Properties

Label 72.7.e.b
Level $72$
Weight $7$
Character orbit 72.e
Analytic conductor $16.564$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.5638940206\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{145})\)
Defining polynomial: \( x^{4} - 2x^{3} - 67x^{2} + 68x + 1446 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 25 \beta_1) q^{5} + ( - \beta_{2} - 108) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 25 \beta_1) q^{5} + ( - \beta_{2} - 108) q^{7} + (6 \beta_{3} + 524 \beta_1) q^{11} + (7 \beta_{2} - 1088) q^{13} + ( - 2 \beta_{3} + 1633 \beta_1) q^{17} + ( - 8 \beta_{2} - 6736) q^{19} + ( - 50 \beta_{3} + 9188 \beta_1) q^{23} + ( - 50 \beta_{2} - 27385) q^{25} + (67 \beta_{3} + 7761 \beta_1) q^{29} + (131 \beta_{2} - 14044) q^{31} + (158 \beta_{3} + 44460 \beta_1) q^{35} + (76 \beta_{2} - 59250) q^{37} + ( - 250 \beta_{3} - 34905 \beta_1) q^{41} + ( - 514 \beta_{2} + 712) q^{43} + ( - 298 \beta_{3} + 50292 \beta_1) q^{47} + (216 \beta_{2} - 22465) q^{49} + (175 \beta_{3} - 127975 \beta_1) q^{53} + (674 \beta_{2} + 276760) q^{55} + (708 \beta_{3} + 32504 \beta_1) q^{59} + ( - 556 \beta_{2} + 36706) q^{61} + (738 \beta_{3} - 265120 \beta_1) q^{65} + (302 \beta_{2} + 386552) q^{67} + ( - 1374 \beta_{3} + 76300 \beta_1) q^{71} + ( - 564 \beta_{2} + 70480) q^{73} + ( - 1696 \beta_{3} - 307152 \beta_1) q^{77} + ( - 1067 \beta_{2} + 314620) q^{79} + ( - 686 \beta_{3} - 126668 \beta_1) q^{83} + (1583 \beta_{2} - 1870) q^{85} + (1716 \beta_{3} + 275415 \beta_1) q^{89} + (332 \beta_{2} - 467136) q^{91} + (7136 \beta_{3} + 502480 \beta_1) q^{95} + (2350 \beta_{2} - 330592) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 432 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 432 q^{7} - 4352 q^{13} - 26944 q^{19} - 109540 q^{25} - 56176 q^{31} - 237000 q^{37} + 2848 q^{43} - 89860 q^{49} + 1107040 q^{55} + 146824 q^{61} + 1546208 q^{67} + 281920 q^{73} + 1258480 q^{79} - 7480 q^{85} - 1868544 q^{91} - 1322368 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 67x^{2} + 68x + 1446 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} - 59\nu + 30 ) / 153 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -32\nu^{3} + 48\nu^{2} + 3392\nu - 1704 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 12\nu^{2} - 12\nu - 408 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 48\beta _1 + 24 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + \beta_{2} + 48\beta _1 + 1656 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{3} + 31\beta_{2} + 5160\beta _1 + 2472 ) / 48 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
6.52080 + 1.41421i
−5.52080 1.41421i
−5.52080 + 1.41421i
6.52080 1.41421i
0 0 0 239.708i 0 −396.998 0 0 0
17.2 0 0 0 168.997i 0 180.998 0 0 0
17.3 0 0 0 168.997i 0 180.998 0 0 0
17.4 0 0 0 239.708i 0 −396.998 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.e.b 4
3.b odd 2 1 inner 72.7.e.b 4
4.b odd 2 1 144.7.e.e 4
8.b even 2 1 576.7.e.m 4
8.d odd 2 1 576.7.e.r 4
12.b even 2 1 144.7.e.e 4
24.f even 2 1 576.7.e.r 4
24.h odd 2 1 576.7.e.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.e.b 4 1.a even 1 1 trivial
72.7.e.b 4 3.b odd 2 1 inner
144.7.e.e 4 4.b odd 2 1
144.7.e.e 4 12.b even 2 1
576.7.e.m 4 8.b even 2 1
576.7.e.m 4 24.h odd 2 1
576.7.e.r 4 8.d odd 2 1
576.7.e.r 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 86020T_{5}^{2} + 1641060100 \) acting on \(S_{7}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 86020 T^{2} + \cdots + 1641060100 \) Copy content Toggle raw display
$7$ \( (T^{2} + 216 T - 71856)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 4105024 T^{2} + \cdots + 910512907264 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2176 T - 2908736)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 11000836 T^{2} + \cdots + 26691048330244 \) Copy content Toggle raw display
$19$ \( (T^{2} + 13472 T + 40028416)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 546477376 T^{2} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} + 615853764 T^{2} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{2} + 28088 T - 1236052784)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 118500 T + 3028150980)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 10093436100 T^{2} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} - 1424 T - 22065142976)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 17534051136 T^{2} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{4} + 68068202500 T^{2} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + 46091609344 T^{2} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 73412 T - 24471708284)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 773104 T + 141805090624)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 180962163520 T^{2} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} - 140960 T - 21599947520)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 629240 T + 3899143120)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 103483306816 T^{2} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{4} + 549351358020 T^{2} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + 661184 T - 351948129536)^{2} \) Copy content Toggle raw display
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