# Properties

 Label 37.11.g.a Level $37$ Weight $11$ Character orbit 37.g Analytic conductor $23.508$ Analytic rank $0$ Dimension $120$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,11,Mod(8,37)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 11, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("37.8");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 37.g (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$23.5082183489$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$30$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10})$$ 120 * q + 6 * q^2 - 6 * q^3 - 3768 * q^4 + 2922 * q^5 - 2052 * q^6 - 2 * q^7 + 20880 * q^8 + 1052964 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9} - 15176 q^{10} - 109222 q^{12} + 880136 q^{13} - 857000 q^{14} - 3258802 q^{15} + 8499956 q^{16} - 2746076 q^{17} - 13504458 q^{18} + 11393476 q^{19} - 5416890 q^{20} - 14333028 q^{21} + 17692060 q^{22} + 4159744 q^{23} - 28379646 q^{24} + 7754160 q^{25} + 55598712 q^{26} + 190427130 q^{28} - 28793628 q^{29} + 86507298 q^{30} - 146926848 q^{31} + 68952080 q^{32} + 13647364 q^{33} + 206758542 q^{34} + 260789890 q^{35} - 91044910 q^{37} + 676632816 q^{38} - 24659180 q^{39} + 51953658 q^{40} + 112692084 q^{41} + 2228088 q^{42} + 813060360 q^{43} - 849019016 q^{44} - 484973908 q^{45} + 884196142 q^{46} - 1973080032 q^{47} - 729122360 q^{49} + 1256181004 q^{50} - 1081895844 q^{51} - 391899384 q^{52} + 281825276 q^{53} + 226976492 q^{54} + 2189957728 q^{55} - 857019742 q^{56} - 1013779940 q^{57} + 1718997972 q^{58} - 1425149824 q^{59} + 1469915184 q^{60} - 540818358 q^{61} + 1697453388 q^{62} - 274044164 q^{63} - 402430740 q^{65} - 13923356888 q^{66} - 6617681724 q^{67} - 4708838492 q^{68} + 2818015996 q^{69} - 4699988362 q^{70} + 4604545270 q^{71} + 21749389052 q^{72} - 13343383028 q^{74} - 3188045312 q^{75} + 25267697676 q^{76} - 6292443600 q^{77} - 45002270004 q^{78} - 13887434436 q^{79} + 17391532784 q^{80} - 2539723136 q^{81} - 30397672 q^{82} + 7809575250 q^{83} - 6859621528 q^{84} + 8295392338 q^{86} + 16626744498 q^{87} - 42989009276 q^{88} - 20098865396 q^{89} + 34381493044 q^{90} + 26099214560 q^{91} + 16840440410 q^{92} + 59641886872 q^{93} + 196358508 q^{94} - 16143270300 q^{95} + 59405033118 q^{96} + 43391222864 q^{97} + 6510054022 q^{98} + 35556194736 q^{99}+O(q^{100})$$ 120 * q + 6 * q^2 - 6 * q^3 - 3768 * q^4 + 2922 * q^5 - 2052 * q^6 - 2 * q^7 + 20880 * q^8 + 1052964 * q^9 - 15176 * q^10 - 109222 * q^12 + 880136 * q^13 - 857000 * q^14 - 3258802 * q^15 + 8499956 * q^16 - 2746076 * q^17 - 13504458 * q^18 + 11393476 * q^19 - 5416890 * q^20 - 14333028 * q^21 + 17692060 * q^22 + 4159744 * q^23 - 28379646 * q^24 + 7754160 * q^25 + 55598712 * q^26 + 190427130 * q^28 - 28793628 * q^29 + 86507298 * q^30 - 146926848 * q^31 + 68952080 * q^32 + 13647364 * q^33 + 206758542 * q^34 + 260789890 * q^35 - 91044910 * q^37 + 676632816 * q^38 - 24659180 * q^39 + 51953658 * q^40 + 112692084 * q^41 + 2228088 * q^42 + 813060360 * q^43 - 849019016 * q^44 - 484973908 * q^45 + 884196142 * q^46 - 1973080032 * q^47 - 729122360 * q^49 + 1256181004 * q^50 - 1081895844 * q^51 - 391899384 * q^52 + 281825276 * q^53 + 226976492 * q^54 + 2189957728 * q^55 - 857019742 * q^56 - 1013779940 * q^57 + 1718997972 * q^58 - 1425149824 * q^59 + 1469915184 * q^60 - 540818358 * q^61 + 1697453388 * q^62 - 274044164 * q^63 - 402430740 * q^65 - 13923356888 * q^66 - 6617681724 * q^67 - 4708838492 * q^68 + 2818015996 * q^69 - 4699988362 * q^70 + 4604545270 * q^71 + 21749389052 * q^72 - 13343383028 * q^74 - 3188045312 * q^75 + 25267697676 * q^76 - 6292443600 * q^77 - 45002270004 * q^78 - 13887434436 * q^79 + 17391532784 * q^80 - 2539723136 * q^81 - 30397672 * q^82 + 7809575250 * q^83 - 6859621528 * q^84 + 8295392338 * q^86 + 16626744498 * q^87 - 42989009276 * q^88 - 20098865396 * q^89 + 34381493044 * q^90 + 26099214560 * q^91 + 16840440410 * q^92 + 59641886872 * q^93 + 196358508 * q^94 - 16143270300 * q^95 + 59405033118 * q^96 + 43391222864 * q^97 + 6510054022 * q^98 + 35556194736 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −57.5818 15.4290i 155.272 + 89.6465i 2190.80 + 1264.86i 1994.02 534.297i −7557.70 7557.70i 1378.57 2387.76i −63470.2 63470.2i −13451.5 23298.7i −123063.
8.2 −54.2077 14.5249i −356.538 205.848i 1840.69 + 1062.72i 1975.98 529.461i 16337.2 + 16337.2i 14659.3 25390.7i −43708.4 43708.4i 55221.9 + 95647.1i −114803.
8.3 −52.9068 14.1763i −219.093 126.493i 1711.35 + 988.050i −2671.29 + 715.769i 9798.29 + 9798.29i −11032.3 + 19108.5i −36875.3 36875.3i 2476.56 + 4289.53i 151476.
8.4 −49.5335 13.2725i 39.3340 + 22.7095i 1390.60 + 802.863i −4045.30 + 1083.93i −1646.94 1646.94i 4712.15 8161.69i −21094.0 21094.0i −28493.1 49351.4i 214764.
8.5 −43.3980 11.6284i 393.244 + 227.040i 861.352 + 497.302i −4035.49 + 1081.31i −14425.9 14425.9i 7702.44 13341.0i 933.920 + 933.920i 73569.6 + 127426.i 187706.
8.6 −41.0206 10.9914i −196.274 113.319i 675.070 + 389.752i 3858.05 1033.76i 6805.74 + 6805.74i −8257.62 + 14302.6i 7342.02 + 7342.02i −3842.19 6654.87i −169622.
8.7 −40.1395 10.7554i 294.444 + 169.997i 608.694 + 351.429i 1914.73 513.049i −9990.46 9990.46i −9723.20 + 16841.1i 9436.46 + 9436.46i 28273.6 + 48971.4i −82374.2
8.8 −33.6907 9.02740i −34.0392 19.6525i 166.761 + 96.2797i 3757.77 1006.89i 969.394 + 969.394i 8651.45 14984.7i 20506.1 + 20506.1i −28752.1 49800.0i −135692.
8.9 −27.9992 7.50237i −35.7432 20.6364i −159.139 91.8788i −1410.80 + 378.024i 845.961 + 845.961i 7821.46 13547.2i 24755.2 + 24755.2i −28672.8 49662.7i 42337.5
8.10 −24.0184 6.43570i −350.545 202.387i −351.346 202.850i −3310.95 + 887.167i 7117.01 + 7117.01i 2258.75 3912.27i 25137.9 + 25137.9i 52396.6 + 90753.7i 85233.2
8.11 −20.4348 5.47548i 129.456 + 74.7417i −499.211 288.219i −3560.99 + 954.164i −2236.17 2236.17i −12767.8 + 22114.5i 23941.4 + 23941.4i −18351.9 31786.4i 77992.5
8.12 −19.8168 5.30990i 278.697 + 160.906i −522.299 301.550i 4955.27 1327.76i −4668.48 4668.48i 8535.82 14784.5i 23604.2 + 23604.2i 22256.7 + 38549.7i −105248.
8.13 −9.38107 2.51365i −275.634 159.137i −805.124 464.839i 2126.03 569.669i 2185.73 + 2185.73i −7724.56 + 13379.3i 13416.7 + 13416.7i 21125.0 + 36589.5i −21376.4
8.14 −2.23188 0.598031i 276.550 + 159.666i −882.186 509.331i −1565.42 + 419.454i −521.741 521.741i 11439.9 19814.4i 3337.40 + 3337.40i 21462.0 + 37173.3i 3744.69
8.15 0.259656 + 0.0695747i 270.672 + 156.273i −886.747 511.964i −656.767 + 175.980i 59.4092 + 59.4092i −1021.79 + 1769.78i −389.274 389.274i 19317.8 + 33459.4i −182.778
8.16 2.54900 + 0.683004i 30.7968 + 17.7805i −880.779 508.518i 2807.41 752.242i 66.3569 + 66.3569i −7543.00 + 13064.9i −3808.57 3808.57i −28892.2 50042.8i 7669.88
8.17 8.41925 + 2.25593i −149.333 86.2176i −821.016 474.014i −1530.72 + 410.155i −1062.77 1062.77i 8137.72 14094.9i −12154.2 12154.2i −14657.6 25387.6i −13812.8
8.18 12.1045 + 3.24340i −326.253 188.362i −750.810 433.480i 4604.88 1233.87i −3338.20 3338.20i 11511.3 19938.2i −16756.0 16756.0i 41436.3 + 71769.7i 59741.8
8.19 15.3889 + 4.12345i −135.779 78.3918i −666.994 385.089i −5456.89 + 1462.17i −1766.24 1766.24i −1683.82 + 2916.47i −20212.3 20212.3i −17233.9 29850.1i −90004.9
8.20 25.9310 + 6.94820i 385.296 + 222.451i −262.669 151.652i 3458.18 926.615i 8445.50 + 8445.50i −8871.91 + 15366.6i −25196.0 25196.0i 69444.4 + 120281.i 96112.4
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.11.g.a 120
37.g odd 12 1 inner 37.11.g.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.11.g.a 120 1.a even 1 1 trivial
37.11.g.a 120 37.g odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(37, [\chi])$$.