# Properties

 Label 2240.2.b.g Level $2240$ Weight $2$ Character orbit 2240.b Analytic conductor $17.886$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} - 1822 x^{3} + 1035 x^{2} - 364 x + 61$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{11} q^{3} + \beta_{9} q^{5} - q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{11} q^{3} + \beta_{9} q^{5} - q^{7} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{9} + ( \beta_{3} + \beta_{9} + \beta_{11} ) q^{11} + ( \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{13} -\beta_{1} q^{15} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( -\beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{19} + \beta_{11} q^{21} + ( -\beta_{2} - \beta_{4} + \beta_{10} ) q^{23} - q^{25} + ( 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{11} ) q^{27} + ( -\beta_{3} + \beta_{6} - \beta_{7} - 3 \beta_{9} + \beta_{11} ) q^{29} + ( 1 - \beta_{4} + \beta_{5} + \beta_{10} ) q^{31} + ( 5 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{33} -\beta_{9} q^{35} + ( \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{11} ) q^{37} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{10} ) q^{39} + ( -\beta_{2} + \beta_{4} + \beta_{10} ) q^{41} + ( -\beta_{3} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{43} + ( \beta_{3} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{45} + ( 2 - \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{10} ) q^{47} + q^{49} + ( -\beta_{3} + 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} - \beta_{11} ) q^{51} + ( -2 \beta_{3} - \beta_{7} + 2 \beta_{9} + 2 \beta_{11} ) q^{53} + ( -1 + \beta_{1} + \beta_{4} ) q^{55} + ( 1 + 2 \beta_{1} - \beta_{4} - \beta_{5} - 3 \beta_{10} ) q^{57} + ( 2 \beta_{3} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{59} + ( \beta_{3} + \beta_{6} - \beta_{7} + 3 \beta_{9} + 2 \beta_{11} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{63} + ( 1 + \beta_{1} - \beta_{2} - \beta_{10} ) q^{65} + ( 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( \beta_{3} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{69} + ( -4 - 4 \beta_{1} + \beta_{5} ) q^{71} + ( 2 \beta_{2} + \beta_{5} + 2 \beta_{10} ) q^{73} + \beta_{11} q^{75} + ( -\beta_{3} - \beta_{9} - \beta_{11} ) q^{77} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{10} ) q^{79} + ( 7 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{5} + 2 \beta_{10} ) q^{81} + ( 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{9} ) q^{83} + ( \beta_{8} + \beta_{9} - \beta_{11} ) q^{85} + ( 2 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{5} - \beta_{10} ) q^{87} + ( 2 - \beta_{2} - \beta_{4} + \beta_{10} ) q^{89} + ( -\beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{91} + ( \beta_{3} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{11} ) q^{93} + ( \beta_{2} - \beta_{4} + \beta_{5} + \beta_{10} ) q^{95} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{5} ) q^{97} + ( 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{9} - 8 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 12q^{7} - 20q^{9} + O(q^{10})$$ $$12q - 12q^{7} - 20q^{9} + 16q^{17} - 8q^{23} - 12q^{25} + 8q^{31} + 72q^{33} + 32q^{39} + 32q^{47} + 12q^{49} - 8q^{55} + 8q^{57} + 20q^{63} + 8q^{65} - 48q^{71} + 8q^{73} + 16q^{79} + 92q^{81} + 32q^{87} + 16q^{89} + 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} - 1822 x^{3} + 1035 x^{2} - 364 x + 61$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{4} + 2 \nu^{3} - 9 \nu^{2} + 8 \nu - 9$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} - 7 \nu^{2} + 6 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$($$$$10 \nu^{11} - 55 \nu^{10} + 251 \nu^{9} - 717 \nu^{8} + 1474 \nu^{7} - 2198 \nu^{6} + 1376 \nu^{5} + 657 \nu^{4} - 4333 \nu^{3} + 5139 \nu^{2} - 3168 \nu + 782$$$$)/286$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{8} + 4 \nu^{7} - 18 \nu^{6} + 40 \nu^{5} - 77 \nu^{4} + 92 \nu^{3} - 68 \nu^{2} + 28 \nu - 3$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{8} - 4 \nu^{7} + 20 \nu^{6} - 46 \nu^{5} + 105 \nu^{4} - 138 \nu^{3} + 152 \nu^{2} - 90 \nu + 29$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-34 \nu^{11} + 187 \nu^{10} - 1025 \nu^{9} + 3210 \nu^{8} - 8844 \nu^{7} + 17283 \nu^{6} - 27301 \nu^{5} + 31600 \nu^{4} - 23506 \nu^{3} + 10441 \nu^{2} + 3421 \nu - 2716$$$$)/572$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{11} - 44 \nu^{10} + 258 \nu^{9} - 831 \nu^{8} + 2552 \nu^{7} - 5362 \nu^{6} + 10310 \nu^{5} - 14089 \nu^{4} + 16668 \nu^{3} - 13106 \nu^{2} + 7590 \nu - 1977$$$$)/143$$ $$\beta_{8}$$ $$=$$ $$($$$$40 \nu^{11} - 220 \nu^{10} + 1290 \nu^{9} - 4155 \nu^{8} + 12188 \nu^{7} - 24808 \nu^{6} + 42112 \nu^{5} - 51855 \nu^{4} + 45874 \nu^{3} - 26920 \nu^{2} + 8492 \nu - 1019$$$$)/572$$ $$\beta_{9}$$ $$=$$ $$($$$$56 \nu^{11} - 308 \nu^{10} + 1806 \nu^{9} - 5817 \nu^{8} + 17292 \nu^{7} - 35532 \nu^{6} + 62732 \nu^{5} - 80033 \nu^{4} + 79210 \nu^{3} - 53132 \nu^{2} + 22528 \nu - 4401$$$$)/572$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{10} - 5 \nu^{9} + 28 \nu^{8} - 82 \nu^{7} + 229 \nu^{6} - 421 \nu^{5} + 682 \nu^{4} - 748 \nu^{3} + 639 \nu^{2} - 323 \nu + 82$$$$)/4$$ $$\beta_{11}$$ $$=$$ $$($$$$162 \nu^{11} - 891 \nu^{10} + 5153 \nu^{9} - 16506 \nu^{8} + 48532 \nu^{7} - 99071 \nu^{6} + 173957 \nu^{5} - 221274 \nu^{4} + 220982 \nu^{3} - 150067 \nu^{2} + 65395 \nu - 13186$$$$)/572$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{9} + 2 \beta_{8} + \beta_{7} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{2} - 4 \beta_{1} - 12$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} + 15 \beta_{9} - 9 \beta_{8} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 3 \beta_{2} - 6 \beta_{1} - 19$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{11} + 32 \beta_{9} - 20 \beta_{8} - 9 \beta_{7} + 4 \beta_{6} + 4 \beta_{3} - 12 \beta_{2} + 16 \beta_{1} + 50$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$14 \beta_{11} - 73 \beta_{9} + 35 \beta_{8} + 13 \beta_{7} - 6 \beta_{6} - 10 \beta_{3} - 35 \beta_{2} + 50 \beta_{1} + 157$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$26 \beta_{11} - 150 \beta_{9} + 78 \beta_{8} + 31 \beta_{7} - 14 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 20 \beta_{3} + 24 \beta_{2} - 22 \beta_{1} - 76$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-48 \beta_{11} + 180 \beta_{9} - 64 \beta_{8} - 7 \beta_{7} + 14 \beta_{5} + 28 \beta_{4} + 24 \beta_{3} + 294 \beta_{2} - 336 \beta_{1} - 1104$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-444 \beta_{11} + 2196 \beta_{9} - 1032 \beta_{8} - 339 \beta_{7} + 140 \beta_{6} - 16 \beta_{5} - 48 \beta_{4} + 292 \beta_{3} - 24 \beta_{2} - 64 \beta_{1} - 138$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-142 \beta_{11} + 1063 \beta_{9} - 609 \beta_{8} - 302 \beta_{7} + 150 \beta_{6} - 156 \beta_{5} - 384 \beta_{4} + 130 \beta_{3} - 2025 \beta_{2} + 1950 \beta_{1} + 6701$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$3004 \beta_{11} + 16 \beta_{10} - 13418 \beta_{9} + 5890 \beta_{8} + 1513 \beta_{7} - 516 \beta_{6} - 100 \beta_{5} - 112 \beta_{4} - 1840 \beta_{3} - 1922 \beta_{2} + 2272 \beta_{1} + 7420$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$4226 \beta_{11} + 88 \beta_{10} - 20663 \beta_{9} + 9613 \beta_{8} + 3127 \beta_{7} - 1290 \beta_{6} + 1034 \beta_{5} + 3212 \beta_{4} - 2682 \beta_{3} + 11627 \beta_{2} - 9658 \beta_{1} - 34593$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1121.1
 0.5 + 0.234551i 0.5 − 2.14588i 0.5 − 0.631151i 0.5 + 2.40029i 0.5 − 1.16542i 0.5 + 2.51441i 0.5 − 2.51441i 0.5 + 1.16542i 0.5 − 2.40029i 0.5 + 0.631151i 0.5 + 2.14588i 0.5 − 0.234551i
0 3.32646i 0 1.00000i 0 −1.00000 0 −8.06534 0
1121.2 0 3.13466i 0 1.00000i 0 −1.00000 0 −6.82607 0
1121.3 0 2.11677i 0 1.00000i 0 −1.00000 0 −1.48073 0
1121.4 0 1.47713i 0 1.00000i 0 −1.00000 0 0.818075 0
1121.5 0 0.639640i 0 1.00000i 0 −1.00000 0 2.59086 0
1121.6 0 0.191804i 0 1.00000i 0 −1.00000 0 2.96321 0
1121.7 0 0.191804i 0 1.00000i 0 −1.00000 0 2.96321 0
1121.8 0 0.639640i 0 1.00000i 0 −1.00000 0 2.59086 0
1121.9 0 1.47713i 0 1.00000i 0 −1.00000 0 0.818075 0
1121.10 0 2.11677i 0 1.00000i 0 −1.00000 0 −1.48073 0
1121.11 0 3.13466i 0 1.00000i 0 −1.00000 0 −6.82607 0
1121.12 0 3.32646i 0 1.00000i 0 −1.00000 0 −8.06534 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1121.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.b.g 12
4.b odd 2 1 2240.2.b.h yes 12
8.b even 2 1 inner 2240.2.b.g 12
8.d odd 2 1 2240.2.b.h yes 12
16.e even 4 1 8960.2.a.cc 6
16.e even 4 1 8960.2.a.ch 6
16.f odd 4 1 8960.2.a.cb 6
16.f odd 4 1 8960.2.a.ce 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.g 12 1.a even 1 1 trivial
2240.2.b.g 12 8.b even 2 1 inner
2240.2.b.h yes 12 4.b odd 2 1
2240.2.b.h yes 12 8.d odd 2 1
8960.2.a.cb 6 16.f odd 4 1
8960.2.a.cc 6 16.e even 4 1
8960.2.a.ce 6 16.f odd 4 1
8960.2.a.ch 6 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{12} + 28 T_{3}^{10} + 270 T_{3}^{8} + 1044 T_{3}^{6} + 1481 T_{3}^{4} + 488 T_{3}^{2} + 16$$ $$T_{23}^{6} + 4 T_{23}^{5} - 80 T_{23}^{4} - 160 T_{23}^{3} + 1504 T_{23}^{2} - 2304 T_{23} + 1024$$ $$T_{31}^{6} - 4 T_{31}^{5} - 80 T_{31}^{4} + 400 T_{31}^{3} + 784 T_{31}^{2} - 4608 T_{31} + 1024$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$16 + 488 T^{2} + 1481 T^{4} + 1044 T^{6} + 270 T^{8} + 28 T^{10} + T^{12}$$
$5$ $$( 1 + T^{2} )^{6}$$
$7$ $$( 1 + T )^{12}$$
$11$ $$7311616 + 3845088 T^{2} + 772417 T^{4} + 76100 T^{6} + 3910 T^{8} + 100 T^{10} + T^{12}$$
$13$ $$595984 + 2485096 T^{2} + 769929 T^{4} + 90252 T^{6} + 4814 T^{8} + 116 T^{10} + T^{12}$$
$17$ $$( 88 - 436 T + 453 T^{2} + 228 T^{3} - 34 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$19$ $$34668544 + 45056000 T^{2} + 8754176 T^{4} + 568320 T^{6} + 16128 T^{8} + 208 T^{10} + T^{12}$$
$23$ $$( 1024 - 2304 T + 1504 T^{2} - 160 T^{3} - 80 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$29$ $$891136 + 33067936 T^{2} + 9077569 T^{4} + 781724 T^{6} + 22150 T^{8} + 252 T^{10} + T^{12}$$
$31$ $$( 1024 - 4608 T + 784 T^{2} + 400 T^{3} - 80 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$37$ $$3421782016 + 612130816 T^{2} + 42692608 T^{4} + 1494272 T^{6} + 27856 T^{8} + 264 T^{10} + T^{12}$$
$41$ $$( -32192 + 3584 T + 4048 T^{2} - 224 T^{3} - 140 T^{4} + T^{6} )^{2}$$
$43$ $$31719424 + 4407066624 T^{2} + 263734272 T^{4} + 6268928 T^{6} + 73920 T^{8} + 432 T^{10} + T^{12}$$
$47$ $$( 50272 - 27656 T - 1243 T^{2} + 1776 T^{3} - 78 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$53$ $$5858983936 + 1129021440 T^{2} + 82769920 T^{4} + 2898944 T^{6} + 49552 T^{8} + 376 T^{10} + T^{12}$$
$59$ $$6461587456 + 7657881600 T^{2} + 454045696 T^{4} + 10342400 T^{6} + 110272 T^{8} + 544 T^{10} + T^{12}$$
$61$ $$28217344 + 126539776 T^{2} + 31433472 T^{4} + 1933056 T^{6} + 40880 T^{8} + 344 T^{10} + T^{12}$$
$67$ $$15352201216 + 4959371264 T^{2} + 352391168 T^{4} + 9252864 T^{6} + 106368 T^{8} + 544 T^{10} + T^{12}$$
$71$ $$( 29248 - 1664 T - 23504 T^{2} - 3904 T^{3} - 20 T^{4} + 24 T^{5} + T^{6} )^{2}$$
$73$ $$( 22528 - 52224 T + 8704 T^{2} + 1024 T^{3} - 200 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$79$ $$( -13628 - 13816 T + 2921 T^{2} + 864 T^{3} - 138 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$83$ $$184913760256 + 37272811520 T^{2} + 1563770624 T^{4} + 26587392 T^{6} + 209904 T^{8} + 760 T^{10} + T^{12}$$
$89$ $$( 11584 - 7552 T + 464 T^{2} + 480 T^{3} - 60 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$97$ $$( 2776 + 5580 T - 7115 T^{2} + 1756 T^{3} - 50 T^{4} - 16 T^{5} + T^{6} )^{2}$$