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Properties

Label 2310.2.g.f
Level $2310$
Weight $2$
Character orbit 2310.g
Analytic conductor $18.445$
Analytic rank $0$
Dimension $12$
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(1121,2310)]
 
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([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2310.1121");
 
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.g (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: \(12\)
Coefficient field: 12.0.308868533518336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 5x^{10} + 2x^{9} - 10x^{8} + 18x^{6} - 40x^{4} + 16x^{3} + 80x^{2} - 128x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{4} q^{3} + q^{4} - \beta_{7} q^{5} + \beta_{4} q^{6} - \beta_{7} q^{7} + q^{8} - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_{4} q^{3} + q^{4} - \beta_{7} q^{5} + \beta_{4} q^{6} - \beta_{7} q^{7} + q^{8} - \beta_{10} q^{9} - \beta_{7} q^{10} + (\beta_{9} - \beta_{8} - \beta_{5}) q^{11} + \beta_{4} q^{12} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{2}) q^{13}+ \cdots + (2 \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 4 q^{3} + 12 q^{4} + 4 q^{6} + 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 4 q^{3} + 12 q^{4} + 4 q^{6} + 12 q^{8} - 4 q^{9} + 4 q^{12} + 12 q^{16} + 12 q^{17} - 4 q^{18} + 4 q^{24} - 12 q^{25} + 4 q^{27} - 48 q^{29} + 40 q^{31} + 12 q^{32} - 16 q^{33} + 12 q^{34} - 12 q^{35} - 4 q^{36} - 12 q^{37} + 20 q^{39} + 16 q^{41} + 4 q^{48} - 12 q^{49} - 12 q^{50} - 16 q^{51} + 4 q^{54} + 40 q^{57} - 48 q^{58} + 40 q^{62} + 12 q^{64} - 12 q^{65} - 16 q^{66} - 20 q^{67} + 12 q^{68} + 52 q^{69} - 12 q^{70} - 4 q^{72} - 12 q^{74} - 4 q^{75} + 20 q^{78} - 36 q^{81} + 16 q^{82} + 56 q^{83} - 16 q^{87} - 12 q^{91} + 32 q^{93} + 4 q^{96} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{11} + 5x^{10} + 2x^{9} - 10x^{8} + 18x^{6} - 40x^{4} + 16x^{3} + 80x^{2} - 128x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3 \nu^{11} + 2 \nu^{10} + 13 \nu^{9} - 16 \nu^{8} - 34 \nu^{7} + 36 \nu^{6} + 66 \nu^{5} + \cdots - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3 \nu^{11} + 6 \nu^{10} - 13 \nu^{9} - 8 \nu^{8} + 18 \nu^{7} + 12 \nu^{6} - 66 \nu^{5} - 12 \nu^{4} + \cdots + 96 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5 \nu^{11} + 14 \nu^{10} + 11 \nu^{9} - 48 \nu^{8} - 14 \nu^{7} + 92 \nu^{6} + 14 \nu^{5} + \cdots - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + 2\nu^{10} - 3\nu^{9} + 8\nu^{7} - 4\nu^{6} - 14\nu^{5} + 4\nu^{4} + 28\nu^{3} - 24\nu^{2} - 80\nu + 112 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{11} - 22 \nu^{10} + \nu^{9} + 48 \nu^{8} - 18 \nu^{7} - 76 \nu^{6} + 42 \nu^{5} + 172 \nu^{4} + \cdots - 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{11} + 26 \nu^{10} - 11 \nu^{9} - 40 \nu^{8} + 46 \nu^{7} + 52 \nu^{6} - 78 \nu^{5} + \cdots + 480 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13 \nu^{11} + 38 \nu^{10} - 29 \nu^{9} - 56 \nu^{8} + 82 \nu^{7} + 92 \nu^{6} - 162 \nu^{5} + \cdots + 672 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23 \nu^{11} + 58 \nu^{10} - 31 \nu^{9} - 80 \nu^{8} + 110 \nu^{7} + 132 \nu^{6} - 214 \nu^{5} + \cdots + 1120 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + 8 \nu^{10} - 5 \nu^{9} - 12 \nu^{8} + 16 \nu^{7} + 18 \nu^{6} - 30 \nu^{5} - 36 \nu^{4} + \cdots + 152 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{11} + 4 \nu^{10} - 7 \nu^{8} + 2 \nu^{7} + 13 \nu^{6} - 6 \nu^{5} - 26 \nu^{4} + 20 \nu^{3} + \cdots + 52 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4 \nu^{11} + 11 \nu^{10} - 5 \nu^{9} - 17 \nu^{8} + 17 \nu^{7} + 28 \nu^{6} - 34 \nu^{5} - 58 \nu^{4} + \cdots + 168 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{7} + \beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{10} + \beta_{9} - 2\beta_{8} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{5} - 2 \beta_{4} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{10} + \beta_{9} + 2\beta_{7} - 2\beta_{6} + 4\beta_{5} - 6\beta_{4} - 2\beta_{2} - 2\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} - 3 \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - 8 \beta_{4} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4\beta_{11} - \beta_{10} - 11\beta_{9} + 8\beta_{8} + 4\beta_{7} + 8\beta_{5} - 2\beta_{4} - 2\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4 \beta_{11} - \beta_{10} + \beta_{9} + 16 \beta_{8} - 20 \beta_{7} + 10 \beta_{6} + 8 \beta_{5} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 24 \beta_{11} - \beta_{10} - 7 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} + 8 \beta_{6} + 16 \beta_{5} + \cdots - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20 \beta_{11} - 9 \beta_{10} + \beta_{9} - 28 \beta_{7} + 6 \beta_{6} + 36 \beta_{5} + 6 \beta_{4} + \cdots + 40 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
0.713644 + 1.22095i
1.41093 + 0.0963767i
1.41093 − 0.0963767i
0.713644 − 1.22095i
1.31089 + 0.530639i
−0.884069 + 1.10382i
−0.884069 − 1.10382i
1.31089 − 0.530639i
−1.35920 − 0.390597i
0.807816 + 1.16079i
0.807816 − 1.16079i
−1.35920 + 0.390597i
1.00000 −1.12457 − 1.31732i 1.00000 − 1.00000i −1.12457 − 1.31732i − 1.00000i 1.00000 −0.470683 + 2.96285i − 1.00000i
1121.2 1.00000 −1.12457 − 1.31732i 1.00000 1.00000i −1.12457 − 1.31732i 1.00000i 1.00000 −0.470683 + 2.96285i 1.00000i
1121.3 1.00000 −1.12457 + 1.31732i 1.00000 − 1.00000i −1.12457 + 1.31732i − 1.00000i 1.00000 −0.470683 − 2.96285i − 1.00000i
1121.4 1.00000 −1.12457 + 1.31732i 1.00000 1.00000i −1.12457 + 1.31732i 1.00000i 1.00000 −0.470683 − 2.96285i 1.00000i
1121.5 1.00000 0.573183 − 1.63446i 1.00000 − 1.00000i 0.573183 − 1.63446i − 1.00000i 1.00000 −2.34292 − 1.87369i − 1.00000i
1121.6 1.00000 0.573183 − 1.63446i 1.00000 1.00000i 0.573183 − 1.63446i 1.00000i 1.00000 −2.34292 − 1.87369i 1.00000i
1121.7 1.00000 0.573183 + 1.63446i 1.00000 − 1.00000i 0.573183 + 1.63446i − 1.00000i 1.00000 −2.34292 + 1.87369i − 1.00000i
1121.8 1.00000 0.573183 + 1.63446i 1.00000 1.00000i 0.573183 + 1.63446i 1.00000i 1.00000 −2.34292 + 1.87369i 1.00000i
1121.9 1.00000 1.55139 − 0.770193i 1.00000 − 1.00000i 1.55139 − 0.770193i − 1.00000i 1.00000 1.81361 − 2.38973i − 1.00000i
1121.10 1.00000 1.55139 − 0.770193i 1.00000 1.00000i 1.55139 − 0.770193i 1.00000i 1.00000 1.81361 − 2.38973i 1.00000i
1121.11 1.00000 1.55139 + 0.770193i 1.00000 − 1.00000i 1.55139 + 0.770193i − 1.00000i 1.00000 1.81361 + 2.38973i − 1.00000i
1121.12 1.00000 1.55139 + 0.770193i 1.00000 1.00000i 1.55139 + 0.770193i 1.00000i 1.00000 1.81361 + 2.38973i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d 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.g.f yes 12
3.b odd 2 1 2310.2.g.e ✓ 12
11.b odd 2 1 2310.2.g.e ✓ 12
33.d even 2 1 inner 2310.2.g.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.g.e ✓ 12 3.b odd 2 1
2310.2.g.e ✓ 12 11.b odd 2 1
2310.2.g.f yes 12 1.a even 1 1 trivial
2310.2.g.f yes 12 33.d 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 \(S_{2}^{\mathrm{new}}(2310, [\chi])\):

\( T_{13}^{12} + 96T_{13}^{10} + 3396T_{13}^{8} + 53504T_{13}^{6} + 364544T_{13}^{4} + 1048576T_{13}^{2} + 1048576 \) Copy content Toggle raw display
\( T_{17}^{6} - 6T_{17}^{5} - 30T_{17}^{4} + 112T_{17}^{3} + 288T_{17}^{2} - 224T_{17} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} + 3 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 96 T^{10} + \cdots + 1048576 \) Copy content Toggle raw display
$17$ \( (T^{6} - 6 T^{5} - 30 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 152 T^{10} + \cdots + 473344 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 5219773504 \) Copy content Toggle raw display
$29$ \( (T^{6} + 24 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 10 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} - 30 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} + \cdots + 15056)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 60523872256 \) Copy content Toggle raw display
$47$ \( (T^{6} + 60 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 296 T^{10} + \cdots + 215296 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 220463104 \) Copy content Toggle raw display
$61$ \( T^{12} + 320 T^{10} + \cdots + 4194304 \) Copy content Toggle raw display
$67$ \( (T^{6} + 10 T^{5} + \cdots - 352)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 196 T^{4} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 736362496 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 3380724736 \) Copy content Toggle raw display
$83$ \( (T^{6} - 28 T^{5} + \cdots - 40448)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 146989425664 \) Copy content Toggle raw display
$97$ \( (T^{6} - 400 T^{4} + \cdots - 855136)^{2} \) Copy content Toggle raw display
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