# Properties

 Label 1274.2.g.g Level 1274 Weight 2 Character orbit 1274.g Analytic conductor 10.173 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1274 = 2 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1274.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1729412175$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$885$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −3.00000 −0.500000 0.866025i 0 −1.00000 1.00000 + 1.73205i −1.50000 + 2.59808i
393.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −3.00000 −0.500000 + 0.866025i 0 −1.00000 1.00000 1.73205i −1.50000 2.59808i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.g.g 2
7.b odd 2 1 182.2.g.b 2
7.c even 3 1 1274.2.e.i 2
7.c even 3 1 1274.2.h.i 2
7.d odd 6 1 1274.2.e.d 2
7.d odd 6 1 1274.2.h.j 2
13.c even 3 1 inner 1274.2.g.g 2
21.c even 2 1 1638.2.r.c 2
28.d even 2 1 1456.2.s.e 2
91.g even 3 1 1274.2.e.i 2
91.h even 3 1 1274.2.h.i 2
91.m odd 6 1 1274.2.e.d 2
91.n odd 6 1 182.2.g.b 2
91.n odd 6 1 2366.2.a.f 1
91.t odd 6 1 2366.2.a.n 1
91.v odd 6 1 1274.2.h.j 2
91.bc even 12 2 2366.2.d.f 2
273.bn even 6 1 1638.2.r.c 2
364.v even 6 1 1456.2.s.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.b 2 7.b odd 2 1
182.2.g.b 2 91.n odd 6 1
1274.2.e.d 2 7.d odd 6 1
1274.2.e.d 2 91.m odd 6 1
1274.2.e.i 2 7.c even 3 1
1274.2.e.i 2 91.g even 3 1
1274.2.g.g 2 1.a even 1 1 trivial
1274.2.g.g 2 13.c even 3 1 inner
1274.2.h.i 2 7.c even 3 1
1274.2.h.i 2 91.h even 3 1
1274.2.h.j 2 7.d odd 6 1
1274.2.h.j 2 91.v odd 6 1
1456.2.s.e 2 28.d even 2 1
1456.2.s.e 2 364.v even 6 1
1638.2.r.c 2 21.c even 2 1
1638.2.r.c 2 273.bn even 6 1
2366.2.a.f 1 91.n odd 6 1
2366.2.a.n 1 91.t odd 6 1
2366.2.d.f 2 91.bc even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1274, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5} + 3$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$1 + 5 T + 13 T^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 10 T + 31 T^{2} )^{2}$$
$37$ $$1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4}$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$( 1 + 6 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 - 11 T + 60 T^{2} - 671 T^{3} + 3721 T^{4}$$
$67$ $$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$1 - 3 T - 62 T^{2} - 213 T^{3} + 5041 T^{4}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4}$$