↑

Properties

Label 345.2.x.a
Level $345$
Weight $2$
Character orbit 345.x
Analytic conductor $2.755$
Analytic rank $0$
Dimension $880$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [345,2,Mod(2,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 11, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.x (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: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(880\)
Relative dimension: \(44\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 880 q - 22 q^{3} - 44 q^{6} - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 880 q - 22 q^{3} - 44 q^{6} - 36 q^{7} - 36 q^{10} - 22 q^{12} - 28 q^{13} - 34 q^{15} - 82 q^{18} - 52 q^{21} - 112 q^{22} - 44 q^{25} - 34 q^{27} - 20 q^{28} - 38 q^{30} - 72 q^{31} - 48 q^{33} - 44 q^{36} - 84 q^{37} - 108 q^{40} + 10 q^{42} - 12 q^{43} - 20 q^{45} - 88 q^{46} - 94 q^{48} - 20 q^{51} - 296 q^{52} - 20 q^{55} + 18 q^{57} - 52 q^{58} - 42 q^{60} - 16 q^{61} - 90 q^{63} + 60 q^{66} - 60 q^{67} + 16 q^{70} - 58 q^{72} - 68 q^{73} - 74 q^{75} + 72 q^{76} + 78 q^{78} + 52 q^{81} - 60 q^{82} - 76 q^{85} + 18 q^{87} + 12 q^{88} - 436 q^{90} - 80 q^{91} + 12 q^{93} + 212 q^{96} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.62033 + 0.570019i 1.66581 + 0.474430i 4.72197 − 2.15645i 2.14313 − 0.637968i −4.63541 − 0.293624i −0.379942 + 0.0271740i −6.85043 + 5.12817i 2.54983 + 1.58062i −5.25206 + 2.89331i
2.2 −2.41836 + 0.526082i −1.72749 + 0.125593i 3.75243 − 1.71368i 2.09129 + 0.791528i 4.11162 − 1.21253i 2.30751 − 0.165036i −4.21063 + 3.15204i 2.96845 − 0.433922i −5.47389 − 0.814011i
2.3 −2.41718 + 0.525826i 1.14869 − 1.29635i 3.74701 − 1.71120i −1.80608 − 1.31836i −2.09493 + 3.73752i −0.738208 + 0.0527977i −4.19680 + 3.14168i −0.361042 − 2.97820i 5.05885 + 2.23704i
2.4 −2.38979 + 0.519866i −1.32155 + 1.11960i 3.62155 − 1.65391i −0.657218 − 2.13730i 2.57619 − 3.36263i −3.63868 + 0.260244i −3.87920 + 2.90393i 0.493002 − 2.95921i 2.68172 + 4.76603i
2.5 −2.33013 + 0.506888i −1.39379 − 1.02827i 3.35329 − 1.53140i −1.89999 + 1.17900i 3.76893 + 1.68950i −1.07867 + 0.0771477i −3.21937 + 2.40999i 0.885324 + 2.86639i 3.82960 − 3.71030i
2.6 −2.18883 + 0.476150i 0.355236 + 1.69523i 2.74498 − 1.25359i 1.07204 + 1.96233i −1.58474 − 3.54142i −3.67649 + 0.262947i −1.82494 + 1.36613i −2.74761 + 1.20441i −3.28088 − 3.78474i
2.7 −2.05456 + 0.446941i −0.537387 − 1.64658i 2.20218 − 1.00570i 0.600696 − 2.15387i 1.84001 + 3.14280i 4.50930 − 0.322511i −0.708560 + 0.530421i −2.42243 + 1.76970i −0.271510 + 4.69372i
2.8 −1.95914 + 0.426184i 1.72896 + 0.103370i 1.83733 − 0.839079i −0.889696 + 2.05145i −3.43133 + 0.534342i 1.60823 − 0.115023i −0.0318708 + 0.0238582i 2.97863 + 0.357445i 0.868743 − 4.39825i
2.9 −1.64540 + 0.357934i 1.30730 − 1.13620i 0.759947 − 0.347056i 2.02926 − 0.939214i −1.74434 + 2.33743i −2.71354 + 0.194076i 1.56984 − 1.17517i 0.418082 − 2.97073i −3.00275 + 2.27172i
2.10 −1.61252 + 0.350783i −0.312005 − 1.70372i 0.657916 − 0.300460i 1.52233 + 1.63784i 1.10075 + 2.63784i −2.22999 + 0.159492i 1.68665 − 1.26261i −2.80531 + 1.06314i −3.02931 − 2.10705i
2.11 −1.54168 + 0.335372i 0.346191 + 1.69710i 0.445043 − 0.203244i −2.18573 + 0.471768i −1.10288 − 2.50029i −0.662375 + 0.0473740i 1.90814 − 1.42841i −2.76030 + 1.17504i 3.21149 − 1.46035i
2.12 −1.41621 + 0.308079i −1.52423 + 0.822635i 0.0914867 − 0.0417806i −2.20287 − 0.383868i 1.90520 − 1.63461i 3.61486 − 0.258540i 2.20381 − 1.64975i 1.64654 − 2.50777i 3.23800 − 0.135018i
2.13 −1.39323 + 0.303078i −0.943819 + 1.45231i 0.0299605 − 0.0136825i 1.43027 + 1.71882i 0.874791 − 2.30945i 2.53246 − 0.181125i 2.24525 − 1.68077i −1.21841 − 2.74144i −2.51362 − 1.96122i
2.14 −1.34607 + 0.292820i 1.36669 + 1.06403i −0.0931036 + 0.0425190i −0.460136 − 2.18821i −2.15123 − 1.03207i −1.32276 + 0.0946055i 2.31845 − 1.73557i 0.735672 + 2.90840i 1.26013 + 2.81075i
2.15 −1.07923 + 0.234772i −1.67962 − 0.422940i −0.709642 + 0.324083i 1.29708 − 1.82142i 1.91199 + 0.0621221i −1.04970 + 0.0750763i 2.45813 − 1.84014i 2.64224 + 1.42076i −0.972231 + 2.27025i
2.16 −0.888745 + 0.193335i 0.381551 − 1.68950i −1.06678 + 0.487180i −1.43783 + 1.71250i −0.0124623 + 1.57530i 2.76708 − 0.197905i 2.31014 − 1.72935i −2.70884 − 1.28926i 0.946778 − 1.79996i
2.17 −0.789544 + 0.171755i 1.30508 + 1.13876i −1.22538 + 0.559614i 2.23598 + 0.0200893i −1.22600 − 0.674944i 4.39726 − 0.314498i 2.16507 − 1.62075i 0.406468 + 2.97234i −1.76885 + 0.368178i
2.18 −0.677104 + 0.147295i −1.66578 + 0.474524i −1.38249 + 0.631362i −0.759201 + 2.10324i 1.05801 − 0.566663i −4.75773 + 0.340279i 1.95255 − 1.46166i 2.54965 − 1.58091i 0.204262 − 1.53594i
2.19 −0.458596 + 0.0997614i 1.50511 − 0.857107i −1.61891 + 0.739330i 0.116123 − 2.23305i −0.604733 + 0.543218i 1.89498 − 0.135532i 1.42009 − 1.06307i 1.53074 − 2.58009i 0.169519 + 1.03565i
2.20 −0.189741 + 0.0412755i 1.72537 + 0.151959i −1.78497 + 0.815167i −2.22559 + 0.216227i −0.333645 + 0.0423829i −4.24164 + 0.303368i 0.615929 − 0.461079i 2.95382 + 0.524371i 0.413360 − 0.132889i
See next 80 embeddings (of 880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
115.k odd 44 1 inner
345.x even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.x.a ✓ 880
3.b odd 2 1 inner 345.2.x.a ✓ 880
5.c odd 4 1 inner 345.2.x.a ✓ 880
15.e even 4 1 inner 345.2.x.a ✓ 880
23.c even 11 1 inner 345.2.x.a ✓ 880
69.h odd 22 1 inner 345.2.x.a ✓ 880
115.k odd 44 1 inner 345.2.x.a ✓ 880
345.x even 44 1 inner 345.2.x.a ✓ 880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.x.a ✓ 880 1.a even 1 1 trivial
345.2.x.a ✓ 880 3.b odd 2 1 inner
345.2.x.a ✓ 880 5.c odd 4 1 inner
345.2.x.a ✓ 880 15.e even 4 1 inner
345.2.x.a ✓ 880 23.c even 11 1 inner
345.2.x.a ✓ 880 69.h odd 22 1 inner
345.2.x.a ✓ 880 115.k odd 44 1 inner
345.2.x.a ✓ 880 345.x even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(345, [\chi])\).