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Properties

Label 1890.2.be.e
Level $1890$
Weight $2$
Character orbit 1890.be
Analytic conductor $15.092$
Analytic rank $0$
Dimension $8$
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] = [1890,2,Mod(971,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.971");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.be (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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.4330692864.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 11x^{6} - 8x^{5} - 32x^{4} + 102x^{3} - 81x^{2} - 594x + 1089 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{3} q^{4} + (\beta_{3} + 1) q^{5} + ( - \beta_{6} + \beta_{5} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - \beta_{4} - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_{3} q^{4} + (\beta_{3} + 1) q^{5} + ( - \beta_{6} + \beta_{5} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{5} - 6 q^{7} - 18 q^{11} + 6 q^{14} - 4 q^{16} - 6 q^{17} - 6 q^{19} + 8 q^{20} + 6 q^{23} - 4 q^{25} - 4 q^{26} - 12 q^{31} - 6 q^{35} + 6 q^{37} - 16 q^{38} - 16 q^{41} + 68 q^{43} - 18 q^{44} + 2 q^{46} + 10 q^{47} + 18 q^{49} + 6 q^{52} + 12 q^{53} - 10 q^{58} - 18 q^{59} - 30 q^{61} - 28 q^{62} - 8 q^{64} - 6 q^{65} + 6 q^{67} + 6 q^{68} + 6 q^{70} - 12 q^{73} + 6 q^{74} - 4 q^{77} + 12 q^{79} + 4 q^{80} - 12 q^{82} + 8 q^{83} - 12 q^{85} - 6 q^{86} + 14 q^{89} - 38 q^{91} + 6 q^{94} - 6 q^{95} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 11x^{6} - 8x^{5} - 32x^{4} + 102x^{3} - 81x^{2} - 594x + 1089 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8900 \nu^{7} - 11659 \nu^{6} + 68328 \nu^{5} + 126928 \nu^{4} + 70962 \nu^{3} + 2319952 \nu^{2} + \cdots - 1691811 ) / 11818686 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -89\nu^{7} + 122\nu^{6} - 640\nu^{5} - 1226\nu^{4} + 2980\nu^{3} - 7662\nu^{2} - 3891\nu + 37092 ) / 35706 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39249 \nu^{7} - 71114 \nu^{6} + 143744 \nu^{5} + 402170 \nu^{4} - 1302278 \nu^{3} + \cdots - 21820590 ) / 11818686 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 44500 \nu^{7} - 58295 \nu^{6} + 341640 \nu^{5} + 634640 \nu^{4} + 354810 \nu^{3} + \cdots - 20277741 ) / 11818686 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46633 \nu^{7} + 216881 \nu^{6} - 572418 \nu^{5} + 448480 \nu^{4} + 1767498 \nu^{3} + \cdots + 32740257 ) / 11818686 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -145\nu^{7} + 217\nu^{6} - 1298\nu^{5} + 1358\nu^{4} - 1564\nu^{3} - 3438\nu^{2} + 23823\nu + 24123 ) / 35706 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 5\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} + \beta_{4} + 11\beta_{3} + 12\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{7} - \beta_{6} + 12\beta_{5} - 3\beta_{4} - \beta_{3} + 4\beta_{2} - 12\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} - 8\beta_{6} + 16\beta_{5} - 71\beta_{4} - 80\beta_{3} - 79\beta_{2} + \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -80\beta_{7} + 72\beta_{6} - 80\beta_{5} - 90\beta_{4} - 254\beta_{3} - 98\beta_{2} + 72\beta _1 - 199 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -254\beta_{7} + 170\beta_{6} - 170\beta_{5} + 462\beta_{4} + 208\beta_{3} + 350\beta_{2} + 127\beta _1 + 96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
971.1
2.04677 + 0.280482i
−1.91280 − 0.780482i
1.60788 − 2.26865i
0.258143 + 2.76865i
2.04677 − 0.280482i
−1.91280 + 0.780482i
1.60788 + 2.26865i
0.258143 − 2.76865i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −2.14651 + 1.54677i − 1.00000i 0 −0.866025 + 0.500000i
971.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −1.08554 − 2.41280i − 1.00000i 0 −0.866025 + 0.500000i
971.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −2.40262 − 1.10788i 1.00000i 0 0.866025 − 0.500000i
971.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.500000 − 0.866025i 0 2.63467 + 0.241857i 1.00000i 0 0.866025 − 0.500000i
1781.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −2.14651 − 1.54677i 1.00000i 0 −0.866025 − 0.500000i
1781.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −1.08554 + 2.41280i 1.00000i 0 −0.866025 − 0.500000i
1781.3 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −2.40262 + 1.10788i − 1.00000i 0 0.866025 + 0.500000i
1781.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.500000 + 0.866025i 0 2.63467 − 0.241857i − 1.00000i 0 0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 971.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.be.e yes 8
3.b odd 2 1 1890.2.be.c ✓ 8
7.d odd 6 1 1890.2.be.c ✓ 8
21.g even 6 1 inner 1890.2.be.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.be.c ✓ 8 3.b odd 2 1
1890.2.be.c ✓ 8 7.d odd 6 1
1890.2.be.e yes 8 1.a even 1 1 trivial
1890.2.be.e yes 8 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 18T_{11}^{7} + 131T_{11}^{6} + 414T_{11}^{5} + 357T_{11}^{4} - 828T_{11}^{3} - 580T_{11}^{2} + 1584T_{11} + 1936 \) acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$13$ \( T^{8} + 42 T^{6} + \cdots + 484 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 256036 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 16384 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 952576 \) Copy content Toggle raw display
$29$ \( T^{8} + 130 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{4} + 8 T^{3} + \cdots - 416)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 34 T^{3} + \cdots + 3544)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 1401856 \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} + \cdots + 123904 \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 39337984 \) Copy content Toggle raw display
$61$ \( T^{8} + 30 T^{7} + \cdots + 135424 \) Copy content Toggle raw display
$67$ \( T^{8} - 6 T^{7} + \cdots + 627264 \) Copy content Toggle raw display
$71$ \( T^{8} + 174 T^{6} + \cdots + 20164 \) Copy content Toggle raw display
$73$ \( T^{8} + 12 T^{7} + \cdots + 440896 \) Copy content Toggle raw display
$79$ \( T^{8} - 12 T^{7} + \cdots + 4648336 \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{3} + \cdots - 299)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 14 T^{7} + \cdots + 17707264 \) Copy content Toggle raw display
$97$ \( T^{8} + 170 T^{6} + \cdots + 59536 \) Copy content Toggle raw display
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