0.276191	+	0.961103i	$0.410928\pi$
$968$	9.63646	0.309728
$969$	5.95019	0.191148
$970$	26.1198	0.838656
$971$	58.2348	1.86885	0.934423	−	0.356166i	$-0.115916\pi$
$971$	58.2348	1.86885	0.934423	+	0.356166i	$0.115916\pi$
$972$	2.51167	0.0805620
$973$	13.1094	0.420270
$974$	78.4547	2.51385
$975$	1.00000	0.0320256
$976$	34.1044	1.09166
$977$	21.9588	0.702523	0.351262	−	0.936277i	$-0.385753\pi$
$977$	21.9588	0.702523	0.351262	+	0.936277i	$0.385753\pi$
$978$	4.67274	0.149418
$979$	−5.93388	−0.189648
$980$	1.15666	0.0369481
$981$	0.914767	0.0292062
$982$	−24.0752	−0.768272
$983$	−19.6905	−0.628030	−0.314015	−	0.949418i	$-0.601674\pi$
$983$	−19.6905	−0.628030	−0.314015	+	0.949418i	$0.601674\pi$
$984$	−6.15938	−0.196354
$985$	−18.8794	−0.601548
$986$	−34.5328	−1.09975
$987$	−21.9814	−0.699677
$988$	−2.62161	−0.0834044
$989$	−22.8192	−0.725608
$990$	3.10248	0.0986034
$991$	−19.1056	−0.606908	−0.303454	−	0.952846i	$-0.598140\pi$
$991$	−19.1056	−0.606908	−0.303454	+	0.952846i	$0.598140\pi$
$992$	7.94018	0.252101
$993$	−16.5894	−0.526448
$994$	77.3193	2.45242
$995$	2.31495	0.0733889
$996$	32.7700	1.03836
$997$	18.8939	0.598377	0.299188	−	0.954194i	$-0.403284\pi$
$997$	18.8939	0.598377	0.299188	+	0.954194i	$0.403284\pi$
$998$	−44.8812	−1.42069
$999$	8.41324	0.266183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.t.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.t.1.3	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12407 q^{2} +1.00000 q^{3} +2.51167 q^{4} +1.00000 q^{5} -2.12407 q^{6} +2.73139 q^{7} -1.08683 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.12407 q^{2} +1.00000 q^{3} +2.51167 q^{4} +1.00000 q^{5} -2.12407 q^{6} +2.73139 q^{7} -1.08683 q^{8} +1.00000 q^{9} -2.12407 q^{10} -1.46063 q^{11} +2.51167 q^{12} +1.00000 q^{13} -5.80167 q^{14} +1.00000 q^{15} -2.71484 q^{16} -5.70068 q^{17} -2.12407 q^{18} -1.04377 q^{19} +2.51167 q^{20} +2.73139 q^{21} +3.10248 q^{22} -8.86066 q^{23} -1.08683 q^{24} +1.00000 q^{25} -2.12407 q^{26} +1.00000 q^{27} +6.86037 q^{28} -2.85192 q^{29} -2.12407 q^{30} +1.00000 q^{31} +7.94018 q^{32} -1.46063 q^{33} +12.1086 q^{34} +2.73139 q^{35} +2.51167 q^{36} +8.41324 q^{37} +2.21704 q^{38} +1.00000 q^{39} -1.08683 q^{40} +5.66728 q^{41} -5.80167 q^{42} +2.57534 q^{43} -3.66863 q^{44} +1.00000 q^{45} +18.8207 q^{46} -8.04770 q^{47} -2.71484 q^{48} +0.460513 q^{49} -2.12407 q^{50} -5.70068 q^{51} +2.51167 q^{52} +6.47549 q^{53} -2.12407 q^{54} -1.46063 q^{55} -2.96857 q^{56} -1.04377 q^{57} +6.05767 q^{58} -4.11628 q^{59} +2.51167 q^{60} -12.5622 q^{61} -2.12407 q^{62} +2.73139 q^{63} -11.4358 q^{64} +1.00000 q^{65} +3.10248 q^{66} -1.65069 q^{67} -14.3182 q^{68} -8.86066 q^{69} -5.80167 q^{70} -13.3271 q^{71} -1.08683 q^{72} -13.6038 q^{73} -17.8703 q^{74} +1.00000 q^{75} -2.62161 q^{76} -3.98956 q^{77} -2.12407 q^{78} -3.70559 q^{79} -2.71484 q^{80} +1.00000 q^{81} -12.0377 q^{82} +13.0471 q^{83} +6.86037 q^{84} -5.70068 q^{85} -5.47019 q^{86} -2.85192 q^{87} +1.58746 q^{88} +4.06255 q^{89} -2.12407 q^{90} +2.73139 q^{91} -22.2551 q^{92} +1.00000 q^{93} +17.0939 q^{94} -1.04377 q^{95} +7.94018 q^{96} -12.2970 q^{97} -0.978163 q^{98} -1.46063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 9 q^{3} + 8 q^{4} + 9 q^{5} - 6 q^{6} - 12 q^{7} - 21 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q - 6 * q^2 + 9 * q^3 + 8 * q^4 + 9 * q^5 - 6 * q^6 - 12 * q^7 - 21 * q^8 + 9 * q^9	\( 9 q - 6 q^{2} + 9 q^{3} + 8 q^{4} + 9 q^{5} - 6 q^{6} - 12 q^{7} - 21 q^{8} + 9 q^{9} - 6 q^{10} - 10 q^{11} + 8 q^{12} + 9 q^{13} - 9 q^{14} + 9 q^{15} + 30 q^{16} - 17 q^{17} - 6 q^{18} + 6 q^{19} + 8 q^{20} - 12 q^{21} + 15 q^{22} - 23 q^{23} - 21 q^{24} + 9 q^{25} - 6 q^{26} + 9 q^{27} + 2 q^{28} - 4 q^{29} - 6 q^{30} + 9 q^{31} - 38 q^{32} - 10 q^{33} + 15 q^{34} - 12 q^{35} + 8 q^{36} + 11 q^{37} - 16 q^{38} + 9 q^{39} - 21 q^{40} - 8 q^{41} - 9 q^{42} - 15 q^{43} - q^{44} + 9 q^{45} + 26 q^{46} - 33 q^{47} + 30 q^{48} + 15 q^{49} - 6 q^{50} - 17 q^{51} + 8 q^{52} - 38 q^{53} - 6 q^{54} - 10 q^{55} + 37 q^{56} + 6 q^{57} - 26 q^{58} - 25 q^{59} + 8 q^{60} - 10 q^{61} - 6 q^{62} - 12 q^{63} + 47 q^{64} + 9 q^{65} + 15 q^{66} - 19 q^{67} + 7 q^{68} - 23 q^{69} - 9 q^{70} - 43 q^{71} - 21 q^{72} + q^{73} + 4 q^{74} + 9 q^{75} - 26 q^{76} + 2 q^{77} - 6 q^{78} - 9 q^{79} + 30 q^{80} + 9 q^{81} + 15 q^{82} - 12 q^{83} + 2 q^{84} - 17 q^{85} - 30 q^{86} - 4 q^{87} + q^{88} - 5 q^{89} - 6 q^{90} - 12 q^{91} - 57 q^{92} + 9 q^{93} + 40 q^{94} + 6 q^{95} - 38 q^{96} - 4 q^{97} + 34 q^{98} - 10 q^{99}+O(q^{100}) \) 9 * q - 6 * q^2 + 9 * q^3 + 8 * q^4 + 9 * q^5 - 6 * q^6 - 12 * q^7 - 21 * q^8 + 9 * q^9 - 6 * q^10 - 10 * q^11 + 8 * q^12 + 9 * q^13 - 9 * q^14 + 9 * q^15 + 30 * q^16 - 17 * q^17 - 6 * q^18 + 6 * q^19 + 8 * q^20 - 12 * q^21 + 15 * q^22 - 23 * q^23 - 21 * q^24 + 9 * q^25 - 6 * q^26 + 9 * q^27 + 2 * q^28 - 4 * q^29 - 6 * q^30 + 9 * q^31 - 38 * q^32 - 10 * q^33 + 15 * q^34 - 12 * q^35 + 8 * q^36 + 11 * q^37 - 16 * q^38 + 9 * q^39 - 21 * q^40 - 8 * q^41 - 9 * q^42 - 15 * q^43 - q^44 + 9 * q^45 + 26 * q^46 - 33 * q^47 + 30 * q^48 + 15 * q^49 - 6 * q^50 - 17 * q^51 + 8 * q^52 - 38 * q^53 - 6 * q^54 - 10 * q^55 + 37 * q^56 + 6 * q^57 - 26 * q^58 - 25 * q^59 + 8 * q^60 - 10 * q^61 - 6 * q^62 - 12 * q^63 + 47 * q^64 + 9 * q^65 + 15 * q^66 - 19 * q^67 + 7 * q^68 - 23 * q^69 - 9 * q^70 - 43 * q^71 - 21 * q^72 + q^73 + 4 * q^74 + 9 * q^75 - 26 * q^76 + 2 * q^77 - 6 * q^78 - 9 * q^79 + 30 * q^80 + 9 * q^81 + 15 * q^82 - 12 * q^83 + 2 * q^84 - 17 * q^85 - 30 * q^86 - 4 * q^87 + q^88 - 5 * q^89 - 6 * q^90 - 12 * q^91 - 57 * q^92 + 9 * q^93 + 40 * q^94 + 6 * q^95 - 38 * q^96 - 4 * q^97 + 34 * q^98 - 10 * q^99

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