Properties

Label 2288.4.a.u
Level $2288$
Weight $4$
Character orbit 2288.a
Self dual yes
Analytic conductor $134.996$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2288,4,Mod(1,2288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2288.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2288.a (trivial)

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: yes
Analytic conductor: \(134.996370093\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 64 x^{9} + 268 x^{8} + 1564 x^{7} - 4963 x^{6} - 16942 x^{5} + 37082 x^{4} + 68209 x^{3} - 90926 x^{2} - 1672 x + 16256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 143)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} - \beta_{4} q^{5} + (\beta_{3} + \beta_{2} - 4) q^{7} + (\beta_{10} - \beta_{9} + \beta_{7} + \beta_{4} - \beta_1 + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} - \beta_{4} q^{5} + (\beta_{3} + \beta_{2} - 4) q^{7} + (\beta_{10} - \beta_{9} + \beta_{7} + \beta_{4} - \beta_1 + 11) q^{9} - 11 q^{11} + 13 q^{13} + (\beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{3} + 3 \beta_{2} + \beta_1 + 13) q^{15} + (\beta_{10} - 2 \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_1 + 25) q^{17} + ( - 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - 4 \beta_1 - 12) q^{19} + (\beta_{10} + \beta_{9} - \beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + 8 \beta_{2} + \cdots - 22) q^{21}+ \cdots + ( - 11 \beta_{10} + 11 \beta_{9} - 11 \beta_{7} - 11 \beta_{4} + 11 \beta_1 - 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 6 q^{3} - 4 q^{5} - 45 q^{7} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 6 q^{3} - 4 q^{5} - 45 q^{7} + 135 q^{9} - 121 q^{11} + 143 q^{13} + 125 q^{15} + 265 q^{17} - 127 q^{19} - 287 q^{21} - 42 q^{23} + 737 q^{25} - 69 q^{27} + 435 q^{29} + 174 q^{31} + 66 q^{33} - 844 q^{35} + 187 q^{37} - 78 q^{39} + 128 q^{41} - 696 q^{43} - 1537 q^{45} + 355 q^{47} + 1758 q^{49} + 25 q^{51} - 693 q^{53} + 44 q^{55} + 99 q^{57} + 609 q^{59} + 1625 q^{61} - 1365 q^{63} - 52 q^{65} - 633 q^{67} - 2192 q^{69} + 1937 q^{71} + 404 q^{73} - 1781 q^{75} + 495 q^{77} - 1670 q^{79} + 2619 q^{81} - 785 q^{83} + 3189 q^{85} - 46 q^{87} + 1464 q^{89} - 585 q^{91} + 1826 q^{93} + 2356 q^{95} + 1184 q^{97} - 1485 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5 x^{10} - 64 x^{9} + 268 x^{8} + 1564 x^{7} - 4963 x^{6} - 16942 x^{5} + 37082 x^{4} + 68209 x^{3} - 90926 x^{2} - 1672 x + 16256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 833811 \nu^{10} - 2410905 \nu^{9} - 50444340 \nu^{8} + 75562068 \nu^{7} + 1065164652 \nu^{6} - 240341729 \nu^{5} - 7571043660 \nu^{4} + \cdots + 1290598552 ) / 601708456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 499260 \nu^{10} - 759762 \nu^{9} + 34818189 \nu^{8} + 92778516 \nu^{7} - 721641868 \nu^{6} - 2879270108 \nu^{5} + 2636047738 \nu^{4} + \cdots - 10415480070 ) / 150427114 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3965935 \nu^{10} + 6054403 \nu^{9} + 281696508 \nu^{8} - 144928752 \nu^{7} - 6964105264 \nu^{6} - 1548015215 \nu^{5} + \cdots + 35528598512 ) / 601708456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2064684 \nu^{10} + 309163 \nu^{9} - 162196584 \nu^{8} - 115884092 \nu^{7} + 4273976216 \nu^{6} + 4488971444 \nu^{5} - 41231773801 \nu^{4} + \cdots - 33457025880 ) / 300854228 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2422251 \nu^{10} - 454644 \nu^{9} - 176135856 \nu^{8} - 134762840 \nu^{7} + 4291932836 \nu^{6} + 6062756911 \nu^{5} - 35495084327 \nu^{4} + \cdots - 18432195952 ) / 300854228 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3137893 \nu^{10} - 5667591 \nu^{9} - 207388836 \nu^{8} + 122476220 \nu^{7} + 4657016800 \nu^{6} + 1910520389 \nu^{5} - 35010813736 \nu^{4} + \cdots - 2849585352 ) / 300854228 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3420771 \nu^{10} - 1064880 \nu^{9} + 245772234 \nu^{8} + 320319872 \nu^{7} - 5735216572 \nu^{6} - 11821297127 \nu^{5} + \cdots - 18644892500 ) / 300854228 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3680375 \nu^{10} - 9616457 \nu^{9} - 230235544 \nu^{8} + 292922582 \nu^{7} + 5017437234 \nu^{6} - 351639767 \nu^{5} - 37929130814 \nu^{4} + \cdots - 16035719200 ) / 300854228 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4037174 \nu^{10} - 11323297 \nu^{9} - 271053874 \nu^{8} + 453462554 \nu^{7} + 6570292662 \nu^{6} - 4646616028 \nu^{5} - 63025711533 \nu^{4} + \cdots - 49255062368 ) / 300854228 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{6} - \beta_{3} + \beta _1 + 56 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{8} - 2\beta_{7} + 2\beta_{6} - \beta_{5} - 4\beta_{4} - 5\beta_{3} + 2\beta_{2} + 23\beta _1 + 110 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{10} - 2 \beta_{9} + 30 \beta_{8} - 8 \beta_{7} + 31 \beta_{6} - 9 \beta_{5} - 16 \beta_{4} - 38 \beta_{3} + 2 \beta_{2} + 61 \beta _1 + 1334 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{9} + 139 \beta_{8} - 96 \beta_{7} + 106 \beta_{6} - 63 \beta_{5} - 198 \beta_{4} - 229 \beta_{3} + 16 \beta_{2} + 647 \beta _1 + 4608 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 26 \beta_{10} - 32 \beta_{9} + 991 \beta_{8} - 450 \beta_{7} + 1002 \beta_{6} - 473 \beta_{5} - 1014 \beta_{4} - 1379 \beta_{3} - 56 \beta_{2} + 2628 \beta _1 + 38260 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 240 \beta_{10} + 360 \beta_{9} + 5377 \beta_{8} - 3632 \beta_{7} + 4475 \beta_{6} - 2950 \beta_{5} - 8392 \beta_{4} - 8509 \beta_{3} - 1084 \beta_{2} + 20594 \beta _1 + 174702 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1704 \beta_{10} + 928 \beta_{9} + 34382 \beta_{8} - 19206 \beta_{7} + 33891 \beta_{6} - 20283 \beta_{5} - 48356 \beta_{4} - 49076 \beta_{3} - 9026 \beta_{2} + 102641 \beta _1 + 1221992 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 21314 \beta_{10} + 19202 \beta_{9} + 198160 \beta_{8} - 132218 \beta_{7} + 174518 \beta_{6} - 125184 \beta_{5} - 343348 \beta_{4} - 300474 \beta_{3} - 88784 \beta_{2} + 701943 \beta _1 + 6396116 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 158022 \beta_{10} + 103488 \beta_{9} + 1219206 \beta_{8} - 749670 \beta_{7} + 1183837 \beta_{6} - 821917 \beta_{5} - 2082982 \beta_{4} - 1732226 \beta_{3} - 598216 \beta_{2} + \cdots + 41290712 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.390260
5.65354
−4.35762
3.59635
−3.37601
−3.16556
3.58491
6.09923
0.636678
0.778412
−4.05967
0 −8.98097 0 6.95692 0 −9.63375 0 53.6578 0
1.2 0 −8.62383 0 −21.5680 0 −9.18213 0 47.3704 0
1.3 0 −7.47772 0 −10.6979 0 32.3181 0 28.9163 0
1.4 0 −4.20666 0 11.3514 0 −3.36670 0 −9.30403 0
1.5 0 −4.08546 0 1.30050 0 −14.9266 0 −10.3090 0
1.6 0 0.469632 0 20.3915 0 −2.83667 0 −26.7794 0
1.7 0 2.54183 0 −20.1048 0 29.2165 0 −20.5391 0
1.8 0 3.26338 0 18.2348 0 −29.9777 0 −16.3503 0
1.9 0 3.74669 0 −11.8947 0 −28.3874 0 −12.9623 0
1.10 0 7.13795 0 6.37060 0 23.0480 0 23.9503 0
1.11 0 10.2152 0 −4.34036 0 −31.2716 0 77.3494 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2288.4.a.u 11
4.b odd 2 1 143.4.a.d 11
12.b even 2 1 1287.4.a.m 11
44.c even 2 1 1573.4.a.f 11
52.b odd 2 1 1859.4.a.e 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.d 11 4.b odd 2 1
1287.4.a.m 11 12.b even 2 1
1573.4.a.f 11 44.c even 2 1
1859.4.a.e 11 52.b odd 2 1
2288.4.a.u 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} + 6 T_{3}^{10} - 198 T_{3}^{9} - 1129 T_{3}^{8} + 12882 T_{3}^{7} + 62128 T_{3}^{6} - 360777 T_{3}^{5} - 1149514 T_{3}^{4} + 4884358 T_{3}^{3} + 6141153 T_{3}^{2} - 26381750 T_{3} + 10592776 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2288))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} \) Copy content Toggle raw display
$3$ \( T^{11} + 6 T^{10} - 198 T^{9} + \cdots + 10592776 \) Copy content Toggle raw display
$5$ \( T^{11} + 4 T^{10} + \cdots + 58263405696 \) Copy content Toggle raw display
$7$ \( T^{11} + 45 T^{10} + \cdots - 7302898028448 \) Copy content Toggle raw display
$11$ \( (T + 11)^{11} \) Copy content Toggle raw display
$13$ \( (T - 13)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} - 265 T^{10} + \cdots - 24\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{11} + 127 T^{10} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{11} + 42 T^{10} + \cdots - 72\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{11} - 435 T^{10} + \cdots - 37\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{11} - 174 T^{10} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{11} - 187 T^{10} + \cdots - 21\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( T^{11} - 128 T^{10} + \cdots - 61\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{11} + 696 T^{10} + \cdots + 91\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{11} - 355 T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{11} + 693 T^{10} + \cdots - 58\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{11} - 609 T^{10} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{11} - 1625 T^{10} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{11} + 633 T^{10} + \cdots + 30\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{11} - 1937 T^{10} + \cdots + 34\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{11} - 404 T^{10} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{11} + 1670 T^{10} + \cdots - 58\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{11} + 785 T^{10} + \cdots - 53\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{11} - 1464 T^{10} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{11} - 1184 T^{10} + \cdots - 23\!\cdots\!28 \) Copy content Toggle raw display
show more
show less