# Properties

 Label 37.10.h.a Level $37$ Weight $10$ Character orbit 37.h Analytic conductor $19.056$ Analytic rank $0$ Dimension $168$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,10,Mod(3,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([13]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("37.3");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 37.h (of order $$18$$, degree $$6$$, 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: $$19.0563259381$$ Analytic rank: $$0$$ Dimension: $$168$$ Relative dimension: $$28$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10})$$ 168 * q + 45 * q^2 + 216 * q^3 + 249 * q^4 + 177 * q^5 - 5820 * q^7 - 74520 * q^8 - 30198 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9} - 60003 q^{10} + 175689 q^{11} - 307257 q^{12} - 713598 q^{13} + 290583 q^{14} - 385260 q^{15} + 292473 q^{16} + 951645 q^{17} - 2634174 q^{18} + 87108 q^{19} - 6701172 q^{20} + 1892136 q^{21} - 1410435 q^{22} - 9 q^{23} + 6919605 q^{24} - 1830519 q^{25} - 12361155 q^{26} + 12227553 q^{27} - 14971305 q^{28} - 8346069 q^{29} + 32076480 q^{30} + 25901316 q^{32} + 29661003 q^{33} + 17761257 q^{34} - 25104648 q^{35} - 302330892 q^{36} - 89711496 q^{37} + 137852370 q^{38} + 3073230 q^{39} + 72127485 q^{40} + 40690731 q^{41} + 229015377 q^{42} + 187195086 q^{44} - 293199021 q^{45} - 286926930 q^{46} - 134086527 q^{47} + 86156133 q^{48} + 67568640 q^{49} + 134153859 q^{50} - 528106959 q^{51} + 402280539 q^{52} + 147470676 q^{53} + 422122437 q^{54} - 55011861 q^{55} - 616228263 q^{56} - 449072832 q^{57} + 529606191 q^{58} + 773443290 q^{59} + 1433547972 q^{60} + 469275144 q^{61} - 914552304 q^{62} - 673379979 q^{63} + 1146585192 q^{64} - 243884745 q^{65} + 1805055381 q^{66} + 214516206 q^{67} + 722409993 q^{69} - 2605801512 q^{70} - 1455384636 q^{71} - 4689293691 q^{72} + 930909090 q^{73} - 1261741251 q^{74} + 3013150476 q^{75} + 195486513 q^{76} + 1827454371 q^{77} + 1710012453 q^{78} + 1230271476 q^{79} + 2262925611 q^{81} - 3346032141 q^{82} - 2122951164 q^{83} - 6124511730 q^{84} - 39169146 q^{85} - 2603533938 q^{86} + 1185003879 q^{87} - 261497646 q^{88} + 1603969074 q^{89} + 7288796094 q^{90} - 4883515182 q^{91} - 7533267018 q^{92} + 1209552852 q^{93} + 1712695200 q^{94} + 2036912394 q^{95} + 7661250639 q^{96} - 7583351877 q^{97} - 10074558573 q^{98} - 7520510685 q^{99}+O(q^{100})$$ 168 * q + 45 * q^2 + 216 * q^3 + 249 * q^4 + 177 * q^5 - 5820 * q^7 - 74520 * q^8 - 30198 * q^9 - 60003 * q^10 + 175689 * q^11 - 307257 * q^12 - 713598 * q^13 + 290583 * q^14 - 385260 * q^15 + 292473 * q^16 + 951645 * q^17 - 2634174 * q^18 + 87108 * q^19 - 6701172 * q^20 + 1892136 * q^21 - 1410435 * q^22 - 9 * q^23 + 6919605 * q^24 - 1830519 * q^25 - 12361155 * q^26 + 12227553 * q^27 - 14971305 * q^28 - 8346069 * q^29 + 32076480 * q^30 + 25901316 * q^32 + 29661003 * q^33 + 17761257 * q^34 - 25104648 * q^35 - 302330892 * q^36 - 89711496 * q^37 + 137852370 * q^38 + 3073230 * q^39 + 72127485 * q^40 + 40690731 * q^41 + 229015377 * q^42 + 187195086 * q^44 - 293199021 * q^45 - 286926930 * q^46 - 134086527 * q^47 + 86156133 * q^48 + 67568640 * q^49 + 134153859 * q^50 - 528106959 * q^51 + 402280539 * q^52 + 147470676 * q^53 + 422122437 * q^54 - 55011861 * q^55 - 616228263 * q^56 - 449072832 * q^57 + 529606191 * q^58 + 773443290 * q^59 + 1433547972 * q^60 + 469275144 * q^61 - 914552304 * q^62 - 673379979 * q^63 + 1146585192 * q^64 - 243884745 * q^65 + 1805055381 * q^66 + 214516206 * q^67 + 722409993 * q^69 - 2605801512 * q^70 - 1455384636 * q^71 - 4689293691 * q^72 + 930909090 * q^73 - 1261741251 * q^74 + 3013150476 * q^75 + 195486513 * q^76 + 1827454371 * q^77 + 1710012453 * q^78 + 1230271476 * q^79 + 2262925611 * q^81 - 3346032141 * q^82 - 2122951164 * q^83 - 6124511730 * q^84 - 39169146 * q^85 - 2603533938 * q^86 + 1185003879 * q^87 - 261497646 * q^88 + 1603969074 * q^89 + 7288796094 * q^90 - 4883515182 * q^91 - 7533267018 * q^92 + 1209552852 * q^93 + 1712695200 * q^94 + 2036912394 * q^95 + 7661250639 * q^96 - 7583351877 * q^97 - 10074558573 * q^98 - 7520510685 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −27.9981 + 33.3669i 65.8541 55.2582i −240.545 1364.20i −399.807 + 1098.46i 3744.47i 1891.41 + 688.417i 32940.3 + 19018.1i −2134.62 + 12106.0i −25458.3 44095.1i
3.2 −26.0193 + 31.0086i −128.286 + 107.645i −195.621 1109.42i 759.168 2085.80i 6778.81i −10701.4 3894.99i 21542.9 + 12437.8i 1452.00 8234.72i 44924.6 + 77811.7i
3.3 −23.4612 + 27.9600i −156.305 + 131.155i −142.425 807.730i −731.487 + 2009.75i 7447.34i −678.538 246.968i 9741.66 + 5624.35i 3811.56 21616.4i −39030.9 67603.5i
3.4 −22.9919 + 27.4007i 183.862 154.278i −133.262 755.768i 816.313 2242.80i 8585.09i 8119.51 + 2955.26i 7912.34 + 4568.19i 6585.41 37347.7i 42685.7 + 73933.9i
3.5 −22.5902 + 26.9219i −90.5435 + 75.9750i −125.566 712.119i 323.614 889.121i 4153.89i 8597.44 + 3129.21i 6425.09 + 3709.53i −991.995 + 5625.89i 16626.4 + 28797.7i
3.6 −22.1667 + 26.4172i 114.438 96.0253i −117.600 666.941i 92.1806 253.264i 5151.70i −10964.7 3990.81i 4934.58 + 2848.98i 457.392 2594.00i 4647.19 + 8049.17i
3.7 −17.6863 + 21.0777i 49.4482 41.4920i −42.5571 241.353i 60.4467 166.076i 1776.10i 276.139 + 100.506i −6360.44 3672.20i −2694.37 + 15280.6i 2431.42 + 4211.35i
3.8 −15.3360 + 18.2767i −37.8456 + 31.7562i −9.93794 56.3609i −278.257 + 764.505i 1178.71i −396.292 144.238i −9396.50 5425.07i −2994.09 + 16980.3i −9705.29 16810.1i
3.9 −13.9806 + 16.6614i 158.307 132.835i 6.76230 + 38.3509i −726.809 + 1996.89i 4494.73i 3057.98 + 1113.01i −10377.5 5991.47i 3997.96 22673.6i −23109.8 40027.3i
3.10 −9.94057 + 11.8467i −168.752 + 141.600i 47.3783 + 268.696i −172.317 + 473.436i 3406.75i −6406.43 2331.75i −10511.3 6068.69i 5008.88 28406.8i −3895.74 6747.61i
3.11 −8.27130 + 9.85735i 1.31238 1.10122i 60.1549 + 341.155i 785.378 2157.81i 22.0451i −1325.67 482.503i −9566.12 5523.00i −3417.41 + 19381.1i 14774.2 + 25589.6i
3.12 −7.09899 + 8.46024i −176.851 + 148.396i 67.7278 + 384.103i 403.268 1107.97i 2549.67i 4613.96 + 1679.34i −8627.40 4981.03i 5837.15 33104.1i 6510.90 + 11277.2i
3.13 −3.30493 + 3.93866i 198.043 166.177i 84.3174 + 478.188i 299.908 823.990i 1329.23i −9977.77 3631.61i −4441.87 2564.52i 8188.00 46436.5i 2254.24 + 3904.46i
3.14 −1.46466 + 1.74551i −23.1371 + 19.4143i 88.0063 + 499.108i −648.756 + 1782.44i 68.8214i −9128.06 3322.34i −2010.44 1160.73i −3259.51 + 18485.6i −2161.07 3743.08i
3.15 −1.15965 + 1.38201i 114.860 96.3791i 88.3427 + 501.016i 184.041 505.649i 270.504i 6332.05 + 2304.68i −1594.80 920.758i 486.007 2756.29i 485.390 + 840.720i
3.16 −0.0979000 + 0.116673i −67.1156 + 56.3167i 88.9038 + 504.199i −720.391 + 1979.26i 13.3440i 10808.8 + 3934.07i −135.063 77.9786i −2084.98 + 11824.5i −160.399 277.819i
3.17 6.70758 7.99378i 25.8863 21.7212i 69.9990 + 396.984i 99.8626 274.370i 352.626i −3442.46 1252.95i 8269.92 + 4774.64i −3219.63 + 18259.4i −1523.42 2638.64i
3.18 9.72159 11.5857i −144.436 + 121.196i 49.1878 + 278.958i −28.8177 + 79.1760i 2851.61i 3077.98 + 1120.29i 10416.2 + 6013.81i 2755.31 15626.1i 637.158 + 1103.59i
3.19 10.9936 13.1017i −104.954 + 88.0672i 38.1135 + 216.153i 606.182 1665.47i 2343.25i −6557.17 2386.61i 10834.5 + 6255.32i −158.323 + 897.896i −15156.3 26251.5i
3.20 12.3034 14.6627i 154.895 129.973i 25.2886 + 143.419i −383.475 + 1053.59i 3870.29i 1863.39 + 678.219i 10901.1 + 6293.78i 3681.76 20880.3i 10730.4 + 18585.5i
See next 80 embeddings (of 168 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 30.28 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.h even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.10.h.a 168
37.h even 18 1 inner 37.10.h.a 168

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.10.h.a 168 1.a even 1 1 trivial
37.10.h.a 168 37.h even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(37, [\chi])$$.