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Properties

Label 175.3.i.c
Level $175$
Weight $3$
Character orbit 175.i
Analytic conductor $4.768$
Analytic rank $0$
Dimension $12$
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] = [175,3,Mod(26,175)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 175.i (of order \(6\), 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: \(4.76840462631\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 15x^{10} + 180x^{8} + 669x^{6} + 1980x^{4} + 135x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{11} + \beta_1) q^{3} + (\beta_{7} - \beta_{3} - 1) q^{4} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{6}+ \cdots + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{11} + \beta_1) q^{3} + (\beta_{7} - \beta_{3} - 1) q^{4} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{6}+ \cdots + (3 \beta_{6} + 3 \beta_{5} - 4 \beta_{2} - 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{4} + 18 q^{9} + 6 q^{11} + 66 q^{14} - 6 q^{16} - 18 q^{19} + 12 q^{21} - 216 q^{24} + 18 q^{26} - 108 q^{31} - 204 q^{36} + 240 q^{39} + 42 q^{44} + 114 q^{46} + 324 q^{49} + 180 q^{51} - 252 q^{54} + 336 q^{56} - 396 q^{59} - 108 q^{61} - 372 q^{64} - 108 q^{66} + 192 q^{71} + 594 q^{74} + 192 q^{79} + 294 q^{81} + 1200 q^{84} - 384 q^{86} - 684 q^{89} - 72 q^{91} - 990 q^{94} - 540 q^{96} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 15x^{10} + 180x^{8} + 669x^{6} + 1980x^{4} + 135x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} - 12\nu^{8} - 144\nu^{6} - 132\nu^{4} - 9\nu^{2} + 7767 ) / 1575 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -12\nu^{10} - 179\nu^{8} - 2148\nu^{6} - 7884\nu^{4} - 23628\nu^{2} - 1611 ) / 1575 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{11} - 179\nu^{9} - 2148\nu^{7} - 7884\nu^{5} - 23628\nu^{3} - 36\nu ) / 1575 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\nu^{10} + 811\nu^{8} + 9767\nu^{6} + 37971\nu^{4} + 111777\nu^{2} + 15069 ) / 3150 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -62\nu^{10} - 919\nu^{8} - 10958\nu^{6} - 39159\nu^{4} - 111858\nu^{2} + 7269 ) / 3150 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -12\nu^{10} - 179\nu^{8} - 2148\nu^{6} - 7884\nu^{4} - 23313\nu^{2} - 36 ) / 315 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -53\nu^{11} - 801\nu^{9} - 9627\nu^{7} - 36471\nu^{5} - 108207\nu^{3} - 15939\nu ) / 1350 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 404\nu^{11} + 6003\nu^{9} + 71826\nu^{7} + 259653\nu^{5} + 757746\nu^{3} - 58743\nu ) / 9450 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -583\nu^{11} - 8781\nu^{9} - 105477\nu^{7} - 396681\nu^{5} - 1179567\nu^{3} - 168489\nu ) / 9450 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -622\nu^{11} - 9249\nu^{9} - 110778\nu^{7} - 401829\nu^{5} - 1179918\nu^{3} + 86229\nu ) / 9450 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 5\beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{10} + 2\beta_{9} - \beta_{8} - 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} + 2\beta_{6} - 4\beta_{5} + 45\beta_{3} + 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{11} - 28\beta_{10} - 16\beta_{9} + 32\beta_{8} + 93\beta_{4} - 91\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{6} + 30\beta_{5} - 135\beta_{2} + 453 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{11} + 165\beta_{10} - 195\beta_{9} - 195\beta_{8} + 933\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1488\beta_{7} - 720\beta_{6} + 360\beta_{5} - 4785\beta_{3} - 4785 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3696\beta_{11} + 1848\beta_{10} + 4416\beta_{9} - 2208\beta_{8} - 10737\beta_{4} + 360\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -16281\beta_{7} + 4056\beta_{6} - 8112\beta_{5} + 51525\beta_{3} + 16281\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -20337\beta_{11} - 40674\beta_{10} - 24393\beta_{9} + 48786\beta_{8} + 116649\beta_{4} - 112593\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.65037 + 2.85853i
−1.00460 + 1.74002i
−0.130586 + 0.226181i
0.130586 − 0.226181i
1.00460 − 1.74002i
1.65037 − 2.85853i
−1.65037 − 2.85853i
−1.00460 − 1.74002i
−0.130586 − 0.226181i
0.130586 + 0.226181i
1.00460 + 1.74002i
1.65037 + 2.85853i
−1.65037 + 2.85853i −3.81198 + 2.20085i −3.44745 − 5.97116i 0 − 14.5289i 4.54742 − 5.32175i 9.55533 5.18747 − 8.98496i 0
26.2 −1.00460 + 1.74002i 1.35935 − 0.784824i −0.0184439 − 0.0319457i 0 3.15374i −6.86137 − 1.38623i −7.96269 −3.26810 + 5.66052i 0
26.3 −0.130586 + 0.226181i 3.25898 − 1.88157i 1.96589 + 3.40503i 0 0.982826i 6.80019 − 1.66053i −2.07156 2.58064 − 4.46979i 0
26.4 0.130586 − 0.226181i −3.25898 + 1.88157i 1.96589 + 3.40503i 0 0.982826i −6.80019 + 1.66053i 2.07156 2.58064 − 4.46979i 0
26.5 1.00460 − 1.74002i −1.35935 + 0.784824i −0.0184439 − 0.0319457i 0 3.15374i 6.86137 + 1.38623i 7.96269 −3.26810 + 5.66052i 0
26.6 1.65037 − 2.85853i 3.81198 − 2.20085i −3.44745 − 5.97116i 0 − 14.5289i −4.54742 + 5.32175i −9.55533 5.18747 − 8.98496i 0
101.1 −1.65037 − 2.85853i −3.81198 − 2.20085i −3.44745 + 5.97116i 0 14.5289i 4.54742 + 5.32175i 9.55533 5.18747 + 8.98496i 0
101.2 −1.00460 − 1.74002i 1.35935 + 0.784824i −0.0184439 + 0.0319457i 0 − 3.15374i −6.86137 + 1.38623i −7.96269 −3.26810 − 5.66052i 0
101.3 −0.130586 − 0.226181i 3.25898 + 1.88157i 1.96589 − 3.40503i 0 − 0.982826i 6.80019 + 1.66053i −2.07156 2.58064 + 4.46979i 0
101.4 0.130586 + 0.226181i −3.25898 − 1.88157i 1.96589 − 3.40503i 0 − 0.982826i −6.80019 − 1.66053i 2.07156 2.58064 + 4.46979i 0
101.5 1.00460 + 1.74002i −1.35935 − 0.784824i −0.0184439 + 0.0319457i 0 − 3.15374i 6.86137 − 1.38623i 7.96269 −3.26810 − 5.66052i 0
101.6 1.65037 + 2.85853i 3.81198 + 2.20085i −3.44745 + 5.97116i 0 14.5289i −4.54742 − 5.32175i −9.55533 5.18747 + 8.98496i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.3.i.c 12
5.b even 2 1 inner 175.3.i.c 12
5.c odd 4 2 35.3.i.a ✓ 12
7.d odd 6 1 inner 175.3.i.c 12
15.e even 4 2 315.3.bi.c 12
20.e even 4 2 560.3.br.a 12
35.f even 4 2 245.3.i.d 12
35.i odd 6 1 inner 175.3.i.c 12
35.k even 12 2 35.3.i.a ✓ 12
35.k even 12 2 245.3.c.a 12
35.l odd 12 2 245.3.c.a 12
35.l odd 12 2 245.3.i.d 12
105.w odd 12 2 315.3.bi.c 12
140.x odd 12 2 560.3.br.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.i.a ✓ 12 5.c odd 4 2
35.3.i.a ✓ 12 35.k even 12 2
175.3.i.c 12 1.a even 1 1 trivial
175.3.i.c 12 5.b even 2 1 inner
175.3.i.c 12 7.d odd 6 1 inner
175.3.i.c 12 35.i odd 6 1 inner
245.3.c.a 12 35.k even 12 2
245.3.c.a 12 35.l odd 12 2
245.3.i.d 12 35.f even 4 2
245.3.i.d 12 35.l odd 12 2
315.3.bi.c 12 15.e even 4 2
315.3.bi.c 12 105.w odd 12 2
560.3.br.a 12 20.e even 4 2
560.3.br.a 12 140.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 15T_{2}^{10} + 180T_{2}^{8} + 669T_{2}^{6} + 1980T_{2}^{4} + 135T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 15 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} - 36 T^{10} + \cdots + 456976 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 489 T^{4} + \cdots + 4104676)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 119168138285056 \) Copy content Toggle raw display
$19$ \( (T^{6} + 9 T^{5} + \cdots + 2333772)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 59129994297609 \) Copy content Toggle raw display
$29$ \( (T^{3} - 543 T + 4508)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 54 T^{5} + \cdots + 12288)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 79\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + 1077 T^{4} + \cdots + 30432675)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 6792 T^{4} + \cdots - 2881200)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + 198 T^{5} + \cdots + 139491478272)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 54 T^{5} + \cdots + 4280171952)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{3} - 48 T^{2} + \cdots + 15508)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} - 96 T^{5} + \cdots + 330803344)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 23520 T^{4} + \cdots + 118402057216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 342 T^{5} + \cdots + 285265069488)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 18228 T^{4} + \cdots + 207129664)^{2} \) Copy content Toggle raw display
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