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Properties

Label 825.2.n.l
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
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] = [825,2,Mod(301,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
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("825.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.n (of order \(5\), degree \(4\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.819390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{3} + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{4} + \beta_{7} q^{6} + (\beta_{7} - \beta_{5} + \cdots + 2 \beta_1) q^{7} + ( - 2 \beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{8}+ \cdots + (\beta_{6} + \beta_{3} + \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} - 2 q^{3} - 7 q^{4} + 3 q^{6} + 7 q^{7} - 16 q^{8} - 2 q^{9} + 2 q^{11} + 18 q^{12} - 5 q^{13} - 2 q^{14} - 11 q^{16} + 8 q^{17} - 2 q^{18} - 5 q^{19} - 8 q^{21} - 8 q^{22} - 2 q^{23} + 19 q^{24}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -256\nu^{7} + 341\nu^{6} + 3310\nu^{5} - 16865\nu^{4} + 32996\nu^{3} - 59433\nu^{2} + 33270\nu - 118459 ) / 171589 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -620\nu^{7} + 2532\nu^{6} - 9045\nu^{5} + 18870\nu^{4} - 41955\nu^{3} + 81515\nu^{2} - 102225\nu - 16104 ) / 171589 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -672\nu^{7} + 2845\nu^{6} - 10810\nu^{5} + 23975\nu^{4} - 77175\nu^{3} + 52625\nu^{2} - 72556\nu - 75020 ) / 171589 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1187 \nu^{7} + 4445 \nu^{6} - 17191 \nu^{5} + 14238 \nu^{4} - 12015 \nu^{3} - 32510 \nu^{2} + \cdots - 143748 ) / 171589 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1188 \nu^{7} - 4751 \nu^{6} + 16325 \nu^{5} - 32635 \nu^{4} + 48690 \nu^{3} - 20331 \nu^{2} + \cdots + 144881 ) / 171589 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1432 \nu^{7} + 1420 \nu^{6} - 7808 \nu^{5} - 2207 \nu^{4} - 27965 \nu^{3} - 33635 \nu^{2} + \cdots - 249744 ) / 171589 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + 3\beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 5\beta_{4} + 2\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 15\beta_{6} - 8\beta_{5} - 8\beta_{4} - 5\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 25\beta_{6} - 28\beta_{5} + 25\beta_{2} - 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{7} + 55\beta_{4} + 22\beta_{3} + 110\beta_{2} - 45\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{7} + 175\beta_{6} + 165\beta_{5} + 188\beta_{4} - 80\beta_{3} - 188\beta _1 - 175 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
−0.755243 − 0.548716i
2.06426 + 1.49977i
−0.575405 + 1.77091i
0.766388 − 2.35870i
−0.575405 − 1.77091i
0.766388 + 2.35870i
−0.755243 + 0.548716i
2.06426 − 1.49977i
−0.755243 − 0.548716i 0.309017 − 0.951057i −0.348732 − 1.07329i 0 −0.755243 + 0.548716i −1.37328 − 4.22651i −0.902506 + 2.77763i −0.809017 − 0.587785i 0
301.2 2.06426 + 1.49977i 0.309017 − 0.951057i 1.39382 + 4.28973i 0 2.06426 − 1.49977i 1.44623 + 4.45102i −1.97946 + 6.09215i −0.809017 − 0.587785i 0
526.1 −0.575405 + 1.77091i −0.809017 + 0.587785i −1.18701 − 0.862413i 0 −0.575405 − 1.77091i 1.04263 + 0.757515i −0.802588 + 0.583114i 0.309017 − 0.951057i 0
526.2 0.766388 − 2.35870i −0.809017 + 0.587785i −3.35808 − 2.43978i 0 0.766388 + 2.35870i 2.38442 + 1.73238i −4.31545 + 3.13535i 0.309017 − 0.951057i 0
676.1 −0.575405 − 1.77091i −0.809017 − 0.587785i −1.18701 + 0.862413i 0 −0.575405 + 1.77091i 1.04263 − 0.757515i −0.802588 − 0.583114i 0.309017 + 0.951057i 0
676.2 0.766388 + 2.35870i −0.809017 − 0.587785i −3.35808 + 2.43978i 0 0.766388 − 2.35870i 2.38442 − 1.73238i −4.31545 − 3.13535i 0.309017 + 0.951057i 0
751.1 −0.755243 + 0.548716i 0.309017 + 0.951057i −0.348732 + 1.07329i 0 −0.755243 − 0.548716i −1.37328 + 4.22651i −0.902506 − 2.77763i −0.809017 + 0.587785i 0
751.2 2.06426 − 1.49977i 0.309017 + 0.951057i 1.39382 − 4.28973i 0 2.06426 + 1.49977i 1.44623 − 4.45102i −1.97946 − 6.09215i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.l yes 8
5.b even 2 1 825.2.n.h ✓ 8
5.c odd 4 2 825.2.bx.i 16
11.c even 5 1 inner 825.2.n.l yes 8
11.c even 5 1 9075.2.a.cp 4
11.d odd 10 1 9075.2.a.dh 4
55.h odd 10 1 9075.2.a.cn 4
55.j even 10 1 825.2.n.h ✓ 8
55.j even 10 1 9075.2.a.de 4
55.k odd 20 2 825.2.bx.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.h ✓ 8 5.b even 2 1
825.2.n.h ✓ 8 55.j even 10 1
825.2.n.l yes 8 1.a even 1 1 trivial
825.2.n.l yes 8 11.c even 5 1 inner
825.2.bx.i 16 5.c odd 4 2
825.2.bx.i 16 55.k odd 20 2
9075.2.a.cn 4 55.h odd 10 1
9075.2.a.cp 4 11.c even 5 1
9075.2.a.de 4 55.j even 10 1
9075.2.a.dh 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{8} - 3T_{2}^{7} + 10T_{2}^{6} - 13T_{2}^{5} + 29T_{2}^{4} - 7T_{2}^{3} + 80T_{2}^{2} + 143T_{2} + 121 \) Copy content Toggle raw display
\( T_{13}^{8} + 5T_{13}^{7} + 38T_{13}^{6} + 255T_{13}^{5} + 1069T_{13}^{4} + 2185T_{13}^{3} + 2722T_{13}^{2} + 2015T_{13} + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 7 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} - 9 T^{2} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$23$ \( (T^{4} + T^{3} - 39 T^{2} + \cdots + 341)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$31$ \( T^{8} - 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{8} + 19 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( T^{8} + 18 T^{7} + \cdots + 32761 \) Copy content Toggle raw display
$43$ \( (T^{4} - 26 T^{3} + \cdots - 1089)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 18 T^{7} + \cdots + 17161 \) Copy content Toggle raw display
$53$ \( T^{8} - 31 T^{7} + \cdots + 101761 \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$61$ \( T^{8} + 22 T^{7} + \cdots + 8202496 \) Copy content Toggle raw display
$67$ \( (T^{4} + 19 T^{3} + \cdots + 2351)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 303 T^{6} + \cdots + 138039001 \) Copy content Toggle raw display
$73$ \( T^{8} + 18 T^{7} + \cdots + 1234321 \) Copy content Toggle raw display
$79$ \( T^{8} + 18 T^{7} + \cdots + 60543961 \) Copy content Toggle raw display
$83$ \( T^{8} + 28 T^{7} + \cdots + 83156161 \) Copy content Toggle raw display
$89$ \( (T^{4} - 17 T^{3} + \cdots + 99)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 11 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
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