# Properties

 Label 14.12.a.c Level $14$ Weight $12$ Character orbit 14.a Self dual yes Analytic conductor $10.757$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{153169})$$ Defining polynomial: $$x^{2} - x - 38292$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{153169}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -32 q^{2} + ( 175 - \beta ) q^{3} + 1024 q^{4} + ( 133 - 21 \beta ) q^{5} + ( -5600 + 32 \beta ) q^{6} -16807 q^{7} -32768 q^{8} + ( 6647 - 350 \beta ) q^{9} +O(q^{10})$$ $$q -32 q^{2} + ( 175 - \beta ) q^{3} + 1024 q^{4} + ( 133 - 21 \beta ) q^{5} + ( -5600 + 32 \beta ) q^{6} -16807 q^{7} -32768 q^{8} + ( 6647 - 350 \beta ) q^{9} + ( -4256 + 672 \beta ) q^{10} + ( 386750 + 1386 \beta ) q^{11} + ( 179200 - 1024 \beta ) q^{12} + ( 1637811 - 2199 \beta ) q^{13} + 537824 q^{14} + ( 3239824 - 3808 \beta ) q^{15} + 1048576 q^{16} + ( 4688264 - 4050 \beta ) q^{17} + ( -212704 + 11200 \beta ) q^{18} + ( 870821 + 41757 \beta ) q^{19} + ( 136192 - 21504 \beta ) q^{20} + ( -2941225 + 16807 \beta ) q^{21} + ( -12376000 - 44352 \beta ) q^{22} + ( -5996536 - 94248 \beta ) q^{23} + ( -5734400 + 32768 \beta ) q^{24} + ( 18737093 - 5586 \beta ) q^{25} + ( -52409952 + 70368 \beta ) q^{26} + ( 23771650 + 109250 \beta ) q^{27} -17210368 q^{28} + ( -11677952 - 179214 \beta ) q^{29} + ( -103674368 + 121856 \beta ) q^{30} + ( 71174754 - 11286 \beta ) q^{31} -33554432 q^{32} + ( -144610984 - 144200 \beta ) q^{33} + ( -150024448 + 129600 \beta ) q^{34} + ( -2235331 + 352947 \beta ) q^{35} + ( 6806528 - 358400 \beta ) q^{36} + ( 265064864 + 1151850 \beta ) q^{37} + ( -27866272 - 1336224 \beta ) q^{38} + ( 623435556 - 2022636 \beta ) q^{39} + ( -4358144 + 688128 \beta ) q^{40} + ( -103232556 + 2202114 \beta ) q^{41} + ( 94119200 - 537824 \beta ) q^{42} + ( 518322722 - 1838802 \beta ) q^{43} + ( 396032000 + 1419264 \beta ) q^{44} + ( 1126676201 - 186137 \beta ) q^{45} + ( 191889152 + 3015936 \beta ) q^{46} + ( -1423801722 + 1709598 \beta ) q^{47} + ( 183500800 - 1048576 \beta ) q^{48} + 282475249 q^{49} + ( -599586976 + 178752 \beta ) q^{50} + ( 1440780650 - 5397014 \beta ) q^{51} + ( 1677118464 - 2251776 \beta ) q^{52} + ( -2438042658 + 2408448 \beta ) q^{53} + ( -760692800 - 3496000 \beta ) q^{54} + ( -4406699164 - 7937412 \beta ) q^{55} + 550731776 q^{56} + ( -6243484258 + 6436654 \beta ) q^{57} + ( 373694464 + 5734848 \beta ) q^{58} + ( -3361608971 + 16765893 \beta ) q^{59} + ( 3317579776 - 3899392 \beta ) q^{60} + ( -4862139219 + 677643 \beta ) q^{61} + ( -2277592128 + 361152 \beta ) q^{62} + ( -111716129 + 5882450 \beta ) q^{63} + 1073741824 q^{64} + ( 7291020114 - 34686498 \beta ) q^{65} + ( 4627551488 + 4614400 \beta ) q^{66} + ( 4224305644 + 11839800 \beta ) q^{67} + ( 4800782336 - 4147200 \beta ) q^{68} + ( 13386478112 - 10496864 \beta ) q^{69} + ( 71530592 - 11294304 \beta ) q^{70} + ( -13140841756 + 22375164 \beta ) q^{71} + ( -217808896 + 11468800 \beta ) q^{72} + ( 30569898814 - 8431548 \beta ) q^{73} + ( -8482075648 - 36859200 \beta ) q^{74} + ( 4134593309 - 19714643 \beta ) q^{75} + ( 891720704 + 42759168 \beta ) q^{76} + ( -6500107250 - 23294502 \beta ) q^{77} + ( -19949937792 + 64724352 \beta ) q^{78} + ( 18945232548 + 58710036 \beta ) q^{79} + ( 139460608 - 22020096 \beta ) q^{80} + ( -13751170609 + 57348550 \beta ) q^{81} + ( 3303441792 - 70467648 \beta ) q^{82} + ( -23281885375 - 61936551 \beta ) q^{83} + ( -3011814400 + 17210368 \beta ) q^{84} + ( 13650562562 - 98992194 \beta ) q^{85} + ( -16586327104 + 58841664 \beta ) q^{86} + ( 25406387566 - 19684498 \beta ) q^{87} + ( -12673024000 - 45416448 \beta ) q^{88} + ( 40344662610 + 153100200 \beta ) q^{89} + ( -36053638432 + 5956384 \beta ) q^{90} + ( -27526689477 + 36958593 \beta ) q^{91} + ( -6140452864 - 96509952 \beta ) q^{92} + ( 14184247284 - 73149804 \beta ) q^{93} + ( 45561655104 - 54707136 \beta ) q^{94} + ( -134197617400 - 12733560 \beta ) q^{95} + ( -5872025600 + 33554432 \beta ) q^{96} + ( -4972084208 + 143371446 \beta ) q^{97} -9039207968 q^{98} + ( -71731554650 - 126149758 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 64q^{2} + 350q^{3} + 2048q^{4} + 266q^{5} - 11200q^{6} - 33614q^{7} - 65536q^{8} + 13294q^{9} + O(q^{10})$$ $$2q - 64q^{2} + 350q^{3} + 2048q^{4} + 266q^{5} - 11200q^{6} - 33614q^{7} - 65536q^{8} + 13294q^{9} - 8512q^{10} + 773500q^{11} + 358400q^{12} + 3275622q^{13} + 1075648q^{14} + 6479648q^{15} + 2097152q^{16} + 9376528q^{17} - 425408q^{18} + 1741642q^{19} + 272384q^{20} - 5882450q^{21} - 24752000q^{22} - 11993072q^{23} - 11468800q^{24} + 37474186q^{25} - 104819904q^{26} + 47543300q^{27} - 34420736q^{28} - 23355904q^{29} - 207348736q^{30} + 142349508q^{31} - 67108864q^{32} - 289221968q^{33} - 300048896q^{34} - 4470662q^{35} + 13613056q^{36} + 530129728q^{37} - 55732544q^{38} + 1246871112q^{39} - 8716288q^{40} - 206465112q^{41} + 188238400q^{42} + 1036645444q^{43} + 792064000q^{44} + 2253352402q^{45} + 383778304q^{46} - 2847603444q^{47} + 367001600q^{48} + 564950498q^{49} - 1199173952q^{50} + 2881561300q^{51} + 3354236928q^{52} - 4876085316q^{53} - 1521385600q^{54} - 8813398328q^{55} + 1101463552q^{56} - 12486968516q^{57} + 747388928q^{58} - 6723217942q^{59} + 6635159552q^{60} - 9724278438q^{61} - 4555184256q^{62} - 223432258q^{63} + 2147483648q^{64} + 14582040228q^{65} + 9255102976q^{66} + 8448611288q^{67} + 9601564672q^{68} + 26772956224q^{69} + 143061184q^{70} - 26281683512q^{71} - 435617792q^{72} + 61139797628q^{73} - 16964151296q^{74} + 8269186618q^{75} + 1783441408q^{76} - 13000214500q^{77} - 39899875584q^{78} + 37890465096q^{79} + 278921216q^{80} - 27502341218q^{81} + 6606883584q^{82} - 46563770750q^{83} - 6023628800q^{84} + 27301125124q^{85} - 33172654208q^{86} + 50812775132q^{87} - 25346048000q^{88} + 80689325220q^{89} - 72107276864q^{90} - 55053378954q^{91} - 12280905728q^{92} + 28368494568q^{93} + 91123310208q^{94} - 268395234800q^{95} - 11744051200q^{96} - 9944168416q^{97} - 18078415936q^{98} - 143463109300q^{99} + O(q^{100})$$

## 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
 196.184 −195.184
−32.0000 −216.368 1024.00 −8085.73 6923.78 −16807.0 −32768.0 −130332. 258743.
1.2 −32.0000 566.368 1024.00 8351.73 −18123.8 −16807.0 −32768.0 143626. −267255.
 $$n$$: 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.c 2
3.b odd 2 1 126.12.a.l 2
4.b odd 2 1 112.12.a.c 2
7.b odd 2 1 98.12.a.c 2
7.c even 3 2 98.12.c.i 4
7.d odd 6 2 98.12.c.k 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 350 T_{3} - 122544$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(14))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 32 + T )^{2}$$
$3$ $$-122544 - 350 T + T^{2}$$
$5$ $$-67529840 - 266 T + T^{2}$$
$7$ $$( 16807 + T )^{2}$$
$11$ $$-144661473824 - 773500 T + T^{2}$$
$13$ $$1941760702152 - 3275622 T + T^{2}$$
$17$ $$19467464811196 - 9376528 T + T^{2}$$
$19$ $$-266314345634240 - 1741642 T + T^{2}$$
$23$ $$-1324593611962880 + 11993072 T + T^{2}$$
$29$ $$-4783054964041220 + 23355904 T + T^{2}$$
$31$ $$5046335890000992 - 142349508 T + T^{2}$$
$37$ $$-132958878688564004 - 530129728 T + T^{2}$$
$41$ $$-732106400663755188 + 206465112 T + T^{2}$$
$43$ $$-249235475107112192 - 1036645444 T + T^{2}$$
$47$ $$1579540428785402208 + 2847603444 T + T^{2}$$
$53$ $$5055576566537081988 + 4876085316 T + T^{2}$$
$59$ $$-31754650926878797040 + 6723217942 T + T^{2}$$
$61$ $$23570062574708242080 + 9724278438 T + T^{2}$$
$67$ $$-3626604590212505264 - 8448611288 T + T^{2}$$
$71$ $$95997974054197530112 + 26281683512 T + T^{2}$$
$73$ $$92\!\cdots\!20$$$$- 61139797628 T + T^{2}$$
$79$ $$-$$$$16\!\cdots\!20$$$$- 37890465096 T + T^{2}$$
$83$ $$-45530981944140138944 + 46563770750 T + T^{2}$$
$89$ $$-$$$$19\!\cdots\!00$$$$- 80689325220 T + T^{2}$$
$97$ $$-$$$$31\!\cdots\!40$$$$+ 9944168416 T + T^{2}$$