Properties

Label 770.6.a.a
Level $770$
Weight $6$
Character orbit 770.a
Self dual yes
Analytic conductor $123.496$
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] = [770,6,Mod(1,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, 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("770.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 770.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: \(123.495541256\)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} - 30 q^{3} + 16 q^{4} + 25 q^{5} + 120 q^{6} - 49 q^{7} - 64 q^{8} + 657 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 30 q^{3} + 16 q^{4} + 25 q^{5} + 120 q^{6} - 49 q^{7} - 64 q^{8} + 657 q^{9} - 100 q^{10} + 121 q^{11} - 480 q^{12} + 408 q^{13} + 196 q^{14} - 750 q^{15} + 256 q^{16} - 252 q^{17} - 2628 q^{18} - 372 q^{19} + 400 q^{20} + 1470 q^{21} - 484 q^{22} - 2314 q^{23} + 1920 q^{24} + 625 q^{25} - 1632 q^{26} - 12420 q^{27} - 784 q^{28} + 3266 q^{29} + 3000 q^{30} - 1808 q^{31} - 1024 q^{32} - 3630 q^{33} + 1008 q^{34} - 1225 q^{35} + 10512 q^{36} - 1542 q^{37} + 1488 q^{38} - 12240 q^{39} - 1600 q^{40} - 10570 q^{41} - 5880 q^{42} - 6564 q^{43} + 1936 q^{44} + 16425 q^{45} + 9256 q^{46} + 1722 q^{47} - 7680 q^{48} + 2401 q^{49} - 2500 q^{50} + 7560 q^{51} + 6528 q^{52} - 15006 q^{53} + 49680 q^{54} + 3025 q^{55} + 3136 q^{56} + 11160 q^{57} - 13064 q^{58} + 49148 q^{59} - 12000 q^{60} + 5666 q^{61} + 7232 q^{62} - 32193 q^{63} + 4096 q^{64} + 10200 q^{65} + 14520 q^{66} - 26318 q^{67} - 4032 q^{68} + 69420 q^{69} + 4900 q^{70} - 1612 q^{71} - 42048 q^{72} + 63332 q^{73} + 6168 q^{74} - 18750 q^{75} - 5952 q^{76} - 5929 q^{77} + 48960 q^{78} - 40360 q^{79} + 6400 q^{80} + 212949 q^{81} + 42280 q^{82} + 63972 q^{83} + 23520 q^{84} - 6300 q^{85} + 26256 q^{86} - 97980 q^{87} - 7744 q^{88} + 85830 q^{89} - 65700 q^{90} - 19992 q^{91} - 37024 q^{92} + 54240 q^{93} - 6888 q^{94} - 9300 q^{95} + 30720 q^{96} + 13526 q^{97} - 9604 q^{98} + 79497 q^{99}+O(q^{100}) \) 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
−4.00000 −30.0000 16.0000 25.0000 120.000 −49.0000 −64.0000 657.000 −100.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 770.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 30 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(770))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T + 30 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T - 408 \) Copy content Toggle raw display
$17$ \( T + 252 \) Copy content Toggle raw display
$19$ \( T + 372 \) Copy content Toggle raw display
$23$ \( T + 2314 \) Copy content Toggle raw display
$29$ \( T - 3266 \) Copy content Toggle raw display
$31$ \( T + 1808 \) Copy content Toggle raw display
$37$ \( T + 1542 \) Copy content Toggle raw display
$41$ \( T + 10570 \) Copy content Toggle raw display
$43$ \( T + 6564 \) Copy content Toggle raw display
$47$ \( T - 1722 \) Copy content Toggle raw display
$53$ \( T + 15006 \) Copy content Toggle raw display
$59$ \( T - 49148 \) Copy content Toggle raw display
$61$ \( T - 5666 \) Copy content Toggle raw display
$67$ \( T + 26318 \) Copy content Toggle raw display
$71$ \( T + 1612 \) Copy content Toggle raw display
$73$ \( T - 63332 \) Copy content Toggle raw display
$79$ \( T + 40360 \) Copy content Toggle raw display
$83$ \( T - 63972 \) Copy content Toggle raw display
$89$ \( T - 85830 \) Copy content Toggle raw display
$97$ \( T - 13526 \) Copy content Toggle raw display
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