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Properties

Label 175.2.k.a
Level $175$
Weight $2$
Character orbit 175.k
Analytic conductor $1.397$
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] = [175,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.74");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.39738203537\)
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: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - \beta_1 + 1) q^{4} - q^{6} + ( - \beta_{7} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{8} + \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - \beta_1 + 1) q^{4} - q^{6} + ( - \beta_{7} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{8} + \beta_{5} q^{9} + ( - \beta_{6} + 2 \beta_1 - 2) q^{11} + (2 \beta_{7} - \beta_{3}) q^{12} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{13} + ( - \beta_{5} + 4 \beta_1 - 1) q^{14} - 3 \beta_1 q^{16} - 2 \beta_{2} q^{17} + (\beta_{4} - 3 \beta_{2}) q^{18} - \beta_{5} q^{19} + (\beta_{6} + \beta_{5} - \beta_1 + 4) q^{21} + ( - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2}) q^{22} - \beta_{7} q^{23} + (\beta_{6} - \beta_1 + 1) q^{24} + ( - 2 \beta_{5} + 6 \beta_1) q^{26} + ( - \beta_{3} + \beta_{2}) q^{27} + (2 \beta_{7} - 3 \beta_{4} - \beta_{3} + 5 \beta_{2}) q^{28} + q^{29} + ( - 6 \beta_1 + 6) q^{31} + ( - 2 \beta_{4} + \beta_{2}) q^{32} + (5 \beta_{7} - \beta_{3}) q^{33} + (2 \beta_{6} + 2 \beta_{5} - 6) q^{34} + (\beta_{6} + \beta_{5} - 8) q^{36} + ( - \beta_{4} + 3 \beta_{2}) q^{38} + ( - 2 \beta_1 + 2) q^{39} + (\beta_{6} + \beta_{5} - 5) q^{41} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{42} + ( - 3 \beta_{7} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{43} + ( - \beta_{5} - 6 \beta_1) q^{44} + (\beta_1 - 1) q^{46} + ( - \beta_{7} + \beta_{3}) q^{47} + ( - 3 \beta_{7} + 3 \beta_{4}) q^{48} + (2 \beta_{6} + \beta_{5} - 5 \beta_1) q^{49} + 2 \beta_1 q^{51} + ( - 2 \beta_{4} + 8 \beta_{2}) q^{52} + (3 \beta_{4} - \beta_{2}) q^{53} + ( - \beta_{5} + 3 \beta_1) q^{54} + ( - 2 \beta_{6} - 3 \beta_{5} - \beta_1 + 5) q^{56} + ( - 3 \beta_{7} + 3 \beta_{4} + \beta_{3} - \beta_{2}) q^{57} + \beta_{3} q^{58} + ( - 3 \beta_{6} + 4 \beta_1 - 4) q^{59} + (3 \beta_{5} + 3 \beta_1) q^{61} + (6 \beta_{3} - 6 \beta_{2}) q^{62} + ( - \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{63} + ( - \beta_{6} - \beta_{5} + 7) q^{64} + (\beta_{6} - 2 \beta_1 + 2) q^{66} + (6 \beta_{4} - 5 \beta_{2}) q^{67} + (2 \beta_{7} - 8 \beta_{3}) q^{68} + (\beta_{6} + \beta_{5} + 3) q^{69} + ( - 3 \beta_{6} - 3 \beta_{5} - 4) q^{71} + ( - \beta_{7} - 5 \beta_{3}) q^{72} - 2 \beta_{4} q^{73} + ( - \beta_{6} - \beta_{5} + 8) q^{76} + (5 \beta_{7} - 6 \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{77} + (2 \beta_{3} - 2 \beta_{2}) q^{78} + ( - \beta_{5} + 12 \beta_1) q^{79} + (3 \beta_{6} - \beta_1 + 1) q^{81} + (\beta_{7} - 8 \beta_{3}) q^{82} + (4 \beta_{7} - 4 \beta_{4} + 5 \beta_{3} - 5 \beta_{2}) q^{83} + ( - 2 \beta_{6} + \beta_{5} + 4 \beta_1 - 5) q^{84} + (2 \beta_{5} - 3 \beta_1) q^{86} - \beta_{4} q^{87} + ( - 3 \beta_{4} - \beta_{2}) q^{88} + (2 \beta_{5} - 3 \beta_1) q^{89} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_1 - 6) q^{91} + (2 \beta_{7} - 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{92} - 6 \beta_{7} q^{93} + ( - \beta_{6} - 2 \beta_1 + 2) q^{94} + (2 \beta_{5} + 5 \beta_1) q^{96} + (\beta_{7} - \beta_{4} - 5 \beta_{3} + 5 \beta_{2}) q^{97} + (2 \beta_{7} - \beta_{4} - 6 \beta_{3} - 2 \beta_{2}) q^{98} + ( - 2 \beta_{6} - 2 \beta_{5} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 8 q^{6} - 8 q^{11} + 8 q^{14} - 12 q^{16} + 28 q^{21} + 4 q^{24} + 24 q^{26} + 8 q^{29} + 24 q^{31} - 48 q^{34} - 64 q^{36} + 8 q^{39} - 40 q^{41} - 24 q^{44} - 4 q^{46} - 20 q^{49} + 8 q^{51} + 12 q^{54} + 36 q^{56} - 16 q^{59} + 12 q^{61} + 56 q^{64} + 8 q^{66} + 24 q^{69} - 32 q^{71} + 64 q^{76} + 48 q^{79} + 4 q^{81} - 24 q^{84} - 12 q^{86} - 12 q^{89} - 56 q^{91} + 8 q^{94} + 20 q^{96} - 64 q^{99}+O(q^{100}) \) 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
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} \) 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 \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{3} ) / 4 \) 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
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−2.09077 + 1.20711i 0.358719 + 0.207107i 1.91421 − 3.31552i 0 −1.00000 0.358719 − 2.62132i 4.41421i −1.41421 − 2.44949i 0
74.2 −0.358719 + 0.207107i 2.09077 + 1.20711i −0.914214 + 1.58346i 0 −1.00000 2.09077 − 1.62132i − 1.58579i 1.41421 + 2.44949i 0
74.3 0.358719 − 0.207107i −2.09077 − 1.20711i −0.914214 + 1.58346i 0 −1.00000 −2.09077 + 1.62132i 1.58579i 1.41421 + 2.44949i 0
74.4 2.09077 − 1.20711i −0.358719 − 0.207107i 1.91421 − 3.31552i 0 −1.00000 −0.358719 + 2.62132i − 4.41421i −1.41421 − 2.44949i 0
149.1 −2.09077 − 1.20711i 0.358719 − 0.207107i 1.91421 + 3.31552i 0 −1.00000 0.358719 + 2.62132i − 4.41421i −1.41421 + 2.44949i 0
149.2 −0.358719 − 0.207107i 2.09077 − 1.20711i −0.914214 − 1.58346i 0 −1.00000 2.09077 + 1.62132i 1.58579i 1.41421 − 2.44949i 0
149.3 0.358719 + 0.207107i −2.09077 + 1.20711i −0.914214 − 1.58346i 0 −1.00000 −2.09077 − 1.62132i − 1.58579i 1.41421 − 2.44949i 0
149.4 2.09077 + 1.20711i −0.358719 + 0.207107i 1.91421 + 3.31552i 0 −1.00000 −0.358719 − 2.62132i 4.41421i −1.41421 + 2.44949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
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 175.2.k.a 8
5.b even 2 1 inner 175.2.k.a 8
5.c odd 4 1 35.2.e.a ✓ 4
5.c odd 4 1 175.2.e.c 4
7.c even 3 1 inner 175.2.k.a 8
7.c even 3 1 1225.2.b.g 4
7.d odd 6 1 1225.2.b.h 4
15.e even 4 1 315.2.j.e 4
20.e even 4 1 560.2.q.k 4
35.f even 4 1 245.2.e.e 4
35.i odd 6 1 1225.2.b.h 4
35.j even 6 1 inner 175.2.k.a 8
35.j even 6 1 1225.2.b.g 4
35.k even 12 1 245.2.a.g 2
35.k even 12 1 245.2.e.e 4
35.k even 12 1 1225.2.a.m 2
35.l odd 12 1 35.2.e.a ✓ 4
35.l odd 12 1 175.2.e.c 4
35.l odd 12 1 245.2.a.h 2
35.l odd 12 1 1225.2.a.k 2
105.w odd 12 1 2205.2.a.q 2
105.x even 12 1 315.2.j.e 4
105.x even 12 1 2205.2.a.n 2
140.w even 12 1 560.2.q.k 4
140.w even 12 1 3920.2.a.bq 2
140.x odd 12 1 3920.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a ✓ 4 5.c odd 4 1
35.2.e.a ✓ 4 35.l odd 12 1
175.2.e.c 4 5.c odd 4 1
175.2.e.c 4 35.l odd 12 1
175.2.k.a 8 1.a even 1 1 trivial
175.2.k.a 8 5.b even 2 1 inner
175.2.k.a 8 7.c even 3 1 inner
175.2.k.a 8 35.j even 6 1 inner
245.2.a.g 2 35.k even 12 1
245.2.a.h 2 35.l odd 12 1
245.2.e.e 4 35.f even 4 1
245.2.e.e 4 35.k even 12 1
315.2.j.e 4 15.e even 4 1
315.2.j.e 4 105.x even 12 1
560.2.q.k 4 20.e even 4 1
560.2.q.k 4 140.w even 12 1
1225.2.a.k 2 35.l odd 12 1
1225.2.a.m 2 35.k even 12 1
1225.2.b.g 4 7.c even 3 1
1225.2.b.g 4 35.j even 6 1
1225.2.b.h 4 7.d odd 6 1
1225.2.b.h 4 35.i odd 6 1
2205.2.a.n 2 105.x even 12 1
2205.2.a.q 2 105.w odd 12 1
3920.2.a.bq 2 140.w even 12 1
3920.2.a.bv 2 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + 51 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} + 10 T + 17)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 54 T^{2} + 529)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 48 T^{6} + 2240 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + 120 T^{2} - 448 T + 3136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + 99 T^{2} + 378 T + 3969)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 246 T^{6} + \cdots + 200533921 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 56)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + 440 T^{2} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 326 T^{2} + 25921)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 6 T^{3} + 59 T^{2} - 138 T + 529)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 136 T^{2} + 16)^{2} \) Copy content Toggle raw display
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