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Properties

Label 196.6.a.f.1.1
Level $196$
Weight $6$
Character 196.1
Self dual yes
Analytic conductor $31.435$
Analytic rank $1$
Dimension $1$
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] = [196,6,Mod(1,196)] 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("196.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 196.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,16] 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: \(31.4352286833\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 196.1

$q$-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)
 
\(f(q)\) \(=\) \(q+16.0000 q^{3} -16.0000 q^{5} +13.0000 q^{9} -76.0000 q^{11} -880.000 q^{13} -256.000 q^{15} +1056.00 q^{17} -1936.00 q^{19} +936.000 q^{23} -2869.00 q^{25} -3680.00 q^{27} -3982.00 q^{29} -1568.00 q^{31} -1216.00 q^{33} +4938.00 q^{37} -14080.0 q^{39} +15840.0 q^{41} -16412.0 q^{43} -208.000 q^{45} +20768.0 q^{47} +16896.0 q^{51} -37402.0 q^{53} +1216.00 q^{55} -30976.0 q^{57} -21136.0 q^{59} +2992.00 q^{61} +14080.0 q^{65} -45836.0 q^{67} +14976.0 q^{69} -49840.0 q^{71} +56320.0 q^{73} -45904.0 q^{75} +40744.0 q^{79} -62039.0 q^{81} -112464. q^{83} -16896.0 q^{85} -63712.0 q^{87} -64256.0 q^{89} -25088.0 q^{93} +30976.0 q^{95} +2272.00 q^{97} -988.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 16.0000 1.02640 0.513200 − 0.858269i \(-0.328460\pi\)
0.513200 + 0.858269i \(0.328460\pi\)
\(4\) 0 0
\(5\) −16.0000 −0.286217 −0.143108 − 0.989707i \(-0.545710\pi\)
−0.143108 + 0.989707i \(0.545710\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 13.0000 0.0534979
\(10\) 0 0
\(11\) −76.0000 −0.189379 −0.0946895 − 0.995507i \(-0.530186\pi\)
−0.0946895 + 0.995507i \(0.530186\pi\)
\(12\) 0 0
\(13\) −880.000 −1.44419 −0.722095 − 0.691794i \(-0.756821\pi\)
−0.722095 + 0.691794i \(0.756821\pi\)
\(14\) 0 0
\(15\) −256.000 −0.293773
\(16\) 0 0
\(17\) 1056.00 0.886220 0.443110 − 0.896467i \(-0.353875\pi\)
0.443110 + 0.896467i \(0.353875\pi\)
\(18\) 0 0
\(19\) −1936.00 −1.23033 −0.615165 − 0.788399i \(-0.710910\pi\)
−0.615165 + 0.788399i \(0.710910\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 936.000 0.368940 0.184470 − 0.982838i \(-0.440943\pi\)
0.184470 + 0.982838i \(0.440943\pi\)
\(24\) 0 0
\(25\) −2869.00 −0.918080
\(26\) 0 0
\(27\) −3680.00 −0.971490
\(28\) 0 0
\(29\) −3982.00 −0.879238 −0.439619 − 0.898184i \(-0.644886\pi\)
−0.439619 + 0.898184i \(0.644886\pi\)
\(30\) 0 0
\(31\) −1568.00 −0.293050 −0.146525 − 0.989207i \(-0.546809\pi\)
−0.146525 + 0.989207i \(0.546809\pi\)
\(32\) 0 0
\(33\) −1216.00 −0.194379
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4938.00 0.592989 0.296495 − 0.955035i \(-0.404182\pi\)
0.296495 + 0.955035i \(0.404182\pi\)
\(38\) 0 0
\(39\) −14080.0 −1.48232
\(40\) 0 0
\(41\) 15840.0 1.47162 0.735810 − 0.677188i \(-0.236802\pi\)
0.735810 + 0.677188i \(0.236802\pi\)
\(42\) 0 0
\(43\) −16412.0 −1.35360 −0.676800 − 0.736167i \(-0.736634\pi\)
−0.676800 + 0.736167i \(0.736634\pi\)
\(44\) 0 0
\(45\) −208.000 −0.0153120
\(46\) 0 0
\(47\) 20768.0 1.37136 0.685678 − 0.727905i \(-0.259506\pi\)
0.685678 + 0.727905i \(0.259506\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 16896.0 0.909617
\(52\) 0 0
\(53\) −37402.0 −1.82896 −0.914482 − 0.404627i \(-0.867401\pi\)
−0.914482 + 0.404627i \(0.867401\pi\)
\(54\) 0 0
\(55\) 1216.00 0.0542034
\(56\) 0 0
\(57\) −30976.0 −1.26281
\(58\) 0 0
\(59\) −21136.0 −0.790483 −0.395242 − 0.918577i \(-0.629339\pi\)
−0.395242 + 0.918577i \(0.629339\pi\)
\(60\) 0 0
\(61\) 2992.00 0.102953 0.0514763 − 0.998674i \(-0.483607\pi\)
0.0514763 + 0.998674i \(0.483607\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14080.0 0.413351
\(66\) 0 0
\(67\) −45836.0 −1.24744 −0.623720 − 0.781648i \(-0.714379\pi\)
−0.623720 + 0.781648i \(0.714379\pi\)
\(68\) 0 0
\(69\) 14976.0 0.378681
\(70\) 0 0
\(71\) −49840.0 −1.17336 −0.586681 − 0.809818i \(-0.699566\pi\)
−0.586681 + 0.809818i \(0.699566\pi\)
\(72\) 0 0
\(73\) 56320.0 1.23696 0.618480 − 0.785801i \(-0.287749\pi\)
0.618480 + 0.785801i \(0.287749\pi\)
\(74\) 0 0
\(75\) −45904.0 −0.942318
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 40744.0 0.734507 0.367253 − 0.930121i \(-0.380298\pi\)
0.367253 + 0.930121i \(0.380298\pi\)
\(80\) 0 0
\(81\) −62039.0 −1.05064
\(82\) 0 0
\(83\) −112464. −1.79192 −0.895959 − 0.444136i \(-0.853511\pi\)
−0.895959 + 0.444136i \(0.853511\pi\)
\(84\) 0 0
\(85\) −16896.0 −0.253651
\(86\) 0 0
\(87\) −63712.0 −0.902450
\(88\) 0 0
\(89\) −64256.0 −0.859882 −0.429941 − 0.902857i \(-0.641466\pi\)
−0.429941 + 0.902857i \(0.641466\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −25088.0 −0.300787
\(94\) 0 0
\(95\) 30976.0 0.352141
\(96\) 0 0
\(97\) 2272.00 0.0245177 0.0122588 − 0.999925i \(-0.496098\pi\)
0.0122588 + 0.999925i \(0.496098\pi\)
\(98\) 0 0
\(99\) −988.000 −0.0101314
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 196.6.a.f.1.1 yes 1
4.3 odd 2 784.6.a.b.1.1 1
7.2 even 3 196.6.e.c.165.1 2
7.3 odd 6 196.6.e.h.177.1 2
7.4 even 3 196.6.e.c.177.1 2
7.5 odd 6 196.6.e.h.165.1 2
7.6 odd 2 196.6.a.c.1.1 ✓ 1
28.27 even 2 784.6.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.6.a.c.1.1 ✓ 1 7.6 odd 2
196.6.a.f.1.1 yes 1 1.1 even 1 trivial
196.6.e.c.165.1 2 7.2 even 3
196.6.e.c.177.1 2 7.4 even 3
196.6.e.h.165.1 2 7.5 odd 6
196.6.e.h.177.1 2 7.3 odd 6
784.6.a.b.1.1 1 4.3 odd 2
784.6.a.j.1.1 1 28.27 even 2