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Properties

Label 168.3.x.b
Level $168$
Weight $3$
Character orbit 168.x
Analytic conductor $4.578$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,3,Mod(61,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.61");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.x (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(4.57766844125\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 2 q^{2} - 2 q^{4} + 26 q^{7} + 32 q^{8} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 2 q^{2} - 2 q^{4} + 26 q^{7} + 32 q^{8} - 90 q^{9} - 42 q^{10} - 14 q^{14} + 12 q^{15} + 6 q^{16} - 36 q^{17} + 6 q^{18} + 28 q^{22} - 28 q^{23} + 102 q^{24} - 204 q^{25} - 42 q^{26} + 186 q^{28} - 24 q^{30} + 18 q^{31} - 28 q^{32} - 30 q^{33} + 12 q^{36} - 414 q^{38} - 36 q^{39} + 18 q^{40} + 120 q^{42} - 48 q^{44} - 160 q^{46} + 828 q^{47} - 126 q^{49} - 332 q^{50} + 36 q^{52} + 36 q^{54} + 256 q^{56} - 312 q^{57} - 94 q^{58} + 150 q^{60} - 12 q^{63} + 988 q^{64} + 36 q^{65} - 108 q^{66} + 312 q^{68} + 222 q^{70} + 760 q^{71} - 48 q^{72} - 648 q^{73} - 294 q^{74} + 396 q^{78} + 114 q^{79} - 900 q^{80} - 270 q^{81} + 876 q^{82} - 96 q^{84} + 6 q^{86} - 174 q^{87} - 262 q^{88} - 72 q^{89} - 592 q^{92} - 540 q^{94} - 492 q^{95} - 258 q^{96} - 628 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.99993 + 0.0162709i −0.866025 − 1.50000i 3.99947 − 0.0650815i 0.530070 − 0.918109i 1.75640 + 2.98581i 5.93864 + 3.70574i −7.99762 + 0.195234i −1.50000 + 2.59808i −1.04517 + 1.84478i
61.2 −1.96845 − 0.353828i 0.866025 + 1.50000i 3.74961 + 1.39299i 3.07593 − 5.32766i −1.17399 − 3.25910i −5.56076 − 4.25182i −6.88805 − 4.06875i −1.50000 + 2.59808i −7.93990 + 9.39890i
61.3 −1.94006 − 0.485962i 0.866025 + 1.50000i 3.52768 + 1.88559i −2.81718 + 4.87950i −0.951200 − 3.33095i 6.86898 + 1.34800i −5.92759 − 5.37249i −1.50000 + 2.59808i 7.83676 − 8.09749i
61.4 −1.92193 + 0.553328i 0.866025 + 1.50000i 3.38766 − 2.12692i 0.359123 − 0.622019i −2.49444 − 2.40370i −2.58077 + 6.50689i −5.33396 + 5.96228i −1.50000 + 2.59808i −0.346029 + 1.39419i
61.5 −1.73572 + 0.993625i −0.866025 − 1.50000i 2.02542 − 3.44930i 4.31368 − 7.47152i 2.99361 + 1.74307i −5.25961 − 4.61915i −0.0882384 + 7.99951i −1.50000 + 2.59808i −0.0634399 + 17.2546i
61.6 −1.45126 − 1.37617i −0.866025 − 1.50000i 0.212337 + 3.99436i −3.83514 + 6.64265i −0.807416 + 3.36869i 2.09396 − 6.67947i 5.18874 − 6.08908i −1.50000 + 2.59808i 14.7072 − 4.36246i
61.7 −1.32774 − 1.49570i 0.866025 + 1.50000i −0.474223 + 3.97179i 4.33205 − 7.50333i 1.09369 − 3.28692i 2.39482 + 6.57760i 6.57024 − 4.56420i −1.50000 + 2.59808i −16.9745 + 3.48302i
61.8 −1.29547 + 1.52373i −0.866025 − 1.50000i −0.643513 − 3.94790i −2.65366 + 4.59628i 3.40751 + 0.623616i 6.60250 − 2.32529i 6.84919 + 4.13384i −1.50000 + 2.59808i −3.56575 − 9.99781i
61.9 −1.04209 + 1.70706i −0.866025 − 1.50000i −1.82810 − 3.55781i −0.624581 + 1.08181i 3.46306 + 0.0847751i −6.08598 + 3.45844i 7.97844 + 0.586870i −1.50000 + 2.59808i −1.19584 − 2.19353i
61.10 −0.988146 − 1.73884i 0.866025 + 1.50000i −2.04713 + 3.43646i −1.12416 + 1.94710i 1.75250 − 2.98810i −4.61632 − 5.26209i 7.99832 + 0.163916i −1.50000 + 2.59808i 4.49654 + 0.0307133i
61.11 −0.957313 + 1.75600i 0.866025 + 1.50000i −2.16710 − 3.36209i 0.624581 − 1.08181i −3.46306 + 0.0847751i −6.08598 + 3.45844i 7.97844 − 0.586870i −1.50000 + 2.59808i 1.30174 + 2.13239i
61.12 −0.839793 − 1.81514i −0.866025 − 1.50000i −2.58950 + 3.04869i 4.13891 − 7.16880i −1.99543 + 2.83165i 5.39710 − 4.45773i 7.70845 + 2.14004i −1.50000 + 2.59808i −16.4882 − 1.49241i
61.13 −0.671855 + 1.88378i 0.866025 + 1.50000i −3.09722 − 2.53125i 2.65366 − 4.59628i −3.40751 + 0.623616i 6.60250 − 2.32529i 6.84919 − 4.13384i −1.50000 + 2.59808i 6.87548 + 8.08694i
61.14 −0.305234 − 1.97657i −0.866025 − 1.50000i −3.81366 + 1.20663i −1.57949 + 2.73576i −2.70052 + 2.16961i 1.86493 + 6.74700i 3.54906 + 7.16967i −1.50000 + 2.59808i 5.88954 + 2.28693i
61.15 −0.238716 − 1.98570i 0.866025 + 1.50000i −3.88603 + 0.948038i −0.760939 + 1.31798i 2.77182 − 2.07774i 6.10002 − 3.43362i 2.81018 + 7.49019i −1.50000 + 2.59808i 2.79877 + 1.19637i
61.16 0.00735333 + 1.99999i 0.866025 + 1.50000i −3.99989 + 0.0294131i −4.31368 + 7.47152i −2.99361 + 1.74307i −5.25961 − 4.61915i −0.0882384 − 7.99951i −1.50000 + 2.59808i −14.9747 − 8.57237i
61.17 0.481770 + 1.94111i −0.866025 − 1.50000i −3.53579 + 1.87034i −0.359123 + 0.622019i 2.49444 − 2.40370i −2.58077 + 6.50689i −5.33396 − 5.96228i −1.50000 + 2.59808i −1.38042 − 0.397425i
61.18 0.711057 − 1.86933i −0.866025 − 1.50000i −2.98880 − 2.65840i −2.09813 + 3.63406i −3.41979 + 0.552302i −3.40960 − 6.11348i −7.09464 + 3.69677i −1.50000 + 2.59808i 5.30138 + 6.50612i
61.19 0.751915 − 1.85327i 0.866025 + 1.50000i −2.86925 − 2.78701i −4.00714 + 6.94057i 3.43109 − 0.477110i −3.24792 + 6.20089i −7.32252 + 3.22191i −1.50000 + 2.59808i 9.84974 + 12.6450i
61.20 0.985876 + 1.74013i 0.866025 + 1.50000i −2.05610 + 3.43110i −0.530070 + 0.918109i −1.75640 + 2.98581i 5.93864 + 3.70574i −7.99762 − 0.195234i −1.50000 + 2.59808i −2.12021 − 0.0172495i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.b even 2 1 inner
56.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.3.x.b ✓ 60
4.b odd 2 1 672.3.bf.b 60
7.d odd 6 1 inner 168.3.x.b ✓ 60
8.b even 2 1 inner 168.3.x.b ✓ 60
8.d odd 2 1 672.3.bf.b 60
28.f even 6 1 672.3.bf.b 60
56.j odd 6 1 inner 168.3.x.b ✓ 60
56.m even 6 1 672.3.bf.b 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.x.b ✓ 60 1.a even 1 1 trivial
168.3.x.b ✓ 60 7.d odd 6 1 inner
168.3.x.b ✓ 60 8.b even 2 1 inner
168.3.x.b ✓ 60 56.j odd 6 1 inner
672.3.bf.b 60 4.b odd 2 1
672.3.bf.b 60 8.d odd 2 1
672.3.bf.b 60 28.f even 6 1
672.3.bf.b 60 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} + 477 T_{5}^{58} + 127332 T_{5}^{56} + 23399695 T_{5}^{54} + 3268335801 T_{5}^{52} + \cdots + 83\!\cdots\!04 \) acting on \(S_{3}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display