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Properties

Label 625.2.d.d
Level $625$
Weight $2$
Character orbit 625.d
Analytic conductor $4.991$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{6} - 3 q^{7} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{8}+ \cdots + ( - 3 \zeta_{10}^{2} + \zeta_{10} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{6} - 3 q^{7} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{8}+ \cdots + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{6} - 12 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 2 q^{6} - 12 q^{7} - 5 q^{8} - 8 q^{9} + 3 q^{11} - 7 q^{12} - 6 q^{13} + 6 q^{14} - 6 q^{16} - 7 q^{17} + 14 q^{18} + 5 q^{19} + 3 q^{21} - 9 q^{22} - 6 q^{23} + 10 q^{24} + 18 q^{26} + 5 q^{27} + 6 q^{28} - 15 q^{29} - 12 q^{31} + 18 q^{32} + 3 q^{33} - 4 q^{34} + 9 q^{36} + 18 q^{37} + 5 q^{38} + 9 q^{39} + 3 q^{41} + 6 q^{42} - 36 q^{43} + 6 q^{44} - 2 q^{46} + 18 q^{47} + 9 q^{48} + 8 q^{49} - 2 q^{51} + 3 q^{52} - 6 q^{53} + 5 q^{54} + 15 q^{56} + 10 q^{57} + 15 q^{58} - 15 q^{59} - 12 q^{61} + q^{62} + 24 q^{63} + 3 q^{64} - 9 q^{66} + 18 q^{67} + 6 q^{68} - 6 q^{69} + 3 q^{71} + 5 q^{72} + 9 q^{73} - 24 q^{74} - 9 q^{77} + 3 q^{78} + 5 q^{79} - 16 q^{81} + 6 q^{82} - 6 q^{83} + 21 q^{84} + 18 q^{86} - 30 q^{87} - 15 q^{88} - 30 q^{89} + 18 q^{91} - 2 q^{92} + 28 q^{93} - 19 q^{94} - 7 q^{96} - 12 q^{97} - 4 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 − 0.587785i
−0.309017 − 0.951057i
−0.500000 − 1.53884i 0.309017 − 0.224514i −0.500000 + 0.363271i 0 −0.500000 − 0.363271i −3.00000 −1.80902 − 1.31433i −0.881966 + 2.71441i 0
251.1 −0.500000 + 0.363271i −0.809017 + 2.48990i −0.500000 + 1.53884i 0 −0.500000 − 1.53884i −3.00000 −0.690983 − 2.12663i −3.11803 − 2.26538i 0
376.1 −0.500000 − 0.363271i −0.809017 − 2.48990i −0.500000 − 1.53884i 0 −0.500000 + 1.53884i −3.00000 −0.690983 + 2.12663i −3.11803 + 2.26538i 0
501.1 −0.500000 + 1.53884i 0.309017 + 0.224514i −0.500000 − 0.363271i 0 −0.500000 + 0.363271i −3.00000 −1.80902 + 1.31433i −0.881966 − 2.71441i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + T_{3}^{3} + 6T_{3}^{2} - 4T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 3)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 15 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( T^{4} - 18 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( (T + 9)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 18 T^{3} + \cdots + 3721 \) Copy content Toggle raw display
$53$ \( T^{4} + 6 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( T^{4} + 15 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$71$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$73$ \( T^{4} - 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$79$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{4} + 30 T^{3} + \cdots + 32400 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
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