# Properties

 Label 3136.2.a.bc Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3136.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.0410860739$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + q^{5} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 + q^5 + 6 * q^9 $$q + 3 q^{3} + q^{5} + 6 q^{9} - q^{11} - 2 q^{13} + 3 q^{15} + 3 q^{17} + 5 q^{19} + 3 q^{23} - 4 q^{25} + 9 q^{27} + 6 q^{29} + q^{31} - 3 q^{33} + 5 q^{37} - 6 q^{39} - 10 q^{41} - 4 q^{43} + 6 q^{45} - q^{47} + 9 q^{51} + 9 q^{53} - q^{55} + 15 q^{57} + 3 q^{59} - 3 q^{61} - 2 q^{65} + 11 q^{67} + 9 q^{69} - 16 q^{71} + 7 q^{73} - 12 q^{75} + 11 q^{79} + 9 q^{81} - 4 q^{83} + 3 q^{85} + 18 q^{87} - 9 q^{89} + 3 q^{93} + 5 q^{95} + 6 q^{97} - 6 q^{99}+O(q^{100})$$ q + 3 * q^3 + q^5 + 6 * q^9 - q^11 - 2 * q^13 + 3 * q^15 + 3 * q^17 + 5 * q^19 + 3 * q^23 - 4 * q^25 + 9 * q^27 + 6 * q^29 + q^31 - 3 * q^33 + 5 * q^37 - 6 * q^39 - 10 * q^41 - 4 * q^43 + 6 * q^45 - q^47 + 9 * q^51 + 9 * q^53 - q^55 + 15 * q^57 + 3 * q^59 - 3 * q^61 - 2 * q^65 + 11 * q^67 + 9 * q^69 - 16 * q^71 + 7 * q^73 - 12 * q^75 + 11 * q^79 + 9 * q^81 - 4 * q^83 + 3 * q^85 + 18 * q^87 - 9 * q^89 + 3 * q^93 + 5 * q^95 + 6 * q^97 - 6 * q^99

## 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
0 3.00000 0 1.00000 0 0 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 3136.2.a.bc 1
4.b odd 2 1 3136.2.a.b 1
7.b odd 2 1 3136.2.a.a 1
7.c even 3 2 448.2.i.a 2
8.b even 2 1 784.2.a.a 1
8.d odd 2 1 392.2.a.f 1
24.f even 2 1 3528.2.a.r 1
24.h odd 2 1 7056.2.a.bi 1
28.d even 2 1 3136.2.a.bb 1
28.g odd 6 2 448.2.i.f 2
40.e odd 2 1 9800.2.a.b 1
56.e even 2 1 392.2.a.a 1
56.h odd 2 1 784.2.a.j 1
56.j odd 6 2 784.2.i.a 2
56.k odd 6 2 56.2.i.a 2
56.m even 6 2 392.2.i.f 2
56.p even 6 2 112.2.i.c 2
168.e odd 2 1 3528.2.a.k 1
168.i even 2 1 7056.2.a.s 1
168.s odd 6 2 1008.2.s.e 2
168.v even 6 2 504.2.s.e 2
168.be odd 6 2 3528.2.s.o 2
280.n even 2 1 9800.2.a.bp 1
280.bi odd 6 2 1400.2.q.g 2
280.br even 12 4 1400.2.bh.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 56.k odd 6 2
112.2.i.c 2 56.p even 6 2
392.2.a.a 1 56.e even 2 1
392.2.a.f 1 8.d odd 2 1
392.2.i.f 2 56.m even 6 2
448.2.i.a 2 7.c even 3 2
448.2.i.f 2 28.g odd 6 2
504.2.s.e 2 168.v even 6 2
784.2.a.a 1 8.b even 2 1
784.2.a.j 1 56.h odd 2 1
784.2.i.a 2 56.j odd 6 2
1008.2.s.e 2 168.s odd 6 2
1400.2.q.g 2 280.bi odd 6 2
1400.2.bh.f 4 280.br even 12 4
3136.2.a.a 1 7.b odd 2 1
3136.2.a.b 1 4.b odd 2 1
3136.2.a.bb 1 28.d even 2 1
3136.2.a.bc 1 1.a even 1 1 trivial
3528.2.a.k 1 168.e odd 2 1
3528.2.a.r 1 24.f even 2 1
3528.2.s.o 2 168.be odd 6 2
7056.2.a.s 1 168.i even 2 1
7056.2.a.bi 1 24.h odd 2 1
9800.2.a.b 1 40.e odd 2 1
9800.2.a.bp 1 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{5} - 1$$ T5 - 1 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 3$$
$19$ $$T - 5$$
$23$ $$T - 3$$
$29$ $$T - 6$$
$31$ $$T - 1$$
$37$ $$T - 5$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T + 1$$
$53$ $$T - 9$$
$59$ $$T - 3$$
$61$ $$T + 3$$
$67$ $$T - 11$$
$71$ $$T + 16$$
$73$ $$T - 7$$
$79$ $$T - 11$$
$83$ $$T + 4$$
$89$ $$T + 9$$
$97$ $$T - 6$$