# Properties

 Label 25.10.a.b Level $25$ Weight $10$ Character orbit 25.a Self dual yes Analytic conductor $12.876$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.8758959041$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1009})$$ Defining polynomial: $$x^{2} - x - 252$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{1009}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 5 - \beta ) q^{2} + ( -130 + 2 \beta ) q^{3} + ( 522 - 10 \beta ) q^{4} + ( -2668 + 140 \beta ) q^{6} + ( -850 + 214 \beta ) q^{7} + ( 10140 - 60 \beta ) q^{8} + ( 1253 - 520 \beta ) q^{9} +O(q^{10})$$ $$q + ( 5 - \beta ) q^{2} + ( -130 + 2 \beta ) q^{3} + ( 522 - 10 \beta ) q^{4} + ( -2668 + 140 \beta ) q^{6} + ( -850 + 214 \beta ) q^{7} + ( 10140 - 60 \beta ) q^{8} + ( 1253 - 520 \beta ) q^{9} + ( 11992 + 1900 \beta ) q^{11} + ( -88040 + 2344 \beta ) q^{12} + ( -57510 + 1352 \beta ) q^{13} + ( -220176 + 1920 \beta ) q^{14} + ( -156024 - 5320 \beta ) q^{16} + ( -206410 - 12856 \beta ) q^{17} + ( 530945 - 3853 \beta ) q^{18} + ( -148260 - 2840 \beta ) q^{19} + ( 542352 - 29520 \beta ) q^{21} + ( -1857140 - 2492 \beta ) q^{22} + ( 524610 - 19398 \beta ) q^{23} + ( -1439280 + 28080 \beta ) q^{24} + ( -1651718 + 64270 \beta ) q^{26} + ( 1346540 + 30740 \beta ) q^{27} + ( -2602960 + 120208 \beta ) q^{28} + ( -1833490 - 106960 \beta ) q^{29} + ( 806572 + 154700 \beta ) q^{31} + ( -603920 + 160144 \beta ) q^{32} + ( 2275240 - 223016 \beta ) q^{33} + ( 11939654 + 142130 \beta ) q^{34} + ( 5900866 - 283970 \beta ) q^{36} + ( 10560970 - 205296 \beta ) q^{37} + ( 2124260 + 134060 \beta ) q^{38} + ( 10204636 - 290780 \beta ) q^{39} + ( -13478638 - 155800 \beta ) q^{41} + ( 32497440 - 689952 \beta ) q^{42} + ( -26444850 - 25798 \beta ) q^{43} + ( -12911176 + 871880 \beta ) q^{44} + ( 22195632 - 621600 \beta ) q^{46} + ( -29206090 + 523334 \beta ) q^{47} + ( 9547360 + 379552 \beta ) q^{48} + ( 6577057 - 363800 \beta ) q^{49} + ( 889892 + 1258460 \beta ) q^{51} + ( -43661900 + 1280844 \beta ) q^{52} + ( 19517570 - 1137448 \beta ) q^{53} + ( -24283960 - 1192840 \beta ) q^{54} + ( -21574560 + 2220960 \beta ) q^{56} + ( 13542680 + 72680 \beta ) q^{57} + ( 98755190 + 1298690 \beta ) q^{58} + ( -27497780 - 1544720 \beta ) q^{59} + ( -137289858 - 692000 \beta ) q^{61} + ( -152059440 - 33072 \beta ) q^{62} + ( -113346570 + 710142 \beta ) q^{63} + ( -84720608 + 4128480 \beta ) q^{64} + ( 236399344 - 3390320 \beta ) q^{66} + ( 159290 - 2416706 \beta ) q^{67} + ( 21971020 - 4646732 \beta ) q^{68} + ( -107344464 + 3570960 \beta ) q^{69} + ( -3565468 - 6278500 \beta ) q^{71} + ( 44186220 - 5347980 \beta ) q^{72} + ( -60429090 + 8830952 \beta ) q^{73} + ( 259948514 - 11587450 \beta ) q^{74} + ( -48736120 + 120 \beta ) q^{76} + ( 400066200 + 951288 \beta ) q^{77} + ( 344420200 - 11658536 \beta ) q^{78} + ( 3438760 + 18775640 \beta ) q^{79} + ( -137679679 + 8932040 \beta ) q^{81} + ( 89809010 + 12699638 \beta ) q^{82} + ( -701174370 + 2748402 \beta ) q^{83} + ( 580964544 - 20832960 \beta ) q^{84} + ( -106194068 + 26315860 \beta ) q^{86} + ( 22508420 + 10237820 \beta ) q^{87} + ( 6572880 + 18546480 \beta ) q^{88} + ( 415044330 - 13381680 \beta ) q^{89} + ( 340815452 - 13456340 \beta ) q^{91} + ( 469572240 - 15371856 \beta ) q^{92} + ( 207330240 - 18497856 \beta ) q^{93} + ( -674074456 + 31822760 \beta ) q^{94} + ( 401680192 - 22026560 \beta ) q^{96} + ( -319197290 - 2622216 \beta ) q^{97} + ( 399959485 - 8396057 \beta ) q^{98} + ( -981866024 - 3855140 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{2} - 260q^{3} + 1044q^{4} - 5336q^{6} - 1700q^{7} + 20280q^{8} + 2506q^{9} + O(q^{10})$$ $$2q + 10q^{2} - 260q^{3} + 1044q^{4} - 5336q^{6} - 1700q^{7} + 20280q^{8} + 2506q^{9} + 23984q^{11} - 176080q^{12} - 115020q^{13} - 440352q^{14} - 312048q^{16} - 412820q^{17} + 1061890q^{18} - 296520q^{19} + 1084704q^{21} - 3714280q^{22} + 1049220q^{23} - 2878560q^{24} - 3303436q^{26} + 2693080q^{27} - 5205920q^{28} - 3666980q^{29} + 1613144q^{31} - 1207840q^{32} + 4550480q^{33} + 23879308q^{34} + 11801732q^{36} + 21121940q^{37} + 4248520q^{38} + 20409272q^{39} - 26957276q^{41} + 64994880q^{42} - 52889700q^{43} - 25822352q^{44} + 44391264q^{46} - 58412180q^{47} + 19094720q^{48} + 13154114q^{49} + 1779784q^{51} - 87323800q^{52} + 39035140q^{53} - 48567920q^{54} - 43149120q^{56} + 27085360q^{57} + 197510380q^{58} - 54995560q^{59} - 274579716q^{61} - 304118880q^{62} - 226693140q^{63} - 169441216q^{64} + 472798688q^{66} + 318580q^{67} + 43942040q^{68} - 214688928q^{69} - 7130936q^{71} + 88372440q^{72} - 120858180q^{73} + 519897028q^{74} - 97472240q^{76} + 800132400q^{77} + 688840400q^{78} + 6877520q^{79} - 275359358q^{81} + 179618020q^{82} - 1402348740q^{83} + 1161929088q^{84} - 212388136q^{86} + 45016840q^{87} + 13145760q^{88} + 830088660q^{89} + 681630904q^{91} + 939144480q^{92} + 414660480q^{93} - 1348148912q^{94} + 803360384q^{96} - 638394580q^{97} + 799918970q^{98} - 1963732048q^{99} + O(q^{100})$$

## 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
 16.3824 −15.3824
−26.7648 −66.4705 204.352 0 1779.07 5947.66 8234.11 −15264.7 0
1.2 36.7648 −193.530 839.648 0 −7115.07 −7647.66 12045.9 17770.7 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 25.10.a.b 2
3.b odd 2 1 225.10.a.h 2
4.b odd 2 1 400.10.a.t 2
5.b even 2 1 5.10.a.b 2
5.c odd 4 2 25.10.b.b 4
15.d odd 2 1 45.10.a.f 2
15.e even 4 2 225.10.b.h 4
20.d odd 2 1 80.10.a.f 2
20.e even 4 2 400.10.c.p 4
35.c odd 2 1 245.10.a.d 2
40.e odd 2 1 320.10.a.s 2
40.f even 2 1 320.10.a.k 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 10 T_{2} - 984$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 40 T^{2} - 5120 T^{3} + 262144 T^{4}$$
$3$ $$1 + 260 T + 52230 T^{2} + 5117580 T^{3} + 387420489 T^{4}$$
$5$ 1
$7$ $$1 + 1700 T + 35221550 T^{2} + 68601131900 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 + 115020 T + 22672043710 T^{2} + 1219729517882460 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 412820 T + 113016614470 T^{2} + 48955447175491540 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 + 296520 T + 659218232758 T^{2} + 95683356145429080 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 - 1049220 T + 3497852029390 T^{2} - 1889805395460208860 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 - 21121940 T + 328931801286510 T^{2} -$$$$27\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 + 26957276 T + 811945448362966 T^{2} +$$$$88\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 52889700 T + 1703843788760950 T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 58412180 T + 2814913257457630 T^{2} +$$$$65\!\cdots\!60$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 - 39035140 T + 5675030678030830 T^{2} -$$$$12\!\cdots\!20$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 + 54995560 T + 15674484224932678 T^{2} +$$$$47\!\cdots\!40$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 + 274579716 T + 41753623519328446 T^{2} +$$$$32\!\cdots\!56$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 - 318580 T + 48520062064444070 T^{2} -$$$$86\!\cdots\!60$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 + 7130936 T + 51935375688707086 T^{2} +$$$$32\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 120858180 T + 42707263689423190 T^{2} +$$$$71\!\cdots\!40$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 - 6877520 T - 115982362290712162 T^{2} -$$$$82\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 + 1402348740 T + 857904310704391270 T^{2} +$$$$26\!\cdots\!20$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 - 830088660 T + 692293619421117718 T^{2} -$$$$29\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 + 638394580 T + 1615411126351062630 T^{2} +$$$$48\!\cdots\!60$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$