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Properties

Label 1088.2.c.e
Level $1088$
Weight $2$
Character orbit 1088.c
Analytic conductor $8.688$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1088,2,Mod(545,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1088.545");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.c (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: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{5} q^{5} + \beta_{3} q^{7} + ( - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{5} q^{5} + \beta_{3} q^{7} + ( - \beta_{2} + \beta_1 - 1) q^{9} + (\beta_{8} + \beta_{6}) q^{11} + ( - \beta_{11} - \beta_{5}) q^{13} + ( - \beta_{7} - \beta_{4} + \beta_{3}) q^{15} + q^{17} + ( - \beta_{9} + \beta_{6}) q^{19} + ( - \beta_{10} - \beta_{5}) q^{21} + ( - \beta_{7} + 2 \beta_{3}) q^{23} + \beta_1 q^{25} + (\beta_{9} + 2 \beta_{8} + \beta_{6}) q^{27} + 3 \beta_{5} q^{29} + ( - \beta_{7} - \beta_{4}) q^{31} + (\beta_{2} - 2 \beta_1 + 5) q^{33} + (\beta_{9} + \beta_{6}) q^{35} + (\beta_{11} - \beta_{10} + \beta_{5}) q^{37} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{39} + (4 \beta_{2} - \beta_1 + 3) q^{41} + ( - \beta_{9} + \beta_{8} + \beta_{6}) q^{43} + (\beta_{11} - 3 \beta_{10} - 3 \beta_{5}) q^{45} + ( - \beta_{7} + 3 \beta_{3}) q^{47} + ( - \beta_{2} - 2) q^{49} - \beta_{6} q^{51} + ( - \beta_{11} - \beta_{10}) q^{53} + (\beta_{7} + 2 \beta_{4} - \beta_{3}) q^{55} + (\beta_1 + 3) q^{57} + (2 \beta_{8} - 4 \beta_{6}) q^{59} + 3 \beta_{5} q^{61} + (\beta_{7} + 2 \beta_{4}) q^{63} + ( - 2 \beta_{2} - \beta_1 + 3) q^{65} + ( - 3 \beta_{8} - 2 \beta_{6}) q^{67} + (\beta_{11} - 2 \beta_{10} - 3 \beta_{5}) q^{69} + ( - 2 \beta_{7} + \beta_{4} - \beta_{3}) q^{71} - 2 \beta_{2} q^{73} + (\beta_{9} + 3 \beta_{8} + 2 \beta_{6}) q^{75} + ( - \beta_{11} + 2 \beta_{10} + \beta_{5}) q^{77} + ( - \beta_{7} - 4 \beta_{3}) q^{79} + ( - \beta_{2} - 2 \beta_1 + 4) q^{81} + (\beta_{9} - \beta_{8} + 3 \beta_{6}) q^{83} + \beta_{5} q^{85} + ( - 3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3}) q^{87} + ( - 5 \beta_{2} + 2 \beta_1 - 3) q^{89} + ( - \beta_{9} + 4 \beta_{8} - 3 \beta_{6}) q^{91} + (\beta_{11} - 2 \beta_{10} - 5 \beta_{5}) q^{93} + (\beta_{4} + 4 \beta_{3}) q^{95} + (4 \beta_{2} + \beta_1 - 3) q^{97} + ( - 2 \beta_{9} - 2 \beta_{8} - 7 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} + 12 q^{17} - 4 q^{25} + 64 q^{33} + 24 q^{41} - 20 q^{49} + 32 q^{57} + 48 q^{65} + 8 q^{73} + 60 q^{81} - 24 q^{89} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 473 \nu^{11} + 516 \nu^{10} - 1118 \nu^{9} + 3014 \nu^{8} + 4564 \nu^{7} + 5676 \nu^{6} + \cdots + 194196 ) / 51972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 748 \nu^{11} + 816 \nu^{10} - 1768 \nu^{9} + 4162 \nu^{8} + 6311 \nu^{7} + 8976 \nu^{6} + \cdots + 98004 ) / 51972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3325 \nu^{11} + 3434 \nu^{10} - 5902 \nu^{9} + 4812 \nu^{8} - 57662 \nu^{7} - 126520 \nu^{6} + \cdots + 354888 ) / 103944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1549 \nu^{11} + 173 \nu^{10} + 134 \nu^{9} - 30 \nu^{8} + 24423 \nu^{7} + 35912 \nu^{6} + \cdots - 112248 ) / 25986 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3941 \nu^{11} - 3060 \nu^{10} + 2086 \nu^{9} - 1372 \nu^{8} - 61953 \nu^{7} - 47292 \nu^{6} + \cdots + 146880 ) / 51972 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10271 \nu^{11} + 6480 \nu^{10} - 5378 \nu^{9} + 4008 \nu^{8} + 162468 \nu^{7} + 140576 \nu^{6} + \cdots - 643320 ) / 103944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12981 \nu^{11} + 4570 \nu^{10} - 10730 \nu^{9} + 8220 \nu^{8} - 212726 \nu^{7} - 380984 \nu^{6} + \cdots + 1131912 ) / 103944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + \cdots + 105120 ) / 12993 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21257 \nu^{11} - 13740 \nu^{10} + 12446 \nu^{9} - 7536 \nu^{8} - 345936 \nu^{7} - 272408 \nu^{6} + \cdots + 1349640 ) / 103944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} + \cdots - 4320 ) / 426 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29059 \nu^{11} + 21864 \nu^{10} - 11304 \nu^{9} + 4470 \nu^{8} + 462430 \nu^{7} + 348708 \nu^{6} + \cdots - 1049472 ) / 51972 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} - 3\beta_{6} - 2\beta_{5} + \beta_{4} + 2\beta_{2} - 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} + 2\beta_{7} - 2\beta_{6} + \beta_{4} - 4\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} + \beta_{7} + 4\beta_{5} + 2\beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + 6\beta_{10} + 7\beta_{5} + \beta_{2} - 3\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} + 7 \beta_{10} - 8 \beta_{9} - 10 \beta_{8} - 6 \beta_{7} - 30 \beta_{6} + 18 \beta_{5} + \cdots + 39 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{9} - 15\beta_{8} - 28\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8 \beta_{11} + 40 \beta_{10} - 35 \beta_{9} - 61 \beta_{8} + 32 \beta_{7} - 153 \beta_{6} + \cdots - 216 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -35\beta_{11} + 129\beta_{10} + 214\beta_{5} - 48\beta_{2} + 83\beta _1 - 223 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -24\beta_{11} + 107\beta_{10} - 83\beta_{7} + 212\beta_{5} - 106\beta_{4} + 131\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -131\beta_{9} - 341\beta_{8} - 472\beta_{7} - 721\beta_{6} - 557\beta_{4} + 800\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 131 \beta_{11} - 557 \beta_{10} - 400 \beta_{9} - 878 \beta_{8} - 426 \beta_{7} - 1986 \beta_{6} + \cdots - 2949 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
2.17840 − 0.583700i
0.583700 − 2.17840i
−0.673288 − 0.180407i
−0.180407 − 0.673288i
−0.403293 + 1.50511i
−1.50511 + 0.403293i
−1.50511 − 0.403293i
−0.403293 − 1.50511i
−0.180407 + 0.673288i
−0.673288 + 0.180407i
0.583700 + 2.17840i
2.17840 + 0.583700i
0 − 3.08613i 0 − 3.18940i 0 −2.15594 0 −6.52420 0
545.2 0 − 3.08613i 0 3.18940i 0 2.15594 0 −6.52420 0
545.3 0 − 1.51414i 0 − 0.985762i 0 1.63680 0 0.707389 0
545.4 0 − 1.51414i 0 0.985762i 0 −1.63680 0 0.707389 0
545.5 0 − 0.428007i 0 − 2.20364i 0 2.94497 0 2.81681 0
545.6 0 − 0.428007i 0 2.20364i 0 −2.94497 0 2.81681 0
545.7 0 0.428007i 0 − 2.20364i 0 −2.94497 0 2.81681 0
545.8 0 0.428007i 0 2.20364i 0 2.94497 0 2.81681 0
545.9 0 1.51414i 0 − 0.985762i 0 −1.63680 0 0.707389 0
545.10 0 1.51414i 0 0.985762i 0 1.63680 0 0.707389 0
545.11 0 3.08613i 0 − 3.18940i 0 2.15594 0 −6.52420 0
545.12 0 3.08613i 0 3.18940i 0 −2.15594 0 −6.52420 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.c.e ✓ 12
4.b odd 2 1 inner 1088.2.c.e ✓ 12
8.b even 2 1 inner 1088.2.c.e ✓ 12
8.d odd 2 1 inner 1088.2.c.e ✓ 12
16.e even 4 1 4352.2.a.x 6
16.e even 4 1 4352.2.a.z 6
16.f odd 4 1 4352.2.a.x 6
16.f odd 4 1 4352.2.a.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1088.2.c.e ✓ 12 1.a even 1 1 trivial
1088.2.c.e ✓ 12 4.b odd 2 1 inner
1088.2.c.e ✓ 12 8.b even 2 1 inner
1088.2.c.e ✓ 12 8.d odd 2 1 inner
4352.2.a.x 6 16.e even 4 1
4352.2.a.x 6 16.f odd 4 1
4352.2.a.z 6 16.e even 4 1
4352.2.a.z 6 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1088, [\chi])\):

\( T_{3}^{6} + 12T_{3}^{4} + 24T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 16T_{7}^{4} + 76T_{7}^{2} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 12 T^{4} + 24 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 16 T^{4} + \cdots + 48)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 16 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 32 T^{4} + \cdots + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 60 T^{4} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$17$ \( (T - 1)^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 76 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 76 T^{4} + \cdots - 4332)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 144 T^{4} + \cdots + 34992)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 84 T^{4} + \cdots - 8748)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 72 T^{4} + \cdots + 432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} + \cdots + 936)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 96 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 148 T^{4} + \cdots - 192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 112 T^{4} + \cdots + 768)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 176 T^{4} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 144 T^{4} + \cdots + 34992)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 204 T^{4} + \cdots + 61504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 232 T^{4} + \cdots - 41772)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 2 T^{2} - 36 T - 24)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 316 T^{4} + \cdots - 920748)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 192 T^{4} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 6 T^{2} + \cdots - 1548)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} + \cdots - 1224)^{4} \) Copy content Toggle raw display
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