# Properties

 Label 37.11.d.a Level $37$ Weight $11$ Character orbit 37.d Analytic conductor $23.508$ Analytic rank $0$ Dimension $60$ 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(6,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3]))

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

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

chi := DirichletCharacter("37.6");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 37.d (of order $$4$$, degree $$2$$, 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: $$60$$ Relative dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 12 q^{2} + 11094 q^{5} + 2046 q^{6} - 4 q^{7} - 20280 q^{8} - 964620 q^{9}+O(q^{10})$$ 60 * q - 12 * q^2 + 11094 * q^5 + 2046 * q^6 - 4 * q^7 - 20280 * q^8 - 964620 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 12 q^{2} + 11094 q^{5} + 2046 q^{6} - 4 q^{7} - 20280 q^{8} - 964620 q^{9} + 15164 q^{10} + 490144 q^{12} - 880142 q^{13} + 2081954 q^{14} + 624178 q^{15} - 16030232 q^{16} + 28766 q^{17} + 7009062 q^{18} - 2321122 q^{19} + 22162356 q^{20} - 17685922 q^{22} - 5164204 q^{23} + 22593300 q^{24} + 9974076 q^{26} + 50191740 q^{29} - 104256828 q^{31} - 72200072 q^{32} + 27294728 q^{33} + 15790284 q^{34} + 53701106 q^{35} - 141352730 q^{37} - 46584600 q^{38} - 262899250 q^{39} - 105833598 q^{42} - 54613578 q^{43} + 1296503888 q^{44} - 389012888 q^{45} - 701984848 q^{46} + 1896514872 q^{47} + 3384188120 q^{49} - 1897736584 q^{50} + 180688878 q^{51} - 2109205116 q^{52} - 939427040 q^{53} - 2601789506 q^{54} - 1016968630 q^{55} - 4290681512 q^{56} - 1055504530 q^{57} + 480978340 q^{59} + 1080064356 q^{60} + 5318536212 q^{61} + 6149823752 q^{63} + 2260119998 q^{66} - 12400898752 q^{68} - 1904155648 q^{69} + 12003714076 q^{70} + 9046257716 q^{71} - 1630441448 q^{72} + 15786417092 q^{74} + 2420230076 q^{75} - 14525032188 q^{76} - 14284235682 q^{79} - 29599600196 q^{80} + 27463874804 q^{81} + 31643179114 q^{82} - 11182213956 q^{83} - 1681148924 q^{84} - 25515304636 q^{86} + 6770194788 q^{87} + 3551705012 q^{88} + 4302323954 q^{89} - 34790967724 q^{90} + 31093789402 q^{91} + 2286888388 q^{92} - 27067845352 q^{93} - 54043756914 q^{94} + 23603051040 q^{96} - 3290287460 q^{97} - 1029183298 q^{98}+O(q^{100})$$ 60 * q - 12 * q^2 + 11094 * q^5 + 2046 * q^6 - 4 * q^7 - 20280 * q^8 - 964620 * q^9 + 15164 * q^10 + 490144 * q^12 - 880142 * q^13 + 2081954 * q^14 + 624178 * q^15 - 16030232 * q^16 + 28766 * q^17 + 7009062 * q^18 - 2321122 * q^19 + 22162356 * q^20 - 17685922 * q^22 - 5164204 * q^23 + 22593300 * q^24 + 9974076 * q^26 + 50191740 * q^29 - 104256828 * q^31 - 72200072 * q^32 + 27294728 * q^33 + 15790284 * q^34 + 53701106 * q^35 - 141352730 * q^37 - 46584600 * q^38 - 262899250 * q^39 - 105833598 * q^42 - 54613578 * q^43 + 1296503888 * q^44 - 389012888 * q^45 - 701984848 * q^46 + 1896514872 * q^47 + 3384188120 * q^49 - 1897736584 * q^50 + 180688878 * q^51 - 2109205116 * q^52 - 939427040 * q^53 - 2601789506 * q^54 - 1016968630 * q^55 - 4290681512 * q^56 - 1055504530 * q^57 + 480978340 * q^59 + 1080064356 * q^60 + 5318536212 * q^61 + 6149823752 * q^63 + 2260119998 * q^66 - 12400898752 * q^68 - 1904155648 * q^69 + 12003714076 * q^70 + 9046257716 * q^71 - 1630441448 * q^72 + 15786417092 * q^74 + 2420230076 * q^75 - 14525032188 * q^76 - 14284235682 * q^79 - 29599600196 * q^80 + 27463874804 * q^81 + 31643179114 * q^82 - 11182213956 * q^83 - 1681148924 * q^84 - 25515304636 * q^86 + 6770194788 * q^87 + 3551705012 * q^88 + 4302323954 * q^89 - 34790967724 * q^90 + 31093789402 * q^91 + 2286888388 * q^92 - 27067845352 * q^93 - 54043756914 * q^94 + 23603051040 * q^96 - 3290287460 * q^97 - 1029183298 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −44.7438 + 44.7438i 69.9886i 2980.01i 3419.76 + 3419.76i −3131.55 3131.55i −14682.2 87519.2 + 87519.2i 54150.6 −306026.
6.2 −39.3471 + 39.3471i 165.913i 2072.39i −2246.50 2246.50i 6528.21 + 6528.21i 12584.4 41250.9 + 41250.9i 31521.7 176787.
6.3 −38.1461 + 38.1461i 393.049i 1886.25i −1487.12 1487.12i −14993.3 14993.3i −8790.45 32891.6 + 32891.6i −95438.7 113456.
6.4 −36.6981 + 36.6981i 471.964i 1669.50i 255.153 + 255.153i 17320.2 + 17320.2i −23325.3 23688.8 + 23688.8i −163701. −18727.3
6.5 −34.6099 + 34.6099i 261.549i 1371.69i 987.224 + 987.224i −9052.19 9052.19i 23552.7 12033.6 + 12033.6i −9358.87 −68335.4
6.6 −29.4663 + 29.4663i 91.0754i 712.523i 1067.16 + 1067.16i 2683.65 + 2683.65i −14396.8 −9178.08 9178.08i 50754.3 −62890.5
6.7 −28.7783 + 28.7783i 308.352i 632.385i 4127.67 + 4127.67i 8873.85 + 8873.85i 28375.7 −11270.0 11270.0i −36031.8 −237575.
6.8 −26.8365 + 26.8365i 144.991i 416.394i −3621.85 3621.85i −3891.06 3891.06i −22976.9 −16306.0 16306.0i 38026.5 194395.
6.9 −21.6828 + 21.6828i 236.801i 83.7140i 1066.61 + 1066.61i −5134.51 5134.51i 10682.7 −24018.3 24018.3i 2974.12 −46254.1
6.10 −15.9821 + 15.9821i 314.319i 513.146i 3627.33 + 3627.33i −5023.47 5023.47i −25310.7 −24566.8 24566.8i −39747.3 −115945.
6.11 −14.5575 + 14.5575i 414.474i 600.159i −2714.51 2714.51i 6033.70 + 6033.70i 11282.5 −23643.7 23643.7i −112740. 79032.8
6.12 −13.2079 + 13.2079i 208.898i 675.105i −342.436 342.436i 2759.10 + 2759.10i −3090.02 −22441.5 22441.5i 15410.7 9045.69
6.13 −10.9958 + 10.9958i 127.791i 782.185i −3825.90 3825.90i −1405.16 1405.16i 26128.7 −19860.5 19860.5i 42718.5 84137.7
6.14 −4.55945 + 4.55945i 467.484i 982.423i −1353.22 1353.22i −2131.47 2131.47i 4509.07 −9148.19 9148.19i −159492. 12339.9
6.15 1.07520 1.07520i 33.9150i 1021.69i 2177.49 + 2177.49i −36.4653 36.4653i 6473.29 2199.52 + 2199.52i 57898.8 4682.46
6.16 2.81409 2.81409i 90.9323i 1008.16i −2198.37 2198.37i 255.892 + 255.892i −24301.2 5718.68 + 5718.68i 50780.3 −12372.8
6.17 3.53840 3.53840i 354.328i 998.959i 3636.07 + 3636.07i −1253.76 1253.76i −25459.9 7158.05 + 7158.05i −66499.6 25731.8
6.18 4.21658 4.21658i 126.107i 988.441i 1887.87 + 1887.87i 531.741 + 531.741i 15808.2 8485.62 + 8485.62i 43146.0 15920.7
6.19 13.7409 13.7409i 378.325i 646.374i 945.559 + 945.559i −5198.53 5198.53i 24801.2 22952.5 + 22952.5i −84080.6 25985.7
6.20 14.9926 14.9926i 263.976i 574.445i −850.730 850.730i 3957.69 + 3957.69i −15042.9 23964.8 + 23964.8i −10634.5 −25509.3
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.30 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.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.11.d.a 60
37.d odd 4 1 inner 37.11.d.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.11.d.a 60 1.a even 1 1 trivial
37.11.d.a 60 37.d odd 4 1 inner

## Hecke kernels

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