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Properties

Label 71.7.b.b
Level $71$
Weight $7$
Character orbit 71.b
Analytic conductor $16.334$
Analytic rank $0$
Dimension $28$
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] = [71,7,Mod(70,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.70");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 71.b (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: \(16.3338399370\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 12 q^{2} - 34 q^{3} + 548 q^{4} + 142 q^{5} + 380 q^{6} - 1836 q^{8} + 3426 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 12 q^{2} - 34 q^{3} + 548 q^{4} + 142 q^{5} + 380 q^{6} - 1836 q^{8} + 3426 q^{9} - 1840 q^{10} - 4180 q^{12} - 7364 q^{15} - 10572 q^{16} + 44968 q^{18} - 6722 q^{19} - 33136 q^{20} - 128284 q^{24} - 2326 q^{25} - 86428 q^{27} + 38942 q^{29} - 145792 q^{30} - 203156 q^{32} - 73976 q^{36} + 117214 q^{37} + 321628 q^{38} - 234704 q^{40} - 37298 q^{43} + 459362 q^{45} + 653196 q^{48} - 1536404 q^{49} + 248148 q^{50} + 191940 q^{54} + 253324 q^{57} + 93560 q^{58} + 1637936 q^{60} - 943604 q^{64} + 1601412 q^{71} - 54600 q^{72} + 590398 q^{73} + 4137272 q^{74} + 495284 q^{75} + 1639548 q^{76} + 1322072 q^{77} - 172466 q^{79} + 554088 q^{80} - 1911976 q^{81} - 2232338 q^{83} - 1995220 q^{86} - 5630050 q^{87} - 3718682 q^{89} + 6577304 q^{90} - 945600 q^{91} + 1428444 q^{95} - 4221140 q^{96} - 2005992 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −14.4944 26.7399 146.087 54.6661 −387.578 − 615.169i −1189.80 −13.9781 −792.351
70.2 −14.4944 26.7399 146.087 54.6661 −387.578 615.169i −1189.80 −13.9781 −792.351
70.3 −13.2953 −18.5071 112.766 −127.054 246.058 − 69.3469i −648.354 −386.487 1689.23
70.4 −13.2953 −18.5071 112.766 −127.054 246.058 69.3469i −648.354 −386.487 1689.23
70.5 −10.7741 −12.3976 52.0815 163.160 133.573 − 261.756i 128.411 −575.300 −1757.90
70.6 −10.7741 −12.3976 52.0815 163.160 133.573 261.756i 128.411 −575.300 −1757.90
70.7 −7.88919 −44.7892 −1.76067 −62.0245 353.351 − 439.971i 518.798 1277.08 489.323
70.8 −7.88919 −44.7892 −1.76067 −62.0245 353.351 439.971i 518.798 1277.08 489.323
70.9 −6.86839 20.1516 −16.8253 10.5959 −138.409 − 117.907i 555.139 −322.914 −72.7767
70.10 −6.86839 20.1516 −16.8253 10.5959 −138.409 117.907i 555.139 −322.914 −72.7767
70.11 −4.14831 −1.00971 −46.7915 −167.043 4.18861 − 630.146i 459.598 −727.980 692.947
70.12 −4.14831 −1.00971 −46.7915 −167.043 4.18861 630.146i 459.598 −727.980 692.947
70.13 −1.92748 −32.0660 −60.2848 148.345 61.8066 − 313.349i 239.556 299.231 −285.931
70.14 −1.92748 −32.0660 −60.2848 148.345 61.8066 313.349i 239.556 299.231 −285.931
70.15 0.995399 23.3039 −63.0092 190.349 23.1966 − 544.071i −126.425 −185.930 189.473
70.16 0.995399 23.3039 −63.0092 190.349 23.1966 544.071i −126.425 −185.930 189.473
70.17 1.92003 40.6008 −60.3135 −105.737 77.9549 − 10.0845i −238.686 919.427 −203.018
70.18 1.92003 40.6008 −60.3135 −105.737 77.9549 10.0845i −238.686 919.427 −203.018
70.19 7.56318 −0.853524 −6.79824 39.3939 −6.45536 − 216.894i −535.460 −728.271 297.944
70.20 7.56318 −0.853524 −6.79824 39.3939 −6.45536 216.894i −535.460 −728.271 297.944
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.7.b.b ✓ 28
71.b odd 2 1 inner 71.7.b.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.7.b.b ✓ 28 1.a even 1 1 trivial
71.7.b.b ✓ 28 71.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 6 T_{2}^{13} - 567 T_{2}^{12} - 2912 T_{2}^{11} + 123506 T_{2}^{10} + 535904 T_{2}^{9} + \cdots - 162057945088 \) acting on \(S_{7}^{\mathrm{new}}(71, [\chi])\). Copy content Toggle raw display