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Properties

Label 751.2.l.a
Level $751$
Weight $2$
Character orbit 751.l
Analytic conductor $5.997$
Analytic rank $0$
Dimension $6100$
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] = [751,2,Mod(8,751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(751, base_ring=CyclotomicField(250))
 
chi = DirichletCharacter(H, H._module([166]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("751.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 751.l (of order \(125\), degree \(100\), 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: \(5.99676519180\)
Analytic rank: \(0\)
Dimension: \(6100\)
Relative dimension: \(61\) over \(\Q(\zeta_{125})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{125}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 6100 q - 75 q^{2} - 100 q^{3} - 75 q^{4} - 100 q^{5} - 100 q^{6} - 100 q^{7} - 75 q^{8} - 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 6100 q - 75 q^{2} - 100 q^{3} - 75 q^{4} - 100 q^{5} - 100 q^{6} - 100 q^{7} - 75 q^{8} - 100 q^{9} - 100 q^{10} - 100 q^{11} - 100 q^{12} + 25 q^{13} - 100 q^{14} - 50 q^{15} + 125 q^{16} - 100 q^{17} - 100 q^{18} - 100 q^{19} - 100 q^{20} - 100 q^{21} - 200 q^{22} - 100 q^{23} - 100 q^{24} - 100 q^{25} - 100 q^{26} - 100 q^{27} - 100 q^{28} - 100 q^{29} - 100 q^{30} + 100 q^{31} - 200 q^{32} - 100 q^{33} - 100 q^{34} - 100 q^{35} - 100 q^{36} - 100 q^{37} - 100 q^{38} + 100 q^{39} - 100 q^{40} - 100 q^{41} - 500 q^{42} - 100 q^{43} - 100 q^{45} - 100 q^{46} - 100 q^{47} - 100 q^{48} - 100 q^{49} + 100 q^{51} - 100 q^{52} - 500 q^{53} - 100 q^{54} - 100 q^{55} + 150 q^{56} + 150 q^{57} - 100 q^{58} - 100 q^{59} + 300 q^{60} - 100 q^{61} - 100 q^{62} + 100 q^{63} + 275 q^{64} - 100 q^{65} - 100 q^{66} - 100 q^{68} - 100 q^{69} + 100 q^{70} + 325 q^{71} - 200 q^{72} - 600 q^{73} - 500 q^{74} - 100 q^{75} - 75 q^{76} + 100 q^{77} + 25 q^{78} + 425 q^{80} - 100 q^{81} - 100 q^{82} + 75 q^{83} - 100 q^{84} - 25 q^{85} + 75 q^{86} - 100 q^{87} + 400 q^{88} - 100 q^{90} - 100 q^{91} + 675 q^{92} - 100 q^{94} - 1100 q^{95} - 100 q^{96} - 50 q^{97} + 250 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −2.09747 − 1.78004i −0.658935 + 1.16371i 0.905555 + 5.49381i −0.109966 + 0.147423i 3.45356 − 1.26793i −1.77631 − 1.93986i 5.04942 − 8.41695i 0.623281 + 1.03896i 0.493070 − 0.113472i
8.2 −2.05161 − 1.74113i 0.0379170 − 0.0669635i 0.852323 + 5.17086i −1.47049 + 1.97137i −0.194383 + 0.0713650i 1.51628 + 1.65589i 4.48594 − 7.47768i 1.54027 + 2.56750i 6.44928 − 1.48419i
8.3 −2.01349 − 1.70877i 1.21890 − 2.15264i 0.808963 + 4.90781i 0.860440 − 1.15353i −6.13261 + 2.25150i −2.80251 − 3.06055i 4.04038 − 6.73497i −1.60483 − 2.67512i −3.70360 + 0.852321i
8.4 −1.93721 − 1.64404i 0.657818 − 1.16174i 0.724656 + 4.39634i 2.11871 − 2.84039i −3.18428 + 1.16906i 0.902676 + 0.985789i 3.20976 − 5.35039i 0.626398 + 1.04415i −8.77410 + 2.01921i
8.5 −1.87215 − 1.58882i −1.66027 + 2.93213i 0.655312 + 3.97564i −2.09013 + 2.80208i 7.76691 − 2.85151i 0.434721 + 0.474747i 2.56337 − 4.27292i −4.29758 − 7.16369i 8.36503 − 1.92507i
8.6 −1.84226 − 1.56346i 1.29474 − 2.28659i 0.624254 + 3.78722i −2.58353 + 3.46355i −5.96024 + 2.18822i −0.935358 − 1.02148i 2.28507 − 3.80902i −2.00881 − 3.34852i 10.1747 − 2.34152i
8.7 −1.72580 − 1.46462i −1.33315 + 2.35442i 0.507996 + 3.08191i 0.954651 − 1.27983i 5.74908 − 2.11069i −2.22702 − 2.43207i 1.30824 − 2.18072i −2.22268 − 3.70501i −3.52200 + 0.810528i
8.8 −1.72422 − 1.46328i −0.656350 + 1.15915i 0.506474 + 3.07267i 1.74492 − 2.33929i 2.82786 − 1.03821i 2.02705 + 2.21369i 1.29616 − 2.16059i 0.630485 + 1.05096i −6.43168 + 1.48014i
8.9 −1.70006 − 1.44277i 0.470060 − 0.830151i 0.483324 + 2.93223i −0.165081 + 0.221312i −1.99685 + 0.733115i −2.52575 − 2.75831i 1.11472 − 1.85814i 1.07512 + 1.79214i 0.599951 − 0.138069i
8.10 −1.66528 − 1.41326i −0.929085 + 1.64082i 0.450576 + 2.73355i −1.00459 + 1.34679i 3.86608 − 1.41938i 2.50601 + 2.73674i 0.865664 − 1.44299i −0.285760 − 0.476337i 3.57628 − 0.823019i
8.11 −1.62239 − 1.37686i 1.55698 − 2.74971i 0.411134 + 2.49426i 0.354345 − 0.475044i −6.31200 + 2.31736i 1.01118 + 1.10428i 0.577899 − 0.963308i −3.59339 − 5.98987i −1.22896 + 0.282824i
8.12 −1.42873 − 1.21251i 0.738886 − 1.30491i 0.245819 + 1.49133i 0.0670818 − 0.0899316i −2.63789 + 0.968465i 2.42006 + 2.64288i −0.470955 + 0.785042i 0.386472 + 0.644216i −0.204885 + 0.0471508i
8.13 −1.41152 − 1.19790i −0.232282 + 0.410224i 0.232143 + 1.40836i −1.05487 + 1.41419i 0.819280 − 0.300787i −1.28915 − 1.40785i −0.545371 + 0.909086i 1.42899 + 2.38200i 3.18304 − 0.732523i
8.14 −1.40337 − 1.19099i −0.149611 + 0.264221i 0.225724 + 1.36942i −1.72763 + 2.31610i 0.524644 − 0.192616i −1.72037 − 1.87877i −0.579590 + 0.966127i 1.49589 + 2.49352i 5.18295 − 1.19277i
8.15 −1.25630 − 1.06617i −1.49501 + 2.64028i 0.116286 + 0.705481i 2.18421 − 2.92821i 4.69317 − 1.72303i 1.95714 + 2.13734i −1.08924 + 1.81566i −3.19268 − 5.32193i −5.86598 + 1.34996i
8.16 −1.16968 − 0.992660i −0.150517 + 0.265820i 0.0574951 + 0.348811i 1.23398 − 1.65431i 0.439925 − 0.161512i −0.385520 − 0.421016i −1.29942 + 2.16602i 1.49531 + 2.49256i −3.08553 + 0.710083i
8.17 −1.10169 − 0.934963i 0.105451 − 0.186232i 0.0142921 + 0.0867073i 1.67637 − 2.24738i −0.290293 + 0.106577i −0.755448 − 0.825004i −1.42136 + 2.36928i 1.51976 + 2.53330i −3.94806 + 0.908578i
8.18 −1.07775 − 0.914649i 1.14561 − 2.02322i −0.000303361 − 0.00184043i −1.00313 + 1.34482i −3.08522 + 1.13270i 1.57359 + 1.71848i −1.45573 + 2.42658i −1.23766 − 2.06307i 2.31117 − 0.531876i
8.19 −0.988352 − 0.838777i 1.47190 − 2.59946i −0.0519812 − 0.315359i 0.871274 − 1.16805i −3.63513 + 1.33459i −2.70677 − 2.95599i −1.54687 + 2.57850i −3.04739 − 5.07973i −1.84086 + 0.423643i
8.20 −0.857866 − 0.728038i 1.04649 − 1.84817i −0.119380 − 0.724253i 2.34201 − 3.13975i −2.24329 + 0.823592i −0.944878 − 1.03188i −1.58252 + 2.63793i −0.777253 − 1.29561i −4.29499 + 0.988419i
See next 80 embeddings (of 6100 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.61
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
751.l even 125 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 751.2.l.a ✓ 6100
751.l even 125 1 inner 751.2.l.a ✓ 6100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
751.2.l.a ✓ 6100 1.a even 1 1 trivial
751.2.l.a ✓ 6100 751.l even 125 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(751, [\chi])\).