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Properties

Label 8011.2.a.b
Level $8011$
Weight $2$
Character orbit 8011.a
Self dual yes
Analytic conductor $63.968$
Analytic rank $0$
Dimension $358$
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: \(0\)
Dimension: \(358\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 358 q + 33 q^{2} + 11 q^{3} + 391 q^{4} + 76 q^{5} + 32 q^{6} + 19 q^{7} + 99 q^{8} + 451 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 358 q + 33 q^{2} + 11 q^{3} + 391 q^{4} + 76 q^{5} + 32 q^{6} + 19 q^{7} + 99 q^{8} + 451 q^{9} + 21 q^{10} + 70 q^{11} + 20 q^{12} + 53 q^{13} + 69 q^{14} + 28 q^{15} + 449 q^{16} + 88 q^{17} + 86 q^{18} + 44 q^{19} + 136 q^{20} + 125 q^{21} + 17 q^{22} + 104 q^{23} + 84 q^{24} + 444 q^{25} + 100 q^{26} + 32 q^{27} + 46 q^{28} + 373 q^{29} + 99 q^{30} + 30 q^{31} + 221 q^{32} + 56 q^{33} + 26 q^{34} + 164 q^{35} + 599 q^{36} + 81 q^{37} + 66 q^{38} + 143 q^{39} + 42 q^{40} + 182 q^{41} + 32 q^{42} + 40 q^{43} + 184 q^{44} + 198 q^{45} + 54 q^{46} + 66 q^{47} + 5 q^{48} + 479 q^{49} + 184 q^{50} + 123 q^{51} + 64 q^{52} + 221 q^{53} + 67 q^{54} + 38 q^{55} + 174 q^{56} + 84 q^{57} + 44 q^{58} + 127 q^{59} + 29 q^{60} + 174 q^{61} + 86 q^{62} + 48 q^{63} + 549 q^{64} + 202 q^{65} + 32 q^{66} + 29 q^{67} + 172 q^{68} + 249 q^{69} + 12 q^{70} + 185 q^{71} + 218 q^{72} + 57 q^{73} + 272 q^{74} + 24 q^{75} + 84 q^{76} + 384 q^{77} + 12 q^{78} + 93 q^{79} + 215 q^{80} + 702 q^{81} + 48 q^{82} + 121 q^{83} + 179 q^{84} + 177 q^{85} + 209 q^{86} + 91 q^{87} + 36 q^{88} + 186 q^{89} + 66 q^{90} + 32 q^{91} + 272 q^{92} + 220 q^{93} + 60 q^{94} + 170 q^{95} + 162 q^{96} + 22 q^{97} + 196 q^{98} + 152 q^{99}+O(q^{100}) \) 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 −2.81343 0.884173 5.91537 1.61259 −2.48756 −3.94545 −11.0156 −2.21824 −4.53689
1.2 −2.79741 3.22245 5.82553 −0.295886 −9.01452 1.71950 −10.7016 7.38416 0.827715
1.3 −2.79545 −2.80135 5.81452 0.0114194 7.83103 2.04645 −10.6633 4.84758 −0.0319223
1.4 −2.75246 −1.71159 5.57605 1.50064 4.71110 0.944355 −9.84296 −0.0704466 −4.13045
1.5 −2.70980 −2.88951 5.34304 −1.42877 7.83002 −3.83323 −9.05899 5.34929 3.87170
1.6 −2.70372 −2.07244 5.31009 4.20593 5.60330 4.19315 −8.94956 1.29501 −11.3717
1.7 −2.69935 1.61559 5.28646 2.80670 −4.36103 2.73763 −8.87130 −0.389878 −7.57626
1.8 −2.69679 2.11156 5.27265 −0.515809 −5.69442 −2.39275 −8.82564 1.45867 1.39103
1.9 −2.67959 −1.76320 5.18019 −3.93630 4.72465 −2.55729 −8.52159 0.108876 10.5477
1.10 −2.66752 1.67561 5.11568 −2.95924 −4.46972 −2.28368 −8.31114 −0.192344 7.89385
1.11 −2.64605 −0.0865598 5.00159 −1.46630 0.229042 −4.80741 −7.94236 −2.99251 3.87990
1.12 −2.62246 −2.12328 4.87729 2.57436 5.56822 −3.69406 −7.54557 1.50833 −6.75115
1.13 −2.61876 −0.901244 4.85788 −1.31265 2.36014 −2.24388 −7.48409 −2.18776 3.43751
1.14 −2.61809 −3.37147 4.85437 3.62871 8.82680 −2.17662 −7.47299 8.36681 −9.50027
1.15 −2.61797 1.26070 4.85374 0.306561 −3.30046 −1.43849 −7.47100 −1.41065 −0.802565
1.16 −2.59714 −2.89463 4.74512 2.71586 7.51774 −3.02593 −7.12945 5.37887 −7.05346
1.17 −2.57524 0.178143 4.63185 −1.43369 −0.458760 4.00928 −6.77765 −2.96827 3.69209
1.18 −2.56990 3.01881 4.60437 3.51758 −7.75803 −1.89430 −6.69296 6.11321 −9.03981
1.19 −2.56153 2.54144 4.56142 −3.00625 −6.50997 −3.61687 −6.56114 3.45892 7.70058
1.20 −2.55695 −3.16452 4.53798 −0.685875 8.09151 1.59461 −6.48950 7.01419 1.75375
See next 80 embeddings (of 358 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.358
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.b ✓ 358
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8011.2.a.b ✓ 358 1.a even 1 1 trivial