Properties

Label 100.9.h.a
Level $100$
Weight $9$
Character orbit 100.h
Analytic conductor $40.738$
Analytic rank $0$
Dimension $472$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,9,Mod(19,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.h (of order \(10\), 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: \(40.7378610061\)
Analytic rank: \(0\)
Dimension: \(472\)
Relative dimension: \(118\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 472 q - 5 q^{2} - 3 q^{4} + 160 q^{5} + 509 q^{6} - 21830 q^{8} - 249324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 472 q - 5 q^{2} - 3 q^{4} + 160 q^{5} + 509 q^{6} - 21830 q^{8} - 249324 q^{9} + 4155 q^{10} - 5 q^{12} - 10 q^{13} - 35241 q^{14} + 194457 q^{16} - 10 q^{17} - 315765 q^{20} - 39372 q^{21} - 674630 q^{22} + 1016144 q^{24} - 124000 q^{25} + 840530 q^{26} - 327685 q^{28} - 68886 q^{29} - 2410045 q^{30} - 10 q^{33} - 1593235 q^{34} + 2844341 q^{36} + 7566990 q^{37} + 10916995 q^{38} + 1665050 q^{40} + 5468394 q^{41} - 36609835 q^{42} + 510580 q^{44} - 4274700 q^{45} - 1721391 q^{46} + 27274970 q^{48} + 348414536 q^{49} + 4672165 q^{50} - 92260780 q^{52} + 15636590 q^{53} - 13767487 q^{54} - 7138896 q^{56} + 105499870 q^{58} - 88218440 q^{60} + 15307594 q^{61} - 82821380 q^{62} - 13937508 q^{64} - 77435650 q^{65} + 63249730 q^{66} - 73645122 q^{69} - 20098010 q^{70} + 238689175 q^{72} - 10 q^{73} - 24707620 q^{74} + 18918920 q^{76} - 57648020 q^{77} + 273881375 q^{78} + 340504710 q^{80} - 349779960 q^{81} - 412235262 q^{84} + 56543390 q^{85} + 340843779 q^{86} + 756329470 q^{88} - 296838366 q^{89} + 660374295 q^{90} + 85285570 q^{92} - 507069831 q^{94} + 640257404 q^{96} + 656715990 q^{97} - 285677340 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −15.9981 + 0.246066i −51.5769 + 37.4728i 255.879 7.87319i 307.753 543.979i 815.912 612.185i −2034.31 −4091.64 + 188.919i −771.496 + 2374.42i −4789.60 + 8778.37i
19.2 −15.9944 0.424130i 13.6982 9.95232i 255.640 + 13.5674i −549.088 + 298.542i −223.315 + 153.371i −3961.64 −4083.05 325.427i −1938.87 + 5967.22i 8908.94 4542.11i
19.3 −15.9857 + 0.675350i −65.2708 + 47.4220i 255.088 21.5919i −41.8229 623.599i 1011.38 802.157i 1732.27 −4063.19 + 517.437i −16.0281 + 49.3293i 1089.72 + 9940.45i
19.4 −15.9643 + 1.06805i 57.9518 42.1045i 253.719 34.1014i 577.872 238.094i −880.191 + 734.064i −3732.20 −4014.02 + 815.391i −441.834 + 1359.82i −8971.03 + 4418.21i
19.5 −15.9343 + 1.44876i −3.51845 + 2.55630i 251.802 46.1700i 410.473 + 471.314i 52.3604 45.8302i 2742.72 −3945.40 + 1100.49i −2021.62 + 6221.89i −7223.41 6915.37i
19.6 −15.9057 + 1.73442i −124.836 + 90.6984i 249.984 55.1744i 547.501 + 301.442i 1828.29 1659.14i −453.886 −3880.47 + 1311.17i 5330.27 16404.9i −9231.22 3845.06i
19.7 −15.8025 2.50649i 77.7929 56.5198i 243.435 + 79.2174i 583.325 224.404i −1370.98 + 698.165i 3405.98 −3648.31 1862.00i 829.778 2553.80i −9780.43 + 2084.04i
19.8 −15.6849 + 3.15953i 89.7914 65.2373i 236.035 99.1141i −309.934 + 542.739i −1202.25 + 1306.94i 1411.69 −3389.04 + 2300.36i 1779.14 5475.62i 3146.50 9492.08i
19.9 −15.6574 + 3.29321i 58.5755 42.5576i 234.309 103.126i −541.627 311.874i −776.990 + 859.244i 859.672 −3329.06 + 2386.33i −407.520 + 1254.22i 9507.54 + 3099.45i
19.10 −15.6545 3.30689i −102.426 + 74.4165i 234.129 + 103.536i −591.033 + 203.236i 1849.51 826.246i 4397.38 −3322.80 2395.04i 2925.72 9004.44i 9924.43 1227.08i
19.11 −15.6006 3.55248i 125.792 91.3930i 230.760 + 110.842i −264.332 566.351i −2287.10 + 978.917i −1245.78 −3206.24 2548.97i 5443.41 16753.1i 2111.80 + 9774.47i
19.12 −15.4366 4.20867i −37.4338 + 27.1973i 220.574 + 129.935i −61.0006 + 622.016i 692.313 262.285i −426.852 −2858.05 2934.07i −1365.86 + 4203.69i 3559.50 9345.05i
19.13 −14.7384 6.22740i −4.07905 + 2.96360i 178.439 + 183.563i −605.190 156.108i 78.5741 18.2768i −138.991 −1486.78 3816.64i −2019.60 + 6215.70i 7947.37 + 6069.54i
19.14 −14.6028 + 6.53893i −58.5755 + 42.5576i 170.485 190.974i −541.627 311.874i 577.086 1004.48i −859.672 −1240.79 + 3903.54i −407.520 + 1254.22i 9948.60 + 1012.58i
19.15 −14.5465 + 6.66326i −89.7914 + 65.2373i 167.202 193.854i −309.934 + 542.739i 871.459 1547.28i −1411.69 −1140.50 + 3934.02i 1779.14 5475.62i 892.048 9960.13i
19.16 −14.3707 7.03434i −76.3834 + 55.4958i 157.036 + 202.177i 620.511 + 74.7731i 1488.06 260.208i −555.761 −834.539 4010.08i 727.184 2238.04i −8391.22 5439.43i
19.17 −14.3520 7.07252i −115.709 + 84.0675i 155.959 + 203.009i −457.366 425.959i 2255.22 388.181i −4711.70 −802.528 4016.61i 4293.77 13214.9i 3551.51 + 9348.09i
19.18 −14.3290 7.11888i 86.2065 62.6327i 154.643 + 204.013i 289.418 + 553.952i −1681.13 + 283.774i −2308.57 −763.544 4024.20i 1481.25 4558.82i −203.568 9997.93i
19.19 −14.3070 7.16304i 23.5532 17.1124i 153.382 + 204.963i −83.4753 619.400i −459.553 + 76.1151i 2418.92 −726.276 4031.10i −1765.54 + 5433.78i −3242.50 + 9459.71i
19.20 −13.8875 + 7.94597i 124.836 90.6984i 129.723 220.699i 547.501 + 301.442i −1012.96 + 2251.51i 453.886 −47.8608 + 4095.72i 5330.27 16404.9i −9998.65 + 164.156i
See next 80 embeddings (of 472 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.118
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.h.a 472
4.b odd 2 1 inner 100.9.h.a 472
25.e even 10 1 inner 100.9.h.a 472
100.h odd 10 1 inner 100.9.h.a 472
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.h.a 472 1.a even 1 1 trivial
100.9.h.a 472 4.b odd 2 1 inner
100.9.h.a 472 25.e even 10 1 inner
100.9.h.a 472 100.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(100, [\chi])\).