↑

Properties

Label 290.2.d.a
Level $290$
Weight $2$
Character orbit 290.d
Analytic conductor $2.316$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [290,2,Mod(289,290)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(290, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("290.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 290 = 2 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 290.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.31566165862\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.75200995984.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 30x^{4} + 27x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} - \beta_1 q^{6} + ( - \beta_{5} - \beta_{4}) q^{7} - q^{8} + (\beta_{2} + \beta_1 + 1) q^{9} - \beta_{6} q^{10} - \beta_{4} q^{11} + \beta_1 q^{12}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots - 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - q^{5} - 4 q^{6} - 8 q^{8} + 8 q^{9} + q^{10} + 4 q^{12} + 3 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} - q^{20} - 4 q^{24} + 5 q^{25} + 10 q^{27} - 4 q^{29} - 3 q^{30}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 10x^{6} + 30x^{4} + 27x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} + 5\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 8\nu^{5} - 16\nu^{3} - 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 30\nu^{3} + 27\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 10\nu^{5} - 28\nu^{3} - 17\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 10\nu^{5} + 16\nu^{4} + 30\nu^{3} + 32\nu^{2} + 23\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} - 16\nu^{4} + 30\nu^{3} - 32\nu^{2} + 23\nu - 10 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 5\beta_{6} + 2\beta_{5} - 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{2} - 5\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{7} - 25\beta_{6} - 14\beta_{5} + 13\beta_{4} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} - 8\beta_{2} + 24\beta _1 - 61 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 127\beta_{7} + 127\beta_{6} + 80\beta_{5} - 63\beta_{4} - 20\beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/290\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.28106i
− 2.28106i
1.83507i
− 1.83507i
1.11627i
− 1.11627i
− 0.428026i
0.428026i
−1.00000 −2.20326 1.00000 −1.26671 − 1.84267i 2.20326 0.463643i −1.00000 1.85434 1.26671 + 1.84267i
289.2 −1.00000 −2.20326 1.00000 −1.26671 + 1.84267i 2.20326 − 0.463643i −1.00000 1.85434 1.26671 − 1.84267i
289.3 −1.00000 −0.367473 1.00000 1.82635 − 1.29013i 0.367473 − 2.99580i −1.00000 −2.86496 −1.82635 + 1.29013i
289.4 −1.00000 −0.367473 1.00000 1.82635 + 1.29013i 0.367473 2.99580i −1.00000 −2.86496 −1.82635 − 1.29013i
289.5 −1.00000 1.75394 1.00000 −2.22518 − 0.220433i −1.75394 − 4.19041i −1.00000 0.0762945 2.22518 + 0.220433i
289.6 −1.00000 1.75394 1.00000 −2.22518 + 0.220433i −1.75394 4.19041i −1.00000 0.0762945 2.22518 − 0.220433i
289.7 −1.00000 2.81679 1.00000 1.16553 − 1.90828i −2.81679 2.06171i −1.00000 4.93433 −1.16553 + 1.90828i
289.8 −1.00000 2.81679 1.00000 1.16553 + 1.90828i −2.81679 − 2.06171i −1.00000 4.93433 −1.16553 − 1.90828i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 290.2.d.a ✓ 8
3.b odd 2 1 2610.2.b.f 8
4.b odd 2 1 2320.2.j.d 8
5.b even 2 1 290.2.d.b yes 8
5.c odd 4 2 1450.2.c.g 16
15.d odd 2 1 2610.2.b.d 8
20.d odd 2 1 2320.2.j.e 8
29.b even 2 1 290.2.d.b yes 8
87.d odd 2 1 2610.2.b.d 8
116.d odd 2 1 2320.2.j.e 8
145.d even 2 1 inner 290.2.d.a ✓ 8
145.h odd 4 2 1450.2.c.g 16
435.b odd 2 1 2610.2.b.f 8
580.e odd 2 1 2320.2.j.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.d.a ✓ 8 1.a even 1 1 trivial
290.2.d.a ✓ 8 145.d even 2 1 inner
290.2.d.b yes 8 5.b even 2 1
290.2.d.b yes 8 29.b even 2 1
1450.2.c.g 16 5.c odd 4 2
1450.2.c.g 16 145.h odd 4 2
2320.2.j.d 8 4.b odd 2 1
2320.2.j.d 8 580.e odd 2 1
2320.2.j.e 8 20.d odd 2 1
2320.2.j.e 8 116.d odd 2 1
2610.2.b.d 8 15.d odd 2 1
2610.2.b.d 8 87.d odd 2 1
2610.2.b.f 8 3.b odd 2 1
2610.2.b.f 8 435.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 9T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(290, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} - 6 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 31 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{8} + 27 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{8} + 82 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$17$ \( (T^{4} - T^{3} - 35 T^{2} + \cdots - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 104 T^{6} + \cdots + 36864 \) Copy content Toggle raw display
$23$ \( T^{8} + 111 T^{6} + \cdots + 21904 \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + \cdots + 707281 \) Copy content Toggle raw display
$31$ \( T^{8} + 138 T^{6} + \cdots + 26244 \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} - 40 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 120 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots + 2932)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 3 T^{3} + \cdots + 192)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 238 T^{6} + \cdots + 2972176 \) Copy content Toggle raw display
$59$ \( (T^{4} + 9 T^{3} - 3 T^{2} + \cdots + 72)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 127 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$67$ \( T^{8} + 232 T^{6} + \cdots + 6718464 \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + \cdots - 3456)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 17 T^{3} + \cdots - 1096)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 98 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$83$ \( T^{8} + 516 T^{6} + \cdots + 8202496 \) Copy content Toggle raw display
$89$ \( T^{8} + 356 T^{6} + \cdots + 2985984 \) Copy content Toggle raw display
$97$ \( (T^{4} + 7 T^{3} + \cdots - 1796)^{2} \) Copy content Toggle raw display
show more
show less