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Properties

Label 115.2.l.a
Level $115$
Weight $2$
Character orbit 115.l
Analytic conductor $0.918$
Analytic rank $0$
Dimension $200$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,2,Mod(7,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([11, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.l (of order \(44\), degree \(20\), 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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(200\)
Relative dimension: \(10\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 200 q - 18 q^{2} - 14 q^{3} - 22 q^{5} - 36 q^{6} - 22 q^{7} - 26 q^{8} - 22 q^{10} - 44 q^{11} - 6 q^{12} - 26 q^{13} - 22 q^{15} - 52 q^{16} - 22 q^{17} + 58 q^{18} - 22 q^{20} - 44 q^{21} + 22 q^{23}+ \cdots - 50 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.48666 + 0.177849i 0.639470 − 2.93959i 4.17220 − 0.599872i 0.983237 + 2.00829i −1.06734 + 7.42350i 1.40495 − 2.57297i −5.39608 + 1.17384i −5.50340 − 2.51332i −2.80215 − 4.81907i
7.2 −2.43652 + 0.174263i −0.128656 + 0.591424i 3.92662 − 0.564562i −1.44483 − 1.70660i 0.210410 − 1.46344i −0.0458364 + 0.0839431i −4.69506 + 1.02135i 2.39567 + 1.09406i 3.81775 + 3.90638i
7.3 −1.30561 + 0.0933793i 0.136266 − 0.626406i −0.283736 + 0.0407951i −1.37401 + 1.76411i −0.119418 + 0.830568i −2.05467 + 3.76286i 2.92471 − 0.636232i 2.35508 + 1.07553i 1.62920 − 2.43155i
7.4 −1.05653 + 0.0755644i 0.0308132 − 0.141646i −0.869101 + 0.124958i 2.18100 − 0.493172i −0.0218516 + 0.151981i 0.717546 − 1.31409i 2.97883 − 0.648004i 2.70978 + 1.23752i −2.26703 + 0.685856i
7.5 −0.643856 + 0.0460495i −0.683154 + 3.14041i −1.56721 + 0.225331i −0.267157 − 2.22005i 0.295239 − 2.05343i −1.03991 + 1.90445i 2.26018 − 0.491672i −6.66656 − 3.04452i 0.274243 + 1.41709i
7.6 −0.107969 + 0.00772212i 0.390334 − 1.79434i −1.96805 + 0.282962i −2.08676 − 0.803398i −0.0282880 + 0.196748i 2.36052 − 4.32296i 0.421846 − 0.0917670i −0.338390 − 0.154538i 0.231510 + 0.0706281i
7.7 0.632652 − 0.0452482i −0.364069 + 1.67360i −1.58144 + 0.227377i 1.17051 + 1.90523i −0.154602 + 1.07528i 0.0937908 − 0.171765i −2.22976 + 0.485055i 0.0605071 + 0.0276327i 0.826736 + 1.15238i
7.8 1.60196 − 0.114574i 0.0509080 − 0.234020i 0.573502 − 0.0824571i 0.888172 − 2.05211i 0.0547398 − 0.380723i −0.562120 + 1.02945i −2.22942 + 0.484980i 2.67672 + 1.22242i 1.18770 − 3.38916i
7.9 1.82025 − 0.130187i 0.475590 − 2.18625i 1.31673 − 0.189318i −1.13658 + 1.92566i 0.581073 − 4.04145i −0.696418 + 1.27539i −1.19425 + 0.259794i −1.82462 − 0.833276i −1.81818 + 3.65316i
7.10 2.13477 − 0.152682i −0.476413 + 2.19004i 2.55431 − 0.367254i −2.17992 − 0.497948i −0.682656 + 4.74797i 1.43707 − 2.63180i 1.21416 − 0.264125i −1.84040 − 0.840481i −4.72966 − 0.730172i
17.1 −2.39977 + 0.895067i −0.939581 + 0.513050i 3.44624 − 2.98618i 1.67567 + 1.48058i 1.79556 − 2.07219i 0.380432 + 0.508198i −3.14238 + 5.75484i −1.00233 + 1.55966i −5.34644 − 2.05322i
17.2 −1.75302 + 0.653844i 0.975483 − 0.532654i 1.13408 − 0.982690i 0.872739 − 2.05872i −1.36177 + 1.57157i −1.17575 − 1.57062i 0.447790 − 0.820066i −0.954075 + 1.48457i −0.183850 + 4.17962i
17.3 −1.54777 + 0.577290i −2.80985 + 1.53430i 0.550844 − 0.477309i −1.14879 − 1.91841i 3.46329 − 3.99685i 1.53995 + 2.05714i 1.00633 − 1.84296i 3.91929 − 6.09853i 2.88555 + 2.30607i
17.4 −1.07057 + 0.399303i −0.256216 + 0.139905i −0.524814 + 0.454754i −1.94143 + 1.10943i 0.218434 − 0.252086i −1.96263 − 2.62176i 1.47546 − 2.70211i −1.57585 + 2.45207i 1.63545 − 1.96295i
17.5 −0.117766 + 0.0439246i −0.632714 + 0.345488i −1.49956 + 1.29938i 2.23369 + 0.103094i 0.0593371 − 0.0684786i 1.63018 + 2.17766i 0.239998 − 0.439523i −1.34096 + 2.08657i −0.267582 + 0.0859729i
17.6 0.0418710 − 0.0156171i 2.62848 − 1.43526i −1.50999 + 1.30841i −0.593814 − 2.15578i 0.0876426 − 0.101145i 1.31081 + 1.75104i −0.0856252 + 0.156811i 3.22702 − 5.02134i −0.0585306 − 0.0809910i
17.7 0.864755 − 0.322537i 1.68339 − 0.919199i −0.867728 + 0.751890i 1.28668 + 1.82879i 1.15924 − 1.33784i −2.26318 − 3.02326i −1.39250 + 2.55018i 0.366945 − 0.570978i 1.70251 + 1.16645i
17.8 1.03990 − 0.387863i −2.64829 + 1.44608i −0.580545 + 0.503045i −0.406014 + 2.19890i −2.19308 + 2.53095i −0.573224 − 0.765738i −1.47241 + 2.69652i 3.30039 − 5.13550i 0.430656 + 2.44411i
17.9 1.79955 − 0.671199i 0.756071 − 0.412846i 1.27639 − 1.10600i −2.21107 − 0.333414i 1.08349 − 1.25041i −0.0833334 − 0.111320i −0.286357 + 0.524424i −1.22072 + 1.89948i −4.20273 + 0.884071i
17.10 2.04194 − 0.761604i −1.49195 + 0.814667i 2.07797 − 1.80057i 1.48286 − 1.67366i −2.42602 + 2.79977i 0.435377 + 0.581596i 0.782870 − 1.43372i −0.0596878 + 0.0928760i 1.75324 − 4.54686i
See next 80 embeddings (of 200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.l.a ✓ 200
5.b even 2 1 575.2.r.b 200
5.c odd 4 1 inner 115.2.l.a ✓ 200
5.c odd 4 1 575.2.r.b 200
23.d odd 22 1 inner 115.2.l.a ✓ 200
115.i odd 22 1 575.2.r.b 200
115.l even 44 1 inner 115.2.l.a ✓ 200
115.l even 44 1 575.2.r.b 200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.l.a ✓ 200 1.a even 1 1 trivial
115.2.l.a ✓ 200 5.c odd 4 1 inner
115.2.l.a ✓ 200 23.d odd 22 1 inner
115.2.l.a ✓ 200 115.l even 44 1 inner
575.2.r.b 200 5.b even 2 1
575.2.r.b 200 5.c odd 4 1
575.2.r.b 200 115.i odd 22 1
575.2.r.b 200 115.l even 44 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(115, [\chi])\).