Properties

Label 3675.2.a.bj
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

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

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

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
0.311108
−1.48119
2.17009
−1.90321 1.00000 1.62222 0 −1.90321 0 0.719004 1.00000 0
1.2 0.193937 1.00000 −1.96239 0 0.193937 0 −0.768452 1.00000 0
1.3 2.70928 1.00000 5.34017 0 2.70928 0 9.04945 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.bj 3
5.b even 2 1 3675.2.a.bi 3
5.c odd 4 2 735.2.d.b 6
7.b odd 2 1 525.2.a.k 3
15.e even 4 2 2205.2.d.l 6
21.c even 2 1 1575.2.a.w 3
28.d even 2 1 8400.2.a.dj 3
35.c odd 2 1 525.2.a.j 3
35.f even 4 2 105.2.d.b 6
35.k even 12 4 735.2.q.e 12
35.l odd 12 4 735.2.q.f 12
105.g even 2 1 1575.2.a.x 3
105.k odd 4 2 315.2.d.e 6
140.c even 2 1 8400.2.a.dg 3
140.j odd 4 2 1680.2.t.k 6
420.w even 4 2 5040.2.t.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 35.f even 4 2
315.2.d.e 6 105.k odd 4 2
525.2.a.j 3 35.c odd 2 1
525.2.a.k 3 7.b odd 2 1
735.2.d.b 6 5.c odd 4 2
735.2.q.e 12 35.k even 12 4
735.2.q.f 12 35.l odd 12 4
1575.2.a.w 3 21.c even 2 1
1575.2.a.x 3 105.g even 2 1
1680.2.t.k 6 140.j odd 4 2
2205.2.d.l 6 15.e even 4 2
3675.2.a.bi 3 5.b even 2 1
3675.2.a.bj 3 1.a even 1 1 trivial
5040.2.t.v 6 420.w even 4 2
8400.2.a.dg 3 140.c even 2 1
8400.2.a.dj 3 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{3} - T_{2}^{2} - 5 T_{2} + 1 \)
\( T_{11} - 2 \)
\( T_{13}^{3} - 6 T_{13}^{2} - 4 T_{13} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T - T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( T^{3} \)
$11$ \( ( -2 + T )^{3} \)
$13$ \( 8 - 4 T - 6 T^{2} + T^{3} \)
$17$ \( 16 - 16 T + T^{3} \)
$19$ \( -40 - 4 T + 6 T^{2} + T^{3} \)
$23$ \( 16 - 8 T - 4 T^{2} + T^{3} \)
$29$ \( 40 - 52 T - 2 T^{2} + T^{3} \)
$31$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$37$ \( -64 - 80 T + 4 T^{2} + T^{3} \)
$41$ \( -200 - 60 T + 2 T^{2} + T^{3} \)
$43$ \( 832 - 144 T - 4 T^{2} + T^{3} \)
$47$ \( -128 - 32 T + 8 T^{2} + T^{3} \)
$53$ \( 296 + 12 T - 14 T^{2} + T^{3} \)
$59$ \( -1280 - 64 T + 16 T^{2} + T^{3} \)
$61$ \( 248 - 52 T - 6 T^{2} + T^{3} \)
$67$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$71$ \( ( -2 + T )^{3} \)
$73$ \( -104 + 92 T - 18 T^{2} + T^{3} \)
$79$ \( 320 - 16 T - 12 T^{2} + T^{3} \)
$83$ \( -256 - 64 T + 8 T^{2} + T^{3} \)
$89$ \( 40 + 52 T + 14 T^{2} + T^{3} \)
$97$ \( 1864 - 36 T - 22 T^{2} + T^{3} \)
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