Properties

Label 2366.2.a.bf
Level $2366$
Weight $2$
Character orbit 2366.a
Self dual yes
Analytic conductor $18.893$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.285686784.1
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 12x^{3} + 21x^{2} + 2x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{5} - \beta_1 + 1) q^{5} - \beta_1 q^{6} - q^{7} - q^{8} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{5} - \beta_1 + 1) q^{5} - \beta_1 q^{6} - q^{7} - q^{8} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{5} + \beta_1 - 1) q^{10} + (\beta_{4} + \beta_{3}) q^{11} + \beta_1 q^{12} + q^{14} + ( - \beta_{5} - \beta_{2} + \beta_1 - 3) q^{15} + q^{16} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{17} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{18} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{5} - \beta_1 + 1) q^{20} - \beta_1 q^{21} + ( - \beta_{4} - \beta_{3}) q^{22} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{23} - \beta_1 q^{24} + (\beta_{3} - 2 \beta_1 + 3) q^{25} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{27} - q^{28} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{29} + (\beta_{5} + \beta_{2} - \beta_1 + 3) q^{30} + (\beta_{4} - \beta_{3}) q^{31} - q^{32} + 2 \beta_{2} q^{33} + (\beta_{5} + \beta_{2} - \beta_1) q^{34} + ( - \beta_{5} + \beta_1 - 1) q^{35} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{36} + (\beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{38} + ( - \beta_{5} + \beta_1 - 1) q^{40} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{41} + \beta_1 q^{42} + ( - \beta_{4} + \beta_{3} + 2 \beta_1 + 4) q^{43} + (\beta_{4} + \beta_{3}) q^{44} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{45} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{46} + (2 \beta_{5} + 2 \beta_{4} + 2) q^{47} + \beta_1 q^{48} + q^{49} + ( - \beta_{3} + 2 \beta_1 - 3) q^{50} + (2 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_1 + 4) q^{51} + (\beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{53} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{54} + (5 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{55} + q^{56} + (\beta_{3} + 6 \beta_1 + 3) q^{57} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{58} + ( - \beta_{2} + 2 \beta_1 - 1) q^{59} + ( - \beta_{5} - \beta_{2} + \beta_1 - 3) q^{60} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 4 \beta_1 + 6) q^{61} + ( - \beta_{4} + \beta_{3}) q^{62} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{63} + q^{64} - 2 \beta_{2} q^{66} + ( - 3 \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{67} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{68} + (2 \beta_{5} - 5 \beta_{4} - \beta_{3} - \beta_1 + 6) q^{69} + (\beta_{5} - \beta_1 + 1) q^{70} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{71} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 1) q^{72} + ( - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{73} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{74} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1 - 9) q^{75} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 + 1) q^{76} + ( - \beta_{4} - \beta_{3}) q^{77} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{79} + (\beta_{5} - \beta_1 + 1) q^{80} + ( - 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 4) q^{81} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{82} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{83} - \beta_1 q^{84} + (\beta_{5} + 2 \beta_{4} - \beta_{2} + 3 \beta_1 - 6) q^{85} + (\beta_{4} - \beta_{3} - 2 \beta_1 - 4) q^{86} + ( - 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 4 \beta_1 - 2) q^{87} + ( - \beta_{4} - \beta_{3}) q^{88} + (2 \beta_{3} - 2 \beta_1 + 2) q^{89} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{90} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{92} + 2 q^{93} + ( - 2 \beta_{5} - 2 \beta_{4} - 2) q^{94} + ( - 2 \beta_{5} - 4 \beta_{4} - \beta_{3} - 2 \beta_1 + 5) q^{95} - \beta_1 q^{96} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_1) q^{97} - q^{98} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 6 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{14} - 14 q^{15} + 6 q^{16} + 4 q^{17} - 6 q^{18} + 4 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 12 q^{25} + 20 q^{27} - 6 q^{28} + 10 q^{29} + 14 q^{30} + 2 q^{31} - 6 q^{32} - 4 q^{34} - 2 q^{35} + 6 q^{36} - 4 q^{38} - 2 q^{40} - 6 q^{41} + 2 q^{42} + 26 q^{43} - 2 q^{44} + 6 q^{45} + 6 q^{46} + 8 q^{47} + 2 q^{48} + 6 q^{49} - 12 q^{50} + 18 q^{51} + 18 q^{53} - 20 q^{54} + 6 q^{55} + 6 q^{56} + 28 q^{57} - 10 q^{58} - 2 q^{59} - 14 q^{60} + 28 q^{61} - 2 q^{62} - 6 q^{63} + 6 q^{64} - 6 q^{67} + 4 q^{68} + 32 q^{69} + 2 q^{70} - 4 q^{71} - 6 q^{72} + 22 q^{73} - 48 q^{75} + 4 q^{76} + 2 q^{77} + 22 q^{79} + 2 q^{80} + 34 q^{81} + 6 q^{82} + 10 q^{83} - 2 q^{84} - 32 q^{85} - 26 q^{86} - 2 q^{87} + 2 q^{88} + 4 q^{89} - 6 q^{90} - 6 q^{92} + 12 q^{93} - 8 q^{94} + 32 q^{95} - 2 q^{96} + 12 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 10x^{4} + 12x^{3} + 21x^{2} + 2x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} + 7\nu^{3} - 26\nu^{2} - 9\nu + 1 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} + 3\nu^{4} + 24\nu^{3} - 17\nu^{2} - 68\nu - 13 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - 7\nu^{4} - 26\nu^{3} + 43\nu^{2} + 37\nu - 3 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} - 11\nu^{4} - 33\nu^{3} + 74\nu^{2} + 41\nu - 24 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} - 5\beta_{4} + \beta_{3} + 11\beta_{2} + 13\beta _1 + 31 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 13\beta_{4} + 11\beta_{3} + 20\beta_{2} + 73\beta _1 + 42 \) Copy content Toggle raw display

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
−2.55629
−0.865515
−0.466545
0.252878
2.29079
3.34469
−1.00000 −2.55629 1.00000 3.48754 2.55629 −1.00000 −1.00000 3.53463 −3.48754
1.2 −1.00000 −0.865515 1.00000 3.71131 0.865515 −1.00000 −1.00000 −2.25088 −3.71131
1.3 −1.00000 −0.466545 1.00000 −3.38938 0.466545 −1.00000 −1.00000 −2.78234 3.38938
1.4 −1.00000 0.252878 1.00000 −1.14776 −0.252878 −1.00000 −1.00000 −2.93605 1.14776
1.5 −1.00000 2.29079 1.00000 0.901839 −2.29079 −1.00000 −1.00000 2.24770 −0.901839
1.6 −1.00000 3.34469 1.00000 −1.56356 −3.34469 −1.00000 −1.00000 8.18694 1.56356
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 2366.2.a.bf 6
13.b even 2 1 2366.2.a.bh 6
13.d odd 4 2 2366.2.d.r 12
13.f odd 12 2 182.2.m.b 12
39.k even 12 2 1638.2.bj.g 12
52.l even 12 2 1456.2.cc.d 12
91.w even 12 2 1274.2.v.d 12
91.x odd 12 2 1274.2.o.d 12
91.ba even 12 2 1274.2.o.e 12
91.bc even 12 2 1274.2.m.c 12
91.bd odd 12 2 1274.2.v.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b 12 13.f odd 12 2
1274.2.m.c 12 91.bc even 12 2
1274.2.o.d 12 91.x odd 12 2
1274.2.o.e 12 91.ba even 12 2
1274.2.v.d 12 91.w even 12 2
1274.2.v.e 12 91.bd odd 12 2
1456.2.cc.d 12 52.l even 12 2
1638.2.bj.g 12 39.k even 12 2
2366.2.a.bf 6 1.a even 1 1 trivial
2366.2.a.bh 6 13.b even 2 1
2366.2.d.r 12 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{6} - 2T_{3}^{5} - 10T_{3}^{4} + 12T_{3}^{3} + 21T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} - 19T_{5}^{4} + 24T_{5}^{3} + 93T_{5}^{2} - 10T_{5} - 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} - 10 T^{4} + 12 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} - 19 T^{4} + 24 T^{3} + \cdots - 71 \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 2 T^{5} - 42 T^{4} - 96 T^{3} + \cdots - 704 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} - 39 T^{4} + 168 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} - 92 T^{4} + \cdots - 19232 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} - 68 T^{4} + \cdots + 3142 \) Copy content Toggle raw display
$29$ \( T^{6} - 10 T^{5} - 39 T^{4} + \cdots - 368 \) Copy content Toggle raw display
$31$ \( T^{6} - 2 T^{5} - 42 T^{4} - 48 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$37$ \( T^{6} - 155 T^{4} + 8 T^{3} + 5752 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} - 95 T^{4} - 352 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$43$ \( T^{6} - 26 T^{5} + 214 T^{4} + \cdots + 2944 \) Copy content Toggle raw display
$47$ \( T^{6} - 8 T^{5} - 104 T^{4} + \cdots + 5632 \) Copy content Toggle raw display
$53$ \( T^{6} - 18 T^{5} - 51 T^{4} + \cdots + 44928 \) Copy content Toggle raw display
$59$ \( T^{6} + 2 T^{5} - 82 T^{4} - 84 T^{3} + \cdots - 2 \) Copy content Toggle raw display
$61$ \( T^{6} - 28 T^{5} + 69 T^{4} + \cdots - 283487 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} - 122 T^{4} - 448 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$71$ \( T^{6} + 4 T^{5} - 154 T^{4} + \cdots - 27392 \) Copy content Toggle raw display
$73$ \( T^{6} - 22 T^{5} - 63 T^{4} + \cdots + 52048 \) Copy content Toggle raw display
$79$ \( T^{6} - 22 T^{5} + 62 T^{4} + \cdots + 8032 \) Copy content Toggle raw display
$83$ \( T^{6} - 10 T^{5} - 182 T^{4} + \cdots + 29656 \) Copy content Toggle raw display
$89$ \( T^{6} - 4 T^{5} - 200 T^{4} + \cdots - 101504 \) Copy content Toggle raw display
$97$ \( T^{6} - 12 T^{5} - 224 T^{4} + \cdots - 80000 \) Copy content Toggle raw display
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