Properties

Label 32.7.c.a
Level $32$
Weight $7$
Character orbit 32.c
Analytic conductor $7.362$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,7,Mod(31,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.31");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 32.c (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: no
Analytic conductor: \(7.36173067584\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 50 q^{5} + 46 \beta q^{7} + 713 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 50 q^{5} + 46 \beta q^{7} + 713 q^{9} + 527 \beta q^{11} + 2242 q^{13} + 50 \beta q^{15} + 2898 q^{17} + 2013 \beta q^{19} - 736 q^{21} - 4646 \beta q^{23} - 13125 q^{25} + 1442 \beta q^{27} - 20990 q^{29} - 11720 \beta q^{31} - 8432 q^{33} + 2300 \beta q^{35} + 402 q^{37} + 2242 \beta q^{39} + 13330 q^{41} + 1495 \beta q^{43} + 35650 q^{45} + 36196 \beta q^{47} + 83793 q^{49} + 2898 \beta q^{51} + 171570 q^{53} + 26350 \beta q^{55} - 32208 q^{57} - 36829 \beta q^{59} - 270878 q^{61} + 32798 \beta q^{63} + 112100 q^{65} - 107903 \beta q^{67} + 74336 q^{69} + 9662 \beta q^{71} - 696606 q^{73} - 13125 \beta q^{75} - 387872 q^{77} - 167524 \beta q^{79} + 496705 q^{81} + 75325 \beta q^{83} + 144900 q^{85} - 20990 \beta q^{87} - 161598 q^{89} + 103132 \beta q^{91} + 187520 q^{93} + 100650 \beta q^{95} + 520306 q^{97} + 375751 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 100 q^{5} + 1426 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 100 q^{5} + 1426 q^{9} + 4484 q^{13} + 5796 q^{17} - 1472 q^{21} - 26250 q^{25} - 41980 q^{29} - 16864 q^{33} + 804 q^{37} + 26660 q^{41} + 71300 q^{45} + 167586 q^{49} + 343140 q^{53} - 64416 q^{57} - 541756 q^{61} + 224200 q^{65} + 148672 q^{69} - 1393212 q^{73} - 775744 q^{77} + 993410 q^{81} + 289800 q^{85} - 323196 q^{89} + 375040 q^{93} + 1040612 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.00000i
1.00000i
0 4.00000i 0 50.0000 0 184.000i 0 713.000 0
31.2 0 4.00000i 0 50.0000 0 184.000i 0 713.000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.7.c.a 2
3.b odd 2 1 288.7.g.a 2
4.b odd 2 1 inner 32.7.c.a 2
8.b even 2 1 64.7.c.b 2
8.d odd 2 1 64.7.c.b 2
12.b even 2 1 288.7.g.a 2
16.e even 4 1 256.7.d.a 2
16.e even 4 1 256.7.d.c 2
16.f odd 4 1 256.7.d.a 2
16.f odd 4 1 256.7.d.c 2
24.f even 2 1 576.7.g.i 2
24.h odd 2 1 576.7.g.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.7.c.a 2 1.a even 1 1 trivial
32.7.c.a 2 4.b odd 2 1 inner
64.7.c.b 2 8.b even 2 1
64.7.c.b 2 8.d odd 2 1
256.7.d.a 2 16.e even 4 1
256.7.d.a 2 16.f odd 4 1
256.7.d.c 2 16.e even 4 1
256.7.d.c 2 16.f odd 4 1
288.7.g.a 2 3.b odd 2 1
288.7.g.a 2 12.b even 2 1
576.7.g.i 2 24.f even 2 1
576.7.g.i 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 16 \) acting on \(S_{7}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T - 50)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 33856 \) Copy content Toggle raw display
$11$ \( T^{2} + 4443664 \) Copy content Toggle raw display
$13$ \( (T - 2242)^{2} \) Copy content Toggle raw display
$17$ \( (T - 2898)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 64834704 \) Copy content Toggle raw display
$23$ \( T^{2} + 345365056 \) Copy content Toggle raw display
$29$ \( (T + 20990)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2197734400 \) Copy content Toggle raw display
$37$ \( (T - 402)^{2} \) Copy content Toggle raw display
$41$ \( (T - 13330)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 35760400 \) Copy content Toggle raw display
$47$ \( T^{2} + 20962406656 \) Copy content Toggle raw display
$53$ \( (T - 171570)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 21702003856 \) Copy content Toggle raw display
$61$ \( (T + 270878)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 186288918544 \) Copy content Toggle raw display
$71$ \( T^{2} + 1493667904 \) Copy content Toggle raw display
$73$ \( (T + 696606)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 449028649216 \) Copy content Toggle raw display
$83$ \( T^{2} + 90781690000 \) Copy content Toggle raw display
$89$ \( (T + 161598)^{2} \) Copy content Toggle raw display
$97$ \( (T - 520306)^{2} \) Copy content Toggle raw display
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