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Properties

Label 605.4.a.o
Level $605$
Weight $4$
Character orbit 605.a
Self dual yes
Analytic conductor $35.696$
Analytic rank $1$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,4,Mod(1,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

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: yes
Analytic conductor: \(35.6961555535\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 35x^{6} + 376x^{4} - 1452x^{2} + 1764 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{4} - 3) q^{3} + ( - \beta_{5} + \beta_{4} + 2) q^{4} + 5 q^{5} + (\beta_{6} - 6 \beta_{3} + \cdots - 7 \beta_1) q^{6} + (\beta_{6} - 4 \beta_{3} - \beta_{2}) q^{7}+ \cdots + ( - 54 \beta_{6} + 143 \beta_{3} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{3} + 18 q^{4} + 40 q^{5} + 104 q^{9} - 362 q^{12} - 70 q^{14} - 120 q^{15} - 42 q^{16} + 90 q^{20} - 572 q^{23} + 200 q^{25} + 692 q^{26} - 1044 q^{27} - 492 q^{31} - 600 q^{34} + 2172 q^{36}+ \cdots - 1864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 35x^{6} + 376x^{4} - 1452x^{2} + 1764 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 49\nu^{5} - 143\nu^{3} - 1086\nu ) / 126 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 154\nu^{5} - 1271\nu^{3} + 2640\nu ) / 378 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 29\nu^{4} - 202\nu^{2} + 276 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 29\nu^{4} + 220\nu^{2} - 438 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 623\nu^{5} - 5548\nu^{3} + 11544\nu ) / 378 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{6} + 67\nu^{4} - 620\nu^{2} + 1380 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 5\beta_{3} + \beta_{2} + 13\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 24\beta_{5} + 20\beta_{4} + 124 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29\beta_{6} - 133\beta_{3} + 19\beta_{2} + 207\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29\beta_{7} + 494\beta_{5} + 360\beta_{4} + 2054 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 639\beta_{6} - 2901\beta_{3} + 331\beta_{2} + 3599\beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.10041
−1.54002
−4.35061
−2.02187
2.02187
4.35061
1.54002
3.10041
−4.83246 −9.98262 15.3527 5.00000 48.2407 23.0737 −35.5316 72.6528 −24.1623
1.2 −3.27207 −0.0390529 2.70645 5.00000 0.127784 −14.1165 17.3209 −26.9985 −16.3604
1.3 −2.61856 −6.39235 −1.14314 5.00000 16.7387 −10.2514 23.9419 13.8621 −13.0928
1.4 −0.289819 4.41402 −7.91600 5.00000 −1.27927 −11.9683 4.61277 −7.51640 −1.44910
1.5 0.289819 4.41402 −7.91600 5.00000 1.27927 11.9683 −4.61277 −7.51640 1.44910
1.6 2.61856 −6.39235 −1.14314 5.00000 −16.7387 10.2514 −23.9419 13.8621 13.0928
1.7 3.27207 −0.0390529 2.70645 5.00000 −0.127784 14.1165 −17.3209 −26.9985 16.3604
1.8 4.83246 −9.98262 15.3527 5.00000 −48.2407 −23.0737 35.5316 72.6528 24.1623
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(11\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.4.a.o ✓ 8
11.b odd 2 1 inner 605.4.a.o ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.4.a.o ✓ 8 1.a even 1 1 trivial
605.4.a.o ✓ 8 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{8} - 41T_{2}^{6} + 487T_{2}^{4} - 1755T_{2}^{2} + 144 \) Copy content Toggle raw display
\( T_{3}^{4} + 12T_{3}^{3} - 8T_{3}^{2} - 282T_{3} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 41 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( (T^{4} + 12 T^{3} + \cdots - 11)^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1597041369 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 6219956192256 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 20351214157824 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11606940354816 \) Copy content Toggle raw display
$23$ \( (T^{4} + 286 T^{3} + \cdots + 5778720)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + 246 T^{3} + \cdots - 84685544)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots - 446406528)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 21\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( (T^{4} + 236 T^{3} + \cdots + 6087784941)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 756 T^{3} + \cdots - 8094368016)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 54 T^{3} + \cdots + 2928268152)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( (T^{4} + 2392 T^{3} + \cdots + 91567885245)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1454 T^{3} + \cdots + 903758616)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + 160 T^{3} + \cdots + 339659726877)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 932 T^{3} + \cdots - 237654925680)^{2} \) Copy content Toggle raw display
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