Properties

 Label 3136.2.f.e Level $3136$ Weight $2$ Character orbit 3136.f Analytic conductor $25.041$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3136.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$25.0410860739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 28) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{2} q^{5}+O(q^{10})$$ q - b3 * q^3 - b2 * q^5 $$q - \beta_{3} q^{3} - \beta_{2} q^{5} + \beta_1 q^{11} - 2 \beta_{2} q^{13} + 3 \beta_1 q^{15} - \beta_{2} q^{17} + 3 \beta_{3} q^{19} + \beta_1 q^{23} + 2 q^{25} + 3 \beta_{3} q^{27} - 4 q^{29} - \beta_{3} q^{31} - \beta_{2} q^{33} - 3 q^{37} + 6 \beta_1 q^{39} - 2 \beta_{2} q^{41} - 2 \beta_1 q^{43} + 5 \beta_{3} q^{47} + 3 \beta_1 q^{51} + q^{53} + \beta_{3} q^{55} - 9 q^{57} + 3 \beta_{3} q^{59} - 3 \beta_{2} q^{61} - 6 q^{65} + 3 \beta_1 q^{67} - \beta_{2} q^{69} - 14 \beta_1 q^{71} + 5 \beta_{2} q^{73} - 2 \beta_{3} q^{75} + 9 \beta_1 q^{79} - 9 q^{81} - 8 \beta_{3} q^{83} - 3 q^{85} + 4 \beta_{3} q^{87} - 9 \beta_{2} q^{89} + 3 q^{93} - 9 \beta_1 q^{95} - 10 \beta_{2} q^{97}+O(q^{100})$$ q - b3 * q^3 - b2 * q^5 + b1 * q^11 - 2*b2 * q^13 + 3*b1 * q^15 - b2 * q^17 + 3*b3 * q^19 + b1 * q^23 + 2 * q^25 + 3*b3 * q^27 - 4 * q^29 - b3 * q^31 - b2 * q^33 - 3 * q^37 + 6*b1 * q^39 - 2*b2 * q^41 - 2*b1 * q^43 + 5*b3 * q^47 + 3*b1 * q^51 + q^53 + b3 * q^55 - 9 * q^57 + 3*b3 * q^59 - 3*b2 * q^61 - 6 * q^65 + 3*b1 * q^67 - b2 * q^69 - 14*b1 * q^71 + 5*b2 * q^73 - 2*b3 * q^75 + 9*b1 * q^79 - 9 * q^81 - 8*b3 * q^83 - 3 * q^85 + 4*b3 * q^87 - 9*b2 * q^89 + 3 * q^93 - 9*b1 * q^95 - 10*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{25} - 16 q^{29} - 12 q^{37} + 4 q^{53} - 36 q^{57} - 24 q^{65} - 36 q^{81} - 12 q^{85} + 12 q^{93}+O(q^{100})$$ 4 * q + 8 * q^25 - 16 * q^29 - 12 * q^37 + 4 * q^53 - 36 * q^57 - 24 * q^65 - 36 * q^81 - 12 * q^85 + 12 * q^93

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 $$\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3135.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
0 −1.73205 0 1.73205i 0 0 0 0 0
3135.2 0 −1.73205 0 1.73205i 0 0 0 0 0
3135.3 0 1.73205 0 1.73205i 0 0 0 0 0
3135.4 0 1.73205 0 1.73205i 0 0 0 0 0
 $$n$$: 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
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.f.e 4
4.b odd 2 1 inner 3136.2.f.e 4
7.b odd 2 1 inner 3136.2.f.e 4
7.c even 3 1 448.2.p.d 4
7.d odd 6 1 448.2.p.d 4
8.b even 2 1 196.2.d.b 4
8.d odd 2 1 196.2.d.b 4
24.f even 2 1 1764.2.b.a 4
24.h odd 2 1 1764.2.b.a 4
28.d even 2 1 inner 3136.2.f.e 4
28.f even 6 1 448.2.p.d 4
28.g odd 6 1 448.2.p.d 4
56.e even 2 1 196.2.d.b 4
56.h odd 2 1 196.2.d.b 4
56.j odd 6 1 28.2.f.a 4
56.j odd 6 1 196.2.f.a 4
56.k odd 6 1 28.2.f.a 4
56.k odd 6 1 196.2.f.a 4
56.m even 6 1 28.2.f.a 4
56.m even 6 1 196.2.f.a 4
56.p even 6 1 28.2.f.a 4
56.p even 6 1 196.2.f.a 4
168.e odd 2 1 1764.2.b.a 4
168.i even 2 1 1764.2.b.a 4
168.s odd 6 1 252.2.bf.e 4
168.v even 6 1 252.2.bf.e 4
168.ba even 6 1 252.2.bf.e 4
168.be odd 6 1 252.2.bf.e 4
280.ba even 6 1 700.2.p.a 4
280.bf even 6 1 700.2.p.a 4
280.bi odd 6 1 700.2.p.a 4
280.bk odd 6 1 700.2.p.a 4
280.bp odd 12 1 700.2.t.a 4
280.bp odd 12 1 700.2.t.b 4
280.br even 12 1 700.2.t.a 4
280.br even 12 1 700.2.t.b 4
280.bt odd 12 1 700.2.t.a 4
280.bt odd 12 1 700.2.t.b 4
280.bv even 12 1 700.2.t.a 4
280.bv even 12 1 700.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 56.j odd 6 1
28.2.f.a 4 56.k odd 6 1
28.2.f.a 4 56.m even 6 1
28.2.f.a 4 56.p even 6 1
196.2.d.b 4 8.b even 2 1
196.2.d.b 4 8.d odd 2 1
196.2.d.b 4 56.e even 2 1
196.2.d.b 4 56.h odd 2 1
196.2.f.a 4 56.j odd 6 1
196.2.f.a 4 56.k odd 6 1
196.2.f.a 4 56.m even 6 1
196.2.f.a 4 56.p even 6 1
252.2.bf.e 4 168.s odd 6 1
252.2.bf.e 4 168.v even 6 1
252.2.bf.e 4 168.ba even 6 1
252.2.bf.e 4 168.be odd 6 1
448.2.p.d 4 7.c even 3 1
448.2.p.d 4 7.d odd 6 1
448.2.p.d 4 28.f even 6 1
448.2.p.d 4 28.g odd 6 1
700.2.p.a 4 280.ba even 6 1
700.2.p.a 4 280.bf even 6 1
700.2.p.a 4 280.bi odd 6 1
700.2.p.a 4 280.bk odd 6 1
700.2.t.a 4 280.bp odd 12 1
700.2.t.a 4 280.br even 12 1
700.2.t.a 4 280.bt odd 12 1
700.2.t.a 4 280.bv even 12 1
700.2.t.b 4 280.bp odd 12 1
700.2.t.b 4 280.br even 12 1
700.2.t.b 4 280.bt odd 12 1
700.2.t.b 4 280.bv even 12 1
1764.2.b.a 4 24.f even 2 1
1764.2.b.a 4 24.h odd 2 1
1764.2.b.a 4 168.e odd 2 1
1764.2.b.a 4 168.i even 2 1
3136.2.f.e 4 1.a even 1 1 trivial
3136.2.f.e 4 4.b odd 2 1 inner
3136.2.f.e 4 7.b odd 2 1 inner
3136.2.f.e 4 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3136, [\chi])$$:

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{5}^{2} + 3$$ T5^2 + 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 1)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} + 3)^{2}$$
$19$ $$(T^{2} - 27)^{2}$$
$23$ $$(T^{2} + 1)^{2}$$
$29$ $$(T + 4)^{4}$$
$31$ $$(T^{2} - 3)^{2}$$
$37$ $$(T + 3)^{4}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} - 75)^{2}$$
$53$ $$(T - 1)^{4}$$
$59$ $$(T^{2} - 27)^{2}$$
$61$ $$(T^{2} + 27)^{2}$$
$67$ $$(T^{2} + 9)^{2}$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$(T^{2} + 75)^{2}$$
$79$ $$(T^{2} + 81)^{2}$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} + 243)^{2}$$
$97$ $$(T^{2} + 300)^{2}$$