Properties

Label 3381.2.a.ba
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.62622704.1
Defining polynomial: \(x^{6} - 3 x^{5} - 5 x^{4} + 13 x^{3} + 9 x^{2} - 5 x - 2\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 5 x^{4} + 13 x^{3} + 9 x^{2} - 5 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 4 \nu^{3} + 12 \nu^{2} + 4 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 11 \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 3 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 41 \beta_{1} + 31\)

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
−1.82095
−0.835848
−0.311423
0.584512
2.52648
2.85723
−2.82095 −1.00000 5.95777 −2.07008 2.82095 0 −11.1647 1.00000 5.83961
1.2 −1.83585 −1.00000 1.37034 2.06079 1.83585 0 1.15597 1.00000 −3.78330
1.3 −1.31142 −1.00000 −0.280171 1.11850 1.31142 0 2.99027 1.00000 −1.46683
1.4 −0.415488 −1.00000 −1.82737 −2.48000 0.415488 0 1.59023 1.00000 1.03041
1.5 1.52648 −1.00000 0.330154 −0.362203 −1.52648 0 −2.54899 1.00000 −0.552896
1.6 1.85723 −1.00000 1.44929 3.73299 −1.85723 0 −1.02280 1.00000 6.93301
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.ba 6
7.b odd 2 1 3381.2.a.bb yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.ba 6 1.a even 1 1 trivial
3381.2.a.bb yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{6} + 3 T_{2}^{5} - 5 T_{2}^{4} - 17 T_{2}^{3} + 3 T_{2}^{2} + 23 T_{2} + 8 \)
\( T_{5}^{6} - 2 T_{5}^{5} - 13 T_{5}^{4} + 16 T_{5}^{3} + 41 T_{5}^{2} - 32 T_{5} - 16 \)
\( T_{11}^{6} + 3 T_{11}^{5} - 38 T_{11}^{4} - 122 T_{11}^{3} + 181 T_{11}^{2} + 827 T_{11} + 596 \)
\( T_{13}^{6} - 41 T_{13}^{4} - 36 T_{13}^{3} + 347 T_{13}^{2} + 480 T_{13} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 23 T + 3 T^{2} - 17 T^{3} - 5 T^{4} + 3 T^{5} + T^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( -16 - 32 T + 41 T^{2} + 16 T^{3} - 13 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 596 + 827 T + 181 T^{2} - 122 T^{3} - 38 T^{4} + 3 T^{5} + T^{6} \)
$13$ \( 64 + 480 T + 347 T^{2} - 36 T^{3} - 41 T^{4} + T^{6} \)
$17$ \( 1480 - 1866 T - 195 T^{2} + 326 T^{3} - 30 T^{4} - 8 T^{5} + T^{6} \)
$19$ \( 26816 - 3035 T - 3223 T^{2} + 542 T^{3} + 70 T^{4} - 19 T^{5} + T^{6} \)
$23$ \( ( 1 + T )^{6} \)
$29$ \( 4744 + 2802 T - 3351 T^{2} - 1226 T^{3} - 66 T^{4} + 12 T^{5} + T^{6} \)
$31$ \( -824 + 702 T + 447 T^{2} - 248 T^{3} - 84 T^{4} + 2 T^{5} + T^{6} \)
$37$ \( -128 + 258 T + 115 T^{2} - 284 T^{3} - 24 T^{4} + 10 T^{5} + T^{6} \)
$41$ \( -81584 - 10723 T + 6725 T^{2} + 418 T^{3} - 162 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( 392 + 2394 T + 2615 T^{2} - 18 T^{3} - 125 T^{4} + 2 T^{5} + T^{6} \)
$47$ \( 256 - 7232 T + 7840 T^{2} + 712 T^{3} - 168 T^{4} - 7 T^{5} + T^{6} \)
$53$ \( 674 + 2495 T + 639 T^{2} - 433 T^{3} - 85 T^{4} + 5 T^{5} + T^{6} \)
$59$ \( 44476 - 15523 T - 3539 T^{2} + 1321 T^{3} + 29 T^{4} - 21 T^{5} + T^{6} \)
$61$ \( 130124 - 8963 T - 16537 T^{2} + 1981 T^{3} + 153 T^{4} - 29 T^{5} + T^{6} \)
$67$ \( 12304 - 6992 T - 6165 T^{2} + 1988 T^{3} - 71 T^{4} - 16 T^{5} + T^{6} \)
$71$ \( -1394528 + 57522 T + 38453 T^{2} - 1176 T^{3} - 345 T^{4} + 6 T^{5} + T^{6} \)
$73$ \( -64 - 240 T + 1055 T^{2} - 56 T^{3} - 100 T^{4} + 8 T^{5} + T^{6} \)
$79$ \( -297056 + 66012 T + 12711 T^{2} - 1808 T^{3} - 188 T^{4} + 12 T^{5} + T^{6} \)
$83$ \( 612464 - 97334 T - 18863 T^{2} + 3978 T^{3} - 34 T^{4} - 24 T^{5} + T^{6} \)
$89$ \( 1864 - 970 T - 1219 T^{2} + 340 T^{3} + 51 T^{4} - 18 T^{5} + T^{6} \)
$97$ \( 71272 - 2582 T - 6903 T^{2} + 908 T^{3} + 94 T^{4} - 22 T^{5} + T^{6} \)
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