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Properties

Label 8820.2.d.d
Level $8820$
Weight $2$
Character orbit 8820.d
Analytic conductor $70.428$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + \cdots + 68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{17} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) 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 \( q + q^{5} + ( - \beta_{12} + \beta_{11} - \beta_{10}) q^{11} + \beta_{15} q^{13} + ( - \beta_{6} - \beta_{4}) q^{17} + (\beta_{14} + \beta_{13} + \beta_{8}) q^{19} + (2 \beta_{11} + \beta_{9} + \beta_{8}) q^{23} + q^{25} + ( - \beta_{15} - \beta_{12} + \cdots + \beta_{8}) q^{29}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + \beta_{8}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{5} + 16 q^{25} + 32 q^{41} - 32 q^{43} - 32 q^{47} - 32 q^{59} - 32 q^{67} - 64 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + \cdots + 68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 563 \nu^{14} - 3941 \nu^{13} + 29695 \nu^{12} - 126937 \nu^{11} + 503911 \nu^{10} - 1449893 \nu^{9} + \cdots + 150044 ) / 56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 31 \nu^{14} + 217 \nu^{13} - 1635 \nu^{12} + 6989 \nu^{11} - 27743 \nu^{10} + 79821 \nu^{9} + \cdots - 8164 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64 \nu^{14} + 448 \nu^{13} - 3376 \nu^{12} + 14432 \nu^{11} - 57300 \nu^{10} + 164884 \nu^{9} + \cdots - 17398 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 933 \nu^{14} - 6531 \nu^{13} + 49213 \nu^{12} - 210375 \nu^{11} + 835197 \nu^{10} - 2403203 \nu^{9} + \cdots + 251108 ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 477 \nu^{14} + 3339 \nu^{13} - 25163 \nu^{12} + 107571 \nu^{11} - 427122 \nu^{10} + 1229122 \nu^{9} + \cdots - 130804 ) / 28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1203 \nu^{14} - 8421 \nu^{13} + 63457 \nu^{12} - 271269 \nu^{11} + 1077003 \nu^{10} - 3099083 \nu^{9} + \cdots + 325876 ) / 56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 911 \nu^{14} - 6377 \nu^{13} + 48053 \nu^{12} - 205417 \nu^{11} + 815524 \nu^{10} - 2346616 \nu^{9} + \cdots + 245380 ) / 28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9578 \nu^{15} + 71835 \nu^{14} - 538751 \nu^{13} + 2412384 \nu^{12} - 9654653 \nu^{11} + \cdots + 1295576 ) / 56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41198 \nu^{15} - 308985 \nu^{14} + 2317365 \nu^{13} - 10376600 \nu^{12} + 41529045 \nu^{11} + \cdots - 5590392 ) / 56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45046 \nu^{15} + 337845 \nu^{14} - 2533879 \nu^{13} + 11346231 \nu^{12} - 45411237 \nu^{11} + \cdots + 6148892 ) / 56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3856 \nu^{15} - 28920 \nu^{14} + 216902 \nu^{13} - 971243 \nu^{12} + 3887188 \nu^{11} - 11660825 \nu^{10} + \cdots - 525468 ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 58612 \nu^{15} + 439590 \nu^{14} - 3296928 \nu^{13} + 14762917 \nu^{12} - 59084708 \nu^{11} + \cdots + 7973036 ) / 56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4336 \nu^{15} + 32520 \nu^{14} - 243898 \nu^{13} + 1092117 \nu^{12} - 4370860 \nu^{11} + \cdots + 588636 ) / 4 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18462 \nu^{15} + 138465 \nu^{14} - 1038487 \nu^{13} + 4650113 \nu^{12} - 18610825 \nu^{11} + \cdots + 2510464 ) / 8 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 198424 \nu^{15} + 1488180 \nu^{14} - 11161258 \nu^{13} + 49977447 \nu^{12} - 200019240 \nu^{11} + \cdots + 26936464 ) / 56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{12} + \beta_{11} + 2\beta_{9} + 2\beta_{8} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{12} + \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + \cdots - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} - \beta_{14} + 6 \beta_{13} - 17 \beta_{12} - 11 \beta_{11} - \beta_{10} - 13 \beta_{9} + \cdots - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} - 2 \beta_{14} + 12 \beta_{13} - 36 \beta_{12} - 23 \beta_{11} - 2 \beta_{10} + \cdots + 90 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{15} + 5 \beta_{14} - 60 \beta_{13} + 133 \beta_{12} + 89 \beta_{11} + 13 \beta_{10} + \cdots + 262 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 20 \beta_{15} + 20 \beta_{14} - 210 \beta_{13} + 490 \beta_{12} + 325 \beta_{11} + 44 \beta_{10} + \cdots - 506 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 19 \beta_{15} + 13 \beta_{14} + 406 \beta_{13} - 789 \beta_{12} - 539 \beta_{11} - 129 \beta_{10} + \cdots - 2714 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 174 \beta_{15} - 46 \beta_{14} + 2632 \beta_{13} - 5528 \beta_{12} - 3727 \beta_{11} - 726 \beta_{10} + \cdots + 1834 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 29 \beta_{15} - 411 \beta_{14} - 1416 \beta_{13} + 1957 \beta_{12} + 1415 \beta_{11} + 769 \beta_{10} + \cdots + 25682 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1600 \beta_{15} - 1560 \beta_{14} - 28350 \beta_{13} + 54858 \beta_{12} + 37419 \beta_{11} + 9608 \beta_{10} + \cdots + 8026 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 911 \beta_{15} + 2791 \beta_{14} - 14498 \beta_{13} + 36049 \beta_{12} + 24079 \beta_{11} + \cdots - 224134 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 15478 \beta_{15} + 32674 \beta_{14} + 273108 \beta_{13} - 489740 \beta_{12} - 336165 \beta_{11} + \cdots - 286698 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22361 \beta_{15} + 18215 \beta_{14} + 422188 \beta_{13} - 830061 \beta_{12} - 567603 \beta_{11} + \cdots + 1784902 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 146204 \beta_{15} - 414660 \beta_{14} - 2342886 \beta_{13} + 3894802 \beta_{12} + 2674609 \beta_{11} + \cdots + 4357318 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 377749 \beta_{15} - 821099 \beta_{14} - 6628866 \beta_{13} + 11837823 \beta_{12} + 8131025 \beta_{11} + \cdots - 12458762 ) / 4 \) 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
0.500000 − 0.0420372i
0.500000 − 0.00906270i
0.500000 + 3.11247i
0.500000 + 2.03007i
0.500000 − 2.65516i
0.500000 − 2.81224i
0.500000 − 0.199114i
0.500000 − 1.09145i
0.500000 + 1.09145i
0.500000 + 0.199114i
0.500000 + 2.81224i
0.500000 + 2.65516i
0.500000 − 2.03007i
0.500000 − 3.11247i
0.500000 + 0.00906270i
0.500000 + 0.0420372i
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
881.13 0 0 0 1.00000 0 0 0 0 0
881.14 0 0 0 1.00000 0 0 0 0 0
881.15 0 0 0 1.00000 0 0 0 0 0
881.16 0 0 0 1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
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.d yes 16
3.b odd 2 1 8820.2.d.c ✓ 16
7.b odd 2 1 8820.2.d.c ✓ 16
21.c even 2 1 inner 8820.2.d.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8820.2.d.c ✓ 16 3.b odd 2 1
8820.2.d.c ✓ 16 7.b odd 2 1
8820.2.d.d yes 16 1.a even 1 1 trivial
8820.2.d.d yes 16 21.c even 2 1 inner

Hecke kernels

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

\( T_{11}^{16} + 128 T_{11}^{14} + 6160 T_{11}^{12} + 137984 T_{11}^{10} + 1442400 T_{11}^{8} + \cdots + 73984 \) Copy content Toggle raw display
\( T_{17}^{8} - 72T_{17}^{6} - 32T_{17}^{5} + 1288T_{17}^{4} + 128T_{17}^{3} - 5088T_{17}^{2} + 2176T_{17} + 1552 \) 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
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 128 T^{14} + \cdots + 73984 \) Copy content Toggle raw display
$13$ \( T^{16} + 112 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( (T^{8} - 72 T^{6} + \cdots + 1552)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 144 T^{14} + \cdots + 1993744 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 570063376 \) Copy content Toggle raw display
$29$ \( T^{16} + 208 T^{14} + \cdots + 31899904 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 49005562384 \) Copy content Toggle raw display
$37$ \( (T^{8} - 128 T^{6} + \cdots - 291776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 16 T^{7} + \cdots + 344576)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 16 T^{7} + \cdots - 32512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 16 T^{7} + \cdots - 6896)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 613546357264 \) Copy content Toggle raw display
$59$ \( (T^{8} + 16 T^{7} + \cdots + 477248)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 193258710544 \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots + 53312)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7539813040384 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 6105722392576 \) Copy content Toggle raw display
$79$ \( (T^{8} - 216 T^{6} + \cdots + 156944)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 168 T^{6} + \cdots + 75536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 32 T^{7} + \cdots - 12224)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 82218758311936 \) Copy content Toggle raw display
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