Properties

Label 32.9.c.a
Level $32$
Weight $9$
Character orbit 32.c
Analytic conductor $13.036$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.31");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(13.0361155220\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} - 182) q^{5} + (22 \beta_{3} + 17 \beta_1) q^{7} + (18 \beta_{2} - 4719) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} - 182) q^{5} + (22 \beta_{3} + 17 \beta_1) q^{7} + (18 \beta_{2} - 4719) q^{9} + ( - 41 \beta_{3} - 175 \beta_1) q^{11} + ( - 115 \beta_{2} - 3158) q^{13} + ( - 326 \beta_{3} - 869 \beta_1) q^{15} + (106 \beta_{2} - 97998) q^{17} + (365 \beta_{3} - 2437 \beta_1) q^{19} + (316 \beta_{2} - 236640) q^{21} + (2034 \beta_{3} + 791 \beta_1) q^{23} + ( - 364 \beta_{2} - 197757) q^{25} + ( - 750 \beta_{3} - 10524 \beta_1) q^{27} + (449 \beta_{2} - 176374) q^{29} + ( - 6824 \beta_{3} + 7616 \beta_1) q^{31} + ( - 2882 \beta_{2} + 771216) q^{33} + ( - 5892 \beta_{3} - 17648 \beta_1) q^{35} + (509 \beta_{2} + 1110762) q^{37} + (13402 \beta_{3} + 75847 \beta_1) q^{39} + (9524 \beta_{2} + 738338) q^{41} + (42527 \beta_{3} + 30507 \beta_1) q^{43} + ( - 7995 \beta_{2} + 3734250) q^{45} + ( - 19916 \beta_{3} + 73218 \beta_1) q^{47} + (5192 \beta_{2} + 709761) q^{49} + ( - 113262 \beta_{3} - 170820 \beta_1) q^{51} + ( - 3403 \beta_{2} - 1125462) q^{53} + (47670 \beta_{3} + 75025 \beta_1) q^{55} + ( - 38262 \beta_{2} + 2338608) q^{57} + (113435 \beta_{3} - 5757 \beta_1) q^{59} + (35469 \beta_{2} - 10039766) q^{61} + ( - 137802 \beta_{3} - 342195 \beta_1) q^{63} + (17772 \beta_{2} - 17795804) q^{65} + (64673 \beta_{3} + 34815 \beta_1) q^{67} + (16724 \beta_{2} - 20079648) q^{69} + (239142 \beta_{3} - 178643 \beta_1) q^{71} + (26234 \beta_{2} - 1480206) q^{73} + ( - 145341 \beta_{3} + 52311 \beta_1) q^{75} + ( - 60124 \beta_{2} + 13750112) q^{77} + ( - 143988 \beta_{3} - 394318 \beta_1) q^{79} + ( - 51786 \beta_{2} + 17937) q^{81} + ( - 189139 \beta_{3} + 530989 \beta_1) q^{83} + ( - 117290 \beta_{2} + 34768500) q^{85} + ( - 241030 \beta_{3} - 484837 \beta_1) q^{87} + (112810 \beta_{2} + 33161970) q^{89} + (147644 \beta_{3} + 1620024 \beta_1) q^{91} + (108208 \beta_{2} + 43704960) q^{93} + (598322 \beta_{3} + 506603 \beta_1) q^{95} + (115170 \beta_{2} + 14111218) q^{97} + (917223 \beta_{3} + 1602975 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 728 q^{5} - 18876 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 728 q^{5} - 18876 q^{9} - 12632 q^{13} - 391992 q^{17} - 946560 q^{21} - 791028 q^{25} - 705496 q^{29} + 3084864 q^{33} + 4443048 q^{37} + 2953352 q^{41} + 14937000 q^{45} + 2839044 q^{49} - 4501848 q^{53} + 9354432 q^{57} - 40159064 q^{61} - 71183216 q^{65} - 80318592 q^{69} - 5920824 q^{73} + 55000448 q^{77} + 71748 q^{81} + 139074000 q^{85} + 132647880 q^{89} + 174819840 q^{93} + 56444872 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 19x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -32\nu^{3} + 288\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -32\nu^{3} + 928\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\nu^{3} + 160\nu^{2} - 126\nu - 1520 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16\beta_{3} + 7\beta _1 + 4864 ) / 512 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{2} - 29\beta_1 ) / 128 \) 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
−3.12250 + 0.500000i
3.12250 0.500000i
3.12250 + 0.500000i
−3.12250 0.500000i
0 135.920i 0 −581.680 0 2670.24i 0 −11913.2 0
31.2 0 63.9200i 0 217.680 0 1726.24i 0 2475.24 0
31.3 0 63.9200i 0 217.680 0 1726.24i 0 2475.24 0
31.4 0 135.920i 0 −581.680 0 2670.24i 0 −11913.2 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 32.9.c.a 4
3.b odd 2 1 288.9.g.b 4
4.b odd 2 1 inner 32.9.c.a 4
8.b even 2 1 64.9.c.f 4
8.d odd 2 1 64.9.c.f 4
12.b even 2 1 288.9.g.b 4
16.e even 4 1 256.9.d.b 4
16.e even 4 1 256.9.d.h 4
16.f odd 4 1 256.9.d.b 4
16.f odd 4 1 256.9.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.9.c.a 4 1.a even 1 1 trivial
32.9.c.a 4 4.b odd 2 1 inner
64.9.c.f 4 8.b even 2 1
64.9.c.f 4 8.d odd 2 1
256.9.d.b 4 16.e even 4 1
256.9.d.b 4 16.f odd 4 1
256.9.d.h 4 16.e even 4 1
256.9.d.h 4 16.f odd 4 1
288.9.g.b 4 3.b odd 2 1
288.9.g.b 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 22560T_{3}^{2} + 75481344 \) acting on \(S_{9}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 22560 T^{2} + 75481344 \) Copy content Toggle raw display
$5$ \( (T^{2} + 364 T - 126620)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 21247232118784 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6316 T - 2102641436)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 195996 T + 7808724420)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 352748 T - 1096762268)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots + 1192405585380)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 13944688274300)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2250924 T - 583236141852)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 100169031635228)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 107748446132028)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 933201241117500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
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