# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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