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Properties

Label 50.12.d.b.11.6
Level $50$
Weight $12$
Character 50.11
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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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] = [50,12,Mod(11,50)] 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("50.11"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 11.6
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.6

$q$-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+(-25.8885 + 18.8091i) q^{2} +(-84.0143 + 258.570i) q^{3} +(316.433 - 973.882i) q^{4} +(3827.94 - 5845.94i) q^{5} +(-2688.46 - 8274.23i) q^{6} +54397.6 q^{7} +(10125.9 + 31164.2i) q^{8} +(83515.1 + 60677.3i) q^{9} +(10857.1 + 223343. i) q^{10} +(152190. - 110573. i) q^{11} +(225231. + 163640. i) q^{12} +(-50210.8 - 36480.3i) q^{13} +(-1.40827e6 + 1.02317e6i) q^{14} +(1.18998e6 + 1.48093e6i) q^{15} +(-848316. - 616338. i) q^{16} +(-2.39588e6 - 7.37377e6i) q^{17} -3.30337e6 q^{18} +(-4.31912e6 - 1.32929e7i) q^{19} +(-4.48196e6 - 5.57781e6i) q^{20} +(-4.57018e6 + 1.40656e7i) q^{21} +(-1.86021e6 + 5.72514e6i) q^{22} +(8.47978e6 - 6.16092e6i) q^{23} -8.90884e6 q^{24} +(-1.95218e7 - 4.47558e7i) q^{25} +1.98605e6 q^{26} +(-6.16697e7 + 4.48056e7i) q^{27} +(1.72132e7 - 5.29768e7i) q^{28} +(-1.87725e6 + 5.77759e6i) q^{29} +(-5.86619e7 - 1.59567e7i) q^{30} +(4.31924e7 + 1.32933e8i) q^{31} +3.35544e7 q^{32} +(1.58046e7 + 4.86415e7i) q^{33} +(2.00720e8 + 1.45832e8i) q^{34} +(2.08231e8 - 3.18005e8i) q^{35} +(8.55195e7 - 6.21336e7i) q^{36} +(-3.29199e8 - 2.39177e8i) q^{37} +(3.61843e8 + 2.62894e8i) q^{38} +(1.36511e7 - 9.91811e6i) q^{39} +(2.20945e8 + 6.00996e7i) q^{40} +(1.82269e8 + 1.32426e8i) q^{41} +(-1.46246e8 - 4.50098e8i) q^{42} +1.79385e8 q^{43} +(-5.95267e7 - 1.83204e8i) q^{44} +(6.74407e8 - 2.55955e8i) q^{45} +(-1.03648e8 + 3.18995e8i) q^{46} +(2.59755e8 - 7.99445e8i) q^{47} +(2.30637e8 - 1.67567e8i) q^{48} +9.81769e8 q^{49} +(1.34721e9 + 7.91474e8i) q^{50} +2.10792e9 q^{51} +(-5.14158e7 + 3.73558e7i) q^{52} +(1.00594e9 - 3.09597e9i) q^{53} +(7.53783e8 - 2.31991e9i) q^{54} +(-6.38256e7 - 1.31296e9i) q^{55} +(5.50823e8 + 1.69526e9i) q^{56} +3.80000e9 q^{57} +(-6.00721e7 - 1.84883e8i) q^{58} +(4.68847e9 + 3.40638e9i) q^{59} +(1.81880e9 - 6.90283e8i) q^{60} +(7.46048e9 - 5.42036e9i) q^{61} +(-3.61853e9 - 2.62902e9i) q^{62} +(4.54302e9 + 3.30070e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-4.05465e8 + 1.53885e8i) q^{65} +(-1.32406e9 - 9.61987e8i) q^{66} +(3.06203e8 + 9.42396e8i) q^{67} -7.93932e9 q^{68} +(8.80603e8 + 2.71022e9i) q^{69} +(5.90602e8 + 1.21493e10i) q^{70} +(6.39672e9 - 1.96871e10i) q^{71} +(-1.04530e9 + 3.21709e9i) q^{72} +(-2.35531e10 + 1.71123e10i) q^{73} +1.30212e10 q^{74} +(1.32126e10 - 1.28762e9i) q^{75} -1.43124e10 q^{76} +(8.27879e9 - 6.01489e9i) q^{77} +(-1.66856e8 + 5.13531e8i) q^{78} +(1.65694e10 - 5.09952e10i) q^{79} +(-6.85037e9 + 2.59990e9i) q^{80} +(-7.53259e8 - 2.31829e9i) q^{81} -7.20950e9 q^{82} +(6.37641e9 + 1.96246e10i) q^{83} +(1.22520e10 + 8.90162e9i) q^{84} +(-5.22779e10 - 1.42202e10i) q^{85} +(-4.64402e9 + 3.37408e9i) q^{86} +(-1.33619e9 - 9.70801e8i) q^{87} +(4.98697e9 + 3.62325e9i) q^{88} +(-3.64769e10 + 2.65020e10i) q^{89} +(-1.26451e10 + 1.93113e10i) q^{90} +(-2.73134e9 - 1.98444e9i) q^{91} +(-3.31672e9 - 1.02078e10i) q^{92} -3.80011e10 q^{93} +(8.31217e9 + 2.55822e10i) q^{94} +(-9.42427e10 - 2.56351e10i) q^{95} +(-2.81905e9 + 8.67615e9i) q^{96} +(4.27303e9 - 1.31510e10i) q^{97} +(-2.54166e10 + 1.84662e10i) q^{98} +1.94195e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 448 q^{2} - 263 q^{3} - 14336 q^{4} + 1770 q^{5} - 8416 q^{6} - 111844 q^{7} - 458752 q^{8} - 1174523 q^{9} + 304960 q^{10} + 207277 q^{11} + 1026048 q^{12} + 893677 q^{13} - 1270048 q^{14} + 4696640 q^{15}+ \cdots - 505737997606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −25.8885 + 18.8091i −0.572061 + 0.415627i
\(3\) −84.0143 + 258.570i −0.199612 + 0.614342i 0.800280 + 0.599627i \(0.204684\pi\)
−0.999892 + 0.0147156i \(0.995316\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) 3827.94 − 5845.94i 0.547810 − 0.836602i
\(6\) −2688.46 − 8274.23i −0.141147 − 0.434406i
\(7\) 54397.6 1.22332 0.611660 − 0.791121i \(-0.290502\pi\)
0.611660 + 0.791121i \(0.290502\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) 83515.1 + 60677.3i 0.471445 + 0.342525i
\(10\) 10857.1 + 223343.i 0.0343333 + 0.706273i
\(11\) 152190. − 110573.i 0.284923 − 0.207009i −0.436139 − 0.899879i \(-0.643654\pi\)
0.721062 + 0.692871i \(0.243654\pi\)
\(12\) 225231. + 163640.i 0.261295 + 0.189842i
\(13\) −50210.8 − 36480.3i −0.0375067 − 0.0272502i 0.568874 − 0.822425i \(-0.307379\pi\)
−0.606381 + 0.795175i \(0.707379\pi\)
\(14\) −1.40827e6 + 1.02317e6i −0.699814 + 0.508445i
\(15\) 1.18998e6 + 1.48093e6i 0.404611 + 0.503539i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) −2.39588e6 − 7.37377e6i −0.409258 − 1.25957i −0.917287 − 0.398226i \(-0.869626\pi\)
0.508030 − 0.861339i \(-0.330374\pi\)
\(18\) −3.30337e6 −0.412058
\(19\) −4.31912e6 − 1.32929e7i −0.400175 − 1.23161i −0.924857 − 0.380315i \(-0.875816\pi\)
0.524682 − 0.851298i \(-0.324184\pi\)
\(20\) −4.48196e6 − 5.57781e6i −0.313187 − 0.389762i
\(21\) −4.57018e6 + 1.40656e7i −0.244189 + 0.751538i
\(22\) −1.86021e6 + 5.72514e6i −0.0769551 + 0.236843i
\(23\) 8.47978e6 − 6.16092e6i 0.274714 − 0.199592i −0.441894 − 0.897067i \(-0.645693\pi\)
0.716609 + 0.697475i \(0.245693\pi\)
\(24\) −8.90884e6 −0.228381
\(25\) −1.95218e7 − 4.47558e7i −0.399807 − 0.916599i
\(26\) 1.98605e6 0.0327820
\(27\) −6.16697e7 + 4.48056e7i −0.827125 + 0.600941i
\(28\) 1.72132e7 − 5.29768e7i 0.189013 − 0.581723i
\(29\) −1.87725e6 + 5.77759e6i −0.0169955 + 0.0523068i −0.959195 − 0.282747i \(-0.908754\pi\)
0.942199 + 0.335054i \(0.108754\pi\)
\(30\) −5.86619e7 − 1.59567e7i −0.440747 − 0.119888i
\(31\) 4.31924e7 + 1.32933e8i 0.270968 + 0.833954i 0.990258 + 0.139244i \(0.0444673\pi\)
−0.719290 + 0.694710i \(0.755533\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 1.58046e7 + 4.86415e7i 0.0703002 + 0.216362i
\(34\) 2.00720e8 + 1.45832e8i 0.757630 + 0.550450i
\(35\) 2.08231e8 − 3.18005e8i 0.670148 − 1.02343i
\(36\) 8.55195e7 − 6.21336e7i 0.235723 − 0.171263i
\(37\) −3.29199e8 − 2.39177e8i −0.780457 − 0.567035i 0.124659 − 0.992200i \(-0.460216\pi\)
−0.905116 + 0.425165i \(0.860216\pi\)
\(38\) 3.61843e8 + 2.62894e8i 0.740816 + 0.538235i
\(39\) 1.36511e7 − 9.91811e6i 0.0242277 − 0.0176025i
\(40\) 2.20945e8 + 6.00996e7i 0.341157 + 0.0927987i
\(41\) 1.82269e8 + 1.32426e8i 0.245698 + 0.178510i 0.703818 − 0.710380i \(-0.251477\pi\)
−0.458120 + 0.888890i \(0.651477\pi\)
\(42\) −1.46246e8 − 4.50098e8i −0.172668 − 0.531417i
\(43\) 1.79385e8 0.186084 0.0930421 − 0.995662i \(-0.470341\pi\)
0.0930421 + 0.995662i \(0.470341\pi\)
\(44\) −5.95267e7 − 1.83204e8i −0.0544154 − 0.167474i
\(45\) 6.74407e8 − 2.55955e8i 0.544820 − 0.206774i
\(46\) −1.03648e8 + 3.18995e8i −0.0741978 + 0.228357i
\(47\) 2.59755e8 − 7.99445e8i 0.165206 − 0.508452i −0.833845 − 0.551998i \(-0.813865\pi\)
0.999051 + 0.0435460i \(0.0138655\pi\)
\(48\) 2.30637e8 − 1.67567e8i 0.130648 − 0.0949211i
\(49\) 9.81769e8 0.496513
\(50\) 1.34721e9 + 7.91474e8i 0.609678 + 0.358180i
\(51\) 2.10792e9 0.855497
\(52\) −5.14158e7 + 3.73558e7i −0.0187533 + 0.0136251i
\(53\) 1.00594e9 − 3.09597e9i 0.330412 − 1.01690i −0.638526 − 0.769600i \(-0.720456\pi\)
0.968938 − 0.247303i \(-0.0795444\pi\)
\(54\) 7.53783e8 − 2.31991e9i 0.223399 − 0.687551i
\(55\) −6.38256e7 − 1.31296e9i −0.0171002 − 0.351769i
\(56\) 5.50823e8 + 1.69526e9i 0.133653 + 0.411341i
\(57\) 3.80000e9 0.836512
\(58\) −6.00721e7 − 1.84883e8i −0.0120176 − 0.0369865i
\(59\) 4.68847e9 + 3.40638e9i 0.853779 + 0.620307i 0.926185 − 0.377069i \(-0.123068\pi\)
−0.0724064 + 0.997375i \(0.523068\pi\)
\(60\) 1.81880e9 − 6.90283e8i 0.301963 − 0.114603i
\(61\) 7.46048e9 − 5.42036e9i 1.13097 − 0.821701i 0.145138 − 0.989411i \(-0.453637\pi\)
0.985836 + 0.167710i \(0.0536373\pi\)
\(62\) −3.61853e9 − 2.62902e9i −0.501624 − 0.364451i
\(63\) 4.54302e9 + 3.30070e9i 0.576729 + 0.419018i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −4.05465e8 + 1.53885e8i −0.0433441 + 0.0164502i
\(66\) −1.32406e9 − 9.61987e8i −0.130142 − 0.0945535i
\(67\) 3.06203e8 + 9.42396e8i 0.0277076 + 0.0852751i 0.963954 − 0.266069i \(-0.0857248\pi\)
−0.936246 + 0.351344i \(0.885725\pi\)
\(68\) −7.93932e9 −0.662193
\(69\) 8.80603e8 + 2.71022e9i 0.0677814 + 0.208610i
\(70\) 5.90602e8 + 1.21493e10i 0.0420006 + 0.863998i
\(71\) 6.39672e9 − 1.96871e10i 0.420762 − 1.29497i −0.486233 − 0.873829i \(-0.661629\pi\)
0.906995 − 0.421142i \(-0.138371\pi\)
\(72\) −1.04530e9 + 3.21709e9i −0.0636665 + 0.195945i
\(73\) −2.35531e10 + 1.71123e10i −1.32976 + 0.966124i −0.330000 + 0.943981i \(0.607049\pi\)
−0.999755 + 0.0221429i \(0.992951\pi\)
\(74\) 1.30212e10 0.682144
\(75\) 1.32126e10 − 1.28762e9i 0.642912 − 0.0626545i
\(76\) −1.43124e10 −0.647497
\(77\) 8.27879e9 − 6.01489e9i 0.348552 − 0.253238i
\(78\) −1.66856e8 + 5.13531e8i −0.00654369 + 0.0201394i
\(79\) 1.65694e10 − 5.09952e10i 0.605838 − 1.86458i 0.114916 − 0.993375i \(-0.463340\pi\)
0.490923 − 0.871203i \(-0.336660\pi\)
\(80\) −6.85037e9 + 2.59990e9i −0.233733 + 0.0887077i
\(81\) −7.53259e8 − 2.31829e9i −0.0240036 − 0.0738755i
\(82\) −7.20950e9 −0.214748
\(83\) 6.37641e9 + 1.96246e10i 0.177683 + 0.546853i 0.999746 − 0.0225441i \(-0.00717660\pi\)
−0.822062 + 0.569397i \(0.807177\pi\)
\(84\) 1.22520e10 + 8.90162e9i 0.319648 + 0.232238i
\(85\) −5.22779e10 − 1.42202e10i −1.27795 − 0.347617i
\(86\) −4.64402e9 + 3.37408e9i −0.106452 + 0.0773416i
\(87\) −1.33619e9 − 9.70801e8i −0.0287418 − 0.0208821i
\(88\) 4.98697e9 + 3.62325e9i 0.100735 + 0.0731886i
\(89\) −3.64769e10 + 2.65020e10i −0.692425 + 0.503076i −0.877456 − 0.479656i \(-0.840761\pi\)
0.185031 + 0.982733i \(0.440761\pi\)
\(90\) −1.26451e10 + 1.93113e10i −0.225730 + 0.344729i
\(91\) −2.73134e9 − 1.98444e9i −0.0458827 − 0.0333357i
\(92\) −3.31672e9 − 1.02078e10i −0.0524658 − 0.161473i
\(93\) −3.80011e10 −0.566422
\(94\) 8.31217e9 + 2.55822e10i 0.116818 + 0.359530i
\(95\) −9.42427e10 − 2.56351e10i −1.24959 − 0.339903i
\(96\) −2.81905e9 + 8.67615e9i −0.0352867 + 0.108601i
\(97\) 4.27303e9 − 1.31510e10i 0.0505233 − 0.155495i −0.922612 − 0.385730i \(-0.873950\pi\)
0.973135 + 0.230235i \(0.0739496\pi\)
\(98\) −2.54166e10 + 1.84662e10i −0.284036 + 0.206364i
\(99\) 1.94195e10 0.205231
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 50.12.d.b.11.6 ✓ 56
25.16 even 5 inner 50.12.d.b.41.6 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.6 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.6 yes 56 25.16 even 5 inner