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Properties

Label 2475.1.ev.b
Level $2475$
Weight $1$
Character orbit 2475.ev
Analytic conductor $1.235$
Analytic rank $0$
Dimension $8$
Projective image $D_{30}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,1,Mod(439,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 9, 15]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.439");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2475.ev (of order \(30\), degree \(8\), 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: \(1.23518590627\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{30}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{30} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{30} q^{3} - \zeta_{30}^{4} q^{4} + \zeta_{30}^{5} q^{5} + \zeta_{30}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{30} q^{3} - \zeta_{30}^{4} q^{4} + \zeta_{30}^{5} q^{5} + \zeta_{30}^{2} q^{9} - \zeta_{30}^{2} q^{11} + \zeta_{30}^{5} q^{12} - \zeta_{30}^{6} q^{15} + \zeta_{30}^{8} q^{16} - \zeta_{30}^{9} q^{20} + ( - \zeta_{30}^{13} + \zeta_{30}) q^{23} + \zeta_{30}^{10} q^{25} - \zeta_{30}^{3} q^{27} + (\zeta_{30}^{9} - \zeta_{30}^{8}) q^{31} + \zeta_{30}^{3} q^{33} - \zeta_{30}^{6} q^{36} + (\zeta_{30}^{13} + \zeta_{30}^{8}) q^{37} + \zeta_{30}^{6} q^{44} + \zeta_{30}^{7} q^{45} + (\zeta_{30}^{9} + \zeta_{30}^{4}) q^{47} - \zeta_{30}^{9} q^{48} - \zeta_{30}^{10} q^{49} + (\zeta_{30}^{2} + \zeta_{30}) q^{53} - \zeta_{30}^{7} q^{55} + ( - \zeta_{30}^{11} + 1) q^{59} + \zeta_{30}^{10} q^{60} - \zeta_{30}^{12} q^{64} + (\zeta_{30}^{2} - 1) q^{67} + (\zeta_{30}^{14} - \zeta_{30}^{2}) q^{69} + ( - \zeta_{30}^{11} - \zeta_{30}^{7}) q^{71} - \zeta_{30}^{11} q^{75} + \zeta_{30}^{13} q^{80} + \zeta_{30}^{4} q^{81} + ( - \zeta_{30}^{6} + \zeta_{30}^{3}) q^{89} + ( - \zeta_{30}^{5} - \zeta_{30}^{2}) q^{92} + ( - \zeta_{30}^{10} + \zeta_{30}^{9}) q^{93} + (\zeta_{30}^{7} + \zeta_{30}^{6}) q^{97} - \zeta_{30}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - q^{4} + 4 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - q^{4} + 4 q^{5} + q^{9} - q^{11} + 4 q^{12} + 2 q^{15} + q^{16} - 2 q^{20} - 4 q^{25} - 2 q^{27} + q^{31} + 2 q^{33} + 2 q^{36} - 2 q^{44} - q^{45} + 3 q^{47} - 2 q^{48} + 4 q^{49} + q^{55} + 9 q^{59} - 4 q^{60} + 2 q^{64} - 7 q^{67} + 2 q^{71} + q^{75} - q^{80} + q^{81} + 4 q^{89} - 5 q^{92} + 6 q^{93} - 3 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(\zeta_{30}^{10}\) \(-1\) \(\zeta_{30}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
0.669131 − 0.743145i
0.913545 − 0.406737i
−0.978148 + 0.207912i
−0.978148 − 0.207912i
0.913545 + 0.406737i
0.669131 + 0.743145i
−0.104528 − 0.994522i
−0.104528 + 0.994522i
0 0.669131 − 0.743145i 0.978148 − 0.207912i 0.500000 − 0.866025i 0 0 0 −0.104528 − 0.994522i 0
769.1 0 0.913545 − 0.406737i 0.104528 + 0.994522i 0.500000 + 0.866025i 0 0 0 0.669131 − 0.743145i 0
934.1 0 −0.978148 + 0.207912i −0.669131 + 0.743145i 0.500000 − 0.866025i 0 0 0 0.913545 − 0.406737i 0
1264.1 0 −0.978148 − 0.207912i −0.669131 − 0.743145i 0.500000 + 0.866025i 0 0 0 0.913545 + 0.406737i 0
1429.1 0 0.913545 + 0.406737i 0.104528 − 0.994522i 0.500000 − 0.866025i 0 0 0 0.669131 + 0.743145i 0
1759.1 0 0.669131 + 0.743145i 0.978148 + 0.207912i 0.500000 + 0.866025i 0 0 0 −0.104528 + 0.994522i 0
2254.1 0 −0.104528 − 0.994522i −0.913545 + 0.406737i 0.500000 + 0.866025i 0 0 0 −0.978148 + 0.207912i 0
2419.1 0 −0.104528 + 0.994522i −0.913545 − 0.406737i 0.500000 − 0.866025i 0 0 0 −0.978148 − 0.207912i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
225.u even 30 1 inner
2475.ev odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.1.ev.b yes 8
9.c even 3 1 2475.1.ev.a ✓ 8
11.b odd 2 1 CM 2475.1.ev.b yes 8
25.e even 10 1 2475.1.ev.a ✓ 8
99.h odd 6 1 2475.1.ev.a ✓ 8
225.u even 30 1 inner 2475.1.ev.b yes 8
275.s odd 10 1 2475.1.ev.a ✓ 8
2475.ev odd 30 1 inner 2475.1.ev.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.1.ev.a ✓ 8 9.c even 3 1
2475.1.ev.a ✓ 8 25.e even 10 1
2475.1.ev.a ✓ 8 99.h odd 6 1
2475.1.ev.a ✓ 8 275.s odd 10 1
2475.1.ev.b yes 8 1.a even 1 1 trivial
2475.1.ev.b yes 8 11.b odd 2 1 CM
2475.1.ev.b yes 8 225.u even 30 1 inner
2475.1.ev.b yes 8 2475.ev odd 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{31}^{8} - T_{31}^{7} - 4T_{31}^{5} + 9T_{31}^{4} - 4T_{31}^{3} + 20T_{31}^{2} + 9T_{31} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - T^{7} - 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 3 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} - 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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