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Properties

Label 102.5.g.c
Level $102$
Weight $5$
Character orbit 102.g
Analytic conductor $10.544$
Analytic rank $0$
Dimension $44$
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] = [102,5,Mod(53,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 102.g (of order \(8\), 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: \(10.5437362346\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 88 q^{2} - 32 q^{3} - 72 q^{5} + 112 q^{6} + 132 q^{7} + 704 q^{8} - 184 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 88 q^{2} - 32 q^{3} - 72 q^{5} + 112 q^{6} + 132 q^{7} + 704 q^{8} - 184 q^{9} + 384 q^{10} + 616 q^{11} - 192 q^{12} - 416 q^{14} - 92 q^{15} - 2816 q^{16} - 424 q^{17} + 1016 q^{18} + 568 q^{19} - 960 q^{20} - 1020 q^{21} + 424 q^{22} - 332 q^{23} - 128 q^{24} + 1888 q^{25} + 1008 q^{26} + 1600 q^{27} + 608 q^{28} + 6688 q^{29} + 2000 q^{30} - 1428 q^{31} + 5632 q^{32} + 2224 q^{33} + 512 q^{34} - 2592 q^{36} + 296 q^{37} - 2272 q^{38} - 1600 q^{39} + 768 q^{40} - 4592 q^{41} + 752 q^{42} - 228 q^{43} - 6624 q^{44} + 12804 q^{45} + 1872 q^{46} - 10712 q^{47} + 2048 q^{48} + 5304 q^{49} - 18284 q^{51} - 4032 q^{52} + 5656 q^{53} - 4496 q^{54} + 896 q^{56} - 8460 q^{57} - 11472 q^{58} - 3620 q^{59} - 7264 q^{60} - 10448 q^{61} + 3536 q^{62} - 24512 q^{63} + 13760 q^{65} + 4056 q^{66} - 18584 q^{67} + 1344 q^{68} + 6500 q^{69} - 9664 q^{70} - 36660 q^{71} + 2240 q^{72} + 18008 q^{73} + 144 q^{74} + 49316 q^{75} + 4544 q^{76} + 5704 q^{77} + 9392 q^{78} - 7324 q^{79} + 4608 q^{80} + 33668 q^{81} + 11160 q^{82} + 12620 q^{83} + 5152 q^{84} - 10896 q^{85} + 15772 q^{87} + 23104 q^{88} + 5040 q^{89} - 43728 q^{90} + 52368 q^{91} - 4832 q^{92} - 2440 q^{93} + 21424 q^{94} - 16032 q^{95} - 7168 q^{96} - 40164 q^{97} - 21216 q^{98} + 21956 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −2.00000 + 2.00000i −8.75155 − 2.10010i − 8.00000i 10.0719 − 4.17192i 21.7033 − 13.3029i 7.83473 − 18.9147i 16.0000 + 16.0000i 72.1792 + 36.7583i −11.8000 + 28.4876i
53.2 −2.00000 + 2.00000i −7.88501 + 4.33896i − 8.00000i 4.86643 − 2.01574i 7.09210 − 24.4479i 5.00624 − 12.0861i 16.0000 + 16.0000i 43.3468 − 68.4255i −5.70137 + 13.7643i
53.3 −2.00000 + 2.00000i −6.42909 − 6.29815i − 8.00000i −7.71818 + 3.19697i 25.4545 − 0.261878i −32.7174 + 78.9869i 16.0000 + 16.0000i 1.66649 + 80.9829i 9.04241 − 21.8303i
53.4 −2.00000 + 2.00000i −6.02226 + 6.68823i − 8.00000i −44.5723 + 18.4624i −1.33193 − 25.4210i −18.9619 + 45.7780i 16.0000 + 16.0000i −8.46471 − 80.5565i 52.2197 − 126.069i
53.5 −2.00000 + 2.00000i −1.33960 − 8.89975i − 8.00000i 40.4335 − 16.7481i 20.4787 + 15.1203i 9.74985 − 23.5382i 16.0000 + 16.0000i −77.4110 + 23.8442i −47.3708 + 114.363i
53.6 −2.00000 + 2.00000i −0.910515 − 8.95382i − 8.00000i −24.3911 + 10.1031i 19.7287 + 16.0866i 18.4989 − 44.6604i 16.0000 + 16.0000i −79.3419 + 16.3052i 28.5759 − 68.9884i
53.7 −2.00000 + 2.00000i −0.180968 + 8.99818i − 8.00000i 13.4598 − 5.57524i −17.6344 − 18.3583i −17.8254 + 43.0343i 16.0000 + 16.0000i −80.9345 − 3.25676i −15.7692 + 38.0701i
53.8 −2.00000 + 2.00000i 1.48465 + 8.87670i − 8.00000i −12.7918 + 5.29855i −20.7227 − 14.7841i 35.6639 − 86.1003i 16.0000 + 16.0000i −76.5916 + 26.3576i 14.9866 − 36.1808i
53.9 −2.00000 + 2.00000i 7.18993 − 5.41340i − 8.00000i 14.6814 − 6.08124i −3.55306 + 25.2067i 11.9385 − 28.8221i 16.0000 + 16.0000i 22.3902 − 77.8439i −17.2004 + 41.5253i
53.10 −2.00000 + 2.00000i 8.43000 + 3.15199i − 8.00000i 36.5016 − 15.1195i −23.1640 + 10.5560i −20.7919 + 50.1961i 16.0000 + 16.0000i 61.1299 + 53.1426i −42.7643 + 103.242i
53.11 −2.00000 + 2.00000i 8.53573 − 2.85330i − 8.00000i −40.0560 + 16.5917i −11.3649 + 22.7781i −2.16514 + 5.22710i 16.0000 + 16.0000i 64.7174 − 48.7100i 46.9285 − 113.295i
59.1 −2.00000 − 2.00000i −8.83972 − 1.69095i 8.00000i −14.7889 + 35.7036i 14.2975 + 21.0613i 33.4426 − 13.8524i 16.0000 − 16.0000i 75.2814 + 29.8951i 100.985 − 41.8294i
59.2 −2.00000 − 2.00000i −8.63126 − 2.54976i 8.00000i 1.58463 − 3.82565i 12.1630 + 22.3620i −65.9383 + 27.3125i 16.0000 − 16.0000i 67.9975 + 44.0153i −10.8206 + 4.48202i
59.3 −2.00000 − 2.00000i −7.47308 + 5.01529i 8.00000i 7.65773 − 18.4874i 24.9767 + 4.91559i 3.64521 − 1.50990i 16.0000 − 16.0000i 30.6938 − 74.9592i −52.2903 + 21.6593i
59.4 −2.00000 − 2.00000i −6.14010 − 6.58021i 8.00000i 13.3978 − 32.3452i −0.880208 + 25.4406i 79.8085 − 33.0578i 16.0000 − 16.0000i −5.59826 + 80.8063i −91.4860 + 37.8947i
59.5 −2.00000 − 2.00000i −3.21510 + 8.40614i 8.00000i −16.1337 + 38.9501i 23.2425 − 10.3821i −77.2612 + 32.0026i 16.0000 − 16.0000i −60.3263 − 54.0531i 110.167 − 45.6329i
59.6 −2.00000 − 2.00000i −1.11167 − 8.93108i 8.00000i −3.88053 + 9.36843i −15.6388 + 20.0855i 15.3463 − 6.35666i 16.0000 − 16.0000i −78.5284 + 19.8568i 26.4979 − 10.9758i
59.7 −2.00000 − 2.00000i 1.50892 + 8.87261i 8.00000i −8.97188 + 21.6600i 14.7274 − 20.7631i 85.5528 − 35.4371i 16.0000 − 16.0000i −76.4463 + 26.7761i 61.2638 − 25.3763i
59.8 −2.00000 − 2.00000i 2.14351 + 8.74102i 8.00000i 13.6292 − 32.9039i 13.1950 − 21.7690i 1.24493 − 0.515667i 16.0000 − 16.0000i −71.8108 + 37.4728i −93.0663 + 38.5493i
59.9 −2.00000 − 2.00000i 5.14497 − 7.38440i 8.00000i −11.1111 + 26.8246i −25.0587 + 4.47885i 27.8275 − 11.5265i 16.0000 − 16.0000i −28.0586 − 75.9850i 75.8714 − 31.4270i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.5.g.c ✓ 44
3.b odd 2 1 102.5.g.d yes 44
17.d even 8 1 102.5.g.d yes 44
51.g odd 8 1 inner 102.5.g.c ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.5.g.c ✓ 44 1.a even 1 1 trivial
102.5.g.c ✓ 44 51.g odd 8 1 inner
102.5.g.d yes 44 3.b odd 2 1
102.5.g.d yes 44 17.d even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{44} + 72 T_{5}^{43} + 1648 T_{5}^{42} - 38112 T_{5}^{41} - 3210880 T_{5}^{40} + \cdots + 24\!\cdots\!12 \) acting on \(S_{5}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display