Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1575.cp (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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.99225.1
Artin image $C_8.A_4$
Artin field Galois closure of 32.0.2294081869728198830572049522437155246734619140625.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-\zeta_{24}^{4}\) \(\zeta_{24}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i 0.707107 + 0.707107i 0 0 0.500000 0.866025i 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
193.2 0.258819 + 0.965926i −0.707107 0.707107i 0 0 0.500000 0.866025i −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
268.1 −0.965926 0.258819i −0.707107 0.707107i 0 0 0.500000 + 0.866025i −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
268.2 0.965926 + 0.258819i 0.707107 + 0.707107i 0 0 0.500000 + 0.866025i 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
382.1 −0.965926 + 0.258819i −0.707107 + 0.707107i 0 0 0.500000 0.866025i −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
382.2 0.965926 0.258819i 0.707107 0.707107i 0 0 0.500000 0.866025i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0
457.1 −0.258819 + 0.965926i 0.707107 0.707107i 0 0 0.500000 + 0.866025i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0
457.2 0.258819 0.965926i −0.707107 + 0.707107i 0 0 0.500000 + 0.866025i −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 457.2
Significant digits:
Format:

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

Hecke kernels

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

Hecke characteristic polynomials

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