Properties

Label 2175.2.a.r
Level $2175$
Weight $2$
Character orbit 2175.a
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
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] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + (\beta + 3) q^{4} - \beta q^{6} - q^{7} + (2 \beta + 5) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + (\beta + 3) q^{4} - \beta q^{6} - q^{7} + (2 \beta + 5) q^{8} + q^{9} + 5 q^{11} + ( - \beta - 3) q^{12} + ( - 2 \beta + 1) q^{13} - \beta q^{14} + (5 \beta + 4) q^{16} + 3 q^{17} + \beta q^{18} - 2 \beta q^{19} + q^{21} + 5 \beta q^{22} + 4 q^{23} + ( - 2 \beta - 5) q^{24} + ( - \beta - 10) q^{26} - q^{27} + ( - \beta - 3) q^{28} + q^{29} + 4 q^{31} + (5 \beta + 15) q^{32} - 5 q^{33} + 3 \beta q^{34} + (\beta + 3) q^{36} + 4 q^{37} + ( - 2 \beta - 10) q^{38} + (2 \beta - 1) q^{39} + (4 \beta - 2) q^{41} + \beta q^{42} + ( - 2 \beta + 6) q^{43} + (5 \beta + 15) q^{44} + 4 \beta q^{46} + (2 \beta - 7) q^{47} + ( - 5 \beta - 4) q^{48} - 6 q^{49} - 3 q^{51} + ( - 7 \beta - 7) q^{52} + ( - 2 \beta - 4) q^{53} - \beta q^{54} + ( - 2 \beta - 5) q^{56} + 2 \beta q^{57} + \beta q^{58} + (2 \beta - 4) q^{59} + ( - 6 \beta + 2) q^{61} + 4 \beta q^{62} - q^{63} + (10 \beta + 17) q^{64} - 5 \beta q^{66} + ( - 4 \beta - 3) q^{67} + (3 \beta + 9) q^{68} - 4 q^{69} + (2 \beta - 6) q^{71} + (2 \beta + 5) q^{72} - 4 q^{73} + 4 \beta q^{74} + ( - 8 \beta - 10) q^{76} - 5 q^{77} + (\beta + 10) q^{78} + ( - 2 \beta + 4) q^{79} + q^{81} + (2 \beta + 20) q^{82} + ( - 2 \beta + 8) q^{83} + (\beta + 3) q^{84} + (4 \beta - 10) q^{86} - q^{87} + (10 \beta + 25) q^{88} + (2 \beta + 5) q^{89} + (2 \beta - 1) q^{91} + (4 \beta + 12) q^{92} - 4 q^{93} + ( - 5 \beta + 10) q^{94} + ( - 5 \beta - 15) q^{96} + (2 \beta - 8) q^{97} - 6 \beta q^{98} + 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26} - 2 q^{27} - 7 q^{28} + 2 q^{29} + 8 q^{31} + 35 q^{32} - 10 q^{33} + 3 q^{34} + 7 q^{36} + 8 q^{37} - 22 q^{38} + q^{42} + 10 q^{43} + 35 q^{44} + 4 q^{46} - 12 q^{47} - 13 q^{48} - 12 q^{49} - 6 q^{51} - 21 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} - 2 q^{61} + 4 q^{62} - 2 q^{63} + 44 q^{64} - 5 q^{66} - 10 q^{67} + 21 q^{68} - 8 q^{69} - 10 q^{71} + 12 q^{72} - 8 q^{73} + 4 q^{74} - 28 q^{76} - 10 q^{77} + 21 q^{78} + 6 q^{79} + 2 q^{81} + 42 q^{82} + 14 q^{83} + 7 q^{84} - 16 q^{86} - 2 q^{87} + 60 q^{88} + 12 q^{89} + 28 q^{92} - 8 q^{93} + 15 q^{94} - 35 q^{96} - 14 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100}) \) 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
−1.79129
2.79129
−1.79129 −1.00000 1.20871 0 1.79129 −1.00000 1.41742 1.00000 0
1.2 2.79129 −1.00000 5.79129 0 −2.79129 −1.00000 10.5826 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.r 2
3.b odd 2 1 6525.2.a.t 2
5.b even 2 1 435.2.a.f 2
5.c odd 4 2 2175.2.c.f 4
15.d odd 2 1 1305.2.a.m 2
20.d odd 2 1 6960.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.f 2 5.b even 2 1
1305.2.a.m 2 15.d odd 2 1
2175.2.a.r 2 1.a even 1 1 trivial
2175.2.c.f 4 5.c odd 4 2
6525.2.a.t 2 3.b odd 2 1
6960.2.a.bw 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 21 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 20 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 84 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 188 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T - 59 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 4 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 12 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 15 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 28 \) Copy content Toggle raw display
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