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Properties

Label 693.2.bz.a
Level $693$
Weight $2$
Character orbit 693.bz
Analytic conductor $5.534$
Analytic rank $0$
Dimension $288$
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(148,693)]
 
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([10, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.148");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bz (of order \(15\), 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: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} - 3 q^{3} + 36 q^{4} - 9 q^{5} - 3 q^{6} - 36 q^{7} - 12 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} - 3 q^{3} + 36 q^{4} - 9 q^{5} - 3 q^{6} - 36 q^{7} - 12 q^{8} + 13 q^{9} - 2 q^{11} - 36 q^{12} + 6 q^{13} - 2 q^{14} - 9 q^{15} + 36 q^{16} + 18 q^{17} + 14 q^{18} - 18 q^{19} - 24 q^{20} - 8 q^{21} - 15 q^{22} + 56 q^{23} - 15 q^{24} + 33 q^{25} + 56 q^{26} - 84 q^{27} + 72 q^{28} + 13 q^{29} + 38 q^{30} - 3 q^{31} - 56 q^{32} - 21 q^{33} - 18 q^{34} - 18 q^{35} + 9 q^{36} + 6 q^{37} + 18 q^{38} + 13 q^{39} - 14 q^{41} - 6 q^{42} - 6 q^{43} - 80 q^{44} - 30 q^{45} - 4 q^{47} + 96 q^{48} + 36 q^{49} - 40 q^{50} - 21 q^{51} - 36 q^{52} + 84 q^{53} + 26 q^{54} + 36 q^{55} + 24 q^{56} + 74 q^{57} + 100 q^{60} + 18 q^{62} - 21 q^{63} - 24 q^{64} - 78 q^{65} - 8 q^{66} + 6 q^{67} + 29 q^{68} + 18 q^{69} + 50 q^{71} - 57 q^{72} - 12 q^{73} + 8 q^{74} + 18 q^{75} + 54 q^{76} - 3 q^{77} + 60 q^{78} - 6 q^{79} - 160 q^{80} + 21 q^{81} - 18 q^{82} - 18 q^{83} + 14 q^{84} - 6 q^{85} + 61 q^{86} + 28 q^{87} - 81 q^{88} - 300 q^{89} - 26 q^{90} - 18 q^{91} - 78 q^{92} + 41 q^{93} + 24 q^{94} + 8 q^{95} + 104 q^{96} + 15 q^{97} + 16 q^{98} + 171 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1 −1.80902 + 2.00912i −0.476986 + 1.66508i −0.554956 − 5.28006i 1.25753 + 1.39663i −2.48247 − 3.97049i −0.913545 − 0.406737i 7.23779 + 5.25856i −2.54497 − 1.58844i −5.08092
148.2 −1.76714 + 1.96261i −1.53697 − 0.798582i −0.519988 − 4.94736i 0.271873 + 0.301945i 4.28334 − 1.60525i −0.913545 − 0.406737i 6.35546 + 4.61751i 1.72453 + 2.45479i −1.07304
148.3 −1.67940 + 1.86517i 1.62298 − 0.604939i −0.449392 − 4.27568i −1.31821 − 1.46402i −1.59732 + 4.04305i −0.913545 − 0.406737i 4.66858 + 3.39192i 2.26810 − 1.96360i 4.94443
148.4 −1.53838 + 1.70854i −1.51677 + 0.836300i −0.343449 − 3.26770i −1.89439 − 2.10394i 0.904515 − 3.87801i −0.913545 − 0.406737i 2.39139 + 1.73744i 1.60120 − 2.53696i 6.50894
148.5 −1.51549 + 1.68312i −0.128866 − 1.72725i −0.327132 − 3.11245i −2.52948 − 2.80928i 3.10246 + 2.40073i −0.913545 − 0.406737i 2.06977 + 1.50378i −2.96679 + 0.445168i 8.56175
148.6 −1.34384 + 1.49248i 1.42401 − 0.986000i −0.212549 − 2.02227i 0.985396 + 1.09439i −0.442051 + 3.45034i −0.913545 − 0.406737i 0.0542795 + 0.0394364i 1.05561 − 2.80815i −2.95758
148.7 −1.32219 + 1.46845i 0.931157 + 1.46046i −0.199078 − 1.89410i 0.159171 + 0.176778i −3.37578 − 0.563660i −0.913545 − 0.406737i −0.152613 − 0.110880i −1.26589 + 2.71984i −0.470044
148.8 −1.12191 + 1.24601i −0.688274 − 1.58943i −0.0847967 − 0.806787i 0.561545 + 0.623659i 2.75263 + 0.925602i −0.913545 − 0.406737i −1.61251 − 1.17156i −2.05256 + 2.18792i −1.40709
148.9 −1.03050 + 1.14449i −0.361534 + 1.69390i −0.0388623 − 0.369750i −1.21976 − 1.35468i −1.56609 − 2.15934i −0.913545 − 0.406737i −2.02865 − 1.47390i −2.73859 − 1.22480i 2.80738
148.10 −0.903972 + 1.00396i 1.68029 + 0.420262i 0.0182814 + 0.173936i −2.91174 − 3.23382i −1.94086 + 1.30704i −0.913545 − 0.406737i −2.37706 − 1.72703i 2.64676 + 1.41233i 5.87877
148.11 −0.902791 + 1.00265i 1.44909 + 0.948762i 0.0187793 + 0.178673i 2.73738 + 3.04017i −2.25950 + 0.596394i −0.913545 − 0.406737i −2.37915 − 1.72856i 1.19970 + 2.74968i −5.51952
148.12 −0.737704 + 0.819303i −0.963393 − 1.43940i 0.0820063 + 0.780238i 2.25112 + 2.50012i 1.89000 + 0.272540i −0.913545 − 0.406737i −2.48360 − 1.80444i −1.14375 + 2.77342i −3.70902
148.13 −0.694075 + 0.770848i −1.59540 + 0.674308i 0.0965898 + 0.918991i 0.663845 + 0.737274i 0.587540 − 1.69783i −0.913545 − 0.406737i −2.45380 − 1.78279i 2.09062 − 2.15158i −1.02908
148.14 −0.539040 + 0.598664i 0.957885 − 1.44307i 0.141222 + 1.34364i −0.831645 − 0.923636i 0.347578 + 1.35132i −0.913545 − 0.406737i −2.18397 − 1.58675i −1.16491 − 2.76459i 1.00124
148.15 −0.429677 + 0.477205i 0.583754 − 1.63072i 0.165955 + 1.57896i −0.823357 − 0.914431i 0.527359 + 0.979251i −0.913545 − 0.406737i −1.86380 − 1.35413i −2.31846 − 1.90387i 0.790148
148.16 −0.397381 + 0.441336i −1.71400 − 0.249402i 0.172191 + 1.63829i 0.969542 + 1.07679i 0.791181 − 0.657343i −0.913545 − 0.406737i −1.75237 − 1.27317i 2.87560 + 0.854949i −0.860502
148.17 −0.0178340 + 0.0198067i −0.913998 + 1.47126i 0.208983 + 1.98834i −0.671269 − 0.745519i −0.0128405 − 0.0443418i −0.913545 − 0.406737i −0.0862341 − 0.0626527i −1.32921 − 2.68946i 0.0267377
148.18 0.104977 − 0.116588i −1.52117 − 0.828269i 0.206484 + 1.96457i −1.72136 − 1.91176i −0.256254 + 0.0904023i −0.913545 − 0.406737i 0.504566 + 0.366589i 1.62794 + 2.51988i −0.403591
148.19 0.160277 − 0.178006i 1.70921 − 0.280352i 0.203060 + 1.93198i 2.11487 + 2.34880i 0.224044 − 0.349184i −0.913545 − 0.406737i 0.764019 + 0.555093i 2.84281 − 0.958361i 0.757066
148.20 0.186097 − 0.206681i 0.792795 + 1.53996i 0.200972 + 1.91212i −2.50010 − 2.77664i 0.465817 + 0.122725i −0.913545 − 0.406737i 0.882602 + 0.641248i −1.74295 + 2.44174i −1.03914
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.bz.a ✓ 288
9.c even 3 1 inner 693.2.bz.a ✓ 288
11.c even 5 1 inner 693.2.bz.a ✓ 288
99.m even 15 1 inner 693.2.bz.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.bz.a ✓ 288 1.a even 1 1 trivial
693.2.bz.a ✓ 288 9.c even 3 1 inner
693.2.bz.a ✓ 288 11.c even 5 1 inner
693.2.bz.a ✓ 288 99.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2 T_{2}^{287} - 52 T_{2}^{286} + 116 T_{2}^{285} + 1197 T_{2}^{284} - 2978 T_{2}^{283} + \cdots + 377801998336 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display