Properties

Label 72.4.d.b
Level $72$
Weight $4$
Character orbit 72.d
Analytic conductor $4.248$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta - 6) q^{4} + 4 \beta q^{5} - 8 q^{7} + ( - 4 \beta - 20) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta - 6) q^{4} + 4 \beta q^{5} - 8 q^{7} + ( - 4 \beta - 20) q^{8} + (4 \beta - 28) q^{10} + 6 \beta q^{11} + 20 \beta q^{13} + ( - 8 \beta - 8) q^{14} + ( - 24 \beta + 8) q^{16} + 14 q^{17} - 14 \beta q^{19} + ( - 24 \beta - 56) q^{20} + (6 \beta - 42) q^{22} + 152 q^{23} + 13 q^{25} + (20 \beta - 140) q^{26} + ( - 16 \beta + 48) q^{28} + 60 \beta q^{29} + 224 q^{31} + ( - 16 \beta + 176) q^{32} + (14 \beta + 14) q^{34} - 32 \beta q^{35} + 92 \beta q^{37} + ( - 14 \beta + 98) q^{38} + ( - 80 \beta + 112) q^{40} + 70 q^{41} - 166 \beta q^{43} + ( - 36 \beta - 84) q^{44} + (152 \beta + 152) q^{46} - 336 q^{47} - 279 q^{49} + (13 \beta + 13) q^{50} + ( - 120 \beta - 280) q^{52} - 12 \beta q^{53} - 168 q^{55} + (32 \beta + 160) q^{56} + (60 \beta - 420) q^{58} - 202 \beta q^{59} + 36 \beta q^{61} + (224 \beta + 224) q^{62} + (160 \beta + 288) q^{64} - 560 q^{65} + 66 \beta q^{67} + (28 \beta - 84) q^{68} + ( - 32 \beta + 224) q^{70} + 72 q^{71} - 294 q^{73} + (92 \beta - 644) q^{74} + (84 \beta + 196) q^{76} - 48 \beta q^{77} - 464 q^{79} + (32 \beta + 672) q^{80} + (70 \beta + 70) q^{82} + 206 \beta q^{83} + 56 \beta q^{85} + ( - 166 \beta + 1162) q^{86} + ( - 120 \beta + 168) q^{88} - 266 q^{89} - 160 \beta q^{91} + (304 \beta - 912) q^{92} + ( - 336 \beta - 336) q^{94} + 392 q^{95} + 994 q^{97} + ( - 279 \beta - 279) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 12 q^{4} - 16 q^{7} - 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 12 q^{4} - 16 q^{7} - 40 q^{8} - 56 q^{10} - 16 q^{14} + 16 q^{16} + 28 q^{17} - 112 q^{20} - 84 q^{22} + 304 q^{23} + 26 q^{25} - 280 q^{26} + 96 q^{28} + 448 q^{31} + 352 q^{32} + 28 q^{34} + 196 q^{38} + 224 q^{40} + 140 q^{41} - 168 q^{44} + 304 q^{46} - 672 q^{47} - 558 q^{49} + 26 q^{50} - 560 q^{52} - 336 q^{55} + 320 q^{56} - 840 q^{58} + 448 q^{62} + 576 q^{64} - 1120 q^{65} - 168 q^{68} + 448 q^{70} + 144 q^{71} - 588 q^{73} - 1288 q^{74} + 392 q^{76} - 928 q^{79} + 1344 q^{80} + 140 q^{82} + 2324 q^{86} + 336 q^{88} - 532 q^{89} - 1824 q^{92} - 672 q^{94} + 784 q^{95} + 1988 q^{97} - 558 q^{98}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 1.32288i
0.500000 + 1.32288i
1.00000 2.64575i 0 −6.00000 5.29150i 10.5830i 0 −8.00000 −20.0000 + 10.5830i 0 −28.0000 10.5830i
37.2 1.00000 + 2.64575i 0 −6.00000 + 5.29150i 10.5830i 0 −8.00000 −20.0000 10.5830i 0 −28.0000 + 10.5830i
\(n\): e.g. 2-40 or 990-1000
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 72.4.d.b 2
3.b odd 2 1 8.4.b.a 2
4.b odd 2 1 288.4.d.a 2
8.b even 2 1 inner 72.4.d.b 2
8.d odd 2 1 288.4.d.a 2
12.b even 2 1 32.4.b.a 2
15.d odd 2 1 200.4.d.a 2
15.e even 4 2 200.4.f.a 4
16.e even 4 2 2304.4.a.bn 2
16.f odd 4 2 2304.4.a.v 2
24.f even 2 1 32.4.b.a 2
24.h odd 2 1 8.4.b.a 2
48.i odd 4 2 256.4.a.l 2
48.k even 4 2 256.4.a.j 2
60.h even 2 1 800.4.d.a 2
60.l odd 4 2 800.4.f.a 4
120.i odd 2 1 200.4.d.a 2
120.m even 2 1 800.4.d.a 2
120.q odd 4 2 800.4.f.a 4
120.w even 4 2 200.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 3.b odd 2 1
8.4.b.a 2 24.h odd 2 1
32.4.b.a 2 12.b even 2 1
32.4.b.a 2 24.f even 2 1
72.4.d.b 2 1.a even 1 1 trivial
72.4.d.b 2 8.b even 2 1 inner
200.4.d.a 2 15.d odd 2 1
200.4.d.a 2 120.i odd 2 1
200.4.f.a 4 15.e even 4 2
200.4.f.a 4 120.w even 4 2
256.4.a.j 2 48.k even 4 2
256.4.a.l 2 48.i odd 4 2
288.4.d.a 2 4.b odd 2 1
288.4.d.a 2 8.d odd 2 1
800.4.d.a 2 60.h even 2 1
800.4.d.a 2 120.m even 2 1
800.4.f.a 4 60.l odd 4 2
800.4.f.a 4 120.q odd 4 2
2304.4.a.v 2 16.f odd 4 2
2304.4.a.bn 2 16.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 112 \) acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 112 \) Copy content Toggle raw display
$7$ \( (T + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 252 \) Copy content Toggle raw display
$13$ \( T^{2} + 2800 \) Copy content Toggle raw display
$17$ \( (T - 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1372 \) Copy content Toggle raw display
$23$ \( (T - 152)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 25200 \) Copy content Toggle raw display
$31$ \( (T - 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 59248 \) Copy content Toggle raw display
$41$ \( (T - 70)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 192892 \) Copy content Toggle raw display
$47$ \( (T + 336)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 1008 \) Copy content Toggle raw display
$59$ \( T^{2} + 285628 \) Copy content Toggle raw display
$61$ \( T^{2} + 9072 \) Copy content Toggle raw display
$67$ \( T^{2} + 30492 \) Copy content Toggle raw display
$71$ \( (T - 72)^{2} \) Copy content Toggle raw display
$73$ \( (T + 294)^{2} \) Copy content Toggle raw display
$79$ \( (T + 464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 297052 \) Copy content Toggle raw display
$89$ \( (T + 266)^{2} \) Copy content Toggle raw display
$97$ \( (T - 994)^{2} \) Copy content Toggle raw display
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