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Properties

Label 882.5.c.j
Level $882$
Weight $5$
Character orbit 882.c
Analytic conductor $91.172$
Analytic rank $0$
Dimension $12$
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] = [882,5,Mod(685,882)]
 
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([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.685");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (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: \(91.1723074400\)
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: \( x^{12} - 2 x^{11} + 37 x^{10} - 26 x^{9} + 936 x^{8} - 694 x^{7} + 9925 x^{6} + 938 x^{5} + 67795 x^{4} + \cdots + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{6}\cdot 7^{8} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 8 q^{4} - \beta_{3} q^{5} - 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 8 q^{4} - \beta_{3} q^{5} - 8 \beta_1 q^{8} + ( - \beta_{8} + \beta_{2}) q^{10} + (\beta_{6} - 4 \beta_1) q^{11} + ( - 3 \beta_{10} + 2 \beta_{8}) q^{13} + 64 q^{16} + ( - 2 \beta_{11} + 2 \beta_{5} + 4 \beta_{3}) q^{17} + ( - \beta_{10} + 8 \beta_{8}) q^{19} - 8 \beta_{3} q^{20} + ( - \beta_{9} + 3 \beta_{7} + 32) q^{22} + ( - 2 \beta_{6} - 4 \beta_{4} + 20 \beta_1) q^{23} + ( - \beta_{7} + 263) q^{25} + (3 \beta_{11} + 2 \beta_{5} + 16 \beta_{3}) q^{26} + ( - 3 \beta_{6} + 5 \beta_{4} - 198 \beta_1) q^{29} + (4 \beta_{10} + 5 \beta_{8} + 40 \beta_{2}) q^{31} - 64 \beta_1 q^{32} + (16 \beta_{10} + 4 \beta_{8} + 12 \beta_{2}) q^{34} + ( - 4 \beta_{9} - 11 \beta_{7} + 683) q^{37} + (\beta_{11} + 8 \beta_{5} + 64 \beta_{3}) q^{38} + ( - 8 \beta_{8} + 8 \beta_{2}) q^{40} + (2 \beta_{11} + 34 \beta_{5} - 34 \beta_{3}) q^{41} + (6 \beta_{9} - 5 \beta_{7} + 171) q^{43} + (8 \beta_{6} - 32 \beta_1) q^{44} + ( - 6 \beta_{9} - 14 \beta_{7} - 160) q^{46} + (12 \beta_{5} - 46 \beta_{3}) q^{47} + ( - 2 \beta_{6} - \beta_{4} - 263 \beta_1) q^{50} + ( - 24 \beta_{10} + 16 \beta_{8}) q^{52} + ( - \beta_{6} + 5 \beta_{4} - 578 \beta_1) q^{53} + ( - 23 \beta_{10} + \cdots - 178 \beta_{2}) q^{55}+ \cdots + (37 \beta_{10} + 22 \beta_{8} - 323 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 96 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 96 q^{4} + 768 q^{16} + 384 q^{22} + 3156 q^{25} + 8196 q^{37} + 2052 q^{43} - 1920 q^{46} + 19008 q^{58} + 6144 q^{64} + 5148 q^{67} + 53652 q^{79} + 13152 q^{85} + 3072 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 37 x^{10} - 26 x^{9} + 936 x^{8} - 694 x^{7} + 9925 x^{6} + 938 x^{5} + 67795 x^{4} + \cdots + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 345294368 \nu^{11} - 465802015 \nu^{10} + 12358168121 \nu^{9} - 759758236 \nu^{8} + \cdots + 1213833414564 ) / 389183569425 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21813775196194 \nu^{11} - 41349240400370 \nu^{10} + 804132724503868 \nu^{9} + \cdots - 36\!\cdots\!63 ) / 36\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5647395023159 \nu^{11} - 10544653464940 \nu^{10} + 208152343802678 \nu^{9} + \cdots - 10\!\cdots\!28 ) / 730860797808855 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4131228834427 \nu^{11} - 5812119126065 \nu^{10} + 148026934696834 \nu^{9} + \cdots + 85\!\cdots\!16 ) / 406033776560475 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22467638 \nu^{11} + 44745295 \nu^{10} - 829516871 \nu^{9} + 575346580 \nu^{8} + \cdots + 30920365116 ) / 782619327 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 363731084496236 \nu^{11} + 494495338604680 \nu^{10} + \cdots - 47\!\cdots\!28 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 320845882656 \nu^{11} + 433832128485 \nu^{10} - 11481235410522 \nu^{9} + \cdots - 559326276753778 ) / 9022972812455 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 140766936776269 \nu^{11} + 283867377067925 \nu^{10} + \cdots + 18\!\cdots\!93 ) / 36\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 378898227243 \nu^{11} + 518887253430 \nu^{10} - 13566380034306 \nu^{9} + \cdots - 23\!\cdots\!74 ) / 9022972812455 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20074371700882 \nu^{11} + 39125780337350 \nu^{10} - 740367533975374 \nu^{9} + \cdots + 30\!\cdots\!29 ) / 406033776560475 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11911681773371 \nu^{11} + 23312747400250 \nu^{10} - 439341113690777 \nu^{9} + \cdots + 17\!\cdots\!52 ) / 81206755312095 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 6\beta_{10} - \beta_{9} + 3\beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - 10\beta_{3} + 3\beta_{2} + 28 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - 4 \beta_{8} + \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 31 \beta_{3} + \cdots - 490 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21\beta_{9} - 11\beta_{7} - 6\beta_{6} + 25\beta_{4} + 42\beta _1 - 952 ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 23 \beta_{11} + 33 \beta_{10} + 12 \beta_{9} + 120 \beta_{8} + 27 \beta_{7} + 21 \beta_{6} + \cdots - 9618 ) / 84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 398 \beta_{11} + 1261 \beta_{10} - 242 \beta_{9} + 38 \beta_{8} + 51 \beta_{7} + 103 \beta_{6} + \cdots + 14434 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -475\beta_{9} - 719\beta_{7} - 380\beta_{6} - 1055\beta_{4} - 53410\beta _1 + 205744 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19551 \beta_{11} - 57974 \beta_{10} - 11605 \beta_{9} - 11809 \beta_{8} - 51 \beta_{7} + 5428 \beta_{6} + \cdots + 809900 ) / 168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 20283 \beta_{11} - 55597 \beta_{10} + 14846 \beta_{9} - 89378 \beta_{8} + 18633 \beta_{7} + \cdots - 4644514 ) / 84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 283435\beta_{9} + 43679\beta_{7} - 132078\beta_{6} + 343735\beta_{4} + 6284586\beta _1 - 22287328 ) / 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 591778 \beta_{11} + 1665197 \beta_{10} + 427599 \beta_{9} + 2318791 \beta_{8} + 472798 \beta_{7} + \cdots - 108599890 ) / 84 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5804591 \beta_{11} + 16569439 \beta_{10} - 3497091 \beta_{9} + 7161155 \beta_{8} - 916016 \beta_{7} + \cdots + 303916550 ) / 84 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
1.91847 − 3.32288i
0.0765877 + 0.132654i
−1.49506 − 2.58951i
−1.49506 + 2.58951i
0.0765877 − 0.132654i
1.91847 + 3.32288i
0.223599 + 0.387284i
2.53510 − 4.39092i
−2.25870 + 3.91218i
−2.25870 − 3.91218i
2.53510 + 4.39092i
0.223599 − 0.387284i
−2.82843 0 8.00000 − 23.2086i 0 0 −22.6274 0 65.6437i
685.2 −2.82843 0 8.00000 − 17.6540i 0 0 −22.6274 0 49.9330i
685.3 −2.82843 0 8.00000 − 15.3525i 0 0 −22.6274 0 43.4235i
685.4 −2.82843 0 8.00000 15.3525i 0 0 −22.6274 0 − 43.4235i
685.5 −2.82843 0 8.00000 17.6540i 0 0 −22.6274 0 − 49.9330i
685.6 −2.82843 0 8.00000 23.2086i 0 0 −22.6274 0 − 65.6437i
685.7 2.82843 0 8.00000 − 23.2086i 0 0 22.6274 0 − 65.6437i
685.8 2.82843 0 8.00000 − 17.6540i 0 0 22.6274 0 − 49.9330i
685.9 2.82843 0 8.00000 − 15.3525i 0 0 22.6274 0 − 43.4235i
685.10 2.82843 0 8.00000 15.3525i 0 0 22.6274 0 43.4235i
685.11 2.82843 0 8.00000 17.6540i 0 0 22.6274 0 49.9330i
685.12 2.82843 0 8.00000 23.2086i 0 0 22.6274 0 65.6437i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.12
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 882.5.c.j 12
3.b odd 2 1 inner 882.5.c.j 12
7.b odd 2 1 inner 882.5.c.j 12
7.c even 3 1 126.5.n.d ✓ 12
7.d odd 6 1 126.5.n.d ✓ 12
21.c even 2 1 inner 882.5.c.j 12
21.g even 6 1 126.5.n.d ✓ 12
21.h odd 6 1 126.5.n.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.5.n.d ✓ 12 7.c even 3 1
126.5.n.d ✓ 12 7.d odd 6 1
126.5.n.d ✓ 12 21.g even 6 1
126.5.n.d ✓ 12 21.h odd 6 1
882.5.c.j 12 1.a even 1 1 trivial
882.5.c.j 12 3.b odd 2 1 inner
882.5.c.j 12 7.b odd 2 1 inner
882.5.c.j 12 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 \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{6} + 1086T_{5}^{4} + 368289T_{5}^{2} + 39567744 \) Copy content Toggle raw display
\( T_{11}^{6} - 77706T_{11}^{4} + 1576646937T_{11}^{2} - 2698852151808 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 1086 T^{4} + \cdots + 39567744)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 2698852151808)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 68617377870768)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 609034717200384)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 56\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 40\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 23483910654603)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2049 T^{2} + \cdots + 3605802080)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 513 T^{2} + \cdots - 1219705360)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 1287 T^{2} + \cdots + 29233629088)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 34\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 13413 T^{2} + \cdots + 323573453617)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 88\!\cdots\!32)^{2} \) Copy content Toggle raw display
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