LMFDB - Embedded newform 855.2.a.m.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,2,Mod(1,855)] 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("855.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,2,0,8,4,0,4,12,0,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.28734$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.63010 q^{2} +4.91744 q^{4} +1.00000 q^{5} -0.574672 q^{7} +7.67316 q^{8} +2.63010 q^{10} -2.57467 q^{11} -0.468387 q^{13} -1.51145 q^{14} +10.3463 q^{16} +4.08612 q^{17} +1.00000 q^{19} +4.91744 q^{20} -6.77165 q^{22} -1.51145 q^{23} +1.00000 q^{25} -1.23191 q^{26} -2.82591 q^{28} +4.08612 q^{29} -9.92099 q^{31} +11.8656 q^{32} +10.7469 q^{34} -0.574672 q^{35} -8.30326 q^{37} +2.63010 q^{38} +7.67316 q^{40} +1.83488 q^{41} -0.574672 q^{43} -12.6608 q^{44} -3.97526 q^{46} -7.09508 q^{47} -6.66975 q^{49} +2.63010 q^{50} -2.30326 q^{52} -4.30326 q^{53} -2.57467 q^{55} -4.40955 q^{56} +10.7469 q^{58} +2.68553 q^{59} +12.4095 q^{61} -26.0932 q^{62} +10.5150 q^{64} -0.468387 q^{65} -2.70570 q^{67} +20.0932 q^{68} -1.51145 q^{70} +7.40058 q^{71} +12.0861 q^{73} -21.8384 q^{74} +4.91744 q^{76} +1.47959 q^{77} -6.68553 q^{79} +10.3463 q^{80} +4.82591 q^{82} +6.66079 q^{83} +4.08612 q^{85} -1.51145 q^{86} -19.7559 q^{88} -14.6065 q^{89} +0.269169 q^{91} -7.43244 q^{92} -18.6608 q^{94} +1.00000 q^{95} +17.4526 q^{97} -17.5421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29}+ \cdots - 38 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

Coefficient data

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

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

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

$2$	2.63010	1.85976	0.929882	−	0.367859i	$-0.119909\pi$
$2$	2.63010	1.85976	0.929882	+	0.367859i	$0.119909\pi$
$3$	0	0
$3$	0	0
$4$	4.91744	2.45872
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.574672	−0.217205	−0.108603	−	0.994085i	$-0.534638\pi$
$7$	−0.574672	−0.217205	−0.108603	+	0.994085i	$0.534638\pi$
$8$	7.67316	2.71287
$9$	0	0
$10$	2.63010	0.831711
$11$	−2.57467	−0.776293	−0.388146	−	0.921598i	$-0.626884\pi$
$11$	−2.57467	−0.776293	−0.388146	+	0.921598i	$0.626884\pi$
$12$	0	0
$13$	−0.468387	−0.129907	−0.0649536	−	0.997888i	$-0.520690\pi$
$13$	−0.468387	−0.129907	−0.0649536	+	0.997888i	$0.520690\pi$
$14$	−1.51145	−0.403951
$15$	0	0
$16$	10.3463	2.58658
$17$	4.08612	0.991029	0.495514	−	0.868600i	$-0.334980\pi$
$17$	4.08612	0.991029	0.495514	+	0.868600i	$0.334980\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	4.91744	1.09957
$21$	0	0
$22$	−6.77165	−1.44372
$23$	−1.51145	−0.315158	−0.157579	−	0.987506i	$-0.550369\pi$
$23$	−1.51145	−0.315158	−0.157579	+	0.987506i	$0.550369\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.23191	−0.241596
$27$	0	0
$28$	−2.82591	−0.534047
$29$	4.08612	0.758773	0.379386	−	0.925238i	$-0.376135\pi$
$29$	4.08612	0.758773	0.379386	+	0.925238i	$0.376135\pi$
$30$	0	0
$31$	−9.92099	−1.78186	−0.890931	−	0.454138i	$-0.849947\pi$
$31$	−9.92099	−1.78186	−0.890931	+	0.454138i	$0.849947\pi$
$32$	11.8656	2.09755
$33$	0	0
$34$	10.7469	1.84308
$35$	−0.574672	−0.0971372
$36$	0	0
$37$	−8.30326	−1.36505	−0.682524	−	0.730863i	$-0.739118\pi$
$37$	−8.30326	−1.36505	−0.682524	+	0.730863i	$0.739118\pi$
$38$	2.63010	0.426659
$39$	0	0
$40$	7.67316	1.21323
$41$	1.83488	0.286560	0.143280	−	0.989682i	$-0.454235\pi$
$41$	1.83488	0.286560	0.143280	+	0.989682i	$0.454235\pi$
$42$	0	0
$43$	−0.574672	−0.0876366	−0.0438183	−	0.999040i	$-0.513952\pi$
$43$	−0.574672	−0.0876366	−0.0438183	+	0.999040i	$0.513952\pi$
$44$	−12.6608	−1.90869
$45$	0	0
$46$	−3.97526	−0.586119
$47$	−7.09508	−1.03492	−0.517462	−	0.855706i	$-0.673123\pi$
$47$	−7.09508	−1.03492	−0.517462	+	0.855706i	$0.673123\pi$
$48$	0	0
$49$	−6.66975	−0.952822
$50$	2.63010	0.371953
$51$	0	0
$52$	−2.30326	−0.319405
$53$	−4.30326	−0.591099	−0.295549	−	0.955327i	$-0.595503\pi$
$53$	−4.30326	−0.591099	−0.295549	+	0.955327i	$0.595503\pi$
$54$	0	0
$55$	−2.57467	−0.347169
$56$	−4.40955	−0.589251
$57$	0	0
$58$	10.7469	1.41114
$59$	2.68553	0.349627	0.174813	−	0.984602i	$-0.444068\pi$
$59$	2.68553	0.349627	0.174813	+	0.984602i	$0.444068\pi$
$60$	0	0
$61$	12.4095	1.58888	0.794440	−	0.607343i	$-0.207765\pi$
$61$	12.4095	1.58888	0.794440	+	0.607343i	$0.207765\pi$
$62$	−26.0932	−3.31384
$63$	0	0
$64$	10.5150	1.31438
$65$	−0.468387	−0.0580962
$66$	0	0
$67$	−2.70570	−0.330554	−0.165277	−	0.986247i	$-0.552852\pi$
$67$	−2.70570	−0.330554	−0.165277	+	0.986247i	$0.552852\pi$
$68$	20.0932	2.43666
$69$	0	0
$70$	−1.51145	−0.180652
$71$	7.40058	0.878288	0.439144	−	0.898417i	$-0.355282\pi$
$71$	7.40058	0.878288	0.439144	+	0.898417i	$0.355282\pi$
$72$	0	0
$73$	12.0861	1.41457	0.707286	−	0.706927i	$-0.249919\pi$
$73$	12.0861	1.41457	0.707286	+	0.706927i	$0.249919\pi$
$74$	−21.8384	−2.53867
$75$	0	0
$76$	4.91744	0.564069
$77$	1.47959	0.168615
$78$	0	0
$79$	−6.68553	−0.752181	−0.376091	−	0.926583i	$-0.622732\pi$
$79$	−6.68553	−0.752181	−0.376091	+	0.926583i	$0.622732\pi$
$80$	10.3463	1.15675
$81$	0	0
$82$	4.82591	0.532933
$83$	6.66079	0.731117	0.365558	−	0.930788i	$-0.380878\pi$
$83$	6.66079	0.731117	0.365558	+	0.930788i	$0.380878\pi$
$84$	0	0
$85$	4.08612	0.443202
$86$	−1.51145	−0.162983
$87$	0	0
$88$	−19.7559	−2.10598
$89$	−14.6065	−1.54829	−0.774144	−	0.633009i	$-0.781820\pi$
$89$	−14.6065	−1.54829	−0.774144	+	0.633009i	$0.781820\pi$
$90$	0	0
$91$	0.269169	0.0282165
$92$	−7.43244	−0.774885
$93$	0	0
$94$	−18.6608	−1.92471
$95$	1.00000	0.102598
$96$	0	0
$97$	17.4526	1.77204	0.886022	−	0.463643i	$-0.153458\pi$
$97$	17.4526	1.77204	0.886022	+	0.463643i	$0.153458\pi$
$98$	−17.5421	−1.77202
$99$	0	0
$100$	4.91744	0.491744
$101$	−14.2831	−1.42122	−0.710611	−	0.703586i	$-0.751581\pi$
$101$	−14.2831	−1.42122	−0.710611	+	0.703586i	$0.751581\pi$
$102$	0	0
$103$	4.79182	0.472152	0.236076	−	0.971735i	$-0.424139\pi$
$103$	4.79182	0.472152	0.236076	+	0.971735i	$0.424139\pi$
$104$	−3.59401	−0.352421
$105$	0	0
$106$	−11.3180	−1.09930
$107$	−9.22611	−0.891922	−0.445961	−	0.895052i	$-0.647138\pi$
$107$	−9.22611	−0.891922	−0.445961	+	0.895052i	$0.647138\pi$
$108$	0	0
$109$	4.89810	0.469153	0.234577	−	0.972098i	$-0.424630\pi$
$109$	4.89810	0.469153	0.234577	+	0.972098i	$0.424630\pi$
$110$	−6.77165	−0.645651
$111$	0	0
$112$	−5.94574	−0.561819
$113$	−1.61773	−0.152183	−0.0760916	−	0.997101i	$-0.524244\pi$
$113$	−1.61773	−0.152183	−0.0760916	+	0.997101i	$0.524244\pi$
$114$	0	0
$115$	−1.51145	−0.140943
$116$	20.0932	1.86561
$117$	0	0
$118$	7.06323	0.650223
$119$	−2.34818	−0.215257
$120$	0	0
$121$	−4.37107	−0.397370
$122$	32.6384	2.95494
$123$	0	0
$124$	−48.7859	−4.38110
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.1292	−1.16503	−0.582513	−	0.812821i	$-0.697930\pi$
$127$	−13.1292	−1.16503	−0.582513	+	0.812821i	$0.697930\pi$
$128$	3.92440	0.346871
$129$	0	0
$130$	−1.23191	−0.108045
$131$	8.17223	0.714011	0.357006	−	0.934102i	$-0.383798\pi$
$131$	8.17223	0.714011	0.357006	+	0.934102i	$0.383798\pi$
$132$	0	0
$133$	−0.574672	−0.0498304
$134$	−7.11627	−0.614752
$135$	0	0
$136$	31.3534	2.68853
$137$	14.6065	1.24792	0.623960	−	0.781456i	$-0.285523\pi$
$137$	14.6065	1.24792	0.623960	+	0.781456i	$0.285523\pi$
$138$	0	0
$139$	−2.07219	−0.175761	−0.0878804	−	0.996131i	$-0.528009\pi$
$139$	−2.07219	−0.175761	−0.0878804	+	0.996131i	$0.528009\pi$
$140$	−2.82591	−0.238833
$141$	0	0
$142$	19.4643	1.63341
$143$	1.20594	0.100846
$144$	0	0
$145$	4.08612	0.339334
$146$	31.7877	2.63077
$147$	0	0
$148$	−40.8308	−3.35627
$149$	−8.91203	−0.730102	−0.365051	−	0.930988i	$-0.618948\pi$
$149$	−8.91203	−0.730102	−0.365051	+	0.930988i	$0.618948\pi$
$150$	0	0
$151$	11.4572	0.932372	0.466186	−	0.884687i	$-0.345628\pi$
$151$	11.4572	0.932372	0.466186	+	0.884687i	$0.345628\pi$
$152$	7.67316	0.622376
$153$	0	0
$154$	3.89148	0.313584
$155$	−9.92099	−0.796873
$156$	0	0
$157$	−6.60653	−0.527258	−0.263629	−	0.964624i	$-0.584919\pi$
$157$	−6.60653	−0.527258	−0.263629	+	0.964624i	$0.584919\pi$
$158$	−17.5836	−1.39888
$159$	0	0
$160$	11.8656	0.938055
$161$	0.868585	0.0684541
$162$	0	0
$163$	20.2444	1.58567	0.792833	−	0.609439i	$-0.208605\pi$
$163$	20.2444	1.58567	0.792833	+	0.609439i	$0.208605\pi$
$164$	9.02289	0.704569
$165$	0	0
$166$	17.5186	1.35970
$167$	5.89372	0.456069	0.228035	−	0.973653i	$-0.426770\pi$
$167$	5.89372	0.456069	0.228035	+	0.973653i	$0.426770\pi$
$168$	0	0
$169$	−12.7806	−0.983124
$170$	10.7469	0.824250
$171$	0	0
$172$	−2.82591	−0.215474
$173$	3.53161	0.268504	0.134252	−	0.990947i	$-0.457137\pi$
$173$	3.53161	0.268504	0.134252	+	0.990947i	$0.457137\pi$
$174$	0	0
$175$	−0.574672	−0.0434411
$176$	−26.6384	−2.00794
$177$	0	0
$178$	−38.4167	−2.87945
$179$	−7.18801	−0.537257	−0.268629	−	0.963244i	$-0.586570\pi$
$179$	−7.18801	−0.537257	−0.268629	+	0.963244i	$0.586570\pi$
$180$	0	0
$181$	15.5433	1.15532	0.577662	−	0.816276i	$-0.303965\pi$
$181$	15.5433	1.15532	0.577662	+	0.816276i	$0.303965\pi$
$182$	0.707941	0.0524761
$183$	0	0
$184$	−11.5976	−0.854984
$185$	−8.30326	−0.610468
$186$	0	0
$187$	−10.5204	−0.769329
$188$	−34.8896	−2.54459
$189$	0	0
$190$	2.63010	0.190808
$191$	−13.3216	−0.963915	−0.481958	−	0.876194i	$-0.660074\pi$
$191$	−13.3216	−0.963915	−0.481958	+	0.876194i	$0.660074\pi$
$192$	0	0
$193$	18.9959	1.36736	0.683678	−	0.729784i	$-0.260379\pi$
$193$	18.9959	1.36736	0.683678	+	0.729784i	$0.260379\pi$
$194$	45.9021	3.29558
$195$	0	0
$196$	−32.7981	−2.34272
$197$	−2.17223	−0.154765	−0.0773826	−	0.997001i	$-0.524656\pi$
$197$	−2.17223	−0.154765	−0.0773826	+	0.997001i	$0.524656\pi$
$198$	0	0
$199$	−1.87355	−0.132812	−0.0664061	−	0.997793i	$-0.521153\pi$
$199$	−1.87355	−0.132812	−0.0664061	+	0.997793i	$0.521153\pi$
$200$	7.67316	0.542574
$201$	0	0
$202$	−37.5660	−2.64313
$203$	−2.34818	−0.164810
$204$	0	0
$205$	1.83488	0.128153
$206$	12.6030	0.878091
$207$	0	0
$208$	−4.84608	−0.336015
$209$	−2.57467	−0.178094
$210$	0	0
$211$	−17.1090	−1.17783	−0.588916	−	0.808194i	$-0.700445\pi$
$211$	−17.1090	−1.17783	−0.588916	+	0.808194i	$0.700445\pi$
$212$	−21.1610	−1.45335
$213$	0	0
$214$	−24.2656	−1.65876
$215$	−0.574672	−0.0391923
$216$	0	0
$217$	5.70131	0.387030
$218$	12.8825	0.872514
$219$	0	0
$220$	−12.6608	−0.853590
$221$	−1.91388	−0.128742
$222$	0	0
$223$	5.12918	0.343475	0.171737	−	0.985143i	$-0.445062\pi$
$223$	5.12918	0.343475	0.171737	+	0.985143i	$0.445062\pi$
$224$	−6.81880	−0.455600
$225$	0	0
$226$	−4.25480	−0.283025
$227$	7.31223	0.485330	0.242665	−	0.970110i	$-0.421978\pi$
$227$	7.31223	0.485330	0.242665	+	0.970110i	$0.421978\pi$
$228$	0	0
$229$	−8.40955	−0.555719	−0.277859	−	0.960622i	$-0.589625\pi$
$229$	−8.40955	−0.555719	−0.277859	+	0.960622i	$0.589625\pi$
$230$	−3.97526	−0.262121
$231$	0	0
$232$	31.3534	2.05845
$233$	14.1722	0.928454	0.464227	−	0.885716i	$-0.346332\pi$
$233$	14.1722	0.928454	0.464227	+	0.885716i	$0.346332\pi$
$234$	0	0
$235$	−7.09508	−0.462832
$236$	13.2059	0.859634
$237$	0	0
$238$	−6.17594	−0.400327
$239$	14.1902	0.917885	0.458943	−	0.888466i	$-0.348228\pi$
$239$	14.1902	0.917885	0.458943	+	0.888466i	$0.348228\pi$
$240$	0	0
$241$	27.8807	1.79595	0.897976	−	0.440045i	$-0.145038\pi$
$241$	27.8807	1.79595	0.897976	+	0.440045i	$0.145038\pi$
$242$	−11.4964	−0.739013
$243$	0	0
$244$	61.0232	3.90661
$245$	−6.66975	−0.426115
$246$	0	0
$247$	−0.468387	−0.0298027
$248$	−76.1254	−4.83397
$249$	0	0
$250$	2.63010	0.166342
$251$	26.1902	1.65311	0.826554	−	0.562857i	$-0.190298\pi$
$251$	26.1902	1.65311	0.826554	+	0.562857i	$0.190298\pi$
$252$	0	0
$253$	3.89148	0.244655
$254$	−34.5311	−2.16667
$255$	0	0
$256$	−10.7084	−0.669276
$257$	−9.01831	−0.562547	−0.281273	−	0.959628i	$-0.590757\pi$
$257$	−9.01831	−0.562547	−0.281273	+	0.959628i	$0.590757\pi$
$258$	0	0
$259$	4.77165	0.296496
$260$	−2.30326	−0.142842
$261$	0	0
$262$	21.4938	1.32789
$263$	9.00896	0.555517	0.277758	−	0.960651i	$-0.410409\pi$
$263$	9.00896	0.555517	0.277758	+	0.960651i	$0.410409\pi$
$264$	0	0
$265$	−4.30326	−0.264347
$266$	−1.51145	−0.0926727
$267$	0	0
$268$	−13.3051	−0.812739
$269$	30.6136	1.86655	0.933273	−	0.359167i	$-0.116939\pi$
$269$	30.6136	1.86655	0.933273	+	0.359167i	$0.116939\pi$
$270$	0	0
$271$	24.1180	1.46506	0.732531	−	0.680733i	$-0.238339\pi$
$271$	24.1180	1.46506	0.732531	+	0.680733i	$0.238339\pi$
$272$	42.2763	2.56338
$273$	0	0
$274$	38.4167	2.32084
$275$	−2.57467	−0.155259
$276$	0	0
$277$	4.56075	0.274029	0.137014	−	0.990569i	$-0.456249\pi$
$277$	4.56075	0.274029	0.137014	+	0.990569i	$0.456249\pi$
$278$	−5.45007	−0.326874
$279$	0	0
$280$	−4.40955	−0.263521
$281$	6.11563	0.364828	0.182414	−	0.983222i	$-0.441609\pi$
$281$	6.11563	0.364828	0.182414	+	0.983222i	$0.441609\pi$
$282$	0	0
$283$	−19.0547	−1.13269	−0.566344	−	0.824169i	$-0.691642\pi$
$283$	−19.0547	−1.13269	−0.566344	+	0.824169i	$0.691642\pi$
$284$	36.3919	2.15946
$285$	0	0
$286$	3.17175	0.187550
$287$	−1.05445	−0.0622423
$288$	0	0
$289$	−0.303649	−0.0178617
$290$	10.7469	0.631080
$291$	0	0
$292$	59.4327	3.47804
$293$	6.43887	0.376163	0.188081	−	0.982153i	$-0.439773\pi$
$293$	6.43887	0.376163	0.188081	+	0.982153i	$0.439773\pi$
$294$	0	0
$295$	2.68553	0.156358
$296$	−63.7123	−3.70320
$297$	0	0
$298$	−23.4395	−1.35782
$299$	0.707941	0.0409413
$300$	0	0
$301$	0.330247	0.0190351
$302$	30.1336	1.73399
$303$	0	0
$304$	10.3463	0.593402
$305$	12.4095	0.710569
$306$	0	0
$307$	−6.77389	−0.386606	−0.193303	−	0.981139i	$-0.561920\pi$
$307$	−6.77389	−0.386606	−0.193303	+	0.981139i	$0.561920\pi$
$308$	7.27580	0.414577
$309$	0	0
$310$	−26.0932	−1.48200
$311$	−20.6205	−1.16928	−0.584639	−	0.811293i	$-0.698764\pi$
$311$	−20.6205	−1.16928	−0.584639	+	0.811293i	$0.698764\pi$
$312$	0	0
$313$	19.3711	1.09492	0.547459	−	0.836833i	$-0.315595\pi$
$313$	19.3711	1.09492	0.547459	+	0.836833i	$0.315595\pi$
$314$	−17.3758	−0.980575
$315$	0	0
$316$	−32.8757	−1.84940
$317$	−3.37360	−0.189480	−0.0947401	−	0.995502i	$-0.530202\pi$
$317$	−3.37360	−0.189480	−0.0947401	+	0.995502i	$0.530202\pi$
$318$	0	0
$319$	−10.5204	−0.589030
$320$	10.5150	0.587806
$321$	0	0
$322$	2.28447	0.127308
$323$	4.08612	0.227358
$324$	0	0
$325$	−0.468387	−0.0259814
$326$	53.2449	2.94896
$327$	0	0
$328$	14.0793	0.777399
$329$	4.07734	0.224791
$330$	0	0
$331$	−32.7788	−1.80168	−0.900842	−	0.434148i	$-0.857050\pi$
$331$	−32.7788	−1.80168	−0.900842	+	0.434148i	$0.857050\pi$
$332$	32.7540	1.79761
$333$	0	0
$334$	15.5011	0.848181
$335$	−2.70570	−0.147828
$336$	0	0
$337$	8.74915	0.476596	0.238298	−	0.971192i	$-0.423410\pi$
$337$	8.74915	0.476596	0.238298	+	0.971192i	$0.423410\pi$
$338$	−33.6143	−1.82838
$339$	0	0
$340$	20.0932	1.08971
$341$	25.5433	1.38325
$342$	0	0
$343$	7.85562	0.424164
$344$	−4.40955	−0.237747
$345$	0	0
$346$	9.28850	0.499353
$347$	18.5028	0.993281	0.496640	−	0.867956i	$-0.334567\pi$
$347$	18.5028	0.993281	0.496640	+	0.867956i	$0.334567\pi$
$348$	0	0
$349$	3.54330	0.189668	0.0948342	−	0.995493i	$-0.469768\pi$
$349$	3.54330	0.189668	0.0948342	+	0.995493i	$0.469768\pi$
$350$	−1.51145	−0.0807901
$351$	0	0
$352$	−30.5499	−1.62832
$353$	3.41140	0.181571	0.0907853	−	0.995870i	$-0.471062\pi$
$353$	3.41140	0.181571	0.0907853	+	0.995870i	$0.471062\pi$
$354$	0	0
$355$	7.40058	0.392782
$356$	−71.8267	−3.80681
$357$	0	0
$358$	−18.9052	−0.999172
$359$	1.70609	0.0900438	0.0450219	−	0.998986i	$-0.485664\pi$
$359$	1.70609	0.0900438	0.0450219	+	0.998986i	$0.485664\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	40.8805	2.14863
$363$	0	0
$364$	1.32362	0.0693765
$365$	12.0861	0.632616
$366$	0	0
$367$	37.6155	1.96351	0.981756	−	0.190144i	$-0.0608953\pi$
$367$	37.6155	1.96351	0.981756	+	0.190144i	$0.0608953\pi$
$368$	−15.6379	−0.815182
$369$	0	0
$370$	−21.8384	−1.13533
$371$	2.47296	0.128390
$372$	0	0
$373$	13.4031	0.693987	0.346994	−	0.937868i	$-0.387203\pi$
$373$	13.4031	0.693987	0.346994	+	0.937868i	$0.387203\pi$
$374$	−27.6698	−1.43077
$375$	0	0
$376$	−54.4417	−2.80762
$377$	−1.91388	−0.0985700
$378$	0	0
$379$	−9.37107	−0.481359	−0.240680	−	0.970605i	$-0.577370\pi$
$379$	−9.37107	−0.481359	−0.240680	+	0.970605i	$0.577370\pi$
$380$	4.91744	0.252259
$381$	0	0
$382$	−35.0371	−1.79265
$383$	2.09917	0.107263	0.0536314	−	0.998561i	$-0.482920\pi$
$383$	2.09917	0.107263	0.0536314	+	0.998561i	$0.482920\pi$
$384$	0	0
$385$	1.47959	0.0754069
$386$	49.9612	2.54296
$387$	0	0
$388$	85.8221	4.35696
$389$	−1.07238	−0.0543718	−0.0271859	−	0.999630i	$-0.508655\pi$
$389$	−1.07238	−0.0543718	−0.0271859	+	0.999630i	$0.508655\pi$
$390$	0	0
$391$	−6.17594	−0.312331
$392$	−51.1781	−2.58488
$393$	0	0
$394$	−5.71320	−0.287827
$395$	−6.68553	−0.336386
$396$	0	0
$397$	−23.5341	−1.18114	−0.590572	−	0.806985i	$-0.701098\pi$
$397$	−23.5341	−1.18114	−0.590572	+	0.806985i	$0.701098\pi$
$398$	−4.92762	−0.246999
$399$	0	0
$400$	10.3463	0.517316
$401$	−18.6469	−0.931180	−0.465590	−	0.885001i	$-0.654158\pi$
$401$	−18.6469	−0.931180	−0.465590	+	0.885001i	$0.654158\pi$
$402$	0	0
$403$	4.64686	0.231477
$404$	−70.2362	−3.49438
$405$	0	0
$406$	−6.17594	−0.306507
$407$	21.3782	1.05968
$408$	0	0
$409$	−6.88017	−0.340203	−0.170101	−	0.985427i	$-0.554410\pi$
$409$	−6.88017	−0.340203	−0.170101	+	0.985427i	$0.554410\pi$
$410$	4.82591	0.238335
$411$	0	0
$412$	23.5635	1.16089
$413$	−1.54330	−0.0759408
$414$	0	0
$415$	6.66079	0.326965
$416$	−5.55767	−0.272487
$417$	0	0
$418$	−6.77165	−0.331212
$419$	−13.4796	−0.658521	−0.329261	−	0.944239i	$-0.606799\pi$
$419$	−13.4796	−0.658521	−0.329261	+	0.944239i	$0.606799\pi$
$420$	0	0
$421$	−5.83488	−0.284374	−0.142187	−	0.989840i	$-0.545414\pi$
$421$	−5.83488	−0.284374	−0.142187	+	0.989840i	$0.545414\pi$
$422$	−44.9984	−2.19049
$423$	0	0
$424$	−33.0196	−1.60357
$425$	4.08612	0.198206
$426$	0	0
$427$	−7.13142	−0.345113
$428$	−45.3688	−2.19298
$429$	0	0
$430$	−1.51145	−0.0728884
$431$	−29.2039	−1.40670	−0.703351	−	0.710843i	$-0.748314\pi$
$431$	−29.2039	−1.40670	−0.703351	+	0.710843i	$0.748314\pi$
$432$	0	0
$433$	−12.5229	−0.601814	−0.300907	−	0.953653i	$-0.597289\pi$
$433$	−12.5229	−0.601814	−0.300907	+	0.953653i	$0.597289\pi$
$434$	14.9950	0.719785
$435$	0	0
$436$	24.0861	1.15352
$437$	−1.51145	−0.0723022
$438$	0	0
$439$	15.6769	0.748216	0.374108	−	0.927385i	$-0.377949\pi$
$439$	15.6769	0.748216	0.374108	+	0.927385i	$0.377949\pi$
$440$	−19.7559	−0.941824
$441$	0	0
$442$	−5.03371	−0.239429
$443$	29.8281	1.41717	0.708587	−	0.705624i	$-0.249333\pi$
$443$	29.8281	1.41717	0.708587	+	0.705624i	$0.249333\pi$
$444$	0	0
$445$	−14.6065	−0.692416
$446$	13.4903	0.638782
$447$	0	0
$448$	−6.04267	−0.285489
$449$	−29.9668	−1.41422	−0.707110	−	0.707104i	$-0.750001\pi$
$449$	−29.9668	−1.41422	−0.707110	+	0.707104i	$0.750001\pi$
$450$	0	0
$451$	−4.72420	−0.222454
$452$	−7.95509	−0.374176
$453$	0	0
$454$	19.2319	0.902598
$455$	0.269169	0.0126188
$456$	0	0
$457$	17.6698	0.826556	0.413278	−	0.910605i	$-0.364384\pi$
$457$	17.6698	0.826556	0.413278	+	0.910605i	$0.364384\pi$
$458$	−22.1180	−1.03350
$459$	0	0
$460$	−7.43244	−0.346539
$461$	−22.3445	−1.04069	−0.520343	−	0.853957i	$-0.674196\pi$
$461$	−22.3445	−1.04069	−0.520343	+	0.853957i	$0.674196\pi$
$462$	0	0
$463$	−6.83302	−0.317557	−0.158779	−	0.987314i	$-0.550756\pi$
$463$	−6.83302	−0.317557	−0.158779	+	0.987314i	$0.550756\pi$
$464$	42.2763	1.96263
$465$	0	0
$466$	37.2744	1.72670
$467$	−9.00896	−0.416885	−0.208443	−	0.978035i	$-0.566839\pi$
$467$	−9.00896	−0.416885	−0.208443	+	0.978035i	$0.566839\pi$
$468$	0	0
$469$	1.55489	0.0717981
$470$	−18.6608	−0.860758
$471$	0	0
$472$	20.6065	0.948492
$473$	1.47959	0.0680317
$474$	0	0
$475$	1.00000	0.0458831
$476$	−11.5470	−0.529256
$477$	0	0
$478$	37.3216	1.70705
$479$	−9.26731	−0.423434	−0.211717	−	0.977331i	$-0.567906\pi$
$479$	−9.26731	−0.423434	−0.211717	+	0.977331i	$0.567906\pi$
$480$	0	0
$481$	3.88914	0.177329
$482$	73.3290	3.34004
$483$	0	0
$484$	−21.4944	−0.977020
$485$	17.4526	0.792482
$486$	0	0
$487$	−38.7694	−1.75681	−0.878405	−	0.477917i	$-0.841392\pi$
$487$	−38.7694	−1.75681	−0.878405	+	0.477917i	$0.841392\pi$
$488$	95.2205	4.31043
$489$	0	0
$490$	−17.5421	−0.792473
$491$	−21.6877	−0.978751	−0.489376	−	0.872073i	$-0.662775\pi$
$491$	−21.6877	−0.978751	−0.489376	+	0.872073i	$0.662775\pi$
$492$	0	0
$493$	16.6964	0.751966
$494$	−1.23191	−0.0554260
$495$	0	0
$496$	−102.646	−4.60893
$497$	−4.25291	−0.190769
$498$	0	0
$499$	−27.8372	−1.24616	−0.623082	−	0.782156i	$-0.714120\pi$
$499$	−27.8372	−1.24616	−0.623082	+	0.782156i	$0.714120\pi$
$500$	4.91744	0.219915
$501$	0	0
$502$	68.8828	3.07439
$503$	41.2449	1.83902	0.919510	−	0.393067i	$-0.128586\pi$
$503$	41.2449	1.83902	0.919510	+	0.393067i	$0.128586\pi$
$504$	0	0
$505$	−14.2831	−0.635589
$506$	10.2350	0.455000
$507$	0	0
$508$	−64.5619	−2.86447
$509$	−0.416364	−0.0184550	−0.00922751	−	0.999957i	$-0.502937\pi$
$509$	−0.416364	−0.0184550	−0.00922751	+	0.999957i	$0.502937\pi$
$510$	0	0
$511$	−6.94555	−0.307253
$512$	−36.0131	−1.59157
$513$	0	0
$514$	−23.7191	−1.04620
$515$	4.79182	0.211153
$516$	0	0
$517$	18.2675	0.803404
$518$	12.5499	0.551412
$519$	0	0
$520$	−3.59401	−0.157608
$521$	−24.0458	−1.05346	−0.526732	−	0.850031i	$-0.676583\pi$
$521$	−24.0458	−1.05346	−0.526732	+	0.850031i	$0.676583\pi$
$522$	0	0
$523$	−5.19736	−0.227265	−0.113632	−	0.993523i	$-0.536249\pi$
$523$	−5.19736	−0.227265	−0.113632	+	0.993523i	$0.536249\pi$
$524$	40.1865	1.75555
$525$	0	0
$526$	23.6945	1.03313
$527$	−40.5383	−1.76588
$528$	0	0
$529$	−20.7155	−0.900675
$530$	−11.3180	−0.491623
$531$	0	0
$532$	−2.82591	−0.122519
$533$	−0.859432	−0.0372261
$534$	0	0
$535$	−9.22611	−0.398880
$536$	−20.7613	−0.896751
$537$	0	0
$538$	80.5170	3.47133
$539$	17.1724	0.739669
$540$	0	0
$541$	−1.93863	−0.0833481	−0.0416741	−	0.999131i	$-0.513269\pi$
$541$	−1.93863	−0.0833481	−0.0416741	+	0.999131i	$0.513269\pi$
$542$	63.4327	2.72467
$543$	0	0
$544$	48.4841	2.07874
$545$	4.89810	0.209812
$546$	0	0
$547$	−12.9075	−0.551883	−0.275941	−	0.961174i	$-0.588990\pi$
$547$	−12.9075	−0.551883	−0.275941	+	0.961174i	$0.588990\pi$
$548$	71.8267	3.06828
$549$	0	0
$550$	−6.77165	−0.288744
$551$	4.08612	0.174074
$552$	0	0
$553$	3.84199	0.163378
$554$	11.9952	0.509628
$555$	0	0
$556$	−10.1899	−0.432147
$557$	34.1040	1.44503	0.722517	−	0.691353i	$-0.242985\pi$
$557$	34.1040	1.44503	0.722517	+	0.691353i	$0.242985\pi$
$558$	0	0
$559$	0.269169	0.0113846
$560$	−5.94574	−0.251253
$561$	0	0
$562$	16.0847	0.678494
$563$	14.4911	0.610727	0.305363	−	0.952236i	$-0.401222\pi$
$563$	14.4911	0.610727	0.305363	+	0.952236i	$0.401222\pi$
$564$	0	0
$565$	−1.61773	−0.0680584
$566$	−50.1159	−2.10653
$567$	0	0
$568$	56.7859	2.38268
$569$	39.2110	1.64381	0.821906	−	0.569624i	$-0.192911\pi$
$569$	39.2110	1.64381	0.821906	+	0.569624i	$0.192911\pi$
$570$	0	0
$571$	21.9915	0.920316	0.460158	−	0.887837i	$-0.347793\pi$
$571$	21.9915	0.920316	0.460158	+	0.887837i	$0.347793\pi$
$572$	5.93015	0.247952
$573$	0	0
$574$	−2.77331	−0.115756
$575$	−1.51145	−0.0630316
$576$	0	0
$577$	23.8735	0.993869	0.496934	−	0.867788i	$-0.334459\pi$
$577$	23.8735	0.993869	0.496934	+	0.867788i	$0.334459\pi$
$578$	−0.798628	−0.0332185
$579$	0	0
$580$	20.0932	0.834326
$581$	−3.82777	−0.158803
$582$	0	0
$583$	11.0795	0.458866
$584$	92.7387	3.83756
$585$	0	0
$586$	16.9349	0.699574
$587$	−17.8281	−0.735843	−0.367921	−	0.929857i	$-0.619930\pi$
$587$	−17.8281	−0.735843	−0.367921	+	0.929857i	$0.619930\pi$
$588$	0	0
$589$	−9.92099	−0.408787
$590$	7.06323	0.290788
$591$	0	0
$592$	−85.9082	−3.53081
$593$	21.2446	0.872412	0.436206	−	0.899847i	$-0.356322\pi$
$593$	21.2446	0.872412	0.436206	+	0.899847i	$0.356322\pi$
$594$	0	0
$595$	−2.34818	−0.0962658
$596$	−43.8244	−1.79512
$597$	0	0
$598$	1.86196	0.0761411
$599$	−24.3374	−0.994397	−0.497199	−	0.867637i	$-0.665638\pi$
$599$	−24.3374	−0.994397	−0.497199	+	0.867637i	$0.665638\pi$
$600$	0	0
$601$	15.7891	0.644051	0.322025	−	0.946731i	$-0.395636\pi$
$601$	15.7891	0.644051	0.322025	+	0.946731i	$0.395636\pi$
$602$	0.868585	0.0354009
$603$	0	0
$604$	56.3400	2.29244
$605$	−4.37107	−0.177709
$606$	0	0
$607$	7.81471	0.317189	0.158595	−	0.987344i	$-0.449304\pi$
$607$	7.81471	0.317189	0.158595	+	0.987344i	$0.449304\pi$
$608$	11.8656	0.481212
$609$	0	0
$610$	32.6384	1.32149
$611$	3.32324	0.134444
$612$	0	0
$613$	12.9547	0.523235	0.261618	−	0.965172i	$-0.415744\pi$
$613$	12.9547	0.523235	0.261618	+	0.965172i	$0.415744\pi$
$614$	−17.8160	−0.718996
$615$	0	0
$616$	11.3531	0.457431
$617$	−18.8873	−0.760373	−0.380187	−	0.924910i	$-0.624140\pi$
$617$	−18.8873	−0.760373	−0.380187	+	0.924910i	$0.624140\pi$
$618$	0	0
$619$	29.2673	1.17635	0.588176	−	0.808733i	$-0.299846\pi$
$619$	29.2673	1.17635	0.588176	+	0.808733i	$0.299846\pi$
$620$	−48.7859	−1.95929
$621$	0	0
$622$	−54.2339	−2.17458
$623$	8.39396	0.336297
$624$	0	0
$625$	1.00000	0.0400000
$626$	50.9479	2.03629
$627$	0	0
$628$	−32.4872	−1.29638
$629$	−33.9281	−1.35280
$630$	0	0
$631$	−5.25309	−0.209122	−0.104561	−	0.994518i	$-0.533344\pi$
$631$	−5.25309	−0.209122	−0.104561	+	0.994518i	$0.533344\pi$
$632$	−51.2992	−2.04057
$633$	0	0
$634$	−8.87291	−0.352388
$635$	−13.1292	−0.521015
$636$	0	0
$637$	3.12402	0.123778
$638$	−27.6698	−1.09546
$639$	0	0
$640$	3.92440	0.155126
$641$	13.0021	0.513554	0.256777	−	0.966471i	$-0.417339\pi$
$641$	13.0021	0.513554	0.256777	+	0.966471i	$0.417339\pi$
$642$	0	0
$643$	−17.3534	−0.684353	−0.342176	−	0.939636i	$-0.611164\pi$
$643$	−17.3534	−0.684353	−0.342176	+	0.939636i	$0.611164\pi$
$644$	4.27121	0.168309
$645$	0	0
$646$	10.7469	0.422831
$647$	−12.4848	−0.490830	−0.245415	−	0.969418i	$-0.578924\pi$
$647$	−12.4848	−0.490830	−0.245415	+	0.969418i	$0.578924\pi$
$648$	0	0
$649$	−6.91437	−0.271413
$650$	−1.23191	−0.0483193
$651$	0	0
$652$	99.5507	3.89871
$653$	28.3532	1.10955	0.554774	−	0.832001i	$-0.312805\pi$
$653$	28.3532	1.10955	0.554774	+	0.832001i	$0.312805\pi$
$654$	0	0
$655$	8.17223	0.319316
$656$	18.9842	0.741209
$657$	0	0
$658$	10.7238	0.418058
$659$	−45.2202	−1.76153	−0.880764	−	0.473556i	$-0.842970\pi$
$659$	−45.2202	−1.76153	−0.880764	+	0.473556i	$0.842970\pi$
$660$	0	0
$661$	−14.6086	−0.568207	−0.284104	−	0.958794i	$-0.591696\pi$
$661$	−14.6086	−0.568207	−0.284104	+	0.958794i	$0.591696\pi$
$662$	−86.2115	−3.35070
$663$	0	0
$664$	51.1093	1.98343
$665$	−0.574672	−0.0222848
$666$	0	0
$667$	−6.17594	−0.239133
$668$	28.9820	1.12135
$669$	0	0
$670$	−7.11627	−0.274926
$671$	−31.9505	−1.23344
$672$	0	0
$673$	−0.440534	−0.0169813	−0.00849067	−	0.999964i	$-0.502703\pi$
$673$	−0.440534	−0.0169813	−0.00849067	+	0.999964i	$0.502703\pi$
$674$	23.0111	0.886356
$675$	0	0
$676$	−62.8479	−2.41723
$677$	−32.8057	−1.26083	−0.630414	−	0.776259i	$-0.717115\pi$
$677$	−32.8057	−1.26083	−0.630414	+	0.776259i	$0.717115\pi$
$678$	0	0
$679$	−10.0295	−0.384898
$680$	31.3534	1.20235
$681$	0	0
$682$	67.1815	2.57251
$683$	39.6092	1.51561	0.757803	−	0.652484i	$-0.226273\pi$
$683$	39.6092	1.51561	0.757803	+	0.652484i	$0.226273\pi$
$684$	0	0
$685$	14.6065	0.558087
$686$	20.6611	0.788844
$687$	0	0
$688$	−5.94574	−0.226679
$689$	2.01559	0.0767879
$690$	0	0
$691$	−19.8962	−0.756889	−0.378444	−	0.925624i	$-0.623541\pi$
$691$	−19.8962	−0.756889	−0.378444	+	0.925624i	$0.623541\pi$
$692$	17.3665	0.660175
$693$	0	0
$694$	48.6642	1.84727
$695$	−2.07219	−0.0786027
$696$	0	0
$697$	7.49752	0.283989
$698$	9.31924	0.352738
$699$	0	0
$700$	−2.82591	−0.106809
$701$	14.1251	0.533497	0.266748	−	0.963766i	$-0.414051\pi$
$701$	14.1251	0.533497	0.266748	+	0.963766i	$0.414051\pi$
$702$	0	0
$703$	−8.30326	−0.313163
$704$	−27.0727	−1.02034
$705$	0	0
$706$	8.97234	0.337678
$707$	8.20809	0.308697
$708$	0	0
$709$	−41.1815	−1.54660	−0.773302	−	0.634038i	$-0.781396\pi$
$709$	−41.1815	−1.54660	−0.773302	+	0.634038i	$0.781396\pi$
$710$	19.4643	0.730482
$711$	0	0
$712$	−112.078	−4.20031
$713$	14.9950	0.561569
$714$	0	0
$715$	1.20594	0.0450997
$716$	−35.3466	−1.32097
$717$	0	0
$718$	4.48718	0.167460
$719$	18.0227	0.672133	0.336067	−	0.941838i	$-0.390903\pi$
$719$	18.0227	0.672133	0.336067	+	0.941838i	$0.390903\pi$
$720$	0	0
$721$	−2.75372	−0.102554
$722$	2.63010	0.0978823
$723$	0	0
$724$	76.4332	2.84062
$725$	4.08612	0.151755
$726$	0	0
$727$	−21.1266	−0.783544	−0.391772	−	0.920062i	$-0.628138\pi$
$727$	−21.1266	−0.783544	−0.391772	+	0.920062i	$0.628138\pi$
$728$	2.06537	0.0765479
$729$	0	0
$730$	31.7877	1.17652
$731$	−2.34818	−0.0868504
$732$	0	0
$733$	−27.6660	−1.02187	−0.510934	−	0.859620i	$-0.670700\pi$
$733$	−27.6660	−1.02187	−0.510934	+	0.859620i	$0.670700\pi$
$734$	98.9326	3.65167
$735$	0	0
$736$	−17.9341	−0.661061
$737$	6.96629	0.256607
$738$	0	0
$739$	−14.2987	−0.525986	−0.262993	−	0.964798i	$-0.584710\pi$
$739$	−14.2987	−0.525986	−0.262993	+	0.964798i	$0.584710\pi$
$740$	−40.8308	−1.50097
$741$	0	0
$742$	6.50415	0.238775
$743$	−49.9438	−1.83226	−0.916130	−	0.400881i	$-0.868704\pi$
$743$	−49.9438	−1.83226	−0.916130	+	0.400881i	$0.868704\pi$
$744$	0	0
$745$	−8.91203	−0.326511
$746$	35.2516	1.29065
$747$	0	0
$748$	−51.7335	−1.89156
$749$	5.30198	0.193730
$750$	0	0
$751$	9.69216	0.353672	0.176836	−	0.984240i	$-0.443414\pi$
$751$	9.69216	0.353672	0.176836	+	0.984240i	$0.443414\pi$
$752$	−73.4080	−2.67691
$753$	0	0
$754$	−5.03371	−0.183317
$755$	11.4572	0.416970
$756$	0	0
$757$	46.6889	1.69694	0.848469	−	0.529245i	$-0.177525\pi$
$757$	46.6889	1.69694	0.848469	+	0.529245i	$0.177525\pi$
$758$	−24.6469	−0.895214
$759$	0	0
$760$	7.67316	0.278335
$761$	38.7361	1.40418	0.702091	−	0.712087i	$-0.252250\pi$
$761$	38.7361	1.40418	0.702091	+	0.712087i	$0.252250\pi$
$762$	0	0
$763$	−2.81480	−0.101903
$764$	−65.5080	−2.37000
$765$	0	0
$766$	5.52104	0.199483
$767$	−1.25787	−0.0454190
$768$	0	0
$769$	−5.38666	−0.194248	−0.0971239	−	0.995272i	$-0.530964\pi$
$769$	−5.38666	−0.194248	−0.0971239	+	0.995272i	$0.530964\pi$
$770$	3.89148	0.140239
$771$	0	0
$772$	93.4112	3.36194
$773$	−7.20137	−0.259015	−0.129508	−	0.991578i	$-0.541340\pi$
$773$	−7.20137	−0.259015	−0.129508	+	0.991578i	$0.541340\pi$
$774$	0	0
$775$	−9.92099	−0.356373
$776$	133.917	4.80733
$777$	0	0
$778$	−2.82047	−0.101119
$779$	1.83488	0.0657413
$780$	0	0
$781$	−19.0541	−0.681808
$782$	−16.2434	−0.580861
$783$	0	0
$784$	−69.0074	−2.46455
$785$	−6.60653	−0.235797
$786$	0	0
$787$	−33.5231	−1.19497	−0.597485	−	0.801880i	$-0.703833\pi$
$787$	−33.5231	−1.19497	−0.597485	+	0.801880i	$0.703833\pi$
$788$	−10.6818	−0.380524
$789$	0	0
$790$	−17.5836	−0.625598
$791$	0.929664	0.0330550
$792$	0	0
$793$	−5.81247	−0.206407
$794$	−61.8972	−2.19665
$795$	0	0
$796$	−9.21305	−0.326548
$797$	10.4979	0.371855	0.185927	−	0.982563i	$-0.440471\pi$
$797$	10.4979	0.371855	0.185927	+	0.982563i	$0.440471\pi$
$798$	0	0
$799$	−28.9913	−1.02564
$800$	11.8656	0.419511
$801$	0	0
$802$	−49.0432	−1.73177
$803$	−31.1178	−1.09812
$804$	0	0
$805$	0.868585	0.0306136
$806$	12.2217	0.430492
$807$	0	0
$808$	−109.596	−3.85559
$809$	13.0724	0.459600	0.229800	−	0.973238i	$-0.426193\pi$
$809$	13.0724	0.459600	0.229800	+	0.973238i	$0.426193\pi$
$810$	0	0
$811$	−10.0790	−0.353922	−0.176961	−	0.984218i	$-0.556627\pi$
$811$	−10.0790	−0.353922	−0.176961	+	0.984218i	$0.556627\pi$
$812$	−11.5470	−0.405221
$813$	0	0
$814$	56.2268	1.97075
$815$	20.2444	0.709131
$816$	0	0
$817$	−0.574672	−0.0201052
$818$	−18.0956	−0.632697
$819$	0	0
$820$	9.02289	0.315093
$821$	−40.2987	−1.40643	−0.703217	−	0.710975i	$-0.748254\pi$
$821$	−40.2987	−1.40643	−0.703217	+	0.710975i	$0.748254\pi$
$822$	0	0
$823$	−5.07715	−0.176978	−0.0884892	−	0.996077i	$-0.528204\pi$
$823$	−5.07715	−0.176978	−0.0884892	+	0.996077i	$0.528204\pi$
$824$	36.7684	1.28089
$825$	0	0
$826$	−4.05904	−0.141232
$827$	42.8077	1.48857	0.744285	−	0.667862i	$-0.232791\pi$
$827$	42.8077	1.48857	0.744285	+	0.667862i	$0.232791\pi$
$828$	0	0
$829$	24.2018	0.840562	0.420281	−	0.907394i	$-0.361932\pi$
$829$	24.2018	0.840562	0.420281	+	0.907394i	$0.361932\pi$
$830$	17.5186	0.608078
$831$	0	0
$832$	−4.92509	−0.170747
$833$	−27.2534	−0.944274
$834$	0	0
$835$	5.89372	0.203960
$836$	−12.6608	−0.437883
$837$	0	0
$838$	−35.4527	−1.22469
$839$	8.63975	0.298277	0.149139	−	0.988816i	$-0.452350\pi$
$839$	8.63975	0.298277	0.149139	+	0.988816i	$0.452350\pi$
$840$	0	0
$841$	−12.3036	−0.424264
$842$	−15.3463	−0.528869
$843$	0	0
$844$	−84.1325	−2.89596
$845$	−12.7806	−0.439666
$846$	0	0
$847$	2.51193	0.0863109
$848$	−44.5229	−1.52892
$849$	0	0
$850$	10.7469	0.368616
$851$	12.5499	0.430206
$852$	0	0
$853$	−13.0229	−0.445895	−0.222948	−	0.974830i	$-0.571568\pi$
$853$	−13.0229	−0.445895	−0.222948	+	0.974830i	$0.571568\pi$
$854$	−18.7564	−0.641829
$855$	0	0
$856$	−70.7934	−2.41967
$857$	2.82367	0.0964548	0.0482274	−	0.998836i	$-0.484643\pi$
$857$	2.82367	0.0964548	0.0482274	+	0.998836i	$0.484643\pi$
$858$	0	0
$859$	24.1227	0.823057	0.411529	−	0.911397i	$-0.364995\pi$
$859$	24.1227	0.823057	0.411529	+	0.911397i	$0.364995\pi$
$860$	−2.82591	−0.0963628
$861$	0	0
$862$	−76.8092	−2.61613
$863$	−32.7307	−1.11417	−0.557084	−	0.830456i	$-0.688080\pi$
$863$	−32.7307	−1.11417	−0.557084	+	0.830456i	$0.688080\pi$
$864$	0	0
$865$	3.53161	0.120078
$866$	−32.9366	−1.11923
$867$	0	0
$868$	28.0359	0.951599
$869$	17.2131	0.583913
$870$	0	0
$871$	1.26731	0.0429413
$872$	37.5839	1.27275
$873$	0	0
$874$	−3.97526	−0.134465
$875$	−0.574672	−0.0194274
$876$	0	0
$877$	−49.5072	−1.67174	−0.835869	−	0.548929i	$-0.815036\pi$
$877$	−49.5072	−1.67174	−0.835869	+	0.548929i	$0.815036\pi$
$878$	41.2318	1.39150
$879$	0	0
$880$	−26.6384	−0.897980
$881$	−40.3152	−1.35826	−0.679128	−	0.734020i	$-0.737642\pi$
$881$	−40.3152	−1.35826	−0.679128	+	0.734020i	$0.737642\pi$
$882$	0	0
$883$	−29.3347	−0.987192	−0.493596	−	0.869691i	$-0.664318\pi$
$883$	−29.3347	−0.987192	−0.493596	+	0.869691i	$0.664318\pi$
$884$	−9.41140	−0.316540
$885$	0	0
$886$	78.4508	2.63561
$887$	−43.3214	−1.45459	−0.727295	−	0.686325i	$-0.759223\pi$
$887$	−43.3214	−1.45459	−0.727295	+	0.686325i	$0.759223\pi$
$888$	0	0
$889$	7.54496	0.253050
$890$	−38.4167	−1.28773
$891$	0	0
$892$	25.2224	0.844508
$893$	−7.09508	−0.237428
$894$	0	0
$895$	−7.18801	−0.240269
$896$	−2.25524	−0.0753424
$897$	0	0
$898$	−78.8157	−2.63011
$899$	−40.5383	−1.35203
$900$	0	0
$901$	−17.5836	−0.585796
$902$	−12.4251	−0.413712
$903$	0	0
$904$	−12.4131	−0.412854
$905$	15.5433	0.516677
$906$	0	0
$907$	14.0731	0.467288	0.233644	−	0.972322i	$-0.424935\pi$
$907$	14.0731	0.467288	0.233644	+	0.972322i	$0.424935\pi$
$908$	35.9574	1.19329
$909$	0	0
$910$	0.707941	0.0234680
$911$	−30.0725	−0.996346	−0.498173	−	0.867078i	$-0.665996\pi$
$911$	−30.0725	−0.996346	−0.498173	+	0.867078i	$0.665996\pi$
$912$	0	0
$913$	−17.1493	−0.567560
$914$	46.4733	1.53720
$915$	0	0
$916$	−41.3534	−1.36636
$917$	−4.69635	−0.155087
$918$	0	0
$919$	29.6518	0.978123	0.489062	−	0.872249i	$-0.337339\pi$
$919$	29.6518	0.978123	0.489062	+	0.872249i	$0.337339\pi$
$920$	−11.5976	−0.382360
$921$	0	0
$922$	−58.7682	−1.93543
$923$	−3.46634	−0.114096
$924$	0	0
$925$	−8.30326	−0.273010
$926$	−17.9715	−0.590582
$927$	0	0
$928$	48.4841	1.59157
$929$	−58.3803	−1.91540	−0.957698	−	0.287775i	$-0.907085\pi$
$929$	−58.3803	−1.91540	−0.957698	+	0.287775i	$0.907085\pi$
$930$	0	0
$931$	−6.66975	−0.218592
$932$	69.6911	2.28281
$933$	0	0
$934$	−23.6945	−0.775308
$935$	−10.5204	−0.344054
$936$	0	0
$937$	−3.15850	−0.103184	−0.0515918	−	0.998668i	$-0.516429\pi$
$937$	−3.15850	−0.103184	−0.0515918	+	0.998668i	$0.516429\pi$
$938$	4.08952	0.133528
$939$	0	0
$940$	−34.8896	−1.13797
$941$	9.66975	0.315225	0.157612	−	0.987501i	$-0.449620\pi$
$941$	9.66975	0.315225	0.157612	+	0.987501i	$0.449620\pi$
$942$	0	0
$943$	−2.77331	−0.0903116
$944$	27.7854	0.904337
$945$	0	0
$946$	3.89148	0.126523
$947$	−31.3714	−1.01943	−0.509716	−	0.860343i	$-0.670250\pi$
$947$	−31.3714	−1.01943	−0.509716	+	0.860343i	$0.670250\pi$
$948$	0	0
$949$	−5.66098	−0.183763
$950$	2.63010	0.0853318
$951$	0	0
$952$	−18.0179	−0.583964
$953$	13.1224	0.425075	0.212537	−	0.977153i	$-0.431827\pi$
$953$	13.1224	0.425075	0.212537	+	0.977153i	$0.431827\pi$
$954$	0	0
$955$	−13.3216	−0.431076
$956$	69.7792	2.25682
$957$	0	0
$958$	−24.3740	−0.787488
$959$	−8.39396	−0.271055
$960$	0	0
$961$	67.4261	2.17504
$962$	10.2288	0.329791
$963$	0	0
$964$	137.101	4.41574
$965$	18.9959	0.611500
$966$	0	0
$967$	−44.4400	−1.42910	−0.714548	−	0.699587i	$-0.753367\pi$
$967$	−44.4400	−1.42910	−0.714548	+	0.699587i	$0.753367\pi$
$968$	−33.5399	−1.07801
$969$	0	0
$970$	45.9021	1.47383
$971$	5.69927	0.182898	0.0914491	−	0.995810i	$-0.470850\pi$
$971$	5.69927	0.182898	0.0914491	+	0.995810i	$0.470850\pi$
$972$	0	0
$973$	1.19083	0.0381762
$974$	−101.968	−3.26725
$975$	0	0
$976$	128.393	4.10977
$977$	23.9164	0.765154	0.382577	−	0.923924i	$-0.375037\pi$
$977$	23.9164	0.765154	0.382577	+	0.923924i	$0.375037\pi$
$978$	0	0
$979$	37.6070	1.20193
$980$	−32.7981	−1.04770
$981$	0	0
$982$	−57.0408	−1.82025
$983$	45.5302	1.45219	0.726095	−	0.687595i	$-0.241333\pi$
$983$	45.5302	1.45219	0.726095	+	0.687595i	$0.241333\pi$
$984$	0	0
$985$	−2.17223	−0.0692131
$986$	43.9131	1.39848
$987$	0	0
$988$	−2.30326	−0.0732766
$989$	0.868585	0.0276194
$990$	0	0
$991$	27.5521	0.875220	0.437610	−	0.899165i	$-0.355825\pi$
$991$	27.5521	0.875220	0.437610	+	0.899165i	$0.355825\pi$
$992$	−117.718	−3.73756
$993$	0	0
$994$	−11.1856	−0.354785
$995$	−1.87355	−0.0593954
$996$	0	0
$997$	16.8557	0.533826	0.266913	−	0.963721i	$-0.413996\pi$
$997$	16.8557	0.533826	0.266913	+	0.963721i	$0.413996\pi$
$998$	−73.2147	−2.31757
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.m.1.4		4
3.2	odd	2		95.2.a.b.1.1	✓	4
5.4	even	2		4275.2.a.bo.1.1		4
12.11	even	2		1520.2.a.t.1.1		4
15.2	even	4		475.2.b.e.324.1		8
15.8	even	4		475.2.b.e.324.8		8
15.14	odd	2		475.2.a.i.1.4		4
21.20	even	2		4655.2.a.y.1.1		4
24.5	odd	2		6080.2.a.cc.1.1		4
24.11	even	2		6080.2.a.ch.1.4		4
57.56	even	2		1805.2.a.p.1.4		4
60.59	even	2		7600.2.a.cf.1.4		4
285.284	even	2		9025.2.a.bf.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.1	✓	4	3.2	odd	2
475.2.a.i.1.4		4	15.14	odd	2
475.2.b.e.324.1		8	15.2	even	4
475.2.b.e.324.8		8	15.8	even	4
855.2.a.m.1.4		4	1.1	even	1	trivial
1520.2.a.t.1.1		4	12.11	even	2
1805.2.a.p.1.4		4	57.56	even	2
4275.2.a.bo.1.1		4	5.4	even	2
4655.2.a.y.1.1		4	21.20	even	2
6080.2.a.cc.1.1		4	24.5	odd	2
6080.2.a.ch.1.4		4	24.11	even	2
7600.2.a.cf.1.4		4	60.59	even	2
9025.2.a.bf.1.1		4	285.284	even	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+2.63010 q^{2} +4.91744 q^{4} +1.00000 q^{5} -0.574672 q^{7} +7.67316 q^{8} +2.63010 q^{10} -2.57467 q^{11} -0.468387 q^{13} -1.51145 q^{14} +10.3463 q^{16} +4.08612 q^{17} +1.00000 q^{19} +4.91744 q^{20} -6.77165 q^{22} -1.51145 q^{23} +1.00000 q^{25} -1.23191 q^{26} -2.82591 q^{28} +4.08612 q^{29} -9.92099 q^{31} +11.8656 q^{32} +10.7469 q^{34} -0.574672 q^{35} -8.30326 q^{37} +2.63010 q^{38} +7.67316 q^{40} +1.83488 q^{41} -0.574672 q^{43} -12.6608 q^{44} -3.97526 q^{46} -7.09508 q^{47} -6.66975 q^{49} +2.63010 q^{50} -2.30326 q^{52} -4.30326 q^{53} -2.57467 q^{55} -4.40955 q^{56} +10.7469 q^{58} +2.68553 q^{59} +12.4095 q^{61} -26.0932 q^{62} +10.5150 q^{64} -0.468387 q^{65} -2.70570 q^{67} +20.0932 q^{68} -1.51145 q^{70} +7.40058 q^{71} +12.0861 q^{73} -21.8384 q^{74} +4.91744 q^{76} +1.47959 q^{77} -6.68553 q^{79} +10.3463 q^{80} +4.82591 q^{82} +6.66079 q^{83} +4.08612 q^{85} -1.51145 q^{86} -19.7559 q^{88} -14.6065 q^{89} +0.269169 q^{91} -7.43244 q^{92} -18.6608 q^{94} +1.00000 q^{95} +17.4526 q^{97} -17.5421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{13} + 8 q^{14} + 4 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 4 q^{26} - 8 q^{28} - 4 q^{29}+ \cdots - 38 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^13 + 8 * q^14 + 4 * q^16 - 4 * q^17 + 4 * q^19 + 8 * q^20 + 4 * q^22 + 8 * q^23 + 4 * q^25 - 4 * q^26 - 8 * q^28 - 4 * q^29 + 4 * q^31 + 6 * q^32 - 4 * q^34 + 4 * q^35 - 6 * q^37 + 2 * q^38 + 12 * q^40 - 16 * q^41 + 4 * q^43 - 24 * q^44 + 12 * q^47 + 20 * q^49 + 2 * q^50 + 18 * q^52 + 10 * q^53 - 4 * q^55 + 12 * q^56 - 4 * q^58 + 20 * q^61 - 20 * q^62 - 4 * q^64 + 2 * q^65 - 18 * q^67 - 4 * q^68 + 8 * q^70 + 20 * q^71 + 28 * q^73 - 32 * q^74 + 8 * q^76 + 40 * q^77 - 16 * q^79 + 4 * q^80 + 16 * q^82 - 4 * q^85 + 8 * q^86 - 12 * q^88 - 4 * q^89 - 36 * q^91 + 28 * q^92 - 48 * q^94 + 4 * q^95 + 30 * q^97 - 38 * q^98

Introduction

L-functions

Modular forms

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Database

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