Properties

Label 3675.2.a.bl
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: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 735)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 3\)

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
0.334904
2.68554
−1.74912
−1.27133
−2.74912 1.00000 5.55765 0 −2.74912 0 −9.78039 1.00000 0
1.2 −2.27133 1.00000 3.15894 0 −2.27133 0 −2.63234 1.00000 0
1.3 −0.665096 1.00000 −1.55765 0 −0.665096 0 2.36618 1.00000 0
1.4 1.68554 1.00000 0.841058 0 1.68554 0 −1.95345 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.bl 4
5.b even 2 1 735.2.a.n 4
7.b odd 2 1 3675.2.a.bk 4
15.d odd 2 1 2205.2.a.bf 4
35.c odd 2 1 735.2.a.o yes 4
35.i odd 6 2 735.2.i.m 8
35.j even 6 2 735.2.i.n 8
105.g even 2 1 2205.2.a.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.a.n 4 5.b even 2 1
735.2.a.o yes 4 35.c odd 2 1
735.2.i.m 8 35.i odd 6 2
735.2.i.n 8 35.j even 6 2
2205.2.a.bf 4 15.d odd 2 1
2205.2.a.bg 4 105.g even 2 1
3675.2.a.bk 4 7.b odd 2 1
3675.2.a.bl 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{4} + 4 T_{2}^{3} - 12 T_{2} - 7 \)
\( T_{11}^{4} - 8 T_{11}^{3} - 8 T_{11}^{2} + 160 T_{11} - 224 \)
\( T_{13}^{4} - 32 T_{13}^{2} - 64 T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 - 12 T + 4 T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( -224 + 160 T - 8 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( 16 - 64 T - 32 T^{2} + T^{4} \)
$17$ \( 128 + 128 T - 16 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( -284 + 176 T - 12 T^{2} - 8 T^{3} + T^{4} \)
$23$ \( 196 + 64 T - 44 T^{2} + T^{4} \)
$29$ \( -368 + 608 T - 72 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( -28 + 48 T - 12 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 784 - 288 T - 56 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 16 - 128 T - 56 T^{2} + T^{4} \)
$43$ \( 64 + 192 T - 32 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( ( -68 - 4 T + T^{2} )^{2} \)
$53$ \( -28 - 80 T - 4 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( -4544 - 1408 T - 48 T^{2} + 16 T^{3} + T^{4} \)
$61$ \( ( 62 - 16 T + T^{2} )^{2} \)
$67$ \( 3136 + 512 T - 176 T^{2} + T^{4} \)
$71$ \( 32 - 96 T - 72 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( 3088 - 128 T - 128 T^{2} + T^{4} \)
$79$ \( 2576 - 384 T - 184 T^{2} + T^{4} \)
$83$ \( ( 92 - 20 T + T^{2} )^{2} \)
$89$ \( ( -4 - 4 T + T^{2} )^{2} \)
$97$ \( -2032 - 1280 T - 192 T^{2} + T^{4} \)
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