# Properties

 Label 1575.2.d.i Level $1575$ Weight $2$ Character orbit 1575.d Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{12}^{2} ) q^{2} - q^{4} -\zeta_{12}^{3} q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{12}^{2} ) q^{2} - q^{4} -\zeta_{12}^{3} q^{7} + ( -1 + 2 \zeta_{12}^{2} ) q^{8} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{11} + 2 \zeta_{12}^{3} q^{13} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{14} -5 q^{16} + ( -2 + 4 \zeta_{12}^{2} ) q^{17} + 4 q^{19} -6 \zeta_{12}^{3} q^{22} + ( -2 + 4 \zeta_{12}^{2} ) q^{23} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{26} + \zeta_{12}^{3} q^{28} -4 q^{31} + ( 3 - 6 \zeta_{12}^{2} ) q^{32} -6 q^{34} -2 \zeta_{12}^{3} q^{37} + ( -4 + 8 \zeta_{12}^{2} ) q^{38} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{41} -4 \zeta_{12}^{3} q^{43} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{44} -6 q^{46} + ( -4 + 8 \zeta_{12}^{2} ) q^{47} - q^{49} -2 \zeta_{12}^{3} q^{52} + ( -4 + 8 \zeta_{12}^{2} ) q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{56} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{59} -10 q^{61} + ( 4 - 8 \zeta_{12}^{2} ) q^{62} - q^{64} + 4 \zeta_{12}^{3} q^{67} + ( 2 - 4 \zeta_{12}^{2} ) q^{68} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{71} + 14 \zeta_{12}^{3} q^{73} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{74} -4 q^{76} + ( -2 + 4 \zeta_{12}^{2} ) q^{77} -8 q^{79} -18 \zeta_{12}^{3} q^{82} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{86} -6 \zeta_{12}^{3} q^{88} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{89} + 2 q^{91} + ( 2 - 4 \zeta_{12}^{2} ) q^{92} -12 q^{94} -14 \zeta_{12}^{3} q^{97} + ( 1 - 2 \zeta_{12}^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} - 20q^{16} + 16q^{19} - 16q^{31} - 24q^{34} - 24q^{46} - 4q^{49} - 40q^{61} - 4q^{64} - 16q^{76} - 32q^{79} + 8q^{91} - 48q^{94} + O(q^{100})$$

## 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
 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
1.73205i 0 −1.00000 0 0 1.00000i 1.73205i 0 0
1324.2 1.73205i 0 −1.00000 0 0 1.00000i 1.73205i 0 0
1324.3 1.73205i 0 −1.00000 0 0 1.00000i 1.73205i 0 0
1324.4 1.73205i 0 −1.00000 0 0 1.00000i 1.73205i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 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.i 4
3.b odd 2 1 inner 1575.2.d.i 4
5.b even 2 1 inner 1575.2.d.i 4
5.c odd 4 1 63.2.a.b 2
5.c odd 4 1 1575.2.a.q 2
15.d odd 2 1 inner 1575.2.d.i 4
15.e even 4 1 63.2.a.b 2
15.e even 4 1 1575.2.a.q 2
20.e even 4 1 1008.2.a.n 2
35.f even 4 1 441.2.a.g 2
35.k even 12 2 441.2.e.i 4
35.l odd 12 2 441.2.e.j 4
40.i odd 4 1 4032.2.a.bt 2
40.k even 4 1 4032.2.a.bq 2
45.k odd 12 2 567.2.f.j 4
45.l even 12 2 567.2.f.j 4
55.e even 4 1 7623.2.a.bi 2
60.l odd 4 1 1008.2.a.n 2
105.k odd 4 1 441.2.a.g 2
105.w odd 12 2 441.2.e.i 4
105.x even 12 2 441.2.e.j 4
120.q odd 4 1 4032.2.a.bq 2
120.w even 4 1 4032.2.a.bt 2
140.j odd 4 1 7056.2.a.cm 2
165.l odd 4 1 7623.2.a.bi 2
420.w even 4 1 7056.2.a.cm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 5.c odd 4 1
63.2.a.b 2 15.e even 4 1
441.2.a.g 2 35.f even 4 1
441.2.a.g 2 105.k odd 4 1
441.2.e.i 4 35.k even 12 2
441.2.e.i 4 105.w odd 12 2
441.2.e.j 4 35.l odd 12 2
441.2.e.j 4 105.x even 12 2
567.2.f.j 4 45.k odd 12 2
567.2.f.j 4 45.l even 12 2
1008.2.a.n 2 20.e even 4 1
1008.2.a.n 2 60.l odd 4 1
1575.2.a.q 2 5.c odd 4 1
1575.2.a.q 2 15.e even 4 1
1575.2.d.i 4 1.a even 1 1 trivial
1575.2.d.i 4 3.b odd 2 1 inner
1575.2.d.i 4 5.b even 2 1 inner
1575.2.d.i 4 15.d odd 2 1 inner
4032.2.a.bq 2 40.k even 4 1
4032.2.a.bq 2 120.q odd 4 1
4032.2.a.bt 2 40.i odd 4 1
4032.2.a.bt 2 120.w even 4 1
7056.2.a.cm 2 140.j odd 4 1
7056.2.a.cm 2 420.w even 4 1
7623.2.a.bi 2 55.e even 4 1
7623.2.a.bi 2 165.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 $$S_{2}^{\mathrm{new}}(1575, [\chi])$$:

 $$T_{2}^{2} + 3$$ $$T_{11}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 3 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( -12 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( 12 + T^{2} )^{2}$$
$19$ $$( -4 + T )^{4}$$
$23$ $$( 12 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( -108 + T^{2} )^{2}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( 48 + T^{2} )^{2}$$
$53$ $$( 48 + T^{2} )^{2}$$
$59$ $$( -48 + T^{2} )^{2}$$
$61$ $$( 10 + T )^{4}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -108 + T^{2} )^{2}$$
$73$ $$( 196 + T^{2} )^{2}$$
$79$ $$( 8 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$( -12 + T^{2} )^{2}$$
$97$ $$( 196 + T^{2} )^{2}$$