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Properties

Label 696.2.ba.b
Level $696$
Weight $2$
Character orbit 696.ba
Analytic conductor $5.558$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [696,2,Mod(121,696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(696, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("696.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 696 = 2^{3} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 696.ba (of order \(14\), 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: \(5.55758798068\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{5} + 12 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{5} + 12 q^{7} + 8 q^{9} - 14 q^{15} + 14 q^{19} - 14 q^{21} + 14 q^{23} + 12 q^{25} + 2 q^{29} - 14 q^{31} + 6 q^{33} - 10 q^{35} + 14 q^{37} - 14 q^{39} - 14 q^{43} + 8 q^{45} + 14 q^{47} + 40 q^{49} + 8 q^{51} + 58 q^{53} - 4 q^{57} + 48 q^{59} - 42 q^{61} + 16 q^{63} + 76 q^{65} + 16 q^{67} + 14 q^{69} + 50 q^{71} - 42 q^{73} + 70 q^{77} - 56 q^{79} - 8 q^{81} - 70 q^{83} + 32 q^{87} - 56 q^{89} + 92 q^{91} - 18 q^{93} + 84 q^{95} - 42 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −0.974928 + 0.222521i 0 −1.53024 − 1.91886i 0 0.376705 + 1.65045i 0 0.900969 − 0.433884i 0
121.2 0 −0.974928 + 0.222521i 0 −0.158505 − 0.198760i 0 0.0546923 + 0.239622i 0 0.900969 − 0.433884i 0
121.3 0 −0.974928 + 0.222521i 0 1.28265 + 1.60839i 0 0.518619 + 2.27222i 0 0.900969 − 0.433884i 0
121.4 0 −0.974928 + 0.222521i 0 1.88102 + 2.35873i 0 −0.602931 − 2.64161i 0 0.900969 − 0.433884i 0
121.5 0 0.974928 − 0.222521i 0 −1.83351 − 2.29915i 0 −0.334317 − 1.46474i 0 0.900969 − 0.433884i 0
121.6 0 0.974928 − 0.222521i 0 −0.917762 − 1.15084i 0 0.879951 + 3.85532i 0 0.900969 − 0.433884i 0
121.7 0 0.974928 − 0.222521i 0 0.179695 + 0.225330i 0 −0.724686 − 3.17506i 0 0.900969 − 0.433884i 0
121.8 0 0.974928 − 0.222521i 0 2.09665 + 2.62912i 0 0.139944 + 0.613137i 0 0.900969 − 0.433884i 0
241.1 0 −0.781831 + 0.623490i 0 −3.17522 + 1.52910i 0 2.94923 + 3.69821i 0 0.222521 − 0.974928i 0
241.2 0 −0.781831 + 0.623490i 0 −0.0554207 + 0.0266892i 0 −0.0952383 − 0.119425i 0 0.222521 − 0.974928i 0
241.3 0 −0.781831 + 0.623490i 0 0.901430 − 0.434106i 0 −1.52611 − 1.91369i 0 0.222521 − 0.974928i 0
241.4 0 −0.781831 + 0.623490i 0 3.61104 − 1.73898i 0 2.41230 + 3.02493i 0 0.222521 − 0.974928i 0
241.5 0 0.781831 − 0.623490i 0 −3.75798 + 1.80975i 0 −0.218571 − 0.274079i 0 0.222521 − 0.974928i 0
241.6 0 0.781831 − 0.623490i 0 −1.31281 + 0.632216i 0 2.60457 + 3.26602i 0 0.222521 − 0.974928i 0
241.7 0 0.781831 − 0.623490i 0 2.13200 − 1.02672i 0 −1.54615 − 1.93882i 0 0.222521 − 0.974928i 0
241.8 0 0.781831 − 0.623490i 0 2.65696 − 1.27952i 0 0.468903 + 0.587986i 0 0.222521 − 0.974928i 0
265.1 0 −0.433884 − 0.900969i 0 −0.540435 + 2.36780i 0 2.56394 − 1.23473i 0 −0.623490 + 0.781831i 0
265.2 0 −0.433884 − 0.900969i 0 −0.0646396 + 0.283205i 0 −1.40255 + 0.675434i 0 −0.623490 + 0.781831i 0
265.3 0 −0.433884 − 0.900969i 0 0.566652 − 2.48266i 0 −2.88827 + 1.39092i 0 −0.623490 + 0.781831i 0
265.4 0 −0.433884 − 0.900969i 0 0.972307 − 4.25995i 0 3.45726 − 1.66493i 0 −0.623490 + 0.781831i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 696.2.ba.b ✓ 48
29.e even 14 1 inner 696.2.ba.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
696.2.ba.b ✓ 48 1.a even 1 1 trivial
696.2.ba.b ✓ 48 29.e even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} - 6 T_{5}^{47} + 32 T_{5}^{46} - 76 T_{5}^{45} + 202 T_{5}^{44} - 632 T_{5}^{43} + \cdots + 226291849 \) acting on \(S_{2}^{\mathrm{new}}(696, [\chi])\). Copy content Toggle raw display