LMFDB - Embedded newform 465.2.a.c.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -2.82843 q^{11} +3.82843 q^{12} -2.58579 q^{13} +1.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} -4.00000 q^{17} -2.41421 q^{18} +2.82843 q^{19} -3.82843 q^{20} -0.585786 q^{21} +6.82843 q^{22} -6.00000 q^{23} -4.41421 q^{24} +1.00000 q^{25} +6.24264 q^{26} +1.00000 q^{27} -2.24264 q^{28} +2.24264 q^{29} +2.41421 q^{30} +1.00000 q^{31} +1.58579 q^{32} -2.82843 q^{33} +9.65685 q^{34} +0.585786 q^{35} +3.82843 q^{36} -1.41421 q^{37} -6.82843 q^{38} -2.58579 q^{39} +4.41421 q^{40} -0.828427 q^{41} +1.41421 q^{42} -11.3137 q^{43} -10.8284 q^{44} -1.00000 q^{45} +14.4853 q^{46} -4.82843 q^{47} +3.00000 q^{48} -6.65685 q^{49} -2.41421 q^{50} -4.00000 q^{51} -9.89949 q^{52} -4.00000 q^{53} -2.41421 q^{54} +2.82843 q^{55} +2.58579 q^{56} +2.82843 q^{57} -5.41421 q^{58} -0.242641 q^{59} -3.82843 q^{60} -10.4853 q^{61} -2.41421 q^{62} -0.585786 q^{63} -9.82843 q^{64} +2.58579 q^{65} +6.82843 q^{66} +3.89949 q^{67} -15.3137 q^{68} -6.00000 q^{69} -1.41421 q^{70} +9.89949 q^{71} -4.41421 q^{72} -5.89949 q^{73} +3.41421 q^{74} +1.00000 q^{75} +10.8284 q^{76} +1.65685 q^{77} +6.24264 q^{78} -14.4853 q^{79} -3.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -0.343146 q^{83} -2.24264 q^{84} +4.00000 q^{85} +27.3137 q^{86} +2.24264 q^{87} +12.4853 q^{88} +5.07107 q^{89} +2.41421 q^{90} +1.51472 q^{91} -22.9706 q^{92} +1.00000 q^{93} +11.6569 q^{94} -2.82843 q^{95} +1.58579 q^{96} +15.6569 q^{97} +16.0711 q^{98} -2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 8 q^{13} - 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{20} - 4 q^{21} + 8 q^{22} - 12 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} + 4 q^{28} - 4 q^{29} + 2 q^{30} + 2 q^{31} + 6 q^{32} + 8 q^{34} + 4 q^{35} + 2 q^{36} - 8 q^{38} - 8 q^{39} + 6 q^{40} + 4 q^{41} - 16 q^{44} - 2 q^{45} + 12 q^{46} - 4 q^{47} + 6 q^{48} - 2 q^{49} - 2 q^{50} - 8 q^{51} - 8 q^{53} - 2 q^{54} + 8 q^{56} - 8 q^{58} + 8 q^{59} - 2 q^{60} - 4 q^{61} - 2 q^{62} - 4 q^{63} - 14 q^{64} + 8 q^{65} + 8 q^{66} - 12 q^{67} - 8 q^{68} - 12 q^{69} - 6 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} + 16 q^{76} - 8 q^{77} + 4 q^{78} - 12 q^{79} - 6 q^{80} + 2 q^{81} + 4 q^{82} - 12 q^{83} + 4 q^{84} + 8 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} - 4 q^{89} + 2 q^{90} + 20 q^{91} - 12 q^{92} + 2 q^{93} + 12 q^{94} + 6 q^{96} + 20 q^{97} + 18 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 8 * q^13 - 2 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 2 * q^20 - 4 * q^21 + 8 * q^22 - 12 * q^23 - 6 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^27 + 4 * q^28 - 4 * q^29 + 2 * q^30 + 2 * q^31 + 6 * q^32 + 8 * q^34 + 4 * q^35 + 2 * q^36 - 8 * q^38 - 8 * q^39 + 6 * q^40 + 4 * q^41 - 16 * q^44 - 2 * q^45 + 12 * q^46 - 4 * q^47 + 6 * q^48 - 2 * q^49 - 2 * q^50 - 8 * q^51 - 8 * q^53 - 2 * q^54 + 8 * q^56 - 8 * q^58 + 8 * q^59 - 2 * q^60 - 4 * q^61 - 2 * q^62 - 4 * q^63 - 14 * q^64 + 8 * q^65 + 8 * q^66 - 12 * q^67 - 8 * q^68 - 12 * q^69 - 6 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 + 16 * q^76 - 8 * q^77 + 4 * q^78 - 12 * q^79 - 6 * q^80 + 2 * q^81 + 4 * q^82 - 12 * q^83 + 4 * q^84 + 8 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 - 4 * q^89 + 2 * q^90 + 20 * q^91 - 12 * q^92 + 2 * q^93 + 12 * q^94 + 6 * q^96 + 20 * q^97 + 18 * q^98

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.41421	−0.985599
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	2.41421	0.763441
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	3.82843	1.10517
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	1.41421	0.377964
$15$	−1.00000	−0.258199
$16$	3.00000	0.750000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−2.41421	−0.569036
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−3.82843	−0.856062
$21$	−0.585786	−0.127829
$22$	6.82843	1.45583
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−4.41421	−0.901048
$25$	1.00000	0.200000
$26$	6.24264	1.22428
$27$	1.00000	0.192450
$28$	−2.24264	−0.423819
$29$	2.24264	0.416448	0.208224	−	0.978081i	$-0.433232\pi$
$29$	2.24264	0.416448	0.208224	+	0.978081i	$0.433232\pi$
$30$	2.41421	0.440773
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.58579	0.280330
$33$	−2.82843	−0.492366
$34$	9.65685	1.65614
$35$	0.585786	0.0990160
$36$	3.82843	0.638071
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	−6.82843	−1.10772
$39$	−2.58579	−0.414057
$40$	4.41421	0.697948
$41$	−0.828427	−0.129379	−0.0646893	−	0.997905i	$-0.520606\pi$
$41$	−0.828427	−0.129379	−0.0646893	+	0.997905i	$0.520606\pi$
$42$	1.41421	0.218218
$43$	−11.3137	−1.72532	−0.862662	−	0.505781i	$-0.831205\pi$
$43$	−11.3137	−1.72532	−0.862662	+	0.505781i	$0.831205\pi$
$44$	−10.8284	−1.63245
$45$	−1.00000	−0.149071
$46$	14.4853	2.13574
$47$	−4.82843	−0.704298	−0.352149	−	0.935944i	$-0.614549\pi$
$47$	−4.82843	−0.704298	−0.352149	+	0.935944i	$0.614549\pi$
$48$	3.00000	0.433013
$49$	−6.65685	−0.950979
$50$	−2.41421	−0.341421
$51$	−4.00000	−0.560112
$52$	−9.89949	−1.37281
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−2.41421	−0.328533
$55$	2.82843	0.381385
$56$	2.58579	0.345540
$57$	2.82843	0.374634
$58$	−5.41421	−0.710921
$59$	−0.242641	−0.0315891	−0.0157946	−	0.999875i	$-0.505028\pi$
$59$	−0.242641	−0.0315891	−0.0157946	+	0.999875i	$0.505028\pi$
$60$	−3.82843	−0.494248
$61$	−10.4853	−1.34250	−0.671251	−	0.741230i	$-0.734243\pi$
$61$	−10.4853	−1.34250	−0.671251	+	0.741230i	$0.734243\pi$
$62$	−2.41421	−0.306605
$63$	−0.585786	−0.0738022
$64$	−9.82843	−1.22855
$65$	2.58579	0.320727
$66$	6.82843	0.840521
$67$	3.89949	0.476399	0.238200	−	0.971216i	$-0.423443\pi$
$67$	3.89949	0.476399	0.238200	+	0.971216i	$0.423443\pi$
$68$	−15.3137	−1.85706
$69$	−6.00000	−0.722315
$70$	−1.41421	−0.169031
$71$	9.89949	1.17485	0.587427	−	0.809277i	$-0.300141\pi$
$71$	9.89949	1.17485	0.587427	+	0.809277i	$0.300141\pi$
$72$	−4.41421	−0.520220
$73$	−5.89949	−0.690484	−0.345242	−	0.938514i	$-0.612203\pi$
$73$	−5.89949	−0.690484	−0.345242	+	0.938514i	$0.612203\pi$
$74$	3.41421	0.396894
$75$	1.00000	0.115470
$76$	10.8284	1.24211
$77$	1.65685	0.188816
$78$	6.24264	0.706840
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−0.343146	−0.0376651	−0.0188326	−	0.999823i	$-0.505995\pi$
$83$	−0.343146	−0.0376651	−0.0188326	+	0.999823i	$0.505995\pi$
$84$	−2.24264	−0.244692
$85$	4.00000	0.433861
$86$	27.3137	2.94531
$87$	2.24264	0.240436
$88$	12.4853	1.33094
$89$	5.07107	0.537532	0.268766	−	0.963205i	$-0.413384\pi$
$89$	5.07107	0.537532	0.268766	+	0.963205i	$0.413384\pi$
$90$	2.41421	0.254480
$91$	1.51472	0.158786
$92$	−22.9706	−2.39485
$93$	1.00000	0.103695
$94$	11.6569	1.20231
$95$	−2.82843	−0.290191
$96$	1.58579	0.161849
$97$	15.6569	1.58971	0.794856	−	0.606798i	$-0.207546\pi$
$97$	15.6569	1.58971	0.794856	+	0.606798i	$0.207546\pi$
$98$	16.0711	1.62342
$99$	−2.82843	−0.284268
$100$	3.82843	0.382843
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	9.65685	0.956171
$103$	−2.24264	−0.220974	−0.110487	−	0.993878i	$-0.535241\pi$
$103$	−2.24264	−0.220974	−0.110487	+	0.993878i	$0.535241\pi$
$104$	11.4142	1.11926
$105$	0.585786	0.0571669
$106$	9.65685	0.937957
$107$	−3.17157	−0.306608	−0.153304	−	0.988179i	$-0.548991\pi$
$107$	−3.17157	−0.306608	−0.153304	+	0.988179i	$0.548991\pi$
$108$	3.82843	0.368391
$109$	19.3137	1.84992	0.924959	−	0.380067i	$-0.124099\pi$
$109$	19.3137	1.84992	0.924959	+	0.380067i	$0.124099\pi$
$110$	−6.82843	−0.651065
$111$	−1.41421	−0.134231
$112$	−1.75736	−0.166055
$113$	17.3137	1.62874	0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137	1.62874	0.814368	+	0.580348i	$0.197084\pi$
$114$	−6.82843	−0.639541
$115$	6.00000	0.559503
$116$	8.58579	0.797170
$117$	−2.58579	−0.239056
$118$	0.585786	0.0539260
$119$	2.34315	0.214796
$120$	4.41421	0.402961
$121$	−3.00000	−0.272727
$122$	25.3137	2.29180
$123$	−0.828427	−0.0746968
$124$	3.82843	0.343803
$125$	−1.00000	−0.0894427
$126$	1.41421	0.125988
$127$	0.485281	0.0430618	0.0215309	−	0.999768i	$-0.493146\pi$
$127$	0.485281	0.0430618	0.0215309	+	0.999768i	$0.493146\pi$
$128$	20.5563	1.81694
$129$	−11.3137	−0.996116
$130$	−6.24264	−0.547516
$131$	8.72792	0.762562	0.381281	−	0.924459i	$-0.375483\pi$
$131$	8.72792	0.762562	0.381281	+	0.924459i	$0.375483\pi$
$132$	−10.8284	−0.942494
$133$	−1.65685	−0.143667
$134$	−9.41421	−0.813264
$135$	−1.00000	−0.0860663
$136$	17.6569	1.51406
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	14.4853	1.23307
$139$	−1.65685	−0.140533	−0.0702663	−	0.997528i	$-0.522385\pi$
$139$	−1.65685	−0.140533	−0.0702663	+	0.997528i	$0.522385\pi$
$140$	2.24264	0.189538
$141$	−4.82843	−0.406627
$142$	−23.8995	−2.00560
$143$	7.31371	0.611603
$144$	3.00000	0.250000
$145$	−2.24264	−0.186241
$146$	14.2426	1.17873
$147$	−6.65685	−0.549048
$148$	−5.41421	−0.445046
$149$	−10.4853	−0.858988	−0.429494	−	0.903070i	$-0.641308\pi$
$149$	−10.4853	−0.858988	−0.429494	+	0.903070i	$0.641308\pi$
$150$	−2.41421	−0.197120
$151$	−11.1716	−0.909130	−0.454565	−	0.890714i	$-0.650205\pi$
$151$	−11.1716	−0.909130	−0.454565	+	0.890714i	$0.650205\pi$
$152$	−12.4853	−1.01269
$153$	−4.00000	−0.323381
$154$	−4.00000	−0.322329
$155$	−1.00000	−0.0803219
$156$	−9.89949	−0.792594
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	34.9706	2.78211
$159$	−4.00000	−0.317221
$160$	−1.58579	−0.125367
$161$	3.51472	0.276999
$162$	−2.41421	−0.189679
$163$	10.2426	0.802266	0.401133	−	0.916020i	$-0.368617\pi$
$163$	10.2426	0.802266	0.401133	+	0.916020i	$0.368617\pi$
$164$	−3.17157	−0.247658
$165$	2.82843	0.220193
$166$	0.828427	0.0642984
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	2.58579	0.199498
$169$	−6.31371	−0.485670
$170$	−9.65685	−0.740647
$171$	2.82843	0.216295
$172$	−43.3137	−3.30264
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−5.41421	−0.410450
$175$	−0.585786	−0.0442813
$176$	−8.48528	−0.639602
$177$	−0.242641	−0.0182380
$178$	−12.2426	−0.917625
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	−3.82843	−0.285354
$181$	−21.7990	−1.62031	−0.810153	−	0.586218i	$-0.800616\pi$
$181$	−21.7990	−1.62031	−0.810153	+	0.586218i	$0.800616\pi$
$182$	−3.65685	−0.271064
$183$	−10.4853	−0.775094
$184$	26.4853	1.95252
$185$	1.41421	0.103975
$186$	−2.41421	−0.177019
$187$	11.3137	0.827340
$188$	−18.4853	−1.34818
$189$	−0.585786	−0.0426097
$190$	6.82843	0.495386
$191$	−0.727922	−0.0526706	−0.0263353	−	0.999653i	$-0.508384\pi$
$191$	−0.727922	−0.0526706	−0.0263353	+	0.999653i	$0.508384\pi$
$192$	−9.82843	−0.709306
$193$	7.65685	0.551152	0.275576	−	0.961279i	$-0.411131\pi$
$193$	7.65685	0.551152	0.275576	+	0.961279i	$0.411131\pi$
$194$	−37.7990	−2.71381
$195$	2.58579	0.185172
$196$	−25.4853	−1.82038
$197$	11.7990	0.840643	0.420322	−	0.907375i	$-0.361917\pi$
$197$	11.7990	0.840643	0.420322	+	0.907375i	$0.361917\pi$
$198$	6.82843	0.485275
$199$	16.1421	1.14429	0.572143	−	0.820154i	$-0.306112\pi$
$199$	16.1421	1.14429	0.572143	+	0.820154i	$0.306112\pi$
$200$	−4.41421	−0.312132
$201$	3.89949	0.275049
$202$	11.6569	0.820173
$203$	−1.31371	−0.0922043
$204$	−15.3137	−1.07217
$205$	0.828427	0.0578599
$206$	5.41421	0.377226
$207$	−6.00000	−0.417029
$208$	−7.75736	−0.537876
$209$	−8.00000	−0.553372
$210$	−1.41421	−0.0975900
$211$	−9.17157	−0.631397	−0.315699	−	0.948860i	$-0.602239\pi$
$211$	−9.17157	−0.631397	−0.315699	+	0.948860i	$0.602239\pi$
$212$	−15.3137	−1.05175
$213$	9.89949	0.678302
$214$	7.65685	0.523412
$215$	11.3137	0.771589
$216$	−4.41421	−0.300349
$217$	−0.585786	−0.0397658
$218$	−46.6274	−3.15801
$219$	−5.89949	−0.398651
$220$	10.8284	0.730052
$221$	10.3431	0.695755
$222$	3.41421	0.229147
$223$	−22.8284	−1.52870	−0.764352	−	0.644799i	$-0.776941\pi$
$223$	−22.8284	−1.52870	−0.764352	+	0.644799i	$0.776941\pi$
$224$	−0.928932	−0.0620669
$225$	1.00000	0.0666667
$226$	−41.7990	−2.78043
$227$	−4.34315	−0.288265	−0.144132	−	0.989558i	$-0.546039\pi$
$227$	−4.34315	−0.288265	−0.144132	+	0.989558i	$0.546039\pi$
$228$	10.8284	0.717130
$229$	26.9706	1.78226	0.891132	−	0.453743i	$-0.149912\pi$
$229$	26.9706	1.78226	0.891132	+	0.453743i	$0.149912\pi$
$230$	−14.4853	−0.955131
$231$	1.65685	0.109013
$232$	−9.89949	−0.649934
$233$	4.48528	0.293841	0.146920	−	0.989148i	$-0.453064\pi$
$233$	4.48528	0.293841	0.146920	+	0.989148i	$0.453064\pi$
$234$	6.24264	0.408094
$235$	4.82843	0.314972
$236$	−0.928932	−0.0604683
$237$	−14.4853	−0.940920
$238$	−5.65685	−0.366679
$239$	−14.3431	−0.927781	−0.463890	−	0.885893i	$-0.653547\pi$
$239$	−14.3431	−0.927781	−0.463890	+	0.885893i	$0.653547\pi$
$240$	−3.00000	−0.193649
$241$	10.9706	0.706676	0.353338	−	0.935496i	$-0.385047\pi$
$241$	10.9706	0.706676	0.353338	+	0.935496i	$0.385047\pi$
$242$	7.24264	0.465575
$243$	1.00000	0.0641500
$244$	−40.1421	−2.56984
$245$	6.65685	0.425291
$246$	2.00000	0.127515
$247$	−7.31371	−0.465360
$248$	−4.41421	−0.280303
$249$	−0.343146	−0.0217460
$250$	2.41421	0.152688
$251$	13.6569	0.862013	0.431006	−	0.902349i	$-0.358159\pi$
$251$	13.6569	0.862013	0.431006	+	0.902349i	$0.358159\pi$
$252$	−2.24264	−0.141273
$253$	16.9706	1.06693
$254$	−1.17157	−0.0735110
$255$	4.00000	0.250490
$256$	−29.9706	−1.87316
$257$	24.4853	1.52735	0.763675	−	0.645601i	$-0.223393\pi$
$257$	24.4853	1.52735	0.763675	+	0.645601i	$0.223393\pi$
$258$	27.3137	1.70048
$259$	0.828427	0.0514760
$260$	9.89949	0.613941
$261$	2.24264	0.138816
$262$	−21.0711	−1.30177
$263$	9.65685	0.595467	0.297734	−	0.954649i	$-0.403769\pi$
$263$	9.65685	0.595467	0.297734	+	0.954649i	$0.403769\pi$
$264$	12.4853	0.768416
$265$	4.00000	0.245718
$266$	4.00000	0.245256
$267$	5.07107	0.310344
$268$	14.9289	0.911930
$269$	−13.0711	−0.796957	−0.398479	−	0.917178i	$-0.630462\pi$
$269$	−13.0711	−0.796957	−0.398479	+	0.917178i	$0.630462\pi$
$270$	2.41421	0.146924
$271$	−11.3137	−0.687259	−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137	−0.687259	−0.343629	+	0.939105i	$0.611656\pi$
$272$	−12.0000	−0.727607
$273$	1.51472	0.0916749
$274$	−6.82843	−0.412520
$275$	−2.82843	−0.170561
$276$	−22.9706	−1.38267
$277$	23.0711	1.38621	0.693103	−	0.720839i	$-0.256243\pi$
$277$	23.0711	1.38621	0.693103	+	0.720839i	$0.256243\pi$
$278$	4.00000	0.239904
$279$	1.00000	0.0598684
$280$	−2.58579	−0.154530
$281$	−23.4558	−1.39926	−0.699629	−	0.714506i	$-0.746651\pi$
$281$	−23.4558	−1.39926	−0.699629	+	0.714506i	$0.746651\pi$
$282$	11.6569	0.694156
$283$	22.7279	1.35103	0.675517	−	0.737344i	$-0.263920\pi$
$283$	22.7279	1.35103	0.675517	+	0.737344i	$0.263920\pi$
$284$	37.8995	2.24892
$285$	−2.82843	−0.167542
$286$	−17.6569	−1.04407
$287$	0.485281	0.0286453
$288$	1.58579	0.0934434
$289$	−1.00000	−0.0588235
$290$	5.41421	0.317934
$291$	15.6569	0.917821
$292$	−22.5858	−1.32173
$293$	−17.1716	−1.00317	−0.501587	−	0.865107i	$-0.667250\pi$
$293$	−17.1716	−1.00317	−0.501587	+	0.865107i	$0.667250\pi$
$294$	16.0711	0.937284
$295$	0.242641	0.0141271
$296$	6.24264	0.362846
$297$	−2.82843	−0.164122
$298$	25.3137	1.46638
$299$	15.5147	0.897239
$300$	3.82843	0.221034
$301$	6.62742	0.381998
$302$	26.9706	1.55198
$303$	−4.82843	−0.277386
$304$	8.48528	0.486664
$305$	10.4853	0.600385
$306$	9.65685	0.552046
$307$	−21.0711	−1.20259	−0.601295	−	0.799027i	$-0.705348\pi$
$307$	−21.0711	−1.20259	−0.601295	+	0.799027i	$0.705348\pi$
$308$	6.34315	0.361434
$309$	−2.24264	−0.127579
$310$	2.41421	0.137118
$311$	−24.2426	−1.37467	−0.687337	−	0.726339i	$-0.741220\pi$
$311$	−24.2426	−1.37467	−0.687337	+	0.726339i	$0.741220\pi$
$312$	11.4142	0.646203
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	−43.4558	−2.45236
$315$	0.585786	0.0330053
$316$	−55.4558	−3.11963
$317$	13.1716	0.739789	0.369895	−	0.929074i	$-0.379394\pi$
$317$	13.1716	0.739789	0.369895	+	0.929074i	$0.379394\pi$
$318$	9.65685	0.541529
$319$	−6.34315	−0.355148
$320$	9.82843	0.549426
$321$	−3.17157	−0.177020
$322$	−8.48528	−0.472866
$323$	−11.3137	−0.629512
$324$	3.82843	0.212690
$325$	−2.58579	−0.143434
$326$	−24.7279	−1.36955
$327$	19.3137	1.06805
$328$	3.65685	0.201916
$329$	2.82843	0.155936
$330$	−6.82843	−0.375893
$331$	−22.4853	−1.23590	−0.617951	−	0.786216i	$-0.712037\pi$
$331$	−22.4853	−1.23590	−0.617951	+	0.786216i	$0.712037\pi$
$332$	−1.31371	−0.0720991
$333$	−1.41421	−0.0774984
$334$	21.6569	1.18501
$335$	−3.89949	−0.213052
$336$	−1.75736	−0.0958718
$337$	22.5858	1.23033	0.615163	−	0.788400i	$-0.289090\pi$
$337$	22.5858	1.23033	0.615163	+	0.788400i	$0.289090\pi$
$338$	15.2426	0.829090
$339$	17.3137	0.940352
$340$	15.3137	0.830502
$341$	−2.82843	−0.153168
$342$	−6.82843	−0.369239
$343$	8.00000	0.431959
$344$	49.9411	2.69265
$345$	6.00000	0.323029
$346$	33.7990	1.81704
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	8.58579	0.460246
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	1.41421	0.0755929
$351$	−2.58579	−0.138019
$352$	−4.48528	−0.239066
$353$	−12.9706	−0.690353	−0.345177	−	0.938538i	$-0.612181\pi$
$353$	−12.9706	−0.690353	−0.345177	+	0.938538i	$0.612181\pi$
$354$	0.585786	0.0311342
$355$	−9.89949	−0.525411
$356$	19.4142	1.02895
$357$	2.34315	0.124012
$358$	−56.2843	−2.97472
$359$	−20.0416	−1.05776	−0.528878	−	0.848698i	$-0.677387\pi$
$359$	−20.0416	−1.05776	−0.528878	+	0.848698i	$0.677387\pi$
$360$	4.41421	0.232649
$361$	−11.0000	−0.578947
$362$	52.6274	2.76604
$363$	−3.00000	−0.157459
$364$	5.79899	0.303950
$365$	5.89949	0.308794
$366$	25.3137	1.32317
$367$	−32.9706	−1.72105	−0.860525	−	0.509409i	$-0.829864\pi$
$367$	−32.9706	−1.72105	−0.860525	+	0.509409i	$0.829864\pi$
$368$	−18.0000	−0.938315
$369$	−0.828427	−0.0431262
$370$	−3.41421	−0.177497
$371$	2.34315	0.121650
$372$	3.82843	0.198495
$373$	−25.7990	−1.33582	−0.667911	−	0.744242i	$-0.732811\pi$
$373$	−25.7990	−1.33582	−0.667911	+	0.744242i	$0.732811\pi$
$374$	−27.3137	−1.41236
$375$	−1.00000	−0.0516398
$376$	21.3137	1.09917
$377$	−5.79899	−0.298663
$378$	1.41421	0.0727393
$379$	−26.8284	−1.37808	−0.689042	−	0.724722i	$-0.741968\pi$
$379$	−26.8284	−1.37808	−0.689042	+	0.724722i	$0.741968\pi$
$380$	−10.8284	−0.555487
$381$	0.485281	0.0248617
$382$	1.75736	0.0899143
$383$	9.31371	0.475908	0.237954	−	0.971276i	$-0.423523\pi$
$383$	9.31371	0.475908	0.237954	+	0.971276i	$0.423523\pi$
$384$	20.5563	1.04901
$385$	−1.65685	−0.0844411
$386$	−18.4853	−0.940876
$387$	−11.3137	−0.575108
$388$	59.9411	3.04305
$389$	21.0711	1.06835	0.534173	−	0.845375i	$-0.320623\pi$
$389$	21.0711	1.06835	0.534173	+	0.845375i	$0.320623\pi$
$390$	−6.24264	−0.316108
$391$	24.0000	1.21373
$392$	29.3848	1.48416
$393$	8.72792	0.440265
$394$	−28.4853	−1.43507
$395$	14.4853	0.728834
$396$	−10.8284	−0.544149
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−38.9706	−1.95342
$399$	−1.65685	−0.0829465
$400$	3.00000	0.150000
$401$	6.24264	0.311743	0.155871	−	0.987777i	$-0.450181\pi$
$401$	6.24264	0.311743	0.155871	+	0.987777i	$0.450181\pi$
$402$	−9.41421	−0.469538
$403$	−2.58579	−0.128807
$404$	−18.4853	−0.919677
$405$	−1.00000	−0.0496904
$406$	3.17157	0.157403
$407$	4.00000	0.198273
$408$	17.6569	0.874145
$409$	29.7990	1.47347	0.736733	−	0.676184i	$-0.236368\pi$
$409$	29.7990	1.47347	0.736733	+	0.676184i	$0.236368\pi$
$410$	−2.00000	−0.0987730
$411$	2.82843	0.139516
$412$	−8.58579	−0.422991
$413$	0.142136	0.00699404
$414$	14.4853	0.711913
$415$	0.343146	0.0168444
$416$	−4.10051	−0.201044
$417$	−1.65685	−0.0811365
$418$	19.3137	0.944664
$419$	27.0711	1.32251	0.661254	−	0.750162i	$-0.270025\pi$
$419$	27.0711	1.32251	0.661254	+	0.750162i	$0.270025\pi$
$420$	2.24264	0.109430
$421$	25.6569	1.25044	0.625219	−	0.780449i	$-0.285010\pi$
$421$	25.6569	1.25044	0.625219	+	0.780449i	$0.285010\pi$
$422$	22.1421	1.07786
$423$	−4.82843	−0.234766
$424$	17.6569	0.857493
$425$	−4.00000	−0.194029
$426$	−23.8995	−1.15793
$427$	6.14214	0.297239
$428$	−12.1421	−0.586912
$429$	7.31371	0.353109
$430$	−27.3137	−1.31718
$431$	−24.9289	−1.20078	−0.600392	−	0.799706i	$-0.704989\pi$
$431$	−24.9289	−1.20078	−0.600392	+	0.799706i	$0.704989\pi$
$432$	3.00000	0.144338
$433$	26.3848	1.26797	0.633986	−	0.773345i	$-0.281418\pi$
$433$	26.3848	1.26797	0.633986	+	0.773345i	$0.281418\pi$
$434$	1.41421	0.0678844
$435$	−2.24264	−0.107526
$436$	73.9411	3.54114
$437$	−16.9706	−0.811812
$438$	14.2426	0.680540
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−12.4853	−0.595212
$441$	−6.65685	−0.316993
$442$	−24.9706	−1.18773
$443$	−11.6569	−0.553834	−0.276917	−	0.960894i	$-0.589313\pi$
$443$	−11.6569	−0.553834	−0.276917	+	0.960894i	$0.589313\pi$
$444$	−5.41421	−0.256947
$445$	−5.07107	−0.240392
$446$	55.1127	2.60966
$447$	−10.4853	−0.495937
$448$	5.75736	0.272010
$449$	−33.3553	−1.57414	−0.787068	−	0.616866i	$-0.788402\pi$
$449$	−33.3553	−1.57414	−0.787068	+	0.616866i	$0.788402\pi$
$450$	−2.41421	−0.113807
$451$	2.34315	0.110334
$452$	66.2843	3.11775
$453$	−11.1716	−0.524886
$454$	10.4853	0.492099
$455$	−1.51472	−0.0710111
$456$	−12.4853	−0.584677
$457$	23.7574	1.11132	0.555661	−	0.831409i	$-0.312465\pi$
$457$	23.7574	1.11132	0.555661	+	0.831409i	$0.312465\pi$
$458$	−65.1127	−3.04252
$459$	−4.00000	−0.186704
$460$	22.9706	1.07101
$461$	30.7279	1.43114	0.715571	−	0.698540i	$-0.246167\pi$
$461$	30.7279	1.43114	0.715571	+	0.698540i	$0.246167\pi$
$462$	−4.00000	−0.186097
$463$	−35.3137	−1.64117	−0.820584	−	0.571527i	$-0.806351\pi$
$463$	−35.3137	−1.64117	−0.820584	+	0.571527i	$0.806351\pi$
$464$	6.72792	0.312336
$465$	−1.00000	−0.0463739
$466$	−10.8284	−0.501617
$467$	−18.4853	−0.855397	−0.427698	−	0.903921i	$-0.640675\pi$
$467$	−18.4853	−0.855397	−0.427698	+	0.903921i	$0.640675\pi$
$468$	−9.89949	−0.457604
$469$	−2.28427	−0.105478
$470$	−11.6569	−0.537691
$471$	18.0000	0.829396
$472$	1.07107	0.0492999
$473$	32.0000	1.47136
$474$	34.9706	1.60625
$475$	2.82843	0.129777
$476$	8.97056	0.411165
$477$	−4.00000	−0.183147
$478$	34.6274	1.58382
$479$	−43.3553	−1.98096	−0.990478	−	0.137671i	$-0.956038\pi$
$479$	−43.3553	−1.98096	−0.990478	+	0.137671i	$0.956038\pi$
$480$	−1.58579	−0.0723809
$481$	3.65685	0.166738
$482$	−26.4853	−1.20637
$483$	3.51472	0.159925
$484$	−11.4853	−0.522058
$485$	−15.6569	−0.710941
$486$	−2.41421	−0.109511
$487$	25.4558	1.15351	0.576757	−	0.816916i	$-0.304318\pi$
$487$	25.4558	1.15351	0.576757	+	0.816916i	$0.304318\pi$
$488$	46.2843	2.09519
$489$	10.2426	0.463188
$490$	−16.0711	−0.726017
$491$	−38.1421	−1.72133	−0.860665	−	0.509171i	$-0.829952\pi$
$491$	−38.1421	−1.72133	−0.860665	+	0.509171i	$0.829952\pi$
$492$	−3.17157	−0.142986
$493$	−8.97056	−0.404014
$494$	17.6569	0.794419
$495$	2.82843	0.127128
$496$	3.00000	0.134704
$497$	−5.79899	−0.260120
$498$	0.828427	0.0371227
$499$	−35.5980	−1.59358	−0.796792	−	0.604253i	$-0.793471\pi$
$499$	−35.5980	−1.59358	−0.796792	+	0.604253i	$0.793471\pi$
$500$	−3.82843	−0.171212
$501$	−8.97056	−0.400775
$502$	−32.9706	−1.47155
$503$	−36.8284	−1.64210	−0.821049	−	0.570857i	$-0.806611\pi$
$503$	−36.8284	−1.64210	−0.821049	+	0.570857i	$0.806611\pi$
$504$	2.58579	0.115180
$505$	4.82843	0.214862
$506$	−40.9706	−1.82136
$507$	−6.31371	−0.280402
$508$	1.85786	0.0824294
$509$	−27.6985	−1.22771	−0.613857	−	0.789417i	$-0.710383\pi$
$509$	−27.6985	−1.22771	−0.613857	+	0.789417i	$0.710383\pi$
$510$	−9.65685	−0.427613
$511$	3.45584	0.152878
$512$	31.2426	1.38074
$513$	2.82843	0.124878
$514$	−59.1127	−2.60735
$515$	2.24264	0.0988226
$516$	−43.3137	−1.90678
$517$	13.6569	0.600628
$518$	−2.00000	−0.0878750
$519$	−14.0000	−0.614532
$520$	−11.4142	−0.500546
$521$	23.4558	1.02762	0.513810	−	0.857904i	$-0.328234\pi$
$521$	23.4558	1.02762	0.513810	+	0.857904i	$0.328234\pi$
$522$	−5.41421	−0.236974
$523$	4.48528	0.196128	0.0980638	−	0.995180i	$-0.468735\pi$
$523$	4.48528	0.196128	0.0980638	+	0.995180i	$0.468735\pi$
$524$	33.4142	1.45971
$525$	−0.585786	−0.0255658
$526$	−23.3137	−1.01653
$527$	−4.00000	−0.174243
$528$	−8.48528	−0.369274
$529$	13.0000	0.565217
$530$	−9.65685	−0.419467
$531$	−0.242641	−0.0105297
$532$	−6.34315	−0.275010
$533$	2.14214	0.0927862
$534$	−12.2426	−0.529791
$535$	3.17157	0.137119
$536$	−17.2132	−0.743497
$537$	23.3137	1.00606
$538$	31.5563	1.36049
$539$	18.8284	0.810998
$540$	−3.82843	−0.164749
$541$	−26.6274	−1.14480	−0.572401	−	0.819974i	$-0.693988\pi$
$541$	−26.6274	−1.14480	−0.572401	+	0.819974i	$0.693988\pi$
$542$	27.3137	1.17322
$543$	−21.7990	−0.935484
$544$	−6.34315	−0.271960
$545$	−19.3137	−0.827308
$546$	−3.65685	−0.156499
$547$	−20.1005	−0.859436	−0.429718	−	0.902963i	$-0.641387\pi$
$547$	−20.1005	−0.859436	−0.429718	+	0.902963i	$0.641387\pi$
$548$	10.8284	0.462567
$549$	−10.4853	−0.447501
$550$	6.82843	0.291165
$551$	6.34315	0.270227
$552$	26.4853	1.12729
$553$	8.48528	0.360831
$554$	−55.6985	−2.36640
$555$	1.41421	0.0600300
$556$	−6.34315	−0.269009
$557$	−33.6569	−1.42609	−0.713043	−	0.701120i	$-0.752684\pi$
$557$	−33.6569	−1.42609	−0.713043	+	0.701120i	$0.752684\pi$
$558$	−2.41421	−0.102202
$559$	29.2548	1.23735
$560$	1.75736	0.0742620
$561$	11.3137	0.477665
$562$	56.6274	2.38868
$563$	23.4558	0.988546	0.494273	−	0.869307i	$-0.335434\pi$
$563$	23.4558	0.988546	0.494273	+	0.869307i	$0.335434\pi$
$564$	−18.4853	−0.778371
$565$	−17.3137	−0.728393
$566$	−54.8701	−2.30636
$567$	−0.585786	−0.0246007
$568$	−43.6985	−1.83355
$569$	30.0416	1.25941	0.629705	−	0.776834i	$-0.283176\pi$
$569$	30.0416	1.25941	0.629705	+	0.776834i	$0.283176\pi$
$570$	6.82843	0.286011
$571$	24.2843	1.01627	0.508133	−	0.861279i	$-0.330336\pi$
$571$	24.2843	1.01627	0.508133	+	0.861279i	$0.330336\pi$
$572$	28.0000	1.17074
$573$	−0.727922	−0.0304094
$574$	−1.17157	−0.0489005
$575$	−6.00000	−0.250217
$576$	−9.82843	−0.409518
$577$	−8.14214	−0.338962	−0.169481	−	0.985533i	$-0.554209\pi$
$577$	−8.14214	−0.338962	−0.169481	+	0.985533i	$0.554209\pi$
$578$	2.41421	0.100418
$579$	7.65685	0.318208
$580$	−8.58579	−0.356505
$581$	0.201010	0.00833931
$582$	−37.7990	−1.56682
$583$	11.3137	0.468566
$584$	26.0416	1.07761
$585$	2.58579	0.106909
$586$	41.4558	1.71253
$587$	21.6569	0.893874	0.446937	−	0.894565i	$-0.352515\pi$
$587$	21.6569	0.893874	0.446937	+	0.894565i	$0.352515\pi$
$588$	−25.4853	−1.05100
$589$	2.82843	0.116543
$590$	−0.585786	−0.0241164
$591$	11.7990	0.485346
$592$	−4.24264	−0.174371
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	6.82843	0.280174
$595$	−2.34315	−0.0960596
$596$	−40.1421	−1.64429
$597$	16.1421	0.660654
$598$	−37.4558	−1.53168
$599$	−4.92893	−0.201391	−0.100695	−	0.994917i	$-0.532107\pi$
$599$	−4.92893	−0.201391	−0.100695	+	0.994917i	$0.532107\pi$
$600$	−4.41421	−0.180210
$601$	−24.1421	−0.984778	−0.492389	−	0.870375i	$-0.663876\pi$
$601$	−24.1421	−0.984778	−0.492389	+	0.870375i	$0.663876\pi$
$602$	−16.0000	−0.652111
$603$	3.89949	0.158800
$604$	−42.7696	−1.74027
$605$	3.00000	0.121967
$606$	11.6569	0.473527
$607$	29.7574	1.20781	0.603907	−	0.797055i	$-0.293610\pi$
$607$	29.7574	1.20781	0.603907	+	0.797055i	$0.293610\pi$
$608$	4.48528	0.181902
$609$	−1.31371	−0.0532342
$610$	−25.3137	−1.02492
$611$	12.4853	0.505100
$612$	−15.3137	−0.619020
$613$	12.9289	0.522195	0.261097	−	0.965312i	$-0.415916\pi$
$613$	12.9289	0.522195	0.261097	+	0.965312i	$0.415916\pi$
$614$	50.8701	2.05295
$615$	0.828427	0.0334054
$616$	−7.31371	−0.294678
$617$	−2.97056	−0.119590	−0.0597952	−	0.998211i	$-0.519045\pi$
$617$	−2.97056	−0.119590	−0.0597952	+	0.998211i	$0.519045\pi$
$618$	5.41421	0.217792
$619$	2.20101	0.0884661	0.0442330	−	0.999021i	$-0.485916\pi$
$619$	2.20101	0.0884661	0.0442330	+	0.999021i	$0.485916\pi$
$620$	−3.82843	−0.153753
$621$	−6.00000	−0.240772
$622$	58.5269	2.34672
$623$	−2.97056	−0.119013
$624$	−7.75736	−0.310543
$625$	1.00000	0.0400000
$626$	30.7279	1.22813
$627$	−8.00000	−0.319489
$628$	68.9117	2.74988
$629$	5.65685	0.225554
$630$	−1.41421	−0.0563436
$631$	2.34315	0.0932792	0.0466396	−	0.998912i	$-0.485149\pi$
$631$	2.34315	0.0932792	0.0466396	+	0.998912i	$0.485149\pi$
$632$	63.9411	2.54344
$633$	−9.17157	−0.364537
$634$	−31.7990	−1.26290
$635$	−0.485281	−0.0192578
$636$	−15.3137	−0.607228
$637$	17.2132	0.682012
$638$	15.3137	0.606276
$639$	9.89949	0.391618
$640$	−20.5563	−0.812561
$641$	29.3553	1.15947	0.579733	−	0.814806i	$-0.303157\pi$
$641$	29.3553	1.15947	0.579733	+	0.814806i	$0.303157\pi$
$642$	7.65685	0.302192
$643$	2.82843	0.111542	0.0557711	−	0.998444i	$-0.482238\pi$
$643$	2.82843	0.111542	0.0557711	+	0.998444i	$0.482238\pi$
$644$	13.4558	0.530235
$645$	11.3137	0.445477
$646$	27.3137	1.07464
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	−4.41421	−0.173407
$649$	0.686292	0.0269393
$650$	6.24264	0.244857
$651$	−0.585786	−0.0229588
$652$	39.2132	1.53571
$653$	−42.4264	−1.66027	−0.830137	−	0.557560i	$-0.811738\pi$
$653$	−42.4264	−1.66027	−0.830137	+	0.557560i	$0.811738\pi$
$654$	−46.6274	−1.82328
$655$	−8.72792	−0.341028
$656$	−2.48528	−0.0970339
$657$	−5.89949	−0.230161
$658$	−6.82843	−0.266200
$659$	24.0416	0.936529	0.468264	−	0.883588i	$-0.344879\pi$
$659$	24.0416	0.936529	0.468264	+	0.883588i	$0.344879\pi$
$660$	10.8284	0.421496
$661$	14.2843	0.555594	0.277797	−	0.960640i	$-0.410396\pi$
$661$	14.2843	0.555594	0.277797	+	0.960640i	$0.410396\pi$
$662$	54.2843	2.10982
$663$	10.3431	0.401694
$664$	1.51472	0.0587825
$665$	1.65685	0.0642501
$666$	3.41421	0.132298
$667$	−13.4558	−0.521012
$668$	−34.3431	−1.32878
$669$	−22.8284	−0.882598
$670$	9.41421	0.363703
$671$	29.6569	1.14489
$672$	−0.928932	−0.0358343
$673$	−12.9289	−0.498374	−0.249187	−	0.968455i	$-0.580163\pi$
$673$	−12.9289	−0.498374	−0.249187	+	0.968455i	$0.580163\pi$
$674$	−54.5269	−2.10030
$675$	1.00000	0.0384900
$676$	−24.1716	−0.929676
$677$	−8.68629	−0.333841	−0.166921	−	0.985970i	$-0.553382\pi$
$677$	−8.68629	−0.333841	−0.166921	+	0.985970i	$0.553382\pi$
$678$	−41.7990	−1.60528
$679$	−9.17157	−0.351973
$680$	−17.6569	−0.677109
$681$	−4.34315	−0.166430
$682$	6.82843	0.261474
$683$	−30.9706	−1.18506	−0.592528	−	0.805550i	$-0.701870\pi$
$683$	−30.9706	−1.18506	−0.592528	+	0.805550i	$0.701870\pi$
$684$	10.8284	0.414035
$685$	−2.82843	−0.108069
$686$	−19.3137	−0.737401
$687$	26.9706	1.02899
$688$	−33.9411	−1.29399
$689$	10.3431	0.394042
$690$	−14.4853	−0.551445
$691$	−14.6274	−0.556453	−0.278227	−	0.960515i	$-0.589747\pi$
$691$	−14.6274	−0.556453	−0.278227	+	0.960515i	$0.589747\pi$
$692$	−53.5980	−2.03749
$693$	1.65685	0.0629387
$694$	28.9706	1.09971
$695$	1.65685	0.0628481
$696$	−9.89949	−0.375239
$697$	3.31371	0.125516
$698$	−48.2843	−1.82759
$699$	4.48528	0.169649
$700$	−2.24264	−0.0847639
$701$	10.9706	0.414352	0.207176	−	0.978304i	$-0.433573\pi$
$701$	10.9706	0.414352	0.207176	+	0.978304i	$0.433573\pi$
$702$	6.24264	0.235613
$703$	−4.00000	−0.150863
$704$	27.7990	1.04771
$705$	4.82843	0.181849
$706$	31.3137	1.17851
$707$	2.82843	0.106374
$708$	−0.928932	−0.0349114
$709$	−10.9706	−0.412008	−0.206004	−	0.978551i	$-0.566046\pi$
$709$	−10.9706	−0.412008	−0.206004	+	0.978551i	$0.566046\pi$
$710$	23.8995	0.896932
$711$	−14.4853	−0.543240
$712$	−22.3848	−0.838905
$713$	−6.00000	−0.224702
$714$	−5.65685	−0.211702
$715$	−7.31371	−0.273517
$716$	89.2548	3.33561
$717$	−14.3431	−0.535655
$718$	48.3848	1.80570
$719$	43.7990	1.63343	0.816713	−	0.577044i	$-0.195794\pi$
$719$	43.7990	1.63343	0.816713	+	0.577044i	$0.195794\pi$
$720$	−3.00000	−0.111803
$721$	1.31371	0.0489251
$722$	26.5563	0.988325
$723$	10.9706	0.408000
$724$	−83.4558	−3.10161
$725$	2.24264	0.0832896
$726$	7.24264	0.268800
$727$	−47.0122	−1.74359	−0.871793	−	0.489875i	$-0.837043\pi$
$727$	−47.0122	−1.74359	−0.871793	+	0.489875i	$0.837043\pi$
$728$	−6.68629	−0.247810
$729$	1.00000	0.0370370
$730$	−14.2426	−0.527144
$731$	45.2548	1.67381
$732$	−40.1421	−1.48370
$733$	−8.62742	−0.318661	−0.159330	−	0.987225i	$-0.550934\pi$
$733$	−8.62742	−0.318661	−0.159330	+	0.987225i	$0.550934\pi$
$734$	79.5980	2.93802
$735$	6.65685	0.245542
$736$	−9.51472	−0.350717
$737$	−11.0294	−0.406275
$738$	2.00000	0.0736210
$739$	48.4264	1.78139	0.890697	−	0.454597i	$-0.150217\pi$
$739$	48.4264	1.78139	0.890697	+	0.454597i	$0.150217\pi$
$740$	5.41421	0.199030
$741$	−7.31371	−0.268676
$742$	−5.65685	−0.207670
$743$	16.2843	0.597412	0.298706	−	0.954345i	$-0.403445\pi$
$743$	16.2843	0.597412	0.298706	+	0.954345i	$0.403445\pi$
$744$	−4.41421	−0.161833
$745$	10.4853	0.384151
$746$	62.2843	2.28039
$747$	−0.343146	−0.0125550
$748$	43.3137	1.58371
$749$	1.85786	0.0678849
$750$	2.41421	0.0881546
$751$	−43.3137	−1.58054	−0.790270	−	0.612759i	$-0.790060\pi$
$751$	−43.3137	−1.58054	−0.790270	+	0.612759i	$0.790060\pi$
$752$	−14.4853	−0.528224
$753$	13.6569	0.497683
$754$	14.0000	0.509850
$755$	11.1716	0.406575
$756$	−2.24264	−0.0815641
$757$	20.0416	0.728425	0.364213	−	0.931316i	$-0.381338\pi$
$757$	20.0416	0.728425	0.364213	+	0.931316i	$0.381338\pi$
$758$	64.7696	2.35254
$759$	16.9706	0.615992
$760$	12.4853	0.452889
$761$	−14.2426	−0.516295	−0.258148	−	0.966105i	$-0.583112\pi$
$761$	−14.2426	−0.516295	−0.258148	+	0.966105i	$0.583112\pi$
$762$	−1.17157	−0.0424416
$763$	−11.3137	−0.409584
$764$	−2.78680	−0.100823
$765$	4.00000	0.144620
$766$	−22.4853	−0.812426
$767$	0.627417	0.0226547
$768$	−29.9706	−1.08147
$769$	15.3137	0.552226	0.276113	−	0.961125i	$-0.410954\pi$
$769$	15.3137	0.552226	0.276113	+	0.961125i	$0.410954\pi$
$770$	4.00000	0.144150
$771$	24.4853	0.881816
$772$	29.3137	1.05502
$773$	−6.62742	−0.238372	−0.119186	−	0.992872i	$-0.538028\pi$
$773$	−6.62742	−0.238372	−0.119186	+	0.992872i	$0.538028\pi$
$774$	27.3137	0.981771
$775$	1.00000	0.0359211
$776$	−69.1127	−2.48100
$777$	0.828427	0.0297197
$778$	−50.8701	−1.82378
$779$	−2.34315	−0.0839519
$780$	9.89949	0.354459
$781$	−28.0000	−1.00192
$782$	−57.9411	−2.07197
$783$	2.24264	0.0801454
$784$	−19.9706	−0.713234
$785$	−18.0000	−0.642448
$786$	−21.0711	−0.751580
$787$	14.8284	0.528576	0.264288	−	0.964444i	$-0.414863\pi$
$787$	14.8284	0.528576	0.264288	+	0.964444i	$0.414863\pi$
$788$	45.1716	1.60917
$789$	9.65685	0.343793
$790$	−34.9706	−1.24420
$791$	−10.1421	−0.360613
$792$	12.4853	0.443645
$793$	27.1127	0.962800
$794$	−82.0833	−2.91303
$795$	4.00000	0.141865
$796$	61.7990	2.19041
$797$	−6.82843	−0.241875	−0.120938	−	0.992660i	$-0.538590\pi$
$797$	−6.82843	−0.241875	−0.120938	+	0.992660i	$0.538590\pi$
$798$	4.00000	0.141598
$799$	19.3137	0.683270
$800$	1.58579	0.0560660
$801$	5.07107	0.179177
$802$	−15.0711	−0.532178
$803$	16.6863	0.588846
$804$	14.9289	0.526503
$805$	−3.51472	−0.123878
$806$	6.24264	0.219888
$807$	−13.0711	−0.460123
$808$	21.3137	0.749814
$809$	24.8701	0.874385	0.437192	−	0.899368i	$-0.355973\pi$
$809$	24.8701	0.874385	0.437192	+	0.899368i	$0.355973\pi$
$810$	2.41421	0.0848268
$811$	−9.45584	−0.332040	−0.166020	−	0.986122i	$-0.553092\pi$
$811$	−9.45584	−0.332040	−0.166020	+	0.986122i	$0.553092\pi$
$812$	−5.02944	−0.176499
$813$	−11.3137	−0.396789
$814$	−9.65685	−0.338473
$815$	−10.2426	−0.358784
$816$	−12.0000	−0.420084
$817$	−32.0000	−1.11954
$818$	−71.9411	−2.51536
$819$	1.51472	0.0529286
$820$	3.17157	0.110756
$821$	−22.9289	−0.800225	−0.400113	−	0.916466i	$-0.631029\pi$
$821$	−22.9289	−0.800225	−0.400113	+	0.916466i	$0.631029\pi$
$822$	−6.82843	−0.238169
$823$	−41.9411	−1.46198	−0.730988	−	0.682390i	$-0.760940\pi$
$823$	−41.9411	−1.46198	−0.730988	+	0.682390i	$0.760940\pi$
$824$	9.89949	0.344865
$825$	−2.82843	−0.0984732
$826$	−0.343146	−0.0119396
$827$	−14.9706	−0.520577	−0.260289	−	0.965531i	$-0.583818\pi$
$827$	−14.9706	−0.520577	−0.260289	+	0.965531i	$0.583818\pi$
$828$	−22.9706	−0.798282
$829$	27.1716	0.943708	0.471854	−	0.881677i	$-0.343585\pi$
$829$	27.1716	0.943708	0.471854	+	0.881677i	$0.343585\pi$
$830$	−0.828427	−0.0287551
$831$	23.0711	0.800326
$832$	25.4142	0.881079
$833$	26.6274	0.922585
$834$	4.00000	0.138509
$835$	8.97056	0.310439
$836$	−30.6274	−1.05927
$837$	1.00000	0.0345651
$838$	−65.3553	−2.25766
$839$	11.5563	0.398969	0.199485	−	0.979901i	$-0.436073\pi$
$839$	11.5563	0.398969	0.199485	+	0.979901i	$0.436073\pi$
$840$	−2.58579	−0.0892181
$841$	−23.9706	−0.826571
$842$	−61.9411	−2.13463
$843$	−23.4558	−0.807862
$844$	−35.1127	−1.20863
$845$	6.31371	0.217198
$846$	11.6569	0.400771
$847$	1.75736	0.0603836
$848$	−12.0000	−0.412082
$849$	22.7279	0.780020
$850$	9.65685	0.331227
$851$	8.48528	0.290872
$852$	37.8995	1.29842
$853$	−15.4558	−0.529198	−0.264599	−	0.964359i	$-0.585240\pi$
$853$	−15.4558	−0.529198	−0.264599	+	0.964359i	$0.585240\pi$
$854$	−14.8284	−0.507418
$855$	−2.82843	−0.0967302
$856$	14.0000	0.478510
$857$	−50.9706	−1.74112	−0.870561	−	0.492061i	$-0.836244\pi$
$857$	−50.9706	−1.74112	−0.870561	+	0.492061i	$0.836244\pi$
$858$	−17.6569	−0.602795
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	43.3137	1.47699
$861$	0.485281	0.0165383
$862$	60.1838	2.04987
$863$	−11.3137	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$864$	1.58579	0.0539496
$865$	14.0000	0.476014
$866$	−63.6985	−2.16456
$867$	−1.00000	−0.0339618
$868$	−2.24264	−0.0761202
$869$	40.9706	1.38983
$870$	5.41421	0.183559
$871$	−10.0833	−0.341658
$872$	−85.2548	−2.88709
$873$	15.6569	0.529904
$874$	40.9706	1.38585
$875$	0.585786	0.0198032
$876$	−22.5858	−0.763103
$877$	−35.4558	−1.19726	−0.598629	−	0.801026i	$-0.704288\pi$
$877$	−35.4558	−1.19726	−0.598629	+	0.801026i	$0.704288\pi$
$878$	0	0
$879$	−17.1716	−0.579183
$880$	8.48528	0.286039
$881$	−46.2426	−1.55795	−0.778977	−	0.627052i	$-0.784261\pi$
$881$	−46.2426	−1.55795	−0.778977	+	0.627052i	$0.784261\pi$
$882$	16.0711	0.541141
$883$	21.4558	0.722047	0.361023	−	0.932557i	$-0.382427\pi$
$883$	21.4558	0.722047	0.361023	+	0.932557i	$0.382427\pi$
$884$	39.5980	1.33182
$885$	0.242641	0.00815628
$886$	28.1421	0.945454
$887$	−46.9706	−1.57712	−0.788559	−	0.614960i	$-0.789172\pi$
$887$	−46.9706	−1.57712	−0.788559	+	0.614960i	$0.789172\pi$
$888$	6.24264	0.209489
$889$	−0.284271	−0.00953415
$890$	12.2426	0.410374
$891$	−2.82843	−0.0947559
$892$	−87.3970	−2.92627
$893$	−13.6569	−0.457009
$894$	25.3137	0.846617
$895$	−23.3137	−0.779291
$896$	−12.0416	−0.402283
$897$	15.5147	0.518021
$898$	80.5269	2.68722
$899$	2.24264	0.0747963
$900$	3.82843	0.127614
$901$	16.0000	0.533037
$902$	−5.65685	−0.188353
$903$	6.62742	0.220547
$904$	−76.4264	−2.54190
$905$	21.7990	0.724623
$906$	26.9706	0.896037
$907$	2.92893	0.0972536	0.0486268	−	0.998817i	$-0.484516\pi$
$907$	2.92893	0.0972536	0.0486268	+	0.998817i	$0.484516\pi$
$908$	−16.6274	−0.551800
$909$	−4.82843	−0.160149
$910$	3.65685	0.121224
$911$	−1.45584	−0.0482343	−0.0241171	−	0.999709i	$-0.507677\pi$
$911$	−1.45584	−0.0482343	−0.0241171	+	0.999709i	$0.507677\pi$
$912$	8.48528	0.280976
$913$	0.970563	0.0321209
$914$	−57.3553	−1.89715
$915$	10.4853	0.346633
$916$	103.255	3.41164
$917$	−5.11270	−0.168836
$918$	9.65685	0.318724
$919$	−28.7696	−0.949020	−0.474510	−	0.880250i	$-0.657375\pi$
$919$	−28.7696	−0.949020	−0.474510	+	0.880250i	$0.657375\pi$
$920$	−26.4853	−0.873194
$921$	−21.0711	−0.694315
$922$	−74.1838	−2.44311
$923$	−25.5980	−0.842568
$924$	6.34315	0.208674
$925$	−1.41421	−0.0464991
$926$	85.2548	2.80165
$927$	−2.24264	−0.0736580
$928$	3.55635	0.116743
$929$	36.1838	1.18715	0.593575	−	0.804778i	$-0.297716\pi$
$929$	36.1838	1.18715	0.593575	+	0.804778i	$0.297716\pi$
$930$	2.41421	0.0791652
$931$	−18.8284	−0.617077
$932$	17.1716	0.562474
$933$	−24.2426	−0.793668
$934$	44.6274	1.46025
$935$	−11.3137	−0.369998
$936$	11.4142	0.373085
$937$	39.9411	1.30482	0.652410	−	0.757866i	$-0.273758\pi$
$937$	39.9411	1.30482	0.652410	+	0.757866i	$0.273758\pi$
$938$	5.51472	0.180062
$939$	−12.7279	−0.415360
$940$	18.4853	0.602923
$941$	−15.2132	−0.495936	−0.247968	−	0.968768i	$-0.579763\pi$
$941$	−15.2132	−0.495936	−0.247968	+	0.968768i	$0.579763\pi$
$942$	−43.4558	−1.41587
$943$	4.97056	0.161864
$944$	−0.727922	−0.0236918
$945$	0.585786	0.0190556
$946$	−77.2548	−2.51177
$947$	−13.3137	−0.432637	−0.216319	−	0.976323i	$-0.569405\pi$
$947$	−13.3137	−0.432637	−0.216319	+	0.976323i	$0.569405\pi$
$948$	−55.4558	−1.80112
$949$	15.2548	0.495193
$950$	−6.82843	−0.221543
$951$	13.1716	0.427118
$952$	−10.3431	−0.335223
$953$	−10.1421	−0.328536	−0.164268	−	0.986416i	$-0.552526\pi$
$953$	−10.1421	−0.328536	−0.164268	+	0.986416i	$0.552526\pi$
$954$	9.65685	0.312652
$955$	0.727922	0.0235550
$956$	−54.9117	−1.77597
$957$	−6.34315	−0.205045
$958$	104.669	3.38170
$959$	−1.65685	−0.0535026
$960$	9.82843	0.317211
$961$	1.00000	0.0322581
$962$	−8.82843	−0.284640
$963$	−3.17157	−0.102203
$964$	42.0000	1.35273
$965$	−7.65685	−0.246483
$966$	−8.48528	−0.273009
$967$	38.4264	1.23571	0.617855	−	0.786292i	$-0.288002\pi$
$967$	38.4264	1.23571	0.617855	+	0.786292i	$0.288002\pi$
$968$	13.2426	0.425635
$969$	−11.3137	−0.363449
$970$	37.7990	1.21365
$971$	−7.55635	−0.242495	−0.121247	−	0.992622i	$-0.538689\pi$
$971$	−7.55635	−0.242495	−0.121247	+	0.992622i	$0.538689\pi$
$972$	3.82843	0.122797
$973$	0.970563	0.0311148
$974$	−61.4558	−1.96917
$975$	−2.58579	−0.0828114
$976$	−31.4558	−1.00688
$977$	−49.5980	−1.58678	−0.793390	−	0.608714i	$-0.791686\pi$
$977$	−49.5980	−1.58678	−0.793390	+	0.608714i	$0.791686\pi$
$978$	−24.7279	−0.790712
$979$	−14.3431	−0.458409
$980$	25.4853	0.814097
$981$	19.3137	0.616639
$982$	92.0833	2.93849
$983$	33.6569	1.07349	0.536743	−	0.843745i	$-0.319654\pi$
$983$	33.6569	1.07349	0.536743	+	0.843745i	$0.319654\pi$
$984$	3.65685	0.116576
$985$	−11.7990	−0.375947
$986$	21.6569	0.689695
$987$	2.82843	0.0900298
$988$	−28.0000	−0.890799
$989$	67.8823	2.15853
$990$	−6.82843	−0.217022
$991$	−0.970563	−0.0308309	−0.0154155	−	0.999881i	$-0.504907\pi$
$991$	−0.970563	−0.0308309	−0.0154155	+	0.999881i	$0.504907\pi$
$992$	1.58579	0.0503488
$993$	−22.4853	−0.713549
$994$	14.0000	0.444053
$995$	−16.1421	−0.511740
$996$	−1.31371	−0.0416264
$997$	−15.8579	−0.502224	−0.251112	−	0.967958i	$-0.580796\pi$
$997$	−15.8579	−0.502224	−0.251112	+	0.967958i	$0.580796\pi$
$998$	85.9411	2.72042
$999$	−1.41421	−0.0447437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.c.1.1	✓	2
3.2	odd	2		1395.2.a.g.1.2		2
4.3	odd	2		7440.2.a.be.1.1		2
5.2	odd	4		2325.2.c.i.1024.1		4
5.3	odd	4		2325.2.c.i.1024.4		4
5.4	even	2		2325.2.a.n.1.2		2
15.14	odd	2		6975.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.c.1.1	✓	2	1.1	even	1	trivial
1395.2.a.g.1.2		2	3.2	odd	2
2325.2.a.n.1.2		2	5.4	even	2
2325.2.c.i.1024.1		4	5.2	odd	4
2325.2.c.i.1024.4		4	5.3	odd	4
6975.2.a.u.1.1		2	15.14	odd	2
7440.2.a.be.1.1		2	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)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.41421 q^{2} +1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} -0.585786 q^{7} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -2.82843 q^{11} +3.82843 q^{12} -2.58579 q^{13} +1.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} -4.00000 q^{17} -2.41421 q^{18} +2.82843 q^{19} -3.82843 q^{20} -0.585786 q^{21} +6.82843 q^{22} -6.00000 q^{23} -4.41421 q^{24} +1.00000 q^{25} +6.24264 q^{26} +1.00000 q^{27} -2.24264 q^{28} +2.24264 q^{29} +2.41421 q^{30} +1.00000 q^{31} +1.58579 q^{32} -2.82843 q^{33} +9.65685 q^{34} +0.585786 q^{35} +3.82843 q^{36} -1.41421 q^{37} -6.82843 q^{38} -2.58579 q^{39} +4.41421 q^{40} -0.828427 q^{41} +1.41421 q^{42} -11.3137 q^{43} -10.8284 q^{44} -1.00000 q^{45} +14.4853 q^{46} -4.82843 q^{47} +3.00000 q^{48} -6.65685 q^{49} -2.41421 q^{50} -4.00000 q^{51} -9.89949 q^{52} -4.00000 q^{53} -2.41421 q^{54} +2.82843 q^{55} +2.58579 q^{56} +2.82843 q^{57} -5.41421 q^{58} -0.242641 q^{59} -3.82843 q^{60} -10.4853 q^{61} -2.41421 q^{62} -0.585786 q^{63} -9.82843 q^{64} +2.58579 q^{65} +6.82843 q^{66} +3.89949 q^{67} -15.3137 q^{68} -6.00000 q^{69} -1.41421 q^{70} +9.89949 q^{71} -4.41421 q^{72} -5.89949 q^{73} +3.41421 q^{74} +1.00000 q^{75} +10.8284 q^{76} +1.65685 q^{77} +6.24264 q^{78} -14.4853 q^{79} -3.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -0.343146 q^{83} -2.24264 q^{84} +4.00000 q^{85} +27.3137 q^{86} +2.24264 q^{87} +12.4853 q^{88} +5.07107 q^{89} +2.41421 q^{90} +1.51472 q^{91} -22.9706 q^{92} +1.00000 q^{93} +11.6569 q^{94} -2.82843 q^{95} +1.58579 q^{96} +15.6569 q^{97} +16.0711 q^{98} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 8 q^{13} - 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{20} - 4 q^{21} + 8 q^{22} - 12 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} + 4 q^{28} - 4 q^{29} + 2 q^{30} + 2 q^{31} + 6 q^{32} + 8 q^{34} + 4 q^{35} + 2 q^{36} - 8 q^{38} - 8 q^{39} + 6 q^{40} + 4 q^{41} - 16 q^{44} - 2 q^{45} + 12 q^{46} - 4 q^{47} + 6 q^{48} - 2 q^{49} - 2 q^{50} - 8 q^{51} - 8 q^{53} - 2 q^{54} + 8 q^{56} - 8 q^{58} + 8 q^{59} - 2 q^{60} - 4 q^{61} - 2 q^{62} - 4 q^{63} - 14 q^{64} + 8 q^{65} + 8 q^{66} - 12 q^{67} - 8 q^{68} - 12 q^{69} - 6 q^{72} + 8 q^{73} + 4 q^{74} + 2 q^{75} + 16 q^{76} - 8 q^{77} + 4 q^{78} - 12 q^{79} - 6 q^{80} + 2 q^{81} + 4 q^{82} - 12 q^{83} + 4 q^{84} + 8 q^{85} + 32 q^{86} - 4 q^{87} + 8 q^{88} - 4 q^{89} + 2 q^{90} + 20 q^{91} - 12 q^{92} + 2 q^{93} + 12 q^{94} + 6 q^{96} + 20 q^{97} + 18 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 8 * q^13 - 2 * q^15 + 6 * q^16 - 8 * q^17 - 2 * q^18 - 2 * q^20 - 4 * q^21 + 8 * q^22 - 12 * q^23 - 6 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^27 + 4 * q^28 - 4 * q^29 + 2 * q^30 + 2 * q^31 + 6 * q^32 + 8 * q^34 + 4 * q^35 + 2 * q^36 - 8 * q^38 - 8 * q^39 + 6 * q^40 + 4 * q^41 - 16 * q^44 - 2 * q^45 + 12 * q^46 - 4 * q^47 + 6 * q^48 - 2 * q^49 - 2 * q^50 - 8 * q^51 - 8 * q^53 - 2 * q^54 + 8 * q^56 - 8 * q^58 + 8 * q^59 - 2 * q^60 - 4 * q^61 - 2 * q^62 - 4 * q^63 - 14 * q^64 + 8 * q^65 + 8 * q^66 - 12 * q^67 - 8 * q^68 - 12 * q^69 - 6 * q^72 + 8 * q^73 + 4 * q^74 + 2 * q^75 + 16 * q^76 - 8 * q^77 + 4 * q^78 - 12 * q^79 - 6 * q^80 + 2 * q^81 + 4 * q^82 - 12 * q^83 + 4 * q^84 + 8 * q^85 + 32 * q^86 - 4 * q^87 + 8 * q^88 - 4 * q^89 + 2 * q^90 + 20 * q^91 - 12 * q^92 + 2 * q^93 + 12 * q^94 + 6 * q^96 + 20 * q^97 + 18 * q^98

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