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Properties

Label 32.15.d.b
Level $32$
Weight $15$
Character orbit 32.d
Analytic conductor $39.785$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,15,Mod(15,32)]
 
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, 1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 32.d (of order \(2\), degree \(1\), not 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: \(39.7852698086\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 4349 x^{10} - 33891 x^{9} + 12151288 x^{8} - 474141530 x^{7} + 82897017850 x^{6} + \cdots + 37\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{134}\cdot 3^{6}\cdot 5^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 252) q^{3} - \beta_{3} q^{5} + ( - 2 \beta_{3} - \beta_{2}) q^{7} + (\beta_{5} - 397 \beta_1 + 1099003) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 252) q^{3} - \beta_{3} q^{5} + ( - 2 \beta_{3} - \beta_{2}) q^{7} + (\beta_{5} - 397 \beta_1 + 1099003) q^{9} + (\beta_{6} + \beta_{5} + \cdots + 2354727) q^{11}+ \cdots + ( - 378675 \beta_{7} + \cdots + 18633903104948) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3024 q^{3} + 13188036 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3024 q^{3} + 13188036 q^{9} + 28256720 q^{11} + 270339544 q^{17} + 2481505872 q^{19} - 15857276820 q^{25} + 16574868000 q^{27} - 136227597840 q^{33} - 149949623040 q^{35} + 264287409880 q^{41} - 32253127344 q^{43} - 646589230644 q^{49} - 4755867895776 q^{51} - 7479401742480 q^{57} - 1223083947184 q^{59} - 8069319822720 q^{65} + 9309378171216 q^{67} + 3619334364696 q^{73} - 9079078926000 q^{75} + 56467107312444 q^{81} + 18774355695824 q^{83} + 54781416936088 q^{89} - 36699395136768 q^{91} + 73839238696536 q^{97} + 223606851712368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 4349 x^{10} - 33891 x^{9} + 12151288 x^{8} - 474141530 x^{7} + 82897017850 x^{6} + \cdots + 37\!\cdots\!50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 636923 \nu^{11} - 78759886 \nu^{10} - 245335731 \nu^{9} - 64007142854 \nu^{8} + \cdots - 82\!\cdots\!70 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 440271 \nu^{11} + 115828246 \nu^{10} - 14836129369 \nu^{9} + 180799613486 \nu^{8} + \cdots + 61\!\cdots\!50 ) / 47\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62163383 \nu^{11} - 2172450522 \nu^{10} - 52480928817 \nu^{9} + 3160065082238 \nu^{8} + \cdots - 14\!\cdots\!50 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 359475209 \nu^{11} - 26036033626 \nu^{10} + 226003651599 \nu^{9} - 23820935857154 \nu^{8} + \cdots - 33\!\cdots\!86 ) / 56\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1837257289 \nu^{11} + 245468020442 \nu^{10} + 2203497666609 \nu^{9} + \cdots + 24\!\cdots\!82 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3245992993 \nu^{11} + 350258561354 \nu^{10} + 1329712543689 \nu^{9} + \cdots + 37\!\cdots\!42 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3297699787 \nu^{11} + 279537264110 \nu^{10} - 2643509381373 \nu^{9} + \cdots + 35\!\cdots\!70 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20145900537 \nu^{11} - 727765905478 \nu^{10} - 14535566713503 \nu^{9} + \cdots - 55\!\cdots\!50 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18689079649 \nu^{11} - 786735360566 \nu^{10} - 5744307697911 \nu^{9} + \cdots - 73\!\cdots\!50 ) / 94\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 201934826927 \nu^{11} - 6117514841898 \nu^{10} - 104092673924473 \nu^{9} + \cdots - 48\!\cdots\!50 ) / 56\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 341146077347 \nu^{11} - 11116149678498 \nu^{10} + 91441123829147 \nu^{9} + \cdots - 92\!\cdots\!50 ) / 56\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 16\beta_{7} - 16\beta_{6} + 16\beta_{5} - 710\beta_{3} - 82\beta_{2} + 34448\beta _1 + 699040 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16 \beta_{11} - 16 \beta_{10} - 15 \beta_{9} + 48 \beta_{8} - 80 \beta_{7} + 208 \beta_{6} + \cdots - 6079643552 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 160 \beta_{11} + 6304 \beta_{10} + 223 \beta_{9} - 20960 \beta_{8} + 11184 \beta_{7} + \cdots + 61954782176 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18384 \beta_{11} + 86064 \beta_{10} + 3429 \beta_{9} + 1148784 \beta_{8} + 589936 \beta_{7} + \cdots - 7451449101856 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 814688 \beta_{11} - 17084000 \beta_{10} + 3467431 \beta_{9} - 88028384 \beta_{8} + \cdots + 11\!\cdots\!76 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8665680 \beta_{11} - 710765136 \beta_{10} + 190721061 \beta_{9} + 2786807536 \beta_{8} + \cdots - 23\!\cdots\!12 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7650563104 \beta_{11} + 47998155808 \beta_{10} - 6790676073 \beta_{9} + 72446399392 \beta_{8} + \cdots - 38\!\cdots\!72 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 55893450704 \beta_{11} + 353691854896 \beta_{10} - 2711399891691 \beta_{9} - 25183112712336 \beta_{8} + \cdots - 37\!\cdots\!24 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 28631101078368 \beta_{11} - 193671377529696 \beta_{10} + 37981664900391 \beta_{9} + \cdots + 11\!\cdots\!36 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 47\!\cdots\!92 \beta_{11} + \cdots - 50\!\cdots\!20 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 41\!\cdots\!72 \beta_{11} + \cdots - 24\!\cdots\!12 ) / 8388608 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
41.6607 − 39.1087i
41.6607 + 39.1087i
−37.2836 − 57.4111i
−37.2836 + 57.4111i
4.85446 + 62.4824i
4.85446 − 62.4824i
50.2622 + 24.1659i
50.2622 − 24.1659i
−64.5010 + 31.8690i
−64.5010 − 31.8690i
5.50725 − 62.3341i
5.50725 + 62.3341i
0 −3476.15 0 − 78301.9i 0 − 1.22276e6i 0 7.30063e6 0
15.2 0 −3476.15 0 78301.9i 0 1.22276e6i 0 7.30063e6 0
15.3 0 −1525.47 0 − 71626.9i 0 647418.i 0 −2.45590e6 0
15.4 0 −1525.47 0 71626.9i 0 − 647418.i 0 −2.45590e6 0
15.5 0 −563.873 0 − 134561.i 0 255103.i 0 −4.46502e6 0
15.6 0 −563.873 0 134561.i 0 − 255103.i 0 −4.46502e6 0
15.7 0 1152.97 0 − 9668.61i 0 − 1.34658e6i 0 −3.45362e6 0
15.8 0 1152.97 0 9668.61i 0 1.34658e6i 0 −3.45362e6 0
15.9 0 2044.69 0 − 54892.6i 0 432198.i 0 −602230. 0
15.10 0 2044.69 0 54892.6i 0 − 432198.i 0 −602230. 0
15.11 0 3879.84 0 − 109885.i 0 − 642807.i 0 1.02702e7 0
15.12 0 3879.84 0 109885.i 0 642807.i 0 1.02702e7 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.15.d.b 12
4.b odd 2 1 8.15.d.b ✓ 12
8.b even 2 1 8.15.d.b ✓ 12
8.d odd 2 1 inner 32.15.d.b 12
12.b even 2 1 72.15.b.b 12
24.h odd 2 1 72.15.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.15.d.b ✓ 12 4.b odd 2 1
8.15.d.b ✓ 12 8.b even 2 1
32.15.d.b 12 1.a even 1 1 trivial
32.15.d.b 12 8.d odd 2 1 inner
72.15.b.b 12 12.b even 2 1
72.15.b.b 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 1512 T_{3}^{5} - 16502844 T_{3}^{4} + 18520805952 T_{3}^{3} + 47859959296944 T_{3}^{2} + \cdots - 27\!\cdots\!00 \) acting on \(S_{15}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 12\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 13\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
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