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Properties

Label 242.2.g.a
Level $242$
Weight $2$
Character orbit 242.g
Analytic conductor $1.932$
Analytic rank $0$
Dimension $200$
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: \(200\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 200 q - 5 q^{2} + 10 q^{3} + 5 q^{4} - q^{5} + q^{6} - 5 q^{8} - 40 q^{9} - 4 q^{10} - 11 q^{11} + 4 q^{12} - 32 q^{13} - 11 q^{14} - 24 q^{15} + 5 q^{16} + 2 q^{17} - 4 q^{18} + q^{19} - q^{20} - 8 q^{21} - 22 q^{22} + 41 q^{23} + q^{24} + 2 q^{25} - q^{26} - 59 q^{27} + 3 q^{29} + 13 q^{30} + 18 q^{31} + 20 q^{32} + 8 q^{34} + 4 q^{35} + 4 q^{36} + 17 q^{37} - 34 q^{38} - 172 q^{39} - 10 q^{40} + 6 q^{41} + 42 q^{42} - 20 q^{43} + 11 q^{44} + 64 q^{45} - 9 q^{46} + 7 q^{47} - q^{48} - 43 q^{49} - 2 q^{50} - 135 q^{51} + q^{52} - 80 q^{53} - 16 q^{54} - 22 q^{55} + 11 q^{56} + 41 q^{57} - 157 q^{58} + 30 q^{59} + 31 q^{60} + 40 q^{61} + 4 q^{62} - 45 q^{63} + 5 q^{64} - 7 q^{65} + 55 q^{66} - 19 q^{67} + 2 q^{68} - 8 q^{69} - 70 q^{70} + 5 q^{71} - 4 q^{72} + 34 q^{73} + 5 q^{74} - 11 q^{75} - 15 q^{76} - 198 q^{77} + 16 q^{78} - 47 q^{79} - q^{80} - 18 q^{81} - 28 q^{82} + 9 q^{83} - 174 q^{84} - 100 q^{85} - 5 q^{86} + 37 q^{87} - 11 q^{88} - 67 q^{89} - 28 q^{90} - 166 q^{91} - 46 q^{92} - 47 q^{93} - 18 q^{94} + 129 q^{95} + q^{96} - 3 q^{97} - 40 q^{98} + 22 q^{99}+O(q^{100}) \) 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 −2.13866 − 1.55383i −0.466667 − 0.884433i −1.07424 + 0.878398i 2.43502 − 1.02905i −1.06439 + 1.38033i 0.998369 + 0.0570888i 1.23243 + 3.79304i −0.197483 − 1.37352i
5.2 −0.516397 + 0.856349i −0.433449 − 0.314919i −0.466667 − 0.884433i −1.12246 + 0.917831i 0.493513 − 0.208560i −1.54587 + 2.00472i 0.998369 + 0.0570888i −0.838347 − 2.58017i −0.206348 − 1.43518i
5.3 −0.516397 + 0.856349i −0.324490 − 0.235756i −0.466667 − 0.884433i 1.42498 − 1.16520i 0.369456 − 0.156133i 0.331985 − 0.430526i 0.998369 + 0.0570888i −0.877338 − 2.70017i 0.261962 + 1.82199i
5.4 −0.516397 + 0.856349i 1.49802 + 1.08837i −0.466667 − 0.884433i 1.49869 − 1.22548i −1.70560 + 0.720792i 1.72885 − 2.24201i 0.998369 + 0.0570888i 0.132446 + 0.407627i 0.275513 + 1.91624i
5.5 −0.516397 + 0.856349i 2.20760 + 1.60392i −0.466667 − 0.884433i −3.30271 + 2.70061i −2.51351 + 1.06222i 1.14932 − 1.49047i 0.998369 + 0.0570888i 1.37390 + 4.22844i −0.607156 − 4.22286i
15.1 −0.0855750 − 0.996332i −0.871571 + 2.68242i −0.985354 + 0.170522i 1.17984 − 0.0674655i 2.74716 + 0.638826i 0.234620 + 0.343121i 0.254218 + 0.967147i −4.00869 − 2.91248i −0.168183 − 1.16974i
15.2 −0.0855750 − 0.996332i −0.287239 + 0.884029i −0.985354 + 0.170522i 1.80964 − 0.103479i 0.905367 + 0.210534i −2.83890 − 4.15177i 0.254218 + 0.967147i 1.72805 + 1.25550i −0.257959 − 1.79415i
15.3 −0.0855750 − 0.996332i −0.194167 + 0.597585i −0.985354 + 0.170522i −0.859975 + 0.0491752i 0.612009 + 0.142317i 1.98344 + 2.90069i 0.254218 + 0.967147i 2.10764 + 1.53129i 0.122587 + 0.852612i
15.4 −0.0855750 − 0.996332i 0.292119 − 0.899048i −0.985354 + 0.170522i 4.07280 − 0.232891i −0.920748 − 0.214111i 1.25754 + 1.83909i 0.254218 + 0.967147i 1.70410 + 1.23810i −0.580567 − 4.03793i
15.5 −0.0855750 − 0.996332i 0.751841 − 2.31393i −0.985354 + 0.170522i −1.44457 + 0.0826036i −2.36978 − 0.551069i −0.642307 − 0.939344i 0.254218 + 0.967147i −2.36195 − 1.71606i 0.205920 + 1.43220i
25.1 0.466667 + 0.884433i −0.816622 − 2.51330i −0.564443 + 0.825472i −0.759633 + 3.74894i 1.84176 − 1.89512i 0.840551 + 3.19779i −0.993482 − 0.113991i −3.22277 + 2.34148i −3.67018 + 1.07766i
25.2 0.466667 + 0.884433i −0.632586 − 1.94690i −0.564443 + 0.825472i 0.184594 − 0.911006i 1.42669 − 1.46803i −0.597482 − 2.27306i −0.993482 − 0.113991i −0.963202 + 0.699807i 0.891867 − 0.261876i
25.3 0.466667 + 0.884433i −0.0758337 − 0.233392i −0.564443 + 0.825472i 0.588996 − 2.90681i 0.171031 − 0.175986i 1.08715 + 4.13596i −0.993482 − 0.113991i 2.37833 − 1.72796i 2.84574 − 0.835586i
25.4 0.466667 + 0.884433i 0.313512 + 0.964891i −0.564443 + 0.825472i −0.337582 + 1.66603i −0.707075 + 0.727563i 0.220471 + 0.838759i −0.993482 − 0.113991i 1.59433 − 1.15835i −1.63103 + 0.478915i
25.5 0.466667 + 0.884433i 0.902513 + 2.77765i −0.564443 + 0.825472i −0.0878091 + 0.433355i −2.03547 + 2.09445i −0.274842 − 1.04561i −0.993482 − 0.113991i −4.47375 + 3.25037i −0.424251 + 0.124571i
31.1 −0.774142 + 0.633012i −1.86596 + 1.35570i 0.198590 − 0.980083i −0.339686 + 0.167012i 0.586344 − 2.23068i −0.378964 − 4.41220i 0.466667 + 0.884433i 0.716842 − 2.20621i 0.157244 − 0.344317i
31.2 −0.774142 + 0.633012i −1.59601 + 1.15957i 0.198590 − 0.980083i 1.05809 − 0.520226i 0.501516 − 1.90796i 0.173538 + 2.02047i 0.466667 + 0.884433i 0.275594 − 0.848191i −0.489799 + 1.07251i
31.3 −0.774142 + 0.633012i 0.108872 − 0.0791004i 0.198590 − 0.980083i 0.362789 − 0.178371i −0.0342111 + 0.130152i 0.152815 + 1.77919i 0.466667 + 0.884433i −0.921455 + 2.83595i −0.167939 + 0.367735i
31.4 −0.774142 + 0.633012i 1.89410 − 1.37614i 0.198590 − 0.980083i 2.54151 − 1.24958i −0.595185 + 2.26432i 0.000169160 0.00196950i 0.466667 + 0.884433i 0.766788 − 2.35993i −1.17649 + 2.57616i
31.5 −0.774142 + 0.633012i 2.26802 − 1.64781i 0.198590 − 0.980083i −2.60086 + 1.27876i −0.712682 + 2.71132i −0.380327 − 4.42807i 0.466667 + 0.884433i 1.50157 − 4.62137i 1.20396 − 2.63631i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.5
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.a ✓ 200
121.g even 55 1 inner 242.2.g.a ✓ 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.2.g.a ✓ 200 1.a even 1 1 trivial
242.2.g.a ✓ 200 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{200} - 10 T_{3}^{199} + 145 T_{3}^{198} - 1117 T_{3}^{197} + 10002 T_{3}^{196} + \cdots + 10\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(242, [\chi])\). Copy content Toggle raw display