Properties

Label 4235.2.a.be
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.173513.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1\)
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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + \beta_{2} q^{3} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{4} - q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{6} - q^{7} + ( -2 - \beta_{3} - \beta_{4} ) q^{8} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{2} + \beta_{2} q^{3} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{4} - q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{6} - q^{7} + ( -2 - \beta_{3} - \beta_{4} ) q^{8} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} + \beta_{4} q^{10} + ( 1 - 2 \beta_{3} - \beta_{4} ) q^{12} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + \beta_{4} q^{14} -\beta_{2} q^{15} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{16} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{18} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{20} -\beta_{2} q^{21} + ( 5 - \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{23} + ( 2 - \beta_{1} + \beta_{2} ) q^{24} + q^{25} + ( -5 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{26} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{27} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{28} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} ) q^{30} + ( -2 + \beta_{1} - 3 \beta_{3} - 2 \beta_{4} ) q^{31} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{32} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{34} + q^{35} + ( -3 + \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{36} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{37} + ( -5 + \beta_{2} - 3 \beta_{3} - 5 \beta_{4} ) q^{38} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{39} + ( 2 + \beta_{3} + \beta_{4} ) q^{40} + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -1 - \beta_{2} + \beta_{3} ) q^{42} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{43} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{45} + ( -6 + \beta_{1} - \beta_{2} - \beta_{3} - 7 \beta_{4} ) q^{46} + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{48} + q^{49} -\beta_{4} q^{50} + ( -\beta_{1} + 6 \beta_{2} + \beta_{3} + \beta_{4} ) q^{51} + ( 6 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} + 6 \beta_{4} ) q^{52} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{53} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{54} + ( 2 + \beta_{3} + \beta_{4} ) q^{56} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{57} + ( -10 - 5 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} ) q^{58} + ( -2 + \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} ) q^{59} + ( -1 + 2 \beta_{3} + \beta_{4} ) q^{60} + ( 6 - \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{61} + ( 7 - \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} ) q^{62} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{63} + ( -6 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{64} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{65} + ( -1 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{67} + ( 3 - \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{68} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{69} -\beta_{4} q^{70} + ( -4 + 2 \beta_{1} - \beta_{2} - 4 \beta_{4} ) q^{71} + ( 8 + \beta_{2} + 2 \beta_{4} ) q^{72} + ( -1 + 5 \beta_{1} - 3 \beta_{3} ) q^{73} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{74} + \beta_{2} q^{75} + ( 15 - \beta_{1} + 7 \beta_{2} + 6 \beta_{4} ) q^{76} + ( 7 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{78} + ( 7 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{79} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{80} + ( 3 + 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{81} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{82} + ( -\beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{83} + ( -1 + 2 \beta_{3} + \beta_{4} ) q^{84} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( 9 - 5 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{86} + ( -6 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{87} + ( -5 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 5 \beta_{4} ) q^{89} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{90} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{91} + ( 16 - 4 \beta_{1} + 5 \beta_{2} + 9 \beta_{4} ) q^{92} + ( 5 - 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{93} + ( 8 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{94} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{95} + ( -2 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{96} + ( 3 + \beta_{1} + \beta_{2} + 7 \beta_{3} + 4 \beta_{4} ) q^{97} -\beta_{4} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{2} - 2q^{3} + 4q^{4} - 5q^{5} + 5q^{6} - 5q^{7} - 6q^{8} + 3q^{9} + O(q^{10}) \) \( 5q + 2q^{2} - 2q^{3} + 4q^{4} - 5q^{5} + 5q^{6} - 5q^{7} - 6q^{8} + 3q^{9} - 2q^{10} + 11q^{12} + 12q^{13} - 2q^{14} + 2q^{15} + 2q^{16} + 14q^{17} + 7q^{18} + 9q^{19} - 4q^{20} + 2q^{21} + 17q^{23} + 6q^{24} + 5q^{25} - 11q^{26} - 11q^{27} - 4q^{28} + 3q^{29} - 5q^{30} + 2q^{31} - 5q^{32} + 16q^{34} + 5q^{35} - 15q^{36} + 4q^{37} - 11q^{38} + 2q^{39} + 6q^{40} - 15q^{41} - 5q^{42} + 4q^{43} - 3q^{45} - 10q^{46} - 2q^{47} - 10q^{48} + 5q^{49} + 2q^{50} - 18q^{51} + 4q^{52} + 6q^{53} + 4q^{54} + 6q^{56} - 32q^{58} - 6q^{59} - 11q^{60} + 20q^{61} + 21q^{62} - 3q^{63} - 26q^{64} - 12q^{65} + 3q^{67} + 5q^{68} + 2q^{70} - 6q^{71} + 34q^{72} + 11q^{73} - 15q^{74} - 2q^{75} + 47q^{76} + 31q^{78} + 19q^{79} - 2q^{80} + 33q^{81} + 8q^{83} - 11q^{84} - 14q^{85} + 27q^{86} - 30q^{87} + q^{89} - 7q^{90} - 12q^{91} + 44q^{92} + 3q^{93} + 28q^{94} - 9q^{95} - 4q^{96} - 7q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 3 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + 2 \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-8 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 23 \beta_{1} + 14\)

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
0.293545
0.859039
3.18986
−1.52979
−0.812660
−2.40663 −0.321225 3.79186 −1.00000 0.773069 −1.00000 −4.31233 −2.89681 2.40663
1.2 −0.164091 −3.27813 −1.97307 −1.00000 0.537912 −1.00000 0.651947 7.74611 0.164091
1.3 0.686507 0.347661 −1.52871 −1.00000 0.238671 −1.00000 −2.42248 −2.87913 −0.686507
1.4 1.65369 −1.14142 0.734678 −1.00000 −1.88755 −1.00000 −2.09245 −1.69716 −1.65369
1.5 2.23053 2.39311 2.97525 −1.00000 5.33790 −1.00000 2.17531 2.72699 −2.23053
\(n\): 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.be yes 5
11.b odd 2 1 4235.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.y 5 11.b odd 2 1
4235.2.a.be yes 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{5} - 2 T_{2}^{4} - 5 T_{2}^{3} + 12 T_{2}^{2} - 4 T_{2} - 1 \)
\( T_{3}^{5} + 2 T_{3}^{4} - 7 T_{3}^{3} - 9 T_{3}^{2} + T_{3} + 1 \)
\( T_{13}^{5} - 12 T_{13}^{4} + 12 T_{13}^{3} + 325 T_{13}^{2} - 1368 T_{13} + 1571 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 4 T + 12 T^{2} - 5 T^{3} - 2 T^{4} + T^{5} \)
$3$ \( 1 + T - 9 T^{2} - 7 T^{3} + 2 T^{4} + T^{5} \)
$5$ \( ( 1 + T )^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( T^{5} \)
$13$ \( 1571 - 1368 T + 325 T^{2} + 12 T^{3} - 12 T^{4} + T^{5} \)
$17$ \( -83 - 286 T + 35 T^{2} + 50 T^{3} - 14 T^{4} + T^{5} \)
$19$ \( -2259 - 303 T + 446 T^{2} - 37 T^{3} - 9 T^{4} + T^{5} \)
$23$ \( -67 + 199 T - 207 T^{2} + 92 T^{3} - 17 T^{4} + T^{5} \)
$29$ \( -3197 + 2257 T + 193 T^{2} - 104 T^{3} - 3 T^{4} + T^{5} \)
$31$ \( 1359 + 1869 T + 55 T^{2} - 85 T^{3} - 2 T^{4} + T^{5} \)
$37$ \( 1051 - 1211 T + 473 T^{2} - 59 T^{3} - 4 T^{4} + T^{5} \)
$41$ \( 4617 - 891 T - 999 T^{2} - 36 T^{3} + 15 T^{4} + T^{5} \)
$43$ \( -10971 + 2391 T + 559 T^{2} - 119 T^{3} - 4 T^{4} + T^{5} \)
$47$ \( 251 + 566 T - 260 T^{2} - 123 T^{3} + 2 T^{4} + T^{5} \)
$53$ \( -22913 + 4025 T + 739 T^{2} - 129 T^{3} - 6 T^{4} + T^{5} \)
$59$ \( 68011 + 11074 T - 1511 T^{2} - 242 T^{3} + 6 T^{4} + T^{5} \)
$61$ \( -1867 - 1336 T + 266 T^{2} + 87 T^{3} - 20 T^{4} + T^{5} \)
$67$ \( 2309 + 721 T - 420 T^{2} - 133 T^{3} - 3 T^{4} + T^{5} \)
$71$ \( 13 - 73 T + 127 T^{2} - 75 T^{3} + 6 T^{4} + T^{5} \)
$73$ \( 1097 - 2993 T + 2176 T^{2} - 173 T^{3} - 11 T^{4} + T^{5} \)
$79$ \( 20143 - 11379 T + 1708 T^{2} + 13 T^{3} - 19 T^{4} + T^{5} \)
$83$ \( -279 - 1296 T + 707 T^{2} - 74 T^{3} - 8 T^{4} + T^{5} \)
$89$ \( -6047 + 3656 T + 571 T^{2} - 213 T^{3} - T^{4} + T^{5} \)
$97$ \( -271673 + 62192 T - 475 T^{2} - 463 T^{3} + 7 T^{4} + T^{5} \)
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