Properties

Label 1859.4.a.o
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $1$
Dimension $39$
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: \(1\)
Dimension: \(39\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 39 q - 23 q^{3} + 114 q^{4} + 23 q^{5} + 77 q^{6} - 4 q^{7} - 21 q^{8} + 260 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 39 q - 23 q^{3} + 114 q^{4} + 23 q^{5} + 77 q^{6} - 4 q^{7} - 21 q^{8} + 260 q^{9} - 158 q^{10} - 429 q^{11} - 351 q^{12} - 176 q^{14} + 30 q^{15} + 230 q^{16} - 244 q^{17} + 21 q^{18} - 70 q^{19} + 366 q^{20} - 142 q^{21} - 47 q^{23} + 846 q^{24} + 322 q^{25} - 416 q^{27} + 1131 q^{28} - 838 q^{29} - 293 q^{30} + 507 q^{31} - 1433 q^{32} + 253 q^{33} + 166 q^{34} - 498 q^{35} + 815 q^{36} + 89 q^{37} + 81 q^{38} - 2917 q^{40} + 618 q^{41} - 318 q^{42} - 1064 q^{43} - 1254 q^{44} + 238 q^{45} - 1331 q^{46} + 1499 q^{47} - 1460 q^{48} - 413 q^{49} - 2459 q^{50} - 2350 q^{51} - 2745 q^{53} - 845 q^{54} - 253 q^{55} - 2904 q^{56} + 1450 q^{57} - 2509 q^{58} + 2285 q^{59} - 3566 q^{60} - 6218 q^{61} - 911 q^{62} - 1930 q^{63} + 67 q^{64} - 847 q^{66} + 546 q^{67} - 170 q^{68} - 5254 q^{69} - 2195 q^{70} - 263 q^{71} - 2393 q^{72} - 1148 q^{73} + 775 q^{74} - 5385 q^{75} - 7247 q^{76} + 44 q^{77} - 3666 q^{79} + 5594 q^{80} - 1901 q^{81} - 4414 q^{82} + 2722 q^{83} - 9971 q^{84} + 1858 q^{85} + 2478 q^{86} - 2284 q^{87} + 231 q^{88} + 13 q^{89} - 6771 q^{90} - 2232 q^{92} - 1082 q^{93} - 7330 q^{94} - 2352 q^{95} + 5770 q^{96} - 1197 q^{97} + 6813 q^{98} - 2860 q^{99}+O(q^{100}) \) 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 −5.49304 −8.44279 22.1734 19.5698 46.3765 18.0683 −77.8552 44.2807 −107.497
1.2 −5.13075 4.07791 18.3246 9.39838 −20.9227 18.3779 −52.9729 −10.3707 −48.2207
1.3 −5.06016 −3.81429 17.6052 13.6056 19.3009 −15.5193 −48.6040 −12.4512 −68.8465
1.4 −4.86773 −7.77085 15.6948 −6.79003 37.8264 12.1965 −37.4563 33.3861 33.0520
1.5 −4.62264 0.223641 13.3688 −5.07801 −1.03381 13.5633 −24.8183 −26.9500 23.4738
1.6 −3.79953 4.80997 6.43642 1.29866 −18.2756 −1.62019 5.94086 −3.86420 −4.93429
1.7 −3.55980 −3.90055 4.67220 −19.3357 13.8852 18.4314 11.8463 −11.7857 68.8313
1.8 −3.48032 −2.00729 4.11265 −5.42082 6.98601 −27.8265 13.5292 −22.9708 18.8662
1.9 −3.20015 −9.16277 2.24095 −10.3410 29.3222 5.43136 18.4298 56.9564 33.0929
1.10 −2.90122 6.23880 0.417058 15.8001 −18.1001 −9.46578 21.9998 11.9227 −45.8396
1.11 −2.83656 8.17655 0.0460816 −2.30923 −23.1933 3.15838 22.5618 39.8559 6.55028
1.12 −2.40771 −4.57703 −2.20293 18.4699 11.0201 11.7306 24.5657 −6.05084 −44.4701
1.13 −2.02818 −9.13071 −3.88650 −2.96871 18.5187 −15.4258 24.1079 56.3698 6.02107
1.14 −1.78462 −0.436688 −4.81514 13.8087 0.779321 −6.26730 22.8701 −26.8093 −24.6433
1.15 −1.63650 −2.06438 −5.32187 −14.3735 3.37836 −27.7797 21.8012 −22.7383 23.5223
1.16 −1.18198 −2.60560 −6.60292 −7.12046 3.07976 33.7625 17.2604 −20.2109 8.41624
1.17 −1.12752 9.49735 −6.72870 7.22240 −10.7085 −1.17565 16.6069 63.1996 −8.14341
1.18 −1.00172 −3.02985 −6.99655 20.0383 3.03507 −1.63570 15.0224 −17.8200 −20.0728
1.19 −0.532806 1.85539 −7.71612 −18.1107 −0.988563 −20.2803 8.37365 −23.5575 9.64951
1.20 −0.169295 6.82637 −7.97134 −1.80831 −1.15567 16.1277 2.70387 19.5993 0.306138
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
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 1859.4.a.o yes 39
13.b even 2 1 1859.4.a.n 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.4.a.n 39 13.b even 2 1
1859.4.a.o yes 39 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{39} - 213 T_{2}^{37} + 7 T_{2}^{36} + 20695 T_{2}^{35} - 1070 T_{2}^{34} - 1215940 T_{2}^{33} + 69288 T_{2}^{32} + 48297883 T_{2}^{31} - 2389752 T_{2}^{30} - 1373688710 T_{2}^{29} + \cdots - 91431516471296 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\). Copy content Toggle raw display