Properties

Label 768.4.f.e
Level $768$
Weight $4$
Character orbit 768.f
Analytic conductor $45.313$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{8} - \beta_{5} + \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} - \beta_1) q^{7} + ( - \beta_{8} - \beta_{5} + \beta_{3} - \beta_1) q^{9} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + 2 \beta_1) q^{11} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_1 - 1) q^{13} + (\beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 4 \beta_1 - 7) q^{15} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + \beta_1) q^{17}+ \cdots + (2 \beta_{11} - 16 \beta_{10} - 12 \beta_{9} + 11 \beta_{8} + 18 \beta_{7} + \cdots + 243) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 12 q^{5} - 84 q^{15} - 180 q^{19} - 156 q^{21} - 120 q^{23} + 300 q^{25} + 130 q^{27} - 588 q^{29} - 116 q^{33} + 620 q^{39} + 372 q^{43} + 740 q^{45} + 1248 q^{47} - 948 q^{49} + 360 q^{51} + 948 q^{53} + 172 q^{57} - 2744 q^{63} - 2292 q^{67} - 3280 q^{69} - 2040 q^{71} + 216 q^{73} + 2522 q^{75} - 4824 q^{77} - 1076 q^{81} + 4156 q^{87} - 3480 q^{91} + 4180 q^{93} + 5448 q^{95} - 48 q^{97} + 3048 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} - 22\nu^{9} + 12\nu^{7} + 2399\nu^{5} + 11762\nu^{3} + 9468\nu ) / 612 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{10} + 61\nu^{8} + 539\nu^{6} + 1091\nu^{4} - 2504\nu^{2} - 3250 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{11} - 533\nu^{9} + 521\nu^{7} + 56558\nu^{5} + 242404\nu^{3} + 79280\nu ) / 3672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\nu^{11} + 559\nu^{9} + 3755\nu^{7} - 5803\nu^{5} - 90026\nu^{3} - 121156\nu ) / 1836 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 115 \nu^{11} - 68 \nu^{10} - 4145 \nu^{9} - 2176 \nu^{8} - 51439 \nu^{7} - 21488 \nu^{6} - 268540 \nu^{5} - 68612 \nu^{4} - 568472 \nu^{3} - 34000 \nu^{2} - 352984 \nu - 6800 ) / 7344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55 \nu^{11} - 68 \nu^{10} + 1601 \nu^{9} - 4216 \nu^{8} + 11767 \nu^{7} - 89624 \nu^{6} - 12248 \nu^{5} - 791996 \nu^{4} - 299872 \nu^{3} - 2646832 \nu^{2} - 454712 \nu - 1819136 ) / 7344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 121 \nu^{11} + 136 \nu^{10} + 4277 \nu^{9} + 4556 \nu^{8} + 51367 \nu^{7} + 49300 \nu^{6} + 254146 \nu^{5} + 195364 \nu^{4} + 497900 \nu^{3} + 223040 \nu^{2} + 281488 \nu + 48688 ) / 3672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 175 \nu^{11} + 1564 \nu^{10} - 6689 \nu^{9} + 53312 \nu^{8} - 91111 \nu^{7} + 600304 \nu^{6} - 549328 \nu^{5} + 2640508 \nu^{4} - 1436816 \nu^{3} + 4217360 \nu^{2} + \cdots + 2154784 ) / 7344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 91 \nu^{11} - 884 \nu^{10} + 3005 \nu^{9} - 30124 \nu^{8} + 31531 \nu^{7} - 338708 \nu^{6} + 113752 \nu^{5} - 1481312 \nu^{4} + 63728 \nu^{3} - 2314720 \nu^{2} + \cdots - 1122680 ) / 3672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7 \nu^{11} + 4 \nu^{10} - 249 \nu^{9} + 128 \nu^{8} - 3023 \nu^{7} + 1264 \nu^{6} - 15232 \nu^{5} + 4036 \nu^{4} - 30672 \nu^{3} + 2000 \nu^{2} - 18536 \nu + 256 ) / 144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 123 \nu^{11} - 136 \nu^{10} + 4321 \nu^{9} - 4556 \nu^{8} + 51343 \nu^{7} - 49300 \nu^{6} + 249348 \nu^{5} - 195364 \nu^{4} + 474376 \nu^{3} - 223040 \nu^{2} + 262552 \nu - 47464 ) / 1224 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} - 2\beta_{10} - 3\beta_{7} - 6\beta_{5} - 5\beta _1 - 1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{9} - 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} + \beta _1 - 52 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{11} + 20 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 36 \beta_{7} + 66 \beta_{5} - 9 \beta_{4} - 12 \beta_{3} + 107 \beta _1 + 10 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{11} + 8 \beta_{10} + 17 \beta_{9} + 31 \beta_{7} - 17 \beta_{6} - 41 \beta_{5} + 11 \beta_{2} - 24 \beta _1 + 565 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 124 \beta_{11} - 251 \beta_{10} - 147 \beta_{9} - 147 \beta_{8} - 519 \beta_{7} - 900 \beta_{5} + 162 \beta_{4} + 258 \beta_{3} - 2081 \beta _1 - 127 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 255 \beta_{11} - 198 \beta_{10} - 270 \beta_{9} + 6 \beta_{8} - 495 \beta_{7} + 264 \beta_{6} + 852 \beta_{5} - 102 \beta_{2} + 453 \beta _1 - 7463 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1756 \beta_{11} + 3593 \beta_{10} + 2727 \beta_{9} + 2727 \beta_{8} + 7995 \beta_{7} + 13506 \beta_{5} - 2493 \beta_{4} - 4734 \beta_{3} + 37598 \beta _1 + 1837 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4129 \beta_{11} + 3762 \beta_{10} + 4285 \beta_{9} - 186 \beta_{8} + 8102 \beta_{7} - 4099 \beta_{6} - 15199 \beta_{5} + 787 \beta_{2} - 7891 \beta _1 + 108400 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 26632 \beta_{11} - 54920 \beta_{10} - 46770 \beta_{9} - 46770 \beta_{8} - 126666 \beta_{7} - 211530 \beta_{5} + 37629 \beta_{4} + 82128 \beta_{3} - 649889 \beta _1 - 28288 ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 67192 \beta_{11} - 65744 \beta_{10} - 68453 \beta_{9} + 4056 \beta_{8} - 133123 \beta_{7} + 64397 \beta_{6} + 257573 \beta_{5} - 3551 \beta_{2} + 132936 \beta _1 - 1655233 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 417652 \beta_{11} + 865511 \beta_{10} + 779583 \beta_{9} + 779583 \beta_{8} + 2032539 \beta_{7} + 3376116 \beta_{5} - 574974 \beta_{4} - 1385826 \beta_{3} + 10952921 \beta _1 + 447859 ) / 24 \) Copy content Toggle raw display

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
4.03251i
4.03251i
1.08600i
1.08600i
0.910871i
0.910871i
2.29679i
2.29679i
3.14286i
3.14286i
2.36157i
2.36157i
0 −4.54315 2.52186i 0 8.01133 0 12.6015i 0 14.2805 + 22.9144i 0
383.2 0 −4.54315 + 2.52186i 0 8.01133 0 12.6015i 0 14.2805 22.9144i 0
383.3 0 −4.21320 3.04120i 0 −9.33303 0 36.3792i 0 8.50216 + 25.6264i 0
383.4 0 −4.21320 + 3.04120i 0 −9.33303 0 36.3792i 0 8.50216 25.6264i 0
383.5 0 −1.98169 4.80343i 0 11.9846 0 22.6995i 0 −19.1458 + 19.0378i 0
383.6 0 −1.98169 + 4.80343i 0 11.9846 0 22.6995i 0 −19.1458 19.0378i 0
383.7 0 1.40813 5.00172i 0 −5.86626 0 5.92149i 0 −23.0343 14.0861i 0
383.8 0 1.40813 + 5.00172i 0 −5.86626 0 5.92149i 0 −23.0343 + 14.0861i 0
383.9 0 3.16369 4.12202i 0 −21.4043 0 20.9034i 0 −6.98212 26.0816i 0
383.10 0 3.16369 + 4.12202i 0 −21.4043 0 20.9034i 0 −6.98212 + 26.0816i 0
383.11 0 5.16622 0.556921i 0 10.6077 0 7.90379i 0 26.3797 5.75435i 0
383.12 0 5.16622 + 0.556921i 0 10.6077 0 7.90379i 0 26.3797 + 5.75435i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 768.4.f.e 12
3.b odd 2 1 768.4.f.f 12
4.b odd 2 1 768.4.f.g 12
8.b even 2 1 768.4.f.h 12
8.d odd 2 1 768.4.f.f 12
12.b even 2 1 768.4.f.h 12
16.e even 4 1 384.4.c.a 12
16.e even 4 1 384.4.c.c yes 12
16.f odd 4 1 384.4.c.b yes 12
16.f odd 4 1 384.4.c.d yes 12
24.f even 2 1 inner 768.4.f.e 12
24.h odd 2 1 768.4.f.g 12
48.i odd 4 1 384.4.c.b yes 12
48.i odd 4 1 384.4.c.d yes 12
48.k even 4 1 384.4.c.a 12
48.k even 4 1 384.4.c.c yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.c.a 12 16.e even 4 1
384.4.c.a 12 48.k even 4 1
384.4.c.b yes 12 16.f odd 4 1
384.4.c.b yes 12 48.i odd 4 1
384.4.c.c yes 12 16.e even 4 1
384.4.c.c yes 12 48.k even 4 1
384.4.c.d yes 12 16.f odd 4 1
384.4.c.d yes 12 48.i odd 4 1
768.4.f.e 12 1.a even 1 1 trivial
768.4.f.e 12 24.f even 2 1 inner
768.4.f.f 12 3.b odd 2 1
768.4.f.f 12 8.d odd 2 1
768.4.f.g 12 4.b odd 2 1
768.4.f.g 12 24.h odd 2 1
768.4.f.h 12 8.b even 2 1
768.4.f.h 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{6} + 6T_{5}^{5} - 432T_{5}^{4} - 200T_{5}^{3} + 43968T_{5}^{2} - 26016T_{5} - 1193536 \) Copy content Toggle raw display
\( T_{19}^{6} + 90T_{19}^{5} - 18024T_{19}^{4} - 1956216T_{19}^{3} + 27304032T_{19}^{2} + 6866489760T_{19} + 146229576256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + 2 T^{10} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{6} + 6 T^{5} - 432 T^{4} + \cdots - 1193536)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 103646082371584 \) Copy content Toggle raw display
$11$ \( T^{12} + 8628 T^{10} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 226854686396416 \) Copy content Toggle raw display
$17$ \( T^{12} + 27792 T^{10} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{6} + 90 T^{5} + \cdots + 146229576256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 60 T^{5} + \cdots - 261412163584)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 294 T^{5} + \cdots + 4470831889088)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 169044 T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{12} + 317976 T^{10} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{12} + 354192 T^{10} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} - 186 T^{5} + \cdots - 58326025091776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 624 T^{5} + \cdots - 144142667350016)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 474 T^{5} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 1799556 T^{10} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{12} + 1802904 T^{10} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{6} + 1146 T^{5} + \cdots + 2132115742656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 1020 T^{5} + \cdots - 379600692064256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 108 T^{5} + \cdots - 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 1129140 T^{10} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{12} + 2756052 T^{10} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + 3691104 T^{10} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{6} + 24 T^{5} + \cdots + 63\!\cdots\!68)^{2} \) Copy content Toggle raw display
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