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Properties

Label 625.2.d.k
Level $625$
Weight $2$
Character orbit 625.d
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(126,625)]
 
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([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.d (of order \(5\), 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 \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.9
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: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_1) q^{2} + (\beta_{7} + \beta_{6} - \beta_1) q^{3} + ( - 3 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}) q^{4} + (3 \beta_{3} + \beta_{2} + 3) q^{6} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6}) q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_1) q^{2} + (\beta_{7} + \beta_{6} - \beta_1) q^{3} + ( - 3 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}) q^{4} + (3 \beta_{3} + \beta_{2} + 3) q^{6} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6}) q^{8} + ( - \beta_{2} - 1) q^{9} + 2 \beta_{3} q^{11} + ( - 2 \beta_{6} + 2 \beta_{4} + 3 \beta_1) q^{12} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{13} + ( - \beta_{5} - 3 \beta_{3} + 1) q^{14} + (3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 1) q^{16} - 2 \beta_{6} q^{17} + \beta_{4} q^{18} + (2 \beta_{3} + 6 \beta_{2} + 2) q^{19} + (\beta_{5} + 4 \beta_{3} + 4 \beta_{2}) q^{21} + (2 \beta_{4} + 2 \beta_1) q^{22} + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_1) q^{23} + ( - \beta_{5} + \beta_{2} - 7) q^{24} + ( - 4 \beta_{5} + 4 \beta_{2} - 6) q^{26} + (2 \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{27}+ \cdots + ( - 2 \beta_{5} + 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{4} + 16 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{4} + 16 q^{6} - 6 q^{9} - 4 q^{11} + 12 q^{14} + 18 q^{16} - 14 q^{21} - 60 q^{24} - 64 q^{26} + 20 q^{29} - 4 q^{31} - 28 q^{34} - 2 q^{36} + 28 q^{39} + 26 q^{41} - 28 q^{44} - 34 q^{46} - 4 q^{49} + 56 q^{51} - 50 q^{54} + 30 q^{56} + 20 q^{59} - 24 q^{61} - 34 q^{64} - 28 q^{66} + 18 q^{69} + 56 q^{71} + 72 q^{74} + 40 q^{76} - 20 q^{79} + 8 q^{81} + 2 q^{84} + 46 q^{86} + 20 q^{89} - 24 q^{91} + 52 q^{94} + 16 q^{96} + 8 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
0.476925 − 1.46782i
−0.476925 + 1.46782i
−1.73855 − 1.26313i
1.73855 + 1.26313i
−1.73855 + 1.26313i
1.73855 − 1.26313i
0.476925 + 1.46782i
−0.476925 − 1.46782i
−0.771681 − 2.37499i −1.24861 + 0.907165i −3.42705 + 2.48990i 0 3.11803 + 2.26538i 0.953850 4.51750 + 3.28216i −0.190983 + 0.587785i 0
126.2 0.771681 + 2.37499i 1.24861 − 0.907165i −3.42705 + 2.48990i 0 3.11803 + 2.26538i −0.953850 −4.51750 − 3.28216i −0.190983 + 0.587785i 0
251.1 −1.07448 + 0.780656i 0.664066 − 2.04378i −0.0729490 + 0.224514i 0 0.881966 + 2.71441i −3.47709 −0.917716 − 2.82444i −1.30902 − 0.951057i 0
251.2 1.07448 − 0.780656i −0.664066 + 2.04378i −0.0729490 + 0.224514i 0 0.881966 + 2.71441i 3.47709 0.917716 + 2.82444i −1.30902 − 0.951057i 0
376.1 −1.07448 − 0.780656i 0.664066 + 2.04378i −0.0729490 − 0.224514i 0 0.881966 − 2.71441i −3.47709 −0.917716 + 2.82444i −1.30902 + 0.951057i 0
376.2 1.07448 + 0.780656i −0.664066 − 2.04378i −0.0729490 − 0.224514i 0 0.881966 − 2.71441i 3.47709 0.917716 − 2.82444i −1.30902 + 0.951057i 0
501.1 −0.771681 + 2.37499i −1.24861 − 0.907165i −3.42705 − 2.48990i 0 3.11803 − 2.26538i 0.953850 4.51750 − 3.28216i −0.190983 − 0.587785i 0
501.2 0.771681 − 2.37499i 1.24861 + 0.907165i −3.42705 − 2.48990i 0 3.11803 − 2.26538i −0.953850 −4.51750 + 3.28216i −0.190983 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 126.2
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.d.k 8
5.b even 2 1 inner 625.2.d.k 8
5.c odd 4 2 625.2.e.h 8
25.d even 5 1 125.2.a.c ✓ 4
25.d even 5 1 inner 625.2.d.k 8
25.d even 5 2 625.2.d.l 8
25.e even 10 1 125.2.a.c ✓ 4
25.e even 10 1 inner 625.2.d.k 8
25.e even 10 2 625.2.d.l 8
25.f odd 20 2 125.2.b.a 4
25.f odd 20 4 625.2.e.b 8
25.f odd 20 2 625.2.e.h 8
75.h odd 10 1 1125.2.a.k 4
75.j odd 10 1 1125.2.a.k 4
75.l even 20 2 1125.2.b.a 4
100.h odd 10 1 2000.2.a.o 4
100.j odd 10 1 2000.2.a.o 4
100.l even 20 2 2000.2.c.c 4
175.l odd 10 1 6125.2.a.o 4
175.m odd 10 1 6125.2.a.o 4
200.n odd 10 1 8000.2.a.bk 4
200.o even 10 1 8000.2.a.bj 4
200.s odd 10 1 8000.2.a.bk 4
200.t even 10 1 8000.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.c ✓ 4 25.d even 5 1
125.2.a.c ✓ 4 25.e even 10 1
125.2.b.a 4 25.f odd 20 2
625.2.d.k 8 1.a even 1 1 trivial
625.2.d.k 8 5.b even 2 1 inner
625.2.d.k 8 25.d even 5 1 inner
625.2.d.k 8 25.e even 10 1 inner
625.2.d.l 8 25.d even 5 2
625.2.d.l 8 25.e even 10 2
625.2.e.b 8 25.f odd 20 4
625.2.e.h 8 5.c odd 4 2
625.2.e.h 8 25.f odd 20 2
1125.2.a.k 4 75.h odd 10 1
1125.2.a.k 4 75.j odd 10 1
1125.2.b.a 4 75.l even 20 2
2000.2.a.o 4 100.h odd 10 1
2000.2.a.o 4 100.j odd 10 1
2000.2.c.c 4 100.l even 20 2
6125.2.a.o 4 175.l odd 10 1
6125.2.a.o 4 175.m odd 10 1
8000.2.a.bj 4 200.o even 10 1
8000.2.a.bj 4 200.t even 10 1
8000.2.a.bk 4 200.n odd 10 1
8000.2.a.bk 4 200.s odd 10 1

Hecke kernels

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

\( T_{2}^{8} + 9T_{2}^{6} + 31T_{2}^{4} - 11T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{3}^{8} + 6T_{3}^{6} + 16T_{3}^{4} + 11T_{3}^{2} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 9 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{6} + \cdots + 121 \) 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
$13$ \( T^{8} + 36 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$17$ \( T^{8} + 24 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$19$ \( (T^{4} + 40 T^{2} + \cdots + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 26 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$29$ \( (T^{4} - 10 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 104 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$41$ \( (T^{4} - 13 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 107 T^{2} + 1331)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 26 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{8} + 96 T^{6} + \cdots + 7929856 \) Copy content Toggle raw display
$59$ \( (T^{4} - 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + \cdots + 961)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$71$ \( (T^{4} - 28 T^{3} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 44 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 11 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 204 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
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