# Properties

 Label 1575.2.d.b Level $1575$ Weight $2$ Character orbit 1575.d Analytic conductor $12.576$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.5764383184$$ 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 105) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} -i q^{7} + 3 i q^{8} +O(q^{10})$$ $$q + i q^{2} + q^{4} -i q^{7} + 3 i q^{8} -6 i q^{13} + q^{14} - q^{16} + 2 i q^{17} + 8 q^{19} -8 i q^{23} + 6 q^{26} -i q^{28} -2 q^{29} + 4 q^{31} + 5 i q^{32} -2 q^{34} + 2 i q^{37} + 8 i q^{38} + 6 q^{41} + 4 i q^{43} + 8 q^{46} + 8 i q^{47} - q^{49} -6 i q^{52} -10 i q^{53} + 3 q^{56} -2 i q^{58} + 4 q^{59} -2 q^{61} + 4 i q^{62} -7 q^{64} -4 i q^{67} + 2 i q^{68} + 12 q^{71} -2 i q^{73} -2 q^{74} + 8 q^{76} -8 q^{79} + 6 i q^{82} + 4 i q^{83} -4 q^{86} -6 q^{89} -6 q^{91} -8 i q^{92} -8 q^{94} + 18 i q^{97} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} + 2q^{14} - 2q^{16} + 16q^{19} + 12q^{26} - 4q^{29} + 8q^{31} - 4q^{34} + 12q^{41} + 16q^{46} - 2q^{49} + 6q^{56} + 8q^{59} - 4q^{61} - 14q^{64} + 24q^{71} - 4q^{74} + 16q^{76} - 16q^{79} - 8q^{86} - 12q^{89} - 12q^{91} - 16q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1324.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 0 0 1.00000i 3.00000i 0 0
1324.2 1.00000i 0 1.00000 0 0 1.00000i 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 1575.2.d.b 2
3.b odd 2 1 525.2.d.b 2
5.b even 2 1 inner 1575.2.d.b 2
5.c odd 4 1 315.2.a.a 1
5.c odd 4 1 1575.2.a.h 1
15.d odd 2 1 525.2.d.b 2
15.e even 4 1 105.2.a.a 1
15.e even 4 1 525.2.a.a 1
20.e even 4 1 5040.2.a.d 1
35.f even 4 1 2205.2.a.b 1
60.l odd 4 1 1680.2.a.f 1
60.l odd 4 1 8400.2.a.co 1
105.k odd 4 1 735.2.a.f 1
105.k odd 4 1 3675.2.a.f 1
105.w odd 12 2 735.2.i.b 2
105.x even 12 2 735.2.i.a 2
120.q odd 4 1 6720.2.a.bk 1
120.w even 4 1 6720.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 15.e even 4 1
315.2.a.a 1 5.c odd 4 1
525.2.a.a 1 15.e even 4 1
525.2.d.b 2 3.b odd 2 1
525.2.d.b 2 15.d odd 2 1
735.2.a.f 1 105.k odd 4 1
735.2.i.a 2 105.x even 12 2
735.2.i.b 2 105.w odd 12 2
1575.2.a.h 1 5.c odd 4 1
1575.2.d.b 2 1.a even 1 1 trivial
1575.2.d.b 2 5.b even 2 1 inner
1680.2.a.f 1 60.l odd 4 1
2205.2.a.b 1 35.f even 4 1
3675.2.a.f 1 105.k odd 4 1
5040.2.a.d 1 20.e even 4 1
6720.2.a.p 1 120.w even 4 1
6720.2.a.bk 1 120.q odd 4 1
8400.2.a.co 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1575, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -8 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$324 + T^{2}$$