Properties

Label 1575.2.bc.a
Level $1575$
Weight $2$
Character orbit 1575.bc
Analytic conductor $12.576$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + ( 3 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + ( 3 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{11} + ( 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{13} + ( 3 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{14} + ( 4 - 4 \zeta_{24}^{4} ) q^{16} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( -1 - \zeta_{24}^{4} ) q^{19} -2 \zeta_{24}^{6} q^{22} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{23} + ( 6 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{26} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{29} + ( 2 - \zeta_{24}^{4} ) q^{31} + ( -4 + 8 \zeta_{24}^{4} ) q^{34} + \zeta_{24}^{2} q^{37} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{38} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{41} + \zeta_{24}^{6} q^{43} + 8 \zeta_{24}^{4} q^{46} + ( -10 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{47} + ( 8 - 3 \zeta_{24}^{4} ) q^{49} + ( 2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{53} + ( 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{56} + 4 \zeta_{24}^{2} q^{58} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{59} + ( -2 - 2 \zeta_{24}^{4} ) q^{61} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{62} + 8 q^{64} + ( 11 \zeta_{24}^{2} - 11 \zeta_{24}^{6} ) q^{67} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{71} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{73} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{74} + ( \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{77} + ( 5 - 5 \zeta_{24}^{4} ) q^{79} + ( 6 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{82} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{83} + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{86} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{88} + ( 4 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{89} + ( 15 - 3 \zeta_{24}^{4} ) q^{91} + ( -20 + 10 \zeta_{24}^{4} ) q^{94} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{97} + ( -3 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{16} - 12q^{19} + 12q^{31} + 32q^{46} + 52q^{49} - 24q^{61} + 64q^{64} + 20q^{79} + 108q^{91} - 120q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
899.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.707107 1.22474i 0 0 0 0 −2.59808 0.500000i −2.82843 0 0
899.2 −0.707107 1.22474i 0 0 0 0 2.59808 + 0.500000i −2.82843 0 0
899.3 0.707107 + 1.22474i 0 0 0 0 −2.59808 0.500000i 2.82843 0 0
899.4 0.707107 + 1.22474i 0 0 0 0 2.59808 + 0.500000i 2.82843 0 0
1349.1 −0.707107 + 1.22474i 0 0 0 0 −2.59808 + 0.500000i −2.82843 0 0
1349.2 −0.707107 + 1.22474i 0 0 0 0 2.59808 0.500000i −2.82843 0 0
1349.3 0.707107 1.22474i 0 0 0 0 −2.59808 + 0.500000i 2.82843 0 0
1349.4 0.707107 1.22474i 0 0 0 0 2.59808 0.500000i 2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.4
Significant digits:
Format:

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
7.d odd 6 1 inner
15.d odd 2 1 inner
21.g even 6 1 inner
35.i odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.bc.a 8
3.b odd 2 1 inner 1575.2.bc.a 8
5.b even 2 1 inner 1575.2.bc.a 8
5.c odd 4 1 63.2.p.a 4
5.c odd 4 1 1575.2.bk.c 4
7.d odd 6 1 inner 1575.2.bc.a 8
15.d odd 2 1 inner 1575.2.bc.a 8
15.e even 4 1 63.2.p.a 4
15.e even 4 1 1575.2.bk.c 4
20.e even 4 1 1008.2.bt.b 4
21.g even 6 1 inner 1575.2.bc.a 8
35.f even 4 1 441.2.p.a 4
35.i odd 6 1 inner 1575.2.bc.a 8
35.k even 12 1 63.2.p.a 4
35.k even 12 1 441.2.c.a 4
35.k even 12 1 1575.2.bk.c 4
35.l odd 12 1 441.2.c.a 4
35.l odd 12 1 441.2.p.a 4
45.k odd 12 1 567.2.i.d 4
45.k odd 12 1 567.2.s.d 4
45.l even 12 1 567.2.i.d 4
45.l even 12 1 567.2.s.d 4
60.l odd 4 1 1008.2.bt.b 4
105.k odd 4 1 441.2.p.a 4
105.p even 6 1 inner 1575.2.bc.a 8
105.w odd 12 1 63.2.p.a 4
105.w odd 12 1 441.2.c.a 4
105.w odd 12 1 1575.2.bk.c 4
105.x even 12 1 441.2.c.a 4
105.x even 12 1 441.2.p.a 4
140.w even 12 1 7056.2.k.b 4
140.x odd 12 1 1008.2.bt.b 4
140.x odd 12 1 7056.2.k.b 4
315.bs even 12 1 567.2.s.d 4
315.bu odd 12 1 567.2.s.d 4
315.bw odd 12 1 567.2.i.d 4
315.cg even 12 1 567.2.i.d 4
420.bp odd 12 1 7056.2.k.b 4
420.br even 12 1 1008.2.bt.b 4
420.br even 12 1 7056.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 5.c odd 4 1
63.2.p.a 4 15.e even 4 1
63.2.p.a 4 35.k even 12 1
63.2.p.a 4 105.w odd 12 1
441.2.c.a 4 35.k even 12 1
441.2.c.a 4 35.l odd 12 1
441.2.c.a 4 105.w odd 12 1
441.2.c.a 4 105.x even 12 1
441.2.p.a 4 35.f even 4 1
441.2.p.a 4 35.l odd 12 1
441.2.p.a 4 105.k odd 4 1
441.2.p.a 4 105.x even 12 1
567.2.i.d 4 45.k odd 12 1
567.2.i.d 4 45.l even 12 1
567.2.i.d 4 315.bw odd 12 1
567.2.i.d 4 315.cg even 12 1
567.2.s.d 4 45.k odd 12 1
567.2.s.d 4 45.l even 12 1
567.2.s.d 4 315.bs even 12 1
567.2.s.d 4 315.bu odd 12 1
1008.2.bt.b 4 20.e even 4 1
1008.2.bt.b 4 60.l odd 4 1
1008.2.bt.b 4 140.x odd 12 1
1008.2.bt.b 4 420.br even 12 1
1575.2.bc.a 8 1.a even 1 1 trivial
1575.2.bc.a 8 3.b odd 2 1 inner
1575.2.bc.a 8 5.b even 2 1 inner
1575.2.bc.a 8 7.d odd 6 1 inner
1575.2.bc.a 8 15.d odd 2 1 inner
1575.2.bc.a 8 21.g even 6 1 inner
1575.2.bc.a 8 35.i odd 6 1 inner
1575.2.bc.a 8 105.p even 6 1 inner
1575.2.bk.c 4 5.c odd 4 1
1575.2.bk.c 4 15.e even 4 1
1575.2.bk.c 4 35.k even 12 1
1575.2.bk.c 4 105.w odd 12 1
7056.2.k.b 4 140.w even 12 1
7056.2.k.b 4 140.x odd 12 1
7056.2.k.b 4 420.bp odd 12 1
7056.2.k.b 4 420.br even 12 1

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{2} + 4 \)
\( T_{11}^{4} - 2 T_{11}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 49 - 13 T^{2} + T^{4} )^{2} \)
$11$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$13$ \( ( -27 + T^{2} )^{4} \)
$17$ \( ( 576 - 24 T^{2} + T^{4} )^{2} \)
$19$ \( ( 3 + 3 T + T^{2} )^{4} \)
$23$ \( ( 1024 + 32 T^{2} + T^{4} )^{2} \)
$29$ \( ( 8 + T^{2} )^{4} \)
$31$ \( ( 3 - 3 T + T^{2} )^{4} \)
$37$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$41$ \( ( -54 + T^{2} )^{4} \)
$43$ \( ( 1 + T^{2} )^{4} \)
$47$ \( ( 22500 - 150 T^{2} + T^{4} )^{2} \)
$53$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$59$ \( ( 576 + 24 T^{2} + T^{4} )^{2} \)
$61$ \( ( 12 + 6 T + T^{2} )^{4} \)
$67$ \( ( 14641 - 121 T^{2} + T^{4} )^{2} \)
$71$ \( ( 50 + T^{2} )^{4} \)
$73$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$79$ \( ( 25 - 5 T + T^{2} )^{4} \)
$83$ \( ( 54 + T^{2} )^{4} \)
$89$ \( ( 576 + 24 T^{2} + T^{4} )^{2} \)
$97$ \( ( -108 + T^{2} )^{4} \)
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