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Properties

Label 717.4.a.c
Level $717$
Weight $4$
Character orbit 717.a
Self dual yes
Analytic conductor $42.304$
Analytic rank $0$
Dimension $32$
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: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{2} - 96 q^{3} + 151 q^{4} - 14 q^{5} - 9 q^{6} + 72 q^{7} + 57 q^{8} + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 3 q^{2} - 96 q^{3} + 151 q^{4} - 14 q^{5} - 9 q^{6} + 72 q^{7} + 57 q^{8} + 288 q^{9} + 32 q^{10} + 154 q^{11} - 453 q^{12} + 100 q^{13} + 42 q^{14} + 42 q^{15} + 719 q^{16} + 32 q^{17} + 27 q^{18} + 202 q^{19} - 132 q^{20} - 216 q^{21} + 265 q^{22} + 552 q^{23} - 171 q^{24} + 1086 q^{25} - 280 q^{26} - 864 q^{27} + 390 q^{28} - 154 q^{29} - 96 q^{30} + 560 q^{31} + 444 q^{32} - 462 q^{33} + 156 q^{34} + 394 q^{35} + 1359 q^{36} + 914 q^{37} + 111 q^{38} - 300 q^{39} + 257 q^{40} - 914 q^{41} - 126 q^{42} + 1722 q^{43} + 1243 q^{44} - 126 q^{45} + 584 q^{46} + 380 q^{47} - 2157 q^{48} + 2446 q^{49} - 454 q^{50} - 96 q^{51} + 1552 q^{52} + 370 q^{53} - 81 q^{54} + 918 q^{55} - 499 q^{56} - 606 q^{57} + 2446 q^{58} + 492 q^{59} + 396 q^{60} + 668 q^{61} + 578 q^{62} + 648 q^{63} + 6475 q^{64} + 736 q^{65} - 795 q^{66} + 4548 q^{67} + 5253 q^{68} - 1656 q^{69} + 7793 q^{70} + 258 q^{71} + 513 q^{72} + 3096 q^{73} + 449 q^{74} - 3258 q^{75} + 6814 q^{76} + 3804 q^{77} + 840 q^{78} + 2864 q^{79} - 1052 q^{80} + 2592 q^{81} + 14145 q^{82} + 2364 q^{83} - 1170 q^{84} + 3088 q^{85} + 2811 q^{86} + 462 q^{87} + 8329 q^{88} - 4172 q^{89} + 288 q^{90} + 7350 q^{91} + 13644 q^{92} - 1680 q^{93} + 6122 q^{94} + 3336 q^{95} - 1332 q^{96} + 6370 q^{97} + 1572 q^{98} + 1386 q^{99}+O(q^{100}) \) 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 −5.50779 −3.00000 22.3357 −17.7046 16.5234 13.2402 −78.9579 9.00000 97.5130
1.2 −5.21887 −3.00000 19.2366 15.9312 15.6566 27.7991 −58.6425 9.00000 −83.1428
1.3 −4.99597 −3.00000 16.9597 4.15481 14.9879 −1.12589 −44.7623 9.00000 −20.7573
1.4 −4.82034 −3.00000 15.2357 −17.0250 14.4610 −0.894290 −34.8785 9.00000 82.0662
1.5 −4.48359 −3.00000 12.1026 16.9253 13.4508 −28.5300 −18.3942 9.00000 −75.8862
1.6 −3.93268 −3.00000 7.46593 −8.10545 11.7980 −18.6867 2.10031 9.00000 31.8761
1.7 −3.63482 −3.00000 5.21194 19.5442 10.9045 −3.64725 10.1341 9.00000 −71.0396
1.8 −3.45734 −3.00000 3.95318 −16.3211 10.3720 31.8168 13.9912 9.00000 56.4275
1.9 −2.55583 −3.00000 −1.46775 −2.17194 7.66748 −33.4848 24.1979 9.00000 5.55111
1.10 −2.48373 −3.00000 −1.83107 −6.58803 7.45120 7.57709 24.4178 9.00000 16.3629
1.11 −2.43690 −3.00000 −2.06152 −1.94921 7.31070 −13.0213 24.5189 9.00000 4.75002
1.12 −2.17667 −3.00000 −3.26211 7.52900 6.53001 −1.76751 24.5139 9.00000 −16.3881
1.13 −1.64226 −3.00000 −5.30297 −14.9594 4.92679 8.99120 21.8470 9.00000 24.5672
1.14 −0.770960 −3.00000 −7.40562 6.93927 2.31288 33.2057 11.8771 9.00000 −5.34989
1.15 −0.646340 −3.00000 −7.58224 −13.9777 1.93902 32.2049 10.0714 9.00000 9.03432
1.16 −0.319477 −3.00000 −7.89793 10.3242 0.958432 16.6816 5.07903 9.00000 −3.29834
1.17 −0.0917574 −3.00000 −7.99158 10.9396 0.275272 −5.88604 1.46735 9.00000 −1.00379
1.18 0.568479 −3.00000 −7.67683 0.988402 −1.70544 −12.3996 −8.91196 9.00000 0.561886
1.19 1.44213 −3.00000 −5.92026 −4.95653 −4.32639 −22.8735 −20.0748 9.00000 −7.14796
1.20 2.35460 −3.00000 −2.45587 −5.04318 −7.06379 7.28879 −24.6194 9.00000 −11.8747
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
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.c ✓ 32
3.b odd 2 1 2151.4.a.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.4.a.c ✓ 32 1.a even 1 1 trivial
2151.4.a.e 32 3.b odd 2 1