LMFDB - Embedded newform 4031.2.a.b.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.18971 q^{2} -0.875363 q^{3} +2.79481 q^{4} -1.80760 q^{5} +1.91679 q^{6} -0.359878 q^{7} -1.74040 q^{8} -2.23374 q^{9} +O(q^{10})$	\(q-2.18971 q^{2} -0.875363 q^{3} +2.79481 q^{4} -1.80760 q^{5} +1.91679 q^{6} -0.359878 q^{7} -1.74040 q^{8} -2.23374 q^{9} +3.95812 q^{10} -5.08211 q^{11} -2.44647 q^{12} -6.16853 q^{13} +0.788028 q^{14} +1.58231 q^{15} -1.77865 q^{16} +4.36892 q^{17} +4.89123 q^{18} +4.11616 q^{19} -5.05191 q^{20} +0.315024 q^{21} +11.1283 q^{22} -1.78666 q^{23} +1.52348 q^{24} -1.73257 q^{25} +13.5073 q^{26} +4.58142 q^{27} -1.00579 q^{28} +1.00000 q^{29} -3.46479 q^{30} +2.73782 q^{31} +7.37553 q^{32} +4.44869 q^{33} -9.56664 q^{34} +0.650517 q^{35} -6.24288 q^{36} +7.58575 q^{37} -9.01319 q^{38} +5.39970 q^{39} +3.14596 q^{40} +7.83139 q^{41} -0.689810 q^{42} +1.78071 q^{43} -14.2035 q^{44} +4.03772 q^{45} +3.91226 q^{46} +1.99606 q^{47} +1.55696 q^{48} -6.87049 q^{49} +3.79381 q^{50} -3.82439 q^{51} -17.2399 q^{52} +7.34454 q^{53} -10.0320 q^{54} +9.18644 q^{55} +0.626334 q^{56} -3.60314 q^{57} -2.18971 q^{58} +12.5535 q^{59} +4.42226 q^{60} +8.70223 q^{61} -5.99503 q^{62} +0.803875 q^{63} -12.5929 q^{64} +11.1503 q^{65} -9.74132 q^{66} -9.64926 q^{67} +12.2103 q^{68} +1.56397 q^{69} -1.42444 q^{70} +3.35712 q^{71} +3.88761 q^{72} -16.3117 q^{73} -16.6106 q^{74} +1.51663 q^{75} +11.5039 q^{76} +1.82894 q^{77} -11.8238 q^{78} -2.90329 q^{79} +3.21510 q^{80} +2.69082 q^{81} -17.1484 q^{82} -14.2498 q^{83} +0.880433 q^{84} -7.89727 q^{85} -3.89923 q^{86} -0.875363 q^{87} +8.84492 q^{88} -1.03778 q^{89} -8.84141 q^{90} +2.21992 q^{91} -4.99337 q^{92} -2.39659 q^{93} -4.37078 q^{94} -7.44039 q^{95} -6.45626 q^{96} +7.81251 q^{97} +15.0443 q^{98} +11.3521 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.18971	−1.54836	−0.774178	−	0.632968i	$-0.781836\pi$
$2$	−2.18971	−1.54836	−0.774178	+	0.632968i	$0.781836\pi$
$3$	−0.875363	−0.505391	−0.252695	−	0.967546i	$-0.581317\pi$
$3$	−0.875363	−0.505391	−0.252695	+	0.967546i	$0.581317\pi$
$4$	2.79481	1.39741
$5$	−1.80760	−0.808385	−0.404193	−	0.914674i	$-0.632447\pi$
$5$	−1.80760	−0.808385	−0.404193	+	0.914674i	$0.632447\pi$
$6$	1.91679	0.782525
$7$	−0.359878	−0.136021	−0.0680106	−	0.997685i	$-0.521665\pi$
$7$	−0.359878	−0.136021	−0.0680106	+	0.997685i	$0.521665\pi$
$8$	−1.74040	−0.615326
$9$	−2.23374	−0.744580
$10$	3.95812	1.25167
$11$	−5.08211	−1.53231	−0.766157	−	0.642653i	$-0.777834\pi$
$11$	−5.08211	−1.53231	−0.766157	+	0.642653i	$0.777834\pi$
$12$	−2.44647	−0.706236
$13$	−6.16853	−1.71084	−0.855421	−	0.517933i	$-0.826702\pi$
$13$	−6.16853	−1.71084	−0.855421	+	0.517933i	$0.826702\pi$
$14$	0.788028	0.210609
$15$	1.58231	0.408550
$16$	−1.77865	−0.444663
$17$	4.36892	1.05962	0.529809	−	0.848117i	$-0.322264\pi$
$17$	4.36892	1.05962	0.529809	+	0.848117i	$0.322264\pi$
$18$	4.89123	1.15287
$19$	4.11616	0.944313	0.472156	−	0.881515i	$-0.343476\pi$
$19$	4.11616	0.944313	0.472156	+	0.881515i	$0.343476\pi$
$20$	−5.05191	−1.12964
$21$	0.315024	0.0687439
$22$	11.1283	2.37257
$23$	−1.78666	−0.372544	−0.186272	−	0.982498i	$-0.559641\pi$
$23$	−1.78666	−0.372544	−0.186272	+	0.982498i	$0.559641\pi$
$24$	1.52348	0.310980
$25$	−1.73257	−0.346514
$26$	13.5073	2.64899
$27$	4.58142	0.881695
$28$	−1.00579	−0.190077
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.46479	−0.632582
$31$	2.73782	0.491728	0.245864	−	0.969304i	$-0.420928\pi$
$31$	2.73782	0.491728	0.245864	+	0.969304i	$0.420928\pi$
$32$	7.37553	1.30382
$33$	4.44869	0.774418
$34$	−9.56664	−1.64067
$35$	0.650517	0.109958
$36$	−6.24288	−1.04048
$37$	7.58575	1.24709	0.623545	−	0.781788i	$-0.285692\pi$
$37$	7.58575	1.24709	0.623545	+	0.781788i	$0.285692\pi$
$38$	−9.01319	−1.46213
$39$	5.39970	0.864644
$40$	3.14596	0.497420
$41$	7.83139	1.22306	0.611528	−	0.791222i	$-0.290555\pi$
$41$	7.83139	1.22306	0.611528	+	0.791222i	$0.290555\pi$
$42$	−0.689810	−0.106440
$43$	1.78071	0.271556	0.135778	−	0.990739i	$-0.456647\pi$
$43$	1.78071	0.271556	0.135778	+	0.990739i	$0.456647\pi$
$44$	−14.2035	−2.14126
$45$	4.03772	0.601907
$46$	3.91226	0.576831
$47$	1.99606	0.291155	0.145577	−	0.989347i	$-0.453496\pi$
$47$	1.99606	0.291155	0.145577	+	0.989347i	$0.453496\pi$
$48$	1.55696	0.224729
$49$	−6.87049	−0.981498
$50$	3.79381	0.536526
$51$	−3.82439	−0.535521
$52$	−17.2399	−2.39074
$53$	7.34454	1.00885	0.504425	−	0.863455i	$-0.331704\pi$
$53$	7.34454	1.00885	0.504425	+	0.863455i	$0.331704\pi$
$54$	−10.0320	−1.36518
$55$	9.18644	1.23870
$56$	0.626334	0.0836973
$57$	−3.60314	−0.477247
$58$	−2.18971	−0.287522
$59$	12.5535	1.63433	0.817165	−	0.576404i	$-0.195544\pi$
$59$	12.5535	1.63433	0.817165	+	0.576404i	$0.195544\pi$
$60$	4.42226	0.570911
$61$	8.70223	1.11421	0.557103	−	0.830443i	$-0.311913\pi$
$61$	8.70223	1.11421	0.557103	+	0.830443i	$0.311913\pi$
$62$	−5.99503	−0.761369
$63$	0.803875	0.101279
$64$	−12.5929	−1.57412
$65$	11.1503	1.38302
$66$	−9.74132	−1.19907
$67$	−9.64926	−1.17884	−0.589422	−	0.807825i	$-0.700644\pi$
$67$	−9.64926	−1.17884	−0.589422	+	0.807825i	$0.700644\pi$
$68$	12.2103	1.48072
$69$	1.56397	0.188280
$70$	−1.42444	−0.170253
$71$	3.35712	0.398417	0.199208	−	0.979957i	$-0.436163\pi$
$71$	3.35712	0.398417	0.199208	+	0.979957i	$0.436163\pi$
$72$	3.88761	0.458159
$73$	−16.3117	−1.90915	−0.954573	−	0.297977i	$-0.903688\pi$
$73$	−16.3117	−1.90915	−0.954573	+	0.297977i	$0.903688\pi$
$74$	−16.6106	−1.93094
$75$	1.51663	0.175125
$76$	11.5039	1.31959
$77$	1.82894	0.208427
$78$	−11.8238	−1.33878
$79$	−2.90329	−0.326645	−0.163323	−	0.986573i	$-0.552221\pi$
$79$	−2.90329	−0.326645	−0.163323	+	0.986573i	$0.552221\pi$
$80$	3.21510	0.359459
$81$	2.69082	0.298979
$82$	−17.1484	−1.89373
$83$	−14.2498	−1.56412	−0.782059	−	0.623204i	$-0.785831\pi$
$83$	−14.2498	−1.56412	−0.782059	+	0.623204i	$0.785831\pi$
$84$	0.880433	0.0960631
$85$	−7.89727	−0.856579
$86$	−3.89923	−0.420465
$87$	−0.875363	−0.0938487
$88$	8.84492	0.942872
$89$	−1.03778	−0.110004	−0.0550022	−	0.998486i	$-0.517517\pi$
$89$	−1.03778	−0.110004	−0.0550022	+	0.998486i	$0.517517\pi$
$90$	−8.84141	−0.931967
$91$	2.21992	0.232711
$92$	−4.99337	−0.520595
$93$	−2.39659	−0.248515
$94$	−4.37078	−0.450811
$95$	−7.44039	−0.763368
$96$	−6.45626	−0.658940
$97$	7.81251	0.793240	0.396620	−	0.917983i	$-0.370183\pi$
$97$	7.81251	0.793240	0.396620	+	0.917983i	$0.370183\pi$
$98$	15.0443	1.51971
$99$	11.3521	1.14093
$100$	−4.84220	−0.484220
$101$	4.08963	0.406933	0.203467	−	0.979082i	$-0.434779\pi$
$101$	4.08963	0.406933	0.203467	+	0.979082i	$0.434779\pi$
$102$	8.37428	0.829177
$103$	−2.78804	−0.274714	−0.137357	−	0.990522i	$-0.543861\pi$
$103$	−2.78804	−0.274714	−0.137357	+	0.990522i	$0.543861\pi$
$104$	10.7357	1.05273
$105$	−0.569439	−0.0555715
$106$	−16.0824	−1.56206
$107$	17.4220	1.68425	0.842125	−	0.539282i	$-0.181304\pi$
$107$	17.4220	1.68425	0.842125	+	0.539282i	$0.181304\pi$
$108$	12.8042	1.23209
$109$	−16.9435	−1.62290	−0.811448	−	0.584424i	$-0.801320\pi$
$109$	−16.9435	−1.62290	−0.811448	+	0.584424i	$0.801320\pi$
$110$	−20.1156	−1.91795
$111$	−6.64028	−0.630268
$112$	0.640098	0.0604836
$113$	−10.2323	−0.962573	−0.481286	−	0.876563i	$-0.659830\pi$
$113$	−10.2323	−0.962573	−0.481286	+	0.876563i	$0.659830\pi$
$114$	7.88981	0.738948
$115$	3.22957	0.301159
$116$	2.79481	0.259492
$117$	13.7789	1.27386
$118$	−27.4885	−2.53052
$119$	−1.57228	−0.144131
$120$	−2.75386	−0.251392
$121$	14.8279	1.34799
$122$	−19.0553	−1.72519
$123$	−6.85530	−0.618122
$124$	7.65170	0.687143
$125$	12.1698	1.08850
$126$	−1.76025	−0.156815
$127$	1.78471	0.158368	0.0791839	−	0.996860i	$-0.474769\pi$
$127$	1.78471	0.158368	0.0791839	+	0.996860i	$0.474769\pi$
$128$	12.8238	1.13347
$129$	−1.55877	−0.137242
$130$	−24.4158	−2.14141
$131$	−8.78511	−0.767559	−0.383779	−	0.923425i	$-0.625378\pi$
$131$	−8.78511	−0.767559	−0.383779	+	0.923425i	$0.625378\pi$
$132$	12.4333	1.08218
$133$	−1.48132	−0.128447
$134$	21.1290	1.82527
$135$	−8.28140	−0.712749
$136$	−7.60368	−0.652010
$137$	19.6655	1.68014	0.840069	−	0.542480i	$-0.182515\pi$
$137$	19.6655	1.68014	0.840069	+	0.542480i	$0.182515\pi$
$138$	−3.42464	−0.291525
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	1.81807	0.153655
$141$	−1.74727	−0.147147
$142$	−7.35110	−0.616891
$143$	31.3491	2.62155
$144$	3.97304	0.331087
$145$	−1.80760	−0.150113
$146$	35.7179	2.95604
$147$	6.01417	0.496040
$148$	21.2007	1.74269
$149$	−2.41040	−0.197468	−0.0987340	−	0.995114i	$-0.531479\pi$
$149$	−2.41040	−0.197468	−0.0987340	+	0.995114i	$0.531479\pi$
$150$	−3.32096	−0.271155
$151$	−17.1119	−1.39255	−0.696275	−	0.717775i	$-0.745161\pi$
$151$	−17.1119	−1.39255	−0.696275	+	0.717775i	$0.745161\pi$
$152$	−7.16379	−0.581060
$153$	−9.75902	−0.788970
$154$	−4.00484	−0.322720
$155$	−4.94890	−0.397505
$156$	15.0911	1.20826
$157$	10.3068	0.822574	0.411287	−	0.911506i	$-0.365079\pi$
$157$	10.3068	0.822574	0.411287	+	0.911506i	$0.365079\pi$
$158$	6.35734	0.505763
$159$	−6.42914	−0.509864
$160$	−13.3320	−1.05399
$161$	0.642980	0.0506739
$162$	−5.89209	−0.462927
$163$	20.3615	1.59484	0.797419	−	0.603426i	$-0.206198\pi$
$163$	20.3615	1.59484	0.797419	+	0.603426i	$0.206198\pi$
$164$	21.8872	1.70911
$165$	−8.04147	−0.626028
$166$	31.2029	2.42181
$167$	−8.77491	−0.679023	−0.339511	−	0.940602i	$-0.610262\pi$
$167$	−8.77491	−0.679023	−0.339511	+	0.940602i	$0.610262\pi$
$168$	−0.548269	−0.0422999
$169$	25.0508	1.92698
$170$	17.2927	1.32629
$171$	−9.19444	−0.703116
$172$	4.97675	0.379473
$173$	−13.1032	−0.996217	−0.498108	−	0.867115i	$-0.665972\pi$
$173$	−13.1032	−0.996217	−0.498108	+	0.867115i	$0.665972\pi$
$174$	1.91679	0.145311
$175$	0.623514	0.0471332
$176$	9.03930	0.681363
$177$	−10.9889	−0.825976
$178$	2.27243	0.170326
$179$	−21.0435	−1.57287	−0.786433	−	0.617675i	$-0.788075\pi$
$179$	−21.0435	−1.57287	−0.786433	+	0.617675i	$0.788075\pi$
$180$	11.2847	0.841109
$181$	−0.499764	−0.0371472	−0.0185736	−	0.999827i	$-0.505912\pi$
$181$	−0.499764	−0.0371472	−0.0185736	+	0.999827i	$0.505912\pi$
$182$	−4.86097	−0.360319
$183$	−7.61761	−0.563110
$184$	3.10951	0.229236
$185$	−13.7120	−1.00813
$186$	5.24782	0.384789
$187$	−22.2033	−1.62367
$188$	5.57860	0.406861
$189$	−1.64875	−0.119929
$190$	16.2923	1.18197
$191$	7.48151	0.541343	0.270671	−	0.962672i	$-0.412754\pi$
$191$	7.48151	0.541343	0.270671	+	0.962672i	$0.412754\pi$
$192$	11.0234	0.795545
$193$	−3.72362	−0.268032	−0.134016	−	0.990979i	$-0.542787\pi$
$193$	−3.72362	−0.268032	−0.134016	+	0.990979i	$0.542787\pi$
$194$	−17.1071	−1.22822
$195$	−9.76052	−0.698965
$196$	−19.2017	−1.37155
$197$	17.0545	1.21508	0.607540	−	0.794289i	$-0.292156\pi$
$197$	17.0545	1.21508	0.607540	+	0.794289i	$0.292156\pi$
$198$	−24.8578	−1.76657
$199$	4.77447	0.338453	0.169227	−	0.985577i	$-0.445873\pi$
$199$	4.77447	0.338453	0.169227	+	0.985577i	$0.445873\pi$
$200$	3.01537	0.213219
$201$	8.44660	0.595777
$202$	−8.95508	−0.630077
$203$	−0.359878	−0.0252585
$204$	−10.6884	−0.748341
$205$	−14.1560	−0.988701
$206$	6.10499	0.425355
$207$	3.99093	0.277389
$208$	10.9717	0.760748
$209$	−20.9188	−1.44698
$210$	1.24690	0.0860445
$211$	19.3595	1.33277	0.666383	−	0.745610i	$-0.267842\pi$
$211$	19.3595	1.33277	0.666383	+	0.745610i	$0.267842\pi$
$212$	20.5266	1.40977
$213$	−2.93870	−0.201356
$214$	−38.1491	−2.60782
$215$	−3.21882	−0.219521
$216$	−7.97352	−0.542529
$217$	−0.985283	−0.0668854
$218$	37.1014	2.51282
$219$	14.2787	0.964865
$220$	25.6744	1.73097
$221$	−26.9498	−1.81284
$222$	14.5403	0.975878
$223$	−9.35073	−0.626171	−0.313085	−	0.949725i	$-0.601363\pi$
$223$	−9.35073	−0.626171	−0.313085	+	0.949725i	$0.601363\pi$
$224$	−2.65429	−0.177347
$225$	3.87011	0.258007
$226$	22.4057	1.49041
$227$	15.7237	1.04362	0.521811	−	0.853061i	$-0.325257\pi$
$227$	15.7237	1.04362	0.521811	+	0.853061i	$0.325257\pi$
$228$	−10.0701	−0.666908
$229$	−24.2227	−1.60068	−0.800341	−	0.599545i	$-0.795348\pi$
$229$	−24.2227	−1.60068	−0.800341	+	0.599545i	$0.795348\pi$
$230$	−7.07181	−0.466301
$231$	−1.60099	−0.105337
$232$	−1.74040	−0.114263
$233$	0.0533028	0.00349198	0.00174599	−	0.999998i	$-0.499444\pi$
$233$	0.0533028	0.00349198	0.00174599	+	0.999998i	$0.499444\pi$
$234$	−30.1717	−1.97239
$235$	−3.60808	−0.235365
$236$	35.0847	2.28382
$237$	2.54143	0.165084
$238$	3.44283	0.223165
$239$	−3.86015	−0.249692	−0.124846	−	0.992176i	$-0.539844\pi$
$239$	−3.86015	−0.249692	−0.124846	+	0.992176i	$0.539844\pi$
$240$	−2.81438	−0.181667
$241$	10.4740	0.674688	0.337344	−	0.941381i	$-0.390471\pi$
$241$	10.4740	0.674688	0.337344	+	0.941381i	$0.390471\pi$
$242$	−32.4686	−2.08716
$243$	−16.0997	−1.03280
$244$	24.3211	1.55700
$245$	12.4191	0.793429
$246$	15.0111	0.957073
$247$	−25.3907	−1.61557
$248$	−4.76492	−0.302573
$249$	12.4737	0.790491
$250$	−26.6483	−1.68539
$251$	3.42252	0.216028	0.108014	−	0.994149i	$-0.465551\pi$
$251$	3.42252	0.216028	0.108014	+	0.994149i	$0.465551\pi$
$252$	2.24668	0.141527
$253$	9.08000	0.570854
$254$	−3.90800	−0.245210
$255$	6.91298	0.432907
$256$	−2.89441	−0.180901
$257$	−18.0764	−1.12757	−0.563787	−	0.825920i	$-0.690656\pi$
$257$	−18.0764	−1.12757	−0.563787	+	0.825920i	$0.690656\pi$
$258$	3.41324	0.212499
$259$	−2.72995	−0.169631
$260$	31.1629	1.93264
$261$	−2.23374	−0.138265
$262$	19.2368	1.18845
$263$	23.4268	1.44456	0.722279	−	0.691602i	$-0.243095\pi$
$263$	23.4268	1.44456	0.722279	+	0.691602i	$0.243095\pi$
$264$	−7.74252	−0.476519
$265$	−13.2760	−0.815540
$266$	3.24365	0.198881
$267$	0.908434	0.0555953
$268$	−26.9679	−1.64732
$269$	−11.6268	−0.708897	−0.354448	−	0.935076i	$-0.615331\pi$
$269$	−11.6268	−0.708897	−0.354448	+	0.935076i	$0.615331\pi$
$270$	18.1338	1.10359
$271$	−18.2561	−1.10898	−0.554489	−	0.832191i	$-0.687086\pi$
$271$	−18.2561	−1.10898	−0.554489	+	0.832191i	$0.687086\pi$
$272$	−7.77078	−0.471173
$273$	−1.94324	−0.117610
$274$	−43.0617	−2.60145
$275$	8.80510	0.530968
$276$	4.37101	0.263104
$277$	21.1477	1.27065	0.635323	−	0.772247i	$-0.280867\pi$
$277$	21.1477	1.27065	0.635323	+	0.772247i	$0.280867\pi$
$278$	−2.18971	−0.131330
$279$	−6.11559	−0.366131
$280$	−1.13216	−0.0676597
$281$	11.9546	0.713154	0.356577	−	0.934266i	$-0.383944\pi$
$281$	11.9546	0.713154	0.356577	+	0.934266i	$0.383944\pi$
$282$	3.82601	0.227836
$283$	−9.16538	−0.544825	−0.272413	−	0.962181i	$-0.587822\pi$
$283$	−9.16538	−0.544825	−0.272413	+	0.962181i	$0.587822\pi$
$284$	9.38251	0.556750
$285$	6.51304	0.385799
$286$	−68.6454	−4.05909
$287$	−2.81835	−0.166362
$288$	−16.4750	−0.970800
$289$	2.08743	0.122790
$290$	3.95812	0.232429
$291$	−6.83878	−0.400896
$292$	−45.5883	−2.66785
$293$	12.1324	0.708783	0.354391	−	0.935097i	$-0.384688\pi$
$293$	12.1324	0.708783	0.354391	+	0.935097i	$0.384688\pi$
$294$	−13.1693	−0.768047
$295$	−22.6918	−1.32117
$296$	−13.2023	−0.767366
$297$	−23.2833	−1.35103
$298$	5.27808	0.305751
$299$	11.0211	0.637364
$300$	4.23868	0.244720
$301$	−0.640839	−0.0369373
$302$	37.4701	2.15616
$303$	−3.57991	−0.205660
$304$	−7.32122	−0.419901
$305$	−15.7302	−0.900708
$306$	21.3694	1.22161
$307$	−4.92700	−0.281199	−0.140599	−	0.990067i	$-0.544903\pi$
$307$	−4.92700	−0.281199	−0.140599	+	0.990067i	$0.544903\pi$
$308$	5.11155	0.291257
$309$	2.44055	0.138838
$310$	10.8366	0.615480
$311$	−3.03286	−0.171978	−0.0859889	−	0.996296i	$-0.527405\pi$
$311$	−3.03286	−0.171978	−0.0859889	+	0.996296i	$0.527405\pi$
$312$	−9.39766	−0.532038
$313$	−8.75398	−0.494804	−0.247402	−	0.968913i	$-0.579577\pi$
$313$	−8.75398	−0.494804	−0.247402	+	0.968913i	$0.579577\pi$
$314$	−22.5689	−1.27364
$315$	−1.45309	−0.0818722
$316$	−8.11414	−0.456456
$317$	−30.9883	−1.74047	−0.870237	−	0.492633i	$-0.836035\pi$
$317$	−30.9883	−1.74047	−0.870237	+	0.492633i	$0.836035\pi$
$318$	14.0779	0.789451
$319$	−5.08211	−0.284544
$320$	22.7630	1.27249
$321$	−15.2506	−0.851205
$322$	−1.40794	−0.0784612
$323$	17.9832	1.00061
$324$	7.52032	0.417796
$325$	10.6874	0.592830
$326$	−44.5858	−2.46938
$327$	14.8317	0.820197
$328$	−13.6298	−0.752578
$329$	−0.718337	−0.0396032
$330$	17.6085	0.969314
$331$	−26.3363	−1.44758	−0.723788	−	0.690022i	$-0.757601\pi$
$331$	−26.3363	−1.44758	−0.723788	+	0.690022i	$0.757601\pi$
$332$	−39.8255	−2.18571
$333$	−16.9446	−0.928558
$334$	19.2145	1.05137
$335$	17.4420	0.952960
$336$	−0.560318	−0.0305678
$337$	18.5913	1.01274	0.506368	−	0.862318i	$-0.330988\pi$
$337$	18.5913	1.01274	0.506368	+	0.862318i	$0.330988\pi$
$338$	−54.8538	−2.98365
$339$	8.95697	0.486476
$340$	−22.0714	−1.19699
$341$	−13.9139	−0.753481
$342$	20.1331	1.08867
$343$	4.99169	0.269526
$344$	−3.09915	−0.167095
$345$	−2.82705	−0.152203
$346$	28.6921	1.54250
$347$	13.8757	0.744888	0.372444	−	0.928055i	$-0.378520\pi$
$347$	13.8757	0.744888	0.372444	+	0.928055i	$0.378520\pi$
$348$	−2.44647	−0.131145
$349$	2.08040	0.111361	0.0556807	−	0.998449i	$-0.482267\pi$
$349$	2.08040	0.111361	0.0556807	+	0.998449i	$0.482267\pi$
$350$	−1.36531	−0.0729790
$351$	−28.2606	−1.50844
$352$	−37.4833	−1.99786
$353$	2.67800	0.142536	0.0712679	−	0.997457i	$-0.477295\pi$
$353$	2.67800	0.142536	0.0712679	+	0.997457i	$0.477295\pi$
$354$	24.0624	1.27890
$355$	−6.06834	−0.322074
$356$	−2.90040	−0.153721
$357$	1.37631	0.0728423
$358$	46.0791	2.43536
$359$	4.62330	0.244008	0.122004	−	0.992530i	$-0.461068\pi$
$359$	4.62330	0.244008	0.122004	+	0.992530i	$0.461068\pi$
$360$	−7.02726	−0.370369
$361$	−2.05720	−0.108274
$362$	1.09434	0.0575171
$363$	−12.9797	−0.681260
$364$	6.20426	0.325191
$365$	29.4852	1.54333
$366$	16.6803	0.871894
$367$	−19.9802	−1.04296	−0.521479	−	0.853264i	$-0.674620\pi$
$367$	−19.9802	−1.04296	−0.521479	+	0.853264i	$0.674620\pi$
$368$	3.17784	0.165656
$369$	−17.4933	−0.910664
$370$	30.0253	1.56094
$371$	−2.64314	−0.137225
$372$	−6.69801	−0.347276
$373$	7.69833	0.398604	0.199302	−	0.979938i	$-0.436132\pi$
$373$	7.69833	0.398604	0.199302	+	0.979938i	$0.436132\pi$
$374$	48.6187	2.51402
$375$	−10.6530	−0.550119
$376$	−3.47394	−0.179155
$377$	−6.16853	−0.317695
$378$	3.61029	0.185693
$379$	−2.63526	−0.135364	−0.0676822	−	0.997707i	$-0.521560\pi$
$379$	−2.63526	−0.135364	−0.0676822	+	0.997707i	$0.521560\pi$
$380$	−20.7945	−1.06674
$381$	−1.56227	−0.0800376
$382$	−16.3823	−0.838191
$383$	−9.11532	−0.465771	−0.232886	−	0.972504i	$-0.574817\pi$
$383$	−9.11532	−0.465771	−0.232886	+	0.972504i	$0.574817\pi$
$384$	−11.2255	−0.572846
$385$	−3.30600	−0.168489
$386$	8.15363	0.415009
$387$	−3.97764	−0.202195
$388$	21.8345	1.10848
$389$	−1.62600	−0.0824415	−0.0412208	−	0.999150i	$-0.513125\pi$
$389$	−1.62600	−0.0824415	−0.0412208	+	0.999150i	$0.513125\pi$
$390$	21.3727	1.08225
$391$	−7.80576	−0.394754
$392$	11.9574	0.603941
$393$	7.69016	0.387917
$394$	−37.3443	−1.88138
$395$	5.24799	0.264055
$396$	31.7270	1.59434
$397$	−0.604089	−0.0303183	−0.0151592	−	0.999885i	$-0.504825\pi$
$397$	−0.604089	−0.0303183	−0.0151592	+	0.999885i	$0.504825\pi$
$398$	−10.4547	−0.524046
$399$	1.29669	0.0649157
$400$	3.08163	0.154082
$401$	−2.45053	−0.122374	−0.0611868	−	0.998126i	$-0.519489\pi$
$401$	−2.45053	−0.122374	−0.0611868	+	0.998126i	$0.519489\pi$
$402$	−18.4956	−0.922475
$403$	−16.8883	−0.841268
$404$	11.4297	0.568651
$405$	−4.86393	−0.241691
$406$	0.788028	0.0391092
$407$	−38.5516	−1.91093
$408$	6.65598	0.329520
$409$	−9.83631	−0.486374	−0.243187	−	0.969979i	$-0.578193\pi$
$409$	−9.83631	−0.486374	−0.243187	+	0.969979i	$0.578193\pi$
$410$	30.9976	1.53086
$411$	−17.2145	−0.849126
$412$	−7.79205	−0.383887
$413$	−4.51774	−0.222304
$414$	−8.73896	−0.429497
$415$	25.7580	1.26441
$416$	−45.4962	−2.23063
$417$	−0.875363	−0.0428667
$418$	45.8060	2.24045
$419$	−33.2113	−1.62248	−0.811239	−	0.584715i	$-0.801206\pi$
$419$	−33.2113	−1.62248	−0.811239	+	0.584715i	$0.801206\pi$
$420$	−1.59147	−0.0776560
$421$	−40.8221	−1.98955	−0.994773	−	0.102114i	$-0.967439\pi$
$421$	−40.8221	−1.98955	−0.994773	+	0.102114i	$0.967439\pi$
$422$	−42.3917	−2.06360
$423$	−4.45867	−0.216788
$424$	−12.7825	−0.620771
$425$	−7.56944	−0.367172
$426$	6.43488	0.311771
$427$	−3.13174	−0.151556
$428$	48.6913	2.35358
$429$	−27.4419	−1.32491
$430$	7.04826	0.339897
$431$	22.4523	1.08149	0.540745	−	0.841187i	$-0.318142\pi$
$431$	22.4523	1.08149	0.540745	+	0.841187i	$0.318142\pi$
$432$	−8.14875	−0.392057
$433$	40.1348	1.92876	0.964378	−	0.264529i	$-0.0852165\pi$
$433$	40.1348	1.92876	0.964378	+	0.264529i	$0.0852165\pi$
$434$	2.15748	0.103562
$435$	1.58231	0.0758659
$436$	−47.3540	−2.26784
$437$	−7.35418	−0.351798
$438$	−31.2661	−1.49395
$439$	−34.0249	−1.62392	−0.811959	−	0.583714i	$-0.801599\pi$
$439$	−34.0249	−1.62392	−0.811959	+	0.583714i	$0.801599\pi$
$440$	−15.9881	−0.762204
$441$	15.3469	0.730804
$442$	59.0121	2.80692
$443$	9.83967	0.467497	0.233749	−	0.972297i	$-0.424901\pi$
$443$	9.83967	0.467497	0.233749	+	0.972297i	$0.424901\pi$
$444$	−18.5583	−0.880740
$445$	1.87590	0.0889260
$446$	20.4753	0.969535
$447$	2.10998	0.0997985
$448$	4.53193	0.214113
$449$	−5.53220	−0.261081	−0.130540	−	0.991443i	$-0.541671\pi$
$449$	−5.53220	−0.261081	−0.130540	+	0.991443i	$0.541671\pi$
$450$	−8.47439	−0.399487
$451$	−39.8000	−1.87411
$452$	−28.5973	−1.34511
$453$	14.9792	0.703782
$454$	−34.4304	−1.61590
$455$	−4.01274	−0.188120
$456$	6.27091	0.293662
$457$	−8.67373	−0.405740	−0.202870	−	0.979206i	$-0.565027\pi$
$457$	−8.67373	−0.405740	−0.202870	+	0.979206i	$0.565027\pi$
$458$	53.0406	2.47843
$459$	20.0158	0.934260
$460$	9.02604	0.420841
$461$	20.3769	0.949049	0.474524	−	0.880242i	$-0.342620\pi$
$461$	20.3769	0.949049	0.474524	+	0.880242i	$0.342620\pi$
$462$	3.50569	0.163100
$463$	28.4062	1.32015	0.660075	−	0.751200i	$-0.270524\pi$
$463$	28.4062	1.32015	0.660075	+	0.751200i	$0.270524\pi$
$464$	−1.77865	−0.0825718
$465$	4.33208	0.200896
$466$	−0.116717	−0.00540683
$467$	35.3240	1.63460	0.817301	−	0.576211i	$-0.195469\pi$
$467$	35.3240	1.63460	0.817301	+	0.576211i	$0.195469\pi$
$468$	38.5094	1.78010
$469$	3.47256	0.160348
$470$	7.90063	0.364429
$471$	−9.02221	−0.415721
$472$	−21.8482	−1.00565
$473$	−9.04976	−0.416108
$474$	−5.56498	−0.255608
$475$	−7.13153	−0.327217
$476$	−4.39422	−0.201409
$477$	−16.4058	−0.751170
$478$	8.45259	0.386613
$479$	−3.31063	−0.151266	−0.0756332	−	0.997136i	$-0.524098\pi$
$479$	−3.31063	−0.151266	−0.0756332	+	0.997136i	$0.524098\pi$
$480$	11.6704	0.532677
$481$	−46.7929	−2.13357
$482$	−22.9349	−1.04466
$483$	−0.562840	−0.0256101
$484$	41.4411	1.88368
$485$	−14.1219	−0.641243
$486$	35.2536	1.59914
$487$	−1.84195	−0.0834666	−0.0417333	−	0.999129i	$-0.513288\pi$
$487$	−1.84195	−0.0834666	−0.0417333	+	0.999129i	$0.513288\pi$
$488$	−15.1454	−0.685600
$489$	−17.8237	−0.806016
$490$	−27.1942	−1.22851
$491$	32.6584	1.47385	0.736926	−	0.675973i	$-0.236277\pi$
$491$	32.6584	1.47385	0.736926	+	0.675973i	$0.236277\pi$
$492$	−19.1593	−0.863767
$493$	4.36892	0.196766
$494$	55.5981	2.50148
$495$	−20.5201	−0.922311
$496$	−4.86963	−0.218653
$497$	−1.20815	−0.0541931
$498$	−27.3138	−1.22396
$499$	−15.5562	−0.696392	−0.348196	−	0.937422i	$-0.613206\pi$
$499$	−15.5562	−0.696392	−0.348196	+	0.937422i	$0.613206\pi$
$500$	34.0123	1.52108
$501$	7.68123	0.343172
$502$	−7.49431	−0.334488
$503$	−0.843886	−0.0376270	−0.0188135	−	0.999823i	$-0.505989\pi$
$503$	−0.843886	−0.0376270	−0.0188135	+	0.999823i	$0.505989\pi$
$504$	−1.39907	−0.0623194
$505$	−7.39243	−0.328959
$506$	−19.8825	−0.883886
$507$	−21.9285	−0.973879
$508$	4.98794	0.221304
$509$	−11.5769	−0.513137	−0.256569	−	0.966526i	$-0.582592\pi$
$509$	−11.5769	−0.513137	−0.256569	+	0.966526i	$0.582592\pi$
$510$	−15.1374	−0.670295
$511$	5.87024	0.259684
$512$	−19.3096	−0.853373
$513$	18.8579	0.832596
$514$	39.5820	1.74589
$515$	5.03967	0.222074
$516$	−4.35646	−0.191782
$517$	−10.1442	−0.446140
$518$	5.97778	0.262649
$519$	11.4700	0.503479
$520$	−19.4060	−0.851007
$521$	−3.69299	−0.161793	−0.0808964	−	0.996723i	$-0.525778\pi$
$521$	−3.69299	−0.161793	−0.0808964	+	0.996723i	$0.525778\pi$
$522$	4.89123	0.214083
$523$	29.0468	1.27013	0.635064	−	0.772460i	$-0.280974\pi$
$523$	29.0468	1.27013	0.635064	+	0.772460i	$0.280974\pi$
$524$	−24.5527	−1.07259
$525$	−0.545800	−0.0238207
$526$	−51.2978	−2.23669
$527$	11.9613	0.521043
$528$	−7.91267	−0.344355
$529$	−19.8079	−0.861211
$530$	29.0706	1.26275
$531$	−28.0413	−1.21689
$532$	−4.14000	−0.179492
$533$	−48.3081	−2.09246
$534$	−1.98920	−0.0860813
$535$	−31.4921	−1.36152
$536$	16.7936	0.725373
$537$	18.4207	0.794913
$538$	25.4592	1.09762
$539$	34.9166	1.50396
$540$	−23.1449	−0.996000
$541$	38.7817	1.66735	0.833677	−	0.552252i	$-0.186231\pi$
$541$	38.7817	1.66735	0.833677	+	0.552252i	$0.186231\pi$
$542$	39.9755	1.71709
$543$	0.437475	0.0187739
$544$	32.2231	1.38155
$545$	30.6272	1.31193
$546$	4.25511	0.182102
$547$	−8.77919	−0.375371	−0.187686	−	0.982229i	$-0.560099\pi$
$547$	−8.77919	−0.375371	−0.187686	+	0.982229i	$0.560099\pi$
$548$	54.9614	2.34783
$549$	−19.4385	−0.829616
$550$	−19.2806	−0.822127
$551$	4.11616	0.175354
$552$	−2.72195	−0.115854
$553$	1.04483	0.0444307
$554$	−46.3073	−1.96741
$555$	12.0030	0.509499
$556$	2.79481	0.118526
$557$	6.39112	0.270801	0.135400	−	0.990791i	$-0.456768\pi$
$557$	6.39112	0.270801	0.135400	+	0.990791i	$0.456768\pi$
$558$	13.3913	0.566900
$559$	−10.9844	−0.464589
$560$	−1.15704	−0.0488940
$561$	19.4360	0.820587
$562$	−26.1771	−1.10422
$563$	−19.8014	−0.834529	−0.417265	−	0.908785i	$-0.637011\pi$
$563$	−19.8014	−0.834529	−0.417265	+	0.908785i	$0.637011\pi$
$564$	−4.88330	−0.205624
$565$	18.4959	0.778130
$566$	20.0695	0.843584
$567$	−0.968366	−0.0406676
$568$	−5.84274	−0.245156
$569$	−14.9997	−0.628819	−0.314410	−	0.949287i	$-0.601807\pi$
$569$	−14.9997	−0.628819	−0.314410	+	0.949287i	$0.601807\pi$
$570$	−14.2616	−0.597355
$571$	−45.3805	−1.89911	−0.949557	−	0.313595i	$-0.898467\pi$
$571$	−45.3805	−1.89911	−0.949557	+	0.313595i	$0.898467\pi$
$572$	87.6150	3.66337
$573$	−6.54903	−0.273590
$574$	6.17135	0.257587
$575$	3.09551	0.129092
$576$	28.1294	1.17206
$577$	−12.3856	−0.515619	−0.257809	−	0.966196i	$-0.583001\pi$
$577$	−12.3856	−0.515619	−0.257809	+	0.966196i	$0.583001\pi$
$578$	−4.57087	−0.190123
$579$	3.25952	0.135461
$580$	−5.05191	−0.209769
$581$	5.12819	0.212753
$582$	14.9749	0.620730
$583$	−37.3258	−1.54588
$584$	28.3890	1.17475
$585$	−24.9068	−1.02977
$586$	−26.5664	−1.09745
$587$	15.6988	0.647957	0.323979	−	0.946064i	$-0.394979\pi$
$587$	15.6988	0.647957	0.323979	+	0.946064i	$0.394979\pi$
$588$	16.8085	0.693170
$589$	11.2693	0.464345
$590$	49.6884	2.04564
$591$	−14.9288	−0.614091
$592$	−13.4924	−0.554534
$593$	6.90751	0.283657	0.141829	−	0.989891i	$-0.454702\pi$
$593$	6.90751	0.283657	0.141829	+	0.989891i	$0.454702\pi$
$594$	50.9836	2.09188
$595$	2.84206	0.116513
$596$	−6.73662	−0.275943
$597$	−4.17940	−0.171051
$598$	−24.1329	−0.986866
$599$	17.1384	0.700255	0.350127	−	0.936702i	$-0.386138\pi$
$599$	17.1384	0.700255	0.350127	+	0.936702i	$0.386138\pi$
$600$	−2.63954	−0.107759
$601$	−37.5712	−1.53256	−0.766280	−	0.642507i	$-0.777894\pi$
$601$	−37.5712	−1.53256	−0.766280	+	0.642507i	$0.777894\pi$
$602$	1.40325	0.0571921
$603$	21.5539	0.877744
$604$	−47.8247	−1.94596
$605$	−26.8029	−1.08969
$606$	7.83894	0.318435
$607$	−20.7269	−0.841278	−0.420639	−	0.907228i	$-0.638194\pi$
$607$	−20.7269	−0.841278	−0.420639	+	0.907228i	$0.638194\pi$
$608$	30.3589	1.23122
$609$	0.315024	0.0127654
$610$	34.4445	1.39462
$611$	−12.3127	−0.498120
$612$	−27.2746	−1.10251
$613$	29.3636	1.18598	0.592992	−	0.805208i	$-0.297946\pi$
$613$	29.3636	1.18598	0.592992	+	0.805208i	$0.297946\pi$
$614$	10.7887	0.435395
$615$	12.3917	0.499680
$616$	−3.18310	−0.128251
$617$	−31.9448	−1.28605	−0.643024	−	0.765846i	$-0.722320\pi$
$617$	−31.9448	−1.28605	−0.643024	+	0.765846i	$0.722320\pi$
$618$	−5.34408	−0.214970
$619$	13.0175	0.523217	0.261608	−	0.965174i	$-0.415747\pi$
$619$	13.0175	0.523217	0.261608	+	0.965174i	$0.415747\pi$
$620$	−13.8312	−0.555476
$621$	−8.18543	−0.328470
$622$	6.64107	0.266283
$623$	0.373475	0.0149629
$624$	−9.60418	−0.384475
$625$	−13.3354	−0.533415
$626$	19.1686	0.766133
$627$	18.3115	0.731292
$628$	28.8056	1.14947
$629$	33.1415	1.32144
$630$	3.18183	0.126767
$631$	35.9592	1.43151	0.715756	−	0.698350i	$-0.246082\pi$
$631$	35.9592	1.43151	0.715756	+	0.698350i	$0.246082\pi$
$632$	5.05289	0.200993
$633$	−16.9466	−0.673568
$634$	67.8552	2.69487
$635$	−3.22606	−0.128022
$636$	−17.9682	−0.712487
$637$	42.3808	1.67919
$638$	11.1283	0.440575
$639$	−7.49893	−0.296653
$640$	−23.1803	−0.916282
$641$	−45.2837	−1.78860	−0.894299	−	0.447470i	$-0.852325\pi$
$641$	−45.2837	−1.78860	−0.894299	+	0.447470i	$0.852325\pi$
$642$	33.3943	1.31797
$643$	−18.0654	−0.712430	−0.356215	−	0.934404i	$-0.615933\pi$
$643$	−18.0654	−0.712430	−0.356215	+	0.934404i	$0.615933\pi$
$644$	1.79701	0.0708120
$645$	2.81763	0.110944
$646$	−39.3779	−1.54930
$647$	−8.60041	−0.338117	−0.169059	−	0.985606i	$-0.554073\pi$
$647$	−8.60041	−0.338117	−0.169059	+	0.985606i	$0.554073\pi$
$648$	−4.68311	−0.183970
$649$	−63.7984	−2.50431
$650$	−23.4022	−0.917912
$651$	0.862480	0.0338033
$652$	56.9066	2.22864
$653$	−22.0998	−0.864834	−0.432417	−	0.901674i	$-0.642339\pi$
$653$	−22.0998	−0.864834	−0.432417	+	0.901674i	$0.642339\pi$
$654$	−32.4771	−1.26996
$655$	15.8800	0.620483
$656$	−13.9293	−0.543848
$657$	36.4362	1.42151
$658$	1.57295	0.0613199
$659$	17.3911	0.677459	0.338730	−	0.940884i	$-0.390003\pi$
$659$	17.3911	0.677459	0.338730	+	0.940884i	$0.390003\pi$
$660$	−22.4744	−0.874815
$661$	−11.3455	−0.441291	−0.220645	−	0.975354i	$-0.570816\pi$
$661$	−11.3455	−0.441291	−0.220645	+	0.975354i	$0.570816\pi$
$662$	57.6688	2.24136
$663$	23.5908	0.916192
$664$	24.8004	0.962443
$665$	2.67764	0.103834
$666$	37.1037	1.43774
$667$	−1.78666	−0.0691797
$668$	−24.5242	−0.948870
$669$	8.18528	0.316461
$670$	−38.1929	−1.47552
$671$	−44.2257	−1.70731
$672$	2.32347	0.0896298
$673$	−23.4499	−0.903929	−0.451964	−	0.892036i	$-0.649277\pi$
$673$	−23.4499	−0.903929	−0.451964	+	0.892036i	$0.649277\pi$
$674$	−40.7096	−1.56807
$675$	−7.93762	−0.305519
$676$	70.0121	2.69277
$677$	−33.8425	−1.30067	−0.650337	−	0.759646i	$-0.725372\pi$
$677$	−33.8425	−1.30067	−0.650337	+	0.759646i	$0.725372\pi$
$678$	−19.6131	−0.753237
$679$	−2.81155	−0.107897
$680$	13.7444	0.527075
$681$	−13.7640	−0.527437
$682$	30.4674	1.16666
$683$	−26.4818	−1.01330	−0.506648	−	0.862153i	$-0.669116\pi$
$683$	−26.4818	−1.01330	−0.506648	+	0.862153i	$0.669116\pi$
$684$	−25.6967	−0.982539
$685$	−35.5474	−1.35820
$686$	−10.9303	−0.417322
$687$	21.2037	0.808970
$688$	−3.16726	−0.120751
$689$	−45.3050	−1.72598
$690$	6.19040	0.235664
$691$	−2.69801	−0.102637	−0.0513186	−	0.998682i	$-0.516342\pi$
$691$	−2.69801	−0.102637	−0.0513186	+	0.998682i	$0.516342\pi$
$692$	−36.6209	−1.39212
$693$	−4.08538	−0.155191
$694$	−30.3837	−1.15335
$695$	−1.80760	−0.0685663
$696$	1.52348	0.0577475
$697$	34.2147	1.29597
$698$	−4.55547	−0.172427
$699$	−0.0466593	−0.00176482
$700$	1.74260	0.0658642
$701$	46.7282	1.76490	0.882449	−	0.470408i	$-0.155893\pi$
$701$	46.7282	1.76490	0.882449	+	0.470408i	$0.155893\pi$
$702$	61.8825	2.33560
$703$	31.2242	1.17764
$704$	63.9987	2.41204
$705$	3.15838	0.118951
$706$	−5.86404	−0.220696
$707$	−1.47177	−0.0553515
$708$	−30.7119	−1.15422
$709$	−10.7915	−0.405283	−0.202641	−	0.979253i	$-0.564953\pi$
$709$	−10.7915	−0.405283	−0.202641	+	0.979253i	$0.564953\pi$
$710$	13.2879	0.498685
$711$	6.48519	0.243214
$712$	1.80616	0.0676886
$713$	−4.89155	−0.183190
$714$	−3.01372	−0.112786
$715$	−56.6668	−2.11922
$716$	−58.8127	−2.19793
$717$	3.37903	0.126192
$718$	−10.1237	−0.377812
$719$	−16.5832	−0.618449	−0.309224	−	0.950989i	$-0.600069\pi$
$719$	−16.5832	−0.618449	−0.309224	+	0.950989i	$0.600069\pi$
$720$	−7.18169	−0.267646
$721$	1.00335	0.0373669
$722$	4.50466	0.167646
$723$	−9.16853	−0.340981
$724$	−1.39675	−0.0519097
$725$	−1.73257	−0.0643459
$726$	28.4218	1.05483
$727$	44.5544	1.65243	0.826215	−	0.563355i	$-0.190490\pi$
$727$	44.5544	1.65243	0.826215	+	0.563355i	$0.190490\pi$
$728$	−3.86356	−0.143193
$729$	6.02063	0.222986
$730$	−64.5639	−2.38962
$731$	7.77977	0.287745
$732$	−21.2898	−0.786893
$733$	−29.4755	−1.08870	−0.544350	−	0.838858i	$-0.683224\pi$
$733$	−29.4755	−1.08870	−0.544350	+	0.838858i	$0.683224\pi$
$734$	43.7508	1.61487
$735$	−10.8712	−0.400992
$736$	−13.1776	−0.485731
$737$	49.0386	1.80636
$738$	38.3051	1.41003
$739$	−28.2555	−1.03940	−0.519698	−	0.854350i	$-0.673955\pi$
$739$	−28.2555	−1.03940	−0.519698	+	0.854350i	$0.673955\pi$
$740$	−38.3225	−1.40876
$741$	22.2260	0.816494
$742$	5.78770	0.212473
$743$	30.8949	1.13342	0.566711	−	0.823916i	$-0.308215\pi$
$743$	30.8949	1.13342	0.566711	+	0.823916i	$0.308215\pi$
$744$	4.17103	0.152917
$745$	4.35706	0.159630
$746$	−16.8571	−0.617182
$747$	31.8304	1.16461
$748$	−62.0541	−2.26892
$749$	−6.26981	−0.229094
$750$	23.3269	0.851780
$751$	−36.0862	−1.31681	−0.658403	−	0.752666i	$-0.728768\pi$
$751$	−36.0862	−1.31681	−0.658403	+	0.752666i	$0.728768\pi$
$752$	−3.55029	−0.129466
$753$	−2.99595	−0.109178
$754$	13.5073	0.491906
$755$	30.9316	1.12572
$756$	−4.60796	−0.167590
$757$	17.6994	0.643296	0.321648	−	0.946859i	$-0.395763\pi$
$757$	17.6994	0.643296	0.321648	+	0.946859i	$0.395763\pi$
$758$	5.77045	0.209592
$759$	−7.94829	−0.288505
$760$	12.9493	0.469720
$761$	13.1835	0.477902	0.238951	−	0.971032i	$-0.423197\pi$
$761$	13.1835	0.477902	0.238951	+	0.971032i	$0.423197\pi$
$762$	3.42092	0.123927
$763$	6.09761	0.220748
$764$	20.9094	0.756476
$765$	17.6405	0.637792
$766$	19.9599	0.721180
$767$	−77.4368	−2.79608
$768$	2.53366	0.0914256
$769$	−2.02109	−0.0728823	−0.0364412	−	0.999336i	$-0.511602\pi$
$769$	−2.02109	−0.0728823	−0.0364412	+	0.999336i	$0.511602\pi$
$770$	7.23917	0.260882
$771$	15.8234	0.569866
$772$	−10.4068	−0.374549
$773$	−27.0305	−0.972220	−0.486110	−	0.873898i	$-0.661585\pi$
$773$	−27.0305	−0.972220	−0.486110	+	0.873898i	$0.661585\pi$
$774$	8.70986	0.313070
$775$	−4.74346	−0.170390
$776$	−13.5969	−0.488101
$777$	2.38969	0.0857298
$778$	3.56046	0.127649
$779$	32.2353	1.15495
$780$	−27.2788	−0.976738
$781$	−17.0612	−0.610499
$782$	17.0923	0.611220
$783$	4.58142	0.163727
$784$	12.2202	0.436436
$785$	−18.6307	−0.664957
$786$	−16.8392	−0.600634
$787$	25.0194	0.891845	0.445922	−	0.895072i	$-0.352876\pi$
$787$	25.0194	0.891845	0.445922	+	0.895072i	$0.352876\pi$
$788$	47.6640	1.69796
$789$	−20.5069	−0.730067
$790$	−11.4916	−0.408851
$791$	3.68238	0.130930
$792$	−19.7573	−0.702044
$793$	−53.6800	−1.90623
$794$	1.32278	0.0469436
$795$	11.6213	0.412166
$796$	13.3438	0.472957
$797$	43.0848	1.52614	0.763072	−	0.646314i	$-0.223690\pi$
$797$	43.0848	1.52614	0.763072	+	0.646314i	$0.223690\pi$
$798$	−2.83937	−0.100513
$799$	8.72060	0.308513
$800$	−12.7786	−0.451792
$801$	2.31813	0.0819071
$802$	5.36594	0.189478
$803$	82.8981	2.92541
$804$	23.6067	0.832542
$805$	−1.16225	−0.0409640
$806$	36.9805	1.30258
$807$	10.1776	0.358270
$808$	−7.11760	−0.250396
$809$	−33.2005	−1.16727	−0.583635	−	0.812016i	$-0.698370\pi$
$809$	−33.2005	−1.16727	−0.583635	+	0.812016i	$0.698370\pi$
$810$	10.6506	0.374223
$811$	−20.7601	−0.728986	−0.364493	−	0.931206i	$-0.618758\pi$
$811$	−20.7601	−0.728986	−0.364493	+	0.931206i	$0.618758\pi$
$812$	−1.00579	−0.0352964
$813$	15.9807	0.560468
$814$	84.4167	2.95880
$815$	−36.8056	−1.28924
$816$	6.80225	0.238126
$817$	7.32969	0.256433
$818$	21.5386	0.753081
$819$	−4.95872	−0.173272
$820$	−39.5635	−1.38162
$821$	5.84606	0.204029	0.102014	−	0.994783i	$-0.467471\pi$
$821$	5.84606	0.204029	0.102014	+	0.994783i	$0.467471\pi$
$822$	37.6946	1.31475
$823$	19.3982	0.676179	0.338089	−	0.941114i	$-0.390219\pi$
$823$	19.3982	0.676179	0.338089	+	0.941114i	$0.390219\pi$
$824$	4.85231	0.169038
$825$	−7.70766	−0.268346
$826$	9.89253	0.344205
$827$	28.3453	0.985663	0.492831	−	0.870125i	$-0.335962\pi$
$827$	28.3453	0.985663	0.492831	+	0.870125i	$0.335962\pi$
$828$	11.1539	0.387625
$829$	36.9099	1.28194	0.640968	−	0.767568i	$-0.278533\pi$
$829$	36.9099	1.28194	0.640968	+	0.767568i	$0.278533\pi$
$830$	−56.4024	−1.95776
$831$	−18.5120	−0.642172
$832$	77.6799	2.69307
$833$	−30.0166	−1.04001
$834$	1.91679	0.0663729
$835$	15.8616	0.548912
$836$	−58.4641	−2.02202
$837$	12.5431	0.433554
$838$	72.7230	2.51217
$839$	−1.95438	−0.0674725	−0.0337363	−	0.999431i	$-0.510741\pi$
$839$	−1.95438	−0.0674725	−0.0337363	+	0.999431i	$0.510741\pi$
$840$	0.991053	0.0341946
$841$	1.00000	0.0344828
$842$	89.3883	3.08052
$843$	−10.4646	−0.360422
$844$	54.1063	1.86242
$845$	−45.2818	−1.55774
$846$	9.76318	0.335665
$847$	−5.33622	−0.183355
$848$	−13.0634	−0.448598
$849$	8.02303	0.275350
$850$	16.5749	0.568513
$851$	−13.5531	−0.464596
$852$	−8.21310	−0.281376
$853$	22.4090	0.767268	0.383634	−	0.923485i	$-0.374672\pi$
$853$	22.4090	0.767268	0.383634	+	0.923485i	$0.374672\pi$
$854$	6.85760	0.234662
$855$	16.6199	0.568389
$856$	−30.3213	−1.03636
$857$	−2.93089	−0.100117	−0.0500587	−	0.998746i	$-0.515941\pi$
$857$	−2.93089	−0.100117	−0.0500587	+	0.998746i	$0.515941\pi$
$858$	60.0896	2.05143
$859$	46.6058	1.59017	0.795085	−	0.606499i	$-0.207426\pi$
$859$	46.6058	1.59017	0.795085	+	0.606499i	$0.207426\pi$
$860$	−8.99599	−0.306761
$861$	2.46707	0.0840777
$862$	−49.1640	−1.67453
$863$	37.2086	1.26659	0.633297	−	0.773909i	$-0.281701\pi$
$863$	37.2086	1.26659	0.633297	+	0.773909i	$0.281701\pi$
$864$	33.7904	1.14957
$865$	23.6854	0.805327
$866$	−87.8834	−2.98640
$867$	−1.82726	−0.0620571
$868$	−2.75368	−0.0934660
$869$	14.7548	0.500523
$870$	−3.46479	−0.117467
$871$	59.5217	2.01682
$872$	29.4886	0.998610
$873$	−17.4511	−0.590630
$874$	16.1035	0.544708
$875$	−4.37965	−0.148059
$876$	39.9063	1.34831
$877$	31.5531	1.06547	0.532737	−	0.846281i	$-0.321164\pi$
$877$	31.5531	1.06547	0.532737	+	0.846281i	$0.321164\pi$
$878$	74.5045	2.51440
$879$	−10.6203	−0.358212
$880$	−16.3395	−0.550804
$881$	−16.9986	−0.572697	−0.286348	−	0.958126i	$-0.592441\pi$
$881$	−16.9986	−0.572697	−0.286348	+	0.958126i	$0.592441\pi$
$882$	−33.6052	−1.13154
$883$	−10.3266	−0.347516	−0.173758	−	0.984788i	$-0.555591\pi$
$883$	−10.3266	−0.347516	−0.173758	+	0.984788i	$0.555591\pi$
$884$	−75.3196	−2.53327
$885$	19.8636	0.667706
$886$	−21.5460	−0.723852
$887$	−19.6133	−0.658551	−0.329276	−	0.944234i	$-0.606805\pi$
$887$	−19.6133	−0.658551	−0.329276	+	0.944234i	$0.606805\pi$
$888$	11.5568	0.387820
$889$	−0.642280	−0.0215414
$890$	−4.10766	−0.137689
$891$	−13.6750	−0.458130
$892$	−26.1335	−0.875015
$893$	8.21609	0.274941
$894$	−4.62023	−0.154524
$895$	38.0384	1.27148
$896$	−4.61500	−0.154176
$897$	−9.64742	−0.322118
$898$	12.1139	0.404246
$899$	2.73782	0.0913115
$900$	10.8162	0.360541
$901$	32.0877	1.06900
$902$	87.1502	2.90179
$903$	0.560966	0.0186678
$904$	17.8083	0.592296
$905$	0.903376	0.0300292
$906$	−32.8000	−1.08971
$907$	57.3806	1.90529	0.952646	−	0.304082i	$-0.0983496\pi$
$907$	57.3806	1.90529	0.952646	+	0.304082i	$0.0983496\pi$
$908$	43.9449	1.45836
$909$	−9.13516	−0.302994
$910$	8.78671	0.291277
$911$	28.9907	0.960503	0.480252	−	0.877131i	$-0.340545\pi$
$911$	28.9907	0.960503	0.480252	+	0.877131i	$0.340545\pi$
$912$	6.40872	0.212214
$913$	72.4191	2.39672
$914$	18.9929	0.628230
$915$	13.7696	0.455210
$916$	−67.6980	−2.23680
$917$	3.16157	0.104404
$918$	−43.8288	−1.44657
$919$	−38.5139	−1.27046	−0.635228	−	0.772324i	$-0.719094\pi$
$919$	−38.5139	−1.27046	−0.635228	+	0.772324i	$0.719094\pi$
$920$	−5.62076	−0.185311
$921$	4.31291	0.142115
$922$	−44.6195	−1.46947
$923$	−20.7085	−0.681628
$924$	−4.47446	−0.147199
$925$	−13.1428	−0.432133
$926$	−62.2013	−2.04406
$927$	6.22776	0.204546
$928$	7.37553	0.242114
$929$	−14.3629	−0.471231	−0.235615	−	0.971846i	$-0.575711\pi$
$929$	−14.3629	−0.471231	−0.235615	+	0.971846i	$0.575711\pi$
$930$	−9.48599	−0.311058
$931$	−28.2800	−0.926841
$932$	0.148971	0.00487971
$933$	2.65485	0.0869160
$934$	−77.3493	−2.53094
$935$	40.1348	1.31255
$936$	−23.9808	−0.783838
$937$	31.6451	1.03380	0.516900	−	0.856046i	$-0.327086\pi$
$937$	31.6451	1.03380	0.516900	+	0.856046i	$0.327086\pi$
$938$	−7.60388	−0.248275
$939$	7.66290	0.250069
$940$	−10.0839	−0.328901
$941$	42.4088	1.38249	0.691243	−	0.722622i	$-0.257063\pi$
$941$	42.4088	1.38249	0.691243	+	0.722622i	$0.257063\pi$
$942$	19.7560	0.643685
$943$	−13.9920	−0.455643
$944$	−22.3283	−0.726726
$945$	2.98029	0.0969490
$946$	19.8163	0.644284
$947$	0.00359040	0.000116672 0	5.83361e−5	−	1.00000i	$-0.499981\pi$
$947$	0.00359040	0.000116672 0	5.83361e−5		1.00000i	$0.499981\pi$
$948$	7.10282	0.230689
$949$	100.619	3.26625
$950$	15.6160	0.506648
$951$	27.1260	0.879620
$952$	2.73640	0.0886872
$953$	−24.9788	−0.809142	−0.404571	−	0.914507i	$-0.632579\pi$
$953$	−24.9788	−0.809142	−0.404571	+	0.914507i	$0.632579\pi$
$954$	35.9239	1.16308
$955$	−13.5236	−0.437613
$956$	−10.7884	−0.348922
$957$	4.44869	0.143806
$958$	7.24930	0.234214
$959$	−7.07719	−0.228534
$960$	−19.9259	−0.643106
$961$	−23.5043	−0.758204
$962$	102.463	3.30353
$963$	−38.9163	−1.25406
$964$	29.2728	0.942813
$965$	6.73083	0.216673
$966$	1.23245	0.0396536
$967$	49.0436	1.57714	0.788568	−	0.614947i	$-0.210823\pi$
$967$	49.0436	1.57714	0.788568	+	0.614947i	$0.210823\pi$
$968$	−25.8064	−0.829451
$969$	−15.7418	−0.505699
$970$	30.9228	0.992873
$971$	−19.6578	−0.630848	−0.315424	−	0.948951i	$-0.602147\pi$
$971$	−19.6578	−0.630848	−0.315424	+	0.948951i	$0.602147\pi$
$972$	−44.9956	−1.44324
$973$	−0.359878	−0.0115372
$974$	4.03332	0.129236
$975$	−9.35535	−0.299611
$976$	−15.4782	−0.495446
$977$	26.6380	0.852224	0.426112	−	0.904670i	$-0.359883\pi$
$977$	26.6380	0.852224	0.426112	+	0.904670i	$0.359883\pi$
$978$	39.0287	1.24800
$979$	5.27411	0.168561
$980$	34.7091	1.10874
$981$	37.8475	1.20838
$982$	−71.5123	−2.28205
$983$	41.0046	1.30784	0.653922	−	0.756562i	$-0.273122\pi$
$983$	41.0046	1.30784	0.653922	+	0.756562i	$0.273122\pi$
$984$	11.9310	0.380346
$985$	−30.8277	−0.982253
$986$	−9.56664	−0.304664
$987$	0.628806	0.0200151
$988$	−70.9622	−2.25761
$989$	−3.18152	−0.101166
$990$	44.9330	1.42807
$991$	−40.5849	−1.28922	−0.644612	−	0.764510i	$-0.722981\pi$
$991$	−40.5849	−1.28922	−0.644612	+	0.764510i	$0.722981\pi$
$992$	20.1929	0.641125
$993$	23.0538	0.731592
$994$	2.64550	0.0839102
$995$	−8.63036	−0.273601
$996$	34.8618	1.10464
$997$	21.1371	0.669420	0.334710	−	0.942321i	$-0.391362\pi$
$997$	21.1371	0.669420	0.334710	+	0.942321i	$0.391362\pi$
$998$	34.0635	1.07826
$999$	34.7535	1.09955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.9	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-2.18971 q^{2} -0.875363 q^{3} +2.79481 q^{4} -1.80760 q^{5} +1.91679 q^{6} -0.359878 q^{7} -1.74040 q^{8} -2.23374 q^{9} +O(q^{10})\)	\(q-2.18971 q^{2} -0.875363 q^{3} +2.79481 q^{4} -1.80760 q^{5} +1.91679 q^{6} -0.359878 q^{7} -1.74040 q^{8} -2.23374 q^{9} +3.95812 q^{10} -5.08211 q^{11} -2.44647 q^{12} -6.16853 q^{13} +0.788028 q^{14} +1.58231 q^{15} -1.77865 q^{16} +4.36892 q^{17} +4.89123 q^{18} +4.11616 q^{19} -5.05191 q^{20} +0.315024 q^{21} +11.1283 q^{22} -1.78666 q^{23} +1.52348 q^{24} -1.73257 q^{25} +13.5073 q^{26} +4.58142 q^{27} -1.00579 q^{28} +1.00000 q^{29} -3.46479 q^{30} +2.73782 q^{31} +7.37553 q^{32} +4.44869 q^{33} -9.56664 q^{34} +0.650517 q^{35} -6.24288 q^{36} +7.58575 q^{37} -9.01319 q^{38} +5.39970 q^{39} +3.14596 q^{40} +7.83139 q^{41} -0.689810 q^{42} +1.78071 q^{43} -14.2035 q^{44} +4.03772 q^{45} +3.91226 q^{46} +1.99606 q^{47} +1.55696 q^{48} -6.87049 q^{49} +3.79381 q^{50} -3.82439 q^{51} -17.2399 q^{52} +7.34454 q^{53} -10.0320 q^{54} +9.18644 q^{55} +0.626334 q^{56} -3.60314 q^{57} -2.18971 q^{58} +12.5535 q^{59} +4.42226 q^{60} +8.70223 q^{61} -5.99503 q^{62} +0.803875 q^{63} -12.5929 q^{64} +11.1503 q^{65} -9.74132 q^{66} -9.64926 q^{67} +12.2103 q^{68} +1.56397 q^{69} -1.42444 q^{70} +3.35712 q^{71} +3.88761 q^{72} -16.3117 q^{73} -16.6106 q^{74} +1.51663 q^{75} +11.5039 q^{76} +1.82894 q^{77} -11.8238 q^{78} -2.90329 q^{79} +3.21510 q^{80} +2.69082 q^{81} -17.1484 q^{82} -14.2498 q^{83} +0.880433 q^{84} -7.89727 q^{85} -3.89923 q^{86} -0.875363 q^{87} +8.84492 q^{88} -1.03778 q^{89} -8.84141 q^{90} +2.21992 q^{91} -4.99337 q^{92} -2.39659 q^{93} -4.37078 q^{94} -7.44039 q^{95} -6.45626 q^{96} +7.81251 q^{97} +15.0443 q^{98} +11.3521 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more