# Properties

 Label 49.26.a.h Level $49$ Weight $26$ Character orbit 49.a Self dual yes Analytic conductor $194.038$ Analytic rank $1$ Dimension $24$ CM no 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: $$1$$ Dimension: $$24$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 16290 q^{2} + 333378474 q^{4} - 798202294830 q^{8} + 5073152845720 q^{9}+O(q^{10})$$ 24 * q - 16290 * q^2 + 333378474 * q^4 - 798202294830 * q^8 + 5073152845720 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 16290 q^{2} + 333378474 q^{4} - 798202294830 q^{8} + 5073152845720 q^{9} - 29647806315264 q^{11} + 13\!\cdots\!04 q^{15}+ \cdots - 25\!\cdots\!24 q^{99}+O(q^{100})$$ 24 * q - 16290 * q^2 + 333378474 * q^4 - 798202294830 * q^8 + 5073152845720 * q^9 - 29647806315264 * q^11 + 1390302766599104 * q^15 + 716690983493922 * q^16 - 23991856792402270 * q^18 + 279113049135428820 * q^22 - 277588399475977920 * q^23 + 338211957111767064 * q^25 - 1332491118023885184 * q^29 - 14304112572263314392 * q^30 + 5605670320652716290 * q^32 + 117337039125685642390 * q^36 + 163615550858414490240 * q^37 - 219653501389580081600 * q^39 - 697964208110325610560 * q^43 - 1917200651945960036340 * q^44 - 1500833765795777372952 * q^46 - 9787901213658866981670 * q^50 - 12133976065055882795104 * q^51 - 9312003410122910258640 * q^53 - 68191669708446868149600 * q^57 - 15577066681051268419140 * q^58 + 73873591939575232909720 * q^60 - 227237974326721186555134 * q^64 - 308511558879546554049168 * q^65 - 139005593267187739767840 * q^67 - 267652788156171889739136 * q^71 - 998472544602701330744850 * q^72 - 591209348848539701301972 * q^74 + 3661103265441916581215040 * q^78 + 1445780179286245017093696 * q^79 + 5005641224332350461971768 * q^81 + 7911374137900543226318256 * q^85 + 5755348715719209900524460 * q^86 + 7051849138058806239311580 * q^88 + 12572656813255691697522840 * q^92 + 4311885378480881745927680 * q^93 - 22477102443512669915525376 * q^95 - 25386041685406293662767424 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −9947.40 −1.77441e6 6.53963e7 −5.10763e8 1.76507e10 0 −3.16744e11 2.30123e12 5.08076e12
1.2 −9947.40 1.77441e6 6.53963e7 5.10763e8 −1.76507e10 0 −3.16744e11 2.30123e12 −5.08076e12
1.3 −9621.83 −44215.1 5.90252e7 9.55920e8 4.25430e8 0 −2.45075e11 −8.45334e11 −9.19770e12
1.4 −9621.83 44215.1 5.90252e7 −9.55920e8 −4.25430e8 0 −2.45075e11 −8.45334e11 9.19770e12
1.5 −9304.30 −644120. 5.30155e7 5.39746e8 5.99309e9 0 −1.81072e11 −4.32397e11 −5.02196e12
1.6 −9304.30 644120. 5.30155e7 −5.39746e8 −5.99309e9 0 −1.81072e11 −4.32397e11 5.02196e12
1.7 −4742.65 −1.50280e6 −1.10617e7 −3.39227e8 7.12727e9 0 2.11599e11 1.41112e12 1.60884e12
1.8 −4742.65 1.50280e6 −1.10617e7 3.39227e8 −7.12727e9 0 2.11599e11 1.41112e12 −1.60884e12
1.9 −4573.51 −853606. −1.26374e7 3.17192e8 3.90397e9 0 2.11259e11 −1.18646e11 −1.45068e12
1.10 −4573.51 853606. −1.26374e7 −3.17192e8 −3.90397e9 0 2.11259e11 −1.18646e11 1.45068e12
1.11 −3011.53 −462530. −2.44851e7 −4.78350e8 1.39292e9 0 1.74788e11 −6.33355e11 1.44057e12
1.12 −3011.53 462530. −2.44851e7 4.78350e8 −1.39292e9 0 1.74788e11 −6.33355e11 −1.44057e12
1.13 1174.18 −942587. −3.21757e7 4.94390e8 −1.10677e9 0 −7.71793e10 4.11813e10 5.80505e11
1.14 1174.18 942587. −3.21757e7 −4.94390e8 1.10677e9 0 −7.71793e10 4.11813e10 −5.80505e11
1.15 3451.49 −304783. −2.16417e7 −7.74843e8 −1.05195e9 0 −1.90509e11 −7.54396e11 −2.67436e12
1.16 3451.49 304783. −2.16417e7 7.74843e8 1.05195e9 0 −1.90509e11 −7.54396e11 2.67436e12
1.17 3683.76 −1.60504e6 −1.99843e7 2.32545e8 −5.91257e9 0 −1.97224e11 1.72885e12 8.56642e11
1.18 3683.76 1.60504e6 −1.99843e7 −2.32545e8 5.91257e9 0 −1.97224e11 1.72885e12 −8.56642e11
1.19 6114.28 −603398. 3.83000e6 −9.40230e8 −3.68934e9 0 −1.81744e11 −4.83200e11 −5.74883e12
1.20 6114.28 603398. 3.83000e6 9.40230e8 3.68934e9 0 −1.81744e11 −4.83200e11 5.74883e12
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.26.a.h 24
7.b odd 2 1 inner 49.26.a.h 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.26.a.h 24 1.a even 1 1 trivial
49.26.a.h 24 7.b odd 2 1 inner

## Hecke kernels

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

 $$T_{2}^{12} + 8145 T_{2}^{11} - 251500698 T_{2}^{10} - 1913354786520 T_{2}^{9} + \cdots + 45\!\cdots\!24$$ T2^12 + 8145*T2^11 - 251500698*T2^10 - 1913354786520*T2^9 + 22787264989057536*T2^8 + 154809742872833395200*T2^7 - 921924670642970317922304*T2^6 - 5169219123335375010270412800*T2^5 + 17272046006749453072917841575936*T2^4 + 70981057573615258178673096869806080*T2^3 - 149324526896265699514246957668363141120*T2^2 - 329462138319004438677021043285911750574080*T2 + 458388530976700971142680508850431898077364224 $$T_{3}^{24} - 12704039736176 T_{3}^{22} + \cdots + 42\!\cdots\!56$$ T3^24 - 12704039736176*T3^22 + 67606308185677644325215088*T3^20 - 197283359445173803031992448195810070720*T3^18 + 348169577688730524069191236305482777543010895723104*T3^16 - 388249873220579066163154742455617240293525695501671180052246272*T3^14 + 278623100546226991866353691846675585353076297838661948966774995387211407104*T3^12 - 128658583166762256322197852671962986321366811658695201549577005650443789873162432418816*T3^10 + 37452563681801456082248055128814053091626547701522425417720354921672020533998548578967842497962240*T3^8 - 6527289721552843884527203396814478444110655635896955008835678658281989679158389016409474828083160570445176832*T3^6 + 608296741725988050488041297100455156416048955780148638966618024273143034240583521172793181130872470120735586197766766592*T3^4 - 22775344365279508988044320077592849896599252547139604743846078834678135807004338306024189338427828645572847755566610721386109861888*T3^2 + 42248600405962844148012745564370062406328533191142936986544633338360931227200650933070854282426692654755506114825077535073499534327054073856