Properties

Label 1224.4.c.e
Level $1224$
Weight $4$
Character orbit 1224.c
Analytic conductor $72.218$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1224.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.2183378470\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{7} + 5) q^{13} + ( - \beta_{6} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{17} + (\beta_{7} - \beta_{5} - 2 \beta_{3} + 4) q^{19} + ( - \beta_{6} + 5 \beta_1) q^{23} + ( - \beta_{7} + 3 \beta_{5} - 63) q^{25} + (2 \beta_{6} - 5 \beta_{4} - 18 \beta_{2}) q^{29} + (3 \beta_{6} - 4 \beta_{4} + 14 \beta_{2} + 5 \beta_1) q^{31} + ( - 2 \beta_{7} - 8 \beta_{5} + 130) q^{35} + (\beta_{4} - 28 \beta_{2} + 4 \beta_1) q^{37} + (6 \beta_{6} + 16 \beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{41} + ( - \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 2) q^{43} + (3 \beta_{7} - \beta_{5} - 2 \beta_{3} - 42) q^{47} + (\beta_{7} + 10 \beta_{5} + 8 \beta_{3} - 132) q^{49} + (3 \beta_{7} + 7 \beta_{5} - 6 \beta_{3} - 60) q^{53} + (7 \beta_{7} - 5 \beta_{5} + 6 \beta_{3} + 174) q^{55} + ( - \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 10) q^{59} + ( - 6 \beta_{6} + 5 \beta_{4} - 74 \beta_{2} - 16 \beta_1) q^{61} + (10 \beta_{6} + 16 \beta_{4} - 78 \beta_{2} - 20 \beta_1) q^{65} + ( - 9 \beta_{7} + \beta_{5} - 6 \beta_{3} - 76) q^{67} + (6 \beta_{6} - 27 \beta_{4} + 31 \beta_{2} - 29 \beta_1) q^{71} + (6 \beta_{6} - 26 \beta_{4} - 6 \beta_{2} + 20 \beta_1) q^{73} + ( - 3 \beta_{7} - 2 \beta_{5} - 8 \beta_{3} - 211) q^{77} + ( - 11 \beta_{6} - 16 \beta_{4} - 64 \beta_{2} - 5 \beta_1) q^{79} + ( - 9 \beta_{7} + 15 \beta_{5} - 2 \beta_{3} - 298) q^{83} + (7 \beta_{7} - 2 \beta_{6} + 11 \beta_{5} - 19 \beta_{4} + 94 \beta_{2} + \cdots - 268) q^{85}+ \cdots + (16 \beta_{6} - 22 \beta_{4} + 140 \beta_{2} - 24 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 284\nu^{5} - 2159\nu^{3} - 3402\nu ) / 172 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} + 327\nu^{4} + 6158\nu^{2} + 17420 ) / 172 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 59\nu^{7} + 5485\nu^{5} + 33588\nu^{3} + 47900\nu ) / 1032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -41\nu^{6} - 3781\nu^{4} - 20634\nu^{2} - 22844 ) / 172 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -109\nu^{7} - 9989\nu^{5} - 48702\nu^{3} - 15160\nu ) / 1032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\nu^{6} + 4217\nu^{4} + 28386\nu^{2} + 35292 ) / 172 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 3\beta_{5} + 4\beta_{3} - 188 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} + 23\beta_{4} + 33\beta_{2} - 78\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 277\beta_{7} + 279\beta_{5} - 342\beta_{3} + 14856 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -619\beta_{6} - 2075\beta_{4} - 3053\beta_{2} + 6708\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -24035\beta_{7} - 24253\beta_{5} + 29526\beta_{3} - 1279856 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 53561\beta_{6} + 179881\beta_{4} + 265039\beta_{2} - 580024\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.03229i
2.20783i
9.30031i
1.60125i
1.60125i
9.30031i
2.20783i
1.03229i
0 0 0 18.2701i 0 13.8757i 0 0 0
577.2 0 0 0 16.4090i 0 34.5010i 0 0 0
577.3 0 0 0 11.5318i 0 23.0485i 0 0 0
577.4 0 0 0 4.89575i 0 4.46235i 0 0 0
577.5 0 0 0 4.89575i 0 4.46235i 0 0 0
577.6 0 0 0 11.5318i 0 23.0485i 0 0 0
577.7 0 0 0 16.4090i 0 34.5010i 0 0 0
577.8 0 0 0 18.2701i 0 13.8757i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.4.c.e 8
3.b odd 2 1 136.4.b.b 8
12.b even 2 1 272.4.b.f 8
17.b even 2 1 inner 1224.4.c.e 8
51.c odd 2 1 136.4.b.b 8
51.f odd 4 2 2312.4.a.k 8
204.h even 2 1 272.4.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.b 8 3.b odd 2 1
136.4.b.b 8 51.c odd 2 1
272.4.b.f 8 12.b even 2 1
272.4.b.f 8 204.h even 2 1
1224.4.c.e 8 1.a even 1 1 trivial
1224.4.c.e 8 17.b even 2 1 inner
2312.4.a.k 8 51.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1224, [\chi])\):

\( T_{5}^{8} + 760T_{5}^{6} + 187712T_{5}^{4} + 16028160T_{5}^{2} + 286466048 \) Copy content Toggle raw display
\( T_{47}^{4} + 156T_{47}^{3} - 57344T_{47}^{2} - 3407872T_{47} + 134217728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 760 T^{6} + \cdots + 286466048 \) Copy content Toggle raw display
$7$ \( T^{8} + 1934 T^{6} + \cdots + 2424307712 \) Copy content Toggle raw display
$11$ \( T^{8} + 7406 T^{6} + \cdots + 1063158272 \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{3} - 5836 T^{2} + \cdots + 8525216)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 582622237229761 \) Copy content Toggle raw display
$19$ \( (T^{4} - 24 T^{3} - 21056 T^{2} + \cdots + 44946176)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 36734 T^{6} + \cdots + 2935871578112 \) Copy content Toggle raw display
$29$ \( T^{8} + 106232 T^{6} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{8} + 155918 T^{6} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{8} + 166072 T^{6} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{8} + 419392 T^{6} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 60592 T^{2} + \cdots + 112195072)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 156 T^{3} - 57344 T^{2} + \cdots + 134217728)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 236 T^{3} + \cdots - 14761769616)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 36 T^{3} - 60112 T^{2} + \cdots + 70465536)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 1597240 T^{6} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( (T^{4} + 312 T^{3} + \cdots + 30967766784)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 1607086 T^{6} + \cdots + 14\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{8} + 1779648 T^{6} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{8} + 1578638 T^{6} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( (T^{4} + 1236 T^{3} + \cdots - 54225864704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 34 T^{3} - 1202908 T^{2} + \cdots + 21605388512)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 5185024 T^{6} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
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