Properties

Label 315.3.h.a
Level $315$
Weight $3$
Character orbit 315.h
Analytic conductor $8.583$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,3,Mod(181,315)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.181");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.h (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta q^{5} + ( - 3 \beta - 2) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + \beta q^{5} + ( - 3 \beta - 2) q^{7} + 8 q^{8} - 2 \beta q^{10} + q^{11} + 9 \beta q^{13} + (6 \beta + 4) q^{14} - 16 q^{16} - 3 \beta q^{17} - 6 \beta q^{19} - 2 q^{22} - 8 q^{23} - 5 q^{25} - 18 \beta q^{26} - 41 q^{29} - 18 \beta q^{31} + 6 \beta q^{34} + ( - 2 \beta + 15) q^{35} - 28 q^{37} + 12 \beta q^{38} + 8 \beta q^{40} + 6 \beta q^{41} - 82 q^{43} + 16 q^{46} + 9 \beta q^{47} + (12 \beta - 41) q^{49} + 10 q^{50} - 74 q^{53} + \beta q^{55} + ( - 24 \beta - 16) q^{56} + 82 q^{58} - 42 \beta q^{59} - 36 \beta q^{61} + 36 \beta q^{62} + 64 q^{64} - 45 q^{65} + 2 q^{67} + (4 \beta - 30) q^{70} - 14 q^{71} + 30 \beta q^{73} + 56 q^{74} + ( - 3 \beta - 2) q^{77} - 19 q^{79} - 16 \beta q^{80} - 12 \beta q^{82} + 42 \beta q^{83} + 15 q^{85} + 164 q^{86} + 8 q^{88} + 48 \beta q^{89} + ( - 18 \beta + 135) q^{91} - 18 \beta q^{94} + 30 q^{95} + 27 \beta q^{97} + ( - 24 \beta + 82) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 4 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 4 q^{7} + 16 q^{8} + 2 q^{11} + 8 q^{14} - 32 q^{16} - 4 q^{22} - 16 q^{23} - 10 q^{25} - 82 q^{29} + 30 q^{35} - 56 q^{37} - 164 q^{43} + 32 q^{46} - 82 q^{49} + 20 q^{50} - 148 q^{53} - 32 q^{56} + 164 q^{58} + 128 q^{64} - 90 q^{65} + 4 q^{67} - 60 q^{70} - 28 q^{71} + 112 q^{74} - 4 q^{77} - 38 q^{79} + 30 q^{85} + 328 q^{86} + 16 q^{88} + 270 q^{91} + 60 q^{95} + 164 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
2.23607i
2.23607i
−2.00000 0 0 2.23607i 0 −2.00000 + 6.70820i 8.00000 0 4.47214i
181.2 −2.00000 0 0 2.23607i 0 −2.00000 6.70820i 8.00000 0 4.47214i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.h.a 2
3.b odd 2 1 35.3.d.b 2
7.b odd 2 1 inner 315.3.h.a 2
12.b even 2 1 560.3.f.b 2
15.d odd 2 1 175.3.d.c 2
15.e even 4 2 175.3.c.c 4
21.c even 2 1 35.3.d.b 2
21.g even 6 2 245.3.h.a 4
21.h odd 6 2 245.3.h.a 4
84.h odd 2 1 560.3.f.b 2
105.g even 2 1 175.3.d.c 2
105.k odd 4 2 175.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 3.b odd 2 1
35.3.d.b 2 21.c even 2 1
175.3.c.c 4 15.e even 4 2
175.3.c.c 4 105.k odd 4 2
175.3.d.c 2 15.d odd 2 1
175.3.d.c 2 105.g even 2 1
245.3.h.a 4 21.g even 6 2
245.3.h.a 4 21.h odd 6 2
315.3.h.a 2 1.a even 1 1 trivial
315.3.h.a 2 7.b odd 2 1 inner
560.3.f.b 2 12.b even 2 1
560.3.f.b 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 2 \) acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 49 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 405 \) Copy content Toggle raw display
$17$ \( T^{2} + 45 \) Copy content Toggle raw display
$19$ \( T^{2} + 180 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T + 41)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1620 \) Copy content Toggle raw display
$37$ \( (T + 28)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 180 \) Copy content Toggle raw display
$43$ \( (T + 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 405 \) Copy content Toggle raw display
$53$ \( (T + 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8820 \) Copy content Toggle raw display
$61$ \( T^{2} + 6480 \) Copy content Toggle raw display
$67$ \( (T - 2)^{2} \) Copy content Toggle raw display
$71$ \( (T + 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4500 \) Copy content Toggle raw display
$79$ \( (T + 19)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 8820 \) Copy content Toggle raw display
$89$ \( T^{2} + 11520 \) Copy content Toggle raw display
$97$ \( T^{2} + 3645 \) Copy content Toggle raw display
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