Properties

Label 605.2.a.g
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $0$
Dimension $3$
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] = [605,2,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + \beta_1 + 3) q^{4} - q^{5} + ( - 2 \beta_1 - 1) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{2} - \beta_1 + 4) q^{8} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + \beta_1 + 3) q^{4} - q^{5} + ( - 2 \beta_1 - 1) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{2} - \beta_1 + 4) q^{8} + (\beta_{2} + \beta_1) q^{9} + \beta_{2} q^{10} + (\beta_{2} + 2 \beta_1 + 2) q^{12} + ( - \beta_{2} + \beta_1 - 2) q^{13} + ( - \beta_{2} - 3 \beta_1 + 3) q^{14} - \beta_1 q^{15} + ( - 4 \beta_{2} + 2 \beta_1 + 5) q^{16} + (2 \beta_1 - 2) q^{17} + (\beta_{2} - 3 \beta_1 - 6) q^{18} + (\beta_{2} + \beta_1 - 2) q^{19} + (\beta_{2} - \beta_1 - 3) q^{20} + (2 \beta_{2} + 5) q^{21} + (\beta_{2} - \beta_1 - 2) q^{23} + ( - \beta_{2} - \beta_1 - 5) q^{24} + q^{25} + (\beta_{2} - \beta_1 + 4) q^{26} + (\beta_{2} + 4) q^{27} + ( - 2 \beta_{2} + 3 \beta_1 + 8) q^{28} - 2 \beta_{2} q^{29} + (2 \beta_1 + 1) q^{30} + (\beta_{2} + \beta_1 + 4) q^{31} + ( - 5 \beta_{2} + 2 \beta_1 + 10) q^{32} + (2 \beta_{2} - 4 \beta_1 - 2) q^{34} + (\beta_{2} - 2 \beta_1) q^{35} + (5 \beta_{2} + 3 \beta_1 - 2) q^{36} + ( - 2 \beta_{2} + 2) q^{37} + (3 \beta_{2} - 3 \beta_1 - 6) q^{38} + (\beta_{2} - 3 \beta_1 + 2) q^{39} + (2 \beta_{2} + \beta_1 - 4) q^{40} + (3 \beta_{2} - 3 \beta_1 - 3) q^{41} + ( - 3 \beta_{2} - 2 \beta_1 - 10) q^{42} + (\beta_{2} - 2 \beta_1 - 2) q^{43} + ( - \beta_{2} - \beta_1) q^{45} + (3 \beta_{2} + \beta_1 - 4) q^{46} + ( - 4 \beta_{2} + \beta_1 - 4) q^{47} + (2 \beta_{2} - \beta_1 + 2) q^{48} + (3 \beta_{2} - 3 \beta_1 + 6) q^{49} - \beta_{2} q^{50} + (2 \beta_{2} + 6) q^{51} + ( - \beta_{2} - \beta_1) q^{52} + (\beta_{2} - \beta_1 - 2) q^{53} + ( - 3 \beta_{2} - \beta_1 - 5) q^{54} + ( - 8 \beta_{2} + 2 \beta_1 + 1) q^{56} + (\beta_{2} + \beta_1 + 4) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 + 10) q^{58} + (3 \beta_{2} + \beta_1 + 2) q^{59} + ( - \beta_{2} - 2 \beta_1 - 2) q^{60} + (\beta_{2} + 5 \beta_1 - 3) q^{61} + ( - 3 \beta_{2} - 3 \beta_1 - 6) q^{62} + (3 \beta_{2} + 3 \beta_1 + 2) q^{63} + ( - 7 \beta_{2} - 3 \beta_1 + 13) q^{64} + (\beta_{2} - \beta_1 + 2) q^{65} + (2 \beta_{2} - \beta_1 + 6) q^{67} + (4 \beta_{2} + 2 \beta_1 - 2) q^{68} + ( - \beta_{2} - \beta_1 - 2) q^{69} + (\beta_{2} + 3 \beta_1 - 3) q^{70} + ( - \beta_{2} + \beta_1 + 2) q^{71} + (5 \beta_{2} - 5 \beta_1 - 16) q^{72} + (2 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - 4 \beta_{2} + 2 \beta_1 + 10) q^{74} + \beta_1 q^{75} + (7 \beta_{2} + \beta_1 - 8) q^{76} + ( - \beta_{2} + 5 \beta_1 - 2) q^{78} + (2 \beta_{2} - 4 \beta_1) q^{79} + (4 \beta_{2} - 2 \beta_1 - 5) q^{80} + ( - 3 \beta_{2} + 3 \beta_1 + 1) q^{81} + (6 \beta_{2} + 3 \beta_1 - 12) q^{82} + (3 \beta_{2} - 3 \beta_1 - 6) q^{83} + (3 \beta_{2} + 7 \beta_1 + 7) q^{84} + ( - 2 \beta_1 + 2) q^{85} + (3 \beta_{2} + 3 \beta_1 - 3) q^{86} + ( - 4 \beta_1 - 2) q^{87} + ( - 4 \beta_{2} + 6 \beta_1 + 3) q^{89} + ( - \beta_{2} + 3 \beta_1 + 6) q^{90} + (3 \beta_{2} - 7 \beta_1 + 8) q^{91} + (5 \beta_{2} - 3 \beta_1 - 12) q^{92} + (\beta_{2} + 7 \beta_1 + 4) q^{93} + (2 \beta_1 + 19) q^{94} + ( - \beta_{2} - \beta_1 + 2) q^{95} + (2 \beta_{2} + 2 \beta_1 + 1) q^{96} + ( - 3 \beta_{2} + 7 \beta_1 - 2) q^{97} + ( - 3 \beta_{2} + 3 \beta_1 - 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + q^{3} + 9 q^{4} - 3 q^{5} - 5 q^{6} + q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + q^{3} + 9 q^{4} - 3 q^{5} - 5 q^{6} + q^{7} + 9 q^{8} + 2 q^{9} + q^{10} + 9 q^{12} - 6 q^{13} + 5 q^{14} - q^{15} + 13 q^{16} - 4 q^{17} - 20 q^{18} - 4 q^{19} - 9 q^{20} + 17 q^{21} - 6 q^{23} - 17 q^{24} + 3 q^{25} + 12 q^{26} + 13 q^{27} + 25 q^{28} - 2 q^{29} + 5 q^{30} + 14 q^{31} + 27 q^{32} - 8 q^{34} - q^{35} + 2 q^{36} + 4 q^{37} - 18 q^{38} + 4 q^{39} - 9 q^{40} - 9 q^{41} - 35 q^{42} - 7 q^{43} - 2 q^{45} - 8 q^{46} - 15 q^{47} + 7 q^{48} + 18 q^{49} - q^{50} + 20 q^{51} - 2 q^{52} - 6 q^{53} - 19 q^{54} - 3 q^{56} + 14 q^{57} + 30 q^{58} + 10 q^{59} - 9 q^{60} - 3 q^{61} - 24 q^{62} + 12 q^{63} + 29 q^{64} + 6 q^{65} + 19 q^{67} - 8 q^{69} - 5 q^{70} + 6 q^{71} - 48 q^{72} + 12 q^{73} + 28 q^{74} + q^{75} - 16 q^{76} - 2 q^{78} - 2 q^{79} - 13 q^{80} + 3 q^{81} - 27 q^{82} - 18 q^{83} + 31 q^{84} + 4 q^{85} - 3 q^{86} - 10 q^{87} + 11 q^{89} + 20 q^{90} + 20 q^{91} - 34 q^{92} + 20 q^{93} + 59 q^{94} + 4 q^{95} + 7 q^{96} - 2 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 5x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.86620
−1.65544
−0.210756
−2.34889 2.86620 3.51730 −1.00000 −6.73240 3.38350 −3.56399 5.21509 2.34889
1.2 −1.39593 −1.65544 −0.0513742 −1.00000 2.31088 −4.70682 2.86358 −0.259511 1.39593
1.3 2.74483 −0.210756 5.53407 −1.00000 −0.578488 2.32331 9.70041 −2.95558 −2.74483
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)

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 605.2.a.g 3
3.b odd 2 1 5445.2.a.bd 3
4.b odd 2 1 9680.2.a.bz 3
5.b even 2 1 3025.2.a.u 3
11.b odd 2 1 605.2.a.h yes 3
11.c even 5 4 605.2.g.p 12
11.d odd 10 4 605.2.g.o 12
33.d even 2 1 5445.2.a.bb 3
44.c even 2 1 9680.2.a.cb 3
55.d odd 2 1 3025.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.g 3 1.a even 1 1 trivial
605.2.a.h yes 3 11.b odd 2 1
605.2.g.o 12 11.d odd 10 4
605.2.g.p 12 11.c even 5 4
3025.2.a.p 3 55.d odd 2 1
3025.2.a.u 3 5.b even 2 1
5445.2.a.bb 3 33.d even 2 1
5445.2.a.bd 3 3.b odd 2 1
9680.2.a.bz 3 4.b odd 2 1
9680.2.a.cb 3 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{3} + T_{2}^{2} - 7T_{2} - 9 \) Copy content Toggle raw display
\( T_{3}^{3} - T_{3}^{2} - 5T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 7T - 9 \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 5T - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} - 19 T + 37 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 6 T^{2} + 4 T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 16 T - 48 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} - 12 T - 36 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} + 4 T - 12 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 28 T - 72 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 48 T - 36 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} - 24 T - 16 \) Copy content Toggle raw display
$41$ \( T^{3} + 9 T^{2} - 45 T - 297 \) Copy content Toggle raw display
$43$ \( T^{3} + 7 T^{2} - 3 T - 63 \) Copy content Toggle raw display
$47$ \( T^{3} + 15 T^{2} - 29 T - 801 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} + 4 T - 12 \) Copy content Toggle raw display
$59$ \( T^{3} - 10 T^{2} - 52 T + 348 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 161 T - 919 \) Copy content Toggle raw display
$67$ \( T^{3} - 19 T^{2} + 95 T - 59 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} + 4 T + 12 \) Copy content Toggle raw display
$73$ \( T^{3} - 12 T^{2} + 16 T + 32 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 76 T - 296 \) Copy content Toggle raw display
$83$ \( T^{3} + 18 T^{2} + 36 T - 324 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} - 157 T + 1719 \) Copy content Toggle raw display
$97$ \( T^{3} + 2 T^{2} - 228 T + 932 \) Copy content Toggle raw display
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