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Properties

Label 145.2.b.c
Level $145$
Weight $2$
Character orbit 145.b
Analytic conductor $1.158$
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] = [145,2,Mod(59,145)] 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("145.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-14] 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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.84345856.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 13x^{4} + 41x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - \beta_1) q^{3} + (\beta_{4} + \beta_{3} - 3) q^{4} - \beta_{2} q^{5} + ( - \beta_{4} - \beta_{2} + 2) q^{6} + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} + \cdots - 2 \beta_1) q^{8}+ \cdots + (5 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 14 q^{4} + 3 q^{5} + 14 q^{6} - 12 q^{9} - 3 q^{10} - 10 q^{11} + 8 q^{14} + 7 q^{15} + 42 q^{16} - 16 q^{19} + 13 q^{20} - 16 q^{21} - 26 q^{24} + 11 q^{25} - 46 q^{26} + 6 q^{29} + 25 q^{30} + 22 q^{31}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 13x^{4} + 41x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 14\nu^{3} + 6\nu^{2} + 47\nu - 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} + 14\nu^{3} - 6\nu^{2} + 47\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 14\nu^{3} + 10\nu^{2} - 47\nu + 15 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 12\nu^{3} + 35\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{3} + \beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} - 8\beta_{3} + 2\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{5} - 12\beta_{3} - 12\beta_{2} + 37\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
− 2.77035i
− 2.30229i
− 0.156785i
0.156785i
2.30229i
2.77035i
− 2.77035i 0.269894i −5.67486 −1.96358 − 1.06975i 0.747703 − 1.86960i 10.1807i 2.92716 −2.96358 + 5.43981i
59.2 − 2.30229i 2.89028i −3.30056 2.17686 + 0.511167i 6.65427 3.91261i 2.99427i −5.35371 1.17686 − 5.01177i
59.3 − 0.156785i − 2.56387i 1.97542 1.28672 + 1.82876i −0.401976 1.09364i − 0.623285i −3.57344 0.286721 − 0.201738i
59.4 0.156785i 2.56387i 1.97542 1.28672 − 1.82876i −0.401976 − 1.09364i 0.623285i −3.57344 0.286721 + 0.201738i
59.5 2.30229i − 2.89028i −3.30056 2.17686 − 0.511167i 6.65427 − 3.91261i − 2.99427i −5.35371 1.17686 + 5.01177i
59.6 2.77035i − 0.269894i −5.67486 −1.96358 + 1.06975i 0.747703 1.86960i − 10.1807i 2.92716 −2.96358 − 5.43981i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.6
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 145.2.b.c ✓ 6
3.b odd 2 1 1305.2.c.h 6
4.b odd 2 1 2320.2.d.g 6
5.b even 2 1 inner 145.2.b.c ✓ 6
5.c odd 4 2 725.2.a.l 6
15.d odd 2 1 1305.2.c.h 6
15.e even 4 2 6525.2.a.bt 6
20.d odd 2 1 2320.2.d.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.c ✓ 6 1.a even 1 1 trivial
145.2.b.c ✓ 6 5.b even 2 1 inner
725.2.a.l 6 5.c odd 4 2
1305.2.c.h 6 3.b odd 2 1
1305.2.c.h 6 15.d odd 2 1
2320.2.d.g 6 4.b odd 2 1
2320.2.d.g 6 20.d odd 2 1
6525.2.a.bt 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 13T_{2}^{4} + 41T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 13 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 15 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} - 6 T - 38)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 59 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{4} + \cdots + 784 \) Copy content Toggle raw display
$19$ \( (T^{3} + 8 T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 56 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T - 1)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 11 T^{2} + \cdots + 22)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 127 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$47$ \( T^{6} + 63 T^{4} + \cdots + 8836 \) Copy content Toggle raw display
$53$ \( T^{6} + 187 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$59$ \( (T + 10)^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 64 T - 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 140 T^{4} + \cdots + 92416 \) Copy content Toggle raw display
$71$ \( (T - 2)^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 188 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 54 T + 98)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 48 T^{4} + \cdots + 784 \) Copy content Toggle raw display
$89$ \( (T^{3} - 16 T^{2} + \cdots + 304)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 476 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
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