LMFDB - Embedded newform 8042.2.a.a.1.33

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.33
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.366053 q^{3} +1.00000 q^{4} -0.762289 q^{5} -0.366053 q^{6} -0.714001 q^{7} +1.00000 q^{8} -2.86601 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.366053 q^{3} +1.00000 q^{4} -0.762289 q^{5} -0.366053 q^{6} -0.714001 q^{7} +1.00000 q^{8} -2.86601 q^{9} -0.762289 q^{10} +1.16475 q^{11} -0.366053 q^{12} +2.61076 q^{13} -0.714001 q^{14} +0.279038 q^{15} +1.00000 q^{16} +3.97608 q^{17} -2.86601 q^{18} -2.89336 q^{19} -0.762289 q^{20} +0.261362 q^{21} +1.16475 q^{22} -3.77218 q^{23} -0.366053 q^{24} -4.41892 q^{25} +2.61076 q^{26} +2.14727 q^{27} -0.714001 q^{28} -0.0757303 q^{29} +0.279038 q^{30} -1.16339 q^{31} +1.00000 q^{32} -0.426360 q^{33} +3.97608 q^{34} +0.544275 q^{35} -2.86601 q^{36} -1.98779 q^{37} -2.89336 q^{38} -0.955674 q^{39} -0.762289 q^{40} +5.72873 q^{41} +0.261362 q^{42} +6.75381 q^{43} +1.16475 q^{44} +2.18472 q^{45} -3.77218 q^{46} +4.90581 q^{47} -0.366053 q^{48} -6.49020 q^{49} -4.41892 q^{50} -1.45546 q^{51} +2.61076 q^{52} -4.94868 q^{53} +2.14727 q^{54} -0.887877 q^{55} -0.714001 q^{56} +1.05912 q^{57} -0.0757303 q^{58} -8.74954 q^{59} +0.279038 q^{60} -7.69919 q^{61} -1.16339 q^{62} +2.04633 q^{63} +1.00000 q^{64} -1.99015 q^{65} -0.426360 q^{66} +8.21929 q^{67} +3.97608 q^{68} +1.38082 q^{69} +0.544275 q^{70} -11.5597 q^{71} -2.86601 q^{72} +1.18715 q^{73} -1.98779 q^{74} +1.61756 q^{75} -2.89336 q^{76} -0.831634 q^{77} -0.955674 q^{78} +5.56802 q^{79} -0.762289 q^{80} +7.81200 q^{81} +5.72873 q^{82} -16.6677 q^{83} +0.261362 q^{84} -3.03092 q^{85} +6.75381 q^{86} +0.0277213 q^{87} +1.16475 q^{88} +0.193362 q^{89} +2.18472 q^{90} -1.86408 q^{91} -3.77218 q^{92} +0.425861 q^{93} +4.90581 q^{94} +2.20557 q^{95} -0.366053 q^{96} +10.5655 q^{97} -6.49020 q^{98} -3.33819 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−0.366053	−0.211341	−0.105670	−	0.994401i	$-0.533699\pi$
$3$	−0.366053	−0.211341	−0.105670	+	0.994401i	$0.533699\pi$
$4$	1.00000	0.500000
$5$	−0.762289	−0.340906	−0.170453	−	0.985366i	$-0.554523\pi$
$5$	−0.762289	−0.340906	−0.170453	+	0.985366i	$0.554523\pi$
$6$	−0.366053	−0.149440
$7$	−0.714001	−0.269867	−0.134933	−	0.990855i	$-0.543082\pi$
$7$	−0.714001	−0.269867	−0.134933	+	0.990855i	$0.543082\pi$
$8$	1.00000	0.353553
$9$	−2.86601	−0.955335
$10$	−0.762289	−0.241057
$11$	1.16475	0.351186	0.175593	−	0.984463i	$-0.443816\pi$
$11$	1.16475	0.351186	0.175593	+	0.984463i	$0.443816\pi$
$12$	−0.366053	−0.105670
$13$	2.61076	0.724094	0.362047	−	0.932160i	$-0.382078\pi$
$13$	2.61076	0.724094	0.362047	+	0.932160i	$0.382078\pi$
$14$	−0.714001	−0.190825
$15$	0.279038	0.0720472
$16$	1.00000	0.250000
$17$	3.97608	0.964342	0.482171	−	0.876077i	$-0.339848\pi$
$17$	3.97608	0.964342	0.482171	+	0.876077i	$0.339848\pi$
$18$	−2.86601	−0.675524
$19$	−2.89336	−0.663782	−0.331891	−	0.943318i	$-0.607687\pi$
$19$	−2.89336	−0.663782	−0.331891	+	0.943318i	$0.607687\pi$
$20$	−0.762289	−0.170453
$21$	0.261362	0.0570338
$22$	1.16475	0.248326
$23$	−3.77218	−0.786555	−0.393277	−	0.919420i	$-0.628659\pi$
$23$	−3.77218	−0.786555	−0.393277	+	0.919420i	$0.628659\pi$
$24$	−0.366053	−0.0747202
$25$	−4.41892	−0.883783
$26$	2.61076	0.512011
$27$	2.14727	0.413242
$28$	−0.714001	−0.134933
$29$	−0.0757303	−0.0140628	−0.00703139	−	0.999975i	$-0.502238\pi$
$29$	−0.0757303	−0.0140628	−0.00703139	+	0.999975i	$0.502238\pi$
$30$	0.279038	0.0509451
$31$	−1.16339	−0.208950	−0.104475	−	0.994527i	$-0.533316\pi$
$31$	−1.16339	−0.208950	−0.104475	+	0.994527i	$0.533316\pi$
$32$	1.00000	0.176777
$33$	−0.426360	−0.0742198
$34$	3.97608	0.681893
$35$	0.544275	0.0919992
$36$	−2.86601	−0.477668
$37$	−1.98779	−0.326791	−0.163395	−	0.986561i	$-0.552245\pi$
$37$	−1.98779	−0.326791	−0.163395	+	0.986561i	$0.552245\pi$
$38$	−2.89336	−0.469365
$39$	−0.955674	−0.153030
$40$	−0.762289	−0.120528
$41$	5.72873	0.894677	0.447339	−	0.894365i	$-0.352372\pi$
$41$	5.72873	0.894677	0.447339	+	0.894365i	$0.352372\pi$
$42$	0.261362	0.0403290
$43$	6.75381	1.02995	0.514973	−	0.857206i	$-0.327802\pi$
$43$	6.75381	1.02995	0.514973	+	0.857206i	$0.327802\pi$
$44$	1.16475	0.175593
$45$	2.18472	0.325679
$46$	−3.77218	−0.556178
$47$	4.90581	0.715587	0.357793	−	0.933801i	$-0.383529\pi$
$47$	4.90581	0.715587	0.357793	+	0.933801i	$0.383529\pi$
$48$	−0.366053	−0.0528351
$49$	−6.49020	−0.927172
$50$	−4.41892	−0.624929
$51$	−1.45546	−0.203804
$52$	2.61076	0.362047
$53$	−4.94868	−0.679753	−0.339877	−	0.940470i	$-0.610385\pi$
$53$	−4.94868	−0.679753	−0.339877	+	0.940470i	$0.610385\pi$
$54$	2.14727	0.292206
$55$	−0.887877	−0.119721
$56$	−0.714001	−0.0954124
$57$	1.05912	0.140284
$58$	−0.0757303	−0.00994388
$59$	−8.74954	−1.13909	−0.569547	−	0.821959i	$-0.692881\pi$
$59$	−8.74954	−1.13909	−0.569547	+	0.821959i	$0.692881\pi$
$60$	0.279038	0.0360236
$61$	−7.69919	−0.985781	−0.492890	−	0.870091i	$-0.664060\pi$
$61$	−7.69919	−0.985781	−0.492890	+	0.870091i	$0.664060\pi$
$62$	−1.16339	−0.147750
$63$	2.04633	0.257813
$64$	1.00000	0.125000
$65$	−1.99015	−0.246848
$66$	−0.426360	−0.0524813
$67$	8.21929	1.00415	0.502073	−	0.864825i	$-0.332571\pi$
$67$	8.21929	1.00415	0.502073	+	0.864825i	$0.332571\pi$
$68$	3.97608	0.482171
$69$	1.38082	0.166231
$70$	0.544275	0.0650533
$71$	−11.5597	−1.37189	−0.685944	−	0.727655i	$-0.740610\pi$
$71$	−11.5597	−1.37189	−0.685944	+	0.727655i	$0.740610\pi$
$72$	−2.86601	−0.337762
$73$	1.18715	0.138946	0.0694728	−	0.997584i	$-0.477868\pi$
$73$	1.18715	0.138946	0.0694728	+	0.997584i	$0.477868\pi$
$74$	−1.98779	−0.231076
$75$	1.61756	0.186779
$76$	−2.89336	−0.331891
$77$	−0.831634	−0.0947735
$78$	−0.955674	−0.108209
$79$	5.56802	0.626451	0.313226	−	0.949679i	$-0.398590\pi$
$79$	5.56802	0.626451	0.313226	+	0.949679i	$0.398590\pi$
$80$	−0.762289	−0.0852265
$81$	7.81200	0.868001
$82$	5.72873	0.632632
$83$	−16.6677	−1.82952	−0.914759	−	0.404000i	$-0.867620\pi$
$83$	−16.6677	−1.82952	−0.914759	+	0.404000i	$0.867620\pi$
$84$	0.261362	0.0285169
$85$	−3.03092	−0.328750
$86$	6.75381	0.728282
$87$	0.0277213	0.00297203
$88$	1.16475	0.124163
$89$	0.193362	0.0204964	0.0102482	−	0.999947i	$-0.496738\pi$
$89$	0.193362	0.0204964	0.0102482	+	0.999947i	$0.496738\pi$
$90$	2.18472	0.230290
$91$	−1.86408	−0.195409
$92$	−3.77218	−0.393277
$93$	0.425861	0.0441597
$94$	4.90581	0.505996
$95$	2.20557	0.226287
$96$	−0.366053	−0.0373601
$97$	10.5655	1.07277	0.536384	−	0.843974i	$-0.319790\pi$
$97$	10.5655	1.07277	0.536384	+	0.843974i	$0.319790\pi$
$98$	−6.49020	−0.655609
$99$	−3.33819	−0.335500
$100$	−4.41892	−0.441892
$101$	7.52869	0.749133	0.374566	−	0.927200i	$-0.377792\pi$
$101$	7.52869	0.749133	0.374566	+	0.927200i	$0.377792\pi$
$102$	−1.45546	−0.144112
$103$	−15.2778	−1.50537	−0.752684	−	0.658382i	$-0.771241\pi$
$103$	−15.2778	−1.50537	−0.752684	+	0.658382i	$0.771241\pi$
$104$	2.61076	0.256006
$105$	−0.199233	−0.0194432
$106$	−4.94868	−0.480658
$107$	16.7947	1.62360	0.811800	−	0.583935i	$-0.198488\pi$
$107$	16.7947	1.62360	0.811800	+	0.583935i	$0.198488\pi$
$108$	2.14727	0.206621
$109$	−20.0640	−1.92179	−0.960894	−	0.276918i	$-0.910687\pi$
$109$	−20.0640	−1.92179	−0.960894	+	0.276918i	$0.910687\pi$
$110$	−0.887877	−0.0846558
$111$	0.727636	0.0690641
$112$	−0.714001	−0.0674667
$113$	−8.01114	−0.753624	−0.376812	−	0.926290i	$-0.622980\pi$
$113$	−8.01114	−0.753624	−0.376812	+	0.926290i	$0.622980\pi$
$114$	1.05912	0.0991958
$115$	2.87549	0.268141
$116$	−0.0757303	−0.00703139
$117$	−7.48244	−0.691752
$118$	−8.74954	−0.805460
$119$	−2.83893	−0.260244
$120$	0.279038	0.0254725
$121$	−9.64335	−0.876668
$122$	−7.69919	−0.697052
$123$	−2.09702	−0.189082
$124$	−1.16339	−0.104475
$125$	7.17993	0.642193
$126$	2.04633	0.182302
$127$	16.5367	1.46740	0.733699	−	0.679475i	$-0.237792\pi$
$127$	16.5367	1.46740	0.733699	+	0.679475i	$0.237792\pi$
$128$	1.00000	0.0883883
$129$	−2.47225	−0.217669
$130$	−1.99015	−0.174548
$131$	−7.74937	−0.677065	−0.338533	−	0.940955i	$-0.609931\pi$
$131$	−7.74937	−0.677065	−0.338533	+	0.940955i	$0.609931\pi$
$132$	−0.426360	−0.0371099
$133$	2.06586	0.179133
$134$	8.21929	0.710038
$135$	−1.63684	−0.140876
$136$	3.97608	0.340946
$137$	−1.51665	−0.129576	−0.0647880	−	0.997899i	$-0.520637\pi$
$137$	−1.51665	−0.129576	−0.0647880	+	0.997899i	$0.520637\pi$
$138$	1.38082	0.117543
$139$	−8.70387	−0.738253	−0.369127	−	0.929379i	$-0.620343\pi$
$139$	−8.70387	−0.738253	−0.369127	+	0.929379i	$0.620343\pi$
$140$	0.544275	0.0459996
$141$	−1.79579	−0.151232
$142$	−11.5597	−0.970071
$143$	3.04088	0.254291
$144$	−2.86601	−0.238834
$145$	0.0577284	0.00479408
$146$	1.18715	0.0982493
$147$	2.37575	0.195949
$148$	−1.98779	−0.163395
$149$	−14.8964	−1.22036	−0.610178	−	0.792264i	$-0.708902\pi$
$149$	−14.8964	−1.22036	−0.610178	+	0.792264i	$0.708902\pi$
$150$	1.61756	0.132073
$151$	−10.6613	−0.867606	−0.433803	−	0.901008i	$-0.642829\pi$
$151$	−10.6613	−0.867606	−0.433803	+	0.901008i	$0.642829\pi$
$152$	−2.89336	−0.234682
$153$	−11.3955	−0.921270
$154$	−0.831634	−0.0670150
$155$	0.886837	0.0712324
$156$	−0.955674	−0.0765152
$157$	2.38133	0.190051	0.0950255	−	0.995475i	$-0.469707\pi$
$157$	2.38133	0.190051	0.0950255	+	0.995475i	$0.469707\pi$
$158$	5.56802	0.442968
$159$	1.81148	0.143659
$160$	−0.762289	−0.0602642
$161$	2.69334	0.212265
$162$	7.81200	0.613769
$163$	11.0574	0.866084	0.433042	−	0.901374i	$-0.357440\pi$
$163$	11.0574	0.866084	0.433042	+	0.901374i	$0.357440\pi$
$164$	5.72873	0.447339
$165$	0.325010	0.0253020
$166$	−16.6677	−1.29366
$167$	−17.6296	−1.36422	−0.682110	−	0.731250i	$-0.738937\pi$
$167$	−17.6296	−1.36422	−0.682110	+	0.731250i	$0.738937\pi$
$168$	0.261362	0.0201645
$169$	−6.18395	−0.475689
$170$	−3.03092	−0.232461
$171$	8.29238	0.634134
$172$	6.75381	0.514973
$173$	−7.51117	−0.571064	−0.285532	−	0.958369i	$-0.592170\pi$
$173$	−7.51117	−0.571064	−0.285532	+	0.958369i	$0.592170\pi$
$174$	0.0277213	0.00210154
$175$	3.15511	0.238504
$176$	1.16475	0.0877965
$177$	3.20279	0.240737
$178$	0.193362	0.0144931
$179$	4.51789	0.337683	0.168841	−	0.985643i	$-0.445997\pi$
$179$	4.51789	0.337683	0.168841	+	0.985643i	$0.445997\pi$
$180$	2.18472	0.162840
$181$	4.90501	0.364587	0.182293	−	0.983244i	$-0.441648\pi$
$181$	4.90501	0.364587	0.182293	+	0.983244i	$0.441648\pi$
$182$	−1.86408	−0.138175
$183$	2.81831	0.208335
$184$	−3.77218	−0.278089
$185$	1.51527	0.111405
$186$	0.425861	0.0312256
$187$	4.63115	0.338663
$188$	4.90581	0.357793
$189$	−1.53315	−0.111520
$190$	2.20557	0.160009
$191$	3.74010	0.270624	0.135312	−	0.990803i	$-0.456796\pi$
$191$	3.74010	0.270624	0.135312	+	0.990803i	$0.456796\pi$
$192$	−0.366053	−0.0264176
$193$	14.1248	1.01672	0.508362	−	0.861144i	$-0.330251\pi$
$193$	14.1248	1.01672	0.508362	+	0.861144i	$0.330251\pi$
$194$	10.5655	0.758561
$195$	0.728499	0.0521689
$196$	−6.49020	−0.463586
$197$	−8.72855	−0.621884	−0.310942	−	0.950429i	$-0.600644\pi$
$197$	−8.72855	−0.621884	−0.310942	+	0.950429i	$0.600644\pi$
$198$	−3.33819	−0.237235
$199$	−27.3514	−1.93889	−0.969445	−	0.245309i	$-0.921111\pi$
$199$	−27.3514	−1.93889	−0.969445	+	0.245309i	$0.921111\pi$
$200$	−4.41892	−0.312465
$201$	−3.00869	−0.212217
$202$	7.52869	0.529717
$203$	0.0540715	0.00379508
$204$	−1.45546	−0.101902
$205$	−4.36695	−0.305001
$206$	−15.2778	−1.06446
$207$	10.8111	0.751423
$208$	2.61076	0.181023
$209$	−3.37005	−0.233111
$210$	−0.199233	−0.0137484
$211$	3.57528	0.246133	0.123066	−	0.992398i	$-0.460727\pi$
$211$	3.57528	0.246133	0.123066	+	0.992398i	$0.460727\pi$
$212$	−4.94868	−0.339877
$213$	4.23147	0.289935
$214$	16.7947	1.14806
$215$	−5.14835	−0.351115
$216$	2.14727	0.146103
$217$	0.830659	0.0563888
$218$	−20.0640	−1.35891
$219$	−0.434560	−0.0293648
$220$	−0.887877	−0.0598607
$221$	10.3806	0.698274
$222$	0.727636	0.0488357
$223$	11.6187	0.778045	0.389022	−	0.921228i	$-0.372813\pi$
$223$	11.6187	0.778045	0.389022	+	0.921228i	$0.372813\pi$
$224$	−0.714001	−0.0477062
$225$	12.6646	0.844309
$226$	−8.01114	−0.532893
$227$	−10.5358	−0.699288	−0.349644	−	0.936883i	$-0.613697\pi$
$227$	−10.5358	−0.699288	−0.349644	+	0.936883i	$0.613697\pi$
$228$	1.05912	0.0701420
$229$	−3.83373	−0.253340	−0.126670	−	0.991945i	$-0.540429\pi$
$229$	−3.83373	−0.253340	−0.126670	+	0.991945i	$0.540429\pi$
$230$	2.87549	0.189604
$231$	0.304422	0.0200295
$232$	−0.0757303	−0.00497194
$233$	−6.53082	−0.427848	−0.213924	−	0.976850i	$-0.568624\pi$
$233$	−6.53082	−0.427848	−0.213924	+	0.976850i	$0.568624\pi$
$234$	−7.48244	−0.489143
$235$	−3.73965	−0.243948
$236$	−8.74954	−0.569547
$237$	−2.03819	−0.132394
$238$	−2.83893	−0.184020
$239$	−27.2043	−1.75970	−0.879850	−	0.475252i	$-0.842357\pi$
$239$	−27.2043	−1.75970	−0.879850	+	0.475252i	$0.842357\pi$
$240$	0.279038	0.0180118
$241$	−16.9487	−1.09176	−0.545880	−	0.837863i	$-0.683805\pi$
$241$	−16.9487	−1.09176	−0.545880	+	0.837863i	$0.683805\pi$
$242$	−9.64335	−0.619898
$243$	−9.30140	−0.596685
$244$	−7.69919	−0.492890
$245$	4.94741	0.316078
$246$	−2.09702	−0.133701
$247$	−7.55386	−0.480640
$248$	−1.16339	−0.0738751
$249$	6.10125	0.386651
$250$	7.17993	0.454099
$251$	12.3431	0.779091	0.389546	−	0.921007i	$-0.372632\pi$
$251$	12.3431	0.779091	0.389546	+	0.921007i	$0.372632\pi$
$252$	2.04633	0.128907
$253$	−4.39366	−0.276227
$254$	16.5367	1.03761
$255$	1.10948	0.0694781
$256$	1.00000	0.0625000
$257$	−0.937983	−0.0585098	−0.0292549	−	0.999572i	$-0.509313\pi$
$257$	−0.937983	−0.0585098	−0.0292549	+	0.999572i	$0.509313\pi$
$258$	−2.47225	−0.153915
$259$	1.41928	0.0881900
$260$	−1.99015	−0.123424
$261$	0.217044	0.0134347
$262$	−7.74937	−0.478758
$263$	−13.4179	−0.827385	−0.413693	−	0.910417i	$-0.635761\pi$
$263$	−13.4179	−0.827385	−0.413693	+	0.910417i	$0.635761\pi$
$264$	−0.426360	−0.0262407
$265$	3.77232	0.231732
$266$	2.06586	0.126666
$267$	−0.0707807	−0.00433171
$268$	8.21929	0.502073
$269$	−6.13587	−0.374111	−0.187055	−	0.982349i	$-0.559894\pi$
$269$	−6.13587	−0.374111	−0.187055	+	0.982349i	$0.559894\pi$
$270$	−1.63684	−0.0996147
$271$	16.0099	0.972530	0.486265	−	0.873811i	$-0.338359\pi$
$271$	16.0099	0.972530	0.486265	+	0.873811i	$0.338359\pi$
$272$	3.97608	0.241085
$273$	0.682352	0.0412978
$274$	−1.51665	−0.0916241
$275$	−5.14694	−0.310372
$276$	1.38082	0.0831154
$277$	20.9201	1.25697	0.628485	−	0.777822i	$-0.283675\pi$
$277$	20.9201	1.25697	0.628485	+	0.777822i	$0.283675\pi$
$278$	−8.70387	−0.522024
$279$	3.33427	0.199618
$280$	0.544275	0.0325266
$281$	−12.0796	−0.720606	−0.360303	−	0.932835i	$-0.617327\pi$
$281$	−12.0796	−0.720606	−0.360303	+	0.932835i	$0.617327\pi$
$282$	−1.79579	−0.106937
$283$	−10.7221	−0.637363	−0.318682	−	0.947862i	$-0.603240\pi$
$283$	−10.7221	−0.637363	−0.318682	+	0.947862i	$0.603240\pi$
$284$	−11.5597	−0.685944
$285$	−0.807356	−0.0478237
$286$	3.04088	0.179811
$287$	−4.09032	−0.241444
$288$	−2.86601	−0.168881
$289$	−1.19077	−0.0700451
$290$	0.0577284	0.00338993
$291$	−3.86754	−0.226719
$292$	1.18715	0.0694728
$293$	−27.5763	−1.61102	−0.805512	−	0.592580i	$-0.798109\pi$
$293$	−27.5763	−1.61102	−0.805512	+	0.592580i	$0.798109\pi$
$294$	2.37575	0.138557
$295$	6.66968	0.388324
$296$	−1.98779	−0.115538
$297$	2.50103	0.145125
$298$	−14.8964	−0.862923
$299$	−9.84825	−0.569539
$300$	1.61756	0.0933896
$301$	−4.82222	−0.277948
$302$	−10.6613	−0.613490
$303$	−2.75590	−0.158322
$304$	−2.89336	−0.165946
$305$	5.86901	0.336058
$306$	−11.3955	−0.651436
$307$	−27.0200	−1.54211	−0.771057	−	0.636767i	$-0.780271\pi$
$307$	−27.0200	−1.54211	−0.771057	+	0.636767i	$0.780271\pi$
$308$	−0.831634	−0.0473867
$309$	5.59248	0.318145
$310$	0.886837	0.0503689
$311$	−14.9217	−0.846133	−0.423067	−	0.906099i	$-0.639046\pi$
$311$	−14.9217	−0.846133	−0.423067	+	0.906099i	$0.639046\pi$
$312$	−0.955674	−0.0541044
$313$	−30.1083	−1.70182	−0.850912	−	0.525308i	$-0.823950\pi$
$313$	−30.1083	−1.70182	−0.850912	+	0.525308i	$0.823950\pi$
$314$	2.38133	0.134386
$315$	−1.55989	−0.0878901
$316$	5.56802	0.313226
$317$	−6.69412	−0.375979	−0.187989	−	0.982171i	$-0.560197\pi$
$317$	−6.69412	−0.375979	−0.187989	+	0.982171i	$0.560197\pi$
$318$	1.81148	0.101583
$319$	−0.0882071	−0.00493865
$320$	−0.762289	−0.0426132
$321$	−6.14772	−0.343133
$322$	2.69334	0.150094
$323$	−11.5042	−0.640113
$324$	7.81200	0.434000
$325$	−11.5367	−0.639942
$326$	11.0574	0.612414
$327$	7.34449	0.406152
$328$	5.72873	0.316316
$329$	−3.50276	−0.193113
$330$	0.325010	0.0178912
$331$	8.28568	0.455422	0.227711	−	0.973729i	$-0.426876\pi$
$331$	8.28568	0.455422	0.227711	+	0.973729i	$0.426876\pi$
$332$	−16.6677	−0.914759
$333$	5.69702	0.312195
$334$	−17.6296	−0.964649
$335$	−6.26547	−0.342319
$336$	0.261362	0.0142585
$337$	1.79247	0.0976422	0.0488211	−	0.998808i	$-0.484454\pi$
$337$	1.79247	0.0976422	0.0488211	+	0.998808i	$0.484454\pi$
$338$	−6.18395	−0.336363
$339$	2.93250	0.159271
$340$	−3.03092	−0.164375
$341$	−1.35506	−0.0733804
$342$	8.29238	0.448401
$343$	9.63202	0.520080
$344$	6.75381	0.364141
$345$	−1.05258	−0.0566691
$346$	−7.51117	−0.403803
$347$	14.4211	0.774166	0.387083	−	0.922045i	$-0.373483\pi$
$347$	14.4211	0.774166	0.387083	+	0.922045i	$0.373483\pi$
$348$	0.0277213	0.00148602
$349$	−24.0262	−1.28609	−0.643047	−	0.765827i	$-0.722330\pi$
$349$	−24.0262	−1.28609	−0.643047	+	0.765827i	$0.722330\pi$
$350$	3.15511	0.168648
$351$	5.60599	0.299226
$352$	1.16475	0.0620815
$353$	−3.96026	−0.210783	−0.105392	−	0.994431i	$-0.533610\pi$
$353$	−3.96026	−0.210783	−0.105392	+	0.994431i	$0.533610\pi$
$354$	3.20279	0.170226
$355$	8.81185	0.467685
$356$	0.193362	0.0102482
$357$	1.03920	0.0550001
$358$	4.51789	0.238778
$359$	17.7950	0.939185	0.469593	−	0.882883i	$-0.344401\pi$
$359$	17.7950	0.939185	0.469593	+	0.882883i	$0.344401\pi$
$360$	2.18472	0.115145
$361$	−10.6285	−0.559393
$362$	4.90501	0.257802
$363$	3.52997	0.185276
$364$	−1.86408	−0.0977045
$365$	−0.904952	−0.0473674
$366$	2.81831	0.147315
$367$	18.3670	0.958748	0.479374	−	0.877611i	$-0.340864\pi$
$367$	18.3670	0.958748	0.479374	+	0.877611i	$0.340864\pi$
$368$	−3.77218	−0.196639
$369$	−16.4186	−0.854717
$370$	1.51527	0.0787752
$371$	3.53336	0.183443
$372$	0.425861	0.0220798
$373$	27.6163	1.42992	0.714959	−	0.699167i	$-0.246445\pi$
$373$	27.6163	1.42992	0.714959	+	0.699167i	$0.246445\pi$
$374$	4.63115	0.239471
$375$	−2.62823	−0.135721
$376$	4.90581	0.252998
$377$	−0.197713	−0.0101828
$378$	−1.53315	−0.0788567
$379$	1.70965	0.0878189	0.0439095	−	0.999036i	$-0.486019\pi$
$379$	1.70965	0.0878189	0.0439095	+	0.999036i	$0.486019\pi$
$380$	2.20557	0.113144
$381$	−6.05331	−0.310121
$382$	3.74010	0.191360
$383$	30.9189	1.57988	0.789941	−	0.613183i	$-0.210111\pi$
$383$	30.9189	1.57988	0.789941	+	0.613183i	$0.210111\pi$
$384$	−0.366053	−0.0186800
$385$	0.633945	0.0323088
$386$	14.1248	0.718932
$387$	−19.3564	−0.983943
$388$	10.5655	0.536384
$389$	24.9617	1.26561	0.632804	−	0.774312i	$-0.281904\pi$
$389$	24.9617	1.26561	0.632804	+	0.774312i	$0.281904\pi$
$390$	0.728499	0.0368890
$391$	−14.9985	−0.758507
$392$	−6.49020	−0.327805
$393$	2.83668	0.143091
$394$	−8.72855	−0.439738
$395$	−4.24444	−0.213561
$396$	−3.33819	−0.167750
$397$	−18.4794	−0.927453	−0.463727	−	0.885978i	$-0.653488\pi$
$397$	−18.4794	−0.927453	−0.463727	+	0.885978i	$0.653488\pi$
$398$	−27.3514	−1.37100
$399$	−0.756214	−0.0378580
$400$	−4.41892	−0.220946
$401$	−36.2861	−1.81204	−0.906020	−	0.423236i	$-0.860894\pi$
$401$	−36.2861	−1.81204	−0.906020	+	0.423236i	$0.860894\pi$
$402$	−3.00869	−0.150060
$403$	−3.03732	−0.151300
$404$	7.52869	0.374566
$405$	−5.95500	−0.295906
$406$	0.0540715	0.00268352
$407$	−2.31528	−0.114764
$408$	−1.45546	−0.0720558
$409$	−0.449474	−0.0222251	−0.0111125	−	0.999938i	$-0.503537\pi$
$409$	−0.449474	−0.0222251	−0.0111125	+	0.999938i	$0.503537\pi$
$410$	−4.36695	−0.215668
$411$	0.555173	0.0273847
$412$	−15.2778	−0.752684
$413$	6.24718	0.307404
$414$	10.8111	0.531336
$415$	12.7056	0.623693
$416$	2.61076	0.128003
$417$	3.18608	0.156023
$418$	−3.37005	−0.164834
$419$	4.54819	0.222193	0.111097	−	0.993810i	$-0.464564\pi$
$419$	4.54819	0.222193	0.111097	+	0.993810i	$0.464564\pi$
$420$	−0.199233	−0.00972158
$421$	−30.7947	−1.50084	−0.750421	−	0.660960i	$-0.770149\pi$
$421$	−30.7947	−1.50084	−0.750421	+	0.660960i	$0.770149\pi$
$422$	3.57528	0.174042
$423$	−14.0601	−0.683625
$424$	−4.94868	−0.240329
$425$	−17.5700	−0.852269
$426$	4.23147	0.205015
$427$	5.49723	0.266030
$428$	16.7947	0.811800
$429$	−1.11312	−0.0537421
$430$	−5.14835	−0.248275
$431$	−3.82029	−0.184017	−0.0920083	−	0.995758i	$-0.529329\pi$
$431$	−3.82029	−0.184017	−0.0920083	+	0.995758i	$0.529329\pi$
$432$	2.14727	0.103310
$433$	−5.59373	−0.268818	−0.134409	−	0.990926i	$-0.542914\pi$
$433$	−5.59373	−0.268818	−0.134409	+	0.990926i	$0.542914\pi$
$434$	0.830659	0.0398729
$435$	−0.0211316	−0.00101318
$436$	−20.0640	−0.960894
$437$	10.9143	0.522101
$438$	−0.434560	−0.0207641
$439$	25.0616	1.19613	0.598063	−	0.801449i	$-0.295937\pi$
$439$	25.0616	1.19613	0.598063	+	0.801449i	$0.295937\pi$
$440$	−0.887877	−0.0423279
$441$	18.6010	0.885760
$442$	10.3806	0.493754
$443$	20.2978	0.964379	0.482189	−	0.876067i	$-0.339842\pi$
$443$	20.2978	0.964379	0.482189	+	0.876067i	$0.339842\pi$
$444$	0.727636	0.0345321
$445$	−0.147398	−0.00698733
$446$	11.6187	0.550161
$447$	5.45285	0.257911
$448$	−0.714001	−0.0337334
$449$	38.4511	1.81462	0.907311	−	0.420461i	$-0.138132\pi$
$449$	38.4511	1.81462	0.907311	+	0.420461i	$0.138132\pi$
$450$	12.6646	0.597017
$451$	6.67255	0.314198
$452$	−8.01114	−0.376812
$453$	3.90260	0.183360
$454$	−10.5358	−0.494471
$455$	1.42097	0.0666161
$456$	1.05912	0.0495979
$457$	19.4116	0.908035	0.454017	−	0.890993i	$-0.349990\pi$
$457$	19.4116	0.908035	0.454017	+	0.890993i	$0.349990\pi$
$458$	−3.83373	−0.179138
$459$	8.53771	0.398506
$460$	2.87549	0.134071
$461$	3.00582	0.139995	0.0699976	−	0.997547i	$-0.477701\pi$
$461$	3.00582	0.139995	0.0699976	+	0.997547i	$0.477701\pi$
$462$	0.304422	0.0141630
$463$	−15.9421	−0.740894	−0.370447	−	0.928854i	$-0.620795\pi$
$463$	−15.9421	−0.740894	−0.370447	+	0.928854i	$0.620795\pi$
$464$	−0.0757303	−0.00351569
$465$	−0.324629	−0.0150543
$466$	−6.53082	−0.302534
$467$	−21.7026	−1.00428	−0.502139	−	0.864787i	$-0.667453\pi$
$467$	−21.7026	−1.00428	−0.502139	+	0.864787i	$0.667453\pi$
$468$	−7.48244	−0.345876
$469$	−5.86858	−0.270986
$470$	−3.73965	−0.172497
$471$	−0.871692	−0.0401655
$472$	−8.74954	−0.402730
$473$	7.86651	0.361702
$474$	−2.03819	−0.0936170
$475$	12.7855	0.586639
$476$	−2.83893	−0.130122
$477$	14.1829	0.649392
$478$	−27.2043	−1.24430
$479$	20.7111	0.946315	0.473157	−	0.880978i	$-0.343114\pi$
$479$	20.7111	0.946315	0.473157	+	0.880978i	$0.343114\pi$
$480$	0.279038	0.0127363
$481$	−5.18964	−0.236627
$482$	−16.9487	−0.771991
$483$	−0.985905	−0.0448602
$484$	−9.64335	−0.438334
$485$	−8.05399	−0.365713
$486$	−9.30140	−0.421920
$487$	14.5359	0.658684	0.329342	−	0.944211i	$-0.393173\pi$
$487$	14.5359	0.658684	0.329342	+	0.944211i	$0.393173\pi$
$488$	−7.69919	−0.348526
$489$	−4.04760	−0.183039
$490$	4.94741	0.223501
$491$	−11.2827	−0.509180	−0.254590	−	0.967049i	$-0.581941\pi$
$491$	−11.2827	−0.509180	−0.254590	+	0.967049i	$0.581941\pi$
$492$	−2.09702	−0.0945408
$493$	−0.301110	−0.0135613
$494$	−7.55386	−0.339864
$495$	2.54466	0.114374
$496$	−1.16339	−0.0522376
$497$	8.25366	0.370227
$498$	6.10125	0.273404
$499$	−23.5183	−1.05282	−0.526412	−	0.850230i	$-0.676463\pi$
$499$	−23.5183	−1.05282	−0.526412	+	0.850230i	$0.676463\pi$
$500$	7.17993	0.321096
$501$	6.45336	0.288315
$502$	12.3431	0.550901
$503$	−2.16481	−0.0965243	−0.0482622	−	0.998835i	$-0.515368\pi$
$503$	−2.16481	−0.0965243	−0.0482622	+	0.998835i	$0.515368\pi$
$504$	2.04633	0.0911508
$505$	−5.73904	−0.255384
$506$	−4.39366	−0.195322
$507$	2.26365	0.100532
$508$	16.5367	0.733699
$509$	13.0283	0.577469	0.288734	−	0.957409i	$-0.406766\pi$
$509$	13.0283	0.577469	0.288734	+	0.957409i	$0.406766\pi$
$510$	1.10948	0.0491285
$511$	−0.847627	−0.0374968
$512$	1.00000	0.0441942
$513$	−6.21281	−0.274302
$514$	−0.937983	−0.0413727
$515$	11.6461	0.513189
$516$	−2.47225	−0.108835
$517$	5.71406	0.251304
$518$	1.41928	0.0623598
$519$	2.74948	0.120689
$520$	−1.99015	−0.0872739
$521$	10.6509	0.466624	0.233312	−	0.972402i	$-0.425044\pi$
$521$	10.6509	0.466624	0.233312	+	0.972402i	$0.425044\pi$
$522$	0.217044	0.00949974
$523$	32.8888	1.43813	0.719064	−	0.694944i	$-0.244571\pi$
$523$	32.8888	1.43813	0.719064	+	0.694944i	$0.244571\pi$
$524$	−7.74937	−0.338533
$525$	−1.15494	−0.0504055
$526$	−13.4179	−0.585050
$527$	−4.62572	−0.201500
$528$	−0.426360	−0.0185550
$529$	−8.77063	−0.381332
$530$	3.77232	0.163859
$531$	25.0762	1.08822
$532$	2.06586	0.0895664
$533$	14.9563	0.647830
$534$	−0.0707807	−0.00306298
$535$	−12.8024	−0.553495
$536$	8.21929	0.355019
$537$	−1.65378	−0.0713661
$538$	−6.13587	−0.264536
$539$	−7.55948	−0.325610
$540$	−1.63684	−0.0704382
$541$	−17.6791	−0.760083	−0.380041	−	0.924970i	$-0.624090\pi$
$541$	−17.6791	−0.760083	−0.380041	+	0.924970i	$0.624090\pi$
$542$	16.0099	0.687682
$543$	−1.79549	−0.0770519
$544$	3.97608	0.170473
$545$	15.2946	0.655149
$546$	0.682352	0.0292020
$547$	−28.4865	−1.21799	−0.608997	−	0.793173i	$-0.708428\pi$
$547$	−28.4865	−1.21799	−0.608997	+	0.793173i	$0.708428\pi$
$548$	−1.51665	−0.0647880
$549$	22.0659	0.941751
$550$	−5.14694	−0.219466
$551$	0.219115	0.00933461
$552$	1.38082	0.0587715
$553$	−3.97557	−0.169058
$554$	20.9201	0.888812
$555$	−0.554669	−0.0235444
$556$	−8.70387	−0.369127
$557$	−2.72203	−0.115336	−0.0576680	−	0.998336i	$-0.518366\pi$
$557$	−2.72203	−0.115336	−0.0576680	+	0.998336i	$0.518366\pi$
$558$	3.33427	0.141151
$559$	17.6325	0.745777
$560$	0.544275	0.0229998
$561$	−1.69524	−0.0715733
$562$	−12.0796	−0.509546
$563$	30.6783	1.29293	0.646467	−	0.762942i	$-0.276246\pi$
$563$	30.6783	1.29293	0.646467	+	0.762942i	$0.276246\pi$
$564$	−1.79579	−0.0756162
$565$	6.10680	0.256915
$566$	−10.7221	−0.450684
$567$	−5.57778	−0.234245
$568$	−11.5597	−0.485036
$569$	24.9547	1.04616	0.523078	−	0.852285i	$-0.324784\pi$
$569$	24.9547	1.04616	0.523078	+	0.852285i	$0.324784\pi$
$570$	−0.807356	−0.0338164
$571$	−6.29675	−0.263511	−0.131755	−	0.991282i	$-0.542061\pi$
$571$	−6.29675	−0.263511	−0.131755	+	0.991282i	$0.542061\pi$
$572$	3.04088	0.127146
$573$	−1.36907	−0.0571939
$574$	−4.09032	−0.170727
$575$	16.6690	0.695144
$576$	−2.86601	−0.119417
$577$	33.2918	1.38595	0.692977	−	0.720959i	$-0.256299\pi$
$577$	33.2918	1.38595	0.692977	+	0.720959i	$0.256299\pi$
$578$	−1.19077	−0.0495293
$579$	−5.17041	−0.214875
$580$	0.0577284	0.00239704
$581$	11.9008	0.493726
$582$	−3.86754	−0.160315
$583$	−5.76398	−0.238720
$584$	1.18715	0.0491247
$585$	5.70378	0.235822
$586$	−27.5763	−1.13917
$587$	−21.9708	−0.906830	−0.453415	−	0.891299i	$-0.649795\pi$
$587$	−21.9708	−0.906830	−0.453415	+	0.891299i	$0.649795\pi$
$588$	2.37575	0.0979745
$589$	3.36610	0.138698
$590$	6.66968	0.274586
$591$	3.19511	0.131429
$592$	−1.98779	−0.0816977
$593$	9.39375	0.385755	0.192878	−	0.981223i	$-0.438218\pi$
$593$	9.39375	0.385755	0.192878	+	0.981223i	$0.438218\pi$
$594$	2.50103	0.102619
$595$	2.16408	0.0887187
$596$	−14.8964	−0.610178
$597$	10.0121	0.409766
$598$	−9.84825	−0.402725
$599$	−21.0849	−0.861506	−0.430753	−	0.902470i	$-0.641752\pi$
$599$	−21.0849	−0.861506	−0.430753	+	0.902470i	$0.641752\pi$
$600$	1.61756	0.0660364
$601$	19.2393	0.784788	0.392394	−	0.919797i	$-0.371647\pi$
$601$	19.2393	0.784788	0.392394	+	0.919797i	$0.371647\pi$
$602$	−4.82222	−0.196539
$603$	−23.5565	−0.959296
$604$	−10.6613	−0.433803
$605$	7.35102	0.298861
$606$	−2.75590	−0.111951
$607$	11.4222	0.463611	0.231806	−	0.972762i	$-0.425537\pi$
$607$	11.4222	0.463611	0.231806	+	0.972762i	$0.425537\pi$
$608$	−2.89336	−0.117341
$609$	−0.0197930	−0.000802054 0
$610$	5.86901	0.237629
$611$	12.8079	0.518152
$612$	−11.3955	−0.460635
$613$	−18.6788	−0.754429	−0.377214	−	0.926126i	$-0.623118\pi$
$613$	−18.6788	−0.754429	−0.377214	+	0.926126i	$0.623118\pi$
$614$	−27.0200	−1.09044
$615$	1.59853	0.0644590
$616$	−0.831634	−0.0335075
$617$	20.1916	0.812883	0.406442	−	0.913677i	$-0.366769\pi$
$617$	20.1916	0.812883	0.406442	+	0.913677i	$0.366769\pi$
$618$	5.59248	0.224963
$619$	26.1720	1.05194	0.525971	−	0.850503i	$-0.323702\pi$
$619$	26.1720	1.05194	0.525971	+	0.850503i	$0.323702\pi$
$620$	0.886837	0.0356162
$621$	−8.09988	−0.325037
$622$	−14.9217	−0.598307
$623$	−0.138061	−0.00553129
$624$	−0.955674	−0.0382576
$625$	16.6214	0.664856
$626$	−30.1083	−1.20337
$627$	1.23361	0.0492658
$628$	2.38133	0.0950255
$629$	−7.90362	−0.315138
$630$	−1.55989	−0.0621477
$631$	−5.63919	−0.224493	−0.112246	−	0.993680i	$-0.535805\pi$
$631$	−5.63919	−0.224493	−0.112246	+	0.993680i	$0.535805\pi$
$632$	5.56802	0.221484
$633$	−1.30874	−0.0520178
$634$	−6.69412	−0.265857
$635$	−12.6058	−0.500245
$636$	1.81148	0.0718297
$637$	−16.9443	−0.671359
$638$	−0.0882071	−0.00349215
$639$	33.1303	1.31061
$640$	−0.762289	−0.0301321
$641$	30.6822	1.21187	0.605937	−	0.795512i	$-0.292798\pi$
$641$	30.6822	1.21187	0.605937	+	0.795512i	$0.292798\pi$
$642$	−6.14772	−0.242631
$643$	24.3324	0.959576	0.479788	−	0.877384i	$-0.340714\pi$
$643$	24.3324	0.959576	0.479788	+	0.877384i	$0.340714\pi$
$644$	2.69334	0.106133
$645$	1.88457	0.0742047
$646$	−11.5042	−0.452628
$647$	30.9915	1.21840	0.609200	−	0.793017i	$-0.291491\pi$
$647$	30.9915	1.21840	0.609200	+	0.793017i	$0.291491\pi$
$648$	7.81200	0.306885
$649$	−10.1910	−0.400033
$650$	−11.5367	−0.452507
$651$	−0.304065	−0.0119172
$652$	11.0574	0.433042
$653$	3.62702	0.141936	0.0709681	−	0.997479i	$-0.477391\pi$
$653$	3.62702	0.141936	0.0709681	+	0.997479i	$0.477391\pi$
$654$	7.34449	0.287192
$655$	5.90726	0.230816
$656$	5.72873	0.223669
$657$	−3.40238	−0.132740
$658$	−3.50276	−0.136552
$659$	−39.1351	−1.52449	−0.762244	−	0.647290i	$-0.775902\pi$
$659$	−39.1351	−1.52449	−0.762244	+	0.647290i	$0.775902\pi$
$660$	0.325010	0.0126510
$661$	−10.8368	−0.421502	−0.210751	−	0.977540i	$-0.567591\pi$
$661$	−10.8368	−0.421502	−0.210751	+	0.977540i	$0.567591\pi$
$662$	8.28568	0.322032
$663$	−3.79984	−0.147574
$664$	−16.6677	−0.646832
$665$	−1.57478	−0.0610674
$666$	5.69702	0.220755
$667$	0.285669	0.0110611
$668$	−17.6296	−0.682110
$669$	−4.25305	−0.164432
$670$	−6.26547	−0.242056
$671$	−8.96765	−0.346192
$672$	0.261362	0.0100823
$673$	−3.81889	−0.147207	−0.0736037	−	0.997288i	$-0.523450\pi$
$673$	−3.81889	−0.147207	−0.0736037	+	0.997288i	$0.523450\pi$
$674$	1.79247	0.0690434
$675$	−9.48859	−0.365216
$676$	−6.18395	−0.237844
$677$	5.71813	0.219766	0.109883	−	0.993945i	$-0.464952\pi$
$677$	5.71813	0.219766	0.109883	+	0.993945i	$0.464952\pi$
$678$	2.93250	0.112622
$679$	−7.54380	−0.289505
$680$	−3.03092	−0.116231
$681$	3.85667	0.147788
$682$	−1.35506	−0.0518878
$683$	6.83738	0.261625	0.130813	−	0.991407i	$-0.458241\pi$
$683$	6.83738	0.261625	0.130813	+	0.991407i	$0.458241\pi$
$684$	8.29238	0.317067
$685$	1.15612	0.0441732
$686$	9.63202	0.367752
$687$	1.40335	0.0535410
$688$	6.75381	0.257486
$689$	−12.9198	−0.492205
$690$	−1.05258	−0.0400711
$691$	−16.4693	−0.626522	−0.313261	−	0.949667i	$-0.601421\pi$
$691$	−16.4693	−0.626522	−0.313261	+	0.949667i	$0.601421\pi$
$692$	−7.51117	−0.285532
$693$	2.38347	0.0905404
$694$	14.4211	0.547418
$695$	6.63487	0.251675
$696$	0.0277213	0.00105077
$697$	22.7779	0.862775
$698$	−24.0262	−0.909406
$699$	2.39062	0.0904216
$700$	3.15511	0.119252
$701$	4.30990	0.162783	0.0813913	−	0.996682i	$-0.474064\pi$
$701$	4.30990	0.162783	0.0813913	+	0.996682i	$0.474064\pi$
$702$	5.60599	0.211584
$703$	5.75139	0.216918
$704$	1.16475	0.0438982
$705$	1.36891	0.0515560
$706$	−3.96026	−0.149046
$707$	−5.37549	−0.202166
$708$	3.20279	0.120368
$709$	11.8797	0.446151	0.223075	−	0.974801i	$-0.428390\pi$
$709$	11.8797	0.446151	0.223075	+	0.974801i	$0.428390\pi$
$710$	8.81185	0.330703
$711$	−15.9580	−0.598471
$712$	0.193362	0.00724656
$713$	4.38851	0.164351
$714$	1.03920	0.0388909
$715$	−2.31803	−0.0866895
$716$	4.51789	0.168841
$717$	9.95820	0.371896
$718$	17.7950	0.664104
$719$	−3.98604	−0.148654	−0.0743270	−	0.997234i	$-0.523681\pi$
$719$	−3.98604	−0.148654	−0.0743270	+	0.997234i	$0.523681\pi$
$720$	2.18472	0.0814198
$721$	10.9084	0.406249
$722$	−10.6285	−0.395551
$723$	6.20410	0.230733
$724$	4.90501	0.182293
$725$	0.334646	0.0124284
$726$	3.52997	0.131010
$727$	−37.3853	−1.38655	−0.693273	−	0.720675i	$-0.743832\pi$
$727$	−37.3853	−1.38655	−0.693273	+	0.720675i	$0.743832\pi$
$728$	−1.86408	−0.0690875
$729$	−20.0312	−0.741897
$730$	−0.904952	−0.0334938
$731$	26.8537	0.993220
$732$	2.81831	0.104168
$733$	20.1138	0.742920	0.371460	−	0.928449i	$-0.378857\pi$
$733$	20.1138	0.742920	0.371460	+	0.928449i	$0.378857\pi$
$734$	18.3670	0.677937
$735$	−1.81101	−0.0668002
$736$	−3.77218	−0.139045
$737$	9.57343	0.352642
$738$	−16.4186	−0.604376
$739$	3.56214	0.131035	0.0655176	−	0.997851i	$-0.479130\pi$
$739$	3.56214	0.131035	0.0655176	+	0.997851i	$0.479130\pi$
$740$	1.51527	0.0557024
$741$	2.76511	0.101579
$742$	3.53336	0.129714
$743$	9.12462	0.334750	0.167375	−	0.985893i	$-0.446471\pi$
$743$	9.12462	0.334750	0.167375	+	0.985893i	$0.446471\pi$
$744$	0.425861	0.0156128
$745$	11.3553	0.416027
$746$	27.6163	1.01110
$747$	47.7697	1.74780
$748$	4.63115	0.169332
$749$	−11.9914	−0.438156
$750$	−2.62823	−0.0959695
$751$	−13.7844	−0.502999	−0.251500	−	0.967857i	$-0.580924\pi$
$751$	−13.7844	−0.502999	−0.251500	+	0.967857i	$0.580924\pi$
$752$	4.90581	0.178897
$753$	−4.51823	−0.164653
$754$	−0.197713	−0.00720030
$755$	8.12701	0.295772
$756$	−1.53315	−0.0557601
$757$	−29.7678	−1.08193	−0.540964	−	0.841046i	$-0.681941\pi$
$757$	−29.7678	−1.08193	−0.540964	+	0.841046i	$0.681941\pi$
$758$	1.70965	0.0620974
$759$	1.60831	0.0583779
$760$	2.20557	0.0800046
$761$	24.0368	0.871334	0.435667	−	0.900108i	$-0.356513\pi$
$761$	24.0368	0.871334	0.435667	+	0.900108i	$0.356513\pi$
$762$	−6.05331	−0.219288
$763$	14.3257	0.518627
$764$	3.74010	0.135312
$765$	8.68664	0.314066
$766$	30.9189	1.11715
$767$	−22.8429	−0.824810
$768$	−0.366053	−0.0132088
$769$	36.8634	1.32933	0.664664	−	0.747143i	$-0.268575\pi$
$769$	36.8634	1.32933	0.664664	+	0.747143i	$0.268575\pi$
$770$	0.633945	0.0228458
$771$	0.343351	0.0123655
$772$	14.1248	0.508362
$773$	22.6501	0.814667	0.407333	−	0.913279i	$-0.366459\pi$
$773$	22.6501	0.814667	0.407333	+	0.913279i	$0.366459\pi$
$774$	−19.3564	−0.695753
$775$	5.14091	0.184667
$776$	10.5655	0.379281
$777$	−0.519533	−0.0186381
$778$	24.9617	0.894920
$779$	−16.5753	−0.593871
$780$	0.728499	0.0260845
$781$	−13.4642	−0.481788
$782$	−14.9985	−0.536346
$783$	−0.162613	−0.00581132
$784$	−6.49020	−0.231793
$785$	−1.81526	−0.0647895
$786$	2.83668	0.101181
$787$	−21.5458	−0.768025	−0.384012	−	0.923328i	$-0.625458\pi$
$787$	−21.5458	−0.768025	−0.384012	+	0.923328i	$0.625458\pi$
$788$	−8.72855	−0.310942
$789$	4.91167	0.174860
$790$	−4.24444	−0.151010
$791$	5.71996	0.203378
$792$	−3.33819	−0.118617
$793$	−20.1007	−0.713797
$794$	−18.4794	−0.655808
$795$	−1.38087	−0.0489743
$796$	−27.3514	−0.969445
$797$	−2.69511	−0.0954656	−0.0477328	−	0.998860i	$-0.515200\pi$
$797$	−2.69511	−0.0954656	−0.0477328	+	0.998860i	$0.515200\pi$
$798$	−0.756214	−0.0267697
$799$	19.5059	0.690070
$800$	−4.41892	−0.156232
$801$	−0.554177	−0.0195809
$802$	−36.2861	−1.28131
$803$	1.38274	0.0487957
$804$	−3.00869	−0.106108
$805$	−2.05310	−0.0723624
$806$	−3.03732	−0.106985
$807$	2.24605	0.0790648
$808$	7.52869	0.264858
$809$	47.3138	1.66347	0.831733	−	0.555176i	$-0.187349\pi$
$809$	47.3138	1.66347	0.831733	+	0.555176i	$0.187349\pi$
$810$	−5.95500	−0.209237
$811$	−43.1970	−1.51685	−0.758426	−	0.651759i	$-0.774031\pi$
$811$	−43.1970	−1.51685	−0.758426	+	0.651759i	$0.774031\pi$
$812$	0.0540715	0.00189754
$813$	−5.86045	−0.205535
$814$	−2.31528	−0.0811506
$815$	−8.42895	−0.295253
$816$	−1.45546	−0.0509511
$817$	−19.5412	−0.683659
$818$	−0.449474	−0.0157155
$819$	5.34247	0.186681
$820$	−4.36695	−0.152500
$821$	−45.7311	−1.59603	−0.798013	−	0.602641i	$-0.794115\pi$
$821$	−45.7311	−1.59603	−0.798013	+	0.602641i	$0.794115\pi$
$822$	0.555173	0.0193639
$823$	−49.9103	−1.73976	−0.869882	−	0.493261i	$-0.835805\pi$
$823$	−49.9103	−1.73976	−0.869882	+	0.493261i	$0.835805\pi$
$824$	−15.2778	−0.532228
$825$	1.88405	0.0655942
$826$	6.24718	0.217367
$827$	−43.8073	−1.52333	−0.761665	−	0.647971i	$-0.775618\pi$
$827$	−43.8073	−1.52333	−0.761665	+	0.647971i	$0.775618\pi$
$828$	10.8111	0.375712
$829$	10.6327	0.369287	0.184644	−	0.982806i	$-0.440887\pi$
$829$	10.6327	0.369287	0.184644	+	0.982806i	$0.440887\pi$
$830$	12.7056	0.441018
$831$	−7.65787	−0.265649
$832$	2.61076	0.0905117
$833$	−25.8056	−0.894110
$834$	3.18608	0.110325
$835$	13.4388	0.465070
$836$	−3.37005	−0.116555
$837$	−2.49810	−0.0863470
$838$	4.54819	0.157115
$839$	20.2173	0.697977	0.348989	−	0.937127i	$-0.386525\pi$
$839$	20.2173	0.697977	0.348989	+	0.937127i	$0.386525\pi$
$840$	−0.199233	−0.00687420
$841$	−28.9943	−0.999802
$842$	−30.7947	−1.06126
$843$	4.42176	0.152293
$844$	3.57528	0.123066
$845$	4.71396	0.162165
$846$	−14.0601	−0.483396
$847$	6.88536	0.236584
$848$	−4.94868	−0.169938
$849$	3.92485	0.134701
$850$	−17.5700	−0.602645
$851$	7.49831	0.257039
$852$	4.23147	0.144968
$853$	−6.50309	−0.222662	−0.111331	−	0.993783i	$-0.535511\pi$
$853$	−6.50309	−0.222662	−0.111331	+	0.993783i	$0.535511\pi$
$854$	5.49723	0.188111
$855$	−6.32119	−0.216180
$856$	16.7947	0.574029
$857$	8.81368	0.301070	0.150535	−	0.988605i	$-0.451900\pi$
$857$	8.81368	0.301070	0.150535	+	0.988605i	$0.451900\pi$
$858$	−1.11312	−0.0380014
$859$	−28.1115	−0.959153	−0.479577	−	0.877500i	$-0.659210\pi$
$859$	−28.1115	−0.959153	−0.479577	+	0.877500i	$0.659210\pi$
$860$	−5.14835	−0.175557
$861$	1.49727	0.0510269
$862$	−3.82029	−0.130119
$863$	−14.9801	−0.509930	−0.254965	−	0.966950i	$-0.582064\pi$
$863$	−14.9801	−0.509930	−0.254965	+	0.966950i	$0.582064\pi$
$864$	2.14727	0.0730515
$865$	5.72568	0.194679
$866$	−5.59373	−0.190083
$867$	0.435883	0.0148034
$868$	0.830659	0.0281944
$869$	6.48536	0.220001
$870$	−0.0211316	−0.000716429 0
$871$	21.4586	0.727096
$872$	−20.0640	−0.679454
$873$	−30.2809	−1.02485
$874$	10.9143	0.369181
$875$	−5.12648	−0.173307
$876$	−0.434560	−0.0146824
$877$	38.4568	1.29860	0.649298	−	0.760534i	$-0.275063\pi$
$877$	38.4568	1.29860	0.649298	+	0.760534i	$0.275063\pi$
$878$	25.0616	0.845788
$879$	10.0944	0.340474
$880$	−0.887877	−0.0299303
$881$	−22.9504	−0.773219	−0.386610	−	0.922243i	$-0.626354\pi$
$881$	−22.9504	−0.773219	−0.386610	+	0.922243i	$0.626354\pi$
$882$	18.6010	0.626327
$883$	−21.6551	−0.728753	−0.364377	−	0.931252i	$-0.618718\pi$
$883$	−21.6551	−0.728753	−0.364377	+	0.931252i	$0.618718\pi$
$884$	10.3806	0.349137
$885$	−2.44145	−0.0820685
$886$	20.2978	0.681919
$887$	29.8467	1.00215	0.501077	−	0.865403i	$-0.332937\pi$
$887$	29.8467	1.00215	0.501077	+	0.865403i	$0.332937\pi$
$888$	0.727636	0.0244179
$889$	−11.8072	−0.396002
$890$	−0.147398	−0.00494079
$891$	9.09905	0.304830
$892$	11.6187	0.389022
$893$	−14.1943	−0.474994
$894$	5.45285	0.182371
$895$	−3.44394	−0.115118
$896$	−0.714001	−0.0238531
$897$	3.60498	0.120367
$898$	38.4511	1.28313
$899$	0.0881037	0.00293842
$900$	12.6646	0.422155
$901$	−19.6764	−0.655515
$902$	6.67255	0.222172
$903$	1.76519	0.0587417
$904$	−8.01114	−0.266446
$905$	−3.73903	−0.124290
$906$	3.90260	0.129655
$907$	17.5047	0.581235	0.290617	−	0.956839i	$-0.406139\pi$
$907$	17.5047	0.581235	0.290617	+	0.956839i	$0.406139\pi$
$908$	−10.5358	−0.349644
$909$	−21.5773	−0.715673
$910$	1.42097	0.0471047
$911$	−32.1164	−1.06406	−0.532032	−	0.846724i	$-0.678572\pi$
$911$	−32.1164	−1.06406	−0.532032	+	0.846724i	$0.678572\pi$
$912$	1.05912	0.0350710
$913$	−19.4137	−0.642501
$914$	19.4116	0.642077
$915$	−2.14837	−0.0710228
$916$	−3.83373	−0.126670
$917$	5.53306	0.182718
$918$	8.53771	0.281786
$919$	24.9752	0.823857	0.411929	−	0.911216i	$-0.364855\pi$
$919$	24.9752	0.823857	0.411929	+	0.911216i	$0.364855\pi$
$920$	2.87549	0.0948022
$921$	9.89074	0.325911
$922$	3.00582	0.0989916
$923$	−30.1796	−0.993375
$924$	0.304422	0.0100147
$925$	8.78388	0.288812
$926$	−15.9421	−0.523891
$927$	43.7863	1.43813
$928$	−0.0757303	−0.00248597
$929$	−14.8056	−0.485756	−0.242878	−	0.970057i	$-0.578091\pi$
$929$	−14.8056	−0.485756	−0.242878	+	0.970057i	$0.578091\pi$
$930$	−0.324629	−0.0106450
$931$	18.7785	0.615440
$932$	−6.53082	−0.213924
$933$	5.46214	0.178822
$934$	−21.7026	−0.710132
$935$	−3.53027	−0.115452
$936$	−7.48244	−0.244571
$937$	31.4562	1.02763	0.513814	−	0.857902i	$-0.328232\pi$
$937$	31.4562	1.02763	0.513814	+	0.857902i	$0.328232\pi$
$938$	−5.86858	−0.191616
$939$	11.0212	0.359664
$940$	−3.73965	−0.121974
$941$	−18.4015	−0.599871	−0.299935	−	0.953960i	$-0.596965\pi$
$941$	−18.4015	−0.599871	−0.299935	+	0.953960i	$0.596965\pi$
$942$	−0.871692	−0.0284013
$943$	−21.6098	−0.703713
$944$	−8.74954	−0.284773
$945$	1.16870	0.0380179
$946$	7.86651	0.255762
$947$	−14.3260	−0.465531	−0.232765	−	0.972533i	$-0.574777\pi$
$947$	−14.3260	−0.465531	−0.232765	+	0.972533i	$0.574777\pi$
$948$	−2.03819	−0.0661972
$949$	3.09936	0.100610
$950$	12.7855	0.414817
$951$	2.45040	0.0794596
$952$	−2.83893	−0.0920101
$953$	3.16507	0.102527	0.0512634	−	0.998685i	$-0.483675\pi$
$953$	3.16507	0.102527	0.0512634	+	0.998685i	$0.483675\pi$
$954$	14.1829	0.459190
$955$	−2.85104	−0.0922575
$956$	−27.2043	−0.879850
$957$	0.0322884	0.00104374
$958$	20.7111	0.669146
$959$	1.08289	0.0349683
$960$	0.279038	0.00900590
$961$	−29.6465	−0.956340
$962$	−5.18964	−0.167321
$963$	−48.1336	−1.55108
$964$	−16.9487	−0.545880
$965$	−10.7672	−0.346607
$966$	−0.985905	−0.0317210
$967$	15.4000	0.495230	0.247615	−	0.968859i	$-0.420353\pi$
$967$	15.4000	0.495230	0.247615	+	0.968859i	$0.420353\pi$
$968$	−9.64335	−0.309949
$969$	4.21115	0.135282
$970$	−8.05399	−0.258598
$971$	−33.0340	−1.06011	−0.530056	−	0.847963i	$-0.677829\pi$
$971$	−33.0340	−1.06011	−0.530056	+	0.847963i	$0.677829\pi$
$972$	−9.30140	−0.298343
$973$	6.21457	0.199230
$974$	14.5359	0.465760
$975$	4.22304	0.135246
$976$	−7.69919	−0.246445
$977$	32.8193	1.04998	0.524992	−	0.851107i	$-0.324068\pi$
$977$	32.8193	1.04998	0.524992	+	0.851107i	$0.324068\pi$
$978$	−4.04760	−0.129428
$979$	0.225219	0.00719803
$980$	4.94741	0.158039
$981$	57.5037	1.83595
$982$	−11.2827	−0.360045
$983$	−6.76117	−0.215648	−0.107824	−	0.994170i	$-0.534388\pi$
$983$	−6.76117	−0.215648	−0.107824	+	0.994170i	$0.534388\pi$
$984$	−2.09702	−0.0668504
$985$	6.65368	0.212004
$986$	−0.301110	−0.00958930
$987$	1.28219	0.0408126
$988$	−7.55386	−0.240320
$989$	−25.4766	−0.810109
$990$	2.54466	0.0808746
$991$	30.7269	0.976071	0.488035	−	0.872824i	$-0.337714\pi$
$991$	30.7269	0.976071	0.488035	+	0.872824i	$0.337714\pi$
$992$	−1.16339	−0.0369376
$993$	−3.03300	−0.0962492
$994$	8.25366	0.261790
$995$	20.8497	0.660979
$996$	6.10125	0.193326
$997$	39.6500	1.25573	0.627864	−	0.778323i	$-0.283929\pi$
$997$	39.6500	1.25573	0.627864	+	0.778323i	$0.283929\pi$
$998$	−23.5183	−0.744458
$999$	−4.26832	−0.135044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.33	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.33	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.366053 q^{3} +1.00000 q^{4} -0.762289 q^{5} -0.366053 q^{6} -0.714001 q^{7} +1.00000 q^{8} -2.86601 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.366053 q^{3} +1.00000 q^{4} -0.762289 q^{5} -0.366053 q^{6} -0.714001 q^{7} +1.00000 q^{8} -2.86601 q^{9} -0.762289 q^{10} +1.16475 q^{11} -0.366053 q^{12} +2.61076 q^{13} -0.714001 q^{14} +0.279038 q^{15} +1.00000 q^{16} +3.97608 q^{17} -2.86601 q^{18} -2.89336 q^{19} -0.762289 q^{20} +0.261362 q^{21} +1.16475 q^{22} -3.77218 q^{23} -0.366053 q^{24} -4.41892 q^{25} +2.61076 q^{26} +2.14727 q^{27} -0.714001 q^{28} -0.0757303 q^{29} +0.279038 q^{30} -1.16339 q^{31} +1.00000 q^{32} -0.426360 q^{33} +3.97608 q^{34} +0.544275 q^{35} -2.86601 q^{36} -1.98779 q^{37} -2.89336 q^{38} -0.955674 q^{39} -0.762289 q^{40} +5.72873 q^{41} +0.261362 q^{42} +6.75381 q^{43} +1.16475 q^{44} +2.18472 q^{45} -3.77218 q^{46} +4.90581 q^{47} -0.366053 q^{48} -6.49020 q^{49} -4.41892 q^{50} -1.45546 q^{51} +2.61076 q^{52} -4.94868 q^{53} +2.14727 q^{54} -0.887877 q^{55} -0.714001 q^{56} +1.05912 q^{57} -0.0757303 q^{58} -8.74954 q^{59} +0.279038 q^{60} -7.69919 q^{61} -1.16339 q^{62} +2.04633 q^{63} +1.00000 q^{64} -1.99015 q^{65} -0.426360 q^{66} +8.21929 q^{67} +3.97608 q^{68} +1.38082 q^{69} +0.544275 q^{70} -11.5597 q^{71} -2.86601 q^{72} +1.18715 q^{73} -1.98779 q^{74} +1.61756 q^{75} -2.89336 q^{76} -0.831634 q^{77} -0.955674 q^{78} +5.56802 q^{79} -0.762289 q^{80} +7.81200 q^{81} +5.72873 q^{82} -16.6677 q^{83} +0.261362 q^{84} -3.03092 q^{85} +6.75381 q^{86} +0.0277213 q^{87} +1.16475 q^{88} +0.193362 q^{89} +2.18472 q^{90} -1.86408 q^{91} -3.77218 q^{92} +0.425861 q^{93} +4.90581 q^{94} +2.20557 q^{95} -0.366053 q^{96} +10.5655 q^{97} -6.49020 q^{98} -3.33819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more