# Properties

 Label 1274.2.n.d Level $1274$ Weight $2$ Character orbit 1274.n Analytic conductor $10.173$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1274 = 2 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1274.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1729412175$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} + 3 \zeta_{12} q^{5} -\zeta_{12}^{3} q^{6} -\zeta_{12}^{3} q^{8} + ( 2 - 2 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} + 3 \zeta_{12} q^{5} -\zeta_{12}^{3} q^{6} -\zeta_{12}^{3} q^{8} + ( 2 - 2 \zeta_{12}^{2} ) q^{9} -3 \zeta_{12}^{2} q^{10} + ( -1 + \zeta_{12}^{2} ) q^{12} + ( 2 + 3 \zeta_{12}^{3} ) q^{13} + 3 \zeta_{12}^{3} q^{15} + ( -1 + \zeta_{12}^{2} ) q^{16} -3 \zeta_{12}^{2} q^{17} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{18} + 6 \zeta_{12} q^{19} + 3 \zeta_{12}^{3} q^{20} + ( 6 - 6 \zeta_{12}^{2} ) q^{23} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{24} + 4 \zeta_{12}^{2} q^{25} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{26} + 5 q^{27} + ( 3 - 3 \zeta_{12}^{2} ) q^{30} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{32} + 3 \zeta_{12}^{3} q^{34} + 2 q^{36} -3 \zeta_{12} q^{37} -6 \zeta_{12}^{2} q^{38} + ( -3 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{39} + ( 3 - 3 \zeta_{12}^{2} ) q^{40} - q^{43} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{45} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{46} -3 \zeta_{12} q^{47} - q^{48} -4 \zeta_{12}^{3} q^{50} + ( 3 - 3 \zeta_{12}^{2} ) q^{51} + ( -3 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{52} + 6 \zeta_{12}^{2} q^{53} -5 \zeta_{12} q^{54} + 6 \zeta_{12}^{3} q^{57} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{60} + ( 8 - 8 \zeta_{12}^{2} ) q^{61} - q^{64} + ( -9 + 6 \zeta_{12} + 9 \zeta_{12}^{2} ) q^{65} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{67} + ( 3 - 3 \zeta_{12}^{2} ) q^{68} + 6 q^{69} + 15 \zeta_{12}^{3} q^{71} -2 \zeta_{12} q^{72} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + 3 \zeta_{12}^{2} q^{74} + ( -4 + 4 \zeta_{12}^{2} ) q^{75} + 6 \zeta_{12}^{3} q^{76} + ( 3 - 2 \zeta_{12}^{3} ) q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + 6 \zeta_{12}^{3} q^{83} -9 \zeta_{12}^{3} q^{85} + \zeta_{12} q^{86} + 6 \zeta_{12} q^{89} -6 q^{90} + 6 q^{92} + 3 \zeta_{12}^{2} q^{94} + 18 \zeta_{12}^{2} q^{95} + \zeta_{12} q^{96} -12 \zeta_{12}^{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 2 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q + 2 q^{3} + 2 q^{4} + 4 q^{9} - 6 q^{10} - 2 q^{12} + 8 q^{13} - 2 q^{16} - 6 q^{17} + 12 q^{23} + 8 q^{25} + 6 q^{26} + 20 q^{27} + 6 q^{30} + 8 q^{36} - 12 q^{38} + 4 q^{39} + 6 q^{40} - 4 q^{43} - 4 q^{48} + 6 q^{51} + 4 q^{52} + 12 q^{53} + 16 q^{61} - 4 q^{64} - 18 q^{65} + 6 q^{68} + 24 q^{69} + 6 q^{74} - 8 q^{75} + 12 q^{78} - 20 q^{79} - 2 q^{81} - 24 q^{90} + 24 q^{92} + 6 q^{94} + 36 q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$885$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
753.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 2.59808 1.50000i 1.00000i 0 1.00000i 1.00000 + 1.73205i −1.50000 + 2.59808i
753.2 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −2.59808 + 1.50000i 1.00000i 0 1.00000i 1.00000 + 1.73205i −1.50000 + 2.59808i
961.1 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 2.59808 + 1.50000i 1.00000i 0 1.00000i 1.00000 1.73205i −1.50000 2.59808i
961.2 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −2.59808 1.50000i 1.00000i 0 1.00000i 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
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.n.d 4
7.b odd 2 1 1274.2.n.c 4
7.c even 3 1 26.2.b.a 2
7.c even 3 1 inner 1274.2.n.d 4
7.d odd 6 1 1274.2.d.c 2
7.d odd 6 1 1274.2.n.c 4
13.b even 2 1 inner 1274.2.n.d 4
21.h odd 6 1 234.2.b.b 2
28.g odd 6 1 208.2.f.a 2
35.j even 6 1 650.2.d.b 2
35.l odd 12 1 650.2.c.a 2
35.l odd 12 1 650.2.c.d 2
56.k odd 6 1 832.2.f.b 2
56.p even 6 1 832.2.f.d 2
84.n even 6 1 1872.2.c.f 2
91.b odd 2 1 1274.2.n.c 4
91.g even 3 1 338.2.e.c 4
91.h even 3 1 338.2.e.c 4
91.k even 6 1 338.2.e.c 4
91.r even 6 1 26.2.b.a 2
91.r even 6 1 inner 1274.2.n.d 4
91.s odd 6 1 1274.2.d.c 2
91.s odd 6 1 1274.2.n.c 4
91.u even 6 1 338.2.e.c 4
91.x odd 12 1 338.2.c.b 2
91.x odd 12 1 338.2.c.f 2
91.z odd 12 1 338.2.a.b 1
91.z odd 12 1 338.2.a.d 1
91.bd odd 12 1 338.2.c.b 2
91.bd odd 12 1 338.2.c.f 2
273.w odd 6 1 234.2.b.b 2
273.cd even 12 1 3042.2.a.g 1
273.cd even 12 1 3042.2.a.j 1
364.bl odd 6 1 208.2.f.a 2
364.ce even 12 1 2704.2.a.j 1
364.ce even 12 1 2704.2.a.k 1
455.bh even 6 1 650.2.d.b 2
455.cv odd 12 1 650.2.c.a 2
455.cv odd 12 1 650.2.c.d 2
455.di odd 12 1 8450.2.a.h 1
455.di odd 12 1 8450.2.a.u 1
728.bs odd 6 1 832.2.f.b 2
728.cj even 6 1 832.2.f.d 2
1092.by even 6 1 1872.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 7.c even 3 1
26.2.b.a 2 91.r even 6 1
208.2.f.a 2 28.g odd 6 1
208.2.f.a 2 364.bl odd 6 1
234.2.b.b 2 21.h odd 6 1
234.2.b.b 2 273.w odd 6 1
338.2.a.b 1 91.z odd 12 1
338.2.a.d 1 91.z odd 12 1
338.2.c.b 2 91.x odd 12 1
338.2.c.b 2 91.bd odd 12 1
338.2.c.f 2 91.x odd 12 1
338.2.c.f 2 91.bd odd 12 1
338.2.e.c 4 91.g even 3 1
338.2.e.c 4 91.h even 3 1
338.2.e.c 4 91.k even 6 1
338.2.e.c 4 91.u even 6 1
650.2.c.a 2 35.l odd 12 1
650.2.c.a 2 455.cv odd 12 1
650.2.c.d 2 35.l odd 12 1
650.2.c.d 2 455.cv odd 12 1
650.2.d.b 2 35.j even 6 1
650.2.d.b 2 455.bh even 6 1
832.2.f.b 2 56.k odd 6 1
832.2.f.b 2 728.bs odd 6 1
832.2.f.d 2 56.p even 6 1
832.2.f.d 2 728.cj even 6 1
1274.2.d.c 2 7.d odd 6 1
1274.2.d.c 2 91.s odd 6 1
1274.2.n.c 4 7.b odd 2 1
1274.2.n.c 4 7.d odd 6 1
1274.2.n.c 4 91.b odd 2 1
1274.2.n.c 4 91.s odd 6 1
1274.2.n.d 4 1.a even 1 1 trivial
1274.2.n.d 4 7.c even 3 1 inner
1274.2.n.d 4 13.b even 2 1 inner
1274.2.n.d 4 91.r even 6 1 inner
1872.2.c.f 2 84.n even 6 1
1872.2.c.f 2 1092.by even 6 1
2704.2.a.j 1 364.ce even 12 1
2704.2.a.k 1 364.ce even 12 1
3042.2.a.g 1 273.cd even 12 1
3042.2.a.j 1 273.cd even 12 1
8450.2.a.h 1 455.di odd 12 1
8450.2.a.u 1 455.di odd 12 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{4} - 9 T_{5}^{2} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$81 - 9 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 13 - 4 T + T^{2} )^{2}$$
$17$ $$( 9 + 3 T + T^{2} )^{2}$$
$19$ $$1296 - 36 T^{2} + T^{4}$$
$23$ $$( 36 - 6 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$81 - 9 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 1 + T )^{4}$$
$47$ $$81 - 9 T^{2} + T^{4}$$
$53$ $$( 36 - 6 T + T^{2} )^{2}$$
$59$ $$1296 - 36 T^{2} + T^{4}$$
$61$ $$( 64 - 8 T + T^{2} )^{2}$$
$67$ $$20736 - 144 T^{2} + T^{4}$$
$71$ $$( 225 + T^{2} )^{2}$$
$73$ $$1296 - 36 T^{2} + T^{4}$$
$79$ $$( 100 + 10 T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$1296 - 36 T^{2} + T^{4}$$
$97$ $$( 144 + T^{2} )^{2}$$