LMFDB - Embedded newform 2300.2.a.o.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,2,Mod(1,2300)] 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("2300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.143376304.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 22x^{2} - 6x - 1 $ x^6 - 12x^4 + 22x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.16223$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.21923 q^{3} -2.43185 q^{7} +7.36343 q^{9} -0.884969 q^{11} +5.10522 q^{13} +0.366626 q^{17} -2.79847 q^{19} -7.82867 q^{21} +1.00000 q^{23} +14.0469 q^{27} +8.02431 q^{29} +7.24179 q^{31} -2.84892 q^{33} -3.10036 q^{37} +16.4349 q^{39} -3.47185 q^{41} +8.56841 q^{43} +5.25528 q^{47} -1.08612 q^{49} +1.18025 q^{51} -11.6413 q^{53} -9.00892 q^{57} -9.33209 q^{59} +5.46699 q^{61} -17.9067 q^{63} +1.49020 q^{67} +3.21923 q^{69} +8.29949 q^{71} -10.2409 q^{73} +2.15211 q^{77} +6.06522 q^{79} +23.1298 q^{81} +16.2520 q^{83} +25.8321 q^{87} -17.6033 q^{89} -12.4151 q^{91} +23.3130 q^{93} -6.55618 q^{97} -6.51641 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{17} + 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} + 9 q^{31} - 10 q^{33} + 21 q^{37} - 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49}+ \cdots - 16 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^17 + 4 * q^19 + 6 * q^23 + 22 * q^27 + 5 * q^29 + 9 * q^31 - 10 * q^33 + 21 * q^37 - 8 * q^39 - q^41 + 16 * q^43 + 16 * q^47 + 19 * q^49 - 12 * q^51 - q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 19 * q^63 + 25 * q^67 + 4 * q^69 - 17 * q^71 - 14 * q^73 + 20 * q^77 + 10 * q^79 + 14 * q^81 + 21 * q^83 + 64 * q^87 - 24 * q^89 - 4 * q^91 + 4 * q^97 - 16 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.21923	1.85862	0.929311	−	0.369298i	$-0.120402\pi$
$3$	3.21923	1.85862	0.929311	+	0.369298i	$0.120402\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.43185	−0.919152	−0.459576	−	0.888138i	$-0.651999\pi$
$7$	−2.43185	−0.919152	−0.459576	+	0.888138i	$0.651999\pi$
$8$	0	0
$9$	7.36343	2.45448
$10$	0	0
$11$	−0.884969	−0.266828	−0.133414	−	0.991060i	$-0.542594\pi$
$11$	−0.884969	−0.266828	−0.133414	+	0.991060i	$0.542594\pi$
$12$	0	0
$13$	5.10522	1.41593	0.707967	−	0.706245i	$-0.249612\pi$
$13$	5.10522	1.41593	0.707967	+	0.706245i	$0.249612\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.366626	0.0889198	0.0444599	−	0.999011i	$-0.485843\pi$
$17$	0.366626	0.0889198	0.0444599	+	0.999011i	$0.485843\pi$
$18$	0	0
$19$	−2.79847	−0.642014	−0.321007	−	0.947077i	$-0.604021\pi$
$19$	−2.79847	−0.642014	−0.321007	+	0.947077i	$0.604021\pi$
$20$	0	0
$21$	−7.82867	−1.70836
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.0469	2.70332
$28$	0	0
$29$	8.02431	1.49008	0.745038	−	0.667022i	$-0.232431\pi$
$29$	8.02431	1.49008	0.745038	+	0.667022i	$0.232431\pi$
$30$	0	0
$31$	7.24179	1.30066	0.650332	−	0.759650i	$-0.274630\pi$
$31$	7.24179	1.30066	0.650332	+	0.759650i	$0.274630\pi$
$32$	0	0
$33$	−2.84892	−0.495933
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.10036	−0.509697	−0.254848	−	0.966981i	$-0.582026\pi$
$37$	−3.10036	−0.509697	−0.254848	+	0.966981i	$0.582026\pi$
$38$	0	0
$39$	16.4349	2.63169
$40$	0	0
$41$	−3.47185	−0.542212	−0.271106	−	0.962550i	$-0.587389\pi$
$41$	−3.47185	−0.542212	−0.271106	+	0.962550i	$0.587389\pi$
$42$	0	0
$43$	8.56841	1.30667	0.653335	−	0.757069i	$-0.273369\pi$
$43$	8.56841	1.30667	0.653335	+	0.757069i	$0.273369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.25528	0.766561	0.383281	−	0.923632i	$-0.374794\pi$
$47$	5.25528	0.766561	0.383281	+	0.923632i	$0.374794\pi$
$48$	0	0
$49$	−1.08612	−0.155160
$50$	0	0
$51$	1.18025	0.165268
$52$	0	0
$53$	−11.6413	−1.59905	−0.799526	−	0.600632i	$-0.794916\pi$
$53$	−11.6413	−1.59905	−0.799526	+	0.600632i	$0.794916\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.00892	−1.19326
$58$	0	0
$59$	−9.33209	−1.21493	−0.607467	−	0.794345i	$-0.707814\pi$
$59$	−9.33209	−1.21493	−0.607467	+	0.794345i	$0.707814\pi$
$60$	0	0
$61$	5.46699	0.699976	0.349988	−	0.936754i	$-0.386186\pi$
$61$	5.46699	0.699976	0.349988	+	0.936754i	$0.386186\pi$
$62$	0	0
$63$	−17.9067	−2.25604
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.49020	0.182057	0.0910285	−	0.995848i	$-0.470985\pi$
$67$	1.49020	0.182057	0.0910285	+	0.995848i	$0.470985\pi$
$68$	0	0
$69$	3.21923	0.387549
$70$	0	0
$71$	8.29949	0.984969	0.492484	−	0.870321i	$-0.336089\pi$
$71$	8.29949	0.984969	0.492484	+	0.870321i	$0.336089\pi$
$72$	0	0
$73$	−10.2409	−1.19860	−0.599302	−	0.800523i	$-0.704555\pi$
$73$	−10.2409	−1.19860	−0.599302	+	0.800523i	$0.704555\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.15211	0.245256
$78$	0	0
$79$	6.06522	0.682391	0.341195	−	0.939992i	$-0.389168\pi$
$79$	6.06522	0.682391	0.341195	+	0.939992i	$0.389168\pi$
$80$	0	0
$81$	23.1298	2.56998
$82$	0	0
$83$	16.2520	1.78389	0.891943	−	0.452148i	$-0.149342\pi$
$83$	16.2520	1.78389	0.891943	+	0.452148i	$0.149342\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	25.8321	2.76949
$88$	0	0
$89$	−17.6033	−1.86594	−0.932972	−	0.359949i	$-0.882794\pi$
$89$	−17.6033	−1.86594	−0.932972	+	0.359949i	$0.882794\pi$
$90$	0	0
$91$	−12.4151	−1.30146
$92$	0	0
$93$	23.3130	2.41744
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.55618	−0.665679	−0.332839	−	0.942984i	$-0.608007\pi$
$97$	−6.55618	−0.665679	−0.332839	+	0.942984i	$0.608007\pi$
$98$	0	0
$99$	−6.51641	−0.654924
$100$	0	0
$101$	13.4912	1.34243	0.671213	−	0.741265i	$-0.265774\pi$
$101$	13.4912	1.34243	0.671213	+	0.741265i	$0.265774\pi$
$102$	0	0
$103$	12.5460	1.23619	0.618096	−	0.786103i	$-0.287904\pi$
$103$	12.5460	1.23619	0.618096	+	0.786103i	$0.287904\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.46269	0.141404	0.0707020	−	0.997497i	$-0.477476\pi$
$107$	1.46269	0.141404	0.0707020	+	0.997497i	$0.477476\pi$
$108$	0	0
$109$	−19.2173	−1.84069	−0.920343	−	0.391113i	$-0.872090\pi$
$109$	−19.2173	−1.84069	−0.920343	+	0.391113i	$0.872090\pi$
$110$	0	0
$111$	−9.98078	−0.947334
$112$	0	0
$113$	−0.834901	−0.0785409	−0.0392705	−	0.999229i	$-0.512503\pi$
$113$	−0.834901	−0.0785409	−0.0392705	+	0.999229i	$0.512503\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	37.5920	3.47538
$118$	0	0
$119$	−0.891578	−0.0817308
$120$	0	0
$121$	−10.2168	−0.928803
$122$	0	0
$123$	−11.1767	−1.00777
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.87453	0.698751	0.349376	−	0.936983i	$-0.386394\pi$
$127$	7.87453	0.698751	0.349376	+	0.936983i	$0.386394\pi$
$128$	0	0
$129$	27.5837	2.42861
$130$	0	0
$131$	−12.8740	−1.12481	−0.562403	−	0.826863i	$-0.690123\pi$
$131$	−12.8740	−1.12481	−0.562403	+	0.826863i	$0.690123\pi$
$132$	0	0
$133$	6.80546	0.590108
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.26675	0.279097	0.139549	−	0.990215i	$-0.455435\pi$
$137$	3.26675	0.279097	0.139549	+	0.990215i	$0.455435\pi$
$138$	0	0
$139$	−12.5578	−1.06514	−0.532570	−	0.846386i	$-0.678774\pi$
$139$	−12.5578	−1.06514	−0.532570	+	0.846386i	$0.678774\pi$
$140$	0	0
$141$	16.9179	1.42475
$142$	0	0
$143$	−4.51797	−0.377811
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.49647	−0.288384
$148$	0	0
$149$	−0.188265	−0.0154233	−0.00771164	−	0.999970i	$-0.502455\pi$
$149$	−0.188265	−0.0154233	−0.00771164	+	0.999970i	$0.502455\pi$
$150$	0	0
$151$	−18.6708	−1.51941	−0.759704	−	0.650269i	$-0.774656\pi$
$151$	−18.6708	−1.51941	−0.759704	+	0.650269i	$0.774656\pi$
$152$	0	0
$153$	2.69962	0.218252
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.0277	1.59839	0.799193	−	0.601074i	$-0.205260\pi$
$157$	20.0277	1.59839	0.799193	+	0.601074i	$0.205260\pi$
$158$	0	0
$159$	−37.4759	−2.97203
$160$	0	0
$161$	−2.43185	−0.191656
$162$	0	0
$163$	3.35607	0.262867	0.131434	−	0.991325i	$-0.458042\pi$
$163$	3.35607	0.262867	0.131434	+	0.991325i	$0.458042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.57745	−0.199449	−0.0997246	−	0.995015i	$-0.531796\pi$
$167$	−2.57745	−0.199449	−0.0997246	+	0.995015i	$0.531796\pi$
$168$	0	0
$169$	13.0633	1.00487
$170$	0	0
$171$	−20.6064	−1.57581
$172$	0	0
$173$	−19.4182	−1.47634	−0.738170	−	0.674615i	$-0.764310\pi$
$173$	−19.4182	−1.47634	−0.738170	+	0.674615i	$0.764310\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−30.0421	−2.25810
$178$	0	0
$179$	−20.3628	−1.52199	−0.760993	−	0.648760i	$-0.775288\pi$
$179$	−20.3628	−1.52199	−0.760993	+	0.648760i	$0.775288\pi$
$180$	0	0
$181$	−22.5629	−1.67709	−0.838543	−	0.544835i	$-0.816592\pi$
$181$	−22.5629	−1.67709	−0.838543	+	0.544835i	$0.816592\pi$
$182$	0	0
$183$	17.5995	1.30099
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.324453	−0.0237263
$188$	0	0
$189$	−34.1598	−2.48476
$190$	0	0
$191$	−16.6554	−1.20514	−0.602571	−	0.798065i	$-0.705857\pi$
$191$	−16.6554	−1.20514	−0.602571	+	0.798065i	$0.705857\pi$
$192$	0	0
$193$	14.5214	1.04527	0.522636	−	0.852556i	$-0.324949\pi$
$193$	14.5214	1.04527	0.522636	+	0.852556i	$0.324949\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.3479	−0.950995	−0.475498	−	0.879717i	$-0.657732\pi$
$197$	−13.3479	−0.950995	−0.475498	+	0.879717i	$0.657732\pi$
$198$	0	0
$199$	18.7856	1.33168	0.665838	−	0.746097i	$-0.268074\pi$
$199$	18.7856	1.33168	0.665838	+	0.746097i	$0.268074\pi$
$200$	0	0
$201$	4.79730	0.338375
$202$	0	0
$203$	−19.5139	−1.36961
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.36343	0.511794
$208$	0	0
$209$	2.47656	0.171307
$210$	0	0
$211$	17.1078	1.17775	0.588874	−	0.808225i	$-0.299571\pi$
$211$	17.1078	1.17775	0.588874	+	0.808225i	$0.299571\pi$
$212$	0	0
$213$	26.7180	1.83068
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−17.6109	−1.19551
$218$	0	0
$219$	−32.9677	−2.22775
$220$	0	0
$221$	1.87171	0.125905
$222$	0	0
$223$	−13.5184	−0.905262	−0.452631	−	0.891698i	$-0.649515\pi$
$223$	−13.5184	−0.905262	−0.452631	+	0.891698i	$0.649515\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.29581	0.550612	0.275306	−	0.961357i	$-0.411221\pi$
$227$	8.29581	0.550612	0.275306	+	0.961357i	$0.411221\pi$
$228$	0	0
$229$	−6.22318	−0.411239	−0.205620	−	0.978632i	$-0.565921\pi$
$229$	−6.22318	−0.411239	−0.205620	+	0.978632i	$0.565921\pi$
$230$	0	0
$231$	6.92813	0.455838
$232$	0	0
$233$	−13.6347	−0.893242	−0.446621	−	0.894723i	$-0.647373\pi$
$233$	−13.6347	−0.893242	−0.446621	+	0.894723i	$0.647373\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	19.5253	1.26831
$238$	0	0
$239$	11.5281	0.745693	0.372847	−	0.927893i	$-0.378382\pi$
$239$	11.5281	0.745693	0.372847	+	0.927893i	$0.378382\pi$
$240$	0	0
$241$	−17.5952	−1.13341	−0.566704	−	0.823921i	$-0.691782\pi$
$241$	−17.5952	−1.13341	−0.566704	+	0.823921i	$0.691782\pi$
$242$	0	0
$243$	32.3195	2.07329
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.2868	−0.909049
$248$	0	0
$249$	52.3188	3.31557
$250$	0	0
$251$	12.5471	0.791968	0.395984	−	0.918257i	$-0.370404\pi$
$251$	12.5471	0.791968	0.395984	+	0.918257i	$0.370404\pi$
$252$	0	0
$253$	−0.884969	−0.0556375
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.88072	0.304451	0.152225	−	0.988346i	$-0.451356\pi$
$257$	4.88072	0.304451	0.152225	+	0.988346i	$0.451356\pi$
$258$	0	0
$259$	7.53961	0.468489
$260$	0	0
$261$	59.0864	3.65736
$262$	0	0
$263$	−11.6045	−0.715566	−0.357783	−	0.933805i	$-0.616467\pi$
$263$	−11.6045	−0.715566	−0.357783	+	0.933805i	$0.616467\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−56.6690	−3.46808
$268$	0	0
$269$	−6.04327	−0.368465	−0.184232	−	0.982883i	$-0.558980\pi$
$269$	−6.04327	−0.368465	−0.184232	+	0.982883i	$0.558980\pi$
$270$	0	0
$271$	3.93401	0.238974	0.119487	−	0.992836i	$-0.461875\pi$
$271$	3.93401	0.238974	0.119487	+	0.992836i	$0.461875\pi$
$272$	0	0
$273$	−39.9671	−2.41892
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.92174	0.475971	0.237986	−	0.971269i	$-0.423513\pi$
$277$	7.92174	0.475971	0.237986	+	0.971269i	$0.423513\pi$
$278$	0	0
$279$	53.3244	3.19245
$280$	0	0
$281$	−6.00419	−0.358180	−0.179090	−	0.983833i	$-0.557315\pi$
$281$	−6.00419	−0.358180	−0.179090	+	0.983833i	$0.557315\pi$
$282$	0	0
$283$	7.65299	0.454923	0.227461	−	0.973787i	$-0.426957\pi$
$283$	7.65299	0.454923	0.227461	+	0.973787i	$0.426957\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.44301	0.498375
$288$	0	0
$289$	−16.8656	−0.992093
$290$	0	0
$291$	−21.1058	−1.23725
$292$	0	0
$293$	−13.5480	−0.791485	−0.395743	−	0.918361i	$-0.629513\pi$
$293$	−13.5480	−0.791485	−0.395743	+	0.918361i	$0.629513\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−12.4310	−0.721323
$298$	0	0
$299$	5.10522	0.295243
$300$	0	0
$301$	−20.8371	−1.20103
$302$	0	0
$303$	43.4313	2.49506
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.0870	0.575697	0.287848	−	0.957676i	$-0.407060\pi$
$307$	10.0870	0.575697	0.287848	+	0.957676i	$0.407060\pi$
$308$	0	0
$309$	40.3883	2.29761
$310$	0	0
$311$	−28.2481	−1.60180	−0.800902	−	0.598795i	$-0.795646\pi$
$311$	−28.2481	−1.60180	−0.800902	+	0.598795i	$0.795646\pi$
$312$	0	0
$313$	−14.4504	−0.816786	−0.408393	−	0.912806i	$-0.633911\pi$
$313$	−14.4504	−0.816786	−0.408393	+	0.912806i	$0.633911\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.7835	0.830324	0.415162	−	0.909748i	$-0.363725\pi$
$317$	14.7835	0.830324	0.415162	+	0.909748i	$0.363725\pi$
$318$	0	0
$319$	−7.10127	−0.397595
$320$	0	0
$321$	4.70875	0.262817
$322$	0	0
$323$	−1.02599	−0.0570877
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−61.8649	−3.42114
$328$	0	0
$329$	−12.7800	−0.704586
$330$	0	0
$331$	−20.1494	−1.10751	−0.553757	−	0.832679i	$-0.686806\pi$
$331$	−20.1494	−1.10751	−0.553757	+	0.832679i	$0.686806\pi$
$332$	0	0
$333$	−22.8293	−1.25104
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.0660	1.31096	0.655479	−	0.755214i	$-0.272467\pi$
$337$	24.0660	1.31096	0.655479	+	0.755214i	$0.272467\pi$
$338$	0	0
$339$	−2.68774	−0.145978
$340$	0	0
$341$	−6.40876	−0.347054
$342$	0	0
$343$	19.6642	1.06177
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.3788	−1.41609	−0.708045	−	0.706168i	$-0.750422\pi$
$347$	−26.3788	−1.41609	−0.708045	+	0.706168i	$0.750422\pi$
$348$	0	0
$349$	0.535518	0.0286656	0.0143328	−	0.999897i	$-0.495438\pi$
$349$	0.535518	0.0286656	0.0143328	+	0.999897i	$0.495438\pi$
$350$	0	0
$351$	71.7124	3.82773
$352$	0	0
$353$	−5.27606	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$353$	−5.27606	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.87019	−0.151907
$358$	0	0
$359$	20.4351	1.07852	0.539261	−	0.842139i	$-0.318704\pi$
$359$	20.4351	1.07852	0.539261	+	0.842139i	$0.318704\pi$
$360$	0	0
$361$	−11.1685	−0.587818
$362$	0	0
$363$	−32.8903	−1.72629
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.4382	0.649268	0.324634	−	0.945840i	$-0.394759\pi$
$367$	12.4382	0.649268	0.324634	+	0.945840i	$0.394759\pi$
$368$	0	0
$369$	−25.5647	−1.33085
$370$	0	0
$371$	28.3098	1.46977
$372$	0	0
$373$	2.99999	0.155334	0.0776669	−	0.996979i	$-0.475253\pi$
$373$	2.99999	0.155334	0.0776669	+	0.996979i	$0.475253\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.9659	2.10985
$378$	0	0
$379$	−3.02165	−0.155212	−0.0776059	−	0.996984i	$-0.524728\pi$
$379$	−3.02165	−0.155212	−0.0776059	+	0.996984i	$0.524728\pi$
$380$	0	0
$381$	25.3499	1.29871
$382$	0	0
$383$	8.15719	0.416813	0.208406	−	0.978042i	$-0.433172\pi$
$383$	8.15719	0.416813	0.208406	+	0.978042i	$0.433172\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	63.0929	3.20719
$388$	0	0
$389$	11.7789	0.597214	0.298607	−	0.954376i	$-0.403478\pi$
$389$	11.7789	0.597214	0.298607	+	0.954376i	$0.403478\pi$
$390$	0	0
$391$	0.366626	0.0185411
$392$	0	0
$393$	−41.4443	−2.09059
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.7004	−0.838167	−0.419083	−	0.907948i	$-0.637649\pi$
$397$	−16.7004	−0.838167	−0.419083	+	0.907948i	$0.637649\pi$
$398$	0	0
$399$	21.9083	1.09679
$400$	0	0
$401$	9.99936	0.499344	0.249672	−	0.968330i	$-0.419677\pi$
$401$	9.99936	0.499344	0.249672	+	0.968330i	$0.419677\pi$
$402$	0	0
$403$	36.9710	1.84165
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.74373	0.136002
$408$	0	0
$409$	2.20305	0.108934	0.0544668	−	0.998516i	$-0.482654\pi$
$409$	2.20305	0.108934	0.0544668	+	0.998516i	$0.482654\pi$
$410$	0	0
$411$	10.5164	0.518736
$412$	0	0
$413$	22.6942	1.11671
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−40.4264	−1.97969
$418$	0	0
$419$	4.12814	0.201673	0.100836	−	0.994903i	$-0.467848\pi$
$419$	4.12814	0.201673	0.100836	+	0.994903i	$0.467848\pi$
$420$	0	0
$421$	−15.7354	−0.766899	−0.383449	−	0.923562i	$-0.625264\pi$
$421$	−15.7354	−0.766899	−0.383449	+	0.923562i	$0.625264\pi$
$422$	0	0
$423$	38.6969	1.88151
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13.2949	−0.643385
$428$	0	0
$429$	−14.5444	−0.702209
$430$	0	0
$431$	−1.35158	−0.0651034	−0.0325517	−	0.999470i	$-0.510363\pi$
$431$	−1.35158	−0.0651034	−0.0325517	+	0.999470i	$0.510363\pi$
$432$	0	0
$433$	39.4114	1.89399	0.946996	−	0.321244i	$-0.104101\pi$
$433$	39.4114	1.89399	0.946996	+	0.321244i	$0.104101\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.79847	−0.133869
$438$	0	0
$439$	10.1059	0.482326	0.241163	−	0.970485i	$-0.422471\pi$
$439$	10.1059	0.482326	0.241163	+	0.970485i	$0.422471\pi$
$440$	0	0
$441$	−7.99756	−0.380836
$442$	0	0
$443$	6.25526	0.297197	0.148598	−	0.988898i	$-0.452524\pi$
$443$	6.25526	0.297197	0.148598	+	0.988898i	$0.452524\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.606068	−0.0286660
$448$	0	0
$449$	2.28779	0.107968	0.0539839	−	0.998542i	$-0.482808\pi$
$449$	2.28779	0.107968	0.0539839	+	0.998542i	$0.482808\pi$
$450$	0	0
$451$	3.07248	0.144677
$452$	0	0
$453$	−60.1055	−2.82400
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.3062	−0.669217	−0.334609	−	0.942357i	$-0.608604\pi$
$457$	−14.3062	−0.669217	−0.334609	+	0.942357i	$0.608604\pi$
$458$	0	0
$459$	5.14995	0.240379
$460$	0	0
$461$	−2.38995	−0.111311	−0.0556556	−	0.998450i	$-0.517725\pi$
$461$	−2.38995	−0.111311	−0.0556556	+	0.998450i	$0.517725\pi$
$462$	0	0
$463$	−9.52232	−0.442540	−0.221270	−	0.975213i	$-0.571020\pi$
$463$	−9.52232	−0.442540	−0.221270	+	0.975213i	$0.571020\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.14501	−0.284357	−0.142178	−	0.989841i	$-0.545411\pi$
$467$	−6.14501	−0.284357	−0.142178	+	0.989841i	$0.545411\pi$
$468$	0	0
$469$	−3.62394	−0.167338
$470$	0	0
$471$	64.4738	2.97080
$472$	0	0
$473$	−7.58278	−0.348657
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−85.7197	−3.92483
$478$	0	0
$479$	−1.40986	−0.0644180	−0.0322090	−	0.999481i	$-0.510254\pi$
$479$	−1.40986	−0.0644180	−0.0322090	+	0.999481i	$0.510254\pi$
$480$	0	0
$481$	−15.8281	−0.721697
$482$	0	0
$483$	−7.82867	−0.356217
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.2993	1.41831	0.709154	−	0.705054i	$-0.249077\pi$
$487$	31.2993	1.41831	0.709154	+	0.705054i	$0.249077\pi$
$488$	0	0
$489$	10.8039	0.488571
$490$	0	0
$491$	−25.9254	−1.17000	−0.584999	−	0.811034i	$-0.698905\pi$
$491$	−25.9254	−1.17000	−0.584999	+	0.811034i	$0.698905\pi$
$492$	0	0
$493$	2.94192	0.132497
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.1831	−0.905336
$498$	0	0
$499$	−8.04562	−0.360172	−0.180086	−	0.983651i	$-0.557638\pi$
$499$	−8.04562	−0.360172	−0.180086	+	0.983651i	$0.557638\pi$
$500$	0	0
$501$	−8.29740	−0.370701
$502$	0	0
$503$	−5.50490	−0.245451	−0.122726	−	0.992441i	$-0.539164\pi$
$503$	−5.50490	−0.245451	−0.122726	+	0.992441i	$0.539164\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	42.0538	1.86767
$508$	0	0
$509$	−8.26586	−0.366378	−0.183189	−	0.983078i	$-0.558642\pi$
$509$	−8.26586	−0.366378	−0.183189	+	0.983078i	$0.558642\pi$
$510$	0	0
$511$	24.9043	1.10170
$512$	0	0
$513$	−39.3098	−1.73557
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.65076	−0.204540
$518$	0	0
$519$	−62.5116	−2.74396
$520$	0	0
$521$	11.9776	0.524747	0.262374	−	0.964966i	$-0.415495\pi$
$521$	11.9776	0.524747	0.262374	+	0.964966i	$0.415495\pi$
$522$	0	0
$523$	−0.711546	−0.0311137	−0.0155569	−	0.999879i	$-0.504952\pi$
$523$	−0.711546	−0.0311137	−0.0155569	+	0.999879i	$0.504952\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.65503	0.115655
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−68.7162	−2.98203
$532$	0	0
$533$	−17.7246	−0.767737
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−65.5524	−2.82880
$538$	0	0
$539$	0.961182	0.0414011
$540$	0	0
$541$	34.9533	1.50276	0.751381	−	0.659869i	$-0.229388\pi$
$541$	34.9533	1.50276	0.751381	+	0.659869i	$0.229388\pi$
$542$	0	0
$543$	−72.6351	−3.11707
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.8478	0.763118	0.381559	−	0.924345i	$-0.375387\pi$
$547$	17.8478	0.763118	0.381559	+	0.924345i	$0.375387\pi$
$548$	0	0
$549$	40.2558	1.71808
$550$	0	0
$551$	−22.4558	−0.956650
$552$	0	0
$553$	−14.7497	−0.627221
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.29661	0.139682	0.0698410	−	0.997558i	$-0.477751\pi$
$557$	3.29661	0.139682	0.0698410	+	0.997558i	$0.477751\pi$
$558$	0	0
$559$	43.7437	1.85016
$560$	0	0
$561$	−1.04449	−0.0440983
$562$	0	0
$563$	4.44153	0.187188	0.0935941	−	0.995610i	$-0.470164\pi$
$563$	4.44153	0.187188	0.0935941	+	0.995610i	$0.470164\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−56.2481	−2.36220
$568$	0	0
$569$	−6.00088	−0.251570	−0.125785	−	0.992058i	$-0.540145\pi$
$569$	−6.00088	−0.251570	−0.125785	+	0.992058i	$0.540145\pi$
$570$	0	0
$571$	18.1609	0.760008	0.380004	−	0.924985i	$-0.375923\pi$
$571$	18.1609	0.760008	0.380004	+	0.924985i	$0.375923\pi$
$572$	0	0
$573$	−53.6175	−2.23990
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.0474	−0.418279	−0.209140	−	0.977886i	$-0.567066\pi$
$577$	−10.0474	−0.418279	−0.209140	+	0.977886i	$0.567066\pi$
$578$	0	0
$579$	46.7476	1.94277
$580$	0	0
$581$	−39.5223	−1.63966
$582$	0	0
$583$	10.3022	0.426672
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−42.3593	−1.74836	−0.874178	−	0.485605i	$-0.838599\pi$
$587$	−42.3593	−1.74836	−0.874178	+	0.485605i	$0.838599\pi$
$588$	0	0
$589$	−20.2660	−0.835044
$590$	0	0
$591$	−42.9698	−1.76754
$592$	0	0
$593$	19.6741	0.807917	0.403958	−	0.914777i	$-0.367634\pi$
$593$	19.6741	0.807917	0.403958	+	0.914777i	$0.367634\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	60.4751	2.47508
$598$	0	0
$599$	−18.2740	−0.746655	−0.373328	−	0.927700i	$-0.621783\pi$
$599$	−18.2740	−0.746655	−0.373328	+	0.927700i	$0.621783\pi$
$600$	0	0
$601$	−8.17793	−0.333585	−0.166792	−	0.985992i	$-0.553341\pi$
$601$	−8.17793	−0.333585	−0.166792	+	0.985992i	$0.553341\pi$
$602$	0	0
$603$	10.9730	0.446855
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.7927	0.884538	0.442269	−	0.896883i	$-0.354174\pi$
$607$	21.7927	0.884538	0.442269	+	0.896883i	$0.354174\pi$
$608$	0	0
$609$	−62.8197	−2.54558
$610$	0	0
$611$	26.8294	1.08540
$612$	0	0
$613$	−15.2379	−0.615455	−0.307727	−	0.951475i	$-0.599568\pi$
$613$	−15.2379	−0.615455	−0.307727	+	0.951475i	$0.599568\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.2945	0.817024	0.408512	−	0.912753i	$-0.366048\pi$
$617$	20.2945	0.817024	0.408512	+	0.912753i	$0.366048\pi$
$618$	0	0
$619$	2.08510	0.0838073	0.0419037	−	0.999122i	$-0.486658\pi$
$619$	2.08510	0.0838073	0.0419037	+	0.999122i	$0.486658\pi$
$620$	0	0
$621$	14.0469	0.563681
$622$	0	0
$623$	42.8085	1.71509
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.97262	0.318396
$628$	0	0
$629$	−1.13667	−0.0453221
$630$	0	0
$631$	39.0512	1.55461	0.777303	−	0.629127i	$-0.216587\pi$
$631$	39.0512	1.55461	0.777303	+	0.629127i	$0.216587\pi$
$632$	0	0
$633$	55.0738	2.18899
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.54488	−0.219696
$638$	0	0
$639$	61.1127	2.41758
$640$	0	0
$641$	−21.8361	−0.862475	−0.431238	−	0.902238i	$-0.641923\pi$
$641$	−21.8361	−0.862475	−0.431238	+	0.902238i	$0.641923\pi$
$642$	0	0
$643$	−16.3727	−0.645676	−0.322838	−	0.946454i	$-0.604637\pi$
$643$	−16.3727	−0.645676	−0.322838	+	0.946454i	$0.604637\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.9991	−0.825560	−0.412780	−	0.910831i	$-0.635442\pi$
$647$	−20.9991	−0.825560	−0.412780	+	0.910831i	$0.635442\pi$
$648$	0	0
$649$	8.25861	0.324179
$650$	0	0
$651$	−56.6936	−2.22200
$652$	0	0
$653$	−23.2756	−0.910846	−0.455423	−	0.890275i	$-0.650512\pi$
$653$	−23.2756	−0.910846	−0.455423	+	0.890275i	$0.650512\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−75.4080	−2.94195
$658$	0	0
$659$	−2.09639	−0.0816637	−0.0408318	−	0.999166i	$-0.513001\pi$
$659$	−2.09639	−0.0816637	−0.0408318	+	0.999166i	$0.513001\pi$
$660$	0	0
$661$	−31.3184	−1.21814	−0.609072	−	0.793115i	$-0.708458\pi$
$661$	−31.3184	−1.21814	−0.609072	+	0.793115i	$0.708458\pi$
$662$	0	0
$663$	6.02545	0.234009
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.02431	0.310703
$668$	0	0
$669$	−43.5190	−1.68254
$670$	0	0
$671$	−4.83812	−0.186773
$672$	0	0
$673$	44.3377	1.70909	0.854545	−	0.519377i	$-0.173836\pi$
$673$	44.3377	1.70909	0.854545	+	0.519377i	$0.173836\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.03772	0.116749	0.0583746	−	0.998295i	$-0.481408\pi$
$677$	3.03772	0.116749	0.0583746	+	0.998295i	$0.481408\pi$
$678$	0	0
$679$	15.9436	0.611860
$680$	0	0
$681$	26.7061	1.02338
$682$	0	0
$683$	−22.6750	−0.867635	−0.433817	−	0.901001i	$-0.642834\pi$
$683$	−22.6750	−0.867635	−0.433817	+	0.901001i	$0.642834\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−20.0338	−0.764338
$688$	0	0
$689$	−59.4313	−2.26415
$690$	0	0
$691$	−18.2394	−0.693860	−0.346930	−	0.937891i	$-0.612776\pi$
$691$	−18.2394	−0.693860	−0.346930	+	0.937891i	$0.612776\pi$
$692$	0	0
$693$	15.8469	0.601974
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.27287	−0.0482134
$698$	0	0
$699$	−43.8934	−1.66020
$700$	0	0
$701$	17.4315	0.658378	0.329189	−	0.944264i	$-0.393225\pi$
$701$	17.4315	0.658378	0.329189	+	0.944264i	$0.393225\pi$
$702$	0	0
$703$	8.67629	0.327232
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−32.8086	−1.23389
$708$	0	0
$709$	−32.4207	−1.21758	−0.608792	−	0.793330i	$-0.708346\pi$
$709$	−32.4207	−1.21758	−0.608792	+	0.793330i	$0.708346\pi$
$710$	0	0
$711$	44.6608	1.67491
$712$	0	0
$713$	7.24179	0.271207
$714$	0	0
$715$	0	0
$716$	0	0
$717$	37.1117	1.38596
$718$	0	0
$719$	0.494269	0.0184331	0.00921656	−	0.999958i	$-0.497066\pi$
$719$	0.494269	0.0184331	0.00921656	+	0.999958i	$0.497066\pi$
$720$	0	0
$721$	−30.5099	−1.13625
$722$	0	0
$723$	−56.6430	−2.10658
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.25774	−0.157911	−0.0789555	−	0.996878i	$-0.525158\pi$
$727$	−4.25774	−0.157911	−0.0789555	+	0.996878i	$0.525158\pi$
$728$	0	0
$729$	34.6543	1.28349
$730$	0	0
$731$	3.14140	0.116189
$732$	0	0
$733$	14.8525	0.548589	0.274295	−	0.961646i	$-0.411556\pi$
$733$	14.8525	0.548589	0.274295	+	0.961646i	$0.411556\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.31878	−0.0485780
$738$	0	0
$739$	−42.1389	−1.55010	−0.775052	−	0.631898i	$-0.782276\pi$
$739$	−42.1389	−1.55010	−0.775052	+	0.631898i	$0.782276\pi$
$740$	0	0
$741$	−45.9926	−1.68958
$742$	0	0
$743$	3.97014	0.145650	0.0728252	−	0.997345i	$-0.476798\pi$
$743$	3.97014	0.145650	0.0728252	+	0.997345i	$0.476798\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	119.670	4.37851
$748$	0	0
$749$	−3.55705	−0.129972
$750$	0	0
$751$	51.5382	1.88066	0.940328	−	0.340269i	$-0.110518\pi$
$751$	51.5382	1.88066	0.940328	+	0.340269i	$0.110518\pi$
$752$	0	0
$753$	40.3921	1.47197
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39.9808	−1.45313	−0.726564	−	0.687099i	$-0.758884\pi$
$757$	−39.9808	−1.45313	−0.726564	+	0.687099i	$0.758884\pi$
$758$	0	0
$759$	−2.84892	−0.103409
$760$	0	0
$761$	−20.6683	−0.749226	−0.374613	−	0.927181i	$-0.622224\pi$
$761$	−20.6683	−0.749226	−0.374613	+	0.927181i	$0.622224\pi$
$762$	0	0
$763$	46.7336	1.69187
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−47.6424	−1.72027
$768$	0	0
$769$	−0.0670558	−0.00241809	−0.00120905	−	0.999999i	$-0.500385\pi$
$769$	−0.0670558	−0.00241809	−0.00120905	+	0.999999i	$0.500385\pi$
$770$	0	0
$771$	15.7121	0.565859
$772$	0	0
$773$	21.2074	0.762778	0.381389	−	0.924415i	$-0.375446\pi$
$773$	21.2074	0.762778	0.381389	+	0.924415i	$0.375446\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.2717	0.870744
$778$	0	0
$779$	9.71588	0.348108
$780$	0	0
$781$	−7.34480	−0.262817
$782$	0	0
$783$	112.716	4.02816
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19.1047	0.681010	0.340505	−	0.940243i	$-0.389402\pi$
$787$	19.1047	0.681010	0.340505	+	0.940243i	$0.389402\pi$
$788$	0	0
$789$	−37.3577	−1.32997
$790$	0	0
$791$	2.03035	0.0721910
$792$	0	0
$793$	27.9102	0.991121
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−51.0751	−1.80917	−0.904587	−	0.426290i	$-0.859820\pi$
$797$	−51.0751	−1.80917	−0.904587	+	0.426290i	$0.859820\pi$
$798$	0	0
$799$	1.92672	0.0681625
$800$	0	0
$801$	−129.621	−4.57992
$802$	0	0
$803$	9.06286	0.319822
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.4547	−0.684837
$808$	0	0
$809$	−5.02800	−0.176775	−0.0883876	−	0.996086i	$-0.528171\pi$
$809$	−5.02800	−0.176775	−0.0883876	+	0.996086i	$0.528171\pi$
$810$	0	0
$811$	51.3571	1.80339	0.901696	−	0.432371i	$-0.142323\pi$
$811$	51.3571	1.80339	0.901696	+	0.432371i	$0.142323\pi$
$812$	0	0
$813$	12.6645	0.444162
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−23.9785	−0.838900
$818$	0	0
$819$	−91.4179	−3.19440
$820$	0	0
$821$	13.7483	0.479818	0.239909	−	0.970795i	$-0.422882\pi$
$821$	13.7483	0.479818	0.239909	+	0.970795i	$0.422882\pi$
$822$	0	0
$823$	−9.07220	−0.316237	−0.158119	−	0.987420i	$-0.550543\pi$
$823$	−9.07220	−0.316237	−0.158119	+	0.987420i	$0.550543\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.2010	−1.67611	−0.838057	−	0.545583i	$-0.816308\pi$
$827$	−48.2010	−1.67611	−0.838057	+	0.545583i	$0.816308\pi$
$828$	0	0
$829$	41.0601	1.42608	0.713038	−	0.701125i	$-0.247319\pi$
$829$	41.0601	1.42608	0.713038	+	0.701125i	$0.247319\pi$
$830$	0	0
$831$	25.5019	0.884651
$832$	0	0
$833$	−0.398199	−0.0137968
$834$	0	0
$835$	0	0
$836$	0	0
$837$	101.724	3.51611
$838$	0	0
$839$	29.3970	1.01490	0.507449	−	0.861682i	$-0.330588\pi$
$839$	29.3970	1.01490	0.507449	+	0.861682i	$0.330588\pi$
$840$	0	0
$841$	35.3895	1.22033
$842$	0	0
$843$	−19.3288	−0.665721
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24.8458	0.853711
$848$	0	0
$849$	24.6367	0.845530
$850$	0	0
$851$	−3.10036	−0.106279
$852$	0	0
$853$	−1.39913	−0.0479053	−0.0239527	−	0.999713i	$-0.507625\pi$
$853$	−1.39913	−0.0479053	−0.0239527	+	0.999713i	$0.507625\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.68808	−0.125982	−0.0629912	−	0.998014i	$-0.520064\pi$
$857$	−3.68808	−0.125982	−0.0629912	+	0.998014i	$0.520064\pi$
$858$	0	0
$859$	16.0281	0.546871	0.273435	−	0.961890i	$-0.411840\pi$
$859$	16.0281	0.546871	0.273435	+	0.961890i	$0.411840\pi$
$860$	0	0
$861$	27.1800	0.926291
$862$	0	0
$863$	−35.0979	−1.19475	−0.597374	−	0.801963i	$-0.703789\pi$
$863$	−35.0979	−1.19475	−0.597374	+	0.801963i	$0.703789\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−54.2942	−1.84393
$868$	0	0
$869$	−5.36753	−0.182081
$870$	0	0
$871$	7.60781	0.257781
$872$	0	0
$873$	−48.2759	−1.63389
$874$	0	0
$875$	0	0
$876$	0	0
$877$	53.2491	1.79809	0.899046	−	0.437853i	$-0.144261\pi$
$877$	53.2491	1.79809	0.899046	+	0.437853i	$0.144261\pi$
$878$	0	0
$879$	−43.6142	−1.47107
$880$	0	0
$881$	40.8449	1.37610	0.688050	−	0.725663i	$-0.258467\pi$
$881$	40.8449	1.37610	0.688050	+	0.725663i	$0.258467\pi$
$882$	0	0
$883$	−14.9288	−0.502393	−0.251196	−	0.967936i	$-0.580824\pi$
$883$	−14.9288	−0.502393	−0.251196	+	0.967936i	$0.580824\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.9975	1.17510	0.587550	−	0.809188i	$-0.300092\pi$
$887$	34.9975	1.17510	0.587550	+	0.809188i	$0.300092\pi$
$888$	0	0
$889$	−19.1496	−0.642259
$890$	0	0
$891$	−20.4692	−0.685742
$892$	0	0
$893$	−14.7068	−0.492143
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.4349	0.548745
$898$	0	0
$899$	58.1104	1.93809
$900$	0	0
$901$	−4.26799	−0.142187
$902$	0	0
$903$	−67.0793	−2.23226
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.3920	−0.544288	−0.272144	−	0.962257i	$-0.587733\pi$
$907$	−16.3920	−0.544288	−0.272144	+	0.962257i	$0.587733\pi$
$908$	0	0
$909$	99.3416	3.29495
$910$	0	0
$911$	−44.1111	−1.46147	−0.730733	−	0.682663i	$-0.760822\pi$
$911$	−44.1111	−1.46147	−0.730733	+	0.682663i	$0.760822\pi$
$912$	0	0
$913$	−14.3825	−0.475991
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31.3076	1.03387
$918$	0	0
$919$	28.5629	0.942203	0.471101	−	0.882079i	$-0.343857\pi$
$919$	28.5629	0.942203	0.471101	+	0.882079i	$0.343857\pi$
$920$	0	0
$921$	32.4724	1.07000
$922$	0	0
$923$	42.3708	1.39465
$924$	0	0
$925$	0	0
$926$	0	0
$927$	92.3813	3.03420
$928$	0	0
$929$	16.2273	0.532400	0.266200	−	0.963918i	$-0.414232\pi$
$929$	16.2273	0.532400	0.266200	+	0.963918i	$0.414232\pi$
$930$	0	0
$931$	3.03948	0.0996148
$932$	0	0
$933$	−90.9372	−2.97715
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.65418	−0.184714	−0.0923571	−	0.995726i	$-0.529440\pi$
$937$	−5.65418	−0.184714	−0.0923571	+	0.995726i	$0.529440\pi$
$938$	0	0
$939$	−46.5192	−1.51810
$940$	0	0
$941$	−25.6246	−0.835338	−0.417669	−	0.908599i	$-0.637153\pi$
$941$	−25.6246	−0.835338	−0.417669	+	0.908599i	$0.637153\pi$
$942$	0	0
$943$	−3.47185	−0.113059
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.248718	0.00808223	0.00404112	−	0.999992i	$-0.498714\pi$
$947$	0.248718	0.00808223	0.00404112	+	0.999992i	$0.498714\pi$
$948$	0	0
$949$	−52.2820	−1.69715
$950$	0	0
$951$	47.5914	1.54326
$952$	0	0
$953$	−39.1737	−1.26896	−0.634481	−	0.772939i	$-0.718786\pi$
$953$	−39.1737	−1.26896	−0.634481	+	0.772939i	$0.718786\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−22.8606	−0.738978
$958$	0	0
$959$	−7.94423	−0.256533
$960$	0	0
$961$	21.4435	0.691726
$962$	0	0
$963$	10.7704	0.347073
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.04114	0.0656386	0.0328193	−	0.999461i	$-0.489551\pi$
$967$	2.04114	0.0656386	0.0328193	+	0.999461i	$0.489551\pi$
$968$	0	0
$969$	−3.30290	−0.106105
$970$	0	0
$971$	−13.6713	−0.438734	−0.219367	−	0.975642i	$-0.570399\pi$
$971$	−13.6713	−0.438734	−0.219367	+	0.975642i	$0.570399\pi$
$972$	0	0
$973$	30.5387	0.979025
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.8777	1.46776	0.733878	−	0.679281i	$-0.237708\pi$
$977$	45.8777	1.46776	0.733878	+	0.679281i	$0.237708\pi$
$978$	0	0
$979$	15.5784	0.497887
$980$	0	0
$981$	−141.505	−4.51792
$982$	0	0
$983$	−59.6356	−1.90208	−0.951039	−	0.309070i	$-0.899982\pi$
$983$	−59.6356	−1.90208	−0.951039	+	0.309070i	$0.899982\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−41.1418	−1.30956
$988$	0	0
$989$	8.56841	0.272460
$990$	0	0
$991$	−42.3396	−1.34496	−0.672480	−	0.740115i	$-0.734771\pi$
$991$	−42.3396	−1.34496	−0.672480	+	0.740115i	$0.734771\pi$
$992$	0	0
$993$	−64.8656	−2.05845
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.02379	−0.222446	−0.111223	−	0.993795i	$-0.535477\pi$
$997$	−7.02379	−0.222446	−0.111223	+	0.993795i	$0.535477\pi$
$998$	0	0
$999$	−43.5504	−1.37787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.o.1.6		6
4.3	odd	2		9200.2.a.cx.1.1		6
5.2	odd	4		460.2.c.a.369.1	✓	12
5.3	odd	4		460.2.c.a.369.12	yes	12
5.4	even	2		2300.2.a.n.1.1		6
15.2	even	4		4140.2.f.b.829.3		12
15.8	even	4		4140.2.f.b.829.4		12
20.3	even	4		1840.2.e.f.369.1		12
20.7	even	4		1840.2.e.f.369.12		12
20.19	odd	2		9200.2.a.cy.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.1	✓	12	5.2	odd	4
460.2.c.a.369.12	yes	12	5.3	odd	4
1840.2.e.f.369.1		12	20.3	even	4
1840.2.e.f.369.12		12	20.7	even	4
2300.2.a.n.1.1		6	5.4	even	2
2300.2.a.o.1.6		6	1.1	even	1	trivial
4140.2.f.b.829.3		12	15.2	even	4
4140.2.f.b.829.4		12	15.8	even	4
9200.2.a.cx.1.1		6	4.3	odd	2
9200.2.a.cy.1.6		6	20.19	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q+3.21923 q^{3} -2.43185 q^{7} +7.36343 q^{9} -0.884969 q^{11} +5.10522 q^{13} +0.366626 q^{17} -2.79847 q^{19} -7.82867 q^{21} +1.00000 q^{23} +14.0469 q^{27} +8.02431 q^{29} +7.24179 q^{31} -2.84892 q^{33} -3.10036 q^{37} +16.4349 q^{39} -3.47185 q^{41} +8.56841 q^{43} +5.25528 q^{47} -1.08612 q^{49} +1.18025 q^{51} -11.6413 q^{53} -9.00892 q^{57} -9.33209 q^{59} +5.46699 q^{61} -17.9067 q^{63} +1.49020 q^{67} +3.21923 q^{69} +8.29949 q^{71} -10.2409 q^{73} +2.15211 q^{77} +6.06522 q^{79} +23.1298 q^{81} +16.2520 q^{83} +25.8321 q^{87} -17.6033 q^{89} -12.4151 q^{91} +23.3130 q^{93} -6.55618 q^{97} -6.51641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{17} + 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} + 9 q^{31} - 10 q^{33} + 21 q^{37} - 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49}+ \cdots - 16 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^17 + 4 * q^19 + 6 * q^23 + 22 * q^27 + 5 * q^29 + 9 * q^31 - 10 * q^33 + 21 * q^37 - 8 * q^39 - q^41 + 16 * q^43 + 16 * q^47 + 19 * q^49 - 12 * q^51 - q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 19 * q^63 + 25 * q^67 + 4 * q^69 - 17 * q^71 - 14 * q^73 + 20 * q^77 + 10 * q^79 + 14 * q^81 + 21 * q^83 + 64 * q^87 - 24 * q^89 - 4 * q^91 + 4 * q^97 - 16 * q^99

Introduction

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