Properties

Label 495.4.a.d
Level $495$
Weight $4$
Character orbit 495.a
Self dual yes
Analytic conductor $29.206$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + (4 \beta - 4) q^{7} + ( - 11 \beta + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta - 4) q^{4} + 5 q^{5} + (4 \beta - 4) q^{7} + ( - 11 \beta + 4) q^{8} + 5 \beta q^{10} + 11 q^{11} + ( - 2 \beta - 44) q^{13} + 16 q^{14} + ( - 15 \beta - 12) q^{16} + ( - 44 \beta + 30) q^{17} + ( - 22 \beta - 74) q^{19} + (5 \beta - 20) q^{20} + 11 \beta q^{22} + (60 \beta + 32) q^{23} + 25 q^{25} + ( - 46 \beta - 8) q^{26} + ( - 16 \beta + 32) q^{28} + ( - 34 \beta + 96) q^{29} + ( - 12 \beta + 36) q^{31} + (61 \beta - 92) q^{32} + ( - 14 \beta - 176) q^{34} + (20 \beta - 20) q^{35} + ( - 112 \beta - 130) q^{37} + ( - 96 \beta - 88) q^{38} + ( - 55 \beta + 20) q^{40} + (154 \beta - 96) q^{41} + ( - 124 \beta - 196) q^{43} + (11 \beta - 44) q^{44} + (92 \beta + 240) q^{46} + ( - 216 \beta - 4) q^{47} + ( - 16 \beta - 263) q^{49} + 25 \beta q^{50} + ( - 38 \beta + 168) q^{52} + (196 \beta - 334) q^{53} + 55 q^{55} + (16 \beta - 192) q^{56} + (62 \beta - 136) q^{58} + ( - 240 \beta - 4) q^{59} + (364 \beta - 146) q^{61} + (24 \beta - 48) q^{62} + (89 \beta + 340) q^{64} + ( - 10 \beta - 220) q^{65} + (16 \beta - 380) q^{67} + (162 \beta - 296) q^{68} + 80 q^{70} + ( - 44 \beta - 1008) q^{71} + (58 \beta - 272) q^{73} + ( - 242 \beta - 448) q^{74} + ( - 8 \beta + 208) q^{76} + (44 \beta - 44) q^{77} + ( - 306 \beta + 474) q^{79} + ( - 75 \beta - 60) q^{80} + (58 \beta + 616) q^{82} + (426 \beta - 70) q^{83} + ( - 220 \beta + 150) q^{85} + ( - 320 \beta - 496) q^{86} + ( - 121 \beta + 44) q^{88} + (128 \beta - 186) q^{89} + ( - 176 \beta + 144) q^{91} + ( - 148 \beta + 112) q^{92} + ( - 220 \beta - 864) q^{94} + ( - 110 \beta - 370) q^{95} + (428 \beta - 298) q^{97} + ( - 279 \beta - 64) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 7 q^{4} + 10 q^{5} - 4 q^{7} - 3 q^{8} + 5 q^{10} + 22 q^{11} - 90 q^{13} + 32 q^{14} - 39 q^{16} + 16 q^{17} - 170 q^{19} - 35 q^{20} + 11 q^{22} + 124 q^{23} + 50 q^{25} - 62 q^{26} + 48 q^{28} + 158 q^{29} + 60 q^{31} - 123 q^{32} - 366 q^{34} - 20 q^{35} - 372 q^{37} - 272 q^{38} - 15 q^{40} - 38 q^{41} - 516 q^{43} - 77 q^{44} + 572 q^{46} - 224 q^{47} - 542 q^{49} + 25 q^{50} + 298 q^{52} - 472 q^{53} + 110 q^{55} - 368 q^{56} - 210 q^{58} - 248 q^{59} + 72 q^{61} - 72 q^{62} + 769 q^{64} - 450 q^{65} - 744 q^{67} - 430 q^{68} + 160 q^{70} - 2060 q^{71} - 486 q^{73} - 1138 q^{74} + 408 q^{76} - 44 q^{77} + 642 q^{79} - 195 q^{80} + 1290 q^{82} + 286 q^{83} + 80 q^{85} - 1312 q^{86} - 33 q^{88} - 244 q^{89} + 112 q^{91} + 76 q^{92} - 1948 q^{94} - 850 q^{95} - 168 q^{97} - 407 q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 0 −5.56155 5.00000 0 −10.2462 21.1771 0 −7.80776
1.2 2.56155 0 −1.43845 5.00000 0 6.24621 −24.1771 0 12.8078
\(n\): 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.d 2
3.b odd 2 1 165.4.a.c 2
5.b even 2 1 2475.4.a.n 2
15.d odd 2 1 825.4.a.m 2
15.e even 4 2 825.4.c.j 4
33.d even 2 1 1815.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 3.b odd 2 1
495.4.a.d 2 1.a even 1 1 trivial
825.4.a.m 2 15.d odd 2 1
825.4.c.j 4 15.e even 4 2
1815.4.a.n 2 33.d even 2 1
2475.4.a.n 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(495))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 90T + 2008 \) Copy content Toggle raw display
$17$ \( T^{2} - 16T - 8164 \) Copy content Toggle raw display
$19$ \( T^{2} + 170T + 5168 \) Copy content Toggle raw display
$23$ \( T^{2} - 124T - 11456 \) Copy content Toggle raw display
$29$ \( T^{2} - 158T + 1328 \) Copy content Toggle raw display
$31$ \( T^{2} - 60T + 288 \) Copy content Toggle raw display
$37$ \( T^{2} + 372T - 18716 \) Copy content Toggle raw display
$41$ \( T^{2} + 38T - 100432 \) Copy content Toggle raw display
$43$ \( T^{2} + 516T + 1216 \) Copy content Toggle raw display
$47$ \( T^{2} + 224T - 185744 \) Copy content Toggle raw display
$53$ \( T^{2} + 472T - 107572 \) Copy content Toggle raw display
$59$ \( T^{2} + 248T - 229424 \) Copy content Toggle raw display
$61$ \( T^{2} - 72T - 561812 \) Copy content Toggle raw display
$67$ \( T^{2} + 744T + 137296 \) Copy content Toggle raw display
$71$ \( T^{2} + 2060 T + 1052672 \) Copy content Toggle raw display
$73$ \( T^{2} + 486T + 44752 \) Copy content Toggle raw display
$79$ \( T^{2} - 642T - 294912 \) Copy content Toggle raw display
$83$ \( T^{2} - 286T - 750824 \) Copy content Toggle raw display
$89$ \( T^{2} + 244T - 54748 \) Copy content Toggle raw display
$97$ \( T^{2} + 168T - 771476 \) Copy content Toggle raw display
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