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Properties

Label 396.4.a.h.1.1
Level $396$
Weight $4$
Character 396.1
Self dual yes
Analytic conductor $23.365$
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] = [396,4,Mod(1,396)] 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("396.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 396.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-12,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.3647563623\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.56776\) of defining polynomial
Character \(\chi\) \(=\) 396.1

$q$-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)
 
\(f(q)\) \(=\) \(q-17.1355 q^{5} -23.1355 q^{7} +11.0000 q^{11} -59.6776 q^{13} +80.8132 q^{17} -11.7289 q^{19} +56.0513 q^{23} +168.626 q^{25} +85.4579 q^{29} -99.8974 q^{31} +396.440 q^{35} +402.982 q^{37} -27.0842 q^{41} -74.0000 q^{43} +408.491 q^{47} +192.253 q^{49} -463.099 q^{53} -188.491 q^{55} +498.813 q^{59} -635.912 q^{61} +1022.61 q^{65} -701.421 q^{67} -27.7435 q^{71} +619.626 q^{73} -254.491 q^{77} -208.322 q^{79} +1300.98 q^{83} -1384.78 q^{85} +1391.18 q^{89} +1380.67 q^{91} +200.982 q^{95} +1051.76 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5} - 24 q^{7} + 22 q^{11} - 8 q^{13} + 28 q^{17} - 68 q^{19} + 268 q^{23} + 70 q^{25} + 260 q^{29} + 112 q^{31} + 392 q^{35} + 316 q^{37} + 124 q^{41} - 148 q^{43} + 572 q^{47} - 150 q^{49}+ \cdots + 500 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −17.1355 −1.53265 −0.766324 − 0.642454i \(-0.777916\pi\)
−0.766324 + 0.642454i \(0.777916\pi\)
\(6\) 0 0
\(7\) −23.1355 −1.24920 −0.624601 − 0.780944i \(-0.714738\pi\)
−0.624601 + 0.780944i \(0.714738\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) −59.6776 −1.27320 −0.636600 − 0.771194i \(-0.719660\pi\)
−0.636600 + 0.771194i \(0.719660\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 80.8132 1.15295 0.576473 − 0.817116i \(-0.304429\pi\)
0.576473 + 0.817116i \(0.304429\pi\)
\(18\) 0 0
\(19\) −11.7289 −0.141621 −0.0708106 − 0.997490i \(-0.522559\pi\)
−0.0708106 + 0.997490i \(0.522559\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 56.0513 0.508152 0.254076 − 0.967184i \(-0.418229\pi\)
0.254076 + 0.967184i \(0.418229\pi\)
\(24\) 0 0
\(25\) 168.626 1.34901
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 85.4579 0.547211 0.273606 − 0.961842i \(-0.411784\pi\)
0.273606 + 0.961842i \(0.411784\pi\)
\(30\) 0 0
\(31\) −99.8974 −0.578778 −0.289389 − 0.957212i \(-0.593452\pi\)
−0.289389 + 0.957212i \(0.593452\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 396.440 1.91459
\(36\) 0 0
\(37\) 402.982 1.79053 0.895267 − 0.445530i \(-0.146985\pi\)
0.895267 + 0.445530i \(0.146985\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −27.0842 −0.103167 −0.0515835 − 0.998669i \(-0.516427\pi\)
−0.0515835 + 0.998669i \(0.516427\pi\)
\(42\) 0 0
\(43\) −74.0000 −0.262439 −0.131220 − 0.991353i \(-0.541889\pi\)
−0.131220 + 0.991353i \(0.541889\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 408.491 1.26776 0.633878 − 0.773433i \(-0.281462\pi\)
0.633878 + 0.773433i \(0.281462\pi\)
\(48\) 0 0
\(49\) 192.253 0.560503
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −463.099 −1.20022 −0.600109 − 0.799919i \(-0.704876\pi\)
−0.600109 + 0.799919i \(0.704876\pi\)
\(54\) 0 0
\(55\) −188.491 −0.462111
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 498.813 1.10068 0.550339 − 0.834942i \(-0.314499\pi\)
0.550339 + 0.834942i \(0.314499\pi\)
\(60\) 0 0
\(61\) −635.912 −1.33476 −0.667379 − 0.744719i \(-0.732584\pi\)
−0.667379 + 0.744719i \(0.732584\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1022.61 1.95137
\(66\) 0 0
\(67\) −701.421 −1.27899 −0.639494 − 0.768796i \(-0.720856\pi\)
−0.639494 + 0.768796i \(0.720856\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −27.7435 −0.0463739 −0.0231870 − 0.999731i \(-0.507381\pi\)
−0.0231870 + 0.999731i \(0.507381\pi\)
\(72\) 0 0
\(73\) 619.626 0.993449 0.496725 − 0.867908i \(-0.334536\pi\)
0.496725 + 0.867908i \(0.334536\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −254.491 −0.376648
\(78\) 0 0
\(79\) −208.322 −0.296685 −0.148342 − 0.988936i \(-0.547394\pi\)
−0.148342 + 0.988936i \(0.547394\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1300.98 1.72050 0.860249 − 0.509875i \(-0.170308\pi\)
0.860249 + 0.509875i \(0.170308\pi\)
\(84\) 0 0
\(85\) −1384.78 −1.76706
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1391.18 1.65691 0.828453 − 0.560058i \(-0.189221\pi\)
0.828453 + 0.560058i \(0.189221\pi\)
\(90\) 0 0
\(91\) 1380.67 1.59048
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 200.982 0.217056
\(96\) 0 0
\(97\) 1051.76 1.10093 0.550463 − 0.834859i \(-0.314451\pi\)
0.550463 + 0.834859i \(0.314451\pi\)
\(98\) 0 0
\(99\) 0 0
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 396.4.a.h.1.1 ✓ 2
3.2 odd 2 396.4.a.j.1.2 yes 2
4.3 odd 2 1584.4.a.y.1.1 2
12.11 even 2 1584.4.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
396.4.a.h.1.1 ✓ 2 1.1 even 1 trivial
396.4.a.j.1.2 yes 2 3.2 odd 2
1584.4.a.y.1.1 2 4.3 odd 2
1584.4.a.bi.1.2 2 12.11 even 2