# Properties

 Label 4368.2.h.q Level $4368$ Weight $2$ Character orbit 4368.h Analytic conductor $34.879$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4$$ x^8 + 15*x^6 + 67*x^4 + 77*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 273) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( - \beta_{7} + \beta_{2}) q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (-b7 + b2) * q^5 - b2 * q^7 + q^9 $$q + q^{3} + ( - \beta_{7} + \beta_{2}) q^{5} - \beta_{2} q^{7} + q^{9} + (\beta_{3} + \beta_{2}) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_{2} - 1) q^{13} + ( - \beta_{7} + \beta_{2}) q^{15} + ( - \beta_{5} + \beta_{4} + 2) q^{17} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{19} - \beta_{2} q^{21} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_1) q^{23} + ( - \beta_{6} + \beta_1 - 5) q^{25} + q^{27} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_1 - 2) q^{29} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3}) q^{31} + (\beta_{3} + \beta_{2}) q^{33} + ( - \beta_1 + 1) q^{35} + (2 \beta_{3} + 2 \beta_{2}) q^{37} + ( - \beta_{6} + \beta_{5} + \beta_{2} - 1) q^{39} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 7 \beta_{2}) q^{41} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_1 - 4) q^{43} + ( - \beta_{7} + \beta_{2}) q^{45} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{47} - q^{49} + ( - \beta_{5} + \beta_{4} + 2) q^{51} + ( - 3 \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 - 2) q^{53} + (2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_1 - 2) q^{55} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{57} + (\beta_{5} + \beta_{4} - \beta_{3} + 7 \beta_{2}) q^{59} + (2 \beta_{6} - \beta_{5} + \beta_{4} - 2) q^{61} - \beta_{2} q^{63} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{65} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 6 \beta_{2}) q^{67} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_1) q^{69} + ( - \beta_{3} - \beta_{2}) q^{71} + (\beta_{7} + \beta_{3} + 8 \beta_{2}) q^{73} + ( - \beta_{6} + \beta_1 - 5) q^{75} + (\beta_{6} + 1) q^{77} + ( - \beta_{6} - \beta_1) q^{79} + q^{81} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{83} + ( - 2 \beta_{7} - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3}) q^{85} + (\beta_{6} - \beta_{5} + \beta_{4} - \beta_1 - 2) q^{87} + ( - \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{89} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{91} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3}) q^{93} + (\beta_{6} + 3 \beta_{5} - 3 \beta_{4} - \beta_1 + 12) q^{95} + ( - \beta_{7} + \beta_{3} - 10 \beta_{2}) q^{97} + (\beta_{3} + \beta_{2}) q^{99}+O(q^{100})$$ q + q^3 + (-b7 + b2) * q^5 - b2 * q^7 + q^9 + (b3 + b2) * q^11 + (-b6 + b5 + b2 - 1) * q^13 + (-b7 + b2) * q^15 + (-b5 + b4 + 2) * q^17 + (b7 - b5 - b4 + b3 - 2*b2) * q^19 - b2 * q^21 + (-b6 - b5 + b4 - b1) * q^23 + (-b6 + b1 - 5) * q^25 + q^27 + (b6 - b5 + b4 - b1 - 2) * q^29 + (b7 + b5 + b4 - b3) * q^31 + (b3 + b2) * q^33 + (-b1 + 1) * q^35 + (2*b3 + 2*b2) * q^37 + (-b6 + b5 + b2 - 1) * q^39 + (-b5 - b4 + b3 + 7*b2) * q^41 + (b6 - b5 + b4 - b1 - 4) * q^43 + (-b7 + b2) * q^45 + (-b7 + 2*b5 + 2*b4 - 2*b3 + b2) * q^47 - q^49 + (-b5 + b4 + 2) * q^51 + (-3*b6 + b5 - b4 - b1 - 2) * q^53 + (2*b6 - b5 + b4 + 2*b1 - 2) * q^55 + (b7 - b5 - b4 + b3 - 2*b2) * q^57 + (b5 + b4 - b3 + 7*b2) * q^59 + (2*b6 - b5 + b4 - 2) * q^61 - b2 * q^63 + (2*b7 + b6 - b5 + 3*b4 + b3 - b2 + b1 - 2) * q^65 + (-2*b7 - 2*b5 - 2*b4 - 6*b2) * q^67 + (-b6 - b5 + b4 - b1) * q^69 + (-b3 - b2) * q^71 + (b7 + b3 + 8*b2) * q^73 + (-b6 + b1 - 5) * q^75 + (b6 + 1) * q^77 + (-b6 - b1) * q^79 + q^81 + (b7 - 2*b5 - 2*b4 + 2*b3 - b2) * q^83 + (-2*b7 - 4*b5 - 4*b4 + 2*b3) * q^85 + (b6 - b5 + b4 - b1 - 2) * q^87 + (-b7 - 2*b5 - 2*b4 + 2*b3 - b2) * q^89 + (-b4 + b3 + b2 + 1) * q^91 + (b7 + b5 + b4 - b3) * q^93 + (b6 + 3*b5 - 3*b4 - b1 + 12) * q^95 + (-b7 + b3 - 10*b2) * q^97 + (b3 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{3} + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^3 + 8 * q^9 $$8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100})$$ 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 8\nu^{3} + 9\nu ) / 2$$ (v^5 + 8*v^3 + 9*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 10\nu^{3} + 23\nu ) / 2$$ (v^5 + 10*v^3 + 23*v) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 8\nu^{4} + 9\nu^{2} + 2\nu ) / 2$$ (v^6 + 8*v^4 + 9*v^2 + 2*v) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} - 8\nu^{4} - 9\nu^{2} + 2\nu ) / 2$$ (-v^6 - 8*v^4 - 9*v^2 + 2*v) / 2 $$\beta_{6}$$ $$=$$ $$\nu^{4} + 7\nu^{2} + 3$$ v^4 + 7*v^2 + 3 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 36\nu ) / 2$$ (v^7 + 12*v^5 + 41*v^3 + 36*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 - 4$$ b1 - 4 $$\nu^{3}$$ $$=$$ $$( -7\beta_{5} - 7\beta_{4} + 2\beta_{3} - 2\beta_{2} ) / 2$$ (-7*b5 - 7*b4 + 2*b3 - 2*b2) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{6} - 7\beta _1 + 25$$ b6 - 7*b1 + 25 $$\nu^{5}$$ $$=$$ $$( 47\beta_{5} + 47\beta_{4} - 16\beta_{3} + 20\beta_{2} ) / 2$$ (47*b5 + 47*b4 - 16*b3 + 20*b2) / 2 $$\nu^{6}$$ $$=$$ $$-8\beta_{6} - \beta_{5} + \beta_{4} + 47\beta _1 - 164$$ -8*b6 - b5 + b4 + 47*b1 - 164 $$\nu^{7}$$ $$=$$ $$( 4\beta_{7} - 313\beta_{5} - 313\beta_{4} + 110\beta_{3} - 158\beta_{2} ) / 2$$ (4*b7 - 313*b5 - 313*b4 + 110*b3 - 158*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$$.

 $$n$$ $$1093$$ $$1249$$ $$1457$$ $$2017$$ $$3823$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 2.60520i 2.54814i 0.233455i − 1.29051i 1.29051i − 0.233455i − 2.54814i 2.60520i
0 1.00000 0 3.78706i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 3.49301i 0 1.00000i 0 1.00000 0
337.3 0 1.00000 0 2.94550i 0 1.00000i 0 1.00000 0
337.4 0 1.00000 0 1.33457i 0 1.00000i 0 1.00000 0
337.5 0 1.00000 0 1.33457i 0 1.00000i 0 1.00000 0
337.6 0 1.00000 0 2.94550i 0 1.00000i 0 1.00000 0
337.7 0 1.00000 0 3.49301i 0 1.00000i 0 1.00000 0
337.8 0 1.00000 0 3.78706i 0 1.00000i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.q 8
4.b odd 2 1 273.2.c.c 8
12.b even 2 1 819.2.c.d 8
13.b even 2 1 inner 4368.2.h.q 8
28.d even 2 1 1911.2.c.l 8
52.b odd 2 1 273.2.c.c 8
52.f even 4 1 3549.2.a.v 4
52.f even 4 1 3549.2.a.x 4
156.h even 2 1 819.2.c.d 8
364.h even 2 1 1911.2.c.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.c 8 4.b odd 2 1
273.2.c.c 8 52.b odd 2 1
819.2.c.d 8 12.b even 2 1
819.2.c.d 8 156.h even 2 1
1911.2.c.l 8 28.d even 2 1
1911.2.c.l 8 364.h even 2 1
3549.2.a.v 4 52.f even 4 1
3549.2.a.x 4 52.f even 4 1
4368.2.h.q 8 1.a even 1 1 trivial
4368.2.h.q 8 13.b 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 $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{8} + 37T_{5}^{6} + 468T_{5}^{4} + 2240T_{5}^{2} + 2704$$ T5^8 + 37*T5^6 + 468*T5^4 + 2240*T5^2 + 2704 $$T_{11}^{8} + 44T_{11}^{6} + 576T_{11}^{4} + 2320T_{11}^{2} + 1024$$ T11^8 + 44*T11^6 + 576*T11^4 + 2320*T11^2 + 1024 $$T_{17}^{4} - 10T_{17}^{3} + 8T_{17}^{2} + 112T_{17} - 160$$ T17^4 - 10*T17^3 + 8*T17^2 + 112*T17 - 160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T - 1)^{8}$$
$5$ $$T^{8} + 37 T^{6} + 468 T^{4} + \cdots + 2704$$
$7$ $$(T^{2} + 1)^{4}$$
$11$ $$T^{8} + 44 T^{6} + 576 T^{4} + \cdots + 1024$$
$13$ $$T^{8} + 6 T^{7} - 78 T^{5} + \cdots + 28561$$
$17$ $$(T^{4} - 10 T^{3} + 8 T^{2} + 112 T - 160)^{2}$$
$19$ $$T^{8} + 97 T^{6} + 1984 T^{4} + \cdots + 1024$$
$23$ $$(T^{4} - 3 T^{3} - 78 T^{2} + 140 T + 1352)^{2}$$
$29$ $$(T^{4} + 9 T^{3} - 14 T^{2} - 244 T - 440)^{2}$$
$31$ $$T^{8} + 89 T^{6} + 2032 T^{4} + \cdots + 4096$$
$37$ $$T^{8} + 176 T^{6} + 9216 T^{4} + \cdots + 262144$$
$41$ $$T^{8} + 248 T^{6} + 17376 T^{4} + \cdots + 1784896$$
$43$ $$(T^{4} + 17 T^{3} + 64 T^{2} - 160 T - 896)^{2}$$
$47$ $$T^{8} + 241 T^{6} + 18984 T^{4} + \cdots + 6739216$$
$53$ $$(T^{4} + 5 T^{3} - 142 T^{2} - 356 T + 712)^{2}$$
$59$ $$T^{8} + 248 T^{6} + 18384 T^{4} + \cdots + 1364224$$
$61$ $$(T^{4} + 10 T^{3} - 28 T^{2} - 136 T + 320)^{2}$$
$67$ $$T^{8} + 452 T^{6} + \cdots + 66064384$$
$71$ $$T^{8} + 44 T^{6} + 576 T^{4} + \cdots + 1024$$
$73$ $$T^{8} + 305 T^{6} + 22384 T^{4} + \cdots + 5234944$$
$79$ $$(T^{4} - T^{3} - 32 T^{2} + 104 T - 80)^{2}$$
$83$ $$T^{8} + 241 T^{6} + 18984 T^{4} + \cdots + 6739216$$
$89$ $$T^{8} + 253 T^{6} + 19060 T^{4} + \cdots + 150544$$
$97$ $$T^{8} + 553 T^{6} + \cdots + 82882816$$