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Properties

Label 36.17.d.d
Level $36$
Weight $17$
Character orbit 36.d
Analytic conductor $58.437$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 17 \)
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: \(58.4368357884\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 83552 x^{14} - 1250532 x^{13} + 2808691818 x^{12} + 87176344944 x^{11} + \cdots + 22\!\cdots\!21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{113}\cdot 3^{52} \)
Twist minimal: no (minimal twist has level 12)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 12) q^{2} + ( - \beta_{3} + 10 \beta_1 + 8540) q^{4} + ( - \beta_{4} - 44 \beta_1 - 22151) q^{5} + ( - \beta_{7} - 22 \beta_{3} + \cdots - 304) q^{7}+ \cdots + ( - \beta_{13} - \beta_{7} + \cdots - 914472) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 12) q^{2} + ( - \beta_{3} + 10 \beta_1 + 8540) q^{4} + ( - \beta_{4} - 44 \beta_1 - 22151) q^{5} + ( - \beta_{7} - 22 \beta_{3} + \cdots - 304) q^{7}+ \cdots + (20483472 \beta_{15} + \cdots + 20\!\cdots\!00) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 186 q^{2} + 136588 q^{4} - 354144 q^{5} - 14683680 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 186 q^{2} + 136588 q^{4} - 354144 q^{5} - 14683680 q^{8} - 49800172 q^{10} - 906419296 q^{13} + 806064072 q^{14} - 2108540816 q^{16} - 12240765600 q^{17} - 1002788712 q^{20} + 216706355928 q^{22} + 206381182512 q^{25} - 1054507182588 q^{26} - 1526063922288 q^{28} - 327679573728 q^{29} + 5158730488416 q^{32} + 9473293385948 q^{34} - 8149494749152 q^{37} - 23318999782920 q^{38} - 28671795971776 q^{40} + 25536724613472 q^{41} + 11442227373552 q^{44} + 9929654732736 q^{46} - 93287012964080 q^{49} + 133601044957998 q^{50} + 302261844234872 q^{52} + 86928436629792 q^{53} - 530930989929024 q^{56} - 189801665049916 q^{58} + 476028596468000 q^{61} + 419080420491096 q^{62} + 305944925720704 q^{64} + 12\!\cdots\!92 q^{65}+ \cdots + 33\!\cdots\!90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 83552 x^{14} - 1250532 x^{13} + 2808691818 x^{12} + 87176344944 x^{11} + \cdots + 22\!\cdots\!21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 45\!\cdots\!14 \nu^{15} + \cdots + 10\!\cdots\!44 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 86\!\cdots\!18 \nu^{15} + \cdots + 17\!\cdots\!68 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!48 \nu^{15} + \cdots - 20\!\cdots\!39 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23\!\cdots\!61 \nu^{15} + \cdots - 44\!\cdots\!66 ) / 81\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!24 \nu^{15} + \cdots + 11\!\cdots\!65 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 63\!\cdots\!26 \nu^{15} + \cdots + 22\!\cdots\!83 ) / 31\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 30\!\cdots\!15 \nu^{15} + \cdots - 27\!\cdots\!28 ) / 81\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 62\!\cdots\!77 \nu^{15} + \cdots + 12\!\cdots\!46 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!91 \nu^{15} + \cdots + 14\!\cdots\!62 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 90\!\cdots\!95 \nu^{15} + \cdots - 12\!\cdots\!34 ) / 81\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!11 \nu^{15} + \cdots - 54\!\cdots\!14 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!46 \nu^{15} + \cdots + 12\!\cdots\!23 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 25\!\cdots\!22 \nu^{15} + \cdots - 40\!\cdots\!17 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 64\!\cdots\!27 \nu^{15} + \cdots + 13\!\cdots\!27 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!46 \nu^{15} + \cdots + 36\!\cdots\!57 ) / 40\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 9 \beta_{15} - 33 \beta_{14} - 36 \beta_{13} + 6 \beta_{10} + 9 \beta_{9} + 405 \beta_{8} + \cdots + 8092756 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 7335 \beta_{15} - 543 \beta_{14} - 9828 \beta_{13} - 648 \beta_{12} + 2088 \beta_{11} + \cdots + 561241608850 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 52329 \beta_{15} - 41951 \beta_{14} - 107142 \beta_{13} + 18414 \beta_{12} + \cdots + 1323893680376 ) / 4478976 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 174571893 \beta_{15} - 24480525 \beta_{14} - 284508684 \beta_{13} - 20569464 \beta_{12} + \cdots + 92\!\cdots\!70 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18972996507 \beta_{15} - 9620438667 \beta_{14} - 39843386244 \beta_{13} + \cdots + 56\!\cdots\!92 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 167298041775 \beta_{15} - 28992296861 \beta_{14} - 321911490390 \beta_{13} + \cdots + 74\!\cdots\!70 ) / 2239488 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 511546242037509 \beta_{15} - 193931252158461 \beta_{14} + \cdots + 16\!\cdots\!16 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 94\!\cdots\!19 \beta_{15} + \cdots + 38\!\cdots\!90 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 11\!\cdots\!43 \beta_{15} + \cdots + 35\!\cdots\!08 ) / 4478976 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 23\!\cdots\!37 \beta_{15} + \cdots + 85\!\cdots\!30 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 34\!\cdots\!39 \beta_{15} + \cdots + 11\!\cdots\!20 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 11\!\cdots\!55 \beta_{15} + \cdots + 41\!\cdots\!82 ) / 1119744 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 86\!\cdots\!17 \beta_{15} + \cdots + 28\!\cdots\!04 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 14\!\cdots\!63 \beta_{15} + \cdots + 48\!\cdots\!34 ) / 53747712 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 18\!\cdots\!49 \beta_{15} + \cdots + 59\!\cdots\!36 ) / 4478976 \) 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
127.667 − 0.866025i
127.667 + 0.866025i
65.4083 − 0.866025i
65.4083 + 0.866025i
159.117 + 0.866025i
159.117 − 0.866025i
−54.6412 − 0.866025i
−54.6412 + 0.866025i
19.1198 + 0.866025i
19.1198 − 0.866025i
−118.725 − 0.866025i
−118.725 + 0.866025i
−75.3219 + 0.866025i
−75.3219 − 0.866025i
−120.624 − 0.866025i
−120.624 + 0.866025i
−252.543 − 41.9299i 0 62019.8 + 21178.2i 380576. 0 8.07663e6i −1.47746e7 − 7.94889e6i 0 −9.61118e7 − 1.59575e7i
19.2 −252.543 + 41.9299i 0 62019.8 − 21178.2i 380576. 0 − 8.07663e6i −1.47746e7 + 7.94889e6i 0 −9.61118e7 + 1.59575e7i
19.3 −197.709 − 162.626i 0 12641.5 + 64305.2i −551815. 0 − 6.23463e6i 7.95836e6 − 1.47695e7i 0 1.09099e8 + 8.97395e7i
19.4 −197.709 + 162.626i 0 12641.5 − 64305.2i −551815. 0 6.23463e6i 7.95836e6 + 1.47695e7i 0 1.09099e8 − 8.97395e7i
19.5 −196.501 − 164.084i 0 11689.0 + 64485.1i 45026.4 0 3.24149e6i 8.28406e6 − 1.45894e7i 0 −8.84773e6 − 7.38811e6i
19.6 −196.501 + 164.084i 0 11689.0 − 64485.1i 45026.4 0 − 3.24149e6i 8.28406e6 + 1.45894e7i 0 −8.84773e6 + 7.38811e6i
19.7 −11.3734 − 255.747i 0 −65277.3 + 5817.42i −86591.3 0 1.07917e7i 2.23021e6 + 1.66283e7i 0 984836. + 2.21455e7i
19.8 −11.3734 + 255.747i 0 −65277.3 − 5817.42i −86591.3 0 − 1.07917e7i 2.23021e6 − 1.66283e7i 0 984836. − 2.21455e7i
19.9 119.478 − 226.409i 0 −36986.2 − 54101.7i 481576. 0 − 515341.i −1.66681e7 + 1.91007e6i 0 5.75375e7 − 1.09033e8i
19.10 119.478 + 226.409i 0 −36986.2 + 54101.7i 481576. 0 515341.i −1.66681e7 − 1.91007e6i 0 5.75375e7 + 1.09033e8i
19.11 141.143 − 213.576i 0 −25693.4 − 60289.4i −110106. 0 − 7.35151e6i −1.65028e7 − 3.02193e6i 0 −1.55406e7 + 2.35159e7i
19.12 141.143 + 213.576i 0 −25693.4 + 60289.4i −110106. 0 7.35151e6i −1.65028e7 + 3.02193e6i 0 −1.55406e7 − 2.35159e7i
19.13 235.541 − 100.281i 0 45423.4 − 47240.7i −699204. 0 − 4.77594e6i 5.96172e6 − 1.56822e7i 0 −1.64691e8 + 7.01170e7i
19.14 235.541 + 100.281i 0 45423.4 + 47240.7i −699204. 0 4.77594e6i 5.96172e6 + 1.56822e7i 0 −1.64691e8 − 7.01170e7i
19.15 254.964 − 23.0089i 0 64477.2 − 11732.9i 363465. 0 − 2.07768e6i 1.61694e7 − 4.47501e6i 0 9.26704e7 − 8.36292e6i
19.16 254.964 + 23.0089i 0 64477.2 + 11732.9i 363465. 0 2.07768e6i 1.61694e7 + 4.47501e6i 0 9.26704e7 + 8.36292e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
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 36.17.d.d 16
3.b odd 2 1 12.17.d.a ✓ 16
4.b odd 2 1 inner 36.17.d.d 16
12.b even 2 1 12.17.d.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.17.d.a ✓ 16 3.b odd 2 1
12.17.d.a ✓ 16 12.b even 2 1
36.17.d.d 16 1.a even 1 1 trivial
36.17.d.d 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 177072 T_{5}^{7} - 646269611536 T_{5}^{6} + \cdots + 11\!\cdots\!00 \) acting on \(S_{17}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 38\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 55\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 48\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 64\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 14\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
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