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Properties

Label 8009.2.a.b
Level $8009$
Weight $2$
Character orbit 8009.a
Self dual yes
Analytic conductor $63.952$
Analytic rank $0$
Dimension $361$
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: \(0\)
Dimension: \(361\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 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 −2.82633 2.37476 5.98811 1.91635 −6.71185 −2.63274 −11.2717 2.63950 −5.41622
1.2 −2.82341 −3.01045 5.97167 2.99967 8.49976 4.35875 −11.2137 6.06283 −8.46931
1.3 −2.80716 0.197946 5.88014 −1.81269 −0.555666 −3.48648 −10.8922 −2.96082 5.08851
1.4 −2.80354 −2.63483 5.85984 1.64291 7.38685 0.300217 −10.8212 3.94232 −4.60596
1.5 −2.78850 0.327361 5.77575 −3.12369 −0.912847 1.40614 −10.5287 −2.89283 8.71042
1.6 −2.76744 2.44821 5.65873 −3.94684 −6.77527 −1.82730 −10.1253 2.99372 10.9226
1.7 −2.74210 −3.01470 5.51909 −3.17594 8.26659 1.49397 −9.64969 6.08840 8.70874
1.8 −2.74193 −0.421446 5.51817 −1.04239 1.15558 0.747202 −9.64659 −2.82238 2.85815
1.9 −2.71747 −2.51708 5.38462 −3.75245 6.84009 −1.33469 −9.19758 3.33572 10.1972
1.10 −2.70375 −0.429580 5.31028 −1.54181 1.16148 3.87074 −8.95019 −2.81546 4.16866
1.11 −2.70301 1.53742 5.30627 −0.0405993 −4.15566 1.54013 −8.93687 −0.636347 0.109740
1.12 −2.69966 0.961365 5.28818 0.760429 −2.59536 −4.15994 −8.87697 −2.07578 −2.05290
1.13 −2.69652 −0.708536 5.27124 1.66526 1.91059 4.37208 −8.82098 −2.49798 −4.49040
1.14 −2.69232 2.96337 5.24860 −1.91075 −7.97835 3.66239 −8.74628 5.78157 5.14435
1.15 −2.65249 1.26544 5.03571 2.87387 −3.35656 2.78404 −8.05219 −1.39867 −7.62292
1.16 −2.64190 −1.64895 4.97966 −2.14113 4.35636 −4.48395 −7.87198 −0.280971 5.65667
1.17 −2.63785 2.50173 4.95827 3.39410 −6.59921 0.113907 −7.80348 3.25867 −8.95313
1.18 −2.63069 −2.68243 4.92052 0.701022 7.05664 −5.11853 −7.68299 4.19544 −1.84417
1.19 −2.63067 1.36161 4.92043 −3.62909 −3.58196 4.78654 −7.68268 −1.14601 9.54694
1.20 −2.62631 −2.69396 4.89749 2.82018 7.07518 −2.28331 −7.60969 4.25744 −7.40666
See next 80 embeddings (of 361 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.361
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.b ✓ 361
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8009.2.a.b ✓ 361 1.a even 1 1 trivial