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 $1$

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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:

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

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