LMFDB - Embedded newform 1690.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} -3.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} -3.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +3.00000 q^{11} -2.73205 q^{12} -3.00000 q^{14} +2.73205 q^{15} +1.00000 q^{16} +2.19615 q^{17} +4.46410 q^{18} +6.46410 q^{19} -1.00000 q^{20} +8.19615 q^{21} +3.00000 q^{22} -2.53590 q^{23} -2.73205 q^{24} +1.00000 q^{25} -4.00000 q^{27} -3.00000 q^{28} -9.46410 q^{29} +2.73205 q^{30} +1.26795 q^{31} +1.00000 q^{32} -8.19615 q^{33} +2.19615 q^{34} +3.00000 q^{35} +4.46410 q^{36} -11.1962 q^{37} +6.46410 q^{38} -1.00000 q^{40} -10.3923 q^{41} +8.19615 q^{42} -2.00000 q^{43} +3.00000 q^{44} -4.46410 q^{45} -2.53590 q^{46} -3.00000 q^{47} -2.73205 q^{48} +2.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -6.46410 q^{53} -4.00000 q^{54} -3.00000 q^{55} -3.00000 q^{56} -17.6603 q^{57} -9.46410 q^{58} +10.3923 q^{59} +2.73205 q^{60} -4.19615 q^{61} +1.26795 q^{62} -13.3923 q^{63} +1.00000 q^{64} -8.19615 q^{66} +2.19615 q^{68} +6.92820 q^{69} +3.00000 q^{70} -6.00000 q^{71} +4.46410 q^{72} +5.66025 q^{73} -11.1962 q^{74} -2.73205 q^{75} +6.46410 q^{76} -9.00000 q^{77} +6.19615 q^{79} -1.00000 q^{80} -2.46410 q^{81} -10.3923 q^{82} +2.19615 q^{83} +8.19615 q^{84} -2.19615 q^{85} -2.00000 q^{86} +25.8564 q^{87} +3.00000 q^{88} +17.1962 q^{89} -4.46410 q^{90} -2.53590 q^{92} -3.46410 q^{93} -3.00000 q^{94} -6.46410 q^{95} -2.73205 q^{96} -15.1244 q^{97} +2.00000 q^{98} +13.3923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 2 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 - 6 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 6 q^{11} - 2 q^{12} - 6 q^{14} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{20} + 6 q^{21} + 6 q^{22} - 12 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{27} - 6 q^{28} - 12 q^{29} + 2 q^{30} + 6 q^{31} + 2 q^{32} - 6 q^{33} - 6 q^{34} + 6 q^{35} + 2 q^{36} - 12 q^{37} + 6 q^{38} - 2 q^{40} + 6 q^{42} - 4 q^{43} + 6 q^{44} - 2 q^{45} - 12 q^{46} - 6 q^{47} - 2 q^{48} + 4 q^{49} + 2 q^{50} - 12 q^{51} - 6 q^{53} - 8 q^{54} - 6 q^{55} - 6 q^{56} - 18 q^{57} - 12 q^{58} + 2 q^{60} + 2 q^{61} + 6 q^{62} - 6 q^{63} + 2 q^{64} - 6 q^{66} - 6 q^{68} + 6 q^{70} - 12 q^{71} + 2 q^{72} - 6 q^{73} - 12 q^{74} - 2 q^{75} + 6 q^{76} - 18 q^{77} + 2 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{83} + 6 q^{84} + 6 q^{85} - 4 q^{86} + 24 q^{87} + 6 q^{88} + 24 q^{89} - 2 q^{90} - 12 q^{92} - 6 q^{94} - 6 q^{95} - 2 q^{96} - 6 q^{97} + 4 q^{98} + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 6 * q^11 - 2 * q^12 - 6 * q^14 + 2 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 + 6 * q^19 - 2 * q^20 + 6 * q^21 + 6 * q^22 - 12 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^27 - 6 * q^28 - 12 * q^29 + 2 * q^30 + 6 * q^31 + 2 * q^32 - 6 * q^33 - 6 * q^34 + 6 * q^35 + 2 * q^36 - 12 * q^37 + 6 * q^38 - 2 * q^40 + 6 * q^42 - 4 * q^43 + 6 * q^44 - 2 * q^45 - 12 * q^46 - 6 * q^47 - 2 * q^48 + 4 * q^49 + 2 * q^50 - 12 * q^51 - 6 * q^53 - 8 * q^54 - 6 * q^55 - 6 * q^56 - 18 * q^57 - 12 * q^58 + 2 * q^60 + 2 * q^61 + 6 * q^62 - 6 * q^63 + 2 * q^64 - 6 * q^66 - 6 * q^68 + 6 * q^70 - 12 * q^71 + 2 * q^72 - 6 * q^73 - 12 * q^74 - 2 * q^75 + 6 * q^76 - 18 * q^77 + 2 * q^79 - 2 * q^80 + 2 * q^81 - 6 * q^83 + 6 * q^84 + 6 * q^85 - 4 * q^86 + 24 * q^87 + 6 * q^88 + 24 * q^89 - 2 * q^90 - 12 * q^92 - 6 * q^94 - 6 * q^95 - 2 * q^96 - 6 * q^97 + 4 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.73205	−1.11536
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	4.46410	1.48803
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−2.73205	−0.788675
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	2.73205	0.705412
$16$	1.00000	0.250000
$17$	2.19615	0.532645	0.266323	−	0.963884i	$-0.414191\pi$
$17$	2.19615	0.532645	0.266323	+	0.963884i	$0.414191\pi$
$18$	4.46410	1.05220
$19$	6.46410	1.48297	0.741483	−	0.670971i	$-0.234123\pi$
$19$	6.46410	1.48297	0.741483	+	0.670971i	$0.234123\pi$
$20$	−1.00000	−0.223607
$21$	8.19615	1.78855
$22$	3.00000	0.639602
$23$	−2.53590	−0.528771	−0.264386	−	0.964417i	$-0.585169\pi$
$23$	−2.53590	−0.528771	−0.264386	+	0.964417i	$0.585169\pi$
$24$	−2.73205	−0.557678
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	−3.00000	−0.566947
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	2.73205	0.498802
$31$	1.26795	0.227730	0.113865	−	0.993496i	$-0.463677\pi$
$31$	1.26795	0.227730	0.113865	+	0.993496i	$0.463677\pi$
$32$	1.00000	0.176777
$33$	−8.19615	−1.42677
$34$	2.19615	0.376637
$35$	3.00000	0.507093
$36$	4.46410	0.744017
$37$	−11.1962	−1.84064	−0.920318	−	0.391171i	$-0.872070\pi$
$37$	−11.1962	−1.84064	−0.920318	+	0.391171i	$0.872070\pi$
$38$	6.46410	1.04862
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−10.3923	−1.62301	−0.811503	−	0.584349i	$-0.801350\pi$
$41$	−10.3923	−1.62301	−0.811503	+	0.584349i	$0.801350\pi$
$42$	8.19615	1.26469
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	3.00000	0.452267
$45$	−4.46410	−0.665469
$46$	−2.53590	−0.373898
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−2.73205	−0.394338
$49$	2.00000	0.285714
$50$	1.00000	0.141421
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−6.46410	−0.887913	−0.443956	−	0.896048i	$-0.646425\pi$
$53$	−6.46410	−0.887913	−0.443956	+	0.896048i	$0.646425\pi$
$54$	−4.00000	−0.544331
$55$	−3.00000	−0.404520
$56$	−3.00000	−0.400892
$57$	−17.6603	−2.33916
$58$	−9.46410	−1.24270
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	2.73205	0.352706
$61$	−4.19615	−0.537262	−0.268631	−	0.963243i	$-0.586571\pi$
$61$	−4.19615	−0.537262	−0.268631	+	0.963243i	$0.586571\pi$
$62$	1.26795	0.161030
$63$	−13.3923	−1.68727
$64$	1.00000	0.125000
$65$	0	0
$66$	−8.19615	−1.00888
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	2.19615	0.266323
$69$	6.92820	0.834058
$70$	3.00000	0.358569
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	4.46410	0.526099
$73$	5.66025	0.662483	0.331241	−	0.943546i	$-0.392533\pi$
$73$	5.66025	0.662483	0.331241	+	0.943546i	$0.392533\pi$
$74$	−11.1962	−1.30153
$75$	−2.73205	−0.315470
$76$	6.46410	0.741483
$77$	−9.00000	−1.02565
$78$	0	0
$79$	6.19615	0.697122	0.348561	−	0.937286i	$-0.386670\pi$
$79$	6.19615	0.697122	0.348561	+	0.937286i	$0.386670\pi$
$80$	−1.00000	−0.111803
$81$	−2.46410	−0.273789
$82$	−10.3923	−1.14764
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	8.19615	0.894274
$85$	−2.19615	−0.238206
$86$	−2.00000	−0.215666
$87$	25.8564	2.77210
$88$	3.00000	0.319801
$89$	17.1962	1.82279	0.911394	−	0.411534i	$-0.135007\pi$
$89$	17.1962	1.82279	0.911394	+	0.411534i	$0.135007\pi$
$90$	−4.46410	−0.470558
$91$	0	0
$92$	−2.53590	−0.264386
$93$	−3.46410	−0.359211
$94$	−3.00000	−0.309426
$95$	−6.46410	−0.663203
$96$	−2.73205	−0.278839
$97$	−15.1244	−1.53565	−0.767823	−	0.640662i	$-0.778660\pi$
$97$	−15.1244	−1.53565	−0.767823	+	0.640662i	$0.778660\pi$
$98$	2.00000	0.202031
$99$	13.3923	1.34598
$100$	1.00000	0.100000
$101$	−7.26795	−0.723188	−0.361594	−	0.932336i	$-0.617767\pi$
$101$	−7.26795	−0.723188	−0.361594	+	0.932336i	$0.617767\pi$
$102$	−6.00000	−0.594089
$103$	−1.19615	−0.117860	−0.0589302	−	0.998262i	$-0.518769\pi$
$103$	−1.19615	−0.117860	−0.0589302	+	0.998262i	$0.518769\pi$
$104$	0	0
$105$	−8.19615	−0.799863
$106$	−6.46410	−0.627849
$107$	0.339746	0.0328445	0.0164222	−	0.999865i	$-0.494772\pi$
$107$	0.339746	0.0328445	0.0164222	+	0.999865i	$0.494772\pi$
$108$	−4.00000	−0.384900
$109$	−15.4641	−1.48119	−0.740596	−	0.671950i	$-0.765457\pi$
$109$	−15.4641	−1.48119	−0.740596	+	0.671950i	$0.765457\pi$
$110$	−3.00000	−0.286039
$111$	30.5885	2.90333
$112$	−3.00000	−0.283473
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	−17.6603	−1.65403
$115$	2.53590	0.236474
$116$	−9.46410	−0.878720
$117$	0	0
$118$	10.3923	0.956689
$119$	−6.58846	−0.603963
$120$	2.73205	0.249401
$121$	−2.00000	−0.181818
$122$	−4.19615	−0.379902
$123$	28.3923	2.56005
$124$	1.26795	0.113865
$125$	−1.00000	−0.0894427
$126$	−13.3923	−1.19308
$127$	−21.1962	−1.88085	−0.940427	−	0.339995i	$-0.889575\pi$
$127$	−21.1962	−1.88085	−0.940427	+	0.339995i	$0.889575\pi$
$128$	1.00000	0.0883883
$129$	5.46410	0.481087
$130$	0	0
$131$	−18.1244	−1.58353	−0.791766	−	0.610824i	$-0.790838\pi$
$131$	−18.1244	−1.58353	−0.791766	+	0.610824i	$0.790838\pi$
$132$	−8.19615	−0.713384
$133$	−19.3923	−1.68153
$134$	0	0
$135$	4.00000	0.344265
$136$	2.19615	0.188319
$137$	8.19615	0.700245	0.350122	−	0.936704i	$-0.386140\pi$
$137$	8.19615	0.700245	0.350122	+	0.936704i	$0.386140\pi$
$138$	6.92820	0.589768
$139$	−9.19615	−0.780007	−0.390004	−	0.920813i	$-0.627526\pi$
$139$	−9.19615	−0.780007	−0.390004	+	0.920813i	$0.627526\pi$
$140$	3.00000	0.253546
$141$	8.19615	0.690241
$142$	−6.00000	−0.503509
$143$	0	0
$144$	4.46410	0.372008
$145$	9.46410	0.785951
$146$	5.66025	0.468446
$147$	−5.46410	−0.450672
$148$	−11.1962	−0.920318
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−2.73205	−0.223071
$151$	6.33975	0.515921	0.257961	−	0.966155i	$-0.416950\pi$
$151$	6.33975	0.515921	0.257961	+	0.966155i	$0.416950\pi$
$152$	6.46410	0.524308
$153$	9.80385	0.792594
$154$	−9.00000	−0.725241
$155$	−1.26795	−0.101844
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	6.19615	0.492939
$159$	17.6603	1.40055
$160$	−1.00000	−0.0790569
$161$	7.60770	0.599570
$162$	−2.46410	−0.193598
$163$	7.26795	0.569270	0.284635	−	0.958636i	$-0.408128\pi$
$163$	7.26795	0.569270	0.284635	+	0.958636i	$0.408128\pi$
$164$	−10.3923	−0.811503
$165$	8.19615	0.638070
$166$	2.19615	0.170454
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	8.19615	0.632347
$169$	0	0
$170$	−2.19615	−0.168437
$171$	28.8564	2.20670
$172$	−2.00000	−0.152499
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	25.8564	1.96017
$175$	−3.00000	−0.226779
$176$	3.00000	0.226134
$177$	−28.3923	−2.13410
$178$	17.1962	1.28891
$179$	2.53590	0.189542	0.0947710	−	0.995499i	$-0.469788\pi$
$179$	2.53590	0.189542	0.0947710	+	0.995499i	$0.469788\pi$
$180$	−4.46410	−0.332734
$181$	16.5885	1.23301	0.616505	−	0.787351i	$-0.288548\pi$
$181$	16.5885	1.23301	0.616505	+	0.787351i	$0.288548\pi$
$182$	0	0
$183$	11.4641	0.847451
$184$	−2.53590	−0.186949
$185$	11.1962	0.823157
$186$	−3.46410	−0.254000
$187$	6.58846	0.481796
$188$	−3.00000	−0.218797
$189$	12.0000	0.872872
$190$	−6.46410	−0.468955
$191$	−19.2679	−1.39418	−0.697090	−	0.716984i	$-0.745522\pi$
$191$	−19.2679	−1.39418	−0.697090	+	0.716984i	$0.745522\pi$
$192$	−2.73205	−0.197169
$193$	4.39230	0.316165	0.158083	−	0.987426i	$-0.449469\pi$
$193$	4.39230	0.316165	0.158083	+	0.987426i	$0.449469\pi$
$194$	−15.1244	−1.08587
$195$	0	0
$196$	2.00000	0.142857
$197$	9.58846	0.683149	0.341575	−	0.939855i	$-0.389040\pi$
$197$	9.58846	0.683149	0.341575	+	0.939855i	$0.389040\pi$
$198$	13.3923	0.951750
$199$	−14.3923	−1.02024	−0.510122	−	0.860102i	$-0.670400\pi$
$199$	−14.3923	−1.02024	−0.510122	+	0.860102i	$0.670400\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−7.26795	−0.511371
$203$	28.3923	1.99275
$204$	−6.00000	−0.420084
$205$	10.3923	0.725830
$206$	−1.19615	−0.0833399
$207$	−11.3205	−0.786830
$208$	0	0
$209$	19.3923	1.34139
$210$	−8.19615	−0.565588
$211$	13.5885	0.935468	0.467734	−	0.883869i	$-0.345071\pi$
$211$	13.5885	0.935468	0.467734	+	0.883869i	$0.345071\pi$
$212$	−6.46410	−0.443956
$213$	16.3923	1.12318
$214$	0.339746	0.0232246
$215$	2.00000	0.136399
$216$	−4.00000	−0.272166
$217$	−3.80385	−0.258222
$218$	−15.4641	−1.04736
$219$	−15.4641	−1.04497
$220$	−3.00000	−0.202260
$221$	0	0
$222$	30.5885	2.05296
$223$	−0.464102	−0.0310785	−0.0155393	−	0.999879i	$-0.504947\pi$
$223$	−0.464102	−0.0310785	−0.0155393	+	0.999879i	$0.504947\pi$
$224$	−3.00000	−0.200446
$225$	4.46410	0.297607
$226$	−6.92820	−0.460857
$227$	−4.39230	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$227$	−4.39230	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$228$	−17.6603	−1.16958
$229$	4.73205	0.312703	0.156351	−	0.987701i	$-0.450027\pi$
$229$	4.73205	0.312703	0.156351	+	0.987701i	$0.450027\pi$
$230$	2.53590	0.167212
$231$	24.5885	1.61780
$232$	−9.46410	−0.621349
$233$	−1.26795	−0.0830661	−0.0415331	−	0.999137i	$-0.513224\pi$
$233$	−1.26795	−0.0830661	−0.0415331	+	0.999137i	$0.513224\pi$
$234$	0	0
$235$	3.00000	0.195698
$236$	10.3923	0.676481
$237$	−16.9282	−1.09960
$238$	−6.58846	−0.427066
$239$	−8.19615	−0.530165	−0.265083	−	0.964226i	$-0.585399\pi$
$239$	−8.19615	−0.530165	−0.265083	+	0.964226i	$0.585399\pi$
$240$	2.73205	0.176353
$241$	−8.66025	−0.557856	−0.278928	−	0.960312i	$-0.589979\pi$
$241$	−8.66025	−0.557856	−0.278928	+	0.960312i	$0.589979\pi$
$242$	−2.00000	−0.128565
$243$	18.7321	1.20166
$244$	−4.19615	−0.268631
$245$	−2.00000	−0.127775
$246$	28.3923	1.81023
$247$	0	0
$248$	1.26795	0.0805149
$249$	−6.00000	−0.380235
$250$	−1.00000	−0.0632456
$251$	−6.80385	−0.429455	−0.214728	−	0.976674i	$-0.568886\pi$
$251$	−6.80385	−0.429455	−0.214728	+	0.976674i	$0.568886\pi$
$252$	−13.3923	−0.843636
$253$	−7.60770	−0.478292
$254$	−21.1962	−1.32996
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	−13.8564	−0.864339	−0.432169	−	0.901792i	$-0.642252\pi$
$257$	−13.8564	−0.864339	−0.432169	+	0.901792i	$0.642252\pi$
$258$	5.46410	0.340180
$259$	33.5885	2.08709
$260$	0	0
$261$	−42.2487	−2.61513
$262$	−18.1244	−1.11973
$263$	−27.5885	−1.70118	−0.850589	−	0.525832i	$-0.823754\pi$
$263$	−27.5885	−1.70118	−0.850589	+	0.525832i	$0.823754\pi$
$264$	−8.19615	−0.504438
$265$	6.46410	0.397087
$266$	−19.3923	−1.18902
$267$	−46.9808	−2.87518
$268$	0	0
$269$	−2.87564	−0.175331	−0.0876656	−	0.996150i	$-0.527941\pi$
$269$	−2.87564	−0.175331	−0.0876656	+	0.996150i	$0.527941\pi$
$270$	4.00000	0.243432
$271$	2.53590	0.154045	0.0770224	−	0.997029i	$-0.475459\pi$
$271$	2.53590	0.154045	0.0770224	+	0.997029i	$0.475459\pi$
$272$	2.19615	0.133161
$273$	0	0
$274$	8.19615	0.495148
$275$	3.00000	0.180907
$276$	6.92820	0.417029
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−9.19615	−0.551549
$279$	5.66025	0.338871
$280$	3.00000	0.179284
$281$	−10.3923	−0.619953	−0.309976	−	0.950744i	$-0.600321\pi$
$281$	−10.3923	−0.619953	−0.309976	+	0.950744i	$0.600321\pi$
$282$	8.19615	0.488074
$283$	30.3923	1.80663	0.903317	−	0.428973i	$-0.141124\pi$
$283$	30.3923	1.80663	0.903317	+	0.428973i	$0.141124\pi$
$284$	−6.00000	−0.356034
$285$	17.6603	1.04610
$286$	0	0
$287$	31.1769	1.84032
$288$	4.46410	0.263050
$289$	−12.1769	−0.716289
$290$	9.46410	0.555751
$291$	41.3205	2.42225
$292$	5.66025	0.331241
$293$	−0.803848	−0.0469613	−0.0234806	−	0.999724i	$-0.507475\pi$
$293$	−0.803848	−0.0469613	−0.0234806	+	0.999724i	$0.507475\pi$
$294$	−5.46410	−0.318673
$295$	−10.3923	−0.605063
$296$	−11.1962	−0.650763
$297$	−12.0000	−0.696311
$298$	6.00000	0.347571
$299$	0	0
$300$	−2.73205	−0.157735
$301$	6.00000	0.345834
$302$	6.33975	0.364811
$303$	19.8564	1.14072
$304$	6.46410	0.370742
$305$	4.19615	0.240271
$306$	9.80385	0.560449
$307$	20.5359	1.17205	0.586023	−	0.810295i	$-0.300693\pi$
$307$	20.5359	1.17205	0.586023	+	0.810295i	$0.300693\pi$
$308$	−9.00000	−0.512823
$309$	3.26795	0.185907
$310$	−1.26795	−0.0720147
$311$	−15.1244	−0.857624	−0.428812	−	0.903394i	$-0.641068\pi$
$311$	−15.1244	−0.857624	−0.428812	+	0.903394i	$0.641068\pi$
$312$	0	0
$313$	5.60770	0.316966	0.158483	−	0.987362i	$-0.449340\pi$
$313$	5.60770	0.316966	0.158483	+	0.987362i	$0.449340\pi$
$314$	−13.0000	−0.733632
$315$	13.3923	0.754571
$316$	6.19615	0.348561
$317$	−12.8038	−0.719136	−0.359568	−	0.933119i	$-0.617076\pi$
$317$	−12.8038	−0.719136	−0.359568	+	0.933119i	$0.617076\pi$
$318$	17.6603	0.990338
$319$	−28.3923	−1.58966
$320$	−1.00000	−0.0559017
$321$	−0.928203	−0.0518073
$322$	7.60770	0.423960
$323$	14.1962	0.789895
$324$	−2.46410	−0.136895
$325$	0	0
$326$	7.26795	0.402534
$327$	42.2487	2.33636
$328$	−10.3923	−0.573819
$329$	9.00000	0.496186
$330$	8.19615	0.451183
$331$	−0.928203	−0.0510187	−0.0255093	−	0.999675i	$-0.508121\pi$
$331$	−0.928203	−0.0510187	−0.0255093	+	0.999675i	$0.508121\pi$
$332$	2.19615	0.120530
$333$	−49.9808	−2.73893
$334$	3.00000	0.164153
$335$	0	0
$336$	8.19615	0.447137
$337$	4.19615	0.228579	0.114289	−	0.993447i	$-0.463541\pi$
$337$	4.19615	0.228579	0.114289	+	0.993447i	$0.463541\pi$
$338$	0	0
$339$	18.9282	1.02804
$340$	−2.19615	−0.119103
$341$	3.80385	0.205990
$342$	28.8564	1.56038
$343$	15.0000	0.809924
$344$	−2.00000	−0.107833
$345$	−6.92820	−0.373002
$346$	15.0000	0.806405
$347$	−25.2679	−1.35645	−0.678227	−	0.734852i	$-0.737252\pi$
$347$	−25.2679	−1.35645	−0.678227	+	0.734852i	$0.737252\pi$
$348$	25.8564	1.38605
$349$	34.0526	1.82279	0.911396	−	0.411531i	$-0.135006\pi$
$349$	34.0526	1.82279	0.911396	+	0.411531i	$0.135006\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	3.00000	0.159901
$353$	−20.1962	−1.07493	−0.537466	−	0.843285i	$-0.680618\pi$
$353$	−20.1962	−1.07493	−0.537466	+	0.843285i	$0.680618\pi$
$354$	−28.3923	−1.50903
$355$	6.00000	0.318447
$356$	17.1962	0.911394
$357$	18.0000	0.952661
$358$	2.53590	0.134026
$359$	−22.3923	−1.18182	−0.590910	−	0.806737i	$-0.701231\pi$
$359$	−22.3923	−1.18182	−0.590910	+	0.806737i	$0.701231\pi$
$360$	−4.46410	−0.235279
$361$	22.7846	1.19919
$362$	16.5885	0.871870
$363$	5.46410	0.286791
$364$	0	0
$365$	−5.66025	−0.296271
$366$	11.4641	0.599238
$367$	−26.3923	−1.37767	−0.688834	−	0.724920i	$-0.741877\pi$
$367$	−26.3923	−1.37767	−0.688834	+	0.724920i	$0.741877\pi$
$368$	−2.53590	−0.132193
$369$	−46.3923	−2.41509
$370$	11.1962	0.582060
$371$	19.3923	1.00680
$372$	−3.46410	−0.179605
$373$	20.3923	1.05587	0.527937	−	0.849284i	$-0.322966\pi$
$373$	20.3923	1.05587	0.527937	+	0.849284i	$0.322966\pi$
$374$	6.58846	0.340681
$375$	2.73205	0.141082
$376$	−3.00000	−0.154713
$377$	0	0
$378$	12.0000	0.617213
$379$	11.7846	0.605335	0.302667	−	0.953096i	$-0.402123\pi$
$379$	11.7846	0.605335	0.302667	+	0.953096i	$0.402123\pi$
$380$	−6.46410	−0.331601
$381$	57.9090	2.96677
$382$	−19.2679	−0.985834
$383$	1.60770	0.0821494	0.0410747	−	0.999156i	$-0.486922\pi$
$383$	1.60770	0.0821494	0.0410747	+	0.999156i	$0.486922\pi$
$384$	−2.73205	−0.139419
$385$	9.00000	0.458682
$386$	4.39230	0.223562
$387$	−8.92820	−0.453846
$388$	−15.1244	−0.767823
$389$	−19.2679	−0.976924	−0.488462	−	0.872585i	$-0.662442\pi$
$389$	−19.2679	−0.976924	−0.488462	+	0.872585i	$0.662442\pi$
$390$	0	0
$391$	−5.56922	−0.281648
$392$	2.00000	0.101015
$393$	49.5167	2.49779
$394$	9.58846	0.483059
$395$	−6.19615	−0.311762
$396$	13.3923	0.672989
$397$	0.803848	0.0403440	0.0201720	−	0.999797i	$-0.493579\pi$
$397$	0.803848	0.0403440	0.0201720	+	0.999797i	$0.493579\pi$
$398$	−14.3923	−0.721421
$399$	52.9808	2.65236
$400$	1.00000	0.0500000
$401$	5.19615	0.259483	0.129742	−	0.991548i	$-0.458585\pi$
$401$	5.19615	0.259483	0.129742	+	0.991548i	$0.458585\pi$
$402$	0	0
$403$	0	0
$404$	−7.26795	−0.361594
$405$	2.46410	0.122442
$406$	28.3923	1.40909
$407$	−33.5885	−1.66492
$408$	−6.00000	−0.297044
$409$	−19.7321	−0.975687	−0.487844	−	0.872931i	$-0.662216\pi$
$409$	−19.7321	−0.975687	−0.487844	+	0.872931i	$0.662216\pi$
$410$	10.3923	0.513239
$411$	−22.3923	−1.10453
$412$	−1.19615	−0.0589302
$413$	−31.1769	−1.53412
$414$	−11.3205	−0.556373
$415$	−2.19615	−0.107805
$416$	0	0
$417$	25.1244	1.23034
$418$	19.3923	0.948509
$419$	17.3205	0.846162	0.423081	−	0.906092i	$-0.360949\pi$
$419$	17.3205	0.846162	0.423081	+	0.906092i	$0.360949\pi$
$420$	−8.19615	−0.399931
$421$	27.1244	1.32196	0.660980	−	0.750403i	$-0.270141\pi$
$421$	27.1244	1.32196	0.660980	+	0.750403i	$0.270141\pi$
$422$	13.5885	0.661476
$423$	−13.3923	−0.651156
$424$	−6.46410	−0.313925
$425$	2.19615	0.106529
$426$	16.3923	0.794210
$427$	12.5885	0.609198
$428$	0.339746	0.0164222
$429$	0	0
$430$	2.00000	0.0964486
$431$	−2.19615	−0.105785	−0.0528925	−	0.998600i	$-0.516844\pi$
$431$	−2.19615	−0.105785	−0.0528925	+	0.998600i	$0.516844\pi$
$432$	−4.00000	−0.192450
$433$	12.3923	0.595536	0.297768	−	0.954638i	$-0.403758\pi$
$433$	12.3923	0.595536	0.297768	+	0.954638i	$0.403758\pi$
$434$	−3.80385	−0.182591
$435$	−25.8564	−1.23972
$436$	−15.4641	−0.740596
$437$	−16.3923	−0.784150
$438$	−15.4641	−0.738903
$439$	34.5885	1.65082	0.825408	−	0.564536i	$-0.190945\pi$
$439$	34.5885	1.65082	0.825408	+	0.564536i	$0.190945\pi$
$440$	−3.00000	−0.143019
$441$	8.92820	0.425153
$442$	0	0
$443$	−1.60770	−0.0763839	−0.0381920	−	0.999270i	$-0.512160\pi$
$443$	−1.60770	−0.0763839	−0.0381920	+	0.999270i	$0.512160\pi$
$444$	30.5885	1.45166
$445$	−17.1962	−0.815176
$446$	−0.464102	−0.0219758
$447$	−16.3923	−0.775329
$448$	−3.00000	−0.141737
$449$	15.5885	0.735665	0.367832	−	0.929892i	$-0.380100\pi$
$449$	15.5885	0.735665	0.367832	+	0.929892i	$0.380100\pi$
$450$	4.46410	0.210440
$451$	−31.1769	−1.46806
$452$	−6.92820	−0.325875
$453$	−17.3205	−0.813788
$454$	−4.39230	−0.206141
$455$	0	0
$456$	−17.6603	−0.827017
$457$	−35.6603	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$457$	−35.6603	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$458$	4.73205	0.221114
$459$	−8.78461	−0.410030
$460$	2.53590	0.118237
$461$	0.588457	0.0274072	0.0137036	−	0.999906i	$-0.495638\pi$
$461$	0.588457	0.0274072	0.0137036	+	0.999906i	$0.495638\pi$
$462$	24.5885	1.14396
$463$	0.928203	0.0431373	0.0215686	−	0.999767i	$-0.493134\pi$
$463$	0.928203	0.0431373	0.0215686	+	0.999767i	$0.493134\pi$
$464$	−9.46410	−0.439360
$465$	3.46410	0.160644
$466$	−1.26795	−0.0587366
$467$	10.1436	0.469390	0.234695	−	0.972069i	$-0.424591\pi$
$467$	10.1436	0.469390	0.234695	+	0.972069i	$0.424591\pi$
$468$	0	0
$469$	0	0
$470$	3.00000	0.138380
$471$	35.5167	1.63652
$472$	10.3923	0.478345
$473$	−6.00000	−0.275880
$474$	−16.9282	−0.777538
$475$	6.46410	0.296593
$476$	−6.58846	−0.301981
$477$	−28.8564	−1.32124
$478$	−8.19615	−0.374883
$479$	10.9808	0.501724	0.250862	−	0.968023i	$-0.419286\pi$
$479$	10.9808	0.501724	0.250862	+	0.968023i	$0.419286\pi$
$480$	2.73205	0.124700
$481$	0	0
$482$	−8.66025	−0.394464
$483$	−20.7846	−0.945732
$484$	−2.00000	−0.0909091
$485$	15.1244	0.686762
$486$	18.7321	0.849703
$487$	33.2487	1.50664	0.753321	−	0.657652i	$-0.228450\pi$
$487$	33.2487	1.50664	0.753321	+	0.657652i	$0.228450\pi$
$488$	−4.19615	−0.189951
$489$	−19.8564	−0.897938
$490$	−2.00000	−0.0903508
$491$	−3.33975	−0.150721	−0.0753603	−	0.997156i	$-0.524011\pi$
$491$	−3.33975	−0.150721	−0.0753603	+	0.997156i	$0.524011\pi$
$492$	28.3923	1.28002
$493$	−20.7846	−0.936092
$494$	0	0
$495$	−13.3923	−0.601939
$496$	1.26795	0.0569326
$497$	18.0000	0.807410
$498$	−6.00000	−0.268866
$499$	−1.85641	−0.0831042	−0.0415521	−	0.999136i	$-0.513230\pi$
$499$	−1.85641	−0.0831042	−0.0415521	+	0.999136i	$0.513230\pi$
$500$	−1.00000	−0.0447214
$501$	−8.19615	−0.366177
$502$	−6.80385	−0.303671
$503$	−9.33975	−0.416439	−0.208219	−	0.978082i	$-0.566767\pi$
$503$	−9.33975	−0.416439	−0.208219	+	0.978082i	$0.566767\pi$
$504$	−13.3923	−0.596541
$505$	7.26795	0.323419
$506$	−7.60770	−0.338203
$507$	0	0
$508$	−21.1962	−0.940427
$509$	1.60770	0.0712598	0.0356299	−	0.999365i	$-0.488656\pi$
$509$	1.60770	0.0712598	0.0356299	+	0.999365i	$0.488656\pi$
$510$	6.00000	0.265684
$511$	−16.9808	−0.751185
$512$	1.00000	0.0441942
$513$	−25.8564	−1.14159
$514$	−13.8564	−0.611180
$515$	1.19615	0.0527088
$516$	5.46410	0.240544
$517$	−9.00000	−0.395820
$518$	33.5885	1.47579
$519$	−40.9808	−1.79886
$520$	0	0
$521$	−0.464102	−0.0203327	−0.0101663	−	0.999948i	$-0.503236\pi$
$521$	−0.464102	−0.0203327	−0.0101663	+	0.999948i	$0.503236\pi$
$522$	−42.2487	−1.84918
$523$	18.3923	0.804239	0.402120	−	0.915587i	$-0.368274\pi$
$523$	18.3923	0.804239	0.402120	+	0.915587i	$0.368274\pi$
$524$	−18.1244	−0.791766
$525$	8.19615	0.357709
$526$	−27.5885	−1.20291
$527$	2.78461	0.121300
$528$	−8.19615	−0.356692
$529$	−16.5692	−0.720401
$530$	6.46410	0.280783
$531$	46.3923	2.01325
$532$	−19.3923	−0.840763
$533$	0	0
$534$	−46.9808	−2.03306
$535$	−0.339746	−0.0146885
$536$	0	0
$537$	−6.92820	−0.298974
$538$	−2.87564	−0.123978
$539$	6.00000	0.258438
$540$	4.00000	0.172133
$541$	16.0526	0.690153	0.345077	−	0.938574i	$-0.387853\pi$
$541$	16.0526	0.690153	0.345077	+	0.938574i	$0.387853\pi$
$542$	2.53590	0.108926
$543$	−45.3205	−1.94489
$544$	2.19615	0.0941593
$545$	15.4641	0.662409
$546$	0	0
$547$	34.7846	1.48728	0.743641	−	0.668579i	$-0.233097\pi$
$547$	34.7846	1.48728	0.743641	+	0.668579i	$0.233097\pi$
$548$	8.19615	0.350122
$549$	−18.7321	−0.799464
$550$	3.00000	0.127920
$551$	−61.1769	−2.60622
$552$	6.92820	0.294884
$553$	−18.5885	−0.790462
$554$	1.00000	0.0424859
$555$	−30.5885	−1.29841
$556$	−9.19615	−0.390004
$557$	−17.1962	−0.728624	−0.364312	−	0.931277i	$-0.618696\pi$
$557$	−17.1962	−0.728624	−0.364312	+	0.931277i	$0.618696\pi$
$558$	5.66025	0.239618
$559$	0	0
$560$	3.00000	0.126773
$561$	−18.0000	−0.759961
$562$	−10.3923	−0.438373
$563$	−21.4641	−0.904604	−0.452302	−	0.891865i	$-0.649397\pi$
$563$	−21.4641	−0.904604	−0.452302	+	0.891865i	$0.649397\pi$
$564$	8.19615	0.345120
$565$	6.92820	0.291472
$566$	30.3923	1.27748
$567$	7.39230	0.310448
$568$	−6.00000	−0.251754
$569$	−21.2487	−0.890792	−0.445396	−	0.895334i	$-0.646937\pi$
$569$	−21.2487	−0.890792	−0.445396	+	0.895334i	$0.646937\pi$
$570$	17.6603	0.739707
$571$	−34.3731	−1.43847	−0.719234	−	0.694768i	$-0.755507\pi$
$571$	−34.3731	−1.43847	−0.719234	+	0.694768i	$0.755507\pi$
$572$	0	0
$573$	52.6410	2.19911
$574$	31.1769	1.30130
$575$	−2.53590	−0.105754
$576$	4.46410	0.186004
$577$	32.4449	1.35070	0.675349	−	0.737499i	$-0.263993\pi$
$577$	32.4449	1.35070	0.675349	+	0.737499i	$0.263993\pi$
$578$	−12.1769	−0.506493
$579$	−12.0000	−0.498703
$580$	9.46410	0.392975
$581$	−6.58846	−0.273335
$582$	41.3205	1.71279
$583$	−19.3923	−0.803147
$584$	5.66025	0.234223
$585$	0	0
$586$	−0.803848	−0.0332066
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	−5.46410	−0.225336
$589$	8.19615	0.337717
$590$	−10.3923	−0.427844
$591$	−26.1962	−1.07757
$592$	−11.1962	−0.460159
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	−12.0000	−0.492366
$595$	6.58846	0.270100
$596$	6.00000	0.245770
$597$	39.3205	1.60928
$598$	0	0
$599$	7.85641	0.321004	0.160502	−	0.987036i	$-0.448689\pi$
$599$	7.85641	0.321004	0.160502	+	0.987036i	$0.448689\pi$
$600$	−2.73205	−0.111536
$601$	3.78461	0.154377	0.0771887	−	0.997016i	$-0.475406\pi$
$601$	3.78461	0.154377	0.0771887	+	0.997016i	$0.475406\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	6.33975	0.257961
$605$	2.00000	0.0813116
$606$	19.8564	0.806611
$607$	−6.41154	−0.260236	−0.130118	−	0.991498i	$-0.541536\pi$
$607$	−6.41154	−0.260236	−0.130118	+	0.991498i	$0.541536\pi$
$608$	6.46410	0.262154
$609$	−77.5692	−3.14326
$610$	4.19615	0.169897
$611$	0	0
$612$	9.80385	0.396297
$613$	26.9090	1.08684	0.543421	−	0.839460i	$-0.317129\pi$
$613$	26.9090	1.08684	0.543421	+	0.839460i	$0.317129\pi$
$614$	20.5359	0.828761
$615$	−28.3923	−1.14489
$616$	−9.00000	−0.362620
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	3.26795	0.131456
$619$	10.6077	0.426359	0.213180	−	0.977013i	$-0.431618\pi$
$619$	10.6077	0.426359	0.213180	+	0.977013i	$0.431618\pi$
$620$	−1.26795	−0.0509221
$621$	10.1436	0.407048
$622$	−15.1244	−0.606431
$623$	−51.5885	−2.06685
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.60770	0.224129
$627$	−52.9808	−2.11585
$628$	−13.0000	−0.518756
$629$	−24.5885	−0.980406
$630$	13.3923	0.533562
$631$	−20.7846	−0.827422	−0.413711	−	0.910408i	$-0.635768\pi$
$631$	−20.7846	−0.827422	−0.413711	+	0.910408i	$0.635768\pi$
$632$	6.19615	0.246470
$633$	−37.1244	−1.47556
$634$	−12.8038	−0.508506
$635$	21.1962	0.841144
$636$	17.6603	0.700275
$637$	0	0
$638$	−28.3923	−1.12406
$639$	−26.7846	−1.05958
$640$	−1.00000	−0.0395285
$641$	−45.9282	−1.81405	−0.907027	−	0.421071i	$-0.861654\pi$
$641$	−45.9282	−1.81405	−0.907027	+	0.421071i	$0.861654\pi$
$642$	−0.928203	−0.0366333
$643$	−7.26795	−0.286620	−0.143310	−	0.989678i	$-0.545775\pi$
$643$	−7.26795	−0.286620	−0.143310	+	0.989678i	$0.545775\pi$
$644$	7.60770	0.299785
$645$	−5.46410	−0.215149
$646$	14.1962	0.558540
$647$	11.1962	0.440166	0.220083	−	0.975481i	$-0.429367\pi$
$647$	11.1962	0.440166	0.220083	+	0.975481i	$0.429367\pi$
$648$	−2.46410	−0.0967991
$649$	31.1769	1.22380
$650$	0	0
$651$	10.3923	0.407307
$652$	7.26795	0.284635
$653$	−19.3923	−0.758880	−0.379440	−	0.925216i	$-0.623883\pi$
$653$	−19.3923	−0.758880	−0.379440	+	0.925216i	$0.623883\pi$
$654$	42.2487	1.65206
$655$	18.1244	0.708177
$656$	−10.3923	−0.405751
$657$	25.2679	0.985797
$658$	9.00000	0.350857
$659$	29.3205	1.14216	0.571082	−	0.820893i	$-0.306524\pi$
$659$	29.3205	1.14216	0.571082	+	0.820893i	$0.306524\pi$
$660$	8.19615	0.319035
$661$	−29.9090	−1.16332	−0.581662	−	0.813431i	$-0.697597\pi$
$661$	−29.9090	−1.16332	−0.581662	+	0.813431i	$0.697597\pi$
$662$	−0.928203	−0.0360756
$663$	0	0
$664$	2.19615	0.0852272
$665$	19.3923	0.752001
$666$	−49.9808	−1.93672
$667$	24.0000	0.929284
$668$	3.00000	0.116073
$669$	1.26795	0.0490217
$670$	0	0
$671$	−12.5885	−0.485972
$672$	8.19615	0.316173
$673$	−3.60770	−0.139066	−0.0695332	−	0.997580i	$-0.522151\pi$
$673$	−3.60770	−0.139066	−0.0695332	+	0.997580i	$0.522151\pi$
$674$	4.19615	0.161630
$675$	−4.00000	−0.153960
$676$	0	0
$677$	1.85641	0.0713475	0.0356737	−	0.999363i	$-0.488642\pi$
$677$	1.85641	0.0713475	0.0356737	+	0.999363i	$0.488642\pi$
$678$	18.9282	0.726933
$679$	45.3731	1.74126
$680$	−2.19615	−0.0842186
$681$	12.0000	0.459841
$682$	3.80385	0.145657
$683$	31.1769	1.19295	0.596476	−	0.802631i	$-0.296567\pi$
$683$	31.1769	1.19295	0.596476	+	0.802631i	$0.296567\pi$
$684$	28.8564	1.10335
$685$	−8.19615	−0.313159
$686$	15.0000	0.572703
$687$	−12.9282	−0.493242
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	−6.92820	−0.263752
$691$	−18.7128	−0.711869	−0.355934	−	0.934511i	$-0.615837\pi$
$691$	−18.7128	−0.711869	−0.355934	+	0.934511i	$0.615837\pi$
$692$	15.0000	0.570214
$693$	−40.1769	−1.52619
$694$	−25.2679	−0.959158
$695$	9.19615	0.348830
$696$	25.8564	0.980085
$697$	−22.8231	−0.864486
$698$	34.0526	1.28891
$699$	3.46410	0.131024
$700$	−3.00000	−0.113389
$701$	29.9090	1.12965	0.564823	−	0.825212i	$-0.308944\pi$
$701$	29.9090	1.12965	0.564823	+	0.825212i	$0.308944\pi$
$702$	0	0
$703$	−72.3731	−2.72960
$704$	3.00000	0.113067
$705$	−8.19615	−0.308685
$706$	−20.1962	−0.760092
$707$	21.8038	0.820018
$708$	−28.3923	−1.06705
$709$	40.9808	1.53906	0.769532	−	0.638608i	$-0.220489\pi$
$709$	40.9808	1.53906	0.769532	+	0.638608i	$0.220489\pi$
$710$	6.00000	0.225176
$711$	27.6603	1.03734
$712$	17.1962	0.644453
$713$	−3.21539	−0.120417
$714$	18.0000	0.673633
$715$	0	0
$716$	2.53590	0.0947710
$717$	22.3923	0.836256
$718$	−22.3923	−0.835673
$719$	1.85641	0.0692323	0.0346161	−	0.999401i	$-0.488979\pi$
$719$	1.85641	0.0692323	0.0346161	+	0.999401i	$0.488979\pi$
$720$	−4.46410	−0.166367
$721$	3.58846	0.133641
$722$	22.7846	0.847955
$723$	23.6603	0.879934
$724$	16.5885	0.616505
$725$	−9.46410	−0.351488
$726$	5.46410	0.202792
$727$	38.3731	1.42318	0.711589	−	0.702596i	$-0.247976\pi$
$727$	38.3731	1.42318	0.711589	+	0.702596i	$0.247976\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	−5.66025	−0.209495
$731$	−4.39230	−0.162455
$732$	11.4641	0.423725
$733$	−50.9090	−1.88037	−0.940183	−	0.340670i	$-0.889346\pi$
$733$	−50.9090	−1.88037	−0.940183	+	0.340670i	$0.889346\pi$
$734$	−26.3923	−0.974158
$735$	5.46410	0.201546
$736$	−2.53590	−0.0934745
$737$	0	0
$738$	−46.3923	−1.70772
$739$	−50.5692	−1.86022	−0.930109	−	0.367283i	$-0.880288\pi$
$739$	−50.5692	−1.86022	−0.930109	+	0.367283i	$0.880288\pi$
$740$	11.1962	0.411579
$741$	0	0
$742$	19.3923	0.711914
$743$	34.3923	1.26173	0.630866	−	0.775892i	$-0.282700\pi$
$743$	34.3923	1.26173	0.630866	+	0.775892i	$0.282700\pi$
$744$	−3.46410	−0.127000
$745$	−6.00000	−0.219823
$746$	20.3923	0.746615
$747$	9.80385	0.358704
$748$	6.58846	0.240898
$749$	−1.01924	−0.0372421
$750$	2.73205	0.0997604
$751$	−0.392305	−0.0143154	−0.00715770	−	0.999974i	$-0.502278\pi$
$751$	−0.392305	−0.0143154	−0.00715770	+	0.999974i	$0.502278\pi$
$752$	−3.00000	−0.109399
$753$	18.5885	0.677401
$754$	0	0
$755$	−6.33975	−0.230727
$756$	12.0000	0.436436
$757$	3.78461	0.137554	0.0687770	−	0.997632i	$-0.478090\pi$
$757$	3.78461	0.137554	0.0687770	+	0.997632i	$0.478090\pi$
$758$	11.7846	0.428036
$759$	20.7846	0.754434
$760$	−6.46410	−0.234478
$761$	−29.1962	−1.05836	−0.529180	−	0.848510i	$-0.677500\pi$
$761$	−29.1962	−1.05836	−0.529180	+	0.848510i	$0.677500\pi$
$762$	57.9090	2.09782
$763$	46.3923	1.67951
$764$	−19.2679	−0.697090
$765$	−9.80385	−0.354459
$766$	1.60770	0.0580884
$767$	0	0
$768$	−2.73205	−0.0985844
$769$	−38.1051	−1.37411	−0.687053	−	0.726607i	$-0.741096\pi$
$769$	−38.1051	−1.37411	−0.687053	+	0.726607i	$0.741096\pi$
$770$	9.00000	0.324337
$771$	37.8564	1.36337
$772$	4.39230	0.158083
$773$	33.5885	1.20809	0.604046	−	0.796949i	$-0.293554\pi$
$773$	33.5885	1.20809	0.604046	+	0.796949i	$0.293554\pi$
$774$	−8.92820	−0.320918
$775$	1.26795	0.0455461
$776$	−15.1244	−0.542933
$777$	−91.7654	−3.29206
$778$	−19.2679	−0.690789
$779$	−67.1769	−2.40686
$780$	0	0
$781$	−18.0000	−0.644091
$782$	−5.56922	−0.199155
$783$	37.8564	1.35288
$784$	2.00000	0.0714286
$785$	13.0000	0.463990
$786$	49.5167	1.76620
$787$	−36.8372	−1.31310	−0.656552	−	0.754281i	$-0.727986\pi$
$787$	−36.8372	−1.31310	−0.656552	+	0.754281i	$0.727986\pi$
$788$	9.58846	0.341575
$789$	75.3731	2.68335
$790$	−6.19615	−0.220449
$791$	20.7846	0.739016
$792$	13.3923	0.475875
$793$	0	0
$794$	0.803848	0.0285275
$795$	−17.6603	−0.626345
$796$	−14.3923	−0.510122
$797$	−0.928203	−0.0328786	−0.0164393	−	0.999865i	$-0.505233\pi$
$797$	−0.928203	−0.0328786	−0.0164393	+	0.999865i	$0.505233\pi$
$798$	52.9808	1.87550
$799$	−6.58846	−0.233083
$800$	1.00000	0.0353553
$801$	76.7654	2.71237
$802$	5.19615	0.183483
$803$	16.9808	0.599238
$804$	0	0
$805$	−7.60770	−0.268136
$806$	0	0
$807$	7.85641	0.276559
$808$	−7.26795	−0.255686
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	2.46410	0.0865797
$811$	22.6077	0.793864	0.396932	−	0.917848i	$-0.370075\pi$
$811$	22.6077	0.793864	0.396932	+	0.917848i	$0.370075\pi$
$812$	28.3923	0.996375
$813$	−6.92820	−0.242983
$814$	−33.5885	−1.17727
$815$	−7.26795	−0.254585
$816$	−6.00000	−0.210042
$817$	−12.9282	−0.452301
$818$	−19.7321	−0.689915
$819$	0	0
$820$	10.3923	0.362915
$821$	−9.80385	−0.342157	−0.171078	−	0.985257i	$-0.554725\pi$
$821$	−9.80385	−0.342157	−0.171078	+	0.985257i	$0.554725\pi$
$822$	−22.3923	−0.781021
$823$	16.8038	0.585745	0.292873	−	0.956151i	$-0.405389\pi$
$823$	16.8038	0.585745	0.292873	+	0.956151i	$0.405389\pi$
$824$	−1.19615	−0.0416699
$825$	−8.19615	−0.285353
$826$	−31.1769	−1.08478
$827$	11.4115	0.396818	0.198409	−	0.980119i	$-0.436423\pi$
$827$	11.4115	0.396818	0.198409	+	0.980119i	$0.436423\pi$
$828$	−11.3205	−0.393415
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	−2.19615	−0.0762296
$831$	−2.73205	−0.0947738
$832$	0	0
$833$	4.39230	0.152184
$834$	25.1244	0.869985
$835$	−3.00000	−0.103819
$836$	19.3923	0.670697
$837$	−5.07180	−0.175307
$838$	17.3205	0.598327
$839$	−32.1962	−1.11153	−0.555767	−	0.831338i	$-0.687575\pi$
$839$	−32.1962	−1.11153	−0.555767	+	0.831338i	$0.687575\pi$
$840$	−8.19615	−0.282794
$841$	60.5692	2.08859
$842$	27.1244	0.934767
$843$	28.3923	0.977883
$844$	13.5885	0.467734
$845$	0	0
$846$	−13.3923	−0.460437
$847$	6.00000	0.206162
$848$	−6.46410	−0.221978
$849$	−83.0333	−2.84970
$850$	2.19615	0.0753274
$851$	28.3923	0.973276
$852$	16.3923	0.561591
$853$	13.8564	0.474434	0.237217	−	0.971457i	$-0.423765\pi$
$853$	13.8564	0.474434	0.237217	+	0.971457i	$0.423765\pi$
$854$	12.5885	0.430768
$855$	−28.8564	−0.986868
$856$	0.339746	0.0116123
$857$	13.2679	0.453225	0.226612	−	0.973985i	$-0.427235\pi$
$857$	13.2679	0.453225	0.226612	+	0.973985i	$0.427235\pi$
$858$	0	0
$859$	52.3731	1.78695	0.893473	−	0.449117i	$-0.148261\pi$
$859$	52.3731	1.78695	0.893473	+	0.449117i	$0.148261\pi$
$860$	2.00000	0.0681994
$861$	−85.1769	−2.90282
$862$	−2.19615	−0.0748012
$863$	−19.1769	−0.652790	−0.326395	−	0.945234i	$-0.605834\pi$
$863$	−19.1769	−0.652790	−0.326395	+	0.945234i	$0.605834\pi$
$864$	−4.00000	−0.136083
$865$	−15.0000	−0.510015
$866$	12.3923	0.421108
$867$	33.2679	1.12984
$868$	−3.80385	−0.129111
$869$	18.5885	0.630570
$870$	−25.8564	−0.876614
$871$	0	0
$872$	−15.4641	−0.523681
$873$	−67.5167	−2.28509
$874$	−16.3923	−0.554478
$875$	3.00000	0.101419
$876$	−15.4641	−0.522484
$877$	−20.5359	−0.693448	−0.346724	−	0.937967i	$-0.612706\pi$
$877$	−20.5359	−0.693448	−0.346724	+	0.937967i	$0.612706\pi$
$878$	34.5885	1.16730
$879$	2.19615	0.0740744
$880$	−3.00000	−0.101130
$881$	8.32051	0.280325	0.140163	−	0.990129i	$-0.455237\pi$
$881$	8.32051	0.280325	0.140163	+	0.990129i	$0.455237\pi$
$882$	8.92820	0.300628
$883$	26.5885	0.894773	0.447386	−	0.894341i	$-0.352355\pi$
$883$	26.5885	0.894773	0.447386	+	0.894341i	$0.352355\pi$
$884$	0	0
$885$	28.3923	0.954397
$886$	−1.60770	−0.0540116
$887$	52.7654	1.77169	0.885844	−	0.463983i	$-0.153580\pi$
$887$	52.7654	1.77169	0.885844	+	0.463983i	$0.153580\pi$
$888$	30.5885	1.02648
$889$	63.5885	2.13269
$890$	−17.1962	−0.576416
$891$	−7.39230	−0.247652
$892$	−0.464102	−0.0155393
$893$	−19.3923	−0.648939
$894$	−16.3923	−0.548241
$895$	−2.53590	−0.0847657
$896$	−3.00000	−0.100223
$897$	0	0
$898$	15.5885	0.520194
$899$	−12.0000	−0.400222
$900$	4.46410	0.148803
$901$	−14.1962	−0.472942
$902$	−31.1769	−1.03808
$903$	−16.3923	−0.545502
$904$	−6.92820	−0.230429
$905$	−16.5885	−0.551419
$906$	−17.3205	−0.575435
$907$	26.5885	0.882855	0.441428	−	0.897297i	$-0.354472\pi$
$907$	26.5885	0.882855	0.441428	+	0.897297i	$0.354472\pi$
$908$	−4.39230	−0.145764
$909$	−32.4449	−1.07613
$910$	0	0
$911$	−14.5359	−0.481596	−0.240798	−	0.970575i	$-0.577409\pi$
$911$	−14.5359	−0.481596	−0.240798	+	0.970575i	$0.577409\pi$
$912$	−17.6603	−0.584789
$913$	6.58846	0.218046
$914$	−35.6603	−1.17954
$915$	−11.4641	−0.378992
$916$	4.73205	0.156351
$917$	54.3731	1.79556
$918$	−8.78461	−0.289935
$919$	−10.7846	−0.355751	−0.177876	−	0.984053i	$-0.556923\pi$
$919$	−10.7846	−0.355751	−0.177876	+	0.984053i	$0.556923\pi$
$920$	2.53590	0.0836061
$921$	−56.1051	−1.84873
$922$	0.588457	0.0193798
$923$	0	0
$924$	24.5885	0.808901
$925$	−11.1962	−0.368127
$926$	0.928203	0.0305027
$927$	−5.33975	−0.175380
$928$	−9.46410	−0.310674
$929$	−44.7846	−1.46934	−0.734668	−	0.678427i	$-0.762662\pi$
$929$	−44.7846	−1.46934	−0.734668	+	0.678427i	$0.762662\pi$
$930$	3.46410	0.113592
$931$	12.9282	0.423705
$932$	−1.26795	−0.0415331
$933$	41.3205	1.35277
$934$	10.1436	0.331909
$935$	−6.58846	−0.215466
$936$	0	0
$937$	−30.3923	−0.992873	−0.496437	−	0.868073i	$-0.665359\pi$
$937$	−30.3923	−0.992873	−0.496437	+	0.868073i	$0.665359\pi$
$938$	0	0
$939$	−15.3205	−0.499966
$940$	3.00000	0.0978492
$941$	44.7846	1.45994	0.729968	−	0.683481i	$-0.239535\pi$
$941$	44.7846	1.45994	0.729968	+	0.683481i	$0.239535\pi$
$942$	35.5167	1.15720
$943$	26.3538	0.858199
$944$	10.3923	0.338241
$945$	−12.0000	−0.390360
$946$	−6.00000	−0.195077
$947$	−57.3731	−1.86437	−0.932187	−	0.361977i	$-0.882102\pi$
$947$	−57.3731	−1.86437	−0.932187	+	0.361977i	$0.882102\pi$
$948$	−16.9282	−0.549802
$949$	0	0
$950$	6.46410	0.209723
$951$	34.9808	1.13433
$952$	−6.58846	−0.213533
$953$	−24.5885	−0.796498	−0.398249	−	0.917277i	$-0.630382\pi$
$953$	−24.5885	−0.796498	−0.398249	+	0.917277i	$0.630382\pi$
$954$	−28.8564	−0.934261
$955$	19.2679	0.623496
$956$	−8.19615	−0.265083
$957$	77.5692	2.50746
$958$	10.9808	0.354772
$959$	−24.5885	−0.794003
$960$	2.73205	0.0881766
$961$	−29.3923	−0.948139
$962$	0	0
$963$	1.51666	0.0488737
$964$	−8.66025	−0.278928
$965$	−4.39230	−0.141393
$966$	−20.7846	−0.668734
$967$	38.5692	1.24030	0.620151	−	0.784482i	$-0.287071\pi$
$967$	38.5692	1.24030	0.620151	+	0.784482i	$0.287071\pi$
$968$	−2.00000	−0.0642824
$969$	−38.7846	−1.24594
$970$	15.1244	0.485614
$971$	−46.7654	−1.50077	−0.750386	−	0.661000i	$-0.770132\pi$
$971$	−46.7654	−1.50077	−0.750386	+	0.661000i	$0.770132\pi$
$972$	18.7321	0.600831
$973$	27.5885	0.884445
$974$	33.2487	1.06536
$975$	0	0
$976$	−4.19615	−0.134316
$977$	4.39230	0.140522	0.0702611	−	0.997529i	$-0.477617\pi$
$977$	4.39230	0.140522	0.0702611	+	0.997529i	$0.477617\pi$
$978$	−19.8564	−0.634938
$979$	51.5885	1.64877
$980$	−2.00000	−0.0638877
$981$	−69.0333	−2.20406
$982$	−3.33975	−0.106576
$983$	38.5692	1.23017	0.615084	−	0.788462i	$-0.289122\pi$
$983$	38.5692	1.23017	0.615084	+	0.788462i	$0.289122\pi$
$984$	28.3923	0.905114
$985$	−9.58846	−0.305514
$986$	−20.7846	−0.661917
$987$	−24.5885	−0.782659
$988$	0	0
$989$	5.07180	0.161274
$990$	−13.3923	−0.425635
$991$	51.1769	1.62569	0.812844	−	0.582481i	$-0.197918\pi$
$991$	51.1769	1.62569	0.812844	+	0.582481i	$0.197918\pi$
$992$	1.26795	0.0402574
$993$	2.53590	0.0804743
$994$	18.0000	0.570925
$995$	14.3923	0.456267
$996$	−6.00000	−0.190117
$997$	−42.5692	−1.34818	−0.674090	−	0.738649i	$-0.735464\pi$
$997$	−42.5692	−1.34818	−0.674090	+	0.738649i	$0.735464\pi$
$998$	−1.85641	−0.0587635
$999$	44.7846	1.41692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.m.1.1		2
5.4	even	2		8450.2.a.bf.1.2		2
13.2	odd	12		130.2.l.a.121.2	yes	4
13.3	even	3		1690.2.e.l.191.2		4
13.4	even	6		1690.2.e.n.991.2		4
13.5	odd	4		1690.2.d.f.1351.1		4
13.6	odd	12		1690.2.l.g.361.1		4
13.7	odd	12		130.2.l.a.101.2	✓	4
13.8	odd	4		1690.2.d.f.1351.3		4
13.9	even	3		1690.2.e.l.991.2		4
13.10	even	6		1690.2.e.n.191.2		4
13.11	odd	12		1690.2.l.g.1161.1		4
13.12	even	2		1690.2.a.j.1.1		2
39.2	even	12		1170.2.bs.c.901.1		4
39.20	even	12		1170.2.bs.c.361.1		4
52.7	even	12		1040.2.da.a.881.1		4
52.15	even	12		1040.2.da.a.641.1		4
65.2	even	12		650.2.n.a.199.1		4
65.7	even	12		650.2.n.b.49.2		4
65.28	even	12		650.2.n.b.199.2		4
65.33	even	12		650.2.n.a.49.1		4
65.54	odd	12		650.2.m.a.251.1		4
65.59	odd	12		650.2.m.a.101.1		4
65.64	even	2		8450.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.l.a.101.2	✓	4	13.7	odd	12
130.2.l.a.121.2	yes	4	13.2	odd	12
650.2.m.a.101.1		4	65.59	odd	12
650.2.m.a.251.1		4	65.54	odd	12
650.2.n.a.49.1		4	65.33	even	12
650.2.n.a.199.1		4	65.2	even	12
650.2.n.b.49.2		4	65.7	even	12
650.2.n.b.199.2		4	65.28	even	12
1040.2.da.a.641.1		4	52.15	even	12
1040.2.da.a.881.1		4	52.7	even	12
1170.2.bs.c.361.1		4	39.20	even	12
1170.2.bs.c.901.1		4	39.2	even	12
1690.2.a.j.1.1		2	13.12	even	2
1690.2.a.m.1.1		2	1.1	even	1	trivial
1690.2.d.f.1351.1		4	13.5	odd	4
1690.2.d.f.1351.3		4	13.8	odd	4
1690.2.e.l.191.2		4	13.3	even	3
1690.2.e.l.991.2		4	13.9	even	3
1690.2.e.n.191.2		4	13.10	even	6
1690.2.e.n.991.2		4	13.4	even	6
1690.2.l.g.361.1		4	13.6	odd	12
1690.2.l.g.1161.1		4	13.11	odd	12
8450.2.a.bf.1.2		2	5.4	even	2
8450.2.a.bm.1.2		2	65.64	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:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} -3.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.73205 q^{6} -3.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} -1.00000 q^{10} +3.00000 q^{11} -2.73205 q^{12} -3.00000 q^{14} +2.73205 q^{15} +1.00000 q^{16} +2.19615 q^{17} +4.46410 q^{18} +6.46410 q^{19} -1.00000 q^{20} +8.19615 q^{21} +3.00000 q^{22} -2.53590 q^{23} -2.73205 q^{24} +1.00000 q^{25} -4.00000 q^{27} -3.00000 q^{28} -9.46410 q^{29} +2.73205 q^{30} +1.26795 q^{31} +1.00000 q^{32} -8.19615 q^{33} +2.19615 q^{34} +3.00000 q^{35} +4.46410 q^{36} -11.1962 q^{37} +6.46410 q^{38} -1.00000 q^{40} -10.3923 q^{41} +8.19615 q^{42} -2.00000 q^{43} +3.00000 q^{44} -4.46410 q^{45} -2.53590 q^{46} -3.00000 q^{47} -2.73205 q^{48} +2.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -6.46410 q^{53} -4.00000 q^{54} -3.00000 q^{55} -3.00000 q^{56} -17.6603 q^{57} -9.46410 q^{58} +10.3923 q^{59} +2.73205 q^{60} -4.19615 q^{61} +1.26795 q^{62} -13.3923 q^{63} +1.00000 q^{64} -8.19615 q^{66} +2.19615 q^{68} +6.92820 q^{69} +3.00000 q^{70} -6.00000 q^{71} +4.46410 q^{72} +5.66025 q^{73} -11.1962 q^{74} -2.73205 q^{75} +6.46410 q^{76} -9.00000 q^{77} +6.19615 q^{79} -1.00000 q^{80} -2.46410 q^{81} -10.3923 q^{82} +2.19615 q^{83} +8.19615 q^{84} -2.19615 q^{85} -2.00000 q^{86} +25.8564 q^{87} +3.00000 q^{88} +17.1962 q^{89} -4.46410 q^{90} -2.53590 q^{92} -3.46410 q^{93} -3.00000 q^{94} -6.46410 q^{95} -2.73205 q^{96} -15.1244 q^{97} +2.00000 q^{98} +13.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 6 q^{11} - 2 q^{12} - 6 q^{14} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} + 6 q^{19} - 2 q^{20} + 6 q^{21} + 6 q^{22} - 12 q^{23} - 2 q^{24} + 2 q^{25} - 8 q^{27} - 6 q^{28} - 12 q^{29} + 2 q^{30} + 6 q^{31} + 2 q^{32} - 6 q^{33} - 6 q^{34} + 6 q^{35} + 2 q^{36} - 12 q^{37} + 6 q^{38} - 2 q^{40} + 6 q^{42} - 4 q^{43} + 6 q^{44} - 2 q^{45} - 12 q^{46} - 6 q^{47} - 2 q^{48} + 4 q^{49} + 2 q^{50} - 12 q^{51} - 6 q^{53} - 8 q^{54} - 6 q^{55} - 6 q^{56} - 18 q^{57} - 12 q^{58} + 2 q^{60} + 2 q^{61} + 6 q^{62} - 6 q^{63} + 2 q^{64} - 6 q^{66} - 6 q^{68} + 6 q^{70} - 12 q^{71} + 2 q^{72} - 6 q^{73} - 12 q^{74} - 2 q^{75} + 6 q^{76} - 18 q^{77} + 2 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{83} + 6 q^{84} + 6 q^{85} - 4 q^{86} + 24 q^{87} + 6 q^{88} + 24 q^{89} - 2 q^{90} - 12 q^{92} - 6 q^{94} - 6 q^{95} - 2 q^{96} - 6 q^{97} + 4 q^{98} + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 6 * q^11 - 2 * q^12 - 6 * q^14 + 2 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 + 6 * q^19 - 2 * q^20 + 6 * q^21 + 6 * q^22 - 12 * q^23 - 2 * q^24 + 2 * q^25 - 8 * q^27 - 6 * q^28 - 12 * q^29 + 2 * q^30 + 6 * q^31 + 2 * q^32 - 6 * q^33 - 6 * q^34 + 6 * q^35 + 2 * q^36 - 12 * q^37 + 6 * q^38 - 2 * q^40 + 6 * q^42 - 4 * q^43 + 6 * q^44 - 2 * q^45 - 12 * q^46 - 6 * q^47 - 2 * q^48 + 4 * q^49 + 2 * q^50 - 12 * q^51 - 6 * q^53 - 8 * q^54 - 6 * q^55 - 6 * q^56 - 18 * q^57 - 12 * q^58 + 2 * q^60 + 2 * q^61 + 6 * q^62 - 6 * q^63 + 2 * q^64 - 6 * q^66 - 6 * q^68 + 6 * q^70 - 12 * q^71 + 2 * q^72 - 6 * q^73 - 12 * q^74 - 2 * q^75 + 6 * q^76 - 18 * q^77 + 2 * q^79 - 2 * q^80 + 2 * q^81 - 6 * q^83 + 6 * q^84 + 6 * q^85 - 4 * q^86 + 24 * q^87 + 6 * q^88 + 24 * q^89 - 2 * q^90 - 12 * q^92 - 6 * q^94 - 6 * q^95 - 2 * q^96 - 6 * q^97 + 4 * q^98 + 6 * q^99

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