# Properties

 Label 775.2.f.c Level $775$ Weight $2$ Character orbit 775.f Analytic conductor $6.188$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

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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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.3317760000.2 Defining polynomial: $$x^{8} - 25x^{4} + 625$$ x^8 - 25*x^4 + 625 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} - \beta_{3} q^{3} - 4 \beta_{2} q^{4} - \beta_{7} q^{6} + \beta_{6} q^{7} + 2 \beta_{4} q^{8} - 2 \beta_{2} q^{9}+O(q^{10})$$ q - b6 * q^2 - b3 * q^3 - 4*b2 * q^4 - b7 * q^6 + b6 * q^7 + 2*b4 * q^8 - 2*b2 * q^9 $$q - \beta_{6} q^{2} - \beta_{3} q^{3} - 4 \beta_{2} q^{4} - \beta_{7} q^{6} + \beta_{6} q^{7} + 2 \beta_{4} q^{8} - 2 \beta_{2} q^{9} - \beta_{7} q^{11} + 4 \beta_1 q^{12} + 6 \beta_{2} q^{14} - 4 q^{16} - \beta_1 q^{17} + 2 \beta_{4} q^{18} + 7 \beta_{2} q^{19} + \beta_{7} q^{21} + 6 \beta_1 q^{22} - 2 \beta_{3} q^{23} - 2 \beta_{5} q^{24} - \beta_1 q^{27} - 4 \beta_{4} q^{28} - \beta_{5} q^{29} + (\beta_{7} - 1) q^{31} + 5 \beta_{4} q^{33} + \beta_{5} q^{34} - 8 q^{36} + 3 \beta_1 q^{37} - 7 \beta_{4} q^{38} + 9 q^{41} - 6 \beta_1 q^{42} - 3 \beta_{3} q^{43} - 4 \beta_{5} q^{44} - 2 \beta_{7} q^{46} + 4 \beta_{3} q^{48} + \beta_{2} q^{49} + 5 q^{51} - 5 \beta_{3} q^{53} + \beta_{5} q^{54} + 12 q^{56} - 7 \beta_1 q^{57} + 6 \beta_{3} q^{58} + 3 \beta_{2} q^{59} + 2 \beta_{7} q^{61} + (\beta_{6} - 6 \beta_1) q^{62} - 2 \beta_{4} q^{63} - 8 \beta_{2} q^{64} - 30 q^{66} + 5 \beta_{6} q^{67} - 4 \beta_{3} q^{68} - 10 \beta_{2} q^{69} - 9 q^{71} + 4 \beta_{6} q^{72} + 3 \beta_{3} q^{73} - 3 \beta_{5} q^{74} + 28 q^{76} - 6 \beta_1 q^{77} + \beta_{5} q^{79} + 11 q^{81} - 9 \beta_{6} q^{82} + 5 \beta_{3} q^{83} + 4 \beta_{5} q^{84} - 3 \beta_{7} q^{86} + 5 \beta_{6} q^{87} + 12 \beta_{3} q^{88} + \beta_{5} q^{89} + 8 \beta_1 q^{92} + ( - 5 \beta_{4} + \beta_{3}) q^{93} - 6 \beta_{6} q^{97} - \beta_{4} q^{98} - 2 \beta_{5} q^{99}+O(q^{100})$$ q - b6 * q^2 - b3 * q^3 - 4*b2 * q^4 - b7 * q^6 + b6 * q^7 + 2*b4 * q^8 - 2*b2 * q^9 - b7 * q^11 + 4*b1 * q^12 + 6*b2 * q^14 - 4 * q^16 - b1 * q^17 + 2*b4 * q^18 + 7*b2 * q^19 + b7 * q^21 + 6*b1 * q^22 - 2*b3 * q^23 - 2*b5 * q^24 - b1 * q^27 - 4*b4 * q^28 - b5 * q^29 + (b7 - 1) * q^31 + 5*b4 * q^33 + b5 * q^34 - 8 * q^36 + 3*b1 * q^37 - 7*b4 * q^38 + 9 * q^41 - 6*b1 * q^42 - 3*b3 * q^43 - 4*b5 * q^44 - 2*b7 * q^46 + 4*b3 * q^48 + b2 * q^49 + 5 * q^51 - 5*b3 * q^53 + b5 * q^54 + 12 * q^56 - 7*b1 * q^57 + 6*b3 * q^58 + 3*b2 * q^59 + 2*b7 * q^61 + (b6 - 6*b1) * q^62 - 2*b4 * q^63 - 8*b2 * q^64 - 30 * q^66 + 5*b6 * q^67 - 4*b3 * q^68 - 10*b2 * q^69 - 9 * q^71 + 4*b6 * q^72 + 3*b3 * q^73 - 3*b5 * q^74 + 28 * q^76 - 6*b1 * q^77 + b5 * q^79 + 11 * q^81 - 9*b6 * q^82 + 5*b3 * q^83 + 4*b5 * q^84 - 3*b7 * q^86 + 5*b6 * q^87 + 12*b3 * q^88 + b5 * q^89 + 8*b1 * q^92 + (-5*b4 + b3) * q^93 - 6*b6 * q^97 - b4 * q^98 - 2*b5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 32 q^{16} - 8 q^{31} - 64 q^{36} + 72 q^{41} + 40 q^{51} + 96 q^{56} - 240 q^{66} - 72 q^{71} + 224 q^{76} + 88 q^{81}+O(q^{100})$$ 8 * q - 32 * q^16 - 8 * q^31 - 64 * q^36 + 72 * q^41 + 40 * q^51 + 96 * q^56 - 240 * q^66 - 72 * q^71 + 224 * q^76 + 88 * q^81

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 25x^{4} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{6} ) / 125$$ (v^6) / 125 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 25\nu ) / 25$$ (-v^5 + 25*v) / 25 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 10\nu^{4} + 50\nu^{2} - 125 ) / 125$$ (-v^6 + 10*v^4 + 50*v^2 - 125) / 125 $$\beta_{5}$$ $$=$$ $$( -2\nu^{7} + 5\nu^{5} + 25\nu^{3} + 125\nu ) / 125$$ (-2*v^7 + 5*v^5 + 25*v^3 + 125*v) / 125 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 10\nu^{4} + 50\nu^{2} + 125 ) / 125$$ (-v^6 - 10*v^4 + 50*v^2 + 125) / 125 $$\beta_{7}$$ $$=$$ $$( 2\nu^{7} + 5\nu^{5} - 25\nu^{3} + 125\nu ) / 125$$ (2*v^7 + 5*v^5 - 25*v^3 + 125*v) / 125
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{5} + 2\beta_{3} ) / 4$$ (b7 + b5 + 2*b3) / 4 $$\nu^{2}$$ $$=$$ $$( 5\beta_{6} + 5\beta_{4} + 10\beta_{2} ) / 4$$ (5*b6 + 5*b4 + 10*b2) / 4 $$\nu^{3}$$ $$=$$ $$5\beta_1$$ 5*b1 $$\nu^{4}$$ $$=$$ $$( -25\beta_{6} + 25\beta_{4} + 50 ) / 4$$ (-25*b6 + 25*b4 + 50) / 4 $$\nu^{5}$$ $$=$$ $$( 25\beta_{7} + 25\beta_{5} - 50\beta_{3} ) / 4$$ (25*b7 + 25*b5 - 50*b3) / 4 $$\nu^{6}$$ $$=$$ $$125\beta_{2}$$ 125*b2 $$\nu^{7}$$ $$=$$ $$( 125\beta_{7} - 125\beta_{5} + 250\beta_1 ) / 4$$ (125*b7 - 125*b5 + 250*b1) / 4

## 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
 2.15988 − 0.578737i −2.15988 + 0.578737i −0.578737 + 2.15988i 0.578737 − 2.15988i 2.15988 + 0.578737i −2.15988 − 0.578737i −0.578737 − 2.15988i 0.578737 + 2.15988i
−1.73205 1.73205i −1.58114 1.58114i 4.00000i 0 5.47723i 1.73205 + 1.73205i 3.46410 3.46410i 2.00000i 0
557.2 −1.73205 1.73205i 1.58114 + 1.58114i 4.00000i 0 5.47723i 1.73205 + 1.73205i 3.46410 3.46410i 2.00000i 0
557.3 1.73205 + 1.73205i −1.58114 1.58114i 4.00000i 0 5.47723i −1.73205 1.73205i −3.46410 + 3.46410i 2.00000i 0
557.4 1.73205 + 1.73205i 1.58114 + 1.58114i 4.00000i 0 5.47723i −1.73205 1.73205i −3.46410 + 3.46410i 2.00000i 0
743.1 −1.73205 + 1.73205i −1.58114 + 1.58114i 4.00000i 0 5.47723i 1.73205 1.73205i 3.46410 + 3.46410i 2.00000i 0
743.2 −1.73205 + 1.73205i 1.58114 1.58114i 4.00000i 0 5.47723i 1.73205 1.73205i 3.46410 + 3.46410i 2.00000i 0
743.3 1.73205 1.73205i −1.58114 + 1.58114i 4.00000i 0 5.47723i −1.73205 + 1.73205i −3.46410 3.46410i 2.00000i 0
743.4 1.73205 1.73205i 1.58114 1.58114i 4.00000i 0 5.47723i −1.73205 + 1.73205i −3.46410 3.46410i 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 743.4 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.b even 2 1 inner
5.c odd 4 2 inner
31.b odd 2 1 inner
155.c odd 2 1 inner
155.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.f.c 8
5.b even 2 1 inner 775.2.f.c 8
5.c odd 4 2 inner 775.2.f.c 8
31.b odd 2 1 inner 775.2.f.c 8
155.c odd 2 1 inner 775.2.f.c 8
155.f even 4 2 inner 775.2.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.f.c 8 1.a even 1 1 trivial
775.2.f.c 8 5.b even 2 1 inner
775.2.f.c 8 5.c odd 4 2 inner
775.2.f.c 8 31.b odd 2 1 inner
775.2.f.c 8 155.c odd 2 1 inner
775.2.f.c 8 155.f even 4 2 inner

## Hecke kernels

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

 $$T_{2}^{4} + 36$$ T2^4 + 36 $$T_{3}^{4} + 25$$ T3^4 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 36)^{2}$$
$3$ $$(T^{4} + 25)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 36)^{2}$$
$11$ $$(T^{2} + 30)^{4}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 25)^{2}$$
$19$ $$(T^{2} + 49)^{4}$$
$23$ $$(T^{4} + 400)^{2}$$
$29$ $$(T^{2} - 30)^{4}$$
$31$ $$(T^{2} + 2 T + 31)^{4}$$
$37$ $$(T^{4} + 2025)^{2}$$
$41$ $$(T - 9)^{8}$$
$43$ $$(T^{4} + 2025)^{2}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + 15625)^{2}$$
$59$ $$(T^{2} + 9)^{4}$$
$61$ $$(T^{2} + 120)^{4}$$
$67$ $$(T^{4} + 22500)^{2}$$
$71$ $$(T + 9)^{8}$$
$73$ $$(T^{4} + 2025)^{2}$$
$79$ $$(T^{2} - 30)^{4}$$
$83$ $$(T^{4} + 15625)^{2}$$
$89$ $$(T^{2} - 30)^{4}$$
$97$ $$(T^{4} + 46656)^{2}$$
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