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Properties

Label 175.4.k.c
Level $175$
Weight $4$
Character orbit 175.k
Analytic conductor $10.325$
Analytic rank $0$
Dimension $8$
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] = [175,4,Mod(74,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.74");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), 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: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{7} - \beta_{3}) q^{2} + ( - \beta_{5} - 2 \beta_{2}) q^{3} + ( - 3 \beta_{6} + 3 \beta_{4} + \cdots + 3) q^{4}+ \cdots + ( - 3 \beta_{6} - 8 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{7} - \beta_{3}) q^{2} + ( - \beta_{5} - 2 \beta_{2}) q^{3} + ( - 3 \beta_{6} + 3 \beta_{4} + \cdots + 3) q^{4}+ \cdots + (2 \beta_{4} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 24 q^{6} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} - 24 q^{6} - 32 q^{9} - 56 q^{11} - 168 q^{14} + 28 q^{16} + 320 q^{19} + 268 q^{21} - 324 q^{24} - 296 q^{26} + 1000 q^{29} + 264 q^{31} - 1776 q^{34} - 768 q^{36} - 168 q^{39} - 424 q^{41} - 312 q^{44} + 804 q^{46} + 460 q^{49} + 1144 q^{51} + 804 q^{54} - 1428 q^{56} + 1680 q^{59} - 196 q^{61} + 2408 q^{64} - 472 q^{66} - 5704 q^{69} + 4256 q^{71} + 3584 q^{74} - 576 q^{76} - 2480 q^{79} + 1508 q^{81} + 312 q^{84} - 4036 q^{86} + 1300 q^{89} - 1992 q^{91} + 664 q^{94} - 3444 q^{96} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−0.258819 + 0.965926i
0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−3.82282 + 2.20711i 2.80821 + 1.62132i 5.74264 − 9.94655i 0 −14.3137 9.14207 + 16.1066i 15.3848i −8.24264 − 14.2767i 0
74.2 −1.37333 + 0.792893i −4.54026 − 2.62132i −2.74264 + 4.75039i 0 8.31371 −17.8023 − 5.10660i − 21.3848i 0.242641 + 0.420266i 0
74.3 1.37333 − 0.792893i 4.54026 + 2.62132i −2.74264 + 4.75039i 0 8.31371 17.8023 + 5.10660i 21.3848i 0.242641 + 0.420266i 0
74.4 3.82282 − 2.20711i −2.80821 − 1.62132i 5.74264 − 9.94655i 0 −14.3137 −9.14207 − 16.1066i − 15.3848i −8.24264 − 14.2767i 0
149.1 −3.82282 − 2.20711i 2.80821 − 1.62132i 5.74264 + 9.94655i 0 −14.3137 9.14207 − 16.1066i − 15.3848i −8.24264 + 14.2767i 0
149.2 −1.37333 − 0.792893i −4.54026 + 2.62132i −2.74264 − 4.75039i 0 8.31371 −17.8023 + 5.10660i 21.3848i 0.242641 − 0.420266i 0
149.3 1.37333 + 0.792893i 4.54026 − 2.62132i −2.74264 − 4.75039i 0 8.31371 17.8023 − 5.10660i − 21.3848i 0.242641 − 0.420266i 0
149.4 3.82282 + 2.20711i −2.80821 + 1.62132i 5.74264 + 9.94655i 0 −14.3137 −9.14207 + 16.1066i 15.3848i −8.24264 + 14.2767i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.k.c 8
5.b even 2 1 inner 175.4.k.c 8
5.c odd 4 1 35.4.e.b ✓ 4
5.c odd 4 1 175.4.e.c 4
7.c even 3 1 inner 175.4.k.c 8
15.e even 4 1 315.4.j.c 4
20.e even 4 1 560.4.q.i 4
35.f even 4 1 245.4.e.l 4
35.j even 6 1 inner 175.4.k.c 8
35.k even 12 1 245.4.a.h 2
35.k even 12 1 245.4.e.l 4
35.k even 12 1 1225.4.a.v 2
35.l odd 12 1 35.4.e.b ✓ 4
35.l odd 12 1 175.4.e.c 4
35.l odd 12 1 245.4.a.g 2
35.l odd 12 1 1225.4.a.x 2
105.w odd 12 1 2205.4.a.bg 2
105.x even 12 1 315.4.j.c 4
105.x even 12 1 2205.4.a.bf 2
140.w even 12 1 560.4.q.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b ✓ 4 5.c odd 4 1
35.4.e.b ✓ 4 35.l odd 12 1
175.4.e.c 4 5.c odd 4 1
175.4.e.c 4 35.l odd 12 1
175.4.k.c 8 1.a even 1 1 trivial
175.4.k.c 8 5.b even 2 1 inner
175.4.k.c 8 7.c even 3 1 inner
175.4.k.c 8 35.j even 6 1 inner
245.4.a.g 2 35.l odd 12 1
245.4.a.h 2 35.k even 12 1
245.4.e.l 4 35.f even 4 1
245.4.e.l 4 35.k even 12 1
315.4.j.c 4 15.e even 4 1
315.4.j.c 4 105.x even 12 1
560.4.q.i 4 20.e even 4 1
560.4.q.i 4 140.w even 12 1
1225.4.a.v 2 35.k even 12 1
1225.4.a.x 2 35.l odd 12 1
2205.4.a.bf 2 105.x even 12 1
2205.4.a.bg 2 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 22T_{2}^{6} + 435T_{2}^{4} - 1078T_{2}^{2} + 2401 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 22 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$3$ \( T^{8} - 38 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1048 T^{2} + 15376)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 370737464647936 \) Copy content Toggle raw display
$19$ \( (T^{4} - 160 T^{3} + \cdots + 25482304)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 55\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{2} - 250 T + 8425)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 132 T^{3} + \cdots + 244734736)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{2} + 106 T + 1009)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 221878 T^{2} + 12285283921)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 3975015137536 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{4} - 840 T^{3} + \cdots + 1217172544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 98 T^{3} + \cdots + 419125465201)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 26\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1064 T + 38024)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + 1240 T^{3} + \cdots + 85736524864)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1816518 T^{2} + 824752769281)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 650 T^{3} + \cdots + 10884122929)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1197192 T^{2} + 161125171216)^{2} \) Copy content Toggle raw display
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