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Properties

Label 148.4.n.a
Level $148$
Weight $4$
Character orbit 148.n
Analytic conductor $8.732$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [148,4,Mod(21,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.21");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 148.n (of order \(18\), degree \(6\), 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: \(8.73228268085\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{3} + 18 q^{5} - 48 q^{7} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{3} + 18 q^{5} - 48 q^{7} + 48 q^{9} + 66 q^{11} - 48 q^{13} + 108 q^{15} - 90 q^{17} - 162 q^{19} + 54 q^{21} + 492 q^{25} - 222 q^{27} - 180 q^{29} + 312 q^{33} + 162 q^{35} + 1038 q^{37} + 942 q^{39} - 540 q^{41} - 1692 q^{45} - 222 q^{47} - 342 q^{49} - 570 q^{53} + 1242 q^{55} - 1866 q^{57} + 2784 q^{59} - 276 q^{61} + 2082 q^{63} - 108 q^{65} + 2010 q^{67} - 3264 q^{69} + 2016 q^{71} - 5460 q^{73} - 7812 q^{75} - 4824 q^{77} - 594 q^{79} - 2136 q^{81} + 3000 q^{83} + 1032 q^{85} + 11388 q^{87} + 1758 q^{89} + 1890 q^{91} - 6912 q^{93} - 5556 q^{95} - 8316 q^{97} + 2310 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 0 −9.39651 + 3.42005i 0 17.0597 + 3.00809i 0 0.600666 − 3.40655i 0 55.9144 − 46.9178i 0
21.2 0 −8.17218 + 2.97443i 0 −20.9642 − 3.69655i 0 −6.09667 + 34.5759i 0 37.2541 − 31.2599i 0
21.3 0 −5.04381 + 1.83580i 0 −5.17928 − 0.913247i 0 3.31971 − 18.8270i 0 1.38671 − 1.16359i 0
21.4 0 −3.51873 + 1.28071i 0 2.47525 + 0.436454i 0 0.598381 − 3.39359i 0 −9.94199 + 8.34232i 0
21.5 0 1.30625 − 0.475435i 0 20.1929 + 3.56055i 0 −2.86278 + 16.2357i 0 −19.2030 + 16.1132i 0
21.6 0 1.66719 − 0.606806i 0 −1.67784 − 0.295848i 0 −4.20518 + 23.8488i 0 −18.2719 + 15.3319i 0
21.7 0 2.76709 − 1.00714i 0 −16.0240 − 2.82546i 0 1.39845 − 7.93103i 0 −14.0407 + 11.7816i 0
21.8 0 5.82688 − 2.12081i 0 12.1624 + 2.14456i 0 5.21235 − 29.5607i 0 8.77147 − 7.36013i 0
21.9 0 5.93974 − 2.16189i 0 −13.7217 − 2.41950i 0 1.60730 − 9.11545i 0 9.92359 − 8.32688i 0
21.10 0 9.27679 − 3.37648i 0 2.89564 + 0.510580i 0 −3.60906 + 20.4680i 0 53.9751 − 45.2905i 0
25.1 0 −7.09331 − 5.95200i 0 3.03119 + 8.32812i 0 −13.0611 + 4.75385i 0 10.2003 + 57.8490i 0
25.2 0 −4.72354 − 3.96352i 0 −1.62070 − 4.45283i 0 21.8596 − 7.95624i 0 1.91383 + 10.8538i 0
25.3 0 −4.21536 − 3.53711i 0 −7.12590 − 19.5783i 0 −18.0659 + 6.57545i 0 0.569641 + 3.23059i 0
25.4 0 −1.96100 − 1.64547i 0 1.67453 + 4.60074i 0 8.21332 − 2.98941i 0 −3.55057 − 20.1363i 0
25.5 0 −0.277516 − 0.232864i 0 6.65985 + 18.2978i 0 −17.0708 + 6.21327i 0 −4.66571 − 26.4606i 0
25.6 0 1.29640 + 1.08781i 0 −0.872592 − 2.39743i 0 −24.5122 + 8.92172i 0 −4.19118 − 23.7694i 0
25.7 0 2.47714 + 2.07857i 0 0.228929 + 0.628977i 0 17.0128 − 6.19214i 0 −2.87272 − 16.2920i 0
25.8 0 4.05954 + 3.40636i 0 −6.46791 − 17.7704i 0 12.0659 − 4.39163i 0 0.188088 + 1.06670i 0
25.9 0 6.38717 + 5.35947i 0 5.69697 + 15.6523i 0 15.7563 − 5.73484i 0 7.38350 + 41.8739i 0
25.10 0 6.92987 + 5.81485i 0 −1.47294 − 4.04687i 0 −23.0063 + 8.37361i 0 9.52209 + 54.0024i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.4.n.a ✓ 60
37.h even 18 1 inner 148.4.n.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.4.n.a ✓ 60 1.a even 1 1 trivial
148.4.n.a ✓ 60 37.h even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(148, [\chi])\).