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Properties

Label 625.2.e.b
Level $625$
Weight $2$
Character orbit 625.e
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(124,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.e (of order \(10\), degree \(4\), 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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.484000000.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 125)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} - \beta_1 q^{3} + (3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 1) q^{4} + ( - 3 \beta_{5} + 2 \beta_{3} - 3) q^{6} + (\beta_{7} + \beta_{4} + \beta_1) q^{7} + (2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} - \beta_1 q^{3} + (3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 1) q^{4} + ( - 3 \beta_{5} + 2 \beta_{3} - 3) q^{6} + (\beta_{7} + \beta_{4} + \beta_1) q^{7} + (2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} - 14 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} - 14 q^{6} - 4 q^{9} - 4 q^{11} - 2 q^{14} - 22 q^{16} + 20 q^{19} + 26 q^{21} + 60 q^{24} - 64 q^{26} + 10 q^{29} - 4 q^{31} - 32 q^{34} + 8 q^{36} + 32 q^{39} - 24 q^{41} - 12 q^{44} + 16 q^{46} + 4 q^{49} + 56 q^{51} - 30 q^{54} - 40 q^{56} + 20 q^{59} + 26 q^{61} - 26 q^{64} + 32 q^{66} + 2 q^{69} - 44 q^{71} - 72 q^{74} + 40 q^{76} + 28 q^{81} + 18 q^{84} - 84 q^{86} - 10 q^{89} - 4 q^{91} + 18 q^{94} - 14 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} - 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} + 75\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 + \beta_{2} - \beta_{3} + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−1.26313 + 1.73855i
1.26313 − 1.73855i
−1.46782 + 0.476925i
1.46782 − 0.476925i
−1.46782 − 0.476925i
1.46782 + 0.476925i
−1.26313 − 1.73855i
1.26313 + 1.73855i
−1.26313 − 0.410415i 1.26313 − 1.73855i −0.190983 − 0.138757i 0 −2.30902 + 1.67760i 3.47709i 1.74560 + 2.40261i −0.500000 − 1.53884i 0
124.2 1.26313 + 0.410415i −1.26313 + 1.73855i −0.190983 − 0.138757i 0 −2.30902 + 1.67760i − 3.47709i −1.74560 − 2.40261i −0.500000 − 1.53884i 0
249.1 −1.46782 + 2.02029i 1.46782 − 0.476925i −1.30902 − 4.02874i 0 −1.19098 + 3.66547i 0.953850i 5.31064 + 1.72553i −0.500000 + 0.363271i 0
249.2 1.46782 − 2.02029i −1.46782 + 0.476925i −1.30902 − 4.02874i 0 −1.19098 + 3.66547i − 0.953850i −5.31064 − 1.72553i −0.500000 + 0.363271i 0
374.1 −1.46782 − 2.02029i 1.46782 + 0.476925i −1.30902 + 4.02874i 0 −1.19098 − 3.66547i − 0.953850i 5.31064 − 1.72553i −0.500000 − 0.363271i 0
374.2 1.46782 + 2.02029i −1.46782 − 0.476925i −1.30902 + 4.02874i 0 −1.19098 − 3.66547i 0.953850i −5.31064 + 1.72553i −0.500000 − 0.363271i 0
499.1 −1.26313 + 0.410415i 1.26313 + 1.73855i −0.190983 + 0.138757i 0 −2.30902 − 1.67760i − 3.47709i 1.74560 − 2.40261i −0.500000 + 1.53884i 0
499.2 1.26313 − 0.410415i −1.26313 − 1.73855i −0.190983 + 0.138757i 0 −2.30902 − 1.67760i 3.47709i −1.74560 + 2.40261i −0.500000 + 1.53884i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.b 8
5.b even 2 1 inner 625.2.e.b 8
5.c odd 4 2 625.2.d.l 8
25.d even 5 1 125.2.b.a 4
25.d even 5 1 inner 625.2.e.b 8
25.d even 5 2 625.2.e.h 8
25.e even 10 1 125.2.b.a 4
25.e even 10 1 inner 625.2.e.b 8
25.e even 10 2 625.2.e.h 8
25.f odd 20 2 125.2.a.c ✓ 4
25.f odd 20 4 625.2.d.k 8
25.f odd 20 2 625.2.d.l 8
75.h odd 10 1 1125.2.b.a 4
75.j odd 10 1 1125.2.b.a 4
75.l even 20 2 1125.2.a.k 4
100.h odd 10 1 2000.2.c.c 4
100.j odd 10 1 2000.2.c.c 4
100.l even 20 2 2000.2.a.o 4
175.s even 20 2 6125.2.a.o 4
200.v even 20 2 8000.2.a.bk 4
200.x odd 20 2 8000.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.c ✓ 4 25.f odd 20 2
125.2.b.a 4 25.d even 5 1
125.2.b.a 4 25.e even 10 1
625.2.d.k 8 25.f odd 20 4
625.2.d.l 8 5.c odd 4 2
625.2.d.l 8 25.f odd 20 2
625.2.e.b 8 1.a even 1 1 trivial
625.2.e.b 8 5.b even 2 1 inner
625.2.e.b 8 25.d even 5 1 inner
625.2.e.b 8 25.e even 10 1 inner
625.2.e.h 8 25.d even 5 2
625.2.e.h 8 25.e even 10 2
1125.2.a.k 4 75.l even 20 2
1125.2.b.a 4 75.h odd 10 1
1125.2.b.a 4 75.j odd 10 1
2000.2.a.o 4 100.l even 20 2
2000.2.c.c 4 100.h odd 10 1
2000.2.c.c 4 100.j odd 10 1
6125.2.a.o 4 175.s even 20 2
8000.2.a.bj 4 200.x odd 20 2
8000.2.a.bk 4 200.v even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{8} + T_{2}^{6} + 31T_{2}^{4} - 99T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{6} + 16T_{3}^{4} - 66T_{3}^{2} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 13 T^{2} + 11)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$19$ \( (T^{4} - 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 9 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$29$ \( (T^{4} - 5 T^{3} + 40 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 36 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 107 T^{2} + 1331)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 69 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{6} + \cdots + 7929856 \) Copy content Toggle raw display
$59$ \( (T^{4} - 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 13 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$71$ \( (T^{4} + 22 T^{3} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 396 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$79$ \( (T^{4} + 40 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 66 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$89$ \( (T^{4} + 5 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 76 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
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