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Properties

Label 289.2.h.a
Level $289$
Weight $2$
Character orbit 289.h
Analytic conductor $2.308$
Analytic rank $0$
Dimension $768$
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(4,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(68))
 
chi = DirichletCharacter(H, H._module([27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.h (of order \(68\), degree \(32\), 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: \(768\)
Relative dimension: \(24\) over \(\Q(\zeta_{68})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{68}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 34 q^{2} - 34 q^{3} + 14 q^{4} - 34 q^{5} - 34 q^{6} - 34 q^{7} - 34 q^{8} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 768 q - 34 q^{2} - 34 q^{3} + 14 q^{4} - 34 q^{5} - 34 q^{6} - 34 q^{7} - 34 q^{8} - 34 q^{9} + 34 q^{10} - 34 q^{11} - 34 q^{12} - 30 q^{13} - 34 q^{14} - 34 q^{15} - 82 q^{16} - 34 q^{17} - 42 q^{18} - 34 q^{19} - 34 q^{20} - 26 q^{21} - 34 q^{22} - 34 q^{23} - 204 q^{24} + 170 q^{25} - 34 q^{26} - 34 q^{27} - 34 q^{28} - 34 q^{29} - 14 q^{30} - 34 q^{31} - 34 q^{32} + 56 q^{33} - 34 q^{34} - 42 q^{35} - 34 q^{36} - 34 q^{37} + 144 q^{38} - 170 q^{39} - 34 q^{40} - 34 q^{41} - 34 q^{42} - 34 q^{43} - 34 q^{44} - 34 q^{45} + 102 q^{46} - 30 q^{47} - 34 q^{48} - 34 q^{49} - 62 q^{50} + 238 q^{51} - 50 q^{52} + 136 q^{54} - 14 q^{55} - 34 q^{56} - 34 q^{57} - 34 q^{58} - 34 q^{59} - 34 q^{60} - 34 q^{61} - 238 q^{62} - 34 q^{63} + 38 q^{64} - 34 q^{65} + 476 q^{66} + 82 q^{67} - 34 q^{68} - 62 q^{69} - 34 q^{70} - 34 q^{71} + 10 q^{72} - 34 q^{73} - 34 q^{74} + 102 q^{75} - 102 q^{76} - 374 q^{77} - 34 q^{78} + 68 q^{79} - 34 q^{80} + 30 q^{81} - 34 q^{82} + 102 q^{83} - 82 q^{84} - 204 q^{85} - 62 q^{86} - 34 q^{87} + 170 q^{88} - 22 q^{89} - 34 q^{90} - 34 q^{91} + 34 q^{92} - 34 q^{93} + 238 q^{94} - 34 q^{95} + 544 q^{96} - 34 q^{97} + 10 q^{98} - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.495373 − 2.65001i 0.0417369 + 0.124526i −4.91222 + 1.90301i 4.04829 − 0.952148i 0.309320 − 0.172290i 0.175906 − 1.26103i 4.63794 + 7.49053i 2.38029 − 1.79751i −4.52862 − 10.2563i
4.2 −0.465724 − 2.49141i 0.237320 + 0.708065i −4.12526 + 1.59813i −1.86724 + 0.439171i 1.65355 − 0.921022i −0.599825 + 4.30001i 3.23428 + 5.22355i 1.94902 − 1.47183i 1.96377 + 4.44752i
4.3 −0.444275 − 2.37666i −0.847134 − 2.52750i −3.58618 + 1.38929i −0.799186 + 0.187967i −5.63066 + 3.13625i 0.328894 − 2.35776i 2.34949 + 3.79455i −3.27659 + 2.47437i 0.801791 + 1.81588i
4.4 −0.436480 − 2.33496i 0.728742 + 2.17427i −3.39658 + 1.31584i −0.795588 + 0.187121i 4.75876 − 2.65061i 0.476955 − 3.41918i 2.05399 + 3.31731i −1.80234 + 1.36106i 0.784177 + 1.77599i
4.5 −0.326276 − 1.74542i 0.00851047 + 0.0253918i −1.07509 + 0.416493i −3.35489 + 0.789062i 0.0415426 − 0.0231391i 0.272424 − 1.95295i −0.791791 − 1.27879i 2.39348 − 1.80747i 2.47186 + 5.59824i
4.6 −0.314878 − 1.68445i 0.886481 + 2.64490i −0.873267 + 0.338305i 3.81483 − 0.897240i 4.17606 − 2.32605i −0.350525 + 2.51283i −0.959384 − 1.54946i −3.81560 + 2.88141i −2.71256 − 6.14336i
4.7 −0.295430 − 1.58041i −0.160324 − 0.478342i −0.545470 + 0.211316i 0.976959 − 0.229779i −0.708611 + 0.394694i −0.0564785 + 0.404881i −1.19766 − 1.93430i 2.19094 − 1.65452i −0.651767 − 1.47611i
4.8 −0.218654 − 1.16970i −0.752999 − 2.24664i 0.544563 − 0.210965i 2.24018 − 0.526886i −2.46325 + 1.37202i 0.0788133 − 0.564994i −1.61870 − 2.61429i −2.08635 + 1.57553i −1.10612 − 2.50513i
4.9 −0.188681 − 1.00935i −0.936628 − 2.79452i 0.881751 − 0.341592i −2.83658 + 0.667157i −2.64393 + 1.47266i −0.648220 + 4.64694i −1.59228 − 2.57162i −4.53801 + 3.42695i 1.20861 + 2.73723i
4.10 −0.140244 − 0.750239i 0.660805 + 1.97157i 1.32175 − 0.512050i 0.945058 − 0.222276i 1.38648 − 0.772263i 0.561910 − 4.02820i −1.37311 − 2.21765i −1.05639 + 0.797746i −0.299299 − 0.677846i
4.11 −0.135906 − 0.727030i 0.960801 + 2.86664i 1.35484 − 0.524868i −3.80500 + 0.894928i 1.95356 − 1.08812i −0.386108 + 2.76792i −1.34445 − 2.17136i −4.90044 + 3.70064i 1.16776 + 2.64473i
4.12 −0.0231486 − 0.123834i 0.0321710 + 0.0959852i 1.85015 − 0.716750i −1.72849 + 0.406537i 0.0111415 − 0.00620579i −0.123494 + 0.885299i −0.264226 − 0.426739i 2.38587 − 1.80173i 0.0903554 + 0.204636i
4.13 0.0369306 + 0.197561i −0.449913 − 1.34236i 1.82728 − 0.707891i 2.94745 − 0.693233i 0.248582 − 0.138459i −0.267814 + 1.91990i 0.418943 + 0.676616i 0.794552 − 0.600018i 0.245807 + 0.556700i
4.14 0.0663695 + 0.355045i 0.616109 + 1.83822i 1.74329 − 0.675355i 1.10788 − 0.260571i −0.611760 + 0.340748i −0.313876 + 2.25010i 0.735773 + 1.18831i −0.605410 + 0.457185i 0.166044 + 0.376054i
4.15 0.0830535 + 0.444297i −0.264019 − 0.787727i 1.67444 − 0.648683i −2.48282 + 0.583953i 0.328057 − 0.182726i 0.396099 − 2.83954i 0.903163 + 1.45866i 1.84324 − 1.39195i −0.465655 − 1.05461i
4.16 0.199794 + 1.06880i −0.925410 − 2.76105i 0.762526 − 0.295404i 1.95497 − 0.459804i 2.76612 − 1.54072i 0.0494501 − 0.354496i 1.61287 + 2.60488i −4.37295 + 3.30230i 0.882029 + 1.99761i
4.17 0.256760 + 1.37354i 0.592956 + 1.76914i 0.0442480 − 0.0171418i −0.851598 + 0.200294i −2.27775 + 1.26870i −0.0713949 + 0.511813i 1.50611 + 2.43245i −0.384213 + 0.290144i −0.493769 − 1.11828i
4.18 0.278847 + 1.49170i −0.555741 − 1.65811i −0.282466 + 0.109428i −2.12983 + 0.500930i 2.31843 − 1.29136i −0.562783 + 4.03446i 1.35576 + 2.18963i −0.0464138 + 0.0350501i −1.34113 − 3.03738i
4.19 0.304577 + 1.62934i −0.0195801 − 0.0584192i −0.697041 + 0.270035i 1.38109 − 0.324830i 0.0892211 − 0.0496958i 0.578833 − 4.14952i 1.09291 + 1.76511i 2.39102 − 1.80562i 0.949907 + 2.15133i
4.20 0.404493 + 2.16385i 0.674814 + 2.01337i −2.65367 + 1.02804i −2.86311 + 0.673396i −4.08367 + 2.27459i 0.0725688 − 0.520229i −0.980205 − 1.58309i −1.20424 + 0.909398i −2.61523 − 5.92293i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
289.h even 68 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.2.h.a ✓ 768
289.h even 68 1 inner 289.2.h.a ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.2.h.a ✓ 768 1.a even 1 1 trivial
289.2.h.a ✓ 768 289.h even 68 1 inner

Hecke kernels

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