# Properties

 Label 95.11.h.a Level $95$ Weight $11$ Character orbit 95.h Analytic conductor $60.359$ Analytic rank $0$ Dimension $196$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,11,Mod(69,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("95.69");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 95.h (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: $$60.3589390040$$ Analytic rank: $$0$$ Dimension: $$196$$ Relative dimension: $$98$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$196 q - 48544 q^{4} - 1112 q^{5} + 6524 q^{6} - 1850204 q^{9}+O(q^{10})$$ 196 * q - 48544 * q^4 - 1112 * q^5 + 6524 * q^6 - 1850204 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$196 q - 48544 q^{4} - 1112 q^{5} + 6524 q^{6} - 1850204 q^{9} - 314796 q^{10} - 213068 q^{11} + 3607068 q^{14} + 465384 q^{15} - 24515188 q^{16} + 4733396 q^{19} + 19628028 q^{20} + 2526708 q^{21} + 25784146 q^{24} + 2396384 q^{25} - 66927172 q^{26} + 51260694 q^{29} - 100952916 q^{30} + 142070514 q^{34} - 51319676 q^{35} - 829288052 q^{36} - 384252652 q^{39} + 875823864 q^{40} - 329731386 q^{41} + 966857082 q^{44} + 457201812 q^{45} - 7045011312 q^{49} + 2434599804 q^{51} + 2592949836 q^{54} + 415896124 q^{55} - 4872445266 q^{59} - 2319080274 q^{60} + 1636469048 q^{61} + 17884092636 q^{64} + 4107010578 q^{66} - 12384717444 q^{70} - 9591268956 q^{71} - 9770187120 q^{74} - 9402631914 q^{76} + 9876538494 q^{79} - 9138687352 q^{80} - 15506290458 q^{81} - 3743847174 q^{85} - 21594282036 q^{86} + 18798288204 q^{89} + 27226287060 q^{90} - 27553181472 q^{91} + 26371782018 q^{95} - 55043776340 q^{96} - 3826747138 q^{99}+O(q^{100})$$ 196 * q - 48544 * q^4 - 1112 * q^5 + 6524 * q^6 - 1850204 * q^9 - 314796 * q^10 - 213068 * q^11 + 3607068 * q^14 + 465384 * q^15 - 24515188 * q^16 + 4733396 * q^19 + 19628028 * q^20 + 2526708 * q^21 + 25784146 * q^24 + 2396384 * q^25 - 66927172 * q^26 + 51260694 * q^29 - 100952916 * q^30 + 142070514 * q^34 - 51319676 * q^35 - 829288052 * q^36 - 384252652 * q^39 + 875823864 * q^40 - 329731386 * q^41 + 966857082 * q^44 + 457201812 * q^45 - 7045011312 * q^49 + 2434599804 * q^51 + 2592949836 * q^54 + 415896124 * q^55 - 4872445266 * q^59 - 2319080274 * q^60 + 1636469048 * q^61 + 17884092636 * q^64 + 4107010578 * q^66 - 12384717444 * q^70 - 9591268956 * q^71 - 9770187120 * q^74 - 9402631914 * q^76 + 9876538494 * q^79 - 9138687352 * q^80 - 15506290458 * q^81 - 3743847174 * q^85 - 21594282036 * q^86 + 18798288204 * q^89 + 27226287060 * q^90 - 27553181472 * q^91 + 26371782018 * q^95 - 55043776340 * q^96 - 3826747138 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1 −31.0972 53.8620i 141.463 + 245.021i −1422.08 + 2463.11i 2370.23 2036.58i 8798.20 15238.9i 24743.9i 113204. −10498.9 + 18184.6i −183402. 64333.1i
69.2 −30.5593 52.9303i −145.722 252.398i −1355.74 + 2348.22i −3041.00 + 719.688i −8906.32 + 15426.2i 6339.92i 103137. −12945.2 + 22421.7i 131024. + 138968.i
69.3 −30.1860 52.2837i −44.5939 77.2389i −1310.39 + 2269.66i −536.928 3078.53i −2692.22 + 4663.07i 1063.31i 96401.0 25547.3 44249.2i −144749. + 121001.i
69.4 −30.1474 52.2168i 183.751 + 318.267i −1305.73 + 2261.59i −2287.13 2129.47i 11079.3 19189.8i 27908.2i 95715.7 −38004.7 + 65826.1i −42243.0 + 183625.i
69.5 −29.6998 51.4415i 38.6395 + 66.9256i −1252.15 + 2168.79i −1777.71 + 2570.09i 2295.17 3975.35i 3017.28i 87929.4 26538.5 45966.0i 185007. + 15116.7i
69.6 −29.6027 51.2733i −175.958 304.768i −1240.63 + 2148.84i 3075.45 + 554.288i −10417.6 + 18043.9i 28006.3i 86278.0 −32397.8 + 56114.7i −62621.3 174097.i
69.7 −28.9573 50.1555i −99.3809 172.133i −1165.05 + 2017.93i 2278.49 + 2138.72i −5755.60 + 9969.00i 25373.5i 75642.4 9771.39 16924.5i 41289.9 176210.i
69.8 −27.2030 47.1170i −214.985 372.364i −968.006 + 1676.64i 498.372 3085.00i −11696.4 + 20258.8i 16403.3i 49618.9 −62912.2 + 108967.i −158913. + 60439.6i
69.9 −27.1737 47.0662i 229.028 + 396.688i −964.816 + 1671.11i 882.048 + 2997.94i 12447.1 21559.0i 9710.93i 49218.7 −75383.3 + 130568.i 117133. 122980.i
69.10 −26.9081 46.6062i 161.496 + 279.720i −936.091 + 1621.36i 3001.11 871.185i 8691.11 15053.4i 26377.2i 45645.9 −22637.5 + 39209.3i −121357. 116428.i
69.11 −26.6844 46.2187i −6.98703 12.1019i −912.114 + 1579.83i 2971.32 967.934i −372.889 + 645.863i 7051.48i 42707.2 29426.9 50968.8i −124024. 111502.i
69.12 −26.6569 46.1711i 90.8504 + 157.358i −909.179 + 1574.74i 1236.97 + 2869.76i 4843.58 8389.32i 6004.35i 42350.2 13016.9 22545.9i 99526.1 133611.i
69.13 −25.9569 44.9586i 134.905 + 233.663i −835.518 + 1447.16i −3106.25 341.819i 7003.43 12130.3i 20025.3i 33590.0 −6874.35 + 11906.7i 65260.8 + 148525.i
69.14 −23.6126 40.8982i −82.5829 143.038i −603.111 + 1044.62i −3123.84 85.0498i −3900.00 + 6754.99i 29811.8i 8605.49 15884.6 27513.0i 70283.7 + 129768.i
69.15 −23.6124 40.8978i 8.67290 + 15.0219i −603.089 + 1044.58i −1663.86 2645.22i 409.575 709.405i 3668.45i 8603.30 29374.1 50877.4i −68896.1 + 130508.i
69.16 −23.3148 40.3824i −199.526 345.589i −575.160 + 996.207i −582.168 + 3070.29i −9303.81 + 16114.7i 8841.39i 5890.30 −50096.6 + 86769.8i 137559. 48073.9i
69.17 −22.9027 39.6686i −186.130 322.386i −537.064 + 930.222i −2851.95 1277.51i −8525.73 + 14767.0i 18942.0i 2296.10 −39764.0 + 68873.2i 14640.3 + 142391.i
69.18 −22.4656 38.9116i −44.1029 76.3884i −497.410 + 861.540i 579.286 3070.84i −1981.60 + 3432.23i 14758.8i −1311.06 25634.4 44400.0i −132505. + 46447.4i
69.19 −21.6887 37.5659i −95.0552 164.640i −428.799 + 742.702i 904.938 + 2991.11i −4123.25 + 7141.67i 25401.6i −7218.06 11453.5 19838.1i 92736.7 98868.0i
69.20 −20.6104 35.6982i 208.009 + 360.282i −337.577 + 584.700i −985.086 2965.68i 8574.29 14851.1i 939.631i −14379.7 −57010.9 + 98745.7i −85566.4 + 96289.6i
See next 80 embeddings (of 196 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.98 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.d odd 6 1 inner
95.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.11.h.a 196
5.b even 2 1 inner 95.11.h.a 196
19.d odd 6 1 inner 95.11.h.a 196
95.h odd 6 1 inner 95.11.h.a 196

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.11.h.a 196 1.a even 1 1 trivial
95.11.h.a 196 5.b even 2 1 inner
95.11.h.a 196 19.d odd 6 1 inner
95.11.h.a 196 95.h odd 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(95, [\chi])$$.