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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 $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [619,2,Mod(1,619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 619.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.94273988512\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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:

Atkin-Lehner signs

\( p \) Sign
\(619\) \(-1\)

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - 9 T_{2}^{29} - 6 T_{2}^{28} + 276 T_{2}^{27} - 458 T_{2}^{26} - 3470 T_{2}^{25} + \cdots - 288 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(619))\). Copy content Toggle raw display