LMFDB - Embedded newform 4650.2.a.by.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(1,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,-2,2,0,2,-1,-2,2,0,7,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.1304369399$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 8$ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.627719 q^{11} -1.00000 q^{12} +2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} +4.74456 q^{17} -1.00000 q^{18} -0.627719 q^{19} +3.37228 q^{21} -0.627719 q^{22} -3.37228 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -3.37228 q^{28} -8.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} -0.627719 q^{33} -4.74456 q^{34} +1.00000 q^{36} +0.744563 q^{37} +0.627719 q^{38} -2.00000 q^{39} +0.744563 q^{41} -3.37228 q^{42} -0.627719 q^{43} +0.627719 q^{44} +3.37228 q^{46} +6.74456 q^{47} -1.00000 q^{48} +4.37228 q^{49} -4.74456 q^{51} +2.00000 q^{52} -1.37228 q^{53} +1.00000 q^{54} +3.37228 q^{56} +0.627719 q^{57} +8.74456 q^{58} +2.74456 q^{59} +11.4891 q^{61} +1.00000 q^{62} -3.37228 q^{63} +1.00000 q^{64} +0.627719 q^{66} -10.7446 q^{67} +4.74456 q^{68} +3.37228 q^{69} +3.37228 q^{71} -1.00000 q^{72} +8.11684 q^{73} -0.744563 q^{74} -0.627719 q^{76} -2.11684 q^{77} +2.00000 q^{78} -4.62772 q^{79} +1.00000 q^{81} -0.744563 q^{82} +12.0000 q^{83} +3.37228 q^{84} +0.627719 q^{86} +8.74456 q^{87} -0.627719 q^{88} -1.37228 q^{89} -6.74456 q^{91} -3.37228 q^{92} +1.00000 q^{93} -6.74456 q^{94} +1.00000 q^{96} -2.00000 q^{97} -4.37228 q^{98} +0.627719 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{12} + 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 7 q^{19} + q^{21} - 7 q^{22} - q^{23} + 2 q^{24}+ \cdots + 7 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 7 * q^11 - 2 * q^12 + 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 7 * q^19 + q^21 - 7 * q^22 - q^23 + 2 * q^24 - 4 * q^26 - 2 * q^27 - q^28 - 6 * q^29 - 2 * q^31 - 2 * q^32 - 7 * q^33 + 2 * q^34 + 2 * q^36 - 10 * q^37 + 7 * q^38 - 4 * q^39 - 10 * q^41 - q^42 - 7 * q^43 + 7 * q^44 + q^46 + 2 * q^47 - 2 * q^48 + 3 * q^49 + 2 * q^51 + 4 * q^52 + 3 * q^53 + 2 * q^54 + q^56 + 7 * q^57 + 6 * q^58 - 6 * q^59 + 2 * q^62 - q^63 + 2 * q^64 + 7 * q^66 - 10 * q^67 - 2 * q^68 + q^69 + q^71 - 2 * q^72 - q^73 + 10 * q^74 - 7 * q^76 + 13 * q^77 + 4 * q^78 - 15 * q^79 + 2 * q^81 + 10 * q^82 + 24 * q^83 + q^84 + 7 * q^86 + 6 * q^87 - 7 * q^88 + 3 * q^89 - 2 * q^91 - q^92 + 2 * q^93 - 2 * q^94 + 2 * q^96 - 4 * q^97 - 3 * q^98 + 7 * q^99

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
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0.627719	0.189264	0.0946322	−	0.995512i	$-0.469833\pi$
$11$	0.627719	0.189264	0.0946322	+	0.995512i	$0.469833\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	3.37228	0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	4.74456	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$17$	4.74456	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$18$	−1.00000	−0.235702
$19$	−0.627719	−0.144009	−0.0720043	−	0.997404i	$-0.522940\pi$
$19$	−0.627719	−0.144009	−0.0720043	+	0.997404i	$0.522940\pi$
$20$	0	0
$21$	3.37228	0.735892
$22$	−0.627719	−0.133830
$23$	−3.37228	−0.703169	−0.351585	−	0.936156i	$-0.614357\pi$
$23$	−3.37228	−0.703169	−0.351585	+	0.936156i	$0.614357\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−3.37228	−0.637301
$29$	−8.74456	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−0.627719	−0.109272
$34$	−4.74456	−0.813686
$35$	0	0
$36$	1.00000	0.166667
$37$	0.744563	0.122405	0.0612027	−	0.998125i	$-0.480506\pi$
$37$	0.744563	0.122405	0.0612027	+	0.998125i	$0.480506\pi$
$38$	0.627719	0.101829
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0.744563	0.116281	0.0581406	−	0.998308i	$-0.481483\pi$
$41$	0.744563	0.116281	0.0581406	+	0.998308i	$0.481483\pi$
$42$	−3.37228	−0.520354
$43$	−0.627719	−0.0957262	−0.0478631	−	0.998854i	$-0.515241\pi$
$43$	−0.627719	−0.0957262	−0.0478631	+	0.998854i	$0.515241\pi$
$44$	0.627719	0.0946322
$45$	0	0
$46$	3.37228	0.497216
$47$	6.74456	0.983796	0.491898	−	0.870653i	$-0.336303\pi$
$47$	6.74456	0.983796	0.491898	+	0.870653i	$0.336303\pi$
$48$	−1.00000	−0.144338
$49$	4.37228	0.624612
$50$	0	0
$51$	−4.74456	−0.664372
$52$	2.00000	0.277350
$53$	−1.37228	−0.188497	−0.0942487	−	0.995549i	$-0.530045\pi$
$53$	−1.37228	−0.188497	−0.0942487	+	0.995549i	$0.530045\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.37228	0.450640
$57$	0.627719	0.0831434
$58$	8.74456	1.14822
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	11.4891	1.47103	0.735516	−	0.677507i	$-0.236940\pi$
$61$	11.4891	1.47103	0.735516	+	0.677507i	$0.236940\pi$
$62$	1.00000	0.127000
$63$	−3.37228	−0.424868
$64$	1.00000	0.125000
$65$	0	0
$66$	0.627719	0.0772668
$67$	−10.7446	−1.31266	−0.656329	−	0.754475i	$-0.727892\pi$
$67$	−10.7446	−1.31266	−0.656329	+	0.754475i	$0.727892\pi$
$68$	4.74456	0.575363
$69$	3.37228	0.405975
$70$	0	0
$71$	3.37228	0.400216	0.200108	−	0.979774i	$-0.435871\pi$
$71$	3.37228	0.400216	0.200108	+	0.979774i	$0.435871\pi$
$72$	−1.00000	−0.117851
$73$	8.11684	0.950005	0.475002	−	0.879985i	$-0.342447\pi$
$73$	8.11684	0.950005	0.475002	+	0.879985i	$0.342447\pi$
$74$	−0.744563	−0.0865536
$75$	0	0
$76$	−0.627719	−0.0720043
$77$	−2.11684	−0.241237
$78$	2.00000	0.226455
$79$	−4.62772	−0.520659	−0.260330	−	0.965520i	$-0.583831\pi$
$79$	−4.62772	−0.520659	−0.260330	+	0.965520i	$0.583831\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.744563	−0.0822232
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	3.37228	0.367946
$85$	0	0
$86$	0.627719	0.0676886
$87$	8.74456	0.937516
$88$	−0.627719	−0.0669150
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	−6.74456	−0.707022
$92$	−3.37228	−0.351585
$93$	1.00000	0.103695
$94$	−6.74456	−0.695649
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−4.37228	−0.441667
$99$	0.627719	0.0630881
$100$	0	0
$101$	8.11684	0.807656	0.403828	−	0.914835i	$-0.367679\pi$
$101$	8.11684	0.807656	0.403828	+	0.914835i	$0.367679\pi$
$102$	4.74456	0.469782
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	1.37228	0.133288
$107$	−0.627719	−0.0606839	−0.0303419	−	0.999540i	$-0.509660\pi$
$107$	−0.627719	−0.0606839	−0.0303419	+	0.999540i	$0.509660\pi$
$108$	−1.00000	−0.0962250
$109$	4.74456	0.454447	0.227223	−	0.973843i	$-0.427035\pi$
$109$	4.74456	0.454447	0.227223	+	0.973843i	$0.427035\pi$
$110$	0	0
$111$	−0.744563	−0.0706708
$112$	−3.37228	−0.318651
$113$	2.62772	0.247195	0.123597	−	0.992332i	$-0.460557\pi$
$113$	2.62772	0.247195	0.123597	+	0.992332i	$0.460557\pi$
$114$	−0.627719	−0.0587912
$115$	0	0
$116$	−8.74456	−0.811912
$117$	2.00000	0.184900
$118$	−2.74456	−0.252657
$119$	−16.0000	−1.46672
$120$	0	0
$121$	−10.6060	−0.964179
$122$	−11.4891	−1.04018
$123$	−0.744563	−0.0671350
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	3.37228	0.300427
$127$	−13.4891	−1.19697	−0.598483	−	0.801135i	$-0.704230\pi$
$127$	−13.4891	−1.19697	−0.598483	+	0.801135i	$0.704230\pi$
$128$	−1.00000	−0.0883883
$129$	0.627719	0.0552675
$130$	0	0
$131$	−16.2337	−1.41834	−0.709172	−	0.705036i	$-0.750931\pi$
$131$	−16.2337	−1.41834	−0.709172	+	0.705036i	$0.750931\pi$
$132$	−0.627719	−0.0546359
$133$	2.11684	0.183554
$134$	10.7446	0.928189
$135$	0	0
$136$	−4.74456	−0.406843
$137$	3.48913	0.298096	0.149048	−	0.988830i	$-0.452379\pi$
$137$	3.48913	0.298096	0.149048	+	0.988830i	$0.452379\pi$
$138$	−3.37228	−0.287068
$139$	−10.7446	−0.911342	−0.455671	−	0.890148i	$-0.650601\pi$
$139$	−10.7446	−0.911342	−0.455671	+	0.890148i	$0.650601\pi$
$140$	0	0
$141$	−6.74456	−0.567995
$142$	−3.37228	−0.282996
$143$	1.25544	0.104985
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.11684	−0.671755
$147$	−4.37228	−0.360620
$148$	0.744563	0.0612027
$149$	−12.1168	−0.992651	−0.496325	−	0.868137i	$-0.665318\pi$
$149$	−12.1168	−0.992651	−0.496325	+	0.868137i	$0.665318\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0.627719	0.0509147
$153$	4.74456	0.383575
$154$	2.11684	0.170580
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−9.37228	−0.747989	−0.373995	−	0.927431i	$-0.622012\pi$
$157$	−9.37228	−0.747989	−0.373995	+	0.927431i	$0.622012\pi$
$158$	4.62772	0.368162
$159$	1.37228	0.108829
$160$	0	0
$161$	11.3723	0.896261
$162$	−1.00000	−0.0785674
$163$	−24.2337	−1.89813	−0.949064	−	0.315082i	$-0.897968\pi$
$163$	−24.2337	−1.89813	−0.949064	+	0.315082i	$0.897968\pi$
$164$	0.744563	0.0581406
$165$	0	0
$166$	−12.0000	−0.931381
$167$	12.6277	0.977162	0.488581	−	0.872518i	$-0.337515\pi$
$167$	12.6277	0.977162	0.488581	+	0.872518i	$0.337515\pi$
$168$	−3.37228	−0.260177
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−0.627719	−0.0480028
$172$	−0.627719	−0.0478631
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−8.74456	−0.662924
$175$	0	0
$176$	0.627719	0.0473161
$177$	−2.74456	−0.206294
$178$	1.37228	0.102857
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.8614	1.10464	0.552320	−	0.833632i	$-0.313743\pi$
$181$	14.8614	1.10464	0.552320	+	0.833632i	$0.313743\pi$
$182$	6.74456	0.499940
$183$	−11.4891	−0.849301
$184$	3.37228	0.248608
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	2.97825	0.217791
$188$	6.74456	0.491898
$189$	3.37228	0.245297
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−1.00000	−0.0721688
$193$	−15.4891	−1.11493	−0.557466	−	0.830200i	$-0.688226\pi$
$193$	−15.4891	−1.11493	−0.557466	+	0.830200i	$0.688226\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	4.37228	0.312306
$197$	−19.4891	−1.38854	−0.694271	−	0.719713i	$-0.744273\pi$
$197$	−19.4891	−1.38854	−0.694271	+	0.719713i	$0.744273\pi$
$198$	−0.627719	−0.0446100
$199$	−12.6277	−0.895155	−0.447578	−	0.894245i	$-0.647713\pi$
$199$	−12.6277	−0.895155	−0.447578	+	0.894245i	$0.647713\pi$
$200$	0	0
$201$	10.7446	0.757863
$202$	−8.11684	−0.571099
$203$	29.4891	2.06973
$204$	−4.74456	−0.332186
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−3.37228	−0.234390
$208$	2.00000	0.138675
$209$	−0.394031	−0.0272557
$210$	0	0
$211$	−0.627719	−0.0432139	−0.0216070	−	0.999767i	$-0.506878\pi$
$211$	−0.627719	−0.0432139	−0.0216070	+	0.999767i	$0.506878\pi$
$212$	−1.37228	−0.0942487
$213$	−3.37228	−0.231065
$214$	0.627719	0.0429100
$215$	0	0
$216$	1.00000	0.0680414
$217$	3.37228	0.228925
$218$	−4.74456	−0.321342
$219$	−8.11684	−0.548485
$220$	0	0
$221$	9.48913	0.638308
$222$	0.744563	0.0499718
$223$	−9.25544	−0.619790	−0.309895	−	0.950771i	$-0.600294\pi$
$223$	−9.25544	−0.619790	−0.309895	+	0.950771i	$0.600294\pi$
$224$	3.37228	0.225320
$225$	0	0
$226$	−2.62772	−0.174793
$227$	−6.11684	−0.405989	−0.202995	−	0.979180i	$-0.565067\pi$
$227$	−6.11684	−0.405989	−0.202995	+	0.979180i	$0.565067\pi$
$228$	0.627719	0.0415717
$229$	3.88316	0.256606	0.128303	−	0.991735i	$-0.459047\pi$
$229$	3.88316	0.256606	0.128303	+	0.991735i	$0.459047\pi$
$230$	0	0
$231$	2.11684	0.139278
$232$	8.74456	0.574109
$233$	14.8614	0.973603	0.486802	−	0.873513i	$-0.338163\pi$
$233$	14.8614	0.973603	0.486802	+	0.873513i	$0.338163\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	2.74456	0.178656
$237$	4.62772	0.300603
$238$	16.0000	1.03713
$239$	−29.4891	−1.90749	−0.953746	−	0.300612i	$-0.902809\pi$
$239$	−29.4891	−1.90749	−0.953746	+	0.300612i	$0.902809\pi$
$240$	0	0
$241$	4.51087	0.290571	0.145285	−	0.989390i	$-0.453590\pi$
$241$	4.51087	0.290571	0.145285	+	0.989390i	$0.453590\pi$
$242$	10.6060	0.681778
$243$	−1.00000	−0.0641500
$244$	11.4891	0.735516
$245$	0	0
$246$	0.744563	0.0474716
$247$	−1.25544	−0.0798816
$248$	1.00000	0.0635001
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−3.37228	−0.212434
$253$	−2.11684	−0.133085
$254$	13.4891	0.846383
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.3723	−0.834140	−0.417070	−	0.908874i	$-0.636943\pi$
$257$	−13.3723	−0.834140	−0.417070	+	0.908874i	$0.636943\pi$
$258$	−0.627719	−0.0390801
$259$	−2.51087	−0.156018
$260$	0	0
$261$	−8.74456	−0.541275
$262$	16.2337	1.00292
$263$	18.9783	1.17025	0.585125	−	0.810943i	$-0.301046\pi$
$263$	18.9783	1.17025	0.585125	+	0.810943i	$0.301046\pi$
$264$	0.627719	0.0386334
$265$	0	0
$266$	−2.11684	−0.129792
$267$	1.37228	0.0839823
$268$	−10.7446	−0.656329
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−13.8832	−0.843342	−0.421671	−	0.906749i	$-0.638556\pi$
$271$	−13.8832	−0.843342	−0.421671	+	0.906749i	$0.638556\pi$
$272$	4.74456	0.287681
$273$	6.74456	0.408200
$274$	−3.48913	−0.210786
$275$	0	0
$276$	3.37228	0.202987
$277$	−15.2554	−0.916610	−0.458305	−	0.888795i	$-0.651543\pi$
$277$	−15.2554	−0.916610	−0.458305	+	0.888795i	$0.651543\pi$
$278$	10.7446	0.644416
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	16.7446	0.998897	0.499448	−	0.866344i	$-0.333536\pi$
$281$	16.7446	0.998897	0.499448	+	0.866344i	$0.333536\pi$
$282$	6.74456	0.401633
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	3.37228	0.200108
$285$	0	0
$286$	−1.25544	−0.0742356
$287$	−2.51087	−0.148212
$288$	−1.00000	−0.0589256
$289$	5.51087	0.324169
$290$	0	0
$291$	2.00000	0.117242
$292$	8.11684	0.475002
$293$	7.48913	0.437519	0.218760	−	0.975779i	$-0.429799\pi$
$293$	7.48913	0.437519	0.218760	+	0.975779i	$0.429799\pi$
$294$	4.37228	0.254997
$295$	0	0
$296$	−0.744563	−0.0432768
$297$	−0.627719	−0.0364239
$298$	12.1168	0.701910
$299$	−6.74456	−0.390048
$300$	0	0
$301$	2.11684	0.122013
$302$	8.00000	0.460348
$303$	−8.11684	−0.466301
$304$	−0.627719	−0.0360021
$305$	0	0
$306$	−4.74456	−0.271229
$307$	−30.9783	−1.76802	−0.884011	−	0.467466i	$-0.845167\pi$
$307$	−30.9783	−1.76802	−0.884011	+	0.467466i	$0.845167\pi$
$308$	−2.11684	−0.120618
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	2.00000	0.113228
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	9.37228	0.528908
$315$	0	0
$316$	−4.62772	−0.260330
$317$	24.7446	1.38979	0.694897	−	0.719110i	$-0.255450\pi$
$317$	24.7446	1.38979	0.694897	+	0.719110i	$0.255450\pi$
$318$	−1.37228	−0.0769537
$319$	−5.48913	−0.307332
$320$	0	0
$321$	0.627719	0.0350358
$322$	−11.3723	−0.633752
$323$	−2.97825	−0.165714
$324$	1.00000	0.0555556
$325$	0	0
$326$	24.2337	1.34218
$327$	−4.74456	−0.262375
$328$	−0.744563	−0.0411116
$329$	−22.7446	−1.25395
$330$	0	0
$331$	−1.48913	−0.0818497	−0.0409249	−	0.999162i	$-0.513030\pi$
$331$	−1.48913	−0.0818497	−0.0409249	+	0.999162i	$0.513030\pi$
$332$	12.0000	0.658586
$333$	0.744563	0.0408018
$334$	−12.6277	−0.690958
$335$	0	0
$336$	3.37228	0.183973
$337$	−28.9783	−1.57855	−0.789273	−	0.614043i	$-0.789542\pi$
$337$	−28.9783	−1.57855	−0.789273	+	0.614043i	$0.789542\pi$
$338$	9.00000	0.489535
$339$	−2.62772	−0.142718
$340$	0	0
$341$	−0.627719	−0.0339929
$342$	0.627719	0.0339431
$343$	8.86141	0.478471
$344$	0.627719	0.0338443
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−36.4674	−1.95767	−0.978836	−	0.204648i	$-0.934395\pi$
$347$	−36.4674	−1.95767	−0.978836	+	0.204648i	$0.934395\pi$
$348$	8.74456	0.468758
$349$	7.25544	0.388375	0.194187	−	0.980964i	$-0.437793\pi$
$349$	7.25544	0.388375	0.194187	+	0.980964i	$0.437793\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−0.627719	−0.0334575
$353$	−15.4891	−0.824403	−0.412201	−	0.911093i	$-0.635240\pi$
$353$	−15.4891	−0.824403	−0.412201	+	0.911093i	$0.635240\pi$
$354$	2.74456	0.145872
$355$	0	0
$356$	−1.37228	−0.0727308
$357$	16.0000	0.846810
$358$	12.0000	0.634220
$359$	3.37228	0.177982	0.0889911	−	0.996032i	$-0.471636\pi$
$359$	3.37228	0.177982	0.0889911	+	0.996032i	$0.471636\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	−14.8614	−0.781098
$363$	10.6060	0.556669
$364$	−6.74456	−0.353511
$365$	0	0
$366$	11.4891	0.600546
$367$	20.2337	1.05619	0.528095	−	0.849185i	$-0.322907\pi$
$367$	20.2337	1.05619	0.528095	+	0.849185i	$0.322907\pi$
$368$	−3.37228	−0.175792
$369$	0.744563	0.0387604
$370$	0	0
$371$	4.62772	0.240259
$372$	1.00000	0.0518476
$373$	−14.8614	−0.769494	−0.384747	−	0.923022i	$-0.625711\pi$
$373$	−14.8614	−0.769494	−0.384747	+	0.923022i	$0.625711\pi$
$374$	−2.97825	−0.154002
$375$	0	0
$376$	−6.74456	−0.347824
$377$	−17.4891	−0.900736
$378$	−3.37228	−0.173451
$379$	−28.8614	−1.48251	−0.741255	−	0.671223i	$-0.765769\pi$
$379$	−28.8614	−1.48251	−0.741255	+	0.671223i	$0.765769\pi$
$380$	0	0
$381$	13.4891	0.691069
$382$	−16.0000	−0.818631
$383$	−13.4891	−0.689262	−0.344631	−	0.938738i	$-0.611996\pi$
$383$	−13.4891	−0.689262	−0.344631	+	0.938738i	$0.611996\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	15.4891	0.788376
$387$	−0.627719	−0.0319087
$388$	−2.00000	−0.101535
$389$	16.9783	0.860831	0.430416	−	0.902631i	$-0.358367\pi$
$389$	16.9783	0.860831	0.430416	+	0.902631i	$0.358367\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	−4.37228	−0.220834
$393$	16.2337	0.818881
$394$	19.4891	0.981848
$395$	0	0
$396$	0.627719	0.0315441
$397$	1.60597	0.0806013	0.0403006	−	0.999188i	$-0.487168\pi$
$397$	1.60597	0.0806013	0.0403006	+	0.999188i	$0.487168\pi$
$398$	12.6277	0.632970
$399$	−2.11684	−0.105975
$400$	0	0
$401$	22.6277	1.12997	0.564987	−	0.825100i	$-0.308881\pi$
$401$	22.6277	1.12997	0.564987	+	0.825100i	$0.308881\pi$
$402$	−10.7446	−0.535890
$403$	−2.00000	−0.0996271
$404$	8.11684	0.403828
$405$	0	0
$406$	−29.4891	−1.46352
$407$	0.467376	0.0231670
$408$	4.74456	0.234891
$409$	−10.2337	−0.506023	−0.253012	−	0.967463i	$-0.581421\pi$
$409$	−10.2337	−0.506023	−0.253012	+	0.967463i	$0.581421\pi$
$410$	0	0
$411$	−3.48913	−0.172106
$412$	8.00000	0.394132
$413$	−9.25544	−0.455430
$414$	3.37228	0.165739
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	10.7446	0.526163
$418$	0.394031	0.0192727
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	19.4891	0.949842	0.474921	−	0.880028i	$-0.342477\pi$
$421$	19.4891	0.949842	0.474921	+	0.880028i	$0.342477\pi$
$422$	0.627719	0.0305569
$423$	6.74456	0.327932
$424$	1.37228	0.0666439
$425$	0	0
$426$	3.37228	0.163388
$427$	−38.7446	−1.87498
$428$	−0.627719	−0.0303419
$429$	−1.25544	−0.0606131
$430$	0	0
$431$	−26.9783	−1.29950	−0.649748	−	0.760149i	$-0.725126\pi$
$431$	−26.9783	−1.29950	−0.649748	+	0.760149i	$0.725126\pi$
$432$	−1.00000	−0.0481125
$433$	−36.1168	−1.73566	−0.867832	−	0.496857i	$-0.834487\pi$
$433$	−36.1168	−1.73566	−0.867832	+	0.496857i	$0.834487\pi$
$434$	−3.37228	−0.161875
$435$	0	0
$436$	4.74456	0.227223
$437$	2.11684	0.101262
$438$	8.11684	0.387838
$439$	14.7446	0.703720	0.351860	−	0.936053i	$-0.385549\pi$
$439$	14.7446	0.703720	0.351860	+	0.936053i	$0.385549\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	−9.48913	−0.451352
$443$	−34.3505	−1.63204	−0.816022	−	0.578022i	$-0.803825\pi$
$443$	−34.3505	−1.63204	−0.816022	+	0.578022i	$0.803825\pi$
$444$	−0.744563	−0.0353354
$445$	0	0
$446$	9.25544	0.438258
$447$	12.1168	0.573107
$448$	−3.37228	−0.159325
$449$	−8.97825	−0.423710	−0.211855	−	0.977301i	$-0.567950\pi$
$449$	−8.97825	−0.423710	−0.211855	+	0.977301i	$0.567950\pi$
$450$	0	0
$451$	0.467376	0.0220079
$452$	2.62772	0.123597
$453$	8.00000	0.375873
$454$	6.11684	0.287078
$455$	0	0
$456$	−0.627719	−0.0293956
$457$	−34.4674	−1.61232	−0.806158	−	0.591700i	$-0.798457\pi$
$457$	−34.4674	−1.61232	−0.806158	+	0.591700i	$0.798457\pi$
$458$	−3.88316	−0.181448
$459$	−4.74456	−0.221457
$460$	0	0
$461$	−19.7228	−0.918583	−0.459291	−	0.888286i	$-0.651897\pi$
$461$	−19.7228	−0.918583	−0.459291	+	0.888286i	$0.651897\pi$
$462$	−2.11684	−0.0984845
$463$	−2.51087	−0.116690	−0.0583451	−	0.998296i	$-0.518582\pi$
$463$	−2.51087	−0.116690	−0.0583451	+	0.998296i	$0.518582\pi$
$464$	−8.74456	−0.405956
$465$	0	0
$466$	−14.8614	−0.688441
$467$	−6.51087	−0.301287	−0.150644	−	0.988588i	$-0.548135\pi$
$467$	−6.51087	−0.301287	−0.150644	+	0.988588i	$0.548135\pi$
$468$	2.00000	0.0924500
$469$	36.2337	1.67312
$470$	0	0
$471$	9.37228	0.431852
$472$	−2.74456	−0.126329
$473$	−0.394031	−0.0181176
$474$	−4.62772	−0.212558
$475$	0	0
$476$	−16.0000	−0.733359
$477$	−1.37228	−0.0628324
$478$	29.4891	1.34880
$479$	8.86141	0.404888	0.202444	−	0.979294i	$-0.435112\pi$
$479$	8.86141	0.404888	0.202444	+	0.979294i	$0.435112\pi$
$480$	0	0
$481$	1.48913	0.0678983
$482$	−4.51087	−0.205465
$483$	−11.3723	−0.517457
$484$	−10.6060	−0.482090
$485$	0	0
$486$	1.00000	0.0453609
$487$	37.4891	1.69879	0.849397	−	0.527754i	$-0.176966\pi$
$487$	37.4891	1.69879	0.849397	+	0.527754i	$0.176966\pi$
$488$	−11.4891	−0.520088
$489$	24.2337	1.09589
$490$	0	0
$491$	27.6060	1.24584	0.622920	−	0.782286i	$-0.285946\pi$
$491$	27.6060	1.24584	0.622920	+	0.782286i	$0.285946\pi$
$492$	−0.744563	−0.0335675
$493$	−41.4891	−1.86858
$494$	1.25544	0.0564848
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−11.3723	−0.510117
$498$	12.0000	0.537733
$499$	33.4891	1.49918	0.749590	−	0.661903i	$-0.230251\pi$
$499$	33.4891	1.49918	0.749590	+	0.661903i	$0.230251\pi$
$500$	0	0
$501$	−12.6277	−0.564165
$502$	4.00000	0.178529
$503$	−5.48913	−0.244748	−0.122374	−	0.992484i	$-0.539051\pi$
$503$	−5.48913	−0.244748	−0.122374	+	0.992484i	$0.539051\pi$
$504$	3.37228	0.150213
$505$	0	0
$506$	2.11684	0.0941052
$507$	9.00000	0.399704
$508$	−13.4891	−0.598483
$509$	−27.2554	−1.20808	−0.604038	−	0.796956i	$-0.706442\pi$
$509$	−27.2554	−1.20808	−0.604038	+	0.796956i	$0.706442\pi$
$510$	0	0
$511$	−27.3723	−1.21088
$512$	−1.00000	−0.0441942
$513$	0.627719	0.0277145
$514$	13.3723	0.589826
$515$	0	0
$516$	0.627719	0.0276338
$517$	4.23369	0.186197
$518$	2.51087	0.110322
$519$	−18.0000	−0.790112
$520$	0	0
$521$	36.9783	1.62005	0.810023	−	0.586398i	$-0.199454\pi$
$521$	36.9783	1.62005	0.810023	+	0.586398i	$0.199454\pi$
$522$	8.74456	0.382739
$523$	12.8614	0.562390	0.281195	−	0.959651i	$-0.409269\pi$
$523$	12.8614	0.562390	0.281195	+	0.959651i	$0.409269\pi$
$524$	−16.2337	−0.709172
$525$	0	0
$526$	−18.9783	−0.827491
$527$	−4.74456	−0.206676
$528$	−0.627719	−0.0273179
$529$	−11.6277	−0.505553
$530$	0	0
$531$	2.74456	0.119104
$532$	2.11684	0.0917768
$533$	1.48913	0.0645012
$534$	−1.37228	−0.0593844
$535$	0	0
$536$	10.7446	0.464094
$537$	12.0000	0.517838
$538$	2.00000	0.0862261
$539$	2.74456	0.118217
$540$	0	0
$541$	−15.4891	−0.665930	−0.332965	−	0.942939i	$-0.608049\pi$
$541$	−15.4891	−0.665930	−0.332965	+	0.942939i	$0.608049\pi$
$542$	13.8832	0.596333
$543$	−14.8614	−0.637764
$544$	−4.74456	−0.203421
$545$	0	0
$546$	−6.74456	−0.288641
$547$	2.74456	0.117349	0.0586745	−	0.998277i	$-0.481313\pi$
$547$	2.74456	0.117349	0.0586745	+	0.998277i	$0.481313\pi$
$548$	3.48913	0.149048
$549$	11.4891	0.490344
$550$	0	0
$551$	5.48913	0.233845
$552$	−3.37228	−0.143534
$553$	15.6060	0.663633
$554$	15.2554	0.648141
$555$	0	0
$556$	−10.7446	−0.455671
$557$	−11.8832	−0.503505	−0.251753	−	0.967792i	$-0.581007\pi$
$557$	−11.8832	−0.503505	−0.251753	+	0.967792i	$0.581007\pi$
$558$	1.00000	0.0423334
$559$	−1.25544	−0.0530993
$560$	0	0
$561$	−2.97825	−0.125742
$562$	−16.7446	−0.706327
$563$	6.97825	0.294098	0.147049	−	0.989129i	$-0.453022\pi$
$563$	6.97825	0.294098	0.147049	+	0.989129i	$0.453022\pi$
$564$	−6.74456	−0.283997
$565$	0	0
$566$	12.0000	0.504398
$567$	−3.37228	−0.141623
$568$	−3.37228	−0.141498
$569$	−1.37228	−0.0575290	−0.0287645	−	0.999586i	$-0.509157\pi$
$569$	−1.37228	−0.0575290	−0.0287645	+	0.999586i	$0.509157\pi$
$570$	0	0
$571$	−26.7446	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$571$	−26.7446	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$572$	1.25544	0.0524925
$573$	−16.0000	−0.668410
$574$	2.51087	0.104802
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.23369	0.0929896	0.0464948	−	0.998919i	$-0.485195\pi$
$577$	2.23369	0.0929896	0.0464948	+	0.998919i	$0.485195\pi$
$578$	−5.51087	−0.229222
$579$	15.4891	0.643706
$580$	0	0
$581$	−40.4674	−1.67887
$582$	−2.00000	−0.0829027
$583$	−0.861407	−0.0356758
$584$	−8.11684	−0.335877
$585$	0	0
$586$	−7.48913	−0.309373
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−4.37228	−0.180310
$589$	0.627719	0.0258647
$590$	0	0
$591$	19.4891	0.801675
$592$	0.744563	0.0306013
$593$	−44.9783	−1.84704	−0.923518	−	0.383556i	$-0.874699\pi$
$593$	−44.9783	−1.84704	−0.923518	+	0.383556i	$0.874699\pi$
$594$	0.627719	0.0257556
$595$	0	0
$596$	−12.1168	−0.496325
$597$	12.6277	0.516818
$598$	6.74456	0.275806
$599$	10.1168	0.413363	0.206682	−	0.978408i	$-0.433734\pi$
$599$	10.1168	0.413363	0.206682	+	0.978408i	$0.433734\pi$
$600$	0	0
$601$	12.5109	0.510329	0.255165	−	0.966898i	$-0.417870\pi$
$601$	12.5109	0.510329	0.255165	+	0.966898i	$0.417870\pi$
$602$	−2.11684	−0.0862761
$603$	−10.7446	−0.437552
$604$	−8.00000	−0.325515
$605$	0	0
$606$	8.11684	0.329724
$607$	4.62772	0.187833	0.0939167	−	0.995580i	$-0.470061\pi$
$607$	4.62772	0.187833	0.0939167	+	0.995580i	$0.470061\pi$
$608$	0.627719	0.0254574
$609$	−29.4891	−1.19496
$610$	0	0
$611$	13.4891	0.545712
$612$	4.74456	0.191788
$613$	−35.4891	−1.43339	−0.716696	−	0.697386i	$-0.754347\pi$
$613$	−35.4891	−1.43339	−0.716696	+	0.697386i	$0.754347\pi$
$614$	30.9783	1.25018
$615$	0	0
$616$	2.11684	0.0852901
$617$	−7.88316	−0.317364	−0.158682	−	0.987330i	$-0.550724\pi$
$617$	−7.88316	−0.317364	−0.158682	+	0.987330i	$0.550724\pi$
$618$	8.00000	0.321807
$619$	16.2337	0.652487	0.326244	−	0.945286i	$-0.394217\pi$
$619$	16.2337	0.652487	0.326244	+	0.945286i	$0.394217\pi$
$620$	0	0
$621$	3.37228	0.135325
$622$	−8.00000	−0.320771
$623$	4.62772	0.185406
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	10.0000	0.399680
$627$	0.394031	0.0157361
$628$	−9.37228	−0.373995
$629$	3.53262	0.140855
$630$	0	0
$631$	−46.3505	−1.84519	−0.922593	−	0.385775i	$-0.873934\pi$
$631$	−46.3505	−1.84519	−0.922593	+	0.385775i	$0.873934\pi$
$632$	4.62772	0.184081
$633$	0.627719	0.0249496
$634$	−24.7446	−0.982732
$635$	0	0
$636$	1.37228	0.0544145
$637$	8.74456	0.346472
$638$	5.48913	0.217317
$639$	3.37228	0.133405
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−0.627719	−0.0247741
$643$	25.0951	0.989654	0.494827	−	0.868992i	$-0.335231\pi$
$643$	25.0951	0.989654	0.494827	+	0.868992i	$0.335231\pi$
$644$	11.3723	0.448131
$645$	0	0
$646$	2.97825	0.117178
$647$	−50.5842	−1.98867	−0.994335	−	0.106287i	$-0.966104\pi$
$647$	−50.5842	−1.98867	−0.994335	+	0.106287i	$0.966104\pi$
$648$	−1.00000	−0.0392837
$649$	1.72281	0.0676263
$650$	0	0
$651$	−3.37228	−0.132170
$652$	−24.2337	−0.949064
$653$	−40.9783	−1.60360	−0.801801	−	0.597591i	$-0.796125\pi$
$653$	−40.9783	−1.60360	−0.801801	+	0.597591i	$0.796125\pi$
$654$	4.74456	0.185527
$655$	0	0
$656$	0.744563	0.0290703
$657$	8.11684	0.316668
$658$	22.7446	0.886675
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−4.97825	−0.193632	−0.0968158	−	0.995302i	$-0.530866\pi$
$661$	−4.97825	−0.193632	−0.0968158	+	0.995302i	$0.530866\pi$
$662$	1.48913	0.0578765
$663$	−9.48913	−0.368527
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−0.744563	−0.0288512
$667$	29.4891	1.14182
$668$	12.6277	0.488581
$669$	9.25544	0.357836
$670$	0	0
$671$	7.21194	0.278414
$672$	−3.37228	−0.130089
$673$	0.510875	0.0196928	0.00984639	−	0.999952i	$-0.496866\pi$
$673$	0.510875	0.0196928	0.00984639	+	0.999952i	$0.496866\pi$
$674$	28.9783	1.11620
$675$	0	0
$676$	−9.00000	−0.346154
$677$	41.6060	1.59905	0.799524	−	0.600635i	$-0.205085\pi$
$677$	41.6060	1.59905	0.799524	+	0.600635i	$0.205085\pi$
$678$	2.62772	0.100917
$679$	6.74456	0.258833
$680$	0	0
$681$	6.11684	0.234398
$682$	0.627719	0.0240366
$683$	12.8614	0.492128	0.246064	−	0.969254i	$-0.420863\pi$
$683$	12.8614	0.492128	0.246064	+	0.969254i	$0.420863\pi$
$684$	−0.627719	−0.0240014
$685$	0	0
$686$	−8.86141	−0.338330
$687$	−3.88316	−0.148152
$688$	−0.627719	−0.0239316
$689$	−2.74456	−0.104560
$690$	0	0
$691$	−19.1386	−0.728066	−0.364033	−	0.931386i	$-0.618601\pi$
$691$	−19.1386	−0.728066	−0.364033	+	0.931386i	$0.618601\pi$
$692$	18.0000	0.684257
$693$	−2.11684	−0.0804123
$694$	36.4674	1.38428
$695$	0	0
$696$	−8.74456	−0.331462
$697$	3.53262	0.133808
$698$	−7.25544	−0.274622
$699$	−14.8614	−0.562110
$700$	0	0
$701$	45.6060	1.72251	0.861257	−	0.508170i	$-0.169678\pi$
$701$	45.6060	1.72251	0.861257	+	0.508170i	$0.169678\pi$
$702$	2.00000	0.0754851
$703$	−0.467376	−0.0176274
$704$	0.627719	0.0236580
$705$	0	0
$706$	15.4891	0.582941
$707$	−27.3723	−1.02944
$708$	−2.74456	−0.103147
$709$	40.1168	1.50662	0.753310	−	0.657666i	$-0.228456\pi$
$709$	40.1168	1.50662	0.753310	+	0.657666i	$0.228456\pi$
$710$	0	0
$711$	−4.62772	−0.173553
$712$	1.37228	0.0514284
$713$	3.37228	0.126293
$714$	−16.0000	−0.598785
$715$	0	0
$716$	−12.0000	−0.448461
$717$	29.4891	1.10129
$718$	−3.37228	−0.125852
$719$	38.7446	1.44493	0.722464	−	0.691408i	$-0.243009\pi$
$719$	38.7446	1.44493	0.722464	+	0.691408i	$0.243009\pi$
$720$	0	0
$721$	−26.9783	−1.00472
$722$	18.6060	0.692442
$723$	−4.51087	−0.167761
$724$	14.8614	0.552320
$725$	0	0
$726$	−10.6060	−0.393624
$727$	−19.3723	−0.718478	−0.359239	−	0.933246i	$-0.616964\pi$
$727$	−19.3723	−0.718478	−0.359239	+	0.933246i	$0.616964\pi$
$728$	6.74456	0.249970
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.97825	−0.110155
$732$	−11.4891	−0.424650
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−20.2337	−0.746839
$735$	0	0
$736$	3.37228	0.124304
$737$	−6.74456	−0.248439
$738$	−0.744563	−0.0274077
$739$	−22.9783	−0.845269	−0.422634	−	0.906300i	$-0.638895\pi$
$739$	−22.9783	−0.845269	−0.422634	+	0.906300i	$0.638895\pi$
$740$	0	0
$741$	1.25544	0.0461196
$742$	−4.62772	−0.169889
$743$	35.3723	1.29768	0.648842	−	0.760924i	$-0.275254\pi$
$743$	35.3723	1.29768	0.648842	+	0.760924i	$0.275254\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	14.8614	0.544115
$747$	12.0000	0.439057
$748$	2.97825	0.108896
$749$	2.11684	0.0773478
$750$	0	0
$751$	−38.7446	−1.41381	−0.706905	−	0.707309i	$-0.749909\pi$
$751$	−38.7446	−1.41381	−0.706905	+	0.707309i	$0.749909\pi$
$752$	6.74456	0.245949
$753$	4.00000	0.145768
$754$	17.4891	0.636916
$755$	0	0
$756$	3.37228	0.122649
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	28.8614	1.04829
$759$	2.11684	0.0768366
$760$	0	0
$761$	−46.8614	−1.69872	−0.849362	−	0.527810i	$-0.823013\pi$
$761$	−46.8614	−1.69872	−0.849362	+	0.527810i	$0.823013\pi$
$762$	−13.4891	−0.488659
$763$	−16.0000	−0.579239
$764$	16.0000	0.578860
$765$	0	0
$766$	13.4891	0.487382
$767$	5.48913	0.198201
$768$	−1.00000	−0.0360844
$769$	−9.37228	−0.337973	−0.168987	−	0.985618i	$-0.554049\pi$
$769$	−9.37228	−0.337973	−0.168987	+	0.985618i	$0.554049\pi$
$770$	0	0
$771$	13.3723	0.481591
$772$	−15.4891	−0.557466
$773$	−21.6060	−0.777113	−0.388556	−	0.921425i	$-0.627026\pi$
$773$	−21.6060	−0.777113	−0.388556	+	0.921425i	$0.627026\pi$
$774$	0.627719	0.0225629
$775$	0	0
$776$	2.00000	0.0717958
$777$	2.51087	0.0900771
$778$	−16.9783	−0.608700
$779$	−0.467376	−0.0167455
$780$	0	0
$781$	2.11684	0.0757466
$782$	16.0000	0.572159
$783$	8.74456	0.312505
$784$	4.37228	0.156153
$785$	0	0
$786$	−16.2337	−0.579036
$787$	9.88316	0.352296	0.176148	−	0.984364i	$-0.443636\pi$
$787$	9.88316	0.352296	0.176148	+	0.984364i	$0.443636\pi$
$788$	−19.4891	−0.694271
$789$	−18.9783	−0.675644
$790$	0	0
$791$	−8.86141	−0.315075
$792$	−0.627719	−0.0223050
$793$	22.9783	0.815982
$794$	−1.60597	−0.0569937
$795$	0	0
$796$	−12.6277	−0.447578
$797$	20.5109	0.726532	0.363266	−	0.931685i	$-0.381662\pi$
$797$	20.5109	0.726532	0.363266	+	0.931685i	$0.381662\pi$
$798$	2.11684	0.0749355
$799$	32.0000	1.13208
$800$	0	0
$801$	−1.37228	−0.0484872
$802$	−22.6277	−0.799013
$803$	5.09509	0.179802
$804$	10.7446	0.378932
$805$	0	0
$806$	2.00000	0.0704470
$807$	2.00000	0.0704033
$808$	−8.11684	−0.285550
$809$	9.60597	0.337728	0.168864	−	0.985639i	$-0.445990\pi$
$809$	9.60597	0.337728	0.168864	+	0.985639i	$0.445990\pi$
$810$	0	0
$811$	−40.6277	−1.42663	−0.713316	−	0.700842i	$-0.752808\pi$
$811$	−40.6277	−1.42663	−0.713316	+	0.700842i	$0.752808\pi$
$812$	29.4891	1.03487
$813$	13.8832	0.486904
$814$	−0.467376	−0.0163815
$815$	0	0
$816$	−4.74456	−0.166093
$817$	0.394031	0.0137854
$818$	10.2337	0.357813
$819$	−6.74456	−0.235674
$820$	0	0
$821$	15.2554	0.532418	0.266209	−	0.963915i	$-0.414229\pi$
$821$	15.2554	0.532418	0.266209	+	0.963915i	$0.414229\pi$
$822$	3.48913	0.121697
$823$	17.2554	0.601487	0.300743	−	0.953705i	$-0.402765\pi$
$823$	17.2554	0.601487	0.300743	+	0.953705i	$0.402765\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	9.25544	0.322038
$827$	17.4891	0.608156	0.304078	−	0.952647i	$-0.401652\pi$
$827$	17.4891	0.608156	0.304078	+	0.952647i	$0.401652\pi$
$828$	−3.37228	−0.117195
$829$	−31.0951	−1.07998	−0.539989	−	0.841672i	$-0.681571\pi$
$829$	−31.0951	−1.07998	−0.539989	+	0.841672i	$0.681571\pi$
$830$	0	0
$831$	15.2554	0.529205
$832$	2.00000	0.0693375
$833$	20.7446	0.718756
$834$	−10.7446	−0.372054
$835$	0	0
$836$	−0.394031	−0.0136278
$837$	1.00000	0.0345651
$838$	12.0000	0.414533
$839$	16.8614	0.582120	0.291060	−	0.956705i	$-0.405992\pi$
$839$	16.8614	0.582120	0.291060	+	0.956705i	$0.405992\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	−19.4891	−0.671640
$843$	−16.7446	−0.576713
$844$	−0.627719	−0.0216070
$845$	0	0
$846$	−6.74456	−0.231883
$847$	35.7663	1.22895
$848$	−1.37228	−0.0471243
$849$	12.0000	0.411839
$850$	0	0
$851$	−2.51087	−0.0860717
$852$	−3.37228	−0.115532
$853$	39.0951	1.33859	0.669295	−	0.742997i	$-0.266596\pi$
$853$	39.0951	1.33859	0.669295	+	0.742997i	$0.266596\pi$
$854$	38.7446	1.32581
$855$	0	0
$856$	0.627719	0.0214550
$857$	−7.48913	−0.255824	−0.127912	−	0.991786i	$-0.540827\pi$
$857$	−7.48913	−0.255824	−0.127912	+	0.991786i	$0.540827\pi$
$858$	1.25544	0.0428599
$859$	−17.4891	−0.596721	−0.298361	−	0.954453i	$-0.596440\pi$
$859$	−17.4891	−0.596721	−0.298361	+	0.954453i	$0.596440\pi$
$860$	0	0
$861$	2.51087	0.0855704
$862$	26.9783	0.918883
$863$	6.35053	0.216175	0.108087	−	0.994141i	$-0.465527\pi$
$863$	6.35053	0.216175	0.108087	+	0.994141i	$0.465527\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	36.1168	1.22730
$867$	−5.51087	−0.187159
$868$	3.37228	0.114463
$869$	−2.90491	−0.0985422
$870$	0	0
$871$	−21.4891	−0.728131
$872$	−4.74456	−0.160671
$873$	−2.00000	−0.0676897
$874$	−2.11684	−0.0716033
$875$	0	0
$876$	−8.11684	−0.274243
$877$	44.9783	1.51881	0.759404	−	0.650620i	$-0.225491\pi$
$877$	44.9783	1.51881	0.759404	+	0.650620i	$0.225491\pi$
$878$	−14.7446	−0.497605
$879$	−7.48913	−0.252602
$880$	0	0
$881$	−3.02175	−0.101805	−0.0509027	−	0.998704i	$-0.516210\pi$
$881$	−3.02175	−0.101805	−0.0509027	+	0.998704i	$0.516210\pi$
$882$	−4.37228	−0.147222
$883$	−44.8614	−1.50971	−0.754853	−	0.655894i	$-0.772292\pi$
$883$	−44.8614	−1.50971	−0.754853	+	0.655894i	$0.772292\pi$
$884$	9.48913	0.319154
$885$	0	0
$886$	34.3505	1.15403
$887$	30.7446	1.03230	0.516151	−	0.856498i	$-0.327364\pi$
$887$	30.7446	1.03230	0.516151	+	0.856498i	$0.327364\pi$
$888$	0.744563	0.0249859
$889$	45.4891	1.52566
$890$	0	0
$891$	0.627719	0.0210294
$892$	−9.25544	−0.309895
$893$	−4.23369	−0.141675
$894$	−12.1168	−0.405248
$895$	0	0
$896$	3.37228	0.112660
$897$	6.74456	0.225194
$898$	8.97825	0.299608
$899$	8.74456	0.291647
$900$	0	0
$901$	−6.51087	−0.216909
$902$	−0.467376	−0.0155619
$903$	−2.11684	−0.0704442
$904$	−2.62772	−0.0873966
$905$	0	0
$906$	−8.00000	−0.265782
$907$	17.4891	0.580717	0.290358	−	0.956918i	$-0.406225\pi$
$907$	17.4891	0.580717	0.290358	+	0.956918i	$0.406225\pi$
$908$	−6.11684	−0.202995
$909$	8.11684	0.269219
$910$	0	0
$911$	−2.51087	−0.0831890	−0.0415945	−	0.999135i	$-0.513244\pi$
$911$	−2.51087	−0.0831890	−0.0415945	+	0.999135i	$0.513244\pi$
$912$	0.627719	0.0207858
$913$	7.53262	0.249293
$914$	34.4674	1.14008
$915$	0	0
$916$	3.88316	0.128303
$917$	54.7446	1.80782
$918$	4.74456	0.156594
$919$	−12.2337	−0.403552	−0.201776	−	0.979432i	$-0.564671\pi$
$919$	−12.2337	−0.403552	−0.201776	+	0.979432i	$0.564671\pi$
$920$	0	0
$921$	30.9783	1.02077
$922$	19.7228	0.649536
$923$	6.74456	0.222000
$924$	2.11684	0.0696391
$925$	0	0
$926$	2.51087	0.0825125
$927$	8.00000	0.262754
$928$	8.74456	0.287054
$929$	−16.1168	−0.528776	−0.264388	−	0.964416i	$-0.585170\pi$
$929$	−16.1168	−0.528776	−0.264388	+	0.964416i	$0.585170\pi$
$930$	0	0
$931$	−2.74456	−0.0899494
$932$	14.8614	0.486802
$933$	−8.00000	−0.261908
$934$	6.51087	0.213042
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	31.2554	1.02107	0.510535	−	0.859857i	$-0.329447\pi$
$937$	31.2554	1.02107	0.510535	+	0.859857i	$0.329447\pi$
$938$	−36.2337	−1.18307
$939$	10.0000	0.326338
$940$	0	0
$941$	−19.7228	−0.642945	−0.321473	−	0.946919i	$-0.604178\pi$
$941$	−19.7228	−0.642945	−0.321473	+	0.946919i	$0.604178\pi$
$942$	−9.37228	−0.305365
$943$	−2.51087	−0.0817653
$944$	2.74456	0.0893279
$945$	0	0
$946$	0.394031	0.0128110
$947$	2.74456	0.0891863	0.0445932	−	0.999005i	$-0.485801\pi$
$947$	2.74456	0.0891863	0.0445932	+	0.999005i	$0.485801\pi$
$948$	4.62772	0.150301
$949$	16.2337	0.526968
$950$	0	0
$951$	−24.7446	−0.802397
$952$	16.0000	0.518563
$953$	−11.7228	−0.379739	−0.189870	−	0.981809i	$-0.560807\pi$
$953$	−11.7228	−0.379739	−0.189870	+	0.981809i	$0.560807\pi$
$954$	1.37228	0.0444292
$955$	0	0
$956$	−29.4891	−0.953746
$957$	5.48913	0.177438
$958$	−8.86141	−0.286299
$959$	−11.7663	−0.379954
$960$	0	0
$961$	1.00000	0.0322581
$962$	−1.48913	−0.0480113
$963$	−0.627719	−0.0202280
$964$	4.51087	0.145285
$965$	0	0
$966$	11.3723	0.365897
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	10.6060	0.340889
$969$	2.97825	0.0956752
$970$	0	0
$971$	−48.7011	−1.56289	−0.781446	−	0.623973i	$-0.785517\pi$
$971$	−48.7011	−1.56289	−0.781446	+	0.623973i	$0.785517\pi$
$972$	−1.00000	−0.0320750
$973$	36.2337	1.16160
$974$	−37.4891	−1.20123
$975$	0	0
$976$	11.4891	0.367758
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	−24.2337	−0.774908
$979$	−0.861407	−0.0275307
$980$	0	0
$981$	4.74456	0.151482
$982$	−27.6060	−0.880942
$983$	2.97825	0.0949914	0.0474957	−	0.998871i	$-0.484876\pi$
$983$	2.97825	0.0949914	0.0474957	+	0.998871i	$0.484876\pi$
$984$	0.744563	0.0237358
$985$	0	0
$986$	41.4891	1.32128
$987$	22.7446	0.723967
$988$	−1.25544	−0.0399408
$989$	2.11684	0.0673117
$990$	0	0
$991$	−47.6060	−1.51225	−0.756127	−	0.654425i	$-0.772911\pi$
$991$	−47.6060	−1.51225	−0.756127	+	0.654425i	$0.772911\pi$
$992$	1.00000	0.0317500
$993$	1.48913	0.0472560
$994$	11.3723	0.360707
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	−33.4891	−1.06008
$999$	−0.744563	−0.0235569

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.by.1.1		2
5.2	odd	4		4650.2.d.bh.3349.1		4
5.3	odd	4		4650.2.d.bh.3349.4		4
5.4	even	2		930.2.a.r.1.2	✓	2
15.14	odd	2		2790.2.a.bd.1.2		2
20.19	odd	2		7440.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.r.1.2	✓	2	5.4	even	2
2790.2.a.bd.1.2		2	15.14	odd	2
4650.2.a.by.1.1		2	1.1	even	1	trivial
4650.2.d.bh.3349.1		4	5.2	odd	4
4650.2.d.bh.3349.4		4	5.3	odd	4
7440.2.a.bg.1.1		2	20.19	odd	2

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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

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L-functions

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4650.2.a.by.1.1

Level	$4650$
Weight	$2$
Character	4650.1
Self dual	yes
Analytic conductor	$37.130$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$