# Properties

 Label 1575.2.b.d Level 1575 Weight 2 Character orbit 1575.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{2} q^{7} -2 \beta_{1} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{2} q^{7} -2 \beta_{1} q^{8} -\beta_{1} q^{11} + ( 1 - 2 \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{3} ) q^{14} -4 q^{16} + ( \beta_{1} - 2 \beta_{3} ) q^{17} + ( -1 + 2 \beta_{2} ) q^{19} -2 q^{22} + 5 \beta_{1} q^{23} + ( \beta_{1} - 2 \beta_{3} ) q^{26} -\beta_{1} q^{29} + ( 1 - 2 \beta_{2} ) q^{31} + ( -2 + 4 \beta_{2} ) q^{34} + 8 q^{37} + ( -\beta_{1} + 2 \beta_{3} ) q^{38} + 5 q^{43} + 10 q^{46} + ( \beta_{1} - 2 \beta_{3} ) q^{47} + ( -7 + \beta_{2} ) q^{49} -4 \beta_{1} q^{53} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{56} -2 q^{58} + ( \beta_{1} - 2 \beta_{3} ) q^{59} + ( -1 + 2 \beta_{2} ) q^{61} + ( \beta_{1} - 2 \beta_{3} ) q^{62} -8 q^{64} -7 q^{67} + 2 \beta_{1} q^{71} + ( 2 - 4 \beta_{2} ) q^{73} -8 \beta_{1} q^{74} + ( \beta_{1} - \beta_{3} ) q^{77} + 14 q^{79} + ( -\beta_{1} + 2 \beta_{3} ) q^{83} -5 \beta_{1} q^{86} -4 q^{88} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{89} + ( -14 + \beta_{2} ) q^{91} + ( -2 + 4 \beta_{2} ) q^{94} + ( -1 + 2 \beta_{2} ) q^{97} + ( 6 \beta_{1} + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{7} + O(q^{10})$$ $$4q - 2q^{7} - 16q^{16} - 8q^{22} + 32q^{37} + 20q^{43} + 40q^{46} - 26q^{49} - 8q^{58} - 32q^{64} - 28q^{67} + 56q^{79} - 16q^{88} - 54q^{91} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.22474 + 0.707107i −1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i
1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.2 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 0 0
251.3 1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.4 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 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
7.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.d 4
3.b odd 2 1 inner 1575.2.b.d 4
5.b even 2 1 1575.2.b.e yes 4
5.c odd 4 2 1575.2.g.b 8
7.b odd 2 1 inner 1575.2.b.d 4
15.d odd 2 1 1575.2.b.e yes 4
15.e even 4 2 1575.2.g.b 8
21.c even 2 1 inner 1575.2.b.d 4
35.c odd 2 1 1575.2.b.e yes 4
35.f even 4 2 1575.2.g.b 8
105.g even 2 1 1575.2.b.e yes 4
105.k odd 4 2 1575.2.g.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.b.d 4 1.a even 1 1 trivial
1575.2.b.d 4 3.b odd 2 1 inner
1575.2.b.d 4 7.b odd 2 1 inner
1575.2.b.d 4 21.c even 2 1 inner
1575.2.b.e yes 4 5.b even 2 1
1575.2.b.e yes 4 15.d odd 2 1
1575.2.b.e yes 4 35.c odd 2 1
1575.2.b.e yes 4 105.g even 2 1
1575.2.g.b 8 5.c odd 4 2
1575.2.g.b 8 15.e even 4 2
1575.2.g.b 8 35.f even 4 2
1575.2.g.b 8 105.k odd 4 2

## Hecke kernels

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

 $$T_{2}^{2} + 2$$ $$T_{37} - 8$$ $$T_{47}^{2} - 54$$ $$T_{67} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ 1
$5$ 1
$7$ $$( 1 + T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2}$$
$17$ $$( 1 - 20 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 4 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 56 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 35 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 40 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 74 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 64 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 95 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 7 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 134 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 38 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 112 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 38 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2}$$