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Properties

Label 124.5.b.a
Level $124$
Weight $5$
Character orbit 124.b
Analytic conductor $12.818$
Analytic rank $0$
Dimension $60$
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] = [124,5,Mod(63,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.63");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 124.b (of order \(2\), degree \(1\), 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: \(12.8178754224\)
Analytic rank: \(0\)
Dimension: \(60\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} - 6 q^{4} + 24 q^{5} + 45 q^{8} - 1732 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} - 6 q^{4} + 24 q^{5} + 45 q^{8} - 1732 q^{9} + 53 q^{10} + 130 q^{12} + 120 q^{13} - 231 q^{14} - 590 q^{16} - 648 q^{17} + 230 q^{18} + 1113 q^{20} + 608 q^{21} + 1080 q^{22} - 1028 q^{24} + 8340 q^{25} - 1554 q^{26} - 165 q^{28} - 168 q^{29} - 2238 q^{30} - 1674 q^{32} - 1120 q^{33} + 1844 q^{34} + 1966 q^{36} - 2248 q^{37} - 5055 q^{38} - 1716 q^{40} + 6072 q^{41} + 5794 q^{42} - 120 q^{44} - 4040 q^{45} + 8850 q^{46} + 2276 q^{48} - 17604 q^{49} - 4539 q^{50} + 5896 q^{52} + 3480 q^{53} + 5148 q^{54} - 396 q^{56} - 10912 q^{57} - 7484 q^{58} + 22812 q^{60} + 2200 q^{61} - 19299 q^{64} - 9168 q^{65} - 468 q^{66} - 21930 q^{68} + 6496 q^{69} - 9615 q^{70} - 10079 q^{72} + 13752 q^{73} - 2106 q^{74} + 20099 q^{76} + 16608 q^{77} + 39460 q^{78} - 5787 q^{80} + 20732 q^{81} - 30525 q^{82} - 21760 q^{84} - 21200 q^{85} - 13398 q^{86} + 34690 q^{88} + 22296 q^{89} - 25419 q^{90} - 18852 q^{92} + 3196 q^{94} + 20790 q^{96} + 12120 q^{97} + 21921 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 −3.98801 − 0.309498i 16.5469i 15.8084 + 2.46856i 5.81989 5.12124 − 65.9893i − 22.5351i −62.2801 − 14.7373i −192.801 −23.2098 − 1.80124i
63.2 −3.98801 + 0.309498i − 16.5469i 15.8084 − 2.46856i 5.81989 5.12124 + 65.9893i 22.5351i −62.2801 + 14.7373i −192.801 −23.2098 + 1.80124i
63.3 −3.93534 − 0.716318i − 7.84728i 14.9738 + 5.63791i 18.4905 −5.62115 + 30.8817i − 83.1629i −54.8883 − 32.9131i 19.4202 −72.7665 − 13.2451i
63.4 −3.93534 + 0.716318i 7.84728i 14.9738 − 5.63791i 18.4905 −5.62115 − 30.8817i 83.1629i −54.8883 + 32.9131i 19.4202 −72.7665 + 13.2451i
63.5 −3.71949 − 1.47153i 6.32028i 11.6692 + 10.9467i −34.2840 9.30050 − 23.5082i − 45.8539i −27.2950 − 57.8877i 41.0540 127.519 + 50.4501i
63.6 −3.71949 + 1.47153i − 6.32028i 11.6692 − 10.9467i −34.2840 9.30050 + 23.5082i 45.8539i −27.2950 + 57.8877i 41.0540 127.519 − 50.4501i
63.7 −3.64645 − 1.64420i − 1.82065i 10.5932 + 11.9910i 43.7825 −2.99351 + 6.63891i 53.1613i −18.9122 − 61.1419i 77.6852 −159.651 − 71.9870i
63.8 −3.64645 + 1.64420i 1.82065i 10.5932 − 11.9910i 43.7825 −2.99351 − 6.63891i − 53.1613i −18.9122 + 61.1419i 77.6852 −159.651 + 71.9870i
63.9 −3.40182 − 2.10420i − 9.06141i 7.14471 + 14.3162i 5.08300 −19.0670 + 30.8253i 38.2255i 5.81911 − 63.7349i −1.10919 −17.2914 − 10.6956i
63.10 −3.40182 + 2.10420i 9.06141i 7.14471 − 14.3162i 5.08300 −19.0670 − 30.8253i − 38.2255i 5.81911 + 63.7349i −1.10919 −17.2914 + 10.6956i
63.11 −3.23786 − 2.34867i 10.1900i 4.96753 + 15.2093i −24.8949 23.9329 − 32.9938i 72.3138i 19.6374 − 60.9128i −22.8358 80.6064 + 58.4699i
63.12 −3.23786 + 2.34867i − 10.1900i 4.96753 − 15.2093i −24.8949 23.9329 + 32.9938i − 72.3138i 19.6374 + 60.9128i −22.8358 80.6064 − 58.4699i
63.13 −3.22841 − 2.36165i − 13.9390i 4.84523 + 15.2487i −45.4494 −32.9189 + 45.0007i − 19.7116i 20.3698 − 60.6718i −113.295 146.729 + 107.335i
63.14 −3.22841 + 2.36165i 13.9390i 4.84523 − 15.2487i −45.4494 −32.9189 − 45.0007i 19.7116i 20.3698 + 60.6718i −113.295 146.729 − 107.335i
63.15 −2.98311 − 2.66478i 10.1030i 1.79785 + 15.8987i 29.6620 26.9222 − 30.1382i − 35.0486i 37.0033 − 52.2183i −21.0701 −88.4850 − 79.0429i
63.16 −2.98311 + 2.66478i − 10.1030i 1.79785 − 15.8987i 29.6620 26.9222 + 30.1382i 35.0486i 37.0033 + 52.2183i −21.0701 −88.4850 + 79.0429i
63.17 −2.06596 − 3.42518i 0.636071i −7.46365 + 14.1525i −5.80492 2.17865 − 1.31409i − 53.1144i 63.8944 − 3.67417i 80.5954 11.9927 + 19.8829i
63.18 −2.06596 + 3.42518i − 0.636071i −7.46365 − 14.1525i −5.80492 2.17865 + 1.31409i 53.1144i 63.8944 + 3.67417i 80.5954 11.9927 − 19.8829i
63.19 −1.86128 − 3.54057i − 8.93980i −9.07128 + 13.1800i 0.767532 −31.6520 + 16.6395i 40.1374i 63.5488 + 7.58587i 1.07989 −1.42859 − 2.71750i
63.20 −1.86128 + 3.54057i 8.93980i −9.07128 − 13.1800i 0.767532 −31.6520 − 16.6395i − 40.1374i 63.5488 − 7.58587i 1.07989 −1.42859 + 2.71750i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.60
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.5.b.a ✓ 60
4.b odd 2 1 inner 124.5.b.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.5.b.a ✓ 60 1.a even 1 1 trivial
124.5.b.a ✓ 60 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(124, [\chi])\).