Properties

Label 50.6.d.b
Level $50$
Weight $6$
Character orbit 50.d
Analytic conductor $8.019$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.01919099065\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28q + 28q^{2} + 7q^{3} - 112q^{4} - 145q^{5} - 28q^{6} - 236q^{7} + 448q^{8} - 446q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 28q^{2} + 7q^{3} - 112q^{4} - 145q^{5} - 28q^{6} - 236q^{7} + 448q^{8} - 446q^{9} - 220q^{10} - 389q^{11} + 192q^{12} + 1047q^{13} + 164q^{14} - 640q^{15} - 1792q^{16} + 909q^{17} - 11576q^{18} + 260q^{19} - 1040q^{20} - 1849q^{21} + 556q^{22} + 817q^{23} + 2432q^{24} + 16055q^{25} - 4528q^{26} + 1675q^{27} + 2544q^{28} - 2445q^{29} - 5440q^{30} + 16731q^{31} - 28672q^{32} + 29629q^{33} - 7576q^{34} - 825q^{35} - 7136q^{36} + 39819q^{37} + 3100q^{38} + 26008q^{39} + 10231q^{41} + 7396q^{42} - 86148q^{43} - 2224q^{44} - 7870q^{45} - 11708q^{46} + 10019q^{47} + 3072q^{48} + 147076q^{49} + 16520q^{50} - 108074q^{51} + 16752q^{52} + 74512q^{53} + 4260q^{54} + 2735q^{55} - 10176q^{56} - 176840q^{57} + 9780q^{58} - 13430q^{59} + 5760q^{60} - 29774q^{61} + 63956q^{62} - 206818q^{63} - 28672q^{64} + 51235q^{65} + 23264q^{66} - 85061q^{67} - 89696q^{68} - 185227q^{69} - 25260q^{70} + 148501q^{71} + 64064q^{72} + 189082q^{73} + 305304q^{74} + 93630q^{75} + 16480q^{76} + 127243q^{77} + 32028q^{78} + 46930q^{79} + 39680q^{80} - 179812q^{81} - 338864q^{82} + 66807q^{83} - 34464q^{84} - 877380q^{85} + 121332q^{86} - 226000q^{87} + 24896q^{88} - 347190q^{89} + 628660q^{90} + 197001q^{91} + 46832q^{92} - 64906q^{93} - 40076q^{94} + 85970q^{95} - 7168q^{96} + 502364q^{97} + 185056q^{98} + 1151018q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 3.23607 2.35114i −9.01407 + 27.7425i 4.94427 15.2169i −53.0123 17.7397i 36.0563 + 110.970i 66.5258 −19.7771 60.8676i −491.799 357.313i −213.260 + 67.2326i
11.2 3.23607 2.35114i −4.67506 + 14.3884i 4.94427 15.2169i 27.5965 + 48.6152i 18.7003 + 57.5535i −181.803 −19.7771 60.8676i 11.4223 + 8.29879i 203.605 + 92.4389i
11.3 3.23607 2.35114i −4.00985 + 12.3411i 4.94427 15.2169i 32.6779 45.3558i 16.0394 + 49.3642i 24.1971 −19.7771 60.8676i 60.3684 + 43.8602i −0.889977 223.605i
11.4 3.23607 2.35114i −0.839907 + 2.58497i 4.94427 15.2169i −19.5514 + 52.3712i 3.35963 + 10.3399i 200.961 −19.7771 60.8676i 190.615 + 138.490i 59.8626 + 215.445i
11.5 3.23607 2.35114i 1.57970 4.86182i 4.94427 15.2169i −52.9218 18.0078i −6.31881 19.4473i −251.763 −19.7771 60.8676i 175.449 + 127.471i −213.597 + 66.1524i
11.6 3.23607 2.35114i 5.47253 16.8427i 4.94427 15.2169i 53.8479 15.0136i −21.8901 67.3709i −2.11162 −19.7771 60.8676i −57.1374 41.5128i 138.956 175.189i
11.7 3.23607 2.35114i 8.20551 25.2540i 4.94427 15.2169i −55.6327 5.47754i −32.8220 101.016i 129.715 −19.7771 60.8676i −373.841 271.611i −192.910 + 113.075i
21.1 −1.23607 + 3.80423i −23.0554 16.7507i −12.9443 9.40456i −50.3632 24.2601i 92.2216 67.0029i 20.3938 51.7771 37.6183i 175.874 + 541.284i 154.543 161.606i
21.2 −1.23607 + 3.80423i −14.1625 10.2897i −12.9443 9.40456i 55.6267 + 5.53762i 56.6502 41.1588i −46.1918 51.7771 37.6183i 19.6088 + 60.3498i −89.8248 + 204.772i
21.3 −1.23607 + 3.80423i −7.20148 5.23218i −12.9443 9.40456i −5.94028 + 55.5852i 28.8059 20.9287i −16.8909 51.7771 37.6183i −50.6055 155.748i −204.116 91.3052i
21.4 −1.23607 + 3.80423i 4.87185 + 3.53960i −12.9443 9.40456i −25.7617 49.6118i −19.4874 + 14.1584i 192.464 51.7771 37.6183i −63.8850 196.618i 220.578 36.6799i
21.5 −1.23607 + 3.80423i 6.10314 + 4.43419i −12.9443 9.40456i 17.4261 53.1162i −24.4126 + 17.7368i −227.988 51.7771 37.6183i −57.5049 176.982i 180.526 + 131.948i
21.6 −1.23607 + 3.80423i 17.4616 + 12.6866i −12.9443 9.40456i 53.8689 + 14.9378i −69.8465 + 50.7465i 157.525 51.7771 37.6183i 68.8673 + 211.952i −123.412 + 186.465i
21.7 −1.23607 + 3.80423i 22.7640 + 16.5390i −12.9443 9.40456i −50.3607 + 24.2652i −91.0559 + 66.1560i −183.034 51.7771 37.6183i 169.569 + 521.880i −30.0609 221.577i
31.1 −1.23607 3.80423i −23.0554 + 16.7507i −12.9443 + 9.40456i −50.3632 + 24.2601i 92.2216 + 67.0029i 20.3938 51.7771 + 37.6183i 175.874 541.284i 154.543 + 161.606i
31.2 −1.23607 3.80423i −14.1625 + 10.2897i −12.9443 + 9.40456i 55.6267 5.53762i 56.6502 + 41.1588i −46.1918 51.7771 + 37.6183i 19.6088 60.3498i −89.8248 204.772i
31.3 −1.23607 3.80423i −7.20148 + 5.23218i −12.9443 + 9.40456i −5.94028 55.5852i 28.8059 + 20.9287i −16.8909 51.7771 + 37.6183i −50.6055 + 155.748i −204.116 + 91.3052i
31.4 −1.23607 3.80423i 4.87185 3.53960i −12.9443 + 9.40456i −25.7617 + 49.6118i −19.4874 14.1584i 192.464 51.7771 + 37.6183i −63.8850 + 196.618i 220.578 + 36.6799i
31.5 −1.23607 3.80423i 6.10314 4.43419i −12.9443 + 9.40456i 17.4261 + 53.1162i −24.4126 17.7368i −227.988 51.7771 + 37.6183i −57.5049 + 176.982i 180.526 131.948i
31.6 −1.23607 3.80423i 17.4616 12.6866i −12.9443 + 9.40456i 53.8689 14.9378i −69.8465 50.7465i 157.525 51.7771 + 37.6183i 68.8673 211.952i −123.412 186.465i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.6.d.b 28
25.d even 5 1 inner 50.6.d.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.d.b 28 1.a even 1 1 trivial
50.6.d.b 28 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(48\!\cdots\!25\)\( T_{3}^{19} + \)\(26\!\cdots\!10\)\( T_{3}^{18} - \)\(93\!\cdots\!20\)\( T_{3}^{17} + \)\(10\!\cdots\!80\)\( T_{3}^{16} - \)\(85\!\cdots\!65\)\( T_{3}^{15} + \)\(32\!\cdots\!25\)\( T_{3}^{14} - \)\(14\!\cdots\!15\)\( T_{3}^{13} + \)\(69\!\cdots\!80\)\( T_{3}^{12} - \)\(23\!\cdots\!45\)\( T_{3}^{11} + \)\(55\!\cdots\!85\)\( T_{3}^{10} - \)\(46\!\cdots\!50\)\( T_{3}^{9} + \)\(44\!\cdots\!05\)\( T_{3}^{8} - \)\(99\!\cdots\!10\)\( T_{3}^{7} + \)\(28\!\cdots\!40\)\( T_{3}^{6} - \)\(27\!\cdots\!20\)\( T_{3}^{5} + \)\(21\!\cdots\!00\)\( T_{3}^{4} - \)\(82\!\cdots\!24\)\( T_{3}^{3} + \)\(29\!\cdots\!68\)\( T_{3}^{2} - \)\(26\!\cdots\!52\)\( T_{3} + \)\(21\!\cdots\!76\)\( \)">\(T_{3}^{28} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(50, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database