Properties

Label 3381.2.a.be
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} - 15 x^{6} + 11 x^{5} + 75 x^{4} - 35 x^{3} - 141 x^{2} + 37 x + 80\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 15 x^{6} + 11 x^{5} + 75 x^{4} - 35 x^{3} - 141 x^{2} + 37 x + 80\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 10 \nu^{4} + 8 \nu^{3} + 25 \nu^{2} - 13 \nu - 14 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 12 \nu^{4} + 8 \nu^{3} + 41 \nu^{2} - 11 \nu - 34 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - \nu^{5} - 12 \nu^{4} + 10 \nu^{3} + 39 \nu^{2} - 21 \nu - 30 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 12 \nu^{5} + 8 \nu^{4} + 41 \nu^{3} - 9 \nu^{2} - 36 \nu - 10 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 12 \nu^{4} + 18 \nu^{3} + 41 \nu^{2} - 32 \nu - 34 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{4} + \beta_{3} + 8 \beta_{2} + \beta_{1} + 22\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 10 \beta_{5} - 8 \beta_{4} + 10 \beta_{2} + 29 \beta_{1} + 20\)
\(\nu^{6}\)\(=\)\(-2 \beta_{7} + 2 \beta_{5} - 10 \beta_{4} + 12 \beta_{3} + 57 \beta_{2} + 12 \beta_{1} + 138\)
\(\nu^{7}\)\(=\)\(-26 \beta_{7} + 2 \beta_{6} + 81 \beta_{5} - 57 \beta_{4} + 4 \beta_{3} + 81 \beta_{2} + 183 \beta_{1} + 166\)

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
−2.55222
−1.87992
−1.67781
−0.818544
1.19194
1.47391
2.51679
2.74585
−2.55222 −1.00000 4.51383 −0.648159 2.55222 0 −6.41586 1.00000 1.65425
1.2 −1.87992 −1.00000 1.53408 2.65555 1.87992 0 0.875886 1.00000 −4.99220
1.3 −1.67781 −1.00000 0.815035 −2.94824 1.67781 0 1.98814 1.00000 4.94658
1.4 −0.818544 −1.00000 −1.32999 −3.72041 0.818544 0 2.72574 1.00000 3.04532
1.5 1.19194 −1.00000 −0.579274 3.78820 −1.19194 0 −3.07435 1.00000 4.51532
1.6 1.47391 −1.00000 0.172415 −3.62341 −1.47391 0 −2.69370 1.00000 −5.34058
1.7 2.51679 −1.00000 4.33421 −2.41834 −2.51679 0 5.87470 1.00000 −6.08644
1.8 2.74585 −1.00000 5.53968 1.91481 −2.74585 0 9.71944 1.00000 5.25777
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.be 8
7.b odd 2 1 3381.2.a.bf 8
7.d odd 6 2 483.2.i.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.g 16 7.d odd 6 2
3381.2.a.be 8 1.a even 1 1 trivial
3381.2.a.bf 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\(T_{2}^{8} - \cdots\)
\(T_{5}^{8} + \cdots\)
\(T_{11}^{8} - \cdots\)
\(T_{13}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 80 + 37 T - 141 T^{2} - 35 T^{3} + 75 T^{4} + 11 T^{5} - 15 T^{6} - T^{7} + T^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( -1200 - 2013 T + 291 T^{2} + 933 T^{3} + 81 T^{4} - 127 T^{5} - 21 T^{6} + 5 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( -4384 + 3056 T + 2704 T^{2} - 1864 T^{3} - 304 T^{4} + 256 T^{5} - 4 T^{6} - 10 T^{7} + T^{8} \)
$13$ \( 2127 - 2370 T - 4602 T^{2} - 930 T^{3} + 660 T^{4} + 158 T^{5} - 38 T^{6} - 6 T^{7} + T^{8} \)
$17$ \( 67182 + 59053 T + 5571 T^{2} - 8711 T^{3} - 2711 T^{4} + 73 T^{5} + 139 T^{6} + 21 T^{7} + T^{8} \)
$19$ \( -99108 - 102984 T - 11692 T^{2} + 13384 T^{3} + 2482 T^{4} - 474 T^{5} - 94 T^{6} + 5 T^{7} + T^{8} \)
$23$ \( ( 1 + T )^{8} \)
$29$ \( 34688 + 26816 T - 22272 T^{2} - 4960 T^{3} + 2736 T^{4} + 232 T^{5} - 108 T^{6} - 2 T^{7} + T^{8} \)
$31$ \( 10896 + 18064 T - 3328 T^{2} - 8176 T^{3} - 372 T^{4} + 672 T^{5} - 22 T^{6} - 13 T^{7} + T^{8} \)
$37$ \( -19776 + 43072 T - 12928 T^{2} - 13936 T^{3} + 1176 T^{4} + 816 T^{5} - 52 T^{6} - 13 T^{7} + T^{8} \)
$41$ \( 256 - 1520 T + 784 T^{2} + 1504 T^{3} - 728 T^{4} - 272 T^{5} + 40 T^{6} + 16 T^{7} + T^{8} \)
$43$ \( -937796 - 141104 T + 322712 T^{2} - 62812 T^{3} - 5446 T^{4} + 2214 T^{5} - 84 T^{6} - 15 T^{7} + T^{8} \)
$47$ \( -1290798 + 898557 T - 86919 T^{2} - 51171 T^{3} + 9453 T^{4} + 651 T^{5} - 185 T^{6} - T^{7} + T^{8} \)
$53$ \( 29772 - 14205 T - 21675 T^{2} + 2895 T^{3} + 3651 T^{4} + 99 T^{5} - 113 T^{6} - 3 T^{7} + T^{8} \)
$59$ \( 1087200 + 1287600 T + 428928 T^{2} - 2352 T^{3} - 20208 T^{4} - 2292 T^{5} + 100 T^{6} + 26 T^{7} + T^{8} \)
$61$ \( 146592 + 343728 T - 63792 T^{2} - 80376 T^{3} + 6744 T^{4} + 2080 T^{5} - 152 T^{6} - 14 T^{7} + T^{8} \)
$67$ \( -2418489 - 2320950 T + 569984 T^{2} + 122768 T^{3} - 36866 T^{4} + 906 T^{5} + 404 T^{6} - 38 T^{7} + T^{8} \)
$71$ \( 44639898 + 10497209 T - 2988199 T^{2} - 332779 T^{3} + 57905 T^{4} + 3195 T^{5} - 421 T^{6} - 9 T^{7} + T^{8} \)
$73$ \( -302829 - 711338 T - 452466 T^{2} - 22898 T^{3} + 20332 T^{4} + 946 T^{5} - 254 T^{6} - 6 T^{7} + T^{8} \)
$79$ \( -14818292 + 4166328 T + 1226628 T^{2} - 320616 T^{3} - 17614 T^{4} + 6626 T^{5} - 178 T^{6} - 23 T^{7} + T^{8} \)
$83$ \( 9331200 - 5526144 T + 105408 T^{2} + 285408 T^{3} - 13104 T^{4} - 4824 T^{5} + 36 T^{6} + 30 T^{7} + T^{8} \)
$89$ \( 111729608 - 74213988 T - 9203060 T^{2} + 1365020 T^{3} + 132516 T^{4} - 7274 T^{5} - 640 T^{6} + 12 T^{7} + T^{8} \)
$97$ \( 1304736 - 1800688 T - 354656 T^{2} + 190152 T^{3} + 18488 T^{4} - 3884 T^{5} - 300 T^{6} + 14 T^{7} + T^{8} \)
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