LMFDB - Embedded newform 4015.2.a.f.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.42903 q^{2} -2.96305 q^{3} +3.90019 q^{4} -1.00000 q^{5} +7.19735 q^{6} +2.52750 q^{7} -4.61563 q^{8} +5.77969 q^{9} +O(q^{10})$	\(q-2.42903 q^{2} -2.96305 q^{3} +3.90019 q^{4} -1.00000 q^{5} +7.19735 q^{6} +2.52750 q^{7} -4.61563 q^{8} +5.77969 q^{9} +2.42903 q^{10} +1.00000 q^{11} -11.5565 q^{12} +4.42384 q^{13} -6.13938 q^{14} +2.96305 q^{15} +3.41112 q^{16} -1.29895 q^{17} -14.0391 q^{18} +4.83821 q^{19} -3.90019 q^{20} -7.48913 q^{21} -2.42903 q^{22} -0.787723 q^{23} +13.6764 q^{24} +1.00000 q^{25} -10.7456 q^{26} -8.23639 q^{27} +9.85775 q^{28} +8.42044 q^{29} -7.19735 q^{30} +4.23922 q^{31} +0.945545 q^{32} -2.96305 q^{33} +3.15520 q^{34} -2.52750 q^{35} +22.5419 q^{36} -10.4958 q^{37} -11.7522 q^{38} -13.1081 q^{39} +4.61563 q^{40} -7.20804 q^{41} +18.1913 q^{42} -8.06892 q^{43} +3.90019 q^{44} -5.77969 q^{45} +1.91340 q^{46} -7.46880 q^{47} -10.1073 q^{48} -0.611733 q^{49} -2.42903 q^{50} +3.84887 q^{51} +17.2538 q^{52} -3.51874 q^{53} +20.0064 q^{54} -1.00000 q^{55} -11.6660 q^{56} -14.3359 q^{57} -20.4535 q^{58} -6.78145 q^{59} +11.5565 q^{60} -3.99500 q^{61} -10.2972 q^{62} +14.6082 q^{63} -9.11899 q^{64} -4.42384 q^{65} +7.19735 q^{66} +6.32760 q^{67} -5.06616 q^{68} +2.33407 q^{69} +6.13938 q^{70} -15.0441 q^{71} -26.6769 q^{72} +1.00000 q^{73} +25.4947 q^{74} -2.96305 q^{75} +18.8700 q^{76} +2.52750 q^{77} +31.8399 q^{78} +16.7010 q^{79} -3.41112 q^{80} +7.06579 q^{81} +17.5086 q^{82} -15.8195 q^{83} -29.2090 q^{84} +1.29895 q^{85} +19.5997 q^{86} -24.9502 q^{87} -4.61563 q^{88} +0.538301 q^{89} +14.0391 q^{90} +11.1813 q^{91} -3.07227 q^{92} -12.5610 q^{93} +18.1420 q^{94} -4.83821 q^{95} -2.80170 q^{96} -9.67525 q^{97} +1.48592 q^{98} +5.77969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.42903	−1.71758	−0.858792	−	0.512324i	$-0.828785\pi$
$2$	−2.42903	−1.71758	−0.858792	+	0.512324i	$0.828785\pi$
$3$	−2.96305	−1.71072	−0.855360	−	0.518034i	$-0.826664\pi$
$3$	−2.96305	−1.71072	−0.855360	+	0.518034i	$0.826664\pi$
$4$	3.90019	1.95010
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	7.19735	2.93831
$7$	2.52750	0.955306	0.477653	−	0.878549i	$-0.341488\pi$
$7$	2.52750	0.955306	0.477653	+	0.878549i	$0.341488\pi$
$8$	−4.61563	−1.63187
$9$	5.77969	1.92656
$10$	2.42903	0.768127
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−11.5565	−3.33607
$13$	4.42384	1.22695	0.613476	−	0.789713i	$-0.289771\pi$
$13$	4.42384	1.22695	0.613476	+	0.789713i	$0.289771\pi$
$14$	−6.13938	−1.64082
$15$	2.96305	0.765058
$16$	3.41112	0.852779
$17$	−1.29895	−0.315042	−0.157521	−	0.987516i	$-0.550350\pi$
$17$	−1.29895	−0.315042	−0.157521	+	0.987516i	$0.550350\pi$
$18$	−14.0391	−3.30904
$19$	4.83821	1.10996	0.554981	−	0.831863i	$-0.312725\pi$
$19$	4.83821	1.10996	0.554981	+	0.831863i	$0.312725\pi$
$20$	−3.90019	−0.872110
$21$	−7.48913	−1.63426
$22$	−2.42903	−0.517871
$23$	−0.787723	−0.164252	−0.0821258	−	0.996622i	$-0.526171\pi$
$23$	−0.787723	−0.164252	−0.0821258	+	0.996622i	$0.526171\pi$
$24$	13.6764	2.79167
$25$	1.00000	0.200000
$26$	−10.7456	−2.10740
$27$	−8.23639	−1.58509
$28$	9.85775	1.86294
$29$	8.42044	1.56364	0.781818	−	0.623506i	$-0.214292\pi$
$29$	8.42044	1.56364	0.781818	+	0.623506i	$0.214292\pi$
$30$	−7.19735	−1.31405
$31$	4.23922	0.761387	0.380693	−	0.924701i	$-0.375685\pi$
$31$	4.23922	0.761387	0.380693	+	0.924701i	$0.375685\pi$
$32$	0.945545	0.167150
$33$	−2.96305	−0.515802
$34$	3.15520	0.541112
$35$	−2.52750	−0.427226
$36$	22.5419	3.75699
$37$	−10.4958	−1.72550	−0.862752	−	0.505628i	$-0.831261\pi$
$37$	−10.4958	−1.72550	−0.862752	+	0.505628i	$0.831261\pi$
$38$	−11.7522	−1.90645
$39$	−13.1081	−2.09897
$40$	4.61563	0.729795
$41$	−7.20804	−1.12571	−0.562853	−	0.826557i	$-0.690296\pi$
$41$	−7.20804	−1.12571	−0.562853	+	0.826557i	$0.690296\pi$
$42$	18.1913	2.80698
$43$	−8.06892	−1.23050	−0.615249	−	0.788333i	$-0.710945\pi$
$43$	−8.06892	−1.23050	−0.615249	+	0.788333i	$0.710945\pi$
$44$	3.90019	0.587976
$45$	−5.77969	−0.861586
$46$	1.91340	0.282116
$47$	−7.46880	−1.08944	−0.544718	−	0.838619i	$-0.683363\pi$
$47$	−7.46880	−1.08944	−0.544718	+	0.838619i	$0.683363\pi$
$48$	−10.1073	−1.45887
$49$	−0.611733	−0.0873904
$50$	−2.42903	−0.343517
$51$	3.84887	0.538949
$52$	17.2538	2.39268
$53$	−3.51874	−0.483336	−0.241668	−	0.970359i	$-0.577695\pi$
$53$	−3.51874	−0.483336	−0.241668	+	0.970359i	$0.577695\pi$
$54$	20.0064	2.72253
$55$	−1.00000	−0.134840
$56$	−11.6660	−1.55894
$57$	−14.3359	−1.89884
$58$	−20.4535	−2.68568
$59$	−6.78145	−0.882869	−0.441435	−	0.897293i	$-0.645530\pi$
$59$	−6.78145	−0.882869	−0.441435	+	0.897293i	$0.645530\pi$
$60$	11.5565	1.49194
$61$	−3.99500	−0.511508	−0.255754	−	0.966742i	$-0.582324\pi$
$61$	−3.99500	−0.511508	−0.255754	+	0.966742i	$0.582324\pi$
$62$	−10.2972	−1.30775
$63$	14.6082	1.84046
$64$	−9.11899	−1.13987
$65$	−4.42384	−0.548710
$66$	7.19735	0.885933
$67$	6.32760	0.773039	0.386519	−	0.922281i	$-0.373677\pi$
$67$	6.32760	0.773039	0.386519	+	0.922281i	$0.373677\pi$
$68$	−5.06616	−0.614363
$69$	2.33407	0.280989
$70$	6.13938	0.733796
$71$	−15.0441	−1.78541	−0.892703	−	0.450645i	$-0.851194\pi$
$71$	−15.0441	−1.78541	−0.892703	+	0.450645i	$0.851194\pi$
$72$	−26.6769	−3.14390
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	25.4947	2.96370
$75$	−2.96305	−0.342144
$76$	18.8700	2.16453
$77$	2.52750	0.288036
$78$	31.8399	3.60516
$79$	16.7010	1.87901	0.939505	−	0.342536i	$-0.111286\pi$
$79$	16.7010	1.87901	0.939505	+	0.342536i	$0.111286\pi$
$80$	−3.41112	−0.381374
$81$	7.06579	0.785088
$82$	17.5086	1.93350
$83$	−15.8195	−1.73641	−0.868205	−	0.496205i	$-0.834726\pi$
$83$	−15.8195	−1.73641	−0.868205	+	0.496205i	$0.834726\pi$
$84$	−29.2090	−3.18697
$85$	1.29895	0.140891
$86$	19.5997	2.11349
$87$	−24.9502	−2.67494
$88$	−4.61563	−0.492027
$89$	0.538301	0.0570597	0.0285299	−	0.999593i	$-0.490917\pi$
$89$	0.538301	0.0570597	0.0285299	+	0.999593i	$0.490917\pi$
$90$	14.0391	1.47985
$91$	11.1813	1.17212
$92$	−3.07227	−0.320307
$93$	−12.5610	−1.30252
$94$	18.1420	1.87120
$95$	−4.83821	−0.496390
$96$	−2.80170	−0.285947
$97$	−9.67525	−0.982372	−0.491186	−	0.871055i	$-0.663437\pi$
$97$	−9.67525	−0.982372	−0.491186	+	0.871055i	$0.663437\pi$
$98$	1.48592	0.150100
$99$	5.77969	0.580881
$100$	3.90019	0.390019
$101$	12.9593	1.28950	0.644751	−	0.764392i	$-0.276961\pi$
$101$	12.9593	1.28950	0.644751	+	0.764392i	$0.276961\pi$
$102$	−9.34902	−0.925691
$103$	−7.49266	−0.738274	−0.369137	−	0.929375i	$-0.620347\pi$
$103$	−7.49266	−0.738274	−0.369137	+	0.929375i	$0.620347\pi$
$104$	−20.4188	−2.00223
$105$	7.48913	0.730864
$106$	8.54713	0.830171
$107$	0.0195881	0.00189365	0.000946826	−	1.00000i	$-0.499699\pi$
$107$	0.0195881	0.00189365	0.000946826		1.00000i	$0.499699\pi$
$108$	−32.1235	−3.09109
$109$	−6.59909	−0.632078	−0.316039	−	0.948746i	$-0.602353\pi$
$109$	−6.59909	−0.632078	−0.316039	+	0.948746i	$0.602353\pi$
$110$	2.42903	0.231599
$111$	31.0997	2.95185
$112$	8.62161	0.814665
$113$	2.55733	0.240573	0.120287	−	0.992739i	$-0.461619\pi$
$113$	2.55733	0.240573	0.120287	+	0.992739i	$0.461619\pi$
$114$	34.8223	3.26141
$115$	0.787723	0.0734556
$116$	32.8413	3.04924
$117$	25.5685	2.36380
$118$	16.4723	1.51640
$119$	−3.28310	−0.300962
$120$	−13.6764	−1.24847
$121$	1.00000	0.0909091
$122$	9.70399	0.878558
$123$	21.3578	1.92577
$124$	16.5338	1.48478
$125$	−1.00000	−0.0894427
$126$	−35.4838	−3.16114
$127$	2.28970	0.203178	0.101589	−	0.994826i	$-0.467607\pi$
$127$	2.28970	0.203178	0.101589	+	0.994826i	$0.467607\pi$
$128$	20.2592	1.79068
$129$	23.9087	2.10504
$130$	10.7456	0.942456
$131$	16.6376	1.45364	0.726819	−	0.686829i	$-0.240998\pi$
$131$	16.6376	1.45364	0.726819	+	0.686829i	$0.240998\pi$
$132$	−11.5565	−1.00586
$133$	12.2286	1.06035
$134$	−15.3699	−1.32776
$135$	8.23639	0.708875
$136$	5.99548	0.514108
$137$	−12.6303	−1.07908	−0.539538	−	0.841961i	$-0.681401\pi$
$137$	−12.6303	−1.07908	−0.539538	+	0.841961i	$0.681401\pi$
$138$	−5.66952	−0.482622
$139$	0.480814	0.0407821	0.0203911	−	0.999792i	$-0.493509\pi$
$139$	0.480814	0.0407821	0.0203911	+	0.999792i	$0.493509\pi$
$140$	−9.85775	−0.833132
$141$	22.1305	1.86372
$142$	36.5426	3.06659
$143$	4.42384	0.369940
$144$	19.7152	1.64293
$145$	−8.42044	−0.699279
$146$	−2.42903	−0.201028
$147$	1.81260	0.149500
$148$	−40.9358	−3.36490
$149$	−23.6229	−1.93526	−0.967630	−	0.252372i	$-0.918789\pi$
$149$	−23.6229	−1.93526	−0.967630	+	0.252372i	$0.918789\pi$
$150$	7.19735	0.587661
$151$	14.3448	1.16736	0.583682	−	0.811983i	$-0.301612\pi$
$151$	14.3448	1.16736	0.583682	+	0.811983i	$0.301612\pi$
$152$	−22.3314	−1.81131
$153$	−7.50755	−0.606949
$154$	−6.13938	−0.494725
$155$	−4.23922	−0.340502
$156$	−51.1241	−4.09320
$157$	6.96307	0.555713	0.277857	−	0.960623i	$-0.410376\pi$
$157$	6.96307	0.555713	0.277857	+	0.960623i	$0.410376\pi$
$158$	−40.5672	−3.22736
$159$	10.4262	0.826853
$160$	−0.945545	−0.0747519
$161$	−1.99097	−0.156911
$162$	−17.1630	−1.34845
$163$	10.0230	0.785065	0.392532	−	0.919738i	$-0.371599\pi$
$163$	10.0230	0.785065	0.392532	+	0.919738i	$0.371599\pi$
$164$	−28.1127	−2.19524
$165$	2.96305	0.230674
$166$	38.4259	2.98243
$167$	−15.0425	−1.16402	−0.582012	−	0.813180i	$-0.697734\pi$
$167$	−15.0425	−1.16402	−0.582012	+	0.813180i	$0.697734\pi$
$168$	34.5670	2.66690
$169$	6.57037	0.505413
$170$	−3.15520	−0.241992
$171$	27.9634	2.13841
$172$	−31.4703	−2.39959
$173$	0.998550	0.0759184	0.0379592	−	0.999279i	$-0.487914\pi$
$173$	0.998550	0.0759184	0.0379592	+	0.999279i	$0.487914\pi$
$174$	60.6049	4.59444
$175$	2.52750	0.191061
$176$	3.41112	0.257123
$177$	20.0938	1.51034
$178$	−1.30755	−0.0980049
$179$	−24.4789	−1.82964	−0.914820	−	0.403861i	$-0.867668\pi$
$179$	−24.4789	−1.82964	−0.914820	+	0.403861i	$0.867668\pi$
$180$	−22.5419	−1.68018
$181$	−18.8990	−1.40475	−0.702376	−	0.711806i	$-0.747878\pi$
$181$	−18.8990	−1.40475	−0.702376	+	0.711806i	$0.747878\pi$
$182$	−27.1597	−2.01321
$183$	11.8374	0.875047
$184$	3.63584	0.268037
$185$	10.4958	0.771669
$186$	30.5112	2.23719
$187$	−1.29895	−0.0949888
$188$	−29.1298	−2.12451
$189$	−20.8175	−1.51425
$190$	11.7522	0.852592
$191$	−17.5878	−1.27261	−0.636305	−	0.771438i	$-0.719538\pi$
$191$	−17.5878	−1.27261	−0.636305	+	0.771438i	$0.719538\pi$
$192$	27.0201	1.95001
$193$	−0.266572	−0.0191883	−0.00959413	−	0.999954i	$-0.503054\pi$
$193$	−0.266572	−0.0191883	−0.00959413	+	0.999954i	$0.503054\pi$
$194$	23.5015	1.68731
$195$	13.1081	0.938689
$196$	−2.38587	−0.170420
$197$	4.58709	0.326817	0.163408	−	0.986558i	$-0.447751\pi$
$197$	4.58709	0.326817	0.163408	+	0.986558i	$0.447751\pi$
$198$	−14.0391	−0.997712
$199$	−1.75995	−0.124760	−0.0623799	−	0.998052i	$-0.519869\pi$
$199$	−1.75995	−0.124760	−0.0623799	+	0.998052i	$0.519869\pi$
$200$	−4.61563	−0.326374
$201$	−18.7490	−1.32245
$202$	−31.4786	−2.21483
$203$	21.2827	1.49375
$204$	15.0113	1.05100
$205$	7.20804	0.503431
$206$	18.1999	1.26805
$207$	−4.55280	−0.316442
$208$	15.0902	1.04632
$209$	4.83821	0.334666
$210$	−18.1913	−1.25532
$211$	−18.2352	−1.25536	−0.627681	−	0.778470i	$-0.715996\pi$
$211$	−18.2352	−1.25536	−0.627681	+	0.778470i	$0.715996\pi$
$212$	−13.7238	−0.942552
$213$	44.5765	3.05433
$214$	−0.0475801	−0.00325251
$215$	8.06892	0.550296
$216$	38.0161	2.58667
$217$	10.7146	0.727357
$218$	16.0294	1.08565
$219$	−2.96305	−0.200225
$220$	−3.90019	−0.262951
$221$	−5.74636	−0.386542
$222$	−75.5422	−5.07006
$223$	8.28947	0.555104	0.277552	−	0.960711i	$-0.410477\pi$
$223$	8.28947	0.555104	0.277552	+	0.960711i	$0.410477\pi$
$224$	2.38987	0.159680
$225$	5.77969	0.385313
$226$	−6.21183	−0.413205
$227$	3.84593	0.255263	0.127632	−	0.991822i	$-0.459262\pi$
$227$	3.84593	0.255263	0.127632	+	0.991822i	$0.459262\pi$
$228$	−55.9127	−3.70291
$229$	4.60150	0.304076	0.152038	−	0.988375i	$-0.451416\pi$
$229$	4.60150	0.304076	0.152038	+	0.988375i	$0.451416\pi$
$230$	−1.91340	−0.126166
$231$	−7.48913	−0.492748
$232$	−38.8656	−2.55165
$233$	23.6084	1.54664	0.773320	−	0.634016i	$-0.218595\pi$
$233$	23.6084	1.54664	0.773320	+	0.634016i	$0.218595\pi$
$234$	−62.1066	−4.06003
$235$	7.46880	0.487211
$236$	−26.4489	−1.72168
$237$	−49.4860	−3.21446
$238$	7.97476	0.516927
$239$	−1.30400	−0.0843486	−0.0421743	−	0.999110i	$-0.513428\pi$
$239$	−1.30400	−0.0843486	−0.0421743	+	0.999110i	$0.513428\pi$
$240$	10.1073	0.652425
$241$	−24.4590	−1.57554	−0.787772	−	0.615967i	$-0.788766\pi$
$241$	−24.4590	−1.57554	−0.787772	+	0.615967i	$0.788766\pi$
$242$	−2.42903	−0.156144
$243$	3.77285	0.242028
$244$	−15.5813	−0.997489
$245$	0.611733	0.0390822
$246$	−51.8788	−3.30767
$247$	21.4035	1.36187
$248$	−19.5667	−1.24248
$249$	46.8739	2.97051
$250$	2.42903	0.153625
$251$	16.6873	1.05329	0.526646	−	0.850084i	$-0.323449\pi$
$251$	16.6873	1.05329	0.526646	+	0.850084i	$0.323449\pi$
$252$	56.9748	3.58907
$253$	−0.787723	−0.0495237
$254$	−5.56176	−0.348976
$255$	−3.84887	−0.241025
$256$	−30.9723	−1.93577
$257$	18.4660	1.15188	0.575939	−	0.817493i	$-0.304636\pi$
$257$	18.4660	1.15188	0.575939	+	0.817493i	$0.304636\pi$
$258$	−58.0749	−3.61558
$259$	−26.5282	−1.64838
$260$	−17.2538	−1.07004
$261$	48.6676	3.01245
$262$	−40.4133	−2.49674
$263$	16.9007	1.04214	0.521070	−	0.853514i	$-0.325533\pi$
$263$	16.9007	1.04214	0.521070	+	0.853514i	$0.325533\pi$
$264$	13.6764	0.841722
$265$	3.51874	0.216155
$266$	−29.7036	−1.82125
$267$	−1.59501	−0.0976133
$268$	24.6788	1.50750
$269$	−23.5248	−1.43433	−0.717166	−	0.696902i	$-0.754561\pi$
$269$	−23.5248	−1.43433	−0.717166	+	0.696902i	$0.754561\pi$
$270$	−20.0064	−1.21755
$271$	24.7533	1.50365	0.751827	−	0.659361i	$-0.229173\pi$
$271$	24.7533	1.50365	0.751827	+	0.659361i	$0.229173\pi$
$272$	−4.43088	−0.268661
$273$	−33.1307	−2.00516
$274$	30.6793	1.85340
$275$	1.00000	0.0603023
$276$	9.10331	0.547955
$277$	3.45589	0.207644	0.103822	−	0.994596i	$-0.466893\pi$
$277$	3.45589	0.207644	0.103822	+	0.994596i	$0.466893\pi$
$278$	−1.16791	−0.0700467
$279$	24.5014	1.46686
$280$	11.6660	0.697177
$281$	5.41489	0.323025	0.161513	−	0.986871i	$-0.448363\pi$
$281$	5.41489	0.323025	0.161513	+	0.986871i	$0.448363\pi$
$282$	−53.7556	−3.20110
$283$	6.52679	0.387977	0.193989	−	0.981004i	$-0.437857\pi$
$283$	6.52679	0.387977	0.193989	+	0.981004i	$0.437857\pi$
$284$	−58.6749	−3.48171
$285$	14.3359	0.849185
$286$	−10.7456	−0.635404
$287$	−18.2183	−1.07539
$288$	5.46496	0.322026
$289$	−15.3127	−0.900748
$290$	20.4535	1.20107
$291$	28.6683	1.68056
$292$	3.90019	0.228242
$293$	−17.0297	−0.994884	−0.497442	−	0.867497i	$-0.665727\pi$
$293$	−17.0297	−0.994884	−0.497442	+	0.867497i	$0.665727\pi$
$294$	−4.40285	−0.256780
$295$	6.78145	0.394831
$296$	48.4448	2.81580
$297$	−8.23639	−0.477924
$298$	57.3807	3.32397
$299$	−3.48476	−0.201529
$300$	−11.5565	−0.667214
$301$	−20.3942	−1.17550
$302$	−34.8440	−2.00504
$303$	−38.3992	−2.20598
$304$	16.5037	0.946553
$305$	3.99500	0.228753
$306$	18.2361	1.04249
$307$	−29.4324	−1.67980	−0.839898	−	0.542744i	$-0.817385\pi$
$307$	−29.4324	−1.67980	−0.839898	+	0.542744i	$0.817385\pi$
$308$	9.85775	0.561697
$309$	22.2012	1.26298
$310$	10.2972	0.584842
$311$	−2.16322	−0.122665	−0.0613324	−	0.998117i	$-0.519535\pi$
$311$	−2.16322	−0.122665	−0.0613324	+	0.998117i	$0.519535\pi$
$312$	60.5020	3.42525
$313$	−17.4563	−0.986688	−0.493344	−	0.869834i	$-0.664226\pi$
$313$	−17.4563	−0.986688	−0.493344	+	0.869834i	$0.664226\pi$
$314$	−16.9135	−0.954484
$315$	−14.6082	−0.823078
$316$	65.1371	3.66425
$317$	−14.1911	−0.797053	−0.398526	−	0.917157i	$-0.630478\pi$
$317$	−14.1911	−0.797053	−0.398526	+	0.917157i	$0.630478\pi$
$318$	−25.3256	−1.42019
$319$	8.42044	0.471454
$320$	9.11899	0.509767
$321$	−0.0580406	−0.00323951
$322$	4.83613	0.269507
$323$	−6.28461	−0.349685
$324$	27.5579	1.53100
$325$	4.42384	0.245391
$326$	−24.3463	−1.34842
$327$	19.5535	1.08131
$328$	33.2696	1.83701
$329$	−18.8774	−1.04075
$330$	−7.19735	−0.396201
$331$	3.11496	0.171213	0.0856067	−	0.996329i	$-0.472717\pi$
$331$	3.11496	0.171213	0.0856067	+	0.996329i	$0.472717\pi$
$332$	−61.6989	−3.38617
$333$	−60.6627	−3.32430
$334$	36.5387	1.99931
$335$	−6.32760	−0.345714
$336$	−25.5463	−1.39366
$337$	−10.3003	−0.561092	−0.280546	−	0.959841i	$-0.590515\pi$
$337$	−10.3003	−0.561092	−0.280546	+	0.959841i	$0.590515\pi$
$338$	−15.9596	−0.868090
$339$	−7.57751	−0.411554
$340$	5.06616	0.274751
$341$	4.23922	0.229567
$342$	−67.9240	−3.67291
$343$	−19.2387	−1.03879
$344$	37.2431	2.00801
$345$	−2.33407	−0.125662
$346$	−2.42551	−0.130396
$347$	18.6135	0.999226	0.499613	−	0.866249i	$-0.333476\pi$
$347$	18.6135	0.999226	0.499613	+	0.866249i	$0.333476\pi$
$348$	−97.3107	−5.21640
$349$	−19.8653	−1.06336	−0.531682	−	0.846944i	$-0.678440\pi$
$349$	−19.8653	−1.06336	−0.531682	+	0.846944i	$0.678440\pi$
$350$	−6.13938	−0.328164
$351$	−36.4365	−1.94484
$352$	0.945545	0.0503977
$353$	−13.1765	−0.701317	−0.350658	−	0.936503i	$-0.614042\pi$
$353$	−13.1765	−0.701317	−0.350658	+	0.936503i	$0.614042\pi$
$354$	−48.8085	−2.59414
$355$	15.0441	0.798458
$356$	2.09948	0.111272
$357$	9.72802	0.514861
$358$	59.4601	3.14256
$359$	−16.4005	−0.865584	−0.432792	−	0.901494i	$-0.642472\pi$
$359$	−16.4005	−0.865584	−0.432792	+	0.901494i	$0.642472\pi$
$360$	26.6769	1.40600
$361$	4.40831	0.232016
$362$	45.9063	2.41278
$363$	−2.96305	−0.155520
$364$	43.6091	2.28574
$365$	−1.00000	−0.0523424
$366$	−28.7534	−1.50297
$367$	11.1682	0.582973	0.291486	−	0.956575i	$-0.405850\pi$
$367$	11.1682	0.582973	0.291486	+	0.956575i	$0.405850\pi$
$368$	−2.68702	−0.140070
$369$	−41.6603	−2.16875
$370$	−25.4947	−1.32541
$371$	−8.89362	−0.461734
$372$	−48.9905	−2.54004
$373$	−9.81737	−0.508324	−0.254162	−	0.967162i	$-0.581800\pi$
$373$	−9.81737	−0.508324	−0.254162	+	0.967162i	$0.581800\pi$
$374$	3.15520	0.163151
$375$	2.96305	0.153012
$376$	34.4732	1.77782
$377$	37.2507	1.91851
$378$	50.5663	2.60085
$379$	15.3923	0.790650	0.395325	−	0.918541i	$-0.370632\pi$
$379$	15.3923	0.790650	0.395325	+	0.918541i	$0.370632\pi$
$380$	−18.8700	−0.968009
$381$	−6.78451	−0.347581
$382$	42.7214	2.18582
$383$	−4.81246	−0.245905	−0.122953	−	0.992413i	$-0.539236\pi$
$383$	−4.81246	−0.245905	−0.122953	+	0.992413i	$0.539236\pi$
$384$	−60.0292	−3.06335
$385$	−2.52750	−0.128813
$386$	0.647511	0.0329575
$387$	−46.6359	−2.37064
$388$	−37.7353	−1.91572
$389$	21.0040	1.06495	0.532473	−	0.846447i	$-0.321263\pi$
$389$	21.0040	1.06495	0.532473	+	0.846447i	$0.321263\pi$
$390$	−31.8399	−1.61228
$391$	1.02322	0.0517462
$392$	2.82353	0.142610
$393$	−49.2982	−2.48677
$394$	−11.1422	−0.561336
$395$	−16.7010	−0.840318
$396$	22.5419	1.13277
$397$	−4.29194	−0.215406	−0.107703	−	0.994183i	$-0.534350\pi$
$397$	−4.29194	−0.215406	−0.107703	+	0.994183i	$0.534350\pi$
$398$	4.27498	0.214285
$399$	−36.2340	−1.81397
$400$	3.41112	0.170556
$401$	−3.37410	−0.168495	−0.0842473	−	0.996445i	$-0.526849\pi$
$401$	−3.37410	−0.168495	−0.0842473	+	0.996445i	$0.526849\pi$
$402$	45.5419	2.27143
$403$	18.7536	0.934185
$404$	50.5439	2.51465
$405$	−7.06579	−0.351102
$406$	−51.6963	−2.56564
$407$	−10.4958	−0.520259
$408$	−17.7649	−0.879495
$409$	−27.6406	−1.36674	−0.683371	−	0.730072i	$-0.739487\pi$
$409$	−27.6406	−1.36674	−0.683371	+	0.730072i	$0.739487\pi$
$410$	−17.5086	−0.864686
$411$	37.4242	1.84600
$412$	−29.2228	−1.43970
$413$	−17.1401	−0.843410
$414$	11.0589	0.543515
$415$	15.8195	0.776546
$416$	4.18294	0.205086
$417$	−1.42468	−0.0697668
$418$	−11.7522	−0.574817
$419$	−18.5407	−0.905772	−0.452886	−	0.891568i	$-0.649606\pi$
$419$	−18.5407	−0.905772	−0.452886	+	0.891568i	$0.649606\pi$
$420$	29.2090	1.42526
$421$	−5.16887	−0.251915	−0.125958	−	0.992036i	$-0.540200\pi$
$421$	−5.16887	−0.251915	−0.125958	+	0.992036i	$0.540200\pi$
$422$	44.2939	2.15619
$423$	−43.1674	−2.09887
$424$	16.2412	0.788742
$425$	−1.29895	−0.0630084
$426$	−108.278	−5.24607
$427$	−10.0974	−0.488646
$428$	0.0763973	0.00369280
$429$	−13.1081	−0.632864
$430$	−19.5997	−0.945179
$431$	−14.4425	−0.695670	−0.347835	−	0.937556i	$-0.613083\pi$
$431$	−14.4425	−0.695670	−0.347835	+	0.937556i	$0.613083\pi$
$432$	−28.0953	−1.35173
$433$	−10.0845	−0.484630	−0.242315	−	0.970198i	$-0.577907\pi$
$433$	−10.0845	−0.484630	−0.242315	+	0.970198i	$0.577907\pi$
$434$	−26.0262	−1.24930
$435$	24.9502	1.19627
$436$	−25.7377	−1.23261
$437$	−3.81117	−0.182313
$438$	7.19735	0.343903
$439$	−2.68445	−0.128122	−0.0640610	−	0.997946i	$-0.520405\pi$
$439$	−2.68445	−0.128122	−0.0640610	+	0.997946i	$0.520405\pi$
$440$	4.61563	0.220041
$441$	−3.53563	−0.168363
$442$	13.9581	0.663918
$443$	10.4729	0.497581	0.248791	−	0.968557i	$-0.419967\pi$
$443$	10.4729	0.497581	0.248791	+	0.968557i	$0.419967\pi$
$444$	121.295	5.75640
$445$	−0.538301	−0.0255179
$446$	−20.1354	−0.953438
$447$	69.9958	3.31069
$448$	−23.0483	−1.08893
$449$	−24.3515	−1.14922	−0.574608	−	0.818429i	$-0.694846\pi$
$449$	−24.3515	−1.14922	−0.574608	+	0.818429i	$0.694846\pi$
$450$	−14.0391	−0.661808
$451$	−7.20804	−0.339413
$452$	9.97408	0.469141
$453$	−42.5044	−1.99703
$454$	−9.34188	−0.438436
$455$	−11.1813	−0.524186
$456$	66.1691	3.09865
$457$	27.9060	1.30539	0.652695	−	0.757621i	$-0.273638\pi$
$457$	27.9060	1.30539	0.652695	+	0.757621i	$0.273638\pi$
$458$	−11.1772	−0.522276
$459$	10.6987	0.499371
$460$	3.07227	0.143245
$461$	26.9752	1.25636	0.628181	−	0.778067i	$-0.283800\pi$
$461$	26.9752	1.25636	0.628181	+	0.778067i	$0.283800\pi$
$462$	18.1913	0.846337
$463$	−35.8346	−1.66538	−0.832688	−	0.553742i	$-0.813199\pi$
$463$	−35.8346	−1.66538	−0.832688	+	0.553742i	$0.813199\pi$
$464$	28.7231	1.33344
$465$	12.5610	0.582505
$466$	−57.3456	−2.65648
$467$	36.8535	1.70538	0.852688	−	0.522420i	$-0.174971\pi$
$467$	36.8535	1.70538	0.852688	+	0.522420i	$0.174971\pi$
$468$	99.7219	4.60965
$469$	15.9930	0.738489
$470$	−18.1420	−0.836826
$471$	−20.6320	−0.950670
$472$	31.3006	1.44073
$473$	−8.06892	−0.371009
$474$	120.203	5.52111
$475$	4.83821	0.221992
$476$	−12.8047	−0.586904
$477$	−20.3372	−0.931179
$478$	3.16745	0.144876
$479$	−24.9062	−1.13799	−0.568996	−	0.822340i	$-0.692668\pi$
$479$	−24.9062	−1.13799	−0.568996	+	0.822340i	$0.692668\pi$
$480$	2.80170	0.127880
$481$	−46.4319	−2.11711
$482$	59.4117	2.70613
$483$	5.89936	0.268430
$484$	3.90019	0.177281
$485$	9.67525	0.439330
$486$	−9.16437	−0.415704
$487$	−8.72806	−0.395506	−0.197753	−	0.980252i	$-0.563364\pi$
$487$	−8.72806	−0.395506	−0.197753	+	0.980252i	$0.563364\pi$
$488$	18.4394	0.834714
$489$	−29.6988	−1.34303
$490$	−1.48592	−0.0671269
$491$	0.958751	0.0432678	0.0216339	−	0.999766i	$-0.493113\pi$
$491$	0.958751	0.0432678	0.0216339	+	0.999766i	$0.493113\pi$
$492$	83.2996	3.75544
$493$	−10.9377	−0.492611
$494$	−51.9897	−2.33913
$495$	−5.77969	−0.259778
$496$	14.4605	0.649295
$497$	−38.0240	−1.70561
$498$	−113.858	−5.10211
$499$	9.39150	0.420421	0.210211	−	0.977656i	$-0.432585\pi$
$499$	9.39150	0.420421	0.210211	+	0.977656i	$0.432585\pi$
$500$	−3.90019	−0.174422
$501$	44.5718	1.99132
$502$	−40.5340	−1.80912
$503$	−20.3215	−0.906089	−0.453045	−	0.891488i	$-0.649662\pi$
$503$	−20.3215	−0.906089	−0.453045	+	0.891488i	$0.649662\pi$
$504$	−67.4260	−3.00339
$505$	−12.9593	−0.576683
$506$	1.91340	0.0850612
$507$	−19.4684	−0.864621
$508$	8.93028	0.396217
$509$	27.8843	1.23595	0.617975	−	0.786198i	$-0.287953\pi$
$509$	27.8843	1.23595	0.617975	+	0.786198i	$0.287953\pi$
$510$	9.34902	0.413981
$511$	2.52750	0.111810
$512$	34.7142	1.53417
$513$	−39.8494	−1.75939
$514$	−44.8545	−1.97845
$515$	7.49266	0.330166
$516$	93.2483	4.10503
$517$	−7.46880	−0.328478
$518$	64.4379	2.83124
$519$	−2.95876	−0.129875
$520$	20.4188	0.895424
$521$	−38.7582	−1.69803	−0.849013	−	0.528372i	$-0.822803\pi$
$521$	−38.7582	−1.69803	−0.849013	+	0.528372i	$0.822803\pi$
$522$	−118.215	−5.17413
$523$	5.71169	0.249755	0.124877	−	0.992172i	$-0.460146\pi$
$523$	5.71169	0.249755	0.124877	+	0.992172i	$0.460146\pi$
$524$	64.8900	2.83473
$525$	−7.48913	−0.326852
$526$	−41.0523	−1.78996
$527$	−5.50655	−0.239869
$528$	−10.1073	−0.439865
$529$	−22.3795	−0.973021
$530$	−8.54713	−0.371264
$531$	−39.1947	−1.70090
$532$	47.6939	2.06779
$533$	−31.8872	−1.38119
$534$	3.87434	0.167659
$535$	−0.0195881	−0.000846867 0
$536$	−29.2058	−1.26150
$537$	72.5324	3.13000
$538$	57.1425	2.46359
$539$	−0.611733	−0.0263492
$540$	32.1235	1.38238
$541$	40.4412	1.73870	0.869352	−	0.494193i	$-0.164536\pi$
$541$	40.4412	1.73870	0.869352	+	0.494193i	$0.164536\pi$
$542$	−60.1264	−2.58265
$543$	55.9988	2.40314
$544$	−1.22822	−0.0526594
$545$	6.59909	0.282674
$546$	80.4755	3.44404
$547$	−33.2110	−1.42000	−0.709999	−	0.704202i	$-0.751305\pi$
$547$	−33.2110	−1.42000	−0.709999	+	0.704202i	$0.751305\pi$
$548$	−49.2605	−2.10430
$549$	−23.0899	−0.985453
$550$	−2.42903	−0.103574
$551$	40.7399	1.73558
$552$	−10.7732	−0.458537
$553$	42.2118	1.79503
$554$	−8.39446	−0.356646
$555$	−31.0997	−1.32011
$556$	1.87527	0.0795290
$557$	18.3553	0.777737	0.388869	−	0.921293i	$-0.372866\pi$
$557$	18.3553	0.777737	0.388869	+	0.921293i	$0.372866\pi$
$558$	−59.5147	−2.51946
$559$	−35.6956	−1.50976
$560$	−8.62161	−0.364329
$561$	3.84887	0.162499
$562$	−13.1529	−0.554823
$563$	22.6342	0.953918	0.476959	−	0.878926i	$-0.341739\pi$
$563$	22.6342	0.953918	0.476959	+	0.878926i	$0.341739\pi$
$564$	86.3131	3.63444
$565$	−2.55733	−0.107588
$566$	−15.8538	−0.666384
$567$	17.8588	0.749999
$568$	69.4380	2.91355
$569$	30.4819	1.27787	0.638934	−	0.769261i	$-0.279376\pi$
$569$	30.4819	1.27787	0.638934	+	0.769261i	$0.279376\pi$
$570$	−34.8223	−1.45855
$571$	45.3968	1.89980	0.949899	−	0.312556i	$-0.101185\pi$
$571$	45.3968	1.89980	0.949899	+	0.312556i	$0.101185\pi$
$572$	17.2538	0.721419
$573$	52.1137	2.17708
$574$	44.2529	1.84708
$575$	−0.787723	−0.0328503
$576$	−52.7050	−2.19604
$577$	−3.15396	−0.131301	−0.0656505	−	0.997843i	$-0.520912\pi$
$577$	−3.15396	−0.131301	−0.0656505	+	0.997843i	$0.520912\pi$
$578$	37.1951	1.54711
$579$	0.789867	0.0328258
$580$	−32.8413	−1.36366
$581$	−39.9837	−1.65880
$582$	−69.6362	−2.88651
$583$	−3.51874	−0.145731
$584$	−4.61563	−0.190996
$585$	−25.5685	−1.05713
$586$	41.3656	1.70880
$587$	15.7542	0.650246	0.325123	−	0.945672i	$-0.394594\pi$
$587$	15.7542	0.650246	0.325123	+	0.945672i	$0.394594\pi$
$588$	7.06948	0.291540
$589$	20.5103	0.845110
$590$	−16.4723	−0.678156
$591$	−13.5918	−0.559092
$592$	−35.8025	−1.47147
$593$	−3.39836	−0.139554	−0.0697769	−	0.997563i	$-0.522229\pi$
$593$	−3.39836	−0.139554	−0.0697769	+	0.997563i	$0.522229\pi$
$594$	20.0064	0.820874
$595$	3.28310	0.134594
$596$	−92.1337	−3.77394
$597$	5.21484	0.213429
$598$	8.46460	0.346143
$599$	6.92361	0.282891	0.141446	−	0.989946i	$-0.454825\pi$
$599$	6.92361	0.282891	0.141446	+	0.989946i	$0.454825\pi$
$600$	13.6764	0.558335
$601$	−33.1011	−1.35022	−0.675111	−	0.737717i	$-0.735904\pi$
$601$	−33.1011	−1.35022	−0.675111	+	0.737717i	$0.735904\pi$
$602$	49.5382	2.01903
$603$	36.5716	1.48931
$604$	55.9475	2.27647
$605$	−1.00000	−0.0406558
$606$	93.2730	3.78895
$607$	25.4521	1.03307	0.516535	−	0.856266i	$-0.327222\pi$
$607$	25.4521	1.03307	0.516535	+	0.856266i	$0.327222\pi$
$608$	4.57475	0.185531
$609$	−63.0618	−2.55539
$610$	−9.70399	−0.392903
$611$	−33.0408	−1.33669
$612$	−29.2809	−1.18361
$613$	−14.9066	−0.602072	−0.301036	−	0.953613i	$-0.597332\pi$
$613$	−14.9066	−0.602072	−0.301036	+	0.953613i	$0.597332\pi$
$614$	71.4923	2.88519
$615$	−21.3578	−0.861230
$616$	−11.6660	−0.470037
$617$	36.1099	1.45373	0.726865	−	0.686781i	$-0.240977\pi$
$617$	36.1099	1.45373	0.726865	+	0.686781i	$0.240977\pi$
$618$	−53.9273	−2.16928
$619$	−18.9168	−0.760329	−0.380164	−	0.924919i	$-0.624133\pi$
$619$	−18.9168	−0.760329	−0.380164	+	0.924919i	$0.624133\pi$
$620$	−16.5338	−0.664013
$621$	6.48800	0.260354
$622$	5.25452	0.210687
$623$	1.36056	0.0545095
$624$	−44.7132	−1.78996
$625$	1.00000	0.0400000
$626$	42.4019	1.69472
$627$	−14.3359	−0.572520
$628$	27.1573	1.08369
$629$	13.6336	0.543606
$630$	35.4838	1.41371
$631$	13.5236	0.538364	0.269182	−	0.963089i	$-0.413247\pi$
$631$	13.5236	0.538364	0.269182	+	0.963089i	$0.413247\pi$
$632$	−77.0856	−3.06630
$633$	54.0319	2.14758
$634$	34.4707	1.36901
$635$	−2.28970	−0.0908640
$636$	40.6643	1.61244
$637$	−2.70621	−0.107224
$638$	−20.4535	−0.809762
$639$	−86.9503	−3.43970
$640$	−20.2592	−0.800816
$641$	0.496219	0.0195995	0.00979974	−	0.999952i	$-0.496881\pi$
$641$	0.496219	0.0195995	0.00979974	+	0.999952i	$0.496881\pi$
$642$	0.140982	0.00556413
$643$	38.5264	1.51933	0.759666	−	0.650314i	$-0.225362\pi$
$643$	38.5264	1.51933	0.759666	+	0.650314i	$0.225362\pi$
$644$	−7.76518	−0.305991
$645$	−23.9087	−0.941402
$646$	15.2655	0.600613
$647$	−20.7676	−0.816457	−0.408229	−	0.912880i	$-0.633853\pi$
$647$	−20.7676	−0.816457	−0.408229	+	0.912880i	$0.633853\pi$
$648$	−32.6130	−1.28116
$649$	−6.78145	−0.266195
$650$	−10.7456	−0.421479
$651$	−31.7481	−1.24430
$652$	39.0918	1.53095
$653$	−21.8569	−0.855328	−0.427664	−	0.903938i	$-0.640663\pi$
$653$	−21.8569	−0.855328	−0.427664	+	0.903938i	$0.640663\pi$
$654$	−47.4960	−1.85724
$655$	−16.6376	−0.650086
$656$	−24.5875	−0.959979
$657$	5.77969	0.225487
$658$	45.8538	1.78757
$659$	39.2345	1.52836	0.764179	−	0.645004i	$-0.223144\pi$
$659$	39.2345	1.52836	0.764179	+	0.645004i	$0.223144\pi$
$660$	11.5565	0.449836
$661$	24.6131	0.957338	0.478669	−	0.877995i	$-0.341119\pi$
$661$	24.6131	0.957338	0.478669	+	0.877995i	$0.341119\pi$
$662$	−7.56632	−0.294074
$663$	17.0268	0.661265
$664$	73.0167	2.83360
$665$	−12.2286	−0.474205
$666$	147.352	5.70976
$667$	−6.63298	−0.256830
$668$	−58.6686	−2.26996
$669$	−24.5622	−0.949628
$670$	15.3699	0.593792
$671$	−3.99500	−0.154225
$672$	−7.08131	−0.273167
$673$	−22.5771	−0.870281	−0.435141	−	0.900362i	$-0.643301\pi$
$673$	−22.5771	−0.870281	−0.435141	+	0.900362i	$0.643301\pi$
$674$	25.0197	0.963722
$675$	−8.23639	−0.317019
$676$	25.6257	0.985604
$677$	11.6231	0.446713	0.223357	−	0.974737i	$-0.428299\pi$
$677$	11.6231	0.446713	0.223357	+	0.974737i	$0.428299\pi$
$678$	18.4060	0.706878
$679$	−24.4542	−0.938466
$680$	−5.99548	−0.229916
$681$	−11.3957	−0.436684
$682$	−10.2972	−0.394300
$683$	38.9033	1.48859	0.744296	−	0.667850i	$-0.232785\pi$
$683$	38.9033	1.48859	0.744296	+	0.667850i	$0.232785\pi$
$684$	109.063	4.17011
$685$	12.6303	0.482578
$686$	46.7313	1.78421
$687$	−13.6345	−0.520189
$688$	−27.5240	−1.04934
$689$	−15.5663	−0.593031
$690$	5.66952	0.215835
$691$	35.8188	1.36261	0.681305	−	0.731999i	$-0.261413\pi$
$691$	35.8188	1.36261	0.681305	+	0.731999i	$0.261413\pi$
$692$	3.89454	0.148048
$693$	14.6082	0.554919
$694$	−45.2128	−1.71625
$695$	−0.480814	−0.0182383
$696$	115.161	4.36516
$697$	9.36290	0.354645
$698$	48.2534	1.82642
$699$	−69.9531	−2.64587
$700$	9.85775	0.372588
$701$	34.1798	1.29095	0.645476	−	0.763780i	$-0.276659\pi$
$701$	34.1798	1.29095	0.645476	+	0.763780i	$0.276659\pi$
$702$	88.5053	3.34042
$703$	−50.7811	−1.91524
$704$	−9.11899	−0.343685
$705$	−22.1305	−0.833482
$706$	32.0062	1.20457
$707$	32.7548	1.23187
$708$	78.3697	2.94531
$709$	15.9871	0.600407	0.300204	−	0.953875i	$-0.402945\pi$
$709$	15.9871	0.600407	0.300204	+	0.953875i	$0.402945\pi$
$710$	−36.5426	−1.37142
$711$	96.5267	3.62003
$712$	−2.48459	−0.0931141
$713$	−3.33933	−0.125059
$714$	−23.6297	−0.884318
$715$	−4.42384	−0.165442
$716$	−95.4725	−3.56798
$717$	3.86382	0.144297
$718$	39.8373	1.48671
$719$	−52.8611	−1.97139	−0.985694	−	0.168545i	$-0.946093\pi$
$719$	−52.8611	−1.97139	−0.985694	+	0.168545i	$0.946093\pi$
$720$	−19.7152	−0.734743
$721$	−18.9377	−0.705277
$722$	−10.7079	−0.398507
$723$	72.4734	2.69532
$724$	−73.7098	−2.73940
$725$	8.42044	0.312727
$726$	7.19735	0.267119
$727$	4.66791	0.173123	0.0865617	−	0.996246i	$-0.472412\pi$
$727$	4.66791	0.173123	0.0865617	+	0.996246i	$0.472412\pi$
$728$	−51.6086	−1.91274
$729$	−32.3765	−1.19913
$730$	2.42903	0.0899025
$731$	10.4811	0.387659
$732$	46.1682	1.70643
$733$	−41.3602	−1.52768	−0.763838	−	0.645408i	$-0.776687\pi$
$733$	−41.3602	−1.52768	−0.763838	+	0.645408i	$0.776687\pi$
$734$	−27.1278	−1.00131
$735$	−1.81260	−0.0668586
$736$	−0.744828	−0.0274547
$737$	6.32760	0.233080
$738$	101.194	3.72501
$739$	−11.0741	−0.407367	−0.203684	−	0.979037i	$-0.565291\pi$
$739$	−11.0741	−0.407367	−0.203684	+	0.979037i	$0.565291\pi$
$740$	40.9358	1.50483
$741$	−63.4197	−2.32978
$742$	21.6029	0.793067
$743$	−38.9928	−1.43051	−0.715253	−	0.698866i	$-0.753689\pi$
$743$	−38.9928	−1.43051	−0.715253	+	0.698866i	$0.753689\pi$
$744$	57.9771	2.12554
$745$	23.6229	0.865475
$746$	23.8467	0.873090
$747$	−91.4316	−3.34531
$748$	−5.06616	−0.185237
$749$	0.0495089	0.00180902
$750$	−7.19735	−0.262810
$751$	34.4197	1.25599	0.627997	−	0.778216i	$-0.283875\pi$
$751$	34.4197	1.25599	0.627997	+	0.778216i	$0.283875\pi$
$752$	−25.4770	−0.929049
$753$	−49.4454	−1.80189
$754$	−90.4831	−3.29520
$755$	−14.3448	−0.522061
$756$	−81.1922	−2.95293
$757$	11.9541	0.434478	0.217239	−	0.976118i	$-0.430295\pi$
$757$	11.9541	0.434478	0.217239	+	0.976118i	$0.430295\pi$
$758$	−37.3884	−1.35801
$759$	2.33407	0.0847213
$760$	22.3314	0.810045
$761$	−51.3219	−1.86042	−0.930208	−	0.367033i	$-0.880374\pi$
$761$	−51.3219	−1.86042	−0.930208	+	0.367033i	$0.880374\pi$
$762$	16.4798	0.597000
$763$	−16.6792	−0.603828
$764$	−68.5959	−2.48171
$765$	7.50755	0.271436
$766$	11.6896	0.422363
$767$	−30.0000	−1.08324
$768$	91.7726	3.31156
$769$	−39.1684	−1.41245	−0.706224	−	0.707989i	$-0.749603\pi$
$769$	−39.1684	−1.41245	−0.706224	+	0.707989i	$0.749603\pi$
$770$	6.13938	0.221248
$771$	−54.7158	−1.97054
$772$	−1.03968	−0.0374190
$773$	20.5994	0.740910	0.370455	−	0.928850i	$-0.379202\pi$
$773$	20.5994	0.740910	0.370455	+	0.928850i	$0.379202\pi$
$774$	113.280	4.07177
$775$	4.23922	0.152277
$776$	44.6573	1.60310
$777$	78.6046	2.81992
$778$	−51.0195	−1.82914
$779$	−34.8740	−1.24949
$780$	51.1241	1.83053
$781$	−15.0441	−0.538320
$782$	−2.48542	−0.0888785
$783$	−69.3540	−2.47851
$784$	−2.08669	−0.0745247
$785$	−6.96307	−0.248523
$786$	119.747	4.27123
$787$	46.4208	1.65472	0.827361	−	0.561670i	$-0.189841\pi$
$787$	46.4208	1.65472	0.827361	+	0.561670i	$0.189841\pi$
$788$	17.8906	0.637325
$789$	−50.0776	−1.78281
$790$	40.5672	1.44332
$791$	6.46366	0.229821
$792$	−26.6769	−0.947923
$793$	−17.6733	−0.627596
$794$	10.4253	0.369979
$795$	−10.4262	−0.369780
$796$	−6.86415	−0.243293
$797$	15.9114	0.563611	0.281805	−	0.959472i	$-0.409067\pi$
$797$	15.9114	0.563611	0.281805	+	0.959472i	$0.409067\pi$
$798$	88.0135	3.11564
$799$	9.70162	0.343219
$800$	0.945545	0.0334301
$801$	3.11121	0.109929
$802$	8.19580	0.289404
$803$	1.00000	0.0352892
$804$	−73.1248	−2.57891
$805$	1.99097	0.0701726
$806$	−45.5532	−1.60454
$807$	69.7053	2.45374
$808$	−59.8155	−2.10430
$809$	48.2416	1.69608	0.848042	−	0.529929i	$-0.177781\pi$
$809$	48.2416	1.69608	0.848042	+	0.529929i	$0.177781\pi$
$810$	17.1630	0.603047
$811$	3.81635	0.134010	0.0670051	−	0.997753i	$-0.478656\pi$
$811$	3.81635	0.134010	0.0670051	+	0.997753i	$0.478656\pi$
$812$	83.0065	2.91296
$813$	−73.3453	−2.57233
$814$	25.4947	0.893589
$815$	−10.0230	−0.351092
$816$	13.1289	0.459605
$817$	−39.0392	−1.36581
$818$	67.1400	2.34749
$819$	64.6243	2.25816
$820$	28.1127	0.981740
$821$	−37.4503	−1.30702	−0.653512	−	0.756916i	$-0.726705\pi$
$821$	−37.4503	−1.30702	−0.653512	+	0.756916i	$0.726705\pi$
$822$	−90.9045	−3.17066
$823$	−3.95188	−0.137754	−0.0688769	−	0.997625i	$-0.521942\pi$
$823$	−3.95188	−0.137754	−0.0688769	+	0.997625i	$0.521942\pi$
$824$	34.5833	1.20477
$825$	−2.96305	−0.103160
$826$	41.6339	1.44863
$827$	−11.3982	−0.396356	−0.198178	−	0.980166i	$-0.563502\pi$
$827$	−11.3982	−0.396356	−0.198178	+	0.980166i	$0.563502\pi$
$828$	−17.7568	−0.617091
$829$	−46.7686	−1.62434	−0.812170	−	0.583421i	$-0.801714\pi$
$829$	−46.7686	−1.62434	−0.812170	+	0.583421i	$0.801714\pi$
$830$	−38.4259	−1.33378
$831$	−10.2400	−0.355221
$832$	−40.3410	−1.39857
$833$	0.794611	0.0275316
$834$	3.46059	0.119830
$835$	15.0425	0.520567
$836$	18.8700	0.652631
$837$	−34.9159	−1.20687
$838$	45.0359	1.55574
$839$	26.9537	0.930545	0.465273	−	0.885167i	$-0.345956\pi$
$839$	26.9537	0.930545	0.465273	+	0.885167i	$0.345956\pi$
$840$	−34.5670	−1.19268
$841$	41.9038	1.44496
$842$	12.5553	0.432686
$843$	−16.0446	−0.552606
$844$	−71.1208	−2.44808
$845$	−6.57037	−0.226028
$846$	104.855	3.60499
$847$	2.52750	0.0868460
$848$	−12.0028	−0.412179
$849$	−19.3392	−0.663721
$850$	3.15520	0.108222
$851$	8.26781	0.283417
$852$	173.857	5.95624
$853$	11.7800	0.403339	0.201670	−	0.979454i	$-0.435363\pi$
$853$	11.7800	0.403339	0.201670	+	0.979454i	$0.435363\pi$
$854$	24.5268	0.839291
$855$	−27.9634	−0.956328
$856$	−0.0904113	−0.00309020
$857$	−36.1281	−1.23411	−0.617056	−	0.786919i	$-0.711675\pi$
$857$	−36.1281	−1.23411	−0.617056	+	0.786919i	$0.711675\pi$
$858$	31.8399	1.08700
$859$	−13.8933	−0.474034	−0.237017	−	0.971505i	$-0.576170\pi$
$859$	−13.8933	−0.474034	−0.237017	+	0.971505i	$0.576170\pi$
$860$	31.4703	1.07313
$861$	53.9819	1.83970
$862$	35.0812	1.19487
$863$	23.8207	0.810867	0.405433	−	0.914125i	$-0.367121\pi$
$863$	23.8207	0.810867	0.405433	+	0.914125i	$0.367121\pi$
$864$	−7.78787	−0.264949
$865$	−0.998550	−0.0339517
$866$	24.4956	0.832393
$867$	45.3724	1.54093
$868$	41.7892	1.41842
$869$	16.7010	0.566543
$870$	−60.6049	−2.05470
$871$	27.9923	0.948482
$872$	30.4589	1.03147
$873$	−55.9200	−1.89260
$874$	9.25746	0.313138
$875$	−2.52750	−0.0854452
$876$	−11.5565	−0.390457
$877$	7.77061	0.262395	0.131197	−	0.991356i	$-0.458118\pi$
$877$	7.77061	0.262395	0.131197	+	0.991356i	$0.458118\pi$
$878$	6.52062	0.220060
$879$	50.4599	1.70197
$880$	−3.41112	−0.114989
$881$	−39.5096	−1.33111	−0.665556	−	0.746348i	$-0.731805\pi$
$881$	−39.5096	−1.33111	−0.665556	+	0.746348i	$0.731805\pi$
$882$	8.58815	0.289178
$883$	55.8971	1.88109	0.940544	−	0.339672i	$-0.110316\pi$
$883$	55.8971	1.88109	0.940544	+	0.339672i	$0.110316\pi$
$884$	−22.4119	−0.753794
$885$	−20.0938	−0.675446
$886$	−25.4389	−0.854638
$887$	−22.4863	−0.755014	−0.377507	−	0.926007i	$-0.623219\pi$
$887$	−22.4863	−0.755014	−0.377507	+	0.926007i	$0.623219\pi$
$888$	−143.545	−4.81705
$889$	5.78722	0.194097
$890$	1.30755	0.0438291
$891$	7.06579	0.236713
$892$	32.3305	1.08251
$893$	−36.1357	−1.20923
$894$	−170.022	−5.68639
$895$	24.4789	0.818240
$896$	51.2052	1.71065
$897$	10.3255	0.344760
$898$	59.1505	1.97388
$899$	35.6961	1.19053
$900$	22.5419	0.751397
$901$	4.57068	0.152271
$902$	17.5086	0.582971
$903$	60.4292	2.01096
$904$	−11.8037	−0.392585
$905$	18.8990	0.628224
$906$	103.245	3.43007
$907$	−49.3097	−1.63730	−0.818651	−	0.574291i	$-0.805278\pi$
$907$	−49.3097	−1.63730	−0.818651	+	0.574291i	$0.805278\pi$
$908$	14.9999	0.497788
$909$	74.9010	2.48431
$910$	27.1597	0.900334
$911$	−2.55305	−0.0845864	−0.0422932	−	0.999105i	$-0.513466\pi$
$911$	−2.55305	−0.0845864	−0.0422932	+	0.999105i	$0.513466\pi$
$912$	−48.9014	−1.61929
$913$	−15.8195	−0.523548
$914$	−67.7846	−2.24212
$915$	−11.8374	−0.391333
$916$	17.9467	0.592977
$917$	42.0517	1.38867
$918$	−25.9874	−0.857713
$919$	−8.91125	−0.293955	−0.146977	−	0.989140i	$-0.546954\pi$
$919$	−8.91125	−0.293955	−0.146977	+	0.989140i	$0.546954\pi$
$920$	−3.63584	−0.119870
$921$	87.2099	2.87366
$922$	−65.5237	−2.15791
$923$	−66.5527	−2.19061
$924$	−29.2090	−0.960907
$925$	−10.4958	−0.345101
$926$	87.0434	2.86042
$927$	−43.3053	−1.42233
$928$	7.96190	0.261362
$929$	−31.1733	−1.02276	−0.511382	−	0.859353i	$-0.670866\pi$
$929$	−31.1733	−1.02276	−0.511382	+	0.859353i	$0.670866\pi$
$930$	−30.5112	−1.00050
$931$	−2.95969	−0.0970000
$932$	92.0774	3.01610
$933$	6.40973	0.209845
$934$	−89.5183	−2.92913
$935$	1.29895	0.0424803
$936$	−118.014	−3.85742
$937$	6.99467	0.228506	0.114253	−	0.993452i	$-0.463553\pi$
$937$	6.99467	0.228506	0.114253	+	0.993452i	$0.463553\pi$
$938$	−38.8475	−1.26842
$939$	51.7239	1.68795
$940$	29.1298	0.950108
$941$	−45.7053	−1.48995	−0.744975	−	0.667092i	$-0.767539\pi$
$941$	−45.7053	−1.48995	−0.744975	+	0.667092i	$0.767539\pi$
$942$	50.1157	1.63286
$943$	5.67794	0.184899
$944$	−23.1323	−0.752892
$945$	20.8175	0.677193
$946$	19.5997	0.637240
$947$	−3.99247	−0.129738	−0.0648689	−	0.997894i	$-0.520663\pi$
$947$	−3.99247	−0.129738	−0.0648689	+	0.997894i	$0.520663\pi$
$948$	−193.005	−6.26851
$949$	4.42384	0.143604
$950$	−11.7522	−0.381291
$951$	42.0491	1.36353
$952$	15.1536	0.491131
$953$	−61.0142	−1.97644	−0.988222	−	0.153025i	$-0.951099\pi$
$953$	−61.0142	−1.97644	−0.988222	+	0.153025i	$0.951099\pi$
$954$	49.3998	1.59938
$955$	17.5878	0.569129
$956$	−5.08584	−0.164488
$957$	−24.9502	−0.806526
$958$	60.4979	1.95460
$959$	−31.9230	−1.03085
$960$	−27.0201	−0.872069
$961$	−13.0290	−0.420290
$962$	112.784	3.63632
$963$	0.113213	0.00364824
$964$	−95.3949	−3.07246
$965$	0.266572	0.00858125
$966$	−14.3297	−0.461052
$967$	20.8815	0.671504	0.335752	−	0.941950i	$-0.391010\pi$
$967$	20.8815	0.671504	0.335752	+	0.941950i	$0.391010\pi$
$968$	−4.61563	−0.148352
$969$	18.6216	0.598213
$970$	−23.5015	−0.754587
$971$	36.2144	1.16217	0.581087	−	0.813841i	$-0.302627\pi$
$971$	36.2144	1.16217	0.581087	+	0.813841i	$0.302627\pi$
$972$	14.7148	0.471978
$973$	1.21526	0.0389594
$974$	21.2007	0.679315
$975$	−13.1081	−0.419795
$976$	−13.6274	−0.436203
$977$	−3.29813	−0.105516	−0.0527582	−	0.998607i	$-0.516801\pi$
$977$	−3.29813	−0.105516	−0.0527582	+	0.998607i	$0.516801\pi$
$978$	72.1393	2.30676
$979$	0.538301	0.0172042
$980$	2.38587	0.0762140
$981$	−38.1407	−1.21774
$982$	−2.32884	−0.0743161
$983$	60.9957	1.94546	0.972730	−	0.231940i	$-0.0745072\pi$
$983$	60.9957	1.94546	0.972730	+	0.231940i	$0.0745072\pi$
$984$	−98.5797	−3.14261
$985$	−4.58709	−0.146157
$986$	26.5681	0.846102
$987$	55.9348	1.78043
$988$	83.4777	2.65578
$989$	6.35608	0.202111
$990$	14.0391	0.446191
$991$	10.5301	0.334500	0.167250	−	0.985915i	$-0.446511\pi$
$991$	10.5301	0.334500	0.167250	+	0.985915i	$0.446511\pi$
$992$	4.00837	0.127266
$993$	−9.22978	−0.292898
$994$	92.3615	2.92953
$995$	1.75995	0.0557942
$996$	182.817	5.79279
$997$	20.5529	0.650918	0.325459	−	0.945556i	$-0.394481\pi$
$997$	20.5529	0.650918	0.325459	+	0.945556i	$0.394481\pi$
$998$	−22.8122	−0.722109
$999$	86.4477	2.73509

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.f.1.5	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.f.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.42903 q^{2} -2.96305 q^{3} +3.90019 q^{4} -1.00000 q^{5} +7.19735 q^{6} +2.52750 q^{7} -4.61563 q^{8} +5.77969 q^{9} +O(q^{10})\)	\(q-2.42903 q^{2} -2.96305 q^{3} +3.90019 q^{4} -1.00000 q^{5} +7.19735 q^{6} +2.52750 q^{7} -4.61563 q^{8} +5.77969 q^{9} +2.42903 q^{10} +1.00000 q^{11} -11.5565 q^{12} +4.42384 q^{13} -6.13938 q^{14} +2.96305 q^{15} +3.41112 q^{16} -1.29895 q^{17} -14.0391 q^{18} +4.83821 q^{19} -3.90019 q^{20} -7.48913 q^{21} -2.42903 q^{22} -0.787723 q^{23} +13.6764 q^{24} +1.00000 q^{25} -10.7456 q^{26} -8.23639 q^{27} +9.85775 q^{28} +8.42044 q^{29} -7.19735 q^{30} +4.23922 q^{31} +0.945545 q^{32} -2.96305 q^{33} +3.15520 q^{34} -2.52750 q^{35} +22.5419 q^{36} -10.4958 q^{37} -11.7522 q^{38} -13.1081 q^{39} +4.61563 q^{40} -7.20804 q^{41} +18.1913 q^{42} -8.06892 q^{43} +3.90019 q^{44} -5.77969 q^{45} +1.91340 q^{46} -7.46880 q^{47} -10.1073 q^{48} -0.611733 q^{49} -2.42903 q^{50} +3.84887 q^{51} +17.2538 q^{52} -3.51874 q^{53} +20.0064 q^{54} -1.00000 q^{55} -11.6660 q^{56} -14.3359 q^{57} -20.4535 q^{58} -6.78145 q^{59} +11.5565 q^{60} -3.99500 q^{61} -10.2972 q^{62} +14.6082 q^{63} -9.11899 q^{64} -4.42384 q^{65} +7.19735 q^{66} +6.32760 q^{67} -5.06616 q^{68} +2.33407 q^{69} +6.13938 q^{70} -15.0441 q^{71} -26.6769 q^{72} +1.00000 q^{73} +25.4947 q^{74} -2.96305 q^{75} +18.8700 q^{76} +2.52750 q^{77} +31.8399 q^{78} +16.7010 q^{79} -3.41112 q^{80} +7.06579 q^{81} +17.5086 q^{82} -15.8195 q^{83} -29.2090 q^{84} +1.29895 q^{85} +19.5997 q^{86} -24.9502 q^{87} -4.61563 q^{88} +0.538301 q^{89} +14.0391 q^{90} +11.1813 q^{91} -3.07227 q^{92} -12.5610 q^{93} +18.1420 q^{94} -4.83821 q^{95} -2.80170 q^{96} -9.67525 q^{97} +1.48592 q^{98} +5.77969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) \) 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