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Properties

Label 50.12.d.b.21.2
Level $50$
Weight $12$
Character 50.21
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 21.2
Character \(\chi\) \(=\) 50.21
Dual form 50.12.d.b.31.2

$q$-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+(9.88854 - 30.4338i) q^{2} +(-505.161 - 367.021i) q^{3} +(-828.433 - 601.892i) q^{4} +(6683.35 - 2039.85i) q^{5} +(-16165.1 + 11744.7i) q^{6} +20699.7 q^{7} +(-26509.9 + 19260.5i) q^{8} +(65741.6 + 202332. i) q^{9} +(4008.11 - 223571. i) q^{10} +(257517. - 792557. i) q^{11} +(197585. + 608104. i) q^{12} +(-583905. - 1.79707e6i) q^{13} +(204690. - 629972. i) q^{14} +(-4.12483e6 - 1.42247e6i) q^{15} +(324028. + 997255. i) q^{16} +(5.70275e6 - 4.14329e6i) q^{17} +6.80782e6 q^{18} +(1.54639e7 - 1.12352e7i) q^{19} +(-6.76448e6 - 2.33277e6i) q^{20} +(-1.04567e7 - 7.59723e6i) q^{21} +(-2.15741e7 - 1.56745e7i) q^{22} +(-4.58761e6 + 1.41192e7i) q^{23} +2.04608e7 q^{24} +(4.05061e7 - 2.72661e7i) q^{25} -6.04658e7 q^{26} +(6.86866e6 - 2.11396e7i) q^{27} +(-1.71483e7 - 1.24590e7i) q^{28} +(-1.17265e7 - 8.51978e6i) q^{29} +(-8.40799e7 + 1.11468e8i) q^{30} +(-1.76201e8 + 1.28017e8i) q^{31} +3.35544e7 q^{32} +(-4.20973e8 + 3.05854e8i) q^{33} +(-6.97042e7 - 2.14528e8i) q^{34} +(1.38343e8 - 4.22244e7i) q^{35} +(6.73194e7 - 2.07188e8i) q^{36} +(2.34229e7 + 7.20884e7i) q^{37} +(-1.89014e8 - 5.81724e8i) q^{38} +(-3.64597e8 + 1.12212e9i) q^{39} +(-1.37886e8 + 1.82801e8i) q^{40} +(-3.32362e8 - 1.02290e9i) q^{41} +(-3.34614e8 + 2.43111e8i) q^{42} +1.62758e9 q^{43} +(-6.90370e8 + 5.01583e8i) q^{44} +(8.52101e8 + 1.21815e9i) q^{45} +(3.84337e8 + 2.79237e8i) q^{46} +(1.08023e9 + 7.84831e8i) q^{47} +(2.02327e8 - 6.22699e8i) q^{48} -1.54885e9 q^{49} +(-4.29264e8 - 1.50238e9i) q^{50} -4.40148e9 q^{51} +(-5.97918e8 + 1.84020e9i) q^{52} +(1.35951e9 + 9.87739e8i) q^{53} +(-5.75436e8 - 4.18079e8i) q^{54} +(1.04379e8 - 5.82223e9i) q^{55} +(-5.48747e8 + 3.98688e8i) q^{56} -1.19353e10 q^{57} +(-3.75247e8 + 2.72633e8i) q^{58} +(2.17481e9 + 6.69336e9i) q^{59} +(2.56097e9 + 3.66113e9i) q^{60} +(-4.80838e8 + 1.47987e9i) q^{61} +(2.15369e9 + 6.62837e9i) q^{62} +(1.36083e9 + 4.18822e9i) q^{63} +(3.31804e8 - 1.02119e9i) q^{64} +(-7.56820e9 - 1.08194e10i) q^{65} +(5.14551e9 + 1.58363e10i) q^{66} +(-1.82625e9 + 1.32685e9i) q^{67} -7.21817e9 q^{68} +(7.49953e9 - 5.44873e9i) q^{69} +(8.29668e7 - 4.62786e9i) q^{70} +(-1.22888e10 - 8.92833e9i) q^{71} +(-5.63983e9 - 4.09757e9i) q^{72} +(-2.58455e9 + 7.95444e9i) q^{73} +2.42554e9 q^{74} +(-3.04693e10 - 1.09284e9i) q^{75} -1.95732e10 q^{76} +(5.33054e9 - 1.64057e10i) q^{77} +(3.05449e10 + 2.21922e10i) q^{78} +(-7.61907e9 - 5.53558e9i) q^{79} +(4.19984e9 + 6.00403e9i) q^{80} +(1.92610e10 - 1.39939e10i) q^{81} -3.44174e10 q^{82} +(-5.41348e10 + 3.93312e10i) q^{83} +(4.08996e9 + 1.25876e10i) q^{84} +(2.96618e10 - 3.93238e10i) q^{85} +(1.60944e10 - 4.95334e10i) q^{86} +(2.79682e9 + 8.60772e9i) q^{87} +(8.43833e9 + 2.59705e10i) q^{88} +(1.37308e10 - 4.22589e10i) q^{89} +(4.54990e10 - 1.38869e10i) q^{90} +(-1.20867e10 - 3.71989e10i) q^{91} +(1.22988e10 - 8.93559e9i) q^{92} +1.35995e11 q^{93} +(3.45673e10 - 2.51146e10i) q^{94} +(8.04324e10 - 1.06633e11i) q^{95} +(-1.69504e10 - 1.23152e10i) q^{96} +(6.17829e10 + 4.48879e10i) q^{97} +(-1.53159e10 + 4.71373e10i) q^{98} +1.77289e11 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 \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.88854 − 30.4338i 0.218508 − 0.672499i
\(3\) −505.161 − 367.021i −1.20022 − 0.872014i −0.205918 − 0.978569i \(-0.566018\pi\)
−0.994307 + 0.106555i \(0.966018\pi\)
\(4\) −828.433 − 601.892i −0.404508 − 0.293893i
\(5\) 6683.35 − 2039.85i 0.956443 − 0.291920i
\(6\) −16165.1 + 11744.7i −0.848687 + 0.616607i
\(7\) 20699.7 0.465506 0.232753 − 0.972536i \(-0.425227\pi\)
0.232753 + 0.972536i \(0.425227\pi\)
\(8\) −26509.9 + 19260.5i −0.286031 + 0.207813i
\(9\) 65741.6 + 202332.i 0.371113 + 1.14217i
\(10\) 4008.11 − 223571.i 0.0126748 − 0.706993i
\(11\) 257517. − 792557.i 0.482111 − 1.48378i −0.354011 − 0.935241i \(-0.615183\pi\)
0.836122 − 0.548543i \(-0.184817\pi\)
\(12\) 197585. + 608104.i 0.229223 + 0.705474i
\(13\) −583905. − 1.79707e6i −0.436168 − 1.34239i −0.891885 − 0.452262i \(-0.850617\pi\)
0.455718 − 0.890124i \(-0.349383\pi\)
\(14\) 204690. − 629972.i 0.101717 − 0.313052i
\(15\) −4.12483e6 − 1.42247e6i −1.40250 − 0.483662i
\(16\) 324028. + 997255.i 0.0772542 + 0.237764i
\(17\) 5.70275e6 − 4.14329e6i 0.974127 − 0.707745i 0.0177384 − 0.999843i \(-0.494353\pi\)
0.956388 + 0.292098i \(0.0943534\pi\)
\(18\) 6.80782e6 0.849199
\(19\) 1.54639e7 − 1.12352e7i 1.43276 − 1.04096i 0.443267 − 0.896390i \(-0.353819\pi\)
0.989494 − 0.144573i \(-0.0461807\pi\)
\(20\) −6.76448e6 − 2.33277e6i −0.472682 − 0.163007i
\(21\) −1.04567e7 − 7.59723e6i −0.558712 − 0.405928i
\(22\) −2.15741e7 − 1.56745e7i −0.892498 − 0.648438i
\(23\) −4.58761e6 + 1.41192e7i −0.148622 + 0.457412i −0.997459 − 0.0712429i \(-0.977303\pi\)
0.848837 + 0.528655i \(0.177303\pi\)
\(24\) 2.04608e7 0.524517
\(25\) 4.05061e7 − 2.72661e7i 0.829566 − 0.558409i
\(26\) −6.04658e7 −0.998059
\(27\) 6.86866e6 − 2.11396e7i 0.0921237 − 0.283528i
\(28\) −1.71483e7 − 1.24590e7i −0.188301 − 0.136809i
\(29\) −1.17265e7 − 8.51978e6i −0.106164 − 0.0771329i 0.533437 − 0.845840i \(-0.320900\pi\)
−0.639601 + 0.768707i \(0.720900\pi\)
\(30\) −8.40799e7 + 1.11468e8i −0.631721 + 0.837498i
\(31\) −1.76201e8 + 1.28017e8i −1.10540 + 0.803119i −0.981933 − 0.189230i \(-0.939401\pi\)
−0.123465 + 0.992349i \(0.539401\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −4.20973e8 + 3.05854e8i −1.87252 + 1.36047i
\(34\) −6.97042e7 − 2.14528e8i −0.263103 − 0.809747i
\(35\) 1.38343e8 − 4.22244e7i 0.445230 − 0.135890i
\(36\) 6.73194e7 − 2.07188e8i 0.185557 − 0.571085i
\(37\) 2.34229e7 + 7.20884e7i 0.0555305 + 0.170905i 0.974975 − 0.222314i \(-0.0713611\pi\)
−0.919444 + 0.393220i \(0.871361\pi\)
\(38\) −1.89014e8 − 5.81724e8i −0.386976 − 1.19099i
\(39\) −3.64597e8 + 1.12212e9i −0.647081 + 1.99151i
\(40\) −1.37886e8 + 1.82801e8i −0.212907 + 0.282260i
\(41\) −3.32362e8 − 1.02290e9i −0.448022 − 1.37887i −0.879135 − 0.476573i \(-0.841879\pi\)
0.431113 − 0.902298i \(-0.358121\pi\)
\(42\) −3.34614e8 + 2.43111e8i −0.395069 + 0.287034i
\(43\) 1.62758e9 1.68836 0.844180 − 0.536060i \(-0.180088\pi\)
0.844180 + 0.536060i \(0.180088\pi\)
\(44\) −6.90370e8 + 5.01583e8i −0.631091 + 0.458515i
\(45\) 8.52101e8 + 1.21815e9i 0.688371 + 0.984085i
\(46\) 3.84337e8 + 2.79237e8i 0.275134 + 0.199896i
\(47\) 1.08023e9 + 7.84831e8i 0.687032 + 0.499158i 0.875683 − 0.482886i \(-0.160412\pi\)
−0.188651 + 0.982044i \(0.560412\pi\)
\(48\) 2.02327e8 − 6.22699e8i 0.114611 − 0.352737i
\(49\) −1.54885e9 −0.783304
\(50\) −4.29264e8 − 1.50238e9i −0.194263 − 0.679899i
\(51\) −4.40148e9 −1.78633
\(52\) −5.97918e8 + 1.84020e9i −0.218084 + 0.671193i
\(53\) 1.35951e9 + 9.87739e8i 0.446544 + 0.324433i 0.788230 − 0.615381i \(-0.210998\pi\)
−0.341686 + 0.939814i \(0.610998\pi\)
\(54\) −5.75436e8 − 4.18079e8i −0.170542 − 0.123906i
\(55\) 1.04379e8 − 5.82223e9i 0.0279653 − 1.55989i
\(56\) −5.48747e8 + 3.98688e8i −0.133149 + 0.0967384i
\(57\) −1.19353e10 −2.62737
\(58\) −3.75247e8 + 2.72633e8i −0.0750695 + 0.0545412i
\(59\) 2.17481e9 + 6.69336e9i 0.396036 + 1.21887i 0.928152 + 0.372201i \(0.121397\pi\)
−0.532116 + 0.846671i \(0.678603\pi\)
\(60\) 2.56097e9 + 3.66113e9i 0.425180 + 0.607831i
\(61\) −4.80838e8 + 1.47987e9i −0.0728929 + 0.224341i −0.980865 − 0.194690i \(-0.937630\pi\)
0.907972 + 0.419031i \(0.137630\pi\)
\(62\) 2.15369e9 + 6.62837e9i 0.298558 + 0.918866i
\(63\) 1.36083e9 + 4.18822e9i 0.172756 + 0.531687i
\(64\) 3.31804e8 − 1.02119e9i 0.0386271 − 0.118882i
\(65\) −7.56820e9 − 1.08194e10i −0.809038 − 1.15659i
\(66\) 5.14551e9 + 1.58363e10i 0.505751 + 1.55654i
\(67\) −1.82625e9 + 1.32685e9i −0.165253 + 0.120063i −0.667338 − 0.744755i \(-0.732566\pi\)
0.502085 + 0.864818i \(0.332566\pi\)
\(68\) −7.21817e9 −0.602043
\(69\) 7.49953e9 − 5.44873e9i 0.577250 − 0.419397i
\(70\) 8.29668e7 − 4.62786e9i 0.00590018 − 0.329110i
\(71\) −1.22888e10 − 8.92833e9i −0.808329 − 0.587286i 0.105016 − 0.994470i \(-0.466510\pi\)
−0.913346 + 0.407185i \(0.866510\pi\)
\(72\) −5.63983e9 − 4.09757e9i −0.343508 − 0.249573i
\(73\) −2.58455e9 + 7.95444e9i −0.145918 + 0.449090i −0.997128 − 0.0757355i \(-0.975870\pi\)
0.851210 + 0.524826i \(0.175870\pi\)
\(74\) 2.42554e9 0.127068
\(75\) −3.04693e10 − 1.09284e9i −1.48261 − 0.0531765i
\(76\) −1.95732e10 −0.885495
\(77\) 5.33054e9 − 1.64057e10i 0.224426 − 0.690711i
\(78\) 3.05449e10 + 2.21922e10i 1.19789 + 0.870321i
\(79\) −7.61907e9 − 5.53558e9i −0.278582 − 0.202402i 0.439717 − 0.898137i \(-0.355079\pi\)
−0.718299 + 0.695735i \(0.755079\pi\)
\(80\) 4.19984e9 + 6.00403e9i 0.143297 + 0.204856i
\(81\) 1.92610e10 − 1.39939e10i 0.613778 − 0.445936i
\(82\) −3.44174e10 −1.02519
\(83\) −5.41348e10 + 3.93312e10i −1.50851 + 1.09599i −0.541671 + 0.840591i \(0.682208\pi\)
−0.966835 + 0.255403i \(0.917792\pi\)
\(84\) 4.08996e9 + 1.25876e10i 0.106704 + 0.328403i
\(85\) 2.96618e10 − 3.93238e10i 0.725092 − 0.961284i
\(86\) 1.60944e10 − 4.95334e10i 0.368920 − 1.13542i
\(87\) 2.79682e9 + 8.60772e9i 0.0601600 + 0.185154i
\(88\) 8.43833e9 + 2.59705e10i 0.170452 + 0.524597i
\(89\) 1.37308e10 − 4.22589e10i 0.260645 − 0.802183i −0.732020 − 0.681283i \(-0.761422\pi\)
0.992665 − 0.120899i \(-0.0385779\pi\)
\(90\) 4.54990e10 − 1.38869e10i 0.812210 − 0.247898i
\(91\) −1.20867e10 − 3.71989e10i −0.203039 − 0.624889i
\(92\) 1.22988e10 − 8.93559e9i 0.194549 − 0.141348i
\(93\) 1.35995e11 2.02706
\(94\) 3.45673e10 − 2.51146e10i 0.485805 − 0.352958i
\(95\) 8.04324e10 − 1.06633e11i 1.06648 − 1.41387i
\(96\) −1.69504e10 − 1.23152e10i −0.212172 − 0.154152i
\(97\) 6.17829e10 + 4.48879e10i 0.730506 + 0.530744i 0.889724 − 0.456500i \(-0.150897\pi\)
−0.159217 + 0.987244i \(0.550897\pi\)
\(98\) −1.53159e10 + 4.71373e10i −0.171158 + 0.526771i
\(99\) 1.77289e11 1.87365
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.21.2 ✓ 56
25.6 even 5 inner 50.12.d.b.31.2 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.21.2 ✓ 56 1.1 even 1 trivial
50.12.d.b.31.2 yes 56 25.6 even 5 inner