Properties

Label 2175.2.c.g
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} - 3 q^{4} - \beta_{3} q^{6} - 2 \beta_1 q^{7} - \beta_{2} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} - 3 q^{4} - \beta_{3} q^{6} - 2 \beta_1 q^{7} - \beta_{2} q^{8} - q^{9} - 2 q^{11} - 3 \beta_1 q^{12} + 2 \beta_1 q^{13} + 2 \beta_{3} q^{14} - q^{16} - 2 \beta_{2} q^{17} - \beta_{2} q^{18} - 2 q^{19} + 2 q^{21} - 2 \beta_{2} q^{22} - 2 \beta_1 q^{23} + \beta_{3} q^{24} - 2 \beta_{3} q^{26} - \beta_1 q^{27} + 6 \beta_1 q^{28} + q^{29} - 2 q^{31} - 3 \beta_{2} q^{32} - 2 \beta_1 q^{33} + 10 q^{34} + 3 q^{36} + ( - 2 \beta_{2} - 4 \beta_1) q^{37} - 2 \beta_{2} q^{38} - 2 q^{39} + 2 q^{41} + 2 \beta_{2} q^{42} + 4 \beta_1 q^{43} + 6 q^{44} + 2 \beta_{3} q^{46} + ( - 4 \beta_{2} - 4 \beta_1) q^{47} - \beta_1 q^{48} + 3 q^{49} + 2 \beta_{3} q^{51} - 6 \beta_1 q^{52} + 2 \beta_1 q^{53} + \beta_{3} q^{54} - 2 \beta_{3} q^{56} - 2 \beta_1 q^{57} + \beta_{2} q^{58} - 8 q^{59} + (4 \beta_{3} - 2) q^{61} - 2 \beta_{2} q^{62} + 2 \beta_1 q^{63} + 13 q^{64} + 2 \beta_{3} q^{66} + ( - 4 \beta_{2} + 6 \beta_1) q^{67} + 6 \beta_{2} q^{68} + 2 q^{69} + 4 q^{71} + \beta_{2} q^{72} + ( - 2 \beta_{2} + 8 \beta_1) q^{73} + (4 \beta_{3} + 10) q^{74} + 6 q^{76} + 4 \beta_1 q^{77} - 2 \beta_{2} q^{78} + (4 \beta_{3} - 6) q^{79} + q^{81} + 2 \beta_{2} q^{82} + ( - 4 \beta_{2} - 6 \beta_1) q^{83} - 6 q^{84} - 4 \beta_{3} q^{86} + \beta_1 q^{87} + 2 \beta_{2} q^{88} + 6 q^{89} + 4 q^{91} + 6 \beta_1 q^{92} - 2 \beta_1 q^{93} + (4 \beta_{3} + 20) q^{94} + 3 \beta_{3} q^{96} + (6 \beta_{2} + 4 \beta_1) q^{97} + 3 \beta_{2} q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 4 q^{9} - 8 q^{11} - 4 q^{16} - 8 q^{19} + 8 q^{21} + 4 q^{29} - 8 q^{31} + 40 q^{34} + 12 q^{36} - 8 q^{39} + 8 q^{41} + 24 q^{44} + 12 q^{49} - 32 q^{59} - 8 q^{61} + 52 q^{64} + 8 q^{69} + 16 q^{71} + 40 q^{74} + 24 q^{76} - 24 q^{79} + 4 q^{81} - 24 q^{84} + 24 q^{89} + 16 q^{91} + 80 q^{94} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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
0.618034i
1.61803i
1.61803i
0.618034i
2.23607i 1.00000i −3.00000 0 −2.23607 2.00000i 2.23607i −1.00000 0
349.2 2.23607i 1.00000i −3.00000 0 2.23607 2.00000i 2.23607i −1.00000 0
349.3 2.23607i 1.00000i −3.00000 0 2.23607 2.00000i 2.23607i −1.00000 0
349.4 2.23607i 1.00000i −3.00000 0 −2.23607 2.00000i 2.23607i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
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.g 4
5.b even 2 1 inner 2175.2.c.g 4
5.c odd 4 1 435.2.a.g 2
5.c odd 4 1 2175.2.a.o 2
15.e even 4 1 1305.2.a.j 2
15.e even 4 1 6525.2.a.y 2
20.e even 4 1 6960.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.g 2 5.c odd 4 1
1305.2.a.j 2 15.e even 4 1
2175.2.a.o 2 5.c odd 4 1
2175.2.c.g 4 1.a even 1 1 trivial
2175.2.c.g 4 5.b even 2 1 inner
6525.2.a.y 2 15.e even 4 1
6960.2.a.bp 2 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 \(S_{2}^{\mathrm{new}}(2175, [\chi])\):

\( T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T + 8)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$71$ \( (T - 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T - 44)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 392 T^{2} + 26896 \) Copy content Toggle raw display
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