# Properties

 Label 49.26.a.a Level $49$ Weight $26$ Character orbit 49.a Self dual yes Analytic conductor $194.038$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$26$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$194.038422177$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 48 q^{2} + 195804 q^{3} - 33552128 q^{4} + 741989850 q^{5} - 9398592 q^{6} + 3221114880 q^{8} - 808949403027 q^{9}+O(q^{10})$$ q - 48 * q^2 + 195804 * q^3 - 33552128 * q^4 + 741989850 * q^5 - 9398592 * q^6 + 3221114880 * q^8 - 808949403027 * q^9 $$q - 48 q^{2} + 195804 q^{3} - 33552128 q^{4} + 741989850 q^{5} - 9398592 q^{6} + 3221114880 q^{8} - 808949403027 q^{9} - 35615512800 q^{10} + 8419515299052 q^{11} - 6569640870912 q^{12} + 81651045335314 q^{13} + 145284580589400 q^{15} + 11\!\cdots\!56 q^{16}+ \cdots - 68\!\cdots\!04 q^{99}+O(q^{100})$$ q - 48 * q^2 + 195804 * q^3 - 33552128 * q^4 + 741989850 * q^5 - 9398592 * q^6 + 3221114880 * q^8 - 808949403027 * q^9 - 35615512800 * q^10 + 8419515299052 * q^11 - 6569640870912 * q^12 + 81651045335314 * q^13 + 145284580589400 * q^15 + 1125667983917056 * q^16 + 2519900028948078 * q^17 + 38829571345296 * q^18 + 6082056370308940 * q^19 - 24895338421900800 * q^20 - 404136734354496 * q^22 - 94995280296320424 * q^23 + 630707177963520 * q^24 + 252525713626069375 * q^25 - 3919250176095072 * q^26 - 324298027793675880 * q^27 - 271246959476737410 * q^29 - 6973659868291200 * q^30 - 4291666067521509152 * q^31 - 162114743433166848 * q^32 + 1648574773615577808 * q^33 - 120955201389507744 * q^34 + 27141973915885491456 * q^36 + 20301484446109126982 * q^37 - 291938705774829120 * q^38 + 15987601280835822456 * q^39 + 2390034546643968000 * q^40 + 183744249574071224598 * q^41 + 300901824185586335756 * q^43 - 282492655011750982656 * q^44 - 600232246209593275950 * q^45 + 4559773454223380352 * q^46 + 924361048064704868688 * q^47 + 220410293922895233024 * q^48 - 12121234254051330000 * q^50 + 493406505268149464712 * q^51 - 2739566324424258248192 * q^52 - 990292205554990470954 * q^53 + 15566305334096442240 * q^54 + 6247194893816298622200 * q^55 + 1190890965531971687760 * q^57 + 13019854054883395680 * q^58 - 13052569416454201837980 * q^59 - 4874606844361864243200 * q^60 - 9015451224701414617502 * q^61 + 205999971241032439296 * q^62 - 37763368313237157183488 * q^64 + 60584246880692834562900 * q^65 - 79131589133547734784 * q^66 - 26689067808908579702428 * q^67 - 84548008318469618409984 * q^68 - 18600455863140724300896 * q^69 - 192390516186217637440248 * q^71 - 2605718959257386741760 * q^72 - 42404584838092453858826 * q^73 - 974471253413238095136 * q^74 + 49445544830838887902500 * q^75 - 204065933839820954424320 * q^76 - 767404861480119477888 * q^78 - 271681055025772277197360 * q^79 + 835234218536418793881600 * q^80 + 621914763766378892976441 * q^81 - 8819723979555418780704 * q^82 + 931454457307013524361484 * q^83 + 1869740244494180053008300 * q^85 - 14443287560908144116288 * q^86 - 53111239653383091827640 * q^87 + 27120226012164047093760 * q^88 + 1763635518049807316502630 * q^89 + 28811147818060477245600 * q^90 + 3187293803898020795062272 * q^92 - 840325382684981577998208 * q^93 - 44369330307105833697024 * q^94 + 4512824093897074844259000 * q^95 - 31742715223187801505792 * q^96 - 2829240869926872086187362 * q^97 - 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 7.41990e8 −9.39859e6 0 3.22111e9 −8.08949e11 −3.56155e10
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

## 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 49.26.a.a 1
7.b odd 2 1 1.26.a.a 1
21.c even 2 1 9.26.a.a 1
28.d even 2 1 16.26.a.b 1
35.c odd 2 1 25.26.a.a 1
35.f even 4 2 25.26.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.26.a.a 1 7.b odd 2 1
9.26.a.a 1 21.c even 2 1
16.26.a.b 1 28.d even 2 1
25.26.a.a 1 35.c odd 2 1
25.26.b.a 2 35.f even 4 2
49.26.a.a 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{26}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2} + 48$$ T2 + 48 $$T_{3} - 195804$$ T3 - 195804

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 48$$
$3$ $$T - 195804$$
$5$ $$T - 741989850$$
$7$ $$T$$
$11$ $$T - 8419515299052$$
$13$ $$T - 81651045335314$$
$17$ $$T - 2519900028948078$$
$19$ $$T - 6082056370308940$$
$23$ $$T + 94\!\cdots\!24$$
$29$ $$T + 27\!\cdots\!10$$
$31$ $$T + 42\!\cdots\!52$$
$37$ $$T - 20\!\cdots\!82$$
$41$ $$T - 18\!\cdots\!98$$
$43$ $$T - 30\!\cdots\!56$$
$47$ $$T - 92\!\cdots\!88$$
$53$ $$T + 99\!\cdots\!54$$
$59$ $$T + 13\!\cdots\!80$$
$61$ $$T + 90\!\cdots\!02$$
$67$ $$T + 26\!\cdots\!28$$
$71$ $$T + 19\!\cdots\!48$$
$73$ $$T + 42\!\cdots\!26$$
$79$ $$T + 27\!\cdots\!60$$
$83$ $$T - 93\!\cdots\!84$$
$89$ $$T - 17\!\cdots\!30$$
$97$ $$T + 28\!\cdots\!62$$