Properties

 Label 1922.2.a.c Level $1922$ Weight $2$ Character orbit 1922.a Self dual yes Analytic conductor $15.347$ 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] = [1922,2,Mod(1,1922)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1922, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1922.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1922 = 2 \cdot 31^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1922.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: $$15.3472472685$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 62) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 3 q^{3} + q^{4} + q^{5} - 3 q^{6} - 3 q^{7} + q^{8} + 6 q^{9}+O(q^{10})$$ q + q^2 - 3 * q^3 + q^4 + q^5 - 3 * q^6 - 3 * q^7 + q^8 + 6 * q^9 $$q + q^{2} - 3 q^{3} + q^{4} + q^{5} - 3 q^{6} - 3 q^{7} + q^{8} + 6 q^{9} + q^{10} + 3 q^{11} - 3 q^{12} - 5 q^{13} - 3 q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 6 q^{18} + 7 q^{19} + q^{20} + 9 q^{21} + 3 q^{22} + 4 q^{23} - 3 q^{24} - 4 q^{25} - 5 q^{26} - 9 q^{27} - 3 q^{28} - 2 q^{29} - 3 q^{30} + q^{32} - 9 q^{33} - 3 q^{34} - 3 q^{35} + 6 q^{36} - q^{37} + 7 q^{38} + 15 q^{39} + q^{40} - 9 q^{41} + 9 q^{42} + q^{43} + 3 q^{44} + 6 q^{45} + 4 q^{46} - 8 q^{47} - 3 q^{48} + 2 q^{49} - 4 q^{50} + 9 q^{51} - 5 q^{52} + 3 q^{53} - 9 q^{54} + 3 q^{55} - 3 q^{56} - 21 q^{57} - 2 q^{58} + 3 q^{59} - 3 q^{60} - 6 q^{61} - 18 q^{63} + q^{64} - 5 q^{65} - 9 q^{66} - 3 q^{67} - 3 q^{68} - 12 q^{69} - 3 q^{70} - q^{71} + 6 q^{72} - 7 q^{73} - q^{74} + 12 q^{75} + 7 q^{76} - 9 q^{77} + 15 q^{78} - q^{79} + q^{80} + 9 q^{81} - 9 q^{82} - 5 q^{83} + 9 q^{84} - 3 q^{85} + q^{86} + 6 q^{87} + 3 q^{88} - 6 q^{89} + 6 q^{90} + 15 q^{91} + 4 q^{92} - 8 q^{94} + 7 q^{95} - 3 q^{96} + 14 q^{97} + 2 q^{98} + 18 q^{99}+O(q^{100})$$ q + q^2 - 3 * q^3 + q^4 + q^5 - 3 * q^6 - 3 * q^7 + q^8 + 6 * q^9 + q^10 + 3 * q^11 - 3 * q^12 - 5 * q^13 - 3 * q^14 - 3 * q^15 + q^16 - 3 * q^17 + 6 * q^18 + 7 * q^19 + q^20 + 9 * q^21 + 3 * q^22 + 4 * q^23 - 3 * q^24 - 4 * q^25 - 5 * q^26 - 9 * q^27 - 3 * q^28 - 2 * q^29 - 3 * q^30 + q^32 - 9 * q^33 - 3 * q^34 - 3 * q^35 + 6 * q^36 - q^37 + 7 * q^38 + 15 * q^39 + q^40 - 9 * q^41 + 9 * q^42 + q^43 + 3 * q^44 + 6 * q^45 + 4 * q^46 - 8 * q^47 - 3 * q^48 + 2 * q^49 - 4 * q^50 + 9 * q^51 - 5 * q^52 + 3 * q^53 - 9 * q^54 + 3 * q^55 - 3 * q^56 - 21 * q^57 - 2 * q^58 + 3 * q^59 - 3 * q^60 - 6 * q^61 - 18 * q^63 + q^64 - 5 * q^65 - 9 * q^66 - 3 * q^67 - 3 * q^68 - 12 * q^69 - 3 * q^70 - q^71 + 6 * q^72 - 7 * q^73 - q^74 + 12 * q^75 + 7 * q^76 - 9 * q^77 + 15 * q^78 - q^79 + q^80 + 9 * q^81 - 9 * q^82 - 5 * q^83 + 9 * q^84 - 3 * q^85 + q^86 + 6 * q^87 + 3 * q^88 - 6 * q^89 + 6 * q^90 + 15 * q^91 + 4 * q^92 - 8 * q^94 + 7 * q^95 - 3 * q^96 + 14 * q^97 + 2 * q^98 + 18 * 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 −3.00000 1.00000 1.00000 −3.00000 −3.00000 1.00000 6.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$31$$ $$-1$$

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 1922.2.a.c 1
31.b odd 2 1 1922.2.a.e 1
31.e odd 6 2 62.2.c.b 2
93.g even 6 2 558.2.e.b 2
124.g even 6 2 496.2.i.g 2
155.i odd 6 2 1550.2.e.d 2
155.p even 12 4 1550.2.p.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.2.c.b 2 31.e odd 6 2
496.2.i.g 2 124.g even 6 2
558.2.e.b 2 93.g even 6 2
1550.2.e.d 2 155.i odd 6 2
1550.2.p.a 4 155.p even 12 4
1922.2.a.c 1 1.a even 1 1 trivial
1922.2.a.e 1 31.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1922))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 3$$
$5$ $$T - 1$$
$7$ $$T + 3$$
$11$ $$T - 3$$
$13$ $$T + 5$$
$17$ $$T + 3$$
$19$ $$T - 7$$
$23$ $$T - 4$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 1$$
$41$ $$T + 9$$
$43$ $$T - 1$$
$47$ $$T + 8$$
$53$ $$T - 3$$
$59$ $$T - 3$$
$61$ $$T + 6$$
$67$ $$T + 3$$
$71$ $$T + 1$$
$73$ $$T + 7$$
$79$ $$T + 1$$
$83$ $$T + 5$$
$89$ $$T + 6$$
$97$ $$T - 14$$