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Properties

Label 1152.4.c.b
Level $1152$
Weight $4$
Character orbit 1152.c
Analytic conductor $67.970$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(1151,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.1151");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.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: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 4 x^{10} + 12 x^{9} + 88 x^{8} + 356 x^{7} + 1278 x^{6} + 3320 x^{5} + 8177 x^{4} + \cdots + 98 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{33}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} - \beta_{7} q^{7} + (\beta_{4} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{13} + ( - \beta_{11} + \beta_{8} + \cdots + 2 \beta_{6}) q^{17} + ( - \beta_{10} - 4 \beta_{9} + \cdots - 5 \beta_{6}) q^{19}+ \cdots + (6 \beta_{5} + 10 \beta_{4} + 21 \beta_{3} + \cdots + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 240 q^{23} - 300 q^{25} + 864 q^{35} - 264 q^{37} - 624 q^{47} + 132 q^{49} + 1632 q^{59} - 312 q^{61} - 4080 q^{71} + 432 q^{73} + 3744 q^{83} + 1704 q^{85} - 5856 q^{95} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 4 x^{10} + 12 x^{9} + 88 x^{8} + 356 x^{7} + 1278 x^{6} + 3320 x^{5} + 8177 x^{4} + \cdots + 98 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 31\!\cdots\!24 \nu^{11} + \cdots - 15\!\cdots\!52 ) / 77\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 58\!\cdots\!86 \nu^{11} + \cdots - 39\!\cdots\!38 ) / 77\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 205422544184 \nu^{11} + 976420265916 \nu^{10} + 327105386208 \nu^{9} + \cdots + 10\!\cdots\!36 ) / 32247935360533 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52\!\cdots\!20 \nu^{11} + \cdots - 11\!\cdots\!44 ) / 77\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65\!\cdots\!76 \nu^{11} + \cdots + 28\!\cdots\!12 ) / 77\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 54512090104 \nu^{11} - 211105124627 \nu^{10} - 248574560260 \nu^{9} + 637387911231 \nu^{8} + \cdots - 47487255309242 ) / 5192575793062 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 57\!\cdots\!54 \nu^{11} + \cdots + 52\!\cdots\!34 ) / 54\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 682713622216 \nu^{11} - 2695113490060 \nu^{10} - 2824084391672 \nu^{9} + \cdots - 612625091208704 ) / 35302542901367 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61\!\cdots\!12 \nu^{11} + \cdots - 53\!\cdots\!98 ) / 15\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!52 \nu^{11} + \cdots - 13\!\cdots\!86 ) / 10\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!84 \nu^{11} + \cdots + 22\!\cdots\!78 ) / 10\!\cdots\!42 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + \cdots + 64 ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{10} - 8 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} + 38 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 22 \beta_{11} - 92 \beta_{10} - 188 \beta_{9} + 526 \beta_{8} + 224 \beta_{7} + 678 \beta_{6} + \cdots + 1216 ) / 192 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - 19\beta_{10} - 44\beta_{9} + 144\beta_{8} + 41\beta_{7} + 166\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 370 \beta_{11} - 2216 \beta_{10} - 4964 \beta_{9} + 13708 \beta_{8} + 4580 \beta_{7} + 18718 \beta_{6} + \cdots - 33856 ) / 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 192 \beta_{11} - 1386 \beta_{10} - 3320 \beta_{9} + 8681 \beta_{8} + 2578 \beta_{7} + 12894 \beta_{6} + \cdots - 54880 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3530 \beta_{11} - 22900 \beta_{10} - 51364 \beta_{9} + 138242 \beta_{8} + 41488 \beta_{7} + \cdots - 2149888 ) / 192 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 63370\beta_{5} + 38960\beta_{4} - 40293\beta_{3} + 6628\beta_{2} - 48064\beta _1 - 974432 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 80698 \beta_{11} + 551936 \beta_{10} + 1264580 \beta_{9} - 3442072 \beta_{8} - 1055132 \beta_{7} + \cdots - 52910144 ) / 192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 119680 \beta_{11} + 836914 \beta_{10} + 1932720 \beta_{9} - 5276455 \beta_{8} - 1623818 \beta_{7} + \cdots - 33424736 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4664114 \beta_{11} + 32492956 \beta_{10} + 74847748 \beta_{9} - 204979694 \beta_{8} - 63330232 \beta_{7} + \cdots - 536969600 ) / 192 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
−0.843160 + 2.03557i
0.0928179 + 0.0384464i
−0.552426 − 1.33367i
4.57521 − 1.89511i
−2.25381 + 0.933560i
0.981372 − 2.36924i
0.981372 + 2.36924i
−2.25381 − 0.933560i
4.57521 + 1.89511i
−0.552426 + 1.33367i
0.0928179 − 0.0384464i
−0.843160 − 2.03557i
0 0 0 − 19.4649i 0 22.2014i 0 0 0
1151.2 0 0 0 − 15.9893i 0 4.39346i 0 0 0
1151.3 0 0 0 − 12.9056i 0 − 14.2604i 0 0 0
1151.4 0 0 0 − 9.77328i 0 16.6123i 0 0 0
1151.5 0 0 0 − 1.63027i 0 − 15.3904i 0 0 0
1151.6 0 0 0 − 0.854915i 0 − 27.6333i 0 0 0
1151.7 0 0 0 0.854915i 0 27.6333i 0 0 0
1151.8 0 0 0 1.63027i 0 15.3904i 0 0 0
1151.9 0 0 0 9.77328i 0 − 16.6123i 0 0 0
1151.10 0 0 0 12.9056i 0 14.2604i 0 0 0
1151.11 0 0 0 15.9893i 0 − 4.39346i 0 0 0
1151.12 0 0 0 19.4649i 0 − 22.2014i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1152.4.c.b yes 12
3.b odd 2 1 1152.4.c.c yes 12
4.b odd 2 1 1152.4.c.c yes 12
8.b even 2 1 1152.4.c.a ✓ 12
8.d odd 2 1 1152.4.c.d yes 12
12.b even 2 1 inner 1152.4.c.b yes 12
16.e even 4 1 2304.4.f.j 12
16.e even 4 1 2304.4.f.l 12
16.f odd 4 1 2304.4.f.i 12
16.f odd 4 1 2304.4.f.k 12
24.f even 2 1 1152.4.c.a ✓ 12
24.h odd 2 1 1152.4.c.d yes 12
48.i odd 4 1 2304.4.f.i 12
48.i odd 4 1 2304.4.f.k 12
48.k even 4 1 2304.4.f.j 12
48.k even 4 1 2304.4.f.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.c.a ✓ 12 8.b even 2 1
1152.4.c.a ✓ 12 24.f even 2 1
1152.4.c.b yes 12 1.a even 1 1 trivial
1152.4.c.b yes 12 12.b even 2 1 inner
1152.4.c.c yes 12 3.b odd 2 1
1152.4.c.c yes 12 4.b odd 2 1
1152.4.c.d yes 12 8.d odd 2 1
1152.4.c.d yes 12 24.h odd 2 1
2304.4.f.i 12 16.f odd 4 1
2304.4.f.i 12 48.i odd 4 1
2304.4.f.j 12 16.e even 4 1
2304.4.f.j 12 48.k even 4 1
2304.4.f.k 12 16.f odd 4 1
2304.4.f.k 12 48.i odd 4 1
2304.4.f.l 12 16.e even 4 1
2304.4.f.l 12 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1152, [\chi])\):

\( T_{11}^{6} - 4440T_{11}^{4} + 18432T_{11}^{3} + 2362560T_{11}^{2} + 17547264T_{11} - 9179648 \) Copy content Toggle raw display
\( T_{13}^{6} - 5976T_{13}^{4} - 24576T_{13}^{3} + 4408512T_{13}^{2} + 14352384T_{13} - 433607168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 2993402944 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 96575117725696 \) Copy content Toggle raw display
$11$ \( (T^{6} - 4440 T^{4} + \cdots - 9179648)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 5976 T^{4} + \cdots - 433607168)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{6} + 120 T^{5} + \cdots - 705453121024)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 17955598011328)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 3631402471936)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 55328206716928)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 433942746165248)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 83\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 53\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
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