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Properties

Label 684.2.cc.a
Level $684$
Weight $2$
Character orbit 684.cc
Analytic conductor $5.462$
Analytic rank $0$
Dimension $696$
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(211,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.211");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.cc (of order \(18\), degree \(6\), 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: \(696\)
Relative dimension: \(116\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 696 q - 3 q^{2} - 3 q^{4} - 6 q^{5} - 6 q^{6} - 18 q^{8} - 12 q^{10} - 9 q^{12} - 6 q^{13} - 15 q^{14} - 3 q^{16} - 24 q^{17} - 6 q^{20} - 30 q^{21} + 21 q^{22} + 24 q^{24} - 6 q^{25} - 30 q^{26} - 6 q^{29}+ \cdots - 63 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1 −1.41419 − 0.00795020i 0.932010 − 1.45992i 1.99987 + 0.0224862i −0.221464 + 1.25599i −1.32965 + 2.05719i 2.39632 + 1.38352i −2.82802 − 0.0476992i −1.26271 − 2.72131i 0.323178 − 1.77444i
211.2 −1.41416 − 0.0122028i −1.57703 + 0.716213i 1.99970 + 0.0345136i −0.487934 + 2.76721i 2.23892 − 0.993596i −1.71049 − 0.987554i −2.82748 − 0.0732098i 1.97408 − 2.25899i 0.723785 − 3.90733i
211.3 −1.41415 + 0.0137416i 1.44144 − 0.960342i 1.99962 − 0.0388654i −0.428293 + 2.42897i −2.02521 + 1.37787i −2.99634 − 1.72994i −2.82723 + 0.0824394i 1.15549 − 2.76855i 0.572291 − 3.44081i
211.4 −1.41400 + 0.0246459i 0.412982 + 1.68210i 1.99879 − 0.0696986i 0.504900 − 2.86343i −0.625413 − 2.36830i −0.170714 − 0.0985616i −2.82456 + 0.147816i −2.65889 + 1.38935i −0.643356 + 4.06133i
211.5 −1.40991 − 0.110292i −0.721709 − 1.57453i 1.97567 + 0.311003i 0.251320 − 1.42531i 0.843885 + 2.29953i −0.441456 − 0.254875i −2.75121 − 0.656387i −1.95827 + 2.27270i −0.511538 + 1.98183i
211.6 −1.39629 + 0.224416i 1.71078 − 0.270635i 1.89927 − 0.626702i 0.541604 − 3.07159i −2.32801 + 0.761812i 2.74289 + 1.58361i −2.51130 + 1.30129i 2.85351 − 0.925992i −0.0669241 + 4.41039i
211.7 −1.38230 − 0.298732i 1.72024 + 0.201885i 1.82152 + 0.825876i 0.221266 − 1.25486i −2.31759 − 0.792958i −3.98072 − 2.29827i −2.27117 − 1.68576i 2.91848 + 0.694584i −0.680724 + 1.66850i
211.8 −1.37521 + 0.329836i −1.73191 + 0.0221199i 1.78242 − 0.907188i 0.0398193 − 0.225827i 2.37445 − 0.601665i 4.04735 + 2.33674i −2.15198 + 1.83548i 2.99902 − 0.0766194i 0.0197256 + 0.323693i
211.9 −1.37319 + 0.338140i −1.45909 − 0.933306i 1.77132 − 0.928663i 0.279827 − 1.58698i 2.31920 + 0.788234i −2.00663 − 1.15853i −2.11835 + 1.87419i 1.25788 + 2.72355i 0.152364 + 2.27385i
211.10 −1.36818 − 0.357897i 0.308394 + 1.70437i 1.74382 + 0.979332i −0.488317 + 2.76938i 0.188052 − 2.44226i 1.81309 + 1.04679i −2.03536 − 1.96401i −2.80979 + 1.05124i 1.65926 − 3.61424i
211.11 −1.36543 + 0.368223i 1.40642 + 1.01093i 1.72882 − 1.00557i −0.560736 + 3.18009i −2.29263 − 0.862475i −0.491177 − 0.283581i −1.99032 + 2.00963i 0.956059 + 2.84358i −0.405334 − 4.54868i
211.12 −1.35669 + 0.399256i −0.883299 + 1.48989i 1.68119 − 1.08333i 0.190011 − 1.07761i 0.603510 − 2.37398i −2.34296 − 1.35271i −1.84832 + 2.14096i −1.43957 − 2.63204i 0.172456 + 1.53784i
211.13 −1.33319 − 0.471815i −1.43900 + 0.963992i 1.55478 + 1.25804i 0.0439475 − 0.249239i 2.37328 − 0.606241i 0.176578 + 0.101947i −1.47925 − 2.41077i 1.14144 − 2.77437i −0.176185 + 0.311547i
211.14 −1.32811 + 0.485935i −0.901949 − 1.47868i 1.52773 − 1.29075i −0.665002 + 3.77142i 1.91643 + 1.52555i −1.66340 − 0.960363i −1.40178 + 2.45663i −1.37298 + 2.66738i −0.949468 − 5.33199i
211.15 −1.30169 − 0.552805i −1.31885 − 1.12278i 1.38881 + 1.43917i −0.612489 + 3.47360i 1.09605 + 2.19059i 3.65674 + 2.11122i −1.01223 − 2.64110i 0.478708 + 2.96156i 2.71750 − 4.18297i
211.16 −1.29054 − 0.578360i −1.72737 − 0.127210i 1.33100 + 1.49280i 0.745816 − 4.22973i 2.15568 + 1.16321i 0.448734 + 0.259077i −0.854340 − 2.69631i 2.96764 + 0.439479i −3.40881 + 5.02730i
211.17 −1.28452 − 0.591617i −0.0622019 − 1.73093i 1.29998 + 1.51989i 0.494013 − 2.80169i −0.944151 + 2.26022i 2.08413 + 1.20327i −0.770655 − 2.72141i −2.99226 + 0.215335i −2.29210 + 3.30655i
211.18 −1.27379 − 0.614383i 1.70410 + 0.309904i 1.24507 + 1.56519i −0.202951 + 1.15099i −1.98026 − 1.44172i 1.91680 + 1.10667i −0.624325 − 2.75866i 2.80792 + 1.05622i 0.965667 − 1.34143i
211.19 −1.25457 + 0.652729i 0.528477 − 1.64946i 1.14789 − 1.63779i 0.600364 − 3.40483i 0.413638 + 2.41431i −2.98853 − 1.72543i −0.371075 + 2.80398i −2.44142 − 1.74340i 1.46924 + 4.66348i
211.20 −1.24172 + 0.676851i 1.50688 + 0.853991i 1.08375 − 1.68092i 0.189825 − 1.07655i −2.44916 − 0.0404831i 0.295212 + 0.170441i −0.207978 + 2.82077i 1.54140 + 2.57373i 0.492954 + 1.46526i
See next 80 embeddings (of 696 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
171.bc odd 18 1 inner
684.cc even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.cc.a yes 696
4.b odd 2 1 inner 684.2.cc.a yes 696
9.c even 3 1 684.2.bt.a ✓ 696
19.f odd 18 1 684.2.bt.a ✓ 696
36.f odd 6 1 684.2.bt.a ✓ 696
76.k even 18 1 684.2.bt.a ✓ 696
171.bc odd 18 1 inner 684.2.cc.a yes 696
684.cc even 18 1 inner 684.2.cc.a yes 696
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.bt.a ✓ 696 9.c even 3 1
684.2.bt.a ✓ 696 19.f odd 18 1
684.2.bt.a ✓ 696 36.f odd 6 1
684.2.bt.a ✓ 696 76.k even 18 1
684.2.cc.a yes 696 1.a even 1 1 trivial
684.2.cc.a yes 696 4.b odd 2 1 inner
684.2.cc.a yes 696 171.bc odd 18 1 inner
684.2.cc.a yes 696 684.cc even 18 1 inner

Hecke kernels

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