# Properties

 Label 775.2.b.a 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - i q^{3} - 2 q^{4} + 2 q^{6} + 2 i q^{7} + 2 q^{9} +O(q^{10})$$ q + 2*i * q^2 - i * q^3 - 2 * q^4 + 2 * q^6 + 2*i * q^7 + 2 * q^9 $$q + 2 i q^{2} - i q^{3} - 2 q^{4} + 2 q^{6} + 2 i q^{7} + 2 q^{9} + 2 q^{11} + 2 i q^{12} - 6 i q^{13} - 4 q^{14} - 4 q^{16} + 7 i q^{17} + 4 i q^{18} + 5 q^{19} + 2 q^{21} + 4 i q^{22} + 4 i q^{23} + 12 q^{26} - 5 i q^{27} - 4 i q^{28} + q^{31} - 8 i q^{32} - 2 i q^{33} - 14 q^{34} - 4 q^{36} + 7 i q^{37} + 10 i q^{38} - 6 q^{39} - 3 q^{41} + 4 i q^{42} + 9 i q^{43} - 4 q^{44} - 8 q^{46} + 2 i q^{47} + 4 i q^{48} + 3 q^{49} + 7 q^{51} + 12 i q^{52} + 9 i q^{53} + 10 q^{54} - 5 i q^{57} + 5 q^{59} - 8 q^{61} + 2 i q^{62} + 4 i q^{63} + 8 q^{64} + 4 q^{66} - 8 i q^{67} - 14 i q^{68} + 4 q^{69} - 3 q^{71} - i q^{73} - 14 q^{74} - 10 q^{76} + 4 i q^{77} - 12 i q^{78} + q^{81} - 6 i q^{82} - 11 i q^{83} - 4 q^{84} - 18 q^{86} - 10 q^{89} + 12 q^{91} - 8 i q^{92} - i q^{93} - 4 q^{94} - 8 q^{96} - 18 i q^{97} + 6 i q^{98} + 4 q^{99} +O(q^{100})$$ q + 2*i * q^2 - i * q^3 - 2 * q^4 + 2 * q^6 + 2*i * q^7 + 2 * q^9 + 2 * q^11 + 2*i * q^12 - 6*i * q^13 - 4 * q^14 - 4 * q^16 + 7*i * q^17 + 4*i * q^18 + 5 * q^19 + 2 * q^21 + 4*i * q^22 + 4*i * q^23 + 12 * q^26 - 5*i * q^27 - 4*i * q^28 + q^31 - 8*i * q^32 - 2*i * q^33 - 14 * q^34 - 4 * q^36 + 7*i * q^37 + 10*i * q^38 - 6 * q^39 - 3 * q^41 + 4*i * q^42 + 9*i * q^43 - 4 * q^44 - 8 * q^46 + 2*i * q^47 + 4*i * q^48 + 3 * q^49 + 7 * q^51 + 12*i * q^52 + 9*i * q^53 + 10 * q^54 - 5*i * q^57 + 5 * q^59 - 8 * q^61 + 2*i * q^62 + 4*i * q^63 + 8 * q^64 + 4 * q^66 - 8*i * q^67 - 14*i * q^68 + 4 * q^69 - 3 * q^71 - i * q^73 - 14 * q^74 - 10 * q^76 + 4*i * q^77 - 12*i * q^78 + q^81 - 6*i * q^82 - 11*i * q^83 - 4 * q^84 - 18 * q^86 - 10 * q^89 + 12 * q^91 - 8*i * q^92 - i * q^93 - 4 * q^94 - 8 * q^96 - 18*i * q^97 + 6*i * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^6 + 4 * q^9 $$2 q - 4 q^{4} + 4 q^{6} + 4 q^{9} + 4 q^{11} - 8 q^{14} - 8 q^{16} + 10 q^{19} + 4 q^{21} + 24 q^{26} + 2 q^{31} - 28 q^{34} - 8 q^{36} - 12 q^{39} - 6 q^{41} - 8 q^{44} - 16 q^{46} + 6 q^{49} + 14 q^{51} + 20 q^{54} + 10 q^{59} - 16 q^{61} + 16 q^{64} + 8 q^{66} + 8 q^{69} - 6 q^{71} - 28 q^{74} - 20 q^{76} + 2 q^{81} - 8 q^{84} - 36 q^{86} - 20 q^{89} + 24 q^{91} - 8 q^{94} - 16 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^6 + 4 * q^9 + 4 * q^11 - 8 * q^14 - 8 * q^16 + 10 * q^19 + 4 * q^21 + 24 * q^26 + 2 * q^31 - 28 * q^34 - 8 * q^36 - 12 * q^39 - 6 * q^41 - 8 * q^44 - 16 * q^46 + 6 * q^49 + 14 * q^51 + 20 * q^54 + 10 * q^59 - 16 * q^61 + 16 * q^64 + 8 * q^66 + 8 * q^69 - 6 * q^71 - 28 * q^74 - 20 * q^76 + 2 * q^81 - 8 * q^84 - 36 * q^86 - 20 * q^89 + 24 * q^91 - 8 * q^94 - 16 * q^96 + 8 * 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
2.00000i 1.00000i −2.00000 0 2.00000 2.00000i 0 2.00000 0
249.2 2.00000i 1.00000i −2.00000 0 2.00000 2.00000i 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.a 2
5.b even 2 1 inner 775.2.b.a 2
5.c odd 4 1 155.2.a.a 1
5.c odd 4 1 775.2.a.c 1
15.e even 4 1 1395.2.a.e 1
15.e even 4 1 6975.2.a.b 1
20.e even 4 1 2480.2.a.m 1
35.f even 4 1 7595.2.a.b 1
40.i odd 4 1 9920.2.a.x 1
40.k even 4 1 9920.2.a.j 1
155.f even 4 1 4805.2.a.b 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 49$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 81$$
$59$ $$(T - 5)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 1$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 121$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 324$$