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Properties

Label 670.2.o.a
Level $670$
Weight $2$
Character orbit 670.o
Analytic conductor $5.350$
Analytic rank $0$
Dimension $340$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [670,2,Mod(9,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.o (of order \(22\), degree \(10\), 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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(340\)
Relative dimension: \(34\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 340 q + 34 q^{4} + 4 q^{5} - 4 q^{6} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 340 q + 34 q^{4} + 4 q^{5} - 4 q^{6} + 30 q^{9} + 4 q^{11} + 4 q^{14} - 36 q^{15} - 34 q^{16} + 88 q^{19} - 4 q^{20} - 2 q^{21} - 18 q^{24} - 4 q^{25} + 12 q^{29} + 36 q^{31} - 4 q^{35} - 30 q^{36} + 24 q^{39} - 104 q^{41} - 4 q^{44} - 62 q^{45} - 12 q^{46} + 86 q^{49} - 16 q^{50} + 8 q^{51} + 40 q^{54} - 26 q^{55} - 4 q^{56} - 160 q^{59} + 36 q^{60} - 74 q^{61} + 34 q^{64} + 56 q^{65} + 104 q^{66} - 32 q^{69} + 150 q^{70} - 192 q^{71} - 12 q^{74} + 228 q^{75} - 88 q^{76} - 32 q^{79} - 18 q^{80} - 134 q^{81} - 64 q^{84} + 4 q^{85} - 20 q^{86} + 184 q^{89} + 4 q^{90} - 8 q^{91} - 64 q^{94} - 106 q^{95} - 4 q^{96} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.540641 + 0.841254i −0.875484 + 2.98162i −0.415415 − 0.909632i 0.634176 + 2.14425i −2.03498 − 2.34849i −0.494544 + 0.769526i 0.989821 + 0.142315i −5.59985 − 3.59881i −2.14672 − 0.625768i
9.2 −0.540641 + 0.841254i −0.845975 + 2.88113i −0.415415 − 0.909632i −0.291316 − 2.21701i −1.96639 − 2.26933i −1.78490 + 2.77736i 0.989821 + 0.142315i −5.06146 − 3.25280i 2.02257 + 0.953536i
9.3 −0.540641 + 0.841254i −0.745927 + 2.54039i −0.415415 − 0.909632i 2.15874 − 0.582942i −1.73384 − 2.00095i 1.67525 − 2.60674i 0.989821 + 0.142315i −3.37343 − 2.16797i −0.676703 + 2.13121i
9.4 −0.540641 + 0.841254i −0.582012 + 1.98215i −0.415415 − 0.909632i −1.78198 − 1.35075i −1.35283 − 1.56125i −0.102317 + 0.159209i 0.989821 + 0.142315i −1.06642 − 0.685349i 2.09974 − 0.768828i
9.5 −0.540641 + 0.841254i −0.315284 + 1.07376i −0.415415 − 0.909632i −1.84761 − 1.25950i −0.732849 − 0.845752i 2.66226 − 4.14256i 0.989821 + 0.142315i 1.47020 + 0.944843i 2.05845 − 0.873375i
9.6 −0.540641 + 0.841254i −0.278858 + 0.949704i −0.415415 − 0.909632i −0.464004 + 2.18740i −0.648180 − 0.748039i 0.770972 − 1.19966i 0.989821 + 0.142315i 1.69958 + 1.09226i −1.58929 − 1.57294i
9.7 −0.540641 + 0.841254i −0.248316 + 0.845687i −0.415415 − 0.909632i −1.78230 + 1.35033i −0.577187 − 0.666110i −2.47530 + 3.85164i 0.989821 + 0.142315i 1.87024 + 1.20193i −0.172383 − 2.22941i
9.8 −0.540641 + 0.841254i −0.204353 + 0.695962i −0.415415 − 0.909632i 2.23447 + 0.0844229i −0.474999 − 0.548178i −1.91882 + 2.98574i 0.989821 + 0.142315i 2.08116 + 1.33748i −1.27907 + 1.83412i
9.9 −0.540641 + 0.841254i −0.123607 + 0.420968i −0.415415 − 0.909632i −0.505362 + 2.17821i −0.287313 − 0.331577i 2.10528 − 3.27589i 0.989821 + 0.142315i 2.36183 + 1.51785i −1.55921 − 1.60277i
9.10 −0.540641 + 0.841254i 0.0892774 − 0.304051i −0.415415 − 0.909632i 0.791296 − 2.09138i 0.207517 + 0.239487i 0.0688643 − 0.107155i 0.989821 + 0.142315i 2.43928 + 1.56763i 1.33157 + 1.79636i
9.11 −0.540641 + 0.841254i 0.289617 − 0.986346i −0.415415 − 0.909632i 1.61319 − 1.54842i 0.673188 + 0.776901i 1.31888 − 2.05221i 0.989821 + 0.142315i 1.63476 + 1.05060i 0.430459 + 2.19424i
9.12 −0.540641 + 0.841254i 0.323497 − 1.10173i −0.415415 − 0.909632i −2.23592 − 0.0260743i 0.751938 + 0.867783i −0.365620 + 0.568915i 0.989821 + 0.142315i 1.41460 + 0.909110i 1.23076 − 1.86688i
9.13 −0.540641 + 0.841254i 0.418836 − 1.42642i −0.415415 − 0.909632i 1.09254 + 1.95099i 0.973544 + 1.12353i −0.718561 + 1.11810i 0.989821 + 0.142315i 0.664502 + 0.427049i −2.23195 − 0.135682i
9.14 −0.540641 + 0.841254i 0.495306 − 1.68686i −0.415415 − 0.909632i 1.95654 + 1.08257i 1.15129 + 1.32866i 1.33984 − 2.08484i 0.989821 + 0.142315i −0.0764034 − 0.0491015i −1.96850 + 1.06067i
9.15 −0.540641 + 0.841254i 0.782914 − 2.66636i −0.415415 − 0.909632i −1.43037 − 1.71873i 1.81981 + 2.10017i −1.35168 + 2.10326i 0.989821 + 0.142315i −3.97277 − 2.55314i 2.21921 − 0.274087i
9.16 −0.540641 + 0.841254i 0.793104 − 2.70106i −0.415415 − 0.909632i −0.378763 − 2.20376i 1.84349 + 2.12751i 2.05948 − 3.20462i 0.989821 + 0.142315i −4.14297 − 2.66253i 2.05869 + 0.872804i
9.17 −0.540641 + 0.841254i 0.793193 − 2.70137i −0.415415 − 0.909632i 0.0502795 + 2.23550i 1.84370 + 2.12775i −1.75161 + 2.72556i 0.989821 + 0.142315i −4.14447 − 2.66349i −1.90781 − 1.16631i
9.18 0.540641 − 0.841254i −0.793193 + 2.70137i −0.415415 − 0.909632i −2.21990 + 0.268377i 1.84370 + 2.12775i 1.75161 − 2.72556i −0.989821 − 0.142315i −4.14447 − 2.66349i −0.974397 + 2.01260i
9.19 0.540641 − 0.841254i −0.793104 + 2.70106i −0.415415 − 0.909632i 2.23523 + 0.0612811i 1.84349 + 2.12751i −2.05948 + 3.20462i −0.989821 − 0.142315i −4.14297 − 2.66253i 1.26001 − 1.84726i
9.20 0.540641 − 0.841254i −0.782914 + 2.66636i −0.415415 − 0.909632i 1.90480 + 1.17121i 1.81981 + 2.10017i 1.35168 − 2.10326i −0.989821 − 0.142315i −3.97277 − 2.55314i 2.01510 − 0.969217i
See next 80 embeddings (of 340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
67.e even 11 1 inner
335.o even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.o.a ✓ 340
5.b even 2 1 inner 670.2.o.a ✓ 340
67.e even 11 1 inner 670.2.o.a ✓ 340
335.o even 22 1 inner 670.2.o.a ✓ 340
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.o.a ✓ 340 1.a even 1 1 trivial
670.2.o.a ✓ 340 5.b even 2 1 inner
670.2.o.a ✓ 340 67.e even 11 1 inner
670.2.o.a ✓ 340 335.o even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(670, [\chi])\).