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Properties

Label 931.2.w.f
Level $931$
Weight $2$
Character orbit 931.w
Analytic conductor $7.434$
Analytic rank $0$
Dimension $66$
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(99,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(18))
 
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("931.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.w (of order \(9\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} - 12 q^{6} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 6 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} - 12 q^{6} - 6 q^{8} + 6 q^{9} - 3 q^{10} + 27 q^{12} - 12 q^{13} - 9 q^{15} - 9 q^{17} + 36 q^{18} + 24 q^{19} - 12 q^{22} + 24 q^{23} + 9 q^{24} + 24 q^{25} + 9 q^{26} + 30 q^{27} + 6 q^{29} + 3 q^{30} + 39 q^{31} + 6 q^{32} + 24 q^{33} - 24 q^{34} + 51 q^{36} - 12 q^{37} + 27 q^{38} + 48 q^{39} - 15 q^{40} - 30 q^{41} + 12 q^{43} - 21 q^{44} - 3 q^{45} + 6 q^{46} - 12 q^{47} - 21 q^{48} + 27 q^{50} + 6 q^{51} + 54 q^{52} - 36 q^{53} + 30 q^{54} - 27 q^{55} + 6 q^{57} + 84 q^{58} + 24 q^{59} - 42 q^{60} - 72 q^{61} - 27 q^{62} + 24 q^{64} + 21 q^{65} + 6 q^{66} - 27 q^{67} - 15 q^{68} + 42 q^{69} + 30 q^{71} + 48 q^{72} + 12 q^{73} + 30 q^{74} - 114 q^{75} + 42 q^{76} - 51 q^{78} + 72 q^{79} - 66 q^{80} - 15 q^{81} - 189 q^{82} - 18 q^{83} - 24 q^{85} - 30 q^{86} - 9 q^{87} - 45 q^{88} - 48 q^{89} - 48 q^{90} + 12 q^{92} - 39 q^{93} - 144 q^{94} - 42 q^{95} - 138 q^{96} + 3 q^{97} - 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −2.50426 − 0.911478i −0.313582 − 1.77841i 3.90846 + 3.27959i 0.0836518 − 0.0701922i −0.835690 + 4.73943i 0 −4.13357 − 7.15955i −0.245331 + 0.0892931i −0.273465 + 0.0995331i
99.2 −1.92393 − 0.700254i 0.0531995 + 0.301709i 1.67907 + 1.40891i −1.47302 + 1.23601i 0.108921 − 0.617722i 0 −0.196424 − 0.340216i 2.73088 − 0.993959i 3.69951 − 1.34651i
99.3 −1.27118 − 0.462670i −0.369619 − 2.09621i −0.130264 − 0.109304i 2.77326 − 2.32704i −0.500005 + 2.83567i 0 1.46777 + 2.54226i −1.43842 + 0.523541i −4.60196 + 1.67498i
99.4 −0.815932 − 0.296975i 0.342341 + 1.94151i −0.954538 − 0.800953i −0.516233 + 0.433171i 0.297254 − 1.68581i 0 1.40927 + 2.44093i −0.833194 + 0.303258i 0.549852 − 0.200130i
99.5 −0.557252 − 0.202823i −0.217861 − 1.23555i −1.26270 − 1.05953i −0.220861 + 0.185325i −0.129195 + 0.732703i 0 1.08176 + 1.87366i 1.33995 − 0.487702i 0.160664 − 0.0584768i
99.6 0.597648 + 0.217526i 0.447973 + 2.54058i −1.22222 − 1.02557i 2.05895 − 1.72766i −0.284912 + 1.61582i 0 −1.14338 − 1.98038i −3.43479 + 1.25016i 1.60634 − 0.584659i
99.7 0.983908 + 0.358113i 0.335565 + 1.90309i −0.692258 − 0.580874i −3.08613 + 2.58957i −0.351355 + 1.99263i 0 −1.52015 − 2.63298i −0.690051 + 0.251158i −3.96383 + 1.44272i
99.8 1.31593 + 0.478959i 0.00500293 + 0.0283730i −0.0298210 − 0.0250228i 2.30156 − 1.93124i −0.00700601 + 0.0397331i 0 −1.42764 − 2.47275i 2.81830 − 1.02578i 3.95367 − 1.43902i
99.9 1.52346 + 0.554493i −0.591495 − 3.35454i 0.481373 + 0.403920i 1.50056 − 1.25912i 0.958950 − 5.43848i 0 −1.11185 − 1.92578i −8.08397 + 2.94232i 2.98421 − 1.08616i
99.10 1.94162 + 0.706692i −0.221217 − 1.25458i 1.73839 + 1.45868i −2.09611 + 1.75884i 0.457084 − 2.59225i 0 0.278223 + 0.481897i 1.29404 − 0.470992i −5.31281 + 1.93370i
99.11 2.47604 + 0.901204i 0.0690552 + 0.391631i 3.78651 + 3.17726i 0.614065 − 0.515262i −0.181957 + 1.03193i 0 3.87723 + 6.71557i 2.67047 − 0.971972i 1.98480 − 0.722410i
442.1 −2.50426 + 0.911478i −0.313582 + 1.77841i 3.90846 − 3.27959i 0.0836518 + 0.0701922i −0.835690 − 4.73943i 0 −4.13357 + 7.15955i −0.245331 − 0.0892931i −0.273465 − 0.0995331i
442.2 −1.92393 + 0.700254i 0.0531995 − 0.301709i 1.67907 − 1.40891i −1.47302 − 1.23601i 0.108921 + 0.617722i 0 −0.196424 + 0.340216i 2.73088 + 0.993959i 3.69951 + 1.34651i
442.3 −1.27118 + 0.462670i −0.369619 + 2.09621i −0.130264 + 0.109304i 2.77326 + 2.32704i −0.500005 − 2.83567i 0 1.46777 − 2.54226i −1.43842 − 0.523541i −4.60196 − 1.67498i
442.4 −0.815932 + 0.296975i 0.342341 − 1.94151i −0.954538 + 0.800953i −0.516233 − 0.433171i 0.297254 + 1.68581i 0 1.40927 − 2.44093i −0.833194 − 0.303258i 0.549852 + 0.200130i
442.5 −0.557252 + 0.202823i −0.217861 + 1.23555i −1.26270 + 1.05953i −0.220861 − 0.185325i −0.129195 − 0.732703i 0 1.08176 − 1.87366i 1.33995 + 0.487702i 0.160664 + 0.0584768i
442.6 0.597648 − 0.217526i 0.447973 − 2.54058i −1.22222 + 1.02557i 2.05895 + 1.72766i −0.284912 − 1.61582i 0 −1.14338 + 1.98038i −3.43479 − 1.25016i 1.60634 + 0.584659i
442.7 0.983908 − 0.358113i 0.335565 − 1.90309i −0.692258 + 0.580874i −3.08613 − 2.58957i −0.351355 − 1.99263i 0 −1.52015 + 2.63298i −0.690051 − 0.251158i −3.96383 − 1.44272i
442.8 1.31593 − 0.478959i 0.00500293 − 0.0283730i −0.0298210 + 0.0250228i 2.30156 + 1.93124i −0.00700601 − 0.0397331i 0 −1.42764 + 2.47275i 2.81830 + 1.02578i 3.95367 + 1.43902i
442.9 1.52346 − 0.554493i −0.591495 + 3.35454i 0.481373 − 0.403920i 1.50056 + 1.25912i 0.958950 + 5.43848i 0 −1.11185 + 1.92578i −8.08397 − 2.94232i 2.98421 + 1.08616i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.w.f 66
7.b odd 2 1 931.2.w.e 66
7.c even 3 1 133.2.u.a ✓ 66
7.c even 3 1 133.2.w.a yes 66
7.d odd 6 1 931.2.v.h 66
7.d odd 6 1 931.2.x.h 66
19.e even 9 1 inner 931.2.w.f 66
133.u even 9 1 133.2.w.a yes 66
133.w even 9 1 133.2.u.a ✓ 66
133.x odd 18 1 931.2.x.h 66
133.y odd 18 1 931.2.w.e 66
133.z odd 18 1 931.2.v.h 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.u.a ✓ 66 7.c even 3 1
133.2.u.a ✓ 66 133.w even 9 1
133.2.w.a yes 66 7.c even 3 1
133.2.w.a yes 66 133.u even 9 1
931.2.v.h 66 7.d odd 6 1
931.2.v.h 66 133.z odd 18 1
931.2.w.e 66 7.b odd 2 1
931.2.w.e 66 133.y odd 18 1
931.2.w.f 66 1.a even 1 1 trivial
931.2.w.f 66 19.e even 9 1 inner
931.2.x.h 66 7.d odd 6 1
931.2.x.h 66 133.x odd 18 1

Hecke kernels

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

\( T_{2}^{66} - 6 T_{2}^{65} + 15 T_{2}^{64} - 16 T_{2}^{63} - 3 T_{2}^{62} + 54 T_{2}^{61} + 174 T_{2}^{60} + \cdots + 29241 \) Copy content Toggle raw display
\( T_{3}^{66} - 6 T_{3}^{65} + 15 T_{3}^{64} - 46 T_{3}^{63} + 183 T_{3}^{62} - 459 T_{3}^{61} + 1989 T_{3}^{60} + \cdots + 361 \) Copy content Toggle raw display