Properties

Label 37.9.d.a
Level $37$
Weight $9$
Character orbit 37.d
Analytic conductor $15.073$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 37.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.0730085723\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 50q + 4q^{2} + 124q^{5} + 510q^{6} - 4q^{7} - 5256q^{8} - 93534q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 50q + 4q^{2} + 124q^{5} + 510q^{6} - 4q^{7} - 5256q^{8} - 93534q^{9} - 10756q^{10} - 45440q^{12} + 63592q^{13} - 80206q^{14} - 124346q^{15} - 639128q^{16} - 178796q^{17} - 45882q^{18} + 264774q^{19} + 469204q^{20} + 714574q^{22} - 654272q^{23} - 1573644q^{24} + 1841660q^{26} - 1683686q^{29} + 1766336q^{31} - 4332632q^{32} - 1438888q^{33} - 5838452q^{34} - 650482q^{35} - 2870078q^{37} - 2281528q^{38} + 11036546q^{39} - 7453902q^{42} + 5097830q^{43} - 14992464q^{44} - 15078446q^{45} + 28776464q^{46} + 13992320q^{47} + 4370314q^{49} + 7603288q^{50} + 19417938q^{51} + 54318452q^{52} + 28075952q^{53} - 19197122q^{54} - 6104306q^{55} + 47803128q^{56} + 17462426q^{57} - 677072q^{59} + 24953316q^{60} + 4206386q^{61} - 18098032q^{63} - 56061730q^{66} - 52579760q^{68} + 88118984q^{69} - 295493636q^{70} - 37172476q^{71} - 74582504q^{72} - 224754524q^{74} - 144364996q^{75} + 189294548q^{76} + 271163438q^{79} - 312998084q^{80} + 41535626q^{81} - 177494710q^{82} - 74250268q^{83} - 82088924q^{84} + 112964036q^{86} - 142771908q^{87} + 224951860q^{88} + 84928780q^{89} + 716592116q^{90} - 40969098q^{91} - 701292428q^{92} - 83761576q^{93} + 553881406q^{94} - 7259376q^{96} + 467792674q^{97} + 538649310q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −21.1416 + 21.1416i 117.640i 637.931i 228.288 + 228.288i 2487.09 + 2487.09i 1007.48 8074.63 + 8074.63i −7278.14 −9652.73
6.2 −20.8978 + 20.8978i 69.3010i 617.432i −503.319 503.319i −1448.23 1448.23i −665.450 7553.12 + 7553.12i 1758.37 21036.5
6.3 −18.0071 + 18.0071i 19.6784i 392.513i 258.631 + 258.631i −354.351 354.351i 308.826 2458.20 + 2458.20i 6173.76 −9314.40
6.4 −17.3562 + 17.3562i 155.578i 346.477i 713.128 + 713.128i −2700.25 2700.25i 792.085 1570.34 + 1570.34i −17643.6 −24754.4
6.5 −15.5660 + 15.5660i 38.2524i 228.599i 431.867 + 431.867i 595.436 + 595.436i −3114.53 −426.526 426.526i 5097.75 −13444.9
6.6 −13.8808 + 13.8808i 94.2533i 129.356i −267.305 267.305i 1308.32 + 1308.32i 3639.81 −1757.93 1757.93i −2322.69 7420.84
6.7 −13.4597 + 13.4597i 99.3332i 106.325i −869.321 869.321i 1336.99 + 1336.99i −3726.79 −2014.58 2014.58i −3306.09 23401.5
6.8 −12.0383 + 12.0383i 50.1945i 33.8405i −357.237 357.237i −604.256 604.256i 3070.05 −2674.42 2674.42i 4041.51 8601.04
6.9 −7.80165 + 7.80165i 110.709i 134.269i −218.780 218.780i −863.711 863.711i −1962.21 −3044.74 3044.74i −5695.45 3413.69
6.10 −6.40563 + 6.40563i 7.09124i 173.936i 773.631 + 773.631i −45.4238 45.4238i 1906.19 −2754.01 2754.01i 6510.71 −9911.18
6.11 −5.51546 + 5.51546i 143.656i 195.159i 357.883 + 357.883i 792.331 + 792.331i −623.716 −2488.35 2488.35i −14076.1 −3947.78
6.12 −2.21071 + 2.21071i 26.6774i 246.226i 1.29434 + 1.29434i 58.9759 + 58.9759i −2354.04 −1110.28 1110.28i 5849.32 −5.72281
6.13 2.06067 2.06067i 90.4071i 247.507i −149.663 149.663i −186.299 186.299i 2623.40 1037.56 + 1037.56i −1612.44 −616.811
6.14 2.65225 2.65225i 19.9229i 241.931i −726.637 726.637i −52.8405 52.8405i 711.744 1320.64 + 1320.64i 6164.08 −3854.44
6.15 3.37322 3.37322i 108.831i 233.243i 154.950 + 154.950i 367.112 + 367.112i 3840.48 1650.32 + 1650.32i −5283.27 1045.36
6.16 6.39144 6.39144i 92.4393i 174.299i 596.761 + 596.761i 590.820 + 590.820i −3526.08 2750.23 + 2750.23i −1984.02 7628.33
6.17 10.0496 10.0496i 127.375i 54.0099i −863.145 863.145i 1280.08 + 1280.08i −165.087 3115.48 + 3115.48i −9663.48 −17348.6
6.18 11.2848 11.2848i 11.8532i 1.30831i −14.3499 14.3499i 133.760 + 133.760i −997.164 2903.66 + 2903.66i 6420.50 −323.870
6.19 11.7867 11.7867i 78.9085i 21.8525i 740.066 + 740.066i −930.071 930.071i 1535.91 2759.83 + 2759.83i 334.442 17445.9
6.20 11.9313 11.9313i 129.954i 28.7121i −176.820 176.820i −1550.52 1550.52i −2761.63 2711.84 + 2711.84i −10327.0 −4219.39
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.9.d.a 50
37.d odd 4 1 inner 37.9.d.a 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.9.d.a 50 1.a even 1 1 trivial
37.9.d.a 50 37.d odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(37, [\chi])\).