LMFDB - Embedded newform 2300.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1,2300)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$18.3655924649$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.239123$ 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)

$f(q)$	$=$	$q+0.760877 q^{3} +0.478247 q^{7} -2.42107 q^{9} +O(q^{10})$	$q+0.760877 q^{3} +0.478247 q^{7} -2.42107 q^{9} -1.52175 q^{11} -5.42107 q^{13} +4.36389 q^{17} +3.88564 q^{19} +0.363887 q^{21} -1.00000 q^{23} -4.12476 q^{27} +6.06758 q^{29} -9.70370 q^{31} -1.15787 q^{33} -0.478247 q^{37} -4.12476 q^{39} -9.96690 q^{41} +6.84213 q^{43} -6.82846 q^{47} -6.77128 q^{49} +3.32038 q^{51} -14.2495 q^{53} +2.95649 q^{57} -8.32038 q^{59} -12.8421 q^{61} -1.15787 q^{63} +16.1352 q^{67} -0.760877 q^{69} -7.05718 q^{71} +2.29630 q^{73} -0.727773 q^{77} -1.15787 q^{79} +4.12476 q^{81} -8.36389 q^{83} +4.61668 q^{87} -3.32038 q^{89} -2.59261 q^{91} -7.38332 q^{93} -2.84213 q^{97} +3.68427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^7 + q^9	$ 3 q + 2 q^{3} + 2 q^{7} + q^{9} - 4 q^{11} - 8 q^{13} - 4 q^{17} - 6 q^{19} - 16 q^{21} - 3 q^{23} + 5 q^{27} - 8 q^{29} - 20 q^{31} - 20 q^{33} - 2 q^{37} + 5 q^{39} + 4 q^{41} + 4 q^{43} + 6 q^{47} + 15 q^{49} - 6 q^{51} - 8 q^{53} + 10 q^{57} - 9 q^{59} - 22 q^{61} - 20 q^{63} - 4 q^{67} - 2 q^{69} - 30 q^{71} + 16 q^{73} + 32 q^{77} - 20 q^{79} - 5 q^{81} - 8 q^{83} + 7 q^{87} + 6 q^{89} - 26 q^{91} - 29 q^{93} + 8 q^{97} - 22 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^7 + q^9 - 4 * q^11 - 8 * q^13 - 4 * q^17 - 6 * q^19 - 16 * q^21 - 3 * q^23 + 5 * q^27 - 8 * q^29 - 20 * q^31 - 20 * q^33 - 2 * q^37 + 5 * q^39 + 4 * q^41 + 4 * q^43 + 6 * q^47 + 15 * q^49 - 6 * q^51 - 8 * q^53 + 10 * q^57 - 9 * q^59 - 22 * q^61 - 20 * q^63 - 4 * q^67 - 2 * q^69 - 30 * q^71 + 16 * q^73 + 32 * q^77 - 20 * q^79 - 5 * q^81 - 8 * q^83 + 7 * q^87 + 6 * q^89 - 26 * q^91 - 29 * q^93 + 8 * q^97 - 22 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.760877	0.439292	0.219646	−	0.975580i	$-0.429510\pi$
$3$	0.760877	0.439292	0.219646	+	0.975580i	$0.429510\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.478247	0.180760	0.0903801	−	0.995907i	$-0.471192\pi$
$7$	0.478247	0.180760	0.0903801	+	0.995907i	$0.471192\pi$
$8$	0	0
$9$	−2.42107	−0.807022
$10$	0	0
$11$	−1.52175	−0.458826	−0.229413	−	0.973329i	$-0.573681\pi$
$11$	−1.52175	−0.458826	−0.229413	+	0.973329i	$0.573681\pi$
$12$	0	0
$13$	−5.42107	−1.50353	−0.751767	−	0.659429i	$-0.770798\pi$
$13$	−5.42107	−1.50353	−0.751767	+	0.659429i	$0.770798\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.36389	1.05840	0.529199	−	0.848498i	$-0.322493\pi$
$17$	4.36389	1.05840	0.529199	+	0.848498i	$0.322493\pi$
$18$	0	0
$19$	3.88564	0.891427	0.445713	−	0.895176i	$-0.352950\pi$
$19$	3.88564	0.891427	0.445713	+	0.895176i	$0.352950\pi$
$20$	0	0
$21$	0.363887	0.0794066
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.12476	−0.793811
$28$	0	0
$29$	6.06758	1.12672	0.563361	−	0.826211i	$-0.309508\pi$
$29$	6.06758	1.12672	0.563361	+	0.826211i	$0.309508\pi$
$30$	0	0
$31$	−9.70370	−1.74284	−0.871418	−	0.490542i	$-0.836799\pi$
$31$	−9.70370	−1.74284	−0.871418	+	0.490542i	$0.836799\pi$
$32$	0	0
$33$	−1.15787	−0.201559
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.478247	−0.0786232	−0.0393116	−	0.999227i	$-0.512517\pi$
$37$	−0.478247	−0.0786232	−0.0393116	+	0.999227i	$0.512517\pi$
$38$	0	0
$39$	−4.12476	−0.660491
$40$	0	0
$41$	−9.96690	−1.55657	−0.778284	−	0.627913i	$-0.783910\pi$
$41$	−9.96690	−1.55657	−0.778284	+	0.627913i	$0.783910\pi$
$42$	0	0
$43$	6.84213	1.04342	0.521708	−	0.853124i	$-0.325295\pi$
$43$	6.84213	1.04342	0.521708	+	0.853124i	$0.325295\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.82846	−0.996033	−0.498017	−	0.867167i	$-0.665938\pi$
$47$	−6.82846	−0.996033	−0.498017	+	0.867167i	$0.665938\pi$
$48$	0	0
$49$	−6.77128	−0.967326
$50$	0	0
$51$	3.32038	0.464946
$52$	0	0
$53$	−14.2495	−1.95732	−0.978662	−	0.205479i	$-0.934125\pi$
$53$	−14.2495	−1.95732	−0.978662	+	0.205479i	$0.934125\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.95649	0.391597
$58$	0	0
$59$	−8.32038	−1.08322	−0.541611	−	0.840630i	$-0.682185\pi$
$59$	−8.32038	−1.08322	−0.541611	+	0.840630i	$0.682185\pi$
$60$	0	0
$61$	−12.8421	−1.64427	−0.822133	−	0.569295i	$-0.807216\pi$
$61$	−12.8421	−1.64427	−0.822133	+	0.569295i	$0.807216\pi$
$62$	0	0
$63$	−1.15787	−0.145878
$64$	0	0
$65$	0	0
$66$	0	0
$67$	16.1352	1.97122	0.985612	−	0.169023i	$-0.0540611\pi$
$67$	16.1352	1.97122	0.985612	+	0.169023i	$0.0540611\pi$
$68$	0	0
$69$	−0.760877	−0.0915988
$70$	0	0
$71$	−7.05718	−0.837533	−0.418767	−	0.908094i	$-0.637537\pi$
$71$	−7.05718	−0.837533	−0.418767	+	0.908094i	$0.637537\pi$
$72$	0	0
$73$	2.29630	0.268762	0.134381	−	0.990930i	$-0.457095\pi$
$73$	2.29630	0.268762	0.134381	+	0.990930i	$0.457095\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.727773	−0.0829375
$78$	0	0
$79$	−1.15787	−0.130270	−0.0651351	−	0.997876i	$-0.520748\pi$
$79$	−1.15787	−0.130270	−0.0651351	+	0.997876i	$0.520748\pi$
$80$	0	0
$81$	4.12476	0.458307
$82$	0	0
$83$	−8.36389	−0.918056	−0.459028	−	0.888422i	$-0.651802\pi$
$83$	−8.36389	−0.918056	−0.459028	+	0.888422i	$0.651802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.61668	0.494960
$88$	0	0
$89$	−3.32038	−0.351960	−0.175980	−	0.984394i	$-0.556309\pi$
$89$	−3.32038	−0.351960	−0.175980	+	0.984394i	$0.556309\pi$
$90$	0	0
$91$	−2.59261	−0.271779
$92$	0	0
$93$	−7.38332	−0.765614
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.84213	−0.288575	−0.144287	−	0.989536i	$-0.546089\pi$
$97$	−2.84213	−0.288575	−0.144287	+	0.989536i	$0.546089\pi$
$98$	0	0
$99$	3.68427	0.370283
$100$	0	0
$101$	10.7713	1.07178	0.535891	−	0.844287i	$-0.319976\pi$
$101$	10.7713	1.07178	0.535891	+	0.844287i	$0.319976\pi$
$102$	0	0
$103$	11.0435	1.08815	0.544075	−	0.839037i	$-0.316881\pi$
$103$	11.0435	1.08815	0.544075	+	0.839037i	$0.316881\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.63611	0.351516	0.175758	−	0.984433i	$-0.443762\pi$
$107$	3.63611	0.351516	0.175758	+	0.984433i	$0.443762\pi$
$108$	0	0
$109$	−4.08701	−0.391465	−0.195732	−	0.980657i	$-0.562708\pi$
$109$	−4.08701	−0.391465	−0.195732	+	0.980657i	$0.562708\pi$
$110$	0	0
$111$	−0.363887	−0.0345386
$112$	0	0
$113$	2.24953	0.211618	0.105809	−	0.994386i	$-0.466257\pi$
$113$	2.24953	0.211618	0.105809	+	0.994386i	$0.466257\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	13.1248	1.21338
$118$	0	0
$119$	2.08701	0.191316
$120$	0	0
$121$	−8.68427	−0.789479
$122$	0	0
$123$	−7.58358	−0.683788
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.4315	0.925643	0.462822	−	0.886451i	$-0.346837\pi$
$127$	10.4315	0.925643	0.462822	+	0.886451i	$0.346837\pi$
$128$	0	0
$129$	5.20602	0.458364
$130$	0	0
$131$	−5.32614	−0.465347	−0.232673	−	0.972555i	$-0.574747\pi$
$131$	−5.32614	−0.465347	−0.232673	+	0.972555i	$0.574747\pi$
$132$	0	0
$133$	1.85829	0.161135
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.0917	1.63111	0.815555	−	0.578679i	$-0.196432\pi$
$137$	19.0917	1.63111	0.815555	+	0.578679i	$0.196432\pi$
$138$	0	0
$139$	−13.8993	−1.17892	−0.589462	−	0.807796i	$-0.700660\pi$
$139$	−13.8993	−1.17892	−0.589462	+	0.807796i	$0.700660\pi$
$140$	0	0
$141$	−5.19562	−0.437550
$142$	0	0
$143$	8.24953	0.689860
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.15211	−0.424939
$148$	0	0
$149$	21.6843	1.77644	0.888222	−	0.459414i	$-0.151941\pi$
$149$	21.6843	1.77644	0.888222	+	0.459414i	$0.151941\pi$
$150$	0	0
$151$	−18.8285	−1.53224	−0.766119	−	0.642699i	$-0.777815\pi$
$151$	−18.8285	−1.53224	−0.766119	+	0.642699i	$0.777815\pi$
$152$	0	0
$153$	−10.5653	−0.854151
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.2060	−1.05396	−0.526978	−	0.849879i	$-0.676675\pi$
$157$	−13.2060	−1.05396	−0.526978	+	0.849879i	$0.676675\pi$
$158$	0	0
$159$	−10.8421	−0.859837
$160$	0	0
$161$	−0.478247	−0.0376911
$162$	0	0
$163$	−19.7518	−1.54708	−0.773542	−	0.633745i	$-0.781517\pi$
$163$	−19.7518	−1.54708	−0.773542	+	0.633745i	$0.781517\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.27687	−0.563101	−0.281551	−	0.959546i	$-0.590849\pi$
$167$	−7.27687	−0.563101	−0.281551	+	0.959546i	$0.590849\pi$
$168$	0	0
$169$	16.3880	1.26061
$170$	0	0
$171$	−9.40739	−0.719401
$172$	0	0
$173$	−7.72777	−0.587532	−0.293766	−	0.955877i	$-0.594909\pi$
$173$	−7.72777	−0.587532	−0.293766	+	0.955877i	$0.594909\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.33078	−0.475851
$178$	0	0
$179$	−17.8389	−1.33334	−0.666670	−	0.745353i	$-0.732281\pi$
$179$	−17.8389	−1.33334	−0.666670	+	0.745353i	$0.732281\pi$
$180$	0	0
$181$	5.68427	0.422508	0.211254	−	0.977431i	$-0.432245\pi$
$181$	5.68427	0.422508	0.211254	+	0.977431i	$0.432245\pi$
$182$	0	0
$183$	−9.77128	−0.722314
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.64076	−0.485620
$188$	0	0
$189$	−1.97265	−0.143489
$190$	0	0
$191$	20.2495	1.46520	0.732602	−	0.680657i	$-0.238306\pi$
$191$	20.2495	1.46520	0.732602	+	0.680657i	$0.238306\pi$
$192$	0	0
$193$	11.3261	0.815273	0.407637	−	0.913144i	$-0.366353\pi$
$193$	11.3261	0.815273	0.407637	+	0.913144i	$0.366353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.9234	1.63322	0.816612	−	0.577187i	$-0.195850\pi$
$197$	22.9234	1.63322	0.816612	+	0.577187i	$0.195850\pi$
$198$	0	0
$199$	−20.2495	−1.43545	−0.717725	−	0.696326i	$-0.754817\pi$
$199$	−20.2495	−1.43545	−0.717725	+	0.696326i	$0.754817\pi$
$200$	0	0
$201$	12.2769	0.865944
$202$	0	0
$203$	2.90180	0.203666
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.42107	0.168276
$208$	0	0
$209$	−5.91299	−0.409010
$210$	0	0
$211$	−13.2769	−0.914018	−0.457009	−	0.889462i	$-0.651079\pi$
$211$	−13.2769	−0.914018	−0.457009	+	0.889462i	$0.651079\pi$
$212$	0	0
$213$	−5.36964	−0.367922
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.64076	−0.315035
$218$	0	0
$219$	1.74720	0.118065
$220$	0	0
$221$	−23.6569	−1.59134
$222$	0	0
$223$	3.36389	0.225263	0.112631	−	0.993637i	$-0.464072\pi$
$223$	3.36389	0.225263	0.112631	+	0.993637i	$0.464072\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3365	0.686060	0.343030	−	0.939324i	$-0.388547\pi$
$227$	10.3365	0.686060	0.343030	+	0.939324i	$0.388547\pi$
$228$	0	0
$229$	−16.0482	−1.06049	−0.530246	−	0.847844i	$-0.677900\pi$
$229$	−16.0482	−1.06049	−0.530246	+	0.847844i	$0.677900\pi$
$230$	0	0
$231$	−0.553746	−0.0364338
$232$	0	0
$233$	9.21969	0.604002	0.302001	−	0.953308i	$-0.402345\pi$
$233$	9.21969	0.604002	0.302001	+	0.953308i	$0.402345\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.880994	−0.0572267
$238$	0	0
$239$	12.7609	0.825432	0.412716	−	0.910860i	$-0.364580\pi$
$239$	12.7609	0.825432	0.412716	+	0.910860i	$0.364580\pi$
$240$	0	0
$241$	27.4555	1.76857	0.884284	−	0.466950i	$-0.154647\pi$
$241$	27.4555	1.76857	0.884284	+	0.466950i	$0.154647\pi$
$242$	0	0
$243$	15.5127	0.995142
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−21.0643	−1.34029
$248$	0	0
$249$	−6.36389	−0.403295
$250$	0	0
$251$	−7.40739	−0.467551	−0.233775	−	0.972291i	$-0.575108\pi$
$251$	−7.40739	−0.467551	−0.233775	+	0.972291i	$0.575108\pi$
$252$	0	0
$253$	1.52175	0.0956718
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.5322	0.906491	0.453246	−	0.891386i	$-0.350266\pi$
$257$	14.5322	0.906491	0.453246	+	0.891386i	$0.350266\pi$
$258$	0	0
$259$	−0.228720	−0.0142120
$260$	0	0
$261$	−14.6900	−0.909290
$262$	0	0
$263$	6.22872	0.384079	0.192040	−	0.981387i	$-0.438490\pi$
$263$	6.22872	0.384079	0.192040	+	0.981387i	$0.438490\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.52640	−0.154613
$268$	0	0
$269$	2.37756	0.144962	0.0724812	−	0.997370i	$-0.476908\pi$
$269$	2.37756	0.144962	0.0724812	+	0.997370i	$0.476908\pi$
$270$	0	0
$271$	17.8629	1.08510	0.542548	−	0.840025i	$-0.317460\pi$
$271$	17.8629	1.08510	0.542548	+	0.840025i	$0.317460\pi$
$272$	0	0
$273$	−1.97265	−0.119390
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.03310	−0.122157	−0.0610787	−	0.998133i	$-0.519454\pi$
$277$	−2.03310	−0.122157	−0.0610787	+	0.998133i	$0.519454\pi$
$278$	0	0
$279$	23.4933	1.40651
$280$	0	0
$281$	24.3639	1.45343	0.726714	−	0.686940i	$-0.241047\pi$
$281$	24.3639	1.45343	0.726714	+	0.686940i	$0.241047\pi$
$282$	0	0
$283$	21.3412	1.26860	0.634301	−	0.773086i	$-0.281288\pi$
$283$	21.3412	1.26860	0.634301	+	0.773086i	$0.281288\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.76663	−0.281366
$288$	0	0
$289$	2.04351	0.120206
$290$	0	0
$291$	−2.16251	−0.126769
$292$	0	0
$293$	−15.4074	−0.900110	−0.450055	−	0.893001i	$-0.648596\pi$
$293$	−15.4074	−0.900110	−0.450055	+	0.893001i	$0.648596\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.27687	0.364221
$298$	0	0
$299$	5.42107	0.313508
$300$	0	0
$301$	3.27223	0.188608
$302$	0	0
$303$	8.19562	0.470826
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.7278	0.726412	0.363206	−	0.931709i	$-0.381682\pi$
$307$	12.7278	0.726412	0.363206	+	0.931709i	$0.381682\pi$
$308$	0	0
$309$	8.40275	0.478016
$310$	0	0
$311$	−20.7472	−1.17647	−0.588233	−	0.808692i	$-0.700176\pi$
$311$	−20.7472	−1.17647	−0.588233	+	0.808692i	$0.700176\pi$
$312$	0	0
$313$	−16.2495	−0.918478	−0.459239	−	0.888313i	$-0.651878\pi$
$313$	−16.2495	−0.918478	−0.459239	+	0.888313i	$0.651878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.6408	0.709976	0.354988	−	0.934871i	$-0.384485\pi$
$317$	12.6408	0.709976	0.354988	+	0.934871i	$0.384485\pi$
$318$	0	0
$319$	−9.23337	−0.516969
$320$	0	0
$321$	2.76663	0.154418
$322$	0	0
$323$	16.9565	0.943485
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.10971	−0.171968
$328$	0	0
$329$	−3.26569	−0.180043
$330$	0	0
$331$	5.55486	0.305323	0.152661	−	0.988279i	$-0.451216\pi$
$331$	5.55486	0.305323	0.152661	+	0.988279i	$0.451216\pi$
$332$	0	0
$333$	1.15787	0.0634507
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.0435	−0.819472	−0.409736	−	0.912204i	$-0.634379\pi$
$337$	−15.0435	−0.819472	−0.409736	+	0.912204i	$0.634379\pi$
$338$	0	0
$339$	1.71161	0.0929620
$340$	0	0
$341$	14.7666	0.799658
$342$	0	0
$343$	−6.58607	−0.355614
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.3639	−1.36160	−0.680802	−	0.732467i	$-0.738369\pi$
$347$	−25.3639	−1.36160	−0.680802	+	0.732467i	$0.738369\pi$
$348$	0	0
$349$	4.46457	0.238983	0.119492	−	0.992835i	$-0.461874\pi$
$349$	4.46457	0.238983	0.119492	+	0.992835i	$0.461874\pi$
$350$	0	0
$351$	22.3606	1.19352
$352$	0	0
$353$	27.0974	1.44225	0.721125	−	0.692805i	$-0.243625\pi$
$353$	27.0974	1.44225	0.721125	+	0.692805i	$0.243625\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.58796	0.0840438
$358$	0	0
$359$	6.16251	0.325245	0.162622	−	0.986688i	$-0.448005\pi$
$359$	6.16251	0.325245	0.162622	+	0.986688i	$0.448005\pi$
$360$	0	0
$361$	−3.90180	−0.205358
$362$	0	0
$363$	−6.60766	−0.346812
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−27.1787	−1.41872	−0.709358	−	0.704849i	$-0.751015\pi$
$367$	−27.1787	−1.41872	−0.709358	+	0.704849i	$0.751015\pi$
$368$	0	0
$369$	24.1305	1.25618
$370$	0	0
$371$	−6.81479	−0.353806
$372$	0	0
$373$	−15.3204	−0.793259	−0.396630	−	0.917979i	$-0.629820\pi$
$373$	−15.3204	−0.793259	−0.396630	+	0.917979i	$0.629820\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−32.8928	−1.69406
$378$	0	0
$379$	−13.7713	−0.707383	−0.353692	−	0.935362i	$-0.615074\pi$
$379$	−13.7713	−0.707383	−0.353692	+	0.935362i	$0.615074\pi$
$380$	0	0
$381$	7.93706	0.406628
$382$	0	0
$383$	−25.0917	−1.28212	−0.641062	−	0.767489i	$-0.721506\pi$
$383$	−25.0917	−1.28212	−0.641062	+	0.767489i	$0.721506\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16.5653	−0.842060
$388$	0	0
$389$	25.1787	1.27661	0.638305	−	0.769784i	$-0.279636\pi$
$389$	25.1787	1.27661	0.638305	+	0.769784i	$0.279636\pi$
$390$	0	0
$391$	−4.36389	−0.220691
$392$	0	0
$393$	−4.05253	−0.204423
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.9371	−0.850047	−0.425023	−	0.905182i	$-0.639734\pi$
$397$	−16.9371	−0.850047	−0.425023	+	0.905182i	$0.639734\pi$
$398$	0	0
$399$	1.41393	0.0707852
$400$	0	0
$401$	8.04815	0.401906	0.200953	−	0.979601i	$-0.435596\pi$
$401$	8.04815	0.401906	0.200953	+	0.979601i	$0.435596\pi$
$402$	0	0
$403$	52.6044	2.62041
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.727773	0.0360744
$408$	0	0
$409$	33.7382	1.66825	0.834123	−	0.551579i	$-0.185974\pi$
$409$	33.7382	1.66825	0.834123	+	0.551579i	$0.185974\pi$
$410$	0	0
$411$	14.5264	0.716534
$412$	0	0
$413$	−3.97919	−0.195803
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.5757	−0.517892
$418$	0	0
$419$	−15.3204	−0.748450	−0.374225	−	0.927338i	$-0.622091\pi$
$419$	−15.3204	−0.748450	−0.374225	+	0.927338i	$0.622091\pi$
$420$	0	0
$421$	−2.08701	−0.101715	−0.0508574	−	0.998706i	$-0.516195\pi$
$421$	−2.08701	−0.101715	−0.0508574	+	0.998706i	$0.516195\pi$
$422$	0	0
$423$	16.5322	0.803821
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.14171	−0.297218
$428$	0	0
$429$	6.27687	0.303050
$430$	0	0
$431$	−16.9018	−0.814131	−0.407066	−	0.913399i	$-0.633448\pi$
$431$	−16.9018	−0.814131	−0.407066	+	0.913399i	$0.633448\pi$
$432$	0	0
$433$	−17.3412	−0.833364	−0.416682	−	0.909052i	$-0.636807\pi$
$433$	−17.3412	−0.833364	−0.416682	+	0.909052i	$0.636807\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.88564	−0.185875
$438$	0	0
$439$	−8.98960	−0.429050	−0.214525	−	0.976718i	$-0.568820\pi$
$439$	−8.98960	−0.429050	−0.214525	+	0.976718i	$0.568820\pi$
$440$	0	0
$441$	16.3937	0.780653
$442$	0	0
$443$	−15.5562	−0.739099	−0.369549	−	0.929211i	$-0.620488\pi$
$443$	−15.5562	−0.739099	−0.369549	+	0.929211i	$0.620488\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.4991	0.780378
$448$	0	0
$449$	13.3157	0.628408	0.314204	−	0.949355i	$-0.398262\pi$
$449$	13.3157	0.628408	0.314204	+	0.949355i	$0.398262\pi$
$450$	0	0
$451$	15.1672	0.714194
$452$	0	0
$453$	−14.3261	−0.673101
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.4120	0.674167	0.337083	−	0.941475i	$-0.390560\pi$
$457$	14.4120	0.674167	0.337083	+	0.941475i	$0.390560\pi$
$458$	0	0
$459$	−18.0000	−0.840168
$460$	0	0
$461$	33.9201	1.57982	0.789909	−	0.613224i	$-0.210128\pi$
$461$	33.9201	1.57982	0.789909	+	0.613224i	$0.210128\pi$
$462$	0	0
$463$	−3.13517	−0.145704	−0.0728518	−	0.997343i	$-0.523210\pi$
$463$	−3.13517	−0.145704	−0.0728518	+	0.997343i	$0.523210\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5218	0.718261	0.359131	−	0.933287i	$-0.383073\pi$
$467$	15.5218	0.718261	0.359131	+	0.933287i	$0.383073\pi$
$468$	0	0
$469$	7.71659	0.356319
$470$	0	0
$471$	−10.0482	−0.462994
$472$	0	0
$473$	−10.4120	−0.478746
$474$	0	0
$475$	0	0
$476$	0	0
$477$	34.4991	1.57960
$478$	0	0
$479$	11.8583	0.541819	0.270910	−	0.962605i	$-0.412676\pi$
$479$	11.8583	0.541819	0.270910	+	0.962605i	$0.412676\pi$
$480$	0	0
$481$	2.59261	0.118213
$482$	0	0
$483$	−0.363887	−0.0165574
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.0298	−0.862324	−0.431162	−	0.902275i	$-0.641896\pi$
$487$	−19.0298	−0.862324	−0.431162	+	0.902275i	$0.641896\pi$
$488$	0	0
$489$	−15.0287	−0.679622
$490$	0	0
$491$	−7.58358	−0.342242	−0.171121	−	0.985250i	$-0.554739\pi$
$491$	−7.58358	−0.342242	−0.171121	+	0.985250i	$0.554739\pi$
$492$	0	0
$493$	26.4782	1.19252
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.37507	−0.151393
$498$	0	0
$499$	−4.59836	−0.205851	−0.102926	−	0.994689i	$-0.532820\pi$
$499$	−4.59836	−0.205851	−0.102926	+	0.994689i	$0.532820\pi$
$500$	0	0
$501$	−5.53680	−0.247366
$502$	0	0
$503$	35.8468	1.59833	0.799164	−	0.601112i	$-0.205276\pi$
$503$	35.8468	1.59833	0.799164	+	0.601112i	$0.205276\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.4692	0.553777
$508$	0	0
$509$	8.28263	0.367121	0.183561	−	0.983008i	$-0.441238\pi$
$509$	8.28263	0.367121	0.183561	+	0.983008i	$0.441238\pi$
$510$	0	0
$511$	1.09820	0.0485815
$512$	0	0
$513$	−16.0273	−0.707625
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.3912	0.457006
$518$	0	0
$519$	−5.87988	−0.258098
$520$	0	0
$521$	25.6843	1.12525	0.562624	−	0.826713i	$-0.309792\pi$
$521$	25.6843	1.12525	0.562624	+	0.826713i	$0.309792\pi$
$522$	0	0
$523$	−2.84213	−0.124278	−0.0621389	−	0.998068i	$-0.519792\pi$
$523$	−2.84213	−0.124278	−0.0621389	+	0.998068i	$0.519792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.3458	−1.84461
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	20.1442	0.874184
$532$	0	0
$533$	54.0312	2.34035
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−13.5732	−0.585726
$538$	0	0
$539$	10.3042	0.443834
$540$	0	0
$541$	−3.48865	−0.149989	−0.0749944	−	0.997184i	$-0.523894\pi$
$541$	−3.48865	−0.149989	−0.0749944	+	0.997184i	$0.523894\pi$
$542$	0	0
$543$	4.32503	0.185605
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−26.8011	−1.14593	−0.572966	−	0.819579i	$-0.694207\pi$
$547$	−26.8011	−1.14593	−0.572966	+	0.819579i	$0.694207\pi$
$548$	0	0
$549$	31.0917	1.32696
$550$	0	0
$551$	23.5764	1.00439
$552$	0	0
$553$	−0.553746	−0.0235477
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.6361	0.493038	0.246519	−	0.969138i	$-0.420713\pi$
$557$	11.6361	0.493038	0.246519	+	0.969138i	$0.420713\pi$
$558$	0	0
$559$	−37.0917	−1.56881
$560$	0	0
$561$	−5.05280	−0.213329
$562$	0	0
$563$	27.8309	1.17293	0.586467	−	0.809973i	$-0.300518\pi$
$563$	27.8309	1.17293	0.586467	+	0.809973i	$0.300518\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.97265	0.0828437
$568$	0	0
$569$	−2.70043	−0.113208	−0.0566039	−	0.998397i	$-0.518027\pi$
$569$	−2.70043	−0.113208	−0.0566039	+	0.998397i	$0.518027\pi$
$570$	0	0
$571$	40.4717	1.69369	0.846844	−	0.531841i	$-0.178500\pi$
$571$	40.4717	1.69369	0.846844	+	0.531841i	$0.178500\pi$
$572$	0	0
$573$	15.4074	0.643653
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.63036	−0.234395	−0.117197	−	0.993109i	$-0.537391\pi$
$577$	−5.63036	−0.234395	−0.117197	+	0.993109i	$0.537391\pi$
$578$	0	0
$579$	8.61779	0.358143
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	21.6843	0.898071
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.9155	−1.19347	−0.596735	−	0.802439i	$-0.703536\pi$
$587$	−28.9155	−1.19347	−0.596735	+	0.802439i	$0.703536\pi$
$588$	0	0
$589$	−37.7051	−1.55361
$590$	0	0
$591$	17.4419	0.717463
$592$	0	0
$593$	4.31573	0.177226	0.0886130	−	0.996066i	$-0.471757\pi$
$593$	4.31573	0.177226	0.0886130	+	0.996066i	$0.471757\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.4074	−0.630583
$598$	0	0
$599$	−3.82133	−0.156135	−0.0780676	−	0.996948i	$-0.524875\pi$
$599$	−3.82133	−0.156135	−0.0780676	+	0.996948i	$0.524875\pi$
$600$	0	0
$601$	−18.6602	−0.761165	−0.380583	−	0.924747i	$-0.624276\pi$
$601$	−18.6602	−0.761165	−0.380583	+	0.924747i	$0.624276\pi$
$602$	0	0
$603$	−39.0643	−1.59082
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.402747	−0.0163470	−0.00817350	−	0.999967i	$-0.502602\pi$
$607$	−0.402747	−0.0163470	−0.00817350	+	0.999967i	$0.502602\pi$
$608$	0	0
$609$	2.20791	0.0894691
$610$	0	0
$611$	37.0175	1.49757
$612$	0	0
$613$	22.8903	0.924530	0.462265	−	0.886742i	$-0.347037\pi$
$613$	22.8903	0.924530	0.462265	+	0.886742i	$0.347037\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.45090	−0.259703	−0.129852	−	0.991533i	$-0.541450\pi$
$617$	−6.45090	−0.259703	−0.129852	+	0.991533i	$0.541450\pi$
$618$	0	0
$619$	−41.2750	−1.65898	−0.829491	−	0.558520i	$-0.811369\pi$
$619$	−41.2750	−1.65898	−0.829491	+	0.558520i	$0.811369\pi$
$620$	0	0
$621$	4.12476	0.165521
$622$	0	0
$623$	−1.58796	−0.0636203
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.49905	−0.179675
$628$	0	0
$629$	−2.08701	−0.0832147
$630$	0	0
$631$	−45.3412	−1.80500	−0.902502	−	0.430686i	$-0.858272\pi$
$631$	−45.3412	−1.80500	−0.902502	+	0.430686i	$0.858272\pi$
$632$	0	0
$633$	−10.1021	−0.401521
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.7076	1.45441
$638$	0	0
$639$	17.0859	0.675908
$640$	0	0
$641$	−20.0755	−0.792935	−0.396467	−	0.918049i	$-0.629764\pi$
$641$	−20.0755	−0.792935	−0.396467	+	0.918049i	$0.629764\pi$
$642$	0	0
$643$	28.6408	1.12948	0.564741	−	0.825268i	$-0.308976\pi$
$643$	28.6408	1.12948	0.564741	+	0.825268i	$0.308976\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.1111	0.987219	0.493609	−	0.869684i	$-0.335677\pi$
$647$	25.1111	0.987219	0.493609	+	0.869684i	$0.335677\pi$
$648$	0	0
$649$	12.6616	0.497010
$650$	0	0
$651$	−3.53105	−0.138393
$652$	0	0
$653$	−44.4796	−1.74062	−0.870311	−	0.492502i	$-0.836082\pi$
$653$	−44.4796	−1.74062	−0.870311	+	0.492502i	$0.836082\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.55950	−0.216897
$658$	0	0
$659$	29.7051	1.15715	0.578573	−	0.815631i	$-0.303610\pi$
$659$	29.7051	1.15715	0.578573	+	0.815631i	$0.303610\pi$
$660$	0	0
$661$	8.89029	0.345792	0.172896	−	0.984940i	$-0.444688\pi$
$661$	8.89029	0.345792	0.172896	+	0.984940i	$0.444688\pi$
$662$	0	0
$663$	−18.0000	−0.699062
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.06758	−0.234938
$668$	0	0
$669$	2.55950	0.0989561
$670$	0	0
$671$	19.5426	0.754432
$672$	0	0
$673$	43.2599	1.66755	0.833774	−	0.552106i	$-0.186176\pi$
$673$	43.2599	1.66755	0.833774	+	0.552106i	$0.186176\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.3157	0.550198	0.275099	−	0.961416i	$-0.411289\pi$
$677$	14.3157	0.550198	0.275099	+	0.961416i	$0.411289\pi$
$678$	0	0
$679$	−1.35924	−0.0521629
$680$	0	0
$681$	7.86483	0.301381
$682$	0	0
$683$	−37.8389	−1.44786	−0.723932	−	0.689871i	$-0.757667\pi$
$683$	−37.8389	−1.44786	−0.723932	+	0.689871i	$0.757667\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−12.2107	−0.465866
$688$	0	0
$689$	77.2476	2.94290
$690$	0	0
$691$	14.5814	0.554703	0.277352	−	0.960768i	$-0.410543\pi$
$691$	14.5814	0.554703	0.277352	+	0.960768i	$0.410543\pi$
$692$	0	0
$693$	1.76199	0.0669324
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−43.4944	−1.64747
$698$	0	0
$699$	7.01505	0.265334
$700$	0	0
$701$	−42.6889	−1.61234	−0.806169	−	0.591685i	$-0.798463\pi$
$701$	−42.6889	−1.61234	−0.806169	+	0.591685i	$0.798463\pi$
$702$	0	0
$703$	−1.85829	−0.0700869
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.15133	0.193736
$708$	0	0
$709$	−39.5426	−1.48505	−0.742526	−	0.669817i	$-0.766372\pi$
$709$	−39.5426	−1.48505	−0.742526	+	0.669817i	$0.766372\pi$
$710$	0	0
$711$	2.80327	0.105131
$712$	0	0
$713$	9.70370	0.363406
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.70945	0.362606
$718$	0	0
$719$	3.50095	0.130563	0.0652816	−	0.997867i	$-0.479205\pi$
$719$	3.50095	0.130563	0.0652816	+	0.997867i	$0.479205\pi$
$720$	0	0
$721$	5.28152	0.196694
$722$	0	0
$723$	20.8903	0.776918
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.51711	−0.0933543	−0.0466772	−	0.998910i	$-0.514863\pi$
$727$	−2.51711	−0.0933543	−0.0466772	+	0.998910i	$0.514863\pi$
$728$	0	0
$729$	−0.571018	−0.0211488
$730$	0	0
$731$	29.8583	1.10435
$732$	0	0
$733$	−35.3412	−1.30536	−0.652678	−	0.757635i	$-0.726355\pi$
$733$	−35.3412	−1.30536	−0.652678	+	0.757635i	$0.726355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.5537	−0.904449
$738$	0	0
$739$	18.6602	0.686426	0.343213	−	0.939258i	$-0.388485\pi$
$739$	18.6602	0.686426	0.343213	+	0.939258i	$0.388485\pi$
$740$	0	0
$741$	−16.0273	−0.588779
$742$	0	0
$743$	−42.4328	−1.55671	−0.778355	−	0.627824i	$-0.783946\pi$
$743$	−42.4328	−1.55671	−0.778355	+	0.627824i	$0.783946\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.2495	0.740891
$748$	0	0
$749$	1.73896	0.0635402
$750$	0	0
$751$	12.5537	0.458093	0.229046	−	0.973416i	$-0.426439\pi$
$751$	12.5537	0.458093	0.229046	+	0.973416i	$0.426439\pi$
$752$	0	0
$753$	−5.63611	−0.205391
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.2722	0.700461	0.350230	−	0.936664i	$-0.386103\pi$
$757$	19.2722	0.700461	0.350230	+	0.936664i	$0.386103\pi$
$758$	0	0
$759$	1.15787	0.0420279
$760$	0	0
$761$	31.9806	1.15929	0.579647	−	0.814867i	$-0.303190\pi$
$761$	31.9806	1.15929	0.579647	+	0.814867i	$0.303190\pi$
$762$	0	0
$763$	−1.95460	−0.0707613
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.1053	1.62866
$768$	0	0
$769$	−4.20137	−0.151505	−0.0757527	−	0.997127i	$-0.524136\pi$
$769$	−4.20137	−0.151505	−0.0757527	+	0.997127i	$0.524136\pi$
$770$	0	0
$771$	11.0572	0.398215
$772$	0	0
$773$	6.04815	0.217537	0.108768	−	0.994067i	$-0.465309\pi$
$773$	6.04815	0.217537	0.108768	+	0.994067i	$0.465309\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.174028	−0.00624320
$778$	0	0
$779$	−38.7278	−1.38757
$780$	0	0
$781$	10.7393	0.384282
$782$	0	0
$783$	−25.0273	−0.894404
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.8676	−1.13596	−0.567978	−	0.823043i	$-0.692274\pi$
$787$	−31.8676	−1.13596	−0.567978	+	0.823043i	$0.692274\pi$
$788$	0	0
$789$	4.73929	0.168723
$790$	0	0
$791$	1.07583	0.0382521
$792$	0	0
$793$	69.6181	2.47221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.9018	0.457005	0.228503	−	0.973543i	$-0.426617\pi$
$797$	12.9018	0.457005	0.228503	+	0.973543i	$0.426617\pi$
$798$	0	0
$799$	−29.7986	−1.05420
$800$	0	0
$801$	8.03886	0.284039
$802$	0	0
$803$	−3.49441	−0.123315
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.80903	0.0636809
$808$	0	0
$809$	29.3250	1.03101	0.515507	−	0.856886i	$-0.327604\pi$
$809$	29.3250	1.03101	0.515507	+	0.856886i	$0.327604\pi$
$810$	0	0
$811$	−45.9683	−1.61416	−0.807082	−	0.590439i	$-0.798955\pi$
$811$	−45.9683	−1.61416	−0.807082	+	0.590439i	$0.798955\pi$
$812$	0	0
$813$	13.5915	0.476675
$814$	0	0
$815$	0	0
$816$	0	0
$817$	26.5861	0.930129
$818$	0	0
$819$	6.27687	0.219332
$820$	0	0
$821$	−8.21753	−0.286794	−0.143397	−	0.989665i	$-0.545803\pi$
$821$	−8.21753	−0.286794	−0.143397	+	0.989665i	$0.545803\pi$
$822$	0	0
$823$	49.9877	1.74246	0.871231	−	0.490873i	$-0.163322\pi$
$823$	49.9877	1.74246	0.871231	+	0.490873i	$0.163322\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.40739	0.188033	0.0940167	−	0.995571i	$-0.470029\pi$
$827$	5.40739	0.188033	0.0940167	+	0.995571i	$0.470029\pi$
$828$	0	0
$829$	−37.5861	−1.30542	−0.652709	−	0.757609i	$-0.726368\pi$
$829$	−37.5861	−1.30542	−0.652709	+	0.757609i	$0.726368\pi$
$830$	0	0
$831$	−1.54694	−0.0536628
$832$	0	0
$833$	−29.5491	−1.02382
$834$	0	0
$835$	0	0
$836$	0	0
$837$	40.0255	1.38348
$838$	0	0
$839$	24.6523	0.851091	0.425546	−	0.904937i	$-0.360082\pi$
$839$	24.6523	0.851091	0.425546	+	0.904937i	$0.360082\pi$
$840$	0	0
$841$	7.81557	0.269502
$842$	0	0
$843$	18.5379	0.638480
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.15322	−0.142706
$848$	0	0
$849$	16.2380	0.557287
$850$	0	0
$851$	0.478247	0.0163941
$852$	0	0
$853$	−14.9565	−0.512101	−0.256050	−	0.966663i	$-0.582421\pi$
$853$	−14.9565	−0.512101	−0.256050	+	0.966663i	$0.582421\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.0722	0.719814	0.359907	−	0.932988i	$-0.382808\pi$
$857$	21.0722	0.719814	0.359907	+	0.932988i	$0.382808\pi$
$858$	0	0
$859$	28.4796	0.971712	0.485856	−	0.874039i	$-0.338508\pi$
$859$	28.4796	0.971712	0.485856	+	0.874039i	$0.338508\pi$
$860$	0	0
$861$	−3.62682	−0.123602
$862$	0	0
$863$	−15.0766	−0.513214	−0.256607	−	0.966516i	$-0.582605\pi$
$863$	−15.0766	−0.513214	−0.256607	+	0.966516i	$0.582605\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.55486	0.0528057
$868$	0	0
$869$	1.76199	0.0597713
$870$	0	0
$871$	−87.4698	−2.96380
$872$	0	0
$873$	6.88099	0.232886
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.3685	0.687796	0.343898	−	0.939007i	$-0.388252\pi$
$877$	20.3685	0.687796	0.343898	+	0.939007i	$0.388252\pi$
$878$	0	0
$879$	−11.7231	−0.395411
$880$	0	0
$881$	33.9611	1.14418	0.572090	−	0.820191i	$-0.306133\pi$
$881$	33.9611	1.14418	0.572090	+	0.820191i	$0.306133\pi$
$882$	0	0
$883$	−18.9518	−0.637780	−0.318890	−	0.947792i	$-0.603310\pi$
$883$	−18.9518	−0.637780	−0.318890	+	0.947792i	$0.603310\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.05391	−0.203270	−0.101635	−	0.994822i	$-0.532407\pi$
$887$	−6.05391	−0.203270	−0.101635	+	0.994822i	$0.532407\pi$
$888$	0	0
$889$	4.98881	0.167319
$890$	0	0
$891$	−6.27687	−0.210283
$892$	0	0
$893$	−26.5329	−0.887891
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.12476	0.137722
$898$	0	0
$899$	−58.8780	−1.96369
$900$	0	0
$901$	−62.1833	−2.07163
$902$	0	0
$903$	2.48976	0.0828541
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.7940	0.358408	0.179204	−	0.983812i	$-0.442648\pi$
$907$	10.7940	0.358408	0.179204	+	0.983812i	$0.442648\pi$
$908$	0	0
$909$	−26.0780	−0.864952
$910$	0	0
$911$	−8.58607	−0.284469	−0.142235	−	0.989833i	$-0.545429\pi$
$911$	−8.58607	−0.284469	−0.142235	+	0.989833i	$0.545429\pi$
$912$	0	0
$913$	12.7278	0.421228
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.54721	−0.0841162
$918$	0	0
$919$	−29.6843	−0.979194	−0.489597	−	0.871949i	$-0.662856\pi$
$919$	−29.6843	−0.979194	−0.489597	+	0.871949i	$0.662856\pi$
$920$	0	0
$921$	9.68427	0.319107
$922$	0	0
$923$	38.2574	1.25926
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−26.7371	−0.878160
$928$	0	0
$929$	−44.9787	−1.47570	−0.737851	−	0.674963i	$-0.764159\pi$
$929$	−44.9787	−1.47570	−0.737851	+	0.674963i	$0.764159\pi$
$930$	0	0
$931$	−26.3108	−0.862300
$932$	0	0
$933$	−15.7861	−0.516813
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.8468	−0.583029	−0.291514	−	0.956566i	$-0.594159\pi$
$937$	−17.8468	−0.583029	−0.291514	+	0.956566i	$0.594159\pi$
$938$	0	0
$939$	−12.3639	−0.403480
$940$	0	0
$941$	−17.6843	−0.576491	−0.288245	−	0.957557i	$-0.593072\pi$
$941$	−17.6843	−0.576491	−0.288245	+	0.957557i	$0.593072\pi$
$942$	0	0
$943$	9.96690	0.324567
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.7083	1.45283	0.726413	−	0.687258i	$-0.241186\pi$
$947$	44.7083	1.45283	0.726413	+	0.687258i	$0.241186\pi$
$948$	0	0
$949$	−12.4484	−0.404093
$950$	0	0
$951$	9.61806	0.311887
$952$	0	0
$953$	−56.7212	−1.83738	−0.918691	−	0.394978i	$-0.870752\pi$
$953$	−56.7212	−1.83738	−0.918691	+	0.394978i	$0.870752\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.02545	−0.227101
$958$	0	0
$959$	9.13052	0.294840
$960$	0	0
$961$	63.1617	2.03748
$962$	0	0
$963$	−8.80327	−0.283681
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.5674	0.404141	0.202070	−	0.979371i	$-0.435233\pi$
$967$	12.5674	0.404141	0.202070	+	0.979371i	$0.435233\pi$
$968$	0	0
$969$	12.9018	0.414466
$970$	0	0
$971$	−40.5199	−1.30034	−0.650172	−	0.759787i	$-0.725303\pi$
$971$	−40.5199	−1.30034	−0.650172	+	0.759787i	$0.725303\pi$
$972$	0	0
$973$	−6.64730	−0.213103
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.2222	0.838922	0.419461	−	0.907773i	$-0.362219\pi$
$977$	26.2222	0.838922	0.419461	+	0.907773i	$0.362219\pi$
$978$	0	0
$979$	5.05280	0.161488
$980$	0	0
$981$	9.89493	0.315921
$982$	0	0
$983$	−22.1144	−0.705339	−0.352669	−	0.935748i	$-0.614726\pi$
$983$	−22.1144	−0.705339	−0.352669	+	0.935748i	$0.614726\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.48479	−0.0790916
$988$	0	0
$989$	−6.84213	−0.217567
$990$	0	0
$991$	30.0046	0.953129	0.476564	−	0.879140i	$-0.341882\pi$
$991$	30.0046	0.953129	0.476564	+	0.879140i	$0.341882\pi$
$992$	0	0
$993$	4.22656	0.134126
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−43.0416	−1.36314	−0.681571	−	0.731752i	$-0.738703\pi$
$997$	−43.0416	−1.36314	−0.681571	+	0.731752i	$0.738703\pi$
$998$	0	0
$999$	1.97265	0.0624120

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.k.1.2	yes	3
4.3	odd	2		9200.2.a.cc.1.2		3
5.2	odd	4		2300.2.c.i.1749.3		6
5.3	odd	4		2300.2.c.i.1749.4		6
5.4	even	2		2300.2.a.j.1.2	✓	3
20.19	odd	2		9200.2.a.cg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2300.2.a.j.1.2	✓	3	5.4	even	2
2300.2.a.k.1.2	yes	3	1.1	even	1	trivial
2300.2.c.i.1749.3		6	5.2	odd	4
2300.2.c.i.1749.4		6	5.3	odd	4
9200.2.a.cc.1.2		3	4.3	odd	2
9200.2.a.cg.1.2		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.760877 q^{3} +0.478247 q^{7} -2.42107 q^{9} +O(q^{10})\)	\(q+0.760877 q^{3} +0.478247 q^{7} -2.42107 q^{9} -1.52175 q^{11} -5.42107 q^{13} +4.36389 q^{17} +3.88564 q^{19} +0.363887 q^{21} -1.00000 q^{23} -4.12476 q^{27} +6.06758 q^{29} -9.70370 q^{31} -1.15787 q^{33} -0.478247 q^{37} -4.12476 q^{39} -9.96690 q^{41} +6.84213 q^{43} -6.82846 q^{47} -6.77128 q^{49} +3.32038 q^{51} -14.2495 q^{53} +2.95649 q^{57} -8.32038 q^{59} -12.8421 q^{61} -1.15787 q^{63} +16.1352 q^{67} -0.760877 q^{69} -7.05718 q^{71} +2.29630 q^{73} -0.727773 q^{77} -1.15787 q^{79} +4.12476 q^{81} -8.36389 q^{83} +4.61668 q^{87} -3.32038 q^{89} -2.59261 q^{91} -7.38332 q^{93} -2.84213 q^{97} +3.68427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^7 + q^9	\( 3 q + 2 q^{3} + 2 q^{7} + q^{9} - 4 q^{11} - 8 q^{13} - 4 q^{17} - 6 q^{19} - 16 q^{21} - 3 q^{23} + 5 q^{27} - 8 q^{29} - 20 q^{31} - 20 q^{33} - 2 q^{37} + 5 q^{39} + 4 q^{41} + 4 q^{43} + 6 q^{47} + 15 q^{49} - 6 q^{51} - 8 q^{53} + 10 q^{57} - 9 q^{59} - 22 q^{61} - 20 q^{63} - 4 q^{67} - 2 q^{69} - 30 q^{71} + 16 q^{73} + 32 q^{77} - 20 q^{79} - 5 q^{81} - 8 q^{83} + 7 q^{87} + 6 q^{89} - 26 q^{91} - 29 q^{93} + 8 q^{97} - 22 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^7 + q^9 - 4 * q^11 - 8 * q^13 - 4 * q^17 - 6 * q^19 - 16 * q^21 - 3 * q^23 + 5 * q^27 - 8 * q^29 - 20 * q^31 - 20 * q^33 - 2 * q^37 + 5 * q^39 + 4 * q^41 + 4 * q^43 + 6 * q^47 + 15 * q^49 - 6 * q^51 - 8 * q^53 + 10 * q^57 - 9 * q^59 - 22 * q^61 - 20 * q^63 - 4 * q^67 - 2 * q^69 - 30 * q^71 + 16 * q^73 + 32 * q^77 - 20 * q^79 - 5 * q^81 - 8 * q^83 + 7 * q^87 + 6 * q^89 - 26 * q^91 - 29 * q^93 + 8 * q^97 - 22 * q^99

Introduction

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