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Properties

Label 574.2.bf.b
Level $574$
Weight $2$
Character orbit 574.bf
Analytic conductor $4.583$
Analytic rank $0$
Dimension $448$
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(17,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(120))
 
chi = DirichletCharacter(H, H._module([20, 99]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.bf (of order \(120\), degree \(32\), 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: \(448\)
Relative dimension: \(14\) over \(\Q(\zeta_{120})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{120}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 448 q + 8 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q + 8 q^{7} + 4 q^{9} - 8 q^{11} + 12 q^{12} - 4 q^{14} + 24 q^{15} - 56 q^{16} + 60 q^{17} - 56 q^{18} + 12 q^{19} - 32 q^{21} - 16 q^{22} + 60 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{28} - 24 q^{30} - 132 q^{33} - 48 q^{35} - 8 q^{36} - 84 q^{37} + 40 q^{39} - 48 q^{42} - 120 q^{43} - 16 q^{44} + 12 q^{46} + 264 q^{47} - 52 q^{49} - 16 q^{50} - 12 q^{51} - 36 q^{53} - 36 q^{54} - 4 q^{56} - 16 q^{57} + 24 q^{59} + 36 q^{60} - 36 q^{61} + 20 q^{65} - 88 q^{67} + 140 q^{70} + 80 q^{71} - 36 q^{73} - 144 q^{75} - 72 q^{77} + 120 q^{78} - 36 q^{79} - 48 q^{82} + 16 q^{84} + 16 q^{85} + 12 q^{87} + 4 q^{88} - 168 q^{89} + 112 q^{91} + 32 q^{92} - 56 q^{93} + 88 q^{95} - 48 q^{98} - 240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0.777146 − 0.629320i −0.361624 + 2.74681i 0.207912 − 0.978148i −1.02987 + 0.0539730i 1.44759 + 2.36225i −2.53994 + 0.740746i −0.453990 − 0.891007i −4.51640 − 1.21017i −0.766390 + 0.690061i
17.2 0.777146 − 0.629320i −0.343623 + 2.61007i 0.207912 − 0.978148i 0.600037 − 0.0314466i 1.37553 + 2.24466i 1.02381 − 2.43963i −0.453990 − 0.891007i −3.79663 − 1.01730i 0.446527 − 0.402054i
17.3 0.777146 − 0.629320i −0.342686 + 2.60296i 0.207912 − 0.978148i 2.86687 − 0.150246i 1.37178 + 2.23854i 0.736006 + 2.54132i −0.453990 − 0.891007i −3.76018 − 1.00754i 2.13342 − 1.92094i
17.4 0.777146 − 0.629320i −0.220533 + 1.67511i 0.207912 − 0.978148i −2.80685 + 0.147101i 0.882796 + 1.44059i −0.458946 − 2.60564i −0.453990 − 0.891007i 0.140415 + 0.0376240i −2.08876 + 1.88073i
17.5 0.777146 − 0.629320i −0.143408 + 1.08930i 0.207912 − 0.978148i −2.44836 + 0.128313i 0.574066 + 0.936791i 2.40552 + 1.10157i −0.453990 − 0.891007i 1.73178 + 0.464029i −1.82199 + 1.64052i
17.6 0.777146 − 0.629320i −0.113033 + 0.858572i 0.207912 − 0.978148i 3.91207 − 0.205023i 0.452473 + 0.738370i 1.31443 − 2.29614i −0.453990 − 0.891007i 2.17341 + 0.582363i 2.91122 − 2.62128i
17.7 0.777146 − 0.629320i −0.0362140 + 0.275073i 0.207912 − 0.978148i −3.20087 + 0.167751i 0.144965 + 0.236562i −1.79613 + 1.94266i −0.453990 − 0.891007i 2.82342 + 0.756534i −2.38198 + 2.14474i
17.8 0.777146 − 0.629320i −0.0249993 + 0.189889i 0.207912 − 0.978148i 1.80274 − 0.0944776i 0.100073 + 0.163304i −2.23945 − 1.40886i −0.453990 − 0.891007i 2.86234 + 0.766963i 1.34153 − 1.20792i
17.9 0.777146 − 0.629320i 0.0897830 − 0.681970i 0.207912 − 0.978148i 2.51507 − 0.131809i −0.359403 − 0.586492i 0.385996 + 2.61744i −0.453990 − 0.891007i 2.44076 + 0.653998i 1.87162 − 1.68522i
17.10 0.777146 − 0.629320i 0.155838 − 1.18371i 0.207912 − 0.978148i −0.448561 + 0.0235081i −0.623824 − 1.01799i 2.64566 + 0.0222632i −0.453990 − 0.891007i 1.52089 + 0.407522i −0.333803 + 0.300558i
17.11 0.777146 − 0.629320i 0.255491 − 1.94065i 0.207912 − 0.978148i −2.65185 + 0.138977i −1.02273 − 1.66895i −0.226376 + 2.63605i −0.453990 − 0.891007i −0.803057 − 0.215178i −1.97341 + 1.77687i
17.12 0.777146 − 0.629320i 0.303857 − 2.30802i 0.207912 − 0.978148i 0.125656 − 0.00658536i −1.21635 − 1.98490i 0.973165 − 2.46027i −0.453990 − 0.891007i −2.33687 − 0.626163i 0.0935088 − 0.0841957i
17.13 0.777146 − 0.629320i 0.339224 − 2.57666i 0.207912 − 0.978148i 3.45011 − 0.180813i −1.35792 − 2.21592i −2.60764 + 0.447461i −0.453990 − 0.891007i −3.62634 − 0.971676i 2.56745 − 2.31174i
17.14 0.777146 − 0.629320i 0.406451 − 3.08730i 0.207912 − 0.978148i −2.68619 + 0.140777i −1.62703 − 2.65507i −2.53666 − 0.751903i −0.453990 − 0.891007i −6.46843 − 1.73321i −1.99897 + 1.79988i
19.1 0.0523360 − 0.998630i −2.26311 − 1.73655i −0.994522 − 0.104528i 2.51470 + 0.965301i −1.85261 + 2.16913i 1.60855 − 2.10061i −0.156434 + 0.987688i 1.32963 + 4.96224i 1.09559 − 2.46073i
19.2 0.0523360 − 0.998630i −1.89881 − 1.45701i −0.994522 − 0.104528i −3.60809 − 1.38502i −1.55439 + 1.81995i −2.34017 + 1.23434i −0.156434 + 0.987688i 0.706150 + 2.63539i −1.57195 + 3.53066i
19.3 0.0523360 − 0.998630i −1.84742 − 1.41758i −0.994522 − 0.104528i 0.301417 + 0.115703i −1.51232 + 1.77070i 0.745388 + 2.53858i −0.156434 + 0.987688i 0.626991 + 2.33996i 0.131320 − 0.294949i
19.4 0.0523360 − 0.998630i −1.51204 − 1.16023i −0.994522 − 0.104528i −1.06250 − 0.407856i −1.23777 + 1.44924i −1.03331 − 2.43563i −0.156434 + 0.987688i 0.163674 + 0.610839i −0.462905 + 1.03970i
19.5 0.0523360 − 0.998630i −1.22682 − 0.941371i −0.994522 − 0.104528i 2.49915 + 0.959332i −1.00429 + 1.17587i −2.45009 − 0.998519i −0.156434 + 0.987688i −0.157552 − 0.587993i 1.08881 − 2.44551i
19.6 0.0523360 − 0.998630i 0.00698683 + 0.00536118i −0.994522 − 0.104528i −0.851120 − 0.326714i 0.00571950 − 0.00669667i 2.31950 + 1.27275i −0.156434 + 0.987688i −0.776437 − 2.89770i −0.370811 + 0.832854i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
41.h odd 40 1 inner
287.be even 120 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.bf.b ✓ 448
7.d odd 6 1 inner 574.2.bf.b ✓ 448
41.h odd 40 1 inner 574.2.bf.b ✓ 448
287.be even 120 1 inner 574.2.bf.b ✓ 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.bf.b ✓ 448 1.a even 1 1 trivial
574.2.bf.b ✓ 448 7.d odd 6 1 inner
574.2.bf.b ✓ 448 41.h odd 40 1 inner
574.2.bf.b ✓ 448 287.be even 120 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{448} - 2 T_{3}^{446} + 2 T_{3}^{444} + 264 T_{3}^{443} + 428 T_{3}^{442} + 1368 T_{3}^{441} + \cdots + 73\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display