Properties

Label 684.7.h.c
Level $684$
Weight $7$
Character orbit 684.h
Analytic conductor $157.357$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(157.356993196\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{7} q^{5} + ( 46 + \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{7} +O(q^{10})\) \( q -\beta_{7} q^{5} + ( 46 + \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{7} + ( -112 - 2 \beta_{2} - 6 \beta_{6} - 9 \beta_{7} ) q^{11} + ( 2 \beta_{1} - \beta_{3} ) q^{13} + ( -206 - \beta_{2} - 27 \beta_{6} + 22 \beta_{7} ) q^{17} + ( 771 + 3 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 23 \beta_{6} + 9 \beta_{7} ) q^{19} + ( 2328 - 3 \beta_{2} - 48 \beta_{6} + 50 \beta_{7} ) q^{23} + ( -1475 - 10 \beta_{2} + 80 \beta_{6} - 99 \beta_{7} ) q^{25} + ( -14 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{29} + ( 24 \beta_{1} - 7 \beta_{3} + 7 \beta_{5} ) q^{31} + ( 12710 - 110 \beta_{2} - 150 \beta_{6} - 109 \beta_{7} ) q^{35} + ( -4 \beta_{1} + 11 \beta_{3} - 2 \beta_{4} + 13 \beta_{5} ) q^{37} + ( -64 \beta_{1} + 35 \beta_{3} + 4 \beta_{4} + 9 \beta_{5} ) q^{41} + ( -42068 + 100 \beta_{2} - 360 \beta_{6} + 391 \beta_{7} ) q^{43} + ( 71560 - 22 \beta_{2} + 1020 \beta_{6} - 23 \beta_{7} ) q^{47} + ( 55951 - 199 \beta_{2} + 555 \beta_{6} + 842 \beta_{7} ) q^{49} + ( 12 \beta_{1} - 60 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{53} + ( 136650 + 130 \beta_{2} + 1220 \beta_{6} - 781 \beta_{7} ) q^{55} + ( 99 \beta_{1} - 9 \beta_{3} - 20 \beta_{4} + 11 \beta_{5} ) q^{59} + ( -79408 - 180 \beta_{2} - 1702 \beta_{6} - 77 \beta_{7} ) q^{61} + ( -334 \beta_{1} + 62 \beta_{3} - 20 \beta_{4} + 12 \beta_{5} ) q^{65} + ( 209 \beta_{1} - 86 \beta_{3} + 48 \beta_{4} - 8 \beta_{5} ) q^{67} + ( 464 \beta_{1} + 74 \beta_{3} - 48 \beta_{4} + 40 \beta_{5} ) q^{71} + ( -106742 + 895 \beta_{2} - 789 \beta_{6} - 290 \beta_{7} ) q^{73} + ( -231402 - 924 \beta_{2} - 3306 \beta_{6} - 2963 \beta_{7} ) q^{77} + ( -854 \beta_{1} + 111 \beta_{3} + 48 \beta_{4} + 31 \beta_{5} ) q^{79} + ( -54774 - 882 \beta_{2} + 1644 \beta_{6} + 140 \beta_{7} ) q^{83} + ( -229610 + 570 \beta_{2} - 1030 \beta_{6} + 3223 \beta_{7} ) q^{85} + ( 1360 \beta_{1} - 41 \beta_{3} - 16 \beta_{4} - 141 \beta_{5} ) q^{89} + ( 1887 \beta_{1} - 349 \beta_{3} + 46 \beta_{4} - 73 \beta_{5} ) q^{91} + ( -78420 + 1126 \beta_{1} + 720 \beta_{2} - 68 \beta_{3} + 30 \beta_{4} + 122 \beta_{5} + 690 \beta_{6} + 395 \beta_{7} ) q^{95} + ( 778 \beta_{1} - 96 \beta_{3} - 60 \beta_{4} + 112 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{5} + 362q^{7} + O(q^{10}) \) \( 8q - 2q^{5} + 362q^{7} - 902q^{11} - 1550q^{17} + 6232q^{19} + 18820q^{23} - 12158q^{25} + 101762q^{35} - 335042q^{43} + 570394q^{47} + 448182q^{49} + 1089198q^{55} - 632014q^{61} - 852938q^{73} - 1850530q^{77} - 441200q^{83} - 1828374q^{85} - 627950q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5090 x^{6} + 8905881 x^{4} + 5831691048 x^{2} + 827887219200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 9 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{6} - 46859 \nu^{4} - 67416687 \nu^{2} - 4315657896 \)\()/21128796\)
\(\beta_{3}\)\(=\)\((\)\( 55 \nu^{7} + 204128 \nu^{5} + 211681665 \nu^{3} + 41835088584 \nu \)\()/42257592\)
\(\beta_{4}\)\(=\)\((\)\( 157 \nu^{7} + 518666 \nu^{5} + 500211621 \nu^{3} + 121606800264 \nu \)\()/84515184\)
\(\beta_{5}\)\(=\)\((\)\( 106 \nu^{7} + 361397 \nu^{5} + 334817847 \nu^{3} + 48274060356 \nu \)\()/21128796\)
\(\beta_{6}\)\(=\)\((\)\( -55 \nu^{6} - 204128 \nu^{4} - 211681665 \nu^{2} - 40102527312 \)\()/42257592\)
\(\beta_{7}\)\(=\)\((\)\( -89 \nu^{6} - 308974 \nu^{4} - 293772453 \nu^{2} - 46015049520 \)\()/14085864\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/9\)
\(\nu^{2}\)\(=\)\(\beta_{7} - 5 \beta_{6} + \beta_{2} - 1274\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{5} + 18 \beta_{4} + 9 \beta_{3} - 1583 \beta_{1}\)\()/9\)
\(\nu^{4}\)\(=\)\(-1625 \beta_{7} + 8221 \beta_{6} - 2285 \beta_{2} + 2026546\)
\(\nu^{5}\)\(=\)\((\)\(14625 \beta_{5} - 41130 \beta_{4} + 2331 \beta_{3} + 2606527 \beta_{1}\)\()/9\)
\(\nu^{6}\)\(=\)\(2182297 \beta_{7} - 12036101 \beta_{6} + 4631833 \beta_{2} - 3347179418\)
\(\nu^{7}\)\(=\)\((\)\(-19640673 \beta_{5} + 83372994 \beta_{4} - 36375255 \beta_{3} - 4341966479 \beta_{1}\)\()/9\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
42.5965i
42.5965i
38.1507i
38.1507i
13.8790i
13.8790i
40.3415i
40.3415i
0 0 0 −103.439 0 −619.422 0 0 0
37.2 0 0 0 −103.439 0 −619.422 0 0 0
37.3 0 0 0 −102.472 0 512.609 0 0 0
37.4 0 0 0 −102.472 0 512.609 0 0 0
37.5 0 0 0 18.1155 0 85.5200 0 0 0
37.6 0 0 0 18.1155 0 85.5200 0 0 0
37.7 0 0 0 186.796 0 202.293 0 0 0
37.8 0 0 0 186.796 0 202.293 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.7.h.c 8
3.b odd 2 1 76.7.c.b 8
12.b even 2 1 304.7.e.c 8
19.b odd 2 1 inner 684.7.h.c 8
57.d even 2 1 76.7.c.b 8
228.b odd 2 1 304.7.e.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.c.b 8 3.b odd 2 1
76.7.c.b 8 57.d even 2 1
304.7.e.c 8 12.b even 2 1
304.7.e.c 8 228.b odd 2 1
684.7.h.c 8 1.a even 1 1 trivial
684.7.h.c 8 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + T_{5}^{3} - 28210 T_{5}^{2} - 1475200 T_{5} + 35868000 \) acting on \(S_{7}^{\mathrm{new}}(684, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 35868000 - 1475200 T - 28210 T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( ( -5493140806 + 93234493 T - 330963 T^{2} - 181 T^{3} + T^{4} )^{2} \)
$11$ \( ( -70045150776 + 1452421460 T - 4134886 T^{2} + 451 T^{3} + T^{4} )^{2} \)
$13$ \( \)\(40\!\cdots\!00\)\( + 21740366264471651808 T^{2} + 90968814465705 T^{4} + 20512698 T^{6} + T^{8} \)
$17$ \( ( 56395405380078 + 6203796653 T - 20243875 T^{2} + 775 T^{3} + T^{4} )^{2} \)
$19$ \( \)\(48\!\cdots\!21\)\( - \)\(64\!\cdots\!12\)\( T + \)\(30\!\cdots\!08\)\( T^{2} - 33132962622494782952 T^{3} + 9232740094257854 T^{4} - 704269149992 T^{5} + 140039728 T^{6} - 6232 T^{7} + T^{8} \)
$23$ \( ( -329689946145300 + 563918448080 T - 54276811 T^{2} - 9410 T^{3} + T^{4} )^{2} \)
$29$ \( \)\(14\!\cdots\!00\)\( + \)\(29\!\cdots\!48\)\( T^{2} + 1586649112632837801 T^{4} + 2802614130 T^{6} + T^{8} \)
$31$ \( \)\(25\!\cdots\!00\)\( + \)\(90\!\cdots\!92\)\( T^{2} + 11248102188712326096 T^{4} + 5739788496 T^{6} + T^{8} \)
$37$ \( \)\(85\!\cdots\!00\)\( + \)\(45\!\cdots\!52\)\( T^{2} + 43719014470366470096 T^{4} + 13281482256 T^{6} + T^{8} \)
$41$ \( \)\(29\!\cdots\!00\)\( + \)\(18\!\cdots\!28\)\( T^{2} + \)\(38\!\cdots\!52\)\( T^{4} + 33188295504 T^{6} + T^{8} \)
$43$ \( ( -4346662716173125856 - 316371500354144 T + 2774402490 T^{2} + 167521 T^{3} + T^{4} )^{2} \)
$47$ \( ( -\)\(15\!\cdots\!12\)\( + 2432258045959076 T + 9897704330 T^{2} - 285197 T^{3} + T^{4} )^{2} \)
$53$ \( \)\(14\!\cdots\!00\)\( + \)\(94\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!53\)\( T^{4} + 70469336370 T^{6} + T^{8} \)
$59$ \( \)\(72\!\cdots\!00\)\( + \)\(24\!\cdots\!72\)\( T^{2} + \)\(77\!\cdots\!13\)\( T^{4} + 54182885322 T^{6} + T^{8} \)
$61$ \( ( -7254911790602711680 - 3823129093392224 T - 23927920458 T^{2} + 316007 T^{3} + T^{4} )^{2} \)
$67$ \( \)\(22\!\cdots\!00\)\( + \)\(10\!\cdots\!52\)\( T^{2} + \)\(49\!\cdots\!57\)\( T^{4} + 439547155026 T^{6} + T^{8} \)
$71$ \( \)\(77\!\cdots\!00\)\( + \)\(43\!\cdots\!72\)\( T^{2} + \)\(74\!\cdots\!28\)\( T^{4} + 474354989544 T^{6} + T^{8} \)
$73$ \( ( \)\(16\!\cdots\!62\)\( - 17773225199911541 T - 193182556047 T^{2} + 426469 T^{3} + T^{4} )^{2} \)
$79$ \( \)\(30\!\cdots\!00\)\( + \)\(30\!\cdots\!92\)\( T^{2} + \)\(30\!\cdots\!52\)\( T^{4} + 1008954259536 T^{6} + T^{8} \)
$83$ \( ( \)\(46\!\cdots\!88\)\( - 34337886760600384 T - 287093649748 T^{2} + 220600 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(16\!\cdots\!00\)\( + \)\(31\!\cdots\!32\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{4} + 2584899501672 T^{6} + T^{8} \)
$97$ \( \)\(24\!\cdots\!00\)\( + \)\(45\!\cdots\!68\)\( T^{2} + \)\(65\!\cdots\!76\)\( T^{4} + 1538138511624 T^{6} + T^{8} \)
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