Properties

Label 975.2.bb.d
Level $975$
Weight $2$
Character orbit 975.bb
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM no
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.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{4} + \zeta_{12}^{2} q^{6} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} - 3 \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{4} + \zeta_{12}^{2} q^{6} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} - 3 \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( - 2 \zeta_{12}^{2} + 2) q^{11} - \zeta_{12}^{3} q^{12} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} - 2 q^{14} + ( - \zeta_{12}^{2} + 1) q^{16} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{17} + \zeta_{12}^{3} q^{18} - 6 \zeta_{12}^{2} q^{19} - 2 q^{21} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{22} + 6 \zeta_{12} q^{23} + ( - 3 \zeta_{12}^{2} + 3) q^{24} + ( - 3 \zeta_{12}^{2} + 4) q^{26} + \zeta_{12}^{3} q^{27} + 2 \zeta_{12} q^{28} + (\zeta_{12}^{2} - 1) q^{29} + 4 q^{31} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{33} + 7 q^{34} + ( - \zeta_{12}^{2} + 1) q^{36} + \zeta_{12} q^{37} - 6 \zeta_{12}^{3} q^{38} + ( - 3 \zeta_{12}^{2} + 4) q^{39} + (9 \zeta_{12}^{2} - 9) q^{41} - 2 \zeta_{12} q^{42} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{43} - 2 q^{44} + 6 \zeta_{12}^{2} q^{46} - 6 \zeta_{12}^{3} q^{47} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{48} + (3 \zeta_{12}^{2} - 3) q^{49} + 7 q^{51} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{52} - 9 \zeta_{12}^{3} q^{53} + (\zeta_{12}^{2} - 1) q^{54} + 6 \zeta_{12}^{2} q^{56} - 6 \zeta_{12}^{3} q^{57} + (\zeta_{12}^{3} - \zeta_{12}) q^{58} - \zeta_{12}^{2} q^{61} + 4 \zeta_{12} q^{62} - 2 \zeta_{12} q^{63} - 7 q^{64} + 2 q^{66} - 2 \zeta_{12} q^{67} - 7 \zeta_{12} q^{68} + 6 \zeta_{12}^{2} q^{69} - 6 \zeta_{12}^{2} q^{71} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{72} + 11 \zeta_{12}^{3} q^{73} + \zeta_{12}^{2} q^{74} + (6 \zeta_{12}^{2} - 6) q^{76} + 4 \zeta_{12}^{3} q^{77} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{78} + 4 q^{79} + (\zeta_{12}^{2} - 1) q^{81} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{82} - 14 \zeta_{12}^{3} q^{83} + 2 \zeta_{12}^{2} q^{84} + 6 q^{86} + (\zeta_{12}^{3} - \zeta_{12}) q^{87} - 6 \zeta_{12} q^{88} + (14 \zeta_{12}^{2} - 14) q^{89} + (8 \zeta_{12}^{2} - 2) q^{91} - 6 \zeta_{12}^{3} q^{92} + 4 \zeta_{12} q^{93} + ( - 6 \zeta_{12}^{2} + 6) q^{94} - 5 q^{96} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{97} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{98} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 2 q^{6} + 2 q^{9} + 4 q^{11} - 8 q^{14} + 2 q^{16} - 12 q^{19} - 8 q^{21} + 6 q^{24} + 10 q^{26} - 2 q^{29} + 16 q^{31} + 28 q^{34} + 2 q^{36} + 10 q^{39} - 18 q^{41} - 8 q^{44} + 12 q^{46} - 6 q^{49} + 28 q^{51} - 2 q^{54} + 12 q^{56} - 2 q^{61} - 28 q^{64} + 8 q^{66} + 12 q^{69} - 12 q^{71} + 2 q^{74} - 12 q^{76} + 16 q^{79} - 2 q^{81} + 4 q^{84} + 24 q^{86} - 28 q^{89} + 8 q^{91} + 12 q^{94} - 20 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.866025 0.500000i −0.500000 0.866025i 0 0.500000 + 0.866025i 1.73205 1.00000i 3.00000i 0.500000 + 0.866025i 0
724.2 0.866025 + 0.500000i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.500000 + 0.866025i −1.73205 + 1.00000i 3.00000i 0.500000 + 0.866025i 0
874.1 −0.866025 + 0.500000i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.500000 0.866025i 1.73205 + 1.00000i 3.00000i 0.500000 0.866025i 0
874.2 0.866025 0.500000i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.500000 0.866025i −1.73205 1.00000i 3.00000i 0.500000 0.866025i 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bb.d 4
5.b even 2 1 inner 975.2.bb.d 4
5.c odd 4 1 39.2.e.a 2
5.c odd 4 1 975.2.i.f 2
13.c even 3 1 inner 975.2.bb.d 4
15.e even 4 1 117.2.g.b 2
20.e even 4 1 624.2.q.c 2
60.l odd 4 1 1872.2.t.j 2
65.f even 4 1 507.2.j.d 4
65.h odd 4 1 507.2.e.c 2
65.k even 4 1 507.2.j.d 4
65.n even 6 1 inner 975.2.bb.d 4
65.o even 12 1 507.2.b.b 2
65.o even 12 1 507.2.j.d 4
65.q odd 12 1 39.2.e.a 2
65.q odd 12 1 507.2.a.c 1
65.q odd 12 1 975.2.i.f 2
65.r odd 12 1 507.2.a.b 1
65.r odd 12 1 507.2.e.c 2
65.t even 12 1 507.2.b.b 2
65.t even 12 1 507.2.j.d 4
195.bc odd 12 1 1521.2.b.c 2
195.bf even 12 1 1521.2.a.d 1
195.bl even 12 1 117.2.g.b 2
195.bl even 12 1 1521.2.a.a 1
195.bn odd 12 1 1521.2.b.c 2
260.bg even 12 1 8112.2.a.bc 1
260.bj even 12 1 624.2.q.c 2
260.bj even 12 1 8112.2.a.w 1
780.cj odd 12 1 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 5.c odd 4 1
39.2.e.a 2 65.q odd 12 1
117.2.g.b 2 15.e even 4 1
117.2.g.b 2 195.bl even 12 1
507.2.a.b 1 65.r odd 12 1
507.2.a.c 1 65.q odd 12 1
507.2.b.b 2 65.o even 12 1
507.2.b.b 2 65.t even 12 1
507.2.e.c 2 65.h odd 4 1
507.2.e.c 2 65.r odd 12 1
507.2.j.d 4 65.f even 4 1
507.2.j.d 4 65.k even 4 1
507.2.j.d 4 65.o even 12 1
507.2.j.d 4 65.t even 12 1
624.2.q.c 2 20.e even 4 1
624.2.q.c 2 260.bj even 12 1
975.2.i.f 2 5.c odd 4 1
975.2.i.f 2 65.q odd 12 1
975.2.bb.d 4 1.a even 1 1 trivial
975.2.bb.d 4 5.b even 2 1 inner
975.2.bb.d 4 13.c even 3 1 inner
975.2.bb.d 4 65.n even 6 1 inner
1521.2.a.a 1 195.bl even 12 1
1521.2.a.d 1 195.bf even 12 1
1521.2.b.c 2 195.bc odd 12 1
1521.2.b.c 2 195.bn odd 12 1
1872.2.t.j 2 60.l odd 4 1
1872.2.t.j 2 780.cj odd 12 1
8112.2.a.w 1 260.bj even 12 1
8112.2.a.bc 1 260.bg even 12 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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