Properties

Label 3525.2.a.u
Level $3525$
Weight $2$
Character orbit 3525.a
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.14656.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 4 x + 2\)
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 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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{6} + ( -1 + \beta_{1} - \beta_{3} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + ( -1 + \beta_{1} ) q^{6} + ( -1 + \beta_{1} - \beta_{3} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( 3 - \beta_{1} + \beta_{2} ) q^{11} + ( 2 - \beta_{1} + \beta_{2} ) q^{12} + ( 1 + \beta_{1} ) q^{13} + ( 3 + \beta_{3} ) q^{14} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{17} + ( -1 + \beta_{1} ) q^{18} + ( -1 - 2 \beta_{2} - \beta_{3} ) q^{19} + ( -1 + \beta_{1} - \beta_{3} ) q^{21} + ( -5 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{22} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{23} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{24} + ( 2 + \beta_{1} + \beta_{2} ) q^{26} + q^{27} + ( \beta_{2} + \beta_{3} ) q^{28} + 5 q^{29} + ( 1 + 3 \beta_{2} + \beta_{3} ) q^{31} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{32} + ( 3 - \beta_{1} + \beta_{2} ) q^{33} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{34} + ( 2 - \beta_{1} + \beta_{2} ) q^{36} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{37} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{38} + ( 1 + \beta_{1} ) q^{39} + ( 4 + \beta_{1} - \beta_{2} ) q^{41} + ( 3 + \beta_{3} ) q^{42} + ( 6 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( 11 - 5 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{44} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{46} - q^{47} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{48} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{51} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{52} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -4 + \beta_{2} - 2 \beta_{3} ) q^{56} + ( -1 - 2 \beta_{2} - \beta_{3} ) q^{57} + ( -5 + 5 \beta_{1} ) q^{58} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{59} + ( -5 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{62} + ( -1 + \beta_{1} - \beta_{3} ) q^{63} + ( 4 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{64} + ( -5 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{66} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{67} + ( 4 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{68} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{69} + ( 1 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{71} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{72} + ( -1 - 2 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{73} + ( 5 + 3 \beta_{2} - \beta_{3} ) q^{74} + ( -4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{76} + ( -1 + \beta_{1} + \beta_{2} ) q^{77} + ( 2 + \beta_{1} + \beta_{2} ) q^{78} + ( 1 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{79} + q^{81} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{82} + ( 8 - 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( \beta_{2} + \beta_{3} ) q^{84} + ( -14 + 6 \beta_{1} - 2 \beta_{2} ) q^{86} + 5 q^{87} + ( -15 + 8 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{88} + ( -1 - 2 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{89} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{91} + ( 10 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{92} + ( 1 + 3 \beta_{2} + \beta_{3} ) q^{93} + ( 1 - \beta_{1} ) q^{94} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{96} + ( 6 - 4 \beta_{1} - 2 \beta_{3} ) q^{97} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{98} + ( 3 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{3} + 4q^{4} - 2q^{6} - 6q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{3} + 4q^{4} - 2q^{6} - 6q^{8} + 4q^{9} + 8q^{11} + 4q^{12} + 6q^{13} + 10q^{14} + 4q^{16} + 2q^{17} - 2q^{18} + 2q^{19} - 12q^{22} - 8q^{23} - 6q^{24} + 8q^{26} + 4q^{27} - 4q^{28} + 20q^{29} - 4q^{31} - 14q^{32} + 8q^{33} + 8q^{34} + 4q^{36} - 8q^{38} + 6q^{39} + 20q^{41} + 10q^{42} + 8q^{43} + 32q^{44} - 2q^{46} - 4q^{47} + 4q^{48} + 2q^{51} - 2q^{52} - 10q^{53} - 2q^{54} - 14q^{56} + 2q^{57} - 10q^{58} + 10q^{59} - 20q^{61} + 12q^{62} + 4q^{64} - 12q^{66} + 2q^{68} - 8q^{69} + 18q^{71} - 6q^{72} + 4q^{73} + 16q^{74} - 26q^{76} - 4q^{77} + 8q^{78} + 20q^{79} + 4q^{81} - 2q^{82} + 28q^{83} - 4q^{84} - 40q^{86} + 20q^{87} - 40q^{88} + 10q^{91} + 36q^{92} - 4q^{93} + 2q^{94} - 14q^{96} + 20q^{97} - 4q^{98} + 8q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 4\)

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
−1.59286
−0.385537
1.15244
2.82596
−2.59286 1.00000 4.72294 0 −2.59286 −0.255601 −7.06020 1.00000 0
1.2 −1.38554 1.00000 −0.0802864 0 −1.38554 −4.18757 2.88231 1.00000 0
1.3 0.152445 1.00000 −1.97676 0 0.152445 2.73544 −0.606236 1.00000 0
1.4 1.82596 1.00000 1.33411 0 1.82596 1.70773 −1.21588 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.u 4
5.b even 2 1 705.2.a.j 4
15.d odd 2 1 2115.2.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.a.j 4 5.b even 2 1
2115.2.a.p 4 15.d odd 2 1
3525.2.a.u 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):

\( T_{2}^{4} + 2 T_{2}^{3} - 4 T_{2}^{2} - 6 T_{2} + 1 \)
\( T_{7}^{4} - 14 T_{7}^{2} + 16 T_{7} + 5 \)
\( T_{11}^{4} - 8 T_{11}^{3} + 12 T_{11}^{2} + 8 T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T - 4 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( 5 + 16 T - 14 T^{2} + T^{4} \)
$11$ \( -12 + 8 T + 12 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( -3 + 2 T + 8 T^{2} - 6 T^{3} + T^{4} \)
$17$ \( -123 + 142 T - 40 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( 41 + 10 T - 40 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 85 - 216 T - 30 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( ( -5 + T )^{4} \)
$31$ \( 564 - 136 T - 80 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( 564 - 8 T - 52 T^{2} + T^{4} \)
$41$ \( 289 - 372 T + 138 T^{2} - 20 T^{3} + T^{4} \)
$43$ \( 576 + 704 T - 96 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( ( 1 + T )^{4} \)
$53$ \( 293 - 198 T - 16 T^{2} + 10 T^{3} + T^{4} \)
$59$ \( -15 + 98 T + 8 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( -9847 - 2364 T - 34 T^{2} + 20 T^{3} + T^{4} \)
$67$ \( 244 - 224 T - 88 T^{2} + T^{4} \)
$71$ \( -8031 + 2690 T - 96 T^{2} - 18 T^{3} + T^{4} \)
$73$ \( 17540 + 568 T - 264 T^{2} - 4 T^{3} + T^{4} \)
$79$ \( 2612 + 1024 T - 4 T^{2} - 20 T^{3} + T^{4} \)
$83$ \( -12080 + 1584 T + 144 T^{2} - 28 T^{3} + T^{4} \)
$89$ \( 8644 + 368 T - 288 T^{2} + T^{4} \)
$97$ \( -1200 + 656 T + 16 T^{2} - 20 T^{3} + T^{4} \)
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