↑

Properties

Label 882.4.d.c
Level $882$
Weight $4$
Character orbit 882.d
Analytic conductor $52.040$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \( x^{16} - 26x^{14} + 451x^{12} - 4330x^{10} + 30081x^{8} - 130232x^{6} + 401200x^{4} - 595840x^{2} + 614656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{12}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - 4 q^{4} + ( - \beta_{11} - \beta_{5} + \beta_{2}) q^{5} - 4 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - 4 q^{4} + ( - \beta_{11} - \beta_{5} + \beta_{2}) q^{5} - 4 \beta_{4} q^{8} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_1) q^{10} + ( - \beta_{13} + \beta_{12} + \cdots + \beta_{4}) q^{11}+ \cdots + ( - 8 \beta_{9} - 75 \beta_{7} + \cdots + 12 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} + 256 q^{16} - 48 q^{22} + 424 q^{25} - 2504 q^{37} - 1112 q^{43} - 1344 q^{46} - 1104 q^{58} - 1024 q^{64} - 136 q^{67} - 5488 q^{79} - 672 q^{85} + 192 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26x^{14} + 451x^{12} - 4330x^{10} + 30081x^{8} - 130232x^{6} + 401200x^{4} - 595840x^{2} + 614656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 142677179 \nu^{14} - 3449734804 \nu^{12} + 58606224029 \nu^{10} - 522506995056 \nu^{8} + \cdots - 46485795713168 ) / 21344035382768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32388 \nu^{15} + 519129 \nu^{13} - 6905658 \nu^{11} + 11173011 \nu^{9} + \cdots - 30203812464 \nu ) / 3440822336 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 490653599 \nu^{15} - 17485826480 \nu^{13} + 349329715265 \nu^{11} + \cdots - 15\!\cdots\!08 \nu ) / 35154881806912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 340499 \nu^{15} - 9803476 \nu^{13} + 166547501 \nu^{11} - 1626535168 \nu^{9} + \cdots - 71447997600 \nu ) / 24253601344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5989 \nu^{15} - 118817 \nu^{13} + 1730937 \nu^{11} - 10214199 \nu^{9} + 40479335 \nu^{7} + \cdots + 1037684368 \nu ) / 270459616 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 505094349 \nu^{14} + 12981639550 \nu^{12} - 222950371535 \nu^{10} + \cdots + 173543877574368 ) / 6098295823648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3267 \nu^{14} - 37902 \nu^{12} + 492633 \nu^{10} + 624450 \nu^{8} - 1471341 \nu^{6} + \cdots - 126923328 ) / 37158464 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 617698575 \nu^{14} - 13757301678 \nu^{12} + 226486732173 \nu^{10} - 1789607292798 \nu^{8} + \cdots + 130438133166656 ) / 6098295823648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21799456891 \nu^{14} - 1221476362270 \nu^{12} + 23312343609281 \nu^{10} + \cdots - 22\!\cdots\!00 ) / 170752283062144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 56994500329 \nu^{15} + 857264024252 \nu^{13} - 14563730347927 \nu^{11} + \cdots + 62\!\cdots\!00 \nu ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4898212841 \nu^{15} - 108823931705 \nu^{13} + 1694109345605 \nu^{11} + \cdots - 15\!\cdots\!20 \nu ) / 85376141531072 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 496431 \nu^{15} - 12659364 \nu^{13} + 215065089 \nu^{11} - 1987862400 \nu^{9} + \cdots - 92261786400 \nu ) / 7625641856 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 116590638015 \nu^{15} - 2771475031140 \nu^{13} + 47083528385265 \nu^{11} + \cdots - 20\!\cdots\!00 \nu ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 350325 \nu^{14} + 8300970 \nu^{12} - 128450943 \nu^{10} + 1014967818 \nu^{8} + \cdots + 48089692224 ) / 1029335104 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12480297075 \nu^{14} + 291464157126 \nu^{12} - 4576053459513 \nu^{10} + \cdots - 63860644841280 ) / 24393183294592 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{13} + 2\beta_{11} + 2\beta_{10} - 9\beta_{5} - 11\beta_{4} - 4\beta_{3} ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{15} - 7\beta_{14} - 6\beta_{9} + 3\beta_{8} - 21\beta_{7} - 3\beta_{6} + 411\beta _1 + 408 ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 96\beta_{13} + 56\beta_{12} + 90\beta_{10} - 621\beta_{4} ) / 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -30\beta_{15} + 35\beta_{14} - 30\beta_{9} - 29\beta_{8} - 105\beta_{7} - 57\beta_{6} + 1173\beta _1 - 1172 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 936 \beta_{13} + 1120 \beta_{12} - 2358 \beta_{11} + 1098 \beta_{10} + 7083 \beta_{5} + \cdots + 3360 \beta_{2} ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1182\beta_{15} + 1351\beta_{14} - 1431\beta_{8} - 35604 ) / 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10176 \beta_{13} - 16520 \beta_{12} - 29406 \beta_{11} - 13194 \beta_{10} + 90243 \beta_{5} + \cdots + 49560 \beta_{2} ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5014 \beta_{15} + 5607 \beta_{14} + 5014 \beta_{9} - 6637 \beta_{8} + 16821 \beta_{7} + \cdots - 131764 ) / 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -39176\beta_{13} - 73360\beta_{12} - 53230\beta_{10} + 431575\beta_{4} ) / 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 188838 \beta_{15} - 207683 \beta_{14} + 188838 \beta_{9} + 259479 \beta_{8} + 623049 \beta_{7} + \cdots + 4618980 ) / 126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1402224 \beta_{13} - 2812376 \beta_{12} + 4439406 \beta_{11} - 1947210 \beta_{10} + \cdots - 8437128 \beta_{2} ) / 252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2353098\beta_{15} - 2559473\beta_{14} + 3285831\beta_{8} + 55455276 ) / 63 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 17013960 \beta_{13} + 35280896 \beta_{12} + 54577494 \beta_{11} + 23863818 \beta_{10} + \cdots - 105842688 \beta_{2} ) / 252 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 9732410 \beta_{15} - 10514133 \beta_{14} - 9732410 \beta_{9} + 13686429 \beta_{8} - 31542399 \beta_{7} + \cdots + 225025020 ) / 42 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 208205856\beta_{13} + 438827816\beta_{12} + 293357322\beta_{10} - 2476128429\beta_{4} ) / 126 \) 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
−2.03494 + 1.17487i
3.04238 + 1.75652i
2.17636 − 1.25652i
−1.16892 − 0.674875i
1.16892 − 0.674875i
−2.17636 − 1.25652i
−3.04238 + 1.75652i
2.03494 + 1.17487i
−2.03494 − 1.17487i
3.04238 − 1.75652i
2.17636 + 1.25652i
−1.16892 + 0.674875i
1.16892 + 0.674875i
−2.17636 + 1.25652i
−3.04238 − 1.75652i
2.03494 − 1.17487i
− 2.00000i 0 −4.00000 −19.7628 0 0 8.00000i 0 39.5256i
881.2 − 2.00000i 0 −4.00000 −11.8430 0 0 8.00000i 0 23.6860i
881.3 − 2.00000i 0 −4.00000 −7.24119 0 0 8.00000i 0 14.4824i
881.4 − 2.00000i 0 −4.00000 −4.76869 0 0 8.00000i 0 9.53738i
881.5 − 2.00000i 0 −4.00000 4.76869 0 0 8.00000i 0 − 9.53738i
881.6 − 2.00000i 0 −4.00000 7.24119 0 0 8.00000i 0 − 14.4824i
881.7 − 2.00000i 0 −4.00000 11.8430 0 0 8.00000i 0 − 23.6860i
881.8 − 2.00000i 0 −4.00000 19.7628 0 0 8.00000i 0 − 39.5256i
881.9 2.00000i 0 −4.00000 −19.7628 0 0 − 8.00000i 0 − 39.5256i
881.10 2.00000i 0 −4.00000 −11.8430 0 0 − 8.00000i 0 − 23.6860i
881.11 2.00000i 0 −4.00000 −7.24119 0 0 − 8.00000i 0 − 14.4824i
881.12 2.00000i 0 −4.00000 −4.76869 0 0 − 8.00000i 0 − 9.53738i
881.13 2.00000i 0 −4.00000 4.76869 0 0 − 8.00000i 0 9.53738i
881.14 2.00000i 0 −4.00000 7.24119 0 0 − 8.00000i 0 14.4824i
881.15 2.00000i 0 −4.00000 11.8430 0 0 − 8.00000i 0 23.6860i
881.16 2.00000i 0 −4.00000 19.7628 0 0 − 8.00000i 0 39.5256i
\(n\): 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.c 16
3.b odd 2 1 inner 882.4.d.c 16
7.b odd 2 1 inner 882.4.d.c 16
7.c even 3 1 126.4.k.a ✓ 16
7.c even 3 1 882.4.k.a 16
7.d odd 6 1 126.4.k.a ✓ 16
7.d odd 6 1 882.4.k.a 16
21.c even 2 1 inner 882.4.d.c 16
21.g even 6 1 126.4.k.a ✓ 16
21.g even 6 1 882.4.k.a 16
21.h odd 6 1 126.4.k.a ✓ 16
21.h odd 6 1 882.4.k.a 16
28.f even 6 1 1008.4.bt.c 16
28.g odd 6 1 1008.4.bt.c 16
84.j odd 6 1 1008.4.bt.c 16
84.n even 6 1 1008.4.bt.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.k.a ✓ 16 7.c even 3 1
126.4.k.a ✓ 16 7.d odd 6 1
126.4.k.a ✓ 16 21.g even 6 1
126.4.k.a ✓ 16 21.h odd 6 1
882.4.d.c 16 1.a even 1 1 trivial
882.4.d.c 16 3.b odd 2 1 inner
882.4.d.c 16 7.b odd 2 1 inner
882.4.d.c 16 21.c even 2 1 inner
882.4.k.a 16 7.c even 3 1
882.4.k.a 16 7.d odd 6 1
882.4.k.a 16 21.g even 6 1
882.4.k.a 16 21.h odd 6 1
1008.4.bt.c 16 28.f even 6 1
1008.4.bt.c 16 28.g odd 6 1
1008.4.bt.c 16 84.j odd 6 1
1008.4.bt.c 16 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 606T_{5}^{6} + 95877T_{5}^{4} - 4751028T_{5}^{2} + 65318724 \) 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
$5$ \( (T^{8} - 606 T^{6} + \cdots + 65318724)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4230 T^{6} + \cdots + 72056812356)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 12498 T^{6} + \cdots + 138656437956)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 11316 T^{6} + \cdots + 2354966784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 95\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 29\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 35\!\cdots\!41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 626 T^{3} + \cdots + 195825358)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 399206154260736)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 278 T^{3} + \cdots - 863420834)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 62\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 75\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 80\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 34 T^{3} + \cdots + 58819069444)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1372 T^{3} + \cdots - 6319604609)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
show more
show less