# Properties

 Label 1014.2.i.c Level $1014$ Weight $2$ Character orbit 1014.i Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} - \zeta_{12}^{2} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + 2 \zeta_{12}^{3} q^{5} - \zeta_{12} q^{6} + 2 \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9} +O(q^{10})$$ q + (-z^3 + z) * q^2 - z^2 * q^3 + (-z^2 + 1) * q^4 + 2*z^3 * q^5 - z * q^6 + 2*z * q^7 - z^3 * q^8 + (z^2 - 1) * q^9 $$q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} - \zeta_{12}^{2} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + 2 \zeta_{12}^{3} q^{5} - \zeta_{12} q^{6} + 2 \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} + (\zeta_{12}^{2} - 1) q^{9} + 2 \zeta_{12}^{2} q^{10} - q^{12} + 2 q^{14} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{15} - \zeta_{12}^{2} q^{16} + ( - 2 \zeta_{12}^{2} + 2) q^{17} + \zeta_{12}^{3} q^{18} + 6 \zeta_{12} q^{19} + 2 \zeta_{12} q^{20} - 2 \zeta_{12}^{3} q^{21} - 4 \zeta_{12}^{2} q^{23} + (\zeta_{12}^{3} - \zeta_{12}) q^{24} + q^{25} + q^{27} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} + 10 \zeta_{12}^{2} q^{29} + ( - 2 \zeta_{12}^{2} + 2) q^{30} + 10 \zeta_{12}^{3} q^{31} - \zeta_{12} q^{32} - 2 \zeta_{12}^{3} q^{34} + (4 \zeta_{12}^{2} - 4) q^{35} + \zeta_{12}^{2} q^{36} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{37} + 6 q^{38} + 2 q^{40} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{41} - 2 \zeta_{12}^{2} q^{42} + (4 \zeta_{12}^{2} - 4) q^{43} - 2 \zeta_{12} q^{45} - 4 \zeta_{12} q^{46} - 12 \zeta_{12}^{3} q^{47} + (\zeta_{12}^{2} - 1) q^{48} - 3 \zeta_{12}^{2} q^{49} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{50} - 2 q^{51} - 6 q^{53} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{54} + ( - 2 \zeta_{12}^{2} + 2) q^{56} - 6 \zeta_{12}^{3} q^{57} + 10 \zeta_{12} q^{58} - 4 \zeta_{12} q^{59} - 2 \zeta_{12}^{3} q^{60} + (2 \zeta_{12}^{2} - 2) q^{61} + 10 \zeta_{12}^{2} q^{62} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{63} - q^{64} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{67} - 2 \zeta_{12}^{2} q^{68} + (4 \zeta_{12}^{2} - 4) q^{69} + 4 \zeta_{12}^{3} q^{70} + \zeta_{12} q^{72} - 4 \zeta_{12}^{3} q^{73} + ( - 8 \zeta_{12}^{2} + 8) q^{74} - \zeta_{12}^{2} q^{75} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{76} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{80} - \zeta_{12}^{2} q^{81} + ( - 10 \zeta_{12}^{2} + 10) q^{82} - 4 \zeta_{12}^{3} q^{83} - 2 \zeta_{12} q^{84} + 4 \zeta_{12} q^{85} + 4 \zeta_{12}^{3} q^{86} + ( - 10 \zeta_{12}^{2} + 10) q^{87} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} - 2 q^{90} - 4 q^{92} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{93} - 12 \zeta_{12}^{2} q^{94} + (12 \zeta_{12}^{2} - 12) q^{95} + \zeta_{12}^{3} q^{96} + 12 \zeta_{12} q^{97} - 3 \zeta_{12} q^{98} +O(q^{100})$$ q + (-z^3 + z) * q^2 - z^2 * q^3 + (-z^2 + 1) * q^4 + 2*z^3 * q^5 - z * q^6 + 2*z * q^7 - z^3 * q^8 + (z^2 - 1) * q^9 + 2*z^2 * q^10 - q^12 + 2 * q^14 + (-2*z^3 + 2*z) * q^15 - z^2 * q^16 + (-2*z^2 + 2) * q^17 + z^3 * q^18 + 6*z * q^19 + 2*z * q^20 - 2*z^3 * q^21 - 4*z^2 * q^23 + (z^3 - z) * q^24 + q^25 + q^27 + (-2*z^3 + 2*z) * q^28 + 10*z^2 * q^29 + (-2*z^2 + 2) * q^30 + 10*z^3 * q^31 - z * q^32 - 2*z^3 * q^34 + (4*z^2 - 4) * q^35 + z^2 * q^36 + (-8*z^3 + 8*z) * q^37 + 6 * q^38 + 2 * q^40 + (-10*z^3 + 10*z) * q^41 - 2*z^2 * q^42 + (4*z^2 - 4) * q^43 - 2*z * q^45 - 4*z * q^46 - 12*z^3 * q^47 + (z^2 - 1) * q^48 - 3*z^2 * q^49 + (-z^3 + z) * q^50 - 2 * q^51 - 6 * q^53 + (-z^3 + z) * q^54 + (-2*z^2 + 2) * q^56 - 6*z^3 * q^57 + 10*z * q^58 - 4*z * q^59 - 2*z^3 * q^60 + (2*z^2 - 2) * q^61 + 10*z^2 * q^62 + (2*z^3 - 2*z) * q^63 - q^64 + (2*z^3 - 2*z) * q^67 - 2*z^2 * q^68 + (4*z^2 - 4) * q^69 + 4*z^3 * q^70 + z * q^72 - 4*z^3 * q^73 + (-8*z^2 + 8) * q^74 - z^2 * q^75 + (-6*z^3 + 6*z) * q^76 + (-2*z^3 + 2*z) * q^80 - z^2 * q^81 + (-10*z^2 + 10) * q^82 - 4*z^3 * q^83 - 2*z * q^84 + 4*z * q^85 + 4*z^3 * q^86 + (-10*z^2 + 10) * q^87 + (6*z^3 - 6*z) * q^89 - 2 * q^90 - 4 * q^92 + (-10*z^3 + 10*z) * q^93 - 12*z^2 * q^94 + (12*z^2 - 12) * q^95 + z^3 * q^96 + 12*z * q^97 - 3*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^4 - 2 * q^9 $$4 q - 2 q^{3} + 2 q^{4} - 2 q^{9} + 4 q^{10} - 4 q^{12} + 8 q^{14} - 2 q^{16} + 4 q^{17} - 8 q^{23} + 4 q^{25} + 4 q^{27} + 20 q^{29} + 4 q^{30} - 8 q^{35} + 2 q^{36} + 24 q^{38} + 8 q^{40} - 4 q^{42} - 8 q^{43} - 2 q^{48} - 6 q^{49} - 8 q^{51} - 24 q^{53} + 4 q^{56} - 4 q^{61} + 20 q^{62} - 4 q^{64} - 4 q^{68} - 8 q^{69} + 16 q^{74} - 2 q^{75} - 2 q^{81} + 20 q^{82} + 20 q^{87} - 8 q^{90} - 16 q^{92} - 24 q^{94} - 24 q^{95}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^4 - 2 * q^9 + 4 * q^10 - 4 * q^12 + 8 * q^14 - 2 * q^16 + 4 * q^17 - 8 * q^23 + 4 * q^25 + 4 * q^27 + 20 * q^29 + 4 * q^30 - 8 * q^35 + 2 * q^36 + 24 * q^38 + 8 * q^40 - 4 * q^42 - 8 * q^43 - 2 * q^48 - 6 * q^49 - 8 * q^51 - 24 * q^53 + 4 * q^56 - 4 * q^61 + 20 * q^62 - 4 * q^64 - 4 * q^68 - 8 * q^69 + 16 * q^74 - 2 * q^75 - 2 * q^81 + 20 * q^82 + 20 * q^87 - 8 * q^90 - 16 * q^92 - 24 * q^94 - 24 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$1 - \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −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 2.00000i 0.866025 + 0.500000i −1.73205 1.00000i 1.00000i −0.500000 + 0.866025i 1.00000 + 1.73205i
361.2 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 2.00000i −0.866025 0.500000i 1.73205 + 1.00000i 1.00000i −0.500000 + 0.866025i 1.00000 + 1.73205i
823.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.00000i 0.866025 0.500000i −1.73205 + 1.00000i 1.00000i −0.500000 0.866025i 1.00000 1.73205i
823.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.00000i −0.866025 + 0.500000i 1.73205 1.00000i 1.00000i −0.500000 0.866025i 1.00000 1.73205i
 $$n$$: 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.c 4
13.b even 2 1 inner 1014.2.i.c 4
13.c even 3 1 78.2.b.a 2
13.c even 3 1 inner 1014.2.i.c 4
13.d odd 4 1 1014.2.e.b 2
13.d odd 4 1 1014.2.e.e 2
13.e even 6 1 78.2.b.a 2
13.e even 6 1 inner 1014.2.i.c 4
13.f odd 12 1 1014.2.a.b 1
13.f odd 12 1 1014.2.a.g 1
13.f odd 12 1 1014.2.e.b 2
13.f odd 12 1 1014.2.e.e 2
39.h odd 6 1 234.2.b.a 2
39.i odd 6 1 234.2.b.a 2
39.k even 12 1 3042.2.a.c 1
39.k even 12 1 3042.2.a.n 1
52.i odd 6 1 624.2.c.a 2
52.j odd 6 1 624.2.c.a 2
52.l even 12 1 8112.2.a.g 1
52.l even 12 1 8112.2.a.j 1
65.l even 6 1 1950.2.b.c 2
65.n even 6 1 1950.2.b.c 2
65.q odd 12 1 1950.2.f.d 2
65.q odd 12 1 1950.2.f.g 2
65.r odd 12 1 1950.2.f.d 2
65.r odd 12 1 1950.2.f.g 2
91.n odd 6 1 3822.2.c.d 2
91.t odd 6 1 3822.2.c.d 2
104.n odd 6 1 2496.2.c.m 2
104.p odd 6 1 2496.2.c.m 2
104.r even 6 1 2496.2.c.f 2
104.s even 6 1 2496.2.c.f 2
156.p even 6 1 1872.2.c.b 2
156.r even 6 1 1872.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 13.c even 3 1
78.2.b.a 2 13.e even 6 1
234.2.b.a 2 39.h odd 6 1
234.2.b.a 2 39.i odd 6 1
624.2.c.a 2 52.i odd 6 1
624.2.c.a 2 52.j odd 6 1
1014.2.a.b 1 13.f odd 12 1
1014.2.a.g 1 13.f odd 12 1
1014.2.e.b 2 13.d odd 4 1
1014.2.e.b 2 13.f odd 12 1
1014.2.e.e 2 13.d odd 4 1
1014.2.e.e 2 13.f odd 12 1
1014.2.i.c 4 1.a even 1 1 trivial
1014.2.i.c 4 13.b even 2 1 inner
1014.2.i.c 4 13.c even 3 1 inner
1014.2.i.c 4 13.e even 6 1 inner
1872.2.c.b 2 156.p even 6 1
1872.2.c.b 2 156.r even 6 1
1950.2.b.c 2 65.l even 6 1
1950.2.b.c 2 65.n even 6 1
1950.2.f.d 2 65.q odd 12 1
1950.2.f.d 2 65.r odd 12 1
1950.2.f.g 2 65.q odd 12 1
1950.2.f.g 2 65.r odd 12 1
2496.2.c.f 2 104.r even 6 1
2496.2.c.f 2 104.s even 6 1
2496.2.c.m 2 104.n odd 6 1
2496.2.c.m 2 104.p odd 6 1
3042.2.a.c 1 39.k even 12 1
3042.2.a.n 1 39.k even 12 1
3822.2.c.d 2 91.n odd 6 1
3822.2.c.d 2 91.t odd 6 1
8112.2.a.g 1 52.l even 12 1
8112.2.a.j 1 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{7}^{4} - 4T_{7}^{2} + 16$$ T7^4 - 4*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + 4)^{2}$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 2 T + 4)^{2}$$
$19$ $$T^{4} - 36T^{2} + 1296$$
$23$ $$(T^{2} + 4 T + 16)^{2}$$
$29$ $$(T^{2} - 10 T + 100)^{2}$$
$31$ $$(T^{2} + 100)^{2}$$
$37$ $$T^{4} - 64T^{2} + 4096$$
$41$ $$T^{4} - 100 T^{2} + 10000$$
$43$ $$(T^{2} + 4 T + 16)^{2}$$
$47$ $$(T^{2} + 144)^{2}$$
$53$ $$(T + 6)^{4}$$
$59$ $$T^{4} - 16T^{2} + 256$$
$61$ $$(T^{2} + 2 T + 4)^{2}$$
$67$ $$T^{4} - 4T^{2} + 16$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 16)^{2}$$
$89$ $$T^{4} - 36T^{2} + 1296$$
$97$ $$T^{4} - 144 T^{2} + 20736$$