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Properties

Label 8009.2.a.a
Level $8009$
Weight $2$
Character orbit 8009.a
Self dual yes
Analytic conductor $63.952$
Analytic rank $1$
Dimension $306$
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] = [8009,2,Mod(1,8009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8009, 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("8009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8009 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8009.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: \(63.9521869788\)
Analytic rank: \(1\)
Dimension: \(306\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 306 q - 13 q^{2} - 25 q^{3} + 253 q^{4} - 25 q^{5} - 49 q^{6} - 102 q^{7} - 33 q^{8} + 251 q^{9} - 61 q^{10} - 43 q^{11} - 50 q^{12} - 89 q^{13} - 40 q^{14} - 61 q^{15} + 151 q^{16} - 52 q^{17} - 57 q^{18} - 185 q^{19} - 66 q^{20} - 63 q^{21} - 55 q^{22} - 62 q^{23} - 131 q^{24} + 209 q^{25} - 57 q^{26} - 88 q^{27} - 182 q^{28} - 67 q^{29} - 68 q^{30} - 240 q^{31} - 64 q^{32} - 52 q^{33} - 128 q^{34} - 99 q^{35} + 106 q^{36} - 49 q^{37} - 45 q^{38} - 190 q^{39} - 158 q^{40} - 72 q^{41} - 36 q^{42} - 141 q^{43} - 80 q^{44} - 100 q^{45} - 91 q^{46} - 105 q^{47} - 85 q^{48} + 116 q^{49} - 51 q^{50} - 145 q^{51} - 237 q^{52} - 48 q^{53} - 156 q^{54} - 420 q^{55} - 116 q^{56} - 35 q^{57} - 43 q^{58} - 139 q^{59} - 73 q^{60} - 233 q^{61} - 58 q^{62} - 252 q^{63} - 3 q^{64} - 45 q^{65} - 127 q^{66} - 108 q^{67} - 85 q^{68} - 164 q^{69} - 56 q^{70} - 131 q^{71} - 117 q^{72} - 118 q^{73} - 47 q^{74} - 112 q^{75} - 389 q^{76} - 36 q^{77} + 9 q^{78} - 382 q^{79} - 119 q^{80} + 102 q^{81} - 131 q^{82} - 59 q^{83} - 144 q^{84} - 140 q^{85} - 38 q^{86} - 301 q^{87} - 131 q^{88} - 98 q^{89} - 138 q^{90} - 176 q^{91} - 97 q^{92} - 60 q^{93} - 342 q^{94} - 154 q^{95} - 243 q^{96} - 109 q^{97} - 21 q^{98} - 173 q^{99}+O(q^{100}) \) 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 −2.77719 −1.72357 5.71279 −0.359137 4.78667 −0.363859 −10.3111 −0.0293152 0.997391
1.2 −2.76073 2.52985 5.62166 0.0311099 −6.98425 0.584590 −9.99844 3.40015 −0.0858861
1.3 −2.72646 1.03232 5.43357 3.59624 −2.81456 2.40174 −9.36147 −1.93433 −9.80500
1.4 −2.72316 −0.885304 5.41558 2.13071 2.41082 −0.787494 −9.30115 −2.21624 −5.80224
1.5 −2.70598 −0.529374 5.32233 3.62605 1.43248 −4.04119 −8.99016 −2.71976 −9.81202
1.6 −2.66884 3.40767 5.12268 0.00216831 −9.09450 −2.11405 −8.33393 8.61218 −0.00578687
1.7 −2.66627 −0.991276 5.10897 −2.13375 2.64301 1.36601 −8.28935 −2.01737 5.68914
1.8 −2.64630 2.23042 5.00289 −0.684253 −5.90235 0.773793 −7.94655 1.97477 1.81074
1.9 −2.63809 −1.56073 4.95951 2.27175 4.11735 −0.895063 −7.80745 −0.564115 −5.99307
1.10 −2.63380 −2.43471 4.93693 −0.366932 6.41255 4.18826 −7.73529 2.92781 0.966426
1.11 −2.60787 1.96510 4.80101 3.32724 −5.12474 0.944759 −7.30469 0.861625 −8.67702
1.12 −2.59381 0.956385 4.72785 −4.27828 −2.48068 1.62544 −7.07552 −2.08533 11.0970
1.13 −2.58891 −2.43363 4.70248 −1.53347 6.30046 1.06858 −6.99648 2.92255 3.97002
1.14 −2.58035 2.52546 4.65823 0.386293 −6.51658 −3.83630 −6.85917 3.37795 −0.996774
1.15 −2.56583 −0.0859559 4.58350 0.644754 0.220548 −0.863168 −6.62882 −2.99261 −1.65433
1.16 −2.56252 1.99928 4.56650 −3.64858 −5.12319 −3.78692 −6.57671 0.997123 9.34955
1.17 −2.55906 0.342269 4.54881 −2.51856 −0.875888 −3.84856 −6.52258 −2.88285 6.44516
1.18 −2.52963 0.577486 4.39902 −1.57582 −1.46082 −2.02955 −6.06864 −2.66651 3.98624
1.19 −2.52232 −2.77220 4.36210 3.44597 6.99239 −1.87628 −5.95798 4.68511 −8.69185
1.20 −2.50471 3.07842 4.27357 4.01883 −7.71055 −2.99037 −5.69464 6.47666 −10.0660
See next 80 embeddings (of 306 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.306
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(8009\) \(1\)

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 8009.2.a.a ✓ 306
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8009.2.a.a ✓ 306 1.a even 1 1 trivial