Properties

Label 29.6.d.a
Level $29$
Weight $6$
Character orbit 29.d
Analytic conductor $4.651$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.65113077458\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66q - 11q^{2} - 5q^{3} - 171q^{4} + 29q^{5} + 445q^{6} + 17q^{7} + 1046q^{8} - 732q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 66q - 11q^{2} - 5q^{3} - 171q^{4} + 29q^{5} + 445q^{6} + 17q^{7} + 1046q^{8} - 732q^{9} - 141q^{10} + 1125q^{11} - 248q^{12} - 1789q^{13} + 1233q^{14} + 1061q^{15} + 2629q^{16} + 676q^{17} - 10896q^{18} - 1903q^{19} + 4081q^{20} - 439q^{21} - 2980q^{22} + 9945q^{23} + 10741q^{24} - 22114q^{25} - 1641q^{26} - 4427q^{27} + 7508q^{28} - 13100q^{29} - 40018q^{30} + 8541q^{31} + 18996q^{32} + 20725q^{33} - 8839q^{34} + 14127q^{35} + 70172q^{36} + 3791q^{37} + 36322q^{38} + 25843q^{39} - 72621q^{40} - 18720q^{41} - 72614q^{42} - 10419q^{43} + 35796q^{44} + 23163q^{45} + 15848q^{46} - 40549q^{47} - 4194q^{48} + 19808q^{49} + 201932q^{50} + 51998q^{51} + 8330q^{52} - 72275q^{53} - 197108q^{54} + 42183q^{55} + 31599q^{56} - 40162q^{57} - 258031q^{58} - 333512q^{59} + 26920q^{60} - 175281q^{61} + 213745q^{62} + 354423q^{63} - 19790q^{64} - 79593q^{65} + 103783q^{66} + 135179q^{67} + 334644q^{68} + 73673q^{69} - 511678q^{70} - 187563q^{71} - 17551q^{72} - 56675q^{73} + 159980q^{74} + 538208q^{75} - 215041q^{76} + 106099q^{77} + 37481q^{78} + 273775q^{79} + 571652q^{80} - 61356q^{81} - 37092q^{82} - 237831q^{83} + 127523q^{84} - 148844q^{85} - 348400q^{86} - 110493q^{87} - 985638q^{88} + 74239q^{89} - 630714q^{90} + 100309q^{91} + 132694q^{92} - 307921q^{93} + 206074q^{94} + 37965q^{95} + 687094q^{96} + 136055q^{97} - 115704q^{98} + 438608q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.41509 10.5812i 0.690047 + 0.332309i −77.2985 + 37.2250i 19.1233 + 83.7844i 1.84971 8.10409i −158.450 76.3053i 364.027 + 456.475i −151.142 189.526i 840.357 404.695i
7.2 −1.89325 8.29487i 14.6426 + 7.05150i −36.3895 + 17.5242i −19.7959 86.7313i 30.7692 134.809i 23.9398 + 11.5288i 44.5031 + 55.8051i 13.1736 + 16.5192i −681.946 + 328.408i
7.3 −1.73099 7.58398i −20.8543 10.0429i −25.6894 + 12.3713i −4.14155 18.1453i −40.0664 + 175.543i 89.9242 + 43.3052i −16.9122 21.2072i 182.533 + 228.889i −130.445 + 62.8189i
7.4 −1.10867 4.85738i 13.3571 + 6.43244i 6.46597 3.11385i 14.7871 + 64.7867i 16.4363 72.0120i 140.749 + 67.7811i −121.699 152.606i −14.4722 18.1476i 298.300 143.654i
7.5 −0.594851 2.60621i −6.88516 3.31572i 22.3925 10.7837i −1.88513 8.25929i −4.54582 + 19.9165i −133.434 64.2584i −94.7603 118.826i −115.097 144.327i −20.4041 + 9.82609i
7.6 0.402095 + 1.76169i 22.1010 + 10.6433i 25.8891 12.4675i −5.52960 24.2268i −9.86349 + 43.2148i −77.8191 37.4757i 68.4265 + 85.8041i 223.667 + 280.469i 40.4567 19.4829i
7.7 0.522777 + 2.29044i −22.0166 10.6026i 23.8582 11.4895i 18.8051 + 82.3907i 12.7749 55.9704i 123.108 + 59.2858i 85.6617 + 107.416i 220.807 + 276.883i −178.880 + 86.1439i
7.8 0.813916 + 3.56600i −6.23245 3.00139i 16.7771 8.07943i −22.0176 96.4653i 5.63027 24.6678i 166.222 + 80.0483i 115.444 + 144.762i −121.673 152.573i 326.075 157.029i
7.9 1.27150 + 5.57079i 5.91325 + 2.84767i −0.585948 + 0.282178i 13.3453 + 58.4697i −8.34510 + 36.5623i −1.93045 0.929655i 111.688 + 140.052i −124.651 156.307i −308.754 + 148.688i
7.10 1.95082 + 8.54710i −16.4262 7.91043i −40.4162 + 19.4634i −4.08126 17.8812i 35.5667 155.828i −139.508 67.1835i −70.2862 88.1361i 55.7366 + 69.8915i 144.870 69.7659i
7.11 2.19360 + 9.61081i 14.0872 + 6.78404i −58.7248 + 28.2804i 0.197313 + 0.864484i −34.2984 + 150.271i 69.1061 + 33.2798i −203.933 255.724i 0.918137 + 1.15131i −7.87556 + 3.79267i
16.1 −9.29727 4.47733i −16.9028 + 21.1954i 46.4410 + 58.2352i −29.3057 14.1129i 252.049 121.380i −125.979 + 157.972i −97.5569 427.425i −109.469 479.616i 209.275 + 262.422i
16.2 −9.01987 4.34374i 7.86720 9.86516i 42.5383 + 53.3413i 61.9634 + 29.8400i −113.813 + 54.8094i 56.6689 71.0605i −80.7019 353.578i 18.6441 + 81.6852i −429.284 538.306i
16.3 −6.13336 2.95367i 1.76173 2.20914i 8.94225 + 11.2132i −38.8583 18.7131i −17.3304 + 8.34587i 0.981087 1.23024i 26.7482 + 117.192i 52.2960 + 229.124i 183.059 + 229.549i
16.4 −3.39646 1.63565i −13.4299 + 16.8406i −11.0911 13.9078i 21.4922 + 10.3501i 73.1594 35.2317i 143.536 179.989i 41.7656 + 182.987i −49.1696 215.426i −56.0684 70.3076i
16.5 −3.07802 1.48230i 18.8759 23.6696i −12.6747 15.8935i 15.8039 + 7.61077i −93.1857 + 44.8759i −39.8314 + 49.9470i 39.7806 + 174.290i −149.879 656.661i −37.3634 46.8523i
16.6 −1.37713 0.663191i −4.45689 + 5.58876i −18.4950 23.1920i 71.8910 + 34.6209i 9.84412 4.74068i −153.020 + 191.881i 20.9732 + 91.8897i 42.7022 + 187.091i −76.0429 95.3548i
16.7 1.69866 + 0.818031i 5.27812 6.61856i −17.7354 22.2395i −69.8948 33.6596i 14.3799 6.92500i 10.7481 13.4777i −25.3589 111.105i 38.1259 + 167.040i −91.1929 114.352i
16.8 3.74798 + 1.80493i −13.9558 + 17.5000i −9.16211 11.4889i −40.2504 19.3836i −83.8924 + 40.4005i −30.7029 + 38.5002i −43.2242 189.378i −57.4139 251.547i −115.872 145.298i
16.9 3.98347 + 1.91834i 6.52648 8.18394i −7.76363 9.73529i 64.0881 + 30.8632i 41.6976 20.0805i 85.0547 106.655i −43.7334 191.609i 29.6906 + 130.083i 196.087 + 245.886i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.6.d.a 66
29.d even 7 1 inner 29.6.d.a 66
29.d even 7 1 841.6.a.i 33
29.e even 14 1 841.6.a.h 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.d.a 66 1.a even 1 1 trivial
29.6.d.a 66 29.d even 7 1 inner
841.6.a.h 33 29.e even 14 1
841.6.a.i 33 29.d even 7 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(29, [\chi])\).