Properties

Label 2175.2.c.b
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $2$
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: \(2\)
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 \( q + i q^{2} - i q^{3} + q^{4} + q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9} - 4 q^{11} - i q^{12} - 6 i q^{13} - 4 q^{14} - q^{16} + 6 i q^{17} - i q^{18} + 4 q^{19} + 4 q^{21} - 4 i q^{22} + 4 i q^{23} + 3 q^{24} + 6 q^{26} + i q^{27} + 4 i q^{28} - q^{29} - 8 q^{31} + 5 i q^{32} + 4 i q^{33} - 6 q^{34} - q^{36} + 2 i q^{37} + 4 i q^{38} - 6 q^{39} - 6 q^{41} + 4 i q^{42} - 4 i q^{43} - 4 q^{44} - 4 q^{46} + i q^{48} - 9 q^{49} + 6 q^{51} - 6 i q^{52} + 10 i q^{53} - q^{54} - 12 q^{56} - 4 i q^{57} - i q^{58} + 12 q^{59} - 10 q^{61} - 8 i q^{62} - 4 i q^{63} - 7 q^{64} - 4 q^{66} + 8 i q^{67} + 6 i q^{68} + 4 q^{69} - 8 q^{71} - 3 i q^{72} + 2 i q^{73} - 2 q^{74} + 4 q^{76} - 16 i q^{77} - 6 i q^{78} + q^{81} - 6 i q^{82} - 8 i q^{83} + 4 q^{84} + 4 q^{86} + i q^{87} - 12 i q^{88} + 6 q^{89} + 24 q^{91} + 4 i q^{92} + 8 i q^{93} + 5 q^{96} - 2 i q^{97} - 9 i q^{98} + 4 q^{99} +O(q^{100}) \) 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 \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 8 q^{14} - 2 q^{16} + 8 q^{19} + 8 q^{21} + 6 q^{24} + 12 q^{26} - 2 q^{29} - 16 q^{31} - 12 q^{34} - 2 q^{36} - 12 q^{39} - 12 q^{41} - 8 q^{44} - 8 q^{46} - 18 q^{49} + 12 q^{51} - 2 q^{54} - 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{66} + 8 q^{69} - 16 q^{71} - 4 q^{74} + 8 q^{76} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 12 q^{89} + 48 q^{91} + 10 q^{96} + 8 q^{99}+O(q^{100}) \) 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.b 2
5.b even 2 1 inner 2175.2.c.b 2
5.c odd 4 1 435.2.a.d 1
5.c odd 4 1 2175.2.a.b 1
15.e even 4 1 1305.2.a.b 1
15.e even 4 1 6525.2.a.j 1
20.e even 4 1 6960.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.d 1 5.c odd 4 1
1305.2.a.b 1 15.e even 4 1
2175.2.a.b 1 5.c odd 4 1
2175.2.c.b 2 1.a even 1 1 trivial
2175.2.c.b 2 5.b even 2 1 inner
6525.2.a.j 1 15.e even 4 1
6960.2.a.l 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
\( T_{11} + 4 \) 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
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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