Properties

Label 384.6.d.j
Level $384$
Weight $6$
Character orbit 384.d
Analytic conductor $61.587$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{8} q^{5} + \beta_{4} q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{8} q^{5} + \beta_{4} q^{7} - 81 q^{9} + ( - \beta_{11} + \beta_{6}) q^{11} + ( - 5 \beta_{8} - \beta_{7}) q^{13} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{15} + (\beta_{2} - 407) q^{17} + ( - 3 \beta_{11} + \beta_{10} + 14 \beta_{6}) q^{19} + ( - \beta_{9} + 7 \beta_{8} + 4 \beta_{7}) q^{21} + ( - 2 \beta_{4} + 4 \beta_{3} + 6 \beta_1) q^{23} + (\beta_{5} + 2 \beta_{2} - 1554) q^{25} - 81 \beta_{6} q^{27} + (5 \beta_{9} - 4 \beta_{8} - 5 \beta_{7}) q^{29} + ( - 13 \beta_{4} - 4 \beta_{3} + 25 \beta_1) q^{31} + ( - \beta_{5} + \beta_{2} - 60) q^{33} + (11 \beta_{11} - 6 \beta_{10} + 303 \beta_{6}) q^{35} + ( - 2 \beta_{9} - \beta_{8} + 29 \beta_{7}) q^{37} + (14 \beta_{4} + 5 \beta_{3} + 4 \beta_1) q^{39} + ( - 2 \beta_{5} - 3 \beta_{2} - 2093) q^{41} + (21 \beta_{11} + 3 \beta_{10} - 452 \beta_{6}) q^{43} - 81 \beta_{8} q^{45} + (30 \beta_{4} - 24 \beta_{3} + 52 \beta_1) q^{47} + (5 \beta_{5} - 8 \beta_{2} + 11296) q^{49} + (9 \beta_{11} + 9 \beta_{10} - 412 \beta_{6}) q^{51} + (9 \beta_{9} - 94 \beta_{8} + 27 \beta_{7}) q^{53} + (68 \beta_{4} + 28 \beta_{3} + 23 \beta_1) q^{55} + ( - 4 \beta_{5} - 5 \beta_{2} - 1095) q^{57} + ( - 32 \beta_{11} - 18 \beta_{10} + 1214 \beta_{6}) q^{59} + (20 \beta_{9} + 35 \beta_{8} + 57 \beta_{7}) q^{61} - 81 \beta_{4} q^{63} + ( - 6 \beta_{5} - 7 \beta_{2} + 21185) q^{65} + ( - 60 \beta_{11} + 10 \beta_{10} - 386 \beta_{6}) q^{67} + (16 \beta_{9} + 212 \beta_{8} + 26 \beta_{7}) q^{69} + ( - 190 \beta_{4} + 20 \beta_{3} - 78 \beta_1) q^{71} + (5 \beta_{5} - 8 \beta_{2} - 15587) q^{73} + ( - 54 \beta_{11} + 27 \beta_{10} - 1548 \beta_{6}) q^{75} + ( - 5 \beta_{9} - 245 \beta_{8} + 347 \beta_{7}) q^{77} + ( - 173 \beta_{4} - 44 \beta_{3} - 40 \beta_1) q^{79} + 6561 q^{81} + ( - 7 \beta_{11} - 30 \beta_{10} + 5621 \beta_{6}) q^{83} + (92 \beta_{9} - 1326 \beta_{8} + 116 \beta_{7}) q^{85} + (239 \beta_{4} - 31 \beta_{3} - 104 \beta_1) q^{87} + (8 \beta_{5} + 22 \beta_{2} - 47952) q^{89} + (57 \beta_{11} + 25 \beta_{10} - 4874 \beta_{6}) q^{91} + (30 \beta_{9} - 534 \beta_{8} + 69 \beta_{7}) q^{93} + (636 \beta_{4} + 132 \beta_{3} - 412 \beta_1) q^{95} + ( - 14 \beta_{5} + 62 \beta_{2} + 7838) q^{97} + (81 \beta_{11} - 81 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 501856 \nu^{10} - 128371584 \nu^{8} - 9748135264 \nu^{6} - 138948034496 \nu^{4} + 6898201181664 \nu^{2} + \cdots + 100166634492736 ) / 228958809975 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 978928 \nu^{10} - 55789152 \nu^{8} + 25346631296 \nu^{6} + 2989391325712 \nu^{4} + 103137790028736 \nu^{2} + \cdots + 899529825156571 ) / 125454890799 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78708328 \nu^{10} - 20962386192 \nu^{8} - 1968575165032 \nu^{6} - 81171924503648 \nu^{4} + \cdots + 67215280640368 ) / 9409116809925 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8166241501 \nu^{10} - 2267974529364 \nu^{8} - 221787567202369 \nu^{6} + \cdots - 78\!\cdots\!94 ) / 395182906016850 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 144829696 \nu^{10} - 34237610880 \nu^{8} - 2552278683136 \nu^{6} - 66770917381760 \nu^{4} - 538239146767872 \nu^{2} + \cdots - 26\!\cdots\!23 ) / 878184235593 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1551427 \nu^{11} + 374874240 \nu^{9} + 28047726409 \nu^{7} + 593227138388 \nu^{5} - 6428215193541 \nu^{3} + \cdots - 158055405918862 \nu ) / 25847648495928 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 153683077859 \nu^{11} + 32357600561226 \nu^{9} + \cdots - 19\!\cdots\!54 \nu ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 163026402997 \nu^{11} - 46810444439958 \nu^{9} + \cdots - 27\!\cdots\!18 \nu ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 126051978287 \nu^{11} + 34430061527418 \nu^{9} + \cdots - 70\!\cdots\!22 \nu ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 707802624817 \nu^{11} - 149633466506112 \nu^{9} + \cdots - 22\!\cdots\!82 \nu ) / 31\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1650016024369 \nu^{11} - 481063311351552 \nu^{9} + \cdots - 27\!\cdots\!14 \nu ) / 31\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -9\beta_{10} + 9\beta_{9} + 9\beta_{8} - 27\beta_{7} - 83\beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 64\beta_{4} + 152\beta_{3} - 8\beta_{2} + 127\beta _1 - 129795 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 216\beta_{11} + 1368\beta_{10} - 2199\beta_{9} - 2175\beta_{8} + 1389\beta_{7} + 69904\beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -251\beta_{5} + 10368\beta_{4} - 17928\beta_{3} + 2528\beta_{2} - 19440\beta _1 + 10420215 ) / 2304 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 77112 \beta_{11} - 132066 \beta_{10} + 232027 \beta_{9} + 527603 \beta_{8} + 79127 \beta_{7} - 10638206 \beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 53528\beta_{5} - 1370048\beta_{4} + 1952464\beta_{3} - 399704\beta_{2} + 2860439\beta _1 - 970240704 ) / 2304 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6980904 \beta_{11} + 6871167 \beta_{10} - 12541321 \beta_{9} - 46005689 \beta_{8} - 11632181 \beta_{7} + 680775997 \beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2511861 \beta_{5} + 56030336 \beta_{4} - 71582536 \beta_{3} + 17976768 \beta_{2} - 130050800 \beta _1 + 32688443289 ) / 768 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2051442504 \beta_{11} - 1486228572 \beta_{10} + 2790312729 \beta_{9} + 13379824785 \beta_{8} + 3595812861 \beta_{7} - 166302107596 \beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 943566077 \beta_{5} - 20134847104 \beta_{4} + 24059081432 \beta_{3} - 6840998480 \beta_{2} + 50313706612 \beta _1 - 10447619634033 ) / 2304 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 273663797400 \beta_{11} + 165010354002 \beta_{10} - 316766850363 \beta_{9} - 1778470066323 \beta_{8} - 478276862391 \beta_{7} + 19986140244334 \beta_{6} ) / 4608 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
4.65977i
2.80773i
6.60695i
8.37346i
10.8525i
8.76697i
8.76697i
10.8525i
8.37346i
6.60695i
2.80773i
4.65977i
0 9.00000i 0 107.066i 0 150.154 0 −81.0000 0
193.2 0 9.00000i 0 46.1613i 0 −59.9278 0 −81.0000 0
193.3 0 9.00000i 0 21.1198i 0 −241.194 0 −81.0000 0
193.4 0 9.00000i 0 21.1198i 0 241.194 0 −81.0000 0
193.5 0 9.00000i 0 46.1613i 0 59.9278 0 −81.0000 0
193.6 0 9.00000i 0 107.066i 0 −150.154 0 −81.0000 0
193.7 0 9.00000i 0 107.066i 0 −150.154 0 −81.0000 0
193.8 0 9.00000i 0 46.1613i 0 59.9278 0 −81.0000 0
193.9 0 9.00000i 0 21.1198i 0 241.194 0 −81.0000 0
193.10 0 9.00000i 0 21.1198i 0 −241.194 0 −81.0000 0
193.11 0 9.00000i 0 46.1613i 0 −59.9278 0 −81.0000 0
193.12 0 9.00000i 0 107.066i 0 150.154 0 −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.d.j 12
4.b odd 2 1 inner 384.6.d.j 12
8.b even 2 1 inner 384.6.d.j 12
8.d odd 2 1 inner 384.6.d.j 12
16.e even 4 1 768.6.a.be 6
16.e even 4 1 768.6.a.bf 6
16.f odd 4 1 768.6.a.be 6
16.f odd 4 1 768.6.a.bf 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.d.j 12 1.a even 1 1 trivial
384.6.d.j 12 4.b odd 2 1 inner
384.6.d.j 12 8.b even 2 1 inner
384.6.d.j 12 8.d odd 2 1 inner
768.6.a.be 6 16.e even 4 1
768.6.a.be 6 16.f odd 4 1
768.6.a.bf 6 16.e even 4 1
768.6.a.bf 6 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{6} + 14040T_{5}^{4} + 30489792T_{5}^{2} + 10895241728 \) Copy content Toggle raw display
\( T_{7}^{6} - 84312T_{7}^{4} + 1601504448T_{7}^{2} - 4710436295168 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 14040 T^{4} + \cdots + 10895241728)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 84312 T^{4} + \cdots - 4710436295168)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 621360 T^{4} + \cdots + 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 478560 T^{4} + \cdots + 31\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 1222 T^{2} + \cdots - 2972835064)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 10626864 T^{4} + \cdots + 83\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 16145760 T^{4} + \cdots - 30\!\cdots\!28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 77332440 T^{4} + \cdots + 14\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 94022424 T^{4} + \cdots - 33\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 183716352 T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 6274 T^{2} + \cdots - 560226749800)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 398631600 T^{4} + \cdots + 55\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 982944096 T^{4} + \cdots - 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 485370072 T^{4} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 3145023792 T^{4} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 1528967040 T^{4} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 2669402928 T^{4} + \cdots + 20\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 4329842016 T^{4} + \cdots - 14\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 46758 T^{2} + \cdots - 12280634576248)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 4272026904 T^{4} + \cdots - 20\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 13191609648 T^{4} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 143886 T^{2} + \cdots + 28858879919016)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} - 23466 T^{2} + \cdots - 592920131000504)^{4} \) Copy content Toggle raw display
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