Properties

 Label 975.2.i.f Level $975$ Weight $2$ Character orbit 975.i Analytic conductor $7.785$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.i (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

Character values

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

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i 3.00000 −0.500000 0.866025i 0
601.1 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 0 0.500000 0.866025i 1.00000 1.73205i 3.00000 −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
13.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.i.f 2
5.b even 2 1 39.2.e.a 2
5.c odd 4 2 975.2.bb.d 4
13.c even 3 1 inner 975.2.i.f 2
15.d odd 2 1 117.2.g.b 2
20.d odd 2 1 624.2.q.c 2
60.h even 2 1 1872.2.t.j 2
65.d even 2 1 507.2.e.c 2
65.g odd 4 2 507.2.j.d 4
65.l even 6 1 507.2.a.b 1
65.l even 6 1 507.2.e.c 2
65.n even 6 1 39.2.e.a 2
65.n even 6 1 507.2.a.c 1
65.q odd 12 2 975.2.bb.d 4
65.s odd 12 2 507.2.b.b 2
65.s odd 12 2 507.2.j.d 4
195.x odd 6 1 117.2.g.b 2
195.x odd 6 1 1521.2.a.a 1
195.y odd 6 1 1521.2.a.d 1
195.bh even 12 2 1521.2.b.c 2
260.v odd 6 1 624.2.q.c 2
260.v odd 6 1 8112.2.a.w 1
260.w odd 6 1 8112.2.a.bc 1
780.br even 6 1 1872.2.t.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 5.b even 2 1
39.2.e.a 2 65.n even 6 1
117.2.g.b 2 15.d odd 2 1
117.2.g.b 2 195.x odd 6 1
507.2.a.b 1 65.l even 6 1
507.2.a.c 1 65.n even 6 1
507.2.b.b 2 65.s odd 12 2
507.2.e.c 2 65.d even 2 1
507.2.e.c 2 65.l even 6 1
507.2.j.d 4 65.g odd 4 2
507.2.j.d 4 65.s odd 12 2
624.2.q.c 2 20.d odd 2 1
624.2.q.c 2 260.v odd 6 1
975.2.i.f 2 1.a even 1 1 trivial
975.2.i.f 2 13.c even 3 1 inner
975.2.bb.d 4 5.c odd 4 2
975.2.bb.d 4 65.q odd 12 2
1521.2.a.a 1 195.x odd 6 1
1521.2.a.d 1 195.y odd 6 1
1521.2.b.c 2 195.bh even 12 2
1872.2.t.j 2 60.h even 2 1
1872.2.t.j 2 780.br even 6 1
8112.2.a.w 1 260.v odd 6 1
8112.2.a.bc 1 260.w odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

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