Properties

Label 4235.2.a.bd
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: 5.5.288385.1
Defining polynomial: \(x^{5} - 7 x^{3} - 2 x^{2} + 10 x + 3\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 7 x^{3} - 2 x^{2} + 10 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 2 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 6 \beta_{2} + 2 \beta_{1} + 12\)

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
−1.90452
−1.58597
−0.300703
1.35681
2.43438
−1.90452 0.214998 1.62718 1.00000 −0.409467 1.00000 0.710046 −2.95378 −1.90452
1.2 −1.58597 −3.01851 0.515286 1.00000 4.78725 1.00000 2.35471 6.11138 −1.58597
1.3 −0.300703 2.68115 −1.90958 1.00000 −0.806228 1.00000 1.17562 4.18855 −0.300703
1.4 1.35681 −0.242974 −0.159077 1.00000 −0.329669 1.00000 −2.92945 −2.94096 1.35681
1.5 2.43438 2.36534 3.92619 1.00000 5.75812 1.00000 4.68908 2.59481 2.43438
\(n\): 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.bd yes 5
11.b odd 2 1 4235.2.a.bc 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.bc 5 11.b odd 2 1
4235.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{5} - 7 T_{2}^{3} - 2 T_{2}^{2} + 10 T_{2} + 3 \)
\( T_{3}^{5} - 2 T_{3}^{4} - 9 T_{3}^{3} + 19 T_{3}^{2} + T_{3} - 1 \)
\( T_{13}^{5} - 6 T_{13}^{4} - 8 T_{13}^{3} + 119 T_{13}^{2} - 260 T_{13} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + 10 T - 2 T^{2} - 7 T^{3} + T^{5} \)
$3$ \( -1 + T + 19 T^{2} - 9 T^{3} - 2 T^{4} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( ( -1 + T )^{5} \)
$11$ \( T^{5} \)
$13$ \( 169 - 260 T + 119 T^{2} - 8 T^{3} - 6 T^{4} + T^{5} \)
$17$ \( 75 - 170 T + 123 T^{2} - 28 T^{3} - 2 T^{4} + T^{5} \)
$19$ \( -73 + 9 T + 56 T^{2} - T^{3} - 7 T^{4} + T^{5} \)
$23$ \( 81 - 339 T + 223 T^{2} - 34 T^{3} - 5 T^{4} + T^{5} \)
$29$ \( -81 - 243 T + 349 T^{2} - 16 T^{3} - 11 T^{4} + T^{5} \)
$31$ \( -327 + 1313 T - 181 T^{2} - 89 T^{3} + 2 T^{4} + T^{5} \)
$37$ \( -9801 + 2651 T + 529 T^{2} - 139 T^{3} - 2 T^{4} + T^{5} \)
$41$ \( 2475 - 1935 T + 409 T^{2} + 16 T^{3} - 13 T^{4} + T^{5} \)
$43$ \( 965 - 5325 T + 1761 T^{2} - 123 T^{3} - 10 T^{4} + T^{5} \)
$47$ \( 12609 + 3198 T - 442 T^{2} - 127 T^{3} + 2 T^{4} + T^{5} \)
$53$ \( -309 - 413 T - 67 T^{2} + 45 T^{3} + 14 T^{4} + T^{5} \)
$59$ \( 81 + 640 T + 563 T^{2} - 100 T^{3} - 6 T^{4} + T^{5} \)
$61$ \( -393 + 778 T - 510 T^{2} + 149 T^{3} - 20 T^{4} + T^{5} \)
$67$ \( 43677 + 1039 T - 2386 T^{2} - 219 T^{3} + 9 T^{4} + T^{5} \)
$71$ \( 3753 + 1191 T - 421 T^{2} - 71 T^{3} + 10 T^{4} + T^{5} \)
$73$ \( 103 - 1067 T + 386 T^{2} + 13 T^{3} - 15 T^{4} + T^{5} \)
$79$ \( -41845 - 4755 T + 2612 T^{2} - 23 T^{3} - 23 T^{4} + T^{5} \)
$83$ \( 26499 + 10186 T + 265 T^{2} - 198 T^{3} - 8 T^{4} + T^{5} \)
$89$ \( 1587 - 3380 T + 1589 T^{2} - 195 T^{3} - 7 T^{4} + T^{5} \)
$97$ \( 1297 - 1506 T + 209 T^{2} + 101 T^{3} - 21 T^{4} + T^{5} \)
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