# Properties

 Label 100.9.j.b Level $100$ Weight $9$ Character orbit 100.j Analytic conductor $40.738$ Analytic rank $0$ Dimension $464$ CM no Inner twists $4$

# Related objects

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

 $$\operatorname{Tr}(f)(q) =$$ $$464 q + 29 q^{2} + 509 q^{4} - 1230 q^{5} - 515 q^{6} + 21284 q^{8} + 262434 q^{9}+O(q^{10})$$ 464 * q + 29 * q^2 + 509 * q^4 - 1230 * q^5 - 515 * q^6 + 21284 * q^8 + 262434 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$464 q + 29 q^{2} + 509 q^{4} - 1230 q^{5} - 515 q^{6} + 21284 q^{8} + 262434 q^{9} + 3445 q^{10} - 35805 q^{12} - 29522 q^{13} - 33705 q^{14} - 63391 q^{16} - 319922 q^{17} - 569696 q^{18} + 1244935 q^{20} + 39360 q^{21} + 140220 q^{22} + 989900 q^{24} - 431710 q^{25} - 855842 q^{26} - 2358605 q^{28} - 2884562 q^{29} + 7229675 q^{30} - 7078666 q^{32} - 1866570 q^{33} + 1448973 q^{34} + 777029 q^{36} - 7317972 q^{37} + 4012465 q^{38} - 4154830 q^{40} + 1687438 q^{41} - 21337315 q^{42} + 510580 q^{44} + 2596050 q^{45} + 1721385 q^{46} + 41153690 q^{48} - 396068336 q^{49} - 60015715 q^{50} - 8419912 q^{52} + 26717388 q^{53} - 13729145 q^{54} + 7138890 q^{56} - 19745300 q^{57} - 41444182 q^{58} + 96768050 q^{60} + 26136558 q^{61} - 87505940 q^{62} + 19616924 q^{64} + 22030140 q^{65} - 69968200 q^{66} + 276958 q^{68} - 73697610 q^{69} + 39639840 q^{70} - 145994351 q^{72} - 39004722 q^{73} + 4922908 q^{74} - 19181080 q^{76} + 111211200 q^{77} - 197089615 q^{78} - 395680960 q^{80} - 539852286 q^{81} + 186999258 q^{82} - 377672700 q^{84} - 283475540 q^{85} - 340843785 q^{86} - 29231040 q^{88} - 206040612 q^{89} + 351285695 q^{90} + 43044280 q^{92} + 241823660 q^{93} - 442380075 q^{94} - 474931550 q^{96} + 2388718 q^{97} + 209675864 q^{98}+O(q^{100})$$ 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 + 39360 * q^21 + 140220 * q^22 + 989900 * q^24 - 431710 * q^25 - 855842 * q^26 - 2358605 * q^28 - 2884562 * q^29 + 7229675 * q^30 - 7078666 * q^32 - 1866570 * q^33 + 1448973 * q^34 + 777029 * q^36 - 7317972 * q^37 + 4012465 * q^38 - 4154830 * q^40 + 1687438 * q^41 - 21337315 * q^42 + 510580 * q^44 + 2596050 * q^45 + 1721385 * q^46 + 41153690 * q^48 - 396068336 * q^49 - 60015715 * q^50 - 8419912 * q^52 + 26717388 * q^53 - 13729145 * q^54 + 7138890 * q^56 - 19745300 * q^57 - 41444182 * q^58 + 96768050 * q^60 + 26136558 * q^61 - 87505940 * q^62 + 19616924 * q^64 + 22030140 * q^65 - 69968200 * q^66 + 276958 * q^68 - 73697610 * q^69 + 39639840 * q^70 - 145994351 * q^72 - 39004722 * q^73 + 4922908 * q^74 - 19181080 * q^76 + 111211200 * q^77 - 197089615 * q^78 - 395680960 * q^80 - 539852286 * q^81 + 186999258 * q^82 - 377672700 * q^84 - 283475540 * q^85 - 340843785 * q^86 - 29231040 * q^88 - 206040612 * q^89 + 351285695 * q^90 + 43044280 * q^92 + 241823660 * q^93 - 442380075 * q^94 - 474931550 * q^96 + 2388718 * q^97 + 209675864 * q^98

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{9}^{\mathrm{new}}(100, [\chi])$$.