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Properties

Label 14.11.b.a
Level $14$
Weight $11$
Character orbit 14.b
Analytic conductor $8.895$
Analytic rank $0$
Dimension $8$
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] = [14,11,Mod(13,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.13");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 14.b (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: \(8.89500153743\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14130x^{6} + 61043589x^{4} + 87066375930x^{2} + 12363031798119 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + 512 q^{4} + ( - \beta_{5} - 3 \beta_{3}) q^{5} + (\beta_{6} - 6 \beta_{3}) q^{6} + ( - \beta_{6} + 2 \beta_{5} + \cdots + 2297) q^{7}+ \cdots + ( - \beta_{4} + 2 \beta_{2} + \cdots - 30807) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + 512 q^{4} + ( - \beta_{5} - 3 \beta_{3}) q^{5} + (\beta_{6} - 6 \beta_{3}) q^{6} + ( - \beta_{6} + 2 \beta_{5} + \cdots + 2297) q^{7}+ \cdots + ( - 39015 \beta_{7} + \cdots - 2041422102) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4096 q^{4} + 18376 q^{7} - 246456 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4096 q^{4} + 18376 q^{7} - 246456 q^{9} + 430800 q^{11} - 136704 q^{14} - 1896960 q^{15} + 2097152 q^{16} - 4512768 q^{18} + 5339136 q^{21} + 13228032 q^{22} + 6265488 q^{23} - 28719160 q^{25} + 9408512 q^{28} - 46431408 q^{29} + 28584960 q^{30} + 184450560 q^{35} - 126185472 q^{36} + 360932816 q^{37} - 836120064 q^{39} + 308382720 q^{42} + 32112848 q^{43} + 220569600 q^{44} - 769191936 q^{46} + 853888904 q^{49} - 53836800 q^{50} + 1737904128 q^{51} - 1132258608 q^{53} - 69992448 q^{56} - 2040889344 q^{57} + 352352256 q^{58} - 971243520 q^{60} + 2661283080 q^{63} + 1073741824 q^{64} - 143001600 q^{65} - 2254742192 q^{67} + 402662400 q^{70} + 2121911184 q^{71} - 2310537216 q^{72} + 4970207232 q^{74} - 17516185008 q^{77} + 1916728320 q^{78} - 5257367792 q^{79} + 24706423944 q^{81} + 2733637632 q^{84} + 4331212800 q^{85} - 14424637440 q^{86} + 6772752384 q^{88} - 8536548864 q^{91} + 3207929856 q^{92} - 20748386304 q^{93} + 30330078720 q^{95} - 802977792 q^{98} - 16331376816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14130x^{6} + 61043589x^{4} + 87066375930x^{2} + 12363031798119 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 112\nu^{6} + 2401680\nu^{4} + 11978938512\nu^{2} + 9593162648640 ) / 342403471431 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3097802 \nu^{7} + 15569032100 \nu^{6} + 2760165368 \nu^{5} + 169289762761080 \nu^{4} + \cdots + 75\!\cdots\!20 ) / 36\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11146238\nu^{7} + 122551384920\nu^{5} + 276650035210764\nu^{3} - 64222112704095558\nu ) / 10886033567205783 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6195604 \nu^{7} - 100371772720 \nu^{6} + 5520330736 \nu^{5} + \cdots - 68\!\cdots\!84 ) / 36\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 207118360 \nu^{7} + 2385383668032 \nu^{5} + \cdots + 44\!\cdots\!88 \nu ) / 10\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 268726988 \nu^{7} - 3098810634096 \nu^{5} + \cdots - 43\!\cdots\!28 \nu ) / 10\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 583660556 \nu^{7} - 5123164586832 \nu^{5} + \cdots + 23\!\cdots\!08 \nu ) / 10\!\cdots\!83 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{6} + 4\beta_{5} - 2\beta_{3} ) / 448 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} + 3\beta_{5} - 5\beta_{4} - 6\beta_{3} - 32\beta_{2} - 3075\beta _1 - 791280 ) / 224 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -654\beta_{7} - 18825\beta_{6} - 17274\beta_{5} - 167118\beta_{3} ) / 448 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 22959 \beta_{7} + 22959 \beta_{6} - 22959 \beta_{5} + 38853 \beta_{4} + 45918 \beta_{3} + \cdots + 4343904432 ) / 224 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9822357\beta_{7} + 126413433\beta_{6} + 96109623\beta_{5} + 1776164832\beta_{3} ) / 448 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6123519 \beta_{7} - 6123519 \beta_{6} + 6123519 \beta_{5} - 10656180 \beta_{4} - 12247038 \beta_{3} + \cdots - 989426774160 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -91763187768\beta_{7} - 905376148947\beta_{6} - 604923943284\beta_{5} - 14954800050990\beta_{3} ) / 448 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
86.7731i
− 12.6133i
12.6133i
− 86.7731i
− 54.6993i
− 58.7309i
58.7309i
54.6993i
−22.6274 − 339.118i 512.000 − 3624.45i 7673.36i 16745.5 − 1436.05i −11585.2 −55951.8 82012.0i
13.2 −22.6274 − 121.867i 512.000 3602.72i 2757.55i −10641.2 − 13009.3i −11585.2 44197.3 − 81520.2i
13.3 −22.6274 121.867i 512.000 − 3602.72i − 2757.55i −10641.2 + 13009.3i −11585.2 44197.3 81520.2i
13.4 −22.6274 339.118i 512.000 3624.45i − 7673.36i 16745.5 + 1436.05i −11585.2 −55951.8 − 82012.0i
13.5 22.6274 − 469.394i 512.000 − 1370.18i − 10621.2i −12242.6 + 11514.9i 11585.2 −161282. − 31003.6i
13.6 22.6274 − 96.1268i 512.000 5042.67i − 2175.10i 15326.2 − 6897.94i 11585.2 49808.6 114102.i
13.7 22.6274 96.1268i 512.000 − 5042.67i 2175.10i 15326.2 + 6897.94i 11585.2 49808.6 − 114102.i
13.8 22.6274 469.394i 512.000 1370.18i 10621.2i −12242.6 − 11514.9i 11585.2 −161282. 31003.6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.11.b.a ✓ 8
3.b odd 2 1 126.11.c.a 8
4.b odd 2 1 112.11.c.d 8
7.b odd 2 1 inner 14.11.b.a ✓ 8
7.c even 3 2 98.11.d.c 16
7.d odd 6 2 98.11.d.c 16
21.c even 2 1 126.11.c.a 8
28.d even 2 1 112.11.c.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.11.b.a ✓ 8 1.a even 1 1 trivial
14.11.b.a ✓ 8 7.b odd 2 1 inner
98.11.d.c 16 7.c even 3 2
98.11.d.c 16 7.d odd 6 2
112.11.c.d 8 4.b odd 2 1
112.11.c.d 8 28.d even 2 1
126.11.c.a 8 3.b odd 2 1
126.11.c.a 8 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 512)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 85\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 22\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 16\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 52\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 74\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 65\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
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