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Properties

Label 2368.2.g.p
Level $2368$
Weight $2$
Character orbit 2368.g
Analytic conductor $18.909$
Analytic rank $0$
Dimension $10$
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(961,2368)]
 
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, 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("2368.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.g (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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 6x^{7} + 53x^{6} - 46x^{5} + 18x^{4} + 12x^{3} + 196x^{2} - 112x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 296)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{7} q^{5} + \beta_{3} q^{7} + (\beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{7} q^{5} + \beta_{3} q^{7} + (\beta_1 + 1) q^{9} + \beta_{5} q^{11} + \beta_{4} q^{13} + ( - \beta_{9} + \beta_{7}) q^{15} + ( - \beta_{9} - \beta_{4}) q^{17} + ( - \beta_{6} - \beta_{4}) q^{19} + (\beta_{5} + \beta_{3} + \beta_1) q^{21} + (\beta_{9} - \beta_{8} + \cdots + \beta_{4}) q^{23}+ \cdots + ( - \beta_{5} + 2 \beta_{3} - 3 \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} + 4 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{21} + 4 q^{25} + 8 q^{27} + 4 q^{33} + 6 q^{37} - 26 q^{41} - 8 q^{47} + 14 q^{49} + 20 q^{53} + 12 q^{63} - 4 q^{65} - 2 q^{67} + 4 q^{71} - 14 q^{73} - 48 q^{75} + 18 q^{81} - 36 q^{83} - 8 q^{85} - 4 q^{95} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 6x^{7} + 53x^{6} - 46x^{5} + 18x^{4} + 12x^{3} + 196x^{2} - 112x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8394 \nu^{9} + 522953 \nu^{8} + 372086 \nu^{7} + 170662 \nu^{6} - 3911172 \nu^{5} + \cdots + 50678928 ) / 10819708 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14595 \nu^{9} - 249836 \nu^{8} - 240502 \nu^{7} + 916 \nu^{6} + 1186653 \nu^{5} + \cdots - 31958752 ) / 10819708 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25917 \nu^{9} + 39435 \nu^{8} + 63448 \nu^{7} - 122834 \nu^{6} + 1165407 \nu^{5} + 1155951 \nu^{4} + \cdots + 18017600 ) / 5409854 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 82826 \nu^{9} + 17066 \nu^{8} + 92346 \nu^{7} - 128815 \nu^{6} + 4718228 \nu^{5} - 3274916 \nu^{4} + \cdots - 6239328 ) / 5409854 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 167514 \nu^{9} - 321579 \nu^{8} - 648682 \nu^{7} + 714406 \nu^{6} - 5385300 \nu^{5} + \cdots + 17347792 ) / 10819708 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 287075 \nu^{9} - 69536 \nu^{8} + 26952 \nu^{7} + 2204940 \nu^{6} - 14066739 \nu^{5} + \cdots + 18030008 ) / 10819708 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 517347 \nu^{9} + 128413 \nu^{8} + 205098 \nu^{7} - 2805784 \nu^{6} + 27406513 \nu^{5} + \cdots - 33324560 ) / 10819708 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 656817 \nu^{9} - 205680 \nu^{8} + 393408 \nu^{7} + 5583404 \nu^{6} - 31317849 \nu^{5} + \cdots + 31656552 ) / 10819708 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 618220 \nu^{9} - 187342 \nu^{8} + 89418 \nu^{7} + 3764909 \nu^{6} - 31405794 \nu^{5} + \cdots + 32344528 ) / 5409854 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 3\beta_{4} - 2\beta_{3} - 4\beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + 2\beta_{8} - 2\beta_{6} - 3\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{9} - 9 \beta_{8} - 8 \beta_{7} - \beta_{6} - 3 \beta_{5} + 23 \beta_{4} + 6 \beta_{3} + \cdots + 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{3} - 10\beta_{2} - 4\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 35 \beta_{9} + 83 \beta_{8} + 72 \beta_{7} - 21 \beta_{6} - 19 \beta_{5} - 193 \beta_{4} + 14 \beta_{3} + \cdots + 208 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 41\beta_{9} - 96\beta_{8} - 48\beta_{7} + 76\beta_{6} + 207\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 295 \beta_{9} + 739 \beta_{8} + 584 \beta_{7} - 309 \beta_{6} + 169 \beta_{5} - 1661 \beta_{4} + \cdots - 2096 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -125\beta_{5} - 111\beta_{3} + 834\beta_{2} + 359\beta _1 + 1352 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2569 \beta_{9} - 6441 \beta_{8} - 4760 \beta_{7} + 3215 \beta_{6} + 1569 \beta_{5} + 14339 \beta_{4} + \cdots - 19312 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
−2.07287 + 2.07287i
−2.07287 − 2.07287i
1.23896 − 1.23896i
1.23896 + 1.23896i
−0.934650 − 0.934650i
−0.934650 + 0.934650i
1.48872 + 1.48872i
1.48872 − 1.48872i
0.279838 − 0.279838i
0.279838 + 0.279838i
0 −2.82032 0 − 0.855471i 0 −0.447854 0 4.95418 0
961.2 0 −2.82032 0 0.855471i 0 −0.447854 0 4.95418 0
961.3 0 −0.706809 0 − 1.32106i 0 −1.54799 0 −2.50042 0
961.4 0 −0.706809 0 1.32106i 0 −1.54799 0 −2.50042 0
961.5 0 −0.352411 0 − 2.78743i 0 4.12216 0 −2.87581 0
961.6 0 −0.352411 0 2.78743i 0 4.12216 0 −2.87581 0
961.7 0 1.93128 0 − 1.58785i 0 −3.41002 0 0.729857 0
961.8 0 1.93128 0 1.58785i 0 −3.41002 0 0.729857 0
961.9 0 2.94825 0 − 3.19874i 0 3.28371 0 5.69219 0
961.10 0 2.94825 0 3.19874i 0 3.28371 0 5.69219 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.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.g.p 10
4.b odd 2 1 2368.2.g.o 10
8.b even 2 1 592.2.g.d 10
8.d odd 2 1 296.2.g.a ✓ 10
24.f even 2 1 2664.2.h.c 10
24.h odd 2 1 5328.2.h.q 10
37.b even 2 1 inner 2368.2.g.p 10
148.b odd 2 1 2368.2.g.o 10
296.e even 2 1 592.2.g.d 10
296.h odd 2 1 296.2.g.a ✓ 10
888.c even 2 1 2664.2.h.c 10
888.i odd 2 1 5328.2.h.q 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.g.a ✓ 10 8.d odd 2 1
296.2.g.a ✓ 10 296.h odd 2 1
592.2.g.d 10 8.b even 2 1
592.2.g.d 10 296.e even 2 1
2368.2.g.o 10 4.b odd 2 1
2368.2.g.o 10 148.b odd 2 1
2368.2.g.p 10 1.a even 1 1 trivial
2368.2.g.p 10 37.b even 2 1 inner
2664.2.h.c 10 24.f even 2 1
2664.2.h.c 10 888.c even 2 1
5328.2.h.q 10 24.h odd 2 1
5328.2.h.q 10 888.i odd 2 1

Hecke kernels

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

\( T_{3}^{5} - T_{3}^{4} - 10T_{3}^{3} + 7T_{3}^{2} + 15T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{10} + 23T_{5}^{8} + 177T_{5}^{6} + 536T_{5}^{4} + 656T_{5}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} - T^{4} - 10 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + 23 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{5} - 2 T^{4} - 19 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - T^{4} - 28 T^{3} + \cdots + 116)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 55 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{10} + 84 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$19$ \( T^{10} + 88 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{10} + 171 T^{8} + \cdots + 1401856 \) Copy content Toggle raw display
$29$ \( T^{10} + 207 T^{8} + \cdots + 22429696 \) Copy content Toggle raw display
$31$ \( T^{10} + 223 T^{8} + \cdots + 22429696 \) Copy content Toggle raw display
$37$ \( T^{10} - 6 T^{9} + \cdots + 69343957 \) Copy content Toggle raw display
$41$ \( (T^{5} + 13 T^{4} + \cdots - 118)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 196 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$47$ \( (T^{5} + 4 T^{4} + \cdots - 11584)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 10 T^{4} + \cdots + 2264)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 384 T^{8} + \cdots + 89718784 \) Copy content Toggle raw display
$61$ \( T^{10} + 327 T^{8} + \cdots + 5607424 \) Copy content Toggle raw display
$67$ \( (T^{5} + T^{4} - 163 T^{3} + \cdots - 2368)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 2 T^{4} + \cdots - 11296)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 7 T^{4} + \cdots + 7858)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 331 T^{8} + \cdots + 23104 \) Copy content Toggle raw display
$83$ \( (T^{5} + 18 T^{4} + \cdots - 3776)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 6237208576 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 5742002176 \) Copy content Toggle raw display
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