Properties

Label 36.23.d.c
Level $36$
Weight $23$
Character orbit 36.d
Analytic conductor $110.415$
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] = [36,23,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, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 23 \)
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: \(110.414676543\)
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: \( x^{10} - 5 x^{9} - 63342 x^{8} - 45742928 x^{7} + 34835133568 x^{6} + 12622768560288 x^{5} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{16} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 154) q^{2} + (\beta_{2} - 147 \beta_1 + 226446) q^{4} + ( - \beta_{5} - \beta_{2} + 3886 \beta_1 + 1709110) q^{5} + ( - \beta_{8} - \beta_{7} - 3 \beta_{4} + 7 \beta_{3} - 10 \beta_{2} - 141318 \beta_1 + 3) q^{7} + ( - \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 37 \beta_{5} + \cdots - 980443086) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 154) q^{2} + (\beta_{2} - 147 \beta_1 + 226446) q^{4} + ( - \beta_{5} - \beta_{2} + 3886 \beta_1 + 1709110) q^{5} + ( - \beta_{8} - \beta_{7} - 3 \beta_{4} + 7 \beta_{3} - 10 \beta_{2} - 141318 \beta_1 + 3) q^{7} + ( - \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 37 \beta_{5} + \cdots - 980443086) q^{8}+ \cdots + ( - 706431641440 \beta_{9} + \cdots + 23\!\cdots\!14) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 1540 q^{2} + 2264464 q^{4} + 17091100 q^{5} - 9804431680 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 1540 q^{2} + 2264464 q^{4} + 17091100 q^{5} - 9804431680 q^{8} + 159414035240 q^{10} - 531230356540 q^{13} + 5894008940736 q^{14} - 27717620084480 q^{16} - 14058178115540 q^{17} - 233643631625120 q^{20} + 120589650366240 q^{22} + 77\!\cdots\!70 q^{25}+ \cdots + 23\!\cdots\!20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} - 63342 x^{8} - 45742928 x^{7} + 34835133568 x^{6} + 12622768560288 x^{5} + \cdots + 11\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 43 \nu^{8} - 61708 \nu^{7} - 43398024 \nu^{6} + 36484258480 \nu^{5} + 11236366738048 \nu^{4} + \cdots - 16\!\cdots\!72 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1393 \nu^{9} - 988677 \nu^{8} + 170893900 \nu^{7} + 121931458312 \nu^{6} - 7652610062256 \nu^{5} + \cdots + 23\!\cdots\!48 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6096753 \nu^{9} + 489570299 \nu^{8} + 2146652109388 \nu^{7} + \cdots + 27\!\cdots\!68 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12086699 \nu^{9} + 1020160953 \nu^{8} + 4283018714884 \nu^{7} + \cdots + 55\!\cdots\!12 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3804831 \nu^{9} + 2031061835 \nu^{8} + 549515927948 \nu^{7} - 126807295357304 \nu^{6} + \cdots - 53\!\cdots\!60 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 614770089 \nu^{9} - 430529850525 \nu^{8} - 166860558967572 \nu^{7} + \cdots + 36\!\cdots\!36 ) / 34\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 242467523 \nu^{9} + 518055938367 \nu^{8} + 351133130116060 \nu^{7} + \cdots + 16\!\cdots\!60 ) / 34\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1603734067 \nu^{9} - 407218243951 \nu^{8} + 134260940242276 \nu^{7} + \cdots + 45\!\cdots\!92 ) / 34\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 512356365 \nu^{9} + 710043676113 \nu^{8} + 427558858031716 \nu^{7} + \cdots + 15\!\cdots\!36 ) / 85\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} + 3\beta_{2} - 3187\beta _1 + 131071 ) / 262144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16\beta_{6} + 112\beta_{5} - 27\beta_{4} + 262\beta_{3} - 2993\beta_{2} - 3471071\beta _1 + 3321601643 ) / 262144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1024 \beta_{9} - 6144 \beta_{8} + 15360 \beta_{7} - 2064 \beta_{6} - 140400 \beta_{5} + 24631 \beta_{4} - 261438 \beta_{3} - 1392635 \beta_{2} + 3280817739 \beta _1 + 3622281111929 ) / 262144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 173056 \beta_{9} + 9709568 \beta_{8} - 287744 \beta_{7} + 143824 \beta_{6} + 86675632 \beta_{5} + 9699529 \beta_{4} - 543518210 \beta_{3} + \cdots - 34\!\cdots\!65 ) / 262144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 662658048 \beta_{9} - 2819631104 \beta_{8} - 957193216 \beta_{7} - 50467920 \beta_{6} + 14157888976 \beta_{5} - 3574012801 \beta_{4} + \cdots - 12\!\cdots\!27 ) / 262144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 588186385408 \beta_{9} - 650997143552 \beta_{8} + 116501527552 \beta_{7} + 39909369808 \beta_{6} - 31944420709712 \beta_{5} + \cdots - 22\!\cdots\!37 ) / 262144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 213620913249280 \beta_{9} + 955599907215360 \beta_{8} + 187983575589888 \beta_{7} - 43366557800016 \beta_{6} + \cdots - 34\!\cdots\!59 ) / 262144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 65\!\cdots\!48 \beta_{9} + \cdots - 15\!\cdots\!25 ) / 262144 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 51\!\cdots\!60 \beta_{9} + \cdots + 34\!\cdots\!97 ) / 262144 \) Copy content Toggle raw display

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
408.476 250.605i
408.476 + 250.605i
407.912 251.607i
407.912 + 251.607i
1.30158 510.489i
1.30158 + 510.489i
−313.209 431.746i
−313.209 + 431.746i
−501.982 216.483i
−501.982 + 216.483i
−1785.90 1002.42i 0 2.18461e6 + 3.58046e6i 6.52379e7 0 3.27904e9i −3.12381e8 8.58425e9i 0 −1.16509e11 6.53958e10i
19.2 −1785.90 + 1002.42i 0 2.18461e6 3.58046e6i 6.52379e7 0 3.27904e9i −3.12381e8 + 8.58425e9i 0 −1.16509e11 + 6.53958e10i
19.3 −1783.65 1006.43i 0 2.16851e6 + 3.59023e6i −8.57934e7 0 2.80689e8i −2.54549e8 8.58616e9i 0 1.53025e11 + 8.63449e10i
19.4 −1783.65 + 1006.43i 0 2.16851e6 3.59023e6i −8.57934e7 0 2.80689e8i −2.54549e8 + 8.58616e9i 0 1.53025e11 8.63449e10i
19.5 −157.206 2041.96i 0 −4.14488e6 + 642017.i −1.73209e7 0 2.02930e9i 1.96257e9 + 8.36273e9i 0 2.72295e9 + 3.53685e10i
19.6 −157.206 + 2041.96i 0 −4.14488e6 642017.i −1.73209e7 0 2.02930e9i 1.96257e9 8.36273e9i 0 2.72295e9 3.53685e10i
19.7 1100.83 1726.98i 0 −1.77063e6 3.80224e6i 6.05072e7 0 1.97415e9i −8.51558e9 1.12779e9i 0 6.66084e10 1.04495e11i
19.8 1100.83 + 1726.98i 0 −1.77063e6 + 3.80224e6i 6.05072e7 0 1.97415e9i −8.51558e9 + 1.12779e9i 0 6.66084e10 + 1.04495e11i
19.9 1855.93 865.934i 0 2.69462e6 3.21422e6i −1.40852e7 0 7.81741e8i 2.21772e9 8.29872e9i 0 −2.61411e10 + 1.21969e10i
19.10 1855.93 + 865.934i 0 2.69462e6 + 3.21422e6i −1.40852e7 0 7.81741e8i 2.21772e9 + 8.29872e9i 0 −2.61411e10 1.21969e10i
\(n\): 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 36.23.d.c 10
3.b odd 2 1 4.23.b.a 10
4.b odd 2 1 inner 36.23.d.c 10
12.b even 2 1 4.23.b.a 10
24.f even 2 1 64.23.c.e 10
24.h odd 2 1 64.23.c.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.23.b.a 10 3.b odd 2 1
4.23.b.a 10 12.b even 2 1
36.23.d.c 10 1.a even 1 1 trivial
36.23.d.c 10 4.b odd 2 1 inner
64.23.c.e 10 24.f even 2 1
64.23.c.e 10 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 8545550 T_{5}^{4} + \cdots + 82\!\cdots\!00 \) acting on \(S_{23}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 1540 T^{9} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - 8545550 T^{4} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + 265615178270 T^{4} + \cdots + 41\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 7029089057770 T^{4} + \cdots - 27\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 79\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 22\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 80\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 46\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 16\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 22\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
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