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Properties

Label 605.2.a.k
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 3x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu + 1 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.35567
−0.477260
0.737640
2.09529
−1.35567 0.575493 −0.162147 −1.00000 −0.780181 3.64941 2.93117 −2.66881 1.35567
1.2 −0.477260 0.323071 −1.77222 −1.00000 −0.154189 −2.68522 1.80033 −2.89563 0.477260
1.3 0.737640 −2.81156 −1.45589 −1.00000 −2.07392 −1.03138 −2.54920 4.90488 −0.737640
1.4 2.09529 1.91300 2.39026 −1.00000 4.00829 3.06719 0.817703 0.659557 −2.09529
\(n\): 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.k 4
3.b odd 2 1 5445.2.a.bi 4
4.b odd 2 1 9680.2.a.cm 4
5.b even 2 1 3025.2.a.w 4
11.b odd 2 1 605.2.a.j 4
11.c even 5 2 605.2.g.e 8
11.c even 5 2 605.2.g.k 8
11.d odd 10 2 55.2.g.b ✓ 8
11.d odd 10 2 605.2.g.m 8
33.d even 2 1 5445.2.a.bp 4
33.f even 10 2 495.2.n.e 8
44.c even 2 1 9680.2.a.cn 4
44.g even 10 2 880.2.bo.h 8
55.d odd 2 1 3025.2.a.bd 4
55.h odd 10 2 275.2.h.a 8
55.l even 20 4 275.2.z.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b ✓ 8 11.d odd 10 2
275.2.h.a 8 55.h odd 10 2
275.2.z.a 16 55.l even 20 4
495.2.n.e 8 33.f even 10 2
605.2.a.j 4 11.b odd 2 1
605.2.a.k 4 1.a even 1 1 trivial
605.2.g.e 8 11.c even 5 2
605.2.g.k 8 11.c even 5 2
605.2.g.m 8 11.d odd 10 2
880.2.bo.h 8 44.g even 10 2
3025.2.a.w 4 5.b even 2 1
3025.2.a.bd 4 55.d odd 2 1
5445.2.a.bi 4 3.b odd 2 1
5445.2.a.bp 4 33.d even 2 1
9680.2.a.cm 4 4.b odd 2 1
9680.2.a.cn 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{4} - T_{2}^{3} - 3T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 6T_{3}^{2} + 5T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 3 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots + 31 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + \cdots + 139 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + \cdots + 19 \) Copy content Toggle raw display
$19$ \( T^{4} - 20 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{4} - 5 T^{3} + \cdots - 11 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + \cdots - 451 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$37$ \( T^{4} - 7 T^{3} + \cdots - 1151 \) Copy content Toggle raw display
$41$ \( T^{4} - 11 T^{3} + \cdots - 319 \) Copy content Toggle raw display
$43$ \( T^{4} - 19 T^{3} + \cdots + 211 \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + \cdots + 941 \) Copy content Toggle raw display
$59$ \( T^{4} - 9 T^{3} + \cdots - 829 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots - 169 \) Copy content Toggle raw display
$67$ \( T^{4} + 19 T^{3} + \cdots - 4079 \) Copy content Toggle raw display
$71$ \( T^{4} - 5 T^{3} + \cdots - 131 \) Copy content Toggle raw display
$73$ \( T^{4} - 11 T^{3} + \cdots - 11 \) Copy content Toggle raw display
$79$ \( T^{4} - 34 T^{3} + \cdots - 6779 \) Copy content Toggle raw display
$83$ \( T^{4} + 11 T^{3} + \cdots - 1699 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} + \cdots + 1861 \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + \cdots - 3011 \) Copy content Toggle raw display
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