Properties

Label 2738.2.a.h
Level $2738$
Weight $2$
Character orbit 2738.a
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} + (\beta + 1) q^{5} - \beta q^{6} - 2 q^{7} - q^{8} + (\beta + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta q^{3} + q^{4} + (\beta + 1) q^{5} - \beta q^{6} - 2 q^{7} - q^{8} + (\beta + 2) q^{9} + ( - \beta - 1) q^{10} + ( - \beta - 1) q^{11} + \beta q^{12} + ( - \beta + 2) q^{13} + 2 q^{14} + (2 \beta + 5) q^{15} + q^{16} + (2 \beta - 4) q^{17} + ( - \beta - 2) q^{18} + (2 \beta + 2) q^{19} + (\beta + 1) q^{20} - 2 \beta q^{21} + (\beta + 1) q^{22} + ( - \beta + 2) q^{23} - \beta q^{24} + (3 \beta + 1) q^{25} + (\beta - 2) q^{26} + 5 q^{27} - 2 q^{28} + ( - \beta + 2) q^{29} + ( - 2 \beta - 5) q^{30} + (3 \beta - 3) q^{31} - q^{32} + ( - 2 \beta - 5) q^{33} + ( - 2 \beta + 4) q^{34} + ( - 2 \beta - 2) q^{35} + (\beta + 2) q^{36} + ( - 2 \beta - 2) q^{38} + (\beta - 5) q^{39} + ( - \beta - 1) q^{40} + (\beta - 8) q^{41} + 2 \beta q^{42} + 6 q^{43} + ( - \beta - 1) q^{44} + (4 \beta + 7) q^{45} + (\beta - 2) q^{46} + (2 \beta - 4) q^{47} + \beta q^{48} - 3 q^{49} + ( - 3 \beta - 1) q^{50} + ( - 2 \beta + 10) q^{51} + ( - \beta + 2) q^{52} + (2 \beta + 2) q^{53} - 5 q^{54} + ( - 3 \beta - 6) q^{55} + 2 q^{56} + (4 \beta + 10) q^{57} + (\beta - 2) q^{58} + (2 \beta + 2) q^{59} + (2 \beta + 5) q^{60} + ( - \beta + 11) q^{61} + ( - 3 \beta + 3) q^{62} + ( - 2 \beta - 4) q^{63} + q^{64} - 3 q^{65} + (2 \beta + 5) q^{66} + (3 \beta - 1) q^{67} + (2 \beta - 4) q^{68} + (\beta - 5) q^{69} + (2 \beta + 2) q^{70} + (4 \beta - 2) q^{71} + ( - \beta - 2) q^{72} + ( - 3 \beta - 1) q^{73} + (4 \beta + 15) q^{75} + (2 \beta + 2) q^{76} + (2 \beta + 2) q^{77} + ( - \beta + 5) q^{78} + ( - \beta - 10) q^{79} + (\beta + 1) q^{80} + (2 \beta - 6) q^{81} + ( - \beta + 8) q^{82} + ( - 4 \beta + 8) q^{83} - 2 \beta q^{84} + 6 q^{85} - 6 q^{86} + (\beta - 5) q^{87} + (\beta + 1) q^{88} + 6 q^{89} + ( - 4 \beta - 7) q^{90} + (2 \beta - 4) q^{91} + ( - \beta + 2) q^{92} + 15 q^{93} + ( - 2 \beta + 4) q^{94} + (6 \beta + 12) q^{95} - \beta q^{96} + ( - 2 \beta + 10) q^{97} + 3 q^{98} + ( - 4 \beta - 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 4 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 4 q^{7} - 2 q^{8} + 5 q^{9} - 3 q^{10} - 3 q^{11} + q^{12} + 3 q^{13} + 4 q^{14} + 12 q^{15} + 2 q^{16} - 6 q^{17} - 5 q^{18} + 6 q^{19} + 3 q^{20} - 2 q^{21} + 3 q^{22} + 3 q^{23} - q^{24} + 5 q^{25} - 3 q^{26} + 10 q^{27} - 4 q^{28} + 3 q^{29} - 12 q^{30} - 3 q^{31} - 2 q^{32} - 12 q^{33} + 6 q^{34} - 6 q^{35} + 5 q^{36} - 6 q^{38} - 9 q^{39} - 3 q^{40} - 15 q^{41} + 2 q^{42} + 12 q^{43} - 3 q^{44} + 18 q^{45} - 3 q^{46} - 6 q^{47} + q^{48} - 6 q^{49} - 5 q^{50} + 18 q^{51} + 3 q^{52} + 6 q^{53} - 10 q^{54} - 15 q^{55} + 4 q^{56} + 24 q^{57} - 3 q^{58} + 6 q^{59} + 12 q^{60} + 21 q^{61} + 3 q^{62} - 10 q^{63} + 2 q^{64} - 6 q^{65} + 12 q^{66} + q^{67} - 6 q^{68} - 9 q^{69} + 6 q^{70} - 5 q^{72} - 5 q^{73} + 34 q^{75} + 6 q^{76} + 6 q^{77} + 9 q^{78} - 21 q^{79} + 3 q^{80} - 10 q^{81} + 15 q^{82} + 12 q^{83} - 2 q^{84} + 12 q^{85} - 12 q^{86} - 9 q^{87} + 3 q^{88} + 12 q^{89} - 18 q^{90} - 6 q^{91} + 3 q^{92} + 30 q^{93} + 6 q^{94} + 30 q^{95} - q^{96} + 18 q^{97} + 6 q^{98} - 18 q^{99}+O(q^{100}) \) 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
−1.79129
2.79129
−1.00000 −1.79129 1.00000 −0.791288 1.79129 −2.00000 −1.00000 0.208712 0.791288
1.2 −1.00000 2.79129 1.00000 3.79129 −2.79129 −2.00000 −1.00000 4.79129 −3.79129
\(n\): e.g. 2-40 or 990-1000
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.h 2
37.b even 2 1 2738.2.a.k 2
37.d odd 4 2 74.2.b.a 4
111.g even 4 2 666.2.c.b 4
148.g even 4 2 592.2.g.c 4
185.f even 4 2 1850.2.c.h 4
185.j odd 4 2 1850.2.d.e 4
185.k even 4 2 1850.2.c.g 4
296.j even 4 2 2368.2.g.h 4
296.m odd 4 2 2368.2.g.j 4
444.j odd 4 2 5328.2.h.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 37.d odd 4 2
592.2.g.c 4 148.g even 4 2
666.2.c.b 4 111.g even 4 2
1850.2.c.g 4 185.k even 4 2
1850.2.c.h 4 185.f even 4 2
1850.2.d.e 4 185.j odd 4 2
2368.2.g.h 4 296.j even 4 2
2368.2.g.j 4 296.m odd 4 2
2738.2.a.h 2 1.a even 1 1 trivial
2738.2.a.k 2 37.b even 2 1
5328.2.h.m 4 444.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):

\( T_{3}^{2} - T_{3} - 5 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 3 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 3 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 45 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 15T + 51 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$61$ \( T^{2} - 21T + 105 \) Copy content Toggle raw display
$67$ \( T^{2} - T - 47 \) Copy content Toggle raw display
$71$ \( T^{2} - 84 \) Copy content Toggle raw display
$73$ \( T^{2} + 5T - 41 \) Copy content Toggle raw display
$79$ \( T^{2} + 21T + 105 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 48 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 18T + 60 \) Copy content Toggle raw display
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