Properties

 Label 495.2.d.a Level $495$ Weight $2$ Character orbit 495.d Analytic conductor $3.953$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{2} + q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{7} -3 \zeta_{8}^{2} q^{8} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{2} + q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{7} -3 \zeta_{8}^{2} q^{8} + ( -\zeta_{8} - 2 \zeta_{8}^{3} ) q^{10} + ( -3 + \zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{14} - q^{16} -2 \zeta_{8}^{2} q^{17} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{19} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{20} + ( \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{28} -6 q^{29} -5 \zeta_{8}^{2} q^{32} -2 q^{34} + ( 3 + \zeta_{8}^{2} ) q^{35} -6 \zeta_{8}^{2} q^{37} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{38} + ( -3 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{40} + 6 q^{41} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{43} + ( -3 + \zeta_{8} + \zeta_{8}^{3} ) q^{44} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{47} -5 q^{49} + ( 3 - 4 \zeta_{8}^{2} ) q^{50} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{52} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{53} + ( -1 - 6 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{55} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{56} + 6 \zeta_{8}^{2} q^{58} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{59} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{61} -7 q^{64} + ( 6 + 2 \zeta_{8}^{2} ) q^{65} -2 \zeta_{8}^{2} q^{68} + ( 1 - 3 \zeta_{8}^{2} ) q^{70} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{73} -6 q^{74} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{77} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{79} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} -6 \zeta_{8}^{2} q^{82} + 14 \zeta_{8}^{2} q^{83} + ( -2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{85} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{86} + ( 3 \zeta_{8} + 9 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{88} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{89} + 4 q^{91} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{94} + ( -3 + 9 \zeta_{8}^{2} ) q^{95} -12 \zeta_{8}^{2} q^{97} + 5 \zeta_{8}^{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + O(q^{10})$$ $$4 q + 4 q^{4} - 12 q^{11} - 4 q^{16} + 16 q^{25} - 24 q^{29} - 8 q^{34} + 12 q^{35} + 24 q^{41} - 12 q^{44} - 20 q^{49} + 12 q^{50} - 4 q^{55} - 28 q^{64} + 24 q^{65} + 4 q^{70} - 24 q^{74} + 16 q^{91} - 12 q^{95} + O(q^{100})$$

Character values

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

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
494.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000i 0 1.00000 −2.12132 0.707107i 0 −1.41421 3.00000i 0 −0.707107 + 2.12132i
494.2 1.00000i 0 1.00000 2.12132 + 0.707107i 0 1.41421 3.00000i 0 0.707107 2.12132i
494.3 1.00000i 0 1.00000 −2.12132 + 0.707107i 0 −1.41421 3.00000i 0 −0.707107 2.12132i
494.4 1.00000i 0 1.00000 2.12132 0.707107i 0 1.41421 3.00000i 0 0.707107 + 2.12132i
 $$n$$: 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
5.b even 2 1 inner
33.d even 2 1 inner
165.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.d.a 4
3.b odd 2 1 495.2.d.b yes 4
5.b even 2 1 inner 495.2.d.a 4
5.c odd 4 1 2475.2.f.a 2
5.c odd 4 1 2475.2.f.c 2
11.b odd 2 1 495.2.d.b yes 4
15.d odd 2 1 495.2.d.b yes 4
15.e even 4 1 2475.2.f.b 2
15.e even 4 1 2475.2.f.d 2
33.d even 2 1 inner 495.2.d.a 4
55.d odd 2 1 495.2.d.b yes 4
55.e even 4 1 2475.2.f.b 2
55.e even 4 1 2475.2.f.d 2
165.d even 2 1 inner 495.2.d.a 4
165.l odd 4 1 2475.2.f.a 2
165.l odd 4 1 2475.2.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.d.a 4 1.a even 1 1 trivial
495.2.d.a 4 5.b even 2 1 inner
495.2.d.a 4 33.d even 2 1 inner
495.2.d.a 4 165.d even 2 1 inner
495.2.d.b yes 4 3.b odd 2 1
495.2.d.b yes 4 11.b odd 2 1
495.2.d.b yes 4 15.d odd 2 1
495.2.d.b yes 4 55.d odd 2 1
2475.2.f.a 2 5.c odd 4 1
2475.2.f.a 2 165.l odd 4 1
2475.2.f.b 2 15.e even 4 1
2475.2.f.b 2 55.e even 4 1
2475.2.f.c 2 5.c odd 4 1
2475.2.f.c 2 165.l odd 4 1
2475.2.f.d 2 15.e even 4 1
2475.2.f.d 2 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{29} + 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$25 - 8 T^{2} + T^{4}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$( 11 + 6 T + T^{2} )^{2}$$
$13$ $$( -8 + T^{2} )^{2}$$
$17$ $$( 4 + T^{2} )^{2}$$
$19$ $$( 18 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( 36 + T^{2} )^{2}$$
$41$ $$( -6 + T )^{4}$$
$43$ $$( -2 + T^{2} )^{2}$$
$47$ $$( -72 + T^{2} )^{2}$$
$53$ $$( -18 + T^{2} )^{2}$$
$59$ $$( 72 + T^{2} )^{2}$$
$61$ $$( 72 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( 72 + T^{2} )^{2}$$
$73$ $$( -128 + T^{2} )^{2}$$
$79$ $$( 162 + T^{2} )^{2}$$
$83$ $$( 196 + T^{2} )^{2}$$
$89$ $$( 50 + T^{2} )^{2}$$
$97$ $$( 144 + T^{2} )^{2}$$