# Properties

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

# Learn more

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: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 - q^4 + q^6 - 3 * q^7 - 3 * q^8 + q^9 $$q + q^{2} + q^{3} - q^{4} + q^{6} - 3 q^{7} - 3 q^{8} + q^{9} - q^{11} - q^{12} + q^{13} - 3 q^{14} - q^{16} - 5 q^{17} + q^{18} - 8 q^{19} - 3 q^{21} - q^{22} - 3 q^{24} + q^{26} + q^{27} + 3 q^{28} + q^{29} + 3 q^{31} + 5 q^{32} - q^{33} - 5 q^{34} - q^{36} - 8 q^{37} - 8 q^{38} + q^{39} - 2 q^{41} - 3 q^{42} + 8 q^{43} + q^{44} - 11 q^{47} - q^{48} + 2 q^{49} - 5 q^{51} - q^{52} - 11 q^{53} + q^{54} + 9 q^{56} - 8 q^{57} + q^{58} + 5 q^{59} + q^{61} + 3 q^{62} - 3 q^{63} + 7 q^{64} - q^{66} + 3 q^{67} + 5 q^{68} + 16 q^{71} - 3 q^{72} - 4 q^{73} - 8 q^{74} + 8 q^{76} + 3 q^{77} + q^{78} + 12 q^{79} + q^{81} - 2 q^{82} - 3 q^{83} + 3 q^{84} + 8 q^{86} + q^{87} + 3 q^{88} - 3 q^{91} + 3 q^{93} - 11 q^{94} + 5 q^{96} - 2 q^{97} + 2 q^{98} - q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + q^6 - 3 * q^7 - 3 * q^8 + q^9 - q^11 - q^12 + q^13 - 3 * q^14 - q^16 - 5 * q^17 + q^18 - 8 * q^19 - 3 * q^21 - q^22 - 3 * q^24 + q^26 + q^27 + 3 * q^28 + q^29 + 3 * q^31 + 5 * q^32 - q^33 - 5 * q^34 - q^36 - 8 * q^37 - 8 * q^38 + q^39 - 2 * q^41 - 3 * q^42 + 8 * q^43 + q^44 - 11 * q^47 - q^48 + 2 * q^49 - 5 * q^51 - q^52 - 11 * q^53 + q^54 + 9 * q^56 - 8 * q^57 + q^58 + 5 * q^59 + q^61 + 3 * q^62 - 3 * q^63 + 7 * q^64 - q^66 + 3 * q^67 + 5 * q^68 + 16 * q^71 - 3 * q^72 - 4 * q^73 - 8 * q^74 + 8 * q^76 + 3 * q^77 + q^78 + 12 * q^79 + q^81 - 2 * q^82 - 3 * q^83 + 3 * q^84 + 8 * q^86 + q^87 + 3 * q^88 - 3 * q^91 + 3 * q^93 - 11 * q^94 + 5 * q^96 - 2 * q^97 + 2 * 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
1.00000 1.00000 −1.00000 0 1.00000 −3.00000 −3.00000 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.k yes 1
3.b odd 2 1 2925.2.a.b 1
5.b even 2 1 975.2.a.d 1
5.c odd 4 2 975.2.c.g 2
15.d odd 2 1 2925.2.a.o 1
15.e even 4 2 2925.2.c.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.a.d 1 5.b even 2 1
975.2.a.k yes 1 1.a even 1 1 trivial
975.2.c.g 2 5.c odd 4 2
2925.2.a.b 1 3.b odd 2 1
2925.2.a.o 1 15.d odd 2 1
2925.2.c.i 2 15.e even 4 2

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{7} + 3$$ T7 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T + 5$$
$19$ $$T + 8$$
$23$ $$T$$
$29$ $$T - 1$$
$31$ $$T - 3$$
$37$ $$T + 8$$
$41$ $$T + 2$$
$43$ $$T - 8$$
$47$ $$T + 11$$
$53$ $$T + 11$$
$59$ $$T - 5$$
$61$ $$T - 1$$
$67$ $$T - 3$$
$71$ $$T - 16$$
$73$ $$T + 4$$
$79$ $$T - 12$$
$83$ $$T + 3$$
$89$ $$T$$
$97$ $$T + 2$$
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