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Properties

Label 465.2.i.d
Level $465$
Weight $2$
Character orbit 465.i
Analytic conductor $3.713$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [465,2,Mod(211,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.211");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.i (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.363706692561.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{6} - 2x^{5} + 45x^{4} - 7x^{3} + 29x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{2} + 2) q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + (2 \beta_{3} + \beta_1 + 2) q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - 1) q^{8} + \beta_{3} q^{9}+ \cdots + (\beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 12 q^{4} - 4 q^{5} + 8 q^{7} - 6 q^{8} - 4 q^{9} + 4 q^{11} - 6 q^{12} + 3 q^{13} + 14 q^{14} + 8 q^{15} + 28 q^{16} - 3 q^{17} + q^{19} - 6 q^{20} + 8 q^{21} - 26 q^{23} + 3 q^{24} - 4 q^{25}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{6} - 2x^{5} + 45x^{4} - 7x^{3} + 29x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -49\nu^{7} + 180\nu^{6} - 315\nu^{5} + 49\nu^{4} - 2385\nu^{3} - 28\nu^{2} - 112\nu - 28732 ) / 8443 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -343\nu^{7} + 1260\nu^{6} - 2205\nu^{5} + 8786\nu^{4} - 16695\nu^{3} + 58905\nu^{2} - 9227\nu + 1508 ) / 33772 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -377\nu^{7} - 1372\nu^{6} + 2401\nu^{5} - 8066\nu^{4} + 18179\nu^{3} - 64141\nu^{2} + 190915\nu - 38416 ) / 33772 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 294\nu^{7} - 1080\nu^{6} + 1890\nu^{5} - 8737\nu^{4} + 14310\nu^{3} - 50490\nu^{2} + 9115\nu - 30240 ) / 8443 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 315\nu^{7} + 49\nu^{6} + 2025\nu^{5} - 315\nu^{4} + 14126\nu^{3} + 180\nu^{2} + 720\nu + 1372 ) / 8443 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7183 \nu^{7} + 196 \nu^{6} - 51001 \nu^{5} + 15626 \nu^{4} - 323431 \nu^{3} + 59821 \nu^{2} + \cdots - 28284 ) / 33772 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 6\beta_{6} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 24\beta_{3} + \beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} - 38\beta_{6} + \beta_{5} + 11\beta_{3} - \beta_{2} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 13\beta_{6} + \beta_{4} + 45\beta_{2} + 153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{5} - 45\beta_{4} - 97\beta_{3} + 243\beta _1 - 97 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
−1.30336 − 2.25748i
−0.355810 − 0.616281i
0.443420 + 0.768026i
1.21575 + 2.10573i
−1.30336 + 2.25748i
−0.355810 + 0.616281i
0.443420 − 0.768026i
1.21575 − 2.10573i
−2.60671 −0.500000 − 0.866025i 4.79495 −0.500000 + 0.866025i 1.30336 + 2.25748i −0.303356 − 0.525429i −7.28564 −0.500000 + 0.866025i 1.30336 − 2.25748i
211.2 −0.711620 −0.500000 − 0.866025i −1.49360 −0.500000 + 0.866025i 0.355810 + 0.616281i 0.644190 + 1.11577i 2.48611 −0.500000 + 0.866025i 0.355810 − 0.616281i
211.3 0.886840 −0.500000 − 0.866025i −1.21351 −0.500000 + 0.866025i −0.443420 − 0.768026i 1.44342 + 2.50008i −2.84987 −0.500000 + 0.866025i −0.443420 + 0.768026i
211.4 2.43149 −0.500000 − 0.866025i 3.91216 −0.500000 + 0.866025i −1.21575 − 2.10573i 2.21575 + 3.83779i 4.64940 −0.500000 + 0.866025i −1.21575 + 2.10573i
346.1 −2.60671 −0.500000 + 0.866025i 4.79495 −0.500000 − 0.866025i 1.30336 − 2.25748i −0.303356 + 0.525429i −7.28564 −0.500000 − 0.866025i 1.30336 + 2.25748i
346.2 −0.711620 −0.500000 + 0.866025i −1.49360 −0.500000 − 0.866025i 0.355810 − 0.616281i 0.644190 − 1.11577i 2.48611 −0.500000 − 0.866025i 0.355810 + 0.616281i
346.3 0.886840 −0.500000 + 0.866025i −1.21351 −0.500000 − 0.866025i −0.443420 + 0.768026i 1.44342 − 2.50008i −2.84987 −0.500000 − 0.866025i −0.443420 − 0.768026i
346.4 2.43149 −0.500000 + 0.866025i 3.91216 −0.500000 − 0.866025i −1.21575 + 2.10573i 2.21575 − 3.83779i 4.64940 −0.500000 − 0.866025i −1.21575 − 2.10573i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.i.d ✓ 8
31.c even 3 1 inner 465.2.i.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.i.d ✓ 8 1.a even 1 1 trivial
465.2.i.d ✓ 8 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\):

\( T_{2}^{4} - 7T_{2}^{2} + T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} - 8T_{7}^{7} + 47T_{7}^{6} - 130T_{7}^{5} + 275T_{7}^{4} - 211T_{7}^{3} + 179T_{7}^{2} + 30T_{7} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 7 T^{2} + T + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 20164 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$23$ \( (T^{4} + 13 T^{3} + \cdots - 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - T^{2} + \cdots - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 633616 \) Copy content Toggle raw display
$41$ \( T^{8} - 13 T^{7} + \cdots + 166464 \) Copy content Toggle raw display
$43$ \( T^{8} + 14 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$47$ \( (T^{4} - 9 T^{3} + \cdots - 522)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + \cdots + 1336336 \) Copy content Toggle raw display
$59$ \( T^{8} + 167 T^{6} + \cdots + 3916441 \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} - 87 T^{2} + \cdots - 87)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$71$ \( T^{8} - 16 T^{7} + \cdots + 12131289 \) Copy content Toggle raw display
$73$ \( T^{8} + 15 T^{7} + \cdots + 419904 \) Copy content Toggle raw display
$79$ \( T^{8} + 30 T^{7} + \cdots + 6405961 \) Copy content Toggle raw display
$83$ \( T^{8} - 31 T^{7} + \cdots + 388957284 \) Copy content Toggle raw display
$89$ \( (T^{4} - 115 T^{2} + \cdots - 969)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 9 T^{3} + \cdots + 10044)^{2} \) Copy content Toggle raw display
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