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Properties

Label 325.2.e.a
Level $325$
Weight $2$
Character orbit 325.e
Analytic conductor $2.595$
Analytic rank $0$
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] = [325,2,Mod(126,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.e (of order \(3\), degree \(2\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1 + 1) q^{6} + \beta_{2} q^{7} + 3 q^{8} - 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_1 + 1) q^{6} + \beta_{2} q^{7} + 3 q^{8} - 2 \beta_{2} q^{9} + (3 \beta_{2} - 2 \beta_1 + 3) q^{11} + (\beta_{3} - 2) q^{12} + ( - 2 \beta_{3} + 1) q^{13} + ( - \beta_{3} + 1) q^{14} + (2 \beta_{2} - \beta_1 + 2) q^{16} + ( - 2 \beta_{3} - 3 \beta_{2} + \cdots + 2) q^{17}+ \cdots + (4 \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 2 q^{3} - 3 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 2 q^{3} - 3 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 4 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{14} + 3 q^{16} + 8 q^{17} - 4 q^{18} + 4 q^{19} - 4 q^{21} - 11 q^{22} - 6 q^{23} + 6 q^{24} - 13 q^{26} + 20 q^{27} - 3 q^{28} - 2 q^{29} - 16 q^{31} + 7 q^{32} - 4 q^{33} - 34 q^{34} + 6 q^{36} - 30 q^{38} - 6 q^{41} + q^{42} - 6 q^{43} + 14 q^{44} - 3 q^{46} - 8 q^{47} - 3 q^{48} + 12 q^{49} + 16 q^{51} - 13 q^{52} - 16 q^{53} - 5 q^{54} - 6 q^{56} + 8 q^{57} + 25 q^{58} + 2 q^{61} + 4 q^{62} + 4 q^{63} - 8 q^{64} - 22 q^{66} - 14 q^{67} + 25 q^{68} + 6 q^{69} + 12 q^{71} + 12 q^{72} + 32 q^{73} + 13 q^{74} + 19 q^{76} - 8 q^{77} + 13 q^{78} - 8 q^{79} - 2 q^{81} - 3 q^{82} + 8 q^{83} + 3 q^{84} + 58 q^{86} + 2 q^{87} + 12 q^{88} - 2 q^{89} + 18 q^{92} - 8 q^{93} + 28 q^{94} + 14 q^{96} - 24 q^{97} + 6 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
1.15139 − 1.99426i
−0.651388 + 1.12824i
1.15139 + 1.99426i
−0.651388 − 1.12824i
−1.15139 + 1.99426i 0.500000 − 0.866025i −1.65139 − 2.86029i 0 1.15139 + 1.99426i −0.500000 − 0.866025i 3.00000 1.00000 + 1.73205i 0
126.2 0.651388 − 1.12824i 0.500000 − 0.866025i 0.151388 + 0.262211i 0 −0.651388 − 1.12824i −0.500000 − 0.866025i 3.00000 1.00000 + 1.73205i 0
276.1 −1.15139 − 1.99426i 0.500000 + 0.866025i −1.65139 + 2.86029i 0 1.15139 − 1.99426i −0.500000 + 0.866025i 3.00000 1.00000 − 1.73205i 0
276.2 0.651388 + 1.12824i 0.500000 + 0.866025i 0.151388 − 0.262211i 0 −0.651388 + 1.12824i −0.500000 + 0.866025i 3.00000 1.00000 − 1.73205i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.a 4
5.b even 2 1 65.2.e.b ✓ 4
5.c odd 4 2 325.2.o.b 8
13.c even 3 1 inner 325.2.e.a 4
13.c even 3 1 4225.2.a.x 2
13.e even 6 1 4225.2.a.t 2
15.d odd 2 1 585.2.j.d 4
20.d odd 2 1 1040.2.q.o 4
65.d even 2 1 845.2.e.d 4
65.g odd 4 2 845.2.m.d 8
65.l even 6 1 845.2.a.f 2
65.l even 6 1 845.2.e.d 4
65.n even 6 1 65.2.e.b ✓ 4
65.n even 6 1 845.2.a.c 2
65.q odd 12 2 325.2.o.b 8
65.s odd 12 2 845.2.c.d 4
65.s odd 12 2 845.2.m.d 8
195.x odd 6 1 585.2.j.d 4
195.x odd 6 1 7605.2.a.bg 2
195.y odd 6 1 7605.2.a.bb 2
260.v odd 6 1 1040.2.q.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b ✓ 4 5.b even 2 1
65.2.e.b ✓ 4 65.n even 6 1
325.2.e.a 4 1.a even 1 1 trivial
325.2.e.a 4 13.c even 3 1 inner
325.2.o.b 8 5.c odd 4 2
325.2.o.b 8 65.q odd 12 2
585.2.j.d 4 15.d odd 2 1
585.2.j.d 4 195.x odd 6 1
845.2.a.c 2 65.n even 6 1
845.2.a.f 2 65.l even 6 1
845.2.c.d 4 65.s odd 12 2
845.2.e.d 4 65.d even 2 1
845.2.e.d 4 65.l even 6 1
845.2.m.d 8 65.g odd 4 2
845.2.m.d 8 65.s odd 12 2
1040.2.q.o 4 20.d odd 2 1
1040.2.q.o 4 260.v odd 6 1
4225.2.a.t 2 13.e even 6 1
4225.2.a.x 2 13.c even 3 1
7605.2.a.bb 2 195.y odd 6 1
7605.2.a.bg 2 195.x odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 117 T^{2} + 13689 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( (T^{2} - 16 T + 12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$97$ \( T^{4} + 24 T^{3} + \cdots + 17161 \) Copy content Toggle raw display
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