Properties

Label 4235.2.a.z
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.270017.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 7 x^{2} + 7 x - 3\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 7 x^{2} + 7 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 7 \beta_{1} + 7\)

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
−1.64458
2.51766
−1.11334
1.89725
0.343016
−2.34922 −1.64458 3.51886 1.00000 3.86349 −1.00000 −3.56813 −0.295356 −2.34922
1.2 −1.82093 2.51766 1.31579 1.00000 −4.58448 −1.00000 1.24589 3.33859 −1.82093
1.3 −0.352882 −1.11334 −1.87547 1.00000 0.392879 −1.00000 1.36758 −1.76046 −0.352882
1.4 0.297682 1.89725 −1.91139 1.00000 0.564779 −1.00000 −1.16435 0.599571 0.297682
1.5 2.22536 0.343016 2.95221 1.00000 0.763334 −1.00000 2.11901 −2.88234 2.22536
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.z 5
11.b odd 2 1 4235.2.a.bf yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.z 5 1.a even 1 1 trivial
4235.2.a.bf yes 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{5} + 2 T_{2}^{4} - 5 T_{2}^{3} - 10 T_{2}^{2} + 1 \)
\( T_{3}^{5} - 2 T_{3}^{4} - 5 T_{3}^{3} + 7 T_{3}^{2} + 7 T_{3} - 3 \)
\( T_{13}^{5} + 8 T_{13}^{4} + 4 T_{13}^{3} - 45 T_{13}^{2} + 40 T_{13} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T^{2} - 5 T^{3} + 2 T^{4} + T^{5} \)
$3$ \( -3 + 7 T + 7 T^{2} - 5 T^{3} - 2 T^{4} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( T^{5} \)
$13$ \( -9 + 40 T - 45 T^{2} + 4 T^{3} + 8 T^{4} + T^{5} \)
$17$ \( 3465 + 1012 T - 311 T^{2} - 62 T^{3} + 6 T^{4} + T^{5} \)
$19$ \( 287 + 179 T - 124 T^{2} - 19 T^{3} + 7 T^{4} + T^{5} \)
$23$ \( 31 - 27 T - 101 T^{2} - 48 T^{3} - 3 T^{4} + T^{5} \)
$29$ \( -89 + 17 T + 161 T^{2} + 92 T^{3} + 17 T^{4} + T^{5} \)
$31$ \( -1823 - 471 T + 461 T^{2} - 19 T^{3} - 12 T^{4} + T^{5} \)
$37$ \( -411 + 155 T + 157 T^{2} - 73 T^{3} + 2 T^{4} + T^{5} \)
$41$ \( -1805 + 1349 T - 91 T^{2} - 70 T^{3} + 5 T^{4} + T^{5} \)
$43$ \( -1 - 45 T + 163 T^{2} - 77 T^{3} + 4 T^{4} + T^{5} \)
$47$ \( -2573 + 1544 T + 40 T^{2} - 91 T^{3} + T^{5} \)
$53$ \( 5889 - 4663 T + 1217 T^{2} - 89 T^{3} - 8 T^{4} + T^{5} \)
$59$ \( 1833 - 838 T - 457 T^{2} + 14 T^{3} + 16 T^{4} + T^{5} \)
$61$ \( 2475 - 2596 T - 332 T^{2} + 135 T^{3} + 24 T^{4} + T^{5} \)
$67$ \( 67 - 337 T - 546 T^{2} - 55 T^{3} + 9 T^{4} + T^{5} \)
$71$ \( 102033 + 14767 T - 1913 T^{2} - 255 T^{3} + 10 T^{4} + T^{5} \)
$73$ \( -1671 - 2009 T - 708 T^{2} - 47 T^{3} + 11 T^{4} + T^{5} \)
$79$ \( 151 + 309 T - 224 T^{2} - 147 T^{3} + 9 T^{4} + T^{5} \)
$83$ \( -7719 + 17722 T - 2815 T^{2} - 352 T^{3} + 10 T^{4} + T^{5} \)
$89$ \( -3181 - 3498 T - 1017 T^{2} - 69 T^{3} + 9 T^{4} + T^{5} \)
$97$ \( -47833 + 8426 T + 1493 T^{2} - 227 T^{3} - 11 T^{4} + T^{5} \)
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