Properties

Label 14.10.a.c
Level $14$
Weight $10$
Character orbit 14.a
Self dual yes
Analytic conductor $7.211$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.21050170629\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
Defining polynomial: \(x^{2} - x - 576\)
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 \(\beta = \sqrt{2305}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -16 q^{2} + ( -7 - 5 \beta ) q^{3} + 256 q^{4} + ( -1365 - 21 \beta ) q^{5} + ( 112 + 80 \beta ) q^{6} + 2401 q^{7} -4096 q^{8} + ( 37991 + 70 \beta ) q^{9} +O(q^{10})\) \( q -16 q^{2} + ( -7 - 5 \beta ) q^{3} + 256 q^{4} + ( -1365 - 21 \beta ) q^{5} + ( 112 + 80 \beta ) q^{6} + 2401 q^{7} -4096 q^{8} + ( 37991 + 70 \beta ) q^{9} + ( 21840 + 336 \beta ) q^{10} + ( 22470 - 1050 \beta ) q^{11} + ( -1792 - 1280 \beta ) q^{12} + ( 50141 + 225 \beta ) q^{13} -38416 q^{14} + ( 251580 + 6972 \beta ) q^{15} + 65536 q^{16} + ( -435204 + 1590 \beta ) q^{17} + ( -607856 - 1120 \beta ) q^{18} + ( 254387 - 13455 \beta ) q^{19} + ( -349440 - 5376 \beta ) q^{20} + ( -16807 - 12005 \beta ) q^{21} + ( -359520 + 16800 \beta ) q^{22} + ( 39900 - 7140 \beta ) q^{23} + ( 28672 + 20480 \beta ) q^{24} + ( 926605 + 57330 \beta ) q^{25} + ( -802256 - 3600 \beta ) q^{26} + ( -934906 - 92030 \beta ) q^{27} + 614656 q^{28} + ( 1003164 + 119490 \beta ) q^{29} + ( -4025280 - 111552 \beta ) q^{30} + ( 1094366 + 39330 \beta ) q^{31} -1048576 q^{32} + ( 11943960 - 105000 \beta ) q^{33} + ( 6963264 - 25440 \beta ) q^{34} + ( -3277365 - 50421 \beta ) q^{35} + ( 9725696 + 17920 \beta ) q^{36} + ( -10361788 + 143010 \beta ) q^{37} + ( -4070192 + 215280 \beta ) q^{38} + ( -2944112 - 252280 \beta ) q^{39} + ( 5591040 + 86016 \beta ) q^{40} + ( 9508296 + 517890 \beta ) q^{41} + ( 268912 + 192080 \beta ) q^{42} + ( 2096858 - 345870 \beta ) q^{43} + ( 5752320 - 268800 \beta ) q^{44} + ( -55246065 - 893361 \beta ) q^{45} + ( -638400 + 114240 \beta ) q^{46} + ( -37271262 + 200430 \beta ) q^{47} + ( -458752 - 327680 \beta ) q^{48} + 5764801 q^{49} + ( -14825680 - 917280 \beta ) q^{50} + ( -15278322 + 2164890 \beta ) q^{51} + ( 12836096 + 57600 \beta ) q^{52} + ( -1619874 - 464520 \beta ) q^{53} + ( 14958496 + 1472480 \beta ) q^{54} + ( 20153700 + 961380 \beta ) q^{55} -9834496 q^{56} + ( 153288166 - 1177750 \beta ) q^{57} + ( -16050624 - 1911840 \beta ) q^{58} + ( -66821181 - 1119735 \beta ) q^{59} + ( 64404480 + 1784832 \beta ) q^{60} + ( 113900843 - 866205 \beta ) q^{61} + ( -17509856 - 629280 \beta ) q^{62} + ( 91216391 + 168070 \beta ) q^{63} + 16777216 q^{64} + ( -79333590 - 1360086 \beta ) q^{65} + ( -191103360 + 1680000 \beta ) q^{66} + ( 166465136 - 1654380 \beta ) q^{67} + ( -111412224 + 407040 \beta ) q^{68} + ( 82009200 - 149520 \beta ) q^{69} + ( 52437840 + 806736 \beta ) q^{70} + ( -83992860 + 6323940 \beta ) q^{71} + ( -155611136 - 286720 \beta ) q^{72} + ( -22342138 + 6043140 \beta ) q^{73} + ( 165788608 - 2288160 \beta ) q^{74} + ( -667214485 - 5034335 \beta ) q^{75} + ( 65123072 - 3444480 \beta ) q^{76} + ( 53950470 - 2521050 \beta ) q^{77} + ( 47105792 + 4036480 \beta ) q^{78} + ( 134821388 - 2213820 \beta ) q^{79} + ( -89456640 - 1376256 \beta ) q^{80} + ( 319413239 + 3940930 \beta ) q^{81} + ( -152132736 - 8286240 \beta ) q^{82} + ( -91552881 - 6075435 \beta ) q^{83} + ( -4302592 - 3073280 \beta ) q^{84} + ( 517089510 + 6968934 \beta ) q^{85} + ( -33549728 + 5533920 \beta ) q^{86} + ( -1384144398 - 5852250 \beta ) q^{87} + ( -92037120 + 4300800 \beta ) q^{88} + ( 395828874 + 6785040 \beta ) q^{89} + ( 883937040 + 14293776 \beta ) q^{90} + ( 120388541 + 540225 \beta ) q^{91} + ( 10214400 - 1827840 \beta ) q^{92} + ( -460938812 - 5747140 \beta ) q^{93} + ( 596340192 - 3206880 \beta ) q^{94} + ( 304051020 + 13023948 \beta ) q^{95} + ( 7340032 + 5242880 \beta ) q^{96} + ( -2084740 - 13989690 \beta ) q^{97} -92236816 q^{98} + ( 684240270 - 38317650 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} - 14 q^{3} + 512 q^{4} - 2730 q^{5} + 224 q^{6} + 4802 q^{7} - 8192 q^{8} + 75982 q^{9} + O(q^{10}) \) \( 2 q - 32 q^{2} - 14 q^{3} + 512 q^{4} - 2730 q^{5} + 224 q^{6} + 4802 q^{7} - 8192 q^{8} + 75982 q^{9} + 43680 q^{10} + 44940 q^{11} - 3584 q^{12} + 100282 q^{13} - 76832 q^{14} + 503160 q^{15} + 131072 q^{16} - 870408 q^{17} - 1215712 q^{18} + 508774 q^{19} - 698880 q^{20} - 33614 q^{21} - 719040 q^{22} + 79800 q^{23} + 57344 q^{24} + 1853210 q^{25} - 1604512 q^{26} - 1869812 q^{27} + 1229312 q^{28} + 2006328 q^{29} - 8050560 q^{30} + 2188732 q^{31} - 2097152 q^{32} + 23887920 q^{33} + 13926528 q^{34} - 6554730 q^{35} + 19451392 q^{36} - 20723576 q^{37} - 8140384 q^{38} - 5888224 q^{39} + 11182080 q^{40} + 19016592 q^{41} + 537824 q^{42} + 4193716 q^{43} + 11504640 q^{44} - 110492130 q^{45} - 1276800 q^{46} - 74542524 q^{47} - 917504 q^{48} + 11529602 q^{49} - 29651360 q^{50} - 30556644 q^{51} + 25672192 q^{52} - 3239748 q^{53} + 29916992 q^{54} + 40307400 q^{55} - 19668992 q^{56} + 306576332 q^{57} - 32101248 q^{58} - 133642362 q^{59} + 128808960 q^{60} + 227801686 q^{61} - 35019712 q^{62} + 182432782 q^{63} + 33554432 q^{64} - 158667180 q^{65} - 382206720 q^{66} + 332930272 q^{67} - 222824448 q^{68} + 164018400 q^{69} + 104875680 q^{70} - 167985720 q^{71} - 311222272 q^{72} - 44684276 q^{73} + 331577216 q^{74} - 1334428970 q^{75} + 130246144 q^{76} + 107900940 q^{77} + 94211584 q^{78} + 269642776 q^{79} - 178913280 q^{80} + 638826478 q^{81} - 304265472 q^{82} - 183105762 q^{83} - 8605184 q^{84} + 1034179020 q^{85} - 67099456 q^{86} - 2768288796 q^{87} - 184074240 q^{88} + 791657748 q^{89} + 1767874080 q^{90} + 240777082 q^{91} + 20428800 q^{92} - 921877624 q^{93} + 1192680384 q^{94} + 608102040 q^{95} + 14680064 q^{96} - 4169480 q^{97} - 184473632 q^{98} + 1368480540 q^{99} + O(q^{100}) \)

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
24.5052
−23.5052
−16.0000 −247.052 256.000 −2373.22 3952.83 2401.00 −4096.00 41351.7 37971.5
1.2 −16.0000 233.052 256.000 −356.781 −3728.83 2401.00 −4096.00 34630.3 5708.50
\(n\): 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 14.10.a.c 2
3.b odd 2 1 126.10.a.o 2
4.b odd 2 1 112.10.a.c 2
5.b even 2 1 350.10.a.j 2
5.c odd 4 2 350.10.c.j 4
7.b odd 2 1 98.10.a.e 2
7.c even 3 2 98.10.c.j 4
7.d odd 6 2 98.10.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 1.a even 1 1 trivial
98.10.a.e 2 7.b odd 2 1
98.10.c.h 4 7.d odd 6 2
98.10.c.j 4 7.c even 3 2
112.10.a.c 2 4.b odd 2 1
126.10.a.o 2 3.b odd 2 1
350.10.a.j 2 5.b even 2 1
350.10.c.j 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 14 T_{3} - 57576 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + T )^{2} \)
$3$ \( -57576 + 14 T + T^{2} \)
$5$ \( 846720 + 2730 T + T^{2} \)
$7$ \( ( -2401 + T )^{2} \)
$11$ \( -2036361600 - 44940 T + T^{2} \)
$13$ \( 2397429256 - 100282 T + T^{2} \)
$17$ \( 183575251116 + 870408 T + T^{2} \)
$19$ \( -352577596856 - 508774 T + T^{2} \)
$23$ \( -115915968000 - 79800 T + T^{2} \)
$29$ \( -31904129519604 - 2006328 T + T^{2} \)
$31$ \( -2367849772544 - 2188732 T + T^{2} \)
$37$ \( 60225113026444 + 20723576 T + T^{2} \)
$41$ \( -527816477266884 - 19016592 T + T^{2} \)
$43$ \( -271341247682336 - 4193716 T + T^{2} \)
$47$ \( 1296550084878144 + 74542524 T + T^{2} \)
$53$ \( -494746212296124 + 3239748 T + T^{2} \)
$59$ \( 1575046316366136 + 133642362 T + T^{2} \)
$61$ \( 11243934945943024 - 227801686 T + T^{2} \)
$67$ \( 21401918313456496 - 332930272 T + T^{2} \)
$71$ \( -85127259938918400 + 167985720 T + T^{2} \)
$73$ \( -83678371011966956 + 44684276 T + T^{2} \)
$79$ \( 6880003984764544 - 269642776 T + T^{2} \)
$83$ \( -76697718543013464 + 183105762 T + T^{2} \)
$89$ \( 50565747709419876 - 791657748 T + T^{2} \)
$97$ \( -451110491471642900 + 4169480 T + T^{2} \)
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