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Properties

Label 3700.2.d.h
Level $3700$
Weight $2$
Character orbit 3700.d
Analytic conductor $29.545$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3700,2,Mod(149,3700)] 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("3700.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3700.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,2,0,-8,0,0,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.5446487479\)
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: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 740)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{5} + 2 \beta_{4}) q^{7} + ( - \beta_{2} + \beta_1) q^{9} + ( - 2 \beta_{2} - 2) q^{11} + ( - \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{13} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{17}+ \cdots + ( - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{9} - 8 q^{11} - 36 q^{19} + 24 q^{21} + 4 q^{29} - 4 q^{39} - 12 q^{41} - 46 q^{49} - 44 q^{51} - 16 q^{59} + 20 q^{61} + 8 q^{69} - 64 q^{71} + 16 q^{79} - 26 q^{81} + 12 q^{89} - 60 q^{91}+ \cdots + 16 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.45161 + 1.45161i
0.403032 − 0.403032i
−0.854638 + 0.854638i
−0.854638 − 0.854638i
0.403032 + 0.403032i
1.45161 − 1.45161i
0 − 2.21432i 0 0 0 2.83654i 0 −1.90321 0
149.2 0 − 1.67513i 0 0 0 4.63752i 0 0.193937 0
149.3 0 − 0.539189i 0 0 0 − 3.80098i 0 2.70928 0
149.4 0 0.539189i 0 0 0 3.80098i 0 2.70928 0
149.5 0 1.67513i 0 0 0 − 4.63752i 0 0.193937 0
149.6 0 2.21432i 0 0 0 − 2.83654i 0 −1.90321 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.6
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 3700.2.d.h 6
5.b even 2 1 inner 3700.2.d.h 6
5.c odd 4 1 740.2.a.e ✓ 3
5.c odd 4 1 3700.2.a.i 3
15.e even 4 1 6660.2.a.q 3
20.e even 4 1 2960.2.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.e ✓ 3 5.c odd 4 1
2960.2.a.t 3 20.e even 4 1
3700.2.a.i 3 5.c odd 4 1
3700.2.d.h 6 1.a even 1 1 trivial
3700.2.d.h 6 5.b even 2 1 inner
6660.2.a.q 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3700, [\chi])\):

\( T_{3}^{6} + 8T_{3}^{4} + 16T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 44T_{7}^{4} + 600T_{7}^{2} + 2500 \) Copy content Toggle raw display
\( T_{13}^{6} + 68T_{13}^{4} + 1456T_{13}^{2} + 10000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 44 T^{4} + \cdots + 2500 \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 68 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$17$ \( T^{6} + 100 T^{4} + \cdots + 21904 \) Copy content Toggle raw display
$19$ \( (T^{3} + 18 T^{2} + \cdots + 194)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 28 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 36 T - 54)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} - 4 T - 40)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 140 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{6} + 196 T^{4} + \cdots + 5476 \) Copy content Toggle raw display
$53$ \( T^{6} + 236 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} + \cdots - 158)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 10 T^{2} + \cdots + 2056)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 496 T^{4} + \cdots + 3422500 \) Copy content Toggle raw display
$71$ \( (T^{3} + 32 T^{2} + \cdots + 736)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 28 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} + \cdots + 262)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 128 T^{4} + \cdots + 5476 \) Copy content Toggle raw display
$89$ \( (T^{3} - 6 T^{2} + \cdots + 216)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 140 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
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