LMFDB - Embedded newform 4031.2.a.c.1.56

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.56
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23234 q^{2} +0.123197 q^{3} +2.98336 q^{4} -0.906941 q^{5} +0.275018 q^{6} +3.44118 q^{7} +2.19519 q^{8} -2.98482 q^{9} +O(q^{10})$	\(q+2.23234 q^{2} +0.123197 q^{3} +2.98336 q^{4} -0.906941 q^{5} +0.275018 q^{6} +3.44118 q^{7} +2.19519 q^{8} -2.98482 q^{9} -2.02460 q^{10} -3.80304 q^{11} +0.367541 q^{12} -2.63768 q^{13} +7.68189 q^{14} -0.111733 q^{15} -1.06629 q^{16} -5.71777 q^{17} -6.66315 q^{18} +0.538595 q^{19} -2.70573 q^{20} +0.423944 q^{21} -8.48969 q^{22} +1.42141 q^{23} +0.270441 q^{24} -4.17746 q^{25} -5.88822 q^{26} -0.737313 q^{27} +10.2663 q^{28} -1.00000 q^{29} -0.249426 q^{30} -6.77685 q^{31} -6.77072 q^{32} -0.468524 q^{33} -12.7640 q^{34} -3.12095 q^{35} -8.90479 q^{36} +7.93208 q^{37} +1.20233 q^{38} -0.324955 q^{39} -1.99091 q^{40} -1.96018 q^{41} +0.946388 q^{42} -1.79661 q^{43} -11.3458 q^{44} +2.70706 q^{45} +3.17307 q^{46} +6.05827 q^{47} -0.131364 q^{48} +4.84171 q^{49} -9.32552 q^{50} -0.704414 q^{51} -7.86916 q^{52} -3.27251 q^{53} -1.64594 q^{54} +3.44913 q^{55} +7.55405 q^{56} +0.0663534 q^{57} -2.23234 q^{58} +3.07364 q^{59} -0.333338 q^{60} -1.55126 q^{61} -15.1283 q^{62} -10.2713 q^{63} -12.9820 q^{64} +2.39222 q^{65} -1.04591 q^{66} -15.1747 q^{67} -17.0582 q^{68} +0.175113 q^{69} -6.96702 q^{70} +6.28579 q^{71} -6.55226 q^{72} -4.05743 q^{73} +17.7071 q^{74} -0.514651 q^{75} +1.60682 q^{76} -13.0869 q^{77} -0.725412 q^{78} +9.75365 q^{79} +0.967066 q^{80} +8.86363 q^{81} -4.37580 q^{82} +8.28308 q^{83} +1.26478 q^{84} +5.18569 q^{85} -4.01066 q^{86} -0.123197 q^{87} -8.34840 q^{88} +9.18444 q^{89} +6.04308 q^{90} -9.07674 q^{91} +4.24056 q^{92} -0.834889 q^{93} +13.5241 q^{94} -0.488474 q^{95} -0.834133 q^{96} +11.2984 q^{97} +10.8084 q^{98} +11.3514 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	2.23234	1.57851	0.789253	−	0.614069i	$-0.210468\pi$
$2$	2.23234	1.57851	0.789253	+	0.614069i	$0.210468\pi$
$3$	0.123197	0.0711279	0.0355640	−	0.999367i	$-0.488677\pi$
$3$	0.123197	0.0711279	0.0355640	+	0.999367i	$0.488677\pi$
$4$	2.98336	1.49168
$5$	−0.906941	−0.405596	−0.202798	−	0.979221i	$-0.565004\pi$
$5$	−0.906941	−0.405596	−0.202798	+	0.979221i	$0.565004\pi$
$6$	0.275018	0.112276
$7$	3.44118	1.30064	0.650322	−	0.759659i	$-0.274634\pi$
$7$	3.44118	1.30064	0.650322	+	0.759659i	$0.274634\pi$
$8$	2.19519	0.776117
$9$	−2.98482	−0.994941
$10$	−2.02460	−0.640236
$11$	−3.80304	−1.14666	−0.573330	−	0.819325i	$-0.694349\pi$
$11$	−3.80304	−1.14666	−0.573330	+	0.819325i	$0.694349\pi$
$12$	0.367541	0.106100
$13$	−2.63768	−0.731562	−0.365781	−	0.930701i	$-0.619198\pi$
$13$	−2.63768	−0.731562	−0.365781	+	0.930701i	$0.619198\pi$
$14$	7.68189	2.05307
$15$	−0.111733	−0.0288492
$16$	−1.06629	−0.266573
$17$	−5.71777	−1.38676	−0.693382	−	0.720570i	$-0.743880\pi$
$17$	−5.71777	−1.38676	−0.693382	+	0.720570i	$0.743880\pi$
$18$	−6.66315	−1.57052
$19$	0.538595	0.123562	0.0617811	−	0.998090i	$-0.480322\pi$
$19$	0.538595	0.123562	0.0617811	+	0.998090i	$0.480322\pi$
$20$	−2.70573	−0.605020
$21$	0.423944	0.0925121
$22$	−8.48969	−1.81001
$23$	1.42141	0.296384	0.148192	−	0.988959i	$-0.452655\pi$
$23$	1.42141	0.296384	0.148192	+	0.988959i	$0.452655\pi$
$24$	0.270441	0.0552036
$25$	−4.17746	−0.835492
$26$	−5.88822	−1.15477
$27$	−0.737313	−0.141896
$28$	10.2663	1.94014
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−0.249426	−0.0455387
$31$	−6.77685	−1.21716	−0.608579	−	0.793493i	$-0.708260\pi$
$31$	−6.77685	−1.21716	−0.608579	+	0.793493i	$0.708260\pi$
$32$	−6.77072	−1.19690
$33$	−0.468524	−0.0815595
$34$	−12.7640	−2.18901
$35$	−3.12095	−0.527536
$36$	−8.90479	−1.48413
$37$	7.93208	1.30403	0.652013	−	0.758208i	$-0.273925\pi$
$37$	7.93208	1.30403	0.652013	+	0.758208i	$0.273925\pi$
$38$	1.20233	0.195044
$39$	−0.324955	−0.0520345
$40$	−1.99091	−0.314790
$41$	−1.96018	−0.306129	−0.153065	−	0.988216i	$-0.548914\pi$
$41$	−1.96018	−0.306129	−0.153065	+	0.988216i	$0.548914\pi$
$42$	0.946388	0.146031
$43$	−1.79661	−0.273981	−0.136991	−	0.990572i	$-0.543743\pi$
$43$	−1.79661	−0.273981	−0.136991	+	0.990572i	$0.543743\pi$
$44$	−11.3458	−1.71045
$45$	2.70706	0.403544
$46$	3.17307	0.467843
$47$	6.05827	0.883690	0.441845	−	0.897091i	$-0.354324\pi$
$47$	6.05827	0.883690	0.441845	+	0.897091i	$0.354324\pi$
$48$	−0.131364	−0.0189608
$49$	4.84171	0.691673
$50$	−9.32552	−1.31883
$51$	−0.704414	−0.0986377
$52$	−7.86916	−1.09126
$53$	−3.27251	−0.449514	−0.224757	−	0.974415i	$-0.572159\pi$
$53$	−3.27251	−0.449514	−0.224757	+	0.974415i	$0.572159\pi$
$54$	−1.64594	−0.223984
$55$	3.44913	0.465081
$56$	7.55405	1.00945
$57$	0.0663534	0.00878872
$58$	−2.23234	−0.293121
$59$	3.07364	0.400154	0.200077	−	0.979780i	$-0.435881\pi$
$59$	3.07364	0.400154	0.200077	+	0.979780i	$0.435881\pi$
$60$	−0.333338	−0.0430338
$61$	−1.55126	−0.198619	−0.0993094	−	0.995057i	$-0.531663\pi$
$61$	−1.55126	−0.198619	−0.0993094	+	0.995057i	$0.531663\pi$
$62$	−15.1283	−1.92129
$63$	−10.2713	−1.29406
$64$	−12.9820	−1.62275
$65$	2.39222	0.296719
$66$	−1.04591	−0.128742
$67$	−15.1747	−1.85389	−0.926943	−	0.375202i	$-0.877573\pi$
$67$	−15.1747	−1.85389	−0.926943	+	0.375202i	$0.877573\pi$
$68$	−17.0582	−2.06861
$69$	0.175113	0.0210812
$70$	−6.96702	−0.832719
$71$	6.28579	0.745987	0.372993	−	0.927834i	$-0.378331\pi$
$71$	6.28579	0.745987	0.372993	+	0.927834i	$0.378331\pi$
$72$	−6.55226	−0.772191
$73$	−4.05743	−0.474886	−0.237443	−	0.971401i	$-0.576309\pi$
$73$	−4.05743	−0.474886	−0.237443	+	0.971401i	$0.576309\pi$
$74$	17.7071	2.05841
$75$	−0.514651	−0.0594268
$76$	1.60682	0.184315
$77$	−13.0869	−1.49140
$78$	−0.725412	−0.0821367
$79$	9.75365	1.09737	0.548686	−	0.836029i	$-0.315128\pi$
$79$	9.75365	1.09737	0.548686	+	0.836029i	$0.315128\pi$
$80$	0.967066	0.108121
$81$	8.86363	0.984848
$82$	−4.37580	−0.483226
$83$	8.28308	0.909186	0.454593	−	0.890699i	$-0.349785\pi$
$83$	8.28308	0.909186	0.454593	+	0.890699i	$0.349785\pi$
$84$	1.26478	0.137998
$85$	5.18569	0.562467
$86$	−4.01066	−0.432481
$87$	−0.123197	−0.0132081
$88$	−8.34840	−0.889942
$89$	9.18444	0.973549	0.486775	−	0.873528i	$-0.338173\pi$
$89$	9.18444	0.973549	0.486775	+	0.873528i	$0.338173\pi$
$90$	6.04308	0.636997
$91$	−9.07674	−0.951501
$92$	4.24056	0.442109
$93$	−0.834889	−0.0865740
$94$	13.5241	1.39491
$95$	−0.488474	−0.0501164
$96$	−0.834133	−0.0851334
$97$	11.2984	1.14718	0.573588	−	0.819144i	$-0.305551\pi$
$97$	11.2984	1.14718	0.573588	+	0.819144i	$0.305551\pi$
$98$	10.8084	1.09181
$99$	11.3514	1.14086
$100$	−12.4628	−1.24628
$101$	0.332673	0.0331022	0.0165511	−	0.999863i	$-0.494731\pi$
$101$	0.332673	0.0331022	0.0165511	+	0.999863i	$0.494731\pi$
$102$	−1.57249	−0.155700
$103$	−11.9037	−1.17291	−0.586454	−	0.809983i	$-0.699476\pi$
$103$	−11.9037	−1.17291	−0.586454	+	0.809983i	$0.699476\pi$
$104$	−5.79022	−0.567778
$105$	−0.384492	−0.0375226
$106$	−7.30537	−0.709560
$107$	19.4416	1.87949	0.939746	−	0.341873i	$-0.111061\pi$
$107$	19.4416	1.87949	0.939746	+	0.341873i	$0.111061\pi$
$108$	−2.19967	−0.211663
$109$	−10.0837	−0.965840	−0.482920	−	0.875664i	$-0.660424\pi$
$109$	−10.0837	−0.965840	−0.482920	+	0.875664i	$0.660424\pi$
$110$	7.69965	0.734133
$111$	0.977210	0.0927526
$112$	−3.66931	−0.346717
$113$	−5.07000	−0.476946	−0.238473	−	0.971149i	$-0.576647\pi$
$113$	−5.07000	−0.476946	−0.238473	+	0.971149i	$0.576647\pi$
$114$	0.148124	0.0138730
$115$	−1.28913	−0.120212
$116$	−2.98336	−0.276998
$117$	7.87302	0.727861
$118$	6.86142	0.631645
$119$	−19.6759	−1.80369
$120$	−0.245274	−0.0223904
$121$	3.46311	0.314828
$122$	−3.46295	−0.313521
$123$	−0.241489	−0.0217743
$124$	−20.2178	−1.81561
$125$	8.32341	0.744469
$126$	−22.9291	−2.04269
$127$	−6.02460	−0.534597	−0.267298	−	0.963614i	$-0.586131\pi$
$127$	−6.02460	−0.534597	−0.267298	+	0.963614i	$0.586131\pi$
$128$	−15.4388	−1.36461
$129$	−0.221338	−0.0194877
$130$	5.34027	0.468372
$131$	−18.8897	−1.65040	−0.825202	−	0.564838i	$-0.808939\pi$
$131$	−18.8897	−1.65040	−0.825202	+	0.564838i	$0.808939\pi$
$132$	−1.39777	−0.121661
$133$	1.85340	0.160710
$134$	−33.8752	−2.92637
$135$	0.668700	0.0575525
$136$	−12.5516	−1.07629
$137$	9.00427	0.769287	0.384643	−	0.923065i	$-0.374324\pi$
$137$	9.00427	0.769287	0.384643	+	0.923065i	$0.374324\pi$
$138$	0.390913	0.0332767
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−9.31090	−0.786915
$141$	0.746362	0.0628551
$142$	14.0321	1.17754
$143$	10.0312	0.838853
$144$	3.18270	0.265225
$145$	0.906941	0.0753174
$146$	−9.05757	−0.749610
$147$	0.596485	0.0491973
$148$	23.6642	1.94519
$149$	10.7407	0.879911	0.439956	−	0.898019i	$-0.354994\pi$
$149$	10.7407	0.879911	0.439956	+	0.898019i	$0.354994\pi$
$150$	−1.14888	−0.0938055
$151$	0.869291	0.0707419	0.0353710	−	0.999374i	$-0.488739\pi$
$151$	0.869291	0.0707419	0.0353710	+	0.999374i	$0.488739\pi$
$152$	1.18232	0.0958987
$153$	17.0665	1.37975
$154$	−29.2145	−2.35417
$155$	6.14621	0.493675
$156$	−0.969458	−0.0776188
$157$	2.62527	0.209520	0.104760	−	0.994498i	$-0.466593\pi$
$157$	2.62527	0.209520	0.104760	+	0.994498i	$0.466593\pi$
$158$	21.7735	1.73221
$159$	−0.403164	−0.0319730
$160$	6.14064	0.485460
$161$	4.89131	0.385490
$162$	19.7867	1.55459
$163$	2.98126	0.233510	0.116755	−	0.993161i	$-0.462751\pi$
$163$	2.98126	0.233510	0.116755	+	0.993161i	$0.462751\pi$
$164$	−5.84793	−0.456646
$165$	0.424924	0.0330803
$166$	18.4907	1.43515
$167$	−11.4385	−0.885136	−0.442568	−	0.896735i	$-0.645933\pi$
$167$	−11.4385	−0.885136	−0.442568	+	0.896735i	$0.645933\pi$
$168$	0.930637	0.0718002
$169$	−6.04262	−0.464817
$170$	11.5762	0.887856
$171$	−1.60761	−0.122937
$172$	−5.35994	−0.408692
$173$	15.5764	1.18425	0.592126	−	0.805845i	$-0.298289\pi$
$173$	15.5764	1.18425	0.592126	+	0.805845i	$0.298289\pi$
$174$	−0.275018	−0.0208491
$175$	−14.3754	−1.08668
$176$	4.05516	0.305669
$177$	0.378664	0.0284621
$178$	20.5028	1.53675
$179$	−13.8627	−1.03615	−0.518074	−	0.855336i	$-0.673351\pi$
$179$	−13.8627	−1.03615	−0.518074	+	0.855336i	$0.673351\pi$
$180$	8.07612	0.601959
$181$	−18.9171	−1.40609	−0.703047	−	0.711143i	$-0.748178\pi$
$181$	−18.9171	−1.40609	−0.703047	+	0.711143i	$0.748178\pi$
$182$	−20.2624	−1.50195
$183$	−0.191111	−0.0141273
$184$	3.12026	0.230029
$185$	−7.19393	−0.528908
$186$	−1.86376	−0.136658
$187$	21.7449	1.59015
$188$	18.0740	1.31818
$189$	−2.53723	−0.184556
$190$	−1.09044	−0.0791090
$191$	−21.5904	−1.56223	−0.781114	−	0.624388i	$-0.785348\pi$
$191$	−21.5904	−1.56223	−0.781114	+	0.624388i	$0.785348\pi$
$192$	−1.59934	−0.115423
$193$	−21.2221	−1.52760	−0.763799	−	0.645454i	$-0.776668\pi$
$193$	−21.2221	−1.52760	−0.763799	+	0.645454i	$0.776668\pi$
$194$	25.2218	1.81082
$195$	0.294715	0.0211050
$196$	14.4445	1.03175
$197$	−1.27122	−0.0905710	−0.0452855	−	0.998974i	$-0.514420\pi$
$197$	−1.27122	−0.0905710	−0.0452855	+	0.998974i	$0.514420\pi$
$198$	25.3402	1.80085
$199$	−9.82152	−0.696229	−0.348115	−	0.937452i	$-0.613178\pi$
$199$	−9.82152	−0.696229	−0.348115	+	0.937452i	$0.613178\pi$
$200$	−9.17032	−0.648439
$201$	−1.86948	−0.131863
$202$	0.742640	0.0522520
$203$	−3.44118	−0.241523
$204$	−2.10152	−0.147136
$205$	1.77777	0.124165
$206$	−26.5732	−1.85144
$207$	−4.24265	−0.294884
$208$	2.81255	0.195015
$209$	−2.04830	−0.141684
$210$	−0.858318	−0.0592296
$211$	−4.50599	−0.310205	−0.155103	−	0.987898i	$-0.549571\pi$
$211$	−4.50599	−0.310205	−0.155103	+	0.987898i	$0.549571\pi$
$212$	−9.76307	−0.670530
$213$	0.774392	0.0530605
$214$	43.4004	2.96679
$215$	1.62942	0.111126
$216$	−1.61854	−0.110128
$217$	−23.3204	−1.58309
$218$	−22.5102	−1.52458
$219$	−0.499864	−0.0337777
$220$	10.2900	0.693751
$221$	15.0817	1.01450
$222$	2.18147	0.146411
$223$	26.0844	1.74674	0.873369	−	0.487059i	$-0.161930\pi$
$223$	26.0844	1.74674	0.873369	+	0.487059i	$0.161930\pi$
$224$	−23.2992	−1.55675
$225$	12.4690	0.831265
$226$	−11.3180	−0.752861
$227$	−26.3096	−1.74623	−0.873115	−	0.487514i	$-0.837904\pi$
$227$	−26.3096	−1.74623	−0.873115	+	0.487514i	$0.837904\pi$
$228$	0.197956	0.0131100
$229$	11.5611	0.763976	0.381988	−	0.924167i	$-0.375240\pi$
$229$	11.5611	0.763976	0.381988	+	0.924167i	$0.375240\pi$
$230$	−2.87779	−0.189756
$231$	−1.61227	−0.106080
$232$	−2.19519	−0.144121
$233$	−2.63812	−0.172829	−0.0864146	−	0.996259i	$-0.527541\pi$
$233$	−2.63812	−0.172829	−0.0864146	+	0.996259i	$0.527541\pi$
$234$	17.5753	1.14893
$235$	−5.49450	−0.358422
$236$	9.16977	0.596901
$237$	1.20162	0.0780538
$238$	−43.9233	−2.84713
$239$	26.9104	1.74069	0.870345	−	0.492442i	$-0.163896\pi$
$239$	26.9104	1.74069	0.870345	+	0.492442i	$0.163896\pi$
$240$	0.119140	0.00769044
$241$	−4.54333	−0.292662	−0.146331	−	0.989236i	$-0.546746\pi$
$241$	−4.54333	−0.292662	−0.146331	+	0.989236i	$0.546746\pi$
$242$	7.73085	0.496958
$243$	3.30392	0.211946
$244$	−4.62797	−0.296275
$245$	−4.39115	−0.280540
$246$	−0.539087	−0.0343709
$247$	−1.42064	−0.0903934
$248$	−14.8765	−0.944658
$249$	1.02045	0.0646685
$250$	18.5807	1.17515
$251$	26.5222	1.67407	0.837034	−	0.547151i	$-0.184288\pi$
$251$	26.5222	1.67407	0.837034	+	0.547151i	$0.184288\pi$
$252$	−30.6430	−1.93033
$253$	−5.40567	−0.339851
$254$	−13.4490	−0.843863
$255$	0.638862	0.0400071
$256$	−8.50075	−0.531297
$257$	16.6008	1.03553	0.517766	−	0.855522i	$-0.326764\pi$
$257$	16.6008	1.03553	0.517766	+	0.855522i	$0.326764\pi$
$258$	−0.494102	−0.0307615
$259$	27.2957	1.69607
$260$	7.13686	0.442609
$261$	2.98482	0.184756
$262$	−42.1684	−2.60517
$263$	−13.4717	−0.830701	−0.415350	−	0.909661i	$-0.636341\pi$
$263$	−13.4717	−0.830701	−0.415350	+	0.909661i	$0.636341\pi$
$264$	−1.02850	−0.0632998
$265$	2.96797	0.182321
$266$	4.13743	0.253682
$267$	1.13150	0.0692465
$268$	−45.2716	−2.76540
$269$	−3.43394	−0.209371	−0.104686	−	0.994505i	$-0.533384\pi$
$269$	−3.43394	−0.209371	−0.104686	+	0.994505i	$0.533384\pi$
$270$	1.49277	0.0908470
$271$	−19.2649	−1.17026	−0.585129	−	0.810940i	$-0.698956\pi$
$271$	−19.2649	−1.17026	−0.585129	+	0.810940i	$0.698956\pi$
$272$	6.09683	0.369674
$273$	−1.11823	−0.0676783
$274$	20.1006	1.21432
$275$	15.8870	0.958024
$276$	0.522426	0.0314463
$277$	5.18417	0.311487	0.155743	−	0.987798i	$-0.450223\pi$
$277$	5.18417	0.311487	0.155743	+	0.987798i	$0.450223\pi$
$278$	−2.23234	−0.133887
$279$	20.2277	1.21100
$280$	−6.85107	−0.409430
$281$	−21.7721	−1.29882	−0.649408	−	0.760441i	$-0.724983\pi$
$281$	−21.7721	−1.29882	−0.649408	+	0.760441i	$0.724983\pi$
$282$	1.66614	0.0992170
$283$	−1.54790	−0.0920129	−0.0460065	−	0.998941i	$-0.514649\pi$
$283$	−1.54790	−0.0920129	−0.0460065	+	0.998941i	$0.514649\pi$
$284$	18.7528	1.11277
$285$	−0.0601786	−0.00356467
$286$	22.3931	1.32413
$287$	−6.74534	−0.398165
$288$	20.2094	1.19085
$289$	15.6929	0.923114
$290$	2.02460	0.118889
$291$	1.39193	0.0815962
$292$	−12.1048	−0.708377
$293$	−2.90166	−0.169517	−0.0847584	−	0.996402i	$-0.527012\pi$
$293$	−2.90166	−0.169517	−0.0847584	+	0.996402i	$0.527012\pi$
$294$	1.33156	0.0776581
$295$	−2.78761	−0.162301
$296$	17.4124	1.01208
$297$	2.80403	0.162706
$298$	23.9769	1.38894
$299$	−3.74922	−0.216823
$300$	−1.53539	−0.0886457
$301$	−6.18247	−0.356352
$302$	1.94056	0.111667
$303$	0.0409844	0.00235449
$304$	−0.574300	−0.0329384
$305$	1.40690	0.0805591
$306$	38.0984	2.17794
$307$	−20.0827	−1.14618	−0.573090	−	0.819493i	$-0.694255\pi$
$307$	−20.0827	−1.14618	−0.573090	+	0.819493i	$0.694255\pi$
$308$	−39.0430	−2.22468
$309$	−1.46650	−0.0834265
$310$	13.7204	0.779269
$311$	−11.3197	−0.641884	−0.320942	−	0.947099i	$-0.603999\pi$
$311$	−11.3197	−0.641884	−0.320942	+	0.947099i	$0.603999\pi$
$312$	−0.713339	−0.0403849
$313$	−3.28456	−0.185654	−0.0928272	−	0.995682i	$-0.529590\pi$
$313$	−3.28456	−0.185654	−0.0928272	+	0.995682i	$0.529590\pi$
$314$	5.86051	0.330728
$315$	9.31547	0.524867
$316$	29.0986	1.63693
$317$	20.2864	1.13940	0.569699	−	0.821853i	$-0.307060\pi$
$317$	20.2864	1.13940	0.569699	+	0.821853i	$0.307060\pi$
$318$	−0.900001	−0.0504695
$319$	3.80304	0.212929
$320$	11.7739	0.658180
$321$	2.39515	0.133684
$322$	10.9191	0.608497
$323$	−3.07956	−0.171352
$324$	26.4434	1.46908
$325$	11.0188	0.611214
$326$	6.65519	0.368597
$327$	−1.24228	−0.0686982
$328$	−4.30298	−0.237592
$329$	20.8476	1.14937
$330$	0.948575	0.0522174
$331$	−27.8371	−1.53007	−0.765033	−	0.643992i	$-0.777277\pi$
$331$	−27.8371	−1.53007	−0.765033	+	0.643992i	$0.777277\pi$
$332$	24.7114	1.35621
$333$	−23.6758	−1.29743
$334$	−25.5346	−1.39719
$335$	13.7626	0.751930
$336$	−0.452048	−0.0246613
$337$	−11.0780	−0.603458	−0.301729	−	0.953394i	$-0.597564\pi$
$337$	−11.0780	−0.603458	−0.301729	+	0.953394i	$0.597564\pi$
$338$	−13.4892	−0.733716
$339$	−0.624610	−0.0339242
$340$	15.4708	0.839019
$341$	25.7726	1.39567
$342$	−3.58874	−0.194057
$343$	−7.42706	−0.401024
$344$	−3.94391	−0.212642
$345$	−0.158818	−0.00855045
$346$	34.7719	1.86935
$347$	−16.0184	−0.859914	−0.429957	−	0.902849i	$-0.641471\pi$
$347$	−16.0184	−0.859914	−0.429957	+	0.902849i	$0.641471\pi$
$348$	−0.367541	−0.0197023
$349$	−14.7131	−0.787576	−0.393788	−	0.919201i	$-0.628836\pi$
$349$	−14.7131	−0.787576	−0.393788	+	0.919201i	$0.628836\pi$
$350$	−32.0908	−1.71532
$351$	1.94480	0.103806
$352$	25.7493	1.37244
$353$	21.1220	1.12421	0.562107	−	0.827065i	$-0.309991\pi$
$353$	21.1220	1.12421	0.562107	+	0.827065i	$0.309991\pi$
$354$	0.845308	0.0449276
$355$	−5.70085	−0.302569
$356$	27.4005	1.45222
$357$	−2.42401	−0.128292
$358$	−30.9463	−1.63557
$359$	5.47560	0.288991	0.144496	−	0.989505i	$-0.453844\pi$
$359$	5.47560	0.288991	0.144496	+	0.989505i	$0.453844\pi$
$360$	5.94251	0.313198
$361$	−18.7099	−0.984732
$362$	−42.2294	−2.21953
$363$	0.426646	0.0223931
$364$	−27.0792	−1.41933
$365$	3.67985	0.192612
$366$	−0.426626	−0.0223001
$367$	−36.8508	−1.92359	−0.961797	−	0.273762i	$-0.911732\pi$
$367$	−36.8508	−1.92359	−0.961797	+	0.273762i	$0.911732\pi$
$368$	−1.51564	−0.0790080
$369$	5.85080	0.304580
$370$	−16.0593	−0.834884
$371$	−11.2613	−0.584657
$372$	−2.49077	−0.129141
$373$	−3.57989	−0.185360	−0.0926798	−	0.995696i	$-0.529543\pi$
$373$	−3.57989	−0.185360	−0.0926798	+	0.995696i	$0.529543\pi$
$374$	48.5421	2.51005
$375$	1.02542	0.0529525
$376$	13.2991	0.685847
$377$	2.63768	0.135848
$378$	−5.66396	−0.291323
$379$	15.5723	0.799896	0.399948	−	0.916538i	$-0.369028\pi$
$379$	15.5723	0.799896	0.399948	+	0.916538i	$0.369028\pi$
$380$	−1.45729	−0.0747575
$381$	−0.742214	−0.0380248
$382$	−48.1972	−2.46599
$383$	−38.9424	−1.98987	−0.994933	−	0.100542i	$-0.967942\pi$
$383$	−38.9424	−1.98987	−0.994933	+	0.100542i	$0.967942\pi$
$384$	−1.90202	−0.0970619
$385$	11.8691	0.604905
$386$	−47.3750	−2.41132
$387$	5.36258	0.272595
$388$	33.7071	1.71122
$389$	18.6937	0.947810	0.473905	−	0.880576i	$-0.342844\pi$
$389$	18.6937	0.947810	0.473905	+	0.880576i	$0.342844\pi$
$390$	0.657906	0.0333144
$391$	−8.12728	−0.411014
$392$	10.6285	0.536819
$393$	−2.32716	−0.117390
$394$	−2.83781	−0.142967
$395$	−8.84599	−0.445090
$396$	33.8653	1.70179
$397$	−23.3441	−1.17161	−0.585803	−	0.810453i	$-0.699221\pi$
$397$	−23.3441	−1.17161	−0.585803	+	0.810453i	$0.699221\pi$
$398$	−21.9250	−1.09900
$399$	0.228334	0.0114310
$400$	4.45440	0.222720
$401$	31.5155	1.57381	0.786905	−	0.617074i	$-0.211682\pi$
$401$	31.5155	1.57381	0.786905	+	0.617074i	$0.211682\pi$
$402$	−4.17333	−0.208147
$403$	17.8752	0.890427
$404$	0.992482	0.0493778
$405$	−8.03879	−0.399451
$406$	−7.68189	−0.381246
$407$	−30.1660	−1.49527
$408$	−1.54632	−0.0765544
$409$	19.5703	0.967689	0.483844	−	0.875154i	$-0.339240\pi$
$409$	19.5703	0.967689	0.483844	+	0.875154i	$0.339240\pi$
$410$	3.96859	0.195995
$411$	1.10930	0.0547178
$412$	−35.5130	−1.74960
$413$	10.5769	0.520457
$414$	−9.47105	−0.465476
$415$	−7.51226	−0.368762
$416$	17.8590	0.875610
$417$	−0.123197	−0.00603299
$418$	−4.57250	−0.223649
$419$	9.38913	0.458689	0.229345	−	0.973345i	$-0.426342\pi$
$419$	9.38913	0.458689	0.229345	+	0.973345i	$0.426342\pi$
$420$	−1.14708	−0.0559716
$421$	33.6967	1.64228	0.821138	−	0.570730i	$-0.193340\pi$
$421$	33.6967	1.64228	0.821138	+	0.570730i	$0.193340\pi$
$422$	−10.0589	−0.489661
$423$	−18.0829	−0.879219
$424$	−7.18379	−0.348875
$425$	23.8858	1.15863
$426$	1.72871	0.0837563
$427$	−5.33817	−0.258332
$428$	58.0013	2.80360
$429$	1.23582	0.0596659
$430$	3.63743	0.175413
$431$	3.62245	0.174487	0.0872437	−	0.996187i	$-0.472194\pi$
$431$	3.62245	0.174487	0.0872437	+	0.996187i	$0.472194\pi$
$432$	0.786193	0.0378257
$433$	5.34939	0.257075	0.128538	−	0.991705i	$-0.458972\pi$
$433$	5.34939	0.257075	0.128538	+	0.991705i	$0.458972\pi$
$434$	−52.0591	−2.49891
$435$	0.111733	0.00535717
$436$	−30.0832	−1.44072
$437$	0.765563	0.0366218
$438$	−1.11587	−0.0533182
$439$	32.7182	1.56155	0.780776	−	0.624811i	$-0.214824\pi$
$439$	32.7182	1.56155	0.780776	+	0.624811i	$0.214824\pi$
$440$	7.57151	0.360957
$441$	−14.4516	−0.688173
$442$	33.6675	1.60140
$443$	18.2638	0.867740	0.433870	−	0.900976i	$-0.357148\pi$
$443$	18.2638	0.867740	0.433870	+	0.900976i	$0.357148\pi$
$444$	2.91537	0.138357
$445$	−8.32975	−0.394868
$446$	58.2293	2.75724
$447$	1.32322	0.0625863
$448$	−44.6733	−2.11062
$449$	−34.7767	−1.64121	−0.820606	−	0.571494i	$-0.806364\pi$
$449$	−34.7767	−1.64121	−0.820606	+	0.571494i	$0.806364\pi$
$450$	27.8350	1.31216
$451$	7.45465	0.351026
$452$	−15.1256	−0.711450
$453$	0.107094	0.00503173
$454$	−58.7321	−2.75643
$455$	8.23207	0.385926
$456$	0.145658	0.00682108
$457$	36.7265	1.71799	0.858997	−	0.511981i	$-0.171088\pi$
$457$	36.7265	1.71799	0.858997	+	0.511981i	$0.171088\pi$
$458$	25.8082	1.20594
$459$	4.21579	0.196776
$460$	−3.84594	−0.179318
$461$	−18.1242	−0.844129	−0.422065	−	0.906566i	$-0.638695\pi$
$461$	−18.1242	−0.844129	−0.422065	+	0.906566i	$0.638695\pi$
$462$	−3.59915	−0.167448
$463$	−23.9385	−1.11251	−0.556257	−	0.831010i	$-0.687763\pi$
$463$	−23.9385	−1.11251	−0.556257	+	0.831010i	$0.687763\pi$
$464$	1.06629	0.0495014
$465$	0.757196	0.0351141
$466$	−5.88919	−0.272812
$467$	19.2503	0.890797	0.445398	−	0.895332i	$-0.353062\pi$
$467$	19.2503	0.890797	0.445398	+	0.895332i	$0.353062\pi$
$468$	23.4880	1.08573
$469$	−52.2189	−2.41124
$470$	−12.2656	−0.565770
$471$	0.323426	0.0149027
$472$	6.74723	0.310566
$473$	6.83260	0.314163
$474$	2.68243	0.123208
$475$	−2.24996	−0.103235
$476$	−58.7002	−2.69052
$477$	9.76786	0.447240
$478$	60.0733	2.74769
$479$	−21.4673	−0.980868	−0.490434	−	0.871478i	$-0.663162\pi$
$479$	−21.4673	−0.980868	−0.490434	+	0.871478i	$0.663162\pi$
$480$	0.756510	0.0345298
$481$	−20.9223	−0.953975
$482$	−10.1423	−0.461968
$483$	0.602596	0.0274191
$484$	10.3317	0.469623
$485$	−10.2470	−0.465290
$486$	7.37547	0.334558
$487$	−26.7180	−1.21071	−0.605355	−	0.795956i	$-0.706969\pi$
$487$	−26.7180	−1.21071	−0.605355	+	0.795956i	$0.706969\pi$
$488$	−3.40532	−0.154152
$489$	0.367283	0.0166091
$490$	−9.80254	−0.442834
$491$	27.3770	1.23551	0.617754	−	0.786371i	$-0.288043\pi$
$491$	27.3770	1.23551	0.617754	+	0.786371i	$0.288043\pi$
$492$	−0.720448	−0.0324803
$493$	5.71777	0.257516
$494$	−3.17136	−0.142686
$495$	−10.2951	−0.462728
$496$	7.22612	0.324462
$497$	21.6305	0.970262
$498$	2.27800	0.102080
$499$	−34.6319	−1.55034	−0.775169	−	0.631754i	$-0.782335\pi$
$499$	−34.6319	−1.55034	−0.775169	+	0.631754i	$0.782335\pi$
$500$	24.8317	1.11051
$501$	−1.40919	−0.0629579
$502$	59.2067	2.64253
$503$	18.7703	0.836928	0.418464	−	0.908233i	$-0.362569\pi$
$503$	18.7703	0.836928	0.418464	+	0.908233i	$0.362569\pi$
$504$	−22.5475	−1.00434
$505$	−0.301715	−0.0134261
$506$	−12.0673	−0.536457
$507$	−0.744434	−0.0330615
$508$	−17.9735	−0.797446
$509$	4.91888	0.218026	0.109013	−	0.994040i	$-0.465231\pi$
$509$	4.91888	0.218026	0.109013	+	0.994040i	$0.465231\pi$
$510$	1.42616	0.0631514
$511$	−13.9623	−0.617657
$512$	11.9010	0.525955
$513$	−0.397113	−0.0175330
$514$	37.0588	1.63459
$515$	10.7960	0.475727
$516$	−0.660330	−0.0290694
$517$	−23.0399	−1.01329
$518$	60.9334	2.67726
$519$	1.91897	0.0842334
$520$	5.25139	0.230289
$521$	31.4373	1.37729	0.688647	−	0.725096i	$-0.258205\pi$
$521$	31.4373	1.37729	0.688647	+	0.725096i	$0.258205\pi$
$522$	6.66315	0.291638
$523$	42.7463	1.86917	0.934583	−	0.355746i	$-0.115773\pi$
$523$	42.7463	1.86917	0.934583	+	0.355746i	$0.115773\pi$
$524$	−56.3548	−2.46187
$525$	−1.77101	−0.0772931
$526$	−30.0735	−1.31127
$527$	38.7485	1.68791
$528$	0.499584	0.0217416
$529$	−20.9796	−0.912157
$530$	6.62554	0.287795
$531$	−9.17427	−0.398129
$532$	5.52936	0.239728
$533$	5.17034	0.223952
$534$	2.52589	0.109306
$535$	−17.6324	−0.762316
$536$	−33.3114	−1.43883
$537$	−1.70785	−0.0736991
$538$	−7.66574	−0.330493
$539$	−18.4132	−0.793113
$540$	1.99497	0.0858499
$541$	9.03448	0.388423	0.194211	−	0.980960i	$-0.437785\pi$
$541$	9.03448	0.388423	0.194211	+	0.980960i	$0.437785\pi$
$542$	−43.0058	−1.84726
$543$	−2.33053	−0.100013
$544$	38.7134	1.65982
$545$	9.14530	0.391741
$546$	−2.49627	−0.106831
$547$	−32.9973	−1.41086	−0.705431	−	0.708779i	$-0.749246\pi$
$547$	−32.9973	−1.41086	−0.705431	+	0.708779i	$0.749246\pi$
$548$	26.8630	1.14753
$549$	4.63024	0.197614
$550$	35.4653	1.51225
$551$	−0.538595	−0.0229449
$552$	0.384407	0.0163615
$553$	33.5641	1.42729
$554$	11.5729	0.491684
$555$	−0.886272	−0.0376201
$556$	−2.98336	−0.126523
$557$	−19.6528	−0.832715	−0.416358	−	0.909201i	$-0.636694\pi$
$557$	−19.6528	−0.832715	−0.416358	+	0.909201i	$0.636694\pi$
$558$	45.1552	1.91157
$559$	4.73890	0.200434
$560$	3.32785	0.140627
$561$	2.67891	0.113104
$562$	−48.6028	−2.05019
$563$	−10.3952	−0.438107	−0.219053	−	0.975713i	$-0.570297\pi$
$563$	−10.3952	−0.438107	−0.219053	+	0.975713i	$0.570297\pi$
$564$	2.22667	0.0937595
$565$	4.59819	0.193447
$566$	−3.45544	−0.145243
$567$	30.5013	1.28094
$568$	13.7985	0.578973
$569$	12.1169	0.507967	0.253983	−	0.967209i	$-0.418259\pi$
$569$	12.1169	0.507967	0.253983	+	0.967209i	$0.418259\pi$
$570$	−0.134339	−0.00562686
$571$	37.0364	1.54993	0.774963	−	0.632006i	$-0.217768\pi$
$571$	37.0364	1.54993	0.774963	+	0.632006i	$0.217768\pi$
$572$	29.9267	1.25130
$573$	−2.65988	−0.111118
$574$	−15.0579	−0.628505
$575$	−5.93787	−0.247626
$576$	38.7489	1.61454
$577$	−25.8502	−1.07616	−0.538079	−	0.842894i	$-0.680850\pi$
$577$	−25.8502	−1.07616	−0.538079	+	0.842894i	$0.680850\pi$
$578$	35.0320	1.45714
$579$	−2.61450	−0.108655
$580$	2.70573	0.112349
$581$	28.5035	1.18253
$582$	3.10726	0.128800
$583$	12.4455	0.515439
$584$	−8.90683	−0.368567
$585$	−7.14037	−0.295218
$586$	−6.47750	−0.267583
$587$	10.4181	0.430000	0.215000	−	0.976614i	$-0.431025\pi$
$587$	10.4181	0.430000	0.215000	+	0.976614i	$0.431025\pi$
$588$	1.77953	0.0733865
$589$	−3.64998	−0.150395
$590$	−6.22291	−0.256193
$591$	−0.156611	−0.00644213
$592$	−8.45792	−0.347618
$593$	33.7984	1.38794	0.693968	−	0.720006i	$-0.255861\pi$
$593$	33.7984	1.38794	0.693968	+	0.720006i	$0.255861\pi$
$594$	6.25956	0.256833
$595$	17.8449	0.731568
$596$	32.0433	1.31255
$597$	−1.20998	−0.0495214
$598$	−8.36955	−0.342256
$599$	−26.6414	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$599$	−26.6414	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$600$	−1.12976	−0.0461222
$601$	22.8881	0.933627	0.466813	−	0.884356i	$-0.345402\pi$
$601$	22.8881	0.933627	0.466813	+	0.884356i	$0.345402\pi$
$602$	−13.8014	−0.562503
$603$	45.2938	1.84451
$604$	2.59341	0.105524
$605$	−3.14084	−0.127693
$606$	0.0914912	0.00371658
$607$	16.5193	0.670499	0.335250	−	0.942129i	$-0.391179\pi$
$607$	16.5193	0.670499	0.335250	+	0.942129i	$0.391179\pi$
$608$	−3.64667	−0.147892
$609$	−0.423944	−0.0171791
$610$	3.14069	0.127163
$611$	−15.9798	−0.646474
$612$	50.9156	2.05814
$613$	−15.2662	−0.616598	−0.308299	−	0.951290i	$-0.599760\pi$
$613$	−15.2662	−0.616598	−0.308299	+	0.951290i	$0.599760\pi$
$614$	−44.8314	−1.80925
$615$	0.219016	0.00883159
$616$	−28.7283	−1.15750
$617$	11.5251	0.463985	0.231992	−	0.972718i	$-0.425476\pi$
$617$	11.5251	0.463985	0.231992	+	0.972718i	$0.425476\pi$
$618$	−3.27374	−0.131689
$619$	20.4831	0.823283	0.411642	−	0.911346i	$-0.364956\pi$
$619$	20.4831	0.823283	0.411642	+	0.911346i	$0.364956\pi$
$620$	18.3363	0.736405
$621$	−1.04802	−0.0420557
$622$	−25.2696	−1.01322
$623$	31.6053	1.26624
$624$	0.346498	0.0138710
$625$	13.3384	0.533538
$626$	−7.33227	−0.293057
$627$	−0.252345	−0.0100777
$628$	7.83213	0.312536
$629$	−45.3538	−1.80838
$630$	20.7953	0.828506
$631$	28.0558	1.11688	0.558441	−	0.829544i	$-0.311400\pi$
$631$	28.0558	1.11688	0.558441	+	0.829544i	$0.311400\pi$
$632$	21.4111	0.851689
$633$	−0.555126	−0.0220643
$634$	45.2862	1.79855
$635$	5.46396	0.216830
$636$	−1.20278	−0.0476934
$637$	−12.7709	−0.506002
$638$	8.48969	0.336110
$639$	−18.7620	−0.742212
$640$	14.0021	0.553481
$641$	31.6205	1.24894	0.624468	−	0.781051i	$-0.285316\pi$
$641$	31.6205	1.24894	0.624468	+	0.781051i	$0.285316\pi$
$642$	5.34681	0.211022
$643$	21.3803	0.843157	0.421579	−	0.906792i	$-0.361476\pi$
$643$	21.3803	0.843157	0.421579	+	0.906792i	$0.361476\pi$
$644$	14.5925	0.575027
$645$	0.200740	0.00790415
$646$	−6.87465	−0.270479
$647$	7.79253	0.306356	0.153178	−	0.988199i	$-0.451049\pi$
$647$	7.79253	0.306356	0.153178	+	0.988199i	$0.451049\pi$
$648$	19.4574	0.764358
$649$	−11.6892	−0.458840
$650$	24.5978	0.964804
$651$	−2.87300	−0.112602
$652$	8.89416	0.348322
$653$	−18.7394	−0.733330	−0.366665	−	0.930353i	$-0.619500\pi$
$653$	−18.7394	−0.733330	−0.366665	+	0.930353i	$0.619500\pi$
$654$	−2.77320	−0.108441
$655$	17.1319	0.669398
$656$	2.09013	0.0816059
$657$	12.1107	0.472483
$658$	46.5390	1.81428
$659$	18.7981	0.732272	0.366136	−	0.930561i	$-0.380681\pi$
$659$	18.7981	0.732272	0.366136	+	0.930561i	$0.380681\pi$
$660$	1.26770	0.0493451
$661$	−45.1543	−1.75630	−0.878149	−	0.478387i	$-0.841222\pi$
$661$	−45.1543	−1.75630	−0.878149	+	0.478387i	$0.841222\pi$
$662$	−62.1420	−2.41522
$663$	1.85802	0.0721596
$664$	18.1829	0.705635
$665$	−1.68093	−0.0651835
$666$	−52.8526	−2.04800
$667$	−1.42141	−0.0550371
$668$	−34.1251	−1.32034
$669$	3.21352	0.124242
$670$	30.7228	1.18692
$671$	5.89951	0.227748
$672$	−2.87040	−0.110728
$673$	−21.0662	−0.812044	−0.406022	−	0.913863i	$-0.633084\pi$
$673$	−21.0662	−0.812044	−0.406022	+	0.913863i	$0.633084\pi$
$674$	−24.7300	−0.952562
$675$	3.08010	0.118553
$676$	−18.0273	−0.693357
$677$	2.88801	0.110995	0.0554976	−	0.998459i	$-0.482325\pi$
$677$	2.88801	0.110995	0.0554976	+	0.998459i	$0.482325\pi$
$678$	−1.39434	−0.0535495
$679$	38.8797	1.49207
$680$	11.3836	0.436540
$681$	−3.24127	−0.124206
$682$	57.5334	2.20307
$683$	−11.8548	−0.453613	−0.226806	−	0.973940i	$-0.572828\pi$
$683$	−11.8548	−0.453613	−0.226806	+	0.973940i	$0.572828\pi$
$684$	−4.79608	−0.183383
$685$	−8.16634	−0.312020
$686$	−16.5798	−0.633018
$687$	1.42429	0.0543400
$688$	1.91572	0.0730361
$689$	8.63185	0.328847
$690$	−0.354535	−0.0134969
$691$	22.5418	0.857531	0.428765	−	0.903416i	$-0.358949\pi$
$691$	22.5418	0.857531	0.428765	+	0.903416i	$0.358949\pi$
$692$	46.4700	1.76652
$693$	39.0622	1.48385
$694$	−35.7586	−1.35738
$695$	0.906941	0.0344022
$696$	−0.270441	−0.0102511
$697$	11.2079	0.424529
$698$	−32.8448	−1.24319
$699$	−0.325009	−0.0122930
$700$	−42.8869	−1.62097
$701$	−1.04870	−0.0396088	−0.0198044	−	0.999804i	$-0.506304\pi$
$701$	−1.04870	−0.0396088	−0.0198044	+	0.999804i	$0.506304\pi$
$702$	4.34146	0.163858
$703$	4.27218	0.161128
$704$	49.3710	1.86074
$705$	−0.676907	−0.0254938
$706$	47.1517	1.77458
$707$	1.14479	0.0430542
$708$	1.12969	0.0424563
$709$	−23.5487	−0.884389	−0.442195	−	0.896919i	$-0.645800\pi$
$709$	−23.5487	−0.884389	−0.442195	+	0.896919i	$0.645800\pi$
$710$	−12.7262	−0.477608
$711$	−29.1129	−1.09182
$712$	20.1616	0.755588
$713$	−9.63267	−0.360746
$714$	−5.41123	−0.202510
$715$	−9.09773	−0.340236
$716$	−41.3574	−1.54560
$717$	3.31529	0.123812
$718$	12.2234	0.456174
$719$	10.3122	0.384579	0.192289	−	0.981338i	$-0.438409\pi$
$719$	10.3122	0.384579	0.192289	+	0.981338i	$0.438409\pi$
$720$	−2.88652	−0.107574
$721$	−40.9628	−1.52553
$722$	−41.7670	−1.55441
$723$	−0.559726	−0.0208164
$724$	−56.4364	−2.09744
$725$	4.17746	0.155147
$726$	0.952420	0.0353476
$727$	−21.7658	−0.807248	−0.403624	−	0.914925i	$-0.632250\pi$
$727$	−21.7658	−0.807248	−0.403624	+	0.914925i	$0.632250\pi$
$728$	−19.9252	−0.738477
$729$	−26.1839	−0.969773
$730$	8.21468	0.304039
$731$	10.2726	0.379947
$732$	−0.570153	−0.0210735
$733$	6.78919	0.250764	0.125382	−	0.992109i	$-0.459984\pi$
$733$	6.78919	0.250764	0.125382	+	0.992109i	$0.459984\pi$
$734$	−82.2636	−3.03640
$735$	−0.540977	−0.0199542
$736$	−9.62394	−0.354743
$737$	57.7100	2.12578
$738$	13.0610	0.480782
$739$	−14.9045	−0.548270	−0.274135	−	0.961691i	$-0.588392\pi$
$739$	−14.9045	−0.548270	−0.274135	+	0.961691i	$0.588392\pi$
$740$	−21.4621	−0.788961
$741$	−0.175019	−0.00642950
$742$	−25.1391	−0.922884
$743$	−50.8001	−1.86368	−0.931838	−	0.362876i	$-0.881795\pi$
$743$	−50.8001	−1.86368	−0.931838	+	0.362876i	$0.881795\pi$
$744$	−1.83274	−0.0671916
$745$	−9.74117	−0.356889
$746$	−7.99154	−0.292591
$747$	−24.7235	−0.904586
$748$	64.8729	2.37199
$749$	66.9021	2.44455
$750$	2.28909	0.0835859
$751$	−26.1878	−0.955608	−0.477804	−	0.878466i	$-0.658567\pi$
$751$	−26.1878	−0.955608	−0.477804	+	0.878466i	$0.658567\pi$
$752$	−6.45990	−0.235568
$753$	3.26746	0.119073
$754$	5.88822	0.214436
$755$	−0.788396	−0.0286927
$756$	−7.56946	−0.275298
$757$	−2.55412	−0.0928309	−0.0464155	−	0.998922i	$-0.514780\pi$
$757$	−2.55412	−0.0928309	−0.0464155	+	0.998922i	$0.514780\pi$
$758$	34.7627	1.26264
$759$	−0.665963	−0.0241729
$760$	−1.07229	−0.0388962
$761$	−34.5260	−1.25157	−0.625783	−	0.779997i	$-0.715220\pi$
$761$	−34.5260	−1.25157	−0.625783	+	0.779997i	$0.715220\pi$
$762$	−1.65688	−0.0600223
$763$	−34.6997	−1.25621
$764$	−64.4119	−2.33034
$765$	−15.4783	−0.559621
$766$	−86.9329	−3.14101
$767$	−8.10729	−0.292737
$768$	−1.04727	−0.0377900
$769$	15.3879	0.554903	0.277451	−	0.960740i	$-0.410510\pi$
$769$	15.3879	0.554903	0.277451	+	0.960740i	$0.410510\pi$
$770$	26.4959	0.954845
$771$	2.04518	0.0736553
$772$	−63.3131	−2.27869
$773$	35.5114	1.27726	0.638629	−	0.769515i	$-0.279502\pi$
$773$	35.5114	1.27726	0.638629	+	0.769515i	$0.279502\pi$
$774$	11.9711	0.430293
$775$	28.3100	1.01693
$776$	24.8021	0.890343
$777$	3.36275	0.120638
$778$	41.7308	1.49612
$779$	−1.05574	−0.0378260
$780$	0.879241	0.0314819
$781$	−23.9051	−0.855393
$782$	−18.1429	−0.648788
$783$	0.737313	0.0263494
$784$	−5.16268	−0.184382
$785$	−2.38097	−0.0849804
$786$	−5.19503	−0.185300
$787$	−3.96281	−0.141259	−0.0706294	−	0.997503i	$-0.522501\pi$
$787$	−3.96281	−0.141259	−0.0706294	+	0.997503i	$0.522501\pi$
$788$	−3.79252	−0.135103
$789$	−1.65968	−0.0590860
$790$	−19.7473	−0.702577
$791$	−17.4468	−0.620336
$792$	24.9185	0.885440
$793$	4.09174	0.145302
$794$	−52.1120	−1.84939
$795$	0.365646	0.0129681
$796$	−29.3011	−1.03855
$797$	−3.00393	−0.106405	−0.0532024	−	0.998584i	$-0.516943\pi$
$797$	−3.00393	−0.106405	−0.0532024	+	0.998584i	$0.516943\pi$
$798$	0.509720	0.0180439
$799$	−34.6398	−1.22547
$800$	28.2844	1.00000
$801$	−27.4139	−0.968624
$802$	70.3535	2.48427
$803$	15.4306	0.544532
$804$	−5.57733	−0.196697
$805$	−4.43613	−0.156353
$806$	39.9036	1.40554
$807$	−0.423052	−0.0148921
$808$	0.730281	0.0256912
$809$	1.34005	0.0471135	0.0235567	−	0.999723i	$-0.492501\pi$
$809$	1.34005	0.0471135	0.0235567	+	0.999723i	$0.492501\pi$
$810$	−17.9453	−0.630535
$811$	52.9054	1.85776	0.928881	−	0.370379i	$-0.120772\pi$
$811$	52.9054	1.85776	0.928881	+	0.370379i	$0.120772\pi$
$812$	−10.2663	−0.360275
$813$	−2.37338	−0.0832381
$814$	−67.3409	−2.36030
$815$	−2.70383	−0.0947109
$816$	0.751112	0.0262942
$817$	−0.967648	−0.0338537
$818$	43.6876	1.52750
$819$	27.0925	0.946687
$820$	5.30373	0.185214
$821$	−42.5519	−1.48507	−0.742536	−	0.669806i	$-0.766377\pi$
$821$	−42.5519	−1.48507	−0.742536	+	0.669806i	$0.766377\pi$
$822$	2.47634	0.0863723
$823$	−15.4188	−0.537464	−0.268732	−	0.963215i	$-0.586605\pi$
$823$	−15.4188	−0.537464	−0.268732	+	0.963215i	$0.586605\pi$
$824$	−26.1309	−0.910314
$825$	1.95724	0.0681423
$826$	23.6114	0.821545
$827$	5.78592	0.201196	0.100598	−	0.994927i	$-0.467924\pi$
$827$	5.78592	0.201196	0.100598	+	0.994927i	$0.467924\pi$
$828$	−12.6573	−0.439873
$829$	−6.72870	−0.233697	−0.116849	−	0.993150i	$-0.537279\pi$
$829$	−6.72870	−0.233697	−0.116849	+	0.993150i	$0.537279\pi$
$830$	−16.7700	−0.582093
$831$	0.638676	0.0221554
$832$	34.2424	1.18714
$833$	−27.6838	−0.959187
$834$	−0.275018	−0.00952311
$835$	10.3740	0.359008
$836$	−6.11081	−0.211347
$837$	4.99666	0.172710
$838$	20.9598	0.724043
$839$	27.0993	0.935573	0.467787	−	0.883841i	$-0.345052\pi$
$839$	27.0993	0.935573	0.467787	+	0.883841i	$0.345052\pi$
$840$	−0.844033	−0.0291219
$841$	1.00000	0.0344828
$842$	75.2226	2.59234
$843$	−2.68226	−0.0923820
$844$	−13.4430	−0.462727
$845$	5.48030	0.188528
$846$	−40.3672	−1.38785
$847$	11.9172	0.409479
$848$	3.48946	0.119828
$849$	−0.190697	−0.00654469
$850$	53.3212	1.82890
$851$	11.2747	0.386492
$852$	2.31029	0.0791492
$853$	−20.1636	−0.690390	−0.345195	−	0.938531i	$-0.612187\pi$
$853$	−20.1636	−0.690390	−0.345195	+	0.938531i	$0.612187\pi$
$854$	−11.9166	−0.407779
$855$	1.45801	0.0498628
$856$	42.6781	1.45871
$857$	−7.86854	−0.268784	−0.134392	−	0.990928i	$-0.542908\pi$
$857$	−7.86854	−0.268784	−0.134392	+	0.990928i	$0.542908\pi$
$858$	2.75877	0.0941829
$859$	−38.5526	−1.31540	−0.657698	−	0.753281i	$-0.728470\pi$
$859$	−38.5526	−1.31540	−0.657698	+	0.753281i	$0.728470\pi$
$860$	4.86115	0.165764
$861$	−0.831007	−0.0283206
$862$	8.08655	0.275429
$863$	31.9653	1.08811	0.544056	−	0.839049i	$-0.316888\pi$
$863$	31.9653	1.08811	0.544056	+	0.839049i	$0.316888\pi$
$864$	4.99214	0.169836
$865$	−14.1269	−0.480328
$866$	11.9417	0.405795
$867$	1.93333	0.0656592
$868$	−69.5730	−2.36146
$869$	−37.0935	−1.25831
$870$	0.249426	0.00845632
$871$	40.0261	1.35623
$872$	−22.1356	−0.749605
$873$	−33.7236	−1.14137
$874$	1.70900	0.0578077
$875$	28.6424	0.968288
$876$	−1.49127	−0.0503854
$877$	−32.7563	−1.10610	−0.553050	−	0.833148i	$-0.686536\pi$
$877$	−32.7563	−1.10610	−0.553050	+	0.833148i	$0.686536\pi$
$878$	73.0382	2.46492
$879$	−0.357477	−0.0120574
$880$	−3.67779	−0.123978
$881$	13.1888	0.444341	0.222171	−	0.975008i	$-0.428686\pi$
$881$	13.1888	0.444341	0.222171	+	0.975008i	$0.428686\pi$
$882$	−32.2610	−1.08629
$883$	−17.2069	−0.579060	−0.289530	−	0.957169i	$-0.593499\pi$
$883$	−17.2069	−0.579060	−0.289530	+	0.957169i	$0.593499\pi$
$884$	44.9941	1.51331
$885$	−0.343426	−0.0115441
$886$	40.7711	1.36973
$887$	43.3381	1.45515	0.727575	−	0.686028i	$-0.240647\pi$
$887$	43.3381	1.45515	0.727575	+	0.686028i	$0.240647\pi$
$888$	2.14516	0.0719869
$889$	−20.7317	−0.695319
$890$	−18.5949	−0.623301
$891$	−33.7087	−1.12929
$892$	77.8190	2.60557
$893$	3.26296	0.109191
$894$	2.95389	0.0987928
$895$	12.5727	0.420258
$896$	−53.1277	−1.77487
$897$	−0.461894	−0.0154222
$898$	−77.6334	−2.59066
$899$	6.77685	0.226021
$900$	37.1994	1.23998
$901$	18.7115	0.623370
$902$	16.6413	0.554096
$903$	−0.761663	−0.0253466
$904$	−11.1296	−0.370166
$905$	17.1567	0.570307
$906$	0.239071	0.00794261
$907$	24.4640	0.812315	0.406158	−	0.913803i	$-0.366868\pi$
$907$	24.4640	0.812315	0.406158	+	0.913803i	$0.366868\pi$
$908$	−78.4910	−2.60481
$909$	−0.992970	−0.0329347
$910$	18.3768	0.609185
$911$	33.1228	1.09741	0.548704	−	0.836017i	$-0.315122\pi$
$911$	33.1228	1.09741	0.548704	+	0.836017i	$0.315122\pi$
$912$	−0.0707522	−0.00234284
$913$	−31.5009	−1.04253
$914$	81.9862	2.71186
$915$	0.173327	0.00573000
$916$	34.4907	1.13961
$917$	−65.0029	−2.14659
$918$	9.41110	0.310612
$919$	−29.0940	−0.959724	−0.479862	−	0.877344i	$-0.659313\pi$
$919$	−29.0940	−0.959724	−0.479862	+	0.877344i	$0.659313\pi$
$920$	−2.82989	−0.0932988
$921$	−2.47413	−0.0815254
$922$	−40.4595	−1.33246
$923$	−16.5799	−0.545736
$924$	−4.80999	−0.158237
$925$	−33.1359	−1.08950
$926$	−53.4389	−1.75611
$927$	35.5305	1.16697
$928$	6.77072	0.222260
$929$	17.8697	0.586284	0.293142	−	0.956069i	$-0.405299\pi$
$929$	17.8697	0.586284	0.293142	+	0.956069i	$0.405299\pi$
$930$	1.69032	0.0554278
$931$	2.60772	0.0854646
$932$	−7.87046	−0.257806
$933$	−1.39456	−0.0456559
$934$	42.9732	1.40613
$935$	−19.7214	−0.644958
$936$	17.2828	0.564906
$937$	−39.7668	−1.29912	−0.649562	−	0.760309i	$-0.725048\pi$
$937$	−39.7668	−1.29912	−0.649562	+	0.760309i	$0.725048\pi$
$938$	−116.570	−3.80616
$939$	−0.404649	−0.0132052
$940$	−16.3921	−0.534650
$941$	10.9764	0.357820	0.178910	−	0.983865i	$-0.442743\pi$
$941$	10.9764	0.357820	0.178910	+	0.983865i	$0.442743\pi$
$942$	0.721999	0.0235240
$943$	−2.78622	−0.0907317
$944$	−3.27740	−0.106670
$945$	2.30112	0.0748553
$946$	15.2527	0.495908
$947$	−55.1556	−1.79232	−0.896158	−	0.443734i	$-0.853653\pi$
$947$	−55.1556	−1.79232	−0.896158	+	0.443734i	$0.853653\pi$
$948$	3.58487	0.116431
$949$	10.7022	0.347409
$950$	−5.02268	−0.162957
$951$	2.49923	0.0810430
$952$	−43.1923	−1.39987
$953$	−47.7619	−1.54716	−0.773579	−	0.633699i	$-0.781536\pi$
$953$	−47.7619	−1.54716	−0.773579	+	0.633699i	$0.781536\pi$
$954$	21.8052	0.705970
$955$	19.5812	0.633634
$956$	80.2834	2.59655
$957$	0.468524	0.0151452
$958$	−47.9225	−1.54830
$959$	30.9853	1.00057
$960$	1.45051	0.0468150
$961$	14.9257	0.481475
$962$	−46.7058	−1.50586
$963$	−58.0298	−1.86998
$964$	−13.5544	−0.436558
$965$	19.2472	0.619589
$966$	1.34520	0.0432812
$967$	−39.7206	−1.27733	−0.638664	−	0.769486i	$-0.720512\pi$
$967$	−39.7206	−1.27733	−0.638664	+	0.769486i	$0.720512\pi$
$968$	7.60219	0.244344
$969$	−0.379394	−0.0121879
$970$	−22.8747	−0.734463
$971$	38.0627	1.22149	0.610746	−	0.791827i	$-0.290870\pi$
$971$	38.0627	1.22149	0.610746	+	0.791827i	$0.290870\pi$
$972$	9.85676	0.316156
$973$	−3.44118	−0.110319
$974$	−59.6438	−1.91111
$975$	1.35749	0.0434744
$976$	1.65410	0.0529465
$977$	27.1955	0.870060	0.435030	−	0.900416i	$-0.356738\pi$
$977$	27.1955	0.870060	0.435030	+	0.900416i	$0.356738\pi$
$978$	0.819901	0.0262176
$979$	−34.9288	−1.11633
$980$	−13.1004	−0.418476
$981$	30.0980	0.960954
$982$	61.1149	1.95026
$983$	−22.0205	−0.702344	−0.351172	−	0.936311i	$-0.614217\pi$
$983$	−22.0205	−0.702344	−0.351172	+	0.936311i	$0.614217\pi$
$984$	−0.530115	−0.0168994
$985$	1.15293	0.0367353
$986$	12.7640	0.406490
$987$	2.56837	0.0817520
$988$	−4.23829	−0.134838
$989$	−2.55372	−0.0812036
$990$	−22.9821	−0.730419
$991$	−1.01160	−0.0321345	−0.0160673	−	0.999871i	$-0.505115\pi$
$991$	−1.01160	−0.0321345	−0.0160673	+	0.999871i	$0.505115\pi$
$992$	45.8842	1.45682
$993$	−3.42945	−0.108830
$994$	48.2868	1.53156
$995$	8.90754	0.282388
$996$	3.04437	0.0964646
$997$	−46.9298	−1.48628	−0.743140	−	0.669136i	$-0.766664\pi$
$997$	−46.9298	−1.48628	−0.743140	+	0.669136i	$0.766664\pi$
$998$	−77.3103	−2.44722
$999$	−5.84843	−0.185036

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.56	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.56	✓	61	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23234 q^{2} +0.123197 q^{3} +2.98336 q^{4} -0.906941 q^{5} +0.275018 q^{6} +3.44118 q^{7} +2.19519 q^{8} -2.98482 q^{9} +O(q^{10})\)	\(q+2.23234 q^{2} +0.123197 q^{3} +2.98336 q^{4} -0.906941 q^{5} +0.275018 q^{6} +3.44118 q^{7} +2.19519 q^{8} -2.98482 q^{9} -2.02460 q^{10} -3.80304 q^{11} +0.367541 q^{12} -2.63768 q^{13} +7.68189 q^{14} -0.111733 q^{15} -1.06629 q^{16} -5.71777 q^{17} -6.66315 q^{18} +0.538595 q^{19} -2.70573 q^{20} +0.423944 q^{21} -8.48969 q^{22} +1.42141 q^{23} +0.270441 q^{24} -4.17746 q^{25} -5.88822 q^{26} -0.737313 q^{27} +10.2663 q^{28} -1.00000 q^{29} -0.249426 q^{30} -6.77685 q^{31} -6.77072 q^{32} -0.468524 q^{33} -12.7640 q^{34} -3.12095 q^{35} -8.90479 q^{36} +7.93208 q^{37} +1.20233 q^{38} -0.324955 q^{39} -1.99091 q^{40} -1.96018 q^{41} +0.946388 q^{42} -1.79661 q^{43} -11.3458 q^{44} +2.70706 q^{45} +3.17307 q^{46} +6.05827 q^{47} -0.131364 q^{48} +4.84171 q^{49} -9.32552 q^{50} -0.704414 q^{51} -7.86916 q^{52} -3.27251 q^{53} -1.64594 q^{54} +3.44913 q^{55} +7.55405 q^{56} +0.0663534 q^{57} -2.23234 q^{58} +3.07364 q^{59} -0.333338 q^{60} -1.55126 q^{61} -15.1283 q^{62} -10.2713 q^{63} -12.9820 q^{64} +2.39222 q^{65} -1.04591 q^{66} -15.1747 q^{67} -17.0582 q^{68} +0.175113 q^{69} -6.96702 q^{70} +6.28579 q^{71} -6.55226 q^{72} -4.05743 q^{73} +17.7071 q^{74} -0.514651 q^{75} +1.60682 q^{76} -13.0869 q^{77} -0.725412 q^{78} +9.75365 q^{79} +0.967066 q^{80} +8.86363 q^{81} -4.37580 q^{82} +8.28308 q^{83} +1.26478 q^{84} +5.18569 q^{85} -4.01066 q^{86} -0.123197 q^{87} -8.34840 q^{88} +9.18444 q^{89} +6.04308 q^{90} -9.07674 q^{91} +4.24056 q^{92} -0.834889 q^{93} +13.5241 q^{94} -0.488474 q^{95} -0.834133 q^{96} +11.2984 q^{97} +10.8084 q^{98} +11.3514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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