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Properties

Label 1936.4.a.ba.1.2
Level $1936$
Weight $4$
Character 1936.1
Self dual yes
Analytic conductor $114.228$
Analytic rank $1$
Dimension $2$
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] = [1936,4,Mod(1,1936)] 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("1936.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,10,0,10,0,0] 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: \(114.227697771\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{26}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 121)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.09902\) of defining polynomial
Character \(\chi\) \(=\) 1936.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{3} +5.00000 q^{5} +20.3961 q^{7} -2.00000 q^{9} -61.1882 q^{13} +25.0000 q^{15} +20.3961 q^{17} -101.980 q^{19} +101.980 q^{21} -35.0000 q^{23} -100.000 q^{25} -145.000 q^{27} +203.961 q^{29} -15.0000 q^{31} +101.980 q^{35} -265.000 q^{37} -305.941 q^{39} +101.980 q^{41} -448.714 q^{43} -10.0000 q^{45} -380.000 q^{47} +73.0000 q^{49} +101.980 q^{51} +510.000 q^{53} -509.902 q^{57} -21.0000 q^{59} -203.961 q^{61} -40.7922 q^{63} -305.941 q^{65} -585.000 q^{67} -175.000 q^{69} -313.000 q^{71} -469.110 q^{73} -500.000 q^{75} -611.882 q^{79} -671.000 q^{81} +652.674 q^{83} +101.980 q^{85} +1019.80 q^{87} -185.000 q^{89} -1248.00 q^{91} -75.0000 q^{93} -509.902 q^{95} +785.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{3} + 10 q^{5} - 4 q^{9} + 50 q^{15} - 70 q^{23} - 200 q^{25} - 290 q^{27} - 30 q^{31} - 530 q^{37} - 20 q^{45} - 760 q^{47} + 146 q^{49} + 1020 q^{53} - 42 q^{59} - 1170 q^{67} - 350 q^{69}+ \cdots + 1570 q^{97}+O(q^{100}) \) 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
\(3\) 5.00000 0.962250 0.481125 − 0.876652i \(-0.340228\pi\)
0.481125 + 0.876652i \(0.340228\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 20.3961 1.10128 0.550642 − 0.834741i \(-0.314383\pi\)
0.550642 + 0.834741i \(0.314383\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.0740741
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −61.1882 −1.30543 −0.652714 − 0.757604i \(-0.726370\pi\)
−0.652714 + 0.757604i \(0.726370\pi\)
\(14\) 0 0
\(15\) 25.0000 0.430331
\(16\) 0 0
\(17\) 20.3961 0.290987 0.145493 − 0.989359i \(-0.453523\pi\)
0.145493 + 0.989359i \(0.453523\pi\)
\(18\) 0 0
\(19\) −101.980 −1.23136 −0.615682 − 0.787995i \(-0.711119\pi\)
−0.615682 + 0.787995i \(0.711119\pi\)
\(20\) 0 0
\(21\) 101.980 1.05971
\(22\) 0 0
\(23\) −35.0000 −0.317305 −0.158652 − 0.987335i \(-0.550715\pi\)
−0.158652 + 0.987335i \(0.550715\pi\)
\(24\) 0 0
\(25\) −100.000 −0.800000
\(26\) 0 0
\(27\) −145.000 −1.03353
\(28\) 0 0
\(29\) 203.961 1.30602 0.653010 − 0.757349i \(-0.273506\pi\)
0.653010 + 0.757349i \(0.273506\pi\)
\(30\) 0 0
\(31\) −15.0000 −0.0869058 −0.0434529 − 0.999055i \(-0.513836\pi\)
−0.0434529 + 0.999055i \(0.513836\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 101.980 0.492509
\(36\) 0 0
\(37\) −265.000 −1.17745 −0.588726 − 0.808333i \(-0.700370\pi\)
−0.588726 + 0.808333i \(0.700370\pi\)
\(38\) 0 0
\(39\) −305.941 −1.25615
\(40\) 0 0
\(41\) 101.980 0.388455 0.194228 − 0.980956i \(-0.437780\pi\)
0.194228 + 0.980956i \(0.437780\pi\)
\(42\) 0 0
\(43\) −448.714 −1.59135 −0.795677 − 0.605721i \(-0.792885\pi\)
−0.795677 + 0.605721i \(0.792885\pi\)
\(44\) 0 0
\(45\) −10.0000 −0.0331269
\(46\) 0 0
\(47\) −380.000 −1.17933 −0.589667 − 0.807646i \(-0.700741\pi\)
−0.589667 + 0.807646i \(0.700741\pi\)
\(48\) 0 0
\(49\) 73.0000 0.212828
\(50\) 0 0
\(51\) 101.980 0.280002
\(52\) 0 0
\(53\) 510.000 1.32177 0.660886 − 0.750487i \(-0.270181\pi\)
0.660886 + 0.750487i \(0.270181\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −509.902 −1.18488
\(58\) 0 0
\(59\) −21.0000 −0.0463384 −0.0231692 − 0.999732i \(-0.507376\pi\)
−0.0231692 + 0.999732i \(0.507376\pi\)
\(60\) 0 0
\(61\) −203.961 −0.428107 −0.214053 − 0.976822i \(-0.568667\pi\)
−0.214053 + 0.976822i \(0.568667\pi\)
\(62\) 0 0
\(63\) −40.7922 −0.0815766
\(64\) 0 0
\(65\) −305.941 −0.583805
\(66\) 0 0
\(67\) −585.000 −1.06670 −0.533352 − 0.845894i \(-0.679068\pi\)
−0.533352 + 0.845894i \(0.679068\pi\)
\(68\) 0 0
\(69\) −175.000 −0.305326
\(70\) 0 0
\(71\) −313.000 −0.523187 −0.261593 − 0.965178i \(-0.584248\pi\)
−0.261593 + 0.965178i \(0.584248\pi\)
\(72\) 0 0
\(73\) −469.110 −0.752125 −0.376063 − 0.926594i \(-0.622722\pi\)
−0.376063 + 0.926594i \(0.622722\pi\)
\(74\) 0 0
\(75\) −500.000 −0.769800
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −611.882 −0.871420 −0.435710 − 0.900087i \(-0.643503\pi\)
−0.435710 + 0.900087i \(0.643503\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 0 0
\(83\) 652.674 0.863137 0.431568 − 0.902080i \(-0.357960\pi\)
0.431568 + 0.902080i \(0.357960\pi\)
\(84\) 0 0
\(85\) 101.980 0.130133
\(86\) 0 0
\(87\) 1019.80 1.25672
\(88\) 0 0
\(89\) −185.000 −0.220337 −0.110168 − 0.993913i \(-0.535139\pi\)
−0.110168 + 0.993913i \(0.535139\pi\)
\(90\) 0 0
\(91\) −1248.00 −1.43765
\(92\) 0 0
\(93\) −75.0000 −0.0836251
\(94\) 0 0
\(95\) −509.902 −0.550682
\(96\) 0 0
\(97\) 785.000 0.821698 0.410849 − 0.911703i \(-0.365232\pi\)
0.410849 + 0.911703i \(0.365232\pi\)
\(98\) 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 1936.4.a.ba.1.2 2
4.3 odd 2 121.4.a.d.1.1 ✓ 2
11.10 odd 2 inner 1936.4.a.ba.1.1 2
12.11 even 2 1089.4.a.r.1.2 2
44.3 odd 10 121.4.c.e.9.1 8
44.7 even 10 121.4.c.e.27.2 8
44.15 odd 10 121.4.c.e.27.1 8
44.19 even 10 121.4.c.e.9.2 8
44.27 odd 10 121.4.c.e.3.2 8
44.31 odd 10 121.4.c.e.81.2 8
44.35 even 10 121.4.c.e.81.1 8
44.39 even 10 121.4.c.e.3.1 8
44.43 even 2 121.4.a.d.1.2 yes 2
132.131 odd 2 1089.4.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.a.d.1.1 ✓ 2 4.3 odd 2
121.4.a.d.1.2 yes 2 44.43 even 2
121.4.c.e.3.1 8 44.39 even 10
121.4.c.e.3.2 8 44.27 odd 10
121.4.c.e.9.1 8 44.3 odd 10
121.4.c.e.9.2 8 44.19 even 10
121.4.c.e.27.1 8 44.15 odd 10
121.4.c.e.27.2 8 44.7 even 10
121.4.c.e.81.1 8 44.35 even 10
121.4.c.e.81.2 8 44.31 odd 10
1089.4.a.r.1.1 2 132.131 odd 2
1089.4.a.r.1.2 2 12.11 even 2
1936.4.a.ba.1.1 2 11.10 odd 2 inner
1936.4.a.ba.1.2 2 1.1 even 1 trivial