Properties

Label 475.2.a.i
Level $475$
Weight $2$
Character orbit 475.a
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

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: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + (2 \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_{2} + 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + (2 \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_{2} + 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9} + 2 \beta_1 q^{11} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{12} + ( - \beta_{3} - 2 \beta_1) q^{13} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{14} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{16} + ( - 2 \beta_{3} - 2) q^{17} + ( - 3 \beta_{2} + 2 \beta_1 + 6) q^{18} + q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{22}+ \cdots + ( - 4 \beta_{3} - 4 \beta_{2} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} + 4 q^{27} + 8 q^{28} + 4 q^{29} + 4 q^{31} + 6 q^{32} - 8 q^{33} - 4 q^{34} + 40 q^{36} + 6 q^{37} + 2 q^{38} - 12 q^{39} + 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} - 38 q^{48} + 20 q^{49} - 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 12 q^{56} - 2 q^{57} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} + 18 q^{67} - 4 q^{68} + 28 q^{69} - 20 q^{71} + 52 q^{72} - 28 q^{73} + 32 q^{74} + 8 q^{76} + 40 q^{77} - 12 q^{78} - 16 q^{79} + 16 q^{81} - 16 q^{82} - 44 q^{84} - 8 q^{86} + 36 q^{87} + 12 q^{88} + 4 q^{89} - 36 q^{91} + 28 q^{92} + 40 q^{93} - 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.78165
−1.51658
−0.552409
1.28734
−1.95594 −0.296842 1.82571 0 0.580605 3.56331 0.340899 −2.91188 0
1.2 −0.816594 −1.53844 −1.33317 0 1.25628 −5.03316 2.72185 −0.633188 0
1.3 2.14243 2.87834 2.59002 0 6.16666 −3.10482 1.26409 5.28487 0
1.4 2.63010 −3.04306 4.91744 0 −8.00355 0.574672 7.67316 6.26020 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(19\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.a.i 4
3.b odd 2 1 4275.2.a.bo 4
4.b odd 2 1 7600.2.a.cf 4
5.b even 2 1 95.2.a.b 4
5.c odd 4 2 475.2.b.e 8
15.d odd 2 1 855.2.a.m 4
19.b odd 2 1 9025.2.a.bf 4
20.d odd 2 1 1520.2.a.t 4
35.c odd 2 1 4655.2.a.y 4
40.e odd 2 1 6080.2.a.ch 4
40.f even 2 1 6080.2.a.cc 4
95.d odd 2 1 1805.2.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 5.b even 2 1
475.2.a.i 4 1.a even 1 1 trivial
475.2.b.e 8 5.c odd 4 2
855.2.a.m 4 15.d odd 2 1
1520.2.a.t 4 20.d odd 2 1
1805.2.a.p 4 95.d odd 2 1
4275.2.a.bo 4 3.b odd 2 1
4655.2.a.y 4 35.c odd 2 1
6080.2.a.cc 4 40.f even 2 1
6080.2.a.ch 4 40.e odd 2 1
7600.2.a.cf 4 4.b odd 2 1
9025.2.a.bf 4 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 8T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 20 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + \cdots + 288 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 48 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots - 640 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + \cdots - 240 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots + 32 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 1056 \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} + \cdots - 348 \) Copy content Toggle raw display
$59$ \( T^{4} - 64 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + \cdots - 2656 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + \cdots - 1076 \) Copy content Toggle raw display
$71$ \( T^{4} + 20 T^{3} + \cdots - 4224 \) Copy content Toggle raw display
$73$ \( T^{4} + 28 T^{3} + \cdots + 176 \) Copy content Toggle raw display
$79$ \( T^{4} + 16 T^{3} + \cdots - 1856 \) Copy content Toggle raw display
$83$ \( T^{4} - 72 T^{2} + \cdots + 480 \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} + \cdots + 240 \) Copy content Toggle raw display
$97$ \( T^{4} + 30 T^{3} + \cdots - 1388 \) Copy content Toggle raw display
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