Properties

Label 1840.2.i.c
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: \(x^{16} + 8 x^{14} + 44 x^{12} + 162 x^{10} + 404 x^{8} - 84 x^{6} - 79 x^{4} - 4 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{5} q^{5} + ( -\beta_{10} + \beta_{14} ) q^{7} + ( 1 - \beta_{1} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + \beta_{5} q^{5} + ( -\beta_{10} + \beta_{14} ) q^{7} + ( 1 - \beta_{1} - \beta_{4} ) q^{9} + ( \beta_{11} - \beta_{13} ) q^{11} + ( -\beta_{1} - \beta_{4} + 2 \beta_{8} ) q^{13} -\beta_{10} q^{15} + ( \beta_{6} + \beta_{12} - \beta_{15} ) q^{17} + ( \beta_{10} - 2 \beta_{11} + \beta_{14} ) q^{19} + ( -3 \beta_{5} + \beta_{6} ) q^{21} + ( \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{23} - q^{25} + ( -2 \beta_{7} - \beta_{9} ) q^{27} + ( -5 - \beta_{1} + \beta_{8} ) q^{29} + ( \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} ) q^{31} + ( \beta_{5} - \beta_{12} - \beta_{15} ) q^{33} + ( -\beta_{3} + \beta_{9} ) q^{35} + ( 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{12} ) q^{37} + ( -2 \beta_{3} + 2 \beta_{7} - \beta_{9} ) q^{39} + ( 1 + 2 \beta_{1} - \beta_{4} ) q^{41} + ( 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{13} ) q^{43} + ( \beta_{5} + \beta_{6} - \beta_{12} ) q^{45} + ( -5 \beta_{2} - 4 \beta_{3} - \beta_{7} ) q^{47} + ( 3 + \beta_{1} - 3 \beta_{4} + \beta_{8} ) q^{49} + ( -\beta_{10} + 2 \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{51} + ( 2 \beta_{6} - 2 \beta_{15} ) q^{53} + ( \beta_{2} + \beta_{3} - \beta_{7} ) q^{55} + ( -\beta_{5} + \beta_{6} + 2 \beta_{12} ) q^{57} -2 \beta_{2} q^{59} + ( 5 \beta_{6} - \beta_{12} + 3 \beta_{15} ) q^{61} + ( \beta_{10} + \beta_{11} + \beta_{13} + 3 \beta_{14} ) q^{63} + ( \beta_{6} - \beta_{12} + 2 \beta_{15} ) q^{65} + ( -2 \beta_{10} + 2 \beta_{11} + 4 \beta_{13} - 2 \beta_{14} ) q^{67} + ( 1 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{12} + \beta_{15} ) q^{69} + ( -4 \beta_{2} - 3 \beta_{3} + 4 \beta_{7} - 2 \beta_{9} ) q^{71} + ( 5 - 3 \beta_{1} + 3 \beta_{8} ) q^{73} -\beta_{3} q^{75} + ( -3 - 3 \beta_{1} + 2 \beta_{4} + 2 \beta_{8} ) q^{77} + ( 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} ) q^{79} + ( 2 - \beta_{1} + 2 \beta_{8} ) q^{81} + ( -4 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} ) q^{83} + ( \beta_{1} - \beta_{4} + \beta_{8} ) q^{85} + ( -\beta_{2} - 6 \beta_{3} + \beta_{7} ) q^{87} + ( 2 \beta_{5} + 2 \beta_{12} ) q^{89} + ( 5 \beta_{11} - 3 \beta_{13} + 4 \beta_{14} ) q^{91} + ( -2 - 2 \beta_{1} - \beta_{8} ) q^{93} + ( -2 \beta_{2} - \beta_{3} + \beta_{9} ) q^{95} + ( -2 \beta_{5} - \beta_{6} + \beta_{12} + \beta_{15} ) q^{97} + ( 3 \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 8q^{13} - 16q^{25} - 84q^{29} + 24q^{41} + 36q^{49} - 12q^{69} + 68q^{73} - 48q^{77} + 32q^{81} + 4q^{85} - 52q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 8 x^{14} + 44 x^{12} + 162 x^{10} + 404 x^{8} - 84 x^{6} - 79 x^{4} - 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -35 \nu^{14} - 828 \nu^{12} - 1561 \nu^{10} - 5803 \nu^{8} + 1072 \nu^{6} + 1099 \nu^{4} + 84 \nu^{2} - 1863616 \)\()/1193292\)
\(\beta_{2}\)\(=\)\((\)\( 293 \nu^{14} + 3459 \nu^{12} + 21907 \nu^{10} + 97195 \nu^{8} + 303239 \nu^{6} + 441599 \nu^{4} - 58158 \nu^{2} - 48044 \)\()/44196\)
\(\beta_{3}\)\(=\)\((\)\( 11062 \nu^{14} + 74178 \nu^{12} + 374036 \nu^{10} + 1177769 \nu^{8} + 2201194 \nu^{6} - 6433136 \nu^{4} + 749091 \nu^{2} + 511628 \)\()/1193292\)
\(\beta_{4}\)\(=\)\((\)\( 4767 \nu^{14} + 38377 \nu^{12} + 211135 \nu^{10} + 774900 \nu^{8} + 1930469 \nu^{6} - 401601 \nu^{4} - 775295 \nu^{2} + 204280 \)\()/397764\)
\(\beta_{5}\)\(=\)\((\)\( -85 \nu^{15} - 942 \nu^{13} - 5966 \nu^{11} - 26360 \nu^{9} - 82582 \nu^{7} - 120262 \nu^{5} - 27027 \nu^{3} + 25120 \nu \)\()/18792\)
\(\beta_{6}\)\(=\)\((\)\( -8459 \nu^{15} - 93414 \nu^{13} - 591622 \nu^{11} - 2609053 \nu^{9} - 8189294 \nu^{7} - 11925854 \nu^{5} - 2680074 \nu^{3} + 2491040 \nu \)\()/1193292\)
\(\beta_{7}\)\(=\)\((\)\( -23473 \nu^{14} - 177744 \nu^{12} - 956294 \nu^{10} - 3387989 \nu^{8} - 8017342 \nu^{6} + 5490332 \nu^{4} - 549150 \nu^{2} - 261674 \)\()/596646\)
\(\beta_{8}\)\(=\)\((\)\( -50557 \nu^{14} - 408714 \nu^{12} - 2246003 \nu^{10} - 8289539 \nu^{8} - 20627170 \nu^{6} + 4292285 \nu^{4} + 8284338 \nu^{2} - 989684 \)\()/1193292\)
\(\beta_{9}\)\(=\)\((\)\( 59564 \nu^{14} + 502149 \nu^{12} + 2837758 \nu^{10} + 10883380 \nu^{8} + 28818305 \nu^{6} + 7636250 \nu^{4} - 1353924 \nu^{2} - 1501916 \)\()/1193292\)
\(\beta_{10}\)\(=\)\((\)\( -30172 \nu^{15} - 227943 \nu^{13} - 1226342 \nu^{11} - 4346207 \nu^{9} - 10267249 \nu^{7} + 7033190 \nu^{5} - 703227 \nu^{3} + 428758 \nu \)\()/1193292\)
\(\beta_{11}\)\(=\)\((\)\( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} - 5493308 \nu^{5} - 5166507 \nu^{3} - 2648236 \nu \)\()/2386584\)
\(\beta_{12}\)\(=\)\((\)\( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} - 5493308 \nu^{5} - 5166507 \nu^{3} + 2124932 \nu \)\()/2386584\)
\(\beta_{13}\)\(=\)\((\)\( 88865 \nu^{15} + 643824 \nu^{13} + 3399880 \nu^{11} + 11651146 \nu^{9} + 26182376 \nu^{7} - 30534400 \nu^{5} + 8769561 \nu^{3} + 450832 \nu \)\()/2386584\)
\(\beta_{14}\)\(=\)\((\)\( 107167 \nu^{15} + 882966 \nu^{13} + 4914446 \nu^{11} + 18437858 \nu^{9} + 47147158 \nu^{7} - 3938 \nu^{5} - 14640789 \nu^{3} - 5926288 \nu \)\()/2386584\)
\(\beta_{15}\)\(=\)\((\)\( -243817 \nu^{15} - 1984836 \nu^{13} - 11009036 \nu^{11} - 41094428 \nu^{9} - 104586772 \nu^{7} + 4294748 \nu^{5} + 14968749 \nu^{3} - 4378520 \nu \)\()/2386584\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} - \beta_{11}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{14} + \beta_{13} + 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{6} - 5 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 7 \beta_{4} + 7 \beta_{3} - \beta_{2} - 4 \beta_{1} - 8\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{15} - 9 \beta_{14} - 11 \beta_{13} - 7 \beta_{12} + 12 \beta_{11} - 19 \beta_{10} - 11 \beta_{6} + 19 \beta_{5}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(11 \beta_{9} - 11 \beta_{7} - 78 \beta_{3} - 39 \beta_{2} + 7 \beta_{1} + 11\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-2 \beta_{15} + 48 \beta_{14} + 50 \beta_{13} - 7 \beta_{12} - 73 \beta_{11} + 80 \beta_{10} - 50 \beta_{6} + 80 \beta_{5}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(9 \beta_{9} - 48 \beta_{8} + 57 \beta_{7} - 171 \beta_{4} + 171 \beta_{3} + 34 \beta_{2} + 57 \beta_{1} + 137\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(57 \beta_{14} - 57 \beta_{13} - 203 \beta_{11} - 203 \beta_{10} + 381 \beta_{6} - 595 \beta_{5}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-422 \beta_{9} + 162 \beta_{8} - 260 \beta_{7} + 577 \beta_{4} + 577 \beta_{3} + 821 \beta_{2} + 260 \beta_{1} + 244\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(422 \beta_{15} - 739 \beta_{14} - 317 \beta_{13} + 1503 \beta_{12} + 1576 \beta_{11} - 73 \beta_{10} - 317 \beta_{6} + 73 \beta_{5}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(317 \beta_{9} - 317 \beta_{7} - 2258 \beta_{3} - 1129 \beta_{2} - 4167 \beta_{1} - 6507\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-2242 \beta_{15} - 796 \beta_{14} + 1446 \beta_{13} - 7985 \beta_{12} + 3485 \beta_{11} + 4500 \beta_{10} - 1446 \beta_{6} + 4500 \beta_{5}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(10227 \beta_{9} + 796 \beta_{8} + 9431 \beta_{7} + 2835 \beta_{4} - 2835 \beta_{3} - 16766 \beta_{2} + 9431 \beta_{1} + 13931\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(9431 \beta_{14} - 9431 \beta_{13} - 33589 \beta_{11} - 33589 \beta_{10} - 16693 \beta_{6} + 26067 \beta_{5}\)\()/2\)

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
−0.605201 2.04824i
0.605201 + 2.04824i
0.215960 0.625946i
−0.215960 + 0.625946i
−1.47123 + 1.54824i
1.47123 1.54824i
0.650065 + 0.125946i
−0.650065 0.125946i
−0.650065 + 0.125946i
0.650065 0.125946i
1.47123 + 1.54824i
−1.47123 1.54824i
−0.215960 0.625946i
0.215960 + 0.625946i
0.605201 2.04824i
−0.605201 + 2.04824i
0 2.94245i 0 1.00000i 0 −1.89011 0 −5.65803 0
1471.2 0 2.94245i 0 1.00000i 0 1.89011 0 −5.65803 0
1471.3 0 1.30013i 0 1.00000i 0 −1.10639 0 1.30966 0
1471.4 0 1.30013i 0 1.00000i 0 1.10639 0 1.30966 0
1471.5 0 1.21040i 0 1.00000i 0 −4.59480 0 1.53493 0
1471.6 0 1.21040i 0 1.00000i 0 4.59480 0 1.53493 0
1471.7 0 0.431920i 0 1.00000i 0 −3.33035 0 2.81344 0
1471.8 0 0.431920i 0 1.00000i 0 3.33035 0 2.81344 0
1471.9 0 0.431920i 0 1.00000i 0 3.33035 0 2.81344 0
1471.10 0 0.431920i 0 1.00000i 0 −3.33035 0 2.81344 0
1471.11 0 1.21040i 0 1.00000i 0 4.59480 0 1.53493 0
1471.12 0 1.21040i 0 1.00000i 0 −4.59480 0 1.53493 0
1471.13 0 1.30013i 0 1.00000i 0 1.10639 0 1.30966 0
1471.14 0 1.30013i 0 1.00000i 0 −1.10639 0 1.30966 0
1471.15 0 2.94245i 0 1.00000i 0 1.89011 0 −5.65803 0
1471.16 0 2.94245i 0 1.00000i 0 −1.89011 0 −5.65803 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.c 16
4.b odd 2 1 inner 1840.2.i.c 16
23.b odd 2 1 inner 1840.2.i.c 16
92.b even 2 1 inner 1840.2.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.c 16 1.a even 1 1 trivial
1840.2.i.c 16 4.b odd 2 1 inner
1840.2.i.c 16 23.b odd 2 1 inner
1840.2.i.c 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 12 T_{3}^{6} + 32 T_{3}^{4} + 27 T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 4 + 27 T^{2} + 32 T^{4} + 12 T^{6} + T^{8} )^{2} \)
$5$ \( ( 1 + T^{2} )^{8} \)
$7$ \( ( 1024 - 1264 T^{2} + 393 T^{4} - 37 T^{6} + T^{8} )^{2} \)
$11$ \( ( 1024 - 908 T^{2} + 261 T^{4} - 29 T^{6} + T^{8} )^{2} \)
$13$ \( ( -2 - 37 T - 36 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$17$ \( ( 1024 + 3364 T^{2} + 1173 T^{4} + 67 T^{6} + T^{8} )^{2} \)
$19$ \( ( 16 - 904 T^{2} + 537 T^{4} - 73 T^{6} + T^{8} )^{2} \)
$23$ \( ( 279841 + 8464 T^{2} + 510 T^{4} + 16 T^{6} + T^{8} )^{2} \)
$29$ \( ( 376 + 432 T + 152 T^{2} + 21 T^{3} + T^{4} )^{4} \)
$31$ \( ( 13924 + 7067 T^{2} + 1020 T^{4} + 56 T^{6} + T^{8} )^{2} \)
$37$ \( ( 262144 + 76352 T^{2} + 6720 T^{4} + 176 T^{6} + T^{8} )^{2} \)
$41$ \( ( 118 + 27 T - 34 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$43$ \( ( 5683456 - 513472 T^{2} + 16512 T^{4} - 220 T^{6} + T^{8} )^{2} \)
$47$ \( ( 11999296 + 923008 T^{2} + 25296 T^{4} + 283 T^{6} + T^{8} )^{2} \)
$53$ \( ( 82944 + 191808 T^{2} + 17136 T^{4} + 252 T^{6} + T^{8} )^{2} \)
$59$ \( ( 12 + T^{2} )^{8} \)
$61$ \( ( 19254544 + 1949144 T^{2} + 48705 T^{4} + 395 T^{6} + T^{8} )^{2} \)
$67$ \( ( 5308416 - 801792 T^{2} + 29376 T^{4} - 348 T^{6} + T^{8} )^{2} \)
$71$ \( ( 69455556 + 3182139 T^{2} + 53244 T^{4} + 384 T^{6} + T^{8} )^{2} \)
$73$ \( ( 376 + 544 T - 12 T^{2} - 17 T^{3} + T^{4} )^{4} \)
$79$ \( ( 16384 - 15040 T^{2} + 2880 T^{4} - 112 T^{6} + T^{8} )^{2} \)
$83$ \( ( 4096 - 130304 T^{2} + 10128 T^{4} - 200 T^{6} + T^{8} )^{2} \)
$89$ \( ( 82944 + 46656 T^{2} + 4608 T^{4} + 144 T^{6} + T^{8} )^{2} \)
$97$ \( ( 82944 + 34128 T^{2} + 3537 T^{4} + 111 T^{6} + T^{8} )^{2} \)
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