# Properties

 Label 95.11.o.a Level $95$ Weight $11$ Character orbit 95.o Analytic conductor $60.359$ Analytic rank $0$ Dimension $588$ 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(14,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 7]))

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

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

chi := DirichletCharacter("95.14");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 95.o (of order $$18$$, degree $$6$$, 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: $$588$$ Relative dimension: $$98$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$588 q - 1842 q^{4} - 6 q^{5} - 7302 q^{6} - 12 q^{9}+O(q^{10})$$ 588 * q - 1842 * q^4 - 6 * q^5 - 7302 * q^6 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$588 q - 1842 q^{4} - 6 q^{5} - 7302 q^{6} - 12 q^{9} + 314787 q^{10} + 343644 q^{11} - 1769646 q^{14} + 2373333 q^{15} + 6815562 q^{16} + 12524508 q^{19} - 5795274 q^{20} + 3990534 q^{21} - 48233604 q^{24} + 54962148 q^{25} + 159714 q^{26} - 51260712 q^{29} - 73197330 q^{30} - 18 q^{31} - 142070532 q^{34} - 153871836 q^{35} - 143737566 q^{36} + 249898896 q^{39} - 162362148 q^{40} + 329731368 q^{41} + 43247604 q^{44} - 343666425 q^{45} - 2934201618 q^{46} + 11044492350 q^{49} - 5706705726 q^{50} - 479014572 q^{51} - 3435679536 q^{54} + 2281950426 q^{55} + 4872445248 q^{59} - 3353868486 q^{60} - 1563443832 q^{61} - 38093728782 q^{64} + 2760985836 q^{65} + 9000466164 q^{66} + 7642077822 q^{69} + 18561088251 q^{70} - 15771730512 q^{71} + 13695176790 q^{74} - 8678922192 q^{76} + 15824124468 q^{79} + 11740810599 q^{80} - 3403661598 q^{81} - 109354282758 q^{84} - 3017143236 q^{85} + 1995559458 q^{86} + 37596576408 q^{89} + 21441348066 q^{90} + 51398917794 q^{91} + 18379983615 q^{95} - 166641665256 q^{96} - 138152381466 q^{99}+O(q^{100})$$ 588 * q - 1842 * q^4 - 6 * q^5 - 7302 * q^6 - 12 * q^9 + 314787 * q^10 + 343644 * q^11 - 1769646 * q^14 + 2373333 * q^15 + 6815562 * q^16 + 12524508 * q^19 - 5795274 * q^20 + 3990534 * q^21 - 48233604 * q^24 + 54962148 * q^25 + 159714 * q^26 - 51260712 * q^29 - 73197330 * q^30 - 18 * q^31 - 142070532 * q^34 - 153871836 * q^35 - 143737566 * q^36 + 249898896 * q^39 - 162362148 * q^40 + 329731368 * q^41 + 43247604 * q^44 - 343666425 * q^45 - 2934201618 * q^46 + 11044492350 * q^49 - 5706705726 * q^50 - 479014572 * q^51 - 3435679536 * q^54 + 2281950426 * q^55 + 4872445248 * q^59 - 3353868486 * q^60 - 1563443832 * q^61 - 38093728782 * q^64 + 2760985836 * q^65 + 9000466164 * q^66 + 7642077822 * q^69 + 18561088251 * q^70 - 15771730512 * q^71 + 13695176790 * q^74 - 8678922192 * q^76 + 15824124468 * q^79 + 11740810599 * q^80 - 3403661598 * q^81 - 109354282758 * q^84 - 3017143236 * q^85 + 1995559458 * q^86 + 37596576408 * q^89 + 21441348066 * q^90 + 51398917794 * q^91 + 18379983615 * q^95 - 166641665256 * q^96 - 138152381466 * 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 −60.0818 + 21.8680i −57.8803 + 328.256i 2347.19 1969.52i 3025.95 + 780.555i −3700.74 20987.9i −15063.3 + 8696.78i −65217.5 + 112960.i −48913.8 17803.2i −198874. + 19274.3i
14.2 −58.7309 + 21.3763i 77.9277 441.950i 2207.94 1852.68i −3124.89 26.5577i 4870.49 + 27621.9i 4083.93 2357.86i −58070.7 + 100581.i −133759. 48684.3i 184095. 65238.7i
14.3 −56.6254 + 20.6100i 29.8981 169.560i 1997.24 1675.88i 2870.07 1236.26i 1801.64 + 10217.6i 18208.5 10512.7i −47701.8 + 82622.0i 27631.1 + 10056.9i −137040. + 129156.i
14.4 −56.5359 + 20.5774i 7.49325 42.4964i 1988.45 1668.50i −1014.68 2955.68i 450.826 + 2556.76i −12222.5 + 7056.68i −47281.0 + 81893.1i 53738.1 + 19559.1i 118186. + 146223.i
14.5 −55.5010 + 20.2007i −66.6891 + 378.213i 1887.87 1584.11i −3012.75 830.042i −3938.86 22338.4i 282.737 163.238i −42538.2 + 73678.3i −83109.4 30249.4i 183978. 14791.5i
14.6 −54.6793 + 19.9017i −24.3306 + 137.986i 1809.33 1518.20i −445.628 + 3093.06i −1415.76 8029.19i 6642.36 3834.97i −38925.4 + 67420.8i 37039.8 + 13481.4i −37190.4 177995.i
14.7 −54.3413 + 19.7786i 31.4181 178.181i 1777.35 1491.38i 165.517 + 3120.61i 1816.87 + 10304.0i 9291.27 5364.32i −37477.9 + 64913.6i 24726.6 + 8999.73i −70715.8 166304.i
14.8 −52.0045 + 18.9281i −9.70442 + 55.0365i 1561.77 1310.48i −2404.05 1996.53i −537.062 3045.83i 24992.6 14429.5i −28079.0 + 48634.2i 52553.1 + 19127.8i 162812. + 58324.6i
14.9 −51.0497 + 18.5806i 46.5101 263.772i 1476.40 1238.85i 2310.54 + 2104.05i 2526.71 + 14329.7i −11690.5 + 6749.50i −24536.5 + 42498.5i −11924.6 4340.21i −157047. 64480.0i
14.10 −50.4477 + 18.3614i 8.30012 47.0723i 1423.39 1194.37i −2581.11 + 1761.68i 445.594 + 2527.09i −27105.4 + 15649.3i −22389.7 + 38780.1i 53341.0 + 19414.5i 97863.8 136265.i
14.11 −49.5545 + 18.0364i 72.1457 409.159i 1345.91 1129.35i 2734.31 1513.00i 3804.59 + 21576.9i −14159.9 + 8175.21i −19326.2 + 33474.0i −106718. 38842.1i −108208. + 124293.i
14.12 −48.4927 + 17.6499i −50.5569 + 286.723i 1255.59 1053.57i −2510.27 + 1861.23i −2608.98 14796.3i 1004.07 579.697i −15869.9 + 27487.5i −24165.9 8795.67i 88879.3 134562.i
14.13 −47.3075 + 17.2185i −28.4010 + 161.070i 1157.09 970.917i 2701.98 1570.00i −1429.81 8108.85i 9830.11 5675.41i −12245.5 + 21209.9i 30351.0 + 11046.8i −100791. + 120797.i
14.14 −47.0573 + 17.1275i −23.4653 + 133.078i 1136.61 953.731i 1823.26 2537.98i −1175.08 6664.22i −14012.0 + 8089.82i −11511.3 + 19938.2i 38328.6 + 13950.5i −42328.3 + 150658.i
14.15 −45.2762 + 16.4792i −64.0259 + 363.109i 993.941 834.015i 2685.52 + 1598.00i −3084.89 17495.3i 27021.6 15600.9i −6588.82 + 11412.2i −72260.9 26300.8i −147924. 28096.0i
14.16 −44.7155 + 16.2751i −69.6806 + 395.179i 950.164 797.283i 292.669 3111.27i −3315.77 18804.7i 431.688 249.235i −5147.57 + 8915.85i −95822.8 34876.6i 37549.3 + 143885.i
14.17 −43.4596 + 15.8180i 48.3680 274.309i 854.096 716.672i −3007.27 + 849.673i 2236.96 + 12686.4i −798.679 + 461.118i −2103.00 + 3642.50i −17417.9 6339.58i 117255. 84495.4i
14.18 −43.0116 + 15.6549i 40.6903 230.766i 820.488 688.472i −1284.24 2848.92i 1862.47 + 10562.6i −9704.24 + 5602.75i −1077.28 + 1865.91i 3890.60 + 1416.06i 99836.7 + 102432.i
14.19 −39.9647 + 14.5460i 63.6154 360.781i 601.166 504.438i 142.147 3121.77i 2705.54 + 15343.9i 18046.6 10419.2i 5087.26 8811.39i −70628.0 25706.5i 39728.3 + 126828.i
14.20 −39.5911 + 14.4100i −31.6778 + 179.654i 575.376 482.798i 2283.06 + 2133.84i −1334.65 7569.16i −23263.4 + 13431.1i 5748.90 9957.38i 24215.9 + 8813.87i −121137. 51582.4i
See next 80 embeddings (of 588 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.98 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.f odd 18 1 inner
95.o odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.11.o.a 588
5.b even 2 1 inner 95.11.o.a 588
19.f odd 18 1 inner 95.11.o.a 588
95.o odd 18 1 inner 95.11.o.a 588

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.11.o.a 588 1.a even 1 1 trivial
95.11.o.a 588 5.b even 2 1 inner
95.11.o.a 588 19.f odd 18 1 inner
95.11.o.a 588 95.o odd 18 1 inner

## Hecke kernels

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