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Properties

Label 175.6.b.c
Level $175$
Weight $6$
Character orbit 175.b
Analytic conductor $28.067$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,6,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.b (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: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{57})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 29x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + (5 \beta_{2} + \beta_1) q^{2} + (6 \beta_{2} + 6 \beta_1) q^{3} + ( - 9 \beta_{3} + 2) q^{4} + ( - 30 \beta_{3} - 84) q^{6} + 49 \beta_{2} q^{7} + ( - \beta_{2} - 11 \beta_1) q^{8} + ( - 36 \beta_{3} - 261) q^{9}+ \cdots + (27468 \beta_{3} - 5364) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} - 396 q^{6} - 1116 q^{9} + 792 q^{11} - 882 q^{14} + 226 q^{16} + 6532 q^{19} - 588 q^{21} + 3708 q^{24} + 4032 q^{26} - 13392 q^{29} - 40 q^{31} - 11868 q^{34} + 21258 q^{36} + 40992 q^{39}+ \cdots + 33480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 29x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 15\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 14\beta_{2} - 15\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
− 3.27492i
4.27492i
− 4.27492i
3.27492i
− 8.27492i − 25.6495i −36.4743 0 −212.248 − 49.0000i 37.0241i −414.897 0
99.2 − 0.725083i 19.6495i 31.4743 0 14.2475 − 49.0000i − 46.0241i −143.103 0
99.3 0.725083i − 19.6495i 31.4743 0 14.2475 49.0000i 46.0241i −143.103 0
99.4 8.27492i 25.6495i −36.4743 0 −212.248 49.0000i − 37.0241i −414.897 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 175.6.b.c 4
5.b even 2 1 inner 175.6.b.c 4
5.c odd 4 1 7.6.a.b ✓ 2
5.c odd 4 1 175.6.a.c 2
15.e even 4 1 63.6.a.f 2
20.e even 4 1 112.6.a.h 2
35.f even 4 1 49.6.a.f 2
35.k even 12 2 49.6.c.d 4
35.l odd 12 2 49.6.c.e 4
40.i odd 4 1 448.6.a.w 2
40.k even 4 1 448.6.a.u 2
55.e even 4 1 847.6.a.c 2
60.l odd 4 1 1008.6.a.bq 2
105.k odd 4 1 441.6.a.l 2
140.j odd 4 1 784.6.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b ✓ 2 5.c odd 4 1
49.6.a.f 2 35.f even 4 1
49.6.c.d 4 35.k even 12 2
49.6.c.e 4 35.l odd 12 2
63.6.a.f 2 15.e even 4 1
112.6.a.h 2 20.e even 4 1
175.6.a.c 2 5.c odd 4 1
175.6.b.c 4 1.a even 1 1 trivial
175.6.b.c 4 5.b even 2 1 inner
441.6.a.l 2 105.k odd 4 1
448.6.a.u 2 40.k even 4 1
448.6.a.w 2 40.i odd 4 1
784.6.a.v 2 140.j odd 4 1
847.6.a.c 2 55.e even 4 1
1008.6.a.bq 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 69T_{2}^{2} + 36 \) acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 69T^{2} + 36 \) Copy content Toggle raw display
$3$ \( T^{4} + 1044 T^{2} + 254016 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 396 T - 179904)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 38262489664 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 529535646864 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3266 T + 2662072)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 12302247591936 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6696 T + 10304172)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 20 T - 4155200)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 30848648872336 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6048 T - 7848036)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 474989071531536 \) Copy content Toggle raw display
$59$ \( (T^{2} - 43938 T + 422751336)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 64754 T + 719128816)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 99\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} - 97416 T + 2121099264)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{2} + 51256 T - 2508546944)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} + 84276 T - 5252421468)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
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