Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
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.

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

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
−1.93185
−0.517638
0.517638
1.93185
−1.93185 1.00000 1.73205 0 −1.93185 0 0.517638 1.00000 0
1.2 −0.517638 1.00000 −1.73205 0 −0.517638 0 1.93185 1.00000 0
1.3 0.517638 1.00000 −1.73205 0 0.517638 0 −1.93185 1.00000 0
1.4 1.93185 1.00000 1.73205 0 1.93185 0 −0.517638 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3675.2.a.bu 4
5.b even 2 1 3675.2.a.bs 4
5.c odd 4 2 735.2.d.f 8
7.b odd 2 1 3675.2.a.bs 4
15.e even 4 2 2205.2.d.t 8
35.c odd 2 1 inner 3675.2.a.bu 4
35.f even 4 2 735.2.d.f 8
35.k even 12 2 735.2.q.c 8
35.k even 12 2 735.2.q.d 8
35.l odd 12 2 735.2.q.c 8
35.l odd 12 2 735.2.q.d 8
105.k odd 4 2 2205.2.d.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.d.f 8 5.c odd 4 2
735.2.d.f 8 35.f even 4 2
735.2.q.c 8 35.k even 12 2
735.2.q.c 8 35.l odd 12 2
735.2.q.d 8 35.k even 12 2
735.2.q.d 8 35.l odd 12 2
2205.2.d.t 8 15.e even 4 2
2205.2.d.t 8 105.k odd 4 2
3675.2.a.bs 4 5.b even 2 1
3675.2.a.bs 4 7.b odd 2 1
3675.2.a.bu 4 1.a even 1 1 trivial
3675.2.a.bu 4 35.c odd 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 4 T_{2}^{2} + 1 \)
\( T_{11}^{2} - 12 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -12 + T^{2} )^{2} \)
$13$ \( ( 4 + T )^{4} \)
$17$ \( ( 4 + T )^{4} \)
$19$ \( 4 - 28 T^{2} + T^{4} \)
$23$ \( 484 - 52 T^{2} + T^{4} \)
$29$ \( ( -44 - 4 T + T^{2} )^{2} \)
$31$ \( ( -54 + T^{2} )^{2} \)
$37$ \( 64 - 112 T^{2} + T^{4} \)
$41$ \( ( -72 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( ( 6 + T )^{4} \)
$53$ \( ( -54 + T^{2} )^{2} \)
$59$ \( 64 - 112 T^{2} + T^{4} \)
$61$ \( 36 - 84 T^{2} + T^{4} \)
$67$ \( 7744 - 208 T^{2} + T^{4} \)
$71$ \( ( -92 - 8 T + T^{2} )^{2} \)
$73$ \( ( -32 + 8 T + T^{2} )^{2} \)
$79$ \( ( 52 + 16 T + T^{2} )^{2} \)
$83$ \( ( 6 + T )^{4} \)
$89$ \( 4096 - 256 T^{2} + T^{4} \)
$97$ \( ( 96 + 24 T + T^{2} )^{2} \)
show more
show less