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Properties

Label 175.2.e.e
Level $175$
Weight $2$
Character orbit 175.e
Analytic conductor $1.397$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{3} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{6} + ( - \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{3} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{4} + ( - \beta_{3} + 2 \beta_{2}) q^{6} + ( - \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{7}+ \cdots + ( - 3 \beta_{3} + 2 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 3 q^{3} - 3 q^{4} - 4 q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 3 q^{3} - 3 q^{4} - 4 q^{6} - 2 q^{7} - 4 q^{9} + 2 q^{11} - 13 q^{12} + 8 q^{13} + 17 q^{14} + 5 q^{16} - 10 q^{17} + 11 q^{18} + 4 q^{19} - 11 q^{21} + 34 q^{22} - q^{23} + 10 q^{24} - 15 q^{26} + 12 q^{27} - 11 q^{28} - 16 q^{29} - 5 q^{31} + 6 q^{32} - 6 q^{33} - 24 q^{34} + 30 q^{36} + 10 q^{37} - 8 q^{38} - 12 q^{39} - 2 q^{41} - 41 q^{42} + 18 q^{43} - 6 q^{44} + 10 q^{46} - 28 q^{47} - 28 q^{48} + 12 q^{49} - 11 q^{51} - 12 q^{52} - 10 q^{53} + 27 q^{54} - 21 q^{56} + 30 q^{57} + 8 q^{58} + 5 q^{59} - 5 q^{61} + 18 q^{62} - 25 q^{63} - 32 q^{64} - 26 q^{66} + 12 q^{67} - 11 q^{68} + 24 q^{69} + 14 q^{71} + 11 q^{72} + 12 q^{73} + 15 q^{74} + 30 q^{76} - 23 q^{77} + 44 q^{78} + 7 q^{79} + 13 q^{81} - 6 q^{82} + 52 q^{83} - 13 q^{84} + 30 q^{86} - 13 q^{87} + 8 q^{88} + 6 q^{89} - 3 q^{91} + 24 q^{92} + 16 q^{93} + 17 q^{94} - 17 q^{96} - 14 q^{97} + 34 q^{98} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 19\nu^{2} + 12\nu - 60 ) / 83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 48\nu - 240 ) / 83 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu + 204 ) / 249 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} - 3\nu^{4} - 68\nu^{3} - 28\nu^{2} - 275\nu - 36 ) / 83 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 12\beta_{4} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} + 3\beta_{4} + 6\beta_{3} - 17\beta_{2} - 17\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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
−0.956115 + 1.65604i
0.356769 − 0.617942i
1.09935 − 1.90412i
−0.956115 − 1.65604i
0.356769 + 0.617942i
1.09935 + 1.90412i
−0.956115 + 1.65604i −0.828310 − 1.43468i −0.828310 − 1.43468i 0 3.16784 −0.799494 − 2.52206i −0.656620 0.127804 − 0.221364i 0
51.2 0.356769 − 0.617942i 0.745432 + 1.29113i 0.745432 + 1.29113i 0 1.06379 −2.63409 − 0.248083i 2.49086 0.388663 − 0.673184i 0
51.3 1.09935 − 1.90412i −1.41712 − 2.45453i −1.41712 − 2.45453i 0 −6.23163 2.43359 + 1.03810i −1.83424 −2.51647 + 4.35865i 0
151.1 −0.956115 − 1.65604i −0.828310 + 1.43468i −0.828310 + 1.43468i 0 3.16784 −0.799494 + 2.52206i −0.656620 0.127804 + 0.221364i 0
151.2 0.356769 + 0.617942i 0.745432 − 1.29113i 0.745432 − 1.29113i 0 1.06379 −2.63409 + 0.248083i 2.49086 0.388663 + 0.673184i 0
151.3 1.09935 + 1.90412i −1.41712 + 2.45453i −1.41712 + 2.45453i 0 −6.23163 2.43359 − 1.03810i −1.83424 −2.51647 − 4.35865i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.3
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.e yes 6
5.b even 2 1 175.2.e.d ✓ 6
5.c odd 4 2 175.2.k.b 12
7.c even 3 1 inner 175.2.e.e yes 6
7.c even 3 1 1225.2.a.x 3
7.d odd 6 1 1225.2.a.w 3
35.i odd 6 1 1225.2.a.z 3
35.j even 6 1 175.2.e.d ✓ 6
35.j even 6 1 1225.2.a.y 3
35.k even 12 2 1225.2.b.m 6
35.l odd 12 2 175.2.k.b 12
35.l odd 12 2 1225.2.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.e.d ✓ 6 5.b even 2 1
175.2.e.d ✓ 6 35.j even 6 1
175.2.e.e yes 6 1.a even 1 1 trivial
175.2.e.e yes 6 7.c even 3 1 inner
175.2.k.b 12 5.c odd 4 2
175.2.k.b 12 35.l odd 12 2
1225.2.a.w 3 7.d odd 6 1
1225.2.a.x 3 7.c even 3 1
1225.2.a.y 3 35.j even 6 1
1225.2.a.z 3 35.i odd 6 1
1225.2.b.l 6 35.l odd 12 2
1225.2.b.m 6 35.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 19T_{2}^{2} - 12T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 5 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots + 2025 \) Copy content Toggle raw display
$13$ \( (T^{3} - 4 T^{2} - 15 T - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{5} + \cdots + 441 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{3} + 8 T^{2} - 5 T - 75)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 5 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$37$ \( T^{6} - 10 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$41$ \( (T^{3} + T^{2} - 46 T + 105)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T^{2} - 20 T + 53)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 28 T^{5} + \cdots + 540225 \) Copy content Toggle raw display
$53$ \( T^{6} + 10 T^{5} + \cdots + 15129 \) Copy content Toggle raw display
$59$ \( T^{6} - 5 T^{5} + \cdots + 11025 \) Copy content Toggle raw display
$61$ \( T^{6} + 5 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$67$ \( T^{6} - 12 T^{5} + \cdots + 1600 \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} + \cdots + 423)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 12 T^{5} + \cdots + 200704 \) Copy content Toggle raw display
$79$ \( T^{6} - 7 T^{5} + \cdots + 22801 \) Copy content Toggle raw display
$83$ \( (T^{3} - 26 T^{2} + \cdots - 399)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots + 603729 \) Copy content Toggle raw display
$97$ \( (T^{3} + 7 T^{2} + \cdots + 259)^{2} \) Copy content Toggle raw display
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