Properties

Label 1143.4.a.d
Level $1143$
Weight $4$
Character orbit 1143.a
Self dual yes
Analytic conductor $67.439$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1143,4,Mod(1,1143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1143.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1143.a (trivial)

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: yes
Analytic conductor: \(67.4391831366\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 55 x^{11} + 264 x^{10} + 1126 x^{9} - 5085 x^{8} - 10823 x^{7} + 44242 x^{6} + 52047 x^{5} - 178872 x^{4} - 113160 x^{3} + 298560 x^{2} + \cdots - 130048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 127)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{9} + \beta_{8} - \beta_1 + 3) q^{4} + ( - \beta_{12} + \beta_{9} + \beta_1 + 3) q^{5} + ( - \beta_{12} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_1 - 2) q^{7} + (\beta_{11} + 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{9} + \beta_{8} - \beta_1 + 3) q^{4} + ( - \beta_{12} + \beta_{9} + \beta_1 + 3) q^{5} + ( - \beta_{12} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_1 - 2) q^{7} + (\beta_{11} + 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 9) q^{8} + ( - \beta_{12} + \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{10}+ \cdots + (71 \beta_{12} - 147 \beta_{11} - 11 \beta_{10} - 39 \beta_{9} - 57 \beta_{8} + \cdots + 222) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 8 q^{2} + 34 q^{4} + 46 q^{5} - 26 q^{7} + 117 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 8 q^{2} + 34 q^{4} + 46 q^{5} - 26 q^{7} + 117 q^{8} - 25 q^{10} + 53 q^{11} - 75 q^{13} + 152 q^{14} + 82 q^{16} + 479 q^{17} - 209 q^{19} + 533 q^{20} - 407 q^{22} + 376 q^{23} + 9 q^{25} + 622 q^{26} - 258 q^{28} + 158 q^{29} - 307 q^{31} + 609 q^{32} + 948 q^{34} + 512 q^{35} - 171 q^{37} - 712 q^{38} + 1897 q^{40} + 641 q^{41} + 530 q^{43} - 1328 q^{44} + 2051 q^{46} + 555 q^{47} + 357 q^{49} - 808 q^{50} + 2473 q^{52} + 1640 q^{53} + 540 q^{55} - 551 q^{56} + 1328 q^{58} + 860 q^{59} + 191 q^{61} - 367 q^{62} + 1915 q^{64} + 1584 q^{65} + 912 q^{67} + 1873 q^{68} + 2329 q^{70} - 115 q^{71} - 2563 q^{73} - 470 q^{74} - 192 q^{76} + 3338 q^{77} + 169 q^{79} + 3194 q^{80} + 54 q^{82} + 1688 q^{83} + 480 q^{85} + 485 q^{86} - 1192 q^{88} + 2752 q^{89} - 398 q^{91} + 846 q^{92} + 531 q^{94} + 1766 q^{95} - 2124 q^{97} + 1479 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 55 x^{11} + 264 x^{10} + 1126 x^{9} - 5085 x^{8} - 10823 x^{7} + 44242 x^{6} + 52047 x^{5} - 178872 x^{4} - 113160 x^{3} + 298560 x^{2} + \cdots - 130048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1002737 \nu^{12} + 723467 \nu^{11} + 81924597 \nu^{10} - 45464246 \nu^{9} - 2529028770 \nu^{8} + 1058101817 \nu^{7} + \cdots - 379711735296 ) / 12616105984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60894341 \nu^{12} - 421156103 \nu^{11} - 2418602233 \nu^{10} + 20568960398 \nu^{9} + 21152228554 \nu^{8} - 344700083501 \nu^{7} + \cdots - 6445760160256 ) / 227089907712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 66841603 \nu^{12} - 437650993 \nu^{11} - 2909259215 \nu^{10} + 21534874690 \nu^{9} + 38546908742 \nu^{8} - 371720539483 \nu^{7} + \cdots - 1992359022080 ) / 227089907712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10515013 \nu^{12} - 85125187 \nu^{11} - 357280253 \nu^{10} + 4134258130 \nu^{9} + 1066320074 \nu^{8} - 68894662885 \nu^{7} + \cdots - 585397755392 ) / 28386238464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13771177 \nu^{12} - 116883187 \nu^{11} - 491339021 \nu^{10} + 5838860294 \nu^{9} + 2693117458 \nu^{8} - 101387078577 \nu^{7} + \cdots - 1040420694528 ) / 25232211968 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31415069 \nu^{12} + 251048519 \nu^{11} + 1092161593 \nu^{10} - 12184503782 \nu^{9} - 4544529658 \nu^{8} + 203212168949 \nu^{7} + \cdots + 2433094191616 ) / 56772476928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52572899 \nu^{12} - 363365393 \nu^{11} - 2167612399 \nu^{10} + 17815473986 \nu^{9} + 24037086214 \nu^{8} - 303502668731 \nu^{7} + \cdots - 2608004468224 ) / 75696635904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 52572899 \nu^{12} + 363365393 \nu^{11} + 2167612399 \nu^{10} - 17815473986 \nu^{9} - 24037086214 \nu^{8} + 303502668731 \nu^{7} + \cdots + 1851038109184 ) / 75696635904 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46441789 \nu^{12} + 285190399 \nu^{11} + 2043676673 \nu^{10} - 13889131726 \nu^{9} - 26502226490 \nu^{8} + 233194913989 \nu^{7} + \cdots + 2901811569152 ) / 56772476928 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 56602447 \nu^{12} - 379328077 \nu^{11} - 2373552179 \nu^{10} + 18649757362 \nu^{9} + 26883180254 \nu^{8} - 317653202743 \nu^{7} + \cdots - 2727849532928 ) / 56772476928 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 227904101 \nu^{12} + 1762596455 \nu^{11} + 8164090777 \nu^{10} - 85107637262 \nu^{9} - 44927733898 \nu^{8} + \cdots + 19413978915328 ) / 227089907712 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} + 3\beta_{8} + 2\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{3} + 17\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 27 \beta_{9} + 29 \beta_{8} + 7 \beta_{7} - \beta_{6} + 5 \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{2} + 35 \beta _1 + 169 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{12} - 18 \beta_{11} + 2 \beta_{10} + 54 \beta_{9} + 110 \beta_{8} + 74 \beta_{7} + 65 \beta_{5} - 41 \beta_{4} + 10 \beta_{3} + \beta_{2} + 372 \beta _1 + 224 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 83 \beta_{12} + 101 \beta_{11} + 87 \beta_{10} + 704 \beta_{9} + 818 \beta_{8} + 307 \beta_{7} - 27 \beta_{6} + 208 \beta_{5} - 207 \beta_{4} - 93 \beta_{3} - 18 \beta_{2} + 1108 \beta _1 + 3491 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 330 \beta_{12} - 178 \beta_{11} + 95 \beta_{10} + 2101 \beta_{9} + 3561 \beta_{8} + 2257 \beta_{7} - 9 \beta_{6} + 1835 \beta_{5} - 1485 \beta_{4} - 280 \beta_{3} + 47 \beta_{2} + 9259 \beta _1 + 7297 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2629 \beta_{12} + 2916 \beta_{11} + 2081 \beta_{10} + 18992 \beta_{9} + 23556 \beta_{8} + 10315 \beta_{7} - 649 \beta_{6} + 6960 \beta_{5} - 7914 \beta_{4} - 4502 \beta_{3} - 92 \beta_{2} + 34434 \beta _1 + 80379 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11356 \beta_{12} + 2104 \beta_{11} + 3276 \beta_{10} + 71883 \beta_{9} + 111675 \beta_{8} + 65852 \beta_{7} - 624 \beta_{6} + 51034 \beta_{5} - 49850 \beta_{4} - 21298 \beta_{3} + 2410 \beta_{2} + \cdots + 229436 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 77280 \beta_{12} + 84015 \beta_{11} + 48318 \beta_{10} + 530049 \beta_{9} + 689571 \beta_{8} + 320248 \beta_{7} - 16418 \beta_{6} + 218278 \beta_{5} - 270669 \beta_{4} - 174945 \beta_{3} + \cdots + 1991626 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 355376 \beta_{12} + 206755 \beta_{11} + 101151 \beta_{10} + 2316705 \beta_{9} + 3448947 \beta_{8} + 1909403 \beta_{7} - 30669 \beta_{6} + 1435905 \beta_{5} - 1604644 \beta_{4} + \cdots + 7068295 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2230037 \beta_{12} + 2467992 \beta_{11} + 1135346 \beta_{10} + 15172180 \beta_{9} + 20396200 \beta_{8} + 9678938 \beta_{7} - 446116 \beta_{6} + 6675729 \beta_{5} + \cdots + 52123626 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.48248
4.55672
3.56231
3.15095
2.38890
1.40898
0.744781
−0.842259
−2.11792
−2.19365
−2.58112
−4.22658
−4.33359
−4.48248 0 12.0927 5.99266 0 −20.2899 −18.3452 0 −26.8620
1.2 −3.55672 0 4.65028 21.6593 0 3.10281 11.9140 0 −77.0363
1.3 −2.56231 0 −1.43456 −3.57484 0 −0.237956 24.1743 0 9.15985
1.4 −2.15095 0 −3.37340 −1.85550 0 −13.5146 24.4637 0 3.99109
1.5 −1.38890 0 −6.07096 6.99532 0 −31.6914 19.5431 0 −9.71578
1.6 −0.408981 0 −7.83273 18.6969 0 22.7740 6.47528 0 −7.64665
1.7 0.255219 0 −7.93486 −14.2191 0 17.9778 −4.06688 0 −3.62899
1.8 1.84226 0 −4.60608 0.253207 0 −11.4184 −23.2237 0 0.466472
1.9 3.11792 0 1.72143 −6.07736 0 8.24382 −19.5761 0 −18.9487
1.10 3.19365 0 2.19938 −11.5549 0 −29.5975 −18.5251 0 −36.9023
1.11 3.58112 0 4.82445 8.90410 0 34.6915 −11.3721 0 31.8867
1.12 5.22658 0 19.3172 5.57653 0 −8.92344 59.1502 0 29.1462
1.13 5.33359 0 20.4472 15.2037 0 2.88328 66.3885 0 81.0905
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(127\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1143.4.a.d 13
3.b odd 2 1 127.4.a.b 13
12.b even 2 1 2032.4.a.g 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.4.a.b 13 3.b odd 2 1
1143.4.a.d 13 1.a even 1 1 trivial
2032.4.a.g 13 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 8 T_{2}^{12} - 37 T_{2}^{11} + 385 T_{2}^{10} + 356 T_{2}^{9} - 6666 T_{2}^{8} + 319 T_{2}^{7} + 52189 T_{2}^{6} - 14223 T_{2}^{5} - 187644 T_{2}^{4} + 29472 T_{2}^{3} + 250576 T_{2}^{2} + 28656 T_{2} - 23328 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1143))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 8 T^{12} - 37 T^{11} + \cdots - 23328 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 46 T^{12} + \cdots + 21492717120 \) Copy content Toggle raw display
$7$ \( T^{13} + 26 T^{12} + \cdots + 6532521113600 \) Copy content Toggle raw display
$11$ \( T^{13} - 53 T^{12} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{13} + 75 T^{12} + \cdots - 87\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{13} - 479 T^{12} + \cdots + 29\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{13} + 209 T^{12} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{13} - 376 T^{12} + \cdots - 12\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{13} - 158 T^{12} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{13} + 307 T^{12} + \cdots - 12\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{13} + 171 T^{12} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{13} - 641 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{13} - 530 T^{12} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{13} - 555 T^{12} + \cdots + 40\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{13} - 1640 T^{12} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{13} - 860 T^{12} + \cdots - 72\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{13} - 191 T^{12} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{13} - 912 T^{12} + \cdots + 93\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{13} + 115 T^{12} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{13} + 2563 T^{12} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{13} - 169 T^{12} + \cdots - 19\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{13} - 1688 T^{12} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{13} - 2752 T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{13} + 2124 T^{12} + \cdots - 59\!\cdots\!68 \) Copy content Toggle raw display
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