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Properties

Label 50.12.d.b.11.12
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 11.12
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.12

$q$-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} +(179.007 - 550.928i) q^{3} +(316.433 - 973.882i) q^{4} +(-4657.15 - 5209.52i) q^{5} +(5728.23 + 17629.7i) q^{6} -160.425 q^{7} +(10125.9 + 31164.2i) q^{8} +(-128163. - 93115.7i) q^{9} +(218553. + 47270.1i) q^{10} +(272639. - 198084. i) q^{11} +(-479895. - 348664. i) q^{12} +(-1.89998e6 - 1.38042e6i) q^{13} +(4153.17 - 3017.46i) q^{14} +(-3.70373e6 + 1.63321e6i) q^{15} +(-848316. - 616338. i) q^{16} +(-747912. - 2.30184e6i) q^{17} +5.06937e6 q^{18} +(-3.69575e6 - 1.13744e7i) q^{19} +(-6.54714e6 + 2.88704e6i) q^{20} +(-28717.3 + 88382.7i) q^{21} +(-3.33245e6 + 1.02562e7i) q^{22} +(-1.05028e7 + 7.63075e6i) q^{23} +1.89818e7 q^{24} +(-5.45012e6 + 4.85230e7i) q^{25} +7.51523e7 q^{26} +(8.77735e6 - 6.37712e6i) q^{27} +(-50763.9 + 156235. i) q^{28} +(3.08446e7 - 9.49299e7i) q^{29} +(6.51650e7 - 1.11945e8i) q^{30} +(4.28073e7 + 1.31747e8i) q^{31} +3.35544e7 q^{32} +(-6.03256e7 - 1.85663e8i) q^{33} +(6.26579e7 + 4.55236e7i) q^{34} +(747123. + 835739. i) q^{35} +(-1.31239e8 + 9.53505e7i) q^{36} +(5.31370e8 + 3.86063e8i) q^{37} +(3.09619e8 + 2.24952e8i) q^{38} +(-1.10062e9 + 7.99649e8i) q^{39} +(1.15193e8 - 1.97887e8i) q^{40} +(2.01450e8 + 1.46362e8i) q^{41} +(-918953. - 2.82825e6i) q^{42} -1.29048e9 q^{43} +(-1.06638e8 - 3.28199e8i) q^{44} +(1.11784e8 + 1.10132e9i) q^{45} +(1.28375e8 - 3.95098e8i) q^{46} +(-2.92767e8 + 9.01044e8i) q^{47} +(-4.91412e8 + 3.57032e8i) q^{48} -1.97730e9 q^{49} +(-7.71580e8 - 1.35870e9i) q^{50} -1.40203e9 q^{51} +(-1.94558e9 + 1.41355e9i) q^{52} +(-1.14564e9 + 3.52591e9i) q^{53} +(-1.07285e8 + 3.30189e8i) q^{54} +(-2.30165e9 - 4.97815e8i) q^{55} +(-1.62444e6 - 4.99953e6i) q^{56} -6.92802e9 q^{57} +(9.87027e8 + 3.03776e9i) q^{58} +(-2.10577e9 - 1.52993e9i) q^{59} +(4.18567e8 + 4.12380e9i) q^{60} +(9.01536e9 - 6.55004e9i) q^{61} +(-3.58627e9 - 2.60558e9i) q^{62} +(2.05605e7 + 1.49381e7i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(1.65718e9 + 1.63268e10i) q^{65} +(5.05390e9 + 3.67188e9i) q^{66} +(5.54077e9 + 1.70527e10i) q^{67} -2.47838e9 q^{68} +(2.32391e9 + 7.15226e9i) q^{69} +(-3.50615e7 - 7.58332e6i) q^{70} +(-1.21556e9 + 3.74112e9i) q^{71} +(1.60412e9 - 4.93697e9i) q^{72} +(1.64598e10 - 1.19588e10i) q^{73} -2.10179e10 q^{74} +(2.57571e10 + 1.16886e10i) q^{75} -1.22467e10 q^{76} +(-4.37382e7 + 3.17777e7i) q^{77} +(1.34528e10 - 4.14035e10i) q^{78} +(2.46214e9 - 7.57769e9i) q^{79} +(7.39906e8 + 7.28969e9i) q^{80} +(-1.06141e10 - 3.26670e10i) q^{81} -7.96818e9 q^{82} +(1.24958e9 + 3.84580e9i) q^{83} +(7.69872e7 + 5.59345e7i) q^{84} +(-8.50833e9 + 1.46163e10i) q^{85} +(3.34087e10 - 2.42728e10i) q^{86} +(-4.67781e10 - 3.39863e10i) q^{87} +(8.93385e9 + 6.49082e9i) q^{88} +(-2.58301e10 + 1.87667e10i) q^{89} +(-2.36088e10 - 2.64090e10i) q^{90} +(3.04805e8 + 2.21454e8i) q^{91} +(4.10800e9 + 1.26431e10i) q^{92} +8.02460e10 q^{93} +(-9.36854e9 - 2.88334e10i) q^{94} +(-4.20433e10 + 7.22252e10i) q^{95} +(6.00649e9 - 1.84861e10i) q^{96} +(-2.19358e10 + 6.75114e10i) q^{97} +(5.11894e10 - 3.71913e10i) q^{98} -5.33870e10 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
\(3\) 179.007 − 550.928i 0.425308 − 1.30896i −0.477391 − 0.878691i \(-0.658417\pi\)
0.902699 − 0.430273i \(-0.141583\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) −4657.15 − 5209.52i −0.666476 − 0.745526i
\(6\) 5728.23 + 17629.7i 0.300738 + 0.925577i
\(7\) −160.425 −0.00360772 −0.00180386 − 0.999998i \(-0.500574\pi\)
−0.00180386 + 0.999998i \(0.500574\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) −128163. − 93115.7i −0.723482 − 0.525641i
\(10\) 218553. + 47270.1i 0.691126 + 0.149481i
\(11\) 272639. − 198084.i 0.510421 − 0.370843i −0.302562 − 0.953130i \(-0.597842\pi\)
0.812983 + 0.582287i \(0.197842\pi\)
\(12\) −479895. − 348664.i −0.556736 − 0.404492i
\(13\) −1.89998e6 − 1.38042e6i −1.41926 − 1.03115i −0.991893 − 0.127073i \(-0.959442\pi\)
−0.427365 − 0.904079i \(-0.640558\pi\)
\(14\) 4153.17 − 3017.46i 0.00206384 − 0.00149947i
\(15\) −3.70373e6 + 1.63321e6i −1.25932 + 0.555315i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) −747912. − 2.30184e6i −0.127756 − 0.393193i 0.866637 − 0.498939i \(-0.166277\pi\)
−0.994393 + 0.105746i \(0.966277\pi\)
\(18\) 5.06937e6 0.632347
\(19\) −3.69575e6 − 1.13744e7i −0.342419 − 1.05386i −0.962951 − 0.269677i \(-0.913083\pi\)
0.620532 − 0.784181i \(-0.286917\pi\)
\(20\) −6.54714e6 + 2.88704e6i −0.457495 + 0.201738i
\(21\) −28717.3 + 88382.7i −0.00153439 + 0.00472238i
\(22\) −3.33245e6 + 1.02562e7i −0.137860 + 0.424290i
\(23\) −1.05028e7 + 7.63075e6i −0.340254 + 0.247209i −0.744769 − 0.667322i \(-0.767440\pi\)
0.404515 + 0.914531i \(0.367440\pi\)
\(24\) 1.89818e7 0.486605
\(25\) −5.45012e6 + 4.85230e7i −0.111618 + 0.993751i
\(26\) 7.51523e7 1.24048
\(27\) 8.77735e6 − 6.37712e6i 0.117723 − 0.0855311i
\(28\) −50763.9 + 156235.i −0.000557424 + 0.00171557i
\(29\) 3.08446e7 − 9.49299e7i 0.279248 − 0.859437i −0.708816 − 0.705393i \(-0.750770\pi\)
0.988064 − 0.154043i \(-0.0492296\pi\)
\(30\) 6.51650e7 − 1.11945e8i 0.489607 − 0.841084i
\(31\) 4.28073e7 + 1.31747e8i 0.268552 + 0.826518i 0.990854 + 0.134940i \(0.0430841\pi\)
−0.722302 + 0.691578i \(0.756916\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −6.03256e7 − 1.85663e8i −0.268334 − 0.825846i
\(34\) 6.26579e7 + 4.55236e7i 0.236506 + 0.171832i
\(35\) 747123. + 835739.i 0.00240446 + 0.00268965i
\(36\) −1.31239e8 + 9.53505e7i −0.361741 + 0.262820i
\(37\) 5.31370e8 + 3.86063e8i 1.25976 + 0.915269i 0.998746 − 0.0500657i \(-0.0159431\pi\)
0.261014 + 0.965335i \(0.415943\pi\)
\(38\) 3.09619e8 + 2.24952e8i 0.633897 + 0.460553i
\(39\) −1.10062e9 + 7.99649e8i −1.95336 + 1.41920i
\(40\) 1.15193e8 − 1.97887e8i 0.177867 − 0.305554i
\(41\) 2.01450e8 + 1.46362e8i 0.271553 + 0.197295i 0.715225 − 0.698894i \(-0.246324\pi\)
−0.443671 + 0.896190i \(0.646324\pi\)
\(42\) −918953. − 2.82825e6i −0.00108498 − 0.00333923i
\(43\) −1.29048e9 −1.33868 −0.669338 − 0.742958i \(-0.733422\pi\)
−0.669338 + 0.742958i \(0.733422\pi\)
\(44\) −1.06638e8 − 3.28199e8i −0.0974818 − 0.300018i
\(45\) 1.11784e8 + 1.10132e9i 0.0903050 + 0.889702i
\(46\) 1.28375e8 − 3.95098e8i 0.0918994 − 0.282837i
\(47\) −2.92767e8 + 9.01044e8i −0.186202 + 0.573070i −0.999967 − 0.00812375i \(-0.997414\pi\)
0.813765 + 0.581194i \(0.197414\pi\)
\(48\) −4.91412e8 + 3.57032e8i −0.278368 + 0.202246i
\(49\) −1.97730e9 −0.999987
\(50\) −7.71580e8 − 1.35870e9i −0.349177 − 0.614878i
\(51\) −1.40203e9 −0.569011
\(52\) −1.94558e9 + 1.41355e9i −0.709629 + 0.515576i
\(53\) −1.14564e9 + 3.52591e9i −0.376297 + 1.15812i 0.566303 + 0.824197i \(0.308373\pi\)
−0.942600 + 0.333925i \(0.891627\pi\)
\(54\) −1.07285e8 + 3.30189e8i −0.0317960 + 0.0978581i
\(55\) −2.30165e9 − 4.97815e8i −0.616657 − 0.133375i
\(56\) −1.62444e6 − 4.99953e6i −0.000394158 − 0.00121309i
\(57\) −6.92802e9 −1.52510
\(58\) 9.87027e8 + 3.03776e9i 0.197458 + 0.607714i
\(59\) −2.10577e9 − 1.52993e9i −0.383464 − 0.278603i 0.379308 − 0.925271i \(-0.376162\pi\)
−0.762772 + 0.646668i \(0.776162\pi\)
\(60\) 4.18567e8 + 4.12380e9i 0.0694917 + 0.684645i
\(61\) 9.01536e9 − 6.55004e9i 1.36669 − 0.992956i 0.368699 − 0.929549i \(-0.379803\pi\)
0.997988 − 0.0634071i \(-0.0201967\pi\)
\(62\) −3.58627e9 − 2.60558e9i −0.497151 − 0.361201i
\(63\) 2.05605e7 + 1.49381e7i 0.00261013 + 0.00189637i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) 1.65718e9 + 1.63268e10i 0.177152 + 1.74533i
\(66\) 5.05390e9 + 3.67188e9i 0.496747 + 0.360908i
\(67\) 5.54077e9 + 1.70527e10i 0.501370 + 1.54306i 0.806788 + 0.590841i \(0.201204\pi\)
−0.305418 + 0.952218i \(0.598796\pi\)
\(68\) −2.47838e9 −0.206714
\(69\) 2.32391e9 + 7.15226e9i 0.178875 + 0.550520i
\(70\) −3.50615e7 − 7.58332e6i −0.00249339 − 0.000539287i
\(71\) −1.21556e9 + 3.74112e9i −0.0799571 + 0.246083i −0.983042 − 0.183379i \(-0.941297\pi\)
0.903085 + 0.429461i \(0.141297\pi\)
\(72\) 1.60412e9 − 4.93697e9i 0.0977030 − 0.300699i
\(73\) 1.64598e10 − 1.19588e10i 0.929286 − 0.675166i −0.0165317 − 0.999863i \(-0.505262\pi\)
0.945818 + 0.324697i \(0.105262\pi\)
\(74\) −2.10179e10 −1.10107
\(75\) 2.57571e10 + 1.16886e10i 1.25331 + 0.568755i
\(76\) −1.22467e10 −0.554046
\(77\) −4.37382e7 + 3.17777e7i −0.00184146 + 0.00133790i
\(78\) 1.34528e10 − 4.14035e10i 0.527585 − 1.62374i
\(79\) 2.46214e9 − 7.57769e9i 0.0900251 − 0.277069i −0.895900 − 0.444256i \(-0.853468\pi\)
0.985925 + 0.167187i \(0.0534682\pi\)
\(80\) 7.39906e8 + 7.28969e9i 0.0252454 + 0.248722i
\(81\) −1.06141e10 − 3.26670e10i −0.338234 − 1.04098i
\(82\) −7.96818e9 −0.237346
\(83\) 1.24958e9 + 3.84580e9i 0.0348204 + 0.107166i 0.966956 − 0.254943i \(-0.0820567\pi\)
−0.932136 + 0.362109i \(0.882057\pi\)
\(84\) 7.69872e7 + 5.59345e7i 0.00200855 + 0.00145930i
\(85\) −8.50833e9 + 1.46163e10i −0.207989 + 0.357299i
\(86\) 3.34087e10 − 2.42728e10i 0.765805 − 0.556390i
\(87\) −4.67781e10 − 3.39863e10i −1.00621 − 0.731051i
\(88\) 8.93385e9 + 6.49082e9i 0.180461 + 0.131113i
\(89\) −2.58301e10 + 1.87667e10i −0.490322 + 0.356240i −0.805308 − 0.592856i \(-0.798000\pi\)
0.314986 + 0.949096i \(0.398000\pi\)
\(90\) −2.36088e10 − 2.64090e10i −0.421444 − 0.471431i
\(91\) 3.04805e8 + 2.21454e8i 0.00512029 + 0.00372011i
\(92\) 4.10800e9 + 1.26431e10i 0.0649827 + 0.199996i
\(93\) 8.02460e10 1.19610
\(94\) −9.36854e9 − 2.88334e10i −0.131664 − 0.405222i
\(95\) −4.20433e10 + 7.22252e10i −0.557464 + 0.957654i
\(96\) 6.00649e9 − 1.84861e10i 0.0751846 − 0.231394i
\(97\) −2.19358e10 + 6.75114e10i −0.259363 + 0.798238i 0.733575 + 0.679608i \(0.237850\pi\)
−0.992939 + 0.118630i \(0.962150\pi\)
\(98\) 5.11894e10 − 3.71913e10i 0.572054 − 0.415622i
\(99\) −5.33870e10 −0.564211
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.11.12 ✓ 56
25.16 even 5 inner 50.12.d.b.41.12 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.12 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.12 yes 56 25.16 even 5 inner