Properties

Label 475.2.a.d
Level $475$
Weight $2$
Character orbit 475.a
Self dual yes
Analytic conductor $3.793$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} + 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{6} + (\beta_{2} + \beta_1) q^{7} + ( - 4 \beta_{2} - \beta_1 - 4) q^{8} + ( - 2 \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} + 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{6} + (\beta_{2} + \beta_1) q^{7} + ( - 4 \beta_{2} - \beta_1 - 4) q^{8} + ( - 2 \beta_{2} + \beta_1 - 2) q^{9} + ( - 3 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{2} - 2) q^{13} + ( - 3 \beta_{2} - \beta_1 - 3) q^{14} + (7 \beta_{2} + \beta_1 + 8) q^{16} + ( - \beta_{2} + 3 \beta_1 - 2) q^{17} + (3 \beta_{2} + \beta_1 + 2) q^{18} - q^{19} + ( - 2 \beta_{2} + \beta_1 - 1) q^{21} + (5 \beta_{2} + 2) q^{22} + (\beta_1 - 3) q^{23} + (4 \beta_{2} + 1) q^{24} + (2 \beta_{2} + 2 \beta_1 + 3) q^{26} + ( - \beta_{2} + 3 \beta_1 - 1) q^{27} + (5 \beta_{2} + 2 \beta_1 + 8) q^{28} + (5 \beta_{2} - 5 \beta_1 + 1) q^{29} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{31} + ( - 7 \beta_{2} - 7 \beta_1 - 9) q^{32} + (5 \beta_{2} - 2 \beta_1 - 1) q^{33} + ( - \beta_{2} - \beta_1 - 3) q^{34} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{36} + ( - \beta_{2} + 3 \beta_1 - 3) q^{37} + (\beta_1 + 1) q^{38} + \beta_1 q^{39} + (8 \beta_{2} - 3 \beta_1 + 4) q^{41} + (3 \beta_{2} + 1) q^{42} + ( - 6 \beta_{2} + 4 \beta_1 - 5) q^{43} + ( - 4 \beta_{2} - 4 \beta_1 - 5) q^{44} + ( - \beta_{2} + 2 \beta_1 + 1) q^{46} + (\beta_{2} - 2 \beta_1) q^{47} + ( - 6 \beta_{2} - \beta_1 - 1) q^{48} + (2 \beta_{2} + \beta_1 - 2) q^{49} + (\beta_1 - 3) q^{51} + ( - 4 \beta_{2} - 5 \beta_1 - 5) q^{52} + (4 \beta_{2} - 6 \beta_1 - 3) q^{53} + ( - \beta_{2} - 2 \beta_1 - 4) q^{54} + ( - 6 \beta_{2} - 8 \beta_1 - 11) q^{56} + ( - \beta_{2} + \beta_1) q^{57} + ( - 5 \beta_{2} + 4 \beta_1 + 4) q^{58} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{59} + (\beta_{2} - 4 \beta_1 - 4) q^{61} + (8 \beta_{2} + 4 \beta_1 + 9) q^{62} + ( - 4 \beta_1 - 1) q^{63} + (7 \beta_{2} + 14 \beta_1 + 14) q^{64} + ( - 8 \beta_{2} + 3 \beta_1) q^{66} + (5 \beta_{2} - 9 \beta_1 + 5) q^{67} + (5 \beta_{2} - 2 \beta_1 + 10) q^{68} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{69} + (2 \beta_{2} + 4 \beta_1 - 7) q^{71} + (5 \beta_{2} + 6 \beta_1 + 12) q^{72} + ( - 8 \beta_{2} - \beta_1 - 2) q^{73} + ( - \beta_{2} - 2) q^{74} + ( - \beta_{2} - 2 \beta_1 - 1) q^{76} + (\beta_{2} - 4 \beta_1 - 3) q^{77} + ( - \beta_{2} - \beta_1 - 2) q^{78} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{79} + (7 \beta_{2} - 3 \beta_1 + 3) q^{81} + ( - 13 \beta_{2} - \beta_1 - 6) q^{82} + (5 \beta_1 - 6) q^{83} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{84} + (8 \beta_{2} + \beta_1 + 3) q^{86} + ( - 9 \beta_{2} + 4 \beta_1 + 5) q^{87} + (2 \beta_{2} + 9 \beta_1 + 13) q^{88} + ( - 6 \beta_{2} - \beta_1 - 1) q^{89} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{91} + ( - 5 \beta_1 + 2) q^{92} + (4 \beta_{2} - \beta_1 + 2) q^{93} + (2 \beta_1 + 3) q^{94} + (5 \beta_{2} + 2 \beta_1 + 7) q^{96} + ( - \beta_{2} - 2) q^{97} + ( - 5 \beta_{2} + \beta_1 - 2) q^{98} + ( - 2 \beta_{2} + 3 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27} + 21 q^{28} - 7 q^{29} - 5 q^{31} - 27 q^{32} - 10 q^{33} - 9 q^{34} - 11 q^{36} - 5 q^{37} + 4 q^{38} + q^{39} + q^{41} - 5 q^{43} - 15 q^{44} + 6 q^{46} - 3 q^{47} + 2 q^{48} - 7 q^{49} - 8 q^{51} - 16 q^{52} - 19 q^{53} - 13 q^{54} - 35 q^{56} + 2 q^{57} + 21 q^{58} + 10 q^{59} - 17 q^{61} + 23 q^{62} - 7 q^{63} + 49 q^{64} + 11 q^{66} + q^{67} + 23 q^{68} + 3 q^{69} - 19 q^{71} + 37 q^{72} + q^{73} - 5 q^{74} - 4 q^{76} - 14 q^{77} - 6 q^{78} + 18 q^{79} - q^{81} - 6 q^{82} - 13 q^{83} - 7 q^{84} + 2 q^{86} + 28 q^{87} + 46 q^{88} + 2 q^{89} - 7 q^{91} + q^{92} + q^{93} + 11 q^{94} + 18 q^{96} - 5 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−2.80194 −0.554958 5.85086 0 1.55496 3.04892 −10.7899 −2.69202 0
1.2 −1.44504 −2.24698 0.0881460 0 3.24698 −1.35690 2.76271 2.04892 0
1.3 0.246980 0.801938 −1.93900 0 0.198062 −1.69202 −0.972853 −2.35690 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.d 3
3.b odd 2 1 4275.2.a.bn 3
4.b odd 2 1 7600.2.a.bw 3
5.b even 2 1 475.2.a.h yes 3
5.c odd 4 2 475.2.b.c 6
15.d odd 2 1 4275.2.a.z 3
19.b odd 2 1 9025.2.a.be 3
20.d odd 2 1 7600.2.a.bn 3
95.d odd 2 1 9025.2.a.w 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.d 3 1.a even 1 1 trivial
475.2.a.h yes 3 5.b even 2 1
475.2.b.c 6 5.c odd 4 2
4275.2.a.z 3 15.d odd 2 1
4275.2.a.bn 3 3.b odd 2 1
7600.2.a.bn 3 20.d odd 2 1
7600.2.a.bw 3 4.b odd 2 1
9025.2.a.w 3 95.d odd 2 1
9025.2.a.be 3 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 4T_{2}^{2} + 3T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 4 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 7T - 7 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} + \cdots - 13 \) Copy content Toggle raw display
$13$ \( T^{3} + 5 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} + \cdots - 91 \) Copy content Toggle raw display
$31$ \( T^{3} + 5 T^{2} + \cdots + 43 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} + \cdots + 421 \) Copy content Toggle raw display
$43$ \( T^{3} + 5 T^{2} + \cdots - 293 \) Copy content Toggle raw display
$47$ \( T^{3} + 3 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$53$ \( T^{3} + 19 T^{2} + \cdots - 307 \) Copy content Toggle raw display
$59$ \( T^{3} - 10 T^{2} + \cdots + 328 \) Copy content Toggle raw display
$61$ \( T^{3} + 17 T^{2} + \cdots - 41 \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} + \cdots - 559 \) Copy content Toggle raw display
$71$ \( T^{3} + 19 T^{2} + \cdots - 307 \) Copy content Toggle raw display
$73$ \( T^{3} - T^{2} + \cdots + 463 \) Copy content Toggle raw display
$79$ \( T^{3} - 18 T^{2} + \cdots - 97 \) Copy content Toggle raw display
$83$ \( T^{3} + 13 T^{2} + \cdots - 139 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} + \cdots + 281 \) Copy content Toggle raw display
$97$ \( T^{3} + 5 T^{2} + \cdots + 1 \) Copy content Toggle raw display
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