Properties

Label 2175.2.c.m
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.14077504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 11x^{4} + 33x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 2) q^{4} - \beta_{5} q^{6} + ( - \beta_{4} - 2 \beta_{2} + \beta_1) q^{7} + ( - \beta_{4} - \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 2) q^{4} - \beta_{5} q^{6} + ( - \beta_{4} - 2 \beta_{2} + \beta_1) q^{7} + ( - \beta_{4} - \beta_1) q^{8} - q^{9} + 3 q^{11} + ( - \beta_{4} - 2 \beta_{2}) q^{12} + ( - \beta_{4} - 2 \beta_{2} - \beta_1) q^{13} + (3 \beta_{5} - 4) q^{14} + \beta_{5} q^{16} + ( - \beta_{4} + \beta_1) q^{17} - \beta_1 q^{18} + ( - 2 \beta_{3} + 2) q^{19} + ( - \beta_{5} - \beta_{3} + 2) q^{21} + 3 \beta_1 q^{22} + (\beta_{4} - \beta_{2} + 3 \beta_1) q^{23} + (\beta_{5} - \beta_{3}) q^{24} + (3 \beta_{5} - 2 \beta_{3} + 4) q^{26} - \beta_{2} q^{27} + (\beta_{4} + 8 \beta_{2} - 2 \beta_1) q^{28} + q^{29} + ( - \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{32} + 3 \beta_{2} q^{33} + (\beta_{5} - 4) q^{34} + ( - \beta_{3} + 2) q^{36} + ( - \beta_{4} + 5 \beta_{2} - 3 \beta_1) q^{37} + (2 \beta_{4} + 4 \beta_1) q^{38} + (\beta_{5} - \beta_{3} + 2) q^{39} + ( - \beta_{5} - \beta_{3} + 5) q^{41} + ( - 4 \beta_{2} + 3 \beta_1) q^{42} + ( - \beta_{4} + 3 \beta_{2} + 3 \beta_1) q^{43} + (3 \beta_{3} - 6) q^{44} + (4 \beta_{3} - 12) q^{46} + (3 \beta_{4} + 2 \beta_{2} - \beta_1) q^{47} + \beta_1 q^{48} + (5 \beta_{5} + \beta_{3} - 5) q^{49} + ( - \beta_{5} - \beta_{3}) q^{51} + (3 \beta_{4} + 8 \beta_{2} + 4 \beta_1) q^{52} + (3 \beta_{4} + \beta_{2} + 3 \beta_1) q^{53} + \beta_{5} q^{54} + ( - 3 \beta_{5} - \beta_{3}) q^{56} + (2 \beta_{4} + 2 \beta_{2}) q^{57} + \beta_1 q^{58} + (2 \beta_{5} - 8) q^{59} + ( - 2 \beta_{3} + 4) q^{61} + (\beta_{4} + 2 \beta_{2} - \beta_1) q^{63} + ( - \beta_{5} - 3 \beta_{3} + 8) q^{64} - 3 \beta_{5} q^{66} + (\beta_{4} - 10 \beta_{2} + 3 \beta_1) q^{67} + ( - \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{68} + ( - 3 \beta_{5} + \beta_{3} + 1) q^{69} + (6 \beta_{5} - 2) q^{71} + (\beta_{4} + \beta_1) q^{72} + ( - 5 \beta_{4} - 3 \beta_{2} + \beta_1) q^{73} + ( - 4 \beta_{5} - 4 \beta_{3} + 12) q^{74} + ( - 2 \beta_{5} + 2 \beta_{3} - 12) q^{76} + ( - 3 \beta_{4} - 6 \beta_{2} + 3 \beta_1) q^{77} + (2 \beta_{4} + 4 \beta_{2} + 3 \beta_1) q^{78} + ( - 2 \beta_{5} + 4 \beta_{3}) q^{79} + q^{81} + ( - 4 \beta_{2} + 6 \beta_1) q^{82} + ( - \beta_{4} + 5 \beta_{2} - \beta_1) q^{83} + (2 \beta_{5} + \beta_{3} - 8) q^{84} + ( - 2 \beta_{5} + 2 \beta_{3} - 12) q^{86} + \beta_{2} q^{87} + ( - 3 \beta_{4} - 3 \beta_1) q^{88} + (3 \beta_{5} - 3 \beta_{3} - 10) q^{89} + ( - \beta_{5} + \beta_{3} - 4) q^{91} + ( - 2 \beta_{4} - 2 \beta_{2} - 10 \beta_1) q^{92} + ( - 5 \beta_{5} + 2 \beta_{3} + 4) q^{94} + (2 \beta_{5} - \beta_{3} - 4) q^{96} + ( - \beta_{4} + \beta_{2} - 5 \beta_1) q^{97} + (4 \beta_{4} + 20 \beta_{2} - 6 \beta_1) q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 18 q^{11} - 18 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{21} + 26 q^{26} + 6 q^{29} - 22 q^{34} + 10 q^{36} + 12 q^{39} + 26 q^{41} - 30 q^{44} - 64 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 44 q^{59} + 20 q^{61} + 40 q^{64} - 6 q^{66} + 2 q^{69} + 56 q^{74} - 72 q^{76} + 4 q^{79} + 6 q^{81} - 42 q^{84} - 72 q^{86} - 60 q^{89} - 24 q^{91} + 18 q^{94} - 22 q^{96} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 11x^{4} + 33x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 9\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 6\nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 6\beta_{3} + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{4} + 4\beta_{2} + 26\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
2.39138i
2.16425i
0.772866i
0.772866i
2.16425i
2.39138i
2.39138i 1.00000i −3.71871 0 −2.39138 1.32733i 4.11009i −1.00000 0
349.2 2.16425i 1.00000i −2.68397 0 2.16425 4.84822i 1.48028i −1.00000 0
349.3 0.772866i 1.00000i 1.40268 0 −0.772866 2.17554i 2.62981i −1.00000 0
349.4 0.772866i 1.00000i 1.40268 0 −0.772866 2.17554i 2.62981i −1.00000 0
349.5 2.16425i 1.00000i −2.68397 0 2.16425 4.84822i 1.48028i −1.00000 0
349.6 2.39138i 1.00000i −3.71871 0 −2.39138 1.32733i 4.11009i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.2.c.m 6
5.b even 2 1 inner 2175.2.c.m 6
5.c odd 4 1 435.2.a.i 3
5.c odd 4 1 2175.2.a.u 3
15.e even 4 1 1305.2.a.q 3
15.e even 4 1 6525.2.a.bf 3
20.e even 4 1 6960.2.a.cl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.i 3 5.c odd 4 1
1305.2.a.q 3 15.e even 4 1
2175.2.a.u 3 5.c odd 4 1
2175.2.c.m 6 1.a even 1 1 trivial
2175.2.c.m 6 5.b even 2 1 inner
6525.2.a.bf 3 15.e even 4 1
6960.2.a.cl 3 20.e even 4 1

Hecke kernels

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

\( T_{2}^{6} + 11T_{2}^{4} + 33T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{6} + 30T_{7}^{4} + 161T_{7}^{2} + 196 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 30 T^{4} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( (T - 3)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 38 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{6} + 26 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 24 T + 88)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 113 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$29$ \( (T - 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} + 169 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{3} - 13 T^{2} + \cdots - 28)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 165 T^{4} + \cdots + 94864 \) Copy content Toggle raw display
$47$ \( T^{6} + 142 T^{4} + \cdots + 70756 \) Copy content Toggle raw display
$53$ \( T^{6} + 237 T^{4} + \cdots + 99856 \) Copy content Toggle raw display
$59$ \( (T^{3} + 22 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 10 T^{2} + \cdots + 112)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 374 T^{4} + \cdots + 37636 \) Copy content Toggle raw display
$71$ \( (T^{3} - 192 T + 488)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 377 T^{4} + \cdots + 1364224 \) Copy content Toggle raw display
$79$ \( (T^{3} - 2 T^{2} + \cdots + 224)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 101 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( (T^{3} + 30 T^{2} + \cdots - 602)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 285 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
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