Properties

Label 12.17.c.b
Level 12
Weight 17
Character orbit 12.c
Analytic conductor 19.479
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(19.4789452628\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{11}\cdot 5 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -3105 - \beta_{1} ) q^{3} \) \( + ( -8 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} \) \( + ( -262570 + 462 \beta_{1} - 77 \beta_{3} ) q^{7} \) \( + ( -2530359 + 4428 \beta_{1} + 27 \beta_{2} - 657 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -3105 - \beta_{1} ) q^{3} \) \( + ( -8 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} \) \( + ( -262570 + 462 \beta_{1} - 77 \beta_{3} ) q^{7} \) \( + ( -2530359 + 4428 \beta_{1} + 27 \beta_{2} - 657 \beta_{3} ) q^{9} \) \( + ( -68 \beta_{1} - 292 \beta_{2} - 103 \beta_{3} ) q^{11} \) \( + ( -68715790 - 65208 \beta_{1} + 10868 \beta_{3} ) q^{13} \) \( + ( -266318280 + 19008 \beta_{1} + 6156 \beta_{2} + 27351 \beta_{3} ) q^{15} \) \( + ( 475376 \beta_{1} - 13640 \beta_{2} + 35068 \beta_{3} ) q^{17} \) \( + ( -10337184946 + 194370 \beta_{1} - 32395 \beta_{3} ) q^{19} \) \( + ( -13900098366 + 1497496 \beta_{1} - 18711 \beta_{2} + 455301 \beta_{3} ) q^{21} \) \( + ( 15330632 \beta_{1} + 114160 \beta_{2} + 1315606 \beta_{3} ) q^{23} \) \( + ( -131409689135 - 28014120 \beta_{1} + 4669020 \beta_{3} ) q^{25} \) \( + ( -114356556945 + 12844251 \beta_{1} - 335340 \beta_{2} + 3376971 \beta_{3} ) q^{27} \) \( + ( 73218376 \beta_{1} + 181709 \beta_{2} + 6162101 \beta_{3} ) q^{29} \) \( + ( -428084121946 + 46484550 \beta_{1} - 7747425 \beta_{3} ) q^{31} \) \( + ( -238324680 - 1981692 \beta_{1} + 1751625 \beta_{2} + 9104049 \beta_{3} ) q^{33} \) \( + ( -121292248 \beta_{1} - 3966116 \beta_{2} - 11429726 \beta_{3} ) q^{35} \) \( + ( 1314667670930 - 166270344 \beta_{1} + 27711724 \beta_{3} ) q^{37} \) \( + ( 2290333053294 - 105585194 \beta_{1} + 2640924 \beta_{2} - 64262484 \beta_{3} ) q^{39} \) \( + ( -1024582064 \beta_{1} + 10630994 \beta_{2} - 81838174 \beta_{3} ) q^{41} \) \( + ( 4742542432430 + 1485494802 \beta_{1} - 247582467 \beta_{3} ) q^{43} \) \( + ( 7696697027760 - 318858768 \beta_{1} - 35363061 \beta_{2} - 230015781 \beta_{3} ) q^{45} \) \( + ( -1482840528 \beta_{1} + 1863960 \beta_{2} - 122948724 \beta_{3} ) q^{47} \) \( + ( -12768473357325 - 242614680 \beta_{1} + 40435780 \beta_{3} ) q^{49} \) \( + ( 16347134374560 - 1681842096 \beta_{1} + 72131796 \beta_{2} + 661078044 \beta_{3} ) q^{51} \) \( + ( 6521633368 \beta_{1} + 2390555 \beta_{2} + 544266299 \beta_{3} ) q^{53} \) \( + ( -82021278501360 - 7857409320 \beta_{1} + 1309568220 \beta_{3} ) q^{55} \) \( + ( 25905989422170 + 10856735956 \beta_{1} - 7871985 \beta_{2} + 191551635 \beta_{3} ) q^{57} \) \( + ( 11146798748 \beta_{1} - 268706288 \beta_{2} + 839331133 \beta_{3} ) q^{59} \) \( + ( -246049263264046 - 307878120 \beta_{1} + 51313020 \beta_{3} ) q^{61} \) \( + ( 183429393805350 + 711620910 \beta_{1} + 109105920 \beta_{2} + 659686797 \beta_{3} ) q^{63} \) \( + ( 17965739792 \beta_{1} + 665564614 \beta_{2} + 1718999854 \beta_{3} ) q^{65} \) \( + ( -199502864250370 + 4695287562 \beta_{1} - 782547927 \beta_{3} ) q^{67} \) \( + ( 523228498963920 - 50049979272 \beta_{1} - 994720338 \beta_{2} + 3981760038 \beta_{3} ) q^{69} \) \( + ( -96672468232 \beta_{1} + 67234312 \beta_{2} - 8033627582 \beta_{3} ) q^{71} \) \( + ( -532037079878590 + 12239273376 \beta_{1} - 2039878896 \beta_{3} ) q^{73} \) \( + ( 1300317931492335 + 56527946375 \beta_{1} + 1134571860 \beta_{2} - 27607915260 \beta_{3} ) q^{75} \) \( + ( -45094266448 \beta_{1} - 1094451050 \beta_{2} - 4122672554 \beta_{3} ) q^{77} \) \( + ( -487453484173018 + 169093966470 \beta_{1} - 28182327745 \beta_{3} ) q^{79} \) \( + ( 1435313770082721 + 11182148496 \beta_{1} + 1945822014 \beta_{2} + 12056139306 \beta_{3} ) q^{81} \) \( + ( -171905078796 \beta_{1} - 2341342740 \beta_{2} - 15105870813 \beta_{3} ) q^{83} \) \( + ( -3640234033846080 - 298944660960 \beta_{1} + 49824110160 \beta_{3} ) q^{85} \) \( + ( 2501511669878760 - 241784506656 \beta_{1} - 2571835860 \beta_{2} + 30392135037 \beta_{3} ) q^{87} \) \( + ( 765134253824 \beta_{1} + 4970953426 \beta_{2} + 65418172294 \beta_{3} ) q^{89} \) \( + ( -2860638443146484 - 14625030420 \beta_{1} + 2437505070 \beta_{3} ) q^{91} \) \( + ( -151399958177070 + 552337324096 \beta_{1} - 1882624275 \beta_{2} + 45810524025 \beta_{3} ) q^{93} \) \( + ( 30784362488 \beta_{1} + 8558116336 \beta_{2} + 5418068986 \beta_{3} ) q^{95} \) \( + ( -7962992039410510 - 149161958328 \beta_{1} + 24860326388 \beta_{3} ) q^{97} \) \( + ( 2214600258602160 - 188177303628 \beta_{1} - 9755600472 \beta_{2} - 72908835723 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 12420q^{3} \) \(\mathstrut -\mathstrut 1050280q^{7} \) \(\mathstrut -\mathstrut 10121436q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 12420q^{3} \) \(\mathstrut -\mathstrut 1050280q^{7} \) \(\mathstrut -\mathstrut 10121436q^{9} \) \(\mathstrut -\mathstrut 274863160q^{13} \) \(\mathstrut -\mathstrut 1065273120q^{15} \) \(\mathstrut -\mathstrut 41348739784q^{19} \) \(\mathstrut -\mathstrut 55600393464q^{21} \) \(\mathstrut -\mathstrut 525638756540q^{25} \) \(\mathstrut -\mathstrut 457426227780q^{27} \) \(\mathstrut -\mathstrut 1712336487784q^{31} \) \(\mathstrut -\mathstrut 953298720q^{33} \) \(\mathstrut +\mathstrut 5258670683720q^{37} \) \(\mathstrut +\mathstrut 9161332213176q^{39} \) \(\mathstrut +\mathstrut 18970169729720q^{43} \) \(\mathstrut +\mathstrut 30786788111040q^{45} \) \(\mathstrut -\mathstrut 51073893429300q^{49} \) \(\mathstrut +\mathstrut 65388537498240q^{51} \) \(\mathstrut -\mathstrut 328085114005440q^{55} \) \(\mathstrut +\mathstrut 103623957688680q^{57} \) \(\mathstrut -\mathstrut 984197053056184q^{61} \) \(\mathstrut +\mathstrut 733717575221400q^{63} \) \(\mathstrut -\mathstrut 798011457001480q^{67} \) \(\mathstrut +\mathstrut 2092913995855680q^{69} \) \(\mathstrut -\mathstrut 2128148319514360q^{73} \) \(\mathstrut +\mathstrut 5201271725969340q^{75} \) \(\mathstrut -\mathstrut 1949813936692072q^{79} \) \(\mathstrut +\mathstrut 5741255080330884q^{81} \) \(\mathstrut -\mathstrut 14560936135384320q^{85} \) \(\mathstrut +\mathstrut 10006046679515040q^{87} \) \(\mathstrut -\mathstrut 11442553772585936q^{91} \) \(\mathstrut -\mathstrut 605599832708280q^{93} \) \(\mathstrut -\mathstrut 31851968157642040q^{97} \) \(\mathstrut +\mathstrut 8858401034408640q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(15630\) \(x^{2}\mathstrut +\mathstrut \) \(12922000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 54 \nu^{2} + 6210 \nu + 422010 \)\()/115\)
\(\beta_{2}\)\(=\)\((\)\( 216 \nu^{3} + 216 \nu^{2} + 3004560 \nu + 1688040 \)\()/115\)
\(\beta_{3}\)\(=\)\((\)\( -648 \nu^{2} + 37260 \nu - 5064120 \)\()/115\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\)\()/972\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(115\) \(\beta_{3}\mathstrut +\mathstrut \) \(690\) \(\beta_{1}\mathstrut -\mathstrut \) \(7596180\)\()/972\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(27590\) \(\beta_{3}\mathstrut +\mathstrut \) \(1035\) \(\beta_{2}\mathstrut -\mathstrut \) \(335220\) \(\beta_{1}\)\()/1944\)

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
29.5942i
29.5942i
121.467i
121.467i
0 −6363.40 1598.09i 0 746888.i 0 4.25357e6 0 3.79389e7 + 2.03386e7i 0
5.2 0 −6363.40 + 1598.09i 0 746888.i 0 4.25357e6 0 3.79389e7 2.03386e7i 0
5.3 0 153.398 6559.21i 0 100768.i 0 −4.77871e6 0 −4.29997e7 2.01234e6i 0
5.4 0 153.398 + 6559.21i 0 100768.i 0 −4.77871e6 0 −4.29997e7 + 2.01234e6i 0
\(n\): e.g. 2-40 or 990-1000
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 \(T_{5}^{4} \) \(\mathstrut +\mathstrut 567995159520 T_{5}^{2} \) \(\mathstrut +\mathstrut \)\(56\!\cdots\!00\)\( \) acting on \(S_{17}^{\mathrm{new}}(12, [\chi])\).