Properties

Label 3675.2.a.bm
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} - q^{3} + \beta_{2} q^{4} + ( 1 - \beta_{1} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} - q^{3} + \beta_{2} q^{4} + ( 1 - \beta_{1} ) q^{6} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} -\beta_{2} q^{12} + ( \beta_{1} - \beta_{2} ) q^{13} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{16} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} ) q^{18} + ( \beta_{1} + 3 \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{22} + ( -1 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{23} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{24} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{26} - q^{27} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{29} + ( 4 - 4 \beta_{1} + 3 \beta_{3} ) q^{31} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{32} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{33} + ( 3 \beta_{1} + \beta_{2} ) q^{34} + \beta_{2} q^{36} + ( -5 + 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{38} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( 8 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -1 - 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{43} + ( 4 - \beta_{2} - 3 \beta_{3} ) q^{44} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{46} + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{48} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{51} + ( -4 + \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{52} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 1 - \beta_{1} ) q^{54} + ( -\beta_{1} - 3 \beta_{2} ) q^{57} + ( 1 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{58} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 8 - 2 \beta_{3} ) q^{61} + ( -8 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{62} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{64} + ( 1 + \beta_{2} ) q^{66} + ( -7 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{67} + ( 2 + 2 \beta_{1} - \beta_{3} ) q^{68} + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{69} + ( -3 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{71} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{72} + ( -4 + 7 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{73} + ( 7 - \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 8 + \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{76} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{78} + ( -5 + 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{79} + q^{81} + ( -8 + 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{82} + ( 2 + 6 \beta_{1} - 7 \beta_{3} ) q^{83} + ( -3 - 3 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{86} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{87} + ( -1 + 2 \beta_{3} ) q^{88} + ( 2 + 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{92} + ( -4 + 4 \beta_{1} - 3 \beta_{3} ) q^{93} + ( -4 - 2 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{94} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{96} + ( -8 + 4 \beta_{2} - \beta_{3} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 4q^{3} + 2q^{4} + 2q^{6} + 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 4q^{3} + 2q^{4} + 2q^{6} + 4q^{9} - 4q^{11} - 2q^{12} - 6q^{16} + 4q^{17} - 2q^{18} + 8q^{19} - 6q^{22} + 12q^{26} - 4q^{27} - 4q^{29} + 8q^{31} - 2q^{32} + 4q^{33} + 8q^{34} + 2q^{36} - 16q^{37} - 4q^{38} + 24q^{41} - 20q^{43} + 14q^{44} - 6q^{46} - 8q^{47} + 6q^{48} - 4q^{51} - 16q^{52} + 20q^{53} + 2q^{54} - 8q^{57} + 6q^{58} + 8q^{59} + 32q^{61} - 28q^{62} - 12q^{64} + 6q^{66} - 12q^{67} + 12q^{68} + 4q^{71} + 34q^{74} + 40q^{76} - 12q^{78} + 4q^{81} - 16q^{82} + 20q^{83} - 14q^{86} + 4q^{87} - 4q^{88} + 8q^{89} + 10q^{92} - 8q^{93} - 32q^{94} + 2q^{96} - 24q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 1\)

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
−1.22833
−0.360409
0.814115
2.77462
−2.22833 −1.00000 2.96545 0 2.22833 0 −2.15133 1.00000 0
1.2 −1.36041 −1.00000 −0.149286 0 1.36041 0 2.92391 1.00000 0
1.3 −0.185885 −1.00000 −1.96545 0 0.185885 0 0.737118 1.00000 0
1.4 1.77462 −1.00000 1.14929 0 −1.77462 0 −1.50970 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 3675.2.a.bm 4
5.b even 2 1 3675.2.a.ca yes 4
7.b odd 2 1 3675.2.a.bo yes 4
35.c odd 2 1 3675.2.a.by yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3675.2.a.bm 4 1.a even 1 1 trivial
3675.2.a.bo yes 4 7.b odd 2 1
3675.2.a.by yes 4 35.c odd 2 1
3675.2.a.ca yes 4 5.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 3 T_{2}^{2} - 6 T_{2} - 1 \)
\( T_{11}^{4} + 4 T_{11}^{3} - 8 T_{11}^{2} - 8 T_{11} + 7 \)
\( T_{13}^{4} - 14 T_{13}^{2} + 16 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 6 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 7 - 8 T - 8 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( 4 + 16 T - 14 T^{2} + T^{4} \)
$17$ \( -4 + 16 T - 10 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 196 + 224 T - 30 T^{2} - 8 T^{3} + T^{4} \)
$23$ \( 79 + 20 T - 32 T^{2} + T^{4} \)
$29$ \( 463 - 64 T - 48 T^{2} + 4 T^{3} + T^{4} \)
$31$ \( 452 + 144 T - 36 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( -623 - 192 T + 46 T^{2} + 16 T^{3} + T^{4} \)
$41$ \( 448 - 512 T + 184 T^{2} - 24 T^{3} + T^{4} \)
$43$ \( -9575 - 1980 T - 2 T^{2} + 20 T^{3} + T^{4} \)
$47$ \( -1028 - 904 T - 102 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( -784 + 80 T + 92 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( 316 + 248 T - 78 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( ( 56 - 16 T + T^{2} )^{2} \)
$67$ \( 1489 - 404 T - 74 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 3383 + 504 T - 168 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( 2884 - 160 T - 158 T^{2} + T^{4} \)
$79$ \( 2009 - 202 T^{2} + T^{4} \)
$83$ \( -6692 + 1976 T - 40 T^{2} - 20 T^{3} + T^{4} \)
$89$ \( -6692 + 2504 T - 214 T^{2} - 8 T^{3} + T^{4} \)
$97$ \( -3676 - 688 T + 108 T^{2} + 24 T^{3} + T^{4} \)
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