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Properties

Label 348.2.s.a
Level $348$
Weight $2$
Character orbit 348.s
Analytic conductor $2.779$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,2,Mod(23,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 7, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 348.s (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: \(2.77879399034\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(56\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 10 q^{4} - 7 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 10 q^{4} - 7 q^{6} - 10 q^{9} - 18 q^{10} - 24 q^{12} - 20 q^{13} - 26 q^{16} + 32 q^{18} - 14 q^{21} - 12 q^{22} - 13 q^{24} - 76 q^{28} - 10 q^{30} + 2 q^{33} - 12 q^{34} + 9 q^{36} - 36 q^{37} - 14 q^{40} + 40 q^{42} - 38 q^{45} - 96 q^{46} - 34 q^{48} + 4 q^{49} - 76 q^{52} + 21 q^{54} - 36 q^{57} - 98 q^{58} + 62 q^{60} - 4 q^{61} - 106 q^{64} + 27 q^{66} - 6 q^{69} - 208 q^{70} - 106 q^{72} - 28 q^{73} + 32 q^{76} + 23 q^{78} - 50 q^{81} + 26 q^{82} + 111 q^{84} + 96 q^{85} + 4 q^{88} - 107 q^{90} - 142 q^{93} + 52 q^{94} + 14 q^{96} + 20 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.41018 + 0.106715i −0.511869 − 1.65469i 1.97722 − 0.300974i −2.98648 − 2.38164i 0.898408 + 2.27879i −1.19936 − 0.273747i −2.75613 + 0.635427i −2.47598 + 1.69397i 4.46564 + 3.03984i
23.2 −1.40856 + 0.126301i 1.07607 + 1.35723i 1.96810 − 0.355805i 1.33698 + 1.06621i −1.68712 − 1.77584i 1.59311 + 0.363617i −2.72725 + 0.749746i −0.684167 + 2.92094i −2.01788 − 1.33296i
23.3 −1.39891 + 0.207486i 1.42985 − 0.977519i 1.91390 − 0.580509i −0.175942 − 0.140309i −1.79740 + 1.66413i −1.15932 − 0.264607i −2.55693 + 1.20919i 1.08892 − 2.79540i 0.275239 + 0.159774i
23.4 −1.37815 + 0.317349i −0.992514 − 1.41948i 1.79858 − 0.874708i 2.67203 + 2.13087i 1.81830 + 1.64128i 2.59678 + 0.592698i −2.20112 + 1.77625i −1.02983 + 2.81770i −4.35868 − 2.08869i
23.5 −1.36951 − 0.352773i −1.72103 − 0.195078i 1.75110 + 0.966250i −0.514108 − 0.409987i 2.28815 + 0.874293i 0.165219 + 0.0377101i −2.05728 − 1.94103i 2.92389 + 0.671470i 0.559442 + 0.742844i
23.6 −1.32538 − 0.493324i 1.38338 + 1.04224i 1.51326 + 1.30768i −0.965048 − 0.769600i −1.31933 − 2.06382i −4.99754 − 1.14066i −1.36054 − 2.47970i 0.827454 + 2.88363i 0.899393 + 1.49609i
23.7 −1.30114 + 0.554114i −0.921364 + 1.46666i 1.38591 − 1.44196i −0.945183 − 0.753758i 0.386124 − 2.41887i 1.56852 + 0.358004i −1.00425 + 2.64414i −1.30218 − 2.70265i 1.64748 + 0.457003i
23.8 −1.27523 − 0.611391i −0.609325 + 1.62133i 1.25240 + 1.55932i −1.94372 − 1.55007i 1.76830 − 1.69503i 0.949440 + 0.216703i −0.643740 − 2.75420i −2.25745 − 1.97584i 1.53099 + 3.16506i
23.9 −1.26741 − 0.627441i 1.54699 − 0.778994i 1.21263 + 1.59045i 1.75747 + 1.40154i −2.44943 + 0.0166575i 1.97438 + 0.450639i −0.538988 − 2.77660i 1.78634 − 2.41019i −1.34805 − 2.87903i
23.10 −1.21794 + 0.718766i −1.73121 + 0.0540225i 0.966751 − 1.75083i 0.913425 + 0.728432i 2.06968 − 1.31013i −3.85681 − 0.880291i 0.0809899 + 2.82727i 2.99416 − 0.187048i −1.63607 − 0.230647i
23.11 −1.18418 − 0.773115i −0.410048 − 1.68281i 0.804588 + 1.83102i 2.41802 + 1.92830i −0.815435 + 2.30978i −3.56005 − 0.812558i 0.462809 − 2.79031i −2.66372 + 1.38007i −1.37258 − 4.15287i
23.12 −1.03970 + 0.958652i 1.69448 + 0.358803i 0.161972 − 1.99343i 3.00035 + 2.39270i −2.10573 + 1.25137i −1.36080 − 0.310593i 1.74260 + 2.22785i 2.74252 + 1.21597i −5.41325 + 0.388591i
23.13 −1.02174 + 0.977779i 0.731482 − 1.57001i 0.0878951 − 1.99807i −0.459747 − 0.366636i 0.787741 + 2.31937i 4.01016 + 0.915293i 1.86386 + 2.12744i −1.92987 − 2.29687i 0.828229 − 0.0749253i
23.14 −1.01724 − 0.982461i 0.410048 + 1.68281i 0.0695419 + 1.99879i 2.41802 + 1.92830i 1.23618 − 2.11468i 3.56005 + 0.812558i 1.89299 − 2.10157i −2.66372 + 1.38007i −0.565213 − 4.33715i
23.15 −0.975310 + 1.02409i 0.527427 + 1.64979i −0.0975396 − 1.99762i −2.37257 − 1.89206i −2.20395 − 1.06893i −3.10697 − 0.709146i 2.14088 + 1.84841i −2.44364 + 1.74029i 4.25164 − 0.584389i
23.16 −0.893734 − 1.09601i −1.54699 + 0.778994i −0.402477 + 1.95908i 1.75747 + 1.40154i 2.23638 + 0.999300i −1.97438 − 0.450639i 2.50688 − 1.30978i 1.78634 − 2.41019i −0.0346141 − 3.17881i
23.17 −0.879827 − 1.10721i 0.609325 − 1.62133i −0.451810 + 1.94830i −1.94372 − 1.55007i −2.33125 + 0.751844i −0.949440 − 0.216703i 2.55468 − 1.21392i −2.25745 − 1.97584i −0.00610475 + 3.51589i
23.18 −0.781391 + 1.17874i −1.19101 + 1.25757i −0.778855 − 1.84211i 2.37257 + 1.89206i −0.551701 − 2.38655i 3.10697 + 0.709146i 2.77996 + 0.521346i −0.162972 − 2.99557i −4.08415 + 1.31820i
23.19 −0.775880 − 1.18238i −1.38338 − 1.04224i −0.796022 + 1.83476i −0.965048 − 0.769600i −0.158991 + 2.44432i 4.99754 + 1.14066i 2.78699 − 0.482358i 0.827454 + 2.88363i −0.161195 + 1.73817i
23.20 −0.725906 + 1.21370i 0.0221596 − 1.73191i −0.946120 − 1.76206i 0.459747 + 0.366636i 2.08593 + 1.28410i −4.01016 − 0.915293i 2.82540 + 0.130788i −2.99902 − 0.0767568i −0.778717 + 0.291850i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
29.d even 7 1 inner
87.j odd 14 1 inner
116.j odd 14 1 inner
348.s even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.2.s.a ✓ 336
3.b odd 2 1 inner 348.2.s.a ✓ 336
4.b odd 2 1 inner 348.2.s.a ✓ 336
12.b even 2 1 inner 348.2.s.a ✓ 336
29.d even 7 1 inner 348.2.s.a ✓ 336
87.j odd 14 1 inner 348.2.s.a ✓ 336
116.j odd 14 1 inner 348.2.s.a ✓ 336
348.s even 14 1 inner 348.2.s.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.2.s.a ✓ 336 1.a even 1 1 trivial
348.2.s.a ✓ 336 3.b odd 2 1 inner
348.2.s.a ✓ 336 4.b odd 2 1 inner
348.2.s.a ✓ 336 12.b even 2 1 inner
348.2.s.a ✓ 336 29.d even 7 1 inner
348.2.s.a ✓ 336 87.j odd 14 1 inner
348.2.s.a ✓ 336 116.j odd 14 1 inner
348.2.s.a ✓ 336 348.s even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(348, [\chi])\).