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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Newspace parameters

Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1334.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6520436296\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.978400.1
Defining polynomial: \(x^{5} - 2 x^{4} - 7 x^{3} + 6 x^{2} + 14 x + 2\)
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

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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 4 \nu^{3} + \nu^{2} - 11 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 7 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(4 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} + 14 \beta_{1} + 15\)

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.

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\( T_{3}^{5} - T_{3}^{4} - 13 T_{3}^{3} + 5 T_{3}^{2} + 40 T_{3} + 20 \)
\( T_{5}^{5} - 3 T_{5}^{4} - 7 T_{5}^{3} + 25 T_{5}^{2} - 10 T_{5} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{5} \)
$3$ \( 20 + 40 T + 5 T^{2} - 13 T^{3} - T^{4} + T^{5} \)
$5$ \( -10 - 10 T + 25 T^{2} - 7 T^{3} - 3 T^{4} + T^{5} \)
$7$ \( -16 + 22 T^{2} - 4 T^{3} - 4 T^{4} + T^{5} \)
$11$ \( -244 + 272 T + 115 T^{2} - 43 T^{3} - 3 T^{4} + T^{5} \)
$13$ \( -4 - 24 T - 35 T^{2} - 11 T^{3} + 3 T^{4} + T^{5} \)
$17$ \( -524 + 516 T - 66 T^{2} - 40 T^{3} + 4 T^{4} + T^{5} \)
$19$ \( 400 - 400 T + 28 T^{2} + 52 T^{3} - 14 T^{4} + T^{5} \)
$23$ \( ( -1 + T )^{5} \)
$29$ \( ( 1 + T )^{5} \)
$31$ \( -400 + 200 T + 135 T^{2} - 27 T^{3} - 7 T^{4} + T^{5} \)
$37$ \( -10000 + 2400 T + 406 T^{2} - 112 T^{3} - 2 T^{4} + T^{5} \)
$41$ \( -4424 + 3144 T + 170 T^{2} - 136 T^{3} - 4 T^{4} + T^{5} \)
$43$ \( -20 + 680 T - 285 T^{2} - 39 T^{3} + 9 T^{4} + T^{5} \)
$47$ \( 11864 + 152 T - 825 T^{2} - 43 T^{3} + 13 T^{4} + T^{5} \)
$53$ \( -2186 - 26 T + 585 T^{2} - 71 T^{3} - 11 T^{4} + T^{5} \)
$59$ \( -6656 + 1152 T + 584 T^{2} - 124 T^{3} - 2 T^{4} + T^{5} \)
$61$ \( -24416 + 26416 T - 7698 T^{2} + 928 T^{3} - 50 T^{4} + T^{5} \)
$67$ \( -268 + 700 T - 526 T^{2} + 164 T^{3} - 22 T^{4} + T^{5} \)
$71$ \( 3232 + 1520 T - 96 T^{2} - 96 T^{3} - 2 T^{4} + T^{5} \)
$73$ \( -9784 + 9784 T - 3534 T^{2} + 568 T^{3} - 40 T^{4} + T^{5} \)
$79$ \( 122 + 374 T - 9 T^{2} - 87 T^{3} + 3 T^{4} + T^{5} \)
$83$ \( -3152 - 3736 T - 1060 T^{2} - 14 T^{3} + 18 T^{4} + T^{5} \)
$89$ \( -203612 + 17820 T + 3686 T^{2} - 308 T^{3} - 12 T^{4} + T^{5} \)
$97$ \( -36224 + 17696 T + 488 T^{2} - 322 T^{3} + T^{5} \)
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