Properties

Label 384.5.e.d
Level $384$
Weight $5$
Character orbit 384.e
Analytic conductor $39.694$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + 391324 x^{9} + 724325 x^{8} - 2262056 x^{7} + 45109352 x^{6} + \cdots + 21479188203 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + \beta_{6} q^{5} + (\beta_{3} + \beta_1 + 5) q^{7} + (\beta_{7} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{3} + \beta_{6} q^{5} + (\beta_{3} + \beta_1 + 5) q^{7} + (\beta_{7} - \beta_{3}) q^{9} - \beta_{9} q^{11} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{13} + ( - \beta_{15} - \beta_{8} + \beta_{6} + \beta_{3} + \beta_1 + 26) q^{15} + (\beta_{15} + \beta_{9} - \beta_{6} - \beta_{2}) q^{17} + ( - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_1 - 51) q^{19} + (\beta_{12} + \beta_{10} - \beta_{9} + \beta_{4} - 4 \beta_{3} + 2 \beta_1 - 36) q^{21} + ( - \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots - 3) q^{23}+ \cdots + ( - \beta_{15} + 4 \beta_{14} - 5 \beta_{13} - 24 \beta_{12} + 11 \beta_{11} + \cdots + 1642) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} + 80 q^{7} + 416 q^{15} - 816 q^{19} - 608 q^{21} - 2000 q^{25} - 280 q^{27} + 592 q^{31} - 496 q^{33} - 2240 q^{37} - 16 q^{39} - 368 q^{43} - 800 q^{45} + 3984 q^{49} + 352 q^{51} + 1920 q^{55} + 560 q^{57} + 3520 q^{61} - 816 q^{63} + 3536 q^{67} + 10784 q^{69} + 3680 q^{73} - 5112 q^{75} - 14448 q^{79} - 624 q^{81} + 11136 q^{85} - 14944 q^{87} + 22944 q^{91} - 13760 q^{93} + 3264 q^{97} + 26976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 32 x^{14} + 356 x^{13} + 1348 x^{12} - 8992 x^{11} + 22064 x^{10} + 391324 x^{9} + 724325 x^{8} - 2262056 x^{7} + 45109352 x^{6} + \cdots + 21479188203 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\!\cdots\!48 \nu^{15} + \cdots - 45\!\cdots\!28 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\!\cdots\!64 \nu^{15} + \cdots - 14\!\cdots\!64 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!28 \nu^{15} + \cdots + 28\!\cdots\!33 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!72 \nu^{15} + \cdots + 58\!\cdots\!33 ) / 16\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!28 \nu^{15} + \cdots - 90\!\cdots\!67 ) / 72\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 68\!\cdots\!44 \nu^{15} + \cdots + 94\!\cdots\!56 ) / 21\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!24 \nu^{15} + \cdots + 19\!\cdots\!44 ) / 72\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 75\!\cdots\!04 \nu^{15} + \cdots + 35\!\cdots\!53 ) / 14\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43\!\cdots\!52 \nu^{15} + \cdots + 16\!\cdots\!58 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46\!\cdots\!68 \nu^{15} + \cdots + 44\!\cdots\!32 ) / 67\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!68 \nu^{15} + \cdots + 87\!\cdots\!92 ) / 21\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 86\!\cdots\!88 \nu^{15} + \cdots + 42\!\cdots\!73 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!16 \nu^{15} + \cdots - 34\!\cdots\!49 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!12 \nu^{15} + \cdots - 23\!\cdots\!08 ) / 65\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 24\!\cdots\!92 \nu^{15} + \cdots + 17\!\cdots\!18 ) / 72\!\cdots\!45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{13} + 3 \beta_{12} - 9 \beta_{11} - 6 \beta_{10} - 3 \beta_{8} + 6 \beta_{7} + 3 \beta_{6} - 10 \beta_{5} - 3 \beta_{4} + 72 \beta_{3} - 3 \beta_{2} + 18 \beta _1 + 112 ) / 576 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 18 \beta_{14} - 9 \beta_{13} - 3 \beta_{12} - 12 \beta_{11} - 78 \beta_{10} + 21 \beta_{9} + 36 \beta_{7} - 36 \beta_{6} + 11 \beta_{5} - 12 \beta_{4} - 294 \beta_{3} - 33 \beta _1 + 1567 ) / 288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 399 \beta_{15} - 24 \beta_{14} - 93 \beta_{13} - 192 \beta_{12} - 111 \beta_{11} - 1239 \beta_{10} + 225 \beta_{9} + 27 \beta_{8} + 681 \beta_{7} - 2964 \beta_{6} - 47 \beta_{5} + 165 \beta_{4} - 4485 \beta_{3} + \cdots - 42649 ) / 1152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 504 \beta_{15} - 162 \beta_{14} + 90 \beta_{13} + 210 \beta_{12} + 24 \beta_{11} - 2727 \beta_{10} + 1410 \beta_{9} + 612 \beta_{8} + 576 \beta_{7} - 3060 \beta_{6} - 1018 \beta_{5} + 12 \beta_{4} - 11808 \beta_{3} + \cdots - 117170 ) / 288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23835 \beta_{15} + 822 \beta_{14} + 9267 \beta_{13} - 1482 \beta_{12} + 2697 \beta_{11} - 4095 \beta_{10} - 13173 \beta_{9} + 22203 \beta_{8} + 13983 \beta_{7} - 157854 \beta_{6} - 25217 \beta_{5} + \cdots - 2541451 ) / 1152 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 25695 \beta_{15} - 20070 \beta_{14} + 47565 \beta_{13} + 29682 \beta_{12} + 5313 \beta_{11} + 21333 \beta_{10} - 14229 \beta_{9} + 76527 \beta_{8} + 13617 \beta_{7} - 20592 \beta_{6} + \cdots - 12550003 ) / 576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69963 \beta_{15} + 3003 \beta_{14} + 383475 \beta_{13} - 94905 \beta_{12} + 160743 \beta_{11} + 1434873 \beta_{10} - 919068 \beta_{9} + 475767 \beta_{8} + 207282 \beta_{7} + 975945 \beta_{6} + \cdots - 52056770 ) / 576 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 260559 \beta_{15} + 59688 \beta_{14} + 236736 \beta_{13} - 75558 \beta_{12} + 179187 \beta_{11} + 1133697 \beta_{10} - 621375 \beta_{9} + 155187 \beta_{8} - 428958 \beta_{7} + \cdots - 46146125 ) / 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 42954225 \beta_{15} + 24039156 \beta_{14} + 14825337 \beta_{13} - 17000310 \beta_{12} + 28946697 \beta_{11} + 161307858 \beta_{10} - 70251453 \beta_{9} + \cdots + 1687531457 ) / 1152 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 185312709 \beta_{15} + 111978990 \beta_{14} - 38738799 \beta_{13} - 42483702 \beta_{12} + 62346909 \beta_{11} + 481446126 \beta_{10} - 148548129 \beta_{9} + \cdots + 10877942909 ) / 576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2596057755 \beta_{15} + 1107324282 \beta_{14} - 1404857001 \beta_{13} + 9575256 \beta_{12} + 517420167 \beta_{11} + 3646733652 \beta_{10} + \cdots + 425153221169 ) / 1152 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1408718718 \beta_{15} + 696853548 \beta_{14} - 1908838827 \beta_{13} + 185930514 \beta_{12} - 318258924 \beta_{11} + 954713379 \beta_{10} + 1384340652 \beta_{9} + \cdots + 373014824522 ) / 144 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 29974297578 \beta_{15} - 21184407888 \beta_{14} - 109916664456 \beta_{13} + 41550296328 \beta_{12} - 47039296458 \beta_{11} - 237455847273 \beta_{10} + \cdots + 21514937631830 ) / 1152 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 107681168628 \beta_{15} - 82002501531 \beta_{14} - 149942222739 \beta_{13} + 56627239623 \beta_{12} - 124433511276 \beta_{11} - 634678808685 \beta_{10} + \cdots + 25098831316387 ) / 288 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5820299923323 \beta_{15} - 4301167909614 \beta_{14} - 1787943120429 \beta_{13} + 1949654378148 \beta_{12} - 3964444728267 \beta_{11} + \cdots + 114192210569339 ) / 1152 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−1.10373 0.840249i
−1.10373 + 0.840249i
2.18273 + 4.51404i
2.18273 4.51404i
4.43019 + 3.93201i
4.43019 3.93201i
−3.40000 + 2.59434i
−3.40000 2.59434i
2.27178 3.46189i
2.27178 + 3.46189i
6.34189 3.61720i
6.34189 + 3.61720i
−3.05642 3.40211i
−3.05642 + 3.40211i
−5.66644 2.02063i
−5.66644 + 2.02063i
0 −8.70713 2.27724i 0 28.9306i 0 −2.36161 0 70.6284 + 39.6564i 0
257.2 0 −8.70713 + 2.27724i 0 28.9306i 0 −2.36161 0 70.6284 39.6564i 0
257.3 0 −6.57180 6.14911i 0 9.58700i 0 −22.6799 0 5.37699 + 80.8213i 0
257.4 0 −6.57180 + 6.14911i 0 9.58700i 0 −22.6799 0 5.37699 80.8213i 0
257.5 0 −5.70733 6.95891i 0 34.7105i 0 89.5593 0 −15.8529 + 79.4335i 0
257.6 0 −5.70733 + 6.95891i 0 34.7105i 0 89.5593 0 −15.8529 79.4335i 0
257.7 0 0.622561 8.97844i 0 0.562050i 0 −55.8203 0 −80.2248 11.1793i 0
257.8 0 0.622561 + 8.97844i 0 0.562050i 0 −55.8203 0 −80.2248 + 11.1793i 0
257.9 0 3.32132 8.36474i 0 25.4639i 0 36.1251 0 −58.9376 55.5640i 0
257.10 0 3.32132 + 8.36474i 0 25.4639i 0 36.1251 0 −58.9376 + 55.5640i 0
257.11 0 4.08506 8.01949i 0 47.0883i 0 −13.4570 0 −47.6246 65.5202i 0
257.12 0 4.08506 + 8.01949i 0 47.0883i 0 −13.4570 0 −47.6246 + 65.5202i 0
257.13 0 7.95805 4.20350i 0 19.3338i 0 67.6413 0 45.6611 66.9034i 0
257.14 0 7.95805 + 4.20350i 0 19.3338i 0 67.6413 0 45.6611 + 66.9034i 0
257.15 0 8.99926 0.115244i 0 25.0292i 0 −59.0069 0 80.9734 2.07423i 0
257.16 0 8.99926 + 0.115244i 0 25.0292i 0 −59.0069 0 80.9734 + 2.07423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.e.d yes 16
3.b odd 2 1 inner 384.5.e.d yes 16
4.b odd 2 1 384.5.e.a 16
8.b even 2 1 384.5.e.b yes 16
8.d odd 2 1 384.5.e.c yes 16
12.b even 2 1 384.5.e.a 16
24.f even 2 1 384.5.e.c yes 16
24.h odd 2 1 384.5.e.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.e.a 16 4.b odd 2 1
384.5.e.a 16 12.b even 2 1
384.5.e.b yes 16 8.b even 2 1
384.5.e.b yes 16 24.h odd 2 1
384.5.e.c yes 16 8.d odd 2 1
384.5.e.c yes 16 24.f even 2 1
384.5.e.d yes 16 1.a even 1 1 trivial
384.5.e.d yes 16 3.b 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 \(S_{5}^{\mathrm{new}}(384, [\chi])\):

\( T_{7}^{8} - 40 T_{7}^{7} - 9800 T_{7}^{6} + 186176 T_{7}^{5} + 29617840 T_{7}^{4} - 12942592 T_{7}^{3} - 22111959680 T_{7}^{2} - 271760728576 T_{7} - 519546316544 \) Copy content Toggle raw display
\( T_{13}^{8} - 107088 T_{13}^{6} + 4006400 T_{13}^{5} + 3129196896 T_{13}^{4} - 218176770048 T_{13}^{3} - 13278939766016 T_{13}^{2} + \cdots + 115283511800064 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{15} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$5$ \( T^{16} + 6000 T^{14} + \cdots + 98\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( (T^{8} - 40 T^{7} + \cdots - 519546316544)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 133456 T^{14} + \cdots + 83\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} - 107088 T^{6} + \cdots + 115283511800064)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 693696 T^{14} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + 408 T^{7} + \cdots - 30\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 2222656 T^{14} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{16} + 5824624 T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} - 296 T^{7} + \cdots - 45\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 1120 T^{7} + \cdots - 37\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 20659648 T^{14} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{8} + 184 T^{7} + \cdots + 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 43866624 T^{14} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + 46166640 T^{14} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + 110124688 T^{14} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{8} - 1760 T^{7} + \cdots - 33\!\cdots\!40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 1768 T^{7} + \cdots - 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 289956160 T^{14} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{8} - 1840 T^{7} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 7224 T^{7} + \cdots - 56\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 554373456 T^{14} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + 720972928 T^{14} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{8} - 1632 T^{7} + \cdots - 33\!\cdots\!84)^{2} \) Copy content Toggle raw display
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