Properties

Label 2175.2.a.ba
Level $2175$
Weight $2$
Character orbit 2175.a
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{6} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{6} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} - 7 q^{3} + 10 q^{4} + 2 q^{6} + q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{2} - 7 q^{3} + 10 q^{4} + 2 q^{6} + q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} - 10 q^{12} + q^{13} + 15 q^{14} + 12 q^{16} - 8 q^{17} - 2 q^{18} + 15 q^{19} - q^{21} - 3 q^{22} - 14 q^{23} + 3 q^{24} + 6 q^{26} - 7 q^{27} + 24 q^{28} - 7 q^{29} + 5 q^{31} - 18 q^{32} - 4 q^{33} + 7 q^{34} + 10 q^{36} + 6 q^{37} + 18 q^{38} - q^{39} + 22 q^{41} - 15 q^{42} + 19 q^{43} + 15 q^{44} - 4 q^{46} - 22 q^{47} - 12 q^{48} + 12 q^{49} + 8 q^{51} - 11 q^{52} - 10 q^{53} + 2 q^{54} + 14 q^{56} - 15 q^{57} + 2 q^{58} + 6 q^{59} + 23 q^{61} - 40 q^{62} + q^{63} + 5 q^{64} + 3 q^{66} + 13 q^{67} + 7 q^{68} + 14 q^{69} + 26 q^{71} - 3 q^{72} + 24 q^{73} - 10 q^{74} + 46 q^{76} - 4 q^{77} - 6 q^{78} + 14 q^{79} + 7 q^{81} - 16 q^{82} - 10 q^{83} - 24 q^{84} + 44 q^{86} + 7 q^{87} + 66 q^{88} + 14 q^{89} + 13 q^{91} - 58 q^{92} - 5 q^{93} - 3 q^{94} + 18 q^{96} + 31 q^{97} + 59 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 10x^{5} + 19x^{4} + 24x^{3} - 44x^{2} - 3x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + \nu^{5} + 19\nu^{4} - 7\nu^{3} - 41\nu^{2} + 9\nu + 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + \nu^{5} + 24\nu^{4} - 7\nu^{3} - 76\nu^{2} + 9\nu + 37 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{6} - 4\nu^{5} - 31\nu^{4} + 33\nu^{3} + 84\nu^{2} - 56\nu - 38 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{6} - \beta_{5} - 2\beta_{4} + 9\beta_{3} + 2\beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{6} + 9\beta_{5} - 13\beta_{4} + \beta_{3} + 47\beta_{2} + \beta _1 + 85 \) 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
2.66356
2.26695
1.14889
0.792771
−0.560139
−1.83397
−2.47806
−2.66356 −1.00000 5.09453 0 2.66356 0.0170416 −8.24244 1.00000 0
1.2 −2.26695 −1.00000 3.13907 0 2.26695 −0.159887 −2.58223 1.00000 0
1.3 −1.14889 −1.00000 −0.680059 0 1.14889 −3.52165 3.07908 1.00000 0
1.4 −0.792771 −1.00000 −1.37151 0 0.792771 2.01830 2.67284 1.00000 0
1.5 0.560139 −1.00000 −1.68624 0 −0.560139 −4.36313 −2.06481 1.00000 0
1.6 1.83397 −1.00000 1.36343 0 −1.83397 4.17416 −1.16745 1.00000 0
1.7 2.47806 −1.00000 4.14079 0 −2.47806 2.83516 5.30500 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.ba 7
3.b odd 2 1 6525.2.a.bx 7
5.b even 2 1 2175.2.a.bb yes 7
5.c odd 4 2 2175.2.c.o 14
15.d odd 2 1 6525.2.a.bu 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2175.2.a.ba 7 1.a even 1 1 trivial
2175.2.a.bb yes 7 5.b even 2 1
2175.2.c.o 14 5.c odd 4 2
6525.2.a.bu 7 15.d odd 2 1
6525.2.a.bx 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2175))\):

\( T_{2}^{7} + 2T_{2}^{6} - 10T_{2}^{5} - 19T_{2}^{4} + 24T_{2}^{3} + 44T_{2}^{2} - 3T_{2} - 14 \) Copy content Toggle raw display
\( T_{7}^{7} - T_{7}^{6} - 30T_{7}^{5} + 38T_{7}^{4} + 217T_{7}^{3} - 337T_{7}^{2} - 53T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 2 T^{6} + \cdots - 14 \) Copy content Toggle raw display
$3$ \( (T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - T^{6} - 30 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{7} - 4 T^{6} + \cdots + 10746 \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} + \cdots + 6803 \) Copy content Toggle raw display
$17$ \( T^{7} + 8 T^{6} + \cdots - 86134 \) Copy content Toggle raw display
$19$ \( T^{7} - 15 T^{6} + \cdots - 15680 \) Copy content Toggle raw display
$23$ \( T^{7} + 14 T^{6} + \cdots + 27008 \) Copy content Toggle raw display
$29$ \( (T + 1)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} - 5 T^{6} + \cdots - 12096 \) Copy content Toggle raw display
$37$ \( T^{7} - 6 T^{6} + \cdots + 13696 \) Copy content Toggle raw display
$41$ \( T^{7} - 22 T^{6} + \cdots - 626632 \) Copy content Toggle raw display
$43$ \( T^{7} - 19 T^{6} + \cdots + 42304 \) Copy content Toggle raw display
$47$ \( T^{7} + 22 T^{6} + \cdots + 2842 \) Copy content Toggle raw display
$53$ \( T^{7} + 10 T^{6} + \cdots + 1229824 \) Copy content Toggle raw display
$59$ \( T^{7} - 6 T^{6} + \cdots - 87680 \) Copy content Toggle raw display
$61$ \( T^{7} - 23 T^{6} + \cdots + 51776 \) Copy content Toggle raw display
$67$ \( T^{7} - 13 T^{6} + \cdots - 37259 \) Copy content Toggle raw display
$71$ \( T^{7} - 26 T^{6} + \cdots - 34048 \) Copy content Toggle raw display
$73$ \( T^{7} - 24 T^{6} + \cdots - 1658752 \) Copy content Toggle raw display
$79$ \( T^{7} - 14 T^{6} + \cdots + 2560 \) Copy content Toggle raw display
$83$ \( T^{7} + 10 T^{6} + \cdots + 5248 \) Copy content Toggle raw display
$89$ \( T^{7} - 14 T^{6} + \cdots + 231800 \) Copy content Toggle raw display
$97$ \( T^{7} - 31 T^{6} + \cdots - 6565312 \) Copy content Toggle raw display
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