# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{3} - q^{4} + 2 i q^{5} + i q^{6} - 4 i q^{7} + i q^{8} + q^{9} +O(q^{10})$$ q - i * q^2 - q^3 - q^4 + 2*i * q^5 + i * q^6 - 4*i * q^7 + i * q^8 + q^9 $$q - i q^{2} - q^{3} - q^{4} + 2 i q^{5} + i q^{6} - 4 i q^{7} + i q^{8} + q^{9} + 2 q^{10} + 4 i q^{11} + q^{12} - 4 q^{14} - 2 i q^{15} + q^{16} - 2 q^{17} - i q^{18} - 8 i q^{19} - 2 i q^{20} + 4 i q^{21} + 4 q^{22} - i q^{24} + q^{25} - q^{27} + 4 i q^{28} + 6 q^{29} - 2 q^{30} - 4 i q^{31} - i q^{32} - 4 i q^{33} + 2 i q^{34} + 8 q^{35} - q^{36} + 2 i q^{37} - 8 q^{38} - 2 q^{40} - 10 i q^{41} + 4 q^{42} - 4 q^{43} - 4 i q^{44} + 2 i q^{45} - 8 i q^{47} - q^{48} - 9 q^{49} - i q^{50} + 2 q^{51} - 10 q^{53} + i q^{54} - 8 q^{55} + 4 q^{56} + 8 i q^{57} - 6 i q^{58} - 4 i q^{59} + 2 i q^{60} - 2 q^{61} - 4 q^{62} - 4 i q^{63} - q^{64} - 4 q^{66} - 16 i q^{67} + 2 q^{68} - 8 i q^{70} - 8 i q^{71} + i q^{72} - 2 i q^{73} + 2 q^{74} - q^{75} + 8 i q^{76} + 16 q^{77} + 8 q^{79} + 2 i q^{80} + q^{81} - 10 q^{82} + 12 i q^{83} - 4 i q^{84} - 4 i q^{85} + 4 i q^{86} - 6 q^{87} - 4 q^{88} - 14 i q^{89} + 2 q^{90} + 4 i q^{93} - 8 q^{94} + 16 q^{95} + i q^{96} + 10 i q^{97} + 9 i q^{98} + 4 i q^{99} +O(q^{100})$$ q - i * q^2 - q^3 - q^4 + 2*i * q^5 + i * q^6 - 4*i * q^7 + i * q^8 + q^9 + 2 * q^10 + 4*i * q^11 + q^12 - 4 * q^14 - 2*i * q^15 + q^16 - 2 * q^17 - i * q^18 - 8*i * q^19 - 2*i * q^20 + 4*i * q^21 + 4 * q^22 - i * q^24 + q^25 - q^27 + 4*i * q^28 + 6 * q^29 - 2 * q^30 - 4*i * q^31 - i * q^32 - 4*i * q^33 + 2*i * q^34 + 8 * q^35 - q^36 + 2*i * q^37 - 8 * q^38 - 2 * q^40 - 10*i * q^41 + 4 * q^42 - 4 * q^43 - 4*i * q^44 + 2*i * q^45 - 8*i * q^47 - q^48 - 9 * q^49 - i * q^50 + 2 * q^51 - 10 * q^53 + i * q^54 - 8 * q^55 + 4 * q^56 + 8*i * q^57 - 6*i * q^58 - 4*i * q^59 + 2*i * q^60 - 2 * q^61 - 4 * q^62 - 4*i * q^63 - q^64 - 4 * q^66 - 16*i * q^67 + 2 * q^68 - 8*i * q^70 - 8*i * q^71 + i * q^72 - 2*i * q^73 + 2 * q^74 - q^75 + 8*i * q^76 + 16 * q^77 + 8 * q^79 + 2*i * q^80 + q^81 - 10 * q^82 + 12*i * q^83 - 4*i * q^84 - 4*i * q^85 + 4*i * q^86 - 6 * q^87 - 4 * q^88 - 14*i * q^89 + 2 * q^90 + 4*i * q^93 - 8 * q^94 + 16 * q^95 + i * q^96 + 10*i * q^97 + 9*i * q^98 + 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} - 8 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{22} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 4 q^{30} + 16 q^{35} - 2 q^{36} - 16 q^{38} - 4 q^{40} + 8 q^{42} - 8 q^{43} - 2 q^{48} - 18 q^{49} + 4 q^{51} - 20 q^{53} - 16 q^{55} + 8 q^{56} - 4 q^{61} - 8 q^{62} - 2 q^{64} - 8 q^{66} + 4 q^{68} + 4 q^{74} - 2 q^{75} + 32 q^{77} + 16 q^{79} + 2 q^{81} - 20 q^{82} - 12 q^{87} - 8 q^{88} + 4 q^{90} - 16 q^{94} + 32 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^10 + 2 * q^12 - 8 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^22 + 2 * q^25 - 2 * q^27 + 12 * q^29 - 4 * q^30 + 16 * q^35 - 2 * q^36 - 16 * q^38 - 4 * q^40 + 8 * q^42 - 8 * q^43 - 2 * q^48 - 18 * q^49 + 4 * q^51 - 20 * q^53 - 16 * q^55 + 8 * q^56 - 4 * q^61 - 8 * q^62 - 2 * q^64 - 8 * q^66 + 4 * q^68 + 4 * q^74 - 2 * q^75 + 32 * q^77 + 16 * q^79 + 2 * q^81 - 20 * q^82 - 12 * q^87 - 8 * q^88 + 4 * q^90 - 16 * q^94 + 32 * q^95

## Character values

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

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 1.00000i − 1.00000i
1.00000i −1.00000 −1.00000 2.00000i 1.00000i 4.00000i 1.00000i 1.00000 2.00000
337.2 1.00000i −1.00000 −1.00000 2.00000i 1.00000i 4.00000i 1.00000i 1.00000 2.00000
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.b.b 2
3.b odd 2 1 3042.2.b.g 2
13.b even 2 1 inner 1014.2.b.b 2
13.c even 3 2 1014.2.i.d 4
13.d odd 4 1 78.2.a.a 1
13.d odd 4 1 1014.2.a.d 1
13.e even 6 2 1014.2.i.d 4
13.f odd 12 2 1014.2.e.c 2
13.f odd 12 2 1014.2.e.f 2
39.d odd 2 1 3042.2.b.g 2
39.f even 4 1 234.2.a.c 1
39.f even 4 1 3042.2.a.f 1
52.f even 4 1 624.2.a.h 1
52.f even 4 1 8112.2.a.v 1
65.f even 4 1 1950.2.e.i 2
65.g odd 4 1 1950.2.a.w 1
65.k even 4 1 1950.2.e.i 2
91.i even 4 1 3822.2.a.j 1
104.j odd 4 1 2496.2.a.t 1
104.m even 4 1 2496.2.a.b 1
117.y odd 12 2 2106.2.e.q 2
117.z even 12 2 2106.2.e.j 2
143.g even 4 1 9438.2.a.t 1
156.l odd 4 1 1872.2.a.c 1
195.j odd 4 1 5850.2.e.bb 2
195.n even 4 1 5850.2.a.d 1
195.u odd 4 1 5850.2.e.bb 2
312.w odd 4 1 7488.2.a.bk 1
312.y even 4 1 7488.2.a.bz 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 13.d odd 4 1
234.2.a.c 1 39.f even 4 1
624.2.a.h 1 52.f even 4 1
1014.2.a.d 1 13.d odd 4 1
1014.2.b.b 2 1.a even 1 1 trivial
1014.2.b.b 2 13.b even 2 1 inner
1014.2.e.c 2 13.f odd 12 2
1014.2.e.f 2 13.f odd 12 2
1014.2.i.d 4 13.c even 3 2
1014.2.i.d 4 13.e even 6 2
1872.2.a.c 1 156.l odd 4 1
1950.2.a.w 1 65.g odd 4 1
1950.2.e.i 2 65.f even 4 1
1950.2.e.i 2 65.k even 4 1
2106.2.e.j 2 117.z even 12 2
2106.2.e.q 2 117.y odd 12 2
2496.2.a.b 1 104.m even 4 1
2496.2.a.t 1 104.j odd 4 1
3042.2.a.f 1 39.f even 4 1
3042.2.b.g 2 3.b odd 2 1
3042.2.b.g 2 39.d odd 2 1
3822.2.a.j 1 91.i even 4 1
5850.2.a.d 1 195.n even 4 1
5850.2.e.bb 2 195.j odd 4 1
5850.2.e.bb 2 195.u odd 4 1
7488.2.a.bk 1 312.w odd 4 1
7488.2.a.bz 1 312.y even 4 1
8112.2.a.v 1 52.f even 4 1
9438.2.a.t 1 143.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 100$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$T^{2} + 64$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 100$$