# Properties

 Label 1425.2.c.d Level $1425$ Weight $2$ Character orbit 1425.c Analytic conductor $11.379$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.3786822880$$ 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: no (minimal twist has level 285) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} + q^{4} - q^{6} -2 i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} + q^{4} - q^{6} -2 i q^{7} + 3 i q^{8} - q^{9} -2 q^{11} + i q^{12} + 4 i q^{13} + 2 q^{14} - q^{16} + 2 i q^{17} -i q^{18} + q^{19} + 2 q^{21} -2 i q^{22} + 4 i q^{23} -3 q^{24} -4 q^{26} -i q^{27} -2 i q^{28} -4 q^{29} + 5 i q^{32} -2 i q^{33} -2 q^{34} - q^{36} + i q^{38} -4 q^{39} + 2 i q^{42} + 10 i q^{43} -2 q^{44} -4 q^{46} + 12 i q^{47} -i q^{48} + 3 q^{49} -2 q^{51} + 4 i q^{52} + 2 i q^{53} + q^{54} + 6 q^{56} + i q^{57} -4 i q^{58} -4 q^{59} + 2 q^{61} + 2 i q^{63} -7 q^{64} + 2 q^{66} -16 i q^{67} + 2 i q^{68} -4 q^{69} -3 i q^{72} + 2 i q^{73} + q^{76} + 4 i q^{77} -4 i q^{78} + 8 q^{79} + q^{81} + 12 i q^{83} + 2 q^{84} -10 q^{86} -4 i q^{87} -6 i q^{88} + 8 q^{91} + 4 i q^{92} -12 q^{94} -5 q^{96} -16 i q^{97} + 3 i q^{98} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 4 q^{14} - 2 q^{16} + 2 q^{19} + 4 q^{21} - 6 q^{24} - 8 q^{26} - 8 q^{29} - 4 q^{34} - 2 q^{36} - 8 q^{39} - 4 q^{44} - 8 q^{46} + 6 q^{49} - 4 q^{51} + 2 q^{54} + 12 q^{56} - 8 q^{59} + 4 q^{61} - 14 q^{64} + 4 q^{66} - 8 q^{69} + 2 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 20 q^{86} + 16 q^{91} - 24 q^{94} - 10 q^{96} + 4 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 2.00000i 3.00000i −1.00000 0
799.2 1.00000i 1.00000i 1.00000 0 −1.00000 2.00000i 3.00000i −1.00000 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 1425.2.c.d 2
5.b even 2 1 inner 1425.2.c.d 2
5.c odd 4 1 285.2.a.b 1
5.c odd 4 1 1425.2.a.d 1
15.e even 4 1 855.2.a.b 1
15.e even 4 1 4275.2.a.o 1
20.e even 4 1 4560.2.a.v 1
95.g even 4 1 5415.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.b 1 5.c odd 4 1
855.2.a.b 1 15.e even 4 1
1425.2.a.d 1 5.c odd 4 1
1425.2.c.d 2 1.a even 1 1 trivial
1425.2.c.d 2 5.b even 2 1 inner
4275.2.a.o 1 15.e even 4 1
4560.2.a.v 1 20.e even 4 1
5415.2.a.c 1 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1425, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{7}^{2} + 4$$ $$T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$256 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$256 + T^{2}$$