LMFDB - Embedded newform 1785.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1785,2,Mod(1,1785)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1785, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1785.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1785 = 3 \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1785.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:	$14.2532967608$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
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			$2.51414$ of defining polynomial
Character	$\chi$	$=$	1785.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51414 q^{2} +1.00000 q^{3} +4.32088 q^{4} +1.00000 q^{5} -2.51414 q^{6} +1.00000 q^{7} -5.83502 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.51414 q^{2} +1.00000 q^{3} +4.32088 q^{4} +1.00000 q^{5} -2.51414 q^{6} +1.00000 q^{7} -5.83502 q^{8} +1.00000 q^{9} -2.51414 q^{10} -1.70739 q^{11} +4.32088 q^{12} -1.32088 q^{13} -2.51414 q^{14} +1.00000 q^{15} +6.02827 q^{16} -1.00000 q^{17} -2.51414 q^{18} +2.00000 q^{19} +4.32088 q^{20} +1.00000 q^{21} +4.29261 q^{22} -6.64177 q^{23} -5.83502 q^{24} +1.00000 q^{25} +3.32088 q^{26} +1.00000 q^{27} +4.32088 q^{28} +2.34916 q^{29} -2.51414 q^{30} +4.34916 q^{31} -3.48586 q^{32} -1.70739 q^{33} +2.51414 q^{34} +1.00000 q^{35} +4.32088 q^{36} +2.00000 q^{37} -5.02827 q^{38} -1.32088 q^{39} -5.83502 q^{40} +6.00000 q^{41} -2.51414 q^{42} +12.0565 q^{43} -7.37743 q^{44} +1.00000 q^{45} +16.6983 q^{46} -0.641769 q^{47} +6.02827 q^{48} +1.00000 q^{49} -2.51414 q^{50} -1.00000 q^{51} -5.70739 q^{52} +2.67912 q^{53} -2.51414 q^{54} -1.70739 q^{55} -5.83502 q^{56} +2.00000 q^{57} -5.90611 q^{58} +7.61350 q^{59} +4.32088 q^{60} +6.93438 q^{61} -10.9344 q^{62} +1.00000 q^{63} -3.29261 q^{64} -1.32088 q^{65} +4.29261 q^{66} +8.64177 q^{67} -4.32088 q^{68} -6.64177 q^{69} -2.51414 q^{70} +8.34916 q^{71} -5.83502 q^{72} +0.386505 q^{73} -5.02827 q^{74} +1.00000 q^{75} +8.64177 q^{76} -1.70739 q^{77} +3.32088 q^{78} +6.38650 q^{79} +6.02827 q^{80} +1.00000 q^{81} -15.0848 q^{82} -6.00000 q^{83} +4.32088 q^{84} -1.00000 q^{85} -30.3118 q^{86} +2.34916 q^{87} +9.96265 q^{88} -6.97173 q^{89} -2.51414 q^{90} -1.32088 q^{91} -28.6983 q^{92} +4.34916 q^{93} +1.61350 q^{94} +2.00000 q^{95} -3.48586 q^{96} -6.25526 q^{97} -2.51414 q^{98} -1.70739 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} + 5 q^{12} + 4 q^{13} - q^{14} + 3 q^{15} + 5 q^{16} - 3 q^{17} - q^{18} + 6 q^{19} + 5 q^{20} + 3 q^{21} + 18 q^{22} - 4 q^{23} - 3 q^{24} + 3 q^{25} + 2 q^{26} + 3 q^{27} + 5 q^{28} - 14 q^{29} - q^{30} - 8 q^{31} - 17 q^{32} + q^{34} + 3 q^{35} + 5 q^{36} + 6 q^{37} - 2 q^{38} + 4 q^{39} - 3 q^{40} + 18 q^{41} - q^{42} + 10 q^{43} + 12 q^{44} + 3 q^{45} + 8 q^{46} + 14 q^{47} + 5 q^{48} + 3 q^{49} - q^{50} - 3 q^{51} - 12 q^{52} + 16 q^{53} - q^{54} - 3 q^{56} + 6 q^{57} - 20 q^{58} + 20 q^{59} + 5 q^{60} + 10 q^{61} - 22 q^{62} + 3 q^{63} - 15 q^{64} + 4 q^{65} + 18 q^{66} + 10 q^{67} - 5 q^{68} - 4 q^{69} - q^{70} + 4 q^{71} - 3 q^{72} + 4 q^{73} - 2 q^{74} + 3 q^{75} + 10 q^{76} + 2 q^{78} + 22 q^{79} + 5 q^{80} + 3 q^{81} - 6 q^{82} - 18 q^{83} + 5 q^{84} - 3 q^{85} - 46 q^{86} - 14 q^{87} + 6 q^{88} - 34 q^{89} - q^{90} + 4 q^{91} - 44 q^{92} - 8 q^{93} + 2 q^{94} + 6 q^{95} - 17 q^{96} - q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - q^10 + 5 * q^12 + 4 * q^13 - q^14 + 3 * q^15 + 5 * q^16 - 3 * q^17 - q^18 + 6 * q^19 + 5 * q^20 + 3 * q^21 + 18 * q^22 - 4 * q^23 - 3 * q^24 + 3 * q^25 + 2 * q^26 + 3 * q^27 + 5 * q^28 - 14 * q^29 - q^30 - 8 * q^31 - 17 * q^32 + q^34 + 3 * q^35 + 5 * q^36 + 6 * q^37 - 2 * q^38 + 4 * q^39 - 3 * q^40 + 18 * q^41 - q^42 + 10 * q^43 + 12 * q^44 + 3 * q^45 + 8 * q^46 + 14 * q^47 + 5 * q^48 + 3 * q^49 - q^50 - 3 * q^51 - 12 * q^52 + 16 * q^53 - q^54 - 3 * q^56 + 6 * q^57 - 20 * q^58 + 20 * q^59 + 5 * q^60 + 10 * q^61 - 22 * q^62 + 3 * q^63 - 15 * q^64 + 4 * q^65 + 18 * q^66 + 10 * q^67 - 5 * q^68 - 4 * q^69 - q^70 + 4 * q^71 - 3 * q^72 + 4 * q^73 - 2 * q^74 + 3 * q^75 + 10 * q^76 + 2 * q^78 + 22 * q^79 + 5 * q^80 + 3 * q^81 - 6 * q^82 - 18 * q^83 + 5 * q^84 - 3 * q^85 - 46 * q^86 - 14 * q^87 + 6 * q^88 - 34 * q^89 - q^90 + 4 * q^91 - 44 * q^92 - 8 * q^93 + 2 * q^94 + 6 * q^95 - 17 * q^96 - 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 $

$2$	−2.51414	−1.77776	−0.888882	−	0.458137i	$-0.848517\pi$
$2$	−2.51414	−1.77776	−0.888882	+	0.458137i	$0.848517\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.32088	2.16044
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.51414	−1.02639
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−5.83502	−2.06299
$9$	1.00000	0.333333
$10$	−2.51414	−0.795040
$11$	−1.70739	−0.514797	−0.257399	−	0.966305i	$-0.582865\pi$
$11$	−1.70739	−0.514797	−0.257399	+	0.966305i	$0.582865\pi$
$12$	4.32088	1.24733
$13$	−1.32088	−0.366347	−0.183174	−	0.983081i	$-0.558637\pi$
$13$	−1.32088	−0.366347	−0.183174	+	0.983081i	$0.558637\pi$
$14$	−2.51414	−0.671931
$15$	1.00000	0.258199
$16$	6.02827	1.50707
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.51414	−0.592588
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	4.32088	0.966179
$21$	1.00000	0.218218
$22$	4.29261	0.915188
$23$	−6.64177	−1.38490	−0.692452	−	0.721464i	$-0.743470\pi$
$23$	−6.64177	−1.38490	−0.692452	+	0.721464i	$0.743470\pi$
$24$	−5.83502	−1.19107
$25$	1.00000	0.200000
$26$	3.32088	0.651279
$27$	1.00000	0.192450
$28$	4.32088	0.816570
$29$	2.34916	0.436228	0.218114	−	0.975923i	$-0.430010\pi$
$29$	2.34916	0.436228	0.218114	+	0.975923i	$0.430010\pi$
$30$	−2.51414	−0.459017
$31$	4.34916	0.781132	0.390566	−	0.920575i	$-0.372279\pi$
$31$	4.34916	0.781132	0.390566	+	0.920575i	$0.372279\pi$
$32$	−3.48586	−0.616219
$33$	−1.70739	−0.297218
$34$	2.51414	0.431171
$35$	1.00000	0.169031
$36$	4.32088	0.720147
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−5.02827	−0.815694
$39$	−1.32088	−0.211511
$40$	−5.83502	−0.922598
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.51414	−0.387940
$43$	12.0565	1.83861	0.919303	−	0.393550i	$-0.128753\pi$
$43$	12.0565	1.83861	0.919303	+	0.393550i	$0.128753\pi$
$44$	−7.37743	−1.11219
$45$	1.00000	0.149071
$46$	16.6983	2.46203
$47$	−0.641769	−0.0936116	−0.0468058	−	0.998904i	$-0.514904\pi$
$47$	−0.641769	−0.0936116	−0.0468058	+	0.998904i	$0.514904\pi$
$48$	6.02827	0.870106
$49$	1.00000	0.142857
$50$	−2.51414	−0.355553
$51$	−1.00000	−0.140028
$52$	−5.70739	−0.791472
$53$	2.67912	0.368005	0.184002	−	0.982926i	$-0.441095\pi$
$53$	2.67912	0.368005	0.184002	+	0.982926i	$0.441095\pi$
$54$	−2.51414	−0.342131
$55$	−1.70739	−0.230224
$56$	−5.83502	−0.779738
$57$	2.00000	0.264906
$58$	−5.90611	−0.775510
$59$	7.61350	0.991193	0.495596	−	0.868553i	$-0.334950\pi$
$59$	7.61350	0.991193	0.495596	+	0.868553i	$0.334950\pi$
$60$	4.32088	0.557824
$61$	6.93438	0.887856	0.443928	−	0.896062i	$-0.353585\pi$
$61$	6.93438	0.887856	0.443928	+	0.896062i	$0.353585\pi$
$62$	−10.9344	−1.38867
$63$	1.00000	0.125988
$64$	−3.29261	−0.411576
$65$	−1.32088	−0.163836
$66$	4.29261	0.528384
$67$	8.64177	1.05576	0.527880	−	0.849319i	$-0.322987\pi$
$67$	8.64177	1.05576	0.527880	+	0.849319i	$0.322987\pi$
$68$	−4.32088	−0.523984
$69$	−6.64177	−0.799575
$70$	−2.51414	−0.300497
$71$	8.34916	0.990863	0.495431	−	0.868647i	$-0.335010\pi$
$71$	8.34916	0.990863	0.495431	+	0.868647i	$0.335010\pi$
$72$	−5.83502	−0.687664
$73$	0.386505	0.0452370	0.0226185	−	0.999744i	$-0.492800\pi$
$73$	0.386505	0.0452370	0.0226185	+	0.999744i	$0.492800\pi$
$74$	−5.02827	−0.584525
$75$	1.00000	0.115470
$76$	8.64177	0.991279
$77$	−1.70739	−0.194575
$78$	3.32088	0.376016
$79$	6.38650	0.718538	0.359269	−	0.933234i	$-0.383026\pi$
$79$	6.38650	0.718538	0.359269	+	0.933234i	$0.383026\pi$
$80$	6.02827	0.673982
$81$	1.00000	0.111111
$82$	−15.0848	−1.66584
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	4.32088	0.471447
$85$	−1.00000	−0.108465
$86$	−30.3118	−3.26861
$87$	2.34916	0.251856
$88$	9.96265	1.06202
$89$	−6.97173	−0.739001	−0.369501	−	0.929230i	$-0.620471\pi$
$89$	−6.97173	−0.739001	−0.369501	+	0.929230i	$0.620471\pi$
$90$	−2.51414	−0.265013
$91$	−1.32088	−0.138466
$92$	−28.6983	−2.99201
$93$	4.34916	0.450987
$94$	1.61350	0.166419
$95$	2.00000	0.205196
$96$	−3.48586	−0.355774
$97$	−6.25526	−0.635126	−0.317563	−	0.948237i	$-0.602864\pi$
$97$	−6.25526	−0.635126	−0.317563	+	0.948237i	$0.602864\pi$
$98$	−2.51414	−0.253966
$99$	−1.70739	−0.171599
$100$	4.32088	0.432088
$101$	8.25526	0.821429	0.410715	−	0.911764i	$-0.365279\pi$
$101$	8.25526	0.821429	0.410715	+	0.911764i	$0.365279\pi$
$102$	2.51414	0.248937
$103$	−2.93438	−0.289133	−0.144567	−	0.989495i	$-0.546179\pi$
$103$	−2.93438	−0.289133	−0.144567	+	0.989495i	$0.546179\pi$
$104$	7.70739	0.755772
$105$	1.00000	0.0975900
$106$	−6.73566	−0.654225
$107$	−19.9253	−1.92625	−0.963126	−	0.269051i	$-0.913290\pi$
$107$	−19.9253	−1.92625	−0.963126	+	0.269051i	$0.913290\pi$
$108$	4.32088	0.415777
$109$	7.35823	0.704791	0.352395	−	0.935851i	$-0.385367\pi$
$109$	7.35823	0.704791	0.352395	+	0.935851i	$0.385367\pi$
$110$	4.29261	0.409284
$111$	2.00000	0.189832
$112$	6.02827	0.569618
$113$	−0.971726	−0.0914123	−0.0457062	−	0.998955i	$-0.514554\pi$
$113$	−0.971726	−0.0914123	−0.0457062	+	0.998955i	$0.514554\pi$
$114$	−5.02827	−0.470941
$115$	−6.64177	−0.619348
$116$	10.1504	0.942445
$117$	−1.32088	−0.122116
$118$	−19.1414	−1.76211
$119$	−1.00000	−0.0916698
$120$	−5.83502	−0.532662
$121$	−8.08482	−0.734984
$122$	−17.4340	−1.57840
$123$	6.00000	0.541002
$124$	18.7922	1.68759
$125$	1.00000	0.0894427
$126$	−2.51414	−0.223977
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	15.2498	1.34790
$129$	12.0565	1.06152
$130$	3.32088	0.291261
$131$	8.44305	0.737673	0.368836	−	0.929494i	$-0.379756\pi$
$131$	8.44305	0.737673	0.368836	+	0.929494i	$0.379756\pi$
$132$	−7.37743	−0.642123
$133$	2.00000	0.173422
$134$	−21.7266	−1.87689
$135$	1.00000	0.0860663
$136$	5.83502	0.500349
$137$	0.735663	0.0628520	0.0314260	−	0.999506i	$-0.489995\pi$
$137$	0.735663	0.0628520	0.0314260	+	0.999506i	$0.489995\pi$
$138$	16.6983	1.42146
$139$	1.12217	0.0951811	0.0475905	−	0.998867i	$-0.484846\pi$
$139$	1.12217	0.0951811	0.0475905	+	0.998867i	$0.484846\pi$
$140$	4.32088	0.365181
$141$	−0.641769	−0.0540467
$142$	−20.9909	−1.76152
$143$	2.25526	0.188595
$144$	6.02827	0.502356
$145$	2.34916	0.195087
$146$	−0.971726	−0.0804206
$147$	1.00000	0.0824786
$148$	8.64177	0.710349
$149$	0.641769	0.0525758	0.0262879	−	0.999654i	$-0.491631\pi$
$149$	0.641769	0.0525758	0.0262879	+	0.999654i	$0.491631\pi$
$150$	−2.51414	−0.205278
$151$	16.1131	1.31127	0.655633	−	0.755080i	$-0.272402\pi$
$151$	16.1131	1.31127	0.655633	+	0.755080i	$0.272402\pi$
$152$	−11.6700	−0.946565
$153$	−1.00000	−0.0808452
$154$	4.29261	0.345908
$155$	4.34916	0.349333
$156$	−5.70739	−0.456957
$157$	−23.3774	−1.86572	−0.932861	−	0.360236i	$-0.882696\pi$
$157$	−23.3774	−1.86572	−0.932861	+	0.360236i	$0.882696\pi$
$158$	−16.0565	−1.27739
$159$	2.67912	0.212468
$160$	−3.48586	−0.275582
$161$	−6.64177	−0.523445
$162$	−2.51414	−0.197529
$163$	6.05655	0.474385	0.237193	−	0.971463i	$-0.423773\pi$
$163$	6.05655	0.474385	0.237193	+	0.971463i	$0.423773\pi$
$164$	25.9253	2.02443
$165$	−1.70739	−0.132920
$166$	15.0848	1.17081
$167$	−9.08482	−0.703005	−0.351502	−	0.936187i	$-0.614329\pi$
$167$	−9.08482	−0.703005	−0.351502	+	0.936187i	$0.614329\pi$
$168$	−5.83502	−0.450182
$169$	−11.2553	−0.865790
$170$	2.51414	0.192826
$171$	2.00000	0.152944
$172$	52.0950	3.97220
$173$	24.9536	1.89719	0.948593	−	0.316499i	$-0.102507\pi$
$173$	24.9536	1.89719	0.948593	+	0.316499i	$0.102507\pi$
$174$	−5.90611	−0.447741
$175$	1.00000	0.0755929
$176$	−10.2926	−0.775835
$177$	7.61350	0.572265
$178$	17.5279	1.31377
$179$	22.8861	1.71059	0.855294	−	0.518143i	$-0.173377\pi$
$179$	22.8861	1.71059	0.855294	+	0.518143i	$0.173377\pi$
$180$	4.32088	0.322060
$181$	13.5761	1.00911	0.504554	−	0.863380i	$-0.331657\pi$
$181$	13.5761	1.00911	0.504554	+	0.863380i	$0.331657\pi$
$182$	3.32088	0.246160
$183$	6.93438	0.512604
$184$	38.7549	2.85705
$185$	2.00000	0.147043
$186$	−10.9344	−0.801748
$187$	1.70739	0.124857
$188$	−2.77301	−0.202243
$189$	1.00000	0.0727393
$190$	−5.02827	−0.364789
$191$	−2.77301	−0.200648	−0.100324	−	0.994955i	$-0.531988\pi$
$191$	−2.77301	−0.200648	−0.100324	+	0.994955i	$0.531988\pi$
$192$	−3.29261	−0.237624
$193$	−8.84049	−0.636352	−0.318176	−	0.948032i	$-0.603070\pi$
$193$	−8.84049	−0.636352	−0.318176	+	0.948032i	$0.603070\pi$
$194$	15.7266	1.12910
$195$	−1.32088	−0.0945905
$196$	4.32088	0.308635
$197$	−2.44305	−0.174060	−0.0870301	−	0.996206i	$-0.527738\pi$
$197$	−2.44305	−0.174060	−0.0870301	+	0.996206i	$0.527738\pi$
$198$	4.29261	0.305063
$199$	11.6508	0.825906	0.412953	−	0.910752i	$-0.364497\pi$
$199$	11.6508	0.825906	0.412953	+	0.910752i	$0.364497\pi$
$200$	−5.83502	−0.412598
$201$	8.64177	0.609543
$202$	−20.7549	−1.46031
$203$	2.34916	0.164879
$204$	−4.32088	−0.302522
$205$	6.00000	0.419058
$206$	7.37743	0.514010
$207$	−6.64177	−0.461635
$208$	−7.96265	−0.552111
$209$	−3.41478	−0.236205
$210$	−2.51414	−0.173492
$211$	−5.61350	−0.386449	−0.193224	−	0.981155i	$-0.561895\pi$
$211$	−5.61350	−0.386449	−0.193224	+	0.981155i	$0.561895\pi$
$212$	11.5761	0.795053
$213$	8.34916	0.572075
$214$	50.0950	3.42442
$215$	12.0565	0.822250
$216$	−5.83502	−0.397023
$217$	4.34916	0.295240
$218$	−18.4996	−1.25295
$219$	0.386505	0.0261176
$220$	−7.37743	−0.497386
$221$	1.32088	0.0888523
$222$	−5.02827	−0.337476
$223$	−12.9909	−0.869937	−0.434968	−	0.900446i	$-0.643240\pi$
$223$	−12.9909	−0.869937	−0.434968	+	0.900446i	$0.643240\pi$
$224$	−3.48586	−0.232909
$225$	1.00000	0.0666667
$226$	2.44305	0.162509
$227$	−10.0565	−0.667477	−0.333738	−	0.942666i	$-0.608310\pi$
$227$	−10.0565	−0.667477	−0.333738	+	0.942666i	$0.608310\pi$
$228$	8.64177	0.572315
$229$	−4.82956	−0.319146	−0.159573	−	0.987186i	$-0.551012\pi$
$229$	−4.82956	−0.319146	−0.159573	+	0.987186i	$0.551012\pi$
$230$	16.6983	1.10105
$231$	−1.70739	−0.112338
$232$	−13.7074	−0.899934
$233$	24.4996	1.60502	0.802511	−	0.596637i	$-0.203497\pi$
$233$	24.4996	1.60502	0.802511	+	0.596637i	$0.203497\pi$
$234$	3.32088	0.217093
$235$	−0.641769	−0.0418644
$236$	32.8970	2.14141
$237$	6.38650	0.414848
$238$	2.51414	0.162967
$239$	−13.9253	−0.900753	−0.450377	−	0.892839i	$-0.648710\pi$
$239$	−13.9253	−0.900753	−0.450377	+	0.892839i	$0.648710\pi$
$240$	6.02827	0.389123
$241$	−4.40571	−0.283796	−0.141898	−	0.989881i	$-0.545321\pi$
$241$	−4.40571	−0.283796	−0.141898	+	0.989881i	$0.545321\pi$
$242$	20.3263	1.30663
$243$	1.00000	0.0641500
$244$	29.9627	1.91816
$245$	1.00000	0.0638877
$246$	−15.0848	−0.961773
$247$	−2.64177	−0.168092
$248$	−25.3774	−1.61147
$249$	−6.00000	−0.380235
$250$	−2.51414	−0.159008
$251$	3.22699	0.203686	0.101843	−	0.994800i	$-0.467526\pi$
$251$	3.22699	0.203686	0.101843	+	0.994800i	$0.467526\pi$
$252$	4.32088	0.272190
$253$	11.3401	0.712945
$254$	−5.02827	−0.315502
$255$	−1.00000	−0.0626224
$256$	−31.7549	−1.98468
$257$	17.3401	1.08164	0.540822	−	0.841137i	$-0.318113\pi$
$257$	17.3401	1.08164	0.540822	+	0.841137i	$0.318113\pi$
$258$	−30.3118	−1.88713
$259$	2.00000	0.124274
$260$	−5.70739	−0.353957
$261$	2.34916	0.145409
$262$	−21.2270	−1.31141
$263$	22.2745	1.37350	0.686751	−	0.726893i	$-0.259036\pi$
$263$	22.2745	1.37350	0.686751	+	0.726893i	$0.259036\pi$
$264$	9.96265	0.613159
$265$	2.67912	0.164577
$266$	−5.02827	−0.308303
$267$	−6.97173	−0.426663
$268$	37.3401	2.28091
$269$	20.7549	1.26545	0.632723	−	0.774378i	$-0.281937\pi$
$269$	20.7549	1.26545	0.632723	+	0.774378i	$0.281937\pi$
$270$	−2.51414	−0.153006
$271$	5.22699	0.317517	0.158759	−	0.987317i	$-0.449251\pi$
$271$	5.22699	0.317517	0.158759	+	0.987317i	$0.449251\pi$
$272$	−6.02827	−0.365518
$273$	−1.32088	−0.0799436
$274$	−1.84956	−0.111736
$275$	−1.70739	−0.102959
$276$	−28.6983	−1.72744
$277$	12.0565	0.724408	0.362204	−	0.932099i	$-0.382024\pi$
$277$	12.0565	0.724408	0.362204	+	0.932099i	$0.382024\pi$
$278$	−2.82128	−0.169209
$279$	4.34916	0.260377
$280$	−5.83502	−0.348709
$281$	−11.3582	−0.677575	−0.338788	−	0.940863i	$-0.610017\pi$
$281$	−11.3582	−0.677575	−0.338788	+	0.940863i	$0.610017\pi$
$282$	1.61350	0.0960822
$283$	4.58522	0.272563	0.136282	−	0.990670i	$-0.456485\pi$
$283$	4.58522	0.272563	0.136282	+	0.990670i	$0.456485\pi$
$284$	36.0757	2.14070
$285$	2.00000	0.118470
$286$	−5.67004	−0.335277
$287$	6.00000	0.354169
$288$	−3.48586	−0.205406
$289$	1.00000	0.0588235
$290$	−5.90611	−0.346818
$291$	−6.25526	−0.366690
$292$	1.67004	0.0977319
$293$	17.3401	1.01302	0.506509	−	0.862234i	$-0.330936\pi$
$293$	17.3401	1.01302	0.506509	+	0.862234i	$0.330936\pi$
$294$	−2.51414	−0.146627
$295$	7.61350	0.443275
$296$	−11.6700	−0.678307
$297$	−1.70739	−0.0990728
$298$	−1.61350	−0.0934673
$299$	8.77301	0.507356
$300$	4.32088	0.249466
$301$	12.0565	0.694928
$302$	−40.5105	−2.33112
$303$	8.25526	0.474253
$304$	12.0565	0.691490
$305$	6.93438	0.397061
$306$	2.51414	0.143724
$307$	−8.48040	−0.484002	−0.242001	−	0.970276i	$-0.577804\pi$
$307$	−8.48040	−0.484002	−0.242001	+	0.970276i	$0.577804\pi$
$308$	−7.37743	−0.420368
$309$	−2.93438	−0.166931
$310$	−10.9344	−0.621031
$311$	−24.9536	−1.41499	−0.707494	−	0.706719i	$-0.750174\pi$
$311$	−24.9536	−1.41499	−0.707494	+	0.706719i	$0.750174\pi$
$312$	7.70739	0.436345
$313$	−21.8578	−1.23548	−0.617739	−	0.786383i	$-0.711951\pi$
$313$	−21.8578	−1.23548	−0.617739	+	0.786383i	$0.711951\pi$
$314$	58.7741	3.31681
$315$	1.00000	0.0563436
$316$	27.5953	1.55236
$317$	−34.5561	−1.94087	−0.970433	−	0.241369i	$-0.922403\pi$
$317$	−34.5561	−1.94087	−0.970433	+	0.241369i	$0.922403\pi$
$318$	−6.73566	−0.377717
$319$	−4.01093	−0.224569
$320$	−3.29261	−0.184063
$321$	−19.9253	−1.11212
$322$	16.6983	0.930561
$323$	−2.00000	−0.111283
$324$	4.32088	0.240049
$325$	−1.32088	−0.0732695
$326$	−15.2270	−0.843345
$327$	7.35823	0.406911
$328$	−35.0101	−1.93311
$329$	−0.641769	−0.0353819
$330$	4.29261	0.236300
$331$	−10.8296	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$331$	−10.8296	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$332$	−25.9253	−1.42284
$333$	2.00000	0.109599
$334$	22.8405	1.24978
$335$	8.64177	0.472150
$336$	6.02827	0.328869
$337$	−4.82956	−0.263083	−0.131541	−	0.991311i	$-0.541993\pi$
$337$	−4.82956	−0.263083	−0.131541	+	0.991311i	$0.541993\pi$
$338$	28.2973	1.53917
$339$	−0.971726	−0.0527769
$340$	−4.32088	−0.234333
$341$	−7.42571	−0.402125
$342$	−5.02827	−0.271898
$343$	1.00000	0.0539949
$344$	−70.3502	−3.79303
$345$	−6.64177	−0.357581
$346$	−62.7367	−3.37275
$347$	16.5105	0.886332	0.443166	−	0.896440i	$-0.353855\pi$
$347$	16.5105	0.886332	0.443166	+	0.896440i	$0.353855\pi$
$348$	10.1504	0.544121
$349$	−21.3401	−1.14231	−0.571154	−	0.820843i	$-0.693504\pi$
$349$	−21.3401	−1.14231	−0.571154	+	0.820843i	$0.693504\pi$
$350$	−2.51414	−0.134386
$351$	−1.32088	−0.0705036
$352$	5.95173	0.317228
$353$	19.2835	1.02636	0.513180	−	0.858281i	$-0.328467\pi$
$353$	19.2835	1.02636	0.513180	+	0.858281i	$0.328467\pi$
$354$	−19.1414	−1.01735
$355$	8.34916	0.443127
$356$	−30.1240	−1.59657
$357$	−1.00000	−0.0529256
$358$	−57.5388	−3.04102
$359$	10.5105	0.554724	0.277362	−	0.960765i	$-0.410540\pi$
$359$	10.5105	0.554724	0.277362	+	0.960765i	$0.410540\pi$
$360$	−5.83502	−0.307533
$361$	−15.0000	−0.789474
$362$	−34.1323	−1.79395
$363$	−8.08482	−0.424343
$364$	−5.70739	−0.299148
$365$	0.386505	0.0202306
$366$	−17.4340	−0.911289
$367$	22.5671	1.17799	0.588996	−	0.808136i	$-0.299523\pi$
$367$	22.5671	1.17799	0.588996	+	0.808136i	$0.299523\pi$
$368$	−40.0384	−2.08715
$369$	6.00000	0.312348
$370$	−5.02827	−0.261408
$371$	2.67912	0.139093
$372$	18.7922	0.974331
$373$	−31.2088	−1.61593	−0.807966	−	0.589229i	$-0.799432\pi$
$373$	−31.2088	−1.61593	−0.807966	+	0.589229i	$0.799432\pi$
$374$	−4.29261	−0.221966
$375$	1.00000	0.0516398
$376$	3.74474	0.193120
$377$	−3.10297	−0.159811
$378$	−2.51414	−0.129313
$379$	4.91518	0.252476	0.126238	−	0.992000i	$-0.459710\pi$
$379$	4.91518	0.252476	0.126238	+	0.992000i	$0.459710\pi$
$380$	8.64177	0.443313
$381$	2.00000	0.102463
$382$	6.97173	0.356705
$383$	−19.2835	−0.985343	−0.492671	−	0.870215i	$-0.663980\pi$
$383$	−19.2835	−0.985343	−0.492671	+	0.870215i	$0.663980\pi$
$384$	15.2498	0.778213
$385$	−1.70739	−0.0870166
$386$	22.2262	1.13128
$387$	12.0565	0.612869
$388$	−27.0283	−1.37215
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	3.32088	0.168160
$391$	6.64177	0.335889
$392$	−5.83502	−0.294713
$393$	8.44305	0.425896
$394$	6.14217	0.309438
$395$	6.38650	0.321340
$396$	−7.37743	−0.370730
$397$	0.386505	0.0193981	0.00969906	−	0.999953i	$-0.496913\pi$
$397$	0.386505	0.0193981	0.00969906	+	0.999953i	$0.496913\pi$
$398$	−29.2918	−1.46827
$399$	2.00000	0.100125
$400$	6.02827	0.301414
$401$	−8.80314	−0.439608	−0.219804	−	0.975544i	$-0.570542\pi$
$401$	−8.80314	−0.439608	−0.219804	+	0.975544i	$0.570542\pi$
$402$	−21.7266	−1.08362
$403$	−5.74474	−0.286166
$404$	35.6700	1.77465
$405$	1.00000	0.0496904
$406$	−5.90611	−0.293115
$407$	−3.41478	−0.169264
$408$	5.83502	0.288877
$409$	−2.07469	−0.102587	−0.0512935	−	0.998684i	$-0.516334\pi$
$409$	−2.07469	−0.102587	−0.0512935	+	0.998684i	$0.516334\pi$
$410$	−15.0848	−0.744986
$411$	0.735663	0.0362876
$412$	−12.6791	−0.624655
$413$	7.61350	0.374636
$414$	16.6983	0.820677
$415$	−6.00000	−0.294528
$416$	4.60442	0.225750
$417$	1.12217	0.0549528
$418$	8.58522	0.419917
$419$	−29.8397	−1.45776	−0.728882	−	0.684639i	$-0.759960\pi$
$419$	−29.8397	−1.45776	−0.728882	+	0.684639i	$0.759960\pi$
$420$	4.32088	0.210838
$421$	−29.1414	−1.42026	−0.710132	−	0.704069i	$-0.751365\pi$
$421$	−29.1414	−1.42026	−0.710132	+	0.704069i	$0.751365\pi$
$422$	14.1131	0.687015
$423$	−0.641769	−0.0312039
$424$	−15.6327	−0.759191
$425$	−1.00000	−0.0485071
$426$	−20.9909	−1.01701
$427$	6.93438	0.335578
$428$	−86.0950	−4.16156
$429$	2.25526	0.108885
$430$	−30.3118	−1.46177
$431$	6.21792	0.299507	0.149753	−	0.988723i	$-0.452152\pi$
$431$	6.21792	0.299507	0.149753	+	0.988723i	$0.452152\pi$
$432$	6.02827	0.290035
$433$	−2.60442	−0.125161	−0.0625803	−	0.998040i	$-0.519933\pi$
$433$	−2.60442	−0.125161	−0.0625803	+	0.998040i	$0.519933\pi$
$434$	−10.9344	−0.524867
$435$	2.34916	0.112634
$436$	31.7941	1.52266
$437$	−13.2835	−0.635438
$438$	−0.971726	−0.0464309
$439$	−13.8205	−0.659616	−0.329808	−	0.944048i	$-0.606984\pi$
$439$	−13.8205	−0.659616	−0.329808	+	0.944048i	$0.606984\pi$
$440$	9.96265	0.474951
$441$	1.00000	0.0476190
$442$	−3.32088	−0.157958
$443$	17.9517	0.852912	0.426456	−	0.904508i	$-0.359762\pi$
$443$	17.9517	0.852912	0.426456	+	0.904508i	$0.359762\pi$
$444$	8.64177	0.410120
$445$	−6.97173	−0.330492
$446$	32.6610	1.54654
$447$	0.641769	0.0303546
$448$	−3.29261	−0.155561
$449$	−39.8205	−1.87924	−0.939622	−	0.342213i	$-0.888824\pi$
$449$	−39.8205	−1.87924	−0.939622	+	0.342213i	$0.888824\pi$
$450$	−2.51414	−0.118518
$451$	−10.2443	−0.482387
$452$	−4.19872	−0.197491
$453$	16.1131	0.757059
$454$	25.2835	1.18662
$455$	−1.32088	−0.0619240
$456$	−11.6700	−0.546500
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	12.1422	0.567366
$459$	−1.00000	−0.0466760
$460$	−28.6983	−1.33807
$461$	1.80128	0.0838941	0.0419471	−	0.999120i	$-0.486644\pi$
$461$	1.80128	0.0838941	0.0419471	+	0.999120i	$0.486644\pi$
$462$	4.29261	0.199710
$463$	11.8688	0.551588	0.275794	−	0.961217i	$-0.411059\pi$
$463$	11.8688	0.551588	0.275794	+	0.961217i	$0.411059\pi$
$464$	14.1614	0.657425
$465$	4.34916	0.201687
$466$	−61.5953	−2.85335
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−5.70739	−0.263824
$469$	8.64177	0.399040
$470$	1.61350	0.0744250
$471$	−23.3774	−1.07718
$472$	−44.4249	−2.04482
$473$	−20.5852	−0.946509
$474$	−16.0565	−0.737502
$475$	2.00000	0.0917663
$476$	−4.32088	−0.198047
$477$	2.67912	0.122668
$478$	35.0101	1.60133
$479$	2.89703	0.132369	0.0661844	−	0.997807i	$-0.478917\pi$
$479$	2.89703	0.132369	0.0661844	+	0.997807i	$0.478917\pi$
$480$	−3.48586	−0.159107
$481$	−2.64177	−0.120454
$482$	11.0765	0.504523
$483$	−6.64177	−0.302211
$484$	−34.9336	−1.58789
$485$	−6.25526	−0.284037
$486$	−2.51414	−0.114044
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−40.4623	−1.83164
$489$	6.05655	0.273887
$490$	−2.51414	−0.113577
$491$	30.8114	1.39050	0.695250	−	0.718768i	$-0.255294\pi$
$491$	30.8114	1.39050	0.695250	+	0.718768i	$0.255294\pi$
$492$	25.9253	1.16880
$493$	−2.34916	−0.105801
$494$	6.64177	0.298827
$495$	−1.70739	−0.0767414
$496$	26.2179	1.17722
$497$	8.34916	0.374511
$498$	15.0848	0.675967
$499$	−33.6519	−1.50647	−0.753233	−	0.657754i	$-0.771507\pi$
$499$	−33.6519	−1.50647	−0.753233	+	0.657754i	$0.771507\pi$
$500$	4.32088	0.193236
$501$	−9.08482	−0.405880
$502$	−8.11310	−0.362105
$503$	36.6236	1.63297	0.816483	−	0.577369i	$-0.195921\pi$
$503$	36.6236	1.63297	0.816483	+	0.577369i	$0.195921\pi$
$504$	−5.83502	−0.259913
$505$	8.25526	0.367354
$506$	−28.5105	−1.26745
$507$	−11.2553	−0.499864
$508$	8.64177	0.383416
$509$	−41.6519	−1.84619	−0.923094	−	0.384575i	$-0.874348\pi$
$509$	−41.6519	−1.84619	−0.923094	+	0.384575i	$0.874348\pi$
$510$	2.51414	0.111328
$511$	0.386505	0.0170980
$512$	49.3365	2.18038
$513$	2.00000	0.0883022
$514$	−43.5953	−1.92291
$515$	−2.93438	−0.129304
$516$	52.0950	2.29335
$517$	1.09575	0.0481910
$518$	−5.02827	−0.220930
$519$	24.9536	1.09534
$520$	7.70739	0.337991
$521$	9.41478	0.412469	0.206234	−	0.978503i	$-0.433879\pi$
$521$	9.41478	0.412469	0.206234	+	0.978503i	$0.433879\pi$
$522$	−5.90611	−0.258503
$523$	−0.367304	−0.0160611	−0.00803053	−	0.999968i	$-0.502556\pi$
$523$	−0.367304	−0.0160611	−0.00803053	+	0.999968i	$0.502556\pi$
$524$	36.4815	1.59370
$525$	1.00000	0.0436436
$526$	−56.0011	−2.44176
$527$	−4.34916	−0.189452
$528$	−10.2926	−0.447928
$529$	21.1131	0.917961
$530$	−6.73566	−0.292579
$531$	7.61350	0.330398
$532$	8.64177	0.374668
$533$	−7.92531	−0.343283
$534$	17.5279	0.758505
$535$	−19.9253	−0.861446
$536$	−50.4249	−2.17802
$537$	22.8861	0.987608
$538$	−52.1806	−2.24966
$539$	−1.70739	−0.0735425
$540$	4.32088	0.185941
$541$	25.1523	1.08138	0.540691	−	0.841221i	$-0.318163\pi$
$541$	25.1523	1.08138	0.540691	+	0.841221i	$0.318163\pi$
$542$	−13.1414	−0.564470
$543$	13.5761	0.582608
$544$	3.48586	0.149455
$545$	7.35823	0.315192
$546$	3.32088	0.142121
$547$	14.4540	0.618008	0.309004	−	0.951061i	$-0.400004\pi$
$547$	14.4540	0.618008	0.309004	+	0.951061i	$0.400004\pi$
$548$	3.17872	0.135788
$549$	6.93438	0.295952
$550$	4.29261	0.183038
$551$	4.69832	0.200155
$552$	38.7549	1.64952
$553$	6.38650	0.271582
$554$	−30.3118	−1.28783
$555$	2.00000	0.0848953
$556$	4.84876	0.205633
$557$	−9.69646	−0.410852	−0.205426	−	0.978673i	$-0.565858\pi$
$557$	−9.69646	−0.410852	−0.205426	+	0.978673i	$0.565858\pi$
$558$	−10.9344	−0.462889
$559$	−15.9253	−0.673569
$560$	6.02827	0.254741
$561$	1.70739	0.0720860
$562$	28.5561	1.20457
$563$	−13.9253	−0.586882	−0.293441	−	0.955977i	$-0.594800\pi$
$563$	−13.9253	−0.586882	−0.293441	+	0.955977i	$0.594800\pi$
$564$	−2.77301	−0.116765
$565$	−0.971726	−0.0408808
$566$	−11.5279	−0.484553
$567$	1.00000	0.0419961
$568$	−48.7175	−2.04414
$569$	−14.1131	−0.591652	−0.295826	−	0.955242i	$-0.595595\pi$
$569$	−14.1131	−0.591652	−0.295826	+	0.955242i	$0.595595\pi$
$570$	−5.02827	−0.210611
$571$	13.6882	0.572833	0.286416	−	0.958105i	$-0.407536\pi$
$571$	13.6882	0.572833	0.286416	+	0.958105i	$0.407536\pi$
$572$	9.74474	0.407448
$573$	−2.77301	−0.115844
$574$	−15.0848	−0.629628
$575$	−6.64177	−0.276981
$576$	−3.29261	−0.137192
$577$	3.37743	0.140604	0.0703022	−	0.997526i	$-0.477604\pi$
$577$	3.37743	0.140604	0.0703022	+	0.997526i	$0.477604\pi$
$578$	−2.51414	−0.104574
$579$	−8.84049	−0.367398
$580$	10.1504	0.421474
$581$	−6.00000	−0.248922
$582$	15.7266	0.651888
$583$	−4.57429	−0.189448
$584$	−2.25526	−0.0933235
$585$	−1.32088	−0.0546119
$586$	−43.5953	−1.80091
$587$	−2.77301	−0.114454	−0.0572272	−	0.998361i	$-0.518226\pi$
$587$	−2.77301	−0.114454	−0.0572272	+	0.998361i	$0.518226\pi$
$588$	4.32088	0.178190
$589$	8.69832	0.358408
$590$	−19.1414	−0.788038
$591$	−2.44305	−0.100494
$592$	12.0565	0.495521
$593$	39.5844	1.62554	0.812769	−	0.582587i	$-0.197959\pi$
$593$	39.5844	1.62554	0.812769	+	0.582587i	$0.197959\pi$
$594$	4.29261	0.176128
$595$	−1.00000	−0.0409960
$596$	2.77301	0.113587
$597$	11.6508	0.476837
$598$	−22.0565	−0.901959
$599$	−13.3017	−0.543492	−0.271746	−	0.962369i	$-0.587601\pi$
$599$	−13.3017	−0.543492	−0.271746	+	0.962369i	$0.587601\pi$
$600$	−5.83502	−0.238214
$601$	−15.1222	−0.616846	−0.308423	−	0.951249i	$-0.599801\pi$
$601$	−15.1222	−0.616846	−0.308423	+	0.951249i	$0.599801\pi$
$602$	−30.3118	−1.23542
$603$	8.64177	0.351920
$604$	69.6228	2.83291
$605$	−8.08482	−0.328695
$606$	−20.7549	−0.843109
$607$	−35.2654	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$607$	−35.2654	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$608$	−6.97173	−0.282741
$609$	2.34916	0.0951927
$610$	−17.4340	−0.705881
$611$	0.847703	0.0342944
$612$	−4.32088	−0.174661
$613$	−24.9427	−1.00742	−0.503712	−	0.863872i	$-0.668033\pi$
$613$	−24.9427	−1.00742	−0.503712	+	0.863872i	$0.668033\pi$
$614$	21.3209	0.860441
$615$	6.00000	0.241943
$616$	9.96265	0.401407
$617$	32.4249	1.30538	0.652689	−	0.757626i	$-0.273641\pi$
$617$	32.4249	1.30538	0.652689	+	0.757626i	$0.273641\pi$
$618$	7.37743	0.296764
$619$	17.4449	0.701170	0.350585	−	0.936531i	$-0.385983\pi$
$619$	17.4449	0.701170	0.350585	+	0.936531i	$0.385983\pi$
$620$	18.7922	0.754713
$621$	−6.64177	−0.266525
$622$	62.7367	2.51551
$623$	−6.97173	−0.279316
$624$	−7.96265	−0.318761
$625$	1.00000	0.0400000
$626$	54.9536	2.19639
$627$	−3.41478	−0.136373
$628$	−101.011	−4.03079
$629$	−2.00000	−0.0797452
$630$	−2.51414	−0.100166
$631$	36.3865	1.44852	0.724262	−	0.689525i	$-0.242181\pi$
$631$	36.3865	1.44852	0.724262	+	0.689525i	$0.242181\pi$
$632$	−37.2654	−1.48234
$633$	−5.61350	−0.223116
$634$	86.8789	3.45040
$635$	2.00000	0.0793676
$636$	11.5761	0.459024
$637$	−1.32088	−0.0523353
$638$	10.0840	0.399230
$639$	8.34916	0.330288
$640$	15.2498	0.602801
$641$	30.3876	1.20024	0.600118	−	0.799911i	$-0.295120\pi$
$641$	30.3876	1.20024	0.600118	+	0.799911i	$0.295120\pi$
$642$	50.0950	1.97709
$643$	−12.1131	−0.477694	−0.238847	−	0.971057i	$-0.576769\pi$
$643$	−12.1131	−0.477694	−0.238847	+	0.971057i	$0.576769\pi$
$644$	−28.6983	−1.13087
$645$	12.0565	0.474726
$646$	5.02827	0.197835
$647$	34.2262	1.34557	0.672785	−	0.739838i	$-0.265098\pi$
$647$	34.2262	1.34557	0.672785	+	0.739838i	$0.265098\pi$
$648$	−5.83502	−0.229221
$649$	−12.9992	−0.510263
$650$	3.32088	0.130256
$651$	4.34916	0.170457
$652$	26.1696	1.02488
$653$	−0.971726	−0.0380266	−0.0190133	−	0.999819i	$-0.506052\pi$
$653$	−0.971726	−0.0380266	−0.0190133	+	0.999819i	$0.506052\pi$
$654$	−18.4996	−0.723392
$655$	8.44305	0.329897
$656$	36.1696	1.41219
$657$	0.386505	0.0150790
$658$	1.61350	0.0629006
$659$	17.1523	0.668159	0.334079	−	0.942545i	$-0.391575\pi$
$659$	17.1523	0.668159	0.334079	+	0.942545i	$0.391575\pi$
$660$	−7.37743	−0.287166
$661$	−23.0957	−0.898321	−0.449160	−	0.893451i	$-0.648277\pi$
$661$	−23.0957	−0.898321	−0.449160	+	0.893451i	$0.648277\pi$
$662$	27.2270	1.05821
$663$	1.32088	0.0512989
$664$	35.0101	1.35866
$665$	2.00000	0.0775567
$666$	−5.02827	−0.194842
$667$	−15.6026	−0.604134
$668$	−39.2545	−1.51880
$669$	−12.9909	−0.502258
$670$	−21.7266	−0.839371
$671$	−11.8397	−0.457066
$672$	−3.48586	−0.134470
$673$	31.5097	1.21461	0.607305	−	0.794468i	$-0.292250\pi$
$673$	31.5097	1.21461	0.607305	+	0.794468i	$0.292250\pi$
$674$	12.1422	0.467699
$675$	1.00000	0.0384900
$676$	−48.6327	−1.87049
$677$	−34.3502	−1.32019	−0.660093	−	0.751184i	$-0.729483\pi$
$677$	−34.3502	−1.32019	−0.660093	+	0.751184i	$0.729483\pi$
$678$	2.44305	0.0938249
$679$	−6.25526	−0.240055
$680$	5.83502	0.223763
$681$	−10.0565	−0.385368
$682$	18.6692	0.714882
$683$	8.11310	0.310439	0.155219	−	0.987880i	$-0.450392\pi$
$683$	8.11310	0.310439	0.155219	+	0.987880i	$0.450392\pi$
$684$	8.64177	0.330426
$685$	0.735663	0.0281082
$686$	−2.51414	−0.0959902
$687$	−4.82956	−0.184259
$688$	72.6802	2.77091
$689$	−3.53880	−0.134818
$690$	16.6983	0.635694
$691$	5.00907	0.190554	0.0952771	−	0.995451i	$-0.469626\pi$
$691$	5.00907	0.190554	0.0952771	+	0.995451i	$0.469626\pi$
$692$	107.822	4.09876
$693$	−1.70739	−0.0648584
$694$	−41.5097	−1.57569
$695$	1.12217	0.0425663
$696$	−13.7074	−0.519577
$697$	−6.00000	−0.227266
$698$	53.6519	2.03075
$699$	24.4996	0.926660
$700$	4.32088	0.163314
$701$	37.4532	1.41459	0.707294	−	0.706920i	$-0.249916\pi$
$701$	37.4532	1.41459	0.707294	+	0.706920i	$0.249916\pi$
$702$	3.32088	0.125339
$703$	4.00000	0.150863
$704$	5.62177	0.211878
$705$	−0.641769	−0.0241704
$706$	−48.4815	−1.82462
$707$	8.25526	0.310471
$708$	32.8970	1.23635
$709$	−18.1131	−0.680252	−0.340126	−	0.940380i	$-0.610470\pi$
$709$	−18.1131	−0.680252	−0.340126	+	0.940380i	$0.610470\pi$
$710$	−20.9909	−0.787775
$711$	6.38650	0.239513
$712$	40.6802	1.52455
$713$	−28.8861	−1.08179
$714$	2.51414	0.0940892
$715$	2.25526	0.0843421
$716$	98.8882	3.69563
$717$	−13.9253	−0.520050
$718$	−26.4249	−0.986169
$719$	6.97173	0.260002	0.130001	−	0.991514i	$-0.458502\pi$
$719$	6.97173	0.260002	0.130001	+	0.991514i	$0.458502\pi$
$720$	6.02827	0.224661
$721$	−2.93438	−0.109282
$722$	37.7121	1.40350
$723$	−4.40571	−0.163850
$724$	58.6610	2.18012
$725$	2.34916	0.0872456
$726$	20.3263	0.754382
$727$	−39.3702	−1.46016	−0.730080	−	0.683361i	$-0.760517\pi$
$727$	−39.3702	−1.46016	−0.730080	+	0.683361i	$0.760517\pi$
$728$	7.70739	0.285655
$729$	1.00000	0.0370370
$730$	−0.971726	−0.0359652
$731$	−12.0565	−0.445928
$732$	29.9627	1.10745
$733$	−12.0373	−0.444610	−0.222305	−	0.974977i	$-0.571358\pi$
$733$	−12.0373	−0.444610	−0.222305	+	0.974977i	$0.571358\pi$
$734$	−56.7367	−2.09419
$735$	1.00000	0.0368856
$736$	23.1523	0.853405
$737$	−14.7549	−0.543502
$738$	−15.0848	−0.555280
$739$	−21.6700	−0.797145	−0.398573	−	0.917137i	$-0.630494\pi$
$739$	−21.6700	−0.797145	−0.398573	+	0.917137i	$0.630494\pi$
$740$	8.64177	0.317678
$741$	−2.64177	−0.0970478
$742$	−6.73566	−0.247274
$743$	21.3966	0.784966	0.392483	−	0.919759i	$-0.371616\pi$
$743$	21.3966	0.784966	0.392483	+	0.919759i	$0.371616\pi$
$744$	−25.3774	−0.930382
$745$	0.641769	0.0235126
$746$	78.4633	2.87275
$747$	−6.00000	−0.219529
$748$	7.37743	0.269746
$749$	−19.9253	−0.728055
$750$	−2.51414	−0.0918033
$751$	−13.7266	−0.500890	−0.250445	−	0.968131i	$-0.580577\pi$
$751$	−13.7266	−0.500890	−0.250445	+	0.968131i	$0.580577\pi$
$752$	−3.86876	−0.141079
$753$	3.22699	0.117598
$754$	7.80128	0.284106
$755$	16.1131	0.586416
$756$	4.32088	0.157149
$757$	−33.5279	−1.21859	−0.609296	−	0.792943i	$-0.708548\pi$
$757$	−33.5279	−1.21859	−0.609296	+	0.792943i	$0.708548\pi$
$758$	−12.3574	−0.448842
$759$	11.3401	0.411619
$760$	−11.6700	−0.423317
$761$	24.8296	0.900071	0.450035	−	0.893011i	$-0.351411\pi$
$761$	24.8296	0.900071	0.450035	+	0.893011i	$0.351411\pi$
$762$	−5.02827	−0.182155
$763$	7.35823	0.266386
$764$	−11.9819	−0.433488
$765$	−1.00000	−0.0361551
$766$	48.4815	1.75171
$767$	−10.0565	−0.363121
$768$	−31.7549	−1.14585
$769$	−17.2654	−0.622606	−0.311303	−	0.950311i	$-0.600765\pi$
$769$	−17.2654	−0.622606	−0.311303	+	0.950311i	$0.600765\pi$
$770$	4.29261	0.154695
$771$	17.3401	0.624488
$772$	−38.1987	−1.37480
$773$	12.6418	0.454693	0.227346	−	0.973814i	$-0.426995\pi$
$773$	12.6418	0.454693	0.227346	+	0.973814i	$0.426995\pi$
$774$	−30.3118	−1.08954
$775$	4.34916	0.156226
$776$	36.4996	1.31026
$777$	2.00000	0.0717496
$778$	15.0848	0.540817
$779$	12.0000	0.429945
$780$	−5.70739	−0.204357
$781$	−14.2553	−0.510093
$782$	−16.6983	−0.597131
$783$	2.34916	0.0839521
$784$	6.02827	0.215295
$785$	−23.3774	−0.834376
$786$	−21.2270	−0.757142
$787$	1.35823	0.0484157	0.0242079	−	0.999707i	$-0.492294\pi$
$787$	1.35823	0.0484157	0.0242079	+	0.999707i	$0.492294\pi$
$788$	−10.5561	−0.376047
$789$	22.2745	0.792992
$790$	−16.0565	−0.571266
$791$	−0.971726	−0.0345506
$792$	9.96265	0.354007
$793$	−9.15951	−0.325264
$794$	−0.971726	−0.0344853
$795$	2.67912	0.0950184
$796$	50.3419	1.78432
$797$	−11.1704	−0.395677	−0.197839	−	0.980235i	$-0.563392\pi$
$797$	−11.1704	−0.395677	−0.197839	+	0.980235i	$0.563392\pi$
$798$	−5.02827	−0.177999
$799$	0.641769	0.0227042
$800$	−3.48586	−0.123244
$801$	−6.97173	−0.246334
$802$	22.1323	0.781519
$803$	−0.659914	−0.0232879
$804$	37.3401	1.31688
$805$	−6.64177	−0.234092
$806$	14.4431	0.508735
$807$	20.7549	0.730606
$808$	−48.1696	−1.69460
$809$	−32.8031	−1.15330	−0.576648	−	0.816992i	$-0.695640\pi$
$809$	−32.8031	−1.15330	−0.576648	+	0.816992i	$0.695640\pi$
$810$	−2.51414	−0.0883378
$811$	28.3492	0.995474	0.497737	−	0.867328i	$-0.334165\pi$
$811$	28.3492	0.995474	0.497737	+	0.867328i	$0.334165\pi$
$812$	10.1504	0.356211
$813$	5.22699	0.183319
$814$	8.58522	0.300912
$815$	6.05655	0.212152
$816$	−6.02827	−0.211032
$817$	24.1131	0.843610
$818$	5.21606	0.182375
$819$	−1.32088	−0.0461554
$820$	25.9253	0.905351
$821$	−27.8205	−0.970942	−0.485471	−	0.874253i	$-0.661352\pi$
$821$	−27.8205	−0.970942	−0.485471	+	0.874253i	$0.661352\pi$
$822$	−1.84956	−0.0645107
$823$	19.4039	0.676376	0.338188	−	0.941079i	$-0.390186\pi$
$823$	19.4039	0.676376	0.338188	+	0.941079i	$0.390186\pi$
$824$	17.1222	0.596479
$825$	−1.70739	−0.0594437
$826$	−19.1414	−0.666013
$827$	−39.8506	−1.38574	−0.692871	−	0.721062i	$-0.743654\pi$
$827$	−39.8506	−1.38574	−0.692871	+	0.721062i	$0.743654\pi$
$828$	−28.6983	−0.997335
$829$	55.3219	1.92141	0.960705	−	0.277571i	$-0.0895293\pi$
$829$	55.3219	1.92141	0.960705	+	0.277571i	$0.0895293\pi$
$830$	15.0848	0.523602
$831$	12.0565	0.418237
$832$	4.34916	0.150780
$833$	−1.00000	−0.0346479
$834$	−2.82128	−0.0976931
$835$	−9.08482	−0.314393
$836$	−14.7549	−0.510308
$837$	4.34916	0.150329
$838$	75.0211	2.59156
$839$	−36.4815	−1.25948	−0.629740	−	0.776806i	$-0.716839\pi$
$839$	−36.4815	−1.25948	−0.629740	+	0.776806i	$0.716839\pi$
$840$	−5.83502	−0.201327
$841$	−23.4815	−0.809705
$842$	73.2654	2.52489
$843$	−11.3582	−0.391198
$844$	−24.2553	−0.834901
$845$	−11.2553	−0.387193
$846$	1.61350	0.0554731
$847$	−8.08482	−0.277798
$848$	16.1504	0.554608
$849$	4.58522	0.157364
$850$	2.51414	0.0862342
$851$	−13.2835	−0.455354
$852$	36.0757	1.23593
$853$	52.8970	1.81116	0.905580	−	0.424176i	$-0.139436\pi$
$853$	52.8970	1.81116	0.905580	+	0.424176i	$0.139436\pi$
$854$	−17.4340	−0.596579
$855$	2.00000	0.0683986
$856$	116.265	3.97384
$857$	−10.0109	−0.341967	−0.170983	−	0.985274i	$-0.554694\pi$
$857$	−10.0109	−0.341967	−0.170983	+	0.985274i	$0.554694\pi$
$858$	−5.67004	−0.193572
$859$	−0.567076	−0.0193484	−0.00967419	−	0.999953i	$-0.503079\pi$
$859$	−0.567076	−0.0193484	−0.00967419	+	0.999953i	$0.503079\pi$
$860$	52.0950	1.77642
$861$	6.00000	0.204479
$862$	−15.6327	−0.532452
$863$	32.8031	1.11663	0.558316	−	0.829628i	$-0.311448\pi$
$863$	32.8031	1.11663	0.558316	+	0.829628i	$0.311448\pi$
$864$	−3.48586	−0.118591
$865$	24.9536	0.848447
$866$	6.54787	0.222506
$867$	1.00000	0.0339618
$868$	18.7922	0.637849
$869$	−10.9043	−0.369901
$870$	−5.90611	−0.200236
$871$	−11.4148	−0.386775
$872$	−42.9354	−1.45398
$873$	−6.25526	−0.211709
$874$	33.3966	1.12966
$875$	1.00000	0.0338062
$876$	1.67004	0.0564255
$877$	−52.1696	−1.76164	−0.880822	−	0.473448i	$-0.843009\pi$
$877$	−52.1696	−1.76164	−0.880822	+	0.473448i	$0.843009\pi$
$878$	34.7466	1.17264
$879$	17.3401	0.584867
$880$	−10.2926	−0.346964
$881$	−40.6802	−1.37055	−0.685275	−	0.728284i	$-0.740318\pi$
$881$	−40.6802	−1.37055	−0.685275	+	0.728284i	$0.740318\pi$
$882$	−2.51414	−0.0846554
$883$	−42.1131	−1.41722	−0.708609	−	0.705601i	$-0.750677\pi$
$883$	−42.1131	−1.41722	−0.708609	+	0.705601i	$0.750677\pi$
$884$	5.70739	0.191960
$885$	7.61350	0.255925
$886$	−45.1331	−1.51628
$887$	54.1696	1.81884	0.909419	−	0.415880i	$-0.136527\pi$
$887$	54.1696	1.81884	0.909419	+	0.415880i	$0.136527\pi$
$888$	−11.6700	−0.391621
$889$	2.00000	0.0670778
$890$	17.5279	0.587536
$891$	−1.70739	−0.0571997
$892$	−56.1323	−1.87945
$893$	−1.28354	−0.0429520
$894$	−1.61350	−0.0539633
$895$	22.8861	0.764998
$896$	15.2498	0.509460
$897$	8.77301	0.292922
$898$	100.114	3.34085
$899$	10.2169	0.340751
$900$	4.32088	0.144029
$901$	−2.67912	−0.0892543
$902$	25.7557	0.857570
$903$	12.0565	0.401217
$904$	5.67004	0.188583
$905$	13.5761	0.451286
$906$	−40.5105	−1.34587
$907$	36.8223	1.22267	0.611333	−	0.791374i	$-0.290634\pi$
$907$	36.8223	1.22267	0.611333	+	0.791374i	$0.290634\pi$
$908$	−43.4532	−1.44204
$909$	8.25526	0.273810
$910$	3.32088	0.110086
$911$	−1.70739	−0.0565683	−0.0282842	−	0.999600i	$-0.509004\pi$
$911$	−1.70739	−0.0565683	−0.0282842	+	0.999600i	$0.509004\pi$
$912$	12.0565	0.399232
$913$	10.2443	0.339038
$914$	−65.3676	−2.16217
$915$	6.93438	0.229244
$916$	−20.8680	−0.689497
$917$	8.44305	0.278814
$918$	2.51414	0.0829789
$919$	23.8506	0.786759	0.393380	−	0.919376i	$-0.371306\pi$
$919$	23.8506	0.786759	0.393380	+	0.919376i	$0.371306\pi$
$920$	38.7549	1.27771
$921$	−8.48040	−0.279439
$922$	−4.52867	−0.149144
$923$	−11.0283	−0.363000
$924$	−7.37743	−0.242700
$925$	2.00000	0.0657596
$926$	−29.8397	−0.980593
$927$	−2.93438	−0.0963777
$928$	−8.18884	−0.268812
$929$	−8.75486	−0.287238	−0.143619	−	0.989633i	$-0.545874\pi$
$929$	−8.75486	−0.287238	−0.143619	+	0.989633i	$0.545874\pi$
$930$	−10.9344	−0.358552
$931$	2.00000	0.0655474
$932$	105.860	3.46756
$933$	−24.9536	−0.816944
$934$	45.2545	1.48077
$935$	1.70739	0.0558376
$936$	7.70739	0.251924
$937$	−16.5479	−0.540596	−0.270298	−	0.962777i	$-0.587122\pi$
$937$	−16.5479	−0.540596	−0.270298	+	0.962777i	$0.587122\pi$
$938$	−21.7266	−0.709398
$939$	−21.8578	−0.713303
$940$	−2.77301	−0.0904456
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	58.7741	1.91496
$943$	−39.8506	−1.29771
$944$	45.8962	1.49380
$945$	1.00000	0.0325300
$946$	51.7541	1.68267
$947$	−7.92531	−0.257538	−0.128769	−	0.991675i	$-0.541103\pi$
$947$	−7.92531	−0.257538	−0.128769	+	0.991675i	$0.541103\pi$
$948$	27.5953	0.896255
$949$	−0.510528	−0.0165724
$950$	−5.02827	−0.163139
$951$	−34.5561	−1.12056
$952$	5.83502	0.189114
$953$	2.20699	0.0714914	0.0357457	−	0.999361i	$-0.488619\pi$
$953$	2.20699	0.0714914	0.0357457	+	0.999361i	$0.488619\pi$
$954$	−6.73566	−0.218075
$955$	−2.77301	−0.0897325
$956$	−60.1696	−1.94603
$957$	−4.01093	−0.129655
$958$	−7.28354	−0.235320
$959$	0.735663	0.0237558
$960$	−3.29261	−0.106269
$961$	−12.0848	−0.389833
$962$	6.64177	0.214139
$963$	−19.9253	−0.642084
$964$	−19.0365	−0.613126
$965$	−8.84049	−0.284585
$966$	16.6983	0.537260
$967$	−33.9637	−1.09220	−0.546100	−	0.837720i	$-0.683888\pi$
$967$	−33.9637	−1.09220	−0.546100	+	0.837720i	$0.683888\pi$
$968$	47.1751	1.51627
$969$	−2.00000	−0.0642493
$970$	15.7266	0.504950
$971$	−57.8962	−1.85798	−0.928989	−	0.370107i	$-0.879321\pi$
$971$	−57.8962	−1.85798	−0.928989	+	0.370107i	$0.879321\pi$
$972$	4.32088	0.138592
$973$	1.12217	0.0359751
$974$	40.2262	1.28893
$975$	−1.32088	−0.0423022
$976$	41.8023	1.33806
$977$	−52.9619	−1.69440	−0.847200	−	0.531274i	$-0.821713\pi$
$977$	−52.9619	−1.69440	−0.847200	+	0.531274i	$0.821713\pi$
$978$	−15.2270	−0.486905
$979$	11.9035	0.380436
$980$	4.32088	0.138026
$981$	7.35823	0.234930
$982$	−77.4641	−2.47198
$983$	−38.8789	−1.24004	−0.620022	−	0.784584i	$-0.712876\pi$
$983$	−38.8789	−1.24004	−0.620022	+	0.784584i	$0.712876\pi$
$984$	−35.0101	−1.11608
$985$	−2.44305	−0.0778421
$986$	5.90611	0.188089
$987$	−0.641769	−0.0204277
$988$	−11.4148	−0.363152
$989$	−80.0768	−2.54629
$990$	4.29261	0.136428
$991$	−22.4996	−0.714723	−0.357362	−	0.933966i	$-0.616324\pi$
$991$	−22.4996	−0.714723	−0.357362	+	0.933966i	$0.616324\pi$
$992$	−15.1606	−0.481349
$993$	−10.8296	−0.343666
$994$	−20.9909	−0.665792
$995$	11.6508	0.369357
$996$	−25.9253	−0.821475
$997$	18.5561	0.587679	0.293840	−	0.955855i	$-0.405067\pi$
$997$	18.5561	0.587679	0.293840	+	0.955855i	$0.405067\pi$
$998$	84.6055	2.67814
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1785.2.a.x.1.1	✓	3
3.2	odd	2		5355.2.a.bd.1.3		3
5.4	even	2		8925.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.a.x.1.1	✓	3	1.1	even	1	trivial
5355.2.a.bd.1.3		3	3.2	odd	2
8925.2.a.bp.1.3		3	5.4	even	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:	\( N \)	\(=\)	\( 1785 = 3 \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1785.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51414 q^{2} +1.00000 q^{3} +4.32088 q^{4} +1.00000 q^{5} -2.51414 q^{6} +1.00000 q^{7} -5.83502 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.51414 q^{2} +1.00000 q^{3} +4.32088 q^{4} +1.00000 q^{5} -2.51414 q^{6} +1.00000 q^{7} -5.83502 q^{8} +1.00000 q^{9} -2.51414 q^{10} -1.70739 q^{11} +4.32088 q^{12} -1.32088 q^{13} -2.51414 q^{14} +1.00000 q^{15} +6.02827 q^{16} -1.00000 q^{17} -2.51414 q^{18} +2.00000 q^{19} +4.32088 q^{20} +1.00000 q^{21} +4.29261 q^{22} -6.64177 q^{23} -5.83502 q^{24} +1.00000 q^{25} +3.32088 q^{26} +1.00000 q^{27} +4.32088 q^{28} +2.34916 q^{29} -2.51414 q^{30} +4.34916 q^{31} -3.48586 q^{32} -1.70739 q^{33} +2.51414 q^{34} +1.00000 q^{35} +4.32088 q^{36} +2.00000 q^{37} -5.02827 q^{38} -1.32088 q^{39} -5.83502 q^{40} +6.00000 q^{41} -2.51414 q^{42} +12.0565 q^{43} -7.37743 q^{44} +1.00000 q^{45} +16.6983 q^{46} -0.641769 q^{47} +6.02827 q^{48} +1.00000 q^{49} -2.51414 q^{50} -1.00000 q^{51} -5.70739 q^{52} +2.67912 q^{53} -2.51414 q^{54} -1.70739 q^{55} -5.83502 q^{56} +2.00000 q^{57} -5.90611 q^{58} +7.61350 q^{59} +4.32088 q^{60} +6.93438 q^{61} -10.9344 q^{62} +1.00000 q^{63} -3.29261 q^{64} -1.32088 q^{65} +4.29261 q^{66} +8.64177 q^{67} -4.32088 q^{68} -6.64177 q^{69} -2.51414 q^{70} +8.34916 q^{71} -5.83502 q^{72} +0.386505 q^{73} -5.02827 q^{74} +1.00000 q^{75} +8.64177 q^{76} -1.70739 q^{77} +3.32088 q^{78} +6.38650 q^{79} +6.02827 q^{80} +1.00000 q^{81} -15.0848 q^{82} -6.00000 q^{83} +4.32088 q^{84} -1.00000 q^{85} -30.3118 q^{86} +2.34916 q^{87} +9.96265 q^{88} -6.97173 q^{89} -2.51414 q^{90} -1.32088 q^{91} -28.6983 q^{92} +4.34916 q^{93} +1.61350 q^{94} +2.00000 q^{95} -3.48586 q^{96} -6.25526 q^{97} -2.51414 q^{98} -1.70739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} + 5 q^{12} + 4 q^{13} - q^{14} + 3 q^{15} + 5 q^{16} - 3 q^{17} - q^{18} + 6 q^{19} + 5 q^{20} + 3 q^{21} + 18 q^{22} - 4 q^{23} - 3 q^{24} + 3 q^{25} + 2 q^{26} + 3 q^{27} + 5 q^{28} - 14 q^{29} - q^{30} - 8 q^{31} - 17 q^{32} + q^{34} + 3 q^{35} + 5 q^{36} + 6 q^{37} - 2 q^{38} + 4 q^{39} - 3 q^{40} + 18 q^{41} - q^{42} + 10 q^{43} + 12 q^{44} + 3 q^{45} + 8 q^{46} + 14 q^{47} + 5 q^{48} + 3 q^{49} - q^{50} - 3 q^{51} - 12 q^{52} + 16 q^{53} - q^{54} - 3 q^{56} + 6 q^{57} - 20 q^{58} + 20 q^{59} + 5 q^{60} + 10 q^{61} - 22 q^{62} + 3 q^{63} - 15 q^{64} + 4 q^{65} + 18 q^{66} + 10 q^{67} - 5 q^{68} - 4 q^{69} - q^{70} + 4 q^{71} - 3 q^{72} + 4 q^{73} - 2 q^{74} + 3 q^{75} + 10 q^{76} + 2 q^{78} + 22 q^{79} + 5 q^{80} + 3 q^{81} - 6 q^{82} - 18 q^{83} + 5 q^{84} - 3 q^{85} - 46 q^{86} - 14 q^{87} + 6 q^{88} - 34 q^{89} - q^{90} + 4 q^{91} - 44 q^{92} - 8 q^{93} + 2 q^{94} + 6 q^{95} - 17 q^{96} - q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - q^10 + 5 * q^12 + 4 * q^13 - q^14 + 3 * q^15 + 5 * q^16 - 3 * q^17 - q^18 + 6 * q^19 + 5 * q^20 + 3 * q^21 + 18 * q^22 - 4 * q^23 - 3 * q^24 + 3 * q^25 + 2 * q^26 + 3 * q^27 + 5 * q^28 - 14 * q^29 - q^30 - 8 * q^31 - 17 * q^32 + q^34 + 3 * q^35 + 5 * q^36 + 6 * q^37 - 2 * q^38 + 4 * q^39 - 3 * q^40 + 18 * q^41 - q^42 + 10 * q^43 + 12 * q^44 + 3 * q^45 + 8 * q^46 + 14 * q^47 + 5 * q^48 + 3 * q^49 - q^50 - 3 * q^51 - 12 * q^52 + 16 * q^53 - q^54 - 3 * q^56 + 6 * q^57 - 20 * q^58 + 20 * q^59 + 5 * q^60 + 10 * q^61 - 22 * q^62 + 3 * q^63 - 15 * q^64 + 4 * q^65 + 18 * q^66 + 10 * q^67 - 5 * q^68 - 4 * q^69 - q^70 + 4 * q^71 - 3 * q^72 + 4 * q^73 - 2 * q^74 + 3 * q^75 + 10 * q^76 + 2 * q^78 + 22 * q^79 + 5 * q^80 + 3 * q^81 - 6 * q^82 - 18 * q^83 + 5 * q^84 - 3 * q^85 - 46 * q^86 - 14 * q^87 + 6 * q^88 - 34 * q^89 - q^90 + 4 * q^91 - 44 * q^92 - 8 * q^93 + 2 * q^94 + 6 * q^95 - 17 * q^96 - q^98

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