LMFDB - Embedded newform 6840.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
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			1.3
Root			$-3.32803$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.20191	1.21021	0.605104	−	0.796146i	$-0.293131\pi$
$7$	3.20191	1.21021	0.605104	+	0.796146i	$0.293131\pi$
$8$	0	0
$9$	0	0
$10$	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
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	3.20191	0.594580	0.297290	−	0.954787i	$-0.403917\pi$
$29$	3.20191	0.594580	0.297290	+	0.954787i	$0.403917\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	9.85797	1.53956	0.769778	−	0.638311i	$-0.220367\pi$
$41$	9.85797	1.53956	0.769778	+	0.638311i	$0.220367\pi$
$42$	0	0
$43$	−3.45415	−0.526753	−0.263377	−	0.964693i	$-0.584836\pi$
$43$	−3.45415	−0.526753	−0.263377	+	0.964693i	$0.584836\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	10.6561	1.46372	0.731861	−	0.681454i	$-0.238652\pi$
$53$	10.6561	1.46372	0.731861	+	0.681454i	$0.238652\pi$
$54$	0	0
$55$	5.20191	0.701426
$56$	0	0
$57$	0	0
$58$	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
$63$	0	0
$64$	0	0
$65$	−5.45415	−0.676504
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	−16.6561	−1.89813
$78$	0	0
$79$	−6.15158	−0.692107	−0.346054	−	0.938215i	$-0.612478\pi$
$79$	−6.15158	−0.692107	−0.346054	+	0.938215i	$0.612478\pi$
$80$	0	0
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	−13.8580	−1.46894	−0.734471	−	0.678640i	$-0.762570\pi$
$89$	−13.8580	−1.46894	−0.734471	+	0.678640i	$0.762570\pi$
$90$	0	0
$91$	17.4637	1.83069
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	−4.40382	−0.438197	−0.219098	−	0.975703i	$-0.570312\pi$
$101$	−4.40382	−0.438197	−0.219098	+	0.975703i	$0.570312\pi$
$102$	0	0
$103$	−12.8076	−1.26197	−0.630987	−	0.775793i	$-0.717350\pi$
$103$	−12.8076	−1.26197	−0.630987	+	0.775793i	$0.717350\pi$
$104$	0	0
$105$	0	0
$106$	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
$111$	0	0
$112$	0	0
$113$	−5.05989	−0.475994	−0.237997	−	0.971266i	$-0.576491\pi$
$113$	−5.05989	−0.475994	−0.237997	+	0.971266i	$0.576491\pi$
$114$	0	0
$115$	−8.65606	−0.807182
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.8076	−1.17408
$120$	0	0
$121$	16.0599	1.45999
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.49552	0.310177	0.155089	−	0.987901i	$-0.450434\pi$
$127$	3.49552	0.310177	0.155089	+	0.987901i	$0.450434\pi$
$128$	0	0
$129$	0	0
$130$	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
$135$	0	0
$136$	0	0
$137$	0.656063	0.0560512	0.0280256	−	0.999607i	$-0.491078\pi$
$137$	0.656063	0.0560512	0.0280256	+	0.999607i	$0.491078\pi$
$138$	0	0
$139$	6.40382	0.543165	0.271583	−	0.962415i	$-0.412453\pi$
$139$	6.40382	0.543165	0.271583	+	0.962415i	$0.412453\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−28.3720	−2.37259
$144$	0	0
$145$	−3.20191	−0.265904
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.3121	1.90980	0.954902	−	0.296922i	$-0.0959600\pi$
$149$	23.3121	1.90980	0.954902	+	0.296922i	$0.0959600\pi$
$150$	0	0
$151$	−23.3121	−1.89711	−0.948557	−	0.316607i	$-0.897456\pi$
$151$	−23.3121	−1.89711	−0.948557	+	0.316607i	$0.897456\pi$
$152$	0	0
$153$	0	0
$154$	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
$159$	0	0
$160$	0	0
$161$	27.7159	2.18432
$162$	0	0
$163$	9.85797	0.772136	0.386068	−	0.922470i	$-0.373833\pi$
$163$	9.85797	0.772136	0.386068	+	0.922470i	$0.373833\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	0	0
$172$	0	0
$173$	−9.05989	−0.688810	−0.344405	−	0.938821i	$-0.611919\pi$
$173$	−9.05989	−0.688810	−0.344405	+	0.938821i	$0.611919\pi$
$174$	0	0
$175$	3.20191	0.242042
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	11.8580	0.871816
$186$	0	0
$187$	20.8076	1.52161
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	−15.4637	−1.10174	−0.550872	−	0.834590i	$-0.685705\pi$
$197$	−15.4637	−1.10174	−0.550872	+	0.834590i	$0.685705\pi$
$198$	0	0
$199$	2.90830	0.206164	0.103082	−	0.994673i	$-0.467130\pi$
$199$	2.90830	0.206164	0.103082	+	0.994673i	$0.467130\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.2522	0.719566
$204$	0	0
$205$	−9.85797	−0.688511
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.20191	−0.359824
$210$	0	0
$211$	−3.49552	−0.240642	−0.120321	−	0.992735i	$-0.538392\pi$
$211$	−3.49552	−0.240642	−0.120321	+	0.992735i	$0.538392\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.45415	0.235571
$216$	0	0
$217$	−6.40382	−0.434720
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−21.8166	−1.46754
$222$	0	0
$223$	0.504478	0.0337823	0.0168912	−	0.999857i	$-0.494623\pi$
$223$	0.504478	0.0337823	0.0168912	+	0.999857i	$0.494623\pi$
$224$	0	0
$225$	0	0
$226$	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
$231$	0	0
$232$	0	0
$233$	17.1605	1.12422	0.562112	−	0.827061i	$-0.309989\pi$
$233$	17.1605	1.12422	0.562112	+	0.827061i	$0.309989\pi$
$234$	0	0
$235$	8.65606	0.564659
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	−3.25224	−0.207778
$246$	0	0
$247$	5.45415	0.347039
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	−21.5644	−1.34515	−0.672574	−	0.740030i	$-0.734811\pi$
$257$	−21.5644	−1.34515	−0.672574	+	0.740030i	$0.734811\pi$
$258$	0	0
$259$	−37.9682	−2.35923
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	−16.5140	−1.00688	−0.503439	−	0.864031i	$-0.667932\pi$
$269$	−16.5140	−1.00688	−0.503439	+	0.864031i	$0.667932\pi$
$270$	0	0
$271$	−22.4038	−1.36094	−0.680468	−	0.732778i	$-0.738223\pi$
$271$	−22.4038	−1.36094	−0.680468	+	0.732778i	$0.738223\pi$
$272$	0	0
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	1.85797	0.110837	0.0554187	−	0.998463i	$-0.482351\pi$
$281$	1.85797	0.110837	0.0554187	+	0.998463i	$0.482351\pi$
$282$	0	0
$283$	1.05033	0.0624355	0.0312177	−	0.999513i	$-0.490061\pi$
$283$	1.05033	0.0624355	0.0312177	+	0.999513i	$0.490061\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.5644	1.86319
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.7478	0.919994	0.459997	−	0.887920i	$-0.347850\pi$
$293$	15.7478	0.919994	0.459997	+	0.887920i	$0.347850\pi$
$294$	0	0
$295$	−13.0599	−0.760376
$296$	0	0
$297$	0	0
$298$	0	0
$299$	47.2115	2.73031
$300$	0	0
$301$	−11.0599	−0.637481
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.74776	−0.214596
$306$	0	0
$307$	17.5962	1.00427	0.502133	−	0.864790i	$-0.332549\pi$
$307$	17.5962	1.00427	0.502133	+	0.864790i	$0.332549\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	28.4727	1.59918	0.799592	−	0.600543i	$-0.205049\pi$
$317$	28.4727	1.59918	0.799592	+	0.600543i	$0.205049\pi$
$318$	0	0
$319$	−16.6561	−0.932560
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	5.45415	0.302542
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−27.7159	−1.52803
$330$	0	0
$331$	21.9682	1.20748	0.603740	−	0.797181i	$-0.293676\pi$
$331$	21.9682	1.20748	0.603740	+	0.797181i	$0.293676\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	10.4038	0.563399
$342$	0	0
$343$	−12.0000	−0.647939
$344$	0	0
$345$	0	0
$346$	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
$351$	0	0
$352$	0	0
$353$	−13.4637	−0.716601	−0.358300	−	0.933606i	$-0.616644\pi$
$353$	−13.4637	−0.716601	−0.358300	+	0.933606i	$0.616644\pi$
$354$	0	0
$355$	6.90830	0.366655
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	16.5140	0.862026	0.431013	−	0.902346i	$-0.358156\pi$
$367$	16.5140	0.862026	0.431013	+	0.902346i	$0.358156\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	17.4637	0.899427
$378$	0	0
$379$	5.46371	0.280652	0.140326	−	0.990105i	$-0.455185\pi$
$379$	5.46371	0.280652	0.140326	+	0.990105i	$0.455185\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−34.6243	−1.75102
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	37.0694	1.85116	0.925580	−	0.378552i	$-0.123578\pi$
$401$	37.0694	1.85116	0.925580	+	0.378552i	$0.123578\pi$
$402$	0	0
$403$	−10.9083	−0.543381
$404$	0	0
$405$	0	0
$406$	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	0
$412$	0	0
$413$	41.8166	2.05766
$414$	0	0
$415$	−8.40382	−0.412527
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	12.0000	0.580721
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	8.65606	0.414076
$438$	0	0
$439$	−30.1516	−1.43906	−0.719528	−	0.694463i	$-0.755642\pi$
$439$	−30.1516	−1.43906	−0.719528	+	0.694463i	$0.755642\pi$
$440$	0	0
$441$	0	0
$442$	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
$447$	0	0
$448$	0	0
$449$	0.545849	0.0257602	0.0128801	−	0.999917i	$-0.495900\pi$
$449$	0.545849	0.0257602	0.0128801	+	0.999917i	$0.495900\pi$
$450$	0	0
$451$	−51.2803	−2.41470
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	0.687875	0.0320375	0.0160188	−	0.999872i	$-0.494901\pi$
$461$	0.687875	0.0320375	0.0160188	+	0.999872i	$0.494901\pi$
$462$	0	0
$463$	−4.79809	−0.222986	−0.111493	−	0.993765i	$-0.535563\pi$
$463$	−4.79809	−0.222986	−0.111493	+	0.993765i	$0.535563\pi$
$464$	0	0
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	17.9682	0.826178
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−0.142026	−0.00644905
$486$	0	0
$487$	18.7070	0.847695	0.423847	−	0.905734i	$-0.360679\pi$
$487$	18.7070	0.847695	0.423847	+	0.905734i	$0.360679\pi$
$488$	0	0
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	−22.1198	−0.992207
$498$	0	0
$499$	−1.59618	−0.0714547	−0.0357273	−	0.999362i	$-0.511375\pi$
$499$	−1.59618	−0.0714547	−0.0357273	+	0.999362i	$0.511375\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	−0.210868	−0.00934655	−0.00467328	−	0.999989i	$-0.501488\pi$
$509$	−0.210868	−0.00934655	−0.00467328	+	0.999989i	$0.501488\pi$
$510$	0	0
$511$	19.2115	0.849865
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.8076	0.564372
$516$	0	0
$517$	45.0281	1.98033
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.04137	0.352299	0.176149	−	0.984363i	$-0.443636\pi$
$521$	8.04137	0.352299	0.176149	+	0.984363i	$0.443636\pi$
$522$	0	0
$523$	5.39487	0.235901	0.117951	−	0.993019i	$-0.462368\pi$
$523$	5.39487	0.235901	0.117951	+	0.993019i	$0.462368\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	51.9274	2.25771
$530$	0	0
$531$	0	0
$532$	0	0
$533$	53.7669	2.32890
$534$	0	0
$535$	−12.8076	−0.553723
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	−15.0599	−0.645095
$546$	0	0
$547$	9.59618	0.410303	0.205151	−	0.978730i	$-0.434231\pi$
$547$	9.59618	0.410303	0.205151	+	0.978730i	$0.434231\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.20191	0.136406
$552$	0	0
$553$	−19.6968	−0.837594
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−18.8395	−0.796824
$560$	0	0
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	16.5458	0.693638	0.346819	−	0.937932i	$-0.387262\pi$
$569$	16.5458	0.693638	0.346819	+	0.937932i	$0.387262\pi$
$570$	0	0
$571$	−31.7159	−1.32727	−0.663636	−	0.748056i	$-0.730987\pi$
$571$	−31.7159	−1.32727	−0.663636	+	0.748056i	$0.730987\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	26.9083	1.11634
$582$	0	0
$583$	−55.4319	−2.29575
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	24.9592	1.02495	0.512476	−	0.858701i	$-0.328728\pi$
$593$	24.9592	1.02495	0.512476	+	0.858701i	$0.328728\pi$
$594$	0	0
$595$	12.8076	0.525062
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	−16.0599	−0.652927
$606$	0	0
$607$	−1.39487	−0.0566159	−0.0283080	−	0.999599i	$-0.509012\pi$
$607$	−1.39487	−0.0566159	−0.0283080	+	0.999599i	$0.509012\pi$
$608$	0	0
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	−16.3032	−0.656341	−0.328170	−	0.944619i	$-0.606432\pi$
$617$	−16.3032	−0.656341	−0.328170	+	0.944619i	$0.606432\pi$
$618$	0	0
$619$	−36.5236	−1.46801	−0.734004	−	0.679146i	$-0.762351\pi$
$619$	−36.5236	−1.46801	−0.734004	+	0.679146i	$0.762351\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−44.3720	−1.77773
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47.4319	1.89123
$630$	0	0
$631$	10.6243	0.422945	0.211472	−	0.977384i	$-0.432174\pi$
$631$	10.6243	0.422945	0.211472	+	0.977384i	$0.432174\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.49552	−0.138716
$636$	0	0
$637$	17.7382	0.702813
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.1611	−0.717322	−0.358661	−	0.933468i	$-0.616767\pi$
$641$	−18.1611	−0.717322	−0.358661	+	0.933468i	$0.616767\pi$
$642$	0	0
$643$	27.4542	1.08269	0.541343	−	0.840802i	$-0.317916\pi$
$643$	27.4542	1.08269	0.541343	+	0.840802i	$0.317916\pi$
$644$	0	0
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	38.1516	1.49299	0.746493	−	0.665393i	$-0.231736\pi$
$653$	38.1516	1.49299	0.746493	+	0.665393i	$0.231736\pi$
$654$	0	0
$655$	−15.6057	−0.609767
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	−3.20191	−0.124165
$666$	0	0
$667$	27.7159	1.07317
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	31.9682	1.22864	0.614319	−	0.789058i	$-0.289431\pi$
$677$	31.9682	1.22864	0.614319	+	0.789058i	$0.289431\pi$
$678$	0	0
$679$	0.454754	0.0174518
$680$	0	0
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	58.1198	2.21419
$690$	0	0
$691$	−26.9083	−1.02364	−0.511820	−	0.859093i	$-0.671029\pi$
$691$	−26.9083	−1.02364	−0.511820	+	0.859093i	$0.671029\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.40382	−0.242911
$696$	0	0
$697$	−39.4319	−1.49359
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.6243	−1.23220	−0.616100	−	0.787668i	$-0.711288\pi$
$701$	−32.6243	−1.23220	−0.616100	+	0.787668i	$0.711288\pi$
$702$	0	0
$703$	−11.8580	−0.447232
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	−17.3121	−0.648344
$714$	0	0
$715$	28.3720	1.06105
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	3.20191	0.118916
$726$	0	0
$727$	−46.1293	−1.71084	−0.855421	−	0.517933i	$-0.826702\pi$
$727$	−46.1293	−1.71084	−0.855421	+	0.517933i	$0.826702\pi$
$728$	0	0
$729$	0	0
$730$	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
$735$	0	0
$736$	0	0
$737$	−20.8076	−0.766460
$738$	0	0
$739$	−25.8166	−0.949679	−0.474840	−	0.880072i	$-0.657494\pi$
$739$	−25.8166	−0.949679	−0.474840	+	0.880072i	$0.657494\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	41.0090	1.49843
$750$	0	0
$751$	7.31213	0.266823	0.133412	−	0.991061i	$-0.457407\pi$
$751$	7.31213	0.266823	0.133412	+	0.991061i	$0.457407\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	12.3211	0.446639	0.223319	−	0.974745i	$-0.428311\pi$
$761$	12.3211	0.446639	0.223319	+	0.974745i	$0.428311\pi$
$762$	0	0
$763$	48.2204	1.74570
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	17.3439	0.623818	0.311909	−	0.950112i	$-0.399032\pi$
$773$	17.3439	0.623818	0.311909	+	0.950112i	$0.399032\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.85797	0.353199
$780$	0	0
$781$	35.9364	1.28590
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.25224	−0.223152
$786$	0	0
$787$	−28.2204	−1.00595	−0.502975	−	0.864301i	$-0.667761\pi$
$787$	−28.2204	−1.00595	−0.502975	+	0.864301i	$0.667761\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.2013	−0.576052
$792$	0	0
$793$	20.4409	0.725876
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.56436	−0.0554126	−0.0277063	−	0.999616i	$-0.508820\pi$
$797$	−1.56436	−0.0554126	−0.0277063	+	0.999616i	$0.508820\pi$
$798$	0	0
$799$	34.6243	1.22492
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−31.2115	−1.10143
$804$	0	0
$805$	−27.7159	−0.976859
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.80765	0.239344	0.119672	−	0.992813i	$-0.461816\pi$
$809$	6.80765	0.239344	0.119672	+	0.992813i	$0.461816\pi$
$810$	0	0
$811$	13.5134	0.474521	0.237260	−	0.971446i	$-0.423751\pi$
$811$	13.5134	0.474521	0.237260	+	0.971446i	$0.423751\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.85797	−0.345310
$816$	0	0
$817$	−3.45415	−0.120845
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.5326	1.10049	0.550247	−	0.835002i	$-0.314534\pi$
$821$	31.5326	1.10049	0.550247	+	0.835002i	$0.314534\pi$
$822$	0	0
$823$	−38.1102	−1.32844	−0.664219	−	0.747538i	$-0.731236\pi$
$823$	−38.1102	−1.32844	−0.664219	+	0.747538i	$0.731236\pi$
$824$	0	0
$825$	0	0
$826$	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
$831$	0	0
$832$	0	0
$833$	−13.0090	−0.450734
$834$	0	0
$835$	−6.25224	−0.216368
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	−16.7478	−0.576140
$846$	0	0
$847$	51.4223	1.76689
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	−53.7848	−1.83725	−0.918627	−	0.395126i	$-0.870701\pi$
$857$	−53.7848	−1.83725	−0.918627	+	0.395126i	$0.870701\pi$
$858$	0	0
$859$	5.31213	0.181247	0.0906237	−	0.995885i	$-0.471114\pi$
$859$	5.31213	0.181247	0.0906237	+	0.995885i	$0.471114\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	21.8166	0.739227
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	59.0281	1.98871	0.994353	−	0.106122i	$-0.0338433\pi$
$881$	59.0281	1.98871	0.994353	+	0.106122i	$0.0338433\pi$
$882$	0	0
$883$	39.1701	1.31818	0.659089	−	0.752065i	$-0.270942\pi$
$883$	39.1701	1.31818	0.659089	+	0.752065i	$0.270942\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	−8.65606	−0.289664
$894$	0	0
$895$	−4.25224	−0.142137
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.40382	−0.213579
$900$	0	0
$901$	−42.6243	−1.42002
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.5554	−0.616803
$906$	0	0
$907$	18.9083	0.627840	0.313920	−	0.949449i	$-0.398358\pi$
$907$	18.9083	0.627840	0.313920	+	0.949449i	$0.398358\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	49.9682	1.65009
$918$	0	0
$919$	−38.1198	−1.25746	−0.628728	−	0.777626i	$-0.716424\pi$
$919$	−38.1198	−1.25746	−0.628728	+	0.777626i	$0.716424\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−37.6789	−1.24022
$924$	0	0
$925$	−11.8580	−0.389888
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.5146	−1.09958	−0.549790	−	0.835303i	$-0.685292\pi$
$929$	−33.5146	−1.09958	−0.549790	+	0.835303i	$0.685292\pi$
$930$	0	0
$931$	3.25224	0.106588
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	−52.8172	−1.72179	−0.860896	−	0.508781i	$-0.830096\pi$
$941$	−52.8172	−1.72179	−0.860896	+	0.508781i	$0.830096\pi$
$942$	0	0
$943$	85.3312	2.77877
$944$	0	0
$945$	0	0
$946$	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
$951$	0	0
$952$	0	0
$953$	−46.1707	−1.49562	−0.747808	−	0.663915i	$-0.768894\pi$
$953$	−46.1707	−1.49562	−0.747808	+	0.663915i	$0.768894\pi$
$954$	0	0
$955$	−22.5140	−0.728537
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.10065	0.0678337
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.7573	0.571628
$966$	0	0
$967$	−25.5230	−0.820764	−0.410382	−	0.911914i	$-0.634605\pi$
$967$	−25.5230	−0.820764	−0.410382	+	0.911914i	$0.634605\pi$
$968$	0	0
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	1.34394	0.0429964	0.0214982	−	0.999769i	$-0.493156\pi$
$977$	1.34394	0.0429964	0.0214982	+	0.999769i	$0.493156\pi$
$978$	0	0
$979$	72.0880	2.30394
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	−29.8993	−0.950744
$990$	0	0
$991$	−9.28031	−0.294799	−0.147399	−	0.989077i	$-0.547090\pi$
$991$	−9.28031	−0.294799	−0.147399	+	0.989077i	$0.547090\pi$
$992$	0	0
$993$	0	0
$994$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.u.1.3	✓	3	3.2	odd	2
4560.2.a.bs.1.1		3	12.11	even	2
6840.2.a.bh.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

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

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