Properties

Label 495.6.a.e
Level $495$
Weight $6$
Character orbit 495.a
Self dual yes
Analytic conductor $79.390$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.3899908074\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.34253.1
Defining polynomial: \( x^{3} - x^{2} - 52x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 2) q^{2} + (\beta_{2} + 4 \beta_1 + 7) q^{4} - 25 q^{5} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + (7 \beta_{2} + 77) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 2) q^{2} + (\beta_{2} + 4 \beta_1 + 7) q^{4} - 25 q^{5} + (\beta_{2} + 11 \beta_1 - 61) q^{7} + (7 \beta_{2} + 77) q^{8} + ( - 25 \beta_1 - 50) q^{10} + 121 q^{11} + ( - 28 \beta_{2} - 24 \beta_1 - 210) q^{13} + (14 \beta_{2} - 22 \beta_1 + 250) q^{14} + ( - 11 \beta_{2} + 68 \beta_1 - 161) q^{16} + ( - 39 \beta_{2} - 37 \beta_1 + 801) q^{17} + ( - 29 \beta_{2} - 67 \beta_1 - 935) q^{19} + ( - 25 \beta_{2} - 100 \beta_1 - 175) q^{20} + (121 \beta_1 + 242) q^{22} + (20 \beta_{2} - 220 \beta_1 - 684) q^{23} + 625 q^{25} + ( - 108 \beta_{2} - 734 \beta_1 - 896) q^{26} + ( - 12 \beta_{2} + 92 \beta_1 + 1500) q^{28} + ( - 77 \beta_{2} - 107 \beta_1 + 2615) q^{29} + (38 \beta_{2} + 130 \beta_1 + 146) q^{31} + ( - 189 \beta_{2} - 212 \beta_1 - 263) q^{32} + ( - 154 \beta_{2} + 64 \beta_1 + 814) q^{34} + ( - 25 \beta_{2} - 275 \beta_1 + 1525) q^{35} + (272 \beta_{2} - 528 \beta_1 - 2866) q^{37} + ( - 154 \beta_{2} - 1562 \beta_1 - 3838) q^{38} + ( - 175 \beta_{2} - 1925) q^{40} + (473 \beta_{2} - 977 \beta_1 + 3245) q^{41} + (341 \beta_{2} - 809 \beta_1 - 4621) q^{43} + (121 \beta_{2} + 484 \beta_1 + 847) q^{44} + ( - 160 \beta_{2} - 784 \beta_1 - 9328) q^{46} + (422 \beta_{2} - 222 \beta_1 + 6538) q^{47} + (4 \beta_{2} - 964 \beta_1 - 8555) q^{49} + (625 \beta_1 + 1250) q^{50} + ( - 162 \beta_{2} - 3432 \beta_1 - 19358) q^{52} + ( - 586 \beta_{2} + 962 \beta_1 + 1212) q^{53} - 3025 q^{55} + ( - 392 \beta_{2} + 2184 \beta_1 - 1624) q^{56} + ( - 338 \beta_{2} + 1092 \beta_1 + 2486) q^{58} + ( - 356 \beta_{2} + 2748 \beta_1 + 2200) q^{59} + (364 \beta_{2} - 3300 \beta_1 - 18926) q^{61} + (244 \beta_{2} + 1052 \beta_1 + 4348) q^{62} + ( - 427 \beta_{2} - 6076 \beta_1 - 337) q^{64} + (700 \beta_{2} + 600 \beta_1 + 5250) q^{65} + (680 \beta_{2} - 2144 \beta_1 - 12108) q^{67} + (850 \beta_{2} - 492 \beta_1 - 19762) q^{68} + ( - 350 \beta_{2} + 550 \beta_1 - 6250) q^{70} + ( - 980 \beta_{2} - 2612 \beta_1 + 25548) q^{71} + (428 \beta_{2} + 2808 \beta_1 - 15750) q^{73} + (288 \beta_{2} + 702 \beta_1 - 27748) q^{74} + ( - 1096 \beta_{2} - 7436 \beta_1 - 30424) q^{76} + (121 \beta_{2} + 1331 \beta_1 - 7381) q^{77} + ( - 775 \beta_{2} - 5417 \beta_1 - 34233) q^{79} + (275 \beta_{2} - 1700 \beta_1 + 4025) q^{80} + (442 \beta_{2} + 9332 \beta_1 - 33854) q^{82} + ( - 1474 \beta_{2} - 14234 \beta_1 + 32042) q^{83} + (975 \beta_{2} + 925 \beta_1 - 20025) q^{85} + (214 \beta_{2} - 442 \beta_1 - 41990) q^{86} + (847 \beta_{2} + 9317) q^{88} + (132 \beta_{2} + 2140 \beta_1 - 56494) q^{89} + (1378 \beta_{2} - 6602 \beta_1 - 8410) q^{91} + ( - 1904 \beta_{2} - 6576 \beta_1 - 22128) q^{92} + (1044 \beta_{2} + 13268 \beta_1 - 180) q^{94} + (725 \beta_{2} + 1675 \beta_1 + 23375) q^{95} + ( - 1664 \beta_{2} - 8760 \beta_1 + 56154) q^{97} + ( - 952 \beta_{2} - 10415 \beta_1 - 50902) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 7 q^{2} + 25 q^{4} - 75 q^{5} - 172 q^{7} + 231 q^{8} - 175 q^{10} + 363 q^{11} - 654 q^{13} + 728 q^{14} - 415 q^{16} + 2366 q^{17} - 2872 q^{19} - 625 q^{20} + 847 q^{22} - 2272 q^{23} + 1875 q^{25} - 3422 q^{26} + 4592 q^{28} + 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 2506 q^{34} + 4300 q^{35} - 9126 q^{37} - 13076 q^{38} - 5775 q^{40} + 8758 q^{41} - 14672 q^{43} + 3025 q^{44} - 28768 q^{46} + 19392 q^{47} - 26629 q^{49} + 4375 q^{50} - 61506 q^{52} + 4598 q^{53} - 9075 q^{55} - 2688 q^{56} + 8550 q^{58} + 9348 q^{59} - 60078 q^{61} + 14096 q^{62} - 7087 q^{64} + 16350 q^{65} - 38468 q^{67} - 59778 q^{68} - 18200 q^{70} + 74032 q^{71} - 44442 q^{73} - 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 108116 q^{79} + 10375 q^{80} - 92230 q^{82} + 81892 q^{83} - 59150 q^{85} - 126412 q^{86} + 27951 q^{88} - 167342 q^{89} - 31832 q^{91} - 72960 q^{92} + 12728 q^{94} + 71800 q^{95} + 159702 q^{97} - 163121 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 52x + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 35 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 35 \) 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
−7.17710
0.921799
7.25531
−5.17710 0 −5.19759 −25.0000 0 −123.437 192.576 0 129.428
1.2 2.92180 0 −23.4631 −25.0000 0 −85.0105 −162.052 0 −73.0450
1.3 9.25531 0 53.6607 −25.0000 0 36.4478 200.476 0 −231.383
\(n\): 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 495.6.a.e 3
3.b odd 2 1 165.6.a.a 3
15.d odd 2 1 825.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.a 3 3.b odd 2 1
495.6.a.e 3 1.a even 1 1 trivial
825.6.a.j 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 7T_{2}^{2} - 36T_{2} + 140 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 7 T^{2} - 36 T + 140 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 172 T^{2} + 2896 T - 382464 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 654 T^{2} + \cdots - 317918392 \) Copy content Toggle raw display
$17$ \( T^{3} - 2366 T^{2} + \cdots + 137826264 \) Copy content Toggle raw display
$19$ \( T^{3} + 2872 T^{2} + \cdots + 11543616 \) Copy content Toggle raw display
$23$ \( T^{3} + 2272 T^{2} + \cdots - 3706904576 \) Copy content Toggle raw display
$29$ \( T^{3} - 7738 T^{2} + \cdots - 5220625848 \) Copy content Toggle raw display
$31$ \( T^{3} - 568 T^{2} + \cdots - 289787904 \) Copy content Toggle raw display
$37$ \( T^{3} + 9126 T^{2} + \cdots - 129972509048 \) Copy content Toggle raw display
$41$ \( T^{3} - 8758 T^{2} + \cdots + 1122652557432 \) Copy content Toggle raw display
$43$ \( T^{3} + 14672 T^{2} + \cdots - 518908872384 \) Copy content Toggle raw display
$47$ \( T^{3} - 19392 T^{2} + \cdots + 1508908531200 \) Copy content Toggle raw display
$53$ \( T^{3} - 4598 T^{2} + \cdots - 728896505288 \) Copy content Toggle raw display
$59$ \( T^{3} - 9348 T^{2} + \cdots + 6267836310080 \) Copy content Toggle raw display
$61$ \( T^{3} + 60078 T^{2} + \cdots - 13500896397400 \) Copy content Toggle raw display
$67$ \( T^{3} + 38468 T^{2} + \cdots - 8479952260160 \) Copy content Toggle raw display
$71$ \( T^{3} - 74032 T^{2} + \cdots + 17011990639616 \) Copy content Toggle raw display
$73$ \( T^{3} + 44442 T^{2} + \cdots - 9750515676328 \) Copy content Toggle raw display
$79$ \( T^{3} + 108116 T^{2} + \cdots + 9069370346752 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 739830059345664 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 158914472576552 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 352998320493112 \) Copy content Toggle raw display
show more
show less