LMFDB - Embedded newform 8450.2.a.cy.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,9,0,9,0,0,-4,9,15,0,-1,0,0,-4,0,9,2] 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:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 21x^{7} - 3x^{6} + 133x^{5} + 28x^{4} - 249x^{3} + 21x^{2} + 126x - 43 $ x^9 - 21x^7 - 3x^6 + 133x^5 + 28x^4 - 249x^3 + 21x^2 + 126*x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-3.19756$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +3.19756 q^{3} +1.00000 q^{4} +3.19756 q^{6} -4.88245 q^{7} +1.00000 q^{8} +7.22438 q^{9} +2.67335 q^{11} +3.19756 q^{12} -4.88245 q^{14} +1.00000 q^{16} +0.251383 q^{17} +7.22438 q^{18} +6.00402 q^{19} -15.6119 q^{21} +2.67335 q^{22} +8.21459 q^{23} +3.19756 q^{24} +13.5077 q^{27} -4.88245 q^{28} -4.32537 q^{29} +0.311545 q^{31} +1.00000 q^{32} +8.54820 q^{33} +0.251383 q^{34} +7.22438 q^{36} -9.29183 q^{37} +6.00402 q^{38} -3.48438 q^{41} -15.6119 q^{42} +3.39417 q^{43} +2.67335 q^{44} +8.21459 q^{46} +0.622582 q^{47} +3.19756 q^{48} +16.8383 q^{49} +0.803810 q^{51} -3.61716 q^{53} +13.5077 q^{54} -4.88245 q^{56} +19.1982 q^{57} -4.32537 q^{58} +2.01655 q^{59} -2.09255 q^{61} +0.311545 q^{62} -35.2727 q^{63} +1.00000 q^{64} +8.54820 q^{66} +4.59674 q^{67} +0.251383 q^{68} +26.2666 q^{69} +15.2210 q^{71} +7.22438 q^{72} -12.0033 q^{73} -9.29183 q^{74} +6.00402 q^{76} -13.0525 q^{77} +5.70863 q^{79} +21.5185 q^{81} -3.48438 q^{82} -2.32437 q^{83} -15.6119 q^{84} +3.39417 q^{86} -13.8306 q^{87} +2.67335 q^{88} +10.3329 q^{89} +8.21459 q^{92} +0.996184 q^{93} +0.622582 q^{94} +3.19756 q^{96} -2.13228 q^{97} +16.8383 q^{98} +19.3133 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{7} + 9 q^{8} + 15 q^{9} - q^{11} - 4 q^{14} + 9 q^{16} + 2 q^{17} + 15 q^{18} + 10 q^{19} + 3 q^{21} - q^{22} - q^{23} - 9 q^{27} - 4 q^{28} - q^{29} - 4 q^{31} + 9 q^{32}+ \cdots - 62 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^7 + 9 * q^8 + 15 * q^9 - q^11 - 4 * q^14 + 9 * q^16 + 2 * q^17 + 15 * q^18 + 10 * q^19 + 3 * q^21 - q^22 - q^23 - 9 * q^27 - 4 * q^28 - q^29 - 4 * q^31 + 9 * q^32 + 30 * q^33 + 2 * q^34 + 15 * q^36 + 18 * q^37 + 10 * q^38 - 2 * q^41 + 3 * q^42 - 5 * q^43 - q^44 - q^46 - 10 * q^47 + 39 * q^49 + 38 * q^51 + 3 * q^53 - 9 * q^54 - 4 * q^56 + 14 * q^57 - q^58 - 20 * q^59 + 40 * q^61 - 4 * q^62 - 45 * q^63 + 9 * q^64 + 30 * q^66 + 15 * q^67 + 2 * q^68 + 68 * q^69 + 46 * q^71 + 15 * q^72 + 13 * q^73 + 18 * q^74 + 10 * q^76 + 13 * q^77 + 37 * q^79 + 53 * q^81 - 2 * q^82 + 2 * q^83 + 3 * q^84 - 5 * q^86 + q^87 - q^88 + 14 * q^89 - q^92 + 2 * q^93 - 10 * q^94 + 23 * q^97 + 39 * q^98 - 62 * 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
$3$	3.19756	1.84611	0.923055	−	0.384667i	$-0.125684\pi$
$3$	3.19756	1.84611	0.923055	+	0.384667i	$0.125684\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.19756	1.30540
$7$	−4.88245	−1.84539	−0.922696	−	0.385528i	$-0.874019\pi$
$7$	−4.88245	−1.84539	−0.922696	+	0.385528i	$0.874019\pi$
$8$	1.00000	0.353553
$9$	7.22438	2.40813
$10$	0	0
$11$	2.67335	0.806047	0.403023	−	0.915190i	$-0.367959\pi$
$11$	2.67335	0.806047	0.403023	+	0.915190i	$0.367959\pi$
$12$	3.19756	0.923055
$13$	0	0
$13$	0	0
$14$	−4.88245	−1.30489
$15$	0	0
$16$	1.00000	0.250000
$17$	0.251383	0.0609692	0.0304846	−	0.999535i	$-0.490295\pi$
$17$	0.251383	0.0609692	0.0304846	+	0.999535i	$0.490295\pi$
$18$	7.22438	1.70280
$19$	6.00402	1.37742	0.688708	−	0.725039i	$-0.258178\pi$
$19$	6.00402	1.37742	0.688708	+	0.725039i	$0.258178\pi$
$20$	0	0
$21$	−15.6119	−3.40680
$22$	2.67335	0.569961
$23$	8.21459	1.71286	0.856430	−	0.516263i	$-0.172677\pi$
$23$	8.21459	1.71286	0.856430	+	0.516263i	$0.172677\pi$
$24$	3.19756	0.652699
$25$	0	0
$26$	0	0
$27$	13.5077	2.59956
$28$	−4.88245	−0.922696
$29$	−4.32537	−0.803201	−0.401600	−	0.915815i	$-0.631546\pi$
$29$	−4.32537	−0.803201	−0.401600	+	0.915815i	$0.631546\pi$
$30$	0	0
$31$	0.311545	0.0559552	0.0279776	−	0.999609i	$-0.491093\pi$
$31$	0.311545	0.0559552	0.0279776	+	0.999609i	$0.491093\pi$
$32$	1.00000	0.176777
$33$	8.54820	1.48805
$34$	0.251383	0.0431117
$35$	0	0
$36$	7.22438	1.20406
$37$	−9.29183	−1.52757	−0.763784	−	0.645472i	$-0.776661\pi$
$37$	−9.29183	−1.52757	−0.763784	+	0.645472i	$0.776661\pi$
$38$	6.00402	0.973980
$39$	0	0
$40$	0	0
$41$	−3.48438	−0.544169	−0.272085	−	0.962273i	$-0.587713\pi$
$41$	−3.48438	−0.544169	−0.272085	+	0.962273i	$0.587713\pi$
$42$	−15.6119	−2.40897
$43$	3.39417	0.517606	0.258803	−	0.965930i	$-0.416672\pi$
$43$	3.39417	0.517606	0.258803	+	0.965930i	$0.416672\pi$
$44$	2.67335	0.403023
$45$	0	0
$46$	8.21459	1.21117
$47$	0.622582	0.0908129	0.0454065	−	0.998969i	$-0.485542\pi$
$47$	0.622582	0.0908129	0.0454065	+	0.998969i	$0.485542\pi$
$48$	3.19756	0.461528
$49$	16.8383	2.40547
$50$	0	0
$51$	0.803810	0.112556
$52$	0	0
$53$	−3.61716	−0.496855	−0.248427	−	0.968651i	$-0.579914\pi$
$53$	−3.61716	−0.496855	−0.248427	+	0.968651i	$0.579914\pi$
$54$	13.5077	1.83816
$55$	0	0
$56$	−4.88245	−0.652445
$57$	19.1982	2.54286
$58$	−4.32537	−0.567949
$59$	2.01655	0.262533	0.131267	−	0.991347i	$-0.458096\pi$
$59$	2.01655	0.262533	0.131267	+	0.991347i	$0.458096\pi$
$60$	0	0
$61$	−2.09255	−0.267924	−0.133962	−	0.990986i	$-0.542770\pi$
$61$	−2.09255	−0.267924	−0.133962	+	0.990986i	$0.542770\pi$
$62$	0.311545	0.0395663
$63$	−35.2727	−4.44394
$64$	1.00000	0.125000
$65$	0	0
$66$	8.54820	1.05221
$67$	4.59674	0.561581	0.280791	−	0.959769i	$-0.409403\pi$
$67$	4.59674	0.561581	0.280791	+	0.959769i	$0.409403\pi$
$68$	0.251383	0.0304846
$69$	26.2666	3.16213
$70$	0	0
$71$	15.2210	1.80640	0.903202	−	0.429215i	$-0.141210\pi$
$71$	15.2210	1.80640	0.903202	+	0.429215i	$0.141210\pi$
$72$	7.22438	0.851401
$73$	−12.0033	−1.40489	−0.702443	−	0.711740i	$-0.747907\pi$
$73$	−12.0033	−1.40489	−0.702443	+	0.711740i	$0.747907\pi$
$74$	−9.29183	−1.08015
$75$	0	0
$76$	6.00402	0.688708
$77$	−13.0525	−1.48747
$78$	0	0
$79$	5.70863	0.642271	0.321136	−	0.947033i	$-0.395935\pi$
$79$	5.70863	0.642271	0.321136	+	0.947033i	$0.395935\pi$
$80$	0	0
$81$	21.5185	2.39094
$82$	−3.48438	−0.384786
$83$	−2.32437	−0.255133	−0.127566	−	0.991830i	$-0.540717\pi$
$83$	−2.32437	−0.255133	−0.127566	+	0.991830i	$0.540717\pi$
$84$	−15.6119	−1.70340
$85$	0	0
$86$	3.39417	0.366003
$87$	−13.8306	−1.48280
$88$	2.67335	0.284981
$89$	10.3329	1.09529	0.547644	−	0.836711i	$-0.315525\pi$
$89$	10.3329	1.09529	0.547644	+	0.836711i	$0.315525\pi$
$90$	0	0
$91$	0	0
$92$	8.21459	0.856430
$93$	0.996184	0.103299
$94$	0.622582	0.0642144
$95$	0	0
$96$	3.19756	0.326349
$97$	−2.13228	−0.216500	−0.108250	−	0.994124i	$-0.534525\pi$
$97$	−2.13228	−0.216500	−0.108250	+	0.994124i	$0.534525\pi$
$98$	16.8383	1.70093
$99$	19.3133	1.94106
$100$	0	0
$101$	−14.6401	−1.45675	−0.728374	−	0.685180i	$-0.759724\pi$
$101$	−14.6401	−1.45675	−0.728374	+	0.685180i	$0.759724\pi$
$102$	0.803810	0.0795891
$103$	13.9235	1.37192	0.685960	−	0.727640i	$-0.259383\pi$
$103$	13.9235	1.37192	0.685960	+	0.727640i	$0.259383\pi$
$104$	0	0
$105$	0	0
$106$	−3.61716	−0.351329
$107$	2.94584	0.284785	0.142392	−	0.989810i	$-0.454521\pi$
$107$	2.94584	0.284785	0.142392	+	0.989810i	$0.454521\pi$
$108$	13.5077	1.29978
$109$	1.81954	0.174280	0.0871401	−	0.996196i	$-0.472227\pi$
$109$	1.81954	0.174280	0.0871401	+	0.996196i	$0.472227\pi$
$110$	0	0
$111$	−29.7112	−2.82006
$112$	−4.88245	−0.461348
$113$	15.1589	1.42603	0.713014	−	0.701150i	$-0.247330\pi$
$113$	15.1589	1.42603	0.713014	+	0.701150i	$0.247330\pi$
$114$	19.1982	1.79808
$115$	0	0
$116$	−4.32537	−0.401600
$117$	0	0
$118$	2.01655	0.185639
$119$	−1.22736	−0.112512
$120$	0	0
$121$	−3.85318	−0.350289
$122$	−2.09255	−0.189451
$123$	−11.1415	−1.00460
$124$	0.311545	0.0279776
$125$	0	0
$126$	−35.2727	−3.14234
$127$	0.572035	0.0507599	0.0253799	−	0.999678i	$-0.491920\pi$
$127$	0.572035	0.0507599	0.0253799	+	0.999678i	$0.491920\pi$
$128$	1.00000	0.0883883
$129$	10.8531	0.955558
$130$	0	0
$131$	14.3728	1.25576	0.627878	−	0.778312i	$-0.283924\pi$
$131$	14.3728	1.25576	0.627878	+	0.778312i	$0.283924\pi$
$132$	8.54820	0.744026
$133$	−29.3143	−2.54187
$134$	4.59674	0.397098
$135$	0	0
$136$	0.251383	0.0215559
$137$	−5.10073	−0.435785	−0.217892	−	0.975973i	$-0.569918\pi$
$137$	−5.10073	−0.435785	−0.217892	+	0.975973i	$0.569918\pi$
$138$	26.2666	2.23596
$139$	21.3219	1.80850	0.904248	−	0.427007i	$-0.140432\pi$
$139$	21.3219	1.80850	0.904248	+	0.427007i	$0.140432\pi$
$140$	0	0
$141$	1.99074	0.167651
$142$	15.2210	1.27732
$143$	0	0
$144$	7.22438	0.602031
$145$	0	0
$146$	−12.0033	−0.993404
$147$	53.8415	4.44077
$148$	−9.29183	−0.763784
$149$	−8.05447	−0.659848	−0.329924	−	0.944008i	$-0.607023\pi$
$149$	−8.05447	−0.659848	−0.329924	+	0.944008i	$0.607023\pi$
$150$	0	0
$151$	14.1403	1.15072	0.575360	−	0.817900i	$-0.304862\pi$
$151$	14.1403	1.15072	0.575360	+	0.817900i	$0.304862\pi$
$152$	6.00402	0.486990
$153$	1.81608	0.146822
$154$	−13.0525	−1.05180
$155$	0	0
$156$	0	0
$157$	−10.3364	−0.824934	−0.412467	−	0.910973i	$-0.635333\pi$
$157$	−10.3364	−0.824934	−0.412467	+	0.910973i	$0.635333\pi$
$158$	5.70863	0.454154
$159$	−11.5661	−0.917249
$160$	0	0
$161$	−40.1073	−3.16090
$162$	21.5185	1.69065
$163$	−1.30284	−0.102046	−0.0510231	−	0.998697i	$-0.516248\pi$
$163$	−1.30284	−0.102046	−0.0510231	+	0.998697i	$0.516248\pi$
$164$	−3.48438	−0.272085
$165$	0	0
$166$	−2.32437	−0.180406
$167$	−10.4567	−0.809160	−0.404580	−	0.914503i	$-0.632582\pi$
$167$	−10.4567	−0.809160	−0.404580	+	0.914503i	$0.632582\pi$
$168$	−15.6119	−1.20449
$169$	0	0
$170$	0	0
$171$	43.3753	3.31699
$172$	3.39417	0.258803
$173$	8.37489	0.636731	0.318366	−	0.947968i	$-0.396866\pi$
$173$	8.37489	0.636731	0.318366	+	0.947968i	$0.396866\pi$
$174$	−13.8306	−1.04850
$175$	0	0
$176$	2.67335	0.201512
$177$	6.44805	0.484665
$178$	10.3329	0.774486
$179$	−7.12772	−0.532751	−0.266375	−	0.963869i	$-0.585826\pi$
$179$	−7.12772	−0.532751	−0.266375	+	0.963869i	$0.585826\pi$
$180$	0	0
$181$	1.22784	0.0912643	0.0456321	−	0.998958i	$-0.485470\pi$
$181$	1.22784	0.0912643	0.0456321	+	0.998958i	$0.485470\pi$
$182$	0	0
$183$	−6.69106	−0.494617
$184$	8.21459	0.605587
$185$	0	0
$186$	0.996184	0.0730438
$187$	0.672035	0.0491440
$188$	0.622582	0.0454065
$189$	−65.9506	−4.79720
$190$	0	0
$191$	18.1199	1.31111	0.655555	−	0.755147i	$-0.272435\pi$
$191$	18.1199	1.31111	0.655555	+	0.755147i	$0.272435\pi$
$192$	3.19756	0.230764
$193$	−23.7221	−1.70756	−0.853778	−	0.520638i	$-0.825694\pi$
$193$	−23.7221	−1.70756	−0.853778	+	0.520638i	$0.825694\pi$
$194$	−2.13228	−0.153089
$195$	0	0
$196$	16.8383	1.20274
$197$	7.73716	0.551250	0.275625	−	0.961265i	$-0.411115\pi$
$197$	7.73716	0.551250	0.275625	+	0.961265i	$0.411115\pi$
$198$	19.3133	1.37254
$199$	−7.14002	−0.506142	−0.253071	−	0.967448i	$-0.581441\pi$
$199$	−7.14002	−0.506142	−0.253071	+	0.967448i	$0.581441\pi$
$200$	0	0
$201$	14.6983	1.03674
$202$	−14.6401	−1.03008
$203$	21.1184	1.48222
$204$	0.803810	0.0562780
$205$	0	0
$206$	13.9235	0.970093
$207$	59.3453	4.12478
$208$	0	0
$209$	16.0509	1.11026
$210$	0	0
$211$	−25.6074	−1.76289	−0.881444	−	0.472289i	$-0.843428\pi$
$211$	−25.6074	−1.76289	−0.881444	+	0.472289i	$0.843428\pi$
$212$	−3.61716	−0.248427
$213$	48.6701	3.33482
$214$	2.94584	0.201373
$215$	0	0
$216$	13.5077	0.919082
$217$	−1.52110	−0.103259
$218$	1.81954	0.123235
$219$	−38.3814	−2.59357
$220$	0	0
$221$	0	0
$222$	−29.7112	−1.99408
$223$	3.02615	0.202646	0.101323	−	0.994854i	$-0.467692\pi$
$223$	3.02615	0.202646	0.101323	+	0.994854i	$0.467692\pi$
$224$	−4.88245	−0.326222
$225$	0	0
$226$	15.1589	1.00835
$227$	10.9527	0.726959	0.363480	−	0.931602i	$-0.381589\pi$
$227$	10.9527	0.726959	0.363480	+	0.931602i	$0.381589\pi$
$228$	19.1982	1.27143
$229$	−14.6073	−0.965281	−0.482640	−	0.875819i	$-0.660322\pi$
$229$	−14.6073	−0.965281	−0.482640	+	0.875819i	$0.660322\pi$
$230$	0	0
$231$	−41.7362	−2.74604
$232$	−4.32537	−0.283974
$233$	−23.2146	−1.52084	−0.760421	−	0.649431i	$-0.775007\pi$
$233$	−23.2146	−1.52084	−0.760421	+	0.649431i	$0.775007\pi$
$234$	0	0
$235$	0	0
$236$	2.01655	0.131267
$237$	18.2537	1.18570
$238$	−1.22736	−0.0795581
$239$	−10.3141	−0.667161	−0.333580	−	0.942722i	$-0.608257\pi$
$239$	−10.3141	−0.667161	−0.333580	+	0.942722i	$0.608257\pi$
$240$	0	0
$241$	5.45201	0.351195	0.175597	−	0.984462i	$-0.443814\pi$
$241$	5.45201	0.351195	0.175597	+	0.984462i	$0.443814\pi$
$242$	−3.85318	−0.247692
$243$	28.2835	1.81439
$244$	−2.09255	−0.133962
$245$	0	0
$246$	−11.1415	−0.710357
$247$	0	0
$248$	0.311545	0.0197831
$249$	−7.43231	−0.471003
$250$	0	0
$251$	6.94123	0.438127	0.219063	−	0.975711i	$-0.429700\pi$
$251$	6.94123	0.438127	0.219063	+	0.975711i	$0.429700\pi$
$252$	−35.2727	−2.22197
$253$	21.9605	1.38064
$254$	0.572035	0.0358927
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.46921	−0.0916466	−0.0458233	−	0.998950i	$-0.514591\pi$
$257$	−1.46921	−0.0916466	−0.0458233	+	0.998950i	$0.514591\pi$
$258$	10.8531	0.675682
$259$	45.3669	2.81896
$260$	0	0
$261$	−31.2481	−1.93421
$262$	14.3728	0.887954
$263$	−17.6267	−1.08691	−0.543456	−	0.839438i	$-0.682884\pi$
$263$	−17.6267	−1.08691	−0.543456	+	0.839438i	$0.682884\pi$
$264$	8.54820	0.526106
$265$	0	0
$266$	−29.3143	−1.79738
$267$	33.0401	2.02202
$268$	4.59674	0.280791
$269$	−14.7647	−0.900217	−0.450109	−	0.892974i	$-0.648615\pi$
$269$	−14.7647	−0.900217	−0.450109	+	0.892974i	$0.648615\pi$
$270$	0	0
$271$	0.418154	0.0254011	0.0127005	−	0.999919i	$-0.495957\pi$
$271$	0.418154	0.0254011	0.0127005	+	0.999919i	$0.495957\pi$
$272$	0.251383	0.0152423
$273$	0	0
$274$	−5.10073	−0.308146
$275$	0	0
$276$	26.2666	1.58106
$277$	−17.6433	−1.06008	−0.530040	−	0.847972i	$-0.677823\pi$
$277$	−17.6433	−1.06008	−0.530040	+	0.847972i	$0.677823\pi$
$278$	21.3219	1.27880
$279$	2.25072	0.134747
$280$	0	0
$281$	7.14422	0.426188	0.213094	−	0.977032i	$-0.431646\pi$
$281$	7.14422	0.426188	0.213094	+	0.977032i	$0.431646\pi$
$282$	1.99074	0.118547
$283$	−25.9638	−1.54339	−0.771693	−	0.635996i	$-0.780590\pi$
$283$	−25.9638	−1.54339	−0.771693	+	0.635996i	$0.780590\pi$
$284$	15.2210	0.903202
$285$	0	0
$286$	0	0
$287$	17.0123	1.00421
$288$	7.22438	0.425700
$289$	−16.9368	−0.996283
$290$	0	0
$291$	−6.81808	−0.399683
$292$	−12.0033	−0.702443
$293$	19.4593	1.13682	0.568411	−	0.822744i	$-0.307558\pi$
$293$	19.4593	1.13682	0.568411	+	0.822744i	$0.307558\pi$
$294$	53.8415	3.14010
$295$	0	0
$296$	−9.29183	−0.540077
$297$	36.1108	2.09536
$298$	−8.05447	−0.466583
$299$	0	0
$300$	0	0
$301$	−16.5719	−0.955187
$302$	14.1403	0.813682
$303$	−46.8127	−2.68932
$304$	6.00402	0.344354
$305$	0	0
$306$	1.81608	0.103818
$307$	28.8189	1.64478	0.822390	−	0.568924i	$-0.192640\pi$
$307$	28.8189	1.64478	0.822390	+	0.568924i	$0.192640\pi$
$308$	−13.0525	−0.743736
$309$	44.5211	2.53271
$310$	0	0
$311$	1.28730	0.0729962	0.0364981	−	0.999334i	$-0.488380\pi$
$311$	1.28730	0.0729962	0.0364981	+	0.999334i	$0.488380\pi$
$312$	0	0
$313$	−12.0688	−0.682167	−0.341084	−	0.940033i	$-0.610794\pi$
$313$	−12.0688	−0.682167	−0.341084	+	0.940033i	$0.610794\pi$
$314$	−10.3364	−0.583316
$315$	0	0
$316$	5.70863	0.321136
$317$	−5.72558	−0.321581	−0.160790	−	0.986989i	$-0.551404\pi$
$317$	−5.72558	−0.321581	−0.160790	+	0.986989i	$0.551404\pi$
$318$	−11.5661	−0.648593
$319$	−11.5632	−0.647417
$320$	0	0
$321$	9.41948	0.525744
$322$	−40.1073	−2.23509
$323$	1.50931	0.0839800
$324$	21.5185	1.19547
$325$	0	0
$326$	−1.30284	−0.0721576
$327$	5.81808	0.321741
$328$	−3.48438	−0.192393
$329$	−3.03973	−0.167585
$330$	0	0
$331$	−17.2506	−0.948178	−0.474089	−	0.880477i	$-0.657222\pi$
$331$	−17.2506	−0.948178	−0.474089	+	0.880477i	$0.657222\pi$
$332$	−2.32437	−0.127566
$333$	−67.1277	−3.67857
$334$	−10.4567	−0.572163
$335$	0	0
$336$	−15.6119	−0.851700
$337$	−1.82457	−0.0993905	−0.0496953	−	0.998764i	$-0.515825\pi$
$337$	−1.82457	−0.0993905	−0.0496953	+	0.998764i	$0.515825\pi$
$338$	0	0
$339$	48.4714	2.63260
$340$	0	0
$341$	0.832871	0.0451025
$342$	43.3753	2.34547
$343$	−48.0351	−2.59365
$344$	3.39417	0.183001
$345$	0	0
$346$	8.37489	0.450237
$347$	−28.9749	−1.55546	−0.777728	−	0.628601i	$-0.783628\pi$
$347$	−28.9749	−1.55546	−0.777728	+	0.628601i	$0.783628\pi$
$348$	−13.8306	−0.741399
$349$	−0.850671	−0.0455354	−0.0227677	−	0.999741i	$-0.507248\pi$
$349$	−0.850671	−0.0455354	−0.0227677	+	0.999741i	$0.507248\pi$
$350$	0	0
$351$	0	0
$352$	2.67335	0.142490
$353$	24.3387	1.29542	0.647710	−	0.761887i	$-0.275727\pi$
$353$	24.3387	1.29542	0.647710	+	0.761887i	$0.275727\pi$
$354$	6.44805	0.342710
$355$	0	0
$356$	10.3329	0.547644
$357$	−3.92456	−0.207710
$358$	−7.12772	−0.376712
$359$	−7.71757	−0.407318	−0.203659	−	0.979042i	$-0.565283\pi$
$359$	−7.71757	−0.407318	−0.203659	+	0.979042i	$0.565283\pi$
$360$	0	0
$361$	17.0482	0.897276
$362$	1.22784	0.0645336
$363$	−12.3208	−0.646672
$364$	0	0
$365$	0	0
$366$	−6.69106	−0.349747
$367$	−10.1218	−0.528354	−0.264177	−	0.964474i	$-0.585100\pi$
$367$	−10.1218	−0.528354	−0.264177	+	0.964474i	$0.585100\pi$
$368$	8.21459	0.428215
$369$	−25.1725	−1.31043
$370$	0	0
$371$	17.6606	0.916892
$372$	0.996184	0.0516497
$373$	23.4158	1.21242	0.606211	−	0.795303i	$-0.292688\pi$
$373$	23.4158	1.21242	0.606211	+	0.795303i	$0.292688\pi$
$374$	0.672035	0.0347501
$375$	0	0
$376$	0.622582	0.0321072
$377$	0	0
$378$	−65.9506	−3.39213
$379$	−6.59880	−0.338958	−0.169479	−	0.985534i	$-0.554208\pi$
$379$	−6.59880	−0.338958	−0.169479	+	0.985534i	$0.554208\pi$
$380$	0	0
$381$	1.82911	0.0937084
$382$	18.1199	0.927095
$383$	−25.1726	−1.28626	−0.643131	−	0.765756i	$-0.722365\pi$
$383$	−25.1726	−1.28626	−0.643131	+	0.765756i	$0.722365\pi$
$384$	3.19756	0.163175
$385$	0	0
$386$	−23.7221	−1.20742
$387$	24.5208	1.24646
$388$	−2.13228	−0.108250
$389$	−14.8655	−0.753708	−0.376854	−	0.926273i	$-0.622994\pi$
$389$	−14.8655	−0.753708	−0.376854	+	0.926273i	$0.622994\pi$
$390$	0	0
$391$	2.06500	0.104432
$392$	16.8383	0.850463
$393$	45.9578	2.31826
$394$	7.73716	0.389792
$395$	0	0
$396$	19.3133	0.970531
$397$	−17.7608	−0.891388	−0.445694	−	0.895185i	$-0.647043\pi$
$397$	−17.7608	−0.891388	−0.445694	+	0.895185i	$0.647043\pi$
$398$	−7.14002	−0.357897
$399$	−93.7342	−4.69258
$400$	0	0
$401$	16.2630	0.812138	0.406069	−	0.913842i	$-0.366899\pi$
$401$	16.2630	0.812138	0.406069	+	0.913842i	$0.366899\pi$
$402$	14.6983	0.733087
$403$	0	0
$404$	−14.6401	−0.728374
$405$	0	0
$406$	21.1184	1.04809
$407$	−24.8404	−1.23129
$408$	0.803810	0.0397945
$409$	−20.1556	−0.996631	−0.498316	−	0.866996i	$-0.666048\pi$
$409$	−20.1556	−0.996631	−0.498316	+	0.866996i	$0.666048\pi$
$410$	0	0
$411$	−16.3099	−0.804507
$412$	13.9235	0.685960
$413$	−9.84573	−0.484477
$414$	59.3453	2.91666
$415$	0	0
$416$	0	0
$417$	68.1779	3.33869
$418$	16.0509	0.785074
$419$	28.6680	1.40052	0.700261	−	0.713887i	$-0.253067\pi$
$419$	28.6680	1.40052	0.700261	+	0.713887i	$0.253067\pi$
$420$	0	0
$421$	21.0686	1.02682	0.513409	−	0.858144i	$-0.328382\pi$
$421$	21.0686	1.02682	0.513409	+	0.858144i	$0.328382\pi$
$422$	−25.6074	−1.24655
$423$	4.49777	0.218689
$424$	−3.61716	−0.175665
$425$	0	0
$426$	48.6701	2.35808
$427$	10.2168	0.494425
$428$	2.94584	0.142392
$429$	0	0
$430$	0	0
$431$	−5.38784	−0.259523	−0.129762	−	0.991545i	$-0.541421\pi$
$431$	−5.38784	−0.259523	−0.129762	+	0.991545i	$0.541421\pi$
$432$	13.5077	0.649889
$433$	5.87863	0.282509	0.141255	−	0.989973i	$-0.454886\pi$
$433$	5.87863	0.282509	0.141255	+	0.989973i	$0.454886\pi$
$434$	−1.52110	−0.0730153
$435$	0	0
$436$	1.81954	0.0871401
$437$	49.3205	2.35932
$438$	−38.3814	−1.83393
$439$	−21.5346	−1.02779	−0.513896	−	0.857853i	$-0.671798\pi$
$439$	−21.5346	−1.02779	−0.513896	+	0.857853i	$0.671798\pi$
$440$	0	0
$441$	121.646	5.79268
$442$	0	0
$443$	37.3549	1.77478	0.887391	−	0.461017i	$-0.152515\pi$
$443$	37.3549	1.77478	0.887391	+	0.461017i	$0.152515\pi$
$444$	−29.7112	−1.41003
$445$	0	0
$446$	3.02615	0.143292
$447$	−25.7546	−1.21815
$448$	−4.88245	−0.230674
$449$	10.4914	0.495122	0.247561	−	0.968872i	$-0.420371\pi$
$449$	10.4914	0.495122	0.247561	+	0.968872i	$0.420371\pi$
$450$	0	0
$451$	−9.31499	−0.438626
$452$	15.1589	0.713014
$453$	45.2144	2.12436
$454$	10.9527	0.514038
$455$	0	0
$456$	19.1982	0.899038
$457$	16.7814	0.785002	0.392501	−	0.919751i	$-0.371610\pi$
$457$	16.7814	0.785002	0.392501	+	0.919751i	$0.371610\pi$
$458$	−14.6073	−0.682557
$459$	3.39560	0.158493
$460$	0	0
$461$	24.8690	1.15826	0.579132	−	0.815234i	$-0.303392\pi$
$461$	24.8690	1.15826	0.579132	+	0.815234i	$0.303392\pi$
$462$	−41.7362	−1.94174
$463$	−33.4890	−1.55637	−0.778184	−	0.628037i	$-0.783859\pi$
$463$	−33.4890	−1.55637	−0.778184	+	0.628037i	$0.783859\pi$
$464$	−4.32537	−0.200800
$465$	0	0
$466$	−23.2146	−1.07540
$467$	−8.28758	−0.383504	−0.191752	−	0.981443i	$-0.561417\pi$
$467$	−8.28758	−0.383504	−0.191752	+	0.981443i	$0.561417\pi$
$468$	0	0
$469$	−22.4434	−1.03634
$470$	0	0
$471$	−33.0512	−1.52292
$472$	2.01655	0.0928195
$473$	9.07382	0.417215
$474$	18.2537	0.838419
$475$	0	0
$476$	−1.22736	−0.0562561
$477$	−26.1317	−1.19649
$478$	−10.3141	−0.471754
$479$	21.8070	0.996387	0.498194	−	0.867066i	$-0.333997\pi$
$479$	21.8070	0.996387	0.498194	+	0.867066i	$0.333997\pi$
$480$	0	0
$481$	0	0
$482$	5.45201	0.248332
$483$	−128.245	−5.83537
$484$	−3.85318	−0.175144
$485$	0	0
$486$	28.2835	1.28297
$487$	18.0233	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$487$	18.0233	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$488$	−2.09255	−0.0947254
$489$	−4.16591	−0.188389
$490$	0	0
$491$	−1.31023	−0.0591299	−0.0295649	−	0.999563i	$-0.509412\pi$
$491$	−1.31023	−0.0591299	−0.0295649	+	0.999563i	$0.509412\pi$
$492$	−11.1415	−0.502298
$493$	−1.08732	−0.0489705
$494$	0	0
$495$	0	0
$496$	0.311545	0.0139888
$497$	−74.3159	−3.33353
$498$	−7.43231	−0.333050
$499$	−29.8490	−1.33623	−0.668113	−	0.744060i	$-0.732898\pi$
$499$	−29.8490	−1.33623	−0.668113	+	0.744060i	$0.732898\pi$
$500$	0	0
$501$	−33.4358	−1.49380
$502$	6.94123	0.309802
$503$	−35.8660	−1.59919	−0.799593	−	0.600543i	$-0.794951\pi$
$503$	−35.8660	−1.59919	−0.799593	+	0.600543i	$0.794951\pi$
$504$	−35.2727	−1.57117
$505$	0	0
$506$	21.9605	0.976263
$507$	0	0
$508$	0.572035	0.0253799
$509$	20.4116	0.904730	0.452365	−	0.891833i	$-0.350580\pi$
$509$	20.4116	0.904730	0.452365	+	0.891833i	$0.350580\pi$
$510$	0	0
$511$	58.6057	2.59256
$512$	1.00000	0.0441942
$513$	81.1004	3.58067
$514$	−1.46921	−0.0648039
$515$	0	0
$516$	10.8531	0.477779
$517$	1.66438	0.0731994
$518$	45.3669	1.99331
$519$	26.7792	1.17548
$520$	0	0
$521$	−35.7922	−1.56809	−0.784043	−	0.620706i	$-0.786846\pi$
$521$	−35.7922	−1.56809	−0.784043	+	0.620706i	$0.786846\pi$
$522$	−31.2481	−1.36769
$523$	5.60042	0.244889	0.122445	−	0.992475i	$-0.460927\pi$
$523$	5.60042	0.244889	0.122445	+	0.992475i	$0.460927\pi$
$524$	14.3728	0.627878
$525$	0	0
$526$	−17.6267	−0.768562
$527$	0.0783170	0.00341154
$528$	8.54820	0.372013
$529$	44.4794	1.93389
$530$	0	0
$531$	14.5683	0.632212
$532$	−29.3143	−1.27094
$533$	0	0
$534$	33.0401	1.42979
$535$	0	0
$536$	4.59674	0.198549
$537$	−22.7913	−0.983517
$538$	−14.7647	−0.636550
$539$	45.0148	1.93892
$540$	0	0
$541$	−13.5312	−0.581753	−0.290877	−	0.956761i	$-0.593947\pi$
$541$	−13.5312	−0.581753	−0.290877	+	0.956761i	$0.593947\pi$
$542$	0.418154	0.0179613
$543$	3.92608	0.168484
$544$	0.251383	0.0107779
$545$	0	0
$546$	0	0
$547$	19.9690	0.853814	0.426907	−	0.904296i	$-0.359603\pi$
$547$	19.9690	0.853814	0.426907	+	0.904296i	$0.359603\pi$
$548$	−5.10073	−0.217892
$549$	−15.1174	−0.645194
$550$	0	0
$551$	−25.9696	−1.10634
$552$	26.2666	1.11798
$553$	−27.8721	−1.18524
$554$	−17.6433	−0.749590
$555$	0	0
$556$	21.3219	0.904248
$557$	11.5258	0.488365	0.244182	−	0.969729i	$-0.421480\pi$
$557$	11.5258	0.488365	0.244182	+	0.969729i	$0.421480\pi$
$558$	2.25072	0.0952806
$559$	0	0
$560$	0	0
$561$	2.14887	0.0907253
$562$	7.14422	0.301361
$563$	−28.3639	−1.19540	−0.597699	−	0.801721i	$-0.703918\pi$
$563$	−28.3639	−1.19540	−0.597699	+	0.801721i	$0.703918\pi$
$564$	1.99074	0.0838253
$565$	0	0
$566$	−25.9638	−1.09134
$567$	−105.063	−4.41223
$568$	15.2210	0.638660
$569$	33.6641	1.41127	0.705636	−	0.708574i	$-0.250661\pi$
$569$	33.6641	1.41127	0.705636	+	0.708574i	$0.250661\pi$
$570$	0	0
$571$	−32.6070	−1.36456	−0.682281	−	0.731090i	$-0.739012\pi$
$571$	−32.6070	−1.36456	−0.682281	+	0.731090i	$0.739012\pi$
$572$	0	0
$573$	57.9394	2.42045
$574$	17.0123	0.710080
$575$	0	0
$576$	7.22438	0.301016
$577$	−42.2328	−1.75817	−0.879087	−	0.476662i	$-0.841847\pi$
$577$	−42.2328	−1.75817	−0.879087	+	0.476662i	$0.841847\pi$
$578$	−16.9368	−0.704478
$579$	−75.8528	−3.15234
$580$	0	0
$581$	11.3486	0.470820
$582$	−6.81808	−0.282619
$583$	−9.66995	−0.400488
$584$	−12.0033	−0.496702
$585$	0	0
$586$	19.4593	0.803855
$587$	−28.6716	−1.18340	−0.591702	−	0.806157i	$-0.701544\pi$
$587$	−28.6716	−1.18340	−0.591702	+	0.806157i	$0.701544\pi$
$588$	53.8415	2.22039
$589$	1.87052	0.0770736
$590$	0	0
$591$	24.7400	1.01767
$592$	−9.29183	−0.381892
$593$	−8.89057	−0.365092	−0.182546	−	0.983197i	$-0.558434\pi$
$593$	−8.89057	−0.365092	−0.182546	+	0.983197i	$0.558434\pi$
$594$	36.1108	1.48165
$595$	0	0
$596$	−8.05447	−0.329924
$597$	−22.8306	−0.934395
$598$	0	0
$599$	−7.24674	−0.296094	−0.148047	−	0.988980i	$-0.547299\pi$
$599$	−7.24674	−0.296094	−0.148047	+	0.988980i	$0.547299\pi$
$600$	0	0
$601$	−16.8042	−0.685457	−0.342728	−	0.939435i	$-0.611351\pi$
$601$	−16.8042	−0.685457	−0.342728	+	0.939435i	$0.611351\pi$
$602$	−16.5719	−0.675419
$603$	33.2086	1.35236
$604$	14.1403	0.575360
$605$	0	0
$606$	−46.8127	−1.90164
$607$	−0.510157	−0.0207066	−0.0103533	−	0.999946i	$-0.503296\pi$
$607$	−0.510157	−0.0207066	−0.0103533	+	0.999946i	$0.503296\pi$
$608$	6.00402	0.243495
$609$	67.5273	2.73634
$610$	0	0
$611$	0	0
$612$	1.81608	0.0734108
$613$	−13.2522	−0.535253	−0.267626	−	0.963523i	$-0.586239\pi$
$613$	−13.2522	−0.535253	−0.267626	+	0.963523i	$0.586239\pi$
$614$	28.8189	1.16304
$615$	0	0
$616$	−13.0525	−0.525901
$617$	12.9669	0.522029	0.261014	−	0.965335i	$-0.415943\pi$
$617$	12.9669	0.522029	0.261014	+	0.965335i	$0.415943\pi$
$618$	44.5211	1.79090
$619$	−35.9106	−1.44337	−0.721685	−	0.692221i	$-0.756632\pi$
$619$	−35.9106	−1.44337	−0.721685	+	0.692221i	$0.756632\pi$
$620$	0	0
$621$	110.960	4.45267
$622$	1.28730	0.0516161
$623$	−50.4500	−2.02124
$624$	0	0
$625$	0	0
$626$	−12.0688	−0.482365
$627$	51.3236	2.04967
$628$	−10.3364	−0.412467
$629$	−2.33580	−0.0931346
$630$	0	0
$631$	20.3134	0.808664	0.404332	−	0.914612i	$-0.367504\pi$
$631$	20.3134	0.808664	0.404332	+	0.914612i	$0.367504\pi$
$632$	5.70863	0.227077
$633$	−81.8812	−3.25449
$634$	−5.72558	−0.227392
$635$	0	0
$636$	−11.5661	−0.458625
$637$	0	0
$638$	−11.5632	−0.457793
$639$	109.962	4.35005
$640$	0	0
$641$	−4.01466	−0.158569	−0.0792847	−	0.996852i	$-0.525264\pi$
$641$	−4.01466	−0.158569	−0.0792847	+	0.996852i	$0.525264\pi$
$642$	9.41948	0.371757
$643$	−19.2085	−0.757508	−0.378754	−	0.925497i	$-0.623647\pi$
$643$	−19.2085	−0.757508	−0.378754	+	0.925497i	$0.623647\pi$
$644$	−40.1073	−1.58045
$645$	0	0
$646$	1.50931	0.0593828
$647$	−42.6054	−1.67499	−0.837495	−	0.546445i	$-0.815981\pi$
$647$	−42.6054	−1.67499	−0.837495	+	0.546445i	$0.815981\pi$
$648$	21.5185	0.845326
$649$	5.39096	0.211614
$650$	0	0
$651$	−4.86382	−0.190628
$652$	−1.30284	−0.0510231
$653$	26.6810	1.04411	0.522054	−	0.852913i	$-0.325166\pi$
$653$	26.6810	1.04411	0.522054	+	0.852913i	$0.325166\pi$
$654$	5.81808	0.227505
$655$	0	0
$656$	−3.48438	−0.136042
$657$	−86.7167	−3.38314
$658$	−3.03973	−0.118501
$659$	−27.4957	−1.07108	−0.535540	−	0.844510i	$-0.679892\pi$
$659$	−27.4957	−1.07108	−0.535540	+	0.844510i	$0.679892\pi$
$660$	0	0
$661$	−29.8055	−1.15930	−0.579649	−	0.814866i	$-0.696810\pi$
$661$	−29.8055	−1.15930	−0.579649	+	0.814866i	$0.696810\pi$
$662$	−17.2506	−0.670463
$663$	0	0
$664$	−2.32437	−0.0902030
$665$	0	0
$666$	−67.1277	−2.60114
$667$	−35.5311	−1.37577
$668$	−10.4567	−0.404580
$669$	9.67630	0.374107
$670$	0	0
$671$	−5.59413	−0.215959
$672$	−15.6119	−0.602243
$673$	−16.0319	−0.617986	−0.308993	−	0.951064i	$-0.599992\pi$
$673$	−16.0319	−0.617986	−0.308993	+	0.951064i	$0.599992\pi$
$674$	−1.82457	−0.0702797
$675$	0	0
$676$	0	0
$677$	−21.7965	−0.837707	−0.418853	−	0.908054i	$-0.637568\pi$
$677$	−21.7965	−0.837707	−0.418853	+	0.908054i	$0.637568\pi$
$678$	48.4714	1.86153
$679$	10.4107	0.399528
$680$	0	0
$681$	35.0220	1.34205
$682$	0.832871	0.0318923
$683$	−37.7285	−1.44364	−0.721820	−	0.692081i	$-0.756694\pi$
$683$	−37.7285	−1.44364	−0.721820	+	0.692081i	$0.756694\pi$
$684$	43.3753	1.65850
$685$	0	0
$686$	−48.0351	−1.83399
$687$	−46.7078	−1.78202
$688$	3.39417	0.129402
$689$	0	0
$690$	0	0
$691$	44.8284	1.70535	0.852677	−	0.522438i	$-0.174977\pi$
$691$	44.8284	1.70535	0.852677	+	0.522438i	$0.174977\pi$
$692$	8.37489	0.318366
$693$	−94.2963	−3.58202
$694$	−28.9749	−1.09987
$695$	0	0
$696$	−13.8306	−0.524248
$697$	−0.875913	−0.0331776
$698$	−0.850671	−0.0321984
$699$	−74.2301	−2.80764
$700$	0	0
$701$	−33.3165	−1.25835	−0.629173	−	0.777265i	$-0.716606\pi$
$701$	−33.3165	−1.25835	−0.629173	+	0.777265i	$0.716606\pi$
$702$	0	0
$703$	−55.7883	−2.10410
$704$	2.67335	0.100756
$705$	0	0
$706$	24.3387	0.916001
$707$	71.4798	2.68827
$708$	6.44805	0.242333
$709$	−0.811295	−0.0304688	−0.0152344	−	0.999884i	$-0.504849\pi$
$709$	−0.811295	−0.0304688	−0.0152344	+	0.999884i	$0.504849\pi$
$710$	0	0
$711$	41.2413	1.54667
$712$	10.3329	0.387243
$713$	2.55922	0.0958434
$714$	−3.92456	−0.146873
$715$	0	0
$716$	−7.12772	−0.266375
$717$	−32.9798	−1.23165
$718$	−7.71757	−0.288017
$719$	18.1297	0.676123	0.338061	−	0.941124i	$-0.390229\pi$
$719$	18.1297	0.676123	0.338061	+	0.941124i	$0.390229\pi$
$720$	0	0
$721$	−67.9806	−2.53173
$722$	17.0482	0.634470
$723$	17.4331	0.648345
$724$	1.22784	0.0456321
$725$	0	0
$726$	−12.3208	−0.457266
$727$	−40.3661	−1.49710	−0.748548	−	0.663080i	$-0.769249\pi$
$727$	−40.3661	−1.49710	−0.748548	+	0.663080i	$0.769249\pi$
$728$	0	0
$729$	25.8827	0.958620
$730$	0	0
$731$	0.853235	0.0315580
$732$	−6.69106	−0.247309
$733$	27.4536	1.01402	0.507010	−	0.861940i	$-0.330751\pi$
$733$	27.4536	1.01402	0.507010	+	0.861940i	$0.330751\pi$
$734$	−10.1218	−0.373603
$735$	0	0
$736$	8.21459	0.302794
$737$	12.2887	0.452661
$738$	−25.1725	−0.926612
$739$	15.8917	0.584584	0.292292	−	0.956329i	$-0.405582\pi$
$739$	15.8917	0.584584	0.292292	+	0.956329i	$0.405582\pi$
$740$	0	0
$741$	0	0
$742$	17.6606	0.648341
$743$	−0.856474	−0.0314210	−0.0157105	−	0.999877i	$-0.505001\pi$
$743$	−0.856474	−0.0314210	−0.0157105	+	0.999877i	$0.505001\pi$
$744$	0.996184	0.0365219
$745$	0	0
$746$	23.4158	0.857313
$747$	−16.7921	−0.614392
$748$	0.672035	0.0245720
$749$	−14.3829	−0.525540
$750$	0	0
$751$	−20.6326	−0.752892	−0.376446	−	0.926438i	$-0.622854\pi$
$751$	−20.6326	−0.752892	−0.376446	+	0.926438i	$0.622854\pi$
$752$	0.622582	0.0227032
$753$	22.1950	0.808830
$754$	0	0
$755$	0	0
$756$	−65.9506	−2.39860
$757$	−17.3062	−0.629003	−0.314502	−	0.949257i	$-0.601837\pi$
$757$	−17.3062	−0.629003	−0.314502	+	0.949257i	$0.601837\pi$
$758$	−6.59880	−0.239679
$759$	70.2200	2.54882
$760$	0	0
$761$	−32.7721	−1.18799	−0.593994	−	0.804469i	$-0.702450\pi$
$761$	−32.7721	−1.18799	−0.593994	+	0.804469i	$0.702450\pi$
$762$	1.82911	0.0662618
$763$	−8.88381	−0.321615
$764$	18.1199	0.655555
$765$	0	0
$766$	−25.1726	−0.909525
$767$	0	0
$768$	3.19756	0.115382
$769$	−35.5603	−1.28234	−0.641169	−	0.767400i	$-0.721550\pi$
$769$	−35.5603	−1.28234	−0.641169	+	0.767400i	$0.721550\pi$
$770$	0	0
$771$	−4.69787	−0.169190
$772$	−23.7221	−0.853778
$773$	13.6078	0.489440	0.244720	−	0.969594i	$-0.421304\pi$
$773$	13.6078	0.489440	0.244720	+	0.969594i	$0.421304\pi$
$774$	24.5208	0.881381
$775$	0	0
$776$	−2.13228	−0.0765443
$777$	145.063	5.20412
$778$	−14.8655	−0.532952
$779$	−20.9203	−0.749547
$780$	0	0
$781$	40.6912	1.45605
$782$	2.06500	0.0738444
$783$	−58.4257	−2.08796
$784$	16.8383	0.601368
$785$	0	0
$786$	45.9578	1.63926
$787$	14.4420	0.514803	0.257401	−	0.966305i	$-0.417134\pi$
$787$	14.4420	0.514803	0.257401	+	0.966305i	$0.417134\pi$
$788$	7.73716	0.275625
$789$	−56.3625	−2.00656
$790$	0	0
$791$	−74.0125	−2.63158
$792$	19.3133	0.686269
$793$	0	0
$794$	−17.7608	−0.630307
$795$	0	0
$796$	−7.14002	−0.253071
$797$	24.1395	0.855064	0.427532	−	0.904000i	$-0.359383\pi$
$797$	24.1395	0.855064	0.427532	+	0.904000i	$0.359383\pi$
$798$	−93.7342	−3.31816
$799$	0.156506	0.00553679
$800$	0	0
$801$	74.6489	2.63759
$802$	16.2630	0.574268
$803$	−32.0892	−1.13240
$804$	14.6983	0.518371
$805$	0	0
$806$	0	0
$807$	−47.2109	−1.66190
$808$	−14.6401	−0.515038
$809$	−16.5090	−0.580425	−0.290212	−	0.956962i	$-0.593726\pi$
$809$	−16.5090	−0.580425	−0.290212	+	0.956962i	$0.593726\pi$
$810$	0	0
$811$	31.7149	1.11366	0.556831	−	0.830626i	$-0.312017\pi$
$811$	31.7149	1.11366	0.556831	+	0.830626i	$0.312017\pi$
$812$	21.1184	0.741110
$813$	1.33707	0.0468932
$814$	−24.8404	−0.870654
$815$	0	0
$816$	0.803810	0.0281390
$817$	20.3787	0.712959
$818$	−20.1556	−0.704725
$819$	0	0
$820$	0	0
$821$	21.3196	0.744059	0.372030	−	0.928221i	$-0.378662\pi$
$821$	21.3196	0.744059	0.372030	+	0.928221i	$0.378662\pi$
$822$	−16.3099	−0.568872
$823$	−28.3509	−0.988249	−0.494124	−	0.869391i	$-0.664511\pi$
$823$	−28.3509	−0.988249	−0.494124	+	0.869391i	$0.664511\pi$
$824$	13.9235	0.485047
$825$	0	0
$826$	−9.84573	−0.342577
$827$	1.39404	0.0484755	0.0242377	−	0.999706i	$-0.492284\pi$
$827$	1.39404	0.0484755	0.0242377	+	0.999706i	$0.492284\pi$
$828$	59.3453	2.06239
$829$	15.9922	0.555432	0.277716	−	0.960663i	$-0.410423\pi$
$829$	15.9922	0.555432	0.277716	+	0.960663i	$0.410423\pi$
$830$	0	0
$831$	−56.4153	−1.95703
$832$	0	0
$833$	4.23286	0.146660
$834$	68.1779	2.36081
$835$	0	0
$836$	16.0509	0.555131
$837$	4.20826	0.145459
$838$	28.6680	0.990318
$839$	50.8074	1.75407	0.877033	−	0.480431i	$-0.159520\pi$
$839$	50.8074	1.75407	0.877033	+	0.480431i	$0.159520\pi$
$840$	0	0
$841$	−10.2912	−0.354869
$842$	21.0686	0.726070
$843$	22.8440	0.786791
$844$	−25.6074	−0.881444
$845$	0	0
$846$	4.49777	0.154636
$847$	18.8129	0.646421
$848$	−3.61716	−0.124214
$849$	−83.0206	−2.84926
$850$	0	0
$851$	−76.3285	−2.61651
$852$	48.6701	1.66741
$853$	15.4878	0.530293	0.265146	−	0.964208i	$-0.414580\pi$
$853$	15.4878	0.530293	0.265146	+	0.964208i	$0.414580\pi$
$854$	10.2168	0.349611
$855$	0	0
$856$	2.94584	0.100687
$857$	33.8502	1.15630	0.578150	−	0.815931i	$-0.303775\pi$
$857$	33.8502	1.15630	0.578150	+	0.815931i	$0.303775\pi$
$858$	0	0
$859$	36.9014	1.25906	0.629529	−	0.776977i	$-0.283248\pi$
$859$	36.9014	1.25906	0.629529	+	0.776977i	$0.283248\pi$
$860$	0	0
$861$	54.3979	1.85387
$862$	−5.38784	−0.183511
$863$	−35.7631	−1.21739	−0.608694	−	0.793405i	$-0.708306\pi$
$863$	−35.7631	−1.21739	−0.608694	+	0.793405i	$0.708306\pi$
$864$	13.5077	0.459541
$865$	0	0
$866$	5.87863	0.199764
$867$	−54.1564	−1.83925
$868$	−1.52110	−0.0516296
$869$	15.2612	0.517700
$870$	0	0
$871$	0	0
$872$	1.81954	0.0616174
$873$	−15.4044	−0.521359
$874$	49.3205	1.66829
$875$	0	0
$876$	−38.3814	−1.29679
$877$	−20.0585	−0.677329	−0.338665	−	0.940907i	$-0.609975\pi$
$877$	−20.0585	−0.677329	−0.338665	+	0.940907i	$0.609975\pi$
$878$	−21.5346	−0.726758
$879$	62.2221	2.09870
$880$	0	0
$881$	−28.7097	−0.967253	−0.483627	−	0.875274i	$-0.660681\pi$
$881$	−28.7097	−0.967253	−0.483627	+	0.875274i	$0.660681\pi$
$882$	121.646	4.09604
$883$	−20.4699	−0.688865	−0.344433	−	0.938811i	$-0.611929\pi$
$883$	−20.4699	−0.688865	−0.344433	+	0.938811i	$0.611929\pi$
$884$	0	0
$885$	0	0
$886$	37.3549	1.25496
$887$	−34.0271	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$887$	−34.0271	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$888$	−29.7112	−0.997041
$889$	−2.79293	−0.0936719
$890$	0	0
$891$	57.5265	1.92721
$892$	3.02615	0.101323
$893$	3.73799	0.125087
$894$	−25.7546	−0.861364
$895$	0	0
$896$	−4.88245	−0.163111
$897$	0	0
$898$	10.4914	0.350104
$899$	−1.34755	−0.0449432
$900$	0	0
$901$	−0.909290	−0.0302929
$902$	−9.31499	−0.310155
$903$	−52.9895	−1.76338
$904$	15.1589	0.504177
$905$	0	0
$906$	45.2144	1.50215
$907$	−43.9743	−1.46014	−0.730071	−	0.683371i	$-0.760513\pi$
$907$	−43.9743	−1.46014	−0.730071	+	0.683371i	$0.760513\pi$
$908$	10.9527	0.363480
$909$	−105.766	−3.50803
$910$	0	0
$911$	43.5382	1.44249	0.721243	−	0.692682i	$-0.243571\pi$
$911$	43.5382	1.44249	0.721243	+	0.692682i	$0.243571\pi$
$912$	19.1982	0.635716
$913$	−6.21386	−0.205649
$914$	16.7814	0.555081
$915$	0	0
$916$	−14.6073	−0.482640
$917$	−70.1744	−2.31736
$918$	3.39560	0.112071
$919$	−21.9336	−0.723523	−0.361762	−	0.932271i	$-0.617825\pi$
$919$	−21.9336	−0.723523	−0.361762	+	0.932271i	$0.617825\pi$
$920$	0	0
$921$	92.1500	3.03645
$922$	24.8690	0.819016
$923$	0	0
$924$	−41.7362	−1.37302
$925$	0	0
$926$	−33.4890	−1.10052
$927$	100.588	3.30375
$928$	−4.32537	−0.141987
$929$	−7.76056	−0.254616	−0.127308	−	0.991863i	$-0.540634\pi$
$929$	−7.76056	−0.254616	−0.127308	+	0.991863i	$0.540634\pi$
$930$	0	0
$931$	101.098	3.31334
$932$	−23.2146	−0.760421
$933$	4.11623	0.134759
$934$	−8.28758	−0.271178
$935$	0	0
$936$	0	0
$937$	37.2577	1.21716	0.608578	−	0.793494i	$-0.291740\pi$
$937$	37.2577	1.21716	0.608578	+	0.793494i	$0.291740\pi$
$938$	−22.4434	−0.732802
$939$	−38.5906	−1.25936
$940$	0	0
$941$	−46.2849	−1.50885	−0.754423	−	0.656389i	$-0.772083\pi$
$941$	−46.2849	−1.50885	−0.754423	+	0.656389i	$0.772083\pi$
$942$	−33.0512	−1.07687
$943$	−28.6228	−0.932085
$944$	2.01655	0.0656333
$945$	0	0
$946$	9.07382	0.295015
$947$	29.9853	0.974392	0.487196	−	0.873293i	$-0.338020\pi$
$947$	29.9853	0.974392	0.487196	+	0.873293i	$0.338020\pi$
$948$	18.2537	0.592852
$949$	0	0
$950$	0	0
$951$	−18.3079	−0.593674
$952$	−1.22736	−0.0397791
$953$	48.1719	1.56044	0.780220	−	0.625505i	$-0.215107\pi$
$953$	48.1719	1.56044	0.780220	+	0.625505i	$0.215107\pi$
$954$	−26.1317	−0.846045
$955$	0	0
$956$	−10.3141	−0.333580
$957$	−36.9741	−1.19520
$958$	21.8070	0.704552
$959$	24.9040	0.804194
$960$	0	0
$961$	−30.9029	−0.996869
$962$	0	0
$963$	21.2818	0.685797
$964$	5.45201	0.175597
$965$	0	0
$966$	−128.245	−4.12623
$967$	−40.6180	−1.30619	−0.653094	−	0.757277i	$-0.726529\pi$
$967$	−40.6180	−1.30619	−0.653094	+	0.757277i	$0.726529\pi$
$968$	−3.85318	−0.123846
$969$	4.82609	0.155036
$970$	0	0
$971$	−40.2598	−1.29200	−0.646000	−	0.763337i	$-0.723559\pi$
$971$	−40.2598	−1.29200	−0.646000	+	0.763337i	$0.723559\pi$
$972$	28.2835	0.907194
$973$	−104.103	−3.33739
$974$	18.0233	0.577504
$975$	0	0
$976$	−2.09255	−0.0669810
$977$	21.7806	0.696823	0.348411	−	0.937342i	$-0.386721\pi$
$977$	21.7806	0.696823	0.348411	+	0.937342i	$0.386721\pi$
$978$	−4.16591	−0.133211
$979$	27.6236	0.882853
$980$	0	0
$981$	13.1450	0.419689
$982$	−1.31023	−0.0418111
$983$	−37.9727	−1.21114	−0.605571	−	0.795792i	$-0.707055\pi$
$983$	−37.9727	−1.21114	−0.605571	+	0.795792i	$0.707055\pi$
$984$	−11.1415	−0.355178
$985$	0	0
$986$	−1.08732	−0.0346274
$987$	−9.71970	−0.309381
$988$	0	0
$989$	27.8817	0.886587
$990$	0	0
$991$	−4.92433	−0.156426	−0.0782132	−	0.996937i	$-0.524921\pi$
$991$	−4.92433	−0.156426	−0.0782132	+	0.996937i	$0.524921\pi$
$992$	0.311545	0.00989157
$993$	−55.1597	−1.75044
$994$	−74.3159	−2.35716
$995$	0	0
$996$	−7.43231	−0.235502
$997$	33.0666	1.04723	0.523615	−	0.851955i	$-0.324583\pi$
$997$	33.0666	1.04723	0.523615	+	0.851955i	$0.324583\pi$
$998$	−29.8490	−0.944855
$999$	−125.511	−3.97100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cy.1.9	yes	9
5.4	even	2		8450.2.a.cu.1.1	✓	9
13.12	even	2		8450.2.a.cv.1.9	yes	9
65.64	even	2		8450.2.a.cz.1.1	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8450.2.a.cu.1.1	✓	9	5.4	even	2
8450.2.a.cv.1.9	yes	9	13.12	even	2
8450.2.a.cy.1.9	yes	9	1.1	even	1	trivial
8450.2.a.cz.1.1	yes	9	65.64	even	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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.19756 q^{3} +1.00000 q^{4} +3.19756 q^{6} -4.88245 q^{7} +1.00000 q^{8} +7.22438 q^{9} +2.67335 q^{11} +3.19756 q^{12} -4.88245 q^{14} +1.00000 q^{16} +0.251383 q^{17} +7.22438 q^{18} +6.00402 q^{19} -15.6119 q^{21} +2.67335 q^{22} +8.21459 q^{23} +3.19756 q^{24} +13.5077 q^{27} -4.88245 q^{28} -4.32537 q^{29} +0.311545 q^{31} +1.00000 q^{32} +8.54820 q^{33} +0.251383 q^{34} +7.22438 q^{36} -9.29183 q^{37} +6.00402 q^{38} -3.48438 q^{41} -15.6119 q^{42} +3.39417 q^{43} +2.67335 q^{44} +8.21459 q^{46} +0.622582 q^{47} +3.19756 q^{48} +16.8383 q^{49} +0.803810 q^{51} -3.61716 q^{53} +13.5077 q^{54} -4.88245 q^{56} +19.1982 q^{57} -4.32537 q^{58} +2.01655 q^{59} -2.09255 q^{61} +0.311545 q^{62} -35.2727 q^{63} +1.00000 q^{64} +8.54820 q^{66} +4.59674 q^{67} +0.251383 q^{68} +26.2666 q^{69} +15.2210 q^{71} +7.22438 q^{72} -12.0033 q^{73} -9.29183 q^{74} +6.00402 q^{76} -13.0525 q^{77} +5.70863 q^{79} +21.5185 q^{81} -3.48438 q^{82} -2.32437 q^{83} -15.6119 q^{84} +3.39417 q^{86} -13.8306 q^{87} +2.67335 q^{88} +10.3329 q^{89} +8.21459 q^{92} +0.996184 q^{93} +0.622582 q^{94} +3.19756 q^{96} -2.13228 q^{97} +16.8383 q^{98} +19.3133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{7} + 9 q^{8} + 15 q^{9} - q^{11} - 4 q^{14} + 9 q^{16} + 2 q^{17} + 15 q^{18} + 10 q^{19} + 3 q^{21} - q^{22} - q^{23} - 9 q^{27} - 4 q^{28} - q^{29} - 4 q^{31} + 9 q^{32}+ \cdots - 62 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^7 + 9 * q^8 + 15 * q^9 - q^11 - 4 * q^14 + 9 * q^16 + 2 * q^17 + 15 * q^18 + 10 * q^19 + 3 * q^21 - q^22 - q^23 - 9 * q^27 - 4 * q^28 - q^29 - 4 * q^31 + 9 * q^32 + 30 * q^33 + 2 * q^34 + 15 * q^36 + 18 * q^37 + 10 * q^38 - 2 * q^41 + 3 * q^42 - 5 * q^43 - q^44 - q^46 - 10 * q^47 + 39 * q^49 + 38 * q^51 + 3 * q^53 - 9 * q^54 - 4 * q^56 + 14 * q^57 - q^58 - 20 * q^59 + 40 * q^61 - 4 * q^62 - 45 * q^63 + 9 * q^64 + 30 * q^66 + 15 * q^67 + 2 * q^68 + 68 * q^69 + 46 * q^71 + 15 * q^72 + 13 * q^73 + 18 * q^74 + 10 * q^76 + 13 * q^77 + 37 * q^79 + 53 * q^81 - 2 * q^82 + 2 * q^83 + 3 * q^84 - 5 * q^86 + q^87 - q^88 + 14 * q^89 - q^92 + 2 * q^93 - 10 * q^94 + 23 * q^97 + 39 * q^98 - 62 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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