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Properties

Label 483.2.bb.a
Level $483$
Weight $2$
Character orbit 483.bb
Analytic conductor $3.857$
Analytic rank $0$
Dimension $1200$
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] = [483,2,Mod(26,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 55, 48]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.bb (of order \(66\), degree \(20\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(1200\)
Relative dimension: \(60\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1200 q - 27 q^{3} - 74 q^{4} - 36 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1200 q - 27 q^{3} - 74 q^{4} - 36 q^{7} - 13 q^{9} - 54 q^{10} - 15 q^{12} - 64 q^{15} + 38 q^{16} + 43 q^{18} - 54 q^{19} - 70 q^{21} - 160 q^{22} - 66 q^{24} + 34 q^{25} - 76 q^{28} - 23 q^{30} - 54 q^{31} - 3 q^{33} - 28 q^{36} + 26 q^{37} + 5 q^{39} - 30 q^{40} - 68 q^{42} - 176 q^{43} - 96 q^{45} - 22 q^{46} - 128 q^{49} - 23 q^{51} - 102 q^{52} + 21 q^{54} - 36 q^{57} + 58 q^{58} - 43 q^{60} - 126 q^{61} + 41 q^{63} + 40 q^{64} - 132 q^{66} - 10 q^{67} + 20 q^{70} + 59 q^{72} - 30 q^{73} - 81 q^{75} + 164 q^{78} - 14 q^{79} - 97 q^{81} - 66 q^{82} + 24 q^{84} - 56 q^{85} - 399 q^{87} - 74 q^{88} - 156 q^{91} + 8 q^{93} - 126 q^{94} - 117 q^{96} + 196 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.01420 + 2.53335i −1.73203 − 0.00746686i −3.94179 − 3.75849i 0.322502 + 0.931809i 1.77555 − 4.38028i 2.53241 − 0.766108i 8.55491 − 3.90689i 2.99989 + 0.0258657i −2.68768 − 0.128030i
26.2 −0.981829 + 2.45249i 0.192945 + 1.72127i −3.60325 − 3.43569i 0.892911 + 2.57990i −4.41084 − 1.21680i −1.97669 − 1.75861i 7.15778 − 3.26885i −2.92554 + 0.664220i −7.20385 − 0.343162i
26.3 −0.980902 + 2.45017i 1.36067 − 1.07171i −3.59372 − 3.42660i 1.30920 + 3.78269i 1.29119 + 4.38513i 1.14218 + 2.38651i 7.11941 − 3.25132i 0.702868 − 2.91650i −10.5524 − 0.502675i
26.4 −0.971305 + 2.42620i 1.35467 + 1.07929i −3.49556 − 3.33301i −0.822695 − 2.37702i −3.93437 + 2.23840i 2.00942 − 1.72112i 6.72732 − 3.07226i 0.670288 + 2.92416i 6.56622 + 0.312788i
26.5 −0.954248 + 2.38360i 0.773902 − 1.54954i −3.32348 − 3.16893i −0.808015 − 2.33461i 2.95498 + 3.32332i −2.64462 − 0.0772305i 6.05389 − 2.76472i −1.80215 − 2.39839i 6.33581 + 0.301812i
26.6 −0.918036 + 2.29314i −0.970596 + 1.43455i −2.96825 − 2.83022i −0.975897 − 2.81967i −2.39859 − 3.54269i −0.366602 + 2.62023i 4.72132 − 2.15615i −1.11589 − 2.78474i 7.36181 + 0.350686i
26.7 −0.901659 + 2.25223i −1.16696 − 1.27992i −2.81210 − 2.68133i 0.232309 + 0.671212i 3.93489 − 1.47421i −2.47888 + 0.924746i 4.16099 − 1.90026i −0.276410 + 2.98724i −1.72119 − 0.0819904i
26.8 −0.821614 + 2.05229i 1.54581 − 0.781324i −2.08939 − 1.99223i −0.851870 − 2.46132i 0.333444 + 3.81440i 2.30607 + 1.29693i 1.78355 − 0.814520i 1.77907 − 2.41556i 5.75125 + 0.273966i
26.9 −0.813318 + 2.03157i −0.0960674 − 1.72938i −2.01832 − 1.92447i 0.284667 + 0.822493i 3.59150 + 1.21137i 1.39865 − 2.24583i 1.57009 − 0.717034i −2.98154 + 0.332275i −1.90248 − 0.0906261i
26.10 −0.725531 + 1.81229i −1.09457 + 1.34236i −1.31052 − 1.24958i −0.346765 − 1.00191i −1.63860 − 2.95759i 0.180768 − 2.63957i −0.335997 + 0.153445i −0.603850 − 2.93860i 2.06734 + 0.0984795i
26.11 −0.703607 + 1.75752i 1.73195 − 0.0186537i −1.14636 − 1.09305i 0.971199 + 2.80609i −1.18583 + 3.05707i 0.190502 − 2.63888i −0.716451 + 0.327192i 2.99930 − 0.0646145i −5.61512 − 0.267481i
26.12 −0.703178 + 1.75645i −1.58538 + 0.697538i −1.14320 − 1.09004i 1.14174 + 3.29884i −0.110386 − 3.27514i −2.47316 + 0.939942i −0.723536 + 0.330428i 2.02688 − 2.21173i −6.59711 − 0.314259i
26.13 −0.666844 + 1.66570i −1.67749 − 0.431318i −0.882393 − 0.841360i −0.572245 − 1.65339i 1.83707 − 2.50656i 0.792612 + 2.52424i −1.27429 + 0.581949i 2.62793 + 1.44706i 3.13565 + 0.149369i
26.14 −0.660745 + 1.65046i 0.966959 + 1.43701i −0.839971 − 0.800910i 0.513695 + 1.48422i −3.01064 + 0.646433i 2.47761 + 0.928150i −1.35743 + 0.619916i −1.12998 + 2.77905i −2.78908 − 0.132860i
26.15 −0.633650 + 1.58278i 1.63108 + 0.582744i −0.656216 − 0.625700i 0.289228 + 0.835670i −1.95589 + 2.21238i −2.05798 + 1.66274i −1.69552 + 0.774317i 2.32082 + 1.90100i −1.50595 − 0.0717373i
26.16 −0.620544 + 1.55004i 0.712920 + 1.57853i −0.570095 − 0.543585i −1.09580 − 3.16611i −2.88918 + 0.125512i −2.61610 − 0.394990i −1.84117 + 0.840836i −1.98349 + 2.25072i 5.58761 + 0.266171i
26.17 −0.556499 + 1.39007i −0.00816795 − 1.73203i −0.175125 − 0.166982i −0.460513 − 1.33056i 2.41218 + 0.952519i 1.82653 + 1.91411i −2.39445 + 1.09351i −2.99987 + 0.0282943i 2.10585 + 0.100314i
26.18 −0.540355 + 1.34974i 1.46964 − 0.916597i −0.0823512 − 0.0785217i −0.663517 − 1.91711i 0.443041 + 2.47893i −2.13757 − 1.55910i −2.49452 + 1.13921i 1.31970 − 2.69414i 2.94614 + 0.140342i
26.19 −0.397865 + 0.993819i −1.24011 − 1.20919i 0.618088 + 0.589346i 1.43145 + 4.13589i 1.69511 − 0.751347i 2.25080 + 1.39065i −2.77914 + 1.26919i 0.0757244 + 2.99904i −4.67985 − 0.222929i
26.20 −0.375451 + 0.937831i −1.72861 + 0.109178i 0.708904 + 0.675938i 0.119266 + 0.344595i 0.546616 − 1.66213i −1.73289 − 1.99928i −2.73788 + 1.25035i 2.97616 − 0.377452i −0.367951 − 0.0175277i
See next 80 embeddings (of 1200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
161.n odd 66 1 inner
483.bb even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.bb.a ✓ 1200
3.b odd 2 1 inner 483.2.bb.a ✓ 1200
7.d odd 6 1 inner 483.2.bb.a ✓ 1200
21.g even 6 1 inner 483.2.bb.a ✓ 1200
23.c even 11 1 inner 483.2.bb.a ✓ 1200
69.h odd 22 1 inner 483.2.bb.a ✓ 1200
161.n odd 66 1 inner 483.2.bb.a ✓ 1200
483.bb even 66 1 inner 483.2.bb.a ✓ 1200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.bb.a ✓ 1200 1.a even 1 1 trivial
483.2.bb.a ✓ 1200 3.b odd 2 1 inner
483.2.bb.a ✓ 1200 7.d odd 6 1 inner
483.2.bb.a ✓ 1200 21.g even 6 1 inner
483.2.bb.a ✓ 1200 23.c even 11 1 inner
483.2.bb.a ✓ 1200 69.h odd 22 1 inner
483.2.bb.a ✓ 1200 161.n odd 66 1 inner
483.2.bb.a ✓ 1200 483.bb even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(483, [\chi])\).