# Properties

 Label 775.2.f.a Level $775$ Weight $2$ Character orbit 775.f Analytic conductor $6.188$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.18840615665$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{6} + 2 \beta_1 q^{7} - \beta_{3} q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - 1) * q^3 + b2 * q^4 + (-b3 - b1) * q^6 + 2*b1 * q^7 - b3 * q^8 - b2 * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{6} + 2 \beta_1 q^{7} - \beta_{3} q^{8} - \beta_{2} q^{9} + (2 \beta_{3} + 2 \beta_1) q^{11} + ( - \beta_{2} + 1) q^{12} + ( - 3 \beta_{2} - 3) q^{13} + 6 \beta_{2} q^{14} + 5 q^{16} + ( - 5 \beta_{2} + 5) q^{17} - \beta_{3} q^{18} + 2 \beta_{2} q^{19} + ( - 2 \beta_{3} - 2 \beta_1) q^{21} + (6 \beta_{2} - 6) q^{22} + (\beta_{2} + 1) q^{23} + (\beta_{3} - \beta_1) q^{24} + ( - 3 \beta_{3} - 3 \beta_1) q^{26} + (4 \beta_{2} - 4) q^{27} + 2 \beta_{3} q^{28} + ( - 4 \beta_{3} + 4 \beta_1) q^{29} + (\beta_{3} + \beta_1 + 5) q^{31} + 3 \beta_1 q^{32} - 4 \beta_{3} q^{33} + ( - 5 \beta_{3} + 5 \beta_1) q^{34} + q^{36} + ( - 3 \beta_{2} + 3) q^{37} + 2 \beta_{3} q^{38} + 6 \beta_{2} q^{39} - 6 q^{41} + ( - 6 \beta_{2} + 6) q^{42} + (3 \beta_{2} + 3) q^{43} + (2 \beta_{3} - 2 \beta_1) q^{44} + (\beta_{3} + \beta_1) q^{46} - 6 \beta_1 q^{47} + ( - 5 \beta_{2} - 5) q^{48} + 5 \beta_{2} q^{49} - 10 q^{51} + ( - 3 \beta_{2} + 3) q^{52} + (\beta_{2} + 1) q^{53} + (4 \beta_{3} - 4 \beta_1) q^{54} + 6 q^{56} + ( - 2 \beta_{2} + 2) q^{57} + (12 \beta_{2} + 12) q^{58} + 12 \beta_{2} q^{59} + (2 \beta_{3} + 2 \beta_1) q^{61} + (3 \beta_{2} + 5 \beta_1 - 3) q^{62} - 2 \beta_{3} q^{63} - \beta_{2} q^{64} + 12 q^{66} - 2 \beta_1 q^{67} + (5 \beta_{2} + 5) q^{68} - 2 \beta_{2} q^{69} + 6 q^{71} - \beta_1 q^{72} + ( - 3 \beta_{2} - 3) q^{73} + ( - 3 \beta_{3} + 3 \beta_1) q^{74} - 2 q^{76} + (12 \beta_{2} - 12) q^{77} + 6 \beta_{3} q^{78} + (4 \beta_{3} - 4 \beta_1) q^{79} + 5 q^{81} - 6 \beta_1 q^{82} + ( - 7 \beta_{2} - 7) q^{83} + ( - 2 \beta_{3} + 2 \beta_1) q^{84} + (3 \beta_{3} + 3 \beta_1) q^{86} - 8 \beta_1 q^{87} + (6 \beta_{2} + 6) q^{88} + (4 \beta_{3} - 4 \beta_1) q^{89} + ( - 6 \beta_{3} - 6 \beta_1) q^{91} + (\beta_{2} - 1) q^{92} + ( - 2 \beta_{3} - 5 \beta_{2} - 5) q^{93} - 18 \beta_{2} q^{94} + ( - 3 \beta_{3} - 3 \beta_1) q^{96} + 5 \beta_{3} q^{98} + ( - 2 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - 1) * q^3 + b2 * q^4 + (-b3 - b1) * q^6 + 2*b1 * q^7 - b3 * q^8 - b2 * q^9 + (2*b3 + 2*b1) * q^11 + (-b2 + 1) * q^12 + (-3*b2 - 3) * q^13 + 6*b2 * q^14 + 5 * q^16 + (-5*b2 + 5) * q^17 - b3 * q^18 + 2*b2 * q^19 + (-2*b3 - 2*b1) * q^21 + (6*b2 - 6) * q^22 + (b2 + 1) * q^23 + (b3 - b1) * q^24 + (-3*b3 - 3*b1) * q^26 + (4*b2 - 4) * q^27 + 2*b3 * q^28 + (-4*b3 + 4*b1) * q^29 + (b3 + b1 + 5) * q^31 + 3*b1 * q^32 - 4*b3 * q^33 + (-5*b3 + 5*b1) * q^34 + q^36 + (-3*b2 + 3) * q^37 + 2*b3 * q^38 + 6*b2 * q^39 - 6 * q^41 + (-6*b2 + 6) * q^42 + (3*b2 + 3) * q^43 + (2*b3 - 2*b1) * q^44 + (b3 + b1) * q^46 - 6*b1 * q^47 + (-5*b2 - 5) * q^48 + 5*b2 * q^49 - 10 * q^51 + (-3*b2 + 3) * q^52 + (b2 + 1) * q^53 + (4*b3 - 4*b1) * q^54 + 6 * q^56 + (-2*b2 + 2) * q^57 + (12*b2 + 12) * q^58 + 12*b2 * q^59 + (2*b3 + 2*b1) * q^61 + (3*b2 + 5*b1 - 3) * q^62 - 2*b3 * q^63 - b2 * q^64 + 12 * q^66 - 2*b1 * q^67 + (5*b2 + 5) * q^68 - 2*b2 * q^69 + 6 * q^71 - b1 * q^72 + (-3*b2 - 3) * q^73 + (-3*b3 + 3*b1) * q^74 - 2 * q^76 + (12*b2 - 12) * q^77 + 6*b3 * q^78 + (4*b3 - 4*b1) * q^79 + 5 * q^81 - 6*b1 * q^82 + (-7*b2 - 7) * q^83 + (-2*b3 + 2*b1) * q^84 + (3*b3 + 3*b1) * q^86 - 8*b1 * q^87 + (6*b2 + 6) * q^88 + (4*b3 - 4*b1) * q^89 + (-6*b3 - 6*b1) * q^91 + (b2 - 1) * q^92 + (-2*b3 - 5*b2 - 5) * q^93 - 18*b2 * q^94 + (-3*b3 - 3*b1) * q^96 + 5*b3 * q^98 + (-2*b3 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3}+O(q^{10})$$ 4 * q - 4 * q^3 $$4 q - 4 q^{3} + 4 q^{12} - 12 q^{13} + 20 q^{16} + 20 q^{17} - 24 q^{22} + 4 q^{23} - 16 q^{27} + 20 q^{31} + 4 q^{36} + 12 q^{37} - 24 q^{41} + 24 q^{42} + 12 q^{43} - 20 q^{48} - 40 q^{51} + 12 q^{52} + 4 q^{53} + 24 q^{56} + 8 q^{57} + 48 q^{58} - 12 q^{62} + 48 q^{66} + 20 q^{68} + 24 q^{71} - 12 q^{73} - 8 q^{76} - 48 q^{77} + 20 q^{81} - 28 q^{83} + 24 q^{88} - 4 q^{92} - 20 q^{93}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^12 - 12 * q^13 + 20 * q^16 + 20 * q^17 - 24 * q^22 + 4 * q^23 - 16 * q^27 + 20 * q^31 + 4 * q^36 + 12 * q^37 - 24 * q^41 + 24 * q^42 + 12 * q^43 - 20 * q^48 - 40 * q^51 + 12 * q^52 + 4 * q^53 + 24 * q^56 + 8 * q^57 + 48 * q^58 - 12 * q^62 + 48 * q^66 + 20 * q^68 + 24 * q^71 - 12 * q^73 - 8 * q^76 - 48 * q^77 + 20 * q^81 - 28 * q^83 + 24 * q^88 - 4 * q^92 - 20 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

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

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−1.22474 1.22474i −1.00000 1.00000i 1.00000i 0 2.44949i −2.44949 2.44949i −1.22474 + 1.22474i 1.00000i 0
557.2 1.22474 + 1.22474i −1.00000 1.00000i 1.00000i 0 2.44949i 2.44949 + 2.44949i 1.22474 1.22474i 1.00000i 0
743.1 −1.22474 + 1.22474i −1.00000 + 1.00000i 1.00000i 0 2.44949i −2.44949 + 2.44949i −1.22474 1.22474i 1.00000i 0
743.2 1.22474 1.22474i −1.00000 + 1.00000i 1.00000i 0 2.44949i 2.44949 2.44949i 1.22474 + 1.22474i 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
5.c odd 4 1 inner
155.c odd 2 1 inner
155.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.f.a 4
5.b even 2 1 775.2.f.b yes 4
5.c odd 4 1 inner 775.2.f.a 4
5.c odd 4 1 775.2.f.b yes 4
31.b odd 2 1 775.2.f.b yes 4
155.c odd 2 1 inner 775.2.f.a 4
155.f even 4 1 inner 775.2.f.a 4
155.f even 4 1 775.2.f.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.f.a 4 1.a even 1 1 trivial
775.2.f.a 4 5.c odd 4 1 inner
775.2.f.a 4 155.c odd 2 1 inner
775.2.f.a 4 155.f even 4 1 inner
775.2.f.b yes 4 5.b even 2 1
775.2.f.b yes 4 5.c odd 4 1
775.2.f.b yes 4 31.b odd 2 1
775.2.f.b yes 4 155.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(775, [\chi])$$:

 $$T_{2}^{4} + 9$$ T2^4 + 9 $$T_{3}^{2} + 2T_{3} + 2$$ T3^2 + 2*T3 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9$$
$3$ $$(T^{2} + 2 T + 2)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 144$$
$11$ $$(T^{2} + 24)^{2}$$
$13$ $$(T^{2} + 6 T + 18)^{2}$$
$17$ $$(T^{2} - 10 T + 50)^{2}$$
$19$ $$(T^{2} + 4)^{2}$$
$23$ $$(T^{2} - 2 T + 2)^{2}$$
$29$ $$(T^{2} - 96)^{2}$$
$31$ $$(T^{2} - 10 T + 31)^{2}$$
$37$ $$(T^{2} - 6 T + 18)^{2}$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} - 6 T + 18)^{2}$$
$47$ $$T^{4} + 11664$$
$53$ $$(T^{2} - 2 T + 2)^{2}$$
$59$ $$(T^{2} + 144)^{2}$$
$61$ $$(T^{2} + 24)^{2}$$
$67$ $$T^{4} + 144$$
$71$ $$(T - 6)^{4}$$
$73$ $$(T^{2} + 6 T + 18)^{2}$$
$79$ $$(T^{2} - 96)^{2}$$
$83$ $$(T^{2} + 14 T + 98)^{2}$$
$89$ $$(T^{2} - 96)^{2}$$
$97$ $$T^{4}$$
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