LMFDB - Embedded newform 338.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	338.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:	$2.69894358832$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.35690 q^{3} +1.00000 q^{4} +0.890084 q^{5} +2.35690 q^{6} -4.49396 q^{7} +1.00000 q^{8} +2.55496 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.35690 q^{3} +1.00000 q^{4} +0.890084 q^{5} +2.35690 q^{6} -4.49396 q^{7} +1.00000 q^{8} +2.55496 q^{9} +0.890084 q^{10} +2.69202 q^{11} +2.35690 q^{12} -4.49396 q^{14} +2.09783 q^{15} +1.00000 q^{16} +3.58211 q^{17} +2.55496 q^{18} -2.93900 q^{19} +0.890084 q^{20} -10.5918 q^{21} +2.69202 q^{22} -6.09783 q^{23} +2.35690 q^{24} -4.20775 q^{25} -1.04892 q^{27} -4.49396 q^{28} +2.98792 q^{29} +2.09783 q^{30} -2.39612 q^{31} +1.00000 q^{32} +6.34481 q^{33} +3.58211 q^{34} -4.00000 q^{35} +2.55496 q^{36} -1.50604 q^{37} -2.93900 q^{38} +0.890084 q^{40} -3.65279 q^{41} -10.5918 q^{42} +0.170915 q^{43} +2.69202 q^{44} +2.27413 q^{45} -6.09783 q^{46} +5.20775 q^{47} +2.35690 q^{48} +13.1957 q^{49} -4.20775 q^{50} +8.44265 q^{51} +6.09783 q^{53} -1.04892 q^{54} +2.39612 q^{55} -4.49396 q^{56} -6.92692 q^{57} +2.98792 q^{58} +3.07069 q^{59} +2.09783 q^{60} -13.9758 q^{61} -2.39612 q^{62} -11.4819 q^{63} +1.00000 q^{64} +6.34481 q^{66} +11.0707 q^{67} +3.58211 q^{68} -14.3720 q^{69} -4.00000 q^{70} -10.0978 q^{71} +2.55496 q^{72} +10.9487 q^{73} -1.50604 q^{74} -9.91723 q^{75} -2.93900 q^{76} -12.0978 q^{77} +2.81163 q^{79} +0.890084 q^{80} -10.1371 q^{81} -3.65279 q^{82} +2.93900 q^{83} -10.5918 q^{84} +3.18837 q^{85} +0.170915 q^{86} +7.04221 q^{87} +2.69202 q^{88} +12.1806 q^{89} +2.27413 q^{90} -6.09783 q^{92} -5.64742 q^{93} +5.20775 q^{94} -2.61596 q^{95} +2.35690 q^{96} +12.9661 q^{97} +13.1957 q^{98} +6.87800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} + 3 q^{12} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} + q^{19} + 2 q^{20} - 4 q^{21} + 3 q^{22} + 3 q^{24} + 5 q^{25} + 6 q^{27} - 4 q^{28} - 10 q^{29} - 12 q^{30} - 16 q^{31} + 3 q^{32} - 4 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 14 q^{37} + q^{38} + 2 q^{40} + 7 q^{41} - 4 q^{42} + 11 q^{43} + 3 q^{44} - 4 q^{45} - 2 q^{47} + 3 q^{48} + 3 q^{49} + 5 q^{50} - 16 q^{51} + 6 q^{54} + 16 q^{55} - 4 q^{56} + 8 q^{57} - 10 q^{58} - 3 q^{59} - 12 q^{60} - 4 q^{61} - 16 q^{62} - 6 q^{63} + 3 q^{64} - 4 q^{66} + 21 q^{67} + 5 q^{68} - 14 q^{69} - 12 q^{70} - 12 q^{71} + 8 q^{72} + q^{73} - 14 q^{74} - 23 q^{75} + q^{76} - 18 q^{77} - 18 q^{79} + 2 q^{80} - 25 q^{81} + 7 q^{82} - q^{83} - 4 q^{84} + 36 q^{85} + 11 q^{86} - 10 q^{87} + 3 q^{88} + 25 q^{89} - 4 q^{90} - 2 q^{93} - 2 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 3 q^{98} + q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 + 3 * q^12 - 4 * q^14 - 12 * q^15 + 3 * q^16 + 5 * q^17 + 8 * q^18 + q^19 + 2 * q^20 - 4 * q^21 + 3 * q^22 + 3 * q^24 + 5 * q^25 + 6 * q^27 - 4 * q^28 - 10 * q^29 - 12 * q^30 - 16 * q^31 + 3 * q^32 - 4 * q^33 + 5 * q^34 - 12 * q^35 + 8 * q^36 - 14 * q^37 + q^38 + 2 * q^40 + 7 * q^41 - 4 * q^42 + 11 * q^43 + 3 * q^44 - 4 * q^45 - 2 * q^47 + 3 * q^48 + 3 * q^49 + 5 * q^50 - 16 * q^51 + 6 * q^54 + 16 * q^55 - 4 * q^56 + 8 * q^57 - 10 * q^58 - 3 * q^59 - 12 * q^60 - 4 * q^61 - 16 * q^62 - 6 * q^63 + 3 * q^64 - 4 * q^66 + 21 * q^67 + 5 * q^68 - 14 * q^69 - 12 * q^70 - 12 * q^71 + 8 * q^72 + q^73 - 14 * q^74 - 23 * q^75 + q^76 - 18 * q^77 - 18 * q^79 + 2 * q^80 - 25 * q^81 + 7 * q^82 - q^83 - 4 * q^84 + 36 * q^85 + 11 * q^86 - 10 * q^87 + 3 * q^88 + 25 * q^89 - 4 * q^90 - 2 * q^93 - 2 * q^94 - 18 * q^95 + 3 * q^96 + 23 * q^97 + 3 * q^98 + 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
$3$	2.35690	1.36075	0.680377	−	0.732862i	$-0.261816\pi$
$3$	2.35690	1.36075	0.680377	+	0.732862i	$0.261816\pi$
$4$	1.00000	0.500000
$5$	0.890084	0.398058	0.199029	−	0.979994i	$-0.436221\pi$
$5$	0.890084	0.398058	0.199029	+	0.979994i	$0.436221\pi$
$6$	2.35690	0.962199
$7$	−4.49396	−1.69856	−0.849278	−	0.527945i	$-0.822963\pi$
$7$	−4.49396	−1.69856	−0.849278	+	0.527945i	$0.822963\pi$
$8$	1.00000	0.353553
$9$	2.55496	0.851653
$10$	0.890084	0.281469
$11$	2.69202	0.811675	0.405838	−	0.913945i	$-0.366980\pi$
$11$	2.69202	0.811675	0.405838	+	0.913945i	$0.366980\pi$
$12$	2.35690	0.680377
$13$	0	0
$13$	0	0
$14$	−4.49396	−1.20106
$15$	2.09783	0.541659
$16$	1.00000	0.250000
$17$	3.58211	0.868788	0.434394	−	0.900723i	$-0.356963\pi$
$17$	3.58211	0.868788	0.434394	+	0.900723i	$0.356963\pi$
$18$	2.55496	0.602209
$19$	−2.93900	−0.674253	−0.337127	−	0.941459i	$-0.609455\pi$
$19$	−2.93900	−0.674253	−0.337127	+	0.941459i	$0.609455\pi$
$20$	0.890084	0.199029
$21$	−10.5918	−2.31132
$22$	2.69202	0.573941
$23$	−6.09783	−1.27149	−0.635743	−	0.771901i	$-0.719306\pi$
$23$	−6.09783	−1.27149	−0.635743	+	0.771901i	$0.719306\pi$
$24$	2.35690	0.481099
$25$	−4.20775	−0.841550
$26$	0	0
$27$	−1.04892	−0.201864
$28$	−4.49396	−0.849278
$29$	2.98792	0.554843	0.277421	−	0.960748i	$-0.410520\pi$
$29$	2.98792	0.554843	0.277421	+	0.960748i	$0.410520\pi$
$30$	2.09783	0.383010
$31$	−2.39612	−0.430357	−0.215178	−	0.976575i	$-0.569033\pi$
$31$	−2.39612	−0.430357	−0.215178	+	0.976575i	$0.569033\pi$
$32$	1.00000	0.176777
$33$	6.34481	1.10449
$34$	3.58211	0.614326
$35$	−4.00000	−0.676123
$36$	2.55496	0.425826
$37$	−1.50604	−0.247592	−0.123796	−	0.992308i	$-0.539507\pi$
$37$	−1.50604	−0.247592	−0.123796	+	0.992308i	$0.539507\pi$
$38$	−2.93900	−0.476769
$39$	0	0
$40$	0.890084	0.140735
$41$	−3.65279	−0.570470	−0.285235	−	0.958458i	$-0.592072\pi$
$41$	−3.65279	−0.570470	−0.285235	+	0.958458i	$0.592072\pi$
$42$	−10.5918	−1.63435
$43$	0.170915	0.0260643	0.0130322	−	0.999915i	$-0.495852\pi$
$43$	0.170915	0.0260643	0.0130322	+	0.999915i	$0.495852\pi$
$44$	2.69202	0.405838
$45$	2.27413	0.339007
$46$	−6.09783	−0.899077
$47$	5.20775	0.759629	0.379814	−	0.925063i	$-0.375988\pi$
$47$	5.20775	0.759629	0.379814	+	0.925063i	$0.375988\pi$
$48$	2.35690	0.340189
$49$	13.1957	1.88510
$50$	−4.20775	−0.595066
$51$	8.44265	1.18221
$52$	0	0
$53$	6.09783	0.837602	0.418801	−	0.908078i	$-0.362450\pi$
$53$	6.09783	0.837602	0.418801	+	0.908078i	$0.362450\pi$
$54$	−1.04892	−0.142740
$55$	2.39612	0.323093
$56$	−4.49396	−0.600531
$57$	−6.92692	−0.917493
$58$	2.98792	0.392333
$59$	3.07069	0.399769	0.199885	−	0.979819i	$-0.435943\pi$
$59$	3.07069	0.399769	0.199885	+	0.979819i	$0.435943\pi$
$60$	2.09783	0.270829
$61$	−13.9758	−1.78942	−0.894711	−	0.446645i	$-0.852619\pi$
$61$	−13.9758	−1.78942	−0.894711	+	0.446645i	$0.852619\pi$
$62$	−2.39612	−0.304308
$63$	−11.4819	−1.44658
$64$	1.00000	0.125000
$65$	0	0
$66$	6.34481	0.780993
$67$	11.0707	1.35250	0.676250	−	0.736672i	$-0.263604\pi$
$67$	11.0707	1.35250	0.676250	+	0.736672i	$0.263604\pi$
$68$	3.58211	0.434394
$69$	−14.3720	−1.73018
$70$	−4.00000	−0.478091
$71$	−10.0978	−1.19839	−0.599196	−	0.800602i	$-0.704513\pi$
$71$	−10.0978	−1.19839	−0.599196	+	0.800602i	$0.704513\pi$
$72$	2.55496	0.301105
$73$	10.9487	1.28145	0.640724	−	0.767772i	$-0.278634\pi$
$73$	10.9487	1.28145	0.640724	+	0.767772i	$0.278634\pi$
$74$	−1.50604	−0.175074
$75$	−9.91723	−1.14514
$76$	−2.93900	−0.337127
$77$	−12.0978	−1.37868
$78$	0	0
$79$	2.81163	0.316333	0.158166	−	0.987412i	$-0.449442\pi$
$79$	2.81163	0.316333	0.158166	+	0.987412i	$0.449442\pi$
$80$	0.890084	0.0995144
$81$	−10.1371	−1.12634
$82$	−3.65279	−0.403383
$83$	2.93900	0.322597	0.161299	−	0.986906i	$-0.448432\pi$
$83$	2.93900	0.322597	0.161299	+	0.986906i	$0.448432\pi$
$84$	−10.5918	−1.15566
$85$	3.18837	0.345828
$86$	0.170915	0.0184303
$87$	7.04221	0.755004
$88$	2.69202	0.286970
$89$	12.1806	1.29114	0.645571	−	0.763700i	$-0.276620\pi$
$89$	12.1806	1.29114	0.645571	+	0.763700i	$0.276620\pi$
$90$	2.27413	0.239714
$91$	0	0
$92$	−6.09783	−0.635743
$93$	−5.64742	−0.585610
$94$	5.20775	0.537138
$95$	−2.61596	−0.268392
$96$	2.35690	0.240550
$97$	12.9661	1.31651	0.658256	−	0.752794i	$-0.271294\pi$
$97$	12.9661	1.31651	0.658256	+	0.752794i	$0.271294\pi$
$98$	13.1957	1.33296
$99$	6.87800	0.691265
$100$	−4.20775	−0.420775
$101$	−10.1957	−1.01451	−0.507254	−	0.861797i	$-0.669339\pi$
$101$	−10.1957	−1.01451	−0.507254	+	0.861797i	$0.669339\pi$
$102$	8.44265	0.835947
$103$	10.6703	1.05137	0.525686	−	0.850679i	$-0.323809\pi$
$103$	10.6703	1.05137	0.525686	+	0.850679i	$0.323809\pi$
$104$	0	0
$105$	−9.42758	−0.920038
$106$	6.09783	0.592274
$107$	−20.5623	−1.98783	−0.993914	−	0.110158i	$-0.964864\pi$
$107$	−20.5623	−1.98783	−0.993914	+	0.110158i	$0.964864\pi$
$108$	−1.04892	−0.100932
$109$	2.71379	0.259934	0.129967	−	0.991518i	$-0.458513\pi$
$109$	2.71379	0.259934	0.129967	+	0.991518i	$0.458513\pi$
$110$	2.39612	0.228462
$111$	−3.54958	−0.336911
$112$	−4.49396	−0.424639
$113$	20.6504	1.94263	0.971313	−	0.237804i	$-0.0764277\pi$
$113$	20.6504	1.94263	0.971313	+	0.237804i	$0.0764277\pi$
$114$	−6.92692	−0.648765
$115$	−5.42758	−0.506125
$116$	2.98792	0.277421
$117$	0	0
$118$	3.07069	0.282680
$119$	−16.0978	−1.47569
$120$	2.09783	0.191505
$121$	−3.75302	−0.341184
$122$	−13.9758	−1.26531
$123$	−8.60925	−0.776270
$124$	−2.39612	−0.215178
$125$	−8.19567	−0.733043
$126$	−11.4819	−1.02289
$127$	12.7681	1.13298	0.566492	−	0.824067i	$-0.308300\pi$
$127$	12.7681	1.13298	0.566492	+	0.824067i	$0.308300\pi$
$128$	1.00000	0.0883883
$129$	0.402829	0.0354671
$130$	0	0
$131$	−5.15883	−0.450729	−0.225365	−	0.974274i	$-0.572357\pi$
$131$	−5.15883	−0.450729	−0.225365	+	0.974274i	$0.572357\pi$
$132$	6.34481	0.552245
$133$	13.2078	1.14526
$134$	11.0707	0.956362
$135$	−0.933624	−0.0803536
$136$	3.58211	0.307163
$137$	−3.30127	−0.282047	−0.141023	−	0.990006i	$-0.545039\pi$
$137$	−3.30127	−0.282047	−0.141023	+	0.990006i	$0.545039\pi$
$138$	−14.3720	−1.22342
$139$	−6.49157	−0.550607	−0.275304	−	0.961357i	$-0.588778\pi$
$139$	−6.49157	−0.550607	−0.275304	+	0.961357i	$0.588778\pi$
$140$	−4.00000	−0.338062
$141$	12.2741	1.03367
$142$	−10.0978	−0.847391
$143$	0	0
$144$	2.55496	0.212913
$145$	2.65950	0.220859
$146$	10.9487	0.906120
$147$	31.1008	2.56515
$148$	−1.50604	−0.123796
$149$	5.03146	0.412193	0.206097	−	0.978532i	$-0.433924\pi$
$149$	5.03146	0.412193	0.206097	+	0.978532i	$0.433924\pi$
$150$	−9.91723	−0.809739
$151$	−1.72587	−0.140450	−0.0702248	−	0.997531i	$-0.522372\pi$
$151$	−1.72587	−0.140450	−0.0702248	+	0.997531i	$0.522372\pi$
$152$	−2.93900	−0.238384
$153$	9.15213	0.739906
$154$	−12.0978	−0.974871
$155$	−2.13275	−0.171307
$156$	0	0
$157$	15.5060	1.23752	0.618758	−	0.785581i	$-0.287636\pi$
$157$	15.5060	1.23752	0.618758	+	0.785581i	$0.287636\pi$
$158$	2.81163	0.223681
$159$	14.3720	1.13977
$160$	0.890084	0.0703673
$161$	27.4034	2.15969
$162$	−10.1371	−0.796443
$163$	−19.7235	−1.54486	−0.772431	−	0.635099i	$-0.780959\pi$
$163$	−19.7235	−1.54486	−0.772431	+	0.635099i	$0.780959\pi$
$164$	−3.65279	−0.285235
$165$	5.64742	0.439651
$166$	2.93900	0.228111
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	−10.5918	−0.817175
$169$	0	0
$170$	3.18837	0.244537
$171$	−7.50902	−0.574229
$172$	0.170915	0.0130322
$173$	1.20775	0.0918236	0.0459118	−	0.998945i	$-0.485381\pi$
$173$	1.20775	0.0918236	0.0459118	+	0.998945i	$0.485381\pi$
$174$	7.04221	0.533869
$175$	18.9095	1.42942
$176$	2.69202	0.202919
$177$	7.23729	0.543988
$178$	12.1806	0.912975
$179$	16.5157	1.23444	0.617222	−	0.786789i	$-0.288258\pi$
$179$	16.5157	1.23444	0.617222	+	0.786789i	$0.288258\pi$
$180$	2.27413	0.169503
$181$	6.37196	0.473624	0.236812	−	0.971555i	$-0.423897\pi$
$181$	6.37196	0.473624	0.236812	+	0.971555i	$0.423897\pi$
$182$	0	0
$183$	−32.9396	−2.43496
$184$	−6.09783	−0.449538
$185$	−1.34050	−0.0985557
$186$	−5.64742	−0.414089
$187$	9.64310	0.705174
$188$	5.20775	0.379814
$189$	4.71379	0.342878
$190$	−2.61596	−0.189781
$191$	−2.49396	−0.180457	−0.0902283	−	0.995921i	$-0.528760\pi$
$191$	−2.49396	−0.180457	−0.0902283	+	0.995921i	$0.528760\pi$
$192$	2.35690	0.170094
$193$	−4.00538	−0.288313	−0.144157	−	0.989555i	$-0.546047\pi$
$193$	−4.00538	−0.288313	−0.144157	+	0.989555i	$0.546047\pi$
$194$	12.9661	0.930915
$195$	0	0
$196$	13.1957	0.942548
$197$	23.6340	1.68385	0.841927	−	0.539592i	$-0.181422\pi$
$197$	23.6340	1.68385	0.841927	+	0.539592i	$0.181422\pi$
$198$	6.87800	0.488798
$199$	16.5375	1.17231	0.586156	−	0.810198i	$-0.300641\pi$
$199$	16.5375	1.17231	0.586156	+	0.810198i	$0.300641\pi$
$200$	−4.20775	−0.297533
$201$	26.0925	1.84042
$202$	−10.1957	−0.717365
$203$	−13.4276	−0.942432
$204$	8.44265	0.591104
$205$	−3.25129	−0.227080
$206$	10.6703	0.743432
$207$	−15.5797	−1.08286
$208$	0	0
$209$	−7.91185	−0.547274
$210$	−9.42758	−0.650565
$211$	−1.66056	−0.114318	−0.0571589	−	0.998365i	$-0.518204\pi$
$211$	−1.66056	−0.114318	−0.0571589	+	0.998365i	$0.518204\pi$
$212$	6.09783	0.418801
$213$	−23.7995	−1.63072
$214$	−20.5623	−1.40561
$215$	0.152129	0.0103751
$216$	−1.04892	−0.0713698
$217$	10.7681	0.730985
$218$	2.71379	0.183801
$219$	25.8049	1.74374
$220$	2.39612	0.161547
$221$	0	0
$222$	−3.54958	−0.238232
$223$	−0.792249	−0.0530529	−0.0265265	−	0.999648i	$-0.508445\pi$
$223$	−0.792249	−0.0530529	−0.0265265	+	0.999648i	$0.508445\pi$
$224$	−4.49396	−0.300265
$225$	−10.7506	−0.716708
$226$	20.6504	1.37364
$227$	−21.7603	−1.44428	−0.722141	−	0.691745i	$-0.756842\pi$
$227$	−21.7603	−1.44428	−0.722141	+	0.691745i	$0.756842\pi$
$228$	−6.92692	−0.458746
$229$	−1.97584	−0.130567	−0.0652835	−	0.997867i	$-0.520795\pi$
$229$	−1.97584	−0.130567	−0.0652835	+	0.997867i	$0.520795\pi$
$230$	−5.42758	−0.357884
$231$	−28.5133	−1.87604
$232$	2.98792	0.196166
$233$	18.2349	1.19461	0.597304	−	0.802015i	$-0.296239\pi$
$233$	18.2349	1.19461	0.597304	+	0.802015i	$0.296239\pi$
$234$	0	0
$235$	4.63533	0.302376
$236$	3.07069	0.199885
$237$	6.62671	0.430451
$238$	−16.0978	−1.04347
$239$	−22.0737	−1.42783	−0.713914	−	0.700234i	$-0.753079\pi$
$239$	−22.0737	−1.42783	−0.713914	+	0.700234i	$0.753079\pi$
$240$	2.09783	0.135415
$241$	6.98792	0.450131	0.225066	−	0.974344i	$-0.427740\pi$
$241$	6.98792	0.450131	0.225066	+	0.974344i	$0.427740\pi$
$242$	−3.75302	−0.241253
$243$	−20.7453	−1.33081
$244$	−13.9758	−0.894711
$245$	11.7453	0.750377
$246$	−8.60925	−0.548906
$247$	0	0
$248$	−2.39612	−0.152154
$249$	6.92692	0.438976
$250$	−8.19567	−0.518340
$251$	−12.5593	−0.792734	−0.396367	−	0.918092i	$-0.629729\pi$
$251$	−12.5593	−0.792734	−0.396367	+	0.918092i	$0.629729\pi$
$252$	−11.4819	−0.723290
$253$	−16.4155	−1.03203
$254$	12.7681	0.801141
$255$	7.51466	0.470587
$256$	1.00000	0.0625000
$257$	−16.9933	−1.06001	−0.530006	−	0.847994i	$-0.677810\pi$
$257$	−16.9933	−1.06001	−0.530006	+	0.847994i	$0.677810\pi$
$258$	0.402829	0.0250791
$259$	6.76809	0.420548
$260$	0	0
$261$	7.63401	0.472533
$262$	−5.15883	−0.318714
$263$	−4.39612	−0.271077	−0.135538	−	0.990772i	$-0.543276\pi$
$263$	−4.39612	−0.271077	−0.135538	+	0.990772i	$0.543276\pi$
$264$	6.34481	0.390496
$265$	5.42758	0.333414
$266$	13.2078	0.809819
$267$	28.7084	1.75693
$268$	11.0707	0.676250
$269$	15.5603	0.948730	0.474365	−	0.880328i	$-0.342678\pi$
$269$	15.5603	0.948730	0.474365	+	0.880328i	$0.342678\pi$
$270$	−0.933624	−0.0568186
$271$	−21.9952	−1.33611	−0.668057	−	0.744110i	$-0.732874\pi$
$271$	−21.9952	−1.33611	−0.668057	+	0.744110i	$0.732874\pi$
$272$	3.58211	0.217197
$273$	0	0
$274$	−3.30127	−0.199437
$275$	−11.3274	−0.683065
$276$	−14.3720	−0.865090
$277$	−1.87800	−0.112838	−0.0564191	−	0.998407i	$-0.517968\pi$
$277$	−1.87800	−0.112838	−0.0564191	+	0.998407i	$0.517968\pi$
$278$	−6.49157	−0.389338
$279$	−6.12200	−0.366514
$280$	−4.00000	−0.239046
$281$	9.20536	0.549146	0.274573	−	0.961566i	$-0.411464\pi$
$281$	9.20536	0.549146	0.274573	+	0.961566i	$0.411464\pi$
$282$	12.2741	0.730914
$283$	4.70841	0.279886	0.139943	−	0.990160i	$-0.455308\pi$
$283$	4.70841	0.279886	0.139943	+	0.990160i	$0.455308\pi$
$284$	−10.0978	−0.599196
$285$	−6.16554	−0.365215
$286$	0	0
$287$	16.4155	0.968976
$288$	2.55496	0.150552
$289$	−4.16852	−0.245207
$290$	2.65950	0.156171
$291$	30.5599	1.79145
$292$	10.9487	0.640724
$293$	−13.7017	−0.800462	−0.400231	−	0.916414i	$-0.631070\pi$
$293$	−13.7017	−0.800462	−0.400231	+	0.916414i	$0.631070\pi$
$294$	31.1008	1.81384
$295$	2.73317	0.159131
$296$	−1.50604	−0.0875368
$297$	−2.82371	−0.163848
$298$	5.03146	0.291465
$299$	0	0
$300$	−9.91723	−0.572572
$301$	−0.768086	−0.0442717
$302$	−1.72587	−0.0993128
$303$	−24.0301	−1.38049
$304$	−2.93900	−0.168563
$305$	−12.4397	−0.712293
$306$	9.15213	0.523192
$307$	8.03252	0.458440	0.229220	−	0.973375i	$-0.426382\pi$
$307$	8.03252	0.458440	0.229220	+	0.973375i	$0.426382\pi$
$308$	−12.0978	−0.689338
$309$	25.1487	1.43066
$310$	−2.13275	−0.121132
$311$	4.09783	0.232367	0.116183	−	0.993228i	$-0.462934\pi$
$311$	4.09783	0.232367	0.116183	+	0.993228i	$0.462934\pi$
$312$	0	0
$313$	4.37435	0.247253	0.123627	−	0.992329i	$-0.460548\pi$
$313$	4.37435	0.247253	0.123627	+	0.992329i	$0.460548\pi$
$314$	15.5060	0.875057
$315$	−10.2198	−0.575822
$316$	2.81163	0.158166
$317$	−20.3612	−1.14360	−0.571800	−	0.820393i	$-0.693755\pi$
$317$	−20.3612	−1.14360	−0.571800	+	0.820393i	$0.693755\pi$
$318$	14.3720	0.805940
$319$	8.04354	0.450352
$320$	0.890084	0.0497572
$321$	−48.4631	−2.70495
$322$	27.4034	1.52713
$323$	−10.5278	−0.585783
$324$	−10.1371	−0.563170
$325$	0	0
$326$	−19.7235	−1.09238
$327$	6.39612	0.353706
$328$	−3.65279	−0.201692
$329$	−23.4034	−1.29027
$330$	5.64742	0.310880
$331$	6.13275	0.337087	0.168543	−	0.985694i	$-0.446094\pi$
$331$	6.13275	0.337087	0.168543	+	0.985694i	$0.446094\pi$
$332$	2.93900	0.161299
$333$	−3.84787	−0.210862
$334$	−14.0000	−0.766046
$335$	9.85384	0.538373
$336$	−10.5918	−0.577830
$337$	−27.8485	−1.51700	−0.758501	−	0.651672i	$-0.774068\pi$
$337$	−27.8485	−1.51700	−0.758501	+	0.651672i	$0.774068\pi$
$338$	0	0
$339$	48.6708	2.64344
$340$	3.18837	0.172914
$341$	−6.45042	−0.349310
$342$	−7.50902	−0.406042
$343$	−27.8431	−1.50339
$344$	0.170915	0.00921513
$345$	−12.7922	−0.688712
$346$	1.20775	0.0649291
$347$	4.43967	0.238334	0.119167	−	0.992874i	$-0.461978\pi$
$347$	4.43967	0.238334	0.119167	+	0.992874i	$0.461978\pi$
$348$	7.04221	0.377502
$349$	−19.9215	−1.06638	−0.533188	−	0.845997i	$-0.679006\pi$
$349$	−19.9215	−1.06638	−0.533188	+	0.845997i	$0.679006\pi$
$350$	18.9095	1.01075
$351$	0	0
$352$	2.69202	0.143485
$353$	−30.5894	−1.62811	−0.814055	−	0.580788i	$-0.802744\pi$
$353$	−30.5894	−1.62811	−0.814055	+	0.580788i	$0.802744\pi$
$354$	7.23729	0.384658
$355$	−8.98792	−0.477029
$356$	12.1806	0.645571
$357$	−37.9409	−2.00805
$358$	16.5157	0.872883
$359$	−21.6039	−1.14021	−0.570104	−	0.821572i	$-0.693097\pi$
$359$	−21.6039	−1.14021	−0.570104	+	0.821572i	$0.693097\pi$
$360$	2.27413	0.119857
$361$	−10.3623	−0.545383
$362$	6.37196	0.334903
$363$	−8.84548	−0.464267
$364$	0	0
$365$	9.74525	0.510090
$366$	−32.9396	−1.72178
$367$	18.7681	0.979686	0.489843	−	0.871811i	$-0.337054\pi$
$367$	18.7681	0.979686	0.489843	+	0.871811i	$0.337054\pi$
$368$	−6.09783	−0.317872
$369$	−9.33273	−0.485843
$370$	−1.34050	−0.0696894
$371$	−27.4034	−1.42271
$372$	−5.64742	−0.292805
$373$	−9.42758	−0.488142	−0.244071	−	0.969757i	$-0.578483\pi$
$373$	−9.42758	−0.488142	−0.244071	+	0.969757i	$0.578483\pi$
$374$	9.64310	0.498633
$375$	−19.3163	−0.997491
$376$	5.20775	0.268569
$377$	0	0
$378$	4.71379	0.242451
$379$	32.0103	1.64426	0.822129	−	0.569302i	$-0.192786\pi$
$379$	32.0103	1.64426	0.822129	+	0.569302i	$0.192786\pi$
$380$	−2.61596	−0.134196
$381$	30.0930	1.54171
$382$	−2.49396	−0.127602
$383$	15.9517	0.815092	0.407546	−	0.913185i	$-0.366385\pi$
$383$	15.9517	0.815092	0.407546	+	0.913185i	$0.366385\pi$
$384$	2.35690	0.120275
$385$	−10.7681	−0.548792
$386$	−4.00538	−0.203868
$387$	0.436681	0.0221978
$388$	12.9661	0.658256
$389$	−18.8659	−0.956540	−0.478270	−	0.878213i	$-0.658736\pi$
$389$	−18.8659	−0.956540	−0.478270	+	0.878213i	$0.658736\pi$
$390$	0	0
$391$	−21.8431	−1.10465
$392$	13.1957	0.666482
$393$	−12.1588	−0.613332
$394$	23.6340	1.19066
$395$	2.50258	0.125919
$396$	6.87800	0.345633
$397$	37.6969	1.89195	0.945977	−	0.324233i	$-0.105106\pi$
$397$	37.6969	1.89195	0.945977	+	0.324233i	$0.105106\pi$
$398$	16.5375	0.828950
$399$	31.1293	1.55841
$400$	−4.20775	−0.210388
$401$	−3.22952	−0.161275	−0.0806373	−	0.996744i	$-0.525696\pi$
$401$	−3.22952	−0.161275	−0.0806373	+	0.996744i	$0.525696\pi$
$402$	26.0925	1.30137
$403$	0	0
$404$	−10.1957	−0.507254
$405$	−9.02284	−0.448348
$406$	−13.4276	−0.666400
$407$	−4.05429	−0.200964
$408$	8.44265	0.417973
$409$	−3.54527	−0.175302	−0.0876511	−	0.996151i	$-0.527936\pi$
$409$	−3.54527	−0.175302	−0.0876511	+	0.996151i	$0.527936\pi$
$410$	−3.25129	−0.160570
$411$	−7.78076	−0.383797
$412$	10.6703	0.525686
$413$	−13.7995	−0.679031
$414$	−15.5797	−0.765701
$415$	2.61596	0.128412
$416$	0	0
$417$	−15.2999	−0.749242
$418$	−7.91185	−0.386981
$419$	−14.4155	−0.704243	−0.352122	−	0.935954i	$-0.614540\pi$
$419$	−14.4155	−0.704243	−0.352122	+	0.935954i	$0.614540\pi$
$420$	−9.42758	−0.460019
$421$	10.0978	0.492138	0.246069	−	0.969252i	$-0.420861\pi$
$421$	10.0978	0.492138	0.246069	+	0.969252i	$0.420861\pi$
$422$	−1.66056	−0.0808349
$423$	13.3056	0.646940
$424$	6.09783	0.296137
$425$	−15.0726	−0.731129
$426$	−23.7995	−1.15309
$427$	62.8068	3.03944
$428$	−20.5623	−0.993914
$429$	0	0
$430$	0.152129	0.00733630
$431$	−20.2198	−0.973955	−0.486978	−	0.873414i	$-0.661901\pi$
$431$	−20.2198	−0.973955	−0.486978	+	0.873414i	$0.661901\pi$
$432$	−1.04892	−0.0504661
$433$	36.4849	1.75335	0.876675	−	0.481083i	$-0.159756\pi$
$433$	36.4849	1.75335	0.876675	+	0.481083i	$0.159756\pi$
$434$	10.7681	0.516885
$435$	6.26816	0.300535
$436$	2.71379	0.129967
$437$	17.9215	0.857304
$438$	25.8049	1.23301
$439$	−28.3612	−1.35361	−0.676803	−	0.736164i	$-0.736635\pi$
$439$	−28.3612	−1.35361	−0.676803	+	0.736164i	$0.736635\pi$
$440$	2.39612	0.114231
$441$	33.7144	1.60545
$442$	0	0
$443$	2.80061	0.133061	0.0665305	−	0.997784i	$-0.478807\pi$
$443$	2.80061	0.133061	0.0665305	+	0.997784i	$0.478807\pi$
$444$	−3.54958	−0.168456
$445$	10.8418	0.513949
$446$	−0.792249	−0.0375141
$447$	11.8586	0.560894
$448$	−4.49396	−0.212320
$449$	−13.2760	−0.626535	−0.313268	−	0.949665i	$-0.601424\pi$
$449$	−13.2760	−0.626535	−0.313268	+	0.949665i	$0.601424\pi$
$450$	−10.7506	−0.506789
$451$	−9.83340	−0.463037
$452$	20.6504	0.971313
$453$	−4.06770	−0.191117
$454$	−21.7603	−1.02126
$455$	0	0
$456$	−6.92692	−0.324383
$457$	−13.6474	−0.638399	−0.319200	−	0.947688i	$-0.603414\pi$
$457$	−13.6474	−0.638399	−0.319200	+	0.947688i	$0.603414\pi$
$458$	−1.97584	−0.0923248
$459$	−3.75733	−0.175377
$460$	−5.42758	−0.253062
$461$	12.1655	0.566606	0.283303	−	0.959031i	$-0.408570\pi$
$461$	12.1655	0.566606	0.283303	+	0.959031i	$0.408570\pi$
$462$	−28.5133	−1.32656
$463$	−4.24996	−0.197513	−0.0987563	−	0.995112i	$-0.531486\pi$
$463$	−4.24996	−0.197513	−0.0987563	+	0.995112i	$0.531486\pi$
$464$	2.98792	0.138711
$465$	−5.02667	−0.233106
$466$	18.2349	0.844715
$467$	8.21552	0.380169	0.190084	−	0.981768i	$-0.439124\pi$
$467$	8.21552	0.380169	0.190084	+	0.981768i	$0.439124\pi$
$468$	0	0
$469$	−49.7512	−2.29730
$470$	4.63533	0.213812
$471$	36.5461	1.68396
$472$	3.07069	0.141340
$473$	0.460107	0.0211558
$474$	6.62671	0.304375
$475$	12.3666	0.567418
$476$	−16.0978	−0.737843
$477$	15.5797	0.713346
$478$	−22.0737	−1.00963
$479$	31.0267	1.41764	0.708822	−	0.705387i	$-0.249227\pi$
$479$	31.0267	1.41764	0.708822	+	0.705387i	$0.249227\pi$
$480$	2.09783	0.0957526
$481$	0	0
$482$	6.98792	0.318291
$483$	64.5870	2.93881
$484$	−3.75302	−0.170592
$485$	11.5410	0.524048
$486$	−20.7453	−0.941024
$487$	−10.9987	−0.498397	−0.249199	−	0.968452i	$-0.580167\pi$
$487$	−10.9987	−0.498397	−0.249199	+	0.968452i	$0.580167\pi$
$488$	−13.9758	−0.632656
$489$	−46.4862	−2.10218
$490$	11.7453	0.530596
$491$	21.2336	0.958258	0.479129	−	0.877745i	$-0.340953\pi$
$491$	21.2336	0.958258	0.479129	+	0.877745i	$0.340953\pi$
$492$	−8.60925	−0.388135
$493$	10.7030	0.482041
$494$	0	0
$495$	6.12200	0.275163
$496$	−2.39612	−0.107589
$497$	45.3793	2.03554
$498$	6.92692	0.310403
$499$	−16.8635	−0.754915	−0.377458	−	0.926027i	$-0.623202\pi$
$499$	−16.8635	−0.754915	−0.377458	+	0.926027i	$0.623202\pi$
$500$	−8.19567	−0.366521
$501$	−32.9965	−1.47418
$502$	−12.5593	−0.560548
$503$	20.3806	0.908725	0.454363	−	0.890817i	$-0.349867\pi$
$503$	20.3806	0.908725	0.454363	+	0.890817i	$0.349867\pi$
$504$	−11.4819	−0.511443
$505$	−9.07500	−0.403832
$506$	−16.4155	−0.729758
$507$	0	0
$508$	12.7681	0.566492
$509$	21.9215	0.971655	0.485828	−	0.874055i	$-0.338518\pi$
$509$	21.9215	0.971655	0.485828	+	0.874055i	$0.338518\pi$
$510$	7.51466	0.332755
$511$	−49.2030	−2.17661
$512$	1.00000	0.0441942
$513$	3.08277	0.136108
$514$	−16.9933	−0.749542
$515$	9.49742	0.418506
$516$	0.402829	0.0177336
$517$	14.0194	0.616572
$518$	6.76809	0.297373
$519$	2.84654	0.124949
$520$	0	0
$521$	26.1564	1.14593	0.572967	−	0.819578i	$-0.305792\pi$
$521$	26.1564	1.14593	0.572967	+	0.819578i	$0.305792\pi$
$522$	7.63401	0.334131
$523$	4.91185	0.214780	0.107390	−	0.994217i	$-0.465751\pi$
$523$	4.91185	0.214780	0.107390	+	0.994217i	$0.465751\pi$
$524$	−5.15883	−0.225365
$525$	44.5676	1.94509
$526$	−4.39612	−0.191680
$527$	−8.58317	−0.373889
$528$	6.34481	0.276123
$529$	14.1836	0.616678
$530$	5.42758	0.235759
$531$	7.84548	0.340465
$532$	13.2078	0.572629
$533$	0	0
$534$	28.7084	1.24233
$535$	−18.3021	−0.791270
$536$	11.0707	0.478181
$537$	38.9259	1.67977
$538$	15.5603	0.670854
$539$	35.5230	1.53009
$540$	−0.933624	−0.0401768
$541$	0.459042	0.0197358	0.00986789	−	0.999951i	$-0.496859\pi$
$541$	0.459042	0.0197358	0.00986789	+	0.999951i	$0.496859\pi$
$542$	−21.9952	−0.944775
$543$	15.0180	0.644486
$544$	3.58211	0.153581
$545$	2.41550	0.103469
$546$	0	0
$547$	−20.3327	−0.869365	−0.434682	−	0.900584i	$-0.643139\pi$
$547$	−20.3327	−0.869365	−0.434682	+	0.900584i	$0.643139\pi$
$548$	−3.30127	−0.141023
$549$	−35.7077	−1.52397
$550$	−11.3274	−0.483000
$551$	−8.78150	−0.374104
$552$	−14.3720	−0.611711
$553$	−12.6353	−0.537309
$554$	−1.87800	−0.0797887
$555$	−3.15942	−0.134110
$556$	−6.49157	−0.275304
$557$	−43.9469	−1.86209	−0.931045	−	0.364905i	$-0.881101\pi$
$557$	−43.9469	−1.86209	−0.931045	+	0.364905i	$0.881101\pi$
$558$	−6.12200	−0.259165
$559$	0	0
$560$	−4.00000	−0.169031
$561$	22.7278	0.959568
$562$	9.20536	0.388305
$563$	43.5991	1.83748	0.918741	−	0.394860i	$-0.129207\pi$
$563$	43.5991	1.83748	0.918741	+	0.394860i	$0.129207\pi$
$564$	12.2741	0.516834
$565$	18.3806	0.773277
$566$	4.70841	0.197909
$567$	45.5555	1.91315
$568$	−10.0978	−0.423696
$569$	8.28919	0.347501	0.173751	−	0.984790i	$-0.444411\pi$
$569$	8.28919	0.347501	0.173751	+	0.984790i	$0.444411\pi$
$570$	−6.16554	−0.258246
$571$	−22.9836	−0.961834	−0.480917	−	0.876766i	$-0.659696\pi$
$571$	−22.9836	−0.961834	−0.480917	+	0.876766i	$0.659696\pi$
$572$	0	0
$573$	−5.87800	−0.245557
$574$	16.4155	0.685170
$575$	25.6582	1.07002
$576$	2.55496	0.106457
$577$	0.553630	0.0230479	0.0115240	−	0.999934i	$-0.496332\pi$
$577$	0.553630	0.0230479	0.0115240	+	0.999934i	$0.496332\pi$
$578$	−4.16852	−0.173388
$579$	−9.44026	−0.392324
$580$	2.65950	0.110430
$581$	−13.2078	−0.547950
$582$	30.5599	1.26675
$583$	16.4155	0.679861
$584$	10.9487	0.453060
$585$	0	0
$586$	−13.7017	−0.566012
$587$	−16.0355	−0.661856	−0.330928	−	0.943656i	$-0.607362\pi$
$587$	−16.0355	−0.661856	−0.330928	+	0.943656i	$0.607362\pi$
$588$	31.1008	1.28258
$589$	7.04221	0.290169
$590$	2.73317	0.112523
$591$	55.7029	2.29131
$592$	−1.50604	−0.0618979
$593$	25.9976	1.06759	0.533797	−	0.845613i	$-0.320765\pi$
$593$	25.9976	1.06759	0.533797	+	0.845613i	$0.320765\pi$
$594$	−2.82371	−0.115858
$595$	−14.3284	−0.587408
$596$	5.03146	0.206097
$597$	38.9772	1.59523
$598$	0	0
$599$	−16.2150	−0.662529	−0.331264	−	0.943538i	$-0.607475\pi$
$599$	−16.2150	−0.662529	−0.331264	+	0.943538i	$0.607475\pi$
$600$	−9.91723	−0.404869
$601$	−29.5200	−1.20415	−0.602074	−	0.798440i	$-0.705659\pi$
$601$	−29.5200	−1.20415	−0.602074	+	0.798440i	$0.705659\pi$
$602$	−0.768086	−0.0313048
$603$	28.2851	1.15186
$604$	−1.72587	−0.0702248
$605$	−3.34050	−0.135811
$606$	−24.0301	−0.976157
$607$	37.4228	1.51894	0.759472	−	0.650540i	$-0.225457\pi$
$607$	37.4228	1.51894	0.759472	+	0.650540i	$0.225457\pi$
$608$	−2.93900	−0.119192
$609$	−31.6474	−1.28242
$610$	−12.4397	−0.503667
$611$	0	0
$612$	9.15213	0.369953
$613$	−38.0737	−1.53778	−0.768891	−	0.639380i	$-0.779191\pi$
$613$	−38.0737	−1.53778	−0.768891	+	0.639380i	$0.779191\pi$
$614$	8.03252	0.324166
$615$	−7.66296	−0.309000
$616$	−12.0978	−0.487436
$617$	−41.6383	−1.67630	−0.838148	−	0.545443i	$-0.816361\pi$
$617$	−41.6383	−1.67630	−0.838148	+	0.545443i	$0.816361\pi$
$618$	25.1487	1.01163
$619$	1.67158	0.0671864	0.0335932	−	0.999436i	$-0.489305\pi$
$619$	1.67158	0.0671864	0.0335932	+	0.999436i	$0.489305\pi$
$620$	−2.13275	−0.0856534
$621$	6.39612	0.256668
$622$	4.09783	0.164308
$623$	−54.7391	−2.19308
$624$	0	0
$625$	13.7439	0.549757
$626$	4.37435	0.174834
$627$	−18.6474	−0.744706
$628$	15.5060	0.618758
$629$	−5.39480	−0.215105
$630$	−10.2198	−0.407168
$631$	−8.70304	−0.346462	−0.173231	−	0.984881i	$-0.555421\pi$
$631$	−8.70304	−0.346462	−0.173231	+	0.984881i	$0.555421\pi$
$632$	2.81163	0.111840
$633$	−3.91377	−0.155559
$634$	−20.3612	−0.808647
$635$	11.3647	0.450993
$636$	14.3720	0.569885
$637$	0	0
$638$	8.04354	0.318447
$639$	−25.7995	−1.02061
$640$	0.890084	0.0351836
$641$	−19.9075	−0.786301	−0.393150	−	0.919474i	$-0.628615\pi$
$641$	−19.9075	−0.786301	−0.393150	+	0.919474i	$0.628615\pi$
$642$	−48.4631	−1.91269
$643$	−27.0756	−1.06776	−0.533879	−	0.845561i	$-0.679266\pi$
$643$	−27.0756	−1.06776	−0.533879	+	0.845561i	$0.679266\pi$
$644$	27.4034	1.07985
$645$	0.358552	0.0141180
$646$	−10.5278	−0.414211
$647$	9.43237	0.370825	0.185412	−	0.982661i	$-0.440638\pi$
$647$	9.43237	0.370825	0.185412	+	0.982661i	$0.440638\pi$
$648$	−10.1371	−0.398221
$649$	8.26636	0.324483
$650$	0	0
$651$	25.3793	0.994692
$652$	−19.7235	−0.772431
$653$	−37.6292	−1.47255	−0.736273	−	0.676685i	$-0.763416\pi$
$653$	−37.6292	−1.47255	−0.736273	+	0.676685i	$0.763416\pi$
$654$	6.39612	0.250108
$655$	−4.59179	−0.179416
$656$	−3.65279	−0.142618
$657$	27.9734	1.09135
$658$	−23.4034	−0.912360
$659$	−32.7724	−1.27663	−0.638316	−	0.769775i	$-0.720369\pi$
$659$	−32.7724	−1.27663	−0.638316	+	0.769775i	$0.720369\pi$
$660$	5.64742	0.219825
$661$	20.1957	0.785520	0.392760	−	0.919641i	$-0.371520\pi$
$661$	20.1957	0.785520	0.392760	+	0.919641i	$0.371520\pi$
$662$	6.13275	0.238356
$663$	0	0
$664$	2.93900	0.114055
$665$	11.7560	0.455878
$666$	−3.84787	−0.149102
$667$	−18.2198	−0.705475
$668$	−14.0000	−0.541676
$669$	−1.86725	−0.0721920
$670$	9.85384	0.380687
$671$	−37.6233	−1.45243
$672$	−10.5918	−0.408587
$673$	−30.3435	−1.16966	−0.584828	−	0.811158i	$-0.698838\pi$
$673$	−30.3435	−1.16966	−0.584828	+	0.811158i	$0.698838\pi$
$674$	−27.8485	−1.07268
$675$	4.41358	0.169879
$676$	0	0
$677$	−21.1642	−0.813407	−0.406703	−	0.913560i	$-0.633322\pi$
$677$	−21.1642	−0.813407	−0.406703	+	0.913560i	$0.633322\pi$
$678$	48.6708	1.86919
$679$	−58.2693	−2.23617
$680$	3.18837	0.122269
$681$	−51.2868	−1.96531
$682$	−6.45042	−0.246999
$683$	35.4873	1.35788	0.678941	−	0.734193i	$-0.262439\pi$
$683$	35.4873	1.35788	0.678941	+	0.734193i	$0.262439\pi$
$684$	−7.50902	−0.287115
$685$	−2.93841	−0.112271
$686$	−27.8431	−1.06305
$687$	−4.65684	−0.177670
$688$	0.170915	0.00651608
$689$	0	0
$690$	−12.7922	−0.486993
$691$	−36.7472	−1.39793	−0.698964	−	0.715157i	$-0.746355\pi$
$691$	−36.7472	−1.39793	−0.698964	+	0.715157i	$0.746355\pi$
$692$	1.20775	0.0459118
$693$	−30.9095	−1.17415
$694$	4.43967	0.168527
$695$	−5.77804	−0.219173
$696$	7.04221	0.266934
$697$	−13.0847	−0.495618
$698$	−19.9215	−0.754042
$699$	42.9778	1.62557
$700$	18.9095	0.714710
$701$	19.7668	0.746580	0.373290	−	0.927715i	$-0.378230\pi$
$701$	19.7668	0.746580	0.373290	+	0.927715i	$0.378230\pi$
$702$	0	0
$703$	4.42626	0.166939
$704$	2.69202	0.101459
$705$	10.9250	0.411459
$706$	−30.5894	−1.15125
$707$	45.8189	1.72320
$708$	7.23729	0.271994
$709$	−11.6280	−0.436700	−0.218350	−	0.975871i	$-0.570067\pi$
$709$	−11.6280	−0.436700	−0.218350	+	0.975871i	$0.570067\pi$
$710$	−8.98792	−0.337311
$711$	7.18359	0.269406
$712$	12.1806	0.456487
$713$	14.6112	0.547193
$714$	−37.9409	−1.41990
$715$	0	0
$716$	16.5157	0.617222
$717$	−52.0253	−1.94292
$718$	−21.6039	−0.806249
$719$	−1.06638	−0.0397691	−0.0198846	−	0.999802i	$-0.506330\pi$
$719$	−1.06638	−0.0397691	−0.0198846	+	0.999802i	$0.506330\pi$
$720$	2.27413	0.0847517
$721$	−47.9517	−1.78581
$722$	−10.3623	−0.385644
$723$	16.4698	0.612518
$724$	6.37196	0.236812
$725$	−12.5724	−0.466928
$726$	−8.84548	−0.328286
$727$	−1.26205	−0.0468067	−0.0234033	−	0.999726i	$-0.507450\pi$
$727$	−1.26205	−0.0468067	−0.0234033	+	0.999726i	$0.507450\pi$
$728$	0	0
$729$	−18.4832	−0.684563
$730$	9.74525	0.360688
$731$	0.612236	0.0226444
$732$	−32.9396	−1.21748
$733$	20.6789	0.763792	0.381896	−	0.924205i	$-0.375271\pi$
$733$	20.6789	0.763792	0.381896	+	0.924205i	$0.375271\pi$
$734$	18.7681	0.692743
$735$	27.6823	1.02108
$736$	−6.09783	−0.224769
$737$	29.8025	1.09779
$738$	−9.33273	−0.343543
$739$	−5.17331	−0.190303	−0.0951516	−	0.995463i	$-0.530334\pi$
$739$	−5.17331	−0.190303	−0.0951516	+	0.995463i	$0.530334\pi$
$740$	−1.34050	−0.0492778
$741$	0	0
$742$	−27.4034	−1.00601
$743$	10.4397	0.382994	0.191497	−	0.981493i	$-0.438666\pi$
$743$	10.4397	0.382994	0.191497	+	0.981493i	$0.438666\pi$
$744$	−5.64742	−0.207044
$745$	4.47842	0.164077
$746$	−9.42758	−0.345168
$747$	7.50902	0.274741
$748$	9.64310	0.352587
$749$	92.4059	3.37644
$750$	−19.3163	−0.705333
$751$	−14.0435	−0.512456	−0.256228	−	0.966616i	$-0.582480\pi$
$751$	−14.0435	−0.512456	−0.256228	+	0.966616i	$0.582480\pi$
$752$	5.20775	0.189907
$753$	−29.6009	−1.07872
$754$	0	0
$755$	−1.53617	−0.0559070
$756$	4.71379	0.171439
$757$	−9.30559	−0.338217	−0.169109	−	0.985597i	$-0.554089\pi$
$757$	−9.30559	−0.338217	−0.169109	+	0.985597i	$0.554089\pi$
$758$	32.0103	1.16267
$759$	−38.6896	−1.40434
$760$	−2.61596	−0.0948907
$761$	9.56273	0.346649	0.173324	−	0.984865i	$-0.444549\pi$
$761$	9.56273	0.346649	0.173324	+	0.984865i	$0.444549\pi$
$762$	30.0930	1.09016
$763$	−12.1957	−0.441513
$764$	−2.49396	−0.0902283
$765$	8.14616	0.294525
$766$	15.9517	0.576357
$767$	0	0
$768$	2.35690	0.0850472
$769$	31.9299	1.15142	0.575711	−	0.817653i	$-0.304725\pi$
$769$	31.9299	1.15142	0.575711	+	0.817653i	$0.304725\pi$
$770$	−10.7681	−0.388055
$771$	−40.0514	−1.44242
$772$	−4.00538	−0.144157
$773$	34.1172	1.22711	0.613555	−	0.789652i	$-0.289739\pi$
$773$	34.1172	1.22711	0.613555	+	0.789652i	$0.289739\pi$
$774$	0.436681	0.0156962
$775$	10.0823	0.362167
$776$	12.9661	0.465458
$777$	15.9517	0.572263
$778$	−18.8659	−0.676376
$779$	10.7356	0.384641
$780$	0	0
$781$	−27.1836	−0.972705
$782$	−21.8431	−0.781107
$783$	−3.13408	−0.112003
$784$	13.1957	0.471274
$785$	13.8017	0.492603
$786$	−12.1588	−0.433691
$787$	37.3467	1.33127	0.665634	−	0.746279i	$-0.268161\pi$
$787$	37.3467	1.33127	0.665634	+	0.746279i	$0.268161\pi$
$788$	23.6340	0.841927
$789$	−10.3612	−0.368869
$790$	2.50258	0.0890379
$791$	−92.8021	−3.29966
$792$	6.87800	0.244399
$793$	0	0
$794$	37.6969	1.33781
$795$	12.7922	0.453694
$796$	16.5375	0.586156
$797$	−14.9831	−0.530730	−0.265365	−	0.964148i	$-0.585492\pi$
$797$	−14.9831	−0.530730	−0.265365	+	0.964148i	$0.585492\pi$
$798$	31.1293	1.10197
$799$	18.6547	0.659956
$800$	−4.20775	−0.148766
$801$	31.1209	1.09960
$802$	−3.22952	−0.114038
$803$	29.4741	1.04012
$804$	26.0925	0.920210
$805$	24.3913	0.859682
$806$	0	0
$807$	36.6741	1.29099
$808$	−10.1957	−0.358682
$809$	25.6770	0.902754	0.451377	−	0.892333i	$-0.350933\pi$
$809$	25.6770	0.902754	0.451377	+	0.892333i	$0.350933\pi$
$810$	−9.02284	−0.317030
$811$	−8.66786	−0.304370	−0.152185	−	0.988352i	$-0.548631\pi$
$811$	−8.66786	−0.304370	−0.152185	+	0.988352i	$0.548631\pi$
$812$	−13.4276	−0.471216
$813$	−51.8404	−1.81812
$814$	−4.05429	−0.142103
$815$	−17.5555	−0.614944
$816$	8.44265	0.295552
$817$	−0.502320	−0.0175739
$818$	−3.54527	−0.123957
$819$	0	0
$820$	−3.25129	−0.113540
$821$	54.6547	1.90746	0.953731	−	0.300660i	$-0.0972070\pi$
$821$	54.6547	1.90746	0.953731	+	0.300660i	$0.0972070\pi$
$822$	−7.78076	−0.271385
$823$	−6.33704	−0.220895	−0.110448	−	0.993882i	$-0.535228\pi$
$823$	−6.33704	−0.220895	−0.110448	+	0.993882i	$0.535228\pi$
$824$	10.6703	0.371716
$825$	−26.6974	−0.929484
$826$	−13.7995	−0.480148
$827$	−35.5405	−1.23586	−0.617932	−	0.786232i	$-0.712029\pi$
$827$	−35.5405	−1.23586	−0.617932	+	0.786232i	$0.712029\pi$
$828$	−15.5797	−0.541432
$829$	−41.5555	−1.44328	−0.721642	−	0.692267i	$-0.756612\pi$
$829$	−41.5555	−1.44328	−0.721642	+	0.692267i	$0.756612\pi$
$830$	2.61596	0.0908012
$831$	−4.42626	−0.153545
$832$	0	0
$833$	47.2683	1.63775
$834$	−15.2999	−0.529794
$835$	−12.4612	−0.431237
$836$	−7.91185	−0.273637
$837$	2.51334	0.0868736
$838$	−14.4155	−0.497975
$839$	−17.5496	−0.605879	−0.302939	−	0.953010i	$-0.597968\pi$
$839$	−17.5496	−0.605879	−0.302939	+	0.953010i	$0.597968\pi$
$840$	−9.42758	−0.325283
$841$	−20.0723	−0.692150
$842$	10.0978	0.347994
$843$	21.6961	0.747252
$844$	−1.66056	−0.0571589
$845$	0	0
$846$	13.3056	0.457455
$847$	16.8659	0.579520
$848$	6.09783	0.209401
$849$	11.0972	0.380856
$850$	−15.0726	−0.516986
$851$	9.18359	0.314809
$852$	−23.7995	−0.815359
$853$	−20.1414	−0.689628	−0.344814	−	0.938671i	$-0.612058\pi$
$853$	−20.1414	−0.689628	−0.344814	+	0.938671i	$0.612058\pi$
$854$	62.8068	2.14921
$855$	−6.68366	−0.228576
$856$	−20.5623	−0.702803
$857$	−8.85756	−0.302568	−0.151284	−	0.988490i	$-0.548341\pi$
$857$	−8.85756	−0.302568	−0.151284	+	0.988490i	$0.548341\pi$
$858$	0	0
$859$	28.8810	0.985407	0.492703	−	0.870197i	$-0.336009\pi$
$859$	28.8810	0.985407	0.492703	+	0.870197i	$0.336009\pi$
$860$	0.152129	0.00518755
$861$	38.6896	1.31854
$862$	−20.2198	−0.688690
$863$	−3.90813	−0.133034	−0.0665172	−	0.997785i	$-0.521189\pi$
$863$	−3.90813	−0.133034	−0.0665172	+	0.997785i	$0.521189\pi$
$864$	−1.04892	−0.0356849
$865$	1.07500	0.0365511
$866$	36.4849	1.23981
$867$	−9.82477	−0.333667
$868$	10.7681	0.365493
$869$	7.56896	0.256759
$870$	6.26816	0.212510
$871$	0	0
$872$	2.71379	0.0919006
$873$	33.1280	1.12121
$874$	17.9215	0.606205
$875$	36.8310	1.24512
$876$	25.8049	0.871868
$877$	44.9879	1.51913	0.759567	−	0.650429i	$-0.225411\pi$
$877$	44.9879	1.51913	0.759567	+	0.650429i	$0.225411\pi$
$878$	−28.3612	−0.957144
$879$	−32.2935	−1.08923
$880$	2.39612	0.0807733
$881$	−14.1933	−0.478184	−0.239092	−	0.970997i	$-0.576850\pi$
$881$	−14.1933	−0.478184	−0.239092	+	0.970997i	$0.576850\pi$
$882$	33.7144	1.13522
$883$	48.4626	1.63090	0.815448	−	0.578830i	$-0.196490\pi$
$883$	48.4626	1.63090	0.815448	+	0.578830i	$0.196490\pi$
$884$	0	0
$885$	6.44179	0.216539
$886$	2.80061	0.0940883
$887$	30.8611	1.03622	0.518108	−	0.855315i	$-0.326637\pi$
$887$	30.8611	1.03622	0.518108	+	0.855315i	$0.326637\pi$
$888$	−3.54958	−0.119116
$889$	−57.3793	−1.92444
$890$	10.8418	0.363417
$891$	−27.2892	−0.914222
$892$	−0.792249	−0.0265265
$893$	−15.3056	−0.512182
$894$	11.8586	0.396612
$895$	14.7004	0.491380
$896$	−4.49396	−0.150133
$897$	0	0
$898$	−13.2760	−0.443027
$899$	−7.15942	−0.238780
$900$	−10.7506	−0.358354
$901$	21.8431	0.727699
$902$	−9.83340	−0.327416
$903$	−1.81030	−0.0602430
$904$	20.6504	0.686822
$905$	5.67158	0.188530
$906$	−4.06770	−0.135140
$907$	3.94139	0.130872	0.0654359	−	0.997857i	$-0.479156\pi$
$907$	3.94139	0.130872	0.0654359	+	0.997857i	$0.479156\pi$
$908$	−21.7603	−0.722141
$909$	−26.0495	−0.864008
$910$	0	0
$911$	−37.1943	−1.23230	−0.616152	−	0.787627i	$-0.711309\pi$
$911$	−37.1943	−1.23230	−0.616152	+	0.787627i	$0.711309\pi$
$912$	−6.92692	−0.229373
$913$	7.91185	0.261844
$914$	−13.6474	−0.451416
$915$	−29.3190	−0.969256
$916$	−1.97584	−0.0652835
$917$	23.1836	0.765590
$918$	−3.75733	−0.124010
$919$	0.681005	0.0224643	0.0112321	−	0.999937i	$-0.496425\pi$
$919$	0.681005	0.0224643	0.0112321	+	0.999937i	$0.496425\pi$
$920$	−5.42758	−0.178942
$921$	18.9318	0.623825
$922$	12.1655	0.400651
$923$	0	0
$924$	−28.5133	−0.938020
$925$	6.33704	0.208361
$926$	−4.24996	−0.139662
$927$	27.2620	0.895403
$928$	2.98792	0.0980832
$929$	−32.4355	−1.06417	−0.532087	−	0.846690i	$-0.678592\pi$
$929$	−32.4355	−1.06417	−0.532087	+	0.846690i	$0.678592\pi$
$930$	−5.02667	−0.164831
$931$	−38.7821	−1.27103
$932$	18.2349	0.597304
$933$	9.65817	0.316194
$934$	8.21552	0.268820
$935$	8.58317	0.280700
$936$	0	0
$937$	−14.6165	−0.477502	−0.238751	−	0.971081i	$-0.576738\pi$
$937$	−14.6165	−0.477502	−0.238751	+	0.971081i	$0.576738\pi$
$938$	−49.7512	−1.62443
$939$	10.3099	0.336451
$940$	4.63533	0.151188
$941$	30.1763	0.983719	0.491860	−	0.870675i	$-0.336317\pi$
$941$	30.1763	0.983719	0.491860	+	0.870675i	$0.336317\pi$
$942$	36.5461	1.19074
$943$	22.2741	0.725345
$944$	3.07069	0.0999424
$945$	4.19567	0.136485
$946$	0.460107	0.0149594
$947$	20.6708	0.671712	0.335856	−	0.941913i	$-0.390974\pi$
$947$	20.6708	0.671712	0.335856	+	0.941913i	$0.390974\pi$
$948$	6.62671	0.215226
$949$	0	0
$950$	12.3666	0.401225
$951$	−47.9892	−1.55616
$952$	−16.0978	−0.521734
$953$	−47.2411	−1.53029	−0.765145	−	0.643858i	$-0.777333\pi$
$953$	−47.2411	−1.53029	−0.765145	+	0.643858i	$0.777333\pi$
$954$	15.5797	0.504412
$955$	−2.21983	−0.0718321
$956$	−22.0737	−0.713914
$957$	18.9578	0.612818
$958$	31.0267	1.00243
$959$	14.8358	0.479073
$960$	2.09783	0.0677073
$961$	−25.2586	−0.814793
$962$	0	0
$963$	−52.5357	−1.69294
$964$	6.98792	0.225066
$965$	−3.56512	−0.114765
$966$	64.5870	2.07805
$967$	53.9517	1.73497	0.867484	−	0.497464i	$-0.165735\pi$
$967$	53.9517	1.73497	0.867484	+	0.497464i	$0.165735\pi$
$968$	−3.75302	−0.120627
$969$	−24.8130	−0.797107
$970$	11.5410	0.370558
$971$	49.8920	1.60111	0.800555	−	0.599259i	$-0.204538\pi$
$971$	49.8920	1.60111	0.800555	+	0.599259i	$0.204538\pi$
$972$	−20.7453	−0.665404
$973$	29.1728	0.935238
$974$	−10.9987	−0.352420
$975$	0	0
$976$	−13.9758	−0.447356
$977$	29.1299	0.931948	0.465974	−	0.884799i	$-0.345704\pi$
$977$	29.1299	0.931948	0.465974	+	0.884799i	$0.345704\pi$
$978$	−46.4862	−1.48646
$979$	32.7904	1.04799
$980$	11.7453	0.375188
$981$	6.93362	0.221374
$982$	21.2336	0.677590
$983$	−41.3309	−1.31825	−0.659126	−	0.752033i	$-0.729074\pi$
$983$	−41.3309	−1.31825	−0.659126	+	0.752033i	$0.729074\pi$
$984$	−8.60925	−0.274453
$985$	21.0362	0.670270
$986$	10.7030	0.340854
$987$	−55.1594	−1.75574
$988$	0	0
$989$	−1.04221	−0.0331404
$990$	6.12200	0.194570
$991$	19.2185	0.610496	0.305248	−	0.952273i	$-0.401261\pi$
$991$	19.2185	0.610496	0.305248	+	0.952273i	$0.401261\pi$
$992$	−2.39612	−0.0760770
$993$	14.4543	0.458692
$994$	45.3793	1.43934
$995$	14.7198	0.466648
$996$	6.92692	0.219488
$997$	22.4940	0.712391	0.356195	−	0.934411i	$-0.384074\pi$
$997$	22.4940	0.712391	0.356195	+	0.934411i	$0.384074\pi$
$998$	−16.8635	−0.533806
$999$	1.57971	0.0499799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.2.a.h.1.2	yes	3
3.2	odd	2		3042.2.a.z.1.2		3
4.3	odd	2		2704.2.a.w.1.2		3
5.4	even	2		8450.2.a.bn.1.2		3
13.2	odd	12		338.2.e.e.147.5		12
13.3	even	3		338.2.c.h.191.2		6
13.4	even	6		338.2.c.i.315.2		6
13.5	odd	4		338.2.b.d.337.2		6
13.6	odd	12		338.2.e.e.23.2		12
13.7	odd	12		338.2.e.e.23.5		12
13.8	odd	4		338.2.b.d.337.5		6
13.9	even	3		338.2.c.h.315.2		6
13.10	even	6		338.2.c.i.191.2		6
13.11	odd	12		338.2.e.e.147.2		12
13.12	even	2		338.2.a.g.1.2	✓	3
39.5	even	4		3042.2.b.n.1351.5		6
39.8	even	4		3042.2.b.n.1351.2		6
39.38	odd	2		3042.2.a.bi.1.2		3
52.31	even	4		2704.2.f.m.337.3		6
52.47	even	4		2704.2.f.m.337.4		6
52.51	odd	2		2704.2.a.v.1.2		3
65.64	even	2		8450.2.a.bx.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.2	✓	3	13.12	even	2
338.2.a.h.1.2	yes	3	1.1	even	1	trivial
338.2.b.d.337.2		6	13.5	odd	4
338.2.b.d.337.5		6	13.8	odd	4
338.2.c.h.191.2		6	13.3	even	3
338.2.c.h.315.2		6	13.9	even	3
338.2.c.i.191.2		6	13.10	even	6
338.2.c.i.315.2		6	13.4	even	6
338.2.e.e.23.2		12	13.6	odd	12
338.2.e.e.23.5		12	13.7	odd	12
338.2.e.e.147.2		12	13.11	odd	12
338.2.e.e.147.5		12	13.2	odd	12
2704.2.a.v.1.2		3	52.51	odd	2
2704.2.a.w.1.2		3	4.3	odd	2
2704.2.f.m.337.3		6	52.31	even	4
2704.2.f.m.337.4		6	52.47	even	4
3042.2.a.z.1.2		3	3.2	odd	2
3042.2.a.bi.1.2		3	39.38	odd	2
3042.2.b.n.1351.2		6	39.8	even	4
3042.2.b.n.1351.5		6	39.5	even	4
8450.2.a.bn.1.2		3	5.4	even	2
8450.2.a.bx.1.2		3	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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.35690 q^{3} +1.00000 q^{4} +0.890084 q^{5} +2.35690 q^{6} -4.49396 q^{7} +1.00000 q^{8} +2.55496 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.35690 q^{3} +1.00000 q^{4} +0.890084 q^{5} +2.35690 q^{6} -4.49396 q^{7} +1.00000 q^{8} +2.55496 q^{9} +0.890084 q^{10} +2.69202 q^{11} +2.35690 q^{12} -4.49396 q^{14} +2.09783 q^{15} +1.00000 q^{16} +3.58211 q^{17} +2.55496 q^{18} -2.93900 q^{19} +0.890084 q^{20} -10.5918 q^{21} +2.69202 q^{22} -6.09783 q^{23} +2.35690 q^{24} -4.20775 q^{25} -1.04892 q^{27} -4.49396 q^{28} +2.98792 q^{29} +2.09783 q^{30} -2.39612 q^{31} +1.00000 q^{32} +6.34481 q^{33} +3.58211 q^{34} -4.00000 q^{35} +2.55496 q^{36} -1.50604 q^{37} -2.93900 q^{38} +0.890084 q^{40} -3.65279 q^{41} -10.5918 q^{42} +0.170915 q^{43} +2.69202 q^{44} +2.27413 q^{45} -6.09783 q^{46} +5.20775 q^{47} +2.35690 q^{48} +13.1957 q^{49} -4.20775 q^{50} +8.44265 q^{51} +6.09783 q^{53} -1.04892 q^{54} +2.39612 q^{55} -4.49396 q^{56} -6.92692 q^{57} +2.98792 q^{58} +3.07069 q^{59} +2.09783 q^{60} -13.9758 q^{61} -2.39612 q^{62} -11.4819 q^{63} +1.00000 q^{64} +6.34481 q^{66} +11.0707 q^{67} +3.58211 q^{68} -14.3720 q^{69} -4.00000 q^{70} -10.0978 q^{71} +2.55496 q^{72} +10.9487 q^{73} -1.50604 q^{74} -9.91723 q^{75} -2.93900 q^{76} -12.0978 q^{77} +2.81163 q^{79} +0.890084 q^{80} -10.1371 q^{81} -3.65279 q^{82} +2.93900 q^{83} -10.5918 q^{84} +3.18837 q^{85} +0.170915 q^{86} +7.04221 q^{87} +2.69202 q^{88} +12.1806 q^{89} +2.27413 q^{90} -6.09783 q^{92} -5.64742 q^{93} +5.20775 q^{94} -2.61596 q^{95} +2.35690 q^{96} +12.9661 q^{97} +13.1957 q^{98} +6.87800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 3 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} + 3 q^{12} - 4 q^{14} - 12 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} + q^{19} + 2 q^{20} - 4 q^{21} + 3 q^{22} + 3 q^{24} + 5 q^{25} + 6 q^{27} - 4 q^{28} - 10 q^{29} - 12 q^{30} - 16 q^{31} + 3 q^{32} - 4 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 14 q^{37} + q^{38} + 2 q^{40} + 7 q^{41} - 4 q^{42} + 11 q^{43} + 3 q^{44} - 4 q^{45} - 2 q^{47} + 3 q^{48} + 3 q^{49} + 5 q^{50} - 16 q^{51} + 6 q^{54} + 16 q^{55} - 4 q^{56} + 8 q^{57} - 10 q^{58} - 3 q^{59} - 12 q^{60} - 4 q^{61} - 16 q^{62} - 6 q^{63} + 3 q^{64} - 4 q^{66} + 21 q^{67} + 5 q^{68} - 14 q^{69} - 12 q^{70} - 12 q^{71} + 8 q^{72} + q^{73} - 14 q^{74} - 23 q^{75} + q^{76} - 18 q^{77} - 18 q^{79} + 2 q^{80} - 25 q^{81} + 7 q^{82} - q^{83} - 4 q^{84} + 36 q^{85} + 11 q^{86} - 10 q^{87} + 3 q^{88} + 25 q^{89} - 4 q^{90} - 2 q^{93} - 2 q^{94} - 18 q^{95} + 3 q^{96} + 23 q^{97} + 3 q^{98} + q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 + 3 * q^6 - 4 * q^7 + 3 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 + 3 * q^12 - 4 * q^14 - 12 * q^15 + 3 * q^16 + 5 * q^17 + 8 * q^18 + q^19 + 2 * q^20 - 4 * q^21 + 3 * q^22 + 3 * q^24 + 5 * q^25 + 6 * q^27 - 4 * q^28 - 10 * q^29 - 12 * q^30 - 16 * q^31 + 3 * q^32 - 4 * q^33 + 5 * q^34 - 12 * q^35 + 8 * q^36 - 14 * q^37 + q^38 + 2 * q^40 + 7 * q^41 - 4 * q^42 + 11 * q^43 + 3 * q^44 - 4 * q^45 - 2 * q^47 + 3 * q^48 + 3 * q^49 + 5 * q^50 - 16 * q^51 + 6 * q^54 + 16 * q^55 - 4 * q^56 + 8 * q^57 - 10 * q^58 - 3 * q^59 - 12 * q^60 - 4 * q^61 - 16 * q^62 - 6 * q^63 + 3 * q^64 - 4 * q^66 + 21 * q^67 + 5 * q^68 - 14 * q^69 - 12 * q^70 - 12 * q^71 + 8 * q^72 + q^73 - 14 * q^74 - 23 * q^75 + q^76 - 18 * q^77 - 18 * q^79 + 2 * q^80 - 25 * q^81 + 7 * q^82 - q^83 - 4 * q^84 + 36 * q^85 + 11 * q^86 - 10 * q^87 + 3 * q^88 + 25 * q^89 - 4 * q^90 - 2 * q^93 - 2 * q^94 - 18 * q^95 + 3 * q^96 + 23 * q^97 + 3 * q^98 + q^99

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