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Properties

Label 50.12.d.b.11.10
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 11.10
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.10

$q$-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} +(53.1095 - 163.454i) q^{3} +(316.433 - 973.882i) q^{4} +(1.40075 + 6987.71i) q^{5} +(1699.50 + 5230.53i) q^{6} +86927.7 q^{7} +(10125.9 + 31164.2i) q^{8} +(119418. + 86762.5i) q^{9} +(-131469. - 180875. i) q^{10} +(325375. - 236399. i) q^{11} +(-142379. - 103445. i) q^{12} +(-401204. - 291492. i) q^{13} +(-2.25043e6 + 1.63503e6i) q^{14} +(1.14224e6 + 370885. i) q^{15} +(-848316. - 616338. i) q^{16} +(-1.40205e6 - 4.31505e6i) q^{17} -4.72349e6 q^{18} +(4.11591e6 + 1.26675e7i) q^{19} +(6.80565e6 + 2.20978e6i) q^{20} +(4.61668e6 - 1.42087e7i) q^{21} +(-3.97703e6 + 1.22401e7i) q^{22} +(-7.52744e6 + 5.46900e6i) q^{23} +5.63170e6 q^{24} +(-4.88281e7 + 19576.0i) q^{25} +1.58693e7 q^{26} +(4.51549e7 - 3.28069e7i) q^{27} +(2.75068e7 - 8.46573e7i) q^{28} +(4.54609e7 - 1.39914e8i) q^{29} +(-3.65471e7 + 1.18830e7i) q^{30} +(-5.96126e7 - 1.83469e8i) q^{31} +3.35544e7 q^{32} +(-2.13599e7 - 6.57390e7i) q^{33} +(1.17459e8 + 8.53392e7i) q^{34} +(121764. + 6.07426e8i) q^{35} +(1.22284e8 - 8.88448e7i) q^{36} +(4.79874e8 + 3.48649e8i) q^{37} +(-3.44819e8 - 2.50526e8i) q^{38} +(-6.89533e7 + 5.00975e7i) q^{39} +(-2.17752e8 + 7.08003e7i) q^{40} +(5.75203e8 + 4.17909e8i) q^{41} +(1.47734e8 + 4.54678e8i) q^{42} +7.54343e7 q^{43} +(-1.27265e8 - 3.91682e8i) q^{44} +(-6.06104e8 + 8.34582e8i) q^{45} +(9.20072e7 - 2.83169e8i) q^{46} +(-6.67166e8 + 2.05333e9i) q^{47} +(-1.45797e8 + 1.05927e8i) q^{48} +5.57909e9 q^{49} +(1.26372e9 - 9.18921e8i) q^{50} -7.79775e8 q^{51} +(-4.10833e8 + 2.98488e8i) q^{52} +(-2.10264e8 + 6.47125e8i) q^{53} +(-5.51924e8 + 1.69865e9i) q^{54} +(1.65234e9 + 2.27330e9i) q^{55} +(8.80218e8 + 2.70903e9i) q^{56} +2.28915e9 q^{57} +(1.45475e9 + 4.47725e9i) q^{58} +(-3.10573e8 - 2.25645e8i) q^{59} +(7.22642e8 - 9.95051e8i) q^{60} +(-3.14882e9 + 2.28775e9i) q^{61} +(4.99417e9 + 3.62848e9i) q^{62} +(1.03808e10 + 7.54206e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(2.03630e9 - 2.80391e9i) q^{65} +(1.78947e9 + 1.30013e9i) q^{66} +(-5.25978e9 - 1.61879e10i) q^{67} -4.64600e9 q^{68} +(4.94153e8 + 1.52085e9i) q^{69} +(-1.14283e10 - 1.57231e10i) q^{70} +(-7.67557e9 + 2.36230e10i) q^{71} +(-1.49467e9 + 4.60012e9i) q^{72} +(1.99578e10 - 1.45002e10i) q^{73} -1.89810e10 q^{74} +(-2.59004e9 + 7.98220e9i) q^{75} +1.36390e10 q^{76} +(2.82841e10 - 2.05496e10i) q^{77} +(8.42810e8 - 2.59390e9i) q^{78} +(6.17475e9 - 1.90039e10i) q^{79} +(4.30560e9 - 5.92865e9i) q^{80} +(5.11606e9 + 1.57456e10i) q^{81} -2.27517e10 q^{82} +(6.83944e9 + 2.10496e10i) q^{83} +(-1.23767e10 - 8.99221e9i) q^{84} +(3.01504e10 - 9.80313e9i) q^{85} +(-1.95289e9 + 1.41885e9i) q^{86} +(-2.04551e10 - 1.48615e10i) q^{87} +(1.06619e10 + 7.74632e9i) q^{88} +(-2.57740e10 + 1.87259e10i) q^{89} +(-6.61641e6 - 3.30064e10i) q^{90} +(-3.48758e10 - 2.53387e10i) q^{91} +(2.94423e9 + 9.06141e9i) q^{92} -3.31547e10 q^{93} +(-2.13493e10 - 6.57064e10i) q^{94} +(-8.85109e10 + 2.87786e10i) q^{95} +(1.78206e9 - 5.48461e9i) q^{96} +(4.16971e9 - 1.28331e10i) q^{97} +(-1.44435e11 + 1.04938e11i) q^{98} +5.93663e10 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
\(3\) 53.1095 − 163.454i 0.126184 − 0.388355i −0.867931 − 0.496685i \(-0.834550\pi\)
0.994115 + 0.108330i \(0.0345503\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) 1.40075 + 6987.71i 0.000200458 + 1.00000i
\(6\) 1699.50 + 5230.53i 0.0892257 + 0.274609i
\(7\) 86927.7 1.95487 0.977437 − 0.211227i \(-0.0677459\pi\)
0.977437 + 0.211227i \(0.0677459\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) 119418. + 86762.5i 0.674120 + 0.489777i
\(10\) −131469. − 180875.i −0.415742 − 0.571978i
\(11\) 325375. − 236399.i 0.609151 − 0.442574i −0.239964 − 0.970782i \(-0.577136\pi\)
0.849115 + 0.528208i \(0.177136\pi\)
\(12\) −142379. − 103445.i −0.165177 − 0.120008i
\(13\) −401204. − 291492.i −0.299693 − 0.217740i 0.427768 − 0.903889i \(-0.359300\pi\)
−0.727461 + 0.686149i \(0.759300\pi\)
\(14\) −2.25043e6 + 1.63503e6i −1.11831 + 0.812498i
\(15\) 1.14224e6 + 370885.i 0.388380 + 0.126106i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) −1.40205e6 − 4.31505e6i −0.239493 − 0.737084i −0.996494 − 0.0836696i \(-0.973336\pi\)
0.757000 − 0.653414i \(-0.226664\pi\)
\(18\) −4.72349e6 −0.589202
\(19\) 4.11591e6 + 1.26675e7i 0.381348 + 1.17367i 0.939095 + 0.343657i \(0.111666\pi\)
−0.557747 + 0.830011i \(0.688334\pi\)
\(20\) 6.80565e6 + 2.20978e6i 0.475559 + 0.154413i
\(21\) 4.61668e6 − 1.42087e7i 0.246674 − 0.759185i
\(22\) −3.97703e6 + 1.22401e7i −0.164526 + 0.506359i
\(23\) −7.52744e6 + 5.46900e6i −0.243862 + 0.177176i −0.703002 − 0.711188i \(-0.748158\pi\)
0.459140 + 0.888364i \(0.348158\pi\)
\(24\) 5.63170e6 0.144370
\(25\) −4.88281e7 + 19576.0i −1.00000 + 0.000400917i
\(26\) 1.58693e7 0.261942
\(27\) 4.51549e7 − 3.28069e7i 0.605625 − 0.440012i
\(28\) 2.75068e7 − 8.46573e7i 0.302045 − 0.929598i
\(29\) 4.54609e7 − 1.39914e8i 0.411575 − 1.26670i −0.503704 − 0.863876i \(-0.668030\pi\)
0.915279 − 0.402820i \(-0.131970\pi\)
\(30\) −3.65471e7 + 1.18830e7i −0.274591 + 0.0892808i
\(31\) −5.96126e7 − 1.83469e8i −0.373980 − 1.15099i −0.944164 − 0.329476i \(-0.893128\pi\)
0.570184 − 0.821517i \(-0.306872\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −2.13599e7 − 6.57390e7i −0.0950107 − 0.292413i
\(34\) 1.17459e8 + 8.53392e7i 0.443357 + 0.322117i
\(35\) 121764. + 6.07426e8i 0.000391871 + 1.95487i
\(36\) 1.22284e8 − 8.88448e7i 0.337060 − 0.244888i
\(37\) 4.79874e8 + 3.48649e8i 1.13767 + 0.826568i 0.986793 − 0.161984i \(-0.0517894\pi\)
0.150880 + 0.988552i \(0.451789\pi\)
\(38\) −3.44819e8 − 2.50526e8i −0.705962 − 0.512912i
\(39\) −6.89533e7 + 5.00975e7i −0.122377 + 0.0889121i
\(40\) −2.17752e8 + 7.08003e7i −0.336227 + 0.109321i
\(41\) 5.75203e8 + 4.17909e8i 0.775371 + 0.563340i 0.903586 − 0.428406i \(-0.140925\pi\)
−0.128215 + 0.991746i \(0.540925\pi\)
\(42\) 1.47734e8 + 4.54678e8i 0.174425 + 0.536825i
\(43\) 7.54343e7 0.0782515 0.0391257 − 0.999234i \(-0.487543\pi\)
0.0391257 + 0.999234i \(0.487543\pi\)
\(44\) −1.27265e8 − 3.91682e8i −0.116337 − 0.358050i
\(45\) −6.06104e8 + 8.34582e8i −0.489642 + 0.674218i
\(46\) 9.20072e7 − 2.83169e8i 0.0658649 − 0.202711i
\(47\) −6.67166e8 + 2.05333e9i −0.424322 + 1.30593i 0.479320 + 0.877640i \(0.340883\pi\)
−0.903642 + 0.428289i \(0.859117\pi\)
\(48\) −1.45797e8 + 1.05927e8i −0.0825886 + 0.0600042i
\(49\) 5.57909e9 2.82153
\(50\) 1.26372e9 − 9.18921e8i 0.571895 − 0.415856i
\(51\) −7.79775e8 −0.316471
\(52\) −4.10833e8 + 2.98488e8i −0.149847 + 0.108870i
\(53\) −2.10264e8 + 6.47125e8i −0.0690632 + 0.212555i −0.979631 − 0.200804i \(-0.935644\pi\)
0.910568 + 0.413359i \(0.135644\pi\)
\(54\) −5.51924e8 + 1.69865e9i −0.163574 + 0.503428i
\(55\) 1.65234e9 + 2.27330e9i 0.442696 + 0.609062i
\(56\) 8.80218e8 + 2.70903e9i 0.213578 + 0.657325i
\(57\) 2.28915e9 0.503920
\(58\) 1.45475e9 + 4.47725e9i 0.291027 + 0.895690i
\(59\) −3.10573e8 − 2.25645e8i −0.0565559 − 0.0410903i 0.559148 − 0.829068i \(-0.311128\pi\)
−0.615704 + 0.787977i \(0.711128\pi\)
\(60\) 7.22642e8 − 9.95051e8i 0.119975 − 0.165201i
\(61\) −3.14882e9 + 2.28775e9i −0.477346 + 0.346812i −0.800297 − 0.599603i \(-0.795325\pi\)
0.322951 + 0.946416i \(0.395325\pi\)
\(62\) 4.99417e9 + 3.62848e9i 0.692324 + 0.503002i
\(63\) 1.03808e10 + 7.54206e9i 1.31782 + 0.957452i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) 2.03630e9 − 2.80391e9i 0.217680 − 0.299737i
\(66\) 1.78947e9 + 1.30013e9i 0.175887 + 0.127789i
\(67\) −5.25978e9 − 1.61879e10i −0.475944 − 1.46481i −0.844680 − 0.535271i \(-0.820209\pi\)
0.368736 − 0.929534i \(-0.379791\pi\)
\(68\) −4.64600e9 −0.387508
\(69\) 4.94153e8 + 1.52085e9i 0.0380357 + 0.117062i
\(70\) −1.14283e10 − 1.57231e10i −0.812723 − 1.11815i
\(71\) −7.67557e9 + 2.36230e10i −0.504882 + 1.55387i 0.296087 + 0.955161i \(0.404318\pi\)
−0.800969 + 0.598706i \(0.795682\pi\)
\(72\) −1.49467e9 + 4.60012e9i −0.0910368 + 0.280182i
\(73\) 1.99578e10 − 1.45002e10i 1.12678 − 0.818651i 0.141554 − 0.989931i \(-0.454790\pi\)
0.985222 + 0.171280i \(0.0547902\pi\)
\(74\) −1.89810e10 −0.994363
\(75\) −2.59004e9 + 7.98220e9i −0.126029 + 0.388406i
\(76\) 1.36390e10 0.617034
\(77\) 2.82841e10 − 2.05496e10i 1.19081 − 0.865177i
\(78\) 8.42810e8 − 2.59390e9i 0.0330529 − 0.101726i
\(79\) 6.17475e9 − 1.90039e10i 0.225772 − 0.694856i −0.772440 − 0.635088i \(-0.780964\pi\)
0.998212 − 0.0597679i \(-0.0190361\pi\)
\(80\) 4.30560e9 − 5.92865e9i 0.146906 − 0.202284i
\(81\) 5.11606e9 + 1.57456e10i 0.163030 + 0.501755i
\(82\) −2.27517e10 −0.677699
\(83\) 6.83944e9 + 2.10496e10i 0.190586 + 0.586563i 1.00000 0.000675467i \(-0.000215008\pi\)
−0.809414 + 0.587239i \(0.800215\pi\)
\(84\) −1.23767e10 − 8.99221e9i −0.322901 − 0.234601i
\(85\) 3.01504e10 − 9.80313e9i 0.737036 − 0.239641i
\(86\) −1.95289e9 + 1.41885e9i −0.0447647 + 0.0325234i
\(87\) −2.04551e10 − 1.48615e10i −0.439994 − 0.319674i
\(88\) 1.06619e10 + 7.74632e9i 0.215367 + 0.156474i
\(89\) −2.57740e10 + 1.87259e10i −0.489257 + 0.355466i −0.804899 − 0.593412i \(-0.797780\pi\)
0.315642 + 0.948879i \(0.397780\pi\)
\(90\) −6.61641e6 − 3.30064e10i −0.000118111 − 0.589202i
\(91\) −3.48758e10 − 2.53387e10i −0.585863 − 0.425654i
\(92\) 2.94423e9 + 9.06141e9i 0.0465735 + 0.143338i
\(93\) −3.31547e10 −0.494185
\(94\) −2.13493e10 − 6.57064e10i −0.300041 − 0.923432i
\(95\) −8.85109e10 + 2.87786e10i −1.17359 + 0.381583i
\(96\) 1.78206e9 − 5.48461e9i 0.0223064 − 0.0686521i
\(97\) 4.16971e9 − 1.28331e10i 0.0493016 − 0.151735i −0.923375 − 0.383900i \(-0.874581\pi\)
0.972676 + 0.232165i \(0.0745809\pi\)
\(98\) −1.44435e11 + 1.04938e11i −1.61409 + 1.17271i
\(99\) 5.93663e10 0.627403
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.11.10 ✓ 56
25.16 even 5 inner 50.12.d.b.41.10 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.10 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.10 yes 56 25.16 even 5 inner