Properties

Label 2310.2.e.h
Level $2310$
Weight $2$
Character orbit 2310.e
Analytic conductor $18.445$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2310,2,Mod(1849,2310)]
 
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, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2310.1849");
 
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.e (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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_1 q^{3} - q^{4} + \beta_{3} q^{5} - q^{6} - \beta_1 q^{7} + \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_1 q^{3} - q^{4} + \beta_{3} q^{5} - q^{6} - \beta_1 q^{7} + \beta_1 q^{8} - q^{9} - \beta_{2} q^{10} + q^{11} + \beta_1 q^{12} + 2 \beta_{2} q^{13} - q^{14} - \beta_{2} q^{15} + q^{16} + 2 \beta_{2} q^{17} + \beta_1 q^{18} + (2 \beta_{3} - 4) q^{19} - \beta_{3} q^{20} - q^{21} - \beta_1 q^{22} - 8 \beta_1 q^{23} + q^{24} + 5 q^{25} + 2 \beta_{3} q^{26} + \beta_1 q^{27} + \beta_1 q^{28} + 8 q^{29} - \beta_{3} q^{30} + 4 q^{31} - \beta_1 q^{32} - \beta_1 q^{33} + 2 \beta_{3} q^{34} - \beta_{2} q^{35} + q^{36} + ( - 4 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{2} + 4 \beta_1) q^{38} + 2 \beta_{3} q^{39} + \beta_{2} q^{40} + (2 \beta_{3} - 6) q^{41} + \beta_1 q^{42} + ( - 2 \beta_{2} - 6 \beta_1) q^{43} - q^{44} - \beta_{3} q^{45} - 8 q^{46} + ( - 2 \beta_{2} + 8 \beta_1) q^{47} - \beta_1 q^{48} - q^{49} - 5 \beta_1 q^{50} + 2 \beta_{3} q^{51} - 2 \beta_{2} q^{52} - 2 \beta_1 q^{53} + q^{54} + \beta_{3} q^{55} + q^{56} + ( - 2 \beta_{2} + 4 \beta_1) q^{57} - 8 \beta_1 q^{58} + (4 \beta_{3} + 4) q^{59} + \beta_{2} q^{60} + (2 \beta_{3} - 8) q^{61} - 4 \beta_1 q^{62} + \beta_1 q^{63} - q^{64} + 10 \beta_1 q^{65} - q^{66} + (4 \beta_{2} + 2 \beta_1) q^{67} - 2 \beta_{2} q^{68} - 8 q^{69} - \beta_{3} q^{70} + (2 \beta_{3} + 6) q^{71} - \beta_1 q^{72} + (2 \beta_{2} + 8 \beta_1) q^{73} + ( - 4 \beta_{3} + 2) q^{74} - 5 \beta_1 q^{75} + ( - 2 \beta_{3} + 4) q^{76} - \beta_1 q^{77} - 2 \beta_{2} q^{78} + (2 \beta_{3} - 8) q^{79} + \beta_{3} q^{80} + q^{81} + ( - 2 \beta_{2} + 6 \beta_1) q^{82} + ( - 4 \beta_{2} - 4 \beta_1) q^{83} + q^{84} + 10 \beta_1 q^{85} + ( - 2 \beta_{3} - 6) q^{86} - 8 \beta_1 q^{87} + \beta_1 q^{88} + ( - 2 \beta_{3} + 8) q^{89} + \beta_{2} q^{90} + 2 \beta_{3} q^{91} + 8 \beta_1 q^{92} - 4 \beta_1 q^{93} + ( - 2 \beta_{3} + 8) q^{94} + ( - 4 \beta_{3} + 10) q^{95} - q^{96} + ( - 2 \beta_{2} + 4 \beta_1) q^{97} + \beta_1 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} - 16 q^{19} - 4 q^{21} + 4 q^{24} + 20 q^{25} + 32 q^{29} + 16 q^{31} + 4 q^{36} - 24 q^{41} - 4 q^{44} - 32 q^{46} - 4 q^{49} + 4 q^{54} + 4 q^{56} + 16 q^{59} - 32 q^{61} - 4 q^{64} - 4 q^{66} - 32 q^{69} + 24 q^{71} + 8 q^{74} + 16 q^{76} - 32 q^{79} + 4 q^{81} + 4 q^{84} - 24 q^{86} + 32 q^{89} + 32 q^{94} + 40 q^{95} - 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.61803i
0.618034i
1.61803i
0.618034i
1.00000i 1.00000i −1.00000 −2.23607 −1.00000 1.00000i 1.00000i −1.00000 2.23607i
1849.2 1.00000i 1.00000i −1.00000 2.23607 −1.00000 1.00000i 1.00000i −1.00000 2.23607i
1849.3 1.00000i 1.00000i −1.00000 −2.23607 −1.00000 1.00000i 1.00000i −1.00000 2.23607i
1849.4 1.00000i 1.00000i −1.00000 2.23607 −1.00000 1.00000i 1.00000i −1.00000 2.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.e.h 4
5.b even 2 1 inner 2310.2.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.e.h 4 1.a even 1 1 trivial
2310.2.e.h 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2310, [\chi])\):

\( T_{13}^{2} + 20 \) Copy content Toggle raw display
\( T_{17}^{2} + 20 \) Copy content Toggle raw display
\( T_{19}^{2} + 8T_{19} - 4 \) 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
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T - 8)^{4} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 112T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 16 T + 44)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( (T^{2} + 16 T + 44)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
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