LMFDB - Embedded newform 8041.2.a.j.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00119 q^{2} -0.510214 q^{3} +2.00475 q^{4} +2.33538 q^{5} +1.02103 q^{6} -3.93653 q^{7} -0.00949577 q^{8} -2.73968 q^{9} +O(q^{10})$	\(q-2.00119 q^{2} -0.510214 q^{3} +2.00475 q^{4} +2.33538 q^{5} +1.02103 q^{6} -3.93653 q^{7} -0.00949577 q^{8} -2.73968 q^{9} -4.67353 q^{10} +1.00000 q^{11} -1.02285 q^{12} +5.23418 q^{13} +7.87773 q^{14} -1.19154 q^{15} -3.99049 q^{16} +1.00000 q^{17} +5.48261 q^{18} -2.97160 q^{19} +4.68184 q^{20} +2.00847 q^{21} -2.00119 q^{22} +0.0593703 q^{23} +0.00484488 q^{24} +0.453998 q^{25} -10.4746 q^{26} +2.92847 q^{27} -7.89174 q^{28} -9.76766 q^{29} +2.38450 q^{30} -0.151436 q^{31} +8.00470 q^{32} -0.510214 q^{33} -2.00119 q^{34} -9.19330 q^{35} -5.49236 q^{36} -1.80139 q^{37} +5.94671 q^{38} -2.67055 q^{39} -0.0221762 q^{40} +7.48468 q^{41} -4.01933 q^{42} +1.00000 q^{43} +2.00475 q^{44} -6.39820 q^{45} -0.118811 q^{46} -1.63126 q^{47} +2.03600 q^{48} +8.49628 q^{49} -0.908534 q^{50} -0.510214 q^{51} +10.4932 q^{52} +2.27586 q^{53} -5.86041 q^{54} +2.33538 q^{55} +0.0373804 q^{56} +1.51615 q^{57} +19.5469 q^{58} +4.09795 q^{59} -2.38874 q^{60} -7.28320 q^{61} +0.303052 q^{62} +10.7848 q^{63} -8.03792 q^{64} +12.2238 q^{65} +1.02103 q^{66} +12.8220 q^{67} +2.00475 q^{68} -0.0302916 q^{69} +18.3975 q^{70} -9.00003 q^{71} +0.0260154 q^{72} -15.6354 q^{73} +3.60492 q^{74} -0.231636 q^{75} -5.95729 q^{76} -3.93653 q^{77} +5.34427 q^{78} +13.8794 q^{79} -9.31930 q^{80} +6.72490 q^{81} -14.9782 q^{82} -5.94659 q^{83} +4.02648 q^{84} +2.33538 q^{85} -2.00119 q^{86} +4.98360 q^{87} -0.00949577 q^{88} -9.06518 q^{89} +12.8040 q^{90} -20.6045 q^{91} +0.119022 q^{92} +0.0772650 q^{93} +3.26444 q^{94} -6.93980 q^{95} -4.08411 q^{96} +11.3857 q^{97} -17.0026 q^{98} -2.73968 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) $ 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * 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 $

$2$	−2.00119	−1.41505	−0.707526	−	0.706687i	$-0.750189\pi$
$2$	−2.00119	−1.41505	−0.707526	+	0.706687i	$0.750189\pi$
$3$	−0.510214	−0.294572	−0.147286	−	0.989094i	$-0.547054\pi$
$3$	−0.510214	−0.294572	−0.147286	+	0.989094i	$0.547054\pi$
$4$	2.00475	1.00237
$5$	2.33538	1.04441	0.522207	−	0.852819i	$-0.325109\pi$
$5$	2.33538	1.04441	0.522207	+	0.852819i	$0.325109\pi$
$6$	1.02103	0.416835
$7$	−3.93653	−1.48787	−0.743935	−	0.668252i	$-0.767043\pi$
$7$	−3.93653	−1.48787	−0.743935	+	0.668252i	$0.767043\pi$
$8$	−0.00949577	−0.00335726
$9$	−2.73968	−0.913227
$10$	−4.67353	−1.47790
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.02285	−0.295271
$13$	5.23418	1.45170	0.725850	−	0.687853i	$-0.241447\pi$
$13$	5.23418	1.45170	0.725850	+	0.687853i	$0.241447\pi$
$14$	7.87773	2.10541
$15$	−1.19154	−0.307655
$16$	−3.99049	−0.997622
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	5.48261	1.29226
$19$	−2.97160	−0.681731	−0.340865	−	0.940112i	$-0.610720\pi$
$19$	−2.97160	−0.681731	−0.340865	+	0.940112i	$0.610720\pi$
$20$	4.68184	1.04689
$21$	2.00847	0.438285
$22$	−2.00119	−0.426654
$23$	0.0593703	0.0123796	0.00618978	−	0.999981i	$-0.498030\pi$
$23$	0.0593703	0.0123796	0.00618978	+	0.999981i	$0.498030\pi$
$24$	0.00484488	0.000988957 0
$25$	0.453998	0.0907996
$26$	−10.4746	−2.05423
$27$	2.92847	0.563584
$28$	−7.89174	−1.49140
$29$	−9.76766	−1.81381	−0.906904	−	0.421337i	$-0.861561\pi$
$29$	−9.76766	−1.81381	−0.906904	+	0.421337i	$0.861561\pi$
$30$	2.38450	0.435348
$31$	−0.151436	−0.0271988	−0.0135994	−	0.999908i	$-0.504329\pi$
$31$	−0.151436	−0.0271988	−0.0135994	+	0.999908i	$0.504329\pi$
$32$	8.00470	1.41504
$33$	−0.510214	−0.0888169
$34$	−2.00119	−0.343201
$35$	−9.19330	−1.55395
$36$	−5.49236	−0.915394
$37$	−1.80139	−0.296147	−0.148073	−	0.988976i	$-0.547307\pi$
$37$	−1.80139	−0.296147	−0.148073	+	0.988976i	$0.547307\pi$
$38$	5.94671	0.964684
$39$	−2.67055	−0.427631
$40$	−0.0221762	−0.00350637
$41$	7.48468	1.16891	0.584456	−	0.811426i	$-0.301308\pi$
$41$	7.48468	1.16891	0.584456	+	0.811426i	$0.301308\pi$
$42$	−4.01933	−0.620196
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	2.00475	0.302227
$45$	−6.39820	−0.953787
$46$	−0.118811	−0.0175177
$47$	−1.63126	−0.237943	−0.118972	−	0.992898i	$-0.537960\pi$
$47$	−1.63126	−0.237943	−0.118972	+	0.992898i	$0.537960\pi$
$48$	2.03600	0.293872
$49$	8.49628	1.21375
$50$	−0.908534	−0.128486
$51$	−0.510214	−0.0714443
$52$	10.4932	1.45514
$53$	2.27586	0.312613	0.156307	−	0.987709i	$-0.450041\pi$
$53$	2.27586	0.312613	0.156307	+	0.987709i	$0.450041\pi$
$54$	−5.86041	−0.797500
$55$	2.33538	0.314903
$56$	0.0373804	0.00499517
$57$	1.51615	0.200819
$58$	19.5469	2.56663
$59$	4.09795	0.533507	0.266754	−	0.963765i	$-0.414049\pi$
$59$	4.09795	0.533507	0.266754	+	0.963765i	$0.414049\pi$
$60$	−2.38874	−0.308385
$61$	−7.28320	−0.932518	−0.466259	−	0.884648i	$-0.654399\pi$
$61$	−7.28320	−0.932518	−0.466259	+	0.884648i	$0.654399\pi$
$62$	0.303052	0.0384877
$63$	10.7848	1.35876
$64$	−8.03792	−1.00474
$65$	12.2238	1.51618
$66$	1.02103	0.125681
$67$	12.8220	1.56645	0.783227	−	0.621736i	$-0.213572\pi$
$67$	12.8220	1.56645	0.783227	+	0.621736i	$0.213572\pi$
$68$	2.00475	0.243111
$69$	−0.0302916	−0.00364668
$70$	18.3975	2.19892
$71$	−9.00003	−1.06811	−0.534054	−	0.845451i	$-0.679332\pi$
$71$	−9.00003	−1.06811	−0.534054	+	0.845451i	$0.679332\pi$
$72$	0.0260154	0.00306594
$73$	−15.6354	−1.82999	−0.914995	−	0.403466i	$-0.867805\pi$
$73$	−15.6354	−1.82999	−0.914995	+	0.403466i	$0.867805\pi$
$74$	3.60492	0.419063
$75$	−0.231636	−0.0267470
$76$	−5.95729	−0.683348
$77$	−3.93653	−0.448609
$78$	5.34427	0.605120
$79$	13.8794	1.56156	0.780778	−	0.624808i	$-0.214823\pi$
$79$	13.8794	1.56156	0.780778	+	0.624808i	$0.214823\pi$
$80$	−9.31930	−1.04193
$81$	6.72490	0.747211
$82$	−14.9782	−1.65407
$83$	−5.94659	−0.652723	−0.326361	−	0.945245i	$-0.605823\pi$
$83$	−5.94659	−0.652723	−0.326361	+	0.945245i	$0.605823\pi$
$84$	4.02648	0.439325
$85$	2.33538	0.253307
$86$	−2.00119	−0.215793
$87$	4.98360	0.534298
$88$	−0.00949577	−0.00101225
$89$	−9.06518	−0.960907	−0.480453	−	0.877020i	$-0.659528\pi$
$89$	−9.06518	−0.960907	−0.480453	+	0.877020i	$0.659528\pi$
$90$	12.8040	1.34966
$91$	−20.6045	−2.15994
$92$	0.119022	0.0124089
$93$	0.0772650	0.00801201
$94$	3.26444	0.336702
$95$	−6.93980	−0.712009
$96$	−4.08411	−0.416833
$97$	11.3857	1.15605	0.578023	−	0.816020i	$-0.303824\pi$
$97$	11.3857	1.15605	0.578023	+	0.816020i	$0.303824\pi$
$98$	−17.0026	−1.71753
$99$	−2.73968	−0.275348
$100$	0.910150	0.0910150
$101$	−4.24687	−0.422579	−0.211290	−	0.977423i	$-0.567766\pi$
$101$	−4.24687	−0.422579	−0.211290	+	0.977423i	$0.567766\pi$
$102$	1.02103	0.101097
$103$	3.21297	0.316583	0.158292	−	0.987392i	$-0.449401\pi$
$103$	3.21297	0.316583	0.158292	+	0.987392i	$0.449401\pi$
$104$	−0.0497026	−0.00487374
$105$	4.69055	0.457751
$106$	−4.55442	−0.442364
$107$	15.6026	1.50836	0.754181	−	0.656666i	$-0.228034\pi$
$107$	15.6026	1.50836	0.754181	+	0.656666i	$0.228034\pi$
$108$	5.87083	0.564921
$109$	−7.44771	−0.713361	−0.356680	−	0.934226i	$-0.616091\pi$
$109$	−7.44771	−0.713361	−0.356680	+	0.934226i	$0.616091\pi$
$110$	−4.67353	−0.445603
$111$	0.919096	0.0872367
$112$	15.7087	1.48433
$113$	−13.7104	−1.28977	−0.644883	−	0.764282i	$-0.723094\pi$
$113$	−13.7104	−1.28977	−0.644883	+	0.764282i	$0.723094\pi$
$114$	−3.03410	−0.284169
$115$	0.138652	0.0129294
$116$	−19.5817	−1.81811
$117$	−14.3400	−1.32573
$118$	−8.20075	−0.754940
$119$	−3.93653	−0.360861
$120$	0.0113146	0.00103288
$121$	1.00000	0.0909091
$122$	14.5750	1.31956
$123$	−3.81879	−0.344329
$124$	−0.303591	−0.0272633
$125$	−10.6166	−0.949581
$126$	−21.5825	−1.92272
$127$	9.22368	0.818469	0.409235	−	0.912429i	$-0.365796\pi$
$127$	9.22368	0.818469	0.409235	+	0.912429i	$0.365796\pi$
$128$	0.0759660	0.00671451
$129$	−0.510214	−0.0449219
$130$	−24.4621	−2.14547
$131$	6.83688	0.597341	0.298671	−	0.954356i	$-0.403457\pi$
$131$	6.83688	0.597341	0.298671	+	0.954356i	$0.403457\pi$
$132$	−1.02285	−0.0890276
$133$	11.6978	1.01433
$134$	−25.6592	−2.21661
$135$	6.83908	0.588615
$136$	−0.00949577	−0.000814256 0
$137$	16.5748	1.41608	0.708039	−	0.706173i	$-0.249580\pi$
$137$	16.5748	1.41608	0.708039	+	0.706173i	$0.249580\pi$
$138$	0.0606191	0.00516024
$139$	−14.8185	−1.25689	−0.628446	−	0.777853i	$-0.716309\pi$
$139$	−14.8185	−1.25689	−0.628446	+	0.777853i	$0.716309\pi$
$140$	−18.4302	−1.55764
$141$	0.832290	0.0700914
$142$	18.0107	1.51143
$143$	5.23418	0.437704
$144$	10.9327	0.911055
$145$	−22.8112	−1.89437
$146$	31.2894	2.58953
$147$	−4.33493	−0.357539
$148$	−3.61133	−0.296850
$149$	−2.86405	−0.234632	−0.117316	−	0.993095i	$-0.537429\pi$
$149$	−2.86405	−0.234632	−0.117316	+	0.993095i	$0.537429\pi$
$150$	0.463547	0.0378485
$151$	−0.818528	−0.0666109	−0.0333054	−	0.999445i	$-0.510603\pi$
$151$	−0.818528	−0.0666109	−0.0333054	+	0.999445i	$0.510603\pi$
$152$	0.0282176	0.00228875
$153$	−2.73968	−0.221490
$154$	7.87773	0.634806
$155$	−0.353661	−0.0284068
$156$	−5.35378	−0.428645
$157$	4.17895	0.333517	0.166758	−	0.985998i	$-0.446670\pi$
$157$	4.17895	0.333517	0.166758	+	0.985998i	$0.446670\pi$
$158$	−27.7753	−2.20968
$159$	−1.16118	−0.0920872
$160$	18.6940	1.47789
$161$	−0.233713	−0.0184192
$162$	−13.4578	−1.05734
$163$	−1.56120	−0.122283	−0.0611413	−	0.998129i	$-0.519474\pi$
$163$	−1.56120	−0.122283	−0.0611413	+	0.998129i	$0.519474\pi$
$164$	15.0049	1.17168
$165$	−1.19154	−0.0927616
$166$	11.9002	0.923637
$167$	8.02859	0.621271	0.310636	−	0.950529i	$-0.399458\pi$
$167$	8.02859	0.621271	0.310636	+	0.950529i	$0.399458\pi$
$168$	−0.0190720	−0.00147144
$169$	14.3966	1.10743
$170$	−4.67353	−0.358443
$171$	8.14122	0.622575
$172$	2.00475	0.152860
$173$	−23.0098	−1.74941	−0.874703	−	0.484659i	$-0.838944\pi$
$173$	−23.0098	−1.74941	−0.874703	+	0.484659i	$0.838944\pi$
$174$	−9.97311	−0.756059
$175$	−1.78718	−0.135098
$176$	−3.99049	−0.300794
$177$	−2.09083	−0.157156
$178$	18.1411	1.35973
$179$	−14.1309	−1.05619	−0.528096	−	0.849185i	$-0.677094\pi$
$179$	−14.1309	−1.05619	−0.528096	+	0.849185i	$0.677094\pi$
$180$	−12.8268	−0.956050
$181$	14.2374	1.05826	0.529130	−	0.848541i	$-0.322518\pi$
$181$	14.2374	1.05826	0.529130	+	0.848541i	$0.322518\pi$
$182$	41.2335	3.05643
$183$	3.71599	0.274694
$184$	−0.000563767 0	−4.15614e−5 0
$185$	−4.20693	−0.309300
$186$	−0.154622	−0.0113374
$187$	1.00000	0.0731272
$188$	−3.27025	−0.238508
$189$	−11.5280	−0.838539
$190$	13.8878	1.00753
$191$	6.93198	0.501580	0.250790	−	0.968041i	$-0.419310\pi$
$191$	6.93198	0.501580	0.250790	+	0.968041i	$0.419310\pi$
$192$	4.10106	0.295968
$193$	−0.666990	−0.0480110	−0.0240055	−	0.999712i	$-0.507642\pi$
$193$	−0.666990	−0.0480110	−0.0240055	+	0.999712i	$0.507642\pi$
$194$	−22.7850	−1.63587
$195$	−6.23676	−0.446623
$196$	17.0329	1.21663
$197$	7.56580	0.539041	0.269520	−	0.962995i	$-0.413135\pi$
$197$	7.56580	0.539041	0.269520	+	0.962995i	$0.413135\pi$
$198$	5.48261	0.389632
$199$	18.7514	1.32925	0.664627	−	0.747176i	$-0.268591\pi$
$199$	18.7514	1.32925	0.664627	+	0.747176i	$0.268591\pi$
$200$	−0.00431106	−0.000304838 0
$201$	−6.54195	−0.461434
$202$	8.49877	0.597972
$203$	38.4507	2.69871
$204$	−1.02285	−0.0716138
$205$	17.4796	1.22083
$206$	−6.42975	−0.447982
$207$	−0.162656	−0.0113054
$208$	−20.8869	−1.44825
$209$	−2.97160	−0.205550
$210$	−9.38666	−0.647741
$211$	9.51422	0.654986	0.327493	−	0.944854i	$-0.393796\pi$
$211$	9.51422	0.654986	0.327493	+	0.944854i	$0.393796\pi$
$212$	4.56252	0.313355
$213$	4.59194	0.314635
$214$	−31.2237	−2.13441
$215$	2.33538	0.159272
$216$	−0.0278081	−0.00189210
$217$	0.596134	0.0404682
$218$	14.9042	1.00944
$219$	7.97742	0.539064
$220$	4.68184	0.315650
$221$	5.23418	0.352089
$222$	−1.83928	−0.123444
$223$	9.20959	0.616720	0.308360	−	0.951270i	$-0.400220\pi$
$223$	9.20959	0.616720	0.308360	+	0.951270i	$0.400220\pi$
$224$	−31.5108	−2.10540
$225$	−1.24381	−0.0829206
$226$	27.4370	1.82508
$227$	0.569489	0.0377983	0.0188992	−	0.999821i	$-0.493984\pi$
$227$	0.569489	0.0377983	0.0188992	+	0.999821i	$0.493984\pi$
$228$	3.03949	0.201295
$229$	6.29506	0.415989	0.207995	−	0.978130i	$-0.433306\pi$
$229$	6.29506	0.415989	0.207995	+	0.978130i	$0.433306\pi$
$230$	−0.277469	−0.0182957
$231$	2.00847	0.132148
$232$	0.0927514	0.00608943
$233$	−15.0067	−0.983119	−0.491559	−	0.870844i	$-0.663573\pi$
$233$	−15.0067	−0.983119	−0.491559	+	0.870844i	$0.663573\pi$
$234$	28.6970	1.87598
$235$	−3.80960	−0.248511
$236$	8.21534	0.534773
$237$	−7.08148	−0.459991
$238$	7.87773	0.510638
$239$	23.2725	1.50537	0.752687	−	0.658379i	$-0.228757\pi$
$239$	23.2725	1.50537	0.752687	+	0.658379i	$0.228757\pi$
$240$	4.75484	0.306924
$241$	5.92917	0.381931	0.190966	−	0.981597i	$-0.438838\pi$
$241$	5.92917	0.381931	0.190966	+	0.981597i	$0.438838\pi$
$242$	−2.00119	−0.128641
$243$	−12.2165	−0.783691
$244$	−14.6010	−0.934731
$245$	19.8421	1.26766
$246$	7.64211	0.487243
$247$	−15.5539	−0.989669
$248$	0.00143800	9.13134e−5 0
$249$	3.03403	0.192274
$250$	21.2459	1.34371
$251$	5.75160	0.363038	0.181519	−	0.983387i	$-0.441899\pi$
$251$	5.75160	0.363038	0.181519	+	0.983387i	$0.441899\pi$
$252$	21.6209	1.36199
$253$	0.0593703	0.00373258
$254$	−18.4583	−1.15818
$255$	−1.19154	−0.0746174
$256$	15.9238	0.995238
$257$	3.40258	0.212247	0.106123	−	0.994353i	$-0.466156\pi$
$257$	3.40258	0.212247	0.106123	+	0.994353i	$0.466156\pi$
$258$	1.02103	0.0635668
$259$	7.09124	0.440628
$260$	24.5056	1.51977
$261$	26.7603	1.65642
$262$	−13.6819	−0.845269
$263$	12.9937	0.801227	0.400613	−	0.916247i	$-0.368797\pi$
$263$	12.9937	0.801227	0.400613	+	0.916247i	$0.368797\pi$
$264$	0.00484488	0.000298182 0
$265$	5.31499	0.326497
$266$	−23.4094	−1.43532
$267$	4.62518	0.283057
$268$	25.7048	1.57017
$269$	−28.8142	−1.75683	−0.878416	−	0.477897i	$-0.841399\pi$
$269$	−28.8142	−1.75683	−0.878416	+	0.477897i	$0.841399\pi$
$270$	−13.6863	−0.832920
$271$	21.0282	1.27737	0.638685	−	0.769468i	$-0.279479\pi$
$271$	21.0282	1.27737	0.638685	+	0.769468i	$0.279479\pi$
$272$	−3.99049	−0.241959
$273$	10.5127	0.636259
$274$	−33.1692	−2.00383
$275$	0.453998	0.0273771
$276$	−0.0607269	−0.00365533
$277$	−15.5587	−0.934830	−0.467415	−	0.884038i	$-0.654815\pi$
$277$	−15.5587	−0.934830	−0.467415	+	0.884038i	$0.654815\pi$
$278$	29.6546	1.77857
$279$	0.414887	0.0248387
$280$	0.0872974	0.00521702
$281$	−4.25632	−0.253911	−0.126955	−	0.991908i	$-0.540520\pi$
$281$	−4.25632	−0.253911	−0.126955	+	0.991908i	$0.540520\pi$
$282$	−1.66557	−0.0991830
$283$	−27.4760	−1.63328	−0.816639	−	0.577148i	$-0.804165\pi$
$283$	−27.4760	−1.63328	−0.816639	+	0.577148i	$0.804165\pi$
$284$	−18.0428	−1.07064
$285$	3.54079	0.209738
$286$	−10.4746	−0.619374
$287$	−29.4637	−1.73919
$288$	−21.9303	−1.29226
$289$	1.00000	0.0588235
$290$	45.6494	2.68063
$291$	−5.80916	−0.340539
$292$	−31.3451	−1.83433
$293$	−22.3199	−1.30394	−0.651972	−	0.758243i	$-0.726058\pi$
$293$	−22.3199	−1.30394	−0.651972	+	0.758243i	$0.726058\pi$
$294$	8.67499	0.505936
$295$	9.57026	0.557202
$296$	0.0171056	0.000994243 0
$297$	2.92847	0.169927
$298$	5.73150	0.332017
$299$	0.310755	0.0179714
$300$	−0.464371	−0.0268105
$301$	−3.93653	−0.226898
$302$	1.63803	0.0942579
$303$	2.16681	0.124480
$304$	11.8581	0.680109
$305$	−17.0090	−0.973935
$306$	5.48261	0.313420
$307$	−20.3931	−1.16390	−0.581949	−	0.813225i	$-0.697710\pi$
$307$	−20.3931	−1.16390	−0.581949	+	0.813225i	$0.697710\pi$
$308$	−7.89174	−0.449674
$309$	−1.63930	−0.0932566
$310$	0.707742	0.0401971
$311$	11.5468	0.654757	0.327379	−	0.944893i	$-0.393835\pi$
$311$	11.5468	0.654757	0.327379	+	0.944893i	$0.393835\pi$
$312$	0.0253590	0.00143567
$313$	4.96935	0.280884	0.140442	−	0.990089i	$-0.455148\pi$
$313$	4.96935	0.280884	0.140442	+	0.990089i	$0.455148\pi$
$314$	−8.36286	−0.471943
$315$	25.1867	1.41911
$316$	27.8247	1.56526
$317$	−16.5534	−0.929730	−0.464865	−	0.885381i	$-0.653897\pi$
$317$	−16.5534	−0.929730	−0.464865	+	0.885381i	$0.653897\pi$
$318$	2.32373	0.130308
$319$	−9.76766	−0.546884
$320$	−18.7716	−1.04936
$321$	−7.96068	−0.444322
$322$	0.467703	0.0260641
$323$	−2.97160	−0.165344
$324$	13.4817	0.748984
$325$	2.37631	0.131814
$326$	3.12425	0.173036
$327$	3.79993	0.210136
$328$	−0.0710728	−0.00392434
$329$	6.42149	0.354028
$330$	2.38450	0.131262
$331$	8.22636	0.452162	0.226081	−	0.974109i	$-0.427409\pi$
$331$	8.22636	0.452162	0.226081	+	0.974109i	$0.427409\pi$
$332$	−11.9214	−0.654271
$333$	4.93524	0.270449
$334$	−16.0667	−0.879131
$335$	29.9442	1.63602
$336$	−8.01479	−0.437243
$337$	−5.61124	−0.305664	−0.152832	−	0.988252i	$-0.548839\pi$
$337$	−5.61124	−0.305664	−0.152832	+	0.988252i	$0.548839\pi$
$338$	−28.8104	−1.56708
$339$	6.99524	0.379929
$340$	4.68184	0.253908
$341$	−0.151436	−0.00820074
$342$	−16.2921	−0.880976
$343$	−5.89017	−0.318039
$344$	−0.00949577	−0.000511978 0
$345$	−0.0707423	−0.00380864
$346$	46.0470	2.47550
$347$	29.3420	1.57516	0.787580	−	0.616213i	$-0.211334\pi$
$347$	29.3420	1.57516	0.787580	+	0.616213i	$0.211334\pi$
$348$	9.99084	0.535565
$349$	−18.4204	−0.986024	−0.493012	−	0.870023i	$-0.664104\pi$
$349$	−18.4204	−0.986024	−0.493012	+	0.870023i	$0.664104\pi$
$350$	3.57647	0.191171
$351$	15.3281	0.818155
$352$	8.00470	0.426652
$353$	24.7243	1.31594	0.657970	−	0.753044i	$-0.271415\pi$
$353$	24.7243	1.31594	0.657970	+	0.753044i	$0.271415\pi$
$354$	4.18414	0.222385
$355$	−21.0185	−1.11555
$356$	−18.1734	−0.963187
$357$	2.00847	0.106300
$358$	28.2785	1.49457
$359$	8.36752	0.441621	0.220810	−	0.975317i	$-0.429130\pi$
$359$	8.36752	0.441621	0.220810	+	0.975317i	$0.429130\pi$
$360$	0.0607558	0.00320211
$361$	−10.1696	−0.535243
$362$	−28.4918	−1.49749
$363$	−0.510214	−0.0267793
$364$	−41.3068	−2.16507
$365$	−36.5147	−1.91127
$366$	−7.43639	−0.388707
$367$	1.01560	0.0530137	0.0265068	−	0.999649i	$-0.491562\pi$
$367$	1.01560	0.0530137	0.0265068	+	0.999649i	$0.491562\pi$
$368$	−0.236916	−0.0123501
$369$	−20.5057	−1.06748
$370$	8.41885	0.437675
$371$	−8.95899	−0.465128
$372$	0.154897	0.00803101
$373$	19.5396	1.01172	0.505861	−	0.862615i	$-0.331175\pi$
$373$	19.5396	1.01172	0.505861	+	0.862615i	$0.331175\pi$
$374$	−2.00119	−0.103479
$375$	5.41676	0.279720
$376$	0.0154900	0.000798837 0
$377$	−51.1257	−2.63311
$378$	23.0697	1.18658
$379$	−9.99246	−0.513278	−0.256639	−	0.966507i	$-0.582615\pi$
$379$	−9.99246	−0.513278	−0.256639	+	0.966507i	$0.582615\pi$
$380$	−13.9125	−0.713698
$381$	−4.70605	−0.241098
$382$	−13.8722	−0.709762
$383$	−6.99998	−0.357682	−0.178841	−	0.983878i	$-0.557235\pi$
$383$	−6.99998	−0.357682	−0.178841	+	0.983878i	$0.557235\pi$
$384$	−0.0387589	−0.00197791
$385$	−9.19330	−0.468534
$386$	1.33477	0.0679381
$387$	−2.73968	−0.139266
$388$	22.8255	1.15879
$389$	22.8231	1.15718	0.578588	−	0.815620i	$-0.303604\pi$
$389$	22.8231	1.15718	0.578588	+	0.815620i	$0.303604\pi$
$390$	12.4809	0.631995
$391$	0.0593703	0.00300248
$392$	−0.0806788	−0.00407489
$393$	−3.48827	−0.175960
$394$	−15.1406	−0.762771
$395$	32.4137	1.63091
$396$	−5.49236	−0.276002
$397$	7.72213	0.387563	0.193781	−	0.981045i	$-0.437925\pi$
$397$	7.72213	0.387563	0.193781	+	0.981045i	$0.437925\pi$
$398$	−37.5251	−1.88096
$399$	−5.96837	−0.298792
$400$	−1.81167	−0.0905836
$401$	38.6727	1.93122	0.965612	−	0.259987i	$-0.0837184\pi$
$401$	38.6727	1.93122	0.965612	+	0.259987i	$0.0837184\pi$
$402$	13.0917	0.652953
$403$	−0.792645	−0.0394845
$404$	−8.51389	−0.423582
$405$	15.7052	0.780397
$406$	−76.9470	−3.81882
$407$	−1.80139	−0.0892917
$408$	0.00484488	0.000239857 0
$409$	−0.587094	−0.0290299	−0.0145150	−	0.999895i	$-0.504620\pi$
$409$	−0.587094	−0.0290299	−0.0145150	+	0.999895i	$0.504620\pi$
$410$	−34.9799	−1.72753
$411$	−8.45669	−0.417138
$412$	6.44118	0.317334
$413$	−16.1317	−0.793789
$414$	0.325504	0.0159977
$415$	−13.8875	−0.681713
$416$	41.8980	2.05422
$417$	7.56063	0.370246
$418$	5.94671	0.290863
$419$	3.61037	0.176378	0.0881890	−	0.996104i	$-0.471892\pi$
$419$	3.61037	0.176378	0.0881890	+	0.996104i	$0.471892\pi$
$420$	9.40336	0.458837
$421$	36.9882	1.80269	0.901346	−	0.433099i	$-0.142580\pi$
$421$	36.9882	1.80269	0.901346	+	0.433099i	$0.142580\pi$
$422$	−19.0397	−0.926839
$423$	4.46912	0.217296
$424$	−0.0216110	−0.00104952
$425$	0.453998	0.0220221
$426$	−9.18933	−0.445225
$427$	28.6706	1.38747
$428$	31.2793	1.51194
$429$	−2.67055	−0.128936
$430$	−4.67353	−0.225378
$431$	3.29861	0.158888	0.0794442	−	0.996839i	$-0.474685\pi$
$431$	3.29861	0.158888	0.0794442	+	0.996839i	$0.474685\pi$
$432$	−11.6860	−0.562243
$433$	13.4354	0.645666	0.322833	−	0.946456i	$-0.395365\pi$
$433$	13.4354	0.645666	0.322833	+	0.946456i	$0.395365\pi$
$434$	−1.19298	−0.0572646
$435$	11.6386	0.558028
$436$	−14.9308	−0.715053
$437$	−0.176424	−0.00843953
$438$	−15.9643	−0.762804
$439$	0.441484	0.0210709	0.0105354	−	0.999945i	$-0.496646\pi$
$439$	0.441484	0.0210709	0.0105354	+	0.999945i	$0.496646\pi$
$440$	−0.0221762	−0.00105721
$441$	−23.2771	−1.10843
$442$	−10.4746	−0.498224
$443$	−1.34419	−0.0638643	−0.0319321	−	0.999490i	$-0.510166\pi$
$443$	−1.34419	−0.0638643	−0.0319321	+	0.999490i	$0.510166\pi$
$444$	1.84255	0.0874437
$445$	−21.1706	−1.00358
$446$	−18.4301	−0.872691
$447$	1.46128	0.0691162
$448$	31.6415	1.49492
$449$	23.9721	1.13131	0.565657	−	0.824641i	$-0.308623\pi$
$449$	23.9721	1.13131	0.565657	+	0.824641i	$0.308623\pi$
$450$	2.48909	0.117337
$451$	7.48468	0.352440
$452$	−27.4858	−1.29283
$453$	0.417625	0.0196217
$454$	−1.13965	−0.0534866
$455$	−48.1194	−2.25587
$456$	−0.0143970	−0.000674202 0
$457$	−15.3473	−0.717917	−0.358959	−	0.933353i	$-0.616868\pi$
$457$	−15.3473	−0.717917	−0.358959	+	0.933353i	$0.616868\pi$
$458$	−12.5976	−0.588647
$459$	2.92847	0.136689
$460$	0.277962	0.0129601
$461$	−40.5155	−1.88700	−0.943498	−	0.331379i	$-0.892486\pi$
$461$	−40.5155	−1.88700	−0.943498	+	0.331379i	$0.892486\pi$
$462$	−4.01933	−0.186996
$463$	24.2300	1.12606	0.563031	−	0.826436i	$-0.309635\pi$
$463$	24.2300	1.12606	0.563031	+	0.826436i	$0.309635\pi$
$464$	38.9777	1.80949
$465$	0.180443	0.00836785
$466$	30.0311	1.39116
$467$	−21.9439	−1.01544	−0.507721	−	0.861521i	$-0.669512\pi$
$467$	−21.9439	−1.01544	−0.507721	+	0.861521i	$0.669512\pi$
$468$	−28.7480	−1.32888
$469$	−50.4741	−2.33068
$470$	7.62372	0.351656
$471$	−2.13216	−0.0982448
$472$	−0.0389132	−0.00179112
$473$	1.00000	0.0459800
$474$	14.1714	0.650912
$475$	−1.34910	−0.0619008
$476$	−7.89174	−0.361717
$477$	−6.23513	−0.285487
$478$	−46.5726	−2.13018
$479$	3.16129	0.144443	0.0722216	−	0.997389i	$-0.476991\pi$
$479$	3.16129	0.144443	0.0722216	+	0.997389i	$0.476991\pi$
$480$	−9.53795	−0.435346
$481$	−9.42881	−0.429917
$482$	−11.8654	−0.540453
$483$	0.119244	0.00542578
$484$	2.00475	0.0911248
$485$	26.5900	1.20739
$486$	24.4476	1.10896
$487$	−14.6023	−0.661694	−0.330847	−	0.943684i	$-0.607334\pi$
$487$	−14.6023	−0.661694	−0.330847	+	0.943684i	$0.607334\pi$
$488$	0.0691596	0.00313071
$489$	0.796546	0.0360210
$490$	−39.7076	−1.79381
$491$	−23.5476	−1.06269	−0.531344	−	0.847156i	$-0.678313\pi$
$491$	−23.5476	−1.06269	−0.531344	+	0.847156i	$0.678313\pi$
$492$	−7.65571	−0.345146
$493$	−9.76766	−0.439913
$494$	31.1262	1.40043
$495$	−6.39820	−0.287578
$496$	0.604305	0.0271341
$497$	35.4289	1.58920
$498$	−6.07167	−0.272078
$499$	−39.9410	−1.78800	−0.894002	−	0.448063i	$-0.852114\pi$
$499$	−39.9410	−1.78800	−0.894002	+	0.448063i	$0.852114\pi$
$500$	−21.2837	−0.951834
$501$	−4.09630	−0.183009
$502$	−11.5100	−0.513717
$503$	19.9446	0.889287	0.444644	−	0.895708i	$-0.353330\pi$
$503$	19.9446	0.889287	0.444644	+	0.895708i	$0.353330\pi$
$504$	−0.102410	−0.00456172
$505$	−9.91805	−0.441347
$506$	−0.118811	−0.00528179
$507$	−7.34537	−0.326220
$508$	18.4911	0.820411
$509$	−24.6342	−1.09189	−0.545946	−	0.837820i	$-0.683830\pi$
$509$	−24.6342	−1.09189	−0.545946	+	0.837820i	$0.683830\pi$
$510$	2.38450	0.105587
$511$	61.5494	2.72278
$512$	−32.0184	−1.41503
$513$	−8.70222	−0.384212
$514$	−6.80919	−0.300340
$515$	7.50350	0.330644
$516$	−1.02285	−0.0450284
$517$	−1.63126	−0.0717425
$518$	−14.1909	−0.623511
$519$	11.7400	0.515327
$520$	−0.116074	−0.00509020
$521$	14.4156	0.631560	0.315780	−	0.948832i	$-0.397734\pi$
$521$	14.4156	0.631560	0.315780	+	0.948832i	$0.397734\pi$
$522$	−53.5523	−2.34392
$523$	−37.9966	−1.66148	−0.830739	−	0.556663i	$-0.812082\pi$
$523$	−37.9966	−1.66148	−0.830739	+	0.556663i	$0.812082\pi$
$524$	13.7062	0.598758
$525$	0.911843	0.0397961
$526$	−26.0028	−1.13378
$527$	−0.151436	−0.00659667
$528$	2.03600	0.0886057
$529$	−22.9965	−0.999847
$530$	−10.6363	−0.462011
$531$	−11.2271	−0.487213
$532$	23.4511	1.01673
$533$	39.1762	1.69691
$534$	−9.25585	−0.400540
$535$	36.4380	1.57535
$536$	−0.121755	−0.00525899
$537$	7.20977	0.311125
$538$	57.6625	2.48601
$539$	8.49628	0.365961
$540$	13.7106	0.590011
$541$	−2.41675	−0.103904	−0.0519521	−	0.998650i	$-0.516544\pi$
$541$	−2.41675	−0.103904	−0.0519521	+	0.998650i	$0.516544\pi$
$542$	−42.0813	−1.80755
$543$	−7.26414	−0.311734
$544$	8.00470	0.343199
$545$	−17.3932	−0.745044
$546$	−21.0379	−0.900339
$547$	4.93835	0.211148	0.105574	−	0.994411i	$-0.466332\pi$
$547$	4.93835	0.211148	0.105574	+	0.994411i	$0.466332\pi$
$548$	33.2282	1.41944
$549$	19.9537	0.851601
$550$	−0.908534	−0.0387400
$551$	29.0255	1.23653
$552$	0.000287642 0	1.22428e−5 0
$553$	−54.6368	−2.32339
$554$	31.1358	1.32283
$555$	2.14644	0.0911112
$556$	−29.7074	−1.25987
$557$	38.2611	1.62117	0.810587	−	0.585619i	$-0.199148\pi$
$557$	38.2611	1.62117	0.810587	+	0.585619i	$0.199148\pi$
$558$	−0.830267	−0.0351480
$559$	5.23418	0.221382
$560$	36.6857	1.55026
$561$	−0.510214	−0.0215413
$562$	8.51768	0.359297
$563$	−3.87484	−0.163305	−0.0816526	−	0.996661i	$-0.526020\pi$
$563$	−3.87484	−0.163305	−0.0816526	+	0.996661i	$0.526020\pi$
$564$	1.66853	0.0702577
$565$	−32.0190	−1.34705
$566$	54.9846	2.31117
$567$	−26.4728	−1.11175
$568$	0.0854622	0.00358592
$569$	−20.1346	−0.844087	−0.422043	−	0.906576i	$-0.638687\pi$
$569$	−20.1346	−0.844087	−0.422043	+	0.906576i	$0.638687\pi$
$570$	−7.08577	−0.296790
$571$	40.9352	1.71308	0.856542	−	0.516077i	$-0.172608\pi$
$571$	40.9352	1.71308	0.856542	+	0.516077i	$0.172608\pi$
$572$	10.4932	0.438743
$573$	−3.53679	−0.147752
$574$	58.9623	2.46104
$575$	0.0269540	0.00112406
$576$	22.0213	0.917555
$577$	30.3027	1.26152	0.630759	−	0.775979i	$-0.282744\pi$
$577$	30.3027	1.26152	0.630759	+	0.775979i	$0.282744\pi$
$578$	−2.00119	−0.0832384
$579$	0.340308	0.0141427
$580$	−45.7306	−1.89886
$581$	23.4089	0.971166
$582$	11.6252	0.481881
$583$	2.27586	0.0942564
$584$	0.148471	0.00614375
$585$	−33.4893	−1.38461
$586$	44.6663	1.84515
$587$	10.5504	0.435461	0.217731	−	0.976009i	$-0.430135\pi$
$587$	10.5504	0.435461	0.217731	+	0.976009i	$0.430135\pi$
$588$	−8.69042	−0.358387
$589$	0.450008	0.0185422
$590$	−19.1519	−0.788470
$591$	−3.86018	−0.158786
$592$	7.18843	0.295443
$593$	5.41032	0.222175	0.111088	−	0.993811i	$-0.464567\pi$
$593$	5.41032	0.222175	0.111088	+	0.993811i	$0.464567\pi$
$594$	−5.86041	−0.240455
$595$	−9.19330	−0.376888
$596$	−5.74169	−0.235189
$597$	−9.56724	−0.391561
$598$	−0.621878	−0.0254305
$599$	40.3259	1.64767	0.823835	−	0.566829i	$-0.191830\pi$
$599$	40.3259	1.64767	0.823835	+	0.566829i	$0.191830\pi$
$600$	0.00219956	8.97968e−5 0
$601$	12.5929	0.513675	0.256837	−	0.966455i	$-0.417320\pi$
$601$	12.5929	0.513675	0.256837	+	0.966455i	$0.417320\pi$
$602$	7.87773	0.321072
$603$	−35.1281	−1.43053
$604$	−1.64094	−0.0667689
$605$	2.33538	0.0949467
$606$	−4.33620	−0.176146
$607$	−33.9938	−1.37976	−0.689882	−	0.723922i	$-0.742338\pi$
$607$	−33.9938	−1.37976	−0.689882	+	0.723922i	$0.742338\pi$
$608$	−23.7867	−0.964679
$609$	−19.6181	−0.794965
$610$	34.0383	1.37817
$611$	−8.53828	−0.345422
$612$	−5.49236	−0.222016
$613$	−3.43514	−0.138744	−0.0693720	−	0.997591i	$-0.522100\pi$
$613$	−3.43514	−0.138744	−0.0693720	+	0.997591i	$0.522100\pi$
$614$	40.8105	1.64698
$615$	−8.91833	−0.359622
$616$	0.0373804	0.00150610
$617$	−6.32568	−0.254662	−0.127331	−	0.991860i	$-0.540641\pi$
$617$	−6.32568	−0.254662	−0.127331	+	0.991860i	$0.540641\pi$
$618$	3.28055	0.131963
$619$	13.9742	0.561673	0.280836	−	0.959756i	$-0.409388\pi$
$619$	13.9742	0.561673	0.280836	+	0.959756i	$0.409388\pi$
$620$	−0.709001	−0.0284742
$621$	0.173864	0.00697692
$622$	−23.1072	−0.926515
$623$	35.6854	1.42970
$624$	10.6568	0.426614
$625$	−27.0639	−1.08255
$626$	−9.94460	−0.397466
$627$	1.51615	0.0605492
$628$	8.37773	0.334308
$629$	−1.80139	−0.0718262
$630$	−50.4033	−2.00811
$631$	−6.84549	−0.272514	−0.136257	−	0.990673i	$-0.543507\pi$
$631$	−6.84549	−0.272514	−0.136257	+	0.990673i	$0.543507\pi$
$632$	−0.131796	−0.00524256
$633$	−4.85429	−0.192941
$634$	33.1264	1.31562
$635$	21.5408	0.854820
$636$	−2.32786	−0.0923057
$637$	44.4711	1.76201
$638$	19.5469	0.773869
$639$	24.6572	0.975425
$640$	0.177409	0.00701272
$641$	22.7772	0.899646	0.449823	−	0.893118i	$-0.351487\pi$
$641$	22.7772	0.899646	0.449823	+	0.893118i	$0.351487\pi$
$642$	15.9308	0.628739
$643$	25.5687	1.00833	0.504166	−	0.863607i	$-0.331800\pi$
$643$	25.5687	1.00833	0.504166	+	0.863607i	$0.331800\pi$
$644$	−0.468535	−0.0184629
$645$	−1.19154	−0.0469170
$646$	5.94671	0.233970
$647$	24.0035	0.943673	0.471837	−	0.881686i	$-0.343591\pi$
$647$	24.0035	0.943673	0.471837	+	0.881686i	$0.343591\pi$
$648$	−0.0638581	−0.00250858
$649$	4.09795	0.160858
$650$	−4.75543	−0.186523
$651$	−0.304156	−0.0119208
$652$	−3.12980	−0.122573
$653$	−39.5512	−1.54776	−0.773878	−	0.633334i	$-0.781686\pi$
$653$	−39.5512	−1.54776	−0.773878	+	0.633334i	$0.781686\pi$
$654$	−7.60436	−0.297354
$655$	15.9667	0.623871
$656$	−29.8675	−1.16613
$657$	42.8361	1.67120
$658$	−12.8506	−0.500968
$659$	12.4586	0.485316	0.242658	−	0.970112i	$-0.421981\pi$
$659$	12.4586	0.485316	0.242658	+	0.970112i	$0.421981\pi$
$660$	−2.38874	−0.0929816
$661$	−6.31587	−0.245659	−0.122829	−	0.992428i	$-0.539197\pi$
$661$	−6.31587	−0.245659	−0.122829	+	0.992428i	$0.539197\pi$
$662$	−16.4625	−0.639833
$663$	−2.67055	−0.103716
$664$	0.0564674	0.00219136
$665$	27.3188	1.05938
$666$	−9.87633	−0.382700
$667$	−0.579909	−0.0224542
$668$	16.0953	0.622745
$669$	−4.69887	−0.181669
$670$	−59.9239	−2.31506
$671$	−7.28320	−0.281165
$672$	16.0772	0.620193
$673$	21.1714	0.816096	0.408048	−	0.912960i	$-0.366210\pi$
$673$	21.1714	0.816096	0.408048	+	0.912960i	$0.366210\pi$
$674$	11.2291	0.432530
$675$	1.32952	0.0511732
$676$	28.8616	1.11006
$677$	31.2475	1.20094	0.600469	−	0.799648i	$-0.294980\pi$
$677$	31.2475	1.20094	0.600469	+	0.799648i	$0.294980\pi$
$678$	−13.9988	−0.537619
$679$	−44.8203	−1.72005
$680$	−0.0221762	−0.000850420 0
$681$	−0.290561	−0.0111343
$682$	0.303052	0.0116045
$683$	39.9494	1.52862	0.764311	−	0.644848i	$-0.223079\pi$
$683$	39.9494	1.52862	0.764311	+	0.644848i	$0.223079\pi$
$684$	16.3211	0.624052
$685$	38.7084	1.47897
$686$	11.7873	0.450042
$687$	−3.21183	−0.122539
$688$	−3.99049	−0.152136
$689$	11.9123	0.453821
$690$	0.141568	0.00538942
$691$	5.13326	0.195278	0.0976392	−	0.995222i	$-0.468871\pi$
$691$	5.13326	0.195278	0.0976392	+	0.995222i	$0.468871\pi$
$692$	−46.1289	−1.75356
$693$	10.7848	0.409682
$694$	−58.7187	−2.22893
$695$	−34.6069	−1.31271
$696$	−0.0473231	−0.00179378
$697$	7.48468	0.283503
$698$	36.8627	1.39527
$699$	7.65661	0.289600
$700$	−3.58283	−0.135418
$701$	48.2753	1.82333	0.911666	−	0.410933i	$-0.134797\pi$
$701$	48.2753	1.82333	0.911666	+	0.410933i	$0.134797\pi$
$702$	−30.6744	−1.15773
$703$	5.35301	0.201892
$704$	−8.03792	−0.302940
$705$	1.94371	0.0732044
$706$	−49.4779	−1.86212
$707$	16.7179	0.628743
$708$	−4.19158	−0.157529
$709$	−1.05698	−0.0396958	−0.0198479	−	0.999803i	$-0.506318\pi$
$709$	−1.05698	−0.0396958	−0.0198479	+	0.999803i	$0.506318\pi$
$710$	42.0619	1.57856
$711$	−38.0252	−1.42606
$712$	0.0860809	0.00322602
$713$	−0.00899082	−0.000336709 0
$714$	−4.01933	−0.150420
$715$	12.2238	0.457144
$716$	−28.3288	−1.05870
$717$	−11.8740	−0.443441
$718$	−16.7450	−0.624917
$719$	20.2170	0.753966	0.376983	−	0.926220i	$-0.376961\pi$
$719$	20.2170	0.753966	0.376983	+	0.926220i	$0.376961\pi$
$720$	25.5319	0.951518
$721$	−12.6480	−0.471034
$722$	20.3513	0.757397
$723$	−3.02515	−0.112506
$724$	28.5424	1.06077
$725$	−4.43449	−0.164693
$726$	1.02103	0.0378941
$727$	49.4689	1.83470	0.917350	−	0.398081i	$-0.130324\pi$
$727$	49.4689	1.83470	0.917350	+	0.398081i	$0.130324\pi$
$728$	0.195656	0.00725149
$729$	−13.9416	−0.516357
$730$	73.0727	2.70454
$731$	1.00000	0.0369863
$732$	7.44962	0.275346
$733$	21.0044	0.775815	0.387908	−	0.921698i	$-0.373198\pi$
$733$	21.0044	0.775815	0.387908	+	0.921698i	$0.373198\pi$
$734$	−2.03240	−0.0750171
$735$	−10.1237	−0.373418
$736$	0.475241	0.0175176
$737$	12.8220	0.472303
$738$	41.0356	1.51054
$739$	41.0547	1.51022	0.755110	−	0.655598i	$-0.227583\pi$
$739$	41.0547	1.51022	0.755110	+	0.655598i	$0.227583\pi$
$740$	−8.43383	−0.310034
$741$	7.93580	0.291529
$742$	17.9286	0.658180
$743$	37.7544	1.38507	0.692537	−	0.721382i	$-0.256493\pi$
$743$	37.7544	1.38507	0.692537	+	0.721382i	$0.256493\pi$
$744$	−0.000733691 0	−2.68984e−5 0
$745$	−6.68865	−0.245053
$746$	−39.1024	−1.43164
$747$	16.2918	0.596084
$748$	2.00475	0.0733007
$749$	−61.4202	−2.24425
$750$	−10.8399	−0.395819
$751$	51.1162	1.86526	0.932629	−	0.360837i	$-0.117509\pi$
$751$	51.1162	1.86526	0.932629	+	0.360837i	$0.117509\pi$
$752$	6.50950	0.237377
$753$	−2.93455	−0.106941
$754$	102.312	3.72598
$755$	−1.91157	−0.0695693
$756$	−23.1107	−0.840528
$757$	18.0813	0.657176	0.328588	−	0.944473i	$-0.393427\pi$
$757$	18.0813	0.657176	0.328588	+	0.944473i	$0.393427\pi$
$758$	19.9968	0.726316
$759$	−0.0302916	−0.00109951
$760$	0.0658988	0.00239040
$761$	12.5547	0.455109	0.227554	−	0.973765i	$-0.426927\pi$
$761$	12.5547	0.455109	0.227554	+	0.973765i	$0.426927\pi$
$762$	9.41769	0.341167
$763$	29.3181	1.06139
$764$	13.8968	0.502770
$765$	−6.39820	−0.231327
$766$	14.0083	0.506139
$767$	21.4494	0.774492
$768$	−8.12455	−0.293170
$769$	−21.6407	−0.780385	−0.390193	−	0.920733i	$-0.627592\pi$
$769$	−21.6407	−0.780385	−0.390193	+	0.920733i	$0.627592\pi$
$770$	18.3975	0.663000
$771$	−1.73604	−0.0625220
$772$	−1.33715	−0.0481249
$773$	−4.75588	−0.171057	−0.0855285	−	0.996336i	$-0.527258\pi$
$773$	−4.75588	−0.171057	−0.0855285	+	0.996336i	$0.527258\pi$
$774$	5.48261	0.197068
$775$	−0.0687518	−0.00246964
$776$	−0.108116	−0.00388115
$777$	−3.61805	−0.129797
$778$	−45.6732	−1.63746
$779$	−22.2415	−0.796883
$780$	−12.5031	−0.447683
$781$	−9.00003	−0.322046
$782$	−0.118811	−0.00424867
$783$	−28.6043	−1.02223
$784$	−33.9043	−1.21087
$785$	9.75944	0.348329
$786$	6.98068	0.248993
$787$	−39.0420	−1.39170	−0.695849	−	0.718189i	$-0.744972\pi$
$787$	−39.0420	−1.39170	−0.695849	+	0.718189i	$0.744972\pi$
$788$	15.1675	0.540320
$789$	−6.62958	−0.236019
$790$	−64.8659	−2.30782
$791$	53.9714	1.91900
$792$	0.0260154	0.000924417 0
$793$	−38.1216	−1.35374
$794$	−15.4534	−0.548421
$795$	−2.71178	−0.0961771
$796$	37.5918	1.33241
$797$	39.6294	1.40375	0.701873	−	0.712302i	$-0.252347\pi$
$797$	39.6294	1.40375	0.701873	+	0.712302i	$0.252347\pi$
$798$	11.9438	0.422807
$799$	−1.63126	−0.0577097
$800$	3.63412	0.128485
$801$	24.8357	0.877526
$802$	−77.3913	−2.73278
$803$	−15.6354	−0.551763
$804$	−13.1149	−0.462529
$805$	−0.545809	−0.0192372
$806$	1.58623	0.0558726
$807$	14.7014	0.517514
$808$	0.0403273	0.00141871
$809$	49.7006	1.74738	0.873689	−	0.486484i	$-0.161721\pi$
$809$	49.7006	1.74738	0.873689	+	0.486484i	$0.161721\pi$
$810$	−31.4290	−1.10430
$811$	−30.6224	−1.07530	−0.537648	−	0.843169i	$-0.680687\pi$
$811$	−30.6224	−1.07530	−0.537648	+	0.843169i	$0.680687\pi$
$812$	77.0838	2.70511
$813$	−10.7289	−0.376278
$814$	3.60492	0.126352
$815$	−3.64599	−0.127713
$816$	2.03600	0.0712744
$817$	−2.97160	−0.103963
$818$	1.17488	0.0410789
$819$	56.4498	1.97252
$820$	35.0421	1.22372
$821$	32.6707	1.14021	0.570107	−	0.821570i	$-0.306902\pi$
$821$	32.6707	1.14021	0.570107	+	0.821570i	$0.306902\pi$
$822$	16.9234	0.590272
$823$	50.9251	1.77514	0.887568	−	0.460676i	$-0.152393\pi$
$823$	50.9251	1.77514	0.887568	+	0.460676i	$0.152393\pi$
$824$	−0.0305096	−0.00106285
$825$	−0.231636	−0.00806453
$826$	32.2825	1.12325
$827$	43.7759	1.52224	0.761119	−	0.648612i	$-0.224650\pi$
$827$	43.7759	1.52224	0.761119	+	0.648612i	$0.224650\pi$
$828$	−0.326083	−0.0113322
$829$	42.0428	1.46021	0.730104	−	0.683337i	$-0.239472\pi$
$829$	42.0428	1.46021	0.730104	+	0.683337i	$0.239472\pi$
$830$	27.7916	0.964659
$831$	7.93826	0.275375
$832$	−42.0719	−1.45858
$833$	8.49628	0.294379
$834$	−15.1302	−0.523917
$835$	18.7498	0.648864
$836$	−5.95729	−0.206037
$837$	−0.443476	−0.0153288
$838$	−7.22502	−0.249584
$839$	−1.03105	−0.0355957	−0.0177979	−	0.999842i	$-0.505666\pi$
$839$	−1.03105	−0.0355957	−0.0177979	+	0.999842i	$0.505666\pi$
$840$	−0.0445404	−0.00153679
$841$	66.4071	2.28990
$842$	−74.0202	−2.55090
$843$	2.17163	0.0747951
$844$	19.0736	0.656540
$845$	33.6216	1.15662
$846$	−8.94354	−0.307485
$847$	−3.93653	−0.135261
$848$	−9.08178	−0.311870
$849$	14.0186	0.481119
$850$	−0.908534	−0.0311625
$851$	−0.106949	−0.00366617
$852$	9.20568	0.315381
$853$	−14.1009	−0.482806	−0.241403	−	0.970425i	$-0.577608\pi$
$853$	−14.1009	−0.482806	−0.241403	+	0.970425i	$0.577608\pi$
$854$	−57.3751	−1.96334
$855$	19.0128	0.650226
$856$	−0.148159	−0.00506397
$857$	−46.9479	−1.60371	−0.801855	−	0.597518i	$-0.796154\pi$
$857$	−46.9479	−1.60371	−0.801855	+	0.597518i	$0.796154\pi$
$858$	5.34427	0.182451
$859$	−27.4488	−0.936541	−0.468271	−	0.883585i	$-0.655123\pi$
$859$	−27.4488	−0.936541	−0.468271	+	0.883585i	$0.655123\pi$
$860$	4.68184	0.159649
$861$	15.0328	0.512316
$862$	−6.60113	−0.224835
$863$	20.6359	0.702456	0.351228	−	0.936290i	$-0.385764\pi$
$863$	20.6359	0.702456	0.351228	+	0.936290i	$0.385764\pi$
$864$	23.4415	0.797496
$865$	−53.7367	−1.82710
$866$	−26.8868	−0.913651
$867$	−0.510214	−0.0173278
$868$	1.19510	0.0405642
$869$	13.8794	0.470827
$870$	−23.2910	−0.789638
$871$	67.1125	2.27402
$872$	0.0707217	0.00239494
$873$	−31.1933	−1.05573
$874$	0.353058	0.0119424
$875$	41.7927	1.41285
$876$	15.9927	0.540343
$877$	−16.5448	−0.558679	−0.279340	−	0.960192i	$-0.590116\pi$
$877$	−16.5448	−0.558679	−0.279340	+	0.960192i	$0.590116\pi$
$878$	−0.883491	−0.0298164
$879$	11.3879	0.384106
$880$	−9.31930	−0.314154
$881$	−8.49121	−0.286076	−0.143038	−	0.989717i	$-0.545687\pi$
$881$	−8.49121	−0.286076	−0.143038	+	0.989717i	$0.545687\pi$
$882$	46.5818	1.56849
$883$	34.8691	1.17344	0.586719	−	0.809790i	$-0.300419\pi$
$883$	34.8691	1.17344	0.586719	+	0.809790i	$0.300419\pi$
$884$	10.4932	0.352924
$885$	−4.88288	−0.164136
$886$	2.68997	0.0903713
$887$	−31.7678	−1.06666	−0.533330	−	0.845907i	$-0.679060\pi$
$887$	−31.7678	−1.06666	−0.533330	+	0.845907i	$0.679060\pi$
$888$	−0.00872752	−0.000292876 0
$889$	−36.3093	−1.21778
$890$	42.3664	1.42012
$891$	6.72490	0.225293
$892$	18.4629	0.618183
$893$	4.84743	0.162213
$894$	−2.92429	−0.0978030
$895$	−33.0010	−1.10310
$896$	−0.299042	−0.00999031
$897$	−0.158552	−0.00529388
$898$	−47.9726	−1.60087
$899$	1.47918	0.0493334
$900$	−2.49352	−0.0831173
$901$	2.27586	0.0758198
$902$	−14.9782	−0.498721
$903$	2.00847	0.0668379
$904$	0.130191	0.00433008
$905$	33.2498	1.10526
$906$	−0.835745	−0.0277658
$907$	9.02766	0.299759	0.149879	−	0.988704i	$-0.452111\pi$
$907$	9.02766	0.299759	0.149879	+	0.988704i	$0.452111\pi$
$908$	1.14168	0.0378880
$909$	11.6351	0.385911
$910$	96.2958	3.19218
$911$	−20.2933	−0.672346	−0.336173	−	0.941800i	$-0.609133\pi$
$911$	−20.2933	−0.672346	−0.336173	+	0.941800i	$0.609133\pi$
$912$	−6.05018	−0.200341
$913$	−5.94659	−0.196803
$914$	30.7128	1.01589
$915$	8.67826	0.286894
$916$	12.6200	0.416976
$917$	−26.9136	−0.888765
$918$	−5.86041	−0.193422
$919$	−13.2130	−0.435858	−0.217929	−	0.975965i	$-0.569930\pi$
$919$	−13.2130	−0.435858	−0.217929	+	0.975965i	$0.569930\pi$
$920$	−0.00131661	−4.34073e−5 0
$921$	10.4049	0.342852
$922$	81.0791	2.67020
$923$	−47.1078	−1.55057
$924$	4.02648	0.132461
$925$	−0.817828	−0.0268900
$926$	−48.4887	−1.59344
$927$	−8.80251	−0.289112
$928$	−78.1872	−2.56662
$929$	51.9918	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$929$	51.9918	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$930$	−0.361100	−0.0118409
$931$	−25.2475	−0.827454
$932$	−30.0845	−0.985451
$933$	−5.89132	−0.192873
$934$	43.9138	1.43690
$935$	2.33538	0.0763751
$936$	0.136169	0.00445083
$937$	42.4970	1.38832	0.694158	−	0.719823i	$-0.255777\pi$
$937$	42.4970	1.38832	0.694158	+	0.719823i	$0.255777\pi$
$938$	101.008	3.29803
$939$	−2.53543	−0.0827408
$940$	−7.63728	−0.249100
$941$	57.6190	1.87833	0.939163	−	0.343471i	$-0.111603\pi$
$941$	57.6190	1.87833	0.939163	+	0.343471i	$0.111603\pi$
$942$	4.26685	0.139021
$943$	0.444368	0.0144706
$944$	−16.3528	−0.532238
$945$	−26.9223	−0.875781
$946$	−2.00119	−0.0650642
$947$	60.6058	1.96942	0.984712	−	0.174189i	$-0.0557305\pi$
$947$	60.6058	1.96942	0.984712	+	0.174189i	$0.0557305\pi$
$948$	−14.1966	−0.461083
$949$	−81.8387	−2.65660
$950$	2.69980	0.0875929
$951$	8.44577	0.273873
$952$	0.0373804	0.00121151
$953$	−47.1823	−1.52839	−0.764193	−	0.644988i	$-0.776862\pi$
$953$	−47.1823	−1.52839	−0.764193	+	0.644988i	$0.776862\pi$
$954$	12.4776	0.403979
$955$	16.1888	0.523857
$956$	46.6555	1.50895
$957$	4.98360	0.161097
$958$	−6.32634	−0.204395
$959$	−65.2471	−2.10694
$960$	9.57753	0.309113
$961$	−30.9771	−0.999260
$962$	18.8688	0.608354
$963$	−42.7462	−1.37748
$964$	11.8865	0.382838
$965$	−1.55768	−0.0501434
$966$	−0.238629	−0.00767776
$967$	−24.6420	−0.792432	−0.396216	−	0.918157i	$-0.629677\pi$
$967$	−24.6420	−0.792432	−0.396216	+	0.918157i	$0.629677\pi$
$968$	−0.00949577	−0.000305206 0
$969$	1.51615	0.0487058
$970$	−53.2116	−1.70852
$971$	11.2730	0.361768	0.180884	−	0.983504i	$-0.442104\pi$
$971$	11.2730	0.361768	0.180884	+	0.983504i	$0.442104\pi$
$972$	−24.4910	−0.785551
$973$	58.3336	1.87009
$974$	29.2220	0.936332
$975$	−1.21243	−0.0388287
$976$	29.0635	0.930301
$977$	13.8289	0.442425	0.221212	−	0.975226i	$-0.428999\pi$
$977$	13.8289	0.442425	0.221212	+	0.975226i	$0.428999\pi$
$978$	−1.59404	−0.0509717
$979$	−9.06518	−0.289724
$980$	39.7783	1.27067
$981$	20.4043	0.651460
$982$	47.1231	1.50376
$983$	−32.5424	−1.03794	−0.518970	−	0.854792i	$-0.673684\pi$
$983$	−32.5424	−1.03794	−0.518970	+	0.854792i	$0.673684\pi$
$984$	0.0362624	0.00115600
$985$	17.6690	0.562981
$986$	19.5469	0.622500
$987$	−3.27633	−0.104287
$988$	−31.1815	−0.992017
$989$	0.0593703	0.00188787
$990$	12.8040	0.406937
$991$	20.3540	0.646565	0.323283	−	0.946302i	$-0.395214\pi$
$991$	20.3540	0.646565	0.323283	+	0.946302i	$0.395214\pi$
$992$	−1.21220	−0.0384875
$993$	−4.19721	−0.133194
$994$	−70.8998	−2.24881
$995$	43.7917	1.38829
$996$	6.08246	0.192730
$997$	−25.2795	−0.800610	−0.400305	−	0.916382i	$-0.631096\pi$
$997$	−25.2795	−0.800610	−0.400305	+	0.916382i	$0.631096\pi$
$998$	79.9294	2.53012
$999$	−5.27532	−0.166904

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.j.1.14	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.j.1.14	✓	82	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00119 q^{2} -0.510214 q^{3} +2.00475 q^{4} +2.33538 q^{5} +1.02103 q^{6} -3.93653 q^{7} -0.00949577 q^{8} -2.73968 q^{9} +O(q^{10})\)	\(q-2.00119 q^{2} -0.510214 q^{3} +2.00475 q^{4} +2.33538 q^{5} +1.02103 q^{6} -3.93653 q^{7} -0.00949577 q^{8} -2.73968 q^{9} -4.67353 q^{10} +1.00000 q^{11} -1.02285 q^{12} +5.23418 q^{13} +7.87773 q^{14} -1.19154 q^{15} -3.99049 q^{16} +1.00000 q^{17} +5.48261 q^{18} -2.97160 q^{19} +4.68184 q^{20} +2.00847 q^{21} -2.00119 q^{22} +0.0593703 q^{23} +0.00484488 q^{24} +0.453998 q^{25} -10.4746 q^{26} +2.92847 q^{27} -7.89174 q^{28} -9.76766 q^{29} +2.38450 q^{30} -0.151436 q^{31} +8.00470 q^{32} -0.510214 q^{33} -2.00119 q^{34} -9.19330 q^{35} -5.49236 q^{36} -1.80139 q^{37} +5.94671 q^{38} -2.67055 q^{39} -0.0221762 q^{40} +7.48468 q^{41} -4.01933 q^{42} +1.00000 q^{43} +2.00475 q^{44} -6.39820 q^{45} -0.118811 q^{46} -1.63126 q^{47} +2.03600 q^{48} +8.49628 q^{49} -0.908534 q^{50} -0.510214 q^{51} +10.4932 q^{52} +2.27586 q^{53} -5.86041 q^{54} +2.33538 q^{55} +0.0373804 q^{56} +1.51615 q^{57} +19.5469 q^{58} +4.09795 q^{59} -2.38874 q^{60} -7.28320 q^{61} +0.303052 q^{62} +10.7848 q^{63} -8.03792 q^{64} +12.2238 q^{65} +1.02103 q^{66} +12.8220 q^{67} +2.00475 q^{68} -0.0302916 q^{69} +18.3975 q^{70} -9.00003 q^{71} +0.0260154 q^{72} -15.6354 q^{73} +3.60492 q^{74} -0.231636 q^{75} -5.95729 q^{76} -3.93653 q^{77} +5.34427 q^{78} +13.8794 q^{79} -9.31930 q^{80} +6.72490 q^{81} -14.9782 q^{82} -5.94659 q^{83} +4.02648 q^{84} +2.33538 q^{85} -2.00119 q^{86} +4.98360 q^{87} -0.00949577 q^{88} -9.06518 q^{89} +12.8040 q^{90} -20.6045 q^{91} +0.119022 q^{92} +0.0772650 q^{93} +3.26444 q^{94} -6.93980 q^{95} -4.08411 q^{96} +11.3857 q^{97} -17.0026 q^{98} -2.73968 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more