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 $4$

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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:

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

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