# Properties

 Label 71.9.e.a Level $71$ Weight $9$ Character orbit 71.e Analytic conductor $28.924$ Analytic rank $0$ Dimension $188$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [71,9,Mod(14,71)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(71, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("71.14");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$71$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 71.e (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: $$28.9238813143$$ Analytic rank: $$0$$ Dimension: $$188$$ Relative dimension: $$47$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$188 q + 6 q^{2} - 73 q^{3} - 6170 q^{4} + 291 q^{5} - 4342 q^{6} - 5 q^{7} - 4231 q^{8} - 103462 q^{9}+O(q^{10})$$ 188 * q + 6 * q^2 - 73 * q^3 - 6170 * q^4 + 291 * q^5 - 4342 * q^6 - 5 * q^7 - 4231 * q^8 - 103462 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$188 q + 6 q^{2} - 73 q^{3} - 6170 q^{4} + 291 q^{5} - 4342 q^{6} - 5 q^{7} - 4231 q^{8} - 103462 q^{9} - 39605 q^{10} + 28465 q^{11} + 87226 q^{12} - 5 q^{13} - 270305 q^{14} - 273057 q^{15} - 607702 q^{16} + 267190 q^{17} - 595620 q^{18} - 164489 q^{19} + 1380836 q^{20} - 5 q^{21} + 1441620 q^{22} + 1594977 q^{24} - 3158228 q^{25} + 2482310 q^{27} + 1860770 q^{28} + 2583153 q^{29} + 780472 q^{30} - 1534580 q^{31} - 6084580 q^{32} - 5198730 q^{33} + 10938955 q^{35} - 12158488 q^{36} - 4992382 q^{37} + 305303 q^{38} - 8681861 q^{40} + 30641485 q^{42} - 9988337 q^{43} - 32861225 q^{44} - 21060496 q^{45} + 19044385 q^{46} + 23015995 q^{47} - 53495300 q^{48} + 25425264 q^{49} - 4315274 q^{50} - 45871475 q^{52} + 75873220 q^{53} + 18140888 q^{54} - 11520710 q^{55} + 42589435 q^{56} + 9074010 q^{57} + 40460737 q^{58} + 32264995 q^{59} - 158115870 q^{60} + 9916960 q^{61} - 41573135 q^{62} - 20329385 q^{63} - 173398207 q^{64} - 29584700 q^{65} + 144000740 q^{66} - 128145130 q^{67} - 164559375 q^{68} - 6139445 q^{69} + 23495669 q^{71} + 775956446 q^{72} + 162484067 q^{73} - 71216283 q^{74} + 39808021 q^{75} + 234840675 q^{76} - 257706146 q^{77} + 42725920 q^{78} + 86434027 q^{79} - 407918718 q^{80} + 89998907 q^{81} - 401066875 q^{82} + 201744669 q^{83} - 650053825 q^{84} - 238358045 q^{85} + 120450107 q^{86} + 243937047 q^{87} + 8458490 q^{88} + 121627626 q^{89} + 542716852 q^{90} + 213723098 q^{91} + 377149290 q^{92} - 564095080 q^{93} - 185902021 q^{95} + 133775573 q^{96} + 803758296 q^{98} - 824163320 q^{99}+O(q^{100})$$ 188 * q + 6 * q^2 - 73 * q^3 - 6170 * q^4 + 291 * q^5 - 4342 * q^6 - 5 * q^7 - 4231 * q^8 - 103462 * q^9 - 39605 * q^10 + 28465 * q^11 + 87226 * q^12 - 5 * q^13 - 270305 * q^14 - 273057 * q^15 - 607702 * q^16 + 267190 * q^17 - 595620 * q^18 - 164489 * q^19 + 1380836 * q^20 - 5 * q^21 + 1441620 * q^22 + 1594977 * q^24 - 3158228 * q^25 + 2482310 * q^27 + 1860770 * q^28 + 2583153 * q^29 + 780472 * q^30 - 1534580 * q^31 - 6084580 * q^32 - 5198730 * q^33 + 10938955 * q^35 - 12158488 * q^36 - 4992382 * q^37 + 305303 * q^38 - 8681861 * q^40 + 30641485 * q^42 - 9988337 * q^43 - 32861225 * q^44 - 21060496 * q^45 + 19044385 * q^46 + 23015995 * q^47 - 53495300 * q^48 + 25425264 * q^49 - 4315274 * q^50 - 45871475 * q^52 + 75873220 * q^53 + 18140888 * q^54 - 11520710 * q^55 + 42589435 * q^56 + 9074010 * q^57 + 40460737 * q^58 + 32264995 * q^59 - 158115870 * q^60 + 9916960 * q^61 - 41573135 * q^62 - 20329385 * q^63 - 173398207 * q^64 - 29584700 * q^65 + 144000740 * q^66 - 128145130 * q^67 - 164559375 * q^68 - 6139445 * q^69 + 23495669 * q^71 + 775956446 * q^72 + 162484067 * q^73 - 71216283 * q^74 + 39808021 * q^75 + 234840675 * q^76 - 257706146 * q^77 + 42725920 * q^78 + 86434027 * q^79 - 407918718 * q^80 + 89998907 * q^81 - 401066875 * q^82 + 201744669 * q^83 - 650053825 * q^84 - 238358045 * q^85 + 120450107 * q^86 + 243937047 * q^87 + 8458490 * q^88 + 121627626 * q^89 + 542716852 * q^90 + 213723098 * q^91 + 377149290 * q^92 - 564095080 * q^93 - 185902021 * q^95 + 133775573 * q^96 + 803758296 * q^98 - 824163320 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −9.55562 + 29.4092i −36.4701 + 112.243i −566.481 411.572i −865.123 + 628.549i −2952.49 2145.11i 1604.44 + 521.316i 11112.7 8073.87i −5960.56 4330.60i −10218.3 31448.7i
14.2 −9.48670 + 29.1971i −18.9267 + 58.2505i −555.363 403.495i 474.158 344.496i −1521.19 1105.21i 399.230 + 129.718i 10691.3 7767.66i 2273.07 + 1651.48i 5560.07 + 17112.2i
14.3 −9.23578 + 28.4248i 34.7597 106.979i −515.562 374.578i 556.251 404.140i 2719.84 + 1976.08i 816.157 + 265.185i 9218.94 6697.95i −4928.39 3580.69i 6350.19 + 19543.9i
14.4 −8.72538 + 26.8540i 13.7218 42.2315i −437.895 318.149i −120.642 + 87.6517i 1014.36 + 736.972i −2163.49 702.961i 6516.48 4734.50i 3712.75 + 2697.47i −1301.15 4004.52i
14.5 −8.44070 + 25.9778i 2.81292 8.65727i −396.492 288.068i −624.581 + 453.784i 201.154 + 146.147i −3601.98 1170.35i 5172.95 3758.37i 5240.92 + 3807.75i −6516.42 20055.5i
14.6 −8.09329 + 24.9086i 31.8997 98.1772i −347.828 252.712i −564.927 + 410.444i 2187.28 + 1589.15i 2666.06 + 866.254i 3685.51 2677.68i −3313.21 2407.19i −5651.45 17393.4i
14.7 −7.67748 + 23.6288i −14.9010 + 45.8605i −292.270 212.347i 443.724 322.384i −969.228 704.186i 1661.54 + 539.868i 2115.83 1537.24i 3426.81 + 2489.73i 4210.89 + 12959.8i
14.8 −7.48640 + 23.0408i −43.7983 + 134.797i −267.723 194.512i 378.287 274.842i −2777.94 2018.29i −2111.84 686.179i 1468.48 1066.91i −10944.1 7951.33i 3500.55 + 10773.6i
14.9 −6.53857 + 20.1236i 1.61600 4.97354i −155.100 112.687i −471.892 + 342.850i 89.5195 + 65.0397i 2413.84 + 784.305i −1100.46 + 799.529i 5285.84 + 3840.38i −3813.89 11737.9i
14.10 −6.45743 + 19.8739i −29.1194 + 89.6202i −146.166 106.196i −428.254 + 311.145i −1593.07 1157.43i 2120.69 + 689.055i −1273.50 + 925.251i −1875.89 1362.91i −3418.25 10520.3i
14.11 −5.93473 + 18.2652i 43.1747 132.878i −91.2892 66.3255i 146.058 106.118i 2170.82 + 1577.19i −1719.22 558.608i −2224.33 + 1616.07i −10484.6 7617.48i 1071.45 + 3297.57i
14.12 −5.83043 + 17.9442i 6.58777 20.2751i −80.8925 58.7719i 675.661 490.897i 325.410 + 236.425i −3087.36 1003.14i −2381.40 + 1730.19i 4940.28 + 3589.32i 4869.36 + 14986.4i
14.13 −5.26657 + 16.2088i 19.4523 59.8680i −27.8812 20.2569i 856.389 622.203i 867.943 + 630.598i 3680.93 + 1196.01i −3054.56 + 2219.27i 2102.18 + 1527.32i 5574.95 + 17157.9i
14.14 −4.97778 + 15.3200i −31.0907 + 95.6873i −2.81636 2.04620i −559.299 + 406.354i −1311.17 952.619i −3234.47 1050.94i −3290.82 + 2390.92i −2881.46 2093.50i −3441.29 10591.2i
14.15 −3.86234 + 11.8870i −7.77024 + 23.9144i 80.7241 + 58.6495i −308.789 + 224.348i −254.260 184.731i −143.740 46.7041i −3597.55 + 2613.78i 4796.44 + 3484.82i −1474.19 4537.10i
14.16 −3.73678 + 11.5006i 33.6182 103.466i 88.8075 + 64.5225i −962.003 + 698.936i 1064.30 + 773.260i −1774.46 576.558i −3578.35 + 2599.83i −4267.09 3100.22i −4443.41 13675.4i
14.17 −3.43660 + 10.5768i −45.3188 + 139.477i 107.051 + 77.7768i 388.730 282.429i −1319.47 958.652i 4438.96 + 1442.30i −3493.78 + 2538.38i −12092.0 8785.38i 1651.28 + 5082.10i
14.18 −2.83536 + 8.72635i 26.1649 80.5273i 138.998 + 100.988i −68.6655 + 49.8884i 628.523 + 456.648i 2633.26 + 855.599i −3175.68 + 2307.26i −492.089 357.524i −240.652 740.650i
14.19 −2.43649 + 7.49875i 16.1739 49.7781i 156.814 + 113.932i −5.52655 + 4.01527i 333.866 + 242.568i −761.080 247.290i −2869.40 + 2084.74i 3091.69 + 2246.25i −16.6441 51.2254i
14.20 −1.99278 + 6.13314i −22.7009 + 69.8663i 173.464 + 126.029i 339.186 246.433i −383.262 278.456i 653.541 + 212.348i −2454.22 + 1783.10i 941.995 + 684.399i 835.485 + 2571.36i
See next 80 embeddings (of 188 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.47 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.e odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.9.e.a 188
71.e odd 10 1 inner 71.9.e.a 188

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.9.e.a 188 1.a even 1 1 trivial
71.9.e.a 188 71.e odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(71, [\chi])$$.