Properties

Label 37.4.h.a
Level $37$
Weight $4$
Character orbit 37.h
Analytic conductor $2.183$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.18307067021\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48q - 3q^{2} + 9q^{4} - 27q^{5} - 108q^{7} + 144q^{8} + 42q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 3q^{2} + 9q^{4} - 27q^{5} - 108q^{7} + 144q^{8} + 42q^{9} + 57q^{10} - 135q^{11} + 111q^{12} - 270q^{13} + 27q^{14} + 84q^{15} - 375q^{16} + 201q^{17} + 378q^{18} + 36q^{19} - 684q^{20} - 132q^{21} - 27q^{22} - 9q^{23} + 693q^{24} - 399q^{25} + 189q^{26} - 207q^{27} - 1161q^{28} - 189q^{29} + 1200q^{30} - 276q^{32} + 387q^{33} + 393q^{34} + 936q^{35} + 852q^{36} + 1116q^{37} - 2526q^{38} + 1422q^{39} + 2997q^{40} - 909q^{41} + 1305q^{42} - 1122q^{44} - 1701q^{45} - 294q^{46} + 1185q^{47} - 2163q^{48} - 708q^{49} - 597q^{50} - 3159q^{51} + 2115q^{52} - 528q^{53} + 2277q^{54} + 531q^{55} - 4935q^{56} - 1596q^{57} + 243q^{58} + 474q^{59} - 4932q^{60} - 432q^{61} - 4248q^{62} - 195q^{63} - 1512q^{64} + 1887q^{65} + 4077q^{66} + 1614q^{67} - 63q^{69} + 3144q^{70} + 1860q^{71} + 5613q^{72} + 7002q^{73} + 2157q^{74} - 5604q^{75} + 6753q^{76} + 6987q^{77} + 2913q^{78} + 1860q^{79} + 2691q^{81} - 5085q^{82} - 1956q^{83} + 8574q^{84} + 726q^{85} - 1986q^{86} - 7473q^{87} - 13950q^{88} - 3546q^{89} - 1110q^{90} + 378q^{91} - 8706q^{92} - 8556q^{93} - 11112q^{94} + 402q^{95} + 4167q^{96} + 3123q^{97} - 8997q^{98} - 6717q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −3.52321 + 4.19880i 5.42588 4.55286i −3.82771 21.7080i 4.86754 13.3735i 38.8228i −13.3663 4.86492i 66.6589 + 38.4856i 4.02320 22.8167i 39.0031 + 67.5553i
3.2 −2.33117 + 2.77818i −5.31333 + 4.45842i −0.894742 5.07433i 2.98312 8.19605i 25.1547i −5.48259 1.99550i −8.94300 5.16324i 3.66553 20.7883i 15.8159 + 27.3940i
3.3 −2.02577 + 2.41422i 1.44698 1.21416i −0.335518 1.90282i −4.98217 + 13.6884i 5.95292i −5.58350 2.03223i −16.5610 9.56148i −4.06894 + 23.0761i −22.9540 39.7575i
3.4 −0.746486 + 0.889628i 3.53224 2.96390i 1.15499 + 6.55027i 3.46112 9.50934i 5.35489i 25.4052 + 9.24674i −14.7354 8.50748i −0.996488 + 5.65136i 5.87610 + 10.1777i
3.5 0.679529 0.809831i −4.59174 + 3.85293i 1.19512 + 6.77785i −2.08766 + 5.73579i 6.33671i 1.40748 + 0.512281i 13.6253 + 7.86654i 1.55053 8.79347i 3.22640 + 5.58829i
3.6 1.20911 1.44096i 6.02403 5.05476i 0.774762 + 4.39389i −1.51038 + 4.14973i 14.7922i −23.8352 8.67530i 20.3005 + 11.7205i 6.04983 34.3103i 4.15339 + 7.19388i
3.7 2.37200 2.82684i −1.46213 + 1.22687i −0.975449 5.53205i 6.54993 17.9958i 7.04333i −0.795732 0.289623i 7.61432 + 4.39613i −4.05590 + 23.0021i −35.3347 61.2016i
3.8 3.09995 3.69437i 1.24007 1.04054i −2.64954 15.0263i −4.06803 + 11.1768i 7.80690i 6.95750 + 2.53232i −30.3137 17.5016i −4.23346 + 24.0091i 28.6807 + 49.6764i
4.1 −4.51144 0.795489i −1.14073 6.46938i 12.2027 + 4.44144i −2.86066 3.40920i 30.0937i −25.5484 + 21.4376i −19.7805 11.4203i −15.1799 + 5.52504i 10.1937 + 17.6560i
4.2 −3.85704 0.680100i 0.216129 + 1.22573i 6.89666 + 2.51018i 1.80772 + 2.15435i 4.87467i 24.0879 20.2122i 2.24108 + 1.29389i 23.9160 8.70471i −5.50725 9.53884i
4.3 −2.35856 0.415877i 1.60672 + 9.11218i −2.12770 0.774419i −12.3995 14.7772i 22.1598i −11.9270 + 10.0079i 21.2889 + 12.2911i −55.0785 + 20.0469i 23.0995 + 40.0095i
4.4 −1.09478 0.193039i 0.594427 + 3.37116i −6.35626 2.31349i 12.8529 + 15.3175i 3.80543i −19.5668 + 16.4185i 14.2140 + 8.20644i 14.3603 5.22672i −11.1142 19.2504i
4.5 0.00515467 0.000908908i −0.799053 4.53165i −7.51752 2.73615i −5.10418 6.08292i 0.0240855i 2.28847 1.92026i −0.0725270 0.0418735i 5.47429 1.99248i −0.0207816 0.0359947i
4.6 2.82396 + 0.497941i 1.06334 + 6.03049i 0.209292 + 0.0761760i 2.04009 + 2.43128i 17.5594i 11.1877 9.38761i −19.3137 11.1508i −9.86447 + 3.59037i 4.55050 + 7.88169i
4.7 3.59444 + 0.633796i −1.19386 6.77069i 5.00074 + 1.82012i 7.95665 + 9.48237i 25.0935i 3.25082 2.72776i −8.46590 4.88779i −19.0453 + 6.93190i 22.5898 + 39.1267i
4.8 4.72461 + 0.833076i 0.102510 + 0.581362i 14.1104 + 5.13576i −9.52827 11.3554i 2.83211i −20.7402 + 17.4031i 29.1495 + 16.8295i 25.0442 9.11535i −35.5575 61.5874i
21.1 −1.72697 + 4.74480i −3.76881 + 1.37173i −13.4023 11.2459i 2.75463 + 0.485716i 20.2512i −0.954954 + 5.41581i 41.5223 23.9729i −8.36093 + 7.01565i −7.06178 + 12.2314i
21.2 −1.05871 + 2.90878i 4.78213 1.74055i −1.21176 1.01679i 7.19342 + 1.26839i 15.7529i −0.149662 + 0.848773i −17.2054 + 9.93356i −0.843972 + 0.708177i −11.3052 + 19.5812i
21.3 −0.699326 + 1.92138i −4.26666 + 1.55294i 2.92571 + 2.45496i −16.6033 2.92761i 9.28388i 0.336001 1.90556i −20.9290 + 12.0833i −4.89045 + 4.10358i 17.2362 29.8539i
21.4 0.236488 0.649746i −6.38447 + 2.32376i 5.76211 + 4.83499i 12.1267 + 2.13827i 4.69783i −5.32451 + 30.1968i 9.29466 5.36628i 14.6784 12.3166i 4.25717 7.37363i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.4.h.a 48
37.h even 18 1 inner 37.4.h.a 48
37.i odd 36 2 1369.4.a.j 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.h.a 48 1.a even 1 1 trivial
37.4.h.a 48 37.h even 18 1 inner
1369.4.a.j 48 37.i odd 36 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(37, [\chi])\).