# Properties

 Label 25.8.a.c Level $25$ Weight $8$ Character orbit 25.a Self dual yes Analytic conductor $7.810$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.80962563710$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{649})$$ Defining polynomial: $$x^{2} - x - 162$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{649})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -7 - \beta ) q^{2} + ( -19 - 2 \beta ) q^{3} + ( 83 + 15 \beta ) q^{4} + ( 457 + 35 \beta ) q^{6} + ( 258 + 84 \beta ) q^{7} + ( -2115 - 75 \beta ) q^{8} + ( -1178 + 80 \beta ) q^{9} +O(q^{10})$$ $$q + ( -7 - \beta ) q^{2} + ( -19 - 2 \beta ) q^{3} + ( 83 + 15 \beta ) q^{4} + ( 457 + 35 \beta ) q^{6} + ( 258 + 84 \beta ) q^{7} + ( -2115 - 75 \beta ) q^{8} + ( -1178 + 80 \beta ) q^{9} + ( 2297 - 250 \beta ) q^{11} + ( -6437 - 481 \beta ) q^{12} + ( -9044 + 408 \beta ) q^{13} + ( -15414 - 930 \beta ) q^{14} + ( 16331 + 795 \beta ) q^{16} + ( -3787 + 704 \beta ) q^{17} + ( -4714 + 538 \beta ) q^{18} + ( 8545 + 1110 \beta ) q^{19} + ( -32118 - 2280 \beta ) q^{21} + ( 24421 - 297 \beta ) q^{22} + ( -7194 - 6732 \beta ) q^{23} + ( 64485 + 5805 \beta ) q^{24} + ( -2788 + 5780 \beta ) q^{26} + ( 38015 + 5050 \beta ) q^{27} + ( 225534 + 12102 \beta ) q^{28} + ( 34480 - 13160 \beta ) q^{29} + ( -148638 - 4500 \beta ) q^{31} + ( 27613 - 13091 \beta ) q^{32} + ( 37357 + 656 \beta ) q^{33} + ( -87539 - 1845 \beta ) q^{34} + ( 96626 - 9830 \beta ) q^{36} + ( -299302 - 11256 \beta ) q^{37} + ( -239635 - 17425 \beta ) q^{38} + ( 39644 + 9520 \beta ) q^{39} + ( -53243 - 2000 \beta ) q^{41} + ( 594186 + 50358 \beta ) q^{42} + ( -493244 + 20088 \beta ) q^{43} + ( -416849 + 9955 \beta ) q^{44} + ( 1140942 + 61050 \beta ) q^{46} + ( -900772 + 13664 \beta ) q^{47} + ( -567869 - 49357 \beta ) q^{48} + ( 386093 + 50400 \beta ) q^{49} + ( -156143 - 7210 \beta ) q^{51} + ( 240788 - 95676 \beta ) q^{52} + ( -87394 + 44048 \beta ) q^{53} + ( -1084205 - 78415 \beta ) q^{54} + ( -1566270 - 203310 \beta ) q^{56} + ( -521995 - 40400 \beta ) q^{57} + ( 1890560 + 70800 \beta ) q^{58} + ( 1077560 - 87520 \beta ) q^{59} + ( 178522 + 225000 \beta ) q^{61} + ( 1769466 + 184638 \beta ) q^{62} + ( 784716 - 71592 \beta ) q^{63} + ( -162917 - 24645 \beta ) q^{64} + ( -367771 - 42605 \beta ) q^{66} + ( -91137 - 73446 \beta ) q^{67} + ( 1396399 + 12187 \beta ) q^{68} + ( 2317854 + 155760 \beta ) q^{69} + ( -2339108 - 50000 \beta ) q^{71} + ( 1519470 - 86850 \beta ) q^{72} + ( 711931 - 84432 \beta ) q^{73} + ( 3918586 + 389350 \beta ) q^{74} + ( 3406535 + 236955 \beta ) q^{76} + ( -2809374 + 107448 \beta ) q^{77} + ( -1819748 - 115804 \beta ) q^{78} + ( -3548970 - 88260 \beta ) q^{79} + ( 217801 - 357040 \beta ) q^{81} + ( 696701 + 69243 \beta ) q^{82} + ( -6059469 + 69378 \beta ) q^{83} + ( -8206194 - 705210 \beta ) q^{84} + ( 198452 + 332540 \beta ) q^{86} + ( 3608720 + 207400 \beta ) q^{87} + ( -1820655 + 375225 \beta ) q^{88} + ( -2774385 - 442080 \beta ) q^{89} + ( 3218712 - 620160 \beta ) q^{91} + ( -16955862 - 767646 \beta ) q^{92} + ( 4282122 + 391776 \beta ) q^{93} + ( 4091836 + 791460 \beta ) q^{94} + ( 3716837 + 219685 \beta ) q^{96} + ( 8621378 - 122736 \beta ) q^{97} + ( -10867451 - 789293 \beta ) q^{98} + ( -5945866 + 458260 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 15q^{2} - 40q^{3} + 181q^{4} + 949q^{6} + 600q^{7} - 4305q^{8} - 2276q^{9} + O(q^{10})$$ $$2q - 15q^{2} - 40q^{3} + 181q^{4} + 949q^{6} + 600q^{7} - 4305q^{8} - 2276q^{9} + 4344q^{11} - 13355q^{12} - 17680q^{13} - 31758q^{14} + 33457q^{16} - 6870q^{17} - 8890q^{18} + 18200q^{19} - 66516q^{21} + 48545q^{22} - 21120q^{23} + 134775q^{24} + 204q^{26} + 81080q^{27} + 463170q^{28} + 55800q^{29} - 301776q^{31} + 42135q^{32} + 75370q^{33} - 176923q^{34} + 183422q^{36} - 609860q^{37} - 496695q^{38} + 88808q^{39} - 108486q^{41} + 1238730q^{42} - 966400q^{43} - 823743q^{44} + 2342934q^{46} - 1787880q^{47} - 1185095q^{48} + 822586q^{49} - 319496q^{51} + 385900q^{52} - 130740q^{53} - 2246825q^{54} - 3335850q^{56} - 1084390q^{57} + 3851920q^{58} + 2067600q^{59} + 582044q^{61} + 3723570q^{62} + 1497840q^{63} - 350479q^{64} - 778147q^{66} - 255720q^{67} + 2804985q^{68} + 4791468q^{69} - 4728216q^{71} + 2952090q^{72} + 1339430q^{73} + 8226522q^{74} + 7050025q^{76} - 5511300q^{77} - 3755300q^{78} - 7186200q^{79} + 78562q^{81} + 1462645q^{82} - 12049560q^{83} - 17117598q^{84} + 729444q^{86} + 7424840q^{87} - 3266085q^{88} - 5990850q^{89} + 5817264q^{91} - 34679370q^{92} + 8956020q^{93} + 8975132q^{94} + 7653359q^{96} + 17120020q^{97} - 22524195q^{98} - 11433472q^{99} + O(q^{100})$$

## 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
 13.2377 −12.2377
−20.2377 −45.4755 281.566 0 920.321 1369.97 −3107.83 −118.981 0
1.2 5.23774 5.47548 −100.566 0 28.6791 −769.970 −1197.17 −2157.02 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$

## 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 25.8.a.c 2
3.b odd 2 1 225.8.a.v 2
4.b odd 2 1 400.8.a.bd 2
5.b even 2 1 25.8.a.e yes 2
5.c odd 4 2 25.8.b.b 4
15.d odd 2 1 225.8.a.k 2
15.e even 4 2 225.8.b.l 4
20.d odd 2 1 400.8.a.v 2
20.e even 4 2 400.8.c.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.8.a.c 2 1.a even 1 1 trivial
25.8.a.e yes 2 5.b even 2 1
25.8.b.b 4 5.c odd 4 2
225.8.a.k 2 15.d odd 2 1
225.8.a.v 2 3.b odd 2 1
225.8.b.l 4 15.e even 4 2
400.8.a.v 2 20.d odd 2 1
400.8.a.bd 2 4.b odd 2 1
400.8.c.s 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 15 T_{2} - 106$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 15 T + 150 T^{2} + 1920 T^{3} + 16384 T^{4}$$
$3$ $$1 + 40 T + 4125 T^{2} + 87480 T^{3} + 4782969 T^{4}$$
$5$ 1
$7$ $$1 - 600 T + 592250 T^{2} - 494125800 T^{3} + 678223072849 T^{4}$$
$11$ $$1 - 4344 T + 33551301 T^{2} - 84652270824 T^{3} + 379749833583241 T^{4}$$
$13$ $$1 + 17680 T + 176633850 T^{2} + 1109393780560 T^{3} + 3937376385699289 T^{4}$$
$17$ $$1 + 6870 T + 752062875 T^{2} + 2819026683510 T^{3} + 168377826559400929 T^{4}$$
$19$ $$1 - 18200 T + 1670645253 T^{2} - 16268465649800 T^{3} + 799006685782884121 T^{4}$$
$23$ $$1 + 21120 T - 431976950 T^{2} + 71909913440640 T^{3} + 11592836324538749809 T^{4}$$
$29$ $$1 - 55800 T + 7178799018 T^{2} - 962543098042200 T^{3} +$$$$29\!\cdots\!81$$$$T^{4}$$
$31$ $$1 + 301776 T + 74506854266 T^{2} + 8302646635961136 T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$37$ $$1 + 609860 T + 262289383950 T^{2} + 57895154588331380 T^{3} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$1 + 108486 T + 391801850811 T^{2} + 21128112156254166 T^{3} +$$$$37\!\cdots\!61$$$$T^{4}$$
$43$ $$1 + 966400 T + 711647085750 T^{2} + 262685505773804800 T^{3} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 1787880 T + 1782082095150 T^{2} + 905781344613388440 T^{3} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 + 130740 T + 2038893798750 T^{2} + 153581734422289380 T^{3} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$1 - 2067600 T + 4803250907238 T^{2} - 5145535810011764400 T^{3} +$$$$61\!\cdots\!61$$$$T^{4}$$
$61$ $$1 - 582044 T - 1843726773474 T^{2} - 1829214611249006924 T^{3} +$$$$98\!\cdots\!41$$$$T^{4}$$
$67$ $$1 + 255720 T + 11262543795125 T^{2} + 1549845171713197560 T^{3} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$1 + 4728216 T + 23373621952446 T^{2} + 43003692654826860456 T^{3} +$$$$82\!\cdots\!81$$$$T^{4}$$
$73$ $$1 - 1339430 T + 21386673483675 T^{2} - 14797216998434094710 T^{3} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$1 + 7186200 T + 50054286054218 T^{2} +$$$$13\!\cdots\!00$$$$T^{3} +$$$$36\!\cdots\!81$$$$T^{4}$$
$83$ $$1 + 12049560 T + 89789116985725 T^{2} +$$$$32\!\cdots\!20$$$$T^{3} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$1 + 5990850 T + 65725956363283 T^{2} +$$$$26\!\cdots\!50$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4}$$
$97$ $$1 - 17120020 T + 232426185762150 T^{2} -$$$$13\!\cdots\!60$$$$T^{3} +$$$$65\!\cdots\!69$$$$T^{4}$$