Properties

Label 37.6.c.a
Level $37$
Weight $6$
Character orbit 37.c
Analytic conductor $5.934$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30q - 4q^{2} + 16q^{3} - 230q^{4} - 51q^{5} + 60q^{6} + 50q^{7} + 24q^{8} - 1085q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 30q - 4q^{2} + 16q^{3} - 230q^{4} - 51q^{5} + 60q^{6} + 50q^{7} + 24q^{8} - 1085q^{9} - 44q^{10} - 1556q^{11} + 708q^{12} - 888q^{13} + 1888q^{14} + 1020q^{15} - 4566q^{16} - 437q^{17} - 7302q^{18} + 4358q^{19} - 1204q^{20} + 2354q^{21} + 7958q^{22} + 2824q^{23} + 13824q^{24} - 620q^{25} - 17604q^{26} - 32q^{27} + 11414q^{28} + 16954q^{29} + 15994q^{30} - 4548q^{31} - 9148q^{32} - 680q^{33} - 4576q^{34} + 11606q^{35} + 5828q^{36} + 12449q^{37} - 84560q^{38} + 13468q^{39} - 23018q^{40} + 32319q^{41} + 26750q^{42} - 49916q^{43} + 12034q^{44} + 81730q^{45} + 5300q^{46} - 57476q^{47} - 14944q^{48} - 52069q^{49} - 19224q^{50} - 78336q^{51} + 1316q^{52} + 65784q^{53} + 72114q^{54} - 24742q^{55} - 81130q^{56} - 3762q^{57} - 73868q^{58} - 49372q^{59} - 288608q^{60} + 137725q^{61} + 118854q^{62} - 81596q^{63} + 427612q^{64} + 35600q^{65} + 289576q^{66} + 64042q^{67} - 60500q^{68} - 141544q^{69} + 5200q^{70} + 136206q^{71} - 294660q^{72} - 270556q^{73} + 133162q^{74} - 240592q^{75} + 209502q^{76} + 152148q^{77} + 152814q^{78} - 61886q^{79} + 264936q^{80} - 104975q^{81} - 28160q^{82} + 202892q^{83} - 197912q^{84} + 576930q^{85} + 154822q^{86} - 312q^{87} - 884284q^{88} - 12065q^{89} - 312764q^{90} + 206252q^{91} - 291924q^{92} - 60752q^{93} - 116814q^{94} + 506q^{95} + 81968q^{96} + 132062q^{97} + 249798q^{98} + 289874q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −5.29373 + 9.16901i −7.75625 13.4342i −40.0472 69.3637i 14.2918 + 24.7542i 164.238 10.7449 + 18.6107i 509.197 1.18106 2.04566i −302.628
10.2 −5.15115 + 8.92206i 13.8137 + 23.9261i −37.0688 64.2050i −33.8746 58.6725i −284.626 −29.2828 50.7192i 434.114 −260.137 + 450.571i 697.972
10.3 −3.72321 + 6.44878i 5.98171 + 10.3606i −11.7245 20.3075i 37.3727 + 64.7314i −89.0845 68.5781 + 118.781i −63.6740 49.9383 86.4957i −556.584
10.4 −3.51823 + 6.09376i −0.728242 1.26135i −8.75591 15.1657i −24.4115 42.2819i 10.2485 −37.5928 65.1127i −101.945 120.439 208.607i 343.541
10.5 −2.35757 + 4.08344i −14.3898 24.9239i 4.88369 + 8.45880i −24.5682 42.5534i 135.700 109.024 + 188.835i −196.939 −292.634 + 506.856i 231.686
10.6 −2.03969 + 3.53285i −10.2415 17.7388i 7.67933 + 13.3010i 36.0654 + 62.4672i 83.5580 −123.244 213.465i −193.194 −88.2770 + 152.900i −294.249
10.7 −1.17581 + 2.03656i 6.67175 + 11.5558i 13.2349 + 22.9236i −7.00965 12.1411i −31.3789 52.0933 + 90.2282i −137.499 32.4755 56.2492i 32.9681
10.8 −0.480658 + 0.832525i 12.5533 + 21.7430i 15.5379 + 26.9125i 14.2475 + 24.6775i −24.1355 −96.5783 167.278i −60.6359 −193.673 + 335.451i −27.3928
10.9 0.684514 1.18561i −2.47268 4.28281i 15.0629 + 26.0897i −49.2630 85.3260i −6.77035 −21.8515 37.8479i 85.0519 109.272 189.264i −134.885
10.10 1.06983 1.85300i −4.46391 7.73172i 13.7109 + 23.7480i 22.0008 + 38.1065i −19.1024 29.8851 + 51.7625i 127.142 81.6470 141.417i 94.1481
10.11 2.93501 5.08359i 12.1402 + 21.0274i −1.22857 2.12795i −29.0869 50.3799i 142.526 97.1808 + 168.322i 173.417 −173.269 + 300.110i −341.481
10.12 3.64161 6.30745i −11.9080 20.6253i −10.5226 18.2257i −4.92186 8.52492i −173.457 −8.42954 14.6004i 79.7857 −162.101 + 280.767i −71.6940
10.13 3.71247 6.43018i 6.84687 + 11.8591i −11.5648 20.0308i 46.2802 + 80.1596i 101.675 3.53945 + 6.13051i 65.8619 27.7406 48.0482i 687.254
10.14 4.11131 7.12099i 3.90426 + 6.76238i −17.8057 30.8404i −21.1612 36.6522i 64.2065 −110.755 191.833i −29.6949 91.0134 157.640i −348.000
10.15 5.58532 9.67406i −1.95144 3.38000i −46.3916 80.3526i −1.46163 2.53162i −43.5978 81.6887 + 141.489i −678.988 113.884 197.252i −32.6548
26.1 −5.29373 9.16901i −7.75625 + 13.4342i −40.0472 + 69.3637i 14.2918 24.7542i 164.238 10.7449 18.6107i 509.197 1.18106 + 2.04566i −302.628
26.2 −5.15115 8.92206i 13.8137 23.9261i −37.0688 + 64.2050i −33.8746 + 58.6725i −284.626 −29.2828 + 50.7192i 434.114 −260.137 450.571i 697.972
26.3 −3.72321 6.44878i 5.98171 10.3606i −11.7245 + 20.3075i 37.3727 64.7314i −89.0845 68.5781 118.781i −63.6740 49.9383 + 86.4957i −556.584
26.4 −3.51823 6.09376i −0.728242 + 1.26135i −8.75591 + 15.1657i −24.4115 + 42.2819i 10.2485 −37.5928 + 65.1127i −101.945 120.439 + 208.607i 343.541
26.5 −2.35757 4.08344i −14.3898 + 24.9239i 4.88369 8.45880i −24.5682 + 42.5534i 135.700 109.024 188.835i −196.939 −292.634 506.856i 231.686
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.6.c.a 30
37.c even 3 1 inner 37.6.c.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.6.c.a 30 1.a even 1 1 trivial
37.6.c.a 30 37.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).