↑

Properties

Label 50.12.b.e.49.4
Level $50$
Weight $12$
Character 50.49
Analytic conductor $38.417$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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

$q$-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)
 
\(f(q)\) \(=\) \(q+32.0000i q^{2} +531.215i q^{3} -1024.00 q^{4} -16998.9 q^{6} +48491.0i q^{7} -32768.0i q^{8} -105042. q^{9} -483056. q^{11} -543964. i q^{12} +1.49607e6i q^{13} -1.55171e6 q^{14} +1.04858e6 q^{16} -6.22735e6i q^{17} -3.36134e6i q^{18} -1.85189e7 q^{19} -2.57591e7 q^{21} -1.54578e7i q^{22} -2.19633e7i q^{23} +1.74068e7 q^{24} -4.78743e7 q^{26} +3.83032e7i q^{27} -4.96548e7i q^{28} -8.27736e7 q^{29} +2.65413e8 q^{31} +3.35544e7i q^{32} -2.56606e8i q^{33} +1.99275e8 q^{34} +1.07563e8 q^{36} +4.36392e8i q^{37} -5.92604e8i q^{38} -7.94736e8 q^{39} +6.58528e8 q^{41} -8.24292e8i q^{42} +8.41846e8i q^{43} +4.94649e8 q^{44} +7.02826e8 q^{46} -2.89979e9i q^{47} +5.57019e8i q^{48} -3.74052e8 q^{49} +3.30806e9 q^{51} -1.53198e9i q^{52} -2.52068e9i q^{53} -1.22570e9 q^{54} +1.58895e9 q^{56} -9.83749e9i q^{57} -2.64876e9i q^{58} -4.10272e9 q^{59} +1.03084e10 q^{61} +8.49323e9i q^{62} -5.09359e9i q^{63} -1.07374e9 q^{64} +8.21140e9 q^{66} +8.74070e9i q^{67} +6.37681e9i q^{68} +1.16672e10 q^{69} -5.10695e9 q^{71} +3.44202e9i q^{72} -2.69800e10i q^{73} -1.39645e10 q^{74} +1.89633e10 q^{76} -2.34239e10i q^{77} -2.54315e10i q^{78} -2.28871e10 q^{79} -3.89551e10 q^{81} +2.10729e10i q^{82} +2.22128e10i q^{83} +2.63774e10 q^{84} -2.69391e10 q^{86} -4.39706e10i q^{87} +1.58288e10i q^{88} -4.89433e10 q^{89} -7.25461e10 q^{91} +2.24904e10i q^{92} +1.40991e11i q^{93} +9.27934e10 q^{94} -1.78246e10 q^{96} +1.20490e10i q^{97} -1.19697e10i q^{98} +5.07412e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4096 q^{4} - 3584 q^{6} - 307448 q^{9} - 199152 q^{11} - 3501568 q^{14} + 4194304 q^{16} - 23490320 q^{19} - 45605672 q^{21} + 3670016 q^{24} - 74783744 q^{26} - 246047280 q^{29} + 245805728 q^{31}+ \cdots + 64145207424 q^{99}+O(q^{100}) \) 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 \(0.217381\pi\)
−0.775731 + 0.631064i \(0.782619\pi\)
\(4\) −1024.00 −0.500000
\(5\) 0 0
\(6\) −16998.9 −0.892459
\(7\) 48491.0i 1.09049i 0.838276 + 0.545245i \(0.183564\pi\)
−0.838276 + 0.545245i \(0.816436\pi\)
\(8\) − 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 \(0.188727\pi\)
−0.829322 + 0.558772i \(0.811273\pi\)
\(14\) −1.55171e6 −0.771094
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) − 6.22735e6i − 1.06374i −0.846827 − 0.531869i \(-0.821490\pi\)
0.846827 − 0.531869i \(-0.178510\pi\)
\(18\) − 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 \(-0.884220\pi\)
0.934575 − 0.355766i \(-0.115780\pi\)
\(24\) 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 \(0.173060\pi\)
−0.855808 + 0.517293i \(0.826940\pi\)
\(38\) − 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 \(0.143833\pi\)
−0.899635 + 0.436643i \(0.856167\pi\)
\(44\) 4.94649e8 0.452176
\(45\) 0 0
\(46\) 7.02826e8 0.503129
\(47\) − 2.89979e9i − 1.84429i −0.386846 − 0.922144i \(-0.626436\pi\)
0.386846 − 0.922144i \(-0.373564\pi\)
\(48\) 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 \(-0.864141\pi\)
0.910290 − 0.413972i \(-0.135859\pi\)
\(54\) −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 \(0.129416\pi\)
−0.918482 + 0.395462i \(0.870584\pi\)
\(68\) 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 \(-0.724406\pi\)
0.648027 − 0.761617i \(-0.275594\pi\)
\(74\) −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 \(0.100157\pi\)
−0.950904 + 0.309487i \(0.899843\pi\)
\(84\) 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 \(0.0226931\pi\)
−0.997460 + 0.0712322i \(0.977307\pi\)
\(98\) − 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.49.4 4
5.2 odd 4 50.12.a.g.1.2 ✓ 2
5.3 odd 4 50.12.a.h.1.1 yes 2
5.4 even 2 inner 50.12.b.e.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.a.g.1.2 ✓ 2 5.2 odd 4
50.12.a.h.1.1 yes 2 5.3 odd 4
50.12.b.e.49.1 4 5.4 even 2 inner
50.12.b.e.49.4 4 1.1 even 1 trivial