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Properties

Label 6001.2.a.d
Level $6001$
Weight $2$
Character orbit 6001.a
Self dual yes
Analytic conductor $47.918$
Analytic rank $0$
Dimension $121$
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] = [6001,2,Mod(1,6001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, 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("6001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6001 = 17 \cdot 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6001.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: \(47.9182262530\)
Analytic rank: \(0\)
Dimension: \(121\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 121 q + 9 q^{2} + 21 q^{3} + 127 q^{4} + 27 q^{5} + 17 q^{6} + 39 q^{7} + 24 q^{8} + 134 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 121 q + 9 q^{2} + 21 q^{3} + 127 q^{4} + 27 q^{5} + 17 q^{6} + 39 q^{7} + 24 q^{8} + 134 q^{9} + 19 q^{10} + 48 q^{11} + 43 q^{12} + 6 q^{13} + 40 q^{14} + 49 q^{15} + 135 q^{16} + 121 q^{17} + 30 q^{19} + 50 q^{20} + 18 q^{21} + 24 q^{22} + 75 q^{23} + 24 q^{24} + 128 q^{25} + 59 q^{26} + 75 q^{27} + 52 q^{28} + 49 q^{29} - 34 q^{30} + 101 q^{31} + 47 q^{32} + 20 q^{33} + 9 q^{34} + 47 q^{35} + 138 q^{36} + 32 q^{37} + 30 q^{38} + 101 q^{39} + 36 q^{40} + 83 q^{41} - 11 q^{42} + 8 q^{43} + 98 q^{44} + 49 q^{45} + 45 q^{46} + 135 q^{47} + 54 q^{48} + 116 q^{49} + 3 q^{50} + 21 q^{51} - 5 q^{52} + 28 q^{53} + 10 q^{54} + 37 q^{55} + 75 q^{56} + 31 q^{58} + 150 q^{59} + 50 q^{60} + 36 q^{61} + 34 q^{62} + 118 q^{63} + 110 q^{64} + 18 q^{65} - 28 q^{66} - 6 q^{67} + 127 q^{68} + 25 q^{69} - 22 q^{70} + 223 q^{71} + q^{72} + 38 q^{73} - 10 q^{74} + 88 q^{75} - 4 q^{76} + 38 q^{77} + 42 q^{78} + 74 q^{79} + 106 q^{80} + 133 q^{81} + 28 q^{82} + 55 q^{83} + 10 q^{84} + 27 q^{85} + 64 q^{86} + 14 q^{87} + 56 q^{88} + 118 q^{89} + 51 q^{90} + 73 q^{91} + 82 q^{92} + 31 q^{93} + 33 q^{94} + 106 q^{95} + 38 q^{96} + 37 q^{97} + 88 q^{98} + 81 q^{99}+O(q^{100}) \) 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 −2.77548 −0.417524 5.70327 0.239845 1.15883 2.35024 −10.2783 −2.82567 −0.665684
1.2 −2.70572 −1.68319 5.32093 1.67453 4.55424 1.84178 −8.98553 −0.166876 −4.53081
1.3 −2.68420 3.02151 5.20492 3.70816 −8.11034 3.11131 −8.60266 6.12953 −9.95344
1.4 −2.66371 −2.14217 5.09536 −3.47946 5.70612 −4.33919 −8.24515 1.58889 9.26828
1.5 −2.63308 2.70578 4.93313 2.79051 −7.12456 −5.09743 −7.72317 4.32127 −7.34766
1.6 −2.58868 −1.67883 4.70126 −1.30271 4.34594 −1.36507 −6.99268 −0.181538 3.37229
1.7 −2.53489 1.89170 4.42568 −3.70427 −4.79525 1.26299 −6.14885 0.578524 9.38992
1.8 −2.51719 0.226046 4.33623 −1.31460 −0.569001 −1.40872 −5.88074 −2.94890 3.30910
1.9 −2.51330 −2.00286 4.31669 3.06325 5.03379 0.908464 −5.82253 1.01144 −7.69887
1.10 −2.48199 1.43528 4.16025 0.277666 −3.56234 3.77920 −5.36171 −0.939973 −0.689164
1.11 −2.43959 2.16595 3.95162 −1.55175 −5.28403 −2.38051 −4.76117 1.69132 3.78564
1.12 −2.40063 1.09881 3.76301 3.50161 −2.63784 −0.487426 −4.23233 −1.79261 −8.40605
1.13 −2.26922 −1.02425 3.14937 −0.261054 2.32424 2.61024 −2.60816 −1.95092 0.592389
1.14 −2.25245 −3.40363 3.07353 −2.33953 7.66650 2.81637 −2.41808 8.58466 5.26966
1.15 −2.24229 2.77352 3.02787 0.258648 −6.21903 3.59133 −2.30477 4.69240 −0.579964
1.16 −2.20655 2.67875 2.86888 −1.00223 −5.91080 −0.364832 −1.91723 4.17568 2.21147
1.17 −2.15090 0.00192410 2.62636 −2.18392 −0.00413853 −0.0660171 −1.34723 −3.00000 4.69738
1.18 −2.15009 −0.229401 2.62287 4.13461 0.493233 −0.301576 −1.33923 −2.94737 −8.88976
1.19 −2.08098 −2.15379 2.33048 2.78537 4.48200 −3.81134 −0.687730 1.63882 −5.79629
1.20 −1.98791 3.32266 1.95179 2.27659 −6.60516 3.45158 0.0958417 8.04010 −4.52565
See next 80 embeddings (of 121 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.121
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(353\) \(1\)

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 6001.2.a.d ✓ 121
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6001.2.a.d ✓ 121 1.a even 1 1 trivial