# Properties

 Label 2475.2.c.c Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} -3 i q^{7} + 3 i q^{8} +O(q^{10})$$ $$q + i q^{2} + q^{4} -3 i q^{7} + 3 i q^{8} - q^{11} -2 i q^{13} + 3 q^{14} - q^{16} + 3 i q^{17} + q^{19} -i q^{22} -i q^{23} + 2 q^{26} -3 i q^{28} + 6 q^{29} + 4 q^{31} + 5 i q^{32} -3 q^{34} + i q^{37} + i q^{38} + 5 q^{41} -4 i q^{43} - q^{44} + q^{46} + 3 i q^{47} -2 q^{49} -2 i q^{52} -10 i q^{53} + 9 q^{56} + 6 i q^{58} + 11 q^{59} + 14 q^{61} + 4 i q^{62} -7 q^{64} + 2 i q^{67} + 3 i q^{68} + 5 q^{71} -2 i q^{73} - q^{74} + q^{76} + 3 i q^{77} -5 q^{79} + 5 i q^{82} -8 i q^{83} + 4 q^{86} -3 i q^{88} -10 q^{89} -6 q^{91} -i q^{92} -3 q^{94} -17 i q^{97} -2 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{11} + 6q^{14} - 2q^{16} + 2q^{19} + 4q^{26} + 12q^{29} + 8q^{31} - 6q^{34} + 10q^{41} - 2q^{44} + 2q^{46} - 4q^{49} + 18q^{56} + 22q^{59} + 28q^{61} - 14q^{64} + 10q^{71} - 2q^{74} + 2q^{76} - 10q^{79} + 8q^{86} - 20q^{89} - 12q^{91} - 6q^{94} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
1.00000i 0 1.00000 0 0 3.00000i 3.00000i 0 0
199.2 1.00000i 0 1.00000 0 0 3.00000i 3.00000i 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.c 2
3.b odd 2 1 2475.2.c.e 2
5.b even 2 1 inner 2475.2.c.c 2
5.c odd 4 1 2475.2.a.d yes 1
5.c odd 4 1 2475.2.a.h yes 1
15.d odd 2 1 2475.2.c.e 2
15.e even 4 1 2475.2.a.b 1
15.e even 4 1 2475.2.a.k yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.b 1 15.e even 4 1
2475.2.a.d yes 1 5.c odd 4 1
2475.2.a.h yes 1 5.c odd 4 1
2475.2.a.k yes 1 15.e even 4 1
2475.2.c.c 2 1.a even 1 1 trivial
2475.2.c.c 2 5.b even 2 1 inner
2475.2.c.e 2 3.b odd 2 1
2475.2.c.e 2 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ $$T_{7}^{2} + 9$$ $$T_{29} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$9 + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$( -5 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$9 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -11 + T )^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( -5 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 5 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$289 + T^{2}$$