Properties

Label 1395.2.a.l
Level $1395$
Weight $2$
Character orbit 1395.a
Self dual yes
Analytic conductor $11.139$
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] = [1395,2,Mod(1,1395)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1395, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1395.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1395.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: \(11.1391310820\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{8} - \beta_{2} q^{10} + (\beta_{3} + \beta_1 + 1) q^{11} + (\beta_{3} - \beta_1 + 3) q^{13} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{14} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{16} + ( - \beta_{3} - 3) q^{17} + (\beta_{2} + 2 \beta_1 - 1) q^{19} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{20} + (2 \beta_{2} + 2 \beta_1 + 2) q^{22} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} + q^{25} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{26} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{28} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{29} - q^{31} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{32} + (\beta_{3} - 4 \beta_{2} - \beta_1 - 1) q^{34} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{35} + ( - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{37} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{38} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{40} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 4) q^{41} + (\beta_{3} - 2 \beta_1 + 5) q^{43} + (2 \beta_{3} + 2 \beta_1 + 6) q^{44} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{46} + (4 \beta_{2} - 2 \beta_1 + 4) q^{47} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 5) q^{49} + \beta_{2} q^{50} + (4 \beta_{3} - 6 \beta_{2} + 4 \beta_1 + 4) q^{52} + ( - 5 \beta_1 - 2) q^{53} + ( - \beta_{3} - \beta_1 - 1) q^{55} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 4) q^{56} + (4 \beta_{2} + 2 \beta_1 - 2) q^{58} + ( - 3 \beta_{3} + \beta_{2} + \beta_1) q^{59} + ( - 2 \beta_1 + 6) q^{61} - \beta_{2} q^{62} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{64} + ( - \beta_{3} + \beta_1 - 3) q^{65} + ( - 3 \beta_{3} - 3 \beta_1 - 3) q^{67} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 6) q^{68} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{70} + ( - \beta_{2} + 5) q^{71} + ( - 2 \beta_{3} + \beta_1 + 4) q^{73} + (3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 1) q^{74} + ( - 2 \beta_{3} + 8 \beta_{2} + 2) q^{76} + (2 \beta_{2} + 2 \beta_1 - 2) q^{77} + ( - \beta_{3} - 2 \beta_{2} + 7 \beta_1 - 3) q^{79} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{80} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 9) q^{82} + (3 \beta_{3} + 2 \beta_{2} + 5) q^{83} + (\beta_{3} + 3) q^{85} + ( - 3 \beta_{3} + 6 \beta_{2} - \beta_1 - 1) q^{86} + 4 \beta_{2} q^{88} + (\beta_{3} - 6 \beta_{2} + \beta_1 + 1) q^{89} + ( - 4 \beta_{3} + 10 \beta_{2} - 6 \beta_1 + 2) q^{91} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{92} + (2 \beta_{3} + 2 \beta_1 + 10) q^{94} + ( - \beta_{2} - 2 \beta_1 + 1) q^{95} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{97} + ( - 8 \beta_{3} + 9 \beta_{2} - 4 \beta_1 - 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 5 q^{4} - 4 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 5 q^{4} - 4 q^{5} + 2 q^{7} - 3 q^{8} + q^{10} + 4 q^{11} + 10 q^{13} + 16 q^{14} + 3 q^{16} - 11 q^{17} - 3 q^{19} - 5 q^{20} + 8 q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{26} - 24 q^{28} + 8 q^{29} - 4 q^{31} - 7 q^{32} - 2 q^{34} - 2 q^{35} + 3 q^{37} + 22 q^{38} + 3 q^{40} + 11 q^{41} + 17 q^{43} + 24 q^{44} + 16 q^{46} + 10 q^{47} + 16 q^{49} - q^{50} + 22 q^{52} - 13 q^{53} - 4 q^{55} + 24 q^{56} - 10 q^{58} + 3 q^{59} + 22 q^{61} + q^{62} - 9 q^{64} - 10 q^{65} - 12 q^{67} - 28 q^{68} - 16 q^{70} + 21 q^{71} + 19 q^{73} + 2 q^{74} + 2 q^{76} - 8 q^{77} - 2 q^{79} - 3 q^{80} + 36 q^{82} + 15 q^{83} + 11 q^{85} - 8 q^{86} - 4 q^{88} + 10 q^{89} - 4 q^{91} - 24 q^{92} + 40 q^{94} + 3 q^{95} + 4 q^{97} - 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) 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
−0.704624
1.31743
−1.89122
2.27841
−2.50350 0 4.26753 −1.00000 0 −4.77104 −5.67678 0 2.50350
1.2 −1.26438 0 −0.401352 −1.00000 0 1.13698 3.03621 0 1.26438
1.3 0.576713 0 −1.66740 −1.00000 0 4.24412 −2.11504 0 −0.576713
1.4 2.19117 0 2.80122 −1.00000 0 1.38995 1.75561 0 −2.19117
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(31\) \(1\)

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 1395.2.a.l 4
3.b odd 2 1 155.2.a.e 4
5.b even 2 1 6975.2.a.bn 4
12.b even 2 1 2480.2.a.x 4
15.d odd 2 1 775.2.a.e 4
15.e even 4 2 775.2.b.f 8
21.c even 2 1 7595.2.a.s 4
24.f even 2 1 9920.2.a.cg 4
24.h odd 2 1 9920.2.a.cb 4
93.c even 2 1 4805.2.a.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.e 4 3.b odd 2 1
775.2.a.e 4 15.d odd 2 1
775.2.b.f 8 15.e even 4 2
1395.2.a.l 4 1.a even 1 1 trivial
2480.2.a.x 4 12.b even 2 1
4805.2.a.n 4 93.c even 2 1
6975.2.a.bn 4 5.b even 2 1
7595.2.a.s 4 21.c even 2 1
9920.2.a.cb 4 24.h odd 2 1
9920.2.a.cg 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1395))\):

\( T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - 4T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 20T_{7}^{2} + 52T_{7} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 20 T^{2} + 52 T - 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} - 8 T^{2} + 12 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + 20 T^{2} + \cdots - 136 \) Copy content Toggle raw display
$17$ \( T^{4} + 11 T^{3} + 35 T^{2} + 13 T - 58 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} - 33 T^{2} - 107 T + 44 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} - 20 T^{2} + 52 T - 32 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} - 20 T^{2} + 292 T - 584 \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} - 81 T^{2} + \cdots + 1538 \) Copy content Toggle raw display
$41$ \( T^{4} - 11 T^{3} - 31 T^{2} + \cdots + 506 \) Copy content Toggle raw display
$43$ \( T^{4} - 17 T^{3} + 73 T^{2} + \cdots - 236 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} - 52 T^{2} + \cdots + 1408 \) Copy content Toggle raw display
$53$ \( T^{4} + 13 T^{3} - 71 T^{2} + \cdots + 1306 \) Copy content Toggle raw display
$59$ \( T^{4} - 3 T^{3} - 97 T^{2} - 129 T - 44 \) Copy content Toggle raw display
$61$ \( T^{4} - 22 T^{3} + 160 T^{2} + \cdots + 352 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 72 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} - 21 T^{3} + 159 T^{2} + \cdots + 584 \) Copy content Toggle raw display
$73$ \( T^{4} - 19 T^{3} + 85 T^{2} - 123 T + 34 \) Copy content Toggle raw display
$79$ \( T^{4} + 2 T^{3} - 260 T^{2} + \cdots + 6592 \) Copy content Toggle raw display
$83$ \( T^{4} - 15 T^{3} - 63 T^{2} + \cdots - 3364 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} - 152 T^{2} + \cdots + 3688 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} - 248 T^{2} + \cdots - 464 \) Copy content Toggle raw display
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