# Properties

 Label 289.10.a.i Level $289$ Weight $10$ Character orbit 289.a Self dual yes Analytic conductor $148.845$ Analytic rank $0$ Dimension $52$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 289.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$148.845356651$$ Analytic rank: $$0$$ Dimension: $$52$$ Twist minimal: no (minimal twist has level 17) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$52 q + 64 q^{2} + 13312 q^{4} + 49152 q^{8} + 341172 q^{9}+O(q^{10})$$ 52 * q + 64 * q^2 + 13312 * q^4 + 49152 * q^8 + 341172 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$52 q + 64 q^{2} + 13312 q^{4} + 49152 q^{8} + 341172 q^{9} + 156200 q^{13} + 1207872 q^{15} + 3407880 q^{16} + 2193336 q^{18} + 1185568 q^{19} + 5198336 q^{21} + 13827692 q^{25} + 3618944 q^{26} - 9167544 q^{30} + 61884888 q^{32} + 1635208 q^{33} + 46992776 q^{35} + 156027320 q^{36} + 84813952 q^{38} - 4635776 q^{42} + 125448912 q^{43} + 164193176 q^{47} + 270850284 q^{49} - 226223888 q^{50} + 103553016 q^{52} + 426167208 q^{53} + 677761520 q^{55} + 375214512 q^{59} + 336918024 q^{60} + 190014416 q^{64} + 1377178928 q^{66} + 311910088 q^{67} + 533688136 q^{69} + 1477690280 q^{70} + 2757942680 q^{72} + 4047975520 q^{76} + 3440336432 q^{77} + 3266558756 q^{81} + 2072890608 q^{83} + 2630025952 q^{84} + 1538547296 q^{86} - 1010436256 q^{87} + 1873849184 q^{89} - 1998451624 q^{93} - 6880776704 q^{94} - 4667454128 q^{98}+O(q^{100})$$ 52 * q + 64 * q^2 + 13312 * q^4 + 49152 * q^8 + 341172 * q^9 + 156200 * q^13 + 1207872 * q^15 + 3407880 * q^16 + 2193336 * q^18 + 1185568 * q^19 + 5198336 * q^21 + 13827692 * q^25 + 3618944 * q^26 - 9167544 * q^30 + 61884888 * q^32 + 1635208 * q^33 + 46992776 * q^35 + 156027320 * q^36 + 84813952 * q^38 - 4635776 * q^42 + 125448912 * q^43 + 164193176 * q^47 + 270850284 * q^49 - 226223888 * q^50 + 103553016 * q^52 + 426167208 * q^53 + 677761520 * q^55 + 375214512 * q^59 + 336918024 * q^60 + 190014416 * q^64 + 1377178928 * q^66 + 311910088 * q^67 + 533688136 * q^69 + 1477690280 * q^70 + 2757942680 * q^72 + 4047975520 * q^76 + 3440336432 * q^77 + 3266558756 * q^81 + 2072890608 * q^83 + 2630025952 * q^84 + 1538547296 * q^86 - 1010436256 * q^87 + 1873849184 * q^89 - 1998451624 * q^93 - 6880776704 * q^94 - 4667454128 * q^98

## 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 −41.2664 −175.384 1190.92 699.952 7237.48 8848.92 −28016.5 11076.6 −28884.5
1.2 −41.2664 175.384 1190.92 −699.952 −7237.48 −8848.92 −28016.5 11076.6 28884.5
1.3 −40.6100 −102.962 1137.17 −769.524 4181.30 1567.27 −25388.3 −9081.79 31250.4
1.4 −40.6100 102.962 1137.17 769.524 −4181.30 −1567.27 −25388.3 −9081.79 −31250.4
1.5 −37.3725 −245.740 884.705 −1549.27 9183.93 −9379.03 −13928.9 40705.3 57900.2
1.6 −37.3725 245.740 884.705 1549.27 −9183.93 9379.03 −13928.9 40705.3 −57900.2
1.7 −35.0759 −118.571 718.321 −1635.44 4158.98 10785.0 −7236.92 −5623.96 57364.5
1.8 −35.0759 118.571 718.321 1635.44 −4158.98 −10785.0 −7236.92 −5623.96 −57364.5
1.9 −31.9341 −159.334 507.789 1397.46 5088.20 −9976.78 134.480 5704.34 −44626.6
1.10 −31.9341 159.334 507.789 −1397.46 −5088.20 9976.78 134.480 5704.34 44626.6
1.11 −26.9890 −111.659 216.409 −2126.40 3013.56 −8475.21 7977.73 −7215.34 57389.5
1.12 −26.9890 111.659 216.409 2126.40 −3013.56 8475.21 7977.73 −7215.34 −57389.5
1.13 −25.4513 −80.7952 135.770 783.444 2056.34 5744.00 9575.55 −13155.1 −19939.7
1.14 −25.4513 80.7952 135.770 −783.444 −2056.34 −5744.00 9575.55 −13155.1 19939.7
1.15 −24.0067 −194.286 64.3195 2543.89 4664.15 −1962.84 10747.3 18063.9 −61070.2
1.16 −24.0067 194.286 64.3195 −2543.89 −4664.15 1962.84 10747.3 18063.9 61070.2
1.17 −14.8865 −140.613 −290.392 1537.70 2093.24 2138.55 11944.8 89.0110 −22891.0
1.18 −14.8865 140.613 −290.392 −1537.70 −2093.24 −2138.55 11944.8 89.0110 22891.0
1.19 −8.91738 −33.9373 −432.480 34.8748 302.632 8539.90 8422.29 −18531.3 −310.992
1.20 −8.91738 33.9373 −432.480 −34.8748 −302.632 −8539.90 8422.29 −18531.3 310.992
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.52 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.10.a.i 52
17.b even 2 1 inner 289.10.a.i 52
17.e odd 16 2 17.10.d.a 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.d.a 52 17.e odd 16 2
289.10.a.i 52 1.a even 1 1 trivial
289.10.a.i 52 17.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$T_{2}^{26} - 32 T_{2}^{25} - 9472 T_{2}^{24} + 294912 T_{2}^{23} + 38993919 T_{2}^{22} - 1179754716 T_{2}^{21} - 91560086140 T_{2}^{20} + 2688404815616 T_{2}^{19} + \cdots - 60\!\cdots\!00$$ T2^26 - 32*T2^25 - 9472*T2^24 + 294912*T2^23 + 38993919*T2^22 - 1179754716*T2^21 - 91560086140*T2^20 + 2688404815616*T2^19 + 135296082676992*T2^18 - 3849720228654208*T2^17 - 131007903986713344*T2^16 + 3603867142622328320*T2^15 + 84021475101232944384*T2^14 - 2222789082278651309056*T2^13 - 35371642253690302116864*T2^12 + 887592346652081765507072*T2^11 + 9563584211619025505566720*T2^10 - 219149950124571639354753024*T2^9 - 1628209961974003896397332480*T2^8 + 30660559064251214953822814208*T2^7 + 176068751980481225950786224128*T2^6 - 2067732999079058227916613615616*T2^5 - 11661098643913522862091837374464*T2^4 + 43812217553763761454775401447424*T2^3 + 305084254732557544741879198777344*T2^2 + 196034048215902394426742108323840*T2 - 603515986974327961514528682803200 $$T_{3}^{52} - 682344 T_{3}^{50} + 216870628408 T_{3}^{48} + \cdots + 32\!\cdots\!88$$ T3^52 - 682344*T3^50 + 216870628408*T3^48 - 42690191892590400*T3^46 + 5837990981607084194160*T3^44 - 589758336087205363159439872*T3^42 + 45690076675983171302364278423360*T3^40 - 2781922751203311657891694933144899840*T3^38 + 135345655109080620265705925674005685406304*T3^36 - 5321230401454703312891373088594944926145106176*T3^34 + 170318113770625062880855124731807853639784729441792*T3^32 - 4457432417931408792278788149146343258474758106121300992*T3^30 + 95557104046889536390272584088342527111720408249826800717568*T3^28 - 1677097604225078731586151524494095914341603062514483420279762944*T3^26 + 24029075318679508175378455877488304754436648475965559913437627882496*T3^24 - 279588274170542524289549276213161398190220871068731664879162734660694016*T3^22 + 2620941893297386673042570981442966720857920337803128666615707706073772564736*T3^20 - 19576706311547330276170341908494745762913658779354663302157435105996995452618752*T3^18 + 114786582686065236207883809191992392702194735183306214063991197142282015128344340480*T3^16 - 518040844729803650476715865052512438161906653859424659989227278831016700787361516945408*T3^14 + 1753864442497931950894080597984988168235729475193567576419717105844974176933892402680070144*T3^12 - 4308785108512032085764234669022560988371372215580652660222940778815984811427348917339632173056*T3^10 + 7362195819520658293317743427267452856450890090755552985788233562374614627018815690307216432431104*T3^8 - 8288016609812272859246922147524308928986025051782293544782742061165055423068926719315699707808841728*T3^6 + 5707009471181846176088219998982877028502231264235955855409963615026614686456564594706724180808608776192*T3^4 - 2128554900896948064099292968049557300422864181450601903541220813368542954978682937661258940932724839415808*T3^2 + 324876949549552388562352700875732783866153389374372562104571235745423039265344134497175832147827485665394688