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Properties

Label 2736.2.bm.t
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(559,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.bm (of order \(6\), degree \(2\), 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 542x^{12} + 5488x^{10} + 40451x^{8} + 151312x^{6} + 395134x^{4} + 52164x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + (\beta_{5} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{12} + \beta_{11}) q^{11} + (\beta_{6} + \beta_1) q^{13} + ( - \beta_{10} - \beta_{9} + \beta_{8}) q^{17} + \beta_{7} q^{19} - \beta_{15} q^{23} + (\beta_{6} - \beta_1) q^{25} + 2 \beta_{10} q^{29} + ( - \beta_{4} + \beta_{3}) q^{31} + (\beta_{15} - \beta_{14}) q^{35} + (2 \beta_{2} + 1) q^{37} + (2 \beta_{13} - \beta_{10} + \beta_{9} + \beta_{8}) q^{41} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{15} + \beta_{11}) q^{47} + ( - \beta_1 - 1) q^{49} + ( - 2 \beta_{13} + \beta_{10} + 2 \beta_{8}) q^{53} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{55} + (\beta_{12} - 2 \beta_{11}) q^{59} + ( - \beta_{6} - 2 \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{13} + 3 \beta_{9} - 4 \beta_{8}) q^{65} + (2 \beta_{7} + \beta_{3}) q^{67} + (\beta_{15} + \beta_{14} - \beta_{12} + 2 \beta_{11}) q^{71} + ( - 4 \beta_{6} - 3 \beta_{2}) q^{73} + ( - \beta_{13} - 4 \beta_{10} + 2 \beta_{9}) q^{77} + ( - \beta_{7} + 3 \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{79} + (\beta_{14} + \beta_{12} - \beta_{11}) q^{83} + ( - 4 \beta_{6} + 2 \beta_{2} + 4 \beta_1 + 2) q^{85} + (\beta_{13} + 2 \beta_{10} - \beta_{8}) q^{89} + (3 \beta_{7} + \beta_{5} + 2 \beta_{3}) q^{91} + (\beta_{15} - \beta_{12} - \beta_{11}) q^{95} + (3 \beta_{6} - 3 \beta_{2} - 6 \beta_1 - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{49} - 16 q^{61} + 24 q^{73} + 16 q^{85} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 542x^{12} + 5488x^{10} + 40451x^{8} + 151312x^{6} + 395134x^{4} + 52164x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10164 \nu^{14} - 255509 \nu^{12} - 4754946 \nu^{10} - 41215272 \nu^{8} - 266357336 \nu^{6} - 462486696 \nu^{4} - 61069302 \nu^{2} + \cdots + 4905120953 ) / 1856785928 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3806323948 \nu^{14} - 105837336802 \nu^{12} - 2044580895560 \nu^{10} - 20543041875661 \nu^{8} - 150969971356232 \nu^{6} + \cdots - 194108471710191 ) / 184747182558867 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 96213205709 \nu^{14} - 2754784164545 \nu^{12} - 53992628470114 \nu^{10} - 561812335230251 \nu^{8} + \cdots - 11\!\cdots\!28 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 120384307763 \nu^{14} - 3391806056156 \nu^{12} - 65300409366145 \nu^{10} - 659826753342743 \nu^{8} + \cdots - 15\!\cdots\!11 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 132220047635 \nu^{14} - 3686953004252 \nu^{12} - 70837432581553 \nu^{10} - 707820971971799 \nu^{8} + \cdots + 52\!\cdots\!45 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77189178416 \nu^{14} + 2176005166862 \nu^{12} + 42180465284398 \nu^{10} + 430495021839779 \nu^{8} + \cdots + 41\!\cdots\!01 ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 65958342944 \nu^{14} - 1804014851009 \nu^{12} - 34720999750351 \nu^{10} - 343729127623994 \nu^{8} + \cdots + 17\!\cdots\!65 ) / 985318306980624 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 990175670 \nu^{15} - 25195594535 \nu^{13} - 463226273255 \nu^{11} - 4015186891660 \nu^{9} - 25480512543977 \nu^{7} + \cdots + 831702838632193 \nu ) / 109479811886736 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2834183765 \nu^{15} - 78805865495 \nu^{13} - 1518941639794 \nu^{11} - 15250342530713 \nu^{9} - 111711718597870 \nu^{7} + \cdots - 281697264791856 \nu ) / 83923010373744 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 909972638167 \nu^{15} + 25176125585668 \nu^{13} + 486901477983041 \nu^{11} + \cdots - 41\!\cdots\!95 \nu ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1890306552809 \nu^{15} - 52424270976620 \nu^{13} + \cdots + 77\!\cdots\!89 \nu ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3394373004184 \nu^{15} - 95065139318341 \nu^{13} + \cdots - 90\!\cdots\!71 \nu ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 46212447101 \nu^{15} + 1289258286323 \nu^{13} + 24906077016940 \nu^{11} + 250804054500443 \nu^{9} + \cdots + 114034611732426 \nu ) / 286060153639536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 78658668889 \nu^{15} - 2212467122437 \nu^{13} - 42911401938458 \nu^{11} - 437082575176471 \nu^{9} + \cdots - 84\!\cdots\!24 \nu ) / 286060153639536 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2811623585401 \nu^{15} + 78035033220436 \nu^{13} + \cdots - 12\!\cdots\!83 \nu ) / 88\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{12} - 2\beta_{11} + \beta_{10} + \beta_{9} - 3\beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{3} - 7\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 22\beta_{15} - 11\beta_{14} - 45\beta_{13} + 5\beta_{12} + 5\beta_{11} - 34\beta_{10} + 17\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{7} - 8\beta_{6} - 14\beta_{4} - 14\beta_{3} + 75\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 139 \beta_{15} + 278 \beta_{14} + 633 \beta_{13} - 14 \beta_{12} + 7 \beta_{11} + 223 \beta_{10} - 446 \beta_{9} + 633 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -197\beta_{5} + 181\beta_{4} + 16\beta_{3} + 168\beta _1 + 889 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1829 \beta_{15} - 1829 \beta_{14} - 361 \beta_{12} + 722 \beta_{11} + 2819 \beta_{10} + 2819 \beta_{9} - 8955 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2324\beta_{7} + 2768\beta_{6} + 3220\beta_{5} + 2772\beta_{3} - 11169\beta_{2} - 2768\beta _1 - 11169 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 49154 \beta_{15} - 24577 \beta_{14} - 126291 \beta_{13} - 8663 \beta_{12} - 8663 \beta_{11} - 71666 \beta_{10} + 35833 \beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -30205\beta_{7} - 42000\beta_{6} - 8672\beta_{5} - 30205\beta_{4} - 47549\beta_{3} + 145327\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 334163 \beta_{15} + 668326 \beta_{14} + 1773045 \beta_{13} + 302198 \beta_{12} - 151099 \beta_{11} + 463361 \beta_{10} - 926722 \beta_{9} + 1773045 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -543466\beta_{5} + 398762\beta_{4} + 144704\beta_{3} + 613688\beta _1 + 1932699 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4575547 \beta_{15} - 4575547 \beta_{14} - 2367377 \beta_{12} + 4734754 \beta_{11} + 6099895 \beta_{10} + 6099895 \beta_{9} - 24798585 \beta_{8} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5337721 \beta_{7} + 8788248 \beta_{6} + 9818329 \beta_{5} + 7578025 \beta_{3} - 26067265 \beta_{2} - 8788248 \beta _1 - 26067265 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 125854618 \beta_{15} - 62927309 \beta_{14} - 345889539 \beta_{13} - 35270329 \beta_{12} - 35270329 \beta_{11} - 163023622 \beta_{10} + 81511811 \beta_{9} ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1 + \beta_{2}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.86197 + 3.22503i
0.181830 − 0.314939i
−1.09560 + 1.89764i
1.51646 − 2.62659i
1.09560 − 1.89764i
−1.51646 + 2.62659i
1.86197 − 3.22503i
−0.181830 + 0.314939i
0.181830 + 0.314939i
−1.86197 − 3.22503i
1.51646 + 2.62659i
−1.09560 − 1.89764i
−1.51646 − 2.62659i
1.09560 + 1.89764i
−0.181830 − 0.314939i
1.86197 + 3.22503i
0 0 0 −1.38255 + 2.39464i 0 − 3.26278i 0 0 0
559.2 0 0 0 −1.38255 + 2.39464i 0 3.26278i 0 0 0
559.3 0 0 0 −0.767178 + 1.32879i 0 − 2.31392i 0 0 0
559.4 0 0 0 −0.767178 + 1.32879i 0 2.31392i 0 0 0
559.5 0 0 0 0.767178 − 1.32879i 0 − 2.31392i 0 0 0
559.6 0 0 0 0.767178 − 1.32879i 0 2.31392i 0 0 0
559.7 0 0 0 1.38255 − 2.39464i 0 − 3.26278i 0 0 0
559.8 0 0 0 1.38255 − 2.39464i 0 3.26278i 0 0 0
1855.1 0 0 0 −1.38255 − 2.39464i 0 − 3.26278i 0 0 0
1855.2 0 0 0 −1.38255 − 2.39464i 0 3.26278i 0 0 0
1855.3 0 0 0 −0.767178 − 1.32879i 0 − 2.31392i 0 0 0
1855.4 0 0 0 −0.767178 − 1.32879i 0 2.31392i 0 0 0
1855.5 0 0 0 0.767178 + 1.32879i 0 − 2.31392i 0 0 0
1855.6 0 0 0 0.767178 + 1.32879i 0 2.31392i 0 0 0
1855.7 0 0 0 1.38255 + 2.39464i 0 − 3.26278i 0 0 0
1855.8 0 0 0 1.38255 + 2.39464i 0 3.26278i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner
76.f even 6 1 inner
228.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.t ✓ 16
3.b odd 2 1 inner 2736.2.bm.t ✓ 16
4.b odd 2 1 inner 2736.2.bm.t ✓ 16
12.b even 2 1 inner 2736.2.bm.t ✓ 16
19.d odd 6 1 inner 2736.2.bm.t ✓ 16
57.f even 6 1 inner 2736.2.bm.t ✓ 16
76.f even 6 1 inner 2736.2.bm.t ✓ 16
228.n odd 6 1 inner 2736.2.bm.t ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.bm.t ✓ 16 1.a even 1 1 trivial
2736.2.bm.t ✓ 16 3.b odd 2 1 inner
2736.2.bm.t ✓ 16 4.b odd 2 1 inner
2736.2.bm.t ✓ 16 12.b even 2 1 inner
2736.2.bm.t ✓ 16 19.d odd 6 1 inner
2736.2.bm.t ✓ 16 57.f even 6 1 inner
2736.2.bm.t ✓ 16 76.f even 6 1 inner
2736.2.bm.t ✓ 16 228.n odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{8} + 10T_{5}^{6} + 82T_{5}^{4} + 180T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{7}^{4} + 16T_{7}^{2} + 57 \) Copy content Toggle raw display
\( T_{11}^{4} + 34T_{11}^{2} + 114 \) Copy content Toggle raw display
\( T_{23}^{8} - 94T_{23}^{6} + 7810T_{23}^{4} - 96444T_{23}^{2} + 1052676 \) Copy content Toggle raw display
\( T_{31}^{4} - 40T_{31}^{2} + 57 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 10 T^{6} + 82 T^{4} + 180 T^{2} + \cdots + 324)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 57)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 34 T^{2} + 114)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 21 T^{2} + 441)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 52 T^{6} + 2056 T^{4} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 24 T^{6} + 418 T^{4} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 94 T^{6} + 7810 T^{4} + \cdots + 1052676)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 72 T^{6} + 4896 T^{4} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 40 T^{2} + 57)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3)^{8} \) Copy content Toggle raw display
$41$ \( (T^{8} - 60 T^{6} + 2952 T^{4} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 120 T^{6} + 13887 T^{4} + \cdots + 263169)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 124 T^{6} + 14920 T^{4} + \cdots + 207936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 162 T^{6} + 19746 T^{4} + \cdots + 42224004)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 102 T^{6} + 9378 T^{4} + \cdots + 1052676)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4 T^{3} + 19 T^{2} - 12 T + 9)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 136 T^{6} + 13879 T^{4} + \cdots + 21316689)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 396 T^{6} + 119880 T^{4} + \cdots + 1364268096)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + 139 T^{2} + 618 T + 10609)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 304 T^{6} + 71839 T^{4} + \cdots + 423412929)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 124 T^{2} + 456)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 78 T^{6} + 4626 T^{4} + \cdots + 2125764)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 18 T^{3} - 54 T^{2} - 2916 T + 26244)^{4} \) Copy content Toggle raw display
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