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Properties

Label 285.2.i.f
Level $285$
Weight $2$
Character orbit 285.i
Analytic conductor $2.276$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [285,2,Mod(106,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.i (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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} - \beta_{4} q^{3} + ( - \beta_{8} - \beta_{4} - \beta_{3} - 1) q^{4} + \beta_{4} q^{5} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{6} + 1) q^{7} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{8}+ \cdots + ( - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} + 5 q^{3} - 7 q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 12 q^{8} - 5 q^{9} + q^{10} + 10 q^{11} - 14 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} - 7 q^{16} - 10 q^{17} - 2 q^{18} + 5 q^{19} + 14 q^{20}+ \cdots - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} - 9\nu^{6} + 15\nu^{5} - 45\nu^{4} + 135\nu^{3} - 86\nu^{2} + 24\nu - 72 ) / 234 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 3\nu^{8} + 9\nu^{7} - 15\nu^{6} + 45\nu^{5} - 135\nu^{4} + 86\nu^{3} - 258\nu^{2} + 72\nu - 702 ) / 234 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{9} - 9 \nu^{8} + 79 \nu^{7} - 12 \nu^{6} + 486 \nu^{5} - 132 \nu^{4} + 1035 \nu^{3} + \cdots - 420 ) / 468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28 \nu^{9} - 44 \nu^{8} + 275 \nu^{7} - 229 \nu^{6} + 1713 \nu^{5} - 1551 \nu^{4} + 4223 \nu^{3} + \cdots - 3932 ) / 702 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31 \nu^{9} - 5 \nu^{8} + 275 \nu^{7} + 158 \nu^{6} + 1830 \nu^{5} + 906 \nu^{4} + 4319 \nu^{3} + \cdots + 2332 ) / 702 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34 \nu^{9} - \nu^{8} - 257 \nu^{7} - 248 \nu^{6} - 1740 \nu^{5} - 1176 \nu^{4} - 3254 \nu^{3} + \cdots - 1306 ) / 702 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{9} + 9 \nu^{8} - 79 \nu^{7} + 12 \nu^{6} - 486 \nu^{5} + 132 \nu^{4} - 1035 \nu^{3} + \cdots + 420 ) / 156 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49 \nu^{9} + 56 \nu^{8} - 428 \nu^{7} + 163 \nu^{6} - 2595 \nu^{5} + 1389 \nu^{4} - 5228 \nu^{3} + \cdots + 4160 ) / 702 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 8\beta_{8} - 2\beta_{6} - 14\beta_{4} - 8\beta_{3} + \beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{9} - 11\beta_{8} - \beta_{6} + 8\beta_{5} - 11\beta_{4} - 28\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} + 3\beta_{7} + 12\beta_{6} + 57\beta_{3} - 24\beta_{2} + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 66 \beta_{9} + 96 \beta_{8} + 54 \beta_{7} + 78 \beta_{6} - 54 \beta_{5} + 96 \beta_{4} + 96 \beta_{3} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -174\beta_{9} + 397\beta_{8} + 66\beta_{6} - 42\beta_{5} + 495\beta_{4} + 156\beta_{2} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -108\beta_{9} - 355\beta_{7} - 463\beta_{6} - 769\beta_{3} + 1181\beta_{2} - 426 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
−1.12375 − 1.94639i
−0.690702 − 1.19633i
0.145349 + 0.251751i
0.823305 + 1.42601i
1.34580 + 2.33099i
−1.12375 + 1.94639i
−0.690702 + 1.19633i
0.145349 − 0.251751i
0.823305 − 1.42601i
1.34580 − 2.33099i
−1.12375 + 1.94639i 0.500000 − 0.866025i −1.52562 − 2.64245i −0.500000 + 0.866025i 1.12375 + 1.94639i 3.16638 2.36264 −0.500000 − 0.866025i −1.12375 − 1.94639i
106.2 −0.690702 + 1.19633i 0.500000 − 0.866025i 0.0458624 + 0.0794360i −0.500000 + 0.866025i 0.690702 + 1.19633i −4.36264 −2.88952 −0.500000 − 0.866025i −0.690702 − 1.19633i
106.3 0.145349 − 0.251751i 0.500000 − 0.866025i 0.957748 + 1.65887i −0.500000 + 0.866025i −0.145349 − 0.251751i −0.486575 1.13822 −0.500000 − 0.866025i 0.145349 + 0.251751i
106.4 0.823305 − 1.42601i 0.500000 − 0.866025i −0.355663 − 0.616027i −0.500000 + 0.866025i −0.823305 − 1.42601i 4.47988 2.12194 −0.500000 − 0.866025i 0.823305 + 1.42601i
106.5 1.34580 − 2.33099i 0.500000 − 0.866025i −2.62233 − 4.54201i −0.500000 + 0.866025i −1.34580 − 2.33099i −0.797044 −8.73329 −0.500000 − 0.866025i 1.34580 + 2.33099i
121.1 −1.12375 − 1.94639i 0.500000 + 0.866025i −1.52562 + 2.64245i −0.500000 − 0.866025i 1.12375 − 1.94639i 3.16638 2.36264 −0.500000 + 0.866025i −1.12375 + 1.94639i
121.2 −0.690702 − 1.19633i 0.500000 + 0.866025i 0.0458624 − 0.0794360i −0.500000 − 0.866025i 0.690702 − 1.19633i −4.36264 −2.88952 −0.500000 + 0.866025i −0.690702 + 1.19633i
121.3 0.145349 + 0.251751i 0.500000 + 0.866025i 0.957748 − 1.65887i −0.500000 − 0.866025i −0.145349 + 0.251751i −0.486575 1.13822 −0.500000 + 0.866025i 0.145349 − 0.251751i
121.4 0.823305 + 1.42601i 0.500000 + 0.866025i −0.355663 + 0.616027i −0.500000 − 0.866025i −0.823305 + 1.42601i 4.47988 2.12194 −0.500000 + 0.866025i 0.823305 − 1.42601i
121.5 1.34580 + 2.33099i 0.500000 + 0.866025i −2.62233 + 4.54201i −0.500000 − 0.866025i −1.34580 + 2.33099i −0.797044 −8.73329 −0.500000 + 0.866025i 1.34580 − 2.33099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.5
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 285.2.i.f ✓ 10
3.b odd 2 1 855.2.k.i 10
19.c even 3 1 inner 285.2.i.f ✓ 10
19.c even 3 1 5415.2.a.y 5
19.d odd 6 1 5415.2.a.z 5
57.h odd 6 1 855.2.k.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.f ✓ 10 1.a even 1 1 trivial
285.2.i.f ✓ 10 19.c even 3 1 inner
855.2.k.i 10 3.b odd 2 1
855.2.k.i 10 57.h odd 6 1
5415.2.a.y 5 19.c even 3 1
5415.2.a.z 5 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - T_{2}^{9} + 9T_{2}^{8} - 2T_{2}^{7} + 56T_{2}^{6} - 18T_{2}^{5} + 125T_{2}^{4} + T_{2}^{3} + 189T_{2}^{2} - 52T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$7$ \( (T^{5} - 2 T^{4} - 23 T^{3} + \cdots + 24)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 5 T^{4} + \cdots - 992)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{10} + 10 T^{9} + \cdots + 25240576 \) Copy content Toggle raw display
$19$ \( T^{10} - 5 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} - 2 T^{9} + \cdots + 147456 \) Copy content Toggle raw display
$29$ \( T^{10} - 7 T^{9} + \cdots + 9048064 \) Copy content Toggle raw display
$31$ \( (T^{5} + 9 T^{4} + \cdots + 117)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 6 T^{4} + \cdots - 9024)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 12 T^{9} + \cdots + 36864 \) Copy content Toggle raw display
$43$ \( T^{10} - 8 T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{10} + 6 T^{9} + \cdots + 4562496 \) Copy content Toggle raw display
$53$ \( T^{10} + 8 T^{9} + \cdots + 18731584 \) Copy content Toggle raw display
$59$ \( T^{10} - T^{9} + \cdots + 20647936 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 591267856 \) Copy content Toggle raw display
$67$ \( T^{10} - 14 T^{9} + \cdots + 70157376 \) Copy content Toggle raw display
$71$ \( T^{10} - 27 T^{9} + \cdots + 37161216 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 2135179264 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 11935125504 \) Copy content Toggle raw display
$83$ \( (T^{5} + 12 T^{4} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 126157824 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 4)^{5} \) Copy content Toggle raw display
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