Properties

 Label 25.26.a.a Level $25$ Weight $26$ Character orbit 25.a Self dual yes Analytic conductor $98.999$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$26$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$98.9991949881$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 48 q^{2} + 195804 q^{3} - 33552128 q^{4} + 9398592 q^{6} - 39080597192 q^{7} - 3221114880 q^{8} - 808949403027 q^{9}+O(q^{10})$$ q + 48 * q^2 + 195804 * q^3 - 33552128 * q^4 + 9398592 * q^6 - 39080597192 * q^7 - 3221114880 * q^8 - 808949403027 * q^9 $$q + 48 q^{2} + 195804 q^{3} - 33552128 q^{4} + 9398592 q^{6} - 39080597192 q^{7} - 3221114880 q^{8} - 808949403027 q^{9} + 8419515299052 q^{11} - 6569640870912 q^{12} + 81651045335314 q^{13} - 1875868665216 q^{14} + 11\!\cdots\!56 q^{16}+ \cdots - 68\!\cdots\!04 q^{99}+O(q^{100})$$ q + 48 * q^2 + 195804 * q^3 - 33552128 * q^4 + 9398592 * q^6 - 39080597192 * q^7 - 3221114880 * q^8 - 808949403027 * q^9 + 8419515299052 * q^11 - 6569640870912 * q^12 + 81651045335314 * q^13 - 1875868665216 * q^14 + 1125667983917056 * q^16 + 2519900028948078 * q^17 - 38829571345296 * q^18 - 6082056370308940 * q^19 - 7652137252582368 * q^21 + 404136734354496 * q^22 + 94995280296320424 * q^23 - 630707177963520 * q^24 + 3919250176095072 * q^26 - 324298027793675880 * q^27 + 1311237199302424576 * q^28 - 271246959476737410 * q^29 + 4291666067521509152 * q^31 + 162114743433166848 * q^32 + 1648574773615577808 * q^33 + 120955201389507744 * q^34 + 27141973915885491456 * q^36 - 20301484446109126982 * q^37 - 291938705774829120 * q^38 + 15987601280835822456 * q^39 - 183744249574071224598 * q^41 - 367302588123953664 * q^42 - 300901824185586335756 * q^43 - 282492655011750982656 * q^44 + 4559773454223380352 * q^46 + 924361048064704868688 * q^47 + 220410293922895233024 * q^48 + 186224457219393384057 * q^49 + 493406505268149464712 * q^51 - 2739566324424258248192 * q^52 + 990292205554990470954 * q^53 - 15566305334096442240 * q^54 + 125883093134437416960 * q^56 - 1190890965531971687760 * q^57 - 13019854054883395680 * q^58 + 13052569416454201837980 * q^59 + 9015451224701414617502 * q^61 + 205999971241032439296 * q^62 + 31614225768407052500184 * q^63 - 37763368313237157183488 * q^64 + 79131589133547734784 * q^66 + 26689067808908579702428 * q^67 - 84548008318469618409984 * q^68 + 18600455863140724300896 * q^69 - 192390516186217637440248 * q^71 + 2605718959257386741760 * q^72 - 42404584838092453858826 * q^73 - 974471253413238095136 * q^74 + 204065933839820954424320 * q^76 - 329039685954132631461984 * q^77 + 767404861480119477888 * q^78 - 271681055025772277197360 * q^79 + 621914763766378892976441 * q^81 - 8819723979555418780704 * q^82 + 931454457307013524361484 * q^83 + 256745488572211941679104 * q^84 - 14443287560908144116288 * q^86 - 53111239653383091827640 * q^87 - 27120226012164047093760 * q^88 - 1763635518049807316502630 * q^89 - 3190971613055137006838288 * q^91 - 3187293803898020795062272 * q^92 + 840325382684981577998208 * q^93 + 44369330307105833697024 * q^94 + 31742715223187801505792 * q^96 - 2829240869926872086187362 * q^97 + 8938773946530882434736 * q^98 - 6810961874944808779030404 * q^99

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
 0
48.0000 195804. −3.35521e7 0 9.39859e6 −3.90806e10 −3.22111e9 −8.08949e11 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 25.26.a.a 1
5.b even 2 1 1.26.a.a 1
5.c odd 4 2 25.26.b.a 2
15.d odd 2 1 9.26.a.a 1
20.d odd 2 1 16.26.a.b 1
35.c odd 2 1 49.26.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.26.a.a 1 5.b even 2 1
9.26.a.a 1 15.d odd 2 1
16.26.a.b 1 20.d odd 2 1
25.26.a.a 1 1.a even 1 1 trivial
25.26.b.a 2 5.c odd 4 2
49.26.a.a 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 48$$ acting on $$S_{26}^{\mathrm{new}}(\Gamma_0(25))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 48$$
$3$ $$T - 195804$$
$5$ $$T$$
$7$ $$T + 39080597192$$
$11$ $$T - 8419515299052$$
$13$ $$T - 81651045335314$$
$17$ $$T - 2519900028948078$$
$19$ $$T + 6082056370308940$$
$23$ $$T - 94\!\cdots\!24$$
$29$ $$T + 27\!\cdots\!10$$
$31$ $$T - 42\!\cdots\!52$$
$37$ $$T + 20\!\cdots\!82$$
$41$ $$T + 18\!\cdots\!98$$
$43$ $$T + 30\!\cdots\!56$$
$47$ $$T - 92\!\cdots\!88$$
$53$ $$T - 99\!\cdots\!54$$
$59$ $$T - 13\!\cdots\!80$$
$61$ $$T - 90\!\cdots\!02$$
$67$ $$T - 26\!\cdots\!28$$
$71$ $$T + 19\!\cdots\!48$$
$73$ $$T + 42\!\cdots\!26$$
$79$ $$T + 27\!\cdots\!60$$
$83$ $$T - 93\!\cdots\!84$$
$89$ $$T + 17\!\cdots\!30$$
$97$ $$T + 28\!\cdots\!62$$