# Properties

 Label 98.4.c.h Level $98$ Weight $4$ Character orbit 98.c Analytic conductor $5.782$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 98.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.78218718056$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \beta_{2} ) q^{2} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{3} + 4 \beta_{2} q^{4} -14 \beta_{1} q^{5} + 10 \beta_{3} q^{6} -8 q^{8} + ( -23 - 23 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 + 2 \beta_{2} ) q^{2} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{3} + 4 \beta_{2} q^{4} -14 \beta_{1} q^{5} + 10 \beta_{3} q^{6} -8 q^{8} + ( -23 - 23 \beta_{2} ) q^{9} + ( -28 \beta_{1} - 28 \beta_{3} ) q^{10} -14 \beta_{2} q^{11} -20 \beta_{1} q^{12} + 36 \beta_{3} q^{13} + 140 q^{15} + ( -16 - 16 \beta_{2} ) q^{16} + ( -\beta_{1} - \beta_{3} ) q^{17} -46 \beta_{2} q^{18} -\beta_{1} q^{19} -56 \beta_{3} q^{20} + 28 q^{22} + ( -140 - 140 \beta_{2} ) q^{23} + ( -40 \beta_{1} - 40 \beta_{3} ) q^{24} + 267 \beta_{2} q^{25} -72 \beta_{1} q^{26} + 20 \beta_{3} q^{27} -286 q^{29} + ( 280 + 280 \beta_{2} ) q^{30} + ( 66 \beta_{1} + 66 \beta_{3} ) q^{31} -32 \beta_{2} q^{32} + 70 \beta_{1} q^{33} -2 \beta_{3} q^{34} + 92 q^{36} + ( 38 + 38 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{38} -360 \beta_{2} q^{39} + 112 \beta_{1} q^{40} -89 \beta_{3} q^{41} -34 q^{43} + ( 56 + 56 \beta_{2} ) q^{44} + ( 322 \beta_{1} + 322 \beta_{3} ) q^{45} -280 \beta_{2} q^{46} + 370 \beta_{1} q^{47} -80 \beta_{3} q^{48} -534 q^{50} + ( 10 + 10 \beta_{2} ) q^{51} + ( -144 \beta_{1} - 144 \beta_{3} ) q^{52} -74 \beta_{2} q^{53} -40 \beta_{1} q^{54} + 196 \beta_{3} q^{55} + 10 q^{57} + ( -572 - 572 \beta_{2} ) q^{58} + ( -307 \beta_{1} - 307 \beta_{3} ) q^{59} + 560 \beta_{2} q^{60} + 10 \beta_{1} q^{61} + 132 \beta_{3} q^{62} + 64 q^{64} + ( 1008 + 1008 \beta_{2} ) q^{65} + ( 140 \beta_{1} + 140 \beta_{3} ) q^{66} + 684 \beta_{2} q^{67} + 4 \beta_{1} q^{68} -700 \beta_{3} q^{69} + 588 q^{71} + ( 184 + 184 \beta_{2} ) q^{72} + ( 191 \beta_{1} + 191 \beta_{3} ) q^{73} + 76 \beta_{2} q^{74} -1335 \beta_{1} q^{75} -4 \beta_{3} q^{76} + 720 q^{78} + ( -1220 - 1220 \beta_{2} ) q^{79} + ( 224 \beta_{1} + 224 \beta_{3} ) q^{80} -821 \beta_{2} q^{81} + 178 \beta_{1} q^{82} + 299 \beta_{3} q^{83} -28 q^{85} + ( -68 - 68 \beta_{2} ) q^{86} + ( -1430 \beta_{1} - 1430 \beta_{3} ) q^{87} + 112 \beta_{2} q^{88} + 437 \beta_{1} q^{89} + 644 \beta_{3} q^{90} + 560 q^{92} + ( -660 - 660 \beta_{2} ) q^{93} + ( 740 \beta_{1} + 740 \beta_{3} ) q^{94} + 28 \beta_{2} q^{95} + 160 \beta_{1} q^{96} + 1049 \beta_{3} q^{97} -322 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 8q^{4} - 32q^{8} - 46q^{9} + O(q^{10})$$ $$4q + 4q^{2} - 8q^{4} - 32q^{8} - 46q^{9} + 28q^{11} + 560q^{15} - 32q^{16} + 92q^{18} + 112q^{22} - 280q^{23} - 534q^{25} - 1144q^{29} + 560q^{30} + 64q^{32} + 368q^{36} + 76q^{37} + 720q^{39} - 136q^{43} + 112q^{44} + 560q^{46} - 2136q^{50} + 20q^{51} + 148q^{53} + 40q^{57} - 1144q^{58} - 1120q^{60} + 256q^{64} + 2016q^{65} - 1368q^{67} + 2352q^{71} + 368q^{72} - 152q^{74} + 2880q^{78} - 2440q^{79} + 1642q^{81} - 112q^{85} - 136q^{86} - 224q^{88} + 2240q^{92} - 1320q^{93} - 56q^{95} - 1288q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
1.00000 + 1.73205i −3.53553 + 6.12372i −2.00000 + 3.46410i −9.89949 17.1464i −14.1421 0 −8.00000 −11.5000 19.9186i 19.7990 34.2929i
67.2 1.00000 + 1.73205i 3.53553 6.12372i −2.00000 + 3.46410i 9.89949 + 17.1464i 14.1421 0 −8.00000 −11.5000 19.9186i −19.7990 + 34.2929i
79.1 1.00000 1.73205i −3.53553 6.12372i −2.00000 3.46410i −9.89949 + 17.1464i −14.1421 0 −8.00000 −11.5000 + 19.9186i 19.7990 + 34.2929i
79.2 1.00000 1.73205i 3.53553 + 6.12372i −2.00000 3.46410i 9.89949 17.1464i 14.1421 0 −8.00000 −11.5000 + 19.9186i −19.7990 34.2929i
 $$n$$: e.g. 2-40 or 990-1000 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 98.4.c.h 4
3.b odd 2 1 882.4.g.ba 4
7.b odd 2 1 inner 98.4.c.h 4
7.c even 3 1 98.4.a.g 2
7.c even 3 1 inner 98.4.c.h 4
7.d odd 6 1 98.4.a.g 2
7.d odd 6 1 inner 98.4.c.h 4
21.c even 2 1 882.4.g.ba 4
21.g even 6 1 882.4.a.bg 2
21.g even 6 1 882.4.g.ba 4
21.h odd 6 1 882.4.a.bg 2
21.h odd 6 1 882.4.g.ba 4
28.f even 6 1 784.4.a.y 2
28.g odd 6 1 784.4.a.y 2
35.i odd 6 1 2450.4.a.bx 2
35.j even 6 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 7.c even 3 1
98.4.a.g 2 7.d odd 6 1
98.4.c.h 4 1.a even 1 1 trivial
98.4.c.h 4 7.b odd 2 1 inner
98.4.c.h 4 7.c even 3 1 inner
98.4.c.h 4 7.d odd 6 1 inner
784.4.a.y 2 28.f even 6 1
784.4.a.y 2 28.g odd 6 1
882.4.a.bg 2 21.g even 6 1
882.4.a.bg 2 21.h odd 6 1
882.4.g.ba 4 3.b odd 2 1
882.4.g.ba 4 21.c even 2 1
882.4.g.ba 4 21.g even 6 1
882.4.g.ba 4 21.h odd 6 1
2450.4.a.bx 2 35.i odd 6 1
2450.4.a.bx 2 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 50 T_{3}^{2} + 2500$$ acting on $$S_{4}^{\mathrm{new}}(98, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{2}$$
$3$ $$2500 + 50 T^{2} + T^{4}$$
$5$ $$153664 + 392 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 196 - 14 T + T^{2} )^{2}$$
$13$ $$( -2592 + T^{2} )^{2}$$
$17$ $$4 + 2 T^{2} + T^{4}$$
$19$ $$4 + 2 T^{2} + T^{4}$$
$23$ $$( 19600 + 140 T + T^{2} )^{2}$$
$29$ $$( 286 + T )^{4}$$
$31$ $$75898944 + 8712 T^{2} + T^{4}$$
$37$ $$( 1444 - 38 T + T^{2} )^{2}$$
$41$ $$( -15842 + T^{2} )^{2}$$
$43$ $$( 34 + T )^{4}$$
$47$ $$74966440000 + 273800 T^{2} + T^{4}$$
$53$ $$( 5476 - 74 T + T^{2} )^{2}$$
$59$ $$35531496004 + 188498 T^{2} + T^{4}$$
$61$ $$40000 + 200 T^{2} + T^{4}$$
$67$ $$( 467856 + 684 T + T^{2} )^{2}$$
$71$ $$( -588 + T )^{4}$$
$73$ $$5323453444 + 72962 T^{2} + T^{4}$$
$79$ $$( 1488400 + 1220 T + T^{2} )^{2}$$
$83$ $$( -178802 + T^{2} )^{2}$$
$89$ $$145876635844 + 381938 T^{2} + T^{4}$$
$97$ $$( -2200802 + T^{2} )^{2}$$