LMFDB - Embedded newform 1520.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.28734$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04306 q^{3} -1.00000 q^{5} +0.574672 q^{7} +6.26020 q^{9} +O(q^{10})$	$q-3.04306 q^{3} -1.00000 q^{5} +0.574672 q^{7} +6.26020 q^{9} -2.57467 q^{11} -0.468387 q^{13} +3.04306 q^{15} -4.08612 q^{17} -1.00000 q^{19} -1.74876 q^{21} -1.51145 q^{23} +1.00000 q^{25} -9.92099 q^{27} -4.08612 q^{29} +9.92099 q^{31} +7.83488 q^{33} -0.574672 q^{35} -8.30326 q^{37} +1.42533 q^{39} -1.83488 q^{41} +0.574672 q^{43} -6.26020 q^{45} -7.09508 q^{47} -6.66975 q^{49} +12.4343 q^{51} +4.30326 q^{53} +2.57467 q^{55} +3.04306 q^{57} +2.68553 q^{59} +12.4095 q^{61} +3.59756 q^{63} +0.468387 q^{65} +2.70570 q^{67} +4.59942 q^{69} +7.40058 q^{71} +12.0861 q^{73} -3.04306 q^{75} -1.47959 q^{77} +6.68553 q^{79} +11.4095 q^{81} +6.66079 q^{83} +4.08612 q^{85} +12.4343 q^{87} +14.6065 q^{89} -0.269169 q^{91} -30.1902 q^{93} +1.00000 q^{95} +17.4526 q^{97} -16.1180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9	$ 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 - 6 * q^37 + 12 * q^39 + 16 * q^41 - 4 * q^43 - 8 * q^45 + 12 * q^47 + 20 * q^49 + 36 * q^51 - 10 * q^53 + 4 * q^55 + 2 * q^57 + 20 * q^61 - 20 * q^63 - 2 * q^65 + 18 * q^67 + 28 * q^69 + 20 * q^71 + 28 * q^73 - 2 * q^75 - 40 * q^77 + 16 * q^79 + 16 * q^81 - 4 * q^85 + 36 * q^87 + 4 * q^89 + 36 * q^91 - 40 * q^93 + 4 * q^95 + 30 * q^97 + 4 * 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
$3$	−3.04306	−1.75691	−0.878455	−	0.477825i	$-0.841425\pi$
$3$	−3.04306	−1.75691	−0.878455	+	0.477825i	$0.841425\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.574672	0.217205	0.108603	−	0.994085i	$-0.465362\pi$
$7$	0.574672	0.217205	0.108603	+	0.994085i	$0.465362\pi$
$8$	0	0
$9$	6.26020	2.08673
$10$	0	0
$11$	−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	$0.520690\pi$
$14$	0	0
$15$	3.04306	0.785714
$16$	0	0
$17$	−4.08612	−0.991029	−0.495514	−	0.868600i	$-0.665020\pi$
$17$	−4.08612	−0.991029	−0.495514	+	0.868600i	$0.665020\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.74876	−0.381611
$22$	0	0
$23$	−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
$26$	0	0
$27$	−9.92099	−1.90930
$28$	0	0
$29$	−4.08612	−0.758773	−0.379386	−	0.925238i	$-0.623865\pi$
$29$	−4.08612	−0.758773	−0.379386	+	0.925238i	$0.623865\pi$
$30$	0	0
$31$	9.92099	1.78186	0.890931	−	0.454138i	$-0.150053\pi$
$31$	9.92099	1.78186	0.890931	+	0.454138i	$0.150053\pi$
$32$	0	0
$33$	7.83488	1.36388
$34$	0	0
$35$	−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	$0.739118\pi$
$38$	0	0
$39$	1.42533	0.228235
$40$	0	0
$41$	−1.83488	−0.286560	−0.143280	−	0.989682i	$-0.545765\pi$
$41$	−1.83488	−0.286560	−0.143280	+	0.989682i	$0.545765\pi$
$42$	0	0
$43$	0.574672	0.0876366	0.0438183	−	0.999040i	$-0.486048\pi$
$43$	0.574672	0.0876366	0.0438183	+	0.999040i	$0.486048\pi$
$44$	0	0
$45$	−6.26020	−0.933216
$46$	0	0
$47$	−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
$50$	0	0
$51$	12.4343	1.74115
$52$	0	0
$53$	4.30326	0.591099	0.295549	−	0.955327i	$-0.404497\pi$
$53$	4.30326	0.591099	0.295549	+	0.955327i	$0.404497\pi$
$54$	0	0
$55$	2.57467	0.347169
$56$	0	0
$57$	3.04306	0.403063
$58$	0	0
$59$	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	$0.207765\pi$
$62$	0	0
$63$	3.59756	0.453250
$64$	0	0
$65$	0.468387	0.0580962
$66$	0	0
$67$	2.70570	0.330554	0.165277	−	0.986247i	$-0.447148\pi$
$67$	2.70570	0.330554	0.165277	+	0.986247i	$0.447148\pi$
$68$	0	0
$69$	4.59942	0.553705
$70$	0	0
$71$	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	$0.249919\pi$
$74$	0	0
$75$	−3.04306	−0.351382
$76$	0	0
$77$	−1.47959	−0.168615
$78$	0	0
$79$	6.68553	0.752181	0.376091	−	0.926583i	$-0.377268\pi$
$79$	6.68553	0.752181	0.376091	+	0.926583i	$0.377268\pi$
$80$	0	0
$81$	11.4095	1.26773
$82$	0	0
$83$	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
$86$	0	0
$87$	12.4343	1.33310
$88$	0	0
$89$	14.6065	1.54829	0.774144	−	0.633009i	$-0.218180\pi$
$89$	14.6065	1.54829	0.774144	+	0.633009i	$0.218180\pi$
$90$	0	0
$91$	−0.269169	−0.0282165
$92$	0	0
$93$	−30.1902	−3.13057
$94$	0	0
$95$	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	$0.153458\pi$
$98$	0	0
$99$	−16.1180	−1.61992
$100$	0	0
$101$	14.2831	1.42122	0.710611	−	0.703586i	$-0.248419\pi$
$101$	14.2831	1.42122	0.710611	+	0.703586i	$0.248419\pi$
$102$	0	0
$103$	−4.79182	−0.472152	−0.236076	−	0.971735i	$-0.575861\pi$
$103$	−4.79182	−0.472152	−0.236076	+	0.971735i	$0.575861\pi$
$104$	0	0
$105$	1.74876	0.170661
$106$	0	0
$107$	−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	$0.424630\pi$
$110$	0	0
$111$	25.2673	2.39827
$112$	0	0
$113$	1.61773	0.152183	0.0760916	−	0.997101i	$-0.475756\pi$
$113$	1.61773	0.152183	0.0760916	+	0.997101i	$0.475756\pi$
$114$	0	0
$115$	1.51145	0.140943
$116$	0	0
$117$	−2.93220	−0.271082
$118$	0	0
$119$	−2.34818	−0.215257
$120$	0	0
$121$	−4.37107	−0.397370
$122$	0	0
$123$	5.58364	0.503459
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.1292	1.16503	0.582513	−	0.812821i	$-0.302070\pi$
$127$	13.1292	1.16503	0.582513	+	0.812821i	$0.302070\pi$
$128$	0	0
$129$	−1.74876	−0.153970
$130$	0	0
$131$	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
$134$	0	0
$135$	9.92099	0.853863
$136$	0	0
$137$	−14.6065	−1.24792	−0.623960	−	0.781456i	$-0.714477\pi$
$137$	−14.6065	−1.24792	−0.623960	+	0.781456i	$0.714477\pi$
$138$	0	0
$139$	2.07219	0.175761	0.0878804	−	0.996131i	$-0.471991\pi$
$139$	2.07219	0.175761	0.0878804	+	0.996131i	$0.471991\pi$
$140$	0	0
$141$	21.5907	1.81827
$142$	0	0
$143$	1.20594	0.100846
$144$	0	0
$145$	4.08612	0.339334
$146$	0	0
$147$	20.2964	1.67402
$148$	0	0
$149$	8.91203	0.730102	0.365051	−	0.930988i	$-0.381052\pi$
$149$	8.91203	0.730102	0.365051	+	0.930988i	$0.381052\pi$
$150$	0	0
$151$	−11.4572	−0.932372	−0.466186	−	0.884687i	$-0.654372\pi$
$151$	−11.4572	−0.932372	−0.466186	+	0.884687i	$0.654372\pi$
$152$	0	0
$153$	−25.5799	−2.06801
$154$	0	0
$155$	−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	$0.584919\pi$
$158$	0	0
$159$	−13.0951	−1.03851
$160$	0	0
$161$	−0.868585	−0.0684541
$162$	0	0
$163$	−20.2444	−1.58567	−0.792833	−	0.609439i	$-0.791395\pi$
$163$	−20.2444	−1.58567	−0.792833	+	0.609439i	$0.791395\pi$
$164$	0	0
$165$	−7.83488	−0.609944
$166$	0	0
$167$	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
$170$	0	0
$171$	−6.26020	−0.478730
$172$	0	0
$173$	−3.53161	−0.268504	−0.134252	−	0.990947i	$-0.542863\pi$
$173$	−3.53161	−0.268504	−0.134252	+	0.990947i	$0.542863\pi$
$174$	0	0
$175$	0.574672	0.0434411
$176$	0	0
$177$	−8.17223	−0.614263
$178$	0	0
$179$	−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	0
$183$	−37.7630	−2.79152
$184$	0	0
$185$	8.30326	0.610468
$186$	0	0
$187$	10.5204	0.769329
$188$	0	0
$189$	−5.70131	−0.414710
$190$	0	0
$191$	−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	$0.260379\pi$
$194$	0	0
$195$	−1.42533	−0.102070
$196$	0	0
$197$	2.17223	0.154765	0.0773826	−	0.997001i	$-0.475344\pi$
$197$	2.17223	0.154765	0.0773826	+	0.997001i	$0.475344\pi$
$198$	0	0
$199$	1.87355	0.132812	0.0664061	−	0.997793i	$-0.478847\pi$
$199$	1.87355	0.132812	0.0664061	+	0.997793i	$0.478847\pi$
$200$	0	0
$201$	−8.23361	−0.580754
$202$	0	0
$203$	−2.34818	−0.164810
$204$	0	0
$205$	1.83488	0.128153
$206$	0	0
$207$	−9.46196	−0.657651
$208$	0	0
$209$	2.57467	0.178094
$210$	0	0
$211$	17.1090	1.17783	0.588916	−	0.808194i	$-0.299555\pi$
$211$	17.1090	1.17783	0.588916	+	0.808194i	$0.299555\pi$
$212$	0	0
$213$	−22.5204	−1.54307
$214$	0	0
$215$	−0.574672	−0.0391923
$216$	0	0
$217$	5.70131	0.387030
$218$	0	0
$219$	−36.7788	−2.48528
$220$	0	0
$221$	1.91388	0.128742
$222$	0	0
$223$	−5.12918	−0.343475	−0.171737	−	0.985143i	$-0.554938\pi$
$223$	−5.12918	−0.343475	−0.171737	+	0.985143i	$0.554938\pi$
$224$	0	0
$225$	6.26020	0.417347
$226$	0	0
$227$	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	$0.589625\pi$
$230$	0	0
$231$	4.50248	0.296242
$232$	0	0
$233$	−14.1722	−0.928454	−0.464227	−	0.885716i	$-0.653668\pi$
$233$	−14.1722	−0.928454	−0.464227	+	0.885716i	$0.653668\pi$
$234$	0	0
$235$	7.09508	0.462832
$236$	0	0
$237$	−20.3445	−1.32152
$238$	0	0
$239$	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	$0.145038\pi$
$242$	0	0
$243$	−4.95694	−0.317988
$244$	0	0
$245$	6.66975	0.426115
$246$	0	0
$247$	0.468387	0.0298027
$248$	0	0
$249$	−20.2692	−1.28451
$250$	0	0
$251$	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
$254$	0	0
$255$	−12.4343	−0.778666
$256$	0	0
$257$	9.01831	0.562547	0.281273	−	0.959628i	$-0.409243\pi$
$257$	9.01831	0.562547	0.281273	+	0.959628i	$0.409243\pi$
$258$	0	0
$259$	−4.77165	−0.296496
$260$	0	0
$261$	−25.5799	−1.58336
$262$	0	0
$263$	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
$266$	0	0
$267$	−44.4485	−2.72020
$268$	0	0
$269$	−30.6136	−1.86655	−0.933273	−	0.359167i	$-0.883061\pi$
$269$	−30.6136	−1.86655	−0.933273	+	0.359167i	$0.883061\pi$
$270$	0	0
$271$	−24.1180	−1.46506	−0.732531	−	0.680733i	$-0.761661\pi$
$271$	−24.1180	−1.46506	−0.732531	+	0.680733i	$0.761661\pi$
$272$	0	0
$273$	0.819096	0.0495739
$274$	0	0
$275$	−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	$0.456249\pi$
$278$	0	0
$279$	62.1074	3.71828
$280$	0	0
$281$	−6.11563	−0.364828	−0.182414	−	0.983222i	$-0.558391\pi$
$281$	−6.11563	−0.364828	−0.182414	+	0.983222i	$0.558391\pi$
$282$	0	0
$283$	19.0547	1.13269	0.566344	−	0.824169i	$-0.308358\pi$
$283$	19.0547	1.13269	0.566344	+	0.824169i	$0.308358\pi$
$284$	0	0
$285$	−3.04306	−0.180255
$286$	0	0
$287$	−1.05445	−0.0622423
$288$	0	0
$289$	−0.303649	−0.0178617
$290$	0	0
$291$	−53.1093	−3.11332
$292$	0	0
$293$	−6.43887	−0.376163	−0.188081	−	0.982153i	$-0.560227\pi$
$293$	−6.43887	−0.376163	−0.188081	+	0.982153i	$0.560227\pi$
$294$	0	0
$295$	−2.68553	−0.156358
$296$	0	0
$297$	25.5433	1.48217
$298$	0	0
$299$	0.707941	0.0409413
$300$	0	0
$301$	0.330247	0.0190351
$302$	0	0
$303$	−43.4643	−2.49696
$304$	0	0
$305$	−12.4095	−0.710569
$306$	0	0
$307$	6.77389	0.386606	0.193303	−	0.981139i	$-0.438080\pi$
$307$	6.77389	0.386606	0.193303	+	0.981139i	$0.438080\pi$
$308$	0	0
$309$	14.5818	0.829529
$310$	0	0
$311$	−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	$0.315595\pi$
$314$	0	0
$315$	−3.59756	−0.202700
$316$	0	0
$317$	3.37360	0.189480	0.0947401	−	0.995502i	$-0.469798\pi$
$317$	3.37360	0.189480	0.0947401	+	0.995502i	$0.469798\pi$
$318$	0	0
$319$	10.5204	0.589030
$320$	0	0
$321$	28.0756	1.56703
$322$	0	0
$323$	4.08612	0.227358
$324$	0	0
$325$	−0.468387	−0.0259814
$326$	0	0
$327$	−14.9052	−0.824260
$328$	0	0
$329$	−4.07734	−0.224791
$330$	0	0
$331$	32.7788	1.80168	0.900842	−	0.434148i	$-0.142950\pi$
$331$	32.7788	1.80168	0.900842	+	0.434148i	$0.142950\pi$
$332$	0	0
$333$	−51.9801	−2.84849
$334$	0	0
$335$	−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	$0.423410\pi$
$338$	0	0
$339$	−4.92285	−0.267372
$340$	0	0
$341$	−25.5433	−1.38325
$342$	0	0
$343$	−7.85562	−0.424164
$344$	0	0
$345$	−4.59942	−0.247624
$346$	0	0
$347$	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	$0.469768\pi$
$350$	0	0
$351$	4.64686	0.248031
$352$	0	0
$353$	−3.41140	−0.181571	−0.0907853	−	0.995870i	$-0.528938\pi$
$353$	−3.41140	−0.181571	−0.0907853	+	0.995870i	$0.528938\pi$
$354$	0	0
$355$	−7.40058	−0.392782
$356$	0	0
$357$	7.14564	0.378187
$358$	0	0
$359$	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
$362$	0	0
$363$	13.3014	0.698143
$364$	0	0
$365$	−12.0861	−0.632616
$366$	0	0
$367$	−37.6155	−1.96351	−0.981756	−	0.190144i	$-0.939105\pi$
$367$	−37.6155	−1.96351	−0.981756	+	0.190144i	$0.939105\pi$
$368$	0	0
$369$	−11.4867	−0.597974
$370$	0	0
$371$	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	$0.387203\pi$
$374$	0	0
$375$	3.04306	0.157143
$376$	0	0
$377$	1.91388	0.0985700
$378$	0	0
$379$	9.37107	0.481359	0.240680	−	0.970605i	$-0.422630\pi$
$379$	9.37107	0.481359	0.240680	+	0.970605i	$0.422630\pi$
$380$	0	0
$381$	−39.9528	−2.04685
$382$	0	0
$383$	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
$386$	0	0
$387$	3.59756	0.182874
$388$	0	0
$389$	1.07238	0.0543718	0.0271859	−	0.999630i	$-0.491345\pi$
$389$	1.07238	0.0543718	0.0271859	+	0.999630i	$0.491345\pi$
$390$	0	0
$391$	6.17594	0.312331
$392$	0	0
$393$	−24.8686	−1.25445
$394$	0	0
$395$	−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	$0.701098\pi$
$398$	0	0
$399$	1.74876	0.0875475
$400$	0	0
$401$	18.6469	0.931180	0.465590	−	0.885001i	$-0.345842\pi$
$401$	18.6469	0.931180	0.465590	+	0.885001i	$0.345842\pi$
$402$	0	0
$403$	−4.64686	−0.231477
$404$	0	0
$405$	−11.4095	−0.566945
$406$	0	0
$407$	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	$0.554410\pi$
$410$	0	0
$411$	44.4485	2.19248
$412$	0	0
$413$	1.54330	0.0759408
$414$	0	0
$415$	−6.66079	−0.326965
$416$	0	0
$417$	−6.30580	−0.308796
$418$	0	0
$419$	−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	$0.545414\pi$
$422$	0	0
$423$	−44.4167	−2.15961
$424$	0	0
$425$	−4.08612	−0.198206
$426$	0	0
$427$	7.13142	0.345113
$428$	0	0
$429$	−3.66975	−0.177177
$430$	0	0
$431$	−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	$0.597289\pi$
$434$	0	0
$435$	−12.4343	−0.596179
$436$	0	0
$437$	1.51145	0.0723022
$438$	0	0
$439$	−15.6769	−0.748216	−0.374108	−	0.927385i	$-0.622051\pi$
$439$	−15.6769	−0.748216	−0.374108	+	0.927385i	$0.622051\pi$
$440$	0	0
$441$	−41.7540	−1.98829
$442$	0	0
$443$	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
$446$	0	0
$447$	−27.1198	−1.28272
$448$	0	0
$449$	29.9668	1.41422	0.707110	−	0.707104i	$-0.249999\pi$
$449$	29.9668	1.41422	0.707110	+	0.707104i	$0.249999\pi$
$450$	0	0
$451$	4.72420	0.222454
$452$	0	0
$453$	34.8649	1.63809
$454$	0	0
$455$	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	$0.364384\pi$
$458$	0	0
$459$	40.5383	1.89217
$460$	0	0
$461$	22.3445	1.04069	0.520343	−	0.853957i	$-0.325804\pi$
$461$	22.3445	1.04069	0.520343	+	0.853957i	$0.325804\pi$
$462$	0	0
$463$	6.83302	0.317557	0.158779	−	0.987314i	$-0.449244\pi$
$463$	6.83302	0.317557	0.158779	+	0.987314i	$0.449244\pi$
$464$	0	0
$465$	30.1902	1.40004
$466$	0	0
$467$	−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
$470$	0	0
$471$	20.1040	0.926345
$472$	0	0
$473$	−1.47959	−0.0680317
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	26.9393	1.23347
$478$	0	0
$479$	−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
$482$	0	0
$483$	2.64315	0.120268
$484$	0	0
$485$	−17.4526	−0.792482
$486$	0	0
$487$	38.7694	1.75681	0.878405	−	0.477917i	$-0.158608\pi$
$487$	38.7694	1.75681	0.878405	+	0.477917i	$0.158608\pi$
$488$	0	0
$489$	61.6050	2.78587
$490$	0	0
$491$	−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
$494$	0	0
$495$	16.1180	0.724449
$496$	0	0
$497$	4.25291	0.190769
$498$	0	0
$499$	27.8372	1.24616	0.623082	−	0.782156i	$-0.285880\pi$
$499$	27.8372	1.24616	0.623082	+	0.782156i	$0.285880\pi$
$500$	0	0
$501$	−17.9349	−0.801273
$502$	0	0
$503$	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
$506$	0	0
$507$	38.8922	1.72726
$508$	0	0
$509$	0.416364	0.0184550	0.00922751	−	0.999957i	$-0.497063\pi$
$509$	0.416364	0.0184550	0.00922751	+	0.999957i	$0.497063\pi$
$510$	0	0
$511$	6.94555	0.307253
$512$	0	0
$513$	9.92099	0.438023
$514$	0	0
$515$	4.79182	0.211153
$516$	0	0
$517$	18.2675	0.803404
$518$	0	0
$519$	10.7469	0.471737
$520$	0	0
$521$	24.0458	1.05346	0.526732	−	0.850031i	$-0.323417\pi$
$521$	24.0458	1.05346	0.526732	+	0.850031i	$0.323417\pi$
$522$	0	0
$523$	5.19736	0.227265	0.113632	−	0.993523i	$-0.463751\pi$
$523$	5.19736	0.227265	0.113632	+	0.993523i	$0.463751\pi$
$524$	0	0
$525$	−1.74876	−0.0763221
$526$	0	0
$527$	−40.5383	−1.76588
$528$	0	0
$529$	−20.7155	−0.900675
$530$	0	0
$531$	16.8120	0.729578
$532$	0	0
$533$	0.859432	0.0372261
$534$	0	0
$535$	9.22611	0.398880
$536$	0	0
$537$	21.8735	0.943913
$538$	0	0
$539$	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	$0.513269\pi$
$542$	0	0
$543$	−47.2992	−2.02980
$544$	0	0
$545$	−4.89810	−0.209812
$546$	0	0
$547$	12.9075	0.551883	0.275941	−	0.961174i	$-0.411010\pi$
$547$	12.9075	0.551883	0.275941	+	0.961174i	$0.411010\pi$
$548$	0	0
$549$	77.6863	3.31557
$550$	0	0
$551$	4.08612	0.174074
$552$	0	0
$553$	3.84199	0.163378
$554$	0	0
$555$	−25.2673	−1.07254
$556$	0	0
$557$	−34.1040	−1.44503	−0.722517	−	0.691353i	$-0.757015\pi$
$557$	−34.1040	−1.44503	−0.722517	+	0.691353i	$0.757015\pi$
$558$	0	0
$559$	−0.269169	−0.0113846
$560$	0	0
$561$	−32.0142	−1.35164
$562$	0	0
$563$	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
$566$	0	0
$567$	6.55674	0.275357
$568$	0	0
$569$	−39.2110	−1.64381	−0.821906	−	0.569624i	$-0.807089\pi$
$569$	−39.2110	−1.64381	−0.821906	+	0.569624i	$0.807089\pi$
$570$	0	0
$571$	−21.9915	−0.920316	−0.460158	−	0.887837i	$-0.652207\pi$
$571$	−21.9915	−0.920316	−0.460158	+	0.887837i	$0.652207\pi$
$572$	0	0
$573$	40.5383	1.69351
$574$	0	0
$575$	−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	0
$579$	−57.8057	−2.40232
$580$	0	0
$581$	3.82777	0.158803
$582$	0	0
$583$	−11.0795	−0.458866
$584$	0	0
$585$	2.93220	0.121231
$586$	0	0
$587$	−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
$590$	0	0
$591$	−6.61023	−0.271909
$592$	0	0
$593$	−21.2446	−0.872412	−0.436206	−	0.899847i	$-0.643678\pi$
$593$	−21.2446	−0.872412	−0.436206	+	0.899847i	$0.643678\pi$
$594$	0	0
$595$	2.34818	0.0962658
$596$	0	0
$597$	−5.70131	−0.233339
$598$	0	0
$599$	−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	0
$603$	16.9382	0.689779
$604$	0	0
$605$	4.37107	0.177709
$606$	0	0
$607$	−7.81471	−0.317189	−0.158595	−	0.987344i	$-0.550696\pi$
$607$	−7.81471	−0.317189	−0.158595	+	0.987344i	$0.550696\pi$
$608$	0	0
$609$	7.14564	0.289556
$610$	0	0
$611$	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	$0.415744\pi$
$614$	0	0
$615$	−5.58364	−0.225154
$616$	0	0
$617$	18.8873	0.760373	0.380187	−	0.924910i	$-0.375860\pi$
$617$	18.8873	0.760373	0.380187	+	0.924910i	$0.375860\pi$
$618$	0	0
$619$	−29.2673	−1.17635	−0.588176	−	0.808733i	$-0.700154\pi$
$619$	−29.2673	−1.17635	−0.588176	+	0.808733i	$0.700154\pi$
$620$	0	0
$621$	14.9950	0.601730
$622$	0	0
$623$	8.39396	0.336297
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.83488	−0.312895
$628$	0	0
$629$	33.9281	1.35280
$630$	0	0
$631$	5.25309	0.209122	0.104561	−	0.994518i	$-0.466656\pi$
$631$	5.25309	0.209122	0.104561	+	0.994518i	$0.466656\pi$
$632$	0	0
$633$	−52.0637	−2.06935
$634$	0	0
$635$	−13.1292	−0.521015
$636$	0	0
$637$	3.12402	0.123778
$638$	0	0
$639$	46.3292	1.83275
$640$	0	0
$641$	−13.0021	−0.513554	−0.256777	−	0.966471i	$-0.582661\pi$
$641$	−13.0021	−0.513554	−0.256777	+	0.966471i	$0.582661\pi$
$642$	0	0
$643$	17.3534	0.684353	0.342176	−	0.939636i	$-0.388836\pi$
$643$	17.3534	0.684353	0.342176	+	0.939636i	$0.388836\pi$
$644$	0	0
$645$	1.74876	0.0688573
$646$	0	0
$647$	−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
$650$	0	0
$651$	−17.3494	−0.679978
$652$	0	0
$653$	−28.3532	−1.10955	−0.554774	−	0.832001i	$-0.687195\pi$
$653$	−28.3532	−1.10955	−0.554774	+	0.832001i	$0.687195\pi$
$654$	0	0
$655$	−8.17223	−0.319316
$656$	0	0
$657$	75.6616	2.95184
$658$	0	0
$659$	−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	$0.591696\pi$
$662$	0	0
$663$	−5.82406	−0.226188
$664$	0	0
$665$	0.574672	0.0222848
$666$	0	0
$667$	6.17594	0.239133
$668$	0	0
$669$	15.6084	0.603455
$670$	0	0
$671$	−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	$0.502703\pi$
$674$	0	0
$675$	−9.92099	−0.381859
$676$	0	0
$677$	32.8057	1.26083	0.630414	−	0.776259i	$-0.282885\pi$
$677$	32.8057	1.26083	0.630414	+	0.776259i	$0.282885\pi$
$678$	0	0
$679$	10.0295	0.384898
$680$	0	0
$681$	−22.2515	−0.852681
$682$	0	0
$683$	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
$686$	0	0
$687$	25.5907	0.976348
$688$	0	0
$689$	−2.01559	−0.0767879
$690$	0	0
$691$	19.8962	0.756889	0.378444	−	0.925624i	$-0.376459\pi$
$691$	19.8962	0.756889	0.378444	+	0.925624i	$0.376459\pi$
$692$	0	0
$693$	−9.26254	−0.351855
$694$	0	0
$695$	−2.07219	−0.0786027
$696$	0	0
$697$	7.49752	0.283989
$698$	0	0
$699$	43.1269	1.63121
$700$	0	0
$701$	−14.1251	−0.533497	−0.266748	−	0.963766i	$-0.585949\pi$
$701$	−14.1251	−0.533497	−0.266748	+	0.963766i	$0.585949\pi$
$702$	0	0
$703$	8.30326	0.313163
$704$	0	0
$705$	−21.5907	−0.813155
$706$	0	0
$707$	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	$0.781396\pi$
$710$	0	0
$711$	41.8528	1.56960
$712$	0	0
$713$	−14.9950	−0.561569
$714$	0	0
$715$	−1.20594	−0.0450997
$716$	0	0
$717$	−43.1815	−1.61264
$718$	0	0
$719$	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
$722$	0	0
$723$	−84.8425	−3.15533
$724$	0	0
$725$	−4.08612	−0.151755
$726$	0	0
$727$	21.1266	0.783544	0.391772	−	0.920062i	$-0.371862\pi$
$727$	21.1266	0.783544	0.391772	+	0.920062i	$0.371862\pi$
$728$	0	0
$729$	−19.1444	−0.709051
$730$	0	0
$731$	−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	$0.670700\pi$
$734$	0	0
$735$	−20.2964	−0.748646
$736$	0	0
$737$	−6.96629	−0.256607
$738$	0	0
$739$	14.2987	0.525986	0.262993	−	0.964798i	$-0.415290\pi$
$739$	14.2987	0.525986	0.262993	+	0.964798i	$0.415290\pi$
$740$	0	0
$741$	−1.42533	−0.0523607
$742$	0	0
$743$	−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
$746$	0	0
$747$	41.6979	1.52565
$748$	0	0
$749$	−5.30198	−0.193730
$750$	0	0
$751$	−9.69216	−0.353672	−0.176836	−	0.984240i	$-0.556586\pi$
$751$	−9.69216	−0.353672	−0.176836	+	0.984240i	$0.556586\pi$
$752$	0	0
$753$	−79.6982	−2.90436
$754$	0	0
$755$	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	$0.177525\pi$
$758$	0	0
$759$	−11.8420	−0.429837
$760$	0	0
$761$	−38.7361	−1.40418	−0.702091	−	0.712087i	$-0.747750\pi$
$761$	−38.7361	−1.40418	−0.702091	+	0.712087i	$0.747750\pi$
$762$	0	0
$763$	2.81480	0.101903
$764$	0	0
$765$	25.5799	0.924844
$766$	0	0
$767$	−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	$0.530964\pi$
$770$	0	0
$771$	−27.4433	−0.988345
$772$	0	0
$773$	7.20137	0.259015	0.129508	−	0.991578i	$-0.458660\pi$
$773$	7.20137	0.259015	0.129508	+	0.991578i	$0.458660\pi$
$774$	0	0
$775$	9.92099	0.356373
$776$	0	0
$777$	14.5204	0.520917
$778$	0	0
$779$	1.83488	0.0657413
$780$	0	0
$781$	−19.0541	−0.681808
$782$	0	0
$783$	40.5383	1.44872
$784$	0	0
$785$	6.60653	0.235797
$786$	0	0
$787$	33.5231	1.19497	0.597485	−	0.801880i	$-0.296167\pi$
$787$	33.5231	1.19497	0.597485	+	0.801880i	$0.296167\pi$
$788$	0	0
$789$	−27.4148	−0.975993
$790$	0	0
$791$	0.929664	0.0330550
$792$	0	0
$793$	−5.81247	−0.206407
$794$	0	0
$795$	13.0951	0.464435
$796$	0	0
$797$	−10.4979	−0.371855	−0.185927	−	0.982563i	$-0.559529\pi$
$797$	−10.4979	−0.371855	−0.185927	+	0.982563i	$0.559529\pi$
$798$	0	0
$799$	28.9913	1.02564
$800$	0	0
$801$	91.4398	3.23087
$802$	0	0
$803$	−31.1178	−1.09812
$804$	0	0
$805$	0.868585	0.0306136
$806$	0	0
$807$	93.1591	3.27936
$808$	0	0
$809$	−13.0724	−0.459600	−0.229800	−	0.973238i	$-0.573807\pi$
$809$	−13.0724	−0.459600	−0.229800	+	0.973238i	$0.573807\pi$
$810$	0	0
$811$	10.0790	0.353922	0.176961	−	0.984218i	$-0.443373\pi$
$811$	10.0790	0.353922	0.176961	+	0.984218i	$0.443373\pi$
$812$	0	0
$813$	73.3924	2.57398
$814$	0	0
$815$	20.2444	0.709131
$816$	0	0
$817$	−0.574672	−0.0201052
$818$	0	0
$819$	−1.68505	−0.0588804
$820$	0	0
$821$	40.2987	1.40643	0.703217	−	0.710975i	$-0.251746\pi$
$821$	40.2987	1.40643	0.703217	+	0.710975i	$0.251746\pi$
$822$	0	0
$823$	5.07715	0.176978	0.0884892	−	0.996077i	$-0.471796\pi$
$823$	5.07715	0.176978	0.0884892	+	0.996077i	$0.471796\pi$
$824$	0	0
$825$	7.83488	0.272775
$826$	0	0
$827$	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	$0.361932\pi$
$830$	0	0
$831$	−13.8786	−0.481444
$832$	0	0
$833$	27.2534	0.944274
$834$	0	0
$835$	−5.89372	−0.203960
$836$	0	0
$837$	−98.4261	−3.40210
$838$	0	0
$839$	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
$842$	0	0
$843$	18.6102	0.640971
$844$	0	0
$845$	12.7806	0.439666
$846$	0	0
$847$	−2.51193	−0.0863109
$848$	0	0
$849$	−57.9847	−1.99003
$850$	0	0
$851$	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	$0.571568\pi$
$854$	0	0
$855$	6.26020	0.214094
$856$	0	0
$857$	−2.82367	−0.0964548	−0.0482274	−	0.998836i	$-0.515357\pi$
$857$	−2.82367	−0.0964548	−0.0482274	+	0.998836i	$0.515357\pi$
$858$	0	0
$859$	−24.1227	−0.823057	−0.411529	−	0.911397i	$-0.635005\pi$
$859$	−24.1227	−0.823057	−0.411529	+	0.911397i	$0.635005\pi$
$860$	0	0
$861$	3.20876	0.109354
$862$	0	0
$863$	−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
$866$	0	0
$867$	0.924022	0.0313814
$868$	0	0
$869$	−17.2131	−0.583913
$870$	0	0
$871$	−1.26731	−0.0429413
$872$	0	0
$873$	109.257	3.69779
$874$	0	0
$875$	−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	$0.815036\pi$
$878$	0	0
$879$	19.5939	0.660884
$880$	0	0
$881$	40.3152	1.35826	0.679128	−	0.734020i	$-0.262358\pi$
$881$	40.3152	1.35826	0.679128	+	0.734020i	$0.262358\pi$
$882$	0	0
$883$	29.3347	0.987192	0.493596	−	0.869691i	$-0.335682\pi$
$883$	29.3347	0.987192	0.493596	+	0.869691i	$0.335682\pi$
$884$	0	0
$885$	8.17223	0.274707
$886$	0	0
$887$	−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
$890$	0	0
$891$	−29.3758	−0.984128
$892$	0	0
$893$	7.09508	0.237428
$894$	0	0
$895$	7.18801	0.240269
$896$	0	0
$897$	−2.15431	−0.0719302
$898$	0	0
$899$	−40.5383	−1.35203
$900$	0	0
$901$	−17.5836	−0.585796
$902$	0	0
$903$	−1.00496	−0.0334431
$904$	0	0
$905$	−15.5433	−0.516677
$906$	0	0
$907$	−14.0731	−0.467288	−0.233644	−	0.972322i	$-0.575065\pi$
$907$	−14.0731	−0.467288	−0.233644	+	0.972322i	$0.575065\pi$
$908$	0	0
$909$	89.4151	2.96571
$910$	0	0
$911$	−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
$914$	0	0
$915$	37.7630	1.24841
$916$	0	0
$917$	4.69635	0.155087
$918$	0	0
$919$	−29.6518	−0.978123	−0.489062	−	0.872249i	$-0.662661\pi$
$919$	−29.6518	−0.978123	−0.489062	+	0.872249i	$0.662661\pi$
$920$	0	0
$921$	−20.6133	−0.679233
$922$	0	0
$923$	−3.46634	−0.114096
$924$	0	0
$925$	−8.30326	−0.273010
$926$	0	0
$927$	−29.9978	−0.985256
$928$	0	0
$929$	58.3803	1.91540	0.957698	−	0.287775i	$-0.0929154\pi$
$929$	58.3803	1.91540	0.957698	+	0.287775i	$0.0929154\pi$
$930$	0	0
$931$	6.66975	0.218592
$932$	0	0
$933$	62.7492	2.05432
$934$	0	0
$935$	−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	$0.516429\pi$
$938$	0	0
$939$	−58.9473	−1.92367
$940$	0	0
$941$	−9.66975	−0.315225	−0.157612	−	0.987501i	$-0.550380\pi$
$941$	−9.66975	−0.315225	−0.157612	+	0.987501i	$0.550380\pi$
$942$	0	0
$943$	2.77331	0.0903116
$944$	0	0
$945$	5.70131	0.185464
$946$	0	0
$947$	−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
$950$	0	0
$951$	−10.2661	−0.332900
$952$	0	0
$953$	−13.1224	−0.425075	−0.212537	−	0.977153i	$-0.568173\pi$
$953$	−13.1224	−0.425075	−0.212537	+	0.977153i	$0.568173\pi$
$954$	0	0
$955$	13.3216	0.431076
$956$	0	0
$957$	−32.0142	−1.03487
$958$	0	0
$959$	−8.39396	−0.271055
$960$	0	0
$961$	67.4261	2.17504
$962$	0	0
$963$	−57.7573	−1.86120
$964$	0	0
$965$	−18.9959	−0.611500
$966$	0	0
$967$	44.4400	1.42910	0.714548	−	0.699587i	$-0.246633\pi$
$967$	44.4400	1.42910	0.714548	+	0.699587i	$0.246633\pi$
$968$	0	0
$969$	−12.4343	−0.399447
$970$	0	0
$971$	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
$974$	0	0
$975$	1.42533	0.0456470
$976$	0	0
$977$	−23.9164	−0.765154	−0.382577	−	0.923924i	$-0.624963\pi$
$977$	−23.9164	−0.765154	−0.382577	+	0.923924i	$0.624963\pi$
$978$	0	0
$979$	−37.6070	−1.20193
$980$	0	0
$981$	30.6631	0.978998
$982$	0	0
$983$	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
$986$	0	0
$987$	12.4076	0.394938
$988$	0	0
$989$	−0.868585	−0.0276194
$990$	0	0
$991$	−27.5521	−0.875220	−0.437610	−	0.899165i	$-0.644175\pi$
$991$	−27.5521	−0.875220	−0.437610	+	0.899165i	$0.644175\pi$
$992$	0	0
$993$	−99.7477	−3.16540
$994$	0	0
$995$	−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	$0.413996\pi$
$998$	0	0
$999$	82.3766	2.60628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.t.1.1		4
4.3	odd	2		95.2.a.b.1.1	✓	4
5.4	even	2		7600.2.a.cf.1.4		4
8.3	odd	2		6080.2.a.cc.1.1		4
8.5	even	2		6080.2.a.ch.1.4		4
12.11	even	2		855.2.a.m.1.4		4
20.3	even	4		475.2.b.e.324.8		8
20.7	even	4		475.2.b.e.324.1		8
20.19	odd	2		475.2.a.i.1.4		4
28.27	even	2		4655.2.a.y.1.1		4
60.59	even	2		4275.2.a.bo.1.1		4
76.75	even	2		1805.2.a.p.1.4		4
380.379	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	4.3	odd	2
475.2.a.i.1.4		4	20.19	odd	2
475.2.b.e.324.1		8	20.7	even	4
475.2.b.e.324.8		8	20.3	even	4
855.2.a.m.1.4		4	12.11	even	2
1520.2.a.t.1.1		4	1.1	even	1	trivial
1805.2.a.p.1.4		4	76.75	even	2
4275.2.a.bo.1.1		4	60.59	even	2
4655.2.a.y.1.1		4	28.27	even	2
6080.2.a.cc.1.1		4	8.3	odd	2
6080.2.a.ch.1.4		4	8.5	even	2
7600.2.a.cf.1.4		4	5.4	even	2
9025.2.a.bf.1.1		4	380.379	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q-3.04306 q^{3} -1.00000 q^{5} +0.574672 q^{7} +6.26020 q^{9} +O(q^{10})\)	\(q-3.04306 q^{3} -1.00000 q^{5} +0.574672 q^{7} +6.26020 q^{9} -2.57467 q^{11} -0.468387 q^{13} +3.04306 q^{15} -4.08612 q^{17} -1.00000 q^{19} -1.74876 q^{21} -1.51145 q^{23} +1.00000 q^{25} -9.92099 q^{27} -4.08612 q^{29} +9.92099 q^{31} +7.83488 q^{33} -0.574672 q^{35} -8.30326 q^{37} +1.42533 q^{39} -1.83488 q^{41} +0.574672 q^{43} -6.26020 q^{45} -7.09508 q^{47} -6.66975 q^{49} +12.4343 q^{51} +4.30326 q^{53} +2.57467 q^{55} +3.04306 q^{57} +2.68553 q^{59} +12.4095 q^{61} +3.59756 q^{63} +0.468387 q^{65} +2.70570 q^{67} +4.59942 q^{69} +7.40058 q^{71} +12.0861 q^{73} -3.04306 q^{75} -1.47959 q^{77} +6.68553 q^{79} +11.4095 q^{81} +6.66079 q^{83} +4.08612 q^{85} +12.4343 q^{87} +14.6065 q^{89} -0.269169 q^{91} -30.1902 q^{93} +1.00000 q^{95} +17.4526 q^{97} -16.1180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9	\( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 - 4 * q^7 + 8 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 + 4 * q^27 + 4 * q^29 - 4 * q^31 + 8 * q^33 + 4 * q^35 - 6 * q^37 + 12 * q^39 + 16 * q^41 - 4 * q^43 - 8 * q^45 + 12 * q^47 + 20 * q^49 + 36 * q^51 - 10 * q^53 + 4 * q^55 + 2 * q^57 + 20 * q^61 - 20 * q^63 - 2 * q^65 + 18 * q^67 + 28 * q^69 + 20 * q^71 + 28 * q^73 - 2 * q^75 - 40 * q^77 + 16 * q^79 + 16 * q^81 - 4 * q^85 + 36 * q^87 + 4 * q^89 + 36 * q^91 - 40 * q^93 + 4 * q^95 + 30 * q^97 + 4 * q^99

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