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Properties

Label 896.2.p.b
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: \( x^{16} - 6 x^{15} + 18 x^{14} - 42 x^{13} + 98 x^{12} - 228 x^{11} + 486 x^{10} - 900 x^{9} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{6} - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{12} + \beta_{8}) q^{7} + ( - \beta_{13} - \beta_{9} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{6} - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{12} + \beta_{8}) q^{7} + ( - \beta_{13} - \beta_{9} - \beta_1) q^{9} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{5} + 28 q^{21} + 16 q^{25} + 16 q^{29} + 48 q^{33} - 20 q^{37} + 72 q^{45} - 24 q^{49} + 4 q^{53} - 16 q^{57} - 12 q^{61} + 40 q^{65} - 96 q^{73} + 68 q^{77} + 8 q^{81} - 8 q^{85} + 96 q^{89} - 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 42 x^{13} + 98 x^{12} - 228 x^{11} + 486 x^{10} - 900 x^{9} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8266 \nu^{15} + 387525 \nu^{14} - 1106040 \nu^{13} + 2504100 \nu^{12} - 5453230 \nu^{11} + \cdots + 977036418 ) / 62010927 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 37178 \nu^{15} - 974184 \nu^{14} + 2458944 \nu^{13} - 5666208 \nu^{12} + 13270760 \nu^{11} + \cdots - 1734242886 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 138674 \nu^{15} - 868002 \nu^{14} + 2501802 \nu^{13} - 5717544 \nu^{12} + 13734920 \nu^{11} + \cdots - 2522553597 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 559220 \nu^{15} + 1163373 \nu^{14} - 2110860 \nu^{13} + 5474676 \nu^{12} - 13711684 \nu^{11} + \cdots - 950085288 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 229276 \nu^{15} - 1073746 \nu^{14} + 2776230 \nu^{13} - 6161262 \nu^{12} + 14469170 \nu^{11} + \cdots - 1043024040 ) / 62010927 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45146 \nu^{15} - 201522 \nu^{14} + 483102 \nu^{13} - 1091154 \nu^{12} + 2587120 \nu^{11} + \cdots - 147226653 ) / 9791199 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 925162 \nu^{15} - 6954801 \nu^{14} + 17922204 \nu^{13} - 39250248 \nu^{12} + 95898908 \nu^{11} + \cdots - 9556912251 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 353482 \nu^{15} + 1443736 \nu^{14} - 3579966 \nu^{13} + 8249937 \nu^{12} - 19687832 \nu^{11} + \cdots + 1046566980 ) / 62010927 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 426742 \nu^{15} + 1864167 \nu^{14} - 4495716 \nu^{13} + 10110741 \nu^{12} - 23954006 \nu^{11} + \cdots + 1542951099 ) / 62010927 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1396294 \nu^{15} - 6693630 \nu^{14} + 16365114 \nu^{13} - 37209372 \nu^{12} + 90187562 \nu^{11} + \cdots - 6566264109 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 212928 \nu^{15} - 869452 \nu^{14} + 2100936 \nu^{13} - 4818186 \nu^{12} + 11538036 \nu^{11} + \cdots - 638868627 ) / 20670309 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2922598 \nu^{15} + 11920356 \nu^{14} - 27651420 \nu^{13} + 63106332 \nu^{12} + \cdots + 7818395967 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2930920 \nu^{15} + 11725134 \nu^{14} - 27695916 \nu^{13} + 63530673 \nu^{12} + \cdots + 7894361412 ) / 186032781 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1010894 \nu^{15} - 4732480 \nu^{14} + 11336478 \nu^{13} - 25678956 \nu^{12} + 61213150 \nu^{11} + \cdots - 4113104022 ) / 62010927 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1305226 \nu^{15} + 6038163 \nu^{14} - 14692104 \nu^{13} + 33381789 \nu^{12} + \cdots + 5232843648 ) / 62010927 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{11} - \beta_{9} + \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{12} + \beta_{11} - \beta_{5} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} + 3\beta_{10} - \beta_{7} + 3\beta_{6} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} - 5 \beta_{14} + 6 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + \cdots + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{12} + 5\beta_{11} + 10\beta_{8} + 8\beta_{5} - 8\beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10 \beta_{15} - 30 \beta_{14} - 15 \beta_{13} - 10 \beta_{11} - 10 \beta_{9} + 15 \beta_{8} + \cdots - 75 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -5\beta_{13} - 5\beta_{10} + 25\beta_{9} + 5\beta_{7} + 21\beta_{6} - 14\beta_{3} - 10\beta _1 - 46 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 57 \beta_{15} + 70 \beta_{14} + 36 \beta_{13} + 70 \beta_{12} - 40 \beta_{11} + 4 \beta_{10} + \cdots - 121 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -87\beta_{15} - 11\beta_{14} + 37\beta_{12} - 125\beta_{11} + \beta_{8} - 12\beta_{5} - 85\beta_{4} - 58\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 98 \beta_{15} + 74 \beta_{14} - 48 \beta_{13} + 77 \beta_{12} - 36 \beta_{11} - 173 \beta_{10} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -46\beta_{13} - 191\beta_{10} - 131\beta_{9} + 135\beta_{7} - 186\beta_{6} + 49\beta_{3} + 226\beta _1 - 288 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 295 \beta_{15} - 45 \beta_{14} - 381 \beta_{13} + 390 \beta_{12} - 625 \beta_{11} - 375 \beta_{10} + \cdots + 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 375 \beta_{15} + 514 \beta_{14} + 505 \beta_{12} + 155 \beta_{11} - 495 \beta_{8} + 505 \beta_{5} + \cdots - 99 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 588 \beta_{15} + 860 \beta_{14} + 720 \beta_{13} - 625 \beta_{12} - 80 \beta_{11} + 280 \beta_{10} + \cdots + 2948 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(1\)

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
0.505958 − 1.65650i
0.788361 + 1.54223i
−1.15921 − 1.28695i
1.60478 + 0.651679i
−0.651679 + 1.60478i
−1.28695 + 1.15921i
1.54223 − 0.788361i
1.65650 + 0.505958i
0.505958 + 1.65650i
0.788361 − 1.54223i
−1.15921 + 1.28695i
1.60478 − 0.651679i
−0.651679 − 1.60478i
−1.28695 − 1.15921i
1.54223 + 0.788361i
1.65650 − 0.505958i
0 −1.33747 + 2.31656i 0 −3.49280 + 2.01657i 0 −1.52439 − 2.16246i 0 −2.07763 − 3.59857i 0
255.2 0 −1.00310 + 1.73742i 0 −0.194252 + 0.112151i 0 −1.25233 + 2.33060i 0 −0.512424 − 0.887544i 0
255.3 0 −0.440195 + 0.762440i 0 −1.72124 + 0.993760i 0 −1.00813 − 2.44616i 0 1.11246 + 1.92683i 0
255.4 0 −0.105830 + 0.183302i 0 2.40830 − 1.39043i 0 −2.46812 − 0.953099i 0 1.47760 + 2.55928i 0
255.5 0 0.105830 − 0.183302i 0 2.40830 − 1.39043i 0 2.46812 + 0.953099i 0 1.47760 + 2.55928i 0
255.6 0 0.440195 − 0.762440i 0 −1.72124 + 0.993760i 0 1.00813 + 2.44616i 0 1.11246 + 1.92683i 0
255.7 0 1.00310 − 1.73742i 0 −0.194252 + 0.112151i 0 1.25233 − 2.33060i 0 −0.512424 − 0.887544i 0
255.8 0 1.33747 − 2.31656i 0 −3.49280 + 2.01657i 0 1.52439 + 2.16246i 0 −2.07763 − 3.59857i 0
383.1 0 −1.33747 − 2.31656i 0 −3.49280 − 2.01657i 0 −1.52439 + 2.16246i 0 −2.07763 + 3.59857i 0
383.2 0 −1.00310 − 1.73742i 0 −0.194252 − 0.112151i 0 −1.25233 − 2.33060i 0 −0.512424 + 0.887544i 0
383.3 0 −0.440195 − 0.762440i 0 −1.72124 − 0.993760i 0 −1.00813 + 2.44616i 0 1.11246 − 1.92683i 0
383.4 0 −0.105830 − 0.183302i 0 2.40830 + 1.39043i 0 −2.46812 + 0.953099i 0 1.47760 − 2.55928i 0
383.5 0 0.105830 + 0.183302i 0 2.40830 + 1.39043i 0 2.46812 − 0.953099i 0 1.47760 − 2.55928i 0
383.6 0 0.440195 + 0.762440i 0 −1.72124 − 0.993760i 0 1.00813 − 2.44616i 0 1.11246 − 1.92683i 0
383.7 0 1.00310 + 1.73742i 0 −0.194252 − 0.112151i 0 1.25233 + 2.33060i 0 −0.512424 + 0.887544i 0
383.8 0 1.33747 + 2.31656i 0 −3.49280 − 2.01657i 0 1.52439 − 2.16246i 0 −2.07763 + 3.59857i 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.b ✓ 16
4.b odd 2 1 inner 896.2.p.b ✓ 16
7.d odd 6 1 inner 896.2.p.b ✓ 16
8.b even 2 1 896.2.p.d yes 16
8.d odd 2 1 896.2.p.d yes 16
28.f even 6 1 inner 896.2.p.b ✓ 16
56.j odd 6 1 896.2.p.d yes 16
56.m even 6 1 896.2.p.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.p.b ✓ 16 1.a even 1 1 trivial
896.2.p.b ✓ 16 4.b odd 2 1 inner
896.2.p.b ✓ 16 7.d odd 6 1 inner
896.2.p.b ✓ 16 28.f even 6 1 inner
896.2.p.d yes 16 8.b even 2 1
896.2.p.d yes 16 8.d odd 2 1
896.2.p.d yes 16 56.j odd 6 1
896.2.p.d yes 16 56.m even 6 1

Hecke kernels

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

\( T_{3}^{16} + 12T_{3}^{14} + 106T_{3}^{12} + 408T_{3}^{10} + 1155T_{3}^{8} + 888T_{3}^{6} + 538T_{3}^{4} + 24T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 6T_{5}^{7} + 4T_{5}^{6} - 48T_{5}^{5} - 15T_{5}^{4} + 336T_{5}^{3} + 628T_{5}^{2} + 210T_{5} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 12 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{8} + 6 T^{7} + 4 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 12 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 72 T^{14} + \cdots + 1500625 \) Copy content Toggle raw display
$13$ \( (T^{8} + 72 T^{6} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 54 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 24 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1945358247121 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} + \cdots - 716)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + 120 T^{14} + \cdots + 130321 \) Copy content Toggle raw display
$37$ \( (T^{8} + 10 T^{7} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 312 T^{6} + \cdots + 28944400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 256 T^{6} + \cdots + 2310400)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 80 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( (T^{8} - 2 T^{7} + \cdots + 1261129)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 108 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6 T^{7} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 90842562801 \) Copy content Toggle raw display
$71$ \( (T^{8} + 160 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 48 T^{7} + \cdots + 466489)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 49\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{8} - 112 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 48 T^{7} + \cdots + 65983129)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 240 T^{6} + \cdots + 35344)^{2} \) Copy content Toggle raw display
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