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Properties

Label 931.2.bj.c
Level $931$
Weight $2$
Character orbit 931.bj
Analytic conductor $7.434$
Analytic rank $0$
Dimension $240$
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] = [931,2,Mod(117,931)]
 
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([15, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.bj (of order \(18\), degree \(6\), not 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: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 48 q^{15} - 144 q^{18} - 48 q^{22} + 96 q^{25} + 24 q^{29} - 48 q^{53} - 48 q^{57} - 144 q^{58} - 144 q^{60} + 120 q^{64} - 96 q^{67} - 96 q^{71} - 168 q^{72} - 72 q^{74} + 48 q^{78} + 120 q^{79} + 48 q^{81} + 48 q^{85} + 48 q^{86} - 216 q^{92} + 96 q^{93} - 192 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1 −0.889729 − 2.44451i −0.231476 − 0.194232i −3.65192 + 3.06432i −2.12158 + 2.52840i −0.268850 + 0.738659i 0 6.23424 + 3.59934i −0.505089 − 2.86450i 8.06833 + 2.93663i
117.2 −0.889729 − 2.44451i 0.231476 + 0.194232i −3.65192 + 3.06432i 2.12158 − 2.52840i 0.268850 − 0.738659i 0 6.23424 + 3.59934i −0.505089 − 2.86450i −8.06833 − 2.93663i
117.3 −0.829773 − 2.27978i −1.00955 − 0.847110i −2.97679 + 2.49783i 0.518235 − 0.617608i −1.09353 + 3.00445i 0 3.96245 + 2.28772i −0.219356 − 1.24403i −1.83803 − 0.668988i
117.4 −0.829773 − 2.27978i 1.00955 + 0.847110i −2.97679 + 2.49783i −0.518235 + 0.617608i 1.09353 − 3.00445i 0 3.96245 + 2.28772i −0.219356 − 1.24403i 1.83803 + 0.668988i
117.5 −0.824558 − 2.26545i −1.54328 − 1.29496i −2.92030 + 2.45042i −2.05337 + 2.44711i −1.66116 + 4.56400i 0 3.78357 + 2.18445i 0.183830 + 1.04255i 7.23695 + 2.63404i
117.6 −0.824558 − 2.26545i 1.54328 + 1.29496i −2.92030 + 2.45042i 2.05337 − 2.44711i 1.66116 − 4.56400i 0 3.78357 + 2.18445i 0.183830 + 1.04255i −7.23695 − 2.63404i
117.7 −0.711521 − 1.95489i −1.70710 − 1.43242i −1.78324 + 1.49632i 2.42220 − 2.88667i −1.58559 + 4.35638i 0 0.590672 + 0.341024i 0.341394 + 1.93614i −7.36656 − 2.68121i
117.8 −0.711521 − 1.95489i 1.70710 + 1.43242i −1.78324 + 1.49632i −2.42220 + 2.88667i 1.58559 − 4.35638i 0 0.590672 + 0.341024i 0.341394 + 1.93614i 7.36656 + 2.68121i
117.9 −0.659521 − 1.81202i −2.42869 − 2.03791i −1.31636 + 1.10455i 0.561253 − 0.668875i −2.09096 + 5.74487i 0 −0.470294 − 0.271524i 1.22450 + 6.94447i −1.58217 − 0.575863i
117.10 −0.659521 − 1.81202i 2.42869 + 2.03791i −1.31636 + 1.10455i −0.561253 + 0.668875i 2.09096 − 5.74487i 0 −0.470294 − 0.271524i 1.22450 + 6.94447i 1.58217 + 0.575863i
117.11 −0.390032 − 1.07160i −0.422450 − 0.354478i 0.535881 − 0.449657i −0.860978 + 1.02607i −0.215091 + 0.590956i 0 −2.66605 − 1.53925i −0.468135 − 2.65492i 1.43535 + 0.522425i
117.12 −0.390032 − 1.07160i 0.422450 + 0.354478i 0.535881 − 0.449657i 0.860978 − 1.02607i 0.215091 − 0.590956i 0 −2.66605 − 1.53925i −0.468135 − 2.65492i −1.43535 − 0.522425i
117.13 −0.387940 − 1.06586i −0.444751 − 0.373190i 0.546539 − 0.458600i −0.567642 + 0.676489i −0.225230 + 0.618816i 0 −2.66542 − 1.53888i −0.462412 − 2.62247i 0.941250 + 0.342587i
117.14 −0.387940 − 1.06586i 0.444751 + 0.373190i 0.546539 − 0.458600i 0.567642 − 0.676489i 0.225230 − 0.618816i 0 −2.66542 − 1.53888i −0.462412 − 2.62247i −0.941250 − 0.342587i
117.15 −0.189085 − 0.519508i −2.09769 − 1.76017i 1.29795 − 1.08911i 2.25871 − 2.69182i −0.517781 + 1.42259i 0 −1.76879 − 1.02121i 0.781161 + 4.43019i −1.82551 − 0.664433i
117.16 −0.189085 − 0.519508i 2.09769 + 1.76017i 1.29795 − 1.08911i −2.25871 + 2.69182i 0.517781 − 1.42259i 0 −1.76879 − 1.02121i 0.781161 + 4.43019i 1.82551 + 0.664433i
117.17 −0.176583 − 0.485157i −1.47950 − 1.24145i 1.32789 − 1.11423i −2.03956 + 2.43065i −0.341044 + 0.937009i 0 −1.66931 − 0.963776i 0.126784 + 0.719029i 1.53940 + 0.560296i
117.18 −0.176583 − 0.485157i 1.47950 + 1.24145i 1.32789 − 1.11423i 2.03956 − 2.43065i 0.341044 − 0.937009i 0 −1.66931 − 0.963776i 0.126784 + 0.719029i −1.53940 − 0.560296i
117.19 −0.137045 − 0.376528i −0.202843 − 0.170205i 1.40910 − 1.18237i 2.38618 − 2.84374i −0.0362885 + 0.0997017i 0 −1.33233 − 0.769219i −0.508769 − 2.88537i −1.39776 − 0.508744i
117.20 −0.137045 − 0.376528i 0.202843 + 0.170205i 1.40910 − 1.18237i −2.38618 + 2.84374i 0.0362885 − 0.0997017i 0 −1.33233 − 0.769219i −0.508769 − 2.88537i 1.39776 + 0.508744i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
133.be odd 18 1 inner
133.bf even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.bj.c 240
7.b odd 2 1 inner 931.2.bj.c 240
7.c even 3 1 931.2.be.c ✓ 240
7.c even 3 1 931.2.bf.c 240
7.d odd 6 1 931.2.be.c ✓ 240
7.d odd 6 1 931.2.bf.c 240
19.f odd 18 1 931.2.bf.c 240
133.ba even 18 1 931.2.bf.c 240
133.bb even 18 1 931.2.be.c ✓ 240
133.bd odd 18 1 931.2.be.c ✓ 240
133.be odd 18 1 inner 931.2.bj.c 240
133.bf even 18 1 inner 931.2.bj.c 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.be.c ✓ 240 7.c even 3 1
931.2.be.c ✓ 240 7.d odd 6 1
931.2.be.c ✓ 240 133.bb even 18 1
931.2.be.c ✓ 240 133.bd odd 18 1
931.2.bf.c 240 7.c even 3 1
931.2.bf.c 240 7.d odd 6 1
931.2.bf.c 240 19.f odd 18 1
931.2.bf.c 240 133.ba even 18 1
931.2.bj.c 240 1.a even 1 1 trivial
931.2.bj.c 240 7.b odd 2 1 inner
931.2.bj.c 240 133.be odd 18 1 inner
931.2.bj.c 240 133.bf even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 870 T_{2}^{114} + 12 T_{2}^{113} - 171 T_{2}^{112} + 1638 T_{2}^{110} - 1140 T_{2}^{109} + \cdots + 244140625 \) acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\). Copy content Toggle raw display