LMFDB - Embedded newform 465.2.a.c.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:	$1$
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			1.2
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)

$f(q)$	$=$	$q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} -3.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} -3.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} +2.82843 q^{11} -1.82843 q^{12} -5.41421 q^{13} -1.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} -4.00000 q^{17} +0.414214 q^{18} -2.82843 q^{19} +1.82843 q^{20} -3.41421 q^{21} +1.17157 q^{22} -6.00000 q^{23} -1.58579 q^{24} +1.00000 q^{25} -2.24264 q^{26} +1.00000 q^{27} +6.24264 q^{28} -6.24264 q^{29} -0.414214 q^{30} +1.00000 q^{31} +4.41421 q^{32} +2.82843 q^{33} -1.65685 q^{34} +3.41421 q^{35} -1.82843 q^{36} +1.41421 q^{37} -1.17157 q^{38} -5.41421 q^{39} +1.58579 q^{40} +4.82843 q^{41} -1.41421 q^{42} +11.3137 q^{43} -5.17157 q^{44} -1.00000 q^{45} -2.48528 q^{46} +0.828427 q^{47} +3.00000 q^{48} +4.65685 q^{49} +0.414214 q^{50} -4.00000 q^{51} +9.89949 q^{52} -4.00000 q^{53} +0.414214 q^{54} -2.82843 q^{55} +5.41421 q^{56} -2.82843 q^{57} -2.58579 q^{58} +8.24264 q^{59} +1.82843 q^{60} +6.48528 q^{61} +0.414214 q^{62} -3.41421 q^{63} -4.17157 q^{64} +5.41421 q^{65} +1.17157 q^{66} -15.8995 q^{67} +7.31371 q^{68} -6.00000 q^{69} +1.41421 q^{70} -9.89949 q^{71} -1.58579 q^{72} +13.8995 q^{73} +0.585786 q^{74} +1.00000 q^{75} +5.17157 q^{76} -9.65685 q^{77} -2.24264 q^{78} +2.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -11.6569 q^{83} +6.24264 q^{84} +4.00000 q^{85} +4.68629 q^{86} -6.24264 q^{87} -4.48528 q^{88} -9.07107 q^{89} -0.414214 q^{90} +18.4853 q^{91} +10.9706 q^{92} +1.00000 q^{93} +0.343146 q^{94} +2.82843 q^{95} +4.41421 q^{96} +4.34315 q^{97} +1.92893 q^{98} +2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.82843	−0.914214
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.414214	0.169102
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	−1.58579	−0.560660
$9$	1.00000	0.333333
$10$	−0.414214	−0.130986
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	−1.82843	−0.527821
$13$	−5.41421	−1.50163	−0.750816	−	0.660511i	$-0.770340\pi$
$13$	−5.41421	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$14$	−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	$0.661206\pi$
$18$	0.414214	0.0976311
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.82843	0.408849
$21$	−3.41421	−0.745042
$22$	1.17157	0.249780
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.58579	−0.323697
$25$	1.00000	0.200000
$26$	−2.24264	−0.439818
$27$	1.00000	0.192450
$28$	6.24264	1.17975
$29$	−6.24264	−1.15923	−0.579615	−	0.814891i	$-0.696797\pi$
$29$	−6.24264	−1.15923	−0.579615	+	0.814891i	$0.696797\pi$
$30$	−0.414214	−0.0756247
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	4.41421	0.780330
$33$	2.82843	0.492366
$34$	−1.65685	−0.284148
$35$	3.41421	0.577107
$36$	−1.82843	−0.304738
$37$	1.41421	0.232495	0.116248	−	0.993220i	$-0.462913\pi$
$37$	1.41421	0.232495	0.116248	+	0.993220i	$0.462913\pi$
$38$	−1.17157	−0.190054
$39$	−5.41421	−0.866968
$40$	1.58579	0.250735
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	−1.41421	−0.218218
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	−5.17157	−0.779644
$45$	−1.00000	−0.149071
$46$	−2.48528	−0.366435
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	3.00000	0.433013
$49$	4.65685	0.665265
$50$	0.414214	0.0585786
$51$	−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	$0.588586\pi$
$54$	0.414214	0.0563673
$55$	−2.82843	−0.381385
$56$	5.41421	0.723505
$57$	−2.82843	−0.374634
$58$	−2.58579	−0.339530
$59$	8.24264	1.07310	0.536550	−	0.843868i	$-0.319727\pi$
$59$	8.24264	1.07310	0.536550	+	0.843868i	$0.319727\pi$
$60$	1.82843	0.236049
$61$	6.48528	0.830355	0.415178	−	0.909740i	$-0.363719\pi$
$61$	6.48528	0.830355	0.415178	+	0.909740i	$0.363719\pi$
$62$	0.414214	0.0526052
$63$	−3.41421	−0.430150
$64$	−4.17157	−0.521447
$65$	5.41421	0.671551
$66$	1.17157	0.144211
$67$	−15.8995	−1.94243	−0.971216	−	0.238200i	$-0.923443\pi$
$67$	−15.8995	−1.94243	−0.971216	+	0.238200i	$0.923443\pi$
$68$	7.31371	0.886917
$69$	−6.00000	−0.722315
$70$	1.41421	0.169031
$71$	−9.89949	−1.17485	−0.587427	−	0.809277i	$-0.699859\pi$
$71$	−9.89949	−1.17485	−0.587427	+	0.809277i	$0.699859\pi$
$72$	−1.58579	−0.186887
$73$	13.8995	1.62681	0.813406	−	0.581696i	$-0.197611\pi$
$73$	13.8995	1.62681	0.813406	+	0.581696i	$0.197611\pi$
$74$	0.585786	0.0680963
$75$	1.00000	0.115470
$76$	5.17157	0.593220
$77$	−9.65685	−1.10050
$78$	−2.24264	−0.253929
$79$	2.48528	0.279616	0.139808	−	0.990179i	$-0.455351\pi$
$79$	2.48528	0.279616	0.139808	+	0.990179i	$0.455351\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−11.6569	−1.27951	−0.639753	−	0.768581i	$-0.720963\pi$
$83$	−11.6569	−1.27951	−0.639753	+	0.768581i	$0.720963\pi$
$84$	6.24264	0.681128
$85$	4.00000	0.433861
$86$	4.68629	0.505336
$87$	−6.24264	−0.669281
$88$	−4.48528	−0.478133
$89$	−9.07107	−0.961531	−0.480766	−	0.876849i	$-0.659641\pi$
$89$	−9.07107	−0.961531	−0.480766	+	0.876849i	$0.659641\pi$
$90$	−0.414214	−0.0436619
$91$	18.4853	1.93778
$92$	10.9706	1.14376
$93$	1.00000	0.103695
$94$	0.343146	0.0353928
$95$	2.82843	0.290191
$96$	4.41421	0.450524
$97$	4.34315	0.440980	0.220490	−	0.975389i	$-0.429234\pi$
$97$	4.34315	0.440980	0.220490	+	0.975389i	$0.429234\pi$
$98$	1.92893	0.194852
$99$	2.82843	0.284268
$100$	−1.82843	−0.182843
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	−1.65685	−0.164053
$103$	6.24264	0.615106	0.307553	−	0.951531i	$-0.400490\pi$
$103$	6.24264	0.615106	0.307553	+	0.951531i	$0.400490\pi$
$104$	8.58579	0.841906
$105$	3.41421	0.333193
$106$	−1.65685	−0.160928
$107$	−8.82843	−0.853476	−0.426738	−	0.904375i	$-0.640337\pi$
$107$	−8.82843	−0.853476	−0.426738	+	0.904375i	$0.640337\pi$
$108$	−1.82843	−0.175940
$109$	−3.31371	−0.317396	−0.158698	−	0.987327i	$-0.550730\pi$
$109$	−3.31371	−0.317396	−0.158698	+	0.987327i	$0.550730\pi$
$110$	−1.17157	−0.111705
$111$	1.41421	0.134231
$112$	−10.2426	−0.967839
$113$	−5.31371	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$114$	−1.17157	−0.109728
$115$	6.00000	0.559503
$116$	11.4142	1.05978
$117$	−5.41421	−0.500544
$118$	3.41421	0.314304
$119$	13.6569	1.25192
$120$	1.58579	0.144762
$121$	−3.00000	−0.272727
$122$	2.68629	0.243205
$123$	4.82843	0.435365
$124$	−1.82843	−0.164198
$125$	−1.00000	−0.0894427
$126$	−1.41421	−0.125988
$127$	−16.4853	−1.46283	−0.731416	−	0.681931i	$-0.761140\pi$
$127$	−16.4853	−1.46283	−0.731416	+	0.681931i	$0.761140\pi$
$128$	−10.5563	−0.933058
$129$	11.3137	0.996116
$130$	2.24264	0.196693
$131$	−16.7279	−1.46153	−0.730763	−	0.682632i	$-0.760835\pi$
$131$	−16.7279	−1.46153	−0.730763	+	0.682632i	$0.760835\pi$
$132$	−5.17157	−0.450128
$133$	9.65685	0.837355
$134$	−6.58579	−0.568925
$135$	−1.00000	−0.0860663
$136$	6.34315	0.543920
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	−2.48528	−0.211561
$139$	9.65685	0.819084	0.409542	−	0.912291i	$-0.365689\pi$
$139$	9.65685	0.819084	0.409542	+	0.912291i	$0.365689\pi$
$140$	−6.24264	−0.527599
$141$	0.828427	0.0697661
$142$	−4.10051	−0.344107
$143$	−15.3137	−1.28060
$144$	3.00000	0.250000
$145$	6.24264	0.518423
$146$	5.75736	0.476482
$147$	4.65685	0.384091
$148$	−2.58579	−0.212550
$149$	6.48528	0.531295	0.265647	−	0.964070i	$-0.414414\pi$
$149$	6.48528	0.531295	0.265647	+	0.964070i	$0.414414\pi$
$150$	0.414214	0.0338204
$151$	−16.8284	−1.36948	−0.684739	−	0.728788i	$-0.740084\pi$
$151$	−16.8284	−1.36948	−0.684739	+	0.728788i	$0.740084\pi$
$152$	4.48528	0.363804
$153$	−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	$0.244931\pi$
$158$	1.02944	0.0818976
$159$	−4.00000	−0.317221
$160$	−4.41421	−0.348974
$161$	20.4853	1.61447
$162$	0.414214	0.0325437
$163$	1.75736	0.137647	0.0688235	−	0.997629i	$-0.478075\pi$
$163$	1.75736	0.137647	0.0688235	+	0.997629i	$0.478075\pi$
$164$	−8.82843	−0.689384
$165$	−2.82843	−0.220193
$166$	−4.82843	−0.374759
$167$	24.9706	1.93228	0.966140	−	0.258018i	$-0.0830694\pi$
$167$	24.9706	1.93228	0.966140	+	0.258018i	$0.0830694\pi$
$168$	5.41421	0.417716
$169$	16.3137	1.25490
$170$	1.65685	0.127075
$171$	−2.82843	−0.216295
$172$	−20.6863	−1.57731
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−2.58579	−0.196028
$175$	−3.41421	−0.258090
$176$	8.48528	0.639602
$177$	8.24264	0.619555
$178$	−3.75736	−0.281626
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	1.82843	0.136283
$181$	17.7990	1.32299	0.661494	−	0.749950i	$-0.269923\pi$
$181$	17.7990	1.32299	0.661494	+	0.749950i	$0.269923\pi$
$182$	7.65685	0.567564
$183$	6.48528	0.479406
$184$	9.51472	0.701434
$185$	−1.41421	−0.103975
$186$	0.414214	0.0303716
$187$	−11.3137	−0.827340
$188$	−1.51472	−0.110472
$189$	−3.41421	−0.248347
$190$	1.17157	0.0849948
$191$	24.7279	1.78925	0.894625	−	0.446818i	$-0.147443\pi$
$191$	24.7279	1.78925	0.894625	+	0.446818i	$0.147443\pi$
$192$	−4.17157	−0.301057
$193$	−3.65685	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$193$	−3.65685	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$194$	1.79899	0.129160
$195$	5.41421	0.387720
$196$	−8.51472	−0.608194
$197$	−27.7990	−1.98060	−0.990298	−	0.138960i	$-0.955624\pi$
$197$	−27.7990	−1.98060	−0.990298	+	0.138960i	$0.955624\pi$
$198$	1.17157	0.0832601
$199$	−12.1421	−0.860733	−0.430367	−	0.902654i	$-0.641616\pi$
$199$	−12.1421	−0.860733	−0.430367	+	0.902654i	$0.641616\pi$
$200$	−1.58579	−0.112132
$201$	−15.8995	−1.12146
$202$	0.343146	0.0241437
$203$	21.3137	1.49593
$204$	7.31371	0.512062
$205$	−4.82843	−0.337232
$206$	2.58579	0.180160
$207$	−6.00000	−0.417029
$208$	−16.2426	−1.12622
$209$	−8.00000	−0.553372
$210$	1.41421	0.0975900
$211$	−14.8284	−1.02083	−0.510416	−	0.859928i	$-0.670508\pi$
$211$	−14.8284	−1.02083	−0.510416	+	0.859928i	$0.670508\pi$
$212$	7.31371	0.502308
$213$	−9.89949	−0.678302
$214$	−3.65685	−0.249977
$215$	−11.3137	−0.771589
$216$	−1.58579	−0.107899
$217$	−3.41421	−0.231772
$218$	−1.37258	−0.0929631
$219$	13.8995	0.939241
$220$	5.17157	0.348667
$221$	21.6569	1.45680
$222$	0.585786	0.0393154
$223$	−17.1716	−1.14989	−0.574947	−	0.818191i	$-0.694977\pi$
$223$	−17.1716	−1.14989	−0.574947	+	0.818191i	$0.694977\pi$
$224$	−15.0711	−1.00698
$225$	1.00000	0.0666667
$226$	−2.20101	−0.146409
$227$	−15.6569	−1.03918	−0.519591	−	0.854415i	$-0.673916\pi$
$227$	−15.6569	−1.03918	−0.519591	+	0.854415i	$0.673916\pi$
$228$	5.17157	0.342496
$229$	−6.97056	−0.460628	−0.230314	−	0.973116i	$-0.573975\pi$
$229$	−6.97056	−0.460628	−0.230314	+	0.973116i	$0.573975\pi$
$230$	2.48528	0.163875
$231$	−9.65685	−0.635374
$232$	9.89949	0.649934
$233$	−12.4853	−0.817938	−0.408969	−	0.912548i	$-0.634112\pi$
$233$	−12.4853	−0.817938	−0.408969	+	0.912548i	$0.634112\pi$
$234$	−2.24264	−0.146606
$235$	−0.828427	−0.0540406
$236$	−15.0711	−0.981043
$237$	2.48528	0.161436
$238$	5.65685	0.366679
$239$	−25.6569	−1.65960	−0.829802	−	0.558058i	$-0.811547\pi$
$239$	−25.6569	−1.65960	−0.829802	+	0.558058i	$0.811547\pi$
$240$	−3.00000	−0.193649
$241$	−22.9706	−1.47966	−0.739832	−	0.672792i	$-0.765095\pi$
$241$	−22.9706	−1.47966	−0.739832	+	0.672792i	$0.765095\pi$
$242$	−1.24264	−0.0798800
$243$	1.00000	0.0641500
$244$	−11.8579	−0.759122
$245$	−4.65685	−0.297516
$246$	2.00000	0.127515
$247$	15.3137	0.974388
$248$	−1.58579	−0.100698
$249$	−11.6569	−0.738723
$250$	−0.414214	−0.0261972
$251$	2.34315	0.147898	0.0739490	−	0.997262i	$-0.476440\pi$
$251$	2.34315	0.147898	0.0739490	+	0.997262i	$0.476440\pi$
$252$	6.24264	0.393249
$253$	−16.9706	−1.06693
$254$	−6.82843	−0.428454
$255$	4.00000	0.250490
$256$	3.97056	0.248160
$257$	7.51472	0.468755	0.234378	−	0.972146i	$-0.424695\pi$
$257$	7.51472	0.468755	0.234378	+	0.972146i	$0.424695\pi$
$258$	4.68629	0.291756
$259$	−4.82843	−0.300024
$260$	−9.89949	−0.613941
$261$	−6.24264	−0.386410
$262$	−6.92893	−0.428071
$263$	−1.65685	−0.102166	−0.0510830	−	0.998694i	$-0.516267\pi$
$263$	−1.65685	−0.102166	−0.0510830	+	0.998694i	$0.516267\pi$
$264$	−4.48528	−0.276050
$265$	4.00000	0.245718
$266$	4.00000	0.245256
$267$	−9.07107	−0.555140
$268$	29.0711	1.77580
$269$	1.07107	0.0653042	0.0326521	−	0.999467i	$-0.489605\pi$
$269$	1.07107	0.0653042	0.0326521	+	0.999467i	$0.489605\pi$
$270$	−0.414214	−0.0252082
$271$	11.3137	0.687259	0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	−12.0000	−0.727607
$273$	18.4853	1.11878
$274$	−1.17157	−0.0707773
$275$	2.82843	0.170561
$276$	10.9706	0.660350
$277$	8.92893	0.536488	0.268244	−	0.963351i	$-0.413557\pi$
$277$	8.92893	0.536488	0.268244	+	0.963351i	$0.413557\pi$
$278$	4.00000	0.239904
$279$	1.00000	0.0598684
$280$	−5.41421	−0.323561
$281$	27.4558	1.63788	0.818939	−	0.573880i	$-0.194563\pi$
$281$	27.4558	1.63788	0.818939	+	0.573880i	$0.194563\pi$
$282$	0.343146	0.0204340
$283$	−2.72792	−0.162158	−0.0810791	−	0.996708i	$-0.525837\pi$
$283$	−2.72792	−0.162158	−0.0810791	+	0.996708i	$0.525837\pi$
$284$	18.1005	1.07407
$285$	2.82843	0.167542
$286$	−6.34315	−0.375078
$287$	−16.4853	−0.973095
$288$	4.41421	0.260110
$289$	−1.00000	−0.0588235
$290$	2.58579	0.151843
$291$	4.34315	0.254600
$292$	−25.4142	−1.48725
$293$	−22.8284	−1.33365	−0.666825	−	0.745214i	$-0.732347\pi$
$293$	−22.8284	−1.33365	−0.666825	+	0.745214i	$0.732347\pi$
$294$	1.92893	0.112498
$295$	−8.24264	−0.479905
$296$	−2.24264	−0.130351
$297$	2.82843	0.164122
$298$	2.68629	0.155613
$299$	32.4853	1.87867
$300$	−1.82843	−0.105564
$301$	−38.6274	−2.22645
$302$	−6.97056	−0.401111
$303$	0.828427	0.0475919
$304$	−8.48528	−0.486664
$305$	−6.48528	−0.371346
$306$	−1.65685	−0.0947161
$307$	−6.92893	−0.395455	−0.197728	−	0.980257i	$-0.563356\pi$
$307$	−6.92893	−0.395455	−0.197728	+	0.980257i	$0.563356\pi$
$308$	17.6569	1.00609
$309$	6.24264	0.355131
$310$	−0.414214	−0.0235257
$311$	−15.7574	−0.893518	−0.446759	−	0.894654i	$-0.647422\pi$
$311$	−15.7574	−0.893518	−0.446759	+	0.894654i	$0.647422\pi$
$312$	8.58579	0.486074
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	7.45584	0.420758
$315$	3.41421	0.192369
$316$	−4.54416	−0.255629
$317$	18.8284	1.05751	0.528755	−	0.848775i	$-0.322659\pi$
$317$	18.8284	1.05751	0.528755	+	0.848775i	$0.322659\pi$
$318$	−1.65685	−0.0929118
$319$	−17.6569	−0.988594
$320$	4.17157	0.233198
$321$	−8.82843	−0.492755
$322$	8.48528	0.472866
$323$	11.3137	0.629512
$324$	−1.82843	−0.101579
$325$	−5.41421	−0.300327
$326$	0.727922	0.0403159
$327$	−3.31371	−0.183248
$328$	−7.65685	−0.422779
$329$	−2.82843	−0.155936
$330$	−1.17157	−0.0644930
$331$	−5.51472	−0.303116	−0.151558	−	0.988448i	$-0.548429\pi$
$331$	−5.51472	−0.303116	−0.151558	+	0.988448i	$0.548429\pi$
$332$	21.3137	1.16974
$333$	1.41421	0.0774984
$334$	10.3431	0.565952
$335$	15.8995	0.868682
$336$	−10.2426	−0.558782
$337$	25.4142	1.38440	0.692200	−	0.721706i	$-0.256642\pi$
$337$	25.4142	1.38440	0.692200	+	0.721706i	$0.256642\pi$
$338$	6.75736	0.367552
$339$	−5.31371	−0.288601
$340$	−7.31371	−0.396642
$341$	2.82843	0.153168
$342$	−1.17157	−0.0633514
$343$	8.00000	0.431959
$344$	−17.9411	−0.967321
$345$	6.00000	0.323029
$346$	−5.79899	−0.311756
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	11.4142	0.611866
$349$	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
$351$	−5.41421	−0.288989
$352$	12.4853	0.665468
$353$	20.9706	1.11615	0.558075	−	0.829790i	$-0.311540\pi$
$353$	20.9706	1.11615	0.558075	+	0.829790i	$0.311540\pi$
$354$	3.41421	0.181463
$355$	9.89949	0.525411
$356$	16.5858	0.879045
$357$	13.6569	0.722797
$358$	0.284271	0.0150242
$359$	28.0416	1.47998	0.739990	−	0.672618i	$-0.234830\pi$
$359$	28.0416	1.47998	0.739990	+	0.672618i	$0.234830\pi$
$360$	1.58579	0.0835783
$361$	−11.0000	−0.578947
$362$	7.37258	0.387494
$363$	−3.00000	−0.157459
$364$	−33.7990	−1.77155
$365$	−13.8995	−0.727533
$366$	2.68629	0.140415
$367$	0.970563	0.0506630	0.0253315	−	0.999679i	$-0.491936\pi$
$367$	0.970563	0.0506630	0.0253315	+	0.999679i	$0.491936\pi$
$368$	−18.0000	−0.938315
$369$	4.82843	0.251358
$370$	−0.585786	−0.0304536
$371$	13.6569	0.709029
$372$	−1.82843	−0.0947995
$373$	13.7990	0.714485	0.357242	−	0.934012i	$-0.383717\pi$
$373$	13.7990	0.714485	0.357242	+	0.934012i	$0.383717\pi$
$374$	−4.68629	−0.242322
$375$	−1.00000	−0.0516398
$376$	−1.31371	−0.0677493
$377$	33.7990	1.74074
$378$	−1.41421	−0.0727393
$379$	−21.1716	−1.08751	−0.543755	−	0.839244i	$-0.682998\pi$
$379$	−21.1716	−1.08751	−0.543755	+	0.839244i	$0.682998\pi$
$380$	−5.17157	−0.265296
$381$	−16.4853	−0.844567
$382$	10.2426	0.524059
$383$	−13.3137	−0.680299	−0.340149	−	0.940371i	$-0.610478\pi$
$383$	−13.3137	−0.680299	−0.340149	+	0.940371i	$0.610478\pi$
$384$	−10.5563	−0.538701
$385$	9.65685	0.492159
$386$	−1.51472	−0.0770971
$387$	11.3137	0.575108
$388$	−7.94113	−0.403150
$389$	6.92893	0.351311	0.175655	−	0.984452i	$-0.443796\pi$
$389$	6.92893	0.351311	0.175655	+	0.984452i	$0.443796\pi$
$390$	2.24264	0.113561
$391$	24.0000	1.21373
$392$	−7.38478	−0.372988
$393$	−16.7279	−0.843812
$394$	−11.5147	−0.580103
$395$	−2.48528	−0.125048
$396$	−5.17157	−0.259881
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−5.02944	−0.252103
$399$	9.65685	0.483447
$400$	3.00000	0.150000
$401$	−2.24264	−0.111992	−0.0559961	−	0.998431i	$-0.517833\pi$
$401$	−2.24264	−0.111992	−0.0559961	+	0.998431i	$0.517833\pi$
$402$	−6.58579	−0.328469
$403$	−5.41421	−0.269701
$404$	−1.51472	−0.0753601
$405$	−1.00000	−0.0496904
$406$	8.82843	0.438147
$407$	4.00000	0.198273
$408$	6.34315	0.314033
$409$	−9.79899	−0.484529	−0.242264	−	0.970210i	$-0.577890\pi$
$409$	−9.79899	−0.484529	−0.242264	+	0.970210i	$0.577890\pi$
$410$	−2.00000	−0.0987730
$411$	−2.82843	−0.139516
$412$	−11.4142	−0.562338
$413$	−28.1421	−1.38478
$414$	−2.48528	−0.122145
$415$	11.6569	0.572212
$416$	−23.8995	−1.17177
$417$	9.65685	0.472898
$418$	−3.31371	−0.162079
$419$	12.9289	0.631620	0.315810	−	0.948823i	$-0.397724\pi$
$419$	12.9289	0.631620	0.315810	+	0.948823i	$0.397724\pi$
$420$	−6.24264	−0.304610
$421$	14.3431	0.699042	0.349521	−	0.936929i	$-0.386344\pi$
$421$	14.3431	0.699042	0.349521	+	0.936929i	$0.386344\pi$
$422$	−6.14214	−0.298994
$423$	0.828427	0.0402795
$424$	6.34315	0.308050
$425$	−4.00000	−0.194029
$426$	−4.10051	−0.198670
$427$	−22.1421	−1.07153
$428$	16.1421	0.780260
$429$	−15.3137	−0.739353
$430$	−4.68629	−0.225993
$431$	−39.0711	−1.88199	−0.940994	−	0.338424i	$-0.890106\pi$
$431$	−39.0711	−1.88199	−0.940994	+	0.338424i	$0.890106\pi$
$432$	3.00000	0.144338
$433$	−10.3848	−0.499061	−0.249530	−	0.968367i	$-0.580276\pi$
$433$	−10.3848	−0.499061	−0.249530	+	0.968367i	$0.580276\pi$
$434$	−1.41421	−0.0678844
$435$	6.24264	0.299312
$436$	6.05887	0.290167
$437$	16.9706	0.811812
$438$	5.75736	0.275097
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	4.48528	0.213827
$441$	4.65685	0.221755
$442$	8.97056	0.426686
$443$	−0.343146	−0.0163033	−0.00815167	−	0.999967i	$-0.502595\pi$
$443$	−0.343146	−0.0163033	−0.00815167	+	0.999967i	$0.502595\pi$
$444$	−2.58579	−0.122716
$445$	9.07107	0.430010
$446$	−7.11270	−0.336796
$447$	6.48528	0.306743
$448$	14.2426	0.672902
$449$	37.3553	1.76291	0.881454	−	0.472270i	$-0.156565\pi$
$449$	37.3553	1.76291	0.881454	+	0.472270i	$0.156565\pi$
$450$	0.414214	0.0195262
$451$	13.6569	0.643076
$452$	9.71573	0.456989
$453$	−16.8284	−0.790668
$454$	−6.48528	−0.304369
$455$	−18.4853	−0.866603
$456$	4.48528	0.210043
$457$	32.2426	1.50825	0.754124	−	0.656733i	$-0.228062\pi$
$457$	32.2426	1.50825	0.754124	+	0.656733i	$0.228062\pi$
$458$	−2.88730	−0.134915
$459$	−4.00000	−0.186704
$460$	−10.9706	−0.511505
$461$	5.27208	0.245545	0.122773	−	0.992435i	$-0.460821\pi$
$461$	5.27208	0.245545	0.122773	+	0.992435i	$0.460821\pi$
$462$	−4.00000	−0.186097
$463$	−12.6863	−0.589582	−0.294791	−	0.955562i	$-0.595250\pi$
$463$	−12.6863	−0.589582	−0.294791	+	0.955562i	$0.595250\pi$
$464$	−18.7279	−0.869422
$465$	−1.00000	−0.0463739
$466$	−5.17157	−0.239568
$467$	−1.51472	−0.0700928	−0.0350464	−	0.999386i	$-0.511158\pi$
$467$	−1.51472	−0.0700928	−0.0350464	+	0.999386i	$0.511158\pi$
$468$	9.89949	0.457604
$469$	54.2843	2.50661
$470$	−0.343146	−0.0158281
$471$	18.0000	0.829396
$472$	−13.0711	−0.601645
$473$	32.0000	1.47136
$474$	1.02944	0.0472836
$475$	−2.82843	−0.129777
$476$	−24.9706	−1.14452
$477$	−4.00000	−0.183147
$478$	−10.6274	−0.486087
$479$	27.3553	1.24990	0.624949	−	0.780666i	$-0.285120\pi$
$479$	27.3553	1.24990	0.624949	+	0.780666i	$0.285120\pi$
$480$	−4.41421	−0.201480
$481$	−7.65685	−0.349123
$482$	−9.51472	−0.433384
$483$	20.4853	0.932113
$484$	5.48528	0.249331
$485$	−4.34315	−0.197212
$486$	0.414214	0.0187891
$487$	−25.4558	−1.15351	−0.576757	−	0.816916i	$-0.695682\pi$
$487$	−25.4558	−1.15351	−0.576757	+	0.816916i	$0.695682\pi$
$488$	−10.2843	−0.465547
$489$	1.75736	0.0794705
$490$	−1.92893	−0.0871403
$491$	−9.85786	−0.444879	−0.222440	−	0.974946i	$-0.571402\pi$
$491$	−9.85786	−0.444879	−0.222440	+	0.974946i	$0.571402\pi$
$492$	−8.82843	−0.398016
$493$	24.9706	1.12462
$494$	6.34315	0.285392
$495$	−2.82843	−0.127128
$496$	3.00000	0.134704
$497$	33.7990	1.51609
$498$	−4.82843	−0.216367
$499$	43.5980	1.95171	0.975857	−	0.218411i	$-0.0700874\pi$
$499$	43.5980	1.95171	0.975857	+	0.218411i	$0.0700874\pi$
$500$	1.82843	0.0817697
$501$	24.9706	1.11560
$502$	0.970563	0.0433183
$503$	−31.1716	−1.38987	−0.694936	−	0.719072i	$-0.744567\pi$
$503$	−31.1716	−1.38987	−0.694936	+	0.719072i	$0.744567\pi$
$504$	5.41421	0.241168
$505$	−0.828427	−0.0368645
$506$	−7.02944	−0.312497
$507$	16.3137	0.724517
$508$	30.1421	1.33734
$509$	31.6985	1.40501	0.702505	−	0.711678i	$-0.252065\pi$
$509$	31.6985	1.40501	0.702505	+	0.711678i	$0.252065\pi$
$510$	1.65685	0.0733667
$511$	−47.4558	−2.09932
$512$	22.7574	1.00574
$513$	−2.82843	−0.124878
$514$	3.11270	0.137295
$515$	−6.24264	−0.275084
$516$	−20.6863	−0.910663
$517$	2.34315	0.103051
$518$	−2.00000	−0.0878750
$519$	−14.0000	−0.614532
$520$	−8.58579	−0.376512
$521$	−27.4558	−1.20286	−0.601431	−	0.798925i	$-0.705403\pi$
$521$	−27.4558	−1.20286	−0.601431	+	0.798925i	$0.705403\pi$
$522$	−2.58579	−0.113177
$523$	−12.4853	−0.545943	−0.272972	−	0.962022i	$-0.588007\pi$
$523$	−12.4853	−0.545943	−0.272972	+	0.962022i	$0.588007\pi$
$524$	30.5858	1.33615
$525$	−3.41421	−0.149008
$526$	−0.686292	−0.0299237
$527$	−4.00000	−0.174243
$528$	8.48528	0.369274
$529$	13.0000	0.565217
$530$	1.65685	0.0719691
$531$	8.24264	0.357700
$532$	−17.6569	−0.765522
$533$	−26.1421	−1.13234
$534$	−3.75736	−0.162597
$535$	8.82843	0.381686
$536$	25.2132	1.08904
$537$	0.686292	0.0296157
$538$	0.443651	0.0191271
$539$	13.1716	0.567340
$540$	1.82843	0.0786830
$541$	18.6274	0.800855	0.400428	−	0.916328i	$-0.368862\pi$
$541$	18.6274	0.800855	0.400428	+	0.916328i	$0.368862\pi$
$542$	4.68629	0.201293
$543$	17.7990	0.763828
$544$	−17.6569	−0.757031
$545$	3.31371	0.141944
$546$	7.65685	0.327683
$547$	−39.8995	−1.70598	−0.852990	−	0.521928i	$-0.825213\pi$
$547$	−39.8995	−1.70598	−0.852990	+	0.521928i	$0.825213\pi$
$548$	5.17157	0.220919
$549$	6.48528	0.276785
$550$	1.17157	0.0499560
$551$	17.6569	0.752207
$552$	9.51472	0.404973
$553$	−8.48528	−0.360831
$554$	3.69848	0.157134
$555$	−1.41421	−0.0600300
$556$	−17.6569	−0.748817
$557$	−22.3431	−0.946709	−0.473355	−	0.880872i	$-0.656957\pi$
$557$	−22.3431	−0.946709	−0.473355	+	0.880872i	$0.656957\pi$
$558$	0.414214	0.0175351
$559$	−61.2548	−2.59080
$560$	10.2426	0.432831
$561$	−11.3137	−0.477665
$562$	11.3726	0.479723
$563$	−27.4558	−1.15713	−0.578563	−	0.815638i	$-0.696386\pi$
$563$	−27.4558	−1.15713	−0.578563	+	0.815638i	$0.696386\pi$
$564$	−1.51472	−0.0637812
$565$	5.31371	0.223549
$566$	−1.12994	−0.0474950
$567$	−3.41421	−0.143383
$568$	15.6985	0.658694
$569$	−18.0416	−0.756344	−0.378172	−	0.925735i	$-0.623447\pi$
$569$	−18.0416	−0.756344	−0.378172	+	0.925735i	$0.623447\pi$
$570$	1.17157	0.0490718
$571$	−32.2843	−1.35105	−0.675527	−	0.737335i	$-0.736084\pi$
$571$	−32.2843	−1.35105	−0.675527	+	0.737335i	$0.736084\pi$
$572$	28.0000	1.17074
$573$	24.7279	1.03302
$574$	−6.82843	−0.285013
$575$	−6.00000	−0.250217
$576$	−4.17157	−0.173816
$577$	20.1421	0.838528	0.419264	−	0.907864i	$-0.362288\pi$
$577$	20.1421	0.838528	0.419264	+	0.907864i	$0.362288\pi$
$578$	−0.414214	−0.0172290
$579$	−3.65685	−0.151974
$580$	−11.4142	−0.473949
$581$	39.7990	1.65114
$582$	1.79899	0.0745705
$583$	−11.3137	−0.468566
$584$	−22.0416	−0.912089
$585$	5.41421	0.223850
$586$	−9.45584	−0.390617
$587$	10.3431	0.426907	0.213454	−	0.976953i	$-0.431529\pi$
$587$	10.3431	0.426907	0.213454	+	0.976953i	$0.431529\pi$
$588$	−8.51472	−0.351141
$589$	−2.82843	−0.116543
$590$	−3.41421	−0.140561
$591$	−27.7990	−1.14350
$592$	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	$0.592809\pi$
$594$	1.17157	0.0480702
$595$	−13.6569	−0.559876
$596$	−11.8579	−0.485717
$597$	−12.1421	−0.496945
$598$	13.4558	0.550250
$599$	−19.0711	−0.779223	−0.389611	−	0.920979i	$-0.627391\pi$
$599$	−19.0711	−0.779223	−0.389611	+	0.920979i	$0.627391\pi$
$600$	−1.58579	−0.0647395
$601$	4.14214	0.168961	0.0844806	−	0.996425i	$-0.473077\pi$
$601$	4.14214	0.168961	0.0844806	+	0.996425i	$0.473077\pi$
$602$	−16.0000	−0.652111
$603$	−15.8995	−0.647477
$604$	30.7696	1.25200
$605$	3.00000	0.121967
$606$	0.343146	0.0139393
$607$	38.2426	1.55222	0.776110	−	0.630597i	$-0.217190\pi$
$607$	38.2426	1.55222	0.776110	+	0.630597i	$0.217190\pi$
$608$	−12.4853	−0.506345
$609$	21.3137	0.863675
$610$	−2.68629	−0.108765
$611$	−4.48528	−0.181455
$612$	7.31371	0.295639
$613$	27.0711	1.09339	0.546695	−	0.837332i	$-0.315886\pi$
$613$	27.0711	1.09339	0.546695	+	0.837332i	$0.315886\pi$
$614$	−2.87006	−0.115826
$615$	−4.82843	−0.194701
$616$	15.3137	0.617007
$617$	30.9706	1.24683	0.623414	−	0.781892i	$-0.285745\pi$
$617$	30.9706	1.24683	0.623414	+	0.781892i	$0.285745\pi$
$618$	2.58579	0.104016
$619$	41.7990	1.68004	0.840022	−	0.542553i	$-0.182542\pi$
$619$	41.7990	1.68004	0.840022	+	0.542553i	$0.182542\pi$
$620$	1.82843	0.0734314
$621$	−6.00000	−0.240772
$622$	−6.52691	−0.261705
$623$	30.9706	1.24081
$624$	−16.2426	−0.650226
$625$	1.00000	0.0400000
$626$	5.27208	0.210715
$627$	−8.00000	−0.319489
$628$	−32.9117	−1.31332
$629$	−5.65685	−0.225554
$630$	1.41421	0.0563436
$631$	13.6569	0.543671	0.271835	−	0.962344i	$-0.412369\pi$
$631$	13.6569	0.543671	0.271835	+	0.962344i	$0.412369\pi$
$632$	−3.94113	−0.156770
$633$	−14.8284	−0.589377
$634$	7.79899	0.309737
$635$	16.4853	0.654198
$636$	7.31371	0.290007
$637$	−25.2132	−0.998983
$638$	−7.31371	−0.289552
$639$	−9.89949	−0.391618
$640$	10.5563	0.417276
$641$	−41.3553	−1.63344	−0.816719	−	0.577036i	$-0.804209\pi$
$641$	−41.3553	−1.63344	−0.816719	+	0.577036i	$0.804209\pi$
$642$	−3.65685	−0.144325
$643$	−2.82843	−0.111542	−0.0557711	−	0.998444i	$-0.517762\pi$
$643$	−2.82843	−0.111542	−0.0557711	+	0.998444i	$0.517762\pi$
$644$	−37.4558	−1.47597
$645$	−11.3137	−0.445477
$646$	4.68629	0.184380
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	−1.58579	−0.0622956
$649$	23.3137	0.915143
$650$	−2.24264	−0.0879636
$651$	−3.41421	−0.133814
$652$	−3.21320	−0.125839
$653$	42.4264	1.66027	0.830137	−	0.557560i	$-0.188262\pi$
$653$	42.4264	1.66027	0.830137	+	0.557560i	$0.188262\pi$
$654$	−1.37258	−0.0536722
$655$	16.7279	0.653614
$656$	14.4853	0.565555
$657$	13.8995	0.542271
$658$	−1.17157	−0.0456727
$659$	−24.0416	−0.936529	−0.468264	−	0.883588i	$-0.655121\pi$
$659$	−24.0416	−0.936529	−0.468264	+	0.883588i	$0.655121\pi$
$660$	5.17157	0.201303
$661$	−42.2843	−1.64467	−0.822334	−	0.569005i	$-0.807328\pi$
$661$	−42.2843	−1.64467	−0.822334	+	0.569005i	$0.807328\pi$
$662$	−2.28427	−0.0887807
$663$	21.6569	0.841083
$664$	18.4853	0.717368
$665$	−9.65685	−0.374477
$666$	0.585786	0.0226988
$667$	37.4558	1.45030
$668$	−45.6569	−1.76652
$669$	−17.1716	−0.663891
$670$	6.58579	0.254431
$671$	18.3431	0.708129
$672$	−15.0711	−0.581379
$673$	−27.0711	−1.04351	−0.521756	−	0.853094i	$-0.674723\pi$
$673$	−27.0711	−1.04351	−0.521756	+	0.853094i	$0.674723\pi$
$674$	10.5269	0.405481
$675$	1.00000	0.0384900
$676$	−29.8284	−1.14725
$677$	−31.3137	−1.20348	−0.601742	−	0.798691i	$-0.705526\pi$
$677$	−31.3137	−1.20348	−0.601742	+	0.798691i	$0.705526\pi$
$678$	−2.20101	−0.0845293
$679$	−14.8284	−0.569063
$680$	−6.34315	−0.243249
$681$	−15.6569	−0.599972
$682$	1.17157	0.0448618
$683$	2.97056	0.113665	0.0568327	−	0.998384i	$-0.481900\pi$
$683$	2.97056	0.113665	0.0568327	+	0.998384i	$0.481900\pi$
$684$	5.17157	0.197740
$685$	2.82843	0.108069
$686$	3.31371	0.126518
$687$	−6.97056	−0.265944
$688$	33.9411	1.29399
$689$	21.6569	0.825060
$690$	2.48528	0.0946130
$691$	30.6274	1.16512	0.582561	−	0.812787i	$-0.302051\pi$
$691$	30.6274	1.16512	0.582561	+	0.812787i	$0.302051\pi$
$692$	25.5980	0.973089
$693$	−9.65685	−0.366834
$694$	−4.97056	−0.188680
$695$	−9.65685	−0.366305
$696$	9.89949	0.375239
$697$	−19.3137	−0.731559
$698$	8.28427	0.313564
$699$	−12.4853	−0.472237
$700$	6.24264	0.235950
$701$	−22.9706	−0.867586	−0.433793	−	0.901013i	$-0.642825\pi$
$701$	−22.9706	−0.867586	−0.433793	+	0.901013i	$0.642825\pi$
$702$	−2.24264	−0.0846430
$703$	−4.00000	−0.150863
$704$	−11.7990	−0.444691
$705$	−0.828427	−0.0312004
$706$	8.68629	0.326913
$707$	−2.82843	−0.106374
$708$	−15.0711	−0.566405
$709$	22.9706	0.862678	0.431339	−	0.902190i	$-0.358041\pi$
$709$	22.9706	0.862678	0.431339	+	0.902190i	$0.358041\pi$
$710$	4.10051	0.153889
$711$	2.48528	0.0932053
$712$	14.3848	0.539092
$713$	−6.00000	−0.224702
$714$	5.65685	0.211702
$715$	15.3137	0.572700
$716$	−1.25483	−0.0468953
$717$	−25.6569	−0.958173
$718$	11.6152	0.433476
$719$	4.20101	0.156671	0.0783356	−	0.996927i	$-0.475039\pi$
$719$	4.20101	0.156671	0.0783356	+	0.996927i	$0.475039\pi$
$720$	−3.00000	−0.111803
$721$	−21.3137	−0.793764
$722$	−4.55635	−0.169570
$723$	−22.9706	−0.854284
$724$	−32.5442	−1.20949
$725$	−6.24264	−0.231846
$726$	−1.24264	−0.0461187
$727$	35.0122	1.29853	0.649265	−	0.760562i	$-0.275077\pi$
$727$	35.0122	1.29853	0.649265	+	0.760562i	$0.275077\pi$
$728$	−29.3137	−1.08644
$729$	1.00000	0.0370370
$730$	−5.75736	−0.213089
$731$	−45.2548	−1.67381
$732$	−11.8579	−0.438279
$733$	36.6274	1.35286	0.676432	−	0.736505i	$-0.263525\pi$
$733$	36.6274	1.35286	0.676432	+	0.736505i	$0.263525\pi$
$734$	0.402020	0.0148388
$735$	−4.65685	−0.171771
$736$	−26.4853	−0.976260
$737$	−44.9706	−1.65651
$738$	2.00000	0.0736210
$739$	−36.4264	−1.33997	−0.669984	−	0.742376i	$-0.733699\pi$
$739$	−36.4264	−1.33997	−0.669984	+	0.742376i	$0.733699\pi$
$740$	2.58579	0.0950554
$741$	15.3137	0.562563
$742$	5.65685	0.207670
$743$	−40.2843	−1.47789	−0.738943	−	0.673768i	$-0.764675\pi$
$743$	−40.2843	−1.47789	−0.738943	+	0.673768i	$0.764675\pi$
$744$	−1.58579	−0.0581378
$745$	−6.48528	−0.237602
$746$	5.71573	0.209268
$747$	−11.6569	−0.426502
$748$	20.6863	0.756366
$749$	30.1421	1.10137
$750$	−0.414214	−0.0151249
$751$	−20.6863	−0.754853	−0.377427	−	0.926039i	$-0.623191\pi$
$751$	−20.6863	−0.754853	−0.377427	+	0.926039i	$0.623191\pi$
$752$	2.48528	0.0906289
$753$	2.34315	0.0853890
$754$	14.0000	0.509850
$755$	16.8284	0.612449
$756$	6.24264	0.227043
$757$	−28.0416	−1.01919	−0.509595	−	0.860414i	$-0.670205\pi$
$757$	−28.0416	−1.01919	−0.509595	+	0.860414i	$0.670205\pi$
$758$	−8.76955	−0.318524
$759$	−16.9706	−0.615992
$760$	−4.48528	−0.162698
$761$	−5.75736	−0.208704	−0.104352	−	0.994540i	$-0.533277\pi$
$761$	−5.75736	−0.208704	−0.104352	+	0.994540i	$0.533277\pi$
$762$	−6.82843	−0.247368
$763$	11.3137	0.409584
$764$	−45.2132	−1.63576
$765$	4.00000	0.144620
$766$	−5.51472	−0.199255
$767$	−44.6274	−1.61140
$768$	3.97056	0.143275
$769$	−7.31371	−0.263739	−0.131870	−	0.991267i	$-0.542098\pi$
$769$	−7.31371	−0.263739	−0.131870	+	0.991267i	$0.542098\pi$
$770$	4.00000	0.144150
$771$	7.51472	0.270636
$772$	6.68629	0.240645
$773$	38.6274	1.38933	0.694666	−	0.719333i	$-0.255552\pi$
$773$	38.6274	1.38933	0.694666	+	0.719333i	$0.255552\pi$
$774$	4.68629	0.168445
$775$	1.00000	0.0359211
$776$	−6.88730	−0.247240
$777$	−4.82843	−0.173219
$778$	2.87006	0.102897
$779$	−13.6569	−0.489308
$780$	−9.89949	−0.354459
$781$	−28.0000	−1.00192
$782$	9.94113	0.355494
$783$	−6.24264	−0.223094
$784$	13.9706	0.498949
$785$	−18.0000	−0.642448
$786$	−6.92893	−0.247147
$787$	9.17157	0.326931	0.163466	−	0.986549i	$-0.447733\pi$
$787$	9.17157	0.326931	0.163466	+	0.986549i	$0.447733\pi$
$788$	50.8284	1.81069
$789$	−1.65685	−0.0589856
$790$	−1.02944	−0.0366257
$791$	18.1421	0.645060
$792$	−4.48528	−0.159378
$793$	−35.1127	−1.24689
$794$	14.0833	0.499796
$795$	4.00000	0.141865
$796$	22.2010	0.786894
$797$	−1.17157	−0.0414992	−0.0207496	−	0.999785i	$-0.506605\pi$
$797$	−1.17157	−0.0414992	−0.0207496	+	0.999785i	$0.506605\pi$
$798$	4.00000	0.141598
$799$	−3.31371	−0.117231
$800$	4.41421	0.156066
$801$	−9.07107	−0.320510
$802$	−0.928932	−0.0328017
$803$	39.3137	1.38735
$804$	29.0711	1.02526
$805$	−20.4853	−0.722011
$806$	−2.24264	−0.0789936
$807$	1.07107	0.0377034
$808$	−1.31371	−0.0462161
$809$	−28.8701	−1.01502	−0.507509	−	0.861647i	$-0.669433\pi$
$809$	−28.8701	−1.01502	−0.507509	+	0.861647i	$0.669433\pi$
$810$	−0.414214	−0.0145540
$811$	41.4558	1.45571	0.727856	−	0.685730i	$-0.240517\pi$
$811$	41.4558	1.45571	0.727856	+	0.685730i	$0.240517\pi$
$812$	−38.9706	−1.36760
$813$	11.3137	0.396789
$814$	1.65685	0.0580727
$815$	−1.75736	−0.0615576
$816$	−12.0000	−0.420084
$817$	−32.0000	−1.11954
$818$	−4.05887	−0.141915
$819$	18.4853	0.645928
$820$	8.82843	0.308302
$821$	−37.0711	−1.29379	−0.646895	−	0.762579i	$-0.723933\pi$
$821$	−37.0711	−1.29379	−0.646895	+	0.762579i	$0.723933\pi$
$822$	−1.17157	−0.0408633
$823$	25.9411	0.904251	0.452125	−	0.891954i	$-0.350666\pi$
$823$	25.9411	0.904251	0.452125	+	0.891954i	$0.350666\pi$
$824$	−9.89949	−0.344865
$825$	2.82843	0.0984732
$826$	−11.6569	−0.405594
$827$	18.9706	0.659671	0.329836	−	0.944038i	$-0.393007\pi$
$827$	18.9706	0.659671	0.329836	+	0.944038i	$0.393007\pi$
$828$	10.9706	0.381253
$829$	32.8284	1.14018	0.570089	−	0.821583i	$-0.306909\pi$
$829$	32.8284	1.14018	0.570089	+	0.821583i	$0.306909\pi$
$830$	4.82843	0.167597
$831$	8.92893	0.309741
$832$	22.5858	0.783021
$833$	−18.6274	−0.645402
$834$	4.00000	0.138509
$835$	−24.9706	−0.864142
$836$	14.6274	0.505900
$837$	1.00000	0.0345651
$838$	5.35534	0.184997
$839$	−19.5563	−0.675160	−0.337580	−	0.941297i	$-0.609608\pi$
$839$	−19.5563	−0.675160	−0.337580	+	0.941297i	$0.609608\pi$
$840$	−5.41421	−0.186808
$841$	9.97056	0.343813
$842$	5.94113	0.204745
$843$	27.4558	0.945630
$844$	27.1127	0.933258
$845$	−16.3137	−0.561209
$846$	0.343146	0.0117976
$847$	10.2426	0.351941
$848$	−12.0000	−0.412082
$849$	−2.72792	−0.0936220
$850$	−1.65685	−0.0568296
$851$	−8.48528	−0.290872
$852$	18.1005	0.620113
$853$	35.4558	1.21398	0.606992	−	0.794708i	$-0.292376\pi$
$853$	35.4558	1.21398	0.606992	+	0.794708i	$0.292376\pi$
$854$	−9.17157	−0.313845
$855$	2.82843	0.0967302
$856$	14.0000	0.478510
$857$	−17.0294	−0.581714	−0.290857	−	0.956766i	$-0.593940\pi$
$857$	−17.0294	−0.581714	−0.290857	+	0.956766i	$0.593940\pi$
$858$	−6.34315	−0.216551
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	20.6863	0.705397
$861$	−16.4853	−0.561817
$862$	−16.1838	−0.551221
$863$	11.3137	0.385123	0.192562	−	0.981285i	$-0.438320\pi$
$863$	11.3137	0.385123	0.192562	+	0.981285i	$0.438320\pi$
$864$	4.41421	0.150175
$865$	14.0000	0.476014
$866$	−4.30152	−0.146171
$867$	−1.00000	−0.0339618
$868$	6.24264	0.211889
$869$	7.02944	0.238457
$870$	2.58579	0.0876664
$871$	86.0833	2.91682
$872$	5.25483	0.177951
$873$	4.34315	0.146993
$874$	7.02944	0.237774
$875$	3.41421	0.115421
$876$	−25.4142	−0.858667
$877$	15.4558	0.521907	0.260953	−	0.965351i	$-0.415963\pi$
$877$	15.4558	0.521907	0.260953	+	0.965351i	$0.415963\pi$
$878$	0	0
$879$	−22.8284	−0.769984
$880$	−8.48528	−0.286039
$881$	−37.7574	−1.27208	−0.636039	−	0.771657i	$-0.719428\pi$
$881$	−37.7574	−1.27208	−0.636039	+	0.771657i	$0.719428\pi$
$882$	1.92893	0.0649505
$883$	−29.4558	−0.991268	−0.495634	−	0.868531i	$-0.665064\pi$
$883$	−29.4558	−0.991268	−0.495634	+	0.868531i	$0.665064\pi$
$884$	−39.5980	−1.33182
$885$	−8.24264	−0.277073
$886$	−0.142136	−0.00477514
$887$	−13.0294	−0.437486	−0.218743	−	0.975783i	$-0.570196\pi$
$887$	−13.0294	−0.437486	−0.218743	+	0.975783i	$0.570196\pi$
$888$	−2.24264	−0.0752581
$889$	56.2843	1.88771
$890$	3.75736	0.125947
$891$	2.82843	0.0947559
$892$	31.3970	1.05125
$893$	−2.34315	−0.0784104
$894$	2.68629	0.0898430
$895$	−0.686292	−0.0229402
$896$	36.0416	1.20407
$897$	32.4853	1.08465
$898$	15.4731	0.516344
$899$	−6.24264	−0.208204
$900$	−1.82843	−0.0609476
$901$	16.0000	0.533037
$902$	5.65685	0.188353
$903$	−38.6274	−1.28544
$904$	8.42641	0.280258
$905$	−17.7990	−0.591658
$906$	−6.97056	−0.231581
$907$	17.0711	0.566836	0.283418	−	0.958997i	$-0.408532\pi$
$907$	17.0711	0.566836	0.283418	+	0.958997i	$0.408532\pi$
$908$	28.6274	0.950034
$909$	0.828427	0.0274772
$910$	−7.65685	−0.253822
$911$	49.4558	1.63855	0.819273	−	0.573404i	$-0.194378\pi$
$911$	49.4558	1.63855	0.819273	+	0.573404i	$0.194378\pi$
$912$	−8.48528	−0.280976
$913$	−32.9706	−1.09117
$914$	13.3553	0.441755
$915$	−6.48528	−0.214397
$916$	12.7452	0.421112
$917$	57.1127	1.88603
$918$	−1.65685	−0.0546843
$919$	44.7696	1.47681	0.738406	−	0.674357i	$-0.235579\pi$
$919$	44.7696	1.47681	0.738406	+	0.674357i	$0.235579\pi$
$920$	−9.51472	−0.313691
$921$	−6.92893	−0.228316
$922$	2.18377	0.0719185
$923$	53.5980	1.76420
$924$	17.6569	0.580868
$925$	1.41421	0.0464991
$926$	−5.25483	−0.172685
$927$	6.24264	0.205035
$928$	−27.5563	−0.904581
$929$	−40.1838	−1.31839	−0.659193	−	0.751974i	$-0.729102\pi$
$929$	−40.1838	−1.31839	−0.659193	+	0.751974i	$0.729102\pi$
$930$	−0.414214	−0.0135826
$931$	−13.1716	−0.431681
$932$	22.8284	0.747770
$933$	−15.7574	−0.515873
$934$	−0.627417	−0.0205297
$935$	11.3137	0.369998
$936$	8.58579	0.280635
$937$	−27.9411	−0.912797	−0.456398	−	0.889776i	$-0.650861\pi$
$937$	−27.9411	−0.912797	−0.456398	+	0.889776i	$0.650861\pi$
$938$	22.4853	0.734170
$939$	12.7279	0.415360
$940$	1.51472	0.0494047
$941$	27.2132	0.887125	0.443563	−	0.896243i	$-0.353714\pi$
$941$	27.2132	0.887125	0.443563	+	0.896243i	$0.353714\pi$
$942$	7.45584	0.242925
$943$	−28.9706	−0.943411
$944$	24.7279	0.804825
$945$	3.41421	0.111064
$946$	13.2548	0.430952
$947$	9.31371	0.302655	0.151327	−	0.988484i	$-0.451645\pi$
$947$	9.31371	0.302655	0.151327	+	0.988484i	$0.451645\pi$
$948$	−4.54416	−0.147587
$949$	−75.2548	−2.44288
$950$	−1.17157	−0.0380108
$951$	18.8284	0.610554
$952$	−21.6569	−0.701903
$953$	18.1421	0.587681	0.293841	−	0.955854i	$-0.405066\pi$
$953$	18.1421	0.587681	0.293841	+	0.955854i	$0.405066\pi$
$954$	−1.65685	−0.0536426
$955$	−24.7279	−0.800177
$956$	46.9117	1.51723
$957$	−17.6569	−0.570765
$958$	11.3310	0.366086
$959$	9.65685	0.311836
$960$	4.17157	0.134637
$961$	1.00000	0.0322581
$962$	−3.17157	−0.102256
$963$	−8.82843	−0.284492
$964$	42.0000	1.35273
$965$	3.65685	0.117718
$966$	8.48528	0.273009
$967$	−46.4264	−1.49297	−0.746486	−	0.665401i	$-0.768261\pi$
$967$	−46.4264	−1.49297	−0.746486	+	0.665401i	$0.768261\pi$
$968$	4.75736	0.152907
$969$	11.3137	0.363449
$970$	−1.79899	−0.0577621
$971$	23.5563	0.755959	0.377980	−	0.925814i	$-0.376619\pi$
$971$	23.5563	0.755959	0.377980	+	0.925814i	$0.376619\pi$
$972$	−1.82843	−0.0586468
$973$	−32.9706	−1.05699
$974$	−10.5442	−0.337857
$975$	−5.41421	−0.173394
$976$	19.4558	0.622766
$977$	29.5980	0.946923	0.473462	−	0.880814i	$-0.343004\pi$
$977$	29.5980	0.946923	0.473462	+	0.880814i	$0.343004\pi$
$978$	0.727922	0.0232764
$979$	−25.6569	−0.819997
$980$	8.51472	0.271993
$981$	−3.31371	−0.105799
$982$	−4.08326	−0.130302
$983$	22.3431	0.712636	0.356318	−	0.934365i	$-0.384032\pi$
$983$	22.3431	0.712636	0.356318	+	0.934365i	$0.384032\pi$
$984$	−7.65685	−0.244092
$985$	27.7990	0.885749
$986$	10.3431	0.329393
$987$	−2.82843	−0.0900298
$988$	−28.0000	−0.890799
$989$	−67.8823	−2.15853
$990$	−1.17157	−0.0372350
$991$	32.9706	1.04734	0.523672	−	0.851920i	$-0.324562\pi$
$991$	32.9706	1.04734	0.523672	+	0.851920i	$0.324562\pi$
$992$	4.41421	0.140151
$993$	−5.51472	−0.175004
$994$	14.0000	0.444053
$995$	12.1421	0.384932
$996$	21.3137	0.675351
$997$	−44.1421	−1.39800	−0.698998	−	0.715124i	$-0.746370\pi$
$997$	−44.1421	−1.39800	−0.698998	+	0.715124i	$0.746370\pi$
$998$	18.0589	0.571644
$999$	1.41421	0.0447437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.c.1.2	✓	2
3.2	odd	2		1395.2.a.g.1.1		2
4.3	odd	2		7440.2.a.be.1.2		2
5.2	odd	4		2325.2.c.i.1024.3		4
5.3	odd	4		2325.2.c.i.1024.2		4
5.4	even	2		2325.2.a.n.1.1		2
15.14	odd	2		6975.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.c.1.2	✓	2	1.1	even	1	trivial
1395.2.a.g.1.1		2	3.2	odd	2
2325.2.a.n.1.1		2	5.4	even	2
2325.2.c.i.1024.2		4	5.3	odd	4
2325.2.c.i.1024.3		4	5.2	odd	4
6975.2.a.u.1.2		2	15.14	odd	2
7440.2.a.be.1.2		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)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} -3.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -1.00000 q^{5} +0.414214 q^{6} -3.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.414214 q^{10} +2.82843 q^{11} -1.82843 q^{12} -5.41421 q^{13} -1.41421 q^{14} -1.00000 q^{15} +3.00000 q^{16} -4.00000 q^{17} +0.414214 q^{18} -2.82843 q^{19} +1.82843 q^{20} -3.41421 q^{21} +1.17157 q^{22} -6.00000 q^{23} -1.58579 q^{24} +1.00000 q^{25} -2.24264 q^{26} +1.00000 q^{27} +6.24264 q^{28} -6.24264 q^{29} -0.414214 q^{30} +1.00000 q^{31} +4.41421 q^{32} +2.82843 q^{33} -1.65685 q^{34} +3.41421 q^{35} -1.82843 q^{36} +1.41421 q^{37} -1.17157 q^{38} -5.41421 q^{39} +1.58579 q^{40} +4.82843 q^{41} -1.41421 q^{42} +11.3137 q^{43} -5.17157 q^{44} -1.00000 q^{45} -2.48528 q^{46} +0.828427 q^{47} +3.00000 q^{48} +4.65685 q^{49} +0.414214 q^{50} -4.00000 q^{51} +9.89949 q^{52} -4.00000 q^{53} +0.414214 q^{54} -2.82843 q^{55} +5.41421 q^{56} -2.82843 q^{57} -2.58579 q^{58} +8.24264 q^{59} +1.82843 q^{60} +6.48528 q^{61} +0.414214 q^{62} -3.41421 q^{63} -4.17157 q^{64} +5.41421 q^{65} +1.17157 q^{66} -15.8995 q^{67} +7.31371 q^{68} -6.00000 q^{69} +1.41421 q^{70} -9.89949 q^{71} -1.58579 q^{72} +13.8995 q^{73} +0.585786 q^{74} +1.00000 q^{75} +5.17157 q^{76} -9.65685 q^{77} -2.24264 q^{78} +2.48528 q^{79} -3.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -11.6569 q^{83} +6.24264 q^{84} +4.00000 q^{85} +4.68629 q^{86} -6.24264 q^{87} -4.48528 q^{88} -9.07107 q^{89} -0.414214 q^{90} +18.4853 q^{91} +10.9706 q^{92} +1.00000 q^{93} +0.343146 q^{94} +2.82843 q^{95} +4.41421 q^{96} +4.34315 q^{97} +1.92893 q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 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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