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Properties

Label 825.4.c.n
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(199,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not 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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.181494784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 12x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 3 \beta_{4} q^{3} + (2 \beta_{2} - 3) q^{4} + 3 \beta_1 q^{6} + (\beta_{5} - 7 \beta_{4} + 4 \beta_{3}) q^{7} + ( - 2 \beta_{5} - \beta_{3}) q^{8} - 9 q^{9} + 11 q^{11} + ( - 6 \beta_{5} + 9 \beta_{4}) q^{12}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 14 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} + 232 q^{14} - 170 q^{16} + 338 q^{19} - 96 q^{21} - 18 q^{24} + 334 q^{26} + 554 q^{29} - 346 q^{31} + 292 q^{34} + 126 q^{36} + 270 q^{39} + 88 q^{41} - 154 q^{44}+ \cdots - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 12x^{3} + 49x^{2} - 14x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + 8\nu^{4} - 27\nu^{3} - 18\nu^{2} + 6\nu - 103 ) / 141 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 29\nu^{4} - 45\nu^{3} - 30\nu^{2} + 10\nu + 737 ) / 141 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -16\nu^{5} + 27\nu^{4} - 3\nu^{3} - 237\nu^{2} - 673\nu + 93 ) / 141 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -19\nu^{5} + 35\nu^{4} - 30\nu^{3} - 255\nu^{2} - 949\nu + 131 ) / 141 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -98\nu^{5} + 183\nu^{4} - 177\nu^{3} - 1152\nu^{2} - 5021\nu + 693 ) / 141 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 6\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} - 17\beta_{4} + 9\beta_{3} + 2\beta_{2} - 9\beta _1 - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{2} - 15\beta _1 - 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -30\beta_{5} + 223\beta_{4} - 91\beta_{3} + 30\beta_{2} - 91\beta _1 - 223 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.51966 + 1.51966i
2.38150 + 2.38150i
0.138157 − 0.138157i
0.138157 + 0.138157i
2.38150 − 2.38150i
−1.51966 − 1.51966i
− 4.03932i − 3.00000i −8.31608 0 −12.1180 6.49923i 1.27677i −9.00000 0
199.2 − 3.76300i 3.00000i −6.16019 0 11.2890 23.6321i − 6.92320i −9.00000 0
199.3 − 0.723686i − 3.00000i 7.47628 0 −2.17106 1.13288i − 11.2000i −9.00000 0
199.4 0.723686i 3.00000i 7.47628 0 −2.17106 − 1.13288i 11.2000i −9.00000 0
199.5 3.76300i − 3.00000i −6.16019 0 11.2890 − 23.6321i 6.92320i −9.00000 0
199.6 4.03932i 3.00000i −8.31608 0 −12.1180 − 6.49923i − 1.27677i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
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 825.4.c.n 6
5.b even 2 1 inner 825.4.c.n 6
5.c odd 4 1 825.4.a.o ✓ 3
5.c odd 4 1 825.4.a.q yes 3
15.e even 4 1 2475.4.a.v 3
15.e even 4 1 2475.4.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.o ✓ 3 5.c odd 4 1
825.4.a.q yes 3 5.c odd 4 1
825.4.c.n 6 1.a even 1 1 trivial
825.4.c.n 6 5.b even 2 1 inner
2475.4.a.v 3 15.e even 4 1
2475.4.a.y 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{6} + 31T_{2}^{4} + 247T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{7}^{6} + 602T_{7}^{4} + 24361T_{7}^{2} + 30276 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 31 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 602 T^{4} + \cdots + 30276 \) Copy content Toggle raw display
$11$ \( (T - 11)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 1465 T^{4} + \cdots + 24364096 \) Copy content Toggle raw display
$17$ \( T^{6} + 1974 T^{4} + \cdots + 5363856 \) Copy content Toggle raw display
$19$ \( (T^{3} - 169 T^{2} + \cdots - 41805)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 1629010321 \) Copy content Toggle raw display
$29$ \( (T^{3} - 277 T^{2} + \cdots + 1432824)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 173 T^{2} + \cdots - 3720456)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 229444393246084 \) Copy content Toggle raw display
$41$ \( (T^{3} - 44 T^{2} + \cdots + 2957382)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 135232707600 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 12279445673616 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 102374895330304 \) Copy content Toggle raw display
$59$ \( (T^{3} - 684 T^{2} + \cdots + 84071174)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 1038 T^{2} + \cdots - 137885416)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 2558285087296 \) Copy content Toggle raw display
$71$ \( (T^{3} + 1459 T^{2} + \cdots - 46945715)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{3} - 506 T^{2} + \cdots - 8353348)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 46134850245696 \) Copy content Toggle raw display
$89$ \( (T^{3} - 607 T^{2} + \cdots + 262736468)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 109633485301201 \) Copy content Toggle raw display
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