Properties

Label 495.4.a.l
Level $495$
Weight $4$
Character orbit 495.a
Self dual yes
Analytic conductor $29.206$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.23612.1
Defining polynomial: \(x^{3} - x^{2} - 20 x + 26\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + 5 q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{1} + 4 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + 5 q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{1} + 4 \beta_{2} ) q^{8} + ( 5 + 5 \beta_{1} ) q^{10} -11 q^{11} + ( -4 + 12 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 22 - 10 \beta_{1} - 4 \beta_{2} ) q^{14} + ( 9 + 23 \beta_{1} + 7 \beta_{2} ) q^{16} + ( 76 - 10 \beta_{1} - 6 \beta_{2} ) q^{17} + ( 48 + 2 \beta_{1} + 4 \beta_{2} ) q^{19} + ( 35 + 5 \beta_{1} + 5 \beta_{2} ) q^{20} + ( -11 - 11 \beta_{1} ) q^{22} + ( 60 + 20 \beta_{1} + 8 \beta_{2} ) q^{23} + 25 q^{25} + ( 168 + 4 \beta_{1} + 18 \beta_{2} ) q^{26} + ( -110 - 10 \beta_{1} - 6 \beta_{2} ) q^{28} + ( -34 + 34 \beta_{1} - 20 \beta_{2} ) q^{29} + ( -24 + 4 \beta_{1} - 16 \beta_{2} ) q^{31} + ( 225 + 13 \beta_{1} + 12 \beta_{2} ) q^{32} + ( -76 + 52 \beta_{1} - 28 \beta_{2} ) q^{34} + ( -10 + 10 \beta_{1} - 10 \beta_{2} ) q^{35} + ( -122 - 24 \beta_{1} - 4 \beta_{2} ) q^{37} + ( 84 + 64 \beta_{1} + 14 \beta_{2} ) q^{38} + ( 75 + 15 \beta_{1} + 20 \beta_{2} ) q^{40} + ( 66 - 2 \beta_{1} + 20 \beta_{2} ) q^{41} + ( -150 - 74 \beta_{1} + 14 \beta_{2} ) q^{43} + ( -77 - 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( 356 + 92 \beta_{1} + 44 \beta_{2} ) q^{46} + ( 36 - 48 \beta_{1} - 28 \beta_{2} ) q^{47} + ( -51 - 4 \beta_{1} - 20 \beta_{2} ) q^{49} + ( 25 + 25 \beta_{1} ) q^{50} + ( 292 + 144 \beta_{1} + 42 \beta_{2} ) q^{52} + ( 42 + 32 \beta_{1} - 52 \beta_{2} ) q^{53} -55 q^{55} + ( -438 - 54 \beta_{1} + 4 \beta_{2} ) q^{56} + ( 402 - 114 \beta_{1} - 26 \beta_{2} ) q^{58} + ( 308 + 120 \beta_{1} + 4 \beta_{2} ) q^{59} + ( 218 - 12 \beta_{1} - 44 \beta_{2} ) q^{61} + ( -88 \beta_{1} - 44 \beta_{2} ) q^{62} + ( 359 + 89 \beta_{1} - 7 \beta_{2} ) q^{64} + ( -20 + 60 \beta_{1} + 10 \beta_{2} ) q^{65} + ( -128 + 148 \beta_{1} - 28 \beta_{2} ) q^{67} + ( -12 - 108 \beta_{1} + 16 \beta_{2} ) q^{68} + ( 110 - 50 \beta_{1} - 20 \beta_{2} ) q^{70} + ( 216 - 104 \beta_{1} + 16 \beta_{2} ) q^{71} + ( 356 - 168 \beta_{1} + 14 \beta_{2} ) q^{73} + ( -466 - 138 \beta_{1} - 36 \beta_{2} ) q^{74} + ( 624 + 124 \beta_{1} + 74 \beta_{2} ) q^{76} + ( 22 - 22 \beta_{1} + 22 \beta_{2} ) q^{77} + ( -524 - 14 \beta_{1} - 52 \beta_{2} ) q^{79} + ( 45 + 115 \beta_{1} + 35 \beta_{2} ) q^{80} + ( 78 + 146 \beta_{1} + 58 \beta_{2} ) q^{82} + ( 582 - 164 \beta_{1} - 70 \beta_{2} ) q^{83} + ( 380 - 50 \beta_{1} - 30 \beta_{2} ) q^{85} + ( -1158 - 94 \beta_{1} - 32 \beta_{2} ) q^{86} + ( -165 - 33 \beta_{1} - 44 \beta_{2} ) q^{88} + ( 730 - 68 \beta_{1} ) q^{89} + ( 56 - 176 \beta_{1} + 4 \beta_{2} ) q^{91} + ( 1252 + 372 \beta_{1} + 160 \beta_{2} ) q^{92} + ( -692 - 76 \beta_{1} - 132 \beta_{2} ) q^{94} + ( 240 + 10 \beta_{1} + 20 \beta_{2} ) q^{95} + ( 286 - 240 \beta_{1} + 64 \beta_{2} ) q^{97} + ( -147 - 131 \beta_{1} - 64 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 4q^{2} + 22q^{4} + 15q^{5} - 4q^{7} + 48q^{8} + O(q^{10}) \) \( 3q + 4q^{2} + 22q^{4} + 15q^{5} - 4q^{7} + 48q^{8} + 20q^{10} - 33q^{11} + 56q^{14} + 50q^{16} + 218q^{17} + 146q^{19} + 110q^{20} - 44q^{22} + 200q^{23} + 75q^{25} + 508q^{26} - 340q^{28} - 68q^{29} - 68q^{31} + 688q^{32} - 176q^{34} - 20q^{35} - 390q^{37} + 316q^{38} + 240q^{40} + 196q^{41} - 524q^{43} - 242q^{44} + 1160q^{46} + 60q^{47} - 157q^{49} + 100q^{50} + 1020q^{52} + 158q^{53} - 165q^{55} - 1368q^{56} + 1092q^{58} + 1044q^{59} + 642q^{61} - 88q^{62} + 1166q^{64} - 236q^{67} - 144q^{68} + 280q^{70} + 544q^{71} + 900q^{73} - 1536q^{74} + 1996q^{76} + 44q^{77} - 1586q^{79} + 250q^{80} + 380q^{82} + 1582q^{83} + 1090q^{85} - 3568q^{86} - 528q^{88} + 2122q^{89} - 8q^{91} + 4128q^{92} - 2152q^{94} + 730q^{95} + 618q^{97} - 572q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 20 x + 26\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 14\)

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
−4.59056
1.32906
4.26150
−3.59056 0 4.89212 5.00000 0 −16.1465 11.1590 0 −17.9528
1.2 2.32906 0 −2.57547 5.00000 0 22.4672 −24.6309 0 11.6453
1.3 5.26150 0 19.6833 5.00000 0 −10.3207 61.4719 0 26.3075
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 495.4.a.l 3
3.b odd 2 1 165.4.a.d 3
5.b even 2 1 2475.4.a.s 3
15.d odd 2 1 825.4.a.s 3
15.e even 4 2 825.4.c.l 6
33.d even 2 1 1815.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 3.b odd 2 1
495.4.a.l 3 1.a even 1 1 trivial
825.4.a.s 3 15.d odd 2 1
825.4.c.l 6 15.e even 4 2
1815.4.a.s 3 33.d even 2 1
2475.4.a.s 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(495))\):

\( T_{2}^{3} - 4 T_{2}^{2} - 15 T_{2} + 44 \)
\( T_{7}^{3} + 4 T_{7}^{2} - 428 T_{7} - 3744 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 44 - 15 T - 4 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( -5 + T )^{3} \)
$7$ \( -3744 - 428 T + 4 T^{2} + T^{3} \)
$11$ \( ( 11 + T )^{3} \)
$13$ \( -34144 - 3560 T + T^{3} \)
$17$ \( 235104 + 9680 T - 218 T^{2} + T^{3} \)
$19$ \( -30960 + 5376 T - 146 T^{2} + T^{3} \)
$23$ \( -1664 - 2672 T - 200 T^{2} + T^{3} \)
$29$ \( -3163056 - 54364 T + 68 T^{2} + T^{3} \)
$31$ \( -1812096 - 23232 T + 68 T^{2} + T^{3} \)
$37$ \( 618952 + 36460 T + 390 T^{2} + T^{3} \)
$41$ \( 4364208 - 26076 T - 196 T^{2} + T^{3} \)
$43$ \( -31273920 - 28668 T + 524 T^{2} + T^{3} \)
$47$ \( 20966976 - 135920 T - 60 T^{2} + T^{3} \)
$53$ \( -39574952 - 260852 T - 158 T^{2} + T^{3} \)
$59$ \( 84227264 + 64144 T - 1044 T^{2} + T^{3} \)
$61$ \( 22757384 - 60548 T - 642 T^{2} + T^{3} \)
$67$ \( 87537664 - 462208 T + 236 T^{2} + T^{3} \)
$71$ \( -6553600 - 129728 T - 544 T^{2} + T^{3} \)
$73$ \( -5609344 - 299576 T - 900 T^{2} + T^{3} \)
$79$ \( -14694992 + 562208 T + 1586 T^{2} + T^{3} \)
$83$ \( 924645384 - 307644 T - 1582 T^{2} + T^{3} \)
$89$ \( -293444632 + 1406940 T - 2122 T^{2} + T^{3} \)
$97$ \( -223543736 - 1291700 T - 618 T^{2} + T^{3} \)
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