Properties

Label 285.2.i.d
Level $285$
Weight $2$
Character orbit 285.i
Analytic conductor $2.276$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + (\beta_{2} + 1) q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + ( - 2 \beta_{3} - 1) q^{7} + (\beta_{3} + 3) q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + (\beta_{2} + 1) q^{3} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + ( - 2 \beta_{3} - 1) q^{7} + (\beta_{3} + 3) q^{8} + \beta_{2} q^{9} + (\beta_{3} - \beta_{2} + \beta_1) q^{10} + 2 \beta_{3} q^{11} + ( - 2 \beta_{3} - 1) q^{12} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + (5 \beta_{2} - 3 \beta_1 + 5) q^{14} + \beta_{2} q^{15} + ( - 3 \beta_{2} - 3) q^{16} + (4 \beta_{2} - 2 \beta_1 + 4) q^{17} + (\beta_{3} + 1) q^{18} + (2 \beta_{2} + 5) q^{19} + ( - 2 \beta_{3} - 1) q^{20} + ( - \beta_{2} + 2 \beta_1 - 1) q^{21} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{23} + (3 \beta_{2} - \beta_1 + 3) q^{24} + \beta_{2} q^{25} + (3 \beta_{3} + 5) q^{26} - q^{27} + (4 \beta_{3} - 9 \beta_{2} + 4 \beta_1) q^{28} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{29} + (\beta_{3} + 1) q^{30} - 5 q^{31} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{32} - 2 \beta_1 q^{33} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{34} + ( - \beta_{2} + 2 \beta_1 - 1) q^{35} + ( - \beta_{2} + 2 \beta_1 - 1) q^{36} + ( - 2 \beta_{3} - 5) q^{37} + (2 \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 3) q^{38} + ( - 2 \beta_{3} - 1) q^{39} + (3 \beta_{2} - \beta_1 + 3) q^{40} - 2 \beta_1 q^{41} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1) q^{42} + ( - 5 \beta_{2} - 2 \beta_1 - 5) q^{43} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{44} - q^{45} + (4 \beta_{3} + 6) q^{46} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{47} - 3 \beta_{2} q^{48} + (4 \beta_{3} + 2) q^{49} + (\beta_{3} + 1) q^{50} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{51} + ( - 9 \beta_{2} + 4 \beta_1 - 9) q^{52} - 2 \beta_{2} q^{53} + (\beta_{2} - \beta_1 + 1) q^{54} - 2 \beta_1 q^{55} + ( - 7 \beta_{3} - 7) q^{56} + (5 \beta_{2} + 3) q^{57} - 4 q^{58} + ( - \beta_{2} + 2 \beta_1 - 1) q^{60} + ( - 8 \beta_{3} + 3 \beta_{2} - 8 \beta_1) q^{61} + (5 \beta_{2} - 5 \beta_1 + 5) q^{62} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{63} + ( - 2 \beta_{3} - 7) q^{64} + ( - 2 \beta_{3} - 1) q^{65} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{66} + (6 \beta_{3} - 3 \beta_{2} + 6 \beta_1) q^{67} + ( - 10 \beta_{3} - 12) q^{68} + ( - 2 \beta_{3} - 2) q^{69} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1) q^{70} + ( - 10 \beta_{2} - 10) q^{71} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{72} + ( - \beta_{2} - 6 \beta_1 - 1) q^{73} + (9 \beta_{2} - 7 \beta_1 + 9) q^{74} - q^{75} + ( - 10 \beta_{3} + 3 \beta_{2} - 6 \beta_1 - 2) q^{76} + ( - 2 \beta_{3} - 8) q^{77} + (5 \beta_{2} - 3 \beta_1 + 5) q^{78} + (9 \beta_{2} - 4 \beta_1 + 9) q^{79} - 3 \beta_{2} q^{80} + ( - \beta_{2} - 1) q^{81} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{82} - 8 q^{83} + (4 \beta_{3} + 9) q^{84} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{85} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{86} + (4 \beta_{3} - 4) q^{87} + (6 \beta_{3} + 4) q^{88} + (6 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{89} + (\beta_{2} - \beta_1 + 1) q^{90} + (4 \beta_{3} - 9 \beta_{2} + 4 \beta_1) q^{91} + ( - 10 \beta_{2} + 6 \beta_1 - 10) q^{92} + ( - 5 \beta_{2} - 5) q^{93} + ( - 8 \beta_{3} - 10) q^{94} + (5 \beta_{2} + 3) q^{95} + ( - \beta_{3} + 3) q^{96} + (6 \beta_{2} + 6) q^{97} + ( - 10 \beta_{2} + 6 \beta_1 - 10) q^{98} + ( - 2 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} - 2 q^{9} + 2 q^{10} - 4 q^{12} - 2 q^{13} + 10 q^{14} - 2 q^{15} - 6 q^{16} + 8 q^{17} + 4 q^{18} + 16 q^{19} - 4 q^{20} - 2 q^{21} - 8 q^{22} - 4 q^{23} + 6 q^{24} - 2 q^{25} + 20 q^{26} - 4 q^{27} + 18 q^{28} - 8 q^{29} + 4 q^{30} - 20 q^{31} + 6 q^{32} + 16 q^{34} - 2 q^{35} - 2 q^{36} - 20 q^{37} - 2 q^{38} - 4 q^{39} + 6 q^{40} - 10 q^{42} - 10 q^{43} - 16 q^{44} - 4 q^{45} + 24 q^{46} + 12 q^{47} + 6 q^{48} + 8 q^{49} + 4 q^{50} - 8 q^{51} - 18 q^{52} + 4 q^{53} + 2 q^{54} - 28 q^{56} + 2 q^{57} - 16 q^{58} - 2 q^{60} - 6 q^{61} + 10 q^{62} + 2 q^{63} - 28 q^{64} - 4 q^{65} + 8 q^{66} + 6 q^{67} - 48 q^{68} - 8 q^{69} - 10 q^{70} - 20 q^{71} - 6 q^{72} - 2 q^{73} + 18 q^{74} - 4 q^{75} - 14 q^{76} - 32 q^{77} + 10 q^{78} + 18 q^{79} + 6 q^{80} - 2 q^{81} + 8 q^{82} - 32 q^{83} + 36 q^{84} - 8 q^{85} - 2 q^{86} - 16 q^{87} + 16 q^{88} - 8 q^{89} + 2 q^{90} + 18 q^{91} - 20 q^{92} - 10 q^{93} - 40 q^{94} + 2 q^{95} + 12 q^{96} + 12 q^{97} - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−1.20711 + 2.09077i 0.500000 0.866025i −1.91421 3.31552i 0.500000 0.866025i 1.20711 + 2.09077i −3.82843 4.41421 −0.500000 0.866025i 1.20711 + 2.09077i
106.2 0.207107 0.358719i 0.500000 0.866025i 0.914214 + 1.58346i 0.500000 0.866025i −0.207107 0.358719i 1.82843 1.58579 −0.500000 0.866025i −0.207107 0.358719i
121.1 −1.20711 2.09077i 0.500000 + 0.866025i −1.91421 + 3.31552i 0.500000 + 0.866025i 1.20711 2.09077i −3.82843 4.41421 −0.500000 + 0.866025i 1.20711 2.09077i
121.2 0.207107 + 0.358719i 0.500000 + 0.866025i 0.914214 1.58346i 0.500000 + 0.866025i −0.207107 + 0.358719i 1.82843 1.58579 −0.500000 + 0.866025i −0.207107 + 0.358719i
\(n\): e.g. 2-40 or 990-1000
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.d 4
3.b odd 2 1 855.2.k.f 4
19.c even 3 1 inner 285.2.i.d 4
19.c even 3 1 5415.2.a.u 2
19.d odd 6 1 5415.2.a.o 2
57.h odd 6 1 855.2.k.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.d 4 1.a even 1 1 trivial
285.2.i.d 4 19.c even 3 1 inner
855.2.k.f 4 3.b odd 2 1
855.2.k.f 4 57.h odd 6 1
5415.2.a.o 2 19.d odd 6 1
5415.2.a.u 2 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + 80 T^{2} - 128 T + 256 \) Copy content Toggle raw display
$31$ \( (T + 5)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + 83 T^{2} + \cdots + 289 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 6 T^{3} + 155 T^{2} + \cdots + 14161 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$71$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$79$ \( T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$83$ \( (T + 8)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} + 120 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
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