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Properties

Label 8820.2.d.a
Level $8820$
Weight $2$
Character orbit 8820.d
Analytic conductor $70.428$
Analytic rank $0$
Dimension $12$
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] = [8820,2,Mod(881,8820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8820, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("8820.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.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: \(70.4280545828\)
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: \( x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1260)
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.

\(f(q)\) \(=\) \( q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + ( - \beta_{7} + \beta_1) q^{11} - \beta_{8} q^{13} + \beta_{5} q^{17} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{19}+ \cdots - \beta_{9} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} + 12 q^{25} + 4 q^{37} + 8 q^{41} + 36 q^{43} - 32 q^{47} - 4 q^{67} - 28 q^{79} - 40 q^{83} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + \cdots + 117649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} + 9 \nu^{9} - 58 \nu^{8} + 78 \nu^{7} + 298 \nu^{6} - 1341 \nu^{5} + \cdots + 33614 ) / 16807 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10 \nu^{11} - 113 \nu^{10} + 748 \nu^{9} + 519 \nu^{8} - 9433 \nu^{7} + 25114 \nu^{6} + \cdots + 4386627 ) / 84035 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 78 \nu^{11} - 16 \nu^{10} - 1423 \nu^{9} + 2088 \nu^{8} + 5319 \nu^{7} - 31616 \nu^{6} + \cdots - 4437048 ) / 420175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 103 \nu^{11} + 66 \nu^{10} + 1648 \nu^{9} - 3538 \nu^{8} - 3369 \nu^{7} + 39066 \nu^{6} + \cdots + 4857223 ) / 420175 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24 \nu^{11} - 111 \nu^{10} - 41 \nu^{9} + 1518 \nu^{8} - 5281 \nu^{7} + 3691 \nu^{6} + \cdots + 1159683 ) / 84035 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 326 \nu^{11} - 1177 \nu^{10} - 561 \nu^{9} + 18586 \nu^{8} - 60582 \nu^{7} + 37343 \nu^{6} + \cdots + 13815354 ) / 420175 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2024 \nu^{11} - 9816 \nu^{10} + 6648 \nu^{9} + 104918 \nu^{8} - 431264 \nu^{7} + 512760 \nu^{6} + \cdots + 87816575 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 538 \nu^{11} + 2910 \nu^{10} - 2256 \nu^{9} - 29188 \nu^{8} + 124669 \nu^{7} - 156972 \nu^{6} + \cdots - 24168466 ) / 252105 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5459 \nu^{11} + 24666 \nu^{10} - 17418 \nu^{9} - 262778 \nu^{8} + 1126019 \nu^{7} + \cdots - 253953770 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5624 \nu^{11} + 26256 \nu^{10} - 19923 \nu^{9} - 275603 \nu^{8} + 1178894 \nu^{7} + \cdots - 249415880 ) / 1260525 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1594 \nu^{11} + 7221 \nu^{10} - 5088 \nu^{9} - 76858 \nu^{8} + 325759 \nu^{7} - 397035 \nu^{6} + \cdots - 69929125 ) / 180075 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{2} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} + 4 \beta_{10} + \beta_{9} + 2 \beta_{8} + 6 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \cdots - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + 15 \beta_{5} + \cdots + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32 \beta_{11} - 42 \beta_{10} - 23 \beta_{9} + 19 \beta_{8} + 24 \beta_{7} - 5 \beta_{6} + 15 \beta_{5} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23 \beta_{11} - 98 \beta_{10} + 38 \beta_{9} + 11 \beta_{8} - 39 \beta_{7} - 8 \beta_{6} + 11 \beta_{5} + \cdots + 271 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 40 \beta_{11} + 40 \beta_{10} - 5 \beta_{9} + 140 \beta_{8} + 540 \beta_{7} - 24 \beta_{6} - 152 \beta_{5} + \cdots + 338 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 568 \beta_{11} + 398 \beta_{10} + 846 \beta_{9} + 58 \beta_{8} + 283 \beta_{7} + 162 \beta_{6} + \cdots + 3065 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 88 \beta_{11} + 376 \beta_{10} - 161 \beta_{9} + 1118 \beta_{8} + 2976 \beta_{7} - 414 \beta_{6} + \cdots + 231 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2298 \beta_{11} - 897 \beta_{10} + 3955 \beta_{9} + 1307 \beta_{8} - 3513 \beta_{7} - 321 \beta_{6} + \cdots - 7762 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2194 \beta_{11} - 34 \beta_{10} - 4386 \beta_{9} + 11933 \beta_{8} + 20418 \beta_{7} - 7409 \beta_{6} + \cdots - 53577 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(4411\) \(7057\) \(7841\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.63107 + 2.08318i
0.260926 + 2.63285i
−2.61674 + 0.390758i
2.61827 + 0.380350i
1.75207 + 1.98249i
−2.64559 + 0.0290059i
−2.64559 − 0.0290059i
1.75207 − 1.98249i
2.61827 − 0.380350i
−2.61674 − 0.390758i
0.260926 − 2.63285i
1.63107 − 2.08318i
0 0 0 −1.00000 0 0 0 0 0
881.2 0 0 0 −1.00000 0 0 0 0 0
881.3 0 0 0 −1.00000 0 0 0 0 0
881.4 0 0 0 −1.00000 0 0 0 0 0
881.5 0 0 0 −1.00000 0 0 0 0 0
881.6 0 0 0 −1.00000 0 0 0 0 0
881.7 0 0 0 −1.00000 0 0 0 0 0
881.8 0 0 0 −1.00000 0 0 0 0 0
881.9 0 0 0 −1.00000 0 0 0 0 0
881.10 0 0 0 −1.00000 0 0 0 0 0
881.11 0 0 0 −1.00000 0 0 0 0 0
881.12 0 0 0 −1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.d.a 12
3.b odd 2 1 8820.2.d.b 12
7.b odd 2 1 8820.2.d.b 12
7.c even 3 1 1260.2.cg.b yes 12
7.d odd 6 1 1260.2.cg.a ✓ 12
21.c even 2 1 inner 8820.2.d.a 12
21.g even 6 1 1260.2.cg.b yes 12
21.h odd 6 1 1260.2.cg.a ✓ 12
35.i odd 6 1 6300.2.ch.c 12
35.j even 6 1 6300.2.ch.b 12
35.k even 12 2 6300.2.dd.c 24
35.l odd 12 2 6300.2.dd.b 24
105.o odd 6 1 6300.2.ch.c 12
105.p even 6 1 6300.2.ch.b 12
105.w odd 12 2 6300.2.dd.b 24
105.x even 12 2 6300.2.dd.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.cg.a ✓ 12 7.d odd 6 1
1260.2.cg.a ✓ 12 21.h odd 6 1
1260.2.cg.b yes 12 7.c even 3 1
1260.2.cg.b yes 12 21.g even 6 1
6300.2.ch.b 12 35.j even 6 1
6300.2.ch.b 12 105.p even 6 1
6300.2.ch.c 12 35.i odd 6 1
6300.2.ch.c 12 105.o odd 6 1
6300.2.dd.b 24 35.l odd 12 2
6300.2.dd.b 24 105.w odd 12 2
6300.2.dd.c 24 35.k even 12 2
6300.2.dd.c 24 105.x even 12 2
8820.2.d.a 12 1.a even 1 1 trivial
8820.2.d.a 12 21.c even 2 1 inner
8820.2.d.b 12 3.b odd 2 1
8820.2.d.b 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8820, [\chi])\):

\( T_{11}^{12} + 68T_{11}^{10} + 1572T_{11}^{8} + 16988T_{11}^{6} + 93880T_{11}^{4} + 256320T_{11}^{2} + 272484 \) Copy content Toggle raw display
\( T_{17}^{6} - 52T_{17}^{4} - 108T_{17}^{3} + 262T_{17}^{2} + 372T_{17} - 450 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 68 T^{10} + \cdots + 272484 \) Copy content Toggle raw display
$13$ \( T^{12} + 102 T^{10} + \cdots + 189225 \) Copy content Toggle raw display
$17$ \( (T^{6} - 52 T^{4} + \cdots - 450)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 162 T^{10} + \cdots + 2064969 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 170302500 \) Copy content Toggle raw display
$29$ \( T^{12} + 148 T^{10} + \cdots + 8100 \) Copy content Toggle raw display
$31$ \( T^{12} + 210 T^{10} + \cdots + 9126441 \) Copy content Toggle raw display
$37$ \( (T^{6} - 2 T^{5} + \cdots - 1415)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots - 450)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 18 T^{5} + \cdots + 142465)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 16 T^{5} + \cdots + 15264)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 168 T^{10} + \cdots + 82944 \) Copy content Toggle raw display
$59$ \( (T^{6} - 166 T^{4} + \cdots - 2610)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1497690000 \) Copy content Toggle raw display
$67$ \( (T^{6} + 2 T^{5} + \cdots - 66951)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 239073444 \) Copy content Toggle raw display
$73$ \( T^{12} + 98 T^{10} + \cdots + 1172889 \) Copy content Toggle raw display
$79$ \( (T^{6} + 14 T^{5} + \cdots - 166655)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 20 T^{5} + \cdots - 13410)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 20 T^{5} + \cdots - 781650)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 92 T^{10} + \cdots + 1089936 \) Copy content Toggle raw display
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