Properties

Label 12.15.c.b
Level 12
Weight 15
Character orbit 12.c
Analytic conductor 14.919
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 15 \)
Character orbit: \([\chi]\) = 12.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(14.9194761782\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{9}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( 537 - \beta_{1} ) q^{3} \) \( + ( \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 427322 + 250 \beta_{1} - 5 \beta_{3} ) q^{7} \) \( + ( -1434231 - 759 \beta_{1} + 45 \beta_{2} + 12 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 537 - \beta_{1} ) q^{3} \) \( + ( \beta_{1} + \beta_{2} ) q^{5} \) \( + ( 427322 + 250 \beta_{1} - 5 \beta_{3} ) q^{7} \) \( + ( -1434231 - 759 \beta_{1} + 45 \beta_{2} + 12 \beta_{3} ) q^{9} \) \( + ( -3284 \beta_{1} - 260 \beta_{2} - 27 \beta_{3} ) q^{11} \) \( + ( -23376262 + 2600 \beta_{1} - 52 \beta_{3} ) q^{13} \) \( + ( 6058800 + 432 \beta_{1} + 972 \beta_{2} - 1053 \beta_{3} ) q^{15} \) \( + ( -228544 \beta_{1} + 1280 \beta_{2} - 2052 \beta_{3} ) q^{17} \) \( + ( -337300174 + 292550 \beta_{1} - 5851 \beta_{3} ) q^{19} \) \( + ( -893188086 - 554897 \beta_{1} - 18225 \beta_{2} - 4860 \beta_{3} ) q^{21} \) \( + ( -1553872 \beta_{1} + 12560 \beta_{2} - 13986 \beta_{3} ) q^{23} \) \( + ( -1010941175 + 617400 \beta_{1} - 12348 \beta_{3} ) q^{25} \) \( + ( 1845437553 + 1714563 \beta_{1} + 96660 \beta_{2} - 33273 \beta_{3} ) q^{27} \) \( + ( 3303547 \beta_{1} - 107525 \beta_{2} + 30456 \beta_{3} ) q^{29} \) \( + ( 11646303434 - 5867550 \beta_{1} + 117351 \beta_{3} ) q^{31} \) \( + ( -15172304400 - 1772253 \beta_{1} - 154305 \beta_{2} + 300024 \beta_{3} ) q^{33} \) \( + ( 29861372 \beta_{1} + 74972 \beta_{2} + 265950 \beta_{3} ) q^{35} \) \( + ( 75218304266 - 2071400 \beta_{1} + 41428 \beta_{3} ) q^{37} \) \( + ( -24228716694 + 22049482 \beta_{1} - 189540 \beta_{2} - 50544 \beta_{3} ) q^{39} \) \( + ( -16233242 \beta_{1} + 1813990 \beta_{2} - 161136 \beta_{3} ) q^{41} \) \( + ( 61612189538 - 22786650 \beta_{1} + 455733 \beta_{3} ) q^{43} \) \( + ( -318914560800 - 44314371 \beta_{1} - 499851 \beta_{2} - 1408752 \beta_{3} ) q^{45} \) \( + ( -161411208 \beta_{1} - 7634760 \beta_{2} - 1373004 \beta_{3} ) q^{47} \) \( + ( 413735618835 + 213661000 \beta_{1} - 4273220 \beta_{3} ) q^{49} \) \( + ( -1025617982400 - 125601948 \beta_{1} + 8723700 \beta_{2} + 646704 \beta_{3} ) q^{51} \) \( + ( -351762443 \beta_{1} + 10246645 \beta_{2} - 3232224 \beta_{3} ) q^{53} \) \( + ( 1876259959200 - 2039400 \beta_{1} + 40788 \beta_{3} ) q^{55} \) \( + ( -1494866925438 + 188011909 \beta_{1} - 21326895 \beta_{2} - 5687172 \beta_{3} ) q^{57} \) \( + ( 1202845376 \beta_{1} + 5707280 \beta_{2} + 10688733 \beta_{3} ) q^{59} \) \( + ( 2002083066074 - 914363400 \beta_{1} + 18287268 \beta_{3} ) q^{61} \) \( + ( -3024352139382 + 588224802 \beta_{1} - 344160 \beta_{2} + 23823069 \beta_{3} ) q^{63} \) \( + ( 282737858 \beta_{1} - 27040702 \beta_{2} + 2765880 \beta_{3} ) q^{65} \) \( + ( 9573727053362 - 159760050 \beta_{1} + 3195201 \beta_{3} ) q^{67} \) \( + ( -6967155967200 - 854419374 \beta_{1} + 63187290 \beta_{2} + 368712 \beta_{3} ) q^{69} \) \( + ( 2567704424 \beta_{1} + 34541960 \beta_{2} + 22617522 \beta_{3} ) q^{71} \) \( + ( 3180351673394 + 2954344800 \beta_{1} - 59086896 \beta_{3} ) q^{73} \) \( + ( -3315396546975 + 695881955 \beta_{1} - 45008460 \beta_{2} - 12002256 \beta_{3} ) q^{75} \) \( + ( -10201400998 \beta_{1} - 99208870 \beta_{2} - 90198144 \beta_{3} ) q^{77} \) \( + ( -7450850832982 - 5438852350 \beta_{1} + 108777047 \beta_{3} ) q^{79} \) \( + ( -5973013444239 - 2662796862 \beta_{1} - 25267950 \beta_{2} - 133575372 \beta_{3} ) q^{81} \) \( + ( 1438567356 \beta_{1} + 111911340 \beta_{2} + 11845143 \beta_{3} ) q^{83} \) \( + ( -7092414172800 + 12835101600 \beta_{1} - 256702032 \beta_{3} ) q^{85} \) \( + ( 14685962029200 + 1825953624 \beta_{1} - 215526420 \beta_{2} + 83620593 \beta_{3} ) q^{87} \) \( + ( -8182407202 \beta_{1} + 404869790 \beta_{2} - 76672116 \beta_{3} ) q^{89} \) \( + ( -531903190364 - 4733028300 \beta_{1} + 94660566 \beta_{3} ) q^{91} \) \( + ( 32603119676058 - 8652092669 \beta_{1} + 427744395 \beta_{2} + 114065172 \beta_{3} ) q^{93} \) \( + ( 34106425136 \beta_{1} - 749620144 \beta_{2} + 311214690 \beta_{3} ) q^{95} \) \( + ( -75406404782878 - 19416952600 \beta_{1} + 388339052 \beta_{3} ) q^{97} \) \( + ( 81336548066400 + 25731838944 \beta_{1} + 341356680 \beta_{2} + 293507469 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2148q^{3} \) \(\mathstrut +\mathstrut 1709288q^{7} \) \(\mathstrut -\mathstrut 5736924q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2148q^{3} \) \(\mathstrut +\mathstrut 1709288q^{7} \) \(\mathstrut -\mathstrut 5736924q^{9} \) \(\mathstrut -\mathstrut 93505048q^{13} \) \(\mathstrut +\mathstrut 24235200q^{15} \) \(\mathstrut -\mathstrut 1349200696q^{19} \) \(\mathstrut -\mathstrut 3572752344q^{21} \) \(\mathstrut -\mathstrut 4043764700q^{25} \) \(\mathstrut +\mathstrut 7381750212q^{27} \) \(\mathstrut +\mathstrut 46585213736q^{31} \) \(\mathstrut -\mathstrut 60689217600q^{33} \) \(\mathstrut +\mathstrut 300873217064q^{37} \) \(\mathstrut -\mathstrut 96914866776q^{39} \) \(\mathstrut +\mathstrut 246448758152q^{43} \) \(\mathstrut -\mathstrut 1275658243200q^{45} \) \(\mathstrut +\mathstrut 1654942475340q^{49} \) \(\mathstrut -\mathstrut 4102471929600q^{51} \) \(\mathstrut +\mathstrut 7505039836800q^{55} \) \(\mathstrut -\mathstrut 5979467701752q^{57} \) \(\mathstrut +\mathstrut 8008332264296q^{61} \) \(\mathstrut -\mathstrut 12097408557528q^{63} \) \(\mathstrut +\mathstrut 38294908213448q^{67} \) \(\mathstrut -\mathstrut 27868623868800q^{69} \) \(\mathstrut +\mathstrut 12721406693576q^{73} \) \(\mathstrut -\mathstrut 13261586187900q^{75} \) \(\mathstrut -\mathstrut 29803403331928q^{79} \) \(\mathstrut -\mathstrut 23892053776956q^{81} \) \(\mathstrut -\mathstrut 28369656691200q^{85} \) \(\mathstrut +\mathstrut 58743848116800q^{87} \) \(\mathstrut -\mathstrut 2127612761456q^{91} \) \(\mathstrut +\mathstrut 130412478704232q^{93} \) \(\mathstrut -\mathstrut 301625619131512q^{97} \) \(\mathstrut +\mathstrut 325346192265600q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(440\) \(x^{2}\mathstrut +\mathstrut \) \(48015\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu^{3} + 60 \nu^{2} + 1350 \nu + 13200 \)
\(\beta_{2}\)\(=\)\( -42 \nu^{3} - 60 \nu^{2} - 3690 \nu - 13200 \)
\(\beta_{3}\)\(=\)\( 300 \nu^{3} - 6720 \nu^{2} + 67500 \nu - 1478400 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(27\) \(\beta_{2}\mathstrut +\mathstrut \) \(139\) \(\beta_{1}\)\()/155520\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(50\) \(\beta_{1}\mathstrut -\mathstrut \) \(2138400\)\()/9720\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(1215\) \(\beta_{2}\mathstrut -\mathstrut \) \(2671\) \(\beta_{1}\)\()/31104\)

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
14.1555i
14.1555i
15.4797i
15.4797i
0 −640.285 2091.17i 0 68988.7i 0 1.38092e6 0 −3.96304e6 + 2.67789e6i 0
5.2 0 −640.285 + 2091.17i 0 68988.7i 0 1.38092e6 0 −3.96304e6 2.67789e6i 0
5.3 0 1714.29 1358.01i 0 97311.2i 0 −526279. 0 1.09458e6 4.65604e6i 0
5.4 0 1714.29 + 1358.01i 0 97311.2i 0 −526279. 0 1.09458e6 + 4.65604e6i 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 \(T_{5}^{4} \) \(\mathstrut +\mathstrut 14228913600 T_{5}^{2} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!00\)\( \) acting on \(S_{15}^{\mathrm{new}}(12, [\chi])\).