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Properties

Label 2304.4.f.l
Level $2304$
Weight $4$
Character orbit 2304.f
Analytic conductor $135.940$
Analytic rank $0$
Dimension $12$
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] = [2304,4,Mod(1151,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1151");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.f (of order \(2\), degree \(1\), not 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: \(135.940400653\)
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: \( 2^{28}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 1152)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 2) q^{5} + ( - \beta_{7} + \beta_{5}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 2) q^{5} + ( - \beta_{7} + \beta_{5}) q^{7} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{7}) q^{11}+ \cdots + (13 \beta_{6} - 27 \beta_{4} + \cdots + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{5} + 240 q^{23} + 300 q^{25} + 168 q^{29} + 624 q^{43} - 624 q^{47} + 132 q^{49} - 1896 q^{53} + 1488 q^{67} + 4080 q^{71} - 432 q^{73} + 3744 q^{77} + 2784 q^{91} - 5856 q^{95} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( 81\!\cdots\!26 \nu^{11} + \cdots + 73\!\cdots\!18 ) / 15\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\!\cdots\!08 \nu^{11} + \cdots - 47\!\cdots\!72 ) / 77\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51355636046 \nu^{11} - 244105066479 \nu^{10} - 81776346552 \nu^{9} + 948910203301 \nu^{8} + \cdots - 273092394489434 ) / 32247935360533 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52\!\cdots\!90 \nu^{11} + \cdots - 21\!\cdots\!18 ) / 15\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 170678405554 \nu^{11} - 673778372515 \nu^{10} - 706021097918 \nu^{9} + \cdots - 153156272802176 ) / 35302542901367 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 76\!\cdots\!22 \nu^{11} + \cdots - 14\!\cdots\!58 ) / 15\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!52 \nu^{11} + \cdots + 28\!\cdots\!46 ) / 54\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 54512090104 \nu^{11} + 211105124627 \nu^{10} + 248574560260 \nu^{9} + \cdots + 47487255309242 ) / 5192575793062 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!80 \nu^{11} + \cdots - 32\!\cdots\!10 ) / 54\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 40\!\cdots\!04 \nu^{11} + \cdots + 35\!\cdots\!56 ) / 54\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 57\!\cdots\!20 \nu^{11} + \cdots + 52\!\cdots\!12 ) / 77\!\cdots\!53 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} + 2 \beta_{10} - 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} + 3 \beta_{4} + \cdots + 16 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{10} - 2\beta_{9} - 20\beta_{8} + 16\beta_{5} + 4\beta_{4} - 10\beta_{3} + \beta_{2} + 32 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 11 \beta_{11} + 46 \beta_{10} - 18 \beta_{9} - 198 \beta_{8} + 33 \beta_{7} + \beta_{6} + 400 \beta_{5} + \cdots + 304 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + 38\beta_{10} - 21\beta_{9} - 186\beta_{8} + 22\beta_{7} + 472\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 185 \beta_{11} + 1108 \beta_{10} - 528 \beta_{9} - 5326 \beta_{8} + 591 \beta_{7} + 101 \beta_{6} + \cdots - 8464 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 192 \beta_{11} + 1386 \beta_{10} - 734 \beta_{9} - 7236 \beta_{8} + 596 \beta_{7} + 394 \beta_{6} + \cdots - 27440 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1765 \beta_{11} + 11450 \beta_{10} - 5538 \beta_{9} - 56754 \beta_{8} + 4647 \beta_{7} + \cdots - 537472 ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7464\beta_{6} - 35188\beta_{4} + 60081\beta_{3} - 12016\beta_{2} - 3314\beta _1 - 487216 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 40349 \beta_{11} - 275968 \beta_{10} + 137898 \beta_{9} + 1378522 \beta_{8} - 125799 \beta_{7} + \cdots - 13227536 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 119680 \beta_{11} - 836914 \beta_{10} + 423340 \beta_{9} + 4196152 \beta_{8} - 393452 \beta_{7} + \cdots - 16712368 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2332057 \beta_{11} - 16246478 \beta_{10} + 8189940 \beta_{9} + 81201678 \beta_{8} - 7709319 \beta_{7} + \cdots - 134242400 ) / 48 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
−0.552426 − 1.33367i
−0.552426 + 1.33367i
4.57521 + 1.89511i
4.57521 − 1.89511i
−2.25381 + 0.933560i
−2.25381 − 0.933560i
0.981372 − 2.36924i
0.981372 + 2.36924i
0.0928179 − 0.0384464i
0.0928179 + 0.0384464i
−0.843160 − 2.03557i
−0.843160 + 2.03557i
0 0 0 −12.9056 0 − 14.2604i 0 0 0
1151.2 0 0 0 −12.9056 0 14.2604i 0 0 0
1151.3 0 0 0 −9.77328 0 − 16.6123i 0 0 0
1151.4 0 0 0 −9.77328 0 16.6123i 0 0 0
1151.5 0 0 0 −1.63027 0 − 15.3904i 0 0 0
1151.6 0 0 0 −1.63027 0 15.3904i 0 0 0
1151.7 0 0 0 0.854915 0 − 27.6333i 0 0 0
1151.8 0 0 0 0.854915 0 27.6333i 0 0 0
1151.9 0 0 0 15.9893 0 − 4.39346i 0 0 0
1151.10 0 0 0 15.9893 0 4.39346i 0 0 0
1151.11 0 0 0 19.4649 0 − 22.2014i 0 0 0
1151.12 0 0 0 19.4649 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
24.f even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{6} - 12T_{5}^{5} - 378T_{5}^{4} + 2320T_{5}^{3} + 41772T_{5}^{2} + 26832T_{5} - 54712 \) Copy content Toggle raw display
\( T_{19}^{6} - 20832T_{19}^{4} + 245760T_{19}^{3} + 102484992T_{19}^{2} - 951582720T_{19} - 136133771264 \) 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
$5$ \( (T^{6} - 12 T^{5} + \cdots - 54712)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 96575117725696 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 84265937403904 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} - 20832 T^{4} + \cdots - 136133771264)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 120 T^{5} + \cdots - 705453121024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 84 T^{5} + \cdots - 628075479352)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{6} - 312 T^{5} + \cdots + 585604886528)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 3631402471936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 191796419084552)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!48)^{2} \) 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
$83$ \( T^{12} + \cdots + 28\!\cdots\!56 \) 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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