Properties

Label 3525.2.a.p
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + q^{4} + \beta q^{6} + q^{7} -\beta q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} + q^{3} + q^{4} + \beta q^{6} + q^{7} -\beta q^{8} + q^{9} -4 q^{11} + q^{12} + ( -4 - \beta ) q^{13} + \beta q^{14} -5 q^{16} + ( 2 - 3 \beta ) q^{17} + \beta q^{18} + \beta q^{19} + q^{21} -4 \beta q^{22} + ( 1 + 2 \beta ) q^{23} -\beta q^{24} + ( -3 - 4 \beta ) q^{26} + q^{27} + q^{28} + ( -3 + 2 \beta ) q^{29} -2 \beta q^{31} -3 \beta q^{32} -4 q^{33} + ( -9 + 2 \beta ) q^{34} + q^{36} + ( -8 - 2 \beta ) q^{37} + 3 q^{38} + ( -4 - \beta ) q^{39} + ( -3 - 2 \beta ) q^{41} + \beta q^{42} + ( 8 - 2 \beta ) q^{43} -4 q^{44} + ( 6 + \beta ) q^{46} + q^{47} -5 q^{48} -6 q^{49} + ( 2 - 3 \beta ) q^{51} + ( -4 - \beta ) q^{52} + ( -2 + 7 \beta ) q^{53} + \beta q^{54} -\beta q^{56} + \beta q^{57} + ( 6 - 3 \beta ) q^{58} + ( -10 - 3 \beta ) q^{59} + ( 1 - 4 \beta ) q^{61} -6 q^{62} + q^{63} + q^{64} -4 \beta q^{66} + ( 2 + 4 \beta ) q^{67} + ( 2 - 3 \beta ) q^{68} + ( 1 + 2 \beta ) q^{69} + ( -2 + 3 \beta ) q^{71} -\beta q^{72} -4 q^{73} + ( -6 - 8 \beta ) q^{74} + \beta q^{76} -4 q^{77} + ( -3 - 4 \beta ) q^{78} + ( 2 + 2 \beta ) q^{79} + q^{81} + ( -6 - 3 \beta ) q^{82} + ( -4 + 4 \beta ) q^{83} + q^{84} + ( -6 + 8 \beta ) q^{86} + ( -3 + 2 \beta ) q^{87} + 4 \beta q^{88} + ( -10 + 2 \beta ) q^{89} + ( -4 - \beta ) q^{91} + ( 1 + 2 \beta ) q^{92} -2 \beta q^{93} + \beta q^{94} -3 \beta q^{96} + ( 2 + 2 \beta ) q^{97} -6 \beta q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{4} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{4} + 2q^{7} + 2q^{9} - 8q^{11} + 2q^{12} - 8q^{13} - 10q^{16} + 4q^{17} + 2q^{21} + 2q^{23} - 6q^{26} + 2q^{27} + 2q^{28} - 6q^{29} - 8q^{33} - 18q^{34} + 2q^{36} - 16q^{37} + 6q^{38} - 8q^{39} - 6q^{41} + 16q^{43} - 8q^{44} + 12q^{46} + 2q^{47} - 10q^{48} - 12q^{49} + 4q^{51} - 8q^{52} - 4q^{53} + 12q^{58} - 20q^{59} + 2q^{61} - 12q^{62} + 2q^{63} + 2q^{64} + 4q^{67} + 4q^{68} + 2q^{69} - 4q^{71} - 8q^{73} - 12q^{74} - 8q^{77} - 6q^{78} + 4q^{79} + 2q^{81} - 12q^{82} - 8q^{83} + 2q^{84} - 12q^{86} - 6q^{87} - 20q^{89} - 8q^{91} + 2q^{92} + 4q^{97} - 8q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 1.00000 1.00000 0 −1.73205 1.00000 1.73205 1.00000 0
1.2 1.73205 1.00000 1.00000 0 1.73205 1.00000 −1.73205 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(47\) \(-1\)

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 3525.2.a.p 2
5.b even 2 1 705.2.a.i 2
15.d odd 2 1 2115.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.i 2 5.b even 2 1
2115.2.a.l 2 15.d odd 2 1
3525.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{2} - 3 \)
\( T_{7} - 1 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 13 + 8 T + T^{2} \)
$17$ \( -23 - 4 T + T^{2} \)
$19$ \( -3 + T^{2} \)
$23$ \( -11 - 2 T + T^{2} \)
$29$ \( -3 + 6 T + T^{2} \)
$31$ \( -12 + T^{2} \)
$37$ \( 52 + 16 T + T^{2} \)
$41$ \( -3 + 6 T + T^{2} \)
$43$ \( 52 - 16 T + T^{2} \)
$47$ \( ( -1 + T )^{2} \)
$53$ \( -143 + 4 T + T^{2} \)
$59$ \( 73 + 20 T + T^{2} \)
$61$ \( -47 - 2 T + T^{2} \)
$67$ \( -44 - 4 T + T^{2} \)
$71$ \( -23 + 4 T + T^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( -8 - 4 T + T^{2} \)
$83$ \( -32 + 8 T + T^{2} \)
$89$ \( 88 + 20 T + T^{2} \)
$97$ \( -8 - 4 T + T^{2} \)
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