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Properties

Label 2997.1.n.b.1997.1
Level $2997$
Weight $1$
Character 2997.1997
Analytic conductor $1.496$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -111, 37
Inner twists $8$

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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] = [2997,1,Mod(998,2997)] 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("2997.998"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2997, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2997 = 3^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2997.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.49569784286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label 1997.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2997.1997
Dual form 2997.1.n.b.998.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{4} +(1.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{25} +2.00000 q^{28} +1.00000 q^{37} +(-1.50000 + 2.59808i) q^{49} -1.00000 q^{64} +(1.00000 - 1.73205i) q^{67} -2.00000 q^{73} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{4} + 2 q^{7} - q^{16} + q^{25} + 4 q^{28} + 2 q^{37} - 3 q^{49} - 2 q^{64} + 2 q^{67} - 4 q^{73}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2997\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1703\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) 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
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 2.00000
\(29\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 1.00000
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 − 1.73205i 1.00000 − 1.73205i 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 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 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 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 2997.1.n.b.1997.1 2
3.2 odd 2 CM 2997.1.n.b.1997.1 2
9.2 odd 6 inner 2997.1.n.b.998.1 2
9.4 even 3 111.1.d.a.110.1 ✓ 1
9.5 odd 6 111.1.d.a.110.1 ✓ 1
9.7 even 3 inner 2997.1.n.b.998.1 2
36.23 even 6 1776.1.n.a.1553.1 1
36.31 odd 6 1776.1.n.a.1553.1 1
37.36 even 2 RM 2997.1.n.b.1997.1 2
45.4 even 6 2775.1.h.a.776.1 1
45.13 odd 12 2775.1.b.a.2774.2 2
45.14 odd 6 2775.1.h.a.776.1 1
45.22 odd 12 2775.1.b.a.2774.1 2
45.23 even 12 2775.1.b.a.2774.2 2
45.32 even 12 2775.1.b.a.2774.1 2
111.110 odd 2 CM 2997.1.n.b.1997.1 2
333.110 odd 6 inner 2997.1.n.b.998.1 2
333.184 even 6 111.1.d.a.110.1 ✓ 1
333.221 odd 6 111.1.d.a.110.1 ✓ 1
333.295 even 6 inner 2997.1.n.b.998.1 2
1332.887 even 6 1776.1.n.a.1553.1 1
1332.1183 odd 6 1776.1.n.a.1553.1 1
1665.184 even 6 2775.1.h.a.776.1 1
1665.517 odd 12 2775.1.b.a.2774.1 2
1665.554 odd 6 2775.1.h.a.776.1 1
1665.887 even 12 2775.1.b.a.2774.1 2
1665.1183 odd 12 2775.1.b.a.2774.2 2
1665.1553 even 12 2775.1.b.a.2774.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
111.1.d.a.110.1 ✓ 1 9.4 even 3
111.1.d.a.110.1 ✓ 1 9.5 odd 6
111.1.d.a.110.1 ✓ 1 333.184 even 6
111.1.d.a.110.1 ✓ 1 333.221 odd 6
1776.1.n.a.1553.1 1 36.23 even 6
1776.1.n.a.1553.1 1 36.31 odd 6
1776.1.n.a.1553.1 1 1332.887 even 6
1776.1.n.a.1553.1 1 1332.1183 odd 6
2775.1.b.a.2774.1 2 45.22 odd 12
2775.1.b.a.2774.1 2 45.32 even 12
2775.1.b.a.2774.1 2 1665.517 odd 12
2775.1.b.a.2774.1 2 1665.887 even 12
2775.1.b.a.2774.2 2 45.13 odd 12
2775.1.b.a.2774.2 2 45.23 even 12
2775.1.b.a.2774.2 2 1665.1183 odd 12
2775.1.b.a.2774.2 2 1665.1553 even 12
2775.1.h.a.776.1 1 45.4 even 6
2775.1.h.a.776.1 1 45.14 odd 6
2775.1.h.a.776.1 1 1665.184 even 6
2775.1.h.a.776.1 1 1665.554 odd 6
2997.1.n.b.998.1 2 9.2 odd 6 inner
2997.1.n.b.998.1 2 9.7 even 3 inner
2997.1.n.b.998.1 2 333.110 odd 6 inner
2997.1.n.b.998.1 2 333.295 even 6 inner
2997.1.n.b.1997.1 2 1.1 even 1 trivial
2997.1.n.b.1997.1 2 3.2 odd 2 CM
2997.1.n.b.1997.1 2 37.36 even 2 RM
2997.1.n.b.1997.1 2 111.110 odd 2 CM