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Properties

Label 36.25.d.b
Level $36$
Weight $25$
Character orbit 36.d
Analytic conductor $131.388$
Analytic rank $0$
Dimension $10$
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] = [36,25,Mod(19,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 36.d (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: \(131.388174813\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} + 1157886 x^{8} - 1182314620 x^{7} + 1715110302918 x^{6} + \cdots + 10\!\cdots\!86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{15}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 4)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 621) q^{2} + (\beta_{2} + 762 \beta_1 - 3319485) q^{4} + ( - \beta_{5} + \beta_{2} + \cdots - 5675511) q^{5}+ \cdots + (2 \beta_{8} - 2 \beta_{7} + \cdots + 15982548531) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 621) q^{2} + (\beta_{2} + 762 \beta_1 - 3319485) q^{4} + ( - \beta_{5} + \beta_{2} + \cdots - 5675511) q^{5}+ \cdots + ( - 4809933085056 \beta_{9} + \cdots - 25\!\cdots\!51) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6212 q^{2} - 33193328 q^{4} - 56758100 q^{5} + 159817477952 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 6212 q^{2} - 33193328 q^{4} - 56758100 q^{5} + 159817477952 q^{8} - 279847745800 q^{10} - 24798065342764 q^{13} + 3829480368768 q^{14} - 588864378801920 q^{16} + 16\!\cdots\!64 q^{17}+ \cdots - 25\!\cdots\!08 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} + 1157886 x^{8} - 1182314620 x^{7} + 1715110302918 x^{6} + \cdots + 10\!\cdots\!86 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 16\nu^{2} + 1912\nu + 3704647 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27976231 \nu^{9} + 40880542738 \nu^{8} - 63474252902296 \nu^{7} + \cdots + 81\!\cdots\!74 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} - 158 \nu^{8} + 1182376 \nu^{7} - 1365582900 \nu^{6} + 1926775652418 \nu^{5} + \cdots - 12\!\cdots\!02 ) / 46\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80345 \nu^{9} - 11160594 \nu^{8} - 1632006248 \nu^{7} - 241321145318380 \nu^{6} + \cdots - 19\!\cdots\!50 ) / 46\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 472925443 \nu^{9} + 515151849510 \nu^{8} + 268803015010296 \nu^{7} + \cdots + 14\!\cdots\!06 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2038482333 \nu^{9} - 2559062968602 \nu^{8} + \cdots - 46\!\cdots\!30 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1168599913 \nu^{9} + 1125545653554 \nu^{8} + \cdots + 18\!\cdots\!22 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7270559531 \nu^{9} + 8051467852938 \nu^{8} + \cdots - 48\!\cdots\!22 ) / 26\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 478\beta _1 - 3705125 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{8} - 2 \beta_{7} - 25 \beta_{6} + 17 \beta_{5} + 121 \beta_{4} + 109 \beta_{3} + \cdots + 22634599831 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 224 \beta_{9} + 2746 \beta_{8} - 618 \beta_{7} + 2083 \beta_{6} + 104773 \beta_{5} + \cdots - 26656795122557 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 195608 \beta_{9} + 73142 \beta_{8} + 306734 \beta_{7} + 1802915 \beta_{6} + 3719529 \beta_{5} + \cdots + 35\!\cdots\!99 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 594896736 \beta_{9} + 183048746 \beta_{8} - 124110874 \beta_{7} - 9538317621 \beta_{6} + \cdots + 12\!\cdots\!45 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 126913999744 \beta_{9} - 1812138951098 \beta_{8} - 1719205617542 \beta_{7} + 11591738277045 \beta_{6} + \cdots - 58\!\cdots\!79 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 445643850469664 \beta_{9} - 279859097184986 \beta_{8} - 854252010678902 \beta_{7} + \cdots + 49\!\cdots\!61 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 29\!\cdots\!80 \beta_{9} + \cdots + 10\!\cdots\!67 ) / 32 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−734.642 − 844.151i
−734.642 + 844.151i
−697.391 − 868.555i
−697.391 + 868.555i
112.122 − 988.545i
112.122 + 988.545i
472.934 − 808.873i
472.934 + 808.873i
848.477 − 203.983i
848.477 + 203.983i
−2318.57 − 3376.61i 0 −6.02571e6 + 1.56578e7i −1.24317e8 0 − 3.87327e9i 6.68411e10 − 1.59571e10i 0 2.88236e11 + 4.19768e11i
19.2 −2318.57 + 3376.61i 0 −6.02571e6 − 1.56578e7i −1.24317e8 0 3.87327e9i 6.68411e10 + 1.59571e10i 0 2.88236e11 − 4.19768e11i
19.3 −2169.56 − 3474.22i 0 −7.36320e6 + 1.50751e7i 2.26723e8 0 − 6.45252e9i 6.83491e10 − 7.12502e9i 0 −4.91891e11 − 7.87687e11i
19.4 −2169.56 + 3474.22i 0 −7.36320e6 − 1.50751e7i 2.26723e8 0 6.45252e9i 6.83491e10 + 7.12502e9i 0 −4.91891e11 + 7.87687e11i
19.5 1068.49 − 3954.18i 0 −1.44939e7 − 8.44998e6i −3.74285e8 0 2.47654e10i −4.88993e10 + 4.82828e10i 0 −3.99918e11 + 1.47999e12i
19.6 1068.49 + 3954.18i 0 −1.44939e7 + 8.44998e6i −3.74285e8 0 − 2.47654e10i −4.88993e10 − 4.82828e10i 0 −3.99918e11 − 1.47999e12i
19.7 2511.74 − 3235.49i 0 −4.15959e6 − 1.62534e7i 3.41994e8 0 − 1.51829e10i −6.30355e10 − 2.73659e10i 0 8.58998e11 − 1.10652e12i
19.8 2511.74 + 3235.49i 0 −4.15959e6 + 1.62534e7i 3.41994e8 0 1.51829e10i −6.30355e10 + 2.73659e10i 0 8.58998e11 + 1.10652e12i
19.9 4013.91 − 815.932i 0 1.54457e7 − 6.55016e6i −9.84947e7 0 − 1.39617e10i 5.66533e10 − 3.88944e10i 0 −3.95349e11 + 8.03650e10i
19.10 4013.91 + 815.932i 0 1.54457e7 + 6.55016e6i −9.84947e7 0 1.39617e10i 5.66533e10 + 3.88944e10i 0 −3.95349e11 − 8.03650e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
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 36.25.d.b 10
3.b odd 2 1 4.25.b.b ✓ 10
4.b odd 2 1 inner 36.25.d.b 10
12.b even 2 1 4.25.b.b ✓ 10
24.f even 2 1 64.25.c.d 10
24.h odd 2 1 64.25.c.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.25.b.b ✓ 10 3.b odd 2 1
4.25.b.b ✓ 10 12.b even 2 1
36.25.d.b 10 1.a even 1 1 trivial
36.25.d.b 10 4.b odd 2 1 inner
64.25.c.d 10 24.f even 2 1
64.25.c.d 10 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 28379050 T_{5}^{4} + \cdots + 35\!\cdots\!00 \) acting on \(S_{25}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 91\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
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