Properties

Label 285.6.a.a
Level $285$
Weight $6$
Character orbit 285.a
Self dual yes
Analytic conductor $45.709$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [285,6,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 285.a (trivial)

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: yes
Analytic conductor: \(45.7093886467\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 9 q^{3} - 28 q^{4} + 25 q^{5} + 18 q^{6} + 22 q^{7} - 120 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 9 q^{3} - 28 q^{4} + 25 q^{5} + 18 q^{6} + 22 q^{7} - 120 q^{8} + 81 q^{9} + 50 q^{10} - 568 q^{11} - 252 q^{12} + 1046 q^{13} + 44 q^{14} + 225 q^{15} + 656 q^{16} - 428 q^{17} + 162 q^{18} - 361 q^{19} - 700 q^{20} + 198 q^{21} - 1136 q^{22} - 4244 q^{23} - 1080 q^{24} + 625 q^{25} + 2092 q^{26} + 729 q^{27} - 616 q^{28} - 5110 q^{29} + 450 q^{30} - 378 q^{31} + 5152 q^{32} - 5112 q^{33} - 856 q^{34} + 550 q^{35} - 2268 q^{36} + 8622 q^{37} - 722 q^{38} + 9414 q^{39} - 3000 q^{40} - 5448 q^{41} + 396 q^{42} - 18724 q^{43} + 15904 q^{44} + 2025 q^{45} - 8488 q^{46} - 13428 q^{47} + 5904 q^{48} - 16323 q^{49} + 1250 q^{50} - 3852 q^{51} - 29288 q^{52} - 16774 q^{53} + 1458 q^{54} - 14200 q^{55} - 2640 q^{56} - 3249 q^{57} - 10220 q^{58} + 11030 q^{59} - 6300 q^{60} + 10682 q^{61} - 756 q^{62} + 1782 q^{63} - 10688 q^{64} + 26150 q^{65} - 10224 q^{66} - 54268 q^{67} + 11984 q^{68} - 38196 q^{69} + 1100 q^{70} - 3528 q^{71} - 9720 q^{72} + 75886 q^{73} + 17244 q^{74} + 5625 q^{75} + 10108 q^{76} - 12496 q^{77} + 18828 q^{78} - 89450 q^{79} + 16400 q^{80} + 6561 q^{81} - 10896 q^{82} - 57294 q^{83} - 5544 q^{84} - 10700 q^{85} - 37448 q^{86} - 45990 q^{87} + 68160 q^{88} + 18540 q^{89} + 4050 q^{90} + 23012 q^{91} + 118832 q^{92} - 3402 q^{93} - 26856 q^{94} - 9025 q^{95} + 46368 q^{96} - 52798 q^{97} - 32646 q^{98} - 46008 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 9.00000 −28.0000 25.0000 18.0000 22.0000 −120.000 81.0000 50.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.6.a.a 1
3.b odd 2 1 855.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.6.a.a 1 1.a even 1 1 trivial
855.6.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(285))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 22 \) Copy content Toggle raw display
$11$ \( T + 568 \) Copy content Toggle raw display
$13$ \( T - 1046 \) Copy content Toggle raw display
$17$ \( T + 428 \) Copy content Toggle raw display
$19$ \( T + 361 \) Copy content Toggle raw display
$23$ \( T + 4244 \) Copy content Toggle raw display
$29$ \( T + 5110 \) Copy content Toggle raw display
$31$ \( T + 378 \) Copy content Toggle raw display
$37$ \( T - 8622 \) Copy content Toggle raw display
$41$ \( T + 5448 \) Copy content Toggle raw display
$43$ \( T + 18724 \) Copy content Toggle raw display
$47$ \( T + 13428 \) Copy content Toggle raw display
$53$ \( T + 16774 \) Copy content Toggle raw display
$59$ \( T - 11030 \) Copy content Toggle raw display
$61$ \( T - 10682 \) Copy content Toggle raw display
$67$ \( T + 54268 \) Copy content Toggle raw display
$71$ \( T + 3528 \) Copy content Toggle raw display
$73$ \( T - 75886 \) Copy content Toggle raw display
$79$ \( T + 89450 \) Copy content Toggle raw display
$83$ \( T + 57294 \) Copy content Toggle raw display
$89$ \( T - 18540 \) Copy content Toggle raw display
$97$ \( T + 52798 \) Copy content Toggle raw display
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