# Properties

 Label 49.20.a.b Level $49$ Weight $20$ Character orbit 49.a Self dual yes Analytic conductor $112.120$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$112.120181313$$ 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

 $$f(q)$$ $$=$$ $$q + 456 q^{2} - 50652 q^{3} - 316352 q^{4} + 2377410 q^{5} - 23097312 q^{6} - 383331840 q^{8} + 1403363637 q^{9}+O(q^{10})$$ q + 456 * q^2 - 50652 * q^3 - 316352 * q^4 + 2377410 * q^5 - 23097312 * q^6 - 383331840 * q^8 + 1403363637 * q^9 $$q + 456 q^{2} - 50652 q^{3} - 316352 q^{4} + 2377410 q^{5} - 23097312 q^{6} - 383331840 q^{8} + 1403363637 q^{9} + 1084098960 q^{10} - 16212108 q^{11} + 16023861504 q^{12} - 50421615062 q^{13} - 120420571320 q^{15} - 8939761664 q^{16} - 225070099506 q^{17} + 639933818472 q^{18} + 1710278572660 q^{19} - 752098408320 q^{20} - 7392721248 q^{22} + 14036534788872 q^{23} + 19416524359680 q^{24} - 13421408020025 q^{25} - 22992256468272 q^{26} - 12212307114840 q^{27} + 1137835269510 q^{29} - 54911780521920 q^{30} + 104626880141728 q^{31} + 196899752411136 q^{32} + 821175694416 q^{33} - 102631965374736 q^{34} - 443956893292224 q^{36} - 169392327370594 q^{37} + 779887029132960 q^{38} + 25\!\cdots\!24 q^{39}+ \cdots - 22\!\cdots\!96 q^{99}+O(q^{100})$$ q + 456 * q^2 - 50652 * q^3 - 316352 * q^4 + 2377410 * q^5 - 23097312 * q^6 - 383331840 * q^8 + 1403363637 * q^9 + 1084098960 * q^10 - 16212108 * q^11 + 16023861504 * q^12 - 50421615062 * q^13 - 120420571320 * q^15 - 8939761664 * q^16 - 225070099506 * q^17 + 639933818472 * q^18 + 1710278572660 * q^19 - 752098408320 * q^20 - 7392721248 * q^22 + 14036534788872 * q^23 + 19416524359680 * q^24 - 13421408020025 * q^25 - 22992256468272 * q^26 - 12212307114840 * q^27 + 1137835269510 * q^29 - 54911780521920 * q^30 + 104626880141728 * q^31 + 196899752411136 * q^32 + 821175694416 * q^33 - 102631965374736 * q^34 - 443956893292224 * q^36 - 169392327370594 * q^37 + 779887029132960 * q^38 + 2553955646120424 * q^39 - 911336949734400 * q^40 + 3309984750560838 * q^41 + 1127913532193492 * q^43 + 5128732790016 * q^44 + 3336370744240170 * q^45 + 6400659863725632 * q^46 - 3498693987674256 * q^47 + 452816807804928 * q^48 - 6120162057131400 * q^50 + 11400250680177912 * q^51 + 15950978768093824 * q^52 + 29956294112980302 * q^53 - 5568812044367040 * q^54 - 38542827680280 * q^55 - 86629030262374320 * q^57 + 518852882896560 * q^58 - 58391397642732420 * q^59 + 38095288578224640 * q^60 - 23373685132672742 * q^61 + 47709857344627968 * q^62 + 94473296862773248 * q^64 - 119872851864549420 * q^65 + 374456116653696 * q^66 - 205102524257382244 * q^67 + 71201376118922112 * q^68 - 710978560125944544 * q^69 - 177902341950417768 * q^71 - 537953965160302080 * q^72 - 299853775038660122 * q^73 - 77242901280990864 * q^74 + 679821159030306300 * q^75 - 541050047018136320 * q^76 + 1164603774630913344 * q^78 - 92227090144007440 * q^79 - 21253478777610240 * q^80 - 1012497699493199799 * q^81 + 1509353046255742128 * q^82 - 1208542823470585932 * q^83 - 535083905266559460 * q^85 + 514328570680232352 * q^86 - 57633632071220520 * q^87 + 6214617189918720 * q^88 - 4371201192290304330 * q^89 + 1521385059373517520 * q^90 - 4440485853529234944 * q^92 - 5299560732938806656 * q^93 - 1595404458379460736 * q^94 + 4066033381427610600 * q^95 - 9973366259128860672 * q^96 + 635013222218448094 * q^97 - 22751482846316796 * 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
456.000 −50652.0 −316352. 2.37741e6 −2.30973e7 0 −3.83332e8 1.40336e9 1.08410e9
 $$n$$: 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 49.20.a.b 1
7.b odd 2 1 1.20.a.a 1
21.c even 2 1 9.20.a.a 1
28.d even 2 1 16.20.a.a 1
35.c odd 2 1 25.20.a.a 1
35.f even 4 2 25.20.b.a 2
56.e even 2 1 64.20.a.h 1
56.h odd 2 1 64.20.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.20.a.a 1 7.b odd 2 1
9.20.a.a 1 21.c even 2 1
16.20.a.a 1 28.d even 2 1
25.20.a.a 1 35.c odd 2 1
25.20.b.a 2 35.f even 4 2
49.20.a.b 1 1.a even 1 1 trivial
64.20.a.b 1 56.h odd 2 1
64.20.a.h 1 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{20}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2} - 456$$ T2 - 456 $$T_{3} + 50652$$ T3 + 50652

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 456$$
$3$ $$T + 50652$$
$5$ $$T - 2377410$$
$7$ $$T$$
$11$ $$T + 16212108$$
$13$ $$T + 50421615062$$
$17$ $$T + 225070099506$$
$19$ $$T - 1710278572660$$
$23$ $$T - 14036534788872$$
$29$ $$T - 1137835269510$$
$31$ $$T - 104626880141728$$
$37$ $$T + 169392327370594$$
$41$ $$T - 3309984750560838$$
$43$ $$T - 1127913532193492$$
$47$ $$T + 3498693987674256$$
$53$ $$T - 29\!\cdots\!02$$
$59$ $$T + 58\!\cdots\!20$$
$61$ $$T + 23\!\cdots\!42$$
$67$ $$T + 20\!\cdots\!44$$
$71$ $$T + 17\!\cdots\!68$$
$73$ $$T + 29\!\cdots\!22$$
$79$ $$T + 92\!\cdots\!40$$
$83$ $$T + 12\!\cdots\!32$$
$89$ $$T + 43\!\cdots\!30$$
$97$ $$T - 63\!\cdots\!94$$