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Properties

Label 574.2.bc.b
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
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 2 q^{3} - 28 q^{4} - 4 q^{6} - 6 q^{7} - 8 q^{10} + 6 q^{11} - 2 q^{12} - 16 q^{13} - 14 q^{14} + 36 q^{15} + 28 q^{16} - 10 q^{17} + 28 q^{18} - 2 q^{19} - 12 q^{22} + 26 q^{23} - 2 q^{24} - 20 q^{25} - 2 q^{26} + 32 q^{27} + 18 q^{30} + 6 q^{31} - 70 q^{33} + 40 q^{34} - 16 q^{35} - 14 q^{37} + 2 q^{38} - 40 q^{39} - 32 q^{40} + 24 q^{41} - 32 q^{42} - 100 q^{43} + 4 q^{44} + 24 q^{45} + 36 q^{47} - 4 q^{48} - 20 q^{49} - 2 q^{51} + 2 q^{52} + 8 q^{53} - 4 q^{54} - 4 q^{55} + 6 q^{56} + 48 q^{57} - 40 q^{59} - 42 q^{60} + 50 q^{61} - 22 q^{63} + 56 q^{64} - 4 q^{65} + 32 q^{66} + 96 q^{67} + 10 q^{68} - 92 q^{69} + 30 q^{70} + 48 q^{71} - 28 q^{72} - 8 q^{75} - 4 q^{76} - 30 q^{77} - 60 q^{78} + 12 q^{79} + 12 q^{81} + 24 q^{82} - 328 q^{83} - 92 q^{85} + 20 q^{86} + 14 q^{88} + 8 q^{89} - 28 q^{92} - 6 q^{93} - 6 q^{94} + 68 q^{95} + 2 q^{96} - 32 q^{97} - 16 q^{98} + 124 q^{99}+O(q^{100}) \) 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 −3.09828 + 0.830181i 0.104528 + 0.994522i −0.538548 + 1.20960i 2.85797 + 1.45621i −0.649838 + 2.56470i 0.587785 − 0.809017i 6.31205 − 3.64426i 1.20960 − 0.538548i
39.2 −0.743145 − 0.669131i −2.82300 + 0.756419i 0.104528 + 0.994522i 1.13937 − 2.55906i 2.60404 + 1.32682i −2.15346 − 1.53707i 0.587785 − 0.809017i 4.79906 − 2.77074i −2.55906 + 1.13937i
39.3 −0.743145 − 0.669131i −2.03564 + 0.545447i 0.104528 + 0.994522i 0.919375 − 2.06495i 1.87775 + 0.956761i 2.55350 + 0.692558i 0.587785 − 0.809017i 1.24823 − 0.720665i −2.06495 + 0.919375i
39.4 −0.743145 − 0.669131i −1.71367 + 0.459176i 0.104528 + 0.994522i −0.523054 + 1.17480i 1.58075 + 0.805434i −0.564481 − 2.58483i 0.587785 − 0.809017i 0.127745 − 0.0737535i 1.17480 − 0.523054i
39.5 −0.743145 − 0.669131i −1.28358 + 0.343934i 0.104528 + 0.994522i −1.36963 + 3.07624i 1.18402 + 0.603289i 2.54213 − 0.733198i 0.587785 − 0.809017i −1.06879 + 0.617067i 3.07624 − 1.36963i
39.6 −0.743145 − 0.669131i −0.469669 + 0.125847i 0.104528 + 0.994522i 0.650622 − 1.46132i 0.433240 + 0.220747i 0.999352 + 2.44975i 0.587785 − 0.809017i −2.39333 + 1.38179i −1.46132 + 0.650622i
39.7 −0.743145 − 0.669131i −0.409154 + 0.109632i 0.104528 + 0.994522i 1.74403 − 3.91715i 0.377419 + 0.192305i −2.41157 + 1.08827i 0.587785 − 0.809017i −2.44269 + 1.41029i −3.91715 + 1.74403i
39.8 −0.743145 − 0.669131i 0.151542 − 0.0406054i 0.104528 + 0.994522i 0.139560 − 0.313458i −0.139788 − 0.0712254i −2.64464 + 0.0768460i 0.587785 − 0.809017i −2.57676 + 1.48769i −0.313458 + 0.139560i
39.9 −0.743145 − 0.669131i 0.989356 − 0.265097i 0.104528 + 0.994522i 0.600689 − 1.34917i −0.912620 − 0.465003i 2.23677 − 1.41310i 0.587785 − 0.809017i −1.68953 + 0.975449i −1.34917 + 0.600689i
39.10 −0.743145 − 0.669131i 1.44973 − 0.388454i 0.104528 + 0.994522i −1.50741 + 3.38571i −1.33728 − 0.681380i −2.14998 − 1.54194i 0.587785 − 0.809017i −0.647260 + 0.373696i 3.38571 − 1.50741i
39.11 −0.743145 − 0.669131i 1.47653 − 0.395634i 0.104528 + 0.994522i −0.317525 + 0.713172i −1.36200 − 0.693976i −1.72554 + 2.00562i 0.587785 − 0.809017i −0.574470 + 0.331670i 0.713172 − 0.317525i
39.12 −0.743145 − 0.669131i 2.31742 − 0.620951i 0.104528 + 0.994522i −0.909575 + 2.04294i −2.13768 − 1.08920i 2.15140 − 1.53996i 0.587785 − 0.809017i 2.38678 − 1.37801i 2.04294 − 0.909575i
39.13 −0.743145 − 0.669131i 2.60158 − 0.697091i 0.104528 + 0.994522i 1.39813 − 3.14025i −2.39979 − 1.22276i −0.765795 − 2.53250i 0.587785 − 0.809017i 3.68420 − 2.12707i −3.14025 + 1.39813i
39.14 −0.743145 − 0.669131i 3.14904 − 0.843782i 0.104528 + 0.994522i 0.200922 − 0.451277i −2.90479 − 1.48007i 0.896092 + 2.48938i 0.587785 − 0.809017i 6.60639 − 3.81420i −0.451277 + 0.200922i
121.1 −0.207912 + 0.978148i −0.843782 + 3.14904i −0.913545 − 0.406737i −0.491279 + 0.0516355i −2.90479 − 1.48007i −1.48724 − 2.18818i 0.587785 − 0.809017i −6.60639 − 3.81420i 0.0516355 − 0.491279i
121.2 −0.207912 + 0.978148i −0.697091 + 2.60158i −0.913545 − 0.406737i −3.41860 + 0.359309i −2.39979 − 1.22276i 1.59871 + 2.10811i 0.587785 − 0.809017i −3.68420 − 2.12707i 0.359309 − 3.41860i
121.3 −0.207912 + 0.978148i −0.620951 + 2.31742i −0.913545 − 0.406737i 2.22403 − 0.233755i −2.13768 − 1.08920i 2.51042 − 0.835353i 0.587785 − 0.809017i −2.38678 − 1.37801i −0.233755 + 2.22403i
121.4 −0.207912 + 0.978148i −0.395634 + 1.47653i −0.913545 − 0.406737i 0.776387 − 0.0816016i −1.36200 − 0.693976i −2.63683 + 0.217115i 0.587785 − 0.809017i 0.574470 + 0.331670i −0.0816016 + 0.776387i
121.5 −0.207912 + 0.978148i −0.388454 + 1.44973i −0.913545 − 0.406737i 3.68582 − 0.387395i −1.33728 − 0.681380i −0.0162702 + 2.64570i 0.587785 − 0.809017i 0.647260 + 0.373696i −0.387395 + 3.68582i
121.6 −0.207912 + 0.978148i −0.265097 + 0.989356i −0.913545 − 0.406737i −1.46876 + 0.154373i −0.912620 − 0.465003i 2.45796 − 0.978983i 0.587785 − 0.809017i 1.68953 + 0.975449i 0.154373 − 1.46876i
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.b ✓ 224
7.c even 3 1 inner 574.2.bc.b ✓ 224
41.g even 20 1 inner 574.2.bc.b ✓ 224
287.bc even 60 1 inner 574.2.bc.b ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.bc.b ✓ 224 1.a even 1 1 trivial
574.2.bc.b ✓ 224 7.c even 3 1 inner
574.2.bc.b ✓ 224 41.g even 20 1 inner
574.2.bc.b ✓ 224 287.bc even 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{224} - 2 T_{3}^{223} + 2 T_{3}^{222} - 20 T_{3}^{221} - 721 T_{3}^{220} + 1554 T_{3}^{219} + \cdots + 34\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display