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Properties

Label 154.3.g.a
Level $154$
Weight $3$
Character orbit 154.g
Analytic conductor $4.196$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [154,3,Mod(65,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 154.g (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: \(4.19619607115\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} + 4 q^{5} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} + 4 q^{5} - 40 q^{9} - 6 q^{11} - 24 q^{14} + 128 q^{15} - 64 q^{16} + 16 q^{20} + 40 q^{22} + 20 q^{23} - 140 q^{25} - 16 q^{26} + 120 q^{27} + 40 q^{31} + 50 q^{33} - 16 q^{34} - 160 q^{36} - 116 q^{37} - 40 q^{38} + 48 q^{42} + 12 q^{44} - 284 q^{45} + 216 q^{47} + 248 q^{49} + 56 q^{53} - 372 q^{55} - 48 q^{56} + 104 q^{58} - 4 q^{59} + 128 q^{60} - 256 q^{64} - 160 q^{66} - 68 q^{67} - 208 q^{69} + 80 q^{70} - 184 q^{71} - 244 q^{75} - 362 q^{77} + 288 q^{78} + 16 q^{80} + 384 q^{81} + 224 q^{82} + 72 q^{86} + 40 q^{88} - 592 q^{89} + 804 q^{91} + 80 q^{92} - 68 q^{93} + 376 q^{97} + 432 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 −1.22474 − 0.707107i −2.62199 − 4.54141i 1.00000 + 1.73205i −4.02718 + 6.97527i 7.41610i 6.90190 − 1.16780i − 2.82843i −9.24963 + 16.0208i 9.86453 − 5.69529i
65.2 −1.22474 − 0.707107i −1.72227 − 2.98307i 1.00000 + 1.73205i 3.17059 − 5.49162i 4.87133i 6.99788 + 0.172128i − 2.82843i −1.43245 + 2.48108i −7.76632 + 4.48389i
65.3 −1.22474 − 0.707107i −1.36264 − 2.36017i 1.00000 + 1.73205i 0.281703 − 0.487923i 3.85414i −6.90499 − 1.14943i − 2.82843i 0.786401 − 1.36209i −0.690028 + 0.398388i
65.4 −1.22474 − 0.707107i −0.248487 − 0.430392i 1.00000 + 1.73205i −2.88194 + 4.99167i 0.702828i −2.81565 − 6.40875i − 2.82843i 4.37651 − 7.58033i 7.05929 − 4.07568i
65.5 −1.22474 − 0.707107i 0.179185 + 0.310358i 1.00000 + 1.73205i 3.00671 − 5.20777i − 0.506812i −0.287101 + 6.99411i − 2.82843i 4.43579 − 7.68301i −7.36489 + 4.25212i
65.6 −1.22474 − 0.707107i 1.77533 + 3.07496i 1.00000 + 1.73205i −0.784929 + 1.35954i − 5.02139i 6.54162 − 2.49144i − 2.82843i −1.80360 + 3.12393i 1.92268 − 1.11006i
65.7 −1.22474 − 0.707107i 1.83765 + 3.18290i 1.00000 + 1.73205i 4.33390 − 7.50654i − 5.19765i −3.85425 − 5.84335i − 2.82843i −2.25390 + 3.90387i −10.6159 + 6.12906i
65.8 −1.22474 − 0.707107i 2.16323 + 3.74682i 1.00000 + 1.73205i −2.09885 + 3.63531i − 6.11853i −4.12992 + 5.65188i − 2.82843i −4.85911 + 8.41622i 5.14111 − 2.96822i
65.9 1.22474 + 0.707107i −2.62199 − 4.54141i 1.00000 + 1.73205i −4.02718 + 6.97527i − 7.41610i −6.90190 + 1.16780i 2.82843i −9.24963 + 16.0208i −9.86453 + 5.69529i
65.10 1.22474 + 0.707107i −1.72227 − 2.98307i 1.00000 + 1.73205i 3.17059 − 5.49162i − 4.87133i −6.99788 − 0.172128i 2.82843i −1.43245 + 2.48108i 7.76632 − 4.48389i
65.11 1.22474 + 0.707107i −1.36264 − 2.36017i 1.00000 + 1.73205i 0.281703 − 0.487923i − 3.85414i 6.90499 + 1.14943i 2.82843i 0.786401 − 1.36209i 0.690028 − 0.398388i
65.12 1.22474 + 0.707107i −0.248487 − 0.430392i 1.00000 + 1.73205i −2.88194 + 4.99167i − 0.702828i 2.81565 + 6.40875i 2.82843i 4.37651 − 7.58033i −7.05929 + 4.07568i
65.13 1.22474 + 0.707107i 0.179185 + 0.310358i 1.00000 + 1.73205i 3.00671 − 5.20777i 0.506812i 0.287101 − 6.99411i 2.82843i 4.43579 − 7.68301i 7.36489 − 4.25212i
65.14 1.22474 + 0.707107i 1.77533 + 3.07496i 1.00000 + 1.73205i −0.784929 + 1.35954i 5.02139i −6.54162 + 2.49144i 2.82843i −1.80360 + 3.12393i −1.92268 + 1.11006i
65.15 1.22474 + 0.707107i 1.83765 + 3.18290i 1.00000 + 1.73205i 4.33390 − 7.50654i 5.19765i 3.85425 + 5.84335i 2.82843i −2.25390 + 3.90387i 10.6159 − 6.12906i
65.16 1.22474 + 0.707107i 2.16323 + 3.74682i 1.00000 + 1.73205i −2.09885 + 3.63531i 6.11853i 4.12992 − 5.65188i 2.82843i −4.85911 + 8.41622i −5.14111 + 2.96822i
109.1 −1.22474 + 0.707107i −2.62199 + 4.54141i 1.00000 − 1.73205i −4.02718 − 6.97527i − 7.41610i 6.90190 + 1.16780i 2.82843i −9.24963 − 16.0208i 9.86453 + 5.69529i
109.2 −1.22474 + 0.707107i −1.72227 + 2.98307i 1.00000 − 1.73205i 3.17059 + 5.49162i − 4.87133i 6.99788 − 0.172128i 2.82843i −1.43245 − 2.48108i −7.76632 − 4.48389i
109.3 −1.22474 + 0.707107i −1.36264 + 2.36017i 1.00000 − 1.73205i 0.281703 + 0.487923i − 3.85414i −6.90499 + 1.14943i 2.82843i 0.786401 + 1.36209i −0.690028 − 0.398388i
109.4 −1.22474 + 0.707107i −0.248487 + 0.430392i 1.00000 − 1.73205i −2.88194 − 4.99167i − 0.702828i −2.81565 + 6.40875i 2.82843i 4.37651 + 7.58033i 7.05929 + 4.07568i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.b odd 2 1 inner
77.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.3.g.a ✓ 32
7.c even 3 1 inner 154.3.g.a ✓ 32
7.c even 3 1 1078.3.d.d 16
7.d odd 6 1 1078.3.d.e 16
11.b odd 2 1 inner 154.3.g.a ✓ 32
77.h odd 6 1 inner 154.3.g.a ✓ 32
77.h odd 6 1 1078.3.d.d 16
77.i even 6 1 1078.3.d.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.3.g.a ✓ 32 1.a even 1 1 trivial
154.3.g.a ✓ 32 7.c even 3 1 inner
154.3.g.a ✓ 32 11.b odd 2 1 inner
154.3.g.a ✓ 32 77.h odd 6 1 inner
1078.3.d.d 16 7.c even 3 1
1078.3.d.d 16 77.h odd 6 1
1078.3.d.e 16 7.d odd 6 1
1078.3.d.e 16 77.i even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(154, [\chi])\).