# Properties

 Label 36.9.f.a Level $36$ Weight $9$ Character orbit 36.f Analytic conductor $14.666$ Analytic rank $0$ Dimension $92$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,9,Mod(7,36)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("36.7");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 36.f (of order $$6$$, degree $$2$$, 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: $$14.6656299622$$ Analytic rank: $$0$$ Dimension: $$92$$ Relative dimension: $$46$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$92 q - q^{2} - q^{4} - 2 q^{5} + 1359 q^{6} - 9958 q^{8} - 1908 q^{9}+O(q^{10})$$ 92 * q - q^2 - q^4 - 2 * q^5 + 1359 * q^6 - 9958 * q^8 - 1908 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$92 q - q^{2} - q^{4} - 2 q^{5} + 1359 q^{6} - 9958 q^{8} - 1908 q^{9} + 508 q^{10} - 25668 q^{12} - 2 q^{13} - 9348 q^{14} - q^{16} + 27544 q^{17} - 558168 q^{18} + 26260 q^{20} + 157722 q^{21} - 513 q^{22} + 36633 q^{24} - 2968752 q^{25} - 2625752 q^{26} - 131076 q^{28} - 632546 q^{29} + 3910260 q^{30} - 4303321 q^{32} + 3423594 q^{33} + 874897 q^{34} + 2094867 q^{36} - 8 q^{37} + 5021955 q^{38} - 101876 q^{40} + 3438958 q^{41} - 1253550 q^{42} + 4513722 q^{44} + 4527390 q^{45} - 8607912 q^{46} - 6410637 q^{48} + 28000460 q^{49} - 4611225 q^{50} + 4952398 q^{52} + 7237432 q^{53} + 1445961 q^{54} + 16638318 q^{56} + 27312 q^{57} - 6359024 q^{58} - 19327032 q^{60} - 2 q^{61} + 64961196 q^{62} - 11896126 q^{64} - 179134 q^{65} - 62514462 q^{66} + 37085155 q^{68} - 93014982 q^{69} + 6546306 q^{70} - 94972077 q^{72} - 32396456 q^{73} - 19652372 q^{74} - 14869203 q^{76} - 11357382 q^{77} - 58314714 q^{78} + 26901520 q^{80} - 109921044 q^{81} + 98754394 q^{82} + 48970518 q^{84} - 781252 q^{85} + 60280257 q^{86} - 15010485 q^{88} + 32208376 q^{89} - 119152692 q^{90} - 66511734 q^{92} + 127902330 q^{93} + 125494500 q^{94} - 151076052 q^{96} - 56298482 q^{97} - 17303818 q^{98}+O(q^{100})$$ 92 * q - q^2 - q^4 - 2 * q^5 + 1359 * q^6 - 9958 * q^8 - 1908 * q^9 + 508 * q^10 - 25668 * q^12 - 2 * q^13 - 9348 * q^14 - q^16 + 27544 * q^17 - 558168 * q^18 + 26260 * q^20 + 157722 * q^21 - 513 * q^22 + 36633 * q^24 - 2968752 * q^25 - 2625752 * q^26 - 131076 * q^28 - 632546 * q^29 + 3910260 * q^30 - 4303321 * q^32 + 3423594 * q^33 + 874897 * q^34 + 2094867 * q^36 - 8 * q^37 + 5021955 * q^38 - 101876 * q^40 + 3438958 * q^41 - 1253550 * q^42 + 4513722 * q^44 + 4527390 * q^45 - 8607912 * q^46 - 6410637 * q^48 + 28000460 * q^49 - 4611225 * q^50 + 4952398 * q^52 + 7237432 * q^53 + 1445961 * q^54 + 16638318 * q^56 + 27312 * q^57 - 6359024 * q^58 - 19327032 * q^60 - 2 * q^61 + 64961196 * q^62 - 11896126 * q^64 - 179134 * q^65 - 62514462 * q^66 + 37085155 * q^68 - 93014982 * q^69 + 6546306 * q^70 - 94972077 * q^72 - 32396456 * q^73 - 19652372 * q^74 - 14869203 * q^76 - 11357382 * q^77 - 58314714 * q^78 + 26901520 * q^80 - 109921044 * q^81 + 98754394 * q^82 + 48970518 * q^84 - 781252 * q^85 + 60280257 * q^86 - 15010485 * q^88 + 32208376 * q^89 - 119152692 * q^90 - 66511734 * q^92 + 127902330 * q^93 + 125494500 * q^94 - 151076052 * q^96 - 56298482 * q^97 - 17303818 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −15.9923 + 0.495568i 79.7883 + 13.9581i 255.509 15.8506i 289.572 + 501.554i −1282.92 183.682i 3250.55 + 1876.71i −4078.32 + 380.110i 6171.34 + 2227.39i −4879.49 7877.51i
7.2 −15.7236 + 2.96105i 68.3969 43.3920i 238.464 93.1167i −467.293 809.376i −946.962 + 884.805i −1883.62 1087.51i −3473.80 + 2170.24i 2795.28 5935.75i 9744.14 + 11342.6i
7.3 −15.6333 3.40608i −38.7984 71.1034i 232.797 + 106.496i −283.328 490.739i 364.361 + 1243.73i 3704.20 + 2138.62i −3276.65 2457.81i −3550.38 + 5517.39i 2757.85 + 8636.88i
7.4 −15.5239 3.87411i 35.3783 + 72.8655i 225.983 + 120.283i −163.731 283.590i −266.920 1268.22i −1882.64 1086.94i −3042.14 2742.73i −4057.76 + 5155.71i 1443.08 + 5036.73i
7.5 −15.3972 + 4.35060i −46.6263 66.2344i 218.145 133.974i 311.483 + 539.504i 1006.07 + 816.969i −2580.46 1489.83i −2775.94 + 3011.87i −2212.98 + 6176.52i −7143.12 6951.69i
7.6 −14.9567 5.68316i −61.5567 + 52.6476i 191.403 + 170.002i 593.427 + 1027.85i 1219.89 437.595i 635.064 + 366.654i −1896.60 3630.44i 1017.46 6481.63i −3034.27 18745.7i
7.7 −14.8590 + 5.93391i −4.99837 + 80.8456i 185.578 176.343i 56.7972 + 98.3756i −405.460 1230.94i −282.931 163.350i −1711.08 + 3721.48i −6511.03 808.193i −1427.70 1124.73i
7.8 −14.6491 6.43465i −78.5798 + 19.6525i 173.190 + 188.523i −381.111 660.104i 1277.58 + 217.742i −2243.02 1295.01i −1324.00 3876.11i 5788.56 3088.58i 1335.38 + 12122.2i
7.9 −14.0033 + 7.74003i −78.2105 + 21.0739i 136.184 216.772i −139.812 242.161i 932.092 900.456i 1136.46 + 656.134i −229.201 + 4089.58i 5672.78 3296.41i 3832.16 + 2308.90i
7.10 −13.5660 8.48314i 33.4010 73.7928i 112.073 + 230.165i 264.138 + 457.500i −1079.11 + 717.726i −1060.07 612.033i 432.142 4073.14i −4329.74 4929.51i 297.745 8447.16i
7.11 −10.9565 + 11.6600i 36.5074 72.3064i −15.9093 255.505i 97.4304 + 168.754i 443.096 + 1217.90i 1574.95 + 909.296i 3153.49 + 2613.95i −3895.43 5279.43i −3035.17 712.926i
7.12 −10.4281 + 12.1349i 77.1237 + 24.7575i −38.5113 253.087i 467.991 + 810.585i −1104.68 + 677.716i −2961.68 1709.92i 3472.78 + 2171.87i 5335.14 + 3818.77i −14716.6 2773.80i
7.13 −10.4259 12.1367i 80.0766 + 12.1960i −38.5996 + 253.073i −83.1442 144.010i −686.854 1099.02i −636.059 367.229i 3473.91 2170.05i 6263.51 + 1953.23i −880.951 + 2510.54i
7.14 −9.81578 12.6353i −8.98529 + 80.5001i −63.3010 + 248.050i −228.834 396.352i 1105.34 676.639i 3521.45 + 2033.11i 3755.54 1634.98i −6399.53 1446.63i −2761.84 + 6781.89i
7.15 −9.20735 + 13.0853i 55.4291 + 59.0645i −86.4493 240.962i −491.818 851.853i −1283.23 + 181.477i 2255.09 + 1301.97i 3949.02 + 1087.40i −416.237 + 6547.78i 15675.1 + 1407.74i
7.16 −7.57892 14.0911i −72.9555 35.1923i −141.120 + 213.591i 134.666 + 233.248i 57.0253 + 1294.74i −62.1498 35.8822i 4079.28 + 369.747i 4084.01 + 5134.94i 2266.10 3665.36i
7.17 −6.72851 + 14.5164i −55.4291 59.0645i −165.454 195.348i −491.818 851.853i 1230.36 407.216i −2255.09 1301.97i 3949.02 1087.40i −416.237 + 6547.78i 15675.1 1407.74i
7.18 −5.29510 + 15.0984i −77.1237 24.7575i −199.924 159.895i 467.991 + 810.585i 782.176 1033.35i 2961.68 + 1709.92i 3472.78 2171.87i 5335.14 + 3818.77i −14716.6 + 2773.80i
7.19 −4.61956 + 15.3186i −36.5074 + 72.3064i −213.319 141.530i 97.4304 + 168.754i −938.985 893.266i −1574.95 909.296i 3153.49 2613.95i −3895.43 5279.43i −3035.17 + 712.926i
7.20 −4.46537 15.3643i 19.6627 78.5772i −216.121 + 137.214i −517.737 896.746i −1295.08 + 48.7744i −28.8505 16.6568i 3073.26 + 2707.82i −5787.76 3090.07i −11466.0 + 11958.9i
See all 92 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.46 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
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.9.f.a 92
3.b odd 2 1 108.9.f.a 92
4.b odd 2 1 inner 36.9.f.a 92
9.c even 3 1 inner 36.9.f.a 92
9.d odd 6 1 108.9.f.a 92
12.b even 2 1 108.9.f.a 92
36.f odd 6 1 inner 36.9.f.a 92
36.h even 6 1 108.9.f.a 92

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.9.f.a 92 1.a even 1 1 trivial
36.9.f.a 92 4.b odd 2 1 inner
36.9.f.a 92 9.c even 3 1 inner
36.9.f.a 92 36.f odd 6 1 inner
108.9.f.a 92 3.b odd 2 1
108.9.f.a 92 9.d odd 6 1
108.9.f.a 92 12.b even 2 1
108.9.f.a 92 36.h even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(36, [\chi])$$.