Properties

Label 1216.2.i.p
Level $1216$
Weight $2$
Character orbit 1216.i
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.39075800976.1
Defining polynomial: \(x^{8} - x^{7} + 12 x^{6} - 13 x^{5} + 125 x^{4} - 116 x^{3} + 232 x^{2} + 96 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 608)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{4} ) q^{3} -\beta_{1} q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{4} ) q^{3} -\beta_{1} q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{9} + ( -\beta_{2} + \beta_{6} ) q^{11} + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{13} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{15} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{17} + ( \beta_{4} - \beta_{7} ) q^{19} + ( -\beta_{6} - \beta_{7} ) q^{23} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{25} + ( -3 \beta_{2} + 2 \beta_{6} ) q^{27} + ( -3 + 3 \beta_{1} + 3 \beta_{5} ) q^{29} -2 \beta_{2} q^{31} + ( -3 - 5 \beta_{1} + 3 \beta_{3} ) q^{33} + ( -4 + 2 \beta_{5} ) q^{37} + ( -4 \beta_{2} + \beta_{6} ) q^{39} + ( -1 + \beta_{3} ) q^{41} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{43} + ( 12 - 4 \beta_{5} ) q^{45} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{47} -7 q^{49} + ( -4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{51} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{53} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{55} + ( 10 - 3 \beta_{1} - 2 \beta_{3} - 5 \beta_{5} ) q^{57} + ( -\beta_{2} + \beta_{4} ) q^{59} + ( -1 + \beta_{1} - 4 \beta_{3} + \beta_{5} ) q^{61} + ( 7 - 3 \beta_{5} ) q^{65} + ( -3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( 1 - 3 \beta_{5} ) q^{69} -\beta_{7} q^{71} + ( 5 + 4 \beta_{1} - 5 \beta_{3} ) q^{73} + ( -\beta_{2} + \beta_{6} ) q^{75} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{79} + ( -5 - 6 \beta_{1} + 5 \beta_{3} ) q^{81} + ( -\beta_{2} - \beta_{6} ) q^{83} + ( 3 - 3 \beta_{1} + 4 \beta_{3} - 3 \beta_{5} ) q^{85} + ( -6 \beta_{2} + 3 \beta_{6} ) q^{87} + ( -3 + 3 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} ) q^{89} + ( -10 - 4 \beta_{1} + 10 \beta_{3} ) q^{93} + ( -2 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{95} + ( -5 + 4 \beta_{1} + 5 \beta_{3} ) q^{97} + ( 10 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{5} - 12q^{9} + O(q^{10}) \) \( 8q - 2q^{5} - 12q^{9} - 10q^{13} + 10q^{17} + 2q^{25} - 6q^{29} - 22q^{33} - 24q^{37} - 4q^{41} + 80q^{45} - 56q^{49} + 10q^{53} + 46q^{57} - 18q^{61} + 44q^{65} - 4q^{69} + 28q^{73} - 32q^{81} + 22q^{85} + 18q^{89} - 48q^{93} - 12q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 12 x^{6} - 13 x^{5} + 125 x^{4} - 116 x^{3} + 232 x^{2} + 96 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 83 \nu^{7} - 325 \nu^{6} + 2470 \nu^{5} - 3543 \nu^{4} + 26065 \nu^{3} - 38870 \nu^{2} + 139144 \nu - 11440 \)\()/47664\)
\(\beta_{2}\)\(=\)\((\)\( -1067 \nu^{7} - 1445 \nu^{6} - 10864 \nu^{5} - 873 \nu^{4} - 106543 \nu^{3} - 31816 \nu^{2} - 15520 \nu + 330448 \)\()/238320\)
\(\beta_{3}\)\(=\)\((\)\( 557 \nu^{7} - 865 \nu^{6} + 6574 \nu^{5} - 10377 \nu^{4} + 69373 \nu^{3} - 103454 \nu^{2} + 120040 \nu + 48992 \)\()/79440\)
\(\beta_{4}\)\(=\)\((\)\( 1138 \nu^{7} - 3140 \nu^{6} + 12941 \nu^{5} - 35178 \nu^{4} + 140612 \nu^{3} - 305041 \nu^{2} + 204320 \nu + 80128 \)\()/119160\)
\(\beta_{5}\)\(=\)\((\)\( 154 \nu^{7} + 55 \nu^{6} + 1568 \nu^{5} + 126 \nu^{4} + 14456 \nu^{3} + 4592 \nu^{2} + 2240 \nu + 37684 \)\()/14895\)
\(\beta_{6}\)\(=\)\((\)\( 649 \nu^{7} + 19 \nu^{6} + 6608 \nu^{5} + 531 \nu^{4} + 71561 \nu^{3} + 19352 \nu^{2} + 9440 \nu + 229456 \)\()/47664\)
\(\beta_{7}\)\(=\)\((\)\( 5429 \nu^{7} - 9055 \nu^{6} + 68818 \nu^{5} - 118329 \nu^{4} + 726211 \nu^{3} - 1082978 \nu^{2} + 1956640 \nu - 318736 \)\()/238320\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - 11 \beta_{3} + \beta_{1} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\(4 \beta_{6} - 7 \beta_{5} - 4 \beta_{2} + 4\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{7} - 37 \beta_{4} + 99 \beta_{3} + 37 \beta_{2} - 9 \beta_{1} - 99\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-75 \beta_{7} - 75 \beta_{6} + 149 \beta_{5} + 79 \beta_{4} + 25 \beta_{3} + 149 \beta_{1} - 149\)\()/2\)
\(\nu^{6}\)\(=\)\(-24 \beta_{6} - 55 \beta_{5} - 200 \beta_{2} + 532\)
\(\nu^{7}\)\(=\)\((\)\(725 \beta_{7} - 849 \beta_{4} + 7 \beta_{3} + 849 \beta_{2} - 1525 \beta_{1} - 7\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.63248 2.82754i
1.51772 + 2.62877i
−0.236942 0.410396i
0.851703 + 1.47519i
−1.63248 + 2.82754i
1.51772 2.62877i
−0.236942 + 0.410396i
0.851703 1.47519i
0 −1.59084 2.75542i 0 −1.28078 2.21837i 0 0 0 −3.56155 + 6.16879i 0
577.2 0 −0.684999 1.18645i 0 0.780776 + 1.35234i 0 0 0 0.561553 0.972638i 0
577.3 0 0.684999 + 1.18645i 0 0.780776 + 1.35234i 0 0 0 0.561553 0.972638i 0
577.4 0 1.59084 + 2.75542i 0 −1.28078 2.21837i 0 0 0 −3.56155 + 6.16879i 0
961.1 0 −1.59084 + 2.75542i 0 −1.28078 + 2.21837i 0 0 0 −3.56155 6.16879i 0
961.2 0 −0.684999 + 1.18645i 0 0.780776 1.35234i 0 0 0 0.561553 + 0.972638i 0
961.3 0 0.684999 1.18645i 0 0.780776 1.35234i 0 0 0 0.561553 + 0.972638i 0
961.4 0 1.59084 2.75542i 0 −1.28078 + 2.21837i 0 0 0 −3.56155 6.16879i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.i.p 8
4.b odd 2 1 inner 1216.2.i.p 8
8.b even 2 1 608.2.i.d 8
8.d odd 2 1 608.2.i.d 8
19.c even 3 1 inner 1216.2.i.p 8
76.g odd 6 1 inner 1216.2.i.p 8
152.k odd 6 1 608.2.i.d 8
152.p even 6 1 608.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.2.i.d 8 8.b even 2 1
608.2.i.d 8 8.d odd 2 1
608.2.i.d 8 152.k odd 6 1
608.2.i.d 8 152.p even 6 1
1216.2.i.p 8 1.a even 1 1 trivial
1216.2.i.p 8 4.b odd 2 1 inner
1216.2.i.p 8 19.c even 3 1 inner
1216.2.i.p 8 76.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{8} + 12 T_{3}^{6} + 125 T_{3}^{4} + 228 T_{3}^{2} + 361 \)
\( T_{5}^{4} + T_{5}^{3} + 5 T_{5}^{2} - 4 T_{5} + 16 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 361 + 228 T^{2} + 125 T^{4} + 12 T^{6} + T^{8} \)
$5$ \( ( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 304 - 37 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$17$ \( ( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$19$ \( 130321 + 12635 T^{2} + 684 T^{4} + 35 T^{6} + T^{8} \)
$23$ \( 5776 + 2052 T^{2} + 653 T^{4} + 27 T^{6} + T^{8} \)
$29$ \( ( 1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$31$ \( ( 304 - 48 T^{2} + T^{4} )^{2} \)
$37$ \( ( -8 + 6 T + T^{2} )^{4} \)
$41$ \( ( 1 + T + T^{2} )^{4} \)
$43$ \( 1478656 + 96064 T^{2} + 5025 T^{4} + 79 T^{6} + T^{8} \)
$47$ \( 92416 + 21584 T^{2} + 4737 T^{4} + 71 T^{6} + T^{8} \)
$53$ \( ( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$59$ \( 361 + 228 T^{2} + 125 T^{4} + 12 T^{6} + T^{8} \)
$61$ \( ( 256 + 144 T + 65 T^{2} + 9 T^{3} + T^{4} )^{2} \)
$67$ \( 47045881 + 1563852 T^{2} + 45125 T^{4} + 228 T^{6} + T^{8} \)
$71$ \( 5776 + 2052 T^{2} + 653 T^{4} + 27 T^{6} + T^{8} \)
$73$ \( ( 361 + 266 T + 215 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$79$ \( 92416 + 21584 T^{2} + 4737 T^{4} + 71 T^{6} + T^{8} \)
$83$ \( ( 76 - 41 T^{2} + T^{4} )^{2} \)
$89$ \( ( 324 + 162 T + 99 T^{2} - 9 T^{3} + T^{4} )^{2} \)
$97$ \( ( 3481 - 354 T + 95 T^{2} + 6 T^{3} + T^{4} )^{2} \)
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