Properties

 Label 2800.2.a.bi Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{7} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 - q^7 + (b + 1) * q^9 $$q - \beta q^{3} - q^{7} + (\beta + 1) q^{9} - \beta q^{11} + ( - \beta - 2) q^{13} + (\beta + 2) q^{17} + ( - 2 \beta + 4) q^{19} + \beta q^{21} - 2 \beta q^{23} + (\beta - 4) q^{27} + ( - 3 \beta + 2) q^{29} + (\beta + 4) q^{33} - 6 q^{37} + (3 \beta + 4) q^{39} + ( - 2 \beta + 2) q^{41} + (2 \beta + 4) q^{43} + (3 \beta - 4) q^{47} + q^{49} + ( - 3 \beta - 4) q^{51} + ( - 2 \beta + 2) q^{53} + ( - 2 \beta + 8) q^{57} + 4 q^{59} + ( - 6 \beta + 6) q^{61} + ( - \beta - 1) q^{63} + ( - 4 \beta + 4) q^{67} + (2 \beta + 8) q^{69} - 8 q^{71} + ( - 4 \beta + 6) q^{73} + \beta q^{77} + (\beta + 4) q^{79} - 7 q^{81} + 4 q^{83} + (\beta + 12) q^{87} + (2 \beta + 2) q^{89} + (\beta + 2) q^{91} + (5 \beta + 2) q^{97} + ( - 2 \beta - 4) q^{99} +O(q^{100})$$ q - b * q^3 - q^7 + (b + 1) * q^9 - b * q^11 + (-b - 2) * q^13 + (b + 2) * q^17 + (-2*b + 4) * q^19 + b * q^21 - 2*b * q^23 + (b - 4) * q^27 + (-3*b + 2) * q^29 + (b + 4) * q^33 - 6 * q^37 + (3*b + 4) * q^39 + (-2*b + 2) * q^41 + (2*b + 4) * q^43 + (3*b - 4) * q^47 + q^49 + (-3*b - 4) * q^51 + (-2*b + 2) * q^53 + (-2*b + 8) * q^57 + 4 * q^59 + (-6*b + 6) * q^61 + (-b - 1) * q^63 + (-4*b + 4) * q^67 + (2*b + 8) * q^69 - 8 * q^71 + (-4*b + 6) * q^73 + b * q^77 + (b + 4) * q^79 - 7 * q^81 + 4 * q^83 + (b + 12) * q^87 + (2*b + 2) * q^89 + (b + 2) * q^91 + (5*b + 2) * q^97 + (-2*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{7} + 3 q^{9} - q^{11} - 5 q^{13} + 5 q^{17} + 6 q^{19} + q^{21} - 2 q^{23} - 7 q^{27} + q^{29} + 9 q^{33} - 12 q^{37} + 11 q^{39} + 2 q^{41} + 10 q^{43} - 5 q^{47} + 2 q^{49} - 11 q^{51} + 2 q^{53} + 14 q^{57} + 8 q^{59} + 6 q^{61} - 3 q^{63} + 4 q^{67} + 18 q^{69} - 16 q^{71} + 8 q^{73} + q^{77} + 9 q^{79} - 14 q^{81} + 8 q^{83} + 25 q^{87} + 6 q^{89} + 5 q^{91} + 9 q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^7 + 3 * q^9 - q^11 - 5 * q^13 + 5 * q^17 + 6 * q^19 + q^21 - 2 * q^23 - 7 * q^27 + q^29 + 9 * q^33 - 12 * q^37 + 11 * q^39 + 2 * q^41 + 10 * q^43 - 5 * q^47 + 2 * q^49 - 11 * q^51 + 2 * q^53 + 14 * q^57 + 8 * q^59 + 6 * q^61 - 3 * q^63 + 4 * q^67 + 18 * q^69 - 16 * q^71 + 8 * q^73 + q^77 + 9 * q^79 - 14 * q^81 + 8 * q^83 + 25 * q^87 + 6 * q^89 + 5 * q^91 + 9 * q^97 - 10 * 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.56155 −1.56155
0 −2.56155 0 0 0 −1.00000 0 3.56155 0
1.2 0 1.56155 0 0 0 −1.00000 0 −0.561553 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

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 2800.2.a.bi 2
4.b odd 2 1 175.2.a.f 2
5.b even 2 1 560.2.a.i 2
5.c odd 4 2 2800.2.g.t 4
12.b even 2 1 1575.2.a.p 2
15.d odd 2 1 5040.2.a.bt 2
20.d odd 2 1 35.2.a.b 2
20.e even 4 2 175.2.b.b 4
28.d even 2 1 1225.2.a.s 2
35.c odd 2 1 3920.2.a.bs 2
40.e odd 2 1 2240.2.a.bh 2
40.f even 2 1 2240.2.a.bd 2
60.h even 2 1 315.2.a.e 2
60.l odd 4 2 1575.2.d.e 4
140.c even 2 1 245.2.a.d 2
140.j odd 4 2 1225.2.b.f 4
140.p odd 6 2 245.2.e.i 4
140.s even 6 2 245.2.e.h 4
220.g even 2 1 4235.2.a.m 2
260.g odd 2 1 5915.2.a.l 2
420.o odd 2 1 2205.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 20.d odd 2 1
175.2.a.f 2 4.b odd 2 1
175.2.b.b 4 20.e even 4 2
245.2.a.d 2 140.c even 2 1
245.2.e.h 4 140.s even 6 2
245.2.e.i 4 140.p odd 6 2
315.2.a.e 2 60.h even 2 1
560.2.a.i 2 5.b even 2 1
1225.2.a.s 2 28.d even 2 1
1225.2.b.f 4 140.j odd 4 2
1575.2.a.p 2 12.b even 2 1
1575.2.d.e 4 60.l odd 4 2
2205.2.a.x 2 420.o odd 2 1
2240.2.a.bd 2 40.f even 2 1
2240.2.a.bh 2 40.e odd 2 1
2800.2.a.bi 2 1.a even 1 1 trivial
2800.2.g.t 4 5.c odd 4 2
3920.2.a.bs 2 35.c odd 2 1
4235.2.a.m 2 220.g even 2 1
5040.2.a.bt 2 15.d odd 2 1
5915.2.a.l 2 260.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2800))$$:

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{11}^{2} + T_{11} - 4$$ T11^2 + T11 - 4 $$T_{13}^{2} + 5T_{13} + 2$$ T13^2 + 5*T13 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + T - 4$$
$13$ $$T^{2} + 5T + 2$$
$17$ $$T^{2} - 5T + 2$$
$19$ $$T^{2} - 6T - 8$$
$23$ $$T^{2} + 2T - 16$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2}$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 2T - 16$$
$43$ $$T^{2} - 10T + 8$$
$47$ $$T^{2} + 5T - 32$$
$53$ $$T^{2} - 2T - 16$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 6T - 144$$
$67$ $$T^{2} - 4T - 64$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} - 8T - 52$$
$79$ $$T^{2} - 9T + 16$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} - 6T - 8$$
$97$ $$T^{2} - 9T - 86$$
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