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

Downloads

Learn more

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
0.500000 + 0.234551i
0.500000 2.14588i
0.500000 0.631151i
0.500000 + 2.40029i
0.500000 1.16542i
0.500000 + 2.51441i
0.500000 2.51441i
0.500000 + 1.16542i
0.500000 2.40029i
0.500000 + 0.631151i
0.500000 + 2.14588i
0.500000 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:

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

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