Properties

Label 1104.6.a.i
Level $1104$
Weight $6$
Character orbit 1104.a
Self dual yes
Analytic conductor $177.064$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(177.063737074\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -9 q^{3} + ( -21 + 4 \beta_{1} + 3 \beta_{2} ) q^{5} + ( 33 + 11 \beta_{1} + 4 \beta_{2} ) q^{7} + 81 q^{9} +O(q^{10})\) \( q -9 q^{3} + ( -21 + 4 \beta_{1} + 3 \beta_{2} ) q^{5} + ( 33 + 11 \beta_{1} + 4 \beta_{2} ) q^{7} + 81 q^{9} + ( 130 - 35 \beta_{1} + 21 \beta_{2} ) q^{11} + ( -264 - 66 \beta_{2} ) q^{13} + ( 189 - 36 \beta_{1} - 27 \beta_{2} ) q^{15} + ( -844 - 52 \beta_{1} + 36 \beta_{2} ) q^{17} + ( 924 + 69 \beta_{1} + 5 \beta_{2} ) q^{19} + ( -297 - 99 \beta_{1} - 36 \beta_{2} ) q^{21} -529 q^{23} + ( 246 - 5 \beta_{1} + 20 \beta_{2} ) q^{25} -729 q^{27} + ( -5431 - 177 \beta_{1} + 100 \beta_{2} ) q^{29} + ( 4939 + 219 \beta_{1} - 280 \beta_{2} ) q^{31} + ( -1170 + 315 \beta_{1} - 189 \beta_{2} ) q^{33} + ( 5962 + 192 \beta_{1} + 442 \beta_{2} ) q^{35} + ( 5278 - 145 \beta_{1} + 185 \beta_{2} ) q^{37} + ( 2376 + 594 \beta_{2} ) q^{39} + ( 3927 + 223 \beta_{1} + 602 \beta_{2} ) q^{41} + ( -896 + 303 \beta_{1} - 769 \beta_{2} ) q^{43} + ( -1701 + 324 \beta_{1} + 243 \beta_{2} ) q^{45} + ( -9619 - 929 \beta_{1} - 142 \beta_{2} ) q^{47} + ( 641 + 1148 \beta_{1} + 1400 \beta_{2} ) q^{49} + ( 7596 + 468 \beta_{1} - 324 \beta_{2} ) q^{51} + ( -15895 + 2018 \beta_{1} + 1583 \beta_{2} ) q^{53} + ( -12775 + 1577 \beta_{1} - 1612 \beta_{2} ) q^{55} + ( -8316 - 621 \beta_{1} - 45 \beta_{2} ) q^{57} + ( 9010 + 1894 \beta_{1} + 376 \beta_{2} ) q^{59} + ( 17828 + 1019 \beta_{1} - 493 \beta_{2} ) q^{61} + ( 2673 + 891 \beta_{1} + 324 \beta_{2} ) q^{63} + ( -16236 - 3498 \beta_{1} + 990 \beta_{2} ) q^{65} + ( -16568 + 3245 \beta_{1} + 3069 \beta_{2} ) q^{67} + 4761 q^{69} + ( -7137 - 2695 \beta_{1} + 682 \beta_{2} ) q^{71} + ( -32533 + 2293 \beta_{1} + 3904 \beta_{2} ) q^{73} + ( -2214 + 45 \beta_{1} - 180 \beta_{2} ) q^{75} + ( -32348 + 1934 \beta_{1} - 2098 \beta_{2} ) q^{77} + ( 14881 + 4771 \beta_{1} - 16 \beta_{2} ) q^{79} + 6561 q^{81} + ( 32258 - 841 \beta_{1} + 8003 \beta_{2} ) q^{83} + ( 4384 - 1628 \beta_{1} - 5636 \beta_{2} ) q^{85} + ( 48879 + 1593 \beta_{1} - 900 \beta_{2} ) q^{87} + ( 32506 + 3028 \beta_{1} - 4434 \beta_{2} ) q^{89} + ( -38874 - 8778 \beta_{1} - 3828 \beta_{2} ) q^{91} + ( -44451 - 1971 \beta_{1} + 2520 \beta_{2} ) q^{93} + ( 15711 + 3329 \beta_{1} + 5466 \beta_{2} ) q^{95} + ( -45436 + 6024 \beta_{1} - 5034 \beta_{2} ) q^{97} + ( 10530 - 2835 \beta_{1} + 1701 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 27q^{3} - 56q^{5} + 114q^{7} + 243q^{9} + O(q^{10}) \) \( 3q - 27q^{3} - 56q^{5} + 114q^{7} + 243q^{9} + 376q^{11} - 858q^{13} + 504q^{15} - 2548q^{17} + 2846q^{19} - 1026q^{21} - 1587q^{23} + 753q^{25} - 2187q^{27} - 16370q^{29} + 14756q^{31} - 3384q^{33} + 18520q^{35} + 15874q^{37} + 7722q^{39} + 12606q^{41} - 3154q^{43} - 4536q^{45} - 29928q^{47} + 4471q^{49} + 22932q^{51} - 44084q^{53} - 38360q^{55} - 25614q^{57} + 29300q^{59} + 54010q^{61} + 9234q^{63} - 51216q^{65} - 43390q^{67} + 14283q^{69} - 23424q^{71} - 91402q^{73} - 6777q^{75} - 97208q^{77} + 49398q^{79} + 19683q^{81} + 103936q^{83} + 5888q^{85} + 147330q^{87} + 96112q^{89} - 129228q^{91} - 132804q^{93} + 55928q^{95} - 135318q^{97} + 30456q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 11 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 15\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.20733
0.714018
3.49331
0 −9.00000 0 −59.5955 0 −96.8268 0 81.0000 0
1.2 0 −9.00000 0 −55.5168 0 −2.50462 0 81.0000 0
1.3 0 −9.00000 0 59.1123 0 213.331 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.6.a.i 3
4.b odd 2 1 69.6.a.b 3
12.b even 2 1 207.6.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.b 3 4.b odd 2 1
207.6.a.c 3 12.b even 2 1
1104.6.a.i 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 56 T_{5}^{2} - 3496 T_{5} - 195576 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1104))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 9 + T )^{3} \)
$5$ \( -195576 - 3496 T + 56 T^{2} + T^{3} \)
$7$ \( -51736 - 20948 T - 114 T^{2} + T^{3} \)
$11$ \( -21141352 - 230672 T - 376 T^{2} + T^{3} \)
$13$ \( -368282376 - 439956 T + 858 T^{2} + T^{3} \)
$17$ \( -129112640 + 1504816 T + 2548 T^{2} + T^{3} \)
$19$ \( 4313168 + 1826204 T - 2846 T^{2} + T^{3} \)
$23$ \( ( 529 + T )^{3} \)
$29$ \( 117835741080 + 82401708 T + 16370 T^{2} + T^{3} \)
$31$ \( -16664141952 + 52692624 T - 14756 T^{2} + T^{3} \)
$37$ \( -103469473312 + 75298092 T - 15874 T^{2} + T^{3} \)
$41$ \( 213582513784 - 15571268 T - 12606 T^{2} + T^{3} \)
$43$ \( -409945701888 - 102023460 T + 3154 T^{2} + T^{3} \)
$47$ \( -524672802816 + 136428688 T + 29928 T^{2} + T^{3} \)
$53$ \( -30187666172280 - 544538824 T + 44084 T^{2} + T^{3} \)
$59$ \( 4070512924224 - 399858640 T - 29300 T^{2} + T^{3} \)
$61$ \( -694379910768 + 755208380 T - 54010 T^{2} + T^{3} \)
$67$ \( -128918418373088 - 2949665124 T + 43390 T^{2} + T^{3} \)
$71$ \( -26245560332032 - 1173004304 T + 23424 T^{2} + T^{3} \)
$73$ \( -119459239092680 - 733701012 T + 91402 T^{2} + T^{3} \)
$79$ \( 93978622829240 - 3312817172 T - 49398 T^{2} + T^{3} \)
$83$ \( 694250483317800 - 6478606576 T - 103936 T^{2} + T^{3} \)
$89$ \( 102645237296960 - 1426041824 T - 96112 T^{2} + T^{3} \)
$97$ \( -82905816948920 - 3897620052 T + 135318 T^{2} + T^{3} \)
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