Properties

Label 1008.3.cg.q
Level $1008$
Weight $3$
Character orbit 1008.cg
Analytic conductor $27.466$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.cg (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} + ( - \beta_{5} + \beta_{2} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} + ( - \beta_{5} + \beta_{2} - 1) q^{7} + (\beta_{13} + \beta_{4}) q^{11} + (\beta_{6} + \beta_{5}) q^{13} + (\beta_{14} - \beta_{7} + \beta_{3}) q^{17} + (\beta_{12} - \beta_{9} - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{19} + ( - 2 \beta_{14} - \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{3}) q^{23} + ( - \beta_{15} + 2 \beta_{12} - 2 \beta_{9} - \beta_{6} - 4 \beta_{5} + 2 \beta_1) q^{25} + (\beta_{14} - \beta_{10} - 2 \beta_{7} - 2 \beta_{4}) q^{29} + ( - 2 \beta_{15} - \beta_{12} - \beta_{9} + 2 \beta_{6} + 3 \beta_{5} - \beta_{2} + 2 \beta_1 + 6) q^{31} + ( - \beta_{14} + 2 \beta_{13} + \beta_{11} - \beta_{8} + 5 \beta_{7} + \beta_{4} - 4 \beta_{3}) q^{35} + (\beta_{15} - 4 \beta_{12} - 2 \beta_{9} - 2 \beta_{6} - 5 \beta_{5} + 2 \beta_{2} + 4 \beta_1 - 4) q^{37} + (2 \beta_{14} - 4 \beta_{13} + 2 \beta_{10} - 2 \beta_{7} - 2 \beta_{4}) q^{41} + (4 \beta_{15} - \beta_{12} - 5 \beta_{9} - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2} + 4 \beta_1 + 1) q^{43} + (2 \beta_{13} - 2 \beta_{11} + \beta_{10} + \beta_{8} + 9 \beta_{7} + 4 \beta_{4} + 2 \beta_{3}) q^{47} + (2 \beta_{15} - 6 \beta_{9} + \beta_{6} - 2 \beta_{2} + 4 \beta_1 - 9) q^{49} + ( - 2 \beta_{13} - 5 \beta_{7} - 2 \beta_{4} - 5 \beta_{3}) q^{53} + ( - 3 \beta_{12} - 4 \beta_{9} - 4 \beta_{6} - 28 \beta_{5} + 7 \beta_{2} + \cdots - 12) q^{55}+ \cdots + (8 \beta_{12} - 12 \beta_{9} + \beta_{6} - 45 \beta_{5} + 4 \beta_{2} + 4 \beta_1 - 23) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 24 q^{19} + 36 q^{25} + 84 q^{31} - 68 q^{37} - 80 q^{43} - 184 q^{49} + 216 q^{61} - 56 q^{67} + 156 q^{73} - 28 q^{79} + 448 q^{85} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!45 \nu^{15} + \cdots + 21\!\cdots\!24 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!90 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!90 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32\!\cdots\!85 \nu^{15} + \cdots - 16\!\cdots\!08 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\!\cdots\!73 \nu^{15} + \cdots - 21\!\cdots\!08 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!11 \nu^{15} + \cdots - 42\!\cdots\!00 ) / 25\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!21 \nu^{15} + \cdots + 91\!\cdots\!12 ) / 18\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38\!\cdots\!09 \nu^{15} + \cdots - 20\!\cdots\!64 ) / 46\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!46 \nu^{15} + \cdots - 10\!\cdots\!40 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 83\!\cdots\!77 \nu^{15} + \cdots + 12\!\cdots\!16 ) / 64\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!43 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 18\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 96\!\cdots\!19 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 64\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26\!\cdots\!95 \nu^{15} + \cdots - 11\!\cdots\!92 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!00 \nu^{15} + \cdots - 18\!\cdots\!16 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30\!\cdots\!35 \nu^{15} + \cdots - 18\!\cdots\!96 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + \beta_{13} + \beta_{10} - 3\beta_{9} - \beta_{8} + 4\beta_{7} - \beta_{4} + \beta _1 - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + 2 \beta_{14} - 19 \beta_{13} + 4 \beta_{12} + \beta_{11} + 5 \beta_{10} + 15 \beta_{9} + 25 \beta_{8} - 6 \beta_{7} + 22 \beta_{6} + 176 \beta_{5} - 8 \beta_{4} + 41 \beta_{3} - 45 \beta_{2} + 2 \beta _1 + 47 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 109 \beta_{15} - 4 \beta_{14} + 16 \beta_{13} + 59 \beta_{12} + 32 \beta_{11} - 50 \beta_{10} + 105 \beta_{9} - 305 \beta_{7} + 27 \beta_{6} - 317 \beta_{5} + 41 \beta_{4} + 135 \beta_{3} - 9 \beta_{2} + 29 \beta _1 + 332 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 279 \beta_{15} - 468 \beta_{14} + 871 \beta_{13} - 442 \beta_{12} + 198 \beta_{11} - 563 \beta_{10} - 1557 \beta_{9} - 941 \beta_{8} + 1315 \beta_{7} - 1338 \beta_{6} - 14216 \beta_{5} + 428 \beta_{4} - 3351 \beta_{3} + \cdots - 6081 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5209 \beta_{15} - 24 \beta_{14} - 1485 \beta_{13} - 5865 \beta_{12} - 3017 \beta_{11} + 1797 \beta_{10} - 2202 \beta_{9} + 707 \beta_{8} + 16955 \beta_{7} - 1410 \beta_{6} + 25140 \beta_{5} - 2985 \beta_{4} + \cdots - 22583 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24675 \beta_{15} + 32956 \beta_{14} - 45339 \beta_{13} + 43890 \beta_{12} - 8589 \beta_{11} + 27685 \beta_{10} + 79415 \beta_{9} + 39298 \beta_{8} - 94080 \beta_{7} + 82082 \beta_{6} + \cdots + 456701 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 281111 \beta_{15} + 7840 \beta_{14} + 70358 \beta_{13} + 314552 \beta_{12} + 189210 \beta_{11} - 87940 \beta_{10} + 59028 \beta_{9} - 58808 \beta_{8} - 929365 \beta_{7} + 47320 \beta_{6} + \cdots + 1501223 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1823989 \beta_{15} - 1925334 \beta_{14} + 2500899 \beta_{13} - 2474216 \beta_{12} + 189113 \beta_{11} - 1228677 \beta_{10} - 4490331 \beta_{9} - 2225317 \beta_{8} + 6002452 \beta_{7} + \cdots - 31098363 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 15761336 \beta_{15} - 406340 \beta_{14} - 4194951 \beta_{13} - 16548277 \beta_{12} - 10315623 \beta_{11} + 5308169 \beta_{10} - 1948744 \beta_{9} + 4285624 \beta_{8} + \cdots - 80877241 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 127051415 \beta_{15} + 107202502 \beta_{14} - 138044043 \beta_{13} + 143357780 \beta_{12} - 3359816 \beta_{11} + 55725989 \beta_{10} + 271490321 \beta_{9} + \cdots + 1912446727 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 869585982 \beta_{15} - 516652 \beta_{14} + 290868584 \beta_{13} + 894758565 \beta_{12} + 576135224 \beta_{11} - 329410426 \beta_{10} + 28956455 \beta_{9} + \cdots + 3998642990 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 8475803371 \beta_{15} - 5956688682 \beta_{14} + 7417316179 \beta_{13} - 9265432204 \beta_{12} - 419542935 \beta_{11} - 2598095535 \beta_{10} - 16031100855 \beta_{9} + \cdots - 112654415615 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 46926047268 \beta_{15} + 2026278086 \beta_{14} - 19502362417 \beta_{13} - 47573182678 \beta_{12} - 32663193668 \beta_{11} + 20102185643 \beta_{10} + \cdots - 192474230932 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 546882905353 \beta_{15} + 333119813700 \beta_{14} - 393537231435 \beta_{13} + 600336340426 \beta_{12} + 70136917089 \beta_{11} + 120588039003 \beta_{10} + \cdots + 6633855205101 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1 + \beta_{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−0.338813 1.51822i
−3.20903 3.57433i
1.75486 4.85286i
2.29298 + 1.67052i
2.29298 + 3.14719i
1.75486 2.33310i
−3.20903 + 1.64043i
−0.338813 + 7.55243i
−0.338813 + 1.51822i
−3.20903 + 3.57433i
1.75486 + 4.85286i
2.29298 1.67052i
2.29298 3.14719i
1.75486 + 2.33310i
−3.20903 1.64043i
−0.338813 7.55243i
0 0 0 −7.85541 4.53532i 0 −1.17763 + 6.90023i 0 0 0
145.2 0 0 0 −4.51611 2.60738i 0 −6.91807 1.06788i 0 0 0
145.3 0 0 0 −2.18217 1.25988i 0 3.00972 6.31993i 0 0 0
145.4 0 0 0 −1.27883 0.738332i 0 4.08597 + 5.68374i 0 0 0
145.5 0 0 0 1.27883 + 0.738332i 0 4.08597 + 5.68374i 0 0 0
145.6 0 0 0 2.18217 + 1.25988i 0 3.00972 6.31993i 0 0 0
145.7 0 0 0 4.51611 + 2.60738i 0 −6.91807 1.06788i 0 0 0
145.8 0 0 0 7.85541 + 4.53532i 0 −1.17763 + 6.90023i 0 0 0
577.1 0 0 0 −7.85541 + 4.53532i 0 −1.17763 6.90023i 0 0 0
577.2 0 0 0 −4.51611 + 2.60738i 0 −6.91807 + 1.06788i 0 0 0
577.3 0 0 0 −2.18217 + 1.25988i 0 3.00972 + 6.31993i 0 0 0
577.4 0 0 0 −1.27883 + 0.738332i 0 4.08597 5.68374i 0 0 0
577.5 0 0 0 1.27883 0.738332i 0 4.08597 5.68374i 0 0 0
577.6 0 0 0 2.18217 1.25988i 0 3.00972 + 6.31993i 0 0 0
577.7 0 0 0 4.51611 2.60738i 0 −6.91807 + 1.06788i 0 0 0
577.8 0 0 0 7.85541 4.53532i 0 −1.17763 6.90023i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.cg.q 16
3.b odd 2 1 inner 1008.3.cg.q 16
4.b odd 2 1 504.3.by.d 16
7.d odd 6 1 inner 1008.3.cg.q 16
12.b even 2 1 504.3.by.d 16
21.g even 6 1 inner 1008.3.cg.q 16
28.f even 6 1 504.3.by.d 16
28.f even 6 1 3528.3.f.i 16
28.g odd 6 1 3528.3.f.i 16
84.j odd 6 1 504.3.by.d 16
84.j odd 6 1 3528.3.f.i 16
84.n even 6 1 3528.3.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.by.d 16 4.b odd 2 1
504.3.by.d 16 12.b even 2 1
504.3.by.d 16 28.f even 6 1
504.3.by.d 16 84.j odd 6 1
1008.3.cg.q 16 1.a even 1 1 trivial
1008.3.cg.q 16 3.b odd 2 1 inner
1008.3.cg.q 16 7.d odd 6 1 inner
1008.3.cg.q 16 21.g even 6 1 inner
3528.3.f.i 16 28.f even 6 1
3528.3.f.i 16 28.g odd 6 1
3528.3.f.i 16 84.j odd 6 1
3528.3.f.i 16 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{16} - 118 T_{5}^{14} + 10739 T_{5}^{12} - 334630 T_{5}^{10} + 7682449 T_{5}^{8} - 58300664 T_{5}^{6} + 325701440 T_{5}^{4} - 638105600 T_{5}^{2} + 959512576 \) Copy content Toggle raw display
\( T_{11}^{16} + 666 T_{11}^{14} + 303067 T_{11}^{12} + 73620906 T_{11}^{10} + 12902674833 T_{11}^{8} + 1144098699168 T_{11}^{6} + 72351409853440 T_{11}^{4} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
\( T_{13}^{8} + 766T_{13}^{6} + 165641T_{13}^{4} + 7200248T_{13}^{2} + 63043600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 118 T^{14} + \cdots + 959512576 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + 48 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 666 T^{14} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{8} + 766 T^{6} + 165641 T^{4} + \cdots + 63043600)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 920 T^{14} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{8} - 12 T^{7} - 95 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 3152 T^{14} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} - 4554 T^{6} + \cdots + 142968684544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 42 T^{7} - 1544 T^{6} + \cdots + 74287318249)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 34 T^{7} + \cdots + 212012360704)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 10520 T^{6} + \cdots + 12478867111936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} - 5901 T^{2} + \cdots + 3352240)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} - 12976 T^{14} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + 7314 T^{14} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} - 10038 T^{14} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} - 108 T^{7} + \cdots + 2040555110400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 28 T^{7} + 6317 T^{6} + \cdots + 2997781504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 6544 T^{6} + \cdots + 21724401664)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 78 T^{7} + \cdots + 999648030976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 14 T^{7} + \cdots + 3096660231289)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 5366 T^{6} + \cdots + 1728025927936)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 46136 T^{14} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{8} + 48470 T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
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