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Properties

Label 50.12.b.e.49.1
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 49.1
Root \(-50.8215i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.12.b.e.49.4

$q$-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)
 
\(f(q)\) \(=\) \(q-32.0000i q^{2} -531.215i q^{3} -1024.00 q^{4} -16998.9 q^{6} -48491.0i q^{7} +32768.0i q^{8} -105042. q^{9} -483056. q^{11} +543964. i q^{12} -1.49607e6i q^{13} -1.55171e6 q^{14} +1.04858e6 q^{16} +6.22735e6i q^{17} +3.36134e6i q^{18} -1.85189e7 q^{19} -2.57591e7 q^{21} +1.54578e7i q^{22} +2.19633e7i q^{23} +1.74068e7 q^{24} -4.78743e7 q^{26} -3.83032e7i q^{27} +4.96548e7i q^{28} -8.27736e7 q^{29} +2.65413e8 q^{31} -3.35544e7i q^{32} +2.56606e8i q^{33} +1.99275e8 q^{34} +1.07563e8 q^{36} -4.36392e8i q^{37} +5.92604e8i q^{38} -7.94736e8 q^{39} +6.58528e8 q^{41} +8.24292e8i q^{42} -8.41846e8i q^{43} +4.94649e8 q^{44} +7.02826e8 q^{46} +2.89979e9i q^{47} -5.57019e8i q^{48} -3.74052e8 q^{49} +3.30806e9 q^{51} +1.53198e9i q^{52} +2.52068e9i q^{53} -1.22570e9 q^{54} +1.58895e9 q^{56} +9.83749e9i q^{57} +2.64876e9i q^{58} -4.10272e9 q^{59} +1.03084e10 q^{61} -8.49323e9i q^{62} +5.09359e9i q^{63} -1.07374e9 q^{64} +8.21140e9 q^{66} -8.74070e9i q^{67} -6.37681e9i q^{68} +1.16672e10 q^{69} -5.10695e9 q^{71} -3.44202e9i q^{72} +2.69800e10i q^{73} -1.39645e10 q^{74} +1.89633e10 q^{76} +2.34239e10i q^{77} +2.54315e10i q^{78} -2.28871e10 q^{79} -3.89551e10 q^{81} -2.10729e10i q^{82} -2.22128e10i q^{83} +2.63774e10 q^{84} -2.69391e10 q^{86} +4.39706e10i q^{87} -1.58288e10i q^{88} -4.89433e10 q^{89} -7.25461e10 q^{91} -2.24904e10i q^{92} -1.40991e11i q^{93} +9.27934e10 q^{94} -1.78246e10 q^{96} -1.20490e10i q^{97} +1.19697e10i q^{98} +5.07412e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4096 q^{4} - 3584 q^{6} - 307448 q^{9} - 199152 q^{11} - 3501568 q^{14} + 4194304 q^{16} - 23490320 q^{19} - 45605672 q^{21} + 3670016 q^{24} - 74783744 q^{26} - 246047280 q^{29} + 245805728 q^{31}+ \cdots + 64145207424 q^{99}+O(q^{100}) \) 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
\(3\) − 531.215i − 1.26213i −0.775731 − 0.631064i \(-0.782619\pi\)
0.775731 − 0.631064i \(-0.217381\pi\)
\(4\) −1024.00 −0.500000
\(5\) 0 0
\(6\) −16998.9 −0.892459
\(7\) − 48491.0i − 1.09049i −0.838276 − 0.545245i \(-0.816436\pi\)
0.838276 − 0.545245i \(-0.183564\pi\)
\(8\) 32768.0i 0.353553i
\(9\) −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.i 0.631064i
\(13\) − 1.49607e6i − 1.11754i −0.829322 − 0.558772i \(-0.811273\pi\)
0.829322 − 0.558772i \(-0.188727\pi\)
\(14\) −1.55171e6 −0.771094
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) 6.22735e6i 1.06374i 0.846827 + 0.531869i \(0.178510\pi\)
−0.846827 + 0.531869i \(0.821490\pi\)
\(18\) 3.36134e6i 0.419290i
\(19\) −1.85189e7 −1.71581 −0.857906 − 0.513807i \(-0.828235\pi\)
−0.857906 + 0.513807i \(0.828235\pi\)
\(20\) 0 0
\(21\) −2.57591e7 −1.37634
\(22\) 1.54578e7i 0.639474i
\(23\) 2.19633e7i 0.711533i 0.934575 + 0.355766i \(0.115780\pi\)
−0.934575 + 0.355766i \(0.884220\pi\)
\(24\) 1.74068e7 0.446229
\(25\) 0 0
\(26\) −4.78743e7 −0.790222
\(27\) − 3.83032e7i − 0.513730i
\(28\) 4.96548e7i 0.545245i
\(29\) −8.27736e7 −0.749382 −0.374691 − 0.927150i \(-0.622251\pi\)
−0.374691 + 0.927150i \(0.622251\pi\)
\(30\) 0 0
\(31\) 2.65413e8 1.66507 0.832537 − 0.553970i \(-0.186888\pi\)
0.832537 + 0.553970i \(0.186888\pi\)
\(32\) − 3.35544e7i − 0.176777i
\(33\) 2.56606e8i 1.14141i
\(34\) 1.99275e8 0.752176
\(35\) 0 0
\(36\) 1.07563e8 0.296483
\(37\) − 4.36392e8i − 1.03459i −0.855808 − 0.517293i \(-0.826940\pi\)
0.855808 − 0.517293i \(-0.173060\pi\)
\(38\) 5.92604e8i 1.21326i
\(39\) −7.94736e8 −1.41048
\(40\) 0 0
\(41\) 6.58528e8 0.887694 0.443847 − 0.896103i \(-0.353613\pi\)
0.443847 + 0.896103i \(0.353613\pi\)
\(42\) 8.24292e8i 0.973218i
\(43\) − 8.41846e8i − 0.873286i −0.899635 − 0.436643i \(-0.856167\pi\)
0.899635 − 0.436643i \(-0.143833\pi\)
\(44\) 4.94649e8 0.452176
\(45\) 0 0
\(46\) 7.02826e8 0.503129
\(47\) 2.89979e9i 1.84429i 0.386846 + 0.922144i \(0.373564\pi\)
−0.386846 + 0.922144i \(0.626436\pi\)
\(48\) − 5.57019e8i − 0.315532i
\(49\) −3.74052e8 −0.189170
\(50\) 0 0
\(51\) 3.30806e9 1.34257
\(52\) 1.53198e9i 0.558772i
\(53\) 2.52068e9i 0.827944i 0.910290 + 0.413972i \(0.135859\pi\)
−0.910290 + 0.413972i \(0.864141\pi\)
\(54\) −1.22570e9 −0.363262
\(55\) 0 0
\(56\) 1.58895e9 0.385547
\(57\) 9.83749e9i 2.16557i
\(58\) 2.64876e9i 0.529893i
\(59\) −4.10272e9 −0.747111 −0.373556 − 0.927608i \(-0.621862\pi\)
−0.373556 + 0.927608i \(0.621862\pi\)
\(60\) 0 0
\(61\) 1.03084e10 1.56270 0.781350 − 0.624093i \(-0.214531\pi\)
0.781350 + 0.624093i \(0.214531\pi\)
\(62\) − 8.49323e9i − 1.17739i
\(63\) 5.09359e9i 0.646623i
\(64\) −1.07374e9 −0.125000
\(65\) 0 0
\(66\) 8.21140e9 0.807097
\(67\) − 8.74070e9i − 0.790924i −0.918482 − 0.395462i \(-0.870584\pi\)
0.918482 − 0.395462i \(-0.129416\pi\)
\(68\) − 6.37681e9i − 0.531869i
\(69\) 1.16672e10 0.898045
\(70\) 0 0
\(71\) −5.10695e9 −0.335924 −0.167962 − 0.985794i \(-0.553719\pi\)
−0.167962 + 0.985794i \(0.553719\pi\)
\(72\) − 3.44202e9i − 0.209645i
\(73\) 2.69800e10i 1.52323i 0.648027 + 0.761617i \(0.275594\pi\)
−0.648027 + 0.761617i \(0.724406\pi\)
\(74\) −1.39645e10 −0.731563
\(75\) 0 0
\(76\) 1.89633e10 0.857906
\(77\) 2.34239e10i 0.986188i
\(78\) 2.54315e10i 0.997361i
\(79\) −2.28871e10 −0.836838 −0.418419 − 0.908254i \(-0.637416\pi\)
−0.418419 + 0.908254i \(0.637416\pi\)
\(80\) 0 0
\(81\) −3.89551e10 −1.24136
\(82\) − 2.10729e10i − 0.627694i
\(83\) − 2.22128e10i − 0.618975i −0.950904 − 0.309487i \(-0.899843\pi\)
0.950904 − 0.309487i \(-0.100157\pi\)
\(84\) 2.63774e10 0.688169
\(85\) 0 0
\(86\) −2.69391e10 −0.617506
\(87\) 4.39706e10i 0.945815i
\(88\) − 1.58288e10i − 0.319737i
\(89\) −4.89433e10 −0.929069 −0.464534 − 0.885555i \(-0.653778\pi\)
−0.464534 + 0.885555i \(0.653778\pi\)
\(90\) 0 0
\(91\) −7.25461e10 −1.21867
\(92\) − 2.24904e10i − 0.355766i
\(93\) − 1.40991e11i − 2.10154i
\(94\) 9.27934e10 1.30411
\(95\) 0 0
\(96\) −1.78246e10 −0.223115
\(97\) − 1.20490e10i − 0.142464i −0.997460 − 0.0712322i \(-0.977307\pi\)
0.997460 − 0.0712322i \(-0.0226931\pi\)
\(98\) 1.19697e10i 0.133764i
\(99\) 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.b.e.49.1 4
5.2 odd 4 50.12.a.h.1.1 yes 2
5.3 odd 4 50.12.a.g.1.2 ✓ 2
5.4 even 2 inner 50.12.b.e.49.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.a.g.1.2 ✓ 2 5.3 odd 4
50.12.a.h.1.1 yes 2 5.2 odd 4
50.12.b.e.49.1 4 1.1 even 1 trivial
50.12.b.e.49.4 4 5.4 even 2 inner