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Properties

Label 8033.2.a.d
Level $8033$
Weight $2$
Character orbit 8033.a
Self dual yes
Analytic conductor $64.144$
Analytic rank $0$
Dimension $168$
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] = [8033,2,Mod(1,8033)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8033, 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("8033.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8033 = 29 \cdot 277 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8033.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: \(64.1438279437\)
Analytic rank: \(0\)
Dimension: \(168\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 12 q^{2} + 35 q^{3} + 184 q^{4} + 12 q^{5} + 10 q^{6} + 74 q^{7} + 39 q^{8} + 183 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q + 12 q^{2} + 35 q^{3} + 184 q^{4} + 12 q^{5} + 10 q^{6} + 74 q^{7} + 39 q^{8} + 183 q^{9} + 41 q^{10} + 29 q^{11} + 82 q^{12} + 62 q^{13} + 23 q^{14} + 31 q^{15} + 204 q^{16} + 56 q^{17} + 35 q^{18} + 83 q^{19} + 6 q^{20} + 30 q^{21} + 56 q^{22} + 54 q^{23} + 28 q^{24} + 210 q^{25} + 21 q^{26} + 140 q^{27} + 151 q^{28} + 168 q^{29} + 29 q^{30} + 72 q^{31} + 40 q^{32} + 32 q^{33} + 34 q^{34} + 18 q^{35} + 152 q^{36} + 42 q^{37} + 29 q^{38} + 70 q^{39} + 97 q^{40} + 41 q^{41} - 20 q^{42} + 119 q^{43} + 37 q^{44} + 22 q^{45} + 24 q^{46} + 119 q^{47} + 135 q^{48} + 216 q^{49} + 38 q^{50} + 18 q^{51} + 154 q^{52} - 7 q^{53} + 35 q^{54} + 224 q^{55} + 46 q^{56} + 12 q^{57} + 12 q^{58} + 25 q^{59} + 13 q^{60} + 82 q^{61} + 27 q^{62} + 211 q^{63} + 217 q^{64} + 8 q^{65} - 6 q^{66} + 76 q^{67} + 132 q^{68} + 36 q^{69} + 39 q^{70} + 32 q^{71} + 39 q^{72} + 89 q^{73} - q^{74} + 123 q^{75} + 180 q^{76} + 68 q^{77} - 54 q^{78} + 176 q^{79} - 11 q^{80} + 192 q^{81} + 51 q^{82} + 76 q^{83} + 86 q^{84} + 65 q^{85} - 72 q^{86} + 35 q^{87} + 178 q^{88} + 55 q^{89} + 2 q^{90} + 80 q^{91} + 44 q^{92} + 39 q^{93} + 89 q^{94} + 77 q^{95} - 68 q^{96} + 82 q^{97} + 80 q^{98} + 67 q^{99}+O(q^{100}) \) 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 −2.81703 3.39159 5.93564 −2.93306 −9.55421 1.32517 −11.0868 8.50291 8.26251
1.2 −2.78974 1.21931 5.78265 −2.52098 −3.40156 3.27965 −10.5526 −1.51328 7.03288
1.3 −2.72891 1.81680 5.44695 0.385312 −4.95790 −3.34669 −9.40642 0.300780 −1.05148
1.4 −2.69929 0.546346 5.28617 −1.10957 −1.47475 1.76751 −8.87031 −2.70151 2.99504
1.5 −2.69133 −1.56780 5.24327 1.58955 4.21948 −0.425755 −8.72870 −0.541992 −4.27800
1.6 −2.62201 −2.34634 4.87492 −4.13917 6.15213 1.37941 −7.53807 2.50532 10.8529
1.7 −2.58554 −1.99550 4.68504 0.489761 5.15945 4.81577 −6.94230 0.982009 −1.26630
1.8 −2.57726 2.47639 4.64228 3.37343 −6.38231 −1.62921 −6.80985 3.13252 −8.69420
1.9 −2.53885 1.12516 4.44574 −4.00247 −2.85660 4.31591 −6.20937 −1.73402 10.1617
1.10 −2.52738 −1.64022 4.38763 1.90507 4.14544 −1.36071 −6.03443 −0.309688 −4.81483
1.11 −2.52129 1.55242 4.35693 3.41413 −3.91411 1.49471 −5.94251 −0.589992 −8.60803
1.12 −2.50804 0.217745 4.29026 −0.0259578 −0.546113 −4.91509 −5.74406 −2.95259 0.0651033
1.13 −2.50746 −1.63037 4.28734 3.92256 4.08808 3.39028 −5.73541 −0.341901 −9.83565
1.14 −2.46662 2.34956 4.08421 0.440108 −5.79546 2.07787 −5.14095 2.52042 −1.08558
1.15 −2.44996 −1.86877 4.00229 −1.18542 4.57841 −4.60968 −4.90553 0.492307 2.90424
1.16 −2.41744 −0.0531053 3.84399 −0.881470 0.128379 1.45309 −4.45773 −2.99718 2.13090
1.17 −2.40677 −1.38895 3.79253 −2.91778 3.34287 3.15140 −4.31420 −1.07083 7.02241
1.18 −2.40306 1.89974 3.77470 −4.37204 −4.56518 −3.13994 −4.26472 0.608994 10.5063
1.19 −2.34777 0.405872 3.51204 −3.82329 −0.952896 0.144242 −3.54992 −2.83527 8.97621
1.20 −2.33211 3.25701 3.43872 3.02617 −7.59569 3.43735 −3.35525 7.60809 −7.05735
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.168
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)
\(277\) \(1\)

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 8033.2.a.d ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8033.2.a.d ✓ 168 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{168} - 12 T_{2}^{167} - 188 T_{2}^{166} + 2803 T_{2}^{165} + 15629 T_{2}^{164} + \cdots + 9648337658368 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8033))\). Copy content Toggle raw display