# 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 $4$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 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.

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

## 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: 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.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 $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{2} + 1$$ T2^4 - T2^2 + 1 $$T_{7}^{4} - 4T_{7}^{2} + 16$$ T7^4 - 4*T7^2 + 16

## Hecke characteristic polynomials

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