Properties

Label 384.10.d.d
Level $384$
Weight $10$
Character orbit 384.d
Analytic conductor $197.774$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 81 \beta_{5} q^{3} + (\beta_{6} + 112 \beta_{5}) q^{5} + ( - \beta_{2} + \beta_1 + 558) q^{7} - 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 \beta_{5} q^{3} + (\beta_{6} + 112 \beta_{5}) q^{5} + ( - \beta_{2} + \beta_1 + 558) q^{7} - 6561 q^{9} + (\beta_{8} - 4 \beta_{7} - 11 \beta_{6} - 4503 \beta_{5}) q^{11} + ( - 4 \beta_{8} - \beta_{7} - 24 \beta_{6} + 5806 \beta_{5}) q^{13} + ( - 81 \beta_{2} - 9072) q^{15} + ( - \beta_{4} + 3 \beta_{3} - 93 \beta_{2} - 5 \beta_1 + 19568) q^{17} + (\beta_{9} + 15 \beta_{8} + 11 \beta_{7} + 31 \beta_{6} - 50574 \beta_{5}) q^{19} + (81 \beta_{7} - 81 \beta_{6} + 45198 \beta_{5}) q^{21} + ( - 4 \beta_{4} - 40 \beta_{3} + 82 \beta_{2} - 34 \beta_1 - 13536) q^{23} + (5 \beta_{4} + 50 \beta_{3} - 826 \beta_{2} + 21 \beta_1 - 131696) q^{25} - 531441 \beta_{5} q^{27} + ( - 4 \beta_{9} - 20 \beta_{8} + 342 \beta_{7} - 15 \beta_{6} + 7484 \beta_{5}) q^{29} + (16 \beta_{4} - 44 \beta_{3} + 1249 \beta_{2} - \beta_1 - 1488762) q^{31} + ( - 81 \beta_{3} + 891 \beta_{2} + 324 \beta_1 + 364743) q^{33} + ( - 14 \beta_{9} + 149 \beta_{8} + 330 \beta_{7} + \cdots - 2115873 \beta_{5}) q^{35}+ \cdots + ( - 6561 \beta_{8} + 26244 \beta_{7} + 72171 \beta_{6} + 29544183 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5576 q^{7} - 65610 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5576 q^{7} - 65610 q^{9} - 90720 q^{15} + 195692 q^{17} - 135152 q^{23} - 1317134 q^{25} - 14887496 q^{31} + 3646296 q^{33} - 4703832 q^{39} + 32469956 q^{41} - 28692880 q^{47} + 80865242 q^{49} + 223324800 q^{55} + 40971096 q^{57} - 36584136 q^{63} + 511472000 q^{65} + 722817008 q^{71} + 642721212 q^{73} - 114892616 q^{79} + 430467210 q^{81} - 5955120 q^{87} + 1709981116 q^{89} - 657134976 q^{95} + 3919129836 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 303938737 \nu^{8} - 3215107672890 \nu^{6} + \cdots - 30\!\cdots\!72 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 308496745 \nu^{8} + 5087199131946 \nu^{6} + \cdots + 16\!\cdots\!24 ) / 56\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6881235923 \nu^{8} + 103669410123246 \nu^{6} + \cdots - 71\!\cdots\!48 ) / 56\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69206986817 \nu^{8} + \cdots - 17\!\cdots\!60 ) / 37\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2120694601 \nu^{9} - 35965121354090 \nu^{7} + \cdots - 10\!\cdots\!56 \nu ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 105159413737129 \nu^{9} + \cdots - 88\!\cdots\!76 \nu ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 692616151606351 \nu^{9} + \cdots + 24\!\cdots\!76 \nu ) / 55\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 125349951870869 \nu^{9} + \cdots + 82\!\cdots\!00 \nu ) / 34\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!27 \nu^{9} + \cdots - 60\!\cdots\!72 \nu ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} + 53\beta_{7} + 148\beta_{6} + 21\beta_{5} ) / 9216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} - 120\beta_{3} + 1652\beta_{2} - 1397\beta _1 - 10682541 ) / 3072 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4733\beta_{9} - 35136\beta_{8} - 467425\beta_{7} - 947588\beta_{6} - 1018767489\beta_{5} ) / 9216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 45961\beta_{4} + 1008216\beta_{3} + 378892\beta_{2} + 15025565\beta _1 + 73666993461 ) / 3072 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 33885109 \beta_{9} + 371750688 \beta_{8} + 4421316473 \beta_{7} + 6168369604 \beta_{6} + 14034606037113 \beta_{5} ) / 9216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 218709739 \beta_{4} - 3328147432 \beta_{3} - 23500742980 \beta_{2} - 49005318759 \beta _1 - 206497978666895 ) / 1024 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 282754258541 \beta_{9} - 3486009452544 \beta_{8} - 42756986004625 \beta_{7} - 46818662199044 \beta_{6} - 15\!\cdots\!01 \beta_{5} ) / 9216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7301243400377 \beta_{4} + 100824151254552 \beta_{3} + 963825255608204 \beta_{2} + \cdots + 56\!\cdots\!05 ) / 3072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25\!\cdots\!97 \beta_{9} + \cdots + 15\!\cdots\!05 \beta_{5} ) / 9216 \) 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
8.96538i
98.3652i
49.7838i
26.9187i
66.5347i
66.5347i
26.9187i
49.7838i
98.3652i
8.96538i
0 81.0000i 0 2526.54i 0 −4839.94 0 −6561.00 0
193.2 0 81.0000i 0 553.862i 0 12491.9 0 −6561.00 0
193.3 0 81.0000i 0 309.546i 0 −5272.51 0 −6561.00 0
193.4 0 81.0000i 0 332.682i 0 −3969.81 0 −6561.00 0
193.5 0 81.0000i 0 1878.17i 0 4378.40 0 −6561.00 0
193.6 0 81.0000i 0 1878.17i 0 4378.40 0 −6561.00 0
193.7 0 81.0000i 0 332.682i 0 −3969.81 0 −6561.00 0
193.8 0 81.0000i 0 309.546i 0 −5272.51 0 −6561.00 0
193.9 0 81.0000i 0 553.862i 0 12491.9 0 −6561.00 0
193.10 0 81.0000i 0 2526.54i 0 −4839.94 0 −6561.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.10.d.d yes 10
4.b odd 2 1 384.10.d.c 10
8.b even 2 1 inner 384.10.d.d yes 10
8.d odd 2 1 384.10.d.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.d.c 10 4.b odd 2 1
384.10.d.c 10 8.d odd 2 1
384.10.d.d yes 10 1.a even 1 1 trivial
384.10.d.d yes 10 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{10} + 10424192 T_{5}^{8} + 27678481125376 T_{5}^{6} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{5} - 2788T_{7}^{4} - 117213856T_{7}^{3} - 236229687680T_{7}^{2} + 1882375258450176T_{7} + 5540757921577073664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6561)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + 10424192 T^{8} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} - 2788 T^{4} + \cdots + 55\!\cdots\!64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 7487555664 T^{8} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{10} + 61996270160 T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} - 97846 T^{4} + \cdots - 76\!\cdots\!48)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 1457745135696 T^{8} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{5} + 67576 T^{4} + \cdots + 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 36404919905216 T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + 7443748 T^{4} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 897755940902736 T^{8} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{5} - 16234978 T^{4} + \cdots - 81\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{5} + 14346440 T^{4} + \cdots + 90\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{5} - 361408504 T^{4} + \cdots - 48\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 321360606 T^{4} + \cdots - 55\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 57446308 T^{4} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{5} - 854990558 T^{4} + \cdots - 65\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 1959564918 T^{4} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
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