Properties

Label 717.4.a.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 13 q^{2} + 84 q^{3} + 99 q^{4} - 74 q^{5} - 39 q^{6} - 82 q^{7} - 135 q^{8} + 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 13 q^{2} + 84 q^{3} + 99 q^{4} - 74 q^{5} - 39 q^{6} - 82 q^{7} - 135 q^{8} + 252 q^{9} - 68 q^{10} - 258 q^{11} + 297 q^{12} - 134 q^{13} - 292 q^{14} - 222 q^{15} + 327 q^{16} - 364 q^{17} - 117 q^{18} - 278 q^{19} - 986 q^{20} - 246 q^{21} - 179 q^{22} - 668 q^{23} - 405 q^{24} + 490 q^{25} - 760 q^{26} + 756 q^{27} - 802 q^{28} - 714 q^{29} - 204 q^{30} - 608 q^{31} - 918 q^{32} - 774 q^{33} - 228 q^{34} - 934 q^{35} + 891 q^{36} - 1080 q^{37} - 1395 q^{38} - 402 q^{39} - 563 q^{40} - 1796 q^{41} - 876 q^{42} - 1934 q^{43} - 3157 q^{44} - 666 q^{45} - 940 q^{46} - 2032 q^{47} + 981 q^{48} + 762 q^{49} - 1754 q^{50} - 1092 q^{51} - 2328 q^{52} - 1790 q^{53} - 351 q^{54} - 478 q^{55} - 3557 q^{56} - 834 q^{57} - 2626 q^{58} - 3622 q^{59} - 2958 q^{60} + 324 q^{61} - 796 q^{62} - 738 q^{63} + 2023 q^{64} - 2200 q^{65} - 537 q^{66} - 2444 q^{67} + 357 q^{68} - 2004 q^{69} + 4305 q^{70} - 1298 q^{71} - 1215 q^{72} - 1368 q^{73} + 813 q^{74} + 1470 q^{75} + 1390 q^{76} - 1408 q^{77} - 2280 q^{78} - 1378 q^{79} - 7684 q^{80} + 2268 q^{81} + 9001 q^{82} - 3524 q^{83} - 2406 q^{84} + 60 q^{85} - 2543 q^{86} - 2142 q^{87} + 1749 q^{88} - 7854 q^{89} - 612 q^{90} + 850 q^{91} - 496 q^{92} - 1824 q^{93} + 6634 q^{94} - 3696 q^{95} - 2754 q^{96} - 1746 q^{97} - 4632 q^{98} - 2322 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −5.49015 3.00000 22.1417 −8.06682 −16.4704 32.6199 −77.6401 9.00000 44.2880
1.2 −5.29084 3.00000 19.9930 −20.9477 −15.8725 −16.3120 −63.4528 9.00000 110.831
1.3 −5.18205 3.00000 18.8537 5.12080 −15.5462 5.19788 −56.2444 9.00000 −26.5363
1.4 −4.77227 3.00000 14.7746 17.3570 −14.3168 −36.1394 −32.3302 9.00000 −82.8322
1.5 −4.22853 3.00000 9.88049 −18.4854 −12.6856 24.6174 −7.95172 9.00000 78.1662
1.6 −3.94640 3.00000 7.57406 3.12472 −11.8392 13.3014 1.68092 9.00000 −12.3314
1.7 −3.72975 3.00000 5.91105 8.83024 −11.1893 −20.7717 7.79125 9.00000 −32.9346
1.8 −3.71471 3.00000 5.79908 −14.6178 −11.1441 −2.58079 8.17580 9.00000 54.3008
1.9 −3.06927 3.00000 1.42044 10.0975 −9.20782 1.48241 20.1945 9.00000 −30.9919
1.10 −2.60026 3.00000 −1.23865 −11.0636 −7.80078 −16.8612 24.0229 9.00000 28.7682
1.11 −2.03138 3.00000 −3.87348 11.5727 −6.09415 −13.0852 24.1196 9.00000 −23.5086
1.12 −1.64745 3.00000 −5.28592 5.57976 −4.94234 35.6078 21.8878 9.00000 −9.19235
1.13 −1.27488 3.00000 −6.37468 −17.5425 −3.82464 −26.4099 18.3260 9.00000 22.3646
1.14 −0.797577 3.00000 −7.36387 −7.16277 −2.39273 0.569206 12.2539 9.00000 5.71286
1.15 −0.608170 3.00000 −7.63013 −1.98658 −1.82451 −5.67520 9.50578 9.00000 1.20818
1.16 0.404541 3.00000 −7.83635 −0.184451 1.21362 12.9030 −6.40644 9.00000 −0.0746179
1.17 0.456628 3.00000 −7.79149 14.2736 1.36988 −13.0550 −7.21084 9.00000 6.51773
1.18 0.686651 3.00000 −7.52851 −17.8875 2.05995 15.8088 −10.6627 9.00000 −12.2825
1.19 1.55579 3.00000 −5.57952 17.3600 4.66737 −1.16704 −21.1269 9.00000 27.0085
1.20 2.01135 3.00000 −3.95446 −6.74089 6.03406 7.15060 −24.0446 9.00000 −13.5583
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.a 28
3.b odd 2 1 2151.4.a.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.4.a.a 28 1.a even 1 1 trivial
2151.4.a.c 28 3.b odd 2 1