Properties

Label 475.2.a.g
Level $475$
Weight $2$
Character orbit 475.a
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{2} + 1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{2} + 1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + \beta_1 q^{9} - \beta_{2} q^{11} + (\beta_{2} - \beta_1 + 3) q^{12} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{13} + ( - \beta_{2} + 2) q^{14} + (\beta_{2} - 2 \beta_1 - 1) q^{16} + (\beta_{2} + 5) q^{17} + ( - \beta_{2} - 3) q^{18} + q^{19} + (2 \beta_{2} - \beta_1 - 1) q^{21} + ( - \beta_{2} + \beta_1 + 1) q^{22} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{23} + 5 q^{24} + ( - 6 \beta_{2} + 4 \beta_1 - 7) q^{26} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{27} + (\beta_{2} - \beta_1 + 1) q^{28} + ( - \beta_{2} - 2) q^{29} + (3 \beta_{2} - \beta_1 + 1) q^{31} + ( - \beta_{2} + 2 \beta_1) q^{32} + (\beta_{2} - \beta_1 - 2) q^{33} + (\beta_{2} - 6 \beta_1 + 4) q^{34} + ( - \beta_{2} + 2 \beta_1 - 2) q^{36} + (5 \beta_{2} - 2 \beta_1 + 4) q^{37} + ( - \beta_1 + 1) q^{38} + (2 \beta_{2} + 3 \beta_1 - 4) q^{39} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{41} + (3 \beta_{2} - \beta_1) q^{42} + (2 \beta_{2} - 6 \beta_1 + 1) q^{43} - q^{44} + ( - 5 \beta_{2} + \beta_1 - 6) q^{46} + (\beta_{2} - 5 \beta_1 + 5) q^{47} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{48} + ( - 4 \beta_{2} + \beta_1 - 4) q^{49} + (4 \beta_{2} + \beta_1 + 7) q^{51} + ( - 4 \beta_{2} + 7 \beta_1 - 11) q^{52} + ( - 2 \beta_{2} + 2 \beta_1 + 9) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 - 5) q^{54} + (4 \beta_{2} - 2 \beta_1 - 1) q^{56} + (\beta_{2} + 1) q^{57} + ( - \beta_{2} + 3 \beta_1 - 1) q^{58} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (\beta_{2} + \beta_1 + 1) q^{61} + (4 \beta_{2} - 4 \beta_1 + 1) q^{62} - q^{63} + ( - 5 \beta_{2} + 5 \beta_1 - 3) q^{64} + (2 \beta_{2} + \beta_1) q^{66} + ( - \beta_{2} + 4 \beta_1 - 6) q^{67} + (5 \beta_{2} - 5 \beta_1 + 11) q^{68} + (3 \beta_{2} + 4 \beta_1) q^{69} + ( - 2 \beta_1 + 3) q^{71} + ( - \beta_{2} + 3 \beta_1 - 1) q^{72} + (2 \beta_{2} - 3 \beta_1 + 2) q^{73} + (7 \beta_{2} - 9 \beta_1 + 5) q^{74} + (\beta_{2} - \beta_1 + 2) q^{76} + ( - 3 \beta_{2} + \beta_1 + 2) q^{77} + ( - \beta_{2} + 2 \beta_1 - 15) q^{78} + (5 \beta_{2} - 4 \beta_1 - 3) q^{79} + (\beta_{2} - 2 \beta_1 - 6) q^{81} + ( - 7 \beta_{2} + 4 \beta_1 - 15) q^{82} + (6 \beta_{2} + 3 \beta_1 + 2) q^{83} + ( - \beta_1 + 2) q^{84} + (8 \beta_{2} - 3 \beta_1 + 17) q^{86} + ( - \beta_{2} - \beta_1 - 4) q^{87} + (2 \beta_{2} - \beta_1 - 3) q^{88} + ( - 2 \beta_{2} + 5 \beta_1 - 9) q^{89} + ( - 8 \beta_{2} + 3 \beta_1 + 2) q^{91} + ( - 2 \beta_{2} + 5 \beta_1 - 6) q^{92} + ( - 2 \beta_{2} + \beta_1 + 6) q^{93} + (6 \beta_{2} - 6 \beta_1 + 19) q^{94} + (\beta_{2} + 3 \beta_1) q^{96} + ( - 3 \beta_{2} - \beta_1 - 5) q^{97} + ( - 5 \beta_{2} + 8 \beta_1 - 3) q^{98} + ( - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} + 7 q^{14} - 6 q^{16} + 14 q^{17} - 8 q^{18} + 3 q^{19} - 6 q^{21} + 5 q^{22} + 8 q^{23} + 15 q^{24} - 11 q^{26} - q^{27} + q^{28} - 5 q^{29} - q^{31} + 3 q^{32} - 8 q^{33} + 5 q^{34} - 3 q^{36} + 5 q^{37} + 2 q^{38} - 11 q^{39} + q^{41} - 4 q^{42} - 5 q^{43} - 3 q^{44} - 12 q^{46} + 9 q^{47} - 4 q^{48} - 7 q^{49} + 18 q^{51} - 22 q^{52} + 31 q^{53} - 5 q^{54} - 9 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} + 3 q^{61} - 5 q^{62} - 3 q^{63} + q^{64} - q^{66} - 13 q^{67} + 23 q^{68} + q^{69} + 7 q^{71} + q^{72} + q^{73} - q^{74} + 4 q^{76} + 10 q^{77} - 42 q^{78} - 18 q^{79} - 21 q^{81} - 34 q^{82} + 3 q^{83} + 5 q^{84} + 40 q^{86} - 12 q^{87} - 12 q^{88} - 20 q^{89} + 17 q^{91} - 11 q^{92} + 21 q^{93} + 45 q^{94} + 2 q^{96} - 13 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) 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

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
2.65109
−0.273891
−1.37720
−1.65109 2.37720 0.726109 0 −3.92498 −0.377203 2.10331 2.65109 0
1.2 1.27389 −1.65109 −0.377203 0 −2.10331 3.65109 −3.02830 −0.273891 0
1.3 2.37720 1.27389 3.65109 0 3.02830 0.726109 3.92498 −1.37720 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.g yes 3
3.b odd 2 1 4275.2.a.ba 3
4.b odd 2 1 7600.2.a.bh 3
5.b even 2 1 475.2.a.e 3
5.c odd 4 2 475.2.b.b 6
15.d odd 2 1 4275.2.a.bm 3
19.b odd 2 1 9025.2.a.y 3
20.d odd 2 1 7600.2.a.cc 3
95.d odd 2 1 9025.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.e 3 5.b even 2 1
475.2.a.g yes 3 1.a even 1 1 trivial
475.2.b.b 6 5.c odd 4 2
4275.2.a.ba 3 3.b odd 2 1
4275.2.a.bm 3 15.d odd 2 1
7600.2.a.bh 3 4.b odd 2 1
7600.2.a.cc 3 20.d odd 2 1
9025.2.a.y 3 19.b odd 2 1
9025.2.a.bc 3 95.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2T_{2}^{2} - 3T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 4T - 1 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} + \cdots + 103 \) Copy content Toggle raw display
$17$ \( T^{3} - 14 T^{2} + \cdots - 79 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$29$ \( T^{3} + 5 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$31$ \( T^{3} + T^{2} + \cdots + 53 \) Copy content Toggle raw display
$37$ \( T^{3} - 5 T^{2} + \cdots + 395 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} + \cdots + 155 \) Copy content Toggle raw display
$43$ \( T^{3} + 5 T^{2} + \cdots - 317 \) Copy content Toggle raw display
$47$ \( T^{3} - 9 T^{2} + \cdots + 311 \) Copy content Toggle raw display
$53$ \( T^{3} - 31 T^{2} + \cdots - 905 \) Copy content Toggle raw display
$59$ \( T^{3} + 6 T^{2} + \cdots - 200 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$67$ \( T^{3} + 13T^{2} - 169 \) Copy content Toggle raw display
$71$ \( T^{3} - 7T^{2} - T + 47 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} + \cdots - 53 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + \cdots - 395 \) Copy content Toggle raw display
$83$ \( T^{3} - 3 T^{2} + \cdots - 131 \) Copy content Toggle raw display
$89$ \( T^{3} + 20 T^{2} + \cdots - 125 \) Copy content Toggle raw display
$97$ \( T^{3} + 13T^{2} - 169 \) Copy content Toggle raw display
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