Properties

Label 1334.2.a.e
Level $1334$
Weight $2$
Character orbit 1334.a
Self dual yes
Analytic conductor $10.652$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 3\)

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
2.68554
−1.27133
−1.74912
0.334904
1.00000 −2.41421 1.00000 −1.38372 −2.41421 0.526602 1.00000 2.82843 −1.38372
1.2 1.00000 −2.41421 1.00000 4.21215 −2.41421 −1.11239 1.00000 2.82843 4.21215
1.3 1.00000 0.414214 1.00000 −2.88784 0.414214 0.808530 1.00000 −2.82843 −2.88784
1.4 1.00000 0.414214 1.00000 0.0594122 0.414214 −4.22274 1.00000 −2.82843 0.0594122
\(n\): e.g. 2-40 or 990-1000
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.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.e 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{5}^{4} - 14 T_{5}^{2} - 16 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + 2 T + T^{2} )^{2} \)
$5$ \( 1 - 16 T - 14 T^{2} + T^{4} \)
$7$ \( 2 - 4 T - 2 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 1 - 12 T - 8 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( -151 - 96 T + 8 T^{3} + T^{4} \)
$17$ \( -14 + 52 T + 46 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 4 + 96 T + 76 T^{2} + 16 T^{3} + T^{4} \)
$23$ \( ( -1 + T )^{4} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( -199 - 132 T + 16 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 322 - 52 T - 58 T^{2} + 4 T^{3} + T^{4} \)
$41$ \( -1198 - 548 T - 18 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( -23 - 196 T - 80 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 1169 + 972 T + 264 T^{2} + 28 T^{3} + T^{4} \)
$53$ \( -127 + 64 T + 50 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( 4424 + 352 T - 140 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( 2098 - 196 T - 90 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( 3986 + 212 T - 198 T^{2} + T^{4} \)
$71$ \( -1264 - 768 T - 128 T^{2} + T^{4} \)
$73$ \( -10462 - 1796 T + 66 T^{2} + 24 T^{3} + T^{4} \)
$79$ \( 1993 + 476 T - 86 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( -56 - 304 T - 20 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( -686 + 2884 T - 174 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( -248 + 384 T - 92 T^{2} - 4 T^{3} + T^{4} \)
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