Properties

Label 1656.2.a.o.1.3
Level $1656$
Weight $2$
Character 1656.1
Self dual yes
Analytic conductor $13.223$
Analytic rank $0$
Dimension $4$
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] = [1656,2,Mod(1,1656)] 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("1656.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1656, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1656.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4,0,2,0,0,0,-2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.2232265747\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{13 +4 \sqrt{7}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.277334\) of defining polynomial
Character \(\chi\) \(=\) 1656.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.64575 q^{5} -1.01109 q^{7} +5.21151 q^{11} +2.55467 q^{13} -3.56576 q^{17} +3.09108 q^{19} +1.00000 q^{23} -2.29150 q^{25} -1.44533 q^{29} +1.44533 q^{31} -1.66401 q^{35} +6.10217 q^{37} -4.73683 q^{41} +11.3319 q^{43} -0.0221841 q^{47} -5.97769 q^{49} -7.66794 q^{53} +8.57685 q^{55} +3.28535 q^{59} +7.21151 q^{61} +4.20435 q^{65} +11.4919 q^{67} -0.736834 q^{71} +11.6862 q^{73} -5.26932 q^{77} -2.12043 q^{79} +9.21151 q^{83} -5.86836 q^{85} +7.96660 q^{89} -2.58301 q^{91} +5.08715 q^{95} -7.31369 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 2 q^{7} - 2 q^{11} + 4 q^{13} - 2 q^{17} + 8 q^{19} + 4 q^{23} + 12 q^{25} - 12 q^{29} + 12 q^{31} + 12 q^{35} + 14 q^{37} - 4 q^{41} + 4 q^{43} + 12 q^{47} + 28 q^{49} - 8 q^{53} + 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
\(5\) 1.64575 0.736002 0.368001 − 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) 0 0
\(7\) −1.01109 −0.382157 −0.191078 − 0.981575i \(-0.561198\pi\)
−0.191078 + 0.981575i \(0.561198\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.21151 1.57133 0.785665 − 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(12\) 0 0
\(13\) 2.55467 0.708538 0.354269 − 0.935144i \(-0.384730\pi\)
0.354269 + 0.935144i \(0.384730\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.56576 −0.864824 −0.432412 − 0.901676i \(-0.642337\pi\)
−0.432412 + 0.901676i \(0.642337\pi\)
\(18\) 0 0
\(19\) 3.09108 0.709143 0.354571 − 0.935029i \(-0.384627\pi\)
0.354571 + 0.935029i \(0.384627\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.44533 −0.268391 −0.134196 − 0.990955i \(-0.542845\pi\)
−0.134196 + 0.990955i \(0.542845\pi\)
\(30\) 0 0
\(31\) 1.44533 0.259589 0.129795 − 0.991541i \(-0.458568\pi\)
0.129795 + 0.991541i \(0.458568\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.66401 −0.281268
\(36\) 0 0
\(37\) 6.10217 1.00319 0.501596 − 0.865102i \(-0.332747\pi\)
0.501596 + 0.865102i \(0.332747\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.73683 −0.739769 −0.369885 − 0.929078i \(-0.620603\pi\)
−0.369885 + 0.929078i \(0.620603\pi\)
\(42\) 0 0
\(43\) 11.3319 1.72810 0.864052 − 0.503402i \(-0.167918\pi\)
0.864052 + 0.503402i \(0.167918\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.0221841 −0.00323589 −0.00161794 − 0.999999i \(-0.500515\pi\)
−0.00161794 + 0.999999i \(0.500515\pi\)
\(48\) 0 0
\(49\) −5.97769 −0.853956
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.66794 −1.05327 −0.526636 − 0.850091i \(-0.676547\pi\)
−0.526636 + 0.850091i \(0.676547\pi\)
\(54\) 0 0
\(55\) 8.57685 1.15650
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.28535 0.427716 0.213858 − 0.976865i \(-0.431397\pi\)
0.213858 + 0.976865i \(0.431397\pi\)
\(60\) 0 0
\(61\) 7.21151 0.923340 0.461670 − 0.887052i \(-0.347251\pi\)
0.461670 + 0.887052i \(0.347251\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.20435 0.521485
\(66\) 0 0
\(67\) 11.4919 1.40396 0.701981 − 0.712196i \(-0.252299\pi\)
0.701981 + 0.712196i \(0.252299\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.736834 −0.0874461 −0.0437230 − 0.999044i \(-0.513922\pi\)
−0.0437230 + 0.999044i \(0.513922\pi\)
\(72\) 0 0
\(73\) 11.6862 1.36777 0.683883 − 0.729592i \(-0.260290\pi\)
0.683883 + 0.729592i \(0.260290\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.26932 −0.600495
\(78\) 0 0
\(79\) −2.12043 −0.238567 −0.119283 − 0.992860i \(-0.538060\pi\)
−0.119283 + 0.992860i \(0.538060\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.21151 1.01109 0.505547 − 0.862799i \(-0.331291\pi\)
0.505547 + 0.862799i \(0.331291\pi\)
\(84\) 0 0
\(85\) −5.86836 −0.636513
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.96660 0.844458 0.422229 − 0.906489i \(-0.361248\pi\)
0.422229 + 0.906489i \(0.361248\pi\)
\(90\) 0 0
\(91\) −2.58301 −0.270773
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.08715 0.521931
\(96\) 0 0
\(97\) −7.31369 −0.742592 −0.371296 − 0.928514i \(-0.621087\pi\)
−0.371296 + 0.928514i \(0.621087\pi\)
\(98\) 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 1656.2.a.o.1.3 ✓ 4
3.2 odd 2 1656.2.a.p.1.1 yes 4
4.3 odd 2 3312.2.a.bg.1.4 4
12.11 even 2 3312.2.a.bh.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1656.2.a.o.1.3 ✓ 4 1.1 even 1 trivial
1656.2.a.p.1.1 yes 4 3.2 odd 2
3312.2.a.bg.1.4 4 4.3 odd 2
3312.2.a.bh.1.2 4 12.11 even 2