# Properties

 Label 49.26.a.b Level $49$ Weight $26$ Character orbit 49.a Self dual yes Analytic conductor $194.038$ Analytic rank $0$ Dimension $1$ CM discriminant -7 Inner twists $2$

# 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: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4411 q^{2} - 14097511 q^{4} - 210192720573 q^{8} - 847288609443 q^{9}+O(q^{10})$$ q + 4411 * q^2 - 14097511 * q^4 - 210192720573 * q^8 - 847288609443 * q^9 $$q + 4411 q^{2} - 14097511 q^{4} - 210192720573 q^{8} - 847288609443 q^{9} - 9505269254876 q^{11} - 454126116228751 q^{16} - 37\!\cdots\!73 q^{18}+ \cdots + 80\!\cdots\!68 q^{99}+O(q^{100})$$ q + 4411 * q^2 - 14097511 * q^4 - 210192720573 * q^8 - 847288609443 * q^9 - 9505269254876 * q^11 - 454126116228751 * q^16 - 3737390056253073 * q^18 - 41927742683258036 * q^22 - 95604060075370552 * q^23 - 298023223876953125 * q^25 - 3807195347450164318 * q^29 + 5049747050676708875 * q^32 + 11944660491797396373 * q^36 - 25773142737840533286 * q^37 + 301435695530660748948 * q^43 + 134000637878576213636 * q^44 - 421709508992459504872 * q^46 - 1314580440521240234375 * q^50 - 553594896014275110250 * q^53 - 16793538677602674806698 * q^58 + 37512378126956684722057 * q^64 - 15003619703277941363036 * q^67 - 36099127417258839104624 * q^71 + 178093897929338228170839 * q^72 - 113685332616614592324546 * q^74 - 820380440578818957445432 * q^79 + 717897987691852588770249 * q^81 + 1329632852985744563609628 * q^86 + 1997938404461278985763948 * q^88 + 1347779288557197185896072 * q^92 + 8053706369345186787394068 * 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
4411.00 0 −1.40975e7 0 0 0 −2.10193e11 −8.47289e11 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
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.26.a.b 1
7.b odd 2 1 CM 49.26.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.26.a.b 1 1.a even 1 1 trivial
49.26.a.b 1 7.b odd 2 1 CM

## Hecke kernels

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

 $$T_{2} - 4411$$ T2 - 4411 $$T_{3}$$ T3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4411$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 9505269254876$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 95\!\cdots\!52$$
$29$ $$T + 38\!\cdots\!18$$
$31$ $$T$$
$37$ $$T + 25\!\cdots\!86$$
$41$ $$T$$
$43$ $$T - 30\!\cdots\!48$$
$47$ $$T$$
$53$ $$T + 55\!\cdots\!50$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T + 15\!\cdots\!36$$
$71$ $$T + 36\!\cdots\!24$$
$73$ $$T$$
$79$ $$T + 82\!\cdots\!32$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$