Properties

 Label 126.4.a.h Level $126$ Weight $4$ Character orbit 126.a Self dual yes Analytic conductor $7.434$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 126.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$7.43424066072$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + 14 q^{5} - 7 q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + 14 * q^5 - 7 * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + 14 q^{5} - 7 q^{7} + 8 q^{8} + 28 q^{10} + 28 q^{11} + 18 q^{13} - 14 q^{14} + 16 q^{16} - 74 q^{17} + 80 q^{19} + 56 q^{20} + 56 q^{22} + 112 q^{23} + 71 q^{25} + 36 q^{26} - 28 q^{28} - 190 q^{29} + 72 q^{31} + 32 q^{32} - 148 q^{34} - 98 q^{35} - 346 q^{37} + 160 q^{38} + 112 q^{40} - 162 q^{41} - 412 q^{43} + 112 q^{44} + 224 q^{46} - 24 q^{47} + 49 q^{49} + 142 q^{50} + 72 q^{52} - 318 q^{53} + 392 q^{55} - 56 q^{56} - 380 q^{58} + 200 q^{59} - 198 q^{61} + 144 q^{62} + 64 q^{64} + 252 q^{65} - 716 q^{67} - 296 q^{68} - 196 q^{70} - 392 q^{71} + 538 q^{73} - 692 q^{74} + 320 q^{76} - 196 q^{77} + 240 q^{79} + 224 q^{80} - 324 q^{82} + 1072 q^{83} - 1036 q^{85} - 824 q^{86} + 224 q^{88} - 810 q^{89} - 126 q^{91} + 448 q^{92} - 48 q^{94} + 1120 q^{95} + 1354 q^{97} + 98 q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + 14 * q^5 - 7 * q^7 + 8 * q^8 + 28 * q^10 + 28 * q^11 + 18 * q^13 - 14 * q^14 + 16 * q^16 - 74 * q^17 + 80 * q^19 + 56 * q^20 + 56 * q^22 + 112 * q^23 + 71 * q^25 + 36 * q^26 - 28 * q^28 - 190 * q^29 + 72 * q^31 + 32 * q^32 - 148 * q^34 - 98 * q^35 - 346 * q^37 + 160 * q^38 + 112 * q^40 - 162 * q^41 - 412 * q^43 + 112 * q^44 + 224 * q^46 - 24 * q^47 + 49 * q^49 + 142 * q^50 + 72 * q^52 - 318 * q^53 + 392 * q^55 - 56 * q^56 - 380 * q^58 + 200 * q^59 - 198 * q^61 + 144 * q^62 + 64 * q^64 + 252 * q^65 - 716 * q^67 - 296 * q^68 - 196 * q^70 - 392 * q^71 + 538 * q^73 - 692 * q^74 + 320 * q^76 - 196 * q^77 + 240 * q^79 + 224 * q^80 - 324 * q^82 + 1072 * q^83 - 1036 * q^85 - 824 * q^86 + 224 * q^88 - 810 * q^89 - 126 * q^91 + 448 * q^92 - 48 * q^94 + 1120 * q^95 + 1354 * q^97 + 98 * q^98

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
 0
2.00000 0 4.00000 14.0000 0 −7.00000 8.00000 0 28.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

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 126.4.a.h 1
3.b odd 2 1 14.4.a.a 1
4.b odd 2 1 1008.4.a.s 1
7.b odd 2 1 882.4.a.i 1
7.c even 3 2 882.4.g.b 2
7.d odd 6 2 882.4.g.k 2
12.b even 2 1 112.4.a.a 1
15.d odd 2 1 350.4.a.l 1
15.e even 4 2 350.4.c.b 2
21.c even 2 1 98.4.a.a 1
21.g even 6 2 98.4.c.f 2
21.h odd 6 2 98.4.c.d 2
24.f even 2 1 448.4.a.o 1
24.h odd 2 1 448.4.a.b 1
33.d even 2 1 1694.4.a.g 1
39.d odd 2 1 2366.4.a.h 1
84.h odd 2 1 784.4.a.s 1
105.g even 2 1 2450.4.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 3.b odd 2 1
98.4.a.a 1 21.c even 2 1
98.4.c.d 2 21.h odd 6 2
98.4.c.f 2 21.g even 6 2
112.4.a.a 1 12.b even 2 1
126.4.a.h 1 1.a even 1 1 trivial
350.4.a.l 1 15.d odd 2 1
350.4.c.b 2 15.e even 4 2
448.4.a.b 1 24.h odd 2 1
448.4.a.o 1 24.f even 2 1
784.4.a.s 1 84.h odd 2 1
882.4.a.i 1 7.b odd 2 1
882.4.g.b 2 7.c even 3 2
882.4.g.k 2 7.d odd 6 2
1008.4.a.s 1 4.b odd 2 1
1694.4.a.g 1 33.d even 2 1
2366.4.a.h 1 39.d odd 2 1
2450.4.a.bo 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 14$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(126))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T - 14$$
$7$ $$T + 7$$
$11$ $$T - 28$$
$13$ $$T - 18$$
$17$ $$T + 74$$
$19$ $$T - 80$$
$23$ $$T - 112$$
$29$ $$T + 190$$
$31$ $$T - 72$$
$37$ $$T + 346$$
$41$ $$T + 162$$
$43$ $$T + 412$$
$47$ $$T + 24$$
$53$ $$T + 318$$
$59$ $$T - 200$$
$61$ $$T + 198$$
$67$ $$T + 716$$
$71$ $$T + 392$$
$73$ $$T - 538$$
$79$ $$T - 240$$
$83$ $$T - 1072$$
$89$ $$T + 810$$
$97$ $$T - 1354$$