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Properties

Label 50.12.d.b.11.13
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 11.13
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.13

$q$-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} +(203.811 - 627.266i) q^{3} +(316.433 - 973.882i) q^{4} +(6676.95 + 2060.70i) q^{5} +(6521.96 + 20072.5i) q^{6} -9341.29 q^{7} +(10125.9 + 31164.2i) q^{8} +(-208609. - 151563. i) q^{9} +(-211616. + 72239.1i) q^{10} +(-227241. + 165100. i) q^{11} +(-546390. - 396976. i) q^{12} +(-73895.9 - 53688.5i) q^{13} +(241832. - 175701. i) q^{14} +(2.65344e6 - 3.76823e6i) q^{15} +(-848316. - 616338. i) q^{16} +(-3.34720e6 - 1.03016e7i) q^{17} +8.25135e6 q^{18} +(-770770. - 2.37218e6i) q^{19} +(4.11969e6 - 5.85048e6i) q^{20} +(-1.90386e6 + 5.85947e6i) q^{21} +(2.77755e6 - 8.54842e6i) q^{22} +(-2.30227e7 + 1.67269e7i) q^{23} +2.16120e7 q^{24} +(4.03352e7 + 2.75184e7i) q^{25} +2.92289e6 q^{26} +(-4.30644e7 + 3.12881e7i) q^{27} +(-2.95589e6 + 9.09731e6i) q^{28} +(-2.96884e7 + 9.13715e7i) q^{29} +(2.18333e6 + 1.47463e8i) q^{30} +(-7.38571e7 - 2.27309e8i) q^{31} +3.35544e7 q^{32} +(5.72476e7 + 1.76190e8i) q^{33} +(2.80419e8 + 2.03736e8i) q^{34} +(-6.23713e7 - 1.92496e7i) q^{35} +(-2.13615e8 + 1.55201e8i) q^{36} +(-3.88193e8 - 2.82039e8i) q^{37} +(6.45728e7 + 4.69149e7i) q^{38} +(-4.87378e7 + 3.54101e7i) q^{39} +(3.38981e6 + 2.28948e8i) q^{40} +(1.59461e8 + 1.15855e8i) q^{41} +(-6.09235e7 - 1.87503e8i) q^{42} -4.30839e8 q^{43} +(8.88816e7 + 2.73549e8i) q^{44} +(-1.08054e9 - 1.44186e9i) q^{45} +(2.81404e8 - 8.66072e8i) q^{46} +(4.60035e8 - 1.41584e9i) q^{47} +(-5.59504e8 + 4.06503e8i) q^{48} -1.89007e9 q^{49} +(-1.56182e9 + 4.62586e7i) q^{50} -7.14405e9 q^{51} +(-7.56694e7 + 5.49770e7i) q^{52} +(1.00425e9 - 3.09076e9i) q^{53} +(5.26372e8 - 1.62001e9i) q^{54} +(-1.85750e9 + 6.34091e8i) q^{55} +(-9.45886e7 - 2.91114e8i) q^{56} -1.64508e9 q^{57} +(-9.50029e8 - 2.92389e9i) q^{58} +(3.68949e7 + 2.68057e7i) q^{59} +(-2.83017e9 - 3.77653e9i) q^{60} +(-3.11634e9 + 2.26415e9i) q^{61} +(6.18753e9 + 4.49550e9i) q^{62} +(1.94868e9 + 1.41580e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-3.82763e8 - 5.10752e8i) q^{65} +(-4.79604e9 - 3.48453e9i) q^{66} +(7.85767e8 + 2.41834e9i) q^{67} -1.10917e10 q^{68} +(5.79997e9 + 1.78505e10i) q^{69} +(1.97677e9 - 6.74806e8i) q^{70} +(-9.18861e9 + 2.82796e10i) q^{71} +(2.61100e9 - 8.03584e9i) q^{72} +(-2.24314e10 + 1.62973e10i) q^{73} +1.53547e10 q^{74} +(2.54821e10 - 1.96923e10i) q^{75} -2.55413e9 q^{76} +(2.12273e9 - 1.54225e9i) q^{77} +(5.95717e8 - 1.83343e9i) q^{78} +(-1.56338e9 + 4.81159e9i) q^{79} +(-4.39407e9 - 5.86338e9i) q^{80} +(-3.26637e9 - 1.00528e10i) q^{81} -6.30734e9 q^{82} +(-1.84838e9 - 5.68873e9i) q^{83} +(5.10399e9 + 3.70827e9i) q^{84} +(-1.12053e9 - 7.56809e10i) q^{85} +(1.11538e10 - 8.10371e9i) q^{86} +(5.12634e10 + 3.72450e10i) q^{87} +(-7.44624e9 - 5.41001e9i) q^{88} +(2.04807e10 - 1.48801e10i) q^{89} +(5.50939e10 + 1.70036e10i) q^{90} +(6.90282e8 + 5.01519e8i) q^{91} +(9.00492e9 + 2.77143e10i) q^{92} -1.57636e11 q^{93} +(1.47211e10 + 4.53070e10i) q^{94} +(-2.58028e8 - 1.74273e10i) q^{95} +(6.83877e9 - 2.10476e10i) q^{96} +(1.01870e9 - 3.13523e9i) q^{97} +(4.89311e10 - 3.55505e10i) q^{98} +7.24277e10 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
\(3\) 203.811 − 627.266i 0.484240 − 1.49034i −0.348838 − 0.937183i \(-0.613424\pi\)
0.833078 − 0.553155i \(-0.186576\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) 6676.95 + 2060.70i 0.955527 + 0.294903i
\(6\) 6521.96 + 20072.5i 0.342410 + 1.05383i
\(7\) −9341.29 −0.210072 −0.105036 − 0.994468i \(-0.533496\pi\)
−0.105036 + 0.994468i \(0.533496\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) −208609. − 151563.i −1.17760 − 0.855579i
\(10\) −211616. + 72239.1i −0.669190 + 0.228440i
\(11\) −227241. + 165100.i −0.425429 + 0.309093i −0.779819 − 0.626005i \(-0.784689\pi\)
0.354389 + 0.935098i \(0.384689\pi\)
\(12\) −546390. − 396976.i −0.633879 − 0.460540i
\(13\) −73895.9 − 53688.5i −0.0551991 − 0.0401045i 0.559844 − 0.828598i \(-0.310861\pi\)
−0.615043 + 0.788494i \(0.710861\pi\)
\(14\) 241832. − 175701.i 0.120174 − 0.0873114i
\(15\) 2.65344e6 − 3.76823e6i 0.902211 − 1.28125i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) −3.34720e6 − 1.03016e7i −0.571758 − 1.75969i −0.646961 − 0.762523i \(-0.723961\pi\)
0.0752032 − 0.997168i \(-0.476039\pi\)
\(18\) 8.25135e6 1.02926
\(19\) −770770. − 2.37218e6i −0.0714134 − 0.219788i 0.908979 − 0.416841i \(-0.136863\pi\)
−0.980393 + 0.197053i \(0.936863\pi\)
\(20\) 4.11969e6 − 5.85048e6i 0.287872 − 0.408815i
\(21\) −1.90386e6 + 5.85947e6i −0.101725 + 0.313078i
\(22\) 2.77755e6 − 8.54842e6i 0.114905 − 0.353640i
\(23\) −2.30227e7 + 1.67269e7i −0.745851 + 0.541893i −0.894538 − 0.446992i \(-0.852495\pi\)
0.148687 + 0.988884i \(0.452495\pi\)
\(24\) 2.16120e7 0.554030
\(25\) 4.03352e7 + 2.75184e7i 0.826064 + 0.563576i
\(26\) 2.92289e6 0.0482457
\(27\) −4.30644e7 + 3.12881e7i −0.577587 + 0.419642i
\(28\) −2.95589e6 + 9.09731e6i −0.0324579 + 0.0998950i
\(29\) −2.96884e7 + 9.13715e7i −0.268780 + 0.827221i 0.722018 + 0.691874i \(0.243215\pi\)
−0.990798 + 0.135347i \(0.956785\pi\)
\(30\) 2.18333e6 + 1.47463e8i 0.0164041 + 1.10794i
\(31\) −7.38571e7 − 2.27309e8i −0.463343 − 1.42602i −0.861054 − 0.508513i \(-0.830195\pi\)
0.397711 − 0.917511i \(-0.369805\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 5.72476e7 + 1.76190e8i 0.254642 + 0.783709i
\(34\) 2.80419e8 + 2.03736e8i 1.05846 + 0.769013i
\(35\) −6.23713e7 − 1.92496e7i −0.200729 − 0.0619508i
\(36\) −2.13615e8 + 1.55201e8i −0.588802 + 0.427789i
\(37\) −3.88193e8 − 2.82039e8i −0.920319 − 0.668651i 0.0232847 − 0.999729i \(-0.492588\pi\)
−0.943603 + 0.331078i \(0.892588\pi\)
\(38\) 6.45728e7 + 4.69149e7i 0.132203 + 0.0960508i
\(39\) −4.87378e7 + 3.54101e7i −0.0864988 + 0.0628451i
\(40\) 3.38981e6 + 2.28948e8i 0.00523413 + 0.353515i
\(41\) 1.59461e8 + 1.15855e8i 0.214953 + 0.156172i 0.690052 − 0.723760i \(-0.257588\pi\)
−0.475099 + 0.879932i \(0.657588\pi\)
\(42\) −6.09235e7 − 1.87503e8i −0.0719306 − 0.221379i
\(43\) −4.30839e8 −0.446929 −0.223465 − 0.974712i \(-0.571737\pi\)
−0.223465 + 0.974712i \(0.571737\pi\)
\(44\) 8.88816e7 + 2.73549e8i 0.0812498 + 0.250061i
\(45\) −1.08054e9 − 1.44186e9i −0.872919 − 1.16481i
\(46\) 2.81404e8 − 8.66072e8i 0.201448 − 0.619992i
\(47\) 4.60035e8 − 1.41584e9i 0.292585 − 0.900485i −0.691436 − 0.722437i \(-0.743022\pi\)
0.984022 − 0.178048i \(-0.0569782\pi\)
\(48\) −5.59504e8 + 4.06503e8i −0.316939 + 0.230270i
\(49\) −1.89007e9 −0.955870
\(50\) −1.56182e9 + 4.62586e7i −0.706797 + 0.0209342i
\(51\) −7.14405e9 −2.89940
\(52\) −7.56694e7 + 5.49770e7i −0.0275995 + 0.0200522i
\(53\) 1.00425e9 − 3.09076e9i 0.329856 − 1.01519i −0.639345 − 0.768920i \(-0.720794\pi\)
0.969201 − 0.246272i \(-0.0792058\pi\)
\(54\) 5.26372e8 − 1.62001e9i 0.156001 − 0.480122i
\(55\) −1.85750e9 + 6.34091e8i −0.497662 + 0.169886i
\(56\) −9.45886e7 − 2.91114e8i −0.0229512 − 0.0706364i
\(57\) −1.64508e9 −0.362140
\(58\) −9.50029e8 − 2.92389e9i −0.190057 − 0.584934i
\(59\) 3.68949e7 + 2.68057e7i 0.00671863 + 0.00488137i 0.591140 − 0.806569i \(-0.298678\pi\)
−0.584421 + 0.811451i \(0.698678\pi\)
\(60\) −2.83017e9 − 3.77653e9i −0.469874 − 0.626991i
\(61\) −3.11634e9 + 2.26415e9i −0.472422 + 0.343235i −0.798385 − 0.602148i \(-0.794312\pi\)
0.325962 + 0.945383i \(0.394312\pi\)
\(62\) 6.18753e9 + 4.49550e9i 0.857755 + 0.623195i
\(63\) 1.94868e9 + 1.41580e9i 0.247381 + 0.179733i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −3.82763e8 − 5.10752e8i −0.0409173 − 0.0545993i
\(66\) −4.79604e9 − 3.48453e9i −0.471402 − 0.342493i
\(67\) 7.85767e8 + 2.41834e9i 0.0711021 + 0.218830i 0.980293 − 0.197550i \(-0.0632986\pi\)
−0.909191 + 0.416380i \(0.863299\pi\)
\(68\) −1.10917e10 −0.925124
\(69\) 5.79997e9 + 1.78505e10i 0.446432 + 1.37398i
\(70\) 1.97677e9 − 6.74806e8i 0.140578 − 0.0479888i
\(71\) −9.18861e9 + 2.82796e10i −0.604406 + 1.86017i −0.103582 + 0.994621i \(0.533030\pi\)
−0.500824 + 0.865549i \(0.666970\pi\)
\(72\) 2.61100e9 − 8.03584e9i 0.159030 − 0.489444i
\(73\) −2.24314e10 + 1.62973e10i −1.26643 + 0.920113i −0.999054 − 0.0434802i \(-0.986155\pi\)
−0.267373 + 0.963593i \(0.586155\pi\)
\(74\) 1.53547e10 0.804388
\(75\) 2.54821e10 − 1.96923e10i 1.23993 − 0.958209i
\(76\) −2.55413e9 −0.115549
\(77\) 2.12273e9 − 1.54225e9i 0.0893706 − 0.0649316i
\(78\) 5.95717e8 − 1.83343e9i 0.0233625 − 0.0719025i
\(79\) −1.56338e9 + 4.81159e9i −0.0571631 + 0.175930i −0.975561 − 0.219727i \(-0.929483\pi\)
0.918398 + 0.395657i \(0.129483\pi\)
\(80\) −4.39407e9 − 5.86338e9i −0.149924 − 0.200057i
\(81\) −3.26637e9 − 1.00528e10i −0.104087 − 0.320347i
\(82\) −6.30734e9 −0.187876
\(83\) −1.84838e9 − 5.68873e9i −0.0515065 − 0.158521i 0.921995 − 0.387202i \(-0.126559\pi\)
−0.973501 + 0.228682i \(0.926559\pi\)
\(84\) 5.10399e9 + 3.70827e9i 0.133160 + 0.0967464i
\(85\) −1.12053e9 − 7.56809e10i −0.0273918 − 1.85005i
\(86\) 1.11538e10 − 8.10371e9i 0.255671 − 0.185756i
\(87\) 5.12634e10 + 3.72450e10i 1.10269 + 0.801148i
\(88\) −7.44624e9 − 5.41001e9i −0.150412 − 0.109281i
\(89\) 2.04807e10 − 1.48801e10i 0.388777 − 0.282463i −0.376177 − 0.926548i \(-0.622762\pi\)
0.764954 + 0.644085i \(0.222762\pi\)
\(90\) 5.50939e10 + 1.70036e10i 0.983489 + 0.303533i
\(91\) 6.90282e8 + 5.01519e8i 0.0115958 + 0.00842481i
\(92\) 9.00492e9 + 2.77143e10i 0.142445 + 0.438400i
\(93\) −1.57636e11 −2.34963
\(94\) 1.47211e10 + 4.53070e10i 0.206889 + 0.636739i
\(95\) −2.58028e8 − 1.74273e10i −0.00342127 − 0.231073i
\(96\) 6.83877e9 − 2.10476e10i 0.0856024 − 0.263457i
\(97\) 1.01870e9 − 3.13523e9i 0.0120448 − 0.0370702i −0.944853 − 0.327493i \(-0.893796\pi\)
0.956898 + 0.290423i \(0.0937961\pi\)
\(98\) 4.89311e10 − 3.55505e10i 0.546816 − 0.397285i
\(99\) 7.24277e10 0.765440
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.11.13 ✓ 56
25.16 even 5 inner 50.12.d.b.41.13 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.13 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.13 yes 56 25.16 even 5 inner