LMFDB - Embedded newform 2277.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2277,2,Mod(1,2277)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2277, 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("2277.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2277 = 3^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2277.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-3,0,5,-3,0,-1,-3,0,-6,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1819365402$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.8639957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 4x^{4} + 10x^{3} + 6x^{2} - 7x - 2 $ x^6 - 3x^5 - 4x^4 + 10x^3 + 6x^2 - 7*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 253)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.901423$ of defining polynomial
Character	$\chi$	$=$	2277.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0985770 q^{2} -1.99028 q^{4} -1.39335 q^{5} -2.57601 q^{7} +0.393350 q^{8} +0.137352 q^{10} -1.00000 q^{11} +4.27358 q^{13} +0.253935 q^{14} +3.94179 q^{16} +7.01179 q^{17} -1.72242 q^{19} +2.77316 q^{20} +0.0985770 q^{22} +1.00000 q^{23} -3.05858 q^{25} -0.421276 q^{26} +5.12698 q^{28} +8.02757 q^{29} -6.10007 q^{31} -1.17527 q^{32} -0.691201 q^{34} +3.58928 q^{35} -6.30964 q^{37} +0.169791 q^{38} -0.548074 q^{40} -4.29179 q^{41} +10.3776 q^{43} +1.99028 q^{44} -0.0985770 q^{46} -4.35702 q^{47} -0.364192 q^{49} +0.301505 q^{50} -8.50563 q^{52} +1.74607 q^{53} +1.39335 q^{55} -1.01327 q^{56} -0.791334 q^{58} -6.74400 q^{59} -1.19907 q^{61} +0.601326 q^{62} -7.76773 q^{64} -5.95459 q^{65} -4.78988 q^{67} -13.9554 q^{68} -0.353820 q^{70} -8.69264 q^{71} -5.16193 q^{73} +0.621985 q^{74} +3.42811 q^{76} +2.57601 q^{77} -7.94736 q^{79} -5.49229 q^{80} +0.423071 q^{82} -1.58243 q^{83} -9.76987 q^{85} -1.02299 q^{86} -0.393350 q^{88} -5.85263 q^{89} -11.0088 q^{91} -1.99028 q^{92} +0.429502 q^{94} +2.39994 q^{95} +12.8485 q^{97} +0.0359010 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 5 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - 6 q^{10} - 6 q^{11} - 3 q^{13} + 8 q^{14} - q^{16} - 5 q^{17} + q^{19} + 7 q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{25} - 15 q^{26} - 6 q^{29} - 8 q^{31}+ \cdots - 34 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 5 * q^4 - 3 * q^5 - q^7 - 3 * q^8 - 6 * q^10 - 6 * q^11 - 3 * q^13 + 8 * q^14 - q^16 - 5 * q^17 + q^19 + 7 * q^20 + 3 * q^22 + 6 * q^23 + 3 * q^25 - 15 * q^26 - 6 * q^29 - 8 * q^31 + 6 * q^32 - 6 * q^34 - 12 * q^35 + 2 * q^37 - 9 * q^38 - 30 * q^40 + 2 * q^41 + 10 * q^43 - 5 * q^44 - 3 * q^46 - 14 * q^47 - 5 * q^49 - 9 * q^50 - 11 * q^52 + 4 * q^53 + 3 * q^55 + 13 * q^56 - 6 * q^58 - 39 * q^59 - 22 * q^61 - 33 * q^64 + 16 * q^65 + 8 * q^67 - 17 * q^68 + 19 * q^70 - 18 * q^71 - 9 * q^73 - 15 * q^74 - 25 * q^76 + q^77 - 15 * q^79 + 16 * q^80 - 8 * q^82 - 43 * q^83 - 21 * q^85 + 55 * q^86 + 3 * q^88 - 17 * q^89 - 9 * q^91 + 5 * q^92 + 63 * q^94 + 21 * q^95 + 19 * q^97 - 34 * 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 $

$2$	−0.0985770	−0.0697044	−0.0348522	−	0.999392i	$-0.511096\pi$
$2$	−0.0985770	−0.0697044	−0.0348522	+	0.999392i	$0.511096\pi$
$3$	0	0
$3$	0	0
$4$	−1.99028	−0.995141
$5$	−1.39335	−0.623125	−0.311563	−	0.950226i	$-0.600852\pi$
$5$	−1.39335	−0.623125	−0.311563	+	0.950226i	$0.600852\pi$
$6$	0	0
$7$	−2.57601	−0.973639	−0.486819	−	0.873503i	$-0.661843\pi$
$7$	−2.57601	−0.973639	−0.486819	+	0.873503i	$0.661843\pi$
$8$	0.393350	0.139070
$9$	0	0
$10$	0.137352	0.0434346
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.27358	1.18528	0.592639	−	0.805468i	$-0.298086\pi$
$13$	4.27358	1.18528	0.592639	+	0.805468i	$0.298086\pi$
$14$	0.253935	0.0678669
$15$	0	0
$16$	3.94179	0.985447
$17$	7.01179	1.70061	0.850304	−	0.526292i	$-0.176418\pi$
$17$	7.01179	1.70061	0.850304	+	0.526292i	$0.176418\pi$
$18$	0	0
$19$	−1.72242	−0.395151	−0.197575	−	0.980288i	$-0.563307\pi$
$19$	−1.72242	−0.395151	−0.197575	+	0.980288i	$0.563307\pi$
$20$	2.77316	0.620097
$21$	0	0
$22$	0.0985770	0.0210167
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.05858	−0.611715
$26$	−0.421276	−0.0826191
$27$	0	0
$28$	5.12698	0.968908
$29$	8.02757	1.49068	0.745341	−	0.666683i	$-0.232287\pi$
$29$	8.02757	1.49068	0.745341	+	0.666683i	$0.232287\pi$
$30$	0	0
$31$	−6.10007	−1.09560	−0.547802	−	0.836608i	$-0.684535\pi$
$31$	−6.10007	−1.09560	−0.547802	+	0.836608i	$0.684535\pi$
$32$	−1.17527	−0.207760
$33$	0	0
$34$	−0.691201	−0.118540
$35$	3.58928	0.606699
$36$	0	0
$37$	−6.30964	−1.03730	−0.518649	−	0.854987i	$-0.673565\pi$
$37$	−6.30964	−1.03730	−0.518649	+	0.854987i	$0.673565\pi$
$38$	0.169791	0.0275437
$39$	0	0
$40$	−0.548074	−0.0866581
$41$	−4.29179	−0.670265	−0.335132	−	0.942171i	$-0.608781\pi$
$41$	−4.29179	−0.670265	−0.335132	+	0.942171i	$0.608781\pi$
$42$	0	0
$43$	10.3776	1.58257	0.791287	−	0.611445i	$-0.209412\pi$
$43$	10.3776	1.58257	0.791287	+	0.611445i	$0.209412\pi$
$44$	1.99028	0.300046
$45$	0	0
$46$	−0.0985770	−0.0145344
$47$	−4.35702	−0.635537	−0.317769	−	0.948168i	$-0.602934\pi$
$47$	−4.35702	−0.635537	−0.317769	+	0.948168i	$0.602934\pi$
$48$	0	0
$49$	−0.364192	−0.0520274
$50$	0.301505	0.0426393
$51$	0	0
$52$	−8.50563	−1.17952
$53$	1.74607	0.239840	0.119920	−	0.992784i	$-0.461736\pi$
$53$	1.74607	0.239840	0.119920	+	0.992784i	$0.461736\pi$
$54$	0	0
$55$	1.39335	0.187879
$56$	−1.01327	−0.135404
$57$	0	0
$58$	−0.791334	−0.103907
$59$	−6.74400	−0.877994	−0.438997	−	0.898488i	$-0.644666\pi$
$59$	−6.74400	−0.877994	−0.438997	+	0.898488i	$0.644666\pi$
$60$	0	0
$61$	−1.19907	−0.153525	−0.0767626	−	0.997049i	$-0.524458\pi$
$61$	−1.19907	−0.153525	−0.0767626	+	0.997049i	$0.524458\pi$
$62$	0.601326	0.0763685
$63$	0	0
$64$	−7.76773	−0.970966
$65$	−5.95459	−0.738576
$66$	0	0
$67$	−4.78988	−0.585177	−0.292589	−	0.956238i	$-0.594517\pi$
$67$	−4.78988	−0.585177	−0.292589	+	0.956238i	$0.594517\pi$
$68$	−13.9554	−1.69235
$69$	0	0
$70$	−0.353820	−0.0422896
$71$	−8.69264	−1.03163	−0.515813	−	0.856701i	$-0.672510\pi$
$71$	−8.69264	−1.03163	−0.515813	+	0.856701i	$0.672510\pi$
$72$	0	0
$73$	−5.16193	−0.604159	−0.302079	−	0.953283i	$-0.597681\pi$
$73$	−5.16193	−0.604159	−0.302079	+	0.953283i	$0.597681\pi$
$74$	0.621985	0.0723043
$75$	0	0
$76$	3.42811	0.393231
$77$	2.57601	0.293563
$78$	0	0
$79$	−7.94736	−0.894148	−0.447074	−	0.894497i	$-0.647534\pi$
$79$	−7.94736	−0.894148	−0.447074	+	0.894497i	$0.647534\pi$
$80$	−5.49229	−0.614057
$81$	0	0
$82$	0.423071	0.0467204
$83$	−1.58243	−0.173694	−0.0868471	−	0.996222i	$-0.527679\pi$
$83$	−1.58243	−0.173694	−0.0868471	+	0.996222i	$0.527679\pi$
$84$	0	0
$85$	−9.76987	−1.05969
$86$	−1.02299	−0.110312
$87$	0	0
$88$	−0.393350	−0.0419312
$89$	−5.85263	−0.620377	−0.310189	−	0.950675i	$-0.600392\pi$
$89$	−5.85263	−0.620377	−0.310189	+	0.950675i	$0.600392\pi$
$90$	0	0
$91$	−11.0088	−1.15403
$92$	−1.99028	−0.207501
$93$	0	0
$94$	0.429502	0.0442998
$95$	2.39994	0.246228
$96$	0	0
$97$	12.8485	1.30456	0.652282	−	0.757977i	$-0.273812\pi$
$97$	12.8485	1.30456	0.652282	+	0.757977i	$0.273812\pi$
$98$	0.0359010	0.00362654
$99$	0	0
$100$	6.08743	0.608743
$101$	1.05074	0.104552	0.0522762	−	0.998633i	$-0.483352\pi$
$101$	1.05074	0.104552	0.0522762	+	0.998633i	$0.483352\pi$
$102$	0	0
$103$	−7.29909	−0.719200	−0.359600	−	0.933106i	$-0.617087\pi$
$103$	−7.29909	−0.719200	−0.359600	+	0.933106i	$0.617087\pi$
$104$	1.68101	0.164837
$105$	0	0
$106$	−0.172122	−0.0167179
$107$	−10.4902	−1.01413	−0.507064	−	0.861908i	$-0.669269\pi$
$107$	−10.4902	−1.01413	−0.507064	+	0.861908i	$0.669269\pi$
$108$	0	0
$109$	10.7094	1.02578	0.512888	−	0.858455i	$-0.328576\pi$
$109$	10.7094	1.02578	0.512888	+	0.858455i	$0.328576\pi$
$110$	−0.137352	−0.0130960
$111$	0	0
$112$	−10.1541	−0.959470
$113$	0.487507	0.0458608	0.0229304	−	0.999737i	$-0.492700\pi$
$113$	0.487507	0.0458608	0.0229304	+	0.999737i	$0.492700\pi$
$114$	0	0
$115$	−1.39335	−0.129931
$116$	−15.9771	−1.48344
$117$	0	0
$118$	0.664803	0.0612001
$119$	−18.0624	−1.65578
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.118201	0.0107014
$123$	0	0
$124$	12.1409	1.09028
$125$	11.2284	1.00430
$126$	0	0
$127$	9.26452	0.822093	0.411046	−	0.911614i	$-0.365163\pi$
$127$	9.26452	0.822093	0.411046	+	0.911614i	$0.365163\pi$
$128$	3.11626	0.275441
$129$	0	0
$130$	0.586985	0.0514820
$131$	−12.0582	−1.05353	−0.526766	−	0.850011i	$-0.676595\pi$
$131$	−12.0582	−1.05353	−0.526766	+	0.850011i	$0.676595\pi$
$132$	0	0
$133$	4.43697	0.384734
$134$	0.472172	0.0407895
$135$	0	0
$136$	2.75809	0.236504
$137$	−9.54165	−0.815198	−0.407599	−	0.913161i	$-0.633634\pi$
$137$	−9.54165	−0.815198	−0.407599	+	0.913161i	$0.633634\pi$
$138$	0	0
$139$	0.752437	0.0638209	0.0319104	−	0.999491i	$-0.489841\pi$
$139$	0.752437	0.0638209	0.0319104	+	0.999491i	$0.489841\pi$
$140$	−7.14368	−0.603751
$141$	0	0
$142$	0.856894	0.0719089
$143$	−4.27358	−0.357375
$144$	0	0
$145$	−11.1852	−0.928882
$146$	0.508848	0.0421125
$147$	0	0
$148$	12.5580	1.03226
$149$	0.762211	0.0624428	0.0312214	−	0.999512i	$-0.490060\pi$
$149$	0.762211	0.0624428	0.0312214	+	0.999512i	$0.490060\pi$
$150$	0	0
$151$	3.02363	0.246060	0.123030	−	0.992403i	$-0.460739\pi$
$151$	3.02363	0.246060	0.123030	+	0.992403i	$0.460739\pi$
$152$	−0.677514	−0.0549537
$153$	0	0
$154$	−0.253935	−0.0204627
$155$	8.49953	0.682699
$156$	0	0
$157$	−4.27555	−0.341226	−0.170613	−	0.985338i	$-0.554575\pi$
$157$	−4.27555	−0.341226	−0.170613	+	0.985338i	$0.554575\pi$
$158$	0.783427	0.0623261
$159$	0	0
$160$	1.63756	0.129461
$161$	−2.57601	−0.203018
$162$	0	0
$163$	7.12165	0.557810	0.278905	−	0.960319i	$-0.410028\pi$
$163$	7.12165	0.557810	0.278905	+	0.960319i	$0.410028\pi$
$164$	8.54187	0.667008
$165$	0	0
$166$	0.155991	0.0121073
$167$	−4.65914	−0.360535	−0.180268	−	0.983618i	$-0.557696\pi$
$167$	−4.65914	−0.360535	−0.180268	+	0.983618i	$0.557696\pi$
$168$	0	0
$169$	5.26347	0.404883
$170$	0.963084	0.0738652
$171$	0	0
$172$	−20.6544	−1.57488
$173$	11.3203	0.860668	0.430334	−	0.902670i	$-0.358396\pi$
$173$	11.3203	0.860668	0.430334	+	0.902670i	$0.358396\pi$
$174$	0	0
$175$	7.87891	0.595590
$176$	−3.94179	−0.297124
$177$	0	0
$178$	0.576934	0.0432430
$179$	0.975679	0.0729257	0.0364628	−	0.999335i	$-0.488391\pi$
$179$	0.975679	0.0729257	0.0364628	+	0.999335i	$0.488391\pi$
$180$	0	0
$181$	−12.2845	−0.913100	−0.456550	−	0.889698i	$-0.650915\pi$
$181$	−12.2845	−0.913100	−0.456550	+	0.889698i	$0.650915\pi$
$182$	1.08521	0.0804412
$183$	0	0
$184$	0.393350	0.0289981
$185$	8.79153	0.646366
$186$	0	0
$187$	−7.01179	−0.512753
$188$	8.67171	0.632449
$189$	0	0
$190$	−0.236578	−0.0171632
$191$	−10.5415	−0.762759	−0.381380	−	0.924418i	$-0.624551\pi$
$191$	−10.5415	−0.762759	−0.381380	+	0.924418i	$0.624551\pi$
$192$	0	0
$193$	−2.78596	−0.200537	−0.100269	−	0.994960i	$-0.531970\pi$
$193$	−2.78596	−0.200537	−0.100269	+	0.994960i	$0.531970\pi$
$194$	−1.26656	−0.0909339
$195$	0	0
$196$	0.724845	0.0517747
$197$	17.9830	1.28123	0.640616	−	0.767861i	$-0.278679\pi$
$197$	17.9830	1.28123	0.640616	+	0.767861i	$0.278679\pi$
$198$	0	0
$199$	0.857794	0.0608074	0.0304037	−	0.999538i	$-0.490321\pi$
$199$	0.857794	0.0608074	0.0304037	+	0.999538i	$0.490321\pi$
$200$	−1.20309	−0.0850714
$201$	0	0
$202$	−0.103579	−0.00728777
$203$	−20.6791	−1.45139
$204$	0	0
$205$	5.97996	0.417659
$206$	0.719522	0.0501315
$207$	0	0
$208$	16.8455	1.16803
$209$	1.72242	0.119142
$210$	0	0
$211$	−20.7330	−1.42732	−0.713658	−	0.700495i	$-0.752963\pi$
$211$	−20.7330	−1.42732	−0.713658	+	0.700495i	$0.752963\pi$
$212$	−3.47516	−0.238675
$213$	0	0
$214$	1.03410	0.0706893
$215$	−14.4597	−0.986141
$216$	0	0
$217$	15.7138	1.06672
$218$	−1.05570	−0.0715012
$219$	0	0
$220$	−2.77316	−0.186966
$221$	29.9654	2.01569
$222$	0	0
$223$	−0.901902	−0.0603958	−0.0301979	−	0.999544i	$-0.509614\pi$
$223$	−0.901902	−0.0603958	−0.0301979	+	0.999544i	$0.509614\pi$
$224$	3.02750	0.202283
$225$	0	0
$226$	−0.0480570	−0.00319670
$227$	−10.8182	−0.718028	−0.359014	−	0.933332i	$-0.616887\pi$
$227$	−10.8182	−0.718028	−0.359014	+	0.933332i	$0.616887\pi$
$228$	0	0
$229$	−27.6949	−1.83013	−0.915066	−	0.403304i	$-0.867862\pi$
$229$	−27.6949	−1.83013	−0.915066	+	0.403304i	$0.867862\pi$
$230$	0.137352	0.00905674
$231$	0	0
$232$	3.15764	0.207310
$233$	−9.85448	−0.645589	−0.322794	−	0.946469i	$-0.604622\pi$
$233$	−9.85448	−0.645589	−0.322794	+	0.946469i	$0.604622\pi$
$234$	0	0
$235$	6.07086	0.396019
$236$	13.4225	0.873728
$237$	0	0
$238$	1.78054	0.115415
$239$	−22.2862	−1.44158	−0.720788	−	0.693156i	$-0.756220\pi$
$239$	−22.2862	−1.44158	−0.720788	+	0.693156i	$0.756220\pi$
$240$	0	0
$241$	−8.26747	−0.532555	−0.266277	−	0.963896i	$-0.585794\pi$
$241$	−8.26747	−0.532555	−0.266277	+	0.963896i	$0.585794\pi$
$242$	−0.0985770	−0.00633677
$243$	0	0
$244$	2.38649	0.152779
$245$	0.507447	0.0324196
$246$	0	0
$247$	−7.36090	−0.468363
$248$	−2.39946	−0.152366
$249$	0	0
$250$	−1.10686	−0.0700042
$251$	−7.07648	−0.446663	−0.223332	−	0.974743i	$-0.571693\pi$
$251$	−7.07648	−0.446663	−0.223332	+	0.974743i	$0.571693\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	−0.913268	−0.0573035
$255$	0	0
$256$	15.2283	0.951766
$257$	−5.69650	−0.355338	−0.177669	−	0.984090i	$-0.556856\pi$
$257$	−5.69650	−0.355338	−0.177669	+	0.984090i	$0.556856\pi$
$258$	0	0
$259$	16.2537	1.00995
$260$	11.8513	0.734988
$261$	0	0
$262$	1.18866	0.0734358
$263$	−2.50258	−0.154316	−0.0771579	−	0.997019i	$-0.524585\pi$
$263$	−2.50258	−0.154316	−0.0771579	+	0.997019i	$0.524585\pi$
$264$	0	0
$265$	−2.43288	−0.149451
$266$	−0.437383	−0.0268177
$267$	0	0
$268$	9.53322	0.582334
$269$	3.23678	0.197350	0.0986749	−	0.995120i	$-0.468540\pi$
$269$	3.23678	0.197350	0.0986749	+	0.995120i	$0.468540\pi$
$270$	0	0
$271$	9.71834	0.590347	0.295173	−	0.955444i	$-0.404623\pi$
$271$	9.71834	0.590347	0.295173	+	0.955444i	$0.404623\pi$
$272$	27.6390	1.67586
$273$	0	0
$274$	0.940587	0.0568229
$275$	3.05858	0.184439
$276$	0	0
$277$	20.3613	1.22339	0.611695	−	0.791094i	$-0.290488\pi$
$277$	20.3613	1.22339	0.611695	+	0.791094i	$0.290488\pi$
$278$	−0.0741730	−0.00444860
$279$	0	0
$280$	1.41184	0.0843737
$281$	−27.8912	−1.66385	−0.831926	−	0.554887i	$-0.812762\pi$
$281$	−27.8912	−1.66385	−0.831926	+	0.554887i	$0.812762\pi$
$282$	0	0
$283$	−32.2777	−1.91871	−0.959356	−	0.282198i	$-0.908936\pi$
$283$	−32.2777	−1.91871	−0.959356	+	0.282198i	$0.908936\pi$
$284$	17.3008	1.02661
$285$	0	0
$286$	0.421276	0.0249106
$287$	11.0557	0.652596
$288$	0	0
$289$	32.1651	1.89207
$290$	1.10260	0.0647472
$291$	0	0
$292$	10.2737	0.601223
$293$	−0.0837545	−0.00489299	−0.00244650	−	0.999997i	$-0.500779\pi$
$293$	−0.0837545	−0.00489299	−0.00244650	+	0.999997i	$0.500779\pi$
$294$	0	0
$295$	9.39676	0.547100
$296$	−2.48190	−0.144257
$297$	0	0
$298$	−0.0751365	−0.00435254
$299$	4.27358	0.247147
$300$	0	0
$301$	−26.7328	−1.54085
$302$	−0.298060	−0.0171514
$303$	0	0
$304$	−6.78942	−0.389400
$305$	1.67072	0.0956654
$306$	0	0
$307$	34.7795	1.98497	0.992485	−	0.122367i	$-0.0390485\pi$
$307$	34.7795	1.98497	0.992485	+	0.122367i	$0.0390485\pi$
$308$	−5.12698	−0.292137
$309$	0	0
$310$	−0.837858	−0.0475871
$311$	25.1850	1.42811	0.714054	−	0.700091i	$-0.246857\pi$
$311$	25.1850	1.42811	0.714054	+	0.700091i	$0.246857\pi$
$312$	0	0
$313$	−8.68839	−0.491097	−0.245548	−	0.969384i	$-0.578968\pi$
$313$	−8.68839	−0.491097	−0.245548	+	0.969384i	$0.578968\pi$
$314$	0.421471	0.0237850
$315$	0	0
$316$	15.8175	0.889804
$317$	−8.22291	−0.461845	−0.230922	−	0.972972i	$-0.574174\pi$
$317$	−8.22291	−0.461845	−0.230922	+	0.972972i	$0.574174\pi$
$318$	0	0
$319$	−8.02757	−0.449458
$320$	10.8232	0.605033
$321$	0	0
$322$	0.253935	0.0141512
$323$	−12.0772	−0.671996
$324$	0	0
$325$	−13.0711	−0.725052
$326$	−0.702030	−0.0388819
$327$	0	0
$328$	−1.68817	−0.0932138
$329$	11.2237	0.618784
$330$	0	0
$331$	−10.9564	−0.602217	−0.301108	−	0.953590i	$-0.597357\pi$
$331$	−10.9564	−0.602217	−0.301108	+	0.953590i	$0.597357\pi$
$332$	3.14948	0.172850
$333$	0	0
$334$	0.459284	0.0251309
$335$	6.67398	0.364639
$336$	0	0
$337$	−32.2285	−1.75560	−0.877798	−	0.479031i	$-0.840988\pi$
$337$	−32.2285	−1.75560	−0.877798	+	0.479031i	$0.840988\pi$
$338$	−0.518857	−0.0282221
$339$	0	0
$340$	19.4448	1.05454
$341$	6.10007	0.330337
$342$	0	0
$343$	18.9702	1.02429
$344$	4.08204	0.220089
$345$	0	0
$346$	−1.11592	−0.0599924
$347$	27.4718	1.47476	0.737382	−	0.675476i	$-0.236062\pi$
$347$	27.4718	1.47476	0.737382	+	0.675476i	$0.236062\pi$
$348$	0	0
$349$	13.8793	0.742941	0.371470	−	0.928445i	$-0.378854\pi$
$349$	13.8793	0.742941	0.371470	+	0.928445i	$0.378854\pi$
$350$	−0.776679	−0.0415152
$351$	0	0
$352$	1.17527	0.0626421
$353$	−3.45719	−0.184008	−0.0920039	−	0.995759i	$-0.529327\pi$
$353$	−3.45719	−0.184008	−0.0920039	+	0.995759i	$0.529327\pi$
$354$	0	0
$355$	12.1119	0.642832
$356$	11.6484	0.617363
$357$	0	0
$358$	−0.0961795	−0.00508324
$359$	19.2010	1.01339	0.506696	−	0.862125i	$-0.330867\pi$
$359$	19.2010	1.01339	0.506696	+	0.862125i	$0.330867\pi$
$360$	0	0
$361$	−16.0333	−0.843856
$362$	1.21097	0.0636471
$363$	0	0
$364$	21.9106	1.14843
$365$	7.19238	0.376466
$366$	0	0
$367$	−4.36951	−0.228087	−0.114043	−	0.993476i	$-0.536380\pi$
$367$	−4.36951	−0.228087	−0.114043	+	0.993476i	$0.536380\pi$
$368$	3.94179	0.205480
$369$	0	0
$370$	−0.866643	−0.0450546
$371$	−4.49787	−0.233518
$372$	0	0
$373$	−18.1184	−0.938135	−0.469068	−	0.883162i	$-0.655410\pi$
$373$	−18.1184	−0.938135	−0.469068	+	0.883162i	$0.655410\pi$
$374$	0.691201	0.0357411
$375$	0	0
$376$	−1.71384	−0.0883843
$377$	34.3065	1.76687
$378$	0	0
$379$	−18.3481	−0.942480	−0.471240	−	0.882005i	$-0.656194\pi$
$379$	−18.3481	−0.942480	−0.471240	+	0.882005i	$0.656194\pi$
$380$	−4.77655	−0.245032
$381$	0	0
$382$	1.03915	0.0531677
$383$	−34.2931	−1.75230	−0.876148	−	0.482042i	$-0.839895\pi$
$383$	−34.2931	−1.75230	−0.876148	+	0.482042i	$0.839895\pi$
$384$	0	0
$385$	−3.58928	−0.182927
$386$	0.274631	0.0139784
$387$	0	0
$388$	−25.5721	−1.29822
$389$	−18.2975	−0.927720	−0.463860	−	0.885908i	$-0.653536\pi$
$389$	−18.2975	−0.927720	−0.463860	+	0.885908i	$0.653536\pi$
$390$	0	0
$391$	7.01179	0.354601
$392$	−0.143255	−0.00723547
$393$	0	0
$394$	−1.77271	−0.0893076
$395$	11.0735	0.557166
$396$	0	0
$397$	−16.7978	−0.843057	−0.421528	−	0.906815i	$-0.638506\pi$
$397$	−16.7978	−0.843057	−0.421528	+	0.906815i	$0.638506\pi$
$398$	−0.0845587	−0.00423855
$399$	0	0
$400$	−12.0563	−0.602813
$401$	−33.6597	−1.68088	−0.840442	−	0.541901i	$-0.817705\pi$
$401$	−33.6597	−1.68088	−0.840442	+	0.541901i	$0.817705\pi$
$402$	0	0
$403$	−26.0691	−1.29860
$404$	−2.09127	−0.104044
$405$	0	0
$406$	2.03848	0.101168
$407$	6.30964	0.312757
$408$	0	0
$409$	−23.8012	−1.17689	−0.588447	−	0.808536i	$-0.700260\pi$
$409$	−23.8012	−1.17689	−0.588447	+	0.808536i	$0.700260\pi$
$410$	−0.589487	−0.0291127
$411$	0	0
$412$	14.5272	0.715706
$413$	17.3726	0.854849
$414$	0	0
$415$	2.20488	0.108233
$416$	−5.02261	−0.246254
$417$	0	0
$418$	−0.169791	−0.00830475
$419$	32.1373	1.57001	0.785006	−	0.619489i	$-0.212660\pi$
$419$	32.1373	1.57001	0.785006	+	0.619489i	$0.212660\pi$
$420$	0	0
$421$	−33.1371	−1.61501	−0.807503	−	0.589864i	$-0.799181\pi$
$421$	−33.1371	−1.61501	−0.807503	+	0.589864i	$0.799181\pi$
$422$	2.04379	0.0994902
$423$	0	0
$424$	0.686815	0.0333547
$425$	−21.4461	−1.04029
$426$	0	0
$427$	3.08881	0.149478
$428$	20.8785	1.00920
$429$	0	0
$430$	1.42539	0.0687384
$431$	−9.06476	−0.436634	−0.218317	−	0.975878i	$-0.570057\pi$
$431$	−9.06476	−0.436634	−0.218317	+	0.975878i	$0.570057\pi$
$432$	0	0
$433$	29.3934	1.41256	0.706279	−	0.707934i	$-0.250373\pi$
$433$	29.3934	1.41256	0.706279	+	0.707934i	$0.250373\pi$
$434$	−1.54902	−0.0743553
$435$	0	0
$436$	−21.3148	−1.02079
$437$	−1.72242	−0.0823946
$438$	0	0
$439$	7.67129	0.366131	0.183065	−	0.983101i	$-0.441398\pi$
$439$	7.67129	0.366131	0.183065	+	0.983101i	$0.441398\pi$
$440$	0.548074	0.0261284
$441$	0	0
$442$	−2.95390	−0.140503
$443$	18.8452	0.895362	0.447681	−	0.894193i	$-0.352250\pi$
$443$	18.8452	0.895362	0.447681	+	0.894193i	$0.352250\pi$
$444$	0	0
$445$	8.15476	0.386573
$446$	0.0889068	0.00420986
$447$	0	0
$448$	20.0097	0.945370
$449$	19.2501	0.908467	0.454234	−	0.890883i	$-0.349913\pi$
$449$	19.2501	0.908467	0.454234	+	0.890883i	$0.349913\pi$
$450$	0	0
$451$	4.29179	0.202092
$452$	−0.970276	−0.0456380
$453$	0	0
$454$	1.06642	0.0500498
$455$	15.3391	0.719106
$456$	0	0
$457$	14.2597	0.667041	0.333520	−	0.942743i	$-0.391763\pi$
$457$	14.2597	0.667041	0.333520	+	0.942743i	$0.391763\pi$
$458$	2.73008	0.127568
$459$	0	0
$460$	2.77316	0.129299
$461$	6.79786	0.316608	0.158304	−	0.987390i	$-0.449397\pi$
$461$	6.79786	0.316608	0.158304	+	0.987390i	$0.449397\pi$
$462$	0	0
$463$	−23.3464	−1.08500	−0.542499	−	0.840056i	$-0.682522\pi$
$463$	−23.3464	−1.08500	−0.542499	+	0.840056i	$0.682522\pi$
$464$	31.6430	1.46899
$465$	0	0
$466$	0.971425	0.0450004
$467$	−20.0130	−0.926093	−0.463047	−	0.886334i	$-0.653244\pi$
$467$	−20.0130	−0.926093	−0.463047	+	0.886334i	$0.653244\pi$
$468$	0	0
$469$	12.3388	0.569751
$470$	−0.598447	−0.0276043
$471$	0	0
$472$	−2.65275	−0.122103
$473$	−10.3776	−0.477164
$474$	0	0
$475$	5.26816	0.241720
$476$	35.9493	1.64773
$477$	0	0
$478$	2.19691	0.100484
$479$	32.3385	1.47758	0.738791	−	0.673934i	$-0.235397\pi$
$479$	32.3385	1.47758	0.738791	+	0.673934i	$0.235397\pi$
$480$	0	0
$481$	−26.9647	−1.22949
$482$	0.814983	0.0371214
$483$	0	0
$484$	−1.99028	−0.0904674
$485$	−17.9024	−0.812906
$486$	0	0
$487$	10.2937	0.466450	0.233225	−	0.972423i	$-0.425072\pi$
$487$	10.2937	0.466450	0.233225	+	0.972423i	$0.425072\pi$
$488$	−0.471654	−0.0213508
$489$	0	0
$490$	−0.0500226	−0.00225979
$491$	2.44398	0.110295	0.0551477	−	0.998478i	$-0.482437\pi$
$491$	2.44398	0.110295	0.0551477	+	0.998478i	$0.482437\pi$
$492$	0	0
$493$	56.2876	2.53507
$494$	0.725615	0.0326470
$495$	0	0
$496$	−24.0452	−1.07966
$497$	22.3923	1.00443
$498$	0	0
$499$	−15.7764	−0.706247	−0.353124	−	0.935577i	$-0.614880\pi$
$499$	−15.7764	−0.706247	−0.353124	+	0.935577i	$0.614880\pi$
$500$	−22.3477	−0.999420
$501$	0	0
$502$	0.697578	0.0311344
$503$	−37.7052	−1.68119	−0.840595	−	0.541664i	$-0.817795\pi$
$503$	−37.7052	−1.68119	−0.840595	+	0.541664i	$0.817795\pi$
$504$	0	0
$505$	−1.46405	−0.0651492
$506$	0.0985770	0.00438228
$507$	0	0
$508$	−18.4390	−0.818099
$509$	15.7070	0.696201	0.348101	−	0.937457i	$-0.386827\pi$
$509$	15.7070	0.696201	0.348101	+	0.937457i	$0.386827\pi$
$510$	0	0
$511$	13.2972	0.588232
$512$	−7.73367	−0.341783
$513$	0	0
$514$	0.561543	0.0247686
$515$	10.1702	0.448152
$516$	0	0
$517$	4.35702	0.191622
$518$	−1.60224	−0.0703982
$519$	0	0
$520$	−2.34224	−0.102714
$521$	29.6576	1.29932	0.649661	−	0.760224i	$-0.274911\pi$
$521$	29.6576	1.29932	0.649661	+	0.760224i	$0.274911\pi$
$522$	0	0
$523$	−1.12098	−0.0490169	−0.0245084	−	0.999700i	$-0.507802\pi$
$523$	−1.12098	−0.0490169	−0.0245084	+	0.999700i	$0.507802\pi$
$524$	23.9993	1.04841
$525$	0	0
$526$	0.246697	0.0107565
$527$	−42.7724	−1.86319
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.239826	0.0104174
$531$	0	0
$532$	−8.83082	−0.382865
$533$	−18.3413	−0.794450
$534$	0	0
$535$	14.6166	0.631929
$536$	−1.88410	−0.0813807
$537$	0	0
$538$	−0.319072	−0.0137562
$539$	0.364192	0.0156869
$540$	0	0
$541$	41.1739	1.77020	0.885102	−	0.465397i	$-0.154088\pi$
$541$	41.1739	1.77020	0.885102	+	0.465397i	$0.154088\pi$
$542$	−0.958004	−0.0411498
$543$	0	0
$544$	−8.24074	−0.353319
$545$	−14.9220	−0.639187
$546$	0	0
$547$	−17.2965	−0.739546	−0.369773	−	0.929122i	$-0.620565\pi$
$547$	−17.2965	−0.739546	−0.369773	+	0.929122i	$0.620565\pi$
$548$	18.9906	0.811237
$549$	0	0
$550$	−0.301505	−0.0128562
$551$	−13.8269	−0.589044
$552$	0	0
$553$	20.4725	0.870577
$554$	−2.00715	−0.0852757
$555$	0	0
$556$	−1.49756	−0.0635108
$557$	−44.5087	−1.88589	−0.942947	−	0.332944i	$-0.891958\pi$
$557$	−44.5087	−1.88589	−0.942947	+	0.332944i	$0.891958\pi$
$558$	0	0
$559$	44.3496	1.87579
$560$	14.1482	0.597870
$561$	0	0
$562$	2.74943	0.115978
$563$	−20.7078	−0.872732	−0.436366	−	0.899769i	$-0.643735\pi$
$563$	−20.7078	−0.872732	−0.436366	+	0.899769i	$0.643735\pi$
$564$	0	0
$565$	−0.679268	−0.0285770
$566$	3.18184	0.133743
$567$	0	0
$568$	−3.41925	−0.143468
$569$	8.30654	0.348228	0.174114	−	0.984725i	$-0.444294\pi$
$569$	8.30654	0.348228	0.174114	+	0.984725i	$0.444294\pi$
$570$	0	0
$571$	12.6846	0.530836	0.265418	−	0.964133i	$-0.414490\pi$
$571$	12.6846	0.530836	0.265418	+	0.964133i	$0.414490\pi$
$572$	8.50563	0.355638
$573$	0	0
$574$	−1.08983	−0.0454888
$575$	−3.05858	−0.127551
$576$	0	0
$577$	−35.3453	−1.47145	−0.735723	−	0.677283i	$-0.763157\pi$
$577$	−35.3453	−1.47145	−0.735723	+	0.677283i	$0.763157\pi$
$578$	−3.17074	−0.131885
$579$	0	0
$580$	22.2617	0.924369
$581$	4.07635	0.169115
$582$	0	0
$583$	−1.74607	−0.0723146
$584$	−2.03045	−0.0840205
$585$	0	0
$586$	0.00825627	0.000341063 0
$587$	−15.9955	−0.660205	−0.330103	−	0.943945i	$-0.607083\pi$
$587$	−15.9955	−0.660205	−0.330103	+	0.943945i	$0.607083\pi$
$588$	0	0
$589$	10.5069	0.432929
$590$	−0.926304	−0.0381353
$591$	0	0
$592$	−24.8713	−1.02220
$593$	33.5080	1.37601	0.688005	−	0.725706i	$-0.258487\pi$
$593$	33.5080	1.37601	0.688005	+	0.725706i	$0.258487\pi$
$594$	0	0
$595$	25.1672	1.03176
$596$	−1.51702	−0.0621394
$597$	0	0
$598$	−0.421276	−0.0172273
$599$	5.35026	0.218606	0.109303	−	0.994008i	$-0.465138\pi$
$599$	5.35026	0.218606	0.109303	+	0.994008i	$0.465138\pi$
$600$	0	0
$601$	−36.3268	−1.48180	−0.740901	−	0.671614i	$-0.765601\pi$
$601$	−36.3268	−1.48180	−0.740901	+	0.671614i	$0.765601\pi$
$602$	2.63524	0.107404
$603$	0	0
$604$	−6.01788	−0.244864
$605$	−1.39335	−0.0566477
$606$	0	0
$607$	36.7889	1.49321	0.746607	−	0.665266i	$-0.231682\pi$
$607$	36.7889	1.49321	0.746607	+	0.665266i	$0.231682\pi$
$608$	2.02431	0.0820966
$609$	0	0
$610$	−0.164695	−0.00666830
$611$	−18.6201	−0.753288
$612$	0	0
$613$	−14.8696	−0.600577	−0.300288	−	0.953848i	$-0.597083\pi$
$613$	−14.8696	−0.600577	−0.300288	+	0.953848i	$0.597083\pi$
$614$	−3.42846	−0.138361
$615$	0	0
$616$	1.01327	0.0408259
$617$	23.7451	0.955944	0.477972	−	0.878375i	$-0.341372\pi$
$617$	23.7451	0.955944	0.477972	+	0.878375i	$0.341372\pi$
$618$	0	0
$619$	43.3459	1.74222	0.871110	−	0.491088i	$-0.163401\pi$
$619$	43.3459	1.74222	0.871110	+	0.491088i	$0.163401\pi$
$620$	−16.9165	−0.679382
$621$	0	0
$622$	−2.48266	−0.0995455
$623$	15.0764	0.604023
$624$	0	0
$625$	−0.352234	−0.0140893
$626$	0.856475	0.0342316
$627$	0	0
$628$	8.50955	0.339568
$629$	−44.2418	−1.76404
$630$	0	0
$631$	19.0544	0.758544	0.379272	−	0.925285i	$-0.376174\pi$
$631$	19.0544	0.758544	0.379272	+	0.925285i	$0.376174\pi$
$632$	−3.12609	−0.124349
$633$	0	0
$634$	0.810590	0.0321926
$635$	−12.9087	−0.512267
$636$	0	0
$637$	−1.55640	−0.0616670
$638$	0.791334	0.0313292
$639$	0	0
$640$	−4.34204	−0.171634
$641$	10.6471	0.420535	0.210268	−	0.977644i	$-0.432566\pi$
$641$	10.6471	0.420535	0.210268	+	0.977644i	$0.432566\pi$
$642$	0	0
$643$	19.0246	0.750258	0.375129	−	0.926973i	$-0.377598\pi$
$643$	19.0246	0.750258	0.375129	+	0.926973i	$0.377598\pi$
$644$	5.12698	0.202031
$645$	0	0
$646$	1.19054	0.0468411
$647$	24.8112	0.975430	0.487715	−	0.873003i	$-0.337831\pi$
$647$	24.8112	0.975430	0.487715	+	0.873003i	$0.337831\pi$
$648$	0	0
$649$	6.74400	0.264725
$650$	1.28851	0.0505394
$651$	0	0
$652$	−14.1741	−0.555100
$653$	22.1826	0.868072	0.434036	−	0.900895i	$-0.357089\pi$
$653$	22.1826	0.868072	0.434036	+	0.900895i	$0.357089\pi$
$654$	0	0
$655$	16.8013	0.656482
$656$	−16.9173	−0.660511
$657$	0	0
$658$	−1.10640	−0.0431320
$659$	−17.1147	−0.666692	−0.333346	−	0.942804i	$-0.608178\pi$
$659$	−17.1147	−0.666692	−0.333346	+	0.942804i	$0.608178\pi$
$660$	0	0
$661$	−25.6537	−0.997815	−0.498908	−	0.866655i	$-0.666265\pi$
$661$	−25.6537	−0.997815	−0.498908	+	0.866655i	$0.666265\pi$
$662$	1.08005	0.0419772
$663$	0	0
$664$	−0.622448	−0.0241557
$665$	−6.18225	−0.239737
$666$	0	0
$667$	8.02757	0.310829
$668$	9.27301	0.358784
$669$	0	0
$670$	−0.657901	−0.0254169
$671$	1.19907	0.0462896
$672$	0	0
$673$	19.0359	0.733780	0.366890	−	0.930264i	$-0.380422\pi$
$673$	19.0359	0.733780	0.366890	+	0.930264i	$0.380422\pi$
$674$	3.17698	0.122373
$675$	0	0
$676$	−10.4758	−0.402915
$677$	12.5375	0.481856	0.240928	−	0.970543i	$-0.422548\pi$
$677$	12.5375	0.481856	0.240928	+	0.970543i	$0.422548\pi$
$678$	0	0
$679$	−33.0977	−1.27017
$680$	−3.84298	−0.147371
$681$	0	0
$682$	−0.601326	−0.0230260
$683$	5.31521	0.203381	0.101690	−	0.994816i	$-0.467575\pi$
$683$	5.31521	0.203381	0.101690	+	0.994816i	$0.467575\pi$
$684$	0	0
$685$	13.2949	0.507970
$686$	−1.87003	−0.0713979
$687$	0	0
$688$	40.9064	1.55954
$689$	7.46195	0.284278
$690$	0	0
$691$	−51.1052	−1.94413	−0.972066	−	0.234706i	$-0.924587\pi$
$691$	−51.1052	−1.94413	−0.972066	+	0.234706i	$0.924587\pi$
$692$	−22.5306	−0.856486
$693$	0	0
$694$	−2.70809	−0.102798
$695$	−1.04841	−0.0397684
$696$	0	0
$697$	−30.0931	−1.13986
$698$	−1.36818	−0.0517863
$699$	0	0
$700$	−15.6813	−0.592696
$701$	−16.8643	−0.636957	−0.318479	−	0.947930i	$-0.603172\pi$
$701$	−16.8643	−0.636957	−0.318479	+	0.947930i	$0.603172\pi$
$702$	0	0
$703$	10.8679	0.409889
$704$	7.76773	0.292757
$705$	0	0
$706$	0.340800	0.0128262
$707$	−2.70671	−0.101796
$708$	0	0
$709$	23.2166	0.871917	0.435958	−	0.899967i	$-0.356410\pi$
$709$	23.2166	0.871917	0.435958	+	0.899967i	$0.356410\pi$
$710$	−1.19395	−0.0448082
$711$	0	0
$712$	−2.30213	−0.0862760
$713$	−6.10007	−0.228449
$714$	0	0
$715$	5.95459	0.222689
$716$	−1.94188	−0.0725714
$717$	0	0
$718$	−1.89278	−0.0706379
$719$	22.5357	0.840439	0.420219	−	0.907423i	$-0.361953\pi$
$719$	22.5357	0.840439	0.420219	+	0.907423i	$0.361953\pi$
$720$	0	0
$721$	18.8025	0.700242
$722$	1.58051	0.0588205
$723$	0	0
$724$	24.4496	0.908664
$725$	−24.5529	−0.911873
$726$	0	0
$727$	3.69172	0.136918	0.0684592	−	0.997654i	$-0.478192\pi$
$727$	3.69172	0.136918	0.0684592	+	0.997654i	$0.478192\pi$
$728$	−4.33030	−0.160491
$729$	0	0
$730$	−0.709003	−0.0262414
$731$	72.7657	2.69134
$732$	0	0
$733$	−6.78702	−0.250684	−0.125342	−	0.992114i	$-0.540003\pi$
$733$	−6.78702	−0.250684	−0.125342	+	0.992114i	$0.540003\pi$
$734$	0.430733	0.0158987
$735$	0	0
$736$	−1.17527	−0.0433210
$737$	4.78988	0.176438
$738$	0	0
$739$	−30.2153	−1.11149	−0.555743	−	0.831354i	$-0.687566\pi$
$739$	−30.2153	−1.11149	−0.555743	+	0.831354i	$0.687566\pi$
$740$	−17.4976	−0.643226
$741$	0	0
$742$	0.443387	0.0162772
$743$	−20.0638	−0.736070	−0.368035	−	0.929812i	$-0.619969\pi$
$743$	−20.0638	−0.736070	−0.368035	+	0.929812i	$0.619969\pi$
$744$	0	0
$745$	−1.06203	−0.0389097
$746$	1.78606	0.0653922
$747$	0	0
$748$	13.9554	0.510261
$749$	27.0229	0.987395
$750$	0	0
$751$	27.7596	1.01296	0.506481	−	0.862251i	$-0.330946\pi$
$751$	27.7596	1.01296	0.506481	+	0.862251i	$0.330946\pi$
$752$	−17.1745	−0.626289
$753$	0	0
$754$	−3.38183	−0.123159
$755$	−4.21298	−0.153326
$756$	0	0
$757$	−5.91530	−0.214995	−0.107498	−	0.994205i	$-0.534284\pi$
$757$	−5.91530	−0.214995	−0.107498	+	0.994205i	$0.534284\pi$
$758$	1.80870	0.0656951
$759$	0	0
$760$	0.944015	0.0342430
$761$	8.67915	0.314619	0.157310	−	0.987549i	$-0.449718\pi$
$761$	8.67915	0.314619	0.157310	+	0.987549i	$0.449718\pi$
$762$	0	0
$763$	−27.5875	−0.998736
$764$	20.9807	0.759053
$765$	0	0
$766$	3.38051	0.122143
$767$	−28.8210	−1.04067
$768$	0	0
$769$	52.7197	1.90112	0.950561	−	0.310538i	$-0.100509\pi$
$769$	52.7197	1.90112	0.950561	+	0.310538i	$0.100509\pi$
$770$	0.353820	0.0127508
$771$	0	0
$772$	5.54484	0.199563
$773$	−46.4791	−1.67174	−0.835869	−	0.548928i	$-0.815036\pi$
$773$	−46.4791	−1.67174	−0.835869	+	0.548928i	$0.815036\pi$
$774$	0	0
$775$	18.6575	0.670198
$776$	5.05394	0.181426
$777$	0	0
$778$	1.80371	0.0646662
$779$	7.39227	0.264855
$780$	0	0
$781$	8.69264	0.311047
$782$	−0.691201	−0.0247173
$783$	0	0
$784$	−1.43557	−0.0512703
$785$	5.95734	0.212626
$786$	0	0
$787$	−4.85692	−0.173131	−0.0865653	−	0.996246i	$-0.527589\pi$
$787$	−4.85692	−0.173131	−0.0865653	+	0.996246i	$0.527589\pi$
$788$	−35.7912	−1.27501
$789$	0	0
$790$	−1.09159	−0.0388369
$791$	−1.25582	−0.0446518
$792$	0	0
$793$	−5.12432	−0.181970
$794$	1.65588	0.0587648
$795$	0	0
$796$	−1.70725	−0.0605120
$797$	6.47756	0.229447	0.114724	−	0.993397i	$-0.463402\pi$
$797$	6.47756	0.229447	0.114724	+	0.993397i	$0.463402\pi$
$798$	0	0
$799$	−30.5505	−1.08080
$800$	3.59465	0.127090
$801$	0	0
$802$	3.31807	0.117165
$803$	5.16193	0.182161
$804$	0	0
$805$	3.58928	0.126505
$806$	2.56981	0.0905179
$807$	0	0
$808$	0.413308	0.0145401
$809$	34.9544	1.22893	0.614466	−	0.788943i	$-0.289371\pi$
$809$	34.9544	1.22893	0.614466	+	0.788943i	$0.289371\pi$
$810$	0	0
$811$	−27.6077	−0.969439	−0.484719	−	0.874670i	$-0.661078\pi$
$811$	−27.6077	−0.969439	−0.484719	+	0.874670i	$0.661078\pi$
$812$	41.1572	1.44433
$813$	0	0
$814$	−0.621985	−0.0218006
$815$	−9.92294	−0.347586
$816$	0	0
$817$	−17.8746	−0.625355
$818$	2.34625	0.0820347
$819$	0	0
$820$	−11.9018	−0.415629
$821$	−18.5526	−0.647491	−0.323745	−	0.946144i	$-0.604942\pi$
$821$	−18.5526	−0.647491	−0.323745	+	0.946144i	$0.604942\pi$
$822$	0	0
$823$	0.337866	0.0117773	0.00588863	−	0.999983i	$-0.498126\pi$
$823$	0.337866	0.0117773	0.00588863	+	0.999983i	$0.498126\pi$
$824$	−2.87110	−0.100019
$825$	0	0
$826$	−1.71254	−0.0595868
$827$	−43.1299	−1.49977	−0.749887	−	0.661566i	$-0.769892\pi$
$827$	−43.1299	−1.49977	−0.749887	+	0.661566i	$0.769892\pi$
$828$	0	0
$829$	0.536280	0.0186258	0.00931289	−	0.999957i	$-0.497036\pi$
$829$	0.536280	0.0186258	0.00931289	+	0.999957i	$0.497036\pi$
$830$	−0.217350	−0.00754433
$831$	0	0
$832$	−33.1960	−1.15086
$833$	−2.55364	−0.0884783
$834$	0	0
$835$	6.49182	0.224659
$836$	−3.42811	−0.118563
$837$	0	0
$838$	−3.16800	−0.109437
$839$	−12.8678	−0.444246	−0.222123	−	0.975019i	$-0.571299\pi$
$839$	−12.8678	−0.444246	−0.222123	+	0.975019i	$0.571299\pi$
$840$	0	0
$841$	35.4419	1.22213
$842$	3.26656	0.112573
$843$	0	0
$844$	41.2644	1.42038
$845$	−7.33386	−0.252293
$846$	0	0
$847$	−2.57601	−0.0885126
$848$	6.88262	0.236350
$849$	0	0
$850$	2.11409	0.0725127
$851$	−6.30964	−0.216292
$852$	0	0
$853$	48.4726	1.65967	0.829835	−	0.558009i	$-0.188434\pi$
$853$	48.4726	1.65967	0.829835	+	0.558009i	$0.188434\pi$
$854$	−0.304486	−0.0104193
$855$	0	0
$856$	−4.12633	−0.141035
$857$	12.2020	0.416811	0.208406	−	0.978042i	$-0.433173\pi$
$857$	12.2020	0.416811	0.208406	+	0.978042i	$0.433173\pi$
$858$	0	0
$859$	24.5683	0.838258	0.419129	−	0.907927i	$-0.362335\pi$
$859$	24.5683	0.838258	0.419129	+	0.907927i	$0.362335\pi$
$860$	28.7788	0.981350
$861$	0	0
$862$	0.893577	0.0304354
$863$	29.4190	1.00143	0.500717	−	0.865611i	$-0.333070\pi$
$863$	29.4190	1.00143	0.500717	+	0.865611i	$0.333070\pi$
$864$	0	0
$865$	−15.7732	−0.536304
$866$	−2.89751	−0.0984615
$867$	0	0
$868$	−31.2749	−1.06154
$869$	7.94736	0.269596
$870$	0	0
$871$	−20.4699	−0.693598
$872$	4.21255	0.142655
$873$	0	0
$874$	0.169791	0.00574327
$875$	−28.9245	−0.977826
$876$	0	0
$877$	−3.38594	−0.114335	−0.0571676	−	0.998365i	$-0.518207\pi$
$877$	−3.38594	−0.114335	−0.0571676	+	0.998365i	$0.518207\pi$
$878$	−0.756212	−0.0255209
$879$	0	0
$880$	5.49229	0.185145
$881$	−20.3444	−0.685420	−0.342710	−	0.939441i	$-0.611345\pi$
$881$	−20.3444	−0.685420	−0.342710	+	0.939441i	$0.611345\pi$
$882$	0	0
$883$	48.8737	1.64473	0.822365	−	0.568960i	$-0.192654\pi$
$883$	48.8737	1.64473	0.822365	+	0.568960i	$0.192654\pi$
$884$	−59.6397	−2.00590
$885$	0	0
$886$	−1.85770	−0.0624107
$887$	17.1414	0.575552	0.287776	−	0.957698i	$-0.407084\pi$
$887$	17.1414	0.575552	0.287776	+	0.957698i	$0.407084\pi$
$888$	0	0
$889$	−23.8655	−0.800422
$890$	−0.803871	−0.0269458
$891$	0	0
$892$	1.79504	0.0601024
$893$	7.50463	0.251133
$894$	0	0
$895$	−1.35946	−0.0454418
$896$	−8.02750	−0.268180
$897$	0	0
$898$	−1.89761	−0.0633242
$899$	−48.9687	−1.63320
$900$	0	0
$901$	12.2430	0.407875
$902$	−0.423071	−0.0140867
$903$	0	0
$904$	0.191761	0.00637787
$905$	17.1166	0.568976
$906$	0	0
$907$	−11.1499	−0.370228	−0.185114	−	0.982717i	$-0.559265\pi$
$907$	−11.1499	−0.370228	−0.185114	+	0.982717i	$0.559265\pi$
$908$	21.5313	0.714540
$909$	0	0
$910$	−1.51208	−0.0501249
$911$	−6.30489	−0.208890	−0.104445	−	0.994531i	$-0.533307\pi$
$911$	−6.30489	−0.208890	−0.104445	+	0.994531i	$0.533307\pi$
$912$	0	0
$913$	1.58243	0.0523708
$914$	−1.40568	−0.0464957
$915$	0	0
$916$	55.1207	1.82124
$917$	31.0620	1.02576
$918$	0	0
$919$	−55.3838	−1.82694	−0.913472	−	0.406902i	$-0.866609\pi$
$919$	−55.3838	−1.82694	−0.913472	+	0.406902i	$0.866609\pi$
$920$	−0.548074	−0.0180695
$921$	0	0
$922$	−0.670112	−0.0220690
$923$	−37.1487	−1.22276
$924$	0	0
$925$	19.2985	0.634531
$926$	2.30142	0.0756292
$927$	0	0
$928$	−9.43456	−0.309705
$929$	−22.6620	−0.743517	−0.371758	−	0.928330i	$-0.621245\pi$
$929$	−22.6620	−0.743517	−0.371758	+	0.928330i	$0.621245\pi$
$930$	0	0
$931$	0.627292	0.0205587
$932$	19.6132	0.642452
$933$	0	0
$934$	1.97283	0.0645528
$935$	9.76987	0.319509
$936$	0	0
$937$	−12.4128	−0.405507	−0.202754	−	0.979230i	$-0.564989\pi$
$937$	−12.4128	−0.405507	−0.202754	+	0.979230i	$0.564989\pi$
$938$	−1.21632	−0.0397142
$939$	0	0
$940$	−12.0827	−0.394095
$941$	6.03145	0.196620	0.0983098	−	0.995156i	$-0.468656\pi$
$941$	6.03145	0.196620	0.0983098	+	0.995156i	$0.468656\pi$
$942$	0	0
$943$	−4.29179	−0.139760
$944$	−26.5834	−0.865217
$945$	0	0
$946$	1.02299	0.0332604
$947$	−41.3238	−1.34284	−0.671422	−	0.741075i	$-0.734316\pi$
$947$	−41.3238	−1.34284	−0.671422	+	0.741075i	$0.734316\pi$
$948$	0	0
$949$	−22.0599	−0.716096
$950$	−0.519319	−0.0168489
$951$	0	0
$952$	−7.10485	−0.230269
$953$	31.9472	1.03487	0.517435	−	0.855722i	$-0.326887\pi$
$953$	31.9472	1.03487	0.517435	+	0.855722i	$0.326887\pi$
$954$	0	0
$955$	14.6881	0.475295
$956$	44.3559	1.43457
$957$	0	0
$958$	−3.18783	−0.102994
$959$	24.5794	0.793709
$960$	0	0
$961$	6.21084	0.200350
$962$	2.65810	0.0857006
$963$	0	0
$964$	16.4546	0.529967
$965$	3.88181	0.124960
$966$	0	0
$967$	−38.5020	−1.23814	−0.619071	−	0.785335i	$-0.712491\pi$
$967$	−38.5020	−1.23814	−0.619071	+	0.785335i	$0.712491\pi$
$968$	0.393350	0.0126427
$969$	0	0
$970$	1.76476	0.0566632
$971$	−46.5026	−1.49234	−0.746169	−	0.665756i	$-0.768109\pi$
$971$	−46.5026	−1.49234	−0.746169	+	0.665756i	$0.768109\pi$
$972$	0	0
$973$	−1.93828	−0.0621385
$974$	−1.01472	−0.0325136
$975$	0	0
$976$	−4.72648	−0.151291
$977$	32.3962	1.03645	0.518224	−	0.855245i	$-0.326593\pi$
$977$	32.3962	1.03645	0.518224	+	0.855245i	$0.326593\pi$
$978$	0	0
$979$	5.85263	0.187051
$980$	−1.00996	−0.0322621
$981$	0	0
$982$	−0.240921	−0.00768809
$983$	−41.4597	−1.32236	−0.661179	−	0.750228i	$-0.729944\pi$
$983$	−41.4597	−1.32236	−0.661179	+	0.750228i	$0.729944\pi$
$984$	0	0
$985$	−25.0566	−0.798368
$986$	−5.54866	−0.176705
$987$	0	0
$988$	14.6503	0.466087
$989$	10.3776	0.329989
$990$	0	0
$991$	5.48208	0.174144	0.0870720	−	0.996202i	$-0.472249\pi$
$991$	5.48208	0.174144	0.0870720	+	0.996202i	$0.472249\pi$
$992$	7.16922	0.227623
$993$	0	0
$994$	−2.20736	−0.0700133
$995$	−1.19521	−0.0378906
$996$	0	0
$997$	34.1456	1.08140	0.540700	−	0.841215i	$-0.318159\pi$
$997$	34.1456	1.08140	0.540700	+	0.841215i	$0.318159\pi$
$998$	1.55519	0.0492286
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2277.2.a.m.1.4		6
3.2	odd	2		253.2.a.d.1.3	✓	6
12.11	even	2		4048.2.a.bc.1.1		6
15.14	odd	2		6325.2.a.m.1.4		6
33.32	even	2		2783.2.a.h.1.4		6
69.68	even	2		5819.2.a.e.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.a.d.1.3	✓	6	3.2	odd	2
2277.2.a.m.1.4		6	1.1	even	1	trivial
2783.2.a.h.1.4		6	33.32	even	2
4048.2.a.bc.1.1		6	12.11	even	2
5819.2.a.e.1.3		6	69.68	even	2
6325.2.a.m.1.4		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2277 = 3^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2277.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0985770 q^{2} -1.99028 q^{4} -1.39335 q^{5} -2.57601 q^{7} +0.393350 q^{8} +0.137352 q^{10} -1.00000 q^{11} +4.27358 q^{13} +0.253935 q^{14} +3.94179 q^{16} +7.01179 q^{17} -1.72242 q^{19} +2.77316 q^{20} +0.0985770 q^{22} +1.00000 q^{23} -3.05858 q^{25} -0.421276 q^{26} +5.12698 q^{28} +8.02757 q^{29} -6.10007 q^{31} -1.17527 q^{32} -0.691201 q^{34} +3.58928 q^{35} -6.30964 q^{37} +0.169791 q^{38} -0.548074 q^{40} -4.29179 q^{41} +10.3776 q^{43} +1.99028 q^{44} -0.0985770 q^{46} -4.35702 q^{47} -0.364192 q^{49} +0.301505 q^{50} -8.50563 q^{52} +1.74607 q^{53} +1.39335 q^{55} -1.01327 q^{56} -0.791334 q^{58} -6.74400 q^{59} -1.19907 q^{61} +0.601326 q^{62} -7.76773 q^{64} -5.95459 q^{65} -4.78988 q^{67} -13.9554 q^{68} -0.353820 q^{70} -8.69264 q^{71} -5.16193 q^{73} +0.621985 q^{74} +3.42811 q^{76} +2.57601 q^{77} -7.94736 q^{79} -5.49229 q^{80} +0.423071 q^{82} -1.58243 q^{83} -9.76987 q^{85} -1.02299 q^{86} -0.393350 q^{88} -5.85263 q^{89} -11.0088 q^{91} -1.99028 q^{92} +0.429502 q^{94} +2.39994 q^{95} +12.8485 q^{97} +0.0359010 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 5 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - 6 q^{10} - 6 q^{11} - 3 q^{13} + 8 q^{14} - q^{16} - 5 q^{17} + q^{19} + 7 q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{25} - 15 q^{26} - 6 q^{29} - 8 q^{31}+ \cdots - 34 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 5 * q^4 - 3 * q^5 - q^7 - 3 * q^8 - 6 * q^10 - 6 * q^11 - 3 * q^13 + 8 * q^14 - q^16 - 5 * q^17 + q^19 + 7 * q^20 + 3 * q^22 + 6 * q^23 + 3 * q^25 - 15 * q^26 - 6 * q^29 - 8 * q^31 + 6 * q^32 - 6 * q^34 - 12 * q^35 + 2 * q^37 - 9 * q^38 - 30 * q^40 + 2 * q^41 + 10 * q^43 - 5 * q^44 - 3 * q^46 - 14 * q^47 - 5 * q^49 - 9 * q^50 - 11 * q^52 + 4 * q^53 + 3 * q^55 + 13 * q^56 - 6 * q^58 - 39 * q^59 - 22 * q^61 - 33 * q^64 + 16 * q^65 + 8 * q^67 - 17 * q^68 + 19 * q^70 - 18 * q^71 - 9 * q^73 - 15 * q^74 - 25 * q^76 + q^77 - 15 * q^79 + 16 * q^80 - 8 * q^82 - 43 * q^83 - 21 * q^85 + 55 * q^86 + 3 * q^88 - 17 * q^89 - 9 * q^91 + 5 * q^92 + 63 * q^94 + 21 * q^95 + 19 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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