# Properties

 Label 975.2.a.b 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 $$q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + q^{9} - 5 q^{11} - 2 q^{12} - q^{13} - 6 q^{14} - 4 q^{16} - 5 q^{17} - 2 q^{18} + 2 q^{19} - 3 q^{21} + 10 q^{22} + q^{23} + 2 q^{26} - q^{27} + 6 q^{28} + 10 q^{29} - 2 q^{31} + 8 q^{32} + 5 q^{33} + 10 q^{34} + 2 q^{36} + 3 q^{37} - 4 q^{38} + q^{39} - 9 q^{41} + 6 q^{42} + 4 q^{43} - 10 q^{44} - 2 q^{46} - 10 q^{47} + 4 q^{48} + 2 q^{49} + 5 q^{51} - 2 q^{52} - 9 q^{53} + 2 q^{54} - 2 q^{57} - 20 q^{58} - 11 q^{61} + 4 q^{62} + 3 q^{63} - 8 q^{64} - 10 q^{66} + 4 q^{67} - 10 q^{68} - q^{69} + 15 q^{71} - 6 q^{73} - 6 q^{74} + 4 q^{76} - 15 q^{77} - 2 q^{78} - 11 q^{79} + q^{81} + 18 q^{82} - 8 q^{83} - 6 q^{84} - 8 q^{86} - 10 q^{87} - 11 q^{89} - 3 q^{91} + 2 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{96} + 9 q^{97} - 4 q^{98} - 5 q^{99}+O(q^{100})$$ q - 2 * q^2 - q^3 + 2 * q^4 + 2 * q^6 + 3 * q^7 + q^9 - 5 * q^11 - 2 * q^12 - q^13 - 6 * q^14 - 4 * q^16 - 5 * q^17 - 2 * q^18 + 2 * q^19 - 3 * q^21 + 10 * q^22 + q^23 + 2 * q^26 - q^27 + 6 * q^28 + 10 * q^29 - 2 * q^31 + 8 * q^32 + 5 * q^33 + 10 * q^34 + 2 * q^36 + 3 * q^37 - 4 * q^38 + q^39 - 9 * q^41 + 6 * q^42 + 4 * q^43 - 10 * q^44 - 2 * q^46 - 10 * q^47 + 4 * q^48 + 2 * q^49 + 5 * q^51 - 2 * q^52 - 9 * q^53 + 2 * q^54 - 2 * q^57 - 20 * q^58 - 11 * q^61 + 4 * q^62 + 3 * q^63 - 8 * q^64 - 10 * q^66 + 4 * q^67 - 10 * q^68 - q^69 + 15 * q^71 - 6 * q^73 - 6 * q^74 + 4 * q^76 - 15 * q^77 - 2 * q^78 - 11 * q^79 + q^81 + 18 * q^82 - 8 * q^83 - 6 * q^84 - 8 * q^86 - 10 * q^87 - 11 * q^89 - 3 * q^91 + 2 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^96 + 9 * q^97 - 4 * q^98 - 5 * 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.b 1
3.b odd 2 1 2925.2.a.t 1
5.b even 2 1 195.2.a.d 1
5.c odd 4 2 975.2.c.b 2
15.d odd 2 1 585.2.a.a 1
15.e even 4 2 2925.2.c.d 2
20.d odd 2 1 3120.2.a.n 1
35.c odd 2 1 9555.2.a.t 1
60.h even 2 1 9360.2.a.w 1
65.d even 2 1 2535.2.a.b 1
195.e odd 2 1 7605.2.a.v 1

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

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 3$$
$11$ $$T + 5$$
$13$ $$T + 1$$
$17$ $$T + 5$$
$19$ $$T - 2$$
$23$ $$T - 1$$
$29$ $$T - 10$$
$31$ $$T + 2$$
$37$ $$T - 3$$
$41$ $$T + 9$$
$43$ $$T - 4$$
$47$ $$T + 10$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T + 11$$
$67$ $$T - 4$$
$71$ $$T - 15$$
$73$ $$T + 6$$
$79$ $$T + 11$$
$83$ $$T + 8$$
$89$ $$T + 11$$
$97$ $$T - 9$$