Properties

Label 1512.2.s.m
Level $1512$
Weight $2$
Character orbit 1512.s
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.9391935744.3
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 12 x^{5} - 76 x^{4} + 84 x^{3} + 245 x^{2} - 1372 x + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{13} + ( 1 + \beta_{3} + 3 \beta_{6} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( 1 + \beta_{3} + 2 \beta_{6} ) q^{25} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{29} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{35} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( -1 - 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{41} + ( -1 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{53} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{55} + ( 5 + \beta_{1} + 5 \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{65} + ( \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{67} + ( 6 - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{71} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{73} + ( 7 - \beta_{1} - \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{7} ) q^{79} + ( -4 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( 8 - 2 \beta_{5} ) q^{85} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{5} - \beta_{6} ) q^{89} + ( -7 + \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{91} + ( -8 + \beta_{1} - 8 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{95} + ( 1 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} - 2q^{7} + O(q^{10}) \) \( 8q - 4q^{5} - 2q^{7} - 2q^{11} - 8q^{13} + 4q^{17} - 6q^{19} + 2q^{23} + 4q^{25} + 16q^{29} - 6q^{31} - 2q^{35} - 16q^{41} - 20q^{47} - 6q^{49} - 10q^{53} + 16q^{55} + 22q^{59} + 2q^{61} - 14q^{65} + 2q^{67} + 44q^{71} - 10q^{73} + 54q^{77} + 8q^{79} - 40q^{83} + 64q^{85} - 16q^{89} - 24q^{91} - 30q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 5 x^{6} + 12 x^{5} - 76 x^{4} + 84 x^{3} + 245 x^{2} - 1372 x + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 3 \nu^{6} - 26 \nu^{5} - 194 \nu^{4} - 1282 \nu^{3} + 3290 \nu^{2} - 1911 \nu + 343 \)\()/12348\)
\(\beta_{3}\)\(=\)\((\)\( 13 \nu^{7} + 18 \nu^{6} + 128 \nu^{5} + 506 \nu^{4} - 1520 \nu^{3} + 574 \nu^{2} + 2205 \nu - 42532 \)\()/37044\)
\(\beta_{4}\)\(=\)\((\)\( 19 \nu^{7} + 15 \nu^{6} + 74 \nu^{5} - 346 \nu^{4} + 334 \nu^{3} - 518 \nu^{2} - 8967 \nu + 3773 \)\()/12348\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} - 4 \nu^{6} - 10 \nu^{5} + 48 \nu^{4} - 146 \nu^{3} - 468 \nu^{2} + 1169 \nu - 1960 \)\()/1764\)
\(\beta_{6}\)\(=\)\((\)\( -43 \nu^{7} + 53 \nu^{6} + 114 \nu^{5} - 866 \nu^{4} + 762 \nu^{3} - 2506 \nu^{2} - 5929 \nu + 27783 \)\()/24696\)
\(\beta_{7}\)\(=\)\((\)\( -124 \nu^{7} + 405 \nu^{6} - 746 \nu^{5} - 2384 \nu^{4} + 5882 \nu^{3} + 224 \nu^{2} - 34398 \nu + 154693 \)\()/37044\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{6} - 4 \beta_{5} + \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 2 \beta_{1} - 7\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 10 \beta_{6} - 4 \beta_{5} - 11 \beta_{4} + 25 \beta_{3} - 11 \beta_{2} - 10 \beta_{1} + 35\)
\(\nu^{5}\)\(=\)\(-17 \beta_{7} + 32 \beta_{6} - 28 \beta_{5} + 14 \beta_{4} + 67 \beta_{3} - 17 \beta_{2} + 14 \beta_{1} + 77\)
\(\nu^{6}\)\(=\)\(74 \beta_{7} - 24 \beta_{6} + 50 \beta_{4} + 362 \beta_{3} - 24 \beta_{2} + 74 \beta_{1} + 119\)
\(\nu^{7}\)\(=\)\(26 \beta_{7} - 272 \beta_{6} + 52 \beta_{5} + 338 \beta_{4} + 14 \beta_{3} + 169 \beta_{1} + 168\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
2.61033 0.431486i
−2.47635 + 0.931486i
1.57052 + 2.12920i
0.295509 2.62920i
2.61033 + 0.431486i
−2.47635 0.931486i
1.57052 2.12920i
0.295509 + 2.62920i
0 0 0 −1.36603 2.36603i 0 −0.931486 + 2.47635i 0 0 0
865.2 0 0 0 −1.36603 2.36603i 0 0.431486 2.61033i 0 0 0
865.3 0 0 0 0.366025 + 0.633975i 0 −2.62920 + 0.295509i 0 0 0
865.4 0 0 0 0.366025 + 0.633975i 0 2.12920 + 1.57052i 0 0 0
1297.1 0 0 0 −1.36603 + 2.36603i 0 −0.931486 2.47635i 0 0 0
1297.2 0 0 0 −1.36603 + 2.36603i 0 0.431486 + 2.61033i 0 0 0
1297.3 0 0 0 0.366025 0.633975i 0 −2.62920 0.295509i 0 0 0
1297.4 0 0 0 0.366025 0.633975i 0 2.12920 1.57052i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1297.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.s.m 8
3.b odd 2 1 1512.2.s.p yes 8
7.c even 3 1 inner 1512.2.s.m 8
21.h odd 6 1 1512.2.s.p yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.m 8 1.a even 1 1 trivial
1512.2.s.m 8 7.c even 3 1 inner
1512.2.s.p yes 8 3.b odd 2 1
1512.2.s.p yes 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{4} + 2 T_{5}^{3} + 6 T_{5}^{2} - 4 T_{5} + 4 \)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{4} + 4 T_{13}^{3} - 35 T_{13}^{2} - 204 T_{13} - 264 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 2 T - 4 T^{2} - 4 T^{3} + 19 T^{4} - 20 T^{5} - 100 T^{6} + 250 T^{7} + 625 T^{8} )^{2} \)
$7$ \( 1 + 2 T + 5 T^{2} + 18 T^{3} + 8 T^{4} + 126 T^{5} + 245 T^{6} + 686 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 31922 T^{9} - 20328 T^{10} + 34606 T^{11} - 849178 T^{12} - 7730448 T^{13} - 24801854 T^{14} + 38974342 T^{15} + 214358881 T^{16} \)
$13$ \( ( 1 + 4 T + 17 T^{2} - 48 T^{3} - 160 T^{4} - 624 T^{5} + 2873 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 2 T - 4 T^{2} + 52 T^{3} - 293 T^{4} + 884 T^{5} - 1156 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( 1 + 6 T - 5 T^{2} + 114 T^{3} + 877 T^{4} - 936 T^{5} + 9706 T^{6} + 54060 T^{7} - 159614 T^{8} + 1027140 T^{9} + 3503866 T^{10} - 6420024 T^{11} + 114291517 T^{12} + 282275286 T^{13} - 235229405 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 2 T - 10 T^{2} + 72 T^{3} - 922 T^{4} + 2222 T^{5} + 760 T^{6} - 57846 T^{7} + 692635 T^{8} - 1330458 T^{9} + 402040 T^{10} + 27035074 T^{11} - 258013402 T^{12} + 463416696 T^{13} - 1480358890 T^{14} - 6809650894 T^{15} + 78310985281 T^{16} \)
$29$ \( ( 1 - 8 T + 36 T^{2} - 24 T^{3} - 182 T^{4} - 696 T^{5} + 30276 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( 1 + 6 T - 38 T^{2} - 348 T^{3} + 517 T^{4} + 10584 T^{5} + 34246 T^{6} - 188550 T^{7} - 2230412 T^{8} - 5845050 T^{9} + 32910406 T^{10} + 315307944 T^{11} + 477460357 T^{12} - 9962944548 T^{13} - 33725139878 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 61 T^{2} - 648 T^{3} + 2017 T^{4} + 31752 T^{5} + 168050 T^{6} - 910440 T^{7} - 7948706 T^{8} - 33686280 T^{9} + 230060450 T^{10} + 1608334056 T^{11} + 3780182737 T^{12} - 44934884136 T^{13} - 156509310949 T^{14} + 3512479453921 T^{16} \)
$41$ \( ( 1 + 8 T + 60 T^{2} + 456 T^{3} + 2554 T^{4} + 18696 T^{5} + 100860 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 14 T^{2} - 72 T^{3} - 1881 T^{4} - 3096 T^{5} + 25886 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 20 T + 196 T^{2} + 1464 T^{3} + 7286 T^{4} + 1508 T^{5} - 284064 T^{6} - 3306668 T^{7} - 27412733 T^{8} - 155413396 T^{9} - 627497376 T^{10} + 156565084 T^{11} + 35553355766 T^{12} + 335761090248 T^{13} + 2112726204484 T^{14} + 10132462409260 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 10 T - 34 T^{2} - 720 T^{3} - 538 T^{4} + 28730 T^{5} + 169816 T^{6} - 937470 T^{7} - 17639549 T^{8} - 49685910 T^{9} + 477013144 T^{10} + 4277236210 T^{11} - 4245078778 T^{12} - 301100754960 T^{13} - 753588278386 T^{14} + 11747111398370 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 22 T + 106 T^{2} - 48 T^{3} + 14486 T^{4} - 158758 T^{5} + 199560 T^{6} - 3891698 T^{7} + 83505403 T^{8} - 229610182 T^{9} + 694668360 T^{10} - 32605559282 T^{11} + 175532091446 T^{12} - 34316366352 T^{13} + 4471136565946 T^{14} - 54750332666018 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 2 T - 90 T^{2} + 116 T^{3} + 4577 T^{4} - 7740 T^{5} + 332338 T^{6} - 11534 T^{7} - 30676068 T^{8} - 703574 T^{9} + 1236629698 T^{10} - 1756832940 T^{11} + 63372414257 T^{12} + 97973170916 T^{13} - 4636833692490 T^{14} - 6285485672042 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 2 T - 145 T^{2} - 310 T^{3} + 10229 T^{4} + 49232 T^{5} - 447054 T^{6} - 1987140 T^{7} + 26208538 T^{8} - 133138380 T^{9} - 2006825406 T^{10} + 14807164016 T^{11} + 206125816709 T^{12} - 418538783170 T^{13} - 13116465414505 T^{14} - 12121423210646 T^{15} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 22 T + 402 T^{2} - 4638 T^{3} + 45946 T^{4} - 329298 T^{5} + 2026482 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( 1 + 10 T - 141 T^{2} - 1714 T^{3} + 12677 T^{4} + 151788 T^{5} - 657782 T^{6} - 4715864 T^{7} + 43753554 T^{8} - 344258072 T^{9} - 3505320278 T^{10} + 59048112396 T^{11} + 360004501157 T^{12} - 3553244710402 T^{13} - 21338125906749 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 8 T - 201 T^{2} + 920 T^{3} + 28853 T^{4} - 55488 T^{5} - 3083690 T^{6} + 2064928 T^{7} + 256704030 T^{8} + 163129312 T^{9} - 19245309290 T^{10} - 27357748032 T^{11} + 1123826687093 T^{12} + 2830891887080 T^{13} - 48860578559721 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} \)
$83$ \( ( 1 + 20 T + 360 T^{2} + 3684 T^{3} + 40318 T^{4} + 305772 T^{5} + 2480040 T^{6} + 11435740 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( 1 + 16 T - 92 T^{2} - 1248 T^{3} + 22646 T^{4} + 83536 T^{5} - 3013152 T^{6} - 6008080 T^{7} + 235094707 T^{8} - 534719120 T^{9} - 23867176992 T^{10} + 58890290384 T^{11} + 1420860789686 T^{12} - 6968906192352 T^{13} - 45722278768412 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \)
$97$ \( ( 1 + 230 T^{2} + 72 T^{3} + 26415 T^{4} + 6984 T^{5} + 2164070 T^{6} + 88529281 T^{8} )^{2} \)
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