LMFDB - Embedded newform 4560.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.32803$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.20191 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.20191 q^{7} +1.00000 q^{9} -5.20191 q^{11} +5.45415 q^{13} -1.00000 q^{15} +4.00000 q^{17} -1.00000 q^{19} +3.20191 q^{21} +8.65606 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.20191 q^{29} +2.00000 q^{31} +5.20191 q^{33} -3.20191 q^{35} -11.8580 q^{37} -5.45415 q^{39} -9.85797 q^{41} +3.45415 q^{43} +1.00000 q^{45} -8.65606 q^{47} +3.25224 q^{49} -4.00000 q^{51} -10.6561 q^{53} -5.20191 q^{55} +1.00000 q^{57} +13.0599 q^{59} +3.74776 q^{61} -3.20191 q^{63} +5.45415 q^{65} -4.00000 q^{67} -8.65606 q^{69} -6.90830 q^{71} +6.00000 q^{73} -1.00000 q^{75} +16.6561 q^{77} +6.15158 q^{79} +1.00000 q^{81} +8.40382 q^{83} +4.00000 q^{85} +3.20191 q^{87} +13.8580 q^{89} -17.4637 q^{91} -2.00000 q^{93} -1.00000 q^{95} +0.142026 q^{97} -5.20191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 6 q^{11} + 4 q^{13} - 3 q^{15} + 12 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{31} + 6 q^{33} - 4 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} + 3 q^{45} - 4 q^{47} + 7 q^{49} - 12 q^{51} - 10 q^{53} - 6 q^{55} + 3 q^{57} - 2 q^{59} + 14 q^{61} + 4 q^{65} - 12 q^{67} - 4 q^{69} + 4 q^{71} + 18 q^{73} - 3 q^{75} + 28 q^{77} + 2 q^{79} + 3 q^{81} + 6 q^{83} + 12 q^{85} + 10 q^{89} + 8 q^{91} - 6 q^{93} - 3 q^{95} + 32 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 6 * q^11 + 4 * q^13 - 3 * q^15 + 12 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^31 + 6 * q^33 - 4 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 + 3 * q^45 - 4 * q^47 + 7 * q^49 - 12 * q^51 - 10 * q^53 - 6 * q^55 + 3 * q^57 - 2 * q^59 + 14 * q^61 + 4 * q^65 - 12 * q^67 - 4 * q^69 + 4 * q^71 + 18 * q^73 - 3 * q^75 + 28 * q^77 + 2 * q^79 + 3 * q^81 + 6 * q^83 + 12 * q^85 + 10 * q^89 + 8 * q^91 - 6 * q^93 - 3 * q^95 + 32 * q^97 - 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.20191	−1.21021	−0.605104	−	0.796146i	$-0.706869\pi$
$7$	−3.20191	−1.21021	−0.605104	+	0.796146i	$0.706869\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.20191	−1.56844	−0.784218	−	0.620486i	$-0.786935\pi$
$11$	−5.20191	−1.56844	−0.784218	+	0.620486i	$0.786935\pi$
$12$	0	0
$13$	5.45415	1.51271	0.756355	−	0.654162i	$-0.226978\pi$
$13$	5.45415	1.51271	0.756355	+	0.654162i	$0.226978\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.20191	0.698714
$22$	0	0
$23$	8.65606	1.80491	0.902457	−	0.430780i	$-0.141762\pi$
$23$	8.65606	1.80491	0.902457	+	0.430780i	$0.141762\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.20191	−0.594580	−0.297290	−	0.954787i	$-0.596083\pi$
$29$	−3.20191	−0.594580	−0.297290	+	0.954787i	$0.596083\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	5.20191	0.905537
$34$	0	0
$35$	−3.20191	−0.541222
$36$	0	0
$37$	−11.8580	−1.94944	−0.974719	−	0.223432i	$-0.928274\pi$
$37$	−11.8580	−1.94944	−0.974719	+	0.223432i	$0.928274\pi$
$38$	0	0
$39$	−5.45415	−0.873363
$40$	0	0
$41$	−9.85797	−1.53956	−0.769778	−	0.638311i	$-0.779633\pi$
$41$	−9.85797	−1.53956	−0.769778	+	0.638311i	$0.779633\pi$
$42$	0	0
$43$	3.45415	0.526753	0.263377	−	0.964693i	$-0.415164\pi$
$43$	3.45415	0.526753	0.263377	+	0.964693i	$0.415164\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−8.65606	−1.26262	−0.631308	−	0.775532i	$-0.717482\pi$
$47$	−8.65606	−1.26262	−0.631308	+	0.775532i	$0.717482\pi$
$48$	0	0
$49$	3.25224	0.464606
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−10.6561	−1.46372	−0.731861	−	0.681454i	$-0.761348\pi$
$53$	−10.6561	−1.46372	−0.731861	+	0.681454i	$0.761348\pi$
$54$	0	0
$55$	−5.20191	−0.701426
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	13.0599	1.70025	0.850126	−	0.526579i	$-0.176526\pi$
$59$	13.0599	1.70025	0.850126	+	0.526579i	$0.176526\pi$
$60$	0	0
$61$	3.74776	0.479852	0.239926	−	0.970791i	$-0.422877\pi$
$61$	3.74776	0.479852	0.239926	+	0.970791i	$0.422877\pi$
$62$	0	0
$63$	−3.20191	−0.403403
$64$	0	0
$65$	5.45415	0.676504
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−8.65606	−1.04207
$70$	0	0
$71$	−6.90830	−0.819865	−0.409932	−	0.912116i	$-0.634448\pi$
$71$	−6.90830	−0.819865	−0.409932	+	0.912116i	$0.634448\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	16.6561	1.89813
$78$	0	0
$79$	6.15158	0.692107	0.346054	−	0.938215i	$-0.387522\pi$
$79$	6.15158	0.692107	0.346054	+	0.938215i	$0.387522\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.40382	0.922439	0.461220	−	0.887286i	$-0.347412\pi$
$83$	8.40382	0.922439	0.461220	+	0.887286i	$0.347412\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	3.20191	0.343281
$88$	0	0
$89$	13.8580	1.46894	0.734471	−	0.678640i	$-0.237430\pi$
$89$	13.8580	1.46894	0.734471	+	0.678640i	$0.237430\pi$
$90$	0	0
$91$	−17.4637	−1.83069
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	0.142026	0.0144205	0.00721026	−	0.999974i	$-0.497705\pi$
$97$	0.142026	0.0144205	0.00721026	+	0.999974i	$0.497705\pi$
$98$	0	0
$99$	−5.20191	−0.522812
$100$	0	0
$101$	4.40382	0.438197	0.219098	−	0.975703i	$-0.429688\pi$
$101$	4.40382	0.438197	0.219098	+	0.975703i	$0.429688\pi$
$102$	0	0
$103$	12.8076	1.26197	0.630987	−	0.775793i	$-0.282650\pi$
$103$	12.8076	1.26197	0.630987	+	0.775793i	$0.282650\pi$
$104$	0	0
$105$	3.20191	0.312475
$106$	0	0
$107$	12.8076	1.23816	0.619081	−	0.785327i	$-0.287505\pi$
$107$	12.8076	1.23816	0.619081	+	0.785327i	$0.287505\pi$
$108$	0	0
$109$	15.0599	1.44248	0.721238	−	0.692688i	$-0.243574\pi$
$109$	15.0599	1.44248	0.721238	+	0.692688i	$0.243574\pi$
$110$	0	0
$111$	11.8580	1.12551
$112$	0	0
$113$	5.05989	0.475994	0.237997	−	0.971266i	$-0.423509\pi$
$113$	5.05989	0.475994	0.237997	+	0.971266i	$0.423509\pi$
$114$	0	0
$115$	8.65606	0.807182
$116$	0	0
$117$	5.45415	0.504236
$118$	0	0
$119$	−12.8076	−1.17408
$120$	0	0
$121$	16.0599	1.45999
$122$	0	0
$123$	9.85797	0.888864
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.49552	−0.310177	−0.155089	−	0.987901i	$-0.549566\pi$
$127$	−3.49552	−0.310177	−0.155089	+	0.987901i	$0.549566\pi$
$128$	0	0
$129$	−3.45415	−0.304121
$130$	0	0
$131$	15.6057	1.36348	0.681740	−	0.731595i	$-0.261224\pi$
$131$	15.6057	1.36348	0.681740	+	0.731595i	$0.261224\pi$
$132$	0	0
$133$	3.20191	0.277641
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−0.656063	−0.0560512	−0.0280256	−	0.999607i	$-0.508922\pi$
$137$	−0.656063	−0.0560512	−0.0280256	+	0.999607i	$0.508922\pi$
$138$	0	0
$139$	−6.40382	−0.543165	−0.271583	−	0.962415i	$-0.587547\pi$
$139$	−6.40382	−0.543165	−0.271583	+	0.962415i	$0.587547\pi$
$140$	0	0
$141$	8.65606	0.728972
$142$	0	0
$143$	−28.3720	−2.37259
$144$	0	0
$145$	−3.20191	−0.265904
$146$	0	0
$147$	−3.25224	−0.268240
$148$	0	0
$149$	−23.3121	−1.90980	−0.954902	−	0.296922i	$-0.904040\pi$
$149$	−23.3121	−1.90980	−0.954902	+	0.296922i	$0.904040\pi$
$150$	0	0
$151$	23.3121	1.89711	0.948557	−	0.316607i	$-0.102544\pi$
$151$	23.3121	1.89711	0.948557	+	0.316607i	$0.102544\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	6.25224	0.498983	0.249491	−	0.968377i	$-0.419737\pi$
$157$	6.25224	0.498983	0.249491	+	0.968377i	$0.419737\pi$
$158$	0	0
$159$	10.6561	0.845081
$160$	0	0
$161$	−27.7159	−2.18432
$162$	0	0
$163$	−9.85797	−0.772136	−0.386068	−	0.922470i	$-0.626167\pi$
$163$	−9.85797	−0.772136	−0.386068	+	0.922470i	$0.626167\pi$
$164$	0	0
$165$	5.20191	0.404968
$166$	0	0
$167$	6.25224	0.483813	0.241906	−	0.970300i	$-0.422227\pi$
$167$	6.25224	0.483813	0.241906	+	0.970300i	$0.422227\pi$
$168$	0	0
$169$	16.7478	1.28829
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	9.05989	0.688810	0.344405	−	0.938821i	$-0.388081\pi$
$173$	9.05989	0.688810	0.344405	+	0.938821i	$0.388081\pi$
$174$	0	0
$175$	−3.20191	−0.242042
$176$	0	0
$177$	−13.0599	−0.981641
$178$	0	0
$179$	4.25224	0.317827	0.158914	−	0.987292i	$-0.449201\pi$
$179$	4.25224	0.317827	0.158914	+	0.987292i	$0.449201\pi$
$180$	0	0
$181$	18.5554	1.37921	0.689606	−	0.724184i	$-0.257784\pi$
$181$	18.5554	1.37921	0.689606	+	0.724184i	$0.257784\pi$
$182$	0	0
$183$	−3.74776	−0.277042
$184$	0	0
$185$	−11.8580	−0.871816
$186$	0	0
$187$	−20.8076	−1.52161
$188$	0	0
$189$	3.20191	0.232905
$190$	0	0
$191$	22.5140	1.62906	0.814529	−	0.580122i	$-0.196995\pi$
$191$	22.5140	1.62906	0.814529	+	0.580122i	$0.196995\pi$
$192$	0	0
$193$	−17.7573	−1.27820	−0.639100	−	0.769124i	$-0.720693\pi$
$193$	−17.7573	−1.27820	−0.639100	+	0.769124i	$0.720693\pi$
$194$	0	0
$195$	−5.45415	−0.390580
$196$	0	0
$197$	15.4637	1.10174	0.550872	−	0.834590i	$-0.314295\pi$
$197$	15.4637	1.10174	0.550872	+	0.834590i	$0.314295\pi$
$198$	0	0
$199$	−2.90830	−0.206164	−0.103082	−	0.994673i	$-0.532870\pi$
$199$	−2.90830	−0.206164	−0.103082	+	0.994673i	$0.532870\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	10.2522	0.719566
$204$	0	0
$205$	−9.85797	−0.688511
$206$	0	0
$207$	8.65606	0.601638
$208$	0	0
$209$	5.20191	0.359824
$210$	0	0
$211$	3.49552	0.240642	0.120321	−	0.992735i	$-0.461608\pi$
$211$	3.49552	0.240642	0.120321	+	0.992735i	$0.461608\pi$
$212$	0	0
$213$	6.90830	0.473349
$214$	0	0
$215$	3.45415	0.235571
$216$	0	0
$217$	−6.40382	−0.434720
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	21.8166	1.46754
$222$	0	0
$223$	−0.504478	−0.0337823	−0.0168912	−	0.999857i	$-0.505377\pi$
$223$	−0.504478	−0.0337823	−0.0168912	+	0.999857i	$0.505377\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−19.5644	−1.29853	−0.649266	−	0.760561i	$-0.724924\pi$
$227$	−19.5644	−1.29853	−0.649266	+	0.760561i	$0.724924\pi$
$228$	0	0
$229$	15.7478	1.04064	0.520321	−	0.853971i	$-0.325812\pi$
$229$	15.7478	1.04064	0.520321	+	0.853971i	$0.325812\pi$
$230$	0	0
$231$	−16.6561	−1.09589
$232$	0	0
$233$	−17.1605	−1.12422	−0.562112	−	0.827061i	$-0.690011\pi$
$233$	−17.1605	−1.12422	−0.562112	+	0.827061i	$0.690011\pi$
$234$	0	0
$235$	−8.65606	−0.564659
$236$	0	0
$237$	−6.15158	−0.399588
$238$	0	0
$239$	17.7064	1.14533	0.572666	−	0.819789i	$-0.305909\pi$
$239$	17.7064	1.14533	0.572666	+	0.819789i	$0.305909\pi$
$240$	0	0
$241$	23.8166	1.53416	0.767081	−	0.641550i	$-0.221708\pi$
$241$	23.8166	1.53416	0.767081	+	0.641550i	$0.221708\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.25224	0.207778
$246$	0	0
$247$	−5.45415	−0.347039
$248$	0	0
$249$	−8.40382	−0.532571
$250$	0	0
$251$	−7.10126	−0.448227	−0.224114	−	0.974563i	$-0.571949\pi$
$251$	−7.10126	−0.448227	−0.224114	+	0.974563i	$0.571949\pi$
$252$	0	0
$253$	−45.0281	−2.83089
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	21.5644	1.34515	0.672574	−	0.740030i	$-0.265189\pi$
$257$	21.5644	1.34515	0.672574	+	0.740030i	$0.265189\pi$
$258$	0	0
$259$	37.9682	2.35923
$260$	0	0
$261$	−3.20191	−0.198193
$262$	0	0
$263$	−6.25224	−0.385530	−0.192765	−	0.981245i	$-0.561745\pi$
$263$	−6.25224	−0.385530	−0.192765	+	0.981245i	$0.561745\pi$
$264$	0	0
$265$	−10.6561	−0.654597
$266$	0	0
$267$	−13.8580	−0.848094
$268$	0	0
$269$	16.5140	1.00688	0.503439	−	0.864031i	$-0.332068\pi$
$269$	16.5140	1.00688	0.503439	+	0.864031i	$0.332068\pi$
$270$	0	0
$271$	22.4038	1.36094	0.680468	−	0.732778i	$-0.261777\pi$
$271$	22.4038	1.36094	0.680468	+	0.732778i	$0.261777\pi$
$272$	0	0
$273$	17.4637	1.05695
$274$	0	0
$275$	−5.20191	−0.313687
$276$	0	0
$277$	7.34394	0.441254	0.220627	−	0.975358i	$-0.429190\pi$
$277$	7.34394	0.441254	0.220627	+	0.975358i	$0.429190\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−1.85797	−0.110837	−0.0554187	−	0.998463i	$-0.517649\pi$
$281$	−1.85797	−0.110837	−0.0554187	+	0.998463i	$0.517649\pi$
$282$	0	0
$283$	−1.05033	−0.0624355	−0.0312177	−	0.999513i	$-0.509939\pi$
$283$	−1.05033	−0.0624355	−0.0312177	+	0.999513i	$0.509939\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	31.5644	1.86319
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−0.142026	−0.00832569
$292$	0	0
$293$	−15.7478	−0.919994	−0.459997	−	0.887920i	$-0.652150\pi$
$293$	−15.7478	−0.919994	−0.459997	+	0.887920i	$0.652150\pi$
$294$	0	0
$295$	13.0599	0.760376
$296$	0	0
$297$	5.20191	0.301846
$298$	0	0
$299$	47.2115	2.73031
$300$	0	0
$301$	−11.0599	−0.637481
$302$	0	0
$303$	−4.40382	−0.252993
$304$	0	0
$305$	3.74776	0.214596
$306$	0	0
$307$	−17.5962	−1.00427	−0.502133	−	0.864790i	$-0.667451\pi$
$307$	−17.5962	−1.00427	−0.502133	+	0.864790i	$0.667451\pi$
$308$	0	0
$309$	−12.8076	−0.728602
$310$	0	0
$311$	−6.51404	−0.369377	−0.184689	−	0.982797i	$-0.559128\pi$
$311$	−6.51404	−0.369377	−0.184689	+	0.982797i	$0.559128\pi$
$312$	0	0
$313$	29.2115	1.65113	0.825565	−	0.564307i	$-0.190857\pi$
$313$	29.2115	1.65113	0.825565	+	0.564307i	$0.190857\pi$
$314$	0	0
$315$	−3.20191	−0.180407
$316$	0	0
$317$	−28.4727	−1.59918	−0.799592	−	0.600543i	$-0.794951\pi$
$317$	−28.4727	−1.59918	−0.799592	+	0.600543i	$0.794951\pi$
$318$	0	0
$319$	16.6561	0.932560
$320$	0	0
$321$	−12.8076	−0.714853
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	5.45415	0.302542
$326$	0	0
$327$	−15.0599	−0.832814
$328$	0	0
$329$	27.7159	1.52803
$330$	0	0
$331$	−21.9682	−1.20748	−0.603740	−	0.797181i	$-0.706324\pi$
$331$	−21.9682	−1.20748	−0.603740	+	0.797181i	$0.706324\pi$
$332$	0	0
$333$	−11.8580	−0.649813
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	11.3535	0.618464	0.309232	−	0.950987i	$-0.399928\pi$
$337$	11.3535	0.618464	0.309232	+	0.950987i	$0.399928\pi$
$338$	0	0
$339$	−5.05989	−0.274815
$340$	0	0
$341$	−10.4038	−0.563399
$342$	0	0
$343$	12.0000	0.647939
$344$	0	0
$345$	−8.65606	−0.466027
$346$	0	0
$347$	−21.2115	−1.13869	−0.569346	−	0.822098i	$-0.692803\pi$
$347$	−21.2115	−1.13869	−0.569346	+	0.822098i	$0.692803\pi$
$348$	0	0
$349$	−11.3121	−0.605524	−0.302762	−	0.953066i	$-0.597909\pi$
$349$	−11.3121	−0.605524	−0.302762	+	0.953066i	$0.597909\pi$
$350$	0	0
$351$	−5.45415	−0.291121
$352$	0	0
$353$	13.4637	0.716601	0.358300	−	0.933606i	$-0.383356\pi$
$353$	13.4637	0.716601	0.358300	+	0.933606i	$0.383356\pi$
$354$	0	0
$355$	−6.90830	−0.366655
$356$	0	0
$357$	12.8076	0.677853
$358$	0	0
$359$	3.88979	0.205295	0.102648	−	0.994718i	$-0.467269\pi$
$359$	3.88979	0.205295	0.102648	+	0.994718i	$0.467269\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−16.0599	−0.842925
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−16.5140	−0.862026	−0.431013	−	0.902346i	$-0.641844\pi$
$367$	−16.5140	−0.862026	−0.431013	+	0.902346i	$0.641844\pi$
$368$	0	0
$369$	−9.85797	−0.513186
$370$	0	0
$371$	34.1198	1.77141
$372$	0	0
$373$	9.45415	0.489517	0.244759	−	0.969584i	$-0.421291\pi$
$373$	9.45415	0.489517	0.244759	+	0.969584i	$0.421291\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−17.4637	−0.899427
$378$	0	0
$379$	−5.46371	−0.280652	−0.140326	−	0.990105i	$-0.544815\pi$
$379$	−5.46371	−0.280652	−0.140326	+	0.990105i	$0.544815\pi$
$380$	0	0
$381$	3.49552	0.179081
$382$	0	0
$383$	22.1198	1.13027	0.565134	−	0.824999i	$-0.308825\pi$
$383$	22.1198	1.13027	0.565134	+	0.824999i	$0.308825\pi$
$384$	0	0
$385$	16.6561	0.848872
$386$	0	0
$387$	3.45415	0.175584
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	34.6243	1.75102
$392$	0	0
$393$	−15.6057	−0.787205
$394$	0	0
$395$	6.15158	0.309520
$396$	0	0
$397$	33.1605	1.66428	0.832140	−	0.554566i	$-0.187116\pi$
$397$	33.1605	1.66428	0.832140	+	0.554566i	$0.187116\pi$
$398$	0	0
$399$	−3.20191	−0.160296
$400$	0	0
$401$	−37.0694	−1.85116	−0.925580	−	0.378552i	$-0.876422\pi$
$401$	−37.0694	−1.85116	−0.925580	+	0.378552i	$0.876422\pi$
$402$	0	0
$403$	10.9083	0.543381
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	61.6841	3.05757
$408$	0	0
$409$	15.5962	0.771181	0.385591	−	0.922670i	$-0.373998\pi$
$409$	15.5962	0.771181	0.385591	+	0.922670i	$0.373998\pi$
$410$	0	0
$411$	0.656063	0.0323612
$412$	0	0
$413$	−41.8166	−2.05766
$414$	0	0
$415$	8.40382	0.412527
$416$	0	0
$417$	6.40382	0.313597
$418$	0	0
$419$	−5.70639	−0.278775	−0.139388	−	0.990238i	$-0.544513\pi$
$419$	−5.70639	−0.278775	−0.139388	+	0.990238i	$0.544513\pi$
$420$	0	0
$421$	3.64711	0.177749	0.0888745	−	0.996043i	$-0.471673\pi$
$421$	3.64711	0.177749	0.0888745	+	0.996043i	$0.471673\pi$
$422$	0	0
$423$	−8.65606	−0.420872
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	28.3720	1.36981
$430$	0	0
$431$	−40.5236	−1.95195	−0.975976	−	0.217876i	$-0.930087\pi$
$431$	−40.5236	−1.95195	−0.975976	+	0.217876i	$0.930087\pi$
$432$	0	0
$433$	−8.14203	−0.391281	−0.195640	−	0.980676i	$-0.562679\pi$
$433$	−8.14203	−0.391281	−0.195640	+	0.980676i	$0.562679\pi$
$434$	0	0
$435$	3.20191	0.153520
$436$	0	0
$437$	−8.65606	−0.414076
$438$	0	0
$439$	30.1516	1.43906	0.719528	−	0.694463i	$-0.244358\pi$
$439$	30.1516	1.43906	0.719528	+	0.694463i	$0.244358\pi$
$440$	0	0
$441$	3.25224	0.154869
$442$	0	0
$443$	−24.6243	−1.16993	−0.584967	−	0.811057i	$-0.698892\pi$
$443$	−24.6243	−1.16993	−0.584967	+	0.811057i	$0.698892\pi$
$444$	0	0
$445$	13.8580	0.656931
$446$	0	0
$447$	23.3121	1.10263
$448$	0	0
$449$	−0.545849	−0.0257602	−0.0128801	−	0.999917i	$-0.504100\pi$
$449$	−0.545849	−0.0257602	−0.0128801	+	0.999917i	$0.504100\pi$
$450$	0	0
$451$	51.2803	2.41470
$452$	0	0
$453$	−23.3121	−1.09530
$454$	0	0
$455$	−17.4637	−0.818711
$456$	0	0
$457$	34.2204	1.60076	0.800382	−	0.599490i	$-0.204630\pi$
$457$	34.2204	1.60076	0.800382	+	0.599490i	$0.204630\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−0.687875	−0.0320375	−0.0160188	−	0.999872i	$-0.505099\pi$
$461$	−0.687875	−0.0320375	−0.0160188	+	0.999872i	$0.505099\pi$
$462$	0	0
$463$	4.79809	0.222986	0.111493	−	0.993765i	$-0.464437\pi$
$463$	4.79809	0.222986	0.111493	+	0.993765i	$0.464437\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	−11.0090	−0.509434	−0.254717	−	0.967016i	$-0.581982\pi$
$467$	−11.0090	−0.509434	−0.254717	+	0.967016i	$0.581982\pi$
$468$	0	0
$469$	12.8076	0.591402
$470$	0	0
$471$	−6.25224	−0.288088
$472$	0	0
$473$	−17.9682	−0.826178
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−10.6561	−0.487908
$478$	0	0
$479$	26.7981	1.22444	0.612218	−	0.790689i	$-0.290278\pi$
$479$	26.7981	1.22444	0.612218	+	0.790689i	$0.290278\pi$
$480$	0	0
$481$	−64.6752	−2.94893
$482$	0	0
$483$	27.7159	1.26112
$484$	0	0
$485$	0.142026	0.00644905
$486$	0	0
$487$	−18.7070	−0.847695	−0.423847	−	0.905734i	$-0.639321\pi$
$487$	−18.7070	−0.847695	−0.423847	+	0.905734i	$0.639321\pi$
$488$	0	0
$489$	9.85797	0.445793
$490$	0	0
$491$	−12.1102	−0.546526	−0.273263	−	0.961939i	$-0.588103\pi$
$491$	−12.1102	−0.546526	−0.273263	+	0.961939i	$0.588103\pi$
$492$	0	0
$493$	−12.8076	−0.576827
$494$	0	0
$495$	−5.20191	−0.233809
$496$	0	0
$497$	22.1198	0.992207
$498$	0	0
$499$	1.59618	0.0714547	0.0357273	−	0.999362i	$-0.488625\pi$
$499$	1.59618	0.0714547	0.0357273	+	0.999362i	$0.488625\pi$
$500$	0	0
$501$	−6.25224	−0.279329
$502$	0	0
$503$	−2.55541	−0.113940	−0.0569700	−	0.998376i	$-0.518144\pi$
$503$	−2.55541	−0.113940	−0.0569700	+	0.998376i	$0.518144\pi$
$504$	0	0
$505$	4.40382	0.195968
$506$	0	0
$507$	−16.7478	−0.743794
$508$	0	0
$509$	0.210868	0.00934655	0.00467328	−	0.999989i	$-0.498512\pi$
$509$	0.210868	0.00934655	0.00467328	+	0.999989i	$0.498512\pi$
$510$	0	0
$511$	−19.2115	−0.849865
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	12.8076	0.564372
$516$	0	0
$517$	45.0281	1.98033
$518$	0	0
$519$	−9.05989	−0.397685
$520$	0	0
$521$	−8.04137	−0.352299	−0.176149	−	0.984363i	$-0.556364\pi$
$521$	−8.04137	−0.352299	−0.176149	+	0.984363i	$0.556364\pi$
$522$	0	0
$523$	−5.39487	−0.235901	−0.117951	−	0.993019i	$-0.537632\pi$
$523$	−5.39487	−0.235901	−0.117951	+	0.993019i	$0.537632\pi$
$524$	0	0
$525$	3.20191	0.139743
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	51.9274	2.25771
$530$	0	0
$531$	13.0599	0.566751
$532$	0	0
$533$	−53.7669	−2.32890
$534$	0	0
$535$	12.8076	0.553723
$536$	0	0
$537$	−4.25224	−0.183498
$538$	0	0
$539$	−16.9179	−0.728704
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−18.5554	−0.796289
$544$	0	0
$545$	15.0599	0.645095
$546$	0	0
$547$	−9.59618	−0.410303	−0.205151	−	0.978730i	$-0.565769\pi$
$547$	−9.59618	−0.410303	−0.205151	+	0.978730i	$0.565769\pi$
$548$	0	0
$549$	3.74776	0.159951
$550$	0	0
$551$	3.20191	0.136406
$552$	0	0
$553$	−19.6968	−0.837594
$554$	0	0
$555$	11.8580	0.503343
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	18.8395	0.796824
$560$	0	0
$561$	20.8076	0.878500
$562$	0	0
$563$	−38.2522	−1.61214	−0.806070	−	0.591820i	$-0.798409\pi$
$563$	−38.2522	−1.61214	−0.806070	+	0.591820i	$0.798409\pi$
$564$	0	0
$565$	5.05989	0.212871
$566$	0	0
$567$	−3.20191	−0.134468
$568$	0	0
$569$	−16.5458	−0.693638	−0.346819	−	0.937932i	$-0.612738\pi$
$569$	−16.5458	−0.693638	−0.346819	+	0.937932i	$0.612738\pi$
$570$	0	0
$571$	31.7159	1.32727	0.663636	−	0.748056i	$-0.269013\pi$
$571$	31.7159	1.32727	0.663636	+	0.748056i	$0.269013\pi$
$572$	0	0
$573$	−22.5140	−0.940537
$574$	0	0
$575$	8.65606	0.360983
$576$	0	0
$577$	20.7070	0.862043	0.431022	−	0.902342i	$-0.358153\pi$
$577$	20.7070	0.862043	0.431022	+	0.902342i	$0.358153\pi$
$578$	0	0
$579$	17.7573	0.737969
$580$	0	0
$581$	−26.9083	−1.11634
$582$	0	0
$583$	55.4319	2.29575
$584$	0	0
$585$	5.45415	0.225501
$586$	0	0
$587$	11.3121	0.466901	0.233451	−	0.972369i	$-0.424998\pi$
$587$	11.3121	0.466901	0.233451	+	0.972369i	$0.424998\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−15.4637	−0.636092
$592$	0	0
$593$	−24.9592	−1.02495	−0.512476	−	0.858701i	$-0.671272\pi$
$593$	−24.9592	−1.02495	−0.512476	+	0.858701i	$0.671272\pi$
$594$	0	0
$595$	−12.8076	−0.525062
$596$	0	0
$597$	2.90830	0.119029
$598$	0	0
$599$	42.4229	1.73335	0.866677	−	0.498869i	$-0.166251\pi$
$599$	42.4229	1.73335	0.866677	+	0.498869i	$0.166251\pi$
$600$	0	0
$601$	16.9083	0.689704	0.344852	−	0.938657i	$-0.387929\pi$
$601$	16.9083	0.689704	0.344852	+	0.938657i	$0.387929\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	16.0599	0.652927
$606$	0	0
$607$	1.39487	0.0566159	0.0283080	−	0.999599i	$-0.490988\pi$
$607$	1.39487	0.0566159	0.0283080	+	0.999599i	$0.490988\pi$
$608$	0	0
$609$	−10.2522	−0.415442
$610$	0	0
$611$	−47.2115	−1.90997
$612$	0	0
$613$	36.8765	1.48943	0.744714	−	0.667384i	$-0.232586\pi$
$613$	36.8765	1.48943	0.744714	+	0.667384i	$0.232586\pi$
$614$	0	0
$615$	9.85797	0.397512
$616$	0	0
$617$	16.3032	0.656341	0.328170	−	0.944619i	$-0.393568\pi$
$617$	16.3032	0.656341	0.328170	+	0.944619i	$0.393568\pi$
$618$	0	0
$619$	36.5236	1.46801	0.734004	−	0.679146i	$-0.237649\pi$
$619$	36.5236	1.46801	0.734004	+	0.679146i	$0.237649\pi$
$620$	0	0
$621$	−8.65606	−0.347356
$622$	0	0
$623$	−44.3720	−1.77773
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.20191	−0.207744
$628$	0	0
$629$	−47.4319	−1.89123
$630$	0	0
$631$	−10.6243	−0.422945	−0.211472	−	0.977384i	$-0.567826\pi$
$631$	−10.6243	−0.422945	−0.211472	+	0.977384i	$0.567826\pi$
$632$	0	0
$633$	−3.49552	−0.138935
$634$	0	0
$635$	−3.49552	−0.138716
$636$	0	0
$637$	17.7382	0.702813
$638$	0	0
$639$	−6.90830	−0.273288
$640$	0	0
$641$	18.1611	0.717322	0.358661	−	0.933468i	$-0.383233\pi$
$641$	18.1611	0.717322	0.358661	+	0.933468i	$0.383233\pi$
$642$	0	0
$643$	−27.4542	−1.08269	−0.541343	−	0.840802i	$-0.682084\pi$
$643$	−27.4542	−1.08269	−0.541343	+	0.840802i	$0.682084\pi$
$644$	0	0
$645$	−3.45415	−0.136007
$646$	0	0
$647$	−19.5644	−0.769155	−0.384577	−	0.923093i	$-0.625653\pi$
$647$	−19.5644	−0.769155	−0.384577	+	0.923093i	$0.625653\pi$
$648$	0	0
$649$	−67.9364	−2.66674
$650$	0	0
$651$	6.40382	0.250986
$652$	0	0
$653$	−38.1516	−1.49299	−0.746493	−	0.665393i	$-0.768264\pi$
$653$	−38.1516	−1.49299	−0.746493	+	0.665393i	$0.768264\pi$
$654$	0	0
$655$	15.6057	0.609767
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	12.4727	0.485866	0.242933	−	0.970043i	$-0.421890\pi$
$659$	12.4727	0.485866	0.242933	+	0.970043i	$0.421890\pi$
$660$	0	0
$661$	11.5644	0.449802	0.224901	−	0.974382i	$-0.427794\pi$
$661$	11.5644	0.449802	0.224901	+	0.974382i	$0.427794\pi$
$662$	0	0
$663$	−21.8166	−0.847287
$664$	0	0
$665$	3.20191	0.124165
$666$	0	0
$667$	−27.7159	−1.07317
$668$	0	0
$669$	0.504478	0.0195042
$670$	0	0
$671$	−19.4955	−0.752616
$672$	0	0
$673$	−6.46311	−0.249134	−0.124567	−	0.992211i	$-0.539754\pi$
$673$	−6.46311	−0.249134	−0.124567	+	0.992211i	$0.539754\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−31.9682	−1.22864	−0.614319	−	0.789058i	$-0.710569\pi$
$677$	−31.9682	−1.22864	−0.614319	+	0.789058i	$0.710569\pi$
$678$	0	0
$679$	−0.454754	−0.0174518
$680$	0	0
$681$	19.5644	0.749708
$682$	0	0
$683$	−2.68787	−0.102849	−0.0514243	−	0.998677i	$-0.516376\pi$
$683$	−2.68787	−0.102849	−0.0514243	+	0.998677i	$0.516376\pi$
$684$	0	0
$685$	−0.656063	−0.0250669
$686$	0	0
$687$	−15.7478	−0.600815
$688$	0	0
$689$	−58.1198	−2.21419
$690$	0	0
$691$	26.9083	1.02364	0.511820	−	0.859093i	$-0.328971\pi$
$691$	26.9083	1.02364	0.511820	+	0.859093i	$0.328971\pi$
$692$	0	0
$693$	16.6561	0.632711
$694$	0	0
$695$	−6.40382	−0.242911
$696$	0	0
$697$	−39.4319	−1.49359
$698$	0	0
$699$	17.1605	0.649071
$700$	0	0
$701$	32.6243	1.23220	0.616100	−	0.787668i	$-0.288712\pi$
$701$	32.6243	1.23220	0.616100	+	0.787668i	$0.288712\pi$
$702$	0	0
$703$	11.8580	0.447232
$704$	0	0
$705$	8.65606	0.326006
$706$	0	0
$707$	−14.1007	−0.530310
$708$	0	0
$709$	−29.3631	−1.10275	−0.551376	−	0.834257i	$-0.685897\pi$
$709$	−29.3631	−1.10275	−0.551376	+	0.834257i	$0.685897\pi$
$710$	0	0
$711$	6.15158	0.230702
$712$	0	0
$713$	17.3121	0.648344
$714$	0	0
$715$	−28.3720	−1.06105
$716$	0	0
$717$	−17.7064	−0.661257
$718$	0	0
$719$	−28.4134	−1.05964	−0.529820	−	0.848110i	$-0.677741\pi$
$719$	−28.4134	−1.05964	−0.529820	+	0.848110i	$0.677741\pi$
$720$	0	0
$721$	−41.0090	−1.52725
$722$	0	0
$723$	−23.8166	−0.885749
$724$	0	0
$725$	−3.20191	−0.118916
$726$	0	0
$727$	46.1293	1.71084	0.855421	−	0.517933i	$-0.173298\pi$
$727$	46.1293	1.71084	0.855421	+	0.517933i	$0.173298\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.8166	0.511026
$732$	0	0
$733$	28.6561	1.05844	0.529218	−	0.848486i	$-0.322485\pi$
$733$	28.6561	1.05844	0.529218	+	0.848486i	$0.322485\pi$
$734$	0	0
$735$	−3.25224	−0.119961
$736$	0	0
$737$	20.8076	0.766460
$738$	0	0
$739$	25.8166	0.949679	0.474840	−	0.880072i	$-0.342506\pi$
$739$	25.8166	0.949679	0.474840	+	0.880072i	$0.342506\pi$
$740$	0	0
$741$	5.45415	0.200363
$742$	0	0
$743$	38.1198	1.39848	0.699239	−	0.714888i	$-0.253522\pi$
$743$	38.1198	1.39848	0.699239	+	0.714888i	$0.253522\pi$
$744$	0	0
$745$	−23.3121	−0.854090
$746$	0	0
$747$	8.40382	0.307480
$748$	0	0
$749$	−41.0090	−1.49843
$750$	0	0
$751$	−7.31213	−0.266823	−0.133412	−	0.991061i	$-0.542593\pi$
$751$	−7.31213	−0.266823	−0.133412	+	0.991061i	$0.542593\pi$
$752$	0	0
$753$	7.10126	0.258784
$754$	0	0
$755$	23.3121	0.848415
$756$	0	0
$757$	−3.86753	−0.140568	−0.0702839	−	0.997527i	$-0.522391\pi$
$757$	−3.86753	−0.140568	−0.0702839	+	0.997527i	$0.522391\pi$
$758$	0	0
$759$	45.0281	1.63442
$760$	0	0
$761$	−12.3211	−0.446639	−0.223319	−	0.974745i	$-0.571689\pi$
$761$	−12.3211	−0.446639	−0.223319	+	0.974745i	$0.571689\pi$
$762$	0	0
$763$	−48.2204	−1.74570
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	71.2306	2.57199
$768$	0	0
$769$	−8.11977	−0.292806	−0.146403	−	0.989225i	$-0.546770\pi$
$769$	−8.11977	−0.292806	−0.146403	+	0.989225i	$0.546770\pi$
$770$	0	0
$771$	−21.5644	−0.776622
$772$	0	0
$773$	−17.3439	−0.623818	−0.311909	−	0.950112i	$-0.600968\pi$
$773$	−17.3439	−0.623818	−0.311909	+	0.950112i	$0.600968\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	−37.9682	−1.36210
$778$	0	0
$779$	9.85797	0.353199
$780$	0	0
$781$	35.9364	1.28590
$782$	0	0
$783$	3.20191	0.114427
$784$	0	0
$785$	6.25224	0.223152
$786$	0	0
$787$	28.2204	1.00595	0.502975	−	0.864301i	$-0.332239\pi$
$787$	28.2204	1.00595	0.502975	+	0.864301i	$0.332239\pi$
$788$	0	0
$789$	6.25224	0.222586
$790$	0	0
$791$	−16.2013	−0.576052
$792$	0	0
$793$	20.4409	0.725876
$794$	0	0
$795$	10.6561	0.377932
$796$	0	0
$797$	1.56436	0.0554126	0.0277063	−	0.999616i	$-0.491180\pi$
$797$	1.56436	0.0554126	0.0277063	+	0.999616i	$0.491180\pi$
$798$	0	0
$799$	−34.6243	−1.22492
$800$	0	0
$801$	13.8580	0.489647
$802$	0	0
$803$	−31.2115	−1.10143
$804$	0	0
$805$	−27.7159	−0.976859
$806$	0	0
$807$	−16.5140	−0.581322
$808$	0	0
$809$	−6.80765	−0.239344	−0.119672	−	0.992813i	$-0.538184\pi$
$809$	−6.80765	−0.239344	−0.119672	+	0.992813i	$0.538184\pi$
$810$	0	0
$811$	−13.5134	−0.474521	−0.237260	−	0.971446i	$-0.576249\pi$
$811$	−13.5134	−0.474521	−0.237260	+	0.971446i	$0.576249\pi$
$812$	0	0
$813$	−22.4038	−0.785736
$814$	0	0
$815$	−9.85797	−0.345310
$816$	0	0
$817$	−3.45415	−0.120845
$818$	0	0
$819$	−17.4637	−0.610231
$820$	0	0
$821$	−31.5326	−1.10049	−0.550247	−	0.835002i	$-0.685466\pi$
$821$	−31.5326	−1.10049	−0.550247	+	0.835002i	$0.685466\pi$
$822$	0	0
$823$	38.1102	1.32844	0.664219	−	0.747538i	$-0.268764\pi$
$823$	38.1102	1.32844	0.664219	+	0.747538i	$0.268764\pi$
$824$	0	0
$825$	5.20191	0.181107
$826$	0	0
$827$	−1.74776	−0.0607756	−0.0303878	−	0.999538i	$-0.509674\pi$
$827$	−1.74776	−0.0607756	−0.0303878	+	0.999538i	$0.509674\pi$
$828$	0	0
$829$	−10.2522	−0.356075	−0.178037	−	0.984024i	$-0.556975\pi$
$829$	−10.2522	−0.356075	−0.178037	+	0.984024i	$0.556975\pi$
$830$	0	0
$831$	−7.34394	−0.254758
$832$	0	0
$833$	13.0090	0.450734
$834$	0	0
$835$	6.25224	0.216368
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−19.7159	−0.680670	−0.340335	−	0.940304i	$-0.610540\pi$
$839$	−19.7159	−0.680670	−0.340335	+	0.940304i	$0.610540\pi$
$840$	0	0
$841$	−18.7478	−0.646475
$842$	0	0
$843$	1.85797	0.0639920
$844$	0	0
$845$	16.7478	0.576140
$846$	0	0
$847$	−51.4223	−1.76689
$848$	0	0
$849$	1.05033	0.0360471
$850$	0	0
$851$	−102.643	−3.51857
$852$	0	0
$853$	22.2522	0.761902	0.380951	−	0.924595i	$-0.375597\pi$
$853$	22.2522	0.761902	0.380951	+	0.924595i	$0.375597\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	53.7848	1.83725	0.918627	−	0.395126i	$-0.129299\pi$
$857$	53.7848	1.83725	0.918627	+	0.395126i	$0.129299\pi$
$858$	0	0
$859$	−5.31213	−0.181247	−0.0906237	−	0.995885i	$-0.528886\pi$
$859$	−5.31213	−0.181247	−0.0906237	+	0.995885i	$0.528886\pi$
$860$	0	0
$861$	−31.5644	−1.07571
$862$	0	0
$863$	−9.31213	−0.316988	−0.158494	−	0.987360i	$-0.550664\pi$
$863$	−9.31213	−0.316988	−0.158494	+	0.987360i	$0.550664\pi$
$864$	0	0
$865$	9.05989	0.308045
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−21.8166	−0.739227
$872$	0	0
$873$	0.142026	0.00480684
$874$	0	0
$875$	−3.20191	−0.108244
$876$	0	0
$877$	24.3625	0.822662	0.411331	−	0.911486i	$-0.365064\pi$
$877$	24.3625	0.822662	0.411331	+	0.911486i	$0.365064\pi$
$878$	0	0
$879$	15.7478	0.531159
$880$	0	0
$881$	−59.0281	−1.98871	−0.994353	−	0.106122i	$-0.966157\pi$
$881$	−59.0281	−1.98871	−0.994353	+	0.106122i	$0.966157\pi$
$882$	0	0
$883$	−39.1701	−1.31818	−0.659089	−	0.752065i	$-0.729058\pi$
$883$	−39.1701	−1.31818	−0.659089	+	0.752065i	$0.729058\pi$
$884$	0	0
$885$	−13.0599	−0.439003
$886$	0	0
$887$	−55.4319	−1.86122	−0.930610	−	0.366011i	$-0.880723\pi$
$887$	−55.4319	−1.86122	−0.930610	+	0.366011i	$0.880723\pi$
$888$	0	0
$889$	11.1924	0.375379
$890$	0	0
$891$	−5.20191	−0.174271
$892$	0	0
$893$	8.65606	0.289664
$894$	0	0
$895$	4.25224	0.142137
$896$	0	0
$897$	−47.2115	−1.57635
$898$	0	0
$899$	−6.40382	−0.213579
$900$	0	0
$901$	−42.6243	−1.42002
$902$	0	0
$903$	11.0599	0.368050
$904$	0	0
$905$	18.5554	0.616803
$906$	0	0
$907$	−18.9083	−0.627840	−0.313920	−	0.949449i	$-0.601642\pi$
$907$	−18.9083	−0.627840	−0.313920	+	0.949449i	$0.601642\pi$
$908$	0	0
$909$	4.40382	0.146066
$910$	0	0
$911$	25.8166	0.855342	0.427671	−	0.903934i	$-0.359334\pi$
$911$	25.8166	0.855342	0.427671	+	0.903934i	$0.359334\pi$
$912$	0	0
$913$	−43.7159	−1.44679
$914$	0	0
$915$	−3.74776	−0.123897
$916$	0	0
$917$	−49.9682	−1.65009
$918$	0	0
$919$	38.1198	1.25746	0.628728	−	0.777626i	$-0.283576\pi$
$919$	38.1198	1.25746	0.628728	+	0.777626i	$0.283576\pi$
$920$	0	0
$921$	17.5962	0.579814
$922$	0	0
$923$	−37.6789	−1.24022
$924$	0	0
$925$	−11.8580	−0.389888
$926$	0	0
$927$	12.8076	0.420658
$928$	0	0
$929$	33.5146	1.09958	0.549790	−	0.835303i	$-0.314708\pi$
$929$	33.5146	1.09958	0.549790	+	0.835303i	$0.314708\pi$
$930$	0	0
$931$	−3.25224	−0.106588
$932$	0	0
$933$	6.51404	0.213260
$934$	0	0
$935$	−20.8076	−0.680483
$936$	0	0
$937$	52.8447	1.72636	0.863180	−	0.504896i	$-0.168469\pi$
$937$	52.8447	1.72636	0.863180	+	0.504896i	$0.168469\pi$
$938$	0	0
$939$	−29.2115	−0.953280
$940$	0	0
$941$	52.8172	1.72179	0.860896	−	0.508781i	$-0.169904\pi$
$941$	52.8172	1.72179	0.860896	+	0.508781i	$0.169904\pi$
$942$	0	0
$943$	−85.3312	−2.77877
$944$	0	0
$945$	3.20191	0.104158
$946$	0	0
$947$	−19.0917	−0.620397	−0.310198	−	0.950672i	$-0.600395\pi$
$947$	−19.0917	−0.620397	−0.310198	+	0.950672i	$0.600395\pi$
$948$	0	0
$949$	32.7249	1.06230
$950$	0	0
$951$	28.4727	0.923289
$952$	0	0
$953$	46.1707	1.49562	0.747808	−	0.663915i	$-0.231106\pi$
$953$	46.1707	1.49562	0.747808	+	0.663915i	$0.231106\pi$
$954$	0	0
$955$	22.5140	0.728537
$956$	0	0
$957$	−16.6561	−0.538414
$958$	0	0
$959$	2.10065	0.0678337
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	12.8076	0.412721
$964$	0	0
$965$	−17.7573	−0.571628
$966$	0	0
$967$	25.5230	0.820764	0.410382	−	0.911914i	$-0.365395\pi$
$967$	25.5230	0.820764	0.410382	+	0.911914i	$0.365395\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−19.9682	−0.640810	−0.320405	−	0.947281i	$-0.603819\pi$
$971$	−19.9682	−0.640810	−0.320405	+	0.947281i	$0.603819\pi$
$972$	0	0
$973$	20.5045	0.657343
$974$	0	0
$975$	−5.45415	−0.174673
$976$	0	0
$977$	−1.34394	−0.0429964	−0.0214982	−	0.999769i	$-0.506844\pi$
$977$	−1.34394	−0.0429964	−0.0214982	+	0.999769i	$0.506844\pi$
$978$	0	0
$979$	−72.0880	−2.30394
$980$	0	0
$981$	15.0599	0.480825
$982$	0	0
$983$	52.3720	1.67041	0.835204	−	0.549940i	$-0.185350\pi$
$983$	52.3720	1.67041	0.835204	+	0.549940i	$0.185350\pi$
$984$	0	0
$985$	15.4637	0.492715
$986$	0	0
$987$	−27.7159	−0.882208
$988$	0	0
$989$	29.8993	0.950744
$990$	0	0
$991$	9.28031	0.294799	0.147399	−	0.989077i	$-0.452910\pi$
$991$	9.28031	0.294799	0.147399	+	0.989077i	$0.452910\pi$
$992$	0	0
$993$	21.9682	0.697139
$994$	0	0
$995$	−2.90830	−0.0921994
$996$	0	0
$997$	13.2433	0.419419	0.209709	−	0.977764i	$-0.432748\pi$
$997$	13.2433	0.419419	0.209709	+	0.977764i	$0.432748\pi$
$998$	0	0
$999$	11.8580	0.375170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bs.1.1		3
4.3	odd	2		2280.2.a.u.1.3	✓	3
12.11	even	2		6840.2.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.u.1.3	✓	3	4.3	odd	2
4560.2.a.bs.1.1		3	1.1	even	1	trivial
6840.2.a.bh.1.3		3	12.11	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.20191 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.20191 q^{7} +1.00000 q^{9} -5.20191 q^{11} +5.45415 q^{13} -1.00000 q^{15} +4.00000 q^{17} -1.00000 q^{19} +3.20191 q^{21} +8.65606 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.20191 q^{29} +2.00000 q^{31} +5.20191 q^{33} -3.20191 q^{35} -11.8580 q^{37} -5.45415 q^{39} -9.85797 q^{41} +3.45415 q^{43} +1.00000 q^{45} -8.65606 q^{47} +3.25224 q^{49} -4.00000 q^{51} -10.6561 q^{53} -5.20191 q^{55} +1.00000 q^{57} +13.0599 q^{59} +3.74776 q^{61} -3.20191 q^{63} +5.45415 q^{65} -4.00000 q^{67} -8.65606 q^{69} -6.90830 q^{71} +6.00000 q^{73} -1.00000 q^{75} +16.6561 q^{77} +6.15158 q^{79} +1.00000 q^{81} +8.40382 q^{83} +4.00000 q^{85} +3.20191 q^{87} +13.8580 q^{89} -17.4637 q^{91} -2.00000 q^{93} -1.00000 q^{95} +0.142026 q^{97} -5.20191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 6 q^{11} + 4 q^{13} - 3 q^{15} + 12 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{31} + 6 q^{33} - 4 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} + 3 q^{45} - 4 q^{47} + 7 q^{49} - 12 q^{51} - 10 q^{53} - 6 q^{55} + 3 q^{57} - 2 q^{59} + 14 q^{61} + 4 q^{65} - 12 q^{67} - 4 q^{69} + 4 q^{71} + 18 q^{73} - 3 q^{75} + 28 q^{77} + 2 q^{79} + 3 q^{81} + 6 q^{83} + 12 q^{85} + 10 q^{89} + 8 q^{91} - 6 q^{93} - 3 q^{95} + 32 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 6 * q^11 + 4 * q^13 - 3 * q^15 + 12 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^31 + 6 * q^33 - 4 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 + 3 * q^45 - 4 * q^47 + 7 * q^49 - 12 * q^51 - 10 * q^53 - 6 * q^55 + 3 * q^57 - 2 * q^59 + 14 * q^61 + 4 * q^65 - 12 * q^67 - 4 * q^69 + 4 * q^71 + 18 * q^73 - 3 * q^75 + 28 * q^77 + 2 * q^79 + 3 * q^81 + 6 * q^83 + 12 * q^85 + 10 * q^89 + 8 * q^91 - 6 * q^93 - 3 * q^95 + 32 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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