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Properties

Label 100.7.h.b
Level $100$
Weight $7$
Character orbit 100.h
Analytic conductor $23.005$
Analytic rank $0$
Dimension $344$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(23.0054083620\)
Analytic rank: \(0\)
Dimension: \(344\)
Relative dimension: \(86\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 344 q - 5 q^{2} - 131 q^{4} - 220 q^{5} + 125 q^{6} + 2410 q^{8} - 21876 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 344 q - 5 q^{2} - 131 q^{4} - 220 q^{5} + 125 q^{6} + 2410 q^{8} - 21876 q^{9} - 2005 q^{10} - 5 q^{12} - 10 q^{13} + 5415 q^{14} - 4231 q^{16} - 10 q^{17} - 45125 q^{20} - 4380 q^{21} + 55570 q^{22} - 1840 q^{24} + 24440 q^{25} - 23982 q^{26} - 20485 q^{28} - 82222 q^{29} + 208595 q^{30} - 10 q^{33} + 179053 q^{34} + 239909 q^{36} - 454110 q^{37} + 351115 q^{38} + 396570 q^{40} + 305578 q^{41} - 44635 q^{42} - 221660 q^{44} + 358920 q^{45} + 237425 q^{46} - 436870 q^{48} + 5858744 q^{49} + 389285 q^{50} + 740980 q^{52} + 223210 q^{53} + 865985 q^{54} - 751920 q^{56} - 1265450 q^{58} - 1622600 q^{60} - 793582 q^{61} - 591180 q^{62} + 974524 q^{64} - 1212410 q^{65} - 2203070 q^{66} + 798990 q^{69} - 2166330 q^{70} - 2930585 q^{72} - 10 q^{73} + 389068 q^{74} - 1628200 q^{76} - 1176500 q^{77} - 3470545 q^{78} - 3523250 q^{80} - 3837816 q^{81} + 830850 q^{84} + 5066810 q^{85} - 756045 q^{86} - 2480450 q^{88} - 3420462 q^{89} - 2355705 q^{90} - 1403470 q^{92} + 2949305 q^{94} - 621100 q^{96} - 5338210 q^{97} + 1400900 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −7.99326 − 0.328251i −2.50917 + 1.82302i 63.7845 + 5.24759i −79.2577 + 96.6603i 20.6548 − 13.7482i 224.379 −508.124 − 62.8827i −222.301 + 684.172i 665.256 − 746.615i
19.2 −7.97624 + 0.616114i 29.7556 − 21.6187i 63.2408 − 9.82855i 2.12630 + 124.982i −224.018 + 190.769i −343.672 −498.368 + 117.358i 192.753 − 593.233i −93.9630 − 995.576i
19.3 −7.95853 + 0.813523i 4.33404 − 3.14887i 62.6764 − 12.9489i −53.8177 − 112.821i −31.9309 + 28.5862i 573.695 −488.277 + 154.043i −216.405 + 666.026i 520.092 + 854.110i
19.4 −7.94276 − 0.955268i 23.1960 − 16.8529i 62.1749 + 15.1749i 94.9684 − 81.2773i −200.340 + 111.700i −66.8084 −479.345 − 179.925i 28.7624 − 88.5216i −831.953 + 554.846i
19.5 −7.89677 + 1.28105i −32.1986 + 23.3937i 60.7178 − 20.2324i −124.901 − 4.97452i 224.296 − 225.983i −206.736 −453.555 + 237.553i 264.215 − 813.169i 992.686 − 120.722i
19.6 −7.87830 + 1.39011i −40.1979 + 29.2055i 60.1352 − 21.9034i 54.8220 − 112.337i 276.092 − 285.969i 176.283 −443.315 + 256.156i 537.637 − 1654.68i −275.743 + 961.231i
19.7 −7.84451 − 1.56959i −24.8814 + 18.0774i 59.0728 + 24.6253i 43.0431 + 117.355i 223.556 − 102.755i −573.312 −424.746 − 285.894i 67.0177 − 206.259i −153.452 − 988.156i
19.8 −7.59346 − 2.51780i 35.8633 − 26.0562i 51.3213 + 38.2377i 88.5385 + 88.2380i −337.931 + 107.560i 474.323 −293.432 − 419.573i 381.977 − 1175.60i −450.148 − 892.954i
19.9 −7.53152 − 2.69743i 23.6846 − 17.2079i 49.4477 + 40.6316i −120.316 − 33.8961i −224.799 + 65.7139i −407.003 −262.815 − 439.400i 39.5770 − 121.805i 814.734 + 579.835i
19.10 −7.48625 − 2.82064i −13.6684 + 9.93066i 48.0880 + 42.2320i 60.6798 − 109.284i 130.336 − 35.7899i −277.486 −240.878 − 451.798i −137.067 + 421.848i −762.515 + 646.971i
19.11 −7.43740 − 2.94704i −27.8191 + 20.2118i 46.6299 + 43.8367i 113.193 + 53.0324i 266.467 − 68.3389i 548.049 −217.616 − 463.451i 140.114 − 431.227i −685.570 − 728.007i
19.12 −7.19076 + 3.50613i 40.1979 − 29.2055i 39.4142 − 50.4234i 54.8220 − 112.337i −186.655 + 350.949i −176.283 −106.627 + 500.774i 537.637 − 1654.68i −0.344831 + 1000.00i
19.13 −7.14160 + 3.60521i 32.1986 − 23.3937i 38.0050 − 51.4939i −124.901 − 4.97452i −145.611 + 283.151i 206.736 −85.7699 + 504.765i 264.215 − 813.169i 909.927 − 414.768i
19.14 −6.91676 + 4.01975i −4.33404 + 3.14887i 31.6832 − 55.6073i −53.8177 − 112.821i 17.3199 − 39.2018i −573.695 4.38264 + 511.981i −216.405 + 666.026i 825.758 + 564.025i
19.15 −6.86014 − 4.11564i −19.9401 + 14.4873i 30.1230 + 56.4677i −103.331 − 70.3393i 196.417 − 17.3188i 106.192 25.7532 − 511.352i −37.5482 + 115.562i 419.377 + 907.812i
19.16 −6.81506 + 4.18987i −29.7556 + 21.6187i 28.8900 − 57.1084i 2.12630 + 124.982i 112.206 − 272.005i 343.672 42.3898 + 510.242i 192.753 − 593.233i −538.149 − 842.850i
19.17 −6.27374 + 4.96388i 2.50917 − 1.82302i 14.7197 − 62.2843i −79.2577 + 96.6603i −6.69263 + 23.8923i −224.379 216.824 + 463.823i −222.301 + 684.172i 17.4318 − 999.848i
19.18 −6.21292 − 5.03981i 8.61023 − 6.25570i 13.2007 + 62.6238i −73.7400 + 100.933i −85.0222 − 4.52777i 305.561 233.597 − 455.605i −190.271 + 585.594i 966.822 − 255.451i
19.19 −5.86434 + 5.44147i −23.1960 + 16.8529i 4.78089 − 63.8212i 94.9684 − 81.2773i 44.3249 − 225.052i 66.8084 319.244 + 400.284i 28.7624 − 88.5216i −114.659 + 993.405i
19.20 −5.82004 − 5.48882i 13.5229 − 9.82497i 3.74564 + 63.8903i 124.573 + 10.3281i −132.631 − 17.0432i −208.013 328.883 − 392.403i −138.934 + 427.596i −668.328 − 743.867i
See next 80 embeddings (of 344 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.86
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 100.7.h.b ✓ 344
4.b odd 2 1 inner 100.7.h.b ✓ 344
25.e even 10 1 inner 100.7.h.b ✓ 344
100.h odd 10 1 inner 100.7.h.b ✓ 344
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.7.h.b ✓ 344 1.a even 1 1 trivial
100.7.h.b ✓ 344 4.b odd 2 1 inner
100.7.h.b ✓ 344 25.e even 10 1 inner
100.7.h.b ✓ 344 100.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{344} + 42285 T_{3}^{342} + 929777265 T_{3}^{340} + 14160815506495 T_{3}^{338} + \cdots + 36\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display