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Properties

Label 50.12.d.b.11.14
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 11.14
Character \(\chi\) \(=\) 50.11
Dual form 50.12.d.b.41.14

$q$-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} +(228.024 - 701.784i) q^{3} +(316.433 - 973.882i) q^{4} +(-4899.58 + 4982.19i) q^{5} +(7296.75 + 22457.1i) q^{6} +8243.01 q^{7} +(10125.9 + 31164.2i) q^{8} +(-297192. - 215922. i) q^{9} +(33132.4 - 221139. i) q^{10} +(227188. - 165062. i) q^{11} +(-611301. - 444136. i) q^{12} +(2.08448e6 + 1.51447e6i) q^{13} +(-213400. + 155044. i) q^{14} +(2.37920e6 + 4.57451e6i) q^{15} +(-848316. - 616338. i) q^{16} +(2.85275e6 + 8.77985e6i) q^{17} +1.17552e7 q^{18} +(-1.93173e6 - 5.94524e6i) q^{19} +(3.30167e6 + 6.34815e6i) q^{20} +(1.87960e6 - 5.78482e6i) q^{21} +(-2.77690e6 + 8.54641e6i) q^{22} +(2.51614e7 - 1.82808e7i) q^{23} +2.41795e7 q^{24} +(-816328. - 4.88213e7i) q^{25} -8.24500e7 q^{26} +(-1.13546e8 + 8.24956e7i) q^{27} +(2.60836e6 - 8.02772e6i) q^{28} +(-3.49733e6 + 1.07637e7i) q^{29} +(-1.47637e8 - 7.36766e7i) q^{30} +(1.96948e7 + 6.06144e7i) q^{31} +3.35544e7 q^{32} +(-6.40335e7 - 1.97075e8i) q^{33} +(-2.38995e8 - 1.73640e8i) q^{34} +(-4.03873e7 + 4.10683e7i) q^{35} +(-3.04324e8 + 2.21104e8i) q^{36} +(4.96228e7 + 3.60531e7i) q^{37} +(1.61834e8 + 1.17580e8i) q^{38} +(1.53814e9 - 1.11752e9i) q^{39} +(-2.04879e8 - 1.02243e8i) q^{40} +(-6.98883e7 - 5.07769e7i) q^{41} +(6.01472e7 + 1.85114e8i) q^{42} -1.23150e8 q^{43} +(-8.88607e7 - 2.73485e8i) q^{44} +(2.53188e9 - 4.22736e8i) q^{45} +(-3.07545e8 + 9.46527e8i) q^{46} +(3.73912e8 - 1.15078e9i) q^{47} +(-6.25972e8 + 4.54795e8i) q^{48} -1.90938e9 q^{49} +(9.39420e8 + 1.24856e9i) q^{50} +6.81206e9 q^{51} +(2.13451e9 - 1.55081e9i) q^{52} +(1.21694e9 - 3.74534e9i) q^{53} +(1.38786e9 - 4.27138e9i) q^{54} +(-2.90757e8 + 1.94063e9i) q^{55} +(8.34677e7 + 2.56887e8i) q^{56} -4.61276e9 q^{57} +(-1.11915e8 - 3.44438e8i) q^{58} +(6.96456e9 + 5.06005e9i) q^{59} +(5.20789e9 - 8.69537e8i) q^{60} +(5.70551e9 - 4.14529e9i) q^{61} +(-1.64997e9 - 1.19878e9i) q^{62} +(-2.44975e9 - 1.77985e9i) q^{63} +(-8.68675e8 + 6.31130e8i) q^{64} +(-1.77584e10 + 2.96504e9i) q^{65} +(5.36454e9 + 3.89756e9i) q^{66} +(-4.82473e9 - 1.48490e10i) q^{67} +9.45324e9 q^{68} +(-7.09180e9 - 2.18263e10i) q^{69} +(2.73110e8 - 1.82285e9i) q^{70} +(-2.17557e9 + 6.69571e9i) q^{71} +(3.71973e9 - 1.14481e10i) q^{72} +(1.36569e10 - 9.92230e9i) q^{73} -1.96279e9 q^{74} +(-3.44482e10 - 1.05595e10i) q^{75} -6.40122e9 q^{76} +(1.87271e9 - 1.36060e9i) q^{77} +(-1.88005e10 + 5.78621e10i) q^{78} +(4.59022e9 - 1.41272e10i) q^{79} +(7.22710e9 - 1.20668e9i) q^{80} +(1.18939e10 + 3.66056e10i) q^{81} +2.76438e9 q^{82} +(2.18840e10 + 6.73521e10i) q^{83} +(-5.03896e9 - 3.66102e9i) q^{84} +(-5.77202e10 - 2.88047e10i) q^{85} +(3.18818e9 - 2.31635e9i) q^{86} +(6.75631e9 + 4.90874e9i) q^{87} +(7.44449e9 + 5.40874e9i) q^{88} +(7.69914e10 - 5.59375e10i) q^{89} +(-5.75954e10 + 5.85665e10i) q^{90} +(1.71824e10 + 1.24838e10i) q^{91} +(-9.84145e9 - 3.02889e10i) q^{92} +4.70291e10 q^{93} +(1.19652e10 + 3.68250e10i) q^{94} +(3.90850e10 + 1.95050e10i) q^{95} +(7.65120e9 - 2.35480e10i) q^{96} +(2.89985e10 - 8.92484e10i) q^{97} +(4.94311e10 - 3.59138e10i) q^{98} -1.03159e11 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
\(3\) 228.024 − 701.784i 0.541767 − 1.66739i −0.186787 − 0.982400i \(-0.559807\pi\)
0.728554 − 0.684988i \(-0.240193\pi\)
\(4\) 316.433 − 973.882i 0.154508 − 0.475528i
\(5\) −4899.58 + 4982.19i −0.701171 + 0.712993i
\(6\) 7296.75 + 22457.1i 0.383087 + 1.17902i
\(7\) 8243.01 0.185373 0.0926866 − 0.995695i \(-0.470455\pi\)
0.0926866 + 0.995695i \(0.470455\pi\)
\(8\) 10125.9 + 31164.2i 0.109254 + 0.336249i
\(9\) −297192. − 215922.i −1.67766 − 1.21889i
\(10\) 33132.4 − 221139.i 0.104774 − 0.699301i
\(11\) 227188. − 165062.i 0.425329 − 0.309020i −0.354449 − 0.935075i \(-0.615332\pi\)
0.779778 + 0.626056i \(0.215332\pi\)
\(12\) −611301. − 444136.i −0.709183 − 0.515251i
\(13\) 2.08448e6 + 1.51447e6i 1.55708 + 1.13128i 0.938356 + 0.345670i \(0.112348\pi\)
0.618720 + 0.785612i \(0.287652\pi\)
\(14\) −213400. + 155044.i −0.106045 + 0.0770461i
\(15\) 2.37920e6 + 4.57451e6i 0.808965 + 1.55540i
\(16\) −848316. − 616338.i −0.202254 − 0.146946i
\(17\) 2.85275e6 + 8.77985e6i 0.487298 + 1.49975i 0.828625 + 0.559803i \(0.189123\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(18\) 1.17552e7 1.46632
\(19\) −1.93173e6 − 5.94524e6i −0.178978 − 0.550839i 0.820814 − 0.571195i \(-0.193520\pi\)
−0.999793 + 0.0203561i \(0.993520\pi\)
\(20\) 3.30167e6 + 6.34815e6i 0.230711 + 0.443590i
\(21\) 1.87960e6 − 5.78482e6i 0.100429 − 0.309089i
\(22\) −2.77690e6 + 8.54641e6i −0.114877 + 0.353556i
\(23\) 2.51614e7 − 1.82808e7i 0.815138 − 0.592233i −0.100177 − 0.994970i \(-0.531941\pi\)
0.915316 + 0.402737i \(0.131941\pi\)
\(24\) 2.41795e7 0.619848
\(25\) −816328. − 4.88213e7i −0.0167184 − 0.999860i
\(26\) −8.24500e7 −1.36093
\(27\) −1.13546e8 + 8.24956e7i −1.52289 + 1.10645i
\(28\) 2.60836e6 − 8.02772e6i 0.0286417 − 0.0881501i
\(29\) −3.49733e6 + 1.07637e7i −0.0316627 + 0.0974477i −0.965639 − 0.259887i \(-0.916315\pi\)
0.933976 + 0.357335i \(0.116315\pi\)
\(30\) −1.47637e8 − 7.36766e7i −1.10924 − 0.553557i
\(31\) 1.96948e7 + 6.06144e7i 0.123556 + 0.380265i 0.993635 − 0.112646i \(-0.0359327\pi\)
−0.870079 + 0.492911i \(0.835933\pi\)
\(32\) 3.35544e7 0.176777
\(33\) −6.40335e7 − 1.97075e8i −0.284826 − 0.876606i
\(34\) −2.38995e8 − 1.73640e8i −0.902100 − 0.655414i
\(35\) −4.03873e7 + 4.10683e7i −0.129978 + 0.132170i
\(36\) −3.04324e8 + 2.21104e8i −0.838828 + 0.609444i
\(37\) 4.96228e7 + 3.60531e7i 0.117645 + 0.0854738i 0.645052 − 0.764139i \(-0.276836\pi\)
−0.527407 + 0.849613i \(0.676836\pi\)
\(38\) 1.61834e8 + 1.17580e8i 0.331330 + 0.240725i
\(39\) 1.53814e9 − 1.11752e9i 2.72986 − 1.98336i
\(40\) −2.04879e8 − 1.02243e8i −0.316349 − 0.157871i
\(41\) −6.98883e7 − 5.07769e7i −0.0942092 − 0.0684470i 0.539683 − 0.841868i \(-0.318544\pi\)
−0.633893 + 0.773421i \(0.718544\pi\)
\(42\) 6.01472e7 + 1.85114e8i 0.0710141 + 0.218559i
\(43\) −1.23150e8 −0.127749 −0.0638747 − 0.997958i \(-0.520346\pi\)
−0.0638747 + 0.997958i \(0.520346\pi\)
\(44\) −8.88607e7 − 2.73485e8i −0.0812306 − 0.250002i
\(45\) 2.53188e9 − 4.22736e8i 2.04538 − 0.341508i
\(46\) −3.07545e8 + 9.46527e8i −0.220161 + 0.677587i
\(47\) 3.73912e8 − 1.15078e9i 0.237811 − 0.731906i −0.758926 − 0.651177i \(-0.774275\pi\)
0.996736 − 0.0807283i \(-0.0257246\pi\)
\(48\) −6.25972e8 + 4.54795e8i −0.354591 + 0.257626i
\(49\) −1.90938e9 −0.965637
\(50\) 9.39420e8 + 1.24856e9i 0.425133 + 0.565033i
\(51\) 6.81206e9 2.76466
\(52\) 2.13451e9 − 1.55081e9i 0.778538 − 0.565641i
\(53\) 1.21694e9 − 3.74534e9i 0.399715 − 1.23020i −0.525514 − 0.850785i \(-0.676127\pi\)
0.925228 − 0.379410i \(-0.123873\pi\)
\(54\) 1.38786e9 − 4.27138e9i 0.411319 − 1.26591i
\(55\) −2.90757e8 + 1.94063e9i −0.0778995 + 0.519932i
\(56\) 8.34677e7 + 2.56887e8i 0.0202528 + 0.0623316i
\(57\) −4.61276e9 −1.01543
\(58\) −1.11915e8 − 3.44438e8i −0.0223889 − 0.0689059i
\(59\) 6.96456e9 + 5.06005e9i 1.26826 + 0.921443i 0.999132 − 0.0416618i \(-0.0132652\pi\)
0.269126 + 0.963105i \(0.413265\pi\)
\(60\) 5.20789e9 − 8.69537e8i 0.864629 − 0.144363i
\(61\) 5.70551e9 − 4.14529e9i 0.864928 − 0.628407i −0.0642931 − 0.997931i \(-0.520479\pi\)
0.929221 + 0.369524i \(0.120479\pi\)
\(62\) −1.64997e9 − 1.19878e9i −0.228730 − 0.166182i
\(63\) −2.44975e9 − 1.77985e9i −0.310992 − 0.225949i
\(64\) −8.68675e8 + 6.31130e8i −0.101127 + 0.0734732i
\(65\) −1.77584e10 + 2.96504e9i −1.89837 + 0.316962i
\(66\) 5.36454e9 + 3.89756e9i 0.527279 + 0.383091i
\(67\) −4.82473e9 − 1.48490e10i −0.436577 − 1.34365i −0.891462 − 0.453096i \(-0.850319\pi\)
0.454884 − 0.890551i \(-0.349681\pi\)
\(68\) 9.45324e9 0.788464
\(69\) −7.09180e9 − 2.18263e10i −0.545867 − 1.68000i
\(70\) 2.73110e8 − 1.82285e9i 0.0194222 − 0.129632i
\(71\) −2.17557e9 + 6.69571e9i −0.143104 + 0.440429i −0.996762 − 0.0804046i \(-0.974379\pi\)
0.853658 + 0.520833i \(0.174379\pi\)
\(72\) 3.71973e9 − 1.14481e10i 0.226560 − 0.697279i
\(73\) 1.36569e10 − 9.92230e9i 0.771038 − 0.560192i −0.131238 − 0.991351i \(-0.541895\pi\)
0.902276 + 0.431159i \(0.141895\pi\)
\(74\) −1.96279e9 −0.102825
\(75\) −3.44482e10 − 1.05595e10i −1.67621 − 0.513816i
\(76\) −6.40122e9 −0.289593
\(77\) 1.87271e9 − 1.36060e9i 0.0788446 − 0.0572839i
\(78\) −1.88005e10 + 5.78621e10i −0.737310 + 2.26921i
\(79\) 4.59022e9 − 1.41272e10i 0.167836 − 0.516545i −0.831398 − 0.555677i \(-0.812459\pi\)
0.999234 + 0.0391318i \(0.0124592\pi\)
\(80\) 7.22710e9 − 1.20668e9i 0.246587 − 0.0411714i
\(81\) 1.18939e10 + 3.66056e10i 0.379015 + 1.16649i
\(82\) 2.76438e9 0.0823419
\(83\) 2.18840e10 + 6.73521e10i 0.609814 + 1.87681i 0.459503 + 0.888176i \(0.348027\pi\)
0.150311 + 0.988639i \(0.451973\pi\)
\(84\) −5.03896e9 − 3.66102e9i −0.131463 − 0.0955137i
\(85\) −5.77202e10 − 2.88047e10i −1.41099 − 0.704140i
\(86\) 3.18818e9 − 2.31635e9i 0.0730805 − 0.0530961i
\(87\) 6.75631e9 + 4.90874e9i 0.145329 + 0.105588i
\(88\) 7.44449e9 + 5.40874e9i 0.150377 + 0.109255i
\(89\) 7.69914e10 − 5.59375e10i 1.46149 − 1.06184i 0.478523 − 0.878075i \(-0.341172\pi\)
0.982971 − 0.183763i \(-0.0588278\pi\)
\(90\) −5.75954e10 + 5.85665e10i −1.02814 + 1.04548i
\(91\) 1.71824e10 + 1.24838e10i 0.288640 + 0.209709i
\(92\) −9.84145e9 − 3.02889e10i −0.155678 − 0.479126i
\(93\) 4.70291e10 0.700988
\(94\) 1.19652e10 + 3.68250e10i 0.168157 + 0.517535i
\(95\) 3.90850e10 + 1.95050e10i 0.518239 + 0.258622i
\(96\) 7.65120e9 − 2.35480e10i 0.0957718 − 0.294755i
\(97\) 2.89985e10 − 8.92484e10i 0.342872 − 1.05525i −0.619842 − 0.784727i \(-0.712803\pi\)
0.962713 − 0.270524i \(-0.0871969\pi\)
\(98\) 4.94311e10 − 3.59138e10i 0.552404 − 0.401345i
\(99\) −1.03159e11 −1.09022
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.11.14 ✓ 56
25.16 even 5 inner 50.12.d.b.41.14 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.11.14 ✓ 56 1.1 even 1 trivial
50.12.d.b.41.14 yes 56 25.16 even 5 inner