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Properties

Label 896.2.p.c
Level $896$
Weight $2$
Character orbit 896.p
Analytic conductor $7.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,2,Mod(255,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("896.255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.p (of order \(6\), degree \(2\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{14} - \beta_{13} + \cdots + \beta_1) q^{3}+ \cdots + ( - \beta_{12} + \beta_{11} + 3 \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} - \beta_{13} + \cdots + \beta_1) q^{3}+ \cdots + (\beta_{15} + \beta_{14} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{5} - 16 q^{9} - 12 q^{21} - 48 q^{29} - 48 q^{33} + 20 q^{37} - 24 q^{45} + 8 q^{49} + 12 q^{53} + 48 q^{57} + 60 q^{61} - 40 q^{65} + 48 q^{73} - 20 q^{77} - 8 q^{81} - 56 q^{85} - 48 q^{89} + 76 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 115452774644 \nu^{15} - 886173093256 \nu^{14} + 3342656190846 \nu^{13} + \cdots + 1622048373702 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 160058287024 \nu^{15} - 771577489740 \nu^{14} + 1641567852568 \nu^{13} - 1696081236240 \nu^{12} + \cdots - 66864361188 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 229802623928 \nu^{15} + 1436997485217 \nu^{14} - 4492702190872 \nu^{13} + \cdots + 3885063043211 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 240987328 \nu^{15} - 1500754791 \nu^{14} + 4684007142 \nu^{13} - 9759606114 \nu^{12} + \cdots + 5426709937 ) / 3810388399 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 394538957168 \nu^{15} + 2231811901383 \nu^{14} - 6199106801734 \nu^{13} + \cdots - 1505816495286 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 478111040426 \nu^{15} + 3200881799663 \nu^{14} - 10758695525452 \nu^{13} + \cdots - 4356828011091 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 554493809494 \nu^{15} - 3465206360700 \nu^{14} + 10839727273718 \nu^{13} + \cdots + 8665374283685 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 555467294122 \nu^{15} + 3482004425340 \nu^{14} - 10897139330910 \nu^{13} + \cdots + 9042032307616 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 911698757420 \nu^{15} - 5291260258946 \nu^{14} + 15209828248580 \nu^{13} + \cdots + 4175440107705 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 785074438 \nu^{15} - 4907305158 \nu^{14} + 15343416116 \nu^{13} - 31997236762 \nu^{12} + \cdots + 3343669588 ) / 1973128213 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2510038757616 \nu^{15} - 15684028307955 \nu^{14} + 49036604704584 \nu^{13} + \cdots + 12080647657008 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2514569665950 \nu^{15} + 15712922094177 \nu^{14} - 49122854319350 \nu^{13} + \cdots - 2988549015669 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2577261808428 \nu^{15} + 15864063233271 \nu^{14} - 48888078330392 \nu^{13} + \cdots - 16417552421604 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2737320095452 \nu^{15} + 16635640723011 \nu^{14} - 50529646182960 \nu^{13} + \cdots - 16350688060416 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3232643076 \nu^{15} + 19877833112 \nu^{14} - 61189084950 \nu^{13} + 125743165020 \nu^{12} + \cdots - 20467949094 ) / 3810388399 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} - \beta_{5} - \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + 5 \beta_{14} - 4 \beta_{13} - 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{12} - \beta_{11} - 5\beta_{10} - 2\beta_{8} + 3\beta_{7} + 3\beta_{4} - 6\beta_{3} + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{15} - 8 \beta_{14} - 8 \beta_{13} - 8 \beta_{12} - 8 \beta_{11} - 6 \beta_{10} - 8 \beta_{9} + \cdots + 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{15} + 7\beta_{14} - 34\beta_{13} - 8\beta_{9} + 16\beta_{6} - 41\beta_{5} - 8\beta_{2} + 34\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 41 \beta_{15} + 98 \beta_{14} - 147 \beta_{13} + 50 \beta_{12} + 49 \beta_{11} - 32 \beta_{10} + \cdots - 79 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50\beta_{12} + 49\beta_{11} - 96\beta_{10} - 99\beta_{8} + 49\beta_{7} + 178\beta_{4} - 89\beta_{3} + 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 228 \beta_{15} - 799 \beta_{14} + 521 \beta_{13} + 10 \beta_{12} - 10 \beta_{11} - 336 \beta_{10} + \cdots + 446 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 660 \beta_{15} - 1027 \beta_{14} + 218 \beta_{13} - 576 \beta_{9} + 288 \beta_{6} - 1027 \beta_{5} + \cdots + 1245 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2490 \beta_{15} - 1533 \beta_{14} - 1533 \beta_{13} + 1533 \beta_{12} + 1603 \beta_{11} + 887 \beta_{10} + \cdots - 4432 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3002\beta_{12} + 3002\beta_{11} - 1603\beta_{8} - 1603\beta_{7} + 2574\beta_{4} + 2574\beta_{3} - 6069 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6751 \beta_{15} - 15075 \beta_{14} + 23429 \beta_{13} + 8782 \beta_{12} + 8354 \beta_{11} + \cdots - 10585 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -36503\beta_{14} + 36503\beta_{13} - 8782\beta_{9} - 8782\beta_{6} + 6323\beta_{5} - 1688\beta_{2} + 6323\beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 36503 \beta_{15} - 126585 \beta_{14} + 81300 \beta_{13} - 2459 \beta_{12} + 2459 \beta_{11} + \cdots - 70547 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(1 - \beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
−0.0499037 + 0.186243i
2.24352 + 0.601150i
−1.29724 − 0.347596i
0.430324 − 1.60599i
1.60599 + 0.430324i
−0.347596 + 1.29724i
0.601150 − 2.24352i
−0.186243 − 0.0499037i
−0.0499037 − 0.186243i
2.24352 − 0.601150i
−1.29724 + 0.347596i
0.430324 + 1.60599i
1.60599 − 0.430324i
−0.347596 − 1.29724i
0.601150 + 2.24352i
−0.186243 + 0.0499037i
0 −1.56352 + 2.70809i 0 1.73615 − 1.00236i 0 −2.33922 + 1.23615i 0 −3.38916 − 5.87020i 0
255.2 0 −1.29759 + 2.24749i 0 −1.34467 + 0.776347i 0 1.89662 + 1.84467i 0 −1.86747 − 3.23456i 0
255.3 0 −0.779618 + 1.35034i 0 3.14484 − 1.81567i 0 −0.0694427 − 2.64484i 0 0.284392 + 0.492581i 0
255.4 0 −0.513691 + 0.889740i 0 −0.536314 + 0.309641i 0 2.43435 − 1.03631i 0 0.972242 + 1.68397i 0
255.5 0 0.513691 − 0.889740i 0 −0.536314 + 0.309641i 0 −2.43435 + 1.03631i 0 0.972242 + 1.68397i 0
255.6 0 0.779618 − 1.35034i 0 3.14484 − 1.81567i 0 0.0694427 + 2.64484i 0 0.284392 + 0.492581i 0
255.7 0 1.29759 − 2.24749i 0 −1.34467 + 0.776347i 0 −1.89662 − 1.84467i 0 −1.86747 − 3.23456i 0
255.8 0 1.56352 − 2.70809i 0 1.73615 − 1.00236i 0 2.33922 − 1.23615i 0 −3.38916 − 5.87020i 0
383.1 0 −1.56352 − 2.70809i 0 1.73615 + 1.00236i 0 −2.33922 − 1.23615i 0 −3.38916 + 5.87020i 0
383.2 0 −1.29759 − 2.24749i 0 −1.34467 − 0.776347i 0 1.89662 − 1.84467i 0 −1.86747 + 3.23456i 0
383.3 0 −0.779618 − 1.35034i 0 3.14484 + 1.81567i 0 −0.0694427 + 2.64484i 0 0.284392 − 0.492581i 0
383.4 0 −0.513691 − 0.889740i 0 −0.536314 − 0.309641i 0 2.43435 + 1.03631i 0 0.972242 − 1.68397i 0
383.5 0 0.513691 + 0.889740i 0 −0.536314 − 0.309641i 0 −2.43435 − 1.03631i 0 0.972242 − 1.68397i 0
383.6 0 0.779618 + 1.35034i 0 3.14484 + 1.81567i 0 0.0694427 − 2.64484i 0 0.284392 − 0.492581i 0
383.7 0 1.29759 + 2.24749i 0 −1.34467 − 0.776347i 0 −1.89662 + 1.84467i 0 −1.86747 + 3.23456i 0
383.8 0 1.56352 + 2.70809i 0 1.73615 + 1.00236i 0 2.33922 + 1.23615i 0 −3.38916 + 5.87020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.p.c yes 16
4.b odd 2 1 inner 896.2.p.c yes 16
7.d odd 6 1 inner 896.2.p.c yes 16
8.b even 2 1 896.2.p.a ✓ 16
8.d odd 2 1 896.2.p.a ✓ 16
28.f even 6 1 inner 896.2.p.c yes 16
56.j odd 6 1 896.2.p.a ✓ 16
56.m even 6 1 896.2.p.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.p.a ✓ 16 8.b even 2 1
896.2.p.a ✓ 16 8.d odd 2 1
896.2.p.a ✓ 16 56.j odd 6 1
896.2.p.a ✓ 16 56.m even 6 1
896.2.p.c yes 16 1.a even 1 1 trivial
896.2.p.c yes 16 4.b odd 2 1 inner
896.2.p.c yes 16 7.d odd 6 1 inner
896.2.p.c yes 16 28.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\):

\( T_{3}^{16} + 20 T_{3}^{14} + 274 T_{3}^{12} + 1976 T_{3}^{10} + 10267 T_{3}^{8} + 27512 T_{3}^{6} + \cdots + 28561 \) Copy content Toggle raw display
\( T_{5}^{8} - 6T_{5}^{7} + 8T_{5}^{6} + 24T_{5}^{5} - 27T_{5}^{4} - 72T_{5}^{3} + 80T_{5}^{2} + 126T_{5} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 20 T^{14} + \cdots + 28561 \) Copy content Toggle raw display
$5$ \( (T^{8} - 6 T^{7} + 8 T^{6} + \cdots + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 16 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{8} + 56 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 22 T^{6} + \cdots + 529)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 5082121521 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 244140625 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots - 540)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 12003612721 \) Copy content Toggle raw display
$37$ \( (T^{8} - 10 T^{7} + \cdots + 819025)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 152 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 64 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 260144641 \) Copy content Toggle raw display
$53$ \( (T^{8} - 6 T^{7} + \cdots + 2634129)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5049013494001 \) Copy content Toggle raw display
$61$ \( (T^{8} - 30 T^{7} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 519554081601 \) Copy content Toggle raw display
$71$ \( (T^{8} + 96 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 24 T^{7} + \cdots + 23377225)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 4216817073121 \) Copy content Toggle raw display
$83$ \( (T^{8} - 176 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 24 T^{7} + \cdots + 1142761)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 432 T^{6} + \cdots + 23619600)^{2} \) Copy content Toggle raw display
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