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Properties

Label 717.4.a.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 11 q^{2} + 96 q^{3} + 147 q^{4} + 66 q^{5} + 33 q^{6} + 58 q^{7} + 153 q^{8} + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 11 q^{2} + 96 q^{3} + 147 q^{4} + 66 q^{5} + 33 q^{6} + 58 q^{7} + 153 q^{8} + 288 q^{9} + 52 q^{10} + 270 q^{11} + 441 q^{12} + 48 q^{13} + 184 q^{14} + 198 q^{15} + 775 q^{16} + 384 q^{17} + 99 q^{18} + 216 q^{19} + 534 q^{20} + 174 q^{21} + 437 q^{22} + 712 q^{23} + 459 q^{24} + 1190 q^{25} + 436 q^{26} + 864 q^{27} + 598 q^{28} + 562 q^{29} + 156 q^{30} + 384 q^{31} + 1770 q^{32} + 810 q^{33} + 452 q^{34} + 1026 q^{35} + 1323 q^{36} + 770 q^{37} + 733 q^{38} + 144 q^{39} + 877 q^{40} + 1648 q^{41} + 552 q^{42} + 1592 q^{43} + 1595 q^{44} + 594 q^{45} + 532 q^{46} + 1540 q^{47} + 2325 q^{48} + 2134 q^{49} + 1646 q^{50} + 1152 q^{51} - 144 q^{52} + 1708 q^{53} + 297 q^{54} + 1282 q^{55} + 2155 q^{56} + 648 q^{57} + 1086 q^{58} + 2396 q^{59} + 1602 q^{60} + 364 q^{61} + 2180 q^{62} + 522 q^{63} + 1663 q^{64} + 1520 q^{65} + 1311 q^{66} + 2728 q^{67} + 1545 q^{68} + 2136 q^{69} - 4609 q^{70} + 3322 q^{71} + 1377 q^{72} - 188 q^{73} + 1111 q^{74} + 3570 q^{75} - 3134 q^{76} + 556 q^{77} + 1308 q^{78} - 462 q^{79} + 6076 q^{80} + 2592 q^{81} - 7965 q^{82} + 4604 q^{83} + 1794 q^{84} - 852 q^{85} + 549 q^{86} + 1686 q^{87} - 1127 q^{88} + 6742 q^{89} + 468 q^{90} + 1390 q^{91} + 1802 q^{92} + 1152 q^{93} - 2796 q^{94} + 448 q^{95} + 5310 q^{96} - 1322 q^{97} + 1000 q^{98} + 2430 q^{99}+O(q^{100}) \) 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 −5.03814 3.00000 17.3829 −1.37408 −15.1144 −13.6849 −47.2724 9.00000 6.92282
1.2 −5.00513 3.00000 17.0513 22.0129 −15.0154 18.0649 −45.3032 9.00000 −110.178
1.3 −4.96520 3.00000 16.6532 −1.79703 −14.8956 −13.6883 −42.9649 9.00000 8.92262
1.4 −4.60028 3.00000 13.1625 −14.5426 −13.8008 −14.6801 −23.7491 9.00000 66.8999
1.5 −4.45062 3.00000 11.8081 8.02977 −13.3519 20.0356 −16.9482 9.00000 −35.7375
1.6 −4.00539 3.00000 8.04312 3.38255 −12.0162 23.4822 −0.172728 9.00000 −13.5484
1.7 −3.55300 3.00000 4.62382 −4.70427 −10.6590 −31.4434 11.9956 9.00000 16.7143
1.8 −2.72055 3.00000 −0.598607 19.6459 −8.16165 −3.08265 23.3929 9.00000 −53.4475
1.9 −2.40778 3.00000 −2.20261 −6.97053 −7.22333 19.5306 24.5656 9.00000 16.7835
1.10 −2.38222 3.00000 −2.32503 −9.46407 −7.14666 23.5031 24.5965 9.00000 22.5455
1.11 −2.01921 3.00000 −3.92279 10.5315 −6.05763 −9.21359 24.0746 9.00000 −21.2653
1.12 −1.76460 3.00000 −4.88618 −16.9871 −5.29381 −0.437244 22.7390 9.00000 29.9755
1.13 −0.752561 3.00000 −7.43365 3.00130 −2.25768 −33.6762 11.6148 9.00000 −2.25866
1.14 −0.560231 3.00000 −7.68614 18.3687 −1.68069 31.3786 8.78786 9.00000 −10.2907
1.15 −0.323742 3.00000 −7.89519 13.7872 −0.971225 10.6908 5.14594 9.00000 −4.46348
1.16 0.197960 3.00000 −7.96081 −0.664309 0.593879 12.2133 −3.15960 9.00000 −0.131506
1.17 0.980358 3.00000 −7.03890 −13.2849 2.94108 4.25684 −14.7435 9.00000 −13.0240
1.18 1.10380 3.00000 −6.78162 20.8736 3.31140 −33.6250 −16.3160 9.00000 23.0403
1.19 1.28682 3.00000 −6.34409 −14.8834 3.86046 −31.0382 −18.4583 9.00000 −19.1522
1.20 2.06669 3.00000 −3.72878 −0.727240 6.20008 31.6777 −24.2398 9.00000 −1.50298
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.d ✓ 32
3.b odd 2 1 2151.4.a.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.4.a.d ✓ 32 1.a even 1 1 trivial
2151.4.a.d 32 3.b odd 2 1