Properties

Label 6001.2.a.b
Level $6001$
Weight $2$
Character orbit 6001.a
Self dual yes
Analytic conductor $47.918$
Analytic rank $1$
Dimension $114$
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: \(1\)
Dimension: \(114\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 114 q - 8 q^{2} - 23 q^{3} + 110 q^{4} - 27 q^{5} - 23 q^{6} - 53 q^{7} - 21 q^{8} + 107 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 114 q - 8 q^{2} - 23 q^{3} + 110 q^{4} - 27 q^{5} - 23 q^{6} - 53 q^{7} - 21 q^{8} + 107 q^{9} - 19 q^{10} - 52 q^{11} - 49 q^{12} - 12 q^{13} - 40 q^{14} - 39 q^{15} + 110 q^{16} + 114 q^{17} - 21 q^{18} - 30 q^{19} - 88 q^{20} - 30 q^{21} - 36 q^{22} - 77 q^{23} - 72 q^{24} + 119 q^{25} - 79 q^{26} - 77 q^{27} - 92 q^{28} - 65 q^{29} - 10 q^{30} - 131 q^{31} - 30 q^{32} - 12 q^{33} - 8 q^{34} - 33 q^{35} + 109 q^{36} - 54 q^{37} - 14 q^{38} - 83 q^{39} - 42 q^{40} - 99 q^{41} + 29 q^{42} + 4 q^{43} - 98 q^{44} - 73 q^{45} - 35 q^{46} - 113 q^{47} - 86 q^{48} + 101 q^{49} - 44 q^{50} - 23 q^{51} - 3 q^{52} - 18 q^{53} - 78 q^{54} - 63 q^{55} - 117 q^{56} - 64 q^{57} - 31 q^{58} - 134 q^{59} - 6 q^{60} - 30 q^{61} - 30 q^{62} - 154 q^{63} + 117 q^{64} - 66 q^{65} - 12 q^{66} - 34 q^{67} + 110 q^{68} - 35 q^{69} + 18 q^{70} - 233 q^{71} + 16 q^{72} - 56 q^{73} - 64 q^{74} - 100 q^{75} - 64 q^{76} - 6 q^{77} + 50 q^{78} - 154 q^{79} - 128 q^{80} + 118 q^{81} + 2 q^{82} - 53 q^{83} - 6 q^{84} - 27 q^{85} - 52 q^{86} - 22 q^{87} - 52 q^{88} - 118 q^{89} - 5 q^{90} - 95 q^{91} - 102 q^{92} + 47 q^{93} - 3 q^{94} - 158 q^{95} - 144 q^{96} - 57 q^{97} + 3 q^{98} - 131 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.79736 1.48474 5.82522 0.639507 −4.15334 0.101329 −10.7005 −0.795561 −1.78893
1.2 −2.75915 3.04798 5.61289 −2.57270 −8.40983 −0.275912 −9.96848 6.29019 7.09847
1.3 −2.75487 −1.38395 5.58934 1.31563 3.81261 −4.81832 −9.88817 −1.08468 −3.62439
1.4 −2.73023 −3.10574 5.45415 −3.71411 8.47938 1.88828 −9.43061 6.64562 10.1404
1.5 −2.69250 0.578947 5.24958 4.35539 −1.55882 −1.96736 −8.74952 −2.66482 −11.7269
1.6 −2.68479 0.238039 5.20809 −3.86690 −0.639084 2.11314 −8.61305 −2.94334 10.3818
1.7 −2.61975 −2.31872 4.86309 0.216177 6.07447 3.38406 −7.50057 2.37647 −0.566328
1.8 −2.56595 1.01181 4.58410 2.09874 −2.59626 3.20271 −6.63067 −1.97623 −5.38525
1.9 −2.39336 0.882221 3.72817 0.123030 −2.11147 −1.77301 −4.13613 −2.22169 −0.294456
1.10 −2.39289 −2.96659 3.72594 3.31044 7.09873 0.664206 −4.12999 5.80065 −7.92153
1.11 −2.38632 −0.731332 3.69453 1.80073 1.74519 −2.93364 −4.04370 −2.46515 −4.29713
1.12 −2.36622 −1.16111 3.59897 −3.78541 2.74743 −1.24944 −3.78352 −1.65183 8.95709
1.13 −2.31496 −0.975130 3.35906 −3.23089 2.25739 5.09566 −3.14618 −2.04912 7.47940
1.14 −2.31236 3.26187 3.34699 −1.15202 −7.54261 −3.58394 −3.11471 7.63982 2.66388
1.15 −2.29894 0.676824 3.28511 −3.14176 −1.55598 −3.61058 −2.95439 −2.54191 7.22271
1.16 −2.28982 −3.10307 3.24328 2.16467 7.10548 −1.63956 −2.84688 6.62906 −4.95671
1.17 −2.18122 1.64876 2.75771 −2.64324 −3.59630 −4.67385 −1.65274 −0.281605 5.76549
1.18 −2.12375 2.77384 2.51031 −3.45627 −5.89093 3.25214 −1.08377 4.69417 7.34024
1.19 −2.10824 −2.66682 2.44469 −3.46040 5.62230 −1.09952 −0.937516 4.11192 7.29537
1.20 −2.09110 −1.81594 2.37271 1.01486 3.79732 2.98461 −0.779370 0.297648 −2.12218
See next 80 embeddings (of 114 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.114
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.b 114
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6001.2.a.b 114 1.a even 1 1 trivial