LMFDB - Embedded newform 4015.2.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.60539 q^{2} +2.52956 q^{3} +0.577267 q^{4} -1.00000 q^{5} -4.06093 q^{6} -3.16259 q^{7} +2.28404 q^{8} +3.39869 q^{9} +O(q^{10})$	\(q-1.60539 q^{2} +2.52956 q^{3} +0.577267 q^{4} -1.00000 q^{5} -4.06093 q^{6} -3.16259 q^{7} +2.28404 q^{8} +3.39869 q^{9} +1.60539 q^{10} +1.00000 q^{11} +1.46023 q^{12} +0.868107 q^{13} +5.07718 q^{14} -2.52956 q^{15} -4.82130 q^{16} -5.38787 q^{17} -5.45621 q^{18} +5.17851 q^{19} -0.577267 q^{20} -7.99997 q^{21} -1.60539 q^{22} -6.99019 q^{23} +5.77762 q^{24} +1.00000 q^{25} -1.39365 q^{26} +1.00851 q^{27} -1.82566 q^{28} +9.43728 q^{29} +4.06093 q^{30} +9.37344 q^{31} +3.17197 q^{32} +2.52956 q^{33} +8.64961 q^{34} +3.16259 q^{35} +1.96195 q^{36} +4.52274 q^{37} -8.31351 q^{38} +2.19593 q^{39} -2.28404 q^{40} -7.12523 q^{41} +12.8431 q^{42} -9.21462 q^{43} +0.577267 q^{44} -3.39869 q^{45} +11.2220 q^{46} -6.83783 q^{47} -12.1958 q^{48} +3.00198 q^{49} -1.60539 q^{50} -13.6290 q^{51} +0.501129 q^{52} -1.44393 q^{53} -1.61905 q^{54} -1.00000 q^{55} -7.22348 q^{56} +13.0994 q^{57} -15.1505 q^{58} +12.1086 q^{59} -1.46023 q^{60} -10.4081 q^{61} -15.0480 q^{62} -10.7487 q^{63} +4.55035 q^{64} -0.868107 q^{65} -4.06093 q^{66} -8.78632 q^{67} -3.11024 q^{68} -17.6821 q^{69} -5.07718 q^{70} +5.49831 q^{71} +7.76273 q^{72} +1.00000 q^{73} -7.26074 q^{74} +2.52956 q^{75} +2.98938 q^{76} -3.16259 q^{77} -3.52532 q^{78} -12.1133 q^{79} +4.82130 q^{80} -7.64498 q^{81} +11.4387 q^{82} +6.71093 q^{83} -4.61812 q^{84} +5.38787 q^{85} +14.7930 q^{86} +23.8722 q^{87} +2.28404 q^{88} -3.73803 q^{89} +5.45621 q^{90} -2.74547 q^{91} -4.03521 q^{92} +23.7107 q^{93} +10.9774 q^{94} -5.17851 q^{95} +8.02370 q^{96} +9.23029 q^{97} -4.81934 q^{98} +3.39869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9 + 7 * q^10 + 31 * q^11 - 4 * q^12 - 24 * q^13 - 9 * q^14 + 4 * q^15 + 43 * q^16 - 49 * q^17 - 35 * q^18 - 22 * q^19 - 39 * q^20 - 8 * q^21 - 7 * q^22 - q^23 - 13 * q^24 + 31 * q^25 - 9 * q^26 - 22 * q^27 - 34 * q^28 - 12 * q^29 + 5 * q^30 + 4 * q^31 - 45 * q^32 - 4 * q^33 + 2 * q^34 + 11 * q^35 + 34 * q^36 - 18 * q^37 - 7 * q^38 - q^39 + 24 * q^40 - 58 * q^41 - 21 * q^42 - 41 * q^43 + 39 * q^44 - 31 * q^45 + 23 * q^46 - 31 * q^47 - 29 * q^48 + 44 * q^49 - 7 * q^50 + 8 * q^51 - 89 * q^52 - 46 * q^53 - 47 * q^54 - 31 * q^55 + 10 * q^56 - 47 * q^57 - 34 * q^58 - 9 * q^59 + 4 * q^60 - 5 * q^61 - 50 * q^62 - 61 * q^63 + 78 * q^64 + 24 * q^65 - 5 * q^66 + q^67 - 115 * q^68 - 19 * q^69 + 9 * q^70 - 8 * q^71 - 93 * q^72 + 31 * q^73 - 19 * q^74 - 4 * q^75 - 7 * q^76 - 11 * q^77 + 57 * q^78 - 43 * q^80 + 43 * q^81 + 20 * q^82 - 29 * q^83 - 32 * q^84 + 49 * q^85 + 25 * q^86 - 62 * q^87 - 24 * q^88 - 77 * q^89 + 35 * q^90 - 11 * q^91 - 25 * q^92 - 38 * q^94 + 22 * q^95 - 23 * q^96 - 39 * q^97 - 65 * q^98 + 31 * 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 $

$2$	−1.60539	−1.13518	−0.567590	−	0.823311i	$-0.692124\pi$
$2$	−1.60539	−1.13518	−0.567590	+	0.823311i	$0.692124\pi$
$3$	2.52956	1.46044	0.730222	−	0.683210i	$-0.239417\pi$
$3$	2.52956	1.46044	0.730222	+	0.683210i	$0.239417\pi$
$4$	0.577267	0.288633
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−4.06093	−1.65787
$7$	−3.16259	−1.19535	−0.597674	−	0.801740i	$-0.703908\pi$
$7$	−3.16259	−1.19535	−0.597674	+	0.801740i	$0.703908\pi$
$8$	2.28404	0.807529
$9$	3.39869	1.13290
$10$	1.60539	0.507668
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.46023	0.421533
$13$	0.868107	0.240770	0.120385	−	0.992727i	$-0.461587\pi$
$13$	0.868107	0.240770	0.120385	+	0.992727i	$0.461587\pi$
$14$	5.07718	1.35693
$15$	−2.52956	−0.653130
$16$	−4.82130	−1.20532
$17$	−5.38787	−1.30675	−0.653375	−	0.757034i	$-0.726647\pi$
$17$	−5.38787	−1.30675	−0.653375	+	0.757034i	$0.726647\pi$
$18$	−5.45621	−1.28604
$19$	5.17851	1.18803	0.594016	−	0.804453i	$-0.297542\pi$
$19$	5.17851	1.18803	0.594016	+	0.804453i	$0.297542\pi$
$20$	−0.577267	−0.129081
$21$	−7.99997	−1.74574
$22$	−1.60539	−0.342270
$23$	−6.99019	−1.45756	−0.728778	−	0.684750i	$-0.759911\pi$
$23$	−6.99019	−1.45756	−0.728778	+	0.684750i	$0.759911\pi$
$24$	5.77762	1.17935
$25$	1.00000	0.200000
$26$	−1.39365	−0.273317
$27$	1.00851	0.194088
$28$	−1.82566	−0.345017
$29$	9.43728	1.75246	0.876229	−	0.481895i	$-0.160051\pi$
$29$	9.43728	1.75246	0.876229	+	0.481895i	$0.160051\pi$
$30$	4.06093	0.741420
$31$	9.37344	1.68352	0.841760	−	0.539852i	$-0.181520\pi$
$31$	9.37344	1.68352	0.841760	+	0.539852i	$0.181520\pi$
$32$	3.17197	0.560731
$33$	2.52956	0.440340
$34$	8.64961	1.48340
$35$	3.16259	0.534575
$36$	1.96195	0.326992
$37$	4.52274	0.743533	0.371767	−	0.928326i	$-0.378752\pi$
$37$	4.52274	0.743533	0.371767	+	0.928326i	$0.378752\pi$
$38$	−8.31351	−1.34863
$39$	2.19593	0.351631
$40$	−2.28404	−0.361138
$41$	−7.12523	−1.11277	−0.556387	−	0.830924i	$-0.687813\pi$
$41$	−7.12523	−1.11277	−0.556387	+	0.830924i	$0.687813\pi$
$42$	12.8431	1.98173
$43$	−9.21462	−1.40522	−0.702609	−	0.711577i	$-0.747981\pi$
$43$	−9.21462	−1.40522	−0.702609	+	0.711577i	$0.747981\pi$
$44$	0.577267	0.0870262
$45$	−3.39869	−0.506647
$46$	11.2220	1.65459
$47$	−6.83783	−0.997399	−0.498700	−	0.866775i	$-0.666189\pi$
$47$	−6.83783	−0.997399	−0.498700	+	0.866775i	$0.666189\pi$
$48$	−12.1958	−1.76031
$49$	3.00198	0.428855
$50$	−1.60539	−0.227036
$51$	−13.6290	−1.90843
$52$	0.501129	0.0694941
$53$	−1.44393	−0.198339	−0.0991693	−	0.995071i	$-0.531619\pi$
$53$	−1.44393	−0.198339	−0.0991693	+	0.995071i	$0.531619\pi$
$54$	−1.61905	−0.220325
$55$	−1.00000	−0.134840
$56$	−7.22348	−0.965278
$57$	13.0994	1.73505
$58$	−15.1505	−1.98936
$59$	12.1086	1.57640	0.788201	−	0.615417i	$-0.211013\pi$
$59$	12.1086	1.57640	0.788201	+	0.615417i	$0.211013\pi$
$60$	−1.46023	−0.188515
$61$	−10.4081	−1.33262	−0.666310	−	0.745675i	$-0.732127\pi$
$61$	−10.4081	−1.33262	−0.666310	+	0.745675i	$0.732127\pi$
$62$	−15.0480	−1.91110
$63$	−10.7487	−1.35420
$64$	4.55035	0.568794
$65$	−0.868107	−0.107675
$66$	−4.06093	−0.499866
$67$	−8.78632	−1.07342	−0.536710	−	0.843767i	$-0.680333\pi$
$67$	−8.78632	−1.07342	−0.536710	+	0.843767i	$0.680333\pi$
$68$	−3.11024	−0.377172
$69$	−17.6821	−2.12868
$70$	−5.07718	−0.606839
$71$	5.49831	0.652529	0.326264	−	0.945279i	$-0.394210\pi$
$71$	5.49831	0.652529	0.326264	+	0.945279i	$0.394210\pi$
$72$	7.76273	0.914847
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	−7.26074	−0.844044
$75$	2.52956	0.292089
$76$	2.98938	0.342906
$77$	−3.16259	−0.360411
$78$	−3.52532	−0.399164
$79$	−12.1133	−1.36286	−0.681429	−	0.731884i	$-0.738641\pi$
$79$	−12.1133	−1.36286	−0.681429	+	0.731884i	$0.738641\pi$
$80$	4.82130	0.539037
$81$	−7.64498	−0.849442
$82$	11.4387	1.26320
$83$	6.71093	0.736621	0.368310	−	0.929703i	$-0.379936\pi$
$83$	6.71093	0.736621	0.368310	+	0.929703i	$0.379936\pi$
$84$	−4.61812	−0.503878
$85$	5.38787	0.584396
$86$	14.7930	1.59517
$87$	23.8722	2.55937
$88$	2.28404	0.243479
$89$	−3.73803	−0.396231	−0.198115	−	0.980179i	$-0.563482\pi$
$89$	−3.73803	−0.396231	−0.198115	+	0.980179i	$0.563482\pi$
$90$	5.45621	0.575135
$91$	−2.74547	−0.287803
$92$	−4.03521	−0.420699
$93$	23.7107	2.45869
$94$	10.9774	1.13223
$95$	−5.17851	−0.531304
$96$	8.02370	0.818916
$97$	9.23029	0.937194	0.468597	−	0.883412i	$-0.344760\pi$
$97$	9.23029	0.937194	0.468597	+	0.883412i	$0.344760\pi$
$98$	−4.81934	−0.486827
$99$	3.39869	0.341581
$100$	0.577267	0.0577267
$101$	3.36180	0.334512	0.167256	−	0.985914i	$-0.446509\pi$
$101$	3.36180	0.334512	0.167256	+	0.985914i	$0.446509\pi$
$102$	21.8797	2.16642
$103$	5.07085	0.499646	0.249823	−	0.968292i	$-0.419628\pi$
$103$	5.07085	0.499646	0.249823	+	0.968292i	$0.419628\pi$
$104$	1.98279	0.194428
$105$	7.99997	0.780718
$106$	2.31806	0.225150
$107$	−18.8398	−1.82131	−0.910656	−	0.413165i	$-0.864423\pi$
$107$	−18.8398	−1.82131	−0.910656	+	0.413165i	$0.864423\pi$
$108$	0.582179	0.0560202
$109$	−13.8041	−1.32220	−0.661098	−	0.750299i	$-0.729909\pi$
$109$	−13.8041	−1.32220	−0.661098	+	0.750299i	$0.729909\pi$
$110$	1.60539	0.153068
$111$	11.4405	1.08589
$112$	15.2478	1.44078
$113$	−15.5558	−1.46337	−0.731685	−	0.681643i	$-0.761266\pi$
$113$	−15.5558	−1.46337	−0.731685	+	0.681643i	$0.761266\pi$
$114$	−21.0295	−1.96960
$115$	6.99019	0.651839
$116$	5.44782	0.505818
$117$	2.95043	0.272767
$118$	−19.4390	−1.78950
$119$	17.0396	1.56202
$120$	−5.77762	−0.527422
$121$	1.00000	0.0909091
$122$	16.7090	1.51276
$123$	−18.0237	−1.62514
$124$	5.41098	0.485920
$125$	−1.00000	−0.0894427
$126$	17.2558	1.53727
$127$	6.26466	0.555899	0.277949	−	0.960596i	$-0.410345\pi$
$127$	6.26466	0.555899	0.277949	+	0.960596i	$0.410345\pi$
$128$	−13.6490	−1.20641
$129$	−23.3090	−2.05224
$130$	1.39365	0.122231
$131$	1.84024	0.160783	0.0803914	−	0.996763i	$-0.474383\pi$
$131$	1.84024	0.160783	0.0803914	+	0.996763i	$0.474383\pi$
$132$	1.46023	0.127097
$133$	−16.3775	−1.42011
$134$	14.1054	1.21852
$135$	−1.00851	−0.0867987
$136$	−12.3061	−1.05524
$137$	−6.22016	−0.531424	−0.265712	−	0.964052i	$-0.585607\pi$
$137$	−6.22016	−0.531424	−0.265712	+	0.964052i	$0.585607\pi$
$138$	28.3867	2.41643
$139$	−4.77890	−0.405341	−0.202670	−	0.979247i	$-0.564962\pi$
$139$	−4.77890	−0.405341	−0.202670	+	0.979247i	$0.564962\pi$
$140$	1.82566	0.154296
$141$	−17.2967	−1.45665
$142$	−8.82691	−0.740738
$143$	0.868107	0.0725948
$144$	−16.3861	−1.36551
$145$	−9.43728	−0.783723
$146$	−1.60539	−0.132863
$147$	7.59371	0.626318
$148$	2.61082	0.214608
$149$	−12.9315	−1.05939	−0.529695	−	0.848188i	$-0.677694\pi$
$149$	−12.9315	−1.05939	−0.529695	+	0.848188i	$0.677694\pi$
$150$	−4.06093	−0.331573
$151$	−18.5847	−1.51240	−0.756199	−	0.654341i	$-0.772946\pi$
$151$	−18.5847	−1.51240	−0.756199	+	0.654341i	$0.772946\pi$
$152$	11.8279	0.959370
$153$	−18.3117	−1.48041
$154$	5.07718	0.409131
$155$	−9.37344	−0.752893
$156$	1.26764	0.101492
$157$	10.6515	0.850085	0.425042	−	0.905173i	$-0.360259\pi$
$157$	10.6515	0.850085	0.425042	+	0.905173i	$0.360259\pi$
$158$	19.4466	1.54709
$159$	−3.65250	−0.289662
$160$	−3.17197	−0.250766
$161$	22.1071	1.74229
$162$	12.2731	0.964270
$163$	−6.54955	−0.513001	−0.256500	−	0.966544i	$-0.582569\pi$
$163$	−6.54955	−0.513001	−0.256500	+	0.966544i	$0.582569\pi$
$164$	−4.11316	−0.321183
$165$	−2.52956	−0.196926
$166$	−10.7736	−0.836197
$167$	−25.0519	−1.93857	−0.969286	−	0.245935i	$-0.920905\pi$
$167$	−25.0519	−1.93857	−0.969286	+	0.245935i	$0.920905\pi$
$168$	−18.2722	−1.40973
$169$	−12.2464	−0.942030
$170$	−8.64961	−0.663395
$171$	17.6001	1.34592
$172$	−5.31930	−0.405592
$173$	17.0205	1.29404	0.647022	−	0.762471i	$-0.276014\pi$
$173$	17.0205	1.29404	0.647022	+	0.762471i	$0.276014\pi$
$174$	−38.3241	−2.90534
$175$	−3.16259	−0.239069
$176$	−4.82130	−0.363419
$177$	30.6294	2.30225
$178$	6.00099	0.449793
$179$	10.2781	0.768222	0.384111	−	0.923287i	$-0.374508\pi$
$179$	10.2781	0.768222	0.384111	+	0.923287i	$0.374508\pi$
$180$	−1.96195	−0.146235
$181$	15.4341	1.14721	0.573605	−	0.819132i	$-0.305545\pi$
$181$	15.4341	1.14721	0.573605	+	0.819132i	$0.305545\pi$
$182$	4.40754	0.326708
$183$	−26.3279	−1.94622
$184$	−15.9659	−1.17702
$185$	−4.52274	−0.332518
$186$	−38.0649	−2.79105
$187$	−5.38787	−0.394000
$188$	−3.94725	−0.287883
$189$	−3.18950	−0.232002
$190$	8.31351	0.603125
$191$	−10.2123	−0.738935	−0.369467	−	0.929244i	$-0.620460\pi$
$191$	−10.2123	−0.738935	−0.369467	+	0.929244i	$0.620460\pi$
$192$	11.5104	0.830692
$193$	−10.9598	−0.788902	−0.394451	−	0.918917i	$-0.629065\pi$
$193$	−10.9598	−0.788902	−0.394451	+	0.918917i	$0.629065\pi$
$194$	−14.8182	−1.06388
$195$	−2.19593	−0.157254
$196$	1.73294	0.123782
$197$	−4.78378	−0.340831	−0.170415	−	0.985372i	$-0.554511\pi$
$197$	−4.78378	−0.340831	−0.170415	+	0.985372i	$0.554511\pi$
$198$	−5.45621	−0.387756
$199$	3.55941	0.252320	0.126160	−	0.992010i	$-0.459735\pi$
$199$	3.55941	0.252320	0.126160	+	0.992010i	$0.459735\pi$
$200$	2.28404	0.161506
$201$	−22.2255	−1.56767
$202$	−5.39700	−0.379731
$203$	−29.8462	−2.09480
$204$	−7.86754	−0.550838
$205$	7.12523	0.497647
$206$	−8.14068	−0.567188
$207$	−23.7575	−1.65126
$208$	−4.18540	−0.290205
$209$	5.17851	0.358205
$210$	−12.8431	−0.886255
$211$	−1.58451	−0.109082	−0.0545409	−	0.998512i	$-0.517370\pi$
$211$	−1.58451	−0.109082	−0.0545409	+	0.998512i	$0.517370\pi$
$212$	−0.833531	−0.0572471
$213$	13.9083	0.952982
$214$	30.2452	2.06752
$215$	9.21462	0.628432
$216$	2.30347	0.156732
$217$	−29.6444	−2.01239
$218$	22.1610	1.50093
$219$	2.52956	0.170932
$220$	−0.577267	−0.0389193
$221$	−4.67725	−0.314626
$222$	−18.3665	−1.23268
$223$	10.8478	0.726419	0.363210	−	0.931707i	$-0.381681\pi$
$223$	10.8478	0.726419	0.363210	+	0.931707i	$0.381681\pi$
$224$	−10.0316	−0.670268
$225$	3.39869	0.226579
$226$	24.9731	1.66119
$227$	−9.23344	−0.612845	−0.306422	−	0.951896i	$-0.599132\pi$
$227$	−9.23344	−0.612845	−0.306422	+	0.951896i	$0.599132\pi$
$228$	7.56183	0.500794
$229$	4.60666	0.304417	0.152208	−	0.988348i	$-0.451362\pi$
$229$	4.60666	0.304417	0.152208	+	0.988348i	$0.451362\pi$
$230$	−11.2220	−0.739954
$231$	−7.99997	−0.526360
$232$	21.5551	1.41516
$233$	−9.05268	−0.593061	−0.296530	−	0.955023i	$-0.595830\pi$
$233$	−9.05268	−0.593061	−0.296530	+	0.955023i	$0.595830\pi$
$234$	−4.73658	−0.309640
$235$	6.83783	0.446051
$236$	6.98988	0.455002
$237$	−30.6415	−1.99038
$238$	−27.3552	−1.77317
$239$	−1.28997	−0.0834415	−0.0417207	−	0.999129i	$-0.513284\pi$
$239$	−1.28997	−0.0834415	−0.0417207	+	0.999129i	$0.513284\pi$
$240$	12.1958	0.787234
$241$	−21.2468	−1.36863	−0.684314	−	0.729188i	$-0.739898\pi$
$241$	−21.2468	−1.36863	−0.684314	+	0.729188i	$0.739898\pi$
$242$	−1.60539	−0.103198
$243$	−22.3640	−1.43465
$244$	−6.00824	−0.384638
$245$	−3.00198	−0.191790
$246$	28.9350	1.84483
$247$	4.49550	0.286042
$248$	21.4093	1.35949
$249$	16.9757	1.07579
$250$	1.60539	0.101534
$251$	15.3109	0.966418	0.483209	−	0.875505i	$-0.339471\pi$
$251$	15.3109	0.966418	0.483209	+	0.875505i	$0.339471\pi$
$252$	−6.20485	−0.390869
$253$	−6.99019	−0.439470
$254$	−10.0572	−0.631045
$255$	13.6290	0.853478
$256$	12.8112	0.800703
$257$	−2.49679	−0.155745	−0.0778726	−	0.996963i	$-0.524813\pi$
$257$	−2.49679	−0.155745	−0.0778726	+	0.996963i	$0.524813\pi$
$258$	37.4199	2.32966
$259$	−14.3036	−0.888780
$260$	−0.501129	−0.0310787
$261$	32.0744	1.98535
$262$	−2.95430	−0.182517
$263$	−19.8484	−1.22390	−0.611952	−	0.790895i	$-0.709615\pi$
$263$	−19.8484	−1.22390	−0.611952	+	0.790895i	$0.709615\pi$
$264$	5.77762	0.355588
$265$	1.44393	0.0886997
$266$	26.2922	1.61208
$267$	−9.45559	−0.578672
$268$	−5.07205	−0.309825
$269$	−13.5468	−0.825964	−0.412982	−	0.910739i	$-0.635513\pi$
$269$	−13.5468	−0.825964	−0.412982	+	0.910739i	$0.635513\pi$
$270$	1.61905	0.0985321
$271$	14.7369	0.895204	0.447602	−	0.894233i	$-0.352278\pi$
$271$	14.7369	0.895204	0.447602	+	0.894233i	$0.352278\pi$
$272$	25.9765	1.57506
$273$	−6.94484	−0.420321
$274$	9.98576	0.603262
$275$	1.00000	0.0603023
$276$	−10.2073	−0.614408
$277$	20.2043	1.21396	0.606978	−	0.794719i	$-0.292382\pi$
$277$	20.2043	1.21396	0.606978	+	0.794719i	$0.292382\pi$
$278$	7.67198	0.460135
$279$	31.8574	1.90725
$280$	7.22348	0.431685
$281$	−23.0623	−1.37578	−0.687890	−	0.725814i	$-0.741463\pi$
$281$	−23.0623	−1.37578	−0.687890	+	0.725814i	$0.741463\pi$
$282$	27.7679	1.65356
$283$	25.6213	1.52303	0.761513	−	0.648150i	$-0.224457\pi$
$283$	25.6213	1.52303	0.761513	+	0.648150i	$0.224457\pi$
$284$	3.17399	0.188342
$285$	−13.0994	−0.775939
$286$	−1.39365	−0.0824081
$287$	22.5342	1.33015
$288$	10.7805	0.635250
$289$	12.0291	0.707595
$290$	15.1505	0.889667
$291$	23.3486	1.36872
$292$	0.577267	0.0337820
$293$	−19.7974	−1.15658	−0.578288	−	0.815832i	$-0.696279\pi$
$293$	−19.7974	−1.15658	−0.578288	+	0.815832i	$0.696279\pi$
$294$	−12.1908	−0.710984
$295$	−12.1086	−0.704989
$296$	10.3301	0.600425
$297$	1.00851	0.0585197
$298$	20.7601	1.20260
$299$	−6.06824	−0.350935
$300$	1.46023	0.0843066
$301$	29.1421	1.67972
$302$	29.8356	1.71684
$303$	8.50390	0.488536
$304$	−24.9671	−1.43196
$305$	10.4081	0.595965
$306$	29.3973	1.68053
$307$	−17.9723	−1.02573	−0.512866	−	0.858469i	$-0.671416\pi$
$307$	−17.9723	−1.02573	−0.512866	+	0.858469i	$0.671416\pi$
$308$	−1.82566	−0.104027
$309$	12.8270	0.729705
$310$	15.0480	0.854669
$311$	−12.5200	−0.709946	−0.354973	−	0.934877i	$-0.615510\pi$
$311$	−12.5200	−0.709946	−0.354973	+	0.934877i	$0.615510\pi$
$312$	5.01559	0.283952
$313$	−10.3248	−0.583595	−0.291797	−	0.956480i	$-0.594253\pi$
$313$	−10.3248	−0.583595	−0.291797	+	0.956480i	$0.594253\pi$
$314$	−17.0998	−0.964999
$315$	10.7487	0.605619
$316$	−6.99263	−0.393366
$317$	30.6489	1.72142	0.860708	−	0.509099i	$-0.170021\pi$
$317$	30.6489	1.72142	0.860708	+	0.509099i	$0.170021\pi$
$318$	5.86368	0.328819
$319$	9.43728	0.528386
$320$	−4.55035	−0.254372
$321$	−47.6565	−2.65993
$322$	−35.4905	−1.97781
$323$	−27.9011	−1.55246
$324$	−4.41319	−0.245177
$325$	0.868107	0.0481539
$326$	10.5146	0.582348
$327$	−34.9184	−1.93099
$328$	−16.2743	−0.898597
$329$	21.6252	1.19224
$330$	4.06093	0.223547
$331$	−22.9112	−1.25932	−0.629658	−	0.776873i	$-0.716805\pi$
$331$	−22.9112	−1.25932	−0.629658	+	0.776873i	$0.716805\pi$
$332$	3.87400	0.212613
$333$	15.3714	0.842346
$334$	40.2180	2.20063
$335$	8.78632	0.480048
$336$	38.5702	2.10418
$337$	−10.7481	−0.585485	−0.292743	−	0.956191i	$-0.594568\pi$
$337$	−10.7481	−0.585485	−0.292743	+	0.956191i	$0.594568\pi$
$338$	19.6602	1.06937
$339$	−39.3495	−2.13717
$340$	3.11024	0.168676
$341$	9.37344	0.507600
$342$	−28.2550	−1.52786
$343$	12.6441	0.682717
$344$	−21.0465	−1.13475
$345$	17.6821	0.951974
$346$	−27.3245	−1.46897
$347$	25.6897	1.37910	0.689548	−	0.724240i	$-0.257809\pi$
$347$	25.6897	1.37910	0.689548	+	0.724240i	$0.257809\pi$
$348$	13.7806	0.738719
$349$	18.0019	0.963622	0.481811	−	0.876275i	$-0.339979\pi$
$349$	18.0019	0.963622	0.481811	+	0.876275i	$0.339979\pi$
$350$	5.07718	0.271387
$351$	0.875495	0.0467304
$352$	3.17197	0.169067
$353$	25.0690	1.33429	0.667144	−	0.744929i	$-0.267517\pi$
$353$	25.0690	1.33429	0.667144	+	0.744929i	$0.267517\pi$
$354$	−49.1721	−2.61347
$355$	−5.49831	−0.291820
$356$	−2.15784	−0.114365
$357$	43.1028	2.28124
$358$	−16.5003	−0.872070
$359$	−7.29279	−0.384899	−0.192449	−	0.981307i	$-0.561643\pi$
$359$	−7.29279	−0.384899	−0.192449	+	0.981307i	$0.561643\pi$
$360$	−7.76273	−0.409132
$361$	7.81696	0.411419
$362$	−24.7777	−1.30229
$363$	2.52956	0.132768
$364$	−1.58487	−0.0830696
$365$	−1.00000	−0.0523424
$366$	42.2665	2.20930
$367$	−7.40737	−0.386661	−0.193331	−	0.981134i	$-0.561929\pi$
$367$	−7.40737	−0.386661	−0.193331	+	0.981134i	$0.561929\pi$
$368$	33.7018	1.75683
$369$	−24.2164	−1.26066
$370$	7.26074	0.377468
$371$	4.56655	0.237083
$372$	13.6874	0.709659
$373$	−20.9557	−1.08504	−0.542522	−	0.840042i	$-0.682530\pi$
$373$	−20.9557	−1.08504	−0.542522	+	0.840042i	$0.682530\pi$
$374$	8.64961	0.447261
$375$	−2.52956	−0.130626
$376$	−15.6178	−0.805429
$377$	8.19257	0.421939
$378$	5.12039	0.263364
$379$	−26.9371	−1.38367	−0.691833	−	0.722058i	$-0.743197\pi$
$379$	−26.9371	−1.38367	−0.691833	+	0.722058i	$0.743197\pi$
$380$	−2.98938	−0.153352
$381$	15.8469	0.811859
$382$	16.3947	0.838824
$383$	−13.5142	−0.690544	−0.345272	−	0.938503i	$-0.612213\pi$
$383$	−13.5142	−0.690544	−0.345272	+	0.938503i	$0.612213\pi$
$384$	−34.5261	−1.76190
$385$	3.16259	0.161181
$386$	17.5947	0.895545
$387$	−31.3176	−1.59197
$388$	5.32834	0.270506
$389$	−25.8750	−1.31191	−0.655956	−	0.754799i	$-0.727734\pi$
$389$	−25.8750	−1.31191	−0.655956	+	0.754799i	$0.727734\pi$
$390$	3.52532	0.178512
$391$	37.6622	1.90466
$392$	6.85664	0.346313
$393$	4.65501	0.234814
$394$	7.67983	0.386904
$395$	12.1133	0.609489
$396$	1.96195	0.0985917
$397$	12.3642	0.620541	0.310271	−	0.950648i	$-0.399580\pi$
$397$	12.3642	0.620541	0.310271	+	0.950648i	$0.399580\pi$
$398$	−5.71423	−0.286428
$399$	−41.4279	−2.07399
$400$	−4.82130	−0.241065
$401$	−30.9815	−1.54714	−0.773570	−	0.633710i	$-0.781531\pi$
$401$	−30.9815	−1.54714	−0.773570	+	0.633710i	$0.781531\pi$
$402$	35.6806	1.77959
$403$	8.13715	0.405341
$404$	1.94066	0.0965513
$405$	7.64498	0.379882
$406$	47.9148	2.37797
$407$	4.52274	0.224184
$408$	−31.1290	−1.54112
$409$	27.3107	1.35043	0.675213	−	0.737623i	$-0.264052\pi$
$409$	27.3107	1.35043	0.675213	+	0.737623i	$0.264052\pi$
$410$	−11.4387	−0.564919
$411$	−15.7343	−0.776115
$412$	2.92724	0.144215
$413$	−38.2945	−1.88435
$414$	38.1400	1.87448
$415$	−6.71093	−0.329427
$416$	2.75361	0.135007
$417$	−12.0885	−0.591977
$418$	−8.31351	−0.406627
$419$	−3.46962	−0.169502	−0.0847510	−	0.996402i	$-0.527009\pi$
$419$	−3.46962	−0.169502	−0.0847510	+	0.996402i	$0.527009\pi$
$420$	4.61812	0.225341
$421$	−8.95107	−0.436248	−0.218124	−	0.975921i	$-0.569994\pi$
$421$	−8.95107	−0.436248	−0.218124	+	0.975921i	$0.569994\pi$
$422$	2.54374	0.123828
$423$	−23.2396	−1.12995
$424$	−3.29798	−0.160164
$425$	−5.38787	−0.261350
$426$	−22.3282	−1.08181
$427$	32.9165	1.59294
$428$	−10.8756	−0.525692
$429$	2.19593	0.106021
$430$	−14.7930	−0.713384
$431$	31.2509	1.50530	0.752651	−	0.658420i	$-0.228775\pi$
$431$	31.2509	1.50530	0.752651	+	0.658420i	$0.228775\pi$
$432$	−4.86232	−0.233939
$433$	33.7474	1.62180	0.810898	−	0.585187i	$-0.198979\pi$
$433$	33.7474	1.62180	0.810898	+	0.585187i	$0.198979\pi$
$434$	47.5907	2.28443
$435$	−23.8722	−1.14458
$436$	−7.96867	−0.381630
$437$	−36.1988	−1.73162
$438$	−4.06093	−0.194039
$439$	5.23572	0.249887	0.124944	−	0.992164i	$-0.460125\pi$
$439$	5.23572	0.249887	0.124944	+	0.992164i	$0.460125\pi$
$440$	−2.28404	−0.108887
$441$	10.2028	0.485848
$442$	7.50879	0.357157
$443$	−31.4388	−1.49370	−0.746850	−	0.664992i	$-0.768435\pi$
$443$	−31.4388	−1.49370	−0.746850	+	0.664992i	$0.768435\pi$
$444$	6.60425	0.313424
$445$	3.73803	0.177200
$446$	−17.4148	−0.824616
$447$	−32.7110	−1.54718
$448$	−14.3909	−0.679906
$449$	7.58450	0.357935	0.178967	−	0.983855i	$-0.442724\pi$
$449$	7.58450	0.357935	0.178967	+	0.983855i	$0.442724\pi$
$450$	−5.45621	−0.257208
$451$	−7.12523	−0.335514
$452$	−8.97987	−0.422377
$453$	−47.0111	−2.20877
$454$	14.8232	0.695689
$455$	2.74547	0.128710
$456$	29.9194	1.40111
$457$	−8.98443	−0.420274	−0.210137	−	0.977672i	$-0.567391\pi$
$457$	−8.98443	−0.420274	−0.210137	+	0.977672i	$0.567391\pi$
$458$	−7.39547	−0.345567
$459$	−5.43372	−0.253624
$460$	4.03521	0.188142
$461$	−29.4553	−1.37187	−0.685934	−	0.727663i	$-0.740606\pi$
$461$	−29.4553	−1.37187	−0.685934	+	0.727663i	$0.740606\pi$
$462$	12.8431	0.597513
$463$	21.7894	1.01264	0.506319	−	0.862346i	$-0.331006\pi$
$463$	21.7894	1.01264	0.506319	+	0.862346i	$0.331006\pi$
$464$	−45.4999	−2.11228
$465$	−23.7107	−1.09956
$466$	14.5331	0.673231
$467$	1.56499	0.0724190	0.0362095	−	0.999344i	$-0.488472\pi$
$467$	1.56499	0.0724190	0.0362095	+	0.999344i	$0.488472\pi$
$468$	1.70318	0.0787297
$469$	27.7875	1.28311
$470$	−10.9774	−0.506348
$471$	26.9437	1.24150
$472$	27.6564	1.27299
$473$	−9.21462	−0.423689
$474$	49.1914	2.25944
$475$	5.17851	0.237606
$476$	9.83641	0.450851
$477$	−4.90746	−0.224697
$478$	2.07091	0.0947211
$479$	4.34391	0.198478	0.0992392	−	0.995064i	$-0.468359\pi$
$479$	4.34391	0.198478	0.0992392	+	0.995064i	$0.468359\pi$
$480$	−8.02370	−0.366230
$481$	3.92622	0.179020
$482$	34.1094	1.55364
$483$	55.9214	2.54451
$484$	0.577267	0.0262394
$485$	−9.23029	−0.419126
$486$	35.9028	1.62859
$487$	13.4124	0.607773	0.303886	−	0.952708i	$-0.401716\pi$
$487$	13.4124	0.607773	0.303886	+	0.952708i	$0.401716\pi$
$488$	−23.7725	−1.07613
$489$	−16.5675	−0.749208
$490$	4.81934	0.217716
$491$	−9.09115	−0.410278	−0.205139	−	0.978733i	$-0.565765\pi$
$491$	−9.09115	−0.410278	−0.205139	+	0.978733i	$0.565765\pi$
$492$	−10.4045	−0.469070
$493$	−50.8468	−2.29002
$494$	−7.21702	−0.324709
$495$	−3.39869	−0.152760
$496$	−45.1922	−2.02919
$497$	−17.3889	−0.779999
$498$	−27.2526	−1.22122
$499$	35.0549	1.56927	0.784635	−	0.619957i	$-0.212850\pi$
$499$	35.0549	1.56927	0.784635	+	0.619957i	$0.212850\pi$
$500$	−0.577267	−0.0258162
$501$	−63.3703	−2.83118
$502$	−24.5800	−1.09706
$503$	23.3624	1.04168	0.520838	−	0.853656i	$-0.325620\pi$
$503$	23.3624	1.04168	0.520838	+	0.853656i	$0.325620\pi$
$504$	−24.5504	−1.09356
$505$	−3.36180	−0.149598
$506$	11.2220	0.498877
$507$	−30.9780	−1.37578
$508$	3.61638	0.160451
$509$	37.9243	1.68097	0.840483	−	0.541838i	$-0.182271\pi$
$509$	37.9243	1.68097	0.840483	+	0.541838i	$0.182271\pi$
$510$	−21.8797	−0.968851
$511$	−3.16259	−0.139905
$512$	6.73103	0.297472
$513$	5.22258	0.230582
$514$	4.00831	0.176799
$515$	−5.07085	−0.223449
$516$	−13.4555	−0.592345
$517$	−6.83783	−0.300727
$518$	22.9628	1.00893
$519$	43.0544	1.88988
$520$	−1.98279	−0.0869511
$521$	−30.7307	−1.34634	−0.673169	−	0.739489i	$-0.735067\pi$
$521$	−30.7307	−1.34634	−0.673169	+	0.739489i	$0.735067\pi$
$522$	−51.4918	−2.25373
$523$	−25.5096	−1.11546	−0.557729	−	0.830023i	$-0.688327\pi$
$523$	−25.5096	−1.11546	−0.557729	+	0.830023i	$0.688327\pi$
$524$	1.06231	0.0464073
$525$	−7.99997	−0.349147
$526$	31.8643	1.38935
$527$	−50.5029	−2.19994
$528$	−12.1958	−0.530753
$529$	25.8628	1.12447
$530$	−2.31806	−0.100690
$531$	41.1533	1.78590
$532$	−9.45419	−0.409891
$533$	−6.18546	−0.267922
$534$	15.1799	0.656897
$535$	18.8398	0.814516
$536$	−20.0683	−0.866818
$537$	25.9991	1.12195
$538$	21.7479	0.937618
$539$	3.00198	0.129305
$540$	−0.582179	−0.0250530
$541$	19.1411	0.822942	0.411471	−	0.911423i	$-0.365015\pi$
$541$	19.1411	0.822942	0.411471	+	0.911423i	$0.365015\pi$
$542$	−23.6584	−1.01622
$543$	39.0416	1.67544
$544$	−17.0902	−0.732735
$545$	13.8041	0.591304
$546$	11.1491	0.477139
$547$	−3.22149	−0.137741	−0.0688706	−	0.997626i	$-0.521940\pi$
$547$	−3.22149	−0.137741	−0.0688706	+	0.997626i	$0.521940\pi$
$548$	−3.59069	−0.153387
$549$	−35.3739	−1.50972
$550$	−1.60539	−0.0684539
$551$	48.8710	2.08198
$552$	−40.3867	−1.71897
$553$	38.3096	1.62909
$554$	−32.4356	−1.37806
$555$	−11.4405	−0.485624
$556$	−2.75870	−0.116995
$557$	−39.1422	−1.65851	−0.829254	−	0.558872i	$-0.811234\pi$
$557$	−39.1422	−1.65851	−0.829254	+	0.558872i	$0.811234\pi$
$558$	−51.1435	−2.16508
$559$	−7.99928	−0.338334
$560$	−15.2478	−0.644337
$561$	−13.6290	−0.575415
$562$	37.0239	1.56176
$563$	2.96302	0.124877	0.0624383	−	0.998049i	$-0.480112\pi$
$563$	2.96302	0.124877	0.0624383	+	0.998049i	$0.480112\pi$
$564$	−9.98481	−0.420437
$565$	15.5558	0.654439
$566$	−41.1320	−1.72891
$567$	24.1779	1.01538
$568$	12.5583	0.526936
$569$	−14.5407	−0.609579	−0.304790	−	0.952420i	$-0.598586\pi$
$569$	−14.5407	−0.609579	−0.304790	+	0.952420i	$0.598586\pi$
$570$	21.0295	0.880831
$571$	−29.6504	−1.24083	−0.620416	−	0.784273i	$-0.713036\pi$
$571$	−29.6504	−1.24083	−0.620416	+	0.784273i	$0.713036\pi$
$572$	0.501129	0.0209533
$573$	−25.8326	−1.07917
$574$	−36.1761	−1.50996
$575$	−6.99019	−0.291511
$576$	15.4652	0.644385
$577$	8.08352	0.336521	0.168261	−	0.985743i	$-0.446185\pi$
$577$	8.08352	0.336521	0.168261	+	0.985743i	$0.446185\pi$
$578$	−19.3114	−0.803248
$579$	−27.7234	−1.15215
$580$	−5.44782	−0.226209
$581$	−21.2239	−0.880517
$582$	−37.4835	−1.55374
$583$	−1.44393	−0.0598013
$584$	2.28404	0.0945141
$585$	−2.95043	−0.121985
$586$	31.7825	1.31292
$587$	30.1087	1.24272	0.621359	−	0.783526i	$-0.286581\pi$
$587$	30.1087	1.24272	0.621359	+	0.783526i	$0.286581\pi$
$588$	4.38359	0.180776
$589$	48.5405	2.00008
$590$	19.4390	0.800289
$591$	−12.1009	−0.497764
$592$	−21.8055	−0.896198
$593$	4.24522	0.174330	0.0871652	−	0.996194i	$-0.472219\pi$
$593$	4.24522	0.174330	0.0871652	+	0.996194i	$0.472219\pi$
$594$	−1.61905	−0.0664304
$595$	−17.0396	−0.698556
$596$	−7.46492	−0.305775
$597$	9.00375	0.368499
$598$	9.74187	0.398375
$599$	−17.1605	−0.701160	−0.350580	−	0.936533i	$-0.614016\pi$
$599$	−17.1605	−0.701160	−0.350580	+	0.936533i	$0.614016\pi$
$600$	5.77762	0.235870
$601$	11.3325	0.462263	0.231132	−	0.972922i	$-0.425757\pi$
$601$	11.3325	0.462263	0.231132	+	0.972922i	$0.425757\pi$
$602$	−46.7843	−1.90679
$603$	−29.8620	−1.21607
$604$	−10.7283	−0.436529
$605$	−1.00000	−0.0406558
$606$	−13.6520	−0.554576
$607$	22.7355	0.922806	0.461403	−	0.887191i	$-0.347346\pi$
$607$	22.7355	0.922806	0.461403	+	0.887191i	$0.347346\pi$
$608$	16.4261	0.666166
$609$	−75.4980	−3.05933
$610$	−16.7090	−0.676528
$611$	−5.93597	−0.240143
$612$	−10.5707	−0.427296
$613$	−22.5128	−0.909284	−0.454642	−	0.890674i	$-0.650233\pi$
$613$	−22.5128	−0.909284	−0.454642	+	0.890674i	$0.650233\pi$
$614$	28.8524	1.16439
$615$	18.0237	0.726786
$616$	−7.22348	−0.291042
$617$	−25.8047	−1.03886	−0.519429	−	0.854514i	$-0.673855\pi$
$617$	−25.8047	−1.03886	−0.519429	+	0.854514i	$0.673855\pi$
$618$	−20.5924	−0.828347
$619$	26.2747	1.05607	0.528034	−	0.849223i	$-0.322929\pi$
$619$	26.2747	1.05607	0.528034	+	0.849223i	$0.322929\pi$
$620$	−5.41098	−0.217310
$621$	−7.04968	−0.282894
$622$	20.0995	0.805916
$623$	11.8219	0.473633
$624$	−10.5872	−0.423829
$625$	1.00000	0.0400000
$626$	16.5754	0.662485
$627$	13.0994	0.523138
$628$	6.14877	0.245363
$629$	−24.3679	−0.971612
$630$	−17.2558	−0.687486
$631$	39.7294	1.58160	0.790801	−	0.612074i	$-0.209664\pi$
$631$	39.7294	1.58160	0.790801	+	0.612074i	$0.209664\pi$
$632$	−27.6673	−1.10055
$633$	−4.00811	−0.159308
$634$	−49.2034	−1.95412
$635$	−6.26466	−0.248606
$636$	−2.10847	−0.0836062
$637$	2.60604	0.103255
$638$	−15.1505	−0.599813
$639$	18.6870	0.739248
$640$	13.6490	0.539525
$641$	−12.8692	−0.508303	−0.254152	−	0.967164i	$-0.581796\pi$
$641$	−12.8692	−0.508303	−0.254152	+	0.967164i	$0.581796\pi$
$642$	76.5071	3.01949
$643$	11.8132	0.465866	0.232933	−	0.972493i	$-0.425168\pi$
$643$	11.8132	0.465866	0.232933	+	0.972493i	$0.425168\pi$
$644$	12.7617	0.502882
$645$	23.3090	0.917790
$646$	44.7921	1.76232
$647$	−26.4400	−1.03946	−0.519731	−	0.854330i	$-0.673968\pi$
$647$	−26.4400	−1.03946	−0.519731	+	0.854330i	$0.673968\pi$
$648$	−17.4614	−0.685949
$649$	12.1086	0.475303
$650$	−1.39365	−0.0546634
$651$	−74.9873	−2.93898
$652$	−3.78084	−0.148069
$653$	−39.4899	−1.54536	−0.772679	−	0.634797i	$-0.781084\pi$
$653$	−39.4899	−1.54536	−0.772679	+	0.634797i	$0.781084\pi$
$654$	56.0576	2.19202
$655$	−1.84024	−0.0719043
$656$	34.3528	1.34125
$657$	3.39869	0.132595
$658$	−34.7169	−1.35341
$659$	−17.8369	−0.694827	−0.347413	−	0.937712i	$-0.612940\pi$
$659$	−17.8369	−0.694827	−0.347413	+	0.937712i	$0.612940\pi$
$660$	−1.46023	−0.0568395
$661$	46.0105	1.78960	0.894800	−	0.446467i	$-0.147318\pi$
$661$	46.0105	1.78960	0.894800	+	0.446467i	$0.147318\pi$
$662$	36.7814	1.42955
$663$	−11.8314	−0.459493
$664$	15.3280	0.594843
$665$	16.3775	0.635093
$666$	−24.6770	−0.956214
$667$	−65.9684	−2.55431
$668$	−14.4616	−0.559537
$669$	27.4401	1.06089
$670$	−14.1054	−0.544941
$671$	−10.4081	−0.401800
$672$	−25.3757	−0.978888
$673$	24.6914	0.951782	0.475891	−	0.879504i	$-0.342126\pi$
$673$	24.6914	0.951782	0.475891	+	0.879504i	$0.342126\pi$
$674$	17.2548	0.664631
$675$	1.00851	0.0388176
$676$	−7.06943	−0.271901
$677$	17.9942	0.691572	0.345786	−	0.938313i	$-0.387612\pi$
$677$	17.9942	0.691572	0.345786	+	0.938313i	$0.387612\pi$
$678$	63.1711	2.42607
$679$	−29.1916	−1.12027
$680$	12.3061	0.471917
$681$	−23.3566	−0.895025
$682$	−15.0480	−0.576218
$683$	−26.8877	−1.02883	−0.514415	−	0.857541i	$-0.671991\pi$
$683$	−26.8877	−1.02883	−0.514415	+	0.857541i	$0.671991\pi$
$684$	10.1600	0.388476
$685$	6.22016	0.237660
$686$	−20.2987	−0.775006
$687$	11.6528	0.444583
$688$	44.4264	1.69374
$689$	−1.25348	−0.0477539
$690$	−28.3867	−1.08066
$691$	−14.1874	−0.539713	−0.269856	−	0.962901i	$-0.586976\pi$
$691$	−14.1874	−0.539713	−0.269856	+	0.962901i	$0.586976\pi$
$692$	9.82537	0.373504
$693$	−10.7487	−0.408308
$694$	−41.2419	−1.56552
$695$	4.77890	0.181274
$696$	54.5250	2.06676
$697$	38.3898	1.45412
$698$	−28.9001	−1.09388
$699$	−22.8993	−0.866132
$700$	−1.82566	−0.0690034
$701$	35.3396	1.33476	0.667380	−	0.744718i	$-0.267416\pi$
$701$	35.3396	1.33476	0.667380	+	0.744718i	$0.267416\pi$
$702$	−1.40551	−0.0530475
$703$	23.4210	0.883341
$704$	4.55035	0.171498
$705$	17.2967	0.651432
$706$	−40.2454	−1.51466
$707$	−10.6320	−0.399858
$708$	17.6813	0.664505
$709$	−13.8647	−0.520701	−0.260350	−	0.965514i	$-0.583838\pi$
$709$	−13.8647	−0.520701	−0.260350	+	0.965514i	$0.583838\pi$
$710$	8.82691	0.331268
$711$	−41.1695	−1.54398
$712$	−8.53780	−0.319968
$713$	−65.5222	−2.45383
$714$	−69.1967	−2.58962
$715$	−0.868107	−0.0324654
$716$	5.93321	0.221735
$717$	−3.26307	−0.121862
$718$	11.7077	0.436929
$719$	−35.4697	−1.32280	−0.661398	−	0.750035i	$-0.730037\pi$
$719$	−35.4697	−1.32280	−0.661398	+	0.750035i	$0.730037\pi$
$720$	16.3861	0.610673
$721$	−16.0370	−0.597251
$722$	−12.5492	−0.467034
$723$	−53.7452	−1.99880
$724$	8.90961	0.331123
$725$	9.43728	0.350492
$726$	−4.06093	−0.150715
$727$	2.79222	0.103558	0.0517788	−	0.998659i	$-0.483511\pi$
$727$	2.79222	0.103558	0.0517788	+	0.998659i	$0.483511\pi$
$728$	−6.27075	−0.232410
$729$	−33.6362	−1.24578
$730$	1.60539	0.0594180
$731$	49.6472	1.83627
$732$	−15.1982	−0.561743
$733$	−14.7760	−0.545763	−0.272881	−	0.962048i	$-0.587977\pi$
$733$	−14.7760	−0.545763	−0.272881	+	0.962048i	$0.587977\pi$
$734$	11.8917	0.438930
$735$	−7.59371	−0.280098
$736$	−22.1727	−0.817296
$737$	−8.78632	−0.323648
$738$	38.8767	1.43107
$739$	−3.69861	−0.136055	−0.0680277	−	0.997683i	$-0.521671\pi$
$739$	−3.69861	−0.136055	−0.0680277	+	0.997683i	$0.521671\pi$
$740$	−2.61082	−0.0959758
$741$	11.3717	0.417748
$742$	−7.33108	−0.269132
$743$	48.1373	1.76599	0.882994	−	0.469385i	$-0.155524\pi$
$743$	48.1373	1.76599	0.882994	+	0.469385i	$0.155524\pi$
$744$	54.1562	1.98546
$745$	12.9315	0.473774
$746$	33.6420	1.23172
$747$	22.8084	0.834515
$748$	−3.11024	−0.113722
$749$	59.5826	2.17710
$750$	4.06093	0.148284
$751$	4.09671	0.149491	0.0747456	−	0.997203i	$-0.476186\pi$
$751$	4.09671	0.149491	0.0747456	+	0.997203i	$0.476186\pi$
$752$	32.9672	1.20219
$753$	38.7300	1.41140
$754$	−13.1522	−0.478976
$755$	18.5847	0.676365
$756$	−1.84119	−0.0669636
$757$	−13.6495	−0.496101	−0.248050	−	0.968747i	$-0.579790\pi$
$757$	−13.6495	−0.496101	−0.248050	+	0.968747i	$0.579790\pi$
$758$	43.2445	1.57071
$759$	−17.6821	−0.641821
$760$	−11.8279	−0.429043
$761$	−0.154306	−0.00559360	−0.00279680	−	0.999996i	$-0.500890\pi$
$761$	−0.154306	−0.00559360	−0.00279680	+	0.999996i	$0.500890\pi$
$762$	−25.4403	−0.921606
$763$	43.6568	1.58048
$764$	−5.89521	−0.213281
$765$	18.3117	0.662061
$766$	21.6955	0.783891
$767$	10.5115	0.379550
$768$	32.4069	1.16938
$769$	−27.7924	−1.00222	−0.501109	−	0.865384i	$-0.667075\pi$
$769$	−27.7924	−1.00222	−0.501109	+	0.865384i	$0.667075\pi$
$770$	−5.07718	−0.182969
$771$	−6.31578	−0.227457
$772$	−6.32671	−0.227703
$773$	16.9445	0.609451	0.304726	−	0.952440i	$-0.401435\pi$
$773$	16.9445	0.609451	0.304726	+	0.952440i	$0.401435\pi$
$774$	50.2769	1.80717
$775$	9.37344	0.336704
$776$	21.0823	0.756812
$777$	−36.1818	−1.29801
$778$	41.5393	1.48926
$779$	−36.8980	−1.32201
$780$	−1.26764	−0.0453887
$781$	5.49831	0.196745
$782$	−60.4625	−2.16213
$783$	9.51758	0.340131
$784$	−14.4735	−0.516909
$785$	−10.6515	−0.380169
$786$	−7.47310	−0.266557
$787$	−7.10485	−0.253261	−0.126630	−	0.991950i	$-0.540416\pi$
$787$	−7.10485	−0.253261	−0.126630	+	0.991950i	$0.540416\pi$
$788$	−2.76152	−0.0983751
$789$	−50.2078	−1.78744
$790$	−19.4466	−0.691879
$791$	49.1968	1.74924
$792$	7.76273	0.275837
$793$	−9.03534	−0.320854
$794$	−19.8493	−0.704426
$795$	3.65250	0.129541
$796$	2.05473	0.0728279
$797$	−12.1848	−0.431608	−0.215804	−	0.976437i	$-0.569237\pi$
$797$	−12.1848	−0.431608	−0.215804	+	0.976437i	$0.569237\pi$
$798$	66.5079	2.35435
$799$	36.8413	1.30335
$800$	3.17197	0.112146
$801$	−12.7044	−0.448888
$802$	49.7372	1.75628
$803$	1.00000	0.0352892
$804$	−12.8301	−0.452482
$805$	−22.1071	−0.779174
$806$	−13.0633	−0.460134
$807$	−34.2675	−1.20627
$808$	7.67849	0.270128
$809$	37.9169	1.33309	0.666543	−	0.745467i	$-0.267773\pi$
$809$	37.9169	1.33309	0.666543	+	0.745467i	$0.267773\pi$
$810$	−12.2731	−0.431234
$811$	20.4629	0.718549	0.359275	−	0.933232i	$-0.383024\pi$
$811$	20.4629	0.718549	0.359275	+	0.933232i	$0.383024\pi$
$812$	−17.2292	−0.604628
$813$	37.2779	1.30739
$814$	−7.26074	−0.254489
$815$	6.54955	0.229421
$816$	65.7092	2.30028
$817$	−47.7180	−1.66944
$818$	−43.8442	−1.53298
$819$	−9.33099	−0.326051
$820$	4.11316	0.143638
$821$	13.1362	0.458458	0.229229	−	0.973372i	$-0.426379\pi$
$821$	13.1362	0.458458	0.229229	+	0.973372i	$0.426379\pi$
$822$	25.2596	0.881030
$823$	−51.4161	−1.79225	−0.896126	−	0.443800i	$-0.853630\pi$
$823$	−51.4161	−1.79225	−0.896126	+	0.443800i	$0.853630\pi$
$824$	11.5820	0.403479
$825$	2.52956	0.0880681
$826$	61.4775	2.13907
$827$	−43.8476	−1.52473	−0.762365	−	0.647148i	$-0.775962\pi$
$827$	−43.8476	−1.52473	−0.762365	+	0.647148i	$0.775962\pi$
$828$	−13.7144	−0.476609
$829$	−0.736851	−0.0255919	−0.0127960	−	0.999918i	$-0.504073\pi$
$829$	−0.736851	−0.0255919	−0.0127960	+	0.999918i	$0.504073\pi$
$830$	10.7736	0.373959
$831$	51.1079	1.77291
$832$	3.95019	0.136948
$833$	−16.1743	−0.560406
$834$	19.4067	0.672001
$835$	25.0519	0.866956
$836$	2.98938	0.103390
$837$	9.45321	0.326751
$838$	5.57008	0.192415
$839$	14.1974	0.490148	0.245074	−	0.969504i	$-0.421188\pi$
$839$	14.1974	0.490148	0.245074	+	0.969504i	$0.421188\pi$
$840$	18.2722	0.630452
$841$	60.0622	2.07111
$842$	14.3699	0.495220
$843$	−58.3375	−2.00925
$844$	−0.914682	−0.0314847
$845$	12.2464	0.421289
$846$	37.3086	1.28270
$847$	−3.16259	−0.108668
$848$	6.96160	0.239062
$849$	64.8106	2.22429
$850$	8.64961	0.296679
$851$	−31.6148	−1.08374
$852$	8.02881	0.275062
$853$	13.1027	0.448628	0.224314	−	0.974517i	$-0.427986\pi$
$853$	13.1027	0.448628	0.224314	+	0.974517i	$0.427986\pi$
$854$	−52.8438	−1.80828
$855$	−17.6001	−0.601912
$856$	−43.0308	−1.47076
$857$	26.3992	0.901779	0.450890	−	0.892580i	$-0.351107\pi$
$857$	26.3992	0.901779	0.450890	+	0.892580i	$0.351107\pi$
$858$	−3.52532	−0.120352
$859$	−10.8080	−0.368765	−0.184383	−	0.982855i	$-0.559029\pi$
$859$	−10.8080	−0.368765	−0.184383	+	0.982855i	$0.559029\pi$
$860$	5.31930	0.181386
$861$	57.0016	1.94261
$862$	−50.1697	−1.70879
$863$	−18.4264	−0.627241	−0.313620	−	0.949548i	$-0.601542\pi$
$863$	−18.4264	−0.627241	−0.313620	+	0.949548i	$0.601542\pi$
$864$	3.19896	0.108831
$865$	−17.0205	−0.578714
$866$	−54.1776	−1.84103
$867$	30.4284	1.03340
$868$	−17.1127	−0.580843
$869$	−12.1133	−0.410917
$870$	38.3241	1.29931
$871$	−7.62747	−0.258447
$872$	−31.5292	−1.06771
$873$	31.3709	1.06174
$874$	58.1130	1.96570
$875$	3.16259	0.106915
$876$	1.46023	0.0493367
$877$	−9.65970	−0.326185	−0.163092	−	0.986611i	$-0.552147\pi$
$877$	−9.65970	−0.326185	−0.163092	+	0.986611i	$0.552147\pi$
$878$	−8.40536	−0.283667
$879$	−50.0788	−1.68912
$880$	4.82130	0.162526
$881$	13.6768	0.460785	0.230392	−	0.973098i	$-0.425999\pi$
$881$	13.6768	0.460785	0.230392	+	0.973098i	$0.425999\pi$
$882$	−16.3795	−0.551525
$883$	41.0828	1.38255	0.691273	−	0.722593i	$-0.257050\pi$
$883$	41.0828	1.38255	0.691273	+	0.722593i	$0.257050\pi$
$884$	−2.70002	−0.0908115
$885$	−30.6294	−1.02960
$886$	50.4714	1.69562
$887$	−1.28933	−0.0432915	−0.0216458	−	0.999766i	$-0.506891\pi$
$887$	−1.28933	−0.0432915	−0.0216458	+	0.999766i	$0.506891\pi$
$888$	26.1306	0.876887
$889$	−19.8126	−0.664492
$890$	−6.00099	−0.201153
$891$	−7.64498	−0.256116
$892$	6.26205	0.209669
$893$	−35.4097	−1.18494
$894$	52.5139	1.75633
$895$	−10.2781	−0.343559
$896$	43.1663	1.44208
$897$	−15.3500	−0.512521
$898$	−12.1761	−0.406320
$899$	88.4598	2.95030
$900$	1.96195	0.0653983
$901$	7.77968	0.259179
$902$	11.4387	0.380868
$903$	73.7168	2.45314
$904$	−35.5301	−1.18171
$905$	−15.4341	−0.513048
$906$	75.4710	2.50736
$907$	−6.18549	−0.205386	−0.102693	−	0.994713i	$-0.532746\pi$
$907$	−6.18549	−0.205386	−0.102693	+	0.994713i	$0.532746\pi$
$908$	−5.33015	−0.176887
$909$	11.4257	0.378967
$910$	−4.40754	−0.146108
$911$	33.2270	1.10086	0.550429	−	0.834882i	$-0.314464\pi$
$911$	33.2270	1.10086	0.550429	+	0.834882i	$0.314464\pi$
$912$	−63.1559	−2.09130
$913$	6.71093	0.222099
$914$	14.4235	0.477086
$915$	26.3279	0.870374
$916$	2.65927	0.0878647
$917$	−5.81994	−0.192191
$918$	8.72322	0.287909
$919$	27.1512	0.895635	0.447817	−	0.894125i	$-0.352202\pi$
$919$	27.1512	0.895635	0.447817	+	0.894125i	$0.352202\pi$
$920$	15.9659	0.526379
$921$	−45.4620	−1.49802
$922$	47.2871	1.55732
$923$	4.77312	0.157109
$924$	−4.61812	−0.151925
$925$	4.52274	0.148707
$926$	−34.9804	−1.14953
$927$	17.2343	0.566047
$928$	29.9348	0.982657
$929$	−51.2075	−1.68006	−0.840031	−	0.542538i	$-0.817463\pi$
$929$	−51.2075	−1.68006	−0.840031	+	0.542538i	$0.817463\pi$
$930$	38.0649	1.24820
$931$	15.5458	0.509493
$932$	−5.22581	−0.171177
$933$	−31.6702	−1.03684
$934$	−2.51241	−0.0822085
$935$	5.38787	0.176202
$936$	6.73888	0.220267
$937$	−2.76418	−0.0903019	−0.0451509	−	0.998980i	$-0.514377\pi$
$937$	−2.76418	−0.0903019	−0.0451509	+	0.998980i	$0.514377\pi$
$938$	−44.6097	−1.45656
$939$	−26.1173	−0.852307
$940$	3.94725	0.128745
$941$	−34.3841	−1.12089	−0.560444	−	0.828192i	$-0.689370\pi$
$941$	−34.3841	−1.12089	−0.560444	+	0.828192i	$0.689370\pi$
$942$	−43.2551	−1.40933
$943$	49.8067	1.62193
$944$	−58.3791	−1.90008
$945$	3.18950	0.103755
$946$	14.7930	0.480963
$947$	−21.2355	−0.690062	−0.345031	−	0.938591i	$-0.612132\pi$
$947$	−21.2355	−0.690062	−0.345031	+	0.938591i	$0.612132\pi$
$948$	−17.6883	−0.574489
$949$	0.868107	0.0281800
$950$	−8.31351	−0.269726
$951$	77.5284	2.51403
$952$	38.9191	1.26138
$953$	37.9204	1.22836	0.614181	−	0.789165i	$-0.289486\pi$
$953$	37.9204	1.22836	0.614181	+	0.789165i	$0.289486\pi$
$954$	7.87837	0.255072
$955$	10.2123	0.330462
$956$	−0.744659	−0.0240840
$957$	23.8722	0.771678
$958$	−6.97366	−0.225309
$959$	19.6718	0.635236
$960$	−11.5104	−0.371497
$961$	56.8615	1.83424
$962$	−6.30310	−0.203220
$963$	−64.0306	−2.06336
$964$	−12.2651	−0.395032
$965$	10.9598	0.352808
$966$	−89.7754	−2.88848
$967$	−8.10103	−0.260512	−0.130256	−	0.991480i	$-0.541580\pi$
$967$	−8.10103	−0.260512	−0.130256	+	0.991480i	$0.541580\pi$
$968$	2.28404	0.0734117
$969$	−70.5777	−2.26728
$970$	14.8182	0.475783
$971$	−53.5637	−1.71894	−0.859470	−	0.511186i	$-0.829206\pi$
$971$	−53.5637	−1.71894	−0.859470	+	0.511186i	$0.829206\pi$
$972$	−12.9100	−0.414088
$973$	15.1137	0.484523
$974$	−21.5320	−0.689931
$975$	2.19593	0.0703261
$976$	50.1805	1.60624
$977$	16.7526	0.535964	0.267982	−	0.963424i	$-0.413643\pi$
$977$	16.7526	0.535964	0.267982	+	0.963424i	$0.413643\pi$
$978$	26.5973	0.850486
$979$	−3.73803	−0.119468
$980$	−1.73294	−0.0553569
$981$	−46.9160	−1.49791
$982$	14.5948	0.465739
$983$	−5.34491	−0.170476	−0.0852380	−	0.996361i	$-0.527165\pi$
$983$	−5.34491	−0.170476	−0.0852380	+	0.996361i	$0.527165\pi$
$984$	−41.1668	−1.31235
$985$	4.78378	0.152424
$986$	81.6288	2.59959
$987$	54.7024	1.74120
$988$	2.59510	0.0825612
$989$	64.4120	2.04818
$990$	5.45621	0.173410
$991$	25.0931	0.797110	0.398555	−	0.917144i	$-0.369512\pi$
$991$	25.0931	0.797110	0.398555	+	0.917144i	$0.369512\pi$
$992$	29.7323	0.944001
$993$	−57.9554	−1.83916
$994$	27.9159	0.885439
$995$	−3.55941	−0.112841
$996$	9.79952	0.310510
$997$	5.66789	0.179504	0.0897520	−	0.995964i	$-0.471393\pi$
$997$	5.66789	0.179504	0.0897520	+	0.995964i	$0.471393\pi$
$998$	−56.2766	−1.78140
$999$	4.56122	0.144311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.f.1.11	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.f.1.11	✓	31	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:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.60539 q^{2} +2.52956 q^{3} +0.577267 q^{4} -1.00000 q^{5} -4.06093 q^{6} -3.16259 q^{7} +2.28404 q^{8} +3.39869 q^{9} +O(q^{10})\)	\(q-1.60539 q^{2} +2.52956 q^{3} +0.577267 q^{4} -1.00000 q^{5} -4.06093 q^{6} -3.16259 q^{7} +2.28404 q^{8} +3.39869 q^{9} +1.60539 q^{10} +1.00000 q^{11} +1.46023 q^{12} +0.868107 q^{13} +5.07718 q^{14} -2.52956 q^{15} -4.82130 q^{16} -5.38787 q^{17} -5.45621 q^{18} +5.17851 q^{19} -0.577267 q^{20} -7.99997 q^{21} -1.60539 q^{22} -6.99019 q^{23} +5.77762 q^{24} +1.00000 q^{25} -1.39365 q^{26} +1.00851 q^{27} -1.82566 q^{28} +9.43728 q^{29} +4.06093 q^{30} +9.37344 q^{31} +3.17197 q^{32} +2.52956 q^{33} +8.64961 q^{34} +3.16259 q^{35} +1.96195 q^{36} +4.52274 q^{37} -8.31351 q^{38} +2.19593 q^{39} -2.28404 q^{40} -7.12523 q^{41} +12.8431 q^{42} -9.21462 q^{43} +0.577267 q^{44} -3.39869 q^{45} +11.2220 q^{46} -6.83783 q^{47} -12.1958 q^{48} +3.00198 q^{49} -1.60539 q^{50} -13.6290 q^{51} +0.501129 q^{52} -1.44393 q^{53} -1.61905 q^{54} -1.00000 q^{55} -7.22348 q^{56} +13.0994 q^{57} -15.1505 q^{58} +12.1086 q^{59} -1.46023 q^{60} -10.4081 q^{61} -15.0480 q^{62} -10.7487 q^{63} +4.55035 q^{64} -0.868107 q^{65} -4.06093 q^{66} -8.78632 q^{67} -3.11024 q^{68} -17.6821 q^{69} -5.07718 q^{70} +5.49831 q^{71} +7.76273 q^{72} +1.00000 q^{73} -7.26074 q^{74} +2.52956 q^{75} +2.98938 q^{76} -3.16259 q^{77} -3.52532 q^{78} -12.1133 q^{79} +4.82130 q^{80} -7.64498 q^{81} +11.4387 q^{82} +6.71093 q^{83} -4.61812 q^{84} +5.38787 q^{85} +14.7930 q^{86} +23.8722 q^{87} +2.28404 q^{88} -3.73803 q^{89} +5.45621 q^{90} -2.74547 q^{91} -4.03521 q^{92} +23.7107 q^{93} +10.9774 q^{94} -5.17851 q^{95} +8.02370 q^{96} +9.23029 q^{97} -4.81934 q^{98} +3.39869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9 + 7 * q^10 + 31 * q^11 - 4 * q^12 - 24 * q^13 - 9 * q^14 + 4 * q^15 + 43 * q^16 - 49 * q^17 - 35 * q^18 - 22 * q^19 - 39 * q^20 - 8 * q^21 - 7 * q^22 - q^23 - 13 * q^24 + 31 * q^25 - 9 * q^26 - 22 * q^27 - 34 * q^28 - 12 * q^29 + 5 * q^30 + 4 * q^31 - 45 * q^32 - 4 * q^33 + 2 * q^34 + 11 * q^35 + 34 * q^36 - 18 * q^37 - 7 * q^38 - q^39 + 24 * q^40 - 58 * q^41 - 21 * q^42 - 41 * q^43 + 39 * q^44 - 31 * q^45 + 23 * q^46 - 31 * q^47 - 29 * q^48 + 44 * q^49 - 7 * q^50 + 8 * q^51 - 89 * q^52 - 46 * q^53 - 47 * q^54 - 31 * q^55 + 10 * q^56 - 47 * q^57 - 34 * q^58 - 9 * q^59 + 4 * q^60 - 5 * q^61 - 50 * q^62 - 61 * q^63 + 78 * q^64 + 24 * q^65 - 5 * q^66 + q^67 - 115 * q^68 - 19 * q^69 + 9 * q^70 - 8 * q^71 - 93 * q^72 + 31 * q^73 - 19 * q^74 - 4 * q^75 - 7 * q^76 - 11 * q^77 + 57 * q^78 - 43 * q^80 + 43 * q^81 + 20 * q^82 - 29 * q^83 - 32 * q^84 + 49 * q^85 + 25 * q^86 - 62 * q^87 - 24 * q^88 - 77 * q^89 + 35 * q^90 - 11 * q^91 - 25 * q^92 - 38 * q^94 + 22 * q^95 - 23 * q^96 - 39 * q^97 - 65 * q^98 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more