# Properties

 Label 975.2.a.c Level $975$ Weight $2$ Character orbit 975.a Self dual yes Analytic conductor $7.785$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.a (trivial)

## 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: yes Analytic conductor: $$7.78541419707$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 3 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 - 3 * q^7 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 3 q^{7} + q^{9} - q^{11} + 2 q^{12} + q^{13} + 6 q^{14} - 4 q^{16} + q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{21} + 2 q^{22} + 3 q^{23} - 2 q^{26} + q^{27} - 6 q^{28} - 2 q^{29} - 6 q^{31} + 8 q^{32} - q^{33} - 2 q^{34} + 2 q^{36} - 11 q^{37} + 4 q^{38} + q^{39} - 5 q^{41} + 6 q^{42} - 4 q^{43} - 2 q^{44} - 6 q^{46} + 10 q^{47} - 4 q^{48} + 2 q^{49} + q^{51} + 2 q^{52} - 11 q^{53} - 2 q^{54} - 2 q^{57} + 4 q^{58} + 8 q^{59} + 13 q^{61} + 12 q^{62} - 3 q^{63} - 8 q^{64} + 2 q^{66} - 12 q^{67} + 2 q^{68} + 3 q^{69} - 5 q^{71} - 10 q^{73} + 22 q^{74} - 4 q^{76} + 3 q^{77} - 2 q^{78} - 3 q^{79} + q^{81} + 10 q^{82} + 12 q^{83} - 6 q^{84} + 8 q^{86} - 2 q^{87} - 15 q^{89} - 3 q^{91} + 6 q^{92} - 6 q^{93} - 20 q^{94} + 8 q^{96} - 17 q^{97} - 4 q^{98} - q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 - 3 * q^7 + q^9 - q^11 + 2 * q^12 + q^13 + 6 * q^14 - 4 * q^16 + q^17 - 2 * q^18 - 2 * q^19 - 3 * q^21 + 2 * q^22 + 3 * q^23 - 2 * q^26 + q^27 - 6 * q^28 - 2 * q^29 - 6 * q^31 + 8 * q^32 - q^33 - 2 * q^34 + 2 * q^36 - 11 * q^37 + 4 * q^38 + q^39 - 5 * q^41 + 6 * q^42 - 4 * q^43 - 2 * q^44 - 6 * q^46 + 10 * q^47 - 4 * q^48 + 2 * q^49 + q^51 + 2 * q^52 - 11 * q^53 - 2 * q^54 - 2 * q^57 + 4 * q^58 + 8 * q^59 + 13 * q^61 + 12 * q^62 - 3 * q^63 - 8 * q^64 + 2 * q^66 - 12 * q^67 + 2 * q^68 + 3 * q^69 - 5 * q^71 - 10 * q^73 + 22 * q^74 - 4 * q^76 + 3 * q^77 - 2 * q^78 - 3 * q^79 + q^81 + 10 * q^82 + 12 * q^83 - 6 * q^84 + 8 * q^86 - 2 * q^87 - 15 * q^89 - 3 * q^91 + 6 * q^92 - 6 * q^93 - 20 * q^94 + 8 * q^96 - 17 * q^97 - 4 * q^98 - q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 1.00000 2.00000 0 −2.00000 −3.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.a.c 1
3.b odd 2 1 2925.2.a.q 1
5.b even 2 1 195.2.a.b 1
5.c odd 4 2 975.2.c.a 2
15.d odd 2 1 585.2.a.b 1
15.e even 4 2 2925.2.c.c 2
20.d odd 2 1 3120.2.a.u 1
35.c odd 2 1 9555.2.a.v 1
60.h even 2 1 9360.2.a.d 1
65.d even 2 1 2535.2.a.a 1
195.e odd 2 1 7605.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.b 1 5.b even 2 1
585.2.a.b 1 15.d odd 2 1
975.2.a.c 1 1.a even 1 1 trivial
975.2.c.a 2 5.c odd 4 2
2535.2.a.a 1 65.d even 2 1
2925.2.a.q 1 3.b odd 2 1
2925.2.c.c 2 15.e even 4 2
3120.2.a.u 1 20.d odd 2 1
7605.2.a.u 1 195.e odd 2 1
9360.2.a.d 1 60.h even 2 1
9555.2.a.v 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(975))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T - 1$$
$19$ $$T + 2$$
$23$ $$T - 3$$
$29$ $$T + 2$$
$31$ $$T + 6$$
$37$ $$T + 11$$
$41$ $$T + 5$$
$43$ $$T + 4$$
$47$ $$T - 10$$
$53$ $$T + 11$$
$59$ $$T - 8$$
$61$ $$T - 13$$
$67$ $$T + 12$$
$71$ $$T + 5$$
$73$ $$T + 10$$
$79$ $$T + 3$$
$83$ $$T - 12$$
$89$ $$T + 15$$
$97$ $$T + 17$$