# Properties

 Label 98.2.g.b Level $98$ Weight $2$ Character orbit 98.g Analytic conductor $0.783$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 98.g (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.782533939809$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$2$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 2q^{2} - 7q^{3} + 2q^{4} - 7q^{6} - 4q^{8} + 19q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 2q^{2} - 7q^{3} + 2q^{4} - 7q^{6} - 4q^{8} + 19q^{9} - 11q^{11} - 14q^{13} + 9q^{15} + 2q^{16} - 7q^{17} - 9q^{18} - 14q^{19} - 7q^{20} - 7q^{21} + q^{22} - 29q^{23} - 8q^{25} - 7q^{26} - 7q^{27} + 14q^{28} + 13q^{29} - 8q^{30} - 28q^{31} + 2q^{32} - 14q^{33} - 7q^{34} - 35q^{35} - 17q^{36} + 20q^{37} + 35q^{38} + 56q^{39} + 14q^{40} + 28q^{41} - 21q^{42} + 6q^{43} + 3q^{44} + 7q^{45} + 34q^{46} + 42q^{47} + 14q^{48} + 28q^{49} + 16q^{50} + 32q^{51} - 7q^{52} - 60q^{53} + 21q^{54} - 14q^{55} + 7q^{56} + 23q^{57} + 18q^{58} + 49q^{59} + 6q^{60} - 14q^{61} - 28q^{63} - 4q^{64} - 28q^{65} + 21q^{66} + 24q^{67} - 14q^{68} + 7q^{69} - 28q^{70} + 6q^{71} - 2q^{72} - 35q^{73} - 15q^{74} - 56q^{75} + 49q^{77} + 6q^{79} - 14q^{80} - 45q^{81} - 14q^{82} - 77q^{83} - 21q^{84} - 33q^{85} - 38q^{86} + 63q^{87} + 3q^{88} - 21q^{90} - 21q^{91} - 5q^{92} - 38q^{93} - 35q^{94} + 86q^{95} + 98q^{97} + 28q^{98} - 106q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −0.733052 0.680173i −2.81578 0.424410i 0.0747301 + 0.997204i 0.904721 + 2.30519i 1.77544 + 2.22633i −1.30835 + 2.29961i 0.623490 0.781831i 4.88176 + 1.50582i 0.904721 2.30519i
9.2 −0.733052 0.680173i 1.10442 + 0.166465i 0.0747301 + 0.997204i 0.578784 + 1.47472i −0.696376 0.873227i 2.58946 0.542854i 0.623490 0.781831i −1.67467 0.516569i 0.578784 1.47472i
11.1 −0.733052 + 0.680173i −2.81578 + 0.424410i 0.0747301 0.997204i 0.904721 2.30519i 1.77544 2.22633i −1.30835 2.29961i 0.623490 + 0.781831i 4.88176 1.50582i 0.904721 + 2.30519i
11.2 −0.733052 + 0.680173i 1.10442 0.166465i 0.0747301 0.997204i 0.578784 1.47472i −0.696376 + 0.873227i 2.58946 + 0.542854i 0.623490 + 0.781831i −1.67467 + 0.516569i 0.578784 + 1.47472i
23.1 0.0747301 0.997204i −2.39605 + 0.739084i −0.988831 0.149042i −2.18585 2.02817i 0.557960 + 2.44458i −2.12971 1.56982i −0.222521 + 0.974928i 2.71611 1.85181i −2.18585 + 2.02817i
23.2 0.0747301 0.997204i 1.95066 0.601698i −0.988831 0.149042i 0.958118 + 0.889004i −0.454243 1.99017i −2.43754 + 1.02877i −0.222521 + 0.974928i 0.964304 0.657452i 0.958118 0.889004i
25.1 0.955573 0.294755i −0.784496 1.99886i 0.826239 0.563320i −2.07983 0.313484i −1.33882 1.67883i 2.53097 + 0.770831i 0.623490 0.781831i −1.18087 + 1.09568i −2.07983 + 0.313484i
25.2 0.955573 0.294755i 0.427316 + 1.08878i 0.826239 0.563320i −0.0821245 0.0123783i 0.729256 + 0.914459i −2.45519 + 0.985929i 0.623490 0.781831i 1.19630 1.11001i −0.0821245 + 0.0123783i
37.1 0.826239 0.563320i −2.19243 + 2.03428i 0.365341 0.930874i 3.17919 + 0.980650i −0.665522 + 2.91584i −0.0867132 + 2.64433i −0.222521 0.974928i 0.444274 5.92843i 3.17919 0.980650i
37.2 0.826239 0.563320i 0.0584109 0.0541974i 0.365341 0.930874i −0.427005 0.131714i 0.0177309 0.0776840i 1.60505 2.10329i −0.222521 0.974928i −0.223716 + 2.98528i −0.427005 + 0.131714i
39.1 −0.988831 0.149042i −0.934966 + 0.637449i 0.955573 + 0.294755i 0.0772188 + 1.03041i 1.01953 0.490980i 1.36748 + 2.26495i −0.900969 0.433884i −0.628203 + 1.60064i 0.0772188 1.03041i
39.2 −0.988831 0.149042i 1.88469 1.28496i 0.955573 + 0.294755i 0.269665 + 3.59844i −2.05516 + 0.989712i 0.415648 2.61290i −0.900969 0.433884i 0.804920 2.05090i 0.269665 3.59844i
51.1 0.955573 + 0.294755i −0.784496 + 1.99886i 0.826239 + 0.563320i −2.07983 + 0.313484i −1.33882 + 1.67883i 2.53097 0.770831i 0.623490 + 0.781831i −1.18087 1.09568i −2.07983 0.313484i
51.2 0.955573 + 0.294755i 0.427316 1.08878i 0.826239 + 0.563320i −0.0821245 + 0.0123783i 0.729256 0.914459i −2.45519 0.985929i 0.623490 + 0.781831i 1.19630 + 1.11001i −0.0821245 0.0123783i
53.1 0.826239 + 0.563320i −2.19243 2.03428i 0.365341 + 0.930874i 3.17919 0.980650i −0.665522 2.91584i −0.0867132 2.64433i −0.222521 + 0.974928i 0.444274 + 5.92843i 3.17919 + 0.980650i
53.2 0.826239 + 0.563320i 0.0584109 + 0.0541974i 0.365341 + 0.930874i −0.427005 + 0.131714i 0.0177309 + 0.0776840i 1.60505 + 2.10329i −0.222521 + 0.974928i −0.223716 2.98528i −0.427005 0.131714i
65.1 0.365341 + 0.930874i 0.0722934 + 0.964688i −0.733052 + 0.680173i −3.07015 + 2.09319i −0.871591 + 0.419736i 2.50800 0.842585i −0.900969 0.433884i 2.04110 0.307646i −3.07015 2.09319i
65.2 0.365341 + 0.930874i 0.125927 + 1.68037i −0.733052 + 0.680173i 1.87725 1.27989i −1.51821 + 0.731131i −2.59910 + 0.494638i −0.900969 0.433884i 0.158698 0.0239199i 1.87725 + 1.27989i
81.1 0.0747301 + 0.997204i −2.39605 0.739084i −0.988831 + 0.149042i −2.18585 + 2.02817i 0.557960 2.44458i −2.12971 + 1.56982i −0.222521 0.974928i 2.71611 + 1.85181i −2.18585 2.02817i
81.2 0.0747301 + 0.997204i 1.95066 + 0.601698i −0.988831 + 0.149042i 0.958118 0.889004i −0.454243 + 1.99017i −2.43754 1.02877i −0.222521 0.974928i 0.964304 + 0.657452i 0.958118 + 0.889004i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 95.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.2.g.b 24
3.b odd 2 1 882.2.z.b 24
4.b odd 2 1 784.2.bg.b 24
7.b odd 2 1 686.2.g.e 24
7.c even 3 1 686.2.e.c 24
7.c even 3 1 686.2.g.f 24
7.d odd 6 1 686.2.e.d 24
7.d odd 6 1 686.2.g.d 24
49.e even 7 1 686.2.g.f 24
49.f odd 14 1 686.2.g.d 24
49.g even 21 1 inner 98.2.g.b 24
49.g even 21 1 686.2.e.c 24
49.g even 21 1 4802.2.a.o 12
49.h odd 42 1 686.2.e.d 24
49.h odd 42 1 686.2.g.e 24
49.h odd 42 1 4802.2.a.l 12
147.n odd 42 1 882.2.z.b 24
196.o odd 42 1 784.2.bg.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.g.b 24 1.a even 1 1 trivial
98.2.g.b 24 49.g even 21 1 inner
686.2.e.c 24 7.c even 3 1
686.2.e.c 24 49.g even 21 1
686.2.e.d 24 7.d odd 6 1
686.2.e.d 24 49.h odd 42 1
686.2.g.d 24 7.d odd 6 1
686.2.g.d 24 49.f odd 14 1
686.2.g.e 24 7.b odd 2 1
686.2.g.e 24 49.h odd 42 1
686.2.g.f 24 7.c even 3 1
686.2.g.f 24 49.e even 7 1
784.2.bg.b 24 4.b odd 2 1
784.2.bg.b 24 196.o odd 42 1
882.2.z.b 24 3.b odd 2 1
882.2.z.b 24 147.n odd 42 1
4802.2.a.l 12 49.h odd 42 1
4802.2.a.o 12 49.g even 21 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(98, [\chi])$$.