Properties

Label 4235.2.a.u
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.15952.1
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 1 \) Copy content Toggle raw display
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + q^{5} + (\beta_1 - 1) q^{6} + q^{7} + ( - \beta_{2} - 2 \beta_1 - 1) q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + q^{5} + (\beta_1 - 1) q^{6} + q^{7} + ( - \beta_{2} - 2 \beta_1 - 1) q^{8} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9} - \beta_1 q^{10} + ( - \beta_{3} + \beta_{2} - 3) q^{12} + ( - \beta_{3} + \beta_{2} - 5) q^{13} - \beta_1 q^{14} - \beta_{2} q^{15} + (2 \beta_1 + 3) q^{16} + (2 \beta_{3} + \beta_{2}) q^{17} + (\beta_{2} + 2 \beta_1 + 1) q^{18} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{19} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{20} - \beta_{2} q^{21} + (\beta_{2} + \beta_1 - 1) q^{23} + ( - \beta_{3} + \beta_1 + 2) q^{24} + q^{25} + ( - \beta_{3} + 5 \beta_1) q^{26} + (\beta_{3} + 3) q^{27} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{28} + (2 \beta_{2} - 2 \beta_1 - 4) q^{29} + (\beta_1 - 1) q^{30} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{31} + ( - 2 \beta_{3} - \beta_1 - 4) q^{32} + (2 \beta_{3} - 3 \beta_1 + 3) q^{34} + q^{35} + ( - 2 \beta_1 - 7) q^{36} + (3 \beta_{2} + 5 \beta_1 - 3) q^{37} + ( - \beta_{3} + 4 \beta_1 - 3) q^{38} + (3 \beta_{3} + 5 \beta_{2} + \beta_1 - 4) q^{39} + ( - \beta_{2} - 2 \beta_1 - 1) q^{40} + (\beta_{3} + 3 \beta_1 - 4) q^{41} + (\beta_1 - 1) q^{42} + (3 \beta_{2} + 3 \beta_1 - 3) q^{43} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{45} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{46} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{47} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{48} + q^{49} - \beta_1 q^{50} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{51} + ( - 4 \beta_{3} - 7 \beta_{2} - 4 \beta_1 - 6) q^{52} + ( - \beta_{2} + \beta_1 + 9) q^{53} + (\beta_{3} - 4 \beta_1 + 1) q^{54} + ( - \beta_{2} - 2 \beta_1 - 1) q^{56} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 8) q^{58} + ( - 3 \beta_{3} + \beta_1 - 4) q^{59} + ( - \beta_{3} + \beta_{2} - 3) q^{60} + ( - 2 \beta_{3} + 3 \beta_1 + 1) q^{61} + (2 \beta_{3} + \beta_{2} + 6) q^{62} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{63} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{64} + ( - \beta_{3} + \beta_{2} - 5) q^{65} + ( - 3 \beta_{2} - 3 \beta_1 + 5) q^{67} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 11) q^{68} + (\beta_{3} + 2 \beta_{2} - 3) q^{69} - \beta_1 q^{70} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{71} + (2 \beta_{3} + 5 \beta_1 + 4) q^{72} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{73} + ( - 5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 - 12) q^{74} - \beta_{2} q^{75} + ( - 3 \beta_{3} - 11) q^{76} + (2 \beta_{3} - \beta_{2} - 5 \beta_1 + 5) q^{78} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{79} + (2 \beta_1 + 3) q^{80} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{81} + ( - 2 \beta_{3} - 3 \beta_{2} - 8) q^{82} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 1) q^{83} + ( - \beta_{3} + \beta_{2} - 3) q^{84} + (2 \beta_{3} + \beta_{2}) q^{85} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 6) q^{86} + (2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 10) q^{87} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{89} + (\beta_{2} + 2 \beta_1 + 1) q^{90} + ( - \beta_{3} + \beta_{2} - 5) q^{91} + ( - \beta_{2} + 3 \beta_1 + 3) q^{92} + (5 \beta_{2} + 3 \beta_1 - 9) q^{93} + ( - 5 \beta_{3} - 4 \beta_{2} - 3 \beta_1 - 14) q^{94} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{95} + (4 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{96} + (\beta_{3} - 2 \beta_{2} - 7) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} - 8 q^{12} - 16 q^{13} - 2 q^{15} + 12 q^{16} - 2 q^{17} + 6 q^{18} - 6 q^{19} + 4 q^{20} - 2 q^{21} - 2 q^{23} + 10 q^{24} + 4 q^{25} + 2 q^{26} + 10 q^{27} + 4 q^{28} - 12 q^{29} - 4 q^{30} - 6 q^{31} - 12 q^{32} + 8 q^{34} + 4 q^{35} - 28 q^{36} - 6 q^{37} - 10 q^{38} - 12 q^{39} - 6 q^{40} - 18 q^{41} - 4 q^{42} - 6 q^{43} + 4 q^{45} - 8 q^{46} + 4 q^{47} + 2 q^{48} + 4 q^{49} - 4 q^{51} - 30 q^{52} + 34 q^{53} + 2 q^{54} - 6 q^{56} + 28 q^{57} + 32 q^{58} - 10 q^{59} - 8 q^{60} + 8 q^{61} + 22 q^{62} + 4 q^{63} - 16 q^{64} - 16 q^{65} + 14 q^{67} + 44 q^{68} - 10 q^{69} + 12 q^{72} - 2 q^{73} - 48 q^{74} - 2 q^{75} - 38 q^{76} + 14 q^{78} + 8 q^{79} + 12 q^{80} - 12 q^{81} - 34 q^{82} + 10 q^{83} - 8 q^{84} - 2 q^{85} - 24 q^{86} - 32 q^{87} + 10 q^{89} + 6 q^{90} - 16 q^{91} + 10 q^{92} - 26 q^{93} - 54 q^{94} - 6 q^{95} - 8 q^{96} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display

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
2.57411
0.275377
−0.637875
−2.21161
−2.57411 −0.611516 4.62605 1.00000 1.57411 1.00000 −6.75974 −2.62605 −2.57411
1.2 −0.275377 2.63138 −1.92417 1.00000 −0.724623 1.00000 1.08063 3.92417 −0.275377
1.3 0.637875 −2.56771 −1.59312 1.00000 −1.63787 1.00000 −2.29196 3.59312 0.637875
1.4 2.21161 −1.45216 2.89124 1.00000 −3.21161 1.00000 1.97107 −0.891236 2.21161
\(n\): 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.u 4
11.b odd 2 1 4235.2.a.v yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.u 4 1.a even 1 1 trivial
4235.2.a.v yes 4 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{4} - 6T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} - 6T_{3}^{2} - 14T_{3} - 6 \) Copy content Toggle raw display
\( T_{13}^{4} + 16T_{13}^{3} + 70T_{13}^{2} - 14T_{13} - 442 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 6 T^{2} - 14 T - 6 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 16 T^{3} + 70 T^{2} + \cdots - 442 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 50 T^{2} - 98 T + 246 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} - 20 T^{2} - 128 T - 136 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 8 T^{2} - 8 T + 12 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} - 16 T^{2} + \cdots + 624 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} - 36 T^{2} - 166 T - 102 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} - 144 T^{2} + \cdots + 1636 \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + 60 T^{2} + \cdots - 1298 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} - 72 T^{2} - 216 T + 972 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} - 130 T^{2} + \cdots - 262 \) Copy content Toggle raw display
$53$ \( T^{4} - 34 T^{3} + 416 T^{2} + \cdots + 4036 \) Copy content Toggle raw display
$59$ \( T^{4} + 10 T^{3} - 96 T^{2} + \cdots - 702 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} - 96 T^{2} + \cdots - 2066 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} - 12 T^{2} + \cdots + 316 \) Copy content Toggle raw display
$71$ \( T^{4} - 120 T^{2} + 160 T + 1872 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} - 122 T^{2} + \cdots + 326 \) Copy content Toggle raw display
$79$ \( T^{4} - 8 T^{3} - 28 T^{2} + 372 T - 788 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} - 256 T^{2} + \cdots - 1304 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} - 136 T^{2} + \cdots + 5448 \) Copy content Toggle raw display
$97$ \( T^{4} + 34 T^{3} + 380 T^{2} + \cdots + 2168 \) Copy content Toggle raw display
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