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Properties

Label 471.4.b.a
Level $471$
Weight $4$
Character orbit 471.b
Analytic conductor $27.790$
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] = [471,4,Mod(313,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.313");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 471.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: \(27.7898996127\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
313.1 − 5.39623i −3.00000 −21.1193 − 16.5163i 16.1887i − 25.8596i 70.7949i 9.00000 −89.1257
313.2 − 5.31164i −3.00000 −20.2135 13.7454i 15.9349i 18.9295i 64.8739i 9.00000 73.0104
313.3 − 5.19343i −3.00000 −18.9718 − 15.3454i 15.5803i 23.0710i 56.9811i 9.00000 −79.6953
313.4 − 5.06578i −3.00000 −17.6621 10.9807i 15.1973i − 24.6884i 48.9461i 9.00000 55.6257
313.5 − 4.67506i −3.00000 −13.8562 − 4.61864i 14.0252i − 11.3073i 27.3780i 9.00000 −21.5924
313.6 − 4.30375i −3.00000 −10.5223 − 4.32730i 12.9113i 21.9616i 10.8553i 9.00000 −18.6236
313.7 − 4.28965i −3.00000 −10.4011 − 1.35162i 12.8690i 9.03182i 10.3000i 9.00000 −5.79797
313.8 − 3.78788i −3.00000 −6.34802 − 9.26420i 11.3636i − 4.30898i − 6.25748i 9.00000 −35.0916
313.9 − 3.68048i −3.00000 −5.54594 17.9346i 11.0414i − 17.3085i − 9.03212i 9.00000 66.0080
313.10 − 3.30421i −3.00000 −2.91779 13.7388i 9.91262i 33.0358i − 16.7927i 9.00000 45.3958
313.11 − 3.18058i −3.00000 −2.11606 − 2.38612i 9.54173i − 14.3272i − 18.7143i 9.00000 −7.58923
313.12 − 2.64007i −3.00000 1.03005 − 13.5204i 7.92020i − 34.8670i − 23.8399i 9.00000 −35.6946
313.13 − 2.63265i −3.00000 1.06918 − 20.9406i 7.89794i 20.4039i − 23.8759i 9.00000 −55.1291
313.14 − 2.09701i −3.00000 3.60255 1.98891i 6.29103i 9.73610i − 24.3307i 9.00000 4.17077
313.15 − 1.46936i −3.00000 5.84098 − 4.94652i 4.40809i 12.8754i − 20.3374i 9.00000 −7.26823
313.16 − 1.38921i −3.00000 6.07011 4.30021i 4.16762i − 36.7534i − 19.5463i 9.00000 5.97388
313.17 − 0.999086i −3.00000 7.00183 10.0353i 2.99726i 4.69571i − 14.9881i 9.00000 10.0261
313.18 − 0.783695i −3.00000 7.38582 11.1397i 2.35108i 5.77258i − 12.0578i 9.00000 8.73009
313.19 − 0.500423i −3.00000 7.74958 10.0481i 1.50127i − 27.3201i − 7.88145i 9.00000 5.02829
313.20 − 0.275605i −3.00000 7.92404 − 19.4532i 0.826816i − 15.2471i − 4.38875i 9.00000 −5.36141
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 313.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.4.b.a ✓ 40
157.b even 2 1 inner 471.4.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.4.b.a ✓ 40 1.a even 1 1 trivial
471.4.b.a ✓ 40 157.b even 2 1 inner