LMFDB - Embedded newform 2277.2.a.m.1.6

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			1.6
Root			$3.14649$ 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)

$f(q)$	$=$	\(q+2.14649 q^{2} +2.60742 q^{4} -2.30383 q^{5} +1.71408 q^{7} +1.30383 q^{8} -4.94515 q^{10} -1.00000 q^{11} -5.20211 q^{13} +3.67927 q^{14} -2.41619 q^{16} -2.40321 q^{17} -5.97011 q^{19} -6.00707 q^{20} -2.14649 q^{22} +1.00000 q^{23} +0.307640 q^{25} -11.1663 q^{26} +4.46935 q^{28} -5.57442 q^{29} +6.55851 q^{31} -7.79399 q^{32} -5.15847 q^{34} -3.94896 q^{35} -0.451431 q^{37} -12.8148 q^{38} -3.00381 q^{40} -0.0529225 q^{41} +6.61909 q^{43} -2.60742 q^{44} +2.14649 q^{46} +12.1993 q^{47} -4.06191 q^{49} +0.660347 q^{50} -13.5641 q^{52} -1.67927 q^{53} +2.30383 q^{55} +2.23488 q^{56} -11.9654 q^{58} -0.751014 q^{59} -12.5170 q^{61} +14.0778 q^{62} -11.8973 q^{64} +11.9848 q^{65} +4.92036 q^{67} -6.26620 q^{68} -8.47641 q^{70} +4.97794 q^{71} +4.33662 q^{73} -0.968992 q^{74} -15.5666 q^{76} -1.71408 q^{77} -14.1995 q^{79} +5.56649 q^{80} -0.113598 q^{82} -11.9442 q^{83} +5.53660 q^{85} +14.2078 q^{86} -1.30383 q^{88} -7.20560 q^{89} -8.91687 q^{91} +2.60742 q^{92} +26.1857 q^{94} +13.7541 q^{95} -0.228005 q^{97} -8.71886 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.14649	1.51780	0.758899	−	0.651208i	$-0.225737\pi$
$2$	2.14649	1.51780	0.758899	+	0.651208i	$0.225737\pi$
$3$	0	0
$3$	0	0
$4$	2.60742	1.30371
$5$	−2.30383	−1.03030	−0.515152	−	0.857099i	$-0.672265\pi$
$5$	−2.30383	−1.03030	−0.515152	+	0.857099i	$0.672265\pi$
$6$	0	0
$7$	1.71408	0.647863	0.323932	−	0.946081i	$-0.394995\pi$
$7$	1.71408	0.647863	0.323932	+	0.946081i	$0.394995\pi$
$8$	1.30383	0.460974
$9$	0	0
$10$	−4.94515	−1.56380
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.20211	−1.44281	−0.721403	−	0.692515i	$-0.756503\pi$
$13$	−5.20211	−1.44281	−0.721403	+	0.692515i	$0.756503\pi$
$14$	3.67927	0.983326
$15$	0	0
$16$	−2.41619	−0.604047
$17$	−2.40321	−0.582865	−0.291432	−	0.956591i	$-0.594132\pi$
$17$	−2.40321	−0.582865	−0.291432	+	0.956591i	$0.594132\pi$
$18$	0	0
$19$	−5.97011	−1.36964	−0.684819	−	0.728713i	$-0.740119\pi$
$19$	−5.97011	−1.36964	−0.684819	+	0.728713i	$0.740119\pi$
$20$	−6.00707	−1.34322
$21$	0	0
$22$	−2.14649	−0.457633
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.307640	0.0615281
$26$	−11.1663	−2.18989
$27$	0	0
$28$	4.46935	0.844627
$29$	−5.57442	−1.03514	−0.517572	−	0.855640i	$-0.673164\pi$
$29$	−5.57442	−1.03514	−0.517572	+	0.855640i	$0.673164\pi$
$30$	0	0
$31$	6.55851	1.17794	0.588972	−	0.808154i	$-0.299533\pi$
$31$	6.55851	1.17794	0.588972	+	0.808154i	$0.299533\pi$
$32$	−7.79399	−1.37779
$33$	0	0
$34$	−5.15847	−0.884671
$35$	−3.94896	−0.667497
$36$	0	0
$37$	−0.451431	−0.0742147	−0.0371074	−	0.999311i	$-0.511814\pi$
$37$	−0.451431	−0.0742147	−0.0371074	+	0.999311i	$0.511814\pi$
$38$	−12.8148	−2.07883
$39$	0	0
$40$	−3.00381	−0.474944
$41$	−0.0529225	−0.00826510	−0.00413255	−	0.999991i	$-0.501315\pi$
$41$	−0.0529225	−0.00826510	−0.00413255	+	0.999991i	$0.501315\pi$
$42$	0	0
$43$	6.61909	1.00940	0.504701	−	0.863294i	$-0.331603\pi$
$43$	6.61909	1.00940	0.504701	+	0.863294i	$0.331603\pi$
$44$	−2.60742	−0.393084
$45$	0	0
$46$	2.14649	0.316483
$47$	12.1993	1.77945	0.889727	−	0.456493i	$-0.150895\pi$
$47$	12.1993	1.77945	0.889727	+	0.456493i	$0.150895\pi$
$48$	0	0
$49$	−4.06191	−0.580273
$50$	0.660347	0.0933872
$51$	0	0
$52$	−13.5641	−1.88101
$53$	−1.67927	−0.230665	−0.115333	−	0.993327i	$-0.536793\pi$
$53$	−1.67927	−0.230665	−0.115333	+	0.993327i	$0.536793\pi$
$54$	0	0
$55$	2.30383	0.310649
$56$	2.23488	0.298648
$57$	0	0
$58$	−11.9654	−1.57114
$59$	−0.751014	−0.0977737	−0.0488868	−	0.998804i	$-0.515567\pi$
$59$	−0.751014	−0.0977737	−0.0488868	+	0.998804i	$0.515567\pi$
$60$	0	0
$61$	−12.5170	−1.60264	−0.801320	−	0.598237i	$-0.795868\pi$
$61$	−12.5170	−1.60264	−0.801320	+	0.598237i	$0.795868\pi$
$62$	14.0778	1.78788
$63$	0	0
$64$	−11.8973	−1.48717
$65$	11.9848	1.48653
$66$	0	0
$67$	4.92036	0.601118	0.300559	−	0.953763i	$-0.402827\pi$
$67$	4.92036	0.601118	0.300559	+	0.953763i	$0.402827\pi$
$68$	−6.26620	−0.759888
$69$	0	0
$70$	−8.47641	−1.01313
$71$	4.97794	0.590773	0.295387	−	0.955378i	$-0.404552\pi$
$71$	4.97794	0.590773	0.295387	+	0.955378i	$0.404552\pi$
$72$	0	0
$73$	4.33662	0.507563	0.253782	−	0.967262i	$-0.418326\pi$
$73$	4.33662	0.507563	0.253782	+	0.967262i	$0.418326\pi$
$74$	−0.968992	−0.112643
$75$	0	0
$76$	−15.5666	−1.78561
$77$	−1.71408	−0.195338
$78$	0	0
$79$	−14.1995	−1.59757	−0.798785	−	0.601617i	$-0.794523\pi$
$79$	−14.1995	−1.59757	−0.798785	+	0.601617i	$0.794523\pi$
$80$	5.56649	0.622352
$81$	0	0
$82$	−0.113598	−0.0125448
$83$	−11.9442	−1.31104	−0.655521	−	0.755177i	$-0.727551\pi$
$83$	−11.9442	−1.31104	−0.655521	+	0.755177i	$0.727551\pi$
$84$	0	0
$85$	5.53660	0.600528
$86$	14.2078	1.53207
$87$	0	0
$88$	−1.30383	−0.138989
$89$	−7.20560	−0.763792	−0.381896	−	0.924205i	$-0.624729\pi$
$89$	−7.20560	−0.763792	−0.381896	+	0.924205i	$0.624729\pi$
$90$	0	0
$91$	−8.91687	−0.934742
$92$	2.60742	0.271843
$93$	0	0
$94$	26.1857	2.70085
$95$	13.7541	1.41114
$96$	0	0
$97$	−0.228005	−0.0231504	−0.0115752	−	0.999933i	$-0.503685\pi$
$97$	−0.228005	−0.0231504	−0.0115752	+	0.999933i	$0.503685\pi$
$98$	−8.71886	−0.880738
$99$	0	0
$100$	0.802149	0.0802149
$101$	−11.9772	−1.19177	−0.595887	−	0.803068i	$-0.703199\pi$
$101$	−11.9772	−1.19177	−0.595887	+	0.803068i	$0.703199\pi$
$102$	0	0
$103$	18.3463	1.80771	0.903855	−	0.427839i	$-0.140725\pi$
$103$	18.3463	1.80771	0.903855	+	0.427839i	$0.140725\pi$
$104$	−6.78268	−0.665097
$105$	0	0
$106$	−3.60453	−0.350103
$107$	3.13620	0.303188	0.151594	−	0.988443i	$-0.451559\pi$
$107$	3.13620	0.303188	0.151594	+	0.988443i	$0.451559\pi$
$108$	0	0
$109$	−12.7472	−1.22096	−0.610480	−	0.792032i	$-0.709023\pi$
$109$	−12.7472	−1.22096	−0.610480	+	0.792032i	$0.709023\pi$
$110$	4.94515	0.471502
$111$	0	0
$112$	−4.14155	−0.391340
$113$	9.89636	0.930971	0.465486	−	0.885055i	$-0.345880\pi$
$113$	9.89636	0.930971	0.465486	+	0.885055i	$0.345880\pi$
$114$	0	0
$115$	−2.30383	−0.214833
$116$	−14.5349	−1.34953
$117$	0	0
$118$	−1.61204	−0.148401
$119$	−4.11931	−0.377617
$120$	0	0
$121$	1.00000	0.0909091
$122$	−26.8677	−2.43248
$123$	0	0
$124$	17.1008	1.53570
$125$	10.8104	0.966912
$126$	0	0
$127$	−13.8315	−1.22734	−0.613672	−	0.789561i	$-0.710308\pi$
$127$	−13.8315	−1.22734	−0.613672	+	0.789561i	$0.710308\pi$
$128$	−9.94959	−0.879427
$129$	0	0
$130$	25.7253	2.25625
$131$	−1.83583	−0.160397	−0.0801986	−	0.996779i	$-0.525555\pi$
$131$	−1.83583	−0.160397	−0.0801986	+	0.996779i	$0.525555\pi$
$132$	0	0
$133$	−10.2333	−0.887338
$134$	10.5615	0.912376
$135$	0	0
$136$	−3.13338	−0.268686
$137$	2.65444	0.226784	0.113392	−	0.993550i	$-0.463828\pi$
$137$	2.65444	0.226784	0.113392	+	0.993550i	$0.463828\pi$
$138$	0	0
$139$	−12.3258	−1.04546	−0.522729	−	0.852499i	$-0.675086\pi$
$139$	−12.3258	−1.04546	−0.522729	+	0.852499i	$0.675086\pi$
$140$	−10.2966	−0.870224
$141$	0	0
$142$	10.6851	0.896675
$143$	5.20211	0.435023
$144$	0	0
$145$	12.8425	1.06651
$146$	9.30852	0.770379
$147$	0	0
$148$	−1.17707	−0.0967547
$149$	0.00605095	0.000495713 0	0.000247857	−	1.00000i	$-0.499921\pi$
$149$	0.00605095	0.000495713 0	0.000247857		1.00000i	$0.499921\pi$
$150$	0	0
$151$	20.1394	1.63892	0.819461	−	0.573135i	$-0.194273\pi$
$151$	20.1394	1.63892	0.819461	+	0.573135i	$0.194273\pi$
$152$	−7.78402	−0.631367
$153$	0	0
$154$	−3.67927	−0.296484
$155$	−15.1097	−1.21364
$156$	0	0
$157$	−4.36068	−0.348020	−0.174010	−	0.984744i	$-0.555673\pi$
$157$	−4.36068	−0.348020	−0.174010	+	0.984744i	$0.555673\pi$
$158$	−30.4791	−2.42479
$159$	0	0
$160$	17.9560	1.41955
$161$	1.71408	0.135089
$162$	0	0
$163$	24.0218	1.88153	0.940765	−	0.339059i	$-0.110109\pi$
$163$	24.0218	1.88153	0.940765	+	0.339059i	$0.110109\pi$
$164$	−0.137991	−0.0107753
$165$	0	0
$166$	−25.6380	−1.98990
$167$	−10.7369	−0.830849	−0.415424	−	0.909628i	$-0.636367\pi$
$167$	−10.7369	−0.830849	−0.415424	+	0.909628i	$0.636367\pi$
$168$	0	0
$169$	14.0620	1.08169
$170$	11.8843	0.911481
$171$	0	0
$172$	17.2588	1.31597
$173$	−16.4113	−1.24773	−0.623864	−	0.781533i	$-0.714438\pi$
$173$	−16.4113	−1.24773	−0.623864	+	0.781533i	$0.714438\pi$
$174$	0	0
$175$	0.527322	0.0398618
$176$	2.41619	0.182127
$177$	0	0
$178$	−15.4668	−1.15928
$179$	7.59999	0.568050	0.284025	−	0.958817i	$-0.408330\pi$
$179$	7.59999	0.568050	0.284025	+	0.958817i	$0.408330\pi$
$180$	0	0
$181$	−7.93248	−0.589616	−0.294808	−	0.955556i	$-0.595256\pi$
$181$	−7.93248	−0.589616	−0.294808	+	0.955556i	$0.595256\pi$
$182$	−19.1400	−1.41875
$183$	0	0
$184$	1.30383	0.0961197
$185$	1.04002	0.0764638
$186$	0	0
$187$	2.40321	0.175740
$188$	31.8088	2.31990
$189$	0	0
$190$	29.5231	2.14183
$191$	19.7120	1.42631	0.713155	−	0.701006i	$-0.247266\pi$
$191$	19.7120	1.42631	0.713155	+	0.701006i	$0.247266\pi$
$192$	0	0
$193$	0.995322	0.0716449	0.0358224	−	0.999358i	$-0.488595\pi$
$193$	0.995322	0.0716449	0.0358224	+	0.999358i	$0.488595\pi$
$194$	−0.489411	−0.0351377
$195$	0	0
$196$	−10.5911	−0.756509
$197$	16.3485	1.16478	0.582392	−	0.812908i	$-0.302117\pi$
$197$	16.3485	1.16478	0.582392	+	0.812908i	$0.302117\pi$
$198$	0	0
$199$	−24.5484	−1.74019	−0.870096	−	0.492883i	$-0.835943\pi$
$199$	−24.5484	−1.74019	−0.870096	+	0.492883i	$0.835943\pi$
$200$	0.401111	0.0283628
$201$	0	0
$202$	−25.7089	−1.80887
$203$	−9.55503	−0.670632
$204$	0	0
$205$	0.121924	0.00851558
$206$	39.3801	2.74374
$207$	0	0
$208$	12.5693	0.871523
$209$	5.97011	0.412961
$210$	0	0
$211$	25.8454	1.77927	0.889635	−	0.456673i	$-0.150959\pi$
$211$	25.8454	1.77927	0.889635	+	0.456673i	$0.150959\pi$
$212$	−4.37857	−0.300721
$213$	0	0
$214$	6.73183	0.460179
$215$	−15.2493	−1.03999
$216$	0	0
$217$	11.2418	0.763146
$218$	−27.3617	−1.85317
$219$	0	0
$220$	6.00707	0.404996
$221$	12.5018	0.840961
$222$	0	0
$223$	26.7181	1.78918	0.894589	−	0.446890i	$-0.147468\pi$
$223$	26.7181	1.78918	0.894589	+	0.446890i	$0.147468\pi$
$224$	−13.3596	−0.892623
$225$	0	0
$226$	21.2424	1.41303
$227$	−26.2689	−1.74353	−0.871764	−	0.489925i	$-0.837024\pi$
$227$	−26.2689	−1.74353	−0.871764	+	0.489925i	$0.837024\pi$
$228$	0	0
$229$	−3.52787	−0.233128	−0.116564	−	0.993183i	$-0.537188\pi$
$229$	−3.52787	−0.233128	−0.116564	+	0.993183i	$0.537188\pi$
$230$	−4.94515	−0.326074
$231$	0	0
$232$	−7.26811	−0.477175
$233$	8.93025	0.585040	0.292520	−	0.956259i	$-0.405506\pi$
$233$	8.93025	0.585040	0.292520	+	0.956259i	$0.405506\pi$
$234$	0	0
$235$	−28.1052	−1.83338
$236$	−1.95821	−0.127469
$237$	0	0
$238$	−8.84206	−0.573146
$239$	−11.0369	−0.713918	−0.356959	−	0.934120i	$-0.616186\pi$
$239$	−11.0369	−0.713918	−0.356959	+	0.934120i	$0.616186\pi$
$240$	0	0
$241$	−22.2940	−1.43608	−0.718041	−	0.696000i	$-0.754961\pi$
$241$	−22.2940	−1.43608	−0.718041	+	0.696000i	$0.754961\pi$
$242$	2.14649	0.137982
$243$	0	0
$244$	−32.6372	−2.08938
$245$	9.35796	0.597858
$246$	0	0
$247$	31.0572	1.97612
$248$	8.55120	0.543001
$249$	0	0
$250$	23.2044	1.46758
$251$	4.15633	0.262345	0.131173	−	0.991360i	$-0.458126\pi$
$251$	4.15633	0.262345	0.131173	+	0.991360i	$0.458126\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	−29.6891	−1.86286
$255$	0	0
$256$	2.43800	0.152375
$257$	−25.5049	−1.59095	−0.795477	−	0.605984i	$-0.792779\pi$
$257$	−25.5049	−1.59095	−0.795477	+	0.605984i	$0.792779\pi$
$258$	0	0
$259$	−0.773790	−0.0480810
$260$	31.2495	1.93801
$261$	0	0
$262$	−3.94059	−0.243451
$263$	−9.02268	−0.556362	−0.278181	−	0.960529i	$-0.589732\pi$
$263$	−9.02268	−0.556362	−0.278181	+	0.960529i	$0.589732\pi$
$264$	0	0
$265$	3.86875	0.237656
$266$	−21.9656	−1.34680
$267$	0	0
$268$	12.8295	0.783685
$269$	20.1309	1.22741	0.613703	−	0.789537i	$-0.289679\pi$
$269$	20.1309	1.22741	0.613703	+	0.789537i	$0.289679\pi$
$270$	0	0
$271$	13.1224	0.797131	0.398565	−	0.917140i	$-0.369508\pi$
$271$	13.1224	0.797131	0.398565	+	0.917140i	$0.369508\pi$
$272$	5.80661	0.352077
$273$	0	0
$274$	5.69773	0.344213
$275$	−0.307640	−0.0185514
$276$	0	0
$277$	−8.77180	−0.527046	−0.263523	−	0.964653i	$-0.584885\pi$
$277$	−8.77180	−0.527046	−0.263523	+	0.964653i	$0.584885\pi$
$278$	−26.4572	−1.58680
$279$	0	0
$280$	−5.14878	−0.307699
$281$	5.97847	0.356646	0.178323	−	0.983972i	$-0.442933\pi$
$281$	5.97847	0.356646	0.178323	+	0.983972i	$0.442933\pi$
$282$	0	0
$283$	18.3711	1.09205	0.546024	−	0.837769i	$-0.316141\pi$
$283$	18.3711	1.09205	0.546024	+	0.837769i	$0.316141\pi$
$284$	12.9796	0.770199
$285$	0	0
$286$	11.1663	0.660277
$287$	−0.0907136	−0.00535466
$288$	0	0
$289$	−11.2246	−0.660269
$290$	27.5664	1.61875
$291$	0	0
$292$	11.3074	0.661716
$293$	24.4454	1.42812	0.714059	−	0.700086i	$-0.246855\pi$
$293$	24.4454	1.42812	0.714059	+	0.700086i	$0.246855\pi$
$294$	0	0
$295$	1.73021	0.100737
$296$	−0.588590	−0.0342111
$297$	0	0
$298$	0.0129883	0.000752393 0
$299$	−5.20211	−0.300846
$300$	0	0
$301$	11.3457	0.653954
$302$	43.2291	2.48755
$303$	0	0
$304$	14.4249	0.827325
$305$	28.8371	1.65121
$306$	0	0
$307$	27.6313	1.57700	0.788500	−	0.615035i	$-0.210858\pi$
$307$	27.6313	1.57700	0.788500	+	0.615035i	$0.210858\pi$
$308$	−4.46935	−0.254665
$309$	0	0
$310$	−32.4329	−1.84206
$311$	7.35608	0.417125	0.208563	−	0.978009i	$-0.433122\pi$
$311$	7.35608	0.417125	0.208563	+	0.978009i	$0.433122\pi$
$312$	0	0
$313$	3.38614	0.191396	0.0956981	−	0.995410i	$-0.469492\pi$
$313$	3.38614	0.191396	0.0956981	+	0.995410i	$0.469492\pi$
$314$	−9.36017	−0.528225
$315$	0	0
$316$	−37.0242	−2.08277
$317$	−14.5602	−0.817782	−0.408891	−	0.912583i	$-0.634084\pi$
$317$	−14.5602	−0.817782	−0.408891	+	0.912583i	$0.634084\pi$
$318$	0	0
$319$	5.57442	0.312108
$320$	27.4095	1.53224
$321$	0	0
$322$	3.67927	0.205038
$323$	14.3474	0.798313
$324$	0	0
$325$	−1.60038	−0.0887731
$326$	51.5625	2.85578
$327$	0	0
$328$	−0.0690020	−0.00381000
$329$	20.9107	1.15284
$330$	0	0
$331$	33.2753	1.82897	0.914487	−	0.404615i	$-0.132594\pi$
$331$	33.2753	1.82897	0.914487	+	0.404615i	$0.132594\pi$
$332$	−31.1435	−1.70922
$333$	0	0
$334$	−23.0467	−1.26106
$335$	−11.3357	−0.619335
$336$	0	0
$337$	−0.512573	−0.0279216	−0.0139608	−	0.999903i	$-0.504444\pi$
$337$	−0.512573	−0.0279216	−0.0139608	+	0.999903i	$0.504444\pi$
$338$	30.1840	1.64179
$339$	0	0
$340$	14.4363	0.782916
$341$	−6.55851	−0.355163
$342$	0	0
$343$	−18.9611	−1.02380
$344$	8.63017	0.465308
$345$	0	0
$346$	−35.2267	−1.89380
$347$	−0.949097	−0.0509502	−0.0254751	−	0.999675i	$-0.508110\pi$
$347$	−0.949097	−0.0509502	−0.0254751	+	0.999675i	$0.508110\pi$
$348$	0	0
$349$	−24.8145	−1.32829	−0.664145	−	0.747604i	$-0.731204\pi$
$349$	−24.8145	−1.32829	−0.664145	+	0.747604i	$0.731204\pi$
$350$	1.13189	0.0605021
$351$	0	0
$352$	7.79399	0.415421
$353$	−14.4798	−0.770680	−0.385340	−	0.922775i	$-0.625916\pi$
$353$	−14.4798	−0.770680	−0.385340	+	0.922775i	$0.625916\pi$
$354$	0	0
$355$	−11.4683	−0.608677
$356$	−18.7881	−0.995765
$357$	0	0
$358$	16.3133	0.862185
$359$	20.0668	1.05909	0.529543	−	0.848283i	$-0.322364\pi$
$359$	20.0668	1.05909	0.529543	+	0.848283i	$0.322364\pi$
$360$	0	0
$361$	16.6422	0.875907
$362$	−17.0270	−0.894919
$363$	0	0
$364$	−23.2501	−1.21863
$365$	−9.99085	−0.522945
$366$	0	0
$367$	−15.2983	−0.798564	−0.399282	−	0.916828i	$-0.630741\pi$
$367$	−15.2983	−0.798564	−0.399282	+	0.916828i	$0.630741\pi$
$368$	−2.41619	−0.125952
$369$	0	0
$370$	2.23239	0.116057
$371$	−2.87841	−0.149440
$372$	0	0
$373$	−10.6040	−0.549052	−0.274526	−	0.961580i	$-0.588521\pi$
$373$	−10.6040	−0.549052	−0.274526	+	0.961580i	$0.588521\pi$
$374$	5.15847	0.266738
$375$	0	0
$376$	15.9059	0.820282
$377$	28.9988	1.49351
$378$	0	0
$379$	−4.01160	−0.206062	−0.103031	−	0.994678i	$-0.532854\pi$
$379$	−4.01160	−0.206062	−0.103031	+	0.994678i	$0.532854\pi$
$380$	35.8629	1.83973
$381$	0	0
$382$	42.3116	2.16485
$383$	−29.4485	−1.50475	−0.752375	−	0.658735i	$-0.771092\pi$
$383$	−29.4485	−1.50475	−0.752375	+	0.658735i	$0.771092\pi$
$384$	0	0
$385$	3.94896	0.201258
$386$	2.13645	0.108742
$387$	0	0
$388$	−0.594506	−0.0301815
$389$	−35.0463	−1.77692	−0.888460	−	0.458954i	$-0.848224\pi$
$389$	−35.0463	−1.77692	−0.888460	+	0.458954i	$0.848224\pi$
$390$	0	0
$391$	−2.40321	−0.121536
$392$	−5.29605	−0.267491
$393$	0	0
$394$	35.0920	1.76791
$395$	32.7133	1.64598
$396$	0	0
$397$	−19.5343	−0.980399	−0.490199	−	0.871610i	$-0.663076\pi$
$397$	−19.5343	−0.980399	−0.490199	+	0.871610i	$0.663076\pi$
$398$	−52.6930	−2.64126
$399$	0	0
$400$	−0.743316	−0.0371658
$401$	−29.2730	−1.46183	−0.730913	−	0.682471i	$-0.760905\pi$
$401$	−29.2730	−1.46183	−0.730913	+	0.682471i	$0.760905\pi$
$402$	0	0
$403$	−34.1181	−1.69955
$404$	−31.2296	−1.55373
$405$	0	0
$406$	−20.5098	−1.01788
$407$	0.451431	0.0223766
$408$	0	0
$409$	−6.31960	−0.312484	−0.156242	−	0.987719i	$-0.549938\pi$
$409$	−6.31960	−0.312484	−0.156242	+	0.987719i	$0.549938\pi$
$410$	0.261710	0.0129249
$411$	0	0
$412$	47.8365	2.35673
$413$	−1.28730	−0.0633440
$414$	0	0
$415$	27.5173	1.35077
$416$	40.5452	1.98789
$417$	0	0
$418$	12.8148	0.626792
$419$	−19.6098	−0.958003	−0.479001	−	0.877814i	$-0.659001\pi$
$419$	−19.6098	−0.958003	−0.479001	+	0.877814i	$0.659001\pi$
$420$	0	0
$421$	0.413874	0.0201710	0.0100855	−	0.999949i	$-0.496790\pi$
$421$	0.413874	0.0201710	0.0100855	+	0.999949i	$0.496790\pi$
$422$	55.4769	2.70057
$423$	0	0
$424$	−2.18948	−0.106331
$425$	−0.739325	−0.0358625
$426$	0	0
$427$	−21.4552	−1.03829
$428$	8.17741	0.395270
$429$	0	0
$430$	−32.7324	−1.57850
$431$	−13.4015	−0.645528	−0.322764	−	0.946479i	$-0.604612\pi$
$431$	−13.4015	−0.645528	−0.322764	+	0.946479i	$0.604612\pi$
$432$	0	0
$433$	−22.5136	−1.08194	−0.540968	−	0.841043i	$-0.681942\pi$
$433$	−22.5136	−1.08194	−0.540968	+	0.841043i	$0.681942\pi$
$434$	24.1305	1.15830
$435$	0	0
$436$	−33.2374	−1.59178
$437$	−5.97011	−0.285589
$438$	0	0
$439$	11.1902	0.534081	0.267041	−	0.963685i	$-0.413954\pi$
$439$	11.1902	0.534081	0.267041	+	0.963685i	$0.413954\pi$
$440$	3.00381	0.143201
$441$	0	0
$442$	26.8350	1.27641
$443$	−20.5087	−0.974397	−0.487199	−	0.873291i	$-0.661981\pi$
$443$	−20.5087	−0.974397	−0.487199	+	0.873291i	$0.661981\pi$
$444$	0	0
$445$	16.6005	0.786939
$446$	57.3502	2.71561
$447$	0	0
$448$	−20.3931	−0.963482
$449$	−10.8036	−0.509853	−0.254927	−	0.966960i	$-0.582051\pi$
$449$	−10.8036	−0.509853	−0.254927	+	0.966960i	$0.582051\pi$
$450$	0	0
$451$	0.0529225	0.00249202
$452$	25.8040	1.21372
$453$	0	0
$454$	−56.3860	−2.64633
$455$	20.5430	0.963069
$456$	0	0
$457$	−17.4139	−0.814590	−0.407295	−	0.913297i	$-0.633528\pi$
$457$	−17.4139	−0.814590	−0.407295	+	0.913297i	$0.633528\pi$
$458$	−7.57254	−0.353841
$459$	0	0
$460$	−6.00707	−0.280081
$461$	−21.5017	−1.00143	−0.500717	−	0.865611i	$-0.666930\pi$
$461$	−21.5017	−1.00143	−0.500717	+	0.865611i	$0.666930\pi$
$462$	0	0
$463$	8.61269	0.400266	0.200133	−	0.979769i	$-0.435863\pi$
$463$	8.61269	0.400266	0.200133	+	0.979769i	$0.435863\pi$
$464$	13.4688	0.625275
$465$	0	0
$466$	19.1687	0.887973
$467$	−21.4571	−0.992917	−0.496458	−	0.868061i	$-0.665366\pi$
$467$	−21.4571	−0.992917	−0.496458	+	0.868061i	$0.665366\pi$
$468$	0	0
$469$	8.43392	0.389442
$470$	−60.3275	−2.78270
$471$	0	0
$472$	−0.979196	−0.0450711
$473$	−6.61909	−0.304346
$474$	0	0
$475$	−1.83665	−0.0842711
$476$	−10.7408	−0.492303
$477$	0	0
$478$	−23.6906	−1.08358
$479$	−1.05146	−0.0480425	−0.0240213	−	0.999711i	$-0.507647\pi$
$479$	−1.05146	−0.0480425	−0.0240213	+	0.999711i	$0.507647\pi$
$480$	0	0
$481$	2.34839	0.107078
$482$	−47.8539	−2.17968
$483$	0	0
$484$	2.60742	0.118519
$485$	0.525285	0.0238520
$486$	0	0
$487$	10.9677	0.496995	0.248497	−	0.968633i	$-0.420063\pi$
$487$	10.9677	0.496995	0.248497	+	0.968633i	$0.420063\pi$
$488$	−16.3201	−0.738775
$489$	0	0
$490$	20.0868	0.907429
$491$	25.4780	1.14981	0.574904	−	0.818221i	$-0.305040\pi$
$491$	25.4780	1.14981	0.574904	+	0.818221i	$0.305040\pi$
$492$	0	0
$493$	13.3965	0.603349
$494$	66.6640	2.99936
$495$	0	0
$496$	−15.8466	−0.711533
$497$	8.53262	0.382740
$498$	0	0
$499$	−17.9068	−0.801618	−0.400809	−	0.916162i	$-0.631271\pi$
$499$	−17.9068	−0.801618	−0.400809	+	0.916162i	$0.631271\pi$
$500$	28.1873	1.26058
$501$	0	0
$502$	8.92152	0.398187
$503$	12.9389	0.576916	0.288458	−	0.957493i	$-0.406857\pi$
$503$	12.9389	0.576916	0.288458	+	0.957493i	$0.406857\pi$
$504$	0	0
$505$	27.5934	1.22789
$506$	−2.14649	−0.0954232
$507$	0	0
$508$	−36.0645	−1.60010
$509$	−12.7018	−0.562998	−0.281499	−	0.959562i	$-0.590832\pi$
$509$	−12.7018	−0.562998	−0.281499	+	0.959562i	$0.590832\pi$
$510$	0	0
$511$	7.43334	0.328832
$512$	25.1323	1.11070
$513$	0	0
$514$	−54.7461	−2.41475
$515$	−42.2667	−1.86249
$516$	0	0
$517$	−12.1993	−0.536526
$518$	−1.66093	−0.0729773
$519$	0	0
$520$	15.6262	0.685252
$521$	−31.4279	−1.37688	−0.688440	−	0.725293i	$-0.741704\pi$
$521$	−31.4279	−1.37688	−0.688440	+	0.725293i	$0.741704\pi$
$522$	0	0
$523$	−28.6135	−1.25118	−0.625590	−	0.780152i	$-0.715142\pi$
$523$	−28.6135	−1.25118	−0.625590	+	0.780152i	$0.715142\pi$
$524$	−4.78679	−0.209112
$525$	0	0
$526$	−19.3671	−0.844446
$527$	−15.7615	−0.686582
$528$	0	0
$529$	1.00000	0.0434783
$530$	8.30424	0.360713
$531$	0	0
$532$	−26.6825	−1.15683
$533$	0.275309	0.0119249
$534$	0	0
$535$	−7.22529	−0.312376
$536$	6.41533	0.277100
$537$	0	0
$538$	43.2109	1.86295
$539$	4.06191	0.174959
$540$	0	0
$541$	34.7309	1.49320	0.746600	−	0.665273i	$-0.231685\pi$
$541$	34.7309	1.49320	0.746600	+	0.665273i	$0.231685\pi$
$542$	28.1672	1.20988
$543$	0	0
$544$	18.7306	0.803068
$545$	29.3674	1.25796
$546$	0	0
$547$	−12.2468	−0.523636	−0.261818	−	0.965117i	$-0.584322\pi$
$547$	−12.2468	−0.523636	−0.261818	+	0.965117i	$0.584322\pi$
$548$	6.92126	0.295661
$549$	0	0
$550$	−0.660347	−0.0281573
$551$	33.2799	1.41777
$552$	0	0
$553$	−24.3392	−1.03501
$554$	−18.8286	−0.799950
$555$	0	0
$556$	−32.1385	−1.36298
$557$	−7.51356	−0.318360	−0.159180	−	0.987250i	$-0.550885\pi$
$557$	−7.51356	−0.318360	−0.159180	+	0.987250i	$0.550885\pi$
$558$	0	0
$559$	−34.4332	−1.45637
$560$	9.54143	0.403199
$561$	0	0
$562$	12.8327	0.541316
$563$	−6.44946	−0.271812	−0.135906	−	0.990722i	$-0.543395\pi$
$563$	−6.44946	−0.271812	−0.135906	+	0.990722i	$0.543395\pi$
$564$	0	0
$565$	−22.7995	−0.959184
$566$	39.4334	1.65751
$567$	0	0
$568$	6.49040	0.272331
$569$	−6.42317	−0.269273	−0.134637	−	0.990895i	$-0.542987\pi$
$569$	−6.42317	−0.269273	−0.134637	+	0.990895i	$0.542987\pi$
$570$	0	0
$571$	32.5196	1.36090	0.680450	−	0.732794i	$-0.261784\pi$
$571$	32.5196	1.36090	0.680450	+	0.732794i	$0.261784\pi$
$572$	13.5641	0.567144
$573$	0	0
$574$	−0.194716	−0.00812729
$575$	0.307640	0.0128295
$576$	0	0
$577$	29.3541	1.22203	0.611014	−	0.791619i	$-0.290762\pi$
$577$	29.3541	1.22203	0.611014	+	0.791619i	$0.290762\pi$
$578$	−24.0934	−1.00215
$579$	0	0
$580$	33.4859	1.39043
$581$	−20.4733	−0.849376
$582$	0	0
$583$	1.67927	0.0695482
$584$	5.65423	0.233973
$585$	0	0
$586$	52.4719	2.16759
$587$	−27.9525	−1.15372	−0.576862	−	0.816841i	$-0.695723\pi$
$587$	−27.9525	−1.15372	−0.576862	+	0.816841i	$0.695723\pi$
$588$	0	0
$589$	−39.1550	−1.61336
$590$	3.71388	0.152898
$591$	0	0
$592$	1.09074	0.0448292
$593$	−8.65121	−0.355263	−0.177631	−	0.984097i	$-0.556843\pi$
$593$	−8.65121	−0.355263	−0.177631	+	0.984097i	$0.556843\pi$
$594$	0	0
$595$	9.49020	0.389060
$596$	0.0157774	0.000646268 0
$597$	0	0
$598$	−11.1663	−0.456624
$599$	42.9367	1.75435	0.877173	−	0.480174i	$-0.159427\pi$
$599$	42.9367	1.75435	0.877173	+	0.480174i	$0.159427\pi$
$600$	0	0
$601$	18.7697	0.765633	0.382816	−	0.923824i	$-0.374954\pi$
$601$	18.7697	0.765633	0.382816	+	0.923824i	$0.374954\pi$
$602$	24.3534	0.992570
$603$	0	0
$604$	52.5120	2.13668
$605$	−2.30383	−0.0936641
$606$	0	0
$607$	−9.77720	−0.396844	−0.198422	−	0.980117i	$-0.563582\pi$
$607$	−9.77720	−0.396844	−0.198422	+	0.980117i	$0.563582\pi$
$608$	46.5310	1.88708
$609$	0	0
$610$	61.8986	2.50620
$611$	−63.4623	−2.56741
$612$	0	0
$613$	−15.6379	−0.631611	−0.315805	−	0.948824i	$-0.602275\pi$
$613$	−15.6379	−0.631611	−0.315805	+	0.948824i	$0.602275\pi$
$614$	59.3103	2.39357
$615$	0	0
$616$	−2.23488	−0.0900458
$617$	−1.08731	−0.0437733	−0.0218866	−	0.999760i	$-0.506967\pi$
$617$	−1.08731	−0.0437733	−0.0218866	+	0.999760i	$0.506967\pi$
$618$	0	0
$619$	21.4376	0.861651	0.430825	−	0.902435i	$-0.358222\pi$
$619$	21.4376	0.861651	0.430825	+	0.902435i	$0.358222\pi$
$620$	−39.3974	−1.58224
$621$	0	0
$622$	15.7898	0.633112
$623$	−12.3510	−0.494833
$624$	0	0
$625$	−26.4436	−1.05774
$626$	7.26833	0.290501
$627$	0	0
$628$	−11.3702	−0.453718
$629$	1.08488	0.0432572
$630$	0	0
$631$	−31.9020	−1.27000	−0.634998	−	0.772513i	$-0.718999\pi$
$631$	−31.9020	−1.27000	−0.634998	+	0.772513i	$0.718999\pi$
$632$	−18.5138	−0.736438
$633$	0	0
$634$	−31.2533	−1.24123
$635$	31.8654	1.26454
$636$	0	0
$637$	21.1305	0.837222
$638$	11.9654	0.473717
$639$	0	0
$640$	22.9222	0.906078
$641$	−20.8702	−0.824322	−0.412161	−	0.911111i	$-0.635226\pi$
$641$	−20.8702	−0.824322	−0.412161	+	0.911111i	$0.635226\pi$
$642$	0	0
$643$	3.35892	0.132463	0.0662314	−	0.997804i	$-0.478902\pi$
$643$	3.35892	0.132463	0.0662314	+	0.997804i	$0.478902\pi$
$644$	4.46935	0.176117
$645$	0	0
$646$	30.7967	1.21168
$647$	−15.4981	−0.609294	−0.304647	−	0.952465i	$-0.598538\pi$
$647$	−15.4981	−0.609294	−0.304647	+	0.952465i	$0.598538\pi$
$648$	0	0
$649$	0.751014	0.0294799
$650$	−3.43520	−0.134740
$651$	0	0
$652$	62.6350	2.45297
$653$	28.2669	1.10617	0.553085	−	0.833125i	$-0.313450\pi$
$653$	28.2669	1.10617	0.553085	+	0.833125i	$0.313450\pi$
$654$	0	0
$655$	4.22945	0.165258
$656$	0.127871	0.00499251
$657$	0	0
$658$	44.8846	1.74978
$659$	−33.7629	−1.31522	−0.657608	−	0.753360i	$-0.728432\pi$
$659$	−33.7629	−1.31522	−0.657608	+	0.753360i	$0.728432\pi$
$660$	0	0
$661$	28.8495	1.12212	0.561058	−	0.827776i	$-0.310394\pi$
$661$	28.8495	1.12212	0.561058	+	0.827776i	$0.310394\pi$
$662$	71.4251	2.77601
$663$	0	0
$664$	−15.5732	−0.604357
$665$	23.5757	0.914228
$666$	0	0
$667$	−5.57442	−0.215842
$668$	−27.9957	−1.08319
$669$	0	0
$670$	−24.3320	−0.940026
$671$	12.5170	0.483214
$672$	0	0
$673$	−41.8888	−1.61469	−0.807347	−	0.590076i	$-0.799098\pi$
$673$	−41.8888	−1.61469	−0.807347	+	0.590076i	$0.799098\pi$
$674$	−1.10023	−0.0423794
$675$	0	0
$676$	36.6656	1.41022
$677$	23.3644	0.897967	0.448984	−	0.893540i	$-0.351786\pi$
$677$	23.3644	0.897967	0.448984	+	0.893540i	$0.351786\pi$
$678$	0	0
$679$	−0.390820	−0.0149983
$680$	7.21879	0.276828
$681$	0	0
$682$	−14.0778	−0.539066
$683$	−16.1543	−0.618128	−0.309064	−	0.951041i	$-0.600016\pi$
$683$	−16.1543	−0.618128	−0.309064	+	0.951041i	$0.600016\pi$
$684$	0	0
$685$	−6.11539	−0.233657
$686$	−40.6997	−1.55392
$687$	0	0
$688$	−15.9929	−0.609725
$689$	8.73575	0.332805
$690$	0	0
$691$	−17.9699	−0.683606	−0.341803	−	0.939772i	$-0.611038\pi$
$691$	−17.9699	−0.683606	−0.341803	+	0.939772i	$0.611038\pi$
$692$	−42.7912	−1.62668
$693$	0	0
$694$	−2.03723	−0.0773321
$695$	28.3965	1.07714
$696$	0	0
$697$	0.127184	0.00481744
$698$	−53.2641	−2.01608
$699$	0	0
$700$	1.37495	0.0519683
$701$	8.99333	0.339673	0.169837	−	0.985472i	$-0.445676\pi$
$701$	8.99333	0.339673	0.169837	+	0.985472i	$0.445676\pi$
$702$	0	0
$703$	2.69509	0.101647
$704$	11.8973	0.448398
$705$	0	0
$706$	−31.0807	−1.16974
$707$	−20.5299	−0.772106
$708$	0	0
$709$	−7.10172	−0.266711	−0.133355	−	0.991068i	$-0.542575\pi$
$709$	−7.10172	−0.266711	−0.133355	+	0.991068i	$0.542575\pi$
$710$	−24.6167	−0.923849
$711$	0	0
$712$	−9.39489	−0.352088
$713$	6.55851	0.245618
$714$	0	0
$715$	−11.9848	−0.448206
$716$	19.8164	0.740574
$717$	0	0
$718$	43.0732	1.60748
$719$	−17.6914	−0.659776	−0.329888	−	0.944020i	$-0.607011\pi$
$719$	−17.6914	−0.659776	−0.329888	+	0.944020i	$0.607011\pi$
$720$	0	0
$721$	31.4470	1.17115
$722$	35.7224	1.32945
$723$	0	0
$724$	−20.6833	−0.768690
$725$	−1.71492	−0.0636904
$726$	0	0
$727$	−42.4019	−1.57260	−0.786299	−	0.617846i	$-0.788006\pi$
$727$	−42.4019	−1.57260	−0.786299	+	0.617846i	$0.788006\pi$
$728$	−11.6261	−0.430892
$729$	0	0
$730$	−21.4453	−0.793725
$731$	−15.9071	−0.588344
$732$	0	0
$733$	6.44691	0.238122	0.119061	−	0.992887i	$-0.462012\pi$
$733$	6.44691	0.238122	0.119061	+	0.992887i	$0.462012\pi$
$734$	−32.8376	−1.21206
$735$	0	0
$736$	−7.79399	−0.287290
$737$	−4.92036	−0.181244
$738$	0	0
$739$	9.24032	0.339911	0.169955	−	0.985452i	$-0.445638\pi$
$739$	9.24032	0.339911	0.169955	+	0.985452i	$0.445638\pi$
$740$	2.71177	0.0996868
$741$	0	0
$742$	−6.17848	−0.226819
$743$	−3.52426	−0.129292	−0.0646462	−	0.997908i	$-0.520592\pi$
$743$	−3.52426	−0.129292	−0.0646462	+	0.997908i	$0.520592\pi$
$744$	0	0
$745$	−0.0139404	−0.000510736 0
$746$	−22.7613	−0.833350
$747$	0	0
$748$	6.26620	0.229115
$749$	5.37572	0.196425
$750$	0	0
$751$	13.1540	0.479996	0.239998	−	0.970773i	$-0.422853\pi$
$751$	13.1540	0.479996	0.239998	+	0.970773i	$0.422853\pi$
$752$	−29.4758	−1.07487
$753$	0	0
$754$	62.2456	2.26685
$755$	−46.3978	−1.68859
$756$	0	0
$757$	28.4151	1.03276	0.516382	−	0.856359i	$-0.327279\pi$
$757$	28.4151	1.03276	0.516382	+	0.856359i	$0.327279\pi$
$758$	−8.61086	−0.312761
$759$	0	0
$760$	17.9331	0.650501
$761$	11.9813	0.434320	0.217160	−	0.976136i	$-0.430321\pi$
$761$	11.9813	0.434320	0.217160	+	0.976136i	$0.430321\pi$
$762$	0	0
$763$	−21.8498	−0.791015
$764$	51.3976	1.85950
$765$	0	0
$766$	−63.2110	−2.28391
$767$	3.90686	0.141069
$768$	0	0
$769$	16.1101	0.580946	0.290473	−	0.956883i	$-0.406187\pi$
$769$	16.1101	0.580946	0.290473	+	0.956883i	$0.406187\pi$
$770$	8.47641	0.305469
$771$	0	0
$772$	2.59523	0.0934043
$773$	45.1851	1.62519	0.812597	−	0.582826i	$-0.198053\pi$
$773$	45.1851	1.62519	0.812597	+	0.582826i	$0.198053\pi$
$774$	0	0
$775$	2.01766	0.0724766
$776$	−0.297280	−0.0106717
$777$	0	0
$778$	−75.2267	−2.69701
$779$	0.315953	0.0113202
$780$	0	0
$781$	−4.97794	−0.178125
$782$	−5.15847	−0.184467
$783$	0	0
$784$	9.81434	0.350512
$785$	10.0463	0.358567
$786$	0	0
$787$	−38.1660	−1.36047	−0.680235	−	0.732994i	$-0.738122\pi$
$787$	−38.1660	−1.36047	−0.680235	+	0.732994i	$0.738122\pi$
$788$	42.6275	1.51854
$789$	0	0
$790$	70.2188	2.49827
$791$	16.9632	0.603142
$792$	0	0
$793$	65.1149	2.31230
$794$	−41.9302	−1.48805
$795$	0	0
$796$	−64.0082	−2.26871
$797$	−52.0399	−1.84335	−0.921674	−	0.387966i	$-0.873178\pi$
$797$	−52.0399	−1.84335	−0.921674	+	0.387966i	$0.873178\pi$
$798$	0	0
$799$	−29.3176	−1.03718
$800$	−2.39774	−0.0847730
$801$	0	0
$802$	−62.8343	−2.21876
$803$	−4.33662	−0.153036
$804$	0	0
$805$	−3.94896	−0.139183
$806$	−73.2343	−2.57957
$807$	0	0
$808$	−15.6162	−0.549377
$809$	−23.5786	−0.828981	−0.414490	−	0.910054i	$-0.636040\pi$
$809$	−23.5786	−0.828981	−0.414490	+	0.910054i	$0.636040\pi$
$810$	0	0
$811$	−21.7514	−0.763796	−0.381898	−	0.924204i	$-0.624730\pi$
$811$	−21.7514	−0.763796	−0.381898	+	0.924204i	$0.624730\pi$
$812$	−24.9140	−0.874311
$813$	0	0
$814$	0.968992	0.0339631
$815$	−55.3421	−1.93855
$816$	0	0
$817$	−39.5167	−1.38251
$818$	−13.5650	−0.474288
$819$	0	0
$820$	0.317909	0.0111019
$821$	18.0492	0.629922	0.314961	−	0.949105i	$-0.398009\pi$
$821$	18.0492	0.629922	0.314961	+	0.949105i	$0.398009\pi$
$822$	0	0
$823$	−14.1733	−0.494051	−0.247025	−	0.969009i	$-0.579453\pi$
$823$	−14.1733	−0.494051	−0.247025	+	0.969009i	$0.579453\pi$
$824$	23.9204	0.833308
$825$	0	0
$826$	−2.76318	−0.0961434
$827$	41.8804	1.45632	0.728162	−	0.685405i	$-0.240375\pi$
$827$	41.8804	1.45632	0.728162	+	0.685405i	$0.240375\pi$
$828$	0	0
$829$	45.6030	1.58386	0.791928	−	0.610614i	$-0.209077\pi$
$829$	45.6030	1.58386	0.791928	+	0.610614i	$0.209077\pi$
$830$	59.0657	2.05020
$831$	0	0
$832$	61.8914	2.14570
$833$	9.76164	0.338221
$834$	0	0
$835$	24.7361	0.856028
$836$	15.5666	0.538383
$837$	0	0
$838$	−42.0923	−1.45406
$839$	11.2025	0.386754	0.193377	−	0.981125i	$-0.438056\pi$
$839$	11.2025	0.386754	0.193377	+	0.981125i	$0.438056\pi$
$840$	0	0
$841$	2.07418	0.0715236
$842$	0.888378	0.0306155
$843$	0	0
$844$	67.3899	2.31966
$845$	−32.3965	−1.11447
$846$	0	0
$847$	1.71408	0.0588967
$848$	4.05742	0.139333
$849$	0	0
$850$	−1.58695	−0.0544321
$851$	−0.451431	−0.0154748
$852$	0	0
$853$	26.0454	0.891776	0.445888	−	0.895089i	$-0.352888\pi$
$853$	26.0454	0.891776	0.445888	+	0.895089i	$0.352888\pi$
$854$	−46.0534	−1.57592
$855$	0	0
$856$	4.08908	0.139762
$857$	11.2871	0.385559	0.192780	−	0.981242i	$-0.438250\pi$
$857$	11.2871	0.385559	0.192780	+	0.981242i	$0.438250\pi$
$858$	0	0
$859$	8.66093	0.295507	0.147754	−	0.989024i	$-0.452796\pi$
$859$	8.66093	0.295507	0.147754	+	0.989024i	$0.452796\pi$
$860$	−39.7613	−1.35585
$861$	0	0
$862$	−28.7662	−0.979782
$863$	−1.68934	−0.0575060	−0.0287530	−	0.999587i	$-0.509154\pi$
$863$	−1.68934	−0.0575060	−0.0287530	+	0.999587i	$0.509154\pi$
$864$	0	0
$865$	37.8089	1.28554
$866$	−48.3253	−1.64216
$867$	0	0
$868$	29.3123	0.994923
$869$	14.1995	0.481685
$870$	0	0
$871$	−25.5963	−0.867298
$872$	−16.6202	−0.562831
$873$	0	0
$874$	−12.8148	−0.433467
$875$	18.5300	0.626427
$876$	0	0
$877$	−23.6638	−0.799068	−0.399534	−	0.916718i	$-0.630828\pi$
$877$	−23.6638	−0.799068	−0.399534	+	0.916718i	$0.630828\pi$
$878$	24.0198	0.810628
$879$	0	0
$880$	−5.56649	−0.187646
$881$	52.6497	1.77381	0.886907	−	0.461949i	$-0.152850\pi$
$881$	52.6497	1.77381	0.886907	+	0.461949i	$0.152850\pi$
$882$	0	0
$883$	0.0150906	0.000507838 0	0.000253919	−	1.00000i	$-0.499919\pi$
$883$	0.0150906	0.000507838 0	0.000253919		1.00000i	$0.499919\pi$
$884$	32.5975	1.09637
$885$	0	0
$886$	−44.0217	−1.47894
$887$	−4.97866	−0.167167	−0.0835835	−	0.996501i	$-0.526637\pi$
$887$	−4.97866	−0.167167	−0.0835835	+	0.996501i	$0.526637\pi$
$888$	0	0
$889$	−23.7083	−0.795151
$890$	35.6328	1.19441
$891$	0	0
$892$	69.6655	2.33257
$893$	−72.8313	−2.43721
$894$	0	0
$895$	−17.5091	−0.585265
$896$	−17.0544	−0.569749
$897$	0	0
$898$	−23.1898	−0.773854
$899$	−36.5599	−1.21934
$900$	0	0
$901$	4.03564	0.134447
$902$	0.113598	0.00378239
$903$	0	0
$904$	12.9032	0.429154
$905$	18.2751	0.607485
$906$	0	0
$907$	−2.61087	−0.0866926	−0.0433463	−	0.999060i	$-0.513802\pi$
$907$	−2.61087	−0.0866926	−0.0433463	+	0.999060i	$0.513802\pi$
$908$	−68.4942	−2.27306
$909$	0	0
$910$	44.0953	1.46174
$911$	10.3836	0.344023	0.172011	−	0.985095i	$-0.444973\pi$
$911$	10.3836	0.344023	0.172011	+	0.985095i	$0.444973\pi$
$912$	0	0
$913$	11.9442	0.395294
$914$	−37.3789	−1.23638
$915$	0	0
$916$	−9.19865	−0.303932
$917$	−3.14677	−0.103916
$918$	0	0
$919$	−50.1160	−1.65318	−0.826588	−	0.562808i	$-0.809721\pi$
$919$	−50.1160	−1.65318	−0.826588	+	0.562808i	$0.809721\pi$
$920$	−3.00381	−0.0990326
$921$	0	0
$922$	−46.1532	−1.51997
$923$	−25.8958	−0.852372
$924$	0	0
$925$	−0.138878	−0.00456629
$926$	18.4871	0.607522
$927$	0	0
$928$	43.4470	1.42622
$929$	18.5166	0.607509	0.303755	−	0.952750i	$-0.401760\pi$
$929$	18.5166	0.607509	0.303755	+	0.952750i	$0.401760\pi$
$930$	0	0
$931$	24.2501	0.794764
$932$	23.2849	0.762724
$933$	0	0
$934$	−46.0575	−1.50705
$935$	−5.53660	−0.181066
$936$	0	0
$937$	−6.40596	−0.209274	−0.104637	−	0.994511i	$-0.533368\pi$
$937$	−6.40596	−0.209274	−0.104637	+	0.994511i	$0.533368\pi$
$938$	18.1033	0.591095
$939$	0	0
$940$	−73.2822	−2.39020
$941$	45.9567	1.49814	0.749072	−	0.662488i	$-0.230500\pi$
$941$	45.9567	1.49814	0.749072	+	0.662488i	$0.230500\pi$
$942$	0	0
$943$	−0.0529225	−0.00172339
$944$	1.81459	0.0590599
$945$	0	0
$946$	−14.2078	−0.461936
$947$	−4.92251	−0.159960	−0.0799801	−	0.996796i	$-0.525486\pi$
$947$	−4.92251	−0.159960	−0.0799801	+	0.996796i	$0.525486\pi$
$948$	0	0
$949$	−22.5596	−0.732316
$950$	−3.94235	−0.127907
$951$	0	0
$952$	−5.37089	−0.174071
$953$	35.5422	1.15132	0.575662	−	0.817688i	$-0.304744\pi$
$953$	35.5422	1.15132	0.575662	+	0.817688i	$0.304744\pi$
$954$	0	0
$955$	−45.4131	−1.46953
$956$	−28.7779	−0.930744
$957$	0	0
$958$	−2.25695	−0.0729189
$959$	4.54994	0.146925
$960$	0	0
$961$	12.0141	0.387551
$962$	5.04081	0.162522
$963$	0	0
$964$	−58.1299	−1.87224
$965$	−2.29305	−0.0738161
$966$	0	0
$967$	37.7483	1.21391	0.606953	−	0.794738i	$-0.292392\pi$
$967$	37.7483	1.21391	0.606953	+	0.794738i	$0.292392\pi$
$968$	1.30383	0.0419067
$969$	0	0
$970$	1.12752	0.0362025
$971$	26.8027	0.860140	0.430070	−	0.902796i	$-0.358489\pi$
$971$	26.8027	0.860140	0.430070	+	0.902796i	$0.358489\pi$
$972$	0	0
$973$	−21.1274	−0.677314
$974$	23.5421	0.754338
$975$	0	0
$976$	30.2434	0.968069
$977$	−22.8876	−0.732239	−0.366120	−	0.930568i	$-0.619314\pi$
$977$	−22.8876	−0.732239	−0.366120	+	0.930568i	$0.619314\pi$
$978$	0	0
$979$	7.20560	0.230292
$980$	24.4002	0.779435
$981$	0	0
$982$	54.6884	1.74518
$983$	51.3842	1.63890	0.819451	−	0.573149i	$-0.194278\pi$
$983$	51.3842	1.63890	0.819451	+	0.573149i	$0.194278\pi$
$984$	0	0
$985$	−37.6642	−1.20008
$986$	28.7555	0.915762
$987$	0	0
$988$	80.9793	2.57630
$989$	6.61909	0.210475
$990$	0	0
$991$	−42.5506	−1.35166	−0.675831	−	0.737056i	$-0.736215\pi$
$991$	−42.5506	−1.35166	−0.675831	+	0.737056i	$0.736215\pi$
$992$	−51.1169	−1.62296
$993$	0	0
$994$	18.3152	0.580923
$995$	56.5554	1.79293
$996$	0	0
$997$	−0.989349	−0.0313330	−0.0156665	−	0.999877i	$-0.504987\pi$
$997$	−0.989349	−0.0313330	−0.0156665	+	0.999877i	$0.504987\pi$
$998$	−38.4368	−1.21669
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2277.2.a.m.1.6		6
3.2	odd	2		253.2.a.d.1.1	✓	6
12.11	even	2		4048.2.a.bc.1.2		6
15.14	odd	2		6325.2.a.m.1.6		6
33.32	even	2		2783.2.a.h.1.6		6
69.68	even	2		5819.2.a.e.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.2.a.d.1.1	✓	6	3.2	odd	2
2277.2.a.m.1.6		6	1.1	even	1	trivial
2783.2.a.h.1.6		6	33.32	even	2
4048.2.a.bc.1.2		6	12.11	even	2
5819.2.a.e.1.1		6	69.68	even	2
6325.2.a.m.1.6		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)

\(f(q)\)	\(=\)	\(q+2.14649 q^{2} +2.60742 q^{4} -2.30383 q^{5} +1.71408 q^{7} +1.30383 q^{8} -4.94515 q^{10} -1.00000 q^{11} -5.20211 q^{13} +3.67927 q^{14} -2.41619 q^{16} -2.40321 q^{17} -5.97011 q^{19} -6.00707 q^{20} -2.14649 q^{22} +1.00000 q^{23} +0.307640 q^{25} -11.1663 q^{26} +4.46935 q^{28} -5.57442 q^{29} +6.55851 q^{31} -7.79399 q^{32} -5.15847 q^{34} -3.94896 q^{35} -0.451431 q^{37} -12.8148 q^{38} -3.00381 q^{40} -0.0529225 q^{41} +6.61909 q^{43} -2.60742 q^{44} +2.14649 q^{46} +12.1993 q^{47} -4.06191 q^{49} +0.660347 q^{50} -13.5641 q^{52} -1.67927 q^{53} +2.30383 q^{55} +2.23488 q^{56} -11.9654 q^{58} -0.751014 q^{59} -12.5170 q^{61} +14.0778 q^{62} -11.8973 q^{64} +11.9848 q^{65} +4.92036 q^{67} -6.26620 q^{68} -8.47641 q^{70} +4.97794 q^{71} +4.33662 q^{73} -0.968992 q^{74} -15.5666 q^{76} -1.71408 q^{77} -14.1995 q^{79} +5.56649 q^{80} -0.113598 q^{82} -11.9442 q^{83} +5.53660 q^{85} +14.2078 q^{86} -1.30383 q^{88} -7.20560 q^{89} -8.91687 q^{91} +2.60742 q^{92} +26.1857 q^{94} +13.7541 q^{95} -0.228005 q^{97} -8.71886 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 5 q^{4} - 3 q^{5} - q^{7} - 3 q^{8} - 6 q^{10} - 6 q^{11} - 3 q^{13} + 8 q^{14} - q^{16} - 5 q^{17} + q^{19} + 7 q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{25} - 15 q^{26} - 6 q^{29} - 8 q^{31}+ \cdots - 34 q^{98}+O(q^{100}) \) 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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