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Properties

Label 2736.3.h.f
Level $2736$
Weight $3$
Character orbit 2736.h
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
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] = [2736,3,Mod(305,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.h (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 300 x^{18} + 35598 x^{16} + 2172160 x^{14} + 75089753 x^{12} + 1521382580 x^{10} + \cdots + 42452481600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 1368)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + \beta_{3} q^{7} + (\beta_{16} - \beta_{8}) q^{11} + \beta_{7} q^{13} + ( - \beta_{15} + \beta_{14} - \beta_1) q^{17} + \beta_{2} q^{19} + (\beta_{17} - \beta_{13} + \beta_{8}) q^{23} + (\beta_{6} + \beta_{4} - \beta_{2} - 5) q^{25} + (\beta_{19} + \beta_{18} + \cdots + 2 \beta_1) q^{29}+ \cdots + (2 \beta_{10} - \beta_{9} - 3 \beta_{7} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 8 q^{7} + 8 q^{13} - 100 q^{25} - 64 q^{31} - 24 q^{37} + 88 q^{43} - 12 q^{49} - 152 q^{55} + 88 q^{61} - 104 q^{67} + 24 q^{73} + 232 q^{79} + 320 q^{85} + 184 q^{91} - 120 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 300 x^{18} + 35598 x^{16} + 2172160 x^{14} + 75089753 x^{12} + 1521382580 x^{10} + \cdots + 42452481600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 164264255322007 \nu^{18} + \cdots - 13\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!07 \nu^{18} + \cdots - 66\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!13 \nu^{18} + \cdots - 37\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2719047908353 \nu^{18} + 774377531266736 \nu^{16} + \cdots + 34\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!89 \nu^{18} + \cdots + 80\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!01 \nu^{18} + \cdots - 60\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 421480406078456 \nu^{19} + \cdots + 11\!\cdots\!20 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!37 \nu^{18} + \cdots - 45\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!87 \nu^{18} + \cdots - 52\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!87 \nu^{18} + \cdots + 30\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!73 \nu^{19} + \cdots + 26\!\cdots\!40 \nu ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 20\!\cdots\!11 \nu^{19} + \cdots - 14\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 51\!\cdots\!79 \nu^{19} + \cdots + 14\!\cdots\!68 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 37\!\cdots\!59 \nu^{19} + \cdots + 32\!\cdots\!40 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!77 \nu^{19} + \cdots + 15\!\cdots\!80 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 79\!\cdots\!23 \nu^{19} + \cdots + 68\!\cdots\!88 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 47\!\cdots\!21 \nu^{19} + \cdots - 12\!\cdots\!20 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 65\!\cdots\!73 \nu^{19} + \cdots + 32\!\cdots\!80 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{2} - 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} - 2 \beta_{18} + 2 \beta_{17} - 6 \beta_{16} - 18 \beta_{15} + 5 \beta_{14} + \cdots - 64 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{11} - 15 \beta_{10} - \beta_{9} - 39 \beta_{7} - 61 \beta_{6} + 23 \beta_{5} - 105 \beta_{4} + \cdots + 1899 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{19} + 150 \beta_{18} - 176 \beta_{17} + 622 \beta_{16} + 2003 \beta_{15} + \cdots + 5047 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1835 \beta_{11} + 2157 \beta_{10} + 103 \beta_{9} + 4658 \beta_{7} + 4280 \beta_{6} - 2858 \beta_{5} + \cdots - 149887 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6163 \beta_{19} - 11236 \beta_{18} + 15306 \beta_{17} - 56372 \beta_{16} - 189286 \beta_{15} + \cdots - 427810 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 184990 \beta_{11} - 223493 \beta_{10} - 10331 \beta_{9} - 454629 \beta_{7} - 336951 \beta_{6} + \cdots + 12734341 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 739216 \beta_{19} + 906474 \beta_{18} - 1371280 \beta_{17} + 4993026 \beta_{16} + \cdots + 37241409 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17245241 \beta_{11} + 20889659 \beta_{10} + 979889 \beta_{9} + 41870738 \beta_{7} + 28210072 \beta_{6} + \cdots - 1109738393 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 71628237 \beta_{19} - 76887116 \beta_{18} + 124672502 \beta_{17} - 440282876 \beta_{16} + \cdots - 3274906290 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1563183822 \beta_{11} - 1879775111 \beta_{10} - 90149769 \beta_{9} - 3775797259 \beta_{7} + \cdots + 97606716639 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6475478668 \beta_{19} + 6683860598 \beta_{18} - 11391181032 \beta_{17} + 38799059534 \beta_{16} + \cdots + 289115756927 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 140189626151 \beta_{11} + 166682697377 \beta_{10} + 8196120003 \beta_{9} + 337686494850 \beta_{7} + \cdots - 8615088469411 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 570242448075 \beta_{19} - 587240725692 \beta_{18} + 1041821333162 \beta_{17} + \cdots - 25562867693298 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 12520311620934 \beta_{11} - 14694463033037 \beta_{10} - 742232198323 \beta_{9} + \cdots + 761433752228653 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 49687748070104 \beta_{19} + 51819756164202 \beta_{18} - 95244377270176 \beta_{17} + \cdots + 22\!\cdots\!41 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 11\!\cdots\!77 \beta_{11} + \cdots - 67\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 43\!\cdots\!53 \beta_{19} + \cdots - 20\!\cdots\!18 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
− 9.42398i
− 9.38255i
− 6.03686i
− 5.73632i
− 4.17640i
− 3.99651i
− 3.50777i
− 2.56614i
− 1.16424i
− 0.384695i
0.384695i
1.16424i
2.56614i
3.50777i
3.99651i
4.17640i
5.73632i
6.03686i
9.38255i
9.42398i
0 0 0 − 9.42398i 0 −2.65729 0 0 0
305.2 0 0 0 − 9.38255i 0 3.30429 0 0 0
305.3 0 0 0 − 6.03686i 0 3.40078 0 0 0
305.4 0 0 0 − 5.73632i 0 −9.16014 0 0 0
305.5 0 0 0 − 4.17640i 0 0.125388 0 0 0
305.6 0 0 0 − 3.99651i 0 5.26928 0 0 0
305.7 0 0 0 − 3.50777i 0 8.42276 0 0 0
305.8 0 0 0 − 2.56614i 0 −9.43527 0 0 0
305.9 0 0 0 − 1.16424i 0 11.6282 0 0 0
305.10 0 0 0 − 0.384695i 0 −6.89804 0 0 0
305.11 0 0 0 0.384695i 0 −6.89804 0 0 0
305.12 0 0 0 1.16424i 0 11.6282 0 0 0
305.13 0 0 0 2.56614i 0 −9.43527 0 0 0
305.14 0 0 0 3.50777i 0 8.42276 0 0 0
305.15 0 0 0 3.99651i 0 5.26928 0 0 0
305.16 0 0 0 4.17640i 0 0.125388 0 0 0
305.17 0 0 0 5.73632i 0 −9.16014 0 0 0
305.18 0 0 0 6.03686i 0 3.40078 0 0 0
305.19 0 0 0 9.38255i 0 3.30429 0 0 0
305.20 0 0 0 9.42398i 0 −2.65729 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.h.f 20
3.b odd 2 1 inner 2736.3.h.f 20
4.b odd 2 1 1368.3.h.b ✓ 20
12.b even 2 1 1368.3.h.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.3.h.b ✓ 20 4.b odd 2 1
1368.3.h.b ✓ 20 12.b even 2 1
2736.3.h.f 20 1.a even 1 1 trivial
2736.3.h.f 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 300 T_{5}^{18} + 35598 T_{5}^{16} + 2172160 T_{5}^{14} + 75089753 T_{5}^{12} + \cdots + 42452481600 \) acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 42452481600 \) Copy content Toggle raw display
$7$ \( (T^{10} - 4 T^{9} + \cdots + 1152000)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 116144936027136 \) Copy content Toggle raw display
$13$ \( (T^{10} - 4 T^{9} + \cdots - 837642240)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{10} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 91\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 8965282590720)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 43379873792000)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 112084025816960)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 99\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
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