Properties

Label 2175.2.c.n
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.1267360000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_1) q^{2} - \beta_{5} q^{3} + ( - \beta_{6} + \beta_{4} - 2) q^{4} + (\beta_{6} + 1) q^{6} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_1) q^{2} - \beta_{5} q^{3} + ( - \beta_{6} + \beta_{4} - 2) q^{4} + (\beta_{6} + 1) q^{6} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{8} - q^{9} + (\beta_{7} - \beta_{4}) q^{11} + (\beta_{5} + \beta_{3} + \beta_1) q^{12} + ( - \beta_{5} - \beta_{3} + \cdots + 2 \beta_1) q^{13}+ \cdots + ( - \beta_{7} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{11} + 6 q^{14} + 22 q^{16} + 4 q^{19} - 4 q^{21} - 24 q^{24} - 14 q^{26} + 8 q^{29} - 8 q^{31} + 2 q^{34} + 10 q^{36} - 16 q^{39} - 24 q^{41} - 18 q^{44} - 16 q^{46} - 12 q^{49} + 20 q^{51} - 6 q^{54} - 4 q^{59} - 52 q^{61} - 68 q^{64} - 6 q^{66} - 24 q^{69} - 20 q^{71} - 96 q^{74} + 32 q^{76} - 44 q^{79} + 8 q^{81} - 18 q^{84} - 8 q^{86} + 8 q^{89} - 16 q^{91} - 78 q^{94} + 34 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 7\nu^{5} + \nu^{3} - 24\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 11\nu^{5} - 33\nu^{3} - 20\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 9\nu^{4} - 19\nu^{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + 29\nu^{5} + 75\nu^{3} + 48\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 21\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 33\nu^{2} + 22 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} - \beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 7\beta_{6} - 6\beta_{4} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} - 9\beta_{3} + 6\beta_{2} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{7} + 44\beta_{6} + 33\beta_{4} - 66 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{5} + 62\beta_{3} - 33\beta_{2} - 119\beta_1 \) 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
1.75660i
0.820249i
2.43828i
1.13856i
1.13856i
2.43828i
0.820249i
1.75660i
2.75660i 1.00000i −5.59883 0 2.75660 0.393832i 9.92054i −1.00000 0
349.2 1.82025i 1.00000i −1.31331 0 1.82025 0.729126i 1.24995i −1.00000 0
349.3 1.43828i 1.00000i −0.0686587 0 −1.43828 2.74301i 2.77782i −1.00000 0
349.4 0.138564i 1.00000i 1.98080 0 −0.138564 5.07830i 0.551597i −1.00000 0
349.5 0.138564i 1.00000i 1.98080 0 −0.138564 5.07830i 0.551597i −1.00000 0
349.6 1.43828i 1.00000i −0.0686587 0 −1.43828 2.74301i 2.77782i −1.00000 0
349.7 1.82025i 1.00000i −1.31331 0 1.82025 0.729126i 1.24995i −1.00000 0
349.8 2.75660i 1.00000i −5.59883 0 2.75660 0.393832i 9.92054i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
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.n 8
5.b even 2 1 inner 2175.2.c.n 8
5.c odd 4 1 435.2.a.j 4
5.c odd 4 1 2175.2.a.v 4
15.e even 4 1 1305.2.a.r 4
15.e even 4 1 6525.2.a.bi 4
20.e even 4 1 6960.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.j 4 5.c odd 4 1
1305.2.a.r 4 15.e even 4 1
2175.2.a.v 4 5.c odd 4 1
2175.2.c.n 8 1.a even 1 1 trivial
2175.2.c.n 8 5.b even 2 1 inner
6525.2.a.bi 4 15.e even 4 1
6960.2.a.co 4 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 \(S_{2}^{\mathrm{new}}(2175, [\chi])\):

\( T_{2}^{8} + 13T_{2}^{6} + 48T_{2}^{4} + 53T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} + 34T_{7}^{6} + 217T_{7}^{4} + 136T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 15T_{11}^{2} + 4T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 13 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 34 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 15 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 58 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$17$ \( T^{8} + 58 T^{6} + \cdots + 13456 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 40 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 168 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 60 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 232 T^{6} + \cdots + 6885376 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 1616)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 236 T^{6} + \cdots + 1478656 \) Copy content Toggle raw display
$47$ \( T^{8} + 146 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 260 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{3} + \cdots + 10496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 26 T^{3} + \cdots - 11344)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 218 T^{6} + \cdots + 15376 \) Copy content Toggle raw display
$71$ \( (T^{4} + 10 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 168 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( (T^{4} + 22 T^{3} + \cdots + 2416)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 372 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + \cdots + 10156)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 564 T^{6} + \cdots + 5683456 \) Copy content Toggle raw display
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