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Properties

Label 213.7.d.a
Level $213$
Weight $7$
Character orbit 213.d
Analytic conductor $49.002$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,7,Mod(70,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.70");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 213.d (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: \(49.0015198110\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 20 q^{2} + 2484 q^{4} + 2872 q^{8} + 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 20 q^{2} + 2484 q^{4} + 2872 q^{8} + 17496 q^{9} + 1332 q^{10} - 1944 q^{12} + 2592 q^{15} + 83244 q^{16} + 4860 q^{18} + 9600 q^{19} - 40796 q^{20} + 225480 q^{25} - 55424 q^{29} - 17496 q^{30} + 222808 q^{32} + 603612 q^{36} - 172512 q^{37} - 163232 q^{38} + 170940 q^{40} - 28656 q^{43} - 377136 q^{48} - 894936 q^{49} - 360912 q^{50} + 225504 q^{57} - 624600 q^{58} + 705996 q^{60} + 1972344 q^{64} + 1729592 q^{71} + 697896 q^{72} - 316224 q^{73} + 4305044 q^{74} - 707616 q^{75} + 356844 q^{76} + 2259344 q^{77} + 317040 q^{79} - 7347348 q^{80} + 4251528 q^{81} + 2242752 q^{83} - 151188 q^{86} - 2240784 q^{87} + 3982800 q^{89} + 323676 q^{90} - 511824 q^{91} - 2454848 q^{95} - 2438100 q^{96} + 3910100 q^{98}+O(q^{100}) \) 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 −15.3258 −15.5885 170.881 −186.706 238.906 639.188i −1638.05 243.000 2861.43
70.2 −15.3258 −15.5885 170.881 −186.706 238.906 − 639.188i −1638.05 243.000 2861.43
70.3 −14.8461 15.5885 156.405 −101.446 −231.427 17.4595i −1371.86 243.000 1506.08
70.4 −14.8461 15.5885 156.405 −101.446 −231.427 − 17.4595i −1371.86 243.000 1506.08
70.5 −14.1013 15.5885 134.846 186.015 −219.817 − 123.911i −999.022 243.000 −2623.05
70.6 −14.1013 15.5885 134.846 186.015 −219.817 123.911i −999.022 243.000 −2623.05
70.7 −13.3818 −15.5885 115.072 13.7535 208.601 − 231.027i −683.434 243.000 −184.047
70.8 −13.3818 −15.5885 115.072 13.7535 208.601 231.027i −683.434 243.000 −184.047
70.9 −12.6905 −15.5885 97.0493 53.4634 197.826 282.278i −419.412 243.000 −678.478
70.10 −12.6905 −15.5885 97.0493 53.4634 197.826 − 282.278i −419.412 243.000 −678.478
70.11 −10.8544 15.5885 53.8187 84.2090 −169.204 388.744i 110.513 243.000 −914.041
70.12 −10.8544 15.5885 53.8187 84.2090 −169.204 − 388.744i 110.513 243.000 −914.041
70.13 −10.8153 15.5885 52.9711 −26.0884 −168.594 − 596.211i 119.281 243.000 282.154
70.14 −10.8153 15.5885 52.9711 −26.0884 −168.594 596.211i 119.281 243.000 282.154
70.15 −10.5971 −15.5885 48.2992 234.073 165.193 530.309i 166.383 243.000 −2480.50
70.16 −10.5971 −15.5885 48.2992 234.073 165.193 − 530.309i 166.383 243.000 −2480.50
70.17 −10.0913 15.5885 37.8349 −161.881 −157.308 − 336.503i 264.040 243.000 1633.59
70.18 −10.0913 15.5885 37.8349 −161.881 −157.308 336.503i 264.040 243.000 1633.59
70.19 −9.95922 −15.5885 35.1861 −180.970 155.249 − 302.703i 286.964 243.000 1802.32
70.20 −9.95922 −15.5885 35.1861 −180.970 155.249 302.703i 286.964 243.000 1802.32
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.72
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 213.7.d.a ✓ 72
71.b odd 2 1 inner 213.7.d.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.7.d.a ✓ 72 1.a even 1 1 trivial
213.7.d.a ✓ 72 71.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(213, [\chi])\).