Properties

Label 1344.4.c.e
Level $1344$
Weight $4$
Character orbit 1344.c
Analytic conductor $79.299$
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] = [1344,4,Mod(673,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.673");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.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: \(79.2985670477\)
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: \( x^{12} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22} \)
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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{5} - 7 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{5} - 7 q^{7} - 9 q^{9} + (\beta_{9} + \beta_{8} + \beta_{3} + 11 \beta_1) q^{11} + ( - \beta_{11} - \beta_{3} - \beta_{2} - 3 \beta_1) q^{13} + ( - 3 \beta_{4} - 3) q^{15} + (5 \beta_{5} - \beta_{4}) q^{17} + (\beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{3} - 2 \beta_{2} + 10 \beta_1) q^{19} - 21 \beta_1 q^{21} + (\beta_{10} - 2 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + 6) q^{23} + (3 \beta_{7} + 4 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} - 47) q^{25} - 27 \beta_1 q^{27} + ( - \beta_{11} + 3 \beta_{9} + \beta_{8} - 5 \beta_{3} + 4 \beta_{2} - 20 \beta_1) q^{29} + (2 \beta_{10} - \beta_{7} + 2 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} - 50) q^{31} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 33) q^{33} + (7 \beta_{3} - 7 \beta_1) q^{35} + ( - 3 \beta_{11} + 9 \beta_{9} - 5 \beta_{8} + 3 \beta_{3} + 62 \beta_1) q^{37} + (3 \beta_{10} + 3 \beta_{5} - 3 \beta_{4} + 9) q^{39} + ( - 6 \beta_{7} + 6 \beta_{6} + \beta_{5} - 11 \beta_{4} + 114) q^{41} + ( - 2 \beta_{11} + \beta_{9} - 7 \beta_{8} + 4 \beta_{3} - 5 \beta_{2} + 39 \beta_1) q^{43} + (9 \beta_{3} - 9 \beta_1) q^{45} + (5 \beta_{10} + 10 \beta_{7} - 8 \beta_{5} + 5 \beta_{4} - 122) q^{47} + 49 q^{49} + (3 \beta_{3} + 15 \beta_{2}) q^{51} + ( - 13 \beta_{9} - \beta_{8} - 6 \beta_{3} - 12 \beta_{2} - 246 \beta_1) q^{53} + (\beta_{7} - 2 \beta_{6} + 12 \beta_{5} + 112) q^{55} + ( - 3 \beta_{10} + 3 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} - 30) q^{57} + ( - 9 \beta_{11} + 6 \beta_{8} + 13 \beta_{3} + 10 \beta_{2} - 8 \beta_1) q^{59} + ( - 7 \beta_{11} + 8 \beta_{9} + 22 \beta_{8} + 11 \beta_{3} + 5 \beta_{2} - 263 \beta_1) q^{61} + 63 q^{63} + (3 \beta_{10} + 10 \beta_{7} + 44 \beta_{6} - 22 \beta_{5} + 11 \beta_{4} + \cdots - 228) q^{65}+ \cdots + ( - 9 \beta_{9} - 9 \beta_{8} - 9 \beta_{3} - 99 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 84 q^{7} - 108 q^{9} - 24 q^{15} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 415171 \nu^{11} + 174672590 \nu^{9} + 26906273167 \nu^{7} + 1820198588256 \nu^{5} + 51364871210556 \nu^{3} + \cdots + 517741765875936 \nu ) / 433631923478208 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 415171 \nu^{11} + 174672590 \nu^{9} + 26906273167 \nu^{7} + 1820198588256 \nu^{5} + 51364871210556 \nu^{3} + \cdots + 13\!\cdots\!52 \nu ) / 433631923478208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 150398 \nu^{11} - 907043 \nu^{9} - 11496542836 \nu^{7} - 1860503278707 \nu^{5} - 103808484356868 \nu^{3} + \cdots - 16\!\cdots\!40 \nu ) / 108407980869552 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 897179 \nu^{10} - 395288260 \nu^{8} - 58374438995 \nu^{6} - 3268567476990 \nu^{4} - 46608391182636 \nu^{2} + \cdots + 227912499930024 ) / 20649139213248 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1886 \nu^{10} - 543931 \nu^{8} - 50881052 \nu^{6} - 1823053635 \nu^{4} - 29324763072 \nu^{2} + 56926306332 ) / 28364202216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -149\nu^{10} - 34288\nu^{8} - 1766423\nu^{6} + 59669136\nu^{4} + 4564567404\nu^{2} + 23304782592 ) / 1412003502 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 725843 \nu^{10} + 180719902 \nu^{8} + 15930365615 \nu^{6} + 713417388048 \nu^{4} + 17320042967436 \nu^{2} + \cdots + 48026973574944 ) / 5162284803312 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 597544 \nu^{11} - 88086239 \nu^{9} + 6707059802 \nu^{7} + 1485391973733 \nu^{5} + 56665055702952 \nu^{3} + \cdots + 308637875104020 \nu ) / 36135993623184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -26\nu^{11} - 9595\nu^{9} - 1316132\nu^{7} - 80656935\nu^{5} - 2006939520\nu^{3} - 11659276188\nu ) / 660235212 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12881299 \nu^{10} + 4283247980 \nu^{8} + 503759280859 \nu^{6} + 24505239122262 \nu^{4} + 429014116275084 \nu^{2} + \cdots + 15\!\cdots\!16 ) / 20649139213248 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19037951 \nu^{11} + 7224960640 \nu^{9} + 991200314147 \nu^{7} + 58547526756570 \nu^{5} + \cdots + 79\!\cdots\!08 \nu ) / 108407980869552 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{7} + 4\beta_{5} + 5\beta_{4} - 128 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 9\beta_{9} - 5\beta_{8} + 5\beta_{3} - 100\beta_{2} + 560\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -115\beta_{10} + 109\beta_{7} + 72\beta_{6} - 634\beta_{5} - 503\beta_{4} + 13142 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 271\beta_{11} - 945\beta_{9} + 635\beta_{8} - 1715\beta_{3} + 11122\beta_{2} - 86246\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13207\beta_{10} - 9385\beta_{7} - 10566\beta_{6} + 86194\beta_{5} + 53003\beta_{4} - 1474790 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -51523\beta_{11} + 63999\beta_{9} - 73367\beta_{8} + 289943\beta_{3} - 1285582\beta_{2} + 11681138\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1521847\beta_{10} + 670873\beta_{7} + 1377972\beta_{6} - 11179762\beta_{5} - 5916587\beta_{4} + 171508166 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8320819 \beta_{11} - 1246689 \beta_{9} + 8889023 \beta_{8} - 41548295 \beta_{3} + 151579342 \beta_{2} - 1515355250 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 178218511 \beta_{10} - 30105001 \beta_{7} - 181957770 \beta_{6} + 1419494314 \beta_{5} + 684909251 \beta_{4} - 20329210046 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1226087659 \beta_{11} - 593510517 \beta_{9} - 1150460591 \beta_{8} + 5590175999 \beta_{3} - 18093843574 \beta_{2} + 192693279482 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1
3.41237i
0.910182i
10.8906i
11.2296i
9.31944i
6.47819i
6.47819i
9.31944i
11.2296i
10.8906i
0.910182i
3.41237i
0 3.00000i 0 20.9786i 0 −7.00000 0 −9.00000 0
673.2 0 3.00000i 0 13.8002i 0 −7.00000 0 −9.00000 0
673.3 0 3.00000i 0 1.28149i 0 −7.00000 0 −9.00000 0
673.4 0 3.00000i 0 6.85537i 0 −7.00000 0 −9.00000 0
673.5 0 3.00000i 0 8.40898i 0 −7.00000 0 −9.00000 0
673.6 0 3.00000i 0 16.7959i 0 −7.00000 0 −9.00000 0
673.7 0 3.00000i 0 16.7959i 0 −7.00000 0 −9.00000 0
673.8 0 3.00000i 0 8.40898i 0 −7.00000 0 −9.00000 0
673.9 0 3.00000i 0 6.85537i 0 −7.00000 0 −9.00000 0
673.10 0 3.00000i 0 1.28149i 0 −7.00000 0 −9.00000 0
673.11 0 3.00000i 0 13.8002i 0 −7.00000 0 −9.00000 0
673.12 0 3.00000i 0 20.9786i 0 −7.00000 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.c.e 12
4.b odd 2 1 1344.4.c.h yes 12
8.b even 2 1 inner 1344.4.c.e 12
8.d odd 2 1 1344.4.c.h yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.e 12 1.a even 1 1 trivial
1344.4.c.e 12 8.b even 2 1 inner
1344.4.c.h yes 12 4.b odd 2 1
1344.4.c.h yes 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{12} + 1032T_{5}^{10} + 374136T_{5}^{8} + 58092608T_{5}^{6} + 3747196176T_{5}^{4} + 84573340800T_{5}^{2} + 129036134656 \) Copy content Toggle raw display
\( T_{23}^{6} - 40T_{23}^{5} - 36968T_{23}^{4} + 638448T_{23}^{3} + 361912548T_{23}^{2} - 2746383552T_{23} - 854202265344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 1032 T^{10} + \cdots + 129036134656 \) Copy content Toggle raw display
$7$ \( (T + 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 6648 T^{10} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{12} + 14824 T^{10} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{6} - 12 T^{5} - 20108 T^{4} + \cdots + 2432891472)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 31184 T^{10} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{6} - 40 T^{5} + \cdots - 854202265344)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 79888 T^{10} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{6} + 320 T^{5} + \cdots - 2770620689408)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 315120 T^{10} + \cdots + 72\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{6} - 708 T^{5} + \cdots + 6866119763664)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 355864 T^{10} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{6} + 768 T^{5} + \cdots + 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 835152 T^{10} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{12} + 1278240 T^{10} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{12} + 3270984 T^{10} + \cdots + 77\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + 1587576 T^{10} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{6} - 696 T^{5} + \cdots + 14258685155328)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 1236 T^{5} + \cdots - 28788786341568)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 272 T^{5} + \cdots - 13\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 2676608 T^{10} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{6} - 444 T^{5} + \cdots + 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 1356 T^{5} + \cdots - 73\!\cdots\!12)^{2} \) Copy content Toggle raw display
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