Properties

Label 1840.2.i.a
Level $1840$
Weight $2$
Character orbit 1840.i
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 32 x^{14} + 400 x^{12} - 2398 x^{10} + 7128 x^{8} - 9200 x^{6} + 4705 x^{4} + 2696 x^{2} + 784\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{8} q^{5} + ( \beta_{3} - \beta_{10} ) q^{7} + ( -2 - \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{8} q^{5} + ( \beta_{3} - \beta_{10} ) q^{7} + ( -2 - \beta_{6} ) q^{9} + ( -2 \beta_{10} + \beta_{12} ) q^{11} + ( 1 + \beta_{6} ) q^{13} -\beta_{3} q^{15} + \beta_{13} q^{17} + ( -\beta_{10} + \beta_{12} + \beta_{15} ) q^{19} + ( -\beta_{4} + 4 \beta_{8} + \beta_{13} ) q^{21} + ( -\beta_{1} + \beta_{3} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{23} - q^{25} + ( -\beta_{1} + \beta_{9} + \beta_{11} ) q^{27} + ( 5 - \beta_{2} - \beta_{6} ) q^{29} + ( \beta_{5} + 3 \beta_{9} ) q^{31} + ( -\beta_{4} - 3 \beta_{8} - \beta_{13} + \beta_{14} ) q^{33} + ( \beta_{1} + \beta_{5} ) q^{35} + ( \beta_{8} + \beta_{13} + \beta_{14} ) q^{37} + ( 3 \beta_{1} - \beta_{9} - \beta_{11} ) q^{39} + ( -\beta_{2} + \beta_{6} - \beta_{7} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{10} ) q^{43} + ( -2 \beta_{8} - \beta_{13} ) q^{45} + ( \beta_{1} + \beta_{9} ) q^{47} + ( -1 - 2 \beta_{2} + \beta_{6} ) q^{49} + ( -2 \beta_{3} - \beta_{10} + \beta_{12} + \beta_{15} ) q^{51} + ( \beta_{8} + \beta_{13} - \beta_{14} ) q^{53} + ( 2 \beta_{5} + \beta_{9} ) q^{55} + ( -\beta_{4} - 3 \beta_{13} + 2 \beta_{14} ) q^{57} + ( -\beta_{1} - 5 \beta_{5} - \beta_{9} - \beta_{11} ) q^{59} + ( -2 \beta_{4} - \beta_{8} - \beta_{14} ) q^{61} + ( -3 \beta_{3} + \beta_{10} + \beta_{12} ) q^{63} + ( \beta_{8} + \beta_{13} ) q^{65} + ( -\beta_{3} - 4 \beta_{10} - \beta_{12} + \beta_{15} ) q^{67} + ( 6 - \beta_{2} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + \beta_{14} ) q^{69} + ( -2 \beta_{1} - \beta_{5} - \beta_{9} - \beta_{11} ) q^{71} + ( -2 + \beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{73} -\beta_{1} q^{75} + ( 2 - \beta_{2} - \beta_{6} ) q^{77} + ( -2 \beta_{3} + 2 \beta_{10} + 4 \beta_{12} ) q^{79} + ( -1 + \beta_{2} + \beta_{6} - 2 \beta_{7} ) q^{81} + ( \beta_{3} + 4 \beta_{10} + \beta_{12} - \beta_{15} ) q^{83} -\beta_{6} q^{85} + ( 3 \beta_{1} + 4 \beta_{5} + \beta_{9} + 2 \beta_{11} ) q^{87} + ( -4 \beta_{8} - 2 \beta_{13} ) q^{89} + ( 2 \beta_{3} - \beta_{12} ) q^{91} + ( 4 - 2 \beta_{2} + 3 \beta_{6} - 3 \beta_{7} ) q^{93} + ( \beta_{9} + \beta_{11} ) q^{95} + ( -2 \beta_{4} - \beta_{8} + 2 \beta_{13} + \beta_{14} ) q^{97} + ( 4 \beta_{3} + 3 \beta_{10} - \beta_{12} - 3 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 32q^{9} + O(q^{10}) \) \( 16q - 32q^{9} + 16q^{13} - 16q^{25} + 72q^{29} - 4q^{41} - 32q^{49} + 92q^{69} - 20q^{73} + 24q^{77} + 60q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 32 x^{14} + 400 x^{12} - 2398 x^{10} + 7128 x^{8} - 9200 x^{6} + 4705 x^{4} + 2696 x^{2} + 784\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -7141 \nu^{14} + 639364 \nu^{12} - 14917580 \nu^{10} + 151546566 \nu^{8} - 729577504 \nu^{6} + 1633121252 \nu^{4} - 989294405 \nu^{2} - 188557796 \)\()/ 547177568 \)
\(\beta_{2}\)\(=\)\((\)\( -5547 \nu^{14} + 173356 \nu^{12} - 2080516 \nu^{10} + 11582258 \nu^{8} - 29821744 \nu^{6} + 24310636 \nu^{4} + 12535373 \nu^{2} + 5858020 \)\()/18868192\)
\(\beta_{3}\)\(=\)\((\)\( -165589 \nu^{15} + 5706972 \nu^{13} - 79149040 \nu^{11} + 555881878 \nu^{9} - 2108936952 \nu^{7} + 4187604864 \nu^{5} - 3971651317 \nu^{3} - 262952740 \nu \)\()/ 1094355136 \)
\(\beta_{4}\)\(=\)\((\)\( -7343 \nu^{15} + 214244 \nu^{13} - 2331136 \nu^{11} + 10946626 \nu^{9} - 19800248 \nu^{7} - 235120 \nu^{5} - 16298863 \nu^{3} - 11846476 \nu \)\()/37736384\)
\(\beta_{5}\)\(=\)\((\)\( 971 \nu^{14} - 32540 \nu^{12} + 432756 \nu^{10} - 2840762 \nu^{8} + 9652416 \nu^{6} - 15727740 \nu^{4} + 10040651 \nu^{2} + 1863596 \)\()/2088464\)
\(\beta_{6}\)\(=\)\((\)\( -9687 \nu^{14} + 298300 \nu^{12} - 3520980 \nu^{10} + 19188922 \nu^{8} - 48266192 \nu^{6} + 39042108 \nu^{4} + 20161121 \nu^{2} - 80965164 \)\()/18868192\)
\(\beta_{7}\)\(=\)\((\)\( -20175 \nu^{14} + 647232 \nu^{12} - 8072756 \nu^{10} + 47481542 \nu^{8} - 129690636 \nu^{6} + 107683996 \nu^{4} + 55334005 \nu^{2} - 86948092 \)\()/18868192\)
\(\beta_{8}\)\(=\)\((\)\( -9589 \nu^{15} + 302708 \nu^{13} - 3710656 \nu^{11} + 21553958 \nu^{9} - 60743728 \nu^{7} + 69774352 \nu^{5} - 30384773 \nu^{3} - 37094388 \nu \)\()/18868192\)
\(\beta_{9}\)\(=\)\((\)\( 760831 \nu^{14} - 24021912 \nu^{12} + 294082508 \nu^{10} - 1701925438 \nu^{8} + 4773358420 \nu^{6} - 5691404292 \nu^{4} + 3858584675 \nu^{2} + 692456380 \)\()/ 547177568 \)
\(\beta_{10}\)\(=\)\((\)\( -345 \nu^{15} + 11264 \nu^{13} - 145224 \nu^{11} + 918310 \nu^{9} - 3009724 \nu^{7} + 4829752 \nu^{5} - 3773569 \nu^{3} - 189996 \nu \)\()/425488\)
\(\beta_{11}\)\(=\)\((\)\( -169285 \nu^{14} + 5459985 \nu^{12} - 69284106 \nu^{10} + 428229415 \nu^{8} - 1354547795 \nu^{6} + 2000680458 \nu^{4} - 1302558772 \nu^{2} - 239476132 \)\()/68397196\)
\(\beta_{12}\)\(=\)\((\)\( 1996395 \nu^{15} - 64300516 \nu^{13} + 811056936 \nu^{11} - 4930874706 \nu^{9} + 14996450792 \nu^{7} - 20448941960 \nu^{5} + 12816032259 \nu^{3} + 361303484 \nu \)\()/ 1094355136 \)
\(\beta_{13}\)\(=\)\((\)\( 80267 \nu^{15} - 2562948 \nu^{13} + 31894496 \nu^{11} - 189379626 \nu^{9} + 550748136 \nu^{7} - 668515696 \nu^{5} + 265063979 \nu^{3} + 336396572 \nu \)\()/37736384\)
\(\beta_{14}\)\(=\)\((\)\( 81585 \nu^{15} - 2642028 \nu^{13} + 33591704 \nu^{11} - 206610582 \nu^{9} + 635359128 \nu^{7} - 838029688 \nu^{5} + 290130617 \nu^{3} + 390867284 \nu \)\()/37736384\)
\(\beta_{15}\)\(=\)\((\)\(-2530093 \nu^{15} + 81970636 \nu^{13} - 1045043272 \nu^{11} + 6493543742 \nu^{9} - 20714457512 \nu^{7} + 31723436072 \nu^{5} - 23466355989 \nu^{3} - 1062631524 \nu\)\()/ 547177568 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{10} - \beta_{8} - 2 \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} + \beta_{5} - 2 \beta_{2} + 2 \beta_{1} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} - 3 \beta_{13} + 2 \beta_{12} + 13 \beta_{10} - 14 \beta_{8} + 3 \beta_{4} - 14 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{11} - 4 \beta_{9} + 14 \beta_{6} + 25 \beta_{5} - 22 \beta_{2} + 30 \beta_{1} + 67\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{15} - 40 \beta_{13} + 12 \beta_{12} + 127 \beta_{10} - 191 \beta_{8} + 60 \beta_{4} - 106 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(47 \beta_{11} - 46 \beta_{9} + 8 \beta_{7} + 91 \beta_{6} + 402 \beta_{5} - 183 \beta_{2} + 394 \beta_{1} + 484\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-122 \beta_{15} - 28 \beta_{14} - 462 \beta_{13} + 56 \beta_{12} + 953 \beta_{10} - 2379 \beta_{8} + 854 \beta_{4} - 706 \beta_{3}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(712 \beta_{11} - 448 \beta_{9} + 80 \beta_{7} + 470 \beta_{6} + 5259 \beta_{5} - 1118 \beta_{2} + 4734 \beta_{1} + 2731\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-607 \beta_{15} - 528 \beta_{14} - 5061 \beta_{13} + 126 \beta_{12} + 4259 \beta_{10} - 27394 \beta_{8} + 10413 \beta_{4} - 2826 \beta_{3}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(8714 \beta_{11} - 4332 \beta_{9} + 280 \beta_{7} + 118 \beta_{6} + 60643 \beta_{5} - 1502 \beta_{2} + 52546 \beta_{1} + 2249\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(2634 \beta_{15} - 6688 \beta_{14} - 52580 \beta_{13} - 2468 \beta_{12} - 24711 \beta_{10} - 291829 \beta_{8} + 113784 \beta_{4} + 21314 \beta_{3}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(92865 \beta_{11} - 41314 \beta_{9} - 4744 \beta_{7} - 48755 \beta_{6} + 630646 \beta_{5} + 98679 \beta_{2} + 537406 \beta_{1} - 261836\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(125478 \beta_{15} - 69836 \beta_{14} - 507910 \beta_{13} - 62296 \beta_{12} - 998513 \beta_{10} - 2850433 \beta_{8} + 1123486 \beta_{4} + 756434 \beta_{3}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(876628 \beta_{11} - 372008 \beta_{9} - 129248 \beta_{7} - 1009946 \beta_{6} + 5895185 \beta_{5} + 2188986 \beta_{2} + 4989146 \beta_{1} - 5610087\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(2251575 \beta_{15} - 626000 \beta_{14} - 4417867 \beta_{13} - 1002078 \beta_{12} - 17543547 \beta_{10} - 24904974 \beta_{8} + 9858843 \beta_{4} + 13032306 \beta_{3}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
−2.25010 + 0.500000i
2.25010 0.500000i
−3.23546 + 0.500000i
3.23546 0.500000i
−1.19389 + 0.500000i
1.19389 0.500000i
0.208533 + 0.500000i
−0.208533 0.500000i
−0.208533 + 0.500000i
0.208533 0.500000i
1.19389 + 0.500000i
−1.19389 0.500000i
3.23546 + 0.500000i
−3.23546 0.500000i
2.25010 + 0.500000i
−2.25010 0.500000i
0 3.11612i 0 1.00000i 0 1.38407 0 −6.71022 0
1471.2 0 3.11612i 0 1.00000i 0 −1.38407 0 −6.71022 0
1471.3 0 2.36943i 0 1.00000i 0 4.10148 0 −2.61421 0
1471.4 0 2.36943i 0 1.00000i 0 −4.10148 0 −2.61421 0
1471.5 0 2.05992i 0 1.00000i 0 0.327869 0 −1.24327 0
1471.6 0 2.05992i 0 1.00000i 0 −0.327869 0 −1.24327 0
1471.7 0 0.657492i 0 1.00000i 0 −1.07456 0 2.56770 0
1471.8 0 0.657492i 0 1.00000i 0 1.07456 0 2.56770 0
1471.9 0 0.657492i 0 1.00000i 0 1.07456 0 2.56770 0
1471.10 0 0.657492i 0 1.00000i 0 −1.07456 0 2.56770 0
1471.11 0 2.05992i 0 1.00000i 0 −0.327869 0 −1.24327 0
1471.12 0 2.05992i 0 1.00000i 0 0.327869 0 −1.24327 0
1471.13 0 2.36943i 0 1.00000i 0 −4.10148 0 −2.61421 0
1471.14 0 2.36943i 0 1.00000i 0 4.10148 0 −2.61421 0
1471.15 0 3.11612i 0 1.00000i 0 −1.38407 0 −6.71022 0
1471.16 0 3.11612i 0 1.00000i 0 1.38407 0 −6.71022 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.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.i.a 16
4.b odd 2 1 inner 1840.2.i.a 16
23.b odd 2 1 inner 1840.2.i.a 16
92.b even 2 1 inner 1840.2.i.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.a 16 1.a even 1 1 trivial
1840.2.i.a 16 4.b odd 2 1 inner
1840.2.i.a 16 23.b odd 2 1 inner
1840.2.i.a 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 20 T_{3}^{6} + 128 T_{3}^{4} + 283 T_{3}^{2} + 100 \) acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 100 + 283 T^{2} + 128 T^{4} + 20 T^{6} + T^{8} )^{2} \)
$5$ \( ( 1 + T^{2} )^{8} \)
$7$ \( ( 4 - 43 T^{2} + 56 T^{4} - 20 T^{6} + T^{8} )^{2} \)
$11$ \( ( 784 - 2728 T^{2} + 1121 T^{4} - 65 T^{6} + T^{8} )^{2} \)
$13$ \( ( -8 + 37 T - 16 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$17$ \( ( 100 + 449 T^{2} + 504 T^{4} + 44 T^{6} + T^{8} )^{2} \)
$19$ \( ( 462400 - 88508 T^{2} + 5549 T^{4} - 133 T^{6} + T^{8} )^{2} \)
$23$ \( 78310985281 + 5329292004 T^{2} + 251856900 T^{4} + 13895772 T^{6} + 598070 T^{8} + 26268 T^{10} + 900 T^{12} + 36 T^{14} + T^{16} \)
$29$ \( ( -1172 + 330 T + 59 T^{2} - 18 T^{3} + T^{4} )^{4} \)
$31$ \( ( 6487209 + 701487 T^{2} + 21816 T^{4} + 255 T^{6} + T^{8} )^{2} \)
$37$ \( ( 153664 + 62720 T^{2} + 5328 T^{4} + 137 T^{6} + T^{8} )^{2} \)
$41$ \( ( -89 + 173 T - 58 T^{2} + T^{3} + T^{4} )^{4} \)
$43$ \( ( 1024 - 2752 T^{2} + 896 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$47$ \( ( 64 + 320 T^{2} + 224 T^{4} + 43 T^{6} + T^{8} )^{2} \)
$53$ \( ( 4096 + 11280 T^{2} + 3512 T^{4} + 129 T^{6} + T^{8} )^{2} \)
$59$ \( ( 462400 + 582992 T^{2} + 45884 T^{4} + 415 T^{6} + T^{8} )^{2} \)
$61$ \( ( 4129024 + 479620 T^{2} + 17757 T^{4} + 247 T^{6} + T^{8} )^{2} \)
$67$ \( ( 20070400 - 2169088 T^{2} + 47972 T^{4} - 383 T^{6} + T^{8} )^{2} \)
$71$ \( ( 1466521 + 228535 T^{2} + 11360 T^{4} + 191 T^{6} + T^{8} )^{2} \)
$73$ \( ( -1640 - 968 T - 136 T^{2} + 5 T^{3} + T^{4} )^{4} \)
$79$ \( ( 333865984 - 11789504 T^{2} + 139328 T^{4} - 640 T^{6} + T^{8} )^{2} \)
$83$ \( ( 20070400 - 2169088 T^{2} + 47972 T^{4} - 383 T^{6} + T^{8} )^{2} \)
$89$ \( ( 802816 + 194112 T^{2} + 11840 T^{4} + 240 T^{6} + T^{8} )^{2} \)
$97$ \( ( 1993744 + 860696 T^{2} + 48753 T^{4} + 431 T^{6} + T^{8} )^{2} \)
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