Properties

Label 3525.2.a.s
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} - q^{3} + ( 1 + 2 \beta ) q^{4} + ( -1 - \beta ) q^{6} + 3 q^{7} + ( 3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} - q^{3} + ( 1 + 2 \beta ) q^{4} + ( -1 - \beta ) q^{6} + 3 q^{7} + ( 3 + \beta ) q^{8} + q^{9} + 2 \beta q^{11} + ( -1 - 2 \beta ) q^{12} + ( 5 + \beta ) q^{13} + ( 3 + 3 \beta ) q^{14} + 3 q^{16} + ( 3 - 3 \beta ) q^{17} + ( 1 + \beta ) q^{18} + ( -3 - 3 \beta ) q^{19} -3 q^{21} + ( 4 + 2 \beta ) q^{22} + ( 3 + 2 \beta ) q^{23} + ( -3 - \beta ) q^{24} + ( 7 + 6 \beta ) q^{26} - q^{27} + ( 3 + 6 \beta ) q^{28} + ( -3 - 2 \beta ) q^{29} + 2 q^{31} + ( -3 + \beta ) q^{32} -2 \beta q^{33} -3 q^{34} + ( 1 + 2 \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} + ( -9 - 6 \beta ) q^{38} + ( -5 - \beta ) q^{39} + ( 1 + 4 \beta ) q^{41} + ( -3 - 3 \beta ) q^{42} + ( 2 + 2 \beta ) q^{43} + ( 8 + 2 \beta ) q^{44} + ( 7 + 5 \beta ) q^{46} + q^{47} -3 q^{48} + 2 q^{49} + ( -3 + 3 \beta ) q^{51} + ( 9 + 11 \beta ) q^{52} + ( -3 + 5 \beta ) q^{53} + ( -1 - \beta ) q^{54} + ( 9 + 3 \beta ) q^{56} + ( 3 + 3 \beta ) q^{57} + ( -7 - 5 \beta ) q^{58} + ( 5 - \beta ) q^{59} -3 q^{61} + ( 2 + 2 \beta ) q^{62} + 3 q^{63} + ( -7 - 2 \beta ) q^{64} + ( -4 - 2 \beta ) q^{66} + ( 2 - 10 \beta ) q^{67} + ( -9 + 3 \beta ) q^{68} + ( -3 - 2 \beta ) q^{69} + ( -9 - 3 \beta ) q^{71} + ( 3 + \beta ) q^{72} + 6 \beta q^{73} + ( -6 - 2 \beta ) q^{74} + ( -15 - 9 \beta ) q^{76} + 6 \beta q^{77} + ( -7 - 6 \beta ) q^{78} -4 q^{79} + q^{81} + ( 9 + 5 \beta ) q^{82} + ( 4 + 4 \beta ) q^{83} + ( -3 - 6 \beta ) q^{84} + ( 6 + 4 \beta ) q^{86} + ( 3 + 2 \beta ) q^{87} + ( 4 + 6 \beta ) q^{88} + ( -4 - 8 \beta ) q^{89} + ( 15 + 3 \beta ) q^{91} + ( 11 + 8 \beta ) q^{92} -2 q^{93} + ( 1 + \beta ) q^{94} + ( 3 - \beta ) q^{96} + ( -4 + 6 \beta ) q^{97} + ( 2 + 2 \beta ) q^{98} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} + 6q^{8} + 2q^{9} - 2q^{12} + 10q^{13} + 6q^{14} + 6q^{16} + 6q^{17} + 2q^{18} - 6q^{19} - 6q^{21} + 8q^{22} + 6q^{23} - 6q^{24} + 14q^{26} - 2q^{27} + 6q^{28} - 6q^{29} + 4q^{31} - 6q^{32} - 6q^{34} + 2q^{36} + 4q^{37} - 18q^{38} - 10q^{39} + 2q^{41} - 6q^{42} + 4q^{43} + 16q^{44} + 14q^{46} + 2q^{47} - 6q^{48} + 4q^{49} - 6q^{51} + 18q^{52} - 6q^{53} - 2q^{54} + 18q^{56} + 6q^{57} - 14q^{58} + 10q^{59} - 6q^{61} + 4q^{62} + 6q^{63} - 14q^{64} - 8q^{66} + 4q^{67} - 18q^{68} - 6q^{69} - 18q^{71} + 6q^{72} - 12q^{74} - 30q^{76} - 14q^{78} - 8q^{79} + 2q^{81} + 18q^{82} + 8q^{83} - 6q^{84} + 12q^{86} + 6q^{87} + 8q^{88} - 8q^{89} + 30q^{91} + 22q^{92} - 4q^{93} + 2q^{94} + 6q^{96} - 8q^{97} + 4q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−0.414214 −1.00000 −1.82843 0 0.414214 3.00000 1.58579 1.00000 0
1.2 2.41421 −1.00000 3.82843 0 −2.41421 3.00000 4.41421 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.s 2
5.b even 2 1 705.2.a.g 2
15.d odd 2 1 2115.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.g 2 5.b even 2 1
2115.2.a.n 2 15.d odd 2 1
3525.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{2} - 2 T_{2} - 1 \)
\( T_{7} - 3 \)
\( T_{11}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( -8 + T^{2} \)
$13$ \( 23 - 10 T + T^{2} \)
$17$ \( -9 - 6 T + T^{2} \)
$19$ \( -9 + 6 T + T^{2} \)
$23$ \( 1 - 6 T + T^{2} \)
$29$ \( 1 + 6 T + T^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( -28 - 4 T + T^{2} \)
$41$ \( -31 - 2 T + T^{2} \)
$43$ \( -4 - 4 T + T^{2} \)
$47$ \( ( -1 + T )^{2} \)
$53$ \( -41 + 6 T + T^{2} \)
$59$ \( 23 - 10 T + T^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( -196 - 4 T + T^{2} \)
$71$ \( 63 + 18 T + T^{2} \)
$73$ \( -72 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -16 - 8 T + T^{2} \)
$89$ \( -112 + 8 T + T^{2} \)
$97$ \( -56 + 8 T + T^{2} \)
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