Properties

Label 1859.4.a.p
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $0$
Dimension $51$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.684550701\)
Analytic rank: \(0\)
Dimension: \(51\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 51 q + 21 q^{3} + 234 q^{4} - 41 q^{5} + 73 q^{6} - 4 q^{7} + 21 q^{8} + 594 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 51 q + 21 q^{3} + 234 q^{4} - 41 q^{5} + 73 q^{6} - 4 q^{7} + 21 q^{8} + 594 q^{9} + 212 q^{10} - 561 q^{11} + 209 q^{12} + 280 q^{14} - 284 q^{15} + 1246 q^{16} + 164 q^{17} + 189 q^{18} - 26 q^{19} - 438 q^{20} - 134 q^{21} + 373 q^{23} + 354 q^{24} + 2048 q^{25} + 1470 q^{27} + 1245 q^{28} + 898 q^{29} + 427 q^{30} - 767 q^{31} - 1127 q^{32} - 231 q^{33} - 206 q^{34} + 54 q^{35} + 3415 q^{36} - 395 q^{37} + 1577 q^{38} + 3253 q^{40} + 354 q^{41} + 942 q^{42} + 484 q^{43} - 2574 q^{44} - 1452 q^{45} + 2117 q^{46} - 1925 q^{47} + 1780 q^{48} + 4535 q^{49} + 1093 q^{50} + 230 q^{51} + 1387 q^{53} + 5271 q^{54} + 451 q^{55} + 2568 q^{56} + 5738 q^{57} - 3695 q^{58} - 1145 q^{59} + 1590 q^{60} + 5382 q^{61} - 395 q^{62} - 710 q^{63} + 9839 q^{64} - 803 q^{66} + 210 q^{67} + 1742 q^{68} + 7028 q^{69} + 6747 q^{70} - 3693 q^{71} + 12481 q^{72} - 968 q^{73} + 1735 q^{74} - 727 q^{75} + 2801 q^{76} + 44 q^{77} + 4234 q^{79} - 2390 q^{80} + 7743 q^{81} + 4770 q^{82} + 2798 q^{83} - 14821 q^{84} + 1802 q^{85} - 6558 q^{86} + 1896 q^{87} - 231 q^{88} - 3927 q^{89} + 1927 q^{90} + 1984 q^{92} + 1332 q^{93} + 7590 q^{94} + 4944 q^{95} + 7280 q^{96} - 3913 q^{97} + 15201 q^{98} - 6534 q^{99} + O(q^{100}) \)

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 −5.64970 −1.71534 23.9192 7.89488 9.69115 −11.5333 −89.9385 −24.0576 −44.6038
1.2 −5.53541 −5.57466 22.6408 −16.4961 30.8581 23.7730 −81.0427 4.07688 91.3127
1.3 −5.50285 4.91472 22.2813 −19.9510 −27.0449 −0.0210524 −78.5880 −2.84554 109.787
1.4 −5.02887 1.98348 17.2895 15.8951 −9.97465 8.39176 −46.7159 −23.0658 −79.9343
1.5 −4.99679 8.59946 16.9679 −10.1780 −42.9697 −9.94954 −44.8107 46.9506 50.8572
1.6 −4.81770 −9.24702 15.2103 −7.67520 44.5494 28.1687 −34.7370 58.5075 36.9768
1.7 −4.60628 7.07074 13.2178 −11.0597 −32.5698 −9.07530 −24.0348 22.9953 50.9440
1.8 −4.57982 0.436116 12.9748 3.32236 −1.99733 −25.3240 −22.7836 −26.8098 −15.2158
1.9 −3.95893 −9.52439 7.67316 16.7554 37.7064 −7.49508 1.29395 63.7139 −66.3336
1.10 −3.79290 −5.52551 6.38608 −13.9792 20.9577 −15.0455 6.12143 3.53129 53.0218
1.11 −3.74164 9.22560 5.99989 −0.452132 −34.5189 −22.7458 7.48368 58.1117 1.69172
1.12 −3.11673 −1.97531 1.71399 2.69678 6.15651 30.4006 19.5918 −23.0981 −8.40514
1.13 −3.06434 −3.83075 1.39021 −4.40166 11.7387 9.87023 20.2547 −12.3254 13.4882
1.14 −3.01034 0.463820 1.06212 13.2599 −1.39626 −19.7000 20.8853 −26.7849 −39.9167
1.15 −2.72755 0.736988 −0.560489 −18.0105 −2.01017 23.4852 23.3491 −26.4568 49.1244
1.16 −2.60372 8.44963 −1.22064 3.90395 −22.0005 31.4455 24.0080 44.3963 −10.1648
1.17 −2.52498 0.684748 −1.62445 14.7550 −1.72898 2.67947 24.3016 −26.5311 −37.2563
1.18 −2.33473 −7.99292 −2.54902 −16.6573 18.6613 −33.9647 24.6291 36.8868 38.8903
1.19 −2.01569 9.83385 −3.93701 16.4444 −19.8220 −5.22049 24.0613 69.7045 −33.1467
1.20 −1.58200 3.04819 −5.49728 −5.74523 −4.82223 −14.7260 21.3527 −17.7085 9.08894
See all 51 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.51
Significant digits:
Format:

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 1859.4.a.p 51
13.b even 2 1 1859.4.a.q yes 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.4.a.p 51 1.a even 1 1 trivial
1859.4.a.q yes 51 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(15\!\cdots\!92\)\( T_{2}^{37} - \)\(37\!\cdots\!28\)\( T_{2}^{36} + \)\(34\!\cdots\!15\)\( T_{2}^{35} + \)\(10\!\cdots\!48\)\( T_{2}^{34} - \)\(65\!\cdots\!85\)\( T_{2}^{33} - \)\(22\!\cdots\!53\)\( T_{2}^{32} + \)\(99\!\cdots\!01\)\( T_{2}^{31} + \)\(39\!\cdots\!86\)\( T_{2}^{30} - \)\(12\!\cdots\!32\)\( T_{2}^{29} - \)\(55\!\cdots\!26\)\( T_{2}^{28} + \)\(12\!\cdots\!17\)\( T_{2}^{27} + \)\(62\!\cdots\!04\)\( T_{2}^{26} - \)\(10\!\cdots\!58\)\( T_{2}^{25} - \)\(55\!\cdots\!71\)\( T_{2}^{24} + \)\(66\!\cdots\!26\)\( T_{2}^{23} + \)\(39\!\cdots\!84\)\( T_{2}^{22} - \)\(35\!\cdots\!44\)\( T_{2}^{21} - \)\(22\!\cdots\!56\)\( T_{2}^{20} + \)\(14\!\cdots\!84\)\( T_{2}^{19} + \)\(96\!\cdots\!96\)\( T_{2}^{18} - \)\(48\!\cdots\!64\)\( T_{2}^{17} - \)\(32\!\cdots\!16\)\( T_{2}^{16} + \)\(12\!\cdots\!68\)\( T_{2}^{15} + \)\(81\!\cdots\!44\)\( T_{2}^{14} - \)\(22\!\cdots\!04\)\( T_{2}^{13} - \)\(14\!\cdots\!80\)\( T_{2}^{12} + \)\(30\!\cdots\!24\)\( T_{2}^{11} + \)\(18\!\cdots\!52\)\( T_{2}^{10} - \)\(28\!\cdots\!28\)\( T_{2}^{9} - \)\(15\!\cdots\!24\)\( T_{2}^{8} + \)\(17\!\cdots\!12\)\( T_{2}^{7} + \)\(76\!\cdots\!12\)\( T_{2}^{6} - \)\(58\!\cdots\!20\)\( T_{2}^{5} - \)\(16\!\cdots\!00\)\( T_{2}^{4} + \)\(89\!\cdots\!68\)\( T_{2}^{3} + \)\(43\!\cdots\!92\)\( T_{2}^{2} - \)\(20\!\cdots\!56\)\( T_{2} - \)\(10\!\cdots\!24\)\( \)">\(T_{2}^{51} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).