# Properties

 Label 2268.2.i.m Level $2268$ Weight $2$ Character orbit 2268.i Analytic conductor $18.110$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.310217769.2 Defining polynomial: $$x^{8} + 4 x^{6} - 2 x^{5} + 15 x^{4} - 4 x^{3} + 5 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{5} q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{5} q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{17} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{23} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{25} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{29} + ( 1 - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{31} + ( -1 + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( -\beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{43} + ( 3 + \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{47} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{49} + ( -\beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( 1 - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{55} + ( 2 - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{59} + ( 4 + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} + ( -3 + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{65} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{67} + ( 1 - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{71} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{73} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{77} + ( 1 + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{79} + ( 2 \beta_{1} + 10 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{83} + ( 5 + \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{85} + ( 6 + 3 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{89} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{91} + ( -5 - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} ) q^{95} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{5} + q^{7} + O(q^{10})$$ $$8 q + 2 q^{5} + q^{7} - 5 q^{11} - 3 q^{13} - 2 q^{17} - 8 q^{19} - 2 q^{23} - 8 q^{25} + 2 q^{29} - 11 q^{35} + 4 q^{37} + 3 q^{41} - 5 q^{43} + 30 q^{47} - 19 q^{49} + 24 q^{53} + 16 q^{55} + 20 q^{59} + 24 q^{61} - 24 q^{65} + 14 q^{67} + 22 q^{71} - 10 q^{73} + 11 q^{77} - 35 q^{83} + 13 q^{85} + 18 q^{89} - 9 q^{91} - 20 q^{95} - 19 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4 x^{6} - 2 x^{5} + 15 x^{4} - 4 x^{3} + 5 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$14 \nu^{7} + 23 \nu^{6} - 92 \nu^{5} - 14 \nu^{4} - 391 \nu^{3} + 437 \nu^{2} - 1586 \nu + 92$$$$)/289$$ $$\beta_{2}$$ $$=$$ $$($$$$64 \nu^{7} - 60 \nu^{6} + 240 \nu^{5} - 353 \nu^{4} + 1020 \nu^{3} - 1140 \nu^{2} + 305 \nu - 240$$$$)/289$$ $$\beta_{3}$$ $$=$$ $$($$$$125 \nu^{7} - 63 \nu^{6} + 541 \nu^{5} - 414 \nu^{4} + 2227 \nu^{3} - 1197 \nu^{2} + 1693 \nu + 326$$$$)/289$$ $$\beta_{4}$$ $$=$$ $$($$$$-237 \nu^{7} - 121 \nu^{6} - 961 \nu^{5} - 52 \nu^{4} - 3434 \nu^{3} - 854 \nu^{2} - 854 \nu - 773$$$$)/289$$ $$\beta_{5}$$ $$=$$ $$($$$$273 \nu^{7} + 15 \nu^{6} + 1096 \nu^{5} - 562 \nu^{4} + 4080 \nu^{3} - 1160 \nu^{2} + 1730 \nu - 229$$$$)/289$$ $$\beta_{6}$$ $$=$$ $$($$$$-344 \nu^{7} - 111 \nu^{6} - 1290 \nu^{5} + 344 \nu^{4} - 4471 \nu^{3} - 86 \nu^{2} - 86 \nu - 444$$$$)/289$$ $$\beta_{7}$$ $$=$$ $$($$$$-412 \nu^{7} + 25 \nu^{6} - 1545 \nu^{5} + 990 \nu^{4} - 5916 \nu^{3} + 1920 \nu^{2} - 970 \nu - 189$$$$)/289$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + \beta_{3} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - \beta_{1} - 6$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 5 \beta_{6} + \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 3$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$9 \beta_{7} + \beta_{5} - 8 \beta_{4} + \beta_{3} + 21 \beta_{2} + 5 \beta_{1}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-6 \beta_{7} - 20 \beta_{6} - 17 \beta_{5} + 17 \beta_{4} + 6 \beta_{3} - 18 \beta_{2} - 20 \beta_{1} - 18$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-7 \beta_{7} + 24 \beta_{6} + 31 \beta_{5} - 7 \beta_{4} - 38 \beta_{3} + 81$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$73 \beta_{7} + 42 \beta_{5} - 31 \beta_{4} + 42 \beta_{3} + 90 \beta_{2} + 80 \beta_{1}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.882007 − 1.52768i −0.198169 + 0.343239i −1.03075 + 1.78531i 0.346911 − 0.600868i 0.882007 + 1.52768i −0.198169 − 0.343239i −1.03075 − 1.78531i 0.346911 + 0.600868i
0 0 0 −1.25300 + 2.17026i 0 1.71031 2.01862i 0 0 0
865.2 0 0 0 −0.705299 + 1.22161i 0 −0.779537 + 2.52830i 0 0 0
865.3 0 0 0 0.951526 1.64809i 0 1.46157 + 2.20541i 0 0 0
865.4 0 0 0 2.00677 3.47583i 0 −1.89234 1.84906i 0 0 0
2053.1 0 0 0 −1.25300 2.17026i 0 1.71031 + 2.01862i 0 0 0
2053.2 0 0 0 −0.705299 1.22161i 0 −0.779537 2.52830i 0 0 0
2053.3 0 0 0 0.951526 + 1.64809i 0 1.46157 2.20541i 0 0 0
2053.4 0 0 0 2.00677 + 3.47583i 0 −1.89234 + 1.84906i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2053.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.m 8
3.b odd 2 1 2268.2.i.l 8
7.c even 3 1 2268.2.l.l 8
9.c even 3 1 2268.2.k.d yes 8
9.c even 3 1 2268.2.l.l 8
9.d odd 6 1 2268.2.k.c 8
9.d odd 6 1 2268.2.l.m 8
21.h odd 6 1 2268.2.l.m 8
63.g even 3 1 2268.2.k.d yes 8
63.h even 3 1 inner 2268.2.i.m 8
63.j odd 6 1 2268.2.i.l 8
63.n odd 6 1 2268.2.k.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.l 8 3.b odd 2 1
2268.2.i.l 8 63.j odd 6 1
2268.2.i.m 8 1.a even 1 1 trivial
2268.2.i.m 8 63.h even 3 1 inner
2268.2.k.c 8 9.d odd 6 1
2268.2.k.c 8 63.n odd 6 1
2268.2.k.d yes 8 9.c even 3 1
2268.2.k.d yes 8 63.g even 3 1
2268.2.l.l 8 7.c even 3 1
2268.2.l.l 8 9.c even 3 1
2268.2.l.m 8 9.d odd 6 1
2268.2.l.m 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{8} - 2 T_{5}^{7} + 16 T_{5}^{6} + 6 T_{5}^{5} + 135 T_{5}^{4} + 405 T_{5}^{2} + 243 T_{5} + 729$$ $$T_{13}^{8} + \cdots$$ $$T_{19}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$729 + 243 T + 405 T^{2} + 135 T^{4} + 6 T^{5} + 16 T^{6} - 2 T^{7} + T^{8}$$
$7$ $$2401 - 343 T + 490 T^{2} - 35 T^{3} + 101 T^{4} - 5 T^{5} + 10 T^{6} - T^{7} + T^{8}$$
$11$ $$729 + 972 T + 1134 T^{2} + 486 T^{3} + 243 T^{4} + 42 T^{5} + 31 T^{6} + 5 T^{7} + T^{8}$$
$13$ $$9 + 48 T + 220 T^{2} + 210 T^{3} + 195 T^{4} - 4 T^{5} + 21 T^{6} + 3 T^{7} + T^{8}$$
$17$ $$35721 - 3402 T + 6561 T^{2} - 162 T^{3} + 936 T^{4} - 30 T^{5} + 37 T^{6} + 2 T^{7} + T^{8}$$
$19$ $$97969 - 66043 T + 53911 T^{2} + 1322 T^{3} + 2275 T^{4} + 182 T^{5} + 94 T^{6} + 8 T^{7} + T^{8}$$
$23$ $$35721 + 39123 T + 32643 T^{2} + 11934 T^{3} + 3519 T^{4} + 306 T^{5} + 58 T^{6} + 2 T^{7} + T^{8}$$
$29$ $$729 + 486 T + 1944 T^{2} - 972 T^{3} + 3609 T^{4} + 84 T^{5} + 64 T^{6} - 2 T^{7} + T^{8}$$
$31$ $$( -423 + 340 T - 75 T^{2} + T^{4} )^{2}$$
$37$ $$361 + 1805 T + 9595 T^{2} - 2698 T^{3} + 1261 T^{4} - 70 T^{5} + 46 T^{6} - 4 T^{7} + T^{8}$$
$41$ $$59049 - 65610 T + 55404 T^{2} - 20898 T^{3} + 6237 T^{4} - 324 T^{5} + 81 T^{6} - 3 T^{7} + T^{8}$$
$43$ $$39601 - 18706 T + 14806 T^{2} + 830 T^{3} + 1171 T^{4} + 38 T^{5} + 55 T^{6} + 5 T^{7} + T^{8}$$
$47$ $$( -81 + 81 T + 45 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$53$ $$66928761 - 12590559 T + 2957553 T^{2} - 281880 T^{3} + 50301 T^{4} - 4806 T^{5} + 504 T^{6} - 24 T^{7} + T^{8}$$
$59$ $$( 2403 + 405 T - 90 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$61$ $$( -4497 + 1594 T - 105 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$67$ $$( 12527 + 970 T - 252 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$71$ $$( -7803 + 2295 T - 141 T^{2} - 11 T^{3} + T^{4} )^{2}$$
$73$ $$2411809 - 1156985 T + 694795 T^{2} + 35990 T^{3} + 13997 T^{4} + 590 T^{5} + 190 T^{6} + 10 T^{7} + T^{8}$$
$79$ $$( 729 - 227 T - 120 T^{2} + T^{4} )^{2}$$
$83$ $$729 + 36450 T + 1812132 T^{2} + 516510 T^{3} + 100179 T^{4} + 10740 T^{5} + 841 T^{6} + 35 T^{7} + T^{8}$$
$89$ $$344065401 - 72619335 T + 12990051 T^{2} - 1161054 T^{3} + 104895 T^{4} - 5562 T^{5} + 450 T^{6} - 18 T^{7} + T^{8}$$
$97$ $$21520321 + 8257420 T + 2917894 T^{2} + 272402 T^{3} + 41375 T^{4} + 2534 T^{5} + 415 T^{6} + 19 T^{7} + T^{8}$$
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