Properties

Label 29.15.c.a
Level $29$
Weight $15$
Character orbit 29.c
Analytic conductor $36.055$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.0554007641\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68q - 312q^{2} - 2q^{3} - 4q^{7} - 689310q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 68q - 312q^{2} - 2q^{3} - 4q^{7} - 689310q^{8} + 23502846q^{10} - 2993734q^{11} - 76269906q^{12} - 3845224q^{14} + 277690070q^{15} - 5490752792q^{16} + 285786056q^{17} + 5809842386q^{18} - 1195066336q^{19} + 1866268668q^{20} - 8197524756q^{21} + 2117392192q^{23} + 8629372824q^{24} - 73846917196q^{25} - 16368356994q^{26} + 33411191086q^{27} + 48687460392q^{29} + 128044102700q^{30} + 73968522614q^{31} - 2657032122q^{32} - 259972090824q^{36} + 95888936640q^{37} - 571710579738q^{39} + 977850700426q^{40} - 57594847104q^{41} + 48472463810q^{43} + 1173476843650q^{44} - 299491373708q^{45} + 656204001636q^{46} + 29961288922q^{47} + 1808198535114q^{48} + 9857850529980q^{49} + 1443642384290q^{50} - 11263919114280q^{52} - 1993070689076q^{53} + 2064324525592q^{54} + 3054165001846q^{55} + 8002123380864q^{56} - 9170547007720q^{58} - 8402401993912q^{59} + 4455428077662q^{60} - 4381209993964q^{61} - 14884429709724q^{65} - 5756218265814q^{66} + 4595908790532q^{68} + 51089269002600q^{69} - 65383337180236q^{70} + 101900024607216q^{72} + 39493186331224q^{73} - 152862151734316q^{74} - 46335428712972q^{75} + 46232026918072q^{76} + 63231072283300q^{77} + 111617680995888q^{78} - 29034273461086q^{79} - 345331621902328q^{81} + 104609665443600q^{82} - 2994621113016q^{83} + 269240332456580q^{84} + 11907997971872q^{85} - 148747542169982q^{87} + 186485775340436q^{88} - 89923791148548q^{89} + 103388070190448q^{90} - 920451476162284q^{94} - 393920660173420q^{95} - 116095608365672q^{97} + 24492650399928q^{98} - 402079041111864q^{99} + O(q^{100}) \)

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 −168.771 168.771i −2665.63 2665.63i 40583.2i 37437.0i 899760.i 644600. 4.08412e6 4.08412e6i 9.42815e6i −6.31827e6 + 6.31827e6i
12.2 −168.212 168.212i 1372.65 + 1372.65i 40206.7i 110532.i 461791.i 806243. 4.00726e6 4.00726e6i 1.01466e6i −1.85928e7 + 1.85928e7i
12.3 −165.801 165.801i −544.841 544.841i 38595.9i 15745.6i 180670.i −1.09500e6 3.68274e6 3.68274e6i 4.18927e6i 2.61063e6 2.61063e6i
12.4 −157.774 157.774i 3020.51 + 3020.51i 33401.0i 76455.0i 953113.i −1.19070e6 2.68483e6 2.68483e6i 1.34640e7i 1.20626e7 1.20626e7i
12.5 −152.203 152.203i −362.559 362.559i 29947.3i 149475.i 110365.i 876221. 2.06437e6 2.06437e6i 4.52007e6i 2.27505e7 2.27505e7i
12.6 −126.672 126.672i 1067.39 + 1067.39i 15707.8i 7103.16i 270419.i −354580. −85651.6 + 85651.6i 2.50431e6i −899774. + 899774.i
12.7 −120.665 120.665i −1296.13 1296.13i 12736.1i 36154.9i 312796.i −444023. −440176. + 440176.i 1.42305e6i −4.36263e6 + 4.36263e6i
12.8 −118.027 118.027i 1825.31 + 1825.31i 11476.7i 35591.1i 430872.i 1.00122e6 −579197. + 579197.i 1.88056e6i 4.20070e6 4.20070e6i
12.9 −99.0246 99.0246i −2875.09 2875.09i 3227.73i 105139.i 569409.i −619848. −1.30279e6 + 1.30279e6i 1.17493e7i 1.04113e7 1.04113e7i
12.10 −94.2182 94.2182i −1537.46 1537.46i 1370.15i 119266.i 289713.i 1.40104e6 −1.41458e6 + 1.41458e6i 55424.1i −1.12371e7 + 1.12371e7i
12.11 −84.8508 84.8508i 1293.97 + 1293.97i 1984.68i 144289.i 219589.i −1.06721e6 −1.55860e6 + 1.55860e6i 1.43426e6i −1.22430e7 + 1.22430e7i
12.12 −59.1433 59.1433i −1441.85 1441.85i 9388.15i 74847.5i 170552.i 888867. −1.52425e6 + 1.52425e6i 625088.i 4.42672e6 4.42672e6i
12.13 −55.9098 55.9098i 1465.84 + 1465.84i 10132.2i 40470.7i 163909.i 363644. −1.48251e6 + 1.48251e6i 485606.i 2.26271e6 2.26271e6i
12.14 −50.6787 50.6787i 910.227 + 910.227i 11247.3i 127648.i 92258.2i −1.54934e6 −1.40032e6 + 1.40032e6i 3.12594e6i 6.46904e6 6.46904e6i
12.15 −48.4316 48.4316i 2910.35 + 2910.35i 11692.8i 56568.4i 281906.i 166969. −1.35980e6 + 1.35980e6i 1.21573e7i −2.73969e6 + 2.73969e6i
12.16 −34.5470 34.5470i −634.913 634.913i 13997.0i 29356.7i 43868.6i 160670. −1.04957e6 + 1.04957e6i 3.97674e6i 1.01419e6 1.01419e6i
12.17 −26.8284 26.8284i −2568.33 2568.33i 14944.5i 116825.i 137808.i −1.09289e6 −840493. + 840493.i 8.40968e6i −3.13424e6 + 3.13424e6i
12.18 12.2697 + 12.2697i −17.2708 17.2708i 16082.9i 67606.8i 423.814i −248594. 398359. 398359.i 4.78237e6i 829515. 829515.i
12.19 13.7439 + 13.7439i 888.946 + 888.946i 16006.2i 102078.i 24435.1i 1.24811e6 445167. 445167.i 3.20252e6i 1.40294e6 1.40294e6i
12.20 18.0493 + 18.0493i −1757.84 1757.84i 15732.4i 22055.5i 63455.4i −823925. 579679. 579679.i 1.39700e6i −398086. + 398086.i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.34
Significant digits:
Format:

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 29.15.c.a 68
29.c odd 4 1 inner 29.15.c.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.15.c.a 68 1.a even 1 1 trivial
29.15.c.a 68 29.c odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(29, [\chi])\).