# Properties

 Label 1274.2.f.p Level $1274$ Weight $2$ Character orbit 1274.f 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.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1729412175$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.50000 2.59808i −1.00000 0 −1.00000 1.00000 1.73205i −1.50000 2.59808i
1145.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.50000 + 2.59808i −1.00000 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
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 1274.2.f.p 2
7.b odd 2 1 1274.2.f.r 2
7.c even 3 1 26.2.a.a 1
7.c even 3 1 inner 1274.2.f.p 2
7.d odd 6 1 1274.2.a.d 1
7.d odd 6 1 1274.2.f.r 2
21.h odd 6 1 234.2.a.e 1
28.g odd 6 1 208.2.a.a 1
35.j even 6 1 650.2.a.j 1
35.l odd 12 2 650.2.b.d 2
56.k odd 6 1 832.2.a.i 1
56.p even 6 1 832.2.a.d 1
63.g even 3 1 2106.2.e.ba 2
63.h even 3 1 2106.2.e.ba 2
63.j odd 6 1 2106.2.e.b 2
63.n odd 6 1 2106.2.e.b 2
77.h odd 6 1 3146.2.a.n 1
84.n even 6 1 1872.2.a.q 1
91.g even 3 1 338.2.c.d 2
91.h even 3 1 338.2.c.d 2
91.k even 6 1 338.2.c.a 2
91.r even 6 1 338.2.a.f 1
91.u even 6 1 338.2.c.a 2
91.x odd 12 2 338.2.e.a 4
91.z odd 12 2 338.2.b.c 2
91.bd odd 12 2 338.2.e.a 4
105.o odd 6 1 5850.2.a.p 1
105.x even 12 2 5850.2.e.a 2
112.u odd 12 2 3328.2.b.j 2
112.w even 12 2 3328.2.b.m 2
119.j even 6 1 7514.2.a.c 1
133.r odd 6 1 9386.2.a.j 1
140.p odd 6 1 5200.2.a.x 1
168.s odd 6 1 7488.2.a.g 1
168.v even 6 1 7488.2.a.h 1
273.w odd 6 1 3042.2.a.a 1
273.cd even 12 2 3042.2.b.a 2
364.bl odd 6 1 2704.2.a.f 1
364.ce even 12 2 2704.2.f.d 2
455.bh even 6 1 8450.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 7.c even 3 1
208.2.a.a 1 28.g odd 6 1
234.2.a.e 1 21.h odd 6 1
338.2.a.f 1 91.r even 6 1
338.2.b.c 2 91.z odd 12 2
338.2.c.a 2 91.k even 6 1
338.2.c.a 2 91.u even 6 1
338.2.c.d 2 91.g even 3 1
338.2.c.d 2 91.h even 3 1
338.2.e.a 4 91.x odd 12 2
338.2.e.a 4 91.bd odd 12 2
650.2.a.j 1 35.j even 6 1
650.2.b.d 2 35.l odd 12 2
832.2.a.d 1 56.p even 6 1
832.2.a.i 1 56.k odd 6 1
1274.2.a.d 1 7.d odd 6 1
1274.2.f.p 2 1.a even 1 1 trivial
1274.2.f.p 2 7.c even 3 1 inner
1274.2.f.r 2 7.b odd 2 1
1274.2.f.r 2 7.d odd 6 1
1872.2.a.q 1 84.n even 6 1
2106.2.e.b 2 63.j odd 6 1
2106.2.e.b 2 63.n odd 6 1
2106.2.e.ba 2 63.g even 3 1
2106.2.e.ba 2 63.h even 3 1
2704.2.a.f 1 364.bl odd 6 1
2704.2.f.d 2 364.ce even 12 2
3042.2.a.a 1 273.w odd 6 1
3042.2.b.a 2 273.cd even 12 2
3146.2.a.n 1 77.h odd 6 1
3328.2.b.j 2 112.u odd 12 2
3328.2.b.m 2 112.w even 12 2
5200.2.a.x 1 140.p odd 6 1
5850.2.a.p 1 105.o odd 6 1
5850.2.e.a 2 105.x even 12 2
7488.2.a.g 1 168.s odd 6 1
7488.2.a.h 1 168.v even 6 1
7514.2.a.c 1 119.j even 6 1
8450.2.a.c 1 455.bh even 6 1
9386.2.a.j 1 133.r odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} + 6T_{11} + 36$$ T11^2 + 6*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T + 10)^{2}$$