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Properties

Label 50.12.a.h.1.1
Level $50$
Weight $12$
Character 50.1
Self dual yes
Analytic conductor $38.417$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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(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, 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([0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,64,-56] 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: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10129}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2532 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(50.8215\) 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)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} -531.215 q^{3} +1024.00 q^{4} -16998.9 q^{6} +48491.0 q^{7} +32768.0 q^{8} +105042. q^{9} -483056. q^{11} -543964. q^{12} -1.49607e6 q^{13} +1.55171e6 q^{14} +1.04858e6 q^{16} -6.22735e6 q^{17} +3.36134e6 q^{18} +1.85189e7 q^{19} -2.57591e7 q^{21} -1.54578e7 q^{22} +2.19633e7 q^{23} -1.74068e7 q^{24} -4.78743e7 q^{26} +3.83032e7 q^{27} +4.96548e7 q^{28} +8.27736e7 q^{29} +2.65413e8 q^{31} +3.35544e7 q^{32} +2.56606e8 q^{33} -1.99275e8 q^{34} +1.07563e8 q^{36} +4.36392e8 q^{37} +5.92604e8 q^{38} +7.94736e8 q^{39} +6.58528e8 q^{41} -8.24292e8 q^{42} -8.41846e8 q^{43} -4.94649e8 q^{44} +7.02826e8 q^{46} -2.89979e9 q^{47} -5.57019e8 q^{48} +3.74052e8 q^{49} +3.30806e9 q^{51} -1.53198e9 q^{52} +2.52068e9 q^{53} +1.22570e9 q^{54} +1.58895e9 q^{56} -9.83749e9 q^{57} +2.64876e9 q^{58} +4.10272e9 q^{59} +1.03084e10 q^{61} +8.49323e9 q^{62} +5.09359e9 q^{63} +1.07374e9 q^{64} +8.21140e9 q^{66} +8.74070e9 q^{67} -6.37681e9 q^{68} -1.16672e10 q^{69} -5.10695e9 q^{71} +3.44202e9 q^{72} +2.69800e10 q^{73} +1.39645e10 q^{74} +1.89633e10 q^{76} -2.34239e10 q^{77} +2.54315e10 q^{78} +2.28871e10 q^{79} -3.89551e10 q^{81} +2.10729e10 q^{82} -2.22128e10 q^{83} -2.63774e10 q^{84} -2.69391e10 q^{86} -4.39706e10 q^{87} -1.58288e10 q^{88} +4.89433e10 q^{89} -7.25461e10 q^{91} +2.24904e10 q^{92} -1.40991e11 q^{93} -9.27934e10 q^{94} -1.78246e10 q^{96} +1.20490e10 q^{97} +1.19697e10 q^{98} -5.07412e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{2} - 56 q^{3} + 2048 q^{4} - 1792 q^{6} + 54712 q^{7} + 65536 q^{8} + 153724 q^{9} - 99576 q^{11} - 57344 q^{12} - 1168496 q^{13} + 1750784 q^{14} + 2097152 q^{16} + 4405002 q^{17} + 4919168 q^{18}+ \cdots - 32072603712 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 0.707107
\(3\) −531.215 −1.26213 −0.631064 − 0.775731i \(-0.717381\pi\)
−0.631064 + 0.775731i \(0.717381\pi\)
\(4\) 1024.00 0.500000
\(5\) 0 0
\(6\) −16998.9 −0.892459
\(7\) 48491.0 1.09049 0.545245 − 0.838276i \(-0.316436\pi\)
0.545245 + 0.838276i \(0.316436\pi\)
\(8\) 32768.0 0.353553
\(9\) 105042. 0.592965
\(10\) 0 0
\(11\) −483056. −0.904352 −0.452176 − 0.891929i \(-0.649352\pi\)
−0.452176 + 0.891929i \(0.649352\pi\)
\(12\) −543964. −0.631064
\(13\) −1.49607e6 −1.11754 −0.558772 − 0.829322i \(-0.688727\pi\)
−0.558772 + 0.829322i \(0.688727\pi\)
\(14\) 1.55171e6 0.771094
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) −6.22735e6 −1.06374 −0.531869 − 0.846827i \(-0.678510\pi\)
−0.531869 + 0.846827i \(0.678510\pi\)
\(18\) 3.36134e6 0.419290
\(19\) 1.85189e7 1.71581 0.857906 − 0.513807i \(-0.171765\pi\)
0.857906 + 0.513807i \(0.171765\pi\)
\(20\) 0 0
\(21\) −2.57591e7 −1.37634
\(22\) −1.54578e7 −0.639474
\(23\) 2.19633e7 0.711533 0.355766 − 0.934575i \(-0.384220\pi\)
0.355766 + 0.934575i \(0.384220\pi\)
\(24\) −1.74068e7 −0.446229
\(25\) 0 0
\(26\) −4.78743e7 −0.790222
\(27\) 3.83032e7 0.513730
\(28\) 4.96548e7 0.545245
\(29\) 8.27736e7 0.749382 0.374691 − 0.927150i \(-0.377749\pi\)
0.374691 + 0.927150i \(0.377749\pi\)
\(30\) 0 0
\(31\) 2.65413e8 1.66507 0.832537 − 0.553970i \(-0.186888\pi\)
0.832537 + 0.553970i \(0.186888\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 2.56606e8 1.14141
\(34\) −1.99275e8 −0.752176
\(35\) 0 0
\(36\) 1.07563e8 0.296483
\(37\) 4.36392e8 1.03459 0.517293 − 0.855808i \(-0.326940\pi\)
0.517293 + 0.855808i \(0.326940\pi\)
\(38\) 5.92604e8 1.21326
\(39\) 7.94736e8 1.41048
\(40\) 0 0
\(41\) 6.58528e8 0.887694 0.443847 − 0.896103i \(-0.353613\pi\)
0.443847 + 0.896103i \(0.353613\pi\)
\(42\) −8.24292e8 −0.973218
\(43\) −8.41846e8 −0.873286 −0.436643 − 0.899635i \(-0.643833\pi\)
−0.436643 + 0.899635i \(0.643833\pi\)
\(44\) −4.94649e8 −0.452176
\(45\) 0 0
\(46\) 7.02826e8 0.503129
\(47\) −2.89979e9 −1.84429 −0.922144 − 0.386846i \(-0.873564\pi\)
−0.922144 + 0.386846i \(0.873564\pi\)
\(48\) −5.57019e8 −0.315532
\(49\) 3.74052e8 0.189170
\(50\) 0 0
\(51\) 3.30806e9 1.34257
\(52\) −1.53198e9 −0.558772
\(53\) 2.52068e9 0.827944 0.413972 − 0.910290i \(-0.364141\pi\)
0.413972 + 0.910290i \(0.364141\pi\)
\(54\) 1.22570e9 0.363262
\(55\) 0 0
\(56\) 1.58895e9 0.385547
\(57\) −9.83749e9 −2.16557
\(58\) 2.64876e9 0.529893
\(59\) 4.10272e9 0.747111 0.373556 − 0.927608i \(-0.378138\pi\)
0.373556 + 0.927608i \(0.378138\pi\)
\(60\) 0 0
\(61\) 1.03084e10 1.56270 0.781350 − 0.624093i \(-0.214531\pi\)
0.781350 + 0.624093i \(0.214531\pi\)
\(62\) 8.49323e9 1.17739
\(63\) 5.09359e9 0.646623
\(64\) 1.07374e9 0.125000
\(65\) 0 0
\(66\) 8.21140e9 0.807097
\(67\) 8.74070e9 0.790924 0.395462 − 0.918482i \(-0.370584\pi\)
0.395462 + 0.918482i \(0.370584\pi\)
\(68\) −6.37681e9 −0.531869
\(69\) −1.16672e10 −0.898045
\(70\) 0 0
\(71\) −5.10695e9 −0.335924 −0.167962 − 0.985794i \(-0.553719\pi\)
−0.167962 + 0.985794i \(0.553719\pi\)
\(72\) 3.44202e9 0.209645
\(73\) 2.69800e10 1.52323 0.761617 − 0.648027i \(-0.224406\pi\)
0.761617 + 0.648027i \(0.224406\pi\)
\(74\) 1.39645e10 0.731563
\(75\) 0 0
\(76\) 1.89633e10 0.857906
\(77\) −2.34239e10 −0.986188
\(78\) 2.54315e10 0.997361
\(79\) 2.28871e10 0.836838 0.418419 − 0.908254i \(-0.362584\pi\)
0.418419 + 0.908254i \(0.362584\pi\)
\(80\) 0 0
\(81\) −3.89551e10 −1.24136
\(82\) 2.10729e10 0.627694
\(83\) −2.22128e10 −0.618975 −0.309487 − 0.950904i \(-0.600157\pi\)
−0.309487 + 0.950904i \(0.600157\pi\)
\(84\) −2.63774e10 −0.688169
\(85\) 0 0
\(86\) −2.69391e10 −0.617506
\(87\) −4.39706e10 −0.945815
\(88\) −1.58288e10 −0.319737
\(89\) 4.89433e10 0.929069 0.464534 − 0.885555i \(-0.346222\pi\)
0.464534 + 0.885555i \(0.346222\pi\)
\(90\) 0 0
\(91\) −7.25461e10 −1.21867
\(92\) 2.24904e10 0.355766
\(93\) −1.40991e11 −2.10154
\(94\) −9.27934e10 −1.30411
\(95\) 0 0
\(96\) −1.78246e10 −0.223115
\(97\) 1.20490e10 0.142464 0.0712322 − 0.997460i \(-0.477307\pi\)
0.0712322 + 0.997460i \(0.477307\pi\)
\(98\) 1.19697e10 0.133764
\(99\) −5.07412e10 −0.536249
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.a.h.1.1 yes 2
5.2 odd 4 50.12.b.e.49.4 4
5.3 odd 4 50.12.b.e.49.1 4
5.4 even 2 50.12.a.g.1.2 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.a.g.1.2 ✓ 2 5.4 even 2
50.12.a.h.1.1 yes 2 1.1 even 1 trivial
50.12.b.e.49.1 4 5.3 odd 4
50.12.b.e.49.4 4 5.2 odd 4