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Properties

Label 70.4.k.a
Level $70$
Weight $4$
Character orbit 70.k
Analytic conductor $4.130$
Analytic rank $0$
Dimension $48$
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] = [70,4,Mod(3,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 70.k (of order \(12\), degree \(4\), 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: \(4.13013370040\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{5} - 4 q^{7} - 72 q^{10} + 44 q^{11} - 504 q^{15} + 384 q^{16} + 288 q^{17} + 16 q^{18} - 84 q^{21} + 624 q^{22} - 252 q^{23} + 152 q^{25} + 456 q^{26} + 112 q^{28} - 120 q^{30} - 528 q^{31} - 1956 q^{33} - 632 q^{35} - 2464 q^{36} - 552 q^{37} - 528 q^{38} - 488 q^{42} + 2176 q^{43} + 4128 q^{45} + 896 q^{46} + 1464 q^{47} + 448 q^{50} - 168 q^{51} + 576 q^{53} + 288 q^{56} - 224 q^{57} + 1024 q^{58} - 352 q^{60} - 5076 q^{61} + 3972 q^{63} - 468 q^{65} - 1936 q^{67} + 1152 q^{68} - 792 q^{70} - 672 q^{71} + 64 q^{72} - 5616 q^{73} + 5232 q^{75} - 6908 q^{77} - 4192 q^{78} + 768 q^{80} + 6400 q^{81} + 864 q^{82} - 776 q^{85} + 776 q^{86} + 15732 q^{87} + 1248 q^{88} + 10512 q^{91} + 2016 q^{92} + 144 q^{93} - 2560 q^{95} - 384 q^{96} - 400 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.93185 − 0.517638i −2.52047 − 9.40652i 3.46410 + 2.00000i 5.48744 − 9.74105i 19.4767i 13.1933 − 12.9976i −5.65685 − 5.65685i −58.7472 + 33.9177i −15.6433 + 15.9778i
3.2 −1.93185 − 0.517638i −0.860315 − 3.21074i 3.46410 + 2.00000i −10.3430 − 4.24524i 6.64801i −6.99963 + 17.1466i −5.65685 − 5.65685i 13.8140 − 7.97550i 17.7837 + 13.5551i
3.3 −1.93185 − 0.517638i −0.707358 − 2.63990i 3.46410 + 2.00000i −2.58639 + 10.8771i 5.46604i 18.4018 − 2.09116i −5.65685 − 5.65685i 16.9140 − 9.76530i 10.6269 − 19.6741i
3.4 −1.93185 − 0.517638i −0.436841 − 1.63031i 3.46410 + 2.00000i 11.0082 − 1.95411i 3.37565i −15.4162 + 10.2635i −5.65685 − 5.65685i 20.9156 − 12.0756i −22.2778 − 1.92323i
3.5 −1.93185 − 0.517638i 1.80856 + 6.74962i 3.46410 + 2.00000i −0.108452 + 11.1798i − 13.9754i −15.7380 − 9.76300i −5.65685 − 5.65685i −18.9039 + 10.9141i 5.99661 − 21.5416i
3.6 −1.93185 − 0.517638i 2.26814 + 8.46482i 3.46410 + 2.00000i 7.56128 − 8.23572i − 17.5268i 13.9430 + 12.1899i −5.65685 − 5.65685i −43.1260 + 24.8988i −18.8704 + 11.9962i
3.7 1.93185 + 0.517638i −2.20465 − 8.22787i 3.46410 + 2.00000i −7.34876 − 8.42589i − 17.0362i −16.4126 + 8.58054i 5.65685 + 5.65685i −39.4546 + 22.7791i −9.83516 − 20.0816i
3.8 1.93185 + 0.517638i −1.44159 − 5.38010i 3.46410 + 2.00000i 9.49407 + 5.90447i − 11.1398i −0.512273 − 18.5132i 5.65685 + 5.65685i −3.48456 + 2.01181i 15.2847 + 16.3210i
3.9 1.93185 + 0.517638i −0.265304 − 0.990126i 3.46410 + 2.00000i 1.54409 − 11.0732i − 2.05011i 18.5182 − 0.277160i 5.65685 + 5.65685i 22.4727 − 12.9746i 8.71486 − 20.5925i
3.10 1.93185 + 0.517638i 0.234331 + 0.874534i 3.46410 + 2.00000i −2.63205 + 10.8661i 1.81077i −0.531755 + 18.5126i 5.65685 + 5.65685i 22.6728 − 13.0901i −10.7094 + 19.6293i
3.11 1.93185 + 0.517638i 1.71158 + 6.38770i 3.46410 + 2.00000i 11.0860 − 1.44897i 13.2261i −17.6263 − 5.68445i 5.65685 + 5.65685i −14.4906 + 8.36613i 22.1666 + 2.93936i
3.12 1.93185 + 0.517638i 2.41392 + 9.00888i 3.46410 + 2.00000i −11.1625 − 0.631474i 18.6534i 16.8408 − 7.70633i 5.65685 + 5.65685i −51.9503 + 29.9935i −21.2374 − 6.99805i
17.1 −0.517638 + 1.93185i −8.22787 + 2.20465i −3.46410 − 2.00000i 3.62266 + 10.5772i − 17.0362i −8.58054 − 16.4126i 5.65685 − 5.65685i 39.4546 − 22.7791i −22.3087 + 1.52329i
17.2 −0.517638 + 1.93185i −5.38010 + 1.44159i −3.46410 − 2.00000i −0.366385 − 11.1743i − 11.1398i 18.5132 − 0.512273i 5.65685 − 5.65685i 3.48456 − 2.01181i 21.7768 + 5.07646i
17.3 −0.517638 + 1.93185i −0.990126 + 0.265304i −3.46410 − 2.00000i 10.3617 + 4.19938i − 2.05011i 0.277160 + 18.5182i 5.65685 − 5.65685i −22.4727 + 12.9746i −13.4762 + 17.8435i
17.4 −0.517638 + 1.93185i 0.874534 − 0.234331i −3.46410 − 2.00000i −10.7263 − 3.15364i 1.81077i −18.5126 − 0.531755i 5.65685 − 5.65685i −22.6728 + 13.0901i 11.6447 − 19.0893i
17.5 −0.517638 + 1.93185i 6.38770 − 1.71158i −3.46410 − 2.00000i 6.79787 − 8.87631i 13.2261i 5.68445 − 17.6263i 5.65685 − 5.65685i 14.4906 − 8.36613i 13.6289 + 17.7272i
17.6 −0.517638 + 1.93185i 9.00888 − 2.41392i −3.46410 − 2.00000i −5.03437 + 9.98274i 18.6534i 7.70633 + 16.8408i 5.65685 − 5.65685i 51.9503 − 29.9935i −16.6792 − 14.8931i
17.7 0.517638 − 1.93185i −9.40652 + 2.52047i −3.46410 − 2.00000i 11.1797 + 0.118266i 19.4767i 12.9976 + 13.1933i −5.65685 + 5.65685i 58.7472 − 33.9177i 6.01552 − 21.5363i
17.8 0.517638 − 1.93185i −3.21074 + 0.860315i −3.46410 − 2.00000i −1.49503 + 11.0799i 6.64801i −17.1466 − 6.99963i −5.65685 + 5.65685i −13.8140 + 7.97550i 20.6309 + 8.62356i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.4.k.a ✓ 48
5.c odd 4 1 inner 70.4.k.a ✓ 48
7.d odd 6 1 inner 70.4.k.a ✓ 48
35.k even 12 1 inner 70.4.k.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.k.a ✓ 48 1.a even 1 1 trivial
70.4.k.a ✓ 48 5.c odd 4 1 inner
70.4.k.a ✓ 48 7.d odd 6 1 inner
70.4.k.a ✓ 48 35.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(70, [\chi])\).