Properties

Label 8011.2.a.a
Level $8011$
Weight $2$
Character orbit 8011.a
Self dual yes
Analytic conductor $63.968$
Analytic rank $1$
Dimension $309$
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] = [8011,2,Mod(1,8011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8011.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: \(63.9681570592\)
Analytic rank: \(1\)
Dimension: \(309\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 309 q - 33 q^{2} - 15 q^{3} + 273 q^{4} - 74 q^{5} - 32 q^{6} - 19 q^{7} - 93 q^{8} + 214 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 309 q - 33 q^{2} - 15 q^{3} + 273 q^{4} - 74 q^{5} - 32 q^{6} - 19 q^{7} - 93 q^{8} + 214 q^{9} - 23 q^{10} - 72 q^{11} - 42 q^{12} - 57 q^{13} - 77 q^{14} - 44 q^{15} + 205 q^{16} - 86 q^{17} - 82 q^{18} - 58 q^{19} - 134 q^{20} - 123 q^{21} - 31 q^{22} - 94 q^{23} - 84 q^{24} + 225 q^{25} - 92 q^{26} - 48 q^{27} - 36 q^{28} - 345 q^{29} - 85 q^{30} - 36 q^{31} - 199 q^{32} - 56 q^{33} - 28 q^{34} - 168 q^{35} + 65 q^{36} - 79 q^{37} - 66 q^{38} - 145 q^{39} - 54 q^{40} - 176 q^{41} - 48 q^{42} - 58 q^{43} - 194 q^{44} - 192 q^{45} - 44 q^{46} - 82 q^{47} - 81 q^{48} + 186 q^{49} - 206 q^{50} - 145 q^{51} - 86 q^{52} - 223 q^{53} - 117 q^{54} - 58 q^{55} - 216 q^{56} - 124 q^{57} - 151 q^{59} - 91 q^{60} - 184 q^{61} - 124 q^{62} - 78 q^{63} + 101 q^{64} - 194 q^{65} - 112 q^{66} - 53 q^{67} - 182 q^{68} - 243 q^{69} - 193 q^{71} - 208 q^{72} - 69 q^{73} - 236 q^{74} - 62 q^{75} - 142 q^{76} - 324 q^{77} - 20 q^{78} - 91 q^{79} - 223 q^{80} - 27 q^{81} + 2 q^{82} - 117 q^{83} - 157 q^{84} - 171 q^{85} - 203 q^{86} - 69 q^{87} - 36 q^{88} - 172 q^{89} - 10 q^{90} - 84 q^{91} - 226 q^{92} - 220 q^{93} - 96 q^{94} - 166 q^{95} - 118 q^{96} - 12 q^{97} - 116 q^{98} - 154 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.81833 1.84881 5.94296 −3.66460 −5.21056 −0.909168 −11.1126 0.418106 10.3280
1.2 −2.78702 −1.95051 5.76751 −0.484177 5.43612 0.734493 −10.5001 0.804484 1.34941
1.3 −2.76917 0.105018 5.66833 −1.01486 −0.290815 4.45070 −10.1582 −2.98897 2.81033
1.4 −2.75364 −0.00102412 5.58256 −3.57698 0.00282007 1.05818 −9.86509 −3.00000 9.84973
1.5 −2.74707 2.55044 5.54642 3.00412 −7.00626 1.74411 −9.74228 3.50477 −8.25255
1.6 −2.74693 2.61520 5.54560 −1.49229 −7.18376 4.10131 −9.73951 3.83927 4.09920
1.7 −2.74292 −0.194612 5.52364 4.36086 0.533805 −1.38725 −9.66507 −2.96213 −11.9615
1.8 −2.69623 −0.432527 5.26965 2.72420 1.16619 −0.628855 −8.81573 −2.81292 −7.34507
1.9 −2.68837 −2.21810 5.22734 1.47525 5.96309 −1.22716 −8.67630 1.91998 −3.96602
1.10 −2.67443 −1.89818 5.15257 −2.43152 5.07654 2.86566 −8.43133 0.603070 6.50294
1.11 −2.67381 −2.54689 5.14927 −3.71609 6.80992 3.32085 −8.42057 3.48667 9.93612
1.12 −2.67151 −0.346621 5.13696 1.63571 0.926001 −0.275244 −8.38042 −2.87985 −4.36981
1.13 −2.65734 −0.429161 5.06145 2.18236 1.14043 3.77949 −8.13530 −2.81582 −5.79927
1.14 −2.64340 2.84192 4.98759 0.696294 −7.51235 −3.08264 −7.89741 5.07651 −1.84059
1.15 −2.61747 −3.23549 4.85115 −2.81151 8.46879 −1.74803 −7.46281 7.46837 7.35905
1.16 −2.61660 −0.249789 4.84659 −1.85842 0.653599 −0.258344 −7.44839 −2.93761 4.86274
1.17 −2.58476 −1.81476 4.68100 2.17232 4.69073 −4.47910 −6.92974 0.293363 −5.61492
1.18 −2.57419 1.43469 4.62646 0.684490 −3.69317 3.09372 −6.76100 −0.941663 −1.76201
1.19 −2.56976 2.00808 4.60368 −0.878358 −5.16030 1.06025 −6.69084 1.03240 2.25717
1.20 −2.56050 2.53687 4.55617 −3.00342 −6.49566 2.39644 −6.54508 3.43571 7.69027
See next 80 embeddings (of 309 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.309
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(8011\) \(1\)

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 8011.2.a.a 309
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8011.2.a.a 309 1.a even 1 1 trivial