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Properties

Label 680.2.bo.a
Level $680$
Weight $2$
Character orbit 680.bo
Analytic conductor $5.430$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(121,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 0, 7])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.bo (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{9} + 8 q^{11} - 8 q^{17} + 8 q^{19} + 8 q^{23} - 24 q^{27} - 32 q^{29} + 32 q^{31} + 16 q^{33} + 16 q^{35} + 16 q^{37} - 24 q^{39} - 16 q^{41} - 8 q^{49} + 16 q^{51} + 16 q^{53} + 48 q^{57}+ \cdots + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −2.77754 + 1.15049i 0 −0.382683 − 0.923880i 0 −1.49876 + 3.61834i 0 4.26976 − 4.26976i 0
121.2 0 −2.58372 + 1.07021i 0 0.382683 + 0.923880i 0 0.740612 − 1.78800i 0 3.40892 − 3.40892i 0
121.3 0 −0.465135 + 0.192665i 0 −0.382683 − 0.923880i 0 0.134218 − 0.324032i 0 −1.94209 + 1.94209i 0
121.4 0 −0.120181 + 0.0497804i 0 0.382683 + 0.923880i 0 0.278826 − 0.673145i 0 −2.10936 + 2.10936i 0
121.5 0 −0.0428807 + 0.0177618i 0 −0.382683 − 0.923880i 0 −1.01246 + 2.44430i 0 −2.11980 + 2.11980i 0
121.6 0 1.46749 − 0.607854i 0 0.382683 + 0.923880i 0 −1.67090 + 4.03391i 0 −0.337282 + 0.337282i 0
121.7 0 2.16029 − 0.894821i 0 0.382683 + 0.923880i 0 0.709723 − 1.71342i 0 1.74482 − 1.74482i 0
121.8 0 2.36167 − 0.978237i 0 −0.382683 − 0.923880i 0 0.904535 − 2.18374i 0 2.49923 − 2.49923i 0
161.1 0 −0.805905 + 1.94563i 0 0.923880 + 0.382683i 0 4.50228 − 1.86491i 0 −1.01466 − 1.01466i 0
161.2 0 −0.597638 + 1.44282i 0 −0.923880 − 0.382683i 0 0.401099 − 0.166141i 0 0.396748 + 0.396748i 0
161.3 0 −0.575026 + 1.38824i 0 0.923880 + 0.382683i 0 −2.70987 + 1.12246i 0 0.524778 + 0.524778i 0
161.4 0 −0.251249 + 0.606569i 0 −0.923880 − 0.382683i 0 −3.28536 + 1.36084i 0 1.81652 + 1.81652i 0
161.5 0 0.221505 − 0.534762i 0 −0.923880 − 0.382683i 0 0.202666 − 0.0839471i 0 1.88442 + 1.88442i 0
161.6 0 0.424268 − 1.02427i 0 0.923880 + 0.382683i 0 −0.208657 + 0.0864287i 0 1.25219 + 1.25219i 0
161.7 0 0.573979 − 1.38571i 0 0.923880 + 0.382683i 0 0.971110 − 0.402247i 0 0.530583 + 0.530583i 0
161.8 0 1.01006 − 2.43851i 0 −0.923880 − 0.382683i 0 1.54095 − 0.638280i 0 −2.80479 − 2.80479i 0
281.1 0 −2.77754 − 1.15049i 0 −0.382683 + 0.923880i 0 −1.49876 − 3.61834i 0 4.26976 + 4.26976i 0
281.2 0 −2.58372 − 1.07021i 0 0.382683 − 0.923880i 0 0.740612 + 1.78800i 0 3.40892 + 3.40892i 0
281.3 0 −0.465135 − 0.192665i 0 −0.382683 + 0.923880i 0 0.134218 + 0.324032i 0 −1.94209 − 1.94209i 0
281.4 0 −0.120181 − 0.0497804i 0 0.382683 − 0.923880i 0 0.278826 + 0.673145i 0 −2.10936 − 2.10936i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.bo.a ✓ 32
17.d even 8 1 inner 680.2.bo.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.bo.a ✓ 32 1.a even 1 1 trivial
680.2.bo.a ✓ 32 17.d even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 8 T_{3}^{30} + 8 T_{3}^{29} + 32 T_{3}^{28} - 128 T_{3}^{27} + 232 T_{3}^{26} + 200 T_{3}^{25} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\). Copy content Toggle raw display