# Properties

 Label 1575.1.cb.a Level $1575$ Weight $1$ Character orbit 1575.cb Analytic conductor $0.786$ Analytic rank $0$ Dimension $8$ Projective image $A_{4}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1575.cb (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.786027394897$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.99225.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{9} q^{2} + \zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + \zeta_{24} q^{7} + \zeta_{24}^{3} q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{9} q^{2} + \zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + \zeta_{24} q^{7} + \zeta_{24}^{3} q^{8} -\zeta_{24}^{2} q^{9} -\zeta_{24}^{7} q^{13} -\zeta_{24}^{10} q^{14} + q^{16} + \zeta_{24}^{5} q^{17} + \zeta_{24}^{11} q^{18} + \zeta_{24}^{10} q^{19} + \zeta_{24}^{8} q^{21} + \zeta_{24}^{10} q^{24} -\zeta_{24}^{4} q^{26} -\zeta_{24}^{9} q^{27} -\zeta_{24}^{2} q^{29} - q^{31} + \zeta_{24}^{2} q^{34} + \zeta_{24} q^{37} + \zeta_{24}^{7} q^{38} + \zeta_{24}^{2} q^{39} -\zeta_{24}^{4} q^{41} + \zeta_{24}^{5} q^{42} -\zeta_{24}^{11} q^{43} + \zeta_{24}^{9} q^{47} + \zeta_{24}^{7} q^{48} + \zeta_{24}^{2} q^{49} - q^{51} -\zeta_{24}^{11} q^{53} -\zeta_{24}^{6} q^{54} + \zeta_{24}^{4} q^{56} -\zeta_{24}^{5} q^{57} + \zeta_{24}^{11} q^{58} + \zeta_{24}^{6} q^{59} + q^{61} + \zeta_{24}^{9} q^{62} -\zeta_{24}^{3} q^{63} + \zeta_{24}^{6} q^{64} -\zeta_{24}^{9} q^{67} -2 q^{71} -\zeta_{24}^{5} q^{72} + \zeta_{24}^{11} q^{73} -\zeta_{24}^{10} q^{74} -\zeta_{24}^{11} q^{78} -\zeta_{24}^{6} q^{79} + \zeta_{24}^{4} q^{81} -\zeta_{24} q^{82} + \zeta_{24}^{11} q^{83} -\zeta_{24}^{8} q^{86} -\zeta_{24}^{9} q^{87} -\zeta_{24}^{10} q^{89} -\zeta_{24}^{8} q^{91} -\zeta_{24}^{7} q^{93} + \zeta_{24}^{6} q^{94} -\zeta_{24}^{5} q^{97} -\zeta_{24}^{11} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{6} + O(q^{10})$$ $$8q + 4q^{6} + 8q^{16} - 4q^{21} - 4q^{26} - 8q^{31} - 4q^{41} - 8q^{51} + 4q^{56} + 8q^{61} - 16q^{71} + 4q^{81} + 4q^{86} + 4q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-\zeta_{24}^{6}$$ $$-\zeta_{24}^{4}$$ $$-\zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
718.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i
−0.707107 + 0.707107i 0.258819 0.965926i 0 0 0.500000 + 0.866025i −0.965926 0.258819i −0.707107 0.707107i −0.866025 0.500000i 0
718.2 0.707107 0.707107i −0.258819 + 0.965926i 0 0 0.500000 + 0.866025i 0.965926 + 0.258819i 0.707107 + 0.707107i −0.866025 0.500000i 0
907.1 −0.707107 0.707107i −0.965926 0.258819i 0 0 0.500000 + 0.866025i 0.258819 0.965926i −0.707107 + 0.707107i 0.866025 + 0.500000i 0
907.2 0.707107 + 0.707107i 0.965926 + 0.258819i 0 0 0.500000 + 0.866025i −0.258819 + 0.965926i 0.707107 0.707107i 0.866025 + 0.500000i 0
1318.1 −0.707107 + 0.707107i −0.965926 + 0.258819i 0 0 0.500000 0.866025i 0.258819 + 0.965926i −0.707107 0.707107i 0.866025 0.500000i 0
1318.2 0.707107 0.707107i 0.965926 0.258819i 0 0 0.500000 0.866025i −0.258819 0.965926i 0.707107 + 0.707107i 0.866025 0.500000i 0
1507.1 −0.707107 0.707107i 0.258819 + 0.965926i 0 0 0.500000 0.866025i −0.965926 + 0.258819i −0.707107 + 0.707107i −0.866025 + 0.500000i 0
1507.2 0.707107 + 0.707107i −0.258819 0.965926i 0 0 0.500000 0.866025i 0.965926 0.258819i 0.707107 0.707107i −0.866025 + 0.500000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1507.2 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
5.c odd 4 2 inner
63.h even 3 1 inner
315.r even 6 1 inner
315.bt odd 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.cb.a 8
5.b even 2 1 inner 1575.1.cb.a 8
5.c odd 4 2 inner 1575.1.cb.a 8
7.c even 3 1 1575.1.cp.a yes 8
9.c even 3 1 1575.1.cp.a yes 8
35.j even 6 1 1575.1.cp.a yes 8
35.l odd 12 2 1575.1.cp.a yes 8
45.j even 6 1 1575.1.cp.a yes 8
45.k odd 12 2 1575.1.cp.a yes 8
63.h even 3 1 inner 1575.1.cb.a 8
315.r even 6 1 inner 1575.1.cb.a 8
315.bt odd 12 2 inner 1575.1.cb.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.cb.a 8 1.a even 1 1 trivial
1575.1.cb.a 8 5.b even 2 1 inner
1575.1.cb.a 8 5.c odd 4 2 inner
1575.1.cb.a 8 63.h even 3 1 inner
1575.1.cb.a 8 315.r even 6 1 inner
1575.1.cb.a 8 315.bt odd 12 2 inner
1575.1.cp.a yes 8 7.c even 3 1
1575.1.cp.a yes 8 9.c even 3 1
1575.1.cp.a yes 8 35.j even 6 1
1575.1.cp.a yes 8 35.l odd 12 2
1575.1.cp.a yes 8 45.j even 6 1
1575.1.cp.a yes 8 45.k odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1575, [\chi])$$.

## Hecke characteristic polynomials

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