Properties

Label 1840.3.k.d
Level $1840$
Weight $3$
Character orbit 1840.k
Analytic conductor $50.136$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1840.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(50.1363686423\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{2} q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{12} + \beta_{10} + \beta_{9} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + \beta_{2} q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{12} + \beta_{10} + \beta_{9} + 5) q^{9} + (\beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{15} - \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} + 2 \beta_{6} - \beta_{4} + 1) q^{13} + ( - \beta_{13} - \beta_{8} + \beta_{5} - \beta_{3} - \beta_{2}) q^{15} + ( - \beta_{13} + \beta_{11} + \beta_{7} + 3 \beta_{5} - 2 \beta_{2}) q^{17} + (\beta_{11} - 2 \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} - 3 \beta_{2} + \beta_1) q^{19} + ( - 3 \beta_{13} - 2 \beta_{11} - 5 \beta_{8} + 2 \beta_{7} + \beta_{5} - 4 \beta_{3} + \cdots - 3 \beta_1) q^{21}+ \cdots + ( - 12 \beta_{11} - 8 \beta_{8} + 6 \beta_{7} + 4 \beta_{5} - 14 \beta_{3} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 249266 \nu^{15} - 19450521 \nu^{13} - 540300528 \nu^{11} - 7075107694 \nu^{9} - 46294879486 \nu^{7} - 143284317912 \nu^{5} + \cdots - 76734676377 \nu ) / 1473983980 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2028139777 \nu^{15} + 159386627629 \nu^{13} + 4483504013264 \nu^{11} + 59956611391678 \nu^{9} + 407018725514790 \nu^{7} + \cdots + 890813001228721 \nu ) / 3401955025840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + \cdots - 93749086185413 \nu ) / 485993575120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 328071 \nu^{14} - 25254583 \nu^{12} - 684448696 \nu^{10} - 8586649314 \nu^{8} - 51832164194 \nu^{6} - 133890163508 \nu^{4} + \cdots - 885659459 ) / 252679840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 229620557 \nu^{15} + 17699955559 \nu^{13} + 480902133464 \nu^{11} + 6059544792898 \nu^{9} + 36877647447520 \nu^{7} + \cdots + 4166619657821 \nu ) / 121498393780 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3312811321 \nu^{15} - 255261199013 \nu^{13} - 6931819756236 \nu^{11} - 87315212918534 \nu^{9} - 531693729398674 \nu^{7} + \cdots - 105891864922749 \nu ) / 1700977512920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14365389855 \nu^{14} - 1105876709199 \nu^{12} - 29972322851736 \nu^{10} - 375976473404450 \nu^{8} + \cdots + 44834433467221 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11655955868 \nu^{15} + 10501228463 \nu^{14} - 902137906104 \nu^{13} + 808854205323 \nu^{12} - 24691247910968 \nu^{11} + \cdots + 41218966537631 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8504894308903 \nu ) / 57175714720 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2913988967 \nu^{15} + 3973258621 \nu^{14} - 225534476526 \nu^{13} + 305594639244 \nu^{12} - 6172811977742 \nu^{11} + \cdots - 2167987268690 ) / 1700977512920 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + \cdots - 392465157363513 \nu ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5827977934 \nu^{15} + 9278078059 \nu^{14} + 451068953052 \nu^{13} + 713586557253 \nu^{12} + 12345623955484 \nu^{11} + \cdots - 1041206773123 ) / 3401955025840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5827977934 \nu^{15} + 19413839169 \nu^{14} - 451068953052 \nu^{13} + 1494915649571 \nu^{12} - 12345623955484 \nu^{11} + \cdots + 25507426770635 ) / 3401955025840 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta _1 - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{13} - 7\beta_{11} + 5\beta_{8} - 5\beta_{7} + 2\beta_{5} + 7\beta_{3} - 10\beta_{2} - 20\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{15} + 14 \beta_{14} - 45 \beta_{13} - 47 \beta_{12} - 45 \beta_{11} - 43 \beta_{10} - 79 \beta_{9} - 90 \beta_{8} + 127 \beta_{6} + 45 \beta_{5} + 117 \beta_{4} - 45 \beta_{3} - 45 \beta_{2} - 90 \beta _1 + 400 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 349 \beta_{13} + 353 \beta_{11} - 223 \beta_{8} + 193 \beta_{7} - 28 \beta_{5} - 313 \beta_{3} + 342 \beta_{2} + 611 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 359 \beta_{15} - 325 \beta_{14} + 868 \beta_{13} + 898 \beta_{12} + 868 \beta_{11} + 872 \beta_{10} + 1235 \beta_{9} + 1736 \beta_{8} - 2404 \beta_{6} - 868 \beta_{5} - 2129 \beta_{4} + 868 \beta_{3} + 868 \beta_{2} + \cdots - 5998 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13910 \beta_{13} - 14070 \beta_{11} + 8514 \beta_{8} - 6966 \beta_{7} + 52 \beta_{5} + 11798 \beta_{3} - 12044 \beta_{2} - 21063 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 28822 \beta_{15} + 25026 \beta_{14} - 64379 \beta_{13} - 66115 \beta_{12} - 64379 \beta_{11} - 66439 \beta_{10} - 84671 \beta_{9} - 128758 \beta_{8} + 176117 \beta_{6} + 64379 \beta_{5} + \cdots + 406476 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 520505 \beta_{13} + 526825 \beta_{11} - 313151 \beta_{8} + 250339 \beta_{7} + 12974 \beta_{5} - 430885 \beta_{3} + 434946 \beta_{2} + 752686 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1081278 \beta_{15} - 923762 \beta_{14} + 2355559 \beta_{13} + 2410481 \beta_{12} + 2355559 \beta_{11} + 2458153 \beta_{10} + 3010165 \beta_{9} + 4711118 \beta_{8} + \cdots - 14401944 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19083917 \beta_{13} - 19330033 \beta_{11} + 11400387 \beta_{8} - 9029821 \beta_{7} - 679532 \beta_{5} + 15643385 \beta_{3} - 15800198 \beta_{2} - 27187083 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19844966 \beta_{15} + 16848718 \beta_{14} - 42873518 \beta_{13} - 43809734 \beta_{12} - 42873518 \beta_{11} - 44933550 \beta_{10} - 54218408 \beta_{9} - 85747036 \beta_{8} + \cdots + 259136963 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 694927048 \beta_{13} + 704244200 \beta_{11} - 413915040 \beta_{8} + 326698624 \beta_{7} + 27487912 \beta_{5} - 567460000 \beta_{3} + 574139472 \beta_{2} + 985154605 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1446079476 \beta_{15} - 1224795972 \beta_{14} + 3115174881 \beta_{13} + 3181503425 \beta_{12} + 3115174881 \beta_{11} + 3270129841 \beta_{10} + \cdots - 18751391662 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 25246432839 \beta_{13} - 25591957271 \beta_{11} + 15018968005 \beta_{8} - 11839142477 \beta_{7} - 1035217254 \beta_{5} + 20585412727 \beta_{3} + \cdots - 35732448956 \beta_1 ) / 2 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
3.47734i
3.47734i
1.00527i
1.00527i
1.01877i
1.01877i
2.26343i
2.26343i
2.98291i
2.98291i
3.68124i
3.68124i
6.02373i
6.02373i
0.0431371i
0.0431371i
0 −4.76369 0 2.23607i 0 7.05858i 0 13.6927 0
321.2 0 −4.76369 0 2.23607i 0 7.05858i 0 13.6927 0
321.3 0 −4.30716 0 2.23607i 0 1.47532i 0 9.55167 0
321.4 0 −4.30716 0 2.23607i 0 1.47532i 0 9.55167 0
321.5 0 −2.34854 0 2.23607i 0 7.61815i 0 −3.48436 0
321.6 0 −2.34854 0 2.23607i 0 7.61815i 0 −3.48436 0
321.7 0 −1.43837 0 2.23607i 0 10.1866i 0 −6.93108 0
321.8 0 −1.43837 0 2.23607i 0 10.1866i 0 −6.93108 0
321.9 0 0.278523 0 2.23607i 0 8.51262i 0 −8.92243 0
321.10 0 0.278523 0 2.23607i 0 8.51262i 0 −8.92243 0
321.11 0 3.36596 0 2.23607i 0 1.16919i 0 2.32968 0
321.12 0 3.36596 0 2.23607i 0 1.16919i 0 2.32968 0
321.13 0 3.79379 0 2.23607i 0 7.10180i 0 5.39287 0
321.14 0 3.79379 0 2.23607i 0 7.10180i 0 5.39287 0
321.15 0 5.41949 0 2.23607i 0 8.24199i 0 20.3709 0
321.16 0 5.41949 0 2.23607i 0 8.24199i 0 20.3709 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.3.k.d 16
4.b odd 2 1 230.3.d.a 16
12.b even 2 1 2070.3.c.a 16
20.d odd 2 1 1150.3.d.b 16
20.e even 4 2 1150.3.c.c 32
23.b odd 2 1 inner 1840.3.k.d 16
92.b even 2 1 230.3.d.a 16
276.h odd 2 1 2070.3.c.a 16
460.g even 2 1 1150.3.d.b 16
460.k odd 4 2 1150.3.c.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.d.a 16 4.b odd 2 1
230.3.d.a 16 92.b even 2 1
1150.3.c.c 32 20.e even 4 2
1150.3.c.c 32 460.k odd 4 2
1150.3.d.b 16 20.d odd 2 1
1150.3.d.b 16 460.g even 2 1
1840.3.k.d 16 1.a even 1 1 trivial
1840.3.k.d 16 23.b odd 2 1 inner
2070.3.c.a 16 12.b even 2 1
2070.3.c.a 16 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 52T_{3}^{6} - 16T_{3}^{5} + 829T_{3}^{4} + 456T_{3}^{3} - 4114T_{3}^{2} - 3704T_{3} + 1336 \) acting on \(S_{3}^{\mathrm{new}}(1840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 52 T^{6} - 16 T^{5} + 829 T^{4} + \cdots + 1336)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 406 T^{14} + \cdots + 221645107264 \) Copy content Toggle raw display
$11$ \( T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{8} - 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + 4 T^{15} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 58 T^{7} + \cdots + 229759835104)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{8} + 78 T^{7} + \cdots - 212194449184)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{8} - 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{8} + 102 T^{7} + \cdots - 42922529206784)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{8} + 118 T^{7} + \cdots - 24390990617024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 56 T^{7} + \cdots + 1317400530416)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04 \) Copy content Toggle raw display
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