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Properties

Label 1859.4.a.m
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1859,4,Mod(1,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1859.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.684550701\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 143)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{2} + 12 q^{3} + 152 q^{4} + 40 q^{5} + 98 q^{6} + 56 q^{7} + 84 q^{8} + 360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{2} + 12 q^{3} + 152 q^{4} + 40 q^{5} + 98 q^{6} + 56 q^{7} + 84 q^{8} + 360 q^{9} - 56 q^{10} + 396 q^{11} + 66 q^{12} + 164 q^{14} + 120 q^{15} + 644 q^{16} + 138 q^{17} - 28 q^{18} + 498 q^{19} + 320 q^{20} + 404 q^{21} + 44 q^{22} + 46 q^{23} + 1340 q^{24} + 818 q^{25} + 24 q^{27} + 1382 q^{28} + 262 q^{29} + 104 q^{30} + 660 q^{31} + 106 q^{32} + 132 q^{33} - 640 q^{34} + 68 q^{35} + 1960 q^{36} + 1372 q^{37} + 224 q^{38} + 108 q^{40} + 1216 q^{41} + 82 q^{42} + 436 q^{43} + 1672 q^{44} + 820 q^{45} + 2512 q^{46} - 140 q^{47} - 1944 q^{48} + 1192 q^{49} + 3484 q^{50} + 712 q^{51} + 146 q^{53} + 3078 q^{54} + 440 q^{55} - 102 q^{56} + 3656 q^{57} - 324 q^{58} + 404 q^{59} - 2944 q^{60} - 590 q^{61} + 1776 q^{62} + 4364 q^{63} + 2514 q^{64} + 1078 q^{66} + 2236 q^{67} - 1530 q^{68} + 12 q^{69} + 4852 q^{70} + 292 q^{71} + 2990 q^{72} + 1936 q^{73} + 1568 q^{74} + 4688 q^{75} + 7398 q^{76} + 616 q^{77} - 1464 q^{79} - 808 q^{80} + 3780 q^{81} - 4072 q^{82} + 4422 q^{83} - 2678 q^{84} + 3700 q^{85} - 2588 q^{86} - 1832 q^{87} + 924 q^{88} + 6856 q^{89} - 3422 q^{90} - 668 q^{92} + 840 q^{93} - 4166 q^{94} - 256 q^{95} + 18290 q^{96} + 2776 q^{97} + 4608 q^{98} + 3960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.60356 −9.06169 23.3999 −0.564007 50.7777 5.19556 −86.2940 55.1142 3.16045
1.2 −5.08212 −1.73174 17.8280 8.37251 8.80093 −4.33663 −49.9469 −24.0011 −42.5501
1.3 −4.85206 3.92771 15.5425 −7.28280 −19.0575 33.6189 −36.5968 −11.5731 35.3366
1.4 −4.68172 1.61215 13.9185 −5.38645 −7.54762 −1.36204 −27.7085 −24.4010 25.2178
1.5 −4.47216 −0.713439 12.0002 8.38905 3.19061 −17.1080 −17.8896 −26.4910 −37.5171
1.6 −4.21733 −9.25269 9.78589 13.5656 39.0217 15.6023 −7.53168 58.6123 −57.2105
1.7 −3.84193 8.79911 6.76044 17.0951 −33.8056 −10.3600 4.76230 50.4244 −65.6783
1.8 −3.44987 −6.27760 3.90158 −11.8272 21.6569 15.4267 14.1390 12.4083 40.8021
1.9 −3.18974 9.92242 2.17445 1.24067 −31.6500 27.3151 18.5820 71.4545 −3.95741
1.10 −3.08237 2.36462 1.50103 −12.3909 −7.28863 −19.5172 20.0323 −21.4086 38.1933
1.11 −3.07095 5.81732 1.43071 4.09794 −17.8647 −12.1390 20.1739 6.84123 −12.5845
1.12 −1.74714 −6.49445 −4.94751 15.6988 11.3467 −25.8156 22.6211 15.1779 −27.4280
1.13 −1.61805 −2.32647 −5.38191 −8.11969 3.76435 −9.06436 21.6526 −21.5875 13.1381
1.14 −1.08936 2.82223 −6.81329 21.6301 −3.07443 −28.6310 16.1371 −19.0350 −23.5631
1.15 −0.956991 −1.98410 −7.08417 15.0434 1.89877 18.0966 14.4354 −23.0633 −14.3964
1.16 −0.869842 −9.93949 −7.24337 −11.3907 8.64579 −27.9317 13.2593 71.7934 9.90807
1.17 −0.677132 5.62189 −7.54149 −21.4343 −3.80677 10.4645 10.5236 4.60569 14.5139
1.18 −0.665052 −3.21477 −7.55771 10.5229 2.13799 28.9946 10.3467 −16.6653 −6.99830
1.19 −0.231915 8.68804 −7.94622 0.297636 −2.01489 7.88297 3.69817 48.4820 −0.0690264
1.20 0.236535 −7.74253 −7.94405 −11.7890 −1.83138 −5.71517 −3.77132 32.9468 −2.78850
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.36
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.m 36
13.b even 2 1 1859.4.a.l 36
13.f odd 12 2 143.4.j.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.j.a ✓ 72 13.f odd 12 2
1859.4.a.l 36 13.b even 2 1
1859.4.a.m 36 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 4 T_{2}^{35} - 212 T_{2}^{34} + 820 T_{2}^{33} + 20423 T_{2}^{32} - 75918 T_{2}^{31} + \cdots - 43206514311168 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\). Copy content Toggle raw display