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Properties

Label 276.5.f.a
Level $276$
Weight $5$
Character orbit 276.f
Analytic conductor $28.530$
Analytic rank $0$
Dimension $88$
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] = [276,5,Mod(139,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.139");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 276.f (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: \(28.5301098111\)
Analytic rank: \(0\)
Dimension: \(88\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 48 q^{5} + 180 q^{8} - 2376 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 48 q^{5} + 180 q^{8} - 2376 q^{9} + 244 q^{10} - 360 q^{12} - 592 q^{13} - 612 q^{14} - 824 q^{16} + 1200 q^{17} + 1956 q^{20} + 288 q^{21} - 796 q^{22} - 1404 q^{24} + 10056 q^{25} + 336 q^{26} - 840 q^{28} + 1680 q^{29} + 1512 q^{30} + 1260 q^{32} - 1440 q^{33} + 7008 q^{34} + 1520 q^{37} - 13140 q^{38} - 2356 q^{40} - 3600 q^{41} - 324 q^{42} + 9168 q^{44} + 1296 q^{45} - 25832 q^{49} + 20760 q^{50} - 376 q^{52} - 8304 q^{53} - 8748 q^{56} - 11232 q^{57} - 2592 q^{58} - 7992 q^{60} - 912 q^{61} - 13200 q^{62} + 26472 q^{64} + 31584 q^{65} - 2304 q^{66} - 15732 q^{68} - 3288 q^{70} - 4860 q^{72} - 17680 q^{73} - 21600 q^{74} + 11228 q^{76} - 8640 q^{77} - 19800 q^{78} - 63780 q^{80} + 64152 q^{81} + 12760 q^{82} + 18792 q^{84} + 9632 q^{85} + 47532 q^{86} + 21420 q^{88} + 40560 q^{89} - 6588 q^{90} + 19872 q^{93} + 16232 q^{94} + 19980 q^{96} - 30928 q^{97} - 16608 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −3.99956 − 0.0592192i − 5.19615i 15.9930 + 0.473702i −32.4152 −0.307712 + 20.7823i 48.8782i −63.9369 − 2.84169i −27.0000 129.647 + 1.91960i
139.2 −3.99956 + 0.0592192i 5.19615i 15.9930 − 0.473702i −32.4152 −0.307712 − 20.7823i − 48.8782i −63.9369 + 2.84169i −27.0000 129.647 − 1.91960i
139.3 −3.98939 − 0.291214i − 5.19615i 15.8304 + 2.32353i 20.5704 −1.51319 + 20.7295i 10.9673i −62.4769 − 13.8795i −27.0000 −82.0631 − 5.99038i
139.4 −3.98939 + 0.291214i 5.19615i 15.8304 − 2.32353i 20.5704 −1.51319 − 20.7295i − 10.9673i −62.4769 + 13.8795i −27.0000 −82.0631 + 5.99038i
139.5 −3.92969 − 0.746694i − 5.19615i 14.8849 + 5.86855i 20.3882 −3.87994 + 20.4193i − 80.2781i −54.1110 − 34.1760i −27.0000 −80.1191 − 15.2237i
139.6 −3.92969 + 0.746694i 5.19615i 14.8849 − 5.86855i 20.3882 −3.87994 − 20.4193i 80.2781i −54.1110 + 34.1760i −27.0000 −80.1191 + 15.2237i
139.7 −3.88898 − 0.935877i 5.19615i 14.2483 + 7.27920i −17.8879 4.86296 − 20.2077i − 46.7136i −48.5987 − 41.6433i −27.0000 69.5658 + 16.7409i
139.8 −3.88898 + 0.935877i − 5.19615i 14.2483 − 7.27920i −17.8879 4.86296 + 20.2077i 46.7136i −48.5987 + 41.6433i −27.0000 69.5658 − 16.7409i
139.9 −3.77266 − 1.32930i 5.19615i 12.4659 + 10.0300i 33.0090 6.90727 − 19.6033i − 31.7411i −33.6966 − 54.4108i −27.0000 −124.532 − 43.8789i
139.10 −3.77266 + 1.32930i − 5.19615i 12.4659 − 10.0300i 33.0090 6.90727 + 19.6033i 31.7411i −33.6966 + 54.4108i −27.0000 −124.532 + 43.8789i
139.11 −3.53501 − 1.87183i 5.19615i 8.99253 + 13.2338i −4.46833 9.72630 − 18.3684i 52.6004i −7.01719 − 63.6141i −27.0000 15.7956 + 8.36394i
139.12 −3.53501 + 1.87183i − 5.19615i 8.99253 − 13.2338i −4.46833 9.72630 + 18.3684i − 52.6004i −7.01719 + 63.6141i −27.0000 15.7956 − 8.36394i
139.13 −3.47007 − 1.98962i − 5.19615i 8.08281 + 13.8083i −13.7720 −10.3384 + 18.0310i − 21.7633i −0.574704 − 63.9974i −27.0000 47.7897 + 27.4010i
139.14 −3.47007 + 1.98962i 5.19615i 8.08281 − 13.8083i −13.7720 −10.3384 − 18.0310i 21.7633i −0.574704 + 63.9974i −27.0000 47.7897 − 27.4010i
139.15 −3.46166 − 2.00422i − 5.19615i 7.96617 + 13.8759i −36.6825 −10.4143 + 17.9873i − 27.4286i 0.234228 − 63.9996i −27.0000 126.982 + 73.5199i
139.16 −3.46166 + 2.00422i 5.19615i 7.96617 − 13.8759i −36.6825 −10.4143 − 17.9873i 27.4286i 0.234228 + 63.9996i −27.0000 126.982 − 73.5199i
139.17 −3.00935 − 2.63511i 5.19615i 2.11241 + 15.8599i −17.7651 13.6924 − 15.6371i 6.95769i 35.4357 − 53.2946i −27.0000 53.4614 + 46.8129i
139.18 −3.00935 + 2.63511i − 5.19615i 2.11241 − 15.8599i −17.7651 13.6924 + 15.6371i − 6.95769i 35.4357 + 53.2946i −27.0000 53.4614 − 46.8129i
139.19 −2.99794 − 2.64808i − 5.19615i 1.97533 + 15.8776i 39.9529 −13.7598 + 15.5778i 18.6548i 36.1233 − 52.8310i −27.0000 −119.777 − 105.799i
139.20 −2.99794 + 2.64808i 5.19615i 1.97533 − 15.8776i 39.9529 −13.7598 − 15.5778i − 18.6548i 36.1233 + 52.8310i −27.0000 −119.777 + 105.799i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.88
Significant digits:
Format:

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 276.5.f.a ✓ 88
4.b odd 2 1 inner 276.5.f.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.5.f.a ✓ 88 1.a even 1 1 trivial
276.5.f.a ✓ 88 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(276, [\chi])\).