Properties

Label 975.2.bp.d
Level $975$
Weight $2$
Character orbit 975.bp
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bp (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - \zeta_{12}^{2} ) q^{3} -2 \zeta_{12} q^{4} + ( 1 - \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( 3 - 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( 2 - \zeta_{12}^{2} ) q^{3} -2 \zeta_{12} q^{4} + ( 1 - \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{7} + ( 3 - 3 \zeta_{12}^{2} ) q^{9} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{12} + ( 1 + 3 \zeta_{12}^{2} ) q^{13} + 4 \zeta_{12}^{2} q^{16} + ( -5 + 5 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( 4 + \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{21} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( 6 - 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{28} + ( -1 + 5 \zeta_{12} - 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{31} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{36} + ( 4 + 7 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{37} + ( 5 + 2 \zeta_{12}^{2} ) q^{39} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{43} + ( 4 + 4 \zeta_{12}^{2} ) q^{48} + ( -6 - 7 \zeta_{12} + 3 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{49} + ( -2 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{52} + ( -7 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{57} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{61} + ( 9 + 6 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( 7 + 2 \zeta_{12} - 9 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{67} + ( 8 - \zeta_{12} + \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{73} + ( -4 + 10 \zeta_{12} - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{76} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 8 - 8 \zeta_{12} - 10 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{84} + ( -5 - 10 \zeta_{12} + 11 \zeta_{12}^{2} + 9 \zeta_{12}^{3} ) q^{91} + ( -7 + 11 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{93} + ( -3 - 3 \zeta_{12} - 8 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{3} + 8q^{7} + 6q^{9} + O(q^{10}) \) \( 4q + 6q^{3} + 8q^{7} + 6q^{9} + 10q^{13} + 8q^{16} - 14q^{19} + 18q^{21} + 16q^{28} - 14q^{31} + 22q^{37} + 24q^{39} + 24q^{48} - 18q^{49} - 12q^{57} + 30q^{63} + 10q^{67} + 34q^{73} - 28q^{76} - 18q^{81} + 12q^{84} + 2q^{91} - 36q^{93} - 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{12}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 1.50000 + 0.866025i 1.73205 1.00000i 0 0 2.86603 + 0.767949i 0 1.50000 + 2.59808i 0
449.1 0 1.50000 + 0.866025i −1.73205 + 1.00000i 0 0 1.13397 4.23205i 0 1.50000 + 2.59808i 0
674.1 0 1.50000 0.866025i 1.73205 + 1.00000i 0 0 2.86603 0.767949i 0 1.50000 2.59808i 0
899.1 0 1.50000 0.866025i −1.73205 1.00000i 0 0 1.13397 + 4.23205i 0 1.50000 2.59808i 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 CM by \(\Q(\sqrt{-3}) \)
65.s odd 12 1 inner
195.bh even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bp.d 4
3.b odd 2 1 CM 975.2.bp.d 4
5.b even 2 1 975.2.bp.a 4
5.c odd 4 1 39.2.k.a 4
5.c odd 4 1 975.2.bo.c 4
13.f odd 12 1 975.2.bp.a 4
15.d odd 2 1 975.2.bp.a 4
15.e even 4 1 39.2.k.a 4
15.e even 4 1 975.2.bo.c 4
20.e even 4 1 624.2.cn.b 4
39.k even 12 1 975.2.bp.a 4
60.l odd 4 1 624.2.cn.b 4
65.f even 4 1 507.2.k.b 4
65.h odd 4 1 507.2.k.c 4
65.k even 4 1 507.2.k.a 4
65.o even 12 1 507.2.f.b 4
65.o even 12 1 507.2.k.c 4
65.o even 12 1 975.2.bo.c 4
65.q odd 12 1 507.2.f.c 4
65.q odd 12 1 507.2.k.b 4
65.r odd 12 1 507.2.f.b 4
65.r odd 12 1 507.2.k.a 4
65.s odd 12 1 inner 975.2.bp.d 4
65.t even 12 1 39.2.k.a 4
65.t even 12 1 507.2.f.c 4
195.j odd 4 1 507.2.k.a 4
195.s even 4 1 507.2.k.c 4
195.u odd 4 1 507.2.k.b 4
195.bc odd 12 1 39.2.k.a 4
195.bc odd 12 1 507.2.f.c 4
195.bf even 12 1 507.2.f.b 4
195.bf even 12 1 507.2.k.a 4
195.bh even 12 1 inner 975.2.bp.d 4
195.bl even 12 1 507.2.f.c 4
195.bl even 12 1 507.2.k.b 4
195.bn odd 12 1 507.2.f.b 4
195.bn odd 12 1 507.2.k.c 4
195.bn odd 12 1 975.2.bo.c 4
260.bl odd 12 1 624.2.cn.b 4
780.cy even 12 1 624.2.cn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 5.c odd 4 1
39.2.k.a 4 15.e even 4 1
39.2.k.a 4 65.t even 12 1
39.2.k.a 4 195.bc odd 12 1
507.2.f.b 4 65.o even 12 1
507.2.f.b 4 65.r odd 12 1
507.2.f.b 4 195.bf even 12 1
507.2.f.b 4 195.bn odd 12 1
507.2.f.c 4 65.q odd 12 1
507.2.f.c 4 65.t even 12 1
507.2.f.c 4 195.bc odd 12 1
507.2.f.c 4 195.bl even 12 1
507.2.k.a 4 65.k even 4 1
507.2.k.a 4 65.r odd 12 1
507.2.k.a 4 195.j odd 4 1
507.2.k.a 4 195.bf even 12 1
507.2.k.b 4 65.f even 4 1
507.2.k.b 4 65.q odd 12 1
507.2.k.b 4 195.u odd 4 1
507.2.k.b 4 195.bl even 12 1
507.2.k.c 4 65.h odd 4 1
507.2.k.c 4 65.o even 12 1
507.2.k.c 4 195.s even 4 1
507.2.k.c 4 195.bn odd 12 1
624.2.cn.b 4 20.e even 4 1
624.2.cn.b 4 60.l odd 4 1
624.2.cn.b 4 260.bl odd 12 1
624.2.cn.b 4 780.cy even 12 1
975.2.bo.c 4 5.c odd 4 1
975.2.bo.c 4 15.e even 4 1
975.2.bo.c 4 65.o even 12 1
975.2.bo.c 4 195.bn odd 12 1
975.2.bp.a 4 5.b even 2 1
975.2.bp.a 4 13.f odd 12 1
975.2.bp.a 4 15.d odd 2 1
975.2.bp.a 4 39.k even 12 1
975.2.bp.d 4 1.a even 1 1 trivial
975.2.bp.d 4 3.b odd 2 1 CM
975.2.bp.d 4 65.s odd 12 1 inner
975.2.bp.d 4 195.bh even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2} \)
\( T_{7}^{4} - 8 T_{7}^{3} + 41 T_{7}^{2} - 130 T_{7} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 - 3 T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 169 - 130 T + 41 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 13 - 5 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( 676 + 52 T + 50 T^{2} + 14 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 169 - 182 T + 98 T^{2} + 14 T^{3} + T^{4} \)
$37$ \( 676 + 52 T + 122 T^{2} - 22 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 9 + 3 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( 5625 + 75 T^{2} + T^{4} \)
$67$ \( 169 - 416 T + 281 T^{2} - 10 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 20449 - 4862 T + 578 T^{2} - 34 T^{3} + T^{4} \)
$79$ \( ( -147 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 28561 + 6422 T + 557 T^{2} + 28 T^{3} + T^{4} \)
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