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Properties

Label 50.12.b.e.49.2
Level $50$
Weight $12$
Character 50.49
Analytic conductor $38.417$
Analytic rank $0$
Dimension $4$
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(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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4096,0,-3584] 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: \(38.4171590280\)
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: \( x^{4} + 5065x^{2} + 6411024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(49.8215i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.12.b.e.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-32.0000i q^{2} +475.215i q^{3} -1024.00 q^{4} +15206.9 q^{6} -6220.98i q^{7} +32768.0i q^{8} -48682.0 q^{9} +383480. q^{11} -486620. i q^{12} +327577. i q^{13} -199071. q^{14} +1.04858e6 q^{16} -1.06324e7i q^{17} +1.55782e6i q^{18} +6.77371e6 q^{19} +2.95630e6 q^{21} -1.22714e7i q^{22} +5.84424e7i q^{23} -1.55718e7 q^{24} +1.04825e7 q^{26} +6.10485e7i q^{27} +6.37029e6i q^{28} -4.02500e7 q^{29} -1.42511e8 q^{31} -3.35544e7i q^{32} +1.82235e8i q^{33} -3.40235e8 q^{34} +4.98503e7 q^{36} +5.87980e8i q^{37} -2.16759e8i q^{38} -1.55669e8 q^{39} +8.88163e8 q^{41} -9.46017e7i q^{42} +8.40247e8i q^{43} -3.92683e8 q^{44} +1.87016e9 q^{46} +1.17537e9i q^{47} +4.98299e8i q^{48} +1.93863e9 q^{49} +5.05265e9 q^{51} -3.35439e8i q^{52} +1.23551e9i q^{53} +1.95355e9 q^{54} +2.03849e8 q^{56} +3.21897e9i q^{57} +1.28800e9i q^{58} -6.61244e9 q^{59} +6.94243e9 q^{61} +4.56034e9i q^{62} +3.02850e8i q^{63} -1.07374e9 q^{64} +5.83153e9 q^{66} +1.42821e10i q^{67} +1.08875e10i q^{68} -2.77727e10 q^{69} -1.71808e9 q^{71} -1.59521e9i q^{72} +4.72457e9i q^{73} +1.88154e10 q^{74} -6.93628e9 q^{76} -2.38562e9i q^{77} +4.98142e9i q^{78} -1.99880e9 q^{79} -3.76350e10 q^{81} -2.84212e10i q^{82} +7.75019e9i q^{83} -3.02725e9 q^{84} +2.68879e10 q^{86} -1.91274e10i q^{87} +1.25659e10i q^{88} +6.32612e10 q^{89} +2.03785e9 q^{91} -5.98450e10i q^{92} -6.77231e10i q^{93} +3.76118e10 q^{94} +1.59456e10 q^{96} +4.67106e10i q^{97} -6.20360e10i q^{98} -1.86686e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4096 q^{4} - 3584 q^{6} - 307448 q^{9} - 199152 q^{11} - 3501568 q^{14} + 4194304 q^{16} - 23490320 q^{19} - 45605672 q^{21} + 3670016 q^{24} - 74783744 q^{26} - 246047280 q^{29} + 245805728 q^{31}+ \cdots + 64145207424 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 32.0000i − 0.707107i
\(3\) 475.215i 1.12908i 0.825407 + 0.564538i \(0.190946\pi\)
−0.825407 + 0.564538i \(0.809054\pi\)
\(4\) −1024.00 −0.500000
\(5\) 0 0
\(6\) 15206.9 0.798377
\(7\) − 6220.98i − 0.139901i −0.997550 − 0.0699503i \(-0.977716\pi\)
0.997550 − 0.0699503i \(-0.0222841\pi\)
\(8\) 32768.0i 0.353553i
\(9\) −48682.0 −0.274811
\(10\) 0 0
\(11\) 383480. 0.717931 0.358966 − 0.933351i \(-0.383130\pi\)
0.358966 + 0.933351i \(0.383130\pi\)
\(12\) − 486620.i − 0.564538i
\(13\) 327577.i 0.244695i 0.992487 + 0.122347i \(0.0390422\pi\)
−0.992487 + 0.122347i \(0.960958\pi\)
\(14\) −199071. −0.0989247
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) − 1.06324e7i − 1.81619i −0.418769 − 0.908093i \(-0.637538\pi\)
0.418769 − 0.908093i \(-0.362462\pi\)
\(18\) 1.55782e6i 0.194321i
\(19\) 6.77371e6 0.627598 0.313799 − 0.949489i \(-0.398398\pi\)
0.313799 + 0.949489i \(0.398398\pi\)
\(20\) 0 0
\(21\) 2.95630e6 0.157958
\(22\) − 1.22714e7i − 0.507654i
\(23\) 5.84424e7i 1.89332i 0.322231 + 0.946661i \(0.395567\pi\)
−0.322231 + 0.946661i \(0.604433\pi\)
\(24\) −1.55718e7 −0.399188
\(25\) 0 0
\(26\) 1.04825e7 0.173025
\(27\) 6.10485e7i 0.818793i
\(28\) 6.37029e6i 0.0699503i
\(29\) −4.02500e7 −0.364399 −0.182199 − 0.983262i \(-0.558322\pi\)
−0.182199 + 0.983262i \(0.558322\pi\)
\(30\) 0 0
\(31\) −1.42511e8 −0.894041 −0.447021 − 0.894524i \(-0.647515\pi\)
−0.447021 + 0.894524i \(0.647515\pi\)
\(32\) − 3.35544e7i − 0.176777i
\(33\) 1.82235e8i 0.810598i
\(34\) −3.40235e8 −1.28424
\(35\) 0 0
\(36\) 4.98503e7 0.137406
\(37\) 5.87980e8i 1.39397i 0.717086 + 0.696985i \(0.245476\pi\)
−0.717086 + 0.696985i \(0.754524\pi\)
\(38\) − 2.16759e8i − 0.443779i
\(39\) −1.55669e8 −0.276279
\(40\) 0 0
\(41\) 8.88163e8 1.19724 0.598620 − 0.801033i \(-0.295716\pi\)
0.598620 + 0.801033i \(0.295716\pi\)
\(42\) − 9.46017e7i − 0.111693i
\(43\) 8.40247e8i 0.871627i 0.900037 + 0.435814i \(0.143539\pi\)
−0.900037 + 0.435814i \(0.856461\pi\)
\(44\) −3.92683e8 −0.358966
\(45\) 0 0
\(46\) 1.87016e9 1.33878
\(47\) 1.17537e9i 0.747543i 0.927521 + 0.373771i \(0.121936\pi\)
−0.927521 + 0.373771i \(0.878064\pi\)
\(48\) 4.98299e8i 0.282269i
\(49\) 1.93863e9 0.980428
\(50\) 0 0
\(51\) 5.05265e9 2.05061
\(52\) − 3.35439e8i − 0.122347i
\(53\) 1.23551e9i 0.405814i 0.979198 + 0.202907i \(0.0650390\pi\)
−0.979198 + 0.202907i \(0.934961\pi\)
\(54\) 1.95355e9 0.578974
\(55\) 0 0
\(56\) 2.03849e8 0.0494624
\(57\) 3.21897e9i 0.708606i
\(58\) 1.28800e9i 0.257669i
\(59\) −6.61244e9 −1.20414 −0.602069 − 0.798444i \(-0.705657\pi\)
−0.602069 + 0.798444i \(0.705657\pi\)
\(60\) 0 0
\(61\) 6.94243e9 1.05244 0.526220 − 0.850349i \(-0.323609\pi\)
0.526220 + 0.850349i \(0.323609\pi\)
\(62\) 4.56034e9i 0.632183i
\(63\) 3.02850e8i 0.0384463i
\(64\) −1.07374e9 −0.125000
\(65\) 0 0
\(66\) 5.83153e9 0.573180
\(67\) 1.42821e10i 1.29235i 0.763188 + 0.646177i \(0.223633\pi\)
−0.763188 + 0.646177i \(0.776367\pi\)
\(68\) 1.08875e10i 0.908093i
\(69\) −2.77727e10 −2.13770
\(70\) 0 0
\(71\) −1.71808e9 −0.113011 −0.0565056 − 0.998402i \(-0.517996\pi\)
−0.0565056 + 0.998402i \(0.517996\pi\)
\(72\) − 1.59521e9i − 0.0971604i
\(73\) 4.72457e9i 0.266739i 0.991066 + 0.133370i \(0.0425797\pi\)
−0.991066 + 0.133370i \(0.957420\pi\)
\(74\) 1.88154e10 0.985686
\(75\) 0 0
\(76\) −6.93628e9 −0.313799
\(77\) − 2.38562e9i − 0.100439i
\(78\) 4.98142e9i 0.195359i
\(79\) −1.99880e9 −0.0730838 −0.0365419 − 0.999332i \(-0.511634\pi\)
−0.0365419 + 0.999332i \(0.511634\pi\)
\(80\) 0 0
\(81\) −3.76350e10 −1.19929
\(82\) − 2.84212e10i − 0.846577i
\(83\) 7.75019e9i 0.215965i 0.994153 + 0.107982i \(0.0344390\pi\)
−0.994153 + 0.107982i \(0.965561\pi\)
\(84\) −3.02725e9 −0.0789792
\(85\) 0 0
\(86\) 2.68879e10 0.616333
\(87\) − 1.91274e10i − 0.411434i
\(88\) 1.25659e10i 0.253827i
\(89\) 6.32612e10 1.20086 0.600430 − 0.799677i \(-0.294996\pi\)
0.600430 + 0.799677i \(0.294996\pi\)
\(90\) 0 0
\(91\) 2.03785e9 0.0342330
\(92\) − 5.98450e10i − 0.946661i
\(93\) − 6.77231e10i − 1.00944i
\(94\) 3.76118e10 0.528593
\(95\) 0 0
\(96\) 1.59456e10 0.199594
\(97\) 4.67106e10i 0.552294i 0.961115 + 0.276147i \(0.0890577\pi\)
−0.961115 + 0.276147i \(0.910942\pi\)
\(98\) − 6.20360e10i − 0.693267i
\(99\) −1.86686e10 −0.197295
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.b.e.49.2 4
5.2 odd 4 50.12.a.h.1.2 yes 2
5.3 odd 4 50.12.a.g.1.1 ✓ 2
5.4 even 2 inner 50.12.b.e.49.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.a.g.1.1 ✓ 2 5.3 odd 4
50.12.a.h.1.2 yes 2 5.2 odd 4
50.12.b.e.49.2 4 1.1 even 1 trivial
50.12.b.e.49.3 4 5.4 even 2 inner