Properties

Label 3626.2.a.a
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9537557729\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.30278
−1.30278
−1.00000 −3.30278 1.00000 2.30278 3.30278 0 −1.00000 7.90833 −2.30278
1.2 −1.00000 0.302776 1.00000 −1.30278 −0.302776 0 −1.00000 −2.90833 1.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(37\) \(-1\)

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 3626.2.a.a 2
7.b odd 2 1 74.2.a.a 2
21.c even 2 1 666.2.a.j 2
28.d even 2 1 592.2.a.f 2
35.c odd 2 1 1850.2.a.u 2
35.f even 4 2 1850.2.b.i 4
56.e even 2 1 2368.2.a.ba 2
56.h odd 2 1 2368.2.a.s 2
77.b even 2 1 8954.2.a.p 2
84.h odd 2 1 5328.2.a.bf 2
259.b odd 2 1 2738.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.a 2 7.b odd 2 1
592.2.a.f 2 28.d even 2 1
666.2.a.j 2 21.c even 2 1
1850.2.a.u 2 35.c odd 2 1
1850.2.b.i 4 35.f even 4 2
2368.2.a.s 2 56.h odd 2 1
2368.2.a.ba 2 56.e even 2 1
2738.2.a.l 2 259.b odd 2 1
3626.2.a.a 2 1.a even 1 1 trivial
5328.2.a.bf 2 84.h odd 2 1
8954.2.a.p 2 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{2} + 3 T_{3} - 1 \)
\( T_{5}^{2} - T_{5} - 3 \)
\( T_{11}^{2} + T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -1 + 3 T + T^{2} \)
$5$ \( -3 - T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -3 + T + T^{2} \)
$13$ \( -3 - T + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( -27 + 3 T + T^{2} \)
$29$ \( -27 - 3 T + T^{2} \)
$31$ \( -1 + 3 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -9 + 9 T + T^{2} \)
$43$ \( -4 + 6 T + T^{2} \)
$47$ \( -12 + 2 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 + 14 T + T^{2} \)
$61$ \( -79 - 3 T + T^{2} \)
$67$ \( -51 - 11 T + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( 107 - 21 T + T^{2} \)
$79$ \( -147 + 7 T + T^{2} \)
$83$ \( 48 + 20 T + T^{2} \)
$89$ \( -48 - 4 T + T^{2} \)
$97$ \( -204 - 4 T + T^{2} \)
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