Properties

Label 175.2.e.a
Level $175$
Weight $2$
Character orbit 175.e
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,2,Mod(51,175)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.e (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - 3 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} - 3 q^{8} + 2 \zeta_{6} q^{9} - \zeta_{6} q^{12} + 2 q^{13} + (2 \zeta_{6} - 3) q^{14} + \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + ( - 2 \zeta_{6} + 2) q^{18} - 6 \zeta_{6} q^{19} + ( - \zeta_{6} - 2) q^{21} + 3 \zeta_{6} q^{23} + (3 \zeta_{6} - 3) q^{24} - 2 \zeta_{6} q^{26} + 5 q^{27} + ( - \zeta_{6} - 2) q^{28} + 7 q^{29} + (2 \zeta_{6} - 2) q^{31} + (5 \zeta_{6} - 5) q^{32} + 2 q^{34} + 2 q^{36} + 8 \zeta_{6} q^{37} + (6 \zeta_{6} - 6) q^{38} + ( - 2 \zeta_{6} + 2) q^{39} + 5 q^{41} + (3 \zeta_{6} - 1) q^{42} + 7 q^{43} + ( - 3 \zeta_{6} + 3) q^{46} + q^{48} + (3 \zeta_{6} - 8) q^{49} + 2 \zeta_{6} q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + (6 \zeta_{6} - 6) q^{53} - 5 \zeta_{6} q^{54} + (9 \zeta_{6} - 3) q^{56} - 6 q^{57} - 7 \zeta_{6} q^{58} + (10 \zeta_{6} - 10) q^{59} - 7 \zeta_{6} q^{61} + 2 q^{62} + ( - 4 \zeta_{6} + 6) q^{63} + 7 q^{64} + ( - 5 \zeta_{6} + 5) q^{67} + 2 \zeta_{6} q^{68} + 3 q^{69} - 2 q^{71} - 6 \zeta_{6} q^{72} + ( - 6 \zeta_{6} + 6) q^{73} + ( - 8 \zeta_{6} + 8) q^{74} - 6 q^{76} - 2 q^{78} + 2 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 5 \zeta_{6} q^{82} - 11 q^{83} + (2 \zeta_{6} - 3) q^{84} - 7 \zeta_{6} q^{86} + ( - 7 \zeta_{6} + 7) q^{87} - 9 \zeta_{6} q^{89} + ( - 6 \zeta_{6} + 2) q^{91} + 3 q^{92} + 2 \zeta_{6} q^{93} + 5 \zeta_{6} q^{96} - 16 q^{97} + (5 \zeta_{6} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} + q^{4} - 2 q^{6} - q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} + q^{4} - 2 q^{6} - q^{7} - 6 q^{8} + 2 q^{9} - q^{12} + 4 q^{13} - 4 q^{14} + q^{16} - 2 q^{17} + 2 q^{18} - 6 q^{19} - 5 q^{21} + 3 q^{23} - 3 q^{24} - 2 q^{26} + 10 q^{27} - 5 q^{28} + 14 q^{29} - 2 q^{31} - 5 q^{32} + 4 q^{34} + 4 q^{36} + 8 q^{37} - 6 q^{38} + 2 q^{39} + 10 q^{41} + q^{42} + 14 q^{43} + 3 q^{46} + 2 q^{48} - 13 q^{49} + 2 q^{51} + 2 q^{52} - 6 q^{53} - 5 q^{54} + 3 q^{56} - 12 q^{57} - 7 q^{58} - 10 q^{59} - 7 q^{61} + 4 q^{62} + 8 q^{63} + 14 q^{64} + 5 q^{67} + 2 q^{68} + 6 q^{69} - 4 q^{71} - 6 q^{72} + 6 q^{73} + 8 q^{74} - 12 q^{76} - 4 q^{78} + 2 q^{79} - q^{81} - 5 q^{82} - 22 q^{83} - 4 q^{84} - 7 q^{86} + 7 q^{87} - 9 q^{89} - 2 q^{91} + 6 q^{92} + 2 q^{93} + 5 q^{96} - 32 q^{97} + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.00000 −0.500000 + 2.59808i −3.00000 1.00000 1.73205i 0
151.1 −0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 0 −1.00000 −0.500000 2.59808i −3.00000 1.00000 + 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.e.a 2
5.b even 2 1 175.2.e.b 2
5.c odd 4 2 35.2.j.a 4
7.c even 3 1 inner 175.2.e.a 2
7.c even 3 1 1225.2.a.f 1
7.d odd 6 1 1225.2.a.g 1
15.e even 4 2 315.2.bf.a 4
20.e even 4 2 560.2.bw.b 4
35.f even 4 2 245.2.j.c 4
35.i odd 6 1 1225.2.a.b 1
35.j even 6 1 175.2.e.b 2
35.j even 6 1 1225.2.a.d 1
35.k even 12 2 245.2.b.b 2
35.k even 12 2 245.2.j.c 4
35.l odd 12 2 35.2.j.a 4
35.l odd 12 2 245.2.b.c 2
105.w odd 12 2 2205.2.d.e 2
105.x even 12 2 315.2.bf.a 4
105.x even 12 2 2205.2.d.d 2
140.w even 12 2 560.2.bw.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.j.a 4 5.c odd 4 2
35.2.j.a 4 35.l odd 12 2
175.2.e.a 2 1.a even 1 1 trivial
175.2.e.a 2 7.c even 3 1 inner
175.2.e.b 2 5.b even 2 1
175.2.e.b 2 35.j even 6 1
245.2.b.b 2 35.k even 12 2
245.2.b.c 2 35.l odd 12 2
245.2.j.c 4 35.f even 4 2
245.2.j.c 4 35.k even 12 2
315.2.bf.a 4 15.e even 4 2
315.2.bf.a 4 105.x even 12 2
560.2.bw.b 4 20.e even 4 2
560.2.bw.b 4 140.w even 12 2
1225.2.a.b 1 35.i odd 6 1
1225.2.a.d 1 35.j even 6 1
1225.2.a.f 1 7.c even 3 1
1225.2.a.g 1 7.d odd 6 1
2205.2.d.d 2 105.x even 12 2
2205.2.d.e 2 105.w odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( (T - 7)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( (T - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 7)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$83$ \( (T + 11)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$97$ \( (T + 16)^{2} \) Copy content Toggle raw display
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