Properties

Label 35.4.f.a
Level $35$
Weight $4$
Character orbit 35.f
Analytic conductor $2.065$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,4,Mod(13,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.f (of order \(4\), degree \(2\), 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: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{2} + 2 \beta_1 + 1) q^{3} + ( - 7 \beta_{3} - 4 \beta_{2} + \cdots + 3) q^{5}+ \cdots - 17 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + (\beta_{2} + 2 \beta_1 + 1) q^{3} + ( - 7 \beta_{3} - 4 \beta_{2} + \cdots + 3) q^{5}+ \cdots - 408 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 42 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} - 42 q^{7} - 64 q^{8} + 96 q^{11} + 140 q^{15} + 256 q^{16} - 136 q^{18} + 140 q^{21} - 192 q^{22} - 236 q^{23} - 140 q^{25} - 320 q^{30} - 70 q^{35} + 372 q^{37} - 280 q^{42} + 884 q^{43} + 944 q^{46} - 680 q^{50} - 1640 q^{51} - 348 q^{53} + 1344 q^{56} + 720 q^{57} + 416 q^{58} - 714 q^{63} + 2660 q^{65} - 1108 q^{67} - 840 q^{70} - 3664 q^{71} + 1088 q^{72} - 1008 q^{77} + 1520 q^{78} - 76 q^{81} + 820 q^{85} - 3536 q^{86} - 1536 q^{88} + 2660 q^{91} - 520 q^{93} + 2160 q^{95} + 784 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.61803i
0.618034i
1.61803i
0.618034i
−2.00000 + 2.00000i −2.23607 + 2.23607i 0 −6.70820 8.94427i 8.94427i −18.3262 + 2.67376i −16.0000 16.0000i 17.0000i 31.3050 + 4.47214i
13.2 −2.00000 + 2.00000i 2.23607 2.23607i 0 6.70820 + 8.94427i 8.94427i −2.67376 + 18.3262i −16.0000 16.0000i 17.0000i −31.3050 4.47214i
27.1 −2.00000 2.00000i −2.23607 2.23607i 0 −6.70820 + 8.94427i 8.94427i −18.3262 2.67376i −16.0000 + 16.0000i 17.0000i 31.3050 4.47214i
27.2 −2.00000 2.00000i 2.23607 + 2.23607i 0 6.70820 8.94427i 8.94427i −2.67376 18.3262i −16.0000 + 16.0000i 17.0000i −31.3050 + 4.47214i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.f.a 4
5.b even 2 1 175.4.f.d 4
5.c odd 4 1 inner 35.4.f.a 4
5.c odd 4 1 175.4.f.d 4
7.b odd 2 1 inner 35.4.f.a 4
7.c even 3 2 245.4.l.a 8
7.d odd 6 2 245.4.l.a 8
35.c odd 2 1 175.4.f.d 4
35.f even 4 1 inner 35.4.f.a 4
35.f even 4 1 175.4.f.d 4
35.k even 12 2 245.4.l.a 8
35.l odd 12 2 245.4.l.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.f.a 4 1.a even 1 1 trivial
35.4.f.a 4 5.c odd 4 1 inner
35.4.f.a 4 7.b odd 2 1 inner
35.4.f.a 4 35.f even 4 1 inner
175.4.f.d 4 5.b even 2 1
175.4.f.d 4 5.c odd 4 1
175.4.f.d 4 35.c odd 2 1
175.4.f.d 4 35.f even 4 1
245.4.l.a 8 7.c even 3 2
245.4.l.a 8 7.d odd 6 2
245.4.l.a 8 35.k even 12 2
245.4.l.a 8 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4T_{2} + 8 \) acting on \(S_{4}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 100 \) Copy content Toggle raw display
$5$ \( T^{4} + 70T^{2} + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 42 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T - 24)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 13032100 \) Copy content Toggle raw display
$17$ \( T^{4} + 282576100 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6480)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 118 T + 6962)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3380)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 186 T + 17298)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 442 T + 97682)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 27984100 \) Copy content Toggle raw display
$53$ \( (T^{2} + 174 T + 15138)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 176720)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 115520)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 554 T + 153458)^{2} \) Copy content Toggle raw display
$71$ \( (T + 916)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 150062500 \) Copy content Toggle raw display
$79$ \( (T^{2} + 228484)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 183603680100 \) Copy content Toggle raw display
$89$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3387108968100 \) Copy content Toggle raw display
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