# Properties

 Label 25.26.b.a Level $25$ Weight $26$ Character orbit 25.b Analytic conductor $98.999$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$26$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$98.9991949881$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 24 \beta q^{2} - 97902 \beta q^{3} + 33552128 q^{4} + 9398592 q^{6} - 19540298596 \beta q^{7} + 1610557440 \beta q^{8} + 808949403027 q^{9} +O(q^{10})$$ q + 24*b * q^2 - 97902*b * q^3 + 33552128 * q^4 + 9398592 * q^6 - 19540298596*b * q^7 + 1610557440*b * q^8 + 808949403027 * q^9 $$q + 24 \beta q^{2} - 97902 \beta q^{3} + 33552128 q^{4} + 9398592 q^{6} - 19540298596 \beta q^{7} + 1610557440 \beta q^{8} + 808949403027 q^{9} + 8419515299052 q^{11} - 3284820435456 \beta q^{12} - 40825522667657 \beta q^{13} + 1875868665216 q^{14} + 11\!\cdots\!56 q^{16} + \cdots + 68\!\cdots\!04 q^{99} +O(q^{100})$$ q + 24*b * q^2 - 97902*b * q^3 + 33552128 * q^4 + 9398592 * q^6 - 19540298596*b * q^7 + 1610557440*b * q^8 + 808949403027 * q^9 + 8419515299052 * q^11 - 3284820435456*b * q^12 - 40825522667657*b * q^13 + 1875868665216 * q^14 + 1125667983917056 * q^16 + 1259950014474039*b * q^17 + 19414785672648*b * q^18 + 6082056370308940 * q^19 - 7652137252582368 * q^21 + 202068367177248*b * q^22 - 47497640148160212*b * q^23 + 630707177963520 * q^24 + 3919250176095072 * q^26 - 162149013896837940*b * q^27 - 655618599651212288*b * q^28 + 271246959476737410 * q^29 + 4291666067521509152 * q^31 + 81057371716583424*b * q^32 - 824287386807788904*b * q^33 - 120955201389507744 * q^34 + 27141973915885491456 * q^36 - 10150742223054563491*b * q^37 + 145969352887414560*b * q^38 - 15987601280835822456 * q^39 - 183744249574071224598 * q^41 - 183651294061976832*b * q^42 + 150450912092793167878*b * q^43 + 282492655011750982656 * q^44 + 4559773454223380352 * q^46 + 462180524032352434344*b * q^47 - 110205146961447616512*b * q^48 - 186224457219393384057 * q^49 + 493406505268149464712 * q^51 - 1369783162212129124096*b * q^52 - 495146102777495235477*b * q^53 + 15566305334096442240 * q^54 + 125883093134437416960 * q^56 - 595445482765985843880*b * q^57 + 6509927027441697840*b * q^58 - 13052569416454201837980 * q^59 + 9015451224701414617502 * q^61 + 102999985620516219648*b * q^62 - 15807112884203526250092*b * q^63 + 37763368313237157183488 * q^64 + 79131589133547734784 * q^66 + 13344533904454289851214*b * q^67 + 42274004159234809204992*b * q^68 - 18600455863140724300896 * q^69 - 192390516186217637440248 * q^71 + 1302859479628693370880*b * q^72 + 21202292419046226929413*b * q^73 + 974471253413238095136 * q^74 + 204065933839820954424320 * q^76 - 164519842977066315730992*b * q^77 - 383702430740059738944*b * q^78 + 271681055025772277197360 * q^79 + 621914763766378892976441 * q^81 - 4409861989777709390352*b * q^82 - 465727228653506762180742*b * q^83 - 256745488572211941679104 * q^84 - 14443287560908144116288 * q^86 - 26555619826691545913820*b * q^87 + 13560113006082023546880*b * q^88 + 1763635518049807316502630 * q^89 - 3190971613055137006838288 * q^91 - 1593646901949010397531136*b * q^92 - 420162691342490788999104*b * q^93 - 44369330307105833697024 * q^94 + 31742715223187801505792 * q^96 - 1414620434963436043093681*b * q^97 - 4469386973265441217368*b * q^98 + 6810961874944808779030404 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 67104256 q^{4} + 18797184 q^{6} + 1617898806054 q^{9}+O(q^{10})$$ 2 * q + 67104256 * q^4 + 18797184 * q^6 + 1617898806054 * q^9 $$2 q + 67104256 q^{4} + 18797184 q^{6} + 1617898806054 q^{9} + 16839030598104 q^{11} + 3751737330432 q^{14} + 22\!\cdots\!12 q^{16}+ \cdots + 13\!\cdots\!08 q^{99}+O(q^{100})$$ 2 * q + 67104256 * q^4 + 18797184 * q^6 + 1617898806054 * q^9 + 16839030598104 * q^11 + 3751737330432 * q^14 + 2251335967834112 * q^16 + 12164112740617880 * q^19 - 15304274505164736 * q^21 + 1261414355927040 * q^24 + 7838500352190144 * q^26 + 542493918953474820 * q^29 + 8583332135043018304 * q^31 - 241910402779015488 * q^34 + 54283947831770982912 * q^36 - 31975202561671644912 * q^39 - 367488499148142449196 * q^41 + 564985310023501965312 * q^44 + 9119546908446760704 * q^46 - 372448914438786768114 * q^49 + 986813010536298929424 * q^51 + 31132610668192884480 * q^54 + 251766186268874833920 * q^56 - 26105138832908403675960 * q^59 + 18030902449402829235004 * q^61 + 75526736626474314366976 * q^64 + 158263178267095469568 * q^66 - 37200911726281448601792 * q^69 - 384781032372435274880496 * q^71 + 1948942506826476190272 * q^74 + 408131867679641908848640 * q^76 + 543362110051544554394720 * q^79 + 1243829527532757785952882 * q^81 - 513490977144423883358208 * q^84 - 28886575121816288232576 * q^86 + 3527271036099614633005260 * q^89 - 6381943226110274013676576 * q^91 - 88738660614211667394048 * q^94 + 63485430446375603011584 * q^96 + 13621923749889617558060808 * q^99

## 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
 − 1.00000i 1.00000i
48.0000i 195804.i 3.35521e7 0 9.39859e6 3.90806e10i 3.22111e9i 8.08949e11 0
24.2 48.0000i 195804.i 3.35521e7 0 9.39859e6 3.90806e10i 3.22111e9i 8.08949e11 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 25.26.b.a 2
5.b even 2 1 inner 25.26.b.a 2
5.c odd 4 1 1.26.a.a 1
5.c odd 4 1 25.26.a.a 1
15.e even 4 1 9.26.a.a 1
20.e even 4 1 16.26.a.b 1
35.f even 4 1 49.26.a.a 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2304$$ acting on $$S_{26}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2304$$
$3$ $$T^{2} + 38339206416$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 15\!\cdots\!64$$
$11$ $$(T - 8419515299052)^{2}$$
$13$ $$T^{2} + 66\!\cdots\!96$$
$17$ $$T^{2} + 63\!\cdots\!84$$
$19$ $$(T - 60\!\cdots\!40)^{2}$$
$23$ $$T^{2} + 90\!\cdots\!76$$
$29$ $$(T - 27\!\cdots\!10)^{2}$$
$31$ $$(T - 42\!\cdots\!52)^{2}$$
$37$ $$T^{2} + 41\!\cdots\!24$$
$41$ $$(T + 18\!\cdots\!98)^{2}$$
$43$ $$T^{2} + 90\!\cdots\!36$$
$47$ $$T^{2} + 85\!\cdots\!44$$
$53$ $$T^{2} + 98\!\cdots\!16$$
$59$ $$(T + 13\!\cdots\!80)^{2}$$
$61$ $$(T - 90\!\cdots\!02)^{2}$$
$67$ $$T^{2} + 71\!\cdots\!84$$
$71$ $$(T + 19\!\cdots\!48)^{2}$$
$73$ $$T^{2} + 17\!\cdots\!76$$
$79$ $$(T - 27\!\cdots\!60)^{2}$$
$83$ $$T^{2} + 86\!\cdots\!56$$
$89$ $$(T - 17\!\cdots\!30)^{2}$$
$97$ $$T^{2} + 80\!\cdots\!44$$