Properties

Label 495.2.n.b
Level $495$
Weight $2$
Character orbit 495.n
Analytic conductor $3.953$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.819390625.1
Defining polynomial: \( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{7} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{4} + \beta_{3} q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{7} + 2 \beta_{6} - 3 \beta_{3} - 3 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{7} + \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{4} + \beta_{3} q^{5} + (\beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{7} + 2 \beta_{6} - 3 \beta_{3} - 3 \beta_{2}) q^{8} - \beta_{5} q^{10} + (\beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 1) q^{11} + ( - 3 \beta_{6} - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4) q^{13} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 4 \beta_{3} + 4 \beta_{2}) q^{14} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{16} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{19} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{20} + (2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 2) q^{22} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} - 2 \beta_1 + 1) q^{23} - \beta_{6} q^{25} + (\beta_{3} + 4 \beta_{2} - 4 \beta_1 + 1) q^{26} + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 3 \beta_1 + 1) q^{28} + ( - \beta_{7} + \beta_{4} - 4 \beta_{3} - 7 \beta_{2} - \beta_1 - 4) q^{29} + (6 \beta_{6} - \beta_{5} - 3 \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{31} + ( - 2 \beta_{7} - \beta_{6} + \beta_{2} - 2 \beta_1 + 4) q^{32} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 8) q^{34} + ( - 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{35} + ( - 5 \beta_{3} - 3 \beta_{2} - \beta_1 - 5) q^{37} + (\beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 3) q^{38} + (3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{40} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{41} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{2} + \beta_1 - 5) q^{43} + ( - 3 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} + 5 \beta_{2} - 2) q^{44} + (\beta_{6} + \beta_{4} - \beta_{3} + 6 \beta_{2} + 6) q^{46} + ( - 3 \beta_{7} + 6 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} - 5 \beta_{2}) q^{47} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 2) q^{49} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_1) q^{50} + (2 \beta_{7} + 6 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{52} + ( - 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 2) q^{53} + (\beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{55} + (9 \beta_{6} + 6 \beta_{5} - 9 \beta_{2} - 3) q^{56} + ( - 6 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - 3 \beta_{2}) q^{58} + (2 \beta_{7} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{59} + (2 \beta_{6} - 3 \beta_{4} - 3 \beta_{3} + 3 \beta_1 - 2) q^{61} + ( - \beta_{7} + \beta_{4} - \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 1) q^{62} + (2 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{64} + (\beta_{6} - \beta_{5} - \beta_{2} - 4) q^{65} + ( - 5 \beta_{6} - \beta_{5} + 5 \beta_{2} + 4) q^{67} + (5 \beta_{6} + \beta_{5} + 7 \beta_{4} - 5 \beta_{3} - \beta_1) q^{68} + ( - \beta_{7} + \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 4) q^{70} + (2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} + 5 \beta_1 - 4) q^{71} + ( - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 14 \beta_{2} - 3 \beta_1 - 2) q^{73} + ( - 3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} + 6 \beta_{4} + \beta_{3} + \beta_{2}) q^{74} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} - 2 \beta_1 - 4) q^{76} + (5 \beta_{7} - 3 \beta_{3} - 6 \beta_{2} + 2 \beta_1 - 5) q^{77} + ( - 6 \beta_{6} + 4 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{79} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{80} + ( - 4 \beta_{7} + 4 \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + \beta_{3} - 3 \beta_1 - 4) q^{82} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 3 \beta_1 + 2) q^{83} + ( - 2 \beta_{7} - \beta_{6}) q^{85} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 5) q^{86} + (2 \beta_{7} - 4 \beta_{6} + \beta_{5} + 5 \beta_{4} - \beta_{3} - 5 \beta_1 + 4) q^{88} + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{2} + 2 \beta_1 - 1) q^{89} + (3 \beta_{7} - 4 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} + 6 \beta_{2}) q^{91} + (4 \beta_{7} - 4 \beta_{4} + \beta_{3} - 6 \beta_{2} + 1) q^{92} + (2 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} - 7 \beta_{3} + 5 \beta_1 + 6) q^{94} + (\beta_{7} - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{95} + ( - 5 \beta_{6} - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 6 \beta_{2} + \beta_1 + 6) q^{97} + ( - 10 \beta_{6} - 7 \beta_{5} + 10 \beta_{2} + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 9 q^{7} + 19 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 9 q^{7} + 19 q^{8} - 2 q^{10} + 3 q^{11} + 10 q^{13} - 24 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{19} - 7 q^{20} - 7 q^{22} + 2 q^{23} - 2 q^{25} - 14 q^{26} + 13 q^{28} - 14 q^{29} - 5 q^{31} + 16 q^{32} - 70 q^{34} - q^{35} - 27 q^{37} + 16 q^{38} + 19 q^{40} - q^{41} - 28 q^{43} - 47 q^{44} + 42 q^{46} + 27 q^{47} - 15 q^{49} - 2 q^{50} + 22 q^{52} + q^{53} - 7 q^{55} + 24 q^{56} + 18 q^{58} - 13 q^{59} - 3 q^{61} - 15 q^{62} + 19 q^{64} - 30 q^{65} + 10 q^{67} + 33 q^{68} - 24 q^{70} - 9 q^{71} + 5 q^{73} + 17 q^{74} - 46 q^{76} - q^{77} - 10 q^{79} - 11 q^{80} - 33 q^{82} + 25 q^{83} - 8 q^{85} - 20 q^{86} + 29 q^{88} - 4 q^{89} - 43 q^{91} + 22 q^{92} + 57 q^{94} - 2 q^{95} + 13 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 10x^{6} - 13x^{5} + 29x^{4} - 7x^{3} + 80x^{2} + 143x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -256\nu^{7} + 341\nu^{6} + 3310\nu^{5} - 16865\nu^{4} + 32996\nu^{3} - 59433\nu^{2} + 33270\nu - 118459 ) / 171589 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -620\nu^{7} + 2532\nu^{6} - 9045\nu^{5} + 18870\nu^{4} - 41955\nu^{3} + 81515\nu^{2} - 102225\nu - 16104 ) / 171589 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -672\nu^{7} + 2845\nu^{6} - 10810\nu^{5} + 23975\nu^{4} - 77175\nu^{3} + 52625\nu^{2} - 72556\nu - 75020 ) / 171589 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1187 \nu^{7} + 4445 \nu^{6} - 17191 \nu^{5} + 14238 \nu^{4} - 12015 \nu^{3} - 32510 \nu^{2} - 25003 \nu - 143748 ) / 171589 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1188 \nu^{7} - 4751 \nu^{6} + 16325 \nu^{5} - 32635 \nu^{4} + 48690 \nu^{3} - 20331 \nu^{2} + 62530 \nu + 144881 ) / 171589 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1432 \nu^{7} + 1420 \nu^{6} - 7808 \nu^{5} - 2207 \nu^{4} - 27965 \nu^{3} - 33635 \nu^{2} - 187297 \nu - 249744 ) / 171589 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + 3\beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} - 5\beta_{4} + 2\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 15\beta_{6} - 8\beta_{5} - 8\beta_{4} - 5\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 25\beta_{6} - 28\beta_{5} + 25\beta_{2} - 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -55\beta_{7} + 55\beta_{4} + 22\beta_{3} + 110\beta_{2} - 45\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{7} + 175\beta_{6} + 165\beta_{5} + 188\beta_{4} - 80\beta_{3} - 188\beta _1 - 175 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0.766388 2.35870i
−0.575405 + 1.77091i
0.766388 + 2.35870i
−0.575405 1.77091i
2.06426 1.49977i
−0.755243 + 0.548716i
2.06426 + 1.49977i
−0.755243 0.548716i
−2.00643 + 1.45776i 0 1.28267 3.94765i −0.809017 0.587785i 0 −1.35808 + 4.17973i 1.64835 + 5.07311i 0 2.48008
91.2 1.50643 1.09448i 0 0.453397 1.39541i −0.809017 0.587785i 0 0.812990 2.50213i 0.306561 + 0.943499i 0 −1.86205
136.1 −2.00643 1.45776i 0 1.28267 + 3.94765i −0.809017 + 0.587785i 0 −1.35808 4.17973i 1.64835 5.07311i 0 2.48008
136.2 1.50643 + 1.09448i 0 0.453397 + 1.39541i −0.809017 + 0.587785i 0 0.812990 + 2.50213i 0.306561 0.943499i 0 −1.86205
181.1 −0.788477 2.42668i 0 −3.64906 + 2.65120i 0.309017 0.951057i 0 3.39382 2.46575i 5.18229 + 3.76516i 0 −2.55157
181.2 0.288477 + 0.887841i 0 0.912991 0.663327i 0.309017 0.951057i 0 1.65127 1.19972i 2.36279 + 1.71667i 0 0.933531
361.1 −0.788477 + 2.42668i 0 −3.64906 2.65120i 0.309017 + 0.951057i 0 3.39382 + 2.46575i 5.18229 3.76516i 0 −2.55157
361.2 0.288477 0.887841i 0 0.912991 + 0.663327i 0.309017 + 0.951057i 0 1.65127 + 1.19972i 2.36279 1.71667i 0 0.933531
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.n.b 8
3.b odd 2 1 165.2.m.b 8
11.c even 5 1 inner 495.2.n.b 8
11.c even 5 1 5445.2.a.bk 4
11.d odd 10 1 5445.2.a.br 4
15.d odd 2 1 825.2.n.i 8
15.e even 4 2 825.2.bx.g 16
33.f even 10 1 1815.2.a.r 4
33.h odd 10 1 165.2.m.b 8
33.h odd 10 1 1815.2.a.v 4
165.o odd 10 1 825.2.n.i 8
165.o odd 10 1 9075.2.a.cq 4
165.r even 10 1 9075.2.a.dg 4
165.v even 20 2 825.2.bx.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.b 8 3.b odd 2 1
165.2.m.b 8 33.h odd 10 1
495.2.n.b 8 1.a even 1 1 trivial
495.2.n.b 8 11.c even 5 1 inner
825.2.n.i 8 15.d odd 2 1
825.2.n.i 8 165.o odd 10 1
825.2.bx.g 16 15.e even 4 2
825.2.bx.g 16 165.v even 20 2
1815.2.a.r 4 33.f even 10 1
1815.2.a.v 4 33.h odd 10 1
5445.2.a.bk 4 11.c even 5 1
5445.2.a.br 4 11.d odd 10 1
9075.2.a.cq 4 165.o odd 10 1
9075.2.a.dg 4 165.r even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 5T_{2}^{6} - 3T_{2}^{5} + 4T_{2}^{4} + 3T_{2}^{3} + 135T_{2}^{2} - 77T_{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + 5 T^{6} - 3 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 9 T^{7} + 55 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + 8 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + 57 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + 21 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( (T^{4} - T^{3} - 39 T^{2} + 29 T + 341)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 14 T^{7} + 235 T^{6} + \cdots + 383161 \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + 95 T^{6} + 1025 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( T^{8} + 27 T^{7} + 415 T^{6} + \cdots + 1324801 \) Copy content Toggle raw display
$41$ \( T^{8} + T^{7} + 45 T^{6} + 279 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( (T^{4} + 14 T^{3} + 50 T^{2} + 57 T + 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 27 T^{7} + 423 T^{6} + \cdots + 12313081 \) Copy content Toggle raw display
$53$ \( T^{8} - T^{7} + 50 T^{6} + \cdots + 2927521 \) Copy content Toggle raw display
$59$ \( T^{8} + 13 T^{7} + 103 T^{6} + \cdots + 9801 \) Copy content Toggle raw display
$61$ \( T^{8} + 3 T^{7} + 110 T^{6} + \cdots + 1234321 \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} - 74 T^{2} + 180 T + 1049)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 9 T^{7} + 269 T^{6} + \cdots + 674041 \) Copy content Toggle raw display
$73$ \( T^{8} - 5 T^{7} + 318 T^{6} + \cdots + 22801 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + 407 T^{6} + \cdots + 707281 \) Copy content Toggle raw display
$83$ \( T^{8} - 25 T^{7} + 335 T^{6} + \cdots + 3900625 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} - 55 T^{2} - 156 T + 99)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 13 T^{7} + 160 T^{6} + \cdots + 793881 \) Copy content Toggle raw display
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