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Properties

Label 648.2.a.e.1.1
Level $648$
Weight $2$
Character 648.1
Self dual yes
Analytic conductor $5.174$
Analytic rank $1$
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] = [648,2,Mod(1,648)] 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("648.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0] 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: \(5.17430605098\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 648.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.73205 q^{5} +3.46410 q^{7} -2.00000 q^{11} -2.46410 q^{13} -2.26795 q^{17} -7.46410 q^{19} +4.92820 q^{23} +8.92820 q^{25} -4.26795 q^{29} -10.9282 q^{31} -12.9282 q^{35} -0.464102 q^{37} -6.92820 q^{41} -4.53590 q^{43} -6.92820 q^{47} +5.00000 q^{49} +10.9282 q^{53} +7.46410 q^{55} -8.00000 q^{59} +10.4641 q^{61} +9.19615 q^{65} -0.535898 q^{67} +2.00000 q^{71} +1.00000 q^{73} -6.92820 q^{77} +0.535898 q^{79} +2.92820 q^{83} +8.46410 q^{85} +5.19615 q^{89} -8.53590 q^{91} +27.8564 q^{95} -11.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 4 q^{11} + 2 q^{13} - 8 q^{17} - 8 q^{19} - 4 q^{23} + 4 q^{25} - 12 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{43} + 10 q^{49} + 8 q^{53} + 8 q^{55} - 16 q^{59} + 14 q^{61} + 8 q^{65}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(5\) −3.73205 −1.66902 −0.834512 − 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 − 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −2.46410 −0.683419 −0.341709 − 0.939806i \(-0.611006\pi\)
−0.341709 + 0.939806i \(0.611006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.26795 −0.550058 −0.275029 − 0.961436i \(-0.588688\pi\)
−0.275029 + 0.961436i \(0.588688\pi\)
\(18\) 0 0
\(19\) −7.46410 −1.71238 −0.856191 − 0.516659i \(-0.827175\pi\)
−0.856191 + 0.516659i \(0.827175\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.92820 1.02760 0.513801 − 0.857910i \(-0.328237\pi\)
0.513801 + 0.857910i \(0.328237\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.26795 −0.792538 −0.396269 − 0.918134i \(-0.629695\pi\)
−0.396269 + 0.918134i \(0.629695\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 − 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.9282 −2.18527
\(36\) 0 0
\(37\) −0.464102 −0.0762978 −0.0381489 − 0.999272i \(-0.512146\pi\)
−0.0381489 + 0.999272i \(0.512146\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 − 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) −4.53590 −0.691718 −0.345859 − 0.938286i \(-0.612412\pi\)
−0.345859 + 0.938286i \(0.612412\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 − 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.9282 1.50110 0.750552 − 0.660811i \(-0.229788\pi\)
0.750552 + 0.660811i \(0.229788\pi\)
\(54\) 0 0
\(55\) 7.46410 1.00646
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 10.4641 1.33979 0.669895 − 0.742455i \(-0.266339\pi\)
0.669895 + 0.742455i \(0.266339\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.19615 1.14064
\(66\) 0 0
\(67\) −0.535898 −0.0654704 −0.0327352 − 0.999464i \(-0.510422\pi\)
−0.0327352 + 0.999464i \(0.510422\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) 0.535898 0.0602933 0.0301466 − 0.999545i \(-0.490403\pi\)
0.0301466 + 0.999545i \(0.490403\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.92820 0.321412 0.160706 − 0.987002i \(-0.448623\pi\)
0.160706 + 0.987002i \(0.448623\pi\)
\(84\) 0 0
\(85\) 8.46410 0.918061
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 − 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −8.53590 −0.894805
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 27.8564 2.85801
\(96\) 0 0
\(97\) −11.8564 −1.20384 −0.601918 − 0.798558i \(-0.705597\pi\)
−0.601918 + 0.798558i \(0.705597\pi\)
\(98\) 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 648.2.a.e.1.1 ✓ 2
3.2 odd 2 648.2.a.h.1.2 yes 2
4.3 odd 2 1296.2.a.m.1.1 2
8.3 odd 2 5184.2.a.bz.1.2 2
8.5 even 2 5184.2.a.cb.1.2 2
9.2 odd 6 648.2.i.i.433.1 4
9.4 even 3 648.2.i.j.217.2 4
9.5 odd 6 648.2.i.i.217.1 4
9.7 even 3 648.2.i.j.433.2 4
12.11 even 2 1296.2.a.q.1.2 2
24.5 odd 2 5184.2.a.bg.1.1 2
24.11 even 2 5184.2.a.bi.1.1 2
36.7 odd 6 1296.2.i.t.433.2 4
36.11 even 6 1296.2.i.r.433.1 4
36.23 even 6 1296.2.i.r.865.1 4
36.31 odd 6 1296.2.i.t.865.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.2.a.e.1.1 ✓ 2 1.1 even 1 trivial
648.2.a.h.1.2 yes 2 3.2 odd 2
648.2.i.i.217.1 4 9.5 odd 6
648.2.i.i.433.1 4 9.2 odd 6
648.2.i.j.217.2 4 9.4 even 3
648.2.i.j.433.2 4 9.7 even 3
1296.2.a.m.1.1 2 4.3 odd 2
1296.2.a.q.1.2 2 12.11 even 2
1296.2.i.r.433.1 4 36.11 even 6
1296.2.i.r.865.1 4 36.23 even 6
1296.2.i.t.433.2 4 36.7 odd 6
1296.2.i.t.865.2 4 36.31 odd 6
5184.2.a.bg.1.1 2 24.5 odd 2
5184.2.a.bi.1.1 2 24.11 even 2
5184.2.a.bz.1.2 2 8.3 odd 2
5184.2.a.cb.1.2 2 8.5 even 2