# Properties

 Label 775.2.b.b 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 2 i q^{3} + q^{4} - 2 q^{6} - 4 i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + 2*i * q^3 + q^4 - 2 * q^6 - 4*i * q^7 + 3*i * q^8 - q^9 $$q + i q^{2} + 2 i q^{3} + q^{4} - 2 q^{6} - 4 i q^{7} + 3 i q^{8} - q^{9} + 4 q^{11} + 2 i q^{12} + 4 q^{14} - q^{16} + 8 i q^{17} - i q^{18} - 4 q^{19} + 8 q^{21} + 4 i q^{22} + 2 i q^{23} - 6 q^{24} + 4 i q^{27} - 4 i q^{28} + 6 q^{29} + q^{31} + 5 i q^{32} + 8 i q^{33} - 8 q^{34} - q^{36} + 4 i q^{37} - 4 i q^{38} - 6 q^{41} + 8 i q^{42} - 6 i q^{43} + 4 q^{44} - 2 q^{46} - 8 i q^{47} - 2 i q^{48} - 9 q^{49} - 16 q^{51} - 12 i q^{53} - 4 q^{54} + 12 q^{56} - 8 i q^{57} + 6 i q^{58} + 4 q^{59} + 10 q^{61} + i q^{62} + 4 i q^{63} - 7 q^{64} - 8 q^{66} - 8 i q^{67} + 8 i q^{68} - 4 q^{69} - 3 i q^{72} - 4 i q^{73} - 4 q^{74} - 4 q^{76} - 16 i q^{77} - 11 q^{81} - 6 i q^{82} + 2 i q^{83} + 8 q^{84} + 6 q^{86} + 12 i q^{87} + 12 i q^{88} - 14 q^{89} + 2 i q^{92} + 2 i q^{93} + 8 q^{94} - 10 q^{96} + 18 i q^{97} - 9 i q^{98} - 4 q^{99} +O(q^{100})$$ q + i * q^2 + 2*i * q^3 + q^4 - 2 * q^6 - 4*i * q^7 + 3*i * q^8 - q^9 + 4 * q^11 + 2*i * q^12 + 4 * q^14 - q^16 + 8*i * q^17 - i * q^18 - 4 * q^19 + 8 * q^21 + 4*i * q^22 + 2*i * q^23 - 6 * q^24 + 4*i * q^27 - 4*i * q^28 + 6 * q^29 + q^31 + 5*i * q^32 + 8*i * q^33 - 8 * q^34 - q^36 + 4*i * q^37 - 4*i * q^38 - 6 * q^41 + 8*i * q^42 - 6*i * q^43 + 4 * q^44 - 2 * q^46 - 8*i * q^47 - 2*i * q^48 - 9 * q^49 - 16 * q^51 - 12*i * q^53 - 4 * q^54 + 12 * q^56 - 8*i * q^57 + 6*i * q^58 + 4 * q^59 + 10 * q^61 + i * q^62 + 4*i * q^63 - 7 * q^64 - 8 * q^66 - 8*i * q^67 + 8*i * q^68 - 4 * q^69 - 3*i * q^72 - 4*i * q^73 - 4 * q^74 - 4 * q^76 - 16*i * q^77 - 11 * q^81 - 6*i * q^82 + 2*i * q^83 + 8 * q^84 + 6 * q^86 + 12*i * q^87 + 12*i * q^88 - 14 * q^89 + 2*i * q^92 + 2*i * q^93 + 8 * q^94 - 10 * q^96 + 18*i * q^97 - 9*i * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 4 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 4 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} - 2 q^{16} - 8 q^{19} + 16 q^{21} - 12 q^{24} + 12 q^{29} + 2 q^{31} - 16 q^{34} - 2 q^{36} - 12 q^{41} + 8 q^{44} - 4 q^{46} - 18 q^{49} - 32 q^{51} - 8 q^{54} + 24 q^{56} + 8 q^{59} + 20 q^{61} - 14 q^{64} - 16 q^{66} - 8 q^{69} - 8 q^{74} - 8 q^{76} - 22 q^{81} + 16 q^{84} + 12 q^{86} - 28 q^{89} + 16 q^{94} - 20 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 4 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 - 2 * q^16 - 8 * q^19 + 16 * q^21 - 12 * q^24 + 12 * q^29 + 2 * q^31 - 16 * q^34 - 2 * q^36 - 12 * q^41 + 8 * q^44 - 4 * q^46 - 18 * q^49 - 32 * q^51 - 8 * q^54 + 24 * q^56 + 8 * q^59 + 20 * q^61 - 14 * q^64 - 16 * q^66 - 8 * q^69 - 8 * q^74 - 8 * q^76 - 22 * q^81 + 16 * q^84 + 12 * q^86 - 28 * q^89 + 16 * q^94 - 20 * 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
1.00000i 2.00000i 1.00000 0 −2.00000 4.00000i 3.00000i −1.00000 0
249.2 1.00000i 2.00000i 1.00000 0 −2.00000 4.00000i 3.00000i −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 775.2.b.b 2
5.b even 2 1 inner 775.2.b.b 2
5.c odd 4 1 155.2.a.b 1
5.c odd 4 1 775.2.a.b 1
15.e even 4 1 1395.2.a.d 1
15.e even 4 1 6975.2.a.d 1
20.e even 4 1 2480.2.a.b 1
35.f even 4 1 7595.2.a.c 1
40.i odd 4 1 9920.2.a.g 1
40.k even 4 1 9920.2.a.bd 1
155.f even 4 1 4805.2.a.d 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 64$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 144$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 324$$