# Properties

 Label 3456.2.i.f Level $3456$ Weight $2$ Character orbit 3456.i Analytic conductor $27.596$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3456 = 2^{7} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3456.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5962989386$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.8528759163648.1 Defining polynomial: $$x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243$$ x^10 - 2*x^9 + x^8 + 9*x^6 - 36*x^5 + 27*x^4 + 27*x^2 - 162*x + 243 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{9} - \beta_{8} + \beta_{2}) q^{5} + (\beta_{9} + \beta_1) q^{7}+O(q^{10})$$ q + (-b9 - b8 + b2) * q^5 + (b9 + b1) * q^7 $$q + ( - \beta_{9} - \beta_{8} + \beta_{2}) q^{5} + (\beta_{9} + \beta_1) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3}) q^{11} + ( - \beta_{4} - \beta_1 - 1) q^{13} + (\beta_{7} - \beta_{6} - \beta_{5} + 1) q^{17} + (\beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{19} + ( - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{23} + ( - \beta_{8} - \beta_{7}) q^{25} + ( - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{29} + (3 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{35} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} + 3) q^{37} + (\beta_{9} - 3 \beta_{8} - \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{41} + (2 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + \cdots + 2 \beta_1) q^{43}+ \cdots + ( - \beta_{9} - 5 \beta_{8} + 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_1) q^{97}+O(q^{100})$$ q + (-b9 - b8 + b2) * q^5 + (b9 + b1) * q^7 + (-b7 + b6 - b4 - b3) * q^11 + (-b4 - b1 - 1) * q^13 + (b7 - b6 - b5 + 1) * q^17 + (b7 - b5 - b3 + b2 + 2) * q^19 + (-b9 - b8 + b5 + b4 + b2 - b1 - 1) * q^23 + (-b8 - b7) * q^25 + (-2*b9 - 2*b8 - 2*b7 + b6 - b4 - b3 + b1) * q^29 + (3*b9 + b8 + b5 - b4 - 2*b3 - b2 - b1 - 1) * q^31 + (-b7 + 2*b6 + b5 - 2*b3 - 2*b2 + 2) * q^35 + (2*b7 - 3*b6 - 2*b5 + b3 + b2 + 3) * q^37 + (b9 - 3*b8 - b4 - 4*b3 + 3*b2 + 2*b1 + 2) * q^41 + (2*b9 + 3*b8 + 2*b7 - 2*b6 + 2*b4 + 2*b3 + 2*b1) * q^43 + (b9 - 2*b7 - b1) * q^47 + (-3*b9 - 2*b8 - b5 - 3*b4 + b3 + 2*b2 + b1 + 1) * q^49 + (-3*b6 + b3 + 3*b2 + 1) * q^53 + (b7 - b6 - b5 + 3*b3 - b2 + 1) * q^55 + (-4*b9 - b8 - 2*b5 + 3*b3 + b2 + 2*b1 + 2) * q^59 + (2*b9 - 2*b8 + 2*b7 + b6 - b4 - b3 + 3*b1) * q^61 + (3*b9 + 2*b8 + b7 - b6 + b4 + b3) * q^65 + (2*b8 - b5 + b4 + 2*b3 - 2*b2 - 4*b1 - 4) * q^67 + (-b7 - 3*b6 + b5 + 2*b3 - b2) * q^71 + (-b7 - b6 + b5 - 3) * q^73 + (3*b9 + b8 - 2*b3 - b2 + 2*b1 + 2) * q^77 + (-3*b9 - 3*b8 + 3*b7 - b6 + b4 + b3 + b1) * q^79 + (2*b8 + b7 + 2*b1) * q^83 + (-b9 - 3*b8 - b4 - 2*b3 + 3*b2 - 3*b1 - 3) * q^85 + (-5*b6 + 3*b3 + b2 - 3) * q^89 + (b6 - 4*b3 - b2 + 4) * q^91 + (-4*b9 - 6*b8 - 2*b5 - 2*b4 - 2*b3 + 6*b2 + 2*b1 + 2) * q^95 + (-b9 - 5*b8 + 3*b6 - 3*b4 - 3*b3 - 4*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 4 q^{7}+O(q^{10})$$ 10 * q - 4 * q^7 $$10 q - 4 q^{7} + q^{11} - 6 q^{13} + 6 q^{17} + 18 q^{19} - 4 q^{23} + q^{25} - 4 q^{29} - 8 q^{31} + 24 q^{35} + 20 q^{37} + 5 q^{41} - 13 q^{43} + 6 q^{47} + 3 q^{49} + 12 q^{55} + 13 q^{59} - 10 q^{61} - 17 q^{67} - 8 q^{71} - 34 q^{73} + 8 q^{77} - 6 q^{79} - 12 q^{83} - 18 q^{85} - 44 q^{89} + 36 q^{91} + 6 q^{95} + 27 q^{97}+O(q^{100})$$ 10 * q - 4 * q^7 + q^11 - 6 * q^13 + 6 * q^17 + 18 * q^19 - 4 * q^23 + q^25 - 4 * q^29 - 8 * q^31 + 24 * q^35 + 20 * q^37 + 5 * q^41 - 13 * q^43 + 6 * q^47 + 3 * q^49 + 12 * q^55 + 13 * q^59 - 10 * q^61 - 17 * q^67 - 8 * q^71 - 34 * q^73 + 8 * q^77 - 6 * q^79 - 12 * q^83 - 18 * q^85 - 44 * q^89 + 36 * q^91 + 6 * q^95 + 27 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{9} + \nu^{8} + 4\nu^{7} + 12\nu^{6} + 45\nu^{5} - 63\nu^{4} + 27\nu - 1053 ) / 486$$ (v^9 + v^8 + 4*v^7 + 12*v^6 + 45*v^5 - 63*v^4 + 27*v - 1053) / 486 $$\beta_{2}$$ $$=$$ $$( -\nu^{9} - \nu^{8} - 4\nu^{7} + 15\nu^{6} - 18\nu^{5} + 9\nu^{4} + 81\nu^{3} + 162\nu^{2} - 270\nu + 81 ) / 243$$ (-v^9 - v^8 - 4*v^7 + 15*v^6 - 18*v^5 + 9*v^4 + 81*v^3 + 162*v^2 - 270*v + 81) / 243 $$\beta_{3}$$ $$=$$ $$( \nu^{9} - 8\nu^{8} - 5\nu^{7} + 3\nu^{6} + 45\nu^{5} - 36\nu^{4} + 81\nu^{3} + 162\nu^{2} + 270\nu - 324 ) / 243$$ (v^9 - 8*v^8 - 5*v^7 + 3*v^6 + 45*v^5 - 36*v^4 + 81*v^3 + 162*v^2 + 270*v - 324) / 243 $$\beta_{4}$$ $$=$$ $$( -\nu^{9} + 5\nu^{8} + 2\nu^{7} + 12\nu^{6} - 27\nu^{5} + 63\nu^{4} - 54\nu^{3} - 351\nu + 243 ) / 162$$ (-v^9 + 5*v^8 + 2*v^7 + 12*v^6 - 27*v^5 + 63*v^4 - 54*v^3 - 351*v + 243) / 162 $$\beta_{5}$$ $$=$$ $$( 2\nu^{9} + 2\nu^{8} + 8\nu^{7} - 30\nu^{6} + 36\nu^{5} - 18\nu^{4} + 81\nu^{3} - 324\nu^{2} + 297\nu - 162 ) / 243$$ (2*v^9 + 2*v^8 + 8*v^7 - 30*v^6 + 36*v^5 - 18*v^4 + 81*v^3 - 324*v^2 + 297*v - 162) / 243 $$\beta_{6}$$ $$=$$ $$( -4\nu^{9} + 5\nu^{8} - 7\nu^{7} + 15\nu^{6} - 45\nu^{5} + 117\nu^{4} + 162\nu^{2} - 108\nu + 567 ) / 243$$ (-4*v^9 + 5*v^8 - 7*v^7 + 15*v^6 - 45*v^5 + 117*v^4 + 162*v^2 - 108*v + 567) / 243 $$\beta_{7}$$ $$=$$ $$( 4\nu^{9} + 4\nu^{8} - 11\nu^{7} - 6\nu^{6} + 45\nu^{5} - 36\nu^{4} - 81\nu^{3} - 162\nu^{2} + 351\nu - 324 ) / 243$$ (4*v^9 + 4*v^8 - 11*v^7 - 6*v^6 + 45*v^5 - 36*v^4 - 81*v^3 - 162*v^2 + 351*v - 324) / 243 $$\beta_{8}$$ $$=$$ $$( \nu^{9} - \nu^{8} + 2\nu^{7} + 4\nu^{6} + 3\nu^{5} - 27\nu^{4} + 18\nu^{3} - 27\nu - 135 ) / 54$$ (v^9 - v^8 + 2*v^7 + 4*v^6 + 3*v^5 - 27*v^4 + 18*v^3 - 27*v - 135) / 54 $$\beta_{9}$$ $$=$$ $$( -19\nu^{9} - \nu^{8} - 4\nu^{7} + 6\nu^{6} - 153\nu^{5} + 333\nu^{4} + 162\nu^{3} + 162\nu^{2} - 513\nu + 2025 ) / 486$$ (-19*v^9 - v^8 - 4*v^7 + 6*v^6 - 153*v^5 + 333*v^4 + 162*v^3 + 162*v^2 - 513*v + 2025) / 486
 $$\nu$$ $$=$$ $$( \beta_{8} + 2\beta_{6} - \beta_{4} - 2\beta_{2} ) / 3$$ (b8 + 2*b6 - b4 - 2*b2) / 3 $$\nu^{2}$$ $$=$$ $$( -3\beta_{9} - 2\beta_{8} - 3\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} ) / 3$$ (-3*b9 - 2*b8 - 3*b7 + 2*b6 - b4 + b2) / 3 $$\nu^{3}$$ $$=$$ $$( \beta_{8} + 2\beta_{6} + 3\beta_{5} - \beta_{4} + 4\beta_{2} ) / 3$$ (b8 + 2*b6 + 3*b5 - b4 + 4*b2) / 3 $$\nu^{4}$$ $$=$$ $$( \beta_{8} + 2\beta_{6} + 3\beta_{5} + 8\beta_{4} + 9\beta_{3} - 5\beta_{2} - 9\beta _1 - 18 ) / 3$$ (b8 + 2*b6 + 3*b5 + 8*b4 + 9*b3 - 5*b2 - 9*b1 - 18) / 3 $$\nu^{5}$$ $$=$$ $$( -8\beta_{8} - 7\beta_{6} + 3\beta_{5} + 8\beta_{4} + 9\beta_{3} + 4\beta_{2} + 18\beta _1 + 36 ) / 3$$ (-8*b8 - 7*b6 + 3*b5 + 8*b4 + 9*b3 + 4*b2 + 18*b1 + 36) / 3 $$\nu^{6}$$ $$=$$ $$( 18 \beta_{9} + 28 \beta_{8} + 18 \beta_{7} + 11 \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 23 \beta_{2} + 18 \beta _1 + 36 ) / 3$$ (18*b9 + 28*b8 + 18*b7 + 11*b6 - 6*b5 + 8*b4 + 9*b3 - 23*b2 + 18*b1 + 36) / 3 $$\nu^{7}$$ $$=$$ $$( - 18 \beta_{9} + 10 \beta_{8} - 45 \beta_{7} + 29 \beta_{6} + 12 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 50 \beta_{2} + 18 \beta _1 + 36 ) / 3$$ (-18*b9 + 10*b8 - 45*b7 + 29*b6 + 12*b5 + 8*b4 + 9*b3 - 50*b2 + 18*b1 + 36) / 3 $$\nu^{8}$$ $$=$$ $$( - 54 \beta_{9} - 35 \beta_{8} - 27 \beta_{7} + 74 \beta_{6} + 30 \beta_{5} - 46 \beta_{4} - 72 \beta_{3} + 58 \beta_{2} + 99 \beta _1 + 117 ) / 3$$ (-54*b9 - 35*b8 - 27*b7 + 74*b6 + 30*b5 - 46*b4 - 72*b3 + 58*b2 + 99*b1 + 117) / 3 $$\nu^{9}$$ $$=$$ $$( - 90 \beta_{9} + 55 \beta_{8} - 9 \beta_{7} + 65 \beta_{6} + 48 \beta_{5} + 89 \beta_{4} + 90 \beta_{3} - 23 \beta_{2} - 306 \beta _1 - 288 ) / 3$$ (-90*b9 + 55*b8 - 9*b7 + 65*b6 + 48*b5 + 89*b4 + 90*b3 - 23*b2 - 306*b1 - 288) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$$.

 $$n$$ $$2053$$ $$2431$$ $$2945$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1153.1
 1.06839 − 1.36328i 0.756905 + 1.55791i −1.41743 + 0.995434i 1.72806 + 0.117480i −1.13593 − 1.30754i 1.06839 + 1.36328i 0.756905 − 1.55791i −1.41743 − 0.995434i 1.72806 − 0.117480i −1.13593 + 1.30754i
0 0 0 −1.34011 2.32114i 0 −2.48656 + 4.30684i 0 0 0
1153.2 0 0 0 −1.07447 1.86104i 0 0.153174 0.265305i 0 0 0
1153.3 0 0 0 0.115851 + 0.200661i 0 −0.230793 + 0.399745i 0 0 0
1153.4 0 0 0 0.705463 + 1.22190i 0 1.17123 2.02864i 0 0 0
1153.5 0 0 0 1.59327 + 2.75962i 0 −0.607060 + 1.05146i 0 0 0
2305.1 0 0 0 −1.34011 + 2.32114i 0 −2.48656 4.30684i 0 0 0
2305.2 0 0 0 −1.07447 + 1.86104i 0 0.153174 + 0.265305i 0 0 0
2305.3 0 0 0 0.115851 0.200661i 0 −0.230793 0.399745i 0 0 0
2305.4 0 0 0 0.705463 1.22190i 0 1.17123 + 2.02864i 0 0 0
2305.5 0 0 0 1.59327 2.75962i 0 −0.607060 1.05146i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2305.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3456.2.i.f 10
3.b odd 2 1 1152.2.i.g yes 10
4.b odd 2 1 3456.2.i.g 10
8.b even 2 1 3456.2.i.e 10
8.d odd 2 1 3456.2.i.h 10
9.c even 3 1 inner 3456.2.i.f 10
9.d odd 6 1 1152.2.i.g yes 10
12.b even 2 1 1152.2.i.f yes 10
24.f even 2 1 1152.2.i.h yes 10
24.h odd 2 1 1152.2.i.e 10
36.f odd 6 1 3456.2.i.g 10
36.h even 6 1 1152.2.i.f yes 10
72.j odd 6 1 1152.2.i.e 10
72.l even 6 1 1152.2.i.h yes 10
72.n even 6 1 3456.2.i.e 10
72.p odd 6 1 3456.2.i.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.e 10 24.h odd 2 1
1152.2.i.e 10 72.j odd 6 1
1152.2.i.f yes 10 12.b even 2 1
1152.2.i.f yes 10 36.h even 6 1
1152.2.i.g yes 10 3.b odd 2 1
1152.2.i.g yes 10 9.d odd 6 1
1152.2.i.h yes 10 24.f even 2 1
1152.2.i.h yes 10 72.l even 6 1
3456.2.i.e 10 8.b even 2 1
3456.2.i.e 10 72.n even 6 1
3456.2.i.f 10 1.a even 1 1 trivial
3456.2.i.f 10 9.c even 3 1 inner
3456.2.i.g 10 4.b odd 2 1
3456.2.i.g 10 36.f odd 6 1
3456.2.i.h 10 8.d odd 2 1
3456.2.i.h 10 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3456, [\chi])$$:

 $$T_{5}^{10} + 12T_{5}^{8} + 4T_{5}^{7} + 117T_{5}^{6} + 18T_{5}^{5} + 328T_{5}^{4} - 198T_{5}^{3} + 717T_{5}^{2} - 162T_{5} + 36$$ T5^10 + 12*T5^8 + 4*T5^7 + 117*T5^6 + 18*T5^5 + 328*T5^4 - 198*T5^3 + 717*T5^2 - 162*T5 + 36 $$T_{7}^{10} + 4T_{7}^{9} + 24T_{7}^{8} + 129T_{7}^{6} + 138T_{7}^{5} + 240T_{7}^{4} + 48T_{7}^{3} + 33T_{7}^{2} - 2T_{7} + 4$$ T7^10 + 4*T7^9 + 24*T7^8 + 129*T7^6 + 138*T7^5 + 240*T7^4 + 48*T7^3 + 33*T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10}$$
$5$ $$T^{10} + 12 T^{8} + 4 T^{7} + 117 T^{6} + \cdots + 36$$
$7$ $$T^{10} + 4 T^{9} + 24 T^{8} + 129 T^{6} + \cdots + 4$$
$11$ $$T^{10} - T^{9} + 21 T^{8} - 80 T^{7} + \cdots + 961$$
$13$ $$T^{10} + 6 T^{9} + 36 T^{8} + \cdots + 2304$$
$17$ $$(T^{5} - 3 T^{4} - 24 T^{3} + 80 T^{2} + \cdots - 108)^{2}$$
$19$ $$(T^{5} - 9 T^{4} + 104 T^{2} - 48 T - 144)^{2}$$
$23$ $$T^{10} + 4 T^{9} + 48 T^{8} + \cdots + 1024$$
$29$ $$T^{10} + 4 T^{9} + 96 T^{8} + \cdots + 34596$$
$31$ $$T^{10} + 8 T^{9} + 132 T^{8} + \cdots + 22810176$$
$37$ $$(T^{5} - 10 T^{4} - 68 T^{3} + 796 T^{2} + \cdots - 344)^{2}$$
$41$ $$T^{10} - 5 T^{9} + 171 T^{8} + \cdots + 319587129$$
$43$ $$T^{10} + 13 T^{9} + 219 T^{8} + \cdots + 720801$$
$47$ $$T^{10} - 6 T^{9} + 120 T^{8} + \cdots + 12194064$$
$53$ $$(T^{5} - 144 T^{3} - 132 T^{2} + \cdots + 9648)^{2}$$
$59$ $$T^{10} - 13 T^{9} + \cdots + 2802325969$$
$61$ $$T^{10} + 10 T^{9} + \cdots + 2249415184$$
$67$ $$T^{10} + 17 T^{9} + 249 T^{8} + \cdots + 63001$$
$71$ $$(T^{5} + 4 T^{4} - 164 T^{3} - 1080 T^{2} + \cdots + 14472)^{2}$$
$73$ $$(T^{5} + 17 T^{4} + 76 T^{3} - 140 T^{2} + \cdots - 2972)^{2}$$
$79$ $$T^{10} + 6 T^{9} + \cdots + 6986619396$$
$83$ $$T^{10} + 12 T^{9} + 162 T^{8} + \cdots + 8088336$$
$89$ $$(T^{5} + 22 T^{4} - 80 T^{3} - 3388 T^{2} + \cdots + 27464)^{2}$$
$97$ $$T^{10} - 27 T^{9} + \cdots + 10710387081$$