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Properties

Label 633.2.q.b
Level $633$
Weight $2$
Character orbit 633.q
Analytic conductor $5.055$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [633,2,Mod(19,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.q (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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + q^{2} + 18 q^{3} + 17 q^{4} - 4 q^{5} + q^{6} - q^{7} - 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + q^{2} + 18 q^{3} + 17 q^{4} - 4 q^{5} + q^{6} - q^{7} - 14 q^{8} + 18 q^{9} + 11 q^{10} - 16 q^{11} + 146 q^{12} - 6 q^{13} + 7 q^{14} - 8 q^{15} + 11 q^{16} + 32 q^{17} + 3 q^{18} - 8 q^{19} + 5 q^{20} - q^{21} + 4 q^{22} + 2 q^{24} - 46 q^{25} - 4 q^{26} - 36 q^{27} - 87 q^{28} + 10 q^{29} + 16 q^{30} - 22 q^{31} - 35 q^{32} - 12 q^{33} - 18 q^{34} - 36 q^{35} + 17 q^{36} + 85 q^{37} - 26 q^{38} + 3 q^{39} - 30 q^{40} + 6 q^{41} + 16 q^{42} - 42 q^{43} - 71 q^{44} + 2 q^{45} - 81 q^{46} - 48 q^{47} + 36 q^{48} - 51 q^{49} + 14 q^{50} - 3 q^{51} - 24 q^{52} - 23 q^{53} + q^{54} - 30 q^{55} + 39 q^{56} + 2 q^{57} + 42 q^{58} + 127 q^{59} - 30 q^{60} - 59 q^{61} - 87 q^{62} + 22 q^{63} - 108 q^{64} + 48 q^{65} - 21 q^{66} + 62 q^{67} - 25 q^{68} - 30 q^{69} + 175 q^{70} - 66 q^{71} + 2 q^{72} - 46 q^{73} - 37 q^{74} + 48 q^{75} - 7 q^{76} + 2 q^{77} - 9 q^{78} - 70 q^{79} + 22 q^{80} + 18 q^{81} + 18 q^{82} - 19 q^{83} + 36 q^{84} - 37 q^{85} - 8 q^{86} - 20 q^{87} + 212 q^{88} - 8 q^{89} - 2 q^{90} + 24 q^{91} - 115 q^{92} - 12 q^{93} - 50 q^{94} + 26 q^{95} - 40 q^{96} - 13 q^{97} - 91 q^{98} - 22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.64551 + 1.82752i −0.104528 + 0.994522i −0.423082 − 4.02536i −2.22586 + 1.61718i −1.64551 − 1.82752i 4.39332 − 0.933829i 4.07359 + 2.95963i −0.978148 − 0.207912i 0.707236 − 6.72890i
19.2 −1.63212 + 1.81265i −0.104528 + 0.994522i −0.412837 − 3.92788i −2.81464 + 2.04496i −1.63212 − 1.81265i −4.85797 + 1.03259i 3.84703 + 2.79503i −0.978148 − 0.207912i 0.887035 − 8.43958i
19.3 −1.52237 + 1.69076i −0.104528 + 0.994522i −0.332010 − 3.15887i 1.40238 − 1.01889i −1.52237 − 1.69076i −2.60948 + 0.554662i 2.16507 + 1.57302i −0.978148 − 0.207912i −0.412242 + 3.92222i
19.4 −1.52178 + 1.69010i −0.104528 + 0.994522i −0.331590 − 3.15487i 1.45639 − 1.05813i −1.52178 − 1.69010i 0.0109142 − 0.00231989i 2.15684 + 1.56704i −0.978148 − 0.207912i −0.427950 + 4.07168i
19.5 −0.955377 + 1.06105i −0.104528 + 0.994522i −0.00403263 − 0.0383679i −0.478642 + 0.347754i −0.955377 − 1.06105i 4.03721 − 0.858136i −2.26565 − 1.64609i −0.978148 − 0.207912i 0.0882981 − 0.840101i
19.6 −0.818642 + 0.909194i −0.104528 + 0.994522i 0.0525978 + 0.500435i 2.92777 − 2.12715i −0.818642 − 0.909194i 0.989286 − 0.210279i −2.47762 − 1.80010i −0.978148 − 0.207912i −0.462804 + 4.40328i
19.7 −0.748330 + 0.831105i −0.104528 + 0.994522i 0.0783197 + 0.745162i 0.584107 − 0.424379i −0.748330 − 0.831105i −3.44816 + 0.732928i −2.48746 − 1.80725i −0.978148 − 0.207912i −0.0844018 + 0.803029i
19.8 −0.703555 + 0.781377i −0.104528 + 0.994522i 0.0934964 + 0.889559i −2.43362 + 1.76813i −0.703555 − 0.781377i 0.148757 − 0.0316193i −2.46214 − 1.78885i −0.978148 − 0.207912i 0.330611 − 3.14556i
19.9 −0.0133579 + 0.0148355i −0.104528 + 0.994522i 0.209015 + 1.98865i −3.19140 + 2.31869i −0.0133579 − 0.0148355i −0.324778 + 0.0690337i −0.0645954 − 0.0469313i −0.978148 − 0.207912i 0.00823162 − 0.0783187i
19.10 0.0174612 − 0.0193926i −0.104528 + 0.994522i 0.208986 + 1.98837i 1.03331 − 0.750745i 0.0174612 + 0.0193926i −3.77723 + 0.802876i 0.0844318 + 0.0613433i −0.978148 − 0.207912i 0.00348394 − 0.0331474i
19.11 0.315183 − 0.350046i −0.104528 + 0.994522i 0.185865 + 1.76839i 1.24859 − 0.907154i 0.315183 + 0.350046i 3.72175 − 0.791083i 1.43975 + 1.04604i −0.978148 − 0.207912i 0.0759886 − 0.722984i
19.12 0.665741 − 0.739380i −0.104528 + 0.994522i 0.105585 + 1.00457i −0.796024 + 0.578345i 0.665741 + 0.739380i −1.54332 + 0.328043i 2.42289 + 1.76033i −0.978148 − 0.207912i −0.102329 + 0.973592i
19.13 1.12369 − 1.24799i −0.104528 + 0.994522i −0.0857304 − 0.815670i −0.719643 + 0.522851i 1.12369 + 1.24799i 1.50465 − 0.319824i 1.60294 + 1.16460i −0.978148 − 0.207912i −0.156146 + 1.48563i
19.14 1.20140 − 1.33429i −0.104528 + 0.994522i −0.127908 − 1.21697i 3.34064 − 2.42712i 1.20140 + 1.33429i 2.80889 − 0.597048i 1.12766 + 0.819294i −0.978148 − 0.207912i 0.774964 − 7.37329i
19.15 1.25407 − 1.39279i −0.104528 + 0.994522i −0.158106 − 1.50428i 1.87774 − 1.36426i 1.25407 + 1.39279i −3.61283 + 0.767932i 0.739069 + 0.536965i −0.978148 − 0.207912i 0.454699 − 4.32617i
19.16 1.56636 − 1.73962i −0.104528 + 0.994522i −0.363733 − 3.46069i −1.38955 + 1.00956i 1.56636 + 1.73962i 1.72021 − 0.365642i −2.80238 − 2.03605i −0.978148 − 0.207912i −0.420272 + 3.99862i
19.17 1.71615 − 1.90598i −0.104528 + 0.994522i −0.478526 − 4.55287i −3.24644 + 2.35868i 1.71615 + 1.90598i −2.38167 + 0.506240i −5.34905 − 3.88631i −0.978148 − 0.207912i −1.07580 + 10.2355i
19.18 1.84086 − 2.04448i −0.104528 + 0.994522i −0.582082 − 5.53814i 1.80687 − 1.31277i 1.84086 + 2.04448i 0.559664 − 0.118960i −7.94274 − 5.77074i −0.978148 − 0.207912i 0.642262 − 6.11071i
100.1 −1.64551 − 1.82752i −0.104528 − 0.994522i −0.423082 + 4.02536i −2.22586 − 1.61718i −1.64551 + 1.82752i 4.39332 + 0.933829i 4.07359 − 2.95963i −0.978148 + 0.207912i 0.707236 + 6.72890i
100.2 −1.63212 − 1.81265i −0.104528 − 0.994522i −0.412837 + 3.92788i −2.81464 − 2.04496i −1.63212 + 1.81265i −4.85797 − 1.03259i 3.84703 − 2.79503i −0.978148 + 0.207912i 0.887035 + 8.43958i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.i even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.q.b ✓ 144
211.i even 15 1 inner 633.2.q.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.q.b ✓ 144 1.a even 1 1 trivial
633.2.q.b ✓ 144 211.i even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - T_{2}^{143} - 26 T_{2}^{142} + 27 T_{2}^{141} + 303 T_{2}^{140} - 298 T_{2}^{139} + \cdots + 518400 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display