Newspace parameters
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.bg (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(26.1581053786\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2\cdot 5 \) |
Twist minimal: | no (minimal twist has level 120) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) :
\(\beta_{1}\) | \(=\) | \( ( - 3096 \nu^{7} + 17905 \nu^{6} - 34598 \nu^{5} - 15150 \nu^{4} - 876904 \nu^{3} + 4737945 \nu^{2} - 10010942 \nu - 2940830 ) / 1277880 \) |
\(\beta_{2}\) | \(=\) | \( ( - 2465 \nu^{7} + 10455 \nu^{6} - 21142 \nu^{5} + 2900 \nu^{4} - 663785 \nu^{3} + 2966245 \nu^{2} - 6107718 \nu - 838660 ) / 958410 \) |
\(\beta_{3}\) | \(=\) | \( ( 12659 \nu^{7} - 84537 \nu^{6} + 213112 \nu^{5} - 71270 \nu^{4} + 3896171 \nu^{3} - 22880593 \nu^{2} + 66899448 \nu - 19263470 ) / 3833640 \) |
\(\beta_{4}\) | \(=\) | \( ( 6953 \nu^{7} - 12966 \nu^{6} - 11750 \nu^{5} - 8180 \nu^{4} + 1740137 \nu^{3} - 3756574 \nu^{2} + 814710 \nu + 9543760 ) / 1916820 \) |
\(\beta_{5}\) | \(=\) | \( ( - 4775 \nu^{7} + 20987 \nu^{6} - 31148 \nu^{5} + 24410 \nu^{4} - 1286695 \nu^{3} + 5631403 \nu^{2} - 9748972 \nu + 227650 ) / 1277880 \) |
\(\beta_{6}\) | \(=\) | \( ( - 39706 \nu^{7} + 125445 \nu^{6} - 335810 \nu^{5} + 103090 \nu^{4} - 10077874 \nu^{3} + 36751685 \nu^{2} - 92366250 \nu - 3625190 ) / 3833640 \) |
\(\beta_{7}\) | \(=\) | \( ( - 6773 \nu^{7} + 25422 \nu^{6} - 63258 \nu^{5} - 4560 \nu^{4} - 1857557 \nu^{3} + 7228198 \nu^{2} - 17092522 \nu - 3992560 ) / 425960 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} - \beta_{6} + 4\beta_{5} + 3\beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 10 \) |
\(\nu^{2}\) | \(=\) | \( ( -3\beta_{7} - 7\beta_{6} - 22\beta_{5} + \beta_{4} + \beta_{3} + 104\beta_{2} - 24\beta _1 + 5 ) / 10 \) |
\(\nu^{3}\) | \(=\) | \( ( -41\beta_{7} + 51\beta_{6} - 24\beta_{5} + 7\beta_{4} + 17\beta_{3} + 28\beta_{2} + 92\beta _1 - 45 ) / 10 \) |
\(\nu^{4}\) | \(=\) | \( ( 3\beta_{7} - 3\beta_{6} + 312\beta_{5} + 99\beta_{4} - 31\beta_{3} - 34\beta_{2} - 346\beta _1 - 1505 ) / 10 \) |
\(\nu^{5}\) | \(=\) | \( ( -279\beta_{7} + 29\beta_{6} - 866\beta_{5} - 837\beta_{4} - 587\beta_{3} + 642\beta_{2} + 308\beta _1 + 55 ) / 10 \) |
\(\nu^{6}\) | \(=\) | \( ( 297 \beta_{7} + 1493 \beta_{6} + 6178 \beta_{5} + 301 \beta_{4} + 301 \beta_{3} - 22396 \beta_{2} + 6776 \beta _1 - 895 ) / 10 \) |
\(\nu^{7}\) | \(=\) | \( ( 8609 \beta_{7} - 13599 \beta_{6} + 4076 \beta_{5} + 457 \beta_{4} - 4533 \beta_{3} + 8228 \beta_{2} - 23008 \beta _1 - 3695 ) / 10 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).
\(n\) | \(511\) | \(577\) | \(641\) | \(901\) |
\(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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193.1 |
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0 | −1.22474 | − | 1.22474i | 0 | −1.88489 | − | 4.63111i | 0 | −6.02356 | + | 6.02356i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
193.2 | 0 | −1.22474 | − | 1.22474i | 0 | 3.66015 | + | 3.40637i | 0 | 2.57407 | − | 2.57407i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
193.3 | 0 | 1.22474 | + | 1.22474i | 0 | 0.862117 | + | 4.92511i | 0 | −7.33876 | + | 7.33876i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
193.4 | 0 | 1.22474 | + | 1.22474i | 0 | 3.36263 | − | 3.70037i | 0 | 8.78825 | − | 8.78825i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||
577.1 | 0 | −1.22474 | + | 1.22474i | 0 | −1.88489 | + | 4.63111i | 0 | −6.02356 | − | 6.02356i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
577.2 | 0 | −1.22474 | + | 1.22474i | 0 | 3.66015 | − | 3.40637i | 0 | 2.57407 | + | 2.57407i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
577.3 | 0 | 1.22474 | − | 1.22474i | 0 | 0.862117 | − | 4.92511i | 0 | −7.33876 | − | 7.33876i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
577.4 | 0 | 1.22474 | − | 1.22474i | 0 | 3.36263 | + | 3.70037i | 0 | 8.78825 | + | 8.78825i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.3.bg.k | 8 | |
4.b | odd | 2 | 1 | 960.3.bg.l | 8 | ||
5.c | odd | 4 | 1 | inner | 960.3.bg.k | 8 | |
8.b | even | 2 | 1 | 240.3.bg.e | 8 | ||
8.d | odd | 2 | 1 | 120.3.u.b | ✓ | 8 | |
20.e | even | 4 | 1 | 960.3.bg.l | 8 | ||
24.f | even | 2 | 1 | 360.3.v.f | 8 | ||
24.h | odd | 2 | 1 | 720.3.bh.o | 8 | ||
40.e | odd | 2 | 1 | 600.3.u.h | 8 | ||
40.f | even | 2 | 1 | 1200.3.bg.q | 8 | ||
40.i | odd | 4 | 1 | 240.3.bg.e | 8 | ||
40.i | odd | 4 | 1 | 1200.3.bg.q | 8 | ||
40.k | even | 4 | 1 | 120.3.u.b | ✓ | 8 | |
40.k | even | 4 | 1 | 600.3.u.h | 8 | ||
120.m | even | 2 | 1 | 1800.3.v.t | 8 | ||
120.q | odd | 4 | 1 | 360.3.v.f | 8 | ||
120.q | odd | 4 | 1 | 1800.3.v.t | 8 | ||
120.w | even | 4 | 1 | 720.3.bh.o | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.3.u.b | ✓ | 8 | 8.d | odd | 2 | 1 | |
120.3.u.b | ✓ | 8 | 40.k | even | 4 | 1 | |
240.3.bg.e | 8 | 8.b | even | 2 | 1 | ||
240.3.bg.e | 8 | 40.i | odd | 4 | 1 | ||
360.3.v.f | 8 | 24.f | even | 2 | 1 | ||
360.3.v.f | 8 | 120.q | odd | 4 | 1 | ||
600.3.u.h | 8 | 40.e | odd | 2 | 1 | ||
600.3.u.h | 8 | 40.k | even | 4 | 1 | ||
720.3.bh.o | 8 | 24.h | odd | 2 | 1 | ||
720.3.bh.o | 8 | 120.w | even | 4 | 1 | ||
960.3.bg.k | 8 | 1.a | even | 1 | 1 | trivial | |
960.3.bg.k | 8 | 5.c | odd | 4 | 1 | inner | |
960.3.bg.l | 8 | 4.b | odd | 2 | 1 | ||
960.3.bg.l | 8 | 20.e | even | 4 | 1 | ||
1200.3.bg.q | 8 | 40.f | even | 2 | 1 | ||
1200.3.bg.q | 8 | 40.i | odd | 4 | 1 | ||
1800.3.v.t | 8 | 120.m | even | 2 | 1 | ||
1800.3.v.t | 8 | 120.q | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 120T_{7}^{5} + 20900T_{7}^{4} + 120000T_{7}^{3} + 320000T_{7}^{2} - 3200000T_{7} + 16000000 \)
acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 9)^{2} \)
$5$
\( T^{8} - 12 T^{7} + 114 T^{6} + \cdots + 390625 \)
$7$
\( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 16000000 \)
$11$
\( (T^{4} - 16 T^{3} - 358 T^{2} + \cdots - 32288)^{2} \)
$13$
\( T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 53824 \)
$17$
\( T^{8} - 52 T^{7} + \cdots + 2570084416 \)
$19$
\( T^{8} + 888 T^{6} + \cdots + 82591744 \)
$23$
\( T^{8} - 40 T^{7} + \cdots + 3064729600 \)
$29$
\( T^{8} + 4500 T^{6} + \cdots + 619810816 \)
$31$
\( (T^{4} + 48 T^{3} - 2104 T^{2} + \cdots - 1815680)^{2} \)
$37$
\( T^{8} - 60 T^{7} + \cdots + 16133718222400 \)
$41$
\( (T^{4} + 76 T^{3} - 948 T^{2} + \cdots - 63872)^{2} \)
$43$
\( T^{8} + 88 T^{7} + \cdots + 949580087296 \)
$47$
\( T^{8} - 16 T^{7} + \cdots + 287296000000 \)
$53$
\( T^{8} + 108 T^{7} + \cdots + 5939943840000 \)
$59$
\( T^{8} + 2348 T^{6} + 301860 T^{4} + \cdots + 861184 \)
$61$
\( (T^{4} + 132 T^{3} - 1012 T^{2} + \cdots - 13929728)^{2} \)
$67$
\( T^{8} + 216 T^{7} + \cdots + 3555366682624 \)
$71$
\( (T^{4} - 120 T^{3} + 176 T^{2} + \cdots - 4371968)^{2} \)
$73$
\( T^{8} + 208 T^{7} + \cdots + 22\!\cdots\!00 \)
$79$
\( T^{8} + 17424 T^{6} + \cdots + 1421016580096 \)
$83$
\( T^{8} - 336 T^{7} + 56448 T^{6} + \cdots + 2166784 \)
$89$
\( T^{8} + 37512 T^{6} + \cdots + 693289369600 \)
$97$
\( T^{8} + 208 T^{7} + \cdots + 10\!\cdots\!76 \)
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