Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4096,96764] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 315576783x^{2} - 792388522562x + 1396510961585520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-16161.7\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{2} +185808. q^{3} +1.04858e6 q^{4} +1.90268e8 q^{6} +4.88079e8 q^{7} +1.07374e9 q^{8} +2.40644e10 q^{9} -7.28578e10 q^{11} +1.94834e11 q^{12} +9.02768e11 q^{13} +4.99793e11 q^{14} +1.09951e12 q^{16} -6.32065e12 q^{17} +2.46419e13 q^{18} +4.36780e13 q^{19} +9.06891e13 q^{21} -7.46064e13 q^{22} -9.00682e13 q^{23} +1.99510e14 q^{24} +9.24434e14 q^{26} +2.52774e15 q^{27} +5.11788e14 q^{28} -1.34854e15 q^{29} -4.56363e14 q^{31} +1.12590e15 q^{32} -1.35376e16 q^{33} -6.47235e15 q^{34} +2.52333e16 q^{36} -2.84931e16 q^{37} +4.47263e16 q^{38} +1.67742e17 q^{39} -1.42955e17 q^{41} +9.28657e16 q^{42} +1.47393e17 q^{43} -7.63969e16 q^{44} -9.22299e16 q^{46} +1.71830e17 q^{47} +2.04298e17 q^{48} -3.20325e17 q^{49} -1.17443e18 q^{51} +9.46621e17 q^{52} +1.36482e18 q^{53} +2.58841e18 q^{54} +5.24071e17 q^{56} +8.11574e18 q^{57} -1.38090e18 q^{58} +6.60455e18 q^{59} +1.48171e18 q^{61} -4.67316e17 q^{62} +1.17453e19 q^{63} +1.15292e18 q^{64} -1.38625e19 q^{66} +2.80036e19 q^{67} -6.62769e18 q^{68} -1.67354e19 q^{69} -4.31882e19 q^{71} +2.58389e19 q^{72} +2.89648e17 q^{73} -2.91770e19 q^{74} +4.57997e19 q^{76} -3.55604e19 q^{77} +1.71768e20 q^{78} -5.22625e19 q^{79} +2.17954e20 q^{81} -1.46386e20 q^{82} -5.91809e19 q^{83} +9.50944e19 q^{84} +1.50931e20 q^{86} -2.50570e20 q^{87} -7.82305e19 q^{88} -4.79550e20 q^{89} +4.40622e20 q^{91} -9.44434e19 q^{92} -8.47961e19 q^{93} +1.75954e20 q^{94} +2.09202e20 q^{96} +8.16242e20 q^{97} -3.28013e20 q^{98} -1.75328e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4096 q^{2} + 96764 q^{3} + 4194304 q^{4} + 99086336 q^{6} + 162760528 q^{7} + 4294967296 q^{8} + 23614761712 q^{9} + 87559254108 q^{11} + 101464408064 q^{12} - 98098343096 q^{13} + 166666780672 q^{14}+ \cdots - 28\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1024.00 0.707107
\(3\) 185808. 1.81674 0.908368 − 0.418171i \(-0.137329\pi\)
0.908368 + 0.418171i \(0.137329\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) 1.90268e8 1.28463
\(7\) 4.88079e8 0.653071 0.326536 − 0.945185i \(-0.394119\pi\)
0.326536 + 0.945185i \(0.394119\pi\)
\(8\) 1.07374e9 0.353553
\(9\) 2.40644e10 2.30053
\(10\) 0 0
\(11\) −7.28578e10 −0.846940 −0.423470 − 0.905910i \(-0.639188\pi\)
−0.423470 + 0.905910i \(0.639188\pi\)
\(12\) 1.94834e11 0.908368
\(13\) 9.02768e11 1.81623 0.908115 − 0.418720i \(-0.137521\pi\)
0.908115 + 0.418720i \(0.137521\pi\)
\(14\) 4.99793e11 0.461791
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) −6.32065e12 −0.760411 −0.380206 − 0.924902i \(-0.624147\pi\)
−0.380206 + 0.924902i \(0.624147\pi\)
\(18\) 2.46419e13 1.62672
\(19\) 4.36780e13 1.63437 0.817184 − 0.576376i \(-0.195534\pi\)
0.817184 + 0.576376i \(0.195534\pi\)
\(20\) 0 0
\(21\) 9.06891e13 1.18646
\(22\) −7.46064e13 −0.598877
\(23\) −9.00682e13 −0.453345 −0.226673 − 0.973971i \(-0.572785\pi\)
−0.226673 + 0.973971i \(0.572785\pi\)
\(24\) 1.99510e14 0.642313
\(25\) 0 0
\(26\) 9.24434e14 1.28427
\(27\) 2.52774e15 2.36273
\(28\) 5.11788e14 0.326536
\(29\) −1.34854e15 −0.595230 −0.297615 − 0.954686i \(-0.596191\pi\)
−0.297615 + 0.954686i \(0.596191\pi\)
\(30\) 0 0
\(31\) −4.56363e14 −0.100003 −0.0500015 − 0.998749i \(-0.515923\pi\)
−0.0500015 + 0.998749i \(0.515923\pi\)
\(32\) 1.12590e15 0.176777
\(33\) −1.35376e16 −1.53867
\(34\) −6.47235e15 −0.537692
\(35\) 0 0
\(36\) 2.52333e16 1.15027
\(37\) −2.84931e16 −0.974142 −0.487071 − 0.873362i \(-0.661935\pi\)
−0.487071 + 0.873362i \(0.661935\pi\)
\(38\) 4.47263e16 1.15567
\(39\) 1.67742e17 3.29961
\(40\) 0 0
\(41\) −1.42955e17 −1.66329 −0.831647 − 0.555304i \(-0.812602\pi\)
−0.831647 + 0.555304i \(0.812602\pi\)
\(42\) 9.28657e16 0.838953
\(43\) 1.47393e17 1.04006 0.520030 − 0.854148i \(-0.325921\pi\)
0.520030 + 0.854148i \(0.325921\pi\)
\(44\) −7.63969e16 −0.423470
\(45\) 0 0
\(46\) −9.22299e16 −0.320564
\(47\) 1.71830e17 0.476508 0.238254 − 0.971203i \(-0.423425\pi\)
0.238254 + 0.971203i \(0.423425\pi\)
\(48\) 2.04298e17 0.454184
\(49\) −3.20325e17 −0.573498
\(50\) 0 0
\(51\) −1.17443e18 −1.38147
\(52\) 9.46621e17 0.908115
\(53\) 1.36482e18 1.07196 0.535979 − 0.844232i \(-0.319943\pi\)
0.535979 + 0.844232i \(0.319943\pi\)
\(54\) 2.58841e18 1.67070
\(55\) 0 0
\(56\) 5.24071e17 0.230896
\(57\) 8.11574e18 2.96922
\(58\) −1.38090e18 −0.420891
\(59\) 6.60455e18 1.68228 0.841138 − 0.540821i \(-0.181886\pi\)
0.841138 + 0.540821i \(0.181886\pi\)
\(60\) 0 0
\(61\) 1.48171e18 0.265950 0.132975 − 0.991119i \(-0.457547\pi\)
0.132975 + 0.991119i \(0.457547\pi\)
\(62\) −4.67316e17 −0.0707128
\(63\) 1.17453e19 1.50241
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) −1.38625e19 −1.08800
\(67\) 2.80036e19 1.87685 0.938424 − 0.345486i \(-0.112286\pi\)
0.938424 + 0.345486i \(0.112286\pi\)
\(68\) −6.62769e18 −0.380206
\(69\) −1.67354e19 −0.823609
\(70\) 0 0
\(71\) −4.31882e19 −1.57454 −0.787269 − 0.616610i \(-0.788505\pi\)
−0.787269 + 0.616610i \(0.788505\pi\)
\(72\) 2.58389e19 0.813361
\(73\) 2.89648e17 0.00788825 0.00394412 − 0.999992i \(-0.498745\pi\)
0.00394412 + 0.999992i \(0.498745\pi\)
\(74\) −2.91770e19 −0.688823
\(75\) 0 0
\(76\) 4.57997e19 0.817184
\(77\) −3.55604e19 −0.553112
\(78\) 1.71768e20 2.33318
\(79\) −5.22625e19 −0.621020 −0.310510 − 0.950570i \(-0.600500\pi\)
−0.310510 + 0.950570i \(0.600500\pi\)
\(80\) 0 0
\(81\) 2.17954e20 1.99192
\(82\) −1.46386e20 −1.17613
\(83\) −5.91809e19 −0.418660 −0.209330 − 0.977845i \(-0.567128\pi\)
−0.209330 + 0.977845i \(0.567128\pi\)
\(84\) 9.50944e19 0.593229
\(85\) 0 0
\(86\) 1.50931e20 0.735434
\(87\) −2.50570e20 −1.08138
\(88\) −7.82305e19 −0.299439
\(89\) −4.79550e20 −1.63019 −0.815097 − 0.579325i \(-0.803316\pi\)
−0.815097 + 0.579325i \(0.803316\pi\)
\(90\) 0 0
\(91\) 4.40622e20 1.18613
\(92\) −9.44434e19 −0.226673
\(93\) −8.47961e19 −0.181679
\(94\) 1.75954e20 0.336942
\(95\) 0 0
\(96\) 2.09202e20 0.321157
\(97\) 8.16242e20 1.12387 0.561935 − 0.827182i \(-0.310057\pi\)
0.561935 + 0.827182i \(0.310057\pi\)
\(98\) −3.28013e20 −0.405524
\(99\) −1.75328e21 −1.94841
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 50.22.a.j.1.4 yes 4
5.2 odd 4 50.22.b.h.49.5 8
5.3 odd 4 50.22.b.h.49.4 8
5.4 even 2 50.22.a.i.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.1 ✓ 4 5.4 even 2
50.22.a.j.1.4 yes 4 1.1 even 1 trivial
50.22.b.h.49.4 8 5.3 odd 4
50.22.b.h.49.5 8 5.2 odd 4