# Properties

 Label 2352.2.bl.t Level $2352$ Weight $2$ Character orbit 2352.bl Analytic conductor $18.781$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.bl (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} + ( \beta_{2} - \beta_{3} - \beta_{6} ) q^{5} + ( -1 - \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} + ( \beta_{2} - \beta_{3} - \beta_{6} ) q^{5} + ( -1 - \beta_{4} ) q^{9} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{5} - \beta_{7} ) q^{13} + ( -\beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{15} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{19} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{23} + ( -4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{25} - q^{27} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{29} + ( 2 \beta_{1} + \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{33} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{37} + ( -\beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{39} + ( -2 + \beta_{1} - \beta_{2} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{41} + ( 2 - 2 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - 6 \beta_{7} ) q^{43} + ( -2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{45} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{51} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( 8 - \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - \beta_{5} - 3 \beta_{7} ) q^{55} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{57} + ( -6 \beta_{1} + \beta_{3} - 6 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( -4 + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{6} ) q^{61} + ( 2 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{65} + ( -8 - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{67} + ( -2 - \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{69} + ( -4 - 3 \beta_{1} + \beta_{2} - 8 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{71} + ( -8 - 4 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{73} + ( -3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{75} + ( -2 + 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{79} + \beta_{4} q^{81} + ( -2 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} ) q^{83} + ( -8 + 4 \beta_{1} + \beta_{2} + 4 \beta_{5} ) q^{85} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{87} + ( -4 + 2 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{89} + ( 4 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{93} + ( 12 - 2 \beta_{3} + 6 \beta_{4} - 8 \beta_{5} - 2 \beta_{7} ) q^{95} + ( -4 - 6 \beta_{1} - 2 \beta_{2} - 8 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{97} + ( -\beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} - 4q^{9} + O(q^{10})$$ $$8q + 4q^{3} - 4q^{9} + 24q^{23} + 12q^{25} - 8q^{27} + 16q^{29} - 16q^{31} - 8q^{47} - 8q^{53} + 64q^{55} + 24q^{59} - 48q^{61} + 8q^{65} - 48q^{67} - 48q^{73} - 12q^{75} - 24q^{79} - 4q^{81} - 64q^{85} + 8q^{87} - 48q^{89} + 16q^{93} + 72q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 20$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 34 \nu$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 2$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} + 7 \nu^{5} - 28 \nu^{3} + 16 \nu$$$$)/14$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 21 \nu^{2} + 2$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{5} - 70 \nu^{3} + 6 \nu$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 3 \beta_{5} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{6} - 6 \beta_{4} + 4 \beta_{2} - 6$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 10 \beta_{5} + 4 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{2} - 20$$ $$\nu^{7}$$ $$=$$ $$14 \beta_{3} - 34 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1 + \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −1.60021 + 0.923880i 1.60021 − 0.923880i 0.662827 − 0.382683i −0.662827 + 0.382683i −1.60021 − 0.923880i 1.60021 + 0.923880i 0.662827 + 0.382683i −0.662827 − 0.382683i
0 0.500000 0.866025i 0 −2.78415 + 1.60743i 0 0 0 −0.500000 0.866025i 0
31.2 0 0.500000 0.866025i 0 −1.45849 + 0.842061i 0 0 0 −0.500000 0.866025i 0
31.3 0 0.500000 0.866025i 0 0.521114 0.300865i 0 0 0 −0.500000 0.866025i 0
31.4 0 0.500000 0.866025i 0 3.72153 2.14862i 0 0 0 −0.500000 0.866025i 0
607.1 0 0.500000 + 0.866025i 0 −2.78415 1.60743i 0 0 0 −0.500000 + 0.866025i 0
607.2 0 0.500000 + 0.866025i 0 −1.45849 0.842061i 0 0 0 −0.500000 + 0.866025i 0
607.3 0 0.500000 + 0.866025i 0 0.521114 + 0.300865i 0 0 0 −0.500000 + 0.866025i 0
607.4 0 0.500000 + 0.866025i 0 3.72153 + 2.14862i 0 0 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.bl.t 8
4.b odd 2 1 2352.2.bl.o 8
7.b odd 2 1 2352.2.bl.q 8
7.c even 3 1 2352.2.b.k 8
7.c even 3 1 2352.2.bl.r 8
7.d odd 6 1 2352.2.b.l yes 8
7.d odd 6 1 2352.2.bl.o 8
21.g even 6 1 7056.2.b.w 8
21.h odd 6 1 7056.2.b.x 8
28.d even 2 1 2352.2.bl.r 8
28.f even 6 1 2352.2.b.k 8
28.f even 6 1 inner 2352.2.bl.t 8
28.g odd 6 1 2352.2.b.l yes 8
28.g odd 6 1 2352.2.bl.q 8
84.j odd 6 1 7056.2.b.x 8
84.n even 6 1 7056.2.b.w 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.k 8 7.c even 3 1
2352.2.b.k 8 28.f even 6 1
2352.2.b.l yes 8 7.d odd 6 1
2352.2.b.l yes 8 28.g odd 6 1
2352.2.bl.o 8 4.b odd 2 1
2352.2.bl.o 8 7.d odd 6 1
2352.2.bl.q 8 7.b odd 2 1
2352.2.bl.q 8 28.g odd 6 1
2352.2.bl.r 8 7.c even 3 1
2352.2.bl.r 8 28.d even 2 1
2352.2.bl.t 8 1.a even 1 1 trivial
2352.2.bl.t 8 28.f even 6 1 inner
7056.2.b.w 8 21.g even 6 1
7056.2.b.w 8 84.n even 6 1
7056.2.b.x 8 21.h odd 6 1
7056.2.b.x 8 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{8} - 16 T_{5}^{6} + 242 T_{5}^{4} + 384 T_{5}^{3} - 32 T_{5}^{2} - 336 T_{5} + 196$$ $$T_{11}^{8} - 20 T_{11}^{6} + 404 T_{11}^{4} + 960 T_{11}^{3} + 848 T_{11}^{2} + 192 T_{11} + 16$$ $$T_{17}^{8} - 32 T_{17}^{6} + 1106 T_{17}^{4} + 5376 T_{17}^{3} + 12032 T_{17}^{2} + 13776 T_{17} + 6724$$ $$T_{19}^{8} + 40 T_{19}^{6} - 192 T_{19}^{5} + 1656 T_{19}^{4} - 3840 T_{19}^{3} + 6976 T_{19}^{2} - 5376 T_{19} + 3136$$ $$T_{31}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - T + T^{2} )^{4}$$
$5$ $$196 - 336 T - 32 T^{2} + 384 T^{3} + 242 T^{4} - 16 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$16 + 192 T + 848 T^{2} + 960 T^{3} + 404 T^{4} - 20 T^{6} + T^{8}$$
$13$ $$( 98 + 20 T^{2} + T^{4} )^{2}$$
$17$ $$6724 + 13776 T + 12032 T^{2} + 5376 T^{3} + 1106 T^{4} - 32 T^{6} + T^{8}$$
$19$ $$3136 - 5376 T + 6976 T^{2} - 3840 T^{3} + 1656 T^{4} - 192 T^{5} + 40 T^{6} + T^{8}$$
$23$ $$38416 - 10192 T^{2} + 2900 T^{4} - 1248 T^{5} + 244 T^{6} - 24 T^{7} + T^{8}$$
$29$ $$( 28 + 32 T - 4 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$678976 + 131840 T + 71744 T^{2} + 17408 T^{3} + 6520 T^{4} + 1216 T^{5} + 200 T^{6} + 16 T^{7} + T^{8}$$
$37$ $$929296 - 185088 T + 133264 T^{2} + 19200 T^{3} + 9036 T^{4} + 384 T^{5} + 100 T^{6} + T^{8}$$
$41$ $$6724 + 10912 T^{2} + 1820 T^{4} + 80 T^{6} + T^{8}$$
$43$ $$20214016 + 1679104 T^{2} + 42848 T^{4} + 368 T^{6} + T^{8}$$
$47$ $$153664 + 87808 T + 43904 T^{2} + 9856 T^{3} + 2440 T^{4} + 320 T^{5} + 80 T^{6} + 8 T^{7} + T^{8}$$
$53$ $$614656 - 25088 T + 70016 T^{2} - 9728 T^{3} + 7216 T^{4} - 640 T^{5} + 152 T^{6} + 8 T^{7} + T^{8}$$
$59$ $$71503936 - 12176640 T + 2750080 T^{2} - 290688 T^{3} + 49416 T^{4} - 4800 T^{5} + 496 T^{6} - 24 T^{7} + T^{8}$$
$61$ $$9604 + 98784 T + 361816 T^{2} + 237888 T^{3} + 71726 T^{4} + 11328 T^{5} + 1004 T^{6} + 48 T^{7} + T^{8}$$
$67$ $$802816 + 1376256 T + 1015808 T^{2} + 393216 T^{3} + 89216 T^{4} + 12288 T^{5} + 1024 T^{6} + 48 T^{7} + T^{8}$$
$71$ $$10265616 + 1314144 T^{2} + 39816 T^{4} + 360 T^{6} + T^{8}$$
$73$ $$4866436 - 741216 T - 447688 T^{2} + 73920 T^{3} + 55982 T^{4} + 10560 T^{5} + 988 T^{6} + 48 T^{7} + T^{8}$$
$79$ $$430336 - 818688 T + 482432 T^{2} + 69888 T^{3} - 7504 T^{4} - 1344 T^{5} + 136 T^{6} + 24 T^{7} + T^{8}$$
$83$ $$( 12256 + 192 T - 256 T^{2} + T^{4} )^{2}$$
$89$ $$1766857156 + 446905488 T + 39024896 T^{2} + 340224 T^{3} - 127054 T^{4} - 1536 T^{5} + 736 T^{6} + 48 T^{7} + T^{8}$$
$97$ $$325658116 + 10546576 T^{2} + 121364 T^{4} + 584 T^{6} + T^{8}$$