# Properties

 Label 1575.2.b.a Level $1575$ Weight $2$ Character orbit 1575.b Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -2 + \beta_{2} ) q^{4} + \beta_{2} q^{7} + ( -2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -2 + \beta_{2} ) q^{4} + \beta_{2} q^{7} + ( -2 \beta_{1} + \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( 3 - 2 \beta_{2} ) q^{16} + ( -5 - \beta_{2} ) q^{22} + ( -3 \beta_{1} + \beta_{3} ) q^{23} + ( 7 - 2 \beta_{2} ) q^{28} + ( -3 \beta_{1} - \beta_{3} ) q^{29} + 3 \beta_{1} q^{32} -4 \beta_{2} q^{37} + 2 \beta_{2} q^{43} + ( -\beta_{1} + \beta_{3} ) q^{44} + ( 11 - 5 \beta_{2} ) q^{46} + 7 q^{49} + ( 5 \beta_{1} - \beta_{3} ) q^{53} + 7 \beta_{1} q^{56} + ( 13 - \beta_{2} ) q^{58} + ( -6 - \beta_{2} ) q^{64} + 4 q^{67} + ( -\beta_{1} + 3 \beta_{3} ) q^{71} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{74} + ( \beta_{1} + 3 \beta_{3} ) q^{77} + 8 q^{79} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{86} + ( -7 - 5 \beta_{2} ) q^{88} + ( 15 \beta_{1} - 3 \beta_{3} ) q^{92} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} + 12q^{16} - 20q^{22} + 28q^{28} + 44q^{46} + 28q^{49} + 52q^{58} - 24q^{64} + 16q^{67} + 32q^{79} - 28q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 − 2.57794i − 1.16372i 1.16372i 2.57794i
2.57794i 0 −4.64575 0 0 −2.64575 6.82058i 0 0
251.2 1.16372i 0 0.645751 0 0 2.64575 3.07892i 0 0
251.3 1.16372i 0 0.645751 0 0 2.64575 3.07892i 0 0
251.4 2.57794i 0 −4.64575 0 0 −2.64575 6.82058i 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c 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.b.a 4
3.b odd 2 1 inner 1575.2.b.a 4
5.b even 2 1 63.2.c.a 4
5.c odd 4 2 1575.2.g.d 8
7.b odd 2 1 CM 1575.2.b.a 4
15.d odd 2 1 63.2.c.a 4
15.e even 4 2 1575.2.g.d 8
20.d odd 2 1 1008.2.k.a 4
21.c even 2 1 inner 1575.2.b.a 4
35.c odd 2 1 63.2.c.a 4
35.f even 4 2 1575.2.g.d 8
35.i odd 6 2 441.2.p.b 8
35.j even 6 2 441.2.p.b 8
40.e odd 2 1 4032.2.k.b 4
40.f even 2 1 4032.2.k.c 4
45.h odd 6 2 567.2.o.f 8
45.j even 6 2 567.2.o.f 8
60.h even 2 1 1008.2.k.a 4
105.g even 2 1 63.2.c.a 4
105.k odd 4 2 1575.2.g.d 8
105.o odd 6 2 441.2.p.b 8
105.p even 6 2 441.2.p.b 8
120.i odd 2 1 4032.2.k.c 4
120.m even 2 1 4032.2.k.b 4
140.c even 2 1 1008.2.k.a 4
280.c odd 2 1 4032.2.k.c 4
280.n even 2 1 4032.2.k.b 4
315.z even 6 2 567.2.o.f 8
315.bg odd 6 2 567.2.o.f 8
420.o odd 2 1 1008.2.k.a 4
840.b odd 2 1 4032.2.k.b 4
840.u even 2 1 4032.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 5.b even 2 1
63.2.c.a 4 15.d odd 2 1
63.2.c.a 4 35.c odd 2 1
63.2.c.a 4 105.g even 2 1
441.2.p.b 8 35.i odd 6 2
441.2.p.b 8 35.j even 6 2
441.2.p.b 8 105.o odd 6 2
441.2.p.b 8 105.p even 6 2
567.2.o.f 8 45.h odd 6 2
567.2.o.f 8 45.j even 6 2
567.2.o.f 8 315.z even 6 2
567.2.o.f 8 315.bg odd 6 2
1008.2.k.a 4 20.d odd 2 1
1008.2.k.a 4 60.h even 2 1
1008.2.k.a 4 140.c even 2 1
1008.2.k.a 4 420.o odd 2 1
1575.2.b.a 4 1.a even 1 1 trivial
1575.2.b.a 4 3.b odd 2 1 inner
1575.2.b.a 4 7.b odd 2 1 CM
1575.2.b.a 4 21.c even 2 1 inner
1575.2.g.d 8 5.c odd 4 2
1575.2.g.d 8 15.e even 4 2
1575.2.g.d 8 35.f even 4 2
1575.2.g.d 8 105.k odd 4 2
4032.2.k.b 4 40.e odd 2 1
4032.2.k.b 4 120.m even 2 1
4032.2.k.b 4 280.n even 2 1
4032.2.k.b 4 840.b odd 2 1
4032.2.k.c 4 40.f even 2 1
4032.2.k.c 4 120.i odd 2 1
4032.2.k.c 4 280.c odd 2 1
4032.2.k.c 4 840.u even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 8 T_{2}^{2} + 9$$ $$T_{37}^{2} - 112$$ $$T_{47}$$ $$T_{67} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 8 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$36 + 44 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$324 + 92 T^{2} + T^{4}$$
$29$ $$2916 + 116 T^{2} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$( -112 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -28 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$36 + 212 T^{2} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$( -4 + T )^{4}$$
$71$ $$12996 + 284 T^{2} + T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -8 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$