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Properties

Label 1000.2.bh.a
Level $1000$
Weight $2$
Character orbit 1000.bh
Analytic conductor $7.985$
Analytic rank $0$
Dimension $5920$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1000,2,Mod(3,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(100))
 
chi = DirichletCharacter(H, H._module([50, 50, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.bh (of order \(100\), degree \(40\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(5920\)
Relative dimension: \(148\) over \(\Q(\zeta_{100})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{100}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 5920 q - 40 q^{2} - 80 q^{3} - 40 q^{4} - 40 q^{6} - 40 q^{8} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 5920 q - 40 q^{2} - 80 q^{3} - 40 q^{4} - 40 q^{6} - 40 q^{8} - 80 q^{9} - 40 q^{10} - 80 q^{11} - 40 q^{12} - 40 q^{14} - 40 q^{16} - 80 q^{17} - 40 q^{18} - 80 q^{19} - 40 q^{20} - 40 q^{22} - 50 q^{24} - 80 q^{25} - 30 q^{26} - 80 q^{27} - 40 q^{28} - 40 q^{30} - 40 q^{32} - 80 q^{33} - 40 q^{34} - 80 q^{35} - 40 q^{36} - 10 q^{38} - 90 q^{40} - 80 q^{41} + 60 q^{42} - 80 q^{43} - 110 q^{44} - 40 q^{46} - 170 q^{48} - 100 q^{49} + 50 q^{50} - 60 q^{51} - 130 q^{52} + 90 q^{54} - 40 q^{56} - 80 q^{57} + 30 q^{58} - 80 q^{59} - 140 q^{60} + 10 q^{62} - 70 q^{64} - 80 q^{65} - 40 q^{66} - 80 q^{67} - 40 q^{68} - 40 q^{70} - 70 q^{72} - 80 q^{73} - 50 q^{74} - 80 q^{75} - 30 q^{76} - 10 q^{78} - 40 q^{80} - 80 q^{81} - 80 q^{82} - 80 q^{83} - 120 q^{84} - 40 q^{86} - 160 q^{88} + 20 q^{89} - 190 q^{90} - 80 q^{91} - 140 q^{92} - 200 q^{94} - 40 q^{96} - 160 q^{97} - 200 q^{98} - 100 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.41421 0.000952239i 0.0994163 + 3.16348i 2.00000 + 0.00269334i −0.374276 + 2.20452i −0.137583 − 4.47393i 0.320216 + 2.02177i −2.82842 − 0.00571343i −7.00363 + 0.440631i 0.531406 − 3.11731i
3.2 −1.41411 − 0.0173282i 0.0604890 + 1.92479i 1.99940 + 0.0490080i 1.70313 + 1.44891i −0.0521847 − 2.72291i −0.402600 − 2.54192i −2.82652 − 0.103949i −0.707084 + 0.0444860i −2.38331 − 2.07842i
3.3 −1.41227 − 0.0741865i 0.0737003 + 2.34518i 1.98899 + 0.209542i −2.22284 + 0.242859i 0.0698963 − 3.31749i −0.342182 − 2.16045i −2.79344 − 0.443486i −2.50036 + 0.157309i 3.15726 − 0.178077i
3.4 −1.41206 + 0.0780424i −0.0996721 − 3.17162i 1.98782 − 0.220401i −2.14902 + 0.617835i 0.388264 + 4.47073i 0.430346 + 2.71710i −2.78972 + 0.466353i −7.05515 + 0.443872i 2.98632 − 1.04013i
3.5 −1.40896 + 0.121780i −0.0389197 − 1.23844i 1.97034 − 0.343166i 1.66003 + 1.49810i 0.205654 + 1.74018i 0.616340 + 3.89142i −2.73434 + 0.723455i 1.46185 − 0.0919718i −2.52135 − 1.90861i
3.6 −1.40887 + 0.122813i 0.0552964 + 1.75956i 1.96983 − 0.346055i −0.327340 − 2.21198i −0.294002 − 2.47220i 0.788680 + 4.97953i −2.73274 + 0.729468i −0.0989170 + 0.00622333i 0.732839 + 3.07619i
3.7 −1.40620 + 0.150299i −0.0530264 − 1.68733i 1.95482 − 0.422702i −0.963350 + 2.01791i 0.328169 + 2.36476i −0.635675 − 4.01349i −2.68535 + 0.888212i 0.149823 − 0.00942604i 1.05138 − 2.98238i
3.8 −1.40550 + 0.156792i 0.00326581 + 0.103920i 1.95083 − 0.440740i −2.02214 − 0.954433i −0.0208838 − 0.145546i 0.125010 + 0.789279i −2.67278 + 0.925333i 2.98329 − 0.187693i 2.99176 + 1.02440i
3.9 −1.40288 − 0.178679i −0.00145641 − 0.0463436i 1.93615 + 0.501330i 1.83463 − 1.27833i −0.00623744 + 0.0652747i 0.125018 + 0.789333i −2.62661 − 1.04925i 2.99193 − 0.188237i −2.80218 + 1.46553i
3.10 −1.40145 − 0.189597i 0.000749809 0.0238593i 1.92811 + 0.531419i 1.24257 − 1.85904i 0.00347282 − 0.0335797i −0.659658 − 4.16492i −2.60138 − 1.11032i 2.99351 − 0.188336i −2.09386 + 2.36975i
3.11 −1.38431 − 0.289274i −0.0828407 − 2.63603i 1.83264 + 0.800891i −0.600574 − 2.15391i −0.647859 + 3.67306i 0.0521515 + 0.329271i −2.30527 − 1.63882i −3.94773 + 0.248370i 0.208312 + 3.15541i
3.12 −1.37573 − 0.327668i −0.0828407 − 2.63603i 1.78527 + 0.901567i 0.600574 + 2.15391i −0.749779 + 3.65362i −0.0521515 − 0.329271i −2.16063 − 1.82529i −3.94773 + 0.248370i −0.120460 − 3.15998i
3.13 −1.35187 + 0.415257i −0.0852496 − 2.71269i 1.65512 − 1.12275i 1.99517 − 1.00960i 1.24171 + 3.63181i 0.303886 + 1.91866i −1.77129 + 2.20512i −4.35731 + 0.274139i −2.27797 + 2.19336i
3.14 −1.34879 − 0.425155i 0.000749809 0.0238593i 1.63849 + 1.14689i −1.24257 + 1.85904i 0.00913256 − 0.0325000i 0.659658 + 4.16492i −1.72237 − 2.24353i 2.99351 − 0.188336i 2.46635 − 1.97917i
3.15 −1.34544 − 0.435644i −0.00145641 − 0.0463436i 1.62043 + 1.17227i −1.83463 + 1.27833i −0.0182298 + 0.0629871i −0.125018 − 0.789333i −1.66950 − 2.28315i 2.99193 − 0.188237i 3.02529 − 0.920667i
3.16 −1.32024 + 0.506911i −0.0342323 − 1.08929i 1.48608 − 1.33849i −2.01493 − 0.969569i 0.597369 + 1.42077i −0.0361753 − 0.228402i −1.28349 + 2.52045i 1.80870 − 0.113794i 3.15168 + 0.258676i
3.17 −1.31150 + 0.529112i −0.0493271 − 1.56961i 1.44008 − 1.38786i 2.22012 + 0.266617i 0.895193 + 2.03245i −0.651315 − 4.11224i −1.15434 + 2.58215i 0.532829 − 0.0335228i −3.05276 + 0.825022i
3.18 −1.30944 − 0.534188i 0.0737003 + 2.34518i 1.42929 + 1.39898i 2.22284 − 0.242859i 1.15626 − 3.11025i 0.342182 + 2.16045i −1.12425 − 2.59539i −2.50036 + 0.157309i −3.04042 − 0.869404i
3.19 −1.30245 + 0.551019i 0.0773283 + 2.46063i 1.39276 − 1.43535i −0.0558491 − 2.23537i −1.45657 − 3.16224i −0.0576180 − 0.363786i −1.02309 + 2.63691i −3.05463 + 0.192181i 1.30447 + 2.88069i
3.20 −1.28799 + 0.584031i 0.0381159 + 1.21287i 1.31782 − 1.50445i −0.820694 + 2.08001i −0.757444 − 1.53989i 0.258237 + 1.63044i −0.818685 + 2.70735i 1.52449 − 0.0959126i −0.157751 − 3.15834i
See next 80 embeddings (of 5920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.148
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
125.i odd 100 1 inner
1000.bh even 100 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.bh.a ✓ 5920
8.d odd 2 1 inner 1000.2.bh.a ✓ 5920
125.i odd 100 1 inner 1000.2.bh.a ✓ 5920
1000.bh even 100 1 inner 1000.2.bh.a ✓ 5920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.bh.a ✓ 5920 1.a even 1 1 trivial
1000.2.bh.a ✓ 5920 8.d odd 2 1 inner
1000.2.bh.a ✓ 5920 125.i odd 100 1 inner
1000.2.bh.a ✓ 5920 1000.bh even 100 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1000, [\chi])\).