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Properties

Label 10.12.a.d.1.1
Level $10$
Weight $12$
Character 10.1
Self dual yes
Analytic conductor $7.683$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [10,12,Mod(1,10)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,64,604] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.68343180560\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1969}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 492 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(22.6867\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} -141.734 q^{3} +1024.00 q^{4} +3125.00 q^{5} -4535.49 q^{6} +85586.9 q^{7} +32768.0 q^{8} -157058. q^{9} +100000. q^{10} +767235. q^{11} -145136. q^{12} +220960. q^{13} +2.73878e6 q^{14} -442919. q^{15} +1.04858e6 q^{16} -930719. q^{17} -5.02587e6 q^{18} -1.77341e7 q^{19} +3.20000e6 q^{20} -1.21306e7 q^{21} +2.45515e7 q^{22} -3.99596e7 q^{23} -4.64434e6 q^{24} +9.76562e6 q^{25} +7.07072e6 q^{26} +4.73683e7 q^{27} +8.76410e7 q^{28} +7.68554e7 q^{29} -1.41734e7 q^{30} -2.96314e7 q^{31} +3.35544e7 q^{32} -1.08743e8 q^{33} -2.97830e7 q^{34} +2.67459e8 q^{35} -1.60828e8 q^{36} +5.40911e7 q^{37} -5.67491e8 q^{38} -3.13176e7 q^{39} +1.02400e8 q^{40} +1.26006e8 q^{41} -3.88179e8 q^{42} -2.88676e8 q^{43} +7.85648e8 q^{44} -4.90808e8 q^{45} -1.27871e9 q^{46} -1.57008e9 q^{47} -1.48619e8 q^{48} +5.34780e9 q^{49} +3.12500e8 q^{50} +1.31915e8 q^{51} +2.26263e8 q^{52} -4.09006e9 q^{53} +1.51579e9 q^{54} +2.39761e9 q^{55} +2.80451e9 q^{56} +2.51353e9 q^{57} +2.45937e9 q^{58} +3.77882e9 q^{59} -4.53549e8 q^{60} -9.64103e9 q^{61} -9.48205e8 q^{62} -1.34422e10 q^{63} +1.07374e9 q^{64} +6.90500e8 q^{65} -3.47979e9 q^{66} -1.63819e10 q^{67} -9.53056e8 q^{68} +5.66364e9 q^{69} +8.55869e9 q^{70} +1.03471e10 q^{71} -5.14649e9 q^{72} +4.27149e9 q^{73} +1.73091e9 q^{74} -1.38412e9 q^{75} -1.81597e10 q^{76} +6.56653e10 q^{77} -1.00216e9 q^{78} -1.96636e10 q^{79} +3.27680e9 q^{80} +2.11087e10 q^{81} +4.03220e9 q^{82} -1.35791e10 q^{83} -1.24217e10 q^{84} -2.90850e9 q^{85} -9.23762e9 q^{86} -1.08930e10 q^{87} +2.51407e10 q^{88} +2.25058e10 q^{89} -1.57058e10 q^{90} +1.89113e10 q^{91} -4.09186e10 q^{92} +4.19978e9 q^{93} -5.02424e10 q^{94} -5.54191e10 q^{95} -4.75581e9 q^{96} -1.08976e11 q^{97} +1.71130e11 q^{98} -1.20501e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{2} + 604 q^{3} + 2048 q^{4} + 6250 q^{5} + 19328 q^{6} + 14092 q^{7} + 65536 q^{8} + 221914 q^{9} + 200000 q^{10} + 421584 q^{11} + 618496 q^{12} + 1730524 q^{13} + 450944 q^{14} + 1887500 q^{15}+ \cdots - 251492724912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 0.707107
\(3\) −141.734 −0.336750 −0.168375 − 0.985723i \(-0.553852\pi\)
−0.168375 + 0.985723i \(0.553852\pi\)
\(4\) 1024.00 0.500000
\(5\) 3125.00 0.447214
\(6\) −4535.49 −0.238118
\(7\) 85586.9 1.92472 0.962362 − 0.271772i \(-0.0876097\pi\)
0.962362 + 0.271772i \(0.0876097\pi\)
\(8\) 32768.0 0.353553
\(9\) −157058. −0.886599
\(10\) 100000. 0.316228
\(11\) 767235. 1.43638 0.718188 − 0.695849i \(-0.244972\pi\)
0.718188 + 0.695849i \(0.244972\pi\)
\(12\) −145136. −0.168375
\(13\) 220960. 0.165054 0.0825268 − 0.996589i \(-0.473701\pi\)
0.0825268 + 0.996589i \(0.473701\pi\)
\(14\) 2.73878e6 1.36098
\(15\) −442919. −0.150599
\(16\) 1.04858e6 0.250000
\(17\) −930719. −0.158983 −0.0794913 − 0.996836i \(-0.525330\pi\)
−0.0794913 + 0.996836i \(0.525330\pi\)
\(18\) −5.02587e6 −0.626920
\(19\) −1.77341e7 −1.64310 −0.821551 − 0.570135i \(-0.806891\pi\)
−0.821551 + 0.570135i \(0.806891\pi\)
\(20\) 3.20000e6 0.223607
\(21\) −1.21306e7 −0.648151
\(22\) 2.45515e7 1.01567
\(23\) −3.99596e7 −1.29455 −0.647274 − 0.762257i \(-0.724091\pi\)
−0.647274 + 0.762257i \(0.724091\pi\)
\(24\) −4.64434e6 −0.119059
\(25\) 9.76562e6 0.200000
\(26\) 7.07072e6 0.116711
\(27\) 4.73683e7 0.635312
\(28\) 8.76410e7 0.962362
\(29\) 7.68554e7 0.695801 0.347901 − 0.937531i \(-0.386895\pi\)
0.347901 + 0.937531i \(0.386895\pi\)
\(30\) −1.41734e7 −0.106490
\(31\) −2.96314e7 −0.185893 −0.0929465 − 0.995671i \(-0.529629\pi\)
−0.0929465 + 0.995671i \(0.529629\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −1.08743e8 −0.483700
\(34\) −2.97830e7 −0.112418
\(35\) 2.67459e8 0.860762
\(36\) −1.60828e8 −0.443300
\(37\) 5.40911e7 0.128238 0.0641189 − 0.997942i \(-0.479576\pi\)
0.0641189 + 0.997942i \(0.479576\pi\)
\(38\) −5.67491e8 −1.16185
\(39\) −3.13176e7 −0.0555818
\(40\) 1.02400e8 0.158114
\(41\) 1.26006e8 0.169856 0.0849280 − 0.996387i \(-0.472934\pi\)
0.0849280 + 0.996387i \(0.472934\pi\)
\(42\) −3.88179e8 −0.458312
\(43\) −2.88676e8 −0.299457 −0.149728 − 0.988727i \(-0.547840\pi\)
−0.149728 + 0.988727i \(0.547840\pi\)
\(44\) 7.85648e8 0.718188
\(45\) −4.90808e8 −0.396499
\(46\) −1.27871e9 −0.915384
\(47\) −1.57008e9 −0.998579 −0.499290 − 0.866435i \(-0.666406\pi\)
−0.499290 + 0.866435i \(0.666406\pi\)
\(48\) −1.48619e8 −0.0841875
\(49\) 5.34780e9 2.70456
\(50\) 3.12500e8 0.141421
\(51\) 1.31915e8 0.0535374
\(52\) 2.26263e8 0.0825268
\(53\) −4.09006e9 −1.34342 −0.671711 − 0.740813i \(-0.734440\pi\)
−0.671711 + 0.740813i \(0.734440\pi\)
\(54\) 1.51579e9 0.449234
\(55\) 2.39761e9 0.642367
\(56\) 2.80451e9 0.680492
\(57\) 2.51353e9 0.553315
\(58\) 2.45937e9 0.492006
\(59\) 3.77882e9 0.688129 0.344065 − 0.938946i \(-0.388196\pi\)
0.344065 + 0.938946i \(0.388196\pi\)
\(60\) −4.53549e8 −0.0752996
\(61\) −9.64103e9 −1.46154 −0.730768 − 0.682626i \(-0.760838\pi\)
−0.730768 + 0.682626i \(0.760838\pi\)
\(62\) −9.48205e8 −0.131446
\(63\) −1.34422e10 −1.70646
\(64\) 1.07374e9 0.125000
\(65\) 6.90500e8 0.0738143
\(66\) −3.47979e9 −0.342028
\(67\) −1.63819e10 −1.48236 −0.741178 − 0.671308i \(-0.765733\pi\)
−0.741178 + 0.671308i \(0.765733\pi\)
\(68\) −9.53056e8 −0.0794913
\(69\) 5.66364e9 0.435939
\(70\) 8.55869e9 0.608651
\(71\) 1.03471e10 0.680609 0.340304 − 0.940315i \(-0.389470\pi\)
0.340304 + 0.940315i \(0.389470\pi\)
\(72\) −5.14649e9 −0.313460
\(73\) 4.27149e9 0.241159 0.120580 − 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(74\) 1.73091e9 0.0906778
\(75\) −1.38412e9 −0.0673500
\(76\) −1.81597e10 −0.821551
\(77\) 6.56653e10 2.76463
\(78\) −1.00216e9 −0.0393023
\(79\) −1.96636e10 −0.718977 −0.359489 − 0.933149i \(-0.617049\pi\)
−0.359489 + 0.933149i \(0.617049\pi\)
\(80\) 3.27680e9 0.111803
\(81\) 2.11087e10 0.672658
\(82\) 4.03220e9 0.120106
\(83\) −1.35791e10 −0.378391 −0.189196 − 0.981939i \(-0.560588\pi\)
−0.189196 + 0.981939i \(0.560588\pi\)
\(84\) −1.24217e10 −0.324075
\(85\) −2.90850e9 −0.0710991
\(86\) −9.23762e9 −0.211748
\(87\) −1.08930e10 −0.234311
\(88\) 2.51407e10 0.507836
\(89\) 2.25058e10 0.427218 0.213609 − 0.976919i \(-0.431478\pi\)
0.213609 + 0.976919i \(0.431478\pi\)
\(90\) −1.57058e10 −0.280367
\(91\) 1.89113e10 0.317683
\(92\) −4.09186e10 −0.647274
\(93\) 4.19978e9 0.0625994
\(94\) −5.02424e10 −0.706102
\(95\) −5.54191e10 −0.734818
\(96\) −4.75581e9 −0.0595296
\(97\) −1.08976e11 −1.28851 −0.644255 − 0.764811i \(-0.722832\pi\)
−0.644255 + 0.764811i \(0.722832\pi\)
\(98\) 1.71130e11 1.91241
\(99\) −1.20501e11 −1.27349
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 10.12.a.d.1.1 ✓ 2
3.2 odd 2 90.12.a.l.1.2 2
4.3 odd 2 80.12.a.g.1.2 2
5.2 odd 4 50.12.b.f.49.4 4
5.3 odd 4 50.12.b.f.49.1 4
5.4 even 2 50.12.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.12.a.d.1.1 ✓ 2 1.1 even 1 trivial
50.12.a.f.1.2 2 5.4 even 2
50.12.b.f.49.1 4 5.3 odd 4
50.12.b.f.49.4 4 5.2 odd 4
80.12.a.g.1.2 2 4.3 odd 2
90.12.a.l.1.2 2 3.2 odd 2