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Properties

Label 770.2.bd.a
Level $770$
Weight $2$
Character orbit 770.bd
Analytic conductor $6.148$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(263,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 8, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.263");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bd (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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{5} + 4 q^{11} + 16 q^{15} + 96 q^{16} + 24 q^{22} - 8 q^{23} + 24 q^{25} - 8 q^{26} - 16 q^{33} - 160 q^{36} - 40 q^{37} - 40 q^{42} + 32 q^{45} - 40 q^{47} + 40 q^{53} + 8 q^{55} + 16 q^{56} + 32 q^{58} - 48 q^{67} - 72 q^{70} + 32 q^{71} - 48 q^{75} - 72 q^{77} - 8 q^{80} + 64 q^{81} + 16 q^{82} - 12 q^{88} - 112 q^{91} - 16 q^{92} + 80 q^{93} - 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1 −0.258819 + 0.965926i −3.03497 + 0.813216i −0.866025 − 0.500000i −1.03794 + 1.98058i − 3.14203i −2.36036 − 1.19528i 0.707107 − 0.707107i 5.95162 − 3.43617i −1.64445 − 1.51518i
263.2 −0.258819 + 0.965926i −2.93076 + 0.785295i −0.866025 − 0.500000i 0.860133 − 2.06402i − 3.03415i −0.604940 + 2.57566i 0.707107 − 0.707107i 5.37460 − 3.10303i 1.77107 + 1.36503i
263.3 −0.258819 + 0.965926i −2.59645 + 0.695717i −0.866025 − 0.500000i 2.04774 + 0.898208i − 2.68804i 1.70855 + 2.02011i 0.707107 − 0.707107i 3.65945 − 2.11279i −1.39760 + 1.74549i
263.4 −0.258819 + 0.965926i −2.38824 + 0.639927i −0.866025 − 0.500000i 0.590421 + 2.15671i − 2.47249i 2.05448 − 1.66707i 0.707107 − 0.707107i 2.69611 − 1.55660i −2.23604 + 0.0121050i
263.5 −0.258819 + 0.965926i −1.88147 + 0.504137i −0.866025 − 0.500000i −1.86717 − 1.23031i − 1.94784i −1.82332 − 1.91716i 0.707107 − 0.707107i 0.687681 − 0.397033i 1.67165 − 1.48512i
263.6 −0.258819 + 0.965926i −1.69921 + 0.455302i −0.866025 − 0.500000i 2.19819 − 0.409847i − 1.75915i −2.07717 − 1.63871i 0.707107 − 0.707107i 0.0819358 − 0.0473057i −0.173050 + 2.22936i
263.7 −0.258819 + 0.965926i −1.66804 + 0.446949i −0.866025 − 0.500000i −2.10823 + 0.745241i − 1.72688i −1.49658 + 2.18180i 0.707107 − 0.707107i −0.0154982 + 0.00894790i −0.174199 − 2.22927i
263.8 −0.258819 + 0.965926i −1.43175 + 0.383635i −0.866025 − 0.500000i −0.782415 − 2.09471i − 1.48225i 1.67839 + 2.04524i 0.707107 − 0.707107i −0.695357 + 0.401464i 2.22584 − 0.213603i
263.9 −0.258819 + 0.965926i −0.886001 + 0.237403i −0.866025 − 0.500000i −1.83005 + 1.28488i − 0.917256i 2.44857 − 1.00226i 0.707107 − 0.707107i −1.86944 + 1.07932i −0.767449 − 2.10024i
263.10 −0.258819 + 0.965926i −0.782217 + 0.209594i −0.866025 − 0.500000i 2.11644 + 0.721579i − 0.809810i −2.52664 + 0.784929i 0.707107 − 0.707107i −2.03014 + 1.17210i −1.24477 + 1.85757i
263.11 −0.258819 + 0.965926i −0.528342 + 0.141569i −0.866025 − 0.500000i 1.04639 − 1.97613i − 0.546979i 2.21126 − 1.45270i 0.707107 − 0.707107i −2.33897 + 1.35041i 1.63797 + 1.52219i
263.12 −0.258819 + 0.965926i 0.00659019 − 0.00176584i −0.866025 − 0.500000i −1.60119 − 1.56083i 0.00682267i −1.96250 + 1.77443i 0.707107 − 0.707107i −2.59804 + 1.49998i 1.92207 − 1.14266i
263.13 −0.258819 + 0.965926i 0.402580 − 0.107871i −0.866025 − 0.500000i 0.481447 + 2.18362i 0.416781i 1.76819 + 1.96812i 0.707107 − 0.707107i −2.44764 + 1.41315i −2.23383 − 0.100121i
263.14 −0.258819 + 0.965926i 0.417007 − 0.111737i −0.866025 − 0.500000i 0.441318 + 2.19209i 0.431718i 0.213128 − 2.63715i 0.707107 − 0.707107i −2.43667 + 1.40681i −2.23161 − 0.141073i
263.15 −0.258819 + 0.965926i 0.854959 − 0.229085i −0.866025 − 0.500000i 2.22654 − 0.206181i 0.885118i 1.06163 + 2.42342i 0.707107 − 0.707107i −1.91960 + 1.10828i −0.377116 + 2.20404i
263.16 −0.258819 + 0.965926i 0.887916 − 0.237917i −0.866025 − 0.500000i −2.09617 − 0.778514i 0.919239i 0.800398 − 2.52178i 0.707107 − 0.707107i −1.86628 + 1.07750i 1.29451 − 1.82325i
263.17 −0.258819 + 0.965926i 1.01597 − 0.272228i −0.866025 − 0.500000i 0.755631 − 2.10452i 1.05181i −1.57788 − 2.12375i 0.707107 − 0.707107i −1.63999 + 0.946852i 1.83724 + 1.27457i
263.18 −0.258819 + 0.965926i 1.61428 − 0.432545i −0.866025 − 0.500000i −1.12130 + 1.93460i 1.67123i −2.48435 − 0.909950i 0.707107 − 0.707107i −0.179268 + 0.103501i −1.57847 − 1.58380i
263.19 −0.258819 + 0.965926i 1.68009 − 0.450179i −0.866025 − 0.500000i −2.17405 − 0.522991i 1.73936i 2.63950 − 0.181786i 0.707107 − 0.707107i 0.0219675 − 0.0126829i 1.06786 − 1.96461i
263.20 −0.258819 + 0.965926i 2.03597 − 0.545536i −0.866025 − 0.500000i −1.98226 + 1.03472i 2.10779i −0.518835 + 2.59438i 0.707107 − 0.707107i 1.24948 − 0.721389i −0.486413 − 2.18252i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
11.b odd 2 1 inner
35.l odd 12 1 inner
55.e even 4 1 inner
77.h odd 6 1 inner
385.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bd.a ✓ 192
5.c odd 4 1 inner 770.2.bd.a ✓ 192
7.c even 3 1 inner 770.2.bd.a ✓ 192
11.b odd 2 1 inner 770.2.bd.a ✓ 192
35.l odd 12 1 inner 770.2.bd.a ✓ 192
55.e even 4 1 inner 770.2.bd.a ✓ 192
77.h odd 6 1 inner 770.2.bd.a ✓ 192
385.bc even 12 1 inner 770.2.bd.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bd.a ✓ 192 1.a even 1 1 trivial
770.2.bd.a ✓ 192 5.c odd 4 1 inner
770.2.bd.a ✓ 192 7.c even 3 1 inner
770.2.bd.a ✓ 192 11.b odd 2 1 inner
770.2.bd.a ✓ 192 35.l odd 12 1 inner
770.2.bd.a ✓ 192 55.e even 4 1 inner
770.2.bd.a ✓ 192 77.h odd 6 1 inner
770.2.bd.a ✓ 192 385.bc even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(770, [\chi])\).