Properties

Label 3575.1.h.f
Level $3575$
Weight $1$
Character orbit 3575.h
Self dual yes
Analytic conductor $1.784$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -143
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3575,1,Mod(2001,3575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3575.2001");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3575 = 5^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3575.h (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.78415742016\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.20449.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.2.1306755003125.1

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 1) q^{2} + ( - \beta + 1) q^{3} + ( - \beta + 1) q^{4} + ( - \beta + 2) q^{6} + \beta q^{7} + q^{8} + ( - \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 1) q^{2} + ( - \beta + 1) q^{3} + ( - \beta + 1) q^{4} + ( - \beta + 2) q^{6} + \beta q^{7} + q^{8} + ( - \beta + 1) q^{9} + q^{11} + ( - \beta + 2) q^{12} - q^{13} - q^{14} + ( - \beta + 2) q^{18} + (\beta - 1) q^{19} - q^{21} + ( - \beta + 1) q^{22} + \beta q^{23} + ( - \beta + 1) q^{24} + (\beta - 1) q^{26} + q^{27} - q^{28} - q^{32} + ( - \beta + 1) q^{33} + ( - \beta + 2) q^{36} + (\beta - 2) q^{38} + (\beta - 1) q^{39} - \beta q^{41} + (\beta - 1) q^{42} + ( - \beta + 1) q^{44} - q^{46} + \beta q^{49} + (\beta - 1) q^{52} + ( - \beta + 1) q^{53} + ( - \beta + 1) q^{54} + \beta q^{56} + (\beta - 2) q^{57} - q^{63} + (\beta - 1) q^{64} + ( - \beta + 2) q^{66} - q^{69} + ( - \beta + 1) q^{72} + ( - \beta + 1) q^{73} + (\beta - 2) q^{76} + \beta q^{77} + (\beta - 2) q^{78} + q^{82} + \beta q^{83} + (\beta - 1) q^{84} + q^{88} - \beta q^{91} - q^{92} + (\beta - 1) q^{96} - q^{98} + ( - \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + q^{4} + 3 q^{6} + q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + q^{4} + 3 q^{6} + q^{7} + 2 q^{8} + q^{9} + 2 q^{11} + 3 q^{12} - 2 q^{13} - 2 q^{14} + 3 q^{18} - q^{19} - 2 q^{21} + q^{22} + q^{23} + q^{24} - q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{32} + q^{33} + 3 q^{36} - 3 q^{38} - q^{39} - q^{41} - q^{42} + q^{44} - 2 q^{46} + q^{49} - q^{52} + q^{53} + q^{54} + q^{56} - 3 q^{57} - 2 q^{63} - q^{64} + 3 q^{66} - 2 q^{69} + q^{72} + q^{73} - 3 q^{76} + q^{77} - 3 q^{78} + 2 q^{82} + q^{83} - q^{84} + 2 q^{88} - q^{91} - 2 q^{92} - q^{96} - 2 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3575\mathbb{Z}\right)^\times\).

\(n\) \(651\) \(1002\) \(1926\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2001.1
1.61803
−0.618034
−0.618034 −0.618034 −0.618034 0 0.381966 1.61803 1.00000 −0.618034 0
2001.2 1.61803 1.61803 1.61803 0 2.61803 −0.618034 1.00000 1.61803 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
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3575.1.h.f 2
5.b even 2 1 143.1.d.a 2
5.c odd 4 2 3575.1.c.d 4
11.b odd 2 1 3575.1.h.e 2
13.b even 2 1 3575.1.h.e 2
15.d odd 2 1 1287.1.g.b 2
20.d odd 2 1 2288.1.m.b 2
55.d odd 2 1 143.1.d.b yes 2
55.e even 4 2 3575.1.c.c 4
55.h odd 10 2 1573.1.l.a 4
55.h odd 10 2 1573.1.l.c 4
55.j even 10 2 1573.1.l.b 4
55.j even 10 2 1573.1.l.d 4
65.d even 2 1 143.1.d.b yes 2
65.g odd 4 2 1859.1.c.c 4
65.h odd 4 2 3575.1.c.c 4
65.l even 6 2 1859.1.i.a 4
65.n even 6 2 1859.1.i.b 4
65.s odd 12 4 1859.1.k.c 8
143.d odd 2 1 CM 3575.1.h.f 2
165.d even 2 1 1287.1.g.a 2
195.e odd 2 1 1287.1.g.a 2
220.g even 2 1 2288.1.m.a 2
260.g odd 2 1 2288.1.m.a 2
715.c odd 2 1 143.1.d.a 2
715.l even 4 2 1859.1.c.c 4
715.q even 4 2 3575.1.c.d 4
715.x odd 6 2 1859.1.i.a 4
715.bb odd 6 2 1859.1.i.b 4
715.bf even 10 2 1573.1.l.a 4
715.bf even 10 2 1573.1.l.c 4
715.bi odd 10 2 1573.1.l.b 4
715.bi odd 10 2 1573.1.l.d 4
715.bt even 12 4 1859.1.k.c 8
2145.k even 2 1 1287.1.g.b 2
2860.p even 2 1 2288.1.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 5.b even 2 1
143.1.d.a 2 715.c odd 2 1
143.1.d.b yes 2 55.d odd 2 1
143.1.d.b yes 2 65.d even 2 1
1287.1.g.a 2 165.d even 2 1
1287.1.g.a 2 195.e odd 2 1
1287.1.g.b 2 15.d odd 2 1
1287.1.g.b 2 2145.k even 2 1
1573.1.l.a 4 55.h odd 10 2
1573.1.l.a 4 715.bf even 10 2
1573.1.l.b 4 55.j even 10 2
1573.1.l.b 4 715.bi odd 10 2
1573.1.l.c 4 55.h odd 10 2
1573.1.l.c 4 715.bf even 10 2
1573.1.l.d 4 55.j even 10 2
1573.1.l.d 4 715.bi odd 10 2
1859.1.c.c 4 65.g odd 4 2
1859.1.c.c 4 715.l even 4 2
1859.1.i.a 4 65.l even 6 2
1859.1.i.a 4 715.x odd 6 2
1859.1.i.b 4 65.n even 6 2
1859.1.i.b 4 715.bb odd 6 2
1859.1.k.c 8 65.s odd 12 4
1859.1.k.c 8 715.bt even 12 4
2288.1.m.a 2 220.g even 2 1
2288.1.m.a 2 260.g odd 2 1
2288.1.m.b 2 20.d odd 2 1
2288.1.m.b 2 2860.p even 2 1
3575.1.c.c 4 55.e even 4 2
3575.1.c.c 4 65.h odd 4 2
3575.1.c.d 4 5.c odd 4 2
3575.1.c.d 4 715.q even 4 2
3575.1.h.e 2 11.b odd 2 1
3575.1.h.e 2 13.b even 2 1
3575.1.h.f 2 1.a even 1 1 trivial
3575.1.h.f 2 143.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3575, [\chi])\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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