Properties

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

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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: 4.4.4400.1
Defining polynomial: \(x^{4} - 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + q^{3} + ( 2 - \beta_{3} ) q^{4} -\beta_{2} q^{6} + ( \beta_{1} - \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + q^{3} + ( 2 - \beta_{3} ) q^{4} -\beta_{2} q^{6} + ( \beta_{1} - \beta_{2} ) q^{8} + q^{9} + 2 \beta_{3} q^{11} + ( 2 - \beta_{3} ) q^{12} + ( 3 - \beta_{3} ) q^{13} + ( 1 - 2 \beta_{3} ) q^{16} + ( 3 + \beta_{3} ) q^{17} -\beta_{2} q^{18} + ( \beta_{1} - 2 \beta_{2} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{22} + \beta_{1} q^{23} + ( \beta_{1} - \beta_{2} ) q^{24} + ( \beta_{1} - 4 \beta_{2} ) q^{26} + q^{27} + 2 q^{29} + ( \beta_{1} + 2 \beta_{2} ) q^{31} -\beta_{2} q^{32} + 2 \beta_{3} q^{33} + ( -\beta_{1} - 2 \beta_{2} ) q^{34} + ( 2 - \beta_{3} ) q^{36} -2 \beta_{1} q^{37} + ( 9 - 5 \beta_{3} ) q^{38} + ( 3 - \beta_{3} ) q^{39} + ( -\beta_{1} - 4 \beta_{2} ) q^{41} + 4 \beta_{2} q^{43} + ( -10 + 4 \beta_{3} ) q^{44} + ( 1 - 3 \beta_{3} ) q^{46} + ( 4 + 4 \beta_{3} ) q^{47} + ( 1 - 2 \beta_{3} ) q^{48} + ( 3 + \beta_{3} ) q^{51} + ( 11 - 5 \beta_{3} ) q^{52} + ( -\beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{2} q^{54} + ( \beta_{1} - 2 \beta_{2} ) q^{57} -2 \beta_{2} q^{58} + 4 \beta_{2} q^{59} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{61} + ( -7 - \beta_{3} ) q^{62} + ( 2 + 3 \beta_{3} ) q^{64} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{66} + 2 \beta_{1} q^{67} + ( 1 - \beta_{3} ) q^{68} + \beta_{1} q^{69} + ( -4 - 2 \beta_{3} ) q^{71} + ( \beta_{1} - \beta_{2} ) q^{72} + ( 5 + \beta_{3} ) q^{73} + ( -2 + 6 \beta_{3} ) q^{74} + ( 3 \beta_{1} - 10 \beta_{2} ) q^{76} + ( \beta_{1} - 4 \beta_{2} ) q^{78} + ( 4 - 4 \beta_{3} ) q^{79} + q^{81} + ( 15 - \beta_{3} ) q^{82} + ( 2 - 2 \beta_{3} ) q^{83} + ( -16 + 4 \beta_{3} ) q^{86} + 2 q^{87} + 10 \beta_{2} q^{88} -3 \beta_{1} q^{89} + ( \beta_{1} - 4 \beta_{2} ) q^{92} + ( \beta_{1} + 2 \beta_{2} ) q^{93} -4 \beta_{1} q^{94} -\beta_{2} q^{96} + ( -1 - 5 \beta_{3} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 8q^{4} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 8q^{4} + 4q^{9} + 8q^{12} + 12q^{13} + 4q^{16} + 12q^{17} + 4q^{27} + 8q^{29} + 8q^{36} + 36q^{38} + 12q^{39} - 40q^{44} + 4q^{46} + 16q^{47} + 4q^{48} + 12q^{51} + 44q^{52} - 28q^{62} + 8q^{64} + 4q^{68} - 16q^{71} + 20q^{73} - 8q^{74} + 16q^{79} + 4q^{81} + 60q^{82} + 8q^{83} - 64q^{86} + 8q^{87} - 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 2 \beta_{1}\)

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
−1.54336
2.14896
−2.14896
1.54336
−2.49721 1.00000 4.23607 0 −2.49721 0 −5.58394 1.00000 0
1.2 −1.32813 1.00000 −0.236068 0 −1.32813 0 2.96979 1.00000 0
1.3 1.32813 1.00000 −0.236068 0 1.32813 0 −2.96979 1.00000 0
1.4 2.49721 1.00000 4.23607 0 2.49721 0 5.58394 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.bv 4
5.b even 2 1 3675.2.a.bt 4
5.c odd 4 2 735.2.d.c 8
7.b odd 2 1 3675.2.a.bt 4
15.e even 4 2 2205.2.d.m 8
35.c odd 2 1 inner 3675.2.a.bv 4
35.f even 4 2 735.2.d.c 8
35.k even 12 4 735.2.q.h 16
35.l odd 12 4 735.2.q.h 16
105.k odd 4 2 2205.2.d.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.d.c 8 5.c odd 4 2
735.2.d.c 8 35.f even 4 2
735.2.q.h 16 35.k even 12 4
735.2.q.h 16 35.l odd 12 4
2205.2.d.m 8 15.e even 4 2
2205.2.d.m 8 105.k odd 4 2
3675.2.a.bt 4 5.b even 2 1
3675.2.a.bt 4 7.b odd 2 1
3675.2.a.bv 4 1.a even 1 1 trivial
3675.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{4} - 8 T_{2}^{2} + 11 \)
\( T_{11}^{2} - 20 \)
\( T_{13}^{2} - 6 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 - 8 T^{2} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -20 + T^{2} )^{2} \)
$13$ \( ( 4 - 6 T + T^{2} )^{2} \)
$17$ \( ( 4 - 6 T + T^{2} )^{2} \)
$19$ \( 176 - 68 T^{2} + T^{4} \)
$23$ \( 176 - 28 T^{2} + T^{4} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( 176 - 52 T^{2} + T^{4} \)
$37$ \( 2816 - 112 T^{2} + T^{4} \)
$41$ \( 4400 - 140 T^{2} + T^{4} \)
$43$ \( 2816 - 128 T^{2} + T^{4} \)
$47$ \( ( -64 - 8 T + T^{2} )^{2} \)
$53$ \( 176 - 52 T^{2} + T^{4} \)
$59$ \( 2816 - 128 T^{2} + T^{4} \)
$61$ \( 2816 - 208 T^{2} + T^{4} \)
$67$ \( 2816 - 112 T^{2} + T^{4} \)
$71$ \( ( -4 + 8 T + T^{2} )^{2} \)
$73$ \( ( 20 - 10 T + T^{2} )^{2} \)
$79$ \( ( -64 - 8 T + T^{2} )^{2} \)
$83$ \( ( -16 - 4 T + T^{2} )^{2} \)
$89$ \( 14256 - 252 T^{2} + T^{4} \)
$97$ \( ( -124 + 2 T + T^{2} )^{2} \)
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