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Properties

Label 50.12.d.b.11.8
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 11.8
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.8

$q$-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} +(22.6894 - 69.8307i) q^{3} +(316.433 - 973.882i) q^{4} +(4847.01 + 5033.35i) q^{5} +(726.060 + 2234.58i) q^{6} -16025.8 q^{7} +(10125.9 + 31164.2i) q^{8} +(138953. + 100956. i) q^{9} +(-220155. - 39137.9i) q^{10} +(-223625. + 162473. i) q^{11} +(-60827.2 - 44193.5i) q^{12} +(-180723. - 131303. i) q^{13} +(414884. - 301431. i) q^{14} +(461458. - 224267. i) q^{15} +(-848316. - 616338. i) q^{16} +(1.58107e6 + 4.86604e6i) q^{17} -5.49619e6 q^{18} +(-2.81569e6 - 8.66580e6i) q^{19} +(6.43564e6 - 3.12770e6i) q^{20} +(-363615. + 1.11909e6i) q^{21} +(2.73335e6 - 8.41239e6i) q^{22} +(3.82229e7 - 2.77705e7i) q^{23} +2.40597e6 q^{24} +(-1.84103e6 + 4.87934e7i) q^{25} +7.14834e6 q^{26} +(2.07254e7 - 1.50579e7i) q^{27} +(-5.07110e6 + 1.56072e7i) q^{28} +(1.74126e6 - 5.35903e6i) q^{29} +(-7.72821e6 + 1.44856e7i) q^{30} +(9.35145e7 + 2.87808e8i) q^{31} +3.35544e7 q^{32} +(6.27171e6 + 1.93023e7i) q^{33} +(-1.32458e8 - 9.62361e7i) q^{34} +(-7.76773e7 - 8.06634e7i) q^{35} +(1.42288e8 - 1.03378e8i) q^{36} +(-1.08175e8 - 7.85937e7i) q^{37} +(2.35890e8 + 1.71384e8i) q^{38} +(-1.32695e7 + 9.64082e6i) q^{39} +(-1.07780e8 + 2.02020e8i) q^{40} +(-1.11110e9 - 8.07262e8i) q^{41} +(-1.16357e7 - 3.58110e7i) q^{42} -3.87848e8 q^{43} +(8.74673e7 + 2.69197e8i) q^{44} +(1.65365e8 + 1.18873e9i) q^{45} +(-4.67195e8 + 1.43788e9i) q^{46} +(-5.85489e8 + 1.80195e9i) q^{47} +(-6.22871e7 + 4.52542e7i) q^{48} -1.72050e9 q^{49} +(-8.70100e8 - 1.29782e9i) q^{50} +3.75672e8 q^{51} +(-1.85060e8 + 1.34454e8i) q^{52} +(-1.61113e9 + 4.95854e9i) q^{53} +(-2.53324e8 + 7.79652e8i) q^{54} +(-1.90170e9 - 3.38073e8i) q^{55} +(-1.62275e8 - 4.99431e8i) q^{56} -6.69026e8 q^{57} +(5.57202e7 + 1.71489e8i) q^{58} +(5.12316e9 + 3.72219e9i) q^{59} +(-7.23889e7 - 5.20371e8i) q^{60} +(-9.85029e9 + 7.15666e9i) q^{61} +(-7.83437e9 - 5.69201e9i) q^{62} +(-2.22684e9 - 1.61789e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-2.15073e8 - 1.54607e9i) q^{65} +(-5.25425e8 - 3.81744e8i) q^{66} +(4.94888e9 + 1.52311e10i) q^{67} +5.23925e9 q^{68} +(-1.07198e9 - 3.29923e9i) q^{69} +(3.52816e9 + 6.27216e8i) q^{70} +(3.81503e9 - 1.17414e10i) q^{71} +(-1.73918e9 + 5.35264e9i) q^{72} +(-3.12397e9 + 2.26970e9i) q^{73} +4.27877e9 q^{74} +(3.36551e9 + 1.23565e9i) q^{75} -9.33045e9 q^{76} +(3.58377e9 - 2.60376e9i) q^{77} +(1.62191e8 - 4.99174e8i) q^{78} +(4.02638e9 - 1.23919e10i) q^{79} +(-1.00956e9 - 7.25726e9i) q^{80} +(8.82091e9 + 2.71480e10i) q^{81} +4.39487e10 q^{82} +(2.49261e9 + 7.67146e9i) q^{83} +(9.74804e8 + 7.08237e8i) q^{84} +(-1.68290e10 + 3.15438e10i) q^{85} +(1.00408e10 - 7.29509e9i) q^{86} +(-3.34717e8 - 2.43186e8i) q^{87} +(-7.32775e9 - 5.32392e9i) q^{88} +(1.45049e10 - 1.05384e10i) q^{89} +(-2.66401e10 - 2.76642e10i) q^{90} +(2.89623e9 + 2.10423e9i) q^{91} +(-1.49502e10 - 4.60121e10i) q^{92} +2.22196e10 q^{93} +(-1.87356e10 - 5.76624e10i) q^{94} +(2.99703e10 - 5.61756e10i) q^{95} +(7.61329e8 - 2.34313e9i) q^{96} +(1.74883e10 - 5.38234e10i) q^{97} +(4.45413e10 - 3.23611e10i) q^{98} -4.74761e10 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
\(3\) 22.6894 − 69.8307i 0.0539083 − 0.165913i −0.920478 − 0.390796i \(-0.872200\pi\)
0.974386 + 0.224883i \(0.0721999\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) 4847.01 + 5033.35i 0.693648 + 0.720314i
\(6\) 726.060 + 2234.58i 0.0381189 + 0.117318i
\(7\) −16025.8 −0.360396 −0.180198 − 0.983630i \(-0.557674\pi\)
−0.180198 + 0.983630i \(0.557674\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) 138953. + 100956.i 0.784396 + 0.569897i
\(10\) −220155. − 39137.9i −0.696191 − 0.123765i
\(11\) −223625. + 162473.i −0.418660 + 0.304174i −0.777098 − 0.629379i \(-0.783309\pi\)
0.358439 + 0.933553i \(0.383309\pi\)
\(12\) −60827.2 − 44193.5i −0.0705669 − 0.0512698i
\(13\) −180723. − 131303.i −0.134997 − 0.0980811i 0.518237 − 0.855237i \(-0.326588\pi\)
−0.653234 + 0.757156i \(0.726588\pi\)
\(14\) 414884. − 301431.i 0.206169 − 0.149790i
\(15\) 461458. − 224267.i 0.156903 − 0.0762542i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) 1.58107e6 + 4.86604e6i 0.270074 + 0.831202i 0.990481 + 0.137650i \(0.0439550\pi\)
−0.720407 + 0.693551i \(0.756045\pi\)
\(18\) −5.49619e6 −0.685587
\(19\) −2.81569e6 − 8.66580e6i −0.260880 − 0.802905i −0.992614 − 0.121315i \(-0.961289\pi\)
0.731734 − 0.681590i \(-0.238711\pi\)
\(20\) 6.43564e6 − 3.12770e6i 0.449704 − 0.218555i
\(21\) −363615. + 1.11909e6i −0.0194284 + 0.0597943i
\(22\) 2.73335e6 − 8.41239e6i 0.113076 − 0.348012i
\(23\) 3.82229e7 − 2.77705e7i 1.23828 − 0.899665i 0.240800 − 0.970575i \(-0.422590\pi\)
0.997483 + 0.0709092i \(0.0225901\pi\)
\(24\) 2.40597e6 0.0616777
\(25\) −1.84103e6 + 4.87934e7i −0.0377043 + 0.999289i
\(26\) 7.14834e6 0.117992
\(27\) 2.07254e7 − 1.50579e7i 0.277972 − 0.201959i
\(28\) −5.07110e6 + 1.56072e7i −0.0556843 + 0.171379i
\(29\) 1.74126e6 − 5.35903e6i 0.0157643 − 0.0485174i −0.942865 − 0.333175i \(-0.891880\pi\)
0.958629 + 0.284658i \(0.0918800\pi\)
\(30\) −7.72821e6 + 1.44856e7i −0.0580647 + 0.108835i
\(31\) 9.35145e7 + 2.87808e8i 0.586664 + 1.80557i 0.592482 + 0.805584i \(0.298148\pi\)
−0.00581725 + 0.999983i \(0.501852\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 6.27171e6 + 1.93023e7i 0.0278971 + 0.0858585i
\(34\) −1.32458e8 − 9.62361e7i −0.499969 − 0.363248i
\(35\) −7.76773e7 − 8.06634e7i −0.249988 − 0.259598i
\(36\) 1.42288e8 − 1.03378e8i 0.392198 − 0.284949i
\(37\) −1.08175e8 − 7.85937e7i −0.256459 − 0.186328i 0.452126 − 0.891954i \(-0.350666\pi\)
−0.708584 + 0.705626i \(0.750666\pi\)
\(38\) 2.35890e8 + 1.71384e8i 0.482948 + 0.350882i
\(39\) −1.32695e7 + 9.64082e6i −0.0235504 + 0.0171103i
\(40\) −1.07780e8 + 2.02020e8i −0.166421 + 0.311936i
\(41\) −1.11110e9 − 8.07262e8i −1.49776 − 1.08819i −0.971263 − 0.238009i \(-0.923505\pi\)
−0.526497 − 0.850177i \(-0.676495\pi\)
\(42\) −1.16357e7 − 3.58110e7i −0.0137379 − 0.0422810i
\(43\) −3.87848e8 −0.402333 −0.201166 − 0.979557i \(-0.564473\pi\)
−0.201166 + 0.979557i \(0.564473\pi\)
\(44\) 8.74673e7 + 2.69197e8i 0.0799569 + 0.246082i
\(45\) 1.65365e8 + 1.18873e9i 0.133590 + 0.960320i
\(46\) −4.67195e8 + 1.43788e9i −0.334449 + 1.02933i
\(47\) −5.85489e8 + 1.80195e9i −0.372375 + 1.14605i 0.572858 + 0.819655i \(0.305835\pi\)
−0.945233 + 0.326397i \(0.894165\pi\)
\(48\) −6.22871e7 + 4.52542e7i −0.0352834 + 0.0256349i
\(49\) −1.72050e9 −0.870114
\(50\) −8.70100e8 − 1.29782e9i −0.393762 − 0.587326i
\(51\) 3.75672e8 0.152466
\(52\) −1.85060e8 + 1.34454e8i −0.0674985 + 0.0490406i
\(53\) −1.61113e9 + 4.95854e9i −0.529191 + 1.62868i 0.226685 + 0.973968i \(0.427211\pi\)
−0.755876 + 0.654714i \(0.772789\pi\)
\(54\) −2.53324e8 + 7.79652e8i −0.0750778 + 0.231066i
\(55\) −1.90170e9 − 3.38073e8i −0.509503 − 0.0905766i
\(56\) −1.62275e8 − 4.99431e8i −0.0393747 − 0.121183i
\(57\) −6.69026e8 −0.147276
\(58\) 5.57202e7 + 1.71489e8i 0.0111470 + 0.0343070i
\(59\) 5.12316e9 + 3.72219e9i 0.932935 + 0.677817i 0.946710 − 0.322088i \(-0.104385\pi\)
−0.0137745 + 0.999905i \(0.504385\pi\)
\(60\) −7.23889e7 − 5.20371e8i −0.0120182 − 0.0863936i
\(61\) −9.85029e9 + 7.15666e9i −1.49326 + 1.08492i −0.520287 + 0.853991i \(0.674175\pi\)
−0.972972 + 0.230925i \(0.925825\pi\)
\(62\) −7.83437e9 − 5.69201e9i −1.08605 − 0.789062i
\(63\) −2.22684e9 − 1.61789e9i −0.282693 − 0.205389i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −2.15073e8 − 1.54607e9i −0.0229913 − 0.165274i
\(66\) −5.25425e8 − 3.81744e8i −0.0516440 − 0.0375215i
\(67\) 4.94888e9 + 1.52311e10i 0.447812 + 1.37822i 0.879370 + 0.476139i \(0.157964\pi\)
−0.431558 + 0.902085i \(0.642036\pi\)
\(68\) 5.23925e9 0.436989
\(69\) −1.07198e9 − 3.29923e9i −0.0825122 − 0.253946i
\(70\) 3.52816e9 + 6.27216e8i 0.250905 + 0.0446044i
\(71\) 3.81503e9 − 1.17414e10i 0.250944 − 0.772326i −0.743658 − 0.668560i \(-0.766911\pi\)
0.994602 − 0.103765i \(-0.0330891\pi\)
\(72\) −1.73918e9 + 5.35264e9i −0.105929 + 0.326016i
\(73\) −3.12397e9 + 2.26970e9i −0.176373 + 0.128142i −0.672469 − 0.740125i \(-0.734766\pi\)
0.496096 + 0.868267i \(0.334766\pi\)
\(74\) 4.27877e9 0.224153
\(75\) 3.36551e9 + 1.23565e9i 0.163762 + 0.0601256i
\(76\) −9.33045e9 −0.422112
\(77\) 3.58377e9 − 2.60376e9i 0.150883 − 0.109623i
\(78\) 1.62191e8 − 4.99174e8i 0.00636074 − 0.0195763i
\(79\) 4.02638e9 − 1.23919e10i 0.147220 − 0.453095i −0.850070 − 0.526670i \(-0.823441\pi\)
0.997290 + 0.0735741i \(0.0234405\pi\)
\(80\) −1.00956e9 − 7.25726e9i −0.0344458 − 0.247616i
\(81\) 8.82091e9 + 2.71480e10i 0.281090 + 0.865106i
\(82\) 4.39487e10 1.30909
\(83\) 2.49261e9 + 7.67146e9i 0.0694583 + 0.213771i 0.979760 − 0.200174i \(-0.0641507\pi\)
−0.910302 + 0.413945i \(0.864151\pi\)
\(84\) 9.74804e8 + 7.08237e8i 0.0254320 + 0.0184775i
\(85\) −1.68290e10 + 3.15438e10i −0.411390 + 0.771099i
\(86\) 1.00408e10 − 7.29509e9i 0.230159 − 0.167220i
\(87\) −3.34717e8 − 2.43186e8i −0.00719983 − 0.00523098i
\(88\) −7.32775e9 − 5.32392e9i −0.148019 − 0.107542i
\(89\) 1.45049e10 − 1.05384e10i 0.275340 − 0.200046i −0.441542 − 0.897240i \(-0.645568\pi\)
0.716882 + 0.697194i \(0.245568\pi\)
\(90\) −2.66401e10 − 2.76642e10i −0.475556 − 0.493838i
\(91\) 2.89623e9 + 2.10423e9i 0.0486524 + 0.0353481i
\(92\) −1.49502e10 − 4.60121e10i −0.236491 − 0.727845i
\(93\) 2.22196e10 0.331193
\(94\) −1.87356e10 − 5.76624e10i −0.263309 − 0.810381i
\(95\) 2.99703e10 − 5.61756e10i 0.397385 − 0.744849i
\(96\) 7.61329e8 − 2.34313e9i 0.00952973 − 0.0293295i
\(97\) 1.74883e10 − 5.38234e10i 0.206777 − 0.636395i −0.792858 − 0.609406i \(-0.791408\pi\)
0.999636 − 0.0269892i \(-0.00859197\pi\)
\(98\) 4.45413e10 − 3.23611e10i 0.497759 − 0.361643i
\(99\) −4.74761e10 −0.501743
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.11.8 ✓ 56
25.16 even 5 inner 50.12.d.b.41.8 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.8 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.8 yes 56 25.16 even 5 inner