# Properties

 Label 126.10.a.o Level $126$ Weight $10$ Character orbit 126.a Self dual yes Analytic conductor $64.895$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 126.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.8945153566$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2305})$$ Defining polynomial: $$x^{2} - x - 576$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{2305}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 q^{2} + 256 q^{4} + ( 1365 - 7 \beta ) q^{5} + 2401 q^{7} + 4096 q^{8} +O(q^{10})$$ $$q + 16 q^{2} + 256 q^{4} + ( 1365 - 7 \beta ) q^{5} + 2401 q^{7} + 4096 q^{8} + ( 21840 - 112 \beta ) q^{10} + ( -22470 - 350 \beta ) q^{11} + ( 50141 - 75 \beta ) q^{13} + 38416 q^{14} + 65536 q^{16} + ( 435204 + 530 \beta ) q^{17} + ( 254387 + 4485 \beta ) q^{19} + ( 349440 - 1792 \beta ) q^{20} + ( -359520 - 5600 \beta ) q^{22} + ( -39900 - 2380 \beta ) q^{23} + ( 926605 - 19110 \beta ) q^{25} + ( 802256 - 1200 \beta ) q^{26} + 614656 q^{28} + ( -1003164 + 39830 \beta ) q^{29} + ( 1094366 - 13110 \beta ) q^{31} + 1048576 q^{32} + ( 6963264 + 8480 \beta ) q^{34} + ( 3277365 - 16807 \beta ) q^{35} + ( -10361788 - 47670 \beta ) q^{37} + ( 4070192 + 71760 \beta ) q^{38} + ( 5591040 - 28672 \beta ) q^{40} + ( -9508296 + 172630 \beta ) q^{41} + ( 2096858 + 115290 \beta ) q^{43} + ( -5752320 - 89600 \beta ) q^{44} + ( -638400 - 38080 \beta ) q^{46} + ( 37271262 + 66810 \beta ) q^{47} + 5764801 q^{49} + ( 14825680 - 305760 \beta ) q^{50} + ( 12836096 - 19200 \beta ) q^{52} + ( 1619874 - 154840 \beta ) q^{53} + ( 20153700 - 320460 \beta ) q^{55} + 9834496 q^{56} + ( -16050624 + 637280 \beta ) q^{58} + ( 66821181 - 373245 \beta ) q^{59} + ( 113900843 + 288735 \beta ) q^{61} + ( 17509856 - 209760 \beta ) q^{62} + 16777216 q^{64} + ( 79333590 - 453362 \beta ) q^{65} + ( 166465136 + 551460 \beta ) q^{67} + ( 111412224 + 135680 \beta ) q^{68} + ( 52437840 - 268912 \beta ) q^{70} + ( 83992860 + 2107980 \beta ) q^{71} + ( -22342138 - 2014380 \beta ) q^{73} + ( -165788608 - 762720 \beta ) q^{74} + ( 65123072 + 1148160 \beta ) q^{76} + ( -53950470 - 840350 \beta ) q^{77} + ( 134821388 + 737940 \beta ) q^{79} + ( 89456640 - 458752 \beta ) q^{80} + ( -152132736 + 2762080 \beta ) q^{82} + ( 91552881 - 2025145 \beta ) q^{83} + ( 517089510 - 2322978 \beta ) q^{85} + ( 33549728 + 1844640 \beta ) q^{86} + ( -92037120 - 1433600 \beta ) q^{88} + ( -395828874 + 2261680 \beta ) q^{89} + ( 120388541 - 180075 \beta ) q^{91} + ( -10214400 - 609280 \beta ) q^{92} + ( 596340192 + 1068960 \beta ) q^{94} + ( -304051020 + 4341316 \beta ) q^{95} + ( -2084740 + 4663230 \beta ) q^{97} + 92236816 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{2} + 512 q^{4} + 2730 q^{5} + 4802 q^{7} + 8192 q^{8} + O(q^{10})$$ $$2 q + 32 q^{2} + 512 q^{4} + 2730 q^{5} + 4802 q^{7} + 8192 q^{8} + 43680 q^{10} - 44940 q^{11} + 100282 q^{13} + 76832 q^{14} + 131072 q^{16} + 870408 q^{17} + 508774 q^{19} + 698880 q^{20} - 719040 q^{22} - 79800 q^{23} + 1853210 q^{25} + 1604512 q^{26} + 1229312 q^{28} - 2006328 q^{29} + 2188732 q^{31} + 2097152 q^{32} + 13926528 q^{34} + 6554730 q^{35} - 20723576 q^{37} + 8140384 q^{38} + 11182080 q^{40} - 19016592 q^{41} + 4193716 q^{43} - 11504640 q^{44} - 1276800 q^{46} + 74542524 q^{47} + 11529602 q^{49} + 29651360 q^{50} + 25672192 q^{52} + 3239748 q^{53} + 40307400 q^{55} + 19668992 q^{56} - 32101248 q^{58} + 133642362 q^{59} + 227801686 q^{61} + 35019712 q^{62} + 33554432 q^{64} + 158667180 q^{65} + 332930272 q^{67} + 222824448 q^{68} + 104875680 q^{70} + 167985720 q^{71} - 44684276 q^{73} - 331577216 q^{74} + 130246144 q^{76} - 107900940 q^{77} + 269642776 q^{79} + 178913280 q^{80} - 304265472 q^{82} + 183105762 q^{83} + 1034179020 q^{85} + 67099456 q^{86} - 184074240 q^{88} - 791657748 q^{89} + 240777082 q^{91} - 20428800 q^{92} + 1192680384 q^{94} - 608102040 q^{95} - 4169480 q^{97} + 184473632 q^{98} + O(q^{100})$$

## 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 0 256.000 356.781 0 2401.00 4096.00 0 5708.50
1.2 16.0000 0 256.000 2373.22 0 2401.00 4096.00 0 37971.5
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 126.10.a.o 2
3.b odd 2 1 14.10.a.c 2
12.b even 2 1 112.10.a.c 2
15.d odd 2 1 350.10.a.j 2
15.e even 4 2 350.10.c.j 4
21.c even 2 1 98.10.a.e 2
21.g even 6 2 98.10.c.h 4
21.h odd 6 2 98.10.c.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 3.b odd 2 1
98.10.a.e 2 21.c even 2 1
98.10.c.h 4 21.g even 6 2
98.10.c.j 4 21.h odd 6 2
112.10.a.c 2 12.b even 2 1
126.10.a.o 2 1.a even 1 1 trivial
350.10.a.j 2 15.d odd 2 1
350.10.c.j 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2730 T_{5} + 846720$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(126))$$.

## Hecke characteristic polynomials

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