Properties

Label 2738.2.a.t
Level $2738$
Weight $2$
Character orbit 2738.a
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8630400734\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.37902897.1
Defining polynomial: \( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + ( - \beta_{4} - \beta_{2} - 1) q^{7} + q^{8} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + ( - \beta_{4} - \beta_{2} - 1) q^{7} + q^{8} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{9} + \beta_{3} q^{10} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{11} - \beta_1 q^{12} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{13} + ( - \beta_{4} - \beta_{2} - 1) q^{14} + ( - \beta_{4} - \beta_{2} - 1) q^{15} + q^{16} + (\beta_{4} + 1) q^{17} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{18} + ( - \beta_{4} - \beta_{2} + 3) q^{19} + \beta_{3} q^{20} + (\beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1 + 1) q^{21} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{22} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{23} - \beta_1 q^{24} + ( - \beta_{3} + \beta_{2} - 3) q^{25} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{26} + ( - \beta_{4} - 5 \beta_{2} - 1) q^{27} + ( - \beta_{4} - \beta_{2} - 1) q^{28} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{29} + ( - \beta_{4} - \beta_{2} - 1) q^{30} + ( - \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{31} + q^{32} + ( - \beta_{5} - 5 \beta_{3} + 5 \beta_{2} + \beta_1) q^{33} + (\beta_{4} + 1) q^{34} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{35} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{36} + ( - \beta_{4} - \beta_{2} + 3) q^{38} + (\beta_{5} + \beta_{3} - 5 \beta_{2} - \beta_1 + 4) q^{39} + \beta_{3} q^{40} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{41} + (\beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1 + 1) q^{42} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 3) q^{43} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{44} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{45} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{46} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{47} - \beta_1 q^{48} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + \beta_1 + 3) q^{49} + ( - \beta_{3} + \beta_{2} - 3) q^{50} + ( - 4 \beta_{3} - \beta_1) q^{51} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{52} + (\beta_1 + 3) q^{53} + ( - \beta_{4} - 5 \beta_{2} - 1) q^{54} + ( - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{4} - \beta_{2} - 1) q^{56} + (\beta_{5} + \beta_{4} + 5 \beta_{3} - 3 \beta_1 + 1) q^{57} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{58} + (\beta_{4} + \beta_{2} + 5) q^{59} + ( - \beta_{4} - \beta_{2} - 1) q^{60} + (\beta_{5} - \beta_{4} - 5 \beta_{2} + \beta_1 - 5) q^{61} + ( - \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{62} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 7) q^{63} + q^{64} + (\beta_{5} - \beta_{2} + \beta_1) q^{65} + ( - \beta_{5} - 5 \beta_{3} + 5 \beta_{2} + \beta_1) q^{66} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - \beta_1 + 2) q^{67} + (\beta_{4} + 1) q^{68} + (\beta_{4} - 4 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 5) q^{69} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{70} + ( - 4 \beta_{2} + 2 \beta_1) q^{71} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{72} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 + 4) q^{73} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1) q^{75} + ( - \beta_{4} - \beta_{2} + 3) q^{76} + (4 \beta_{3} + \beta_1 - 4) q^{77} + (\beta_{5} + \beta_{3} - 5 \beta_{2} - \beta_1 + 4) q^{78} + ( - \beta_{4} + 4 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{79} + \beta_{3} q^{80} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + \beta_1 - 1) q^{81} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{82} + (\beta_{5} + \beta_{3} - \beta_{2} + 4) q^{83} + (\beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1 + 1) q^{84} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{85} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 3) q^{86} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 1) q^{87} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{88} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{89} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{90} + (4 \beta_{3} - 4 \beta_{2} - \beta_1 - 4) q^{91} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{92} + ( - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 11) q^{93} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{94} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{95} - \beta_1 q^{96} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 7) q^{97} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + \beta_1 + 3) q^{98} + ( - 3 \beta_{5} - \beta_{4} - 7 \beta_{3} + 6 \beta_{2} + \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 3 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 6 q^{4} - 3 q^{7} + 6 q^{8} + 12 q^{9} - 3 q^{11} - 3 q^{14} - 3 q^{15} + 6 q^{16} + 3 q^{17} + 12 q^{18} + 21 q^{19} + 6 q^{21} - 3 q^{22} + 21 q^{23} - 18 q^{25} - 3 q^{27} - 3 q^{28} - 6 q^{29} - 3 q^{30} + 21 q^{31} + 6 q^{32} - 3 q^{33} + 3 q^{34} - 3 q^{35} + 12 q^{36} + 21 q^{38} + 27 q^{39} + 18 q^{41} + 6 q^{42} + 18 q^{43} - 3 q^{44} + 6 q^{45} + 21 q^{46} - 9 q^{47} + 15 q^{49} - 18 q^{50} + 18 q^{53} - 3 q^{54} - 3 q^{55} - 3 q^{56} + 6 q^{57} - 6 q^{58} + 27 q^{59} - 3 q^{60} - 24 q^{61} + 21 q^{62} - 36 q^{63} + 6 q^{64} + 3 q^{65} - 3 q^{66} + 9 q^{67} + 3 q^{68} + 27 q^{69} - 3 q^{70} + 12 q^{72} + 27 q^{73} - 3 q^{75} + 21 q^{76} - 24 q^{77} + 27 q^{78} + 21 q^{79} - 6 q^{81} + 18 q^{82} + 27 q^{83} + 6 q^{84} - 3 q^{85} + 18 q^{86} - 3 q^{88} - 21 q^{89} + 6 q^{90} - 24 q^{91} + 21 q^{92} + 54 q^{93} - 9 q^{94} - 3 q^{95} + 42 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 15\nu^{3} + \nu^{2} - 44\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 11\nu^{2} - \nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{5} - 59\nu^{3} - 5\nu^{2} + 124\nu - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{5} - 4\nu^{4} + 59\nu^{3} + 65\nu^{2} - 120\nu - 144 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 5\beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{5} + 11\beta_{4} + 15\beta_{3} + \beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 16\beta_{4} + \beta_{3} + 59\beta_{2} + 46\beta _1 + 20 \) 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
3.14945
1.83388
1.37564
−1.27006
−2.18117
−2.90773
1.00000 −3.14945 1.00000 1.53209 −3.14945 −4.82524 1.00000 6.91903 1.53209
1.2 1.00000 −1.83388 1.00000 −1.87939 −1.83388 3.44656 1.00000 0.363102 −1.87939
1.3 1.00000 −1.37564 1.00000 0.347296 −1.37564 −0.477756 1.00000 −1.10761 0.347296
1.4 1.00000 1.27006 1.00000 1.53209 1.27006 1.94585 1.00000 −1.38694 1.53209
1.5 1.00000 2.18117 1.00000 −1.87939 2.18117 −4.09926 1.00000 1.75751 −1.87939
1.6 1.00000 2.90773 1.00000 0.347296 2.90773 1.00984 1.00000 5.45490 0.347296
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(37\) \(1\)

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 2738.2.a.t 6
37.b even 2 1 2738.2.a.q 6
37.f even 9 2 74.2.f.b 12
111.p odd 18 2 666.2.x.g 12
148.p odd 18 2 592.2.bc.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.b 12 37.f even 9 2
592.2.bc.d 12 148.p odd 18 2
666.2.x.g 12 111.p odd 18 2
2738.2.a.q 6 37.b even 2 1
2738.2.a.t 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):

\( T_{3}^{6} - 15T_{3}^{4} + T_{3}^{3} + 60T_{3}^{2} - 64 \) Copy content Toggle raw display
\( T_{5}^{3} - 3T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 3T_{7}^{5} - 24T_{7}^{4} - 37T_{7}^{3} + 168T_{7}^{2} - 48T_{7} - 64 \) Copy content Toggle raw display
\( T_{13}^{6} - 42T_{13}^{4} - 43T_{13}^{3} + 324T_{13}^{2} + 183T_{13} - 719 \) Copy content Toggle raw display
\( T_{17}^{6} - 3T_{17}^{5} - 24T_{17}^{4} + 69T_{17}^{3} + 120T_{17}^{2} - 390T_{17} + 163 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 15 T^{4} + T^{3} + 60 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} - 24 T^{4} - 37 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} - 24 T^{4} - 37 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{6} - 42 T^{4} - 43 T^{3} + \cdots - 719 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} - 24 T^{4} + 69 T^{3} + \cdots + 163 \) Copy content Toggle raw display
$19$ \( T^{6} - 21 T^{5} + 156 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{6} - 21 T^{5} + 144 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{6} + 6 T^{5} - 24 T^{4} - 84 T^{3} + \cdots + 37 \) Copy content Toggle raw display
$31$ \( T^{6} - 21 T^{5} + 24 T^{4} + \cdots + 130112 \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 18 T^{5} - 27 T^{4} + \cdots + 21608 \) Copy content Toggle raw display
$43$ \( T^{6} - 18 T^{5} + 51 T^{4} + \cdots + 8704 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} - 141 T^{4} + \cdots - 83008 \) Copy content Toggle raw display
$53$ \( T^{6} - 18 T^{5} + 120 T^{4} + \cdots + 17 \) Copy content Toggle raw display
$59$ \( T^{6} - 27 T^{5} + 276 T^{4} + \cdots + 1088 \) Copy content Toggle raw display
$61$ \( T^{6} + 24 T^{5} + 69 T^{4} + \cdots + 52928 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} - 144 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$71$ \( T^{6} - 108 T^{4} + 72 T^{3} + \cdots - 512 \) Copy content Toggle raw display
$73$ \( T^{6} - 27 T^{5} + 108 T^{4} + \cdots + 216289 \) Copy content Toggle raw display
$79$ \( T^{6} - 21 T^{5} + 63 T^{4} + \cdots + 9792 \) Copy content Toggle raw display
$83$ \( T^{6} - 27 T^{5} + 276 T^{4} + \cdots + 1088 \) Copy content Toggle raw display
$89$ \( T^{6} + 21 T^{5} - 75 T^{4} + \cdots - 219419 \) Copy content Toggle raw display
$97$ \( T^{6} - 42 T^{5} + 549 T^{4} + \cdots + 2744 \) Copy content Toggle raw display
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