# Properties

 Label 1078.2.e.v Level $1078$ Weight $2$ Character orbit 1078.e Analytic conductor $8.608$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.60787333789$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} + 4 \beta_{2} q^{9}+O(q^{10})$$ q - b2 * q^2 + b1 * q^3 + (-b2 - 1) * q^4 + (b3 - b2 + b1) * q^5 - b3 * q^6 - q^8 + 4*b2 * q^9 $$q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} + 4 \beta_{2} q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{10} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{3} - \beta_1) q^{12} - 5 q^{13} + ( - \beta_{3} - 7) q^{15} + \beta_{2} q^{16} + (6 \beta_{2} + 6) q^{17} + (4 \beta_{2} + 4) q^{18} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 1) q^{20} - q^{22} + (\beta_{3} - \beta_{2} + \beta_1) q^{23} - \beta_1 q^{24} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{25} + 5 \beta_{2} q^{26} + \beta_{3} q^{27} + ( - 2 \beta_{3} + 1) q^{29} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{30} + ( - 4 \beta_{2} - 4) q^{31} + (\beta_{2} + 1) q^{32} + ( - \beta_{3} - \beta_1) q^{33} + 6 q^{34} + 4 q^{36} + (\beta_{3} + \beta_{2} + \beta_1) q^{37} + (3 \beta_{2} + \beta_1 + 3) q^{38} - 5 \beta_1 q^{39} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{40} + (3 \beta_{3} - 3) q^{41} - 4 q^{43} + \beta_{2} q^{44} + (4 \beta_{2} - 4 \beta_1 + 4) q^{45} + ( - \beta_{2} + \beta_1 - 1) q^{46} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{47} + \beta_{3} q^{48} + ( - 2 \beta_{3} - 3) q^{50} + (6 \beta_{3} + 6 \beta_1) q^{51} + (5 \beta_{2} + 5) q^{52} + (\beta_{2} - \beta_1 + 1) q^{53} + (\beta_{3} + \beta_1) q^{54} + ( - \beta_{3} - 1) q^{55} + (3 \beta_{3} - 7) q^{57} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{58} + (2 \beta_{2} + \beta_1 + 2) q^{59} + (7 \beta_{2} - \beta_1 + 7) q^{60} + ( - 2 \beta_{3} - 9 \beta_{2} - 2 \beta_1) q^{61} - 4 q^{62} + q^{64} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{65} - \beta_1 q^{66} + (4 \beta_{2} + 3 \beta_1 + 4) q^{67} - 6 \beta_{2} q^{68} + ( - \beta_{3} - 7) q^{69} + (\beta_{3} + 7) q^{71} - 4 \beta_{2} q^{72} + (3 \beta_{2} - \beta_1 + 3) q^{73} + (\beta_{2} + \beta_1 + 1) q^{74} + ( - 3 \beta_{3} + 14 \beta_{2} - 3 \beta_1) q^{75} + ( - \beta_{3} + 3) q^{76} + 5 \beta_{3} q^{78} + ( - \beta_{3} - \beta_1) q^{79} + (\beta_{2} - \beta_1 + 1) q^{80} + (5 \beta_{2} + 5) q^{81} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{82} + ( - 2 \beta_{3} - 8) q^{83} + (6 \beta_{3} + 6) q^{85} + 4 \beta_{2} q^{86} + (14 \beta_{2} + \beta_1 + 14) q^{87} + (\beta_{2} + 1) q^{88} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{89} + (4 \beta_{3} + 4) q^{90} + ( - \beta_{3} - 1) q^{92} + ( - 4 \beta_{3} - 4 \beta_1) q^{93} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{94} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{95} + (\beta_{3} + \beta_1) q^{96} + ( - 2 \beta_{3} + 11) q^{97} + 4 q^{99}+O(q^{100})$$ q - b2 * q^2 + b1 * q^3 + (-b2 - 1) * q^4 + (b3 - b2 + b1) * q^5 - b3 * q^6 - q^8 + 4*b2 * q^9 + (-b2 + b1 - 1) * q^10 + (-b2 - 1) * q^11 + (-b3 - b1) * q^12 - 5 * q^13 + (-b3 - 7) * q^15 + b2 * q^16 + (6*b2 + 6) * q^17 + (4*b2 + 4) * q^18 + (b3 + 3*b2 + b1) * q^19 + (-b3 - 1) * q^20 - q^22 + (b3 - b2 + b1) * q^23 - b1 * q^24 + (-3*b2 + 2*b1 - 3) * q^25 + 5*b2 * q^26 + b3 * q^27 + (-2*b3 + 1) * q^29 + (-b3 + 7*b2 - b1) * q^30 + (-4*b2 - 4) * q^31 + (b2 + 1) * q^32 + (-b3 - b1) * q^33 + 6 * q^34 + 4 * q^36 + (b3 + b2 + b1) * q^37 + (3*b2 + b1 + 3) * q^38 - 5*b1 * q^39 + (-b3 + b2 - b1) * q^40 + (3*b3 - 3) * q^41 - 4 * q^43 + b2 * q^44 + (4*b2 - 4*b1 + 4) * q^45 + (-b2 + b1 - 1) * q^46 + (2*b3 - 8*b2 + 2*b1) * q^47 + b3 * q^48 + (-2*b3 - 3) * q^50 + (6*b3 + 6*b1) * q^51 + (5*b2 + 5) * q^52 + (b2 - b1 + 1) * q^53 + (b3 + b1) * q^54 + (-b3 - 1) * q^55 + (3*b3 - 7) * q^57 + (-2*b3 - b2 - 2*b1) * q^58 + (2*b2 + b1 + 2) * q^59 + (7*b2 - b1 + 7) * q^60 + (-2*b3 - 9*b2 - 2*b1) * q^61 - 4 * q^62 + q^64 + (-5*b3 + 5*b2 - 5*b1) * q^65 - b1 * q^66 + (4*b2 + 3*b1 + 4) * q^67 - 6*b2 * q^68 + (-b3 - 7) * q^69 + (b3 + 7) * q^71 - 4*b2 * q^72 + (3*b2 - b1 + 3) * q^73 + (b2 + b1 + 1) * q^74 + (-3*b3 + 14*b2 - 3*b1) * q^75 + (-b3 + 3) * q^76 + 5*b3 * q^78 + (-b3 - b1) * q^79 + (b2 - b1 + 1) * q^80 + (5*b2 + 5) * q^81 + (3*b3 + 3*b2 + 3*b1) * q^82 + (-2*b3 - 8) * q^83 + (6*b3 + 6) * q^85 + 4*b2 * q^86 + (14*b2 + b1 + 14) * q^87 + (b2 + 1) * q^88 + (-4*b3 + 4*b2 - 4*b1) * q^89 + (4*b3 + 4) * q^90 + (-b3 - 1) * q^92 + (-4*b3 - 4*b1) * q^93 + (-8*b2 + 2*b1 - 8) * q^94 + (-4*b2 - 2*b1 - 4) * q^95 + (b3 + b1) * q^96 + (-2*b3 + 11) * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{8} - 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 4 * q^8 - 8 * q^9 $$4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{8} - 8 q^{9} - 2 q^{10} - 2 q^{11} - 20 q^{13} - 28 q^{15} - 2 q^{16} + 12 q^{17} + 8 q^{18} - 6 q^{19} - 4 q^{20} - 4 q^{22} + 2 q^{23} - 6 q^{25} - 10 q^{26} + 4 q^{29} - 14 q^{30} - 8 q^{31} + 2 q^{32} + 24 q^{34} + 16 q^{36} - 2 q^{37} + 6 q^{38} - 2 q^{40} - 12 q^{41} - 16 q^{43} - 2 q^{44} + 8 q^{45} - 2 q^{46} + 16 q^{47} - 12 q^{50} + 10 q^{52} + 2 q^{53} - 4 q^{55} - 28 q^{57} + 2 q^{58} + 4 q^{59} + 14 q^{60} + 18 q^{61} - 16 q^{62} + 4 q^{64} - 10 q^{65} + 8 q^{67} + 12 q^{68} - 28 q^{69} + 28 q^{71} + 8 q^{72} + 6 q^{73} + 2 q^{74} - 28 q^{75} + 12 q^{76} + 2 q^{80} + 10 q^{81} - 6 q^{82} - 32 q^{83} + 24 q^{85} - 8 q^{86} + 28 q^{87} + 2 q^{88} - 8 q^{89} + 16 q^{90} - 4 q^{92} - 16 q^{94} - 8 q^{95} + 44 q^{97} + 16 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 - 4 * q^8 - 8 * q^9 - 2 * q^10 - 2 * q^11 - 20 * q^13 - 28 * q^15 - 2 * q^16 + 12 * q^17 + 8 * q^18 - 6 * q^19 - 4 * q^20 - 4 * q^22 + 2 * q^23 - 6 * q^25 - 10 * q^26 + 4 * q^29 - 14 * q^30 - 8 * q^31 + 2 * q^32 + 24 * q^34 + 16 * q^36 - 2 * q^37 + 6 * q^38 - 2 * q^40 - 12 * q^41 - 16 * q^43 - 2 * q^44 + 8 * q^45 - 2 * q^46 + 16 * q^47 - 12 * q^50 + 10 * q^52 + 2 * q^53 - 4 * q^55 - 28 * q^57 + 2 * q^58 + 4 * q^59 + 14 * q^60 + 18 * q^61 - 16 * q^62 + 4 * q^64 - 10 * q^65 + 8 * q^67 + 12 * q^68 - 28 * q^69 + 28 * q^71 + 8 * q^72 + 6 * q^73 + 2 * q^74 - 28 * q^75 + 12 * q^76 + 2 * q^80 + 10 * q^81 - 6 * q^82 - 32 * q^83 + 24 * q^85 - 8 * q^86 + 28 * q^87 + 2 * q^88 - 8 * q^89 + 16 * q^90 - 4 * q^92 - 16 * q^94 - 8 * q^95 + 44 * q^97 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$981$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0.500000 + 0.866025i −1.32288 + 2.29129i −0.500000 + 0.866025i 1.82288 + 3.15731i −2.64575 0 −1.00000 −2.00000 3.46410i −1.82288 + 3.15731i
67.2 0.500000 + 0.866025i 1.32288 2.29129i −0.500000 + 0.866025i −0.822876 1.42526i 2.64575 0 −1.00000 −2.00000 3.46410i 0.822876 1.42526i
177.1 0.500000 0.866025i −1.32288 2.29129i −0.500000 0.866025i 1.82288 3.15731i −2.64575 0 −1.00000 −2.00000 + 3.46410i −1.82288 3.15731i
177.2 0.500000 0.866025i 1.32288 + 2.29129i −0.500000 0.866025i −0.822876 + 1.42526i 2.64575 0 −1.00000 −2.00000 + 3.46410i 0.822876 + 1.42526i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.e.v 4
7.b odd 2 1 154.2.e.f 4
7.c even 3 1 1078.2.a.n 2
7.c even 3 1 inner 1078.2.e.v 4
7.d odd 6 1 154.2.e.f 4
7.d odd 6 1 1078.2.a.s 2
21.c even 2 1 1386.2.k.s 4
21.g even 6 1 1386.2.k.s 4
21.g even 6 1 9702.2.a.cz 2
21.h odd 6 1 9702.2.a.dr 2
28.d even 2 1 1232.2.q.g 4
28.f even 6 1 1232.2.q.g 4
28.f even 6 1 8624.2.a.ca 2
28.g odd 6 1 8624.2.a.bk 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 7.b odd 2 1
154.2.e.f 4 7.d odd 6 1
1078.2.a.n 2 7.c even 3 1
1078.2.a.s 2 7.d odd 6 1
1078.2.e.v 4 1.a even 1 1 trivial
1078.2.e.v 4 7.c even 3 1 inner
1232.2.q.g 4 28.d even 2 1
1232.2.q.g 4 28.f even 6 1
1386.2.k.s 4 21.c even 2 1
1386.2.k.s 4 21.g even 6 1
8624.2.a.bk 2 28.g odd 6 1
8624.2.a.ca 2 28.f even 6 1
9702.2.a.cz 2 21.g even 6 1
9702.2.a.dr 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1078, [\chi])$$:

 $$T_{3}^{4} + 7T_{3}^{2} + 49$$ T3^4 + 7*T3^2 + 49 $$T_{5}^{4} - 2T_{5}^{3} + 10T_{5}^{2} + 12T_{5} + 36$$ T5^4 - 2*T5^3 + 10*T5^2 + 12*T5 + 36 $$T_{13} + 5$$ T13 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + 7T^{2} + 49$$
$5$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + T + 1)^{2}$$
$13$ $$(T + 5)^{4}$$
$17$ $$(T^{2} - 6 T + 36)^{2}$$
$19$ $$T^{4} + 6 T^{3} + 34 T^{2} + 12 T + 4$$
$23$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$29$ $$(T^{2} - 2 T - 27)^{2}$$
$31$ $$(T^{2} + 4 T + 16)^{2}$$
$37$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$41$ $$(T^{2} + 6 T - 54)^{2}$$
$43$ $$(T + 4)^{4}$$
$47$ $$T^{4} - 16 T^{3} + 220 T^{2} + \cdots + 1296$$
$53$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$59$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$61$ $$T^{4} - 18 T^{3} + 271 T^{2} + \cdots + 2809$$
$67$ $$T^{4} - 8 T^{3} + 111 T^{2} + \cdots + 2209$$
$71$ $$(T^{2} - 14 T + 42)^{2}$$
$73$ $$T^{4} - 6 T^{3} + 34 T^{2} - 12 T + 4$$
$79$ $$T^{4} + 7T^{2} + 49$$
$83$ $$(T^{2} + 16 T + 36)^{2}$$
$89$ $$T^{4} + 8 T^{3} + 160 T^{2} + \cdots + 9216$$
$97$ $$(T^{2} - 22 T + 93)^{2}$$