Properties

Label 1334.2.a.f
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: 5.5.207184.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 2 x^{2} + 7 x - 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 6 x^{3} + 2 x^{2} + 7 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 8\)

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
0.139666
−1.31977
2.62865
1.22974
−1.67828
−1.00000 −2.54471 1.00000 −0.504925 2.54471 2.12016 −1.00000 3.47557 0.504925
1.2 −1.00000 −1.17692 1.00000 −3.87305 1.17692 −1.06158 −1.00000 −1.61485 3.87305
1.3 −1.00000 0.542094 1.00000 0.203674 −0.542094 −2.28116 −1.00000 −2.70613 −0.203674
1.4 −1.00000 1.85405 1.00000 −2.05025 −1.85405 1.71748 −1.00000 0.437490 2.05025
1.5 −1.00000 2.32550 1.00000 1.22455 −2.32550 −2.49491 −1.00000 2.40793 −1.22455
\(n\): 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.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.2.a.f 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):

\( T_{3}^{5} - T_{3}^{4} - 8 T_{3}^{3} + 8 T_{3}^{2} + 11 T_{3} - 7 \)
\( T_{5}^{5} + 5 T_{5}^{4} + 2 T_{5}^{3} - 10 T_{5}^{2} - 3 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{5} \)
$3$ \( -7 + 11 T + 8 T^{2} - 8 T^{3} - T^{4} + T^{5} \)
$5$ \( 1 - 3 T - 10 T^{2} + 2 T^{3} + 5 T^{4} + T^{5} \)
$7$ \( 22 + 16 T - 14 T^{2} - 8 T^{3} + 2 T^{4} + T^{5} \)
$11$ \( -17 + 63 T + 70 T^{2} - 34 T^{3} - T^{4} + T^{5} \)
$13$ \( 583 - 203 T - 134 T^{2} + 16 T^{3} + 11 T^{4} + T^{5} \)
$17$ \( -1318 + 646 T + 190 T^{2} - 56 T^{3} - 4 T^{4} + T^{5} \)
$19$ \( 4 - 4 T - 40 T^{2} + 26 T^{3} + 12 T^{4} + T^{5} \)
$23$ \( ( -1 + T )^{5} \)
$29$ \( ( -1 + T )^{5} \)
$31$ \( -343 - 929 T - 658 T^{2} - 90 T^{3} + 5 T^{4} + T^{5} \)
$37$ \( -34 + 620 T + 82 T^{2} - 70 T^{3} + T^{5} \)
$41$ \( 3446 + 880 T - 938 T^{2} - 138 T^{3} + 6 T^{4} + T^{5} \)
$43$ \( -2237 + 3163 T - 374 T^{2} - 106 T^{3} + 7 T^{4} + T^{5} \)
$47$ \( 49 + 97 T + 16 T^{2} - 40 T^{3} - 5 T^{4} + T^{5} \)
$53$ \( -16403 + 5973 T + 190 T^{2} - 174 T^{3} + T^{4} + T^{5} \)
$59$ \( 2824 + 4312 T - 2176 T^{2} - 204 T^{3} + 12 T^{4} + T^{5} \)
$61$ \( 31358 - 13304 T - 3562 T^{2} - 90 T^{3} + 20 T^{4} + T^{5} \)
$67$ \( -1838 + 1318 T + 300 T^{2} - 146 T^{3} + 4 T^{4} + T^{5} \)
$71$ \( 10304 + 8096 T - 928 T^{2} - 224 T^{3} + 4 T^{4} + T^{5} \)
$73$ \( -14146 + 10436 T + 1412 T^{2} - 278 T^{3} - 4 T^{4} + T^{5} \)
$79$ \( -37073 + 11571 T + 508 T^{2} - 252 T^{3} + T^{4} + T^{5} \)
$83$ \( 37016 - 15840 T + 1852 T^{2} + 56 T^{3} - 22 T^{4} + T^{5} \)
$89$ \( -502 - 346 T + 640 T^{2} - 156 T^{3} - 8 T^{4} + T^{5} \)
$97$ \( 44728 + 28528 T + 6964 T^{2} + 816 T^{3} + 46 T^{4} + T^{5} \)
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