# Properties

 Label 1014.2.b.d Level $1014$ Weight $2$ Character orbit 1014.b Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{3} - q^{4} - \beta_{2} q^{5} - \beta_1 q^{6} + ( - \beta_{2} + 3 \beta_1) q^{7} - \beta_1 q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 - q^3 - q^4 - b2 * q^5 - b1 * q^6 + (-b2 + 3*b1) * q^7 - b1 * q^8 + q^9 $$q + \beta_1 q^{2} - q^{3} - q^{4} - \beta_{2} q^{5} - \beta_1 q^{6} + ( - \beta_{2} + 3 \beta_1) q^{7} - \beta_1 q^{8} + q^{9} + \beta_{3} q^{10} + ( - \beta_{2} + 3 \beta_1) q^{11} + q^{12} + (\beta_{3} - 3) q^{14} + \beta_{2} q^{15} + q^{16} - 3 \beta_{3} q^{17} + \beta_1 q^{18} + ( - \beta_{2} - 3 \beta_1) q^{19} + \beta_{2} q^{20} + (\beta_{2} - 3 \beta_1) q^{21} + (\beta_{3} - 3) q^{22} + (3 \beta_{3} + 3) q^{23} + \beta_1 q^{24} + 2 q^{25} - q^{27} + (\beta_{2} - 3 \beta_1) q^{28} - 3 q^{29} - \beta_{3} q^{30} + ( - 2 \beta_{2} - 6 \beta_1) q^{31} + \beta_1 q^{32} + (\beta_{2} - 3 \beta_1) q^{33} - 3 \beta_{2} q^{34} + (3 \beta_{3} - 3) q^{35} - q^{36} + 3 \beta_1 q^{37} + (\beta_{3} + 3) q^{38} - \beta_{3} q^{40} + ( - 2 \beta_{2} - 3 \beta_1) q^{41} + ( - \beta_{3} + 3) q^{42} + (3 \beta_{3} - 1) q^{43} + (\beta_{2} - 3 \beta_1) q^{44} - \beta_{2} q^{45} + (3 \beta_{2} + 3 \beta_1) q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} - q^{48} + (6 \beta_{3} - 5) q^{49} + 2 \beta_1 q^{50} + 3 \beta_{3} q^{51} + 3 q^{53} - \beta_1 q^{54} + (3 \beta_{3} - 3) q^{55} + ( - \beta_{3} + 3) q^{56} + (\beta_{2} + 3 \beta_1) q^{57} - 3 \beta_1 q^{58} - 8 \beta_{2} q^{59} - \beta_{2} q^{60} + (3 \beta_{3} + 10) q^{61} + (2 \beta_{3} + 6) q^{62} + ( - \beta_{2} + 3 \beta_1) q^{63} - q^{64} + ( - \beta_{3} + 3) q^{66} + (\beta_{2} - 9 \beta_1) q^{67} + 3 \beta_{3} q^{68} + ( - 3 \beta_{3} - 3) q^{69} + (3 \beta_{2} - 3 \beta_1) q^{70} + (3 \beta_{2} - 3 \beta_1) q^{71} - \beta_1 q^{72} + 7 \beta_{2} q^{73} - 3 q^{74} - 2 q^{75} + (\beta_{2} + 3 \beta_1) q^{76} + (6 \beta_{3} - 12) q^{77} + (6 \beta_{3} - 2) q^{79} - \beta_{2} q^{80} + q^{81} + (2 \beta_{3} + 3) q^{82} + ( - 5 \beta_{2} + 3 \beta_1) q^{83} + ( - \beta_{2} + 3 \beta_1) q^{84} + 9 \beta_1 q^{85} + (3 \beta_{2} - \beta_1) q^{86} + 3 q^{87} + ( - \beta_{3} + 3) q^{88} + (2 \beta_{2} + 6 \beta_1) q^{89} + \beta_{3} q^{90} + ( - 3 \beta_{3} - 3) q^{92} + (2 \beta_{2} + 6 \beta_1) q^{93} + (\beta_{3} + 3) q^{94} + ( - 3 \beta_{3} - 3) q^{95} - \beta_1 q^{96} - 6 \beta_1 q^{97} + (6 \beta_{2} - 5 \beta_1) q^{98} + ( - \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 - q^3 - q^4 - b2 * q^5 - b1 * q^6 + (-b2 + 3*b1) * q^7 - b1 * q^8 + q^9 + b3 * q^10 + (-b2 + 3*b1) * q^11 + q^12 + (b3 - 3) * q^14 + b2 * q^15 + q^16 - 3*b3 * q^17 + b1 * q^18 + (-b2 - 3*b1) * q^19 + b2 * q^20 + (b2 - 3*b1) * q^21 + (b3 - 3) * q^22 + (3*b3 + 3) * q^23 + b1 * q^24 + 2 * q^25 - q^27 + (b2 - 3*b1) * q^28 - 3 * q^29 - b3 * q^30 + (-2*b2 - 6*b1) * q^31 + b1 * q^32 + (b2 - 3*b1) * q^33 - 3*b2 * q^34 + (3*b3 - 3) * q^35 - q^36 + 3*b1 * q^37 + (b3 + 3) * q^38 - b3 * q^40 + (-2*b2 - 3*b1) * q^41 + (-b3 + 3) * q^42 + (3*b3 - 1) * q^43 + (b2 - 3*b1) * q^44 - b2 * q^45 + (3*b2 + 3*b1) * q^46 + (-b2 - 3*b1) * q^47 - q^48 + (6*b3 - 5) * q^49 + 2*b1 * q^50 + 3*b3 * q^51 + 3 * q^53 - b1 * q^54 + (3*b3 - 3) * q^55 + (-b3 + 3) * q^56 + (b2 + 3*b1) * q^57 - 3*b1 * q^58 - 8*b2 * q^59 - b2 * q^60 + (3*b3 + 10) * q^61 + (2*b3 + 6) * q^62 + (-b2 + 3*b1) * q^63 - q^64 + (-b3 + 3) * q^66 + (b2 - 9*b1) * q^67 + 3*b3 * q^68 + (-3*b3 - 3) * q^69 + (3*b2 - 3*b1) * q^70 + (3*b2 - 3*b1) * q^71 - b1 * q^72 + 7*b2 * q^73 - 3 * q^74 - 2 * q^75 + (b2 + 3*b1) * q^76 + (6*b3 - 12) * q^77 + (6*b3 - 2) * q^79 - b2 * q^80 + q^81 + (2*b3 + 3) * q^82 + (-5*b2 + 3*b1) * q^83 + (-b2 + 3*b1) * q^84 + 9*b1 * q^85 + (3*b2 - b1) * q^86 + 3 * q^87 + (-b3 + 3) * q^88 + (2*b2 + 6*b1) * q^89 + b3 * q^90 + (-3*b3 - 3) * q^92 + (2*b2 + 6*b1) * q^93 + (b3 + 3) * q^94 + (-3*b3 - 3) * q^95 - b1 * q^96 - 6*b1 * q^97 + (6*b2 - 5*b1) * q^98 + (-b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{12} - 12 q^{14} + 4 q^{16} - 12 q^{22} + 12 q^{23} + 8 q^{25} - 4 q^{27} - 12 q^{29} - 12 q^{35} - 4 q^{36} + 12 q^{38} + 12 q^{42} - 4 q^{43} - 4 q^{48} - 20 q^{49} + 12 q^{53} - 12 q^{55} + 12 q^{56} + 40 q^{61} + 24 q^{62} - 4 q^{64} + 12 q^{66} - 12 q^{69} - 12 q^{74} - 8 q^{75} - 48 q^{77} - 8 q^{79} + 4 q^{81} + 12 q^{82} + 12 q^{87} + 12 q^{88} - 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^12 - 12 * q^14 + 4 * q^16 - 12 * q^22 + 12 * q^23 + 8 * q^25 - 4 * q^27 - 12 * q^29 - 12 * q^35 - 4 * q^36 + 12 * q^38 + 12 * q^42 - 4 * q^43 - 4 * q^48 - 20 * q^49 + 12 * q^53 - 12 * q^55 + 12 * q^56 + 40 * q^61 + 24 * q^62 - 4 * q^64 + 12 * q^66 - 12 * q^69 - 12 * q^74 - 8 * q^75 - 48 * q^77 - 8 * q^79 + 4 * q^81 + 12 * q^82 + 12 * q^87 + 12 * q^88 - 12 * q^92 + 12 * q^94 - 12 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
1.00000i −1.00000 −1.00000 1.73205i 1.00000i 4.73205i 1.00000i 1.00000 −1.73205
337.2 1.00000i −1.00000 −1.00000 1.73205i 1.00000i 1.26795i 1.00000i 1.00000 1.73205
337.3 1.00000i −1.00000 −1.00000 1.73205i 1.00000i 1.26795i 1.00000i 1.00000 1.73205
337.4 1.00000i −1.00000 −1.00000 1.73205i 1.00000i 4.73205i 1.00000i 1.00000 −1.73205
 $$n$$: 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.d 4
3.b odd 2 1 3042.2.b.l 4
13.b even 2 1 inner 1014.2.b.d 4
13.c even 3 1 78.2.i.b 4
13.c even 3 1 1014.2.i.f 4
13.d odd 4 1 1014.2.a.h 2
13.d odd 4 1 1014.2.a.j 2
13.e even 6 1 78.2.i.b 4
13.e even 6 1 1014.2.i.f 4
13.f odd 12 2 1014.2.e.h 4
13.f odd 12 2 1014.2.e.j 4
39.d odd 2 1 3042.2.b.l 4
39.f even 4 1 3042.2.a.s 2
39.f even 4 1 3042.2.a.v 2
39.h odd 6 1 234.2.l.a 4
39.i odd 6 1 234.2.l.a 4
52.f even 4 1 8112.2.a.bq 2
52.f even 4 1 8112.2.a.bx 2
52.i odd 6 1 624.2.bv.d 4
52.j odd 6 1 624.2.bv.d 4
65.l even 6 1 1950.2.bc.c 4
65.n even 6 1 1950.2.bc.c 4
65.q odd 12 1 1950.2.y.a 4
65.q odd 12 1 1950.2.y.h 4
65.r odd 12 1 1950.2.y.a 4
65.r odd 12 1 1950.2.y.h 4
156.p even 6 1 1872.2.by.k 4
156.r even 6 1 1872.2.by.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 13.c even 3 1
78.2.i.b 4 13.e even 6 1
234.2.l.a 4 39.h odd 6 1
234.2.l.a 4 39.i odd 6 1
624.2.bv.d 4 52.i odd 6 1
624.2.bv.d 4 52.j odd 6 1
1014.2.a.h 2 13.d odd 4 1
1014.2.a.j 2 13.d odd 4 1
1014.2.b.d 4 1.a even 1 1 trivial
1014.2.b.d 4 13.b even 2 1 inner
1014.2.e.h 4 13.f odd 12 2
1014.2.e.j 4 13.f odd 12 2
1014.2.i.f 4 13.c even 3 1
1014.2.i.f 4 13.e even 6 1
1872.2.by.k 4 156.p even 6 1
1872.2.by.k 4 156.r even 6 1
1950.2.y.a 4 65.q odd 12 1
1950.2.y.a 4 65.r odd 12 1
1950.2.y.h 4 65.q odd 12 1
1950.2.y.h 4 65.r odd 12 1
1950.2.bc.c 4 65.l even 6 1
1950.2.bc.c 4 65.n even 6 1
3042.2.a.s 2 39.f even 4 1
3042.2.a.v 2 39.f even 4 1
3042.2.b.l 4 3.b odd 2 1
3042.2.b.l 4 39.d odd 2 1
8112.2.a.bq 2 52.f even 4 1
8112.2.a.bx 2 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T + 1)^{4}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$T^{4} + 24T^{2} + 36$$
$11$ $$T^{4} + 24T^{2} + 36$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 27)^{2}$$
$19$ $$T^{4} + 24T^{2} + 36$$
$23$ $$(T^{2} - 6 T - 18)^{2}$$
$29$ $$(T + 3)^{4}$$
$31$ $$T^{4} + 96T^{2} + 576$$
$37$ $$(T^{2} + 9)^{2}$$
$41$ $$T^{4} + 42T^{2} + 9$$
$43$ $$(T^{2} + 2 T - 26)^{2}$$
$47$ $$T^{4} + 24T^{2} + 36$$
$53$ $$(T - 3)^{4}$$
$59$ $$(T^{2} + 192)^{2}$$
$61$ $$(T^{2} - 20 T + 73)^{2}$$
$67$ $$T^{4} + 168T^{2} + 6084$$
$71$ $$T^{4} + 72T^{2} + 324$$
$73$ $$(T^{2} + 147)^{2}$$
$79$ $$(T^{2} + 4 T - 104)^{2}$$
$83$ $$T^{4} + 168T^{2} + 4356$$
$89$ $$T^{4} + 96T^{2} + 576$$
$97$ $$(T^{2} + 36)^{2}$$