Properties

Label 304.10.a.b
Level $304$
Weight $10$
Character orbit 304.a
Self dual yes
Analytic conductor $156.571$
Analytic rank $0$
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] = [304,10,Mod(1,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, 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("304.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 304.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: \(156.570894194\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
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)
 
\(f(q)\) \(=\) \( q + 119 q^{3} - 684 q^{5} - 9149 q^{7} - 5522 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 119 q^{3} - 684 q^{5} - 9149 q^{7} - 5522 q^{9} - 5790 q^{11} - 179881 q^{13} - 81396 q^{15} - 594093 q^{17} - 130321 q^{19} - 1088731 q^{21} + 1744767 q^{23} - 1485269 q^{25} - 2999395 q^{27} + 4314387 q^{29} - 160232 q^{31} - 689010 q^{33} + 6257916 q^{35} - 21943090 q^{37} - 21405839 q^{39} + 294816 q^{41} + 39393148 q^{43} + 3777048 q^{45} - 46596360 q^{47} + 43350594 q^{49} - 70697067 q^{51} + 22121703 q^{53} + 3960360 q^{55} - 15508199 q^{57} - 33070233 q^{59} + 188535938 q^{61} + 50520778 q^{63} + 123038604 q^{65} + 20769067 q^{67} + 207627273 q^{69} + 232299978 q^{71} - 3022183 q^{73} - 176747011 q^{75} + 52972710 q^{77} + 446379406 q^{79} - 248238479 q^{81} - 794022846 q^{83} + 406359612 q^{85} + 513412053 q^{87} + 90999336 q^{89} + 1645731269 q^{91} - 19067608 q^{93} + 89139564 q^{95} - 123974170 q^{97} + 31972380 q^{99}+O(q^{100}) \) 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
0 119.000 0 −684.000 0 −9149.00 0 −5522.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

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 304.10.a.b 1
4.b odd 2 1 38.10.a.a 1
12.b even 2 1 342.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.a 1 4.b odd 2 1
304.10.a.b 1 1.a even 1 1 trivial
342.10.a.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 119 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(304))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 119 \) Copy content Toggle raw display
$5$ \( T + 684 \) Copy content Toggle raw display
$7$ \( T + 9149 \) Copy content Toggle raw display
$11$ \( T + 5790 \) Copy content Toggle raw display
$13$ \( T + 179881 \) Copy content Toggle raw display
$17$ \( T + 594093 \) Copy content Toggle raw display
$19$ \( T + 130321 \) Copy content Toggle raw display
$23$ \( T - 1744767 \) Copy content Toggle raw display
$29$ \( T - 4314387 \) Copy content Toggle raw display
$31$ \( T + 160232 \) Copy content Toggle raw display
$37$ \( T + 21943090 \) Copy content Toggle raw display
$41$ \( T - 294816 \) Copy content Toggle raw display
$43$ \( T - 39393148 \) Copy content Toggle raw display
$47$ \( T + 46596360 \) Copy content Toggle raw display
$53$ \( T - 22121703 \) Copy content Toggle raw display
$59$ \( T + 33070233 \) Copy content Toggle raw display
$61$ \( T - 188535938 \) Copy content Toggle raw display
$67$ \( T - 20769067 \) Copy content Toggle raw display
$71$ \( T - 232299978 \) Copy content Toggle raw display
$73$ \( T + 3022183 \) Copy content Toggle raw display
$79$ \( T - 446379406 \) Copy content Toggle raw display
$83$ \( T + 794022846 \) Copy content Toggle raw display
$89$ \( T - 90999336 \) Copy content Toggle raw display
$97$ \( T + 123974170 \) Copy content Toggle raw display
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