LMFDB - Embedded newform 4950.2.a.ch.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(1,4950)] 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("4950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,0,0,1,-3,0,0,-3,0,-2,-1,0,3,-5,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.5259490005$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.8220.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 12 $ x^3 - x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.70722$ of defining polynomial
Character	$\chi$	$=$	4950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.70722 q^{7} -1.00000 q^{8} -1.00000 q^{11} -2.72536 q^{13} -4.70722 q^{14} +1.00000 q^{16} +2.70722 q^{17} -5.43259 q^{19} +1.00000 q^{22} -7.43259 q^{23} +2.72536 q^{26} +4.70722 q^{28} +5.70722 q^{29} +7.70722 q^{31} -1.00000 q^{32} -2.70722 q^{34} +4.70722 q^{37} +5.43259 q^{38} -12.1579 q^{41} +0.725363 q^{43} -1.00000 q^{44} +7.43259 q^{46} -8.70722 q^{47} +15.1579 q^{49} -2.72536 q^{52} +3.45073 q^{53} -4.70722 q^{56} -5.70722 q^{58} +8.70722 q^{59} +13.4144 q^{61} -7.70722 q^{62} +1.00000 q^{64} +2.00000 q^{67} +2.70722 q^{68} +12.8470 q^{71} +10.8652 q^{73} -4.70722 q^{74} -5.43259 q^{76} -4.70722 q^{77} +5.29278 q^{79} +12.1579 q^{82} +2.25650 q^{83} -0.725363 q^{86} +1.00000 q^{88} +12.6891 q^{89} -12.8289 q^{91} -7.43259 q^{92} +8.70722 q^{94} -0.414446 q^{97} -15.1579 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} - 3 q^{11} - 2 q^{13} - q^{14} + 3 q^{16} - 5 q^{17} + 3 q^{19} + 3 q^{22} - 3 q^{23} + 2 q^{26} + q^{28} + 4 q^{29} + 10 q^{31} - 3 q^{32} + 5 q^{34} + q^{37}+ \cdots - 20 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 - 3 * q^11 - 2 * q^13 - q^14 + 3 * q^16 - 5 * q^17 + 3 * q^19 + 3 * q^22 - 3 * q^23 + 2 * q^26 + q^28 + 4 * q^29 + 10 * q^31 - 3 * q^32 + 5 * q^34 + q^37 - 3 * q^38 - 11 * q^41 - 4 * q^43 - 3 * q^44 + 3 * q^46 - 13 * q^47 + 20 * q^49 - 2 * q^52 - 2 * q^53 - q^56 - 4 * q^58 + 13 * q^59 + 14 * q^61 - 10 * q^62 + 3 * q^64 + 6 * q^67 - 5 * q^68 - 7 * q^71 - 6 * q^73 - q^74 + 3 * q^76 - q^77 + 29 * q^79 + 11 * q^82 + 6 * q^83 + 4 * q^86 + 3 * q^88 + 18 * q^89 + 14 * q^91 - 3 * q^92 + 13 * q^94 + 25 * q^97 - 20 * 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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.70722	1.77916	0.889582	−	0.456776i	$-0.150996\pi$
$7$	4.70722	1.77916	0.889582	+	0.456776i	$0.150996\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.72536	−0.755880	−0.377940	−	0.925830i	$-0.623367\pi$
$13$	−2.72536	−0.755880	−0.377940	+	0.925830i	$0.623367\pi$
$14$	−4.70722	−1.25806
$15$	0	0
$16$	1.00000	0.250000
$17$	2.70722	0.656598	0.328299	−	0.944574i	$-0.393525\pi$
$17$	2.70722	0.656598	0.328299	+	0.944574i	$0.393525\pi$
$18$	0	0
$19$	−5.43259	−1.24632	−0.623160	−	0.782094i	$-0.714152\pi$
$19$	−5.43259	−1.24632	−0.623160	+	0.782094i	$0.714152\pi$
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	−7.43259	−1.54980	−0.774901	−	0.632083i	$-0.782200\pi$
$23$	−7.43259	−1.54980	−0.774901	+	0.632083i	$0.782200\pi$
$24$	0	0
$25$	0	0
$26$	2.72536	0.534488
$27$	0	0
$28$	4.70722	0.889582
$29$	5.70722	1.05980	0.529902	−	0.848059i	$-0.322229\pi$
$29$	5.70722	1.05980	0.529902	+	0.848059i	$0.322229\pi$
$30$	0	0
$31$	7.70722	1.38426	0.692129	−	0.721774i	$-0.256673\pi$
$31$	7.70722	1.38426	0.692129	+	0.721774i	$0.256673\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.70722	−0.464285
$35$	0	0
$36$	0	0
$37$	4.70722	0.773863	0.386931	−	0.922109i	$-0.373535\pi$
$37$	4.70722	0.773863	0.386931	+	0.922109i	$0.373535\pi$
$38$	5.43259	0.881282
$39$	0	0
$40$	0	0
$41$	−12.1579	−1.89875	−0.949376	−	0.314141i	$-0.898284\pi$
$41$	−12.1579	−1.89875	−0.949376	+	0.314141i	$0.898284\pi$
$42$	0	0
$43$	0.725363	0.110617	0.0553084	−	0.998469i	$-0.482386\pi$
$43$	0.725363	0.110617	0.0553084	+	0.998469i	$0.482386\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	7.43259	1.09588
$47$	−8.70722	−1.27008	−0.635040	−	0.772480i	$-0.719016\pi$
$47$	−8.70722	−1.27008	−0.635040	+	0.772480i	$0.719016\pi$
$48$	0	0
$49$	15.1579	2.16542
$50$	0	0
$51$	0	0
$52$	−2.72536	−0.377940
$53$	3.45073	0.473994	0.236997	−	0.971510i	$-0.423837\pi$
$53$	3.45073	0.473994	0.236997	+	0.971510i	$0.423837\pi$
$54$	0	0
$55$	0	0
$56$	−4.70722	−0.629029
$57$	0	0
$58$	−5.70722	−0.749395
$59$	8.70722	1.13358	0.566792	−	0.823861i	$-0.308184\pi$
$59$	8.70722	1.13358	0.566792	+	0.823861i	$0.308184\pi$
$60$	0	0
$61$	13.4144	1.71754	0.858772	−	0.512358i	$-0.171228\pi$
$61$	13.4144	1.71754	0.858772	+	0.512358i	$0.171228\pi$
$62$	−7.70722	−0.978818
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.70722	0.328299
$69$	0	0
$70$	0	0
$71$	12.8470	1.52466	0.762331	−	0.647187i	$-0.224055\pi$
$71$	12.8470	1.52466	0.762331	+	0.647187i	$0.224055\pi$
$72$	0	0
$73$	10.8652	1.27167	0.635836	−	0.771824i	$-0.280655\pi$
$73$	10.8652	1.27167	0.635836	+	0.771824i	$0.280655\pi$
$74$	−4.70722	−0.547204
$75$	0	0
$76$	−5.43259	−0.623160
$77$	−4.70722	−0.536438
$78$	0	0
$79$	5.29278	0.595484	0.297742	−	0.954646i	$-0.403766\pi$
$79$	5.29278	0.595484	0.297742	+	0.954646i	$0.403766\pi$
$80$	0	0
$81$	0	0
$82$	12.1579	1.34262
$83$	2.25650	0.247683	0.123841	−	0.992302i	$-0.460479\pi$
$83$	2.25650	0.247683	0.123841	+	0.992302i	$0.460479\pi$
$84$	0	0
$85$	0	0
$86$	−0.725363	−0.0782179
$87$	0	0
$88$	1.00000	0.106600
$89$	12.6891	1.34504	0.672520	−	0.740079i	$-0.265212\pi$
$89$	12.6891	1.34504	0.672520	+	0.740079i	$0.265212\pi$
$90$	0	0
$91$	−12.8289	−1.34483
$92$	−7.43259	−0.774901
$93$	0	0
$94$	8.70722	0.898081
$95$	0	0
$96$	0	0
$97$	−0.414446	−0.0420806	−0.0210403	−	0.999779i	$-0.506698\pi$
$97$	−0.414446	−0.0420806	−0.0210403	+	0.999779i	$0.506698\pi$
$98$	−15.1579	−1.53118
$99$	0	0
$100$	0	0
$101$	0.450726	0.0448489	0.0224245	−	0.999749i	$-0.492861\pi$
$101$	0.450726	0.0448489	0.0224245	+	0.999749i	$0.492861\pi$
$102$	0	0
$103$	−9.70722	−0.956481	−0.478241	−	0.878229i	$-0.658725\pi$
$103$	−9.70722	−0.956481	−0.478241	+	0.878229i	$0.658725\pi$
$104$	2.72536	0.267244
$105$	0	0
$106$	−3.45073	−0.335164
$107$	14.5724	1.40877	0.704383	−	0.709820i	$-0.251224\pi$
$107$	14.5724	1.40877	0.704383	+	0.709820i	$0.251224\pi$
$108$	0	0
$109$	14.6891	1.40696	0.703479	−	0.710716i	$-0.251629\pi$
$109$	14.6891	1.40696	0.703479	+	0.710716i	$0.251629\pi$
$110$	0	0
$111$	0	0
$112$	4.70722	0.444791
$113$	−5.41445	−0.509348	−0.254674	−	0.967027i	$-0.581968\pi$
$113$	−5.41445	−0.509348	−0.254674	+	0.967027i	$0.581968\pi$
$114$	0	0
$115$	0	0
$116$	5.70722	0.529902
$117$	0	0
$118$	−8.70722	−0.801565
$119$	12.7435	1.16820
$120$	0	0
$121$	1.00000	0.0909091
$122$	−13.4144	−1.21449
$123$	0	0
$124$	7.70722	0.692129
$125$	0	0
$126$	0	0
$127$	−12.1217	−1.07562	−0.537812	−	0.843065i	$-0.680749\pi$
$127$	−12.1217	−1.07562	−0.537812	+	0.843065i	$0.680749\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	8.57240	0.748974	0.374487	−	0.927232i	$-0.377819\pi$
$131$	8.57240	0.748974	0.374487	+	0.927232i	$0.377819\pi$
$132$	0	0
$133$	−25.5724	−2.21741
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−2.70722	−0.232142
$137$	0.689083	0.0588723	0.0294362	−	0.999567i	$-0.490629\pi$
$137$	0.689083	0.0588723	0.0294362	+	0.999567i	$0.490629\pi$
$138$	0	0
$139$	12.7254	1.07935	0.539676	−	0.841873i	$-0.318547\pi$
$139$	12.7254	1.07935	0.539676	+	0.841873i	$0.318547\pi$
$140$	0	0
$141$	0	0
$142$	−12.8470	−1.07810
$143$	2.72536	0.227906
$144$	0	0
$145$	0	0
$146$	−10.8652	−0.899208
$147$	0	0
$148$	4.70722	0.386931
$149$	−22.9868	−1.88316	−0.941578	−	0.336796i	$-0.890657\pi$
$149$	−22.9868	−1.88316	−0.941578	+	0.336796i	$0.890657\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	5.43259	0.440641
$153$	0	0
$154$	4.70722	0.379319
$155$	0	0
$156$	0	0
$157$	2.58555	0.206350	0.103175	−	0.994663i	$-0.467100\pi$
$157$	2.58555	0.206350	0.103175	+	0.994663i	$0.467100\pi$
$158$	−5.29278	−0.421071
$159$	0	0
$160$	0	0
$161$	−34.9868	−2.75735
$162$	0	0
$163$	−7.45073	−0.583586	−0.291793	−	0.956482i	$-0.594252\pi$
$163$	−7.45073	−0.583586	−0.291793	+	0.956482i	$0.594252\pi$
$164$	−12.1579	−0.949376
$165$	0	0
$166$	−2.25650	−0.175138
$167$	−14.8652	−1.15030	−0.575151	−	0.818047i	$-0.695057\pi$
$167$	−14.8652	−1.15030	−0.575151	+	0.818047i	$0.695057\pi$
$168$	0	0
$169$	−5.57240	−0.428646
$170$	0	0
$171$	0	0
$172$	0.725363	0.0553084
$173$	12.4507	0.946611	0.473306	−	0.880898i	$-0.343061\pi$
$173$	12.4507	0.946611	0.473306	+	0.880898i	$0.343061\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−12.6891	−0.951087
$179$	20.1217	1.50396	0.751982	−	0.659184i	$-0.229098\pi$
$179$	20.1217	1.50396	0.751982	+	0.659184i	$0.229098\pi$
$180$	0	0
$181$	−12.1217	−0.900997	−0.450498	−	0.892777i	$-0.648754\pi$
$181$	−12.1217	−0.900997	−0.450498	+	0.892777i	$0.648754\pi$
$182$	12.8289	0.950941
$183$	0	0
$184$	7.43259	0.547938
$185$	0	0
$186$	0	0
$187$	−2.70722	−0.197972
$188$	−8.70722	−0.635040
$189$	0	0
$190$	0	0
$191$	−19.4326	−1.40609	−0.703046	−	0.711144i	$-0.748177\pi$
$191$	−19.4326	−1.40609	−0.703046	+	0.711144i	$0.748177\pi$
$192$	0	0
$193$	−4.58555	−0.330075	−0.165038	−	0.986287i	$-0.552775\pi$
$193$	−4.58555	−0.330075	−0.165038	+	0.986287i	$0.552775\pi$
$194$	0.414446	0.0297555
$195$	0	0
$196$	15.1579	1.08271
$197$	16.3782	1.16690	0.583448	−	0.812150i	$-0.301703\pi$
$197$	16.3782	1.16690	0.583448	+	0.812150i	$0.301703\pi$
$198$	0	0
$199$	25.1217	1.78083	0.890414	−	0.455152i	$-0.150415\pi$
$199$	25.1217	1.78083	0.890414	+	0.455152i	$0.150415\pi$
$200$	0	0
$201$	0	0
$202$	−0.450726	−0.0317130
$203$	26.8652	1.88557
$204$	0	0
$205$	0	0
$206$	9.70722	0.676334
$207$	0	0
$208$	−2.72536	−0.188970
$209$	5.43259	0.375780
$210$	0	0
$211$	−13.4507	−0.925986	−0.462993	−	0.886362i	$-0.653225\pi$
$211$	−13.4507	−0.925986	−0.462993	+	0.886362i	$0.653225\pi$
$212$	3.45073	0.236997
$213$	0	0
$214$	−14.5724	−0.996148
$215$	0	0
$216$	0	0
$217$	36.2796	2.46282
$218$	−14.6891	−0.994870
$219$	0	0
$220$	0	0
$221$	−7.37817	−0.496309
$222$	0	0
$223$	−21.4144	−1.43402	−0.717009	−	0.697064i	$-0.754489\pi$
$223$	−21.4144	−1.43402	−0.717009	+	0.697064i	$0.754489\pi$
$224$	−4.70722	−0.314515
$225$	0	0
$226$	5.41445	0.360164
$227$	−25.0854	−1.66498	−0.832488	−	0.554043i	$-0.813084\pi$
$227$	−25.0854	−1.66498	−0.832488	+	0.554043i	$0.813084\pi$
$228$	0	0
$229$	12.6709	0.837319	0.418660	−	0.908143i	$-0.362500\pi$
$229$	12.6709	0.837319	0.418660	+	0.908143i	$0.362500\pi$
$230$	0	0
$231$	0	0
$232$	−5.70722	−0.374698
$233$	−0.743503	−0.0487085	−0.0243543	−	0.999703i	$-0.507753\pi$
$233$	−0.743503	−0.0487085	−0.0243543	+	0.999703i	$0.507753\pi$
$234$	0	0
$235$	0	0
$236$	8.70722	0.566792
$237$	0	0
$238$	−12.7435	−0.826039
$239$	−1.96372	−0.127022	−0.0635112	−	0.997981i	$-0.520230\pi$
$239$	−1.96372	−0.127022	−0.0635112	+	0.997981i	$0.520230\pi$
$240$	0	0
$241$	5.45073	0.351112	0.175556	−	0.984469i	$-0.443828\pi$
$241$	5.45073	0.351112	0.175556	+	0.984469i	$0.443828\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	13.4144	0.858772
$245$	0	0
$246$	0	0
$247$	14.8058	0.942069
$248$	−7.70722	−0.489409
$249$	0	0
$250$	0	0
$251$	1.96372	0.123949	0.0619745	−	0.998078i	$-0.480260\pi$
$251$	1.96372	0.123949	0.0619745	+	0.998078i	$0.480260\pi$
$252$	0	0
$253$	7.43259	0.467283
$254$	12.1217	0.760581
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.7254	1.04330	0.521650	−	0.853160i	$-0.325317\pi$
$257$	16.7254	1.04330	0.521650	+	0.853160i	$0.325317\pi$
$258$	0	0
$259$	22.1579	1.37683
$260$	0	0
$261$	0	0
$262$	−8.57240	−0.529604
$263$	19.3782	1.19491	0.597454	−	0.801903i	$-0.296179\pi$
$263$	19.3782	1.19491	0.597454	+	0.801903i	$0.296179\pi$
$264$	0	0
$265$	0	0
$266$	25.5724	1.56794
$267$	0	0
$268$	2.00000	0.122169
$269$	−13.9637	−0.851383	−0.425692	−	0.904868i	$-0.639969\pi$
$269$	−13.9637	−0.851383	−0.425692	+	0.904868i	$0.639969\pi$
$270$	0	0
$271$	23.0231	1.39856	0.699278	−	0.714850i	$-0.253505\pi$
$271$	23.0231	1.39856	0.699278	+	0.714850i	$0.253505\pi$
$272$	2.70722	0.164150
$273$	0	0
$274$	−0.689083	−0.0416290
$275$	0	0
$276$	0	0
$277$	−9.41445	−0.565659	−0.282830	−	0.959170i	$-0.591273\pi$
$277$	−9.41445	−0.565659	−0.282830	+	0.959170i	$0.591273\pi$
$278$	−12.7254	−0.763217
$279$	0	0
$280$	0	0
$281$	20.1217	1.20036	0.600179	−	0.799866i	$-0.295096\pi$
$281$	20.1217	1.20036	0.600179	+	0.799866i	$0.295096\pi$
$282$	0	0
$283$	−7.29278	−0.433511	−0.216755	−	0.976226i	$-0.569547\pi$
$283$	−7.29278	−0.433511	−0.216755	+	0.976226i	$0.569547\pi$
$284$	12.8470	0.762331
$285$	0	0
$286$	−2.72536	−0.161154
$287$	−57.2302	−3.37819
$288$	0	0
$289$	−9.67094	−0.568879
$290$	0	0
$291$	0	0
$292$	10.8652	0.635836
$293$	−10.9868	−0.641858	−0.320929	−	0.947103i	$-0.603995\pi$
$293$	−10.9868	−0.641858	−0.320929	+	0.947103i	$0.603995\pi$
$294$	0	0
$295$	0	0
$296$	−4.70722	−0.273602
$297$	0	0
$298$	22.9868	1.33159
$299$	20.2565	1.17146
$300$	0	0
$301$	3.41445	0.196805
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−5.43259	−0.311580
$305$	0	0
$306$	0	0
$307$	16.2796	0.929127	0.464563	−	0.885540i	$-0.346211\pi$
$307$	16.2796	0.929127	0.464563	+	0.885540i	$0.346211\pi$
$308$	−4.70722	−0.268219
$309$	0	0
$310$	0	0
$311$	−6.68908	−0.379303	−0.189652	−	0.981851i	$-0.560736\pi$
$311$	−6.68908	−0.379303	−0.189652	+	0.981851i	$0.560736\pi$
$312$	0	0
$313$	−2.67094	−0.150971	−0.0754853	−	0.997147i	$-0.524051\pi$
$313$	−2.67094	−0.150971	−0.0754853	+	0.997147i	$0.524051\pi$
$314$	−2.58555	−0.145911
$315$	0	0
$316$	5.29278	0.297742
$317$	27.4507	1.54179	0.770893	−	0.636964i	$-0.219810\pi$
$317$	27.4507	1.54179	0.770893	+	0.636964i	$0.219810\pi$
$318$	0	0
$319$	−5.70722	−0.319543
$320$	0	0
$321$	0	0
$322$	34.9868	1.94974
$323$	−14.7072	−0.818332
$324$	0	0
$325$	0	0
$326$	7.45073	0.412658
$327$	0	0
$328$	12.1579	0.671310
$329$	−40.9868	−2.25968
$330$	0	0
$331$	0.512994	0.0281967	0.0140983	−	0.999901i	$-0.495512\pi$
$331$	0.512994	0.0281967	0.0140983	+	0.999901i	$0.495512\pi$
$332$	2.25650	0.123841
$333$	0	0
$334$	14.8652	0.813386
$335$	0	0
$336$	0	0
$337$	4.54927	0.247815	0.123907	−	0.992294i	$-0.460457\pi$
$337$	4.54927	0.247815	0.123907	+	0.992294i	$0.460457\pi$
$338$	5.57240	0.303098
$339$	0	0
$340$	0	0
$341$	−7.70722	−0.417370
$342$	0	0
$343$	38.4013	2.07347
$344$	−0.725363	−0.0391090
$345$	0	0
$346$	−12.4507	−0.669355
$347$	28.2433	1.51618	0.758091	−	0.652149i	$-0.226132\pi$
$347$	28.2433	1.51618	0.758091	+	0.652149i	$0.226132\pi$
$348$	0	0
$349$	−1.23836	−0.0662877	−0.0331439	−	0.999451i	$-0.510552\pi$
$349$	−1.23836	−0.0662877	−0.0331439	+	0.999451i	$0.510552\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	1.86019	0.0990080	0.0495040	−	0.998774i	$-0.484236\pi$
$353$	1.86019	0.0990080	0.0495040	+	0.998774i	$0.484236\pi$
$354$	0	0
$355$	0	0
$356$	12.6891	0.672520
$357$	0	0
$358$	−20.1217	−1.06346
$359$	−21.4507	−1.13213	−0.566063	−	0.824362i	$-0.691534\pi$
$359$	−21.4507	−1.13213	−0.566063	+	0.824362i	$0.691534\pi$
$360$	0	0
$361$	10.5130	0.553315
$362$	12.1217	0.637101
$363$	0	0
$364$	−12.8289	−0.672417
$365$	0	0
$366$	0	0
$367$	26.0231	1.35840	0.679198	−	0.733955i	$-0.262328\pi$
$367$	26.0231	1.35840	0.679198	+	0.733955i	$0.262328\pi$
$368$	−7.43259	−0.387450
$369$	0	0
$370$	0	0
$371$	16.2433	0.843312
$372$	0	0
$373$	−33.4144	−1.73013	−0.865067	−	0.501656i	$-0.832724\pi$
$373$	−33.4144	−1.73013	−0.865067	+	0.501656i	$0.832724\pi$
$374$	2.70722	0.139987
$375$	0	0
$376$	8.70722	0.449041
$377$	−15.5543	−0.801085
$378$	0	0
$379$	−10.9015	−0.559970	−0.279985	−	0.960004i	$-0.590329\pi$
$379$	−10.9015	−0.559970	−0.279985	+	0.960004i	$0.590329\pi$
$380$	0	0
$381$	0	0
$382$	19.4326	0.994258
$383$	26.9687	1.37804	0.689018	−	0.724744i	$-0.258042\pi$
$383$	26.9687	1.37804	0.689018	+	0.724744i	$0.258042\pi$
$384$	0	0
$385$	0	0
$386$	4.58555	0.233399
$387$	0	0
$388$	−0.414446	−0.0210403
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−20.1217	−1.01760
$392$	−15.1579	−0.765592
$393$	0	0
$394$	−16.3782	−0.825120
$395$	0	0
$396$	0	0
$397$	1.41445	0.0709890	0.0354945	−	0.999370i	$-0.488699\pi$
$397$	1.41445	0.0709890	0.0354945	+	0.999370i	$0.488699\pi$
$398$	−25.1217	−1.25924
$399$	0	0
$400$	0	0
$401$	−7.59054	−0.379053	−0.189527	−	0.981876i	$-0.560695\pi$
$401$	−7.59054	−0.379053	−0.189527	+	0.981876i	$0.560695\pi$
$402$	0	0
$403$	−21.0050	−1.04633
$404$	0.450726	0.0224245
$405$	0	0
$406$	−26.8652	−1.33330
$407$	−4.70722	−0.233328
$408$	0	0
$409$	−17.6941	−0.874915	−0.437458	−	0.899239i	$-0.644121\pi$
$409$	−17.6941	−0.874915	−0.437458	+	0.899239i	$0.644121\pi$
$410$	0	0
$411$	0	0
$412$	−9.70722	−0.478241
$413$	40.9868	2.01683
$414$	0	0
$415$	0	0
$416$	2.72536	0.133622
$417$	0	0
$418$	−5.43259	−0.265716
$419$	26.1217	1.27613	0.638064	−	0.769984i	$-0.279736\pi$
$419$	26.1217	1.27613	0.638064	+	0.769984i	$0.279736\pi$
$420$	0	0
$421$	28.1217	1.37057	0.685283	−	0.728277i	$-0.259678\pi$
$421$	28.1217	1.37057	0.685283	+	0.728277i	$0.259678\pi$
$422$	13.4507	0.654771
$423$	0	0
$424$	−3.45073	−0.167582
$425$	0	0
$426$	0	0
$427$	63.1448	3.05579
$428$	14.5724	0.704383
$429$	0	0
$430$	0	0
$431$	−6.31590	−0.304226	−0.152113	−	0.988363i	$-0.548608\pi$
$431$	−6.31590	−0.304226	−0.152113	+	0.988363i	$0.548608\pi$
$432$	0	0
$433$	4.25650	0.204554	0.102277	−	0.994756i	$-0.467387\pi$
$433$	4.25650	0.204554	0.102277	+	0.994756i	$0.467387\pi$
$434$	−36.2796	−1.74148
$435$	0	0
$436$	14.6891	0.703479
$437$	40.3782	1.93155
$438$	0	0
$439$	14.7435	0.703669	0.351835	−	0.936062i	$-0.385558\pi$
$439$	14.7435	0.703669	0.351835	+	0.936062i	$0.385558\pi$
$440$	0	0
$441$	0	0
$442$	7.37817	0.350944
$443$	30.1579	1.43285	0.716424	−	0.697665i	$-0.245778\pi$
$443$	30.1579	1.43285	0.716424	+	0.697665i	$0.245778\pi$
$444$	0	0
$445$	0	0
$446$	21.4144	1.01400
$447$	0	0
$448$	4.70722	0.222395
$449$	−3.55426	−0.167736	−0.0838678	−	0.996477i	$-0.526727\pi$
$449$	−3.55426	−0.167736	−0.0838678	+	0.996477i	$0.526727\pi$
$450$	0	0
$451$	12.1579	0.572495
$452$	−5.41445	−0.254674
$453$	0	0
$454$	25.0854	1.17732
$455$	0	0
$456$	0	0
$457$	−22.9015	−1.07128	−0.535642	−	0.844445i	$-0.679930\pi$
$457$	−22.9015	−1.07128	−0.535642	+	0.844445i	$0.679930\pi$
$458$	−12.6709	−0.592074
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	14.0231	0.651710	0.325855	−	0.945420i	$-0.394348\pi$
$463$	14.0231	0.651710	0.325855	+	0.945420i	$0.394348\pi$
$464$	5.70722	0.264951
$465$	0	0
$466$	0.743503	0.0344421
$467$	28.5592	1.32156	0.660782	−	0.750578i	$-0.270225\pi$
$467$	28.5592	1.32156	0.660782	+	0.750578i	$0.270225\pi$
$468$	0	0
$469$	9.41445	0.434719
$470$	0	0
$471$	0	0
$472$	−8.70722	−0.400782
$473$	−0.725363	−0.0333522
$474$	0	0
$475$	0	0
$476$	12.7435	0.584098
$477$	0	0
$478$	1.96372	0.0898185
$479$	13.4870	0.616237	0.308119	−	0.951348i	$-0.400301\pi$
$479$	13.4870	0.616237	0.308119	+	0.951348i	$0.400301\pi$
$480$	0	0
$481$	−12.8289	−0.584947
$482$	−5.45073	−0.248274
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	−40.0231	−1.81362	−0.906810	−	0.421539i	$-0.861490\pi$
$487$	−40.0231	−1.81362	−0.906810	+	0.421539i	$0.861490\pi$
$488$	−13.4144	−0.607243
$489$	0	0
$490$	0	0
$491$	−14.2565	−0.643387	−0.321693	−	0.946844i	$-0.604252\pi$
$491$	−14.2565	−0.643387	−0.321693	+	0.946844i	$0.604252\pi$
$492$	0	0
$493$	15.4507	0.695866
$494$	−14.8058	−0.666143
$495$	0	0
$496$	7.70722	0.346065
$497$	60.4738	2.71262
$498$	0	0
$499$	16.8652	0.754989	0.377494	−	0.926012i	$-0.376786\pi$
$499$	16.8652	0.754989	0.377494	+	0.926012i	$0.376786\pi$
$500$	0	0
$501$	0	0
$502$	−1.96372	−0.0876451
$503$	−20.8652	−0.930332	−0.465166	−	0.885223i	$-0.654005\pi$
$503$	−20.8652	−0.930332	−0.465166	+	0.885223i	$0.654005\pi$
$504$	0	0
$505$	0	0
$506$	−7.43259	−0.330419
$507$	0	0
$508$	−12.1217	−0.537812
$509$	25.6941	1.13887	0.569435	−	0.822037i	$-0.307162\pi$
$509$	25.6941	1.13887	0.569435	+	0.822037i	$0.307162\pi$
$510$	0	0
$511$	51.1448	2.26251
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.7254	−0.737724
$515$	0	0
$516$	0	0
$517$	8.70722	0.382943
$518$	−22.1579	−0.973564
$519$	0	0
$520$	0	0
$521$	19.5905	0.858277	0.429138	−	0.903239i	$-0.358817\pi$
$521$	19.5905	0.858277	0.429138	+	0.903239i	$0.358817\pi$
$522$	0	0
$523$	−5.43259	−0.237550	−0.118775	−	0.992921i	$-0.537897\pi$
$523$	−5.43259	−0.237550	−0.118775	+	0.992921i	$0.537897\pi$
$524$	8.57240	0.374487
$525$	0	0
$526$	−19.3782	−0.844928
$527$	20.8652	0.908901
$528$	0	0
$529$	32.2433	1.40188
$530$	0	0
$531$	0	0
$532$	−25.5724	−1.10870
$533$	33.1348	1.43523
$534$	0	0
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	0	0
$538$	13.9637	0.602019
$539$	−15.1579	−0.652899
$540$	0	0
$541$	−33.5180	−1.44105	−0.720525	−	0.693429i	$-0.756099\pi$
$541$	−33.5180	−1.44105	−0.720525	+	0.693429i	$0.756099\pi$
$542$	−23.0231	−0.988928
$543$	0	0
$544$	−2.70722	−0.116071
$545$	0	0
$546$	0	0
$547$	−36.5411	−1.56238	−0.781192	−	0.624291i	$-0.785388\pi$
$547$	−36.5411	−1.56238	−0.781192	+	0.624291i	$0.785388\pi$
$548$	0.689083	0.0294362
$549$	0	0
$550$	0	0
$551$	−31.0050	−1.32086
$552$	0	0
$553$	24.9143	1.05946
$554$	9.41445	0.399981
$555$	0	0
$556$	12.7254	0.539676
$557$	−46.8520	−1.98518	−0.992592	−	0.121497i	$-0.961230\pi$
$557$	−46.8520	−1.98518	−0.992592	+	0.121497i	$0.961230\pi$
$558$	0	0
$559$	−1.97688	−0.0836130
$560$	0	0
$561$	0	0
$562$	−20.1217	−0.848781
$563$	−28.2433	−1.19031	−0.595157	−	0.803609i	$-0.702910\pi$
$563$	−28.2433	−1.19031	−0.595157	+	0.803609i	$0.702910\pi$
$564$	0	0
$565$	0	0
$566$	7.29278	0.306538
$567$	0	0
$568$	−12.8470	−0.539050
$569$	−15.6087	−0.654350	−0.327175	−	0.944964i	$-0.606097\pi$
$569$	−15.6087	−0.654350	−0.327175	+	0.944964i	$0.606097\pi$
$570$	0	0
$571$	9.00498	0.376847	0.188423	−	0.982088i	$-0.439662\pi$
$571$	9.00498	0.376847	0.188423	+	0.982088i	$0.439662\pi$
$572$	2.72536	0.113953
$573$	0	0
$574$	57.2302	2.38874
$575$	0	0
$576$	0	0
$577$	2.74350	0.114214	0.0571068	−	0.998368i	$-0.481812\pi$
$577$	2.74350	0.114214	0.0571068	+	0.998368i	$0.481812\pi$
$578$	9.67094	0.402258
$579$	0	0
$580$	0	0
$581$	10.6218	0.440668
$582$	0	0
$583$	−3.45073	−0.142914
$584$	−10.8652	−0.449604
$585$	0	0
$586$	10.9868	0.453862
$587$	2.39132	0.0987005	0.0493503	−	0.998782i	$-0.484285\pi$
$587$	2.39132	0.0987005	0.0493503	+	0.998782i	$0.484285\pi$
$588$	0	0
$589$	−41.8702	−1.72523
$590$	0	0
$591$	0	0
$592$	4.70722	0.193466
$593$	−5.41445	−0.222345	−0.111172	−	0.993801i	$-0.535461\pi$
$593$	−5.41445	−0.222345	−0.111172	+	0.993801i	$0.535461\pi$
$594$	0	0
$595$	0	0
$596$	−22.9868	−0.941578
$597$	0	0
$598$	−20.2565	−0.828350
$599$	−27.6087	−1.12806	−0.564030	−	0.825754i	$-0.690750\pi$
$599$	−27.6087	−1.12806	−0.564030	+	0.825754i	$0.690750\pi$
$600$	0	0
$601$	2.58555	0.105467	0.0527335	−	0.998609i	$-0.483207\pi$
$601$	2.58555	0.105467	0.0527335	+	0.998609i	$0.483207\pi$
$602$	−3.41445	−0.139162
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	−19.3419	−0.785063	−0.392531	−	0.919739i	$-0.628401\pi$
$607$	−19.3419	−0.785063	−0.392531	+	0.919739i	$0.628401\pi$
$608$	5.43259	0.220320
$609$	0	0
$610$	0	0
$611$	23.7303	0.960027
$612$	0	0
$613$	9.17111	0.370418	0.185209	−	0.982699i	$-0.440704\pi$
$613$	9.17111	0.370418	0.185209	+	0.982699i	$0.440704\pi$
$614$	−16.2796	−0.656992
$615$	0	0
$616$	4.70722	0.189659
$617$	35.6268	1.43428	0.717141	−	0.696928i	$-0.245450\pi$
$617$	35.6268	1.43428	0.717141	+	0.696928i	$0.245450\pi$
$618$	0	0
$619$	−48.5955	−1.95322	−0.976609	−	0.215021i	$-0.931018\pi$
$619$	−48.5955	−1.95322	−0.976609	+	0.215021i	$0.931018\pi$
$620$	0	0
$621$	0	0
$622$	6.68908	0.268208
$623$	59.7303	2.39305
$624$	0	0
$625$	0	0
$626$	2.67094	0.106752
$627$	0	0
$628$	2.58555	0.103175
$629$	12.7435	0.508117
$630$	0	0
$631$	−39.1448	−1.55833	−0.779165	−	0.626819i	$-0.784356\pi$
$631$	−39.1448	−1.55833	−0.779165	+	0.626819i	$0.784356\pi$
$632$	−5.29278	−0.210535
$633$	0	0
$634$	−27.4507	−1.09021
$635$	0	0
$636$	0	0
$637$	−41.3109	−1.63680
$638$	5.70722	0.225951
$639$	0	0
$640$	0	0
$641$	27.5543	1.08833	0.544164	−	0.838979i	$-0.316847\pi$
$641$	27.5543	1.08833	0.544164	+	0.838979i	$0.316847\pi$
$642$	0	0
$643$	−17.6941	−0.697786	−0.348893	−	0.937163i	$-0.613442\pi$
$643$	−17.6941	−0.697786	−0.348893	+	0.937163i	$0.613442\pi$
$644$	−34.9868	−1.37867
$645$	0	0
$646$	14.7072	0.578648
$647$	−3.60868	−0.141872	−0.0709358	−	0.997481i	$-0.522599\pi$
$647$	−3.60868	−0.141872	−0.0709358	+	0.997481i	$0.522599\pi$
$648$	0	0
$649$	−8.70722	−0.341788
$650$	0	0
$651$	0	0
$652$	−7.45073	−0.291793
$653$	−20.2796	−0.793603	−0.396801	−	0.917904i	$-0.629880\pi$
$653$	−20.2796	−0.793603	−0.396801	+	0.917904i	$0.629880\pi$
$654$	0	0
$655$	0	0
$656$	−12.1579	−0.474688
$657$	0	0
$658$	40.9868	1.59783
$659$	13.9868	0.544850	0.272425	−	0.962177i	$-0.412174\pi$
$659$	13.9868	0.544850	0.272425	+	0.962177i	$0.412174\pi$
$660$	0	0
$661$	−15.2565	−0.593409	−0.296704	−	0.954969i	$-0.595888\pi$
$661$	−15.2565	−0.593409	−0.296704	+	0.954969i	$0.595888\pi$
$662$	−0.512994	−0.0199381
$663$	0	0
$664$	−2.25650	−0.0875691
$665$	0	0
$666$	0	0
$667$	−42.4194	−1.64249
$668$	−14.8652	−0.575151
$669$	0	0
$670$	0	0
$671$	−13.4144	−0.517859
$672$	0	0
$673$	−21.7303	−0.837643	−0.418822	−	0.908069i	$-0.637557\pi$
$673$	−21.7303	−0.837643	−0.418822	+	0.908069i	$0.637557\pi$
$674$	−4.54927	−0.175231
$675$	0	0
$676$	−5.57240	−0.214323
$677$	10.8058	0.415299	0.207650	−	0.978203i	$-0.433419\pi$
$677$	10.8058	0.415299	0.207650	+	0.978203i	$0.433419\pi$
$678$	0	0
$679$	−1.95089	−0.0748683
$680$	0	0
$681$	0	0
$682$	7.70722	0.295125
$683$	7.05940	0.270121	0.135060	−	0.990837i	$-0.456877\pi$
$683$	7.05940	0.270121	0.135060	+	0.990837i	$0.456877\pi$
$684$	0	0
$685$	0	0
$686$	−38.4013	−1.46617
$687$	0	0
$688$	0.725363	0.0276542
$689$	−9.40448	−0.358282
$690$	0	0
$691$	0.305935	0.0116383	0.00581916	−	0.999983i	$-0.498148\pi$
$691$	0.305935	0.0116383	0.00581916	+	0.999983i	$0.498148\pi$
$692$	12.4507	0.473306
$693$	0	0
$694$	−28.2433	−1.07210
$695$	0	0
$696$	0	0
$697$	−32.9143	−1.24672
$698$	1.23836	0.0468725
$699$	0	0
$700$	0	0
$701$	24.7666	0.935423	0.467711	−	0.883881i	$-0.345079\pi$
$701$	24.7666	0.935423	0.467711	+	0.883881i	$0.345079\pi$
$702$	0	0
$703$	−25.5724	−0.964481
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−1.86019	−0.0700092
$707$	2.12167	0.0797936
$708$	0	0
$709$	48.7172	1.82961	0.914806	−	0.403893i	$-0.132343\pi$
$709$	48.7172	1.82961	0.914806	+	0.403893i	$0.132343\pi$
$710$	0	0
$711$	0	0
$712$	−12.6891	−0.475543
$713$	−57.2846	−2.14533
$714$	0	0
$715$	0	0
$716$	20.1217	0.751982
$717$	0	0
$718$	21.4507	0.800534
$719$	14.5493	0.542596	0.271298	−	0.962495i	$-0.412547\pi$
$719$	14.5493	0.542596	0.271298	+	0.962495i	$0.412547\pi$
$720$	0	0
$721$	−45.6941	−1.70174
$722$	−10.5130	−0.391253
$723$	0	0
$724$	−12.1217	−0.450498
$725$	0	0
$726$	0	0
$727$	−16.8783	−0.625983	−0.312991	−	0.949756i	$-0.601331\pi$
$727$	−16.8783	−0.625983	−0.312991	+	0.949756i	$0.601331\pi$
$728$	12.8289	0.475470
$729$	0	0
$730$	0	0
$731$	1.96372	0.0726308
$732$	0	0
$733$	25.6268	0.946548	0.473274	−	0.880915i	$-0.343072\pi$
$733$	25.6268	0.946548	0.473274	+	0.880915i	$0.343072\pi$
$734$	−26.0231	−0.960531
$735$	0	0
$736$	7.43259	0.273969
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	−11.2202	−0.412742	−0.206371	−	0.978474i	$-0.566165\pi$
$739$	−11.2202	−0.412742	−0.206371	+	0.978474i	$0.566165\pi$
$740$	0	0
$741$	0	0
$742$	−16.2433	−0.596312
$743$	13.4870	0.494790	0.247395	−	0.968915i	$-0.420425\pi$
$743$	13.4870	0.494790	0.247395	+	0.968915i	$0.420425\pi$
$744$	0	0
$745$	0	0
$746$	33.4144	1.22339
$747$	0	0
$748$	−2.70722	−0.0989859
$749$	68.5955	2.50643
$750$	0	0
$751$	−22.6087	−0.825002	−0.412501	−	0.910957i	$-0.635345\pi$
$751$	−22.6087	−0.825002	−0.412501	+	0.910957i	$0.635345\pi$
$752$	−8.70722	−0.317520
$753$	0	0
$754$	15.5543	0.566453
$755$	0	0
$756$	0	0
$757$	27.3782	0.995076	0.497538	−	0.867442i	$-0.334237\pi$
$757$	27.3782	0.995076	0.497538	+	0.867442i	$0.334237\pi$
$758$	10.9015	0.395959
$759$	0	0
$760$	0	0
$761$	10.5130	0.381096	0.190548	−	0.981678i	$-0.438974\pi$
$761$	10.5130	0.381096	0.190548	+	0.981678i	$0.438974\pi$
$762$	0	0
$763$	69.1448	2.50321
$764$	−19.4326	−0.703046
$765$	0	0
$766$	−26.9687	−0.974419
$767$	−23.7303	−0.856853
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	0	0
$772$	−4.58555	−0.165038
$773$	11.4144	0.410549	0.205275	−	0.978704i	$-0.434191\pi$
$773$	11.4144	0.410549	0.205275	+	0.978704i	$0.434191\pi$
$774$	0	0
$775$	0	0
$776$	0.414446	0.0148778
$777$	0	0
$778$	−6.00000	−0.215110
$779$	66.0491	2.36645
$780$	0	0
$781$	−12.8470	−0.459703
$782$	20.1217	0.719549
$783$	0	0
$784$	15.1579	0.541355
$785$	0	0
$786$	0	0
$787$	5.02312	0.179055	0.0895275	−	0.995984i	$-0.471464\pi$
$787$	5.02312	0.179055	0.0895275	+	0.995984i	$0.471464\pi$
$788$	16.3782	0.583448
$789$	0	0
$790$	0	0
$791$	−25.4870	−0.906214
$792$	0	0
$793$	−36.5592	−1.29826
$794$	−1.41445	−0.0501968
$795$	0	0
$796$	25.1217	0.890414
$797$	−26.8652	−0.951613	−0.475807	−	0.879550i	$-0.657844\pi$
$797$	−26.8652	−0.951613	−0.475807	+	0.879550i	$0.657844\pi$
$798$	0	0
$799$	−23.5724	−0.833931
$800$	0	0
$801$	0	0
$802$	7.59054	0.268031
$803$	−10.8652	−0.383424
$804$	0	0
$805$	0	0
$806$	21.0050	0.739869
$807$	0	0
$808$	−0.450726	−0.0158565
$809$	−9.87833	−0.347304	−0.173652	−	0.984807i	$-0.555557\pi$
$809$	−9.87833	−0.347304	−0.173652	+	0.984807i	$0.555557\pi$
$810$	0	0
$811$	55.7795	1.95868	0.979341	−	0.202217i	$-0.0648146\pi$
$811$	55.7795	1.95868	0.979341	+	0.202217i	$0.0648146\pi$
$812$	26.8652	0.942783
$813$	0	0
$814$	4.70722	0.164988
$815$	0	0
$816$	0	0
$817$	−3.94060	−0.137864
$818$	17.6941	0.618658
$819$	0	0
$820$	0	0
$821$	22.8520	0.797541	0.398770	−	0.917051i	$-0.369437\pi$
$821$	22.8520	0.797541	0.398770	+	0.917051i	$0.369437\pi$
$822$	0	0
$823$	−32.2433	−1.12393	−0.561966	−	0.827160i	$-0.689955\pi$
$823$	−32.2433	−1.12393	−0.561966	+	0.827160i	$0.689955\pi$
$824$	9.70722	0.338167
$825$	0	0
$826$	−40.9868	−1.42611
$827$	1.08539	0.0377427	0.0188713	−	0.999822i	$-0.493993\pi$
$827$	1.08539	0.0377427	0.0188713	+	0.999822i	$0.493993\pi$
$828$	0	0
$829$	44.5230	1.54635	0.773173	−	0.634195i	$-0.218668\pi$
$829$	44.5230	1.54635	0.773173	+	0.634195i	$0.218668\pi$
$830$	0	0
$831$	0	0
$832$	−2.72536	−0.0944850
$833$	41.0360	1.42181
$834$	0	0
$835$	0	0
$836$	5.43259	0.187890
$837$	0	0
$838$	−26.1217	−0.902358
$839$	−31.1085	−1.07399	−0.536993	−	0.843587i	$-0.680440\pi$
$839$	−31.1085	−1.07399	−0.536993	+	0.843587i	$0.680440\pi$
$840$	0	0
$841$	3.57240	0.123186
$842$	−28.1217	−0.969137
$843$	0	0
$844$	−13.4507	−0.462993
$845$	0	0
$846$	0	0
$847$	4.70722	0.161742
$848$	3.45073	0.118498
$849$	0	0
$850$	0	0
$851$	−34.9868	−1.19933
$852$	0	0
$853$	26.9015	0.921088	0.460544	−	0.887637i	$-0.347654\pi$
$853$	26.9015	0.921088	0.460544	+	0.887637i	$0.347654\pi$
$854$	−63.1448	−2.16077
$855$	0	0
$856$	−14.5724	−0.498074
$857$	1.32906	0.0453997	0.0226999	−	0.999742i	$-0.492774\pi$
$857$	1.32906	0.0453997	0.0226999	+	0.999742i	$0.492774\pi$
$858$	0	0
$859$	12.8289	0.437716	0.218858	−	0.975757i	$-0.429767\pi$
$859$	12.8289	0.437716	0.218858	+	0.975757i	$0.429767\pi$
$860$	0	0
$861$	0	0
$862$	6.31590	0.215120
$863$	−11.0413	−0.375849	−0.187924	−	0.982183i	$-0.560176\pi$
$863$	−11.0413	−0.375849	−0.187924	+	0.982183i	$0.560176\pi$
$864$	0	0
$865$	0	0
$866$	−4.25650	−0.144642
$867$	0	0
$868$	36.2796	1.23141
$869$	−5.29278	−0.179545
$870$	0	0
$871$	−5.45073	−0.184691
$872$	−14.6891	−0.497435
$873$	0	0
$874$	−40.3782	−1.36581
$875$	0	0
$876$	0	0
$877$	38.6891	1.30644	0.653219	−	0.757169i	$-0.273418\pi$
$877$	38.6891	1.30644	0.653219	+	0.757169i	$0.273418\pi$
$878$	−14.7435	−0.497569
$879$	0	0
$880$	0	0
$881$	43.2746	1.45796	0.728980	−	0.684535i	$-0.239995\pi$
$881$	43.2746	1.45796	0.728980	+	0.684535i	$0.239995\pi$
$882$	0	0
$883$	29.1811	0.982021	0.491011	−	0.871154i	$-0.336628\pi$
$883$	29.1811	0.982021	0.491011	+	0.871154i	$0.336628\pi$
$884$	−7.37817	−0.248155
$885$	0	0
$886$	−30.1579	−1.01318
$887$	−19.1711	−0.643703	−0.321851	−	0.946790i	$-0.604305\pi$
$887$	−19.1711	−0.643703	−0.321851	+	0.946790i	$0.604305\pi$
$888$	0	0
$889$	−57.0594	−1.91371
$890$	0	0
$891$	0	0
$892$	−21.4144	−0.717009
$893$	47.3027	1.58293
$894$	0	0
$895$	0	0
$896$	−4.70722	−0.157257
$897$	0	0
$898$	3.55426	0.118607
$899$	43.9868	1.46704
$900$	0	0
$901$	9.34189	0.311223
$902$	−12.1579	−0.404815
$903$	0	0
$904$	5.41445	0.180082
$905$	0	0
$906$	0	0
$907$	24.8289	0.824430	0.412215	−	0.911087i	$-0.364755\pi$
$907$	24.8289	0.824430	0.412215	+	0.911087i	$0.364755\pi$
$908$	−25.0854	−0.832488
$909$	0	0
$910$	0	0
$911$	3.60868	0.119561	0.0597804	−	0.998212i	$-0.480960\pi$
$911$	3.60868	0.119561	0.0597804	+	0.998212i	$0.480960\pi$
$912$	0	0
$913$	−2.25650	−0.0746791
$914$	22.9015	0.757513
$915$	0	0
$916$	12.6709	0.418660
$917$	40.3522	1.33255
$918$	0	0
$919$	−24.5984	−0.811426	−0.405713	−	0.914001i	$-0.632977\pi$
$919$	−24.5984	−0.811426	−0.405713	+	0.914001i	$0.632977\pi$
$920$	0	0
$921$	0	0
$922$	18.0000	0.592798
$923$	−35.0128	−1.15246
$924$	0	0
$925$	0	0
$926$	−14.0231	−0.460828
$927$	0	0
$928$	−5.70722	−0.187349
$929$	−28.7254	−0.942449	−0.471224	−	0.882013i	$-0.656188\pi$
$929$	−28.7254	−0.942449	−0.471224	+	0.882013i	$0.656188\pi$
$930$	0	0
$931$	−82.3469	−2.69881
$932$	−0.743503	−0.0243543
$933$	0	0
$934$	−28.5592	−0.934487
$935$	0	0
$936$	0	0
$937$	−54.8652	−1.79237	−0.896184	−	0.443684i	$-0.853671\pi$
$937$	−54.8652	−1.79237	−0.896184	+	0.443684i	$0.853671\pi$
$938$	−9.41445	−0.307393
$939$	0	0
$940$	0	0
$941$	−1.64496	−0.0536240	−0.0268120	−	0.999640i	$-0.508536\pi$
$941$	−1.64496	−0.0536240	−0.0268120	+	0.999640i	$0.508536\pi$
$942$	0	0
$943$	90.3650	2.94269
$944$	8.70722	0.283396
$945$	0	0
$946$	0.725363	0.0235836
$947$	12.6347	0.410571	0.205286	−	0.978702i	$-0.434188\pi$
$947$	12.6347	0.410571	0.205286	+	0.978702i	$0.434188\pi$
$948$	0	0
$949$	−29.6115	−0.961231
$950$	0	0
$951$	0	0
$952$	−12.7435	−0.413019
$953$	26.1217	0.846164	0.423082	−	0.906091i	$-0.360948\pi$
$953$	26.1217	0.846164	0.423082	+	0.906091i	$0.360948\pi$
$954$	0	0
$955$	0	0
$956$	−1.96372	−0.0635112
$957$	0	0
$958$	−13.4870	−0.435745
$959$	3.24367	0.104743
$960$	0	0
$961$	28.4013	0.916171
$962$	12.8289	0.413620
$963$	0	0
$964$	5.45073	0.175556
$965$	0	0
$966$	0	0
$967$	−55.7666	−1.79333	−0.896667	−	0.442706i	$-0.854019\pi$
$967$	−55.7666	−1.79333	−0.896667	+	0.442706i	$0.854019\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0	0
$971$	−49.8520	−1.59983	−0.799914	−	0.600115i	$-0.795121\pi$
$971$	−49.8520	−1.59983	−0.799914	+	0.600115i	$0.795121\pi$
$972$	0	0
$973$	59.9011	1.92034
$974$	40.0231	1.28242
$975$	0	0
$976$	13.4144	0.429386
$977$	52.2433	1.67141	0.835706	−	0.549177i	$-0.185059\pi$
$977$	52.2433	1.67141	0.835706	+	0.549177i	$0.185059\pi$
$978$	0	0
$979$	−12.6891	−0.405545
$980$	0	0
$981$	0	0
$982$	14.2565	0.454943
$983$	−3.18925	−0.101721	−0.0508606	−	0.998706i	$-0.516196\pi$
$983$	−3.18925	−0.101721	−0.0508606	+	0.998706i	$0.516196\pi$
$984$	0	0
$985$	0	0
$986$	−15.4507	−0.492051
$987$	0	0
$988$	14.8058	0.471034
$989$	−5.39132	−0.171434
$990$	0	0
$991$	7.14479	0.226962	0.113481	−	0.993540i	$-0.463800\pi$
$991$	7.14479	0.226962	0.113481	+	0.993540i	$0.463800\pi$
$992$	−7.70722	−0.244705
$993$	0	0
$994$	−60.4738	−1.91811
$995$	0	0
$996$	0	0
$997$	43.1448	1.36641	0.683205	−	0.730227i	$-0.260586\pi$
$997$	43.1448	1.36641	0.683205	+	0.730227i	$0.260586\pi$
$998$	−16.8652	−0.533858
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.a.ch.1.3	yes	3
3.2	odd	2		4950.2.a.cj.1.3	yes	3
5.2	odd	4		4950.2.c.bd.199.3		6
5.3	odd	4		4950.2.c.bd.199.4		6
5.4	even	2		4950.2.a.ci.1.1	yes	3
15.2	even	4		4950.2.c.be.199.6		6
15.8	even	4		4950.2.c.be.199.1		6
15.14	odd	2		4950.2.a.cg.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4950.2.a.cg.1.1	✓	3	15.14	odd	2
4950.2.a.ch.1.3	yes	3	1.1	even	1	trivial
4950.2.a.ci.1.1	yes	3	5.4	even	2
4950.2.a.cj.1.3	yes	3	3.2	odd	2
4950.2.c.bd.199.3		6	5.2	odd	4
4950.2.c.bd.199.4		6	5.3	odd	4
4950.2.c.be.199.1		6	15.8	even	4
4950.2.c.be.199.6		6	15.2	even	4

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:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.70722 q^{7} -1.00000 q^{8} -1.00000 q^{11} -2.72536 q^{13} -4.70722 q^{14} +1.00000 q^{16} +2.70722 q^{17} -5.43259 q^{19} +1.00000 q^{22} -7.43259 q^{23} +2.72536 q^{26} +4.70722 q^{28} +5.70722 q^{29} +7.70722 q^{31} -1.00000 q^{32} -2.70722 q^{34} +4.70722 q^{37} +5.43259 q^{38} -12.1579 q^{41} +0.725363 q^{43} -1.00000 q^{44} +7.43259 q^{46} -8.70722 q^{47} +15.1579 q^{49} -2.72536 q^{52} +3.45073 q^{53} -4.70722 q^{56} -5.70722 q^{58} +8.70722 q^{59} +13.4144 q^{61} -7.70722 q^{62} +1.00000 q^{64} +2.00000 q^{67} +2.70722 q^{68} +12.8470 q^{71} +10.8652 q^{73} -4.70722 q^{74} -5.43259 q^{76} -4.70722 q^{77} +5.29278 q^{79} +12.1579 q^{82} +2.25650 q^{83} -0.725363 q^{86} +1.00000 q^{88} +12.6891 q^{89} -12.8289 q^{91} -7.43259 q^{92} +8.70722 q^{94} -0.414446 q^{97} -15.1579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + q^{7} - 3 q^{8} - 3 q^{11} - 2 q^{13} - q^{14} + 3 q^{16} - 5 q^{17} + 3 q^{19} + 3 q^{22} - 3 q^{23} + 2 q^{26} + q^{28} + 4 q^{29} + 10 q^{31} - 3 q^{32} + 5 q^{34} + q^{37}+ \cdots - 20 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + q^7 - 3 * q^8 - 3 * q^11 - 2 * q^13 - q^14 + 3 * q^16 - 5 * q^17 + 3 * q^19 + 3 * q^22 - 3 * q^23 + 2 * q^26 + q^28 + 4 * q^29 + 10 * q^31 - 3 * q^32 + 5 * q^34 + q^37 - 3 * q^38 - 11 * q^41 - 4 * q^43 - 3 * q^44 + 3 * q^46 - 13 * q^47 + 20 * q^49 - 2 * q^52 - 2 * q^53 - q^56 - 4 * q^58 + 13 * q^59 + 14 * q^61 - 10 * q^62 + 3 * q^64 + 6 * q^67 - 5 * q^68 - 7 * q^71 - 6 * q^73 - q^74 + 3 * q^76 - q^77 + 29 * q^79 + 11 * q^82 + 6 * q^83 + 4 * q^86 + 3 * q^88 + 18 * q^89 + 14 * q^91 - 3 * q^92 + 13 * q^94 + 25 * q^97 - 20 * q^98

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