Properties

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

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1296 = 2^{4} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1296.q (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$35.3134422611$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} -4 \beta_{2} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} -4 \beta_{2} q^{7} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{11} + ( -8 + 8 \beta_{2} ) q^{13} -3 \beta_{3} q^{17} + 16 q^{19} -4 \beta_{1} q^{23} -7 \beta_{2} q^{25} + ( \beta_{1} - \beta_{3} ) q^{29} + ( 44 - 44 \beta_{2} ) q^{31} -4 \beta_{3} q^{35} -34 q^{37} -11 \beta_{1} q^{41} -40 \beta_{2} q^{43} + ( 20 \beta_{1} - 20 \beta_{3} ) q^{47} + ( 33 - 33 \beta_{2} ) q^{49} + 9 \beta_{3} q^{53} -72 q^{55} + 8 \beta_{1} q^{59} -50 \beta_{2} q^{61} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{65} + ( 8 - 8 \beta_{2} ) q^{67} + 12 \beta_{3} q^{71} -16 q^{73} + 16 \beta_{1} q^{77} -76 \beta_{2} q^{79} + ( -28 \beta_{1} + 28 \beta_{3} ) q^{83} + ( 54 - 54 \beta_{2} ) q^{85} + 3 \beta_{3} q^{89} + 32 q^{91} + 16 \beta_{1} q^{95} -176 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{7} + O(q^{10})$$ $$4q - 8q^{7} - 16q^{13} + 64q^{19} - 14q^{25} + 88q^{31} - 136q^{37} - 80q^{43} + 66q^{49} - 288q^{55} - 100q^{61} + 16q^{67} - 64q^{73} - 152q^{79} + 108q^{85} + 128q^{91} - 352q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$$$/3$$

Character values

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

 $$n$$ $$325$$ $$1135$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
593.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 0 0 −3.67423 + 2.12132i 0 −2.00000 + 3.46410i 0 0 0
593.2 0 0 0 3.67423 2.12132i 0 −2.00000 + 3.46410i 0 0 0
1025.1 0 0 0 −3.67423 2.12132i 0 −2.00000 3.46410i 0 0 0
1025.2 0 0 0 3.67423 + 2.12132i 0 −2.00000 3.46410i 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
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.q.f 4
3.b odd 2 1 inner 1296.3.q.f 4
4.b odd 2 1 162.3.d.b 4
9.c even 3 1 144.3.e.b 2
9.c even 3 1 inner 1296.3.q.f 4
9.d odd 6 1 144.3.e.b 2
9.d odd 6 1 inner 1296.3.q.f 4
12.b even 2 1 162.3.d.b 4
36.f odd 6 1 18.3.b.a 2
36.f odd 6 1 162.3.d.b 4
36.h even 6 1 18.3.b.a 2
36.h even 6 1 162.3.d.b 4
45.h odd 6 1 3600.3.l.d 2
45.j even 6 1 3600.3.l.d 2
45.k odd 12 2 3600.3.c.b 4
45.l even 12 2 3600.3.c.b 4
72.j odd 6 1 576.3.e.f 2
72.l even 6 1 576.3.e.c 2
72.n even 6 1 576.3.e.f 2
72.p odd 6 1 576.3.e.c 2
144.u even 12 2 2304.3.h.f 4
144.v odd 12 2 2304.3.h.f 4
144.w odd 12 2 2304.3.h.c 4
144.x even 12 2 2304.3.h.c 4
180.n even 6 1 450.3.d.f 2
180.p odd 6 1 450.3.d.f 2
180.v odd 12 2 450.3.b.b 4
180.x even 12 2 450.3.b.b 4
252.n even 6 1 882.3.s.d 4
252.o even 6 1 882.3.s.b 4
252.r odd 6 1 882.3.s.d 4
252.s odd 6 1 882.3.b.a 2
252.u odd 6 1 882.3.s.b 4
252.bb even 6 1 882.3.s.b 4
252.bi even 6 1 882.3.b.a 2
252.bj even 6 1 882.3.s.d 4
252.bl odd 6 1 882.3.s.b 4
252.bn odd 6 1 882.3.s.d 4
396.k even 6 1 2178.3.c.d 2
396.o odd 6 1 2178.3.c.d 2
468.x even 6 1 3042.3.c.e 2
468.bg odd 6 1 3042.3.c.e 2
468.bs even 12 2 3042.3.d.a 4
468.ch odd 12 2 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 36.f odd 6 1
18.3.b.a 2 36.h even 6 1
144.3.e.b 2 9.c even 3 1
144.3.e.b 2 9.d odd 6 1
162.3.d.b 4 4.b odd 2 1
162.3.d.b 4 12.b even 2 1
162.3.d.b 4 36.f odd 6 1
162.3.d.b 4 36.h even 6 1
450.3.b.b 4 180.v odd 12 2
450.3.b.b 4 180.x even 12 2
450.3.d.f 2 180.n even 6 1
450.3.d.f 2 180.p odd 6 1
576.3.e.c 2 72.l even 6 1
576.3.e.c 2 72.p odd 6 1
576.3.e.f 2 72.j odd 6 1
576.3.e.f 2 72.n even 6 1
882.3.b.a 2 252.s odd 6 1
882.3.b.a 2 252.bi even 6 1
882.3.s.b 4 252.o even 6 1
882.3.s.b 4 252.u odd 6 1
882.3.s.b 4 252.bb even 6 1
882.3.s.b 4 252.bl odd 6 1
882.3.s.d 4 252.n even 6 1
882.3.s.d 4 252.r odd 6 1
882.3.s.d 4 252.bj even 6 1
882.3.s.d 4 252.bn odd 6 1
1296.3.q.f 4 1.a even 1 1 trivial
1296.3.q.f 4 3.b odd 2 1 inner
1296.3.q.f 4 9.c even 3 1 inner
1296.3.q.f 4 9.d odd 6 1 inner
2178.3.c.d 2 396.k even 6 1
2178.3.c.d 2 396.o odd 6 1
2304.3.h.c 4 144.w odd 12 2
2304.3.h.c 4 144.x even 12 2
2304.3.h.f 4 144.u even 12 2
2304.3.h.f 4 144.v odd 12 2
3042.3.c.e 2 468.x even 6 1
3042.3.c.e 2 468.bg odd 6 1
3042.3.d.a 4 468.bs even 12 2
3042.3.d.a 4 468.ch odd 12 2
3600.3.c.b 4 45.k odd 12 2
3600.3.c.b 4 45.l even 12 2
3600.3.l.d 2 45.h odd 6 1
3600.3.l.d 2 45.j even 6 1

Hecke kernels

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

 $$T_{5}^{4} - 18 T_{5}^{2} + 324$$ $$T_{7}^{2} + 4 T_{7} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$324 - 18 T^{2} + T^{4}$$
$7$ $$( 16 + 4 T + T^{2} )^{2}$$
$11$ $$82944 - 288 T^{2} + T^{4}$$
$13$ $$( 64 + 8 T + T^{2} )^{2}$$
$17$ $$( 162 + T^{2} )^{2}$$
$19$ $$( -16 + T )^{4}$$
$23$ $$82944 - 288 T^{2} + T^{4}$$
$29$ $$324 - 18 T^{2} + T^{4}$$
$31$ $$( 1936 - 44 T + T^{2} )^{2}$$
$37$ $$( 34 + T )^{4}$$
$41$ $$4743684 - 2178 T^{2} + T^{4}$$
$43$ $$( 1600 + 40 T + T^{2} )^{2}$$
$47$ $$51840000 - 7200 T^{2} + T^{4}$$
$53$ $$( 1458 + T^{2} )^{2}$$
$59$ $$1327104 - 1152 T^{2} + T^{4}$$
$61$ $$( 2500 + 50 T + T^{2} )^{2}$$
$67$ $$( 64 - 8 T + T^{2} )^{2}$$
$71$ $$( 2592 + T^{2} )^{2}$$
$73$ $$( 16 + T )^{4}$$
$79$ $$( 5776 + 76 T + T^{2} )^{2}$$
$83$ $$199148544 - 14112 T^{2} + T^{4}$$
$89$ $$( 162 + T^{2} )^{2}$$
$97$ $$( 30976 + 176 T + T^{2} )^{2}$$