↑

Properties

Label 451.2.r.a
Level $451$
Weight $2$
Character orbit 451.r
Analytic conductor $3.601$
Analytic rank $0$
Dimension $160$
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(168,451)]
 
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([8, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.r (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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 3 q^{2} - 5 q^{3} - 39 q^{4} - q^{5} - 5 q^{7} + q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 3 q^{2} - 5 q^{3} - 39 q^{4} - q^{5} - 5 q^{7} + q^{8} + 27 q^{9} - 8 q^{10} - 5 q^{11} + 10 q^{12} + 10 q^{14} - 10 q^{15} - 23 q^{16} - 5 q^{17} + 12 q^{18} - 20 q^{19} + 18 q^{20} - 44 q^{21} - 40 q^{22} + q^{23} + 15 q^{24} - 47 q^{25} - 5 q^{26} - 5 q^{27} + 10 q^{29} + 40 q^{30} - 8 q^{31} + 34 q^{32} - 10 q^{33} + 5 q^{34} - 20 q^{35} - 146 q^{36} - 8 q^{37} + 5 q^{38} + 64 q^{39} - 39 q^{40} - 10 q^{41} + 26 q^{42} - 14 q^{43} + 5 q^{44} + 13 q^{45} - 13 q^{46} - 20 q^{47} + 65 q^{48} + 43 q^{49} - 2 q^{50} + 4 q^{51} - 65 q^{52} - 5 q^{53} - 25 q^{54} + 15 q^{55} - 65 q^{56} - 84 q^{57} - 25 q^{58} + 53 q^{59} + 40 q^{60} - 17 q^{61} - 27 q^{62} + 85 q^{63} + 43 q^{64} + 15 q^{65} + 121 q^{66} - 10 q^{67} - 5 q^{68} + 25 q^{69} - 35 q^{70} - 45 q^{71} + 45 q^{72} - 21 q^{73} - 82 q^{74} - 75 q^{76} + 33 q^{77} + 97 q^{78} + 5 q^{79} + 36 q^{80} - 101 q^{81} + 29 q^{82} - 9 q^{83} - 80 q^{84} + 90 q^{85} - 8 q^{86} - 9 q^{87} + 40 q^{88} + 55 q^{89} + 36 q^{90} + q^{91} - 30 q^{92} + 70 q^{93} - 35 q^{94} - 50 q^{95} - 70 q^{97} - 34 q^{98} - 25 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1 −0.868048 + 2.67158i 1.39468 − 0.453160i −4.76578 − 3.46254i 0.606708 − 0.440799i 4.11937i −0.372070 − 0.512111i 8.84220 − 6.42424i −0.687262 + 0.499325i 0.650977 + 2.00350i
168.2 −0.807640 + 2.48566i −2.22696 + 0.723584i −3.90819 − 2.83947i −1.55179 + 1.12744i − 6.11987i −0.347295 − 0.478011i 5.98551 − 4.34873i 2.00874 − 1.45943i −1.54915 − 4.76780i
168.3 −0.729441 + 2.24499i −0.596155 + 0.193702i −2.88985 − 2.09960i −2.47680 + 1.79950i − 1.47965i 1.67447 + 2.30471i 3.00215 − 2.18119i −2.10917 + 1.53240i −2.23317 − 6.87300i
168.4 −0.713724 + 2.19662i −0.370729 + 0.120457i −2.69768 − 1.95998i 2.02852 − 1.47380i − 0.900321i −0.597271 − 0.822073i 2.49362 − 1.81172i −2.30412 + 1.67404i 1.78958 + 5.50776i
168.5 −0.694958 + 2.13886i −3.11839 + 1.01323i −2.47372 − 1.79726i 2.59761 − 1.88727i − 7.37395i 1.91639 + 2.63769i 1.92438 − 1.39815i 6.27069 − 4.55592i 2.23138 + 6.86750i
168.6 −0.692553 + 2.13146i 1.57145 − 0.510595i −2.44545 − 1.77673i −2.02181 + 1.46893i 3.70310i −2.54981 − 3.50952i 1.85437 − 1.34728i −0.218301 + 0.158605i −1.73075 − 5.32670i
168.7 −0.663234 + 2.04123i 2.74817 − 0.892934i −2.10869 − 1.53205i 1.53059 − 1.11204i 6.20186i 0.976248 + 1.34369i 1.05308 − 0.765106i 4.32804 − 3.14451i 1.25479 + 3.86183i
168.8 −0.582205 + 1.79184i 1.71731 − 0.557987i −1.25370 − 0.910866i −0.0689163 + 0.0500706i 3.40201i 1.30720 + 1.79921i −0.686425 + 0.498717i 0.210745 − 0.153115i −0.0495952 − 0.152638i
168.9 −0.570282 + 1.75515i −0.748020 + 0.243047i −1.13729 − 0.826289i 2.72489 − 1.97975i − 1.45149i −1.93837 − 2.66793i −0.887202 + 0.644590i −1.92659 + 1.39975i 1.92079 + 5.91160i
168.10 −0.513816 + 1.58136i −1.16330 + 0.377978i −0.618664 − 0.449486i −0.225326 + 0.163709i − 2.03380i 2.35843 + 3.24611i −1.66169 + 1.20729i −1.21666 + 0.883954i −0.143107 − 0.440438i
168.11 −0.497438 + 1.53096i −2.55935 + 0.831582i −0.478350 − 0.347541i −2.22698 + 1.61799i − 4.33191i −2.49191 − 3.42982i −1.83460 + 1.33291i 3.43168 − 2.49326i −1.36929 − 4.21425i
168.12 −0.389620 + 1.19913i −2.35010 + 0.763594i 0.331934 + 0.241164i 0.523406 − 0.380276i − 3.11558i −0.561123 − 0.772319i −2.45859 + 1.78627i 2.51285 − 1.82569i 0.252070 + 0.775792i
168.13 −0.361185 + 1.11161i 2.48219 − 0.806513i 0.512802 + 0.372572i −3.20161 + 2.32611i 3.05054i 0.193622 + 0.266498i −2.49057 + 1.80950i 3.08376 − 2.24048i −1.42936 − 4.39911i
168.14 −0.349884 + 1.07683i 0.323930 − 0.105251i 0.580885 + 0.422038i −1.67116 + 1.21417i 0.385643i 1.14956 + 1.58224i −2.48972 + 1.80889i −2.33320 + 1.69517i −0.722743 − 2.22438i
168.15 −0.317870 + 0.978302i 1.58915 − 0.516347i 0.761999 + 0.553625i 2.96810 − 2.15645i 1.71880i 1.18154 + 1.62626i −2.44822 + 1.77873i −0.168258 + 0.122246i 1.16619 + 3.58917i
168.16 −0.284953 + 0.876996i 2.66699 − 0.866558i 0.930110 + 0.675764i 1.83706 − 1.33470i 2.58587i −3.01590 − 4.15103i −2.34972 + 1.70717i 3.93487 − 2.85885i 0.647051 + 1.99142i
168.17 −0.218761 + 0.673278i −1.23383 + 0.400896i 1.21259 + 0.880996i 0.761675 − 0.553389i − 0.918411i −0.200135 − 0.275462i −2.00387 + 1.45590i −1.06543 + 0.774081i 0.205960 + 0.633879i
168.18 −0.100185 + 0.308338i −2.76255 + 0.897608i 1.53300 + 1.11379i −3.34399 + 2.42955i − 0.941729i 2.71848 + 3.74166i −1.02158 + 0.742224i 4.39895 − 3.19603i −0.414105 − 1.27448i
168.19 −0.0964902 + 0.296966i −0.129857 + 0.0421932i 1.53916 + 1.11826i −1.81110 + 1.31584i − 0.0426345i −2.28567 − 3.14595i −0.985829 + 0.716247i −2.41197 + 1.75240i −0.216008 − 0.664803i
168.20 −0.0191624 + 0.0589758i 0.798997 − 0.259610i 1.61492 + 1.17331i −0.654439 + 0.475478i 0.0520962i −0.369536 − 0.508622i −0.200478 + 0.145656i −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 168.40
Significant digits:
Format:

Inner twists

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

Hecke kernels

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