Properties

 Label 4003.2.a.a Level $4003$ Weight $2$ Character orbit 4003.a Self dual yes Analytic conductor $31.964$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, 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("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4003$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4003.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: $$31.9641159291$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + \beta q^{3} + \beta q^{5} + 2 q^{6} - q^{7} - 2 \beta q^{8} - q^{9} +O(q^{10})$$ q + b * q^2 + b * q^3 + b * q^5 + 2 * q^6 - q^7 - 2*b * q^8 - q^9 $$q + \beta q^{2} + \beta q^{3} + \beta q^{5} + 2 q^{6} - q^{7} - 2 \beta q^{8} - q^{9} + 2 q^{10} + ( - 2 \beta - 2) q^{11} + (2 \beta + 4) q^{13} - \beta q^{14} + 2 q^{15} - 4 q^{16} + ( - 4 \beta - 1) q^{17} - \beta q^{18} + (4 \beta + 1) q^{19} - \beta q^{21} + ( - 2 \beta - 4) q^{22} + (2 \beta - 6) q^{23} - 4 q^{24} - 3 q^{25} + (4 \beta + 4) q^{26} - 4 \beta q^{27} + ( - 2 \beta - 4) q^{29} + 2 \beta q^{30} + ( - 6 \beta + 2) q^{31} + ( - 2 \beta - 4) q^{33} + ( - \beta - 8) q^{34} - \beta q^{35} + ( - 2 \beta - 7) q^{37} + (\beta + 8) q^{38} + (4 \beta + 4) q^{39} - 4 q^{40} + (2 \beta + 8) q^{41} - 2 q^{42} - \beta q^{45} + ( - 6 \beta + 4) q^{46} + (5 \beta + 4) q^{47} - 4 \beta q^{48} - 6 q^{49} - 3 \beta q^{50} + ( - \beta - 8) q^{51} + ( - 8 \beta - 2) q^{53} - 8 q^{54} + ( - 2 \beta - 4) q^{55} + 2 \beta q^{56} + (\beta + 8) q^{57} + ( - 4 \beta - 4) q^{58} + (2 \beta - 6) q^{59} + (2 \beta + 10) q^{61} + (2 \beta - 12) q^{62} + q^{63} + 8 q^{64} + (4 \beta + 4) q^{65} + ( - 4 \beta - 4) q^{66} + ( - 4 \beta + 7) q^{67} + ( - 6 \beta + 4) q^{69} - 2 q^{70} + 6 q^{71} + 2 \beta q^{72} - 3 q^{73} + ( - 7 \beta - 4) q^{74} - 3 \beta q^{75} + (2 \beta + 2) q^{77} + (4 \beta + 8) q^{78} + ( - 6 \beta + 3) q^{79} - 4 \beta q^{80} - 5 q^{81} + (8 \beta + 4) q^{82} + ( - 10 \beta + 3) q^{83} + ( - \beta - 8) q^{85} + ( - 4 \beta - 4) q^{87} + (4 \beta + 8) q^{88} + (2 \beta - 9) q^{89} - 2 q^{90} + ( - 2 \beta - 4) q^{91} + (2 \beta - 12) q^{93} + (4 \beta + 10) q^{94} + (\beta + 8) q^{95} + (3 \beta + 6) q^{97} - 6 \beta q^{98} + (2 \beta + 2) q^{99} +O(q^{100})$$ q + b * q^2 + b * q^3 + b * q^5 + 2 * q^6 - q^7 - 2*b * q^8 - q^9 + 2 * q^10 + (-2*b - 2) * q^11 + (2*b + 4) * q^13 - b * q^14 + 2 * q^15 - 4 * q^16 + (-4*b - 1) * q^17 - b * q^18 + (4*b + 1) * q^19 - b * q^21 + (-2*b - 4) * q^22 + (2*b - 6) * q^23 - 4 * q^24 - 3 * q^25 + (4*b + 4) * q^26 - 4*b * q^27 + (-2*b - 4) * q^29 + 2*b * q^30 + (-6*b + 2) * q^31 + (-2*b - 4) * q^33 + (-b - 8) * q^34 - b * q^35 + (-2*b - 7) * q^37 + (b + 8) * q^38 + (4*b + 4) * q^39 - 4 * q^40 + (2*b + 8) * q^41 - 2 * q^42 - b * q^45 + (-6*b + 4) * q^46 + (5*b + 4) * q^47 - 4*b * q^48 - 6 * q^49 - 3*b * q^50 + (-b - 8) * q^51 + (-8*b - 2) * q^53 - 8 * q^54 + (-2*b - 4) * q^55 + 2*b * q^56 + (b + 8) * q^57 + (-4*b - 4) * q^58 + (2*b - 6) * q^59 + (2*b + 10) * q^61 + (2*b - 12) * q^62 + q^63 + 8 * q^64 + (4*b + 4) * q^65 + (-4*b - 4) * q^66 + (-4*b + 7) * q^67 + (-6*b + 4) * q^69 - 2 * q^70 + 6 * q^71 + 2*b * q^72 - 3 * q^73 + (-7*b - 4) * q^74 - 3*b * q^75 + (2*b + 2) * q^77 + (4*b + 8) * q^78 + (-6*b + 3) * q^79 - 4*b * q^80 - 5 * q^81 + (8*b + 4) * q^82 + (-10*b + 3) * q^83 + (-b - 8) * q^85 + (-4*b - 4) * q^87 + (4*b + 8) * q^88 + (2*b - 9) * q^89 - 2 * q^90 + (-2*b - 4) * q^91 + (2*b - 12) * q^93 + (4*b + 10) * q^94 + (b + 8) * q^95 + (3*b + 6) * q^97 - 6*b * q^98 + (2*b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^6 - 2 * q^7 - 2 * q^9 $$2 q + 4 q^{6} - 2 q^{7} - 2 q^{9} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 2 q^{19} - 8 q^{22} - 12 q^{23} - 8 q^{24} - 6 q^{25} + 8 q^{26} - 8 q^{29} + 4 q^{31} - 8 q^{33} - 16 q^{34} - 14 q^{37} + 16 q^{38} + 8 q^{39} - 8 q^{40} + 16 q^{41} - 4 q^{42} + 8 q^{46} + 8 q^{47} - 12 q^{49} - 16 q^{51} - 4 q^{53} - 16 q^{54} - 8 q^{55} + 16 q^{57} - 8 q^{58} - 12 q^{59} + 20 q^{61} - 24 q^{62} + 2 q^{63} + 16 q^{64} + 8 q^{65} - 8 q^{66} + 14 q^{67} + 8 q^{69} - 4 q^{70} + 12 q^{71} - 6 q^{73} - 8 q^{74} + 4 q^{77} + 16 q^{78} + 6 q^{79} - 10 q^{81} + 8 q^{82} + 6 q^{83} - 16 q^{85} - 8 q^{87} + 16 q^{88} - 18 q^{89} - 4 q^{90} - 8 q^{91} - 24 q^{93} + 20 q^{94} + 16 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^6 - 2 * q^7 - 2 * q^9 + 4 * q^10 - 4 * q^11 + 8 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 2 * q^19 - 8 * q^22 - 12 * q^23 - 8 * q^24 - 6 * q^25 + 8 * q^26 - 8 * q^29 + 4 * q^31 - 8 * q^33 - 16 * q^34 - 14 * q^37 + 16 * q^38 + 8 * q^39 - 8 * q^40 + 16 * q^41 - 4 * q^42 + 8 * q^46 + 8 * q^47 - 12 * q^49 - 16 * q^51 - 4 * q^53 - 16 * q^54 - 8 * q^55 + 16 * q^57 - 8 * q^58 - 12 * q^59 + 20 * q^61 - 24 * q^62 + 2 * q^63 + 16 * q^64 + 8 * q^65 - 8 * q^66 + 14 * q^67 + 8 * q^69 - 4 * q^70 + 12 * q^71 - 6 * q^73 - 8 * q^74 + 4 * q^77 + 16 * q^78 + 6 * q^79 - 10 * q^81 + 8 * q^82 + 6 * q^83 - 16 * q^85 - 8 * q^87 + 16 * q^88 - 18 * q^89 - 4 * q^90 - 8 * q^91 - 24 * q^93 + 20 * q^94 + 16 * q^95 + 12 * q^97 + 4 * 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
 −1.41421 1.41421
−1.41421 −1.41421 0 −1.41421 2.00000 −1.00000 2.82843 −1.00000 2.00000
1.2 1.41421 1.41421 0 1.41421 2.00000 −1.00000 −2.82843 −1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$4003$$ $$+1$$

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 4003.2.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4003.2.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4003))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 2$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 4T - 4$$
$13$ $$T^{2} - 8T + 8$$
$17$ $$T^{2} + 2T - 31$$
$19$ $$T^{2} - 2T - 31$$
$23$ $$T^{2} + 12T + 28$$
$29$ $$T^{2} + 8T + 8$$
$31$ $$T^{2} - 4T - 68$$
$37$ $$T^{2} + 14T + 41$$
$41$ $$T^{2} - 16T + 56$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 8T - 34$$
$53$ $$T^{2} + 4T - 124$$
$59$ $$T^{2} + 12T + 28$$
$61$ $$T^{2} - 20T + 92$$
$67$ $$T^{2} - 14T + 17$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T + 3)^{2}$$
$79$ $$T^{2} - 6T - 63$$
$83$ $$T^{2} - 6T - 191$$
$89$ $$T^{2} + 18T + 73$$
$97$ $$T^{2} - 12T + 18$$