Properties

Label 272.4.a.h
Level $272$
Weight $4$
Character orbit 272.a
Self dual yes
Analytic conductor $16.049$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0485195216\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} - 3) q^{5} + (2 \beta_{2} + \beta_1 - 8) q^{7} + ( - \beta_{2} - 8 \beta_1 + 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} - 3) q^{5} + (2 \beta_{2} + \beta_1 - 8) q^{7} + ( - \beta_{2} - 8 \beta_1 + 20) q^{9} + (\beta_{2} - 11 \beta_1 + 9) q^{11} + (3 \beta_{2} + 8 \beta_1 + 9) q^{13} + (6 \beta_{2} - 2 \beta_1 - 38) q^{15} - 17 q^{17} + (4 \beta_{2} - 22 \beta_1 - 28) q^{19} + (15 \beta_{2} - 12 \beta_1 - 69) q^{21} + (2 \beta_{2} + 39 \beta_1 - 48) q^{23} + ( - 10 \beta_{2} + 4 \beta_1 - 71) q^{25} + ( - 4 \beta_{2} + 40 \beta_1 + 8) q^{27} + (15 \beta_{2} - 16 \beta_1 - 157) q^{29} + ( - 8 \beta_{2} - 39 \beta_1 - 74) q^{31} + ( - 17 \beta_{2} + 76 \beta_1 - 105) q^{33} + ( - 22 \beta_{2} + 10 \beta_1 + 118) q^{35} + ( - 25 \beta_{2} - 28 \beta_1 + 127) q^{37} + (8 \beta_{2} - 36 \beta_1 - 92) q^{39} + ( - 30 \beta_{2} - 52 \beta_1 - 88) q^{41} + ( - 28 \beta_{2} - 2 \beta_1 - 176) q^{43} + (27 \beta_{2} - 20 \beta_1 - 137) q^{45} + ( - 22 \beta_{2} + 48 \beta_1 - 206) q^{47} + ( - 47 \beta_{2} + 20 \beta_1 - 74) q^{49} + (17 \beta_{2} - 17 \beta_1 + 17) q^{51} + ( - 4 \beta_{2} + 116 \beta_1 + 102) q^{53} + (2 \beta_{2} - 18 \beta_1 - 26) q^{55} + (18 \beta_{2} + 108 \beta_1 - 246) q^{57} + (130 \beta_1 - 212) q^{59} + (39 \beta_{2} - 64 \beta_1 - 41) q^{61} + (48 \beta_{2} - 9 \beta_1 - 390) q^{63} + ( - 12 \beta_{2} + 28 \beta_1 + 140) q^{65} + (66 \beta_{2} - 24 \beta_1 - 358) q^{67} + (93 \beta_{2} - 280 \beta_1 + 161) q^{69} + ( - 36 \beta_{2} - 185 \beta_1 + 146) q^{71} + ( - 8 \beta_{2} + 16 \beta_1 + 282) q^{73} + (45 \beta_{2} - 105 \beta_1 + 501) q^{75} + ( - 9 \beta_{2} - 165) q^{77} + ( - 90 \beta_{2} + 267 \beta_1 + 228) q^{79} + (47 \beta_{2} - 20 \beta_1 - 184) q^{81} + ( - 64 \beta_{2} - 82 \beta_1 + 820) q^{83} + ( - 17 \beta_{2} + 51) q^{85} + (186 \beta_{2} - 46 \beta_1 - 538) q^{87} + (55 \beta_{2} - 276 \beta_1 - 75) q^{89} + ( - 22 \beta_{2} + 64 \beta_1 + 346) q^{91} + (11 \beta_{2} + 152 \beta_1 + 207) q^{93} + ( - 56 \beta_{2} - 28 \beta_1 + 176) q^{95} + (60 \beta_{2} + 140 \beta_1 - 110) q^{97} + (103 \beta_{2} - 281 \beta_1 + 939) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9} + 28 q^{11} + 30 q^{13} - 108 q^{15} - 51 q^{17} - 80 q^{19} - 192 q^{21} - 142 q^{23} - 223 q^{25} + 20 q^{27} - 456 q^{29} - 230 q^{31} - 332 q^{33} + 332 q^{35} + 356 q^{37} - 268 q^{39} - 294 q^{41} - 556 q^{43} - 384 q^{45} - 640 q^{47} - 269 q^{49} + 68 q^{51} + 302 q^{53} - 76 q^{55} - 720 q^{57} - 636 q^{59} - 84 q^{61} - 1122 q^{63} + 408 q^{65} - 1008 q^{67} + 576 q^{69} + 402 q^{71} + 838 q^{73} + 1548 q^{75} - 504 q^{77} + 594 q^{79} - 505 q^{81} + 2396 q^{83} + 136 q^{85} - 1428 q^{87} - 170 q^{89} + 1016 q^{91} + 632 q^{93} + 472 q^{95} - 270 q^{97} + 2920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 14x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 9 \) 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
−3.58966
3.87707
−0.287410
0 −8.47535 0 0.885690 0 −3.81828 0 44.8316 0
1.2 0 −3.15463 0 3.03171 0 7.94049 0 −17.0483 0
1.3 0 7.62999 0 −11.9174 0 −26.1222 0 31.2167 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 272.4.a.h 3
3.b odd 2 1 2448.4.a.bi 3
4.b odd 2 1 17.4.a.b 3
8.b even 2 1 1088.4.a.x 3
8.d odd 2 1 1088.4.a.v 3
12.b even 2 1 153.4.a.g 3
20.d odd 2 1 425.4.a.g 3
20.e even 4 2 425.4.b.f 6
28.d even 2 1 833.4.a.d 3
44.c even 2 1 2057.4.a.e 3
68.d odd 2 1 289.4.a.b 3
68.f odd 4 2 289.4.b.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 4.b odd 2 1
153.4.a.g 3 12.b even 2 1
272.4.a.h 3 1.a even 1 1 trivial
289.4.a.b 3 68.d odd 2 1
289.4.b.b 6 68.f odd 4 2
425.4.a.g 3 20.d odd 2 1
425.4.b.f 6 20.e even 4 2
833.4.a.d 3 28.d even 2 1
1088.4.a.v 3 8.d odd 2 1
1088.4.a.x 3 8.b even 2 1
2057.4.a.e 3 44.c even 2 1
2448.4.a.bi 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(272))\):

\( T_{3}^{3} + 4T_{3}^{2} - 62T_{3} - 204 \) Copy content Toggle raw display
\( T_{5}^{3} + 8T_{5}^{2} - 44T_{5} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} - 62 T - 204 \) Copy content Toggle raw display
$5$ \( T^{3} + 8 T^{2} - 44 T + 32 \) Copy content Toggle raw display
$7$ \( T^{3} + 22 T^{2} - 138 T - 792 \) Copy content Toggle raw display
$11$ \( T^{3} - 28 T^{2} - 1366 T + 4692 \) Copy content Toggle raw display
$13$ \( T^{3} - 30 T^{2} - 1472 T - 9392 \) Copy content Toggle raw display
$17$ \( (T + 17)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 80 T^{2} - 4632 T - 340128 \) Copy content Toggle raw display
$23$ \( T^{3} + 142 T^{2} - 15770 T - 1600544 \) Copy content Toggle raw display
$29$ \( T^{3} + 456 T^{2} + 53908 T + 1518624 \) Copy content Toggle raw display
$31$ \( T^{3} + 230 T^{2} - 11586 T + 81608 \) Copy content Toggle raw display
$37$ \( T^{3} - 356 T^{2} - 17964 T + 6176752 \) Copy content Toggle raw display
$41$ \( T^{3} + 294 T^{2} - 86564 T - 1638744 \) Copy content Toggle raw display
$43$ \( T^{3} + 556 T^{2} + 51096 T - 7270272 \) Copy content Toggle raw display
$47$ \( T^{3} + 640 T^{2} + 85328 T + 1671168 \) Copy content Toggle raw display
$53$ \( T^{3} - 302 T^{2} + \cdots + 18162072 \) Copy content Toggle raw display
$59$ \( T^{3} + 636 T^{2} + \cdots - 49419072 \) Copy content Toggle raw display
$61$ \( T^{3} + 84 T^{2} - 124412 T - 6792784 \) Copy content Toggle raw display
$67$ \( T^{3} + 1008 T^{2} + 65040 T + 765952 \) Copy content Toggle raw display
$71$ \( T^{3} - 402 T^{2} + \cdots + 274866016 \) Copy content Toggle raw display
$73$ \( T^{3} - 838 T^{2} + \cdots - 19957512 \) Copy content Toggle raw display
$79$ \( T^{3} - 594 T^{2} + \cdots + 742135824 \) Copy content Toggle raw display
$83$ \( T^{3} - 2396 T^{2} + \cdots - 142080704 \) Copy content Toggle raw display
$89$ \( T^{3} + 170 T^{2} + \cdots - 446571376 \) Copy content Toggle raw display
$97$ \( T^{3} + 270 T^{2} + \cdots - 206623000 \) Copy content Toggle raw display
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