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Properties

Label 2646.2.d.c.2645.4
Level $2646$
Weight $2$
Character 2646.2645
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [2646,2,Mod(2645,2646)] 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("2646.2645"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,12,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2645.4
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2646.2645
Dual form 2646.2.d.c.2645.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} -3.00000i q^{11} +3.46410i q^{13} +1.00000 q^{16} +3.00000 q^{22} -5.00000 q^{25} -3.46410 q^{26} +9.00000i q^{29} -1.73205i q^{31} +1.00000i q^{32} -8.00000 q^{37} +10.3923 q^{41} +4.00000 q^{43} +3.00000i q^{44} +10.3923 q^{47} -5.00000i q^{50} -3.46410i q^{52} +6.00000i q^{53} -9.00000 q^{58} +5.19615 q^{59} +13.8564i q^{61} +1.73205 q^{62} -1.00000 q^{64} +2.00000 q^{67} +12.0000i q^{71} +5.19615i q^{73} -8.00000i q^{74} -13.0000 q^{79} +10.3923i q^{82} -5.19615 q^{83} +4.00000i q^{86} -3.00000 q^{88} -10.3923 q^{89} +10.3923i q^{94} -8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{16} + 12 q^{22} - 20 q^{25} - 32 q^{37} + 16 q^{43} - 36 q^{58} - 4 q^{64} + 8 q^{67} - 52 q^{79} - 12 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.00000i − 0.904534i −0.891883 − 0.452267i \(-0.850615\pi\)
0.891883 − 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) 0 0
\(29\) 9.00000i 1.67126i 0.549294 + 0.835629i \(0.314897\pi\)
−0.549294 + 0.835629i \(0.685103\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 − 0.155543i \(-0.950287\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923 1.62301 0.811503 − 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 3.00000i 0.452267i
\(45\) 0 0
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 − 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 5.00000i − 0.707107i
\(51\) 0 0
\(52\) − 3.46410i − 0.480384i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −9.00000 −1.18176
\(59\) 5.19615 0.676481 0.338241 − 0.941060i \(-0.390168\pi\)
0.338241 + 0.941060i \(0.390168\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 1.73205 0.219971
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) 5.19615i 0.608164i 0.952646 + 0.304082i \(0.0983496\pi\)
−0.952646 + 0.304082i \(0.901650\pi\)
\(74\) − 8.00000i − 0.929981i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.3923i 1.14764i
\(83\) −5.19615 −0.570352 −0.285176 − 0.958475i \(-0.592052\pi\)
−0.285176 + 0.958475i \(0.592052\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000i 0.431331i
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −10.3923 −1.10158 −0.550791 − 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 10.3923i 1.07188i
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.66025i − 0.879316i −0.898165 − 0.439658i \(-0.855100\pi\)
0.898165 − 0.439658i \(-0.144900\pi\)
\(98\) 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 2646.2.d.c.2645.4 4
3.2 odd 2 inner 2646.2.d.c.2645.2 4
7.4 even 3 378.2.k.a.215.1 ✓ 4
7.5 odd 6 378.2.k.a.269.2 yes 4
7.6 odd 2 inner 2646.2.d.c.2645.3 4
21.5 even 6 378.2.k.a.269.1 yes 4
21.11 odd 6 378.2.k.a.215.2 yes 4
21.20 even 2 inner 2646.2.d.c.2645.1 4
63.4 even 3 1134.2.t.a.593.2 4
63.5 even 6 1134.2.l.d.269.1 4
63.11 odd 6 1134.2.l.d.215.1 4
63.25 even 3 1134.2.l.d.215.2 4
63.32 odd 6 1134.2.t.a.593.1 4
63.40 odd 6 1134.2.l.d.269.2 4
63.47 even 6 1134.2.t.a.1025.2 4
63.61 odd 6 1134.2.t.a.1025.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.a.215.1 ✓ 4 7.4 even 3
378.2.k.a.215.2 yes 4 21.11 odd 6
378.2.k.a.269.1 yes 4 21.5 even 6
378.2.k.a.269.2 yes 4 7.5 odd 6
1134.2.l.d.215.1 4 63.11 odd 6
1134.2.l.d.215.2 4 63.25 even 3
1134.2.l.d.269.1 4 63.5 even 6
1134.2.l.d.269.2 4 63.40 odd 6
1134.2.t.a.593.1 4 63.32 odd 6
1134.2.t.a.593.2 4 63.4 even 3
1134.2.t.a.1025.1 4 63.61 odd 6
1134.2.t.a.1025.2 4 63.47 even 6
2646.2.d.c.2645.1 4 21.20 even 2 inner
2646.2.d.c.2645.2 4 3.2 odd 2 inner
2646.2.d.c.2645.3 4 7.6 odd 2 inner
2646.2.d.c.2645.4 4 1.1 even 1 trivial