# Properties

 Label 29.13.c.a Level $29$ Weight $13$ Character orbit 29.c Analytic conductor $26.506$ Analytic rank $0$ Dimension $58$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 29.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5058207010$$ Analytic rank: $$0$$ Dimension: $$58$$ Relative dimension: $$29$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$58q + 88q^{2} - 2q^{3} - 4q^{7} + 79650q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$58q + 88q^{2} - 2q^{3} - 4q^{7} + 79650q^{8} - 1957890q^{10} + 4120990q^{11} + 2920062q^{12} - 1824520q^{14} - 8383600q^{15} - 133743512q^{16} + 33971578q^{17} - 122384158q^{18} + 65838718q^{19} - 59408388q^{20} + 200896236q^{21} + 104539676q^{23} + 163907064q^{24} - 3086882294q^{25} + 607848030q^{26} - 1190867840q^{27} + 817714294q^{29} + 5793833612q^{30} - 1059975938q^{31} + 2323254598q^{32} + 517001400q^{36} - 864725342q^{37} + 18048639408q^{39} - 22547920086q^{40} - 17292603926q^{41} - 3344004962q^{43} - 53750811886q^{44} - 16067938640q^{45} + 43310099300q^{46} - 15159905282q^{47} - 4602803862q^{48} + 32036753022q^{49} - 16057299278q^{50} + 81167587800q^{52} - 69552844564q^{53} + 38996274808q^{54} + 3944882736q^{55} - 156397031424q^{56} + 107434998568q^{58} + 82613255468q^{59} - 147410252946q^{60} + 128229759922q^{61} + 125938412928q^{65} + 364716671994q^{66} - 141670411468q^{68} + 529640675916q^{69} + 518962441956q^{70} - 180699442320q^{72} - 428225274062q^{73} + 307721180948q^{74} - 617987210610q^{75} - 455232145048q^{76} - 963484794004q^{77} + 688403957040q^{78} - 183006289538q^{79} + 1001949265154q^{81} - 1176460419184q^{82} + 361042835756q^{83} - 402324805420q^{84} + 832273178976q^{85} - 1065344596322q^{87} - 1836857960940q^{88} + 1922736257242q^{89} - 1170237151648q^{90} - 2759662014220q^{94} + 5518358548560q^{95} + 1356111950818q^{97} - 2518255928616q^{98} + 3259343912178q^{99} + O(q^{100})$$

## 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 −88.1500 88.1500i 347.445 + 347.445i 11444.8i 14434.6i 61254.6i −35450.1 647800. 647800.i 290005.i 1.27241e6 1.27241e6i
12.2 −79.7284 79.7284i −520.777 520.777i 8617.22i 29504.0i 83041.4i −164320. 360470. 360470.i 10976.2i −2.35231e6 + 2.35231e6i
12.3 −76.2293 76.2293i −132.533 132.533i 7525.81i 2767.06i 20205.8i 132887. 261452. 261452.i 496311.i −210931. + 210931.i
12.4 −70.5170 70.5170i −767.125 767.125i 5849.31i 9904.63i 108191.i 71097.8 123638. 123638.i 645520.i 698445. 698445.i
12.5 −69.9680 69.9680i 814.355 + 814.355i 5695.05i 19209.4i 113958.i −6147.85 111882. 111882.i 794908.i −1.34405e6 + 1.34405e6i
12.6 −58.0743 58.0743i 433.237 + 433.237i 2649.26i 10278.3i 50319.9i −72219.2 −84018.7 + 84018.7i 156053.i 596905. 596905.i
12.7 −54.5113 54.5113i −434.460 434.460i 1846.95i 19548.9i 47365.9i −201373. −122598. + 122598.i 153930.i 1.06564e6 1.06564e6i
12.8 −44.8874 44.8874i 80.4158 + 80.4158i 66.2505i 12643.6i 7219.30i 209784. −186832. + 186832.i 518508.i −567536. + 567536.i
12.9 −44.4183 44.4183i 834.913 + 834.913i 150.028i 28014.5i 74170.9i 122123. −188601. + 188601.i 862719.i 1.24436e6 1.24436e6i
12.10 −33.9863 33.9863i −37.2586 37.2586i 1785.87i 14007.5i 2532.56i −78700.5 −199903. + 199903.i 528665.i −476064. + 476064.i
12.11 −29.6845 29.6845i −878.412 878.412i 2333.66i 15619.8i 52150.4i 10196.8 −190861. + 190861.i 1.01178e6i −463667. + 463667.i
12.12 −15.4217 15.4217i −236.320 236.320i 3620.34i 24342.5i 7288.92i 119306. −118999. + 118999.i 419746.i 375403. 375403.i
12.13 −12.2710 12.2710i 762.089 + 762.089i 3794.84i 6703.86i 18703.2i −134602. −96828.8 + 96828.8i 630118.i −82263.3 + 82263.3i
12.14 −9.03178 9.03178i −704.167 704.167i 3932.85i 8770.94i 12719.8i 37099.6 −72514.8 + 72514.8i 460261.i 79217.2 79217.2i
12.15 2.50807 + 2.50807i 773.536 + 773.536i 4083.42i 10269.1i 3880.17i 192386. 20514.6 20514.6i 665275.i 25755.5 25755.5i
12.16 8.76721 + 8.76721i 327.395 + 327.395i 3942.27i 12032.8i 5740.68i −63314.5 70473.2 70473.2i 317066.i −105494. + 105494.i
12.17 18.1237 + 18.1237i −105.386 105.386i 3439.06i 24861.6i 3819.95i 4681.32 136563. 136563.i 509229.i 450583. 450583.i
12.18 26.4284 + 26.4284i −670.941 670.941i 2699.08i 5810.81i 35463.8i −231201. 179583. 179583.i 368882.i −153570. + 153570.i
12.19 33.1274 + 33.1274i −631.679 631.679i 1901.16i 13469.1i 41851.7i 113718. 198670. 198670.i 266596.i 446197. 446197.i
12.20 43.6471 + 43.6471i −94.5951 94.5951i 285.862i 15326.9i 8257.61i 49832.3 191256. 191256.i 513545.i −668974. + 668974.i
See all 58 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.29 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 29.13.c.a 58
29.c odd 4 1 inner 29.13.c.a 58

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.13.c.a 58 1.a even 1 1 trivial
29.13.c.a 58 29.c odd 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{13}^{\mathrm{new}}(29, [\chi])$$.