Properties

 Label 4004.2.a.f Level 4004 Weight 2 Character orbit 4004.a Self dual yes Analytic conductor 31.972 Analytic rank 1 Dimension 5 CM no Inner twists 1

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Newspace parameters

 Level: $$N$$ = $$4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4004.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$31.9721009693$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.463341.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} -\beta_{2} q^{5} - q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} -\beta_{2} q^{5} - q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} - q^{11} + q^{13} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{15} + ( -3 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} + ( 1 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{27} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{29} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( -1 + \beta_{1} ) q^{33} + \beta_{2} q^{35} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{37} + ( 1 - \beta_{1} ) q^{39} + ( -4 + \beta_{4} ) q^{41} + ( -5 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{43} + ( -7 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{45} + ( 3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{47} + q^{49} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{51} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} + \beta_{2} q^{55} + ( -1 + \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{57} + ( 2 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{59} + ( -5 - \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} -\beta_{2} q^{65} + ( -4 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{67} + ( -5 + 5 \beta_{1} + 3 \beta_{3} ) q^{69} + ( 2 + 4 \beta_{4} ) q^{71} + ( -8 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{73} + ( -3 + \beta_{2} - \beta_{3} ) q^{75} + q^{77} + ( -5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{79} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{81} + ( 2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{83} + ( -1 - 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{85} + ( 2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{87} + ( -5 + \beta_{1} + \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{89} - q^{91} + ( -7 - 3 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -4 - \beta_{1} + 2 \beta_{2} ) q^{95} + ( -5 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{97} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 3q^{3} - 5q^{7} + 2q^{9} + O(q^{10})$$ $$5q + 3q^{3} - 5q^{7} + 2q^{9} - 5q^{11} + 5q^{13} - 3q^{15} - 9q^{17} - 4q^{19} - 3q^{21} - 4q^{23} + q^{25} + 3q^{27} - 8q^{29} + 7q^{31} - 3q^{33} - 10q^{37} + 3q^{39} - 18q^{41} - 22q^{43} - 26q^{45} + 12q^{47} + 5q^{49} - 7q^{51} + 7q^{53} - 6q^{57} - q^{59} - 26q^{61} - 2q^{63} - 8q^{67} - 12q^{69} + 18q^{71} - 25q^{73} - 16q^{75} + 5q^{77} - 6q^{79} - 7q^{81} + 11q^{83} - 7q^{85} - 5q^{87} - 14q^{89} - 5q^{91} - 41q^{93} - 22q^{95} - 33q^{97} - 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 6 x^{3} + 6 x^{2} + 6 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 5 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 6 \nu + 5$$ $$\beta_{4}$$ $$=$$ $$-2 \nu^{4} + 5 \nu^{3} + 10 \nu^{2} - 17 \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 2 \beta_{2} + 7 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 14 \beta_{1} + 17$$

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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.09892 1.53838 −0.231747 −0.466886 −1.93866
0 −2.09892 0 −2.18167 0 −1.00000 0 1.40545 0
1.2 0 −0.538379 0 3.82180 0 −1.00000 0 −2.71015 0
1.3 0 1.23175 0 −0.600512 0 −1.00000 0 −1.48280 0
1.4 0 1.46689 0 1.17328 0 −1.00000 0 −0.848245 0
1.5 0 2.93866 0 −2.21290 0 −1.00000 0 5.63574 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 4004.2.a.f 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.f 5 1.a even 1 1 trivial

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$
$$13$$ $$-1$$

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{5} - 3 T_{3}^{4} - 4 T_{3}^{3} + 14 T_{3}^{2} - 3 T_{3} - 6$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4004))$$.