Properties

Label 2175.2.c.a
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} + q^{4} - q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} + q^{4} - q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9} + i q^{12} + 6 i q^{13} - 4 q^{14} - q^{16} - 2 i q^{17} - i q^{18} - 8 q^{19} - 4 q^{21} - 4 i q^{23} - 3 q^{24} - 6 q^{26} - i q^{27} + 4 i q^{28} - q^{29} + 4 q^{31} + 5 i q^{32} + 2 q^{34} - q^{36} - 6 i q^{37} - 8 i q^{38} - 6 q^{39} + 2 q^{41} - 4 i q^{42} - 4 i q^{43} + 4 q^{46} - i q^{48} - 9 q^{49} + 2 q^{51} + 6 i q^{52} + 6 i q^{53} + q^{54} - 12 q^{56} - 8 i q^{57} - i q^{58} + 12 q^{59} + 6 q^{61} + 4 i q^{62} - 4 i q^{63} - 7 q^{64} + 8 i q^{67} - 2 i q^{68} + 4 q^{69} + 16 q^{71} - 3 i q^{72} - 6 i q^{73} + 6 q^{74} - 8 q^{76} - 6 i q^{78} - 12 q^{79} + q^{81} + 2 i q^{82} - 16 i q^{83} - 4 q^{84} + 4 q^{86} - i q^{87} - 2 q^{89} - 24 q^{91} - 4 i q^{92} + 4 i q^{93} - 5 q^{96} + 14 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} - 2 q^{16} - 16 q^{19} - 8 q^{21} - 6 q^{24} - 12 q^{26} - 2 q^{29} + 8 q^{31} + 4 q^{34} - 2 q^{36} - 12 q^{39} + 4 q^{41} + 8 q^{46} - 18 q^{49} + 4 q^{51} + 2 q^{54} - 24 q^{56} + 24 q^{59} + 12 q^{61} - 14 q^{64} + 8 q^{69} + 32 q^{71} + 12 q^{74} - 16 q^{76} - 24 q^{79} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 4 q^{89} - 48 q^{91} - 10 q^{96}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 4.00000i 3.00000i −1.00000 0
349.2 1.00000i 1.00000i 1.00000 0 −1.00000 4.00000i 3.00000i −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.a 2
5.b even 2 1 inner 2175.2.c.a 2
5.c odd 4 1 435.2.a.a 1
5.c odd 4 1 2175.2.a.h 1
15.e even 4 1 1305.2.a.e 1
15.e even 4 1 6525.2.a.e 1
20.e even 4 1 6960.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.a 1 5.c odd 4 1
1305.2.a.e 1 15.e even 4 1
2175.2.a.h 1 5.c odd 4 1
2175.2.c.a 2 1.a even 1 1 trivial
2175.2.c.a 2 5.b even 2 1 inner
6525.2.a.e 1 15.e even 4 1
6960.2.a.w 1 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 \(S_{2}^{\mathrm{new}}(2175, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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