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Properties

Label 574.2.bc.a
Level $574$
Weight $2$
Character orbit 574.bc
Analytic conductor $4.583$
Analytic rank $0$
Dimension $224$
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] = [574,2,Mod(39,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([40, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.bc (of order \(60\), degree \(16\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 2 q^{3} - 28 q^{4} - 4 q^{6} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q - 2 q^{3} - 28 q^{4} - 4 q^{6} + 6 q^{7} - 2 q^{11} + 2 q^{12} + 16 q^{13} - 14 q^{14} - 12 q^{15} + 28 q^{16} - 2 q^{17} - 28 q^{18} - 2 q^{19} - 4 q^{22} - 26 q^{23} - 2 q^{24} - 36 q^{25} - 2 q^{26} - 8 q^{27} + 32 q^{29} + 6 q^{30} + 10 q^{31} - 150 q^{33} - 56 q^{34} - 20 q^{35} + 26 q^{37} - 2 q^{38} + 40 q^{39} + 40 q^{41} - 64 q^{42} + 20 q^{43} + 12 q^{44} + 16 q^{45} - 140 q^{47} + 4 q^{48} - 20 q^{49} + 86 q^{51} - 2 q^{52} - 4 q^{53} - 16 q^{54} + 12 q^{55} + 6 q^{56} - 16 q^{57} + 16 q^{58} - 8 q^{59} - 6 q^{60} + 110 q^{61} + 38 q^{63} + 56 q^{64} - 20 q^{65} + 16 q^{66} - 64 q^{67} + 2 q^{68} - 52 q^{69} - 130 q^{70} + 40 q^{71} + 28 q^{72} - 16 q^{75} - 4 q^{76} - 130 q^{77} + 60 q^{78} - 48 q^{79} + 204 q^{81} - 8 q^{82} - 24 q^{83} + 44 q^{85} + 20 q^{86} + 18 q^{88} + 32 q^{89} + 28 q^{92} - 26 q^{93} + 50 q^{94} - 24 q^{95} + 2 q^{96} + 16 q^{97} + 32 q^{98} - 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 0.743145 + 0.669131i −3.11616 + 0.834973i 0.104528 + 0.994522i 0.0358811 − 0.0805902i −2.87447 − 1.46461i 1.28729 − 2.31147i −0.587785 + 0.809017i 6.41520 − 3.70382i 0.0805902 − 0.0358811i
39.2 0.743145 + 0.669131i −2.76459 + 0.740769i 0.104528 + 0.994522i 1.07641 − 2.41766i −2.55016 − 1.29937i −1.83265 + 1.90824i −0.587785 + 0.809017i 4.49613 − 2.59584i 2.41766 − 1.07641i
39.3 0.743145 + 0.669131i −2.50751 + 0.671884i 0.104528 + 0.994522i −1.55451 + 3.49149i −2.31302 − 1.17854i 1.48536 + 2.18945i −0.587785 + 0.809017i 3.23808 − 1.86951i −3.49149 + 1.55451i
39.4 0.743145 + 0.669131i −1.66252 + 0.445470i 0.104528 + 0.994522i 0.890005 − 1.99898i −1.53357 − 0.781391i −2.33622 − 1.24181i −0.587785 + 0.809017i −0.0325627 + 0.0188001i 1.99898 − 0.890005i
39.5 0.743145 + 0.669131i −0.965969 + 0.258831i 0.104528 + 0.994522i −0.610838 + 1.37196i −0.891046 − 0.454011i 0.564477 − 2.58483i −0.587785 + 0.809017i −1.73197 + 0.999955i −1.37196 + 0.610838i
39.6 0.743145 + 0.669131i −0.553124 + 0.148209i 0.104528 + 0.994522i −0.0941954 + 0.211566i −0.510222 − 0.259971i −0.0144554 + 2.64571i −0.587785 + 0.809017i −2.31410 + 1.33604i −0.211566 + 0.0941954i
39.7 0.743145 + 0.669131i −0.227573 + 0.0609780i 0.104528 + 0.994522i 1.29602 − 2.91090i −0.209922 − 0.106961i −0.924004 − 2.47916i −0.587785 + 0.809017i −2.55001 + 1.47225i 2.91090 − 1.29602i
39.8 0.743145 + 0.669131i 0.114004 − 0.0305473i 0.104528 + 0.994522i −0.794056 + 1.78348i 0.105162 + 0.0535825i 2.51856 − 0.810460i −0.587785 + 0.809017i −2.58601 + 1.49303i −1.78348 + 0.794056i
39.9 0.743145 + 0.669131i 0.457550 − 0.122600i 0.104528 + 0.994522i 0.585610 − 1.31530i 0.422062 + 0.215051i 0.804836 + 2.52036i −0.587785 + 0.809017i −2.40375 + 1.38781i 1.31530 − 0.585610i
39.10 0.743145 + 0.669131i 0.885156 − 0.237177i 0.104528 + 0.994522i −1.08326 + 2.43305i 0.816501 + 0.416028i −2.61876 + 0.376960i −0.587785 + 0.809017i −1.87083 + 1.08012i −2.43305 + 1.08326i
39.11 0.743145 + 0.669131i 2.35303 − 0.630491i 0.104528 + 0.994522i 1.43304 − 3.21866i 2.17052 + 1.10593i 2.11454 + 1.59020i −0.587785 + 0.809017i 2.54113 − 1.46712i 3.21866 − 1.43304i
39.12 0.743145 + 0.669131i 2.43073 − 0.651311i 0.104528 + 0.994522i 0.0891568 − 0.200250i 2.24219 + 1.14245i 0.567435 − 2.58419i −0.587785 + 0.809017i 2.88614 − 1.66632i 0.200250 − 0.0891568i
39.13 0.743145 + 0.669131i 2.43986 − 0.653759i 0.104528 + 0.994522i −1.62133 + 3.64157i 2.25062 + 1.14675i 2.59601 + 0.510639i −0.587785 + 0.809017i 2.92745 − 1.69016i −3.64157 + 1.62133i
39.14 0.743145 + 0.669131i 2.81491 − 0.754252i 0.104528 + 0.994522i 0.352077 − 0.790778i 2.59657 + 1.32302i −2.52636 + 0.785818i −0.587785 + 0.809017i 4.75672 − 2.74629i 0.790778 − 0.352077i
121.1 0.207912 − 0.978148i −0.754252 + 2.81491i −0.913545 − 0.406737i −0.860872 + 0.0904813i 2.59657 + 1.32302i −2.12070 + 1.58198i −0.587785 + 0.809017i −4.75672 − 2.74629i −0.0904813 + 0.860872i
121.2 0.207912 − 0.978148i −0.653759 + 2.43986i −0.913545 − 0.406737i 3.96436 − 0.416671i 2.25062 + 1.14675i 1.11278 − 2.40036i −0.587785 + 0.809017i −2.92745 − 1.69016i 0.416671 − 3.96436i
121.3 0.207912 − 0.978148i −0.651311 + 2.43073i −0.913545 − 0.406737i −0.218000 + 0.0229127i 2.24219 + 1.14245i 2.42418 + 1.05988i −0.587785 + 0.809017i −2.88614 − 1.66632i −0.0229127 + 0.218000i
121.4 0.207912 − 0.978148i −0.630491 + 2.35303i −0.913545 − 0.406737i −3.50396 + 0.368281i 2.17052 + 1.10593i −0.0436052 − 2.64539i −0.587785 + 0.809017i −2.54113 − 1.46712i −0.368281 + 3.50396i
121.5 0.207912 − 0.978148i −0.237177 + 0.885156i −0.913545 − 0.406737i 2.64872 − 0.278391i 0.816501 + 0.416028i −1.84424 + 1.89705i −0.587785 + 0.809017i 1.87083 + 1.08012i 0.278391 − 2.64872i
121.6 0.207912 − 0.978148i −0.122600 + 0.457550i −0.913545 − 0.406737i −1.43189 + 0.150498i 0.422062 + 0.215051i −1.56595 − 2.13256i −0.587785 + 0.809017i 2.40375 + 1.38781i −0.150498 + 1.43189i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.g even 20 1 inner
287.bc even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.bc.a ✓ 224
7.c even 3 1 inner 574.2.bc.a ✓ 224
41.g even 20 1 inner 574.2.bc.a ✓ 224
287.bc even 60 1 inner 574.2.bc.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.bc.a ✓ 224 1.a even 1 1 trivial
574.2.bc.a ✓ 224 7.c even 3 1 inner
574.2.bc.a ✓ 224 41.g even 20 1 inner
574.2.bc.a ✓ 224 287.bc even 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{224} + 2 T_{3}^{223} + 2 T_{3}^{222} + 12 T_{3}^{221} - 785 T_{3}^{220} - 1646 T_{3}^{219} + \cdots + 17\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display