LMFDB - Embedded newform 8038.2.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.39141 q^{3} +1.00000 q^{4} +3.31334 q^{5} -2.39141 q^{6} -3.74056 q^{7} +1.00000 q^{8} +2.71882 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.39141 q^{3} +1.00000 q^{4} +3.31334 q^{5} -2.39141 q^{6} -3.74056 q^{7} +1.00000 q^{8} +2.71882 q^{9} +3.31334 q^{10} -1.37696 q^{11} -2.39141 q^{12} -4.99110 q^{13} -3.74056 q^{14} -7.92355 q^{15} +1.00000 q^{16} -7.01355 q^{17} +2.71882 q^{18} +2.32703 q^{19} +3.31334 q^{20} +8.94519 q^{21} -1.37696 q^{22} +4.11728 q^{23} -2.39141 q^{24} +5.97824 q^{25} -4.99110 q^{26} +0.672409 q^{27} -3.74056 q^{28} -6.52552 q^{29} -7.92355 q^{30} +0.609271 q^{31} +1.00000 q^{32} +3.29287 q^{33} -7.01355 q^{34} -12.3937 q^{35} +2.71882 q^{36} +0.843333 q^{37} +2.32703 q^{38} +11.9357 q^{39} +3.31334 q^{40} -11.3367 q^{41} +8.94519 q^{42} -6.08288 q^{43} -1.37696 q^{44} +9.00839 q^{45} +4.11728 q^{46} +3.95288 q^{47} -2.39141 q^{48} +6.99176 q^{49} +5.97824 q^{50} +16.7722 q^{51} -4.99110 q^{52} +12.3146 q^{53} +0.672409 q^{54} -4.56234 q^{55} -3.74056 q^{56} -5.56488 q^{57} -6.52552 q^{58} +3.98150 q^{59} -7.92355 q^{60} +6.08501 q^{61} +0.609271 q^{62} -10.1699 q^{63} +1.00000 q^{64} -16.5372 q^{65} +3.29287 q^{66} +8.50696 q^{67} -7.01355 q^{68} -9.84609 q^{69} -12.3937 q^{70} -5.28105 q^{71} +2.71882 q^{72} +5.93222 q^{73} +0.843333 q^{74} -14.2964 q^{75} +2.32703 q^{76} +5.15059 q^{77} +11.9357 q^{78} -1.40003 q^{79} +3.31334 q^{80} -9.76447 q^{81} -11.3367 q^{82} +12.4483 q^{83} +8.94519 q^{84} -23.2383 q^{85} -6.08288 q^{86} +15.6052 q^{87} -1.37696 q^{88} +1.11174 q^{89} +9.00839 q^{90} +18.6695 q^{91} +4.11728 q^{92} -1.45701 q^{93} +3.95288 q^{94} +7.71025 q^{95} -2.39141 q^{96} +4.31189 q^{97} +6.99176 q^{98} -3.74371 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) $ 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.39141	−1.38068	−0.690339	−	0.723486i	$-0.742539\pi$
$3$	−2.39141	−1.38068	−0.690339	+	0.723486i	$0.742539\pi$
$4$	1.00000	0.500000
$5$	3.31334	1.48177	0.740886	−	0.671631i	$-0.234406\pi$
$5$	3.31334	1.48177	0.740886	+	0.671631i	$0.234406\pi$
$6$	−2.39141	−0.976287
$7$	−3.74056	−1.41380	−0.706899	−	0.707315i	$-0.749906\pi$
$7$	−3.74056	−1.41380	−0.706899	+	0.707315i	$0.749906\pi$
$8$	1.00000	0.353553
$9$	2.71882	0.906274
$10$	3.31334	1.04777
$11$	−1.37696	−0.415169	−0.207584	−	0.978217i	$-0.566560\pi$
$11$	−1.37696	−0.415169	−0.207584	+	0.978217i	$0.566560\pi$
$12$	−2.39141	−0.690339
$13$	−4.99110	−1.38428	−0.692141	−	0.721762i	$-0.743332\pi$
$13$	−4.99110	−1.38428	−0.692141	+	0.721762i	$0.743332\pi$
$14$	−3.74056	−0.999706
$15$	−7.92355	−2.04585
$16$	1.00000	0.250000
$17$	−7.01355	−1.70104	−0.850518	−	0.525946i	$-0.823711\pi$
$17$	−7.01355	−1.70104	−0.850518	+	0.525946i	$0.823711\pi$
$18$	2.71882	0.640833
$19$	2.32703	0.533858	0.266929	−	0.963716i	$-0.413991\pi$
$19$	2.32703	0.533858	0.266929	+	0.963716i	$0.413991\pi$
$20$	3.31334	0.740886
$21$	8.94519	1.95200
$22$	−1.37696	−0.293569
$23$	4.11728	0.858512	0.429256	−	0.903183i	$-0.358776\pi$
$23$	4.11728	0.858512	0.429256	+	0.903183i	$0.358776\pi$
$24$	−2.39141	−0.488144
$25$	5.97824	1.19565
$26$	−4.99110	−0.978835
$27$	0.672409	0.129405
$28$	−3.74056	−0.706899
$29$	−6.52552	−1.21176	−0.605879	−	0.795557i	$-0.707178\pi$
$29$	−6.52552	−1.21176	−0.605879	+	0.795557i	$0.707178\pi$
$30$	−7.92355	−1.44664
$31$	0.609271	0.109428	0.0547141	−	0.998502i	$-0.482575\pi$
$31$	0.609271	0.109428	0.0547141	+	0.998502i	$0.482575\pi$
$32$	1.00000	0.176777
$33$	3.29287	0.573215
$34$	−7.01355	−1.20281
$35$	−12.3937	−2.09493
$36$	2.71882	0.453137
$37$	0.843333	0.138643	0.0693216	−	0.997594i	$-0.477917\pi$
$37$	0.843333	0.138643	0.0693216	+	0.997594i	$0.477917\pi$
$38$	2.32703	0.377494
$39$	11.9357	1.91125
$40$	3.31334	0.523886
$41$	−11.3367	−1.77050	−0.885248	−	0.465119i	$-0.846012\pi$
$41$	−11.3367	−1.77050	−0.885248	+	0.465119i	$0.846012\pi$
$42$	8.94519	1.38027
$43$	−6.08288	−0.927631	−0.463815	−	0.885932i	$-0.653520\pi$
$43$	−6.08288	−0.927631	−0.463815	+	0.885932i	$0.653520\pi$
$44$	−1.37696	−0.207584
$45$	9.00839	1.34289
$46$	4.11728	0.607060
$47$	3.95288	0.576586	0.288293	−	0.957542i	$-0.406912\pi$
$47$	3.95288	0.576586	0.288293	+	0.957542i	$0.406912\pi$
$48$	−2.39141	−0.345170
$49$	6.99176	0.998822
$50$	5.97824	0.845451
$51$	16.7722	2.34858
$52$	−4.99110	−0.692141
$53$	12.3146	1.69154	0.845770	−	0.533547i	$-0.179141\pi$
$53$	12.3146	1.69154	0.845770	+	0.533547i	$0.179141\pi$
$54$	0.672409	0.0915033
$55$	−4.56234	−0.615186
$56$	−3.74056	−0.499853
$57$	−5.56488	−0.737086
$58$	−6.52552	−0.856842
$59$	3.98150	0.518347	0.259173	−	0.965831i	$-0.416550\pi$
$59$	3.98150	0.518347	0.259173	+	0.965831i	$0.416550\pi$
$60$	−7.92355	−1.02293
$61$	6.08501	0.779106	0.389553	−	0.921004i	$-0.372630\pi$
$61$	6.08501	0.779106	0.389553	+	0.921004i	$0.372630\pi$
$62$	0.609271	0.0773775
$63$	−10.1699	−1.28129
$64$	1.00000	0.125000
$65$	−16.5372	−2.05119
$66$	3.29287	0.405324
$67$	8.50696	1.03929	0.519645	−	0.854382i	$-0.326064\pi$
$67$	8.50696	1.03929	0.519645	+	0.854382i	$0.326064\pi$
$68$	−7.01355	−0.850518
$69$	−9.84609	−1.18533
$70$	−12.3937	−1.48134
$71$	−5.28105	−0.626745	−0.313373	−	0.949630i	$-0.601459\pi$
$71$	−5.28105	−0.626745	−0.313373	+	0.949630i	$0.601459\pi$
$72$	2.71882	0.320416
$73$	5.93222	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$73$	5.93222	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$74$	0.843333	0.0980355
$75$	−14.2964	−1.65081
$76$	2.32703	0.266929
$77$	5.15059	0.586964
$78$	11.9357	1.35146
$79$	−1.40003	−0.157515	−0.0787577	−	0.996894i	$-0.525095\pi$
$79$	−1.40003	−0.157515	−0.0787577	+	0.996894i	$0.525095\pi$
$80$	3.31334	0.370443
$81$	−9.76447	−1.08494
$82$	−11.3367	−1.25193
$83$	12.4483	1.36637	0.683187	−	0.730244i	$-0.260593\pi$
$83$	12.4483	1.36637	0.683187	+	0.730244i	$0.260593\pi$
$84$	8.94519	0.976000
$85$	−23.2383	−2.52055
$86$	−6.08288	−0.655934
$87$	15.6052	1.67305
$88$	−1.37696	−0.146784
$89$	1.11174	0.117844	0.0589218	−	0.998263i	$-0.481234\pi$
$89$	1.11174	0.117844	0.0589218	+	0.998263i	$0.481234\pi$
$90$	9.00839	0.949568
$91$	18.6695	1.95709
$92$	4.11728	0.429256
$93$	−1.45701	−0.151085
$94$	3.95288	0.407708
$95$	7.71025	0.791055
$96$	−2.39141	−0.244072
$97$	4.31189	0.437806	0.218903	−	0.975747i	$-0.429752\pi$
$97$	4.31189	0.437806	0.218903	+	0.975747i	$0.429752\pi$
$98$	6.99176	0.706274
$99$	−3.74371	−0.376257
$100$	5.97824	0.597824
$101$	−0.127164	−0.0126533	−0.00632664	−	0.999980i	$-0.502014\pi$
$101$	−0.127164	−0.0126533	−0.00632664	+	0.999980i	$0.502014\pi$
$102$	16.7722	1.66070
$103$	2.48638	0.244990	0.122495	−	0.992469i	$-0.460910\pi$
$103$	2.48638	0.244990	0.122495	+	0.992469i	$0.460910\pi$
$104$	−4.99110	−0.489417
$105$	29.6385	2.89242
$106$	12.3146	1.19610
$107$	14.9595	1.44619	0.723093	−	0.690751i	$-0.242720\pi$
$107$	14.9595	1.44619	0.723093	+	0.690751i	$0.242720\pi$
$108$	0.672409	0.0647026
$109$	−12.2769	−1.17591	−0.587955	−	0.808894i	$-0.700067\pi$
$109$	−12.2769	−1.17591	−0.587955	+	0.808894i	$0.700067\pi$
$110$	−4.56234	−0.435002
$111$	−2.01675	−0.191422
$112$	−3.74056	−0.353449
$113$	18.4858	1.73900	0.869501	−	0.493932i	$-0.164441\pi$
$113$	18.4858	1.73900	0.869501	+	0.493932i	$0.164441\pi$
$114$	−5.56488	−0.521198
$115$	13.6420	1.27212
$116$	−6.52552	−0.605879
$117$	−13.5699	−1.25454
$118$	3.98150	0.366527
$119$	26.2346	2.40492
$120$	−7.92355	−0.723318
$121$	−9.10398	−0.827635
$122$	6.08501	0.550911
$123$	27.1107	2.44449
$124$	0.609271	0.0547141
$125$	3.24126	0.289907
$126$	−10.1699	−0.906007
$127$	5.38269	0.477636	0.238818	−	0.971064i	$-0.423240\pi$
$127$	5.38269	0.477636	0.238818	+	0.971064i	$0.423240\pi$
$128$	1.00000	0.0883883
$129$	14.5466	1.28076
$130$	−16.5372	−1.45041
$131$	19.7030	1.72146	0.860731	−	0.509060i	$-0.170007\pi$
$131$	19.7030	1.72146	0.860731	+	0.509060i	$0.170007\pi$
$132$	3.29287	0.286607
$133$	−8.70439	−0.754766
$134$	8.50696	0.734889
$135$	2.22792	0.191749
$136$	−7.01355	−0.601407
$137$	11.3090	0.966193	0.483096	−	0.875567i	$-0.339512\pi$
$137$	11.3090	0.966193	0.483096	+	0.875567i	$0.339512\pi$
$138$	−9.84609	−0.838155
$139$	5.56085	0.471665	0.235833	−	0.971794i	$-0.424218\pi$
$139$	5.56085	0.471665	0.235833	+	0.971794i	$0.424218\pi$
$140$	−12.3937	−1.04746
$141$	−9.45293	−0.796081
$142$	−5.28105	−0.443176
$143$	6.87254	0.574711
$144$	2.71882	0.226569
$145$	−21.6213	−1.79555
$146$	5.93222	0.490954
$147$	−16.7201	−1.37905
$148$	0.843333	0.0693216
$149$	−0.0242801	−0.00198910	−0.000994552	−	1.00000i	$-0.500317\pi$
$149$	−0.0242801	−0.00198910	−0.000994552		1.00000i	$0.500317\pi$
$150$	−14.2964	−1.16730
$151$	−6.25899	−0.509350	−0.254675	−	0.967027i	$-0.581968\pi$
$151$	−6.25899	−0.509350	−0.254675	+	0.967027i	$0.581968\pi$
$152$	2.32703	0.188747
$153$	−19.0686	−1.54160
$154$	5.15059	0.415047
$155$	2.01872	0.162148
$156$	11.9357	0.955624
$157$	−7.25133	−0.578719	−0.289360	−	0.957220i	$-0.593442\pi$
$157$	−7.25133	−0.578719	−0.289360	+	0.957220i	$0.593442\pi$
$158$	−1.40003	−0.111380
$159$	−29.4492	−2.33547
$160$	3.31334	0.261943
$161$	−15.4009	−1.21376
$162$	−9.76447	−0.767169
$163$	−20.2073	−1.58276	−0.791380	−	0.611324i	$-0.790637\pi$
$163$	−20.2073	−1.58276	−0.791380	+	0.611324i	$0.790637\pi$
$164$	−11.3367	−0.885248
$165$	10.9104	0.849374
$166$	12.4483	0.966172
$167$	7.16579	0.554505	0.277253	−	0.960797i	$-0.410576\pi$
$167$	7.16579	0.554505	0.277253	+	0.960797i	$0.410576\pi$
$168$	8.94519	0.690136
$169$	11.9111	0.916236
$170$	−23.2383	−1.78230
$171$	6.32679	0.483821
$172$	−6.08288	−0.463815
$173$	5.43309	0.413070	0.206535	−	0.978439i	$-0.433781\pi$
$173$	5.43309	0.413070	0.206535	+	0.978439i	$0.433781\pi$
$174$	15.6052	1.18302
$175$	−22.3620	−1.69040
$176$	−1.37696	−0.103792
$177$	−9.52138	−0.715671
$178$	1.11174	0.0833281
$179$	6.88238	0.514413	0.257206	−	0.966356i	$-0.417198\pi$
$179$	6.88238	0.514413	0.257206	+	0.966356i	$0.417198\pi$
$180$	9.00839	0.671446
$181$	−4.86140	−0.361345	−0.180672	−	0.983543i	$-0.557827\pi$
$181$	−4.86140	−0.361345	−0.180672	+	0.983543i	$0.557827\pi$
$182$	18.6695	1.38387
$183$	−14.5517	−1.07569
$184$	4.11728	0.303530
$185$	2.79425	0.205438
$186$	−1.45701	−0.106833
$187$	9.65737	0.706217
$188$	3.95288	0.288293
$189$	−2.51518	−0.182953
$190$	7.71025	0.559361
$191$	−20.0453	−1.45043	−0.725214	−	0.688524i	$-0.758259\pi$
$191$	−20.0453	−1.45043	−0.725214	+	0.688524i	$0.758259\pi$
$192$	−2.39141	−0.172585
$193$	−4.65819	−0.335304	−0.167652	−	0.985846i	$-0.553619\pi$
$193$	−4.65819	−0.335304	−0.167652	+	0.985846i	$0.553619\pi$
$194$	4.31189	0.309576
$195$	39.5472	2.83203
$196$	6.99176	0.499411
$197$	0.434106	0.0309288	0.0154644	−	0.999880i	$-0.495077\pi$
$197$	0.434106	0.0309288	0.0154644	+	0.999880i	$0.495077\pi$
$198$	−3.74371	−0.266054
$199$	19.0496	1.35039	0.675196	−	0.737638i	$-0.264059\pi$
$199$	19.0496	1.35039	0.675196	+	0.737638i	$0.264059\pi$
$200$	5.97824	0.422726
$201$	−20.3436	−1.43493
$202$	−0.127164	−0.00894721
$203$	24.4091	1.71318
$204$	16.7722	1.17429
$205$	−37.5624	−2.62347
$206$	2.48638	0.173234
$207$	11.1942	0.778047
$208$	−4.99110	−0.346070
$209$	−3.20423	−0.221641
$210$	29.6385	2.04525
$211$	22.2561	1.53218	0.766088	−	0.642736i	$-0.222201\pi$
$211$	22.2561	1.53218	0.766088	+	0.642736i	$0.222201\pi$
$212$	12.3146	0.845770
$213$	12.6291	0.865334
$214$	14.9595	1.02261
$215$	−20.1547	−1.37454
$216$	0.672409	0.0457517
$217$	−2.27901	−0.154709
$218$	−12.2769	−0.831494
$219$	−14.1864	−0.958625
$220$	−4.56234	−0.307593
$221$	35.0053	2.35471
$222$	−2.01675	−0.135356
$223$	18.5639	1.24313	0.621565	−	0.783363i	$-0.286497\pi$
$223$	18.5639	1.24313	0.621565	+	0.783363i	$0.286497\pi$
$224$	−3.74056	−0.249926
$225$	16.2538	1.08359
$226$	18.4858	1.22966
$227$	−23.4605	−1.55713	−0.778565	−	0.627564i	$-0.784052\pi$
$227$	−23.4605	−1.55713	−0.778565	+	0.627564i	$0.784052\pi$
$228$	−5.56488	−0.368543
$229$	4.80141	0.317286	0.158643	−	0.987336i	$-0.449288\pi$
$229$	4.80141	0.317286	0.158643	+	0.987336i	$0.449288\pi$
$230$	13.6420	0.899524
$231$	−12.3172	−0.810409
$232$	−6.52552	−0.428421
$233$	−28.2427	−1.85024	−0.925121	−	0.379674i	$-0.876036\pi$
$233$	−28.2427	−1.85024	−0.925121	+	0.379674i	$0.876036\pi$
$234$	−13.5699	−0.887093
$235$	13.0972	0.854370
$236$	3.98150	0.259173
$237$	3.34803	0.217478
$238$	26.2346	1.70053
$239$	−1.03765	−0.0671202	−0.0335601	−	0.999437i	$-0.510685\pi$
$239$	−1.03765	−0.0671202	−0.0335601	+	0.999437i	$0.510685\pi$
$240$	−7.92355	−0.511463
$241$	9.61178	0.619149	0.309575	−	0.950875i	$-0.399813\pi$
$241$	9.61178	0.619149	0.309575	+	0.950875i	$0.399813\pi$
$242$	−9.10398	−0.585226
$243$	21.3336	1.36855
$244$	6.08501	0.389553
$245$	23.1661	1.48003
$246$	27.1107	1.72851
$247$	−11.6144	−0.739009
$248$	0.609271	0.0386887
$249$	−29.7688	−1.88652
$250$	3.24126	0.204995
$251$	9.27661	0.585534	0.292767	−	0.956184i	$-0.405424\pi$
$251$	9.27661	0.585534	0.292767	+	0.956184i	$0.405424\pi$
$252$	−10.1699	−0.640644
$253$	−5.66933	−0.356427
$254$	5.38269	0.337740
$255$	55.5722	3.48007
$256$	1.00000	0.0625000
$257$	−22.7370	−1.41830	−0.709148	−	0.705060i	$-0.750920\pi$
$257$	−22.7370	−1.41830	−0.709148	+	0.705060i	$0.750920\pi$
$258$	14.5466	0.905634
$259$	−3.15453	−0.196013
$260$	−16.5372	−1.02560
$261$	−17.7417	−1.09819
$262$	19.7030	1.21726
$263$	−11.5047	−0.709409	−0.354704	−	0.934978i	$-0.615419\pi$
$263$	−11.5047	−0.709409	−0.354704	+	0.934978i	$0.615419\pi$
$264$	3.29287	0.202662
$265$	40.8025	2.50648
$266$	−8.70439	−0.533700
$267$	−2.65861	−0.162704
$268$	8.50696	0.519645
$269$	−2.83826	−0.173052	−0.0865258	−	0.996250i	$-0.527577\pi$
$269$	−2.83826	−0.173052	−0.0865258	+	0.996250i	$0.527577\pi$
$270$	2.22792	0.135587
$271$	−13.0890	−0.795102	−0.397551	−	0.917580i	$-0.630140\pi$
$271$	−13.0890	−0.795102	−0.397551	+	0.917580i	$0.630140\pi$
$272$	−7.01355	−0.425259
$273$	−44.6463	−2.70212
$274$	11.3090	0.683202
$275$	−8.23180	−0.496396
$276$	−9.84609	−0.592665
$277$	12.3013	0.739116	0.369558	−	0.929208i	$-0.379509\pi$
$277$	12.3013	0.739116	0.369558	+	0.929208i	$0.379509\pi$
$278$	5.56085	0.333518
$279$	1.65650	0.0991720
$280$	−12.3937	−0.740668
$281$	−7.31338	−0.436280	−0.218140	−	0.975918i	$-0.569999\pi$
$281$	−7.31338	−0.436280	−0.218140	+	0.975918i	$0.569999\pi$
$282$	−9.45293	−0.562914
$283$	−4.98738	−0.296469	−0.148234	−	0.988952i	$-0.547359\pi$
$283$	−4.98738	−0.296469	−0.148234	+	0.988952i	$0.547359\pi$
$284$	−5.28105	−0.313373
$285$	−18.4383	−1.09219
$286$	6.87254	0.406382
$287$	42.4056	2.50312
$288$	2.71882	0.160208
$289$	32.1899	1.89352
$290$	−21.6213	−1.26965
$291$	−10.3115	−0.604469
$292$	5.93222	0.347157
$293$	14.3501	0.838339	0.419169	−	0.907908i	$-0.362321\pi$
$293$	14.3501	0.838339	0.419169	+	0.907908i	$0.362321\pi$
$294$	−16.7201	−0.975138
$295$	13.1921	0.768072
$296$	0.843333	0.0490177
$297$	−0.925880	−0.0537250
$298$	−0.0242801	−0.00140651
$299$	−20.5497	−1.18842
$300$	−14.2964	−0.825403
$301$	22.7534	1.31148
$302$	−6.25899	−0.360164
$303$	0.304100	0.0174701
$304$	2.32703	0.133464
$305$	20.1617	1.15446
$306$	−19.0686	−1.09008
$307$	−3.88941	−0.221980	−0.110990	−	0.993821i	$-0.535402\pi$
$307$	−3.88941	−0.221980	−0.110990	+	0.993821i	$0.535402\pi$
$308$	5.15059	0.293482
$309$	−5.94594	−0.338253
$310$	2.01872	0.114656
$311$	16.5343	0.937576	0.468788	−	0.883311i	$-0.344691\pi$
$311$	16.5343	0.937576	0.468788	+	0.883311i	$0.344691\pi$
$312$	11.9357	0.675728
$313$	20.1322	1.13794	0.568971	−	0.822358i	$-0.307342\pi$
$313$	20.1322	1.13794	0.568971	+	0.822358i	$0.307342\pi$
$314$	−7.25133	−0.409216
$315$	−33.6964	−1.89858
$316$	−1.40003	−0.0787577
$317$	25.3155	1.42186	0.710931	−	0.703262i	$-0.248274\pi$
$317$	25.3155	1.42186	0.710931	+	0.703262i	$0.248274\pi$
$318$	−29.4492	−1.65143
$319$	8.98537	0.503084
$320$	3.31334	0.185222
$321$	−35.7741	−1.99672
$322$	−15.4009	−0.858259
$323$	−16.3208	−0.908111
$324$	−9.76447	−0.542471
$325$	−29.8380	−1.65511
$326$	−20.2073	−1.11918
$327$	29.3590	1.62355
$328$	−11.3367	−0.625965
$329$	−14.7860	−0.815176
$330$	10.9104	0.600598
$331$	−15.6130	−0.858169	−0.429085	−	0.903264i	$-0.641164\pi$
$331$	−15.6130	−0.858169	−0.429085	+	0.903264i	$0.641164\pi$
$332$	12.4483	0.683187
$333$	2.29287	0.125649
$334$	7.16579	0.392094
$335$	28.1865	1.53999
$336$	8.94519	0.488000
$337$	15.3005	0.833469	0.416734	−	0.909028i	$-0.363175\pi$
$337$	15.3005	0.833469	0.416734	+	0.909028i	$0.363175\pi$
$338$	11.9111	0.647877
$339$	−44.2071	−2.40100
$340$	−23.2383	−1.26027
$341$	−0.838941	−0.0454312
$342$	6.32679	0.342113
$343$	0.0308377	0.00166508
$344$	−6.08288	−0.327967
$345$	−32.6235	−1.75639
$346$	5.43309	0.292085
$347$	−26.9153	−1.44489	−0.722445	−	0.691428i	$-0.756982\pi$
$347$	−26.9153	−1.44489	−0.722445	+	0.691428i	$0.756982\pi$
$348$	15.6052	0.836525
$349$	0.893123	0.0478078	0.0239039	−	0.999714i	$-0.492390\pi$
$349$	0.893123	0.0478078	0.0239039	+	0.999714i	$0.492390\pi$
$350$	−22.3620	−1.19530
$351$	−3.35606	−0.179133
$352$	−1.37696	−0.0733922
$353$	−2.57021	−0.136798	−0.0683992	−	0.997658i	$-0.521789\pi$
$353$	−2.57021	−0.136798	−0.0683992	+	0.997658i	$0.521789\pi$
$354$	−9.52138	−0.506056
$355$	−17.4979	−0.928694
$356$	1.11174	0.0589218
$357$	−62.7375	−3.32042
$358$	6.88238	0.363745
$359$	3.89331	0.205481	0.102741	−	0.994708i	$-0.467239\pi$
$359$	3.89331	0.205481	0.102741	+	0.994708i	$0.467239\pi$
$360$	9.00839	0.474784
$361$	−13.5849	−0.714996
$362$	−4.86140	−0.255509
$363$	21.7713	1.14270
$364$	18.6695	0.978547
$365$	19.6555	1.02882
$366$	−14.5517	−0.760631
$367$	25.8765	1.35074	0.675370	−	0.737479i	$-0.263984\pi$
$367$	25.8765	1.35074	0.675370	+	0.737479i	$0.263984\pi$
$368$	4.11728	0.214628
$369$	−30.8225	−1.60456
$370$	2.79425	0.145266
$371$	−46.0635	−2.39150
$372$	−1.45701	−0.0755426
$373$	−25.6022	−1.32563	−0.662815	−	0.748784i	$-0.730638\pi$
$373$	−25.6022	−1.32563	−0.662815	+	0.748784i	$0.730638\pi$
$374$	9.65737	0.499371
$375$	−7.75116	−0.400268
$376$	3.95288	0.203854
$377$	32.5695	1.67741
$378$	−2.51518	−0.129367
$379$	24.8309	1.27548	0.637738	−	0.770254i	$-0.279870\pi$
$379$	24.8309	1.27548	0.637738	+	0.770254i	$0.279870\pi$
$380$	7.71025	0.395528
$381$	−12.8722	−0.659462
$382$	−20.0453	−1.02561
$383$	−15.5791	−0.796054	−0.398027	−	0.917374i	$-0.630305\pi$
$383$	−15.5791	−0.796054	−0.398027	+	0.917374i	$0.630305\pi$
$384$	−2.39141	−0.122036
$385$	17.0657	0.869748
$386$	−4.65819	−0.237096
$387$	−16.5383	−0.840688
$388$	4.31189	0.218903
$389$	15.0903	0.765107	0.382553	−	0.923933i	$-0.375045\pi$
$389$	15.0903	0.765107	0.382553	+	0.923933i	$0.375045\pi$
$390$	39.5472	2.00255
$391$	−28.8767	−1.46036
$392$	6.99176	0.353137
$393$	−47.1180	−2.37679
$394$	0.434106	0.0218700
$395$	−4.63877	−0.233402
$396$	−3.74371	−0.188128
$397$	4.59496	0.230614	0.115307	−	0.993330i	$-0.463215\pi$
$397$	4.59496	0.230614	0.115307	+	0.993330i	$0.463215\pi$
$398$	19.0496	0.954871
$399$	20.8157	1.04209
$400$	5.97824	0.298912
$401$	−9.30446	−0.464643	−0.232321	−	0.972639i	$-0.574632\pi$
$401$	−9.30446	−0.464643	−0.232321	+	0.972639i	$0.574632\pi$
$402$	−20.3436	−1.01465
$403$	−3.04093	−0.151480
$404$	−0.127164	−0.00632664
$405$	−32.3530	−1.60764
$406$	24.4091	1.21140
$407$	−1.16124	−0.0575603
$408$	16.7722	0.830350
$409$	2.24769	0.111141	0.0555707	−	0.998455i	$-0.482302\pi$
$409$	2.24769	0.111141	0.0555707	+	0.998455i	$0.482302\pi$
$410$	−37.5624	−1.85508
$411$	−27.0444	−1.33400
$412$	2.48638	0.122495
$413$	−14.8930	−0.732837
$414$	11.1942	0.550163
$415$	41.2453	2.02465
$416$	−4.99110	−0.244709
$417$	−13.2983	−0.651218
$418$	−3.20423	−0.156724
$419$	19.3057	0.943146	0.471573	−	0.881827i	$-0.343686\pi$
$419$	19.3057	0.943146	0.471573	+	0.881827i	$0.343686\pi$
$420$	29.6385	1.44621
$421$	3.50045	0.170601	0.0853007	−	0.996355i	$-0.472815\pi$
$421$	3.50045	0.170601	0.0853007	+	0.996355i	$0.472815\pi$
$422$	22.2561	1.08341
$423$	10.7472	0.522545
$424$	12.3146	0.598050
$425$	−41.9287	−2.03384
$426$	12.6291	0.611884
$427$	−22.7613	−1.10150
$428$	14.9595	0.723093
$429$	−16.4350	−0.793491
$430$	−20.1547	−0.971945
$431$	−8.99874	−0.433454	−0.216727	−	0.976232i	$-0.569538\pi$
$431$	−8.99874	−0.433454	−0.216727	+	0.976232i	$0.569538\pi$
$432$	0.672409	0.0323513
$433$	20.7019	0.994872	0.497436	−	0.867501i	$-0.334275\pi$
$433$	20.7019	0.994872	0.497436	+	0.867501i	$0.334275\pi$
$434$	−2.27901	−0.109396
$435$	51.7053	2.47908
$436$	−12.2769	−0.587955
$437$	9.58104	0.458323
$438$	−14.1864	−0.677850
$439$	16.0392	0.765511	0.382755	−	0.923850i	$-0.374975\pi$
$439$	16.0392	0.765511	0.382755	+	0.923850i	$0.374975\pi$
$440$	−4.56234	−0.217501
$441$	19.0093	0.905207
$442$	35.0053	1.66503
$443$	−15.0022	−0.712776	−0.356388	−	0.934338i	$-0.615992\pi$
$443$	−15.0022	−0.712776	−0.356388	+	0.934338i	$0.615992\pi$
$444$	−2.01675	−0.0957108
$445$	3.68356	0.174618
$446$	18.5639	0.879025
$447$	0.0580636	0.00274631
$448$	−3.74056	−0.176725
$449$	−8.99756	−0.424621	−0.212311	−	0.977202i	$-0.568099\pi$
$449$	−8.99756	−0.424621	−0.212311	+	0.977202i	$0.568099\pi$
$450$	16.2538	0.766211
$451$	15.6102	0.735055
$452$	18.4858	0.869501
$453$	14.9678	0.703248
$454$	−23.4605	−1.10106
$455$	61.8584	2.89997
$456$	−5.56488	−0.260599
$457$	37.7294	1.76491	0.882454	−	0.470399i	$-0.155890\pi$
$457$	37.7294	1.76491	0.882454	+	0.470399i	$0.155890\pi$
$458$	4.80141	0.224355
$459$	−4.71598	−0.220123
$460$	13.6420	0.636060
$461$	12.9509	0.603185	0.301592	−	0.953437i	$-0.402482\pi$
$461$	12.9509	0.603185	0.301592	+	0.953437i	$0.402482\pi$
$462$	−12.3172	−0.573046
$463$	18.1281	0.842482	0.421241	−	0.906949i	$-0.361595\pi$
$463$	18.1281	0.842482	0.421241	+	0.906949i	$0.361595\pi$
$464$	−6.52552	−0.302940
$465$	−4.82759	−0.223874
$466$	−28.2427	−1.30832
$467$	30.1463	1.39501	0.697503	−	0.716582i	$-0.254295\pi$
$467$	30.1463	1.39501	0.697503	+	0.716582i	$0.254295\pi$
$468$	−13.5699	−0.627269
$469$	−31.8207	−1.46935
$470$	13.0972	0.604131
$471$	17.3409	0.799026
$472$	3.98150	0.183263
$473$	8.37588	0.385123
$474$	3.34803	0.153780
$475$	13.9116	0.638306
$476$	26.2346	1.20246
$477$	33.4812	1.53300
$478$	−1.03765	−0.0474611
$479$	−33.2213	−1.51792	−0.758959	−	0.651138i	$-0.774292\pi$
$479$	−33.2213	−1.51792	−0.758959	+	0.651138i	$0.774292\pi$
$480$	−7.92355	−0.361659
$481$	−4.20916	−0.191921
$482$	9.61178	0.437805
$483$	36.8298	1.67582
$484$	−9.10398	−0.413817
$485$	14.2868	0.648729
$486$	21.3336	0.967711
$487$	−33.4795	−1.51710	−0.758550	−	0.651614i	$-0.774092\pi$
$487$	−33.4795	−1.51710	−0.758550	+	0.651614i	$0.774092\pi$
$488$	6.08501	0.275455
$489$	48.3239	2.18528
$490$	23.1661	1.04654
$491$	−8.45113	−0.381394	−0.190697	−	0.981649i	$-0.561075\pi$
$491$	−8.45113	−0.381394	−0.190697	+	0.981649i	$0.561075\pi$
$492$	27.1107	1.22224
$493$	45.7670	2.06124
$494$	−11.6144	−0.522559
$495$	−12.4042	−0.557527
$496$	0.609271	0.0273571
$497$	19.7541	0.886091
$498$	−29.7688	−1.33397
$499$	34.0769	1.52549	0.762746	−	0.646698i	$-0.223851\pi$
$499$	34.0769	1.52549	0.762746	+	0.646698i	$0.223851\pi$
$500$	3.24126	0.144953
$501$	−17.1363	−0.765594
$502$	9.27661	0.414035
$503$	38.1452	1.70081	0.850404	−	0.526130i	$-0.176358\pi$
$503$	38.1452	1.70081	0.850404	+	0.526130i	$0.176358\pi$
$504$	−10.1699	−0.453004
$505$	−0.421337	−0.0187493
$506$	−5.66933	−0.252032
$507$	−28.4842	−1.26503
$508$	5.38269	0.238818
$509$	−24.9361	−1.10527	−0.552637	−	0.833422i	$-0.686378\pi$
$509$	−24.9361	−1.10527	−0.552637	+	0.833422i	$0.686378\pi$
$510$	55.5722	2.46078
$511$	−22.1898	−0.981619
$512$	1.00000	0.0441942
$513$	1.56472	0.0690840
$514$	−22.7370	−1.00289
$515$	8.23823	0.363020
$516$	14.5466	0.640380
$517$	−5.44295	−0.239381
$518$	−3.15453	−0.138602
$519$	−12.9927	−0.570317
$520$	−16.5372	−0.725205
$521$	−38.4721	−1.68549	−0.842747	−	0.538310i	$-0.819063\pi$
$521$	−38.4721	−1.68549	−0.842747	+	0.538310i	$0.819063\pi$
$522$	−17.7417	−0.776534
$523$	17.1038	0.747897	0.373949	−	0.927449i	$-0.378004\pi$
$523$	17.1038	0.747897	0.373949	+	0.927449i	$0.378004\pi$
$524$	19.7030	0.860731
$525$	53.4765	2.33391
$526$	−11.5047	−0.501628
$527$	−4.27315	−0.186141
$528$	3.29287	0.143304
$529$	−6.04801	−0.262957
$530$	40.8025	1.77235
$531$	10.8250	0.469764
$532$	−8.70439	−0.377383
$533$	56.5826	2.45087
$534$	−2.65861	−0.115049
$535$	49.5658	2.14292
$536$	8.50696	0.367445
$537$	−16.4586	−0.710239
$538$	−2.83826	−0.122366
$539$	−9.62736	−0.414680
$540$	2.22792	0.0958745
$541$	7.87438	0.338546	0.169273	−	0.985569i	$-0.445858\pi$
$541$	7.87438	0.338546	0.169273	+	0.985569i	$0.445858\pi$
$542$	−13.0890	−0.562222
$543$	11.6256	0.498901
$544$	−7.01355	−0.300703
$545$	−40.6775	−1.74243
$546$	−44.6463	−1.91069
$547$	27.4669	1.17440	0.587199	−	0.809443i	$-0.300231\pi$
$547$	27.4669	1.17440	0.587199	+	0.809443i	$0.300231\pi$
$548$	11.3090	0.483096
$549$	16.5441	0.706083
$550$	−8.23180	−0.351005
$551$	−15.1851	−0.646906
$552$	−9.84609	−0.419077
$553$	5.23688	0.222695
$554$	12.3013	0.522634
$555$	−6.68219	−0.283643
$556$	5.56085	0.235833
$557$	15.8031	0.669599	0.334799	−	0.942289i	$-0.391331\pi$
$557$	15.8031	0.669599	0.334799	+	0.942289i	$0.391331\pi$
$558$	1.65650	0.0701252
$559$	30.3603	1.28410
$560$	−12.3937	−0.523731
$561$	−23.0947	−0.975059
$562$	−7.31338	−0.308496
$563$	41.3133	1.74115	0.870574	−	0.492038i	$-0.163748\pi$
$563$	41.3133	1.74115	0.870574	+	0.492038i	$0.163748\pi$
$564$	−9.45293	−0.398040
$565$	61.2499	2.57680
$566$	−4.98738	−0.209635
$567$	36.5245	1.53389
$568$	−5.28105	−0.221588
$569$	−5.59774	−0.234669	−0.117335	−	0.993092i	$-0.537435\pi$
$569$	−5.59774	−0.234669	−0.117335	+	0.993092i	$0.537435\pi$
$570$	−18.4383	−0.772297
$571$	3.91248	0.163732	0.0818662	−	0.996643i	$-0.473912\pi$
$571$	3.91248	0.163732	0.0818662	+	0.996643i	$0.473912\pi$
$572$	6.87254	0.287355
$573$	47.9365	2.00257
$574$	42.4056	1.76998
$575$	24.6141	1.02648
$576$	2.71882	0.113284
$577$	−30.3436	−1.26322	−0.631611	−	0.775286i	$-0.717606\pi$
$577$	−30.3436	−1.26322	−0.631611	+	0.775286i	$0.717606\pi$
$578$	32.1899	1.33892
$579$	11.1396	0.462947
$580$	−21.6213	−0.897775
$581$	−46.5634	−1.93178
$582$	−10.3115	−0.427424
$583$	−16.9567	−0.702275
$584$	5.93222	0.245477
$585$	−44.9618	−1.85894
$586$	14.3501	0.592795
$587$	−0.418709	−0.0172820	−0.00864098	−	0.999963i	$-0.502751\pi$
$587$	−0.418709	−0.0172820	−0.00864098	+	0.999963i	$0.502751\pi$
$588$	−16.7201	−0.689526
$589$	1.41779	0.0584191
$590$	13.1921	0.543109
$591$	−1.03812	−0.0427027
$592$	0.843333	0.0346608
$593$	32.3470	1.32833	0.664166	−	0.747585i	$-0.268787\pi$
$593$	32.3470	1.32833	0.664166	+	0.747585i	$0.268787\pi$
$594$	−0.925880	−0.0379893
$595$	86.9242	3.56354
$596$	−0.0242801	−0.000994552 0
$597$	−45.5554	−1.86446
$598$	−20.5497	−0.840342
$599$	0.547861	0.0223850	0.0111925	−	0.999937i	$-0.496437\pi$
$599$	0.547861	0.0223850	0.0111925	+	0.999937i	$0.496437\pi$
$600$	−14.2964	−0.583648
$601$	23.6136	0.963219	0.481610	−	0.876386i	$-0.340052\pi$
$601$	23.6136	0.963219	0.481610	+	0.876386i	$0.340052\pi$
$602$	22.7534	0.927358
$603$	23.1289	0.941882
$604$	−6.25899	−0.254675
$605$	−30.1646	−1.22637
$606$	0.304100	0.0123532
$607$	36.7724	1.49255	0.746274	−	0.665639i	$-0.231841\pi$
$607$	36.7724	1.49255	0.746274	+	0.665639i	$0.231841\pi$
$608$	2.32703	0.0943736
$609$	−58.3720	−2.36535
$610$	20.1617	0.816324
$611$	−19.7292	−0.798158
$612$	−19.0686	−0.770802
$613$	−13.8081	−0.557705	−0.278853	−	0.960334i	$-0.589954\pi$
$613$	−13.8081	−0.557705	−0.278853	+	0.960334i	$0.589954\pi$
$614$	−3.88941	−0.156964
$615$	89.8270	3.62217
$616$	5.15059	0.207523
$617$	−20.6456	−0.831162	−0.415581	−	0.909556i	$-0.636422\pi$
$617$	−20.6456	−0.831162	−0.415581	+	0.909556i	$0.636422\pi$
$618$	−5.94594	−0.239181
$619$	41.1982	1.65590	0.827948	−	0.560804i	$-0.189508\pi$
$619$	41.1982	1.65590	0.827948	+	0.560804i	$0.189508\pi$
$620$	2.01872	0.0810739
$621$	2.76850	0.111096
$622$	16.5343	0.662966
$623$	−4.15851	−0.166607
$624$	11.9357	0.477812
$625$	−19.1518	−0.766073
$626$	20.1322	0.804646
$627$	7.66261	0.306015
$628$	−7.25133	−0.289360
$629$	−5.91476	−0.235837
$630$	−33.6964	−1.34250
$631$	−25.5634	−1.01766	−0.508831	−	0.860867i	$-0.669922\pi$
$631$	−25.5634	−1.01766	−0.508831	+	0.860867i	$0.669922\pi$
$632$	−1.40003	−0.0556901
$633$	−53.2235	−2.11544
$634$	25.3155	1.00541
$635$	17.8347	0.707748
$636$	−29.4492	−1.16774
$637$	−34.8965	−1.38265
$638$	8.98537	0.355734
$639$	−14.3582	−0.568003
$640$	3.31334	0.130971
$641$	38.5958	1.52444	0.762221	−	0.647317i	$-0.224109\pi$
$641$	38.5958	1.52444	0.762221	+	0.647317i	$0.224109\pi$
$642$	−35.7741	−1.41189
$643$	37.0942	1.46285	0.731426	−	0.681921i	$-0.238855\pi$
$643$	37.0942	1.46285	0.731426	+	0.681921i	$0.238855\pi$
$644$	−15.4009	−0.606881
$645$	48.1980	1.89780
$646$	−16.3208	−0.642131
$647$	−7.59279	−0.298503	−0.149252	−	0.988799i	$-0.547686\pi$
$647$	−7.59279	−0.298503	−0.149252	+	0.988799i	$0.547686\pi$
$648$	−9.76447	−0.383585
$649$	−5.48236	−0.215201
$650$	−29.8380	−1.17034
$651$	5.45004	0.213604
$652$	−20.2073	−0.791380
$653$	45.8302	1.79347	0.896737	−	0.442563i	$-0.145931\pi$
$653$	45.8302	1.79347	0.896737	+	0.442563i	$0.145931\pi$
$654$	29.3590	1.14803
$655$	65.2829	2.55082
$656$	−11.3367	−0.442624
$657$	16.1287	0.629239
$658$	−14.7860	−0.576417
$659$	−29.8840	−1.16411	−0.582057	−	0.813148i	$-0.697752\pi$
$659$	−29.8840	−1.16411	−0.582057	+	0.813148i	$0.697752\pi$
$660$	10.9104	0.424687
$661$	−4.46531	−0.173680	−0.0868401	−	0.996222i	$-0.527677\pi$
$661$	−4.46531	−0.173680	−0.0868401	+	0.996222i	$0.527677\pi$
$662$	−15.6130	−0.606817
$663$	−83.7119	−3.25110
$664$	12.4483	0.483086
$665$	−28.8406	−1.11839
$666$	2.29287	0.0888470
$667$	−26.8674	−1.04031
$668$	7.16579	0.277253
$669$	−44.3938	−1.71636
$670$	28.1865	1.08894
$671$	−8.37881	−0.323460
$672$	8.94519	0.345068
$673$	−10.2750	−0.396074	−0.198037	−	0.980195i	$-0.563457\pi$
$673$	−10.2750	−0.396074	−0.198037	+	0.980195i	$0.563457\pi$
$674$	15.3005	0.589351
$675$	4.01983	0.154723
$676$	11.9111	0.458118
$677$	−5.48567	−0.210832	−0.105416	−	0.994428i	$-0.533617\pi$
$677$	−5.48567	−0.210832	−0.105416	+	0.994428i	$0.533617\pi$
$678$	−44.2071	−1.69776
$679$	−16.1289	−0.618969
$680$	−23.2383	−0.891148
$681$	56.1036	2.14990
$682$	−0.838941	−0.0321247
$683$	−16.3900	−0.627147	−0.313573	−	0.949564i	$-0.601526\pi$
$683$	−16.3900	−0.627147	−0.313573	+	0.949564i	$0.601526\pi$
$684$	6.32679	0.241911
$685$	37.4706	1.43168
$686$	0.0308377	0.00117739
$687$	−11.4821	−0.438070
$688$	−6.08288	−0.231908
$689$	−61.4634	−2.34157
$690$	−32.6235	−1.24195
$691$	−1.17402	−0.0446619	−0.0223309	−	0.999751i	$-0.507109\pi$
$691$	−1.17402	−0.0446619	−0.0223309	+	0.999751i	$0.507109\pi$
$692$	5.43309	0.206535
$693$	14.0035	0.531951
$694$	−26.9153	−1.02169
$695$	18.4250	0.698901
$696$	15.6052	0.591512
$697$	79.5106	3.01168
$698$	0.893123	0.0338052
$699$	67.5398	2.55459
$700$	−22.3620	−0.845202
$701$	33.7163	1.27345	0.636724	−	0.771092i	$-0.280289\pi$
$701$	33.7163	1.27345	0.636724	+	0.771092i	$0.280289\pi$
$702$	−3.35606	−0.126666
$703$	1.96246	0.0740157
$704$	−1.37696	−0.0518961
$705$	−31.3208	−1.17961
$706$	−2.57021	−0.0967310
$707$	0.475663	0.0178892
$708$	−9.52138	−0.357835
$709$	−17.0093	−0.638799	−0.319400	−	0.947620i	$-0.603481\pi$
$709$	−17.0093	−0.638799	−0.319400	+	0.947620i	$0.603481\pi$
$710$	−17.4979	−0.656686
$711$	−3.80643	−0.142752
$712$	1.11174	0.0416640
$713$	2.50854	0.0939455
$714$	−62.7375	−2.34789
$715$	22.7711	0.851590
$716$	6.88238	0.257206
$717$	2.48145	0.0926714
$718$	3.89331	0.145297
$719$	−42.3785	−1.58045	−0.790226	−	0.612816i	$-0.790037\pi$
$719$	−42.3785	−1.58045	−0.790226	+	0.612816i	$0.790037\pi$
$720$	9.00839	0.335723
$721$	−9.30044	−0.346366
$722$	−13.5849	−0.505579
$723$	−22.9857	−0.854846
$724$	−4.86140	−0.180672
$725$	−39.0111	−1.44884
$726$	21.7713	0.808009
$727$	23.5254	0.872511	0.436255	−	0.899823i	$-0.356304\pi$
$727$	23.5254	0.872511	0.436255	+	0.899823i	$0.356304\pi$
$728$	18.6695	0.691937
$729$	−21.7239	−0.804587
$730$	19.6555	0.727482
$731$	42.6626	1.57793
$732$	−14.5517	−0.537847
$733$	−48.9468	−1.80789	−0.903946	−	0.427646i	$-0.859343\pi$
$733$	−48.9468	−1.80789	−0.903946	+	0.427646i	$0.859343\pi$
$734$	25.8765	0.955118
$735$	−55.3995	−2.04344
$736$	4.11728	0.151765
$737$	−11.7137	−0.431481
$738$	−30.8225	−1.13459
$739$	−15.9078	−0.585178	−0.292589	−	0.956238i	$-0.594517\pi$
$739$	−15.9078	−0.585178	−0.292589	+	0.956238i	$0.594517\pi$
$740$	2.79425	0.102719
$741$	27.7749	1.02033
$742$	−46.0635	−1.69104
$743$	2.67275	0.0980538	0.0490269	−	0.998797i	$-0.484388\pi$
$743$	2.67275	0.0980538	0.0490269	+	0.998797i	$0.484388\pi$
$744$	−1.45701	−0.0534167
$745$	−0.0804483	−0.00294740
$746$	−25.6022	−0.937361
$747$	33.8446	1.23831
$748$	9.65737	0.353109
$749$	−55.9567	−2.04461
$750$	−7.75116	−0.283032
$751$	−37.9202	−1.38373	−0.691865	−	0.722027i	$-0.743211\pi$
$751$	−37.9202	−1.38373	−0.691865	+	0.722027i	$0.743211\pi$
$752$	3.95288	0.144147
$753$	−22.1841	−0.808435
$754$	32.5695	1.18611
$755$	−20.7382	−0.754740
$756$	−2.51518	−0.0914764
$757$	−8.72785	−0.317219	−0.158610	−	0.987341i	$-0.550701\pi$
$757$	−8.72785	−0.317219	−0.158610	+	0.987341i	$0.550701\pi$
$758$	24.8309	0.901897
$759$	13.5577	0.492112
$760$	7.71025	0.279680
$761$	18.2763	0.662515	0.331258	−	0.943540i	$-0.392527\pi$
$761$	18.2763	0.662515	0.331258	+	0.943540i	$0.392527\pi$
$762$	−12.8722	−0.466310
$763$	45.9223	1.66250
$764$	−20.0453	−0.725214
$765$	−63.1808	−2.28431
$766$	−15.5791	−0.562895
$767$	−19.8721	−0.717538
$768$	−2.39141	−0.0862924
$769$	−10.2970	−0.371319	−0.185659	−	0.982614i	$-0.559442\pi$
$769$	−10.2970	−0.371319	−0.185659	+	0.982614i	$0.559442\pi$
$770$	17.0657	0.615004
$771$	54.3734	1.95821
$772$	−4.65819	−0.167652
$773$	32.6842	1.17557	0.587784	−	0.809018i	$-0.300000\pi$
$773$	32.6842	1.17557	0.587784	+	0.809018i	$0.300000\pi$
$774$	−16.5383	−0.594456
$775$	3.64237	0.130838
$776$	4.31189	0.154788
$777$	7.54377	0.270631
$778$	15.0903	0.541012
$779$	−26.3809	−0.945193
$780$	39.5472	1.41602
$781$	7.27179	0.260205
$782$	−28.8767	−1.03263
$783$	−4.38782	−0.156808
$784$	6.99176	0.249706
$785$	−24.0262	−0.857530
$786$	−47.1180	−1.68064
$787$	1.58697	0.0565693	0.0282846	−	0.999600i	$-0.490996\pi$
$787$	1.58697	0.0565693	0.0282846	+	0.999600i	$0.490996\pi$
$788$	0.434106	0.0154644
$789$	27.5124	0.979466
$790$	−4.63877	−0.165040
$791$	−69.1473	−2.45859
$792$	−3.74371	−0.133027
$793$	−30.3709	−1.07850
$794$	4.59496	0.163069
$795$	−97.5754	−3.46064
$796$	19.0496	0.675196
$797$	13.9916	0.495607	0.247804	−	0.968810i	$-0.420291\pi$
$797$	13.9916	0.495607	0.247804	+	0.968810i	$0.420291\pi$
$798$	20.8157	0.736869
$799$	−27.7237	−0.980794
$800$	5.97824	0.211363
$801$	3.02261	0.106799
$802$	−9.30446	−0.328552
$803$	−8.16843	−0.288258
$804$	−20.3436	−0.717463
$805$	−51.0285	−1.79852
$806$	−3.04093	−0.107112
$807$	6.78743	0.238929
$808$	−0.127164	−0.00447361
$809$	46.2211	1.62505	0.812523	−	0.582929i	$-0.198094\pi$
$809$	46.2211	1.62505	0.812523	+	0.582929i	$0.198094\pi$
$810$	−32.3530	−1.13677
$811$	25.7881	0.905543	0.452772	−	0.891627i	$-0.350435\pi$
$811$	25.7881	0.905543	0.452772	+	0.891627i	$0.350435\pi$
$812$	24.4091	0.856590
$813$	31.3012	1.09778
$814$	−1.16124	−0.0407013
$815$	−66.9538	−2.34529
$816$	16.7722	0.587146
$817$	−14.1551	−0.495223
$818$	2.24769	0.0785888
$819$	50.7590	1.77366
$820$	−37.5624	−1.31174
$821$	21.6721	0.756362	0.378181	−	0.925732i	$-0.376550\pi$
$821$	21.6721	0.756362	0.378181	+	0.925732i	$0.376550\pi$
$822$	−27.0444	−0.943282
$823$	25.0951	0.874761	0.437380	−	0.899277i	$-0.355906\pi$
$823$	25.0951	0.874761	0.437380	+	0.899277i	$0.355906\pi$
$824$	2.48638	0.0866171
$825$	19.6856	0.685364
$826$	−14.8930	−0.518194
$827$	−50.1988	−1.74558	−0.872791	−	0.488094i	$-0.837692\pi$
$827$	−50.1988	−1.74558	−0.872791	+	0.488094i	$0.837692\pi$
$828$	11.1942	0.389024
$829$	−8.46058	−0.293848	−0.146924	−	0.989148i	$-0.546937\pi$
$829$	−8.46058	−0.293848	−0.146924	+	0.989148i	$0.546937\pi$
$830$	41.2453	1.43165
$831$	−29.4175	−1.02048
$832$	−4.99110	−0.173035
$833$	−49.0370	−1.69903
$834$	−13.2983	−0.460481
$835$	23.7427	0.821651
$836$	−3.20423	−0.110821
$837$	0.409679	0.0141606
$838$	19.3057	0.666905
$839$	0.279619	0.00965351	0.00482676	−	0.999988i	$-0.498464\pi$
$839$	0.279619	0.00965351	0.00482676	+	0.999988i	$0.498464\pi$
$840$	29.6385	1.02262
$841$	13.5824	0.468358
$842$	3.50045	0.120633
$843$	17.4893	0.602362
$844$	22.2561	0.766088
$845$	39.4655	1.35765
$846$	10.7472	0.369495
$847$	34.0540	1.17011
$848$	12.3146	0.422885
$849$	11.9268	0.409328
$850$	−41.9287	−1.43814
$851$	3.47224	0.119027
$852$	12.6291	0.432667
$853$	−13.2817	−0.454755	−0.227378	−	0.973807i	$-0.573015\pi$
$853$	−13.2817	−0.454755	−0.227378	+	0.973807i	$0.573015\pi$
$854$	−22.7613	−0.778876
$855$	20.9628	0.716913
$856$	14.9595	0.511304
$857$	21.0478	0.718979	0.359489	−	0.933149i	$-0.382951\pi$
$857$	21.0478	0.718979	0.359489	+	0.933149i	$0.382951\pi$
$858$	−16.4350	−0.561083
$859$	−58.0598	−1.98097	−0.990487	−	0.137606i	$-0.956059\pi$
$859$	−58.0598	−1.98097	−0.990487	+	0.137606i	$0.956059\pi$
$860$	−20.1547	−0.687269
$861$	−101.409	−3.45601
$862$	−8.99874	−0.306498
$863$	38.4311	1.30821	0.654105	−	0.756404i	$-0.273045\pi$
$863$	38.4311	1.30821	0.654105	+	0.756404i	$0.273045\pi$
$864$	0.672409	0.0228758
$865$	18.0017	0.612076
$866$	20.7019	0.703481
$867$	−76.9791	−2.61435
$868$	−2.27901	−0.0773547
$869$	1.92778	0.0653955
$870$	51.7053	1.75297
$871$	−42.4591	−1.43867
$872$	−12.2769	−0.415747
$873$	11.7233	0.396772
$874$	9.58104	0.324083
$875$	−12.1241	−0.409869
$876$	−14.1864	−0.479312
$877$	4.62989	0.156340	0.0781701	−	0.996940i	$-0.475092\pi$
$877$	4.62989	0.156340	0.0781701	+	0.996940i	$0.475092\pi$
$878$	16.0392	0.541298
$879$	−34.3168	−1.15748
$880$	−4.56234	−0.153796
$881$	1.68494	0.0567671	0.0283835	−	0.999597i	$-0.490964\pi$
$881$	1.68494	0.0567671	0.0283835	+	0.999597i	$0.490964\pi$
$882$	19.0093	0.640078
$883$	51.3920	1.72948	0.864739	−	0.502222i	$-0.167484\pi$
$883$	51.3920	1.72948	0.864739	+	0.502222i	$0.167484\pi$
$884$	35.0053	1.17736
$885$	−31.5476	−1.06046
$886$	−15.0022	−0.504009
$887$	18.4162	0.618355	0.309178	−	0.951004i	$-0.399946\pi$
$887$	18.4162	0.618355	0.309178	+	0.951004i	$0.399946\pi$
$888$	−2.01675	−0.0676778
$889$	−20.1342	−0.675281
$890$	3.68356	0.123473
$891$	13.4453	0.450434
$892$	18.5639	0.621565
$893$	9.19847	0.307815
$894$	0.0580636	0.00194194
$895$	22.8037	0.762243
$896$	−3.74056	−0.124963
$897$	49.1428	1.64083
$898$	−8.99756	−0.300253
$899$	−3.97581	−0.132601
$900$	16.2538	0.541793
$901$	−86.3691	−2.87737
$902$	15.6102	0.519762
$903$	−54.4125	−1.81074
$904$	18.4858	0.614830
$905$	−16.1075	−0.535431
$906$	14.9678	0.497272
$907$	−0.256929	−0.00853118	−0.00426559	−	0.999991i	$-0.501358\pi$
$907$	−0.256929	−0.00853118	−0.00426559	+	0.999991i	$0.501358\pi$
$908$	−23.4605	−0.778565
$909$	−0.345736	−0.0114673
$910$	61.8584	2.05059
$911$	−46.3824	−1.53672	−0.768358	−	0.640020i	$-0.778926\pi$
$911$	−46.3824	−1.53672	−0.768358	+	0.640020i	$0.778926\pi$
$912$	−5.56488	−0.184271
$913$	−17.1407	−0.567276
$914$	37.7294	1.24798
$915$	−48.2149	−1.59393
$916$	4.80141	0.158643
$917$	−73.7003	−2.43380
$918$	−4.71598	−0.155650
$919$	16.5785	0.546875	0.273438	−	0.961890i	$-0.411839\pi$
$919$	16.5785	0.546875	0.273438	+	0.961890i	$0.411839\pi$
$920$	13.6420	0.449762
$921$	9.30116	0.306484
$922$	12.9509	0.426516
$923$	26.3582	0.867592
$924$	−12.3172	−0.405205
$925$	5.04165	0.165768
$926$	18.1281	0.595725
$927$	6.76002	0.222028
$928$	−6.52552	−0.214211
$929$	−43.3689	−1.42289	−0.711444	−	0.702743i	$-0.751958\pi$
$929$	−43.3689	−1.42289	−0.711444	+	0.702743i	$0.751958\pi$
$930$	−4.82759	−0.158303
$931$	16.2700	0.533229
$932$	−28.2427	−0.925121
$933$	−39.5403	−1.29449
$934$	30.1463	0.986418
$935$	31.9982	1.04645
$936$	−13.5699	−0.443546
$937$	4.60955	0.150587	0.0752937	−	0.997161i	$-0.476011\pi$
$937$	4.60955	0.150587	0.0752937	+	0.997161i	$0.476011\pi$
$938$	−31.8207	−1.03898
$939$	−48.1444	−1.57113
$940$	13.0972	0.427185
$941$	18.5612	0.605077	0.302539	−	0.953137i	$-0.402166\pi$
$941$	18.5612	0.605077	0.302539	+	0.953137i	$0.402166\pi$
$942$	17.3409	0.564997
$943$	−46.6764	−1.51999
$944$	3.98150	0.129587
$945$	−8.33367	−0.271094
$946$	8.37588	0.272323
$947$	−14.2098	−0.461757	−0.230879	−	0.972983i	$-0.574160\pi$
$947$	−14.2098	−0.461757	−0.230879	+	0.972983i	$0.574160\pi$
$948$	3.34803	0.108739
$949$	−29.6083	−0.961126
$950$	13.9116	0.451351
$951$	−60.5397	−1.96313
$952$	26.2346	0.850267
$953$	−17.1330	−0.554991	−0.277496	−	0.960727i	$-0.589504\pi$
$953$	−17.1330	−0.554991	−0.277496	+	0.960727i	$0.589504\pi$
$954$	33.4812	1.08399
$955$	−66.4170	−2.14920
$956$	−1.03765	−0.0335601
$957$	−21.4877	−0.694598
$958$	−33.2213	−1.07333
$959$	−42.3019	−1.36600
$960$	−7.92355	−0.255731
$961$	−30.6288	−0.988025
$962$	−4.20916	−0.135709
$963$	40.6721	1.31064
$964$	9.61178	0.309575
$965$	−15.4342	−0.496844
$966$	36.8298	1.18498
$967$	−40.6903	−1.30851	−0.654256	−	0.756273i	$-0.727018\pi$
$967$	−40.6903	−1.30851	−0.654256	+	0.756273i	$0.727018\pi$
$968$	−9.10398	−0.292613
$969$	39.0295	1.25381
$970$	14.2868	0.458720
$971$	−32.3799	−1.03912	−0.519560	−	0.854434i	$-0.673904\pi$
$971$	−32.3799	−1.03912	−0.519560	+	0.854434i	$0.673904\pi$
$972$	21.3336	0.684275
$973$	−20.8007	−0.666839
$974$	−33.4795	−1.07275
$975$	71.3548	2.28518
$976$	6.08501	0.194776
$977$	12.4071	0.396939	0.198469	−	0.980107i	$-0.436403\pi$
$977$	12.4071	0.396939	0.198469	+	0.980107i	$0.436403\pi$
$978$	48.3239	1.54523
$979$	−1.53081	−0.0489250
$980$	23.1661	0.740014
$981$	−33.3786	−1.06570
$982$	−8.45113	−0.269686
$983$	−15.3179	−0.488566	−0.244283	−	0.969704i	$-0.578553\pi$
$983$	−15.3179	−0.488566	−0.244283	+	0.969704i	$0.578553\pi$
$984$	27.1107	0.864257
$985$	1.43834	0.0458294
$986$	45.7670	1.45752
$987$	35.3592	1.12550
$988$	−11.6144	−0.369505
$989$	−25.0449	−0.796382
$990$	−12.4042	−0.394231
$991$	54.2159	1.72223	0.861113	−	0.508414i	$-0.169768\pi$
$991$	54.2159	1.72223	0.861113	+	0.508414i	$0.169768\pi$
$992$	0.609271	0.0193444
$993$	37.3371	1.18486
$994$	19.7541	0.626561
$995$	63.1179	2.00097
$996$	−29.7688	−0.943262
$997$	−60.5176	−1.91661	−0.958306	−	0.285743i	$-0.907760\pi$
$997$	−60.5176	−1.91661	−0.958306	+	0.285743i	$0.907760\pi$
$998$	34.0769	1.07869
$999$	0.567065	0.0179411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.d.1.14	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.d.1.14	✓	92	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.39141 q^{3} +1.00000 q^{4} +3.31334 q^{5} -2.39141 q^{6} -3.74056 q^{7} +1.00000 q^{8} +2.71882 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.39141 q^{3} +1.00000 q^{4} +3.31334 q^{5} -2.39141 q^{6} -3.74056 q^{7} +1.00000 q^{8} +2.71882 q^{9} +3.31334 q^{10} -1.37696 q^{11} -2.39141 q^{12} -4.99110 q^{13} -3.74056 q^{14} -7.92355 q^{15} +1.00000 q^{16} -7.01355 q^{17} +2.71882 q^{18} +2.32703 q^{19} +3.31334 q^{20} +8.94519 q^{21} -1.37696 q^{22} +4.11728 q^{23} -2.39141 q^{24} +5.97824 q^{25} -4.99110 q^{26} +0.672409 q^{27} -3.74056 q^{28} -6.52552 q^{29} -7.92355 q^{30} +0.609271 q^{31} +1.00000 q^{32} +3.29287 q^{33} -7.01355 q^{34} -12.3937 q^{35} +2.71882 q^{36} +0.843333 q^{37} +2.32703 q^{38} +11.9357 q^{39} +3.31334 q^{40} -11.3367 q^{41} +8.94519 q^{42} -6.08288 q^{43} -1.37696 q^{44} +9.00839 q^{45} +4.11728 q^{46} +3.95288 q^{47} -2.39141 q^{48} +6.99176 q^{49} +5.97824 q^{50} +16.7722 q^{51} -4.99110 q^{52} +12.3146 q^{53} +0.672409 q^{54} -4.56234 q^{55} -3.74056 q^{56} -5.56488 q^{57} -6.52552 q^{58} +3.98150 q^{59} -7.92355 q^{60} +6.08501 q^{61} +0.609271 q^{62} -10.1699 q^{63} +1.00000 q^{64} -16.5372 q^{65} +3.29287 q^{66} +8.50696 q^{67} -7.01355 q^{68} -9.84609 q^{69} -12.3937 q^{70} -5.28105 q^{71} +2.71882 q^{72} +5.93222 q^{73} +0.843333 q^{74} -14.2964 q^{75} +2.32703 q^{76} +5.15059 q^{77} +11.9357 q^{78} -1.40003 q^{79} +3.31334 q^{80} -9.76447 q^{81} -11.3367 q^{82} +12.4483 q^{83} +8.94519 q^{84} -23.2383 q^{85} -6.08288 q^{86} +15.6052 q^{87} -1.37696 q^{88} +1.11174 q^{89} +9.00839 q^{90} +18.6695 q^{91} +4.11728 q^{92} -1.45701 q^{93} +3.95288 q^{94} +7.71025 q^{95} -2.39141 q^{96} +4.31189 q^{97} +6.99176 q^{98} -3.74371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9	\( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 92 * q + 92 * q^2 + 31 * q^3 + 92 * q^4 + 28 * q^5 + 31 * q^6 + 29 * q^7 + 92 * q^8 + 113 * q^9 + 28 * q^10 + 37 * q^11 + 31 * q^12 + 20 * q^13 + 29 * q^14 + 30 * q^15 + 92 * q^16 + 52 * q^17 + 113 * q^18 + 61 * q^19 + 28 * q^20 + 5 * q^21 + 37 * q^22 + 71 * q^23 + 31 * q^24 + 118 * q^25 + 20 * q^26 + 112 * q^27 + 29 * q^28 + 30 * q^29 + 30 * q^30 + 89 * q^31 + 92 * q^32 + 52 * q^33 + 52 * q^34 + 58 * q^35 + 113 * q^36 + 15 * q^37 + 61 * q^38 + 43 * q^39 + 28 * q^40 + 75 * q^41 + 5 * q^42 + 46 * q^43 + 37 * q^44 + 63 * q^45 + 71 * q^46 + 92 * q^47 + 31 * q^48 + 131 * q^49 + 118 * q^50 + 45 * q^51 + 20 * q^52 + 72 * q^53 + 112 * q^54 + 86 * q^55 + 29 * q^56 + 44 * q^57 + 30 * q^58 + 95 * q^59 + 30 * q^60 - 4 * q^61 + 89 * q^62 + 67 * q^63 + 92 * q^64 + 55 * q^65 + 52 * q^66 + 40 * q^67 + 52 * q^68 + 25 * q^69 + 58 * q^70 + 84 * q^71 + 113 * q^72 + 87 * q^73 + 15 * q^74 + 132 * q^75 + 61 * q^76 + 96 * q^77 + 43 * q^78 + 68 * q^79 + 28 * q^80 + 156 * q^81 + 75 * q^82 + 120 * q^83 + 5 * q^84 - 14 * q^85 + 46 * q^86 + 73 * q^87 + 37 * q^88 + 86 * q^89 + 63 * q^90 + 93 * q^91 + 71 * q^92 + 29 * q^93 + 92 * q^94 + 67 * q^95 + 31 * q^96 + 65 * q^97 + 131 * q^98 + 94 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more