Properties

Label 37.10.c.a
Level $37$
Weight $10$
Character orbit 37.c
Analytic conductor $19.056$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.0563259381\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 54q + 32q^{2} + 73q^{3} - 6486q^{4} + 2729q^{5} + 1020q^{6} + 4231q^{7} - 7848q^{8} - 140840q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 54q + 32q^{2} + 73q^{3} - 6486q^{4} + 2729q^{5} + 1020q^{6} + 4231q^{7} - 7848q^{8} - 140840q^{9} - 82404q^{10} - 108196q^{11} + 45492q^{12} - 29827q^{13} - 504992q^{14} + 87135q^{15} - 1561814q^{16} + 116991q^{17} + 1767678q^{18} - 599677q^{19} + 4122556q^{20} - 221017q^{21} - 257914q^{22} + 1752736q^{23} + 1149408q^{24} - 6037260q^{25} - 16731604q^{26} - 1916582q^{27} + 1917318q^{28} - 11336100q^{29} - 20390006q^{30} + 14447580q^{31} - 873740q^{32} - 4494200q^{33} + 13736524q^{34} - 8546699q^{35} + 61692836q^{36} - 1919410q^{37} - 85447920q^{38} - 14035313q^{39} + 3210182q^{40} + 28676571q^{41} - 101904850q^{42} + 56602028q^{43} + 153011074q^{44} + 104234980q^{45} + 99535572q^{46} - 147509100q^{47} - 203670880q^{48} - 201857540q^{49} - 91421384q^{50} + 328228338q^{51} - 69374156q^{52} + 59137361q^{53} + 201615042q^{54} - 219079762q^{55} + 202573590q^{56} - 5989161q^{57} - 157098688q^{58} + 137993791q^{59} + 413427232q^{60} - 244884419q^{61} + 109345558q^{62} - 995908592q^{63} + 184008092q^{64} + 248745395q^{65} + 1965122248q^{66} - 130143913q^{67} + 667610796q^{68} + 246927272q^{69} - 291173360q^{70} + 473387559q^{71} + 366188748q^{72} + 923638972q^{73} - 885443390q^{74} - 2185997392q^{75} - 337459826q^{76} - 586609928q^{77} - 261025842q^{78} + 563110723q^{79} - 4210381784q^{80} + 713813329q^{81} + 3458576264q^{82} - 663594441q^{83} - 2635249016q^{84} - 1417287750q^{85} - 2269028474q^{86} + 314944482q^{87} - 7714172q^{88} + 655027619q^{89} + 588021376q^{90} - 1833870961q^{91} - 2384055268q^{92} - 90092600q^{93} - 1449940302q^{94} + 2787158841q^{95} + 2383567424q^{96} - 2751583248q^{97} + 4970876154q^{98} + 2668760954q^{99} + O(q^{100}) \)

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 −21.2153 + 36.7459i 86.0505 + 149.044i −644.175 1115.74i 1139.39 + 1973.48i −7302.33 4123.02 + 7141.28i 32941.0 −4967.86 + 8604.59i −96689.7
10.2 −20.2191 + 35.0205i −84.5687 146.477i −561.626 972.764i −198.799 344.329i 6839.62 3839.21 + 6649.70i 24717.9 −4462.23 + 7728.80i 16078.1
10.3 −19.4882 + 33.7545i −36.9005 63.9135i −503.578 872.222i −226.455 392.232i 2876.49 −4356.37 7545.46i 19299.4 7118.21 12329.1i 17652.8
10.4 −18.3336 + 31.7547i 82.9316 + 143.642i −416.240 720.949i −705.543 1222.04i −6081.73 −777.430 1346.55i 11751.1 −3913.79 + 6778.88i 51740.5
10.5 −16.2809 + 28.1993i −30.6895 53.1557i −274.135 474.815i 1243.37 + 2153.58i 1998.61 −3568.14 6180.20i 1181.00 7957.81 13783.3i −80972.6
10.6 −11.9270 + 20.6581i 13.2219 + 22.9010i −28.5049 49.3720i −269.425 466.658i −630.788 2392.52 + 4143.97i −10853.3 9491.86 16440.4i 12853.7
10.7 −11.2222 + 19.4374i −124.325 215.338i 4.12627 + 7.14691i 372.627 + 645.409i 5580.79 −110.128 190.748i −11676.7 −21072.0 + 36497.9i −16726.7
10.8 −10.1826 + 17.6368i 107.307 + 185.861i 48.6277 + 84.2257i 603.246 + 1044.85i −4370.67 −3014.80 5221.78i −12407.7 −13188.1 + 22842.4i −24570.5
10.9 −9.89842 + 17.1446i 16.8051 + 29.1073i 60.0426 + 103.997i 507.953 + 879.801i −665.375 4705.54 + 8150.24i −12513.3 9276.68 16067.7i −20111.7
10.10 −9.23482 + 15.9952i −68.0812 117.920i 85.4363 + 147.980i −1305.65 2261.46i 2514.87 −1262.57 2186.84i −12612.4 571.399 989.692i 48229.8
10.11 −3.26260 + 5.65099i 122.847 + 212.777i 234.711 + 406.531i −940.627 1629.21i −1603.20 3055.33 + 5291.98i −6403.98 −20341.1 + 35231.9i 12275.6
10.12 −1.64814 + 2.85467i 22.5023 + 38.9752i 250.567 + 433.995i −142.433 246.702i −148.348 −5192.34 8993.40i −3339.58 8828.79 15291.9i 939.004
10.13 −0.349842 + 0.605944i −41.9286 72.6225i 255.755 + 442.981i 954.706 + 1653.60i 58.6736 1656.50 + 2869.15i −716.134 6325.48 10956.1i −1335.99
10.14 0.152284 0.263764i −87.8341 152.133i 255.954 + 443.325i 209.008 + 362.012i −53.5031 −2618.01 4534.52i 311.850 −5588.16 + 9678.98i 127.314
10.15 3.41387 5.91300i 41.2346 + 71.4204i 232.691 + 403.033i −741.590 1284.47i 563.079 721.322 + 1249.37i 6673.32 6440.92 11156.0i −10126.8
10.16 3.95661 6.85305i 96.3161 + 166.824i 224.690 + 389.175i 891.239 + 1543.67i 1524.34 2975.14 + 5153.10i 7607.62 −8712.07 + 15089.8i 14105.1
10.17 4.81916 8.34702i −93.7353 162.354i 209.551 + 362.954i −535.707 927.872i −1806.90 5666.05 + 9813.89i 8974.26 −7731.11 + 13390.7i −10326.6
10.18 10.3662 17.9548i 86.3103 + 149.494i 41.0839 + 71.1595i 563.169 + 975.437i 3578.84 −1948.40 3374.73i 12318.5 −5057.43 + 8759.72i 23351.7
10.19 11.0930 19.2137i 11.3281 + 19.6208i 9.88860 + 17.1276i −837.494 1450.58i 502.652 1133.95 + 1964.06i 11798.1 9584.85 16601.4i −37161.4
10.20 11.7918 20.4241i −110.187 190.850i −22.0953 38.2702i −512.092 886.969i −5197.24 −4542.03 7867.02i 11032.7 −14440.9 + 25012.4i −24154.0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.27
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 37.10.c.a 54
37.c even 3 1 inner 37.10.c.a 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.10.c.a 54 1.a even 1 1 trivial
37.10.c.a 54 37.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).