Properties

Label 384.10.d.f
Level $384$
Weight $10$
Character orbit 384.d
Analytic conductor $197.774$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{175}\cdot 3^{32} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{7} q^{5} + \beta_{10} q^{7} - 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{7} q^{5} + \beta_{10} q^{7} - 6561 q^{9} + ( - \beta_{6} + 92 \beta_1) q^{11} + (\beta_{14} + 5 \beta_{7}) q^{13} + ( - \beta_{13} - \beta_{11} + \beta_{10}) q^{15} + (\beta_{2} - 45288) q^{17} + ( - \beta_{9} + \beta_{6} - 41 \beta_1) q^{19} + (\beta_{16} + \beta_{15} + \beta_{14} + 20 \beta_{7}) q^{21} + ( - \beta_{17} - 2 \beta_{13} + 12 \beta_{11} + 118 \beta_{10}) q^{23} + ( - \beta_{3} + 2 \beta_{2} - 484130) q^{25} + 6561 \beta_1 q^{27} + ( - \beta_{18} - 4 \beta_{15} - 9 \beta_{14} + 465 \beta_{7}) q^{29} + (\beta_{19} - 4 \beta_{17} - 111 \beta_{11} - 195 \beta_{10}) q^{31} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 602704) q^{33} + (7 \beta_{12} - 12 \beta_{9} + \beta_{8} - 68 \beta_{6} + 7213 \beta_1) q^{35} + (5 \beta_{18} + 5 \beta_{16} - 9 \beta_{15} + 2 \beta_{14} - 2643 \beta_{7}) q^{37} + ( - 3 \beta_{19} + \beta_{13} + 229 \beta_{11} - 130 \beta_{10}) q^{39} + ( - 4 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 14 \beta_{2} + \cdots + 3713193) q^{41}+ \cdots + (6561 \beta_{6} - 603612 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 131220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 131220 q^{9} - 905768 q^{17} - 9682620 q^{25} + 12054096 q^{33} + 74264008 q^{41} + 252775700 q^{49} - 5335632 q^{57} + 245588672 q^{65} - 895193896 q^{73} + 860934420 q^{81} + 882422136 q^{89} + 433683736 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 300700 x^{18} + 6140664 x^{17} + 35387063979 x^{16} - 1130222504088 x^{15} + \cdots + 12\!\cdots\!52 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 18\!\cdots\!92 \nu^{19} + \cdots + 47\!\cdots\!08 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 39\!\cdots\!29 \nu^{19} + \cdots - 10\!\cdots\!28 ) / 22\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!12 \nu^{19} + \cdots - 95\!\cdots\!09 ) / 14\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 73\!\cdots\!39 \nu^{19} + \cdots + 21\!\cdots\!42 ) / 31\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 91\!\cdots\!09 \nu^{19} + \cdots - 73\!\cdots\!32 ) / 28\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!59 \nu^{19} + \cdots - 86\!\cdots\!00 ) / 40\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!44 \nu^{19} + \cdots - 38\!\cdots\!36 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!90 \nu^{19} + \cdots + 12\!\cdots\!56 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!64 \nu^{19} + \cdots - 13\!\cdots\!08 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63\!\cdots\!13 \nu^{19} + \cdots - 16\!\cdots\!20 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!96 \nu^{19} + \cdots + 30\!\cdots\!80 ) / 31\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!68 \nu^{19} + \cdots + 21\!\cdots\!76 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 48\!\cdots\!81 \nu^{19} + \cdots + 10\!\cdots\!60 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!32 \nu^{19} + \cdots - 27\!\cdots\!28 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15\!\cdots\!80 \nu^{19} + \cdots - 55\!\cdots\!76 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 35\!\cdots\!72 \nu^{19} + \cdots - 88\!\cdots\!12 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 51\!\cdots\!13 \nu^{19} + \cdots + 15\!\cdots\!80 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 69\!\cdots\!92 \nu^{19} + \cdots - 23\!\cdots\!56 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 91\!\cdots\!87 \nu^{19} + \cdots - 22\!\cdots\!40 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + 81 \beta_{11} + 79 \beta_{7} - 9 \beta_{5} - 17 \beta_{4} + 91 \beta_{3} - 326 \beta_{2} + 530686 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 192 \beta_{19} - 694 \beta_{16} - 325 \beta_{15} + 4490 \beta_{14} - 1280 \beta_{13} + 3349 \beta_{11} + 29312 \beta_{10} - 101621 \beta_{7} + 15888 \beta_{5} + 3712 \beta_{4} + \cdots + 79820576560 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10368 \beta_{19} + 27216 \beta_{18} + 27216 \beta_{17} + 215168 \beta_{16} - 128968 \beta_{15} - 64768 \beta_{14} - 627744 \beta_{13} - 27 \beta_{12} + 7685286 \beta_{11} + \cdots - 743501056640 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12849408 \beta_{19} - 6912864 \beta_{18} - 6912864 \beta_{17} - 40858082 \beta_{16} - 5030495 \beta_{15} + 306086302 \beta_{14} - 104345664 \beta_{13} + \cdots + 25\!\cdots\!00 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2853705600 \beta_{19} + 7327635840 \beta_{18} + 7327272960 \beta_{17} + 50167925846 \beta_{16} - 26768711083 \beta_{15} - 26885857834 \beta_{14} + \cdots - 18\!\cdots\!20 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2719601510784 \beta_{19} - 1659649479264 \beta_{18} - 1659511221984 \beta_{17} - 9200106658474 \beta_{16} - 295772850043 \beta_{15} + \cdots + 36\!\cdots\!48 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 161649862732512 \beta_{19} + 372952919493504 \beta_{18} + 372901627312704 \beta_{17} + \cdots - 89\!\cdots\!64 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 21\!\cdots\!48 \beta_{19} + \cdots + 21\!\cdots\!98 ) / 221184 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 33\!\cdots\!32 \beta_{19} + \cdots - 16\!\cdots\!08 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 22\!\cdots\!40 \beta_{19} + \cdots + 17\!\cdots\!16 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15\!\cdots\!64 \beta_{19} + \cdots - 68\!\cdots\!72 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 98\!\cdots\!16 \beta_{19} + \cdots + 63\!\cdots\!12 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 29\!\cdots\!56 \beta_{19} + \cdots - 11\!\cdots\!96 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 16\!\cdots\!92 \beta_{19} + \cdots + 89\!\cdots\!24 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12\!\cdots\!44 \beta_{19} + \cdots - 46\!\cdots\!16 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 11\!\cdots\!28 \beta_{19} + \cdots + 53\!\cdots\!62 ) / 221184 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 22\!\cdots\!40 \beta_{19} + \cdots - 73\!\cdots\!44 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 10\!\cdots\!60 \beta_{19} + \cdots + 45\!\cdots\!24 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 95\!\cdots\!40 \beta_{19} + \cdots - 28\!\cdots\!88 ) / 1327104 \) 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
255.957 0.707107i
80.4463 + 0.707107i
5.69422 + 0.707107i
−271.678 0.707107i
−67.7122 + 0.707107i
−69.1264 0.707107i
−270.264 + 0.707107i
4.28001 0.707107i
79.0321 0.707107i
257.371 + 0.707107i
257.371 0.707107i
79.0321 + 0.707107i
4.28001 + 0.707107i
−270.264 0.707107i
−69.1264 + 0.707107i
−67.7122 0.707107i
−271.678 + 0.707107i
5.69422 0.707107i
80.4463 0.707107i
255.957 + 0.707107i
0 81.0000i 0 2358.11i 0 −1310.82 0 −6561.00 0
193.2 0 81.0000i 0 1825.80i 0 10705.0 0 −6561.00 0
193.3 0 81.0000i 0 1655.59i 0 −11370.0 0 −6561.00 0
193.4 0 81.0000i 0 573.512i 0 2185.20 0 −6561.00 0
193.5 0 81.0000i 0 471.295i 0 −3820.46 0 −6561.00 0
193.6 0 81.0000i 0 471.295i 0 3820.46 0 −6561.00 0
193.7 0 81.0000i 0 573.512i 0 −2185.20 0 −6561.00 0
193.8 0 81.0000i 0 1655.59i 0 11370.0 0 −6561.00 0
193.9 0 81.0000i 0 1825.80i 0 −10705.0 0 −6561.00 0
193.10 0 81.0000i 0 2358.11i 0 1310.82 0 −6561.00 0
193.11 0 81.0000i 0 2358.11i 0 1310.82 0 −6561.00 0
193.12 0 81.0000i 0 1825.80i 0 −10705.0 0 −6561.00 0
193.13 0 81.0000i 0 1655.59i 0 11370.0 0 −6561.00 0
193.14 0 81.0000i 0 573.512i 0 −2185.20 0 −6561.00 0
193.15 0 81.0000i 0 471.295i 0 3820.46 0 −6561.00 0
193.16 0 81.0000i 0 471.295i 0 −3820.46 0 −6561.00 0
193.17 0 81.0000i 0 573.512i 0 2185.20 0 −6561.00 0
193.18 0 81.0000i 0 1655.59i 0 −11370.0 0 −6561.00 0
193.19 0 81.0000i 0 1825.80i 0 10705.0 0 −6561.00 0
193.20 0 81.0000i 0 2358.11i 0 −1310.82 0 −6561.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.20
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 384.10.d.f 20
4.b odd 2 1 inner 384.10.d.f 20
8.b even 2 1 inner 384.10.d.f 20
8.d odd 2 1 inner 384.10.d.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.d.f 20 1.a even 1 1 trivial
384.10.d.f 20 4.b odd 2 1 inner
384.10.d.f 20 8.b even 2 1 inner
384.10.d.f 20 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 \(S_{10}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{10} + 12186280 T_{5}^{8} + 49400434717312 T_{5}^{6} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{10} - 264961960 T_{7}^{8} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6561)^{10} \) Copy content Toggle raw display
$5$ \( (T^{10} + 12186280 T^{8} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 264961960 T^{8} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 17985071696 T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 73095742624 T^{8} + \cdots + 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 226442 T^{4} + \cdots + 23\!\cdots\!36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{10} + 1203448472144 T^{8} + \cdots + 44\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 11315448983200 T^{8} + \cdots - 45\!\cdots\!88)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 33712416022696 T^{8} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 174454618599144 T^{8} + \cdots - 14\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 349412744876544 T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 18566002 T^{4} + \cdots - 30\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 66\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 17\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 36\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 86\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 223798474 T^{4} + \cdots - 10\!\cdots\!04)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 18\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 57\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} - 220605534 T^{4} + \cdots + 12\!\cdots\!84)^{4} \) Copy content Toggle raw display
$97$ \( (T^{5} - 108420934 T^{4} + \cdots - 23\!\cdots\!04)^{4} \) Copy content Toggle raw display
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