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Properties

Label 141.7.d.a
Level $141$
Weight $7$
Character orbit 141.d
Analytic conductor $32.438$
Analytic rank $0$
Dimension $48$
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] = [141,7,Mod(46,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, 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("141.46");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 141.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: \(32.4376257904\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 1404 q^{4} + 324 q^{6} - 288 q^{7} - 1932 q^{8} + 11664 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 1404 q^{4} + 324 q^{6} - 288 q^{7} - 1932 q^{8} + 11664 q^{9} + 1944 q^{12} - 18964 q^{14} + 48060 q^{16} - 288 q^{17} + 5184 q^{24} - 187440 q^{25} - 30312 q^{28} - 182548 q^{32} - 109920 q^{34} + 341172 q^{36} - 12096 q^{37} + 215460 q^{42} + 115768 q^{47} + 124416 q^{48} + 779904 q^{49} + 887340 q^{50} - 2592 q^{51} + 63504 q^{53} + 78732 q^{54} + 127248 q^{55} - 2002264 q^{56} + 165216 q^{59} + 955440 q^{61} - 69984 q^{63} + 1815660 q^{64} + 30000 q^{65} + 776648 q^{68} - 1109040 q^{71} - 469476 q^{72} - 376692 q^{74} + 1842912 q^{75} - 1523712 q^{79} + 2834352 q^{81} + 406064 q^{83} - 1423008 q^{84} - 5383968 q^{89} + 5556972 q^{94} + 3809952 q^{95} + 2079432 q^{96} - 2085840 q^{97} - 2907180 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −15.3521 −15.5885 171.687 − 222.642i 239.316 456.270 −1653.23 243.000 3418.02i
46.2 −15.3521 −15.5885 171.687 222.642i 239.316 456.270 −1653.23 243.000 − 3418.02i
46.3 −14.4157 15.5885 143.811 170.303i −224.718 −365.195 −1150.53 243.000 − 2455.04i
46.4 −14.4157 15.5885 143.811 − 170.303i −224.718 −365.195 −1150.53 243.000 2455.04i
46.5 −13.9897 15.5885 131.712 78.4361i −218.078 344.202 −947.267 243.000 − 1097.30i
46.6 −13.9897 15.5885 131.712 − 78.4361i −218.078 344.202 −947.267 243.000 1097.30i
46.7 −12.9077 −15.5885 102.608 − 0.224856i 201.211 −192.904 −498.340 243.000 2.90237i
46.8 −12.9077 −15.5885 102.608 0.224856i 201.211 −192.904 −498.340 243.000 − 2.90237i
46.9 −9.01763 15.5885 17.3177 − 77.0432i −140.571 −263.011 420.964 243.000 694.748i
46.10 −9.01763 15.5885 17.3177 77.0432i −140.571 −263.011 420.964 243.000 − 694.748i
46.11 −8.87833 −15.5885 14.8247 − 243.420i 138.399 −480.503 436.594 243.000 2161.17i
46.12 −8.87833 −15.5885 14.8247 243.420i 138.399 −480.503 436.594 243.000 − 2161.17i
46.13 −8.29908 −15.5885 4.87466 − 80.2800i 129.370 524.661 490.686 243.000 666.250i
46.14 −8.29908 −15.5885 4.87466 80.2800i 129.370 524.661 490.686 243.000 − 666.250i
46.15 −8.03022 15.5885 0.484428 − 140.071i −125.179 375.123 510.044 243.000 1124.80i
46.16 −8.03022 15.5885 0.484428 140.071i −125.179 375.123 510.044 243.000 − 1124.80i
46.17 −5.02088 −15.5885 −38.7908 140.255i 78.2678 −59.7439 516.100 243.000 − 704.203i
46.18 −5.02088 −15.5885 −38.7908 − 140.255i 78.2678 −59.7439 516.100 243.000 704.203i
46.19 −2.68751 −15.5885 −56.7773 46.0471i 41.8942 −242.590 324.591 243.000 − 123.752i
46.20 −2.68751 −15.5885 −56.7773 − 46.0471i 41.8942 −242.590 324.591 243.000 123.752i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.7.d.a ✓ 48
47.b odd 2 1 inner 141.7.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.7.d.a ✓ 48 1.a even 1 1 trivial
141.7.d.a ✓ 48 47.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(141, [\chi])\).