Properties

 Label 1296.3.e.c Level $1296$ Weight $3$ Character orbit 1296.e Analytic conductor $35.313$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 72) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{5} + ( -3 - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{5} + ( -3 - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{11} + ( 7 - \beta_{3} ) q^{13} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{17} + ( -2 - 2 \beta_{3} ) q^{19} + ( -\beta_{1} + 5 \beta_{2} ) q^{23} + ( -10 + 2 \beta_{3} ) q^{25} + ( -3 \beta_{1} + \beta_{2} ) q^{29} + ( 37 - \beta_{3} ) q^{31} -29 \beta_{2} q^{35} + ( -30 + 2 \beta_{3} ) q^{37} + 23 \beta_{2} q^{41} + ( 5 - 4 \beta_{3} ) q^{43} + ( -3 \beta_{1} - 29 \beta_{2} ) q^{47} + ( 56 + 6 \beta_{3} ) q^{49} + ( -8 \beta_{1} - 32 \beta_{2} ) q^{53} + ( -55 - \beta_{3} ) q^{55} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{59} + ( 31 + \beta_{3} ) q^{61} + ( 10 \beta_{1} - 39 \beta_{2} ) q^{65} + ( -11 - 10 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 24 \beta_{2} ) q^{71} + ( 10 + 6 \beta_{3} ) q^{73} + ( -15 \beta_{1} - 73 \beta_{2} ) q^{77} + ( -43 + 7 \beta_{3} ) q^{79} + ( -14 \beta_{1} + 11 \beta_{2} ) q^{83} + ( -88 + 10 \beta_{3} ) q^{85} + ( 6 \beta_{1} - 24 \beta_{2} ) q^{89} + ( 75 - 4 \beta_{3} ) q^{91} + ( 4 \beta_{1} - 62 \beta_{2} ) q^{95} + ( -121 + 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{7} + O(q^{10})$$ $$4q - 12q^{7} + 28q^{13} - 8q^{19} - 40q^{25} + 148q^{31} - 120q^{37} + 20q^{43} + 224q^{49} - 220q^{55} + 124q^{61} - 44q^{67} + 40q^{73} - 172q^{79} - 352q^{85} + 300q^{91} - 484q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3}$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$-2 \nu^{3} + 8 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{1}$$$$/2$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −1.22474 − 0.707107i 1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i
0 0 0 7.38891i 0 6.79796 0 0 0
161.2 0 0 0 3.92480i 0 −12.7980 0 0 0
161.3 0 0 0 3.92480i 0 −12.7980 0 0 0
161.4 0 0 0 7.38891i 0 6.79796 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
3.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.e.c 4
3.b odd 2 1 inner 1296.3.e.c 4
4.b odd 2 1 648.3.e.b 4
9.c even 3 1 144.3.q.d 4
9.c even 3 1 432.3.q.c 4
9.d odd 6 1 144.3.q.d 4
9.d odd 6 1 432.3.q.c 4
12.b even 2 1 648.3.e.b 4
36.f odd 6 1 72.3.m.a 4
36.f odd 6 1 216.3.m.a 4
36.h even 6 1 72.3.m.a 4
36.h even 6 1 216.3.m.a 4
72.j odd 6 1 576.3.q.c 4
72.j odd 6 1 1728.3.q.f 4
72.l even 6 1 576.3.q.h 4
72.l even 6 1 1728.3.q.e 4
72.n even 6 1 576.3.q.c 4
72.n even 6 1 1728.3.q.f 4
72.p odd 6 1 576.3.q.h 4
72.p odd 6 1 1728.3.q.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a 4 36.f odd 6 1
72.3.m.a 4 36.h even 6 1
144.3.q.d 4 9.c even 3 1
144.3.q.d 4 9.d odd 6 1
216.3.m.a 4 36.f odd 6 1
216.3.m.a 4 36.h even 6 1
432.3.q.c 4 9.c even 3 1
432.3.q.c 4 9.d odd 6 1
576.3.q.c 4 72.j odd 6 1
576.3.q.c 4 72.n even 6 1
576.3.q.h 4 72.l even 6 1
576.3.q.h 4 72.p odd 6 1
648.3.e.b 4 4.b odd 2 1
648.3.e.b 4 12.b even 2 1
1296.3.e.c 4 1.a even 1 1 trivial
1296.3.e.c 4 3.b odd 2 1 inner
1728.3.q.e 4 72.l even 6 1
1728.3.q.e 4 72.p odd 6 1
1728.3.q.f 4 72.j odd 6 1
1728.3.q.f 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1296, [\chi])$$:

 $$T_{5}^{4} + 70 T_{5}^{2} + 841$$ $$T_{7}^{2} + 6 T_{7} - 87$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$841 + 70 T^{2} + T^{4}$$
$7$ $$( -87 + 6 T + T^{2} )^{2}$$
$11$ $$10201 + 310 T^{2} + T^{4}$$
$13$ $$( -47 - 14 T + T^{2} )^{2}$$
$17$ $$4096 + 640 T^{2} + T^{4}$$
$19$ $$( -380 + 4 T + T^{2} )^{2}$$
$23$ $$1849 + 214 T^{2} + T^{4}$$
$29$ $$81225 + 582 T^{2} + T^{4}$$
$31$ $$( 1273 - 74 T + T^{2} )^{2}$$
$37$ $$( 516 + 60 T + T^{2} )^{2}$$
$41$ $$( 1587 + T^{2} )^{2}$$
$43$ $$( -1511 - 10 T + T^{2} )^{2}$$
$47$ $$4995225 + 5622 T^{2} + T^{4}$$
$53$ $$1048576 + 10240 T^{2} + T^{4}$$
$59$ $$10201 + 310 T^{2} + T^{4}$$
$61$ $$( 865 - 62 T + T^{2} )^{2}$$
$67$ $$( -9479 + 22 T + T^{2} )^{2}$$
$71$ $$2560000 + 3712 T^{2} + T^{4}$$
$73$ $$( -3356 - 20 T + T^{2} )^{2}$$
$79$ $$( -2855 + 86 T + T^{2} )^{2}$$
$83$ $$34916281 + 13270 T^{2} + T^{4}$$
$89$ $$331776 + 5760 T^{2} + T^{4}$$
$97$ $$( 14257 + 242 T + T^{2} )^{2}$$