Properties

 Label 6900.2.f.e Level $6900$ Weight $2$ Character orbit 6900.f Analytic conductor $55.097$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6900.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$55.0967773947$$ 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 1380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + i * q^7 - q^9 $$q + i q^{3} + i q^{7} - q^{9} - 4 i q^{13} + 3 i q^{17} + 4 q^{19} - q^{21} - i q^{23} - i q^{27} + 3 q^{29} - 7 q^{31} - 11 i q^{37} + 4 q^{39} - 9 q^{41} - 4 i q^{43} - 6 i q^{47} + 6 q^{49} - 3 q^{51} + 9 i q^{53} + 4 i q^{57} - 3 q^{59} - 10 q^{61} - i q^{63} + 13 i q^{67} + q^{69} + 9 q^{71} - 16 i q^{73} - 8 q^{79} + q^{81} - 15 i q^{83} + 3 i q^{87} + 4 q^{91} - 7 i q^{93} - 2 i q^{97} +O(q^{100})$$ q + i * q^3 + i * q^7 - q^9 - 4*i * q^13 + 3*i * q^17 + 4 * q^19 - q^21 - i * q^23 - i * q^27 + 3 * q^29 - 7 * q^31 - 11*i * q^37 + 4 * q^39 - 9 * q^41 - 4*i * q^43 - 6*i * q^47 + 6 * q^49 - 3 * q^51 + 9*i * q^53 + 4*i * q^57 - 3 * q^59 - 10 * q^61 - i * q^63 + 13*i * q^67 + q^69 + 9 * q^71 - 16*i * q^73 - 8 * q^79 + q^81 - 15*i * q^83 + 3*i * q^87 + 4 * q^91 - 7*i * q^93 - 2*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 8 q^{19} - 2 q^{21} + 6 q^{29} - 14 q^{31} + 8 q^{39} - 18 q^{41} + 12 q^{49} - 6 q^{51} - 6 q^{59} - 20 q^{61} + 2 q^{69} + 18 q^{71} - 16 q^{79} + 2 q^{81} + 8 q^{91}+O(q^{100})$$ 2 * q - 2 * q^9 + 8 * q^19 - 2 * q^21 + 6 * q^29 - 14 * q^31 + 8 * q^39 - 18 * q^41 + 12 * q^49 - 6 * q^51 - 6 * q^59 - 20 * q^61 + 2 * q^69 + 18 * q^71 - 16 * q^79 + 2 * q^81 + 8 * q^91

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1201$$ $$3451$$ $$4601$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6349.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
6349.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6900.2.f.e 2
5.b even 2 1 inner 6900.2.f.e 2
5.c odd 4 1 1380.2.a.c 1
5.c odd 4 1 6900.2.a.b 1
15.e even 4 1 4140.2.a.i 1
20.e even 4 1 5520.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.c 1 5.c odd 4 1
4140.2.a.i 1 15.e even 4 1
5520.2.a.g 1 20.e even 4 1
6900.2.a.b 1 5.c odd 4 1
6900.2.f.e 2 1.a even 1 1 trivial
6900.2.f.e 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(6900, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}$$ T11

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 121$$
$41$ $$(T + 9)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 81$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$(T - 9)^{2}$$
$73$ $$T^{2} + 256$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 225$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 4$$