# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(451,975)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("975.451");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{4} - 2 \zeta_{6} q^{6} + 5 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + (-z + 1) * q^3 - 2*z * q^4 - 2*z * q^6 + 5*z * q^7 - z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{4} - 2 \zeta_{6} q^{6} + 5 \zeta_{6} q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - 2 q^{12} + ( - 3 \zeta_{6} + 4) q^{13} + 10 q^{14} + ( - 4 \zeta_{6} + 4) q^{16} + 2 \zeta_{6} q^{17} - 2 q^{18} + 5 q^{21} + 4 \zeta_{6} q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + ( - 8 \zeta_{6} + 2) q^{26} - q^{27} + ( - 10 \zeta_{6} + 10) q^{28} + ( - 4 \zeta_{6} + 4) q^{29} - 7 q^{31} - 8 \zeta_{6} q^{32} + 2 \zeta_{6} q^{33} + 4 q^{34} + (2 \zeta_{6} - 2) q^{36} + (2 \zeta_{6} - 2) q^{37} + ( - 4 \zeta_{6} + 1) q^{39} + (6 \zeta_{6} - 6) q^{41} + ( - 10 \zeta_{6} + 10) q^{42} + \zeta_{6} q^{43} + 4 q^{44} - 12 \zeta_{6} q^{46} + 8 q^{47} - 4 \zeta_{6} q^{48} + (18 \zeta_{6} - 18) q^{49} + 2 q^{51} + ( - 2 \zeta_{6} - 6) q^{52} + 4 q^{53} + (2 \zeta_{6} - 2) q^{54} - 8 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} + 13 \zeta_{6} q^{61} + (14 \zeta_{6} - 14) q^{62} + ( - 5 \zeta_{6} + 5) q^{63} - 8 q^{64} + 4 q^{66} + (7 \zeta_{6} - 7) q^{67} + ( - 4 \zeta_{6} + 4) q^{68} - 6 \zeta_{6} q^{69} - 12 \zeta_{6} q^{71} - 15 q^{73} + 4 \zeta_{6} q^{74} - 10 q^{77} + ( - 2 \zeta_{6} - 6) q^{78} + 3 q^{79} + (\zeta_{6} - 1) q^{81} + 12 \zeta_{6} q^{82} - 8 q^{83} - 10 \zeta_{6} q^{84} + 2 q^{86} - 4 \zeta_{6} q^{87} + (14 \zeta_{6} - 14) q^{89} + (5 \zeta_{6} + 15) q^{91} - 12 q^{92} + (7 \zeta_{6} - 7) q^{93} + ( - 16 \zeta_{6} + 16) q^{94} - 8 q^{96} - 5 \zeta_{6} q^{97} + 36 \zeta_{6} q^{98} + 2 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + (-z + 1) * q^3 - 2*z * q^4 - 2*z * q^6 + 5*z * q^7 - z * q^9 + (2*z - 2) * q^11 - 2 * q^12 + (-3*z + 4) * q^13 + 10 * q^14 + (-4*z + 4) * q^16 + 2*z * q^17 - 2 * q^18 + 5 * q^21 + 4*z * q^22 + (-6*z + 6) * q^23 + (-8*z + 2) * q^26 - q^27 + (-10*z + 10) * q^28 + (-4*z + 4) * q^29 - 7 * q^31 - 8*z * q^32 + 2*z * q^33 + 4 * q^34 + (2*z - 2) * q^36 + (2*z - 2) * q^37 + (-4*z + 1) * q^39 + (6*z - 6) * q^41 + (-10*z + 10) * q^42 + z * q^43 + 4 * q^44 - 12*z * q^46 + 8 * q^47 - 4*z * q^48 + (18*z - 18) * q^49 + 2 * q^51 + (-2*z - 6) * q^52 + 4 * q^53 + (2*z - 2) * q^54 - 8*z * q^58 - 12*z * q^59 + 13*z * q^61 + (14*z - 14) * q^62 + (-5*z + 5) * q^63 - 8 * q^64 + 4 * q^66 + (7*z - 7) * q^67 + (-4*z + 4) * q^68 - 6*z * q^69 - 12*z * q^71 - 15 * q^73 + 4*z * q^74 - 10 * q^77 + (-2*z - 6) * q^78 + 3 * q^79 + (z - 1) * q^81 + 12*z * q^82 - 8 * q^83 - 10*z * q^84 + 2 * q^86 - 4*z * q^87 + (14*z - 14) * q^89 + (5*z + 15) * q^91 - 12 * q^92 + (7*z - 7) * q^93 + (-16*z + 16) * q^94 - 8 * q^96 - 5*z * q^97 + 36*z * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} - 2 q^{4} - 2 q^{6} + 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 - 2 * q^4 - 2 * q^6 + 5 * q^7 - q^9 $$2 q + 2 q^{2} + q^{3} - 2 q^{4} - 2 q^{6} + 5 q^{7} - q^{9} - 2 q^{11} - 4 q^{12} + 5 q^{13} + 20 q^{14} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 10 q^{21} + 4 q^{22} + 6 q^{23} - 4 q^{26} - 2 q^{27} + 10 q^{28} + 4 q^{29} - 14 q^{31} - 8 q^{32} + 2 q^{33} + 8 q^{34} - 2 q^{36} - 2 q^{37} - 2 q^{39} - 6 q^{41} + 10 q^{42} + q^{43} + 8 q^{44} - 12 q^{46} + 16 q^{47} - 4 q^{48} - 18 q^{49} + 4 q^{51} - 14 q^{52} + 8 q^{53} - 2 q^{54} - 8 q^{58} - 12 q^{59} + 13 q^{61} - 14 q^{62} + 5 q^{63} - 16 q^{64} + 8 q^{66} - 7 q^{67} + 4 q^{68} - 6 q^{69} - 12 q^{71} - 30 q^{73} + 4 q^{74} - 20 q^{77} - 14 q^{78} + 6 q^{79} - q^{81} + 12 q^{82} - 16 q^{83} - 10 q^{84} + 4 q^{86} - 4 q^{87} - 14 q^{89} + 35 q^{91} - 24 q^{92} - 7 q^{93} + 16 q^{94} - 16 q^{96} - 5 q^{97} + 36 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 - 2 * q^4 - 2 * q^6 + 5 * q^7 - q^9 - 2 * q^11 - 4 * q^12 + 5 * q^13 + 20 * q^14 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 10 * q^21 + 4 * q^22 + 6 * q^23 - 4 * q^26 - 2 * q^27 + 10 * q^28 + 4 * q^29 - 14 * q^31 - 8 * q^32 + 2 * q^33 + 8 * q^34 - 2 * q^36 - 2 * q^37 - 2 * q^39 - 6 * q^41 + 10 * q^42 + q^43 + 8 * q^44 - 12 * q^46 + 16 * q^47 - 4 * q^48 - 18 * q^49 + 4 * q^51 - 14 * q^52 + 8 * q^53 - 2 * q^54 - 8 * q^58 - 12 * q^59 + 13 * q^61 - 14 * q^62 + 5 * q^63 - 16 * q^64 + 8 * q^66 - 7 * q^67 + 4 * q^68 - 6 * q^69 - 12 * q^71 - 30 * q^73 + 4 * q^74 - 20 * q^77 - 14 * q^78 + 6 * q^79 - q^81 + 12 * q^82 - 16 * q^83 - 10 * q^84 + 4 * q^86 - 4 * q^87 - 14 * q^89 + 35 * q^91 - 24 * q^92 - 7 * q^93 + 16 * q^94 - 16 * q^96 - 5 * q^97 + 36 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
1.00000 1.73205i 0.500000 0.866025i −1.00000 1.73205i 0 −1.00000 1.73205i 2.50000 + 4.33013i 0 −0.500000 0.866025i 0
601.1 1.00000 + 1.73205i 0.500000 + 0.866025i −1.00000 + 1.73205i 0 −1.00000 + 1.73205i 2.50000 4.33013i 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.i 2
5.b even 2 1 195.2.i.a 2
5.c odd 4 2 975.2.bb.f 4
13.c even 3 1 inner 975.2.i.i 2
15.d odd 2 1 585.2.j.b 2
65.l even 6 1 2535.2.a.c 1
65.n even 6 1 195.2.i.a 2
65.n even 6 1 2535.2.a.m 1
65.q odd 12 2 975.2.bb.f 4
195.x odd 6 1 585.2.j.b 2
195.x odd 6 1 7605.2.a.a 1
195.y odd 6 1 7605.2.a.s 1

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

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} + 4$$ T2^2 - 2*T2 + 4 $$T_{7}^{2} - 5T_{7} + 25$$ T7^2 - 5*T7 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 25$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 4T + 16$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - T + 1$$
$47$ $$(T - 8)^{2}$$
$53$ $$(T - 4)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$T^{2} + 12T + 144$$
$73$ $$(T + 15)^{2}$$
$79$ $$(T - 3)^{2}$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} + 14T + 196$$
$97$ $$T^{2} + 5T + 25$$