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Properties

Label 693.2.cg.c
Level $693$
Weight $2$
Character orbit 693.cg
Analytic conductor $5.534$
Analytic rank $0$
Dimension $128$
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] = [693,2,Mod(19,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.cg (of order \(30\), degree \(8\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{30})\)
Twist minimal: no (minimal twist has level 231)
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 12 q^{4} - 12 q^{5} - 10 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 12 q^{4} - 12 q^{5} - 10 q^{7} + 40 q^{8} + 2 q^{11} - 12 q^{14} + 40 q^{16} + 60 q^{17} + 52 q^{22} + 24 q^{23} - 20 q^{25} - 24 q^{26} + 30 q^{28} - 40 q^{29} - 18 q^{31} + 80 q^{35} - 8 q^{37} + 24 q^{38} - 90 q^{40} + 82 q^{44} + 70 q^{46} + 24 q^{47} - 94 q^{49} - 4 q^{53} + 104 q^{56} - 32 q^{58} - 48 q^{59} + 30 q^{61} - 48 q^{64} - 40 q^{67} - 180 q^{68} + 146 q^{70} + 32 q^{71} + 90 q^{73} - 40 q^{74} + 50 q^{79} - 228 q^{80} + 168 q^{82} - 20 q^{85} - 146 q^{86} + 16 q^{88} + 48 q^{91} + 204 q^{92} - 10 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.65417 + 0.278965i 0 5.01052 − 1.06502i 0.0958279 + 0.215233i 0 −2.64356 − 0.107664i −7.92533 + 2.57510i 0 −0.314386 − 0.544533i
19.2 −2.26565 + 0.238129i 0 3.12016 − 0.663211i 0.832978 + 1.87090i 0 −1.73794 + 1.99489i −2.57800 + 0.837642i 0 −2.33275 − 4.04044i
19.3 −1.95149 + 0.205109i 0 1.80993 − 0.384713i −0.888231 − 1.99500i 0 −1.14019 − 2.38746i 0.279246 − 0.0907326i 0 2.14256 + 3.71103i
19.4 −1.80279 + 0.189480i 0 1.25784 − 0.267361i 1.26266 + 2.83599i 0 1.48247 − 2.19141i 1.23104 − 0.399989i 0 −2.81368 − 4.87343i
19.5 −0.884754 + 0.0929913i 0 −1.18215 + 0.251275i −1.63822 − 3.67950i 0 −1.92178 + 1.81845i 2.71472 − 0.882066i 0 1.79158 + 3.10311i
19.6 −0.788039 + 0.0828262i 0 −1.34215 + 0.285283i −0.490588 − 1.10188i 0 2.62289 + 0.347026i 2.54123 − 0.825697i 0 0.477866 + 0.827689i
19.7 0.0732542 − 0.00769933i 0 −1.95099 + 0.414695i 1.46180 + 3.28325i 0 0.114944 − 2.64325i −0.279831 + 0.0909225i 0 0.132362 + 0.229257i
19.8 0.0764050 − 0.00803049i 0 −1.95052 + 0.414596i −0.750633 − 1.68595i 0 −1.66664 + 2.05482i −0.291832 + 0.0948219i 0 −0.0708911 − 0.122787i
19.9 0.373095 − 0.0392138i 0 −1.81863 + 0.386562i −0.294587 − 0.661654i 0 2.60364 − 0.470147i −1.37694 + 0.447395i 0 −0.135855 − 0.235308i
19.10 0.554274 − 0.0582566i 0 −1.65247 + 0.351243i 0.990165 + 2.22395i 0 0.582413 + 2.58085i −1.95556 + 0.635400i 0 0.678382 + 1.17499i
19.11 1.58179 − 0.166253i 0 0.518133 − 0.110133i −1.31309 − 2.94925i 0 −2.10118 − 1.60780i −2.22405 + 0.722639i 0 −2.56736 − 4.44679i
19.12 1.62945 − 0.171263i 0 0.669495 − 0.142305i −0.520050 − 1.16805i 0 1.22463 − 2.34527i −2.04994 + 0.666066i 0 −1.04744 − 1.81422i
19.13 2.08559 − 0.219204i 0 2.34532 − 0.498514i 0.216464 + 0.486186i 0 −0.578918 + 2.58164i 0.793223 − 0.257734i 0 0.558028 + 0.966533i
19.14 2.47627 − 0.260267i 0 4.10789 − 0.873159i 1.61981 + 3.63814i 0 −2.49816 + 0.871324i 5.20890 − 1.69247i 0 4.95797 + 8.58745i
19.15 2.63866 − 0.277334i 0 4.92930 − 1.04776i −0.00599929 − 0.0134746i 0 0.918955 − 2.48103i 7.66950 − 2.49197i 0 −0.0195671 − 0.0338911i
19.16 2.64148 − 0.277631i 0 4.94404 − 1.05089i −1.44910 − 3.25473i 0 1.42250 + 2.23081i 7.71576 − 2.50700i 0 −4.73138 − 8.19500i
73.1 −2.65417 − 0.278965i 0 5.01052 + 1.06502i 0.0958279 − 0.215233i 0 −2.64356 + 0.107664i −7.92533 − 2.57510i 0 −0.314386 + 0.544533i
73.2 −2.26565 − 0.238129i 0 3.12016 + 0.663211i 0.832978 − 1.87090i 0 −1.73794 − 1.99489i −2.57800 − 0.837642i 0 −2.33275 + 4.04044i
73.3 −1.95149 − 0.205109i 0 1.80993 + 0.384713i −0.888231 + 1.99500i 0 −1.14019 + 2.38746i 0.279246 + 0.0907326i 0 2.14256 − 3.71103i
73.4 −1.80279 − 0.189480i 0 1.25784 + 0.267361i 1.26266 − 2.83599i 0 1.48247 + 2.19141i 1.23104 + 0.399989i 0 −2.81368 + 4.87343i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.cg.c 128
3.b odd 2 1 231.2.ba.a ✓ 128
7.d odd 6 1 inner 693.2.cg.c 128
11.d odd 10 1 inner 693.2.cg.c 128
21.g even 6 1 231.2.ba.a ✓ 128
33.f even 10 1 231.2.ba.a ✓ 128
77.n even 30 1 inner 693.2.cg.c 128
231.bf odd 30 1 231.2.ba.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.ba.a ✓ 128 3.b odd 2 1
231.2.ba.a ✓ 128 21.g even 6 1
231.2.ba.a ✓ 128 33.f even 10 1
231.2.ba.a ✓ 128 231.bf odd 30 1
693.2.cg.c 128 1.a even 1 1 trivial
693.2.cg.c 128 7.d odd 6 1 inner
693.2.cg.c 128 11.d odd 10 1 inner
693.2.cg.c 128 77.n even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} + 22 T_{2}^{126} - 40 T_{2}^{125} + 167 T_{2}^{124} - 880 T_{2}^{123} + 322 T_{2}^{122} + \cdots + 13845841 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display