Properties

Label 37.7.d.a
Level $37$
Weight $7$
Character orbit 37.d
Analytic conductor $8.512$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 37.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.51200109393\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36q - 4q^{2} - 256q^{5} + 126q^{6} - 4q^{7} + 168q^{8} - 7140q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 4q^{2} - 256q^{5} + 126q^{6} - 4q^{7} + 168q^{8} - 7140q^{9} - 1876q^{10} + 11104q^{12} - 2904q^{13} - 7678q^{14} + 4288q^{15} - 44824q^{16} + 2064q^{17} + 6222q^{18} - 4954q^{19} - 43084q^{20} - 39810q^{22} + 21966q^{23} + 79188q^{24} + 66604q^{26} + 69500q^{29} - 32418q^{31} - 141992q^{32} + 54752q^{33} + 256508q^{34} - 26164q^{35} + 180242q^{37} + 329032q^{38} - 60088q^{39} - 346014q^{42} - 144514q^{43} - 294896q^{44} + 842152q^{45} - 522736q^{46} - 157076q^{47} + 405652q^{49} + 46176q^{50} - 624984q^{51} - 161884q^{52} - 512988q^{53} + 541150q^{54} + 824640q^{55} - 608488q^{56} - 311992q^{57} + 187582q^{59} - 1156764q^{60} - 1321908q^{61} - 887284q^{63} + 752126q^{66} + 1244832q^{68} + 41960q^{69} - 1029284q^{70} + 1316252q^{71} + 777880q^{72} + 1117260q^{74} - 2148904q^{75} - 814428q^{76} - 56722q^{79} + 3750844q^{80} - 930796q^{81} + 1462554q^{82} - 841484q^{83} + 5065540q^{84} + 3992612q^{86} + 2494224q^{87} - 3796428q^{88} - 1439724q^{89} - 5618044q^{90} - 8644q^{91} + 7826980q^{92} - 54064q^{93} - 2057938q^{94} - 8819328q^{96} + 2578548q^{97} - 6043946q^{98} + O(q^{100}) \)

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 −10.4127 + 10.4127i 29.1399i 152.847i −98.7683 98.7683i 303.424 + 303.424i −196.570 925.130 + 925.130i −120.135 2056.88
6.2 −10.2905 + 10.2905i 12.1509i 147.790i 92.7424 + 92.7424i −125.039 125.039i 599.396 862.241 + 862.241i 581.356 −1908.74
6.3 −8.48840 + 8.48840i 25.9060i 80.1058i 25.7278 + 25.7278i −219.900 219.900i −623.749 136.713 + 136.713i 57.8805 −436.776
6.4 −7.66813 + 7.66813i 49.4380i 53.6006i −147.246 147.246i −379.097 379.097i 416.828 −79.7443 79.7443i −1715.12 2258.21
6.5 −6.59379 + 6.59379i 40.5120i 22.9560i 83.9037 + 83.9037i 267.127 + 267.127i −26.0039 −270.635 270.635i −912.222 −1106.49
6.6 −5.58722 + 5.58722i 13.0839i 1.56600i −61.8977 61.8977i 73.1027 + 73.1027i 184.379 −366.331 366.331i 557.811 691.671
6.7 −3.76006 + 3.76006i 16.7537i 35.7238i 120.059 + 120.059i −62.9949 62.9949i 144.567 −374.968 374.968i 448.314 −902.857
6.8 −1.40537 + 1.40537i 9.49587i 60.0499i −104.306 104.306i −13.3452 13.3452i −165.435 −174.336 174.336i 638.828 293.177
6.9 −0.186040 + 0.186040i 44.0184i 63.9308i 56.4752 + 56.4752i −8.18919 8.18919i 89.8430 −23.8003 23.8003i −1208.62 −21.0133
6.10 1.01641 1.01641i 50.0255i 61.9338i −128.165 128.165i −50.8466 50.8466i 143.000 128.001 + 128.001i −1773.55 −260.537
6.11 1.13198 1.13198i 20.4994i 61.4373i 84.2468 + 84.2468i −23.2048 23.2048i −640.570 141.992 + 141.992i 308.775 190.731
6.12 2.77875 2.77875i 27.0135i 48.5571i 96.7657 + 96.7657i −75.0636 75.0636i 312.208 312.768 + 312.768i −0.728696 537.775
6.13 5.04935 5.04935i 11.9325i 13.0081i −47.3379 47.3379i 60.2515 + 60.2515i 421.079 388.841 + 388.841i 586.615 −478.051
6.14 5.77519 5.77519i 42.6591i 2.70569i −53.2840 53.2840i 246.364 + 246.364i −452.267 353.986 + 353.986i −1090.80 −615.451
6.15 7.44064 7.44064i 17.9951i 46.7261i −84.4779 84.4779i −133.895 133.895i −514.613 128.529 + 128.529i 405.176 −1257.14
6.16 8.60959 8.60959i 22.2519i 84.2501i 144.668 + 144.668i 191.579 + 191.579i −94.2990 −174.345 174.345i 233.855 2491.07
6.17 9.39156 9.39156i 33.9178i 112.403i 19.7999 + 19.7999i −318.541 318.541i 150.778 −454.578 454.578i −421.419 371.904
6.18 11.1987 11.1987i 29.5809i 186.823i −126.906 126.906i 331.268 + 331.268i 249.429 −1375.46 1375.46i −146.027 −2842.36
31.1 −10.4127 10.4127i 29.1399i 152.847i −98.7683 + 98.7683i 303.424 303.424i −196.570 925.130 925.130i −120.135 2056.88
31.2 −10.2905 10.2905i 12.1509i 147.790i 92.7424 92.7424i −125.039 + 125.039i 599.396 862.241 862.241i 581.356 −1908.74
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.18
Significant digits:
Format:

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 37.7.d.a 36
37.d odd 4 1 inner 37.7.d.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.7.d.a 36 1.a even 1 1 trivial
37.7.d.a 36 37.d odd 4 1 inner

Hecke kernels

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