Properties

Label 5415.2.a.y
Level $5415$
Weight $2$
Character orbit 5415.a
Self dual yes
Analytic conductor $43.239$
Analytic rank $0$
Dimension $5$
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] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, 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("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.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: \(43.2389926945\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.8797896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} - \beta_{3} q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} - \beta_{3} q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + q^{9} - \beta_1 q^{10} + (\beta_{4} + 1) q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{3} - 2) q^{13} + (\beta_{4} + \beta_{2} - \beta_1 - 1) q^{14} - q^{15} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{16} + ( - \beta_{4} - \beta_{3} - 2 \beta_1 + 2) q^{17} - \beta_1 q^{18} + (\beta_{2} + 1) q^{20} + \beta_{3} q^{21} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} + ( - 2 \beta_{2} + 2 \beta_1) q^{23} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{24} + q^{25} + (\beta_{4} + \beta_{2} + \beta_1 - 1) q^{26} - q^{27} + ( - \beta_{3} - 2 \beta_1 + 2) q^{28} + (\beta_{4} - 2 \beta_1 - 1) q^{29} + \beta_1 q^{30} + ( - 2 \beta_{2} - 1) q^{31} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{32} + ( - \beta_{4} - 1) q^{33} + (\beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{34} - \beta_{3} q^{35} + (\beta_{2} + 1) q^{36} + ( - \beta_{3} + 2 \beta_{2}) q^{37} + (\beta_{3} + 2) q^{39} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{40} + (2 \beta_1 + 2) q^{41} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{42} + ( - \beta_{3} - 2) q^{43} + (4 \beta_{2} - 2 \beta_1 + 2) q^{44} + q^{45} + (2 \beta_{3} + 4 \beta_1 - 4) q^{46} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{47} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{48} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{49} - \beta_1 q^{50} + (\beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{51} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{52} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{53} + \beta_1 q^{54} + (\beta_{4} + 1) q^{55} + ( - \beta_{4} + \beta_{2} - \beta_1 + 7) q^{56} + ( - 2 \beta_{3} + 2 \beta_{2} + 6) q^{58} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{59} + ( - \beta_{2} - 1) q^{60} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 1) q^{61} + (2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 2) q^{62} - \beta_{3} q^{63} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{64} + ( - \beta_{3} - 2) q^{65} + (2 \beta_{3} + 2 \beta_1) q^{66} + (\beta_{3} + 4 \beta_{2} - 4) q^{67} + ( - 3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 1) q^{68} + (2 \beta_{2} - 2 \beta_1) q^{69} + (\beta_{4} + \beta_{2} - \beta_1 - 1) q^{70} + ( - \beta_{4} - 2 \beta_1 - 5) q^{71} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{72} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 4) q^{73} + (\beta_{4} - 2 \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{74} - q^{75} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 4) q^{77} + ( - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{78} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 5) q^{79} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{80} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{82} + ( - \beta_{4} + \beta_{3} - 2) q^{83} + (\beta_{3} + 2 \beta_1 - 2) q^{84} + ( - \beta_{4} - \beta_{3} - 2 \beta_1 + 2) q^{85} + (\beta_{4} + \beta_{2} + \beta_1 - 1) q^{86} + ( - \beta_{4} + 2 \beta_1 + 1) q^{87} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{88} + ( - \beta_{4} - 4 \beta_1 - 1) q^{89} - \beta_1 q^{90} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_1 + 10) q^{91} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 10) q^{92} + (2 \beta_{2} + 1) q^{93} + (\beta_{4} + 5 \beta_{2} + 7) q^{94} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{96} + 2 q^{97} + (\beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{98} + (\beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} - 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} + 5 q^{11} - 7 q^{12} - 8 q^{13} - 4 q^{14} - 5 q^{15} + 7 q^{16} + 10 q^{17} - q^{18} + 7 q^{20} - 2 q^{21} + 2 q^{22} - 2 q^{23} + 6 q^{24} + 5 q^{25} - 2 q^{26} - 5 q^{27} + 10 q^{28} - 7 q^{29} + q^{30} - 9 q^{31} - 23 q^{32} - 5 q^{33} + 25 q^{34} + 2 q^{35} + 7 q^{36} + 6 q^{37} + 8 q^{39} - 6 q^{40} + 12 q^{41} + 4 q^{42} - 8 q^{43} + 16 q^{44} + 5 q^{45} - 20 q^{46} + 6 q^{47} - 7 q^{48} + 15 q^{49} - q^{50} - 10 q^{51} - 4 q^{52} + 8 q^{53} + q^{54} + 5 q^{55} + 36 q^{56} + 38 q^{58} - q^{59} - 7 q^{60} + 7 q^{61} + 15 q^{62} + 2 q^{63} + 14 q^{64} - 8 q^{65} - 2 q^{66} - 14 q^{67} - q^{68} + 2 q^{69} - 4 q^{70} - 27 q^{71} - 6 q^{72} + 26 q^{73} - 18 q^{74} - 5 q^{75} + 14 q^{77} + 2 q^{78} + 23 q^{79} + 7 q^{80} + 5 q^{81} - 36 q^{82} - 12 q^{83} - 10 q^{84} + 10 q^{85} - 2 q^{86} + 7 q^{87} - 9 q^{89} - q^{90} + 46 q^{91} - 52 q^{92} + 9 q^{93} + 45 q^{94} + 23 q^{96} + 10 q^{97} + 21 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 15 \) 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
2.69159
1.64661
0.290697
−1.38140
−2.24750
−2.69159 −1.00000 5.24466 1.00000 2.69159 −0.797044 −8.73329 1.00000 −2.69159
1.2 −1.64661 −1.00000 0.711327 1.00000 1.64661 4.47988 2.12194 1.00000 −1.64661
1.3 −0.290697 −1.00000 −1.91550 1.00000 0.290697 −0.486575 1.13822 1.00000 −0.290697
1.4 1.38140 −1.00000 −0.0917248 1.00000 −1.38140 −4.36264 −2.88952 1.00000 1.38140
1.5 2.24750 −1.00000 3.05123 1.00000 −2.24750 3.16638 2.36264 1.00000 2.24750
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

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 5415.2.a.y 5
19.b odd 2 1 5415.2.a.z 5
19.c even 3 2 285.2.i.f 10
57.h odd 6 2 855.2.k.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.f 10 19.c even 3 2
855.2.k.i 10 57.h odd 6 2
5415.2.a.y 5 1.a even 1 1 trivial
5415.2.a.z 5 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):

\( T_{2}^{5} + T_{2}^{4} - 8T_{2}^{3} - 5T_{2}^{2} + 13T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{5} - 2T_{7}^{4} - 23T_{7}^{3} + 36T_{7}^{2} + 72T_{7} + 24 \) Copy content Toggle raw display
\( T_{11}^{5} - 5T_{11}^{4} - 32T_{11}^{3} + 148T_{11}^{2} + 256T_{11} - 992 \) Copy content Toggle raw display
\( T_{13}^{5} + 8T_{13}^{4} + T_{13}^{3} - 70T_{13}^{2} - 44T_{13} + 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 8 T^{3} - 5 T^{2} + 13 T + 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( (T - 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 2 T^{4} - 23 T^{3} + 36 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} - 32 T^{3} + 148 T^{2} + \cdots - 992 \) Copy content Toggle raw display
$13$ \( T^{5} + 8 T^{4} + T^{3} - 70 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$17$ \( T^{5} - 10 T^{4} - 35 T^{3} + \cdots - 5024 \) Copy content Toggle raw display
$19$ \( T^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 2 T^{4} - 68 T^{3} - 24 T^{2} + \cdots + 384 \) Copy content Toggle raw display
$29$ \( T^{5} + 7 T^{4} - 62 T^{3} + \cdots + 3008 \) Copy content Toggle raw display
$31$ \( T^{5} + 9 T^{4} - 30 T^{3} - 222 T^{2} + \cdots + 117 \) Copy content Toggle raw display
$37$ \( T^{5} - 6 T^{4} - 87 T^{3} + \cdots - 9024 \) Copy content Toggle raw display
$41$ \( T^{5} - 12 T^{4} + 24 T^{3} + \cdots - 192 \) Copy content Toggle raw display
$43$ \( T^{5} + 8 T^{4} + T^{3} - 70 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$47$ \( T^{5} - 6 T^{4} - 153 T^{3} + \cdots + 2136 \) Copy content Toggle raw display
$53$ \( T^{5} - 8 T^{4} - 95 T^{3} + \cdots - 4328 \) Copy content Toggle raw display
$59$ \( T^{5} + T^{4} - 176 T^{3} - 332 T^{2} + \cdots - 4544 \) Copy content Toggle raw display
$61$ \( T^{5} - 7 T^{4} - 173 T^{3} + \cdots - 24316 \) Copy content Toggle raw display
$67$ \( T^{5} + 14 T^{4} - 167 T^{3} + \cdots + 8376 \) Copy content Toggle raw display
$71$ \( T^{5} + 27 T^{4} + 222 T^{3} + \cdots - 6096 \) Copy content Toggle raw display
$73$ \( T^{5} - 26 T^{4} + 61 T^{3} + \cdots + 46208 \) Copy content Toggle raw display
$79$ \( T^{5} - 23 T^{4} - 41 T^{3} + \cdots + 109248 \) Copy content Toggle raw display
$83$ \( T^{5} + 12 T^{4} - 21 T^{3} + \cdots - 2304 \) Copy content Toggle raw display
$89$ \( T^{5} + 9 T^{4} - 132 T^{3} + \cdots + 11232 \) Copy content Toggle raw display
$97$ \( (T - 2)^{5} \) Copy content Toggle raw display
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