Properties

Label 143.4.a.c
Level $143$
Weight $4$
Character orbit 143.a
Self dual yes
Analytic conductor $8.437$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 143.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.43727313082\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 5) q^{4} + (\beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{5} + (\beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_1 + 4) q^{6} + ( - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{7} + (\beta_{8} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} + 3 \beta_1 + 5) q^{8} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 5) q^{4} + (\beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{5} + (\beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_1 + 4) q^{6} + ( - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{7} + (\beta_{8} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} + 3 \beta_1 + 5) q^{8} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 10) q^{9} + ( - \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_1 - 4) q^{10} - 11 q^{11} + ( - 3 \beta_{8} - \beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 20) q^{12} - 13 q^{13} + (4 \beta_{8} - 2 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_{2} + 6 \beta_1 + 2) q^{14} + ( - 6 \beta_{8} + \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 6 \beta_{2} - \beta_1 + 37) q^{15} + ( - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + \cdots + 11) q^{16}+ \cdots + (22 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} + 11 \beta_{4} + \cdots - 110) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 8 q^{3} + 46 q^{4} + 30 q^{5} + 34 q^{6} + 25 q^{7} + 36 q^{8} + 91 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 8 q^{3} + 46 q^{4} + 30 q^{5} + 34 q^{6} + 25 q^{7} + 36 q^{8} + 91 q^{9} - 22 q^{10} - 99 q^{11} + 181 q^{12} - 117 q^{13} + 351 q^{15} + 130 q^{16} + 53 q^{17} + 33 q^{18} + 69 q^{19} + 282 q^{20} + 463 q^{21} + 216 q^{23} - 121 q^{24} + 617 q^{25} + 275 q^{27} + 279 q^{28} - 91 q^{29} + 29 q^{30} + 636 q^{31} + 663 q^{32} - 88 q^{33} + 423 q^{34} - 358 q^{35} - 252 q^{36} + 967 q^{37} - 101 q^{38} - 104 q^{39} + 652 q^{40} - 226 q^{41} - 1186 q^{42} + 42 q^{43} - 506 q^{44} + 5 q^{45} - 1127 q^{46} - 269 q^{47} - 1820 q^{48} + 228 q^{49} - 1374 q^{50} - 589 q^{51} - 598 q^{52} + 1227 q^{53} - 2438 q^{54} - 330 q^{55} - 659 q^{56} - 71 q^{57} + 471 q^{58} - 613 q^{59} - 859 q^{60} + 427 q^{61} - 1714 q^{62} + 305 q^{63} - 1194 q^{64} - 390 q^{65} - 374 q^{66} - 271 q^{67} - 2835 q^{68} - 846 q^{69} - 102 q^{70} + 2279 q^{71} - 2400 q^{72} + 3602 q^{73} - 4955 q^{74} - 883 q^{75} + 1126 q^{76} - 275 q^{77} - 442 q^{78} - 1182 q^{79} - 2360 q^{80} + 2697 q^{81} + 1007 q^{82} - 1877 q^{83} + 1618 q^{84} - 441 q^{85} + 830 q^{86} + 1942 q^{87} - 396 q^{88} + 1258 q^{89} - 5669 q^{90} - 325 q^{91} + 1046 q^{92} + 1556 q^{93} + 1439 q^{94} + 2032 q^{95} - 3417 q^{96} + 4002 q^{97} - 1855 q^{98} - 1001 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{8} + 357 \nu^{7} - 350 \nu^{6} - 14878 \nu^{5} + 9822 \nu^{4} + 154641 \nu^{3} - 82789 \nu^{2} - 171682 \nu + 322364 ) / 37760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 87 \nu^{8} - 621 \nu^{7} + 6990 \nu^{6} + 29774 \nu^{5} - 168206 \nu^{4} - 418873 \nu^{3} + 1290477 \nu^{2} + 1596466 \nu - 945052 ) / 75520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 89 \nu^{8} + 93 \nu^{7} + 6290 \nu^{6} + 18 \nu^{5} - 148562 \nu^{4} - 109591 \nu^{3} + 1200419 \nu^{2} + 1253102 \nu - 1282084 ) / 75520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53 \nu^{8} - 119 \nu^{7} + 4810 \nu^{6} + 6346 \nu^{5} - 133194 \nu^{4} - 131707 \nu^{3} + 1212663 \nu^{2} + 1011254 \nu - 1429268 ) / 37760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 283 \nu^{8} + 551 \nu^{7} + 14550 \nu^{6} - 21674 \nu^{5} - 237974 \nu^{4} + 240843 \nu^{3} + 1287513 \nu^{2} - 549206 \nu - 612268 ) / 75520 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 311 \nu^{8} + 653 \nu^{7} - 16270 \nu^{6} - 35982 \nu^{5} + 263118 \nu^{4} + 525529 \nu^{3} - 1433101 \nu^{2} - 1848178 \nu + 1310876 ) / 37760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 145 \nu^{8} - 181 \nu^{7} - 7810 \nu^{6} + 5246 \nu^{5} + 134978 \nu^{4} - 6305 \nu^{3} - 783307 \nu^{2} - 483326 \nu + 495876 ) / 15104 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{2} + 19\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} + 2\beta_{7} - \beta_{6} - 5\beta_{5} + 28\beta_{4} - 21\beta_{3} - 29\beta_{2} + 8\beta _1 + 259 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33\beta_{8} - \beta_{7} + 60\beta_{6} - 43\beta_{5} + 127\beta_{4} - 4\beta_{3} - 37\beta_{2} + 404\beta _1 + 232 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 66 \beta_{8} + 90 \beta_{7} - 2 \beta_{6} - 220 \beta_{5} + 804 \beta_{4} - 436 \beta_{3} - 804 \beta_{2} + 391 \beta _1 + 5756 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 928\beta_{8} + 1662\beta_{6} - 1458\beta_{5} + 4309\beta_{4} - 275\beta_{3} - 1287\beta_{2} + 9292\beta _1 + 8519 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1581 \beta_{8} + 3022 \beta_{7} + 814 \beta_{6} - 7503 \beta_{5} + 23557 \beta_{4} - 9536 \beta_{3} - 22023 \beta_{2} + 14755 \beta _1 + 138197 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.87031
−4.11495
−3.76323
−1.62159
0.388321
0.765277
3.71870
4.08298
5.41479
−4.87031 −1.15688 15.7199 −10.2656 5.63435 15.3321 −37.5984 −25.6616 49.9968
1.2 −4.11495 9.51427 8.93279 14.5196 −39.1507 21.0410 −3.83839 63.5214 −59.7475
1.3 −3.76323 −4.70664 6.16188 20.6048 17.7122 −28.6315 6.91727 −4.84752 −77.5406
1.4 −1.62159 −2.83913 −5.37046 −8.40999 4.60389 −9.04976 21.6813 −18.9393 13.6375
1.5 0.388321 3.09988 −7.84921 16.8933 1.20375 26.1569 −6.15457 −17.3907 6.56001
1.6 0.765277 −9.54214 −7.41435 −17.1562 −7.30238 −4.60754 −11.7962 64.0525 −13.1292
1.7 3.71870 7.61710 5.82875 15.9808 28.3257 −21.9580 −8.07423 31.0202 59.4279
1.8 4.08298 7.19985 8.67073 −7.90460 29.3968 23.1330 2.73856 24.8378 −32.2743
1.9 5.41479 −1.18631 21.3199 5.73789 −6.42360 3.58372 72.1247 −25.5927 31.0694
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(13\) \(1\)

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 143.4.a.c 9
3.b odd 2 1 1287.4.a.k 9
4.b odd 2 1 2288.4.a.r 9
11.b odd 2 1 1573.4.a.e 9
13.b even 2 1 1859.4.a.d 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.c 9 1.a even 1 1 trivial
1287.4.a.k 9 3.b odd 2 1
1573.4.a.e 9 11.b odd 2 1
1859.4.a.d 9 13.b even 2 1
2288.4.a.r 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 59T_{2}^{7} - 12T_{2}^{6} + 1144T_{2}^{5} + 345T_{2}^{4} - 7888T_{2}^{3} - 2245T_{2}^{2} + 9710T_{2} - 2988 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(143))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 59 T^{7} - 12 T^{6} + \cdots - 2988 \) Copy content Toggle raw display
$3$ \( T^{9} - 8 T^{8} - 135 T^{7} + \cdots + 283048 \) Copy content Toggle raw display
$5$ \( T^{9} - 30 T^{8} + \cdots - 5425892224 \) Copy content Toggle raw display
$7$ \( T^{9} - 25 T^{8} + \cdots - 18338418984 \) Copy content Toggle raw display
$11$ \( (T + 11)^{9} \) Copy content Toggle raw display
$13$ \( (T + 13)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 53 T^{8} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{9} - 69 T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{9} - 216 T^{8} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{9} + 91 T^{8} + \cdots + 27\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{9} - 636 T^{8} + \cdots + 56\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{9} - 967 T^{8} + \cdots - 72\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{9} + 226 T^{8} + \cdots - 58\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{9} - 42 T^{8} + \cdots + 51\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{9} + 269 T^{8} + \cdots - 51\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{9} - 1227 T^{8} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{9} + 613 T^{8} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{9} - 427 T^{8} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{9} + 271 T^{8} + \cdots + 89\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{9} - 2279 T^{8} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{9} - 3602 T^{8} + \cdots - 24\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{9} + 1182 T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + 1877 T^{8} + \cdots + 41\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{9} - 1258 T^{8} + \cdots - 94\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{9} - 4002 T^{8} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
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