# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 775.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.18840615665$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} + 1) q^{4} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 3) q^{8} + 2 \beta_1 q^{9}+O(q^{10})$$ q + (b3 + 1) * q^2 + (-b3 + b2 - b1) * q^3 + (2*b3 + 1) * q^4 + (-2*b3 + 3*b2 - 2*b1) * q^6 + (b3 - b2 + b1) * q^7 + (b3 + 3) * q^8 + 2*b1 * q^9 $$q + (\beta_{3} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} + 1) q^{4} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 3) q^{8} + 2 \beta_1 q^{9} + (\beta_{2} - 3 \beta_1 + 1) q^{11} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1) q^{12} + ( - \beta_{2} - 2 \beta_1 - 1) q^{13} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{14} + 3 q^{16} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{17} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{18} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{19} + (3 \beta_{2} - 2 \beta_1 + 3) q^{21} + (7 \beta_{2} - 4 \beta_1 + 7) q^{22} + 4 q^{23} + ( - 4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{24} + (3 \beta_{2} - \beta_1 + 3) q^{26} + ( - \beta_{3} + 1) q^{27} + (3 \beta_{3} - 5 \beta_{2} + 3 \beta_1) q^{28} + (2 \beta_{3} - 4) q^{29} + (\beta_{3} + 2 \beta_1 - 5) q^{31} + (\beta_{3} - 3) q^{32} + ( - 4 \beta_{3} - 7) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{34} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{36} - \beta_{2} q^{37} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{38} + ( - \beta_{3} - 3) q^{39} + ( - \beta_{2} + 6 \beta_1 - 1) q^{41} + (7 \beta_{2} - 5 \beta_1 + 7) q^{42} + (7 \beta_{3} + \beta_{2} + 7 \beta_1) q^{43} + (13 \beta_{2} - 5 \beta_1 + 13) q^{44} + (4 \beta_{3} + 4) q^{46} + (4 \beta_{3} - 4) q^{47} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{48} + (4 \beta_{2} + 2 \beta_1 + 4) q^{49} + (\beta_{2} + \beta_1 + 1) q^{51} + (7 \beta_{2} + 7) q^{52} + (3 \beta_{2} + 2 \beta_1 + 3) q^{53} - q^{54} + (4 \beta_{3} - 5 \beta_{2} + 4 \beta_1) q^{56} + ( - \beta_{2} + 2 \beta_1 - 1) q^{57} - 2 \beta_{3} q^{58} + (5 \beta_{3} - 3 \beta_{2} + 5 \beta_1) q^{59} + 2 \beta_{3} q^{61} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 7) q^{62} + ( - 2 \beta_{3} - 4) q^{63} + ( - 2 \beta_{3} - 7) q^{64} + ( - 11 \beta_{3} - 15) q^{66} + ( - \beta_{2} + 3 \beta_1 - 1) q^{67} + ( - 4 \beta_{3} - 5 \beta_{2} - 4 \beta_1) q^{68} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{69} + (7 \beta_{2} - 5 \beta_1 + 7) q^{71} + ( - 4 \beta_{2} + 6 \beta_1 - 4) q^{72} + (\beta_{2} - 2 \beta_1 + 1) q^{73} + (\beta_{3} - \beta_{2} + \beta_1) q^{74} + ( - 5 \beta_{3} - \beta_{2} - 5 \beta_1) q^{76} + (4 \beta_{3} + 7) q^{77} + ( - 4 \beta_{3} - 5) q^{78} + (3 \beta_{3} - 11 \beta_{2} + 3 \beta_1) q^{79} + (6 \beta_{3} - \beta_{2} + 6 \beta_1) q^{81} + ( - 13 \beta_{2} + 7 \beta_1 - 13) q^{82} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{83} + (11 \beta_{2} - 8 \beta_1 + 11) q^{84} + (6 \beta_{3} - 13 \beta_{2} + 6 \beta_1) q^{86} + (2 \beta_{3} + 2 \beta_1) q^{87} + (9 \beta_{2} - 10 \beta_1 + 9) q^{88} + ( - 6 \beta_{3} - 4) q^{89} + (\beta_{3} + 3) q^{91} + (8 \beta_{3} + 4) q^{92} + (6 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 4) q^{93} + 4 q^{94} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{96} + ( - 2 \beta_{3} - 8) q^{97} - 2 \beta_1 q^{98} + (2 \beta_{3} - 12 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^2 + (-b3 + b2 - b1) * q^3 + (2*b3 + 1) * q^4 + (-2*b3 + 3*b2 - 2*b1) * q^6 + (b3 - b2 + b1) * q^7 + (b3 + 3) * q^8 + 2*b1 * q^9 + (b2 - 3*b1 + 1) * q^11 + (-3*b3 + 5*b2 - 3*b1) * q^12 + (-b2 - 2*b1 - 1) * q^13 + (2*b3 - 3*b2 + 2*b1) * q^14 + 3 * q^16 + (2*b3 + 3*b2 + 2*b1) * q^17 + (-4*b2 + 2*b1 - 4) * q^18 + (b3 + 3*b2 + b1) * q^19 + (3*b2 - 2*b1 + 3) * q^21 + (7*b2 - 4*b1 + 7) * q^22 + 4 * q^23 + (-4*b3 + 5*b2 - 4*b1) * q^24 + (3*b2 - b1 + 3) * q^26 + (-b3 + 1) * q^27 + (3*b3 - 5*b2 + 3*b1) * q^28 + (2*b3 - 4) * q^29 + (b3 + 2*b1 - 5) * q^31 + (b3 - 3) * q^32 + (-4*b3 - 7) * q^33 + (-b3 - b2 - b1) * q^34 + (-8*b2 + 2*b1 - 8) * q^36 - b2 * q^37 + (-2*b3 + b2 - 2*b1) * q^38 + (-b3 - 3) * q^39 + (-b2 + 6*b1 - 1) * q^41 + (7*b2 - 5*b1 + 7) * q^42 + (7*b3 + b2 + 7*b1) * q^43 + (13*b2 - 5*b1 + 13) * q^44 + (4*b3 + 4) * q^46 + (4*b3 - 4) * q^47 + (-3*b3 + 3*b2 - 3*b1) * q^48 + (4*b2 + 2*b1 + 4) * q^49 + (b2 + b1 + 1) * q^51 + (7*b2 + 7) * q^52 + (3*b2 + 2*b1 + 3) * q^53 - q^54 + (4*b3 - 5*b2 + 4*b1) * q^56 + (-b2 + 2*b1 - 1) * q^57 - 2*b3 * q^58 + (5*b3 - 3*b2 + 5*b1) * q^59 + 2*b3 * q^61 + (-4*b3 - 4*b2 + 2*b1 - 7) * q^62 + (-2*b3 - 4) * q^63 + (-2*b3 - 7) * q^64 + (-11*b3 - 15) * q^66 + (-b2 + 3*b1 - 1) * q^67 + (-4*b3 - 5*b2 - 4*b1) * q^68 + (-4*b3 + 4*b2 - 4*b1) * q^69 + (7*b2 - 5*b1 + 7) * q^71 + (-4*b2 + 6*b1 - 4) * q^72 + (b2 - 2*b1 + 1) * q^73 + (b3 - b2 + b1) * q^74 + (-5*b3 - b2 - 5*b1) * q^76 + (4*b3 + 7) * q^77 + (-4*b3 - 5) * q^78 + (3*b3 - 11*b2 + 3*b1) * q^79 + (6*b3 - b2 + 6*b1) * q^81 + (-13*b2 + 7*b1 - 13) * q^82 + (-3*b2 - 5*b1 - 3) * q^83 + (11*b2 - 8*b1 + 11) * q^84 + (6*b3 - 13*b2 + 6*b1) * q^86 + (2*b3 + 2*b1) * q^87 + (9*b2 - 10*b1 + 9) * q^88 + (-6*b3 - 4) * q^89 + (b3 + 3) * q^91 + (8*b3 + 4) * q^92 + (6*b3 - 3*b2 + 4*b1 + 4) * q^93 + 4 * q^94 + (2*b3 - b2 + 2*b1) * q^96 + (-2*b3 - 8) * q^97 - 2*b1 * q^98 + (2*b3 - 12*b2 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{6} + 2 q^{7} + 12 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^6 + 2 * q^7 + 12 * q^8 $$4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{11} - 10 q^{12} - 2 q^{13} + 6 q^{14} + 12 q^{16} - 6 q^{17} - 8 q^{18} - 6 q^{19} + 6 q^{21} + 14 q^{22} + 16 q^{23} - 10 q^{24} + 6 q^{26} + 4 q^{27} + 10 q^{28} - 16 q^{29} - 20 q^{31} - 12 q^{32} - 28 q^{33} + 2 q^{34} - 16 q^{36} + 2 q^{37} - 2 q^{38} - 12 q^{39} - 2 q^{41} + 14 q^{42} - 2 q^{43} + 26 q^{44} + 16 q^{46} - 16 q^{47} - 6 q^{48} + 8 q^{49} + 2 q^{51} + 14 q^{52} + 6 q^{53} - 4 q^{54} + 10 q^{56} - 2 q^{57} + 6 q^{59} - 20 q^{62} - 16 q^{63} - 28 q^{64} - 60 q^{66} - 2 q^{67} + 10 q^{68} - 8 q^{69} + 14 q^{71} - 8 q^{72} + 2 q^{73} + 2 q^{74} + 2 q^{76} + 28 q^{77} - 20 q^{78} + 22 q^{79} + 2 q^{81} - 26 q^{82} - 6 q^{83} + 22 q^{84} + 26 q^{86} + 18 q^{88} - 16 q^{89} + 12 q^{91} + 16 q^{92} + 22 q^{93} + 16 q^{94} + 2 q^{96} - 32 q^{97} + 24 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^11 - 10 * q^12 - 2 * q^13 + 6 * q^14 + 12 * q^16 - 6 * q^17 - 8 * q^18 - 6 * q^19 + 6 * q^21 + 14 * q^22 + 16 * q^23 - 10 * q^24 + 6 * q^26 + 4 * q^27 + 10 * q^28 - 16 * q^29 - 20 * q^31 - 12 * q^32 - 28 * q^33 + 2 * q^34 - 16 * q^36 + 2 * q^37 - 2 * q^38 - 12 * q^39 - 2 * q^41 + 14 * q^42 - 2 * q^43 + 26 * q^44 + 16 * q^46 - 16 * q^47 - 6 * q^48 + 8 * q^49 + 2 * q^51 + 14 * q^52 + 6 * q^53 - 4 * q^54 + 10 * q^56 - 2 * q^57 + 6 * q^59 - 20 * q^62 - 16 * q^63 - 28 * q^64 - 60 * q^66 - 2 * q^67 + 10 * q^68 - 8 * q^69 + 14 * q^71 - 8 * q^72 + 2 * q^73 + 2 * q^74 + 2 * q^76 + 28 * q^77 - 20 * q^78 + 22 * q^79 + 2 * q^81 - 26 * q^82 - 6 * q^83 + 22 * q^84 + 26 * q^86 + 18 * q^88 - 16 * q^89 + 12 * q^91 + 16 * q^92 + 22 * q^93 + 16 * q^94 + 2 * q^96 - 32 * q^97 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
501.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
−0.414214 0.207107 0.358719i −1.82843 0 −0.0857864 + 0.148586i −0.207107 + 0.358719i 1.58579 1.41421 + 2.44949i 0
501.2 2.41421 −1.20711 + 2.09077i 3.82843 0 −2.91421 + 5.04757i 1.20711 2.09077i 4.41421 −1.41421 2.44949i 0
676.1 −0.414214 0.207107 + 0.358719i −1.82843 0 −0.0857864 0.148586i −0.207107 0.358719i 1.58579 1.41421 2.44949i 0
676.2 2.41421 −1.20711 2.09077i 3.82843 0 −2.91421 5.04757i 1.20711 + 2.09077i 4.41421 −1.41421 + 2.44949i 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
31.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.e.e 4
5.b even 2 1 31.2.c.a 4
5.c odd 4 2 775.2.o.d 8
15.d odd 2 1 279.2.h.c 4
20.d odd 2 1 496.2.i.h 4
31.c even 3 1 inner 775.2.e.e 4
155.c odd 2 1 961.2.c.a 4
155.i odd 6 1 961.2.a.c 2
155.i odd 6 1 961.2.c.a 4
155.j even 6 1 31.2.c.a 4
155.j even 6 1 961.2.a.a 2
155.m odd 10 4 961.2.g.r 16
155.n even 10 4 961.2.g.o 16
155.o odd 12 2 775.2.o.d 8
155.u even 30 4 961.2.d.l 8
155.u even 30 4 961.2.g.o 16
155.v odd 30 4 961.2.d.i 8
155.v odd 30 4 961.2.g.r 16
465.t even 6 1 8649.2.a.k 2
465.u odd 6 1 279.2.h.c 4
465.u odd 6 1 8649.2.a.l 2
620.o odd 6 1 496.2.i.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 5.b even 2 1
31.2.c.a 4 155.j even 6 1
279.2.h.c 4 15.d odd 2 1
279.2.h.c 4 465.u odd 6 1
496.2.i.h 4 20.d odd 2 1
496.2.i.h 4 620.o odd 6 1
775.2.e.e 4 1.a even 1 1 trivial
775.2.e.e 4 31.c even 3 1 inner
775.2.o.d 8 5.c odd 4 2
775.2.o.d 8 155.o odd 12 2
961.2.a.a 2 155.j even 6 1
961.2.a.c 2 155.i odd 6 1
961.2.c.a 4 155.c odd 2 1
961.2.c.a 4 155.i odd 6 1
961.2.d.i 8 155.v odd 30 4
961.2.d.l 8 155.u even 30 4
961.2.g.o 16 155.n even 10 4
961.2.g.o 16 155.u even 30 4
961.2.g.r 16 155.m odd 10 4
961.2.g.r 16 155.v odd 30 4
8649.2.a.k 2 465.t even 6 1
8649.2.a.l 2 465.u odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T - 1)^{2}$$
$3$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$11$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$13$ $$T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49$$
$17$ $$T^{4} + 6 T^{3} + 35 T^{2} + 6 T + 1$$
$19$ $$T^{4} + 6 T^{3} + 29 T^{2} + 42 T + 49$$
$23$ $$(T - 4)^{4}$$
$29$ $$(T^{2} + 8 T + 8)^{2}$$
$31$ $$(T^{2} + 10 T + 31)^{2}$$
$37$ $$(T^{2} - T + 1)^{2}$$
$41$ $$T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041$$
$43$ $$T^{4} + 2 T^{3} + 101 T^{2} + \cdots + 9409$$
$47$ $$(T^{2} + 8 T - 16)^{2}$$
$53$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$59$ $$T^{4} - 6 T^{3} + 77 T^{2} + \cdots + 1681$$
$61$ $$(T^{2} - 8)^{2}$$
$67$ $$T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289$$
$71$ $$T^{4} - 14 T^{3} + 197 T^{2} + 14 T + 1$$
$73$ $$T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49$$
$79$ $$T^{4} - 22 T^{3} + 381 T^{2} + \cdots + 10609$$
$83$ $$T^{4} + 6 T^{3} + 77 T^{2} + \cdots + 1681$$
$89$ $$(T^{2} + 8 T - 56)^{2}$$
$97$ $$(T^{2} + 16 T + 56)^{2}$$