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Properties

Label 931.2.e.f
Level $931$
Weight $2$
Character orbit 931.e
Analytic conductor $7.434$
Analytic rank $0$
Dimension $24$
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] = [931,2,Mod(197,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.e (of order \(3\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} + 3 q^{3} - 11 q^{4} - 6 q^{6} - 18 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} + 3 q^{3} - 11 q^{4} - 6 q^{6} - 18 q^{8} - 9 q^{9} + 16 q^{10} + 2 q^{11} + 4 q^{12} + 6 q^{13} - 9 q^{15} - 9 q^{16} + 8 q^{17} - 10 q^{18} + 2 q^{19} - 2 q^{22} - 9 q^{23} - 8 q^{24} - 14 q^{25} - 2 q^{26} - 18 q^{27} - 2 q^{29} + 18 q^{30} - 22 q^{31} + 24 q^{32} + 3 q^{33} + 6 q^{34} - 7 q^{36} + 28 q^{37} - 24 q^{38} + 20 q^{39} + 42 q^{40} + 20 q^{41} + 2 q^{43} + 2 q^{44} + 24 q^{45} + 12 q^{46} + 39 q^{48} + 22 q^{50} + 21 q^{51} + 11 q^{52} + 7 q^{53} - 43 q^{54} + 9 q^{55} + 21 q^{57} - 70 q^{58} + 42 q^{59} - 6 q^{60} + 6 q^{61} - 19 q^{62} - 2 q^{64} + 54 q^{65} + 3 q^{66} - 14 q^{67} - 102 q^{68} - 34 q^{69} + q^{71} + 18 q^{72} - 21 q^{73} - 25 q^{74} - 62 q^{75} - 70 q^{76} + 57 q^{78} - 5 q^{79} + 13 q^{80} + 28 q^{81} - 12 q^{82} + 10 q^{83} - 27 q^{85} + 18 q^{86} - 106 q^{87} + 72 q^{88} - q^{89} + 27 q^{90} - 72 q^{92} + 34 q^{93} - 24 q^{94} + 16 q^{95} + 188 q^{96} + 31 q^{97} + 43 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.16102 − 2.01095i 0.638361 + 1.10567i −1.69596 + 2.93748i 2.05173 + 3.55370i 1.48231 − 2.56743i 0 3.23209 0.684991 − 1.18644i 4.76422 − 8.25187i
197.2 −1.14049 − 1.97539i −1.23198 − 2.13385i −1.60144 + 2.77377i 0.527280 + 0.913275i −2.81012 + 4.86728i 0 2.74373 −1.53555 + 2.65966i 1.20271 − 2.08316i
197.3 −0.797467 − 1.38125i 0.741885 + 1.28498i −0.271907 + 0.470957i −0.310056 − 0.537033i 1.18326 − 2.04946i 0 −2.32252 0.399213 − 0.691457i −0.494519 + 0.856532i
197.4 −0.558365 − 0.967117i 0.464784 + 0.805029i 0.376456 − 0.652042i −0.614075 − 1.06361i 0.519038 − 0.899001i 0 −3.07426 1.06795 − 1.84975i −0.685757 + 1.18777i
197.5 −0.545630 − 0.945059i −1.15977 − 2.00878i 0.404576 − 0.700746i −1.44449 − 2.50193i −1.26561 + 2.19210i 0 −3.06551 −1.19014 + 2.06138i −1.57631 + 2.73025i
197.6 −0.0371811 − 0.0643995i −0.698298 − 1.20949i 0.997235 − 1.72726i 1.23136 + 2.13278i −0.0519269 + 0.0899401i 0 −0.297037 0.524760 − 0.908910i 0.0915665 − 0.158598i
197.7 0.143353 + 0.248294i 1.53242 + 2.65422i 0.958900 − 1.66086i 1.79001 + 3.10038i −0.439353 + 0.760981i 0 1.12326 −3.19660 + 5.53667i −0.513205 + 0.888897i
197.8 0.433977 + 0.751669i 1.27972 + 2.21653i 0.623329 − 1.07964i −1.78218 − 3.08682i −1.11073 + 1.92385i 0 2.81795 −1.77535 + 3.07499i 1.54685 − 2.67922i
197.9 0.594925 + 1.03044i −0.0886809 − 0.153600i 0.292129 − 0.505982i −0.0570924 − 0.0988869i 0.105517 − 0.182761i 0 3.07488 1.48427 − 2.57083i 0.0679313 − 0.117661i
197.10 1.06001 + 1.83599i −0.147810 − 0.256015i −1.24724 + 2.16028i 0.809658 + 1.40237i 0.313361 − 0.542756i 0 −1.04829 1.45630 − 2.52239i −1.71649 + 2.97305i
197.11 1.14607 + 1.98506i −1.07625 − 1.86413i −1.62697 + 2.81799i −1.58716 − 2.74903i 2.46693 − 4.27285i 0 −2.87420 −0.816647 + 1.41447i 3.63799 − 6.30119i
197.12 1.36182 + 2.35875i 1.24563 + 2.15750i −2.70912 + 4.69233i −0.614988 − 1.06519i −3.39266 + 5.87626i 0 −9.31007 −1.60320 + 2.77683i 1.67501 − 2.90120i
638.1 −1.16102 + 2.01095i 0.638361 − 1.10567i −1.69596 − 2.93748i 2.05173 − 3.55370i 1.48231 + 2.56743i 0 3.23209 0.684991 + 1.18644i 4.76422 + 8.25187i
638.2 −1.14049 + 1.97539i −1.23198 + 2.13385i −1.60144 − 2.77377i 0.527280 − 0.913275i −2.81012 − 4.86728i 0 2.74373 −1.53555 − 2.65966i 1.20271 + 2.08316i
638.3 −0.797467 + 1.38125i 0.741885 − 1.28498i −0.271907 − 0.470957i −0.310056 + 0.537033i 1.18326 + 2.04946i 0 −2.32252 0.399213 + 0.691457i −0.494519 − 0.856532i
638.4 −0.558365 + 0.967117i 0.464784 − 0.805029i 0.376456 + 0.652042i −0.614075 + 1.06361i 0.519038 + 0.899001i 0 −3.07426 1.06795 + 1.84975i −0.685757 − 1.18777i
638.5 −0.545630 + 0.945059i −1.15977 + 2.00878i 0.404576 + 0.700746i −1.44449 + 2.50193i −1.26561 − 2.19210i 0 −3.06551 −1.19014 − 2.06138i −1.57631 − 2.73025i
638.6 −0.0371811 + 0.0643995i −0.698298 + 1.20949i 0.997235 + 1.72726i 1.23136 − 2.13278i −0.0519269 − 0.0899401i 0 −0.297037 0.524760 + 0.908910i 0.0915665 + 0.158598i
638.7 0.143353 − 0.248294i 1.53242 − 2.65422i 0.958900 + 1.66086i 1.79001 − 3.10038i −0.439353 − 0.760981i 0 1.12326 −3.19660 − 5.53667i −0.513205 − 0.888897i
638.8 0.433977 − 0.751669i 1.27972 − 2.21653i 0.623329 + 1.07964i −1.78218 + 3.08682i −1.11073 − 1.92385i 0 2.81795 −1.77535 − 3.07499i 1.54685 + 2.67922i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.e.f 24
7.b odd 2 1 931.2.e.e 24
7.c even 3 1 133.2.g.a ✓ 24
7.c even 3 1 133.2.h.a yes 24
7.d odd 6 1 931.2.g.h 24
7.d odd 6 1 931.2.h.h 24
19.c even 3 1 inner 931.2.e.f 24
133.g even 3 1 133.2.h.a yes 24
133.h even 3 1 133.2.g.a ✓ 24
133.k odd 6 1 931.2.h.h 24
133.m odd 6 1 931.2.e.e 24
133.t odd 6 1 931.2.g.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.g.a ✓ 24 7.c even 3 1
133.2.g.a ✓ 24 133.h even 3 1
133.2.h.a yes 24 7.c even 3 1
133.2.h.a yes 24 133.g even 3 1
931.2.e.e 24 7.b odd 2 1
931.2.e.e 24 133.m odd 6 1
931.2.e.f 24 1.a even 1 1 trivial
931.2.e.f 24 19.c even 3 1 inner
931.2.g.h 24 7.d odd 6 1
931.2.g.h 24 133.t odd 6 1
931.2.h.h 24 7.d odd 6 1
931.2.h.h 24 133.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{24} - T_{2}^{23} + 18 T_{2}^{22} - 9 T_{2}^{21} + 196 T_{2}^{20} - 66 T_{2}^{19} + 1326 T_{2}^{18} + \cdots + 9 \) Copy content Toggle raw display
\( T_{3}^{24} - 3 T_{3}^{23} + 27 T_{3}^{22} - 54 T_{3}^{21} + 371 T_{3}^{20} - 634 T_{3}^{19} + 3346 T_{3}^{18} + \cdots + 961 \) Copy content Toggle raw display