Properties

Label 384.6.f.e
Level $384$
Weight $6$
Character orbit 384.f
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{87}\cdot 3^{14} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{4} + \beta_{2}) q^{5} + \beta_{5} q^{7} + ( - \beta_{8} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{4} + \beta_{2}) q^{5} + \beta_{5} q^{7} + ( - \beta_{8} - 2) q^{9} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{3} + 5 \beta_1) q^{13} + ( - \beta_{17} - \beta_{15} - 178) q^{15} + ( - \beta_{15} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{5}) q^{17} + (\beta_{16} - \beta_{14} + 2 \beta_{7} - 3 \beta_{4} + 19 \beta_{2}) q^{19} + ( - \beta_{18} - \beta_{16} + \beta_{14} + \beta_{7} - \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 8 \beta_1) q^{21} + ( - \beta_{17} + \beta_{15} - \beta_{13} + \beta_{9} + 5 \beta_{8} + 305) q^{23} + (3 \beta_{17} + 2 \beta_{15} + \beta_{13} - 2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 760) q^{25}+ \cdots + (9 \beta_{19} - 73 \beta_{18} - 42 \beta_{16} + 83 \beta_{14} - 9 \beta_{7} + \cdots - 272 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 44 q^{9} - 3568 q^{15} + 6112 q^{23} + 15228 q^{25} + 7592 q^{33} - 2800 q^{39} + 26112 q^{47} - 81044 q^{49} - 89296 q^{57} - 14816 q^{63} - 72224 q^{71} - 61256 q^{73} + 89588 q^{81} - 145648 q^{87} + 385504 q^{95} + 92808 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1423883557 \nu^{18} - 7392863505464 \nu^{14} + \cdots - 13\!\cdots\!04 \nu^{2} ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 98\!\cdots\!31 \nu^{19} + \cdots + 27\!\cdots\!52 \nu ) / 47\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!31 \nu^{19} + \cdots + 27\!\cdots\!52 \nu ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!39 \nu^{19} + \cdots + 82\!\cdots\!20 \nu ) / 47\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!19 \nu^{19} - 190718048886868 \nu^{17} + \cdots - 43\!\cdots\!08 \nu ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 98\!\cdots\!31 \nu^{19} + \cdots + 27\!\cdots\!52 \nu ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 98\!\cdots\!31 \nu^{19} + \cdots + 27\!\cdots\!52 \nu ) / 53\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!59 \nu^{19} + \cdots - 18\!\cdots\!76 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 91\!\cdots\!43 \nu^{19} + \cdots - 31\!\cdots\!08 ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 98\!\cdots\!31 \nu^{19} - 270188667554884 \nu^{17} - 97754774539128 \nu^{16} + \cdots - 36\!\cdots\!60 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!97 \nu^{19} + \cdots - 21\!\cdots\!60 ) / 59\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!35 \nu^{19} - 588217616893972 \nu^{17} - 97754774539128 \nu^{16} + \cdots - 36\!\cdots\!60 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!53 \nu^{19} + \cdots - 87\!\cdots\!68 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!59 \nu^{19} + \cdots + 74\!\cdots\!80 \nu ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29\!\cdots\!97 \nu^{19} + \cdots - 25\!\cdots\!76 ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 30\!\cdots\!73 \nu^{19} + \cdots - 82\!\cdots\!32 \nu ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!71 \nu^{19} + \cdots + 55\!\cdots\!84 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 45\!\cdots\!51 \nu^{19} + \cdots + 12\!\cdots\!92 \nu ) / 23\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 43\!\cdots\!63 \nu^{19} + \cdots + 11\!\cdots\!64 \nu ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 4 \beta_{19} + 2 \beta_{18} - 4 \beta_{17} - 8 \beta_{16} - 12 \beta_{15} - 2 \beta_{14} - 6 \beta_{12} - 8 \beta_{11} + 9 \beta_{10} + 12 \beta_{9} - 24 \beta_{8} + 2 \beta_{7} - 24 \beta_{5} + 72 \beta_{4} - 58 \beta_{2} + \beta _1 + 4 ) / 4608 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{19} - 12 \beta_{18} - 6 \beta_{16} - 6 \beta_{14} + 16 \beta_{7} + 24 \beta_{6} + 72 \beta_{3} - 312 \beta_{2} - 1821 \beta_1 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 88 \beta_{19} - 2 \beta_{18} - 76 \beta_{17} + 188 \beta_{16} - 276 \beta_{15} - 10 \beta_{14} - 46 \beta_{12} - 184 \beta_{11} + 161 \beta_{10} + 276 \beta_{9} - 504 \beta_{8} - 90 \beta_{7} - 168 \beta_{5} - 864 \beta_{4} + 202 \beta_{2} + \cdots + 76 ) / 2304 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 178 \beta_{17} - 54 \beta_{15} - 132 \beta_{13} + 139 \beta_{12} - 32 \beta_{11} + 139 \beta_{10} - 54 \beta_{9} - 828 \beta_{8} - 398614 ) / 384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8032 \beta_{19} + 1222 \beta_{18} + 7060 \beta_{17} + 17036 \beta_{16} + 25068 \beta_{15} - 2194 \beta_{14} - 1674 \beta_{12} + 14864 \beta_{11} - 8925 \beta_{10} - 25068 \beta_{9} + 46248 \beta_{8} + \cdots - 7060 ) / 4608 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4581 \beta_{19} + 9162 \beta_{18} + 4581 \beta_{16} + 4581 \beta_{14} - 20632 \beta_{7} - 4416 \beta_{6} - 34272 \beta_{3} + 244740 \beta_{2} + 836934 \beta_1 ) / 576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 368708 \beta_{19} - 92174 \beta_{18} + 326852 \beta_{17} - 779272 \beta_{16} + 1147980 \beta_{15} + 134030 \beta_{14} - 267850 \beta_{12} + 625448 \beta_{11} - 230257 \beta_{10} + \cdots - 326852 ) / 4608 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 571154 \beta_{17} + 40194 \beta_{15} + 250743 \beta_{13} - 363389 \beta_{12} + 95119 \beta_{11} - 363389 \beta_{10} + 40194 \beta_{9} + 2084787 \beta_{8} + 773453720 ) / 384 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8488928 \beta_{19} - 2552678 \beta_{18} - 7507100 \beta_{17} - 17959684 \beta_{16} - 26448612 \beta_{15} + 3534506 \beta_{14} + 9354810 \beta_{12} - 13653640 \beta_{11} + \cdots + 7507100 ) / 2304 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 44634324 \beta_{19} - 89268648 \beta_{18} - 44634324 \beta_{16} - 44634324 \beta_{14} + 242551096 \beta_{7} + 13739208 \beta_{6} + 267048576 \beta_{3} + \cdots - 6769261395 \beta_1 ) / 2304 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 783400208 \beta_{19} + 251215430 \beta_{18} - 686063060 \beta_{17} + 1664137564 \beta_{16} - 2447537772 \beta_{15} - 348552578 \beta_{14} + 1084329706 \beta_{12} + \cdots + 686063060 ) / 4608 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 115686442 \beta_{17} - 3085668 \beta_{15} - 41452467 \beta_{13} + 71439844 \beta_{12} - 14259221 \beta_{11} + 71439844 \beta_{10} - 3085668 \beta_{9} - 397768797 \beta_{8} + \cdots - 132898972992 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 36200015860 \beta_{19} + 11607349246 \beta_{18} + 31239087268 \beta_{17} + 77360960312 \beta_{16} + 113560976172 \beta_{15} - 16568277838 \beta_{14} + \cdots - 31239087268 ) / 4608 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 101426598726 \beta_{19} + 202853197452 \beta_{18} + 101426598726 \beta_{16} + 101426598726 \beta_{14} - 639299912272 \beta_{7} + \cdots + 14259000539613 \beta_1 ) / 2304 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 837238734952 \beta_{19} - 257591632462 \beta_{18} + 709507663276 \beta_{17} - 1802208541580 \beta_{16} + 2639447276532 \beta_{15} + \cdots - 709507663276 ) / 2304 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 3116136283570 \beta_{17} + 73302787350 \beta_{15} + 1007072470572 \beta_{13} - 1953561401179 \beta_{12} + 216085951784 \beta_{11} + \cdots + 33\!\cdots\!18 ) / 384 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 77513966969728 \beta_{19} - 22194276830278 \beta_{18} - 64348134431764 \beta_{17} - 168193766477420 \beta_{16} - 245707733447148 \beta_{15} + \cdots + 64348134431764 ) / 4608 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 56218106417985 \beta_{19} - 112436212835970 \beta_{18} - 56218106417985 \beta_{16} - 56218106417985 \beta_{14} + 402462815488456 \beta_{7} + \cdots - 76\!\cdots\!78 \beta_1 ) / 576 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 35\!\cdots\!80 \beta_{19} + 932007071860718 \beta_{18} + \cdots + 29\!\cdots\!76 ) / 4608 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−3.85040 3.85040i
−3.85040 + 3.85040i
0.123732 0.123732i
0.123732 + 0.123732i
0.860472 + 0.860472i
0.860472 0.860472i
4.77687 + 4.77687i
4.77687 4.77687i
4.85801 4.85801i
4.85801 + 4.85801i
−4.85801 + 4.85801i
−4.85801 4.85801i
−4.77687 4.77687i
−4.77687 + 4.77687i
−0.860472 0.860472i
−0.860472 + 0.860472i
−0.123732 + 0.123732i
−0.123732 0.123732i
3.85040 + 3.85040i
3.85040 3.85040i
0 −15.4669 1.94284i 0 101.098 0 200.335i 0 235.451 + 60.0993i 0
191.2 0 −15.4669 + 1.94284i 0 101.098 0 200.335i 0 235.451 60.0993i 0
191.3 0 −14.0881 6.67280i 0 −28.8990 0 44.6454i 0 153.948 + 188.014i 0
191.4 0 −14.0881 + 6.67280i 0 −28.8990 0 44.6454i 0 153.948 188.014i 0
191.5 0 −10.7340 11.3040i 0 4.25229 0 187.352i 0 −12.5625 + 242.675i 0
191.6 0 −10.7340 + 11.3040i 0 4.25229 0 187.352i 0 −12.5625 242.675i 0
191.7 0 −6.91183 13.9723i 0 −56.7417 0 130.269i 0 −147.453 + 193.149i 0
191.8 0 −6.91183 + 13.9723i 0 −56.7417 0 130.269i 0 −147.453 193.149i 0
191.9 0 −1.14396 15.5464i 0 71.6821 0 100.489i 0 −240.383 + 35.5689i 0
191.10 0 −1.14396 + 15.5464i 0 71.6821 0 100.489i 0 −240.383 35.5689i 0
191.11 0 1.14396 15.5464i 0 −71.6821 0 100.489i 0 −240.383 35.5689i 0
191.12 0 1.14396 + 15.5464i 0 −71.6821 0 100.489i 0 −240.383 + 35.5689i 0
191.13 0 6.91183 13.9723i 0 56.7417 0 130.269i 0 −147.453 193.149i 0
191.14 0 6.91183 + 13.9723i 0 56.7417 0 130.269i 0 −147.453 + 193.149i 0
191.15 0 10.7340 11.3040i 0 −4.25229 0 187.352i 0 −12.5625 242.675i 0
191.16 0 10.7340 + 11.3040i 0 −4.25229 0 187.352i 0 −12.5625 + 242.675i 0
191.17 0 14.0881 6.67280i 0 28.8990 0 44.6454i 0 153.948 188.014i 0
191.18 0 14.0881 + 6.67280i 0 28.8990 0 44.6454i 0 153.948 + 188.014i 0
191.19 0 15.4669 1.94284i 0 −101.098 0 200.335i 0 235.451 60.0993i 0
191.20 0 15.4669 + 1.94284i 0 −101.098 0 200.335i 0 235.451 + 60.0993i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.f.e 20
3.b odd 2 1 384.6.f.f yes 20
4.b odd 2 1 384.6.f.f yes 20
8.b even 2 1 inner 384.6.f.e 20
8.d odd 2 1 384.6.f.f yes 20
12.b even 2 1 inner 384.6.f.e 20
24.f even 2 1 inner 384.6.f.e 20
24.h odd 2 1 384.6.f.f yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.f.e 20 1.a even 1 1 trivial
384.6.f.e 20 8.b even 2 1 inner
384.6.f.e 20 12.b even 2 1 inner
384.6.f.e 20 24.f even 2 1 inner
384.6.f.f yes 20 3.b odd 2 1
384.6.f.f yes 20 4.b odd 2 1
384.6.f.f yes 20 8.d odd 2 1
384.6.f.f yes 20 24.h odd 2 1

Hecke kernels

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

\( T_{5}^{10} - 19432T_{5}^{8} + 117835648T_{5}^{6} - 256371264512T_{5}^{4} + 145811351932928T_{5}^{2} - 2553423501099008 \) Copy content Toggle raw display
\( T_{23}^{5} - 1528T_{23}^{4} - 15431552T_{23}^{3} + 8844092416T_{23}^{2} + 65002850471936T_{23} + 26581880445239296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 22 T^{18} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( (T^{10} - 19432 T^{8} + \cdots - 25\!\cdots\!08)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 104296 T^{8} + \cdots + 48\!\cdots\!72)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 910868 T^{8} + \cdots + 74\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 1989584 T^{8} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 8276896 T^{8} + \cdots + 81\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} - 12896104 T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{5} - 1528 T^{4} + \cdots + 26\!\cdots\!96)^{4} \) Copy content Toggle raw display
$29$ \( (T^{10} - 97006312 T^{8} + \cdots - 50\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 161397672 T^{8} + \cdots + 25\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 209940944 T^{8} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 475505664 T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 1117853800 T^{8} + \cdots - 80\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 6528 T^{4} + \cdots + 10\!\cdots\!80)^{4} \) Copy content Toggle raw display
$53$ \( (T^{10} - 2394345768 T^{8} + \cdots - 49\!\cdots\!52)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 2001626292 T^{8} + \cdots + 39\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 6490406864 T^{8} + \cdots + 41\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 6341733160 T^{8} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 18056 T^{4} + \cdots + 96\!\cdots\!12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{5} + 15314 T^{4} + \cdots - 74\!\cdots\!40)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + 9549472360 T^{8} + \cdots + 37\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 9819894324 T^{8} + \cdots + 38\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 41200982944 T^{8} + \cdots + 22\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 23202 T^{4} + \cdots - 60\!\cdots\!04)^{4} \) Copy content Toggle raw display
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