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Properties

Label 8045.2.a.c
Level $8045$
Weight $2$
Character orbit 8045.a
Self dual yes
Analytic conductor $64.240$
Analytic rank $1$
Dimension $127$
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] = [8045,2,Mod(1,8045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8045, 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("8045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8045.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: \(64.2396484261\)
Analytic rank: \(1\)
Dimension: \(127\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 127 q - 20 q^{2} - 31 q^{3} + 116 q^{4} + 127 q^{5} - 17 q^{6} - 63 q^{7} - 57 q^{8} + 122 q^{9} - 20 q^{10} - 32 q^{11} - 65 q^{12} - 49 q^{13} - 4 q^{14} - 31 q^{15} + 98 q^{16} - 53 q^{17} - 60 q^{18} - 126 q^{19} + 116 q^{20} - 24 q^{21} - 46 q^{22} - 121 q^{23} - 51 q^{24} + 127 q^{25} - 9 q^{26} - 112 q^{27} - 123 q^{28} - 6 q^{29} - 17 q^{30} - 54 q^{31} - 119 q^{32} - 57 q^{33} - 53 q^{34} - 63 q^{35} + 133 q^{36} - 61 q^{37} - 27 q^{38} - 33 q^{39} - 57 q^{40} - 8 q^{41} - 46 q^{42} - 208 q^{43} - 54 q^{44} + 122 q^{45} - 34 q^{46} - 116 q^{47} - 106 q^{48} + 110 q^{49} - 20 q^{50} - 54 q^{51} - 142 q^{52} - 60 q^{53} - 62 q^{54} - 32 q^{55} + 33 q^{56} - 56 q^{57} - 87 q^{58} - 53 q^{59} - 65 q^{60} - 76 q^{61} - 84 q^{62} - 215 q^{63} + 67 q^{64} - 49 q^{65} + 9 q^{66} - 145 q^{67} - 133 q^{68} + q^{69} - 4 q^{70} - 4 q^{71} - 167 q^{72} - 155 q^{73} - 14 q^{74} - 31 q^{75} - 199 q^{76} - 97 q^{77} - 24 q^{78} - 73 q^{79} + 98 q^{80} + 127 q^{81} - 69 q^{82} - 225 q^{83} - 59 q^{84} - 53 q^{85} + 30 q^{86} - 179 q^{87} - 119 q^{88} - 25 q^{89} - 60 q^{90} - 160 q^{91} - 188 q^{92} - 44 q^{93} - 32 q^{94} - 126 q^{95} - 43 q^{96} - 72 q^{97} - 111 q^{98} - 141 q^{99}+O(q^{100}) \) 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 −2.82452 −1.42855 5.97791 1.00000 4.03498 −1.55315 −11.2357 −0.959233 −2.82452
1.2 −2.74468 −3.15686 5.53326 1.00000 8.66457 −4.01431 −9.69765 6.96577 −2.74468
1.3 −2.72254 3.22119 5.41223 1.00000 −8.76982 −3.61577 −9.28994 7.37607 −2.72254
1.4 −2.71791 −0.717872 5.38703 1.00000 1.95111 −2.87590 −9.20565 −2.48466 −2.71791
1.5 −2.69637 1.21455 5.27040 1.00000 −3.27487 2.45597 −8.81819 −1.52487 −2.69637
1.6 −2.69512 −2.02504 5.26369 1.00000 5.45774 1.98108 −8.79604 1.10080 −2.69512
1.7 −2.68498 2.75441 5.20913 1.00000 −7.39554 −2.94460 −8.61647 4.58677 −2.68498
1.8 −2.61813 0.911081 4.85463 1.00000 −2.38533 −2.60374 −7.47379 −2.16993 −2.61813
1.9 −2.53777 0.243430 4.44030 1.00000 −0.617771 4.53173 −6.19292 −2.94074 −2.53777
1.10 −2.52291 −2.56612 4.36506 1.00000 6.47407 −1.96305 −5.96681 3.58496 −2.52291
1.11 −2.51055 −0.644369 4.30288 1.00000 1.61772 −0.493808 −5.78150 −2.58479 −2.51055
1.12 −2.49863 2.41476 4.24316 1.00000 −6.03361 1.44699 −5.60483 2.83108 −2.49863
1.13 −2.47054 −3.37029 4.10358 1.00000 8.32645 4.71800 −5.19700 8.35885 −2.47054
1.14 −2.44032 0.296251 3.95518 1.00000 −0.722948 −5.11077 −4.77127 −2.91224 −2.44032
1.15 −2.41609 −2.71250 3.83748 1.00000 6.55363 −4.44734 −4.43951 4.35764 −2.41609
1.16 −2.35235 −3.18470 3.53353 1.00000 7.49152 −1.35488 −3.60740 7.14233 −2.35235
1.17 −2.31881 0.00298266 3.37687 1.00000 −0.00691620 −0.956165 −3.19269 −2.99999 −2.31881
1.18 −2.26786 −1.17957 3.14318 1.00000 2.67510 2.15242 −2.59257 −1.60862 −2.26786
1.19 −2.26133 1.69211 3.11360 1.00000 −3.82641 −1.09235 −2.51822 −0.136771 −2.26133
1.20 −2.24197 1.72389 3.02644 1.00000 −3.86491 −1.20930 −2.30126 −0.0282099 −2.24197
See next 80 embeddings (of 127 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.127
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1609\) \(-1\)

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 8045.2.a.c ✓ 127
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8045.2.a.c ✓ 127 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{127} + 20 T_{2}^{126} + 15 T_{2}^{125} - 2321 T_{2}^{124} - 12760 T_{2}^{123} + 114080 T_{2}^{122} + \cdots + 154823 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\). Copy content Toggle raw display