Properties

Label 975.2.bb.i
Level $975$
Weight $2$
Character orbit 975.bb
Analytic conductor $7.785$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1731891456.1
Defining polynomial: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( 2 + 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{6} + ( \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{7} + ( \beta_{6} - 4 \beta_{7} ) q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( 2 + 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{6} + ( \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{7} + ( \beta_{6} - 4 \beta_{7} ) q^{8} + ( 1 + \beta_{2} ) q^{9} -2 \beta_{2} q^{11} + ( -\beta_{6} + 2 \beta_{7} ) q^{12} + ( 2 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{13} + ( 4 - 2 \beta_{4} ) q^{14} + ( 3 \beta_{2} + 3 \beta_{5} ) q^{16} + ( -\beta_{1} + \beta_{6} ) q^{17} + \beta_{6} q^{18} + ( 4 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{19} + ( -1 + \beta_{4} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{22} -2 \beta_{3} q^{23} + ( -5 \beta_{2} - \beta_{5} ) q^{24} + ( 8 + 5 \beta_{2} - \beta_{4} + \beta_{5} ) q^{26} + \beta_{7} q^{27} + ( 4 \beta_{1} + 6 \beta_{3} ) q^{28} + ( \beta_{2} + 3 \beta_{5} ) q^{29} + ( 1 + \beta_{4} ) q^{31} + ( -\beta_{1} - 4 \beta_{3} + \beta_{6} - 4 \beta_{7} ) q^{32} + ( -2 \beta_{3} - 2 \beta_{7} ) q^{33} + ( -4 + \beta_{4} ) q^{34} + ( 3 \beta_{2} + \beta_{5} ) q^{36} + ( -\beta_{1} + 6 \beta_{3} ) q^{37} + ( 2 \beta_{6} + 8 \beta_{7} ) q^{38} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{39} + ( \beta_{2} + \beta_{5} ) q^{41} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{42} + ( -\beta_{1} + 3 \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{43} + ( 4 - 2 \beta_{4} ) q^{44} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{46} + ( 4 \beta_{6} + 2 \beta_{7} ) q^{47} + ( 3 \beta_{1} - 3 \beta_{6} ) q^{48} + ( -\beta_{2} - 3 \beta_{5} ) q^{49} -\beta_{4} q^{51} + ( 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{6} - 4 \beta_{7} ) q^{52} + ( 3 \beta_{6} - 4 \beta_{7} ) q^{53} + ( -\beta_{2} - \beta_{5} ) q^{54} + ( 8 + 14 \beta_{2} - 6 \beta_{4} + 6 \beta_{5} ) q^{56} + ( 2 \beta_{6} + 4 \beta_{7} ) q^{57} + ( -\beta_{1} - 12 \beta_{3} + \beta_{6} - 12 \beta_{7} ) q^{58} + ( -6 - 8 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{59} + ( -7 - 9 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} -4 \beta_{3} q^{62} + ( \beta_{1} + \beta_{3} ) q^{63} + ( -4 - \beta_{4} ) q^{64} + 2 \beta_{4} q^{66} + ( -\beta_{1} + 3 \beta_{3} ) q^{67} + ( -3 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 2 + 2 \beta_{2} ) q^{69} + ( -14 - 14 \beta_{2} ) q^{71} + ( -\beta_{1} - 4 \beta_{3} + \beta_{6} - 4 \beta_{7} ) q^{72} + ( 2 \beta_{6} + 7 \beta_{7} ) q^{73} + ( -4 + \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{74} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{76} + ( -2 \beta_{6} + 2 \beta_{7} ) q^{77} + ( -4 \beta_{3} - \beta_{6} + 4 \beta_{7} ) q^{78} + ( -7 + \beta_{4} ) q^{79} + \beta_{2} q^{81} + ( -\beta_{1} - 4 \beta_{3} + \beta_{6} - 4 \beta_{7} ) q^{82} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{83} + ( -6 - 10 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} ) q^{84} + ( -4 - 2 \beta_{4} ) q^{86} + ( 3 \beta_{1} - 2 \beta_{3} - 3 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 2 \beta_{1} + 8 \beta_{3} ) q^{88} + ( -10 \beta_{2} - 2 \beta_{5} ) q^{89} + ( 4 - 4 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{91} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{92} + ( \beta_{1} - \beta_{3} ) q^{93} + ( 18 \beta_{2} + 2 \beta_{5} ) q^{94} + ( 4 - \beta_{4} ) q^{96} + ( -\beta_{1} + 7 \beta_{3} + \beta_{6} + 7 \beta_{7} ) q^{97} + ( \beta_{1} + 12 \beta_{3} - \beta_{6} + 12 \beta_{7} ) q^{98} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 10q^{4} - 2q^{6} + 4q^{9} + O(q^{10}) \) \( 8q + 10q^{4} - 2q^{6} + 4q^{9} + 8q^{11} + 40q^{14} - 6q^{16} + 12q^{19} - 12q^{21} + 18q^{24} + 50q^{26} + 2q^{29} + 4q^{31} - 36q^{34} - 10q^{36} + 2q^{39} - 2q^{41} + 40q^{44} - 4q^{46} - 2q^{49} + 4q^{51} + 2q^{54} + 44q^{56} - 28q^{59} - 32q^{61} - 28q^{64} - 8q^{66} + 8q^{69} - 56q^{71} - 6q^{74} + 4q^{76} - 60q^{79} - 4q^{81} - 32q^{84} - 24q^{86} + 36q^{89} + 54q^{91} - 68q^{94} + 36q^{96} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{6} - 65 \nu^{4} + 585 \nu^{2} - 1296 \)\()/1040\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 181 \nu \)\()/260\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 116 \)\()/65\)
\(\beta_{5}\)\(=\)\((\)\( -29 \nu^{6} + 325 \nu^{4} - 1885 \nu^{2} + 4176 \)\()/1040\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 585 \nu^{3} - 256 \nu \)\()/1040\)
\(\beta_{7}\)\(=\)\((\)\( 9 \nu^{7} - 65 \nu^{5} + 377 \nu^{3} - 256 \nu \)\()/832\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 5 \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 5 \beta_{6}\)
\(\nu^{4}\)\(=\)\(9 \beta_{5} + 29 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-36 \beta_{7} + 29 \beta_{6} - 36 \beta_{3} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(65 \beta_{4} - 116\)
\(\nu^{7}\)\(=\)\(-260 \beta_{3} - 181 \beta_{1}\)

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(-1\)

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
−2.21837 1.28078i
−1.35234 0.780776i
1.35234 + 0.780776i
2.21837 + 1.28078i
−2.21837 + 1.28078i
−1.35234 + 0.780776i
1.35234 0.780776i
2.21837 1.28078i
−2.21837 1.28078i 0.866025 + 0.500000i 2.28078 + 3.95042i 0 −1.28078 2.21837i −3.08440 + 1.78078i 6.56155i 0.500000 + 0.866025i 0
724.2 −1.35234 0.780776i −0.866025 0.500000i 0.219224 + 0.379706i 0 0.780776 + 1.35234i −0.486319 + 0.280776i 2.43845i 0.500000 + 0.866025i 0
724.3 1.35234 + 0.780776i 0.866025 + 0.500000i 0.219224 + 0.379706i 0 0.780776 + 1.35234i 0.486319 0.280776i 2.43845i 0.500000 + 0.866025i 0
724.4 2.21837 + 1.28078i −0.866025 0.500000i 2.28078 + 3.95042i 0 −1.28078 2.21837i 3.08440 1.78078i 6.56155i 0.500000 + 0.866025i 0
874.1 −2.21837 + 1.28078i 0.866025 0.500000i 2.28078 3.95042i 0 −1.28078 + 2.21837i −3.08440 1.78078i 6.56155i 0.500000 0.866025i 0
874.2 −1.35234 + 0.780776i −0.866025 + 0.500000i 0.219224 0.379706i 0 0.780776 1.35234i −0.486319 0.280776i 2.43845i 0.500000 0.866025i 0
874.3 1.35234 0.780776i 0.866025 0.500000i 0.219224 0.379706i 0 0.780776 1.35234i 0.486319 + 0.280776i 2.43845i 0.500000 0.866025i 0
874.4 2.21837 1.28078i −0.866025 + 0.500000i 2.28078 3.95042i 0 −1.28078 + 2.21837i 3.08440 + 1.78078i 6.56155i 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 874.4
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.i 8
5.b even 2 1 inner 975.2.bb.i 8
5.c odd 4 1 39.2.e.b 4
5.c odd 4 1 975.2.i.k 4
13.c even 3 1 inner 975.2.bb.i 8
15.e even 4 1 117.2.g.c 4
20.e even 4 1 624.2.q.h 4
60.l odd 4 1 1872.2.t.r 4
65.f even 4 1 507.2.j.g 8
65.h odd 4 1 507.2.e.g 4
65.k even 4 1 507.2.j.g 8
65.n even 6 1 inner 975.2.bb.i 8
65.o even 12 1 507.2.b.d 4
65.o even 12 1 507.2.j.g 8
65.q odd 12 1 39.2.e.b 4
65.q odd 12 1 507.2.a.g 2
65.q odd 12 1 975.2.i.k 4
65.r odd 12 1 507.2.a.d 2
65.r odd 12 1 507.2.e.g 4
65.t even 12 1 507.2.b.d 4
65.t even 12 1 507.2.j.g 8
195.bc odd 12 1 1521.2.b.h 4
195.bf even 12 1 1521.2.a.m 2
195.bl even 12 1 117.2.g.c 4
195.bl even 12 1 1521.2.a.g 2
195.bn odd 12 1 1521.2.b.h 4
260.bg even 12 1 8112.2.a.bo 2
260.bj even 12 1 624.2.q.h 4
260.bj even 12 1 8112.2.a.bk 2
780.cj odd 12 1 1872.2.t.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 5.c odd 4 1
39.2.e.b 4 65.q odd 12 1
117.2.g.c 4 15.e even 4 1
117.2.g.c 4 195.bl even 12 1
507.2.a.d 2 65.r odd 12 1
507.2.a.g 2 65.q odd 12 1
507.2.b.d 4 65.o even 12 1
507.2.b.d 4 65.t even 12 1
507.2.e.g 4 65.h odd 4 1
507.2.e.g 4 65.r odd 12 1
507.2.j.g 8 65.f even 4 1
507.2.j.g 8 65.k even 4 1
507.2.j.g 8 65.o even 12 1
507.2.j.g 8 65.t even 12 1
624.2.q.h 4 20.e even 4 1
624.2.q.h 4 260.bj even 12 1
975.2.i.k 4 5.c odd 4 1
975.2.i.k 4 65.q odd 12 1
975.2.bb.i 8 1.a even 1 1 trivial
975.2.bb.i 8 5.b even 2 1 inner
975.2.bb.i 8 13.c even 3 1 inner
975.2.bb.i 8 65.n even 6 1 inner
1521.2.a.g 2 195.bl even 12 1
1521.2.a.m 2 195.bf even 12 1
1521.2.b.h 4 195.bc odd 12 1
1521.2.b.h 4 195.bn odd 12 1
1872.2.t.r 4 60.l odd 4 1
1872.2.t.r 4 780.cj odd 12 1
8112.2.a.bk 2 260.bj even 12 1
8112.2.a.bo 2 260.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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{8} - 9 T_{2}^{6} + 65 T_{2}^{4} - 144 T_{2}^{2} + 256 \)
\( T_{7}^{8} - 13 T_{7}^{6} + 165 T_{7}^{4} - 52 T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 144 T^{2} + 65 T^{4} - 9 T^{6} + T^{8} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 16 - 52 T^{2} + 165 T^{4} - 13 T^{6} + T^{8} \)
$11$ \( ( 4 - 2 T + T^{2} )^{4} \)
$13$ \( ( 169 - 25 T^{2} + T^{4} )^{2} \)
$17$ \( 256 - 144 T^{2} + 65 T^{4} - 9 T^{6} + T^{8} \)
$19$ \( ( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$23$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1444 + 38 T + 39 T^{2} - T^{3} + T^{4} )^{2} \)
$31$ \( ( -4 - T + T^{2} )^{4} \)
$37$ \( 456976 - 46644 T^{2} + 4085 T^{4} - 69 T^{6} + T^{8} \)
$41$ \( ( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} )^{2} \)
$43$ \( 16 - 84 T^{2} + 437 T^{4} - 21 T^{6} + T^{8} \)
$47$ \( ( 68 + T^{2} )^{4} \)
$53$ \( ( 64 + 137 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1024 + 448 T + 164 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$61$ \( ( 2209 + 752 T + 209 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$67$ \( 16 - 84 T^{2} + 437 T^{4} - 21 T^{6} + T^{8} \)
$71$ \( ( 196 + 14 T + T^{2} )^{4} \)
$73$ \( ( 361 + 106 T^{2} + T^{4} )^{2} \)
$79$ \( ( 52 + 15 T + T^{2} )^{4} \)
$83$ \( ( 64 + 84 T^{2} + T^{4} )^{2} \)
$89$ \( ( 4096 - 1152 T + 260 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$97$ \( 2085136 - 134292 T^{2} + 7205 T^{4} - 93 T^{6} + T^{8} \)
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