Properties

Label 2310.2.p.b
Level $2310$
Weight $2$
Character orbit 2310.p
Analytic conductor $18.445$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2310,2,Mod(1231,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2310.1231");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.p (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: \(18.4454428669\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_1 q^{3} - q^{4} + \beta_1 q^{5} + q^{6} + ( - \beta_{3} - 1) q^{7} + \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_1 q^{3} - q^{4} + \beta_1 q^{5} + q^{6} + ( - \beta_{3} - 1) q^{7} + \beta_1 q^{8} - q^{9} + q^{10} + ( - \beta_{3} + 2 \beta_1 + 1) q^{11} - \beta_1 q^{12} - 4 q^{13} - \beta_{2} q^{14} - q^{15} + q^{16} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{17} + \beta_1 q^{18} + (\beta_{3} - \beta_{2} - \beta_1 - 4) q^{19} - \beta_1 q^{20} + \beta_{2} q^{21} + ( - \beta_{2} - 2 \beta_1 + 2) q^{22} - q^{24} - q^{25} + 4 \beta_1 q^{26} - \beta_1 q^{27} + (\beta_{3} + 1) q^{28} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{29} + \beta_1 q^{30} + (2 \beta_{3} + 2 \beta_{2}) q^{31} - \beta_1 q^{32} + (\beta_{2} + 2 \beta_1 - 2) q^{33} + (\beta_{3} + \beta_{2} - \beta_1) q^{34} + \beta_{2} q^{35} + q^{36} + 4 q^{37} + (\beta_{3} + \beta_{2} + 5 \beta_1) q^{38} - 4 \beta_1 q^{39} - q^{40} - 10 q^{41} + ( - \beta_{3} - 1) q^{42} + 2 \beta_1 q^{43} + (\beta_{3} - 2 \beta_1 - 1) q^{44} - \beta_1 q^{45} + (\beta_{3} + \beta_{2} - \beta_1) q^{47} + \beta_1 q^{48} + (\beta_{3} - \beta_{2} - 7 \beta_1 + 1) q^{49} + \beta_1 q^{50} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{51} + 4 q^{52} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{53} - q^{54} + (\beta_{2} + 2 \beta_1 - 2) q^{55} + \beta_{2} q^{56} + ( - \beta_{3} - \beta_{2} - 5 \beta_1) q^{57} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{58} - 8 \beta_1 q^{59} + q^{60} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{62} + (\beta_{3} + 1) q^{63} - q^{64} - 4 \beta_1 q^{65} + ( - \beta_{3} + 2 \beta_1 + 1) q^{66} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{67} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{68} + ( - \beta_{3} - 1) q^{70} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{71} - \beta_1 q^{72} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 8) q^{73} - 4 \beta_1 q^{74} - \beta_1 q^{75} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{76} + ( - \beta_{3} + \beta_{2} - 7 \beta_1 - 1) q^{77} - 4 q^{78} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{79} + \beta_1 q^{80} + q^{81} + 10 \beta_1 q^{82} + (\beta_{3} - \beta_{2} - \beta_1 + 6) q^{83} - \beta_{2} q^{84} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{85} + 2 q^{86} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{87} + (\beta_{2} + 2 \beta_1 - 2) q^{88} + (\beta_{3} + \beta_{2} - 9 \beta_1) q^{89} - q^{90} + (4 \beta_{3} + 4) q^{91} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{94} + ( - \beta_{3} - \beta_{2} - 5 \beta_1) q^{95} + q^{96} + ( - 2 \beta_{3} - 2 \beta_{2} + 8 \beta_1) q^{97} + (\beta_{3} + \beta_{2} - 6) q^{98} + (\beta_{3} - 2 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 2 q^{7} - 4 q^{9} + 4 q^{10} + 6 q^{11} - 16 q^{13} - 2 q^{14} - 4 q^{15} + 4 q^{16} + 4 q^{17} - 20 q^{19} + 2 q^{21} + 6 q^{22} - 4 q^{24} - 4 q^{25} + 2 q^{28} - 6 q^{33} + 2 q^{35} + 4 q^{36} + 16 q^{37} - 4 q^{40} - 40 q^{41} - 2 q^{42} - 6 q^{44} + 16 q^{52} - 4 q^{53} - 4 q^{54} - 6 q^{55} + 2 q^{56} + 4 q^{58} + 4 q^{60} - 4 q^{61} + 2 q^{63} - 4 q^{64} + 6 q^{66} - 32 q^{67} - 4 q^{68} - 2 q^{70} - 8 q^{71} - 20 q^{73} + 20 q^{76} - 16 q^{78} + 4 q^{81} + 20 q^{83} - 2 q^{84} + 8 q^{86} - 4 q^{87} - 6 q^{88} - 4 q^{90} + 8 q^{91} - 4 q^{94} + 4 q^{96} - 24 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 3\nu^{2} + 7\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} - 2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2310\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(661\) \(1387\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1231.1
2.30278i
1.30278i
2.30278i
1.30278i
1.00000i 1.00000i −1.00000 1.00000i 1.00000 −2.30278 + 1.30278i 1.00000i −1.00000 1.00000
1231.2 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.30278 2.30278i 1.00000i −1.00000 1.00000
1231.3 1.00000i 1.00000i −1.00000 1.00000i 1.00000 −2.30278 1.30278i 1.00000i −1.00000 1.00000
1231.4 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.30278 + 2.30278i 1.00000i −1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2310.2.p.b yes 4
7.b odd 2 1 2310.2.p.a 4
11.b odd 2 1 2310.2.p.a 4
77.b even 2 1 inner 2310.2.p.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.p.a 4 7.b odd 2 1
2310.2.p.a 4 11.b odd 2 1
2310.2.p.b yes 4 1.a even 1 1 trivial
2310.2.p.b yes 4 77.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} + 4 \) acting on \(S_{2}^{\mathrm{new}}(2310, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 2 T^{2} + 14 T + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 18 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$13$ \( (T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$31$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$37$ \( (T - 4)^{4} \) Copy content Toggle raw display
$41$ \( (T + 10)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16 T + 12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 10 T - 92)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$83$ \( (T^{2} - 10 T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 188T^{2} + 4624 \) Copy content Toggle raw display
$97$ \( T^{4} + 232T^{2} + 144 \) Copy content Toggle raw display
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