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Properties

Label 8035.2.a.e
Level $8035$
Weight $2$
Character orbit 8035.a
Self dual yes
Analytic conductor $64.160$
Analytic rank $0$
Dimension $153$
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: \(0\)
Dimension: \(153\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 153 q + 18 q^{2} + 7 q^{3} + 176 q^{4} + 153 q^{5} + 19 q^{6} + 5 q^{7} + 57 q^{8} + 206 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 153 q + 18 q^{2} + 7 q^{3} + 176 q^{4} + 153 q^{5} + 19 q^{6} + 5 q^{7} + 57 q^{8} + 206 q^{9} + 18 q^{10} + 38 q^{11} + 14 q^{12} + 28 q^{13} + 53 q^{14} + 7 q^{15} + 214 q^{16} + 50 q^{17} + 47 q^{18} + 65 q^{19} + 176 q^{20} + 109 q^{21} + 13 q^{22} + 52 q^{23} + 66 q^{24} + 153 q^{25} + 36 q^{26} + 19 q^{27} + 26 q^{28} + 172 q^{29} + 19 q^{30} + 60 q^{31} + 107 q^{32} + 4 q^{33} + 40 q^{34} + 5 q^{35} + 241 q^{36} + 65 q^{37} + 29 q^{38} + 56 q^{39} + 57 q^{40} + 152 q^{41} - 19 q^{42} + 22 q^{43} + 97 q^{44} + 206 q^{45} + 86 q^{46} + 37 q^{47} - 4 q^{48} + 260 q^{49} + 18 q^{50} + 102 q^{51} - 6 q^{52} + 169 q^{53} + 64 q^{54} + 38 q^{55} + 146 q^{56} + 40 q^{57} - 9 q^{58} + 64 q^{59} + 14 q^{60} + 164 q^{61} + 12 q^{62} + 19 q^{63} + 259 q^{64} + 28 q^{65} + 6 q^{66} + 5 q^{67} + 112 q^{68} + 119 q^{69} + 53 q^{70} + 100 q^{71} + 77 q^{72} + 10 q^{73} + 98 q^{74} + 7 q^{75} + 126 q^{76} + 80 q^{77} - 4 q^{78} + 110 q^{79} + 214 q^{80} + 305 q^{81} - 27 q^{82} + 36 q^{83} + 172 q^{84} + 50 q^{85} + 44 q^{86} + 23 q^{87} + 47 q^{88} + 143 q^{89} + 47 q^{90} + 82 q^{91} + 130 q^{92} + 31 q^{93} + 77 q^{94} + 65 q^{95} + 57 q^{96} + 11 q^{97} + 29 q^{98} + 99 q^{99}+O(q^{100}) \) 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 −2.77624 1.54873 5.70752 1.00000 −4.29965 −0.285554 −10.2930 −0.601430 −2.77624
1.2 −2.71462 2.16303 5.36914 1.00000 −5.87179 −4.59135 −9.14592 1.67870 −2.71462
1.3 −2.69988 −0.494659 5.28933 1.00000 1.33552 3.45834 −8.88080 −2.75531 −2.69988
1.4 −2.68656 −0.551136 5.21758 1.00000 1.48066 −4.17471 −8.64421 −2.69625 −2.68656
1.5 −2.65503 −2.98031 5.04918 1.00000 7.91280 −3.81633 −8.09566 5.88223 −2.65503
1.6 −2.62009 −2.60275 4.86488 1.00000 6.81943 0.626563 −7.50624 3.77428 −2.62009
1.7 −2.59466 −3.04599 4.73224 1.00000 7.90331 −0.447166 −7.08923 6.27808 −2.59466
1.8 −2.59020 −1.81242 4.70914 1.00000 4.69454 3.39615 −7.01723 0.284882 −2.59020
1.9 −2.55001 2.17323 4.50255 1.00000 −5.54177 4.37483 −6.38153 1.72295 −2.55001
1.10 −2.54388 1.32249 4.47134 1.00000 −3.36426 2.08615 −6.28679 −1.25102 −2.54388
1.11 −2.52944 −2.42468 4.39805 1.00000 6.13306 0.256968 −6.06570 2.87905 −2.52944
1.12 −2.51924 −1.73731 4.34657 1.00000 4.37670 −5.14815 −5.91158 0.0182398 −2.51924
1.13 −2.41868 3.20867 3.85002 1.00000 −7.76074 −2.07887 −4.47460 7.29555 −2.41868
1.14 −2.41769 −0.844619 3.84521 1.00000 2.04202 0.104455 −4.46113 −2.28662 −2.41769
1.15 −2.33215 −3.21462 3.43894 1.00000 7.49699 −4.86714 −3.35583 7.33378 −2.33215
1.16 −2.32037 2.64444 3.38410 1.00000 −6.13607 4.54944 −3.21162 3.99307 −2.32037
1.17 −2.31112 3.30909 3.34126 1.00000 −7.64768 0.225615 −3.09980 7.95005 −2.31112
1.18 −2.29854 0.376180 3.28327 1.00000 −0.864664 −1.95111 −2.94965 −2.85849 −2.29854
1.19 −2.28006 1.15086 3.19869 1.00000 −2.62404 −0.851041 −2.73309 −1.67552 −2.28006
1.20 −2.27555 1.50447 3.17814 1.00000 −3.42350 2.50382 −2.68091 −0.736575 −2.27555
See next 80 embeddings (of 153 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.153
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.e ✓ 153
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8035.2.a.e ✓ 153 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{153} - 18 T_{2}^{152} - 79 T_{2}^{151} + 3323 T_{2}^{150} - 5477 T_{2}^{149} + \cdots + 145148388552576 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8035))\). Copy content Toggle raw display