LMFDB - Embedded newform 4031.2.a.b.1.20

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:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-1.17292 q^{2} -0.112413 q^{3} -0.624270 q^{4} +4.03608 q^{5} +0.131851 q^{6} +1.76356 q^{7} +3.07805 q^{8} -2.98736 q^{9} +O(q^{10})$	\(q-1.17292 q^{2} -0.112413 q^{3} -0.624270 q^{4} +4.03608 q^{5} +0.131851 q^{6} +1.76356 q^{7} +3.07805 q^{8} -2.98736 q^{9} -4.73397 q^{10} -5.27457 q^{11} +0.0701761 q^{12} -3.78038 q^{13} -2.06850 q^{14} -0.453708 q^{15} -2.36175 q^{16} +0.477807 q^{17} +3.50392 q^{18} -1.38085 q^{19} -2.51960 q^{20} -0.198247 q^{21} +6.18662 q^{22} +5.09681 q^{23} -0.346013 q^{24} +11.2899 q^{25} +4.43407 q^{26} +0.673058 q^{27} -1.10094 q^{28} +1.00000 q^{29} +0.532160 q^{30} +1.83680 q^{31} -3.38596 q^{32} +0.592930 q^{33} -0.560427 q^{34} +7.11785 q^{35} +1.86492 q^{36} +3.72486 q^{37} +1.61962 q^{38} +0.424964 q^{39} +12.4232 q^{40} -6.02230 q^{41} +0.232527 q^{42} -3.47820 q^{43} +3.29275 q^{44} -12.0572 q^{45} -5.97813 q^{46} -7.15004 q^{47} +0.265491 q^{48} -3.88987 q^{49} -13.2421 q^{50} -0.0537117 q^{51} +2.35998 q^{52} +8.17405 q^{53} -0.789440 q^{54} -21.2885 q^{55} +5.42831 q^{56} +0.155225 q^{57} -1.17292 q^{58} -0.165616 q^{59} +0.283236 q^{60} -7.43644 q^{61} -2.15441 q^{62} -5.26839 q^{63} +8.69494 q^{64} -15.2579 q^{65} -0.695457 q^{66} -6.84673 q^{67} -0.298281 q^{68} -0.572948 q^{69} -8.34863 q^{70} +1.51659 q^{71} -9.19524 q^{72} -6.89087 q^{73} -4.36894 q^{74} -1.26913 q^{75} +0.862022 q^{76} -9.30200 q^{77} -0.498447 q^{78} +4.81605 q^{79} -9.53219 q^{80} +8.88643 q^{81} +7.06365 q^{82} -3.99493 q^{83} +0.123760 q^{84} +1.92846 q^{85} +4.07963 q^{86} -0.112413 q^{87} -16.2354 q^{88} +2.33459 q^{89} +14.1421 q^{90} -6.66692 q^{91} -3.18179 q^{92} -0.206481 q^{93} +8.38639 q^{94} -5.57320 q^{95} +0.380627 q^{96} -14.6888 q^{97} +4.56248 q^{98} +15.7570 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * 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 $

$2$	−1.17292	−0.829376	−0.414688	−	0.909964i	$-0.636109\pi$
$2$	−1.17292	−0.829376	−0.414688	+	0.909964i	$0.636109\pi$
$3$	−0.112413	−0.0649017	−0.0324509	−	0.999473i	$-0.510331\pi$
$3$	−0.112413	−0.0649017	−0.0324509	+	0.999473i	$0.510331\pi$
$4$	−0.624270	−0.312135
$5$	4.03608	1.80499	0.902494	−	0.430703i	$-0.141734\pi$
$5$	4.03608	1.80499	0.902494	+	0.430703i	$0.141734\pi$
$6$	0.131851	0.0538279
$7$	1.76356	0.666562	0.333281	−	0.942828i	$-0.391844\pi$
$7$	1.76356	0.666562	0.333281	+	0.942828i	$0.391844\pi$
$8$	3.07805	1.08825
$9$	−2.98736	−0.995788
$10$	−4.73397	−1.49701
$11$	−5.27457	−1.59034	−0.795171	−	0.606386i	$-0.792619\pi$
$11$	−5.27457	−1.59034	−0.795171	+	0.606386i	$0.792619\pi$
$12$	0.0701761	0.0202581
$13$	−3.78038	−1.04849	−0.524245	−	0.851568i	$-0.675652\pi$
$13$	−3.78038	−1.04849	−0.524245	+	0.851568i	$0.675652\pi$
$14$	−2.06850	−0.552831
$15$	−0.453708	−0.117147
$16$	−2.36175	−0.590437
$17$	0.477807	0.115885	0.0579426	−	0.998320i	$-0.481546\pi$
$17$	0.477807	0.115885	0.0579426	+	0.998320i	$0.481546\pi$
$18$	3.50392	0.825883
$19$	−1.38085	−0.316788	−0.158394	−	0.987376i	$-0.550632\pi$
$19$	−1.38085	−0.316788	−0.158394	+	0.987376i	$0.550632\pi$
$20$	−2.51960	−0.563400
$21$	−0.198247	−0.0432610
$22$	6.18662	1.31899
$23$	5.09681	1.06276	0.531380	−	0.847134i	$-0.321674\pi$
$23$	5.09681	1.06276	0.531380	+	0.847134i	$0.321674\pi$
$24$	−0.346013	−0.0706295
$25$	11.2899	2.25798
$26$	4.43407	0.869592
$27$	0.673058	0.129530
$28$	−1.10094	−0.208057
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.532160	0.0971588
$31$	1.83680	0.329900	0.164950	−	0.986302i	$-0.447254\pi$
$31$	1.83680	0.329900	0.164950	+	0.986302i	$0.447254\pi$
$32$	−3.38596	−0.598560
$33$	0.592930	0.103216
$34$	−0.560427	−0.0961124
$35$	7.11785	1.20314
$36$	1.86492	0.310820
$37$	3.72486	0.612363	0.306181	−	0.951973i	$-0.400949\pi$
$37$	3.72486	0.612363	0.306181	+	0.951973i	$0.400949\pi$
$38$	1.61962	0.262737
$39$	0.424964	0.0680487
$40$	12.4232	1.96428
$41$	−6.02230	−0.940525	−0.470263	−	0.882527i	$-0.655841\pi$
$41$	−6.02230	−0.940525	−0.470263	+	0.882527i	$0.655841\pi$
$42$	0.232527	0.0358797
$43$	−3.47820	−0.530420	−0.265210	−	0.964191i	$-0.585441\pi$
$43$	−3.47820	−0.530420	−0.265210	+	0.964191i	$0.585441\pi$
$44$	3.29275	0.496401
$45$	−12.0572	−1.79738
$46$	−5.97813	−0.881427
$47$	−7.15004	−1.04294	−0.521470	−	0.853269i	$-0.674616\pi$
$47$	−7.15004	−1.04294	−0.521470	+	0.853269i	$0.674616\pi$
$48$	0.265491	0.0383203
$49$	−3.88987	−0.555695
$50$	−13.2421	−1.87272
$51$	−0.0537117	−0.00752115
$52$	2.35998	0.327270
$53$	8.17405	1.12279	0.561396	−	0.827547i	$-0.310265\pi$
$53$	8.17405	1.12279	0.561396	+	0.827547i	$0.310265\pi$
$54$	−0.789440	−0.107429
$55$	−21.2885	−2.87055
$56$	5.42831	0.725388
$57$	0.155225	0.0205601
$58$	−1.17292	−0.154011
$59$	−0.165616	−0.0215613	−0.0107807	−	0.999942i	$-0.503432\pi$
$59$	−0.165616	−0.0215613	−0.0107807	+	0.999942i	$0.503432\pi$
$60$	0.283236	0.0365656
$61$	−7.43644	−0.952139	−0.476069	−	0.879408i	$-0.657939\pi$
$61$	−7.43644	−0.952139	−0.476069	+	0.879408i	$0.657939\pi$
$62$	−2.15441	−0.273611
$63$	−5.26839	−0.663754
$64$	8.69494	1.08687
$65$	−15.2579	−1.89251
$66$	−0.695457	−0.0856048
$67$	−6.84673	−0.836461	−0.418231	−	0.908341i	$-0.637350\pi$
$67$	−6.84673	−0.836461	−0.418231	+	0.908341i	$0.637350\pi$
$68$	−0.298281	−0.0361718
$69$	−0.572948	−0.0689749
$70$	−8.34863	−0.997852
$71$	1.51659	0.179986	0.0899929	−	0.995942i	$-0.471316\pi$
$71$	1.51659	0.179986	0.0899929	+	0.995942i	$0.471316\pi$
$72$	−9.19524	−1.08367
$73$	−6.89087	−0.806516	−0.403258	−	0.915086i	$-0.632122\pi$
$73$	−6.89087	−0.806516	−0.403258	+	0.915086i	$0.632122\pi$
$74$	−4.36894	−0.507879
$75$	−1.26913	−0.146547
$76$	0.862022	0.0988807
$77$	−9.30200	−1.06006
$78$	−0.498447	−0.0564380
$79$	4.81605	0.541847	0.270924	−	0.962601i	$-0.412671\pi$
$79$	4.81605	0.541847	0.270924	+	0.962601i	$0.412671\pi$
$80$	−9.53219	−1.06573
$81$	8.88643	0.987381
$82$	7.06365	0.780049
$83$	−3.99493	−0.438501	−0.219251	−	0.975669i	$-0.570361\pi$
$83$	−3.99493	−0.438501	−0.219251	+	0.975669i	$0.570361\pi$
$84$	0.123760	0.0135033
$85$	1.92846	0.209171
$86$	4.07963	0.439918
$87$	−0.112413	−0.0120519
$88$	−16.2354	−1.73069
$89$	2.33459	0.247466	0.123733	−	0.992316i	$-0.460513\pi$
$89$	2.33459	0.247466	0.123733	+	0.992316i	$0.460513\pi$
$90$	14.1421	1.49071
$91$	−6.66692	−0.698883
$92$	−3.18179	−0.331724
$93$	−0.206481	−0.0214110
$94$	8.38639	0.864990
$95$	−5.57320	−0.571799
$96$	0.380627	0.0388475
$97$	−14.6888	−1.49142	−0.745711	−	0.666269i	$-0.767890\pi$
$97$	−14.6888	−1.49142	−0.745711	+	0.666269i	$0.767890\pi$
$98$	4.56248	0.460880
$99$	15.7570	1.58364
$100$	−7.04795	−0.704795
$101$	−16.9756	−1.68914	−0.844568	−	0.535449i	$-0.820142\pi$
$101$	−16.9756	−1.68914	−0.844568	+	0.535449i	$0.820142\pi$
$102$	0.0629993	0.00623786
$103$	6.06587	0.597688	0.298844	−	0.954302i	$-0.403399\pi$
$103$	6.06587	0.597688	0.298844	+	0.954302i	$0.403399\pi$
$104$	−11.6362	−1.14102
$105$	−0.800139	−0.0780856
$106$	−9.58746	−0.931217
$107$	0.637523	0.0616316	0.0308158	−	0.999525i	$-0.490189\pi$
$107$	0.637523	0.0616316	0.0308158	+	0.999525i	$0.490189\pi$
$108$	−0.420170	−0.0404309
$109$	−3.73786	−0.358023	−0.179011	−	0.983847i	$-0.557290\pi$
$109$	−3.73786	−0.358023	−0.179011	+	0.983847i	$0.557290\pi$
$110$	24.9697	2.38076
$111$	−0.418723	−0.0397434
$112$	−4.16507	−0.393563
$113$	−11.1297	−1.04699	−0.523496	−	0.852028i	$-0.675373\pi$
$113$	−11.1297	−1.04699	−0.523496	+	0.852028i	$0.675373\pi$
$114$	−0.182066	−0.0170520
$115$	20.5711	1.91827
$116$	−0.624270	−0.0579620
$117$	11.2934	1.04407
$118$	0.194253	0.0178824
$119$	0.842640	0.0772446
$120$	−1.39653	−0.127485
$121$	16.8210	1.52919
$122$	8.72231	0.789681
$123$	0.676985	0.0610417
$124$	−1.14666	−0.102973
$125$	25.3865	2.27064
$126$	6.17937	0.550502
$127$	20.2193	1.79417	0.897084	−	0.441859i	$-0.145681\pi$
$127$	20.2193	1.79417	0.897084	+	0.441859i	$0.145681\pi$
$128$	−3.42650	−0.302863
$129$	0.390995	0.0344252
$130$	17.8962	1.56960
$131$	−7.33446	−0.640815	−0.320407	−	0.947280i	$-0.603820\pi$
$131$	−7.33446	−0.640815	−0.320407	+	0.947280i	$0.603820\pi$
$132$	−0.370149	−0.0322173
$133$	−2.43520	−0.211159
$134$	8.03063	0.693741
$135$	2.71651	0.233800
$136$	1.47071	0.126112
$137$	−19.6676	−1.68032	−0.840158	−	0.542342i	$-0.817538\pi$
$137$	−19.6676	−1.68032	−0.840158	+	0.542342i	$0.817538\pi$
$138$	0.672020	0.0572061
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	−4.44346	−0.375541
$141$	0.803758	0.0676886
$142$	−1.77883	−0.149276
$143$	19.9399	1.66746
$144$	7.05539	0.587950
$145$	4.03608	0.335178
$146$	8.08241	0.668905
$147$	0.437272	0.0360656
$148$	−2.32532	−0.191140
$149$	−12.3745	−1.01376	−0.506879	−	0.862017i	$-0.669201\pi$
$149$	−12.3745	−1.01376	−0.506879	+	0.862017i	$0.669201\pi$
$150$	1.48858	0.121542
$151$	−7.52197	−0.612129	−0.306064	−	0.952011i	$-0.599012\pi$
$151$	−7.52197	−0.612129	−0.306064	+	0.952011i	$0.599012\pi$
$152$	−4.25031	−0.344746
$153$	−1.42738	−0.115397
$154$	10.9105	0.879189
$155$	7.41347	0.595465
$156$	−0.265292	−0.0212404
$157$	8.54136	0.681675	0.340837	−	0.940122i	$-0.389289\pi$
$157$	8.54136	0.681675	0.340837	+	0.940122i	$0.389289\pi$
$158$	−5.64881	−0.449395
$159$	−0.918870	−0.0728711
$160$	−13.6660	−1.08039
$161$	8.98852	0.708395
$162$	−10.4230	−0.818910
$163$	0.576695	0.0451702	0.0225851	−	0.999745i	$-0.492810\pi$
$163$	0.576695	0.0451702	0.0225851	+	0.999745i	$0.492810\pi$
$164$	3.75954	0.293571
$165$	2.39311	0.186303
$166$	4.68572	0.363682
$167$	−8.71659	−0.674510	−0.337255	−	0.941413i	$-0.609498\pi$
$167$	−8.71659	−0.674510	−0.337255	+	0.941413i	$0.609498\pi$
$168$	−0.610213	−0.0470789
$169$	1.29128	0.0993294
$170$	−2.26192	−0.173482
$171$	4.12509	0.315454
$172$	2.17133	0.165563
$173$	−6.38388	−0.485357	−0.242679	−	0.970107i	$-0.578026\pi$
$173$	−6.38388	−0.485357	−0.242679	+	0.970107i	$0.578026\pi$
$174$	0.131851	0.00999560
$175$	19.9104	1.50508
$176$	12.4572	0.938996
$177$	0.0186174	0.00139937
$178$	−2.73827	−0.205242
$179$	−13.6151	−1.01764	−0.508820	−	0.860873i	$-0.669918\pi$
$179$	−13.6151	−1.01764	−0.508820	+	0.860873i	$0.669918\pi$
$180$	7.52696	0.561027
$181$	−2.18449	−0.162372	−0.0811858	−	0.996699i	$-0.525871\pi$
$181$	−2.18449	−0.162372	−0.0811858	+	0.996699i	$0.525871\pi$
$182$	7.81973	0.579637
$183$	0.835953	0.0617954
$184$	15.6882	1.15655
$185$	15.0338	1.10531
$186$	0.242184	0.0177578
$187$	−2.52022	−0.184297
$188$	4.46356	0.325538
$189$	1.18698	0.0863398
$190$	6.53689	0.474236
$191$	20.7061	1.49824	0.749119	−	0.662435i	$-0.230477\pi$
$191$	20.7061	1.49824	0.749119	+	0.662435i	$0.230477\pi$
$192$	−0.977425	−0.0705396
$193$	−19.7223	−1.41964	−0.709820	−	0.704383i	$-0.751224\pi$
$193$	−19.7223	−1.41964	−0.709820	+	0.704383i	$0.751224\pi$
$194$	17.2287	1.23695
$195$	1.71519	0.122827
$196$	2.42833	0.173452
$197$	−5.77154	−0.411205	−0.205603	−	0.978636i	$-0.565915\pi$
$197$	−5.77154	−0.411205	−0.205603	+	0.978636i	$0.565915\pi$
$198$	−18.4817	−1.31344
$199$	−8.10383	−0.574466	−0.287233	−	0.957861i	$-0.592735\pi$
$199$	−8.10383	−0.574466	−0.287233	+	0.957861i	$0.592735\pi$
$200$	34.7508	2.45726
$201$	0.769662	0.0542878
$202$	19.9109	1.40093
$203$	1.76356	0.123777
$204$	0.0335306	0.00234761
$205$	−24.3065	−1.69764
$206$	−7.11475	−0.495708
$207$	−15.2260	−1.05828
$208$	8.92830	0.619066
$209$	7.28337	0.503801
$210$	0.938495	0.0647623
$211$	−1.36534	−0.0939942	−0.0469971	−	0.998895i	$-0.514965\pi$
$211$	−1.36534	−0.0939942	−0.0469971	+	0.998895i	$0.514965\pi$
$212$	−5.10281	−0.350463
$213$	−0.170484	−0.0116814
$214$	−0.747760	−0.0511158
$215$	−14.0383	−0.957401
$216$	2.07170	0.140962
$217$	3.23931	0.219898
$218$	4.38420	0.296935
$219$	0.774624	0.0523443
$220$	13.2898	0.895998
$221$	−1.80629	−0.121504
$222$	0.491126	0.0329622
$223$	−26.1169	−1.74892	−0.874459	−	0.485099i	$-0.838783\pi$
$223$	−26.1169	−1.74892	−0.874459	+	0.485099i	$0.838783\pi$
$224$	−5.97134	−0.398977
$225$	−33.7270	−2.24847
$226$	13.0542	0.868350
$227$	−5.01529	−0.332876	−0.166438	−	0.986052i	$-0.553227\pi$
$227$	−5.01529	−0.332876	−0.166438	+	0.986052i	$0.553227\pi$
$228$	−0.0969025	−0.00641753
$229$	21.4435	1.41703	0.708514	−	0.705697i	$-0.249366\pi$
$229$	21.4435	1.41703	0.708514	+	0.705697i	$0.249366\pi$
$230$	−24.1282	−1.59097
$231$	1.04567	0.0687998
$232$	3.07805	0.202084
$233$	8.13994	0.533265	0.266633	−	0.963798i	$-0.414089\pi$
$233$	8.13994	0.533265	0.266633	+	0.963798i	$0.414089\pi$
$234$	−13.2462	−0.865929
$235$	−28.8581	−1.88250
$236$	0.103389	0.00673004
$237$	−0.541386	−0.0351668
$238$	−0.988345	−0.0640649
$239$	7.32716	0.473955	0.236977	−	0.971515i	$-0.423843\pi$
$239$	7.32716	0.473955	0.236977	+	0.971515i	$0.423843\pi$
$240$	1.07154	0.0691678
$241$	17.4351	1.12309	0.561547	−	0.827445i	$-0.310206\pi$
$241$	17.4351	1.12309	0.561547	+	0.827445i	$0.310206\pi$
$242$	−19.7297	−1.26827
$243$	−3.01812	−0.193613
$244$	4.64235	0.297196
$245$	−15.6998	−1.00302
$246$	−0.794046	−0.0506265
$247$	5.22013	0.332149
$248$	5.65376	0.359014
$249$	0.449083	0.0284595
$250$	−29.7762	−1.88321
$251$	26.7311	1.68725	0.843626	−	0.536932i	$-0.180417\pi$
$251$	26.7311	1.68725	0.843626	+	0.536932i	$0.180417\pi$
$252$	3.28890	0.207181
$253$	−26.8835	−1.69015
$254$	−23.7155	−1.48804
$255$	−0.216785	−0.0135756
$256$	−13.3709	−0.835681
$257$	−2.65375	−0.165536	−0.0827681	−	0.996569i	$-0.526376\pi$
$257$	−2.65375	−0.165536	−0.0827681	+	0.996569i	$0.526376\pi$
$258$	−0.458603	−0.0285514
$259$	6.56900	0.408178
$260$	9.52505	0.590719
$261$	−2.98736	−0.184913
$262$	8.60270	0.531477
$263$	−28.6942	−1.76936	−0.884680	−	0.466198i	$-0.845623\pi$
$263$	−28.6942	−1.76936	−0.884680	+	0.466198i	$0.845623\pi$
$264$	1.82507	0.112325
$265$	32.9911	2.02663
$266$	2.85629	0.175130
$267$	−0.262438	−0.0160609
$268$	4.27421	0.261089
$269$	−9.38583	−0.572264	−0.286132	−	0.958190i	$-0.592370\pi$
$269$	−9.38583	−0.572264	−0.286132	+	0.958190i	$0.592370\pi$
$270$	−3.18624	−0.193908
$271$	20.4305	1.24107	0.620533	−	0.784180i	$-0.286916\pi$
$271$	20.4305	1.24107	0.620533	+	0.784180i	$0.286916\pi$
$272$	−1.12846	−0.0684228
$273$	0.749449	0.0453587
$274$	23.0684	1.39361
$275$	−59.5493	−3.59096
$276$	0.357675	0.0215295
$277$	−0.716394	−0.0430439	−0.0215220	−	0.999768i	$-0.506851\pi$
$277$	−0.716394	−0.0430439	−0.0215220	+	0.999768i	$0.506851\pi$
$278$	−1.17292	−0.0703468
$279$	−5.48720	−0.328510
$280$	21.9091	1.30932
$281$	−0.996391	−0.0594397	−0.0297198	−	0.999558i	$-0.509462\pi$
$281$	−0.996391	−0.0594397	−0.0297198	+	0.999558i	$0.509462\pi$
$282$	−0.942740	−0.0561394
$283$	3.18076	0.189076	0.0945382	−	0.995521i	$-0.469863\pi$
$283$	3.18076	0.189076	0.0945382	+	0.995521i	$0.469863\pi$
$284$	−0.946760	−0.0561799
$285$	0.626501	0.0371107
$286$	−23.3878	−1.38295
$287$	−10.6207	−0.626918
$288$	10.1151	0.596038
$289$	−16.7717	−0.986571
$290$	−4.73397	−0.277989
$291$	1.65121	0.0967959
$292$	4.30177	0.251742
$293$	−33.3453	−1.94805	−0.974027	−	0.226430i	$-0.927295\pi$
$293$	−33.3453	−1.94805	−0.974027	+	0.226430i	$0.927295\pi$
$294$	−0.512883	−0.0299119
$295$	−0.668437	−0.0389179
$296$	11.4653	0.666406
$297$	−3.55009	−0.205997
$298$	14.5142	0.840786
$299$	−19.2679	−1.11429
$300$	0.792282	0.0457424
$301$	−6.13400	−0.353558
$302$	8.82263	0.507685
$303$	1.90828	0.109628
$304$	3.26121	0.187043
$305$	−30.0140	−1.71860
$306$	1.67420	0.0957076
$307$	1.20286	0.0686507	0.0343254	−	0.999411i	$-0.489072\pi$
$307$	1.20286	0.0686507	0.0343254	+	0.999411i	$0.489072\pi$
$308$	5.80696	0.330882
$309$	−0.681883	−0.0387910
$310$	−8.69538	−0.493864
$311$	−25.1345	−1.42525	−0.712624	−	0.701546i	$-0.752494\pi$
$311$	−25.1345	−1.42525	−0.712624	+	0.701546i	$0.752494\pi$
$312$	1.30806	0.0740543
$313$	1.42894	0.0807684	0.0403842	−	0.999184i	$-0.487142\pi$
$313$	1.42894	0.0807684	0.0403842	+	0.999184i	$0.487142\pi$
$314$	−10.0183	−0.565365
$315$	−21.2636	−1.19807
$316$	−3.00651	−0.169130
$317$	18.3223	1.02908	0.514541	−	0.857465i	$-0.327962\pi$
$317$	18.3223	1.02908	0.514541	+	0.857465i	$0.327962\pi$
$318$	1.07776	0.0604376
$319$	−5.27457	−0.295319
$320$	35.0934	1.96178
$321$	−0.0716659	−0.00400000
$322$	−10.5428	−0.587526
$323$	−0.659778	−0.0367110
$324$	−5.54753	−0.308196
$325$	−42.6801	−2.36747
$326$	−0.676414	−0.0374631
$327$	0.420185	0.0232363
$328$	−18.5369	−1.02353
$329$	−12.6095	−0.695185
$330$	−2.80692	−0.154516
$331$	−6.78985	−0.373204	−0.186602	−	0.982436i	$-0.559747\pi$
$331$	−6.78985	−0.373204	−0.186602	+	0.982436i	$0.559747\pi$
$332$	2.49392	0.136872
$333$	−11.1275	−0.609783
$334$	10.2238	0.559422
$335$	−27.6339	−1.50980
$336$	0.468209	0.0255429
$337$	−31.9887	−1.74254	−0.871269	−	0.490806i	$-0.836702\pi$
$337$	−31.9887	−1.74254	−0.871269	+	0.490806i	$0.836702\pi$
$338$	−1.51456	−0.0823814
$339$	1.25112	0.0679516
$340$	−1.20388	−0.0652897
$341$	−9.68834	−0.524653
$342$	−4.83838	−0.261630
$343$	−19.2049	−1.03697
$344$	−10.7060	−0.577231
$345$	−2.31246	−0.124499
$346$	7.48774	0.402544
$347$	−5.55895	−0.298420	−0.149210	−	0.988806i	$-0.547673\pi$
$347$	−5.55895	−0.298420	−0.149210	+	0.988806i	$0.547673\pi$
$348$	0.0701761	0.00376183
$349$	−15.2030	−0.813800	−0.406900	−	0.913473i	$-0.633390\pi$
$349$	−15.2030	−0.813800	−0.406900	+	0.913473i	$0.633390\pi$
$350$	−23.3532	−1.24828
$351$	−2.54442	−0.135811
$352$	17.8595	0.951914
$353$	27.6137	1.46973	0.734866	−	0.678213i	$-0.237245\pi$
$353$	27.6137	1.46973	0.734866	+	0.678213i	$0.237245\pi$
$354$	−0.0218366	−0.00116060
$355$	6.12106	0.324872
$356$	−1.45741	−0.0772427
$357$	−0.0947237	−0.00501331
$358$	15.9693	0.844006
$359$	3.93976	0.207933	0.103966	−	0.994581i	$-0.466847\pi$
$359$	3.93976	0.207933	0.103966	+	0.994581i	$0.466847\pi$
$360$	−37.1127	−1.95601
$361$	−17.0933	−0.899645
$362$	2.56222	0.134667
$363$	−1.89090	−0.0992468
$364$	4.16196	0.218146
$365$	−27.8121	−1.45575
$366$	−0.980502	−0.0512517
$367$	28.1891	1.47146	0.735730	−	0.677274i	$-0.236839\pi$
$367$	28.1891	1.47146	0.735730	+	0.677274i	$0.236839\pi$
$368$	−12.0374	−0.627492
$369$	17.9908	0.936564
$370$	−17.6334	−0.916715
$371$	14.4154	0.748410
$372$	0.128900	0.00668314
$373$	−1.73410	−0.0897880	−0.0448940	−	0.998992i	$-0.514295\pi$
$373$	−1.73410	−0.0897880	−0.0448940	+	0.998992i	$0.514295\pi$
$374$	2.95601	0.152852
$375$	−2.85378	−0.147368
$376$	−22.0082	−1.13498
$377$	−3.78038	−0.194700
$378$	−1.39222	−0.0716082
$379$	18.4830	0.949407	0.474703	−	0.880146i	$-0.342555\pi$
$379$	18.4830	0.949407	0.474703	+	0.880146i	$0.342555\pi$
$380$	3.47918	0.178478
$381$	−2.27291	−0.116445
$382$	−24.2865	−1.24260
$383$	−8.72751	−0.445955	−0.222978	−	0.974824i	$-0.571578\pi$
$383$	−8.72751	−0.445955	−0.222978	+	0.974824i	$0.571578\pi$
$384$	0.385183	0.0196563
$385$	−37.5436	−1.91340
$386$	23.1325	1.17742
$387$	10.3906	0.528186
$388$	9.16978	0.465525
$389$	36.8999	1.87090	0.935450	−	0.353460i	$-0.114995\pi$
$389$	36.8999	1.87090	0.935450	+	0.353460i	$0.114995\pi$
$390$	−2.01177	−0.101870
$391$	2.43529	0.123158
$392$	−11.9732	−0.604737
$393$	0.824489	0.0415900
$394$	6.76953	0.341044
$395$	19.4379	0.978028
$396$	−9.83665	−0.494310
$397$	27.5939	1.38490	0.692450	−	0.721466i	$-0.256531\pi$
$397$	27.5939	1.38490	0.692450	+	0.721466i	$0.256531\pi$
$398$	9.50511	0.476448
$399$	0.273749	0.0137046
$400$	−26.6639	−1.33319
$401$	28.8989	1.44314	0.721572	−	0.692340i	$-0.243420\pi$
$401$	28.8989	1.44314	0.721572	+	0.692340i	$0.243420\pi$
$402$	−0.902748	−0.0450250
$403$	−6.94382	−0.345896
$404$	10.5974	0.527238
$405$	35.8663	1.78221
$406$	−2.06850	−0.102658
$407$	−19.6470	−0.973866
$408$	−0.165327	−0.00818491
$409$	−27.6936	−1.36936	−0.684681	−	0.728843i	$-0.740058\pi$
$409$	−27.6936	−1.36936	−0.684681	+	0.728843i	$0.740058\pi$
$410$	28.5094	1.40798
$411$	2.21089	0.109055
$412$	−3.78674	−0.186559
$413$	−0.292073	−0.0143720
$414$	17.8588	0.877714
$415$	−16.1239	−0.791489
$416$	12.8002	0.627583
$417$	−0.112413	−0.00550489
$418$	−8.54277	−0.417841
$419$	14.3972	0.703351	0.351675	−	0.936122i	$-0.385612\pi$
$419$	14.3972	0.703351	0.351675	+	0.936122i	$0.385612\pi$
$420$	0.499503	0.0243733
$421$	18.5752	0.905301	0.452650	−	0.891688i	$-0.350479\pi$
$421$	18.5752	0.905301	0.452650	+	0.891688i	$0.350479\pi$
$422$	1.60143	0.0779565
$423$	21.3598	1.03855
$424$	25.1601	1.22188
$425$	5.39439	0.261666
$426$	0.199963	0.00968826
$427$	−13.1146	−0.634659
$428$	−0.397986	−0.0192374
$429$	−2.24150	−0.108221
$430$	16.4657	0.794046
$431$	−13.9066	−0.669859	−0.334930	−	0.942243i	$-0.608713\pi$
$431$	−13.9066	−0.669859	−0.334930	+	0.942243i	$0.608713\pi$
$432$	−1.58959	−0.0764793
$433$	−28.0793	−1.34941	−0.674703	−	0.738089i	$-0.735728\pi$
$433$	−28.0793	−1.34941	−0.674703	+	0.738089i	$0.735728\pi$
$434$	−3.79943	−0.182379
$435$	−0.453708	−0.0217536
$436$	2.33344	0.111751
$437$	−7.03792	−0.336669
$438$	−0.908569	−0.0434131
$439$	17.3756	0.829292	0.414646	−	0.909983i	$-0.363905\pi$
$439$	17.3756	0.829292	0.414646	+	0.909983i	$0.363905\pi$
$440$	−65.5271	−3.12388
$441$	11.6204	0.553354
$442$	2.11863	0.100773
$443$	−14.9975	−0.712552	−0.356276	−	0.934381i	$-0.615954\pi$
$443$	−14.9975	−0.712552	−0.356276	+	0.934381i	$0.615954\pi$
$444$	0.261396	0.0124053
$445$	9.42256	0.446672
$446$	30.6329	1.45051
$447$	1.39105	0.0657946
$448$	15.3340	0.724465
$449$	−25.2489	−1.19157	−0.595784	−	0.803145i	$-0.703158\pi$
$449$	−25.2489	−1.19157	−0.595784	+	0.803145i	$0.703158\pi$
$450$	39.5590	1.86483
$451$	31.7650	1.49576
$452$	6.94793	0.326803
$453$	0.845567	0.0397282
$454$	5.88251	0.276080
$455$	−26.9082	−1.26148
$456$	0.477791	0.0223746
$457$	−22.2919	−1.04277	−0.521387	−	0.853321i	$-0.674585\pi$
$457$	−22.2919	−1.04277	−0.521387	+	0.853321i	$0.674585\pi$
$458$	−25.1514	−1.17525
$459$	0.321592	0.0150106
$460$	−12.8419	−0.598758
$461$	33.1135	1.54225	0.771125	−	0.636684i	$-0.219694\pi$
$461$	33.1135	1.54225	0.771125	+	0.636684i	$0.219694\pi$
$462$	−1.22648	−0.0570609
$463$	−5.29611	−0.246131	−0.123066	−	0.992399i	$-0.539273\pi$
$463$	−5.29611	−0.246131	−0.123066	+	0.992399i	$0.539273\pi$
$464$	−2.36175	−0.109641
$465$	−0.833371	−0.0386467
$466$	−9.54746	−0.442277
$467$	−30.8314	−1.42671	−0.713354	−	0.700804i	$-0.752825\pi$
$467$	−30.8314	−1.42671	−0.713354	+	0.700804i	$0.752825\pi$
$468$	−7.05012	−0.325892
$469$	−12.0746	−0.557553
$470$	33.8481	1.56130
$471$	−0.960160	−0.0442419
$472$	−0.509773	−0.0234642
$473$	18.3460	0.843549
$474$	0.635000	0.0291665
$475$	−15.5896	−0.715301
$476$	−0.526035	−0.0241108
$477$	−24.4188	−1.11806
$478$	−8.59414	−0.393087
$479$	20.9252	0.956097	0.478048	−	0.878333i	$-0.341344\pi$
$479$	20.9252	0.956097	0.478048	+	0.878333i	$0.341344\pi$
$480$	1.53624	0.0701193
$481$	−14.0814	−0.642056
$482$	−20.4499	−0.931467
$483$	−1.01043	−0.0459760
$484$	−10.5009	−0.477312
$485$	−59.2851	−2.69200
$486$	3.54000	0.160578
$487$	19.1028	0.865630	0.432815	−	0.901483i	$-0.357520\pi$
$487$	19.1028	0.865630	0.432815	+	0.901483i	$0.357520\pi$
$488$	−22.8897	−1.03617
$489$	−0.0648280	−0.00293163
$490$	18.4145	0.831883
$491$	2.56820	0.115901	0.0579506	−	0.998319i	$-0.481543\pi$
$491$	2.56820	0.115901	0.0579506	+	0.998319i	$0.481543\pi$
$492$	−0.422622	−0.0190533
$493$	0.477807	0.0215193
$494$	−6.12277	−0.275476
$495$	63.5966	2.85846
$496$	−4.33806	−0.194785
$497$	2.67459	0.119972
$498$	−0.526736	−0.0236036
$499$	4.76268	0.213207	0.106603	−	0.994302i	$-0.466002\pi$
$499$	4.76268	0.213207	0.106603	+	0.994302i	$0.466002\pi$
$500$	−15.8481	−0.708746
$501$	0.979858	0.0437768
$502$	−31.3533	−1.39937
$503$	9.40004	0.419127	0.209564	−	0.977795i	$-0.432796\pi$
$503$	9.40004	0.419127	0.209564	+	0.977795i	$0.432796\pi$
$504$	−16.2163	−0.722333
$505$	−68.5148	−3.04887
$506$	31.5320	1.40177
$507$	−0.145157	−0.00644665
$508$	−12.6223	−0.560023
$509$	5.41636	0.240076	0.120038	−	0.992769i	$-0.461698\pi$
$509$	5.41636	0.240076	0.120038	+	0.992769i	$0.461698\pi$
$510$	0.254270	0.0112593
$511$	−12.1525	−0.537593
$512$	22.5359	0.995956
$513$	−0.929390	−0.0410336
$514$	3.11262	0.137292
$515$	24.4823	1.07882
$516$	−0.244086	−0.0107453
$517$	37.7134	1.65863
$518$	−7.70488	−0.338533
$519$	0.717631	0.0315005
$520$	−46.9645	−2.05953
$521$	−4.90208	−0.214764	−0.107382	−	0.994218i	$-0.534247\pi$
$521$	−4.90208	−0.214764	−0.107382	+	0.994218i	$0.534247\pi$
$522$	3.50392	0.153363
$523$	10.9917	0.480632	0.240316	−	0.970695i	$-0.422749\pi$
$523$	10.9917	0.480632	0.240316	+	0.970695i	$0.422749\pi$
$524$	4.57869	0.200021
$525$	−2.23819	−0.0976825
$526$	33.6559	1.46747
$527$	0.877637	0.0382305
$528$	−1.40035	−0.0609424
$529$	2.97751	0.129457
$530$	−38.6957	−1.68083
$531$	0.494754	0.0214705
$532$	1.52022	0.0659101
$533$	22.7666	0.986131
$534$	0.307817	0.0133206
$535$	2.57309	0.111244
$536$	−21.0746	−0.910282
$537$	1.53051	0.0660465
$538$	11.0088	0.474622
$539$	20.5174	0.883745
$540$	−1.69584	−0.0729772
$541$	1.82799	0.0785914	0.0392957	−	0.999228i	$-0.487489\pi$
$541$	1.82799	0.0785914	0.0392957	+	0.999228i	$0.487489\pi$
$542$	−23.9633	−1.02931
$543$	0.245565	0.0105382
$544$	−1.61784	−0.0693642
$545$	−15.0863	−0.646226
$546$	−0.879040	−0.0376194
$547$	20.0622	0.857796	0.428898	−	0.903353i	$-0.358902\pi$
$547$	20.0622	0.857796	0.428898	+	0.903353i	$0.358902\pi$
$548$	12.2779	0.524486
$549$	22.2154	0.948128
$550$	69.8463	2.97826
$551$	−1.38085	−0.0588261
$552$	−1.76356	−0.0750622
$553$	8.49337	0.361175
$554$	0.840269	0.0356996
$555$	−1.69000	−0.0717363
$556$	−0.624270	−0.0264750
$557$	37.6076	1.59348	0.796742	−	0.604320i	$-0.206555\pi$
$557$	37.6076	1.59348	0.796742	+	0.604320i	$0.206555\pi$
$558$	6.43602	0.272458
$559$	13.1489	0.556139
$560$	−16.8106	−0.710376
$561$	0.283306	0.0119612
$562$	1.16868	0.0492979
$563$	2.83138	0.119328	0.0596642	−	0.998219i	$-0.480997\pi$
$563$	2.83138	0.119328	0.0596642	+	0.998219i	$0.480997\pi$
$564$	−0.501762	−0.0211280
$565$	−44.9202	−1.88981
$566$	−3.73076	−0.156815
$567$	15.6717	0.658151
$568$	4.66812	0.195870
$569$	−0.665576	−0.0279024	−0.0139512	−	0.999903i	$-0.504441\pi$
$569$	−0.665576	−0.0279024	−0.0139512	+	0.999903i	$0.504441\pi$
$570$	−0.734832	−0.0307787
$571$	−39.0640	−1.63478	−0.817389	−	0.576087i	$-0.804579\pi$
$571$	−39.0640	−1.63478	−0.817389	+	0.576087i	$0.804579\pi$
$572$	−12.4479	−0.520471
$573$	−2.32763	−0.0972382
$574$	12.4571	0.519951
$575$	57.5425	2.39969
$576$	−25.9749	−1.08229
$577$	13.9874	0.582304	0.291152	−	0.956677i	$-0.405962\pi$
$577$	13.9874	0.582304	0.291152	+	0.956677i	$0.405962\pi$
$578$	19.6718	0.818238
$579$	2.21704	0.0921370
$580$	−2.51960	−0.104621
$581$	−7.04530	−0.292288
$582$	−1.93673	−0.0802802
$583$	−43.1145	−1.78562
$584$	−21.2104	−0.877694
$585$	45.5809	1.88454
$586$	39.1112	1.61567
$587$	3.73714	0.154248	0.0771241	−	0.997022i	$-0.475426\pi$
$587$	3.73714	0.154248	0.0771241	+	0.997022i	$0.475426\pi$
$588$	−0.272976	−0.0112573
$589$	−2.53634	−0.104508
$590$	0.784020	0.0322776
$591$	0.648796	0.0266879
$592$	−8.79717	−0.361561
$593$	−27.3845	−1.12455	−0.562274	−	0.826951i	$-0.690073\pi$
$593$	−27.3845	−1.12455	−0.562274	+	0.826951i	$0.690073\pi$
$594$	4.16395	0.170849
$595$	3.40096	0.139426
$596$	7.72502	0.316429
$597$	0.910977	0.0372838
$598$	22.5996	0.924167
$599$	−20.0120	−0.817667	−0.408833	−	0.912609i	$-0.634064\pi$
$599$	−20.0120	−0.817667	−0.408833	+	0.912609i	$0.634064\pi$
$600$	−3.90645	−0.159480
$601$	−11.6315	−0.474459	−0.237229	−	0.971454i	$-0.576239\pi$
$601$	−11.6315	−0.474459	−0.237229	+	0.971454i	$0.576239\pi$
$602$	7.19466	0.293232
$603$	20.4537	0.832938
$604$	4.69574	0.191067
$605$	67.8910	2.76016
$606$	−2.23825	−0.0909227
$607$	−44.3676	−1.80082	−0.900412	−	0.435038i	$-0.856735\pi$
$607$	−44.3676	−1.80082	−0.900412	+	0.435038i	$0.856735\pi$
$608$	4.67550	0.189617
$609$	−0.198247	−0.00803337
$610$	35.2039	1.42536
$611$	27.0299	1.09351
$612$	0.891072	0.0360195
$613$	12.0165	0.485341	0.242670	−	0.970109i	$-0.421977\pi$
$613$	12.0165	0.485341	0.242670	+	0.970109i	$0.421977\pi$
$614$	−1.41085	−0.0569373
$615$	2.73236	0.110180
$616$	−28.6320	−1.15362
$617$	17.6741	0.711533	0.355767	−	0.934575i	$-0.384220\pi$
$617$	17.6741	0.711533	0.355767	+	0.934575i	$0.384220\pi$
$618$	0.799791	0.0321723
$619$	26.4019	1.06118	0.530590	−	0.847629i	$-0.321970\pi$
$619$	26.4019	1.06118	0.530590	+	0.847629i	$0.321970\pi$
$620$	−4.62801	−0.185865
$621$	3.43045	0.137659
$622$	29.4807	1.18207
$623$	4.11717	0.164951
$624$	−1.00366	−0.0401785
$625$	46.0124	1.84050
$626$	−1.67602	−0.0669874
$627$	−0.818746	−0.0326976
$628$	−5.33212	−0.212775
$629$	1.77976	0.0709638
$630$	24.9404	0.993649
$631$	−35.5494	−1.41520	−0.707599	−	0.706614i	$-0.750222\pi$
$631$	−35.5494	−1.41520	−0.707599	+	0.706614i	$0.750222\pi$
$632$	14.8240	0.589667
$633$	0.153482	0.00610038
$634$	−21.4905	−0.853497
$635$	81.6064	3.23845
$636$	0.573623	0.0227456
$637$	14.7052	0.582640
$638$	6.18662	0.244931
$639$	−4.53059	−0.179228
$640$	−13.8296	−0.546663
$641$	−1.62953	−0.0643626	−0.0321813	−	0.999482i	$-0.510245\pi$
$641$	−1.62953	−0.0643626	−0.0321813	+	0.999482i	$0.510245\pi$
$642$	0.0840580	0.00331750
$643$	7.19150	0.283605	0.141802	−	0.989895i	$-0.454710\pi$
$643$	7.19150	0.283605	0.141802	+	0.989895i	$0.454710\pi$
$644$	−5.61127	−0.221115
$645$	1.57808	0.0621370
$646$	0.773864	0.0304473
$647$	35.4030	1.39183	0.695917	−	0.718122i	$-0.254998\pi$
$647$	35.4030	1.39183	0.695917	+	0.718122i	$0.254998\pi$
$648$	27.3528	1.07452
$649$	0.873550	0.0342899
$650$	50.0602	1.96352
$651$	−0.364140	−0.0142718
$652$	−0.360013	−0.0140992
$653$	−16.2971	−0.637756	−0.318878	−	0.947796i	$-0.603306\pi$
$653$	−16.2971	−0.637756	−0.318878	+	0.947796i	$0.603306\pi$
$654$	−0.492841	−0.0192716
$655$	−29.6024	−1.15666
$656$	14.2231	0.555321
$657$	20.5855	0.803119
$658$	14.7899	0.576570
$659$	−21.0210	−0.818860	−0.409430	−	0.912342i	$-0.634272\pi$
$659$	−21.0210	−0.818860	−0.409430	+	0.912342i	$0.634272\pi$
$660$	−1.49395	−0.0581518
$661$	3.92204	0.152550	0.0762748	−	0.997087i	$-0.475697\pi$
$661$	3.92204	0.152550	0.0762748	+	0.997087i	$0.475697\pi$
$662$	7.96392	0.309526
$663$	0.203051	0.00788584
$664$	−12.2966	−0.477200
$665$	−9.82866	−0.381139
$666$	13.0516	0.505740
$667$	5.09681	0.197349
$668$	5.44151	0.210538
$669$	2.93588	0.113508
$670$	32.4122	1.25219
$671$	39.2240	1.51423
$672$	0.671257	0.0258943
$673$	−14.0394	−0.541180	−0.270590	−	0.962695i	$-0.587219\pi$
$673$	−14.0394	−0.541180	−0.270590	+	0.962695i	$0.587219\pi$
$674$	37.5201	1.44522
$675$	7.59876	0.292476
$676$	−0.806109	−0.0310042
$677$	−3.96274	−0.152300	−0.0761502	−	0.997096i	$-0.524263\pi$
$677$	−3.96274	−0.152300	−0.0761502	+	0.997096i	$0.524263\pi$
$678$	−1.46746	−0.0563574
$679$	−25.9046	−0.994125
$680$	5.93590	0.227631
$681$	0.563784	0.0216042
$682$	11.3636	0.435135
$683$	33.2889	1.27376	0.636881	−	0.770962i	$-0.280224\pi$
$683$	33.2889	1.27376	0.636881	+	0.770962i	$0.280224\pi$
$684$	−2.57517	−0.0984642
$685$	−79.3799	−3.03295
$686$	22.5257	0.860036
$687$	−2.41053	−0.0919675
$688$	8.21462	0.313179
$689$	−30.9010	−1.17723
$690$	2.71232	0.103256
$691$	−13.1617	−0.500696	−0.250348	−	0.968156i	$-0.580545\pi$
$691$	−13.1617	−0.500696	−0.250348	+	0.968156i	$0.580545\pi$
$692$	3.98526	0.151497
$693$	27.7884	1.05560
$694$	6.52018	0.247502
$695$	4.03608	0.153097
$696$	−0.346013	−0.0131156
$697$	−2.87750	−0.108993
$698$	17.8319	0.674947
$699$	−0.915035	−0.0346098
$700$	−12.4295	−0.469790
$701$	−3.93629	−0.148672	−0.0743359	−	0.997233i	$-0.523684\pi$
$701$	−3.93629	−0.148672	−0.0743359	+	0.997233i	$0.523684\pi$
$702$	2.98438	0.112638
$703$	−5.14346	−0.193989
$704$	−45.8620	−1.72849
$705$	3.24403	0.122177
$706$	−32.3886	−1.21896
$707$	−29.9374	−1.12591
$708$	−0.0116223	−0.000436791 0
$709$	−16.9160	−0.635295	−0.317647	−	0.948209i	$-0.602893\pi$
$709$	−16.9160	−0.635295	−0.317647	+	0.948209i	$0.602893\pi$
$710$	−7.17948	−0.269441
$711$	−14.3873	−0.539565
$712$	7.18596	0.269305
$713$	9.36184	0.350604
$714$	0.111103	0.00415792
$715$	80.4788	3.00974
$716$	8.49949	0.317641
$717$	−0.823669	−0.0307605
$718$	−4.62100	−0.172454
$719$	48.3380	1.80270	0.901352	−	0.433087i	$-0.142576\pi$
$719$	48.3380	1.80270	0.901352	+	0.433087i	$0.142576\pi$
$720$	28.4761	1.06124
$721$	10.6975	0.398396
$722$	20.0489	0.746144
$723$	−1.95993	−0.0728907
$724$	1.36371	0.0506819
$725$	11.2899	0.419296
$726$	2.21787	0.0823129
$727$	−11.9420	−0.442904	−0.221452	−	0.975171i	$-0.571080\pi$
$727$	−11.9420	−0.442904	−0.221452	+	0.975171i	$0.571080\pi$
$728$	−20.5211	−0.760562
$729$	−26.3200	−0.974815
$730$	32.6212	1.20737
$731$	−1.66191	−0.0614678
$732$	−0.521861	−0.0192885
$733$	23.0549	0.851554	0.425777	−	0.904828i	$-0.360001\pi$
$733$	23.0549	0.851554	0.425777	+	0.904828i	$0.360001\pi$
$734$	−33.0635	−1.22039
$735$	1.76486	0.0650979
$736$	−17.2576	−0.636125
$737$	36.1135	1.33026
$738$	−21.1017	−0.776764
$739$	9.14413	0.336372	0.168186	−	0.985755i	$-0.446209\pi$
$739$	9.14413	0.336372	0.168186	+	0.985755i	$0.446209\pi$
$740$	−9.38515	−0.345005
$741$	−0.586811	−0.0215570
$742$	−16.9080	−0.620714
$743$	−25.0270	−0.918150	−0.459075	−	0.888398i	$-0.651819\pi$
$743$	−25.0270	−0.918150	−0.459075	+	0.888398i	$0.651819\pi$
$744$	−0.635557	−0.0233006
$745$	−49.9444	−1.82982
$746$	2.03395	0.0744681
$747$	11.9343	0.436654
$748$	1.57330	0.0575256
$749$	1.12431	0.0410813
$750$	3.34724	0.122224
$751$	−42.8942	−1.56523	−0.782616	−	0.622505i	$-0.786115\pi$
$751$	−42.8942	−1.56523	−0.782616	+	0.622505i	$0.786115\pi$
$752$	16.8866	0.615790
$753$	−3.00492	−0.109506
$754$	4.43407	0.161479
$755$	−30.3592	−1.10489
$756$	−0.740994	−0.0269497
$757$	−42.7166	−1.55256	−0.776280	−	0.630388i	$-0.782896\pi$
$757$	−42.7166	−1.55256	−0.776280	+	0.630388i	$0.782896\pi$
$758$	−21.6790	−0.787415
$759$	3.02205	0.109694
$760$	−17.1546	−0.622262
$761$	−38.8114	−1.40691	−0.703457	−	0.710738i	$-0.748361\pi$
$761$	−38.8114	−1.40691	−0.703457	+	0.710738i	$0.748361\pi$
$762$	2.66593	0.0965764
$763$	−6.59194	−0.238644
$764$	−12.9262	−0.467653
$765$	−5.76102	−0.208290
$766$	10.2366	0.369865
$767$	0.626090	0.0226068
$768$	1.50306	0.0542371
$769$	−2.47076	−0.0890977	−0.0445489	−	0.999007i	$-0.514185\pi$
$769$	−2.47076	−0.0890977	−0.0445489	+	0.999007i	$0.514185\pi$
$770$	44.0354	1.58693
$771$	0.298316	0.0107436
$772$	12.3120	0.443119
$773$	−1.20880	−0.0434775	−0.0217387	−	0.999764i	$-0.506920\pi$
$773$	−1.20880	−0.0434775	−0.0217387	+	0.999764i	$0.506920\pi$
$774$	−12.1873	−0.438065
$775$	20.7373	0.744907
$776$	−45.2128	−1.62305
$777$	−0.738441	−0.0264914
$778$	−43.2805	−1.55168
$779$	8.31588	0.297947
$780$	−1.07074	−0.0383387
$781$	−7.99933	−0.286239
$782$	−2.85639	−0.102144
$783$	0.673058	0.0240531
$784$	9.18688	0.328103
$785$	34.4736	1.23041
$786$	−0.967056	−0.0344937
$787$	50.2634	1.79170	0.895848	−	0.444361i	$-0.146569\pi$
$787$	50.2634	1.79170	0.895848	+	0.444361i	$0.146569\pi$
$788$	3.60300	0.128352
$789$	3.22560	0.114835
$790$	−22.7990	−0.811153
$791$	−19.6278	−0.697885
$792$	48.5009	1.72340
$793$	28.1126	0.998307
$794$	−32.3654	−1.14860
$795$	−3.70863	−0.131531
$796$	5.05898	0.179311
$797$	−21.7007	−0.768679	−0.384340	−	0.923192i	$-0.625571\pi$
$797$	−21.7007	−0.768679	−0.384340	+	0.923192i	$0.625571\pi$
$798$	−0.321084	−0.0113662
$799$	−3.41634	−0.120861
$800$	−38.2272	−1.35154
$801$	−6.97425	−0.246423
$802$	−33.8960	−1.19691
$803$	36.3464	1.28264
$804$	−0.480477	−0.0169451
$805$	36.2783	1.27864
$806$	8.14451	0.286878
$807$	1.05509	0.0371409
$808$	−52.2517	−1.83821
$809$	55.4230	1.94857	0.974285	−	0.225319i	$-0.0723424\pi$
$809$	55.4230	1.94857	0.974285	+	0.225319i	$0.0723424\pi$
$810$	−42.0681	−1.47812
$811$	−12.5580	−0.440969	−0.220485	−	0.975390i	$-0.570764\pi$
$811$	−12.5580	−0.440969	−0.220485	+	0.975390i	$0.570764\pi$
$812$	−1.10094	−0.0386353
$813$	−2.29666	−0.0805473
$814$	23.0443	0.807701
$815$	2.32758	0.0815317
$816$	0.126853	0.00444076
$817$	4.80286	0.168031
$818$	32.4823	1.13572
$819$	19.9165	0.695939
$820$	15.1738	0.529892
$821$	−32.0111	−1.11719	−0.558597	−	0.829439i	$-0.688660\pi$
$821$	−32.0111	−1.11719	−0.558597	+	0.829439i	$0.688660\pi$
$822$	−2.59319	−0.0904479
$823$	−7.83943	−0.273265	−0.136633	−	0.990622i	$-0.543628\pi$
$823$	−7.83943	−0.273265	−0.136633	+	0.990622i	$0.543628\pi$
$824$	18.6710	0.650436
$825$	6.69412	0.233059
$826$	0.342576	0.0119198
$827$	38.9249	1.35355	0.676776	−	0.736189i	$-0.263377\pi$
$827$	38.9249	1.35355	0.676776	+	0.736189i	$0.263377\pi$
$828$	9.50516	0.330327
$829$	−14.1747	−0.492306	−0.246153	−	0.969231i	$-0.579167\pi$
$829$	−14.1747	−0.492306	−0.246153	+	0.969231i	$0.579167\pi$
$830$	18.9119	0.656442
$831$	0.0805320	0.00279362
$832$	−32.8702	−1.13957
$833$	−1.85860	−0.0643968
$834$	0.131851	0.00456563
$835$	−35.1808	−1.21748
$836$	−4.54679	−0.157254
$837$	1.23627	0.0427319
$838$	−16.8867	−0.583342
$839$	−1.11511	−0.0384977	−0.0192489	−	0.999815i	$-0.506127\pi$
$839$	−1.11511	−0.0384977	−0.0192489	+	0.999815i	$0.506127\pi$
$840$	−2.46287	−0.0849769
$841$	1.00000	0.0344828
$842$	−21.7872	−0.750835
$843$	0.112007	0.00385774
$844$	0.852344	0.0293389
$845$	5.21171	0.179288
$846$	−25.0532	−0.861347
$847$	29.6649	1.01930
$848$	−19.3050	−0.662937
$849$	−0.357559	−0.0122714
$850$	−6.32716	−0.217020
$851$	18.9849	0.650794
$852$	0.106428	0.00364617
$853$	−42.0194	−1.43872	−0.719358	−	0.694639i	$-0.755564\pi$
$853$	−42.0194	−1.43872	−0.719358	+	0.694639i	$0.755564\pi$
$854$	15.3823	0.526371
$855$	16.6492	0.569390
$856$	1.96232	0.0670708
$857$	−36.8854	−1.25998	−0.629991	−	0.776602i	$-0.716941\pi$
$857$	−36.8854	−1.25998	−0.629991	+	0.776602i	$0.716941\pi$
$858$	2.62909	0.0897557
$859$	−33.2222	−1.13353	−0.566764	−	0.823880i	$-0.691805\pi$
$859$	−33.2222	−1.13353	−0.566764	+	0.823880i	$0.691805\pi$
$860$	8.76367	0.298839
$861$	1.19390	0.0406881
$862$	16.3113	0.555566
$863$	14.7558	0.502294	0.251147	−	0.967949i	$-0.419192\pi$
$863$	14.7558	0.502294	0.251147	+	0.967949i	$0.419192\pi$
$864$	−2.27895	−0.0775314
$865$	−25.7658	−0.876064
$866$	32.9347	1.11917
$867$	1.88536	0.0640301
$868$	−2.02220	−0.0686380
$869$	−25.4025	−0.861722
$870$	0.532160	0.0180419
$871$	25.8833	0.877021
$872$	−11.5053	−0.389619
$873$	43.8808	1.48514
$874$	8.25488	0.279226
$875$	44.7706	1.51352
$876$	−0.483575	−0.0163385
$877$	−8.64750	−0.292005	−0.146003	−	0.989284i	$-0.546641\pi$
$877$	−8.64750	−0.292005	−0.146003	+	0.989284i	$0.546641\pi$
$878$	−20.3801	−0.687795
$879$	3.74845	0.126432
$880$	50.2781	1.69488
$881$	−29.0312	−0.978085	−0.489043	−	0.872260i	$-0.662654\pi$
$881$	−29.0312	−0.978085	−0.489043	+	0.872260i	$0.662654\pi$
$882$	−13.6298	−0.458939
$883$	−23.0592	−0.776003	−0.388001	−	0.921659i	$-0.626834\pi$
$883$	−23.0592	−0.776003	−0.388001	+	0.921659i	$0.626834\pi$
$884$	1.12761	0.0379258
$885$	0.0751411	0.00252584
$886$	17.5908	0.590974
$887$	−2.98011	−0.100062	−0.0500311	−	0.998748i	$-0.515932\pi$
$887$	−2.98011	−0.100062	−0.0500311	+	0.998748i	$0.515932\pi$
$888$	−1.28885	−0.0432509
$889$	35.6578	1.19592
$890$	−11.0519	−0.370459
$891$	−46.8721	−1.57027
$892$	16.3040	0.545899
$893$	9.87312	0.330391
$894$	−1.63159	−0.0545685
$895$	−54.9515	−1.83683
$896$	−6.04283	−0.201877
$897$	2.16596	0.0723194
$898$	29.6148	0.988258
$899$	1.83680	0.0612608
$900$	21.0548	0.701826
$901$	3.90562	0.130115
$902$	−37.2577	−1.24054
$903$	0.689541	0.0229465
$904$	−34.2577	−1.13939
$905$	−8.81675	−0.293079
$906$	−0.991778	−0.0329496
$907$	22.2168	0.737695	0.368848	−	0.929490i	$-0.379752\pi$
$907$	22.2168	0.737695	0.368848	+	0.929490i	$0.379752\pi$
$908$	3.13089	0.103902
$909$	50.7123	1.68202
$910$	31.5610	1.04624
$911$	2.22221	0.0736252	0.0368126	−	0.999322i	$-0.488280\pi$
$911$	2.22221	0.0736252	0.0368126	+	0.999322i	$0.488280\pi$
$912$	−0.366603	−0.0121394
$913$	21.0715	0.697366
$914$	26.1466	0.864851
$915$	3.37397	0.111540
$916$	−13.3865	−0.442304
$917$	−12.9347	−0.427143
$918$	−0.377200	−0.0124494
$919$	−59.8267	−1.97350	−0.986750	−	0.162249i	$-0.948125\pi$
$919$	−59.8267	−1.97350	−0.986750	+	0.162249i	$0.948125\pi$
$920$	63.3189	2.08756
$921$	−0.135217	−0.00445555
$922$	−38.8393	−1.27911
$923$	−5.73328	−0.188713
$924$	−0.652778	−0.0214748
$925$	42.0533	1.38270
$926$	6.21189	0.204135
$927$	−18.1210	−0.595171
$928$	−3.38596	−0.111150
$929$	15.6062	0.512022	0.256011	−	0.966674i	$-0.417592\pi$
$929$	15.6062	0.512022	0.256011	+	0.966674i	$0.417592\pi$
$930$	0.977474	0.0320526
$931$	5.37131	0.176038
$932$	−5.08152	−0.166451
$933$	2.82545	0.0925011
$934$	36.1627	1.18328
$935$	−10.1718	−0.332654
$936$	34.7615	1.13622
$937$	−7.01624	−0.229211	−0.114605	−	0.993411i	$-0.536560\pi$
$937$	−7.01624	−0.229211	−0.114605	+	0.993411i	$0.536560\pi$
$938$	14.1625	0.462421
$939$	−0.160631	−0.00524201
$940$	18.0153	0.587593
$941$	52.2646	1.70378	0.851888	−	0.523724i	$-0.175458\pi$
$941$	52.2646	1.70378	0.851888	+	0.523724i	$0.175458\pi$
$942$	1.12619	0.0366931
$943$	−30.6945	−0.999552
$944$	0.391142	0.0127306
$945$	4.79072	0.155842
$946$	−21.5183	−0.699619
$947$	12.2656	0.398577	0.199289	−	0.979941i	$-0.436137\pi$
$947$	12.2656	0.398577	0.199289	+	0.979941i	$0.436137\pi$
$948$	0.337971	0.0109768
$949$	26.0501	0.845623
$950$	18.2853	0.593254
$951$	−2.05967	−0.0667892
$952$	2.59368	0.0840618
$953$	13.0409	0.422437	0.211218	−	0.977439i	$-0.432257\pi$
$953$	13.0409	0.422437	0.211218	+	0.977439i	$0.432257\pi$
$954$	28.6412	0.927294
$955$	83.5712	2.70430
$956$	−4.57413	−0.147938
$957$	0.592930	0.0191667
$958$	−24.5435	−0.792964
$959$	−34.6849	−1.12003
$960$	−3.94496	−0.127323
$961$	−27.6262	−0.891166
$962$	16.5163	0.532506
$963$	−1.90451	−0.0613720
$964$	−10.8842	−0.350557
$965$	−79.6005	−2.56243
$966$	1.18515	0.0381314
$967$	−4.29962	−0.138266	−0.0691332	−	0.997607i	$-0.522023\pi$
$967$	−4.29962	−0.138266	−0.0691332	+	0.997607i	$0.522023\pi$
$968$	51.7759	1.66414
$969$	0.0741677	0.00238261
$970$	69.5364	2.23268
$971$	38.1556	1.22447	0.612236	−	0.790675i	$-0.290270\pi$
$971$	38.1556	1.22447	0.612236	+	0.790675i	$0.290270\pi$
$972$	1.88412	0.0604333
$973$	1.76356	0.0565370
$974$	−22.4059	−0.717933
$975$	4.79780	0.153653
$976$	17.5630	0.562178
$977$	59.3556	1.89896	0.949478	−	0.313834i	$-0.101614\pi$
$977$	59.3556	1.89896	0.949478	+	0.313834i	$0.101614\pi$
$978$	0.0760378	0.00243142
$979$	−12.3139	−0.393555
$980$	9.80091	0.313079
$981$	11.1664	0.356514
$982$	−3.01228	−0.0961257
$983$	1.25621	0.0400667	0.0200334	−	0.999799i	$-0.493623\pi$
$983$	1.25621	0.0400667	0.0200334	+	0.999799i	$0.493623\pi$
$984$	2.08379	0.0664288
$985$	−23.2944	−0.742220
$986$	−0.560427	−0.0178476
$987$	1.41747	0.0451187
$988$	−3.25877	−0.103675
$989$	−17.7277	−0.563709
$990$	−74.5934	−2.37073
$991$	−39.2423	−1.24657	−0.623286	−	0.781994i	$-0.714203\pi$
$991$	−39.2423	−1.24657	−0.623286	+	0.781994i	$0.714203\pi$
$992$	−6.21935	−0.197465
$993$	0.763268	0.0242216
$994$	−3.13706	−0.0995016
$995$	−32.7077	−1.03690
$996$	−0.280349	−0.00888320
$997$	26.0486	0.824967	0.412484	−	0.910965i	$-0.364661\pi$
$997$	26.0486	0.824967	0.412484	+	0.910965i	$0.364661\pi$
$998$	−5.58622	−0.176829
$999$	2.50704	0.0793194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.20	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.20	✓	59	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)

\(f(q)\)	\(=\)	\(q-1.17292 q^{2} -0.112413 q^{3} -0.624270 q^{4} +4.03608 q^{5} +0.131851 q^{6} +1.76356 q^{7} +3.07805 q^{8} -2.98736 q^{9} +O(q^{10})\)	\(q-1.17292 q^{2} -0.112413 q^{3} -0.624270 q^{4} +4.03608 q^{5} +0.131851 q^{6} +1.76356 q^{7} +3.07805 q^{8} -2.98736 q^{9} -4.73397 q^{10} -5.27457 q^{11} +0.0701761 q^{12} -3.78038 q^{13} -2.06850 q^{14} -0.453708 q^{15} -2.36175 q^{16} +0.477807 q^{17} +3.50392 q^{18} -1.38085 q^{19} -2.51960 q^{20} -0.198247 q^{21} +6.18662 q^{22} +5.09681 q^{23} -0.346013 q^{24} +11.2899 q^{25} +4.43407 q^{26} +0.673058 q^{27} -1.10094 q^{28} +1.00000 q^{29} +0.532160 q^{30} +1.83680 q^{31} -3.38596 q^{32} +0.592930 q^{33} -0.560427 q^{34} +7.11785 q^{35} +1.86492 q^{36} +3.72486 q^{37} +1.61962 q^{38} +0.424964 q^{39} +12.4232 q^{40} -6.02230 q^{41} +0.232527 q^{42} -3.47820 q^{43} +3.29275 q^{44} -12.0572 q^{45} -5.97813 q^{46} -7.15004 q^{47} +0.265491 q^{48} -3.88987 q^{49} -13.2421 q^{50} -0.0537117 q^{51} +2.35998 q^{52} +8.17405 q^{53} -0.789440 q^{54} -21.2885 q^{55} +5.42831 q^{56} +0.155225 q^{57} -1.17292 q^{58} -0.165616 q^{59} +0.283236 q^{60} -7.43644 q^{61} -2.15441 q^{62} -5.26839 q^{63} +8.69494 q^{64} -15.2579 q^{65} -0.695457 q^{66} -6.84673 q^{67} -0.298281 q^{68} -0.572948 q^{69} -8.34863 q^{70} +1.51659 q^{71} -9.19524 q^{72} -6.89087 q^{73} -4.36894 q^{74} -1.26913 q^{75} +0.862022 q^{76} -9.30200 q^{77} -0.498447 q^{78} +4.81605 q^{79} -9.53219 q^{80} +8.88643 q^{81} +7.06365 q^{82} -3.99493 q^{83} +0.123760 q^{84} +1.92846 q^{85} +4.07963 q^{86} -0.112413 q^{87} -16.2354 q^{88} +2.33459 q^{89} +14.1421 q^{90} -6.66692 q^{91} -3.18179 q^{92} -0.206481 q^{93} +8.38639 q^{94} -5.57320 q^{95} +0.380627 q^{96} -14.6888 q^{97} +4.56248 q^{98} +15.7570 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

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