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Properties

Label 50.12.d.b.11.1
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 11.1
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.1

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

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

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
\(3\) −246.102 + 757.423i −0.584719 + 1.79958i 0.0156750 + 0.999877i \(0.495010\pi\)
−0.600394 + 0.799704i \(0.704990\pi\)
\(4\) 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 −0.511614 − 0.859216i \(-0.670952\pi\)
0.659065 + 0.752086i \(0.270952\pi\)
\(12\) 659766. + 479348.i 0.765408 + 0.556101i
\(13\) 1.11649e6 + 811176.i 0.833999 + 0.605936i 0.920688 − 0.390300i \(-0.127629\pi\)
−0.0866888 + 0.996235i \(0.527629\pi\)
\(14\) −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 0.832704 + 0.553718i \(0.186792\pi\)
−0.348204 + 0.937419i \(0.613208\pi\)
\(18\) 1.46275e7 1.82461
\(19\) 2.00706e6 + 6.17709e6i 0.185958 + 0.572320i 0.999964 − 0.00853713i \(-0.00271749\pi\)
−0.814006 + 0.580857i \(0.802717\pi\)
\(20\) −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 −0.319076 − 0.947729i \(-0.603373\pi\)
0.802744 + 0.596324i \(0.203373\pi\)
\(24\) −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 −0.169666 − 0.985502i \(-0.554269\pi\)
0.716526 − 0.697561i \(-0.245731\pi\)
\(30\) −3.80111e7 + 1.73977e8i −0.285591 + 1.30715i
\(31\) 2.94676e7 + 9.06921e7i 0.184866 + 0.568958i 0.999946 − 0.0103901i \(-0.00330733\pi\)
−0.815080 + 0.579348i \(0.803307\pi\)
\(32\) 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 0.853464 − 0.521152i \(-0.174497\pi\)
−0.231910 + 0.972737i \(0.574497\pi\)
\(38\) −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 0.343675 − 0.939089i \(-0.388328\pi\)
−0.786925 + 0.617049i \(0.788328\pi\)
\(42\) −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 −0.906126 − 0.423009i \(-0.860974\pi\)
0.981709 + 0.190386i \(0.0609740\pi\)
\(48\) 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 0.432823 + 0.901479i \(0.357517\pi\)
−0.880037 + 0.474904i \(0.842483\pi\)
\(54\) −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 0.651283 − 0.758835i \(-0.274231\pi\)
−0.520438 + 0.853900i \(0.674231\pi\)
\(60\) −2.28830e9 − 5.21896e9i −0.379910 − 0.866467i
\(61\) 6.39921e9 − 4.64930e9i 0.970090 − 0.704812i 0.0146179 − 0.999893i \(-0.495347\pi\)
0.955472 + 0.295082i \(0.0953468\pi\)
\(62\) −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 0.999553 − 0.0298944i \(-0.00951709\pi\)
−0.826227 + 0.563338i \(0.809517\pi\)
\(68\) 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 −0.992206 − 0.124605i \(-0.960234\pi\)
0.875953 + 0.482397i \(0.160234\pi\)
\(72\) 4.62862e9 − 1.42454e10i 0.281918 − 0.867654i
\(73\) −1.71554e10 + 1.24641e10i −0.968557 + 0.703698i −0.955122 − 0.296212i \(-0.904276\pi\)
−0.0134344 + 0.999910i \(0.504276\pi\)
\(74\) −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 −0.980547 − 0.196284i \(-0.937113\pi\)
0.908652 + 0.417554i \(0.137113\pi\)
\(80\) 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 −0.829267 − 0.558853i \(-0.811242\pi\)
0.342405 − 0.939552i \(-0.388758\pi\)
\(84\) 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 −0.269655 − 0.962957i \(-0.586910\pi\)
0.832498 + 0.554027i \(0.186910\pi\)
\(90\) −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 −0.934850 − 0.355044i \(-0.884466\pi\)
0.964999 + 0.262254i \(0.0844659\pi\)
\(98\) −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.11.1 ✓ 56
25.16 even 5 inner 50.12.d.b.41.1 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.1 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.1 yes 56 25.16 even 5 inner