# Properties

 Label 975.2.bb.f Level $975$ Weight $2$ Character orbit 975.bb Analytic conductor $7.785$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(724,975)]

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, 3, 4]))

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

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

chi := DirichletCharacter("975.724");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} - \zeta_{12} q^{3} + 2 \zeta_{12}^{2} q^{4} - 2 \zeta_{12}^{2} q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + 2*z * q^2 - z * q^3 + 2*z^2 * q^4 - 2*z^2 * q^6 + (5*z^3 - 5*z) * q^7 + z^2 * q^9 $$q + 2 \zeta_{12} q^{2} - \zeta_{12} q^{3} + 2 \zeta_{12}^{2} q^{4} - 2 \zeta_{12}^{2} q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{2} - 2) q^{11} - 2 \zeta_{12}^{3} q^{12} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{13} - 10 q^{14} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{17} + 2 \zeta_{12}^{3} q^{18} + 5 q^{21} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{22} - 6 \zeta_{12} q^{23} + ( - 8 \zeta_{12}^{2} + 2) q^{26} - \zeta_{12}^{3} q^{27} - 10 \zeta_{12} q^{28} + (4 \zeta_{12}^{2} - 4) q^{29} - 7 q^{31} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{33} - 4 q^{34} + (2 \zeta_{12}^{2} - 2) q^{36} - 2 \zeta_{12} q^{37} + (4 \zeta_{12}^{2} - 1) q^{39} + (6 \zeta_{12}^{2} - 6) q^{41} + 10 \zeta_{12} q^{42} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{43} - 4 q^{44} - 12 \zeta_{12}^{2} q^{46} + 8 \zeta_{12}^{3} q^{47} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{48} + ( - 18 \zeta_{12}^{2} + 18) q^{49} + 2 q^{51} + ( - 8 \zeta_{12}^{3} + 2 \zeta_{12}) q^{52} - 4 \zeta_{12}^{3} q^{53} + ( - 2 \zeta_{12}^{2} + 2) q^{54} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{58} + 12 \zeta_{12}^{2} q^{59} + 13 \zeta_{12}^{2} q^{61} - 14 \zeta_{12} q^{62} - 5 \zeta_{12} q^{63} + 8 q^{64} + 4 q^{66} - 7 \zeta_{12} q^{67} - 4 \zeta_{12} q^{68} + 6 \zeta_{12}^{2} q^{69} - 12 \zeta_{12}^{2} q^{71} + 15 \zeta_{12}^{3} q^{73} - 4 \zeta_{12}^{2} q^{74} - 10 \zeta_{12}^{3} q^{77} + (8 \zeta_{12}^{3} - 2 \zeta_{12}) q^{78} - 3 q^{79} + (\zeta_{12}^{2} - 1) q^{81} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{82} + 8 \zeta_{12}^{3} q^{83} + 10 \zeta_{12}^{2} q^{84} + 2 q^{86} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{87} + ( - 14 \zeta_{12}^{2} + 14) q^{89} + (5 \zeta_{12}^{2} + 15) q^{91} - 12 \zeta_{12}^{3} q^{92} + 7 \zeta_{12} q^{93} + (16 \zeta_{12}^{2} - 16) q^{94} - 8 q^{96} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{97} + ( - 36 \zeta_{12}^{3} + 36 \zeta_{12}) q^{98} - 2 q^{99} +O(q^{100})$$ q + 2*z * q^2 - z * q^3 + 2*z^2 * q^4 - 2*z^2 * q^6 + (5*z^3 - 5*z) * q^7 + z^2 * q^9 + (2*z^2 - 2) * q^11 - 2*z^3 * q^12 + (-z^3 - 3*z) * q^13 - 10 * q^14 + (-4*z^2 + 4) * q^16 + (2*z^3 - 2*z) * q^17 + 2*z^3 * q^18 + 5 * q^21 + (4*z^3 - 4*z) * q^22 - 6*z * q^23 + (-8*z^2 + 2) * q^26 - z^3 * q^27 - 10*z * q^28 + (4*z^2 - 4) * q^29 - 7 * q^31 + (-8*z^3 + 8*z) * q^32 + (-2*z^3 + 2*z) * q^33 - 4 * q^34 + (2*z^2 - 2) * q^36 - 2*z * q^37 + (4*z^2 - 1) * q^39 + (6*z^2 - 6) * q^41 + 10*z * q^42 + (-z^3 + z) * q^43 - 4 * q^44 - 12*z^2 * q^46 + 8*z^3 * q^47 + (4*z^3 - 4*z) * q^48 + (-18*z^2 + 18) * q^49 + 2 * q^51 + (-8*z^3 + 2*z) * q^52 - 4*z^3 * q^53 + (-2*z^2 + 2) * q^54 + (8*z^3 - 8*z) * q^58 + 12*z^2 * q^59 + 13*z^2 * q^61 - 14*z * q^62 - 5*z * q^63 + 8 * q^64 + 4 * q^66 - 7*z * q^67 - 4*z * q^68 + 6*z^2 * q^69 - 12*z^2 * q^71 + 15*z^3 * q^73 - 4*z^2 * q^74 - 10*z^3 * q^77 + (8*z^3 - 2*z) * q^78 - 3 * q^79 + (z^2 - 1) * q^81 + (12*z^3 - 12*z) * q^82 + 8*z^3 * q^83 + 10*z^2 * q^84 + 2 * q^86 + (-4*z^3 + 4*z) * q^87 + (-14*z^2 + 14) * q^89 + (5*z^2 + 15) * q^91 - 12*z^3 * q^92 + 7*z * q^93 + (16*z^2 - 16) * q^94 - 8 * q^96 + (-5*z^3 + 5*z) * q^97 + (-36*z^3 + 36*z) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 4 * q^6 + 2 * q^9 $$4 q + 4 q^{4} - 4 q^{6} + 2 q^{9} - 4 q^{11} - 40 q^{14} + 8 q^{16} + 20 q^{21} - 8 q^{26} - 8 q^{29} - 28 q^{31} - 16 q^{34} - 4 q^{36} + 4 q^{39} - 12 q^{41} - 16 q^{44} - 24 q^{46} + 36 q^{49} + 8 q^{51} + 4 q^{54} + 24 q^{59} + 26 q^{61} + 32 q^{64} + 16 q^{66} + 12 q^{69} - 24 q^{71} - 8 q^{74} - 12 q^{79} - 2 q^{81} + 20 q^{84} + 8 q^{86} + 28 q^{89} + 70 q^{91} - 32 q^{94} - 32 q^{96} - 8 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 - 4 * q^6 + 2 * q^9 - 4 * q^11 - 40 * q^14 + 8 * q^16 + 20 * q^21 - 8 * q^26 - 8 * q^29 - 28 * q^31 - 16 * q^34 - 4 * q^36 + 4 * q^39 - 12 * q^41 - 16 * q^44 - 24 * q^46 + 36 * q^49 + 8 * q^51 + 4 * q^54 + 24 * q^59 + 26 * q^61 + 32 * q^64 + 16 * q^66 + 12 * q^69 - 24 * q^71 - 8 * q^74 - 12 * q^79 - 2 * q^81 + 20 * q^84 + 8 * q^86 + 28 * q^89 + 70 * q^91 - 32 * q^94 - 32 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
724.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.73205 1.00000i 0.866025 + 0.500000i 1.00000 + 1.73205i 0 −1.00000 1.73205i 4.33013 2.50000i 0 0.500000 + 0.866025i 0
724.2 1.73205 + 1.00000i −0.866025 0.500000i 1.00000 + 1.73205i 0 −1.00000 1.73205i −4.33013 + 2.50000i 0 0.500000 + 0.866025i 0
874.1 −1.73205 + 1.00000i 0.866025 0.500000i 1.00000 1.73205i 0 −1.00000 + 1.73205i 4.33013 + 2.50000i 0 0.500000 0.866025i 0
874.2 1.73205 1.00000i −0.866025 + 0.500000i 1.00000 1.73205i 0 −1.00000 + 1.73205i −4.33013 2.50000i 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
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.f 4
5.b even 2 1 inner 975.2.bb.f 4
5.c odd 4 1 195.2.i.a 2
5.c odd 4 1 975.2.i.i 2
13.c even 3 1 inner 975.2.bb.f 4
15.e even 4 1 585.2.j.b 2
65.n even 6 1 inner 975.2.bb.f 4
65.q odd 12 1 195.2.i.a 2
65.q odd 12 1 975.2.i.i 2
65.q odd 12 1 2535.2.a.m 1
65.r odd 12 1 2535.2.a.c 1
195.bf even 12 1 7605.2.a.s 1
195.bl even 12 1 585.2.j.b 2
195.bl even 12 1 7605.2.a.a 1

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

## Hecke kernels

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

 $$T_{2}^{4} - 4T_{2}^{2} + 16$$ T2^4 - 4*T2^2 + 16 $$T_{7}^{4} - 25T_{7}^{2} + 625$$ T7^4 - 25*T7^2 + 625

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 25T^{2} + 625$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$T^{4} - T^{2} + 169$$
$17$ $$T^{4} - 4T^{2} + 16$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} + 4 T + 16)^{2}$$
$31$ $$(T + 7)^{4}$$
$37$ $$T^{4} - 4T^{2} + 16$$
$41$ $$(T^{2} + 6 T + 36)^{2}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T^{2} + 16)^{2}$$
$59$ $$(T^{2} - 12 T + 144)^{2}$$
$61$ $$(T^{2} - 13 T + 169)^{2}$$
$67$ $$T^{4} - 49T^{2} + 2401$$
$71$ $$(T^{2} + 12 T + 144)^{2}$$
$73$ $$(T^{2} + 225)^{2}$$
$79$ $$(T + 3)^{4}$$
$83$ $$(T^{2} + 64)^{2}$$
$89$ $$(T^{2} - 14 T + 196)^{2}$$
$97$ $$T^{4} - 25T^{2} + 625$$