↑

Properties

Label 945.2.j.e
Level $945$
Weight $2$
Character orbit 945.j
Analytic conductor $7.546$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(541,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("945.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.j (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.5883587346987.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 7x^{8} - 2x^{7} + 38x^{6} - 10x^{5} + 78x^{4} - 31x^{3} + 124x^{2} - 33x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{9} - \beta_{6} + \beta_{2}) q^{4} + (\beta_{6} - 1) q^{5} + ( - \beta_{8} + \beta_{5} + \beta_{4}) q^{7} + (\beta_{4} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{9} - \beta_{6} + \beta_{2}) q^{4} + (\beta_{6} - 1) q^{5} + ( - \beta_{8} + \beta_{5} + \beta_{4}) q^{7} + (\beta_{4} + 1) q^{8} + (\beta_{3} - \beta_1) q^{10} + ( - 2 \beta_{9} - 2 \beta_{8} + \cdots + \beta_1) q^{11}+ \cdots + (4 \beta_{9} + 5 \beta_{8} - \beta_{7} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{4} - 5 q^{5} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{4} - 5 q^{5} - 5 q^{7} + 6 q^{8} - 5 q^{11} + 12 q^{13} + 8 q^{14} + 10 q^{16} + q^{17} - 7 q^{19} + 8 q^{20} + 8 q^{22} - 4 q^{23} - 5 q^{25} - 7 q^{26} + q^{28} + 24 q^{29} - 11 q^{31} + 4 q^{32} + 16 q^{34} + q^{35} + 3 q^{38} - 3 q^{40} + 14 q^{41} + 4 q^{43} + 10 q^{44} - 14 q^{46} - 14 q^{47} - 19 q^{49} - 7 q^{52} - 2 q^{53} + 10 q^{55} + 6 q^{56} + 28 q^{58} - 2 q^{59} - 25 q^{61} - 48 q^{62} - 34 q^{64} - 6 q^{65} - 6 q^{67} + 14 q^{68} - 10 q^{70} - 30 q^{71} - 17 q^{73} - 29 q^{74} + 68 q^{76} - 14 q^{77} + 7 q^{79} + 10 q^{80} - 7 q^{82} + 4 q^{83} - 2 q^{85} - 7 q^{86} - 10 q^{88} - 15 q^{89} + 18 q^{91} + 42 q^{92} - 21 q^{94} - 7 q^{95} + 96 q^{97} + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 7x^{8} - 2x^{7} + 38x^{6} - 10x^{5} + 78x^{4} - 31x^{3} + 124x^{2} - 33x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1793 \nu^{9} + 32054 \nu^{8} - 11656 \nu^{7} + 176825 \nu^{6} - 113646 \nu^{5} + 1229708 \nu^{4} + \cdots + 5768001 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4486 \nu^{9} + 1793 \nu^{8} - 652 \nu^{7} + 2684 \nu^{6} - 6357 \nu^{5} + 68786 \nu^{4} + \cdots + 525096 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17944 \nu^{9} + 7172 \nu^{8} - 2608 \nu^{7} + 10736 \nu^{6} - 25428 \nu^{5} + 275144 \nu^{4} + \cdots + 191175 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44234 \nu^{9} - 1791 \nu^{8} - 230768 \nu^{7} + 59788 \nu^{6} - 977182 \nu^{5} + 162710 \nu^{4} + \cdots - 196548 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58344 \nu^{9} + 4486 \nu^{8} - 406615 \nu^{7} + 116036 \nu^{6} - 2214388 \nu^{5} + \cdots + 1887699 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 91653 \nu^{9} - 98911 \nu^{8} - 716145 \nu^{7} - 483834 \nu^{6} - 3641298 \nu^{5} + \cdots - 3046233 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 119460 \nu^{9} - 35102 \nu^{8} + 764877 \nu^{7} - 560106 \nu^{6} + 4116435 \nu^{5} + \cdots - 6262395 ) / 1909209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 173239 \nu^{9} - 18596 \nu^{8} - 1208189 \nu^{7} + 171283 \nu^{6} - 6529518 \nu^{5} + \cdots - 104904 ) / 1909209 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 3\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 4\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} - \beta_{8} + 2\beta_{7} + 13\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - 7\beta_{6} + 7\beta_{5} + 17\beta_{3} + \beta_{2} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{8} - 7\beta_{7} + 8\beta_{4} - \beta_{3} - 31\beta_{2} + 59 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10\beta_{9} - \beta_{8} + 2\beta_{7} + 42\beta_{6} - 39\beta_{5} - 39\beta_{4} - \beta_{3} + 77\beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 152\beta_{9} + 76\beta_{8} - 38\beta_{7} - 274\beta_{6} + 49\beta_{5} + 50\beta_{3} + 152\beta_{2} - 88\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -11\beta_{8} - 11\beta_{7} + 201\beta_{4} - 339\beta_{3} - 72\beta_{2} + 237 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
−1.11349 + 1.92861i
−0.794486 + 1.37609i
0.139675 − 0.241925i
0.732493 − 1.26872i
1.03580 − 1.79406i
−1.11349 − 1.92861i
−0.794486 − 1.37609i
0.139675 + 0.241925i
0.732493 + 1.26872i
1.03580 + 1.79406i
−1.11349 + 1.92861i 0 −1.47970 − 2.56292i −0.500000 + 0.866025i 0 −0.0803747 − 2.64453i 2.13656 0 −1.11349 − 1.92861i
541.2 −0.794486 + 1.37609i 0 −0.262415 − 0.454516i −0.500000 + 0.866025i 0 −1.88711 + 1.85441i −2.34400 0 −0.794486 − 1.37609i
541.3 0.139675 − 0.241925i 0 0.960982 + 1.66447i −0.500000 + 0.866025i 0 2.26893 + 1.36086i 1.09560 0 0.139675 + 0.241925i
541.4 0.732493 − 1.26872i 0 −0.0730927 − 0.126600i −0.500000 + 0.866025i 0 −1.63550 − 2.07970i 2.71581 0 0.732493 + 1.26872i
541.5 1.03580 − 1.79406i 0 −1.14577 − 1.98454i −0.500000 + 0.866025i 0 −1.16595 + 2.37499i −0.603971 0 1.03580 + 1.79406i
676.1 −1.11349 − 1.92861i 0 −1.47970 + 2.56292i −0.500000 − 0.866025i 0 −0.0803747 + 2.64453i 2.13656 0 −1.11349 + 1.92861i
676.2 −0.794486 − 1.37609i 0 −0.262415 + 0.454516i −0.500000 − 0.866025i 0 −1.88711 − 1.85441i −2.34400 0 −0.794486 + 1.37609i
676.3 0.139675 + 0.241925i 0 0.960982 − 1.66447i −0.500000 − 0.866025i 0 2.26893 − 1.36086i 1.09560 0 0.139675 − 0.241925i
676.4 0.732493 + 1.26872i 0 −0.0730927 + 0.126600i −0.500000 − 0.866025i 0 −1.63550 + 2.07970i 2.71581 0 0.732493 − 1.26872i
676.5 1.03580 + 1.79406i 0 −1.14577 + 1.98454i −0.500000 − 0.866025i 0 −1.16595 − 2.37499i −0.603971 0 1.03580 − 1.79406i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.j.e ✓ 10
3.b odd 2 1 945.2.j.g yes 10
7.c even 3 1 inner 945.2.j.e ✓ 10
7.c even 3 1 6615.2.a.br 5
7.d odd 6 1 6615.2.a.bn 5
21.g even 6 1 6615.2.a.bo 5
21.h odd 6 1 945.2.j.g yes 10
21.h odd 6 1 6615.2.a.bk 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.j.e ✓ 10 1.a even 1 1 trivial
945.2.j.e ✓ 10 7.c even 3 1 inner
945.2.j.g yes 10 3.b odd 2 1
945.2.j.g yes 10 21.h odd 6 1
6615.2.a.bk 5 21.h odd 6 1
6615.2.a.bn 5 7.d odd 6 1
6615.2.a.bo 5 21.g even 6 1
6615.2.a.br 5 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 7T_{2}^{8} - 2T_{2}^{7} + 38T_{2}^{6} - 10T_{2}^{5} + 78T_{2}^{4} - 31T_{2}^{3} + 124T_{2}^{2} - 33T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 7 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 5 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 5 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{5} - 6 T^{4} + \cdots + 101)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - T^{9} + \cdots + 441 \) Copy content Toggle raw display
$19$ \( T^{10} + 7 T^{9} + \cdots + 44521 \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + \cdots + 77841 \) Copy content Toggle raw display
$29$ \( (T^{5} - 12 T^{4} + \cdots - 3351)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 11 T^{9} + \cdots + 2893401 \) Copy content Toggle raw display
$37$ \( T^{10} + 99 T^{8} + \cdots + 4173849 \) Copy content Toggle raw display
$41$ \( (T^{5} - 7 T^{4} + \cdots - 1269)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 2 T^{4} + \cdots + 7559)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 14 T^{9} + \cdots + 56169 \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} + \cdots + 19881 \) Copy content Toggle raw display
$59$ \( T^{10} + 2 T^{9} + \cdots + 370881 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 1865462481 \) Copy content Toggle raw display
$67$ \( T^{10} + 6 T^{9} + \cdots + 8649 \) Copy content Toggle raw display
$71$ \( (T^{5} + 15 T^{4} + \cdots - 40857)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 145709041 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 596189889 \) Copy content Toggle raw display
$83$ \( (T^{5} - 2 T^{4} + \cdots - 8397)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 272151009 \) Copy content Toggle raw display
$97$ \( (T^{5} - 48 T^{4} + \cdots + 97921)^{2} \) Copy content Toggle raw display
show more
show less