Properties

Label 384.8.d.e
Level $384$
Weight $8$
Character orbit 384.d
Analytic conductor $119.956$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00 \) 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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} - \beta_{8} q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} - \beta_{8} q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} - 729 q^{9} + (\beta_{5} + \beta_{4} + 10 \beta_1) q^{11} + (\beta_{11} + 12 \beta_{9}) q^{13} + (3 \beta_{10} - 6 \beta_{7} - 9 \beta_{6}) q^{15} + ( - \beta_{3} + 2 \beta_{2} - 6924) q^{17} + ( - 10 \beta_{5} + \beta_{4} + 27 \beta_1) q^{19} + (6 \beta_{11} - 21 \beta_{9} + 15 \beta_{8}) q^{21} + (20 \beta_{10} + 62 \beta_{7} - 4 \beta_{6}) q^{23} + ( - 5 \beta_{3} - 9 \beta_{2} - 10654) q^{25} - 2187 \beta_1 q^{27} + (19 \beta_{11} - 131 \beta_{9} - 183 \beta_{8}) q^{29} + ( - 52 \beta_{10} - 103 \beta_{7} - 318 \beta_{6}) q^{31} + ( - 6 \beta_{3} - 9 \beta_{2} - 2559) q^{33} + ( - 5 \beta_{5} + 25 \beta_{4} - 7000 \beta_1) q^{35} + ( - 65 \beta_{11} + 442 \beta_{9} + 310 \beta_{8}) q^{37} + ( - 39 \beta_{10} - 219 \beta_{7} - 423 \beta_{6}) q^{39} + ( - 47 \beta_{3} - 28 \beta_{2} - 80762) q^{41} + ( - 108 \beta_{5} - 3 \beta_{4} + 23677 \beta_1) q^{43} + 729 \beta_{8} q^{45} + ( - 120 \beta_{10} + 618 \beta_{7} - 1726 \beta_{6}) q^{47} + ( - 31 \beta_{3} - 9 \beta_{2} - 423922) q^{49} + (162 \beta_{5} - 27 \beta_{4} - 20823 \beta_1) q^{51} + ( - 333 \beta_{11} - 855 \beta_{9} - 2285 \beta_{8}) q^{53} + (460 \beta_{10} - 5075 \beta_{6}) q^{55} + ( - 39 \beta_{3} + 90 \beta_{2} - 5766) q^{57} + (458 \beta_{5} + 116 \beta_{4} + 26546 \beta_1) q^{59} + (111 \beta_{11} - 2056 \beta_{9} + 2690 \beta_{8}) q^{61} + (729 \beta_{7} - 729 \beta_{6}) q^{63} + (25 \beta_{3} + 16 \beta_{2} - 20064) q^{65} + ( - 214 \beta_{5} - 584 \beta_{4} - 11338 \beta_1) q^{67} + ( - 138 \beta_{11} + 2616 \beta_{9} + 2382 \beta_{8}) q^{69} + (484 \beta_{10} - 4598 \beta_{7} - 13100 \beta_{6}) q^{71} + (377 \beta_{3} - 225 \beta_{2} + 1506563) q^{73} + ( - 729 \beta_{5} - 648 \beta_{4} - 31590 \beta_1) q^{75} + ( - 313 \beta_{11} + 2123 \beta_{9} - 1530 \beta_{8}) q^{77} + ( - 1532 \beta_{10} + 2545 \beta_{7} - 6561 \beta_{6}) q^{79} + 531441 q^{81} + (229 \beta_{5} + 1819 \beta_{4} + 189624 \beta_1) q^{83} + ( - 1100 \beta_{11} + 380 \beta_{9} - 2086 \beta_{8}) q^{85} + (885 \beta_{10} + 2280 \beta_{7} - 3222 \beta_{6}) q^{87} + (226 \beta_{3} + 804 \beta_{2} + 3630530) q^{89} + (1106 \beta_{5} + 301 \beta_{4} - 331465 \beta_1) q^{91} + ( - 801 \beta_{11} - 6390 \beta_{9} - 5445 \beta_{8}) q^{93} + ( - 1932 \beta_{10} + 5664 \beta_{7} - 5324 \beta_{6}) q^{95} + (700 \beta_{3} + 162 \beta_{2} - 4038140) q^{97} + ( - 729 \beta_{5} - 729 \beta_{4} - 7290 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 123191 \nu^{10} + 856517735 \nu^{8} - 2139973212207 \nu^{6} + \cdots + 19\!\cdots\!00 ) / 26\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 196637216072 \nu^{10} + \cdots - 12\!\cdots\!75 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 227084275736 \nu^{10} + \cdots - 27\!\cdots\!00 ) / 34\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 56\!\cdots\!27 \nu^{10} + \cdots + 48\!\cdots\!00 ) / 86\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!81 \nu^{10} + \cdots + 14\!\cdots\!00 ) / 86\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1574111264 \nu^{11} - 3149091705952 \nu^{9} + \cdots + 31\!\cdots\!08 \nu ) / 19\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!89 \nu^{11} + \cdots - 16\!\cdots\!82 \nu ) / 16\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46\!\cdots\!28 \nu^{11} + \cdots - 14\!\cdots\!94 \nu ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!18 \nu^{11} + \cdots - 18\!\cdots\!24 \nu ) / 28\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 71\!\cdots\!79 \nu^{11} + \cdots + 42\!\cdots\!82 \nu ) / 16\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64\!\cdots\!54 \nu^{11} + \cdots + 85\!\cdots\!82 \nu ) / 19\!\cdots\!25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} - 16\beta_{10} - \beta_{9} + 2\beta_{8} - 112\beta_{7} + 48\beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -18\beta_{5} - 5\beta_{3} - 111\beta_{2} - 122\beta _1 + 1464539 ) / 1152 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6917\beta_{11} - 58144\beta_{10} - 19877\beta_{9} + 1738\beta_{8} - 419104\beta_{7} - 535263\beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -66582\beta_{5} + 756\beta_{4} + 28355\beta_{3} - 325347\beta_{2} + 9720870\beta _1 + 2696760031 ) / 1152 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18325473 \beta_{11} - 143924608 \beta_{10} - 86840673 \beta_{9} - 1834494 \beta_{8} - 877742656 \beta_{7} - 1870249011 \beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 224962497 \beta_{5} - 12158370 \beta_{4} + 126727450 \beta_{3} - 822635634 \beta_{2} + 46369720457 \beta _1 + 5869887545306 ) / 1152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 50475122283 \beta_{11} - 360261281216 \beta_{10} - 302582978283 \beta_{9} + 1571254422 \beta_{8} - 1982935947776 \beta_{7} - 5022226211907 \beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 237215237616 \beta_{5} - 22491699552 \beta_{4} + 123733724075 \beta_{3} - 672315701451 \beta_{2} + 54069123344880 \beta _1 + 45\!\cdots\!43 ) / 384 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 143194143041177 \beta_{11} - 886669336602752 \beta_{10} - 952500999343577 \beta_{9} + 25032138200434 \beta_{8} + \cdots - 12\!\cdots\!99 \beta_{6} ) / 4608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21\!\cdots\!13 \beta_{5} - 243565265769210 \beta_{4} + 964139649084280 \beta_{3} + \cdots + 32\!\cdots\!24 ) / 1152 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 40\!\cdots\!17 \beta_{11} + \cdots - 30\!\cdots\!73 \beta_{6} ) / 4608 \) 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
19.8448 + 0.707107i
−31.5402 0.707107i
−49.2636 + 0.707107i
49.2636 0.707107i
31.5402 + 0.707107i
−19.8448 0.707107i
−19.8448 + 0.707107i
31.5402 0.707107i
49.2636 + 0.707107i
−49.2636 0.707107i
−31.5402 + 0.707107i
19.8448 0.707107i
0 27.0000i 0 467.267i 0 31.2683 0 −729.000 0
193.2 0 27.0000i 0 218.879i 0 −904.541 0 −729.000 0
193.3 0 27.0000i 0 8.99895i 0 −616.197 0 −729.000 0
193.4 0 27.0000i 0 8.99895i 0 616.197 0 −729.000 0
193.5 0 27.0000i 0 218.879i 0 904.541 0 −729.000 0
193.6 0 27.0000i 0 467.267i 0 −31.2683 0 −729.000 0
193.7 0 27.0000i 0 467.267i 0 −31.2683 0 −729.000 0
193.8 0 27.0000i 0 218.879i 0 904.541 0 −729.000 0
193.9 0 27.0000i 0 8.99895i 0 616.197 0 −729.000 0
193.10 0 27.0000i 0 8.99895i 0 −616.197 0 −729.000 0
193.11 0 27.0000i 0 218.879i 0 −904.541 0 −729.000 0
193.12 0 27.0000i 0 467.267i 0 31.2683 0 −729.000 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 384.8.d.e 12
4.b odd 2 1 inner 384.8.d.e 12
8.b even 2 1 inner 384.8.d.e 12
8.d odd 2 1 inner 384.8.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.d.e 12 1.a even 1 1 trivial
384.8.d.e 12 4.b odd 2 1 inner
384.8.d.e 12 8.b even 2 1 inner
384.8.d.e 12 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{6} + 266328T_{5}^{4} + 10481784000T_{5}^{2} + 847081280000 \) Copy content Toggle raw display
\( T_{7}^{6} - 1198872T_{7}^{4} + 311839224000T_{7}^{2} - 303742233920000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 266328 T^{4} + \cdots + 847081280000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 1198872 T^{4} + \cdots - 303742233920000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 40887216 T^{4} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 153501024 T^{4} + \cdots + 21\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 20774 T^{2} + \cdots + 9568373896)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 3438581424 T^{4} + \cdots + 43\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 12104042592 T^{4} + \cdots - 16\!\cdots\!52)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 29283126744 T^{4} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 72678563928 T^{4} + \cdots - 93\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 267723127296 T^{4} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 242258 T^{2} + \cdots + 25\!\cdots\!80)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 491608617264 T^{4} + \cdots + 62\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 825788238432 T^{4} + \cdots - 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 2907470155608 T^{4} + \cdots + 89\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 5758987678128 T^{4} + \cdots + 52\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 6876216682368 T^{4} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 10722197211312 T^{4} + \cdots + 72\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 49555481902176 T^{4} + \cdots - 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 4519914 T^{2} + \cdots + 96\!\cdots\!96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 45949368823128 T^{4} + \cdots - 21\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 122266303705776 T^{4} + \cdots + 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 10890786 T^{2} + \cdots + 21\!\cdots\!96)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + 12114582 T^{2} + \cdots - 19\!\cdots\!08)^{4} \) Copy content Toggle raw display
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