Properties

 Label 775.2.a.e Level $775$ Weight $2$ Character orbit 775.a Self dual yes Analytic conductor $6.188$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.8468.1 Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 3x + 4$$ x^4 - x^3 - 5*x^2 + 3*x + 4 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - \beta_1 - 1) q^{6} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 - b1 * q^3 + (b3 - b2 + b1 + 1) * q^4 + (-b3 - b1 - 1) * q^6 + (b3 - 2*b2 + b1 - 1) * q^7 + (-b3 + b2 + b1 - 1) * q^8 + b2 * q^9 $$q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{3} - \beta_1 - 1) q^{6} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{8} + \beta_{2} q^{9} + ( - \beta_{3} - \beta_1 - 1) q^{11} + ( - 2 \beta_{2} - 2) q^{12} + ( - \beta_{3} + \beta_1 - 3) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{14} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{16} + ( - \beta_{3} - 3) q^{17} + (\beta_{3} - \beta_{2} + \beta_1 + 3) q^{18} + (\beta_{2} + 2 \beta_1 - 1) q^{19} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{21} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{22} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} - 4 q^{24} + (2 \beta_{3} - 4 \beta_{2}) q^{26} + ( - \beta_{3} + 2 \beta_1 - 1) q^{27} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 6) q^{28} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{29} - q^{31} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{32} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{33} + (\beta_{3} - 4 \beta_{2} - \beta_1 - 1) q^{34} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{36} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{37} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{38} + (\beta_{3} + 3 \beta_1 - 3) q^{39} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{41} + (2 \beta_{2} + 2 \beta_1 - 2) q^{42} + ( - \beta_{3} + 2 \beta_1 - 5) q^{43} + ( - 2 \beta_{3} - 2 \beta_1 - 6) q^{44} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{46} + (4 \beta_{2} - 2 \beta_1 + 4) q^{47} + 4 q^{48} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 5) q^{49} + (\beta_{3} + \beta_{2} + 3 \beta_1) q^{51} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1 - 4) q^{52} + ( - 5 \beta_1 - 2) q^{53} + (3 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{54} + ( - 4 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 4) q^{56} + ( - \beta_{3} - 2 \beta_{2} - 7) q^{57} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{58} + (3 \beta_{3} - \beta_{2} - \beta_1) q^{59} + ( - 2 \beta_1 + 6) q^{61} - \beta_{2} q^{62} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{63} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{64} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 8) q^{66} + (3 \beta_{3} + 3 \beta_1 + 3) q^{67} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 6) q^{68} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 1) q^{69} + (\beta_{2} - 5) q^{71} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{72} + (2 \beta_{3} - \beta_1 - 4) q^{73} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 1) q^{74} + ( - 2 \beta_{3} + 8 \beta_{2} + 2) q^{76} + (2 \beta_{2} + 2 \beta_1 - 2) q^{77} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{78} + ( - \beta_{3} - 2 \beta_{2} + 7 \beta_1 - 3) q^{79} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 6) q^{81} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 9) q^{82} + (3 \beta_{3} + 2 \beta_{2} + 5) q^{83} + (2 \beta_{3} - 2 \beta_1 + 10) q^{84} + (3 \beta_{3} - 6 \beta_{2} + \beta_1 + 1) q^{86} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 7) q^{87} - 4 \beta_{2} q^{88} + ( - \beta_{3} + 6 \beta_{2} - \beta_1 - 1) q^{89} + ( - 4 \beta_{3} + 10 \beta_{2} - 6 \beta_1 + 2) q^{91} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{92} + \beta_1 q^{93} + (2 \beta_{3} + 2 \beta_1 + 10) q^{94} + (4 \beta_{2} + 8) q^{96} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{97} + ( - 8 \beta_{3} + 9 \beta_{2} - 4 \beta_1 - 8) q^{98} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{99}+O(q^{100})$$ q + b2 * q^2 - b1 * q^3 + (b3 - b2 + b1 + 1) * q^4 + (-b3 - b1 - 1) * q^6 + (b3 - 2*b2 + b1 - 1) * q^7 + (-b3 + b2 + b1 - 1) * q^8 + b2 * q^9 + (-b3 - b1 - 1) * q^11 + (-2*b2 - 2) * q^12 + (-b3 + b1 - 3) * q^13 + (-2*b3 + 2*b2 - 4) * q^14 + (b3 - b2 - b1 + 1) * q^16 + (-b3 - 3) * q^17 + (b3 - b2 + b1 + 3) * q^18 + (b2 + 2*b1 - 1) * q^19 + (b3 - 2*b2 + 3*b1 - 1) * q^21 + (-2*b2 - 2*b1 - 2) * q^22 + (-b3 + 2*b2 - b1 + 1) * q^23 - 4 * q^24 + (2*b3 - 4*b2) * q^26 + (-b3 + 2*b1 - 1) * q^27 + (2*b3 - 4*b2 - 2*b1 + 6) * q^28 + (-b3 + 2*b2 - 3*b1 - 1) * q^29 - q^31 + (-b3 + b2 - 3*b1 - 1) * q^32 + (b3 + 2*b2 + b1 + 3) * q^33 + (b3 - 4*b2 - b1 - 1) * q^34 + (-b3 + 3*b2 + b1 - 1) * q^36 + (3*b3 - 2*b2 + 2*b1 - 1) * q^37 + (3*b3 - 2*b2 + 3*b1 + 5) * q^38 + (b3 + 3*b1 - 3) * q^39 + (-b3 - 3*b2 + b1 - 4) * q^41 + (2*b2 + 2*b1 - 2) * q^42 + (-b3 + 2*b1 - 5) * q^43 + (-2*b3 - 2*b1 - 6) * q^44 + (2*b3 - 2*b2 + 4) * q^46 + (4*b2 - 2*b1 + 4) * q^47 + 4 * q^48 + (2*b3 - 2*b2 - 4*b1 + 5) * q^49 + (b3 + b2 + 3*b1) * q^51 + (-4*b3 + 6*b2 - 4*b1 - 4) * q^52 + (-5*b1 - 2) * q^53 + (3*b3 - 2*b2 + b1 + 1) * q^54 + (-4*b3 + 8*b2 - 4*b1 - 4) * q^56 + (-b3 - 2*b2 - 7) * q^57 + (-4*b2 - 2*b1 + 2) * q^58 + (3*b3 - b2 - b1) * q^59 + (-2*b1 + 6) * q^61 - b2 * q^62 + (-2*b3 + 2*b2 - 4) * q^63 + (-3*b3 - b2 - b1 - 3) * q^64 + (2*b3 + 2*b2 + 4*b1 + 8) * q^66 + (3*b3 + 3*b1 + 3) * q^67 + (-4*b3 + 4*b2 - 4*b1 - 6) * q^68 + (-b3 + 2*b2 - 3*b1 + 1) * q^69 + (b2 - 5) * q^71 + (3*b3 - 3*b2 + b1 + 3) * q^72 + (2*b3 - b1 - 4) * q^73 + (-3*b3 + 4*b2 + 3*b1 - 1) * q^74 + (-2*b3 + 8*b2 + 2) * q^76 + (2*b2 + 2*b1 - 2) * q^77 + (2*b3 - 2*b2 + 4*b1 + 4) * q^78 + (-b3 - 2*b2 + 7*b1 - 3) * q^79 + (b3 - 4*b2 + b1 - 6) * q^81 + (-b3 - 2*b2 - 3*b1 - 9) * q^82 + (3*b3 + 2*b2 + 5) * q^83 + (2*b3 - 2*b1 + 10) * q^84 + (3*b3 - 6*b2 + b1 + 1) * q^86 + (-b3 + 4*b2 - b1 + 7) * q^87 - 4*b2 * q^88 + (-b3 + 6*b2 - b1 - 1) * q^89 + (-4*b3 + 10*b2 - 6*b1 + 2) * q^91 + (-2*b3 + 4*b2 + 2*b1 - 6) * q^92 + b1 * q^93 + (2*b3 + 2*b1 + 10) * q^94 + (4*b2 + 8) * q^96 + (4*b3 + 2*b2 + 2*b1) * q^97 + (-8*b3 + 9*b2 - 4*b1 - 8) * q^98 + (-2*b2 - 2*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 2 q^{7} - 3 q^{8} - q^{9}+O(q^{10})$$ 4 * q - q^2 - q^3 + 5 * q^4 - 4 * q^6 - 2 * q^7 - 3 * q^8 - q^9 $$4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} - 4 q^{11} - 6 q^{12} - 10 q^{13} - 16 q^{14} + 3 q^{16} - 11 q^{17} + 13 q^{18} - 3 q^{19} - 8 q^{22} + 2 q^{23} - 16 q^{24} + 2 q^{26} - q^{27} + 24 q^{28} - 8 q^{29} - 4 q^{31} - 7 q^{32} + 10 q^{33} - 2 q^{34} - 5 q^{36} - 3 q^{37} + 22 q^{38} - 10 q^{39} - 11 q^{41} - 8 q^{42} - 17 q^{43} - 24 q^{44} + 16 q^{46} + 10 q^{47} + 16 q^{48} + 16 q^{49} + q^{51} - 22 q^{52} - 13 q^{53} + 4 q^{54} - 24 q^{56} - 25 q^{57} + 10 q^{58} - 3 q^{59} + 22 q^{61} + q^{62} - 16 q^{63} - 9 q^{64} + 32 q^{66} + 12 q^{67} - 28 q^{68} - 21 q^{71} + 13 q^{72} - 19 q^{73} - 2 q^{74} + 2 q^{76} - 8 q^{77} + 20 q^{78} - 2 q^{79} - 20 q^{81} - 36 q^{82} + 15 q^{83} + 36 q^{84} + 8 q^{86} + 24 q^{87} + 4 q^{88} - 10 q^{89} - 4 q^{91} - 24 q^{92} + q^{93} + 40 q^{94} + 28 q^{96} - 4 q^{97} - 37 q^{98} - 8 q^{99}+O(q^{100})$$ 4 * q - q^2 - q^3 + 5 * q^4 - 4 * q^6 - 2 * q^7 - 3 * q^8 - q^9 - 4 * q^11 - 6 * q^12 - 10 * q^13 - 16 * q^14 + 3 * q^16 - 11 * q^17 + 13 * q^18 - 3 * q^19 - 8 * q^22 + 2 * q^23 - 16 * q^24 + 2 * q^26 - q^27 + 24 * q^28 - 8 * q^29 - 4 * q^31 - 7 * q^32 + 10 * q^33 - 2 * q^34 - 5 * q^36 - 3 * q^37 + 22 * q^38 - 10 * q^39 - 11 * q^41 - 8 * q^42 - 17 * q^43 - 24 * q^44 + 16 * q^46 + 10 * q^47 + 16 * q^48 + 16 * q^49 + q^51 - 22 * q^52 - 13 * q^53 + 4 * q^54 - 24 * q^56 - 25 * q^57 + 10 * q^58 - 3 * q^59 + 22 * q^61 + q^62 - 16 * q^63 - 9 * q^64 + 32 * q^66 + 12 * q^67 - 28 * q^68 - 21 * q^71 + 13 * q^72 - 19 * q^73 - 2 * q^74 + 2 * q^76 - 8 * q^77 + 20 * q^78 - 2 * q^79 - 20 * q^81 - 36 * q^82 + 15 * q^83 + 36 * q^84 + 8 * q^86 + 24 * q^87 + 4 * q^88 - 10 * q^89 - 4 * q^91 - 24 * q^92 + q^93 + 40 * q^94 + 28 * q^96 - 4 * q^97 - 37 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5x^{2} + 3x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu - 1$$ v^3 - 4*v - 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta _1 + 1$$ b3 + 4*b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.704624 1.31743 −1.89122 2.27841
−2.50350 0.704624 4.26753 0 −1.76403 4.77104 −5.67678 −2.50350 0
1.2 −1.26438 −1.31743 −0.401352 0 1.66573 −1.13698 3.03621 −1.26438 0
1.3 0.576713 1.89122 −1.66740 0 1.09069 −4.24412 −2.11504 0.576713 0
1.4 2.19117 −2.27841 2.80122 0 −4.99239 −1.38995 1.75561 2.19117 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.e 4
3.b odd 2 1 6975.2.a.bn 4
5.b even 2 1 155.2.a.e 4
5.c odd 4 2 775.2.b.f 8
15.d odd 2 1 1395.2.a.l 4
20.d odd 2 1 2480.2.a.x 4
35.c odd 2 1 7595.2.a.s 4
40.e odd 2 1 9920.2.a.cg 4
40.f even 2 1 9920.2.a.cb 4
155.c odd 2 1 4805.2.a.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.e 4 5.b even 2 1
775.2.a.e 4 1.a even 1 1 trivial
775.2.b.f 8 5.c odd 4 2
1395.2.a.l 4 15.d odd 2 1
2480.2.a.x 4 20.d odd 2 1
4805.2.a.n 4 155.c odd 2 1
6975.2.a.bn 4 3.b odd 2 1
7595.2.a.s 4 35.c odd 2 1
9920.2.a.cb 4 40.f even 2 1
9920.2.a.cg 4 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - 4T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(775))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 6 T^{2} - 4 T + 4$$
$3$ $$T^{4} + T^{3} - 5 T^{2} - 3 T + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2 T^{3} - 20 T^{2} - 52 T - 32$$
$11$ $$T^{4} + 4 T^{3} - 8 T^{2} - 12 T + 16$$
$13$ $$T^{4} + 10 T^{3} + 20 T^{2} + \cdots - 136$$
$17$ $$T^{4} + 11 T^{3} + 35 T^{2} + 13 T - 58$$
$19$ $$T^{4} + 3 T^{3} - 33 T^{2} - 107 T + 44$$
$23$ $$T^{4} - 2 T^{3} - 20 T^{2} + 52 T - 32$$
$29$ $$T^{4} + 8 T^{3} - 20 T^{2} - 292 T - 584$$
$31$ $$(T + 1)^{4}$$
$37$ $$T^{4} + 3 T^{3} - 81 T^{2} + \cdots + 1538$$
$41$ $$T^{4} + 11 T^{3} - 31 T^{2} + \cdots + 506$$
$43$ $$T^{4} + 17 T^{3} + 73 T^{2} + \cdots - 236$$
$47$ $$T^{4} - 10 T^{3} - 52 T^{2} + \cdots + 1408$$
$53$ $$T^{4} + 13 T^{3} - 71 T^{2} + \cdots + 1306$$
$59$ $$T^{4} + 3 T^{3} - 97 T^{2} + 129 T - 44$$
$61$ $$T^{4} - 22 T^{3} + 160 T^{2} + \cdots + 352$$
$67$ $$T^{4} - 12 T^{3} - 72 T^{2} + \cdots + 1296$$
$71$ $$T^{4} + 21 T^{3} + 159 T^{2} + \cdots + 584$$
$73$ $$T^{4} + 19 T^{3} + 85 T^{2} + 123 T + 34$$
$79$ $$T^{4} + 2 T^{3} - 260 T^{2} + \cdots + 6592$$
$83$ $$T^{4} - 15 T^{3} - 63 T^{2} + \cdots - 3364$$
$89$ $$T^{4} + 10 T^{3} - 152 T^{2} + \cdots + 3688$$
$97$ $$T^{4} + 4 T^{3} - 248 T^{2} + \cdots - 464$$