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Properties

Label 420.2.t.d
Level $420$
Weight $2$
Character orbit 420.t
Analytic conductor $3.354$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [420,2,Mod(43,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("420.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-4,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{2} + 8 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{10} + 8 q^{12} + 16 q^{13} + 20 q^{16} + 8 q^{17} + 4 q^{18} + 20 q^{20} - 32 q^{21} - 12 q^{22} + 8 q^{25} + 24 q^{26} - 8 q^{28} - 20 q^{30} - 4 q^{32}+ \cdots - 4 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.40332 + 0.175219i 0.707107 − 0.707107i 1.93860 − 0.491776i −2.22417 + 0.230364i −0.868396 + 1.11619i −0.707107 − 0.707107i −2.63430 + 1.02980i − 1.00000i 3.08085 − 0.712992i
43.2 −1.39565 − 0.228374i −0.707107 + 0.707107i 1.89569 + 0.637461i −1.58711 − 1.57514i 1.14836 − 0.825390i 0.707107 + 0.707107i −2.50015 − 1.32260i − 1.00000i 1.85533 + 2.56081i
43.3 −1.38990 − 0.261096i 0.707107 − 0.707107i 1.86366 + 0.725797i 2.19003 + 0.451400i −1.16743 + 0.798186i −0.707107 − 0.707107i −2.40080 − 1.49538i − 1.00000i −2.92607 − 1.19921i
43.4 −1.14378 + 0.831721i 0.707107 − 0.707107i 0.616482 − 1.90262i 2.16706 + 0.551211i −0.220662 + 1.39689i −0.707107 − 0.707107i 0.877323 + 2.68892i − 1.00000i −2.93711 + 1.17193i
43.5 −1.06251 − 0.933318i −0.707107 + 0.707107i 0.257835 + 1.98331i −1.10360 + 1.94476i 1.41126 − 0.0913493i 0.707107 + 0.707107i 1.57711 − 2.34792i − 1.00000i 2.98765 − 1.03631i
43.6 −0.831721 + 1.14378i −0.707107 + 0.707107i −0.616482 − 1.90262i 2.16706 + 0.551211i −0.220662 − 1.39689i 0.707107 + 0.707107i 2.68892 + 0.877323i − 1.00000i −2.43286 + 2.02020i
43.7 −0.557908 − 1.29951i −0.707107 + 0.707107i −1.37748 + 1.45002i 0.667081 − 2.13425i 1.31340 + 0.524395i 0.707107 + 0.707107i 2.65283 + 0.981075i − 1.00000i −3.14565 + 0.323831i
43.8 −0.175219 + 1.40332i −0.707107 + 0.707107i −1.93860 − 0.491776i −2.22417 + 0.230364i −0.868396 − 1.11619i 0.707107 + 0.707107i 1.02980 − 2.63430i − 1.00000i 0.0664433 − 3.16158i
43.9 −0.0768450 − 1.41212i 0.707107 − 0.707107i −1.98819 + 0.217029i 1.48915 − 1.66806i −1.05286 − 0.944185i −0.707107 − 0.707107i 0.459255 + 2.79089i − 1.00000i −2.46994 − 1.97469i
43.10 0.228374 + 1.39565i 0.707107 − 0.707107i −1.89569 + 0.637461i −1.58711 − 1.57514i 1.14836 + 0.825390i −0.707107 − 0.707107i −1.32260 − 2.50015i − 1.00000i 1.83590 − 2.57478i
43.11 0.261096 + 1.38990i −0.707107 + 0.707107i −1.86366 + 0.725797i 2.19003 + 0.451400i −1.16743 − 0.798186i 0.707107 + 0.707107i −1.49538 − 2.40080i − 1.00000i −0.0555919 + 3.16179i
43.12 0.642678 − 1.25975i −0.707107 + 0.707107i −1.17393 − 1.61922i 0.401547 + 2.19972i 0.436334 + 1.34522i 0.707107 + 0.707107i −2.79427 + 0.438216i − 1.00000i 3.02916 + 0.907863i
43.13 0.933318 + 1.06251i 0.707107 − 0.707107i −0.257835 + 1.98331i −1.10360 + 1.94476i 1.41126 + 0.0913493i −0.707107 − 0.707107i −2.34792 + 1.57711i − 1.00000i −3.09632 + 0.642497i
43.14 1.25975 − 0.642678i 0.707107 − 0.707107i 1.17393 − 1.61922i 0.401547 + 2.19972i 0.436334 − 1.34522i −0.707107 − 0.707107i 0.438216 − 2.79427i − 1.00000i 1.91956 + 2.51303i
43.15 1.29951 + 0.557908i 0.707107 − 0.707107i 1.37748 + 1.45002i 0.667081 − 2.13425i 1.31340 − 0.524395i −0.707107 − 0.707107i 0.981075 + 2.65283i − 1.00000i 2.05759 − 2.40131i
43.16 1.41212 + 0.0768450i −0.707107 + 0.707107i 1.98819 + 0.217029i 1.48915 − 1.66806i −1.05286 + 0.944185i 0.707107 + 0.707107i 2.79089 + 0.459255i − 1.00000i 2.23105 − 2.24107i
127.1 −1.40332 − 0.175219i 0.707107 + 0.707107i 1.93860 + 0.491776i −2.22417 − 0.230364i −0.868396 − 1.11619i −0.707107 + 0.707107i −2.63430 − 1.02980i 1.00000i 3.08085 + 0.712992i
127.2 −1.39565 + 0.228374i −0.707107 − 0.707107i 1.89569 − 0.637461i −1.58711 + 1.57514i 1.14836 + 0.825390i 0.707107 − 0.707107i −2.50015 + 1.32260i 1.00000i 1.85533 − 2.56081i
127.3 −1.38990 + 0.261096i 0.707107 + 0.707107i 1.86366 − 0.725797i 2.19003 − 0.451400i −1.16743 − 0.798186i −0.707107 + 0.707107i −2.40080 + 1.49538i 1.00000i −2.92607 + 1.19921i
127.4 −1.14378 − 0.831721i 0.707107 + 0.707107i 0.616482 + 1.90262i 2.16706 − 0.551211i −0.220662 − 1.39689i −0.707107 + 0.707107i 0.877323 − 2.68892i 1.00000i −2.93711 − 1.17193i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 43.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.t.d ✓ 32
4.b odd 2 1 inner 420.2.t.d ✓ 32
5.c odd 4 1 inner 420.2.t.d ✓ 32
20.e even 4 1 inner 420.2.t.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.t.d ✓ 32 1.a even 1 1 trivial
420.2.t.d ✓ 32 4.b odd 2 1 inner
420.2.t.d ✓ 32 5.c odd 4 1 inner
420.2.t.d ✓ 32 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11}^{16} + 124 T_{11}^{14} + 6076 T_{11}^{12} + 152544 T_{11}^{10} + 2125296 T_{11}^{8} + \cdots + 1478656 \) Copy content Toggle raw display
\( T_{13}^{16} - 8 T_{13}^{15} + 32 T_{13}^{14} - 32 T_{13}^{13} + 896 T_{13}^{12} - 7072 T_{13}^{11} + \cdots + 1478656 \) Copy content Toggle raw display