LMFDB - Embedded newform 465.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [465,2,Mod(1,465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(465, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.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:	$3.71304369399$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} -1.00000 q^{5} +1.53919 q^{6} +4.87936 q^{7} -2.51026 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} -1.00000 q^{5} +1.53919 q^{6} +4.87936 q^{7} -2.51026 q^{8} +1.00000 q^{9} -1.53919 q^{10} +4.34017 q^{11} +0.369102 q^{12} -2.53919 q^{13} +7.51026 q^{14} -1.00000 q^{15} -4.60197 q^{16} +2.63090 q^{17} +1.53919 q^{18} -7.41855 q^{19} -0.369102 q^{20} +4.87936 q^{21} +6.68035 q^{22} +2.29072 q^{23} -2.51026 q^{24} +1.00000 q^{25} -3.90829 q^{26} +1.00000 q^{27} +1.80098 q^{28} +6.09171 q^{29} -1.53919 q^{30} -1.00000 q^{31} -2.06278 q^{32} +4.34017 q^{33} +4.04945 q^{34} -4.87936 q^{35} +0.369102 q^{36} -5.80098 q^{37} -11.4186 q^{38} -2.53919 q^{39} +2.51026 q^{40} -0.183417 q^{41} +7.51026 q^{42} -6.49693 q^{43} +1.60197 q^{44} -1.00000 q^{45} +3.52586 q^{46} -9.80817 q^{47} -4.60197 q^{48} +16.8082 q^{49} +1.53919 q^{50} +2.63090 q^{51} -0.937221 q^{52} -1.86603 q^{53} +1.53919 q^{54} -4.34017 q^{55} -12.2485 q^{56} -7.41855 q^{57} +9.37629 q^{58} -7.90829 q^{59} -0.369102 q^{60} -8.15676 q^{61} -1.53919 q^{62} +4.87936 q^{63} +6.02893 q^{64} +2.53919 q^{65} +6.68035 q^{66} -10.4813 q^{67} +0.971071 q^{68} +2.29072 q^{69} -7.51026 q^{70} +3.17009 q^{71} -2.51026 q^{72} -15.5597 q^{73} -8.92881 q^{74} +1.00000 q^{75} -2.73820 q^{76} +21.1773 q^{77} -3.90829 q^{78} -6.23287 q^{79} +4.60197 q^{80} +1.00000 q^{81} -0.282314 q^{82} +6.38962 q^{83} +1.80098 q^{84} -2.63090 q^{85} -10.0000 q^{86} +6.09171 q^{87} -10.8950 q^{88} -7.51026 q^{89} -1.53919 q^{90} -12.3896 q^{91} +0.845512 q^{92} -1.00000 q^{93} -15.0966 q^{94} +7.41855 q^{95} -2.06278 q^{96} +16.2823 q^{97} +25.8710 q^{98} +4.34017 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 - 3 * q^5 + 3 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} + 5 q^{12} - 6 q^{13} + 6 q^{14} - 3 q^{15} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20} + 2 q^{21} - 2 q^{22} + 14 q^{23} + 9 q^{24} + 3 q^{25} - 14 q^{26} + 3 q^{27} - 4 q^{28} + 16 q^{29} - 3 q^{30} - 3 q^{31} + 11 q^{32} + 2 q^{33} - 6 q^{34} - 2 q^{35} + 5 q^{36} - 8 q^{37} - 20 q^{38} - 6 q^{39} - 9 q^{40} + 4 q^{41} + 6 q^{42} - 2 q^{43} - 14 q^{44} - 3 q^{45} + 8 q^{46} + 14 q^{47} + 5 q^{48} + 7 q^{49} + 3 q^{50} + 4 q^{51} - 20 q^{52} + 8 q^{53} + 3 q^{54} - 2 q^{55} - 28 q^{56} - 8 q^{57} - 2 q^{58} - 26 q^{59} - 5 q^{60} - 18 q^{61} - 3 q^{62} + 2 q^{63} + 33 q^{64} + 6 q^{65} - 2 q^{66} - 12 q^{68} + 14 q^{69} - 6 q^{70} + 4 q^{71} + 9 q^{72} - 12 q^{73} + 4 q^{74} + 3 q^{75} - 16 q^{76} + 24 q^{77} - 14 q^{78} + 4 q^{79} - 5 q^{80} + 3 q^{81} + 40 q^{82} - 10 q^{83} - 4 q^{84} - 4 q^{85} - 30 q^{86} + 16 q^{87} - 34 q^{88} - 6 q^{89} - 3 q^{90} - 8 q^{91} + 22 q^{92} - 3 q^{93} + 4 q^{94} + 8 q^{95} + 11 q^{96} + 8 q^{97} + 17 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 - 3 * q^5 + 3 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 + 5 * q^12 - 6 * q^13 + 6 * q^14 - 3 * q^15 + 5 * q^16 + 4 * q^17 + 3 * q^18 - 8 * q^19 - 5 * q^20 + 2 * q^21 - 2 * q^22 + 14 * q^23 + 9 * q^24 + 3 * q^25 - 14 * q^26 + 3 * q^27 - 4 * q^28 + 16 * q^29 - 3 * q^30 - 3 * q^31 + 11 * q^32 + 2 * q^33 - 6 * q^34 - 2 * q^35 + 5 * q^36 - 8 * q^37 - 20 * q^38 - 6 * q^39 - 9 * q^40 + 4 * q^41 + 6 * q^42 - 2 * q^43 - 14 * q^44 - 3 * q^45 + 8 * q^46 + 14 * q^47 + 5 * q^48 + 7 * q^49 + 3 * q^50 + 4 * q^51 - 20 * q^52 + 8 * q^53 + 3 * q^54 - 2 * q^55 - 28 * q^56 - 8 * q^57 - 2 * q^58 - 26 * q^59 - 5 * q^60 - 18 * q^61 - 3 * q^62 + 2 * q^63 + 33 * q^64 + 6 * q^65 - 2 * q^66 - 12 * q^68 + 14 * q^69 - 6 * q^70 + 4 * q^71 + 9 * q^72 - 12 * q^73 + 4 * q^74 + 3 * q^75 - 16 * q^76 + 24 * q^77 - 14 * q^78 + 4 * q^79 - 5 * q^80 + 3 * q^81 + 40 * q^82 - 10 * q^83 - 4 * q^84 - 4 * q^85 - 30 * q^86 + 16 * q^87 - 34 * q^88 - 6 * q^89 - 3 * q^90 - 8 * q^91 + 22 * q^92 - 3 * q^93 + 4 * q^94 + 8 * q^95 + 11 * q^96 + 8 * q^97 + 17 * 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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.369102	0.184551
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.53919	0.628371
$7$	4.87936	1.84423	0.922113	−	0.386921i	$-0.126462\pi$
$7$	4.87936	1.84423	0.922113	+	0.386921i	$0.126462\pi$
$8$	−2.51026	−0.887511
$9$	1.00000	0.333333
$10$	−1.53919	−0.486734
$11$	4.34017	1.30861	0.654306	−	0.756230i	$-0.272961\pi$
$11$	4.34017	1.30861	0.654306	+	0.756230i	$0.272961\pi$
$12$	0.369102	0.106551
$13$	−2.53919	−0.704244	−0.352122	−	0.935954i	$-0.614540\pi$
$13$	−2.53919	−0.704244	−0.352122	+	0.935954i	$0.614540\pi$
$14$	7.51026	2.00720
$15$	−1.00000	−0.258199
$16$	−4.60197	−1.15049
$17$	2.63090	0.638086	0.319043	−	0.947740i	$-0.396638\pi$
$17$	2.63090	0.638086	0.319043	+	0.947740i	$0.396638\pi$
$18$	1.53919	0.362790
$19$	−7.41855	−1.70193	−0.850966	−	0.525221i	$-0.823983\pi$
$19$	−7.41855	−1.70193	−0.850966	+	0.525221i	$0.823983\pi$
$20$	−0.369102	−0.0825338
$21$	4.87936	1.06476
$22$	6.68035	1.42425
$23$	2.29072	0.477649	0.238825	−	0.971063i	$-0.423238\pi$
$23$	2.29072	0.477649	0.238825	+	0.971063i	$0.423238\pi$
$24$	−2.51026	−0.512405
$25$	1.00000	0.200000
$26$	−3.90829	−0.766479
$27$	1.00000	0.192450
$28$	1.80098	0.340354
$29$	6.09171	1.13120	0.565601	−	0.824679i	$-0.308644\pi$
$29$	6.09171	1.13120	0.565601	+	0.824679i	$0.308644\pi$
$30$	−1.53919	−0.281016
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−2.06278	−0.364651
$33$	4.34017	0.755527
$34$	4.04945	0.694475
$35$	−4.87936	−0.824763
$36$	0.369102	0.0615171
$37$	−5.80098	−0.953676	−0.476838	−	0.878991i	$-0.658217\pi$
$37$	−5.80098	−0.953676	−0.476838	+	0.878991i	$0.658217\pi$
$38$	−11.4186	−1.85233
$39$	−2.53919	−0.406596
$40$	2.51026	0.396907
$41$	−0.183417	−0.0286450	−0.0143225	−	0.999897i	$-0.504559\pi$
$41$	−0.183417	−0.0286450	−0.0143225	+	0.999897i	$0.504559\pi$
$42$	7.51026	1.15886
$43$	−6.49693	−0.990772	−0.495386	−	0.868673i	$-0.664973\pi$
$43$	−6.49693	−0.990772	−0.495386	+	0.868673i	$0.664973\pi$
$44$	1.60197	0.241506
$45$	−1.00000	−0.149071
$46$	3.52586	0.519859
$47$	−9.80817	−1.43067	−0.715334	−	0.698782i	$-0.753726\pi$
$47$	−9.80817	−1.43067	−0.715334	+	0.698782i	$0.753726\pi$
$48$	−4.60197	−0.664237
$49$	16.8082	2.40117
$50$	1.53919	0.217674
$51$	2.63090	0.368399
$52$	−0.937221	−0.129969
$53$	−1.86603	−0.256319	−0.128160	−	0.991754i	$-0.540907\pi$
$53$	−1.86603	−0.256319	−0.128160	+	0.991754i	$0.540907\pi$
$54$	1.53919	0.209457
$55$	−4.34017	−0.585229
$56$	−12.2485	−1.63677
$57$	−7.41855	−0.982611
$58$	9.37629	1.23117
$59$	−7.90829	−1.02957	−0.514786	−	0.857319i	$-0.672129\pi$
$59$	−7.90829	−1.02957	−0.514786	+	0.857319i	$0.672129\pi$
$60$	−0.369102	−0.0476509
$61$	−8.15676	−1.04437	−0.522183	−	0.852834i	$-0.674882\pi$
$61$	−8.15676	−1.04437	−0.522183	+	0.852834i	$0.674882\pi$
$62$	−1.53919	−0.195477
$63$	4.87936	0.614742
$64$	6.02893	0.753616
$65$	2.53919	0.314948
$66$	6.68035	0.822294
$67$	−10.4813	−1.28050	−0.640249	−	0.768167i	$-0.721169\pi$
$67$	−10.4813	−1.28050	−0.640249	+	0.768167i	$0.721169\pi$
$68$	0.971071	0.117760
$69$	2.29072	0.275771
$70$	−7.51026	−0.897648
$71$	3.17009	0.376220	0.188110	−	0.982148i	$-0.439764\pi$
$71$	3.17009	0.376220	0.188110	+	0.982148i	$0.439764\pi$
$72$	−2.51026	−0.295837
$73$	−15.5597	−1.82113	−0.910563	−	0.413370i	$-0.864352\pi$
$73$	−15.5597	−1.82113	−0.910563	+	0.413370i	$0.864352\pi$
$74$	−8.92881	−1.03795
$75$	1.00000	0.115470
$76$	−2.73820	−0.314094
$77$	21.1773	2.41337
$78$	−3.90829	−0.442527
$79$	−6.23287	−0.701252	−0.350626	−	0.936516i	$-0.614031\pi$
$79$	−6.23287	−0.701252	−0.350626	+	0.936516i	$0.614031\pi$
$80$	4.60197	0.514516
$81$	1.00000	0.111111
$82$	−0.282314	−0.0311764
$83$	6.38962	0.701352	0.350676	−	0.936497i	$-0.385952\pi$
$83$	6.38962	0.701352	0.350676	+	0.936497i	$0.385952\pi$
$84$	1.80098	0.196503
$85$	−2.63090	−0.285361
$86$	−10.0000	−1.07833
$87$	6.09171	0.653100
$88$	−10.8950	−1.16141
$89$	−7.51026	−0.796086	−0.398043	−	0.917367i	$-0.630311\pi$
$89$	−7.51026	−0.796086	−0.398043	+	0.917367i	$0.630311\pi$
$90$	−1.53919	−0.162245
$91$	−12.3896	−1.29879
$92$	0.845512	0.0881507
$93$	−1.00000	−0.103695
$94$	−15.0966	−1.55710
$95$	7.41855	0.761127
$96$	−2.06278	−0.210532
$97$	16.2823	1.65322	0.826609	−	0.562776i	$-0.190267\pi$
$97$	16.2823	1.65322	0.826609	+	0.562776i	$0.190267\pi$
$98$	25.8710	2.61336
$99$	4.34017	0.436204
$100$	0.369102	0.0369102
$101$	7.23513	0.719923	0.359961	−	0.932967i	$-0.382790\pi$
$101$	7.23513	0.719923	0.359961	+	0.932967i	$0.382790\pi$
$102$	4.04945	0.400955
$103$	17.8999	1.76373	0.881864	−	0.471504i	$-0.156289\pi$
$103$	17.8999	1.76373	0.881864	+	0.471504i	$0.156289\pi$
$104$	6.37402	0.625024
$105$	−4.87936	−0.476177
$106$	−2.87217	−0.278970
$107$	12.9444	1.25138	0.625692	−	0.780071i	$-0.284817\pi$
$107$	12.9444	1.25138	0.625692	+	0.780071i	$0.284817\pi$
$108$	0.369102	0.0355169
$109$	10.2062	0.977577	0.488789	−	0.872402i	$-0.337439\pi$
$109$	10.2062	0.977577	0.488789	+	0.872402i	$0.337439\pi$
$110$	−6.68035	−0.636946
$111$	−5.80098	−0.550605
$112$	−22.4547	−2.12177
$113$	9.60197	0.903277	0.451639	−	0.892201i	$-0.350840\pi$
$113$	9.60197	0.903277	0.451639	+	0.892201i	$0.350840\pi$
$114$	−11.4186	−1.06945
$115$	−2.29072	−0.213611
$116$	2.24846	0.208765
$117$	−2.53919	−0.234748
$118$	−12.1724	−1.12056
$119$	12.8371	1.17678
$120$	2.51026	0.229154
$121$	7.83710	0.712464
$122$	−12.5548	−1.13666
$123$	−0.183417	−0.0165382
$124$	−0.369102	−0.0331464
$125$	−1.00000	−0.0894427
$126$	7.51026	0.669067
$127$	15.6020	1.38445	0.692225	−	0.721681i	$-0.256630\pi$
$127$	15.6020	1.38445	0.692225	+	0.721681i	$0.256630\pi$
$128$	13.4052	1.18487
$129$	−6.49693	−0.572023
$130$	3.90829	0.342780
$131$	−9.53692	−0.833245	−0.416622	−	0.909080i	$-0.636786\pi$
$131$	−9.53692	−0.833245	−0.416622	+	0.909080i	$0.636786\pi$
$132$	1.60197	0.139433
$133$	−36.1978	−3.13875
$134$	−16.1327	−1.39366
$135$	−1.00000	−0.0860663
$136$	−6.60424	−0.566309
$137$	2.94441	0.251558	0.125779	−	0.992058i	$-0.459857\pi$
$137$	2.94441	0.251558	0.125779	+	0.992058i	$0.459857\pi$
$138$	3.52586	0.300141
$139$	7.02052	0.595473	0.297736	−	0.954648i	$-0.403768\pi$
$139$	7.02052	0.595473	0.297736	+	0.954648i	$0.403768\pi$
$140$	−1.80098	−0.152211
$141$	−9.80817	−0.825997
$142$	4.87936	0.409467
$143$	−11.0205	−0.921582
$144$	−4.60197	−0.383497
$145$	−6.09171	−0.505889
$146$	−23.9493	−1.98206
$147$	16.8082	1.38631
$148$	−2.14116	−0.176002
$149$	−8.49693	−0.696096	−0.348048	−	0.937477i	$-0.613155\pi$
$149$	−8.49693	−0.696096	−0.348048	+	0.937477i	$0.613155\pi$
$150$	1.53919	0.125674
$151$	−3.52586	−0.286930	−0.143465	−	0.989655i	$-0.545824\pi$
$151$	−3.52586	−0.286930	−0.143465	+	0.989655i	$0.545824\pi$
$152$	18.6225	1.51048
$153$	2.63090	0.212695
$154$	32.5958	2.62665
$155$	1.00000	0.0803219
$156$	−0.937221	−0.0750377
$157$	4.49693	0.358894	0.179447	−	0.983768i	$-0.442569\pi$
$157$	4.49693	0.358894	0.179447	+	0.983768i	$0.442569\pi$
$158$	−9.59356	−0.763222
$159$	−1.86603	−0.147986
$160$	2.06278	0.163077
$161$	11.1773	0.880893
$162$	1.53919	0.120930
$163$	−15.3763	−1.20436	−0.602182	−	0.798359i	$-0.705702\pi$
$163$	−15.3763	−1.20436	−0.602182	+	0.798359i	$0.705702\pi$
$164$	−0.0676998	−0.00528647
$165$	−4.34017	−0.337882
$166$	9.83483	0.763331
$167$	10.2413	0.792494	0.396247	−	0.918144i	$-0.370312\pi$
$167$	10.2413	0.792494	0.396247	+	0.918144i	$0.370312\pi$
$168$	−12.2485	−0.944990
$169$	−6.55252	−0.504040
$170$	−4.04945	−0.310579
$171$	−7.41855	−0.567311
$172$	−2.39803	−0.182848
$173$	7.07838	0.538159	0.269080	−	0.963118i	$-0.413281\pi$
$173$	7.07838	0.538159	0.269080	+	0.963118i	$0.413281\pi$
$174$	9.37629	0.710815
$175$	4.87936	0.368845
$176$	−19.9733	−1.50555
$177$	−7.90829	−0.594424
$178$	−11.5597	−0.866437
$179$	14.6225	1.09294	0.546468	−	0.837480i	$-0.315972\pi$
$179$	14.6225	1.09294	0.546468	+	0.837480i	$0.315972\pi$
$180$	−0.369102	−0.0275113
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−19.0700	−1.41356
$183$	−8.15676	−0.602965
$184$	−5.75031	−0.423919
$185$	5.80098	0.426497
$186$	−1.53919	−0.112859
$187$	11.4186	0.835007
$188$	−3.62022	−0.264032
$189$	4.87936	0.354921
$190$	11.4186	0.828389
$191$	1.75154	0.126737	0.0633683	−	0.997990i	$-0.479816\pi$
$191$	1.75154	0.126737	0.0633683	+	0.997990i	$0.479816\pi$
$192$	6.02893	0.435101
$193$	−4.86376	−0.350101	−0.175051	−	0.984559i	$-0.556009\pi$
$193$	−4.86376	−0.350101	−0.175051	+	0.984559i	$0.556009\pi$
$194$	25.0616	1.79931
$195$	2.53919	0.181835
$196$	6.20394	0.443138
$197$	−20.3051	−1.44668	−0.723339	−	0.690493i	$-0.757394\pi$
$197$	−20.3051	−1.44668	−0.723339	+	0.690493i	$0.757394\pi$
$198$	6.68035	0.474752
$199$	−2.04945	−0.145282	−0.0726408	−	0.997358i	$-0.523143\pi$
$199$	−2.04945	−0.145282	−0.0726408	+	0.997358i	$0.523143\pi$
$200$	−2.51026	−0.177502
$201$	−10.4813	−0.739296
$202$	11.1362	0.783543
$203$	29.7237	2.08619
$204$	0.971071	0.0679885
$205$	0.183417	0.0128104
$206$	27.5513	1.91959
$207$	2.29072	0.159216
$208$	11.6853	0.810227
$209$	−32.1978	−2.22717
$210$	−7.51026	−0.518257
$211$	12.1834	0.838741	0.419371	−	0.907815i	$-0.362251\pi$
$211$	12.1834	0.838741	0.419371	+	0.907815i	$0.362251\pi$
$212$	−0.688756	−0.0473040
$213$	3.17009	0.217211
$214$	19.9239	1.36197
$215$	6.49693	0.443087
$216$	−2.51026	−0.170802
$217$	−4.87936	−0.331233
$218$	15.7093	1.06397
$219$	−15.5597	−1.05143
$220$	−1.60197	−0.108005
$221$	−6.68035	−0.449369
$222$	−8.92881	−0.599263
$223$	12.2557	0.820699	0.410350	−	0.911928i	$-0.365407\pi$
$223$	12.2557	0.820699	0.410350	+	0.911928i	$0.365407\pi$
$224$	−10.0650	−0.672499
$225$	1.00000	0.0666667
$226$	14.7792	0.983101
$227$	12.1483	0.806314	0.403157	−	0.915131i	$-0.367913\pi$
$227$	12.1483	0.806314	0.403157	+	0.915131i	$0.367913\pi$
$228$	−2.73820	−0.181342
$229$	6.28231	0.415147	0.207574	−	0.978219i	$-0.433443\pi$
$229$	6.28231	0.415147	0.207574	+	0.978219i	$0.433443\pi$
$230$	−3.52586	−0.232488
$231$	21.1773	1.39336
$232$	−15.2918	−1.00395
$233$	15.3112	1.00307	0.501536	−	0.865137i	$-0.332768\pi$
$233$	15.3112	1.00307	0.501536	+	0.865137i	$0.332768\pi$
$234$	−3.90829	−0.255493
$235$	9.80817	0.639815
$236$	−2.91897	−0.190009
$237$	−6.23287	−0.404868
$238$	19.7587	1.28077
$239$	−30.1978	−1.95333	−0.976666	−	0.214762i	$-0.931102\pi$
$239$	−30.1978	−1.95333	−0.976666	+	0.214762i	$0.931102\pi$
$240$	4.60197	0.297056
$241$	−25.6475	−1.65210	−0.826052	−	0.563594i	$-0.809418\pi$
$241$	−25.6475	−1.65210	−0.826052	+	0.563594i	$0.809418\pi$
$242$	12.0628	0.775425
$243$	1.00000	0.0641500
$244$	−3.01068	−0.192739
$245$	−16.8082	−1.07383
$246$	−0.282314	−0.0179997
$247$	18.8371	1.19858
$248$	2.51026	0.159402
$249$	6.38962	0.404926
$250$	−1.53919	−0.0973469
$251$	−6.25565	−0.394853	−0.197427	−	0.980318i	$-0.563258\pi$
$251$	−6.25565	−0.394853	−0.197427	+	0.980318i	$0.563258\pi$
$252$	1.80098	0.113451
$253$	9.94214	0.625057
$254$	24.0144	1.50680
$255$	−2.63090	−0.164753
$256$	8.57531	0.535957
$257$	−13.1545	−0.820554	−0.410277	−	0.911961i	$-0.634568\pi$
$257$	−13.1545	−0.820554	−0.410277	+	0.911961i	$0.634568\pi$
$258$	−10.0000	−0.622573
$259$	−28.3051	−1.75879
$260$	0.937221	0.0581240
$261$	6.09171	0.377067
$262$	−14.6791	−0.906879
$263$	29.4908	1.81848	0.909240	−	0.416273i	$-0.136664\pi$
$263$	29.4908	1.81848	0.909240	+	0.416273i	$0.136664\pi$
$264$	−10.8950	−0.670538
$265$	1.86603	0.114629
$266$	−55.7152	−3.41612
$267$	−7.51026	−0.459620
$268$	−3.86868	−0.236317
$269$	−20.7454	−1.26487	−0.632434	−	0.774614i	$-0.717944\pi$
$269$	−20.7454	−1.26487	−0.632434	+	0.774614i	$0.717944\pi$
$270$	−1.53919	−0.0936721
$271$	0.979481	0.0594992	0.0297496	−	0.999557i	$-0.490529\pi$
$271$	0.979481	0.0594992	0.0297496	+	0.999557i	$0.490529\pi$
$272$	−12.1073	−0.734113
$273$	−12.3896	−0.749854
$274$	4.53200	0.273788
$275$	4.34017	0.261722
$276$	0.845512	0.0508938
$277$	16.3246	0.980849	0.490424	−	0.871484i	$-0.336842\pi$
$277$	16.3246	0.980849	0.490424	+	0.871484i	$0.336842\pi$
$278$	10.8059	0.648095
$279$	−1.00000	−0.0598684
$280$	12.2485	0.731986
$281$	22.2823	1.32925	0.664626	−	0.747176i	$-0.268591\pi$
$281$	22.2823	1.32925	0.664626	+	0.747176i	$0.268591\pi$
$282$	−15.0966	−0.898991
$283$	−2.05664	−0.122254	−0.0611272	−	0.998130i	$-0.519470\pi$
$283$	−2.05664	−0.122254	−0.0611272	+	0.998130i	$0.519470\pi$
$284$	1.17009	0.0694319
$285$	7.41855	0.439437
$286$	−16.9627	−1.00302
$287$	−0.894960	−0.0528278
$288$	−2.06278	−0.121550
$289$	−10.0784	−0.592846
$290$	−9.37629	−0.550595
$291$	16.2823	0.954486
$292$	−5.74313	−0.336091
$293$	22.3630	1.30646	0.653229	−	0.757160i	$-0.273414\pi$
$293$	22.3630	1.30646	0.653229	+	0.757160i	$0.273414\pi$
$294$	25.8710	1.50882
$295$	7.90829	0.460439
$296$	14.5620	0.846398
$297$	4.34017	0.251842
$298$	−13.0784	−0.757610
$299$	−5.81658	−0.336382
$300$	0.369102	0.0213101
$301$	−31.7009	−1.82721
$302$	−5.42696	−0.312287
$303$	7.23513	0.415648
$304$	34.1399	1.95806
$305$	8.15676	0.467054
$306$	4.04945	0.231492
$307$	−9.24620	−0.527708	−0.263854	−	0.964563i	$-0.584994\pi$
$307$	−9.24620	−0.527708	−0.263854	+	0.964563i	$0.584994\pi$
$308$	7.81658	0.445391
$309$	17.8999	1.01829
$310$	1.53919	0.0874201
$311$	9.24232	0.524084	0.262042	−	0.965056i	$-0.415604\pi$
$311$	9.24232	0.524084	0.262042	+	0.965056i	$0.415604\pi$
$312$	6.37402	0.360858
$313$	3.89988	0.220434	0.110217	−	0.993908i	$-0.464845\pi$
$313$	3.89988	0.220434	0.110217	+	0.993908i	$0.464845\pi$
$314$	6.92162	0.390610
$315$	−4.87936	−0.274921
$316$	−2.30057	−0.129417
$317$	3.62475	0.203586	0.101793	−	0.994806i	$-0.467542\pi$
$317$	3.62475	0.203586	0.101793	+	0.994806i	$0.467542\pi$
$318$	−2.87217	−0.161064
$319$	26.4391	1.48030
$320$	−6.02893	−0.337027
$321$	12.9444	0.722486
$322$	17.2039	0.958738
$323$	−19.5174	−1.08598
$324$	0.369102	0.0205057
$325$	−2.53919	−0.140849
$326$	−23.6670	−1.31079
$327$	10.2062	0.564404
$328$	0.460425	0.0254227
$329$	−47.8576	−2.63848
$330$	−6.68035	−0.367741
$331$	−24.1217	−1.32585	−0.662924	−	0.748687i	$-0.730685\pi$
$331$	−24.1217	−1.32585	−0.662924	+	0.748687i	$0.730685\pi$
$332$	2.35842	0.129435
$333$	−5.80098	−0.317892
$334$	15.7633	0.862527
$335$	10.4813	0.572656
$336$	−22.4547	−1.22500
$337$	3.19287	0.173927	0.0869634	−	0.996212i	$-0.472284\pi$
$337$	3.19287	0.173927	0.0869634	+	0.996212i	$0.472284\pi$
$338$	−10.0856	−0.548582
$339$	9.60197	0.521507
$340$	−0.971071	−0.0526637
$341$	−4.34017	−0.235034
$342$	−11.4186	−0.617445
$343$	47.8576	2.58407
$344$	16.3090	0.879321
$345$	−2.29072	−0.123328
$346$	10.8950	0.585717
$347$	−34.6369	−1.85940	−0.929702	−	0.368312i	$-0.879936\pi$
$347$	−34.6369	−1.85940	−0.929702	+	0.368312i	$0.879936\pi$
$348$	2.24846	0.120530
$349$	5.31124	0.284304	0.142152	−	0.989845i	$-0.454598\pi$
$349$	5.31124	0.284304	0.142152	+	0.989845i	$0.454598\pi$
$350$	7.51026	0.401440
$351$	−2.53919	−0.135532
$352$	−8.95282	−0.477187
$353$	33.3523	1.77516	0.887581	−	0.460651i	$-0.152384\pi$
$353$	33.3523	1.77516	0.887581	+	0.460651i	$0.152384\pi$
$354$	−12.1724	−0.646953
$355$	−3.17009	−0.168251
$356$	−2.77205	−0.146919
$357$	12.8371	0.679411
$358$	22.5068	1.18952
$359$	2.33299	0.123130	0.0615651	−	0.998103i	$-0.480391\pi$
$359$	2.33299	0.123130	0.0615651	+	0.998103i	$0.480391\pi$
$360$	2.51026	0.132302
$361$	36.0349	1.89657
$362$	3.07838	0.161796
$363$	7.83710	0.411341
$364$	−4.57304	−0.239692
$365$	15.5597	0.814432
$366$	−12.5548	−0.656249
$367$	12.3135	0.642760	0.321380	−	0.946950i	$-0.395853\pi$
$367$	12.3135	0.642760	0.321380	+	0.946950i	$0.395853\pi$
$368$	−10.5418	−0.549531
$369$	−0.183417	−0.00954833
$370$	8.92881	0.464187
$371$	−9.10504	−0.472710
$372$	−0.369102	−0.0191371
$373$	−5.86991	−0.303932	−0.151966	−	0.988386i	$-0.548560\pi$
$373$	−5.86991	−0.303932	−0.151966	+	0.988386i	$0.548560\pi$
$374$	17.5753	0.908797
$375$	−1.00000	−0.0516398
$376$	24.6211	1.26973
$377$	−15.4680	−0.796642
$378$	7.51026	0.386286
$379$	−2.42469	−0.124548	−0.0622741	−	0.998059i	$-0.519835\pi$
$379$	−2.42469	−0.124548	−0.0622741	+	0.998059i	$0.519835\pi$
$380$	2.73820	0.140467
$381$	15.6020	0.799313
$382$	2.69594	0.137937
$383$	−32.9588	−1.68412	−0.842058	−	0.539388i	$-0.818656\pi$
$383$	−32.9588	−1.68412	−0.842058	+	0.539388i	$0.818656\pi$
$384$	13.4052	0.684082
$385$	−21.1773	−1.07929
$386$	−7.48625	−0.381040
$387$	−6.49693	−0.330257
$388$	6.00984	0.305103
$389$	19.0589	0.966325	0.483162	−	0.875531i	$-0.339488\pi$
$389$	19.0589	0.966325	0.483162	+	0.875531i	$0.339488\pi$
$390$	3.90829	0.197904
$391$	6.02666	0.304781
$392$	−42.1929	−2.13106
$393$	−9.53692	−0.481074
$394$	−31.2534	−1.57452
$395$	6.23287	0.313610
$396$	1.60197	0.0805019
$397$	−27.2618	−1.36823	−0.684115	−	0.729374i	$-0.739811\pi$
$397$	−27.2618	−1.36823	−0.684115	+	0.729374i	$0.739811\pi$
$398$	−3.15449	−0.158120
$399$	−36.1978	−1.81216
$400$	−4.60197	−0.230098
$401$	−23.6404	−1.18054	−0.590271	−	0.807205i	$-0.700979\pi$
$401$	−23.6404	−1.18054	−0.590271	+	0.807205i	$0.700979\pi$
$402$	−16.1327	−0.804628
$403$	2.53919	0.126486
$404$	2.67050	0.132863
$405$	−1.00000	−0.0496904
$406$	45.7503	2.27055
$407$	−25.1773	−1.24799
$408$	−6.60424	−0.326958
$409$	−7.16290	−0.354183	−0.177091	−	0.984194i	$-0.556669\pi$
$409$	−7.16290	−0.354183	−0.177091	+	0.984194i	$0.556669\pi$
$410$	0.282314	0.0139425
$411$	2.94441	0.145237
$412$	6.60689	0.325498
$413$	−38.5874	−1.89876
$414$	3.52586	0.173286
$415$	−6.38962	−0.313654
$416$	5.23779	0.256804
$417$	7.02052	0.343796
$418$	−49.5585	−2.42398
$419$	−4.43188	−0.216512	−0.108256	−	0.994123i	$-0.534527\pi$
$419$	−4.43188	−0.216512	−0.108256	+	0.994123i	$0.534527\pi$
$420$	−1.80098	−0.0878790
$421$	34.4885	1.68087	0.840434	−	0.541914i	$-0.182300\pi$
$421$	34.4885	1.68087	0.840434	+	0.541914i	$0.182300\pi$
$422$	18.7526	0.912861
$423$	−9.80817	−0.476890
$424$	4.68422	0.227486
$425$	2.63090	0.127617
$426$	4.87936	0.236406
$427$	−39.7998	−1.92605
$428$	4.77781	0.230944
$429$	−11.0205	−0.532076
$430$	10.0000	0.482243
$431$	17.6092	0.848203	0.424102	−	0.905615i	$-0.360590\pi$
$431$	17.6092	0.848203	0.424102	+	0.905615i	$0.360590\pi$
$432$	−4.60197	−0.221412
$433$	−36.5536	−1.75665	−0.878326	−	0.478062i	$-0.841339\pi$
$433$	−36.5536	−1.75665	−0.878326	+	0.478062i	$0.841339\pi$
$434$	−7.51026	−0.360504
$435$	−6.09171	−0.292075
$436$	3.76713	0.180413
$437$	−16.9939	−0.812926
$438$	−23.9493	−1.14434
$439$	−12.3135	−0.587692	−0.293846	−	0.955853i	$-0.594935\pi$
$439$	−12.3135	−0.587692	−0.293846	+	0.955853i	$0.594935\pi$
$440$	10.8950	0.519397
$441$	16.8082	0.800389
$442$	−10.2823	−0.489080
$443$	−4.57757	−0.217487	−0.108744	−	0.994070i	$-0.534683\pi$
$443$	−4.57757	−0.217487	−0.108744	+	0.994070i	$0.534683\pi$
$444$	−2.14116	−0.101615
$445$	7.51026	0.356020
$446$	18.8638	0.893225
$447$	−8.49693	−0.401891
$448$	29.4173	1.38984
$449$	−17.7971	−0.839897	−0.419949	−	0.907548i	$-0.637952\pi$
$449$	−17.7971	−0.839897	−0.419949	+	0.907548i	$0.637952\pi$
$450$	1.53919	0.0725581
$451$	−0.796064	−0.0374852
$452$	3.54411	0.166701
$453$	−3.52586	−0.165659
$454$	18.6986	0.877569
$455$	12.3896	0.580834
$456$	18.6225	0.872078
$457$	−1.78661	−0.0835740	−0.0417870	−	0.999127i	$-0.513305\pi$
$457$	−1.78661	−0.0835740	−0.0417870	+	0.999127i	$0.513305\pi$
$458$	9.66967	0.451834
$459$	2.63090	0.122800
$460$	−0.845512	−0.0394222
$461$	−33.9227	−1.57994	−0.789968	−	0.613148i	$-0.789903\pi$
$461$	−33.9227	−1.57994	−0.789968	+	0.613148i	$0.789903\pi$
$462$	32.5958	1.51650
$463$	41.1194	1.91098	0.955491	−	0.295021i	$-0.0953268\pi$
$463$	41.1194	1.91098	0.955491	+	0.295021i	$0.0953268\pi$
$464$	−28.0338	−1.30144
$465$	1.00000	0.0463739
$466$	23.5669	1.09172
$467$	22.4885	1.04064	0.520322	−	0.853970i	$-0.325812\pi$
$467$	22.4885	1.04064	0.520322	+	0.853970i	$0.325812\pi$
$468$	−0.937221	−0.0433230
$469$	−51.1422	−2.36153
$470$	15.0966	0.696356
$471$	4.49693	0.207208
$472$	19.8519	0.913756
$473$	−28.1978	−1.29654
$474$	−9.59356	−0.440647
$475$	−7.41855	−0.340386
$476$	4.73820	0.217175
$477$	−1.86603	−0.0854397
$478$	−46.4801	−2.12595
$479$	−32.2485	−1.47347	−0.736735	−	0.676182i	$-0.763633\pi$
$479$	−32.2485	−1.47347	−0.736735	+	0.676182i	$0.763633\pi$
$480$	2.06278	0.0941526
$481$	14.7298	0.671621
$482$	−39.4764	−1.79810
$483$	11.1773	0.508584
$484$	2.89269	0.131486
$485$	−16.2823	−0.739342
$486$	1.53919	0.0698190
$487$	23.3340	1.05737	0.528683	−	0.848819i	$-0.322686\pi$
$487$	23.3340	1.05737	0.528683	+	0.848819i	$0.322686\pi$
$488$	20.4756	0.926886
$489$	−15.3763	−0.695340
$490$	−25.8710	−1.16873
$491$	23.3340	1.05305	0.526525	−	0.850160i	$-0.323495\pi$
$491$	23.3340	1.05305	0.526525	+	0.850160i	$0.323495\pi$
$492$	−0.0676998	−0.00305214
$493$	16.0267	0.721805
$494$	28.9939	1.30450
$495$	−4.34017	−0.195076
$496$	4.60197	0.206634
$497$	15.4680	0.693835
$498$	9.83483	0.440709
$499$	−3.20394	−0.143428	−0.0717139	−	0.997425i	$-0.522847\pi$
$499$	−3.20394	−0.143428	−0.0717139	+	0.997425i	$0.522847\pi$
$500$	−0.369102	−0.0165068
$501$	10.2413	0.457546
$502$	−9.62863	−0.429747
$503$	7.55252	0.336750	0.168375	−	0.985723i	$-0.446148\pi$
$503$	7.55252	0.336750	0.168375	+	0.985723i	$0.446148\pi$
$504$	−12.2485	−0.545590
$505$	−7.23513	−0.321959
$506$	15.3028	0.680294
$507$	−6.55252	−0.291008
$508$	5.75872	0.255502
$509$	27.4401	1.21626	0.608131	−	0.793837i	$-0.291920\pi$
$509$	27.4401	1.21626	0.608131	+	0.793837i	$0.291920\pi$
$510$	−4.04945	−0.179313
$511$	−75.9214	−3.35857
$512$	−13.6114	−0.601546
$513$	−7.41855	−0.327537
$514$	−20.2472	−0.893068
$515$	−17.8999	−0.788763
$516$	−2.39803	−0.105567
$517$	−42.5692	−1.87219
$518$	−43.5669	−1.91422
$519$	7.07838	0.310706
$520$	−6.37402	−0.279519
$521$	28.0144	1.22733	0.613666	−	0.789566i	$-0.289694\pi$
$521$	28.0144	1.22733	0.613666	+	0.789566i	$0.289694\pi$
$522$	9.37629	0.410389
$523$	3.33403	0.145787	0.0728935	−	0.997340i	$-0.476777\pi$
$523$	3.33403	0.145787	0.0728935	+	0.997340i	$0.476777\pi$
$524$	−3.52010	−0.153776
$525$	4.87936	0.212953
$526$	45.3919	1.97918
$527$	−2.63090	−0.114604
$528$	−19.9733	−0.869228
$529$	−17.7526	−0.771851
$530$	2.87217	0.124759
$531$	−7.90829	−0.343191
$532$	−13.3607	−0.579259
$533$	0.465732	0.0201731
$534$	−11.5597	−0.500237
$535$	−12.9444	−0.559636
$536$	26.3109	1.13646
$537$	14.6225	0.631007
$538$	−31.9311	−1.37665
$539$	72.9504	3.14220
$540$	−0.369102	−0.0158836
$541$	−7.55252	−0.324708	−0.162354	−	0.986733i	$-0.551909\pi$
$541$	−7.55252	−0.324708	−0.162354	+	0.986733i	$0.551909\pi$
$542$	1.50761	0.0647572
$543$	2.00000	0.0858282
$544$	−5.42696	−0.232679
$545$	−10.2062	−0.437186
$546$	−19.0700	−0.816119
$547$	32.4547	1.38766	0.693831	−	0.720138i	$-0.255922\pi$
$547$	32.4547	1.38766	0.693831	+	0.720138i	$0.255922\pi$
$548$	1.08679	0.0464253
$549$	−8.15676	−0.348122
$550$	6.68035	0.284851
$551$	−45.1917	−1.92523
$552$	−5.75031	−0.244750
$553$	−30.4124	−1.29327
$554$	25.1266	1.06753
$555$	5.80098	0.246238
$556$	2.59129	0.109895
$557$	23.0433	0.976376	0.488188	−	0.872738i	$-0.337658\pi$
$557$	23.0433	0.976376	0.488188	+	0.872738i	$0.337658\pi$
$558$	−1.53919	−0.0651591
$559$	16.4969	0.697746
$560$	22.4547	0.948883
$561$	11.4186	0.482092
$562$	34.2967	1.44672
$563$	−39.9793	−1.68493	−0.842463	−	0.538754i	$-0.818895\pi$
$563$	−39.9793	−1.68493	−0.842463	+	0.538754i	$0.818895\pi$
$564$	−3.62022	−0.152439
$565$	−9.60197	−0.403958
$566$	−3.16555	−0.133058
$567$	4.87936	0.204914
$568$	−7.95774	−0.333899
$569$	−19.6670	−0.824484	−0.412242	−	0.911074i	$-0.635254\pi$
$569$	−19.6670	−0.824484	−0.412242	+	0.911074i	$0.635254\pi$
$570$	11.4186	0.478270
$571$	11.8166	0.494509	0.247254	−	0.968951i	$-0.420472\pi$
$571$	11.8166	0.494509	0.247254	+	0.968951i	$0.420472\pi$
$572$	−4.06770	−0.170079
$573$	1.75154	0.0731715
$574$	−1.37751	−0.0574963
$575$	2.29072	0.0955298
$576$	6.02893	0.251205
$577$	−38.4079	−1.59894	−0.799470	−	0.600706i	$-0.794886\pi$
$577$	−38.4079	−1.59894	−0.799470	+	0.600706i	$0.794886\pi$
$578$	−15.5125	−0.645236
$579$	−4.86376	−0.202131
$580$	−2.24846	−0.0933624
$581$	31.1773	1.29345
$582$	25.0616	1.03883
$583$	−8.09890	−0.335422
$584$	39.0589	1.61627
$585$	2.53919	0.104983
$586$	34.4208	1.42191
$587$	11.3874	0.470006	0.235003	−	0.971995i	$-0.424490\pi$
$587$	11.3874	0.470006	0.235003	+	0.971995i	$0.424490\pi$
$588$	6.20394	0.255846
$589$	7.41855	0.305676
$590$	12.1724	0.501128
$591$	−20.3051	−0.835240
$592$	26.6959	1.09720
$593$	31.5441	1.29536	0.647681	−	0.761912i	$-0.275739\pi$
$593$	31.5441	1.29536	0.647681	+	0.761912i	$0.275739\pi$
$594$	6.68035	0.274098
$595$	−12.8371	−0.526270
$596$	−3.13624	−0.128465
$597$	−2.04945	−0.0838783
$598$	−8.95282	−0.366108
$599$	28.9744	1.18386	0.591931	−	0.805989i	$-0.298366\pi$
$599$	28.9744	1.18386	0.591931	+	0.805989i	$0.298366\pi$
$600$	−2.51026	−0.102481
$601$	−23.9565	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$601$	−23.9565	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$602$	−48.7936	−1.98868
$603$	−10.4813	−0.426833
$604$	−1.30140	−0.0529533
$605$	−7.83710	−0.318623
$606$	11.1362	0.452379
$607$	20.1724	0.818771	0.409385	−	0.912362i	$-0.365743\pi$
$607$	20.1724	0.818771	0.409385	+	0.912362i	$0.365743\pi$
$608$	15.3028	0.620612
$609$	29.7237	1.20446
$610$	12.5548	0.508328
$611$	24.9048	1.00754
$612$	0.971071	0.0392532
$613$	−4.46695	−0.180419	−0.0902093	−	0.995923i	$-0.528754\pi$
$613$	−4.46695	−0.180419	−0.0902093	+	0.995923i	$0.528754\pi$
$614$	−14.2316	−0.574342
$615$	0.183417	0.00739611
$616$	−53.1605	−2.14190
$617$	46.0821	1.85519	0.927597	−	0.373582i	$-0.121870\pi$
$617$	46.0821	1.85519	0.927597	+	0.373582i	$0.121870\pi$
$618$	27.5513	1.10828
$619$	−9.55252	−0.383948	−0.191974	−	0.981400i	$-0.561489\pi$
$619$	−9.55252	−0.383948	−0.191974	+	0.981400i	$0.561489\pi$
$620$	0.369102	0.0148235
$621$	2.29072	0.0919236
$622$	14.2257	0.570398
$623$	−36.6453	−1.46816
$624$	11.6853	0.467785
$625$	1.00000	0.0400000
$626$	6.00265	0.239914
$627$	−32.1978	−1.28586
$628$	1.65983	0.0662343
$629$	−15.2618	−0.608528
$630$	−7.51026	−0.299216
$631$	15.8310	0.630221	0.315110	−	0.949055i	$-0.397959\pi$
$631$	15.8310	0.630221	0.315110	+	0.949055i	$0.397959\pi$
$632$	15.6461	0.622369
$633$	12.1834	0.484247
$634$	5.57918	0.221578
$635$	−15.6020	−0.619145
$636$	−0.688756	−0.0273110
$637$	−42.6791	−1.69101
$638$	40.6947	1.61112
$639$	3.17009	0.125407
$640$	−13.4052	−0.529888
$641$	−5.87709	−0.232131	−0.116066	−	0.993242i	$-0.537028\pi$
$641$	−5.87709	−0.232131	−0.116066	+	0.993242i	$0.537028\pi$
$642$	19.9239	0.786333
$643$	19.8166	0.781490	0.390745	−	0.920499i	$-0.372217\pi$
$643$	19.8166	0.781490	0.390745	+	0.920499i	$0.372217\pi$
$644$	4.12556	0.162570
$645$	6.49693	0.255816
$646$	−30.0410	−1.18195
$647$	10.7915	0.424259	0.212129	−	0.977242i	$-0.431960\pi$
$647$	10.7915	0.424259	0.212129	+	0.977242i	$0.431960\pi$
$648$	−2.51026	−0.0986123
$649$	−34.3234	−1.34731
$650$	−3.90829	−0.153296
$651$	−4.87936	−0.191237
$652$	−5.67543	−0.222267
$653$	−8.93987	−0.349844	−0.174922	−	0.984582i	$-0.555967\pi$
$653$	−8.93987	−0.349844	−0.174922	+	0.984582i	$0.555967\pi$
$654$	15.7093	0.614281
$655$	9.53692	0.372638
$656$	0.844081	0.0329558
$657$	−15.5597	−0.607042
$658$	−73.6619	−2.87164
$659$	−12.3330	−0.480425	−0.240212	−	0.970720i	$-0.577217\pi$
$659$	−12.3330	−0.480425	−0.240212	+	0.970720i	$0.577217\pi$
$660$	−1.60197	−0.0623565
$661$	−22.3980	−0.871182	−0.435591	−	0.900145i	$-0.643461\pi$
$661$	−22.3980	−0.871182	−0.435591	+	0.900145i	$0.643461\pi$
$662$	−37.1278	−1.44301
$663$	−6.68035	−0.259443
$664$	−16.0396	−0.622457
$665$	36.1978	1.40369
$666$	−8.92881	−0.345984
$667$	13.9544	0.540318
$668$	3.78008	0.146256
$669$	12.2557	0.473831
$670$	16.1327	0.623262
$671$	−35.4017	−1.36667
$672$	−10.0650	−0.388268
$673$	−15.4875	−0.596998	−0.298499	−	0.954410i	$-0.596486\pi$
$673$	−15.4875	−0.596998	−0.298499	+	0.954410i	$0.596486\pi$
$674$	4.91443	0.189297
$675$	1.00000	0.0384900
$676$	−2.41855	−0.0930212
$677$	−18.5997	−0.714845	−0.357422	−	0.933943i	$-0.616344\pi$
$677$	−18.5997	−0.714845	−0.357422	+	0.933943i	$0.616344\pi$
$678$	14.7792	0.567593
$679$	79.4473	3.04891
$680$	6.60424	0.253261
$681$	12.1483	0.465526
$682$	−6.68035	−0.255804
$683$	14.4040	0.551154	0.275577	−	0.961279i	$-0.411131\pi$
$683$	14.4040	0.551154	0.275577	+	0.961279i	$0.411131\pi$
$684$	−2.73820	−0.104698
$685$	−2.94441	−0.112500
$686$	73.6619	2.81243
$687$	6.28231	0.239685
$688$	29.8987	1.13988
$689$	4.73820	0.180511
$690$	−3.52586	−0.134227
$691$	33.9565	1.29177	0.645883	−	0.763436i	$-0.276489\pi$
$691$	33.9565	1.29177	0.645883	+	0.763436i	$0.276489\pi$
$692$	2.61265	0.0993179
$693$	21.1773	0.804458
$694$	−53.3127	−2.02372
$695$	−7.02052	−0.266303
$696$	−15.2918	−0.579633
$697$	−0.482553	−0.0182780
$698$	8.17501	0.309429
$699$	15.3112	0.579124
$700$	1.80098	0.0680708
$701$	0.267938	0.0101199	0.00505994	−	0.999987i	$-0.498389\pi$
$701$	0.267938	0.0101199	0.00505994	+	0.999987i	$0.498389\pi$
$702$	−3.90829	−0.147509
$703$	43.0349	1.62309
$704$	26.1666	0.986191
$705$	9.80817	0.369397
$706$	51.3355	1.93204
$707$	35.3028	1.32770
$708$	−2.91897	−0.109702
$709$	7.36069	0.276437	0.138218	−	0.990402i	$-0.455862\pi$
$709$	7.36069	0.276437	0.138218	+	0.990402i	$0.455862\pi$
$710$	−4.87936	−0.183119
$711$	−6.23287	−0.233751
$712$	18.8527	0.706535
$713$	−2.29072	−0.0857883
$714$	19.7587	0.739452
$715$	11.0205	0.412144
$716$	5.39719	0.201703
$717$	−30.1978	−1.12776
$718$	3.59090	0.134011
$719$	−30.3279	−1.13104	−0.565520	−	0.824735i	$-0.691324\pi$
$719$	−30.3279	−1.13104	−0.565520	+	0.824735i	$0.691324\pi$
$720$	4.60197	0.171505
$721$	87.3400	3.25271
$722$	55.4645	2.06418
$723$	−25.6475	−0.953842
$724$	0.738205	0.0274352
$725$	6.09171	0.226240
$726$	12.0628	0.447692
$727$	−9.43415	−0.349893	−0.174947	−	0.984578i	$-0.555975\pi$
$727$	−9.43415	−0.349893	−0.174947	+	0.984578i	$0.555975\pi$
$728$	31.1012	1.15269
$729$	1.00000	0.0370370
$730$	23.9493	0.886404
$731$	−17.0928	−0.632198
$732$	−3.01068	−0.111278
$733$	27.9877	1.03375	0.516875	−	0.856061i	$-0.327095\pi$
$733$	27.9877	1.03375	0.516875	+	0.856061i	$0.327095\pi$
$734$	18.9528	0.699561
$735$	−16.8082	−0.619979
$736$	−4.72526	−0.174175
$737$	−45.4908	−1.67567
$738$	−0.282314	−0.0103921
$739$	−6.41628	−0.236027	−0.118013	−	0.993012i	$-0.537653\pi$
$739$	−6.41628	−0.236027	−0.118013	+	0.993012i	$0.537653\pi$
$740$	2.14116	0.0787105
$741$	18.8371	0.691998
$742$	−14.0144	−0.514484
$743$	16.5503	0.607170	0.303585	−	0.952804i	$-0.401816\pi$
$743$	16.5503	0.607170	0.303585	+	0.952804i	$0.401816\pi$
$744$	2.51026	0.0920306
$745$	8.49693	0.311303
$746$	−9.03489	−0.330791
$747$	6.38962	0.233784
$748$	4.21461	0.154102
$749$	63.1605	2.30783
$750$	−1.53919	−0.0562032
$751$	38.1711	1.39288	0.696442	−	0.717613i	$-0.254765\pi$
$751$	38.1711	1.39288	0.696442	+	0.717613i	$0.254765\pi$
$752$	45.1369	1.64597
$753$	−6.25565	−0.227969
$754$	−23.8082	−0.867042
$755$	3.52586	0.128319
$756$	1.80098	0.0655012
$757$	0.512527	0.0186281	0.00931405	−	0.999957i	$-0.497035\pi$
$757$	0.512527	0.0186281	0.00931405	+	0.999957i	$0.497035\pi$
$758$	−3.73206	−0.135555
$759$	9.94214	0.360877
$760$	−18.6225	−0.675509
$761$	−1.92267	−0.0696966	−0.0348483	−	0.999393i	$-0.511095\pi$
$761$	−1.92267	−0.0696966	−0.0348483	+	0.999393i	$0.511095\pi$
$762$	24.0144	0.869949
$763$	49.7998	1.80287
$764$	0.646496	0.0233894
$765$	−2.63090	−0.0951203
$766$	−50.7298	−1.83294
$767$	20.0806	0.725070
$768$	8.57531	0.309435
$769$	−50.8164	−1.83249	−0.916243	−	0.400622i	$-0.868794\pi$
$769$	−50.8164	−1.83249	−0.916243	+	0.400622i	$0.868794\pi$
$770$	−32.5958	−1.17467
$771$	−13.1545	−0.473747
$772$	−1.79523	−0.0646116
$773$	17.9506	0.645636	0.322818	−	0.946461i	$-0.395370\pi$
$773$	17.9506	0.645636	0.322818	+	0.946461i	$0.395370\pi$
$774$	−10.0000	−0.359443
$775$	−1.00000	−0.0359211
$776$	−40.8728	−1.46725
$777$	−28.3051	−1.01544
$778$	29.3353	1.05172
$779$	1.36069	0.0487518
$780$	0.937221	0.0335579
$781$	13.7587	0.492326
$782$	9.27617	0.331715
$783$	6.09171	0.217700
$784$	−77.3507	−2.76252
$785$	−4.49693	−0.160502
$786$	−14.6791	−0.523587
$787$	6.68035	0.238129	0.119064	−	0.992887i	$-0.462011\pi$
$787$	6.68035	0.238129	0.119064	+	0.992887i	$0.462011\pi$
$788$	−7.49466	−0.266986
$789$	29.4908	1.04990
$790$	9.59356	0.341323
$791$	46.8515	1.66585
$792$	−10.8950	−0.387136
$793$	20.7115	0.735488
$794$	−41.9611	−1.48914
$795$	1.86603	0.0661813
$796$	−0.756456	−0.0268119
$797$	−24.8865	−0.881527	−0.440763	−	0.897623i	$-0.645292\pi$
$797$	−24.8865	−0.881527	−0.440763	+	0.897623i	$0.645292\pi$
$798$	−55.7152	−1.97230
$799$	−25.8043	−0.912890
$800$	−2.06278	−0.0729303
$801$	−7.51026	−0.265362
$802$	−36.3870	−1.28487
$803$	−67.5318	−2.38315
$804$	−3.86868	−0.136438
$805$	−11.1773	−0.393947
$806$	3.90829	0.137664
$807$	−20.7454	−0.730272
$808$	−18.1621	−0.638939
$809$	−18.0027	−0.632940	−0.316470	−	0.948603i	$-0.602498\pi$
$809$	−18.0027	−0.632940	−0.316470	+	0.948603i	$0.602498\pi$
$810$	−1.53919	−0.0540816
$811$	30.2700	1.06292	0.531462	−	0.847082i	$-0.321643\pi$
$811$	30.2700	1.06292	0.531462	+	0.847082i	$0.321643\pi$
$812$	10.9711	0.385009
$813$	0.979481	0.0343519
$814$	−38.7526	−1.35828
$815$	15.3763	0.538608
$816$	−12.1073	−0.423841
$817$	48.1978	1.68623
$818$	−11.0251	−0.385482
$819$	−12.3896	−0.432928
$820$	0.0676998	0.00236418
$821$	35.6235	1.24327	0.621635	−	0.783307i	$-0.286469\pi$
$821$	35.6235	1.24327	0.621635	+	0.783307i	$0.286469\pi$
$822$	4.53200	0.158072
$823$	−53.9976	−1.88224	−0.941118	−	0.338078i	$-0.890223\pi$
$823$	−53.9976	−1.88224	−0.941118	+	0.338078i	$0.890223\pi$
$824$	−44.9333	−1.56533
$825$	4.34017	0.151105
$826$	−59.3933	−2.06656
$827$	−4.64527	−0.161532	−0.0807660	−	0.996733i	$-0.525737\pi$
$827$	−4.64527	−0.161532	−0.0807660	+	0.996733i	$0.525737\pi$
$828$	0.845512	0.0293836
$829$	44.6537	1.55089	0.775443	−	0.631417i	$-0.217526\pi$
$829$	44.6537	1.55089	0.775443	+	0.631417i	$0.217526\pi$
$830$	−9.83483	−0.341372
$831$	16.3246	0.566293
$832$	−15.3086	−0.530730
$833$	44.2206	1.53215
$834$	10.8059	0.374178
$835$	−10.2413	−0.354414
$836$	−11.8843	−0.411026
$837$	−1.00000	−0.0345651
$838$	−6.82150	−0.235645
$839$	17.6814	0.610429	0.305215	−	0.952284i	$-0.401272\pi$
$839$	17.6814	0.610429	0.305215	+	0.952284i	$0.401272\pi$
$840$	12.2485	0.422612
$841$	8.10892	0.279618
$842$	53.0843	1.82941
$843$	22.2823	0.767444
$844$	4.49693	0.154791
$845$	6.55252	0.225414
$846$	−15.0966	−0.519033
$847$	38.2401	1.31394
$848$	8.58741	0.294893
$849$	−2.05664	−0.0705836
$850$	4.04945	0.138895
$851$	−13.2885	−0.455522
$852$	1.17009	0.0400865
$853$	−15.2495	−0.522133	−0.261067	−	0.965321i	$-0.584074\pi$
$853$	−15.2495	−0.522133	−0.261067	+	0.965321i	$0.584074\pi$
$854$	−61.2594	−2.09625
$855$	7.41855	0.253709
$856$	−32.4938	−1.11062
$857$	26.7649	0.914270	0.457135	−	0.889397i	$-0.348876\pi$
$857$	26.7649	0.914270	0.457135	+	0.889397i	$0.348876\pi$
$858$	−16.9627	−0.579096
$859$	−42.7747	−1.45945	−0.729727	−	0.683739i	$-0.760353\pi$
$859$	−42.7747	−1.45945	−0.729727	+	0.683739i	$0.760353\pi$
$860$	2.39803	0.0817722
$861$	−0.894960	−0.0305002
$862$	27.1038	0.923160
$863$	34.6681	1.18011	0.590057	−	0.807361i	$-0.299105\pi$
$863$	34.6681	1.18011	0.590057	+	0.807361i	$0.299105\pi$
$864$	−2.06278	−0.0701772
$865$	−7.07838	−0.240672
$866$	−56.2628	−1.91189
$867$	−10.0784	−0.342280
$868$	−1.80098	−0.0611294
$869$	−27.0517	−0.917667
$870$	−9.37629	−0.317886
$871$	26.6141	0.901784
$872$	−25.6202	−0.867610
$873$	16.2823	0.551073
$874$	−26.1568	−0.884765
$875$	−4.87936	−0.164953
$876$	−5.74313	−0.194042
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	−18.9528	−0.639627
$879$	22.3630	0.754284
$880$	19.9733	0.673301
$881$	−12.8254	−0.432098	−0.216049	−	0.976383i	$-0.569317\pi$
$881$	−12.8254	−0.432098	−0.216049	+	0.976383i	$0.569317\pi$
$882$	25.8710	0.871120
$883$	33.5174	1.12795	0.563976	−	0.825791i	$-0.309271\pi$
$883$	33.5174	1.12795	0.563976	+	0.825791i	$0.309271\pi$
$884$	−2.46573	−0.0829315
$885$	7.90829	0.265834
$886$	−7.04575	−0.236707
$887$	7.55252	0.253589	0.126794	−	0.991929i	$-0.459531\pi$
$887$	7.55252	0.253589	0.126794	+	0.991929i	$0.459531\pi$
$888$	14.5620	0.488668
$889$	76.1276	2.55324
$890$	11.5597	0.387482
$891$	4.34017	0.145401
$892$	4.52359	0.151461
$893$	72.7624	2.43490
$894$	−13.0784	−0.437406
$895$	−14.6225	−0.488776
$896$	65.4089	2.18516
$897$	−5.81658	−0.194210
$898$	−27.3931	−0.914120
$899$	−6.09171	−0.203170
$900$	0.369102	0.0123034
$901$	−4.90934	−0.163554
$902$	−1.22529	−0.0407978
$903$	−31.7009	−1.05494
$904$	−24.1034	−0.801668
$905$	−2.00000	−0.0664822
$906$	−5.42696	−0.180299
$907$	−1.97212	−0.0654830	−0.0327415	−	0.999464i	$-0.510424\pi$
$907$	−1.97212	−0.0654830	−0.0327415	+	0.999464i	$0.510424\pi$
$908$	4.48398	0.148806
$909$	7.23513	0.239974
$910$	19.0700	0.632163
$911$	−44.3689	−1.47001	−0.735004	−	0.678063i	$-0.762820\pi$
$911$	−44.3689	−1.47001	−0.735004	+	0.678063i	$0.762820\pi$
$912$	34.1399	1.13049
$913$	27.7321	0.917797
$914$	−2.74993	−0.0909595
$915$	8.15676	0.269654
$916$	2.31882	0.0766159
$917$	−46.5341	−1.53669
$918$	4.04945	0.133652
$919$	12.8494	0.423862	0.211931	−	0.977285i	$-0.432025\pi$
$919$	12.8494	0.423862	0.211931	+	0.977285i	$0.432025\pi$
$920$	5.75031	0.189582
$921$	−9.24620	−0.304673
$922$	−52.2134	−1.71956
$923$	−8.04945	−0.264951
$924$	7.81658	0.257147
$925$	−5.80098	−0.190735
$926$	63.2905	2.07986
$927$	17.8999	0.587909
$928$	−12.5659	−0.412494
$929$	19.2013	0.629974	0.314987	−	0.949096i	$-0.398000\pi$
$929$	19.2013	0.629974	0.314987	+	0.949096i	$0.398000\pi$
$930$	1.53919	0.0504720
$931$	−124.692	−4.08662
$932$	5.65142	0.185118
$933$	9.24232	0.302580
$934$	34.6141	1.13261
$935$	−11.4186	−0.373427
$936$	6.37402	0.208341
$937$	−59.0037	−1.92757	−0.963783	−	0.266686i	$-0.914071\pi$
$937$	−59.0037	−1.92757	−0.963783	+	0.266686i	$0.914071\pi$
$938$	−78.7175	−2.57022
$939$	3.89988	0.127268
$940$	3.62022	0.118079
$941$	−10.7142	−0.349273	−0.174636	−	0.984633i	$-0.555875\pi$
$941$	−10.7142	−0.349273	−0.174636	+	0.984633i	$0.555875\pi$
$942$	6.92162	0.225519
$943$	−0.420159	−0.0136823
$944$	36.3937	1.18451
$945$	−4.87936	−0.158726
$946$	−43.4017	−1.41111
$947$	−7.23901	−0.235236	−0.117618	−	0.993059i	$-0.537526\pi$
$947$	−7.23901	−0.235236	−0.117618	+	0.993059i	$0.537526\pi$
$948$	−2.30057	−0.0747189
$949$	39.5090	1.28252
$950$	−11.4186	−0.370467
$951$	3.62475	0.117541
$952$	−32.2245	−1.04440
$953$	−44.1361	−1.42971	−0.714854	−	0.699274i	$-0.753507\pi$
$953$	−44.1361	−1.42971	−0.714854	+	0.699274i	$0.753507\pi$
$954$	−2.87217	−0.0929901
$955$	−1.75154	−0.0566784
$956$	−11.1461	−0.360490
$957$	26.4391	0.854654
$958$	−49.6365	−1.60368
$959$	14.3668	0.463929
$960$	−6.02893	−0.194583
$961$	1.00000	0.0322581
$962$	22.6719	0.730973
$963$	12.9444	0.417128
$964$	−9.46657	−0.304898
$965$	4.86376	0.156570
$966$	17.2039	0.553528
$967$	−10.3837	−0.333916	−0.166958	−	0.985964i	$-0.553394\pi$
$967$	−10.3837	−0.333916	−0.166958	+	0.985964i	$0.553394\pi$
$968$	−19.6732	−0.632319
$969$	−19.5174	−0.626991
$970$	−25.0616	−0.804678
$971$	−49.9448	−1.60280	−0.801402	−	0.598126i	$-0.795912\pi$
$971$	−49.9448	−1.60280	−0.801402	+	0.598126i	$0.795912\pi$
$972$	0.369102	0.0118390
$973$	34.2557	1.09819
$974$	35.9155	1.15081
$975$	−2.53919	−0.0813191
$976$	37.5371	1.20153
$977$	32.7214	1.04685	0.523425	−	0.852072i	$-0.324654\pi$
$977$	32.7214	1.04685	0.523425	+	0.852072i	$0.324654\pi$
$978$	−23.6670	−0.756788
$979$	−32.5958	−1.04177
$980$	−6.20394	−0.198177
$981$	10.2062	0.325859
$982$	35.9155	1.14611
$983$	−40.3500	−1.28697	−0.643483	−	0.765461i	$-0.722511\pi$
$983$	−40.3500	−1.28697	−0.643483	+	0.765461i	$0.722511\pi$
$984$	0.460425	0.0146778
$985$	20.3051	0.646974
$986$	24.6681	0.785591
$987$	−47.8576	−1.52332
$988$	6.95282	0.221199
$989$	−14.8827	−0.473242
$990$	−6.68035	−0.212315
$991$	7.81658	0.248302	0.124151	−	0.992263i	$-0.460379\pi$
$991$	7.81658	0.248302	0.124151	+	0.992263i	$0.460379\pi$
$992$	2.06278	0.0654933
$993$	−24.1217	−0.765478
$994$	23.8082	0.755149
$995$	2.04945	0.0649719
$996$	2.35842	0.0747295
$997$	−40.2245	−1.27392	−0.636961	−	0.770896i	$-0.719809\pi$
$997$	−40.2245	−1.27392	−0.636961	+	0.770896i	$0.719809\pi$
$998$	−4.93146	−0.156103
$999$	−5.80098	−0.183535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.g.1.2	✓	3
3.2	odd	2		1395.2.a.h.1.2		3
4.3	odd	2		7440.2.a.bm.1.1		3
5.2	odd	4		2325.2.c.l.1024.5		6
5.3	odd	4		2325.2.c.l.1024.2		6
5.4	even	2		2325.2.a.p.1.2		3
15.14	odd	2		6975.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.g.1.2	✓	3	1.1	even	1	trivial
1395.2.a.h.1.2		3	3.2	odd	2
2325.2.a.p.1.2		3	5.4	even	2
2325.2.c.l.1024.2		6	5.3	odd	4
2325.2.c.l.1024.5		6	5.2	odd	4
6975.2.a.bi.1.2		3	15.14	odd	2
7440.2.a.bm.1.1		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} -1.00000 q^{5} +1.53919 q^{6} +4.87936 q^{7} -2.51026 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} -1.00000 q^{5} +1.53919 q^{6} +4.87936 q^{7} -2.51026 q^{8} +1.00000 q^{9} -1.53919 q^{10} +4.34017 q^{11} +0.369102 q^{12} -2.53919 q^{13} +7.51026 q^{14} -1.00000 q^{15} -4.60197 q^{16} +2.63090 q^{17} +1.53919 q^{18} -7.41855 q^{19} -0.369102 q^{20} +4.87936 q^{21} +6.68035 q^{22} +2.29072 q^{23} -2.51026 q^{24} +1.00000 q^{25} -3.90829 q^{26} +1.00000 q^{27} +1.80098 q^{28} +6.09171 q^{29} -1.53919 q^{30} -1.00000 q^{31} -2.06278 q^{32} +4.34017 q^{33} +4.04945 q^{34} -4.87936 q^{35} +0.369102 q^{36} -5.80098 q^{37} -11.4186 q^{38} -2.53919 q^{39} +2.51026 q^{40} -0.183417 q^{41} +7.51026 q^{42} -6.49693 q^{43} +1.60197 q^{44} -1.00000 q^{45} +3.52586 q^{46} -9.80817 q^{47} -4.60197 q^{48} +16.8082 q^{49} +1.53919 q^{50} +2.63090 q^{51} -0.937221 q^{52} -1.86603 q^{53} +1.53919 q^{54} -4.34017 q^{55} -12.2485 q^{56} -7.41855 q^{57} +9.37629 q^{58} -7.90829 q^{59} -0.369102 q^{60} -8.15676 q^{61} -1.53919 q^{62} +4.87936 q^{63} +6.02893 q^{64} +2.53919 q^{65} +6.68035 q^{66} -10.4813 q^{67} +0.971071 q^{68} +2.29072 q^{69} -7.51026 q^{70} +3.17009 q^{71} -2.51026 q^{72} -15.5597 q^{73} -8.92881 q^{74} +1.00000 q^{75} -2.73820 q^{76} +21.1773 q^{77} -3.90829 q^{78} -6.23287 q^{79} +4.60197 q^{80} +1.00000 q^{81} -0.282314 q^{82} +6.38962 q^{83} +1.80098 q^{84} -2.63090 q^{85} -10.0000 q^{86} +6.09171 q^{87} -10.8950 q^{88} -7.51026 q^{89} -1.53919 q^{90} -12.3896 q^{91} +0.845512 q^{92} -1.00000 q^{93} -15.0966 q^{94} +7.41855 q^{95} -2.06278 q^{96} +16.2823 q^{97} +25.8710 q^{98} +4.34017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 - 3 * q^5 + 3 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} + 5 q^{12} - 6 q^{13} + 6 q^{14} - 3 q^{15} + 5 q^{16} + 4 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20} + 2 q^{21} - 2 q^{22} + 14 q^{23} + 9 q^{24} + 3 q^{25} - 14 q^{26} + 3 q^{27} - 4 q^{28} + 16 q^{29} - 3 q^{30} - 3 q^{31} + 11 q^{32} + 2 q^{33} - 6 q^{34} - 2 q^{35} + 5 q^{36} - 8 q^{37} - 20 q^{38} - 6 q^{39} - 9 q^{40} + 4 q^{41} + 6 q^{42} - 2 q^{43} - 14 q^{44} - 3 q^{45} + 8 q^{46} + 14 q^{47} + 5 q^{48} + 7 q^{49} + 3 q^{50} + 4 q^{51} - 20 q^{52} + 8 q^{53} + 3 q^{54} - 2 q^{55} - 28 q^{56} - 8 q^{57} - 2 q^{58} - 26 q^{59} - 5 q^{60} - 18 q^{61} - 3 q^{62} + 2 q^{63} + 33 q^{64} + 6 q^{65} - 2 q^{66} - 12 q^{68} + 14 q^{69} - 6 q^{70} + 4 q^{71} + 9 q^{72} - 12 q^{73} + 4 q^{74} + 3 q^{75} - 16 q^{76} + 24 q^{77} - 14 q^{78} + 4 q^{79} - 5 q^{80} + 3 q^{81} + 40 q^{82} - 10 q^{83} - 4 q^{84} - 4 q^{85} - 30 q^{86} + 16 q^{87} - 34 q^{88} - 6 q^{89} - 3 q^{90} - 8 q^{91} + 22 q^{92} - 3 q^{93} + 4 q^{94} + 8 q^{95} + 11 q^{96} + 8 q^{97} + 17 q^{98} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 5 * q^4 - 3 * q^5 + 3 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 - 3 * q^10 + 2 * q^11 + 5 * q^12 - 6 * q^13 + 6 * q^14 - 3 * q^15 + 5 * q^16 + 4 * q^17 + 3 * q^18 - 8 * q^19 - 5 * q^20 + 2 * q^21 - 2 * q^22 + 14 * q^23 + 9 * q^24 + 3 * q^25 - 14 * q^26 + 3 * q^27 - 4 * q^28 + 16 * q^29 - 3 * q^30 - 3 * q^31 + 11 * q^32 + 2 * q^33 - 6 * q^34 - 2 * q^35 + 5 * q^36 - 8 * q^37 - 20 * q^38 - 6 * q^39 - 9 * q^40 + 4 * q^41 + 6 * q^42 - 2 * q^43 - 14 * q^44 - 3 * q^45 + 8 * q^46 + 14 * q^47 + 5 * q^48 + 7 * q^49 + 3 * q^50 + 4 * q^51 - 20 * q^52 + 8 * q^53 + 3 * q^54 - 2 * q^55 - 28 * q^56 - 8 * q^57 - 2 * q^58 - 26 * q^59 - 5 * q^60 - 18 * q^61 - 3 * q^62 + 2 * q^63 + 33 * q^64 + 6 * q^65 - 2 * q^66 - 12 * q^68 + 14 * q^69 - 6 * q^70 + 4 * q^71 + 9 * q^72 - 12 * q^73 + 4 * q^74 + 3 * q^75 - 16 * q^76 + 24 * q^77 - 14 * q^78 + 4 * q^79 - 5 * q^80 + 3 * q^81 + 40 * q^82 - 10 * q^83 - 4 * q^84 - 4 * q^85 - 30 * q^86 + 16 * q^87 - 34 * q^88 - 6 * q^89 - 3 * q^90 - 8 * q^91 + 22 * q^92 - 3 * q^93 + 4 * q^94 + 8 * q^95 + 11 * q^96 + 8 * q^97 + 17 * q^98 + 2 * q^99

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