# Properties

 Label 25.10.b.b Level $25$ Weight $10$ Character orbit 25.b Analytic conductor $12.876$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8758959041$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{1009})$$ Defining polynomial: $$x^{4} + 505 x^{2} + 63504$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -2 \beta_{1} + 14 \beta_{2} ) q^{3} + ( -522 + \beta_{3} ) q^{4} + ( -2668 + 14 \beta_{3} ) q^{6} + ( 214 \beta_{1} - 192 \beta_{2} ) q^{7} + ( 60 \beta_{1} - 1044 \beta_{2} ) q^{8} + ( -1253 + 52 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -2 \beta_{1} + 14 \beta_{2} ) q^{3} + ( -522 + \beta_{3} ) q^{4} + ( -2668 + 14 \beta_{3} ) q^{6} + ( 214 \beta_{1} - 192 \beta_{2} ) q^{7} + ( 60 \beta_{1} - 1044 \beta_{2} ) q^{8} + ( -1253 + 52 \beta_{3} ) q^{9} + ( 11992 + 190 \beta_{3} ) q^{11} + ( 2344 \beta_{1} - 9976 \beta_{2} ) q^{12} + ( -1352 \beta_{1} + 6427 \beta_{2} ) q^{13} + ( 220176 - 192 \beta_{3} ) q^{14} + ( -156024 - 532 \beta_{3} ) q^{16} + ( -12856 \beta_{1} - 14213 \beta_{2} ) q^{17} + ( 3853 \beta_{1} - 55021 \beta_{2} ) q^{18} + ( 148260 + 284 \beta_{3} ) q^{19} + ( 542352 - 2952 \beta_{3} ) q^{21} + ( -2492 \beta_{1} - 184468 \beta_{2} ) q^{22} + ( 19398 \beta_{1} - 62160 \beta_{2} ) q^{23} + ( 1439280 - 2808 \beta_{3} ) q^{24} + ( -1651718 + 6427 \beta_{3} ) q^{26} + ( 30740 \beta_{1} + 119284 \beta_{2} ) q^{27} + ( -120208 \beta_{1} + 320400 \beta_{2} ) q^{28} + ( 1833490 + 10696 \beta_{3} ) q^{29} + ( 806572 + 15470 \beta_{3} ) q^{31} + ( 160144 \beta_{1} - 140464 \beta_{2} ) q^{32} + ( 223016 \beta_{1} - 339032 \beta_{2} ) q^{33} + ( -11939654 - 14213 \beta_{3} ) q^{34} + ( 5900866 - 28397 \beta_{3} ) q^{36} + ( -205296 \beta_{1} + 1158745 \beta_{2} ) q^{37} + ( -134060 \beta_{1} - 145396 \beta_{2} ) q^{38} + ( -10204636 + 29078 \beta_{3} ) q^{39} + ( -13478638 - 15580 \beta_{3} ) q^{41} + ( -689952 \beta_{1} + 3594720 \beta_{2} ) q^{42} + ( 25798 \beta_{1} + 2631586 \beta_{2} ) q^{43} + ( 12911176 - 87188 \beta_{3} ) q^{44} + ( 22195632 - 62160 \beta_{3} ) q^{46} + ( 523334 \beta_{1} - 3182276 \beta_{2} ) q^{47} + ( -379552 \beta_{1} - 764960 \beta_{2} ) q^{48} + ( -6577057 + 36380 \beta_{3} ) q^{49} + ( 889892 + 125846 \beta_{3} ) q^{51} + ( 1280844 \beta_{1} - 5006612 \beta_{2} ) q^{52} + ( 1137448 \beta_{1} - 2520481 \beta_{2} ) q^{53} + ( 24283960 + 119284 \beta_{3} ) q^{54} + ( -21574560 + 222096 \beta_{3} ) q^{56} + ( 72680 \beta_{1} + 1317928 \beta_{2} ) q^{57} + ( -1298690 \beta_{1} - 9226174 \beta_{2} ) q^{58} + ( 27497780 + 154472 \beta_{3} ) q^{59} + ( -137289858 - 69200 \beta_{3} ) q^{61} + ( -33072 \beta_{1} - 15189408 \beta_{2} ) q^{62} + ( -710142 \beta_{1} + 11689728 \beta_{2} ) q^{63} + ( 84720608 - 412848 \beta_{3} ) q^{64} + ( 236399344 - 339032 \beta_{3} ) q^{66} + ( -2416706 \beta_{1} + 1224282 \beta_{2} ) q^{67} + ( 4646732 \beta_{1} - 4520468 \beta_{2} ) q^{68} + ( 107344464 - 357096 \beta_{3} ) q^{69} + ( -3565468 - 627850 \beta_{3} ) q^{71} + ( -5347980 \beta_{1} + 7092612 \beta_{2} ) q^{72} + ( -8830952 \beta_{1} + 10458385 \beta_{2} ) q^{73} + ( -259948514 + 1158745 \beta_{3} ) q^{74} + ( -48736120 + 12 \beta_{3} ) q^{76} + ( 951288 \beta_{1} + 39530976 \beta_{2} ) q^{77} + ( 11658536 \beta_{1} - 40271288 \beta_{2} ) q^{78} + ( -3438760 - 1877564 \beta_{3} ) q^{79} + ( -137679679 + 893204 \beta_{3} ) q^{81} + ( 12699638 \beta_{1} + 2631082 \beta_{2} ) q^{82} + ( -2748402 \beta_{1} + 71491638 \beta_{2} ) q^{83} + ( -580964544 + 2083296 \beta_{3} ) q^{84} + ( -106194068 + 2631586 \beta_{3} ) q^{86} + ( 10237820 \beta_{1} - 2868068 \beta_{2} ) q^{87} + ( -18546480 \beta_{1} + 8615952 \beta_{2} ) q^{88} + ( -415044330 + 1338168 \beta_{3} ) q^{89} + ( 340815452 - 1345634 \beta_{3} ) q^{91} + ( -15371856 \beta_{1} + 54643152 \beta_{2} ) q^{92} + ( 18497856 \beta_{1} - 29981952 \beta_{2} ) q^{93} + ( 674074456 - 3182276 \beta_{3} ) q^{94} + ( 401680192 - 2202656 \beta_{3} ) q^{96} + ( -2622216 \beta_{1} - 30608621 \beta_{2} ) q^{97} + ( 8396057 \beta_{1} - 44193977 \beta_{2} ) q^{98} + ( 981866024 + 385514 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2088q^{4} - 10672q^{6} - 5012q^{9} + O(q^{10})$$ $$4q - 2088q^{4} - 10672q^{6} - 5012q^{9} + 47968q^{11} + 880704q^{14} - 624096q^{16} + 593040q^{19} + 2169408q^{21} + 5757120q^{24} - 6606872q^{26} + 7333960q^{29} + 3226288q^{31} - 47758616q^{34} + 23603464q^{36} - 40818544q^{39} - 53914552q^{41} + 51644704q^{44} + 88782528q^{46} - 26308228q^{49} + 3559568q^{51} + 97135840q^{54} - 86298240q^{56} + 109991120q^{59} - 549159432q^{61} + 338882432q^{64} + 945597376q^{66} + 429377856q^{69} - 14261872q^{71} - 1039794056q^{74} - 194944480q^{76} - 13755040q^{79} - 550718716q^{81} - 2323858176q^{84} - 424776272q^{86} - 1660177320q^{89} + 1363261808q^{91} + 2696297824q^{94} + 1606720768q^{96} + 3927464096q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 505 x^{2} + 63504$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 337 \nu$$$$)/42$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{3} - 1265 \nu$$$$)/126$$ $$\beta_{3}$$ $$=$$ $$20 \nu^{2} + 5050$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{2} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 5050$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$-1011 \beta_{2} + 1265 \beta_{1}$$$$)/10$$

## 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
 − 16.3824i − 15.3824i 15.3824i 16.3824i
36.7648i 193.530i −839.648 0 −7115.07 7647.66i 12045.9i −17770.7 0
24.2 26.7648i 66.4705i −204.352 0 1779.07 5947.66i 8234.11i 15264.7 0
24.3 26.7648i 66.4705i −204.352 0 1779.07 5947.66i 8234.11i 15264.7 0
24.4 36.7648i 193.530i −839.648 0 −7115.07 7647.66i 12045.9i −17770.7 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 25.10.b.b 4
3.b odd 2 1 225.10.b.h 4
4.b odd 2 1 400.10.c.p 4
5.b even 2 1 inner 25.10.b.b 4
5.c odd 4 1 5.10.a.b 2
5.c odd 4 1 25.10.a.b 2
15.d odd 2 1 225.10.b.h 4
15.e even 4 1 45.10.a.f 2
15.e even 4 1 225.10.a.h 2
20.d odd 2 1 400.10.c.p 4
20.e even 4 1 80.10.a.f 2
20.e even 4 1 400.10.a.t 2
35.f even 4 1 245.10.a.d 2
40.i odd 4 1 320.10.a.k 2
40.k even 4 1 320.10.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.b 2 5.c odd 4 1
25.10.a.b 2 5.c odd 4 1
25.10.b.b 4 1.a even 1 1 trivial
25.10.b.b 4 5.b even 2 1 inner
45.10.a.f 2 15.e even 4 1
80.10.a.f 2 20.e even 4 1
225.10.a.h 2 15.e even 4 1
225.10.b.h 4 3.b odd 2 1
225.10.b.h 4 15.d odd 2 1
245.10.a.d 2 35.f even 4 1
320.10.a.k 2 40.i odd 4 1
320.10.a.s 2 40.k even 4 1
400.10.a.t 2 20.e even 4 1
400.10.c.p 4 4.b odd 2 1
400.10.c.p 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2068 T_{2}^{2} + 968256$$ acting on $$S_{10}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 20 T^{2} + 423488 T^{4} + 5242880 T^{6} + 68719476736 T^{8}$$
$3$ $$1 - 36860 T^{2} + 841672278 T^{4} - 14280319224540 T^{6} + 150094635296999121 T^{8}$$
$5$ 1
$7$ $$1 - 67553100 T^{2} + 4264140931763398 T^{4} -$$$$11\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} )^{2}$$
$13$ $$1 - 32114487020 T^{2} +$$$$45\!\cdots\!58$$$$T^{4} -$$$$36\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!41$$$$T^{8}$$
$17$ $$1 - 55612876540 T^{2} +$$$$47\!\cdots\!18$$$$T^{4} -$$$$78\!\cdots\!60$$$$T^{6} +$$$$19\!\cdots\!81$$$$T^{8}$$
$19$ $$( 1 - 296520 T + 659218232758 T^{2} - 95683356145429080 T^{3} +$$$$10\!\cdots\!41$$$$T^{4} )^{2}$$
$23$ $$1 - 5894841450380 T^{2} +$$$$14\!\cdots\!38$$$$T^{4} -$$$$19\!\cdots\!20$$$$T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 3666980 T + 20832571957438 T^{2} - 53197414150592105620 T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$31$ $$( 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} +$$$$69\!\cdots\!41$$$$T^{4} )^{2}$$
$37$ $$1 - 211727253209420 T^{2} +$$$$26\!\cdots\!58$$$$T^{4} -$$$$35\!\cdots\!80$$$$T^{6} +$$$$28\!\cdots\!41$$$$T^{8}$$
$41$ $$( 1 + 26957276 T + 811945448362966 T^{2} +$$$$88\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} )^{2}$$
$43$ $$1 - 610367211431900 T^{2} +$$$$59\!\cdots\!98$$$$T^{4} -$$$$15\!\cdots\!00$$$$T^{6} +$$$$63\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 2217843742562860 T^{2} +$$$$27\!\cdots\!78$$$$T^{4} -$$$$27\!\cdots\!40$$$$T^{6} +$$$$15\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 9826319201242060 T^{2} +$$$$43\!\cdots\!78$$$$T^{4} -$$$$10\!\cdots\!40$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$59$ $$( 1 - 54995560 T + 15674484224932678 T^{2} -$$$$47\!\cdots\!40$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4} )^{2}$$
$61$ $$( 1 + 274579716 T + 41753623519328446 T^{2} +$$$$32\!\cdots\!56$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4} )^{2}$$
$67$ $$1 - 97040022635671740 T^{2} +$$$$38\!\cdots\!18$$$$T^{4} -$$$$71\!\cdots\!60$$$$T^{6} +$$$$54\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 + 7130936 T + 51935375688707086 T^{2} +$$$$32\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4} )^{2}$$
$73$ $$1 - 70807827705933980 T^{2} +$$$$70\!\cdots\!38$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$12\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 + 6877520 T - 115982362290712162 T^{2} +$$$$82\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4} )^{2}$$
$83$ $$1 + 250773367170805060 T^{2} +$$$$70\!\cdots\!18$$$$T^{4} +$$$$87\!\cdots\!40$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 + 830088660 T + 692293619421117718 T^{2} +$$$$29\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4} )^{2}$$
$97$ $$1 - 2823274612928748860 T^{2} +$$$$31\!\cdots\!78$$$$T^{4} -$$$$16\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!21$$$$T^{8}$$