Properties

Label 1840.2.e.f
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 24 x^{10} + 188 x^{8} + 530 x^{6} + 508 x^{4} + 80 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 460)
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.

\(f(q)\) \(=\) \( q + ( \beta_{5} + \beta_{6} ) q^{3} -\beta_{8} q^{5} + ( -\beta_{1} - \beta_{6} + \beta_{10} ) q^{7} + ( -2 + \beta_{2} + \beta_{7} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( \beta_{5} + \beta_{6} ) q^{3} -\beta_{8} q^{5} + ( -\beta_{1} - \beta_{6} + \beta_{10} ) q^{7} + ( -2 + \beta_{2} + \beta_{7} + \beta_{10} ) q^{9} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{11} + ( \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{13} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{15} + ( -\beta_{5} + \beta_{11} ) q^{17} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} + ( 1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{21} + \beta_{6} q^{23} + ( 1 - \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{25} + ( 2 \beta_{1} - \beta_{5} - 5 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{27} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{29} + ( -2 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{31} + ( -\beta_{1} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{33} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{35} + ( -\beta_{1} + 5 \beta_{6} + 2 \beta_{7} - \beta_{10} - 2 \beta_{11} ) q^{37} + ( -1 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{8} + \beta_{9} ) q^{39} + ( -2 \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{41} + ( -3 \beta_{1} + \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} ) q^{43} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 5 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{45} + ( -\beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{47} + ( -4 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{49} + ( 2 + \beta_{8} - \beta_{9} ) q^{51} + ( -2 \beta_{1} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{53} + ( -3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{55} + ( \beta_{1} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{57} + ( -3 - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{59} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} ) q^{61} + ( 2 \beta_{1} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 4 \beta_{10} - 3 \beta_{11} ) q^{63} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{65} + ( -\beta_{1} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{10} ) q^{67} + ( -1 + \beta_{2} ) q^{69} + ( 4 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} + 2 \beta_{10} ) q^{71} + ( -\beta_{5} + \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{73} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{75} + ( 2 \beta_{1} - 4 \beta_{5} - 4 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{77} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{79} + ( 1 - 2 \beta_{2} + 4 \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{81} + ( -2 \beta_{1} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{83} + ( 1 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{11} ) q^{85} + ( -\beta_{5} - 11 \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{87} + ( 5 - 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{89} + ( 1 - 7 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{91} + ( 4 \beta_{1} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{93} + ( -1 + 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{95} + ( -\beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{97} + ( -4 + 4 \beta_{2} - 2 \beta_{8} + 2 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 20q^{9} + O(q^{10}) \) \( 12q - 20q^{9} - 4q^{11} - 2q^{15} + 8q^{19} + 8q^{25} - 10q^{29} - 18q^{31} + 10q^{35} - 16q^{39} - 2q^{41} + 2q^{45} - 38q^{49} + 24q^{51} - 16q^{55} - 22q^{59} - 8q^{61} + 38q^{65} - 8q^{69} + 34q^{71} - 16q^{75} + 20q^{79} + 28q^{81} + 6q^{85} + 48q^{89} + 8q^{91} - 12q^{95} - 32q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 24 x^{10} + 188 x^{8} + 530 x^{6} + 508 x^{4} + 80 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{10} - 23 \nu^{8} - 8 \nu^{7} - 157 \nu^{6} - 144 \nu^{5} - 261 \nu^{4} - 752 \nu^{3} + 121 \nu^{2} - 1064 \nu + 159 \)\()/96\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{10} + 69 \nu^{8} + 503 \nu^{6} + 1215 \nu^{4} + 869 \nu^{2} + 83 \)\()/48\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{10} + 169 \nu^{8} + 1331 \nu^{6} + 3779 \nu^{4} + 3689 \nu^{2} + 415 \)\()/96\)
\(\beta_{4}\)\(=\)\((\)\( 9 \nu^{10} + 215 \nu^{8} + 1661 \nu^{6} + 4493 \nu^{4} + 3895 \nu^{2} + 497 \)\()/96\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{11} - 169 \nu^{9} - 1339 \nu^{7} - 3875 \nu^{5} - 3913 \nu^{3} - 663 \nu \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{11} + 24 \nu^{9} + 188 \nu^{7} + 529 \nu^{5} + 496 \nu^{3} + 58 \nu \)\()/6\)
\(\beta_{7}\)\(=\)\((\)\( -27 \nu^{11} - \nu^{10} - 645 \nu^{9} - 23 \nu^{8} - 5007 \nu^{7} - 157 \nu^{6} - 13815 \nu^{5} - 261 \nu^{4} - 12597 \nu^{3} + 121 \nu^{2} - 1419 \nu + 159 \)\()/96\)
\(\beta_{8}\)\(=\)\((\)\( 27 \nu^{11} + \nu^{10} + 645 \nu^{9} + 23 \nu^{8} + 5007 \nu^{7} + 173 \nu^{6} + 13815 \nu^{5} + 453 \nu^{4} + 12597 \nu^{3} + 231 \nu^{2} + 1515 \nu - 143 \)\()/96\)
\(\beta_{9}\)\(=\)\((\)\( 27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 173 \nu^{6} + 13815 \nu^{5} - 453 \nu^{4} + 12597 \nu^{3} - 231 \nu^{2} + 1515 \nu + 143 \)\()/96\)
\(\beta_{10}\)\(=\)\((\)\( 27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 157 \nu^{6} + 13815 \nu^{5} - 261 \nu^{4} + 12597 \nu^{3} + 121 \nu^{2} + 1419 \nu + 159 \)\()/96\)
\(\beta_{11}\)\(=\)\((\)\( -37 \nu^{11} - 891 \nu^{9} - 7017 \nu^{7} - 19993 \nu^{5} - 19547 \nu^{3} - 3245 \nu \)\()/96\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + 4 \beta_{10} - 4 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{10} - 7 \beta_{9} + 7 \beta_{8} - 9 \beta_{7} - 22 \beta_{4} + 22 \beta_{3} + 2 \beta_{2} + 66\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(22 \beta_{11} - 73 \beta_{10} + 75 \beta_{9} + 75 \beta_{8} + 51 \beta_{7} - 56 \beta_{6} - 72 \beta_{5} + 22 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(46 \beta_{10} + 28 \beta_{9} - 28 \beta_{8} + 46 \beta_{7} + 110 \beta_{4} - 110 \beta_{3} - 12 \beta_{2} - 309\)
\(\nu^{7}\)\(=\)\((\)\(-208 \beta_{11} + 707 \beta_{10} - 731 \beta_{9} - 731 \beta_{8} - 475 \beta_{7} + 632 \beta_{6} + 732 \beta_{5} - 232 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-955 \beta_{10} - 483 \beta_{9} + 483 \beta_{8} - 955 \beta_{7} - 2146 \beta_{4} + 2170 \beta_{3} + 208 \beta_{2} + 5972\)\()/2\)
\(\nu^{9}\)\(=\)\(941 \beta_{11} - 3488 \beta_{10} + 3592 \beta_{9} + 3592 \beta_{8} + 2271 \beta_{7} - 3376 \beta_{6} - 3597 \beta_{5} + 1217 \beta_{1}\)
\(\nu^{10}\)\(=\)\((\)\(9895 \beta_{10} + 4265 \beta_{9} - 4265 \beta_{8} + 9895 \beta_{7} + 20802 \beta_{4} - 21354 \beta_{3} - 1538 \beta_{2} - 58206\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-16710 \beta_{11} + 69215 \beta_{10} - 70753 \beta_{9} - 70753 \beta_{8} - 43769 \beta_{7} + 70884 \beta_{6} + 70152 \beta_{5} - 25446 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
3.16223i
1.65047i
1.26443i
0.420790i
3.08006i
0.116918i
0.116918i
3.08006i
0.420790i
1.26443i
1.65047i
3.16223i
0 3.21923i 0 1.59013 1.57210i 0 2.43185i 0 −7.36343 0
369.2 0 2.80150i 0 −2.17393 0.523461i 0 4.50896i 0 −4.84843 0
369.3 0 2.40050i 0 −0.817027 + 2.08146i 0 4.41307i 0 −2.76241 0
369.4 0 1.73961i 0 1.77747 + 1.35668i 0 3.32224i 0 −0.0262434 0
369.5 0 0.873449i 0 −1.89824 1.18182i 0 0.992530i 0 2.23709 0
369.6 0 0.486391i 0 1.52160 1.63852i 0 1.80495i 0 2.76342 0
369.7 0 0.486391i 0 1.52160 + 1.63852i 0 1.80495i 0 2.76342 0
369.8 0 0.873449i 0 −1.89824 + 1.18182i 0 0.992530i 0 2.23709 0
369.9 0 1.73961i 0 1.77747 1.35668i 0 3.32224i 0 −0.0262434 0
369.10 0 2.40050i 0 −0.817027 2.08146i 0 4.41307i 0 −2.76241 0
369.11 0 2.80150i 0 −2.17393 + 0.523461i 0 4.50896i 0 −4.84843 0
369.12 0 3.21923i 0 1.59013 + 1.57210i 0 2.43185i 0 −7.36343 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.e.f 12
4.b odd 2 1 460.2.c.a 12
5.b even 2 1 inner 1840.2.e.f 12
5.c odd 4 1 9200.2.a.cx 6
5.c odd 4 1 9200.2.a.cy 6
12.b even 2 1 4140.2.f.b 12
20.d odd 2 1 460.2.c.a 12
20.e even 4 1 2300.2.a.n 6
20.e even 4 1 2300.2.a.o 6
60.h even 2 1 4140.2.f.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 4.b odd 2 1
460.2.c.a 12 20.d odd 2 1
1840.2.e.f 12 1.a even 1 1 trivial
1840.2.e.f 12 5.b even 2 1 inner
2300.2.a.n 6 20.e even 4 1
2300.2.a.o 6 20.e even 4 1
4140.2.f.b 12 12.b even 2 1
4140.2.f.b 12 60.h even 2 1
9200.2.a.cx 6 5.c odd 4 1
9200.2.a.cy 6 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{12} + 28 T_{3}^{10} + 286 T_{3}^{8} + 1296 T_{3}^{6} + 2497 T_{3}^{4} + 1604 T_{3}^{2} + 256 \)
\( T_{7}^{12} + 61 T_{7}^{10} + 1380 T_{7}^{8} + 14312 T_{7}^{6} + 68992 T_{7}^{4} + 139536 T_{7}^{2} + 82944 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 256 + 1604 T^{2} + 2497 T^{4} + 1296 T^{6} + 286 T^{8} + 28 T^{10} + T^{12} \)
$5$ \( 15625 - 2500 T^{2} + 1500 T^{3} + 1075 T^{4} + 60 T^{5} - 48 T^{6} + 12 T^{7} + 43 T^{8} + 12 T^{9} - 4 T^{10} + T^{12} \)
$7$ \( 82944 + 139536 T^{2} + 68992 T^{4} + 14312 T^{6} + 1380 T^{8} + 61 T^{10} + T^{12} \)
$11$ \( ( -256 + 144 T + 212 T^{2} - 28 T^{3} - 32 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$13$ \( 1401856 + 1726756 T^{2} + 551105 T^{4} + 70560 T^{6} + 4126 T^{8} + 108 T^{10} + T^{12} \)
$17$ \( 256 + 2448 T^{2} + 4384 T^{4} + 2620 T^{6} + 628 T^{8} + 57 T^{10} + T^{12} \)
$19$ \( ( 256 - 848 T + 236 T^{2} + 140 T^{3} - 42 T^{4} - 4 T^{5} + T^{6} )^{2} \)
$23$ \( ( 1 + T^{2} )^{6} \)
$29$ \( ( 11862 + 9453 T + 1229 T^{2} - 450 T^{3} - 84 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$31$ \( ( 916 - 1987 T - 2651 T^{2} - 838 T^{3} - 58 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$37$ \( 3996262656 + 1041449616 T^{2} + 90354016 T^{4} + 3059272 T^{6} + 48220 T^{8} + 357 T^{10} + T^{12} \)
$41$ \( ( -2 + 477 T + 257 T^{2} - 158 T^{3} - 64 T^{4} + T^{5} + T^{6} )^{2} \)
$43$ \( 8399355904 + 1503406336 T^{2} + 96250128 T^{4} + 2919296 T^{6} + 44988 T^{8} + 340 T^{10} + T^{12} \)
$47$ \( 215296 + 2925732 T^{2} + 6778057 T^{4} + 585160 T^{6} + 17974 T^{8} + 228 T^{10} + T^{12} \)
$53$ \( 149426176 + 94470400 T^{2} + 17572096 T^{4} + 1019104 T^{6} + 25232 T^{8} + 273 T^{10} + T^{12} \)
$59$ \( ( -360576 + 63024 T + 12832 T^{2} - 1692 T^{3} - 188 T^{4} + 11 T^{5} + T^{6} )^{2} \)
$61$ \( ( -171088 + 10976 T + 10996 T^{2} - 532 T^{3} - 198 T^{4} + 4 T^{5} + T^{6} )^{2} \)
$67$ \( 55115776 + 103780240 T^{2} + 40158752 T^{4} + 2163464 T^{6} + 41820 T^{8} + 341 T^{10} + T^{12} \)
$71$ \( ( -16108 - 20769 T + 569 T^{2} + 1570 T^{3} - 62 T^{4} - 17 T^{5} + T^{6} )^{2} \)
$73$ \( 30294016 + 238969152 T^{2} + 74655361 T^{4} + 3161400 T^{6} + 51998 T^{8} + 376 T^{10} + T^{12} \)
$79$ \( ( -128 - 584 T + 84 T^{2} + 208 T^{3} - 10 T^{4} - 10 T^{5} + T^{6} )^{2} \)
$83$ \( 401124622336 + 40997671440 T^{2} + 1542179872 T^{4} + 26507740 T^{6} + 215860 T^{8} + 777 T^{10} + T^{12} \)
$89$ \( ( 180432 - 167568 T - 3212 T^{2} + 5172 T^{3} - 142 T^{4} - 24 T^{5} + T^{6} )^{2} \)
$97$ \( 2091049984 + 1007200576 T^{2} + 153522128 T^{4} + 8110928 T^{6} + 146088 T^{8} + 704 T^{10} + T^{12} \)
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