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Properties

Label 8003.2.a.c
Level $8003$
Weight $2$
Character orbit 8003.a
Self dual yes
Analytic conductor $63.904$
Analytic rank $0$
Dimension $172$
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] = [8003,2,Mod(1,8003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8003, 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("8003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8003 = 53 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8003.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: \(63.9042767376\)
Analytic rank: \(0\)
Dimension: \(172\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 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 −2.79279 3.40925 5.79969 2.64273 −9.52131 3.16263 −10.6117 8.62295 −7.38060
1.2 −2.76521 1.30852 5.64636 −2.58270 −3.61833 −0.870652 −10.0829 −1.28777 7.14170
1.3 −2.76129 2.33703 5.62470 −2.36515 −6.45322 1.79490 −10.0088 2.46173 6.53087
1.4 −2.74848 0.238468 5.55414 −0.412520 −0.655426 −2.19155 −9.76848 −2.94313 1.13380
1.5 −2.68534 −1.17289 5.21107 0.884422 3.14961 −0.361098 −8.62282 −1.62433 −2.37498
1.6 −2.68137 1.21893 5.18975 2.92126 −3.26842 4.69101 −8.55290 −1.51420 −7.83297
1.7 −2.65004 −0.851556 5.02269 −0.493922 2.25665 4.68994 −8.01024 −2.27485 1.30891
1.8 −2.60343 −2.10556 4.77783 −3.11826 5.48166 −2.87442 −7.23187 1.43337 8.11816
1.9 −2.58576 1.51660 4.68613 −3.91728 −3.92157 −1.90750 −6.94569 −0.699911 10.1291
1.10 −2.56961 −1.95716 4.60287 1.92681 5.02912 3.09633 −6.68835 0.830463 −4.95115
1.11 −2.55384 0.729856 4.52207 2.13397 −1.86393 −2.81962 −6.44096 −2.46731 −5.44980
1.12 −2.53688 −2.79023 4.43578 3.33172 7.07848 −4.26650 −6.17928 4.78538 −8.45219
1.13 −2.52938 −2.71474 4.39775 −1.01620 6.86661 3.17147 −6.06482 4.36984 2.57036
1.14 −2.49180 1.26406 4.20905 −3.18748 −3.14979 4.16576 −5.50450 −1.40214 7.94255
1.15 −2.40631 0.956408 3.79032 0.737910 −2.30141 −3.66716 −4.30806 −2.08528 −1.77564
1.16 −2.39169 −2.70490 3.72019 3.87028 6.46929 −0.448223 −4.11416 4.31650 −9.25652
1.17 −2.35311 −1.80440 3.53712 −3.84522 4.24595 1.90458 −3.61701 0.255862 9.04823
1.18 −2.31012 2.93351 3.33667 2.06855 −6.77677 −0.615068 −3.08788 5.60546 −4.77860
1.19 −2.30956 −1.91926 3.33407 3.27349 4.43265 −1.87972 −3.08112 0.683571 −7.56031
1.20 −2.30137 2.98215 3.29631 1.02491 −6.86304 3.76670 −2.98329 5.89324 −2.35871
See next 80 embeddings (of 172 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.172
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(53\) \(1\)
\(151\) \(-1\)

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 8003.2.a.c ✓ 172
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8003.2.a.c ✓ 172 1.a even 1 1 trivial