Properties

Label 3675.2.a.bc
Level $3675$
Weight $2$
Character orbit 3675.a
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−0.618034 1.00000 −1.61803 0 −0.618034 0 2.23607 1.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 0 −2.23607 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.bc yes 2
5.b even 2 1 3675.2.a.u 2
7.b odd 2 1 3675.2.a.ba yes 2
35.c odd 2 1 3675.2.a.v yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3675.2.a.u 2 5.b even 2 1
3675.2.a.v yes 2 35.c odd 2 1
3675.2.a.ba yes 2 7.b odd 2 1
3675.2.a.bc yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

\( T_{2}^{2} - T_{2} - 1 \)
\( T_{11}^{2} - 5 \)
\( T_{13}^{2} + 4 T_{13} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -5 + T^{2} \)
$13$ \( -16 + 4 T + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( -16 + 4 T + T^{2} \)
$23$ \( -1 - 4 T + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( -19 - 2 T + T^{2} \)
$41$ \( -80 + T^{2} \)
$43$ \( -41 + 4 T + T^{2} \)
$47$ \( 16 + 12 T + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -16 + 4 T + T^{2} \)
$61$ \( ( 12 + T )^{2} \)
$67$ \( -9 + 12 T + T^{2} \)
$71$ \( 31 - 12 T + T^{2} \)
$73$ \( 80 - 20 T + T^{2} \)
$79$ \( -29 - 8 T + T^{2} \)
$83$ \( -16 + 16 T + T^{2} \)
$89$ \( 16 + 12 T + T^{2} \)
$97$ \( ( -4 + T )^{2} \)
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