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Properties

Label 561.2.bi.a
Level $561$
Weight $2$
Character orbit 561.bi
Analytic conductor $4.480$
Analytic rank $0$
Dimension $576$
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] = [561,2,Mod(25,561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(561, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([0, 32, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("561.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 561 = 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 561.bi (of order \(40\), 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: \(4.47960755339\)
Analytic rank: \(0\)
Dimension: \(576\)
Relative dimension: \(36\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q + 16 q^{5} - 8 q^{11} + 144 q^{16} - 48 q^{17} - 112 q^{22} - 16 q^{23} - 16 q^{25} - 32 q^{26} + 16 q^{28} - 56 q^{31} - 80 q^{32} + 8 q^{33} + 32 q^{34} + 16 q^{37} - 32 q^{39} - 168 q^{40} - 32 q^{42} - 16 q^{43} + 32 q^{44} + 32 q^{46} - 24 q^{49} + 192 q^{52} + 16 q^{53} - 48 q^{57} - 192 q^{58} - 64 q^{59} - 136 q^{61} + 80 q^{62} + 64 q^{65} - 64 q^{66} - 176 q^{68} + 24 q^{69} - 176 q^{70} + 64 q^{71} - 16 q^{73} - 32 q^{75} - 256 q^{76} + 240 q^{77} + 48 q^{78} - 32 q^{79} - 72 q^{80} + 96 q^{82} - 16 q^{83} - 56 q^{85} + 8 q^{88} - 32 q^{91} - 80 q^{92} - 144 q^{93} - 24 q^{94} + 40 q^{95} + 96 q^{96} - 32 q^{97} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.436256 + 2.75441i −0.996917 − 0.0784591i −5.49436 − 1.78523i 0.00935302 + 0.0152627i 0.651020 − 2.71169i 0.0577586 + 0.733893i 4.78208 − 9.38535i 0.987688 + 0.156434i −0.0461202 + 0.0191036i
25.2 −0.420520 + 2.65506i 0.996917 + 0.0784591i −4.97040 − 1.61498i 0.155728 + 0.254125i −0.627538 + 2.61388i 0.365723 + 4.64696i 3.93724 − 7.72726i 0.987688 + 0.156434i −0.740205 + 0.306603i
25.3 −0.385382 + 2.43321i 0.996917 + 0.0784591i −3.86985 − 1.25739i −0.410816 − 0.670391i −0.575101 + 2.39547i −0.324691 − 4.12559i 2.31402 − 4.54152i 0.987688 + 0.156434i 1.78952 − 0.741244i
25.4 −0.365083 + 2.30504i −0.996917 − 0.0784591i −3.27783 − 1.06503i −2.11720 − 3.45495i 0.544809 − 2.26929i 0.0266061 + 0.338063i 1.53260 − 3.00790i 0.987688 + 0.156434i 8.73676 − 3.61889i
25.5 −0.361608 + 2.28310i −0.996917 − 0.0784591i −3.17969 − 1.03314i 1.79243 + 2.92497i 0.539624 − 2.24769i 0.0159738 + 0.202966i 1.40972 − 2.76674i 0.987688 + 0.156434i −7.32618 + 3.03460i
25.6 −0.352412 + 2.22504i 0.996917 + 0.0784591i −2.92449 − 0.950225i −1.84460 − 3.01011i −0.525900 + 2.19053i 0.0817059 + 1.03817i 1.09943 − 2.15776i 0.987688 + 0.156434i 7.34766 − 3.04350i
25.7 −0.310636 + 1.96128i −0.996917 − 0.0784591i −1.84801 − 0.600455i 1.04567 + 1.70638i 0.463559 − 1.93086i −0.344440 − 4.37652i −0.0512829 + 0.100648i 0.987688 + 0.156434i −3.67152 + 1.52079i
25.8 −0.310006 + 1.95730i 0.996917 + 0.0784591i −1.83280 − 0.595513i 0.756151 + 1.23393i −0.462618 + 1.92694i 0.0890607 + 1.13162i −0.0655665 + 0.128681i 0.987688 + 0.156434i −2.64957 + 1.09749i
25.9 −0.241215 + 1.52297i 0.996917 + 0.0784591i −0.359144 − 0.116693i 1.77313 + 2.89349i −0.359962 + 1.49935i −0.216390 − 2.74950i −1.13572 + 2.22897i 0.987688 + 0.156434i −4.83441 + 2.00248i
25.10 −0.205237 + 1.29582i 0.996917 + 0.0784591i 0.265099 + 0.0861357i 0.987682 + 1.61175i −0.306273 + 1.27572i 0.249350 + 3.16830i −1.35727 + 2.66378i 0.987688 + 0.156434i −2.29124 + 0.949062i
25.11 −0.181957 + 1.14883i −0.996917 − 0.0784591i 0.615413 + 0.199960i −0.613879 − 1.00176i 0.271532 − 1.13101i −0.228946 − 2.90903i −1.39782 + 2.74337i 0.987688 + 0.156434i 1.26255 − 0.522966i
25.12 −0.172926 + 1.09181i −0.996917 − 0.0784591i 0.739965 + 0.240429i 0.134982 + 0.220271i 0.258055 − 1.07488i 0.0991192 + 1.25943i −1.39416 + 2.73620i 0.987688 + 0.156434i −0.263837 + 0.109285i
25.13 −0.158519 + 1.00085i −0.996917 − 0.0784591i 0.925542 + 0.300727i 1.80646 + 2.94788i 0.236556 − 0.985327i 0.278895 + 3.54369i −1.36778 + 2.68442i 0.987688 + 0.156434i −3.23674 + 1.34070i
25.14 −0.110680 + 0.698809i 0.996917 + 0.0784591i 1.42603 + 0.463345i −1.53910 − 2.51159i −0.165167 + 0.687971i 0.289740 + 3.68149i −1.12404 + 2.20605i 0.987688 + 0.156434i 1.92547 − 0.797555i
25.15 −0.0949296 + 0.599362i 0.996917 + 0.0784591i 1.55189 + 0.504240i −0.972871 − 1.58758i −0.141662 + 0.590066i −0.380047 − 4.82896i −1.00054 + 1.96366i 0.987688 + 0.156434i 1.04389 − 0.432394i
25.16 −0.0936415 + 0.591229i −0.996917 − 0.0784591i 1.56133 + 0.507307i −2.26059 − 3.68895i 0.139740 − 0.582060i 0.0912925 + 1.15998i −0.989657 + 1.94231i 0.987688 + 0.156434i 2.39270 − 0.991090i
25.17 −0.0190103 + 0.120026i −0.996917 − 0.0784591i 1.88807 + 0.613471i 0.732895 + 1.19598i 0.0283689 − 0.118165i 0.215002 + 2.73186i −0.219865 + 0.431510i 0.987688 + 0.156434i −0.157481 + 0.0652308i
25.18 −0.0124077 + 0.0783390i 0.996917 + 0.0784591i 1.89613 + 0.616090i −0.600934 − 0.980635i −0.0185158 + 0.0771240i −0.0482768 − 0.613415i −0.143807 + 0.282238i 0.987688 + 0.156434i 0.0842782 − 0.0349092i
25.19 0.000399085 − 0.00251972i 0.996917 + 0.0784591i 1.90211 + 0.618032i 0.259100 + 0.422813i 0.000595550 − 0.00248065i 0.167171 + 2.12411i 0.00463275 − 0.00909229i 0.987688 + 0.156434i 0.00116878 0.000484123i
25.20 0.0185647 − 0.117213i −0.996917 − 0.0784591i 1.88872 + 0.613682i 1.82196 + 2.97318i −0.0277039 + 0.115395i −0.324838 − 4.12746i 0.214749 − 0.421468i 0.987688 + 0.156434i 0.382319 − 0.158362i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
17.d even 8 1 inner
187.r even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 561.2.bi.a ✓ 576
11.c even 5 1 inner 561.2.bi.a ✓ 576
17.d even 8 1 inner 561.2.bi.a ✓ 576
187.r even 40 1 inner 561.2.bi.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
561.2.bi.a ✓ 576 1.a even 1 1 trivial
561.2.bi.a ✓ 576 11.c even 5 1 inner
561.2.bi.a ✓ 576 17.d even 8 1 inner
561.2.bi.a ✓ 576 187.r even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(561, [\chi])\).