Properties

Label 14.14.a.c
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: \(\Q(\sqrt{100129}) \)
Defining polynomial: \(x^{2} - x - 25032\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{100129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -64 q^{2} + ( 476 - \beta ) q^{3} + 4096 q^{4} + ( 16002 - 63 \beta ) q^{5} + ( -30464 + 64 \beta ) q^{6} + 117649 q^{7} -262144 q^{8} + ( -967231 - 952 \beta ) q^{9} +O(q^{10})\) \( q -64 q^{2} + ( 476 - \beta ) q^{3} + 4096 q^{4} + ( 16002 - 63 \beta ) q^{5} + ( -30464 + 64 \beta ) q^{6} + 117649 q^{7} -262144 q^{8} + ( -967231 - 952 \beta ) q^{9} + ( -1024128 + 4032 \beta ) q^{10} + ( -676368 - 11088 \beta ) q^{11} + ( 1949696 - 4096 \beta ) q^{12} + ( 1755194 + 50895 \beta ) q^{13} -7529536 q^{14} + ( 32849460 - 45990 \beta ) q^{15} + 16777216 q^{16} + ( 108855978 - 128286 \beta ) q^{17} + ( 61902784 + 60928 \beta ) q^{18} + ( 295667876 + 114129 \beta ) q^{19} + ( 65544192 - 258048 \beta ) q^{20} + ( 56000924 - 117649 \beta ) q^{21} + ( 43287552 + 709632 \beta ) q^{22} + ( 420367500 + 1099350 \beta ) q^{23} + ( -124780544 + 262144 \beta ) q^{24} + ( 625008883 - 2016252 \beta ) q^{25} + ( -112332416 - 3257280 \beta ) q^{26} + ( -838008472 + 2108402 \beta ) q^{27} + 481890304 q^{28} + ( -243811770 + 2389086 \beta ) q^{29} + ( -2102365440 + 2943360 \beta ) q^{30} + ( 1096538072 - 3506274 \beta ) q^{31} -1073741824 q^{32} + ( 4118970240 - 4601520 \beta ) q^{33} + ( -6966782592 + 8210304 \beta ) q^{34} + ( 1882619298 - 7411887 \beta ) q^{35} + ( -3961778176 - 3899392 \beta ) q^{36} + ( 202530134 + 9857106 \beta ) q^{37} + ( -18922744064 - 7304256 \beta ) q^{38} + ( -19548789476 + 22470826 \beta ) q^{39} + ( -4194828288 + 16515072 \beta ) q^{40} + ( 4259086314 - 51860862 \beta ) q^{41} + ( -3584059136 + 7529536 \beta ) q^{42} + ( 13112522648 + 36481536 \beta ) q^{43} + ( -2770403328 - 45416448 \beta ) q^{44} + ( 8543717154 + 45701649 \beta ) q^{45} + ( -26903520000 - 70358400 \beta ) q^{46} + ( 77524424880 - 23291442 \beta ) q^{47} + ( 7985954816 - 16777216 \beta ) q^{48} + 13841287201 q^{49} + ( -40000568512 + 129040128 \beta ) q^{50} + ( 103196041104 - 169920114 \beta ) q^{51} + ( 7189274624 + 208465920 \beta ) q^{52} + ( 33003525246 + 339839892 \beta ) q^{53} + ( 53632542208 - 134937728 \beta ) q^{54} + ( 268954807968 - 134818992 \beta ) q^{55} -30840979456 q^{56} + ( 95027418412 - 241342472 \beta ) q^{57} + ( 15603953280 - 152901504 \beta ) q^{58} + ( -238181148492 - 225693891 \beta ) q^{59} + ( 134551388160 - 188375040 \beta ) q^{60} + ( 98689425002 + 401532849 \beta ) q^{61} + ( -70178436608 + 224401536 \beta ) q^{62} + ( -113793759919 - 112001848 \beta ) q^{63} + 68719476736 q^{64} + ( -1256121880272 + 703844568 \beta ) q^{65} + ( -263614095360 + 294497280 \beta ) q^{66} + ( -859366429744 - 840313278 \beta ) q^{67} + ( 445874085888 - 525459456 \beta ) q^{68} + ( -240212334600 + 102923100 \beta ) q^{69} + ( -120487635072 + 474360768 \beta ) q^{70} + ( -347771739168 + 1823602032 \beta ) q^{71} + ( 253553803264 + 249561088 \beta ) q^{72} + ( -233042619670 - 1429760268 \beta ) q^{73} + ( -12961928576 - 630854784 \beta ) q^{74} + ( 1105045414340 - 1584744835 \beta ) q^{75} + ( 1211055620096 + 467472384 \beta ) q^{76} + ( -79574018832 - 1304492112 \beta ) q^{77} + ( 1251122526464 - 1438132864 \beta ) q^{78} + ( -1216008287920 + 2547097812 \beta ) q^{79} + ( 268469010432 - 1056964608 \beta ) q^{80} + ( 298737861509 + 3359403320 \beta ) q^{81} + ( -272581524096 + 3319095168 \beta ) q^{82} + ( -871992247308 + 1600755741 \beta ) q^{83} + ( 229379784704 - 481890304 \beta ) q^{84} + ( 4978890881244 - 8910759186 \beta ) q^{85} + ( -839201449472 - 2334818304 \beta ) q^{86} + ( -1072921570896 + 1381016706 \beta ) q^{87} + ( 177305812992 + 2906652672 \beta ) q^{88} + ( 1511290120242 + 2399150268 \beta ) q^{89} + ( -546797897856 - 2924905536 \beta ) q^{90} + ( 206496818906 + 5987745855 \beta ) q^{91} + ( 1721825280000 + 4502937600 \beta ) q^{92} + ( 1926270959656 - 2765524496 \beta ) q^{93} + ( -4961563192320 + 1490652288 \beta ) q^{94} + ( 1851516446220 - 16800783930 \beta ) q^{95} + ( -511101108224 + 1073741824 \beta ) q^{96} + ( 3880031330546 + 4519008846 \beta ) q^{97} -885842380864 q^{98} + ( 4881961277424 + 11368559664 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 128q^{2} + 952q^{3} + 8192q^{4} + 32004q^{5} - 60928q^{6} + 235298q^{7} - 524288q^{8} - 1934462q^{9} + O(q^{10}) \) \( 2q - 128q^{2} + 952q^{3} + 8192q^{4} + 32004q^{5} - 60928q^{6} + 235298q^{7} - 524288q^{8} - 1934462q^{9} - 2048256q^{10} - 1352736q^{11} + 3899392q^{12} + 3510388q^{13} - 15059072q^{14} + 65698920q^{15} + 33554432q^{16} + 217711956q^{17} + 123805568q^{18} + 591335752q^{19} + 131088384q^{20} + 112001848q^{21} + 86575104q^{22} + 840735000q^{23} - 249561088q^{24} + 1250017766q^{25} - 224664832q^{26} - 1676016944q^{27} + 963780608q^{28} - 487623540q^{29} - 4204730880q^{30} + 2193076144q^{31} - 2147483648q^{32} + 8237940480q^{33} - 13933565184q^{34} + 3765238596q^{35} - 7923556352q^{36} + 405060268q^{37} - 37845488128q^{38} - 39097578952q^{39} - 8389656576q^{40} + 8518172628q^{41} - 7168118272q^{42} + 26225045296q^{43} - 5540806656q^{44} + 17087434308q^{45} - 53807040000q^{46} + 155048849760q^{47} + 15971909632q^{48} + 27682574402q^{49} - 80001137024q^{50} + 206392082208q^{51} + 14378549248q^{52} + 66007050492q^{53} + 107265084416q^{54} + 537909615936q^{55} - 61681958912q^{56} + 190054836824q^{57} + 31207906560q^{58} - 476362296984q^{59} + 269102776320q^{60} + 197378850004q^{61} - 140356873216q^{62} - 227587519838q^{63} + 137438953472q^{64} - 2512243760544q^{65} - 527228190720q^{66} - 1718732859488q^{67} + 891748171776q^{68} - 480424669200q^{69} - 240975270144q^{70} - 695543478336q^{71} + 507107606528q^{72} - 466085239340q^{73} - 25923857152q^{74} + 2210090828680q^{75} + 2422111240192q^{76} - 159148037664q^{77} + 2502245052928q^{78} - 2432016575840q^{79} + 536938020864q^{80} + 597475723018q^{81} - 545163048192q^{82} - 1743984494616q^{83} + 458759569408q^{84} + 9957781762488q^{85} - 1678402898944q^{86} - 2145843141792q^{87} + 354611625984q^{88} + 3022580240484q^{89} - 1093595795712q^{90} + 412993637812q^{91} + 3443650560000q^{92} + 3852541919312q^{93} - 9923126384640q^{94} + 3703032892440q^{95} - 1022202216448q^{96} + 7760062661092q^{97} - 1771684761728q^{98} + 9763922554848q^{99} + O(q^{100}) \)

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
158.716
−157.716
−64.0000 −156.863 4096.00 −23868.4 10039.3 117649. −262144. −1.56972e6 1.52758e6
1.2 −64.0000 1108.86 4096.00 55872.4 −70967.3 117649. −262144. −364745. −3.57583e6
\(n\): 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.c 2
3.b odd 2 1 126.14.a.l 2
4.b odd 2 1 112.14.a.d 2
7.b odd 2 1 98.14.a.e 2
7.c even 3 2 98.14.c.l 4
7.d odd 6 2 98.14.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.a.c 2 1.a even 1 1 trivial
98.14.a.e 2 7.b odd 2 1
98.14.c.l 4 7.c even 3 2
98.14.c.m 4 7.d odd 6 2
112.14.a.d 2 4.b odd 2 1
126.14.a.l 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 952 T_{3} - 173940 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 64 + T )^{2} \)
$3$ \( -173940 - 952 T + T^{2} \)
$5$ \( -1333584000 - 32004 T + T^{2} \)
$7$ \( ( -117649 + T )^{2} \)
$11$ \( -48783462900480 + 1352736 T + T^{2} \)
$13$ \( -1034376299351264 - 3510388 T + T^{2} \)
$17$ \( 5258212862273748 - 217711956 T + T^{2} \)
$19$ \( 82202600320772620 - 591335752 T + T^{2} \)
$23$ \( -307342956281760000 - 840735000 T + T^{2} \)
$29$ \( -2226593776636211436 + 487623540 T + T^{2} \)
$31$ \( -3721530883884270032 - 2193076144 T + T^{2} \)
$37$ \( -38874132892883083820 - 405060268 T + T^{2} \)
$41$ \( -\)\(10\!\cdots\!08\)\( - 8518172628 T + T^{2} \)
$43$ \( -\)\(36\!\cdots\!32\)\( - 26225045296 T + T^{2} \)
$47$ \( \)\(57\!\cdots\!76\)\( - 155048849760 T + T^{2} \)
$53$ \( -\)\(45\!\cdots\!08\)\( - 66007050492 T + T^{2} \)
$59$ \( \)\(36\!\cdots\!68\)\( + 476362296984 T + T^{2} \)
$61$ \( -\)\(54\!\cdots\!12\)\( - 197378850004 T + T^{2} \)
$67$ \( \)\(45\!\cdots\!92\)\( + 1718732859488 T + T^{2} \)
$71$ \( -\)\(12\!\cdots\!60\)\( + 695543478336 T + T^{2} \)
$73$ \( -\)\(76\!\cdots\!84\)\( + 466085239340 T + T^{2} \)
$79$ \( -\)\(11\!\cdots\!04\)\( + 2432016575840 T + T^{2} \)
$83$ \( -\)\(26\!\cdots\!32\)\( + 1743984494616 T + T^{2} \)
$89$ \( -\)\(21\!\cdots\!20\)\( - 3022580240484 T + T^{2} \)
$97$ \( \)\(68\!\cdots\!60\)\( - 7760062661092 T + T^{2} \)
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