Properties

Label 3381.2.a.y
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.3176240.1
Defining polynomial: \(x^{5} - x^{4} - 9 x^{3} + 7 x^{2} + 19 x - 9\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 9 x^{3} + 7 x^{2} + 19 x - 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 6 \nu^{2} - \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 6 \beta_{2} + \beta_{1} + 20\)

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
−2.34113
−1.67261
0.443666
2.08269
2.48738
−2.34113 −1.00000 3.48089 1.12582 2.34113 0 −3.46695 1.00000 −2.63569
1.2 −1.67261 −1.00000 0.797614 −3.68373 1.67261 0 2.01112 1.00000 6.16143
1.3 0.443666 −1.00000 −1.80316 2.13100 −0.443666 0 −1.68733 1.00000 0.945453
1.4 2.08269 −1.00000 2.33760 1.37958 −2.08269 0 0.703109 1.00000 2.87324
1.5 2.48738 −1.00000 4.18706 −2.95267 −2.48738 0 5.44006 1.00000 −7.34443
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 3381.2.a.y 5
7.b odd 2 1 3381.2.a.z yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.y 5 1.a even 1 1 trivial
3381.2.a.z yes 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{5} - T_{2}^{4} - 9 T_{2}^{3} + 7 T_{2}^{2} + 19 T_{2} - 9 \)
\( T_{5}^{5} + 2 T_{5}^{4} - 13 T_{5}^{3} - 8 T_{5}^{2} + 53 T_{5} - 36 \)
\( T_{11}^{5} + 2 T_{11}^{4} - 16 T_{11}^{3} - 26 T_{11}^{2} + 7 T_{11} + 4 \)
\( T_{13}^{5} + 4 T_{13}^{4} - 17 T_{13}^{3} - 104 T_{13}^{2} - 145 T_{13} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -9 + 19 T + 7 T^{2} - 9 T^{3} - T^{4} + T^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( -36 + 53 T - 8 T^{2} - 13 T^{3} + 2 T^{4} + T^{5} \)
$7$ \( T^{5} \)
$11$ \( 4 + 7 T - 26 T^{2} - 16 T^{3} + 2 T^{4} + T^{5} \)
$13$ \( -40 - 145 T - 104 T^{2} - 17 T^{3} + 4 T^{4} + T^{5} \)
$17$ \( -4 - 27 T + 78 T^{2} - 38 T^{3} + 2 T^{4} + T^{5} \)
$19$ \( 336 - 155 T - 192 T^{2} - 14 T^{3} + 8 T^{4} + T^{5} \)
$23$ \( ( -1 + T )^{5} \)
$29$ \( -1162 + 481 T + 202 T^{2} - 58 T^{3} - 4 T^{4} + T^{5} \)
$31$ \( -148 - 157 T + 40 T^{3} + 12 T^{4} + T^{5} \)
$37$ \( -334 + 3219 T - 648 T^{2} - 144 T^{3} + 6 T^{4} + T^{5} \)
$41$ \( -448 - 877 T - 426 T^{2} - 52 T^{3} + 6 T^{4} + T^{5} \)
$43$ \( -1324 + 511 T + 716 T^{2} + 207 T^{3} + 24 T^{4} + T^{5} \)
$47$ \( -5376 - 3776 T - 352 T^{2} + 136 T^{3} + 24 T^{4} + T^{5} \)
$53$ \( -7554 + 1471 T + 560 T^{2} - 123 T^{3} - 2 T^{4} + T^{5} \)
$59$ \( 500 - 125 T - 310 T^{2} + 29 T^{3} + 16 T^{4} + T^{5} \)
$61$ \( 30564 - 3523 T - 1726 T^{2} + 19 T^{3} + 22 T^{4} + T^{5} \)
$67$ \( -1084 - 2965 T - 1664 T^{2} - 83 T^{3} + 16 T^{4} + T^{5} \)
$71$ \( -960 - 535 T + 154 T^{2} + 55 T^{3} - 16 T^{4} + T^{5} \)
$73$ \( 100600 + 25115 T - 576 T^{2} - 328 T^{3} + T^{5} \)
$79$ \( -435816 + 34343 T + 4812 T^{2} - 388 T^{3} - 12 T^{4} + T^{5} \)
$83$ \( -216 + 425 T - 174 T^{2} - 18 T^{3} + 10 T^{4} + T^{5} \)
$89$ \( 7388 + 17757 T + 2022 T^{2} - 209 T^{3} - 16 T^{4} + T^{5} \)
$97$ \( 36 - 275 T - 344 T^{2} - 102 T^{3} - 4 T^{4} + T^{5} \)
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