↑

Properties

Label 451.2.bf.a
Level $451$
Weight $2$
Character orbit 451.bf
Analytic conductor $3.601$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,2,Mod(4,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.bf (of order \(10\), degree \(4\), 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: \(3.60125313116\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{2} + 5 q^{3} + 146 q^{4} - q^{5} - 20 q^{6} - 5 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{2} + 5 q^{3} + 146 q^{4} - q^{5} - 20 q^{6} - 5 q^{7} + 6 q^{8} + 27 q^{9} - 8 q^{10} - 5 q^{11} + 10 q^{12} - 10 q^{14} + 102 q^{16} - 10 q^{17} - 18 q^{18} - 20 q^{19} - 27 q^{20} - 44 q^{21} - 15 q^{22} + q^{23} - 80 q^{24} - 27 q^{25} + 5 q^{27} - 55 q^{28} + 7 q^{31} + 34 q^{32} - 35 q^{33} + 5 q^{34} - 11 q^{36} + 42 q^{37} - 5 q^{38} - 31 q^{39} - 39 q^{40} + 10 q^{41} + 26 q^{42} - 14 q^{43} - 5 q^{44} + 13 q^{45} - 23 q^{46} - 85 q^{47} - 45 q^{48} + 43 q^{49} - 2 q^{50} - q^{51} - 25 q^{53} - 25 q^{54} - 15 q^{55} - 65 q^{56} + 41 q^{57} - 37 q^{59} + 58 q^{61} + 23 q^{62} + 65 q^{63} + 78 q^{64} + 15 q^{65} - 89 q^{66} - 10 q^{67} + 5 q^{68} + 35 q^{69} + 15 q^{71} - 35 q^{72} - 21 q^{73} + 98 q^{74} - 105 q^{75} - 75 q^{76} - 22 q^{77} + 97 q^{78} - 5 q^{79} - 109 q^{80} - 101 q^{81} - 16 q^{82} - 9 q^{83} - 50 q^{84} - 90 q^{85} - 13 q^{86} - 9 q^{87} - 115 q^{88} + 55 q^{89} + 16 q^{90} + q^{91} - 20 q^{92} + 100 q^{93} + 5 q^{94} - 40 q^{95} + 10 q^{97} - 34 q^{98} + 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.80906 −1.39468 − 0.453160i 5.89082 −0.231742 − 0.713228i 3.91775 + 1.27295i 0.602022 + 0.195609i −10.9296 −0.687262 − 0.499325i 0.650977 + 2.00350i
4.2 −2.61358 2.22696 + 0.723584i 4.83079 0.592732 + 1.82424i −5.82034 − 1.89114i 0.561935 + 0.182584i −7.39850 2.00874 + 1.45943i −1.54915 − 4.76780i
4.3 −2.36052 0.596155 + 0.193702i 3.57205 0.946052 + 2.91165i −1.40723 − 0.457238i −2.70935 − 0.880320i −3.71086 −2.10917 − 1.53240i −2.23317 − 6.87300i
4.4 −2.30966 0.370729 + 0.120457i 3.33452 −0.774825 − 2.38467i −0.856256 − 0.278214i 0.966405 + 0.314004i −3.08228 −2.30412 − 1.67404i 1.78958 + 5.50776i
4.5 −2.24893 3.11839 + 1.01323i 3.05769 −0.992198 − 3.05367i −7.01305 − 2.27868i −3.10079 − 1.00751i −2.37867 6.27069 + 4.55592i 2.23138 + 6.86750i
4.6 −2.24115 −1.57145 − 0.510595i 3.02275 0.772261 + 2.37677i 3.52185 + 1.14432i 4.12568 + 1.34052i −2.29213 −0.218301 − 0.158605i −1.73075 − 5.32670i
4.7 −2.14627 −2.74817 − 0.892934i 2.60648 −0.584635 − 1.79932i 5.89832 + 1.91648i −1.57960 − 0.513244i −1.30168 4.32804 + 3.14451i 1.25479 + 3.86183i
4.8 −1.88405 −1.71731 − 0.557987i 1.54966 0.0263237 + 0.0810159i 3.23550 + 1.05128i −2.11510 − 0.687236i 0.848468 0.210745 + 0.153115i −0.0495952 − 0.152638i
4.9 −1.84547 0.748020 + 0.243047i 1.40577 −1.04081 − 3.20330i −1.38045 − 0.448535i 3.13634 + 1.01906i 1.09664 −1.92659 − 1.39975i 1.92079 + 5.91160i
4.10 −1.66274 1.16330 + 0.377978i 0.764711 0.0860669 + 0.264887i −1.93426 − 0.628480i −3.81603 − 1.23990i 2.05397 −1.21666 − 0.883954i −0.143107 − 0.440438i
4.11 −1.60974 2.55935 + 0.831582i 0.591273 0.850629 + 2.61797i −4.11989 − 1.33863i 4.03200 + 1.31008i 2.26769 3.43168 + 2.49326i −1.36929 − 4.21425i
4.12 −1.26084 2.35010 + 0.763594i −0.410293 −0.199923 − 0.615300i −2.96309 − 0.962767i 0.907915 + 0.295000i 3.03898 2.51285 + 1.82569i 0.252070 + 0.775792i
4.13 −1.16882 −2.48219 − 0.806513i −0.633858 1.22291 + 3.76372i 2.90124 + 0.942669i −0.313287 − 0.101793i 3.07851 3.08376 + 2.24048i −1.42936 − 4.39911i
4.14 −1.13225 −0.323930 − 0.105251i −0.718014 0.638326 + 1.96457i 0.366769 + 0.119170i −1.86003 − 0.604361i 3.07747 −2.33320 − 1.69517i −0.722743 − 2.22438i
4.15 −1.02865 −1.58915 − 0.516347i −0.941883 −1.13371 − 3.48921i 1.63468 + 0.531140i −1.91178 − 0.621174i 3.02616 −0.168258 − 0.122246i 1.16619 + 3.58917i
4.16 −0.922129 −2.66699 − 0.866558i −1.14968 −0.701693 − 2.15959i 2.45931 + 0.799078i 4.87983 + 1.58555i 2.90441 3.93487 + 2.85885i 0.647051 + 1.99142i
4.17 −0.707926 1.23383 + 0.400896i −1.49884 −0.290934 − 0.895402i −0.873461 − 0.283805i 0.323825 + 0.105217i 2.47692 −1.06543 − 0.774081i 0.205960 + 0.633879i
4.18 −0.324206 2.76255 + 0.897608i −1.89489 1.27729 + 3.93109i −0.895637 − 0.291010i −4.39859 − 1.42919i 1.26275 4.39895 + 3.19603i −0.414105 − 1.27448i
4.19 −0.312249 0.129857 + 0.0421932i −1.90250 0.691780 + 2.12908i −0.0405478 − 0.0131748i 3.69829 + 1.20165i 1.21855 −2.41197 − 1.75240i −0.216008 − 0.664803i
4.20 −0.0620108 −0.798997 − 0.259610i −1.99615 0.249973 + 0.769339i 0.0495465 + 0.0160986i 0.597921 + 0.194276i 0.247805 −1.85605 − 1.34850i −0.0155011 − 0.0477074i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
451.bf even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 451.2.bf.a yes 160
11.c even 5 1 451.2.r.a ✓ 160
41.f even 10 1 451.2.r.a ✓ 160
451.bf even 10 1 inner 451.2.bf.a yes 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.r.a ✓ 160 11.c even 5 1
451.2.r.a ✓ 160 41.f even 10 1
451.2.bf.a yes 160 1.a even 1 1 trivial
451.2.bf.a yes 160 451.bf even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(451, [\chi])\).