# Properties

 Label 4.14.a.a Level $4$ Weight $14$ Character orbit 4.a Self dual yes Analytic conductor $4.289$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 4.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.28923715808$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 468q^{3} + 56214q^{5} + 333032q^{7} - 1375299q^{9} + O(q^{10})$$ $$q + 468q^{3} + 56214q^{5} + 333032q^{7} - 1375299q^{9} - 6397380q^{11} + 15199742q^{13} + 26308152q^{15} + 43114194q^{17} - 365115484q^{19} + 155858976q^{21} - 57226824q^{23} + 1939310671q^{25} - 1389783096q^{27} - 46418994q^{29} - 5682185824q^{31} - 2993973840q^{33} + 18721060848q^{35} - 1887185098q^{37} + 7113479256q^{39} - 7336802934q^{41} - 26886674980q^{43} - 77311057986q^{45} + 101839834224q^{47} + 14021302617q^{49} + 20177442792q^{51} + 278731884294q^{53} - 359622319320q^{55} - 170874046512q^{57} + 59573945772q^{59} - 27484470418q^{61} - 458018576568q^{63} + 854438296788q^{65} + 784410054932q^{67} - 26782153632q^{69} - 360365227992q^{71} - 1592635413718q^{73} + 907597394028q^{75} - 2130532256160q^{77} - 23161184752q^{79} + 1542252338649q^{81} + 2050158110436q^{83} + 2423621301516q^{85} - 21724089192q^{87} - 3485391237126q^{89} + 5062000477744q^{91} - 2659262965632q^{93} - 20524601817576q^{95} + 6706667416802q^{97} + 8798310316620q^{99} + O(q^{100})$$

## 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
0 468.000 0 56214.0 0 333032. 0 −1.37530e6 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
$$2$$ $$-1$$

## 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 4.14.a.a 1
3.b odd 2 1 36.14.a.a 1
4.b odd 2 1 16.14.a.b 1
5.b even 2 1 100.14.a.a 1
5.c odd 4 2 100.14.c.a 2
7.b odd 2 1 196.14.a.a 1
7.c even 3 2 196.14.e.a 2
7.d odd 6 2 196.14.e.b 2
8.b even 2 1 64.14.a.c 1
8.d odd 2 1 64.14.a.g 1
12.b even 2 1 144.14.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.14.a.a 1 1.a even 1 1 trivial
16.14.a.b 1 4.b odd 2 1
36.14.a.a 1 3.b odd 2 1
64.14.a.c 1 8.b even 2 1
64.14.a.g 1 8.d odd 2 1
100.14.a.a 1 5.b even 2 1
100.14.c.a 2 5.c odd 4 2
144.14.a.a 1 12.b even 2 1
196.14.a.a 1 7.b odd 2 1
196.14.e.a 2 7.c even 3 2
196.14.e.b 2 7.d odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{14}^{\mathrm{new}}(\Gamma_0(4))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-468 + T$$
$5$ $$-56214 + T$$
$7$ $$-333032 + T$$
$11$ $$6397380 + T$$
$13$ $$-15199742 + T$$
$17$ $$-43114194 + T$$
$19$ $$365115484 + T$$
$23$ $$57226824 + T$$
$29$ $$46418994 + T$$
$31$ $$5682185824 + T$$
$37$ $$1887185098 + T$$
$41$ $$7336802934 + T$$
$43$ $$26886674980 + T$$
$47$ $$-101839834224 + T$$
$53$ $$-278731884294 + T$$
$59$ $$-59573945772 + T$$
$61$ $$27484470418 + T$$
$67$ $$-784410054932 + T$$
$71$ $$360365227992 + T$$
$73$ $$1592635413718 + T$$
$79$ $$23161184752 + T$$
$83$ $$-2050158110436 + T$$
$89$ $$3485391237126 + T$$
$97$ $$-6706667416802 + T$$