# Properties

 Label 98.5.d.c Level $98$ Weight $5$ Character orbit 98.d Analytic conductor $10.130$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 98.d (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1302563822$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -2 \beta_{6} q^{2} + ( 4 \beta_{1} + 7 \beta_{3} ) q^{3} + ( -8 - 8 \beta_{4} ) q^{4} + ( 13 \beta_{3} + 5 \beta_{5} + 13 \beta_{7} ) q^{5} + ( 6 \beta_{1} - 6 \beta_{5} - 22 \beta_{7} ) q^{6} -16 \beta_{2} q^{8} + ( -49 \beta_{4} - 89 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{6} q^{2} + ( 4 \beta_{1} + 7 \beta_{3} ) q^{3} + ( -8 - 8 \beta_{4} ) q^{4} + ( 13 \beta_{3} + 5 \beta_{5} + 13 \beta_{7} ) q^{5} + ( 6 \beta_{1} - 6 \beta_{5} - 22 \beta_{7} ) q^{6} -16 \beta_{2} q^{8} + ( -49 \beta_{4} - 89 \beta_{6} ) q^{9} + ( 16 \beta_{1} + 36 \beta_{3} ) q^{10} + ( -6 + 103 \beta_{2} - 6 \beta_{4} + 103 \beta_{6} ) q^{11} + ( -56 \beta_{3} - 32 \beta_{5} - 56 \beta_{7} ) q^{12} + ( -101 \beta_{1} + 101 \beta_{5} - 97 \beta_{7} ) q^{13} + ( 222 + 158 \beta_{2} ) q^{15} + 64 \beta_{4} q^{16} + ( -282 \beta_{1} - 177 \beta_{3} ) q^{17} + ( -356 - 98 \beta_{2} - 356 \beta_{4} - 98 \beta_{6} ) q^{18} + ( 10 \beta_{3} - 207 \beta_{5} + 10 \beta_{7} ) q^{19} + ( 40 \beta_{1} - 40 \beta_{5} - 104 \beta_{7} ) q^{20} + ( 412 - 12 \beta_{2} ) q^{22} + ( 26 \beta_{4} + 144 \beta_{6} ) q^{23} + ( -48 \beta_{1} - 176 \beta_{3} ) q^{24} + ( -237 + 274 \beta_{2} - 237 \beta_{4} + 274 \beta_{6} ) q^{25} + ( 8 \beta_{3} - 396 \beta_{5} + 8 \beta_{7} ) q^{26} + ( 139 \beta_{1} - 139 \beta_{5} - 755 \beta_{7} ) q^{27} + ( -352 - 22 \beta_{2} ) q^{29} + ( -632 \beta_{4} - 444 \beta_{6} ) q^{30} + ( -698 \beta_{1} + 74 \beta_{3} ) q^{31} + ( 128 \beta_{2} + 128 \beta_{6} ) q^{32} + ( 1091 \beta_{3} + 285 \beta_{5} + 1091 \beta_{7} ) q^{33} + ( 210 \beta_{1} - 210 \beta_{5} + 918 \beta_{7} ) q^{34} + ( -392 - 712 \beta_{2} ) q^{36} + ( -848 \beta_{4} - 36 \beta_{6} ) q^{37} + ( 434 \beta_{1} - 394 \beta_{3} ) q^{38} + ( -550 - 764 \beta_{2} - 550 \beta_{4} - 764 \beta_{6} ) q^{39} + ( -288 \beta_{3} - 128 \beta_{5} - 288 \beta_{7} ) q^{40} + ( -251 \beta_{1} + 251 \beta_{5} - 460 \beta_{7} ) q^{41} + ( -506 + 1175 \beta_{2} ) q^{43} + ( 48 \beta_{4} - 824 \beta_{6} ) q^{44} + ( 957 \beta_{1} + 2239 \beta_{3} ) q^{45} + ( 576 + 52 \beta_{2} + 576 \beta_{4} + 52 \beta_{6} ) q^{46} + ( -1418 \beta_{3} + 1732 \beta_{5} - 1418 \beta_{7} ) q^{47} + ( -256 \beta_{1} + 256 \beta_{5} + 448 \beta_{7} ) q^{48} + ( 1096 - 474 \beta_{2} ) q^{50} + ( 4734 \beta_{4} + 2793 \beta_{6} ) q^{51} + ( 808 \beta_{1} - 776 \beta_{3} ) q^{52} + ( 4170 + 706 \beta_{2} + 4170 \beta_{4} + 706 \beta_{6} ) q^{53} + ( -1788 \beta_{3} - 1232 \beta_{5} - 1788 \beta_{7} ) q^{54} + ( -794 \beta_{1} + 794 \beta_{5} + 1776 \beta_{7} ) q^{55} + ( -1516 - 511 \beta_{2} ) q^{57} + ( 88 \beta_{4} + 704 \beta_{6} ) q^{58} + ( 1379 \beta_{1} - 2820 \beta_{3} ) q^{59} + ( -1776 - 1264 \beta_{2} - 1776 \beta_{4} - 1264 \beta_{6} ) q^{60} + ( 365 \beta_{3} + 1523 \beta_{5} + 365 \beta_{7} ) q^{61} + ( 1544 \beta_{1} - 1544 \beta_{5} + 1248 \beta_{7} ) q^{62} + 512 q^{64} + ( -1512 \beta_{4} - 938 \beta_{6} ) q^{65} + ( 1612 \beta_{1} + 2752 \beta_{3} ) q^{66} + ( 5204 + 786 \beta_{2} + 5204 \beta_{4} + 786 \beta_{6} ) q^{67} + ( 1416 \beta_{3} + 2256 \beta_{5} + 1416 \beta_{7} ) q^{68} + ( -536 \beta_{1} + 536 \beta_{5} + 1766 \beta_{7} ) q^{69} + ( -496 + 1244 \beta_{2} ) q^{71} + ( 2848 \beta_{4} + 784 \beta_{6} ) q^{72} + ( -1385 \beta_{1} + 2736 \beta_{3} ) q^{73} + ( -144 - 1696 \beta_{2} - 144 \beta_{4} - 1696 \beta_{6} ) q^{74} + ( 1355 \beta_{3} - 126 \beta_{5} + 1355 \beta_{7} ) q^{75} + ( -1656 \beta_{1} + 1656 \beta_{5} - 80 \beta_{7} ) q^{76} + ( -3056 - 1100 \beta_{2} ) q^{78} + ( -7404 \beta_{4} - 2270 \beta_{6} ) q^{79} + ( -320 \beta_{1} - 832 \beta_{3} ) q^{80} + ( -7713 - 1513 \beta_{2} - 7713 \beta_{4} - 1513 \beta_{6} ) q^{81} + ( -418 \beta_{3} - 1422 \beta_{5} - 418 \beta_{7} ) q^{82} + ( 4703 \beta_{1} - 4703 \beta_{5} - 1164 \beta_{7} ) q^{83} + ( -7422 - 5442 \beta_{2} ) q^{85} + ( -4700 \beta_{4} + 1012 \beta_{6} ) q^{86} + ( -1474 \beta_{1} - 2706 \beta_{3} ) q^{87} + ( -3296 + 96 \beta_{2} - 3296 \beta_{4} + 96 \beta_{6} ) q^{88} + ( 3746 \beta_{3} - 3573 \beta_{5} + 3746 \beta_{7} ) q^{89} + ( 2564 \beta_{1} - 2564 \beta_{5} - 6392 \beta_{7} ) q^{90} + ( 208 + 1152 \beta_{2} ) q^{92} + ( 4548 \beta_{4} + 1280 \beta_{6} ) q^{93} + ( -6300 \beta_{1} + 628 \beta_{3} ) q^{94} + ( -1810 - 1476 \beta_{2} - 1810 \beta_{4} - 1476 \beta_{6} ) q^{95} + ( 1408 \beta_{3} + 384 \beta_{5} + 1408 \beta_{7} ) q^{96} + ( -5324 \beta_{1} + 5324 \beta_{5} - 141 \beta_{7} ) q^{97} + ( 18040 + 4513 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 32q^{4} + 196q^{9} + O(q^{10})$$ $$8q - 32q^{4} + 196q^{9} - 24q^{11} + 1776q^{15} - 256q^{16} - 1424q^{18} + 3296q^{22} - 104q^{23} - 948q^{25} - 2816q^{29} + 2528q^{30} - 3136q^{36} + 3392q^{37} - 2200q^{39} - 4048q^{43} - 192q^{44} + 2304q^{46} + 8768q^{50} - 18936q^{51} + 16680q^{53} - 12128q^{57} - 352q^{58} - 7104q^{60} + 4096q^{64} + 6048q^{65} + 20816q^{67} - 3968q^{71} - 11392q^{72} - 576q^{74} - 24448q^{78} + 29616q^{79} - 30852q^{81} - 59376q^{85} + 18800q^{86} - 13184q^{88} + 1664q^{92} - 18192q^{93} - 7240q^{95} + 144320q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 20$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 34 \nu$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 2$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} + 7 \nu^{5} - 28 \nu^{3} + 16 \nu$$$$)/14$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 21 \nu^{2} + 2$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{5} - 70 \nu^{3} + 6 \nu$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 3 \beta_{5} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{6} - 6 \beta_{4} + 4 \beta_{2} - 6$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 10 \beta_{5} + 4 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{2} - 20$$ $$\nu^{7}$$ $$=$$ $$14 \beta_{3} - 34 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.60021 + 0.923880i 1.60021 − 0.923880i −0.662827 + 0.382683i 0.662827 − 0.382683i −1.60021 − 0.923880i 1.60021 + 0.923880i −0.662827 − 0.382683i 0.662827 + 0.382683i
−1.41421 + 2.44949i −1.76104 + 1.01673i −4.00000 6.92820i 0.615721 + 0.355487i 5.75152i 0 22.6274 −38.4325 + 66.5670i −1.74152 + 1.00547i
19.2 −1.41421 + 2.44949i 1.76104 1.01673i −4.00000 6.92820i −0.615721 0.355487i 5.75152i 0 22.6274 −38.4325 + 66.5670i 1.74152 1.00547i
19.3 1.41421 2.44949i −13.8528 + 7.99789i −4.00000 6.92820i −24.1168 13.9239i 45.2429i 0 −22.6274 87.4325 151.438i −68.2127 + 39.3826i
19.4 1.41421 2.44949i 13.8528 7.99789i −4.00000 6.92820i 24.1168 + 13.9239i 45.2429i 0 −22.6274 87.4325 151.438i 68.2127 39.3826i
31.1 −1.41421 2.44949i −1.76104 1.01673i −4.00000 + 6.92820i 0.615721 0.355487i 5.75152i 0 22.6274 −38.4325 66.5670i −1.74152 1.00547i
31.2 −1.41421 2.44949i 1.76104 + 1.01673i −4.00000 + 6.92820i −0.615721 + 0.355487i 5.75152i 0 22.6274 −38.4325 66.5670i 1.74152 + 1.00547i
31.3 1.41421 + 2.44949i −13.8528 7.99789i −4.00000 + 6.92820i −24.1168 + 13.9239i 45.2429i 0 −22.6274 87.4325 + 151.438i −68.2127 39.3826i
31.4 1.41421 + 2.44949i 13.8528 + 7.99789i −4.00000 + 6.92820i 24.1168 13.9239i 45.2429i 0 −22.6274 87.4325 + 151.438i 68.2127 + 39.3826i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.5.d.c 8
7.b odd 2 1 inner 98.5.d.c 8
7.c even 3 1 98.5.b.a 4
7.c even 3 1 inner 98.5.d.c 8
7.d odd 6 1 98.5.b.a 4
7.d odd 6 1 inner 98.5.d.c 8
21.g even 6 1 882.5.c.c 4
21.h odd 6 1 882.5.c.c 4
28.f even 6 1 784.5.c.a 4
28.g odd 6 1 784.5.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.b.a 4 7.c even 3 1
98.5.b.a 4 7.d odd 6 1
98.5.d.c 8 1.a even 1 1 trivial
98.5.d.c 8 7.b odd 2 1 inner
98.5.d.c 8 7.c even 3 1 inner
98.5.d.c 8 7.d odd 6 1 inner
784.5.c.a 4 28.f even 6 1
784.5.c.a 4 28.g odd 6 1
882.5.c.c 4 21.g even 6 1
882.5.c.c 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 260 T_{3}^{6} + 66542 T_{3}^{4} - 275080 T_{3}^{2} + 1119364$$ acting on $$S_{5}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 64 + 8 T^{2} + T^{4} )^{2}$$
$3$ $$1119364 - 275080 T^{2} + 66542 T^{4} - 260 T^{6} + T^{8}$$
$5$ $$153664 - 304192 T^{2} + 601784 T^{4} - 776 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 448677124 - 254184 T + 21326 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$13$ $$( 707030408 + 78440 T^{2} + T^{4} )^{2}$$
$17$ $$19\!\cdots\!64$$$$- 19431030862921896 T^{2} + 152792584686 T^{4} - 443412 T^{6} + T^{8}$$
$19$ $$8888207896996496644 - 512177091872552 T^{2} + 26532555854 T^{4} - 171796 T^{6} + T^{8}$$
$23$ $$( 1664313616 - 2121392 T + 43500 T^{2} + 52 T^{3} + T^{4} )^{2}$$
$29$ $$( 122936 + 704 T + T^{2} )^{4}$$
$31$ $$82\!\cdots\!04$$$$- 564432826479677440 T^{2} + 3597327870848 T^{4} - 1970720 T^{6} + T^{8}$$
$37$ $$( 513389446144 - 1215204352 T + 2159904 T^{2} - 1696 T^{3} + T^{4} )^{2}$$
$41$ $$( 288069342722 + 1098404 T^{2} + T^{4} )^{2}$$
$43$ $$( -2505214 + 1012 T + T^{2} )^{4}$$
$47$ $$94\!\cdots\!64$$$$-$$$$61\!\cdots\!36$$$$T^{2} + 370911924537056 T^{4} - 20042192 T^{6} + T^{8}$$
$53$ $$( 268698581952784 - 136709513520 T + 53163572 T^{2} - 8340 T^{3} + T^{4} )^{2}$$
$59$ $$14\!\cdots\!84$$$$-$$$$15\!\cdots\!08$$$$T^{2} + 1171189171743374 T^{4} - 39416164 T^{6} + T^{8}$$
$61$ $$47\!\cdots\!84$$$$-$$$$21\!\cdots\!52$$$$T^{2} + 74501186886584 T^{4} - 9811016 T^{6} + T^{8}$$
$67$ $$( 668016956608576 - 269005417792 T + 82480440 T^{2} - 10408 T^{3} + T^{4} )^{2}$$
$71$ $$( -2849056 + 992 T + T^{2} )^{4}$$
$73$ $$11\!\cdots\!44$$$$-$$$$13\!\cdots\!08$$$$T^{2} + 1069295007170894 T^{4} - 37615684 T^{6} + T^{8}$$
$79$ $$( 1981444203989056 - 659154664128 T + 174763448 T^{2} - 14808 T^{3} + T^{4} )^{2}$$
$83$ $$( 2011288822677218 + 93892420 T^{2} + T^{4} )^{2}$$
$89$ $$24\!\cdots\!24$$$$-$$$$16\!\cdots\!40$$$$T^{2} + 9918915492903182 T^{4} - 107195380 T^{6} + T^{8}$$
$97$ $$( 1439024660341058 + 113459428 T^{2} + T^{4} )^{2}$$