↑

Properties

Label 50.12.d.b.41.6
Level $50$
Weight $12$
Character 50.41
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
Inner twists $2$

Related objects

Downloads

Learn more

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 41.6
Character \(\chi\) \(=\) 50.41
Dual form 50.12.d.b.11.6

$q$-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 \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\)

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 −0.999892 − 0.0147156i \(-0.995316\pi\)
0.800280 − 0.599627i \(-0.204684\pi\)
\(4\) 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 0.721062 − 0.692871i \(-0.243654\pi\)
−0.436139 + 0.899879i \(0.643654\pi\)
\(12\) 225231. − 163640.i 0.261295 − 0.189842i
\(13\) −50210.8 + 36480.3i −0.0375067 + 0.0272502i −0.606381 − 0.795175i \(-0.707379\pi\)
0.568874 + 0.822425i \(0.307379\pi\)
\(14\) −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 0.508030 + 0.861339i \(0.330374\pi\)
−0.917287 + 0.398226i \(0.869626\pi\)
\(18\) −3.30337e6 −0.412058
\(19\) −4.31912e6 + 1.32929e7i −0.400175 + 1.23161i 0.524682 + 0.851298i \(0.324184\pi\)
−0.924857 + 0.380315i \(0.875816\pi\)
\(20\) −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 0.716609 − 0.697475i \(-0.245693\pi\)
−0.441894 + 0.897067i \(0.645693\pi\)
\(24\) −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 0.942199 − 0.335054i \(-0.108754\pi\)
−0.959195 + 0.282747i \(0.908754\pi\)
\(30\) −5.86619e7 + 1.59567e7i −0.440747 + 0.119888i
\(31\) 4.31924e7 − 1.32933e8i 0.270968 − 0.833954i −0.719290 − 0.694710i \(-0.755533\pi\)
0.990258 − 0.139244i \(-0.0444673\pi\)
\(32\) 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 −0.905116 − 0.425165i \(-0.860216\pi\)
0.124659 + 0.992200i \(0.460216\pi\)
\(38\) 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 −0.458120 − 0.888890i \(-0.651477\pi\)
0.703818 + 0.710380i \(0.251477\pi\)
\(42\) −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 0.999051 − 0.0435460i \(-0.0138655\pi\)
−0.833845 + 0.551998i \(0.813865\pi\)
\(48\) 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 0.968938 + 0.247303i \(0.0795444\pi\)
−0.638526 + 0.769600i \(0.720456\pi\)
\(54\) 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 −0.0724064 − 0.997375i \(-0.523068\pi\)
0.926185 + 0.377069i \(0.123068\pi\)
\(60\) 1.81880e9 + 6.90283e8i 0.301963 + 0.114603i
\(61\) 7.46048e9 + 5.42036e9i 1.13097 + 0.821701i 0.985836 − 0.167710i \(-0.0536373\pi\)
0.145138 + 0.989411i \(0.453637\pi\)
\(62\) −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 −0.936246 − 0.351344i \(-0.885725\pi\)
0.963954 + 0.266069i \(0.0857248\pi\)
\(68\) −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 0.906995 + 0.421142i \(0.138371\pi\)
−0.486233 + 0.873829i \(0.661629\pi\)
\(72\) −1.04530e9 − 3.21709e9i −0.0636665 − 0.195945i
\(73\) −2.35531e10 − 1.71123e10i −1.32976 − 0.966124i −0.999755 − 0.0221429i \(-0.992951\pi\)
−0.330000 − 0.943981i \(-0.607049\pi\)
\(74\) 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 0.490923 + 0.871203i \(0.336660\pi\)
0.114916 + 0.993375i \(0.463340\pi\)
\(80\) −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 −0.822062 − 0.569397i \(-0.807177\pi\)
0.999746 + 0.0225441i \(0.00717660\pi\)
\(84\) 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 0.185031 − 0.982733i \(-0.440761\pi\)
−0.877456 + 0.479656i \(0.840761\pi\)
\(90\) −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 0.973135 − 0.230235i \(-0.0739496\pi\)
−0.922612 + 0.385730i \(0.873950\pi\)
\(98\) −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.41.6 yes 56
25.11 even 5 inner 50.12.d.b.11.6 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.6 ✓ 56 25.11 even 5 inner
50.12.d.b.41.6 yes 56 1.1 even 1 trivial