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Properties

Label 574.2.z.b
Level $574$
Weight $2$
Character orbit 574.z
Analytic conductor $4.583$
Analytic rank $0$
Dimension $112$
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(23,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.z (of order \(30\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 14 q^{2} + 14 q^{4} - 28 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 14 q^{2} + 14 q^{4} - 28 q^{8} + 54 q^{9} + 5 q^{12} + 10 q^{13} + 10 q^{15} + 14 q^{16} + 10 q^{17} - 11 q^{18} - 11 q^{21} - 20 q^{22} - 10 q^{23} - 5 q^{24} + 18 q^{25} + 10 q^{29} + 5 q^{30} - 21 q^{31} - 56 q^{32} - 15 q^{33} + 30 q^{34} - 40 q^{35} + 32 q^{36} + 21 q^{37} - 14 q^{39} - 38 q^{41} - 30 q^{42} + 46 q^{43} - 28 q^{45} + 5 q^{46} - 25 q^{47} + 28 q^{49} + 104 q^{50} - 29 q^{51} + 5 q^{53} - 10 q^{54} - 10 q^{56} + 40 q^{57} + 5 q^{58} + 6 q^{59} + 25 q^{60} - 54 q^{61} - 8 q^{62} - 5 q^{63} - 28 q^{64} - 50 q^{65} + 10 q^{66} + 45 q^{67} - 10 q^{69} + 5 q^{70} - 30 q^{71} - 16 q^{72} - 46 q^{73} + 21 q^{74} + 105 q^{75} - 15 q^{77} - 22 q^{78} - 52 q^{81} + 14 q^{82} - 140 q^{83} + q^{84} + 17 q^{86} + 44 q^{87} - 10 q^{88} + 15 q^{89} + 56 q^{90} + 24 q^{91} - 10 q^{92} - 40 q^{93} - 10 q^{94} - 65 q^{95} - 40 q^{97} - 13 q^{98} - 120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.978148 − 0.207912i −2.75515 + 1.59069i 0.913545 + 0.406737i −0.366822 − 3.49008i 3.02567 − 0.983099i −2.09238 − 1.61924i −0.809017 − 0.587785i 3.56057 − 6.16709i −0.366822 + 3.49008i
23.2 −0.978148 − 0.207912i −2.08034 + 1.20108i 0.913545 + 0.406737i −0.0339844 − 0.323340i 2.28460 − 0.742311i 2.48318 − 0.913135i −0.809017 − 0.587785i 1.38521 − 2.39925i −0.0339844 + 0.323340i
23.3 −0.978148 − 0.207912i −1.81451 + 1.04761i 0.913545 + 0.406737i −0.00668236 − 0.0635784i 1.99266 − 0.647456i −1.95655 + 1.78099i −0.809017 − 0.587785i 0.694956 − 1.20370i −0.00668236 + 0.0635784i
23.4 −0.978148 − 0.207912i −1.11789 + 0.645412i 0.913545 + 0.406737i 0.233934 + 2.22573i 1.22765 − 0.398886i −1.76827 + 1.96805i −0.809017 − 0.587785i −0.666888 + 1.15508i 0.233934 − 2.22573i
23.5 −0.978148 − 0.207912i −1.02508 + 0.591831i 0.913545 + 0.406737i −0.338316 − 3.21886i 1.12573 − 0.365772i 1.07411 + 2.41791i −0.809017 − 0.587785i −0.799473 + 1.38473i −0.338316 + 3.21886i
23.6 −0.978148 − 0.207912i −1.01227 + 0.584432i 0.913545 + 0.406737i 0.427213 + 4.06466i 1.11166 − 0.361199i 2.64489 − 0.0676476i −0.809017 − 0.587785i −0.816878 + 1.41487i 0.427213 − 4.06466i
23.7 −0.978148 − 0.207912i −0.130239 + 0.0751937i 0.913545 + 0.406737i 0.00933977 + 0.0888620i 0.143027 − 0.0464723i −2.12474 − 1.57654i −0.809017 − 0.587785i −1.48869 + 2.57849i 0.00933977 − 0.0888620i
23.8 −0.978148 − 0.207912i −0.0301469 + 0.0174053i 0.913545 + 0.406737i −0.316409 − 3.01043i 0.0331069 − 0.0107571i 2.05207 − 1.67003i −0.809017 − 0.587785i −1.49939 + 2.59703i −0.316409 + 3.01043i
23.9 −0.978148 − 0.207912i 0.343537 − 0.198341i 0.913545 + 0.406737i 0.276819 + 2.63376i −0.377267 + 0.122582i −0.564532 − 2.58482i −0.809017 − 0.587785i −1.42132 + 2.46180i 0.276819 − 2.63376i
23.10 −0.978148 − 0.207912i 1.52426 − 0.880030i 0.913545 + 0.406737i 0.0940375 + 0.894707i −1.67392 + 0.543888i 2.55357 + 0.692314i −0.809017 − 0.587785i 0.0489047 − 0.0847054i 0.0940375 − 0.894707i
23.11 −0.978148 − 0.207912i 1.70207 − 0.982689i 0.913545 + 0.406737i −0.152523 − 1.45116i −1.86919 + 0.607335i 0.773922 + 2.53003i −0.809017 − 0.587785i 0.431354 − 0.747128i −0.152523 + 1.45116i
23.12 −0.978148 − 0.207912i 2.09449 − 1.20926i 0.913545 + 0.406737i 0.400341 + 3.80899i −2.30014 + 0.747361i −1.69969 + 2.02757i −0.809017 − 0.587785i 1.42460 − 2.46748i 0.400341 − 3.80899i
23.13 −0.978148 − 0.207912i 2.52472 − 1.45765i 0.913545 + 0.406737i −0.109732 − 1.04403i −2.77261 + 0.900876i 2.15469 − 1.53535i −0.809017 − 0.587785i 2.74947 − 4.76223i −0.109732 + 1.04403i
23.14 −0.978148 − 0.207912i 2.79462 − 1.61347i 0.913545 + 0.406737i −0.117216 − 1.11524i −3.06901 + 0.997182i −2.41222 − 1.08683i −0.809017 − 0.587785i 3.70660 − 6.42001i −0.117216 + 1.11524i
25.1 −0.978148 + 0.207912i −2.75515 − 1.59069i 0.913545 − 0.406737i −0.366822 + 3.49008i 3.02567 + 0.983099i −2.09238 + 1.61924i −0.809017 + 0.587785i 3.56057 + 6.16709i −0.366822 − 3.49008i
25.2 −0.978148 + 0.207912i −2.08034 − 1.20108i 0.913545 − 0.406737i −0.0339844 + 0.323340i 2.28460 + 0.742311i 2.48318 + 0.913135i −0.809017 + 0.587785i 1.38521 + 2.39925i −0.0339844 − 0.323340i
25.3 −0.978148 + 0.207912i −1.81451 − 1.04761i 0.913545 − 0.406737i −0.00668236 + 0.0635784i 1.99266 + 0.647456i −1.95655 − 1.78099i −0.809017 + 0.587785i 0.694956 + 1.20370i −0.00668236 − 0.0635784i
25.4 −0.978148 + 0.207912i −1.11789 − 0.645412i 0.913545 − 0.406737i 0.233934 − 2.22573i 1.22765 + 0.398886i −1.76827 − 1.96805i −0.809017 + 0.587785i −0.666888 − 1.15508i 0.233934 + 2.22573i
25.5 −0.978148 + 0.207912i −1.02508 − 0.591831i 0.913545 − 0.406737i −0.338316 + 3.21886i 1.12573 + 0.365772i 1.07411 − 2.41791i −0.809017 + 0.587785i −0.799473 − 1.38473i −0.338316 − 3.21886i
25.6 −0.978148 + 0.207912i −1.01227 − 0.584432i 0.913545 − 0.406737i 0.427213 − 4.06466i 1.11166 + 0.361199i 2.64489 + 0.0676476i −0.809017 + 0.587785i −0.816878 − 1.41487i 0.427213 + 4.06466i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.f even 10 1 inner
287.z even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.z.b ✓ 112
7.c even 3 1 inner 574.2.z.b ✓ 112
41.f even 10 1 inner 574.2.z.b ✓ 112
287.z even 30 1 inner 574.2.z.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.z.b ✓ 112 1.a even 1 1 trivial
574.2.z.b ✓ 112 7.c even 3 1 inner
574.2.z.b ✓ 112 41.f even 10 1 inner
574.2.z.b ✓ 112 287.z even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} - 111 T_{3}^{110} + 6547 T_{3}^{108} - 266532 T_{3}^{106} + 8317695 T_{3}^{104} + \cdots + 81450625 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display