# Properties

 Label 619.2.a.b Level 619 Weight 2 Character orbit 619.a Self dual yes Analytic conductor 4.943 Analytic rank 0 Dimension 30 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$619$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 619.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.94273988512$$ Analytic rank: $$0$$ Dimension: $$30$$ Coefficient ring index: multiple of None Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 9q^{2} + q^{3} + 33q^{4} + 21q^{5} + 6q^{6} + 2q^{7} + 27q^{8} + 43q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 9q^{2} + q^{3} + 33q^{4} + 21q^{5} + 6q^{6} + 2q^{7} + 27q^{8} + 43q^{9} + 5q^{10} + 23q^{11} - 6q^{12} + 9q^{13} + 7q^{14} - 2q^{15} + 35q^{16} + 4q^{17} + 10q^{18} - q^{19} + 29q^{20} + 30q^{21} + 4q^{23} + 4q^{24} + 35q^{25} + q^{26} - 5q^{27} - 13q^{28} + 90q^{29} - 31q^{30} + 2q^{31} + 43q^{32} - 6q^{33} - 9q^{34} + 9q^{35} + 33q^{36} + 19q^{37} + 5q^{38} + 32q^{39} - 12q^{40} + 59q^{41} - 25q^{42} - 4q^{43} + 52q^{44} + 30q^{45} - q^{46} + 4q^{47} - 44q^{48} + 30q^{49} + 31q^{50} - 12q^{52} + 34q^{53} - 28q^{54} - 17q^{55} + 2q^{56} - 8q^{57} + 6q^{58} + 13q^{59} - 64q^{60} + 16q^{61} + 28q^{62} - 40q^{63} + 37q^{64} + 31q^{65} - 59q^{66} - 11q^{67} - 52q^{68} + 6q^{69} - 40q^{70} + 42q^{71} + 6q^{72} - 4q^{73} + 16q^{74} - 52q^{75} - 42q^{76} + 29q^{77} - 56q^{78} + 3q^{79} + 21q^{80} + 30q^{81} - 43q^{82} - 11q^{83} - 36q^{84} + 19q^{85} - 11q^{86} - 20q^{87} - 47q^{88} + 58q^{89} - 33q^{90} - 39q^{91} - 7q^{92} - 15q^{93} - 46q^{94} + 23q^{95} - 70q^{96} - 9q^{97} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.61841 0.252107 4.85606 3.30541 −0.660120 −1.70947 −7.47834 −2.93644 −8.65491
1.2 −2.42956 −3.02694 3.90276 −0.0323024 7.35414 −3.37580 −4.62286 6.16239 0.0784807
1.3 −2.29572 2.37814 3.27033 1.23255 −5.45954 1.58895 −2.91633 2.65553 −2.82958
1.4 −1.98324 −1.72419 1.93325 −1.05779 3.41948 3.75491 0.132389 −0.0271814 2.09786
1.5 −1.73496 −0.518455 1.01008 −1.50291 0.899498 −4.51257 1.71746 −2.73120 2.60748
1.6 −1.67244 −2.69736 0.797052 3.55917 4.51117 2.20002 2.01186 4.27576 −5.95250
1.7 −1.45512 2.41034 0.117377 −2.25507 −3.50733 −1.27109 2.73944 2.80973 3.28140
1.8 −1.37056 −0.725873 −0.121559 −0.225133 0.994855 2.47282 2.90773 −2.47311 0.308559
1.9 −1.26926 3.28637 −0.388969 2.23550 −4.17127 3.15236 3.03223 7.80023 −2.83744
1.10 −0.938541 2.04590 −1.11914 3.68832 −1.92016 −2.71706 2.92744 1.18571 −3.46164
1.11 −0.394547 1.27337 −1.84433 1.92856 −0.502406 3.31459 1.51677 −1.37852 −0.760909
1.12 −0.386745 −1.55323 −1.85043 −3.75067 0.600704 −2.84697 1.48913 −0.587469 1.45055
1.13 −0.192901 −3.10998 −1.96279 2.33409 0.599918 −3.05253 0.764427 6.67195 −0.450249
1.14 0.258847 2.67579 −1.93300 −2.46163 0.692619 1.88069 −1.01804 4.15985 −0.637185
1.15 0.262115 −0.800074 −1.93130 −2.40890 −0.209711 1.00913 −1.03045 −2.35988 −0.631409
1.16 0.271769 −1.01788 −1.92614 4.11968 −0.276629 2.70884 −1.06700 −1.96391 1.11960
1.17 0.431096 −1.56278 −1.81416 1.50079 −0.673708 −4.83036 −1.64427 −0.557720 0.646986
1.18 0.993756 2.38151 −1.01245 2.26626 2.36664 3.26853 −2.99364 2.67158 2.25211
1.19 1.39793 2.10519 −0.0457880 3.90881 2.94291 −1.82243 −2.85987 1.43181 5.46425
1.20 1.45503 −3.26388 0.117111 −2.69944 −4.74905 −1.44249 −2.73966 7.65293 −3.92777
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 619.2.a.b 30
3.b odd 2 1 5571.2.a.g 30
4.b odd 2 1 9904.2.a.n 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
619.2.a.b 30 1.a even 1 1 trivial
5571.2.a.g 30 3.b odd 2 1
9904.2.a.n 30 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$619$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{30} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(619))$$.