Properties

Label 6012.2.a.i
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 31 x^{7} + 24 x^{6} + 293 x^{5} - 101 x^{4} - 864 x^{3} - 278 x^{2} + 24 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( -\beta_{3} - \beta_{6} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( -\beta_{3} - \beta_{6} ) q^{7} + ( 1 + \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{13} + ( -1 - \beta_{1} + \beta_{8} ) q^{17} + ( -1 - \beta_{3} + \beta_{8} ) q^{19} -\beta_{2} q^{23} + ( 2 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( -1 - \beta_{2} - \beta_{3} - \beta_{8} ) q^{29} + ( 3 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{31} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{35} + ( 2 + \beta_{1} - \beta_{5} ) q^{37} + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{41} + ( -\beta_{3} - \beta_{4} ) q^{43} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{47} + ( 3 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{49} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{53} + ( 1 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{55} + ( 2 - \beta_{2} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{59} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{61} + ( -3 + \beta_{6} - \beta_{8} ) q^{65} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{67} + ( -1 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{71} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{73} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{77} + ( 2 - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{83} + ( 5 + 2 \beta_{1} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{85} + ( -5 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{8} ) q^{89} + ( 7 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{91} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{95} + ( 6 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - q^{5} + 2q^{7} + O(q^{10}) \) \( 9q - q^{5} + 2q^{7} + 9q^{11} + 10q^{13} - 7q^{17} - 2q^{19} + 3q^{23} + 18q^{25} - 5q^{29} + 12q^{31} + 6q^{35} + 15q^{37} - 14q^{41} + 6q^{43} + 3q^{47} + 27q^{49} - 9q^{53} + 19q^{55} + 9q^{59} + 30q^{61} - 28q^{65} + 16q^{67} + 3q^{71} + 32q^{73} - 18q^{77} + 24q^{79} + 3q^{83} + 37q^{85} - 46q^{89} + 33q^{91} - 11q^{95} + 43q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 31 x^{7} + 24 x^{6} + 293 x^{5} - 101 x^{4} - 864 x^{3} - 278 x^{2} + 24 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 454 \nu^{8} + 294 \nu^{7} - 14998 \nu^{6} - 10080 \nu^{5} + 154841 \nu^{4} + 125150 \nu^{3} - 499492 \nu^{2} - 541382 \nu - 14306 \)\()/27853\)
\(\beta_{3}\)\(=\)\((\)\( -575 \nu^{8} + 924 \nu^{7} + 16824 \nu^{6} - 25452 \nu^{5} - 142113 \nu^{4} + 175571 \nu^{3} + 337666 \nu^{2} - 190010 \nu - 38190 \)\()/27853\)
\(\beta_{4}\)\(=\)\((\)\( -2488 \nu^{8} + 2646 \nu^{7} + 78500 \nu^{6} - 64078 \nu^{5} - 762703 \nu^{4} + 284359 \nu^{3} + 2349144 \nu^{2} + 659583 \nu - 147092 \)\()/27853\)
\(\beta_{5}\)\(=\)\((\)\( 3597 \nu^{8} - 3731 \nu^{7} - 111199 \nu^{6} + 90552 \nu^{5} + 1044226 \nu^{4} - 410322 \nu^{3} - 3026506 \nu^{2} - 792549 \nu + 6960 \)\()/27853\)
\(\beta_{6}\)\(=\)\((\)\( -577 \nu^{8} + 611 \nu^{7} + 18018 \nu^{6} - 14597 \nu^{5} - 172292 \nu^{4} + 60797 \nu^{3} + 516238 \nu^{2} + 174672 \nu - 3088 \)\()/3979\)
\(\beta_{7}\)\(=\)\((\)\( -5148 \nu^{8} + 5362 \nu^{7} + 158825 \nu^{6} - 128653 \nu^{5} - 1487567 \nu^{4} + 551542 \nu^{3} + 4318881 \nu^{2} + 1355670 \nu - 76455 \)\()/27853\)
\(\beta_{8}\)\(=\)\((\)\( -5294 \nu^{8} + 4830 \nu^{7} + 163120 \nu^{6} - 116641 \nu^{5} - 1527615 \nu^{4} + 492437 \nu^{3} + 4438630 \nu^{2} + 1359676 \nu - 110393 \)\()/27853\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 7\)
\(\nu^{3}\)\(=\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 13 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-4 \beta_{8} + 16 \beta_{7} - 16 \beta_{6} + 10 \beta_{5} + 14 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 2 \beta_{1} + 90\)
\(\nu^{5}\)\(=\)\(-3 \beta_{8} + 6 \beta_{7} - 42 \beta_{6} - 42 \beta_{5} + 17 \beta_{4} - 50 \beta_{3} + 22 \beta_{2} + 187 \beta_{1} + 15\)
\(\nu^{6}\)\(=\)\(-101 \beta_{8} + 246 \beta_{7} - 255 \beta_{6} + 99 \beta_{5} + 218 \beta_{4} + 96 \beta_{3} - 125 \beta_{2} + 31 \beta_{1} + 1271\)
\(\nu^{7}\)\(=\)\(-91 \beta_{8} + 150 \beta_{7} - 711 \beta_{6} - 746 \beta_{5} + 253 \beta_{4} - 943 \beta_{3} + 417 \beta_{2} + 2807 \beta_{1} - 57\)
\(\nu^{8}\)\(=\)\(-1980 \beta_{8} + 3806 \beta_{7} - 3988 \beta_{6} + 1062 \beta_{5} + 3465 \beta_{4} + 1859 \beta_{3} - 2420 \beta_{2} + 285 \beta_{1} + 18844\)

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
3.91239
3.25977
2.94172
0.111665
−0.0546093
−0.419033
−1.74256
−2.94765
−4.06168
0 0 0 −3.91239 0 1.60938 0 0 0
1.2 0 0 0 −3.25977 0 1.95900 0 0 0
1.3 0 0 0 −2.94172 0 −2.78729 0 0 0
1.4 0 0 0 −0.111665 0 −3.78430 0 0 0
1.5 0 0 0 0.0546093 0 3.75281 0 0 0
1.6 0 0 0 0.419033 0 −4.23221 0 0 0
1.7 0 0 0 1.74256 0 2.35372 0 0 0
1.8 0 0 0 2.94765 0 4.65088 0 0 0
1.9 0 0 0 4.06168 0 −1.52198 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(-1\)

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 6012.2.a.i 9
3.b odd 2 1 2004.2.a.c 9
12.b even 2 1 8016.2.a.bc 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.c 9 3.b odd 2 1
6012.2.a.i 9 1.a even 1 1 trivial
8016.2.a.bc 9 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{9} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + T + 14 T^{2} + 16 T^{3} + 108 T^{4} + 81 T^{5} + 686 T^{6} + 298 T^{7} + 3439 T^{8} + 1678 T^{9} + 17195 T^{10} + 7450 T^{11} + 85750 T^{12} + 50625 T^{13} + 337500 T^{14} + 250000 T^{15} + 1093750 T^{16} + 390625 T^{17} + 1953125 T^{18} \)
$7$ \( 1 - 2 T + 20 T^{2} - 31 T^{3} + 268 T^{4} - 467 T^{5} + 2758 T^{6} - 4405 T^{7} + 23261 T^{8} - 36366 T^{9} + 162827 T^{10} - 215845 T^{11} + 945994 T^{12} - 1121267 T^{13} + 4504276 T^{14} - 3647119 T^{15} + 16470860 T^{16} - 11529602 T^{17} + 40353607 T^{18} \)
$11$ \( 1 - 9 T + 56 T^{2} - 300 T^{3} + 1628 T^{4} - 7585 T^{5} + 32230 T^{6} - 124292 T^{7} + 467121 T^{8} - 1604664 T^{9} + 5138331 T^{10} - 15039332 T^{11} + 42898130 T^{12} - 111051985 T^{13} + 262191028 T^{14} - 531468300 T^{15} + 1091281576 T^{16} - 1929229929 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 - 10 T + 92 T^{2} - 624 T^{3} + 3559 T^{4} - 17541 T^{5} + 78209 T^{6} - 312916 T^{7} + 1202007 T^{8} - 4377322 T^{9} + 15626091 T^{10} - 52882804 T^{11} + 171825173 T^{12} - 500988501 T^{13} + 1321431787 T^{14} - 3011928816 T^{15} + 5772863564 T^{16} - 8157307210 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 7 T + 103 T^{2} + 692 T^{3} + 5651 T^{4} + 32089 T^{5} + 198783 T^{6} + 946082 T^{7} + 4759334 T^{8} + 19249834 T^{9} + 80908678 T^{10} + 273417698 T^{11} + 976620879 T^{12} + 2680105369 T^{13} + 8023611907 T^{14} + 16703197748 T^{15} + 42264883319 T^{16} + 48830302087 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 2 T + 82 T^{2} + 142 T^{3} + 3811 T^{4} + 6287 T^{5} + 124015 T^{6} + 184422 T^{7} + 3042753 T^{8} + 4042502 T^{9} + 57812307 T^{10} + 66576342 T^{11} + 850618885 T^{12} + 819328127 T^{13} + 9436413289 T^{14} + 6680515102 T^{15} + 73297482598 T^{16} + 33967126082 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 - 3 T + 146 T^{2} - 480 T^{3} + 9952 T^{4} - 35395 T^{5} + 426894 T^{6} - 1554404 T^{7} + 13049775 T^{8} - 44055692 T^{9} + 300144825 T^{10} - 822279716 T^{11} + 5194019298 T^{12} - 9904972195 T^{13} + 64054485536 T^{14} - 71057226720 T^{15} + 497104515262 T^{16} - 234932955843 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 5 T + 140 T^{2} + 1018 T^{3} + 10260 T^{4} + 83219 T^{5} + 568490 T^{6} + 3901674 T^{7} + 23850973 T^{8} + 128905912 T^{9} + 691678217 T^{10} + 3281307834 T^{11} + 13864902610 T^{12} + 58859217539 T^{13} + 210444388740 T^{14} + 605530140778 T^{15} + 2414982683260 T^{16} + 2501232064805 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 - 12 T + 193 T^{2} - 1789 T^{3} + 18600 T^{4} - 141873 T^{5} + 1141452 T^{6} - 7333927 T^{7} + 48912708 T^{8} - 268589310 T^{9} + 1516293948 T^{10} - 7047903847 T^{11} + 34004996532 T^{12} - 131022694833 T^{13} + 532502208600 T^{14} - 1587744085309 T^{15} + 5309934523423 T^{16} - 10234692449292 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 - 15 T + 362 T^{2} - 3972 T^{3} + 54830 T^{4} - 472427 T^{5} + 4734840 T^{6} - 33133904 T^{7} + 261735687 T^{8} - 1504564436 T^{9} + 9684220419 T^{10} - 45360314576 T^{11} + 239833850520 T^{12} - 885404258747 T^{13} + 3802129162310 T^{14} - 10191065296548 T^{15} + 34365339522146 T^{16} - 52687191808815 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 14 T + 327 T^{2} + 3205 T^{3} + 44586 T^{4} + 348111 T^{5} + 3681236 T^{6} + 24097457 T^{7} + 209386044 T^{8} + 1166849428 T^{9} + 8584827804 T^{10} + 40507825217 T^{11} + 253714466356 T^{12} + 983678487471 T^{13} + 5165564577786 T^{14} + 15224084092405 T^{15} + 63684647559087 T^{16} + 111788953207694 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 6 T + 319 T^{2} - 1839 T^{3} + 47906 T^{4} - 252725 T^{5} + 4413570 T^{6} - 20520067 T^{7} + 273224160 T^{8} - 1081545560 T^{9} + 11748638880 T^{10} - 37941603883 T^{11} + 350909709990 T^{12} - 864016482725 T^{13} + 7042586470358 T^{14} - 11624986647111 T^{15} + 86710136943133 T^{16} - 70129201665606 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - 3 T + 278 T^{2} - 1171 T^{3} + 38707 T^{4} - 177397 T^{5} + 3551703 T^{6} - 15372301 T^{7} + 231098509 T^{8} - 875034452 T^{9} + 10861629923 T^{10} - 33957412909 T^{11} + 368748460569 T^{12} - 865640770357 T^{13} + 8877257185949 T^{14} - 12622461150259 T^{15} + 140841227488714 T^{16} - 71433859985283 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 9 T + 172 T^{2} + 1209 T^{3} + 14217 T^{4} + 51649 T^{5} + 463831 T^{6} - 1181855 T^{7} - 894743 T^{8} - 208766930 T^{9} - 47421379 T^{10} - 3319830695 T^{11} + 69053767787 T^{12} + 407535453169 T^{13} + 5945485323981 T^{14} + 26796712604961 T^{15} + 202050316051964 T^{16} + 560337213702249 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 - 9 T + 210 T^{2} - 1695 T^{3} + 26299 T^{4} - 206235 T^{5} + 2400067 T^{6} - 17846869 T^{7} + 172863813 T^{8} - 1191970608 T^{9} + 10198964967 T^{10} - 62124950989 T^{11} + 492923360393 T^{12} - 2499023945835 T^{13} + 18801794139401 T^{14} - 71496004521495 T^{15} + 522616811811990 T^{16} - 1321473938438889 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 30 T + 786 T^{2} - 13574 T^{3} + 210383 T^{4} - 2611429 T^{5} + 29858725 T^{6} - 291974242 T^{7} + 2675679737 T^{8} - 21519130886 T^{9} + 163216463957 T^{10} - 1086436154482 T^{11} + 6777363259225 T^{12} - 36157430716789 T^{13} + 177688703593283 T^{14} - 699337561576214 T^{15} + 2470195869112506 T^{16} - 5751219389918430 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 16 T + 454 T^{2} - 6618 T^{3} + 104959 T^{4} - 1279883 T^{5} + 15122569 T^{6} - 153614690 T^{7} + 1458313809 T^{8} - 12428852816 T^{9} + 97707025203 T^{10} - 689576343410 T^{11} + 4548309220147 T^{12} - 25791077198843 T^{13} + 141707781105613 T^{14} - 598653573194442 T^{15} + 2751563068816642 T^{16} - 6497082840906256 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 - 3 T + 389 T^{2} - 385 T^{3} + 72285 T^{4} + 22199 T^{5} + 9030607 T^{6} + 7099821 T^{7} + 847401670 T^{8} + 661160352 T^{9} + 60165518570 T^{10} + 35790197661 T^{11} + 3232153581977 T^{12} + 564113906519 T^{13} + 130418718637035 T^{14} - 49318609309585 T^{15} + 3538001741614099 T^{16} - 1937260593737283 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 32 T + 664 T^{2} - 9928 T^{3} + 127791 T^{4} - 1457675 T^{5} + 15824861 T^{6} - 160721244 T^{7} + 1549062291 T^{8} - 13708341698 T^{9} + 113081547243 T^{10} - 856483509276 T^{11} + 6156139951637 T^{12} - 41395405949675 T^{13} + 264919891941063 T^{14} - 1502446198597192 T^{15} + 7335472616680408 T^{16} - 25806722940610592 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 24 T + 693 T^{2} - 12373 T^{3} + 216700 T^{4} - 2985079 T^{5} + 39461338 T^{6} - 437526161 T^{7} + 4632269092 T^{8} - 42200922052 T^{9} + 365949258268 T^{10} - 2730600770801 T^{11} + 19455978626182 T^{12} - 116269068841399 T^{13} + 666798121663300 T^{14} - 3007721087161333 T^{15} + 13308308927408187 T^{16} - 36410611437757464 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 3 T + 312 T^{2} - 843 T^{3} + 47477 T^{4} - 78457 T^{5} + 4769641 T^{6} - 1897577 T^{7} + 385296895 T^{8} + 92865088 T^{9} + 31979642285 T^{10} - 13072407953 T^{11} + 2727218718467 T^{12} - 3723437490697 T^{13} + 187013832607711 T^{14} - 275610734750067 T^{15} + 8466447908763624 T^{16} - 6756876696417123 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 46 T + 1334 T^{2} + 27744 T^{3} + 484163 T^{4} + 7267645 T^{5} + 97930165 T^{6} + 1174238724 T^{7} + 12786451321 T^{8} + 125889131394 T^{9} + 1137994167569 T^{10} + 9301144932804 T^{11} + 69037730489885 T^{12} + 455988334092445 T^{13} + 2703594975006187 T^{14} + 13788248936421984 T^{15} + 59004600750635686 T^{16} + 181083085062295726 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 - 43 T + 1385 T^{2} - 31873 T^{3} + 627071 T^{4} - 10318687 T^{5} + 151150379 T^{6} - 1939378947 T^{7} + 22511211436 T^{8} - 232559681172 T^{9} + 2183587509292 T^{10} - 18247616512323 T^{11} + 137950869853067 T^{12} - 913505940974047 T^{13} + 5384872042297247 T^{14} - 26549316713102017 T^{15} + 111905624002186505 T^{16} - 337009644558209323 T^{17} + 760231058654565217 T^{18} \)
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