# Properties

 Label 2475.2.c.j Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -3 + \beta_{3} ) q^{4} + \beta_{2} q^{7} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -3 + \beta_{3} ) q^{4} + \beta_{2} q^{7} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} + q^{11} + ( \beta_{1} + 3 \beta_{2} ) q^{13} + ( 1 - \beta_{3} ) q^{14} + ( 3 - 3 \beta_{3} ) q^{16} + ( -\beta_{1} - 4 \beta_{2} ) q^{17} -3 q^{19} + \beta_{1} q^{22} + ( -\beta_{1} + 4 \beta_{2} ) q^{23} + ( -2 - 2 \beta_{3} ) q^{26} + ( \beta_{1} - 2 \beta_{2} ) q^{28} + ( 2 - 2 \beta_{3} ) q^{29} -3 \beta_{3} q^{31} + ( \beta_{1} - 4 \beta_{2} ) q^{32} + ( 1 + 3 \beta_{3} ) q^{34} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{37} -3 \beta_{1} q^{38} + ( 3 - 3 \beta_{3} ) q^{41} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -3 + \beta_{3} ) q^{44} + ( 9 - 5 \beta_{3} ) q^{46} + ( -\beta_{1} - 8 \beta_{2} ) q^{47} + 6 q^{49} -2 \beta_{2} q^{52} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{53} + ( -5 + \beta_{3} ) q^{56} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{58} + ( 9 - \beta_{3} ) q^{59} + ( -6 + \beta_{3} ) q^{61} -12 \beta_{2} q^{62} + ( -3 - \beta_{3} ) q^{64} + ( 3 \beta_{1} + \beta_{2} ) q^{67} + ( -\beta_{1} + 4 \beta_{2} ) q^{68} + ( -1 + 5 \beta_{3} ) q^{71} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{73} + ( -23 + 3 \beta_{3} ) q^{74} + ( 9 - 3 \beta_{3} ) q^{76} + \beta_{2} q^{77} + ( 11 - 3 \beta_{3} ) q^{79} + ( 3 \beta_{1} - 12 \beta_{2} ) q^{82} -12 q^{86} + ( -\beta_{1} + 4 \beta_{2} ) q^{88} + ( -14 + 2 \beta_{3} ) q^{89} + ( -2 - \beta_{3} ) q^{91} + ( 7 \beta_{1} - 12 \beta_{2} ) q^{92} + ( -3 + 7 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{97} + 6 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{4} + O(q^{10})$$ $$4q - 10q^{4} + 4q^{11} + 2q^{14} + 6q^{16} - 12q^{19} - 12q^{26} + 4q^{29} - 6q^{31} + 10q^{34} + 6q^{41} - 10q^{44} + 26q^{46} + 24q^{49} - 18q^{56} + 34q^{59} - 22q^{61} - 14q^{64} + 6q^{71} - 86q^{74} + 30q^{76} + 38q^{79} - 48q^{86} - 52q^{89} - 10q^{91} + 2q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
199.2 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.3 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.4 2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.c.j 4
3.b odd 2 1 2475.2.c.i 4
5.b even 2 1 inner 2475.2.c.j 4
5.c odd 4 1 2475.2.a.q yes 2
5.c odd 4 1 2475.2.a.u yes 2
15.d odd 2 1 2475.2.c.i 4
15.e even 4 1 2475.2.a.p 2
15.e even 4 1 2475.2.a.v yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.p 2 15.e even 4 1
2475.2.a.q yes 2 5.c odd 4 1
2475.2.a.u yes 2 5.c odd 4 1
2475.2.a.v yes 2 15.e even 4 1
2475.2.c.i 4 3.b odd 2 1
2475.2.c.i 4 15.d odd 2 1
2475.2.c.j 4 1.a even 1 1 trivial
2475.2.c.j 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{4} + 9 T_{2}^{2} + 16$$ $$T_{7}^{2} + 1$$ $$T_{29}^{2} - 2 T_{29} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 9 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$4 + 21 T^{2} + T^{4}$$
$17$ $$64 + 33 T^{2} + T^{4}$$
$19$ $$( 3 + T )^{4}$$
$23$ $$256 + 49 T^{2} + T^{4}$$
$29$ $$( -16 - 2 T + T^{2} )^{2}$$
$31$ $$( -36 + 3 T + T^{2} )^{2}$$
$37$ $$11236 + 213 T^{2} + T^{4}$$
$41$ $$( -36 - 3 T + T^{2} )^{2}$$
$43$ $$1296 + 81 T^{2} + T^{4}$$
$47$ $$2704 + 121 T^{2} + T^{4}$$
$53$ $$64 + 52 T^{2} + T^{4}$$
$59$ $$( 68 - 17 T + T^{2} )^{2}$$
$61$ $$( 26 + 11 T + T^{2} )^{2}$$
$67$ $$1444 + 77 T^{2} + T^{4}$$
$71$ $$( -104 - 3 T + T^{2} )^{2}$$
$73$ $$16 + 264 T^{2} + T^{4}$$
$79$ $$( 52 - 19 T + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$( 152 + 26 T + T^{2} )^{2}$$
$97$ $$169 + 42 T^{2} + T^{4}$$