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Properties

Label 50.12.d.b.11.5
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 11.5
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.5

$q$-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} +(-123.506 + 380.112i) q^{3} +(316.433 - 973.882i) q^{4} +(-4417.52 + 5414.21i) q^{5} +(-3952.18 - 12163.6i) q^{6} +6670.69 q^{7} +(10125.9 + 31164.2i) q^{8} +(14083.8 + 10232.5i) q^{9} +(12526.6 - 223256. i) q^{10} +(-625209. + 454241. i) q^{11} +(331102. + 240560. i) q^{12} +(-815228. - 592298. i) q^{13} +(-172694. + 125470. i) q^{14} +(-1.51241e6 - 2.34784e6i) q^{15} +(-848316. - 616338. i) q^{16} +(1.47512e6 + 4.53995e6i) q^{17} -557073. q^{18} +(-5.53435e6 - 1.70330e7i) q^{19} +(3.87495e6 + 6.01538e6i) q^{20} +(-823868. + 2.53561e6i) q^{21} +(7.64188e6 - 2.35193e7i) q^{22} +(-4.46662e7 + 3.24519e7i) q^{23} -1.30965e7 q^{24} +(-9.79916e6 - 4.78347e7i) q^{25} +3.22457e7 q^{26} +(-6.29080e7 + 4.57053e7i) q^{27} +(2.11083e6 - 6.49646e6i) q^{28} +(-2.55283e7 + 7.85681e7i) q^{29} +(8.33149e7 + 3.23349e7i) q^{30} +(-3.16895e7 - 9.75302e7i) q^{31} +3.35544e7 q^{32} +(-9.54454e7 - 2.93751e8i) q^{33} +(-1.23581e8 - 8.97869e7i) q^{34} +(-2.94679e7 + 3.61165e7i) q^{35} +(1.44218e7 - 1.04781e7i) q^{36} +(5.58421e8 + 4.05717e8i) q^{37} +(4.63651e8 + 3.36862e8i) q^{38} +(3.25825e8 - 2.36725e8i) q^{39} +(-2.13461e8 - 8.28450e7i) q^{40} +(1.07988e8 + 7.84579e7i) q^{41} +(-2.63638e7 - 8.11394e7i) q^{42} +1.36973e9 q^{43} +(2.44540e8 + 7.52617e8i) q^{44} +(-1.17616e8 + 3.10504e7i) q^{45} +(5.45950e8 - 1.68026e9i) q^{46} +(8.98180e8 - 2.76431e9i) q^{47} +(3.39049e8 - 2.46333e8i) q^{48} -1.93283e9 q^{49} +(1.15342e9 + 1.05406e9i) q^{50} -1.90787e9 q^{51} +(-8.34794e8 + 6.06513e8i) q^{52} +(5.59903e8 - 1.72320e9i) q^{53} +(7.68919e8 - 2.36649e9i) q^{54} +(3.02519e8 - 5.39163e9i) q^{55} +(6.75465e7 + 2.07887e8i) q^{56} +7.15795e9 q^{57} +(-8.16906e8 - 2.51418e9i) q^{58} +(7.37928e9 + 5.36136e9i) q^{59} +(-2.76509e9 + 7.29979e8i) q^{60} +(4.29007e9 - 3.11692e9i) q^{61} +(2.65485e9 + 1.92886e9i) q^{62} +(9.39486e7 + 6.82577e7i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(6.80811e9 - 1.79733e9i) q^{65} +(7.99614e9 + 5.80953e9i) q^{66} +(2.48002e9 + 7.63273e9i) q^{67} +4.88815e9 q^{68} +(-6.81880e9 - 2.09861e10i) q^{69} +(8.35614e7 - 1.48927e9i) q^{70} +(-5.26823e9 + 1.62140e10i) q^{71} +(-1.76276e8 + 5.42523e8i) q^{72} +(1.42826e10 - 1.03769e10i) q^{73} -2.20879e10 q^{74} +(1.93928e10 + 2.18309e9i) q^{75} -1.83394e10 q^{76} +(-4.17058e9 + 3.03010e9i) q^{77} +(-3.98253e9 + 1.22570e10i) q^{78} +(-3.78214e9 + 1.16402e10i) q^{79} +(7.08443e9 - 1.87027e9i) q^{80} +(-8.65066e9 - 2.66240e10i) q^{81} -4.27138e9 q^{82} +(-1.90831e10 - 5.87318e10i) q^{83} +(2.20868e9 + 1.60470e9i) q^{84} +(-3.10966e10 - 1.20687e10i) q^{85} +(-3.54604e10 + 2.57635e10i) q^{86} +(-2.67117e10 - 1.94072e10i) q^{87} +(-2.04869e10 - 1.48846e10i) q^{88} +(-5.27940e10 + 3.83571e10i) q^{89} +(2.46088e9 - 3.01611e9i) q^{90} +(-5.43813e9 - 3.95104e9i) q^{91} +(1.74704e10 + 5.37684e10i) q^{92} +4.09862e10 q^{93} +(2.87418e10 + 8.84581e10i) q^{94} +(1.16668e11 + 4.52794e10i) q^{95} +(-4.14416e9 + 1.27544e10i) q^{96} +(-1.19231e10 + 3.66954e10i) q^{97} +(5.00381e10 - 3.63548e10i) q^{98} -1.34533e10 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
\(3\) −123.506 + 380.112i −0.293441 + 0.903117i 0.690300 + 0.723523i \(0.257478\pi\)
−0.983741 + 0.179594i \(0.942522\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) −4417.52 + 5414.21i −0.632184 + 0.774818i
\(6\) −3952.18 − 12163.6i −0.207494 − 0.638600i
\(7\) 6670.69 0.150014 0.0750070 − 0.997183i \(-0.476102\pi\)
0.0750070 + 0.997183i \(0.476102\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) 14083.8 + 10232.5i 0.0795034 + 0.0577626i
\(10\) 12526.6 − 223256.i 0.0396127 − 0.705996i
\(11\) −625209. + 454241.i −1.17048 + 0.850407i −0.991067 − 0.133366i \(-0.957421\pi\)
−0.179418 + 0.983773i \(0.557421\pi\)
\(12\) 331102. + 240560.i 0.384119 + 0.279079i
\(13\) −815228. − 592298.i −0.608963 − 0.442437i 0.240086 − 0.970752i \(-0.422824\pi\)
−0.849049 + 0.528314i \(0.822824\pi\)
\(14\) −172694. + 125470.i −0.0858172 + 0.0623498i
\(15\) −1.51241e6 − 2.34784e6i −0.514243 − 0.798300i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) 1.47512e6 + 4.53995e6i 0.251975 + 0.775500i 0.994411 + 0.105582i \(0.0336706\pi\)
−0.742435 + 0.669918i \(0.766329\pi\)
\(18\) −557073. −0.0694886
\(19\) −5.53435e6 − 1.70330e7i −0.512769 − 1.57814i −0.787305 − 0.616563i \(-0.788524\pi\)
0.274537 − 0.961577i \(-0.411476\pi\)
\(20\) 3.87495e6 + 6.01538e6i 0.270770 + 0.420337i
\(21\) −823868. + 2.53561e6i −0.0440202 + 0.135480i
\(22\) 7.64188e6 − 2.35193e7i 0.316137 − 0.972970i
\(23\) −4.46662e7 + 3.24519e7i −1.44702 + 1.05132i −0.460508 + 0.887656i \(0.652333\pi\)
−0.986515 + 0.163668i \(0.947667\pi\)
\(24\) −1.30965e7 −0.335732
\(25\) −9.79916e6 − 4.78347e7i −0.200687 − 0.979655i
\(26\) 3.22457e7 0.532253
\(27\) −6.29080e7 + 4.57053e7i −0.843733 + 0.613008i
\(28\) 2.11083e6 − 6.49646e6i 0.0231784 − 0.0713359i
\(29\) −2.55283e7 + 7.85681e7i −0.231118 + 0.711307i 0.766495 + 0.642250i \(0.221999\pi\)
−0.997613 + 0.0690569i \(0.978001\pi\)
\(30\) 8.33149e7 + 3.23349e7i 0.625974 + 0.242943i
\(31\) −3.16895e7 − 9.75302e7i −0.198804 − 0.611857i −0.999911 − 0.0133353i \(-0.995755\pi\)
0.801107 − 0.598521i \(-0.204245\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −9.54454e7 − 2.93751e8i −0.424549 − 1.30663i
\(34\) −1.23581e8 − 8.97869e7i −0.466464 − 0.338906i
\(35\) −2.94679e7 + 3.61165e7i −0.0948364 + 0.116234i
\(36\) 1.44218e7 − 1.04781e7i 0.0397517 − 0.0288813i
\(37\) 5.58421e8 + 4.05717e8i 1.32389 + 0.961864i 0.999875 + 0.0158126i \(0.00503352\pi\)
0.324017 + 0.946051i \(0.394966\pi\)
\(38\) 4.63651e8 + 3.36862e8i 0.949253 + 0.689672i
\(39\) 3.25825e8 − 2.36725e8i 0.578267 − 0.420136i
\(40\) −2.13461e8 − 8.28450e7i −0.329601 − 0.127919i
\(41\) 1.07988e8 + 7.84579e7i 0.145568 + 0.105761i 0.658185 − 0.752856i \(-0.271324\pi\)
−0.512618 + 0.858617i \(0.671324\pi\)
\(42\) −2.63638e7 − 8.11394e7i −0.0311270 − 0.0957989i
\(43\) 1.36973e9 1.42089 0.710444 − 0.703754i \(-0.248494\pi\)
0.710444 + 0.703754i \(0.248494\pi\)
\(44\) 2.44540e8 + 7.52617e8i 0.223543 + 0.687994i
\(45\) −1.17616e8 + 3.10504e7i −0.0950164 + 0.0250841i
\(46\) 5.45950e8 − 1.68026e9i 0.390828 − 1.20284i
\(47\) 8.98180e8 − 2.76431e9i 0.571249 − 1.75812i −0.0773626 − 0.997003i \(-0.524650\pi\)
0.648611 − 0.761120i \(-0.275350\pi\)
\(48\) 3.39049e8 − 2.46333e8i 0.192059 − 0.139539i
\(49\) −1.93283e9 −0.977496
\(50\) 1.15342e9 + 1.05406e9i 0.521976 + 0.477012i
\(51\) −1.90787e9 −0.774307
\(52\) −8.34794e8 + 6.06513e8i −0.304481 + 0.221219i
\(53\) 5.59903e8 − 1.72320e9i 0.183906 − 0.566004i −0.816022 − 0.578021i \(-0.803825\pi\)
0.999928 + 0.0120170i \(0.00382522\pi\)
\(54\) 7.68919e8 − 2.36649e9i 0.227885 − 0.701357i
\(55\) 3.02519e8 − 5.39163e9i 0.0810509 − 1.44453i
\(56\) 6.75465e7 + 2.07887e8i 0.0163896 + 0.0504421i
\(57\) 7.15795e9 1.57571
\(58\) −8.16906e8 − 2.51418e9i −0.163425 − 0.502970i
\(59\) 7.37928e9 + 5.36136e9i 1.34378 + 0.976313i 0.999296 + 0.0375209i \(0.0119461\pi\)
0.344484 + 0.938792i \(0.388054\pi\)
\(60\) −2.76509e9 + 7.29979e8i −0.459069 + 0.121193i
\(61\) 4.29007e9 − 3.11692e9i 0.650355 − 0.472511i −0.213037 − 0.977044i \(-0.568335\pi\)
0.863392 + 0.504533i \(0.168335\pi\)
\(62\) 2.65485e9 + 1.92886e9i 0.368032 + 0.267391i
\(63\) 9.39486e7 + 6.82577e7i 0.0119266 + 0.00866520i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) 6.80811e9 − 1.79733e9i 0.727785 − 0.192134i
\(66\) 7.99614e9 + 5.80953e9i 0.785938 + 0.571018i
\(67\) 2.48002e9 + 7.63273e9i 0.224411 + 0.690667i 0.998351 + 0.0574073i \(0.0182834\pi\)
−0.773940 + 0.633259i \(0.781717\pi\)
\(68\) 4.88815e9 0.407704
\(69\) −6.81880e9 − 2.09861e10i −0.524853 − 1.61533i
\(70\) 8.35614e7 − 1.48927e9i 0.00594246 − 0.105909i
\(71\) −5.26823e9 + 1.62140e10i −0.346533 + 1.06652i 0.614226 + 0.789130i \(0.289468\pi\)
−0.960758 + 0.277387i \(0.910532\pi\)
\(72\) −1.76276e8 + 5.42523e8i −0.0107366 + 0.0330438i
\(73\) 1.42826e10 − 1.03769e10i 0.806366 − 0.585859i −0.106409 − 0.994322i \(-0.533935\pi\)
0.912775 + 0.408463i \(0.133935\pi\)
\(74\) −2.20879e10 −1.15712
\(75\) 1.93928e10 + 2.18309e9i 0.943633 + 0.106227i
\(76\) −1.83394e10 −0.829677
\(77\) −4.17058e9 + 3.03010e9i −0.175589 + 0.127573i
\(78\) −3.98253e9 + 1.22570e10i −0.156185 + 0.480687i
\(79\) −3.78214e9 + 1.16402e10i −0.138289 + 0.425611i −0.996087 − 0.0883765i \(-0.971832\pi\)
0.857798 + 0.513987i \(0.171832\pi\)
\(80\) 7.08443e9 − 1.87027e9i 0.241719 − 0.0638132i
\(81\) −8.65066e9 − 2.66240e10i −0.275665 − 0.848410i
\(82\) −4.27138e9 −0.127231
\(83\) −1.90831e10 − 5.87318e10i −0.531765 − 1.63660i −0.750537 − 0.660829i \(-0.770205\pi\)
0.218772 − 0.975776i \(-0.429795\pi\)
\(84\) 2.20868e9 + 1.60470e9i 0.0576232 + 0.0418657i
\(85\) −3.10966e10 − 1.20687e10i −0.760166 − 0.295024i
\(86\) −3.54604e10 + 2.57635e10i −0.812835 + 0.590559i
\(87\) −2.67117e10 − 1.94072e10i −0.574575 − 0.417453i
\(88\) −2.04869e10 − 1.48846e10i −0.413829 − 0.300664i
\(89\) −5.27940e10 + 3.83571e10i −1.00217 + 0.728116i −0.962551 − 0.271099i \(-0.912613\pi\)
−0.0396140 + 0.999215i \(0.512613\pi\)
\(90\) 2.46088e9 − 3.01611e9i 0.0439296 − 0.0538410i
\(91\) −5.43813e9 − 3.95104e9i −0.0913529 − 0.0663718i
\(92\) 1.74704e10 + 5.37684e10i 0.276357 + 0.850539i
\(93\) 4.09862e10 0.610916
\(94\) 2.87418e10 + 8.84581e10i 0.403934 + 1.24318i
\(95\) 1.16668e11 + 4.52794e10i 1.54694 + 0.600372i
\(96\) −4.14416e9 + 1.27544e10i −0.0518735 + 0.159650i
\(97\) −1.19231e10 + 3.66954e10i −0.140975 + 0.433877i −0.996472 − 0.0839317i \(-0.973252\pi\)
0.855496 + 0.517809i \(0.173252\pi\)
\(98\) 5.00381e10 − 3.63548e10i 0.559188 − 0.406274i
\(99\) −1.34533e10 −0.142179
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.11.5 ✓ 56
25.16 even 5 inner 50.12.d.b.41.5 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.5 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.5 yes 56 25.16 even 5 inner