# Properties

 Label 1575.2.d.c 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} -i q^{7} +O(q^{10})$$ $$q + 2 q^{4} -i q^{7} + 3 q^{11} + 5 i q^{13} + 4 q^{16} + 3 i q^{17} -2 q^{19} + 6 i q^{23} -2 i q^{28} + 3 q^{29} -4 q^{31} -2 i q^{37} + 12 q^{41} -10 i q^{43} + 6 q^{44} + 9 i q^{47} - q^{49} + 10 i q^{52} -12 i q^{53} + 8 q^{61} + 8 q^{64} + 4 i q^{67} + 6 i q^{68} + 2 i q^{73} -4 q^{76} -3 i q^{77} + q^{79} -12 i q^{83} -12 q^{89} + 5 q^{91} + 12 i q^{92} + i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} + O(q^{10})$$ $$2q + 4q^{4} + 6q^{11} + 8q^{16} - 4q^{19} + 6q^{29} - 8q^{31} + 24q^{41} + 12q^{44} - 2q^{49} + 16q^{61} + 16q^{64} - 8q^{76} + 2q^{79} - 24q^{89} + 10q^{91} + O(q^{100})$$

## 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
0 0 2.00000 0 0 1.00000i 0 0 0
1324.2 0 0 2.00000 0 0 1.00000i 0 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.c 2
3.b odd 2 1 175.2.b.a 2
5.b even 2 1 inner 1575.2.d.c 2
5.c odd 4 1 315.2.a.b 1
5.c odd 4 1 1575.2.a.f 1
12.b even 2 1 2800.2.g.l 2
15.d odd 2 1 175.2.b.a 2
15.e even 4 1 35.2.a.a 1
15.e even 4 1 175.2.a.b 1
20.e even 4 1 5040.2.a.v 1
21.c even 2 1 1225.2.b.d 2
35.f even 4 1 2205.2.a.e 1
60.h even 2 1 2800.2.g.l 2
60.l odd 4 1 560.2.a.b 1
60.l odd 4 1 2800.2.a.z 1
105.g even 2 1 1225.2.b.d 2
105.k odd 4 1 245.2.a.c 1
105.k odd 4 1 1225.2.a.e 1
105.w odd 12 2 245.2.e.b 2
105.x even 12 2 245.2.e.a 2
120.q odd 4 1 2240.2.a.u 1
120.w even 4 1 2240.2.a.k 1
165.l odd 4 1 4235.2.a.c 1
195.s even 4 1 5915.2.a.f 1
420.w even 4 1 3920.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 15.e even 4 1
175.2.a.b 1 15.e even 4 1
175.2.b.a 2 3.b odd 2 1
175.2.b.a 2 15.d odd 2 1
245.2.a.c 1 105.k odd 4 1
245.2.e.a 2 105.x even 12 2
245.2.e.b 2 105.w odd 12 2
315.2.a.b 1 5.c odd 4 1
560.2.a.b 1 60.l odd 4 1
1225.2.a.e 1 105.k odd 4 1
1225.2.b.d 2 21.c even 2 1
1225.2.b.d 2 105.g even 2 1
1575.2.a.f 1 5.c odd 4 1
1575.2.d.c 2 1.a even 1 1 trivial
1575.2.d.c 2 5.b even 2 1 inner
2205.2.a.e 1 35.f even 4 1
2240.2.a.k 1 120.w even 4 1
2240.2.a.u 1 120.q odd 4 1
2800.2.a.z 1 60.l odd 4 1
2800.2.g.l 2 12.b even 2 1
2800.2.g.l 2 60.h even 2 1
3920.2.a.ba 1 420.w even 4 1
4235.2.a.c 1 165.l odd 4 1
5040.2.a.v 1 20.e even 4 1
5915.2.a.f 1 195.s even 4 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$25 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$36 + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -12 + T )^{2}$$
$43$ $$100 + T^{2}$$
$47$ $$81 + T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -1 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$1 + T^{2}$$