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Properties

Label 342.3.o.a
Level $342$
Weight $3$
Character orbit 342.o
Analytic conductor $9.319$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,3,Mod(77,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.77");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.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: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{3} + 72 q^{4} + 36 q^{5} + 16 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 4 q^{3} + 72 q^{4} + 36 q^{5} + 16 q^{6} - 20 q^{9} - 72 q^{11} - 8 q^{12} - 72 q^{14} - 80 q^{15} - 144 q^{16} - 16 q^{18} + 72 q^{20} - 36 q^{21} + 108 q^{23} + 16 q^{24} + 192 q^{25} - 20 q^{27} + 72 q^{29} - 56 q^{30} - 60 q^{31} - 124 q^{33} - 32 q^{36} - 168 q^{37} + 36 q^{39} + 144 q^{41} + 8 q^{42} + 104 q^{45} + 48 q^{46} - 36 q^{47} - 32 q^{48} - 204 q^{49} + 424 q^{51} + 40 q^{54} + 216 q^{55} - 144 q^{56} - 48 q^{58} - 180 q^{59} - 8 q^{60} - 96 q^{61} - 272 q^{63} - 576 q^{64} + 108 q^{65} + 96 q^{66} + 156 q^{67} - 232 q^{69} + 216 q^{70} + 32 q^{72} - 216 q^{74} + 168 q^{75} - 684 q^{77} + 16 q^{78} - 96 q^{79} + 580 q^{81} - 96 q^{82} + 288 q^{83} + 168 q^{84} - 216 q^{85} - 288 q^{86} + 104 q^{87} + 512 q^{90} - 336 q^{91} + 216 q^{92} - 116 q^{93} - 168 q^{94} - 32 q^{96} - 12 q^{97} - 464 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −1.22474 + 0.707107i −2.95399 − 0.523385i 1.00000 − 1.73205i 7.85076 + 4.53264i 3.98798 − 1.44777i 3.67226 + 6.36054i 2.82843i 8.45214 + 3.09215i −12.8202
77.2 −1.22474 + 0.707107i −2.95374 + 0.524798i 1.00000 − 1.73205i −4.20893 − 2.43003i 3.24649 − 2.73135i 4.03473 + 6.98835i 2.82843i 8.44917 − 3.10024i 6.87315
77.3 −1.22474 + 0.707107i −2.94466 + 0.573576i 1.00000 − 1.73205i −2.92566 − 1.68913i 3.20087 − 2.78467i −6.13122 − 10.6196i 2.82843i 8.34202 − 3.37797i 4.77759
77.4 −1.22474 + 0.707107i −2.58582 − 1.52103i 1.00000 − 1.73205i −3.52789 − 2.03683i 4.24250 + 0.0344220i 2.31286 + 4.00598i 2.82843i 4.37294 + 7.86622i 5.76102
77.5 −1.22474 + 0.707107i −2.05928 + 2.18160i 1.00000 − 1.73205i 4.90398 + 2.83131i 0.979461 − 4.12803i −1.61102 − 2.79037i 2.82843i −0.518772 − 8.98504i −8.00816
77.6 −1.22474 + 0.707107i −1.90664 − 2.31619i 1.00000 − 1.73205i 0.157740 + 0.0910713i 3.97294 + 1.48854i 0.597857 + 1.03552i 2.82843i −1.72943 + 8.83227i −0.257589
77.7 −1.22474 + 0.707107i −1.58931 + 2.54443i 1.00000 − 1.73205i 1.71286 + 0.988922i 0.147314 − 4.24008i 1.19087 + 2.06264i 2.82843i −3.94821 − 8.08775i −2.79709
77.8 −1.22474 + 0.707107i −0.253326 − 2.98929i 1.00000 − 1.73205i −6.88071 − 3.97258i 2.42400 + 3.48198i −4.24096 − 7.34556i 2.82843i −8.87165 + 1.51453i 11.2362
77.9 −1.22474 + 0.707107i −0.192054 − 2.99385i 1.00000 − 1.73205i 3.00379 + 1.73424i 2.35219 + 3.53089i 2.44013 + 4.22643i 2.82843i −8.92623 + 1.14996i −4.90516
77.10 −1.22474 + 0.707107i 0.325621 − 2.98228i 1.00000 − 1.73205i 6.82844 + 3.94240i 1.70999 + 3.88278i −5.59683 − 9.69399i 2.82843i −8.78794 − 1.94218i −11.1508
77.11 −1.22474 + 0.707107i 0.654543 + 2.92772i 1.00000 − 1.73205i −2.99435 − 1.72879i −2.87186 − 3.12288i −0.735113 − 1.27325i 2.82843i −8.14315 + 3.83265i 4.88975
77.12 −1.22474 + 0.707107i 0.798749 + 2.89171i 1.00000 − 1.73205i 1.81339 + 1.04696i −3.02301 − 2.97681i 6.73480 + 11.6650i 2.82843i −7.72400 + 4.61950i −2.96125
77.13 −1.22474 + 0.707107i 1.21556 + 2.74270i 1.00000 − 1.73205i 6.93730 + 4.00525i −3.42814 − 2.49957i −4.54800 − 7.87737i 2.82843i −6.04481 + 6.66786i −11.3286
77.14 −1.22474 + 0.707107i 1.73465 − 2.44765i 1.00000 − 1.73205i −4.17227 − 2.40886i −0.393757 + 4.22433i 0.644228 + 1.11584i 2.82843i −2.98196 − 8.49164i 6.81329
77.15 −1.22474 + 0.707107i 2.35664 − 1.85641i 1.00000 − 1.73205i 1.73196 + 0.999946i −1.57360 + 3.94002i 2.62568 + 4.54780i 2.82843i 2.10749 − 8.74977i −2.82827
77.16 −1.22474 + 0.707107i 2.94852 − 0.553373i 1.00000 − 1.73205i 0.727123 + 0.419805i −3.21989 + 2.76266i −4.07397 − 7.05632i 2.82843i 8.38756 − 3.26327i −1.18739
77.17 −1.22474 + 0.707107i 2.95917 + 0.493284i 1.00000 − 1.73205i 5.94620 + 3.43304i −3.97303 + 1.48830i 3.95741 + 6.85444i 2.82843i 8.51334 + 2.91942i −9.71010
77.18 −1.22474 + 0.707107i 2.99587 + 0.157350i 1.00000 − 1.73205i −7.90371 − 4.56321i −3.78044 + 1.92569i 6.07478 + 10.5218i 2.82843i 8.95048 + 0.942803i 12.9067
77.19 1.22474 − 0.707107i −2.99120 + 0.229665i 1.00000 − 1.73205i −2.75548 − 1.59088i −3.50105 + 2.39638i −1.69784 − 2.94075i − 2.82843i 8.89451 − 1.37395i −4.49968
77.20 1.22474 − 0.707107i −2.94222 − 0.585975i 1.00000 − 1.73205i 1.61759 + 0.933914i −4.01781 + 1.36279i −0.485140 − 0.840287i − 2.82843i 8.31327 + 3.44813i 2.64151
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.o.a ✓ 72
3.b odd 2 1 1026.3.o.a 72
9.c even 3 1 1026.3.o.a 72
9.d odd 6 1 inner 342.3.o.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.3.o.a ✓ 72 1.a even 1 1 trivial
342.3.o.a ✓ 72 9.d odd 6 1 inner
1026.3.o.a 72 3.b odd 2 1
1026.3.o.a 72 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(342, [\chi])\).