# Properties

 Label 495.2.k.a Level 495 Weight 2 Character orbit 495.k Analytic conductor 3.953 Analytic rank 0 Dimension 4 CM discriminant -11 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} +O(q^{10})$$ $$q -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -2 \beta_{1} + \beta_{2} ) q^{11} -4 q^{16} + ( -2 + 2 \beta_{3} ) q^{20} + ( -4 + \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{23} + ( 2 + 3 \beta_{3} ) q^{25} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{31} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 + 4 \beta_{3} ) q^{44} + ( 5 + 2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{47} -7 \beta_{2} q^{49} + ( 1 - 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 7 + 3 \beta_{3} ) q^{55} + ( -1 - 2 \beta_{3} ) q^{59} + 8 \beta_{2} q^{64} + ( 5 + 3 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} ) q^{67} + 3 q^{71} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{80} + 9 \beta_{2} q^{89} + ( -10 + 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -10 + 3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 16q^{16} - 12q^{20} - 18q^{23} + 2q^{25} - 14q^{37} + 24q^{47} + 12q^{53} + 22q^{55} + 26q^{67} + 12q^{71} - 36q^{92} - 34q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + 2 \beta_{1}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
208.1
 1.65831 + 0.500000i −1.65831 + 0.500000i 1.65831 − 0.500000i −1.65831 − 0.500000i
0 0 2.00000i −1.65831 1.50000i 0 0 0 0 0
208.2 0 0 2.00000i 1.65831 1.50000i 0 0 0 0 0
307.1 0 0 2.00000i −1.65831 + 1.50000i 0 0 0 0 0
307.2 0 0 2.00000i 1.65831 + 1.50000i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
5.c odd 4 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.k.a 4
3.b odd 2 1 55.2.e.b 4
5.c odd 4 1 inner 495.2.k.a 4
11.b odd 2 1 CM 495.2.k.a 4
12.b even 2 1 880.2.bd.d 4
15.d odd 2 1 275.2.e.a 4
15.e even 4 1 55.2.e.b 4
15.e even 4 1 275.2.e.a 4
33.d even 2 1 55.2.e.b 4
33.f even 10 4 605.2.m.a 16
33.h odd 10 4 605.2.m.a 16
55.e even 4 1 inner 495.2.k.a 4
60.l odd 4 1 880.2.bd.d 4
132.d odd 2 1 880.2.bd.d 4
165.d even 2 1 275.2.e.a 4
165.l odd 4 1 55.2.e.b 4
165.l odd 4 1 275.2.e.a 4
165.u odd 20 4 605.2.m.a 16
165.v even 20 4 605.2.m.a 16
660.q even 4 1 880.2.bd.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 3.b odd 2 1
55.2.e.b 4 15.e even 4 1
55.2.e.b 4 33.d even 2 1
55.2.e.b 4 165.l odd 4 1
275.2.e.a 4 15.d odd 2 1
275.2.e.a 4 15.e even 4 1
275.2.e.a 4 165.d even 2 1
275.2.e.a 4 165.l odd 4 1
495.2.k.a 4 1.a even 1 1 trivial
495.2.k.a 4 5.c odd 4 1 inner
495.2.k.a 4 11.b odd 2 1 CM
495.2.k.a 4 55.e even 4 1 inner
605.2.m.a 16 33.f even 10 4
605.2.m.a 16 33.h odd 10 4
605.2.m.a 16 165.u odd 20 4
605.2.m.a 16 165.v even 20 4
880.2.bd.d 4 12.b even 2 1
880.2.bd.d 4 60.l odd 4 1
880.2.bd.d 4 132.d odd 2 1
880.2.bd.d 4 660.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2}$$
$3$ 
$5$ $$1 - T^{2} + 25 T^{4}$$
$7$ $$( 1 + 49 T^{4} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 + 169 T^{4} )^{2}$$
$17$ $$( 1 + 289 T^{4} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 + 9 T + 23 T^{2} )^{2}( 1 + 35 T^{2} + 529 T^{4} )$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 - 37 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 7 T + 37 T^{2} )^{2}( 1 - 25 T^{2} + 1369 T^{4} )$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 + 1849 T^{4} )^{2}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}( 1 + 50 T^{2} + 2209 T^{4} )$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}( 1 - 70 T^{2} + 2809 T^{4} )$$
$59$ $$( 1 - 15 T + 59 T^{2} )^{2}( 1 + 15 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 - 13 T + 67 T^{2} )^{2}( 1 + 35 T^{2} + 4489 T^{4} )$$
$71$ $$( 1 - 3 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 79 T^{2} )^{4}$$
$83$ $$( 1 + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 97 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 17 T + 97 T^{2} )^{2}( 1 + 95 T^{2} + 9409 T^{4} )$$