# Properties

 Label 2520.2.bi.r Level $2520$ Weight $2$ Character orbit 2520.bi Analytic conductor $20.122$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2520.bi (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.1223013094$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 29 x^{8} + 247 x^{6} + 855 x^{4} + 1212 x^{2} + 588$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + \beta_{6} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + \beta_{6} q^{7} + \beta_{3} q^{11} + ( \beta_{4} - \beta_{6} + \beta_{9} ) q^{13} + ( -\beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{17} + ( 1 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{8} ) q^{19} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{23} + ( -1 - \beta_{1} ) q^{25} + ( -\beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} ) q^{31} + ( -\beta_{6} + \beta_{8} ) q^{35} + ( 2 \beta_{1} + \beta_{8} + \beta_{9} ) q^{37} + ( 2 - \beta_{6} - \beta_{7} + \beta_{9} ) q^{41} + ( \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{8} + 3 \beta_{9} ) q^{43} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{47} + ( -3 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{49} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{53} + ( -\beta_{6} + \beta_{7} + \beta_{9} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} - \beta_{9} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{8} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} ) q^{65} + ( -2 - 2 \beta_{1} + 3 \beta_{5} - 3 \beta_{8} + 3 \beta_{9} ) q^{67} + ( -4 + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{71} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{73} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + 5 \beta_{9} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{79} + ( 4 + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{83} + ( \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{85} + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{89} + ( -3 - 2 \beta_{1} - 3 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{91} + ( \beta_{2} + \beta_{5} - \beta_{6} ) q^{95} + ( 6 - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 5q^{5} - q^{7} + O(q^{10})$$ $$10q - 5q^{5} - q^{7} - 2q^{11} + 6q^{13} + 2q^{17} + q^{19} + 8q^{23} - 5q^{25} + 7q^{31} - q^{35} - 11q^{37} + 20q^{41} + 6q^{43} - 23q^{49} - 14q^{53} + 4q^{55} + 4q^{59} - 6q^{61} - 3q^{65} - 7q^{67} - 32q^{71} + 3q^{73} + 8q^{77} - 19q^{79} + 28q^{83} - 4q^{85} - 18q^{89} - 21q^{91} + q^{95} + 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 29 x^{8} + 247 x^{6} + 855 x^{4} + 1212 x^{2} + 588$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$19 \nu^{9} + 467 \nu^{7} + 2607 \nu^{5} + 4121 \nu^{3} + 642 \nu - 812$$$$)/1624$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{9} - 28 \nu^{8} - 171 \nu^{7} - 763 \nu^{6} + 316 \nu^{5} - 5530 \nu^{4} + 4701 \nu^{3} - 13349 \nu^{2} + 3252 \nu - 8190$$$$)/812$$ $$\beta_{3}$$ $$=$$ $$($$$$-19 \nu^{9} + 84 \nu^{8} - 467 \nu^{7} + 2086 \nu^{6} - 2607 \nu^{5} + 12124 \nu^{4} - 4121 \nu^{3} + 22386 \nu^{2} + 1794 \nu + 10360$$$$)/1624$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{8} - 89 \nu^{6} - 781 \nu^{4} - 2583 \nu^{2} - 2400$$$$)/58$$ $$\beta_{5}$$ $$=$$ $$($$$$-53 \nu^{9} - 14 \nu^{8} - 1495 \nu^{7} - 686 \nu^{6} - 11845 \nu^{5} - 9870 \nu^{4} - 34381 \nu^{3} - 39662 \nu^{2} - 28886 \nu - 35560$$$$)/1624$$ $$\beta_{6}$$ $$=$$ $$($$$$-32 \nu^{9} - 49 \nu^{8} - 872 \nu^{7} - 1386 \nu^{6} - 6378 \nu^{5} - 10997 \nu^{4} - 16300 \nu^{3} - 31430 \nu^{2} - 11680 \nu - 25396$$$$)/812$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{8} - 347 \nu^{6} - 2437 \nu^{4} - 6089 \nu^{2} - 4368$$$$)/58$$ $$\beta_{8}$$ $$=$$ $$($$$$53 \nu^{9} - 14 \nu^{8} + 1495 \nu^{7} - 686 \nu^{6} + 11845 \nu^{5} - 9870 \nu^{4} + 34381 \nu^{3} - 39662 \nu^{2} + 28886 \nu - 35560$$$$)/1624$$ $$\beta_{9}$$ $$=$$ $$($$$$-32 \nu^{9} + 49 \nu^{8} - 872 \nu^{7} + 1386 \nu^{6} - 6378 \nu^{5} + 10997 \nu^{4} - 16300 \nu^{3} + 31430 \nu^{2} - 11680 \nu + 25396$$$$)/812$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{3} + 2 \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 6$$ $$\nu^{3}$$ $$=$$ $$($$$$-19 \beta_{9} - 3 \beta_{8} - 10 \beta_{7} + 7 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 20 \beta_{3} - 6 \beta_{2} - 50 \beta_{1} - 22$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-40 \beta_{9} - 18 \beta_{8} - 19 \beta_{7} + 40 \beta_{6} - 18 \beta_{5} - 5 \beta_{4} + 76$$ $$\nu^{5}$$ $$=$$ $$($$$$337 \beta_{9} + 78 \beta_{8} + 148 \beta_{7} - 67 \beta_{6} - 78 \beta_{5} + 54 \beta_{4} + 296 \beta_{3} + 108 \beta_{2} + 884 \beta_{1} + 388$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$716 \beta_{9} + 309 \beta_{8} + 335 \beta_{7} - 716 \beta_{6} + 309 \beta_{5} + 116 \beta_{4} - 1226$$ $$\nu^{7}$$ $$=$$ $$($$$$-5893 \beta_{9} - 1485 \beta_{8} - 2470 \beta_{7} + 901 \beta_{6} + 1485 \beta_{5} - 927 \beta_{4} - 4940 \beta_{3} - 1854 \beta_{2} - 15422 \beta_{1} - 6784$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$-12550 \beta_{9} - 5342 \beta_{8} - 5853 \beta_{7} + 12550 \beta_{6} - 5342 \beta_{5} - 2159 \beta_{4} + 20952$$ $$\nu^{9}$$ $$=$$ $$($$$$102691 \beta_{9} + 26448 \beta_{8} + 42538 \beta_{7} - 14437 \beta_{6} - 26448 \beta_{5} + 16026 \beta_{4} + 85076 \beta_{3} + 32052 \beta_{2} + 268796 \beta_{1} + 118372$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{1}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 1.05780i 2.19149i − 2.03852i − 4.17259i 1.22977i − 1.05780i − 2.19149i 2.03852i 4.17259i − 1.22977i
0 0 0 −0.500000 0.866025i 0 −2.62478 0.332488i 0 0 0
361.2 0 0 0 −0.500000 0.866025i 0 −0.780339 2.52806i 0 0 0
361.3 0 0 0 −0.500000 0.866025i 0 0.194868 + 2.63857i 0 0 0
361.4 0 0 0 −0.500000 0.866025i 0 0.835066 2.51051i 0 0 0
361.5 0 0 0 −0.500000 0.866025i 0 1.87518 + 1.86646i 0 0 0
1801.1 0 0 0 −0.500000 + 0.866025i 0 −2.62478 + 0.332488i 0 0 0
1801.2 0 0 0 −0.500000 + 0.866025i 0 −0.780339 + 2.52806i 0 0 0
1801.3 0 0 0 −0.500000 + 0.866025i 0 0.194868 2.63857i 0 0 0
1801.4 0 0 0 −0.500000 + 0.866025i 0 0.835066 + 2.51051i 0 0 0
1801.5 0 0 0 −0.500000 + 0.866025i 0 1.87518 1.86646i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1801.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.2.bi.r 10
3.b odd 2 1 2520.2.bi.s yes 10
7.c even 3 1 inner 2520.2.bi.r 10
21.h odd 6 1 2520.2.bi.s yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.2.bi.r 10 1.a even 1 1 trivial
2520.2.bi.r 10 7.c even 3 1 inner
2520.2.bi.s yes 10 3.b odd 2 1
2520.2.bi.s yes 10 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 $$S_{2}^{\mathrm{new}}(2520, [\chi])$$:

 $$T_{11}^{10} + \cdots$$ $$T_{13}^{5} - 3 T_{13}^{4} - 42 T_{13}^{3} + 44 T_{13}^{2} + 525 T_{13} + 471$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10}$$
$5$ $$( 1 + T + T^{2} )^{5}$$
$7$ $$16807 + 2401 T + 4116 T^{2} + 1715 T^{3} + 413 T^{4} + 372 T^{5} + 59 T^{6} + 35 T^{7} + 12 T^{8} + T^{9} + T^{10}$$
$11$ $$448900 + 146060 T + 125244 T^{2} + 29652 T^{3} + 21054 T^{4} + 4554 T^{5} + 1695 T^{6} + 150 T^{7} + 45 T^{8} + 2 T^{9} + T^{10}$$
$13$ $$( 471 + 525 T + 44 T^{2} - 42 T^{3} - 3 T^{4} + T^{5} )^{2}$$
$17$ $$473344 - 732032 T + 1199520 T^{2} + 10704 T^{3} + 80580 T^{4} - 3096 T^{5} + 3756 T^{6} - 60 T^{7} + 72 T^{8} - 2 T^{9} + T^{10}$$
$19$ $$49 + 707 T + 8955 T^{2} + 18846 T^{3} + 37953 T^{4} - 10827 T^{5} + 3921 T^{6} - 294 T^{7} + 63 T^{8} - T^{9} + T^{10}$$
$23$ $$9604 + 59584 T + 410040 T^{2} - 259708 T^{3} + 140384 T^{4} - 29190 T^{5} + 6113 T^{6} - 448 T^{7} + 111 T^{8} - 8 T^{9} + T^{10}$$
$29$ $$( -384 + 960 T - 46 T^{2} - 90 T^{3} + T^{5} )^{2}$$
$31$ $$20611600 + 15508640 T + 10329756 T^{2} + 1924800 T^{3} + 463821 T^{4} + 22569 T^{5} + 8850 T^{6} + 117 T^{7} + 150 T^{8} - 7 T^{9} + T^{10}$$
$37$ $$9 - 171 T + 3519 T^{2} + 5034 T^{3} + 8979 T^{4} - 183 T^{5} + 1303 T^{6} + 356 T^{7} + 105 T^{8} + 11 T^{9} + T^{10}$$
$41$ $$( 498 - 1758 T + 720 T^{2} - 53 T^{3} - 10 T^{4} + T^{5} )^{2}$$
$43$ $$( -14424 + 3756 T + 521 T^{2} - 141 T^{3} - 3 T^{4} + T^{5} )^{2}$$
$47$ $$46656 + 54432 T + 92448 T^{2} + 14184 T^{3} + 45928 T^{4} + 15090 T^{5} + 12069 T^{6} + 268 T^{7} + 111 T^{8} + T^{10}$$
$53$ $$112896 + 274176 T + 544896 T^{2} + 302496 T^{3} + 144912 T^{4} + 17832 T^{5} + 6025 T^{6} + 902 T^{7} + 183 T^{8} + 14 T^{9} + T^{10}$$
$59$ $$8809024 - 2683072 T + 1677936 T^{2} - 248336 T^{3} + 149972 T^{4} - 20676 T^{5} + 7652 T^{6} - 236 T^{7} + 102 T^{8} - 4 T^{9} + T^{10}$$
$61$ $$451584 + 1322496 T + 4483200 T^{2} - 1561152 T^{3} + 1151056 T^{4} + 129600 T^{5} + 31704 T^{6} + 808 T^{7} + 204 T^{8} + 6 T^{9} + T^{10}$$
$67$ $$117852736 + 10378336 T + 16362024 T^{2} + 2786604 T^{3} + 2131533 T^{4} + 269265 T^{5} + 45486 T^{6} + 1509 T^{7} + 240 T^{8} + 7 T^{9} + T^{10}$$
$71$ $$( -216 - 648 T - 486 T^{2} + 10 T^{3} + 16 T^{4} + T^{5} )^{2}$$
$73$ $$1056770064 - 177883776 T + 73145916 T^{2} - 7356312 T^{3} + 2899917 T^{4} - 298701 T^{5} + 49140 T^{6} - 1983 T^{7} + 234 T^{8} - 3 T^{9} + T^{10}$$
$79$ $$1256560704 + 742281120 T + 306085320 T^{2} + 69348900 T^{3} + 12006237 T^{4} + 1227147 T^{5} + 107530 T^{6} + 5095 T^{7} + 486 T^{8} + 19 T^{9} + T^{10}$$
$83$ $$( -186128 + 12376 T + 3382 T^{2} - 236 T^{3} - 14 T^{4} + T^{5} )^{2}$$
$89$ $$331776 + 414720 T + 411264 T^{2} + 182304 T^{3} + 75204 T^{4} + 17532 T^{5} + 5832 T^{6} + 1128 T^{7} + 282 T^{8} + 18 T^{9} + T^{10}$$
$97$ $$( -50496 - 9888 T + 2924 T^{2} - 24 T^{4} + T^{5} )^{2}$$