Properties

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

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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: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} + ( - 4 \beta_1 - 8) q^{4} - 25 q^{5} + ( - 7 \beta_{2} - 31 \beta_1 + 38) q^{7} + (28 \beta_{2} + 8 \beta_1 + 20) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} + ( - 4 \beta_1 - 8) q^{4} - 25 q^{5} + ( - 7 \beta_{2} - 31 \beta_1 + 38) q^{7} + (28 \beta_{2} + 8 \beta_1 + 20) q^{8} + (25 \beta_{2} + 25) q^{10} + 121 q^{11} + (2 \beta_{2} - 77 \beta_1 - 207) q^{13} + ( - 138 \beta_{2} + 34 \beta_1 - 32) q^{14} + (32 \beta_{2} + 224 \beta_1 - 368) q^{16} + ( - 119 \beta_{2} + 219 \beta_1 + 138) q^{17} + (149 \beta_{2} + 206 \beta_1 + 721) q^{19} + (100 \beta_1 + 200) q^{20} + ( - 121 \beta_{2} - 121) q^{22} + ( - 150 \beta_{2} + 190 \beta_1 - 1416) q^{23} + 625 q^{25} + ( - 22 \beta_{2} + 162 \beta_1 - 224) q^{26} + (220 \beta_{2} + 372 \beta_1 + 2160) q^{28} + ( - 129 \beta_{2} + 308 \beta_1 - 1801) q^{29} + ( - 32 \beta_{2} - 242 \beta_1 + 2018) q^{31} + (176 \beta_{2} - 576 \beta_1 + 112) q^{32} + (400 \beta_{2} - 914 \beta_1 + 3694) q^{34} + (175 \beta_{2} + 775 \beta_1 - 950) q^{35} + (1076 \beta_{2} + 928 \beta_1 + 6290) q^{37} + (46 \beta_{2} + 184 \beta_1 - 3118) q^{38} + ( - 700 \beta_{2} - 200 \beta_1 - 500) q^{40} + (781 \beta_{2} - 1892 \beta_1 - 8131) q^{41} + (1903 \beta_{2} + 949 \beta_1 + 7808) q^{43} + ( - 484 \beta_1 - 968) q^{44} + (1836 \beta_{2} - 980 \beta_1 + 5816) q^{46} + ( - 520 \beta_{2} + 598 \beta_1 - 1118) q^{47} + ( - 10 \beta_{2} - 196 \beta_1 + 3775) q^{49} + ( - 625 \beta_{2} - 625) q^{50} + (624 \beta_{2} + 2052 \beta_1 + 8164) q^{52} + (3174 \beta_{2} + 2146 \beta_1 - 3606) q^{53} - 3025 q^{55} + (3592 \beta_{2} - 952 \beta_1 - 4336) q^{56} + (2596 \beta_{2} - 1132 \beta_1 + 6308) q^{58} + (1822 \beta_{2} + 2954 \beta_1 - 5732) q^{59} + ( - 2786 \beta_{2} + 1694 \beta_1 + 2410) q^{61} + ( - 2776 \beta_{2} + 356 \beta_1 - 2492) q^{62} + ( - 2688 \beta_{2} - 5312 \beta_1 + 4736) q^{64} + ( - 50 \beta_{2} + 1925 \beta_1 + 5175) q^{65} + ( - 5936 \beta_{2} - 4430 \beta_1 - 31566) q^{67} + ( - 2228 \beta_{2} - 3580 \beta_1 - 21880) q^{68} + (3450 \beta_{2} - 850 \beta_1 + 800) q^{70} + (9528 \beta_{2} - 3010 \beta_1 - 14670) q^{71} + (1244 \beta_{2} + 12157 \beta_1 - 13479) q^{73} + ( - 2430 \beta_{2} + 2448 \beta_1 - 26398) q^{74} + ( - 1052 \beta_{2} - 6776 \beta_1 - 20092) q^{76} + ( - 847 \beta_{2} - 3751 \beta_1 + 4598) q^{77} + (3973 \beta_{2} - 770 \beta_1 + 3189) q^{79} + ( - 800 \beta_{2} - 5600 \beta_1 + 9200) q^{80} + (3236 \beta_{2} + 6908 \beta_1 - 19292) q^{82} + ( - 1382 \beta_{2} - 10151 \beta_1 - 35935) q^{83} + (2975 \beta_{2} - 5475 \beta_1 - 3450) q^{85} + ( - 3058 \beta_{2} + 5714 \beta_1 - 46832) q^{86} + (3388 \beta_{2} + 968 \beta_1 + 2420) q^{88} + (10502 \beta_{2} - 6468 \beta_1 + 14324) q^{89} + (4896 \beta_{2} + 8798 \beta_1 + 39554) q^{91} + ( - 2120 \beta_{2} + 3224 \beta_1 - 7632) q^{92} + (2392 \beta_{2} - 3276 \beta_1 + 16068) q^{94} + ( - 3725 \beta_{2} - 5150 \beta_1 - 18025) q^{95} + (3184 \beta_{2} - 20686 \beta_1 - 40248) q^{97} + ( - 4373 \beta_{2} + 352 \beta_1 - 4525) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 20 q^{4} - 75 q^{5} + 152 q^{7} + 24 q^{8} + 50 q^{10} + 363 q^{11} - 546 q^{13} + 8 q^{14} - 1360 q^{16} + 314 q^{17} + 1808 q^{19} + 500 q^{20} - 242 q^{22} - 4288 q^{23} + 1875 q^{25} - 812 q^{26} + 5888 q^{28} - 5582 q^{29} + 6328 q^{31} + 736 q^{32} + 11596 q^{34} - 3800 q^{35} + 16866 q^{37} - 9584 q^{38} - 600 q^{40} - 23282 q^{41} + 20572 q^{43} - 2420 q^{44} + 16592 q^{46} - 3432 q^{47} + 11531 q^{49} - 1250 q^{50} + 21816 q^{52} - 16138 q^{53} - 9075 q^{55} - 15648 q^{56} + 17460 q^{58} - 21972 q^{59} + 8322 q^{61} - 5056 q^{62} + 22208 q^{64} + 13650 q^{65} - 84332 q^{67} - 59832 q^{68} - 200 q^{70} - 50528 q^{71} - 53838 q^{73} - 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 6364 q^{79} + 34000 q^{80} - 68020 q^{82} - 96272 q^{83} - 7850 q^{85} - 143152 q^{86} + 2904 q^{88} + 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 49088 q^{94} - 45200 q^{95} - 103242 q^{97} - 9554 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 11 ) / 2 \) 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
−1.87740
3.35386
−0.476452
−6.55890 0 11.0192 −25.0000 0 146.487 137.611 0 163.973
1.2 −1.08127 0 −30.8308 −25.0000 0 −139.508 67.9374 0 27.0319
1.3 5.64018 0 −0.188384 −25.0000 0 145.021 −181.548 0 −141.004
\(n\): 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.c 3
3.b odd 2 1 165.6.a.d 3
15.d odd 2 1 825.6.a.h 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.d 3 3.b odd 2 1
495.6.a.c 3 1.a even 1 1 trivial
825.6.a.h 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 2T_{2}^{2} - 36T_{2} - 40 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 2 T^{2} - 36 T - 40 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 152 T^{2} - 19424 T + 2963664 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 546 T^{2} - 76748 T - 7691848 \) Copy content Toggle raw display
$17$ \( T^{3} - 314 T^{2} + \cdots + 1079472216 \) Copy content Toggle raw display
$19$ \( T^{3} - 1808 T^{2} + \cdots + 729480096 \) Copy content Toggle raw display
$23$ \( T^{3} + 4288 T^{2} + \cdots + 857355136 \) Copy content Toggle raw display
$29$ \( T^{3} + 5582 T^{2} + \cdots + 329440872 \) Copy content Toggle raw display
$31$ \( T^{3} - 6328 T^{2} + \cdots - 5126546304 \) Copy content Toggle raw display
$37$ \( T^{3} - 16866 T^{2} + \cdots + 244700027368 \) Copy content Toggle raw display
$41$ \( T^{3} + 23282 T^{2} + \cdots - 945181300968 \) Copy content Toggle raw display
$43$ \( T^{3} - 20572 T^{2} + \cdots + 1240285492944 \) Copy content Toggle raw display
$47$ \( T^{3} + 3432 T^{2} + \cdots + 18016103040 \) Copy content Toggle raw display
$53$ \( T^{3} + 16138 T^{2} + \cdots + 985333601848 \) Copy content Toggle raw display
$59$ \( T^{3} + 21972 T^{2} + \cdots - 2567903224000 \) Copy content Toggle raw display
$61$ \( T^{3} - 8322 T^{2} + \cdots + 4408473611240 \) Copy content Toggle raw display
$67$ \( T^{3} + 84332 T^{2} + \cdots - 41154720036800 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 117803610062464 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 163971863295832 \) Copy content Toggle raw display
$79$ \( T^{3} - 6364 T^{2} + \cdots - 554174036768 \) Copy content Toggle raw display
$83$ \( T^{3} + 96272 T^{2} + \cdots - 3029676562224 \) Copy content Toggle raw display
$89$ \( T^{3} - 38938 T^{2} + \cdots - 96218735089528 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 251540556847112 \) Copy content Toggle raw display
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