# Properties

 Label 975.2.i.d 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{4} - \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 + 2*z * q^4 - z * q^7 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{4} - \zeta_{6} q^{7} - \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} + 2 q^{12} + (3 \zeta_{6} - 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + 4 \zeta_{6} q^{19} - q^{21} + (6 \zeta_{6} - 6) q^{23} - q^{27} + ( - 2 \zeta_{6} + 2) q^{28} + ( - 6 \zeta_{6} + 6) q^{29} + 5 q^{31} + 6 \zeta_{6} q^{33} + ( - 2 \zeta_{6} + 2) q^{36} + ( - 2 \zeta_{6} + 2) q^{37} + (4 \zeta_{6} - 1) q^{39} + 11 \zeta_{6} q^{43} - 12 q^{44} - 6 q^{47} + 4 \zeta_{6} q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + ( - 2 \zeta_{6} - 6) q^{52} + 4 q^{57} - 6 \zeta_{6} q^{59} + \zeta_{6} q^{61} + (\zeta_{6} - 1) q^{63} - 8 q^{64} + ( - 11 \zeta_{6} + 11) q^{67} + 6 \zeta_{6} q^{69} + 6 \zeta_{6} q^{71} - 5 q^{73} + (8 \zeta_{6} - 8) q^{76} + 6 q^{77} + 11 q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} - 2 \zeta_{6} q^{84} - 6 \zeta_{6} q^{87} + (12 \zeta_{6} - 12) q^{89} + (\zeta_{6} + 3) q^{91} - 12 q^{92} + ( - 5 \zeta_{6} + 5) q^{93} + 17 \zeta_{6} q^{97} + 6 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + 2*z * q^4 - z * q^7 - z * q^9 + (6*z - 6) * q^11 + 2 * q^12 + (3*z - 4) * q^13 + (4*z - 4) * q^16 + 4*z * q^19 - q^21 + (6*z - 6) * q^23 - q^27 + (-2*z + 2) * q^28 + (-6*z + 6) * q^29 + 5 * q^31 + 6*z * q^33 + (-2*z + 2) * q^36 + (-2*z + 2) * q^37 + (4*z - 1) * q^39 + 11*z * q^43 - 12 * q^44 - 6 * q^47 + 4*z * q^48 + (-6*z + 6) * q^49 + (-2*z - 6) * q^52 + 4 * q^57 - 6*z * q^59 + z * q^61 + (z - 1) * q^63 - 8 * q^64 + (-11*z + 11) * q^67 + 6*z * q^69 + 6*z * q^71 - 5 * q^73 + (8*z - 8) * q^76 + 6 * q^77 + 11 * q^79 + (z - 1) * q^81 - 12 * q^83 - 2*z * q^84 - 6*z * q^87 + (12*z - 12) * q^89 + (z + 3) * q^91 - 12 * q^92 + (-5*z + 5) * q^93 + 17*z * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{4} - q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^4 - q^7 - q^9 $$2 q + q^{3} + 2 q^{4} - q^{7} - q^{9} - 6 q^{11} + 4 q^{12} - 5 q^{13} - 4 q^{16} + 4 q^{19} - 2 q^{21} - 6 q^{23} - 2 q^{27} + 2 q^{28} + 6 q^{29} + 10 q^{31} + 6 q^{33} + 2 q^{36} + 2 q^{37} + 2 q^{39} + 11 q^{43} - 24 q^{44} - 12 q^{47} + 4 q^{48} + 6 q^{49} - 14 q^{52} + 8 q^{57} - 6 q^{59} + q^{61} - q^{63} - 16 q^{64} + 11 q^{67} + 6 q^{69} + 6 q^{71} - 10 q^{73} - 8 q^{76} + 12 q^{77} + 22 q^{79} - q^{81} - 24 q^{83} - 2 q^{84} - 6 q^{87} - 12 q^{89} + 7 q^{91} - 24 q^{92} + 5 q^{93} + 17 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^4 - q^7 - q^9 - 6 * q^11 + 4 * q^12 - 5 * q^13 - 4 * q^16 + 4 * q^19 - 2 * q^21 - 6 * q^23 - 2 * q^27 + 2 * q^28 + 6 * q^29 + 10 * q^31 + 6 * q^33 + 2 * q^36 + 2 * q^37 + 2 * q^39 + 11 * q^43 - 24 * q^44 - 12 * q^47 + 4 * q^48 + 6 * q^49 - 14 * q^52 + 8 * q^57 - 6 * q^59 + q^61 - q^63 - 16 * q^64 + 11 * q^67 + 6 * q^69 + 6 * q^71 - 10 * q^73 - 8 * q^76 + 12 * q^77 + 22 * q^79 - q^81 - 24 * q^83 - 2 * q^84 - 6 * q^87 - 12 * q^89 + 7 * q^91 - 24 * q^92 + 5 * q^93 + 17 * q^97 + 12 * 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 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
601.1 0 0.500000 + 0.866025i 1.00000 1.73205i 0 0 −0.500000 + 0.866025i 0 −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.d 2
5.b even 2 1 195.2.i.b 2
5.c odd 4 2 975.2.bb.b 4
13.c even 3 1 inner 975.2.i.d 2
15.d odd 2 1 585.2.j.a 2
65.l even 6 1 2535.2.a.i 1
65.n even 6 1 195.2.i.b 2
65.n even 6 1 2535.2.a.h 1
65.q odd 12 2 975.2.bb.b 4
195.x odd 6 1 585.2.j.a 2
195.x odd 6 1 7605.2.a.l 1
195.y odd 6 1 7605.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.b 2 5.b even 2 1
195.2.i.b 2 65.n even 6 1
585.2.j.a 2 15.d odd 2 1
585.2.j.a 2 195.x odd 6 1
975.2.i.d 2 1.a even 1 1 trivial
975.2.i.d 2 13.c even 3 1 inner
975.2.bb.b 4 5.c odd 4 2
975.2.bb.b 4 65.q odd 12 2
2535.2.a.h 1 65.n even 6 1
2535.2.a.i 1 65.l even 6 1
7605.2.a.k 1 195.y odd 6 1
7605.2.a.l 1 195.x odd 6 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} - 2T + 4$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 11T + 121$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 11T + 121$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T + 5)^{2}$$
$79$ $$(T - 11)^{2}$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$T^{2} - 17T + 289$$