Properties

Label 8619.2.a.bw
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $1$
Dimension $24$
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] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(1\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 7 q^{2} - 24 q^{3} + 17 q^{4} - 13 q^{5} - 7 q^{6} - 12 q^{7} + 21 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 7 q^{2} - 24 q^{3} + 17 q^{4} - 13 q^{5} - 7 q^{6} - 12 q^{7} + 21 q^{8} + 24 q^{9} - 4 q^{10} - q^{11} - 17 q^{12} - 18 q^{14} + 13 q^{15} + 35 q^{16} - 24 q^{17} + 7 q^{18} - 16 q^{19} - 18 q^{20} + 12 q^{21} + 8 q^{22} + 3 q^{23} - 21 q^{24} + 23 q^{25} - 24 q^{27} - 30 q^{28} + 5 q^{29} + 4 q^{30} - 46 q^{31} + 44 q^{32} + q^{33} - 7 q^{34} + 11 q^{35} + 17 q^{36} - 38 q^{37} - q^{38} - 59 q^{40} - 31 q^{41} + 18 q^{42} + 3 q^{43} + 3 q^{44} - 13 q^{45} - 36 q^{46} + 49 q^{47} - 35 q^{48} + 30 q^{49} + 15 q^{50} + 24 q^{51} - 11 q^{53} - 7 q^{54} - 10 q^{55} - 38 q^{56} + 16 q^{57} - 53 q^{58} - 23 q^{59} + 18 q^{60} - 2 q^{61} - 26 q^{62} - 12 q^{63} + 57 q^{64} - 8 q^{66} - 17 q^{68} - 3 q^{69} + 10 q^{70} + 29 q^{71} + 21 q^{72} - 48 q^{73} - 66 q^{74} - 23 q^{75} - 38 q^{76} + 51 q^{77} - 26 q^{79} - 52 q^{80} + 24 q^{81} - 45 q^{82} + 8 q^{83} + 30 q^{84} + 13 q^{85} - 38 q^{86} - 5 q^{87} - 34 q^{88} - 42 q^{89} - 4 q^{90} - 10 q^{92} + 46 q^{93} + 107 q^{94} - 44 q^{96} - 60 q^{97} + 19 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.62635 −1.00000 4.89769 0.512917 2.62635 −2.19487 −7.61034 1.00000 −1.34710
1.2 −2.16723 −1.00000 2.69690 0.362944 2.16723 2.98172 −1.51034 1.00000 −0.786585
1.3 −1.86500 −1.00000 1.47821 −4.02513 1.86500 2.21617 0.973140 1.00000 7.50685
1.4 −1.69539 −1.00000 0.874348 2.79728 1.69539 0.609432 1.90842 1.00000 −4.74247
1.5 −1.57761 −1.00000 0.488848 −2.21771 1.57761 −4.61789 2.38401 1.00000 3.49868
1.6 −1.17199 −1.00000 −0.626431 −3.45670 1.17199 −1.76134 3.07816 1.00000 4.05124
1.7 −0.936172 −1.00000 −1.12358 −1.31747 0.936172 0.0103583 2.92421 1.00000 1.23338
1.8 −0.470840 −1.00000 −1.77831 −2.64995 0.470840 3.75195 1.77898 1.00000 1.24770
1.9 −0.433070 −1.00000 −1.81245 2.79890 0.433070 −3.33739 1.65106 1.00000 −1.21212
1.10 −0.294893 −1.00000 −1.91304 2.91911 0.294893 1.40550 1.15393 1.00000 −0.860824
1.11 −0.271243 −1.00000 −1.92643 0.00688268 0.271243 −0.318502 1.06502 1.00000 −0.00186688
1.12 0.324966 −1.00000 −1.89440 0.225941 −0.324966 3.77265 −1.26555 1.00000 0.0734231
1.13 0.342241 −1.00000 −1.88287 −4.18392 −0.342241 −3.67826 −1.32888 1.00000 −1.43191
1.14 0.671598 −1.00000 −1.54896 −1.50727 −0.671598 −4.47027 −2.38347 1.00000 −1.01228
1.15 0.963570 −1.00000 −1.07153 −1.50390 −0.963570 2.76951 −2.95964 1.00000 −1.44911
1.16 1.10514 −1.00000 −0.778661 −0.625764 −1.10514 −4.27315 −3.07082 1.00000 −0.691558
1.17 1.16683 −1.00000 −0.638517 0.385677 −1.16683 3.99174 −3.07869 1.00000 0.450018
1.18 1.46048 −1.00000 0.133014 3.73069 −1.46048 −2.14359 −2.72670 1.00000 5.44861
1.19 2.10278 −1.00000 2.42170 −3.44599 −2.10278 1.48926 0.886753 1.00000 −7.24618
1.20 2.12481 −1.00000 2.51480 1.58682 −2.12481 −2.67484 1.09385 1.00000 3.37169
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(1\)
\(17\) \(1\)

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 8619.2.a.bw yes 24
13.b even 2 1 8619.2.a.bt 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8619.2.a.bt 24 13.b even 2 1
8619.2.a.bw yes 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{24} - 7 T_{2}^{23} - 8 T_{2}^{22} + 154 T_{2}^{21} - 136 T_{2}^{20} - 1343 T_{2}^{19} + 2302 T_{2}^{18} + 5900 T_{2}^{17} - 14226 T_{2}^{16} - 13305 T_{2}^{15} + 46628 T_{2}^{14} + 11811 T_{2}^{13} - 87948 T_{2}^{12} + \cdots - 13 \) Copy content Toggle raw display
\( T_{5}^{24} + 13 T_{5}^{23} + 13 T_{5}^{22} - 498 T_{5}^{21} - 1853 T_{5}^{20} + 6344 T_{5}^{19} + 41405 T_{5}^{18} - 15884 T_{5}^{17} - 421795 T_{5}^{16} - 336736 T_{5}^{15} + 2227486 T_{5}^{14} + 3387472 T_{5}^{13} + \cdots + 343 \) Copy content Toggle raw display
\( T_{7}^{24} + 12 T_{7}^{23} - 27 T_{7}^{22} - 818 T_{7}^{21} - 1075 T_{7}^{20} + 22840 T_{7}^{19} + 62844 T_{7}^{18} - 333768 T_{7}^{17} - 1291597 T_{7}^{16} + 2667478 T_{7}^{15} + 14325310 T_{7}^{14} + \cdots - 810641 \) Copy content Toggle raw display