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Properties

Label 22.7.d.a
Level $22$
Weight $7$
Character orbit 22.d
Analytic conductor $5.061$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [22,7,Mod(7,22)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(22, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 22.d (of order \(10\), 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: \(5.06118983964\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 52 q^{3} + 192 q^{4} - 368 q^{5} + 400 q^{6} + 720 q^{7} - 3814 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 52 q^{3} + 192 q^{4} - 368 q^{5} + 400 q^{6} + 720 q^{7} - 3814 q^{9} + 4366 q^{11} + 3456 q^{12} + 3600 q^{13} - 4288 q^{14} - 2148 q^{15} - 6144 q^{16} - 5880 q^{17} + 18400 q^{18} - 23850 q^{19} + 11776 q^{20} - 1056 q^{22} - 40096 q^{23} + 12800 q^{24} + 69978 q^{25} + 52224 q^{26} + 4222 q^{27} - 32640 q^{28} - 204800 q^{29} - 134160 q^{30} + 71568 q^{31} + 11242 q^{33} + 88896 q^{34} + 469520 q^{35} + 25728 q^{36} + 309120 q^{37} - 73392 q^{38} - 328640 q^{39} - 107520 q^{40} - 541660 q^{41} - 177760 q^{42} + 213888 q^{44} + 811664 q^{45} + 213120 q^{46} - 299820 q^{47} + 53248 q^{48} - 374658 q^{49} - 364480 q^{50} - 1347330 q^{51} - 291840 q^{52} + 612820 q^{53} + 440004 q^{55} + 70656 q^{56} + 1416050 q^{57} + 552576 q^{58} + 927550 q^{59} + 206336 q^{60} - 1184160 q^{61} - 226640 q^{62} - 921200 q^{63} + 196608 q^{64} + 128672 q^{66} - 515100 q^{67} - 188160 q^{68} + 957220 q^{69} + 207984 q^{70} + 217572 q^{71} - 896000 q^{72} - 590340 q^{73} - 610400 q^{74} - 2208402 q^{75} - 2221964 q^{77} + 3228512 q^{78} + 93240 q^{79} + 196608 q^{80} + 1034076 q^{81} + 312960 q^{82} + 4588350 q^{83} + 2357760 q^{84} + 4132920 q^{85} + 1057632 q^{86} - 772608 q^{88} - 4638852 q^{89} - 6535040 q^{90} - 3916044 q^{91} - 2350208 q^{92} - 556628 q^{93} - 2410800 q^{94} + 661320 q^{95} - 342342 q^{97} + 4890070 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.32502 + 4.57649i −16.1429 + 49.6828i −9.88854 − 30.4338i −56.8190 + 41.2814i −173.698 − 239.074i 422.528 − 137.288i 172.160 + 55.9381i −1618.02 − 1175.56i − 397.293i
7.2 −3.32502 + 4.57649i 2.09392 − 6.44443i −9.88854 − 30.4338i 65.2204 − 47.3854i 22.5306 + 31.0106i 234.010 − 76.0344i 172.160 + 55.9381i 552.627 + 401.507i 456.037i
7.3 −3.32502 + 4.57649i 14.1734 − 43.6214i −9.88854 − 30.4338i −186.053 + 135.176i 152.506 + 209.906i −183.893 + 59.7505i 172.160 + 55.9381i −1112.17 − 808.036i − 1300.93i
7.4 3.32502 − 4.57649i −9.01575 + 27.7476i −9.88854 − 30.4338i −129.576 + 94.1428i 97.0093 + 133.522i −294.799 + 95.7861i −172.160 − 55.9381i −98.8743 − 71.8364i 906.032i
7.5 3.32502 − 4.57649i −6.58001 + 20.2512i −9.88854 − 30.4338i 135.508 − 98.4524i 70.8008 + 97.4489i 455.470 − 147.991i −172.160 − 55.9381i 222.959 + 161.989i − 947.508i
7.6 3.32502 − 4.57649i 10.5828 − 32.5704i −9.88854 − 30.4338i −0.778391 + 0.565534i −113.870 − 156.729i −144.738 + 47.0282i −172.160 − 55.9381i −359.065 − 260.876i 5.44271i
13.1 −5.37999 − 1.74806i −31.8069 + 23.1090i 25.8885 + 18.8091i −8.73605 − 26.8868i 211.517 − 68.7259i 205.916 − 283.418i −106.400 − 146.448i 252.375 − 776.731i 159.922i
13.2 −5.37999 − 1.74806i 0.0235888 − 0.0171383i 25.8885 + 18.8091i −27.6593 − 85.1266i −0.156866 + 0.0509689i −308.398 + 424.474i −106.400 − 146.448i −225.273 + 693.319i 506.330i
13.3 −5.37999 − 1.74806i 36.3463 − 26.4071i 25.8885 + 18.8091i 9.06738 + 27.9065i −241.704 + 78.5344i 121.970 − 167.877i −106.400 − 146.448i 398.443 − 1226.28i − 165.987i
13.4 5.37999 + 1.74806i −18.4229 + 13.3850i 25.8885 + 18.8091i 17.5479 + 54.0070i −122.513 + 39.8069i −314.104 + 432.327i 106.400 + 146.448i −65.0282 + 200.136i 321.232i
13.5 5.37999 + 1.74806i 10.9678 − 7.96859i 25.8885 + 18.8091i 46.4416 + 142.932i 72.9363 − 23.6984i 340.329 − 468.423i 106.400 + 146.448i −168.479 + 518.524i 850.157i
13.6 5.37999 + 1.74806i 33.7806 − 24.5431i 25.8885 + 18.8091i −48.1631 − 148.231i 224.642 − 72.9907i −174.289 + 239.888i 106.400 + 146.448i 313.496 − 964.842i − 881.671i
17.1 −5.37999 + 1.74806i −31.8069 − 23.1090i 25.8885 − 18.8091i −8.73605 + 26.8868i 211.517 + 68.7259i 205.916 + 283.418i −106.400 + 146.448i 252.375 + 776.731i − 159.922i
17.2 −5.37999 + 1.74806i 0.0235888 + 0.0171383i 25.8885 − 18.8091i −27.6593 + 85.1266i −0.156866 − 0.0509689i −308.398 − 424.474i −106.400 + 146.448i −225.273 − 693.319i − 506.330i
17.3 −5.37999 + 1.74806i 36.3463 + 26.4071i 25.8885 − 18.8091i 9.06738 − 27.9065i −241.704 − 78.5344i 121.970 + 167.877i −106.400 + 146.448i 398.443 + 1226.28i 165.987i
17.4 5.37999 − 1.74806i −18.4229 − 13.3850i 25.8885 − 18.8091i 17.5479 − 54.0070i −122.513 − 39.8069i −314.104 − 432.327i 106.400 − 146.448i −65.0282 − 200.136i − 321.232i
17.5 5.37999 − 1.74806i 10.9678 + 7.96859i 25.8885 − 18.8091i 46.4416 − 142.932i 72.9363 + 23.6984i 340.329 + 468.423i 106.400 − 146.448i −168.479 − 518.524i − 850.157i
17.6 5.37999 − 1.74806i 33.7806 + 24.5431i 25.8885 − 18.8091i −48.1631 + 148.231i 224.642 + 72.9907i −174.289 − 239.888i 106.400 − 146.448i 313.496 + 964.842i 881.671i
19.1 −3.32502 − 4.57649i −16.1429 − 49.6828i −9.88854 + 30.4338i −56.8190 − 41.2814i −173.698 + 239.074i 422.528 + 137.288i 172.160 − 55.9381i −1618.02 + 1175.56i 397.293i
19.2 −3.32502 − 4.57649i 2.09392 + 6.44443i −9.88854 + 30.4338i 65.2204 + 47.3854i 22.5306 − 31.0106i 234.010 + 76.0344i 172.160 − 55.9381i 552.627 − 401.507i − 456.037i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.7.d.a ✓ 24
11.c even 5 1 242.7.b.e 24
11.d odd 10 1 inner 22.7.d.a ✓ 24
11.d odd 10 1 242.7.b.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.7.d.a ✓ 24 1.a even 1 1 trivial
22.7.d.a ✓ 24 11.d odd 10 1 inner
242.7.b.e 24 11.c even 5 1
242.7.b.e 24 11.d odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(22, [\chi])\).