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Properties

Label 684.2.o.a
Level $684$
Weight $2$
Character orbit 684.o
Analytic conductor $5.462$
Analytic rank $0$
Dimension $232$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(11,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.o (of order \(6\), degree \(2\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 3 q^{2} + q^{4} + 3 q^{6} + 2 q^{10} - q^{12} + 2 q^{13} + q^{16} - 12 q^{17} - 16 q^{18} - 27 q^{20} - 14 q^{21} + 6 q^{22} - 10 q^{24} - 204 q^{25} - 6 q^{28} - 9 q^{30} - 3 q^{32} - 12 q^{33}+ \cdots - 162 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41421 + 0.00455230i 1.10804 − 1.33126i 1.99996 − 0.0128758i − 4.21996i −1.56094 + 1.88772i 1.74097 + 1.00515i −2.82830 + 0.0273134i −0.544497 − 2.95017i 0.0192105 + 5.96789i
11.2 −1.41394 − 0.0277479i −0.908967 − 1.47437i 1.99846 + 0.0784677i 0.306310i 1.24431 + 2.10990i 0.747988 + 0.431851i −2.82353 − 0.166402i −1.34756 + 2.68031i 0.00849945 − 0.433105i
11.3 −1.41338 − 0.0484033i −1.15926 + 1.28691i 1.99531 + 0.136825i − 0.365397i 1.70076 − 1.76278i −3.58119 − 2.06760i −2.81352 − 0.289966i −0.312254 − 2.98371i −0.0176864 + 0.516447i
11.4 −1.41247 − 0.0701983i 1.42043 − 0.991151i 1.99014 + 0.198306i 4.19860i −2.07589 + 1.30026i 3.34657 + 1.93215i −2.79710 − 0.419806i 1.03524 − 2.81572i 0.294734 − 5.93039i
11.5 −1.41241 − 0.0714431i 1.22511 + 1.22438i 1.98979 + 0.201814i 1.44578i −1.64289 − 1.81684i 2.47038 + 1.42627i −2.79598 − 0.427200i 0.00181117 + 3.00000i 0.103291 − 2.04203i
11.6 −1.41239 − 0.0718796i −1.68620 − 0.395888i 1.98967 + 0.203043i 2.95180i 2.35311 + 0.680350i −0.414473 − 0.239296i −2.79558 − 0.429792i 2.68655 + 1.33509i 0.212174 − 4.16908i
11.7 −1.40988 + 0.110618i −0.0126385 + 1.73200i 1.97553 − 0.311917i − 2.54159i −0.173773 − 2.44332i −1.35871 − 0.784451i −2.75075 + 0.658296i −2.99968 − 0.0437799i 0.281146 + 3.58333i
11.8 −1.38778 + 0.272176i 1.69723 + 0.345580i 1.85184 − 0.755437i − 0.372058i −2.44943 − 0.0176435i 0.164660 + 0.0950666i −2.36433 + 1.55240i 2.76115 + 1.17305i 0.101265 + 0.516332i
11.9 −1.36579 + 0.366886i 1.11809 + 1.32283i 1.73079 − 1.00218i 3.10398i −2.01241 − 1.39650i −4.42597 − 2.55533i −1.99622 + 2.00378i −0.499760 + 2.95808i −1.13881 − 4.23939i
11.10 −1.35542 + 0.403535i −1.43572 + 0.968869i 1.67432 − 1.09392i − 2.62337i 1.55503 − 1.89259i 4.39328 + 2.53646i −1.82797 + 2.15836i 1.12259 − 2.78205i 1.05862 + 3.55576i
11.11 −1.33998 + 0.452168i −0.0387989 − 1.73162i 1.59109 − 1.21179i − 1.06785i 0.834971 + 2.30279i 0.640536 + 0.369814i −1.58409 + 2.34321i −2.99699 + 0.134370i 0.482846 + 1.43089i
11.12 −1.33660 − 0.462057i 0.610129 + 1.62103i 1.57301 + 1.23517i − 0.399409i −0.0664904 − 2.44859i 2.94561 + 1.70065i −1.53176 − 2.37775i −2.25549 + 1.97808i −0.184550 + 0.533851i
11.13 −1.33496 − 0.466773i −1.66738 − 0.468866i 1.56425 + 1.24625i − 1.96275i 2.00704 + 1.40421i 2.10937 + 1.21784i −1.50649 − 2.39384i 2.56033 + 1.56356i −0.916161 + 2.62020i
11.14 −1.32793 + 0.486408i −1.56525 − 0.741608i 1.52681 − 1.29184i − 2.79966i 2.43928 + 0.223454i −3.69046 − 2.13069i −1.39915 + 2.45813i 1.90004 + 2.32161i 1.36178 + 3.71777i
11.15 −1.30529 − 0.544263i 1.25014 − 1.19881i 1.40756 + 1.42084i 1.79274i −2.28426 + 0.884390i −2.43772 − 1.40742i −1.06395 − 2.62069i 0.125699 − 2.99737i 0.975721 − 2.34004i
11.16 −1.29647 − 0.564950i 1.46938 + 0.917012i 1.36166 + 1.46488i − 4.02699i −1.38695 − 2.01901i −2.34202 − 1.35217i −0.937772 − 2.66844i 1.31818 + 2.69488i −2.27505 + 5.22087i
11.17 −1.29024 − 0.579026i −1.20544 + 1.24375i 1.32946 + 1.49417i 1.18126i 2.27547 − 0.906761i −0.0841210 − 0.0485673i −0.850165 − 2.69763i −0.0938310 − 2.99853i 0.683980 − 1.52411i
11.18 −1.28188 + 0.597308i 0.151881 − 1.72538i 1.28645 − 1.53136i 2.85550i 0.835889 + 2.30245i −1.95665 − 1.12967i −0.734381 + 2.73143i −2.95386 − 0.524105i −1.70562 − 3.66042i
11.19 −1.23750 − 0.684543i −0.706340 − 1.58148i 1.06280 + 1.69424i − 3.02562i −0.208498 + 2.44060i −3.37115 − 1.94633i −0.155435 − 2.82415i −2.00217 + 2.23413i −2.07117 + 3.74420i
11.20 −1.22733 + 0.702609i −1.62382 + 0.602662i 1.01268 − 1.72467i 0.824073i 1.56953 − 1.88058i −0.328516 − 0.189669i −0.0311255 + 2.82826i 2.27360 − 1.95723i −0.579001 − 1.01141i
See next 80 embeddings (of 232 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
171.n odd 6 1 inner
684.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.o.a ✓ 232
4.b odd 2 1 inner 684.2.o.a ✓ 232
9.d odd 6 1 684.2.bi.a yes 232
19.c even 3 1 684.2.bi.a yes 232
36.h even 6 1 684.2.bi.a yes 232
76.g odd 6 1 684.2.bi.a yes 232
171.n odd 6 1 inner 684.2.o.a ✓ 232
684.o even 6 1 inner 684.2.o.a ✓ 232
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.o.a ✓ 232 1.a even 1 1 trivial
684.2.o.a ✓ 232 4.b odd 2 1 inner
684.2.o.a ✓ 232 171.n odd 6 1 inner
684.2.o.a ✓ 232 684.o even 6 1 inner
684.2.bi.a yes 232 9.d odd 6 1
684.2.bi.a yes 232 19.c even 3 1
684.2.bi.a yes 232 36.h even 6 1
684.2.bi.a yes 232 76.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).