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Properties

Label 882.4.d.b
Level $882$
Weight $4$
Character orbit 882.d
Analytic conductor $52.040$
Analytic rank $0$
Dimension $16$
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] = [882,4,Mod(881,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.881");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(52.0396846251\)
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: \( x^{16} - 408 x^{14} + 83232 x^{12} - 6071448 x^{10} + 20926577 x^{8} + 34235700240 x^{6} + \cdots + 993321036864016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 4 q^{4} + (\beta_{14} + \beta_{11}) q^{5} + 4 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - 4 q^{4} + (\beta_{14} + \beta_{11}) q^{5} + 4 \beta_{3} q^{8} + (\beta_{12} + \beta_{9}) q^{10} + (\beta_{7} - 9 \beta_{3}) q^{11} + (2 \beta_{10} + 4 \beta_{9} + 11 \beta_{4}) q^{13} + 16 q^{16} + (5 \beta_{15} - 6 \beta_{14} + \cdots - 3 \beta_{11}) q^{17}+ \cdots + ( - \beta_{12} - 37 \beta_{10} + \cdots + 276 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} + 256 q^{16} - 608 q^{22} + 1488 q^{25} + 352 q^{37} - 3296 q^{43} + 2016 q^{46} - 992 q^{58} - 1024 q^{64} + 1600 q^{67} - 6048 q^{79} - 11424 q^{85} + 2432 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 408 x^{14} + 83232 x^{12} - 6071448 x^{10} + 20926577 x^{8} + 34235700240 x^{6} + \cdots + 993321036864016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10944206499 \nu^{14} + 2710671053664 \nu^{12} - 336433787471692 \nu^{10} + \cdots - 10\!\cdots\!36 ) / 73\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5396140083 \nu^{14} - 2281047983784 \nu^{12} + 486009599924852 \nu^{10} + \cdots + 22\!\cdots\!76 ) / 11\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!02 \nu^{14} + \cdots + 77\!\cdots\!60 ) / 28\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37\!\cdots\!09 \nu^{15} + \cdots + 51\!\cdots\!56 \nu ) / 32\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!49 \nu^{14} + \cdots + 34\!\cdots\!72 ) / 34\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!87 \nu^{14} + \cdots + 60\!\cdots\!22 ) / 43\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!07 \nu^{14} + \cdots - 11\!\cdots\!80 ) / 68\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 58\!\cdots\!61 \nu^{14} + \cdots - 19\!\cdots\!88 ) / 83\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!21 \nu^{15} + \cdots + 36\!\cdots\!88 \nu ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!29 \nu^{15} + \cdots + 13\!\cdots\!40 \nu ) / 21\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 93\!\cdots\!79 \nu^{15} + \cdots + 28\!\cdots\!84 \nu ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 37\!\cdots\!11 \nu^{15} + \cdots + 49\!\cdots\!60 \nu ) / 54\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 32\!\cdots\!51 \nu^{15} + \cdots - 10\!\cdots\!60 \nu ) / 38\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!20 \nu^{15} + \cdots + 16\!\cdots\!72 \nu ) / 78\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!29 \nu^{15} + \cdots + 47\!\cdots\!68 \nu ) / 15\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{15} - 5\beta_{14} - 7\beta_{13} - 14\beta_{11} + 7\beta_{10} - \beta_{9} - 3\beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} + 4\beta_{7} + 2\beta_{6} - \beta_{5} - 712\beta_{3} + 1463\beta_{2} + 1463\beta _1 + 1428 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 51 \beta_{15} - 211 \beta_{14} - 735 \beta_{13} - 1092 \beta_{12} - 714 \beta_{11} + \cdots - 925 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -520\beta_{8} + 617\beta_{7} - 113277\beta_{3} + 149940\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10463 \beta_{15} - 3947 \beta_{14} + 114317 \beta_{13} - 189644 \beta_{12} + 150654 \beta_{11} + \cdots - 293791 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 176583 \beta_{8} + 257940 \beta_{7} - 128970 \beta_{6} + 176583 \beta_{5} - 27423576 \beta_{3} + \cdots - 55105092 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13065756 \beta_{15} - 18710323 \beta_{14} + 47688361 \beta_{13} - 13888371 \beta_{12} + \cdots - 31200753 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -42300301\beta_{6} + 60183400\beta_{5} - 9948550416\beta _1 - 14105821870 ) / 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4144523786 \beta_{15} - 5871733457 \beta_{14} + 8594272827 \beta_{13} + 2514392055 \beta_{12} + \cdots + 7019908059 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9414735695 \beta_{8} - 13329524156 \beta_{7} - 6664762078 \beta_{6} + 9414735695 \beta_{5} + \cdots - 1797676211772 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 438414213851 \beta_{15} - 618941684045 \beta_{14} + 644523128795 \beta_{13} + \cdots + 3670237739785 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1422507871584\beta_{8} - 2011299495549\beta_{7} + 114325265217397\beta_{3} - 163085743037196\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 101451309825451 \beta_{15} + 143430293717003 \beta_{14} - 117170280960565 \beta_{13} + \cdots + 772619322202183 \beta_{4} ) / 28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 418348371763519 \beta_{8} - 591658036978180 \beta_{7} + 295829018489090 \beta_{6} + \cdots + 59\!\cdots\!36 ) / 28 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 53\!\cdots\!92 \beta_{15} + \cdots + 65\!\cdots\!85 \beta_{4} ) / 28 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(-1\)

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
1.97767 + 4.77452i
11.8078 − 4.89093i
2.36035 + 5.69840i
12.7316 − 5.27362i
−12.7316 + 5.27362i
−2.36035 − 5.69840i
−11.8078 + 4.89093i
−1.97767 − 4.77452i
1.97767 − 4.77452i
11.8078 + 4.89093i
2.36035 − 5.69840i
12.7316 + 5.27362i
−12.7316 − 5.27362i
−2.36035 + 5.69840i
−11.8078 − 4.89093i
−1.97767 + 4.77452i
− 2.00000i 0 −4.00000 −16.8829 0 0 8.00000i 0 33.7657i
881.2 − 2.00000i 0 −4.00000 −16.7638 0 0 8.00000i 0 33.5276i
881.3 − 2.00000i 0 −4.00000 −12.7390 0 0 8.00000i 0 25.4780i
881.4 − 2.00000i 0 −4.00000 −11.9859 0 0 8.00000i 0 23.9718i
881.5 − 2.00000i 0 −4.00000 11.9859 0 0 8.00000i 0 − 23.9718i
881.6 − 2.00000i 0 −4.00000 12.7390 0 0 8.00000i 0 − 25.4780i
881.7 − 2.00000i 0 −4.00000 16.7638 0 0 8.00000i 0 − 33.5276i
881.8 − 2.00000i 0 −4.00000 16.8829 0 0 8.00000i 0 − 33.7657i
881.9 2.00000i 0 −4.00000 −16.8829 0 0 − 8.00000i 0 − 33.7657i
881.10 2.00000i 0 −4.00000 −16.7638 0 0 − 8.00000i 0 − 33.5276i
881.11 2.00000i 0 −4.00000 −12.7390 0 0 − 8.00000i 0 − 25.4780i
881.12 2.00000i 0 −4.00000 −11.9859 0 0 − 8.00000i 0 − 23.9718i
881.13 2.00000i 0 −4.00000 11.9859 0 0 − 8.00000i 0 23.9718i
881.14 2.00000i 0 −4.00000 12.7390 0 0 − 8.00000i 0 25.4780i
881.15 2.00000i 0 −4.00000 16.7638 0 0 − 8.00000i 0 33.5276i
881.16 2.00000i 0 −4.00000 16.8829 0 0 − 8.00000i 0 33.7657i
\(n\): 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
3.b odd 2 1 inner
7.b odd 2 1 inner
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 882.4.d.b ✓ 16
3.b odd 2 1 inner 882.4.d.b ✓ 16
7.b odd 2 1 inner 882.4.d.b ✓ 16
7.c even 3 2 882.4.k.d 32
7.d odd 6 2 882.4.k.d 32
21.c even 2 1 inner 882.4.d.b ✓ 16
21.g even 6 2 882.4.k.d 32
21.h odd 6 2 882.4.k.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.d.b ✓ 16 1.a even 1 1 trivial
882.4.d.b ✓ 16 3.b odd 2 1 inner
882.4.d.b ✓ 16 7.b odd 2 1 inner
882.4.d.b ✓ 16 21.c even 2 1 inner
882.4.k.d 32 7.c even 3 2
882.4.k.d 32 7.d odd 6 2
882.4.k.d 32 21.g even 6 2
882.4.k.d 32 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 872T_{5}^{6} + 276596T_{5}^{4} - 37703248T_{5}^{2} + 1867449796 \) acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 872 T^{6} + \cdots + 1867449796)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 9248605487104)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 12232 T^{6} + \cdots + 692992981444)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 761262949908036)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 23975829766144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 31\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 84\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 88 T^{3} + \cdots + 750699748)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 65\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 824 T^{3} + \cdots - 858651776)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 35\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 94\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 29\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 400 T^{3} + \cdots + 13726098304)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 18\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1512 T^{3} + \cdots + 158538116608)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 67\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
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