# Properties

 Label 21.7.h.a Level $21$ Weight $7$ Character orbit 21.h Analytic conductor $4.831$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,7,Mod(2,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 21.h (of order $$6$$, 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: $$4.83113575602$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9}+O(q^{10})$$ 28 * q - q^3 + 382 * q^4 - 356 * q^6 + 1120 * q^7 + 1031 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9} - 930 q^{10} - 404 q^{12} + 3632 q^{13} + 5450 q^{15} - 13706 q^{16} + 1612 q^{18} + 6830 q^{19} - 11333 q^{21} - 27348 q^{22} - 54498 q^{24} - 7640 q^{25} + 13268 q^{27} + 88522 q^{28} + 32380 q^{30} + 69410 q^{31} - 85459 q^{33} - 31344 q^{34} + 399028 q^{36} - 71830 q^{37} + 104570 q^{39} + 230250 q^{40} - 467782 q^{42} - 498904 q^{43} - 85375 q^{45} - 342660 q^{46} - 324728 q^{48} + 395668 q^{49} + 42177 q^{51} + 218000 q^{52} - 402122 q^{54} + 231660 q^{55} + 786782 q^{57} + 459522 q^{58} + 1021150 q^{60} + 153158 q^{61} + 179011 q^{63} - 1399412 q^{64} + 80896 q^{66} - 977098 q^{67} - 1520742 q^{69} - 139650 q^{70} - 1001352 q^{72} - 488350 q^{73} + 1434610 q^{75} + 6196280 q^{76} + 245176 q^{78} + 380762 q^{79} + 893531 q^{81} - 3665004 q^{82} + 1231496 q^{84} - 912540 q^{85} + 1452416 q^{87} - 1522482 q^{88} - 11713820 q^{90} - 1529416 q^{91} - 1690071 q^{93} - 3034044 q^{94} + 7549346 q^{96} + 7212176 q^{97} + 435866 q^{99}+O(q^{100})$$ 28 * q - q^3 + 382 * q^4 - 356 * q^6 + 1120 * q^7 + 1031 * q^9 - 930 * q^10 - 404 * q^12 + 3632 * q^13 + 5450 * q^15 - 13706 * q^16 + 1612 * q^18 + 6830 * q^19 - 11333 * q^21 - 27348 * q^22 - 54498 * q^24 - 7640 * q^25 + 13268 * q^27 + 88522 * q^28 + 32380 * q^30 + 69410 * q^31 - 85459 * q^33 - 31344 * q^34 + 399028 * q^36 - 71830 * q^37 + 104570 * q^39 + 230250 * q^40 - 467782 * q^42 - 498904 * q^43 - 85375 * q^45 - 342660 * q^46 - 324728 * q^48 + 395668 * q^49 + 42177 * q^51 + 218000 * q^52 - 402122 * q^54 + 231660 * q^55 + 786782 * q^57 + 459522 * q^58 + 1021150 * q^60 + 153158 * q^61 + 179011 * q^63 - 1399412 * q^64 + 80896 * q^66 - 977098 * q^67 - 1520742 * q^69 - 139650 * q^70 - 1001352 * q^72 - 488350 * q^73 + 1434610 * q^75 + 6196280 * q^76 + 245176 * q^78 + 380762 * q^79 + 893531 * q^81 - 3665004 * q^82 + 1231496 * q^84 - 912540 * q^85 + 1452416 * q^87 - 1522482 * q^88 - 11713820 * q^90 - 1529416 * q^91 - 1690071 * q^93 - 3034044 * q^94 + 7549346 * q^96 + 7212176 * q^97 + 435866 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −13.1658 7.60129i 23.2019 13.8084i 83.5593 + 144.729i 82.1651 + 47.4381i −410.434 + 5.43441i 334.938 + 73.9300i 1567.67i 347.657 640.762i −721.181 1249.12i
2.2 −11.1103 6.41456i −14.0470 + 23.0582i 50.2932 + 87.1103i 38.4659 + 22.2083i 303.975 166.080i −244.854 240.199i 469.371i −334.364 647.797i −284.913 493.483i
2.3 −9.23947 5.33441i −22.3946 15.0825i 24.9119 + 43.1487i −152.840 88.2420i 126.458 + 258.817i 342.557 17.4303i 151.243i 274.035 + 675.534i 941.439 + 1630.62i
2.4 −7.42751 4.28827i 19.2952 + 18.8863i 4.77856 + 8.27670i −88.3745 51.0231i −62.3258 223.022i −127.732 + 318.329i 466.932i 15.6130 + 728.833i 437.602 + 757.948i
2.5 −5.69684 3.28907i 2.79228 26.8552i −10.3640 17.9510i 57.4599 + 33.1745i −104.236 + 143.806i −329.948 93.7191i 557.353i −713.406 149.974i −218.227 377.979i
2.6 −2.88516 1.66575i −26.7007 + 4.00908i −26.4506 45.8137i 166.359 + 96.0474i 83.7139 + 32.9098i 157.090 + 304.913i 389.456i 696.855 214.090i −319.982 554.224i
2.7 −1.07584 0.621137i 26.6844 + 4.11626i −31.2284 54.0891i 93.7473 + 54.1250i −26.1514 21.0031i 147.950 309.451i 157.094i 695.113 + 219.680i −67.2381 116.460i
2.8 1.07584 + 0.621137i −9.77741 + 25.1675i −31.2284 54.0891i −93.7473 54.1250i −26.1514 + 21.0031i 147.950 309.451i 157.094i −537.804 492.146i −67.2381 116.460i
2.9 2.88516 + 1.66575i 16.8223 21.1189i −26.4506 45.8137i −166.359 96.0474i 83.7139 32.9098i 157.090 + 304.913i 389.456i −163.020 710.539i −319.982 554.224i
2.10 5.69684 + 3.28907i −24.6534 11.0094i −10.3640 17.9510i −57.4599 33.1745i −104.236 143.806i −329.948 93.7191i 557.353i 486.585 + 542.841i −218.227 377.979i
2.11 7.42751 + 4.28827i 6.70842 + 26.1533i 4.77856 + 8.27670i 88.3745 + 51.0231i −62.3258 + 223.022i −127.732 + 318.329i 466.932i −638.994 + 350.895i 437.602 + 757.948i
2.12 9.23947 + 5.33441i −1.86456 26.9355i 24.9119 + 43.1487i 152.840 + 88.2420i 126.458 258.817i 342.557 17.4303i 151.243i −722.047 + 100.446i 941.439 + 1630.62i
2.13 11.1103 + 6.41456i 26.9925 0.635931i 50.2932 + 87.1103i −38.4659 22.2083i 303.975 + 166.080i −244.854 240.199i 469.371i 728.191 34.3307i −284.913 493.483i
2.14 13.1658 + 7.60129i −23.5594 + 13.1892i 83.5593 + 144.729i −82.1651 47.4381i −410.434 5.43441i 334.938 + 73.9300i 1567.67i 381.088 621.460i −721.181 1249.12i
11.1 −13.1658 + 7.60129i 23.2019 + 13.8084i 83.5593 144.729i 82.1651 47.4381i −410.434 5.43441i 334.938 73.9300i 1567.67i 347.657 + 640.762i −721.181 + 1249.12i
11.2 −11.1103 + 6.41456i −14.0470 23.0582i 50.2932 87.1103i 38.4659 22.2083i 303.975 + 166.080i −244.854 + 240.199i 469.371i −334.364 + 647.797i −284.913 + 493.483i
11.3 −9.23947 + 5.33441i −22.3946 + 15.0825i 24.9119 43.1487i −152.840 + 88.2420i 126.458 258.817i 342.557 + 17.4303i 151.243i 274.035 675.534i 941.439 1630.62i
11.4 −7.42751 + 4.28827i 19.2952 18.8863i 4.77856 8.27670i −88.3745 + 51.0231i −62.3258 + 223.022i −127.732 318.329i 466.932i 15.6130 728.833i 437.602 757.948i
11.5 −5.69684 + 3.28907i 2.79228 + 26.8552i −10.3640 + 17.9510i 57.4599 33.1745i −104.236 143.806i −329.948 + 93.7191i 557.353i −713.406 + 149.974i −218.227 + 377.979i
11.6 −2.88516 + 1.66575i −26.7007 4.00908i −26.4506 + 45.8137i 166.359 96.0474i 83.7139 32.9098i 157.090 304.913i 389.456i 696.855 + 214.090i −319.982 + 554.224i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.h.a 28
3.b odd 2 1 inner 21.7.h.a 28
7.c even 3 1 inner 21.7.h.a 28
7.c even 3 1 147.7.b.e 14
7.d odd 6 1 147.7.b.d 14
21.g even 6 1 147.7.b.d 14
21.h odd 6 1 inner 21.7.h.a 28
21.h odd 6 1 147.7.b.e 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.h.a 28 1.a even 1 1 trivial
21.7.h.a 28 3.b odd 2 1 inner
21.7.h.a 28 7.c even 3 1 inner
21.7.h.a 28 21.h odd 6 1 inner
147.7.b.d 14 7.d odd 6 1
147.7.b.d 14 21.g even 6 1
147.7.b.e 14 7.c even 3 1
147.7.b.e 14 21.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(21, [\chi])$$.