Properties

Label 37.7.i.a
Level $37$
Weight $7$
Character orbit 37.i
Analytic conductor $8.512$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.51200109393\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216q - 12q^{2} - 12q^{3} - 282q^{4} + 186q^{5} - 12q^{6} - 12q^{7} + 1782q^{8} - 972q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 216q - 12q^{2} - 12q^{3} - 282q^{4} + 186q^{5} - 12q^{6} - 12q^{7} + 1782q^{8} - 972q^{9} - 6q^{10} - 18q^{11} - 12300q^{12} - 12q^{13} - 2508q^{14} - 5724q^{15} + 21138q^{16} + 14652q^{17} - 36462q^{18} + 25692q^{19} + 15156q^{20} + 22308q^{21} - 396q^{22} - 45372q^{23} - 130632q^{24} + 54042q^{25} + 48834q^{26} - 18q^{27} - 261132q^{28} - 63852q^{29} + 140076q^{30} + 255300q^{31} + 444198q^{32} + 151134q^{33} + 121740q^{34} + 64380q^{35} - 323412q^{37} - 530748q^{38} - 734844q^{39} - 873996q^{40} + 143244q^{41} + 682356q^{42} + 298788q^{43} + 410502q^{44} - 296748q^{45} + 1696848q^{46} + 97674q^{47} - 18q^{48} - 827124q^{49} - 701574q^{50} + 1150788q^{51} + 452802q^{52} + 147876q^{53} - 1733052q^{54} - 481266q^{55} - 1608972q^{56} + 238596q^{57} + 2880600q^{58} + 1700916q^{59} - 2913120q^{60} - 212628q^{61} - 3373512q^{62} - 1185486q^{63} - 2139876q^{64} - 2189010q^{65} + 4562748q^{66} - 486876q^{67} + 6077478q^{68} + 3324612q^{69} + 6153942q^{70} + 2088276q^{71} + 1397046q^{72} - 212106q^{74} - 5357160q^{75} - 2036850q^{76} - 4623426q^{77} - 7419738q^{78} - 2895996q^{79} - 8321832q^{80} - 337890q^{81} - 5343312q^{82} + 2697348q^{83} + 1737582q^{84} + 6365610q^{85} + 9282138q^{86} + 9292260q^{87} + 11436912q^{88} - 618732q^{89} - 2893080q^{90} - 2103804q^{91} - 1911270q^{92} + 2397036q^{93} + 751476q^{94} - 8155788q^{95} - 5338512q^{96} - 8958252q^{97} + 1386858q^{98} - 12128778q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −14.8980 1.30340i 11.6276 13.8572i 157.222 + 27.7226i 4.78254 10.2562i −191.288 + 191.288i −248.810 + 90.5593i −1381.66 370.215i 69.7682 + 395.675i −84.6179 + 146.563i
2.2 −12.6746 1.10888i −25.8114 + 30.7608i 96.3874 + 16.9957i 50.4999 108.297i 361.258 361.258i 84.5834 30.7859i −416.297 111.547i −153.410 870.029i −760.153 + 1316.62i
2.3 −11.5362 1.00928i −6.04724 + 7.20682i 69.0368 + 12.1730i −53.1787 + 114.042i 77.0357 77.0357i 358.686 130.551i −68.2523 18.2882i 111.220 + 630.762i 728.579 1261.94i
2.4 −8.95954 0.783858i 21.0804 25.1226i 16.6313 + 2.93254i 74.3976 159.546i −208.563 + 208.563i 147.602 53.7227i 409.278 + 109.666i −60.1738 341.262i −791.630 + 1371.14i
2.5 −7.75264 0.678269i 26.0231 31.0131i −3.38424 0.596733i −73.2086 + 156.996i −222.783 + 222.783i −148.093 + 53.9015i 506.925 + 135.830i −158.022 896.190i 674.046 1167.48i
2.6 −7.18053 0.628215i −17.1726 + 20.4655i −11.8624 2.09166i −35.1352 + 75.3476i 136.165 136.165i −623.699 + 227.008i 529.455 + 141.867i 2.65032 + 15.0307i 299.623 518.963i
2.7 −3.71967 0.325429i −9.90022 + 11.7986i −49.2976 8.69250i 68.4388 146.767i 40.6652 40.6652i −17.2823 + 6.29023i 411.368 + 110.226i 85.3964 + 484.307i −302.332 + 523.655i
2.8 −1.08129 0.0946003i 0.982487 1.17088i −61.8675 10.9089i −20.8024 + 44.6109i −1.17312 + 1.17312i 425.722 154.950i 132.964 + 35.6276i 126.184 + 715.624i 26.7136 46.2693i
2.9 −0.744857 0.0651666i −32.3086 + 38.5039i −62.4771 11.0164i −73.6496 + 157.942i 26.5745 26.5745i 281.089 102.308i 92.0410 + 24.6623i −312.115 1770.09i 65.1510 112.845i
2.10 1.35176 + 0.118264i 12.8836 15.3541i −61.2144 10.7938i 3.06542 6.57381i 19.2315 19.2315i −365.776 + 133.132i −165.355 44.3068i 56.8286 + 322.291i 4.92117 8.52371i
2.11 5.23865 + 0.458323i 29.7705 35.4790i −35.7943 6.31149i −1.99257 + 4.27309i 172.218 172.218i 376.658 137.092i −509.708 136.576i −245.893 1394.53i −12.3969 + 21.4720i
2.12 5.48890 + 0.480216i −25.0783 + 29.8871i −33.1303 5.84176i 69.6929 149.457i −152.004 + 152.004i 37.2020 13.5404i −519.659 139.242i −137.731 781.110i 454.309 786.886i
2.13 8.35059 + 0.730582i 1.76500 2.10345i 6.17099 + 1.08811i −100.792 + 216.150i 16.2756 16.2756i −148.724 + 54.1313i −467.463 125.256i 125.280 + 710.500i −999.591 + 1731.34i
2.14 8.51484 + 0.744952i −15.5833 + 18.5715i 8.91992 + 1.57282i −14.3618 + 30.7990i −146.524 + 146.524i −88.0064 + 32.0317i −453.612 121.545i 24.5295 + 139.114i −145.232 + 251.550i
2.15 9.97417 + 0.872627i 16.2625 19.3809i 35.6949 + 6.29397i 95.3651 204.511i 179.118 179.118i −390.942 + 142.291i −268.416 71.9219i 15.4390 + 87.5587i 1129.65 1956.61i
2.16 12.8119 + 1.12090i −2.07304 + 2.47056i 99.8612 + 17.6082i 27.6431 59.2807i −29.3289 + 29.3289i 497.914 181.226i 464.628 + 124.497i 124.783 + 707.682i 420.608 728.515i
2.17 14.3464 + 1.25514i 22.3262 26.6073i 141.215 + 24.9000i −37.3574 + 80.1132i 353.696 353.696i −152.507 + 55.5081i 1104.40 + 295.923i −82.9011 470.155i −636.496 + 1102.44i
2.18 15.1491 + 1.32538i −28.9017 + 34.4437i 164.711 + 29.0431i −13.1400 + 28.1787i −483.486 + 483.486i −429.463 + 156.312i 1516.66 + 406.387i −224.471 1273.04i −236.406 + 409.468i
5.1 −6.52282 + 13.9882i 4.44480 12.2120i −111.985 133.458i −123.914 + 176.967i 141.831 + 141.831i −94.3880 535.301i 1643.16 440.285i 429.070 + 360.033i −1667.19 2887.66i
5.2 −5.53836 + 11.8770i 6.06356 16.6595i −69.2525 82.5319i 60.5521 86.4774i 164.283 + 164.283i 72.0577 + 408.660i 553.647 148.349i 317.674 + 266.560i 691.737 + 1198.12i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.i odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.7.i.a 216
37.i odd 36 1 inner 37.7.i.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.7.i.a 216 1.a even 1 1 trivial
37.7.i.a 216 37.i odd 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(37, [\chi])\).