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Properties

Label 1008.2.k.a.881.2
Level $1008$
Weight $2$
Character 1008.881
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
CM discriminant -7
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] = [1008,2,Mod(881,1008)] 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("1008.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0] 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: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 881.2
Root \(-1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 1008.881
Dual form 1008.2.k.a.881.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.64575 q^{7} +0.913230i q^{11} +9.39851i q^{23} -5.00000 q^{25} +6.06910i q^{29} -10.5830 q^{37} -5.29150 q^{43} +7.00000 q^{49} +14.5544i q^{53} +4.00000 q^{67} -7.57205i q^{71} -2.41618i q^{77} -8.00000 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{25} + 28 q^{49} + 16 q^{67} - 32 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.913230i 0.275349i 0.990478 + 0.137675i \(0.0439628\pi\)
−0.990478 + 0.137675i \(0.956037\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 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
\(22\) 0 0
\(23\) 9.39851i 1.95973i 0.199673 + 0.979863i \(0.436012\pi\)
−0.199673 + 0.979863i \(0.563988\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.06910i 1.12700i 0.826115 + 0.563502i \(0.190546\pi\)
−0.826115 + 0.563502i \(0.809454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.5830 −1.73984 −0.869918 − 0.493197i \(-0.835828\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −5.29150 −0.806947 −0.403473 − 0.914991i \(-0.632197\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.5544i 1.99920i 0.0283132 + 0.999599i \(0.490986\pi\)
−0.0283132 + 0.999599i \(0.509014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.57205i − 0.898637i −0.893372 − 0.449319i \(-0.851667\pi\)
0.893372 − 0.449319i \(-0.148333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.41618i − 0.275349i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 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 1008.2.k.a.881.2 4
3.2 odd 2 inner 1008.2.k.a.881.1 4
4.3 odd 2 63.2.c.a.62.4 yes 4
7.6 odd 2 CM 1008.2.k.a.881.2 4
8.3 odd 2 4032.2.k.c.3905.4 4
8.5 even 2 4032.2.k.b.3905.1 4
12.11 even 2 63.2.c.a.62.1 ✓ 4
20.3 even 4 1575.2.g.d.1574.7 8
20.7 even 4 1575.2.g.d.1574.2 8
20.19 odd 2 1575.2.b.a.251.1 4
21.20 even 2 inner 1008.2.k.a.881.1 4
24.5 odd 2 4032.2.k.b.3905.2 4
24.11 even 2 4032.2.k.c.3905.3 4
28.3 even 6 441.2.p.b.215.1 8
28.11 odd 6 441.2.p.b.215.1 8
28.19 even 6 441.2.p.b.80.4 8
28.23 odd 6 441.2.p.b.80.4 8
28.27 even 2 63.2.c.a.62.4 yes 4
36.7 odd 6 567.2.o.f.377.4 8
36.11 even 6 567.2.o.f.377.1 8
36.23 even 6 567.2.o.f.188.4 8
36.31 odd 6 567.2.o.f.188.1 8
56.13 odd 2 4032.2.k.b.3905.1 4
56.27 even 2 4032.2.k.c.3905.4 4
60.23 odd 4 1575.2.g.d.1574.1 8
60.47 odd 4 1575.2.g.d.1574.8 8
60.59 even 2 1575.2.b.a.251.4 4
84.11 even 6 441.2.p.b.215.4 8
84.23 even 6 441.2.p.b.80.1 8
84.47 odd 6 441.2.p.b.80.1 8
84.59 odd 6 441.2.p.b.215.4 8
84.83 odd 2 63.2.c.a.62.1 ✓ 4
140.27 odd 4 1575.2.g.d.1574.2 8
140.83 odd 4 1575.2.g.d.1574.7 8
140.139 even 2 1575.2.b.a.251.1 4
168.83 odd 2 4032.2.k.c.3905.3 4
168.125 even 2 4032.2.k.b.3905.2 4
252.83 odd 6 567.2.o.f.377.1 8
252.139 even 6 567.2.o.f.188.1 8
252.167 odd 6 567.2.o.f.188.4 8
252.223 even 6 567.2.o.f.377.4 8
420.83 even 4 1575.2.g.d.1574.1 8
420.167 even 4 1575.2.g.d.1574.8 8
420.419 odd 2 1575.2.b.a.251.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.1 ✓ 4 12.11 even 2
63.2.c.a.62.1 ✓ 4 84.83 odd 2
63.2.c.a.62.4 yes 4 4.3 odd 2
63.2.c.a.62.4 yes 4 28.27 even 2
441.2.p.b.80.1 8 84.23 even 6
441.2.p.b.80.1 8 84.47 odd 6
441.2.p.b.80.4 8 28.19 even 6
441.2.p.b.80.4 8 28.23 odd 6
441.2.p.b.215.1 8 28.3 even 6
441.2.p.b.215.1 8 28.11 odd 6
441.2.p.b.215.4 8 84.11 even 6
441.2.p.b.215.4 8 84.59 odd 6
567.2.o.f.188.1 8 36.31 odd 6
567.2.o.f.188.1 8 252.139 even 6
567.2.o.f.188.4 8 36.23 even 6
567.2.o.f.188.4 8 252.167 odd 6
567.2.o.f.377.1 8 36.11 even 6
567.2.o.f.377.1 8 252.83 odd 6
567.2.o.f.377.4 8 36.7 odd 6
567.2.o.f.377.4 8 252.223 even 6
1008.2.k.a.881.1 4 3.2 odd 2 inner
1008.2.k.a.881.1 4 21.20 even 2 inner
1008.2.k.a.881.2 4 1.1 even 1 trivial
1008.2.k.a.881.2 4 7.6 odd 2 CM
1575.2.b.a.251.1 4 20.19 odd 2
1575.2.b.a.251.1 4 140.139 even 2
1575.2.b.a.251.4 4 60.59 even 2
1575.2.b.a.251.4 4 420.419 odd 2
1575.2.g.d.1574.1 8 60.23 odd 4
1575.2.g.d.1574.1 8 420.83 even 4
1575.2.g.d.1574.2 8 20.7 even 4
1575.2.g.d.1574.2 8 140.27 odd 4
1575.2.g.d.1574.7 8 20.3 even 4
1575.2.g.d.1574.7 8 140.83 odd 4
1575.2.g.d.1574.8 8 60.47 odd 4
1575.2.g.d.1574.8 8 420.167 even 4
4032.2.k.b.3905.1 4 8.5 even 2
4032.2.k.b.3905.1 4 56.13 odd 2
4032.2.k.b.3905.2 4 24.5 odd 2
4032.2.k.b.3905.2 4 168.125 even 2
4032.2.k.c.3905.3 4 24.11 even 2
4032.2.k.c.3905.3 4 168.83 odd 2
4032.2.k.c.3905.4 4 8.3 odd 2
4032.2.k.c.3905.4 4 56.27 even 2