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Properties

Label 840.2.dc.e
Level $840$
Weight $2$
Character orbit 840.dc
Analytic conductor $6.707$
Analytic rank $0$
Dimension $704$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [840,2,Mod(53,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(840, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("840.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.dc (of order \(12\), degree \(4\), 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: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(704\)
Relative dimension: \(176\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 704 q - 16 q^{6} - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 704 q - 16 q^{6} - 24 q^{7} + 12 q^{10} + 10 q^{12} + 32 q^{15} - 72 q^{16} + 6 q^{18} + 64 q^{22} - 8 q^{25} + 24 q^{28} + 8 q^{30} - 16 q^{31} + 44 q^{33} - 240 q^{36} + 16 q^{40} - 78 q^{42} - 8 q^{46} - 20 q^{48} - 60 q^{52} + 272 q^{55} + 8 q^{57} - 20 q^{58} - 14 q^{60} + 12 q^{63} - 84 q^{66} + 112 q^{70} - 38 q^{72} + 104 q^{73} - 64 q^{76} - 16 q^{78} - 168 q^{81} - 72 q^{82} - 64 q^{87} + 52 q^{88} - 44 q^{90} + 40 q^{96} - 80 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.41409 − 0.0185834i −1.24311 − 1.20610i 1.99931 + 0.0525572i −2.23234 − 0.129106i 1.73545 + 1.72864i −2.05258 − 1.66941i −2.82623 − 0.111475i 0.0906294 + 2.99863i 3.15433 + 0.224052i
53.2 −1.41407 − 0.0200013i −1.15172 + 1.29365i 1.99920 + 0.0565665i −1.06073 + 1.96846i 1.65449 − 1.80628i −0.870523 + 2.49844i −2.82588 − 0.119976i −0.347078 − 2.97986i 1.53932 − 2.76233i
53.3 −1.41322 − 0.0530451i −0.496690 + 1.65931i 1.99437 + 0.149929i 2.15598 − 0.593069i 0.789949 − 2.31862i −1.93428 + 1.80515i −2.81053 − 0.317673i −2.50660 − 1.64832i −3.07834 + 0.723772i
53.4 −1.40906 + 0.120630i 1.62994 + 0.585913i 1.97090 − 0.339951i 1.18933 + 1.89354i −2.36736 − 0.628966i −0.390444 + 2.61678i −2.73610 + 0.716761i 2.31341 + 1.91001i −1.90426 − 2.52464i
53.5 −1.40857 + 0.126222i 1.11026 − 1.32941i 1.96814 − 0.355585i 1.61503 − 1.54650i −1.39607 + 2.01271i −2.64070 − 0.163401i −2.72737 + 0.749289i −0.534667 − 2.95197i −2.07968 + 2.38221i
53.6 −1.40478 − 0.163037i 0.0253219 − 1.73187i 1.94684 + 0.458064i −1.11326 − 1.93924i −0.317930 + 2.42877i 0.130595 + 2.64253i −2.66021 − 0.960888i −2.99872 − 0.0877081i 1.24772 + 2.90572i
53.7 −1.40361 + 0.172855i 1.72599 + 0.144732i 1.94024 − 0.485243i 1.89185 + 1.19202i −2.44764 + 0.0951994i −0.719395 − 2.54607i −2.63947 + 1.01647i 2.95811 + 0.499614i −2.86146 − 1.34612i
53.8 −1.40002 + 0.199859i −0.347199 + 1.69690i 1.92011 − 0.559612i −0.272779 − 2.21937i 0.146947 − 2.44508i 1.62543 − 2.08758i −2.57635 + 1.16722i −2.75891 − 1.17832i 0.825456 + 3.05264i
53.9 −1.39807 + 0.213054i −1.61554 − 0.624519i 1.90922 − 0.595730i 0.0578016 + 2.23532i 2.39170 + 0.528926i 2.46301 − 0.966209i −2.54230 + 1.23964i 2.21995 + 2.01787i −0.557055 − 3.11283i
53.10 −1.38686 + 0.276798i 1.10919 + 1.33030i 1.84677 − 0.767762i −0.568440 − 2.16261i −1.90652 − 1.53792i −2.56957 − 0.630305i −2.34869 + 1.57596i −0.539398 + 2.95111i 1.38695 + 2.84189i
53.11 −1.38387 + 0.291371i 0.753731 + 1.55945i 1.83021 − 0.806441i −2.12738 + 0.688660i −1.49745 − 1.93847i 2.44598 + 1.00857i −2.29780 + 1.64928i −1.86378 + 2.35082i 2.74337 − 1.57287i
53.12 −1.38192 − 0.300501i −0.579558 − 1.63221i 1.81940 + 0.830536i 2.09359 + 0.785416i 0.310421 + 2.42974i 2.61991 + 0.368854i −2.26468 − 1.69446i −2.32822 + 1.89192i −2.65715 − 1.71451i
53.13 −1.37818 − 0.317218i 0.721785 − 1.57449i 1.79874 + 0.874366i −1.35516 + 1.77864i −1.49421 + 1.94097i 1.31760 − 2.29433i −2.20162 − 1.77563i −1.95805 − 2.27289i 2.43187 − 2.02139i
53.14 −1.37787 + 0.318567i −1.72430 − 0.163657i 1.79703 − 0.877884i 1.61736 − 1.54406i 2.42799 − 0.323807i 1.51537 + 2.16879i −2.19640 + 1.78208i 2.94643 + 0.564388i −1.73662 + 2.64275i
53.15 −1.35215 − 0.414369i 1.41233 + 1.00266i 1.65660 + 1.12058i −1.35516 + 1.77864i −1.49421 − 1.94097i 1.31760 − 2.29433i −1.77563 − 2.20162i 0.989354 + 2.83217i 2.56938 − 1.84344i
53.16 −1.34703 − 0.430718i 0.314193 + 1.70332i 1.62896 + 1.16038i 2.09359 + 0.785416i 0.310421 − 2.42974i 2.61991 + 0.368854i −1.69446 − 2.26468i −2.80256 + 1.07034i −2.48183 − 1.95972i
53.17 −1.33621 + 0.463179i 0.790327 − 1.54123i 1.57093 − 1.23781i −1.62081 − 1.54045i −0.342181 + 2.42547i 1.57518 − 2.12575i −1.52577 + 2.38160i −1.75077 − 2.43615i 2.87925 + 1.30765i
53.18 −1.32190 + 0.502563i 0.000690002 1.73205i 1.49486 − 1.32868i −0.0418931 + 2.23568i −0.871376 − 2.28926i −1.63690 − 2.07860i −1.30832 + 2.50765i −3.00000 + 0.00239024i −1.06819 − 2.97640i
53.19 −1.31402 + 0.522840i −1.66172 + 0.488560i 1.45328 − 1.37404i −1.48160 − 1.67477i 1.92809 − 1.51079i −2.22954 + 1.42448i −1.19123 + 2.56534i 2.52262 − 1.62370i 2.82249 + 1.42603i
53.20 −1.30976 + 0.533423i −1.08568 − 1.34956i 1.43092 − 1.39731i 1.80217 − 1.32371i 2.14186 + 1.18847i −1.12065 − 2.39669i −1.12880 + 2.59342i −0.642612 + 2.93037i −1.65430 + 2.69505i
See next 80 embeddings (of 704 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.176
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
35.l odd 12 1 inner
40.i odd 4 1 inner
56.p even 6 1 inner
105.x even 12 1 inner
120.w even 4 1 inner
168.s odd 6 1 inner
280.bt odd 12 1 inner
840.dc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.dc.e ✓ 704
3.b odd 2 1 inner 840.2.dc.e ✓ 704
5.c odd 4 1 inner 840.2.dc.e ✓ 704
7.c even 3 1 inner 840.2.dc.e ✓ 704
8.b even 2 1 inner 840.2.dc.e ✓ 704
15.e even 4 1 inner 840.2.dc.e ✓ 704
21.h odd 6 1 inner 840.2.dc.e ✓ 704
24.h odd 2 1 inner 840.2.dc.e ✓ 704
35.l odd 12 1 inner 840.2.dc.e ✓ 704
40.i odd 4 1 inner 840.2.dc.e ✓ 704
56.p even 6 1 inner 840.2.dc.e ✓ 704
105.x even 12 1 inner 840.2.dc.e ✓ 704
120.w even 4 1 inner 840.2.dc.e ✓ 704
168.s odd 6 1 inner 840.2.dc.e ✓ 704
280.bt odd 12 1 inner 840.2.dc.e ✓ 704
840.dc even 12 1 inner 840.2.dc.e ✓ 704
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.dc.e ✓ 704 1.a even 1 1 trivial
840.2.dc.e ✓ 704 3.b odd 2 1 inner
840.2.dc.e ✓ 704 5.c odd 4 1 inner
840.2.dc.e ✓ 704 7.c even 3 1 inner
840.2.dc.e ✓ 704 8.b even 2 1 inner
840.2.dc.e ✓ 704 15.e even 4 1 inner
840.2.dc.e ✓ 704 21.h odd 6 1 inner
840.2.dc.e ✓ 704 24.h odd 2 1 inner
840.2.dc.e ✓ 704 35.l odd 12 1 inner
840.2.dc.e ✓ 704 40.i odd 4 1 inner
840.2.dc.e ✓ 704 56.p even 6 1 inner
840.2.dc.e ✓ 704 105.x even 12 1 inner
840.2.dc.e ✓ 704 120.w even 4 1 inner
840.2.dc.e ✓ 704 168.s odd 6 1 inner
840.2.dc.e ✓ 704 280.bt odd 12 1 inner
840.2.dc.e ✓ 704 840.dc even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{176} + 490 T_{11}^{174} + 125192 T_{11}^{172} + 21957292 T_{11}^{170} + 2950731880 T_{11}^{168} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
\( T_{53}^{352} - 212468 T_{53}^{348} + 24576952368 T_{53}^{344} + \cdots + 67\!\cdots\!16 \) Copy content Toggle raw display