Properties

Label 275.6.b.b
Level $275$
Weight $6$
Character orbit 275.b
Analytic conductor $44.106$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,6,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), 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: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.11877512256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{5} - \beta_{4} + 6 \beta_{3}) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 - 30) q^{4} + ( - 13 \beta_{2} - 10 \beta_1 - 72) q^{6} + ( - 10 \beta_{5} - 30 \beta_{4} - 4 \beta_{3}) q^{7} + ( - 26 \beta_{5} - 94 \beta_{3}) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{5} - \beta_{4} + 6 \beta_{3}) q^{3} + ( - 4 \beta_{2} - 6 \beta_1 - 30) q^{4} + ( - 13 \beta_{2} - 10 \beta_1 - 72) q^{6} + ( - 10 \beta_{5} - 30 \beta_{4} - 4 \beta_{3}) q^{7} + ( - 26 \beta_{5} - 94 \beta_{3}) q^{8} + ( - 19 \beta_{2} + 11 \beta_1 + 6) q^{9} + 121 q^{11} + ( - 112 \beta_{5} + \cdots - 142 \beta_{3}) q^{12}+ \cdots + ( - 2299 \beta_{2} + 1331 \beta_1 + 726) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 168 q^{4} - 412 q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 168 q^{4} - 412 q^{6} + 14 q^{9} + 726 q^{11} + 2040 q^{14} + 3984 q^{16} - 2760 q^{19} - 1816 q^{21} + 23496 q^{24} + 24264 q^{26} + 6852 q^{29} - 8196 q^{31} - 50640 q^{34} + 9512 q^{36} + 13120 q^{39} + 11988 q^{41} - 20328 q^{44} - 33612 q^{46} - 97062 q^{49} - 45448 q^{51} - 37628 q^{54} - 84624 q^{56} + 7476 q^{59} + 36972 q^{61} + 40704 q^{64} - 49852 q^{66} - 70084 q^{69} + 78564 q^{71} - 306588 q^{74} + 207840 q^{76} - 250296 q^{79} - 173834 q^{81} - 687232 q^{84} + 486120 q^{86} + 213648 q^{89} - 219264 q^{91} + 149856 q^{94} - 152912 q^{96} + 1694 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -137\nu^{5} + 578\nu^{4} - 4930\nu^{3} + 27490\nu^{2} + 2527\nu + 713224 ) / 138448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 167\nu^{5} + 306\nu^{4} + 1462\nu^{3} - 30478\nu^{2} - 37945\nu - 804728 ) / 138448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{5} - 58\nu^{4} - 1182\nu^{3} + 454\nu^{2} - 15277\nu + 127832 ) / 20360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1761\nu^{5} - 3434\nu^{4} - 66402\nu^{3} - 25354\nu^{2} - 683001\nu + 7434160 ) / 346120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4207\nu^{5} + 3978\nu^{4} + 157454\nu^{3} - 86742\nu^{2} + 2737847\nu - 19126680 ) / 692240 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{4} + 15\beta_{3} - 5\beta_{2} + \beta _1 - 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -21\beta_{5} - \beta_{4} - 100\beta_{3} - 24\beta_{2} - 29\beta _1 + 79 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -60\beta_{5} + 45\beta_{4} - 495\beta_{3} + 181\beta_{2} + 235\beta _1 + 325 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 521\beta_{5} - 759\beta_{4} + 4520\beta_{3} + 624\beta_{2} + 233\beta _1 + 4745 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.874840 6.07300i
3.64914 0.444721i
−3.52398 4.62828i
−3.52398 + 4.62828i
3.64914 + 0.444721i
0.874840 + 6.07300i
10.3963i 20.6466i −76.0833 0 −214.649 164.454i 458.304i −183.283 0
199.2 8.18772i 3.48600i −35.0388 0 −28.5424 145.071i 24.8808i 230.848 0
199.3 2.20859i 16.8394i 27.1221 0 37.1913 225.525i 130.577i −40.5643 0
199.4 2.20859i 16.8394i 27.1221 0 37.1913 225.525i 130.577i −40.5643 0
199.5 8.18772i 3.48600i −35.0388 0 −28.5424 145.071i 24.8808i 230.848 0
199.6 10.3963i 20.6466i −76.0833 0 −214.649 164.454i 458.304i −183.283 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.6.b.b 6
5.b even 2 1 inner 275.6.b.b 6
5.c odd 4 1 11.6.a.b 3
5.c odd 4 1 275.6.a.b 3
15.e even 4 1 99.6.a.g 3
20.e even 4 1 176.6.a.i 3
35.f even 4 1 539.6.a.e 3
40.i odd 4 1 704.6.a.q 3
40.k even 4 1 704.6.a.t 3
55.e even 4 1 121.6.a.d 3
165.l odd 4 1 1089.6.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 5.c odd 4 1
99.6.a.g 3 15.e even 4 1
121.6.a.d 3 55.e even 4 1
176.6.a.i 3 20.e even 4 1
275.6.a.b 3 5.c odd 4 1
275.6.b.b 6 1.a even 1 1 trivial
275.6.b.b 6 5.b even 2 1 inner
539.6.a.e 3 35.f even 4 1
704.6.a.q 3 40.i odd 4 1
704.6.a.t 3 40.k even 4 1
1089.6.a.r 3 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 180T_{2}^{4} + 8100T_{2}^{2} + 35344 \) acting on \(S_{6}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 180 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$3$ \( T^{6} + 722 T^{4} + \cdots + 1468944 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 28949220680704 \) Copy content Toggle raw display
$11$ \( (T - 121)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{3} + 1380 T^{2} + \cdots - 57024000)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{3} - 3426 T^{2} + \cdots + 4029189120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 4098 T^{2} + \cdots + 1094344400)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{3} - 5994 T^{2} + \cdots + 201929821568)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots - 7759637437060)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 15233874751008)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 1290398551704)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 1279883216320)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 90320980174650)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
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