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Properties

Label 62.6.g.a
Level $62$
Weight $6$
Character orbit 62.g
Analytic conductor $9.944$
Analytic rank $0$
Dimension $56$
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] = [62,6,Mod(7,62)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([28]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("62.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 62.g (of order \(15\), degree \(8\), 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: \(9.94379682840\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 56 q^{2} + 11 q^{3} - 224 q^{4} - 33 q^{5} - 136 q^{6} + 104 q^{7} - 896 q^{8} + 524 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 56 q^{2} + 11 q^{3} - 224 q^{4} - 33 q^{5} - 136 q^{6} + 104 q^{7} - 896 q^{8} + 524 q^{9} + 448 q^{10} - 795 q^{11} + 176 q^{12} + 764 q^{13} - 1384 q^{14} - 3208 q^{15} - 3584 q^{16} + 2440 q^{17} + 2716 q^{18} - 2747 q^{19} + 1952 q^{20} + 4909 q^{21} + 7500 q^{22} - 11297 q^{23} + 384 q^{24} - 21385 q^{25} - 14064 q^{26} - 12466 q^{27} + 1424 q^{28} - 11901 q^{29} + 20008 q^{30} - 4745 q^{31} + 57344 q^{32} + 2812 q^{33} + 7160 q^{34} + 17912 q^{35} - 38496 q^{36} - 54498 q^{37} + 9552 q^{38} - 38158 q^{39} + 7808 q^{40} + 56747 q^{41} + 61396 q^{42} + 69670 q^{43} - 22320 q^{44} + 54688 q^{45} + 22412 q^{46} - 63503 q^{47} + 1536 q^{48} + 9721 q^{49} + 16580 q^{50} + 87134 q^{51} + 12224 q^{52} + 48127 q^{53} - 49864 q^{54} - 150047 q^{55} - 6464 q^{56} - 142312 q^{57} - 36764 q^{58} - 105890 q^{59} - 51328 q^{60} + 383664 q^{61} - 10700 q^{62} - 236690 q^{63} - 57344 q^{64} + 284638 q^{65} + 46348 q^{66} - 59674 q^{67} - 75120 q^{68} + 213337 q^{69} + 71648 q^{70} - 186679 q^{71} + 43456 q^{72} - 132007 q^{73} - 29932 q^{74} + 186494 q^{75} + 97728 q^{76} + 208527 q^{77} + 71688 q^{78} - 229841 q^{79} - 82688 q^{80} - 503164 q^{81} - 234012 q^{82} - 634514 q^{83} - 165056 q^{84} + 129149 q^{85} + 60740 q^{86} + 297951 q^{87} - 33600 q^{88} + 78674 q^{89} + 290652 q^{90} + 127680 q^{91} + 182208 q^{92} + 427164 q^{93} + 143528 q^{94} + 293666 q^{95} + 11264 q^{96} - 94321 q^{97} - 291156 q^{98} - 168255 q^{99}+O(q^{100}) \) 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 1.23607 + 3.80423i −27.0434 + 5.74826i −12.9443 + 9.40456i −22.1421 + 38.3512i −55.2952 − 95.7741i −126.485 − 56.3148i −51.7771 − 37.6183i 476.314 − 212.069i −173.266 − 36.8288i
7.2 1.23607 + 3.80423i −17.2972 + 3.67663i −12.9443 + 9.40456i 40.0278 − 69.3302i −35.3672 − 61.2578i 103.163 + 45.9309i −51.7771 − 37.6183i 63.6829 − 28.3535i 313.225 + 66.5780i
7.3 1.23607 + 3.80423i −12.7527 + 2.71067i −12.9443 + 9.40456i −35.7192 + 61.8675i −26.0752 − 45.1636i 142.150 + 63.2892i −51.7771 − 37.6183i −66.7080 + 29.7003i −279.509 − 59.4116i
7.4 1.23607 + 3.80423i 0.364351 − 0.0774453i −12.9443 + 9.40456i 24.1093 − 41.7585i 0.744982 + 1.29035i −145.977 − 64.9929i −51.7771 − 37.6183i −221.865 + 98.7806i 188.659 + 40.1008i
7.5 1.23607 + 3.80423i 9.01184 − 1.91553i −12.9443 + 9.40456i −5.70663 + 9.88417i 18.4263 + 31.9154i 138.364 + 61.6036i −51.7771 − 37.6183i −144.447 + 64.3121i −44.6554 − 9.49180i
7.6 1.23607 + 3.80423i 14.1119 − 2.99958i −12.9443 + 9.40456i −38.5801 + 66.8227i 28.8543 + 49.9772i −121.647 − 54.1607i −51.7771 − 37.6183i −31.8433 + 14.1775i −301.896 − 64.1700i
7.7 1.23607 + 3.80423i 26.3846 − 5.60821i −12.9443 + 9.40456i 32.3363 − 56.0080i 53.9480 + 93.4407i 18.2040 + 8.10493i −51.7771 − 37.6183i 442.702 − 197.103i 253.037 + 53.7847i
9.1 1.23607 − 3.80423i −27.0434 − 5.74826i −12.9443 − 9.40456i −22.1421 − 38.3512i −55.2952 + 95.7741i −126.485 + 56.3148i −51.7771 + 37.6183i 476.314 + 212.069i −173.266 + 36.8288i
9.2 1.23607 − 3.80423i −17.2972 − 3.67663i −12.9443 − 9.40456i 40.0278 + 69.3302i −35.3672 + 61.2578i 103.163 − 45.9309i −51.7771 + 37.6183i 63.6829 + 28.3535i 313.225 − 66.5780i
9.3 1.23607 − 3.80423i −12.7527 − 2.71067i −12.9443 − 9.40456i −35.7192 − 61.8675i −26.0752 + 45.1636i 142.150 − 63.2892i −51.7771 + 37.6183i −66.7080 − 29.7003i −279.509 + 59.4116i
9.4 1.23607 − 3.80423i 0.364351 + 0.0774453i −12.9443 − 9.40456i 24.1093 + 41.7585i 0.744982 − 1.29035i −145.977 + 64.9929i −51.7771 + 37.6183i −221.865 − 98.7806i 188.659 − 40.1008i
9.5 1.23607 − 3.80423i 9.01184 + 1.91553i −12.9443 − 9.40456i −5.70663 − 9.88417i 18.4263 − 31.9154i 138.364 − 61.6036i −51.7771 + 37.6183i −144.447 − 64.3121i −44.6554 + 9.49180i
9.6 1.23607 − 3.80423i 14.1119 + 2.99958i −12.9443 − 9.40456i −38.5801 − 66.8227i 28.8543 − 49.9772i −121.647 + 54.1607i −51.7771 + 37.6183i −31.8433 − 14.1775i −301.896 + 64.1700i
9.7 1.23607 − 3.80423i 26.3846 + 5.60821i −12.9443 − 9.40456i 32.3363 + 56.0080i 53.9480 − 93.4407i 18.2040 − 8.10493i −51.7771 + 37.6183i 442.702 + 197.103i 253.037 − 53.7847i
19.1 −3.23607 − 2.35114i −22.6412 − 10.0805i 4.94427 + 15.2169i −16.0118 + 27.7332i 49.5677 + 85.8537i −88.9543 + 98.7938i 19.7771 − 60.8676i 248.407 + 275.884i 117.020 − 52.1007i
19.2 −3.23607 − 2.35114i −19.7571 − 8.79643i 4.94427 + 15.2169i 43.2265 − 74.8705i 43.2537 + 74.9175i 134.279 − 149.132i 19.7771 − 60.8676i 150.367 + 166.999i −315.915 + 140.655i
19.3 −3.23607 − 2.35114i −3.13695 − 1.39666i 4.94427 + 15.2169i 9.06224 − 15.6963i 6.86764 + 11.8951i −9.78851 + 10.8712i 19.7771 − 60.8676i −154.709 − 171.822i −66.2302 + 29.4876i
19.4 −3.23607 − 2.35114i 0.780866 + 0.347664i 4.94427 + 15.2169i −50.4812 + 87.4360i −1.70953 − 2.96099i 94.0979 − 104.506i 19.7771 − 60.8676i −162.110 − 180.041i 368.935 − 164.260i
19.5 −3.23607 − 2.35114i 11.9179 + 5.30621i 4.94427 + 15.2169i 39.0913 − 67.7080i −26.0916 − 45.1920i −41.6783 + 46.2885i 19.7771 − 60.8676i −48.7172 − 54.1060i −285.693 + 127.199i
19.6 −3.23607 − 2.35114i 17.0150 + 7.57554i 4.94427 + 15.2169i −15.8394 + 27.4346i −37.2504 − 64.5195i −142.903 + 158.710i 19.7771 − 60.8676i 69.5210 + 77.2109i 115.760 − 51.5396i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.6.g.a ✓ 56
31.g even 15 1 inner 62.6.g.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.6.g.a ✓ 56 1.a even 1 1 trivial
62.6.g.a ✓ 56 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 11 T_{3}^{55} - 1052 T_{3}^{54} + 18115 T_{3}^{53} + 280411 T_{3}^{52} + \cdots + 14\!\cdots\!21 \) acting on \(S_{6}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display