Properties

 Label 775.2.b.c Level $775$ Weight $2$ Character orbit 775.b Analytic conductor $6.188$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.18840615665$$ 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 155) 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + 2 q^{4} + 2 q^{9} +O(q^{10})$$ q - i * q^3 + 2 * q^4 + 2 * q^9 $$q - i q^{3} + 2 q^{4} + 2 q^{9} - 4 q^{11} - 2 i q^{12} - 6 i q^{13} + 4 q^{16} - 5 i q^{17} + q^{19} + 8 i q^{23} - 5 i q^{27} + 10 q^{29} - q^{31} + 4 i q^{33} + 4 q^{36} - i q^{37} - 6 q^{39} - 3 q^{41} - 7 i q^{43} - 8 q^{44} + 6 i q^{47} - 4 i q^{48} + 7 q^{49} - 5 q^{51} - 12 i q^{52} + 5 i q^{53} - i q^{57} - 11 q^{59} - 12 q^{61} + 8 q^{64} + 2 i q^{67} - 10 i q^{68} + 8 q^{69} + 9 q^{71} - 9 i q^{73} + 2 q^{76} + 10 q^{79} + q^{81} + 9 i q^{83} - 10 i q^{87} + 16 i q^{92} + i q^{93} + 14 i q^{97} - 8 q^{99} +O(q^{100})$$ q - i * q^3 + 2 * q^4 + 2 * q^9 - 4 * q^11 - 2*i * q^12 - 6*i * q^13 + 4 * q^16 - 5*i * q^17 + q^19 + 8*i * q^23 - 5*i * q^27 + 10 * q^29 - q^31 + 4*i * q^33 + 4 * q^36 - i * q^37 - 6 * q^39 - 3 * q^41 - 7*i * q^43 - 8 * q^44 + 6*i * q^47 - 4*i * q^48 + 7 * q^49 - 5 * q^51 - 12*i * q^52 + 5*i * q^53 - i * q^57 - 11 * q^59 - 12 * q^61 + 8 * q^64 + 2*i * q^67 - 10*i * q^68 + 8 * q^69 + 9 * q^71 - 9*i * q^73 + 2 * q^76 + 10 * q^79 + q^81 + 9*i * q^83 - 10*i * q^87 + 16*i * q^92 + i * q^93 + 14*i * q^97 - 8 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 + 4 * q^9 $$2 q + 4 q^{4} + 4 q^{9} - 8 q^{11} + 8 q^{16} + 2 q^{19} + 20 q^{29} - 2 q^{31} + 8 q^{36} - 12 q^{39} - 6 q^{41} - 16 q^{44} + 14 q^{49} - 10 q^{51} - 22 q^{59} - 24 q^{61} + 16 q^{64} + 16 q^{69} + 18 q^{71} + 4 q^{76} + 20 q^{79} + 2 q^{81} - 16 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 + 4 * q^9 - 8 * q^11 + 8 * q^16 + 2 * q^19 + 20 * q^29 - 2 * q^31 + 8 * q^36 - 12 * q^39 - 6 * q^41 - 16 * q^44 + 14 * q^49 - 10 * q^51 - 22 * q^59 - 24 * q^61 + 16 * q^64 + 16 * q^69 + 18 * q^71 + 4 * q^76 + 20 * q^79 + 2 * q^81 - 16 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 1.00000i − 1.00000i
0 1.00000i 2.00000 0 0 0 0 2.00000 0
249.2 0 1.00000i 2.00000 0 0 0 0 2.00000 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 775.2.b.c 2
5.b even 2 1 inner 775.2.b.c 2
5.c odd 4 1 155.2.a.c 1
5.c odd 4 1 775.2.a.a 1
15.e even 4 1 1395.2.a.b 1
15.e even 4 1 6975.2.a.l 1
20.e even 4 1 2480.2.a.k 1
35.f even 4 1 7595.2.a.h 1
40.i odd 4 1 9920.2.a.ba 1
40.k even 4 1 9920.2.a.l 1
155.f even 4 1 4805.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.c 1 5.c odd 4 1
775.2.a.a 1 5.c odd 4 1
775.2.b.c 2 1.a even 1 1 trivial
775.2.b.c 2 5.b even 2 1 inner
1395.2.a.b 1 15.e even 4 1
2480.2.a.k 1 20.e even 4 1
4805.2.a.e 1 155.f even 4 1
6975.2.a.l 1 15.e even 4 1
7595.2.a.h 1 35.f even 4 1
9920.2.a.l 1 40.k even 4 1
9920.2.a.ba 1 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(775, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 25$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 49$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 25$$
$59$ $$(T + 11)^{2}$$
$61$ $$(T + 12)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T - 9)^{2}$$
$73$ $$T^{2} + 81$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 196$$