# Properties

 Label 2352.3.f.j Level $2352$ Weight $3$ Character orbit 2352.f Analytic conductor $64.087$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2352.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.0873581775$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4$$ x^8 - 4*x^6 + 14*x^4 - 8*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 588) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{5} + \beta_{4}) q^{5} - 3 q^{9}+O(q^{10})$$ q - b2 * q^3 + (b5 + b4) * q^5 - 3 * q^9 $$q - \beta_{2} q^{3} + (\beta_{5} + \beta_{4}) q^{5} - 3 q^{9} + (\beta_{6} - \beta_1) q^{11} + (2 \beta_{7} - 4 \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{3} + 3 \beta_1) q^{15} + ( - 3 \beta_{5} + 3 \beta_{4} + 4 \beta_{2}) q^{17} + ( - \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 8 \beta_{2}) q^{19} + ( - \beta_{6} - 2 \beta_{3} + 9 \beta_1 + 2) q^{23} + ( - 2 \beta_{6} - \beta_1 + 9) q^{25} + 3 \beta_{2} q^{27} + ( - \beta_{6} + 6 \beta_{3} - 5 \beta_1 + 10) q^{29} + (3 \beta_{7} + 14 \beta_{5} + 4 \beta_{4} - 4 \beta_{2}) q^{31} + ( - 3 \beta_{7} + \beta_{5}) q^{33} + (6 \beta_{6} + 4 \beta_{3} + 9 \beta_1 + 16) q^{37} + (2 \beta_{6} + \beta_{3} - 12 \beta_1) q^{39} + ( - 5 \beta_{7} - 13 \beta_{5} + 7 \beta_{4} - 14 \beta_{2}) q^{41} + (2 \beta_{6} - 8 \beta_{3} - 8 \beta_1 + 14) q^{43} + ( - 3 \beta_{5} - 3 \beta_{4}) q^{45} + ( - 7 \beta_{7} - 2 \beta_{5} + 8 \beta_{4} - 22 \beta_{2}) q^{47} + ( - 3 \beta_{3} - 9 \beta_1 + 12) q^{51} + ( - 6 \beta_{6} - 4 \beta_{3} - 26 \beta_1 - 18) q^{53} + ( - \beta_{7} + 10 \beta_{5} + 6 \beta_{4}) q^{55} + ( - \beta_{6} + 2 \beta_{3} - 6 \beta_1 - 24) q^{57} + ( - 5 \beta_{7} + 22 \beta_{5} - 12 \beta_{4} - 14 \beta_{2}) q^{59} + ( - 8 \beta_{7} + 14 \beta_{5} - 5 \beta_{4} + 12 \beta_{2}) q^{61} + (5 \beta_{6} + 4 \beta_{3} - 19 \beta_1 + 30) q^{65} + (6 \beta_{6} - 4 \beta_{3} - 12 \beta_1 + 8) q^{67} + (3 \beta_{7} - 9 \beta_{5} - 6 \beta_{4} - 2 \beta_{2}) q^{69} + ( - \beta_{6} - 8 \beta_{3} - 43 \beta_1 - 28) q^{71} + (8 \beta_{7} - 6 \beta_{5} - 15 \beta_{4} + 28 \beta_{2}) q^{73} + (6 \beta_{7} + \beta_{5} - 9 \beta_{2}) q^{75} + (4 \beta_{6} - 28 \beta_1 + 54) q^{79} + 9 q^{81} + (14 \beta_{7} + 6 \beta_{5} - 6 \beta_{4} - 16 \beta_{2}) q^{83} + (4 \beta_{3} - 15 \beta_1 - 12) q^{85} + (3 \beta_{7} + 5 \beta_{5} + 18 \beta_{4} - 10 \beta_{2}) q^{87} + (16 \beta_{7} + 23 \beta_{5} + 11 \beta_{4} - 8 \beta_{2}) q^{89} + (3 \beta_{6} - 4 \beta_{3} + 42 \beta_1 - 12) q^{93} + (4 \beta_{6} - 10 \beta_{3} + 36 \beta_1 + 34) q^{95} + (4 \beta_{7} - 10 \beta_{5} + 3 \beta_{4} - 28 \beta_{2}) q^{97} + ( - 3 \beta_{6} + 3 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b5 + b4) * q^5 - 3 * q^9 + (b6 - b1) * q^11 + (2*b7 - 4*b5 - b4) * q^13 + (-b3 + 3*b1) * q^15 + (-3*b5 + 3*b4 + 4*b2) * q^17 + (-b7 - 2*b5 - 2*b4 - 8*b2) * q^19 + (-b6 - 2*b3 + 9*b1 + 2) * q^23 + (-2*b6 - b1 + 9) * q^25 + 3*b2 * q^27 + (-b6 + 6*b3 - 5*b1 + 10) * q^29 + (3*b7 + 14*b5 + 4*b4 - 4*b2) * q^31 + (-3*b7 + b5) * q^33 + (6*b6 + 4*b3 + 9*b1 + 16) * q^37 + (2*b6 + b3 - 12*b1) * q^39 + (-5*b7 - 13*b5 + 7*b4 - 14*b2) * q^41 + (2*b6 - 8*b3 - 8*b1 + 14) * q^43 + (-3*b5 - 3*b4) * q^45 + (-7*b7 - 2*b5 + 8*b4 - 22*b2) * q^47 + (-3*b3 - 9*b1 + 12) * q^51 + (-6*b6 - 4*b3 - 26*b1 - 18) * q^53 + (-b7 + 10*b5 + 6*b4) * q^55 + (-b6 + 2*b3 - 6*b1 - 24) * q^57 + (-5*b7 + 22*b5 - 12*b4 - 14*b2) * q^59 + (-8*b7 + 14*b5 - 5*b4 + 12*b2) * q^61 + (5*b6 + 4*b3 - 19*b1 + 30) * q^65 + (6*b6 - 4*b3 - 12*b1 + 8) * q^67 + (3*b7 - 9*b5 - 6*b4 - 2*b2) * q^69 + (-b6 - 8*b3 - 43*b1 - 28) * q^71 + (8*b7 - 6*b5 - 15*b4 + 28*b2) * q^73 + (6*b7 + b5 - 9*b2) * q^75 + (4*b6 - 28*b1 + 54) * q^79 + 9 * q^81 + (14*b7 + 6*b5 - 6*b4 - 16*b2) * q^83 + (4*b3 - 15*b1 - 12) * q^85 + (3*b7 + 5*b5 + 18*b4 - 10*b2) * q^87 + (16*b7 + 23*b5 + 11*b4 - 8*b2) * q^89 + (3*b6 - 4*b3 + 42*b1 - 12) * q^93 + (4*b6 - 10*b3 + 36*b1 + 34) * q^95 + (4*b7 - 10*b5 + 3*b4 - 28*b2) * q^97 + (-3*b6 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{9}+O(q^{10})$$ 8 * q - 24 * q^9 $$8 q - 24 q^{9} + 16 q^{23} + 72 q^{25} + 80 q^{29} + 128 q^{37} + 112 q^{43} + 96 q^{51} - 144 q^{53} - 192 q^{57} + 240 q^{65} + 64 q^{67} - 224 q^{71} + 432 q^{79} + 72 q^{81} - 96 q^{85} - 96 q^{93} + 272 q^{95}+O(q^{100})$$ 8 * q - 24 * q^9 + 16 * q^23 + 72 * q^25 + 80 * q^29 + 128 * q^37 + 112 * q^43 + 96 * q^51 - 144 * q^53 - 192 * q^57 + 240 * q^65 + 64 * q^67 - 224 * q^71 + 432 * q^79 + 72 * q^81 - 96 * q^85 - 96 * q^93 + 272 * q^95

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 20 ) / 14$$ (v^6 + 20) / 14 $$\beta_{2}$$ $$=$$ $$( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 9 ) / 7$$ (-2*v^6 + 7*v^4 - 28*v^2 + 9) / 7 $$\beta_{3}$$ $$=$$ $$( -8\nu^{7} + 35\nu^{5} - 126\nu^{3} + 134\nu ) / 14$$ (-8*v^7 + 35*v^5 - 126*v^3 + 134*v) / 14 $$\beta_{4}$$ $$=$$ $$( 10\nu^{7} - 35\nu^{5} + 126\nu^{3} - 10\nu ) / 14$$ (10*v^7 - 35*v^5 + 126*v^3 - 10*v) / 14 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 ) / 2$$ (-v^6 + 4*v^4 - 12*v^2 + 4) / 2 $$\beta_{6}$$ $$=$$ $$( 5\nu^{7} - 28\nu^{5} + 91\nu^{3} - 96\nu ) / 7$$ (5*v^7 - 28*v^5 + 91*v^3 - 96*v) / 7 $$\beta_{7}$$ $$=$$ $$( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7$$ (8*v^7 - 28*v^5 + 91*v^3 - 8*v) / 7
 $$\nu$$ $$=$$ $$( -\beta_{7} + \beta_{6} + 3\beta_{4} + 3\beta_{3} ) / 14$$ (-b7 + b6 + 3*b4 + 3*b3) / 14 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 2\beta_{2} - \beta _1 + 2 ) / 2$$ (b5 - 2*b2 - b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{7} + 8\beta_{4} ) / 7$$ (-5*b7 + 8*b4) / 7 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{2} + 2\beta _1 - 3$$ 2*b5 - 3*b2 + 2*b1 - 3 $$\nu^{5}$$ $$=$$ $$( -9\beta_{7} - 9\beta_{6} + 13\beta_{4} - 13\beta_{3} ) / 7$$ (-9*b7 - 9*b6 + 13*b4 - 13*b3) / 7 $$\nu^{6}$$ $$=$$ $$14\beta _1 - 20$$ 14*b1 - 20 $$\nu^{7}$$ $$=$$ $$( 31\beta_{7} - 31\beta_{6} - 44\beta_{4} - 44\beta_{3} ) / 7$$ (31*b7 - 31*b6 - 44*b4 - 44*b3) / 7

## Character values

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

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 1.60021 − 0.923880i 0.662827 − 0.382683i −1.60021 + 0.923880i −0.662827 + 0.382683i −0.662827 − 0.382683i −1.60021 − 0.923880i 0.662827 + 0.382683i 1.60021 + 0.923880i
0 1.73205i 0 5.37964i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 0.929003i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 0.480662i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 5.82798i 0 0 0 −3.00000 0
97.5 0 1.73205i 0 5.82798i 0 0 0 −3.00000 0
97.6 0 1.73205i 0 0.480662i 0 0 0 −3.00000 0
97.7 0 1.73205i 0 0.929003i 0 0 0 −3.00000 0
97.8 0 1.73205i 0 5.37964i 0 0 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.f.j 8
4.b odd 2 1 588.3.d.c 8
7.b odd 2 1 inner 2352.3.f.j 8
12.b even 2 1 1764.3.d.h 8
28.d even 2 1 588.3.d.c 8
28.f even 6 1 588.3.m.e 8
28.f even 6 1 588.3.m.f 8
28.g odd 6 1 588.3.m.e 8
28.g odd 6 1 588.3.m.f 8
84.h odd 2 1 1764.3.d.h 8
84.j odd 6 1 1764.3.z.l 8
84.j odd 6 1 1764.3.z.m 8
84.n even 6 1 1764.3.z.l 8
84.n even 6 1 1764.3.z.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c 8 4.b odd 2 1
588.3.d.c 8 28.d even 2 1
588.3.m.e 8 28.f even 6 1
588.3.m.e 8 28.g odd 6 1
588.3.m.f 8 28.f even 6 1
588.3.m.f 8 28.g odd 6 1
1764.3.d.h 8 12.b even 2 1
1764.3.d.h 8 84.h odd 2 1
1764.3.z.l 8 84.j odd 6 1
1764.3.z.l 8 84.n even 6 1
1764.3.z.m 8 84.j odd 6 1
1764.3.z.m 8 84.n even 6 1
2352.3.f.j 8 1.a even 1 1 trivial
2352.3.f.j 8 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{8} + 64T_{5}^{6} + 1052T_{5}^{4} + 1088T_{5}^{2} + 196$$ T5^8 + 64*T5^6 + 1052*T5^4 + 1088*T5^2 + 196 $$T_{11}^{4} - 124T_{11}^{2} + 48T_{11} + 3292$$ T11^4 - 124*T11^2 + 48*T11 + 3292

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 3)^{4}$$
$5$ $$T^{8} + 64 T^{6} + 1052 T^{4} + \cdots + 196$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 124 T^{2} + 48 T + 3292)^{2}$$
$13$ $$T^{8} + 712 T^{6} + 123284 T^{4} + \cdots + 2979076$$
$17$ $$T^{8} + 768 T^{6} + \cdots + 168428484$$
$19$ $$T^{8} + 1136 T^{6} + \cdots + 285745216$$
$23$ $$(T^{4} - 8 T^{3} - 612 T^{2} + 9856 T - 37988)^{2}$$
$29$ $$(T^{4} - 40 T^{3} - 1924 T^{2} + \cdots + 468892)^{2}$$
$31$ $$T^{8} + 6448 T^{6} + \cdots + 2052452416$$
$37$ $$(T^{4} - 64 T^{3} - 3492 T^{2} + \cdots - 546236)^{2}$$
$41$ $$T^{8} + 9808 T^{6} + \cdots + 26697826996036$$
$43$ $$(T^{4} - 56 T^{3} - 3784 T^{2} + \cdots + 192784)^{2}$$
$47$ $$T^{8} + 11488 T^{6} + \cdots + 8881401308224$$
$53$ $$(T^{4} + 72 T^{3} - 5464 T^{2} + \cdots - 8464112)^{2}$$
$59$ $$T^{8} + \cdots + 372481662446656$$
$61$ $$T^{8} + 13192 T^{6} + \cdots + 454276$$
$67$ $$(T^{4} - 32 T^{3} - 6048 T^{2} + \cdots - 553856)^{2}$$
$71$ $$(T^{4} + 112 T^{3} - 6460 T^{2} + \cdots - 7722596)^{2}$$
$73$ $$T^{8} + 22472 T^{6} + \cdots + 3712242251524$$
$79$ $$(T^{4} - 216 T^{3} + 12440 T^{2} + \cdots - 7050224)^{2}$$
$83$ $$T^{8} + 19712 T^{6} + \cdots + 61585579131904$$
$89$ $$T^{8} + 41600 T^{6} + \cdots + 17\!\cdots\!76$$
$97$ $$T^{8} + 13640 T^{6} + \cdots + 5315948141956$$