Properties

Label 2175.2.a.w
Level $2175$
Weight $2$
Character orbit 2175.a
Self dual yes
Analytic conductor $17.367$
Analytic rank $1$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_1 + 1) q^{6} + (\beta_{2} - 2) q^{7} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_1 + 1) q^{6} + (\beta_{2} - 2) q^{7} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{8} + q^{9} + ( - \beta_{3} - \beta_1 + 3) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{12} - \beta_{2} q^{13} + (\beta_{3} + 1) q^{14} + (\beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{16} + (\beta_{4} + 2 \beta_{2} - 1) q^{17} + (\beta_1 - 1) q^{18} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \cdots + 1) q^{19}+ \cdots + ( - \beta_{3} - \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9} + 12 q^{11} - 5 q^{12} - 2 q^{13} + 6 q^{14} + q^{16} - 3 q^{18} - 2 q^{19} + 8 q^{21} - 14 q^{22} - 8 q^{23} + 9 q^{24} - 5 q^{27} + 6 q^{28} + 5 q^{29} + 2 q^{31} - q^{32} - 12 q^{33} + 4 q^{34} + 5 q^{36} - 16 q^{37} + 14 q^{38} + 2 q^{39} - 14 q^{41} - 6 q^{42} + 20 q^{44} - 6 q^{46} - 2 q^{47} - q^{48} - 7 q^{49} - 16 q^{52} - 26 q^{53} + 3 q^{54} - 2 q^{56} + 2 q^{57} - 3 q^{58} + 4 q^{59} - 12 q^{61} - 8 q^{63} - 9 q^{64} + 14 q^{66} - 12 q^{67} + 20 q^{68} + 8 q^{69} + 30 q^{71} - 9 q^{72} + 12 q^{73} + 2 q^{74} - 44 q^{76} - 18 q^{77} - 18 q^{79} + 5 q^{81} - 10 q^{82} - 2 q^{83} - 6 q^{84} + 30 q^{86} - 5 q^{87} - 42 q^{88} - 22 q^{89} - 12 q^{91} - 20 q^{92} - 2 q^{93} - 50 q^{94} + q^{96} - 20 q^{97} + 9 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6 \) 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
−1.51908
−1.15351
0.245526
1.71250
2.71457
−2.51908 −1.00000 4.34577 0 2.51908 −0.173311 −5.90919 1.00000 0
1.2 −2.15351 −1.00000 2.63760 0 2.15351 −1.51591 −1.37308 1.00000 0
1.3 −0.754474 −1.00000 −1.43077 0 0.754474 −4.18524 2.58843 1.00000 0
1.4 0.712495 −1.00000 −1.49235 0 −0.712495 −2.77986 −2.48828 1.00000 0
1.5 1.71457 −1.00000 0.939748 0 −1.71457 0.654317 −1.81788 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(29\) \( -1 \)

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 2175.2.a.w 5
3.b odd 2 1 6525.2.a.bs 5
5.b even 2 1 2175.2.a.z 5
5.c odd 4 2 435.2.c.e 10
15.d odd 2 1 6525.2.a.bl 5
15.e even 4 2 1305.2.c.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.e 10 5.c odd 4 2
1305.2.c.j 10 15.e even 4 2
2175.2.a.w 5 1.a even 1 1 trivial
2175.2.a.z 5 5.b even 2 1
6525.2.a.bl 5 15.d odd 2 1
6525.2.a.bs 5 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\):

\( T_{2}^{5} + 3T_{2}^{4} - 3T_{2}^{3} - 11T_{2}^{2} + T_{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{5} + 8T_{7}^{4} + 18T_{7}^{3} + 6T_{7}^{2} - 11T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 3 T^{4} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 8 T^{4} + \cdots - 2 \) Copy content Toggle raw display
$11$ \( T^{5} - 12 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$13$ \( T^{5} + 2 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{5} - 28 T^{3} + \cdots + 298 \) Copy content Toggle raw display
$19$ \( T^{5} + 2 T^{4} + \cdots - 304 \) Copy content Toggle raw display
$23$ \( T^{5} + 8 T^{4} + \cdots + 40 \) Copy content Toggle raw display
$29$ \( (T - 1)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 2 T^{4} + \cdots - 6304 \) Copy content Toggle raw display
$37$ \( T^{5} + 16 T^{4} + \cdots + 584 \) Copy content Toggle raw display
$41$ \( T^{5} + 14 T^{4} + \cdots + 6176 \) Copy content Toggle raw display
$43$ \( T^{5} - 146 T^{3} + \cdots + 6848 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} + \cdots - 2692 \) Copy content Toggle raw display
$53$ \( T^{5} + 26 T^{4} + \cdots - 15056 \) Copy content Toggle raw display
$59$ \( T^{5} - 4 T^{4} + \cdots + 2000 \) Copy content Toggle raw display
$61$ \( T^{5} + 12 T^{4} + \cdots + 6872 \) Copy content Toggle raw display
$67$ \( T^{5} + 12 T^{4} + \cdots - 1310 \) Copy content Toggle raw display
$71$ \( T^{5} - 30 T^{4} + \cdots + 6592 \) Copy content Toggle raw display
$73$ \( T^{5} - 12 T^{4} + \cdots - 3368 \) Copy content Toggle raw display
$79$ \( T^{5} + 18 T^{4} + \cdots + 52048 \) Copy content Toggle raw display
$83$ \( T^{5} + 2 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$89$ \( T^{5} + 22 T^{4} + \cdots + 40682 \) Copy content Toggle raw display
$97$ \( T^{5} + 20 T^{4} + \cdots + 328 \) Copy content Toggle raw display
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