Properties

 Label 975.2.a.d Level $975$ Weight $2$ Character orbit 975.a Self dual yes Analytic conductor $7.785$ Analytic rank $0$ 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: $$0$$ 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.d 1
3.b odd 2 1 2925.2.a.o 1
5.b even 2 1 975.2.a.k yes 1
5.c odd 4 2 975.2.c.g 2
15.d odd 2 1 2925.2.a.b 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 1.a even 1 1 trivial
975.2.a.k yes 1 5.b even 2 1
975.2.c.g 2 5.c odd 4 2
2925.2.a.b 1 15.d odd 2 1
2925.2.a.o 1 3.b 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

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