LMFDB - Embedded newform 1792.3.g.g.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(127,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.9710629824630784.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 4x^{10} + 12x^{8} + 25x^{6} + 48x^{4} + 64x^{2} + 64 $ x^12 + 4x^10 + 12x^8 + 25x^6 + 48x^4 + 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$-0.746179 - 1.20134i$ of defining polynomial
Character	$\chi$	$=$	1792.127
Dual form			1792.3.g.g.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.98472 q^{3} -6.80536i q^{5} -2.64575i q^{7} -0.0914622 q^{9} +O(q^{10})$	$q-2.98472 q^{3} -6.80536i q^{5} -2.64575i q^{7} -0.0914622 q^{9} +14.3426 q^{11} -4.62243i q^{13} +20.3121i q^{15} +11.6107 q^{17} +22.4428 q^{19} +7.89682i q^{21} -0.853941i q^{23} -21.3129 q^{25} +27.1354 q^{27} +42.8321i q^{29} -1.54982i q^{31} -42.8087 q^{33} -18.0053 q^{35} +45.0151i q^{37} +13.7967i q^{39} +36.6492 q^{41} +27.9894 q^{43} +0.622433i q^{45} +42.1740i q^{47} -7.00000 q^{49} -34.6547 q^{51} +43.4044i q^{53} -97.6068i q^{55} -66.9856 q^{57} +53.9979 q^{59} +15.9498i q^{61} +0.241986i q^{63} -31.4573 q^{65} +91.9453 q^{67} +2.54877i q^{69} -16.3585i q^{71} -9.58728 q^{73} +63.6130 q^{75} -37.9470i q^{77} -56.9826i q^{79} -80.1685 q^{81} +14.3077 q^{83} -79.0151i q^{85} -127.842i q^{87} -100.136 q^{89} -12.2298 q^{91} +4.62579i q^{93} -152.732i q^{95} +68.2834 q^{97} -1.31181 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{9}+O(q^{10}) $ 12 * q + 20 * q^9	$ 12 q + 20 q^{9} - 8 q^{17} + 60 q^{25} + 160 q^{33} + 40 q^{41} - 84 q^{49} - 320 q^{57} - 272 q^{65} + 264 q^{73} - 436 q^{81} - 696 q^{89} + 504 q^{97}+O(q^{100}) $ 12 * q + 20 * q^9 - 8 * q^17 + 60 * q^25 + 160 * q^33 + 40 * q^41 - 84 * q^49 - 320 * q^57 - 272 * q^65 + 264 * q^73 - 436 * q^81 - 696 * q^89 + 504 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−2.98472	−0.994906	−0.497453	−	0.867491i	$-0.665731\pi$
$3$	−2.98472	−0.994906	−0.497453	+	0.867491i	$0.665731\pi$
$4$	0	0
$5$	− 6.80536i	− 1.36107i	−0.732715	−	0.680536i	$-0.761747\pi$
$5$	− 6.80536i	− 1.36107i	0.732715	−	0.680536i	$-0.238253\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−0.0914622	−0.0101625
$10$	0	0
$11$	14.3426	1.30388	0.651938	−	0.758272i	$-0.273956\pi$
$11$	14.3426	1.30388	0.651938	+	0.758272i	$0.273956\pi$
$12$	0	0
$13$	− 4.62243i	− 0.355572i	−0.984069	−	0.177786i	$-0.943107\pi$
$13$	− 4.62243i	− 0.355572i	0.984069	−	0.177786i	$-0.0568934\pi$
$14$	0	0
$15$	20.3121i	1.35414i
$16$	0	0
$17$	11.6107	0.682983	0.341492	−	0.939885i	$-0.389068\pi$
$17$	11.6107	0.682983	0.341492	+	0.939885i	$0.389068\pi$
$18$	0	0
$19$	22.4428	1.18120	0.590601	−	0.806964i	$-0.298891\pi$
$19$	22.4428	1.18120	0.590601	+	0.806964i	$0.298891\pi$
$20$	0	0
$21$	7.89682i	0.376039i
$22$	0	0
$23$	− 0.853941i	− 0.0371279i	−0.999828	−	0.0185639i	$-0.994091\pi$
$23$	− 0.853941i	− 0.0371279i	0.999828	−	0.0185639i	$-0.00590943\pi$
$24$	0	0
$25$	−21.3129	−0.852516
$26$	0	0
$27$	27.1354	1.00502
$28$	0	0
$29$	42.8321i	1.47697i	0.674270	+	0.738485i	$0.264459\pi$
$29$	42.8321i	1.47697i	−0.674270	+	0.738485i	$0.735541\pi$
$30$	0	0
$31$	− 1.54982i	− 0.0499943i	−0.999688	−	0.0249972i	$-0.992042\pi$
$31$	− 1.54982i	− 0.0499943i	0.999688	−	0.0249972i	$-0.00795767\pi$
$32$	0	0
$33$	−42.8087	−1.29723
$34$	0	0
$35$	−18.0053	−0.514437
$36$	0	0
$37$	45.0151i	1.21662i	0.793698	+	0.608312i	$0.208153\pi$
$37$	45.0151i	1.21662i	−0.793698	+	0.608312i	$0.791847\pi$
$38$	0	0
$39$	13.7967i	0.353760i
$40$	0	0
$41$	36.6492	0.893883	0.446942	−	0.894563i	$-0.352513\pi$
$41$	36.6492	0.893883	0.446942	+	0.894563i	$0.352513\pi$
$42$	0	0
$43$	27.9894	0.650916	0.325458	−	0.945557i	$-0.394482\pi$
$43$	27.9894	0.650916	0.325458	+	0.945557i	$0.394482\pi$
$44$	0	0
$45$	0.622433i	0.0138318i
$46$	0	0
$47$	42.1740i	0.897318i	0.893703	+	0.448659i	$0.148098\pi$
$47$	42.1740i	0.897318i	−0.893703	+	0.448659i	$0.851902\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−34.6547	−0.679504
$52$	0	0
$53$	43.4044i	0.818950i	0.912321	+	0.409475i	$0.134288\pi$
$53$	43.4044i	0.818950i	−0.912321	+	0.409475i	$0.865712\pi$
$54$	0	0
$55$	− 97.6068i	− 1.77467i
$56$	0	0
$57$	−66.9856	−1.17519
$58$	0	0
$59$	53.9979	0.915219	0.457609	−	0.889153i	$-0.348706\pi$
$59$	53.9979	0.915219	0.457609	+	0.889153i	$0.348706\pi$
$60$	0	0
$61$	15.9498i	0.261472i	0.991417	+	0.130736i	$0.0417340\pi$
$61$	15.9498i	0.261472i	−0.991417	+	0.130736i	$0.958266\pi$
$62$	0	0
$63$	0.241986i	0.00384105i
$64$	0	0
$65$	−31.4573	−0.483959
$66$	0	0
$67$	91.9453	1.37232	0.686159	−	0.727452i	$-0.259295\pi$
$67$	91.9453	1.37232	0.686159	+	0.727452i	$0.259295\pi$
$68$	0	0
$69$	2.54877i	0.0369387i
$70$	0	0
$71$	− 16.3585i	− 0.230401i	−0.993342	−	0.115201i	$-0.963249\pi$
$71$	− 16.3585i	− 0.230401i	0.993342	−	0.115201i	$-0.0367511\pi$
$72$	0	0
$73$	−9.58728	−0.131333	−0.0656663	−	0.997842i	$-0.520917\pi$
$73$	−9.58728	−0.131333	−0.0656663	+	0.997842i	$0.520917\pi$
$74$	0	0
$75$	63.6130	0.848173
$76$	0	0
$77$	− 37.9470i	− 0.492819i
$78$	0	0
$79$	− 56.9826i	− 0.721299i	−0.932701	−	0.360649i	$-0.882555\pi$
$79$	− 56.9826i	− 0.721299i	0.932701	−	0.360649i	$-0.117445\pi$
$80$	0	0
$81$	−80.1685	−0.989734
$82$	0	0
$83$	14.3077	0.172381	0.0861907	−	0.996279i	$-0.472531\pi$
$83$	14.3077	0.172381	0.0861907	+	0.996279i	$0.472531\pi$
$84$	0	0
$85$	− 79.0151i	− 0.929589i
$86$	0	0
$87$	− 127.842i	− 1.46945i
$88$	0	0
$89$	−100.136	−1.12512	−0.562562	−	0.826755i	$-0.690184\pi$
$89$	−100.136	−1.12512	−0.562562	+	0.826755i	$0.690184\pi$
$90$	0	0
$91$	−12.2298	−0.134394
$92$	0	0
$93$	4.62579i	0.0497396i
$94$	0	0
$95$	− 152.732i	− 1.60770i
$96$	0	0
$97$	68.2834	0.703952	0.351976	−	0.936009i	$-0.385510\pi$
$97$	68.2834	0.703952	0.351976	+	0.936009i	$0.385510\pi$
$98$	0	0
$99$	−1.31181	−0.0132506
$100$	0	0
$101$	− 28.1628i	− 0.278840i	−0.990233	−	0.139420i	$-0.955476\pi$
$101$	− 28.1628i	− 0.278840i	0.990233	−	0.139420i	$-0.0445238\pi$
$102$	0	0
$103$	160.243i	1.55575i	0.628416	+	0.777877i	$0.283703\pi$
$103$	160.243i	1.55575i	−0.628416	+	0.777877i	$0.716297\pi$
$104$	0	0
$105$	53.7407	0.511816
$106$	0	0
$107$	−118.923	−1.11143	−0.555713	−	0.831374i	$-0.687555\pi$
$107$	−118.923	−1.11143	−0.555713	+	0.831374i	$0.687555\pi$
$108$	0	0
$109$	98.2666i	0.901529i	0.892643	+	0.450764i	$0.148849\pi$
$109$	98.2666i	0.901529i	−0.892643	+	0.450764i	$0.851151\pi$
$110$	0	0
$111$	− 134.357i	− 1.21043i
$112$	0	0
$113$	96.9977	0.858387	0.429193	−	0.903213i	$-0.358798\pi$
$113$	96.9977	0.858387	0.429193	+	0.903213i	$0.358798\pi$
$114$	0	0
$115$	−5.81138	−0.0505337
$116$	0	0
$117$	0.422778i	0.00361349i
$118$	0	0
$119$	− 30.7191i	− 0.258143i
$120$	0	0
$121$	84.7112	0.700092
$122$	0	0
$123$	−109.388	−0.889330
$124$	0	0
$125$	− 25.0921i	− 0.200737i
$126$	0	0
$127$	− 71.8712i	− 0.565915i	−0.959132	−	0.282958i	$-0.908684\pi$
$127$	− 71.8712i	− 0.565915i	0.959132	−	0.282958i	$-0.0913156\pi$
$128$	0	0
$129$	−83.5404	−0.647600
$130$	0	0
$131$	209.837	1.60181	0.800906	−	0.598791i	$-0.204352\pi$
$131$	209.837	1.60181	0.800906	+	0.598791i	$0.204352\pi$
$132$	0	0
$133$	− 59.3782i	− 0.446453i
$134$	0	0
$135$	− 184.666i	− 1.36790i
$136$	0	0
$137$	18.1650	0.132591	0.0662954	−	0.997800i	$-0.478882\pi$
$137$	18.1650	0.132591	0.0662954	+	0.997800i	$0.478882\pi$
$138$	0	0
$139$	134.084	0.964635	0.482318	−	0.875996i	$-0.339795\pi$
$139$	134.084	0.964635	0.482318	+	0.875996i	$0.339795\pi$
$140$	0	0
$141$	− 125.877i	− 0.892747i
$142$	0	0
$143$	− 66.2979i	− 0.463621i
$144$	0	0
$145$	291.488	2.01026
$146$	0	0
$147$	20.8930	0.142129
$148$	0	0
$149$	− 20.7618i	− 0.139341i	−0.997570	−	0.0696706i	$-0.977805\pi$
$149$	− 20.7618i	− 0.139341i	0.997570	−	0.0696706i	$-0.0221948\pi$
$150$	0	0
$151$	− 93.1090i	− 0.616616i	−0.951287	−	0.308308i	$-0.900237\pi$
$151$	− 93.1090i	− 0.616616i	0.951287	−	0.308308i	$-0.0997628\pi$
$152$	0	0
$153$	−1.06194	−0.00694080
$154$	0	0
$155$	−10.5471	−0.0680458
$156$	0	0
$157$	− 159.337i	− 1.01489i	−0.861685	−	0.507444i	$-0.830590\pi$
$157$	− 159.337i	− 1.01489i	0.861685	−	0.507444i	$-0.169410\pi$
$158$	0	0
$159$	− 129.550i	− 0.814778i
$160$	0	0
$161$	−2.25932	−0.0140330
$162$	0	0
$163$	45.8277	0.281152	0.140576	−	0.990070i	$-0.455105\pi$
$163$	45.8277	0.281152	0.140576	+	0.990070i	$0.455105\pi$
$164$	0	0
$165$	291.329i	1.76563i
$166$	0	0
$167$	15.5845i	0.0933203i	0.998911	+	0.0466602i	$0.0148578\pi$
$167$	15.5845i	0.0933203i	−0.998911	+	0.0466602i	$0.985142\pi$
$168$	0	0
$169$	147.633	0.873569
$170$	0	0
$171$	−2.05267	−0.0120039
$172$	0	0
$173$	266.021i	1.53770i	0.639432	+	0.768848i	$0.279170\pi$
$173$	266.021i	1.53770i	−0.639432	+	0.768848i	$0.720830\pi$
$174$	0	0
$175$	56.3886i	0.322221i
$176$	0	0
$177$	−161.168	−0.910556
$178$	0	0
$179$	44.5059	0.248636	0.124318	−	0.992242i	$-0.460326\pi$
$179$	44.5059	0.248636	0.124318	+	0.992242i	$0.460326\pi$
$180$	0	0
$181$	− 168.111i	− 0.928787i	−0.885629	−	0.464394i	$-0.846272\pi$
$181$	− 168.111i	− 0.928787i	0.885629	−	0.464394i	$-0.153728\pi$
$182$	0	0
$183$	− 47.6056i	− 0.260140i
$184$	0	0
$185$	306.344	1.65591
$186$	0	0
$187$	166.528	0.890525
$188$	0	0
$189$	− 71.7936i	− 0.379861i
$190$	0	0
$191$	− 353.607i	− 1.85134i	−0.378328	−	0.925672i	$-0.623501\pi$
$191$	− 353.607i	− 1.85134i	0.378328	−	0.925672i	$-0.376499\pi$
$192$	0	0
$193$	−348.574	−1.80608	−0.903042	−	0.429553i	$-0.858671\pi$
$193$	−348.574	−1.80608	−0.903042	+	0.429553i	$0.858671\pi$
$194$	0	0
$195$	93.8912	0.481493
$196$	0	0
$197$	− 26.9314i	− 0.136707i	−0.997661	−	0.0683537i	$-0.978225\pi$
$197$	− 26.9314i	− 0.136707i	0.997661	−	0.0683537i	$-0.0217746\pi$
$198$	0	0
$199$	231.560i	1.16362i	0.813326	+	0.581809i	$0.197655\pi$
$199$	231.560i	1.16362i	−0.813326	+	0.581809i	$0.802345\pi$
$200$	0	0
$201$	−274.431	−1.36533
$202$	0	0
$203$	113.323	0.558242
$204$	0	0
$205$	− 249.411i	− 1.21664i
$206$	0	0
$207$	0.0781034i	0 0.000377311i
$208$	0	0
$209$	321.890	1.54014
$210$	0	0
$211$	79.6903	0.377679	0.188840	−	0.982008i	$-0.439527\pi$
$211$	79.6903	0.377679	0.188840	+	0.982008i	$0.439527\pi$
$212$	0	0
$213$	48.8254i	0.229227i
$214$	0	0
$215$	− 190.478i	− 0.885943i
$216$	0	0
$217$	−4.10045	−0.0188961
$218$	0	0
$219$	28.6153	0.130664
$220$	0	0
$221$	− 53.6698i	− 0.242850i
$222$	0	0
$223$	264.367i	1.18550i	0.805386	+	0.592750i	$0.201958\pi$
$223$	264.367i	1.18550i	−0.805386	+	0.592750i	$0.798042\pi$
$224$	0	0
$225$	1.94932	0.00866367
$226$	0	0
$227$	431.324	1.90011	0.950053	−	0.312088i	$-0.101028\pi$
$227$	431.324	1.90011	0.950053	+	0.312088i	$0.101028\pi$
$228$	0	0
$229$	− 424.901i	− 1.85546i	−0.373247	−	0.927732i	$-0.621756\pi$
$229$	− 424.901i	− 1.85546i	0.373247	−	0.927732i	$-0.378244\pi$
$230$	0	0
$231$	113.261i	0.490308i
$232$	0	0
$233$	−218.828	−0.939176	−0.469588	−	0.882886i	$-0.655597\pi$
$233$	−218.828	−0.939176	−0.469588	+	0.882886i	$0.655597\pi$
$234$	0	0
$235$	287.009	1.22131
$236$	0	0
$237$	170.077i	0.717625i
$238$	0	0
$239$	− 90.1656i	− 0.377262i	−0.982048	−	0.188631i	$-0.939595\pi$
$239$	− 90.1656i	− 0.377262i	0.982048	−	0.188631i	$-0.0604050\pi$
$240$	0	0
$241$	375.944	1.55994	0.779968	−	0.625820i	$-0.215236\pi$
$241$	375.944	1.55994	0.779968	+	0.625820i	$0.215236\pi$
$242$	0	0
$243$	−4.93877	−0.0203242
$244$	0	0
$245$	47.6375i	0.194439i
$246$	0	0
$247$	− 103.741i	− 0.420002i
$248$	0	0
$249$	−42.7043	−0.171503
$250$	0	0
$251$	−319.525	−1.27301	−0.636503	−	0.771274i	$-0.719620\pi$
$251$	−319.525	−1.27301	−0.636503	+	0.771274i	$0.719620\pi$
$252$	0	0
$253$	− 12.2478i	− 0.0484101i
$254$	0	0
$255$	235.838i	0.924854i
$256$	0	0
$257$	−167.466	−0.651618	−0.325809	−	0.945436i	$-0.605637\pi$
$257$	−167.466	−0.651618	−0.325809	+	0.945436i	$0.605637\pi$
$258$	0	0
$259$	119.099	0.459840
$260$	0	0
$261$	− 3.91752i	− 0.0150097i
$262$	0	0
$263$	86.1276i	0.327481i	0.986503	+	0.163741i	$0.0523560\pi$
$263$	86.1276i	0.327481i	−0.986503	+	0.163741i	$0.947644\pi$
$264$	0	0
$265$	295.382	1.11465
$266$	0	0
$267$	298.878	1.11939
$268$	0	0
$269$	− 269.343i	− 1.00128i	−0.865657	−	0.500638i	$-0.833099\pi$
$269$	− 269.343i	− 1.00128i	0.865657	−	0.500638i	$-0.166901\pi$
$270$	0	0
$271$	318.318i	1.17460i	0.809368	+	0.587302i	$0.199810\pi$
$271$	318.318i	1.17460i	−0.809368	+	0.587302i	$0.800190\pi$
$272$	0	0
$273$	36.5025	0.133709
$274$	0	0
$275$	−305.683	−1.11157
$276$	0	0
$277$	356.597i	1.28735i	0.765298	+	0.643677i	$0.222592\pi$
$277$	356.597i	1.28735i	−0.765298	+	0.643677i	$0.777408\pi$
$278$	0	0
$279$	0.141750i	0 0.000508066i
$280$	0	0
$281$	−109.846	−0.390911	−0.195455	−	0.980713i	$-0.562619\pi$
$281$	−109.846	−0.390911	−0.195455	+	0.980713i	$0.562619\pi$
$282$	0	0
$283$	514.582	1.81831	0.909155	−	0.416457i	$-0.136728\pi$
$283$	514.582	1.81831	0.909155	+	0.416457i	$0.136728\pi$
$284$	0	0
$285$	455.861i	1.59951i
$286$	0	0
$287$	− 96.9647i	− 0.337856i
$288$	0	0
$289$	−154.191	−0.533534
$290$	0	0
$291$	−203.807	−0.700366
$292$	0	0
$293$	− 133.002i	− 0.453931i	−0.973903	−	0.226965i	$-0.927120\pi$
$293$	− 133.002i	− 0.453931i	0.973903	−	0.226965i	$-0.0728804\pi$
$294$	0	0
$295$	− 367.475i	− 1.24568i
$296$	0	0
$297$	389.194	1.31042
$298$	0	0
$299$	−3.94729	−0.0132016
$300$	0	0
$301$	− 74.0530i	− 0.246023i
$302$	0	0
$303$	84.0581i	0.277420i
$304$	0	0
$305$	108.544	0.355882
$306$	0	0
$307$	−382.811	−1.24694	−0.623471	−	0.781847i	$-0.714278\pi$
$307$	−382.811	−1.24694	−0.623471	+	0.781847i	$0.714278\pi$
$308$	0	0
$309$	− 478.279i	− 1.54783i
$310$	0	0
$311$	196.374i	0.631426i	0.948855	+	0.315713i	$0.102244\pi$
$311$	196.374i	0.631426i	−0.948855	+	0.315713i	$0.897756\pi$
$312$	0	0
$313$	−427.045	−1.36436	−0.682181	−	0.731183i	$-0.738968\pi$
$313$	−427.045	−1.36436	−0.682181	+	0.731183i	$0.738968\pi$
$314$	0	0
$315$	1.64680	0.00522795
$316$	0	0
$317$	− 111.049i	− 0.350314i	−0.984541	−	0.175157i	$-0.943957\pi$
$317$	− 111.049i	− 0.350314i	0.984541	−	0.175157i	$-0.0560433\pi$
$318$	0	0
$319$	614.326i	1.92579i
$320$	0	0
$321$	354.951	1.10577
$322$	0	0
$323$	260.577	0.806741
$324$	0	0
$325$	98.5174i	0.303131i
$326$	0	0
$327$	− 293.298i	− 0.896936i
$328$	0	0
$329$	111.582	0.339154
$330$	0	0
$331$	119.629	0.361416	0.180708	−	0.983537i	$-0.442161\pi$
$331$	119.629	0.361416	0.180708	+	0.983537i	$0.442161\pi$
$332$	0	0
$333$	− 4.11718i	− 0.0123639i
$334$	0	0
$335$	− 625.721i	− 1.86782i
$336$	0	0
$337$	−57.3636	−0.170218	−0.0851092	−	0.996372i	$-0.527124\pi$
$337$	−57.3636	−0.170218	−0.0851092	+	0.996372i	$0.527124\pi$
$338$	0	0
$339$	−289.511	−0.854014
$340$	0	0
$341$	− 22.2286i	− 0.0651864i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	17.3453	0.0502763
$346$	0	0
$347$	268.774	0.774566	0.387283	−	0.921961i	$-0.373414\pi$
$347$	268.774	0.774566	0.387283	+	0.921961i	$0.373414\pi$
$348$	0	0
$349$	− 97.0498i	− 0.278080i	−0.990287	−	0.139040i	$-0.955598\pi$
$349$	− 97.0498i	− 0.278080i	0.990287	−	0.139040i	$-0.0444016\pi$
$350$	0	0
$351$	− 125.432i	− 0.357356i
$352$	0	0
$353$	−316.928	−0.897813	−0.448907	−	0.893579i	$-0.648186\pi$
$353$	−316.928	−0.897813	−0.448907	+	0.893579i	$0.648186\pi$
$354$	0	0
$355$	−111.325	−0.313592
$356$	0	0
$357$	91.6877i	0.256828i
$358$	0	0
$359$	− 590.194i	− 1.64399i	−0.569492	−	0.821997i	$-0.692860\pi$
$359$	− 590.194i	− 1.64399i	0.569492	−	0.821997i	$-0.307140\pi$
$360$	0	0
$361$	142.681	0.395239
$362$	0	0
$363$	−252.839	−0.696526
$364$	0	0
$365$	65.2449i	0.178753i
$366$	0	0
$367$	175.544i	0.478321i	0.970980	+	0.239160i	$0.0768722\pi$
$367$	175.544i	0.478321i	−0.970980	+	0.239160i	$0.923128\pi$
$368$	0	0
$369$	−3.35202	−0.00908406
$370$	0	0
$371$	114.837	0.309534
$372$	0	0
$373$	554.016i	1.48530i	0.669681	+	0.742649i	$0.266431\pi$
$373$	554.016i	1.48530i	−0.669681	+	0.742649i	$0.733569\pi$
$374$	0	0
$375$	74.8928i	0.199714i
$376$	0	0
$377$	197.989	0.525169
$378$	0	0
$379$	−749.909	−1.97865	−0.989325	−	0.145723i	$-0.953449\pi$
$379$	−749.909	−1.97865	−0.989325	+	0.145723i	$0.953449\pi$
$380$	0	0
$381$	214.515i	0.563032i
$382$	0	0
$383$	− 246.230i	− 0.642899i	−0.946927	−	0.321450i	$-0.895830\pi$
$383$	− 246.230i	− 0.642899i	0.946927	−	0.321450i	$-0.104170\pi$
$384$	0	0
$385$	−258.243	−0.670762
$386$	0	0
$387$	−2.55997	−0.00661491
$388$	0	0
$389$	298.019i	0.766115i	0.923724	+	0.383058i	$0.125129\pi$
$389$	298.019i	0.766115i	−0.923724	+	0.383058i	$0.874871\pi$
$390$	0	0
$391$	− 9.91487i	− 0.0253577i
$392$	0	0
$393$	−626.305	−1.59365
$394$	0	0
$395$	−387.787	−0.981739
$396$	0	0
$397$	− 127.823i	− 0.321971i	−0.986957	−	0.160986i	$-0.948533\pi$
$397$	− 127.823i	− 0.321971i	0.986957	−	0.160986i	$-0.0514673\pi$
$398$	0	0
$399$	177.227i	0.444178i
$400$	0	0
$401$	−774.286	−1.93089	−0.965444	−	0.260609i	$-0.916077\pi$
$401$	−774.286	−1.93089	−0.965444	+	0.260609i	$0.916077\pi$
$402$	0	0
$403$	−7.16396	−0.0177766
$404$	0	0
$405$	545.575i	1.34710i
$406$	0	0
$407$	645.635i	1.58633i
$408$	0	0
$409$	−666.884	−1.63052	−0.815262	−	0.579093i	$-0.803407\pi$
$409$	−666.884	−1.63052	−0.815262	+	0.579093i	$0.803407\pi$
$410$	0	0
$411$	−54.2172	−0.131915
$412$	0	0
$413$	− 142.865i	− 0.345920i
$414$	0	0
$415$	− 97.3687i	− 0.234623i
$416$	0	0
$417$	−400.204	−0.959721
$418$	0	0
$419$	−95.3634	−0.227598	−0.113799	−	0.993504i	$-0.536302\pi$
$419$	−95.3634	−0.227598	−0.113799	+	0.993504i	$0.536302\pi$
$420$	0	0
$421$	− 279.158i	− 0.663083i	−0.943441	−	0.331541i	$-0.892431\pi$
$421$	− 279.158i	− 0.663083i	0.943441	−	0.331541i	$-0.107569\pi$
$422$	0	0
$423$	− 3.85732i	− 0.00911897i
$424$	0	0
$425$	−247.458	−0.582254
$426$	0	0
$427$	42.1991	0.0988270
$428$	0	0
$429$	197.880i	0.461260i
$430$	0	0
$431$	169.374i	0.392978i	0.980506	+	0.196489i	$0.0629540\pi$
$431$	169.374i	0.392978i	−0.980506	+	0.196489i	$0.937046\pi$
$432$	0	0
$433$	−295.214	−0.681787	−0.340894	−	0.940102i	$-0.610730\pi$
$433$	−295.214	−0.681787	−0.340894	+	0.940102i	$0.610730\pi$
$434$	0	0
$435$	−870.010	−2.00002
$436$	0	0
$437$	− 19.1649i	− 0.0438555i
$438$	0	0
$439$	326.773i	0.744357i	0.928161	+	0.372179i	$0.121389\pi$
$439$	326.773i	0.744357i	−0.928161	+	0.372179i	$0.878611\pi$
$440$	0	0
$441$	0.640236	0.00145178
$442$	0	0
$443$	−514.789	−1.16205	−0.581026	−	0.813885i	$-0.697349\pi$
$443$	−514.789	−1.16205	−0.581026	+	0.813885i	$0.697349\pi$
$444$	0	0
$445$	681.462i	1.53137i
$446$	0	0
$447$	61.9682i	0.138631i
$448$	0	0
$449$	602.217	1.34124	0.670620	−	0.741801i	$-0.266028\pi$
$449$	602.217	1.34124	0.670620	+	0.741801i	$0.266028\pi$
$450$	0	0
$451$	525.646	1.16551
$452$	0	0
$453$	277.904i	0.613475i
$454$	0	0
$455$	83.2282i	0.182919i
$456$	0	0
$457$	264.634	0.579069	0.289534	−	0.957168i	$-0.406500\pi$
$457$	264.634	0.579069	0.289534	+	0.957168i	$0.406500\pi$
$458$	0	0
$459$	315.062	0.686409
$460$	0	0
$461$	434.788i	0.943140i	0.881829	+	0.471570i	$0.156313\pi$
$461$	434.788i	0.943140i	−0.881829	+	0.471570i	$0.843687\pi$
$462$	0	0
$463$	− 193.258i	− 0.417403i	−0.977979	−	0.208702i	$-0.933076\pi$
$463$	− 193.258i	− 0.417403i	0.977979	−	0.208702i	$-0.0669237\pi$
$464$	0	0
$465$	31.4801	0.0676992
$466$	0	0
$467$	−655.297	−1.40320	−0.701602	−	0.712569i	$-0.747532\pi$
$467$	−655.297	−1.40320	−0.701602	+	0.712569i	$0.747532\pi$
$468$	0	0
$469$	− 243.264i	− 0.518687i
$470$	0	0
$471$	475.577i	1.00972i
$472$	0	0
$473$	401.442	0.848714
$474$	0	0
$475$	−478.322	−1.00699
$476$	0	0
$477$	− 3.96986i	− 0.00832256i
$478$	0	0
$479$	− 102.660i	− 0.214322i	−0.994242	−	0.107161i	$-0.965824\pi$
$479$	− 102.660i	− 0.214322i	0.994242	−	0.107161i	$-0.0341761\pi$
$480$	0	0
$481$	208.079	0.432597
$482$	0	0
$483$	6.74342	0.0139615
$484$	0	0
$485$	− 464.693i	− 0.958129i
$486$	0	0
$487$	150.652i	0.309347i	0.987966	+	0.154674i	$0.0494326\pi$
$487$	150.652i	0.309347i	−0.987966	+	0.154674i	$0.950567\pi$
$488$	0	0
$489$	−136.783	−0.279720
$490$	0	0
$491$	−118.795	−0.241946	−0.120973	−	0.992656i	$-0.538601\pi$
$491$	−118.795	−0.241946	−0.120973	+	0.992656i	$0.538601\pi$
$492$	0	0
$493$	497.312i	1.00875i
$494$	0	0
$495$	8.92733i	0.0180350i
$496$	0	0
$497$	−43.2805	−0.0870834
$498$	0	0
$499$	−546.431	−1.09505	−0.547526	−	0.836789i	$-0.684430\pi$
$499$	−546.431	−1.09505	−0.547526	+	0.836789i	$0.684430\pi$
$500$	0	0
$501$	− 46.5153i	− 0.0928449i
$502$	0	0
$503$	− 520.718i	− 1.03522i	−0.855615	−	0.517612i	$-0.826821\pi$
$503$	− 520.718i	− 1.03522i	0.855615	−	0.517612i	$-0.173179\pi$
$504$	0	0
$505$	−191.658	−0.379521
$506$	0	0
$507$	−440.643	−0.869119
$508$	0	0
$509$	− 601.427i	− 1.18159i	−0.806823	−	0.590793i	$-0.798815\pi$
$509$	− 601.427i	− 1.18159i	0.806823	−	0.590793i	$-0.201185\pi$
$510$	0	0
$511$	25.3656i	0.0496391i
$512$	0	0
$513$	608.997	1.18713
$514$	0	0
$515$	1090.51	2.11749
$516$	0	0
$517$	604.886i	1.16999i
$518$	0	0
$519$	− 793.998i	− 1.52986i
$520$	0	0
$521$	582.268	1.11760	0.558798	−	0.829304i	$-0.311263\pi$
$521$	582.268	1.11760	0.558798	+	0.829304i	$0.311263\pi$
$522$	0	0
$523$	143.593	0.274557	0.137278	−	0.990533i	$-0.456165\pi$
$523$	143.593	0.274557	0.137278	+	0.990533i	$0.456165\pi$
$524$	0	0
$525$	− 168.304i	− 0.320579i
$526$	0	0
$527$	− 17.9946i	− 0.0341453i
$528$	0	0
$529$	528.271	0.998622
$530$	0	0
$531$	−4.93877	−0.00930088
$532$	0	0
$533$	− 169.409i	− 0.317840i
$534$	0	0
$535$	809.311i	1.51273i
$536$	0	0
$537$	−132.838	−0.247370
$538$	0	0
$539$	−100.398	−0.186268
$540$	0	0
$541$	− 833.097i	− 1.53992i	−0.638092	−	0.769960i	$-0.720276\pi$
$541$	− 833.097i	− 1.53992i	0.638092	−	0.769960i	$-0.279724\pi$
$542$	0	0
$543$	501.762i	0.924056i
$544$	0	0
$545$	668.740	1.22705
$546$	0	0
$547$	−356.858	−0.652391	−0.326196	−	0.945302i	$-0.605767\pi$
$547$	−356.858	−0.652391	−0.326196	+	0.945302i	$0.605767\pi$
$548$	0	0
$549$	− 1.45880i	− 0.00265720i
$550$	0	0
$551$	961.275i	1.74460i
$552$	0	0
$553$	−150.762	−0.272625
$554$	0	0
$555$	−914.349	−1.64748
$556$	0	0
$557$	− 324.859i	− 0.583230i	−0.956536	−	0.291615i	$-0.905807\pi$
$557$	− 324.859i	− 0.583230i	0.956536	−	0.291615i	$-0.0941927\pi$
$558$	0	0
$559$	− 129.379i	− 0.231447i
$560$	0	0
$561$	−497.040	−0.885989
$562$	0	0
$563$	−85.2531	−0.151427	−0.0757133	−	0.997130i	$-0.524123\pi$
$563$	−85.2531	−0.151427	−0.0757133	+	0.997130i	$0.524123\pi$
$564$	0	0
$565$	− 660.104i	− 1.16833i
$566$	0	0
$567$	212.106i	0.374084i
$568$	0	0
$569$	28.7512	0.0505293	0.0252647	−	0.999681i	$-0.491957\pi$
$569$	28.7512	0.0505293	0.0252647	+	0.999681i	$0.491957\pi$
$570$	0	0
$571$	−677.532	−1.18657	−0.593286	−	0.804992i	$-0.702170\pi$
$571$	−677.532	−1.18657	−0.593286	+	0.804992i	$0.702170\pi$
$572$	0	0
$573$	1055.42i	1.84191i
$574$	0	0
$575$	18.2000i	0.0316521i
$576$	0	0
$577$	−20.6675	−0.0358189	−0.0179095	−	0.999840i	$-0.505701\pi$
$577$	−20.6675	−0.0358189	−0.0179095	+	0.999840i	$0.505701\pi$
$578$	0	0
$579$	1040.40	1.79688
$580$	0	0
$581$	− 37.8545i	− 0.0651541i
$582$	0	0
$583$	622.533i	1.06781i
$584$	0	0
$585$	2.87716	0.00491822
$586$	0	0
$587$	551.489	0.939504	0.469752	−	0.882798i	$-0.344343\pi$
$587$	551.489	0.939504	0.469752	+	0.882798i	$0.344343\pi$
$588$	0	0
$589$	− 34.7825i	− 0.0590534i
$590$	0	0
$591$	80.3825i	0.136011i
$592$	0	0
$593$	45.9313	0.0774558	0.0387279	−	0.999250i	$-0.487669\pi$
$593$	45.9313	0.0774558	0.0387279	+	0.999250i	$0.487669\pi$
$594$	0	0
$595$	−209.054	−0.351352
$596$	0	0
$597$	− 691.141i	− 1.15769i
$598$	0	0
$599$	662.449i	1.10593i	0.833206	+	0.552963i	$0.186503\pi$
$599$	662.449i	1.10593i	−0.833206	+	0.552963i	$0.813497\pi$
$600$	0	0
$601$	840.257	1.39810	0.699049	−	0.715074i	$-0.253607\pi$
$601$	840.257	1.39810	0.699049	+	0.715074i	$0.253607\pi$
$602$	0	0
$603$	−8.40952	−0.0139461
$604$	0	0
$605$	− 576.490i	− 0.952876i
$606$	0	0
$607$	1024.92i	1.68850i	0.535951	+	0.844249i	$0.319953\pi$
$607$	1024.92i	1.68850i	−0.535951	+	0.844249i	$0.680047\pi$
$608$	0	0
$609$	−338.238	−0.555399
$610$	0	0
$611$	194.946	0.319061
$612$	0	0
$613$	− 109.148i	− 0.178055i	−0.996029	−	0.0890276i	$-0.971624\pi$
$613$	− 109.148i	− 0.178055i	0.996029	−	0.0890276i	$-0.0283759\pi$
$614$	0	0
$615$	744.422i	1.21044i
$616$	0	0
$617$	−78.9177	−0.127905	−0.0639527	−	0.997953i	$-0.520371\pi$
$617$	−78.9177	−0.127905	−0.0639527	+	0.997953i	$0.520371\pi$
$618$	0	0
$619$	715.995	1.15670	0.578348	−	0.815790i	$-0.303698\pi$
$619$	715.995	1.15670	0.578348	+	0.815790i	$0.303698\pi$
$620$	0	0
$621$	− 23.1721i	− 0.0373141i
$622$	0	0
$623$	264.935i	0.425257i
$624$	0	0
$625$	−703.583	−1.12573
$626$	0	0
$627$	−960.749	−1.53230
$628$	0	0
$629$	522.657i	0.830933i
$630$	0	0
$631$	849.316i	1.34598i	0.739650	+	0.672992i	$0.234991\pi$
$631$	849.316i	1.34598i	−0.739650	+	0.672992i	$0.765009\pi$
$632$	0	0
$633$	−237.853	−0.375755
$634$	0	0
$635$	−489.109	−0.770251
$636$	0	0
$637$	32.3570i	0.0507960i
$638$	0	0
$639$	1.49618i	0.00234144i
$640$	0	0
$641$	488.089	0.761450	0.380725	−	0.924688i	$-0.375674\pi$
$641$	488.089	0.761450	0.380725	+	0.924688i	$0.375674\pi$
$642$	0	0
$643$	257.106	0.399854	0.199927	−	0.979811i	$-0.435930\pi$
$643$	257.106	0.399854	0.199927	+	0.979811i	$0.435930\pi$
$644$	0	0
$645$	568.522i	0.881430i
$646$	0	0
$647$	− 699.536i	− 1.08120i	−0.841280	−	0.540600i	$-0.818197\pi$
$647$	− 699.536i	− 1.08120i	0.841280	−	0.540600i	$-0.181803\pi$
$648$	0	0
$649$	774.472	1.19333
$650$	0	0
$651$	12.2387	0.0187998
$652$	0	0
$653$	− 277.629i	− 0.425160i	−0.977144	−	0.212580i	$-0.931813\pi$
$653$	− 277.629i	− 0.425160i	0.977144	−	0.212580i	$-0.0681866\pi$
$654$	0	0
$655$	− 1428.02i	− 2.18018i
$656$	0	0
$657$	0.876874	0.00133466
$658$	0	0
$659$	522.721	0.793204	0.396602	−	0.917991i	$-0.370189\pi$
$659$	522.721	0.793204	0.396602	+	0.917991i	$0.370189\pi$
$660$	0	0
$661$	515.922i	0.780518i	0.920705	+	0.390259i	$0.127615\pi$
$661$	515.922i	0.780518i	−0.920705	+	0.390259i	$0.872385\pi$
$662$	0	0
$663$	160.189i	0.241612i
$664$	0	0
$665$	−404.090	−0.607654
$666$	0	0
$667$	36.5761	0.0548368
$668$	0	0
$669$	− 789.060i	− 1.17946i
$670$	0	0
$671$	228.762i	0.340927i
$672$	0	0
$673$	−1231.64	−1.83008	−0.915041	−	0.403361i	$-0.867842\pi$
$673$	−1231.64	−1.83008	−0.915041	+	0.403361i	$0.867842\pi$
$674$	0	0
$675$	−578.335	−0.856792
$676$	0	0
$677$	− 576.563i	− 0.851645i	−0.904807	−	0.425822i	$-0.859985\pi$
$677$	− 576.563i	− 0.851645i	0.904807	−	0.425822i	$-0.140015\pi$
$678$	0	0
$679$	− 180.661i	− 0.266069i
$680$	0	0
$681$	−1287.38	−1.89043
$682$	0	0
$683$	−161.395	−0.236302	−0.118151	−	0.992996i	$-0.537697\pi$
$683$	−161.395	−0.236302	−0.118151	+	0.992996i	$0.537697\pi$
$684$	0	0
$685$	− 123.619i	− 0.180466i
$686$	0	0
$687$	1268.21i	1.84601i
$688$	0	0
$689$	200.634	0.291196
$690$	0	0
$691$	113.661	0.164488	0.0822442	−	0.996612i	$-0.473791\pi$
$691$	113.661	0.164488	0.0822442	+	0.996612i	$0.473791\pi$
$692$	0	0
$693$	3.47072i	0.00500826i
$694$	0	0
$695$	− 912.491i	− 1.31294i
$696$	0	0
$697$	425.524	0.610507
$698$	0	0
$699$	653.140	0.934391
$700$	0	0
$701$	− 331.011i	− 0.472198i	−0.971729	−	0.236099i	$-0.924131\pi$
$701$	− 331.011i	− 0.472198i	0.971729	−	0.236099i	$-0.0758689\pi$
$702$	0	0
$703$	1010.27i	1.43708i
$704$	0	0
$705$	−856.640	−1.21509
$706$	0	0
$707$	−74.5119	−0.105392
$708$	0	0
$709$	− 591.351i	− 0.834063i	−0.908892	−	0.417032i	$-0.863070\pi$
$709$	− 591.351i	− 0.834063i	0.908892	−	0.417032i	$-0.136930\pi$
$710$	0	0
$711$	5.21176i	0.00733018i
$712$	0	0
$713$	−1.32346	−0.00185618
$714$	0	0
$715$	−451.181	−0.631022
$716$	0	0
$717$	269.119i	0.375340i
$718$	0	0
$719$	483.600i	0.672600i	0.941755	+	0.336300i	$0.109176\pi$
$719$	483.600i	0.672600i	−0.941755	+	0.336300i	$0.890824\pi$
$720$	0	0
$721$	423.962	0.588020
$722$	0	0
$723$	−1122.09	−1.55199
$724$	0	0
$725$	− 912.877i	− 1.25914i
$726$	0	0
$727$	− 659.508i	− 0.907163i	−0.891215	−	0.453582i	$-0.850146\pi$
$727$	− 659.508i	− 0.907163i	0.891215	−	0.453582i	$-0.149854\pi$
$728$	0	0
$729$	736.257	1.00995
$730$	0	0
$731$	324.977	0.444565
$732$	0	0
$733$	− 211.776i	− 0.288916i	−0.989511	−	0.144458i	$-0.953856\pi$
$733$	− 211.776i	− 0.288916i	0.989511	−	0.144458i	$-0.0461439\pi$
$734$	0	0
$735$	− 142.184i	− 0.193448i
$736$	0	0
$737$	1318.74	1.78933
$738$	0	0
$739$	−246.912	−0.334116	−0.167058	−	0.985947i	$-0.553427\pi$
$739$	−246.912	−0.334116	−0.167058	+	0.985947i	$0.553427\pi$
$740$	0	0
$741$	309.636i	0.417863i
$742$	0	0
$743$	333.126i	0.448352i	0.974549	+	0.224176i	$0.0719691\pi$
$743$	333.126i	0.448352i	−0.974549	+	0.224176i	$0.928031\pi$
$744$	0	0
$745$	−141.292	−0.189653
$746$	0	0
$747$	−1.30861	−0.00175182
$748$	0	0
$749$	314.640i	0.420080i
$750$	0	0
$751$	609.916i	0.812139i	0.913842	+	0.406069i	$0.133101\pi$
$751$	609.916i	0.812139i	−0.913842	+	0.406069i	$0.866899\pi$
$752$	0	0
$753$	953.691	1.26652
$754$	0	0
$755$	−633.640	−0.839259
$756$	0	0
$757$	439.344i	0.580375i	0.956970	+	0.290187i	$0.0937176\pi$
$757$	439.344i	0.580375i	−0.956970	+	0.290187i	$0.906282\pi$
$758$	0	0
$759$	36.5561i	0.0481635i
$760$	0	0
$761$	460.968	0.605740	0.302870	−	0.953032i	$-0.402055\pi$
$761$	460.968	0.605740	0.302870	+	0.953032i	$0.402055\pi$
$762$	0	0
$763$	259.989	0.340746
$764$	0	0
$765$	7.22689i	0.00944692i
$766$	0	0
$767$	− 249.602i	− 0.325426i
$768$	0	0
$769$	1431.37	1.86134	0.930670	−	0.365859i	$-0.119225\pi$
$769$	1431.37	1.86134	0.930670	+	0.365859i	$0.119225\pi$
$770$	0	0
$771$	499.838	0.648299
$772$	0	0
$773$	245.480i	0.317568i	0.987313	+	0.158784i	$0.0507574\pi$
$773$	245.480i	0.317568i	−0.987313	+	0.158784i	$0.949243\pi$
$774$	0	0
$775$	33.0312i	0.0426209i
$776$	0	0
$777$	−355.476	−0.457498
$778$	0	0
$779$	822.513	1.05586
$780$	0	0
$781$	− 234.624i	− 0.300414i
$782$	0	0
$783$	1162.27i	1.48438i
$784$	0	0
$785$	−1084.35	−1.38134
$786$	0	0
$787$	−579.993	−0.736967	−0.368483	−	0.929634i	$-0.620123\pi$
$787$	−579.993	−0.736967	−0.368483	+	0.929634i	$0.620123\pi$
$788$	0	0
$789$	− 257.067i	− 0.325813i
$790$	0	0
$791$	− 256.632i	− 0.324440i
$792$	0	0
$793$	73.7268	0.0929720
$794$	0	0
$795$	−881.632	−1.10897
$796$	0	0
$797$	1348.22i	1.69162i	0.533484	+	0.845810i	$0.320882\pi$
$797$	1348.22i	1.69162i	−0.533484	+	0.845810i	$0.679118\pi$
$798$	0	0
$799$	489.670i	0.612853i
$800$	0	0
$801$	9.15867	0.0114340
$802$	0	0
$803$	−137.507	−0.171241
$804$	0	0
$805$	15.3755i	0.0190999i
$806$	0	0
$807$	803.913i	0.996175i
$808$	0	0
$809$	697.465	0.862132	0.431066	−	0.902320i	$-0.358138\pi$
$809$	697.465	0.862132	0.431066	+	0.902320i	$0.358138\pi$
$810$	0	0
$811$	−482.271	−0.594662	−0.297331	−	0.954774i	$-0.596097\pi$
$811$	−482.271	−0.594662	−0.297331	+	0.954774i	$0.596097\pi$
$812$	0	0
$813$	− 950.089i	− 1.16862i
$814$	0	0
$815$	− 311.874i	− 0.382668i
$816$	0	0
$817$	628.161	0.768863
$818$	0	0
$819$	1.11857	0.00136577
$820$	0	0
$821$	− 386.345i	− 0.470579i	−0.971925	−	0.235289i	$-0.924396\pi$
$821$	− 386.345i	− 0.470579i	0.971925	−	0.235289i	$-0.0756038\pi$
$822$	0	0
$823$	− 354.386i	− 0.430603i	−0.976548	−	0.215301i	$-0.930927\pi$
$823$	− 354.386i	− 0.430603i	0.976548	−	0.215301i	$-0.0690734\pi$
$824$	0	0
$825$	912.377	1.10591
$826$	0	0
$827$	1260.47	1.52414	0.762072	−	0.647493i	$-0.224182\pi$
$827$	1260.47	1.52414	0.762072	+	0.647493i	$0.224182\pi$
$828$	0	0
$829$	1443.69i	1.74148i	0.491746	+	0.870739i	$0.336359\pi$
$829$	1443.69i	1.74148i	−0.491746	+	0.870739i	$0.663641\pi$
$830$	0	0
$831$	− 1064.34i	− 1.28080i
$832$	0	0
$833$	−81.2750	−0.0975690
$834$	0	0
$835$	106.058	0.127016
$836$	0	0
$837$	− 42.0552i	− 0.0502451i
$838$	0	0
$839$	− 44.6511i	− 0.0532194i	−0.999646	−	0.0266097i	$-0.991529\pi$
$839$	− 44.6511i	− 0.0532194i	0.999646	−	0.0266097i	$-0.00847114\pi$
$840$	0	0
$841$	−993.593	−1.18144
$842$	0	0
$843$	327.859	0.388920
$844$	0	0
$845$	− 1004.70i	− 1.18899i
$846$	0	0
$847$	− 224.125i	− 0.264610i
$848$	0	0
$849$	−1535.88	−1.80905
$850$	0	0
$851$	38.4402	0.0451707
$852$	0	0
$853$	− 118.167i	− 0.138531i	−0.997598	−	0.0692655i	$-0.977934\pi$
$853$	− 118.167i	− 0.138531i	0.997598	−	0.0692655i	$-0.0220656\pi$
$854$	0	0
$855$	13.9692i	0.0163382i
$856$	0	0
$857$	153.130	0.178682	0.0893408	−	0.996001i	$-0.471524\pi$
$857$	153.130	0.178682	0.0893408	+	0.996001i	$0.471524\pi$
$858$	0	0
$859$	136.656	0.159087	0.0795437	−	0.996831i	$-0.474654\pi$
$859$	136.656	0.159087	0.0795437	+	0.996831i	$0.474654\pi$
$860$	0	0
$861$	289.412i	0.336135i
$862$	0	0
$863$	− 1462.52i	− 1.69470i	−0.531036	−	0.847349i	$-0.678197\pi$
$863$	− 1462.52i	− 1.69470i	0.531036	−	0.847349i	$-0.321803\pi$
$864$	0	0
$865$	1810.37	2.09291
$866$	0	0
$867$	460.217	0.530816
$868$	0	0
$869$	− 817.281i	− 0.940484i
$870$	0	0
$871$	− 425.011i	− 0.487957i
$872$	0	0
$873$	−6.24535	−0.00715389
$874$	0	0
$875$	−66.3874	−0.0758713
$876$	0	0
$877$	− 29.8881i	− 0.0340799i	−0.999855	−	0.0170399i	$-0.994576\pi$
$877$	− 29.8881i	− 0.0340799i	0.999855	−	0.0170399i	$-0.00542424\pi$
$878$	0	0
$879$	396.972i	0.451618i
$880$	0	0
$881$	−497.832	−0.565076	−0.282538	−	0.959256i	$-0.591176\pi$
$881$	−497.832	−0.565076	−0.282538	+	0.959256i	$0.591176\pi$
$882$	0	0
$883$	−1430.98	−1.62059	−0.810295	−	0.586022i	$-0.800693\pi$
$883$	−1430.98	−1.62059	−0.810295	+	0.586022i	$0.800693\pi$
$884$	0	0
$885$	1096.81i	1.23933i
$886$	0	0
$887$	− 1187.21i	− 1.33846i	−0.743057	−	0.669228i	$-0.766625\pi$
$887$	− 1187.21i	− 1.33846i	0.743057	−	0.669228i	$-0.233375\pi$
$888$	0	0
$889$	−190.153	−0.213896
$890$	0	0
$891$	−1149.83	−1.29049
$892$	0	0
$893$	946.504i	1.05991i
$894$	0	0
$895$	− 302.879i	− 0.338412i
$896$	0	0
$897$	11.7815	0.0131344
$898$	0	0
$899$	66.3823	0.0738401
$900$	0	0
$901$	503.956i	0.559329i
$902$	0	0
$903$	221.027i	0.244770i
$904$	0	0
$905$	−1144.05	−1.26415
$906$	0	0
$907$	−818.659	−0.902601	−0.451301	−	0.892372i	$-0.649040\pi$
$907$	−818.659	−0.902601	−0.451301	+	0.892372i	$0.649040\pi$
$908$	0	0
$909$	2.57584i	0.00283370i
$910$	0	0
$911$	− 774.905i	− 0.850610i	−0.905050	−	0.425305i	$-0.860167\pi$
$911$	− 774.905i	− 0.850610i	0.905050	−	0.425305i	$-0.139833\pi$
$912$	0	0
$913$	205.210	0.224764
$914$	0	0
$915$	−323.973	−0.354069
$916$	0	0
$917$	− 555.177i	− 0.605428i
$918$	0	0
$919$	− 1695.99i	− 1.84547i	−0.385437	−	0.922734i	$-0.625949\pi$
$919$	− 1695.99i	− 1.84547i	0.385437	−	0.922734i	$-0.374051\pi$
$920$	0	0
$921$	1142.58	1.24059
$922$	0	0
$923$	−75.6160	−0.0819241
$924$	0	0
$925$	− 959.401i	− 1.03719i
$926$	0	0
$927$	− 14.6562i	− 0.0158103i
$928$	0	0
$929$	−1173.31	−1.26298	−0.631491	−	0.775383i	$-0.717557\pi$
$929$	−1173.31	−1.26298	−0.631491	+	0.775383i	$0.717557\pi$
$930$	0	0
$931$	−157.100	−0.168743
$932$	0	0
$933$	− 586.120i	− 0.628210i
$934$	0	0
$935$	− 1133.28i	− 1.21207i
$936$	0	0
$937$	315.505	0.336719	0.168359	−	0.985726i	$-0.446153\pi$
$937$	315.505	0.336719	0.168359	+	0.985726i	$0.446153\pi$
$938$	0	0
$939$	1274.61	1.35741
$940$	0	0
$941$	423.505i	0.450058i	0.974352	+	0.225029i	$0.0722477\pi$
$941$	423.505i	0.450058i	−0.974352	+	0.225029i	$0.927752\pi$
$942$	0	0
$943$	− 31.2963i	− 0.0331880i
$944$	0	0
$945$	−488.581	−0.517017
$946$	0	0
$947$	1147.28	1.21149	0.605743	−	0.795660i	$-0.292876\pi$
$947$	1147.28	1.21149	0.605743	+	0.795660i	$0.292876\pi$
$948$	0	0
$949$	44.3166i	0.0466982i
$950$	0	0
$951$	331.451i	0.348529i
$952$	0	0
$953$	−736.494	−0.772816	−0.386408	−	0.922328i	$-0.626284\pi$
$953$	−736.494	−0.772816	−0.386408	+	0.922328i	$0.626284\pi$
$954$	0	0
$955$	−2406.42	−2.51981
$956$	0	0
$957$	− 1833.59i	− 1.91598i
$958$	0	0
$959$	− 48.0599i	− 0.0501146i
$960$	0	0
$961$	958.598	0.997501
$962$	0	0
$963$	10.8769	0.0112948
$964$	0	0
$965$	2372.17i	2.45821i
$966$	0	0
$967$	− 1147.66i	− 1.18682i	−0.804899	−	0.593411i	$-0.797781\pi$
$967$	− 1147.66i	− 1.18682i	0.804899	−	0.593411i	$-0.202219\pi$
$968$	0	0
$969$	−777.750	−0.802632
$970$	0	0
$971$	1108.64	1.14175	0.570876	−	0.821036i	$-0.306604\pi$
$971$	1108.64	1.14175	0.570876	+	0.821036i	$0.306604\pi$
$972$	0	0
$973$	− 354.754i	− 0.364598i
$974$	0	0
$975$	− 294.047i	− 0.301586i
$976$	0	0
$977$	779.026	0.797366	0.398683	−	0.917089i	$-0.369467\pi$
$977$	779.026	0.797366	0.398683	+	0.917089i	$0.369467\pi$
$978$	0	0
$979$	−1436.21	−1.46702
$980$	0	0
$981$	− 8.98769i	− 0.00916176i
$982$	0	0
$983$	79.5940i	0.0809705i	0.999180	+	0.0404853i	$0.0128904\pi$
$983$	79.5940i	0.0809705i	−0.999180	+	0.0404853i	$0.987110\pi$
$984$	0	0
$985$	−183.277	−0.186069
$986$	0	0
$987$	−333.040	−0.337427
$988$	0	0
$989$	− 23.9013i	− 0.0241671i
$990$	0	0
$991$	− 644.244i	− 0.650095i	−0.945698	−	0.325047i	$-0.894620\pi$
$991$	− 644.244i	− 0.650095i	0.945698	−	0.325047i	$-0.105380\pi$
$992$	0	0
$993$	−357.057	−0.359574
$994$	0	0
$995$	1575.85	1.58377
$996$	0	0
$997$	− 1739.99i	− 1.74523i	−0.488409	−	0.872615i	$-0.662422\pi$
$997$	− 1739.99i	− 1.74523i	0.488409	−	0.872615i	$-0.337578\pi$
$998$	0	0
$999$	1221.50i	1.22273i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.g.g.127.3		12
4.3	odd	2	inner	1792.3.g.g.127.9		12
8.3	odd	2	inner	1792.3.g.g.127.4		12
8.5	even	2	inner	1792.3.g.g.127.10		12
16.3	odd	4		28.3.c.a.15.5	✓	6
16.5	even	4		448.3.d.d.127.5		6
16.11	odd	4		448.3.d.d.127.2		6
16.13	even	4		28.3.c.a.15.6	yes	6
48.29	odd	4		252.3.g.a.127.1		6
48.35	even	4		252.3.g.a.127.2		6
112.3	even	12		196.3.g.j.79.3		12
112.13	odd	4		196.3.c.g.99.6		6
112.19	even	12		196.3.g.j.67.1		12
112.45	odd	12		196.3.g.j.79.1		12
112.51	odd	12		196.3.g.k.67.1		12
112.61	odd	12		196.3.g.j.67.3		12
112.67	odd	12		196.3.g.k.79.3		12
112.83	even	4		196.3.c.g.99.5		6
112.93	even	12		196.3.g.k.67.3		12
112.109	even	12		196.3.g.k.79.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.c.a.15.5	✓	6	16.3	odd	4
28.3.c.a.15.6	yes	6	16.13	even	4
196.3.c.g.99.5		6	112.83	even	4
196.3.c.g.99.6		6	112.13	odd	4
196.3.g.j.67.1		12	112.19	even	12
196.3.g.j.67.3		12	112.61	odd	12
196.3.g.j.79.1		12	112.45	odd	12
196.3.g.j.79.3		12	112.3	even	12
196.3.g.k.67.1		12	112.51	odd	12
196.3.g.k.67.3		12	112.93	even	12
196.3.g.k.79.1		12	112.109	even	12
196.3.g.k.79.3		12	112.67	odd	12
252.3.g.a.127.1		6	48.29	odd	4
252.3.g.a.127.2		6	48.35	even	4
448.3.d.d.127.2		6	16.11	odd	4
448.3.d.d.127.5		6	16.5	even	4
1792.3.g.g.127.3		12	1.1	even	1	trivial
1792.3.g.g.127.4		12	8.3	odd	2	inner
1792.3.g.g.127.9		12	4.3	odd	2	inner
1792.3.g.g.127.10		12	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.98472 q^{3} -6.80536i q^{5} -2.64575i q^{7} -0.0914622 q^{9} +O(q^{10})\)	\(q-2.98472 q^{3} -6.80536i q^{5} -2.64575i q^{7} -0.0914622 q^{9} +14.3426 q^{11} -4.62243i q^{13} +20.3121i q^{15} +11.6107 q^{17} +22.4428 q^{19} +7.89682i q^{21} -0.853941i q^{23} -21.3129 q^{25} +27.1354 q^{27} +42.8321i q^{29} -1.54982i q^{31} -42.8087 q^{33} -18.0053 q^{35} +45.0151i q^{37} +13.7967i q^{39} +36.6492 q^{41} +27.9894 q^{43} +0.622433i q^{45} +42.1740i q^{47} -7.00000 q^{49} -34.6547 q^{51} +43.4044i q^{53} -97.6068i q^{55} -66.9856 q^{57} +53.9979 q^{59} +15.9498i q^{61} +0.241986i q^{63} -31.4573 q^{65} +91.9453 q^{67} +2.54877i q^{69} -16.3585i q^{71} -9.58728 q^{73} +63.6130 q^{75} -37.9470i q^{77} -56.9826i q^{79} -80.1685 q^{81} +14.3077 q^{83} -79.0151i q^{85} -127.842i q^{87} -100.136 q^{89} -12.2298 q^{91} +4.62579i q^{93} -152.732i q^{95} +68.2834 q^{97} -1.31181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{9}+O(q^{10}) \) 12 * q + 20 * q^9	\( 12 q + 20 q^{9} - 8 q^{17} + 60 q^{25} + 160 q^{33} + 40 q^{41} - 84 q^{49} - 320 q^{57} - 272 q^{65} + 264 q^{73} - 436 q^{81} - 696 q^{89} + 504 q^{97}+O(q^{100}) \) 12 * q + 20 * q^9 - 8 * q^17 + 60 * q^25 + 160 * q^33 + 40 * q^41 - 84 * q^49 - 320 * q^57 - 272 * q^65 + 264 * q^73 - 436 * q^81 - 696 * q^89 + 504 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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