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Properties

Label 241.2.h.a
Level $241$
Weight $2$
Character orbit 241.h
Analytic conductor $1.924$
Analytic rank $0$
Dimension $72$
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(36,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.h (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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 7 q^{3} + 52 q^{4} - 9 q^{5} - 3 q^{6} - 5 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 7 q^{3} + 52 q^{4} - 9 q^{5} - 3 q^{6} - 5 q^{7} - 5 q^{9} + q^{10} - 20 q^{12} + 20 q^{13} - 15 q^{14} - 8 q^{15} + 12 q^{16} - 25 q^{17} - 23 q^{18} - 22 q^{20} + 20 q^{21} - 5 q^{23} - 19 q^{24} + 9 q^{25} + 10 q^{26} + 23 q^{27} - 35 q^{28} + q^{29} + 20 q^{30} - 35 q^{31} - 20 q^{32} + 20 q^{33} - 50 q^{34} - 20 q^{35} + q^{36} - 5 q^{39} + 6 q^{40} + 18 q^{41} + 55 q^{43} + 32 q^{45} - 35 q^{46} + 3 q^{47} - 13 q^{48} - 5 q^{49} - 53 q^{50} - 10 q^{51} + 35 q^{52} - 31 q^{53} + 49 q^{54} - 35 q^{56} - 10 q^{57} + 38 q^{58} + 9 q^{59} - 64 q^{60} - 63 q^{61} + 25 q^{62} - 20 q^{64} + 15 q^{66} - 12 q^{67} - 55 q^{68} - 35 q^{70} + 15 q^{71} - 152 q^{72} + 5 q^{73} - 60 q^{74} + q^{75} - 17 q^{77} + 50 q^{78} - 8 q^{79} - 77 q^{80} - 14 q^{81} - 5 q^{82} + 16 q^{83} + 115 q^{84} + 40 q^{85} + 55 q^{86} + 25 q^{87} + 13 q^{90} - 11 q^{91} - 90 q^{92} - 25 q^{93} - 20 q^{94} + 75 q^{95} - 30 q^{96} + 14 q^{97} + 38 q^{98} + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −2.62945 −1.18997 − 0.864564i 4.91400 −0.971375 + 0.705745i 3.12897 + 2.27333i −0.591191 − 0.192090i −7.66222 −0.258492 − 0.795557i 2.55418 − 1.85572i
36.2 −2.26282 1.98039 + 1.43884i 3.12033 −1.21463 + 0.882481i −4.48125 − 3.25582i 2.51170 + 0.816100i −2.53511 0.924636 + 2.84574i 2.74849 − 1.99689i
36.3 −1.88639 0.849425 + 0.617144i 1.55846 −0.582912 + 0.423510i −1.60234 − 1.16417i −3.87381 − 1.25868i 0.832922 −0.586394 − 1.80474i 1.09960 − 0.798904i
36.4 −1.82184 −2.59775 − 1.88738i 1.31912 2.33055 − 1.69324i 4.73269 + 3.43850i 2.34671 + 0.762494i 1.24046 2.25906 + 6.95268i −4.24589 + 3.08482i
36.5 −1.37340 −1.73524 − 1.26072i −0.113785 −3.03079 + 2.20200i 2.38317 + 1.73147i −0.735566 − 0.239000i 2.90306 0.494576 + 1.52215i 4.16247 − 3.02421i
36.6 −1.09607 −1.28005 − 0.930012i −0.798632 1.57742 − 1.14606i 1.40303 + 1.01936i −3.14199 − 1.02089i 3.06749 −0.153441 − 0.472244i −1.72896 + 1.25617i
36.7 −1.07312 1.80288 + 1.30987i −0.848406 2.47363 − 1.79719i −1.93471 − 1.40565i 0.403469 + 0.131095i 3.05669 0.607563 + 1.86989i −2.65451 + 1.92861i
36.8 −0.427146 −0.241952 − 0.175789i −1.81755 −0.241200 + 0.175242i 0.103349 + 0.0750875i 3.64405 + 1.18402i 1.63065 −0.899412 − 2.76810i 0.103028 − 0.0748540i
36.9 −0.304995 1.70332 + 1.23754i −1.90698 −2.56254 + 1.86179i −0.519505 − 0.377443i 0.111347 + 0.0361788i 1.19161 0.442764 + 1.36269i 0.781561 − 0.567837i
36.10 0.209617 −0.580838 − 0.422004i −1.95606 1.46434 − 1.06390i −0.121754 − 0.0884593i 0.835988 + 0.271629i −0.829259 −0.767765 − 2.36294i 0.306950 − 0.223012i
36.11 0.768297 −2.58422 − 1.87754i −1.40972 −1.38053 + 1.00301i −1.98545 − 1.44251i 3.73864 + 1.21476i −2.61968 2.22595 + 6.85078i −1.06065 + 0.770611i
36.12 0.910035 2.62415 + 1.90656i −1.17184 0.394962 − 0.286956i 2.38807 + 1.73504i −0.667785 − 0.216977i −2.88648 2.32416 + 7.15304i 0.359429 − 0.261140i
36.13 1.00978 −0.767602 − 0.557696i −0.980349 −2.38554 + 1.73319i −0.775108 − 0.563149i −2.32756 − 0.756270i −3.00949 −0.648862 − 1.99699i −2.40886 + 1.75014i
36.14 1.55557 −2.27723 − 1.65450i 0.419792 1.89002 − 1.37318i −3.54239 − 2.57369i −4.27144 − 1.38788i −2.45812 1.52134 + 4.68221i 2.94006 − 2.13608i
36.15 1.61206 0.771942 + 0.560849i 0.598753 2.19879 − 1.59752i 1.24442 + 0.904124i 0.355614 + 0.115546i −2.25890 −0.645708 − 1.98728i 3.54460 − 2.57530i
36.16 1.76922 1.23157 + 0.894790i 1.13013 −2.57640 + 1.87186i 2.17892 + 1.58308i 4.18240 + 1.35894i −1.53899 −0.210929 − 0.649174i −4.55820 + 3.31173i
36.17 2.48435 −1.56445 − 1.13664i 4.17198 0.269451 − 0.195767i −3.88663 − 2.82380i 1.31873 + 0.428482i 5.39595 0.228499 + 0.703246i 0.669409 − 0.486354i
36.18 2.55630 0.428562 + 0.311369i 4.53468 −1.58031 + 1.14816i 1.09553 + 0.795952i −2.29423 − 0.745441i 6.47941 −0.840336 − 2.58629i −4.03974 + 2.93504i
143.1 −2.59104 −0.0471082 + 0.144984i 4.71350 0.0319737 + 0.0984048i 0.122059 − 0.375660i −2.78492 − 3.83311i −7.03078 2.40825 + 1.74970i −0.0828451 − 0.254971i
143.2 −2.45725 0.521001 − 1.60348i 4.03808 −0.152953 − 0.470741i −1.28023 + 3.94014i 2.29992 + 3.16557i −5.00807 0.127357 + 0.0925300i 0.375844 + 1.15673i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.h even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.h.a ✓ 72
241.h even 10 1 inner 241.2.h.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.h.a ✓ 72 1.a even 1 1 trivial
241.2.h.a ✓ 72 241.h even 10 1 inner

Hecke kernels

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