Properties

Label 37.8.f.a
Level $37$
Weight $8$
Character orbit 37.f
Analytic conductor $11.558$
Analytic rank $0$
Dimension $132$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5582459429\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9} - 6003 q^{10} - 15975 q^{11} - 4635 q^{12} - 15981 q^{13} - 13491 q^{14} + 38493 q^{15} - 74151 q^{16} + 37698 q^{17} - 20100 q^{18} - 66102 q^{19} - 176184 q^{20} - 142530 q^{21} + 132783 q^{22} + 140631 q^{23} - 38793 q^{24} - 544092 q^{25} - 423147 q^{26} + 130557 q^{27} + 603813 q^{28} + 22890 q^{29} - 319860 q^{30} - 1288044 q^{31} - 217866 q^{32} + 112344 q^{33} + 812019 q^{34} + 1479252 q^{35} + 8118132 q^{36} - 1664550 q^{37} + 305010 q^{38} - 1580340 q^{39} - 4681425 q^{40} - 2956593 q^{41} - 3553029 q^{42} - 1023528 q^{43} + 6409524 q^{44} + 1803336 q^{45} + 10219434 q^{46} - 2607204 q^{47} - 2984043 q^{48} - 6819531 q^{49} + 5724015 q^{50} - 1675833 q^{51} + 11768451 q^{52} + 2280177 q^{53} - 7803897 q^{54} - 3542196 q^{55} + 12168843 q^{56} - 85203 q^{57} - 12425487 q^{58} - 7014600 q^{59} + 2356788 q^{60} + 10840332 q^{61} + 6334368 q^{62} - 3331359 q^{63} - 26175624 q^{64} - 573195 q^{65} - 36720249 q^{66} + 6642918 q^{67} - 25277742 q^{68} - 1236438 q^{69} - 27401982 q^{70} + 20311041 q^{71} + 40093005 q^{72} + 38643780 q^{73} - 9531 q^{74} + 16639446 q^{75} + 5787405 q^{76} - 316971 q^{77} + 9363381 q^{78} + 17712534 q^{79} - 74410128 q^{80} - 10355109 q^{81} - 16702371 q^{82} - 31021176 q^{83} - 61626192 q^{84} + 23576445 q^{85} + 18579072 q^{86} + 104996631 q^{87} + 57279702 q^{88} - 13471320 q^{89} + 56306910 q^{90} - 2335911 q^{91} - 178133436 q^{92} + 24366258 q^{93} + 77086020 q^{94} - 26254407 q^{95} + 152523921 q^{96} + 14408415 q^{97} - 521205 q^{98} + 58375668 q^{99} + O(q^{100}) \)

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 −20.8388 + 7.58471i −2.65397 0.965964i 278.675 233.836i 66.0192 + 374.414i 62.6321 −150.560 853.866i −2614.40 + 4528.27i −1669.23 1400.65i −4215.58 7301.60i
7.2 −18.9650 + 6.90268i −58.5849 21.3231i 213.969 179.541i −54.5768 309.520i 1258.25 222.299 + 1260.72i −1526.94 + 2644.75i 1302.17 + 1092.65i 3171.57 + 5493.32i
7.3 −17.7108 + 6.44620i 49.8626 + 18.1485i 174.065 146.058i −63.8672 362.209i −1000.10 43.2399 + 245.226i −935.075 + 1619.60i 481.575 + 404.089i 3466.01 + 6003.30i
7.4 −14.2518 + 5.18723i 85.5870 + 31.1511i 78.1529 65.5781i 80.5878 + 457.036i −1381.36 115.337 + 654.111i 197.001 341.217i 4679.40 + 3926.48i −3519.28 6095.56i
7.5 −13.3313 + 4.85219i 11.3243 + 4.12171i 56.1255 47.0949i 4.98573 + 28.2755i −170.967 7.15684 + 40.5884i 388.247 672.464i −1564.09 1312.43i −203.664 352.756i
7.6 −13.2275 + 4.81442i −57.0958 20.7812i 53.7348 45.0888i −7.87815 44.6792i 855.285 −278.049 1576.89i 407.190 705.274i 1152.74 + 967.261i 319.313 + 553.066i
7.7 −12.0215 + 4.37547i −68.2052 24.8247i 27.3180 22.9225i 75.7434 + 429.562i 928.548 226.548 + 1284.82i 590.646 1023.03i 2360.34 + 1980.56i −2790.09 4832.57i
7.8 −5.31992 + 1.93629i 16.2628 + 5.91916i −73.5013 + 61.6750i 5.57004 + 31.5893i −97.9779 125.391 + 711.126i 633.927 1097.99i −1445.90 1213.25i −90.7982 157.267i
7.9 −4.76396 + 1.73394i 65.1424 + 23.7099i −78.3649 + 65.7560i −43.0745 244.287i −351.448 −294.271 1668.89i 583.772 1011.12i 2006.04 + 1683.27i 628.785 + 1089.09i
7.10 −4.25018 + 1.54694i −41.1038 14.9605i −82.3827 + 69.1273i −92.7474 525.997i 197.841 9.66574 + 54.8172i 532.674 922.618i −209.638 175.907i 1207.88 + 2092.10i
7.11 −0.910359 + 0.331344i 19.9506 + 7.26143i −97.3347 + 81.6735i 96.6240 + 547.982i −20.5683 −202.895 1150.68i 123.550 213.994i −1330.04 1116.04i −269.533 466.845i
7.12 −0.323158 + 0.117620i −38.0853 13.8619i −97.9631 + 82.2008i 13.0043 + 73.7511i 13.9380 126.462 + 717.201i 43.9985 76.2077i −417.005 349.909i −12.8770 22.3037i
7.13 2.24172 0.815920i 67.7355 + 24.6537i −93.6941 + 78.6187i −36.3337 206.059i 171.960 278.285 + 1578.23i −298.567 + 517.134i 2304.96 + 1934.09i −249.578 432.281i
7.14 3.79145 1.37998i −69.9883 25.4737i −85.5829 + 71.8126i 27.7384 + 157.312i −300.510 −147.454 836.251i −483.610 + 837.637i 2574.12 + 2159.94i 322.256 + 558.163i
7.15 7.19462 2.61863i 53.5140 + 19.4775i −53.1484 + 44.5968i 33.7756 + 191.551i 436.017 41.4542 + 235.098i −755.606 + 1308.75i 809.034 + 678.860i 744.604 + 1289.69i
7.16 10.1961 3.71107i 11.1844 + 4.07079i −7.86580 + 6.60019i −64.6668 366.744i 129.144 −94.8119 537.705i −750.134 + 1299.27i −1566.82 1314.72i −2020.36 3499.36i
7.17 10.2566 3.73308i −19.4495 7.07902i −6.79263 + 5.69969i 7.78053 + 44.1256i −225.911 −27.0245 153.264i −746.939 + 1293.74i −1347.17 1130.41i 244.526 + 423.531i
7.18 14.2134 5.17327i −80.6447 29.3523i 77.2054 64.7830i −48.8700 277.155i −1298.08 203.583 + 1154.58i −205.826 + 356.501i 3966.67 + 3328.43i −2128.41 3686.51i
7.19 15.5384 5.65552i −16.3965 5.96785i 111.404 93.4791i 86.0271 + 487.884i −288.528 270.212 + 1532.45i 144.088 249.568i −1442.11 1210.07i 4095.96 + 7094.42i
7.20 16.4964 6.00419i 64.3192 + 23.4103i 138.026 115.818i 23.4987 + 133.267i 1201.59 −96.3778 546.586i 458.016 793.306i 1913.58 + 1605.68i 1187.81 + 2057.34i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.8.f.a 132
37.f even 9 1 inner 37.8.f.a 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.f.a 132 1.a even 1 1 trivial
37.8.f.a 132 37.f even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).