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Properties

Label 50.22.b.c.49.1
Level $50$
Weight $22$
Character 50.49
Analytic conductor $139.739$
Analytic rank $0$
Dimension $2$
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] = [50,22,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,-2097152,0,121479168] 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 2)
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.c.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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +59316.0i q^{3} -1.04858e6 q^{4} +6.07396e7 q^{6} -1.42743e9i q^{7} +1.07374e9i q^{8} +6.94197e9 q^{9} -1.06768e11 q^{11} -6.21973e10i q^{12} -1.50151e11i q^{13} -1.46168e12 q^{14} +1.09951e12 q^{16} +1.12040e13i q^{17} -7.10857e12i q^{18} -1.10241e13 q^{19} +8.46692e13 q^{21} +1.09330e14i q^{22} +1.29503e14i q^{23} -6.36901e13 q^{24} -1.53754e14 q^{26} +1.03224e15i q^{27} +1.49676e15i q^{28} -2.38237e15 q^{29} -8.78553e14 q^{31} -1.12590e15i q^{32} -6.33304e15i q^{33} +1.14729e16 q^{34} -7.27918e15 q^{36} -3.11300e16i q^{37} +1.12886e16i q^{38} +8.90633e15 q^{39} -2.46129e16 q^{41} -8.67013e16i q^{42} -1.33386e17i q^{43} +1.11954e17 q^{44} +1.32611e17 q^{46} +1.92524e17i q^{47} +6.52186e16i q^{48} -1.47900e18 q^{49} -6.64575e17 q^{51} +1.57444e17i q^{52} -5.94166e17i q^{53} +1.05701e18 q^{54} +1.53269e18 q^{56} -6.53903e17i q^{57} +2.43955e18i q^{58} +2.95595e18 q^{59} +7.98415e18 q^{61} +8.99638e17i q^{62} -9.90914e18i q^{63} -1.15292e18 q^{64} -6.48504e18 q^{66} -4.83704e18i q^{67} -1.17482e19i q^{68} -7.68159e18 q^{69} +8.84902e18 q^{71} +7.45388e18i q^{72} +3.66844e19i q^{73} -3.18771e19 q^{74} +1.15596e19 q^{76} +1.52403e20i q^{77} -9.12008e18i q^{78} -3.38406e19 q^{79} +1.13873e19 q^{81} +2.52036e19i q^{82} +2.04215e20i q^{83} -8.87821e19 q^{84} -1.36587e20 q^{86} -1.41313e20i q^{87} -1.14641e20i q^{88} +4.10241e19 q^{89} -2.14329e20 q^{91} -1.35794e20i q^{92} -5.21122e19i q^{93} +1.97145e20 q^{94} +6.67839e19 q^{96} +7.27592e20i q^{97} +1.51449e21i q^{98} -7.41179e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2097152 q^{4} + 121479168 q^{6} + 13883930694 q^{9} - 213535789896 q^{11} - 2923368103936 q^{14} + 2199023255552 q^{16} - 22048111910920 q^{19} + 169338381301824 q^{21} - 127380140064768 q^{24} - 307508358090752 q^{26}+ \cdots - 14\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) 59316.0i 0.579961i 0.957033 + 0.289980i \(0.0936488\pi\)
−0.957033 + 0.289980i \(0.906351\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 6.07396e7 0.410094
\(7\) − 1.42743e9i − 1.90996i −0.296671 − 0.954980i \(-0.595877\pi\)
0.296671 − 0.954980i \(-0.404123\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) 6.94197e9 0.663645
\(10\) 0 0
\(11\) −1.06768e11 −1.24113 −0.620565 − 0.784155i \(-0.713097\pi\)
−0.620565 + 0.784155i \(0.713097\pi\)
\(12\) − 6.21973e10i − 0.289980i
\(13\) − 1.50151e11i − 0.302080i −0.988528 − 0.151040i \(-0.951738\pi\)
0.988528 − 0.151040i \(-0.0482622\pi\)
\(14\) −1.46168e12 −1.35055
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.12040e13i 1.34790i 0.738776 + 0.673952i \(0.235404\pi\)
−0.738776 + 0.673952i \(0.764596\pi\)
\(18\) − 7.10857e12i − 0.469268i
\(19\) −1.10241e13 −0.412504 −0.206252 − 0.978499i \(-0.566127\pi\)
−0.206252 + 0.978499i \(0.566127\pi\)
\(20\) 0 0
\(21\) 8.46692e13 1.10770
\(22\) 1.09330e14i 0.877612i
\(23\) 1.29503e14i 0.651834i 0.945398 + 0.325917i \(0.105673\pi\)
−0.945398 + 0.325917i \(0.894327\pi\)
\(24\) −6.36901e13 −0.205047
\(25\) 0 0
\(26\) −1.53754e14 −0.213603
\(27\) 1.03224e15i 0.964849i
\(28\) 1.49676e15i 0.954980i
\(29\) −2.38237e15 −1.05155 −0.525776 − 0.850623i \(-0.676225\pi\)
−0.525776 + 0.850623i \(0.676225\pi\)
\(30\) 0 0
\(31\) −8.78553e14 −0.192517 −0.0962587 − 0.995356i \(-0.530688\pi\)
−0.0962587 + 0.995356i \(0.530688\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 6.33304e15i − 0.719807i
\(34\) 1.14729e16 0.953111
\(35\) 0 0
\(36\) −7.27918e15 −0.331823
\(37\) − 3.11300e16i − 1.06429i −0.846652 − 0.532146i \(-0.821386\pi\)
0.846652 − 0.532146i \(-0.178614\pi\)
\(38\) 1.12886e16i 0.291685i
\(39\) 8.90633e15 0.175194
\(40\) 0 0
\(41\) −2.46129e16 −0.286373 −0.143187 − 0.989696i \(-0.545735\pi\)
−0.143187 + 0.989696i \(0.545735\pi\)
\(42\) − 8.67013e16i − 0.783263i
\(43\) − 1.33386e17i − 0.941222i −0.882341 − 0.470611i \(-0.844034\pi\)
0.882341 − 0.470611i \(-0.155966\pi\)
\(44\) 1.11954e17 0.620565
\(45\) 0 0
\(46\) 1.32611e17 0.460916
\(47\) 1.92524e17i 0.533897i 0.963711 + 0.266948i \(0.0860153\pi\)
−0.963711 + 0.266948i \(0.913985\pi\)
\(48\) 6.52186e16i 0.144990i
\(49\) −1.47900e18 −2.64794
\(50\) 0 0
\(51\) −6.64575e17 −0.781731
\(52\) 1.57444e17i 0.151040i
\(53\) − 5.94166e17i − 0.466672i −0.972396 − 0.233336i \(-0.925036\pi\)
0.972396 − 0.233336i \(-0.0749641\pi\)
\(54\) 1.05701e18 0.682251
\(55\) 0 0
\(56\) 1.53269e18 0.675273
\(57\) − 6.53903e17i − 0.239236i
\(58\) 2.43955e18i 0.743559i
\(59\) 2.95595e18 0.752925 0.376462 − 0.926432i \(-0.377140\pi\)
0.376462 + 0.926432i \(0.377140\pi\)
\(60\) 0 0
\(61\) 7.98415e18 1.43306 0.716532 − 0.697554i \(-0.245728\pi\)
0.716532 + 0.697554i \(0.245728\pi\)
\(62\) 8.99638e17i 0.136130i
\(63\) − 9.90914e18i − 1.26754i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −6.48504e18 −0.508981
\(67\) − 4.83704e18i − 0.324186i −0.986775 − 0.162093i \(-0.948176\pi\)
0.986775 − 0.162093i \(-0.0518245\pi\)
\(68\) − 1.17482e19i − 0.673952i
\(69\) −7.68159e18 −0.378038
\(70\) 0 0
\(71\) 8.84902e18 0.322613 0.161307 − 0.986904i \(-0.448429\pi\)
0.161307 + 0.986904i \(0.448429\pi\)
\(72\) 7.45388e18i 0.234634i
\(73\) 3.66844e19i 0.999060i 0.866297 + 0.499530i \(0.166494\pi\)
−0.866297 + 0.499530i \(0.833506\pi\)
\(74\) −3.18771e19 −0.752569
\(75\) 0 0
\(76\) 1.15596e19 0.206252
\(77\) 1.52403e20i 2.37051i
\(78\) − 9.12008e18i − 0.123881i
\(79\) −3.38406e19 −0.402118 −0.201059 − 0.979579i \(-0.564438\pi\)
−0.201059 + 0.979579i \(0.564438\pi\)
\(80\) 0 0
\(81\) 1.13873e19 0.104071
\(82\) 2.52036e19i 0.202497i
\(83\) 2.04215e20i 1.44466i 0.691547 + 0.722332i \(0.256930\pi\)
−0.691547 + 0.722332i \(0.743070\pi\)
\(84\) −8.87821e19 −0.553851
\(85\) 0 0
\(86\) −1.36587e20 −0.665544
\(87\) − 1.41313e20i − 0.609858i
\(88\) − 1.14641e20i − 0.438806i
\(89\) 4.10241e19 0.139458 0.0697290 − 0.997566i \(-0.477787\pi\)
0.0697290 + 0.997566i \(0.477787\pi\)
\(90\) 0 0
\(91\) −2.14329e20 −0.576960
\(92\) − 1.35794e20i − 0.325917i
\(93\) − 5.21122e19i − 0.111653i
\(94\) 1.97145e20 0.377522
\(95\) 0 0
\(96\) 6.67839e19 0.102524
\(97\) 7.27592e20i 1.00181i 0.865503 + 0.500905i \(0.166999\pi\)
−0.865503 + 0.500905i \(0.833001\pi\)
\(98\) 1.51449e21i 1.87238i
\(99\) −7.41179e20 −0.823671
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.c.49.1 2
5.2 odd 4 2.22.a.b.1.1 ✓ 1
5.3 odd 4 50.22.a.a.1.1 1
5.4 even 2 inner 50.22.b.c.49.2 2
15.2 even 4 18.22.a.b.1.1 1
20.7 even 4 16.22.a.b.1.1 1
40.27 even 4 64.22.a.e.1.1 1
40.37 odd 4 64.22.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.b.1.1 ✓ 1 5.2 odd 4
16.22.a.b.1.1 1 20.7 even 4
18.22.a.b.1.1 1 15.2 even 4
50.22.a.a.1.1 1 5.3 odd 4
50.22.b.c.49.1 2 1.1 even 1 trivial
50.22.b.c.49.2 2 5.4 even 2 inner
64.22.a.c.1.1 1 40.37 odd 4
64.22.a.e.1.1 1 40.27 even 4