Properties

Label 5415.2.a.p
Level $5415$
Weight $2$
Character orbit 5415.a
Self dual yes
Analytic conductor $43.239$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 1) q^{6} + ( - \beta + 2) q^{7} + (\beta - 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + (\beta - 1) q^{6} + ( - \beta + 2) q^{7} + (\beta - 3) q^{8} + q^{9} + ( - \beta + 1) q^{10} + \beta q^{11} + ( - 2 \beta + 1) q^{12} + ( - \beta - 4) q^{13} + (3 \beta - 4) q^{14} - q^{15} + 3 q^{16} + (2 \beta - 4) q^{17} + (\beta - 1) q^{18} + (2 \beta - 1) q^{20} + ( - \beta + 2) q^{21} + ( - \beta + 2) q^{22} + (4 \beta + 2) q^{23} + (\beta - 3) q^{24} + q^{25} + ( - 3 \beta + 2) q^{26} + q^{27} + ( - 5 \beta + 6) q^{28} + (5 \beta + 2) q^{29} + ( - \beta + 1) q^{30} + ( - 6 \beta + 2) q^{31} + (\beta + 3) q^{32} + \beta q^{33} + ( - 6 \beta + 8) q^{34} + (\beta - 2) q^{35} + ( - 2 \beta + 1) q^{36} + ( - 5 \beta - 4) q^{37} + ( - \beta - 4) q^{39} + ( - \beta + 3) q^{40} + (\beta + 6) q^{41} + (3 \beta - 4) q^{42} + ( - \beta + 2) q^{43} + (\beta - 4) q^{44} - q^{45} + ( - 2 \beta + 6) q^{46} + ( - 4 \beta + 6) q^{47} + 3 q^{48} + ( - 4 \beta - 1) q^{49} + (\beta - 1) q^{50} + (2 \beta - 4) q^{51} + 7 \beta q^{52} - 4 q^{53} + (\beta - 1) q^{54} - \beta q^{55} + (5 \beta - 8) q^{56} + ( - 3 \beta + 8) q^{58} - 6 \beta q^{59} + (2 \beta - 1) q^{60} + 4 \beta q^{61} + (8 \beta - 14) q^{62} + ( - \beta + 2) q^{63} + (2 \beta - 7) q^{64} + (\beta + 4) q^{65} + ( - \beta + 2) q^{66} - 12 q^{67} + (10 \beta - 12) q^{68} + (4 \beta + 2) q^{69} + ( - 3 \beta + 4) q^{70} + (6 \beta - 4) q^{71} + (\beta - 3) q^{72} - 2 q^{73} + (\beta - 6) q^{74} + q^{75} + (2 \beta - 2) q^{77} + ( - 3 \beta + 2) q^{78} + 8 \beta q^{79} - 3 q^{80} + q^{81} + (5 \beta - 4) q^{82} + ( - 6 \beta - 2) q^{83} + ( - 5 \beta + 6) q^{84} + ( - 2 \beta + 4) q^{85} + (3 \beta - 4) q^{86} + (5 \beta + 2) q^{87} + ( - 3 \beta + 2) q^{88} + ( - 9 \beta + 2) q^{89} + ( - \beta + 1) q^{90} + (2 \beta - 6) q^{91} - 14 q^{92} + ( - 6 \beta + 2) q^{93} + (10 \beta - 14) q^{94} + (\beta + 3) q^{96} + 3 \beta q^{97} + (3 \beta - 7) q^{98} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} - 6 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 8 q^{13} - 8 q^{14} - 2 q^{15} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{20} + 4 q^{21} + 4 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{27} + 12 q^{28} + 4 q^{29} + 2 q^{30} + 4 q^{31} + 6 q^{32} + 16 q^{34} - 4 q^{35} + 2 q^{36} - 8 q^{37} - 8 q^{39} + 6 q^{40} + 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 2 q^{45} + 12 q^{46} + 12 q^{47} + 6 q^{48} - 2 q^{49} - 2 q^{50} - 8 q^{51} - 8 q^{53} - 2 q^{54} - 16 q^{56} + 16 q^{58} - 2 q^{60} - 28 q^{62} + 4 q^{63} - 14 q^{64} + 8 q^{65} + 4 q^{66} - 24 q^{67} - 24 q^{68} + 4 q^{69} + 8 q^{70} - 8 q^{71} - 6 q^{72} - 4 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{77} + 4 q^{78} - 6 q^{80} + 2 q^{81} - 8 q^{82} - 4 q^{83} + 12 q^{84} + 8 q^{85} - 8 q^{86} + 4 q^{87} + 4 q^{88} + 4 q^{89} + 2 q^{90} - 12 q^{91} - 28 q^{92} + 4 q^{93} - 28 q^{94} + 6 q^{96} - 14 q^{98}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−2.41421 1.00000 3.82843 −1.00000 −2.41421 3.41421 −4.41421 1.00000 2.41421
1.2 0.414214 1.00000 −1.82843 −1.00000 0.414214 0.585786 −1.58579 1.00000 −0.414214
\(n\): e.g. 2-40 or 990-1000
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.p 2
19.b odd 2 1 285.2.a.f 2
57.d even 2 1 855.2.a.e 2
76.d even 2 1 4560.2.a.bj 2
95.d odd 2 1 1425.2.a.l 2
95.g even 4 2 1425.2.c.j 4
285.b even 2 1 4275.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.f 2 19.b odd 2 1
855.2.a.e 2 57.d even 2 1
1425.2.a.l 2 95.d odd 2 1
1425.2.c.j 4 95.g even 4 2
4275.2.a.x 2 285.b even 2 1
4560.2.a.bj 2 76.d even 2 1
5415.2.a.p 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 46 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 34 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( T^{2} - 32 \) Copy content Toggle raw display
$67$ \( (T + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T - 158 \) Copy content Toggle raw display
$97$ \( T^{2} - 18 \) Copy content Toggle raw display
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