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Properties

Label 8035.2.a.b
Level $8035$
Weight $2$
Character orbit 8035.a
Self dual yes
Analytic conductor $64.160$
Analytic rank $1$
Dimension $114$
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] = [8035,2,Mod(1,8035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8035, 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("8035.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8035 = 5 \cdot 1607 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8035.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: \(64.1597980241\)
Analytic rank: \(1\)
Dimension: \(114\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 114 q - 17 q^{2} - 10 q^{3} + 93 q^{4} + 114 q^{5} - 24 q^{6} - 11 q^{7} - 48 q^{8} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 114 q - 17 q^{2} - 10 q^{3} + 93 q^{4} + 114 q^{5} - 24 q^{6} - 11 q^{7} - 48 q^{8} + 66 q^{9} - 17 q^{10} - 44 q^{11} - 25 q^{12} - 26 q^{13} - 43 q^{14} - 10 q^{15} + 59 q^{16} - 57 q^{17} - 33 q^{18} - 69 q^{19} + 93 q^{20} - 107 q^{21} - 19 q^{22} - 45 q^{23} - 45 q^{24} + 114 q^{25} - 54 q^{26} - 34 q^{27} - 6 q^{28} - 166 q^{29} - 24 q^{30} - 67 q^{31} - 98 q^{32} - 38 q^{33} - 41 q^{34} - 11 q^{35} - 3 q^{36} - 44 q^{37} - 19 q^{38} - 66 q^{39} - 48 q^{40} - 141 q^{41} + q^{42} - 30 q^{43} - 125 q^{44} + 66 q^{45} - 59 q^{46} - 17 q^{47} - 35 q^{48} - 15 q^{49} - 17 q^{50} - 67 q^{51} - 26 q^{52} - 154 q^{53} - 45 q^{54} - 44 q^{55} - 118 q^{56} - 70 q^{57} + 11 q^{58} - 75 q^{59} - 25 q^{60} - 144 q^{61} - 35 q^{62} - 25 q^{63} + 16 q^{64} - 26 q^{65} - 68 q^{66} - 2 q^{67} - 99 q^{68} - 118 q^{69} - 43 q^{70} - 104 q^{71} - 73 q^{72} - 22 q^{73} - 107 q^{74} - 10 q^{75} - 172 q^{76} - 100 q^{77} - 2 q^{78} - 91 q^{79} + 59 q^{80} - 54 q^{81} + 20 q^{82} - 44 q^{83} - 156 q^{84} - 57 q^{85} - 50 q^{86} + 5 q^{87} - 13 q^{88} - 150 q^{89} - 33 q^{90} - 54 q^{91} - 77 q^{92} - 50 q^{93} - 105 q^{94} - 69 q^{95} - 78 q^{96} - 31 q^{97} - 64 q^{98} - 101 q^{99}+O(q^{100}) \) 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 −2.80100 2.46905 5.84558 1.00000 −6.91579 0.471155 −10.7715 3.09620 −2.80100
1.2 −2.78782 −2.79744 5.77196 1.00000 7.79878 4.93098 −10.5155 4.82570 −2.78782
1.3 −2.71845 −0.706981 5.38997 1.00000 1.92189 −1.47518 −9.21545 −2.50018 −2.71845
1.4 −2.70261 −0.768131 5.30408 1.00000 2.07595 0.883099 −8.92963 −2.40998 −2.70261
1.5 −2.64032 0.589008 4.97127 1.00000 −1.55517 2.76981 −7.84509 −2.65307 −2.64032
1.6 −2.56613 −1.08918 4.58500 1.00000 2.79497 0.189836 −6.63344 −1.81369 −2.56613
1.7 −2.55456 3.12928 4.52579 1.00000 −7.99394 −0.305694 −6.45228 6.79240 −2.55456
1.8 −2.52790 2.08229 4.39026 1.00000 −5.26382 −1.56131 −6.04233 1.33594 −2.52790
1.9 −2.52703 0.421733 4.38589 1.00000 −1.06573 −3.93159 −6.02922 −2.82214 −2.52703
1.10 −2.49776 0.530865 4.23883 1.00000 −1.32598 3.05763 −5.59207 −2.71818 −2.49776
1.11 −2.43125 1.69464 3.91096 1.00000 −4.12009 3.32974 −4.64601 −0.128198 −2.43125
1.12 −2.42522 −2.38106 3.88170 1.00000 5.77461 1.76614 −4.56353 2.66946 −2.42522
1.13 −2.41255 0.682970 3.82038 1.00000 −1.64770 −2.72500 −4.39175 −2.53355 −2.41255
1.14 −2.39306 2.37672 3.72672 1.00000 −5.68762 −3.85134 −4.13213 2.64878 −2.39306
1.15 −2.24666 −1.32930 3.04747 1.00000 2.98648 −2.44152 −2.35330 −1.23296 −2.24666
1.16 −2.22296 −1.69364 2.94157 1.00000 3.76490 0.866301 −2.09307 −0.131588 −2.22296
1.17 −2.17370 0.0354109 2.72495 1.00000 −0.0769725 3.07468 −1.57583 −2.99875 −2.17370
1.18 −2.14559 −2.44526 2.60354 1.00000 5.24651 −2.98560 −1.29496 2.97929 −2.14559
1.19 −2.12278 −3.22748 2.50621 1.00000 6.85125 1.25827 −1.07458 7.41664 −2.12278
1.20 −2.11941 −2.29188 2.49192 1.00000 4.85744 5.02774 −1.04258 2.25270 −2.11941
See next 80 embeddings (of 114 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.114
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1607\) \(-1\)

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 8035.2.a.b ✓ 114
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8035.2.a.b ✓ 114 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{114} + 17 T_{2}^{113} - 16 T_{2}^{112} - 1871 T_{2}^{111} - 6557 T_{2}^{110} + 92419 T_{2}^{109} + \cdots - 8540707 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8035))\). Copy content Toggle raw display