Properties

Label 4235.2.a.bb
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: 5.5.580017.1
Defining polynomial: \( x^{5} - 7x^{3} - 4x^{2} + 6x + 3 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + \nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 7\beta_{3} + \beta_{2} + 3\beta _1 + 10 \) 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
2.74137
0.941569
−0.470042
−1.27779
−1.93511
−2.74137 −3.04102 5.51511 −1.00000 8.33658 1.00000 −9.63623 6.24783 2.74137
1.2 −0.941569 1.53997 −1.11345 −1.00000 −1.44999 1.00000 2.93153 −0.628495 0.941569
1.3 0.470042 −0.577925 −1.77906 −1.00000 −0.271649 1.00000 −1.77632 −2.66600 −0.470042
1.4 1.27779 2.68935 −0.367260 −1.00000 3.43642 1.00000 −3.02485 4.23262 −1.27779
1.5 1.93511 −2.61037 1.74465 −1.00000 −5.05136 1.00000 −0.494123 3.81405 −1.93511
\(n\): 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.bb yes 5
11.b odd 2 1 4235.2.a.ba 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.ba 5 11.b odd 2 1
4235.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 7 T^{3} + 4 T^{2} + 6 T - 3 \) Copy content Toggle raw display
$3$ \( T^{5} + 2 T^{4} - 11 T^{3} - 17 T^{2} + \cdots + 19 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 10 T^{4} + 8 T^{3} - 99 T^{2} + \cdots + 133 \) Copy content Toggle raw display
$17$ \( T^{5} + 2 T^{4} - 28 T^{3} - 43 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$19$ \( T^{5} - 3 T^{4} - 51 T^{3} + 114 T^{2} + \cdots + 269 \) Copy content Toggle raw display
$23$ \( T^{5} - 3 T^{4} - 70 T^{3} + \cdots - 3573 \) Copy content Toggle raw display
$29$ \( T^{5} + T^{4} - 36 T^{3} - 115 T^{2} + \cdots + 147 \) Copy content Toggle raw display
$31$ \( T^{5} + 12 T^{4} - 39 T^{3} + \cdots - 5161 \) Copy content Toggle raw display
$37$ \( T^{5} - 145 T^{3} - 243 T^{2} + \cdots + 11545 \) Copy content Toggle raw display
$41$ \( T^{5} - 11 T^{4} - 30 T^{3} + \cdots + 1425 \) Copy content Toggle raw display
$43$ \( T^{5} + 10 T^{4} + 3 T^{3} - 247 T^{2} + \cdots - 761 \) Copy content Toggle raw display
$47$ \( T^{5} - 16 T^{4} + 21 T^{3} + \cdots - 1335 \) Copy content Toggle raw display
$53$ \( T^{5} - 4 T^{4} - 167 T^{3} + \cdots - 2307 \) Copy content Toggle raw display
$59$ \( T^{5} + 32 T^{4} + 380 T^{3} + \cdots + 3267 \) Copy content Toggle raw display
$61$ \( T^{5} + 40 T^{4} + 613 T^{3} + \cdots + 13937 \) Copy content Toggle raw display
$67$ \( T^{5} - 7 T^{4} - 117 T^{3} + 392 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$71$ \( T^{5} - 10 T^{4} - 35 T^{3} + \cdots - 1155 \) Copy content Toggle raw display
$73$ \( T^{5} + 11 T^{4} - 141 T^{3} + \cdots - 121 \) Copy content Toggle raw display
$79$ \( T^{5} + 13 T^{4} - 95 T^{3} - 1120 T^{2} + \cdots + 59 \) Copy content Toggle raw display
$83$ \( T^{5} + 26 T^{4} + 116 T^{3} + \cdots - 23445 \) Copy content Toggle raw display
$89$ \( T^{5} - 5 T^{4} - 95 T^{3} + 377 T^{2} + \cdots - 927 \) Copy content Toggle raw display
$97$ \( T^{5} + 5 T^{4} - 123 T^{3} - 751 T^{2} + \cdots + 49 \) Copy content Toggle raw display
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