Properties

Label 14.12.a.d
Level $14$
Weight $12$
Character orbit 14.a
Self dual yes
Analytic conductor $10.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{352969}) \)
Defining polynomial: \(x^{2} - x - 88242\)
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 \(\beta = \sqrt{352969}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 32 q^{2} + ( -175 - \beta ) q^{3} + 1024 q^{4} + ( 1869 - 3 \beta ) q^{5} + ( -5600 - 32 \beta ) q^{6} + 16807 q^{7} + 32768 q^{8} + ( 206447 + 350 \beta ) q^{9} +O(q^{10})\) \( q + 32 q^{2} + ( -175 - \beta ) q^{3} + 1024 q^{4} + ( 1869 - 3 \beta ) q^{5} + ( -5600 - 32 \beta ) q^{6} + 16807 q^{7} + 32768 q^{8} + ( 206447 + 350 \beta ) q^{9} + ( 59808 - 96 \beta ) q^{10} + ( 476850 + 714 \beta ) q^{11} + ( -179200 - 1024 \beta ) q^{12} + ( -112861 - 1449 \beta ) q^{13} + 537824 q^{14} + ( 731832 - 1344 \beta ) q^{15} + 1048576 q^{16} + ( 2558136 - 750 \beta ) q^{17} + ( 6606304 + 11200 \beta ) q^{18} + ( 8265971 + 7893 \beta ) q^{19} + ( 1913856 - 3072 \beta ) q^{20} + ( -2941225 - 16807 \beta ) q^{21} + ( 15259200 + 22848 \beta ) q^{22} + ( -7050864 - 1848 \beta ) q^{23} + ( -5734400 - 32768 \beta ) q^{24} + ( -42158243 - 11214 \beta ) q^{25} + ( -3611552 - 46368 \beta ) q^{26} + ( -128666650 - 90550 \beta ) q^{27} + 17210368 q^{28} + ( -35446752 + 280014 \beta ) q^{29} + ( 23418624 - 43008 \beta ) q^{30} + ( -2582146 + 415386 \beta ) q^{31} + 33554432 q^{32} + ( -335468616 - 601800 \beta ) q^{33} + ( 81860352 - 24000 \beta ) q^{34} + ( 31412283 - 50421 \beta ) q^{35} + ( 211401728 + 358400 \beta ) q^{36} + ( 53644736 - 790650 \beta ) q^{37} + ( 264511072 + 252576 \beta ) q^{38} + ( 531202756 + 366436 \beta ) q^{39} + ( 61243392 - 98304 \beta ) q^{40} + ( 86095044 - 903714 \beta ) q^{41} + ( -94119200 - 537824 \beta ) q^{42} + ( 439424078 + 2050398 \beta ) q^{43} + ( 488294400 + 731136 \beta ) q^{44} + ( 15231993 + 34809 \beta ) q^{45} + ( -225627648 - 59136 \beta ) q^{46} + ( -649570278 + 1150398 \beta ) q^{47} + ( -183500800 - 1048576 \beta ) q^{48} + 282475249 q^{49} + ( -1349063776 - 358848 \beta ) q^{50} + ( -182947050 - 2426886 \beta ) q^{51} + ( -115569664 - 1483776 \beta ) q^{52} + ( 3719878158 + 2845248 \beta ) q^{53} + ( -4117332800 - 2897600 \beta ) q^{54} + ( 135173052 - 96084 \beta ) q^{55} + 550731776 q^{56} + ( -4232529242 - 9647246 \beta ) q^{57} + ( -1134296064 + 8960448 \beta ) q^{58} + ( 1109635779 + 2904957 \beta ) q^{59} + ( 749395968 - 1376256 \beta ) q^{60} + ( -4869899419 + 6782157 \beta ) q^{61} + ( -82628672 + 13292352 \beta ) q^{62} + ( 3469754729 + 5882450 \beta ) q^{63} + 1073741824 q^{64} + ( 1323419034 - 2369598 \beta ) q^{65} + ( -10734995712 - 19257600 \beta ) q^{66} + ( 1389543956 - 19290600 \beta ) q^{67} + ( 2619531264 - 768000 \beta ) q^{68} + ( 1886187912 + 7374264 \beta ) q^{69} + ( 1005193056 - 1613472 \beta ) q^{70} + ( 10598770044 - 24722964 \beta ) q^{71} + ( 6764855296 + 11468800 \beta ) q^{72} + ( -7007615314 + 13241052 \beta ) q^{73} + ( 1716631552 - 25300800 \beta ) q^{74} + ( 11335886891 + 44120693 \beta ) q^{75} + ( 8464354304 + 8082432 \beta ) q^{76} + ( 8014417950 + 12000198 \beta ) q^{77} + ( 16998488192 + 11725952 \beta ) q^{78} + ( -10854222052 - 39612636 \beta ) q^{79} + ( 1959788544 - 3145728 \beta ) q^{80} + ( 17906539991 + 82511450 \beta ) q^{81} + ( 2755041408 - 28918848 \beta ) q^{82} + ( 5425851375 - 50028951 \beta ) q^{83} + ( -3011814400 - 17210368 \beta ) q^{84} + ( 5575336434 - 9076158 \beta ) q^{85} + ( 14061570496 + 65612736 \beta ) q^{86} + ( -92633079966 - 13555698 \beta ) q^{87} + ( 15625420800 + 23396352 \beta ) q^{88} + ( -62703588990 - 66399000 \beta ) q^{89} + ( 487423776 + 1113888 \beta ) q^{90} + ( -1896854827 - 24353343 \beta ) q^{91} + ( -7220084736 - 1892352 \beta ) q^{92} + ( -146166505484 - 70110404 \beta ) q^{93} + ( -20786248896 + 36812736 \beta ) q^{94} + ( 7091146848 - 10045896 \beta ) q^{95} + ( -5872025600 - 33554432 \beta ) q^{96} + ( 52346110208 - 58568454 \beta ) q^{97} + 9039207968 q^{98} + ( 186651205050 + 314300658 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 64q^{2} - 350q^{3} + 2048q^{4} + 3738q^{5} - 11200q^{6} + 33614q^{7} + 65536q^{8} + 412894q^{9} + O(q^{10}) \) \( 2q + 64q^{2} - 350q^{3} + 2048q^{4} + 3738q^{5} - 11200q^{6} + 33614q^{7} + 65536q^{8} + 412894q^{9} + 119616q^{10} + 953700q^{11} - 358400q^{12} - 225722q^{13} + 1075648q^{14} + 1463664q^{15} + 2097152q^{16} + 5116272q^{17} + 13212608q^{18} + 16531942q^{19} + 3827712q^{20} - 5882450q^{21} + 30518400q^{22} - 14101728q^{23} - 11468800q^{24} - 84316486q^{25} - 7223104q^{26} - 257333300q^{27} + 34420736q^{28} - 70893504q^{29} + 46837248q^{30} - 5164292q^{31} + 67108864q^{32} - 670937232q^{33} + 163720704q^{34} + 62824566q^{35} + 422803456q^{36} + 107289472q^{37} + 529022144q^{38} + 1062405512q^{39} + 122486784q^{40} + 172190088q^{41} - 188238400q^{42} + 878848156q^{43} + 976588800q^{44} + 30463986q^{45} - 451255296q^{46} - 1299140556q^{47} - 367001600q^{48} + 564950498q^{49} - 2698127552q^{50} - 365894100q^{51} - 231139328q^{52} + 7439756316q^{53} - 8234665600q^{54} + 270346104q^{55} + 1101463552q^{56} - 8465058484q^{57} - 2268592128q^{58} + 2219271558q^{59} + 1498791936q^{60} - 9739798838q^{61} - 165257344q^{62} + 6939509458q^{63} + 2147483648q^{64} + 2646838068q^{65} - 21469991424q^{66} + 2779087912q^{67} + 5239062528q^{68} + 3772375824q^{69} + 2010386112q^{70} + 21197540088q^{71} + 13529710592q^{72} - 14015230628q^{73} + 3433263104q^{74} + 22671773782q^{75} + 16928708608q^{76} + 16028835900q^{77} + 33996976384q^{78} - 21708444104q^{79} + 3919577088q^{80} + 35813079982q^{81} + 5510082816q^{82} + 10851702750q^{83} - 6023628800q^{84} + 11150672868q^{85} + 28123140992q^{86} - 185266159932q^{87} + 31250841600q^{88} - 125407177980q^{89} + 974847552q^{90} - 3793709654q^{91} - 14440169472q^{92} - 292333010968q^{93} - 41572497792q^{94} + 14182293696q^{95} - 11744051200q^{96} + 104692220416q^{97} + 18078415936q^{98} + 373302410100q^{99} + O(q^{100}) \)

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
297.556
−296.556
32.0000 −769.112 1024.00 86.6642 −24611.6 16807.0 32768.0 414386. 2773.25
1.2 32.0000 419.112 1024.00 3651.34 13411.6 16807.0 32768.0 −1492.18 116843.
\(n\): 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 14.12.a.d 2
3.b odd 2 1 126.12.a.i 2
4.b odd 2 1 112.12.a.e 2
7.b odd 2 1 98.12.a.g 2
7.c even 3 2 98.12.c.h 4
7.d odd 6 2 98.12.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.12.a.d 2 1.a even 1 1 trivial
98.12.a.g 2 7.b odd 2 1
98.12.c.e 4 7.d odd 6 2
98.12.c.h 4 7.c even 3 2
112.12.a.e 2 4.b odd 2 1
126.12.a.i 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 350 T_{3} - 322344 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -32 + T )^{2} \)
$3$ \( -322344 + 350 T + T^{2} \)
$5$ \( 316440 - 3738 T + T^{2} \)
$7$ \( ( -16807 + T )^{2} \)
$11$ \( 47443738176 - 953700 T + T^{2} \)
$13$ \( -728356460048 + 225722 T + T^{2} \)
$17$ \( 6345514731996 - 5116272 T + T^{2} \)
$19$ \( 46336502358760 - 16531942 T + T^{2} \)
$23$ \( 48509257302720 + 14101728 T + T^{2} \)
$29$ \( -26419064718792420 + 70893504 T + T^{2} \)
$31$ \( -60896555346223808 + 5164292 T + T^{2} \)
$37$ \( -217772843491892804 - 107289472 T + T^{2} \)
$41$ \( -280857070539818388 - 172190088 T + T^{2} \)
$43$ \( -1290834732899751392 - 878848156 T + T^{2} \)
$47$ \( -45183120173304192 + 1299140556 T + T^{2} \)
$53$ \( 10980055496816187588 - 7439756316 T + T^{2} \)
$59$ \( -1747334471595432840 - 2219271558 T + T^{2} \)
$61$ \( 7480174567292192680 + 9739798838 T + T^{2} \)
$67$ \( -\)\(12\!\cdots\!64\)\( - 2779087912 T + T^{2} \)
$71$ \( -\)\(10\!\cdots\!88\)\( - 21197540088 T + T^{2} \)
$73$ \( -12777779219339125580 + 14015230628 T + T^{2} \)
$79$ \( -\)\(43\!\cdots\!20\)\( + 21708444104 T + T^{2} \)
$83$ \( -\)\(85\!\cdots\!44\)\( - 10851702750 T + T^{2} \)
$89$ \( \)\(23\!\cdots\!00\)\( + 125407177980 T + T^{2} \)
$97$ \( \)\(15\!\cdots\!60\)\( - 104692220416 T + T^{2} \)
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