# Properties

 Label 14.12.a.a Level 14 Weight 12 Character orbit 14.a Self dual yes Analytic conductor 10.757 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 14.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.7568045278$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 32q^{2} - 396q^{3} + 1024q^{4} + 7350q^{5} + 12672q^{6} + 16807q^{7} - 32768q^{8} - 20331q^{9} + O(q^{10})$$ $$q - 32q^{2} - 396q^{3} + 1024q^{4} + 7350q^{5} + 12672q^{6} + 16807q^{7} - 32768q^{8} - 20331q^{9} - 235200q^{10} - 108780q^{11} - 405504q^{12} - 635842q^{13} - 537824q^{14} - 2910600q^{15} + 1048576q^{16} - 9225918q^{17} + 650592q^{18} - 7555372q^{19} + 7526400q^{20} - 6655572q^{21} + 3480960q^{22} + 26489400q^{23} + 12976128q^{24} + 5194375q^{25} + 20346944q^{26} + 78201288q^{27} + 17210368q^{28} - 169827594q^{29} + 93139200q^{30} - 51362704q^{31} - 33554432q^{32} + 43076880q^{33} + 295229376q^{34} + 123531450q^{35} - 20818944q^{36} - 251605906q^{37} + 241771904q^{38} + 251793432q^{39} - 240844800q^{40} - 928817814q^{41} + 212978304q^{42} - 1818895756q^{43} - 111390720q^{44} - 149432850q^{45} - 847660800q^{46} + 523343136q^{47} - 415236096q^{48} + 282475249q^{49} - 166220000q^{50} + 3653463528q^{51} - 651102208q^{52} + 4199520078q^{53} - 2502441216q^{54} - 799533000q^{55} - 550731776q^{56} + 2991927312q^{57} + 5434483008q^{58} + 9140129196q^{59} - 2980454400q^{60} - 6639312802q^{61} + 1643606528q^{62} - 341703117q^{63} + 1073741824q^{64} - 4673438700q^{65} - 1378460160q^{66} - 2878139188q^{67} - 9447340032q^{68} - 10489802400q^{69} - 3953006400q^{70} - 4345596360q^{71} + 666206208q^{72} + 23450332826q^{73} + 8051388992q^{74} - 2056972500q^{75} - 7736700928q^{76} - 1828265460q^{77} - 8057389824q^{78} - 28761853648q^{79} + 7707033600q^{80} - 27366134391q^{81} + 29722170048q^{82} - 5577757548q^{83} - 6815305728q^{84} - 67810497300q^{85} + 58204664192q^{86} + 67251727224q^{87} + 3564503040q^{88} + 78002173386q^{89} + 4781851200q^{90} - 10686596494q^{91} + 27125145600q^{92} + 20339630784q^{93} - 16746980352q^{94} - 55531984200q^{95} + 13287555072q^{96} - 26685859630q^{97} - 9039207968q^{98} + 2211606180q^{99} + O(q^{100})$$

## 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
−32.0000 −396.000 1024.00 7350.00 12672.0 16807.0 −32768.0 −20331.0 −235200.
 $$n$$: 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 14.12.a.a 1
3.b odd 2 1 126.12.a.d 1
4.b odd 2 1 112.12.a.b 1
7.b odd 2 1 98.12.a.a 1
7.c even 3 2 98.12.c.d 2
7.d odd 6 2 98.12.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.a.a 1 1.a even 1 1 trivial
98.12.a.a 1 7.b odd 2 1
98.12.c.c 2 7.d odd 6 2
98.12.c.d 2 7.c even 3 2
112.12.a.b 1 4.b odd 2 1
126.12.a.d 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 396$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(14))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 32 T$$
$3$ $$1 + 396 T + 177147 T^{2}$$
$5$ $$1 - 7350 T + 48828125 T^{2}$$
$7$ $$1 - 16807 T$$
$11$ $$1 + 108780 T + 285311670611 T^{2}$$
$13$ $$1 + 635842 T + 1792160394037 T^{2}$$
$17$ $$1 + 9225918 T + 34271896307633 T^{2}$$
$19$ $$1 + 7555372 T + 116490258898219 T^{2}$$
$23$ $$1 - 26489400 T + 952809757913927 T^{2}$$
$29$ $$1 + 169827594 T + 12200509765705829 T^{2}$$
$31$ $$1 + 51362704 T + 25408476896404831 T^{2}$$
$37$ $$1 + 251605906 T + 177917621779460413 T^{2}$$
$41$ $$1 + 928817814 T + 550329031716248441 T^{2}$$
$43$ $$1 + 1818895756 T + 929293739471222707 T^{2}$$
$47$ $$1 - 523343136 T + 2472159215084012303 T^{2}$$
$53$ $$1 - 4199520078 T + 9269035929372191597 T^{2}$$
$59$ $$1 - 9140129196 T + 30155888444737842659 T^{2}$$
$61$ $$1 + 6639312802 T + 43513917611435838661 T^{2}$$
$67$ $$1 + 2878139188 T +$$$$12\!\cdots\!83$$$$T^{2}$$
$71$ $$1 + 4345596360 T +$$$$23\!\cdots\!71$$$$T^{2}$$
$73$ $$1 - 23450332826 T +$$$$31\!\cdots\!77$$$$T^{2}$$
$79$ $$1 + 28761853648 T +$$$$74\!\cdots\!79$$$$T^{2}$$
$83$ $$1 + 5577757548 T +$$$$12\!\cdots\!67$$$$T^{2}$$
$89$ $$1 - 78002173386 T +$$$$27\!\cdots\!89$$$$T^{2}$$
$97$ $$1 + 26685859630 T +$$$$71\!\cdots\!53$$$$T^{2}$$