# Properties

 Label 2880.2.w.p Level $2880$ Weight $2$ Character orbit 2880.w Analytic conductor $22.997$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2880 = 2^{6} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2880.w (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.9969157821$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 27 x^{8} + 107 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 1440) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{5} + ( -1 + \beta_{2} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q -\beta_{6} q^{5} + ( -1 + \beta_{2} + \beta_{5} ) q^{7} + ( \beta_{6} + \beta_{8} + \beta_{10} ) q^{11} -\beta_{4} q^{13} + ( -\beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{17} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{19} + ( -2 \beta_{10} + 2 \beta_{11} ) q^{23} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{25} + ( \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{29} + ( -4 + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{31} + ( \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{35} + ( 3 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{37} + ( -\beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{43} + ( \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} ) q^{47} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{49} + ( \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{53} + ( 5 + \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{55} + ( -4 \beta_{6} + \beta_{7} - 5 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{59} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{61} + ( 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{65} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{67} + ( -2 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{71} + ( 4 + \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{73} + ( -2 \beta_{10} - 2 \beta_{11} ) q^{77} + ( -2 \beta_{1} - 12 \beta_{2} - \beta_{4} - \beta_{5} ) q^{79} + ( \beta_{6} + \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{85} + ( 3 \beta_{6} + \beta_{7} - 3 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{89} + ( 8 - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{91} + ( \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{95} + ( 4 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 8q^{7} + O(q^{10})$$ $$12q - 8q^{7} + 4q^{13} - 16q^{25} - 32q^{31} + 20q^{37} - 16q^{43} + 72q^{55} + 32q^{67} + 44q^{73} - 16q^{85} + 80q^{91} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 27 x^{8} + 107 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{10} + 38 \nu^{6} + 297 \nu^{2}$$$$)/76$$ $$\beta_{2}$$ $$=$$ $$($$$$-7 \nu^{10} - 190 \nu^{6} - 787 \nu^{2}$$$$)/76$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{8} - 19 \nu^{4} + 7$$$$)/19$$ $$\beta_{4}$$ $$=$$ $$($$$$-14 \nu^{10} + \nu^{8} - 380 \nu^{6} + 38 \nu^{4} - 1498 \nu^{2} + 145$$$$)/76$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{10} - \nu^{8} - 380 \nu^{6} - 38 \nu^{4} - 1498 \nu^{2} - 145$$$$)/76$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{11} + 4 \nu^{9} + 190 \nu^{7} + 114 \nu^{5} + 787 \nu^{3} + 542 \nu$$$$)/76$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{11} + 4 \nu^{9} - 190 \nu^{7} + 114 \nu^{5} - 787 \nu^{3} + 542 \nu$$$$)/76$$ $$\beta_{8}$$ $$=$$ $$($$$$-24 \nu^{11} - 3 \nu^{9} - 646 \nu^{7} - 76 \nu^{5} - 2530 \nu^{3} - 245 \nu$$$$)/76$$ $$\beta_{9}$$ $$=$$ $$($$$$-24 \nu^{11} + 3 \nu^{9} - 646 \nu^{7} + 76 \nu^{5} - 2530 \nu^{3} + 245 \nu$$$$)/76$$ $$\beta_{10}$$ $$=$$ $$($$$$26 \nu^{11} + 703 \nu^{7} + 2801 \nu^{3} + 38 \nu$$$$)/38$$ $$\beta_{11}$$ $$=$$ $$($$$$26 \nu^{11} + 703 \nu^{7} + 2801 \nu^{3} - 38 \nu$$$$)/38$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - 4 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 4 \beta_{7} + 4 \beta_{6}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 8$$ $$\nu^{5}$$ $$=$$ $$($$$$17 \beta_{11} - 17 \beta_{10} - 8 \beta_{9} + 8 \beta_{8} + 6 \beta_{7} + 6 \beta_{6}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-17 \beta_{5} - 17 \beta_{4} + 70 \beta_{2} + 14 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$31 \beta_{11} + 31 \beta_{10} + 45 \beta_{9} + 45 \beta_{8} + 76 \beta_{7} - 76 \beta_{6}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$38 \beta_{5} - 38 \beta_{4} - 38 \beta_{3} + 159$$ $$\nu^{9}$$ $$=$$ $$($$$$-349 \beta_{11} + 349 \beta_{10} + 228 \beta_{9} - 228 \beta_{8} - 152 \beta_{7} - 152 \beta_{6}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$349 \beta_{5} + 349 \beta_{4} - 1472 \beta_{2} - 380 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-729 \beta_{11} - 729 \beta_{10} - 1109 \beta_{9} - 1109 \beta_{8} - 1624 \beta_{7} + 1624 \beta_{6}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$641$$ $$901$$ $$2431$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 −1.53448 − 1.53448i 0.219986 + 0.219986i 1.04736 + 1.04736i −1.04736 − 1.04736i −0.219986 − 0.219986i 1.53448 + 1.53448i −1.53448 + 1.53448i 0.219986 − 0.219986i 1.04736 − 1.04736i −1.04736 + 1.04736i −0.219986 + 0.219986i 1.53448 − 1.53448i
0 0 0 −1.91575 + 1.15322i 0 1.70928 + 1.70928i 0 0 0
2177.2 0 0 0 −1.34577 1.78575i 0 −2.90321 2.90321i 0 0 0
2177.3 0 0 0 −0.137134 2.23186i 0 −0.806063 0.806063i 0 0 0
2177.4 0 0 0 0.137134 + 2.23186i 0 −0.806063 0.806063i 0 0 0
2177.5 0 0 0 1.34577 + 1.78575i 0 −2.90321 2.90321i 0 0 0
2177.6 0 0 0 1.91575 1.15322i 0 1.70928 + 1.70928i 0 0 0
2753.1 0 0 0 −1.91575 1.15322i 0 1.70928 1.70928i 0 0 0
2753.2 0 0 0 −1.34577 + 1.78575i 0 −2.90321 + 2.90321i 0 0 0
2753.3 0 0 0 −0.137134 + 2.23186i 0 −0.806063 + 0.806063i 0 0 0
2753.4 0 0 0 0.137134 2.23186i 0 −0.806063 + 0.806063i 0 0 0
2753.5 0 0 0 1.34577 1.78575i 0 −2.90321 + 2.90321i 0 0 0
2753.6 0 0 0 1.91575 + 1.15322i 0 1.70928 1.70928i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2753.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.w.p 12
3.b odd 2 1 inner 2880.2.w.p 12
4.b odd 2 1 2880.2.w.q 12
5.c odd 4 1 inner 2880.2.w.p 12
8.b even 2 1 1440.2.w.f 12
8.d odd 2 1 1440.2.w.g yes 12
12.b even 2 1 2880.2.w.q 12
15.e even 4 1 inner 2880.2.w.p 12
20.e even 4 1 2880.2.w.q 12
24.f even 2 1 1440.2.w.g yes 12
24.h odd 2 1 1440.2.w.f 12
40.i odd 4 1 1440.2.w.f 12
40.k even 4 1 1440.2.w.g yes 12
60.l odd 4 1 2880.2.w.q 12
120.q odd 4 1 1440.2.w.g yes 12
120.w even 4 1 1440.2.w.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.w.f 12 8.b even 2 1
1440.2.w.f 12 24.h odd 2 1
1440.2.w.f 12 40.i odd 4 1
1440.2.w.f 12 120.w even 4 1
1440.2.w.g yes 12 8.d odd 2 1
1440.2.w.g yes 12 24.f even 2 1
1440.2.w.g yes 12 40.k even 4 1
1440.2.w.g yes 12 120.q odd 4 1
2880.2.w.p 12 1.a even 1 1 trivial
2880.2.w.p 12 3.b odd 2 1 inner
2880.2.w.p 12 5.c odd 4 1 inner
2880.2.w.p 12 15.e even 4 1 inner
2880.2.w.q 12 4.b odd 2 1
2880.2.w.q 12 12.b even 2 1
2880.2.w.q 12 20.e even 4 1
2880.2.w.q 12 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2880, [\chi])$$:

 $$T_{7}^{6} + 4 T_{7}^{5} + 8 T_{7}^{4} - 16 T_{7}^{3} + 64 T_{7}^{2} + 128 T_{7} + 128$$ $$T_{11}^{6} + 40 T_{11}^{4} + 192 T_{11}^{2} + 128$$ $$T_{13}^{6} - 2 T_{13}^{5} + 2 T_{13}^{4} + 16 T_{13}^{3} + 100 T_{13}^{2} - 40 T_{13} + 8$$ $$T_{31}^{3} + 8 T_{31}^{2} - 16 T_{31} - 160$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$15625 + 5000 T^{2} + 1075 T^{4} + 272 T^{6} + 43 T^{8} + 8 T^{10} + T^{12}$$
$7$ $$( 128 + 128 T + 64 T^{2} - 16 T^{3} + 8 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$11$ $$( 128 + 192 T^{2} + 40 T^{4} + T^{6} )^{2}$$
$13$ $$( 8 - 40 T + 100 T^{2} + 16 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$17$ $$65536 + 20480 T^{4} + 640 T^{8} + T^{12}$$
$19$ $$( 1024 + 512 T^{2} + 64 T^{4} + T^{6} )^{2}$$
$23$ $$16777216 + 7012352 T^{4} + 6912 T^{8} + T^{12}$$
$29$ $$( -8 + 44 T^{2} - 54 T^{4} + T^{6} )^{2}$$
$31$ $$( -160 - 16 T + 8 T^{2} + T^{3} )^{4}$$
$37$ $$( 4232 + 4600 T + 2500 T^{2} + 592 T^{3} + 50 T^{4} - 10 T^{5} + T^{6} )^{2}$$
$41$ $$( 8 + 652 T^{2} + 70 T^{4} + T^{6} )^{2}$$
$43$ $$( 8192 - 8192 T + 4096 T^{2} - 640 T^{3} + 32 T^{4} + 8 T^{5} + T^{6} )^{2}$$
$47$ $$741637881856 + 254279680 T^{4} + 28160 T^{8} + T^{12}$$
$53$ $$5473632256 + 9961472 T^{4} + 5632 T^{8} + T^{12}$$
$59$ $$( -2473088 + 57024 T^{2} - 424 T^{4} + T^{6} )^{2}$$
$61$ $$( -416 - 112 T + T^{3} )^{4}$$
$67$ $$( 204800 - 20480 T + 1024 T^{2} - 128 T^{3} + 128 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$71$ $$( 131072 + 16384 T^{2} + 256 T^{4} + T^{6} )^{2}$$
$73$ $$( 8 - 216 T + 2916 T^{2} - 1184 T^{3} + 242 T^{4} - 22 T^{5} + T^{6} )^{2}$$
$79$ $$( 541696 + 45312 T^{2} + 416 T^{4} + T^{6} )^{2}$$
$83$ $$1048576 + 109740032 T^{4} + 22272 T^{8} + T^{12}$$
$89$ $$( -897800 + 31244 T^{2} - 326 T^{4} + T^{6} )^{2}$$
$97$ $$( 1805000 - 262200 T + 19044 T^{2} + 32 T^{3} + 98 T^{4} - 14 T^{5} + T^{6} )^{2}$$