Properties

Label 475.2.e.c
Level $475$
Weight $2$
Character orbit 475.e
Analytic conductor $3.793$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(26,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + 4 q^{7} + 3 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 2 \zeta_{6} q^{4} + 4 q^{7} + 3 \zeta_{6} q^{9} - q^{11} - 2 \zeta_{6} q^{13} + ( - 8 \zeta_{6} + 8) q^{14} + ( - 4 \zeta_{6} + 4) q^{16} + (2 \zeta_{6} - 2) q^{17} + 6 q^{18} + ( - 3 \zeta_{6} - 2) q^{19} + (2 \zeta_{6} - 2) q^{22} + 6 \zeta_{6} q^{23} - 4 q^{26} - 8 \zeta_{6} q^{28} - 9 \zeta_{6} q^{29} - 7 q^{31} - 8 \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( - 6 \zeta_{6} + 6) q^{36} - 2 q^{37} + (4 \zeta_{6} - 10) q^{38} + (2 \zeta_{6} - 2) q^{41} + ( - 2 \zeta_{6} + 2) q^{43} + 2 \zeta_{6} q^{44} + 12 q^{46} + 6 \zeta_{6} q^{47} + 9 q^{49} + (4 \zeta_{6} - 4) q^{52} + 4 \zeta_{6} q^{53} - 18 q^{58} + (9 \zeta_{6} - 9) q^{59} + 7 \zeta_{6} q^{61} + (14 \zeta_{6} - 14) q^{62} + 12 \zeta_{6} q^{63} - 8 q^{64} - 10 \zeta_{6} q^{67} + 4 q^{68} + (\zeta_{6} - 1) q^{71} + ( - 10 \zeta_{6} + 10) q^{73} + (4 \zeta_{6} - 4) q^{74} + (10 \zeta_{6} - 6) q^{76} - 4 q^{77} + (\zeta_{6} - 1) q^{79} + (9 \zeta_{6} - 9) q^{81} + 4 \zeta_{6} q^{82} + 6 q^{83} - 4 \zeta_{6} q^{86} + 11 \zeta_{6} q^{89} - 8 \zeta_{6} q^{91} + ( - 12 \zeta_{6} + 12) q^{92} + 12 q^{94} + (6 \zeta_{6} - 6) q^{97} + ( - 18 \zeta_{6} + 18) q^{98} - 3 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} + 8 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} + 8 q^{7} + 3 q^{9} - 2 q^{11} - 2 q^{13} + 8 q^{14} + 4 q^{16} - 2 q^{17} + 12 q^{18} - 7 q^{19} - 2 q^{22} + 6 q^{23} - 8 q^{26} - 8 q^{28} - 9 q^{29} - 14 q^{31} - 8 q^{32} + 4 q^{34} + 6 q^{36} - 4 q^{37} - 16 q^{38} - 2 q^{41} + 2 q^{43} + 2 q^{44} + 24 q^{46} + 6 q^{47} + 18 q^{49} - 4 q^{52} + 4 q^{53} - 36 q^{58} - 9 q^{59} + 7 q^{61} - 14 q^{62} + 12 q^{63} - 16 q^{64} - 10 q^{67} + 8 q^{68} - q^{71} + 10 q^{73} - 4 q^{74} - 2 q^{76} - 8 q^{77} - q^{79} - 9 q^{81} + 4 q^{82} + 12 q^{83} - 4 q^{86} + 11 q^{89} - 8 q^{91} + 12 q^{92} + 24 q^{94} - 6 q^{97} + 18 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 + 1.73205i 0 −1.00000 + 1.73205i 0 0 4.00000 0 1.50000 2.59808i 0
201.1 1.00000 1.73205i 0 −1.00000 1.73205i 0 0 4.00000 0 1.50000 + 2.59808i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.c 2
5.b even 2 1 475.2.e.a 2
5.c odd 4 2 95.2.i.a 4
15.e even 4 2 855.2.be.a 4
19.c even 3 1 inner 475.2.e.c 2
19.c even 3 1 9025.2.a.b 1
19.d odd 6 1 9025.2.a.j 1
95.h odd 6 1 9025.2.a.a 1
95.i even 6 1 475.2.e.a 2
95.i even 6 1 9025.2.a.i 1
95.l even 12 2 1805.2.b.a 2
95.m odd 12 2 95.2.i.a 4
95.m odd 12 2 1805.2.b.b 2
285.v even 12 2 855.2.be.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 5.c odd 4 2
95.2.i.a 4 95.m odd 12 2
475.2.e.a 2 5.b even 2 1
475.2.e.a 2 95.i even 6 1
475.2.e.c 2 1.a even 1 1 trivial
475.2.e.c 2 19.c even 3 1 inner
855.2.be.a 4 15.e even 4 2
855.2.be.a 4 285.v even 12 2
1805.2.b.a 2 95.l even 12 2
1805.2.b.b 2 95.m odd 12 2
9025.2.a.a 1 95.h odd 6 1
9025.2.a.b 1 19.c even 3 1
9025.2.a.i 1 95.i even 6 1
9025.2.a.j 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
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