Properties

Label 1334.2.a.g
Level $1334$
Weight $2$
Character orbit 1334.a
Self dual yes
Analytic conductor $10.652$
Analytic rank $0$
Dimension $5$
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] = [1334,2,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, 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("1334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.978400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 7x^{3} + 6x^{2} + 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 7x^{3} + 6x^{2} + 14x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + 4\nu^{3} + \nu^{2} - 11\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 7\nu + 3 \) 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
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 3\beta_{3} + 9\beta_{2} + 14\beta _1 + 15 \) 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
−1.60577
−1.32108
3.14920
−0.154912
1.93256
−1.00000 −2.79005 1.00000 1.48849 2.79005 −2.09426 −1.00000 4.78440 −1.48849
1.2 −1.00000 −1.38739 1.00000 −2.84168 1.38739 2.52061 −1.00000 −1.07515 2.84168
1.3 −1.00000 −0.619069 1.00000 3.10109 0.619069 1.04811 −1.00000 −2.61675 −3.10109
1.4 −1.00000 2.66618 1.00000 1.70044 −2.66618 −0.855351 −1.00000 4.10851 −1.70044
1.5 −1.00000 3.13033 1.00000 −0.448336 −3.13033 3.38089 −1.00000 6.79899 0.448336
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(23\) \(-1\)
\(29\) \(1\)

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 1334.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.g 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\( T_{3}^{5} - T_{3}^{4} - 13T_{3}^{3} + 5T_{3}^{2} + 40T_{3} + 20 \) Copy content Toggle raw display
\( T_{5}^{5} - 3T_{5}^{4} - 7T_{5}^{3} + 25T_{5}^{2} - 10T_{5} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} - 13 T^{3} + 5 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 7 T^{3} + 25 T^{2} + \cdots - 10 \) Copy content Toggle raw display
$7$ \( T^{5} - 4 T^{4} - 4 T^{3} + 22 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{5} - 3 T^{4} - 43 T^{3} + 115 T^{2} + \cdots - 244 \) Copy content Toggle raw display
$13$ \( T^{5} + 3 T^{4} - 11 T^{3} - 35 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$17$ \( T^{5} + 4 T^{4} - 40 T^{3} - 66 T^{2} + \cdots - 524 \) Copy content Toggle raw display
$19$ \( T^{5} - 14 T^{4} + 52 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( (T - 1)^{5} \) Copy content Toggle raw display
$29$ \( (T + 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 7 T^{4} - 27 T^{3} + 135 T^{2} + \cdots - 400 \) Copy content Toggle raw display
$37$ \( T^{5} - 2 T^{4} - 112 T^{3} + \cdots - 10000 \) Copy content Toggle raw display
$41$ \( T^{5} - 4 T^{4} - 136 T^{3} + \cdots - 4424 \) Copy content Toggle raw display
$43$ \( T^{5} + 9 T^{4} - 39 T^{3} - 285 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$47$ \( T^{5} + 13 T^{4} - 43 T^{3} + \cdots + 11864 \) Copy content Toggle raw display
$53$ \( T^{5} - 11 T^{4} - 71 T^{3} + \cdots - 2186 \) Copy content Toggle raw display
$59$ \( T^{5} - 2 T^{4} - 124 T^{3} + \cdots - 6656 \) Copy content Toggle raw display
$61$ \( T^{5} - 50 T^{4} + 928 T^{3} + \cdots - 24416 \) Copy content Toggle raw display
$67$ \( T^{5} - 22 T^{4} + 164 T^{3} + \cdots - 268 \) Copy content Toggle raw display
$71$ \( T^{5} - 2 T^{4} - 96 T^{3} + \cdots + 3232 \) Copy content Toggle raw display
$73$ \( T^{5} - 40 T^{4} + 568 T^{3} + \cdots - 9784 \) Copy content Toggle raw display
$79$ \( T^{5} + 3 T^{4} - 87 T^{3} - 9 T^{2} + \cdots + 122 \) Copy content Toggle raw display
$83$ \( T^{5} + 18 T^{4} - 14 T^{3} + \cdots - 3152 \) Copy content Toggle raw display
$89$ \( T^{5} - 12 T^{4} - 308 T^{3} + \cdots - 203612 \) Copy content Toggle raw display
$97$ \( T^{5} - 322 T^{3} + 488 T^{2} + \cdots - 36224 \) Copy content Toggle raw display
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