Properties

Label 35.17.c.b
Level $35$
Weight $17$
Character orbit 35.c
Self dual yes
Analytic conductor $56.814$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,17,Mod(34,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 35.c (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: \(56.8135903498\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 3007 q^{3} + 65536 q^{4} + 390625 q^{5} + 5764801 q^{7} - 34004672 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3007 q^{3} + 65536 q^{4} + 390625 q^{5} + 5764801 q^{7} - 34004672 q^{9} + 145028447 q^{11} + 197066752 q^{12} + 1571306207 q^{13} + 1174609375 q^{15} + 4294967296 q^{16} - 4372390753 q^{17} + 25600000000 q^{20} + 17334756607 q^{21} + 152587890625 q^{25} - 231693538751 q^{27} + 377801998336 q^{28} - 993241792513 q^{29} + 436100540129 q^{33} + 2251875390625 q^{35} - 2228530184192 q^{36} + 4724917764449 q^{39} + 9504584302592 q^{44} - 13283075000000 q^{45} - 42819958052353 q^{47} + 12914966659072 q^{48} + 33232930569601 q^{49} - 13147778994271 q^{51} + 102977123581952 q^{52} + 56651737109375 q^{55} + 76979200000000 q^{60} - 196030167150272 q^{63} + 281474976710656 q^{64} + 613791487109375 q^{65} - 286549000388608 q^{68} + 12\!\cdots\!62 q^{71}+ \cdots - 49\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
0
0 3007.00 65536.0 390625. 0 5.76480e6 0 −3.40047e7 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.17.c.b yes 1
5.b even 2 1 35.17.c.a 1
7.b odd 2 1 35.17.c.a 1
35.c odd 2 1 CM 35.17.c.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.17.c.a 1 5.b even 2 1
35.17.c.a 1 7.b odd 2 1
35.17.c.b yes 1 1.a even 1 1 trivial
35.17.c.b yes 1 35.c 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 \(S_{17}^{\mathrm{new}}(35, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3} - 3007 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3007 \) Copy content Toggle raw display
$5$ \( T - 390625 \) Copy content Toggle raw display
$7$ \( T - 5764801 \) Copy content Toggle raw display
$11$ \( T - 145028447 \) Copy content Toggle raw display
$13$ \( T - 1571306207 \) Copy content Toggle raw display
$17$ \( T + 4372390753 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 993241792513 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 42819958052353 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T - 1283316773477762 \) Copy content Toggle raw display
$73$ \( T + 491238137911678 \) Copy content Toggle raw display
$79$ \( T - 1884802582005407 \) Copy content Toggle raw display
$83$ \( T + 1568384056666558 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 11\!\cdots\!07 \) Copy content Toggle raw display
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