Properties

Label 2738.2.a.s
Level $2738$
Weight $2$
Character orbit 2738.a
Self dual yes
Analytic conductor $21.863$
Analytic rank $1$
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: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 11\beta_1 \) 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
−1.28558
1.28558
−0.684040
0.684040
1.96962
−1.96962
1.00000 −1.87939 1.00000 −4.26414 −1.87939 3.61144 1.00000 0.532089 −4.26414
1.2 1.00000 −1.87939 1.00000 −0.800038 −1.87939 0.147334 1.00000 0.532089 −0.800038
1.3 1.00000 0.347296 1.00000 −0.852666 0.347296 1.38475 1.00000 −2.87939 −0.852666
1.4 1.00000 0.347296 1.00000 2.61144 0.347296 −2.07935 1.00000 −2.87939 2.61144
1.5 1.00000 1.53209 1.00000 −3.07935 1.53209 0.199962 1.00000 −0.652704 −3.07935
1.6 1.00000 1.53209 1.00000 0.384754 1.53209 −3.26414 1.00000 −0.652704 0.384754
\(n\): 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.s 6
37.b even 2 1 2738.2.a.r 6
37.i odd 36 2 74.2.h.a 12
111.q even 36 2 666.2.bj.c 12
148.q even 36 2 592.2.bq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a 12 37.i odd 36 2
592.2.bq.b 12 148.q even 36 2
666.2.bj.c 12 111.q even 36 2
2738.2.a.r 6 37.b even 2 1
2738.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):

\( T_{3}^{3} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 6T_{5}^{5} - 42T_{5}^{3} - 45T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 15T_{7}^{4} - 2T_{7}^{3} + 36T_{7}^{2} - 12T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{6} + 12T_{13}^{5} + 39T_{13}^{4} - 30T_{13}^{3} - 288T_{13}^{2} - 90T_{13} + 537 \) Copy content Toggle raw display
\( T_{17}^{6} + 12T_{17}^{5} + 27T_{17}^{4} - 120T_{17}^{3} - 477T_{17}^{2} - 324T_{17} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 3 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} - 42 T^{3} - 45 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 15 T^{4} - 2 T^{3} + 36 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} - 27 T^{4} - 150 T^{3} + \cdots - 639 \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{5} + 39 T^{4} + \cdots + 537 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + 27 T^{4} - 120 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{5} + 9 T^{4} + \cdots - 3231 \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} - 45 T^{4} + 366 T^{3} + \cdots + 657 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} - 18 T^{4} + 114 T^{3} + \cdots + 333 \) Copy content Toggle raw display
$31$ \( T^{6} + 18 T^{5} + 57 T^{4} + \cdots + 17817 \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} - 36 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( T^{6} - 78 T^{4} + 144 T^{3} + \cdots + 1509 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} - 117 T^{4} + \cdots - 45027 \) Copy content Toggle raw display
$53$ \( T^{6} + 24 T^{5} + 9 T^{4} + \cdots + 212229 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} - 135 T^{4} + \cdots + 132201 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} - 72 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$67$ \( T^{6} - 18 T^{5} - 123 T^{4} + \cdots - 244331 \) Copy content Toggle raw display
$71$ \( T^{6} - 24 T^{5} - 144 T^{4} + \cdots + 2032704 \) Copy content Toggle raw display
$73$ \( T^{6} - 366 T^{4} - 322 T^{3} + \cdots + 94609 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + 39 T^{4} + \cdots - 219 \) Copy content Toggle raw display
$83$ \( T^{6} + 6 T^{5} - 315 T^{4} + \cdots + 333 \) Copy content Toggle raw display
$89$ \( T^{6} - 243 T^{4} + 972 T^{3} + \cdots + 26217 \) Copy content Toggle raw display
$97$ \( T^{6} - 12 T^{5} - 240 T^{4} + \cdots + 165696 \) Copy content Toggle raw display
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