Properties

 Label 1600.2.n.d Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.n (of order $$4$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 800) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{3} + ( - i + 1) q^{7} - i q^{9} +O(q^{10})$$ q + (-i - 1) * q^3 + (-i + 1) * q^7 - i * q^9 $$q + ( - i - 1) q^{3} + ( - i + 1) q^{7} - i q^{9} + 4 i q^{11} + (4 i - 4) q^{13} + ( - 4 i - 4) q^{17} + 4 q^{19} - 2 q^{21} + (5 i + 5) q^{23} + (4 i - 4) q^{27} + 2 i q^{29} + 8 i q^{31} + ( - 4 i + 4) q^{33} + 8 q^{39} - 4 q^{41} + ( - 7 i - 7) q^{43} + ( - 3 i + 3) q^{47} + 5 i q^{49} + 8 i q^{51} + (4 i - 4) q^{53} + ( - 4 i - 4) q^{57} + 4 q^{59} + 8 q^{61} + ( - i - 1) q^{63} + ( - 3 i + 3) q^{67} - 10 i q^{69} + 16 i q^{71} + ( - 4 i + 4) q^{73} + (4 i + 4) q^{77} + 8 q^{79} + 5 q^{81} + (5 i + 5) q^{83} + ( - 2 i + 2) q^{87} - 10 i q^{89} + 8 i q^{91} + ( - 8 i + 8) q^{93} + (12 i + 12) q^{97} + 4 q^{99} +O(q^{100})$$ q + (-i - 1) * q^3 + (-i + 1) * q^7 - i * q^9 + 4*i * q^11 + (4*i - 4) * q^13 + (-4*i - 4) * q^17 + 4 * q^19 - 2 * q^21 + (5*i + 5) * q^23 + (4*i - 4) * q^27 + 2*i * q^29 + 8*i * q^31 + (-4*i + 4) * q^33 + 8 * q^39 - 4 * q^41 + (-7*i - 7) * q^43 + (-3*i + 3) * q^47 + 5*i * q^49 + 8*i * q^51 + (4*i - 4) * q^53 + (-4*i - 4) * q^57 + 4 * q^59 + 8 * q^61 + (-i - 1) * q^63 + (-3*i + 3) * q^67 - 10*i * q^69 + 16*i * q^71 + (-4*i + 4) * q^73 + (4*i + 4) * q^77 + 8 * q^79 + 5 * q^81 + (5*i + 5) * q^83 + (-2*i + 2) * q^87 - 10*i * q^89 + 8*i * q^91 + (-8*i + 8) * q^93 + (12*i + 12) * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^7 $$2 q - 2 q^{3} + 2 q^{7} - 8 q^{13} - 8 q^{17} + 8 q^{19} - 4 q^{21} + 10 q^{23} - 8 q^{27} + 8 q^{33} + 16 q^{39} - 8 q^{41} - 14 q^{43} + 6 q^{47} - 8 q^{53} - 8 q^{57} + 8 q^{59} + 16 q^{61} - 2 q^{63} + 6 q^{67} + 8 q^{73} + 8 q^{77} + 16 q^{79} + 10 q^{81} + 10 q^{83} + 4 q^{87} + 16 q^{93} + 24 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^7 - 8 * q^13 - 8 * q^17 + 8 * q^19 - 4 * q^21 + 10 * q^23 - 8 * q^27 + 8 * q^33 + 16 * q^39 - 8 * q^41 - 14 * q^43 + 6 * q^47 - 8 * q^53 - 8 * q^57 + 8 * q^59 + 16 * q^61 - 2 * q^63 + 6 * q^67 + 8 * q^73 + 8 * q^77 + 16 * q^79 + 10 * q^81 + 10 * q^83 + 4 * q^87 + 16 * q^93 + 24 * q^97 + 8 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$i$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1343.1
 − 1.00000i 1.00000i
0 −1.00000 + 1.00000i 0 0 0 1.00000 + 1.00000i 0 1.00000i 0
1407.1 0 −1.00000 1.00000i 0 0 0 1.00000 1.00000i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.n.d 2
4.b odd 2 1 1600.2.n.i 2
5.b even 2 1 1600.2.n.k 2
5.c odd 4 1 1600.2.n.f 2
5.c odd 4 1 1600.2.n.i 2
8.b even 2 1 800.2.n.i yes 2
8.d odd 2 1 800.2.n.d yes 2
20.d odd 2 1 1600.2.n.f 2
20.e even 4 1 inner 1600.2.n.d 2
20.e even 4 1 1600.2.n.k 2
40.e odd 2 1 800.2.n.g yes 2
40.f even 2 1 800.2.n.b 2
40.i odd 4 1 800.2.n.d yes 2
40.i odd 4 1 800.2.n.g yes 2
40.k even 4 1 800.2.n.b 2
40.k even 4 1 800.2.n.i yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.n.b 2 40.f even 2 1
800.2.n.b 2 40.k even 4 1
800.2.n.d yes 2 8.d odd 2 1
800.2.n.d yes 2 40.i odd 4 1
800.2.n.g yes 2 40.e odd 2 1
800.2.n.g yes 2 40.i odd 4 1
800.2.n.i yes 2 8.b even 2 1
800.2.n.i yes 2 40.k even 4 1
1600.2.n.d 2 1.a even 1 1 trivial
1600.2.n.d 2 20.e even 4 1 inner
1600.2.n.f 2 5.c odd 4 1
1600.2.n.f 2 20.d odd 2 1
1600.2.n.i 2 4.b odd 2 1
1600.2.n.i 2 5.c odd 4 1
1600.2.n.k 2 5.b even 2 1
1600.2.n.k 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} + 2$$ T3^2 + 2*T3 + 2 $$T_{7}^{2} - 2T_{7} + 2$$ T7^2 - 2*T7 + 2 $$T_{11}^{2} + 16$$ T11^2 + 16 $$T_{13}^{2} + 8T_{13} + 32$$ T13^2 + 8*T13 + 32

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T + 2$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 8T + 32$$
$17$ $$T^{2} + 8T + 32$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 10T + 50$$
$29$ $$T^{2} + 4$$
$31$ $$T^{2} + 64$$
$37$ $$T^{2}$$
$41$ $$(T + 4)^{2}$$
$43$ $$T^{2} + 14T + 98$$
$47$ $$T^{2} - 6T + 18$$
$53$ $$T^{2} + 8T + 32$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} - 6T + 18$$
$71$ $$T^{2} + 256$$
$73$ $$T^{2} - 8T + 32$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 10T + 50$$
$89$ $$T^{2} + 100$$
$97$ $$T^{2} - 24T + 288$$