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Properties

Label 633.2.q.a
Level $633$
Weight $2$
Character orbit 633.q
Analytic conductor $5.055$
Analytic rank $0$
Dimension $136$
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: \(136\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q + 3 q^{2} - 17 q^{3} + 19 q^{4} + 4 q^{5} - 3 q^{6} - q^{7} + 6 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q + 3 q^{2} - 17 q^{3} + 19 q^{4} + 4 q^{5} - 3 q^{6} - q^{7} + 6 q^{8} + 17 q^{9} + 31 q^{10} - 8 q^{11} - 122 q^{12} + 12 q^{13} - 13 q^{14} - 8 q^{15} - 17 q^{16} - 28 q^{17} + 9 q^{18} + 16 q^{19} - 11 q^{20} - 4 q^{21} - 8 q^{22} + 16 q^{23} - 12 q^{24} - 36 q^{25} + 8 q^{26} + 34 q^{27} + 35 q^{28} + 16 q^{29} - 16 q^{30} - 21 q^{31} - 63 q^{32} - 4 q^{33} + 10 q^{34} - 20 q^{35} + 19 q^{36} - 44 q^{37} - 34 q^{38} + 6 q^{39} + 162 q^{40} - 10 q^{41} + 4 q^{42} - 2 q^{43} - 35 q^{44} - 2 q^{45} - 21 q^{46} + 2 q^{47} - 48 q^{48} + 52 q^{49} + 8 q^{50} + 3 q^{51} - 46 q^{52} + 25 q^{53} - 3 q^{54} + 46 q^{55} + 59 q^{56} - 31 q^{57} - 86 q^{58} - 49 q^{59} - 12 q^{60} + 30 q^{61} + q^{62} + 12 q^{63} - 4 q^{64} + 76 q^{65} + 13 q^{66} - 124 q^{67} - 5 q^{68} - 22 q^{69} - 77 q^{70} - 20 q^{71} + 12 q^{72} + 21 q^{73} - 75 q^{74} + 27 q^{75} - 5 q^{76} - 36 q^{77} - 13 q^{78} + 46 q^{79} + 138 q^{80} + 17 q^{81} - 22 q^{82} - q^{83} + 40 q^{84} - 29 q^{85} + 36 q^{86} + 2 q^{87} - 228 q^{88} + 44 q^{89} - 2 q^{90} + 49 q^{91} - 177 q^{92} + 11 q^{93} - 38 q^{94} + 84 q^{95} + 14 q^{96} - 47 q^{97} + 15 q^{98} - 26 q^{99}+O(q^{100}) \) 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 −1.78368 + 1.98098i 0.104528 − 0.994522i −0.533702 − 5.07784i 0.792691 − 0.575923i 1.78368 + 1.98098i −4.75022 + 1.00969i 6.69790 + 4.86631i −0.978148 − 0.207912i −0.273016 + 2.59757i
19.2 −1.58880 + 1.76454i 0.104528 − 0.994522i −0.380265 − 3.61798i 3.01851 − 2.19308i 1.58880 + 1.76454i 4.26406 − 0.906353i 3.14634 + 2.28595i −0.978148 − 0.207912i −0.926037 + 8.81066i
19.3 −1.57613 + 1.75047i 0.104528 − 0.994522i −0.370903 − 3.52891i −0.672918 + 0.488904i 1.57613 + 1.75047i 2.19216 − 0.465958i 2.95058 + 2.14372i −0.978148 − 0.207912i 0.204796 − 1.94850i
19.4 −1.20001 + 1.33275i 0.104528 − 0.994522i −0.127133 − 1.20959i −2.90689 + 2.11198i 1.20001 + 1.33275i −2.27375 + 0.483300i −1.13712 − 0.826169i −0.978148 − 0.207912i 0.673567 − 6.40856i
19.5 −1.08429 + 1.20422i 0.104528 − 0.994522i −0.0654178 − 0.622408i −0.190273 + 0.138241i 1.08429 + 1.20422i −0.0909217 + 0.0193260i −1.80148 − 1.30885i −0.978148 − 0.207912i 0.0398371 − 0.379024i
19.6 −1.04560 + 1.16125i 0.104528 − 0.994522i −0.0461788 − 0.439362i 3.16762 − 2.30141i 1.04560 + 1.16125i −2.82054 + 0.599525i −1.96988 − 1.43120i −0.978148 − 0.207912i −0.639533 + 6.08475i
19.7 −0.699403 + 0.776766i 0.104528 − 0.994522i 0.0948565 + 0.902499i −2.01643 + 1.46502i 0.699403 + 0.776766i 4.20584 − 0.893978i −2.45861 − 1.78628i −0.978148 − 0.207912i 0.272318 − 2.59093i
19.8 −0.500080 + 0.555395i 0.104528 − 0.994522i 0.150673 + 1.43356i 1.28166 − 0.931182i 0.500080 + 0.555395i −1.41272 + 0.300282i −2.08079 − 1.51178i −0.978148 − 0.207912i −0.123759 + 1.17749i
19.9 −0.0522770 + 0.0580595i 0.104528 − 0.994522i 0.208419 + 1.98297i −2.15539 + 1.56598i 0.0522770 + 0.0580595i −2.29146 + 0.487065i −0.252438 − 0.183407i −0.978148 − 0.207912i 0.0217572 − 0.207006i
19.10 −0.0339564 + 0.0377124i 0.104528 − 0.994522i 0.208788 + 1.98648i 0.913725 − 0.663860i 0.0339564 + 0.0377124i −2.52013 + 0.535671i −0.164115 − 0.119237i −0.978148 − 0.207912i −0.00599106 + 0.0570011i
19.11 0.360842 − 0.400756i 0.104528 − 0.994522i 0.178659 + 1.69982i 2.03845 − 1.48102i −0.360842 − 0.400756i 3.25761 − 0.692427i 1.61824 + 1.17572i −0.978148 − 0.207912i 0.142031 − 1.35133i
19.12 0.403578 − 0.448219i 0.104528 − 0.994522i 0.171032 + 1.62726i −2.24452 + 1.63074i −0.403578 − 0.448219i 1.99961 − 0.425031i 1.77429 + 1.28910i −0.978148 − 0.207912i −0.174911 + 1.66417i
19.13 0.885835 − 0.983820i 0.104528 − 0.994522i 0.0258597 + 0.246039i −1.07994 + 0.784619i −0.885835 − 0.983820i −4.62579 + 0.983242i 2.40702 + 1.74880i −0.978148 − 0.207912i −0.184721 + 1.75751i
19.14 0.934953 − 1.03837i 0.104528 − 0.994522i 0.00498081 + 0.0473892i 0.369980 − 0.268806i −0.934953 − 1.03837i 1.91135 − 0.406270i 2.31469 + 1.68172i −0.978148 − 0.207912i 0.0667935 − 0.635498i
19.15 1.20397 − 1.33715i 0.104528 − 0.994522i −0.129356 − 1.23074i 2.00765 − 1.45864i −1.20397 − 1.33715i −0.213460 + 0.0453722i 1.10993 + 0.806409i −0.978148 − 0.207912i 0.466735 − 4.44069i
19.16 1.71346 − 1.90299i 0.104528 − 0.994522i −0.476366 − 4.53232i 0.283733 − 0.206144i −1.71346 − 1.90299i 4.06922 − 0.864939i −5.29785 − 3.84911i −0.978148 − 0.207912i 0.0938748 − 0.893159i
19.17 1.75636 − 1.95063i 0.104528 − 0.994522i −0.511117 − 4.86296i −0.989626 + 0.719006i −1.75636 − 1.95063i −3.46168 + 0.735803i −6.13647 − 4.45840i −0.978148 − 0.207912i −0.335621 + 3.19322i
100.1 −1.78368 − 1.98098i 0.104528 + 0.994522i −0.533702 + 5.07784i 0.792691 + 0.575923i 1.78368 − 1.98098i −4.75022 − 1.00969i 6.69790 − 4.86631i −0.978148 + 0.207912i −0.273016 − 2.59757i
100.2 −1.58880 − 1.76454i 0.104528 + 0.994522i −0.380265 + 3.61798i 3.01851 + 2.19308i 1.58880 − 1.76454i 4.26406 + 0.906353i 3.14634 − 2.28595i −0.978148 + 0.207912i −0.926037 − 8.81066i
100.3 −1.57613 − 1.75047i 0.104528 + 0.994522i −0.370903 + 3.52891i −0.672918 − 0.488904i 1.57613 − 1.75047i 2.19216 + 0.465958i 2.95058 − 2.14372i −0.978148 + 0.207912i 0.204796 + 1.94850i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.17
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.a ✓ 136
211.i even 15 1 inner 633.2.q.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.q.a ✓ 136 1.a even 1 1 trivial
633.2.q.a ✓ 136 211.i even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{136} - 3 T_{2}^{135} - 22 T_{2}^{134} + 61 T_{2}^{133} + 219 T_{2}^{132} - 424 T_{2}^{131} + \cdots + 448634761 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display