Properties

Label 1104.6.a.r
Level $1104$
Weight $6$
Character orbit 1104.a
Self dual yes
Analytic conductor $177.064$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{3} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 19) q^{5} + ( - \beta_{4} + 2 \beta_{3} - 55) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 19) q^{5} + ( - \beta_{4} + 2 \beta_{3} - 55) q^{7} + 81 q^{9} + ( - 6 \beta_{4} + \beta_{3} - 7 \beta_{2} - 5 \beta_1 - 222) q^{11} + ( - \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 10 \beta_1 - 199) q^{13} + (9 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 171) q^{15} + ( - 5 \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 8 \beta_1 + 511) q^{17} + (23 \beta_{4} - 20 \beta_{3} - 16 \beta_{2} - 9 \beta_1 - 415) q^{19} + (9 \beta_{4} - 18 \beta_{3} + 495) q^{21} + 529 q^{23} + ( - 11 \beta_{4} - 2 \beta_{3} + 75 \beta_{2} - 24 \beta_1 + 2440) q^{25} - 729 q^{27} + (26 \beta_{4} + 44 \beta_{3} + 124 \beta_{2} + 96 \beta_1 + 332) q^{29} + (20 \beta_{4} - 59 \beta_{3} + 35 \beta_{2} - 36 \beta_1 + 1512) q^{31} + (54 \beta_{4} - 9 \beta_{3} + 63 \beta_{2} + 45 \beta_1 + 1998) q^{33} + (21 \beta_{4} + 108 \beta_{3} - 121 \beta_{2} + 2 \beta_1 - 1307) q^{35} + (30 \beta_{4} + 53 \beta_{3} - 57 \beta_{2} + 9 \beta_1 - 1692) q^{37} + (9 \beta_{4} - 18 \beta_{3} + 63 \beta_{2} - 90 \beta_1 + 1791) q^{39} + (204 \beta_{4} - 204 \beta_{3} + 208 \beta_{2} - 34 \beta_1 + 4374) q^{41} + (19 \beta_{4} + 120 \beta_{3} + 208 \beta_{2} + 361 \beta_1 + 941) q^{43} + ( - 81 \beta_{4} - 81 \beta_{3} - 81 \beta_{2} - 81 \beta_1 + 1539) q^{45} + (161 \beta_{4} - 458 \beta_{3} - 107 \beta_{2} - 102 \beta_1 + 1) q^{47} + (65 \beta_{4} - 163 \beta_{3} - 146 \beta_{2} - 116 \beta_1 - 5602) q^{49} + (45 \beta_{4} + 27 \beta_{3} - 99 \beta_{2} + 72 \beta_1 - 4599) q^{51} + (335 \beta_{4} + 31 \beta_{3} - 83 \beta_{2} + 55 \beta_1 - 2409) q^{53} + (138 \beta_{4} + 620 \beta_{3} + 222 \beta_{2} + 144 \beta_1 + 15226) q^{55} + ( - 207 \beta_{4} + 180 \beta_{3} + 144 \beta_{2} + 81 \beta_1 + 3735) q^{57} + (261 \beta_{4} - 280 \beta_{3} - 189 \beta_{2} + 324 \beta_1 - 14951) q^{59} + ( - 44 \beta_{4} - 103 \beta_{3} + 329 \beta_{2} - 133 \beta_1 - 8742) q^{61} + ( - 81 \beta_{4} + 162 \beta_{3} - 4455) q^{63} + (196 \beta_{4} + 902 \beta_{3} + 264 \beta_{2} + 1202 \beta_1 - 5172) q^{65} + ( - 31 \beta_{4} + 468 \beta_{3} - 716 \beta_{2} + 47 \beta_1 - 185) q^{67} - 4761 q^{69} + ( - 64 \beta_{4} + 1229 \beta_{3} + 613 \beta_{2} - 58 \beta_1 - 7816) q^{71} + ( - 64 \beta_{4} + 139 \beta_{3} + 979 \beta_{2} + 88 \beta_1 + 9658) q^{73} + (99 \beta_{4} + 18 \beta_{3} - 675 \beta_{2} + 216 \beta_1 - 21960) q^{75} + (186 \beta_{4} - 285 \beta_{3} - 731 \beta_{2} - 330 \beta_1 + 31186) q^{77} + ( - 251 \beta_{4} - 922 \beta_{3} + 32 \beta_{2} + 624 \beta_1 - 9677) q^{79} + 6561 q^{81} + (332 \beta_{4} + 35 \beta_{3} + 2205 \beta_{2} + 781 \beta_1 - 17116) q^{83} + ( - 421 \beta_{4} - 1534 \beta_{3} - 603 \beta_{2} - 1568 \beta_1 + 30035) q^{85} + ( - 234 \beta_{4} - 396 \beta_{3} - 1116 \beta_{2} - 864 \beta_1 - 2988) q^{87} + ( - 161 \beta_{4} + 1587 \beta_{3} - 601 \beta_{2} + 558 \beta_1 + 55239) q^{89} + (186 \beta_{4} - 998 \beta_{3} + 542 \beta_{2} - 1152 \beta_1 + 6062) q^{91} + ( - 180 \beta_{4} + 531 \beta_{3} - 315 \beta_{2} + 324 \beta_1 - 13608) q^{93} + (519 \beta_{4} + 952 \beta_{3} + 2549 \beta_{2} + 1130 \beta_1 - 40089) q^{95} + (242 \beta_{4} + 1108 \beta_{3} - 528 \beta_{2} + 958 \beta_1 + 17312) q^{97} + ( - 486 \beta_{4} + 81 \beta_{3} - 567 \beta_{2} - 405 \beta_1 - 17982) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 13\nu^{3} + 56\nu^{2} - 49\nu - 4511 ) / 117 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{4} - 13\nu^{3} - 700\nu^{2} - 265\nu + 5707 ) / 117 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{4} + 13\nu^{3} + 700\nu^{2} + 1201\nu - 5707 ) / 117 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{4} + 65\nu^{3} + 944\nu^{2} - 2347\nu - 8840 ) / 117 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 3\beta_{3} + 4\beta_{2} + 4\beta _1 + 181 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28\beta_{4} + 83\beta_{3} + 123\beta_{2} + 28\beta _1 + 1244 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 63\beta_{4} + 198\beta_{3} + 300\beta_{2} + 213\beta _1 + 7997 ) / 2 \) Copy content Toggle raw display

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
11.0973
−5.17654
−7.90234
3.40352
−1.42196
0 −9.00000 0 −92.1306 0 8.98894 0 81.0000 0
1.2 0 −9.00000 0 −37.4928 0 −154.850 0 81.0000 0
1.3 0 −9.00000 0 53.3906 0 −89.8688 0 81.0000 0
1.4 0 −9.00000 0 70.5203 0 89.7132 0 81.0000 0
1.5 0 −9.00000 0 99.7124 0 −125.983 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.r 5
4.b odd 2 1 69.6.a.e 5
12.b even 2 1 207.6.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.6.a.e 5 4.b odd 2 1
207.6.a.f 5 12.b even 2 1
1104.6.a.r 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 94T_{5}^{4} - 9412T_{5}^{3} + 941728T_{5}^{2} + 7019776T_{5} - 1296820224 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1104))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T + 9)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 94 T^{4} + \cdots - 1296820224 \) Copy content Toggle raw display
$7$ \( T^{5} + 272 T^{4} + \cdots + 1413831904 \) Copy content Toggle raw display
$11$ \( T^{5} + 1100 T^{4} + \cdots - 5578745996288 \) Copy content Toggle raw display
$13$ \( T^{5} + 978 T^{4} + \cdots - 2474410088160 \) Copy content Toggle raw display
$17$ \( T^{5} - 2522 T^{4} + \cdots + 13327498013440 \) Copy content Toggle raw display
$19$ \( T^{5} + 2060 T^{4} + \cdots - 30\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( (T - 529)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 1526 T^{4} + \cdots + 32\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{5} - 7392 T^{4} + \cdots - 16\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{5} + 8210 T^{4} + \cdots + 84\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{5} - 21250 T^{4} + \cdots - 22\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{5} - 4548 T^{4} + \cdots + 23\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{5} + 536 T^{4} + \cdots - 94\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{5} + 11482 T^{4} + \cdots + 77\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{5} + 74676 T^{4} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{5} + 44618 T^{4} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{5} - 1412 T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{5} + 37912 T^{4} + \cdots + 42\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{5} - 46546 T^{4} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{5} + 50544 T^{4} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{5} + 89588 T^{4} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{5} - 280410 T^{4} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{5} - 90074 T^{4} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
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