Properties

Label 1859.2.a.s
Level $1859$
Weight $2$
Character orbit 1859.a
Self dual yes
Analytic conductor $14.844$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8441897358\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 6 q^{3} + 32 q^{4} - 7 q^{5} + 19 q^{6} - q^{7} + 3 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 6 q^{3} + 32 q^{4} - 7 q^{5} + 19 q^{6} - q^{7} + 3 q^{8} + 33 q^{9} + 18 q^{10} - 21 q^{11} + 23 q^{12} + 20 q^{14} - 16 q^{15} + 50 q^{16} + 16 q^{17} - 3 q^{18} + 11 q^{19} - 24 q^{20} + 5 q^{21} - 9 q^{23} + 54 q^{24} + 36 q^{25} + 11 q^{28} + 28 q^{29} + 21 q^{30} - 15 q^{31} + 61 q^{32} - 6 q^{33} + 6 q^{34} - 3 q^{35} + 45 q^{36} + 12 q^{37} + q^{38} + 55 q^{40} + 4 q^{41} - 34 q^{42} + 17 q^{43} - 32 q^{44} - 9 q^{45} - 11 q^{46} - 36 q^{47} + 24 q^{48} + 72 q^{49} + 9 q^{50} + 2 q^{51} + 19 q^{53} - q^{54} + 7 q^{55} + 44 q^{56} + 4 q^{57} + 33 q^{58} - 54 q^{59} - 64 q^{60} + 98 q^{61} - 29 q^{62} + 81 q^{63} + 63 q^{64} - 19 q^{66} - 25 q^{67} + 4 q^{68} + 89 q^{69} - 65 q^{70} - 37 q^{71} - 55 q^{72} - 8 q^{73} - 11 q^{74} + 24 q^{75} - 13 q^{76} + q^{77} + 24 q^{79} - 26 q^{80} + 81 q^{81} + 26 q^{82} + 34 q^{83} + 103 q^{84} + 11 q^{85} - 30 q^{86} + 32 q^{87} - 3 q^{88} - 6 q^{89} + 47 q^{90} - 80 q^{92} - 41 q^{93} + 40 q^{94} + 20 q^{95} + 98 q^{96} + 5 q^{98} - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.68791 −2.90985 5.22486 −3.09378 7.82142 −1.70397 −8.66813 5.46725 8.31580
1.2 −2.56428 1.12676 4.57554 −1.24792 −2.88933 −2.50384 −6.60441 −1.73041 3.20001
1.3 −2.26490 3.17598 3.12978 −2.32745 −7.19329 4.39568 −2.55884 7.08687 5.27144
1.4 −2.23194 1.36188 2.98154 −4.28736 −3.03963 −0.384100 −2.19073 −1.14528 9.56911
1.5 −2.14442 −1.72478 2.59856 3.54668 3.69866 2.57025 −1.28356 −0.0251423 −7.60558
1.6 −2.13563 −2.20961 2.56090 1.38469 4.71890 −5.12151 −1.19787 1.88237 −2.95718
1.7 −1.43463 0.655312 0.0581570 −1.45609 −0.940129 −3.21683 2.78582 −2.57057 2.08894
1.8 −1.17546 −1.22926 −0.618298 1.06938 1.44494 3.67782 3.07770 −1.48892 −1.25701
1.9 −0.776244 3.01008 −1.39745 −0.0164567 −2.33656 −0.793722 2.63725 6.06060 0.0127744
1.10 −0.149116 −3.21882 −1.97776 −1.07607 0.479977 3.58386 0.593149 7.36077 0.160459
1.11 0.340522 −1.21327 −1.88404 3.18769 −0.413146 −4.05402 −1.32260 −1.52797 1.08548
1.12 0.392741 1.69948 −1.84575 −1.18432 0.667456 −5.07600 −1.51039 −0.111771 −0.465133
1.13 0.603243 −0.154529 −1.63610 −4.17643 −0.0932187 −2.73997 −2.19345 −2.97612 −2.51940
1.14 0.930026 2.53271 −1.13505 3.38691 2.35548 3.38196 −2.91568 3.41460 3.14991
1.15 1.50636 −0.544988 0.269129 −3.40659 −0.820949 0.714897 −2.60732 −2.70299 −5.13156
1.16 1.53678 −3.28188 0.361702 1.21246 −5.04353 1.98662 −2.51771 7.77071 1.86328
1.17 1.77144 2.92005 1.13800 2.22672 5.17269 −0.178675 −1.52699 5.52669 3.94449
1.18 2.32475 2.00106 3.40448 −3.04764 4.65197 5.08642 3.26506 1.00425 −7.08500
1.19 2.68347 3.12668 5.20101 −2.33476 8.39034 −0.266736 8.58980 6.77612 −6.26526
1.20 2.70607 0.960218 5.32282 3.57279 2.59842 −3.95601 8.99178 −2.07798 9.66821
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.2.a.s 21
13.b even 2 1 1859.2.a.t yes 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.2.a.s 21 1.a even 1 1 trivial
1859.2.a.t yes 21 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):

\( T_{2}^{21} - 37 T_{2}^{19} - T_{2}^{18} + 582 T_{2}^{17} + 20 T_{2}^{16} - 5075 T_{2}^{15} - 99 T_{2}^{14} + 26832 T_{2}^{13} - 510 T_{2}^{12} - 88400 T_{2}^{11} + 7129 T_{2}^{10} + 179407 T_{2}^{9} - 29118 T_{2}^{8} - 212824 T_{2}^{7} + \cdots - 448 \) Copy content Toggle raw display
\( T_{7}^{21} + T_{7}^{20} - 109 T_{7}^{19} - 94 T_{7}^{18} + 4999 T_{7}^{17} + 3763 T_{7}^{16} - 125749 T_{7}^{15} - 85240 T_{7}^{14} + 1893974 T_{7}^{13} + 1215357 T_{7}^{12} - 17460834 T_{7}^{11} - 11281124 T_{7}^{10} + \cdots - 2981888 \) Copy content Toggle raw display