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Properties

Label 275.6.b.d
Level $275$
Weight $6$
Character orbit 275.b
Analytic conductor $44.106$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(199,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-122] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 70x^{6} + 1541x^{4} + 10660x^{2} + 5476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_1) q^{3} + (\beta_{3} - 15) q^{4} + ( - 2 \beta_{4} + \beta_{2} - 39) q^{6} + (\beta_{7} + 7 \beta_{6} + \cdots - 9 \beta_1) q^{7} + (\beta_{7} + 2 \beta_{6} + \cdots + 5 \beta_1) q^{8}+ \cdots + (363 \beta_{4} - 363 \beta_{3} + 484) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 122 q^{4} - 314 q^{6} - 44 q^{9} - 968 q^{11} + 3374 q^{14} - 5342 q^{16} + 6788 q^{19} - 9416 q^{21} - 5442 q^{24} - 7300 q^{26} + 10496 q^{29} + 9464 q^{31} - 19518 q^{34} + 14300 q^{36} + 42160 q^{39}+ \cdots + 5324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 70x^{6} + 1541x^{4} + 10660x^{2} + 5476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -13\nu^{7} - 614\nu^{5} - 3235\nu^{3} + 67510\nu ) / 19536 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{6} - 905\nu^{4} - 9838\nu^{2} - 4784 ) / 264 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 65\nu^{4} + 1612\nu^{2} + 11444 ) / 264 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{6} - 455\nu^{4} - 7324\nu^{2} - 11204 ) / 264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{7} - 2120\nu^{5} - 50365\nu^{3} - 289070\nu ) / 19536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -85\nu^{7} - 4766\nu^{5} - 76003\nu^{3} - 284426\nu ) / 19536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95\nu^{7} + 6428\nu^{5} + 140919\nu^{3} + 1088698\nu ) / 6512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 4\beta_{6} + 3\beta_{5} - 10\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 7\beta_{3} - 261 ) / 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{7} - 55\beta_{6} - 18\beta_{5} + 175\beta_1 ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -21\beta_{4} - 71\beta_{3} + 4\beta_{2} + 2259 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 247\beta_{7} + 1693\beta_{6} - 135\beta_{5} - 5395\beta_1 ) / 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2483\beta_{4} + 6521\beta_{3} - 780\beta_{2} - 191433 ) / 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6581\beta_{7} - 55889\beta_{6} + 18645\beta_{5} + 162755\beta_1 ) / 15 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
5.71109i
4.95665i
− 3.50110i
− 0.746657i
0.746657i
3.50110i
− 4.95665i
− 5.71109i
− 8.50380i 13.0311i −40.3146 0 110.814 217.433i 70.7058i 73.1914 0
199.2 − 7.82466i − 16.3044i −29.2253 0 −127.576 125.436i − 21.7111i −22.8321 0
199.3 − 6.96278i − 22.6701i −16.4803 0 −157.847 − 169.118i − 108.060i −270.932 0
199.4 − 2.64192i 6.66534i 25.0202 0 17.6093 12.8802i − 150.643i 198.573 0
199.5 2.64192i − 6.66534i 25.0202 0 17.6093 − 12.8802i 150.643i 198.573 0
199.6 6.96278i 22.6701i −16.4803 0 −157.847 169.118i 108.060i −270.932 0
199.7 7.82466i 16.3044i −29.2253 0 −127.576 − 125.436i 21.7111i −22.8321 0
199.8 8.50380i − 13.0311i −40.3146 0 110.814 − 217.433i − 70.7058i 73.1914 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 199.8
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.d 8
5.b even 2 1 inner 275.6.b.d 8
5.c odd 4 1 55.6.a.b ✓ 4
5.c odd 4 1 275.6.a.d 4
15.e even 4 1 495.6.a.g 4
20.e even 4 1 880.6.a.n 4
55.e even 4 1 605.6.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.a.b ✓ 4 5.c odd 4 1
275.6.a.d 4 5.c odd 4 1
275.6.b.d 8 1.a even 1 1 trivial
275.6.b.d 8 5.b even 2 1 inner
495.6.a.g 4 15.e even 4 1
605.6.a.c 4 55.e even 4 1
880.6.a.n 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 189T_{2}^{6} + 12172T_{2}^{4} + 290736T_{2}^{2} + 1498176 \) acting on \(S_{6}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 189 T^{6} + \cdots + 1498176 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 1030666816 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( (T + 121)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{4} - 3394 T^{3} + \cdots + 123062234816)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 563529236434956)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 2285702081856)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 17\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 60\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 81\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 56\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
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