# Properties

 Label 25.8.b.b Level $25$ Weight $8$ Character orbit 25.b Analytic conductor $7.810$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.80962563710$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{649})$$ Defining polynomial: $$x^{4} + 325 x^{2} + 26244$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + 2 \beta_{2} ) q^{2} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{3} + ( -92 + 3 \beta_{3} ) q^{4} + ( 478 - 7 \beta_{3} ) q^{6} + ( 84 \beta_{1} - 102 \beta_{2} ) q^{7} + ( 75 \beta_{1} - 468 \beta_{2} ) q^{8} + ( 1130 + 16 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + 2 \beta_{2} ) q^{2} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{3} + ( -92 + 3 \beta_{3} ) q^{4} + ( 478 - 7 \beta_{3} ) q^{6} + ( 84 \beta_{1} - 102 \beta_{2} ) q^{7} + ( 75 \beta_{1} - 468 \beta_{2} ) q^{8} + ( 1130 + 16 \beta_{3} ) q^{9} + ( 2147 + 50 \beta_{3} ) q^{11} + ( -481 \beta_{1} + 1576 \beta_{2} ) q^{12} + ( -408 \beta_{1} - 1564 \beta_{2} ) q^{13} + ( 15972 - 186 \beta_{3} ) q^{14} + ( 16808 - 159 \beta_{3} ) q^{16} + ( 704 \beta_{1} + 335 \beta_{2} ) q^{17} + ( -538 \beta_{1} - 620 \beta_{2} ) q^{18} + ( -9211 + 222 \beta_{3} ) q^{19} + ( -33486 + 456 \beta_{3} ) q^{21} + ( -297 \beta_{1} - 4706 \beta_{2} ) q^{22} + ( 6732 \beta_{1} - 5478 \beta_{2} ) q^{23} + ( -67968 + 1161 \beta_{3} ) q^{24} + ( 680 - 1156 \beta_{3} ) q^{26} + ( 5050 \beta_{1} - 10633 \beta_{2} ) q^{27} + ( -12102 \beta_{1} + 52368 \beta_{2} ) q^{28} + ( -26584 - 2632 \beta_{3} ) q^{29} + ( -151338 + 900 \beta_{3} ) q^{31} + ( -13091 \beta_{1} + 2332 \beta_{2} ) q^{32} + ( -656 \beta_{1} + 7865 \beta_{2} ) q^{33} + ( 88646 - 369 \beta_{3} ) q^{34} + ( 90728 + 1966 \beta_{3} ) q^{36} + ( -11256 \beta_{1} + 66614 \beta_{2} ) q^{37} + ( 17425 \beta_{1} - 58382 \beta_{2} ) q^{38} + ( -45356 + 1904 \beta_{3} ) q^{39} + ( -54443 + 400 \beta_{3} ) q^{41} + ( 50358 \beta_{1} - 149052 \beta_{2} ) q^{42} + ( -20088 \beta_{1} - 86596 \beta_{2} ) q^{43} + ( 410876 + 1991 \beta_{3} ) q^{44} + ( 1177572 - 12210 \beta_{3} ) q^{46} + ( 13664 \beta_{1} + 171956 \beta_{2} ) q^{47} + ( 49357 \beta_{1} - 143188 \beta_{2} ) q^{48} + ( -416333 + 10080 \beta_{3} ) q^{49} + ( -160469 + 1442 \beta_{3} ) q^{51} + ( -95676 \beta_{1} + 9248 \beta_{2} ) q^{52} + ( -44048 \beta_{1} + 8950 \beta_{2} ) q^{53} + ( 1131254 - 15683 \beta_{3} ) q^{54} + ( -1688256 + 40662 \beta_{3} ) q^{56} + ( -40400 \beta_{1} + 128639 \beta_{2} ) q^{57} + ( -70800 \beta_{1} + 420592 \beta_{2} ) q^{58} + ( -1025048 - 17504 \beta_{3} ) q^{59} + ( 313522 - 45000 \beta_{3} ) q^{61} + ( 184638 \beta_{1} - 464676 \beta_{2} ) q^{62} + ( 71592 \beta_{1} + 113988 \beta_{2} ) q^{63} + ( 177704 - 4929 \beta_{3} ) q^{64} + ( -393334 + 8521 \beta_{3} ) q^{66} + ( -73446 \beta_{1} + 62295 \beta_{2} ) q^{67} + ( -12187 \beta_{1} + 286592 \beta_{2} ) q^{68} + ( -2411310 + 31152 \beta_{3} ) q^{69} + ( -2369108 + 10000 \beta_{3} ) q^{71} + ( -86850 \beta_{1} - 251784 \beta_{2} ) q^{72} + ( 84432 \beta_{1} + 91727 \beta_{2} ) q^{73} + ( -4152196 + 77870 \beta_{3} ) q^{74} + ( 3548708 - 47391 \beta_{3} ) q^{76} + ( 107448 \beta_{1} + 497406 \beta_{2} ) q^{77} + ( 115804 \beta_{1} - 433432 \beta_{2} ) q^{78} + ( 3601926 - 17652 \beta_{3} ) q^{79} + ( 3577 + 71408 \beta_{3} ) q^{81} + ( 69243 \beta_{1} - 180886 \beta_{2} ) q^{82} + ( -69378 \beta_{1} - 1170267 \beta_{2} ) q^{83} + ( 8629320 - 141042 \beta_{3} ) q^{84} + ( 397976 - 66508 \beta_{3} ) q^{86} + ( 207400 \beta_{1} - 846184 \beta_{2} ) q^{87} + ( -375225 \beta_{1} - 138996 \beta_{2} ) q^{88} + ( 3039633 - 88416 \beta_{3} ) q^{89} + ( 2846616 + 124032 \beta_{3} ) q^{91} + ( -767646 \beta_{1} + 3851760 \beta_{2} ) q^{92} + ( -391776 \beta_{1} + 1091490 \beta_{2} ) q^{93} + ( -4566712 + 158292 \beta_{3} ) q^{94} + ( 3848648 - 43937 \beta_{3} ) q^{96} + ( -122736 \beta_{1} - 1650634 \beta_{2} ) q^{97} + ( 789293 \beta_{1} - 2647066 \beta_{2} ) q^{98} + ( 5670910 + 91652 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 362q^{4} + 1898q^{6} + 4552q^{9} + O(q^{10})$$ $$4q - 362q^{4} + 1898q^{6} + 4552q^{9} + 8688q^{11} + 63516q^{14} + 66914q^{16} - 36400q^{19} - 133032q^{21} - 269550q^{24} + 408q^{26} - 111600q^{29} - 603552q^{31} + 353846q^{34} + 366844q^{36} - 177616q^{39} - 216972q^{41} + 1647486q^{44} + 4685868q^{46} - 1645172q^{49} - 638992q^{51} + 4493650q^{54} - 6671700q^{56} - 4135200q^{59} + 1164088q^{61} + 700958q^{64} - 1556294q^{66} - 9582936q^{69} - 9456432q^{71} - 16453044q^{74} + 14100050q^{76} + 14372400q^{79} + 157124q^{81} + 34235196q^{84} + 1458888q^{86} + 11981700q^{89} + 11634528q^{91} - 17950264q^{94} + 15306718q^{96} + 22866944q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 325 x^{2} + 26244$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 217 \nu$$$$)/54$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{3} - 815 \nu$$$$)/162$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{2} + 813$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{2} - 5 \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 813$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$-651 \beta_{2} + 815 \beta_{1}$$$$)/5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 13.2377i − 12.2377i 12.2377i 13.2377i
20.2377i 45.4755i −281.566 0 920.321 1369.97i 3107.83i 118.981 0
24.2 5.23774i 5.47548i 100.566 0 28.6791 769.970i 1197.17i 2157.02 0
24.3 5.23774i 5.47548i 100.566 0 28.6791 769.970i 1197.17i 2157.02 0
24.4 20.2377i 45.4755i −281.566 0 920.321 1369.97i 3107.83i 118.981 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 437 T_{2}^{2} + 11236$$ acting on $$S_{8}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 75 T^{2} - 2332 T^{4} - 1228800 T^{6} + 268435456 T^{8}$$
$3$ $$1 - 6650 T^{2} + 19583163 T^{4} - 31806743850 T^{6} + 22876792454961 T^{8}$$
$5$ 1
$7$ $$1 - 824500 T^{2} + 1114255248198 T^{4} - 559194923564000500 T^{6} +$$$$45\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 4344 T + 33551301 T^{2} - 84652270824 T^{3} + 379749833583241 T^{4} )^{2}$$
$13$ $$1 - 40685300 T^{2} - 153894343380522 T^{4} -$$$$16\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$17$ $$1 - 1456928850 T^{2} + 863620794440640083 T^{4} -$$$$24\!\cdots\!50$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$19$ $$( 1 + 18200 T + 1670645253 T^{2} + 16268465649800 T^{3} + 799006685782884121 T^{4} )^{2}$$
$23$ $$1 + 1310008300 T^{2} + 20334801990676168518 T^{4} +$$$$15\!\cdots\!00$$$$T^{6} +$$$$13\!\cdots\!81$$$$T^{8}$$
$29$ $$( 1 + 55800 T + 7178799018 T^{2} + 962543098042200 T^{3} +$$$$29\!\cdots\!81$$$$T^{4} )^{2}$$
$31$ $$( 1 + 301776 T + 74506854266 T^{2} + 8302646635961136 T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$37$ $$1 - 152649548300 T^{2} +$$$$16\!\cdots\!78$$$$T^{4} -$$$$13\!\cdots\!00$$$$T^{6} +$$$$81\!\cdots\!21$$$$T^{8}$$
$41$ $$( 1 + 108486 T + 391801850811 T^{2} + 21128112156254166 T^{3} +$$$$37\!\cdots\!61$$$$T^{4} )^{2}$$
$43$ $$1 - 489365211500 T^{2} +$$$$14\!\cdots\!98$$$$T^{4} -$$$$36\!\cdots\!00$$$$T^{6} +$$$$54\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 367649295900 T^{2} +$$$$45\!\cdots\!38$$$$T^{4} -$$$$94\!\cdots\!00$$$$T^{6} +$$$$65\!\cdots\!61$$$$T^{8}$$
$53$ $$1 - 4060694649900 T^{2} +$$$$68\!\cdots\!38$$$$T^{4} -$$$$56\!\cdots\!00$$$$T^{6} +$$$$19\!\cdots\!61$$$$T^{8}$$
$59$ $$( 1 + 2067600 T + 4803250907238 T^{2} + 5145535810011764400 T^{3} +$$$$61\!\cdots\!61$$$$T^{4} )^{2}$$
$61$ $$( 1 - 582044 T - 1843726773474 T^{2} - 1829214611249006924 T^{3} +$$$$98\!\cdots\!41$$$$T^{4} )^{2}$$
$67$ $$1 - 22459694871850 T^{2} +$$$$19\!\cdots\!83$$$$T^{4} -$$$$82\!\cdots\!50$$$$T^{6} +$$$$13\!\cdots\!41$$$$T^{8}$$
$71$ $$( 1 + 4728216 T + 23373621952446 T^{2} + 43003692654826860456 T^{3} +$$$$82\!\cdots\!81$$$$T^{4} )^{2}$$
$73$ $$1 - 40979274242450 T^{2} +$$$$66\!\cdots\!43$$$$T^{4} -$$$$50\!\cdots\!50$$$$T^{6} +$$$$14\!\cdots\!81$$$$T^{8}$$
$79$ $$( 1 - 7186200 T + 50054286054218 T^{2} -$$$$13\!\cdots\!00$$$$T^{3} +$$$$36\!\cdots\!81$$$$T^{4} )^{2}$$
$83$ $$1 - 34386337777850 T^{2} +$$$$16\!\cdots\!83$$$$T^{4} -$$$$25\!\cdots\!50$$$$T^{6} +$$$$54\!\cdots\!41$$$$T^{8}$$
$89$ $$( 1 - 5990850 T + 65725956363283 T^{2} -$$$$26\!\cdots\!50$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4} )^{2}$$
$97$ $$1 - 171757286723900 T^{2} +$$$$19\!\cdots\!38$$$$T^{4} -$$$$11\!\cdots\!00$$$$T^{6} +$$$$42\!\cdots\!61$$$$T^{8}$$