Properties

Label 285.10.a.a
Level $285$
Weight $10$
Character orbit 285.a
Self dual yes
Analytic conductor $146.785$
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] = [285,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(146.785213307\)
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)
 
\(f(q)\) \(=\) \( q - 6 q^{2} + 81 q^{3} - 476 q^{4} - 625 q^{5} - 486 q^{6} - 5866 q^{7} + 5928 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{2} + 81 q^{3} - 476 q^{4} - 625 q^{5} - 486 q^{6} - 5866 q^{7} + 5928 q^{8} + 6561 q^{9} + 3750 q^{10} - 17982 q^{11} - 38556 q^{12} - 61720 q^{13} + 35196 q^{14} - 50625 q^{15} + 208144 q^{16} + 290010 q^{17} - 39366 q^{18} + 130321 q^{19} + 297500 q^{20} - 475146 q^{21} + 107892 q^{22} - 670584 q^{23} + 480168 q^{24} + 390625 q^{25} + 370320 q^{26} + 531441 q^{27} + 2792216 q^{28} + 2418540 q^{29} + 303750 q^{30} + 1323344 q^{31} - 4284000 q^{32} - 1456542 q^{33} - 1740060 q^{34} + 3666250 q^{35} - 3123036 q^{36} + 10239068 q^{37} - 781926 q^{38} - 4999320 q^{39} - 3705000 q^{40} + 24845016 q^{41} + 2850876 q^{42} - 22280938 q^{43} + 8559432 q^{44} - 4100625 q^{45} + 4023504 q^{46} - 9039288 q^{47} + 16859664 q^{48} - 5943651 q^{49} - 2343750 q^{50} + 23490810 q^{51} + 29378720 q^{52} + 95165394 q^{53} - 3188646 q^{54} + 11238750 q^{55} - 34773648 q^{56} + 10556001 q^{57} - 14511240 q^{58} + 88064508 q^{59} + 24097500 q^{60} + 159150530 q^{61} - 7940064 q^{62} - 38486826 q^{63} - 80865728 q^{64} + 38575000 q^{65} + 8739252 q^{66} - 165219640 q^{67} - 138044760 q^{68} - 54317304 q^{69} - 21997500 q^{70} - 271185360 q^{71} + 38893608 q^{72} - 167588674 q^{73} - 61434408 q^{74} + 31640625 q^{75} - 62032796 q^{76} + 105482412 q^{77} + 29995920 q^{78} - 261759544 q^{79} - 130090000 q^{80} + 43046721 q^{81} - 149070096 q^{82} - 320534784 q^{83} + 226169496 q^{84} - 181256250 q^{85} + 133685628 q^{86} + 195901740 q^{87} - 106597296 q^{88} + 443005080 q^{89} + 24603750 q^{90} + 362049520 q^{91} + 319197984 q^{92} + 107190864 q^{93} + 54235728 q^{94} - 81450625 q^{95} - 347004000 q^{96} - 1246029172 q^{97} + 35661906 q^{98} - 117979902 q^{99}+O(q^{100}) \) 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
−6.00000 81.0000 −476.000 −625.000 −486.000 −5866.00 5928.00 6561.00 3750.00
\(n\): 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 285.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.10.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 6 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(285))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 6 \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T + 625 \) Copy content Toggle raw display
$7$ \( T + 5866 \) Copy content Toggle raw display
$11$ \( T + 17982 \) Copy content Toggle raw display
$13$ \( T + 61720 \) Copy content Toggle raw display
$17$ \( T - 290010 \) Copy content Toggle raw display
$19$ \( T - 130321 \) Copy content Toggle raw display
$23$ \( T + 670584 \) Copy content Toggle raw display
$29$ \( T - 2418540 \) Copy content Toggle raw display
$31$ \( T - 1323344 \) Copy content Toggle raw display
$37$ \( T - 10239068 \) Copy content Toggle raw display
$41$ \( T - 24845016 \) Copy content Toggle raw display
$43$ \( T + 22280938 \) Copy content Toggle raw display
$47$ \( T + 9039288 \) Copy content Toggle raw display
$53$ \( T - 95165394 \) Copy content Toggle raw display
$59$ \( T - 88064508 \) Copy content Toggle raw display
$61$ \( T - 159150530 \) Copy content Toggle raw display
$67$ \( T + 165219640 \) Copy content Toggle raw display
$71$ \( T + 271185360 \) Copy content Toggle raw display
$73$ \( T + 167588674 \) Copy content Toggle raw display
$79$ \( T + 261759544 \) Copy content Toggle raw display
$83$ \( T + 320534784 \) Copy content Toggle raw display
$89$ \( T - 443005080 \) Copy content Toggle raw display
$97$ \( T + 1246029172 \) Copy content Toggle raw display
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