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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,Mod(49,50)] 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("50.49"); 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([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2097152,0,-44900352] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.b.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} -21924.0i q^{3} -1.04858e6 q^{4} -2.24502e7 q^{6} +7.22753e8i q^{7} +1.07374e9i q^{8} +9.97969e9 q^{9} +4.99764e10 q^{11} +2.29890e10i q^{12} -2.43514e10i q^{13} +7.40099e11 q^{14} +1.09951e12 q^{16} +5.76887e12i q^{17} -1.02192e13i q^{18} +3.02502e13 q^{19} +1.58456e13 q^{21} -5.11758e13i q^{22} -1.45464e14i q^{23} +2.35407e13 q^{24} -2.49358e13 q^{26} -4.48128e14i q^{27} -7.57862e14i q^{28} +1.16711e15 q^{29} -7.43191e15 q^{31} -1.12590e15i q^{32} -1.09568e15i q^{33} +5.90733e15 q^{34} -1.04645e16 q^{36} +5.46217e16i q^{37} -3.09762e16i q^{38} -5.33880e14 q^{39} -2.08896e16 q^{41} -1.62259e16i q^{42} +7.61939e16i q^{43} -5.24041e16 q^{44} -1.48955e17 q^{46} -5.08990e17i q^{47} -2.41057e16i q^{48} +3.61736e16 q^{49} +1.26477e17 q^{51} +2.55343e16i q^{52} +1.34149e18i q^{53} -4.58883e17 q^{54} -7.76050e17 q^{56} -6.63206e17i q^{57} -1.19512e18i q^{58} -2.30640e18 q^{59} +5.26834e18 q^{61} +7.61027e18i q^{62} +7.21285e18i q^{63} -1.15292e18 q^{64} -1.12198e18 q^{66} +1.09050e19i q^{67} -6.04910e18i q^{68} -3.18915e18 q^{69} -1.33678e19 q^{71} +1.07156e19i q^{72} -1.79529e19i q^{73} +5.59326e19 q^{74} -3.17197e19 q^{76} +3.61206e19i q^{77} +5.46693e17i q^{78} -1.18985e20 q^{79} +9.45663e19 q^{81} +2.13909e19i q^{82} +1.34353e20i q^{83} -1.66154e19 q^{84} +7.80226e19 q^{86} -2.55877e19i q^{87} +5.36617e19i q^{88} -3.47081e20 q^{89} +1.76001e19 q^{91} +1.52530e20i q^{92} +1.62937e20i q^{93} -5.21205e20 q^{94} -2.46842e19 q^{96} +3.76522e20i q^{97} -3.70418e19i q^{98} +4.98749e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2097152 q^{4} - 44900352 q^{6} + 19959382854 q^{9} + 99952797144 q^{11} + 1480198651904 q^{14} + 2199023255552 q^{16} + 60500451965240 q^{19} + 31691284418304 q^{21} + 47081431498752 q^{24} - 49871667924992 q^{26}+ \cdots + 99\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1024.00i − 0.707107i
\(3\) − 21924.0i − 0.214361i −0.994240 − 0.107181i \(-0.965818\pi\)
0.994240 − 0.107181i \(-0.0341823\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −2.24502e7 −0.151576
\(7\) 7.22753e8i 0.967076i 0.875323 + 0.483538i \(0.160649\pi\)
−0.875323 + 0.483538i \(0.839351\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) 9.97969e9 0.954049
\(10\) 0 0
\(11\) 4.99764e10 0.580954 0.290477 − 0.956882i \(-0.406186\pi\)
0.290477 + 0.956882i \(0.406186\pi\)
\(12\) 2.29890e10i 0.107181i
\(13\) − 2.43514e10i − 0.0489913i −0.999700 − 0.0244956i \(-0.992202\pi\)
0.999700 − 0.0244956i \(-0.00779798\pi\)
\(14\) 7.40099e11 0.683826
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 5.76887e12i 0.694029i 0.937860 + 0.347014i \(0.112804\pi\)
−0.937860 + 0.347014i \(0.887196\pi\)
\(18\) − 1.02192e13i − 0.674615i
\(19\) 3.02502e13 1.13192 0.565960 − 0.824433i \(-0.308506\pi\)
0.565960 + 0.824433i \(0.308506\pi\)
\(20\) 0 0
\(21\) 1.58456e13 0.207304
\(22\) − 5.11758e13i − 0.410797i
\(23\) − 1.45464e14i − 0.732173i −0.930581 − 0.366086i \(-0.880697\pi\)
0.930581 − 0.366086i \(-0.119303\pi\)
\(24\) 2.35407e13 0.0757882
\(25\) 0 0
\(26\) −2.49358e13 −0.0346421
\(27\) − 4.48128e14i − 0.418873i
\(28\) − 7.57862e14i − 0.483538i
\(29\) 1.16711e15 0.515148 0.257574 − 0.966259i \(-0.417077\pi\)
0.257574 + 0.966259i \(0.417077\pi\)
\(30\) 0 0
\(31\) −7.43191e15 −1.62856 −0.814278 − 0.580476i \(-0.802867\pi\)
−0.814278 + 0.580476i \(0.802867\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 1.09568e15i − 0.124534i
\(34\) 5.90733e15 0.490752
\(35\) 0 0
\(36\) −1.04645e16 −0.477025
\(37\) 5.46217e16i 1.86744i 0.358003 + 0.933721i \(0.383458\pi\)
−0.358003 + 0.933721i \(0.616542\pi\)
\(38\) − 3.09762e16i − 0.800388i
\(39\) −5.33880e14 −0.0105018
\(40\) 0 0
\(41\) −2.08896e16 −0.243052 −0.121526 − 0.992588i \(-0.538779\pi\)
−0.121526 + 0.992588i \(0.538779\pi\)
\(42\) − 1.62259e16i − 0.146586i
\(43\) 7.61939e16i 0.537652i 0.963189 + 0.268826i \(0.0866357\pi\)
−0.963189 + 0.268826i \(0.913364\pi\)
\(44\) −5.24041e16 −0.290477
\(45\) 0 0
\(46\) −1.48955e17 −0.517724
\(47\) − 5.08990e17i − 1.41150i −0.708460 − 0.705751i \(-0.750610\pi\)
0.708460 − 0.705751i \(-0.249390\pi\)
\(48\) − 2.41057e16i − 0.0535904i
\(49\) 3.61736e16 0.0647639
\(50\) 0 0
\(51\) 1.26477e17 0.148773
\(52\) 2.55343e16i 0.0244956i
\(53\) 1.34149e18i 1.05364i 0.849978 + 0.526819i \(0.176615\pi\)
−0.849978 + 0.526819i \(0.823385\pi\)
\(54\) −4.58883e17 −0.296188
\(55\) 0 0
\(56\) −7.76050e17 −0.341913
\(57\) − 6.63206e17i − 0.242640i
\(58\) − 1.19512e18i − 0.364265i
\(59\) −2.30640e18 −0.587475 −0.293737 − 0.955886i \(-0.594899\pi\)
−0.293737 + 0.955886i \(0.594899\pi\)
\(60\) 0 0
\(61\) 5.26834e18 0.945607 0.472803 − 0.881168i \(-0.343242\pi\)
0.472803 + 0.881168i \(0.343242\pi\)
\(62\) 7.61027e18i 1.15156i
\(63\) 7.21285e18i 0.922638i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −1.12198e18 −0.0880589
\(67\) 1.09050e19i 0.730869i 0.930837 + 0.365434i \(0.119080\pi\)
−0.930837 + 0.365434i \(0.880920\pi\)
\(68\) − 6.04910e18i − 0.347014i
\(69\) −3.18915e18 −0.156950
\(70\) 0 0
\(71\) −1.33678e19 −0.487357 −0.243679 − 0.969856i \(-0.578354\pi\)
−0.243679 + 0.969856i \(0.578354\pi\)
\(72\) 1.07156e19i 0.337307i
\(73\) − 1.79529e19i − 0.488926i −0.969659 − 0.244463i \(-0.921388\pi\)
0.969659 − 0.244463i \(-0.0786117\pi\)
\(74\) 5.59326e19 1.32048
\(75\) 0 0
\(76\) −3.17197e19 −0.565960
\(77\) 3.61206e19i 0.561827i
\(78\) 5.46693e17i 0.00742592i
\(79\) −1.18985e20 −1.41387 −0.706933 − 0.707281i \(-0.749922\pi\)
−0.706933 + 0.707281i \(0.749922\pi\)
\(80\) 0 0
\(81\) 9.45663e19 0.864259
\(82\) 2.13909e19i 0.171864i
\(83\) 1.34353e20i 0.950447i 0.879865 + 0.475224i \(0.157633\pi\)
−0.879865 + 0.475224i \(0.842367\pi\)
\(84\) −1.66154e19 −0.103652
\(85\) 0 0
\(86\) 7.80226e19 0.380177
\(87\) − 2.55877e19i − 0.110428i
\(88\) 5.36617e19i 0.205398i
\(89\) −3.47081e20 −1.17988 −0.589938 − 0.807449i \(-0.700848\pi\)
−0.589938 + 0.807449i \(0.700848\pi\)
\(90\) 0 0
\(91\) 1.76001e19 0.0473783
\(92\) 1.52530e20i 0.366086i
\(93\) 1.62937e20i 0.349099i
\(94\) −5.21205e20 −0.998083
\(95\) 0 0
\(96\) −2.46842e19 −0.0378941
\(97\) 3.76522e20i 0.518426i 0.965820 + 0.259213i \(0.0834632\pi\)
−0.965820 + 0.259213i \(0.916537\pi\)
\(98\) − 3.70418e19i − 0.0457950i
\(99\) 4.98749e20 0.554259
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.b.b.49.1 2
5.2 odd 4 10.22.a.a.1.1 ✓ 1
5.3 odd 4 50.22.a.b.1.1 1
5.4 even 2 inner 50.22.b.b.49.2 2
20.7 even 4 80.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.a.1.1 ✓ 1 5.2 odd 4
50.22.a.b.1.1 1 5.3 odd 4
50.22.b.b.49.1 2 1.1 even 1 trivial
50.22.b.b.49.2 2 5.4 even 2 inner
80.22.a.a.1.1 1 20.7 even 4