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Properties

Label 48.14.c.b
Level $48$
Weight $14$
Character orbit 48.c
Analytic conductor $51.471$
Analytic rank $0$
Dimension $8$
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] = [48,14,Mod(47,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(51.4708458969\)
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: \( x^{8} - 4 x^{7} + 38458 x^{6} - 115360 x^{5} - 239611967 x^{4} + 479416196 x^{3} + 75414210590908 x^{2} + \cdots + 38\!\cdots\!95 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{21} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{3} - \beta_{2} + 50 \beta_1) q^{7} + (\beta_{5} + 3 \beta_{4} - 346059) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{3} - \beta_{2} + 50 \beta_1) q^{7} + (\beta_{5} + 3 \beta_{4} - 346059) q^{9} + (\beta_{6} - 2 \beta_{3} + 152 \beta_1) q^{11} + (\beta_{7} + 4 \beta_{5} + \cdots - 2076386) q^{13}+ \cdots + ( - 1538190 \beta_{6} + \cdots + 252951093 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2768472 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2768472 q^{9} - 16611088 q^{13} - 631106640 q^{21} - 3705156824 q^{25} + 1947996864 q^{33} + 29796096688 q^{37} + 35663110272 q^{45} + 27143767640 q^{49} - 161665060464 q^{57} - 1081500045200 q^{61} - 58018384512 q^{69} + 3407783142224 q^{73} + 5075114554440 q^{81} - 5984660980224 q^{85} - 2767731867792 q^{93} - 10520456950000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} + 38458 x^{6} - 115360 x^{5} - 239611967 x^{4} + 479416196 x^{3} + 75414210590908 x^{2} + \cdots + 38\!\cdots\!95 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11552 \nu^{7} + 40432 \nu^{6} + 2655117312 \nu^{5} - 6637894360 \nu^{4} + \cdots - 14\!\cdots\!83 ) / 80\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 128 \nu^{7} - 448 \nu^{6} + 4922400 \nu^{5} - 12304880 \nu^{4} - 30676484216 \nu^{3} + \cdots - 35\!\cdots\!45 ) / 22876792454961 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 400 \nu^{6} - 1200 \nu^{5} - 3749976 \nu^{4} + 7501952 \nu^{3} - 47021365740 \nu^{2} + \cdots + 32\!\cdots\!56 ) / 407953774917 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 400 \nu^{6} + 1200 \nu^{5} + 3749976 \nu^{4} - 7501952 \nu^{3} + 4942466664744 \nu^{2} + \cdots + 14\!\cdots\!96 ) / 135984591639 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 127808 \nu^{7} + 447328 \nu^{6} + 1283711424 \nu^{5} - 3210396880 \nu^{4} + \cdots + 53\!\cdots\!22 ) / 205891132094649 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18064 \nu^{6} + 54192 \nu^{5} - 905097240 \nu^{4} + 1810104160 \nu^{3} + \cdots - 10\!\cdots\!20 ) / 407953774917 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 3\beta_{4} + 6\beta _1 - 346050 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{6} + \beta_{5} + 3\beta_{4} - 818\beta_{3} + 702\beta_{2} - 38862\beta _1 - 346056 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 18225 \beta_{7} - 36 \beta_{6} - 14816 \beta_{5} - 867489 \beta_{4} - 3272 \beta_{3} + \cdots + 23495208600 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 273375 \beta_{7} + 8935407 \beta_{6} - 222260 \beta_{5} - 13012395 \beta_{4} + 1003979415 \beta_{3} + \cdots + 352435050144 ) / 288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 341989938 \beta_{7} + 8935587 \beta_{6} + 35380270 \beta_{5} + 82570781190 \beta_{4} + \cdots - 74\!\cdots\!32 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2393291691 \beta_{7} - 246288313215 \beta_{6} + 248180506 \beta_{5} + 578025830613 \beta_{4} + \cdots - 52\!\cdots\!32 ) / 192 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−184.343 − 100.597i
−184.343 + 100.597i
−22.0156 − 209.236i
−22.0156 + 209.236i
23.0156 − 209.236i
23.0156 + 209.236i
185.343 − 100.597i
185.343 + 100.597i
0 −1109.06 − 603.580i 0 7906.21i 0 327038.i 0 865704. + 1.33881e6i 0
47.2 0 −1109.06 + 603.580i 0 − 7906.21i 0 − 327038.i 0 865704. − 1.33881e6i 0
47.3 0 −135.094 − 1255.42i 0 − 57490.8i 0 − 282910.i 0 −1.55782e6 + 339197.i 0
47.4 0 −135.094 + 1255.42i 0 57490.8i 0 282910.i 0 −1.55782e6 − 339197.i 0
47.5 0 135.094 − 1255.42i 0 57490.8i 0 − 282910.i 0 −1.55782e6 − 339197.i 0
47.6 0 135.094 + 1255.42i 0 − 57490.8i 0 282910.i 0 −1.55782e6 + 339197.i 0
47.7 0 1109.06 − 603.580i 0 − 7906.21i 0 327038.i 0 865704. − 1.33881e6i 0
47.8 0 1109.06 + 603.580i 0 7906.21i 0 − 327038.i 0 865704. + 1.33881e6i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.14.c.b ✓ 8
3.b odd 2 1 inner 48.14.c.b ✓ 8
4.b odd 2 1 inner 48.14.c.b ✓ 8
12.b even 2 1 inner 48.14.c.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.14.c.b ✓ 8 1.a even 1 1 trivial
48.14.c.b ✓ 8 3.b odd 2 1 inner
48.14.c.b ✓ 8 4.b odd 2 1 inner
48.14.c.b ✓ 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3367695456T_{5}^{2} + 206601259848192000 \) acting on \(S_{14}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( (T^{4} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 85\!\cdots\!08)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + \cdots - 48549967903100)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 73\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 15\!\cdots\!00)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 40\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 18\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 25\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 15\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 54\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 13\!\cdots\!00)^{4} \) Copy content Toggle raw display
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