↑

Properties

Label 50.12.d.b.11.7
Level $50$
Weight $12$
Character 50.11
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
Inner twists $2$

Related objects

Downloads

Learn more

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 11.7
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.7

$q$-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} +(-18.4264 + 56.7107i) q^{3} +(316.433 - 973.882i) q^{4} +(-6941.67 - 800.879i) q^{5} +(-589.645 - 1814.74i) q^{6} -8223.94 q^{7} +(10125.9 + 31164.2i) q^{8} +(140438. + 102034. i) q^{9} +(194773. - 109833. i) q^{10} +(167387. - 121614. i) q^{11} +(49398.8 + 35890.3i) q^{12} +(402164. + 292189. i) q^{13} +(212906. - 154685. i) q^{14} +(173328. - 378909. i) q^{15} +(-848316. - 616338. i) q^{16} +(-673563. - 2.07301e6i) q^{17} -5.55492e6 q^{18} +(2.89312e6 + 8.90411e6i) q^{19} +(-2.97654e6 + 6.50694e6i) q^{20} +(151538. - 466385. i) q^{21} +(-2.04596e6 + 6.29681e6i) q^{22} +(4.01604e6 - 2.91783e6i) q^{23} -1.95393e6 q^{24} +(4.75453e7 + 1.11189e7i) q^{25} -1.59072e7 q^{26} +(-1.69200e7 + 1.22931e7i) q^{27} +(-2.60233e6 + 8.00914e6i) q^{28} +(-3.01774e7 + 9.28766e7i) q^{29} +(2.63973e6 + 1.30696e7i) q^{30} +(7.09198e6 + 2.18269e7i) q^{31} +3.35544e7 q^{32} +(3.81246e6 + 1.17335e7i) q^{33} +(5.64292e7 + 4.09982e7i) q^{34} +(5.70878e7 + 6.58638e6i) q^{35} +(1.43809e8 - 1.04483e8i) q^{36} +(-3.36684e8 - 2.44615e8i) q^{37} +(-2.42377e8 - 1.76097e8i) q^{38} +(-2.39807e7 + 1.74230e7i) q^{39} +(-4.53316e7 - 2.24441e8i) q^{40} +(-5.28002e8 - 3.83616e8i) q^{41} +(4.84920e6 + 1.49243e7i) q^{42} -6.87682e8 q^{43} +(-6.54707e7 - 2.01498e8i) q^{44} +(-8.93159e8 - 8.20763e8i) q^{45} +(-4.90877e7 + 1.51076e8i) q^{46} +(1.03585e8 - 3.18802e8i) q^{47} +(5.05843e7 - 3.67517e7i) q^{48} -1.90969e9 q^{49} +(-1.44002e9 + 6.06434e8i) q^{50} +1.29973e8 q^{51} +(4.11816e8 - 2.99201e8i) q^{52} +(9.13125e8 - 2.81031e9i) q^{53} +(2.06811e8 - 6.36499e8i) q^{54} +(-1.25934e9 + 7.10146e8i) q^{55} +(-8.32745e7 - 2.56293e8i) q^{56} -5.58268e8 q^{57} +(-9.65678e8 - 2.97205e9i) q^{58} +(-6.55405e9 - 4.76180e9i) q^{59} +(-3.14166e8 - 2.88701e8i) q^{60} +(-3.86466e9 + 2.80784e9i) q^{61} +(-5.94145e8 - 4.31672e8i) q^{62} +(-1.15496e9 - 8.39125e8i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-2.55768e9 - 2.35036e9i) q^{65} +(-3.19397e8 - 2.32055e8i) q^{66} +(-3.44754e9 - 1.06104e10i) q^{67} -2.23201e9 q^{68} +(9.14706e7 + 2.81517e8i) q^{69} +(-1.60180e9 + 9.03260e8i) q^{70} +(5.23011e8 - 1.60966e9i) q^{71} +(-1.75776e9 + 5.40984e9i) q^{72} +(-4.47365e9 + 3.25030e9i) q^{73} +1.33172e10 q^{74} +(-1.50665e9 + 2.49144e9i) q^{75} +9.58703e9 q^{76} +(-1.37658e9 + 1.00015e9i) q^{77} +(2.93113e8 - 9.02111e8i) q^{78} +(-7.34981e9 + 2.26204e10i) q^{79} +(5.39511e9 + 4.95781e9i) q^{80} +(9.11727e9 + 2.80601e10i) q^{81} +2.08847e10 q^{82} +(-2.00694e9 - 6.17673e9i) q^{83} +(-4.06252e8 - 2.95159e8i) q^{84} +(3.01542e9 + 1.49296e10i) q^{85} +(1.78031e10 - 1.29347e10i) q^{86} +(-4.71103e9 - 3.42276e9i) q^{87} +(5.48494e9 + 3.98504e9i) q^{88} +(-3.35914e10 + 2.44056e10i) q^{89} +(3.85604e10 + 4.44882e9i) q^{90} +(-3.30737e9 - 2.40294e9i) q^{91} +(-1.57081e9 - 4.83445e9i) q^{92} -1.36850e9 q^{93} +(3.31472e9 + 1.02017e10i) q^{94} +(-1.29520e10 - 6.41264e10i) q^{95} +(-6.18288e8 + 1.90289e9i) q^{96} +(-1.19297e10 + 3.67160e10i) q^{97} +(4.94392e10 - 3.59197e10i) q^{98} +3.59164e10 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
\(3\) −18.4264 + 56.7107i −0.0437798 + 0.134740i −0.970557 − 0.240871i \(-0.922567\pi\)
0.926777 + 0.375611i \(0.122567\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) −6941.67 − 800.879i −0.993410 − 0.114612i
\(6\) −589.645 − 1814.74i −0.0309570 − 0.0952758i
\(7\) −8223.94 −0.184944 −0.0924721 − 0.995715i \(-0.529477\pi\)
−0.0924721 + 0.995715i \(0.529477\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) 140438. + 102034.i 0.792779 + 0.575987i
\(10\) 194773. − 109833.i 0.615928 − 0.347323i
\(11\) 167387. − 121614.i 0.313374 − 0.227679i −0.419969 − 0.907538i \(-0.637959\pi\)
0.733343 + 0.679859i \(0.237959\pi\)
\(12\) 49398.8 + 35890.3i 0.0573085 + 0.0416371i
\(13\) 402164. + 292189.i 0.300410 + 0.218261i 0.727771 − 0.685821i \(-0.240557\pi\)
−0.427361 + 0.904081i \(0.640557\pi\)
\(14\) 212906. − 154685.i 0.105799 − 0.0768678i
\(15\) 173328. − 378909.i 0.0589342 − 0.128835i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) −673563. − 2.07301e6i −0.115056 − 0.354106i 0.876903 − 0.480668i \(-0.159606\pi\)
−0.991959 + 0.126562i \(0.959606\pi\)
\(18\) −5.55492e6 −0.692914
\(19\) 2.89312e6 + 8.90411e6i 0.268054 + 0.824984i 0.990974 + 0.134053i \(0.0427992\pi\)
−0.722921 + 0.690931i \(0.757201\pi\)
\(20\) −2.97654e6 + 6.50694e6i −0.207992 + 0.454686i
\(21\) 151538. − 466385.i 0.00809682 − 0.0249194i
\(22\) −2.04596e6 + 6.29681e6i −0.0846393 + 0.260493i
\(23\) 4.01604e6 − 2.91783e6i 0.130105 − 0.0945271i −0.520829 − 0.853661i \(-0.674377\pi\)
0.650935 + 0.759134i \(0.274377\pi\)
\(24\) −1.95393e6 −0.0500895
\(25\) 4.75453e7 + 1.11189e7i 0.973728 + 0.227714i
\(26\) −1.59072e7 −0.262568
\(27\) −1.69200e7 + 1.22931e7i −0.226933 + 0.164877i
\(28\) −2.60233e6 + 8.00914e6i −0.0285754 + 0.0879462i
\(29\) −3.01774e7 + 9.28766e7i −0.273208 + 0.840848i 0.716480 + 0.697608i \(0.245752\pi\)
−0.989688 + 0.143240i \(0.954248\pi\)
\(30\) 2.63973e6 + 1.30696e7i 0.0198332 + 0.0981961i
\(31\) 7.09198e6 + 2.18269e7i 0.0444916 + 0.136931i 0.970835 − 0.239750i \(-0.0770654\pi\)
−0.926343 + 0.376681i \(0.877065\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 3.81246e6 + 1.17335e7i 0.0169582 + 0.0521918i
\(34\) 5.64292e7 + 4.09982e7i 0.212995 + 0.154750i
\(35\) 5.70878e7 + 6.58638e6i 0.183725 + 0.0211969i
\(36\) 1.43809e8 − 1.04483e8i 0.396389 − 0.287994i
\(37\) −3.36684e8 − 2.44615e8i −0.798202 − 0.579928i 0.112184 − 0.993687i \(-0.464215\pi\)
−0.910386 + 0.413760i \(0.864215\pi\)
\(38\) −2.42377e8 − 1.76097e8i −0.496229 − 0.360531i
\(39\) −2.39807e7 + 1.74230e7i −0.0425604 + 0.0309219i
\(40\) −4.53316e7 − 2.24441e8i −0.0699957 − 0.346555i
\(41\) −5.28002e8 − 3.83616e8i −0.711744 − 0.517113i 0.171992 − 0.985098i \(-0.444980\pi\)
−0.883736 + 0.467986i \(0.844980\pi\)
\(42\) 4.84920e6 + 1.49243e7i 0.00572531 + 0.0176207i
\(43\) −6.87682e8 −0.713364 −0.356682 − 0.934226i \(-0.616092\pi\)
−0.356682 + 0.934226i \(0.616092\pi\)
\(44\) −6.54707e7 − 2.01498e8i −0.0598490 − 0.184196i
\(45\) −8.93159e8 − 8.20763e8i −0.721539 − 0.663054i
\(46\) −4.90877e7 + 1.51076e8i −0.0351402 + 0.108151i
\(47\) 1.03585e8 − 3.18802e8i 0.0658807 − 0.202760i −0.912697 − 0.408636i \(-0.866004\pi\)
0.978578 + 0.205876i \(0.0660044\pi\)
\(48\) 5.05843e7 − 3.67517e7i 0.0286543 − 0.0208185i
\(49\) −1.90969e9 −0.965796
\(50\) −1.44002e9 + 6.06434e8i −0.651676 + 0.274441i
\(51\) 1.29973e8 0.0527495
\(52\) 4.11816e8 − 2.99201e8i 0.150205 − 0.109130i
\(53\) 9.13125e8 − 2.81031e9i 0.299925 − 0.923075i −0.681597 − 0.731728i \(-0.738714\pi\)
0.981522 − 0.191347i \(-0.0612856\pi\)
\(54\) 2.06811e8 − 6.36499e8i 0.0612926 − 0.188639i
\(55\) −1.25934e9 + 7.10146e8i −0.337403 + 0.190262i
\(56\) −8.32745e7 − 2.56293e8i −0.0202059 − 0.0621873i
\(57\) −5.58268e8 −0.122894
\(58\) −9.65678e8 − 2.97205e9i −0.193187 − 0.594569i
\(59\) −6.55405e9 − 4.76180e9i −1.19350 − 0.867131i −0.199873 − 0.979822i \(-0.564053\pi\)
−0.993630 + 0.112691i \(0.964053\pi\)
\(60\) −3.14166e8 − 2.88701e8i −0.0521587 − 0.0479310i
\(61\) −3.86466e9 + 2.80784e9i −0.585865 + 0.425656i −0.840834 − 0.541294i \(-0.817935\pi\)
0.254969 + 0.966949i \(0.417935\pi\)
\(62\) −5.94145e8 − 4.31672e8i −0.0823642 − 0.0598411i
\(63\) −1.15496e9 − 8.39125e8i −0.146620 − 0.106525i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −2.55768e9 − 2.35036e9i −0.273415 − 0.251253i
\(66\) −3.19397e8 − 2.32055e8i −0.0313934 − 0.0228087i
\(67\) −3.44754e9 − 1.06104e10i −0.311959 − 0.960112i −0.976988 − 0.213294i \(-0.931581\pi\)
0.665029 − 0.746818i \(-0.268419\pi\)
\(68\) −2.23201e9 −0.186165
\(69\) 9.14706e7 + 2.81517e8i 0.00704063 + 0.0216688i
\(70\) −1.60180e9 + 9.03260e8i −0.113912 + 0.0642353i
\(71\) 5.23011e8 − 1.60966e9i 0.0344025 − 0.105880i −0.932381 − 0.361478i \(-0.882272\pi\)
0.966783 + 0.255598i \(0.0822723\pi\)
\(72\) −1.75776e9 + 5.40984e9i −0.107061 + 0.329500i
\(73\) −4.47365e9 + 3.25030e9i −0.252573 + 0.183505i −0.706866 − 0.707347i \(-0.749892\pi\)
0.454293 + 0.890852i \(0.349892\pi\)
\(74\) 1.33172e10 0.697654
\(75\) −1.50665e9 + 2.49144e9i −0.0733120 + 0.121231i
\(76\) 9.58703e9 0.433720
\(77\) −1.37658e9 + 1.00015e9i −0.0579566 + 0.0421079i
\(78\) 2.93113e8 − 9.02111e8i 0.0114952 − 0.0353785i
\(79\) −7.34981e9 + 2.26204e10i −0.268737 + 0.827087i 0.722072 + 0.691818i \(0.243190\pi\)
−0.990809 + 0.135269i \(0.956810\pi\)
\(80\) 5.39511e9 + 4.95781e9i 0.184080 + 0.169159i
\(81\) 9.11727e9 + 2.80601e10i 0.290534 + 0.894172i
\(82\) 2.08847e10 0.622087
\(83\) −2.00694e9 − 6.17673e9i −0.0559249 − 0.172119i 0.919192 − 0.393809i \(-0.128843\pi\)
−0.975117 + 0.221690i \(0.928843\pi\)
\(84\) −4.06252e8 − 2.95159e8i −0.0105989 − 0.00770053i
\(85\) 3.01542e9 + 1.49296e10i 0.0737129 + 0.364959i
\(86\) 1.78031e10 − 1.29347e10i 0.408088 − 0.296493i
\(87\) −4.71103e9 − 3.42276e9i −0.101335 − 0.0736243i
\(88\) 5.48494e9 + 3.98504e9i 0.110794 + 0.0804968i
\(89\) −3.35914e10 + 2.44056e10i −0.637652 + 0.463281i −0.859043 − 0.511904i \(-0.828940\pi\)
0.221391 + 0.975185i \(0.428940\pi\)
\(90\) 3.85604e10 + 4.44882e9i 0.688348 + 0.0794166i
\(91\) −3.30737e9 − 2.40294e9i −0.0555591 − 0.0403660i
\(92\) −1.57081e9 − 4.83445e9i −0.0248479 − 0.0764740i
\(93\) −1.36850e9 −0.0203980
\(94\) 3.31472e9 + 1.02017e10i 0.0465847 + 0.143373i
\(95\) −1.29520e10 − 6.41264e10i −0.171734 − 0.850270i
\(96\) −6.18288e8 + 1.90289e9i −0.00773925 + 0.0238190i
\(97\) −1.19297e10 + 3.67160e10i −0.141054 + 0.434121i −0.996483 − 0.0838008i \(-0.973294\pi\)
0.855428 + 0.517921i \(0.173294\pi\)
\(98\) 4.94392e10 − 3.59197e10i 0.552494 − 0.401411i
\(99\) 3.59164e10 0.379576
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.11.7 ✓ 56
25.16 even 5 inner 50.12.d.b.41.7 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.7 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.7 yes 56 25.16 even 5 inner