Properties

Label 315.10.a.a
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
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] = [315,10,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(162.236288392\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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)
 
\(f(q)\) \(=\) \( q - 28 q^{2} + 272 q^{4} - 625 q^{5} + 2401 q^{7} + 6720 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 28 q^{2} + 272 q^{4} - 625 q^{5} + 2401 q^{7} + 6720 q^{8} + 17500 q^{10} + 25548 q^{11} - 42306 q^{13} - 67228 q^{14} - 327424 q^{16} + 526342 q^{17} - 350060 q^{19} - 170000 q^{20} - 715344 q^{22} + 621976 q^{23} + 390625 q^{25} + 1184568 q^{26} + 653072 q^{28} - 6720430 q^{29} - 6412208 q^{31} + 5727232 q^{32} - 14737576 q^{34} - 1500625 q^{35} - 2317682 q^{37} + 9801680 q^{38} - 4200000 q^{40} + 10224678 q^{41} + 30114004 q^{43} + 6949056 q^{44} - 17415328 q^{46} + 23644912 q^{47} + 5764801 q^{49} - 10937500 q^{50} - 11507232 q^{52} - 57292654 q^{53} - 15967500 q^{55} + 16134720 q^{56} + 188172040 q^{58} - 84934780 q^{59} + 14677822 q^{61} + 179541824 q^{62} + 7278592 q^{64} + 26441250 q^{65} - 244557812 q^{67} + 143165024 q^{68} + 42017500 q^{70} - 61901952 q^{71} - 283763726 q^{73} + 64895096 q^{74} - 95216320 q^{76} + 61340748 q^{77} + 276107480 q^{79} + 204640000 q^{80} - 286290984 q^{82} + 72995956 q^{83} - 328963750 q^{85} - 843192112 q^{86} + 171682560 q^{88} + 896368470 q^{89} - 101576706 q^{91} + 169177472 q^{92} - 662057536 q^{94} + 218787500 q^{95} + 1205809578 q^{97} - 161414428 q^{98}+O(q^{100}) \) 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
−28.0000 0 272.000 −625.000 0 2401.00 6720.00 0 17500.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

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 315.10.a.a 1
3.b odd 2 1 35.10.a.a 1
15.d odd 2 1 175.10.a.a 1
15.e even 4 2 175.10.b.a 2
21.c even 2 1 245.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.a 1 3.b odd 2 1
175.10.a.a 1 15.d odd 2 1
175.10.b.a 2 15.e even 4 2
245.10.a.b 1 21.c even 2 1
315.10.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 28 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 28 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 625 \) Copy content Toggle raw display
$7$ \( T - 2401 \) Copy content Toggle raw display
$11$ \( T - 25548 \) Copy content Toggle raw display
$13$ \( T + 42306 \) Copy content Toggle raw display
$17$ \( T - 526342 \) Copy content Toggle raw display
$19$ \( T + 350060 \) Copy content Toggle raw display
$23$ \( T - 621976 \) Copy content Toggle raw display
$29$ \( T + 6720430 \) Copy content Toggle raw display
$31$ \( T + 6412208 \) Copy content Toggle raw display
$37$ \( T + 2317682 \) Copy content Toggle raw display
$41$ \( T - 10224678 \) Copy content Toggle raw display
$43$ \( T - 30114004 \) Copy content Toggle raw display
$47$ \( T - 23644912 \) Copy content Toggle raw display
$53$ \( T + 57292654 \) Copy content Toggle raw display
$59$ \( T + 84934780 \) Copy content Toggle raw display
$61$ \( T - 14677822 \) Copy content Toggle raw display
$67$ \( T + 244557812 \) Copy content Toggle raw display
$71$ \( T + 61901952 \) Copy content Toggle raw display
$73$ \( T + 283763726 \) Copy content Toggle raw display
$79$ \( T - 276107480 \) Copy content Toggle raw display
$83$ \( T - 72995956 \) Copy content Toggle raw display
$89$ \( T - 896368470 \) Copy content Toggle raw display
$97$ \( T - 1205809578 \) Copy content Toggle raw display
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