# Properties

 Label 775.2.a.g Level $775$ Weight $2$ Character orbit 775.a Self dual yes Analytic conductor $6.188$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.20308.1 Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + 4x + 12$$ x^4 - x^3 - 8*x^2 + 4*x + 12 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{2} + 2) q^{6} - \beta_{2} q^{7} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{8} + ( - 2 \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 - b3 * q^3 + (b2 + b1 + 2) * q^4 + (-b2 + 2) * q^6 - b2 * q^7 + (2*b3 + b2 + 3*b1 + 2) * q^8 + (-2*b2 - b1 + 2) * q^9 $$q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + ( - \beta_{2} + 2) q^{6} - \beta_{2} q^{7} + (2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{8} + ( - 2 \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{2} + 2 \beta_1 - 2) q^{11} + 2 q^{12} + (\beta_{2} - 4) q^{13} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{14} + (2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 2) q^{16} + ( - \beta_{3} - \beta_{2} - 2 \beta_1) q^{17} + ( - 4 \beta_{3} - \beta_{2} - 3 \beta_1) q^{18} + ( - 2 \beta_{3} - \beta_1 + 1) q^{19} + ( - 2 \beta_{3} - \beta_{2}) q^{21} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 6) q^{22} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{23} + (2 \beta_{2} + 2 \beta_1 - 4) q^{24} + (2 \beta_{3} - 2 \beta_1 - 2) q^{26} + ( - 3 \beta_{3} - \beta_{2} - 2) q^{27} + ( - 2 \beta_{2} - 4) q^{28} + (2 \beta_{3} + \beta_{2} + 2) q^{29} + q^{31} + (2 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 6) q^{32} + (4 \beta_{3} - \beta_{2} + 4) q^{33} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 4) q^{34} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 6) q^{36} + (\beta_{3} + \beta_{2} - 2) q^{37} - 3 \beta_{2} q^{38} + (6 \beta_{3} + \beta_{2}) q^{39} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{41} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{42} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{43} + (4 \beta_{3} + 2 \beta_{2} + 8 \beta_1 + 4) q^{44} + (2 \beta_{3} - 4 \beta_{2} - 6) q^{46} + (2 \beta_{3} + 4) q^{47} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{48} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{49} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{51} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{52} + (3 \beta_{3} - 2 \beta_{2} - 2) q^{53} + ( - 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 8) q^{54} - 4 \beta_1 q^{56} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 8) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 6) q^{58} + ( - 4 \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{59} + (2 \beta_{3} + 6) q^{61} + \beta_1 q^{62} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 10) q^{63} + (6 \beta_{3} + 3 \beta_{2} + 13 \beta_1 + 10) q^{64} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{66} + (\beta_{2} + 2 \beta_1 + 2) q^{67} + ( - 4 \beta_{3} - 4 \beta_{2} - 10 \beta_1 - 6) q^{68} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{69} + ( - 2 \beta_{2} - \beta_1 + 1) q^{71} + (2 \beta_{3} - 3 \beta_{2} - 9 \beta_1 - 2) q^{72} + ( - 3 \beta_{3} - 4 \beta_1 - 2) q^{73} + (2 \beta_{3} + \beta_{2} - 4) q^{74} + ( - 2 \beta_{3} - 4 \beta_1 + 4) q^{76} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 14) q^{78} + ( - 3 \beta_{2} - 4) q^{79} + ( - \beta_{2} + 9) q^{81} + (2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 2) q^{82} + ( - \beta_{3} + 3 \beta_{2} + 4) q^{83} - 2 \beta_{2} q^{84} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 4) q^{86} + (5 \beta_{2} + 2 \beta_1 - 10) q^{87} + (8 \beta_{2} + 12 \beta_1 + 8) q^{88} + (\beta_{2} - 4 \beta_1 - 2) q^{89} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{91} + ( - 4 \beta_{3} - 10 \beta_1 + 4) q^{92} - \beta_{3} q^{93} + (2 \beta_{2} + 4 \beta_1 - 4) q^{94} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{96} + (2 \beta_{3} + 2 \beta_1 - 4) q^{97} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{98} + ( - 6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 14) q^{99}+O(q^{100})$$ q + b1 * q^2 - b3 * q^3 + (b2 + b1 + 2) * q^4 + (-b2 + 2) * q^6 - b2 * q^7 + (2*b3 + b2 + 3*b1 + 2) * q^8 + (-2*b2 - b1 + 2) * q^9 + (b2 + 2*b1 - 2) * q^11 + 2 * q^12 + (b2 - 4) * q^13 + (-2*b3 - 2*b1 + 2) * q^14 + (2*b3 + 3*b2 + 5*b1 + 2) * q^16 + (-b3 - b2 - 2*b1) * q^17 + (-4*b3 - b2 - 3*b1) * q^18 + (-2*b3 - b1 + 1) * q^19 + (-2*b3 - b2) * q^21 + (2*b3 + 2*b2 + 2*b1 + 6) * q^22 + (-2*b3 + b2 - 2*b1) * q^23 + (2*b2 + 2*b1 - 4) * q^24 + (2*b3 - 2*b1 - 2) * q^26 + (-3*b3 - b2 - 2) * q^27 + (-2*b2 - 4) * q^28 + (2*b3 + b2 + 2) * q^29 + q^31 + (2*b3 + 5*b2 + 7*b1 + 6) * q^32 + (4*b3 - b2 + 4) * q^33 + (-2*b3 - 3*b2 - 4*b1 - 4) * q^34 + (-2*b3 - 3*b2 - 3*b1 - 6) * q^36 + (b3 + b2 - 2) * q^37 - 3*b2 * q^38 + (6*b3 + b2) * q^39 + (-2*b3 + b2 - b1 + 3) * q^41 + (-2*b3 - 2*b2 - 2*b1 + 6) * q^42 + (b3 + b2 + 2*b1 - 2) * q^43 + (4*b3 + 2*b2 + 8*b1 + 4) * q^44 + (2*b3 - 4*b2 - 6) * q^46 + (2*b3 + 4) * q^47 + (4*b3 + 2*b2 + 2*b1) * q^48 + (-2*b3 - 2*b1 - 1) * q^49 + (-2*b3 - b2 - b1 + 1) * q^51 + (-2*b2 - 4*b1 - 4) * q^52 + (3*b3 - 2*b2 - 2) * q^53 + (-2*b3 - 3*b2 - 4*b1 + 8) * q^54 - 4*b1 * q^56 + (-b3 - 3*b2 - 2*b1 + 8) * q^57 + (2*b3 + 2*b2 + 4*b1 - 6) * q^58 + (-4*b3 - b2 - 3*b1 + 3) * q^59 + (2*b3 + 6) * q^61 + b1 * q^62 + (-2*b3 - 2*b2 - 2*b1 + 10) * q^63 + (6*b3 + 3*b2 + 13*b1 + 10) * q^64 + (-2*b3 + 4*b2 + 2*b1 - 6) * q^66 + (b2 + 2*b1 + 2) * q^67 + (-4*b3 - 4*b2 - 10*b1 - 6) * q^68 + (2*b3 - b2 - 2*b1 + 6) * q^69 + (-2*b2 - b1 + 1) * q^71 + (2*b3 - 3*b2 - 9*b1 - 2) * q^72 + (-3*b3 - 4*b1 - 2) * q^73 + (2*b3 + b2 - 4) * q^74 + (-2*b3 - 4*b1 + 4) * q^76 + (-2*b3 + 2*b2 - 2*b1 - 2) * q^77 + (2*b3 + 6*b2 + 2*b1 - 14) * q^78 + (-3*b2 - 4) * q^79 + (-b2 + 9) * q^81 + (2*b3 - 3*b2 + 4*b1 - 2) * q^82 + (-b3 + 3*b2 + 4) * q^83 - 2*b2 * q^84 + (2*b3 + 3*b2 + 2*b1 + 4) * q^86 + (5*b2 + 2*b1 - 10) * q^87 + (8*b2 + 12*b1 + 8) * q^88 + (b2 - 4*b1 - 2) * q^89 + (2*b3 + 4*b2 + 2*b1 - 6) * q^91 + (-4*b3 - 10*b1 + 4) * q^92 - b3 * q^93 + (2*b2 + 4*b1 - 4) * q^94 + (4*b3 + 2*b2 + 2*b1 + 4) * q^96 + (2*b3 + 2*b1 - 4) * q^97 + (-4*b2 - 3*b1 - 4) * q^98 + (-6*b3 + 4*b2 - 2*b1 - 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{3} + 9 q^{4} + 8 q^{6} + 9 q^{8} + 7 q^{9}+O(q^{10})$$ 4 * q + q^2 + q^3 + 9 * q^4 + 8 * q^6 + 9 * q^8 + 7 * q^9 $$4 q + q^{2} + q^{3} + 9 q^{4} + 8 q^{6} + 9 q^{8} + 7 q^{9} - 6 q^{11} + 8 q^{12} - 16 q^{13} + 8 q^{14} + 11 q^{16} - q^{17} + q^{18} + 5 q^{19} + 2 q^{21} + 24 q^{22} - 14 q^{24} - 12 q^{26} - 5 q^{27} - 16 q^{28} + 6 q^{29} + 4 q^{31} + 29 q^{32} + 12 q^{33} - 18 q^{34} - 25 q^{36} - 9 q^{37} - 6 q^{39} + 13 q^{41} + 24 q^{42} - 7 q^{43} + 20 q^{44} - 26 q^{46} + 14 q^{47} - 2 q^{48} - 4 q^{49} + 5 q^{51} - 20 q^{52} - 11 q^{53} + 30 q^{54} - 4 q^{56} + 31 q^{57} - 22 q^{58} + 13 q^{59} + 22 q^{61} + q^{62} + 40 q^{63} + 47 q^{64} - 20 q^{66} + 10 q^{67} - 30 q^{68} + 20 q^{69} + 3 q^{71} - 19 q^{72} - 9 q^{73} - 18 q^{74} + 14 q^{76} - 8 q^{77} - 56 q^{78} - 16 q^{79} + 36 q^{81} - 6 q^{82} + 17 q^{83} + 16 q^{86} - 38 q^{87} + 44 q^{88} - 12 q^{89} - 24 q^{91} + 10 q^{92} + q^{93} - 12 q^{94} + 14 q^{96} - 16 q^{97} - 19 q^{98} - 52 q^{99}+O(q^{100})$$ 4 * q + q^2 + q^3 + 9 * q^4 + 8 * q^6 + 9 * q^8 + 7 * q^9 - 6 * q^11 + 8 * q^12 - 16 * q^13 + 8 * q^14 + 11 * q^16 - q^17 + q^18 + 5 * q^19 + 2 * q^21 + 24 * q^22 - 14 * q^24 - 12 * q^26 - 5 * q^27 - 16 * q^28 + 6 * q^29 + 4 * q^31 + 29 * q^32 + 12 * q^33 - 18 * q^34 - 25 * q^36 - 9 * q^37 - 6 * q^39 + 13 * q^41 + 24 * q^42 - 7 * q^43 + 20 * q^44 - 26 * q^46 + 14 * q^47 - 2 * q^48 - 4 * q^49 + 5 * q^51 - 20 * q^52 - 11 * q^53 + 30 * q^54 - 4 * q^56 + 31 * q^57 - 22 * q^58 + 13 * q^59 + 22 * q^61 + q^62 + 40 * q^63 + 47 * q^64 - 20 * q^66 + 10 * q^67 - 30 * q^68 + 20 * q^69 + 3 * q^71 - 19 * q^72 - 9 * q^73 - 18 * q^74 + 14 * q^76 - 8 * q^77 - 56 * q^78 - 16 * q^79 + 36 * q^81 - 6 * q^82 + 17 * q^83 + 16 * q^86 - 38 * q^87 + 44 * q^88 - 12 * q^89 - 24 * q^91 + 10 * q^92 + q^93 - 12 * q^94 + 14 * q^96 - 16 * q^97 - 19 * q^98 - 52 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8x^{2} + 4x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2$$ (v^3 - v^2 - 6*v + 2) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + \beta_{2} + 7\beta _1 + 2$$ 2*b3 + b2 + 7*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.27244 −1.15729 1.62946 2.80027
−2.27244 0.632112 3.16400 0 −1.43644 −3.43644 −2.64511 −2.60043 0
1.2 −1.15729 −3.02722 −0.660672 0 3.50338 1.50338 3.07918 6.16405 0
1.3 1.62946 3.05273 0.655151 0 4.97431 2.97431 −2.19138 6.31916 0
1.4 2.80027 0.342376 5.84153 0 0.958747 −1.04125 10.7573 −2.88278 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.a.g 4
3.b odd 2 1 6975.2.a.bj 4
5.b even 2 1 155.2.a.d 4
5.c odd 4 2 775.2.b.e 8
15.d odd 2 1 1395.2.a.m 4
20.d odd 2 1 2480.2.a.z 4
35.c odd 2 1 7595.2.a.q 4
40.e odd 2 1 9920.2.a.cd 4
40.f even 2 1 9920.2.a.ch 4
155.c odd 2 1 4805.2.a.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.d 4 5.b even 2 1
775.2.a.g 4 1.a even 1 1 trivial
775.2.b.e 8 5.c odd 4 2
1395.2.a.m 4 15.d odd 2 1
2480.2.a.z 4 20.d odd 2 1
4805.2.a.j 4 155.c odd 2 1
6975.2.a.bj 4 3.b odd 2 1
7595.2.a.q 4 35.c odd 2 1
9920.2.a.cd 4 40.e odd 2 1
9920.2.a.ch 4 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} - 8T_{2}^{2} + 4T_{2} + 12$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(775))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} - 8 T^{2} + 4 T + 12$$
$3$ $$T^{4} - T^{3} - 9 T^{2} + 9 T - 2$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 12 T^{2} + 4 T + 16$$
$11$ $$T^{4} + 6 T^{3} - 16 T^{2} - 124 T - 144$$
$13$ $$T^{4} + 16 T^{3} + 84 T^{2} + 156 T + 64$$
$17$ $$T^{4} + T^{3} - 25 T^{2} + 49 T - 24$$
$19$ $$T^{4} - 5 T^{3} - 21 T^{2} + 81 T + 108$$
$23$ $$T^{4} - 64 T^{2} - 196 T - 24$$
$29$ $$T^{4} - 6 T^{3} - 40 T^{2} + 308 T - 456$$
$31$ $$(T - 1)^{4}$$
$37$ $$T^{4} + 9 T^{3} + 7 T^{2} - 7 T - 4$$
$41$ $$T^{4} - 13 T^{3} + 17 T^{2} + \cdots - 294$$
$43$ $$T^{4} + 7 T^{3} - 7 T^{2} - 129 T - 214$$
$47$ $$T^{4} - 14 T^{3} + 36 T^{2} + \cdots - 192$$
$53$ $$T^{4} + 11 T^{3} - 75 T^{2} + \cdots - 2892$$
$59$ $$T^{4} - 13 T^{3} - 65 T^{2} + \cdots + 2484$$
$61$ $$T^{4} - 22 T^{3} + 144 T^{2} + \cdots - 32$$
$67$ $$T^{4} - 10 T^{3} + 8 T^{2} + 36 T - 32$$
$71$ $$T^{4} - 3 T^{3} - 37 T^{2} + 59 T + 384$$
$73$ $$T^{4} + 9 T^{3} - 95 T^{2} - 649 T + 452$$
$79$ $$T^{4} + 16 T^{3} - 12 T^{2} + \cdots + 256$$
$83$ $$T^{4} - 17 T^{3} - 3 T^{2} + 455 T + 738$$
$89$ $$T^{4} + 12 T^{3} - 124 T^{2} + \cdots + 1656$$
$97$ $$T^{4} + 16 T^{3} + 56 T^{2} - 48 T - 16$$