Properties

Label 1344.4.c.d
Level $1344$
Weight $4$
Character orbit 1344.c
Analytic conductor $79.299$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 722 x^{3} + 11881 x^{2} + 54936 x + 127008\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + 7 q^{7} -9 q^{9} +O(q^{10})\) \( q -3 \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + 7 q^{7} -9 q^{9} + ( 9 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{11} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{13} + ( 3 - 3 \beta_{2} ) q^{15} + ( -3 - 2 \beta_{2} - \beta_{4} ) q^{17} + ( 22 \beta_{1} - 5 \beta_{3} + \beta_{5} ) q^{19} -21 \beta_{1} q^{21} + ( -9 - 3 \beta_{4} ) q^{23} + ( -25 - 5 \beta_{2} + \beta_{4} ) q^{25} + 27 \beta_{1} q^{27} + ( -18 \beta_{1} + \beta_{3} - \beta_{5} ) q^{29} + ( -92 + 3 \beta_{2} + 3 \beta_{4} ) q^{31} + ( 27 - 6 \beta_{2} - 3 \beta_{4} ) q^{33} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{35} + ( -40 \beta_{1} + \beta_{3} + \beta_{5} ) q^{37} + ( -6 - 3 \beta_{2} - 3 \beta_{4} ) q^{39} + ( -53 - 18 \beta_{2} + 5 \beta_{4} ) q^{41} + ( 46 \beta_{1} - 10 \beta_{3} + 2 \beta_{5} ) q^{43} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{45} + ( -260 + 8 \beta_{2} ) q^{47} + 49 q^{49} + ( 9 \beta_{1} + 6 \beta_{3} + 3 \beta_{5} ) q^{51} + ( -70 \beta_{1} + 5 \beta_{3} + 11 \beta_{5} ) q^{53} + ( -272 + 11 \beta_{2} + 11 \beta_{4} ) q^{55} + ( 66 - 15 \beta_{2} + 3 \beta_{4} ) q^{57} + ( -164 \beta_{1} + 2 \beta_{3} - 6 \beta_{5} ) q^{59} + ( 164 \beta_{1} - 11 \beta_{3} - 5 \beta_{5} ) q^{61} -63 q^{63} + ( -112 + 6 \beta_{2} + 10 \beta_{4} ) q^{65} + ( 92 \beta_{1} - 42 \beta_{3} ) q^{67} + ( 27 \beta_{1} + 9 \beta_{5} ) q^{69} + ( -435 - 26 \beta_{2} + 5 \beta_{4} ) q^{71} + ( 88 + 28 \beta_{2} - 14 \beta_{4} ) q^{73} + ( 75 \beta_{1} + 15 \beta_{3} - 3 \beta_{5} ) q^{75} + ( 63 \beta_{1} - 14 \beta_{3} - 7 \beta_{5} ) q^{77} + ( -582 + 16 \beta_{2} - 2 \beta_{4} ) q^{79} + 81 q^{81} + ( -238 \beta_{1} + 2 \beta_{3} + 8 \beta_{5} ) q^{83} + ( 260 \beta_{1} + \beta_{3} - 11 \beta_{5} ) q^{85} + ( -54 + 3 \beta_{2} - 3 \beta_{4} ) q^{87} + ( 5 - 14 \beta_{2} - 21 \beta_{4} ) q^{89} + ( -14 \beta_{1} - 7 \beta_{3} - 7 \beta_{5} ) q^{91} + ( 276 \beta_{1} - 9 \beta_{3} - 9 \beta_{5} ) q^{93} + ( -802 - 22 \beta_{2} - 4 \beta_{4} ) q^{95} + ( 280 + 62 \beta_{2} - 4 \beta_{4} ) q^{97} + ( -81 \beta_{1} + 18 \beta_{3} + 9 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 42q^{7} - 54q^{9} + O(q^{10}) \) \( 6q + 42q^{7} - 54q^{9} + 24q^{15} - 16q^{17} - 60q^{23} - 138q^{25} - 552q^{31} + 168q^{33} - 36q^{39} - 272q^{41} - 1576q^{47} + 294q^{49} - 1632q^{55} + 432q^{57} - 378q^{63} - 664q^{65} - 2548q^{71} + 444q^{73} - 3528q^{79} + 486q^{81} - 336q^{87} + 16q^{89} - 4776q^{95} + 1548q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 722 x^{3} + 11881 x^{2} + 54936 x + 127008\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -5869 \nu^{5} + 25724 \nu^{4} - 85196 \nu^{3} - 2684972 \nu^{2} - 64680643 \nu - 172672164 \)\()/ 142698276 \)
\(\beta_{2}\)\(=\)\((\)\( -111 \nu^{5} + 583 \nu^{4} - 12321 \nu^{3} - 40071 \nu^{2} - 55944 \nu - 7048478 \)\()/1132526\)
\(\beta_{3}\)\(=\)\((\)\( 19855 \nu^{5} - 99182 \nu^{4} + 1637642 \nu^{3} + 7733918 \nu^{2} + 357126139 \nu + 918082116 \)\()/ 142698276 \)
\(\beta_{4}\)\(=\)\((\)\( -1721 \nu^{5} + 29445 \nu^{4} - 191031 \nu^{3} - 621281 \nu^{2} - 867384 \nu + 78246822 \)\()/1132526\)
\(\beta_{5}\)\(=\)\((\)\( -233885 \nu^{5} + 988654 \nu^{4} + 653126 \nu^{3} - 206179030 \nu^{2} - 2162653733 \nu - 5879361852 \)\()/47566092\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 9 \beta_{3} + 150 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{5} + \beta_{4} + 118 \beta_{3} - 118 \beta_{2} + 763 \beta_{1} - 763\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(111 \beta_{4} - 1721 \beta_{2} - 18380\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(472 \beta_{5} + 472 \beta_{4} - 16851 \beta_{3} - 16851 \beta_{2} - 139347 \beta_{1} - 139347\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1
8.80977 + 8.80977i
−2.93235 2.93235i
−4.87742 4.87742i
−4.87742 + 4.87742i
−2.93235 + 2.93235i
8.80977 8.80977i
0 3.00000i 0 15.6195i 0 7.00000 0 −9.00000 0
673.2 0 3.00000i 0 7.86469i 0 7.00000 0 −9.00000 0
673.3 0 3.00000i 0 11.7548i 0 7.00000 0 −9.00000 0
673.4 0 3.00000i 0 11.7548i 0 7.00000 0 −9.00000 0
673.5 0 3.00000i 0 7.86469i 0 7.00000 0 −9.00000 0
673.6 0 3.00000i 0 15.6195i 0 7.00000 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.c.d yes 6
4.b odd 2 1 1344.4.c.a 6
8.b even 2 1 inner 1344.4.c.d yes 6
8.d odd 2 1 1344.4.c.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.a 6 4.b odd 2 1
1344.4.c.a 6 8.d odd 2 1
1344.4.c.d yes 6 1.a even 1 1 trivial
1344.4.c.d yes 6 8.b 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 \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{6} + 444 T_{5}^{4} + 57348 T_{5}^{2} + 2085136 \)
\( T_{23}^{3} + 30 T_{23}^{2} - 21186 T_{23} + 470448 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 9 + T^{2} )^{3} \)
$5$ \( 2085136 + 57348 T^{2} + 444 T^{4} + T^{6} \)
$7$ \( ( -7 + T )^{6} \)
$11$ \( 906973456 + 6187140 T^{2} + 6396 T^{4} + T^{6} \)
$13$ \( 369869824 + 6460800 T^{2} + 5028 T^{4} + T^{6} \)
$17$ \( ( -66092 - 3046 T + 8 T^{2} + T^{3} )^{2} \)
$19$ \( 71892624384 + 82646928 T^{2} + 18456 T^{4} + T^{6} \)
$23$ \( ( 470448 - 21186 T + 30 T^{2} + T^{3} )^{2} \)
$29$ \( 97140736 + 1651728 T^{2} + 6456 T^{4} + T^{6} \)
$31$ \( ( -913888 + 2820 T + 276 T^{2} + T^{3} )^{2} \)
$37$ \( 488233216 + 10556304 T^{2} + 9816 T^{4} + T^{6} \)
$41$ \( ( -9550804 - 133462 T + 136 T^{2} + T^{3} )^{2} \)
$43$ \( 4829129310784 + 1332972336 T^{2} + 74412 T^{4} + T^{6} \)
$47$ \( ( 13844416 + 192944 T + 788 T^{2} + T^{3} )^{2} \)
$53$ \( 924071477319936 + 88056208656 T^{2} + 591720 T^{4} + T^{6} \)
$59$ \( 4716055035904 + 2193475584 T^{2} + 259344 T^{4} + T^{6} \)
$61$ \( 268125124864 + 519241152 T^{2} + 244260 T^{4} + T^{6} \)
$67$ \( 15703315888888384 + 204446401008 T^{2} + 807516 T^{4} + T^{6} \)
$71$ \( ( -4238528 + 320246 T + 1274 T^{2} + T^{3} )^{2} \)
$73$ \( ( 226444392 - 662124 T - 222 T^{2} + T^{3} )^{2} \)
$79$ \( ( 157706752 + 968376 T + 1764 T^{2} + T^{3} )^{2} \)
$83$ \( 964272413085696 + 44195674176 T^{2} + 471264 T^{4} + T^{6} \)
$89$ \( ( 690068 - 1066774 T - 8 T^{2} + T^{3} )^{2} \)
$97$ \( ( -107008376 - 706092 T - 774 T^{2} + T^{3} )^{2} \)
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