Properties

Label 4235.2.a.k
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \beta ) q^{2} -\beta q^{3} + 3 q^{4} - q^{5} + ( 2 + \beta ) q^{6} + q^{7} + ( 1 - 2 \beta ) q^{8} + ( -2 + \beta ) q^{9} +O(q^{10})\) \( q + ( 1 - 2 \beta ) q^{2} -\beta q^{3} + 3 q^{4} - q^{5} + ( 2 + \beta ) q^{6} + q^{7} + ( 1 - 2 \beta ) q^{8} + ( -2 + \beta ) q^{9} + ( -1 + 2 \beta ) q^{10} -3 \beta q^{12} + ( -4 + \beta ) q^{13} + ( 1 - 2 \beta ) q^{14} + \beta q^{15} - q^{16} + ( -5 + \beta ) q^{17} + ( -4 + 3 \beta ) q^{18} + ( 2 - 2 \beta ) q^{19} -3 q^{20} -\beta q^{21} + ( -2 + 4 \beta ) q^{23} + ( 2 + \beta ) q^{24} + q^{25} + ( -6 + 7 \beta ) q^{26} + ( -1 + 4 \beta ) q^{27} + 3 q^{28} + ( 1 + 3 \beta ) q^{29} + ( -2 - \beta ) q^{30} + ( 2 - 2 \beta ) q^{31} + ( -3 + 6 \beta ) q^{32} + ( -7 + 9 \beta ) q^{34} - q^{35} + ( -6 + 3 \beta ) q^{36} + ( -6 + 2 \beta ) q^{37} + ( 6 - 2 \beta ) q^{38} + ( -1 + 3 \beta ) q^{39} + ( -1 + 2 \beta ) q^{40} + 4 \beta q^{41} + ( 2 + \beta ) q^{42} + ( 4 - 2 \beta ) q^{43} + ( 2 - \beta ) q^{45} -10 q^{46} + ( -3 - 3 \beta ) q^{47} + \beta q^{48} + q^{49} + ( 1 - 2 \beta ) q^{50} + ( -1 + 4 \beta ) q^{51} + ( -12 + 3 \beta ) q^{52} + ( -4 - 4 \beta ) q^{53} + ( -9 - 2 \beta ) q^{54} + ( 1 - 2 \beta ) q^{56} + 2 q^{57} + ( -5 - 5 \beta ) q^{58} + ( -8 + 2 \beta ) q^{59} + 3 \beta q^{60} + ( 6 + 2 \beta ) q^{61} + ( 6 - 2 \beta ) q^{62} + ( -2 + \beta ) q^{63} -13 q^{64} + ( 4 - \beta ) q^{65} + ( 2 - 10 \beta ) q^{67} + ( -15 + 3 \beta ) q^{68} + ( -4 - 2 \beta ) q^{69} + ( -1 + 2 \beta ) q^{70} + ( 12 - 7 \beta ) q^{71} + ( -4 + 3 \beta ) q^{72} + ( -7 - \beta ) q^{73} + ( -10 + 10 \beta ) q^{74} -\beta q^{75} + ( 6 - 6 \beta ) q^{76} + ( -7 - \beta ) q^{78} + ( -4 + 11 \beta ) q^{79} + q^{80} + ( 2 - 6 \beta ) q^{81} + ( -8 - 4 \beta ) q^{82} + ( -4 + 9 \beta ) q^{83} -3 \beta q^{84} + ( 5 - \beta ) q^{85} + ( 8 - 6 \beta ) q^{86} + ( -3 - 4 \beta ) q^{87} + ( 14 + 2 \beta ) q^{89} + ( 4 - 3 \beta ) q^{90} + ( -4 + \beta ) q^{91} + ( -6 + 12 \beta ) q^{92} + 2 q^{93} + ( 3 + 9 \beta ) q^{94} + ( -2 + 2 \beta ) q^{95} + ( -6 - 3 \beta ) q^{96} + ( -11 + 7 \beta ) q^{97} + ( 1 - 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 6q^{4} - 2q^{5} + 5q^{6} + 2q^{7} - 3q^{9} + O(q^{10}) \) \( 2q - q^{3} + 6q^{4} - 2q^{5} + 5q^{6} + 2q^{7} - 3q^{9} - 3q^{12} - 7q^{13} + q^{15} - 2q^{16} - 9q^{17} - 5q^{18} + 2q^{19} - 6q^{20} - q^{21} + 5q^{24} + 2q^{25} - 5q^{26} + 2q^{27} + 6q^{28} + 5q^{29} - 5q^{30} + 2q^{31} - 5q^{34} - 2q^{35} - 9q^{36} - 10q^{37} + 10q^{38} + q^{39} + 4q^{41} + 5q^{42} + 6q^{43} + 3q^{45} - 20q^{46} - 9q^{47} + q^{48} + 2q^{49} + 2q^{51} - 21q^{52} - 12q^{53} - 20q^{54} + 4q^{57} - 15q^{58} - 14q^{59} + 3q^{60} + 14q^{61} + 10q^{62} - 3q^{63} - 26q^{64} + 7q^{65} - 6q^{67} - 27q^{68} - 10q^{69} + 17q^{71} - 5q^{72} - 15q^{73} - 10q^{74} - q^{75} + 6q^{76} - 15q^{78} + 3q^{79} + 2q^{80} - 2q^{81} - 20q^{82} + q^{83} - 3q^{84} + 9q^{85} + 10q^{86} - 10q^{87} + 30q^{89} + 5q^{90} - 7q^{91} + 4q^{93} + 15q^{94} - 2q^{95} - 15q^{96} - 15q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
−2.23607 −1.61803 3.00000 −1.00000 3.61803 1.00000 −2.23607 −0.381966 2.23607
1.2 2.23607 0.618034 3.00000 −1.00000 1.38197 1.00000 2.23607 −2.61803 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 4235.2.a.k 2
11.b odd 2 1 4235.2.a.j 2
11.d odd 10 2 385.2.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.b 4 11.d odd 10 2
4235.2.a.j 2 11.b odd 2 1
4235.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{2} - 5 \)
\( T_{3}^{2} + T_{3} - 1 \)
\( T_{13}^{2} + 7 T_{13} + 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T^{2} \)
$3$ \( -1 + T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 11 + 7 T + T^{2} \)
$17$ \( 19 + 9 T + T^{2} \)
$19$ \( -4 - 2 T + T^{2} \)
$23$ \( -20 + T^{2} \)
$29$ \( -5 - 5 T + T^{2} \)
$31$ \( -4 - 2 T + T^{2} \)
$37$ \( 20 + 10 T + T^{2} \)
$41$ \( -16 - 4 T + T^{2} \)
$43$ \( 4 - 6 T + T^{2} \)
$47$ \( 9 + 9 T + T^{2} \)
$53$ \( 16 + 12 T + T^{2} \)
$59$ \( 44 + 14 T + T^{2} \)
$61$ \( 44 - 14 T + T^{2} \)
$67$ \( -116 + 6 T + T^{2} \)
$71$ \( 11 - 17 T + T^{2} \)
$73$ \( 55 + 15 T + T^{2} \)
$79$ \( -149 - 3 T + T^{2} \)
$83$ \( -101 - T + T^{2} \)
$89$ \( 220 - 30 T + T^{2} \)
$97$ \( -5 + 15 T + T^{2} \)
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