Properties

Label 310.2.a.e
Level $310$
Weight $2$
Character orbit 310.a
Self dual yes
Analytic conductor $2.475$
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] = [310,2,Mod(1,310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(310, 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("310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 310 = 2 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 310.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: \(2.47536246266\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + q^{5} + ( - \beta_1 + 1) q^{6} + ( - \beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + q^{10} + (\beta_{2} + 2 \beta_1 - 1) q^{11} + ( - \beta_1 + 1) q^{12} + (\beta_1 - 3) q^{13} + ( - \beta_{2} + \beta_1) q^{14} + ( - \beta_1 + 1) q^{15} + q^{16} + ( - \beta_{2} + \beta_1) q^{17} + (\beta_{2} - \beta_1 + 1) q^{18} + (2 \beta_{2} + 2) q^{19} + q^{20} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{21} + (\beta_{2} + 2 \beta_1 - 1) q^{22} + ( - \beta_{2} - \beta_1) q^{23} + ( - \beta_1 + 1) q^{24} + q^{25} + (\beta_1 - 3) q^{26} + (2 \beta_{2} + 2) q^{27} + ( - \beta_{2} + \beta_1) q^{28} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{29} + ( - \beta_1 + 1) q^{30} + q^{31} + q^{32} + ( - \beta_{2} - \beta_1 - 6) q^{33} + ( - \beta_{2} + \beta_1) q^{34} + ( - \beta_{2} + \beta_1) q^{35} + (\beta_{2} - \beta_1 + 1) q^{36} + (2 \beta_{2} - \beta_1 - 3) q^{37} + (2 \beta_{2} + 2) q^{38} + ( - \beta_{2} + 3 \beta_1 - 6) q^{39} + q^{40} + (3 \beta_{2} + \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{42} + (2 \beta_{2} - 3 \beta_1 - 3) q^{43} + (\beta_{2} + 2 \beta_1 - 1) q^{44} + (\beta_{2} - \beta_1 + 1) q^{45} + ( - \beta_{2} - \beta_1) q^{46} + ( - \beta_{2} - 3 \beta_1 + 8) q^{47} + ( - \beta_1 + 1) q^{48} + ( - 4 \beta_1 + 5) q^{49} + q^{50} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{51} + (\beta_1 - 3) q^{52} + (2 \beta_{2} - \beta_1 - 7) q^{53} + (2 \beta_{2} + 2) q^{54} + (\beta_{2} + 2 \beta_1 - 1) q^{55} + ( - \beta_{2} + \beta_1) q^{56} + (2 \beta_{2} - 6 \beta_1 + 4) q^{57} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{58} + (2 \beta_1 + 6) q^{59} + ( - \beta_1 + 1) q^{60} + ( - \beta_{2} - 2 \beta_1 - 5) q^{61} + q^{62} + ( - \beta_{2} + 5 \beta_1 - 12) q^{63} + q^{64} + (\beta_1 - 3) q^{65} + ( - \beta_{2} - \beta_1 - 6) q^{66} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{67} + ( - \beta_{2} + \beta_1) q^{68} + (2 \beta_1 + 2) q^{69} + ( - \beta_{2} + \beta_1) q^{70} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{71} + (\beta_{2} - \beta_1 + 1) q^{72} + ( - 3 \beta_{2} + \beta_1 - 6) q^{73} + (2 \beta_{2} - \beta_1 - 3) q^{74} + ( - \beta_1 + 1) q^{75} + (2 \beta_{2} + 2) q^{76} + 4 \beta_{2} q^{77} + ( - \beta_{2} + 3 \beta_1 - 6) q^{78} + (6 \beta_1 - 2) q^{79} + q^{80} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (3 \beta_{2} + \beta_1 - 2) q^{82} + (2 \beta_{2} - \beta_1 + 3) q^{83} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{84} + ( - \beta_{2} + \beta_1) q^{85} + (2 \beta_{2} - 3 \beta_1 - 3) q^{86} + ( - \beta_{2} + 5 \beta_1 + 4) q^{87} + (\beta_{2} + 2 \beta_1 - 1) q^{88} + (2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (\beta_{2} - \beta_1 + 1) q^{90} + (4 \beta_{2} - 4 \beta_1 + 4) q^{91} + ( - \beta_{2} - \beta_1) q^{92} + ( - \beta_1 + 1) q^{93} + ( - \beta_{2} - 3 \beta_1 + 8) q^{94} + (2 \beta_{2} + 2) q^{95} + ( - \beta_1 + 1) q^{96} + ( - \beta_{2} + 3 \beta_1 - 4) q^{97} + ( - 4 \beta_1 + 5) q^{98} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} + 2 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 2 q^{12} - 8 q^{13} + 2 q^{15} + 3 q^{16} + 3 q^{18} + 8 q^{19} + 3 q^{20} - 12 q^{21} - 2 q^{23} + 2 q^{24} + 3 q^{25} - 8 q^{26} + 8 q^{27} - 2 q^{29} + 2 q^{30} + 3 q^{31} + 3 q^{32} - 20 q^{33} + 3 q^{36} - 8 q^{37} + 8 q^{38} - 16 q^{39} + 3 q^{40} - 2 q^{41} - 12 q^{42} - 10 q^{43} + 3 q^{45} - 2 q^{46} + 20 q^{47} + 2 q^{48} + 11 q^{49} + 3 q^{50} - 12 q^{51} - 8 q^{52} - 20 q^{53} + 8 q^{54} + 8 q^{57} - 2 q^{58} + 20 q^{59} + 2 q^{60} - 18 q^{61} + 3 q^{62} - 32 q^{63} + 3 q^{64} - 8 q^{65} - 20 q^{66} - 12 q^{67} + 8 q^{69} + 8 q^{71} + 3 q^{72} - 20 q^{73} - 8 q^{74} + 2 q^{75} + 8 q^{76} + 4 q^{77} - 16 q^{78} + 3 q^{80} - q^{81} - 2 q^{82} + 10 q^{83} - 12 q^{84} - 10 q^{86} + 16 q^{87} + 18 q^{89} + 3 q^{90} + 12 q^{91} - 2 q^{92} + 2 q^{93} + 20 q^{94} + 8 q^{95} + 2 q^{96} - 10 q^{97} + 11 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
2.17009
−1.48119
0.311108
1.00000 −1.70928 1.00000 1.00000 −1.70928 1.07838 1.00000 −0.0783777 1.00000
1.2 1.00000 0.806063 1.00000 1.00000 0.806063 3.35026 1.00000 −2.35026 1.00000
1.3 1.00000 2.90321 1.00000 1.00000 2.90321 −4.42864 1.00000 5.42864 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(31\) \( -1 \)

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 310.2.a.e 3
3.b odd 2 1 2790.2.a.bi 3
4.b odd 2 1 2480.2.a.u 3
5.b even 2 1 1550.2.a.k 3
5.c odd 4 2 1550.2.b.j 6
8.b even 2 1 9920.2.a.bw 3
8.d odd 2 1 9920.2.a.bx 3
31.b odd 2 1 9610.2.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.a.e 3 1.a even 1 1 trivial
1550.2.a.k 3 5.b even 2 1
1550.2.b.j 6 5.c odd 4 2
2480.2.a.u 3 4.b odd 2 1
2790.2.a.bi 3 3.b odd 2 1
9610.2.a.u 3 31.b odd 2 1
9920.2.a.bw 3 8.b even 2 1
9920.2.a.bx 3 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(310))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 16T + 16 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + \cdots + 160 \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} + \cdots - 260 \) Copy content Toggle raw display
$31$ \( (T - 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} + 8 T^{2} + \cdots - 92 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$43$ \( T^{3} + 10 T^{2} + \cdots - 604 \) Copy content Toggle raw display
$47$ \( T^{3} - 20 T^{2} + \cdots + 208 \) Copy content Toggle raw display
$53$ \( T^{3} + 20 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{3} - 20 T^{2} + \cdots - 160 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$67$ \( T^{3} + 12 T^{2} + \cdots - 1184 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$73$ \( T^{3} + 20 T^{2} + \cdots - 464 \) Copy content Toggle raw display
$79$ \( T^{3} - 192T - 160 \) Copy content Toggle raw display
$83$ \( T^{3} - 10 T^{2} + \cdots + 124 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$97$ \( T^{3} + 10 T^{2} + \cdots - 8 \) Copy content Toggle raw display
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