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Properties

Label 462.6.e.b
Level $462$
Weight $6$
Character orbit 462.e
Analytic conductor $74.097$
Analytic rank $0$
Dimension $40$
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] = [462,6,Mod(307,462)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(462, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("462.307");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 462.e (of order \(2\), degree \(1\), 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: \(74.0973247536\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 640 q^{4} + 1440 q^{6} - 324 q^{7} - 3240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 640 q^{4} + 1440 q^{6} - 324 q^{7} - 3240 q^{9} + 176 q^{10} + 296 q^{11} + 704 q^{14} - 396 q^{15} + 10240 q^{16} - 1068 q^{17} + 356 q^{19} - 1584 q^{21} - 2624 q^{22} + 2552 q^{23} - 23040 q^{24} - 31248 q^{25} + 5184 q^{28} + 5904 q^{33} + 19792 q^{35} + 51840 q^{36} + 17984 q^{37} - 2816 q^{40} + 572 q^{41} - 11664 q^{42} - 4736 q^{44} - 8164 q^{49} - 21840 q^{53} - 116640 q^{54} + 67576 q^{55} - 11264 q^{56} + 19488 q^{58} + 6336 q^{60} + 214984 q^{61} - 44048 q^{62} + 26244 q^{63} - 163840 q^{64} + 10656 q^{66} + 115632 q^{67} + 17088 q^{68} - 912 q^{70} + 110440 q^{71} + 202180 q^{73} - 5696 q^{76} - 144520 q^{77} + 262440 q^{81} - 5956 q^{83} + 25344 q^{84} + 201696 q^{86} - 43848 q^{87} + 41984 q^{88} - 14256 q^{90} - 32988 q^{91} - 40832 q^{92} + 99108 q^{93} - 89392 q^{94} + 368640 q^{96} + 218000 q^{98} - 23976 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 − 4.00000i 9.00000i −16.0000 − 97.2677i 36.0000 −128.280 + 18.7432i 64.0000i −81.0000 −389.071
307.2 − 4.00000i 9.00000i −16.0000 − 88.9905i 36.0000 87.8248 + 95.3615i 64.0000i −81.0000 −355.962
307.3 − 4.00000i 9.00000i −16.0000 − 71.1685i 36.0000 125.959 + 30.6792i 64.0000i −81.0000 −284.674
307.4 − 4.00000i 9.00000i −16.0000 − 68.4413i 36.0000 54.1671 − 117.783i 64.0000i −81.0000 −273.765
307.5 − 4.00000i 9.00000i −16.0000 − 68.2130i 36.0000 −121.044 − 46.4258i 64.0000i −81.0000 −272.852
307.6 − 4.00000i 9.00000i −16.0000 − 66.4688i 36.0000 −10.6708 + 129.202i 64.0000i −81.0000 −265.875
307.7 − 4.00000i 9.00000i −16.0000 − 31.5837i 36.0000 −102.553 − 79.3093i 64.0000i −81.0000 −126.335
307.8 − 4.00000i 9.00000i −16.0000 − 13.5812i 36.0000 −72.3343 + 107.586i 64.0000i −81.0000 −54.3249
307.9 − 4.00000i 9.00000i −16.0000 − 11.5893i 36.0000 −4.90454 − 129.549i 64.0000i −81.0000 −46.3570
307.10 − 4.00000i 9.00000i −16.0000 − 10.2355i 36.0000 −67.9842 + 110.386i 64.0000i −81.0000 −40.9420
307.11 − 4.00000i 9.00000i −16.0000 12.8705i 36.0000 −58.4352 − 115.725i 64.0000i −81.0000 51.4820
307.12 − 4.00000i 9.00000i −16.0000 20.9328i 36.0000 129.641 + 0.367215i 64.0000i −81.0000 83.7313
307.13 − 4.00000i 9.00000i −16.0000 23.7996i 36.0000 65.4231 + 111.923i 64.0000i −81.0000 95.1984
307.14 − 4.00000i 9.00000i −16.0000 33.9950i 36.0000 −65.0676 + 112.130i 64.0000i −81.0000 135.980
307.15 − 4.00000i 9.00000i −16.0000 37.0113i 36.0000 129.641 + 0.552677i 64.0000i −81.0000 148.045
307.16 − 4.00000i 9.00000i −16.0000 63.3083i 36.0000 80.7218 − 101.445i 64.0000i −81.0000 253.233
307.17 − 4.00000i 9.00000i −16.0000 68.7237i 36.0000 −29.6730 − 126.200i 64.0000i −81.0000 274.895
307.18 − 4.00000i 9.00000i −16.0000 83.3585i 36.0000 −126.012 + 30.4643i 64.0000i −81.0000 333.434
307.19 − 4.00000i 9.00000i −16.0000 101.397i 36.0000 −118.840 − 51.8072i 64.0000i −81.0000 405.587
307.20 − 4.00000i 9.00000i −16.0000 104.143i 36.0000 70.4196 + 108.849i 64.0000i −81.0000 416.573
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.6.e.b yes 40
7.b odd 2 1 462.6.e.a ✓ 40
11.b odd 2 1 462.6.e.a ✓ 40
77.b even 2 1 inner 462.6.e.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.6.e.a ✓ 40 7.b odd 2 1
462.6.e.a ✓ 40 11.b odd 2 1
462.6.e.b yes 40 1.a even 1 1 trivial
462.6.e.b yes 40 77.b even 2 1 inner