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Properties

Label 340.2.bd.a
Level $340$
Weight $2$
Character orbit 340.bd
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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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] = [340,2,Mod(57,340)] 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("340.57"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bd (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 0 −1.50403 + 2.25094i 0 −2.02389 − 0.950727i 0 0.694602 − 3.49200i 0 −1.65656 − 3.99930i 0
57.2 0 −1.35240 + 2.02401i 0 2.11145 − 0.736056i 0 −0.777549 + 3.90900i 0 −1.11958 − 2.70291i 0
57.3 0 −1.25555 + 1.87906i 0 0.726829 + 2.11464i 0 0.0284166 − 0.142860i 0 −0.806426 − 1.94689i 0
57.4 0 −0.514627 + 0.770194i 0 −1.14734 − 1.91927i 0 −0.415329 + 2.08800i 0 0.819692 + 1.97891i 0
57.5 0 0.286067 − 0.428130i 0 1.53584 + 1.62517i 0 0.632123 − 3.17790i 0 1.04659 + 2.52669i 0
57.6 0 0.349423 − 0.522949i 0 1.09010 − 1.95235i 0 0.0348214 − 0.175059i 0 0.996671 + 2.40618i 0
57.7 0 0.841077 − 1.25876i 0 −0.704990 + 2.12202i 0 −0.993774 + 4.99604i 0 0.270982 + 0.654209i 0
57.8 0 1.27530 − 1.90862i 0 −2.15814 − 0.585194i 0 0.311904 − 1.56805i 0 −0.868385 − 2.09647i 0
57.9 0 1.87474 − 2.80575i 0 2.20111 − 0.393819i 0 −0.0564111 + 0.283598i 0 −3.20952 − 7.74847i 0
73.1 0 −0.529264 + 2.66079i 0 0.427916 + 2.19474i 0 2.51922 + 1.68329i 0 −4.02805 − 1.66847i 0
73.2 0 −0.441472 + 2.21943i 0 −2.23590 − 0.0274503i 0 −2.81028 − 1.87777i 0 −1.95932 − 0.811579i 0
73.3 0 −0.402101 + 2.02150i 0 −0.456223 − 2.18903i 0 2.17863 + 1.45572i 0 −1.15314 − 0.477647i 0
73.4 0 0.0289619 − 0.145601i 0 0.607979 − 2.15183i 0 −1.82288 − 1.21801i 0 2.75128 + 1.13962i 0
73.5 0 0.0352247 − 0.177087i 0 −0.198315 + 2.22726i 0 −3.20617 − 2.14230i 0 2.74152 + 1.13557i 0
73.6 0 0.0357590 − 0.179773i 0 2.03483 + 0.927070i 0 0.528939 + 0.353426i 0 2.74060 + 1.13519i 0
73.7 0 0.197416 − 0.992478i 0 −2.21386 + 0.314337i 0 4.11814 + 2.75165i 0 1.82560 + 0.756188i 0
73.8 0 0.486949 − 2.44806i 0 2.17164 − 0.532896i 0 1.76999 + 1.18267i 0 −2.98423 − 1.23611i 0
73.9 0 0.588527 − 2.95872i 0 −1.22786 + 1.86879i 0 −1.96902 − 1.31565i 0 −5.63604 − 2.33452i 0
133.1 0 −2.36788 + 1.58217i 0 2.10787 − 0.746254i 0 1.12893 − 0.224559i 0 1.95555 − 4.72111i 0
133.2 0 −1.89969 + 1.26933i 0 −1.28747 + 1.82823i 0 2.80650 − 0.558249i 0 0.849575 − 2.05105i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 57.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.o even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 340.2.bd.a ✓ 72
5.c odd 4 1 340.2.bi.a yes 72
17.e odd 16 1 340.2.bi.a yes 72
85.o even 16 1 inner 340.2.bd.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.2.bd.a ✓ 72 1.a even 1 1 trivial
340.2.bd.a ✓ 72 85.o even 16 1 inner
340.2.bi.a yes 72 5.c odd 4 1
340.2.bi.a yes 72 17.e odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(340, [\chi])\).