Properties

Label 4235.2.a.ba
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: 5.5.580017.1
Defining polynomial: \(x^{5} - 7 x^{3} - 4 x^{2} + 6 x + 3\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 7 x^{3} - 4 x^{2} + 6 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-6 \beta_{4} + 7 \beta_{3} + \beta_{2} + 3 \beta_{1} + 10\)

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
−1.93511
−1.27779
−0.470042
0.941569
2.74137
−1.93511 −2.61037 1.74465 −1.00000 5.05136 −1.00000 0.494123 3.81405 1.93511
1.2 −1.27779 2.68935 −0.367260 −1.00000 −3.43642 −1.00000 3.02485 4.23262 1.27779
1.3 −0.470042 −0.577925 −1.77906 −1.00000 0.271649 −1.00000 1.77632 −2.66600 0.470042
1.4 0.941569 1.53997 −1.11345 −1.00000 1.44999 −1.00000 −2.93153 −0.628495 −0.941569
1.5 2.74137 −3.04102 5.51511 −1.00000 −8.33658 −1.00000 9.63623 6.24783 −2.74137
\(n\): 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.ba 5
11.b odd 2 1 4235.2.a.bb yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.ba 5 1.a even 1 1 trivial
4235.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{5} - 7 T_{2}^{3} - 4 T_{2}^{2} + 6 T_{2} + 3 \)
\( T_{3}^{5} + 2 T_{3}^{4} - 11 T_{3}^{3} - 17 T_{3}^{2} + 27 T_{3} + 19 \)
\( T_{13}^{5} - 10 T_{13}^{4} + 8 T_{13}^{3} + 99 T_{13}^{2} - 4 T_{13} - 133 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + 6 T - 4 T^{2} - 7 T^{3} + T^{5} \)
$3$ \( 19 + 27 T - 17 T^{2} - 11 T^{3} + 2 T^{4} + T^{5} \)
$5$ \( ( 1 + T )^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( T^{5} \)
$13$ \( -133 - 4 T + 99 T^{2} + 8 T^{3} - 10 T^{4} + T^{5} \)
$17$ \( 9 + 84 T + 43 T^{2} - 28 T^{3} - 2 T^{4} + T^{5} \)
$19$ \( -269 + 439 T - 114 T^{2} - 51 T^{3} + 3 T^{4} + T^{5} \)
$23$ \( -3573 + 1191 T + 209 T^{2} - 70 T^{3} - 3 T^{4} + T^{5} \)
$29$ \( -147 - 27 T + 115 T^{2} - 36 T^{3} - T^{4} + T^{5} \)
$31$ \( -5161 - 4367 T - 1035 T^{2} - 39 T^{3} + 12 T^{4} + T^{5} \)
$37$ \( 11545 + 4549 T - 243 T^{2} - 145 T^{3} + T^{5} \)
$41$ \( -1425 - 1683 T - 581 T^{2} - 30 T^{3} + 11 T^{4} + T^{5} \)
$43$ \( 761 - 837 T + 247 T^{2} + 3 T^{3} - 10 T^{4} + T^{5} \)
$47$ \( -1335 - 1404 T + 550 T^{2} + 21 T^{3} - 16 T^{4} + T^{5} \)
$53$ \( -2307 + 3159 T + 655 T^{2} - 167 T^{3} - 4 T^{4} + T^{5} \)
$59$ \( 3267 + 4752 T + 2045 T^{2} + 380 T^{3} + 32 T^{4} + T^{5} \)
$61$ \( -13937 + 14094 T - 4400 T^{2} + 613 T^{3} - 40 T^{4} + T^{5} \)
$67$ \( -5 + 3433 T + 392 T^{2} - 117 T^{3} - 7 T^{4} + T^{5} \)
$71$ \( -1155 - 921 T + 547 T^{2} - 35 T^{3} - 10 T^{4} + T^{5} \)
$73$ \( 121 + 1353 T + 918 T^{2} - 141 T^{3} - 11 T^{4} + T^{5} \)
$79$ \( -59 + 741 T + 1120 T^{2} - 95 T^{3} - 13 T^{4} + T^{5} \)
$83$ \( 23445 - 14748 T + 1681 T^{2} + 116 T^{3} - 26 T^{4} + T^{5} \)
$89$ \( -927 + 1626 T + 377 T^{2} - 95 T^{3} - 5 T^{4} + T^{5} \)
$97$ \( 49 + 280 T - 751 T^{2} - 123 T^{3} + 5 T^{4} + T^{5} \)
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