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Properties

Label 2368.2.c.a
Level $2368$
Weight $2$
Character orbit 2368.c
Analytic conductor $18.909$
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] = [2368,2,Mod(1185,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2368.1185");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.c (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: \(18.9085751986\)
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: \( x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 4 x^{8} - 6 x^{7} + 62 x^{6} - 22 x^{5} + 4 x^{4} - 2 x^{3} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} + \beta_{9} q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} + \beta_{9} q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1 - 2) q^{9} + (\beta_{11} + \beta_{8} - \beta_{2}) q^{11} + ( - \beta_{11} - \beta_{8} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - 6 \beta_{11} - 4 \beta_{9} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{7} - 20 q^{9} + 28 q^{15} + 12 q^{17} - 4 q^{23} - 16 q^{25} + 52 q^{31} - 36 q^{33} + 24 q^{39} + 8 q^{41} + 56 q^{47} + 40 q^{49} + 16 q^{55} - 4 q^{57} + 128 q^{63} + 8 q^{65} + 24 q^{73} + 72 q^{79} + 52 q^{81} + 28 q^{87} - 4 q^{89} + 64 q^{95} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} - 14 x^{9} + 4 x^{8} - 6 x^{7} + 62 x^{6} - 22 x^{5} + 4 x^{4} - 2 x^{3} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 492891 \nu^{11} - 967167 \nu^{10} + 13478706 \nu^{9} - 56809484 \nu^{8} + 32626344 \nu^{7} + \cdots - 43994507 ) / 13243157 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1093689 \nu^{11} - 6142699 \nu^{10} + 14741661 \nu^{9} + 6672765 \nu^{8} - 46274373 \nu^{7} + \cdots + 8088117 ) / 26486314 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1399913 \nu^{11} - 8318013 \nu^{10} + 25327335 \nu^{9} - 21595931 \nu^{8} + 14195081 \nu^{7} + \cdots + 18182527 ) / 26486314 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2305831 \nu^{11} - 8161357 \nu^{10} + 10216827 \nu^{9} + 53718837 \nu^{8} - 24360001 \nu^{7} + \cdots + 22771481 ) / 26486314 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2718361 \nu^{11} + 16764927 \nu^{10} - 52651819 \nu^{9} + 51787291 \nu^{8} - 32603311 \nu^{7} + \cdots + 41693975 ) / 26486314 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1914022 \nu^{11} + 11376297 \nu^{10} - 35052187 \nu^{9} + 32085634 \nu^{8} - 26180896 \nu^{7} + \cdots + 3942176 ) / 13243157 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3451937 \nu^{11} + 20052398 \nu^{10} - 57911409 \nu^{9} + 34664876 \nu^{8} + 1287932 \nu^{7} + \cdots + 8224974 ) / 13243157 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8303787 \nu^{11} + 48422809 \nu^{10} - 141150153 \nu^{9} + 90925683 \nu^{8} - 11619217 \nu^{7} + \cdots + 24753735 ) / 26486314 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11103613 \nu^{11} + 65058835 \nu^{10} - 191804823 \nu^{9} + 134117545 \nu^{8} + \cdots + 41361309 ) / 26486314 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5716247 \nu^{11} - 33314340 \nu^{10} + 96312454 \nu^{9} - 59724181 \nu^{8} + 4866258 \nu^{7} + \cdots - 14945569 ) / 13243157 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31832649 \nu^{11} + 185654863 \nu^{10} - 540943097 \nu^{9} + 351167305 \nu^{8} + \cdots + 95551421 ) / 26486314 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 3\beta_{8} + 2\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} - \beta_{10} + 4 \beta_{9} - 11 \beta_{8} + 10 \beta_{7} + \beta_{6} + 4 \beta_{5} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 11\beta_{5} + 3\beta_{4} + 24\beta_{3} + 3\beta _1 - 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17 \beta_{11} + 18 \beta_{10} - 53 \beta_{9} + 127 \beta_{8} - 118 \beta_{7} + 12 \beta_{6} + 53 \beta_{5} + \cdots - 127 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 44\beta_{11} + 47\beta_{10} - 133\beta_{9} + 312\beta_{8} - 291\beta_{7} - 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 224 \beta_{11} + 242 \beta_{10} - 658 \beta_{9} + 1533 \beta_{8} - 1441 \beta_{7} - 148 \beta_{6} + \cdots + 1533 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -367\beta_{6} - 1636\beta_{5} - 609\beta_{4} - 3573\beta_{3} - 562\beta _1 + 3794 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2807 \beta_{11} - 3049 \beta_{10} + 8122 \beta_{9} - 18801 \beta_{8} + 17734 \beta_{7} - 1831 \beta_{6} + \cdots + 18801 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -6989\beta_{11} - 7598\beta_{10} + 20174\beta_{9} - 46650\beta_{8} + 44027\beta_{7} + 4549\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 34761 \beta_{11} - 37810 \beta_{10} + 100201 \beta_{9} - 231595 \beta_{8} + 218650 \beta_{7} + \cdots - 231595 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(1407\) \(1925\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1185.1
0.169751 + 0.169751i
1.22309 − 1.22309i
−0.863163 − 0.863163i
0.523343 − 0.523343i
−0.536632 + 0.536632i
2.48361 + 2.48361i
2.48361 − 2.48361i
−0.536632 − 0.536632i
0.523343 + 0.523343i
−0.863163 + 0.863163i
1.22309 + 1.22309i
0.169751 − 0.169751i
0 − 3.26960i 0 − 1.62137i 0 −5.06889 0 −7.69027 0
1185.2 0 − 3.08286i 0 2.90046i 0 −2.60357 0 −6.50403 0
1185.3 0 − 1.78422i 0 3.94275i 0 −3.12620 0 −0.183448 0
1185.4 0 − 1.63480i 0 2.54560i 0 3.87301 0 0.327415 0
1185.5 0 − 1.14100i 0 − 1.72247i 0 0.975639 0 1.69811 0
1185.6 0 − 0.804848i 0 1.40221i 0 1.95001 0 2.35222 0
1185.7 0 0.804848i 0 − 1.40221i 0 1.95001 0 2.35222 0
1185.8 0 1.14100i 0 1.72247i 0 0.975639 0 1.69811 0
1185.9 0 1.63480i 0 − 2.54560i 0 3.87301 0 0.327415 0
1185.10 0 1.78422i 0 − 3.94275i 0 −3.12620 0 −0.183448 0
1185.11 0 3.08286i 0 − 2.90046i 0 −2.60357 0 −6.50403 0
1185.12 0 3.26960i 0 1.62137i 0 −5.06889 0 −7.69027 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1185.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2368.2.c.a ✓ 12
4.b odd 2 1 2368.2.c.b yes 12
8.b even 2 1 inner 2368.2.c.a ✓ 12
8.d odd 2 1 2368.2.c.b yes 12
16.e even 4 1 9472.2.a.bb 6
16.e even 4 1 9472.2.a.bc 6
16.f odd 4 1 9472.2.a.z 6
16.f odd 4 1 9472.2.a.ba 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2368.2.c.a ✓ 12 1.a even 1 1 trivial
2368.2.c.a ✓ 12 8.b even 2 1 inner
2368.2.c.b yes 12 4.b odd 2 1
2368.2.c.b yes 12 8.d odd 2 1
9472.2.a.z 6 16.f odd 4 1
9472.2.a.ba 6 16.f odd 4 1
9472.2.a.bb 6 16.e even 4 1
9472.2.a.bc 6 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2368, [\chi])\):

\( T_{3}^{12} + 28T_{3}^{10} + 280T_{3}^{8} + 1234T_{3}^{6} + 2552T_{3}^{4} + 2332T_{3}^{2} + 729 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 23T_{7}^{4} - 76T_{7}^{3} + 132T_{7}^{2} + 272T_{7} - 304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 38 T^{10} + \cdots + 12996 \) Copy content Toggle raw display
$7$ \( (T^{6} + 4 T^{5} + \cdots - 304)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 36 T^{10} + \cdots + 1521 \) Copy content Toggle raw display
$13$ \( T^{12} + 70 T^{10} + \cdots + 1444 \) Copy content Toggle raw display
$17$ \( (T^{6} - 6 T^{5} - 14 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 144 T^{10} + \cdots + 1557504 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + \cdots + 532)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 102171664 \) Copy content Toggle raw display
$31$ \( (T^{6} - 26 T^{5} + \cdots - 2888)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots + 37761)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 223323136 \) Copy content Toggle raw display
$47$ \( (T^{6} - 28 T^{5} + \cdots + 912)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 139134968064 \) Copy content Toggle raw display
$59$ \( T^{12} + 208 T^{10} + \cdots + 27541504 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1201038336 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10968791824 \) Copy content Toggle raw display
$71$ \( (T^{6} - 211 T^{4} + \cdots + 58368)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 12 T^{5} + \cdots - 4771)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 36 T^{5} + \cdots - 11666)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 7662551296 \) Copy content Toggle raw display
$89$ \( (T^{6} + 2 T^{5} + \cdots + 34656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 28 T^{5} + \cdots + 937216)^{2} \) Copy content Toggle raw display
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