Properties

Label 115.5.h.a
Level $115$
Weight $5$
Character orbit 115.h
Analytic conductor $11.888$
Analytic rank $0$
Dimension $320$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,5,Mod(11,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 115.h (of order \(22\), degree \(10\), 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: \(11.8875457546\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(32\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9} + 30 q^{12} - 204 q^{13} - 2264 q^{16} + 990 q^{17} - 50 q^{18} + 1914 q^{19} + 2706 q^{21} + 258 q^{23} - 7260 q^{24} + 4000 q^{25} - 8922 q^{26} - 6642 q^{27} + 888 q^{29} + 3844 q^{31} + 10512 q^{32} - 12826 q^{34} - 5850 q^{35} - 11920 q^{36} + 20790 q^{38} + 5044 q^{39} + 24200 q^{40} + 10530 q^{41} + 14432 q^{43} + 1914 q^{44} - 9812 q^{46} + 10272 q^{47} - 55958 q^{48} - 16022 q^{49} - 6750 q^{50} - 34848 q^{51} + 7966 q^{52} - 14784 q^{53} - 42582 q^{54} - 7000 q^{55} - 14454 q^{56} - 5082 q^{57} + 92648 q^{58} + 12330 q^{59} + 34650 q^{60} + 22484 q^{61} + 70986 q^{62} + 55550 q^{63} + 20882 q^{64} + 22528 q^{66} + 5852 q^{67} + 17060 q^{69} + 4800 q^{70} - 42486 q^{71} - 180818 q^{72} - 11284 q^{73} - 111870 q^{74} - 61952 q^{76} - 72522 q^{77} - 72762 q^{78} - 3124 q^{79} + 12332 q^{81} + 83702 q^{82} + 130218 q^{83} + 349492 q^{84} + 12850 q^{85} + 60192 q^{86} + 146516 q^{87} + 88176 q^{88} + 45936 q^{89} + 5790 q^{92} - 36668 q^{93} - 182364 q^{94} - 10800 q^{95} - 67282 q^{96} - 125136 q^{97} - 67860 q^{98} - 283008 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −6.63856 + 4.26634i 0.913905 6.35635i 19.2221 42.0906i −3.14987 10.7275i 21.0513 + 46.0960i −67.6414 + 58.6116i 33.9969 + 236.453i 38.1510 + 11.2021i 66.6775 + 57.7764i
11.2 −6.40115 + 4.11377i −1.77574 + 12.3506i 17.4050 38.1116i −3.14987 10.7275i −39.4406 86.3628i 66.0549 57.2369i 28.0444 + 195.053i −71.6639 21.0424i 64.2931 + 55.7103i
11.3 −5.95787 + 3.82889i −1.40129 + 9.74618i 14.1892 31.0700i 3.14987 + 10.7275i −28.9684 63.4319i −16.5348 + 14.3275i 18.3000 + 127.279i −15.3055 4.49411i −59.8408 51.8523i
11.4 −5.30060 + 3.40649i −0.401632 + 2.79341i 9.84555 21.5587i 3.14987 + 10.7275i −7.38684 16.1749i 1.20039 1.04014i 6.90507 + 48.0258i 70.0771 + 20.5765i −53.2391 46.1320i
11.5 −5.04053 + 3.23935i 1.96068 13.6368i 8.26690 18.1020i 3.14987 + 10.7275i 34.2916 + 75.0881i −27.5585 + 23.8796i 3.32584 + 23.1317i −104.400 30.6545i −50.6270 43.8685i
11.6 −4.84924 + 3.11642i 1.03283 7.18350i 7.15645 15.6704i −3.14987 10.7275i 17.3783 + 38.0533i 22.5454 19.5357i 1.00676 + 7.00216i 27.1830 + 7.98165i 48.7057 + 42.2038i
11.7 −4.32656 + 2.78051i −0.455362 + 3.16711i 4.34125 9.50600i −3.14987 10.7275i −6.83604 14.9688i −11.2058 + 9.70991i −4.06190 28.2511i 67.8957 + 19.9360i 43.4559 + 37.6548i
11.8 −3.32783 + 2.13867i 2.30957 16.0634i −0.146084 + 0.319880i −3.14987 10.7275i 26.6685 + 58.3958i −26.7873 + 23.2114i −9.20547 64.0255i −174.981 51.3790i 33.4247 + 28.9626i
11.9 −3.24867 + 2.08779i −2.37840 + 16.5422i −0.451668 + 0.989014i 3.14987 + 10.7275i −26.8100 58.7057i 1.96580 1.70338i −9.39077 65.3143i −190.268 55.8676i −32.6296 28.2737i
11.10 −2.75822 + 1.77260i −0.284694 + 1.98009i −2.18096 + 4.77564i 3.14987 + 10.7275i −2.72467 5.96619i 52.3980 45.4031i −9.91547 68.9636i 73.8792 + 21.6929i −27.7036 24.0053i
11.11 −2.63066 + 1.69062i 0.551076 3.83282i −2.58448 + 5.65923i 3.14987 + 10.7275i 5.03015 + 11.0145i −52.9290 + 45.8632i −9.88917 68.7807i 63.3321 + 18.5960i −26.4223 22.8950i
11.12 −2.57806 + 1.65682i −1.85705 + 12.9161i −2.74528 + 6.01133i −3.14987 10.7275i −16.6120 36.3753i 54.7272 47.4214i −9.86028 68.5798i −85.6574 25.1513i 25.8940 + 22.4373i
11.13 −1.37551 + 0.883987i −1.60372 + 11.1541i −5.53604 + 12.1222i −3.14987 10.7275i −7.65414 16.7602i −44.1661 + 38.2702i −6.82414 47.4629i −44.1230 12.9557i 13.8156 + 11.9713i
11.14 −1.04486 + 0.671490i 2.02842 14.1080i −6.00581 + 13.1509i 3.14987 + 10.7275i 7.35396 + 16.1029i 35.3618 30.6412i −5.38361 37.4438i −117.202 34.4135i −10.4945 9.09358i
11.15 −0.0293825 + 0.0188830i −0.0510492 + 0.355055i −6.64613 + 14.5530i −3.14987 10.7275i −0.00520454 0.0113963i 22.4414 19.4456i −0.159054 1.10625i 77.5955 + 22.7841i 0.295117 + 0.255720i
11.16 0.562353 0.361402i −1.00918 + 7.01902i −6.46101 + 14.1476i 3.14987 + 10.7275i 1.96917 + 4.31189i 10.6690 9.24478i 3.00175 + 20.8776i 29.4707 + 8.65337i 5.64826 + 4.89425i
11.17 0.777273 0.499523i 0.742802 5.16630i −6.29201 + 13.7776i −3.14987 10.7275i −2.00333 4.38667i −14.1558 + 12.2660i 4.09547 + 28.4846i 51.5800 + 15.1453i −7.80692 6.76473i
11.18 1.06653 0.685415i −1.69116 + 11.7623i −5.97895 + 13.0921i 3.14987 + 10.7275i 6.25839 + 13.7040i −44.3476 + 38.4274i 5.48359 + 38.1392i −57.7727 16.9636i 10.7122 + 9.28216i
11.19 1.27422 0.818893i 1.58968 11.0565i −5.69358 + 12.4672i 3.14987 + 10.7275i −7.02847 15.3902i −22.4817 + 19.4805i 6.40338 + 44.5365i −41.9999 12.3323i 12.7983 + 11.0898i
11.20 1.74399 1.12079i 2.03039 14.1217i −4.86132 + 10.6448i −3.14987 10.7275i −12.2865 26.9037i −63.0090 + 54.5976i 8.17303 + 56.8447i −117.580 34.5247i −17.5166 15.1782i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.5.h.a 320
23.d odd 22 1 inner 115.5.h.a 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.5.h.a 320 1.a even 1 1 trivial
115.5.h.a 320 23.d odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(115, [\chi])\).