Properties

Label 12.21.c.b
Level 12
Weight 21
Character orbit 12.c
Analytic conductor 30.422
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(30.4216518123\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{22}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -14063 + \beta_{1} ) q^{3} \) \( + ( 27 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( -24173422 + 428 \beta_{1} - 4 \beta_{3} - 7 \beta_{4} ) q^{7} \) \( + ( -1291329039 - 20445 \beta_{1} + 21 \beta_{2} - 68 \beta_{3} + 29 \beta_{4} - \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -14063 + \beta_{1} ) q^{3} \) \( + ( 27 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( -24173422 + 428 \beta_{1} - 4 \beta_{3} - 7 \beta_{4} ) q^{7} \) \( + ( -1291329039 - 20445 \beta_{1} + 21 \beta_{2} - 68 \beta_{3} + 29 \beta_{4} - \beta_{5} ) q^{9} \) \( + ( 11811 \beta_{1} - 424 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} + 18 \beta_{5} ) q^{11} \) \( + ( -61160500462 - 1221446 \beta_{1} - 2186 \beta_{3} + 4084 \beta_{4} ) q^{13} \) \( + ( -89040785760 - 386856 \beta_{1} + 13392 \beta_{2} - 9054 \beta_{3} - 11061 \beta_{4} + 126 \beta_{5} ) q^{15} \) \( + ( 1090794 \beta_{1} - 88364 \beta_{2} + 1054 \beta_{3} - 1054 \beta_{4} - 414 \beta_{5} ) q^{17} \) \( + ( 2033665609634 + 28801549 \beta_{1} + 149587 \beta_{3} + 18259 \beta_{4} ) q^{19} \) \( + ( 1760269963634 - 17973385 \beta_{1} + 1495341 \beta_{2} + 143046 \beta_{3} - 107730 \beta_{4} - 3618 \beta_{5} ) q^{21} \) \( + ( -71378706 \beta_{1} - 3777872 \beta_{2} + 165014 \beta_{3} - 165014 \beta_{4} + 3168 \beta_{5} ) q^{23} \) \( + ( -39413782503695 - 25740990 \beta_{1} - 1408530 \beta_{3} - 1505940 \beta_{4} ) q^{25} \) \( + ( -44890231494303 - 1559101428 \beta_{1} + 8966232 \beta_{2} + 1532427 \beta_{3} + 1758315 \beta_{4} + 52302 \beta_{5} ) q^{27} \) \( + ( -5135911659 \beta_{1} - 3849739 \beta_{2} + 4948764 \beta_{3} - 4948764 \beta_{4} + 2052 \beta_{5} ) q^{29} \) \( + ( -107921915831566 + 6256266714 \beta_{1} + 22024542 \beta_{3} - 8266251 \beta_{4} ) q^{31} \) \( + ( -38944461782880 - 180812169 \beta_{1} - 80090829 \beta_{2} + 28873278 \beta_{3} - 59395239 \beta_{4} - 464409 \beta_{5} ) q^{33} \) \( + ( -62032573704 \beta_{1} + 188229688 \beta_{2} + 53696384 \beta_{3} - 53696384 \beta_{4} - 198846 \beta_{5} ) q^{35} \) \( + ( -1936300494457486 + 38120268158 \beta_{1} + 111978770 \beta_{3} - 76330612 \beta_{4} ) q^{37} \) \( + ( -3158926327070926 - 78335322070 \beta_{1} - 302045760 \beta_{2} + 117896148 \beta_{3} - 42217038 \beta_{4} + 2721006 \beta_{5} ) q^{39} \) \( + ( -260257168950 \beta_{1} - 21289630 \beta_{2} + 246855328 \beta_{3} - 246855328 \beta_{4} + 1360656 \beta_{5} ) q^{41} \) \( + ( -12445661405563582 + 81180675009 \beta_{1} - 84613761 \beta_{3} - 540068661 \beta_{4} ) q^{43} \) \( + ( 2831076562720320 - 82035991053 \beta_{1} + 1925292159 \beta_{2} + 707060340 \beta_{3} - 472426830 \beta_{4} - 10527570 \beta_{5} ) q^{45} \) \( + ( -230459463336 \beta_{1} - 3236770992 \beta_{2} + 299177384 \beta_{3} - 299177384 \beta_{4} - 2369052 \beta_{5} ) q^{47} \) \( + ( 33318030420418275 + 432314420510 \beta_{1} + 2279349170 \beta_{3} + 313835060 \beta_{4} ) q^{49} \) \( + ( -3608460806843520 - 15689231334 \beta_{1} + 3137458104 \beta_{2} - 1492197678 \beta_{3} + 255143106 \beta_{4} + 23929758 \beta_{5} ) q^{51} \) \( + ( 1699365706695 \beta_{1} + 2421993851 \beta_{2} - 1674934080 \beta_{3} + 1674934080 \beta_{4} - 20823264 \beta_{5} ) q^{53} \) \( + ( -57067958133414720 - 2999569275450 \beta_{1} - 9517659510 \beta_{3} + 5180828700 \beta_{4} ) q^{55} \) \( + ( 65921150435667458 + 2434903852859 \beta_{1} - 21461356539 \beta_{2} - 6572404152 \beta_{3} + 3515902443 \beta_{4} - 9348399 \beta_{5} ) q^{57} \) \( + ( 7802633648145 \beta_{1} + 18490651360 \beta_{2} - 7773167771 \beta_{3} + 7773167771 \beta_{4} + 162260568 \beta_{5} ) q^{59} \) \( + ( 304870469682921554 - 1171049554698 \beta_{1} - 2133258534 \beta_{3} + 3871733532 \beta_{4} ) q^{61} \) \( + ( 319447766416349298 + 3242106863880 \beta_{1} - 6174021648 \beta_{2} + 1929410900 \beta_{3} + 29832723823 \beta_{4} - 139992248 \beta_{5} ) q^{63} \) \( + ( 9147931302666 \beta_{1} + 35667157918 \beta_{2} - 9749180156 \beta_{3} + 9749180156 \beta_{4} - 494067636 \beta_{5} ) q^{65} \) \( + ( 59149276963569698 + 2420749953231 \beta_{1} - 56240895 \beta_{3} - 13221966009 \beta_{4} ) q^{67} \) \( + ( 233898446030592960 + 976626609678 \beta_{1} + 32990572674 \beta_{2} - 20328509160 \beta_{3} - 54877703454 \beta_{4} + 504270342 \beta_{5} ) q^{69} \) \( + ( -12758988715386 \beta_{1} - 259618565696 \beta_{2} + 18815570878 \beta_{3} - 18815570878 \beta_{4} + 363424932 \beta_{5} ) q^{71} \) \( + ( -3339358906989847006 - 15292925469408 \beta_{1} + 3565583712 \beta_{3} + 87280032432 \beta_{4} ) q^{73} \) \( + ( 473018680466348545 - 39723726554255 \beta_{1} + 390854720520 \beta_{2} + 55187180820 \beta_{3} - 35915606550 \beta_{4} - 704376810 \beta_{5} ) q^{75} \) \( + ( -112044477845058 \beta_{1} + 118256735446 \beta_{2} + 103812821392 \beta_{3} - 103812821392 \beta_{4} + 2514425616 \beta_{5} ) q^{77} \) \( + ( -3030485199246951118 + 103958379363802 \beta_{1} + 190066129246 \beta_{3} - 342903052043 \beta_{4} ) q^{79} \) \( + ( 3374704406503118241 - 33160751100468 \beta_{1} - 839440273014 \beta_{2} + 207280659402 \beta_{3} - 59043684438 \beta_{4} - 466111998 \beta_{5} ) q^{81} \) \( + ( -134280890469651 \beta_{1} + 155864472792 \beta_{2} + 119109196889 \beta_{3} - 119109196889 \beta_{4} - 9110903238 \beta_{5} ) q^{83} \) \( + ( -11753897489165295360 + 66389111359560 \beta_{1} + 96950235000 \beta_{3} - 247526145840 \beta_{4} ) q^{85} \) \( + ( 16892056755443086560 + 72255281043996 \beta_{1} - 33970377024 \beta_{2} + 214154221482 \beta_{3} - 153423325035 \beta_{4} + 4033101366 \beta_{5} ) q^{87} \) \( + ( -35626653487860 \beta_{1} + 2269137775390 \beta_{2} - 20562263858 \beta_{3} + 20562263858 \beta_{4} + 8945900514 \beta_{5} ) q^{89} \) \( + ( -23893451342253547964 - 309348849463836 \beta_{1} - 419388572148 \beta_{3} + 1191197317674 \beta_{4} ) q^{91} \) \( + ( 22075884462338674898 - 20365462370881 \beta_{1} - 1609720771059 \beta_{2} - 1022706458622 \beta_{3} + 494595803268 \beta_{4} - 7883466264 \beta_{5} ) q^{93} \) \( + ( 985070417520018 \beta_{1} - 4623587534384 \beta_{2} - 804433568150 \beta_{3} + 804433568150 \beta_{4} + 19301864400 \beta_{5} ) q^{95} \) \( + ( -32047439247744577342 - 96465740998946 \beta_{1} - 687646223246 \beta_{3} - 279229331516 \beta_{4} ) q^{97} \) \( + ( 79174752908672969280 + 297261643746051 \beta_{1} + 7638926219424 \beta_{2} - 1833787294029 \beta_{3} + 508759510167 \beta_{4} + 3683908836 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 84378q^{3} \) \(\mathstrut -\mathstrut 145040532q^{7} \) \(\mathstrut -\mathstrut 7747974234q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 84378q^{3} \) \(\mathstrut -\mathstrut 145040532q^{7} \) \(\mathstrut -\mathstrut 7747974234q^{9} \) \(\mathstrut -\mathstrut 366963002772q^{13} \) \(\mathstrut -\mathstrut 534244714560q^{15} \) \(\mathstrut +\mathstrut 12201993657804q^{19} \) \(\mathstrut +\mathstrut 10561619781804q^{21} \) \(\mathstrut -\mathstrut 236482695022170q^{25} \) \(\mathstrut -\mathstrut 269341388965818q^{27} \) \(\mathstrut -\mathstrut 647531494989396q^{31} \) \(\mathstrut -\mathstrut 233666770697280q^{33} \) \(\mathstrut -\mathstrut 11617802966744916q^{37} \) \(\mathstrut -\mathstrut 18953557962425556q^{39} \) \(\mathstrut -\mathstrut 74673968433381492q^{43} \) \(\mathstrut +\mathstrut 16986459376321920q^{45} \) \(\mathstrut +\mathstrut 199908182522509650q^{49} \) \(\mathstrut -\mathstrut 21650764841061120q^{51} \) \(\mathstrut -\mathstrut 342407748800488320q^{55} \) \(\mathstrut +\mathstrut 395526902614004748q^{57} \) \(\mathstrut +\mathstrut 1829222818097529324q^{61} \) \(\mathstrut +\mathstrut 1916686598498095788q^{63} \) \(\mathstrut +\mathstrut 354895661781418188q^{67} \) \(\mathstrut +\mathstrut 1403390676183557760q^{69} \) \(\mathstrut -\mathstrut 20036153441939082036q^{73} \) \(\mathstrut +\mathstrut 2838112082798091270q^{75} \) \(\mathstrut -\mathstrut 18182911195481706708q^{79} \) \(\mathstrut +\mathstrut 20248226439018709446q^{81} \) \(\mathstrut -\mathstrut 70523384934991772160q^{85} \) \(\mathstrut +\mathstrut 101352340532658519360q^{87} \) \(\mathstrut -\mathstrut 143360708053521287784q^{91} \) \(\mathstrut +\mathstrut 132455306774032049388q^{93} \) \(\mathstrut -\mathstrut 192284635486467464052q^{97} \) \(\mathstrut +\mathstrut 475048517452037815680q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(24769850\) \(x^{4}\mathstrut +\mathstrut \) \(131733035896000\) \(x^{2}\mathstrut +\mathstrut \) \(250851218720256000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 967 \nu^{5} - 6552 \nu^{4} + 23795771510 \nu^{3} + 304111800720 \nu^{2} + 124100496670796800 \nu + 3275507965906176000 \)\()/\)\(97219198876800\)
\(\beta_{2}\)\(=\)\((\)\( 50121 \nu^{5} - 58968 \nu^{4} + 1267838842410 \nu^{3} + 2737006206480 \nu^{2} + 6973688776158720000 \nu + 29479571693155584000 \)\()/\)\(32406399625600\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(207905\) \(\nu^{5}\mathstrut -\mathstrut \) \(90145944\) \(\nu^{4}\mathstrut -\mathstrut \) \(5116090874650\) \(\nu^{3}\mathstrut -\mathstrut \) \(1354155400629360\) \(\nu^{2}\mathstrut -\mathstrut \) \(26681606784221312000\) \(\nu\mathstrut -\mathstrut \) \(661329470661581568000\)\()/\)\(97219198876800\)
\(\beta_{4}\)\(=\)\((\)\(22241\) \(\nu^{5}\mathstrut -\mathstrut \) \(10400922\) \(\nu^{4}\mathstrut +\mathstrut \) \(547302744730\) \(\nu^{3}\mathstrut -\mathstrut \) \(209526224698980\) \(\nu^{2}\mathstrut +\mathstrut \) \(2854311423428326400\) \(\nu\mathstrut -\mathstrut \) \(516261550819527744000\)\()/\)\(12152399859600\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(5702877\) \(\nu^{5}\mathstrut +\mathstrut \) \(471744\) \(\nu^{4}\mathstrut -\mathstrut \) \(140438836537650\) \(\nu^{3}\mathstrut -\mathstrut \) \(21896049651840\) \(\nu^{2}\mathstrut -\mathstrut \) \(748625648021511321600\) \(\nu\mathstrut -\mathstrut \) \(235836573545244672000\)\()/\)\(16203199812800\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(211\) \(\beta_{4}\mathstrut +\mathstrut \) \(211\) \(\beta_{3}\mathstrut -\mathstrut \) \(54\) \(\beta_{2}\mathstrut -\mathstrut \) \(225879\) \(\beta_{1}\)\()/8398080\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(23548\) \(\beta_{4}\mathstrut +\mathstrut \) \(21091\) \(\beta_{3}\mathstrut +\mathstrut \) \(8867397\) \(\beta_{1}\mathstrut -\mathstrut \) \(1155662121600\)\()/139968\)
\(\nu^{3}\)\(=\)\((\)\(14135373\) \(\beta_{5}\mathstrut +\mathstrut \) \(261813727\) \(\beta_{4}\mathstrut -\mathstrut \) \(261813727\) \(\beta_{3}\mathstrut +\mathstrut \) \(874348398\) \(\beta_{2}\mathstrut +\mathstrut \) \(259759811283\) \(\beta_{1}\)\()/839808\)
\(\nu^{4}\)\(=\)\((\)\(55245686020\) \(\beta_{4}\mathstrut -\mathstrut \) \(74252673565\) \(\beta_{3}\mathstrut -\mathstrut \) \(26129531044155\) \(\beta_{1}\mathstrut +\mathstrut \) \(2722217392864368000\)\()/23328\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(116169420430665\) \(\beta_{5}\mathstrut -\mathstrut \) \(1838438928421235\) \(\beta_{4}\mathstrut +\mathstrut \) \(1838438928421235\) \(\beta_{3}\mathstrut -\mathstrut \) \(10411402283826390\) \(\beta_{2}\mathstrut -\mathstrut \) \(1715984153160822615\) \(\beta_{1}\)\()/419904\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
4127.93i
4127.93i
2779.95i
2779.95i
43.6454i
43.6454i
0 −53241.6 25536.6i 0 9.13793e6i 0 9.41859e7 0 2.18255e9 + 2.71922e9i 0
5.2 0 −53241.6 + 25536.6i 0 9.13793e6i 0 9.41859e7 0 2.18255e9 2.71922e9i 0
5.3 0 −8570.43 58423.7i 0 1.60830e7i 0 −4.81200e8 0 −3.33988e9 + 1.00143e9i 0
5.4 0 −8570.43 + 58423.7i 0 1.60830e7i 0 −4.81200e8 0 −3.33988e9 1.00143e9i 0
5.5 0 19623.0 55693.1i 0 7.88530e6i 0 3.14494e8 0 −2.71666e9 2.18573e9i 0
5.6 0 19623.0 + 55693.1i 0 7.88530e6i 0 3.14494e8 0 −2.71666e9 + 2.18573e9i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{6} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!60\)\( T_{5}^{4} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!00\)\( T_{5}^{2} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\( \) acting on \(S_{21}^{\mathrm{new}}(12, [\chi])\).