# Properties

 Label 1575.2.d.k Level $1575$ Weight $2$ Character orbit 1575.d Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( -1 + 2 \beta_{2} ) q^{11} -2 \beta_{1} q^{13} -\beta_{2} q^{14} + 3 \beta_{2} q^{16} -4 \beta_{1} q^{17} + ( 2 + 4 \beta_{2} ) q^{19} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( 2 - 2 \beta_{2} ) q^{26} + ( \beta_{1} + \beta_{3} ) q^{28} + 5 q^{29} + 6 \beta_{2} q^{31} + ( \beta_{1} + 5 \beta_{3} ) q^{32} + ( 4 - 4 \beta_{2} ) q^{34} + 3 \beta_{3} q^{37} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{38} + ( -6 + 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{43} + ( 1 - \beta_{2} ) q^{44} + ( -2 - 3 \beta_{2} ) q^{46} + 2 \beta_{3} q^{47} - q^{49} -2 \beta_{3} q^{52} + ( 4 \beta_{1} + 6 \beta_{3} ) q^{53} + ( -1 - 2 \beta_{2} ) q^{56} + 5 \beta_{1} q^{58} + ( 8 + 6 \beta_{2} ) q^{59} + ( -6 - 6 \beta_{2} ) q^{61} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{62} + ( -1 + 2 \beta_{2} ) q^{64} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{67} -4 \beta_{3} q^{68} + ( -5 - 6 \beta_{2} ) q^{71} + ( -2 \beta_{1} + 10 \beta_{3} ) q^{73} -3 \beta_{2} q^{74} + ( 6 + 2 \beta_{2} ) q^{76} + ( 2 \beta_{1} - \beta_{3} ) q^{77} + ( 5 + 10 \beta_{2} ) q^{79} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{82} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -2 + 5 \beta_{2} ) q^{86} + ( -4 \beta_{1} + 3 \beta_{3} ) q^{88} + ( 16 + 2 \beta_{2} ) q^{89} + 2 \beta_{2} q^{91} + ( 5 \beta_{1} + 7 \beta_{3} ) q^{92} -2 \beta_{2} q^{94} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} - 8q^{11} + 2q^{14} - 6q^{16} + 12q^{26} + 20q^{29} - 12q^{31} + 24q^{34} - 28q^{41} + 6q^{44} - 2q^{46} - 4q^{49} + 20q^{59} - 12q^{61} - 8q^{64} - 8q^{71} + 6q^{74} + 20q^{76} - 18q^{86} + 60q^{89} - 4q^{91} + 4q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1324.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 0 −0.618034 0 0 1.00000i 2.23607i 0 0
1324.2 0.618034i 0 1.61803 0 0 1.00000i 2.23607i 0 0
1324.3 0.618034i 0 1.61803 0 0 1.00000i 2.23607i 0 0
1324.4 1.61803i 0 −0.618034 0 0 1.00000i 2.23607i 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.k 4
3.b odd 2 1 175.2.b.c 4
5.b even 2 1 inner 1575.2.d.k 4
5.c odd 4 1 1575.2.a.n 2
5.c odd 4 1 1575.2.a.s 2
12.b even 2 1 2800.2.g.s 4
15.d odd 2 1 175.2.b.c 4
15.e even 4 1 175.2.a.d 2
15.e even 4 1 175.2.a.e yes 2
21.c even 2 1 1225.2.b.k 4
60.h even 2 1 2800.2.g.s 4
60.l odd 4 1 2800.2.a.bh 2
60.l odd 4 1 2800.2.a.bp 2
105.g even 2 1 1225.2.b.k 4
105.k odd 4 1 1225.2.a.n 2
105.k odd 4 1 1225.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 15.e even 4 1
175.2.a.e yes 2 15.e even 4 1
175.2.b.c 4 3.b odd 2 1
175.2.b.c 4 15.d odd 2 1
1225.2.a.n 2 105.k odd 4 1
1225.2.a.u 2 105.k odd 4 1
1225.2.b.k 4 21.c even 2 1
1225.2.b.k 4 105.g even 2 1
1575.2.a.n 2 5.c odd 4 1
1575.2.a.s 2 5.c odd 4 1
1575.2.d.k 4 1.a even 1 1 trivial
1575.2.d.k 4 5.b even 2 1 inner
2800.2.a.bh 2 60.l odd 4 1
2800.2.a.bp 2 60.l odd 4 1
2800.2.g.s 4 12.b even 2 1
2800.2.g.s 4 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 3 T_{2}^{2} + 1$$ $$T_{11}^{2} + 4 T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -1 + 4 T + T^{2} )^{2}$$
$13$ $$16 + 12 T^{2} + T^{4}$$
$17$ $$256 + 48 T^{2} + T^{4}$$
$19$ $$( -20 + T^{2} )^{2}$$
$23$ $$121 + 42 T^{2} + T^{4}$$
$29$ $$( -5 + T )^{4}$$
$31$ $$( -36 + 6 T + T^{2} )^{2}$$
$37$ $$( 9 + T^{2} )^{2}$$
$41$ $$( 44 + 14 T + T^{2} )^{2}$$
$43$ $$121 + 42 T^{2} + T^{4}$$
$47$ $$( 4 + T^{2} )^{2}$$
$53$ $$16 + 72 T^{2} + T^{4}$$
$59$ $$( -20 - 10 T + T^{2} )^{2}$$
$61$ $$( -36 + 6 T + T^{2} )^{2}$$
$67$ $$1 + 18 T^{2} + T^{4}$$
$71$ $$( -41 + 4 T + T^{2} )^{2}$$
$73$ $$13456 + 252 T^{2} + T^{4}$$
$79$ $$( -125 + T^{2} )^{2}$$
$83$ $$1936 + 92 T^{2} + T^{4}$$
$89$ $$( 220 - 30 T + T^{2} )^{2}$$
$97$ $$16 + 28 T^{2} + T^{4}$$