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Properties

Label 241.2.o.a
Level $241$
Weight $2$
Character orbit 241.o
Analytic conductor $1.924$
Analytic rank $0$
Dimension $304$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [241,2,Mod(5,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([23]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.o (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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 16 q^{2} - 16 q^{3} - 16 q^{5} - 20 q^{6} - 24 q^{7} + 16 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 16 q^{2} - 16 q^{3} - 16 q^{5} - 20 q^{6} - 24 q^{7} + 16 q^{8} - 20 q^{9} - 4 q^{10} - 16 q^{11} + 4 q^{12} - 16 q^{13} - 12 q^{14} + 32 q^{15} - 224 q^{16} - 56 q^{17} - 36 q^{18} - 4 q^{19} + 56 q^{20} - 8 q^{21} + 8 q^{22} - 24 q^{23} + 12 q^{24} - 20 q^{25} + 12 q^{26} - 4 q^{27} + 112 q^{28} - 20 q^{29} - 24 q^{30} + 56 q^{31} - 4 q^{32} - 100 q^{33} - 60 q^{34} + 16 q^{35} + 136 q^{36} - 44 q^{37} - 68 q^{38} - 56 q^{39} + 60 q^{40} - 4 q^{41} - 24 q^{42} - 36 q^{43} + 40 q^{44} - 72 q^{45} - 112 q^{46} - 24 q^{47} - 96 q^{48} - 108 q^{49} + 196 q^{50} + 48 q^{51} + 96 q^{52} - 4 q^{53} - 92 q^{54} + 28 q^{55} + 76 q^{56} - 32 q^{57} + 52 q^{58} + 12 q^{59} - 148 q^{62} + 28 q^{63} - 12 q^{65} - 228 q^{66} - 12 q^{67} + 84 q^{68} + 88 q^{69} + 20 q^{70} - 8 q^{71} + 112 q^{72} + 68 q^{73} + 176 q^{74} + 36 q^{75} + 124 q^{76} - 48 q^{77} + 176 q^{78} - 8 q^{79} + 164 q^{80} - 20 q^{81} - 140 q^{82} - 20 q^{83} + 136 q^{84} - 32 q^{85} - 12 q^{86} - 100 q^{87} - 4 q^{88} - 108 q^{89} - 140 q^{90} - 228 q^{91} - 116 q^{92} - 64 q^{93} - 60 q^{94} + 160 q^{95} - 60 q^{96} - 140 q^{97} - 40 q^{98} + 180 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.93406 − 1.93406i −0.0388652 + 0.0762773i 5.48119i −1.59228 − 3.12503i 0.222693 − 0.0723572i 0.947284 − 3.94572i 6.73283 − 6.73283i 1.75905 + 2.42112i −2.96443 + 9.12357i
5.2 −1.86369 − 1.86369i −1.45503 + 2.85567i 4.94670i 0.255105 + 0.500672i 8.03382 − 2.61035i −1.18236 + 4.92490i 5.49175 − 5.49175i −4.27434 − 5.88313i 0.457662 − 1.40854i
5.3 −1.57147 − 1.57147i 0.280883 − 0.551263i 2.93904i 0.646785 + 1.26939i −1.30769 + 0.424895i −0.285080 + 1.18744i 1.47567 − 1.47567i 1.53836 + 2.11737i 0.978400 − 3.01121i
5.4 −1.50639 − 1.50639i 1.32615 − 2.60272i 2.53841i −0.593349 − 1.16451i −5.91841 + 1.92301i −0.402570 + 1.67683i 0.811060 − 0.811060i −3.25212 − 4.47616i −0.860395 + 2.64802i
5.5 −1.27202 − 1.27202i −0.799041 + 1.56821i 1.23605i −0.167132 − 0.328015i 3.01118 − 0.978391i 0.414032 − 1.72457i −0.971755 + 0.971755i −0.0574485 − 0.0790710i −0.204646 + 0.629835i
5.6 −1.06512 − 1.06512i 1.15868 − 2.27404i 0.268963i 1.53857 + 3.01962i −3.65627 + 1.18799i 1.08574 − 4.52244i −1.84376 + 1.84376i −2.06537 − 2.84274i 1.57749 − 4.85503i
5.7 −0.597033 − 0.597033i −0.838436 + 1.64552i − 1.28710i −1.82257 − 3.57700i 1.48301 − 0.481858i −0.862220 + 3.59140i −1.96251 + 1.96251i −0.241418 − 0.332283i −1.04745 + 3.22372i
5.8 −0.478159 − 0.478159i −0.00373273 + 0.00732589i − 1.54273i 1.40563 + 2.75871i 0.00528778 − 0.00171810i −1.02101 + 4.25282i −1.69399 + 1.69399i 1.76332 + 2.42700i 0.646985 − 1.99121i
5.9 −0.426280 − 0.426280i 0.380359 − 0.746496i − 1.63657i −0.451408 − 0.885938i −0.480356 + 0.156077i 0.475829 − 1.98197i −1.55020 + 1.55020i 1.35077 + 1.85918i −0.185232 + 0.570084i
5.10 −0.320541 − 0.320541i −1.38300 + 2.71429i − 1.79451i 1.88705 + 3.70354i 1.31335 − 0.426733i 0.342550 − 1.42683i −1.21629 + 1.21629i −3.69132 − 5.08067i 0.582260 − 1.79201i
5.11 0.0355264 + 0.0355264i 1.20132 − 2.35771i − 1.99748i −0.504802 − 0.990730i 0.126440 − 0.0410827i −0.491884 + 2.04884i 0.142016 − 0.142016i −2.35230 − 3.23767i 0.0172633 − 0.0531309i
5.12 0.118546 + 0.118546i −1.17084 + 2.29791i − 1.97189i −0.975937 − 1.91538i −0.411205 + 0.133609i 0.465581 − 1.93928i 0.470851 − 0.470851i −2.14615 − 2.95393i 0.111367 − 0.342753i
5.13 0.620614 + 0.620614i −0.187598 + 0.368182i − 1.22968i 0.808230 + 1.58624i −0.344925 + 0.112073i 0.504718 − 2.10230i 2.00438 − 2.00438i 1.66299 + 2.28891i −0.482844 + 1.48604i
5.14 0.987392 + 0.987392i 0.225654 − 0.442871i − 0.0501159i −1.71563 − 3.36712i 0.660096 − 0.214478i 0.0342959 − 0.142853i 2.02427 − 2.02427i 1.61814 + 2.22718i 1.63066 − 5.01867i
5.15 1.08848 + 1.08848i −0.608428 + 1.19411i 0.369594i 0.323422 + 0.634752i −1.96203 + 0.637502i −0.771165 + 3.21214i 1.77467 − 1.77467i 0.707648 + 0.973994i −0.338877 + 1.04296i
5.16 1.48838 + 1.48838i 1.07184 − 2.10360i 2.43053i 0.426815 + 0.837672i 4.72625 − 1.53565i −0.834651 + 3.47657i −0.640788 + 0.640788i −1.51295 − 2.08240i −0.611510 + 1.88203i
5.17 1.52421 + 1.52421i −1.44404 + 2.83410i 2.64645i −0.501579 − 0.984405i −6.52079 + 2.11873i −0.105325 + 0.438711i −0.985321 + 0.985321i −4.18348 − 5.75807i 0.735928 − 2.26495i
5.18 1.71302 + 1.71302i −0.611289 + 1.19972i 3.86890i 1.05117 + 2.06303i −3.10230 + 1.00800i 1.14611 − 4.77387i −3.20147 + 3.20147i 0.697698 + 0.960298i −1.73335 + 5.33469i
5.19 1.75148 + 1.75148i 1.04153 − 2.04412i 4.13539i −0.603707 − 1.18484i 5.40446 − 1.75602i 0.678585 − 2.82651i −3.74010 + 3.74010i −1.33028 − 1.83097i 1.01785 − 3.13261i
27.1 −1.80450 − 1.80450i −1.57568 − 0.249564i 4.51247i −2.16549 + 0.342980i 2.39299 + 3.29366i 1.12528 + 1.31753i 4.53376 − 4.53376i −0.432676 − 0.140585i 4.52655 + 3.28873i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.o even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.o.a ✓ 304
241.o even 40 1 inner 241.2.o.a ✓ 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.o.a ✓ 304 1.a even 1 1 trivial
241.2.o.a ✓ 304 241.o even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).