Properties

Label 882.5.c.a
Level $882$
Weight $5$
Character orbit 882.c
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + 8 q^{4} + ( -11 \beta_{2} - \beta_{3} ) q^{5} -8 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + 8 q^{4} + ( -11 \beta_{2} - \beta_{3} ) q^{5} -8 \beta_{1} q^{8} + ( -8 \beta_{2} - 11 \beta_{3} ) q^{10} + ( -81 + 6 \beta_{1} ) q^{11} + ( -36 \beta_{2} + 34 \beta_{3} ) q^{13} + 64 q^{16} + ( 34 \beta_{2} - 34 \beta_{3} ) q^{17} + ( 74 \beta_{2} - 34 \beta_{3} ) q^{19} + ( -88 \beta_{2} - 8 \beta_{3} ) q^{20} + ( -48 + 81 \beta_{1} ) q^{22} + ( 156 - 306 \beta_{1} ) q^{23} + ( 238 + 66 \beta_{1} ) q^{25} + ( 272 \beta_{2} - 36 \beta_{3} ) q^{26} + ( -681 - 21 \beta_{1} ) q^{29} + ( -631 \beta_{2} - 87 \beta_{3} ) q^{31} -64 \beta_{1} q^{32} + ( -272 \beta_{2} + 34 \beta_{3} ) q^{34} + ( -698 - 48 \beta_{1} ) q^{37} + ( -272 \beta_{2} + 74 \beta_{3} ) q^{38} + ( -64 \beta_{2} - 88 \beta_{3} ) q^{40} + ( 518 \beta_{2} - 392 \beta_{3} ) q^{41} + ( -158 + 1140 \beta_{1} ) q^{43} + ( -648 + 48 \beta_{1} ) q^{44} + ( 2448 - 156 \beta_{1} ) q^{46} + ( -1316 \beta_{2} - 340 \beta_{3} ) q^{47} + ( -528 - 238 \beta_{1} ) q^{50} + ( -288 \beta_{2} + 272 \beta_{3} ) q^{52} + ( -519 + 591 \beta_{1} ) q^{53} + ( 939 \beta_{2} + 147 \beta_{3} ) q^{55} + ( 168 + 681 \beta_{1} ) q^{58} + ( 161 \beta_{2} - 656 \beta_{3} ) q^{59} + ( -848 \beta_{2} + 164 \beta_{3} ) q^{61} + ( -696 \beta_{2} - 631 \beta_{3} ) q^{62} + 512 q^{64} + ( -372 - 1014 \beta_{1} ) q^{65} + ( -7300 - 486 \beta_{1} ) q^{67} + ( 272 \beta_{2} - 272 \beta_{3} ) q^{68} + ( 2424 - 732 \beta_{1} ) q^{71} + ( 3764 \beta_{2} + 500 \beta_{3} ) q^{73} + ( 384 + 698 \beta_{1} ) q^{74} + ( 592 \beta_{2} - 272 \beta_{3} ) q^{76} + ( -1987 + 2991 \beta_{1} ) q^{79} + ( -704 \beta_{2} - 64 \beta_{3} ) q^{80} + ( -3136 \beta_{2} + 518 \beta_{3} ) q^{82} + ( 2741 \beta_{2} - 710 \beta_{3} ) q^{83} + ( 306 + 1020 \beta_{1} ) q^{85} + ( -9120 + 158 \beta_{1} ) q^{86} + ( -384 + 648 \beta_{1} ) q^{88} + ( -5526 \beta_{2} + 1218 \beta_{3} ) q^{89} + ( 1248 - 2448 \beta_{1} ) q^{92} + ( -2720 \beta_{2} - 1316 \beta_{3} ) q^{94} + ( 1626 + 900 \beta_{1} ) q^{95} + ( 1013 \beta_{2} + 40 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{4} + O(q^{10}) \) \( 4q + 32q^{4} - 324q^{11} + 256q^{16} - 192q^{22} + 624q^{23} + 952q^{25} - 2724q^{29} - 2792q^{37} - 632q^{43} - 2592q^{44} + 9792q^{46} - 2112q^{50} - 2076q^{53} + 672q^{58} + 2048q^{64} - 1488q^{65} - 29200q^{67} + 9696q^{71} + 1536q^{74} - 7948q^{79} + 1224q^{85} - 36480q^{86} - 1536q^{88} + 4992q^{92} + 6504q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
−0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
0.707107 1.22474i
−2.82843 0 8.00000 14.1536i 0 0 −22.6274 0 40.0324i
685.2 −2.82843 0 8.00000 14.1536i 0 0 −22.6274 0 40.0324i
685.3 2.82843 0 8.00000 23.9515i 0 0 22.6274 0 67.7452i
685.4 2.82843 0 8.00000 23.9515i 0 0 22.6274 0 67.7452i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 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.a 4
3.b odd 2 1 294.5.c.a 4
7.b odd 2 1 inner 882.5.c.a 4
7.c even 3 1 126.5.n.b 4
7.d odd 6 1 126.5.n.b 4
21.c even 2 1 294.5.c.a 4
21.g even 6 1 42.5.g.a 4
21.g even 6 1 294.5.g.c 4
21.h odd 6 1 42.5.g.a 4
21.h odd 6 1 294.5.g.c 4
84.j odd 6 1 336.5.bh.d 4
84.n even 6 1 336.5.bh.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.g.a 4 21.g even 6 1
42.5.g.a 4 21.h odd 6 1
126.5.n.b 4 7.c even 3 1
126.5.n.b 4 7.d odd 6 1
294.5.c.a 4 3.b odd 2 1
294.5.c.a 4 21.c even 2 1
294.5.g.c 4 21.g even 6 1
294.5.g.c 4 21.h odd 6 1
336.5.bh.d 4 84.j odd 6 1
336.5.bh.d 4 84.n even 6 1
882.5.c.a 4 1.a even 1 1 trivial
882.5.c.a 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 774 T_{5}^{2} + 114921 \)
\( T_{11}^{2} + 162 T_{11} + 6273 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -8 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 114921 + 774 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 6273 + 162 T + T^{2} )^{2} \)
$13$ \( 569108736 + 63264 T^{2} + T^{4} \)
$17$ \( 589324176 + 62424 T^{2} + T^{4} \)
$19$ \( 128051856 + 88344 T^{2} + T^{4} \)
$23$ \( ( -724752 - 312 T + T^{2} )^{2} \)
$29$ \( ( 460233 + 1362 T + T^{2} )^{2} \)
$31$ \( 1025818531929 + 2752278 T^{2} + T^{4} \)
$37$ \( ( 468772 + 1396 T + T^{2} )^{2} \)
$41$ \( 8311481425296 + 8985816 T^{2} + T^{4} \)
$43$ \( ( -10371836 + 316 T + T^{2} )^{2} \)
$47$ \( 5862054484224 + 15939936 T^{2} + T^{4} \)
$53$ \( ( -2524887 + 1038 T + T^{2} )^{2} \)
$59$ \( 105068670590601 + 20811654 T^{2} + T^{4} \)
$61$ \( 2285563428864 + 5605632 T^{2} + T^{4} \)
$67$ \( ( 51400432 + 14600 T + T^{2} )^{2} \)
$71$ \( ( 1589184 - 4848 T + T^{2} )^{2} \)
$73$ \( 1332475433535744 + 97006176 T^{2} + T^{4} \)
$79$ \( ( -67620479 + 3974 T + T^{2} )^{2} \)
$83$ \( 109011202550649 + 69275286 T^{2} + T^{4} \)
$89$ \( 3136610653724304 + 254429208 T^{2} + T^{4} \)
$97$ \( 9242250571449 + 6233814 T^{2} + T^{4} \)
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