Properties

Label 1334.2.a.h
Level $1334$
Weight $2$
Character orbit 1334.a
Self dual yes
Analytic conductor $10.652$
Analytic rank $1$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.179024.1
Defining polynomial: \(x^{5} - 8 x^{3} + 6 x - 2\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 8 x^{3} + 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} + \nu^{3} - 8 \nu^{2} - 7 \nu + 5 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{4} - \nu^{3} + 15 \nu^{2} + 8 \nu - 6 \)
\(\beta_{4}\)\(=\)\( -4 \nu^{4} - 2 \nu^{3} + 31 \nu^{2} + 16 \nu - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - 2 \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(7 \beta_{4} - 15 \beta_{3} - \beta_{2} + \beta_{1} + 27\)

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
−2.65344
2.69767
0.458358
−1.07149
0.568906
1.00000 −3.07856 1.00000 0.116361 −3.07856 −1.89971 1.00000 6.47756 0.116361
1.2 1.00000 −1.81085 1.00000 −2.65571 −1.81085 1.95629 1.00000 0.279165 −2.65571
1.3 1.00000 −0.366335 1.00000 1.52258 −0.366335 −3.90505 1.00000 −2.86580 1.52258
1.4 1.00000 0.243316 1.00000 −0.634714 0.243316 0.795080 1.00000 −2.94080 −0.634714
1.5 1.00000 2.01243 1.00000 −3.34851 2.01243 −2.94661 1.00000 1.04987 −3.34851
\(n\): 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.h 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\( T_{3}^{5} + 3 T_{3}^{4} - 4 T_{3}^{3} - 12 T_{3}^{2} - T_{3} + 1 \)
\( T_{5}^{5} + 5 T_{5}^{4} + 2 T_{5}^{3} - 14 T_{5}^{2} - 7 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{5} \)
$3$ \( 1 - T - 12 T^{2} - 4 T^{3} + 3 T^{4} + T^{5} \)
$5$ \( 1 - 7 T - 14 T^{2} + 2 T^{3} + 5 T^{4} + T^{5} \)
$7$ \( 34 - 22 T - 32 T^{2} + 2 T^{3} + 6 T^{4} + T^{5} \)
$11$ \( -267 - 563 T - 212 T^{2} - 2 T^{3} + 9 T^{4} + T^{5} \)
$13$ \( -11 + 71 T - 92 T^{2} - 28 T^{3} + 5 T^{4} + T^{5} \)
$17$ \( 486 - 40 T - 172 T^{2} - 24 T^{3} + 6 T^{4} + T^{5} \)
$19$ \( -44 - 236 T - 124 T^{2} + 6 T^{3} + 10 T^{4} + T^{5} \)
$23$ \( ( 1 + T )^{5} \)
$29$ \( ( -1 + T )^{5} \)
$31$ \( 99 + 233 T + 204 T^{2} + 82 T^{3} + 15 T^{4} + T^{5} \)
$37$ \( -794 + 174 T + 336 T^{2} - 104 T^{3} + T^{5} \)
$41$ \( -5918 + 2666 T + 104 T^{2} - 112 T^{3} + 2 T^{4} + T^{5} \)
$43$ \( -2663 + 2633 T - 200 T^{2} - 94 T^{3} + 5 T^{4} + T^{5} \)
$47$ \( 2203 + 1931 T - 450 T^{2} - 76 T^{3} + 9 T^{4} + T^{5} \)
$53$ \( 1289 + 1473 T + 26 T^{2} - 150 T^{3} - 3 T^{4} + T^{5} \)
$59$ \( -3576 - 1312 T + 468 T^{2} + 216 T^{3} + 26 T^{4} + T^{5} \)
$61$ \( 6998 + 2098 T - 464 T^{2} - 124 T^{3} + 8 T^{4} + T^{5} \)
$67$ \( 15446 - 12912 T + 2886 T^{2} - 158 T^{3} - 12 T^{4} + T^{5} \)
$71$ \( -22656 - 15296 T - 3328 T^{2} - 200 T^{3} + 12 T^{4} + T^{5} \)
$73$ \( -4042 + 1382 T + 226 T^{2} - 88 T^{3} - 2 T^{4} + T^{5} \)
$79$ \( -2789 - 9409 T - 2256 T^{2} + 40 T^{3} + 25 T^{4} + T^{5} \)
$83$ \( 3464 - 912 T - 392 T^{2} + 36 T^{3} + 16 T^{4} + T^{5} \)
$89$ \( 40202 - 2216 T - 2070 T^{2} - 40 T^{3} + 20 T^{4} + T^{5} \)
$97$ \( 11368 + 7112 T - 624 T^{2} - 188 T^{3} + 4 T^{4} + T^{5} \)
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