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Properties

Label 242.2.g.b
Level $242$
Weight $2$
Character orbit 242.g
Analytic conductor $1.932$
Analytic rank $0$
Dimension $240$
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] = [242,2,Mod(5,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([74]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 242.g (of order \(55\), degree \(40\), 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: \(1.93237972891\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(6\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{2} - 8 q^{3} + 6 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} - 80 q^{9} - 22 q^{10} - 10 q^{11} - 13 q^{12} - 30 q^{13} - 13 q^{14} - 30 q^{15} + 6 q^{16} - 4 q^{17} + 12 q^{18} + 4 q^{19} + 5 q^{20} + 20 q^{21} - 11 q^{22} - 21 q^{23} - 2 q^{24} - 7 q^{25} - 7 q^{26} + 46 q^{27} - 2 q^{28} - 13 q^{29} - 19 q^{30} - 68 q^{31} - 24 q^{32} - 11 q^{33} + 6 q^{34} - 16 q^{35} + 12 q^{36} - 75 q^{37} - 44 q^{38} + 178 q^{39} - 16 q^{40} - 4 q^{41} - 44 q^{42} + 14 q^{43} - 15 q^{44} + 4 q^{45} + 13 q^{46} - 7 q^{47} + 3 q^{48} - 46 q^{49} + 13 q^{50} + 21 q^{51} - 7 q^{52} + 12 q^{53} + 16 q^{54} - 26 q^{55} - 3 q^{56} - 57 q^{57} + 97 q^{58} - 41 q^{59} - 31 q^{60} - 48 q^{61} - 2 q^{62} - 81 q^{63} + 6 q^{64} - 75 q^{65} - 82 q^{66} - 41 q^{67} - 4 q^{68} - 22 q^{69} + 26 q^{70} - 69 q^{71} - 3 q^{72} - 64 q^{73} + 13 q^{74} - 2 q^{75} - 30 q^{76} + 162 q^{77} + 32 q^{78} - 33 q^{79} - 5 q^{80} - 131 q^{81} - 21 q^{82} + 10 q^{83} + 166 q^{84} - 122 q^{85} - 26 q^{86} - 17 q^{87} - 21 q^{88} - 13 q^{89} - 160 q^{90} - 44 q^{91} - 52 q^{92} - 49 q^{93} - 170 q^{94} - 295 q^{95} + 3 q^{96} - 88 q^{97} - 36 q^{98} - 119 q^{99}+O(q^{100}) \) 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 0.516397 − 0.856349i −2.22762 − 1.61846i −0.466667 − 0.884433i 2.61490 − 2.13819i −2.53630 + 1.07185i 0.238950 − 0.309876i −0.998369 − 0.0570888i 1.41582 + 4.35745i −0.480712 − 3.34343i
5.2 0.516397 − 0.856349i −0.659289 − 0.479001i −0.466667 − 0.884433i −2.95672 + 2.41770i −0.750647 + 0.317226i −2.54441 + 3.29965i −0.998369 − 0.0570888i −0.721832 − 2.22157i 0.543551 + 3.78048i
5.3 0.516397 − 0.856349i −0.321298 − 0.233437i −0.466667 − 0.884433i 0.747785 − 0.611460i −0.365821 + 0.154597i 0.646547 − 0.838457i −0.998369 − 0.0570888i −0.878311 − 2.70316i −0.137469 − 0.956121i
5.4 0.516397 − 0.856349i −0.107640 − 0.0782047i −0.466667 − 0.884433i −0.862795 + 0.705504i −0.122555 + 0.0517923i 2.67441 − 3.46824i −0.998369 − 0.0570888i −0.921581 − 2.83633i 0.158612 + 1.10317i
5.5 0.516397 − 0.856349i 1.68484 + 1.22411i −0.466667 − 0.884433i 2.72636 − 2.22933i 1.91832 − 0.810687i −0.998224 + 1.29452i −0.998369 − 0.0570888i 0.413202 + 1.27170i −0.501202 − 3.48594i
5.6 0.516397 − 0.856349i 2.41632 + 1.75556i −0.466667 − 0.884433i −1.13886 + 0.931243i 2.75115 − 1.16265i 0.346651 − 0.449545i −0.998369 − 0.0570888i 1.82956 + 5.63081i 0.209364 + 1.45616i
15.1 0.0855750 + 0.996332i −1.04353 + 3.21167i −0.985354 + 0.170522i −1.10788 + 0.0633506i −3.28919 − 0.764868i 2.72097 + 3.97928i −0.254218 − 0.967147i −6.79881 − 4.93962i −0.157925 − 1.09839i
15.2 0.0855750 + 0.996332i −0.398506 + 1.22647i −0.985354 + 0.170522i 3.76248 − 0.215146i −1.25608 − 0.292088i −0.343343 − 0.502123i −0.254218 − 0.967147i 1.08162 + 0.785842i 0.536331 + 3.73026i
15.3 0.0855750 + 0.996332i 0.176884 − 0.544392i −0.985354 + 0.170522i −4.00461 + 0.228992i 0.557532 + 0.129648i 2.14856 + 3.14217i −0.254218 − 0.967147i 2.16198 + 1.57077i −0.570847 − 3.97033i
15.4 0.0855750 + 0.996332i 0.416363 − 1.28143i −0.985354 + 0.170522i 0.325006 − 0.0185845i 1.31236 + 0.305177i 0.546419 + 0.799112i −0.254218 − 0.967147i 0.958336 + 0.696272i 0.0463288 + 0.322224i
15.5 0.0855750 + 0.996332i 0.567015 − 1.74509i −0.985354 + 0.170522i −2.46010 + 0.140674i 1.78721 + 0.415599i −2.92983 − 4.28473i −0.254218 − 0.967147i −0.296792 − 0.215632i −0.350681 − 2.43904i
15.6 0.0855750 + 0.996332i 1.02153 − 3.14394i −0.985354 + 0.170522i 2.66237 − 0.152240i 3.21982 + 0.748737i 1.76910 + 2.58723i −0.254218 − 0.967147i −6.41377 − 4.65988i 0.379513 + 2.63957i
25.1 −0.466667 − 0.884433i −0.720100 − 2.21624i −0.564443 + 0.825472i 0.496453 − 2.45009i −1.62407 + 1.67113i −0.475415 − 1.80867i 0.993482 + 0.113991i −1.96613 + 1.42848i −2.39862 + 0.704299i
25.2 −0.466667 − 0.884433i −0.456517 − 1.40501i −0.564443 + 0.825472i −0.588814 + 2.90591i −1.02960 + 1.05943i 0.913321 + 3.47463i 0.993482 + 0.113991i 0.661395 − 0.480532i 2.84487 − 0.835328i
25.3 −0.466667 − 0.884433i −0.0505515 − 0.155581i −0.564443 + 0.825472i −0.0911070 + 0.449631i −0.114011 + 0.117314i 0.380264 + 1.44667i 0.993482 + 0.113991i 2.40540 − 1.74763i 0.440185 − 0.129250i
25.4 −0.466667 − 0.884433i 0.340662 + 1.04845i −0.564443 + 0.825472i 0.456414 − 2.25249i 0.768307 − 0.790569i −0.473473 − 1.80128i 0.993482 + 0.113991i 1.44386 − 1.04902i −2.20517 + 0.647496i
25.5 −0.466667 − 0.884433i 0.845646 + 2.60263i −0.564443 + 0.825472i −0.614800 + 3.03416i 1.90722 − 1.96248i −0.909018 − 3.45826i 0.993482 + 0.113991i −3.63151 + 2.63845i 2.97041 − 0.872192i
25.6 −0.466667 − 0.884433i 0.931970 + 2.86831i −0.564443 + 0.825472i 0.304917 − 1.50482i 2.10191 − 2.16281i 1.11312 + 4.23475i 0.993482 + 0.113991i −4.93157 + 3.58300i −1.47321 + 0.432573i
31.1 0.774142 − 0.633012i −2.45131 + 1.78098i 0.198590 − 0.980083i −1.82825 + 0.898891i −0.770279 + 2.93045i −0.378745 − 4.40965i −0.466667 − 0.884433i 1.90999 − 5.87834i −0.846316 + 1.85317i
31.2 0.774142 − 0.633012i −2.15202 + 1.56354i 0.198590 − 0.980083i 2.23021 − 1.09652i −0.676233 + 2.57266i 0.232047 + 2.70167i −0.466667 − 0.884433i 1.25951 − 3.87637i 1.03239 − 2.26062i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.2.g.b ✓ 240
121.g even 55 1 inner 242.2.g.b ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.2.g.b ✓ 240 1.a even 1 1 trivial
242.2.g.b ✓ 240 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{240} + 8 T_{3}^{239} + 162 T_{3}^{238} + 1136 T_{3}^{237} + 13315 T_{3}^{236} + \cdots + 16\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(242, [\chi])\). Copy content Toggle raw display