Properties

Label 59.7.b.b
Level $59$
Weight $7$
Character orbit 59.b
Self dual yes
Analytic conductor $13.573$
Analytic rank $0$
Dimension $2$
CM discriminant -59
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [59,7,Mod(58,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.58");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 59.b (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(13.5731909336\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (7 \beta - 1) q^{3} + 64 q^{4} + ( - 21 \beta - 85) q^{5} + (87 \beta - 121) q^{7} + (35 \beta + 1428) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (7 \beta - 1) q^{3} + 64 q^{4} + ( - 21 \beta - 85) q^{5} + (87 \beta - 121) q^{7} + (35 \beta + 1428) q^{9} + (448 \beta - 64) q^{12} + ( - 721 \beta - 6383) q^{15} + 4096 q^{16} + 6050 q^{17} + ( - 1785 \beta + 671) q^{19} + ( - 1344 \beta - 5440) q^{20} + ( - 325 \beta + 26917) q^{21} + (4011 \beta + 11004) q^{25} + (5103 \beta + 10081) q^{27} + (5568 \beta - 7744) q^{28} + ( - 5565 \beta + 14531) q^{29} + ( - 6681 \beta - 70103) q^{35} + (2240 \beta + 91392) q^{36} + ( - 525 \beta - 68629) q^{41} + ( - 33698 \beta - 153720) q^{45} + (28672 \beta - 4096) q^{48} + ( - 13485 \beta + 230028) q^{49} + (42350 \beta - 6050) q^{51} + ( - 29757 \beta + 110291) q^{53} + ( - 6013 \beta - 550451) q^{57} - 205379 q^{59} + ( - 46144 \beta - 408512) q^{60} + (123046 \beta - 38808) q^{63} + 262144 q^{64} + 387200 q^{68} - 683422 q^{71} + (101094 \beta + 1224384) q^{75} + ( - 114240 \beta + 42944) q^{76} + ( - 122745 \beta + 205799) q^{79} + ( - 86016 \beta - 348160) q^{80} + (75670 \beta + 520631) q^{81} + ( - 20800 \beta + 1722688) q^{84} + ( - 127050 \beta - 514250) q^{85} + (68327 \beta - 1728551) q^{87} + (175119 \beta + 1592305) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} + 128 q^{4} - 191 q^{5} - 155 q^{7} + 2891 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} + 128 q^{4} - 191 q^{5} - 155 q^{7} + 2891 q^{9} + 320 q^{12} - 13487 q^{15} + 8192 q^{16} + 12100 q^{17} - 443 q^{19} - 12224 q^{20} + 53509 q^{21} + 26019 q^{25} + 25265 q^{27} - 9920 q^{28} + 23497 q^{29} - 146887 q^{35} + 185024 q^{36} - 137783 q^{41} - 341138 q^{45} + 20480 q^{48} + 446571 q^{49} + 30250 q^{51} + 190825 q^{53} - 1106915 q^{57} - 410758 q^{59} - 863168 q^{60} + 45430 q^{63} + 524288 q^{64} + 774400 q^{68} - 1366844 q^{71} + 2549862 q^{75} - 28352 q^{76} + 288853 q^{79} - 782336 q^{80} + 1116932 q^{81} + 3424576 q^{84} - 1155550 q^{85} - 3388775 q^{87} + 3359729 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/59\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1
−6.15207
7.15207
0 −44.0645 64.0000 44.1934 0 −656.230 0 1212.68 0
58.2 0 49.0645 64.0000 −235.193 0 501.230 0 1678.32 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.7.b.b 2
59.b odd 2 1 CM 59.7.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.7.b.b 2 1.a even 1 1 trivial
59.7.b.b 2 59.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(59, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} - 5T_{3} - 2162 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 2162 \) Copy content Toggle raw display
$5$ \( T^{2} + 191T - 10394 \) Copy content Toggle raw display
$7$ \( T^{2} + 155T - 328922 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6050)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 443 T - 140941394 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 1232360954 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 4733842366 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 30078902762 \) Copy content Toggle raw display
$59$ \( (T + 205379)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T + 683422)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 645826310954 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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