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Properties

Label 100.9.j.b
Level $100$
Weight $9$
Character orbit 100.j
Analytic conductor $40.738$
Analytic rank $0$
Dimension $464$
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(11,100)]
 
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, 8]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.11");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.j (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: \(464\)
Relative dimension: \(116\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 464 q + 29 q^{2} + 509 q^{4} - 1230 q^{5} - 515 q^{6} + 21284 q^{8} + 262434 q^{9} + 3445 q^{10} - 35805 q^{12} - 29522 q^{13} - 33705 q^{14} - 63391 q^{16} - 319922 q^{17} - 569696 q^{18} + 1244935 q^{20}+ \cdots + 209675864 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −15.9990 − 0.175160i 65.8158 + 21.3848i 255.939 + 5.60479i −615.488 − 108.624i −1049.24 − 353.665i − 2056.59i −4093.79 − 134.501i −1433.56 − 1041.54i 9828.20 + 1845.69i
11.2 −15.9922 + 0.498717i −80.1941 − 26.0566i 255.503 − 15.9512i 545.455 − 305.129i 1295.48 + 376.709i 1126.51i −4078.10 + 382.518i 444.182 + 322.717i −8570.87 + 5151.72i
11.3 −15.9466 + 1.30632i 25.9701 + 8.43819i 252.587 − 41.6625i 530.105 + 331.079i −425.157 − 100.635i − 1845.30i −3973.48 + 994.333i −4704.72 − 3418.18i −8885.86 − 4587.10i
11.4 −15.8505 − 2.18212i −30.1760 − 9.80478i 246.477 + 69.1753i 134.651 + 610.323i 456.910 + 221.258i 4004.03i −3755.83 − 1634.30i −4493.50 − 3264.72i −802.492 − 9967.75i
11.5 −15.7643 − 2.73604i 14.5454 + 4.72608i 241.028 + 86.2638i −307.256 − 544.260i −216.368 − 114.300i 2038.02i −3563.62 − 2019.35i −5118.73 − 3718.97i 3354.57 + 9420.56i
11.6 −15.6431 − 3.36037i −50.1034 − 16.2796i 233.416 + 105.133i −559.176 + 279.190i 729.069 + 423.029i − 3486.05i −3298.07 − 2428.98i −3062.64 − 2225.14i 9685.45 − 2488.38i
11.7 −15.6202 + 3.46547i −118.336 − 38.4497i 231.981 − 108.263i −243.760 − 575.505i 1981.68 + 190.502i − 2346.07i −3248.41 + 2495.01i 7217.08 + 5243.52i 5801.98 + 8144.75i
11.8 −15.4734 + 4.07127i 127.949 + 41.5732i 222.850 − 125.992i −607.616 + 146.383i −2149.06 − 122.362i 3595.08i −2935.28 + 2856.80i 9334.70 + 6782.05i 8805.89 − 4738.81i
11.9 −15.4497 + 4.16030i 122.140 + 39.6856i 221.384 − 128.550i 139.543 − 609.223i −2052.12 − 104.991i − 4148.65i −2885.49 + 2907.08i 8035.23 + 5837.94i 378.665 + 9992.83i
11.10 −15.4042 + 4.32544i 92.0404 + 29.9058i 218.581 − 133.260i 397.875 − 481.997i −1547.17 − 62.5602i 2081.70i −2790.67 + 2998.23i 2269.13 + 1648.62i −4044.11 + 9145.77i
11.11 −15.2729 − 4.76863i 116.175 + 37.7477i 210.520 + 145.661i 612.823 + 122.770i −1594.33 − 1130.51i 1558.62i −2520.64 − 3228.56i 6763.88 + 4914.25i −8774.12 − 4797.37i
11.12 −15.2304 − 4.90247i −121.370 − 39.4356i 207.932 + 149.333i 300.549 + 547.992i 1655.19 + 1195.64i − 2537.28i −2434.78 − 3293.79i 7867.65 + 5716.18i −1890.97 − 9819.58i
11.13 −15.1733 + 5.07641i 66.9566 + 21.7555i 204.460 − 154.052i −7.73474 + 624.952i −1126.40 + 9.79525i − 539.637i −2320.31 + 3375.41i −1298.07 − 943.105i −3055.15 − 9521.87i
11.14 −15.1302 + 5.20358i −63.1326 − 20.5130i 201.845 − 157.462i −339.833 + 524.536i 1061.95 − 18.1497i − 868.907i −2234.59 + 3432.76i −1743.02 − 1266.38i 2412.27 − 9704.69i
11.15 −14.9394 − 5.72837i −148.141 − 48.1341i 190.372 + 171.157i −603.811 + 161.359i 1937.42 + 1567.70i 2832.14i −1863.59 − 3647.50i 14321.0 + 10404.8i 9944.91 + 1048.25i
11.16 −14.9112 − 5.80137i 113.891 + 37.0055i 188.688 + 173.011i −151.256 + 606.421i −1483.57 − 1212.52i − 1292.83i −1809.87 − 3674.45i 6293.85 + 4572.75i 5773.49 − 8164.98i
11.17 −14.5364 + 6.68529i −78.1238 − 25.3840i 166.614 − 194.360i −624.998 − 1.49949i 1305.34 − 153.289i 3856.33i −1122.61 + 3939.16i 151.027 + 109.727i 9095.25 − 4156.50i
11.18 −14.2154 − 7.34320i 23.9209 + 7.77239i 148.155 + 208.773i 443.111 − 440.770i −282.972 − 286.144i − 2979.31i −573.020 − 4055.72i −4796.16 − 3484.61i −9535.65 + 3011.87i
11.19 −13.9481 + 7.83900i −143.753 − 46.7082i 133.100 − 218.679i 449.832 + 433.908i 2371.23 − 475.388i 1240.30i −142.279 + 4093.53i 13175.3 + 9572.44i −9675.71 − 2525.98i
11.20 −13.7288 − 8.21702i −101.474 − 32.9708i 120.961 + 225.620i −155.142 − 605.439i 1122.19 + 1286.46i − 328.280i 193.269 − 4091.44i 3901.90 + 2834.89i −2844.99 + 9586.76i
See next 80 embeddings (of 464 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.d even 5 1 inner
100.j 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.j.b ✓ 464
4.b odd 2 1 inner 100.9.j.b ✓ 464
25.d even 5 1 inner 100.9.j.b ✓ 464
100.j odd 10 1 inner 100.9.j.b ✓ 464
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.j.b ✓ 464 1.a even 1 1 trivial
100.9.j.b ✓ 464 4.b odd 2 1 inner
100.9.j.b ✓ 464 25.d even 5 1 inner
100.9.j.b ✓ 464 100.j odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{464} - 511755 T_{3}^{462} + 134869684215 T_{3}^{460} + \cdots + 69\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display