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Properties

Label 35.3.c.c
Level $35$
Weight $3$
Character orbit 35.c
Analytic conductor $0.954$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.34");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} - 5 q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{3} - 2 \beta_{2}) q^{6} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_1 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} - 5 q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{3} - 2 \beta_{2}) q^{6} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_1 q^{8} + q^{9} + (4 \beta_{3} + \beta_{2}) q^{10} + 14 q^{11} - 5 \beta_{3} q^{12} - \beta_{3} q^{13} + ( - 2 \beta_{3} + 4 \beta_{2} + 9) q^{14} + (5 \beta_1 - 5) q^{15} - 11 q^{16} + 2 \beta_{3} q^{17} - \beta_1 q^{18} + (3 \beta_{3} - 6 \beta_{2}) q^{19} + (5 \beta_{3} - 5 \beta_{2}) q^{20} + ( - \beta_{3} + 2 \beta_{2} - 20) q^{21} - 14 \beta_1 q^{22} + 4 \beta_1 q^{23} + ( - \beta_{3} + 2 \beta_{2}) q^{24} + ( - 5 \beta_1 - 20) q^{25} + ( - \beta_{3} + 2 \beta_{2}) q^{26} - 8 \beta_{3} q^{27} + (10 \beta_{3} - 5 \beta_1) q^{28} + 14 q^{29} + (5 \beta_1 + 45) q^{30} + ( - 4 \beta_{3} + 8 \beta_{2}) q^{31} + 15 \beta_1 q^{32} + 14 \beta_{3} q^{33} + (2 \beta_{3} - 4 \beta_{2}) q^{34} + ( - 4 \beta_{3} - \beta_{2} + \cdots + 10) q^{35}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{4} + 4 q^{9} + 56 q^{11} + 36 q^{14} - 20 q^{15} - 44 q^{16} - 80 q^{21} - 80 q^{25} + 56 q^{29} + 180 q^{30} + 40 q^{35} - 20 q^{36} - 40 q^{39} - 280 q^{44} + 144 q^{46} + 124 q^{49} - 180 q^{50} + 80 q^{51} - 36 q^{56} + 100 q^{60} + 364 q^{64} + 20 q^{65} - 360 q^{70} - 64 q^{71} - 216 q^{74} - 304 q^{79} - 356 q^{81} + 400 q^{84} - 40 q^{85} + 504 q^{86} + 80 q^{91} + 540 q^{95} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{3} + 5\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.58114 − 1.58114i
1.58114 + 1.58114i
−1.58114 + 1.58114i
1.58114 − 1.58114i
− 3.00000i −3.16228 −5.00000 1.58114 − 4.74342i 9.48683i 6.32456 + 3.00000i 3.00000i 1.00000 −14.2302 − 4.74342i
34.2 − 3.00000i 3.16228 −5.00000 −1.58114 + 4.74342i − 9.48683i −6.32456 + 3.00000i 3.00000i 1.00000 14.2302 + 4.74342i
34.3 3.00000i −3.16228 −5.00000 1.58114 + 4.74342i − 9.48683i 6.32456 − 3.00000i − 3.00000i 1.00000 −14.2302 + 4.74342i
34.4 3.00000i 3.16228 −5.00000 −1.58114 − 4.74342i 9.48683i −6.32456 − 3.00000i − 3.00000i 1.00000 14.2302 − 4.74342i
\(n\): 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
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.c.c ✓ 4
3.b odd 2 1 315.3.e.c 4
4.b odd 2 1 560.3.p.f 4
5.b even 2 1 inner 35.3.c.c ✓ 4
5.c odd 4 1 175.3.d.b 2
5.c odd 4 1 175.3.d.h 2
7.b odd 2 1 inner 35.3.c.c ✓ 4
7.c even 3 2 245.3.i.c 8
7.d odd 6 2 245.3.i.c 8
15.d odd 2 1 315.3.e.c 4
20.d odd 2 1 560.3.p.f 4
21.c even 2 1 315.3.e.c 4
28.d even 2 1 560.3.p.f 4
35.c odd 2 1 inner 35.3.c.c ✓ 4
35.f even 4 1 175.3.d.b 2
35.f even 4 1 175.3.d.h 2
35.i odd 6 2 245.3.i.c 8
35.j even 6 2 245.3.i.c 8
105.g even 2 1 315.3.e.c 4
140.c even 2 1 560.3.p.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.c.c ✓ 4 1.a even 1 1 trivial
35.3.c.c ✓ 4 5.b even 2 1 inner
35.3.c.c ✓ 4 7.b odd 2 1 inner
35.3.c.c ✓ 4 35.c odd 2 1 inner
175.3.d.b 2 5.c odd 4 1
175.3.d.b 2 35.f even 4 1
175.3.d.h 2 5.c odd 4 1
175.3.d.h 2 35.f even 4 1
245.3.i.c 8 7.c even 3 2
245.3.i.c 8 7.d odd 6 2
245.3.i.c 8 35.i odd 6 2
245.3.i.c 8 35.j even 6 2
315.3.e.c 4 3.b odd 2 1
315.3.e.c 4 15.d odd 2 1
315.3.e.c 4 21.c even 2 1
315.3.e.c 4 105.g even 2 1
560.3.p.f 4 4.b odd 2 1
560.3.p.f 4 20.d odd 2 1
560.3.p.f 4 28.d even 2 1
560.3.p.f 4 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(35, [\chi])\):

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{2} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 40T^{2} + 625 \) Copy content Toggle raw display
$7$ \( T^{4} - 62T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 14)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 810)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T - 14)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1440)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 360)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1764)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1960)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2916)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4410)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10404)^{2} \) Copy content Toggle raw display
$71$ \( (T + 16)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4000)^{2} \) Copy content Toggle raw display
$79$ \( (T + 76)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 5290)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3240)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4840)^{2} \) Copy content Toggle raw display
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