LMFDB - Embedded newform 1950.2.a.be.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1950,2,Mod(1,1950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,2,2,0,-2,2,-2,2,0,0,2,2,-2,0,2,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.5708283941$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1950.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.47214 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.23607 q^{14} +1.00000 q^{16} +7.23607 q^{17} -1.00000 q^{18} -2.76393 q^{19} +3.23607 q^{21} -4.47214 q^{22} +2.76393 q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +3.23607 q^{28} -3.70820 q^{29} -4.00000 q^{31} -1.00000 q^{32} +4.47214 q^{33} -7.23607 q^{34} +1.00000 q^{36} -10.9443 q^{37} +2.76393 q^{38} +1.00000 q^{39} +3.52786 q^{41} -3.23607 q^{42} -2.47214 q^{43} +4.47214 q^{44} -2.76393 q^{46} +12.9443 q^{47} +1.00000 q^{48} +3.47214 q^{49} +7.23607 q^{51} +1.00000 q^{52} -0.472136 q^{53} -1.00000 q^{54} -3.23607 q^{56} -2.76393 q^{57} +3.70820 q^{58} -8.47214 q^{59} -10.9443 q^{61} +4.00000 q^{62} +3.23607 q^{63} +1.00000 q^{64} -4.47214 q^{66} +7.23607 q^{68} +2.76393 q^{69} -2.47214 q^{71} -1.00000 q^{72} +13.2361 q^{73} +10.9443 q^{74} -2.76393 q^{76} +14.4721 q^{77} -1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} -3.52786 q^{82} -4.94427 q^{83} +3.23607 q^{84} +2.47214 q^{86} -3.70820 q^{87} -4.47214 q^{88} +0.472136 q^{89} +3.23607 q^{91} +2.76393 q^{92} -4.00000 q^{93} -12.9443 q^{94} -1.00000 q^{96} -3.70820 q^{97} -3.47214 q^{98} +4.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 2 q^{26} + 2 q^{27}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 2 * q^18 - 10 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 2 * q^26 + 2 * q^27 + 2 * q^28 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 10 * q^34 + 2 * q^36 - 4 * q^37 + 10 * q^38 + 2 * q^39 + 16 * q^41 - 2 * q^42 + 4 * q^43 - 10 * q^46 + 8 * q^47 + 2 * q^48 - 2 * q^49 + 10 * q^51 + 2 * q^52 + 8 * q^53 - 2 * q^54 - 2 * q^56 - 10 * q^57 - 6 * q^58 - 8 * q^59 - 4 * q^61 + 8 * q^62 + 2 * q^63 + 2 * q^64 + 10 * q^68 + 10 * q^69 + 4 * q^71 - 2 * q^72 + 22 * q^73 + 4 * q^74 - 10 * q^76 + 20 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^81 - 16 * q^82 + 8 * q^83 + 2 * q^84 - 4 * q^86 + 6 * q^87 - 8 * q^89 + 2 * q^91 + 10 * q^92 - 8 * q^93 - 8 * q^94 - 2 * q^96 + 6 * q^97 + 2 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.23607	−0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	7.23607	1.75500	0.877502	−	0.479573i	$-0.159208\pi$
$17$	7.23607	1.75500	0.877502	+	0.479573i	$0.159208\pi$
$18$	−1.00000	−0.235702
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	−4.47214	−0.953463
$23$	2.76393	0.576320	0.288160	−	0.957582i	$-0.406957\pi$
$23$	2.76393	0.576320	0.288160	+	0.957582i	$0.406957\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	3.23607	0.611559
$29$	−3.70820	−0.688596	−0.344298	−	0.938860i	$-0.611883\pi$
$29$	−3.70820	−0.688596	−0.344298	+	0.938860i	$0.611883\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	4.47214	0.778499
$34$	−7.23607	−1.24098
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	2.76393	0.448369
$39$	1.00000	0.160128
$40$	0	0
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	−3.23607	−0.499336
$43$	−2.47214	−0.376997	−0.188499	−	0.982073i	$-0.560362\pi$
$43$	−2.47214	−0.376997	−0.188499	+	0.982073i	$0.560362\pi$
$44$	4.47214	0.674200
$45$	0	0
$46$	−2.76393	−0.407520
$47$	12.9443	1.88812	0.944058	−	0.329779i	$-0.106974\pi$
$47$	12.9443	1.88812	0.944058	+	0.329779i	$0.106974\pi$
$48$	1.00000	0.144338
$49$	3.47214	0.496019
$50$	0	0
$51$	7.23607	1.01325
$52$	1.00000	0.138675
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.23607	−0.432438
$57$	−2.76393	−0.366092
$58$	3.70820	0.486911
$59$	−8.47214	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$59$	−8.47214	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$60$	0	0
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	4.00000	0.508001
$63$	3.23607	0.407706
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.47214	−0.550482
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	7.23607	0.877502
$69$	2.76393	0.332738
$70$	0	0
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	−1.00000	−0.117851
$73$	13.2361	1.54916	0.774582	−	0.632473i	$-0.217960\pi$
$73$	13.2361	1.54916	0.774582	+	0.632473i	$0.217960\pi$
$74$	10.9443	1.27225
$75$	0	0
$76$	−2.76393	−0.317045
$77$	14.4721	1.64925
$78$	−1.00000	−0.113228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.52786	−0.389587
$83$	−4.94427	−0.542704	−0.271352	−	0.962480i	$-0.587471\pi$
$83$	−4.94427	−0.542704	−0.271352	+	0.962480i	$0.587471\pi$
$84$	3.23607	0.353084
$85$	0	0
$86$	2.47214	0.266577
$87$	−3.70820	−0.397561
$88$	−4.47214	−0.476731
$89$	0.472136	0.0500463	0.0250232	−	0.999687i	$-0.492034\pi$
$89$	0.472136	0.0500463	0.0250232	+	0.999687i	$0.492034\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	2.76393	0.288160
$93$	−4.00000	−0.414781
$94$	−12.9443	−1.33510
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−3.70820	−0.376511	−0.188256	−	0.982120i	$-0.560283\pi$
$97$	−3.70820	−0.376511	−0.188256	+	0.982120i	$0.560283\pi$
$98$	−3.47214	−0.350739
$99$	4.47214	0.449467
$100$	0	0
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	−7.23607	−0.716477
$103$	8.47214	0.834784	0.417392	−	0.908726i	$-0.362944\pi$
$103$	8.47214	0.834784	0.417392	+	0.908726i	$0.362944\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0.472136	0.0458579
$107$	−1.52786	−0.147704	−0.0738521	−	0.997269i	$-0.523529\pi$
$107$	−1.52786	−0.147704	−0.0738521	+	0.997269i	$0.523529\pi$
$108$	1.00000	0.0962250
$109$	−2.29180	−0.219514	−0.109757	−	0.993958i	$-0.535007\pi$
$109$	−2.29180	−0.219514	−0.109757	+	0.993958i	$0.535007\pi$
$110$	0	0
$111$	−10.9443	−1.03878
$112$	3.23607	0.305780
$113$	18.6525	1.75468	0.877339	−	0.479872i	$-0.159317\pi$
$113$	18.6525	1.75468	0.877339	+	0.479872i	$0.159317\pi$
$114$	2.76393	0.258866
$115$	0	0
$116$	−3.70820	−0.344298
$117$	1.00000	0.0924500
$118$	8.47214	0.779923
$119$	23.4164	2.14658
$120$	0	0
$121$	9.00000	0.818182
$122$	10.9443	0.990848
$123$	3.52786	0.318097
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−3.23607	−0.288292
$127$	21.4164	1.90040	0.950199	−	0.311642i	$-0.100879\pi$
$127$	21.4164	1.90040	0.950199	+	0.311642i	$0.100879\pi$
$128$	−1.00000	−0.0883883
$129$	−2.47214	−0.217659
$130$	0	0
$131$	9.70820	0.848210	0.424105	−	0.905613i	$-0.360589\pi$
$131$	9.70820	0.848210	0.424105	+	0.905613i	$0.360589\pi$
$132$	4.47214	0.389249
$133$	−8.94427	−0.775567
$134$	0	0
$135$	0	0
$136$	−7.23607	−0.620488
$137$	−19.8885	−1.69919	−0.849596	−	0.527433i	$-0.823154\pi$
$137$	−19.8885	−1.69919	−0.849596	+	0.527433i	$0.823154\pi$
$138$	−2.76393	−0.235282
$139$	−6.47214	−0.548959	−0.274480	−	0.961593i	$-0.588506\pi$
$139$	−6.47214	−0.548959	−0.274480	+	0.961593i	$0.588506\pi$
$140$	0	0
$141$	12.9443	1.09010
$142$	2.47214	0.207457
$143$	4.47214	0.373979
$144$	1.00000	0.0833333
$145$	0	0
$146$	−13.2361	−1.09542
$147$	3.47214	0.286377
$148$	−10.9443	−0.899614
$149$	13.5279	1.10825	0.554123	−	0.832435i	$-0.313054\pi$
$149$	13.5279	1.10825	0.554123	+	0.832435i	$0.313054\pi$
$150$	0	0
$151$	−19.4164	−1.58008	−0.790042	−	0.613052i	$-0.789942\pi$
$151$	−19.4164	−1.58008	−0.790042	+	0.613052i	$0.789942\pi$
$152$	2.76393	0.224184
$153$	7.23607	0.585001
$154$	−14.4721	−1.16620
$155$	0	0
$156$	1.00000	0.0800641
$157$	−18.9443	−1.51192	−0.755959	−	0.654619i	$-0.772829\pi$
$157$	−18.9443	−1.51192	−0.755959	+	0.654619i	$0.772829\pi$
$158$	4.00000	0.318223
$159$	−0.472136	−0.0374428
$160$	0	0
$161$	8.94427	0.704907
$162$	−1.00000	−0.0785674
$163$	12.9443	1.01387	0.506937	−	0.861983i	$-0.330778\pi$
$163$	12.9443	1.01387	0.506937	+	0.861983i	$0.330778\pi$
$164$	3.52786	0.275480
$165$	0	0
$166$	4.94427	0.383750
$167$	−19.4164	−1.50249	−0.751243	−	0.660025i	$-0.770546\pi$
$167$	−19.4164	−1.50249	−0.751243	+	0.660025i	$0.770546\pi$
$168$	−3.23607	−0.249668
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.76393	−0.211363
$172$	−2.47214	−0.188499
$173$	−22.9443	−1.74442	−0.872210	−	0.489131i	$-0.837314\pi$
$173$	−22.9443	−1.74442	−0.872210	+	0.489131i	$0.837314\pi$
$174$	3.70820	0.281118
$175$	0	0
$176$	4.47214	0.337100
$177$	−8.47214	−0.636805
$178$	−0.472136	−0.0353881
$179$	8.18034	0.611427	0.305714	−	0.952124i	$-0.401105\pi$
$179$	8.18034	0.611427	0.305714	+	0.952124i	$0.401105\pi$
$180$	0	0
$181$	17.4164	1.29455	0.647276	−	0.762256i	$-0.275908\pi$
$181$	17.4164	1.29455	0.647276	+	0.762256i	$0.275908\pi$
$182$	−3.23607	−0.239873
$183$	−10.9443	−0.809024
$184$	−2.76393	−0.203760
$185$	0	0
$186$	4.00000	0.293294
$187$	32.3607	2.36645
$188$	12.9443	0.944058
$189$	3.23607	0.235389
$190$	0	0
$191$	22.4721	1.62603	0.813013	−	0.582245i	$-0.197826\pi$
$191$	22.4721	1.62603	0.813013	+	0.582245i	$0.197826\pi$
$192$	1.00000	0.0721688
$193$	−8.29180	−0.596857	−0.298428	−	0.954432i	$-0.596462\pi$
$193$	−8.29180	−0.596857	−0.298428	+	0.954432i	$0.596462\pi$
$194$	3.70820	0.266234
$195$	0	0
$196$	3.47214	0.248010
$197$	−14.9443	−1.06474	−0.532368	−	0.846513i	$-0.678698\pi$
$197$	−14.9443	−1.06474	−0.532368	+	0.846513i	$0.678698\pi$
$198$	−4.47214	−0.317821
$199$	−0.944272	−0.0669377	−0.0334688	−	0.999440i	$-0.510655\pi$
$199$	−0.944272	−0.0669377	−0.0334688	+	0.999440i	$0.510655\pi$
$200$	0	0
$201$	0	0
$202$	9.23607	0.649847
$203$	−12.0000	−0.842235
$204$	7.23607	0.506626
$205$	0	0
$206$	−8.47214	−0.590282
$207$	2.76393	0.192107
$208$	1.00000	0.0693375
$209$	−12.3607	−0.855006
$210$	0	0
$211$	13.8885	0.956127	0.478063	−	0.878325i	$-0.341339\pi$
$211$	13.8885	0.956127	0.478063	+	0.878325i	$0.341339\pi$
$212$	−0.472136	−0.0324264
$213$	−2.47214	−0.169388
$214$	1.52786	0.104443
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−12.9443	−0.878714
$218$	2.29180	0.155220
$219$	13.2361	0.894411
$220$	0	0
$221$	7.23607	0.486751
$222$	10.9443	0.734531
$223$	16.7639	1.12260	0.561298	−	0.827614i	$-0.310302\pi$
$223$	16.7639	1.12260	0.561298	+	0.827614i	$0.310302\pi$
$224$	−3.23607	−0.216219
$225$	0	0
$226$	−18.6525	−1.24074
$227$	7.05573	0.468305	0.234153	−	0.972200i	$-0.424768\pi$
$227$	7.05573	0.468305	0.234153	+	0.972200i	$0.424768\pi$
$228$	−2.76393	−0.183046
$229$	9.70820	0.641536	0.320768	−	0.947158i	$-0.396059\pi$
$229$	9.70820	0.641536	0.320768	+	0.947158i	$0.396059\pi$
$230$	0	0
$231$	14.4721	0.952197
$232$	3.70820	0.243456
$233$	20.1803	1.32206	0.661029	−	0.750360i	$-0.270120\pi$
$233$	20.1803	1.32206	0.661029	+	0.750360i	$0.270120\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−8.47214	−0.551489
$237$	−4.00000	−0.259828
$238$	−23.4164	−1.51786
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	3.52786	0.227250	0.113625	−	0.993524i	$-0.463754\pi$
$241$	3.52786	0.227250	0.113625	+	0.993524i	$0.463754\pi$
$242$	−9.00000	−0.578542
$243$	1.00000	0.0641500
$244$	−10.9443	−0.700635
$245$	0	0
$246$	−3.52786	−0.224928
$247$	−2.76393	−0.175865
$248$	4.00000	0.254000
$249$	−4.94427	−0.313331
$250$	0	0
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	3.23607	0.203853
$253$	12.3607	0.777109
$254$	−21.4164	−1.34378
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.763932	0.0476528	0.0238264	−	0.999716i	$-0.492415\pi$
$257$	0.763932	0.0476528	0.0238264	+	0.999716i	$0.492415\pi$
$258$	2.47214	0.153908
$259$	−35.4164	−2.20067
$260$	0	0
$261$	−3.70820	−0.229532
$262$	−9.70820	−0.599775
$263$	10.1803	0.627747	0.313873	−	0.949465i	$-0.398373\pi$
$263$	10.1803	0.627747	0.313873	+	0.949465i	$0.398373\pi$
$264$	−4.47214	−0.275241
$265$	0	0
$266$	8.94427	0.548408
$267$	0.472136	0.0288943
$268$	0	0
$269$	6.76393	0.412404	0.206202	−	0.978509i	$-0.433890\pi$
$269$	6.76393	0.412404	0.206202	+	0.978509i	$0.433890\pi$
$270$	0	0
$271$	−11.4164	−0.693497	−0.346749	−	0.937958i	$-0.612714\pi$
$271$	−11.4164	−0.693497	−0.346749	+	0.937958i	$0.612714\pi$
$272$	7.23607	0.438751
$273$	3.23607	0.195856
$274$	19.8885	1.20151
$275$	0	0
$276$	2.76393	0.166369
$277$	20.8328	1.25172	0.625861	−	0.779934i	$-0.284748\pi$
$277$	20.8328	1.25172	0.625861	+	0.779934i	$0.284748\pi$
$278$	6.47214	0.388173
$279$	−4.00000	−0.239474
$280$	0	0
$281$	24.4721	1.45989	0.729943	−	0.683508i	$-0.239547\pi$
$281$	24.4721	1.45989	0.729943	+	0.683508i	$0.239547\pi$
$282$	−12.9443	−0.770820
$283$	−11.4164	−0.678635	−0.339318	−	0.940672i	$-0.610196\pi$
$283$	−11.4164	−0.678635	−0.339318	+	0.940672i	$0.610196\pi$
$284$	−2.47214	−0.146694
$285$	0	0
$286$	−4.47214	−0.264443
$287$	11.4164	0.673889
$288$	−1.00000	−0.0589256
$289$	35.3607	2.08004
$290$	0	0
$291$	−3.70820	−0.217379
$292$	13.2361	0.774582
$293$	2.94427	0.172006	0.0860031	−	0.996295i	$-0.472591\pi$
$293$	2.94427	0.172006	0.0860031	+	0.996295i	$0.472591\pi$
$294$	−3.47214	−0.202499
$295$	0	0
$296$	10.9443	0.636123
$297$	4.47214	0.259500
$298$	−13.5279	−0.783648
$299$	2.76393	0.159842
$300$	0	0
$301$	−8.00000	−0.461112
$302$	19.4164	1.11729
$303$	−9.23607	−0.530598
$304$	−2.76393	−0.158522
$305$	0	0
$306$	−7.23607	−0.413658
$307$	−24.3607	−1.39034	−0.695169	−	0.718847i	$-0.744670\pi$
$307$	−24.3607	−1.39034	−0.695169	+	0.718847i	$0.744670\pi$
$308$	14.4721	0.824626
$309$	8.47214	0.481963
$310$	0	0
$311$	−22.4721	−1.27428	−0.637139	−	0.770749i	$-0.719882\pi$
$311$	−22.4721	−1.27428	−0.637139	+	0.770749i	$0.719882\pi$
$312$	−1.00000	−0.0566139
$313$	−19.4164	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$313$	−19.4164	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$314$	18.9443	1.06909
$315$	0	0
$316$	−4.00000	−0.225018
$317$	13.4164	0.753541	0.376770	−	0.926307i	$-0.377035\pi$
$317$	13.4164	0.753541	0.376770	+	0.926307i	$0.377035\pi$
$318$	0.472136	0.0264761
$319$	−16.5836	−0.928503
$320$	0	0
$321$	−1.52786	−0.0852771
$322$	−8.94427	−0.498445
$323$	−20.0000	−1.11283
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.9443	−0.716917
$327$	−2.29180	−0.126737
$328$	−3.52786	−0.194794
$329$	41.8885	2.30939
$330$	0	0
$331$	−2.76393	−0.151919	−0.0759597	−	0.997111i	$-0.524202\pi$
$331$	−2.76393	−0.151919	−0.0759597	+	0.997111i	$0.524202\pi$
$332$	−4.94427	−0.271352
$333$	−10.9443	−0.599742
$334$	19.4164	1.06242
$335$	0	0
$336$	3.23607	0.176542
$337$	−6.47214	−0.352560	−0.176280	−	0.984340i	$-0.556406\pi$
$337$	−6.47214	−0.352560	−0.176280	+	0.984340i	$0.556406\pi$
$338$	−1.00000	−0.0543928
$339$	18.6525	1.01306
$340$	0	0
$341$	−17.8885	−0.968719
$342$	2.76393	0.149456
$343$	−11.4164	−0.616428
$344$	2.47214	0.133289
$345$	0	0
$346$	22.9443	1.23349
$347$	1.52786	0.0820200	0.0410100	−	0.999159i	$-0.486942\pi$
$347$	1.52786	0.0820200	0.0410100	+	0.999159i	$0.486942\pi$
$348$	−3.70820	−0.198781
$349$	−15.2361	−0.815568	−0.407784	−	0.913078i	$-0.633698\pi$
$349$	−15.2361	−0.815568	−0.407784	+	0.913078i	$0.633698\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−4.47214	−0.238366
$353$	−26.9443	−1.43410	−0.717049	−	0.697022i	$-0.754508\pi$
$353$	−26.9443	−1.43410	−0.717049	+	0.697022i	$0.754508\pi$
$354$	8.47214	0.450289
$355$	0	0
$356$	0.472136	0.0250232
$357$	23.4164	1.23933
$358$	−8.18034	−0.432344
$359$	6.47214	0.341586	0.170793	−	0.985307i	$-0.445367\pi$
$359$	6.47214	0.341586	0.170793	+	0.985307i	$0.445367\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	−17.4164	−0.915386
$363$	9.00000	0.472377
$364$	3.23607	0.169616
$365$	0	0
$366$	10.9443	0.572066
$367$	0.472136	0.0246453	0.0123226	−	0.999924i	$-0.496077\pi$
$367$	0.472136	0.0246453	0.0123226	+	0.999924i	$0.496077\pi$
$368$	2.76393	0.144080
$369$	3.52786	0.183653
$370$	0	0
$371$	−1.52786	−0.0793227
$372$	−4.00000	−0.207390
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−32.3607	−1.67333
$375$	0	0
$376$	−12.9443	−0.667550
$377$	−3.70820	−0.190982
$378$	−3.23607	−0.166445
$379$	−35.1246	−1.80423	−0.902115	−	0.431496i	$-0.857986\pi$
$379$	−35.1246	−1.80423	−0.902115	+	0.431496i	$0.857986\pi$
$380$	0	0
$381$	21.4164	1.09720
$382$	−22.4721	−1.14977
$383$	5.52786	0.282461	0.141230	−	0.989977i	$-0.454894\pi$
$383$	5.52786	0.282461	0.141230	+	0.989977i	$0.454894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	8.29180	0.422041
$387$	−2.47214	−0.125666
$388$	−3.70820	−0.188256
$389$	12.6525	0.641506	0.320753	−	0.947163i	$-0.396064\pi$
$389$	12.6525	0.641506	0.320753	+	0.947163i	$0.396064\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	−3.47214	−0.175369
$393$	9.70820	0.489714
$394$	14.9443	0.752882
$395$	0	0
$396$	4.47214	0.224733
$397$	17.4164	0.874104	0.437052	−	0.899436i	$-0.356022\pi$
$397$	17.4164	0.874104	0.437052	+	0.899436i	$0.356022\pi$
$398$	0.944272	0.0473321
$399$	−8.94427	−0.447774
$400$	0	0
$401$	−24.4721	−1.22208	−0.611040	−	0.791600i	$-0.709249\pi$
$401$	−24.4721	−1.22208	−0.611040	+	0.791600i	$0.709249\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−9.23607	−0.459512
$405$	0	0
$406$	12.0000	0.595550
$407$	−48.9443	−2.42608
$408$	−7.23607	−0.358239
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−19.8885	−0.981030
$412$	8.47214	0.417392
$413$	−27.4164	−1.34907
$414$	−2.76393	−0.135840
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−6.47214	−0.316942
$418$	12.3607	0.604581
$419$	27.5967	1.34819	0.674095	−	0.738645i	$-0.264534\pi$
$419$	27.5967	1.34819	0.674095	+	0.738645i	$0.264534\pi$
$420$	0	0
$421$	−13.7082	−0.668097	−0.334048	−	0.942556i	$-0.608415\pi$
$421$	−13.7082	−0.668097	−0.334048	+	0.942556i	$0.608415\pi$
$422$	−13.8885	−0.676084
$423$	12.9443	0.629372
$424$	0.472136	0.0229289
$425$	0	0
$426$	2.47214	0.119775
$427$	−35.4164	−1.71392
$428$	−1.52786	−0.0738521
$429$	4.47214	0.215917
$430$	0	0
$431$	−14.8328	−0.714472	−0.357236	−	0.934014i	$-0.616281\pi$
$431$	−14.8328	−0.714472	−0.357236	+	0.934014i	$0.616281\pi$
$432$	1.00000	0.0481125
$433$	−9.52786	−0.457880	−0.228940	−	0.973441i	$-0.573526\pi$
$433$	−9.52786	−0.457880	−0.228940	+	0.973441i	$0.573526\pi$
$434$	12.9443	0.621345
$435$	0	0
$436$	−2.29180	−0.109757
$437$	−7.63932	−0.365438
$438$	−13.2361	−0.632444
$439$	−4.94427	−0.235977	−0.117989	−	0.993015i	$-0.537645\pi$
$439$	−4.94427	−0.235977	−0.117989	+	0.993015i	$0.537645\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	−7.23607	−0.344185
$443$	−22.4721	−1.06768	−0.533842	−	0.845584i	$-0.679252\pi$
$443$	−22.4721	−1.06768	−0.533842	+	0.845584i	$0.679252\pi$
$444$	−10.9443	−0.519392
$445$	0	0
$446$	−16.7639	−0.793795
$447$	13.5279	0.639846
$448$	3.23607	0.152890
$449$	0.472136	0.0222815	0.0111407	−	0.999938i	$-0.496454\pi$
$449$	0.472136	0.0222815	0.0111407	+	0.999938i	$0.496454\pi$
$450$	0	0
$451$	15.7771	0.742914
$452$	18.6525	0.877339
$453$	−19.4164	−0.912262
$454$	−7.05573	−0.331142
$455$	0	0
$456$	2.76393	0.129433
$457$	−3.70820	−0.173462	−0.0867312	−	0.996232i	$-0.527642\pi$
$457$	−3.70820	−0.173462	−0.0867312	+	0.996232i	$0.527642\pi$
$458$	−9.70820	−0.453635
$459$	7.23607	0.337751
$460$	0	0
$461$	41.8885	1.95094	0.975472	−	0.220124i	$-0.0706461\pi$
$461$	41.8885	1.95094	0.975472	+	0.220124i	$0.0706461\pi$
$462$	−14.4721	−0.673305
$463$	−22.2918	−1.03599	−0.517994	−	0.855384i	$-0.673321\pi$
$463$	−22.2918	−1.03599	−0.517994	+	0.855384i	$0.673321\pi$
$464$	−3.70820	−0.172149
$465$	0	0
$466$	−20.1803	−0.934836
$467$	6.47214	0.299495	0.149747	−	0.988724i	$-0.452154\pi$
$467$	6.47214	0.299495	0.149747	+	0.988724i	$0.452154\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	−18.9443	−0.872906
$472$	8.47214	0.389962
$473$	−11.0557	−0.508343
$474$	4.00000	0.183726
$475$	0	0
$476$	23.4164	1.07329
$477$	−0.472136	−0.0216176
$478$	21.8885	1.00116
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−10.9443	−0.499016
$482$	−3.52786	−0.160690
$483$	8.94427	0.406978
$484$	9.00000	0.409091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−23.5967	−1.06927	−0.534635	−	0.845083i	$-0.679551\pi$
$487$	−23.5967	−1.06927	−0.534635	+	0.845083i	$0.679551\pi$
$488$	10.9443	0.495424
$489$	12.9443	0.585360
$490$	0	0
$491$	−16.1803	−0.730209	−0.365104	−	0.930967i	$-0.618967\pi$
$491$	−16.1803	−0.730209	−0.365104	+	0.930967i	$0.618967\pi$
$492$	3.52786	0.159048
$493$	−26.8328	−1.20849
$494$	2.76393	0.124355
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−8.00000	−0.358849
$498$	4.94427	0.221558
$499$	−39.1246	−1.75146	−0.875729	−	0.482803i	$-0.839619\pi$
$499$	−39.1246	−1.75146	−0.875729	+	0.482803i	$0.839619\pi$
$500$	0	0
$501$	−19.4164	−0.867461
$502$	−2.29180	−0.102288
$503$	−30.7639	−1.37170	−0.685848	−	0.727745i	$-0.740569\pi$
$503$	−30.7639	−1.37170	−0.685848	+	0.727745i	$0.740569\pi$
$504$	−3.23607	−0.144146
$505$	0	0
$506$	−12.3607	−0.549499
$507$	1.00000	0.0444116
$508$	21.4164	0.950199
$509$	−15.0557	−0.667333	−0.333667	−	0.942691i	$-0.608286\pi$
$509$	−15.0557	−0.667333	−0.333667	+	0.942691i	$0.608286\pi$
$510$	0	0
$511$	42.8328	1.89481
$512$	−1.00000	−0.0441942
$513$	−2.76393	−0.122031
$514$	−0.763932	−0.0336956
$515$	0	0
$516$	−2.47214	−0.108830
$517$	57.8885	2.54594
$518$	35.4164	1.55611
$519$	−22.9443	−1.00714
$520$	0	0
$521$	20.8328	0.912702	0.456351	−	0.889800i	$-0.349156\pi$
$521$	20.8328	0.912702	0.456351	+	0.889800i	$0.349156\pi$
$522$	3.70820	0.162304
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	9.70820	0.424105
$525$	0	0
$526$	−10.1803	−0.443884
$527$	−28.9443	−1.26083
$528$	4.47214	0.194625
$529$	−15.3607	−0.667856
$530$	0	0
$531$	−8.47214	−0.367659
$532$	−8.94427	−0.387783
$533$	3.52786	0.152809
$534$	−0.472136	−0.0204313
$535$	0	0
$536$	0	0
$537$	8.18034	0.353008
$538$	−6.76393	−0.291614
$539$	15.5279	0.668832
$540$	0	0
$541$	−4.76393	−0.204817	−0.102409	−	0.994742i	$-0.532655\pi$
$541$	−4.76393	−0.204817	−0.102409	+	0.994742i	$0.532655\pi$
$542$	11.4164	0.490377
$543$	17.4164	0.747410
$544$	−7.23607	−0.310244
$545$	0	0
$546$	−3.23607	−0.138491
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−19.8885	−0.849596
$549$	−10.9443	−0.467090
$550$	0	0
$551$	10.2492	0.436632
$552$	−2.76393	−0.117641
$553$	−12.9443	−0.550446
$554$	−20.8328	−0.885102
$555$	0	0
$556$	−6.47214	−0.274480
$557$	19.3050	0.817977	0.408989	−	0.912540i	$-0.365882\pi$
$557$	19.3050	0.817977	0.408989	+	0.912540i	$0.365882\pi$
$558$	4.00000	0.169334
$559$	−2.47214	−0.104560
$560$	0	0
$561$	32.3607	1.36627
$562$	−24.4721	−1.03229
$563$	−19.4164	−0.818304	−0.409152	−	0.912466i	$-0.634175\pi$
$563$	−19.4164	−0.818304	−0.409152	+	0.912466i	$0.634175\pi$
$564$	12.9443	0.545052
$565$	0	0
$566$	11.4164	0.479867
$567$	3.23607	0.135902
$568$	2.47214	0.103729
$569$	34.9443	1.46494	0.732470	−	0.680799i	$-0.238367\pi$
$569$	34.9443	1.46494	0.732470	+	0.680799i	$0.238367\pi$
$570$	0	0
$571$	−8.58359	−0.359212	−0.179606	−	0.983739i	$-0.557482\pi$
$571$	−8.58359	−0.359212	−0.179606	+	0.983739i	$0.557482\pi$
$572$	4.47214	0.186989
$573$	22.4721	0.938787
$574$	−11.4164	−0.476512
$575$	0	0
$576$	1.00000	0.0416667
$577$	−43.1246	−1.79530	−0.897651	−	0.440708i	$-0.854727\pi$
$577$	−43.1246	−1.79530	−0.897651	+	0.440708i	$0.854727\pi$
$578$	−35.3607	−1.47081
$579$	−8.29180	−0.344595
$580$	0	0
$581$	−16.0000	−0.663792
$582$	3.70820	0.153710
$583$	−2.11146	−0.0874476
$584$	−13.2361	−0.547712
$585$	0	0
$586$	−2.94427	−0.121627
$587$	−2.11146	−0.0871491	−0.0435746	−	0.999050i	$-0.513875\pi$
$587$	−2.11146	−0.0871491	−0.0435746	+	0.999050i	$0.513875\pi$
$588$	3.47214	0.143188
$589$	11.0557	0.455543
$590$	0	0
$591$	−14.9443	−0.614725
$592$	−10.9443	−0.449807
$593$	3.52786	0.144872	0.0724360	−	0.997373i	$-0.476923\pi$
$593$	3.52786	0.144872	0.0724360	+	0.997373i	$0.476923\pi$
$594$	−4.47214	−0.183494
$595$	0	0
$596$	13.5279	0.554123
$597$	−0.944272	−0.0386465
$598$	−2.76393	−0.113026
$599$	−6.11146	−0.249707	−0.124854	−	0.992175i	$-0.539846\pi$
$599$	−6.11146	−0.249707	−0.124854	+	0.992175i	$0.539846\pi$
$600$	0	0
$601$	13.4164	0.547267	0.273633	−	0.961834i	$-0.411775\pi$
$601$	13.4164	0.547267	0.273633	+	0.961834i	$0.411775\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−19.4164	−0.790042
$605$	0	0
$606$	9.23607	0.375190
$607$	1.41641	0.0574902	0.0287451	−	0.999587i	$-0.490849\pi$
$607$	1.41641	0.0574902	0.0287451	+	0.999587i	$0.490849\pi$
$608$	2.76393	0.112092
$609$	−12.0000	−0.486265
$610$	0	0
$611$	12.9443	0.523669
$612$	7.23607	0.292501
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	24.3607	0.983117
$615$	0	0
$616$	−14.4721	−0.583099
$617$	35.8885	1.44482	0.722409	−	0.691466i	$-0.243035\pi$
$617$	35.8885	1.44482	0.722409	+	0.691466i	$0.243035\pi$
$618$	−8.47214	−0.340799
$619$	−25.2361	−1.01432	−0.507162	−	0.861851i	$-0.669305\pi$
$619$	−25.2361	−1.01432	−0.507162	+	0.861851i	$0.669305\pi$
$620$	0	0
$621$	2.76393	0.110913
$622$	22.4721	0.901051
$623$	1.52786	0.0612126
$624$	1.00000	0.0400320
$625$	0	0
$626$	19.4164	0.776036
$627$	−12.3607	−0.493638
$628$	−18.9443	−0.755959
$629$	−79.1935	−3.15765
$630$	0	0
$631$	−12.5836	−0.500945	−0.250472	−	0.968124i	$-0.580586\pi$
$631$	−12.5836	−0.500945	−0.250472	+	0.968124i	$0.580586\pi$
$632$	4.00000	0.159111
$633$	13.8885	0.552020
$634$	−13.4164	−0.532834
$635$	0	0
$636$	−0.472136	−0.0187214
$637$	3.47214	0.137571
$638$	16.5836	0.656551
$639$	−2.47214	−0.0977962
$640$	0	0
$641$	−23.3050	−0.920490	−0.460245	−	0.887792i	$-0.652238\pi$
$641$	−23.3050	−0.920490	−0.460245	+	0.887792i	$0.652238\pi$
$642$	1.52786	0.0603000
$643$	19.0557	0.751485	0.375742	−	0.926724i	$-0.377388\pi$
$643$	19.0557	0.751485	0.375742	+	0.926724i	$0.377388\pi$
$644$	8.94427	0.352454
$645$	0	0
$646$	20.0000	0.786889
$647$	−3.70820	−0.145785	−0.0728923	−	0.997340i	$-0.523223\pi$
$647$	−3.70820	−0.145785	−0.0728923	+	0.997340i	$0.523223\pi$
$648$	−1.00000	−0.0392837
$649$	−37.8885	−1.48726
$650$	0	0
$651$	−12.9443	−0.507326
$652$	12.9443	0.506937
$653$	−30.3607	−1.18811	−0.594053	−	0.804426i	$-0.702473\pi$
$653$	−30.3607	−1.18811	−0.594053	+	0.804426i	$0.702473\pi$
$654$	2.29180	0.0896163
$655$	0	0
$656$	3.52786	0.137740
$657$	13.2361	0.516388
$658$	−41.8885	−1.63299
$659$	−10.0689	−0.392228	−0.196114	−	0.980581i	$-0.562832\pi$
$659$	−10.0689	−0.392228	−0.196114	+	0.980581i	$0.562832\pi$
$660$	0	0
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	2.76393	0.107423
$663$	7.23607	0.281026
$664$	4.94427	0.191875
$665$	0	0
$666$	10.9443	0.424082
$667$	−10.2492	−0.396852
$668$	−19.4164	−0.751243
$669$	16.7639	0.648131
$670$	0	0
$671$	−48.9443	−1.88947
$672$	−3.23607	−0.124834
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	6.47214	0.249297
$675$	0	0
$676$	1.00000	0.0384615
$677$	27.5279	1.05798	0.528991	−	0.848628i	$-0.322571\pi$
$677$	27.5279	1.05798	0.528991	+	0.848628i	$0.322571\pi$
$678$	−18.6525	−0.716344
$679$	−12.0000	−0.460518
$680$	0	0
$681$	7.05573	0.270376
$682$	17.8885	0.684988
$683$	−41.8885	−1.60282	−0.801410	−	0.598115i	$-0.795917\pi$
$683$	−41.8885	−1.60282	−0.801410	+	0.598115i	$0.795917\pi$
$684$	−2.76393	−0.105682
$685$	0	0
$686$	11.4164	0.435880
$687$	9.70820	0.370391
$688$	−2.47214	−0.0942493
$689$	−0.472136	−0.0179869
$690$	0	0
$691$	0.291796	0.0111004	0.00555022	−	0.999985i	$-0.498233\pi$
$691$	0.291796	0.0111004	0.00555022	+	0.999985i	$0.498233\pi$
$692$	−22.9443	−0.872210
$693$	14.4721	0.549751
$694$	−1.52786	−0.0579969
$695$	0	0
$696$	3.70820	0.140559
$697$	25.5279	0.966937
$698$	15.2361	0.576694
$699$	20.1803	0.763291
$700$	0	0
$701$	−24.0689	−0.909069	−0.454535	−	0.890729i	$-0.650194\pi$
$701$	−24.0689	−0.909069	−0.454535	+	0.890729i	$0.650194\pi$
$702$	−1.00000	−0.0377426
$703$	30.2492	1.14087
$704$	4.47214	0.168550
$705$	0	0
$706$	26.9443	1.01406
$707$	−29.8885	−1.12407
$708$	−8.47214	−0.318402
$709$	32.5410	1.22210	0.611052	−	0.791591i	$-0.290747\pi$
$709$	32.5410	1.22210	0.611052	+	0.791591i	$0.290747\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−0.472136	−0.0176940
$713$	−11.0557	−0.414040
$714$	−23.4164	−0.876337
$715$	0	0
$716$	8.18034	0.305714
$717$	−21.8885	−0.817443
$718$	−6.47214	−0.241538
$719$	20.9443	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$719$	20.9443	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$720$	0	0
$721$	27.4164	1.02104
$722$	11.3607	0.422801
$723$	3.52786	0.131203
$724$	17.4164	0.647276
$725$	0	0
$726$	−9.00000	−0.334021
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	−3.23607	−0.119937
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.8885	−0.661632
$732$	−10.9443	−0.404512
$733$	33.4164	1.23426	0.617132	−	0.786860i	$-0.288295\pi$
$733$	33.4164	1.23426	0.617132	+	0.786860i	$0.288295\pi$
$734$	−0.472136	−0.0174269
$735$	0	0
$736$	−2.76393	−0.101880
$737$	0	0
$738$	−3.52786	−0.129862
$739$	−0.291796	−0.0107339	−0.00536695	−	0.999986i	$-0.501708\pi$
$739$	−0.291796	−0.0107339	−0.00536695	+	0.999986i	$0.501708\pi$
$740$	0	0
$741$	−2.76393	−0.101536
$742$	1.52786	0.0560897
$743$	28.3607	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$743$	28.3607	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	−4.94427	−0.180901
$748$	32.3607	1.18322
$749$	−4.94427	−0.180660
$750$	0	0
$751$	−46.8328	−1.70895	−0.854477	−	0.519489i	$-0.826122\pi$
$751$	−46.8328	−1.70895	−0.854477	+	0.519489i	$0.826122\pi$
$752$	12.9443	0.472029
$753$	2.29180	0.0835177
$754$	3.70820	0.135045
$755$	0	0
$756$	3.23607	0.117695
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	35.1246	1.27578
$759$	12.3607	0.448664
$760$	0	0
$761$	7.52786	0.272885	0.136442	−	0.990648i	$-0.456433\pi$
$761$	7.52786	0.272885	0.136442	+	0.990648i	$0.456433\pi$
$762$	−21.4164	−0.775835
$763$	−7.41641	−0.268492
$764$	22.4721	0.813013
$765$	0	0
$766$	−5.52786	−0.199730
$767$	−8.47214	−0.305911
$768$	1.00000	0.0360844
$769$	−4.83282	−0.174276	−0.0871379	−	0.996196i	$-0.527772\pi$
$769$	−4.83282	−0.174276	−0.0871379	+	0.996196i	$0.527772\pi$
$770$	0	0
$771$	0.763932	0.0275123
$772$	−8.29180	−0.298428
$773$	−2.94427	−0.105898	−0.0529491	−	0.998597i	$-0.516862\pi$
$773$	−2.94427	−0.105898	−0.0529491	+	0.998597i	$0.516862\pi$
$774$	2.47214	0.0888591
$775$	0	0
$776$	3.70820	0.133117
$777$	−35.4164	−1.27056
$778$	−12.6525	−0.453613
$779$	−9.75078	−0.349358
$780$	0	0
$781$	−11.0557	−0.395605
$782$	−20.0000	−0.715199
$783$	−3.70820	−0.132520
$784$	3.47214	0.124005
$785$	0	0
$786$	−9.70820	−0.346280
$787$	28.9443	1.03175	0.515876	−	0.856663i	$-0.327467\pi$
$787$	28.9443	1.03175	0.515876	+	0.856663i	$0.327467\pi$
$788$	−14.9443	−0.532368
$789$	10.1803	0.362430
$790$	0	0
$791$	60.3607	2.14618
$792$	−4.47214	−0.158910
$793$	−10.9443	−0.388642
$794$	−17.4164	−0.618085
$795$	0	0
$796$	−0.944272	−0.0334688
$797$	9.05573	0.320770	0.160385	−	0.987055i	$-0.448726\pi$
$797$	9.05573	0.320770	0.160385	+	0.987055i	$0.448726\pi$
$798$	8.94427	0.316624
$799$	93.6656	3.31365
$800$	0	0
$801$	0.472136	0.0166821
$802$	24.4721	0.864141
$803$	59.1935	2.08889
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	6.76393	0.238102
$808$	9.23607	0.324924
$809$	31.5279	1.10846	0.554230	−	0.832363i	$-0.313013\pi$
$809$	31.5279	1.10846	0.554230	+	0.832363i	$0.313013\pi$
$810$	0	0
$811$	4.29180	0.150705	0.0753527	−	0.997157i	$-0.475992\pi$
$811$	4.29180	0.150705	0.0753527	+	0.997157i	$0.475992\pi$
$812$	−12.0000	−0.421117
$813$	−11.4164	−0.400391
$814$	48.9443	1.71550
$815$	0	0
$816$	7.23607	0.253313
$817$	6.83282	0.239050
$818$	−6.00000	−0.209785
$819$	3.23607	0.113077
$820$	0	0
$821$	−30.4721	−1.06348	−0.531742	−	0.846906i	$-0.678463\pi$
$821$	−30.4721	−1.06348	−0.531742	+	0.846906i	$0.678463\pi$
$822$	19.8885	0.693693
$823$	48.2492	1.68186	0.840931	−	0.541142i	$-0.182008\pi$
$823$	48.2492	1.68186	0.840931	+	0.541142i	$0.182008\pi$
$824$	−8.47214	−0.295141
$825$	0	0
$826$	27.4164	0.953939
$827$	−41.8885	−1.45661	−0.728304	−	0.685254i	$-0.759691\pi$
$827$	−41.8885	−1.45661	−0.728304	+	0.685254i	$0.759691\pi$
$828$	2.76393	0.0960533
$829$	42.7214	1.48377	0.741887	−	0.670525i	$-0.233931\pi$
$829$	42.7214	1.48377	0.741887	+	0.670525i	$0.233931\pi$
$830$	0	0
$831$	20.8328	0.722682
$832$	1.00000	0.0346688
$833$	25.1246	0.870516
$834$	6.47214	0.224112
$835$	0	0
$836$	−12.3607	−0.427503
$837$	−4.00000	−0.138260
$838$	−27.5967	−0.953314
$839$	−51.7771	−1.78754	−0.893772	−	0.448522i	$-0.851951\pi$
$839$	−51.7771	−1.78754	−0.893772	+	0.448522i	$0.851951\pi$
$840$	0	0
$841$	−15.2492	−0.525835
$842$	13.7082	0.472416
$843$	24.4721	0.842865
$844$	13.8885	0.478063
$845$	0	0
$846$	−12.9443	−0.445033
$847$	29.1246	1.00073
$848$	−0.472136	−0.0162132
$849$	−11.4164	−0.391810
$850$	0	0
$851$	−30.2492	−1.03693
$852$	−2.47214	−0.0846940
$853$	41.4164	1.41807	0.709035	−	0.705173i	$-0.249131\pi$
$853$	41.4164	1.41807	0.709035	+	0.705173i	$0.249131\pi$
$854$	35.4164	1.21192
$855$	0	0
$856$	1.52786	0.0522213
$857$	−0.180340	−0.00616029	−0.00308015	−	0.999995i	$-0.500980\pi$
$857$	−0.180340	−0.00616029	−0.00308015	+	0.999995i	$0.500980\pi$
$858$	−4.47214	−0.152676
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	11.4164	0.389070
$862$	14.8328	0.505208
$863$	22.4721	0.764960	0.382480	−	0.923964i	$-0.375070\pi$
$863$	22.4721	0.764960	0.382480	+	0.923964i	$0.375070\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	9.52786	0.323770
$867$	35.3607	1.20091
$868$	−12.9443	−0.439357
$869$	−17.8885	−0.606827
$870$	0	0
$871$	0	0
$872$	2.29180	0.0776100
$873$	−3.70820	−0.125504
$874$	7.63932	0.258404
$875$	0	0
$876$	13.2361	0.447205
$877$	24.4721	0.826365	0.413183	−	0.910648i	$-0.364417\pi$
$877$	24.4721	0.826365	0.413183	+	0.910648i	$0.364417\pi$
$878$	4.94427	0.166861
$879$	2.94427	0.0993078
$880$	0	0
$881$	34.3607	1.15764	0.578820	−	0.815455i	$-0.303513\pi$
$881$	34.3607	1.15764	0.578820	+	0.815455i	$0.303513\pi$
$882$	−3.47214	−0.116913
$883$	25.8885	0.871219	0.435609	−	0.900136i	$-0.356533\pi$
$883$	25.8885	0.871219	0.435609	+	0.900136i	$0.356533\pi$
$884$	7.23607	0.243375
$885$	0	0
$886$	22.4721	0.754966
$887$	−10.1803	−0.341822	−0.170911	−	0.985286i	$-0.554671\pi$
$887$	−10.1803	−0.341822	−0.170911	+	0.985286i	$0.554671\pi$
$888$	10.9443	0.367266
$889$	69.3050	2.32441
$890$	0	0
$891$	4.47214	0.149822
$892$	16.7639	0.561298
$893$	−35.7771	−1.19723
$894$	−13.5279	−0.452439
$895$	0	0
$896$	−3.23607	−0.108109
$897$	2.76393	0.0922850
$898$	−0.472136	−0.0157554
$899$	14.8328	0.494702
$900$	0	0
$901$	−3.41641	−0.113817
$902$	−15.7771	−0.525320
$903$	−8.00000	−0.266223
$904$	−18.6525	−0.620372
$905$	0	0
$906$	19.4164	0.645067
$907$	45.5279	1.51173	0.755864	−	0.654729i	$-0.227217\pi$
$907$	45.5279	1.51173	0.755864	+	0.654729i	$0.227217\pi$
$908$	7.05573	0.234153
$909$	−9.23607	−0.306341
$910$	0	0
$911$	−11.0557	−0.366293	−0.183146	−	0.983086i	$-0.558628\pi$
$911$	−11.0557	−0.366293	−0.183146	+	0.983086i	$0.558628\pi$
$912$	−2.76393	−0.0915229
$913$	−22.1115	−0.731782
$914$	3.70820	0.122656
$915$	0	0
$916$	9.70820	0.320768
$917$	31.4164	1.03746
$918$	−7.23607	−0.238826
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−24.3607	−0.802712
$922$	−41.8885	−1.37953
$923$	−2.47214	−0.0813713
$924$	14.4721	0.476098
$925$	0	0
$926$	22.2918	0.732554
$927$	8.47214	0.278261
$928$	3.70820	0.121728
$929$	32.4721	1.06538	0.532688	−	0.846312i	$-0.321182\pi$
$929$	32.4721	1.06538	0.532688	+	0.846312i	$0.321182\pi$
$930$	0	0
$931$	−9.59675	−0.314521
$932$	20.1803	0.661029
$933$	−22.4721	−0.735705
$934$	−6.47214	−0.211775
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	19.7771	0.646089	0.323045	−	0.946384i	$-0.395294\pi$
$937$	19.7771	0.646089	0.323045	+	0.946384i	$0.395294\pi$
$938$	0	0
$939$	−19.4164	−0.633631
$940$	0	0
$941$	−37.5279	−1.22337	−0.611687	−	0.791100i	$-0.709509\pi$
$941$	−37.5279	−1.22337	−0.611687	+	0.791100i	$0.709509\pi$
$942$	18.9443	0.617238
$943$	9.75078	0.317529
$944$	−8.47214	−0.275745
$945$	0	0
$946$	11.0557	0.359453
$947$	4.94427	0.160667	0.0803336	−	0.996768i	$-0.474401\pi$
$947$	4.94427	0.160667	0.0803336	+	0.996768i	$0.474401\pi$
$948$	−4.00000	−0.129914
$949$	13.2361	0.429661
$950$	0	0
$951$	13.4164	0.435057
$952$	−23.4164	−0.758930
$953$	29.1246	0.943439	0.471719	−	0.881749i	$-0.343634\pi$
$953$	29.1246	0.943439	0.471719	+	0.881749i	$0.343634\pi$
$954$	0.472136	0.0152860
$955$	0	0
$956$	−21.8885	−0.707926
$957$	−16.5836	−0.536071
$958$	36.0000	1.16311
$959$	−64.3607	−2.07831
$960$	0	0
$961$	−15.0000	−0.483871
$962$	10.9443	0.352857
$963$	−1.52786	−0.0492347
$964$	3.52786	0.113625
$965$	0	0
$966$	−8.94427	−0.287777
$967$	−56.5410	−1.81824	−0.909118	−	0.416538i	$-0.863243\pi$
$967$	−56.5410	−1.81824	−0.909118	+	0.416538i	$0.863243\pi$
$968$	−9.00000	−0.289271
$969$	−20.0000	−0.642493
$970$	0	0
$971$	23.2361	0.745681	0.372840	−	0.927895i	$-0.378384\pi$
$971$	23.2361	0.745681	0.372840	+	0.927895i	$0.378384\pi$
$972$	1.00000	0.0320750
$973$	−20.9443	−0.671443
$974$	23.5967	0.756089
$975$	0	0
$976$	−10.9443	−0.350318
$977$	−10.3607	−0.331468	−0.165734	−	0.986171i	$-0.552999\pi$
$977$	−10.3607	−0.331468	−0.165734	+	0.986171i	$0.552999\pi$
$978$	−12.9443	−0.413912
$979$	2.11146	0.0674824
$980$	0	0
$981$	−2.29180	−0.0731714
$982$	16.1803	0.516335
$983$	−7.41641	−0.236547	−0.118273	−	0.992981i	$-0.537736\pi$
$983$	−7.41641	−0.236547	−0.118273	+	0.992981i	$0.537736\pi$
$984$	−3.52786	−0.112464
$985$	0	0
$986$	26.8328	0.854531
$987$	41.8885	1.33333
$988$	−2.76393	−0.0879324
$989$	−6.83282	−0.217271
$990$	0	0
$991$	24.9443	0.792381	0.396190	−	0.918168i	$-0.370332\pi$
$991$	24.9443	0.792381	0.396190	+	0.918168i	$0.370332\pi$
$992$	4.00000	0.127000
$993$	−2.76393	−0.0877107
$994$	8.00000	0.253745
$995$	0	0
$996$	−4.94427	−0.156665
$997$	43.8885	1.38996	0.694982	−	0.719027i	$-0.255412\pi$
$997$	43.8885	1.38996	0.694982	+	0.719027i	$0.255412\pi$
$998$	39.1246	1.23847
$999$	−10.9443	−0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.a.be.1.2		2
3.2	odd	2		5850.2.a.cm.1.2		2
5.2	odd	4		390.2.e.e.79.2	✓	4
5.3	odd	4		390.2.e.e.79.3	yes	4
5.4	even	2		1950.2.a.bf.1.1		2
15.2	even	4		1170.2.e.e.469.3		4
15.8	even	4		1170.2.e.e.469.2		4
15.14	odd	2		5850.2.a.cf.1.1		2
20.3	even	4		3120.2.l.k.1249.1		4
20.7	even	4		3120.2.l.k.1249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.2	✓	4	5.2	odd	4
390.2.e.e.79.3	yes	4	5.3	odd	4
1170.2.e.e.469.2		4	15.8	even	4
1170.2.e.e.469.3		4	15.2	even	4
1950.2.a.be.1.2		2	1.1	even	1	trivial
1950.2.a.bf.1.1		2	5.4	even	2
3120.2.l.k.1249.1		4	20.3	even	4
3120.2.l.k.1249.4		4	20.7	even	4
5850.2.a.cf.1.1		2	15.14	odd	2
5850.2.a.cm.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.47214 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.23607 q^{14} +1.00000 q^{16} +7.23607 q^{17} -1.00000 q^{18} -2.76393 q^{19} +3.23607 q^{21} -4.47214 q^{22} +2.76393 q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +3.23607 q^{28} -3.70820 q^{29} -4.00000 q^{31} -1.00000 q^{32} +4.47214 q^{33} -7.23607 q^{34} +1.00000 q^{36} -10.9443 q^{37} +2.76393 q^{38} +1.00000 q^{39} +3.52786 q^{41} -3.23607 q^{42} -2.47214 q^{43} +4.47214 q^{44} -2.76393 q^{46} +12.9443 q^{47} +1.00000 q^{48} +3.47214 q^{49} +7.23607 q^{51} +1.00000 q^{52} -0.472136 q^{53} -1.00000 q^{54} -3.23607 q^{56} -2.76393 q^{57} +3.70820 q^{58} -8.47214 q^{59} -10.9443 q^{61} +4.00000 q^{62} +3.23607 q^{63} +1.00000 q^{64} -4.47214 q^{66} +7.23607 q^{68} +2.76393 q^{69} -2.47214 q^{71} -1.00000 q^{72} +13.2361 q^{73} +10.9443 q^{74} -2.76393 q^{76} +14.4721 q^{77} -1.00000 q^{78} -4.00000 q^{79} +1.00000 q^{81} -3.52786 q^{82} -4.94427 q^{83} +3.23607 q^{84} +2.47214 q^{86} -3.70820 q^{87} -4.47214 q^{88} +0.472136 q^{89} +3.23607 q^{91} +2.76393 q^{92} -4.00000 q^{93} -12.9443 q^{94} -1.00000 q^{96} -3.70820 q^{97} -3.47214 q^{98} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} - 2 q^{18} - 10 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 2 q^{26} + 2 q^{27}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 - 2 * q^18 - 10 * q^19 + 2 * q^21 + 10 * q^23 - 2 * q^24 - 2 * q^26 + 2 * q^27 + 2 * q^28 + 6 * q^29 - 8 * q^31 - 2 * q^32 - 10 * q^34 + 2 * q^36 - 4 * q^37 + 10 * q^38 + 2 * q^39 + 16 * q^41 - 2 * q^42 + 4 * q^43 - 10 * q^46 + 8 * q^47 + 2 * q^48 - 2 * q^49 + 10 * q^51 + 2 * q^52 + 8 * q^53 - 2 * q^54 - 2 * q^56 - 10 * q^57 - 6 * q^58 - 8 * q^59 - 4 * q^61 + 8 * q^62 + 2 * q^63 + 2 * q^64 + 10 * q^68 + 10 * q^69 + 4 * q^71 - 2 * q^72 + 22 * q^73 + 4 * q^74 - 10 * q^76 + 20 * q^77 - 2 * q^78 - 8 * q^79 + 2 * q^81 - 16 * q^82 + 8 * q^83 + 2 * q^84 - 4 * q^86 + 6 * q^87 - 8 * q^89 + 2 * q^91 + 10 * q^92 - 8 * q^93 - 8 * q^94 - 2 * q^96 + 6 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more