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Properties

Label 717.4.a.b
Level $717$
Weight $4$
Character orbit 717.a
Self dual yes
Analytic conductor $42.304$
Analytic rank $1$
Dimension $28$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [717,4,Mod(1,717)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(717, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("717.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 717 = 3 \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 717.a (trivial)

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: yes
Analytic conductor: \(42.3043694741\)
Analytic rank: \(1\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 5 q^{2} - 84 q^{3} + 103 q^{4} + 6 q^{5} + 15 q^{6} - 68 q^{7} - 39 q^{8} + 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 5 q^{2} - 84 q^{3} + 103 q^{4} + 6 q^{5} + 15 q^{6} - 68 q^{7} - 39 q^{8} + 252 q^{9} - 88 q^{10} - 110 q^{11} - 309 q^{12} - 82 q^{13} + 126 q^{14} - 18 q^{15} + 271 q^{16} + 100 q^{17} - 45 q^{18} - 292 q^{19} - 52 q^{20} + 204 q^{21} - 351 q^{22} - 276 q^{23} + 117 q^{24} + 386 q^{25} + 84 q^{26} - 756 q^{27} - 1010 q^{28} - 38 q^{29} + 264 q^{30} - 432 q^{31} - 452 q^{32} + 330 q^{33} - 524 q^{34} - 166 q^{35} + 927 q^{36} - 936 q^{37} - 41 q^{38} + 246 q^{39} - 1183 q^{40} + 1054 q^{41} - 378 q^{42} - 1804 q^{43} - 341 q^{44} + 54 q^{45} - 888 q^{46} - 560 q^{47} - 813 q^{48} + 1074 q^{49} - 1054 q^{50} - 300 q^{51} - 632 q^{52} - 160 q^{53} + 135 q^{54} - 842 q^{55} + 509 q^{56} + 876 q^{57} - 1266 q^{58} + 846 q^{59} + 156 q^{60} - 2220 q^{61} + 82 q^{62} - 612 q^{63} - 1565 q^{64} + 296 q^{65} + 1053 q^{66} - 4752 q^{67} - 1719 q^{68} + 828 q^{69} - 5601 q^{70} - 802 q^{71} - 351 q^{72} - 2732 q^{73} - 4581 q^{74} - 1158 q^{75} - 5614 q^{76} - 1008 q^{77} - 252 q^{78} - 3172 q^{79} - 732 q^{80} + 2268 q^{81} - 9709 q^{82} - 4780 q^{83} + 3030 q^{84} - 4624 q^{85} - 2009 q^{86} + 114 q^{87} - 9331 q^{88} + 4372 q^{89} - 792 q^{90} - 7398 q^{91} - 6138 q^{92} + 1296 q^{93} - 7068 q^{94} - 3160 q^{95} + 1356 q^{96} - 4846 q^{97} - 3772 q^{98} - 990 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.46382 −3.00000 21.8533 1.11553 16.3914 −33.8981 −75.6918 9.00000 −6.09507
1.2 −5.18584 −3.00000 18.8929 16.4632 15.5575 −3.62014 −56.4890 9.00000 −85.3753
1.3 −4.89329 −3.00000 15.9443 −2.65683 14.6799 −4.74739 −38.8739 9.00000 13.0007
1.4 −4.23906 −3.00000 9.96963 −4.25997 12.7172 23.9867 −8.34939 9.00000 18.0583
1.5 −4.16029 −3.00000 9.30804 −17.1180 12.4809 −25.6697 −5.44184 9.00000 71.2161
1.6 −3.95914 −3.00000 7.67483 3.41097 11.8774 25.2663 1.28740 9.00000 −13.5045
1.7 −3.55424 −3.00000 4.63263 7.97622 10.6627 −13.2372 11.9684 9.00000 −28.3494
1.8 −3.28508 −3.00000 2.79175 9.57892 9.85524 10.7223 17.1095 9.00000 −31.4675
1.9 −3.21293 −3.00000 2.32293 −16.5200 9.63879 −15.7443 18.2400 9.00000 53.0777
1.10 −2.22027 −3.00000 −3.07038 14.7142 6.66082 26.0557 24.5793 9.00000 −32.6696
1.11 −1.62942 −3.00000 −5.34499 14.4675 4.88826 −35.3781 21.7446 9.00000 −23.5737
1.12 −1.15681 −3.00000 −6.66179 −10.4070 3.47043 11.9554 16.9609 9.00000 12.0389
1.13 −0.709237 −3.00000 −7.49698 22.0711 2.12771 −0.837973 10.9910 9.00000 −15.6536
1.14 −0.609904 −3.00000 −7.62802 −11.6127 1.82971 −26.3682 9.53159 9.00000 7.08263
1.15 −0.410040 −3.00000 −7.83187 −19.1512 1.23012 −11.0347 6.49170 9.00000 7.85278
1.16 0.889333 −3.00000 −7.20909 6.34700 −2.66800 −11.3318 −13.5259 9.00000 5.64460
1.17 0.915825 −3.00000 −7.16127 7.54581 −2.74747 24.7091 −13.8851 9.00000 6.91064
1.18 1.36717 −3.00000 −6.13085 −15.7404 −4.10151 15.1668 −19.3193 9.00000 −21.5198
1.19 1.48544 −3.00000 −5.79347 −10.1836 −4.45632 8.86157 −20.4894 9.00000 −15.1271
1.20 1.61692 −3.00000 −5.38557 9.90974 −4.85076 −20.2386 −21.6434 9.00000 16.0233
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(239\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 717.4.a.b ✓ 28
3.b odd 2 1 2151.4.a.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.4.a.b ✓ 28 1.a even 1 1 trivial
2151.4.a.b 28 3.b odd 2 1