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Properties

Label 125.5.f.b
Level $125$
Weight $5$
Character orbit 125.f
Analytic conductor $12.921$
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] = [125,5,Mod(7,125)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(125, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([17]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("125.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 125.f (of order \(20\), degree \(8\), not 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: \(12.9212453855\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 6 q^{6} + 42 q^{7} - 390 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} + 12 q^{3} + 10 q^{4} - 6 q^{6} + 42 q^{7} - 390 q^{8} + 10 q^{9} - 6 q^{11} - 1768 q^{12} - 278 q^{13} + 10 q^{14} + 762 q^{16} + 2 q^{17} - 28 q^{18} + 1310 q^{19} - 6 q^{21} + 5234 q^{22} + 5132 q^{23} - 976 q^{26} + 30 q^{27} + 3288 q^{28} - 3290 q^{29} - 6 q^{31} - 1298 q^{32} - 3346 q^{33} + 8010 q^{34} - 2758 q^{36} - 9998 q^{37} - 3510 q^{38} - 8790 q^{39} + 1434 q^{41} + 2064 q^{42} - 14958 q^{43} + 22060 q^{44} - 6 q^{46} + 7802 q^{47} + 30032 q^{48} - 16 q^{51} - 34608 q^{52} - 8938 q^{53} - 53940 q^{54} - 1030 q^{56} + 34150 q^{57} - 12840 q^{58} - 23240 q^{59} + 2634 q^{61} - 13256 q^{62} + 43792 q^{63} + 66060 q^{64} + 5178 q^{66} - 8198 q^{67} - 71982 q^{68} + 39560 q^{69} + 7974 q^{71} + 63850 q^{72} + 43182 q^{73} - 1040 q^{76} - 53826 q^{77} - 86986 q^{78} - 26590 q^{79} - 14368 q^{81} + 145614 q^{82} - 16258 q^{83} - 149990 q^{84} - 6 q^{86} - 39760 q^{87} + 170370 q^{88} + 93760 q^{89} - 6 q^{91} - 44698 q^{92} - 134396 q^{93} + 130410 q^{94} + 17694 q^{96} + 29752 q^{97} + 83708 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −6.55470 + 3.33979i −5.51921 + 0.874157i 22.4054 − 30.8383i 0 33.2573 − 24.1628i 37.7565 + 37.7565i −25.4540 + 160.710i −47.3380 + 15.3811i 0
7.2 −4.92542 + 2.50963i 9.36327 − 1.48300i 8.55700 − 11.7777i 0 −42.3963 + 30.8027i 0.684509 + 0.684509i 1.24695 − 7.87292i 8.43590 − 2.74099i 0
7.3 −2.88846 + 1.47174i −12.6351 + 2.00120i −3.22739 + 4.44212i 0 33.5507 − 24.3760i −45.6970 − 45.6970i 10.8986 − 68.8109i 78.6049 − 25.5403i 0
7.4 −1.88103 + 0.958433i 8.94397 − 1.41659i −6.78488 + 9.33859i 0 −15.4662 + 11.2368i −27.3860 − 27.3860i 9.09622 − 57.4312i 0.952289 − 0.309417i 0
7.5 −0.613425 + 0.312556i −7.86762 + 1.24611i −9.12596 + 12.5608i 0 4.43671 − 3.22346i 47.0743 + 47.0743i 3.39533 − 21.4373i −16.6890 + 5.42258i 0
7.6 2.17447 − 1.10795i 6.16392 − 0.976269i −5.90379 + 8.12588i 0 12.3216 − 8.95217i 23.7798 + 23.7798i −9.94294 + 62.7773i −39.9948 + 12.9951i 0
7.7 4.20075 − 2.14039i −2.20827 + 0.349755i 3.66045 − 5.03818i 0 −8.52776 + 6.19578i −46.9663 − 46.9663i −7.20746 + 45.5061i −72.2815 + 23.4857i 0
7.8 5.20267 − 2.65089i −14.5558 + 2.30541i 10.6360 − 14.6392i 0 −69.6176 + 50.5802i 28.1685 + 28.1685i 1.91365 − 12.0823i 129.521 − 42.0839i 0
7.9 6.09417 − 3.10514i 13.5218 − 2.14164i 18.0925 − 24.9022i 0 75.7541 − 55.0385i 9.33601 + 9.33601i 15.8148 − 99.8507i 101.217 − 32.8873i 0
18.1 −6.55470 − 3.33979i −5.51921 − 0.874157i 22.4054 + 30.8383i 0 33.2573 + 24.1628i 37.7565 − 37.7565i −25.4540 − 160.710i −47.3380 − 15.3811i 0
18.2 −4.92542 − 2.50963i 9.36327 + 1.48300i 8.55700 + 11.7777i 0 −42.3963 − 30.8027i 0.684509 − 0.684509i 1.24695 + 7.87292i 8.43590 + 2.74099i 0
18.3 −2.88846 − 1.47174i −12.6351 − 2.00120i −3.22739 − 4.44212i 0 33.5507 + 24.3760i −45.6970 + 45.6970i 10.8986 + 68.8109i 78.6049 + 25.5403i 0
18.4 −1.88103 − 0.958433i 8.94397 + 1.41659i −6.78488 − 9.33859i 0 −15.4662 − 11.2368i −27.3860 + 27.3860i 9.09622 + 57.4312i 0.952289 + 0.309417i 0
18.5 −0.613425 − 0.312556i −7.86762 − 1.24611i −9.12596 − 12.5608i 0 4.43671 + 3.22346i 47.0743 − 47.0743i 3.39533 + 21.4373i −16.6890 − 5.42258i 0
18.6 2.17447 + 1.10795i 6.16392 + 0.976269i −5.90379 − 8.12588i 0 12.3216 + 8.95217i 23.7798 − 23.7798i −9.94294 − 62.7773i −39.9948 − 12.9951i 0
18.7 4.20075 + 2.14039i −2.20827 − 0.349755i 3.66045 + 5.03818i 0 −8.52776 − 6.19578i −46.9663 + 46.9663i −7.20746 − 45.5061i −72.2815 − 23.4857i 0
18.8 5.20267 + 2.65089i −14.5558 − 2.30541i 10.6360 + 14.6392i 0 −69.6176 − 50.5802i 28.1685 − 28.1685i 1.91365 + 12.0823i 129.521 + 42.0839i 0
18.9 6.09417 + 3.10514i 13.5218 + 2.14164i 18.0925 + 24.9022i 0 75.7541 + 55.0385i 9.33601 − 9.33601i 15.8148 + 99.8507i 101.217 + 32.8873i 0
32.1 −7.75939 − 1.22897i 1.30683 + 2.56480i 43.4809 + 14.1278i 0 −6.98817 − 21.5074i −9.39459 − 9.39459i −208.025 − 105.994i 42.7402 − 58.8268i 0
32.2 −4.39138 − 0.695527i −6.80679 − 13.3591i 3.58359 + 1.16438i 0 20.5996 + 63.3991i −39.7357 − 39.7357i 48.4573 + 24.6902i −84.5219 + 116.334i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.5.f.b 72
5.b even 2 1 125.5.f.a 72
5.c odd 4 1 25.5.f.a ✓ 72
5.c odd 4 1 125.5.f.c 72
25.d even 5 1 25.5.f.a ✓ 72
25.e even 10 1 125.5.f.c 72
25.f odd 20 1 125.5.f.a 72
25.f odd 20 1 inner 125.5.f.b 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.f.a ✓ 72 5.c odd 4 1
25.5.f.a ✓ 72 25.d even 5 1
125.5.f.a 72 5.b even 2 1
125.5.f.a 72 25.f odd 20 1
125.5.f.b 72 1.a even 1 1 trivial
125.5.f.b 72 25.f odd 20 1 inner
125.5.f.c 72 5.c odd 4 1
125.5.f.c 72 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 2 T_{2}^{71} - 3 T_{2}^{70} + 160 T_{2}^{69} - 4330 T_{2}^{68} + 7592 T_{2}^{67} + \cdots + 13\!\cdots\!76 \) acting on \(S_{5}^{\mathrm{new}}(125, [\chi])\). Copy content Toggle raw display