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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9} - 930 q^{10} - 404 q^{12} + 3632 q^{13} + 5450 q^{15} - 13706 q^{16} + 1612 q^{18} + 6830 q^{19} - 11333 q^{21} - 27348 q^{22} - 54498 q^{24} - 7640 q^{25} + 13268 q^{27} + 88522 q^{28} + 32380 q^{30} + 69410 q^{31} - 85459 q^{33} - 31344 q^{34} + 399028 q^{36} - 71830 q^{37} + 104570 q^{39} + 230250 q^{40} - 467782 q^{42} - 498904 q^{43} - 85375 q^{45} - 342660 q^{46} - 324728 q^{48} + 395668 q^{49} + 42177 q^{51} + 218000 q^{52} - 402122 q^{54} + 231660 q^{55} + 786782 q^{57} + 459522 q^{58} + 1021150 q^{60} + 153158 q^{61} + 179011 q^{63} - 1399412 q^{64} + 80896 q^{66} - 977098 q^{67} - 1520742 q^{69} - 139650 q^{70} - 1001352 q^{72} - 488350 q^{73} + 1434610 q^{75} + 6196280 q^{76} + 245176 q^{78} + 380762 q^{79} + 893531 q^{81} - 3665004 q^{82} + 1231496 q^{84} - 912540 q^{85} + 1452416 q^{87} - 1522482 q^{88} - 11713820 q^{90} - 1529416 q^{91} - 1690071 q^{93} - 3034044 q^{94} + 7549346 q^{96} + 7212176 q^{97} + 435866 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 2.14
Significant digits:
Format:

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 \(S_{7}^{\mathrm{new}}(21, [\chi])\).