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Properties

Label 2366.2.d.q
Level $2366$
Weight $2$
Character orbit 2366.d
Analytic conductor $18.893$
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] = [2366,2,Mod(337,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2366.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.d (of order \(2\), degree \(1\), not 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: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 23x^{10} + 201x^{8} + 847x^{6} + 1809x^{4} + 1863x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{11} q^{3} - q^{4} + ( - \beta_{9} - \beta_{2}) q^{5} + \beta_1 q^{6} - \beta_{8} q^{7} - \beta_{8} q^{8} + ( - \beta_{11} - \beta_{7} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{11} q^{3} - q^{4} + ( - \beta_{9} - \beta_{2}) q^{5} + \beta_1 q^{6} - \beta_{8} q^{7} - \beta_{8} q^{8} + ( - \beta_{11} - \beta_{7} - \beta_{6}) q^{9} + (\beta_{10} + \beta_{6}) q^{10} + ( - \beta_{9} - 2 \beta_{8} + \cdots - \beta_{2}) q^{11}+ \cdots + (\beta_{9} + 3 \beta_{8} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 12 q^{4} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 12 q^{4} + 10 q^{9} - 8 q^{10} - 2 q^{12} + 12 q^{14} + 12 q^{16} + 18 q^{17} + 12 q^{22} - 42 q^{23} - 16 q^{25} + 14 q^{27} - 2 q^{29} + 10 q^{30} + 8 q^{35} - 10 q^{36} - 20 q^{38} + 8 q^{40} + 2 q^{42} - 4 q^{43} + 2 q^{48} - 12 q^{49} + 30 q^{51} + 56 q^{53} - 58 q^{55} - 12 q^{56} - 54 q^{61} - 40 q^{62} - 12 q^{64} + 18 q^{66} - 18 q^{68} + 30 q^{69} + 32 q^{74} + 14 q^{75} - 12 q^{77} + 12 q^{79} - 52 q^{81} - 4 q^{82} - 78 q^{87} - 12 q^{88} - 34 q^{90} + 42 q^{92} - 10 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 23x^{10} + 201x^{8} + 847x^{6} + 1809x^{4} + 1863x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 23\nu^{9} + 201\nu^{7} + 847\nu^{5} + 1809\nu^{3} + 1620\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{11} - 157\nu^{9} - 1068\nu^{7} - 2969\nu^{5} - 3024\nu^{3} - 648\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 23\nu^{9} + 201\nu^{7} + 847\nu^{5} + 1728\nu^{3} + 1215\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} + 23\nu^{8} + 201\nu^{6} + 847\nu^{4} + 1728\nu^{2} + 1215 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} - 23\nu^{8} - 201\nu^{6} - 820\nu^{4} - 1485\nu^{2} - 891 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} - 20\nu^{8} - 141\nu^{6} - 424\nu^{4} - 528\nu^{2} - 216 ) / 9 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -20\nu^{11} - 424\nu^{9} - 3273\nu^{7} - 11324\nu^{5} - 17352\nu^{3} - 9396\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -7\nu^{11} - 152\nu^{9} - 1200\nu^{7} - 4201\nu^{5} - 6336\nu^{3} - 3240\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -8\nu^{10} - 175\nu^{8} - 1401\nu^{6} - 5048\nu^{4} - 8064\nu^{2} - 4536 ) / 81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4\nu^{10} + 83\nu^{8} + 624\nu^{6} + 2092\nu^{4} + 3096\nu^{2} + 1620 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{7} + \beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + 3\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{11} - 9\beta_{7} - 8\beta_{6} + 3\beta_{5} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} + 3\beta_{8} + 13\beta_{4} - 3\beta_{3} - 24\beta_{2} + 33\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 73\beta_{11} + 3\beta_{10} + 72\beta_{7} + 56\beta_{6} - 36\beta_{5} - 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20\beta_{9} - 51\beta_{8} - 128\beta_{4} + 48\beta_{3} + 168\beta_{2} - 244\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -591\beta_{11} - 60\beta_{10} - 568\beta_{7} - 392\beta_{6} + 324\beta_{5} + 672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -259\beta_{9} + 597\beta_{8} + 1151\beta_{4} - 528\beta_{3} - 1176\beta_{2} + 1891\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4815\beta_{11} + 777\beta_{10} + 4487\beta_{7} + 2808\beta_{6} - 2676\beta_{5} - 4896 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2784\beta_{9} - 6021\beta_{8} - 9947\beta_{4} + 5037\beta_{3} + 8424\beta_{2} - 14975\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
2.46434i
1.52377i
1.05140i
− 1.21736i
− 1.96881i
− 2.85334i
− 2.46434i
− 1.52377i
− 1.05140i
1.21736i
1.96881i
2.85334i
− 1.00000i −2.46434 −1.00000 − 3.19361i 2.46434i 1.00000i 1.00000i 3.07299 −3.19361
337.2 − 1.00000i −1.52377 −1.00000 1.45506i 1.52377i 1.00000i 1.00000i −0.678139 1.45506
337.3 − 1.00000i −1.05140 −1.00000 − 2.26985i 1.05140i 1.00000i 1.00000i −1.89456 −2.26985
337.4 − 1.00000i 1.21736 −1.00000 3.44059i − 1.21736i 1.00000i 1.00000i −1.51803 3.44059
337.5 − 1.00000i 1.96881 −1.00000 − 2.90010i − 1.96881i 1.00000i 1.00000i 0.876202 −2.90010
337.6 − 1.00000i 2.85334 −1.00000 − 0.532083i − 2.85334i 1.00000i 1.00000i 5.14154 −0.532083
337.7 1.00000i −2.46434 −1.00000 3.19361i − 2.46434i − 1.00000i − 1.00000i 3.07299 −3.19361
337.8 1.00000i −1.52377 −1.00000 − 1.45506i − 1.52377i − 1.00000i − 1.00000i −0.678139 1.45506
337.9 1.00000i −1.05140 −1.00000 2.26985i − 1.05140i − 1.00000i − 1.00000i −1.89456 −2.26985
337.10 1.00000i 1.21736 −1.00000 − 3.44059i 1.21736i − 1.00000i − 1.00000i −1.51803 3.44059
337.11 1.00000i 1.96881 −1.00000 2.90010i 1.96881i − 1.00000i − 1.00000i 0.876202 −2.90010
337.12 1.00000i 2.85334 −1.00000 0.532083i 2.85334i − 1.00000i − 1.00000i 5.14154 −0.532083
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.d.q 12
13.b even 2 1 inner 2366.2.d.q 12
13.d odd 4 1 2366.2.a.be ✓ 6
13.d odd 4 1 2366.2.a.bg yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.be ✓ 6 13.d odd 4 1
2366.2.a.bg yes 6 13.d odd 4 1
2366.2.d.q 12 1.a even 1 1 trivial
2366.2.d.q 12 13.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 \(S_{2}^{\mathrm{new}}(2366, [\chi])\):

\( T_{3}^{6} - T_{3}^{5} - 11T_{3}^{4} + 7T_{3}^{3} + 33T_{3}^{2} - 9T_{3} - 27 \) Copy content Toggle raw display
\( T_{5}^{12} + 38T_{5}^{10} + 549T_{5}^{8} + 3725T_{5}^{6} + 11732T_{5}^{4} + 14112T_{5}^{2} + 3136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{5} - 11 T^{4} + \cdots - 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 38 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 46 T^{10} + \cdots + 9409 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 9 T^{5} - T^{4} + \cdots + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 134 T^{10} + \cdots + 25110121 \) Copy content Toggle raw display
$23$ \( (T^{6} + 21 T^{5} + \cdots + 14776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + T^{5} - 114 T^{4} + \cdots - 56)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 130 T^{10} + \cdots + 2483776 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 258309184 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1867190521 \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} + \cdots - 455461)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 537868864 \) Copy content Toggle raw display
$53$ \( (T^{6} - 28 T^{5} + \cdots - 7288)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 366 T^{10} + \cdots + 20187049 \) Copy content Toggle raw display
$61$ \( (T^{6} + 27 T^{5} + \cdots + 4936)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 542 T^{10} + \cdots + 15816529 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 935381056 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 43065795529 \) Copy content Toggle raw display
$79$ \( (T^{6} - 6 T^{5} + \cdots + 1576)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 995339401 \) Copy content Toggle raw display
$89$ \( T^{12} + 611 T^{10} + \cdots + 851929 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 904746241 \) Copy content Toggle raw display
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