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Properties

Label 289.2.f.a
Level $289$
Weight $2$
Character orbit 289.f
Analytic conductor $2.308$
Analytic rank $0$
Dimension $384$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,2,Mod(18,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.f (of order \(17\), degree \(16\), 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: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(24\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 13 q^{2} - 13 q^{3} - 33 q^{4} - 43 q^{5} - 5 q^{6} - 13 q^{7} - 5 q^{8} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q - 13 q^{2} - 13 q^{3} - 33 q^{4} - 43 q^{5} - 5 q^{6} - 13 q^{7} - 5 q^{8} - 25 q^{9} - 35 q^{10} - 5 q^{11} + 11 q^{12} - 3 q^{13} - 74 q^{14} + 5 q^{15} - 145 q^{16} - q^{17} + 15 q^{18} + 5 q^{19} + 23 q^{20} - 40 q^{21} + 19 q^{22} + 3 q^{23} - 42 q^{24} - 113 q^{25} + 19 q^{26} - 28 q^{27} + 43 q^{28} + 7 q^{29} + 45 q^{30} + 11 q^{31} + 41 q^{32} - 26 q^{33} + 35 q^{34} + 35 q^{35} + 37 q^{36} + 23 q^{37} - 152 q^{38} - 29 q^{39} + 62 q^{40} + 31 q^{41} + 65 q^{42} + 17 q^{43} - q^{44} - 98 q^{45} - 9 q^{46} + 12 q^{47} - 114 q^{48} + 5 q^{49} + 61 q^{50} - 85 q^{51} - 116 q^{52} + 8 q^{53} - 356 q^{54} + 45 q^{55} + 91 q^{56} + 63 q^{57} - 40 q^{58} + 49 q^{59} + 133 q^{60} + 55 q^{61} - 19 q^{62} + 14 q^{63} + 67 q^{64} + 46 q^{65} - 144 q^{66} - 82 q^{67} + 103 q^{68} - 54 q^{69} + 103 q^{70} - 77 q^{71} + 114 q^{72} + 63 q^{73} - 92 q^{74} - 46 q^{75} - 71 q^{76} - 105 q^{77} + 151 q^{78} - 68 q^{79} - 37 q^{80} - 73 q^{81} + 103 q^{82} - 5 q^{83} + 183 q^{84} - 168 q^{85} + 105 q^{86} + 91 q^{87} - 75 q^{88} + 57 q^{89} + 223 q^{90} + 35 q^{91} - 15 q^{92} + 95 q^{93} - 331 q^{94} + 95 q^{95} - 122 q^{96} + 79 q^{97} - 63 q^{98} + 139 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −2.61790 − 1.01418i 0.923293 − 1.22264i 4.34682 + 3.96265i −1.29217 + 2.59502i −3.65706 + 2.26436i −0.707816 − 2.48772i −4.85789 − 9.75595i 0.178616 + 0.627771i 6.01458 − 5.48301i
18.2 −2.38373 − 0.923463i −0.435224 + 0.576329i 3.35138 + 3.05519i 0.817142 − 1.64104i 1.56968 − 0.971903i 1.12631 + 3.95856i −2.88851 − 5.80092i 0.678253 + 2.38381i −3.46329 + 3.15720i
18.3 −2.11131 − 0.817926i −0.251587 + 0.333155i 2.31061 + 2.10640i 1.29322 − 2.59713i 0.803675 − 0.497614i −1.01998 − 3.58487i −1.13705 − 2.28349i 0.773293 + 2.71784i −4.85465 + 4.42560i
18.4 −2.10828 − 0.816753i −1.98159 + 2.62405i 2.29975 + 2.09650i 0.334720 − 0.672209i 6.32096 − 3.91377i −0.569894 − 2.00297i −1.12061 − 2.25049i −2.13795 − 7.51411i −1.25471 + 1.14382i
18.5 −1.84868 − 0.716181i 1.18332 − 1.56697i 1.42667 + 1.30058i −0.385338 + 0.773862i −3.30981 + 2.04935i 0.825302 + 2.90064i 0.0614005 + 0.123309i −0.234159 − 0.822983i 1.26659 − 1.15465i
18.6 −1.82778 − 0.708084i −0.789998 + 1.04613i 1.36136 + 1.24105i −1.28729 + 2.58523i 2.18468 − 1.35270i −0.250546 − 0.880577i 0.137918 + 0.276977i 0.350705 + 1.23260i 4.18343 − 3.81370i
18.7 −1.51800 − 0.588078i 1.69130 − 2.23964i 0.480485 + 0.438020i −0.834442 + 1.67579i −3.88449 + 2.40517i −0.355217 − 1.24846i 0.979478 + 1.96706i −1.33452 − 4.69034i 2.25218 − 2.05313i
18.8 −1.22801 − 0.475735i 0.0552260 − 0.0731310i −0.196325 − 0.178974i 0.272624 − 0.547503i −0.102609 + 0.0635330i 0.655504 + 2.30386i 1.32997 + 2.67094i 0.818691 + 2.87740i −0.595252 + 0.542644i
18.9 −1.04505 − 0.404853i 1.21681 − 1.61132i −0.549801 − 0.501210i 1.41651 − 2.84473i −1.92397 + 1.19127i −0.842172 − 2.95993i 1.37075 + 2.75284i −0.294727 − 1.03586i −2.63201 + 2.39940i
18.10 −0.671464 − 0.260126i −1.17921 + 1.56152i −1.09482 − 0.998061i 0.411369 − 0.826139i 1.19799 − 0.741762i −0.441491 − 1.55168i 1.11745 + 2.24415i −0.226833 − 0.797235i −0.491120 + 0.447715i
18.11 −0.557841 − 0.216109i −1.83834 + 2.43435i −1.21353 − 1.10628i −0.901182 + 1.80982i 1.55158 − 0.960699i 1.05921 + 3.72272i 0.971198 + 1.95043i −1.72559 − 6.06482i 0.893833 − 0.814837i
18.12 −0.339140 − 0.131384i 0.322138 − 0.426580i −1.38026 − 1.25828i −1.53942 + 3.09157i −0.165295 + 0.102347i −0.484503 − 1.70285i 0.627016 + 1.25922i 0.742792 + 2.61064i 0.928259 − 0.846220i
18.13 −0.0984912 − 0.0381557i 1.14941 − 1.52207i −1.46977 − 1.33988i 1.74083 − 3.49606i −0.171282 + 0.106053i 0.840395 + 2.95368i 0.187797 + 0.377147i −0.174549 − 0.613477i −0.304851 + 0.277909i
18.14 0.232508 + 0.0900742i 0.0332096 − 0.0439766i −1.43207 − 1.30551i −0.291146 + 0.584700i 0.0116827 − 0.00723360i −1.08981 − 3.83029i −0.437662 − 0.878944i 0.820158 + 2.88256i −0.120360 + 0.109723i
18.15 0.511281 + 0.198071i 1.87484 − 2.48269i −1.25584 − 1.14485i −0.432659 + 0.868897i 1.45032 − 0.898001i −0.00240770 − 0.00846219i −0.904129 − 1.81574i −1.82774 − 6.42383i −0.393314 + 0.358553i
18.16 0.713131 + 0.276268i −0.653849 + 0.865836i −1.04579 − 0.953360i 0.217862 − 0.437526i −0.705483 + 0.436817i 1.25732 + 4.41901i −1.16418 − 2.33798i 0.498836 + 1.75323i 0.276239 − 0.251825i
18.17 0.722386 + 0.279854i −1.47543 + 1.95379i −1.03449 − 0.943066i 1.59137 − 3.19590i −1.61260 + 0.998482i −0.240831 − 0.846433i −1.17401 − 2.35773i −0.819394 − 2.87987i 2.04397 − 1.86332i
18.18 1.17719 + 0.456048i −1.22657 + 1.62423i −0.300209 − 0.273677i −1.61300 + 3.23934i −2.18464 + 1.35267i −0.233825 − 0.821811i −1.35404 − 2.71927i −0.312688 − 1.09898i −3.37611 + 3.07773i
18.19 1.29626 + 0.502175i 1.07211 − 1.41971i −0.0498996 − 0.0454895i 0.580940 − 1.16669i 2.10268 − 1.30192i 0.0726098 + 0.255197i −1.28111 − 2.57282i −0.0451521 − 0.158693i 1.33893 − 1.22060i
18.20 1.45664 + 0.564305i −0.162629 + 0.215356i 0.325340 + 0.296587i −1.32072 + 2.65237i −0.358419 + 0.221924i 0.568502 + 1.99808i −1.08606 − 2.18111i 0.801059 + 2.81543i −3.42056 + 3.11825i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
289.f even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.2.f.a ✓ 384
289.f even 17 1 inner 289.2.f.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.2.f.a ✓ 384 1.a even 1 1 trivial
289.2.f.a ✓ 384 289.f even 17 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(289, [\chi])\).