Properties

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

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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.88404.1
Defining polynomial: \(x^{4} - x^{3} - 7 x^{2} + 5 x + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + q^{9} + ( 2 - \beta_{1} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{16} + ( -2 + \beta_{1} + \beta_{3} ) q^{17} + \beta_{1} q^{18} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( -4 + 3 \beta_{1} + \beta_{3} ) q^{22} -2 \beta_{2} q^{23} + ( -2 \beta_{1} - \beta_{3} ) q^{24} + ( -4 - 3 \beta_{1} ) q^{26} - q^{27} + 4 q^{29} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( 4 + 3 \beta_{1} + 2 \beta_{2} ) q^{32} + ( -2 + \beta_{1} - \beta_{3} ) q^{33} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 3 - 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 8 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{38} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( 8 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{44} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{46} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{48} + ( 2 - \beta_{1} - \beta_{3} ) q^{51} + ( -8 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{52} + ( -2 + 3 \beta_{1} - \beta_{3} ) q^{53} -\beta_{1} q^{54} + ( -2 \beta_{1} + \beta_{2} ) q^{57} + 4 \beta_{1} q^{58} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{59} -5 q^{61} + ( 12 + 4 \beta_{2} ) q^{62} + ( 4 + 6 \beta_{1} + \beta_{2} ) q^{64} + ( 4 - 3 \beta_{1} - \beta_{3} ) q^{66} -\beta_{2} q^{67} + ( 7 \beta_{1} + \beta_{3} ) q^{68} + 2 \beta_{2} q^{69} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 \beta_{1} + \beta_{3} ) q^{72} + ( 1 - 2 \beta_{1} + 2 \beta_{3} ) q^{73} + ( -8 + 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{74} + ( -8 + 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( 4 + 3 \beta_{1} ) q^{78} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{79} + q^{81} + ( -4 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{82} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( 4 - 2 \beta_{1} - 2 \beta_{3} ) q^{86} -4 q^{87} + ( 4 + 9 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{88} + ( -2 - \beta_{1} - \beta_{3} ) q^{89} + ( -16 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 4 + 5 \beta_{1} + \beta_{3} ) q^{94} + ( -4 - 3 \beta_{1} - 2 \beta_{2} ) q^{96} + ( -3 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 2 - \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} - 4q^{3} + 7q^{4} - q^{6} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + q^{2} - 4q^{3} + 7q^{4} - q^{6} + 3q^{8} + 4q^{9} + 8q^{11} - 7q^{12} - 7q^{13} + 17q^{16} - 6q^{17} + q^{18} + 3q^{19} - 12q^{22} + 2q^{23} - 3q^{24} - 19q^{26} - 4q^{27} + 16q^{29} + 9q^{31} + 17q^{32} - 8q^{33} + 14q^{34} + 7q^{36} + 8q^{37} + 27q^{38} + 7q^{39} + 4q^{41} + 5q^{43} + 26q^{44} - 6q^{46} + 6q^{47} - 17q^{48} + 6q^{51} - 35q^{52} - 6q^{53} - q^{54} - 3q^{57} + 4q^{58} + 10q^{59} - 20q^{61} + 44q^{62} + 21q^{64} + 12q^{66} + q^{67} + 8q^{68} - 2q^{69} + 22q^{71} + 3q^{72} + 4q^{73} - 21q^{74} - 23q^{76} + 19q^{78} + 8q^{79} + 4q^{81} - 8q^{82} + 2q^{83} + 12q^{86} - 16q^{87} + 28q^{88} - 10q^{89} - 66q^{92} - 9q^{93} + 22q^{94} - 17q^{96} - 12q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 7 x^{2} + 5 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)

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
−2.43736
−0.494173
1.22868
2.70285
−2.43736 −1.00000 3.94072 0 2.43736 0 −4.73024 1.00000 0
1.2 −0.494173 −1.00000 −1.75579 0 0.494173 0 1.85601 1.00000 0
1.3 1.22868 −1.00000 −0.490347 0 −1.22868 0 −3.05984 1.00000 0
1.4 2.70285 −1.00000 5.30542 0 −2.70285 0 8.93406 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.bw 4
5.b even 2 1 3675.2.a.br 4
7.b odd 2 1 3675.2.a.bx 4
7.c even 3 2 525.2.i.i 8
35.c odd 2 1 3675.2.a.bq 4
35.j even 6 2 525.2.i.j yes 8
35.l odd 12 4 525.2.r.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.i 8 7.c even 3 2
525.2.i.j yes 8 35.j even 6 2
525.2.r.h 16 35.l odd 12 4
3675.2.a.bq 4 35.c odd 2 1
3675.2.a.br 4 5.b even 2 1
3675.2.a.bw 4 1.a even 1 1 trivial
3675.2.a.bx 4 7.b odd 2 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} - 7 T_{2}^{2} + 5 T_{2} + 4 \)
\( T_{11}^{4} - 8 T_{11}^{3} - 8 T_{11}^{2} + 180 T_{11} - 328 \)
\( T_{13}^{4} + 7 T_{13}^{3} - 19 T_{13}^{2} - 179 T_{13} - 194 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 5 T - 7 T^{2} - T^{3} + T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( -328 + 180 T - 8 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( -194 - 179 T - 19 T^{2} + 7 T^{3} + T^{4} \)
$17$ \( 40 - 108 T - 20 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( -196 + 175 T - 37 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( 960 + 72 T - 68 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( -768 + 576 T - 56 T^{2} - 9 T^{3} + T^{4} \)
$37$ \( 773 + 144 T - 62 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( -200 + 236 T - 60 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( 160 - 32 T - 52 T^{2} - 5 T^{3} + T^{4} \)
$47$ \( -856 + 500 T - 64 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( 1112 - 292 T - 76 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( -8 + 44 T - 40 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( ( 5 + T )^{4} \)
$67$ \( 60 + 9 T - 17 T^{2} - T^{3} + T^{4} \)
$71$ \( -12736 + 2880 T - 32 T^{2} - 22 T^{3} + T^{4} \)
$73$ \( -1567 + 924 T - 122 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 153 + 132 T - 70 T^{2} - 8 T^{3} + T^{4} \)
$83$ \( -312 - 396 T - 112 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( 24 - 84 T + 4 T^{2} + 10 T^{3} + T^{4} \)
$97$ \( -79 - 196 T - 58 T^{2} + 12 T^{3} + T^{4} \)
show more
show less