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Properties

Label 605.2.g.l
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,2,Mod(81,605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("605.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) 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 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{2}) q^{2} + 2 \beta_1 q^{3} + (2 \beta_{7} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{4} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{5} + ( - 4 \beta_{6} - 4 \beta_{4} + \cdots - 4) q^{6}+ \cdots + ( - 3 \beta_{5} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} - 10 q^{9} + 8 q^{10} - 64 q^{12} - 8 q^{13} + 4 q^{14} - 6 q^{16} + 8 q^{17} + 10 q^{18} + 2 q^{20} - 8 q^{24} - 2 q^{25} - 4 q^{28} + 4 q^{29}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} + 4x^{4} + 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.437016 − 1.34500i
−0.437016 + 1.34500i
1.14412 − 0.831254i
−1.14412 + 0.831254i
0.437016 + 1.34500i
−0.437016 − 1.34500i
1.14412 + 0.831254i
−1.14412 − 0.831254i
−0.335106 − 0.243469i 0.874032 − 2.68999i −0.565015 − 1.73894i 0.809017 − 0.587785i −0.947822 + 0.688633i 0.618034 + 1.90211i −0.490035 + 1.50817i −4.04508 − 2.93893i −0.414214
81.2 1.95314 + 1.41904i −0.874032 + 2.68999i 1.18305 + 3.64105i 0.809017 − 0.587785i −5.52431 + 4.01365i 0.618034 + 1.90211i −1.36407 + 4.19817i −4.04508 − 2.93893i 2.41421
251.1 −0.746033 + 2.29605i 2.28825 − 1.66251i −3.09726 − 2.25029i −0.309017 − 0.951057i 2.11010 + 6.49422i −1.61803 − 1.17557i 3.57117 − 2.59461i 1.54508 − 4.75528i 2.41421
251.2 0.127999 − 0.393941i −2.28825 + 1.66251i 1.47923 + 1.07472i −0.309017 − 0.951057i 0.362036 + 1.11423i −1.61803 − 1.17557i 1.28293 − 0.932102i 1.54508 − 4.75528i −0.414214
366.1 −0.335106 + 0.243469i 0.874032 + 2.68999i −0.565015 + 1.73894i 0.809017 + 0.587785i −0.947822 − 0.688633i 0.618034 − 1.90211i −0.490035 − 1.50817i −4.04508 + 2.93893i −0.414214
366.2 1.95314 − 1.41904i −0.874032 − 2.68999i 1.18305 − 3.64105i 0.809017 + 0.587785i −5.52431 − 4.01365i 0.618034 − 1.90211i −1.36407 − 4.19817i −4.04508 + 2.93893i 2.41421
511.1 −0.746033 − 2.29605i 2.28825 + 1.66251i −3.09726 + 2.25029i −0.309017 + 0.951057i 2.11010 − 6.49422i −1.61803 + 1.17557i 3.57117 + 2.59461i 1.54508 + 4.75528i 2.41421
511.2 0.127999 + 0.393941i −2.28825 − 1.66251i 1.47923 − 1.07472i −0.309017 + 0.951057i 0.362036 − 1.11423i −1.61803 + 1.17557i 1.28293 + 0.932102i 1.54508 + 4.75528i −0.414214
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.l 8
11.b odd 2 1 605.2.g.f 8
11.c even 5 1 605.2.a.d 2
11.c even 5 3 inner 605.2.g.l 8
11.d odd 10 1 55.2.a.b ✓ 2
11.d odd 10 3 605.2.g.f 8
33.f even 10 1 495.2.a.b 2
33.h odd 10 1 5445.2.a.y 2
44.g even 10 1 880.2.a.m 2
44.h odd 10 1 9680.2.a.bn 2
55.h odd 10 1 275.2.a.c 2
55.j even 10 1 3025.2.a.o 2
55.l even 20 2 275.2.b.d 4
77.l even 10 1 2695.2.a.f 2
88.k even 10 1 3520.2.a.bo 2
88.p odd 10 1 3520.2.a.bn 2
132.n odd 10 1 7920.2.a.ch 2
143.l odd 10 1 9295.2.a.g 2
165.r even 10 1 2475.2.a.x 2
165.u odd 20 2 2475.2.c.l 4
220.o even 10 1 4400.2.a.bn 2
220.w odd 20 2 4400.2.b.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b ✓ 2 11.d odd 10 1
275.2.a.c 2 55.h odd 10 1
275.2.b.d 4 55.l even 20 2
495.2.a.b 2 33.f even 10 1
605.2.a.d 2 11.c even 5 1
605.2.g.f 8 11.b odd 2 1
605.2.g.f 8 11.d odd 10 3
605.2.g.l 8 1.a even 1 1 trivial
605.2.g.l 8 11.c even 5 3 inner
880.2.a.m 2 44.g even 10 1
2475.2.a.x 2 165.r even 10 1
2475.2.c.l 4 165.u odd 20 2
2695.2.a.f 2 77.l even 10 1
3025.2.a.o 2 55.j even 10 1
3520.2.a.bn 2 88.p odd 10 1
3520.2.a.bo 2 88.k even 10 1
4400.2.a.bn 2 220.o even 10 1
4400.2.b.q 4 220.w odd 20 2
5445.2.a.y 2 33.h odd 10 1
7920.2.a.ch 2 132.n odd 10 1
9295.2.a.g 2 143.l odd 10 1
9680.2.a.bn 2 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} - 2T_{2}^{7} + 5T_{2}^{6} - 12T_{2}^{5} + 29T_{2}^{4} + 12T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 8T_{3}^{6} + 64T_{3}^{4} + 512T_{3}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 8 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$43$ \( (T - 6)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{8} - 8 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$61$ \( T^{8} - 4 T^{7} + \cdots + 236421376 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 56)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 128 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T - 124)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
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