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Properties

Label 2900.2.c.g
Level $2900$
Weight $2$
Character orbit 2900.c
Analytic conductor $23.157$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [2900,2,Mod(349,2900)] 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("2900.349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.1566165862\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 580)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{9} + ( - \beta_{3} + 1) q^{11} + (\beta_{5} - \beta_{4} - \beta_1) q^{13} + ( - \beta_{5} + 2 \beta_{4}) q^{17} + ( - \beta_{2} - 1) q^{19}+ \cdots + ( - \beta_{3} + 2 \beta_{2} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} + 4 q^{11} - 4 q^{19} - 6 q^{29} - 4 q^{31} + 16 q^{39} + 4 q^{41} + 2 q^{49} + 16 q^{51} - 8 q^{59} - 20 q^{61} + 24 q^{69} - 8 q^{71} + 60 q^{79} - 2 q^{81} - 4 q^{89} - 32 q^{91} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{5} - 9\nu^{4} + 16\nu^{3} + 4\nu^{2} + 38\nu - 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 25\nu^{4} + 24\nu^{3} + 6\nu^{2} - 12\nu - 67 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -32\nu^{5} + 49\nu^{4} - 36\nu^{3} - 78\nu^{2} - 166\nu + 66 ) / 23 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} + 3\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{3} - 3\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} + 6\beta_{4} - 5\beta_{3} - 11\beta_{2} - 11\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\) \(1451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.45161 − 1.45161i
−0.854638 − 0.854638i
0.403032 − 0.403032i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
0 − 2.90321i 0 0 0 − 1.52543i 0 −5.42864 0
349.2 0 − 1.70928i 0 0 0 0.630898i 0 0.0783777 0
349.3 0 − 0.806063i 0 0 0 4.15633i 0 2.35026 0
349.4 0 0.806063i 0 0 0 − 4.15633i 0 2.35026 0
349.5 0 1.70928i 0 0 0 − 0.630898i 0 0.0783777 0
349.6 0 2.90321i 0 0 0 1.52543i 0 −5.42864 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2900.2.c.g 6
5.b even 2 1 inner 2900.2.c.g 6
5.c odd 4 1 580.2.a.d ✓ 3
5.c odd 4 1 2900.2.a.f 3
15.e even 4 1 5220.2.a.w 3
20.e even 4 1 2320.2.a.o 3
40.i odd 4 1 9280.2.a.bh 3
40.k even 4 1 9280.2.a.bt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.a.d ✓ 3 5.c odd 4 1
2320.2.a.o 3 20.e even 4 1
2900.2.a.f 3 5.c odd 4 1
2900.2.c.g 6 1.a even 1 1 trivial
2900.2.c.g 6 5.b even 2 1 inner
5220.2.a.w 3 15.e even 4 1
9280.2.a.bh 3 40.i odd 4 1
9280.2.a.bt 3 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2900, [\chi])\):

\( T_{3}^{6} + 12T_{3}^{4} + 32T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{6} + 20T_{7}^{4} + 48T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} - 2 T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 28 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 52 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{3} + 2 T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T + 1)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} + 2 T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 84 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} + \cdots + 200)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 188 T^{4} + \cdots + 150544 \) Copy content Toggle raw display
$47$ \( T^{6} + 108 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$53$ \( T^{6} + 300 T^{4} + \cdots + 506944 \) Copy content Toggle raw display
$59$ \( (T^{3} + 4 T^{2} - 56 T + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 10 T^{2} + \cdots - 1432)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 88 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( (T^{3} - 30 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 260 T^{4} + \cdots + 300304 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} + \cdots - 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 380 T^{4} + \cdots + 300304 \) Copy content Toggle raw display
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