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Properties

Label 768.5.g.k
Level $768$
Weight $5$
Character orbit 768.g
Analytic conductor $79.388$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [768,5,Mod(511,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.511"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,-432,0,0,0,0,0,0,0,480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.3881316484\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 48x^{14} + 858x^{12} + 7028x^{10} + 25803x^{8} + 34572x^{6} + 14794x^{4} + 708x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} - \beta_{6} q^{7} - 27 q^{9} - \beta_{10} q^{11} + \beta_{4} q^{13} + \beta_{9} q^{15} + (\beta_{5} + 30) q^{17} + ( - \beta_{14} + \beta_{13} + 17 \beta_1) q^{19} + ( - \beta_{3} + \beta_{2}) q^{21}+ \cdots + 27 \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 432 q^{9} + 480 q^{17} + 1328 q^{25} - 1440 q^{41} - 2480 q^{49} - 7488 q^{57} - 2688 q^{65} + 33760 q^{73} + 11664 q^{81} - 31200 q^{89} - 24352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 858x^{12} + 7028x^{10} + 25803x^{8} + 34572x^{6} + 14794x^{4} + 708x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 529 \nu^{15} + 25384 \nu^{13} + 453525 \nu^{11} + 3712183 \nu^{9} + 13614037 \nu^{7} + \cdots + 568323 \nu ) / 11568 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 55309 \nu^{14} + 2653463 \nu^{12} + 47389120 \nu^{10} + 387525133 \nu^{8} + 1417345580 \nu^{6} + \cdots + 18436338 ) / 26028 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 192991 \nu^{14} + 9258977 \nu^{12} + 165367276 \nu^{10} + 1352461483 \nu^{8} + 4948342904 \nu^{6} + \cdots + 68276934 ) / 26028 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 246419 \nu^{14} + 11823349 \nu^{12} + 211197812 \nu^{10} + 1727702183 \nu^{8} + 6324244384 \nu^{6} + \cdots + 91816974 ) / 26028 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43293 \nu^{14} + 2076761 \nu^{12} + 37083482 \nu^{10} + 303170541 \nu^{8} + 1108283942 \nu^{6} + \cdots + 15284952 ) / 4338 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 336745 \nu^{15} - 16155698 \nu^{13} - 288540391 \nu^{11} - 2359732435 \nu^{9} + \cdots - 121236597 \nu ) / 52056 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112823 \nu^{14} + 5412283 \nu^{12} + 96647582 \nu^{10} + 790161335 \nu^{8} + 2888666146 \nu^{6} + \cdots + 39287592 ) / 2169 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 132910 \nu^{14} - 6375814 \nu^{12} - 113851788 \nu^{10} - 930800638 \nu^{8} - 3402741172 \nu^{6} + \cdots - 45127728 ) / 2169 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 389081 \nu^{15} + 18665194 \nu^{13} + 333318791 \nu^{11} + 2725313507 \nu^{9} + 9964765039 \nu^{7} + \cdots + 129465405 \nu ) / 5784 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 152905 \nu^{15} + 7335360 \nu^{13} + 130996683 \nu^{11} + 1071117263 \nu^{9} + 3916761123 \nu^{7} + \cdots + 50478951 \nu ) / 2169 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4408543 \nu^{14} + 211505405 \nu^{12} + 3777500848 \nu^{10} + 30893154127 \nu^{8} + \cdots + 1511991558 ) / 26028 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1447045 \nu^{15} - 69427370 \nu^{13} - 1240084747 \nu^{11} - 10143321799 \nu^{9} + \cdots - 548776737 \nu ) / 17352 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 239664 \nu^{15} - 11497536 \nu^{13} - 205327760 \nu^{11} - 1678930640 \nu^{9} + \cdots - 82081680 \nu ) / 2169 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7585119 \nu^{15} + 363886040 \nu^{13} + 6498454651 \nu^{11} + 53137232729 \nu^{9} + \cdots + 2642350605 \nu ) / 34704 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28848455 \nu^{15} + 1383964654 \nu^{13} + 24715456745 \nu^{11} + 202094796989 \nu^{9} + \cdots + 9827530299 \nu ) / 52056 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -9\beta_{13} + 6\beta_{12} - 16\beta_{9} - 90\beta_{6} ) / 2304 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{11} + 39\beta_{8} + 33\beta_{7} + 78\beta_{5} + 24\beta_{4} + 16\beta_{3} + 266\beta_{2} - 27648 ) / 4608 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 90 \beta_{15} + 216 \beta_{14} + 603 \beta_{13} - 294 \beta_{12} + 216 \beta_{10} + \cdots - 2232 \beta_1 ) / 9216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 66 \beta_{11} - 255 \beta_{8} - 177 \beta_{7} - 654 \beta_{5} - 246 \beta_{4} - 110 \beta_{3} + \cdots + 169344 ) / 2304 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1674 \beta_{15} - 4680 \beta_{14} - 11331 \beta_{13} + 4182 \beta_{12} - 3240 \beta_{10} + \cdots + 94440 \beta_1 ) / 9216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1236 \beta_{11} + 3411 \beta_{8} + 1953 \beta_{7} + 10782 \beta_{5} + 5070 \beta_{4} + \cdots - 2339712 ) / 2304 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 14562 \beta_{15} + 43344 \beta_{14} + 96849 \beta_{13} - 31638 \beta_{12} + 18144 \beta_{10} + \cdots - 1189776 \beta_1 ) / 4608 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10806 \beta_{11} - 24141 \beta_{8} - 11139 \beta_{7} - 90186 \beta_{5} - 47841 \beta_{4} + \cdots + 17219520 ) / 1152 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 504558 \beta_{15} - 1523448 \beta_{14} - 3191301 \beta_{13} + 996810 \beta_{12} + \cdots + 50329944 \beta_1 ) / 9216 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 735018 \beta_{11} + 1430691 \beta_{8} + 513741 \beta_{7} + 6107286 \beta_{5} + 3446556 \beta_{4} + \cdots - 1057667328 ) / 4608 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 8741682 \beta_{15} + 26179560 \beta_{14} + 52049187 \beta_{13} - 16128366 \beta_{12} + \cdots - 976808712 \beta_1 ) / 9216 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1546005 \beta_{11} - 2740977 \beta_{8} - 732699 \beta_{7} - 13011786 \beta_{5} - 7568922 \beta_{4} + \cdots + 2090397888 ) / 576 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 37821672 \beta_{15} - 111360600 \beta_{14} - 212373495 \beta_{13} + 66372666 \beta_{12} + \cdots + 4517151288 \beta_1 ) / 2304 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 207488490 \beta_{11} + 344681409 \beta_{8} + 64670631 \beta_{7} + 1779434466 \beta_{5} + \cdots - 269869142016 ) / 4608 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2612487042 \beta_{15} + 7547756472 \beta_{14} + 13930822215 \beta_{13} - 4418531934 \beta_{12} + \cdots - 324742372632 \beta_1 ) / 9216 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
511.1
− 4.12169i
− 0.163939i
3.55404i
− 0.160317i
− 0.839683i
− 2.55404i
1.16394i
3.12169i
4.12169i
0.163939i
− 3.55404i
0.160317i
0.839683i
2.55404i
− 1.16394i
− 3.12169i
0 − 5.19615i 0 −38.1617 0 − 42.0542i 0 −27.0000 0
511.2 0 − 5.19615i 0 −34.2464 0 − 9.22331i 0 −27.0000 0
511.3 0 − 5.19615i 0 −13.7025 0 88.3075i 0 −27.0000 0
511.4 0 − 5.19615i 0 −3.88698 0 − 23.9200i 0 −27.0000 0
511.5 0 − 5.19615i 0 3.88698 0 23.9200i 0 −27.0000 0
511.6 0 − 5.19615i 0 13.7025 0 − 88.3075i 0 −27.0000 0
511.7 0 − 5.19615i 0 34.2464 0 9.22331i 0 −27.0000 0
511.8 0 − 5.19615i 0 38.1617 0 42.0542i 0 −27.0000 0
511.9 0 5.19615i 0 −38.1617 0 42.0542i 0 −27.0000 0
511.10 0 5.19615i 0 −34.2464 0 9.22331i 0 −27.0000 0
511.11 0 5.19615i 0 −13.7025 0 − 88.3075i 0 −27.0000 0
511.12 0 5.19615i 0 −3.88698 0 23.9200i 0 −27.0000 0
511.13 0 5.19615i 0 3.88698 0 − 23.9200i 0 −27.0000 0
511.14 0 5.19615i 0 13.7025 0 88.3075i 0 −27.0000 0
511.15 0 5.19615i 0 34.2464 0 − 9.22331i 0 −27.0000 0
511.16 0 5.19615i 0 38.1617 0 − 42.0542i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.16
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 768.5.g.k 16
4.b odd 2 1 inner 768.5.g.k 16
8.b even 2 1 inner 768.5.g.k 16
8.d odd 2 1 inner 768.5.g.k 16
16.e even 4 1 24.5.b.a ✓ 8
16.e even 4 1 96.5.b.a 8
16.f odd 4 1 24.5.b.a ✓ 8
16.f odd 4 1 96.5.b.a 8
48.i odd 4 1 72.5.b.d 8
48.i odd 4 1 288.5.b.d 8
48.k even 4 1 72.5.b.d 8
48.k even 4 1 288.5.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.b.a ✓ 8 16.e even 4 1
24.5.b.a ✓ 8 16.f odd 4 1
72.5.b.d 8 48.i odd 4 1
72.5.b.d 8 48.k even 4 1
96.5.b.a 8 16.e even 4 1
96.5.b.a 8 16.f odd 4 1
288.5.b.d 8 48.i odd 4 1
288.5.b.d 8 48.k even 4 1
768.5.g.k 16 1.a even 1 1 trivial
768.5.g.k 16 4.b odd 2 1 inner
768.5.g.k 16 8.b even 2 1 inner
768.5.g.k 16 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2832T_{5}^{6} + 2244192T_{5}^{4} - 353952000T_{5}^{2} + 4845166848 \) acting on \(S_{5}^{\mathrm{new}}(768, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} - 2832 T^{6} + \cdots + 4845166848)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 10224 T^{6} + \cdots + 671288262912)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 37\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 56\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 120 T^{3} + \cdots - 1023349232)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 64\!\cdots\!92)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 22\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 78\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 360 T^{3} + \cdots + 226033840912)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 50\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 86\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 47\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 34\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 94\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 64\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 35664142829552)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 43\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 219207569912848)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 29\!\cdots\!68)^{4} \) Copy content Toggle raw display
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