LMFDB - Embedded newform 3724.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, 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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.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:	$29.7362897127$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.11350832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 8x^{2} + 23x + 4 $ x^5 - 2x^4 - 9x^3 + 8x^2 + 23x + 4
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			1.5
Root			$-1.90555$ of defining polynomial
Character	$\chi$	$=$	3724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90555 q^{3} +1.53668 q^{5} +0.631128 q^{9} -3.44223 q^{11} -4.55935 q^{13} +2.92822 q^{15} -4.36297 q^{17} -1.00000 q^{19} +3.48009 q^{23} -2.63861 q^{25} -4.51401 q^{27} -10.4590 q^{29} +10.1175 q^{31} -6.55935 q^{33} -12.0796 q^{37} -8.68808 q^{39} +2.46490 q^{41} +9.46368 q^{43} +0.969842 q^{45} -5.53668 q^{47} -8.31386 q^{51} -4.53546 q^{53} -5.28961 q^{55} -1.90555 q^{57} -6.16659 q^{59} +6.65538 q^{61} -7.00627 q^{65} +3.57454 q^{67} +6.63149 q^{69} +4.38170 q^{71} +8.24707 q^{73} -5.02802 q^{75} -14.0032 q^{79} -10.4951 q^{81} -1.91145 q^{83} -6.70449 q^{85} -19.9302 q^{87} -13.0734 q^{89} +19.2794 q^{93} -1.53668 q^{95} -14.2119 q^{97} -2.17249 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} + 7 q^{9} + 2 q^{11} - 4 q^{13} - 8 q^{15} - 4 q^{17} - 5 q^{19} + 7 q^{25} - 26 q^{27} - 8 q^{29} - 14 q^{33} - 8 q^{37} - 6 q^{39} - 18 q^{41} + 4 q^{43} + 40 q^{45} - 20 q^{47} + 10 q^{51}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 + 7 * q^9 + 2 * q^11 - 4 * q^13 - 8 * q^15 - 4 * q^17 - 5 * q^19 + 7 * q^25 - 26 * q^27 - 8 * q^29 - 14 * q^33 - 8 * q^37 - 6 * q^39 - 18 * q^41 + 4 * q^43 + 40 * q^45 - 20 * q^47 + 10 * q^51 - 2 * q^53 - 24 * q^55 + 2 * q^57 - 14 * q^59 - 12 * q^61 + 2 * q^65 + 12 * q^67 - 14 * q^69 + 12 * q^71 + 36 * q^73 - 32 * q^75 + 6 * q^79 + 13 * q^81 - 12 * q^83 - 22 * q^85 + 12 * q^87 - 50 * q^89 + 28 * q^93 - 32 * q^97 - 8 * 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
$3$	1.90555	1.10017	0.550085	−	0.835108i	$-0.314595\pi$
$3$	1.90555	1.10017	0.550085	+	0.835108i	$0.314595\pi$
$4$	0	0
$5$	1.53668	0.687224	0.343612	−	0.939112i	$-0.388349\pi$
$5$	1.53668	0.687224	0.343612	+	0.939112i	$0.388349\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.631128	0.210376
$10$	0	0
$11$	−3.44223	−1.03787	−0.518936	−	0.854813i	$-0.673672\pi$
$11$	−3.44223	−1.03787	−0.518936	+	0.854813i	$0.673672\pi$
$12$	0	0
$13$	−4.55935	−1.26454	−0.632268	−	0.774749i	$-0.717876\pi$
$13$	−4.55935	−1.26454	−0.632268	+	0.774749i	$0.717876\pi$
$14$	0	0
$15$	2.92822	0.756064
$16$	0	0
$17$	−4.36297	−1.05818	−0.529088	−	0.848567i	$-0.677466\pi$
$17$	−4.36297	−1.05818	−0.529088	+	0.848567i	$0.677466\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.48009	0.725649	0.362824	−	0.931858i	$-0.381812\pi$
$23$	3.48009	0.725649	0.362824	+	0.931858i	$0.381812\pi$
$24$	0	0
$25$	−2.63861	−0.527723
$26$	0	0
$27$	−4.51401	−0.868721
$28$	0	0
$29$	−10.4590	−1.94219	−0.971094	−	0.238698i	$-0.923279\pi$
$29$	−10.4590	−1.94219	−0.971094	+	0.238698i	$0.923279\pi$
$30$	0	0
$31$	10.1175	1.81715	0.908577	−	0.417718i	$-0.137170\pi$
$31$	10.1175	1.81715	0.908577	+	0.417718i	$0.137170\pi$
$32$	0	0
$33$	−6.55935	−1.14184
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12.0796	−1.98588	−0.992939	−	0.118625i	$-0.962151\pi$
$37$	−12.0796	−1.98588	−0.992939	+	0.118625i	$0.962151\pi$
$38$	0	0
$39$	−8.68808	−1.39121
$40$	0	0
$41$	2.46490	0.384953	0.192477	−	0.981302i	$-0.438348\pi$
$41$	2.46490	0.384953	0.192477	+	0.981302i	$0.438348\pi$
$42$	0	0
$43$	9.46368	1.44320	0.721599	−	0.692311i	$-0.243407\pi$
$43$	9.46368	1.44320	0.721599	+	0.692311i	$0.243407\pi$
$44$	0	0
$45$	0.969842	0.144576
$46$	0	0
$47$	−5.53668	−0.807608	−0.403804	−	0.914846i	$-0.632312\pi$
$47$	−5.53668	−0.807608	−0.403804	+	0.914846i	$0.632312\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8.31386	−1.16417
$52$	0	0
$53$	−4.53546	−0.622993	−0.311497	−	0.950247i	$-0.600830\pi$
$53$	−4.53546	−0.622993	−0.311497	+	0.950247i	$0.600830\pi$
$54$	0	0
$55$	−5.28961	−0.713251
$56$	0	0
$57$	−1.90555	−0.252397
$58$	0	0
$59$	−6.16659	−0.802821	−0.401411	−	0.915898i	$-0.631480\pi$
$59$	−6.16659	−0.802821	−0.401411	+	0.915898i	$0.631480\pi$
$60$	0	0
$61$	6.65538	0.852135	0.426067	−	0.904691i	$-0.359899\pi$
$61$	6.65538	0.852135	0.426067	+	0.904691i	$0.359899\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.00627	−0.869020
$66$	0	0
$67$	3.57454	0.436699	0.218350	−	0.975871i	$-0.429933\pi$
$67$	3.57454	0.436699	0.218350	+	0.975871i	$0.429933\pi$
$68$	0	0
$69$	6.63149	0.798338
$70$	0	0
$71$	4.38170	0.520012	0.260006	−	0.965607i	$-0.416275\pi$
$71$	4.38170	0.520012	0.260006	+	0.965607i	$0.416275\pi$
$72$	0	0
$73$	8.24707	0.965247	0.482623	−	0.875828i	$-0.339684\pi$
$73$	8.24707	0.965247	0.482623	+	0.875828i	$0.339684\pi$
$74$	0	0
$75$	−5.02802	−0.580585
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.0032	−1.57548	−0.787740	−	0.616008i	$-0.788749\pi$
$79$	−14.0032	−1.57548	−0.787740	+	0.616008i	$0.788749\pi$
$80$	0	0
$81$	−10.4951	−1.16612
$82$	0	0
$83$	−1.91145	−0.209809	−0.104905	−	0.994482i	$-0.533454\pi$
$83$	−1.91145	−0.209809	−0.104905	+	0.994482i	$0.533454\pi$
$84$	0	0
$85$	−6.70449	−0.727204
$86$	0	0
$87$	−19.9302	−2.13674
$88$	0	0
$89$	−13.0734	−1.38577	−0.692887	−	0.721046i	$-0.743661\pi$
$89$	−13.0734	−1.38577	−0.692887	+	0.721046i	$0.743661\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	19.2794	1.99918
$94$	0	0
$95$	−1.53668	−0.157660
$96$	0	0
$97$	−14.2119	−1.44300	−0.721501	−	0.692413i	$-0.756548\pi$
$97$	−14.2119	−1.44300	−0.721501	+	0.692413i	$0.756548\pi$
$98$	0	0
$99$	−2.17249	−0.218343
$100$	0	0
$101$	9.42350	0.937673	0.468837	−	0.883285i	$-0.344673\pi$
$101$	9.42350	0.937673	0.468837	+	0.883285i	$0.344673\pi$
$102$	0	0
$103$	−11.6781	−1.15067	−0.575336	−	0.817917i	$-0.695129\pi$
$103$	−11.6781	−1.15067	−0.575336	+	0.817917i	$0.695129\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.3139	0.997079	0.498539	−	0.866867i	$-0.333870\pi$
$107$	10.3139	0.997079	0.498539	+	0.866867i	$0.333870\pi$
$108$	0	0
$109$	16.2717	1.55855	0.779273	−	0.626685i	$-0.215589\pi$
$109$	16.2717	1.55855	0.779273	+	0.626685i	$0.215589\pi$
$110$	0	0
$111$	−23.0184	−2.18481
$112$	0	0
$113$	12.9164	1.21507	0.607537	−	0.794291i	$-0.292158\pi$
$113$	12.9164	1.21507	0.607537	+	0.794291i	$0.292158\pi$
$114$	0	0
$115$	5.34778	0.498683
$116$	0	0
$117$	−2.87754	−0.266028
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.848961	0.0771783
$122$	0	0
$123$	4.69700	0.423514
$124$	0	0
$125$	−11.7381	−1.04989
$126$	0	0
$127$	0.158525	0.0140668	0.00703341	−	0.999975i	$-0.497761\pi$
$127$	0.158525	0.0140668	0.00703341	+	0.999975i	$0.497761\pi$
$128$	0	0
$129$	18.0335	1.58776
$130$	0	0
$131$	−3.63703	−0.317769	−0.158884	−	0.987297i	$-0.550790\pi$
$131$	−3.63703	−0.317769	−0.158884	+	0.987297i	$0.550790\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.93659	−0.597006
$136$	0	0
$137$	17.8143	1.52198	0.760988	−	0.648766i	$-0.224715\pi$
$137$	17.8143	1.52198	0.760988	+	0.648766i	$0.224715\pi$
$138$	0	0
$139$	1.52488	0.129338	0.0646691	−	0.997907i	$-0.479401\pi$
$139$	1.52488	0.129338	0.0646691	+	0.997907i	$0.479401\pi$
$140$	0	0
$141$	−10.5504	−0.888507
$142$	0	0
$143$	15.6943	1.31243
$144$	0	0
$145$	−16.0721	−1.33472
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.08977	0.253124	0.126562	−	0.991959i	$-0.459606\pi$
$149$	3.08977	0.253124	0.126562	+	0.991959i	$0.459606\pi$
$150$	0	0
$151$	−9.61988	−0.782855	−0.391427	−	0.920209i	$-0.628019\pi$
$151$	−9.61988	−0.782855	−0.391427	+	0.920209i	$0.628019\pi$
$152$	0	0
$153$	−2.75359	−0.222615
$154$	0	0
$155$	15.5473	1.24879
$156$	0	0
$157$	14.3326	1.14387	0.571933	−	0.820300i	$-0.306194\pi$
$157$	14.3326	1.14387	0.571933	+	0.820300i	$0.306194\pi$
$158$	0	0
$159$	−8.64255	−0.685399
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.18458	−0.171109	−0.0855547	−	0.996333i	$-0.527266\pi$
$163$	−2.18458	−0.171109	−0.0855547	+	0.996333i	$0.527266\pi$
$164$	0	0
$165$	−10.0796	−0.784698
$166$	0	0
$167$	−18.8646	−1.45979	−0.729893	−	0.683561i	$-0.760430\pi$
$167$	−18.8646	−1.45979	−0.729893	+	0.683561i	$0.760430\pi$
$168$	0	0
$169$	7.78769	0.599053
$170$	0	0
$171$	−0.631128	−0.0482636
$172$	0	0
$173$	7.84464	0.596417	0.298209	−	0.954501i	$-0.403611\pi$
$173$	7.84464	0.596417	0.298209	+	0.954501i	$0.403611\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.7508	−0.883241
$178$	0	0
$179$	−1.73306	−0.129535	−0.0647676	−	0.997900i	$-0.520631\pi$
$179$	−1.73306	−0.129535	−0.0647676	+	0.997900i	$0.520631\pi$
$180$	0	0
$181$	21.8864	1.62680	0.813402	−	0.581702i	$-0.197613\pi$
$181$	21.8864	1.62680	0.813402	+	0.581702i	$0.197613\pi$
$182$	0	0
$183$	12.6822	0.937494
$184$	0	0
$185$	−18.5625	−1.36474
$186$	0	0
$187$	15.0184	1.09825
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.9653	1.44464	0.722320	−	0.691559i	$-0.243076\pi$
$191$	19.9653	1.44464	0.722320	+	0.691559i	$0.243076\pi$
$192$	0	0
$193$	8.54181	0.614853	0.307427	−	0.951572i	$-0.400532\pi$
$193$	8.54181	0.614853	0.307427	+	0.951572i	$0.400532\pi$
$194$	0	0
$195$	−13.3508	−0.956071
$196$	0	0
$197$	−1.77325	−0.126339	−0.0631693	−	0.998003i	$-0.520121\pi$
$197$	−1.77325	−0.126339	−0.0631693	+	0.998003i	$0.520121\pi$
$198$	0	0
$199$	−18.0737	−1.28121	−0.640606	−	0.767870i	$-0.721317\pi$
$199$	−18.0737	−1.28121	−0.640606	+	0.767870i	$0.721317\pi$
$200$	0	0
$201$	6.81147	0.480444
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.78777	0.264549
$206$	0	0
$207$	2.19638	0.152659
$208$	0	0
$209$	3.44223	0.238104
$210$	0	0
$211$	−8.57144	−0.590082	−0.295041	−	0.955485i	$-0.595333\pi$
$211$	−8.57144	−0.590082	−0.295041	+	0.955485i	$0.595333\pi$
$212$	0	0
$213$	8.34956	0.572103
$214$	0	0
$215$	14.5427	0.991801
$216$	0	0
$217$	0	0
$218$	0	0
$219$	15.7152	1.06194
$220$	0	0
$221$	19.8923	1.33810
$222$	0	0
$223$	−0.133048	−0.00890958	−0.00445479	−	0.999990i	$-0.501418\pi$
$223$	−0.133048	−0.00890958	−0.00445479	+	0.999990i	$0.501418\pi$
$224$	0	0
$225$	−1.66530	−0.111020
$226$	0	0
$227$	16.0753	1.06696	0.533478	−	0.845814i	$-0.320885\pi$
$227$	16.0753	1.06696	0.533478	+	0.845814i	$0.320885\pi$
$228$	0	0
$229$	−13.9578	−0.922359	−0.461179	−	0.887307i	$-0.652574\pi$
$229$	−13.9578	−0.922359	−0.461179	+	0.887307i	$0.652574\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.6765	−0.764951	−0.382476	−	0.923966i	$-0.624928\pi$
$233$	−11.6765	−0.764951	−0.382476	+	0.923966i	$0.624928\pi$
$234$	0	0
$235$	−8.50811	−0.555008
$236$	0	0
$237$	−26.6838	−1.73330
$238$	0	0
$239$	−2.01641	−0.130431	−0.0652153	−	0.997871i	$-0.520773\pi$
$239$	−2.01641	−0.130431	−0.0652153	+	0.997871i	$0.520773\pi$
$240$	0	0
$241$	4.70737	0.303229	0.151614	−	0.988440i	$-0.451553\pi$
$241$	4.70737	0.303229	0.151614	+	0.988440i	$0.451553\pi$
$242$	0	0
$243$	−6.45686	−0.414208
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.55935	0.290105
$248$	0	0
$249$	−3.64238	−0.230826
$250$	0	0
$251$	−17.1764	−1.08417	−0.542083	−	0.840325i	$-0.682364\pi$
$251$	−17.1764	−1.08417	−0.542083	+	0.840325i	$0.682364\pi$
$252$	0	0
$253$	−11.9793	−0.753131
$254$	0	0
$255$	−12.7758	−0.800049
$256$	0	0
$257$	−27.2257	−1.69829	−0.849146	−	0.528159i	$-0.822883\pi$
$257$	−27.2257	−1.69829	−0.849146	+	0.528159i	$0.822883\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.60097	−0.408590
$262$	0	0
$263$	27.8193	1.71541	0.857706	−	0.514140i	$-0.171889\pi$
$263$	27.8193	1.71541	0.857706	+	0.514140i	$0.171889\pi$
$264$	0	0
$265$	−6.96955	−0.428136
$266$	0	0
$267$	−24.9120	−1.52459
$268$	0	0
$269$	−1.78301	−0.108712	−0.0543560	−	0.998522i	$-0.517311\pi$
$269$	−1.78301	−0.108712	−0.0543560	+	0.998522i	$0.517311\pi$
$270$	0	0
$271$	−6.33260	−0.384678	−0.192339	−	0.981329i	$-0.561607\pi$
$271$	−6.33260	−0.384678	−0.192339	+	0.981329i	$0.561607\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.08272	0.547709
$276$	0	0
$277$	25.5347	1.53423	0.767115	−	0.641509i	$-0.221691\pi$
$277$	25.5347	1.53423	0.767115	+	0.641509i	$0.221691\pi$
$278$	0	0
$279$	6.38543	0.382286
$280$	0	0
$281$	−16.8384	−1.00449	−0.502247	−	0.864724i	$-0.667493\pi$
$281$	−16.8384	−1.00449	−0.502247	+	0.864724i	$0.667493\pi$
$282$	0	0
$283$	6.43352	0.382433	0.191217	−	0.981548i	$-0.438757\pi$
$283$	6.43352	0.382433	0.191217	+	0.981548i	$0.438757\pi$
$284$	0	0
$285$	−2.92822	−0.173453
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.03550	0.119735
$290$	0	0
$291$	−27.0816	−1.58755
$292$	0	0
$293$	−17.9505	−1.04868	−0.524339	−	0.851509i	$-0.675688\pi$
$293$	−17.9505	−1.04868	−0.524339	+	0.851509i	$0.675688\pi$
$294$	0	0
$295$	−9.47607	−0.551718
$296$	0	0
$297$	15.5383	0.901622
$298$	0	0
$299$	−15.8670	−0.917610
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.9570	1.03160
$304$	0	0
$305$	10.2272	0.585608
$306$	0	0
$307$	15.2618	0.871040	0.435520	−	0.900179i	$-0.356565\pi$
$307$	15.2618	0.871040	0.435520	+	0.900179i	$0.356565\pi$
$308$	0	0
$309$	−22.2531	−1.26594
$310$	0	0
$311$	−25.7509	−1.46020	−0.730101	−	0.683339i	$-0.760527\pi$
$311$	−25.7509	−1.46020	−0.730101	+	0.683339i	$0.760527\pi$
$312$	0	0
$313$	−9.80174	−0.554027	−0.277014	−	0.960866i	$-0.589345\pi$
$313$	−9.80174	−0.554027	−0.277014	+	0.960866i	$0.589345\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.68060	0.487551	0.243775	−	0.969832i	$-0.421614\pi$
$317$	8.68060	0.487551	0.243775	+	0.969832i	$0.421614\pi$
$318$	0	0
$319$	36.0023	2.01574
$320$	0	0
$321$	19.6536	1.09696
$322$	0	0
$323$	4.36297	0.242762
$324$	0	0
$325$	12.0304	0.667325
$326$	0	0
$327$	31.0065	1.71467
$328$	0	0
$329$	0	0
$330$	0	0
$331$	33.0757	1.81800	0.909002	−	0.416793i	$-0.136846\pi$
$331$	33.0757	1.81800	0.909002	+	0.416793i	$0.136846\pi$
$332$	0	0
$333$	−7.62379	−0.417781
$334$	0	0
$335$	5.49292	0.300110
$336$	0	0
$337$	8.88214	0.483841	0.241920	−	0.970296i	$-0.422223\pi$
$337$	8.88214	0.483841	0.241920	+	0.970296i	$0.422223\pi$
$338$	0	0
$339$	24.6129	1.33679
$340$	0	0
$341$	−34.8267	−1.88597
$342$	0	0
$343$	0	0
$344$	0	0
$345$	10.1905	0.548637
$346$	0	0
$347$	−18.3407	−0.984578	−0.492289	−	0.870432i	$-0.663840\pi$
$347$	−18.3407	−0.984578	−0.492289	+	0.870432i	$0.663840\pi$
$348$	0	0
$349$	−4.02219	−0.215303	−0.107652	−	0.994189i	$-0.534333\pi$
$349$	−4.02219	−0.215303	−0.107652	+	0.994189i	$0.534333\pi$
$350$	0	0
$351$	20.5810	1.09853
$352$	0	0
$353$	1.93518	0.102999	0.0514996	−	0.998673i	$-0.483600\pi$
$353$	1.93518	0.102999	0.0514996	+	0.998673i	$0.483600\pi$
$354$	0	0
$355$	6.73328	0.357365
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.1409	−1.74911	−0.874554	−	0.484927i	$-0.838846\pi$
$359$	−33.1409	−1.74911	−0.874554	+	0.484927i	$0.838846\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.61774	0.0849093
$364$	0	0
$365$	12.6731	0.663341
$366$	0	0
$367$	−36.9403	−1.92827	−0.964135	−	0.265412i	$-0.914492\pi$
$367$	−36.9403	−1.92827	−0.964135	+	0.265412i	$0.914492\pi$
$368$	0	0
$369$	1.55567	0.0809850
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−23.2192	−1.20225	−0.601123	−	0.799157i	$-0.705280\pi$
$373$	−23.2192	−1.20225	−0.601123	+	0.799157i	$0.705280\pi$
$374$	0	0
$375$	−22.3676	−1.15506
$376$	0	0
$377$	47.6863	2.45597
$378$	0	0
$379$	−13.7204	−0.704770	−0.352385	−	0.935855i	$-0.614629\pi$
$379$	−13.7204	−0.704770	−0.352385	+	0.935855i	$0.614629\pi$
$380$	0	0
$381$	0.302078	0.0154759
$382$	0	0
$383$	−22.1816	−1.13342	−0.566712	−	0.823916i	$-0.691785\pi$
$383$	−22.1816	−1.13342	−0.566712	+	0.823916i	$0.691785\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.97280	0.303614
$388$	0	0
$389$	9.55541	0.484479	0.242239	−	0.970217i	$-0.422118\pi$
$389$	9.55541	0.484479	0.242239	+	0.970217i	$0.422118\pi$
$390$	0	0
$391$	−15.1835	−0.767864
$392$	0	0
$393$	−6.93055	−0.349600
$394$	0	0
$395$	−21.5184	−1.08271
$396$	0	0
$397$	−3.20378	−0.160793	−0.0803966	−	0.996763i	$-0.525619\pi$
$397$	−3.20378	−0.160793	−0.0803966	+	0.996763i	$0.525619\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.7768	0.837792	0.418896	−	0.908034i	$-0.362417\pi$
$401$	16.7768	0.837792	0.418896	+	0.908034i	$0.362417\pi$
$402$	0	0
$403$	−46.1292	−2.29786
$404$	0	0
$405$	−16.1276	−0.801385
$406$	0	0
$407$	41.5809	2.06109
$408$	0	0
$409$	17.8424	0.882248	0.441124	−	0.897446i	$-0.354580\pi$
$409$	17.8424	0.882248	0.441124	+	0.897446i	$0.354580\pi$
$410$	0	0
$411$	33.9460	1.67443
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.93729	−0.144186
$416$	0	0
$417$	2.90573	0.142294
$418$	0	0
$419$	−25.6187	−1.25156	−0.625778	−	0.780001i	$-0.715219\pi$
$419$	−25.6187	−1.25156	−0.625778	+	0.780001i	$0.715219\pi$
$420$	0	0
$421$	4.04364	0.197075	0.0985376	−	0.995133i	$-0.468584\pi$
$421$	4.04364	0.197075	0.0985376	+	0.995133i	$0.468584\pi$
$422$	0	0
$423$	−3.49435	−0.169901
$424$	0	0
$425$	11.5122	0.558423
$426$	0	0
$427$	0	0
$428$	0	0
$429$	29.9064	1.44389
$430$	0	0
$431$	10.1561	0.489201	0.244601	−	0.969624i	$-0.421343\pi$
$431$	10.1561	0.489201	0.244601	+	0.969624i	$0.421343\pi$
$432$	0	0
$433$	−26.6646	−1.28142	−0.640710	−	0.767783i	$-0.721360\pi$
$433$	−26.6646	−1.28142	−0.640710	+	0.767783i	$0.721360\pi$
$434$	0	0
$435$	−30.6263	−1.46842
$436$	0	0
$437$	−3.48009	−0.166475
$438$	0	0
$439$	17.4358	0.832163	0.416082	−	0.909327i	$-0.363403\pi$
$439$	17.4358	0.832163	0.416082	+	0.909327i	$0.363403\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.72406	0.414493	0.207246	−	0.978289i	$-0.433550\pi$
$443$	8.72406	0.414493	0.207246	+	0.978289i	$0.433550\pi$
$444$	0	0
$445$	−20.0896	−0.952337
$446$	0	0
$447$	5.88771	0.278479
$448$	0	0
$449$	−19.4956	−0.920056	−0.460028	−	0.887904i	$-0.652161\pi$
$449$	−19.4956	−0.920056	−0.460028	+	0.887904i	$0.652161\pi$
$450$	0	0
$451$	−8.48477	−0.399532
$452$	0	0
$453$	−18.3312	−0.861274
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.37583	0.251471	0.125735	−	0.992064i	$-0.459871\pi$
$457$	5.37583	0.251471	0.125735	+	0.992064i	$0.459871\pi$
$458$	0	0
$459$	19.6945	0.919260
$460$	0	0
$461$	30.7537	1.43234	0.716172	−	0.697924i	$-0.245892\pi$
$461$	30.7537	1.43234	0.716172	+	0.697924i	$0.245892\pi$
$462$	0	0
$463$	−8.15132	−0.378824	−0.189412	−	0.981898i	$-0.560658\pi$
$463$	−8.15132	−0.378824	−0.189412	+	0.981898i	$0.560658\pi$
$464$	0	0
$465$	29.6263	1.37388
$466$	0	0
$467$	−10.5999	−0.490506	−0.245253	−	0.969459i	$-0.578871\pi$
$467$	−10.5999	−0.490506	−0.245253	+	0.969459i	$0.578871\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	27.3115	1.25845
$472$	0	0
$473$	−32.5762	−1.49785
$474$	0	0
$475$	2.63861	0.121068
$476$	0	0
$477$	−2.86246	−0.131063
$478$	0	0
$479$	−0.0148228	−0.000677271 0	−0.000338635	−	1.00000i	$-0.500108\pi$
$479$	−0.0148228	−0.000677271 0	−0.000338635		1.00000i	$0.500108\pi$
$480$	0	0
$481$	55.0753	2.51122
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−21.8392	−0.991667
$486$	0	0
$487$	20.3849	0.923727	0.461863	−	0.886951i	$-0.347181\pi$
$487$	20.3849	0.923727	0.461863	+	0.886951i	$0.347181\pi$
$488$	0	0
$489$	−4.16283	−0.188250
$490$	0	0
$491$	33.6589	1.51901	0.759503	−	0.650504i	$-0.225442\pi$
$491$	33.6589	1.51901	0.759503	+	0.650504i	$0.225442\pi$
$492$	0	0
$493$	45.6323	2.05518
$494$	0	0
$495$	−3.33842	−0.150051
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−26.9373	−1.20588	−0.602939	−	0.797787i	$-0.706004\pi$
$499$	−26.9373	−1.20588	−0.602939	+	0.797787i	$0.706004\pi$
$500$	0	0
$501$	−35.9475	−1.60601
$502$	0	0
$503$	12.8135	0.571324	0.285662	−	0.958330i	$-0.407787\pi$
$503$	12.8135	0.571324	0.285662	+	0.958330i	$0.407787\pi$
$504$	0	0
$505$	14.4809	0.644392
$506$	0	0
$507$	14.8398	0.659061
$508$	0	0
$509$	35.6986	1.58231	0.791157	−	0.611613i	$-0.209479\pi$
$509$	35.6986	1.58231	0.791157	+	0.611613i	$0.209479\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.51401	0.199298
$514$	0	0
$515$	−17.9454	−0.790770
$516$	0	0
$517$	19.0585	0.838194
$518$	0	0
$519$	14.9484	0.656161
$520$	0	0
$521$	−45.2674	−1.98320	−0.991601	−	0.129331i	$-0.958717\pi$
$521$	−45.2674	−1.98320	−0.991601	+	0.129331i	$0.958717\pi$
$522$	0	0
$523$	−14.4868	−0.633464	−0.316732	−	0.948515i	$-0.602586\pi$
$523$	−14.4868	−0.633464	−0.316732	+	0.948515i	$0.602586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−44.1423	−1.92287
$528$	0	0
$529$	−10.8890	−0.473434
$530$	0	0
$531$	−3.89191	−0.168894
$532$	0	0
$533$	−11.2384	−0.486788
$534$	0	0
$535$	15.8491	0.685217
$536$	0	0
$537$	−3.30244	−0.142511
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.7173	−0.976693	−0.488347	−	0.872650i	$-0.662400\pi$
$541$	−22.7173	−0.976693	−0.488347	+	0.872650i	$0.662400\pi$
$542$	0	0
$543$	41.7057	1.78976
$544$	0	0
$545$	25.0044	1.07107
$546$	0	0
$547$	−5.61126	−0.239920	−0.119960	−	0.992779i	$-0.538277\pi$
$547$	−5.61126	−0.239920	−0.119960	+	0.992779i	$0.538277\pi$
$548$	0	0
$549$	4.20040	0.179269
$550$	0	0
$551$	10.4590	0.445568
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−35.3718	−1.50145
$556$	0	0
$557$	0.0214507	0.000908895 0	0.000454447	−	1.00000i	$-0.499855\pi$
$557$	0.0214507	0.000908895 0	0.000454447		1.00000i	$0.499855\pi$
$558$	0	0
$559$	−43.1483	−1.82498
$560$	0	0
$561$	28.6183	1.20826
$562$	0	0
$563$	−43.7451	−1.84364	−0.921819	−	0.387621i	$-0.873297\pi$
$563$	−43.7451	−1.84364	−0.921819	+	0.387621i	$0.873297\pi$
$564$	0	0
$565$	19.8484	0.835029
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.6285	−1.36786	−0.683928	−	0.729549i	$-0.739730\pi$
$569$	−32.6285	−1.36786	−0.683928	+	0.729549i	$0.739730\pi$
$570$	0	0
$571$	21.1686	0.885880	0.442940	−	0.896551i	$-0.353935\pi$
$571$	21.1686	0.885880	0.442940	+	0.896551i	$0.353935\pi$
$572$	0	0
$573$	38.0449	1.58935
$574$	0	0
$575$	−9.18261	−0.382941
$576$	0	0
$577$	−27.6078	−1.14933	−0.574665	−	0.818389i	$-0.694867\pi$
$577$	−27.6078	−1.14933	−0.574665	+	0.818389i	$0.694867\pi$
$578$	0	0
$579$	16.2769	0.676443
$580$	0	0
$581$	0	0
$582$	0	0
$583$	15.6121	0.646587
$584$	0	0
$585$	−4.42185	−0.182821
$586$	0	0
$587$	46.9697	1.93865	0.969323	−	0.245790i	$-0.0790473\pi$
$587$	46.9697	1.93865	0.969323	+	0.245790i	$0.0790473\pi$
$588$	0	0
$589$	−10.1175	−0.416884
$590$	0	0
$591$	−3.37901	−0.138994
$592$	0	0
$593$	−31.6249	−1.29868	−0.649340	−	0.760498i	$-0.724955\pi$
$593$	−31.6249	−1.29868	−0.649340	+	0.760498i	$0.724955\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−34.4404	−1.40955
$598$	0	0
$599$	−17.8860	−0.730801	−0.365400	−	0.930850i	$-0.619068\pi$
$599$	−17.8860	−0.730801	−0.365400	+	0.930850i	$0.619068\pi$
$600$	0	0
$601$	−5.24108	−0.213788	−0.106894	−	0.994270i	$-0.534091\pi$
$601$	−5.24108	−0.213788	−0.106894	+	0.994270i	$0.534091\pi$
$602$	0	0
$603$	2.25599	0.0918711
$604$	0	0
$605$	1.30458	0.0530388
$606$	0	0
$607$	25.5164	1.03568	0.517839	−	0.855478i	$-0.326737\pi$
$607$	25.5164	1.03568	0.517839	+	0.855478i	$0.326737\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	25.2437	1.02125
$612$	0	0
$613$	31.1511	1.25818	0.629090	−	0.777332i	$-0.283428\pi$
$613$	31.1511	1.25818	0.629090	+	0.777332i	$0.283428\pi$
$614$	0	0
$615$	7.21779	0.291049
$616$	0	0
$617$	−12.4899	−0.502826	−0.251413	−	0.967880i	$-0.580895\pi$
$617$	−12.4899	−0.502826	−0.251413	+	0.967880i	$0.580895\pi$
$618$	0	0
$619$	−41.8093	−1.68046	−0.840228	−	0.542233i	$-0.817579\pi$
$619$	−41.8093	−1.68046	−0.840228	+	0.542233i	$0.817579\pi$
$620$	0	0
$621$	−15.7092	−0.630387
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.84464	−0.193786
$626$	0	0
$627$	6.55935	0.261955
$628$	0	0
$629$	52.7030	2.10141
$630$	0	0
$631$	12.8830	0.512865	0.256433	−	0.966562i	$-0.417453\pi$
$631$	12.8830	0.512865	0.256433	+	0.966562i	$0.417453\pi$
$632$	0	0
$633$	−16.3333	−0.649191
$634$	0	0
$635$	0.243602	0.00966706
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.76542	0.109398
$640$	0	0
$641$	−31.7456	−1.25388	−0.626938	−	0.779069i	$-0.715692\pi$
$641$	−31.7456	−1.25388	−0.626938	+	0.779069i	$0.715692\pi$
$642$	0	0
$643$	−15.4173	−0.607999	−0.304000	−	0.952672i	$-0.598322\pi$
$643$	−15.4173	−0.607999	−0.304000	+	0.952672i	$0.598322\pi$
$644$	0	0
$645$	27.7118	1.09115
$646$	0	0
$647$	32.2815	1.26912	0.634558	−	0.772875i	$-0.281182\pi$
$647$	32.2815	1.26912	0.634558	+	0.772875i	$0.281182\pi$
$648$	0	0
$649$	21.2268	0.833226
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.9947	−1.05639	−0.528193	−	0.849125i	$-0.677130\pi$
$653$	−26.9947	−1.05639	−0.528193	+	0.849125i	$0.677130\pi$
$654$	0	0
$655$	−5.58895	−0.218378
$656$	0	0
$657$	5.20496	0.203065
$658$	0	0
$659$	−5.32734	−0.207524	−0.103762	−	0.994602i	$-0.533088\pi$
$659$	−5.32734	−0.207524	−0.103762	+	0.994602i	$0.533088\pi$
$660$	0	0
$661$	−30.0867	−1.17024	−0.585119	−	0.810948i	$-0.698952\pi$
$661$	−30.0867	−1.17024	−0.585119	+	0.810948i	$0.698952\pi$
$662$	0	0
$663$	37.9058	1.47214
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36.3983	−1.40935
$668$	0	0
$669$	−0.253531	−0.00980206
$670$	0	0
$671$	−22.9094	−0.884407
$672$	0	0
$673$	−26.3147	−1.01436	−0.507179	−	0.861841i	$-0.669312\pi$
$673$	−26.3147	−1.01436	−0.507179	+	0.861841i	$0.669312\pi$
$674$	0	0
$675$	11.9107	0.458444
$676$	0	0
$677$	10.6619	0.409769	0.204884	−	0.978786i	$-0.434318\pi$
$677$	10.6619	0.409769	0.204884	+	0.978786i	$0.434318\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	30.6323	1.17383
$682$	0	0
$683$	−12.3678	−0.473242	−0.236621	−	0.971602i	$-0.576040\pi$
$683$	−12.3678	−0.473242	−0.236621	+	0.971602i	$0.576040\pi$
$684$	0	0
$685$	27.3748	1.04594
$686$	0	0
$687$	−26.5974	−1.01475
$688$	0	0
$689$	20.6788	0.787798
$690$	0	0
$691$	13.0548	0.496628	0.248314	−	0.968680i	$-0.420124\pi$
$691$	13.0548	0.496628	0.248314	+	0.968680i	$0.420124\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.34325	0.0888844
$696$	0	0
$697$	−10.7543	−0.407348
$698$	0	0
$699$	−22.2501	−0.841577
$700$	0	0
$701$	19.2140	0.725704	0.362852	−	0.931847i	$-0.381803\pi$
$701$	19.2140	0.725704	0.362852	+	0.931847i	$0.381803\pi$
$702$	0	0
$703$	12.0796	0.455592
$704$	0	0
$705$	−16.2126	−0.610603
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.8783	0.671433	0.335717	−	0.941963i	$-0.391022\pi$
$709$	17.8783	0.671433	0.335717	+	0.941963i	$0.391022\pi$
$710$	0	0
$711$	−8.83779	−0.331443
$712$	0	0
$713$	35.2097	1.31862
$714$	0	0
$715$	24.1172	0.901932
$716$	0	0
$717$	−3.84237	−0.143496
$718$	0	0
$719$	−24.3412	−0.907775	−0.453887	−	0.891059i	$-0.649963\pi$
$719$	−24.3412	−0.907775	−0.453887	+	0.891059i	$0.649963\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	8.97014	0.333603
$724$	0	0
$725$	27.5973	1.02494
$726$	0	0
$727$	16.9340	0.628048	0.314024	−	0.949415i	$-0.398323\pi$
$727$	16.9340	0.628048	0.314024	+	0.949415i	$0.398323\pi$
$728$	0	0
$729$	19.1813	0.710419
$730$	0	0
$731$	−41.2898	−1.52716
$732$	0	0
$733$	42.0157	1.55189	0.775943	−	0.630804i	$-0.217275\pi$
$733$	42.0157	1.55189	0.775943	+	0.630804i	$0.217275\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.3044	−0.453238
$738$	0	0
$739$	39.3583	1.44782	0.723909	−	0.689896i	$-0.242344\pi$
$739$	39.3583	1.44782	0.723909	+	0.689896i	$0.242344\pi$
$740$	0	0
$741$	8.68808	0.319165
$742$	0	0
$743$	24.2438	0.889417	0.444709	−	0.895675i	$-0.353307\pi$
$743$	24.2438	0.889417	0.444709	+	0.895675i	$0.353307\pi$
$744$	0	0
$745$	4.74798	0.173953
$746$	0	0
$747$	−1.20637	−0.0441389
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22.0351	0.804073	0.402037	−	0.915624i	$-0.368302\pi$
$751$	22.0351	0.804073	0.402037	+	0.915624i	$0.368302\pi$
$752$	0	0
$753$	−32.7306	−1.19277
$754$	0	0
$755$	−14.7827	−0.537997
$756$	0	0
$757$	1.23976	0.0450599	0.0225300	−	0.999746i	$-0.492828\pi$
$757$	1.23976	0.0450599	0.0225300	+	0.999746i	$0.492828\pi$
$758$	0	0
$759$	−22.8271	−0.828572
$760$	0	0
$761$	−51.7980	−1.87768	−0.938839	−	0.344357i	$-0.888097\pi$
$761$	−51.7980	−1.87768	−0.938839	+	0.344357i	$0.888097\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.23139	−0.152986
$766$	0	0
$767$	28.1156	1.01520
$768$	0	0
$769$	−5.97280	−0.215385	−0.107692	−	0.994184i	$-0.534346\pi$
$769$	−5.97280	−0.215385	−0.107692	+	0.994184i	$0.534346\pi$
$770$	0	0
$771$	−51.8800	−1.86841
$772$	0	0
$773$	43.9400	1.58041	0.790206	−	0.612842i	$-0.209974\pi$
$773$	43.9400	1.58041	0.790206	+	0.612842i	$0.209974\pi$
$774$	0	0
$775$	−26.6961	−0.958953
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.46490	−0.0883143
$780$	0	0
$781$	−15.0828	−0.539706
$782$	0	0
$783$	47.2120	1.68722
$784$	0	0
$785$	22.0246	0.786092
$786$	0	0
$787$	0.492656	0.0175613	0.00878064	−	0.999961i	$-0.497205\pi$
$787$	0.492656	0.0175613	0.00878064	+	0.999961i	$0.497205\pi$
$788$	0	0
$789$	53.0112	1.88725
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−30.3442	−1.07756
$794$	0	0
$795$	−13.2808	−0.471023
$796$	0	0
$797$	−13.4164	−0.475235	−0.237617	−	0.971359i	$-0.576366\pi$
$797$	−13.4164	−0.475235	−0.237617	+	0.971359i	$0.576366\pi$
$798$	0	0
$799$	24.1564	0.854591
$800$	0	0
$801$	−8.25097	−0.291534
$802$	0	0
$803$	−28.3883	−1.00180
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.39761	−0.119602
$808$	0	0
$809$	0.723054	0.0254212	0.0127106	−	0.999919i	$-0.495954\pi$
$809$	0.723054	0.0254212	0.0127106	+	0.999919i	$0.495954\pi$
$810$	0	0
$811$	34.2153	1.20146	0.600731	−	0.799452i	$-0.294876\pi$
$811$	34.2153	1.20146	0.600731	+	0.799452i	$0.294876\pi$
$812$	0	0
$813$	−12.0671	−0.423211
$814$	0	0
$815$	−3.35700	−0.117590
$816$	0	0
$817$	−9.46368	−0.331092
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.25746	0.0787859	0.0393929	−	0.999224i	$-0.487458\pi$
$821$	2.25746	0.0787859	0.0393929	+	0.999224i	$0.487458\pi$
$822$	0	0
$823$	52.7144	1.83751	0.918755	−	0.394829i	$-0.129196\pi$
$823$	52.7144	1.83751	0.918755	+	0.394829i	$0.129196\pi$
$824$	0	0
$825$	17.3076	0.602573
$826$	0	0
$827$	−10.1115	−0.351611	−0.175805	−	0.984425i	$-0.556253\pi$
$827$	−10.1115	−0.351611	−0.175805	+	0.984425i	$0.556253\pi$
$828$	0	0
$829$	11.8135	0.410301	0.205151	−	0.978730i	$-0.434232\pi$
$829$	11.8135	0.410301	0.205151	+	0.978730i	$0.434232\pi$
$830$	0	0
$831$	48.6577	1.68792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−28.9889	−1.00320
$836$	0	0
$837$	−45.6704	−1.57860
$838$	0	0
$839$	18.5323	0.639807	0.319904	−	0.947450i	$-0.396349\pi$
$839$	18.5323	0.639807	0.319904	+	0.947450i	$0.396349\pi$
$840$	0	0
$841$	80.3907	2.77209
$842$	0	0
$843$	−32.0864	−1.10511
$844$	0	0
$845$	11.9672	0.411684
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12.2594	0.420742
$850$	0	0
$851$	−42.0382	−1.44105
$852$	0	0
$853$	−26.3117	−0.900896	−0.450448	−	0.892803i	$-0.648736\pi$
$853$	−26.3117	−0.900896	−0.450448	+	0.892803i	$0.648736\pi$
$854$	0	0
$855$	−0.969842	−0.0331679
$856$	0	0
$857$	1.25719	0.0429450	0.0214725	−	0.999769i	$-0.493165\pi$
$857$	1.25719	0.0429450	0.0214725	+	0.999769i	$0.493165\pi$
$858$	0	0
$859$	0.299060	0.0102038	0.00510189	−	0.999987i	$-0.498376\pi$
$859$	0.299060	0.0102038	0.00510189	+	0.999987i	$0.498376\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−49.8670	−1.69749	−0.848746	−	0.528800i	$-0.822642\pi$
$863$	−49.8670	−1.69749	−0.848746	+	0.528800i	$0.822642\pi$
$864$	0	0
$865$	12.0547	0.409872
$866$	0	0
$867$	3.87876	0.131729
$868$	0	0
$869$	48.2022	1.63515
$870$	0	0
$871$	−16.2976	−0.552222
$872$	0	0
$873$	−8.96955	−0.303573
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.9976	0.573970	0.286985	−	0.957935i	$-0.407347\pi$
$877$	16.9976	0.573970	0.286985	+	0.957935i	$0.407347\pi$
$878$	0	0
$879$	−34.2056	−1.15373
$880$	0	0
$881$	−7.64757	−0.257653	−0.128827	−	0.991667i	$-0.541121\pi$
$881$	−7.64757	−0.257653	−0.128827	+	0.991667i	$0.541121\pi$
$882$	0	0
$883$	−30.8409	−1.03788	−0.518939	−	0.854811i	$-0.673673\pi$
$883$	−30.8409	−1.03788	−0.518939	+	0.854811i	$0.673673\pi$
$884$	0	0
$885$	−18.0571	−0.606984
$886$	0	0
$887$	4.56723	0.153353	0.0766763	−	0.997056i	$-0.475569\pi$
$887$	4.56723	0.153353	0.0766763	+	0.997056i	$0.475569\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	36.1264	1.21028
$892$	0	0
$893$	5.53668	0.185278
$894$	0	0
$895$	−2.66316	−0.0890197
$896$	0	0
$897$	−30.2353	−1.00953
$898$	0	0
$899$	−105.819	−3.52925
$900$	0	0
$901$	19.7881	0.659236
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.6324	1.11798
$906$	0	0
$907$	−21.8675	−0.726099	−0.363049	−	0.931770i	$-0.618264\pi$
$907$	−21.8675	−0.726099	−0.363049	+	0.931770i	$0.618264\pi$
$908$	0	0
$909$	5.94744	0.197264
$910$	0	0
$911$	6.89066	0.228298	0.114149	−	0.993464i	$-0.463586\pi$
$911$	6.89066	0.228298	0.114149	+	0.993464i	$0.463586\pi$
$912$	0	0
$913$	6.57967	0.217755
$914$	0	0
$915$	19.4885	0.644268
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−38.1546	−1.25861	−0.629303	−	0.777160i	$-0.716659\pi$
$919$	−38.1546	−1.25861	−0.629303	+	0.777160i	$0.716659\pi$
$920$	0	0
$921$	29.0822	0.958293
$922$	0	0
$923$	−19.9777	−0.657575
$924$	0	0
$925$	31.8735	1.04799
$926$	0	0
$927$	−7.37035	−0.242074
$928$	0	0
$929$	−1.02943	−0.0337746	−0.0168873	−	0.999857i	$-0.505376\pi$
$929$	−1.02943	−0.0337746	−0.0168873	+	0.999857i	$0.505376\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−49.0697	−1.60647
$934$	0	0
$935$	23.0784	0.754745
$936$	0	0
$937$	29.1888	0.953557	0.476778	−	0.879024i	$-0.341804\pi$
$937$	29.1888	0.953557	0.476778	+	0.879024i	$0.341804\pi$
$938$	0	0
$939$	−18.6777	−0.609525
$940$	0	0
$941$	37.7089	1.22928	0.614638	−	0.788810i	$-0.289302\pi$
$941$	37.7089	1.22928	0.614638	+	0.788810i	$0.289302\pi$
$942$	0	0
$943$	8.57809	0.279341
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.47117	−0.112798	−0.0563989	−	0.998408i	$-0.517962\pi$
$947$	−3.47117	−0.112798	−0.0563989	+	0.998408i	$0.517962\pi$
$948$	0	0
$949$	−37.6013	−1.22059
$950$	0	0
$951$	16.5413	0.536389
$952$	0	0
$953$	−35.8781	−1.16220	−0.581102	−	0.813831i	$-0.697378\pi$
$953$	−35.8781	−1.16220	−0.581102	+	0.813831i	$0.697378\pi$
$954$	0	0
$955$	30.6803	0.992791
$956$	0	0
$957$	68.6043	2.21766
$958$	0	0
$959$	0	0
$960$	0	0
$961$	71.3635	2.30205
$962$	0	0
$963$	6.50937	0.209762
$964$	0	0
$965$	13.1260	0.422542
$966$	0	0
$967$	−12.2707	−0.394598	−0.197299	−	0.980343i	$-0.563217\pi$
$967$	−12.2707	−0.394598	−0.197299	+	0.980343i	$0.563217\pi$
$968$	0	0
$969$	8.31386	0.267080
$970$	0	0
$971$	18.9879	0.609350	0.304675	−	0.952456i	$-0.401452\pi$
$971$	18.9879	0.609350	0.304675	+	0.952456i	$0.401452\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	22.9245	0.734171
$976$	0	0
$977$	51.1351	1.63596	0.817979	−	0.575248i	$-0.195095\pi$
$977$	51.1351	1.63596	0.817979	+	0.575248i	$0.195095\pi$
$978$	0	0
$979$	45.0015	1.43826
$980$	0	0
$981$	10.2695	0.327881
$982$	0	0
$983$	5.47174	0.174522	0.0872608	−	0.996186i	$-0.472189\pi$
$983$	5.47174	0.174522	0.0872608	+	0.996186i	$0.472189\pi$
$984$	0	0
$985$	−2.72491	−0.0868229
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.9345	1.04725
$990$	0	0
$991$	−29.7789	−0.945957	−0.472978	−	0.881074i	$-0.656821\pi$
$991$	−29.7789	−0.945957	−0.472978	+	0.881074i	$0.656821\pi$
$992$	0	0
$993$	63.0274	2.00011
$994$	0	0
$995$	−27.7735	−0.880480
$996$	0	0
$997$	34.0364	1.07794	0.538972	−	0.842324i	$-0.318813\pi$
$997$	34.0364	1.07794	0.538972	+	0.842324i	$0.318813\pi$
$998$	0	0
$999$	54.5275	1.72517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.a.k.1.5	✓	5
7.6	odd	2		3724.2.a.l.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3724.2.a.k.1.5	✓	5	1.1	even	1	trivial
3724.2.a.l.1.1	yes	5	7.6	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:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90555 q^{3} +1.53668 q^{5} +0.631128 q^{9} -3.44223 q^{11} -4.55935 q^{13} +2.92822 q^{15} -4.36297 q^{17} -1.00000 q^{19} +3.48009 q^{23} -2.63861 q^{25} -4.51401 q^{27} -10.4590 q^{29} +10.1175 q^{31} -6.55935 q^{33} -12.0796 q^{37} -8.68808 q^{39} +2.46490 q^{41} +9.46368 q^{43} +0.969842 q^{45} -5.53668 q^{47} -8.31386 q^{51} -4.53546 q^{53} -5.28961 q^{55} -1.90555 q^{57} -6.16659 q^{59} +6.65538 q^{61} -7.00627 q^{65} +3.57454 q^{67} +6.63149 q^{69} +4.38170 q^{71} +8.24707 q^{73} -5.02802 q^{75} -14.0032 q^{79} -10.4951 q^{81} -1.91145 q^{83} -6.70449 q^{85} -19.9302 q^{87} -13.0734 q^{89} +19.2794 q^{93} -1.53668 q^{95} -14.2119 q^{97} -2.17249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} + 7 q^{9} + 2 q^{11} - 4 q^{13} - 8 q^{15} - 4 q^{17} - 5 q^{19} + 7 q^{25} - 26 q^{27} - 8 q^{29} - 14 q^{33} - 8 q^{37} - 6 q^{39} - 18 q^{41} + 4 q^{43} + 40 q^{45} - 20 q^{47} + 10 q^{51}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 + 7 * q^9 + 2 * q^11 - 4 * q^13 - 8 * q^15 - 4 * q^17 - 5 * q^19 + 7 * q^25 - 26 * q^27 - 8 * q^29 - 14 * q^33 - 8 * q^37 - 6 * q^39 - 18 * q^41 + 4 * q^43 + 40 * q^45 - 20 * q^47 + 10 * q^51 - 2 * q^53 - 24 * q^55 + 2 * q^57 - 14 * q^59 - 12 * q^61 + 2 * q^65 + 12 * q^67 - 14 * q^69 + 12 * q^71 + 36 * q^73 - 32 * q^75 + 6 * q^79 + 13 * q^81 - 12 * q^83 - 22 * q^85 + 12 * q^87 - 50 * q^89 + 28 * q^93 - 32 * q^97 - 8 * q^99

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