# Properties

 Label 775.2.a.f Level $775$ Weight $2$ Character orbit 775.a Self dual yes Analytic conductor $6.188$ Analytic rank $1$ 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.18840615665$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 155) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.517638 −1.93185 1.93185 −0.517638
−1.41421 −0.517638 0 0 0.732051 2.44949 2.82843 −2.73205 0
1.2 −1.41421 1.93185 0 0 −2.73205 −2.44949 2.82843 0.732051 0
1.3 1.41421 −1.93185 0 0 −2.73205 2.44949 −2.82843 0.732051 0
1.4 1.41421 0.517638 0 0 0.732051 −2.44949 −2.82843 −2.73205 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$31$$ $$-1$$

## 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.a.f 4
3.b odd 2 1 6975.2.a.bk 4
5.b even 2 1 inner 775.2.a.f 4
5.c odd 4 2 155.2.b.a 4
15.d odd 2 1 6975.2.a.bk 4
15.e even 4 2 1395.2.c.c 4
20.e even 4 2 2480.2.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.b.a 4 5.c odd 4 2
775.2.a.f 4 1.a even 1 1 trivial
775.2.a.f 4 5.b even 2 1 inner
1395.2.c.c 4 15.e even 4 2
2480.2.d.b 4 20.e even 4 2
6975.2.a.bk 4 3.b odd 2 1
6975.2.a.bk 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(775))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} - 4T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 6)^{2}$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$(T^{2} - 6)^{2}$$
$17$ $$T^{4} - 28T^{2} + 169$$
$19$ $$(T^{2} + 4 T - 23)^{2}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$(T^{2} + 6 T - 18)^{2}$$
$31$ $$(T - 1)^{4}$$
$37$ $$T^{4} - 12T^{2} + 9$$
$41$ $$(T^{2} + 18 T + 69)^{2}$$
$43$ $$T^{4} - 156T^{2} + 9$$
$47$ $$T^{4} - 112T^{2} + 2704$$
$53$ $$T^{4} - 76T^{2} + 121$$
$59$ $$(T^{2} + 6 T - 39)^{2}$$
$61$ $$(T^{2} + 8 T - 92)^{2}$$
$67$ $$(T^{2} - 6)^{2}$$
$71$ $$(T^{2} + 12 T + 33)^{2}$$
$73$ $$T^{4} - 84T^{2} + 1089$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} - 268 T^{2} + 11881$$
$89$ $$(T^{2} + 18 T + 54)^{2}$$
$97$ $$T^{4} - 144T^{2} + 1296$$