# Properties

 Label 29.9.c.a Level $29$ Weight $9$ Character orbit 29.c Analytic conductor $11.814$ Analytic rank $0$ Dimension $38$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 29.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8139796918$$ Analytic rank: $$0$$ Dimension: $$38$$ Relative dimension: $$19$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$38q - 8q^{2} - 2q^{3} - 4q^{7} - 3678q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$38q - 8q^{2} - 2q^{3} - 4q^{7} - 3678q^{8} + 15870q^{10} - 8390q^{11} + 5022q^{12} + 12728q^{14} - 35530q^{15} - 245912q^{16} + 76234q^{17} - 465022q^{18} - 164488q^{19} + 4092q^{20} - 514620q^{21} + 733568q^{23} - 482376q^{24} - 1528354q^{25} + 1405470q^{26} - 1279754q^{27} + 1420996q^{29} + 3659852q^{30} + 291062q^{31} + 1592134q^{32} - 10223688q^{36} + 7994574q^{37} - 309930q^{39} + 9093994q^{40} + 9561658q^{41} - 5181022q^{43} - 11400526q^{44} + 9526060q^{45} - 32561820q^{46} + 5445034q^{47} - 17683542q^{48} + 45106458q^{49} + 3122002q^{50} + 56198616q^{52} - 25331452q^{53} - 42307016q^{54} + 3037606q^{55} - 25661184q^{56} - 33846328q^{58} + 27333008q^{59} + 30592014q^{60} - 49572406q^{61} - 121974972q^{65} + 133443354q^{66} + 21192980q^{68} - 28739184q^{69} - 17976124q^{70} - 186114192q^{72} + 45787870q^{73} + 91919252q^{74} - 159074460q^{75} + 243795016q^{76} + 59851916q^{77} + 163580208q^{78} - 99827470q^{79} - 75138586q^{81} - 18178416q^{82} - 177270160q^{83} - 9436684q^{84} + 156307616q^{85} - 20584526q^{87} + 563433236q^{88} - 388451858q^{89} + 306402752q^{90} + 81569204q^{94} - 44386260q^{95} - 121157834q^{97} - 129908712q^{98} - 216447384q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1 −21.4922 21.4922i −26.9761 26.9761i 667.831i 209.913i 1159.55i 2370.17 8851.17 8851.17i 5105.58i −4511.49 + 4511.49i
12.2 −18.8438 18.8438i 49.8418 + 49.8418i 454.181i 1116.67i 1878.42i −1871.68 3734.50 3734.50i 1592.59i 21042.4 21042.4i
12.3 −16.0506 16.0506i −92.3210 92.3210i 259.242i 266.052i 2963.61i −4134.50 52.0309 52.0309i 10485.3i −4270.28 + 4270.28i
12.4 −15.3962 15.3962i 5.11102 + 5.11102i 218.083i 625.365i 157.380i −974.923 −583.770 + 583.770i 6508.75i −9628.22 + 9628.22i
12.5 −15.2859 15.2859i 93.1254 + 93.1254i 211.315i 424.754i 2847.00i 996.548 −683.041 + 683.041i 10783.7i −6492.74 + 6492.74i
12.6 −12.8569 12.8569i −84.9422 84.9422i 74.6002i 733.457i 2184.19i 4371.72 −2332.24 + 2332.24i 7869.35i 9429.98 9429.98i
12.7 −6.63336 6.63336i −21.3611 21.3611i 167.997i 656.237i 283.392i −2102.40 −2812.53 + 2812.53i 5648.41i 4353.06 4353.06i
12.8 −6.50125 6.50125i 25.7599 + 25.7599i 171.468i 37.7478i 334.943i 1420.11 −2779.07 + 2779.07i 5233.85i 245.408 245.408i
12.9 −3.03267 3.03267i −59.6889 59.6889i 237.606i 1116.88i 362.033i 2170.16 −1496.94 + 1496.94i 564.525i −3387.14 + 3387.14i
12.10 0.384989 + 0.384989i 93.7530 + 93.7530i 255.704i 802.752i 72.1877i 2726.98 197.000 197.000i 11018.2i −309.050 + 309.050i
12.11 0.492724 + 0.492724i 72.5537 + 72.5537i 255.514i 679.404i 71.4979i −4572.34 252.036 252.036i 3967.08i 334.759 334.759i
12.12 4.53287 + 4.53287i −83.7812 83.7812i 214.906i 3.13649i 759.538i −1568.18 2134.56 2134.56i 7477.58i 14.2173 14.2173i
12.13 8.30345 + 8.30345i −37.9600 37.9600i 118.106i 774.671i 630.398i −18.6293 3106.37 3106.37i 3679.07i −6432.44 + 6432.44i
12.14 9.93858 + 9.93858i 27.3327 + 27.3327i 58.4494i 447.536i 543.297i 2077.42 3125.18 3125.18i 5066.84i 4447.87 4447.87i
12.15 13.6258 + 13.6258i 70.4012 + 70.4012i 115.327i 729.288i 1918.55i −3525.11 1916.79 1916.79i 3351.66i −9937.16 + 9937.16i
12.16 17.5030 + 17.5030i −100.063 100.063i 356.708i 183.024i 3502.79i 3415.09 −1762.69 + 1762.69i 13464.1i −3203.47 + 3203.47i
12.17 17.9289 + 17.9289i −42.9110 42.9110i 386.891i 832.826i 1538.69i −3297.00 −2346.74 + 2346.74i 2878.30i 14931.7 14931.7i
12.18 19.3086 + 19.3086i 104.909 + 104.909i 489.642i 703.899i 4051.27i 1720.21 −4511.30 + 4511.30i 15450.6i 13591.3 13591.3i
12.19 20.0740 + 20.0740i 6.21683 + 6.21683i 549.928i 611.920i 249.593i 794.354 −5900.31 + 5900.31i 6483.70i −12283.7 + 12283.7i
17.1 −21.4922 + 21.4922i −26.9761 + 26.9761i 667.831i 209.913i 1159.55i 2370.17 8851.17 + 8851.17i 5105.58i −4511.49 4511.49i
See all 38 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.19 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.9.c.a 38
29.c odd 4 1 inner 29.9.c.a 38

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.9.c.a 38 1.a even 1 1 trivial
29.9.c.a 38 29.c odd 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(29, [\chi])$$.