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Properties

Label 882.6.d.a
Level $882$
Weight $6$
Character orbit 882.d
Analytic conductor $141.459$
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] = [882,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.881");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(141.458529075\)
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: \( x^{16} - 648 x^{14} + 209952 x^{12} - 23898888 x^{10} + 10894513 x^{8} + 361629602352 x^{6} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{8}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{3} q^{2} - 16 q^{4} + ( - \beta_{14} - 3 \beta_{10}) q^{5} - 32 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{3} q^{2} - 16 q^{4} + ( - \beta_{14} - 3 \beta_{10}) q^{5} - 32 \beta_{3} q^{8} + (2 \beta_{13} + 10 \beta_{5}) q^{10} + ( - 2 \beta_{8} - 75 \beta_{3}) q^{11} + ( - \beta_{12} + 31 \beta_{5} + 46 \beta_{4}) q^{13} + 256 q^{16} + ( - 11 \beta_{15} + \cdots + 111 \beta_{10}) q^{17}+ \cdots + ( - 74 \beta_{13} + \cdots + 22200 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 256 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 256 q^{4} + 4096 q^{16} + 9472 q^{22} + 2256 q^{25} - 44768 q^{37} + 45088 q^{43} - 83328 q^{46} - 116672 q^{58} - 65536 q^{64} - 330944 q^{67} - 110880 q^{79} + 621600 q^{85} - 151552 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 648 x^{14} + 209952 x^{12} - 23898888 x^{10} + 10894513 x^{8} + 361629602352 x^{6} + \cdots + 52\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2018723157763 \nu^{14} + 795419248929120 \nu^{12} + \cdots - 53\!\cdots\!52 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1018624469011 \nu^{14} + 685045102001976 \nu^{12} + \cdots - 13\!\cdots\!96 ) / 69\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83\!\cdots\!70 \nu^{14} + \cdots + 10\!\cdots\!40 ) / 39\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 55\!\cdots\!03 \nu^{15} + \cdots - 62\!\cdots\!24 \nu ) / 41\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 75\!\cdots\!01 \nu^{15} + \cdots + 44\!\cdots\!52 \nu ) / 20\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!05 \nu^{14} + \cdots + 57\!\cdots\!88 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93\!\cdots\!24 \nu^{14} + \cdots - 10\!\cdots\!66 ) / 14\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 82\!\cdots\!82 \nu^{14} + \cdots - 83\!\cdots\!72 ) / 10\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 99\!\cdots\!07 \nu^{14} + \cdots + 98\!\cdots\!52 ) / 88\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 99\!\cdots\!21 \nu^{15} + \cdots + 70\!\cdots\!16 \nu ) / 34\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 83\!\cdots\!55 \nu^{15} + \cdots - 59\!\cdots\!92 \nu ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 62\!\cdots\!73 \nu^{15} + \cdots + 25\!\cdots\!92 \nu ) / 13\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!30 \nu^{15} + \cdots - 90\!\cdots\!96 \nu ) / 34\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 50\!\cdots\!59 \nu^{15} + \cdots - 41\!\cdots\!32 \nu ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 52\!\cdots\!31 \nu^{15} + \cdots + 43\!\cdots\!64 \nu ) / 31\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 7\beta_{15} + 14\beta_{14} + 7\beta_{12} + 5\beta_{11} + 13\beta_{10} - 3\beta_{5} + 9\beta_{4} ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 4\beta_{8} - 2\beta_{7} - \beta_{6} - 3400\beta_{3} - 7077\beta_{2} + 7077\beta _1 + 6804 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 169 \beta_{15} + 162 \beta_{14} + 250 \beta_{13} + 331 \beta_{12} + 305 \beta_{11} + \cdots + 633 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 836\beta_{9} + 985\beta_{8} - 874840\beta_{3} - 1149876\beta_{2} ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 292957 \beta_{15} - 384426 \beta_{14} + 485170 \beta_{13} + 677383 \beta_{12} + \cdots + 2260533 \beta_{4} ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 64657 \beta_{9} + 94684 \beta_{8} + 47342 \beta_{7} + 64657 \beta_{6} - 48216832 \beta_{3} + \cdots - 96528348 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 194418469 \beta_{15} - 275646728 \beta_{14} + 56595105 \beta_{13} + 81228259 \beta_{12} + \cdots + 385608015 \beta_{4} ) / 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 173666557\beta_{7} + 247255600\beta_{6} - 194391831312\beta _1 - 275702122122 ) / 84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7965420433 \beta_{15} - 11270506028 \beta_{14} - 2330167419 \beta_{13} + \cdots - 19797170427 \beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 61829440447 \beta_{9} - 87553600252 \beta_{8} + 43776800126 \beta_{7} + 61829440447 \beta_{6} + \cdots - 55850417489916 ) / 84 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6642468272923 \beta_{15} - 9388284995754 \beta_{14} - 11336610770800 \beta_{13} + \cdots - 115768999234947 \beta_{4} ) / 84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -2133018137252\beta_{9} - 3015782685163\beta_{8} + 810931866813388\beta_{3} + 1148829244342164\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 19\!\cdots\!53 \beta_{15} + \cdots - 38\!\cdots\!45 \beta_{4} ) / 84 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 70\!\cdots\!59 \beta_{9} + \cdots + 46\!\cdots\!24 ) / 84 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 19\!\cdots\!51 \beta_{15} + \cdots - 75\!\cdots\!29 \beta_{4} ) / 12 \) 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
2.99562 − 7.23207i
15.9985 + 6.62681i
2.61294 − 6.30819i
15.0747 + 6.24413i
−15.0747 − 6.24413i
−2.61294 + 6.30819i
−15.9985 − 6.62681i
−2.99562 + 7.23207i
2.99562 + 7.23207i
15.9985 − 6.62681i
2.61294 + 6.30819i
15.0747 − 6.24413i
−15.0747 + 6.24413i
−2.61294 − 6.30819i
−15.9985 + 6.62681i
−2.99562 − 7.23207i
− 4.00000i 0 −16.0000 −67.8061 0 0 64.0000i 0 271.225i
881.2 − 4.00000i 0 −16.0000 −66.5516 0 0 64.0000i 0 266.206i
881.3 − 4.00000i 0 −16.0000 −47.0868 0 0 64.0000i 0 188.347i
881.4 − 4.00000i 0 −16.0000 −42.6620 0 0 64.0000i 0 170.648i
881.5 − 4.00000i 0 −16.0000 42.6620 0 0 64.0000i 0 − 170.648i
881.6 − 4.00000i 0 −16.0000 47.0868 0 0 64.0000i 0 − 188.347i
881.7 − 4.00000i 0 −16.0000 66.5516 0 0 64.0000i 0 − 266.206i
881.8 − 4.00000i 0 −16.0000 67.8061 0 0 64.0000i 0 − 271.225i
881.9 4.00000i 0 −16.0000 −67.8061 0 0 − 64.0000i 0 − 271.225i
881.10 4.00000i 0 −16.0000 −66.5516 0 0 − 64.0000i 0 − 266.206i
881.11 4.00000i 0 −16.0000 −47.0868 0 0 − 64.0000i 0 − 188.347i
881.12 4.00000i 0 −16.0000 −42.6620 0 0 − 64.0000i 0 − 170.648i
881.13 4.00000i 0 −16.0000 42.6620 0 0 − 64.0000i 0 170.648i
881.14 4.00000i 0 −16.0000 47.0868 0 0 − 64.0000i 0 188.347i
881.15 4.00000i 0 −16.0000 66.5516 0 0 − 64.0000i 0 266.206i
881.16 4.00000i 0 −16.0000 67.8061 0 0 − 64.0000i 0 271.225i
\(n\): 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 882.6.d.a ✓ 16
3.b odd 2 1 inner 882.6.d.a ✓ 16
7.b odd 2 1 inner 882.6.d.a ✓ 16
21.c even 2 1 inner 882.6.d.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.6.d.a ✓ 16 1.a even 1 1 trivial
882.6.d.a ✓ 16 3.b odd 2 1 inner
882.6.d.a ✓ 16 7.b odd 2 1 inner
882.6.d.a ✓ 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 13064T_{5}^{6} + 60842036T_{5}^{4} - 118638538384T_{5}^{2} + 82174333780036 \) acting on \(S_{6}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 82174333780036)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 31\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 68\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 50\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 34\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 67\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 772641196012324)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 33\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 741615318043072)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 25\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 54\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 86\!\cdots\!24)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 24\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 49\!\cdots\!08)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 65\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 87\!\cdots\!56)^{2} \) Copy content Toggle raw display
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