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Properties

Label 1859.2.a.t
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1859,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1859.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(14.8441897358\)
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

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 −2.76912 −0.0832308 5.66800 −1.06756 0.230476 −3.59787 −10.1571 −2.99307 2.95619
1.2 −2.70607 0.960218 5.32282 −3.57279 −2.59842 3.95601 −8.99178 −2.07798 9.66821
1.3 −2.68347 3.12668 5.20101 2.33476 −8.39034 0.266736 −8.58980 6.77612 −6.26526
1.4 −2.32475 2.00106 3.40448 3.04764 −4.65197 −5.08642 −3.26506 1.00425 −7.08500
1.5 −1.77144 2.92005 1.13800 −2.22672 −5.17269 0.178675 1.52699 5.52669 3.94449
1.6 −1.53678 −3.28188 0.361702 −1.21246 5.04353 −1.98662 2.51771 7.77071 1.86328
1.7 −1.50636 −0.544988 0.269129 3.40659 0.820949 −0.714897 2.60732 −2.70299 −5.13156
1.8 −0.930026 2.53271 −1.13505 −3.38691 −2.35548 −3.38196 2.91568 3.41460 3.14991
1.9 −0.603243 −0.154529 −1.63610 4.17643 0.0932187 2.73997 2.19345 −2.97612 −2.51940
1.10 −0.392741 1.69948 −1.84575 1.18432 −0.667456 5.07600 1.51039 −0.111771 −0.465133
1.11 −0.340522 −1.21327 −1.88404 −3.18769 0.413146 4.05402 1.32260 −1.52797 1.08548
1.12 0.149116 −3.21882 −1.97776 1.07607 −0.479977 −3.58386 −0.593149 7.36077 0.160459
1.13 0.776244 3.01008 −1.39745 0.0164567 2.33656 0.793722 −2.63725 6.06060 0.0127744
1.14 1.17546 −1.22926 −0.618298 −1.06938 −1.44494 −3.67782 −3.07770 −1.48892 −1.25701
1.15 1.43463 0.655312 0.0581570 1.45609 0.940129 3.21683 −2.78582 −2.57057 2.08894
1.16 2.13563 −2.20961 2.56090 −1.38469 −4.71890 5.12151 1.19787 1.88237 −2.95718
1.17 2.14442 −1.72478 2.59856 −3.54668 −3.69866 −2.57025 1.28356 −0.0251423 −7.60558
1.18 2.23194 1.36188 2.98154 4.28736 3.03963 0.384100 2.19073 −1.14528 9.56911
1.19 2.26490 3.17598 3.12978 2.32745 7.19329 −4.39568 2.55884 7.08687 5.27144
1.20 2.56428 1.12676 4.57554 1.24792 2.88933 2.50384 6.60441 −1.73041 3.20001
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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 1859.2.a.t yes 21
13.b even 2 1 1859.2.a.s ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.2.a.s ✓ 21 13.b even 2 1
1859.2.a.t yes 21 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{21} - 37 T_{2}^{19} + T_{2}^{18} + 582 T_{2}^{17} - 20 T_{2}^{16} - 5075 T_{2}^{15} + 99 T_{2}^{14} + \cdots + 448 \) Copy content Toggle raw display
\( T_{7}^{21} - T_{7}^{20} - 109 T_{7}^{19} + 94 T_{7}^{18} + 4999 T_{7}^{17} - 3763 T_{7}^{16} + \cdots + 2981888 \) Copy content Toggle raw display