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Properties

Label 213.3.k.a
Level $213$
Weight $3$
Character orbit 213.k
Analytic conductor $5.804$
Analytic rank $0$
Dimension $276$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,3,Mod(20,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.20");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 213.k (of order \(14\), 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: \(5.80382963087\)
Analytic rank: \(0\)
Dimension: \(276\)
Relative dimension: \(46\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 276 q - 10 q^{3} + 78 q^{4} - 7 q^{6} - 10 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 276 q - 10 q^{3} + 78 q^{4} - 7 q^{6} - 10 q^{7} + 24 q^{9} + 22 q^{10} - 66 q^{12} - 14 q^{13} + 56 q^{15} - 302 q^{16} + 30 q^{18} + 10 q^{19} + 148 q^{21} + 14 q^{22} + 12 q^{24} - 1236 q^{25} + 221 q^{27} - 18 q^{28} + 163 q^{30} - 150 q^{31} - 109 q^{33} + 46 q^{34} + 113 q^{36} - 42 q^{37} - 3 q^{39} + 80 q^{40} - 54 q^{42} + 280 q^{43} + 204 q^{45} + 108 q^{46} - 567 q^{48} - 164 q^{49} - 220 q^{51} + 444 q^{52} + 86 q^{54} + 606 q^{55} - 620 q^{57} + 322 q^{58} + 318 q^{60} + 174 q^{61} - 641 q^{63} - 302 q^{64} - 592 q^{66} + 66 q^{67} - 170 q^{69} + 1128 q^{70} + 1542 q^{72} + 90 q^{73} - 10 q^{75} - 1120 q^{76} + 59 q^{78} + 366 q^{79} + 324 q^{81} + 1294 q^{82} + 1384 q^{84} + 744 q^{85} + 472 q^{87} - 416 q^{88} - 3 q^{90} - 466 q^{91} - 60 q^{93} - 394 q^{94} - 686 q^{96} + 62 q^{97} - 641 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −3.69682 + 0.843775i −2.40594 + 1.79205i 9.35066 − 4.50304i − 8.69429i 7.38225 − 8.65497i 1.55871 − 6.82914i −18.9097 + 15.0800i 2.57711 − 8.62314i 7.33603 + 32.1413i
20.2 −3.57736 + 0.816509i −0.363169 + 2.97794i 8.52694 − 4.10636i 6.08074i −1.13232 − 10.9497i −0.962455 + 4.21679i −15.6758 + 12.5010i −8.73622 − 2.16299i −4.96498 − 21.7530i
20.3 −3.56993 + 0.814814i −1.56680 − 2.55835i 8.47663 − 4.08213i − 0.602619i 7.67794 + 7.85649i −0.677346 + 2.96765i −15.4834 + 12.3476i −4.09030 + 8.01682i 0.491023 + 2.15131i
20.4 −3.41784 + 0.780099i 2.93500 + 0.621129i 7.46918 − 3.59697i − 2.22871i −10.5159 + 0.166669i −0.824517 + 3.61245i −11.7589 + 9.37738i 8.22840 + 3.64602i 1.73862 + 7.61737i
20.5 −3.15331 + 0.719722i −2.99434 − 0.184250i 5.82147 − 2.80347i 8.38608i 9.57467 − 1.57409i 1.78906 − 7.83838i −6.22413 + 4.96358i 8.93210 + 1.10341i −6.03564 − 26.4439i
20.6 −3.07131 + 0.701006i 1.61373 + 2.52901i 5.33764 − 2.57047i 0.999591i −6.72912 − 6.63612i 1.72452 − 7.55562i −4.73964 + 3.77973i −3.79173 + 8.16228i −0.700719 − 3.07005i
20.7 −2.88694 + 0.658925i 2.23928 − 1.99640i 4.29636 − 2.06902i 8.78076i −5.14920 + 7.23900i −2.98442 + 13.0756i −1.77943 + 1.41905i 1.02878 − 8.94101i −5.78587 − 25.3495i
20.8 −2.83692 + 0.647509i 0.550124 − 2.94913i 4.02498 − 1.93833i 3.36116i 0.348930 + 8.72266i 1.86432 − 8.16814i −1.06332 + 0.847968i −8.39473 − 3.24477i −2.17638 − 9.53535i
20.9 −2.79759 + 0.638533i 2.05772 − 2.18307i 3.81494 − 1.83718i − 7.53385i −4.36271 + 7.42126i 0.543873 − 2.38286i −0.525562 + 0.419122i −0.531560 − 8.98429i 4.81061 + 21.0767i
20.10 −2.59161 + 0.591517i −2.99803 − 0.108715i 2.76265 − 1.33042i − 0.701696i 7.83402 − 1.49164i −2.28261 + 10.0008i 1.94048 − 1.54748i 8.97636 + 0.651864i 0.415065 + 1.81852i
20.11 −2.45620 + 0.560612i −1.10144 + 2.79049i 2.11476 − 1.01842i − 4.07011i 1.14097 − 7.47148i −1.93497 + 8.47768i 3.25553 − 2.59620i −6.57366 − 6.14711i 2.28175 + 9.99701i
20.12 −1.98918 + 0.454017i −2.36316 + 1.84810i 0.146831 − 0.0707101i 1.45858i 3.86169 − 4.74912i 1.68121 − 7.36586i 6.12083 − 4.88120i 2.16907 − 8.73471i −0.662221 − 2.90138i
20.13 −1.87177 + 0.427219i 2.99719 − 0.129865i −0.282876 + 0.136226i 3.95605i −5.55456 + 1.52353i 1.51952 − 6.65743i 6.47545 − 5.16400i 8.96627 − 0.778459i −1.69010 − 7.40480i
20.14 −1.87064 + 0.426961i −2.33870 − 1.87896i −0.286885 + 0.138157i − 7.60059i 5.17710 + 2.51632i 1.74939 − 7.66458i 6.47821 − 5.16620i 1.93904 + 8.78864i 3.24516 + 14.2180i
20.15 −1.72198 + 0.393031i 1.82873 + 2.37818i −0.793135 + 0.381954i − 7.77947i −4.08374 − 3.37643i −1.28770 + 5.64177i 6.73932 − 5.37443i −2.31146 + 8.69811i 3.05757 + 13.3961i
20.16 −1.62303 + 0.370447i −0.802576 − 2.89065i −1.10687 + 0.533041i 0.664793i 2.37344 + 4.39431i −0.444640 + 1.94809i 6.80531 − 5.42705i −7.71174 + 4.63994i −0.246270 − 1.07898i
20.17 −1.30334 + 0.297479i 1.52670 + 2.58248i −1.99368 + 0.960104i 6.81107i −2.75804 − 2.91168i −0.806806 + 3.53485i 6.49361 − 5.17848i −4.33837 + 7.88533i −2.02615 − 8.87714i
20.18 −0.850994 + 0.194234i 0.650342 − 2.92866i −2.91741 + 1.40495i − 1.47107i 0.0154083 + 2.61859i −2.30719 + 10.1085i 4.93959 − 3.93919i −8.15411 − 3.80926i 0.285731 + 1.25187i
20.19 −0.665865 + 0.151979i −2.65234 − 1.40181i −3.18360 + 1.53314i 3.49296i 1.97915 + 0.530318i −0.620725 + 2.71958i 4.02277 − 3.20805i 5.06983 + 7.43618i −0.530858 − 2.32584i
20.20 −0.664084 + 0.151573i 2.53602 − 1.60268i −3.18584 + 1.53422i − 4.43209i −1.44121 + 1.44871i −0.600385 + 2.63046i 4.01334 − 3.20053i 3.86281 − 8.12888i 0.671785 + 2.94328i
See next 80 embeddings (of 276 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
71.d even 7 1 inner
213.k odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.3.k.a ✓ 276
3.b odd 2 1 inner 213.3.k.a ✓ 276
71.d even 7 1 inner 213.3.k.a ✓ 276
213.k odd 14 1 inner 213.3.k.a ✓ 276
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.3.k.a ✓ 276 1.a even 1 1 trivial
213.3.k.a ✓ 276 3.b odd 2 1 inner
213.3.k.a ✓ 276 71.d even 7 1 inner
213.3.k.a ✓ 276 213.k odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(213, [\chi])\).