# Properties

 Label 14.14.a.d Level $14$ Weight $14$ Character orbit 14.a Self dual yes Analytic conductor $15.012$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$15.0123300533$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{78985})$$ Defining polynomial: $$x^{2} - x - 19746$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{78985}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 64 q^{2} + ( 553 - 5 \beta ) q^{3} + 4096 q^{4} + ( 37765 - 39 \beta ) q^{5} + ( 35392 - 320 \beta ) q^{6} -117649 q^{7} + 262144 q^{8} + ( 686111 - 5530 \beta ) q^{9} +O(q^{10})$$ $$q + 64 q^{2} + ( 553 - 5 \beta ) q^{3} + 4096 q^{4} + ( 37765 - 39 \beta ) q^{5} + ( 35392 - 320 \beta ) q^{6} -117649 q^{7} + 262144 q^{8} + ( 686111 - 5530 \beta ) q^{9} + ( 2416960 - 2496 \beta ) q^{10} + ( 1667930 + 34650 \beta ) q^{11} + ( 2265088 - 20480 \beta ) q^{12} + ( 3999051 + 62475 \beta ) q^{13} -7529536 q^{14} + ( 36286120 - 210392 \beta ) q^{15} + 16777216 q^{16} + ( 19512416 + 358290 \beta ) q^{17} + ( 43911104 - 353920 \beta ) q^{18} + ( 62046467 - 543855 \beta ) q^{19} + ( 154685440 - 159744 \beta ) q^{20} + ( -65059897 + 588245 \beta ) q^{21} + ( 106747520 + 2217600 \beta ) q^{22} + ( 950408000 + 448560 \beta ) q^{23} + ( 144965632 - 1310720 \beta ) q^{24} + ( 325628285 - 2945670 \beta ) q^{25} + ( 255939264 + 3998400 \beta ) q^{26} + ( 1681694014 + 1482970 \beta ) q^{27} -481890304 q^{28} + ( -122567576 - 10680810 \beta ) q^{29} + ( 2322311680 - 13465088 \beta ) q^{30} + ( -5505280074 + 9271530 \beta ) q^{31} + 1073741824 q^{32} + ( -12761785960 + 10821800 \beta ) q^{33} + ( 1248794624 + 22930560 \beta ) q^{34} + ( -4443014485 + 4588311 \beta ) q^{35} + ( 2810310656 - 22650880 \beta ) q^{36} + ( -19815245608 + 36470910 \beta ) q^{37} + ( 3970973888 - 34806720 \beta ) q^{38} + ( -22461464172 + 14553420 \beta ) q^{39} + ( 9899868160 - 10223616 \beta ) q^{40} + ( -1215513684 - 8828610 \beta ) q^{41} + ( -4163833408 + 37647680 \beta ) q^{42} + ( 8911569158 - 177554370 \beta ) q^{43} + ( 6831841280 + 141926400 \beta ) q^{44} + ( 42945676865 - 235598779 \beta ) q^{45} + ( 60826112000 + 28707840 \beta ) q^{46} + ( 32244155538 + 349815630 \beta ) q^{47} + ( 9277800448 - 83886080 \beta ) q^{48} + 13841287201 q^{49} + ( 20840210240 - 188522880 \beta ) q^{50} + ( -130707312202 + 100572290 \beta ) q^{51} + ( 16380112896 + 255897600 \beta ) q^{52} + ( -63252088314 - 317904720 \beta ) q^{53} + ( 107628416896 + 94910080 \beta ) q^{54} + ( -43747003300 + 1243507980 \beta ) q^{55} -30840979456 q^{56} + ( 249093632126 - 610984150 \beta ) q^{57} + ( -7844324864 - 683571840 \beta ) q^{58} + ( 170629980619 - 896167935 \beta ) q^{59} + ( 148627947520 - 861765632 \beta ) q^{60} + ( 223620350373 - 616179855 \beta ) q^{61} + ( -352337924736 + 593377920 \beta ) q^{62} + ( -80720273039 + 650598970 \beta ) q^{63} + 68719476736 q^{64} + ( -41424766110 + 2203405386 \beta ) q^{65} + ( -816754301440 + 692595200 \beta ) q^{66} + ( -1035661145444 + 846793920 \beta ) q^{67} + ( 79922855936 + 1467555840 \beta ) q^{68} + ( 348428066000 - 4503986320 \beta ) q^{69} + ( -284352927040 + 293651904 \beta ) q^{70} + ( 325217232860 + 1552163340 \beta ) q^{71} + ( 179859881984 - 1449656320 \beta ) q^{72} + ( -724904665058 + 2850319740 \beta ) q^{73} + ( -1268175718912 + 2334138240 \beta ) q^{74} + ( 1343391166355 - 3257096935 \beta ) q^{75} + ( 254142328832 - 2227630080 \beta ) q^{76} + ( -196230296570 - 4076537850 \beta ) q^{77} + ( -1437533707008 + 931418880 \beta ) q^{78} + ( 762576198828 - 5153135820 \beta ) q^{79} + ( 633591562240 - 654311424 \beta ) q^{80} + ( -749567685361 + 1228218530 \beta ) q^{81} + ( -77792875776 - 565031040 \beta ) q^{82} + ( 2128758819719 + 10286107965 \beta ) q^{83} + ( -266485338112 + 2409451520 \beta ) q^{84} + ( -366795500110 + 12769837626 \beta ) q^{85} + ( 570340426112 - 11363479680 \beta ) q^{86} + ( 4150339019722 - 5293650050 \beta ) q^{87} + ( 437237841920 + 9083289600 \beta ) q^{88} + ( 6296825611314 - 2018475960 \beta ) q^{89} + ( 2748523319360 - 15078321856 \beta ) q^{90} + ( -470484351099 - 7350121275 \beta ) q^{91} + ( 3892871168000 + 1837301760 \beta ) q^{92} + ( -6705978866172 + 32653556460 \beta ) q^{93} + ( 2063625954432 + 22388200320 \beta ) q^{94} + ( 4018483926080 - 22958496288 \beta ) q^{95} + ( 593779228672 - 5368709120 \beta ) q^{96} + ( -8270785003880 - 22566578790 \beta ) q^{97} + 885842380864 q^{98} + ( -13990286162270 + 14550093250 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 128q^{2} + 1106q^{3} + 8192q^{4} + 75530q^{5} + 70784q^{6} - 235298q^{7} + 524288q^{8} + 1372222q^{9} + O(q^{10})$$ $$2q + 128q^{2} + 1106q^{3} + 8192q^{4} + 75530q^{5} + 70784q^{6} - 235298q^{7} + 524288q^{8} + 1372222q^{9} + 4833920q^{10} + 3335860q^{11} + 4530176q^{12} + 7998102q^{13} - 15059072q^{14} + 72572240q^{15} + 33554432q^{16} + 39024832q^{17} + 87822208q^{18} + 124092934q^{19} + 309370880q^{20} - 130119794q^{21} + 213495040q^{22} + 1900816000q^{23} + 289931264q^{24} + 651256570q^{25} + 511878528q^{26} + 3363388028q^{27} - 963780608q^{28} - 245135152q^{29} + 4644623360q^{30} - 11010560148q^{31} + 2147483648q^{32} - 25523571920q^{33} + 2497589248q^{34} - 8886028970q^{35} + 5620621312q^{36} - 39630491216q^{37} + 7941947776q^{38} - 44922928344q^{39} + 19799736320q^{40} - 2431027368q^{41} - 8327666816q^{42} + 17823138316q^{43} + 13663682560q^{44} + 85891353730q^{45} + 121652224000q^{46} + 64488311076q^{47} + 18555600896q^{48} + 27682574402q^{49} + 41680420480q^{50} - 261414624404q^{51} + 32760225792q^{52} - 126504176628q^{53} + 215256833792q^{54} - 87494006600q^{55} - 61681958912q^{56} + 498187264252q^{57} - 15688649728q^{58} + 341259961238q^{59} + 297255895040q^{60} + 447240700746q^{61} - 704675849472q^{62} - 161440546078q^{63} + 137438953472q^{64} - 82849532220q^{65} - 1633508602880q^{66} - 2071322290888q^{67} + 159845711872q^{68} + 696856132000q^{69} - 568705854080q^{70} + 650434465720q^{71} + 359719763968q^{72} - 1449809330116q^{73} - 2536351437824q^{74} + 2686782332710q^{75} + 508284657664q^{76} - 392460593140q^{77} - 2875067414016q^{78} + 1525152397656q^{79} + 1267183124480q^{80} - 1499135370722q^{81} - 155585751552q^{82} + 4257517639438q^{83} - 532970676224q^{84} - 733591000220q^{85} + 1140680852224q^{86} + 8300678039444q^{87} + 874475683840q^{88} + 12593651222628q^{89} + 5497046638720q^{90} - 940968702198q^{91} + 7785742336000q^{92} - 13411957732344q^{93} + 4127251908864q^{94} + 8036967852160q^{95} + 1187558457344q^{96} - 16541570007760q^{97} + 1771684761728q^{98} - 27980572324540q^{99} + O(q^{100})$$

## 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
 141.021 −140.021
64.0000 −852.214 4096.00 26804.3 −54541.7 −117649. 262144. −868055. 1.71548e6
1.2 64.0000 1958.21 4096.00 48725.7 125326. −117649. 262144. 2.24028e6 3.11844e6
 $$n$$: 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 14.14.a.d 2
3.b odd 2 1 126.14.a.g 2
4.b odd 2 1 112.14.a.c 2
7.b odd 2 1 98.14.a.f 2
7.c even 3 2 98.14.c.i 4
7.d odd 6 2 98.14.c.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.a.d 2 1.a even 1 1 trivial
98.14.a.f 2 7.b odd 2 1
98.14.c.i 4 7.c even 3 2
98.14.c.k 4 7.d odd 6 2
112.14.a.c 2 4.b odd 2 1
126.14.a.g 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 1106 T_{3} - 1668816$$ acting on $$S_{14}^{\mathrm{new}}(\Gamma_0(14))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -64 + T )^{2}$$
$3$ $$-1668816 - 1106 T + T^{2}$$
$5$ $$1306059040 - 75530 T + T^{2}$$
$7$ $$( 117649 + T )^{2}$$
$11$ $$-92049177677600 - 3335860 T + T^{2}$$
$13$ $$-292295968590024 - 7998102 T + T^{2}$$
$17$ $$-9758706249881444 - 39024832 T + T^{2}$$
$19$ $$-19512281879877536 - 124092934 T + T^{2}$$
$23$ $$887383104740704000 - 1900816000 T + T^{2}$$
$29$ $$-8995562472011542724 + 245135152 T + T^{2}$$
$31$ $$23518457897478458976 + 11010560148 T + T^{2}$$
$37$ $$28\!\cdots\!64$$$$+ 39630491216 T + T^{2}$$
$41$ $$-4678961326726666644 + 2431027368 T + T^{2}$$
$43$ $$-$$$$24\!\cdots\!36$$$$- 17823138316 T + T^{2}$$
$47$ $$-$$$$86\!\cdots\!56$$$$- 64488311076 T + T^{2}$$
$53$ $$-$$$$39\!\cdots\!04$$$$+ 126504176628 T + T^{2}$$
$59$ $$-$$$$34\!\cdots\!64$$$$- 341259961238 T + T^{2}$$
$61$ $$20\!\cdots\!04$$$$- 447240700746 T + T^{2}$$
$67$ $$10\!\cdots\!36$$$$+ 2071322290888 T + T^{2}$$
$71$ $$-$$$$84\!\cdots\!00$$$$- 650434465720 T + T^{2}$$
$73$ $$-$$$$11\!\cdots\!36$$$$+ 1449809330116 T + T^{2}$$
$79$ $$-$$$$15\!\cdots\!16$$$$- 1525152397656 T + T^{2}$$
$83$ $$-$$$$38\!\cdots\!64$$$$- 4257517639438 T + T^{2}$$
$89$ $$39\!\cdots\!96$$$$- 12593651222628 T + T^{2}$$
$97$ $$28\!\cdots\!00$$$$+ 16541570007760 T + T^{2}$$