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Properties

Label 50.12.d.b.41.1
Level $50$
Weight $12$
Character 50.41
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 41.1
Character \(\chi\) \(=\) 50.41
Dual form 50.12.d.b.11.1

$q$-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+(-25.8885 - 18.8091i) q^{2} +(-246.102 - 757.423i) q^{3} +(316.433 + 973.882i) q^{4} +(-6031.65 + 3528.08i) q^{5} +(-7875.25 + 24237.5i) q^{6} +73958.0 q^{7} +(10125.9 - 31164.2i) q^{8} +(-369808. + 268681. i) q^{9} +(222511. + 22113.3i) q^{10} +(78760.5 + 57222.8i) q^{11} +(659766. - 479348. i) q^{12} +(1.11649e6 - 811176. i) q^{13} +(-1.91467e6 - 1.39109e6i) q^{14} +(4.15664e6 + 3.70024e6i) q^{15} +(-848316. + 616338. i) q^{16} +(2.83637e6 - 8.72944e6i) q^{17} +1.46275e7 q^{18} +(2.00706e6 - 6.17709e6i) q^{19} +(-5.34455e6 - 4.75771e6i) q^{20} +(-1.82012e7 - 5.60175e7i) q^{21} +(-962683. - 2.96283e6i) q^{22} +(1.49297e7 + 1.08470e7i) q^{23} -2.60965e7 q^{24} +(2.39335e7 - 4.25603e7i) q^{25} -4.41618e7 q^{26} +(1.80380e8 + 1.31053e8i) q^{27} +(2.34028e7 + 7.20264e7i) q^{28} +(6.04039e7 + 1.85904e8i) q^{29} +(-3.80111e7 - 1.73977e8i) q^{30} +(2.94676e7 - 9.06921e7i) q^{31} +3.35544e7 q^{32} +(2.39588e7 - 7.37376e7i) q^{33} +(-2.37623e8 + 1.72643e8i) q^{34} +(-4.46089e8 + 2.60930e8i) q^{35} +(-3.78684e8 - 2.75130e8i) q^{36} +(2.62173e8 - 1.90480e8i) q^{37} +(-1.68145e8 + 1.22165e8i) q^{38} +(-8.89173e8 - 6.46022e8i) q^{39} +(4.88741e7 + 2.23696e8i) q^{40} +(-3.28821e8 + 2.38903e8i) q^{41} +(-5.82438e8 + 1.79256e9i) q^{42} -7.85579e7 q^{43} +(-3.08058e7 + 9.48106e7i) q^{44} +(1.28262e9 - 2.92530e9i) q^{45} +(-1.82484e8 - 5.61628e8i) q^{46} +(1.18841e8 + 3.65755e8i) q^{47} +(6.75600e8 + 4.90852e8i) q^{48} +3.49246e9 q^{49} +(-1.42012e9 + 6.51656e8i) q^{50} -7.30991e9 q^{51} +(1.14328e9 + 8.30644e8i) q^{52} +(-1.36155e9 - 4.19041e9i) q^{53} +(-2.20476e9 - 6.78557e9i) q^{54} +(-6.76942e8 - 6.72750e7i) q^{55} +(7.48889e8 - 2.30484e9i) q^{56} -5.17261e9 q^{57} +(1.93293e9 - 5.94893e9i) q^{58} +(7.18529e8 - 5.22042e8i) q^{59} +(-2.28830e9 + 5.21896e9i) q^{60} +(6.39921e9 + 4.64930e9i) q^{61} +(-2.46871e9 + 1.79362e9i) q^{62} +(-2.73503e10 + 1.98711e10i) q^{63} +(-8.68675e8 - 6.31130e8i) q^{64} +(-3.87237e9 + 8.83179e9i) q^{65} +(-2.00720e9 + 1.45832e9i) q^{66} +(1.91547e9 - 5.89521e9i) q^{67} +9.39897e9 q^{68} +(4.54158e9 - 1.39776e10i) q^{69} +(1.64564e10 + 1.63545e9i) q^{70} +(-1.76737e9 - 5.43941e9i) q^{71} +(4.62862e9 + 1.42454e10i) q^{72} +(-1.71554e10 - 1.24641e10i) q^{73} -1.03701e10 q^{74} +(-3.81262e10 - 7.65360e9i) q^{75} +6.65085e9 q^{76} +(5.82497e9 + 4.23209e9i) q^{77} +(1.08683e10 + 3.34491e10i) q^{78} +(-1.96630e9 - 6.05163e9i) q^{79} +(2.94226e9 - 6.71046e9i) q^{80} +(2.98484e10 - 9.18640e10i) q^{81} +1.30063e10 q^{82} +(-1.74717e10 + 5.37723e10i) q^{83} +(4.87950e10 - 3.54516e10i) q^{84} +(1.36902e10 + 6.26599e10i) q^{85} +(2.03375e9 + 1.47761e9i) q^{86} +(1.25943e11 - 9.15026e10i) q^{87} +(2.58082e9 - 1.87508e9i) q^{88} +(2.96505e10 + 2.15424e10i) q^{89} +(-8.82277e10 + 5.16068e10i) q^{90} +(8.25732e10 - 5.99930e10i) q^{91} +(-5.83949e9 + 1.79721e10i) q^{92} -7.59443e10 q^{93} +(3.80291e9 - 1.17042e10i) q^{94} +(9.68738e9 + 4.43391e10i) q^{95} +(-8.25780e9 - 2.54149e10i) q^{96} +(2.54988e9 + 7.84774e9i) q^{97} +(-9.04147e10 - 6.56902e10i) q^{98} -4.45010e10 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
\(3\) −246.102 − 757.423i −0.584719 − 1.79958i −0.600394 − 0.799704i \(-0.704990\pi\)
0.0156750 − 0.999877i \(-0.495010\pi\)
\(4\) 316.433 + 973.882i 0.154508 + 0.475528i
\(5\) −6031.65 + 3528.08i −0.863179 + 0.504897i
\(6\) −7875.25 + 24237.5i −0.413459 + 1.27250i
\(7\) 73958.0 1.66321 0.831603 − 0.555371i \(-0.187424\pi\)
0.831603 + 0.555371i \(0.187424\pi\)
\(8\) 10125.9 − 31164.2i 0.109254 − 0.336249i
\(9\) −369808. + 268681.i −2.08758 + 1.51671i
\(10\) 222511. + 22113.3i 0.703641 + 0.0699283i
\(11\) 78760.5 + 57222.8i 0.147451 + 0.107130i 0.659065 − 0.752086i \(-0.270952\pi\)
−0.511614 + 0.859216i \(0.670952\pi\)
\(12\) 659766. − 479348.i 0.765408 − 0.556101i
\(13\) 1.11649e6 − 811176.i 0.833999 − 0.605936i −0.0866888 − 0.996235i \(-0.527629\pi\)
0.920688 + 0.390300i \(0.127629\pi\)
\(14\) −1.91467e6 − 1.39109e6i −0.951456 − 0.691273i
\(15\) 4.15664e6 + 3.70024e6i 1.41332 + 1.25814i
\(16\) −848316. + 616338.i −0.202254 + 0.146946i
\(17\) 2.83637e6 − 8.72944e6i 0.484500 − 1.49114i −0.348204 − 0.937419i \(-0.613208\pi\)
0.832704 − 0.553718i \(-0.186792\pi\)
\(18\) 1.46275e7 1.82461
\(19\) 2.00706e6 − 6.17709e6i 0.185958 − 0.572320i −0.814006 − 0.580857i \(-0.802717\pi\)
0.999964 + 0.00853713i \(0.00271749\pi\)
\(20\) −5.34455e6 − 4.75771e6i −0.373462 − 0.332455i
\(21\) −1.82012e7 − 5.60175e7i −0.972509 − 2.99307i
\(22\) −962683. − 2.96283e6i −0.0398252 − 0.122569i
\(23\) 1.49297e7 + 1.08470e7i 0.483668 + 0.351405i 0.802744 − 0.596324i \(-0.203373\pi\)
−0.319076 + 0.947729i \(0.603373\pi\)
\(24\) −2.60965e7 −0.668991
\(25\) 2.39335e7 − 4.25603e7i 0.490157 − 0.871634i
\(26\) −4.41618e7 −0.728942
\(27\) 1.80380e8 + 1.31053e8i 2.41928 + 1.75771i
\(28\) 2.34028e7 + 7.20264e7i 0.256979 + 0.790901i
\(29\) 6.04039e7 + 1.85904e8i 0.546860 + 1.68306i 0.716526 + 0.697561i \(0.245731\pi\)
−0.169666 + 0.985502i \(0.554269\pi\)
\(30\) −3.80111e7 − 1.73977e8i −0.285591 − 1.30715i
\(31\) 2.94676e7 − 9.06921e7i 0.184866 − 0.568958i −0.815080 − 0.579348i \(-0.803307\pi\)
0.999946 + 0.0103901i \(0.00330733\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 2.39588e7 − 7.37376e7i 0.106571 − 0.327991i
\(34\) −2.37623e8 + 1.72643e8i −0.896920 + 0.651651i
\(35\) −4.46089e8 + 2.60930e8i −1.43565 + 0.839748i
\(36\) −3.78684e8 − 2.75130e8i −1.04379 − 0.758357i
\(37\) 2.62173e8 − 1.90480e8i 0.621554 − 0.451586i −0.231910 − 0.972737i \(-0.574497\pi\)
0.853464 + 0.521152i \(0.174497\pi\)
\(38\) −1.68145e8 + 1.22165e8i −0.344251 + 0.250113i
\(39\) −8.89173e8 − 6.46022e8i −1.57809 − 1.14655i
\(40\) 4.88741e7 + 2.23696e8i 0.0754656 + 0.345405i
\(41\) −3.28821e8 + 2.38903e8i −0.443250 + 0.322040i −0.786925 − 0.617049i \(-0.788328\pi\)
0.343675 + 0.939089i \(0.388328\pi\)
\(42\) −5.82438e8 + 1.79256e9i −0.687668 + 2.11642i
\(43\) −7.85579e7 −0.0814917 −0.0407459 − 0.999170i \(-0.512973\pi\)
−0.0407459 + 0.999170i \(0.512973\pi\)
\(44\) −3.08058e7 + 9.48106e7i −0.0281607 + 0.0866697i
\(45\) 1.28262e9 − 2.92530e9i 1.03617 − 2.36321i
\(46\) −1.82484e8 − 5.61628e8i −0.130634 − 0.402051i
\(47\) 1.18841e8 + 3.65755e8i 0.0755837 + 0.232623i 0.981709 − 0.190386i \(-0.0609740\pi\)
−0.906126 + 0.423009i \(0.860974\pi\)
\(48\) 6.75600e8 + 4.90852e8i 0.382704 + 0.278051i
\(49\) 3.49246e9 1.76625
\(50\) −1.42012e9 + 6.51656e8i −0.642675 + 0.294906i
\(51\) −7.30991e9 −2.96672
\(52\) 1.14328e9 + 8.30644e8i 0.417000 + 0.302968i
\(53\) −1.36155e9 − 4.19041e9i −0.447214 − 1.37638i −0.880037 − 0.474904i \(-0.842483\pi\)
0.432823 − 0.901479i \(-0.357517\pi\)
\(54\) −2.20476e9 − 6.78557e9i −0.653426 − 2.01104i
\(55\) −6.76942e8 − 6.72750e7i −0.181366 − 0.0180243i
\(56\) 7.48889e8 − 2.30484e9i 0.181712 − 0.559252i
\(57\) −5.17261e9 −1.13867
\(58\) 1.93293e9 − 5.94893e9i 0.386688 − 1.19010i
\(59\) 7.18529e8 − 5.22042e8i 0.130845 − 0.0950647i −0.520438 − 0.853900i \(-0.674231\pi\)
0.651283 + 0.758835i \(0.274231\pi\)
\(60\) −2.28830e9 + 5.21896e9i −0.379910 + 0.866467i
\(61\) 6.39921e9 + 4.64930e9i 0.970090 + 0.704812i 0.955472 − 0.295082i \(-0.0953468\pi\)
0.0146179 + 0.999893i \(0.495347\pi\)
\(62\) −2.46871e9 + 1.79362e9i −0.342229 + 0.248644i
\(63\) −2.73503e10 + 1.98711e10i −3.47207 + 2.52261i
\(64\) −8.68675e8 − 6.31130e8i −0.101127 − 0.0734732i
\(65\) −3.87237e9 + 8.83179e9i −0.413955 + 0.944115i
\(66\) −2.00720e9 + 1.45832e9i −0.197287 + 0.143337i
\(67\) 1.91547e9 − 5.89521e9i 0.173326 − 0.533443i −0.826227 − 0.563338i \(-0.809517\pi\)
0.999553 + 0.0298944i \(0.00951709\pi\)
\(68\) 9.39897e9 0.783937
\(69\) 4.54158e9 − 1.39776e10i 0.349572 − 1.07587i
\(70\) 1.64564e10 + 1.63545e9i 1.17030 + 0.116305i
\(71\) −1.76737e9 − 5.43941e9i −0.116254 − 0.357792i 0.875953 − 0.482397i \(-0.160234\pi\)
−0.992206 + 0.124605i \(0.960234\pi\)
\(72\) 4.62862e9 + 1.42454e10i 0.281918 + 0.867654i
\(73\) −1.71554e10 − 1.24641e10i −0.968557 − 0.703698i −0.0134344 − 0.999910i \(-0.504276\pi\)
−0.955122 + 0.296212i \(0.904276\pi\)
\(74\) −1.03701e10 −0.543258
\(75\) −3.81262e10 − 7.65360e9i −1.85518 − 0.372416i
\(76\) 6.65085e9 0.300886
\(77\) 5.82497e9 + 4.23209e9i 0.245242 + 0.178179i
\(78\) 1.08683e10 + 3.34491e10i 0.426227 + 1.31179i
\(79\) −1.96630e9 − 6.05163e9i −0.0718952 − 0.221271i 0.908652 − 0.417554i \(-0.137113\pi\)
−0.980547 + 0.196284i \(0.937113\pi\)
\(80\) 2.94226e9 − 6.71046e9i 0.100389 − 0.228959i
\(81\) 2.98484e10 − 9.18640e10i 0.951160 − 2.92737i
\(82\) 1.30063e10 0.387415
\(83\) −1.74717e10 + 5.37723e10i −0.486861 + 1.49841i 0.342405 + 0.939552i \(0.388758\pi\)
−0.829267 + 0.558853i \(0.811242\pi\)
\(84\) 4.87950e10 − 3.54516e10i 1.27303 − 0.924911i
\(85\) 1.36902e10 + 6.26599e10i 0.334661 + 1.53174i
\(86\) 2.03375e9 + 1.47761e9i 0.0466183 + 0.0338702i
\(87\) 1.25943e11 − 9.15026e10i 2.70905 − 1.96824i
\(88\) 2.58082e9 − 1.87508e9i 0.0521319 − 0.0378761i
\(89\) 2.96505e10 + 2.15424e10i 0.562843 + 0.408929i 0.832498 − 0.554027i \(-0.186910\pi\)
−0.269655 + 0.962957i \(0.586910\pi\)
\(90\) −8.82277e10 + 5.16068e10i −1.57497 + 0.921241i
\(91\) 8.25732e10 − 5.99930e10i 1.38711 − 1.00780i
\(92\) −5.83949e9 + 1.79721e10i −0.0923723 + 0.284293i
\(93\) −7.59443e10 −1.13198
\(94\) 3.80291e9 − 1.17042e10i 0.0534457 − 0.164489i
\(95\) 9.68738e9 + 4.43391e10i 0.128448 + 0.587905i
\(96\) −8.25780e9 − 2.54149e10i −0.103365 − 0.318124i
\(97\) 2.54988e9 + 7.84774e9i 0.0301492 + 0.0927897i 0.964999 − 0.262254i \(-0.0844659\pi\)
−0.934850 + 0.355044i \(0.884466\pi\)
\(98\) −9.04147e10 − 6.56902e10i −1.01041 − 0.734103i
\(99\) −4.45010e10 −0.470301
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.41.1 yes 56
25.11 even 5 inner 50.12.d.b.11.1 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.1 ✓ 56 25.11 even 5 inner
50.12.d.b.41.1 yes 56 1.1 even 1 trivial